quantitative carrier transport in quantum dot photovoltaic
TRANSCRIPT
Quantitative Carrier Transport in Quantum Dot Photovoltaic
Solar Cells from Novel Photocarrier Radiometry and Lock-in
Carrierography
by
Lilei Hu
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Department of Mechanical and Industrial Engineering
University of Toronto
© Copyright by Lilei Hu, 2017
Toronto, Canada
ii
Quantitative Carrier Transport in Quantum Dot Photovoltaic
Solar Cells from Novel Photocarrier Radiometry and Lock-in
Carrierography
Lilei Hu
Doctor of Philosophy
Department of Mechanical and Industrial Engineering
University of Toronto
2017
Abstract
Colloidal quantum dots (CQDs) are promising candidates for fabricating large-scale, low-cost,
flexible, and lightweight photovoltaic solar cells. However, their power conversion efficiency
is still insufficient for commercial applications, partly and significantly, due to the not-well-
understood carrier transport mechanisms and the lack of effective characterization techniques.
Addressing these issues, carrier transport kinetics in CQD systems were studied to develop
high-frequency dynamic testing and/or large-area quantitative imaging techniques:
photocarrier radiometry (PCR), and homodyne (HoLIC) and heterodyne (HeLIC) lock-in
carrierographies.
Based on the discrete carrier hopping transport in CQDs, various carrier drift-diffusion
current-voltage (J-V) analytical models and new concepts including the imbalanced carrier
mobilities, reversed Schottky barrier, and double-diode model were developed to
quantitatively interpret carrier transport and J-V characteristics in CQD solar cells. The further
quantitative study of carrier mobility, CQD bandgap energy, phonon-assisted carrier transport,
and open-circuit voltage deficit revealed CQD solar cell efficiency optimization strategies.
Applying these energy transport mechanisms, for the first time, an analytical PCR signal
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generation model for CQD systems was developed from a novel trap-state-mediated carrier
hopping transport theory. Therefore, multiple carrier transport parameters including carrier
hopping lifetime, diffusivity, and diffusion length were extracted to investigate carrier
transport dependencies on temperature, quantum dot size, surface-passivation ligands, and
carrier hopping activation energies. As an imaging extension of PCR, using a heterodyne
method to overcome the limitations of camera frame rate and exposure time of even the state-
of-the-art InGaAs cameras, the first camera-based HeLIC theoretical model for ultrahigh-
frequency (up to 270 kHz) imaging of CQD solar cells was achieved. Therefore, quantitative
imaging of carrier lifetime, diffusivity, and diffusion and drift lengths of CQD solar cells was
accomplished to explore the influences of carrier transport and contact/CQD interface on CQD
solar cells. Also, low-frequency HoLIC large-area imaging evaluated the sample homogeneity
and quality, reflecting preliminary carrier lifetime distribution.
The combination of the novel carrier discrete hopping transport mechanism, J-V models,
PCR, and the lock-in carrierography techniques (HoLIC and HeLIC) shows great potential for
quantitative carrier transport study of CQD solar cells and for fast, all-optical, contactless,
large-area, and nondestructive characterization of commercial photovoltaic materials and
devices.
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Acknowledgements
I am sincerely and deeply indebted to Prof. Andreas Mandelis, my supervisor, inspiration,
and mentor, for his generosity of mind, spirit, and guidance during the course of my Ph.D.
studies. His proficiency, intelligence, and high efficiency ensure the progress of my Ph.D.
research project while these inspirations and impacts on me will benefit me through my whole
life. I would also like to extend my deep appreciation of the freedom that Prof. Mandelis
offered. There is nowhere for me to find a better supervisor than Prof. Mandelis.
I would like to thank all my Ph.D. committee members, Prof. Olivera Kesler and Prof.
Edward H. Sargent, for their constructive feedbacks and encouragement. I would like to thank
Prof. Daniele Fournier and Prof. Axel Guenther for agreeing to act as the external and internal
examiners, respectively.
I would like to thank all my colleagues at the Center for Advanced Diffusion Wave and
Photoacoustic Technologies (CADIPT) for their discussions and friendship. I would especially
like to mention Dr. Xinxin Guo, Dr. Qiming Sun, Dr. Alexander Melnikov, Lixian Liu, Huiting
Huan, Pantea Tavakolian, Dr. Edem Dovlo, Sean Choi, Dr. Wei Wang, Yingcong Zhang, Dr.
Bahman Lashkari, and Dr. Koneswaran Sivagurunathan for their help. Particularly, I would
like to thank Dr. Sun for the very useful and valuable discussions and Dr. Guo for her kind
encouragement and suggestions for my study.
Additionally, I would like to thank Prof. Sargent for inviting me to present my research in
his group meeting and for his new constructive criticism and guidance throughout. I also would
like to thank Prof. Sargent’s team members, Dr. Xinzheng Lan, Dr. Zhenyu Yang, Mengxia
Liu, Grant Walters, Dr. Oleksandr Voznyy, Olivier Ouellette, and Dr. Sjoerd Hoogland for
their kindly collaboration. Dr. Xinzheng Lan, Dr. Zhenyu Yang, and Mengxia Liu are generous
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with their time, materials, devices, insightful comments, and deep knowledge of a variety of
fabrication issues.
Last, but never the least, I would like to express my immense gratitude to my parents, sister,
and grandparents for their support and belief in me.
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Contents
Abstract ................................................................................................................................... ii
Acknowledgements ................................................................................................................ iv
Contents .................................................................................................................................. vi
List of Tables ........................................................................................................................... x
List of Figures ........................................................................................................................ xi
1 Preamble ............................................................................................................................... 1
1.1 The Imperative of Solar Cells ......................................................................................... 1
1.2 The Imperative of Nondestructive Testing of Photovoltaics .......................................... 3
1.3 Objectives of This Work ................................................................................................. 4
1.4 Thesis Outline ................................................................................................................. 7
2 Introduction to Quantum Dots and Colloidal Quantum Dot Solar Cells ..................... 10
2.1 Synthesis of Colloidal Quantum Dots ........................................................................... 10
2.2 Electrical Properties of Colloidal Quantum Dots ......................................................... 13
2.2.1 Effects of Interdot Distance and QD Disorder ....................................................... 14
2.2.2 Effects of Temperature ........................................................................................... 16
2.2.3 Effects of Quantum Dot Size and Polydispersity ................................................... 19
2.3 Colloidal Quantum Dot Solar Cells .............................................................................. 20
2.4 Conclusions ................................................................................................................... 24
3 Non-destructive Testing (NDT) Techniques for Carrier Transport in Quantum Dot
Materials and Solar Cells ..................................................................................................... 26
3.1 Literature Review and Classification ............................................................................ 26
3.2 Traditional Methodologies for CQD Carrier Transport Characterization .................... 30
3.2.1 Short-Circuit Current/Open-Circuit Voltage Decay (SCCD/OCVD) .................... 30
3.2.2 Photoconductance Decay (PCD) ............................................................................ 33
3.2.3 Time-resolved PL (TRPL, transient PL) ................................................................ 35
3.2.4 Carrier Diffusion Length Measurements ................................................................ 38
3.3 Photocarrier Carrier Radiometry (PCR) ....................................................................... 42
3.3.1 Photocarrier Radiometry Instrumentation .............................................................. 42
3.3.2 Theory of Lock-in Amplifier Signal Computation................................................. 44
3.3.3 General Theory of Photocarrier Radiometry .......................................................... 47
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3.4 Homodyne (HoLIC) and Heterodyne (HeLIC) Lock-in Carrierography ...................... 52
3.4.1 Instrumentation and Signal Processing Techniques used in HoLIC and HeLIC
Imaging ............................................................................................................................ 52
3.4.2 Requirements for HeLIC Response to Laser Excitation: Non-linear
Photoluminescence Processes ......................................................................................... 57
3.5 Comparison of Different Techniques and Advantages of PCR and HeLIC ................. 60
3.6 Conclusions ................................................................................................................... 65
4 Quantitative Carrier Transport Study through Current-voltage Characteristics ...... 66
4.1 Introduction ................................................................................................................... 66
4.2 Derivation of Current-voltage Model from Hopping and Discrete Carrier Transport .. 69
4.2.1 Carrier Hopping Diffusivity and Mobility in Quantum Dot Systems .................... 69
4.2.2 Current Density J(x) across CQD Solar Cells ........................................................ 75
4.3 Imbalanced Charge Carrier Mobility and Schottky Junction Induced Anomalous J-V
Characteristics of CQD Solar Cells .................................................................................... 78
4.3.1 CQD Solar Cell Fabrication and Current-voltage Characterization ....................... 79
4.3.2 Double-diode-equivalent Hopping Transport Model ............................................. 82
4.3.3 Origins of Anomalous Current-voltage Curves...................................................... 84
4.3.4 Open-circuit Voltage Origin of CQD Solar Cells .................................................. 89
4.3.5 Temperature-dependent Carrier Hopping Transport and CQD Solar Cell
Performance..................................................................................................................... 93
4.4 Conclusions ................................................................................................................. 101
5 Colloidal Quantum Dot Solar Cell Efficiency Optimization: Impact of Hopping
Mobility, Bandgap Energy, and Electrode-semiconductor Interfaces .......................... 102
5.1 Introduction ................................................................................................................. 102
5.2 Derivation of Carrier Hopping Drift-diffusion J-V Model for CQD Solar Cells........ 104
5.3 Experimental CQD Solar Cell Efficiency Optimization ............................................. 108
5.4 Non-constant Photocurrent in CQD Solar Cells ......................................................... 110
5.5 Impact of Hopping Mobility and Bandgap Energy ..................................................... 118
5.6 Impact of Electrode-semiconductor Interfaces Using Homodyne Lock-in
Carrierography .................................................................................................................. 128
5.7 Conclusions ................................................................................................................. 135
6 Photocarrier Radiometry Study of Quantitative Carrier Transport in CQD Thin
Films ..................................................................................................................................... 138
6.1 Introduction ................................................................................................................. 138
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6.2 PCR Theory for CQDs: Trap-state-mediated Carrier Transport Model ..................... 140
6.3 CQD Thin Film Homogeneity and Optical Properties ............................................... 145
6.4 Carrier Transport Kinetics in Various CQD Thin Films ............................................ 148
6.4.1 Temperature-dependent Carrier Transport Kinetics ............................................. 148
6.4.2 Carrier Hopping Activation Energy and Exciton Binding Energy....................... 153
6.4.3 Ligand- and Size-dependent Carrier Transport Kinetics ...................................... 158
6.5 Fitting Uniqueness and Reliability – Parameter Extraction from PCR ...................... 162
6.6 Conclusions ................................................................................................................. 166
7 Carrier Recombination Mechanism, Energy Band Structure, and Inhomogeneity-
affected Carrier Transport in Perovskite Shelled PbS CQD Thin Films Using PCR and
HoLIC .................................................................................................................................. 168
7.1 Introduction ................................................................................................................. 168
7.2 Experimental Details and CQD Thin Film Synthesis ................................................. 169
7.3 Charge Carrier Recombination Mechanism for PbS CQDs: Nonlinear Response ..... 170
7.4 Energy Band Structure ................................................................................................ 176
7.4.1 Photoluminescence of CQD Thin Films .............................................................. 177
7.4.2 PCR Photothermal Spectra of CQD Thin Films .................................................. 178
7.5 Large-area Imaging and Carrier Transport of CQD Thin Films ................................. 181
7.5.1 Qualitative Large-area Imaging ............................................................................ 181
7.5.2 PCR Characterization of Carrier Transport Parameters ....................................... 183
7.6 Conclusions ................................................................................................................. 188
8 Heterodyne and Homodyne Lock-in Carrierography Imaging of Carrier Transport in
CQD Solar Cells .................................................................................................................. 189
8.1 Introduction ................................................................................................................. 189
8.2 Theories of Homodyne and Heterodyne Lock-in Carrierography .............................. 191
8.3 Carrier Transport Theory of CQD Solar Cells under Modulated Photoexcitation ..... 193
8.4 Quantitative Colloidal Quantum Dot Solar Cell Imaging ........................................... 197
8.4.1 Device Fabrication and Characterization Details ................................................. 197
8.4.2 Quantitative HeLIC Imaging of Carrier Transport in CQD Solar Cells .............. 197
8.5 Further HeLIC Carrier Lifetime Imaging of CQD Solar Cells ................................... 205
8.5.1 HeLIC Imaging at Various Frequencies ............................................................... 205
8.5.2 HeLIC Lifetime Imaging of CQD Solar Cells with/without Plasma Etching ...... 207
8.5.3 HeLIC Lifetime Imaging for Interface Effects on CQD Solar Cells ................... 209
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8.6 Conclusions ................................................................................................................. 210
9 Conclusions and Outlook ................................................................................................ 212
9.1 Conclusions ................................................................................................................. 212
9.1.1 Advances in Carrier Transport and J-V Mechanisms of CQD Systems............... 212
9.1.2 Advances in Ultrahigh-frequency Diagnostics of Carrier Transport ................... 213
9.1.3 Advances in Quantitative Large-area Ultrahigh-frequency Imaging ................... 214
9.2 Outlook ....................................................................................................................... 215
References ............................................................................................................................ 217
Chapter 1 ....................................................................................................................... 217
Chapter 2 ....................................................................................................................... 218
Chapter 3 ....................................................................................................................... 221
Chapter 4 ....................................................................................................................... 226
Chapter 5 ....................................................................................................................... 231
Chapter 6 ....................................................................................................................... 236
Chapter 7 ....................................................................................................................... 242
Chapter 8 ....................................................................................................................... 245
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List of Tables
Table 3.1: Common optoelectronic parameters of colloidal quantum dot materials and
devices from various characterization techniques…............................................................. 26
Table 4.1: Summary of best-fitted parameters using Eq. (4.29)………………………….....91
Table 4.2: Summary of best-fitted parameters…..................................................................100
Table 5.1: Parameters used for the CQD solar cell simulations…………….………….......123
Table 5.2: Optical counterparts of CQD solar cell electrical parameters, obtained through
best-fitting of the experimental data in Figs.5.11(b) and (d) to Eq. (5.25)………….......….133
Table 6.1: Best-fitted parameters for PbS-MAPbI3 CQD thin films at different
temperatures………………………………………………………………………………...150
Table 6.2: Activation energies at different temperatures for PbS CQD thin films passivated
with various ligands…...........................................................................................................155
Table 6.3: Summary of the best-fitted parameters for CQD thin films surface passivated with
various ligands. These parameters were evaluated for 100 K measurement………….....…160
Table 6.4: Summary of the best-fitted parameters for CQD thin films surface passivated with
various ligands. These parameters were evaluated for 300 K measurement…………….…160
Table 7.1: Summary of best-fitted parameters using Eq. (6.18) in Sect. 6.2…………........184
Table 8.1: Summary of the parameters used for heterodyne lock-in carrierography best-fits to
Eq. (8.17)………………………………………………….. ………………………………202
xi
List of Figures
1.1 (a) Global net electricity generation from renewable energy source from 2012 to 2040.
The unit is trillion kilowatthours. It should be noted that the other generation includes waste,
biomass, and tide/wave/ocean. (b) Total jobs of 8.1 million created by renewable energy by
the end of 2015. Moreover, 60% of these solar energy jobs were created in China…………2
1.2 Structure of the research discussed in this thesis. NDT = nondestructive testing; HoLIC =
homodyne lock-in carrierography; HoLIC = heterodyne lock-in carrierography; PCR =
photocarrier radiometry; CQD = colloidal quantum dots; CQDSC = colloidal quantum dot
solar cell….................................................................................................................................5
2.1 (a) Schematic of La Mer and Dinegar model for the synthesis of monodisperse CQDs. (b)
Representation of the apparatus employed for CQD
synthesis……………….………………………………………………………..…..…..……12
2.2: Transmission electron spectroscopy (TEM) imaging of CdSe QDs: low-resolution (a)
and high-resolution (b) and (c) ……………………………………………………………...13
2.3 Different carrier transport mechanisms for QD thin film systems: (a) bulk crystal-like
Bloch state carrier transport; (b) carrier tunneling transport from one dot to another without
phonon assistance; (c) carrier transport with over-the-barrier activation energy mechanism;
and (d) hopping transport with phonon-assistance……………………………...........……...14
2.4 Schematic of charge carrier transport within colloidal quantum dot array. Energy states
including trap states and Fermi levels are represented by solid and dashed lines, respectively.
Nearest-neighbor hopping (I) and variable range hopping (I+II) occur through carriers
transport within quantized states (long solid lines, 1Se and 1Sh for electrons and holes,
respectively) and surface trap states (short solid lines). The variation in quantum dot creates
energy and spatial disorders that weaken interdot coupling effects and disturb carrier
transport………………………………………………………………………………….…..17
2.5 Current-voltage characteristics at different temperatures for PbSe CQD thin films
vacuum-annealed at 473 K (a) and 523 K(b).The low-right insets in both figures show the
Arrhenius plots of conductivity G same as 𝜎 in Eq.(2.1), and the up-left insets depict the
TEM images of PbSe CQD arrays after vacuum annealing……….........................……...…18
2.6 Experimental and modeled electron (a) and hole (b) mobility of PbSe QDs, at a different
interdot distance, as a function of QD diameter…………………..……………...……....….19
2.7: Configuration of different types of CQD solar cells, Schottky (a, b), heterojunction (c,
d), and CQD sensitized solar cells (e, f). The top row (a, c, and e) illustrates the device
structure and the bottom row (b, d, and f) depict the energy band structure with carrier
transport mechanism displayed. Only the lowest energy state levels are shown (i.e., 1S and
1P) for simplification...............................................................................................................21
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2.8: Schematic (a) and energy band diagram (b) of the planar depleted CQD heterojunction
solar cell at short circuit. The energy diagram is plotted along the A-A’ cross-section.
Correspondingly, the schematic (c) and energy diagram (d, a cross-section along B-B’) for
bulk heterojunction. The vertical ZnO nanowires were grown using solution-processed
hydrothermal methods to produce ordered nanowire array within the PbS CQD thin film....23
3.1: A representative of open-circuit voltage decay curve for CQD solar cells, the linear best
fit (dashed red line) is used for the determination of recombination-determined lifetime…..31
3.2: A typical spectrum of the time-resolved PL for PbS CQDs. The inset illustrates the
exciton transport with diffusion and dissociation processes…................................................35
3.3: The mechanism for quasi-static PL imaging (a). (b) The mechanism for low-frequency
modulated PL imaging…………………………………………………………………….....37
3.4: Schematic PL quenching method for diffusion length measurements. An acceptor layer
of CQDs with smaller bandgap should be pre-coated.……………………………………....40
3.5: Schematic of transient photoluminescence study of carrier diffusion length, diffusivity,
and lifetime. At x = L the boundary condition can be either with or without quenching…...40
3.6: Schematic of experimental instrumental setup for photocarrier radiometry…………....43
3.7: (a) Energy diagram of an n-type semiconductor with the illumination of photoexcitation,
and radiative and nonradiative recombination. Defects related states are also depicted to
carry radiative and non-radiative recombination………………………………………….....49
3.8: Schematic of one-dimensional Si wafer where an emission photon distribution is yielded
following laser excitation and carrier-wave generation. (a) A representative semiconductor
slab with thickness dz, centered at z. (b) Reflection photons from backing support material.
(c) Emissive IR photons from backing support materials at temperature Tb. ∆𝑁(𝑧, 𝜔) represents the depth- and frequency- dependent carrer-diffusion-wave, and L is the thickness
of the Si wafer. Other parameters can be found in the text. R1,2,b(λ) are reflectivity of the
front surface, back surface, and the backing support material. It should be noted that the
backing material is used to support the wafer but not necessary to be in contact with the
sample………………………………………………………………………………….….....50
3.9: Experimental setup for homodyne (HoLIC) and heterodyne (HeLIC) lock-in
carrierography………………..................................................................................................53
3.10: Schematic of oversampling (a) and undersampling (b) signal processing methods. For
sampling, 16 samples are taken per one cycle (waveform), and one circle (waveform) is
skipped for undersampling……………………………………….……………………….....54
3.11: Schematic of camera-based HeLIC imaging using an undersampling method (a) and
modulation laser frequency mixing mechanism for HeLIC imaging……………….…….....55
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3.12: The nonlinear dependence of DC (a) and HeLIC (b) signals on photoexcitation laser
power density for CQD solar cells with a typical structure: Au/PbS-EDT/PbS-
PbX2(AA)/ZnO/ITO as discussed in detail in Sect.7.3 ………………………………….......58
4.1: (a) A virtual volume element dV = Adx, resembling a QD, illustrates the discrete
hopping transport of excitons and charge carriers. (b) Schematic of the discrete particle flux
into and out of three adjacent virtual planes. All planes have an area of A across the thickness
direction of a CQD solar cell…………………………………………………………….…..71
4.2: Schematic of one CQD solar cell energy band structure…..............................................75
4.3: Schematic of the double-layer CQD solar cell with the structure: ITO/ZnO/PbS-TBAI
QD/PbS-EDT QD/Au……………………………………………………………………......81
4.4: Device energy band diagram under illumination. PbS-EDT acts as an electron blocking
layer and a Schottky barrier is formed for holes, thus prevent their extraction to the Au
anode….……………...……………………………………………………………………....81
4.5: Equivalent electric circuit of a double-diode model, consisting of a heterojunction diode
between ZnO and PbS-QD layers and a Schottky diode between PbS-EDT and Au………..81
4.6: I-V characteristic curves of a CQD solar cell measured at 300K (a), 250K (b), 230K (c),
200K (d), 150K (e), and 100K (f)……………………………………. ………….……….....85
4.7: (a) Current-voltage curves at various µs, while other parameters are kept constant, and
(b) solar cell FF as a function of µs……………………………………...………………..…86
4.8: (a) Current-voltage curves at various Ds, while other parameters are kept constant, and
(b) solar cell FF as a function of Ds…………………………………………...…..…………88
4.9: (a) Figure of the measured open-circuit voltage (Voc) and short-circuit current (Isc) as a
function of temperature. Equation (4.29) was used for the best-fitting of Voc. (b) Voc at
various temperatures. (c) The ratio 𝛥𝑉𝑜𝑐𝑟𝑎𝑑 / 𝛥𝑉𝑜𝑐
𝑛𝑜𝑛 as a function of temperature…………...92
4.10: Arrhenius plots of (a) the ratio Th(T)/Dh(T) and (b) the ratio Ts(T)/Ds(T). The mobilities
and diffusivities were calculated and fitted for the PbS-TBAI and the PbS-EDT interface,
respectively………………………………………………………………………………......94
4.11: (a) The CQD solar cell FFs measured at various temperatures. (b) Maximum power of
as-studied CQD solar cell measured at various temperatures…………….……………...….95
5.1: Schematic of the as-fabricated CQD solar cell sandwich structure. PbX2 and AA
represent lead halide and ammonium acetate, respectively, acting as exchange-ligands for
PbS CQDs…………………………………………………………………………………..103
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5.2: Schematic of the CQD solar cell energy band structure. PbX2 and AA represent lead
halide and ammonium acetate, respectively, acting as exchange-ligands for PbS CQDs...107
5.3: (a) Experimental data and theoretical best-fits of current density vs. voltage under
illumination and in the dark; (b) The dark current density in (a) amplified. Comparison
between Jdark - Jsc, and Jdark, as well as Jillu, as a function of voltage, is also shown in (a).
Equations (5.9), (5.11) and (5.20) were used for the best-fits of the J-V characteristics. The
best-fitted Jph at Va = 0 (representing Jsc) is 24.9 mA and 7.9×10-7 mA under illumination and
in the dark, respectively …………………………………………………………………....112
5.4: Simulated photocurrent density Jph (a) using Eq.(5.16) without Jph,diff, and (b) using Eqs.
(5.16) and (5.19) with Jph,diff at different effective carrier hopping mobilities; and (c) (Jillu -
Jdark)/𝐽𝑝ℎ𝑚𝑎𝑥 as a function of the external voltage at various effective mobilities using
Eq.(5.20)................................................................................................................................116
5.5: (a) Simulated carrier-mobility-dependent J-V characteristics; (b) open-circuit voltage
Voc, and short-circuit current density Jsc; (c) fill factor FF; and (d) power conversion
efficiency PCE (d). The CQD thin film bandgap used in 1.32 eV same as our experimentally
optimized bandgap for the CQD solar cell in Fig. 5.3. The CQD solar cell carrier hopping
mobility was estimated from our previous study [20]. Equations (5.9), (5.11) and (5.20) were
used for the simulations …………………………………………………………………....119
5.6: Theoretical simulations of CQD solar cell electrical parameters: (a) Voc and Jsc, (b) PCE,
and (c) FF as functions of CQD bandgap energy (Eg) for five different carrier hopping
mobilities. The maximum photocurrent 𝐽𝑝ℎ𝑚𝑎𝑥 is the same as Jsc at the mobility of 0.1 cm2/Vs.
The illumination intensity used for the simulation is AM1.5 spectrum at 1 sun intensity.
Equations (5.9), (5.11) and (5.20) were used for the simulations.……….………………...124
5.7: (a) Simulated CQD solar cell (a) Voc and Jsc, as well as FF and PEC (b), as functions of
the illumination intensity. Equations (5.9), (5.11) and (5.20) were used for the
simulations.............................................................................................................................127
5.8: (a) Experimental J-V characteristics; and (b) output power curves as a function of
photovoltage. Continuous lines are best fits to J-V characteristics and output power using
Eqs. (5.9), (5.11) and (5.20)…..............................................................................................129
5.9: (a) A photograph of a CQD solar cell sample; and (b) its LIC image at open circuit after
the cell was flipped over. The excitation laser was frequency-modulated at 10 Hz at a mean
intensity of 1 sun. The eight Au-coated thin film electrodes on the top in (a) are electrical
contacts while dark brown regions are without Au contact layers. Both regions have an
energy structure as shown in Fig. 5.2. The Au electrode circumscribed with a dashed
rectangle in (a) and also shown in the flipped over orientation in (b) is further studied in Figs.
5.10 and 5.11.……………………………………………………………………….…...…130
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5.10: HoLIC images of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a) at open-
circuit 0.64 V (a), 0.60 V (b), 0.56 V (c), 0.35 V (d), 0.20 V (e), and short-circuit (f). The
excitation laser was frequency-modulated at 10 Hz at a mean intensity of 1 sun.................132
5.11: LIC of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a): (a) LIC (Voc) -
LIC(Vsc); (b) [LIC (Voc) - LIC (V)] vs. V; (c) [LIC (Voc) - LIC(VPM)]VPM ; and (d) [LIC(Voc) -
LIC(V)]V vs. V characteristics. The excitation laser was frequency-modulated at 10 Hz at a
mean intensity of 1 sun. (b) and (d) are best-fitted to Eq. (5.25). Points A, B, C, and the
dashed rectangle region are shown in (a) and (c). It should be noted that values calculated for
the dashed rectangle region are based on averaging the LIC amplitudes over all pixels in this
region.………………………………………………………………………………….…...134
5.12: Open-circuit voltage Voc LIC contour mapping of the circumscribed CQD solar cell Au
electrode in Fig. 5.9.……………………………………………………..……. ……….….135
6.1: (a) Schematic of carrier hopping transport in PbS CQD thin films embedded in a
surface-passivation ligand matrix when excited by a frequency-modulated laser source. (b)
Illustration of carrier generation, dissociation, hopping transport, and trapping processes in a
CQD assembly. Se and Sh are the ground states for electrons and holes, respectively. Ea,1 and
Ea,2 are the activation energies associated with exciton binding energy (Eb) and trap-mediated
transition process, respectively. Eg and Eg, opt are, respectively, the electronic and optical
band gap energy………………………………………………………………………….....140
6.2: Photoluminescence (PL) spectra of four PbS CQD thin films surface passivated with
MAPbI3, EDT, and TBAI…………………………..……………………………….…...…146
6.3: Homodyne lock-in carrierography images of PbS-MAPbI3 (a), PbS-MAPbI3-B (b), PbS-
EDT (c), and PbS-TBAI (d) measured at 10 Hz. Note all the samples were placed on an
aluminum platform for imaging………………………….……………...…………...…….147
6.4: PCR amplitude (a) and phase (b) of MAPbI3-passivated CQD thin films (PbS-MAPbI3)
measured at various frequencies ranging from 10 Hz to 100 kHz and temperatures between
100 K and 300 K……………………………...………………………………………….....149
6.5: Best-fitted hopping diffusivity Dh (a), and Arrhenius plot of Dh for the extraction of the
carrier hopping transport activation energy (b) of the MAPbI3-passivated (PbS-MAPbI3)
CQD thin film. For the same sample, (c)-(e) are the best-fitted effective exciton lifetime 𝜏𝐸,
carrier trapping rate RT , and Arrhenius plot of thermal emission rate 𝑒𝑖, respectively. (f)
Carrier hopping diffusion length Lh calculated from the best-fitted 𝜏𝐸 and Dh values……..151
6.6: Temperature scans of the PCR amplitude for different ligands passivated PbS CQD thin
films. The continuous lines are the best fits to each set of data using Eq. (6.20)………......153
6.7:100 K PCR amplitudes (a) and phases (b) of CQD thin films passivated with four
different ligands, and the best fits to each curve using Eq. (6.18)………….………………159
xvi
6.8: The determinant of diffusivity Dh (a) and effective carrier lifetime 𝜏𝐸 (b) in the PCR
phase channel. Diamonds indicate frequencies at which linear dependence occurs; no such
linear dependencies were found for the amplitude channel of all parameters. (c) and (d) are
the sensitivity coefficients of 𝜏𝐸 in the amplitude and phase channel, respectively. Besides
the measured parameters in this figure, other parameters were also treated similarly to yield
the best-fitted values for all samples as shown in Tables 6.1, 6.3, and 6.4…………….…..163
7.1: Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs. Due to
the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between QD and
MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles……...170
7.2: Near-band-edge photoluminescence via variable radiative and nonradiative transitions.
(a) Free-exciton recombination, (b) and (c) recombination of donor (D)- and acceptor (A)-
bound excitons (DX, AX), (d) donor-acceptor pair recombination (DA), (e) recombination of
a free electron with a neutral acceptor (eA), (f) recombination of a free hole with a neutral
donor (hD)…………………………………………………………….. …………………..171
7.3: PCR amplitude vs. excitation power at three different temperatures for sample A (a) and
sample B (b), at 10 kHz laser modulation frequency………………………...…………….174
7.4: (a) Photoluminescence (PL) spectra of MAPbI3-passivated PbS (MAPbI3−PbS) thin
films (samples A and B) spin-coated on glass substrates. (b) PL spectra of MAPbI3−PbS thin
films fabricated through the same process as that of samples A and B but with different QD
sizes……………………………………………………………………………………...…177
7.5: Photocarrier radiometry (PCR) photothermal spectra of MAPbI3-passivated PbS
(MAPbI3-PbS) thin films spin-coated on glass substrates, samples A (a) and B (b). (c)
Arrhenius plots of the PCR phase troughs, I, II, and III, as shown in (a), and best-fitted to Eq.
(7.9) for the extraction of activation energies for each sub-bandgap trap level…………...180
7.6: (a) Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs. Due
to the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between QD
and MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles (a);
(b) energy band structure (assumed n-type) of a PbS- MAPbI3 nanolayer, sample A, featuring
shallow and deep level trap states. Excitons are excited in the right QD and diffuse through
nearest-neighbor-hopping (NNH) to the next QD, during which process the coupling strength
between two QDs dissociate excitons into free charge carriers. Carriers may experience
radiative recombination or captured by different types of trap states, where non-radiative
recombination or de-trapping may occur………………………………..……….…………181
7.7: Photos of MAPbI3 -PbS thin films, (a) sample A and (b) sample B. 1 kHz homodyne LIC
amplitude images of MAPbI3- PbS thin films, (c) sample A and (d) sample B. 20 kHz
heterodyne LIC amplitude images of MAPbI3-PbS thin films, (e) sample A and (f) sample B.
Note the very different signal strength scales associated with the two samples…………...182
xvii
7.8: Phase diagram of PCR frequency scans in three different regions 1-3 as shown in Fig.7.7
(c)-(d) and the best-fits of experimental data to Eq. (6.18) in Sect.6.2………………….…184
7.9: PCR phase dependence on time over 25 minutes, the duration of a PCR frequency scan.
Sample A at 100 kHz laser modulation frequency…………………………………………187
8.1: Schematic of CQD solar cell sandwich structure (a), and the corresponding band energy
structure (b) also showing the illumination depth profile, the photocarrier density wave
distribution and the intrinsic and external electric fields……………………………..…….194
8.2: (a) A photograph of the CQD solar cell sample under study, and (b) the corresponding
HoLIC image of this solar cell. The dashed-rectangle-circumscribed solar cell A is selected
for further studies as shown in Figs. 8.3 and 8.4. The HoLIC characterization was carried out
at 10 Hz. It should be noted that for carrierographic imaging, the sample was flipped over
with the top Au contact on the bottom, resulting in mirror image positions being assumed in
(a) and (b) by the dashed rectangles and inscribed solar cells……………………….……..198
8.3: Current-density-voltage characteristic of the CQD solar cell shown circumscribed by a
dashed rectangle in Fig.8.2 (a) …..........................................................................................199
8.4: Frequency-dependent PCR phase spectra of the solar cell electrode units A, B, and area
C without Au contact (Figs. 8.2) at 200 K. Equation (8.15) is used for the best fitting of each
curve. The characterization spot area of the single-detector based PCR is the same as the area
of the circular Au contact tip.…............................................................................................200
8.5: High-frequency HeLIC images at 1 kHz (a) and 100 kHz (b) for the CQD solar cell
shown in Fig. 8.2.…...............................................................................................................200
8.6: The frequency-dependent average HeLIC image amplitudes of the CQD solar cell shown
in Fig. 8.2. The HeLIC images in Fig.8.5 are also included.………………….....................201
8.7: (a) 400 Hz HeLIC image of the CQD solar cell region E, Fig. 8.2, and its carrier lifetime
τ image (b) at 200 K. (c) For comparison, the carrier lifetime image of the same electrode E
at 293 K. (d)-(f) are images of carrier diffusivity, diffusion, and drift lengths, respectively, at
temperature 200 K.................................................................................................................203
8.8: HeLIC images of a CQD solar cell at different modulation frequencies 1 kHz (a), 10 kHz
(b), 50 kHz (c), and 100 kHz (d) as labeled. …………………………………………….....206
8.9: 270 kHz HeLIC images of the same CQD solar cell shown in Fig. 8.8………….........206
8.10: (a) Frequency-dependent HeLIC image average amplitude for regions C and D of the
CQD solar cell shown in Fig. 8.8(a) without plasma etching; and (b) lifetime imaging of the
same CQD solar cell. (c) Furthermore, bar-plotted lifetime statistical distribution for the
above CQD solar cell without plasma etching and another CQD solar cell of the same type
except with 15 s plasma etching………………………………………...………...……..…207
xviii
8.11: (a) Lifetime image of two adjacent CQD solar cell units A and D which reveals the
device homogeneity and influence of electrode contacts on carrier hopping transport, and (b)
the barplot of carrier hopping lifetime image in (a). It should be noted that the top Au
contacts as shown in Fig. 8.2 (a) were on the bottom through flipping the sample over for all
the HeLIC imaging…………………………………………………………………………210
1
Chapter 1
Preamble
1.1 The Imperative of Solar Cells
The importance of solar cells arises from its capabilities in solving the global problems of
the energy crisis and greenhouse gas-induced climate change. Originating from scarce fossil
fuels and emitted carbon dioxide from combusting these fossil fuels, the energy crisis and
climate change are two significant and exigent worldwide problems that humanity needs to
solve in the next several decades. Historically, global economic growth significantly relies on
the consumption of fossil fuels including coal, gas, and oil [1]. In the long term, these sources
of energy are scarce in storage, threatening human beings with energy poverty, especially in
developing countries [1]. However, due to their high energy density, fossil fuels remain the
main energy source driving the world economy while the energy crisis and climate change can
be solved with the application of renewable energies such as wind, solar, biomass, and
geothermal energies. Although nuclear energy does not emit carbon dioxide, they are non-
renewable energy resources and the nuclear waste is radioactive, which can be hazardous to
human health for thousands of years. Germany has announced an end to all nuclear energy
production by 2022 and committed to replacing them with renewable energy. Compared with
other types of renewable energy, solar energy is abundant, renewable, compatible with the
environment, and can be installed wherever there is sunlight. Solar energy is, therefore, a more
suitable renewable energy source for solving the global problems of the energy crisis and
climate change.
The solar photovoltaic (PV) market has seen remarkable growth with a five-year (2012-
2016) average growth rate of about 22% [2]. The growth rate in 2015 was as high as 28%. At
CHAPTER 1. 2
the end of 2015, about 22 countries had enough solar PV electricity capacity to meet more than
1% of their electricity demands; for example, Italy at 7.8%, Greece at 6.5%, and Germany at
6.4% [2]. According to the U.S. energy information administration [as shown in Fig.1.1(a)],
renewable energy is projected to increase to more than 10 TWh in 2040, in which year the total
worldwide electricity generation is estimated at ca.35 TWh [3]. Among all the renewable
sources, solar electricity generation is the fastest-growing with an 8.3% growth rate per year.
Additionally, as shown in Fig.1.1(b), the rapidly growing solar PV market increased
employment by 5 % in 2015, totaling 8.1 million jobs in the renewable energy market [2].
Among these, solar PVs have created 2.772 million jobs worldwide, and 60% of which are in
China.
Figure 1.1: (a) Global net electricity generation from renewable energy sources from 2012 to
2040. The unit is trillion kilowatthours. It should be noted that the other generation includes
waste, biomass, and tide/wave/ocean. Adapted from ref. [3]. (b) Total jobs of 8.1 million were
created in the market of renewable energy by the end of 2015. Moreover, 60% of these solar
energy jobs were created in China. Adapted after [2].
Due to the limited roof area, normal residential installations of solar panels are insufficient
for the electricity consumption of traditional living houses. Strategies of combining solar PV,
solar thermal, and mechanical system optimization have been attempted. Comparatively,
solution-processed solar cells (such as colloidal quantum dot solar cells) are more promising
due to their capability of spray coating and the relatively transparent property that can be
CHAPTER 1. 3
deposited on walls and windows, therefore, generating higher energy outputs. The novel
solution-processed colloidal quantum dot solar cells have high potential for the realization of
large-area, flexible, light-weight, and roll-to-roll processed solar cells. These properties can
further decrease solar PV electricity cost, and their large-scale fabrication feasibility facilitates
Swanson’s law while also being suitable for the markets of automobiles and smart zero-energy
buildings.
1.2 The Imperative of Nondestructive Testing of Photovoltaics
The semiconductor associated industries have been growing rapidly for decades, while new
materials, fabrication techniques, device architectures, and new concepts of optoelectronics
and photovoltaics are still emerging. The rapid growth of these semiconductor industries
significantly relies on the increasing abundant knowledge of semiconductor materials and
devices, which relies on various testing and characterization techniques [4, 5]. Testing and
characterization techniques have played and will continue to play essential roles in
semiconductor industries with their applications ranging from raw material fabrication to
device design and manufacturing. With regards to photovoltaics, to guarantee high-quality and
high-stability photovoltaic products, the various solar cell fabrication processes require precise,
non-destructive, and fast monitoring at all fabrication stages. Hence, fast, in-line, non-contact,
and non-destructive characterization techniques are becoming more and more urgent for large-
scale solar cell manufacturing and utility-scale installations as one bad solar panel is fatal to
the entire PV system.
Following Swanson’s law, to decrease solar PV electricity cost, large-area PV devices
prevail nowadays. Therefore, large-area imaging characterization techniques are garnering
significant interest either in in-line industrial photovoltaic manufacturing or off-line
CHAPTER 1. 4
photovoltaic solar panel maintenance. However, advanced knowledge is established on small-
spot rather than large-area based characterization techniques such as electrical current-voltage
characteristics and photoluminescence techniques. Hence, non-destructive and large-area
imaging techniques are in demand for product quality and stability monitoring, and for the
characterization of critical semiconductor electrical and optical properties including carrier life,
mobility, diffusivity, doping concentration, and defect and trap states. In addition to the need
for large-area characterization, due to the fragile nature of new-generation solar cells including
organic, colloidal quantum dot, and polymer solar cells, full optical and non-destructive
techniques are also essential for photovoltaics. Characterization techniques that need to contact
with photovoltaic materials and devices pose potential damages, even to Si-based solar cells.
Therefore, full optical non-destructive techniques have been developed through static
photoluminescence [6, 7] and frequency modulated quasi-steady-state techniques [8] to extract
various solar cell parameters. Although conventional electrical parameters including open-
circuit voltage, resistance, and short-circuit current have been investigated, carrier transport
dynamic properties such as carrier lifetime, diffusivity, mobility, and diffusion length have not
been comprehensively studied. These parameters, however, are indispensable for solar cell
efficiency optimization, and particularly for the understanding of fundamental working
principles of new-generation solar cells. Therefore, nondestructive testing techniques or
photovoltaics are imperative and urgent.
1.3 Objectives of This Work
Figure.1.2 shows the structure of this work that consists of three essential components: the
theoretical studies of carrier hopping transport and current-voltage characteristics in CQD
systems, large-area and/or high-frequency all-optical NDTs (HoLIC, HeLIC, and PCR), and
CHAPTER 1. 5
CQD thin film materials and solar cells. This thesis focuses on the intersection of either
two/three of the above components and starts with the study and development of theoretical
carrier transport and electrical current-voltage models, which are the basis for the theoretical
derivation of signal generation analytical models for NDTs used for characterizing CQD
materials and solar cells in this thesis. The understanding of carrier hopping transport behavior
and NDT signal generation mechanisms plays key roles in interpreting electrical and optical
properties of CQD materials and devices for solar cell efficiency optimization. The results of
CQD solar cell optimization provide feedbacks to theoretical understanding and models,
therefore, forming a complete and self-consistent system.
Figure 1.2: Structure of the research discussed in this thesis. NDT = nondestructive testing;
HoLIC = homodyne lock-in carrierography; HoLIC = heterodyne lock-in carrierography; PCR
= photocarrier radiometry; CQD = colloidal quantum dot; CQDSC = colloidal quantum dot
solar cell.
As shown in Fig. 1.2, the final objective of this research is to improve CQD solar cell
efficiency and two approaches were attempted: first, better understanding of carrier transport
and recombination dynamics, and J-V characteristics in CQD solar cells; second, dynamic
high-frequency PCR and HeLIC study of carrier transport behaviors in CQD materials and
CHAPTER 1. 6
solar cells, as well as the large-area imaging of CQD systems using HoLIC and HeLIC for
homogeneity and quality monitoring. Therefore, the sub-objectives generated within this
project can be summarized as follows
a. Understand how excitons and charge carriers transport and recombine in the emerging
cutting-edge CQD thin films and solar cells, and how they determine the current-voltage
characteristics of these high-efficiency CQD solar cells [9-11].
Solution-processed CQD thin films and solar cells are different from traditional Si
counterparts with respects to confined carriers, strong exciton binding strength, significant
spatial and energy disorder associated discrete energy states, and the more than three orders of
magnitude lower in carrier mobility. These have led to different experimental electrical and
optical properties. To further increase the efficiencies of CQD solar cells, carrier transport and
recombination dynamics, and J-V behaviors in CQD systems should be better understood.
b. Develop dynamic NDT (i.e. HeLIC, PCR, and HoLIC) signal generation mechanisms
and theoretical models for the characterization of CQD thin films and solar cells [12-17].
HoLIC and HeLIC are frequency-modulated large-area imaging techniques that show
enormous potential in industrial inline roll-to-roll solar cell quality monitoring, while PCR and
HeLIC can perform high-frequency characterization of high-rate carrier transport behaviors.
The development of dynamic NDT (i.e. HeLIC, PCR, and HoLIC) signal generation analytical
models for CQD systems are useful for basic physical studies of energy transport and loss
mechanisms in CQD materials and devices and for industrial quality inspection of
photovoltaics.
Attempts for the first sub-objective are discussed in Chapters 4 and 5, while the second
sub-objective is accentuated in Chapters 6-8.
CHAPTER 1. 7
1.4 Thesis Outline
Chapter 2 reviews the fundamentals of QDs and CQD solar cells, starting from the
solution-processed CQD synthesis methods, which are followed by the discussion of CQD
electrical properties and their dependence on interdot distance, temperature, spatial and energy
disorder, QD size, and QD size polydispersity. This chapter ends with the review of CQD solar
cell architectures, working principles, and efficiency-limiting factors.
Chapter 3 first presents a brief overview of current conventional NDTs that have been used
widely in characterizing carrier transport parameters of QD materials and devices. These
conventional techniques include short-circuit current/open-circuit voltage decay,
photoconductance decay, transient photoluminescence, and various diffusion length
characterization methodologies. The differences and connections between these techniques
and the novel PCR and HeLIC are discussed based on the discussion of the instrumentation
and signal generation theories of these two new techniques. Furthermore, nonlinear
photoluminescence response to laser excitation serving as a prerequisite for HeLIC is
experimentally demonstrated and theoretically analyzed through quantitative models. The
advantages of PCR and HeLIC techniques are also addressed.
Chapter 4 is concerned with the discrete carrier hopping transport behavior and the
associated carrier drift-diffusion current-voltage characteristics in CQD solar cells [9, 10].
Novel definitions of carrier hopping diffusivity and mobility are presented from the description
of discrete carrier hopping nature in CQD thin films. Drift-diffusion current-voltage models
are detailed and act as the basis for the development of a double-diode electrical model to
interpret anomalous current-voltage behaviors observed in CQD solar cells, as these behaviors
were found to reduce CQD solar cell efficiency significantly. The mechanism of open-circuit
CHAPTER 1. 8
dissipation at interfaces was quantitatively analyzed, while the phonon-assisted carrier
hopping transport was revealed through the obtained carrier transport dynamics.
Chapter 5 gives an account of a new CQD solar cell efficiency optimization study [11]
based on the theoretical findings in Chapter 4. Common senses for CQD efficiency
optimization such as constant photocurrent, and high mobility and low energy bandgaps
leading to high efficiency were examined to be invalid. The improved current-voltage model
explored the effects of carrier hopping mobility, bandgap energy, and illumination intensity on
CQD solar cell efficiency. Large-area quantitative imaging of open-circuit voltage and carrier
collection efficiency was addressed to show CQD thin film/metal contact interface influence
on CQD solar cell
Chapter 6 focuses on PCR study of phonon-assisted carrier hopping transport dynamics in
CQD thin films [12, 14]. Combing the novel trap-state-mediated carrier transport model, an
analytical expression for PCR signal generation was developed for CQD thin films. The
influence of temperature, dot-size, and interdot linking ligand on carrier transport kinetics was
studied in various CQD thin films. Additionally, the trap-state-mediated carrier hopping
activation energies are also discussed.
Chapter 7 studies the trap-state-mediated exciton and charge carrier transport in
methylammonium lead triiodide (MAPbI3) perovskite-passivated PbS CQD thin films using
PCR [13]. Carrier recombination mechanisms in CQDs were quantitatively described.
Furthermore, the shallow and deep level trap states were characterized through the PCR
photothermal spectra and static photoluminescence, respectively, which result in the
construction of a complete band energy structure of the CQD ensemble. The interpretation of
CHAPTER 1. 9
large-area HoLIC and HeLIC images of CQD thin films for preliminary carrier lifetime and
homogeneity was discussed.
Chapter 8 reports the large-area carrier transport parameter imaging of CQD thin films and
solar cells using HeLIC [15, 16, 17]. The heterodyne signal processing principles of HeLIC
are developed. Furthermore, the novel theoretical models for HeLIC and HoLIC were derived
and were combined with the excess carrier density wave in CQD solar cells to yield the final
demodulated analytical expressions for both PCR and HeLIC. The applications of HoLIC are
principally for material and device homogeneity and quality characterization. Using HeLIC,
quantitative imaging of carrier hopping lifetime, diffusivity, and drift and diffusion lengths
was achieved, further exploring the physical carrier transport dynamics and interface influence
on CQD solar cell efficiency.
Chapter 9 summarizes key contributions and findings and emphasizes future potential
investigations as well as possible applications of the yielded knowledge in this thesis.
10
Chapter 2
Introduction to Quantum Dots and Colloidal Quantum Dot Solar
Cells
Due to the confined particle motion in three spatial dimensions, the unique optical and
electrical properties of QDs make them promising candidates in fabricating low-cost, large-
area, lightweight, flexible, high-efficiency photovoltaic solar cells. Therefore, this chapter will
introduce QDs and CQD solar cells from solution-based CQD fabrication methodologies to
discussing effects of ligand-determined interdot distance, temperature, spatial and energy
disorders, dot size, and dot size polydispersities on QD electrical properties. Lastly, a review
of CQD solar cells is focused on their fundamentals, classification, working principles, and
efficiency-limiting factors.
2.1 Synthesis of Colloidal Quantum Dots
Two main approaches have been developed for synthesizing QDs in the last several
decades: (1) solution-processed colloidal chemistry methods, and (2) lithographic growth
which includes the subsequent processing techniques such as various deposition and etching
methodologies. The colloidal chemistry fabrication starts with the rapid injection of
semiconductor precursors into a hot and vigorously stirred organic solvent, which contains
organic molecules with long chains and can coordinate with the precipitated CQD particles on
the surface. Through proper surface engineering using various ligands, these water-soluble
CQDs are suitable for different optoelectronic and photovoltaic applications. In comparison,
the lithographic growth of QDs is more time-consuming and expensive. Moreover, QDs
fabricated through this technique are more easily contaminated during the fabrication process.
CHAPTER 2. 11
The contamination can introduce various material defect states, a high degree of dot size
polydispersity, and poor interface quality. There are two types of QD epitaxy growth
techniques: the vapor phase epitaxy (VPE) and the liquid phase epitaxy (LPE). LPE is
commonly used for fabricating semiconductor materials on the micro-scale but infrequently
used in QD fabrications. The metalorganic vapor phase epitaxy (MOVPE) using a
metalorganic medium, and the molecule beam epitaxy (MBE) through the Stranski-Krastanov
growth mode are the two main techniques that have been widely used for QD synthesis. For
example, MBE has been widely implemented in investigating single-photon sources and
quantum computation. When compared with lithography growth, although epitaxy growth can
produce QDs with relatively higher quality, it is uncommonly used for large-scale QD
fabrications.
With respect to QD quality and size polydispersity, QD synthesis through the pyrolysis of
metalorganic precursors is the most successful nanoparticle preparation method. A detailed
review of such techniques has been reported by Wang et al. [1]. As shown in Fig.2.1, the QD
preparation mechanism generally can be understood through La Mer and Dinegar’s model in
the way that precursor nucleation occurs through the rapid injection of QD precursors into a
coordinating organic solvent [2]. In the organic solvent, semiconductor precursors thermally
decompose into monomers at a temperature ranging from 120 oC to 360 oC, a process to
increase the monomer concentration in the solvent. When the monomer concentration
surpasses the nucleation threshold concentration, nucleation processes begin and nanoparticles
grow quickly through absorbing monomers from the solution-phase. New nuclei can no longer
be formed once the monomer concentration is smaller than the critical nucleation threshold
concentration, which keeps a constant population of CQDs while the dot size continues to grow
CHAPTER 2. 12
through absorbing more monomers in the solution. This process will continue until the
monomers are depleted. Due to the depletion of monomers in the solution, the CQD growth
process evolves to the Ostwald stage, where smaller CQDs dissolve into monomers because
of their higher surface energy. The dissolved monomers contribute to the further growth of
large CQDs. In other words, the concentration of CQDs in the solution reduces with time,
while the dot size increases, as shown in Fig.2.1. However, La Mer and Dinegar’s model for
CQD synthesis is a simplified mechanism without considering the concurrence of
semiconductor precursor nucleation and the CQD growth. Furthermore, ligands in the solution
may additionally influence the nucleation process.
Figure 2.1: (a) Schematic of the La Mer and Dinegar’s model for the synthesis of
monodispersed CQDs. (b) Representation of the apparatus employed for CQD synthesis.
Adapted from ref. [2].
CHAPTER 2. 13
Figure 2.2: Transmission electron spectroscopy (TEM) imaging of CdSe QDs: low-resolution
(a) and high-resolution (b) and (c). Adapted from ref. [3]. \
In addition, for a better control of the CQD synthesis process, slow temperature ramping
can be used to trigger the precursor supersaturation and nucleation. Furthermore, proper
temperature control can also be implemented to avoid additional nucleation processes. As
shown in Fig.2.1, the CQD size can be feasibly adjusted through changing the hydrothermal
fabrication time. This technique generally can fabricate CQDs with a dot size distribution of <
7 %, which can further be reduced to less than 5 % through various purification methods. As
an example, for CQD dimensions, the transmission electron spectroscopy images of CdSe
CQDs at high- and low-resolution are shown in Fig.2.2.
2.2 Electrical Properties of Colloidal Quantum Dots
Transport of free charge carriers and excitons is of great interest to better understand
fundamental carrier transport dynamics and energy loss mechanisms in QD systems. Both the
QD dimensions and the surrounding environment have a significant influence on electrical
properties of QD systems. Moreover, QD electrical properties play decisive roles in QD-based
photovoltaic devices. Therefore, this section discusses the dependencies of temperature,
interdot distance, dot size, QD size polydispersity, and spatial and energy disorder on carrier
transport in CQD thin films.
CHAPTER 2. 14
2.2.1 Effects of Interdot Distance and QD Disorder
Figure 2.3: Different carrier transport mechanisms for QD ensembles: (a) bulk crystal-like
Bloch state carrier transport; (b) carrier tunneling transport from one dot to another without
phonon assistance; (c) carrier transport with over-the-barrier activation energy mechanism;
and (d) hopping transport with phonon-assistance. Adapted from ref. [4].
Depending on the QD spatial and energy state disorder, there are four potential carrier
transport mechanisms in CQD ensembles as shown in Fig.2.3 [4]. Generally, at room
temperatures, without specifying the quantum dot materials and quantitative characterization
techniques, all four transport mechanisms are possible. First, as shown in Fig.2.3 (a), bulk
crystal-like Bloch states will be formed when the inter-dot distance is sufficiently small with
a quasi-monodispersed dot size distribution in the whole QD matrix. In this case, strong
coupling strength is formed and leads to a short-range continuous energy band in the QD thin
films. Crystal-like Bloch states are not common in real-world QD systems due to the inevitable
dot size polydispersity, which originates even from today’s state-of-the-art techniques as
discussed in Sect.2.1. For crystal-like Bloch states with extended energy bands, due to the
strong interdot coupling strength, excitons dissociate into free electrons and holes immediately
CHAPTER 2. 15
upon generation. Second, Fig.2.3 (b), with the increase of interdot distance or dot size
polydispersity, the tunneling mechanism starts to dominate the carrier transport behavior in
QD thin films. In this situation, charge carriers can transport from one QD to its neighbors
without phonon-assistance as the interdot coupling strength is still sufficiently strong. Third,
Fig. 2.3 (c), the over-the-barrier activation mechanism is dominant when the energy barrier
between two QDs are low enough that charge carriers and excitons can be thermally excited
to transport over the energy barriers. In other words, for CQD systems in this regime, carriers
can transport freely over the energy barrier from one QD to another.
More importantly, fourth, Fig.2.3 (d), phonon-assisted hopping is the most extensively
observed mechanism in QD systems [5-11]. Depending on the interdot distance, coupling
strength, temperature, and QD dimensions, charge carriers can hop from one dot to its
neighboring dots with the assistance of one or multiple phonons. As the population of phonons
is temperature-dependent, phonon-assisted hopping transport is also temperature-dependent.
This temperature-dependent property leads to increased mobility and conductivity with the
increase of temperatures, a contrasting phenomenon to bulk semiconductors, in which carrier
mobility and conductivity decrease with temperatures because of the enhanced carrier
scattering probability with phonons at high temperatures. Therefore, the interdot coupling
strength that increases with reduced ligand-determined interdot distances has a significant
influence on carrier transport behavior in QD ensembles. Through enhancing interdot coupling
strength, it was found that the electrical properties of a PbSe QD ensemble evolve from the
Coulomb blockade dominated insulating regime to a hopping conduction featured
semiconductor regime [5, 12]. Furthermore. high interdot coupling strength can also assist
CHAPTER 2. 16
exciton dissociation into free electron and hole charge carriers during carrier tunneling or
hopping processes [13, 14].
2.2.2 Effects of Temperature
Carrier transport is also temperature-dependent. With the decrease of temperature, carrier
transport mechanism can evolve from the nearest-neighbor-hopping (NNH) to the Efros-
Shklovskii-variable-range-hopping (ES-VRH). The threshold temperature has been reported
as ca. 200 K by refs. [6, 8, 10], while the further investigation found that the threshold
temperatures could be affected by the QD size [8]. NNH and ES-VRH exhibit the same
temperature-dependent influence on solar cell current density and conductivity through
phonon-assisted carrier hopping transport dynamics. In other words, at high temperatures, the
high population of phonons facilitates carrier hopping transport mobility [5, 9]. The
dependence of conductivity on temperature for hopping conduction takes the general form:
𝜎 = 𝜎0𝑒𝑥𝑝[−(𝑇∗/𝑇)𝑧] (2.1)
where 𝜎0 is the conductivity pre-exponential factor, 𝑇∗ is a fitting parameter with a unit of
Kelvin, and z is a parameter associated with different hopping transport mechanisms.
Specifically, as shown in Fig.2.4, z = 1 is for the NNH carrier transport within CQDs, while z
= 0.5 is for the ES-VRH. Moreover, for Mott variable-range-hopping (M-VRH), z is equal to
0.25 in a three-dimensional transport model and is equal to 0.33 for a two-dimensional carrier
transport model.
At low temperatures, due to strong spatial and energy disorders, ES-VRH is the dominant
carrier transport mechanism for CQD systems with localized states. Carriers with an initial
state energy Ei can be thermally activated and hop to a nearby energy state with an energy of
Ef. The hopping probability is determined by the energy difference between these states, i.e.
CHAPTER 2. 17
ΔE = Ei - Ef, and by the hopping distance. In other words, a large state energy difference with
short hopping distance facilitates the ES -VRH hopping process, as shown by the carrier
hopping paths I and II in Fig.2.4. However, the NNH tends to take place at high temperatures,
such as the hopping path I shown in Fig. 2.4, in which carriers hop from a 1Se state in CQD 1
to another 1Se state or a trap state in CQD 2. Moreover, carriers can also hop between trap
states such as the one from CQD 3 to CQD 2, Fig. 2.4. Furthermore, the conduction mechanism
can transfer from ES-VRH to M-VRH if the state energy difference ΔE is equal to the Coulomb
gap energy δ. For weakly coupled QDs, the Coulomb gap energy can be approximated to δ
≈2Ec, in which Ec is the charge energy. The charge energy is the energy required to add or
remove a charge carrier from a QD. For a spherical QD, the charge energy can be obtained
through
𝐸𝑐 =𝑒2
4𝜋𝜀𝑟 (2.2)
where 𝑟 the radius of the QD is, 𝑒 is the elementary charge, and 휀 is the material permittivity.
Figure 2.4: Schematic of charge carrier transport within a colloidal quantum dot array. Energy
states including trap states and Fermi levels are represented by solid and dashed lines,
respectively. Nearest-neighbor hopping (I) and variable range hopping (I+II) occur through
carrier transport within quantized states (long solid lines, 1Se and 1Sh for electrons and holes,
respectively) and surface trap states (short solid lines). The variation in quantum dot creates
energy and spatial disorders that weaken interdot coupling effects and disturb carrier transport.
Adapted from ref. [8].
CHAPTER 2. 18
Figure 2.5: Current-voltage characteristics at different temperatures for PbSe CQD thin films
vacuum-annealed at 473 K (a) and 523 K (b). The lower right insets in both figures show the
Arrhenius plots of the conductivity G which is proportional to 𝜎 in Eq. (2.1). The upper left
insets illustrate the TEM images of PbSe CQD arrays after vacuum annealing. Adapted from
[5].
Romero et al. [5] used the abovementioned model in Fig. 2.4 to study the carrier transport
behavior in PbSe CQDs, which are surface-capped with oleic acid. It was found that for CQD
thin films annealed in vacuum at a lower temperature (373 K), the Coulomb blockade was the
dominant influential factor of the carrier transport and led to an insulating conductivity
property in PbSe CQD thin films. The insulating behavior also originates from the higher
charge energy 𝐸𝑐 of 36 meV than the thermal energy of ~32.5 meV at 373 K. In contrast, under
high annealing temperatures (such as 473 K), the interdot distance was found to be obviously
reduced, leading to a conductive electrical property. As shown in Fig.2.5, the fitting of
conductivity to Eq.(2.1) reveals a 𝑧 value of 0.95~1.05 and 0.48-0.55 for high and low
temperatures, respectively, which indicates that the NNH carrier hopping process dominates
at high temperatures while ES-VRH dominates at low temperatures.
CHAPTER 2. 19
2.2.3 Effects of Quantum Dot Size and Polydispersity
In QD thin films, different charge carriers (electrons and holes) exhibit different transport
dependencies on the QD size. Lee et al. [15] and Liu et al. [7] found that both electron and
hole mobilities increased by 1-2 orders of magnitude with the growth of the PbSe QD size.
Specifically, carrier mobility is generally found to increase with dot size, while further
investigation showed that electron mobility decreased when the quantum dot size was further
increased [Fig. 2.6 (a)], which led to an optimized mobility peaked at a QD diameter of ca. 6
nm. In contrast, hole mobility shows a monotonic increase with the QD dot size as shown in
Fig. 2.6 (b). The increase of carrier mobility with QD diameter can be attributed to the reduced
activation energy in large QDs, while the decrease of electron mobility for further increased
QD size can be ascribed to the weakened electronic coupling strength amongst large QDs as
discussed by Lee et al. [15]. Furthermore, both electron and hole mobilities drop exponentially
with the increase of ligand length, consistent with the framework of the hopping/tunneling
transport mechanism, i.e. short interdot distance results in higher coupling strength and
narrower energy barrier width.
Figure 2.6: Experimental and modeled electron (a) and hole (b) mobilities of PbSe CQDs of
different diameters. Dependence on CQD interdot distances is also illuminated. Adapted from
[15].
CHAPTER 2. 20
In addition, the QD size polydispersity has been found to significantly deteriorate the
interdot coupling strength, therefore, playing key roles in determining charge carrier transport
in CQD ensembles. Within CQD ensembles, due to the dot-size-dependent quantum
confinement effects, CQDs with varied sizes correspond to different bandgap energies, hence,
the size polydispersity results in a bandgap spectrum. Meanwhile, large CQDs with small
energy bandgap can act as trap states for carrier transport. For CQDs with a given/fixed density
of surface trap states, Liu et al. [7] discovered that carrier mobility was independent of CQD
size polydispersity. This finding was also confirmed by Zhitomirsky et al. [16] by showing
that the CQD polydispersity had a negligible impact on photovoltaic device performance with
a fixed concentration of surface trap states. However, through decreasing surface traps to a
very low level, further studies elucidated that an improved photovoltaic device performance is
achievable when CQD size polydispersity can be successfully suppressed [16].
2.3 Colloidal Quantum Dot Solar Cells
Colloidal quantum dot solar cells have attracted considerable attention due to their much
higher theoretical solar energy to electricity conversion efficiency of ~ 65% than the Shockley-
Queisser limit for Si solar cells (33%) [17, 18-20]. This is due to the tunable CQD energy
bandgap through effective dot-size control, and due to solution-oriented fabrication processes,
which are suitable for fabricating low-cost, flexible, light-weight, and large-area photovoltaic
solar cells. CQD-sensitized solar cells and CQD heterojunction solar cells are the two
prevailing CQD solar cell architectures of intense research interest [21]. Although CQD solar
cells have been fabricated with various architectures, typical CQD solar cells have four main
components: a transparent conduction electrode (indium tin oxide and fluorine tin oxide), a
metal oxide semiconductor thin film with a thickness from tens of nanometers to several
CHAPTER 2. 21
hundred nanometers, light absorbing QD thin films of several hundred nanometers or thinner
(depending on the carrier diffusion length), and a metal electrode such as gold (Au). However,
the first generation CQD solar cells are based on Schottky diodes as shown in Fig.2.7 (a) and
(b) [21].
Figure 2.7: Configuration of different types of CQD solar cells, Schottky (a, b), heterojunction
(c, d), and CQD-sensitized solar cells (e, f). The top row (a, c, and e) illustrates the device
structures and the bottom row (b, d, and f) depicts the energy band structures with carrier
transport mechanisms. It should be noted that, for simplification, only the lowest energy state
levels are shown (i.e., 1S and 1P). Adapted from ref. [21].
Formed at the electrode/CQD thin film interface, the Schottky diode induces a depletion
region with an intrinsic electric field to drift electrons and holes to the Al (anode) and ITO
(cathode) electrodes, respectively. CQD thin films are light absorption layers and generate
excitons upon photoexcitation. The photoexcited excitons diffuse along the exciton density
gradient from the ITO layer, where the excitation light impinges. Meanwhile, the interdot
coupling strength can dissociate these electron-hole pairs (i.e. excitons) into free charge
CHAPTER 2. 22
carriers. Furthermore, excitons can also dissociate at the CQD/Al interfaces. However, this
process delimitates the device efficiency as hole charge carriers need to travel all the way back
to the ITO electrode. In addition, the interface-induced carrier recombinations will be
simultaneously enhanced. Figure 2.7 (b) shows that the dissociated electrons diffuse in the
quasi-neutral region and drift in the depletion region to reach the Al electrode. Generally, open-
circuit voltage, Voc is a function of the energy difference between the quasi-fermi levels of
electrons 𝐸𝐹,𝑛 and holes 𝐸𝐹,𝑝 in the form:
𝑉𝑜𝑐 =𝐸𝐹,𝑛−𝐸𝐹,𝑝
𝑞 (2.3)
With the application of metal electrodes, the quasi-fermi levels can be approximated by the
work functions of the corresponding metals, which can also be seen in Fig.2.7. Schottky-diode-
based CQD solar cells endure low fill factors (FF) and Voc for a given short-circuit current
density Jsc. In addition, as shown in Fig.2.7 (b), hole charge carriers can also be injected into
the electron extraction electrode due to the low energy barriers formed at the Schottky diode,
which results in enhanced carrier recombinations and decreased CQD solar cell efficiency. In
comparison, CQD sensitized solar cells consist of a CQD sensitized photoelectrode [titanium
dioxide, TiO2, Figs.2.7 (e) and (f)] and a counter electrode, which are separated by a liquid
electrolyte. In CQD-sensitized solar cells, CQDs act as the light absorbing layers. Due to the
thinness of the CQD layers, solar cells of this type tolerate low Jsc, however, enhanced fill
factor FF and Voc can be obtained. Heterojunction CQD solar cells [Figs.2.7 (c) and (d)] follow
similar carrier extraction mechanisms to Schottky diode based CQD solar cells, except for
additional exciton dissociation sources that arise from the heterojunction interfaces. Metal
oxide materials such as TiO2 and ZnO in nanostructures are often used as an n-type
semiconductor component of the heterojunction pn junction structure. The depleted
CHAPTER 2. 23
heterojunction structure can simultaneously maximize FF, Voc, and Jsc. Therefore,
heterojunction structures have been reported to yield various CQD solar cells with high
efficiencies, although different QD materials or electrodes were used [21-23].
Figure 2.8: Structure schematic (a) and energy band diagram (b) of the planar depleted CQD
heterojunction solar cells at short circuit. The energy diagram is plotted along the A-A’ cross-
section. Correspondingly, the structure schematic (c) and energy diagram (d, a cross-section
along B-B’) for bulk heterojunction is also illuminated. The vertical ZnO nanowires were
grown using solution-processed hydrothermal methods to produce ordered nanowire array
within the PbS CQD thin film. Adapted from ref. [22].
Typically, there are two types of heterojunction CQD solar cells: planar depleted
heterojunction and bulk heterojunction [22, 23]. Figure 2.8 shows the device structures and
working principles of these two categories of heterojunction CQD solar cells. As shown in
Figs.2.7 (e) and (f), planar heterojunction solar cells have the typical structures as discussed
above. Figures 2.8 (a) and (b) also exhibit the structure of planar heterojunction CQD solar
cells, which consist of ZnO nanoparticles and PbS CQDs. The depletion region is formed and
centered at the ZnO/CQD thin film interfaces and extends into the CQD thin films. To harvest
more charge carriers for high Jsc, high-efficiency CQD solar cells need to absorb more light,
which depends directly on the thickness of the CQD light absorber. However, as carriers need
CHAPTER 2. 24
to travel across the entire thin film to be extracted by the respective electrodes, the thickness
of CQD thin films is limited by the exciton or free charge carrier diffusion lengths (generally,
in several hundred nanometers). Thereby, the thinness of light absorber CQD layers, which is
limited by carrier diffusion lengths restricts the further enhancement of short-circuit current
density, Jsc.
To increase Jsc, bulk heterojunction CQD solar cells are designed with the application of
pillars [24] or nanowire arrays [22, 25-30], which are interpenetrated into CQD thin films.
These pillars and nanowires are generally nanostructured metal oxide such as ZnO and TiO2
as shown in Fig. 2.8(c). One of the major advantages of bulk heterojunction CQD solar cells
is their ability to extend the depleted regions to several micrometers, which leads to improved
charge separation and collection efficiency. Using bulk heterojunction CQD solar cells,
evidential improvements of Jsc have been reported as high as 30 mA/cm2 [25, 26, 30]. However,
the overall CQD solar cell efficiency has not progressed substantially due to the loss of Voc
from enhanced carrier recombination, which is additionally augmented through increased
CQD/metal oxide interfacial trap states. Bulk heterojunction CQD solar cells still have great
potential for CQD solar cell performance optimization with respect to the increased Jsc when
CQD and interface quality have been significantly improved in the future.
2.4 Conclusions
This chapter discusses solution-based CQD fabrication processes, which can produce high-
quality CQDs with low dot size polydispersity. High-efficiency solar cells have been achieved
using these solution-processed CQDs. Ligand-exchanges play essential roles in strengthening
CQD interdot coupling, leading to extended mini-bands in CQD ensembles. The temperature-
dependent phonon-assisted hopping transport in CQD ensembles is the dominant carrier
CHAPTER 2. 25
transport mechanism, which has been experimentally demonstrated. Furthermore, large dots
facilitate carrier hopping transport with the exception that electron mobility decreases with the
further increased quantum dot size. CQD size polydispersity acts as trap states to trap carriers,
and should be eliminated or reduced for high-performance solar cell fabrication. The
heterojunction structure can effectively improve CQD solar cell efficiency through
simultaneously increasing Jsc and Voc. The bulk-structured heterojunction can significantly
improve Jsc, but reduced Voc was reported due to enhanced carrier recombinations at
CQD/metal oxide interfaces. The improvements of CQD quality through various methods are
still the main efforts applied by researchers to boost CQD solar cell efficiency.
26
Chapter 3
Non-destructive Testing (NDT) Techniques for Carrier Transport
in Quantum Dot Materials and Solar Cells
3.1 Literature Review and Classification
Charge carrier transport properties such as carrier mobility, diffusion length, doping
density, carrier lifetime, and trap density are essential parameters for CQD solar cell efficiency
optimization. Including transient photoluminescence, short-circuit/open-circuit voltage decay,
and photoconductance decay, many techniques have been developed to measure these carrier
transport properties. This thesis aims to develop novel high-frequency and/or large-area non-
destructive techniques through the derivation of new theoretical models to quantitatively probe
carrier transport dynamics in CQD materials and solar cells. For comparison with our new
techniques (PCR, HoLIC, and HeLIC), several widely-used testing techniques (although not
all of them are NDTs) for carrier transport parameter characterization of CQD systems are
reviewed in this chapter.
Table 3.1: Carrier transport parameters of colloidal quantum dot materials and solar cells
measured by various characterization techniques.
Property Testing method Related
parameters
NDT Devices
or
substrates
References
Carrier
mobility
CELIV J, time No Both [1-3]
TOF J, time No Both [3, 4]
Transient
photovoltage
Voc No Device [2, 5]
FET Id, Vg No Device [2, 6, 7]
CHAPTER 3. 27
CELIV = carrier extraction by linearly increasing voltage, TOF = time of flight, FET = filed
effect transistor, PCR = photocarrier radiometry, PL = photoluminescence.
Table 3.1 summarizes these key carrier transport parameters that are measured by various
commonly used techniques. For the sake of comparison, both NDTs and non-NDTs have been
reviewed and tabulated in Table 3.1. These techniques can be classified into the following
three categories [13]:
1. Steady-state techniques that require contact with samples.
Steady-state techniques are the most widely used and well-developed techniques, including
the surface photovoltage method for minority carrier lifetime characterization [14, 15], two-
point and four-point probe for resistivity measurements [16, 17], secondary ion mass
spectroscopy for ultra-shallow dopant profile characterization [18-21], and Rutherford
backscattering spectrometry (RBS) for measuring dopant/carrier distribution, structure and
composition of materials [22, 23]. Although widely used in characterizing semiconductor
Diffusion
length
PL
quenching
PL intensity,
layer thickness
Yes Substrate [8]
Voc transient decay Voc No Device [4]
PCR Modulated PL Yes Both [9]
Doping
Density
Capacitance-
voltage
Capacitance No Device [2]
Carrier
lifetime
Voc transient decay Voc No [3]
Transient PL PL Yes Both [2, 10]
PCR Modulated PL Yes Both [11, 12]
Trap
density
Voc transient decay Voc, Jsc No Device [4]
Thermal
admittance
spectroscopy
Capacitance,
frequency of ac
signal
Yes Device [4]
CHAPTER 3. 28
materials and devices, a dominant disadvantage of these techniques is the incapability of
measuring optoelectronic dynamic properties due to the steady-state nature. Furthermore,
many of these techniques are expensive, such as the second beam spectroscopy and RBS,
limiting their applications in industrial photovoltaic manufacturing and maintenance.
Additionally, the direct or indirect contact with samples poses potential damages to samples.
At last, the scanning point-by-point mapping method for most small-spot testing based
techniques to get large-area imaging is time-consuming, not suitable for in-line manufacturing
inspection.
2. Transient and quasi-steady-state contact techniques.
Quasi-steady-state (or frequency-modulated measurements) at high modulation
frequencies can detect high-rate dynamic properties of photovoltaic materials and devices with
high signal-to-noise ratio. Including quasi-steady-state photoluminescence (QSSPL) [24, 25]
and microwave photoconductance decay (μ-PCD, a golden standard for carrier lifetime time
measurement), these contactless quasi-steady-state techniques can be used for measuring
effective lifetime of minority charge carriers [26]. Furthermore, modulated photovoltage [27,
28] and electroluminescence (EL) [27] methods were also reported recently for solar cell
quality inspection. These techniques can detect multiple carrier transport dynamic parameters
including carrier lifetime, diffusion length, and surface recombination velocity through fitting
experimental data to theoretical models. However, the requirements for contacting with
samples limit their further applications as discussed for techniques in category 1.
3. Quasi-steady-state (Frequency-modulated) contactless techniques.
Reviewing the development of characterization techniques for semiconductors, the
properties of being all-optical, fast, contactless, nondestructive, and capable of dynamic testing
CHAPTER 3. 29
of high-rate carrier transport behaviors will be the features of future techniques. Up to today,
this type of characterization techniques include frequency-modulated PL for carrier lifetime
and surface recombination characterization [29], coupled photocurrent and photothermal
reflectance techniques that are sensitive to carrier diffusivity and lifetime [30, 31], modulated
free carrier absorption that are capable of measuring multiple carrier transport parameters [32-
34], modulated photothermal reflection (PDT) for detecting photothermal and electro-thermal
responses) [35, 36], photothermal radiometry (PTR) which is capable of measuring carrier
lifetime and surface recombination velocities [37], and photocarrier radiometry (PCR) that can
also qualitatively measure carrier lifetime and surface recombination velocities with higher
accuracy than PTR methods. These emerging techniques have been widely used in
characterizing a wide scope of materials and devices with remarkable success.
In the following sections, conventional characterization techniques including short-circuit
current/open-circuit voltage decay (SCCD/OCVD) — techniques that have been extensively
used in characterizing CQD solar cells although they are not NDTs, static and transient PL,
photoconductance decay (PCD), and various methods which are developed for carrier
diffusion length measurements are reviewed. Our new technique PCR is one of the essential
quasi-steady-state or frequency-modulated contactless NDTs for carrier transport parameter
testing, therefore, the general features of PCR are addressed in this chapter and compared with
the abovementioned techniques elaborated upon. The principles of this technique specific for
CQD thin films will be detailed in Chapter 6. A major disadvantage of the above-mentioned
techniques is the small-spot based characterization, which limits their applications in
characterizing large-area photovoltaic materials and devices. The solution to this limitation
involves the application of the our novel camera-based large-area dynamic imaging techniques,
CHAPTER 3. 30
HoLIC and HeLIC (modulated at ultrahigh frequencies), which are the imaging evolutions of
the PCR technique and are also discussed with the discussion of instrumentation and signal
processing principles, while their particular theory and models for CQD solar cells are
developed in detail in Chapter 8.
3.2 Traditional Methodologies for CQD Carrier Transport
Characterization
3.2.1 Short-Circuit Current/Open-Circuit Voltage Decay
(SCCD/OCVD)
SCCD and OCVD are two valuable techniques for measuring carrier lifetimes in CQD
solar cells [3]. These techniques probe pn junction voltage and short circuit current decay after
the photoexcitation of electron-hole pairs to measure carrier recombination lifetimes [38, 39].
Here, these two techniques are addressed in comparison to the PCR technique for CQD lifetime
measurements. Unlike most techniques that can characterize only one carrier transport
parameter, the combination of SCCD and OCVD can measure both the carrier lifetime, τ𝑟 and
the back-surface recombination velocity, sr, of a solar cell. The theoretical model is derived
from the analysis of minority carrier diffusion within a pn junction, and the solar cell back-
surface is treated through boundary conditions. The differential equation for minority carrier
concentration in the solar cell base can be expressed by
𝜕∆𝑛(𝑥,𝑡)
𝜕𝑡= 𝐷
𝜕2∆𝑛(𝑥,𝑡)
𝜕𝑥2−∆𝑛(𝑥,𝑡)
𝜏𝑟+ 𝐺(𝑥, 𝑡) (3.1)
in which ∆𝑛(𝑥, 𝑡) is the excess minority carrier density, 𝐷 is the diffusivity, and 𝐺(𝑥, 𝑡) is the
generation rate which equals zero after the photoexcitation is turned off. The solution to Eq.
(3.1) is subject to the following boundary equations [39].
1
∆𝑛(𝑥,𝑡)
𝜕∆𝑛(𝑥,𝑡)
𝜕𝑥= −
𝑠𝑟
𝐷𝑛 𝑓𝑜𝑟 𝑥 = 𝑑 (3.2)
CHAPTER 3. 31
with
∆𝑛(0, 𝑡) = 0 (3.3)
for short-circuit current method, and
𝜕∆𝑛(𝑥,𝑡)
𝜕𝑥= 0 𝑓𝑜𝑟 𝑥 = 0 (3.4)
for the open-circuit voltage method. The term d is the device thickness. When considering the
above boundary conditions, the solution to Eq. (3.1) exhibits an exponential short-circuit
current and open-circuit voltage decay profile with time. The decay behavior has a time
constant that is determined by the time-dependent excess carrier density.
Figure 3.1: A representative of open-circuit voltage decay curve for CQD solar cells, the linear
best fit (dashed red line) is used for the determination of recombination-determined carrier
lifetime. Adapted from ref. [3].
For the open-circuit voltage decay method, generally, researchers calculate the minority
carrier lifetime through [3, 40]
CHAPTER 3. 32
𝜏𝑟 = −𝐹𝑘𝑇/𝑞
𝑑𝑉(𝑡)/𝑑𝑡 (3.5)
with 𝑘 the Boltzmann constant, T the absolute temperature, q the elementary charge, and F a
constant, ranging between 1 at low carrier injection levels, and 2 at high carrier injection levels.
Therefore, using the open-circuit voltage decay curve, the carrier lifetime can be resolved.
Figure 3.1 shows a representative curve of Voc decay and the best fit to Eq. (3.5) for the
extraction of the minority carrier lifetime. The minority carrier lifetimes extracted using this
method for PbS CQD solar cells in a Schottky architecture range from 1 ms at low intensities
to 10 µs at high intensities.
The disadvantages of SCCD and OCVD techniques can be summarized as follows: first,
these techniques need to be in contact with solar cell devices. Measurements through contact
with samples are time-consuming and can present damages to photovoltaic samples. Second,
SCCD and OCVD are based on the theoretical model as presented in Eqs. (3.1) and (3.5),
therefore, it is evident that only effective carrier lifetimes can be measured and these
techniques cannot distinguish bulk and surface lifetimes. Although the back surface
recombination velocity is introduced through boundary conditions, the front surface
recombination velocity is forsaken. Third, these techniques cannot detect depth-resolved
carrier transport parameters as the signals of SCCD and OCVD are from the device’s overall
short-circuit current and open-circuit voltage, which are contributed by carriers at different
depths. Fourth, these techniques can only be used for complete solar cells, while not applicable
to semiconductor wafers or incomplete photovoltaic devices, which have no short-circuit
currents and open-circuit voltages.
CHAPTER 3. 33
3.2.2 Photoconductance Decay (PCD)
PCD was developed in 1955 for semiconductor lifetime characterization [17] and it has
evolved to be a golden standard for minority carrier lifetime measurements. Based on the
excess minority carrier decay which is directly associated with carrier lifetime, this technique
can measure effective carrier lifetime with high accuracy and reliability. In PCD, electron-hole
pairs are generated through the pulse photoexcitation, and their time-dependent concentration
decay is monitored with respect to the time following the cessation of the photoexcitation.
During the measurements, sample are in contact with PCD but it can also be made contactless
if microwave is used in reflection and transmission modes, i.e. the μ-PCD technique [41, 42].
For μ-PCD, photoconductivity is monitored through microwave reflection or transmission.
The theory of PCD starts with the expression for conductivity σ,
𝜎 = 𝑞(𝜇𝑛𝑛 + 𝜇𝑝𝑝) (3.6)
in which 𝑞 is the charge element, and n and p are the electron and hole concentrations,
respectively. The term 𝜇𝑛,𝑝 represents the electron (n) or hole (p) mobility, respectively.
Furthermore, n = n0+Δn and p=p0+Δp (n0 and p0 are electron and hole concentrations at
equilibrium, respectively); considering identical electron and hole mobilities, at low injection
levels, i.e. the excess carrier concentration is much lower than the carrier concentration at
equilibrium. For low trapping conditions, i.e. Δn = Δp, the measurements of conductance
change correspond to the measurements of excess carrier changes, which are given by
𝛥𝑛 =𝛥𝜎
𝑞(𝜇𝑛+𝜇𝑝) (3.7)
CHAPTER 3. 34
Therefore, assuming constant mobility, the measurement of 𝛥𝑛 can be carried through
measuring 𝛥𝜎. For the calculation of carrier lifetimes, the dependence of carrier concentration
decay on time is determined by the carrier lifetime τ through [43]
𝛥𝑛(𝑡) = 𝛥𝑛(0)exp (−𝑡
𝜏) (3.8)
PCD measures an effective minority carrier lifetime, 𝜏𝑒𝑓𝑓 and cannot distinguish bulk lifetime,
𝜏𝐵 and the surface lifetime, τs, in other words, 𝜏 in Eq. (3.8) is an effective carrier lifetime
𝜏𝑒𝑓𝑓 and can be expressed by
1
𝜏𝑒𝑓𝑓=
1
𝜏𝐵+
1
𝜏𝑠 (3.9)
Therefore, if either bulk or surface lifetime is of interest, the other carrier lifetime must be
already known.
However, PCD also has many disadvantages [13]. First, PCD has a relatively very complex
instrumentation system, for example, the conventional μ-PCD in the contactless model has
both the photoexcitation and microwave conductance testing systems. In comparison, most of
other carrier lifetime characterization techniques only require a photoexcitation system.
Second, as discussed above, the use of Eqs. (3.8) and (3.9), which depict the definition of
effective minority carrier lifetime, indicates that PCD can only measure effective lifetimes that
originate from the overall effects of the bulk lifetime, surface lifetime, diffusivity, and surface
recombination velocities. Specifically, μ-PCD cannot distinguish these parameters, although
PCD through contact with samples has this capability, thereby compromising the properties of
being all-optical, contactless, and nonconductive. Third, the signals collected by PCD are the
depth integration of the overall signals along the sample thickness, which imply that PCD is
unable to characterize sample properties at different depths. In other words, these trap states
can influence the detected integration signals no matter how deep the trap states lie as long as
CHAPTER 3. 35
carriers exist within the trap state regions. Therefore, although μ-PCD can construct carrier
lifetime images, it cannot perform depth-resolved characterization. This is an essential
drawback or limitation for the characterization of p-n junction based devices.
3.2.3 Time-resolved PL (TRPL, transient PL)
Photoluminescence techniques detect only radiative recombination induced photoemission.
For example, depending on the specific semiconductor energy bandgap, the center
photoemission wavelength is 1.2 µm for Si semiconductors. Therefore, InGaAs is the most
commonly used material for PL detectors. PL based techniques can be further divided into
steady-state and time-resolved PL (TRPL). The former can characterize semiconductor
material optical and electrical properties including energy bandgap and trap states. In
comparison, TRPL is capable of measuring carrier lifetimes, back and front recombination
velocities, and diffusivity with proper theoretical models.
Figure 3.2: A typical spectrum of the time-resolved PL for PbS CQDs. The inset illustrates the
exciton transport with diffusion and dissociation processes. Adapted from ref. [45].
CHAPTER 3. 36
With regards to the TRPL spectra at different decay time ranges, a PL vs. time curve may
have different slopes that correspond to different carrier recombination mechanisms. Therefore,
different TRPL theoretical models can be applied within different decay time ranges for the
extraction of carrier lifetime and other carrier transport parameters, which correspond to
different carrier transport mechanisms [29, 44]. As an example, a typical spectrum of TRPL is
shown in Fig. 3.2. Upon the generation of excitons in PbS CQDs, these excitons diffuse
through hopping or tunneling and dissociate into free electrons and holes as described in the
inset of Fig.3.2. Figure 3.2 reveals that there are two different carrier decay mechanisms of
excitons and charge carriers in CQD thin films, i.e. the fast PL emission decay component that
corresponds to the exciton dissociation process and the slow exponential decay component
originated from free charge carrier trapping in CQD surface states [45].
The excess carrier decay in TRPL is also described by Eq. (3.8). The PL intensity is the
depth integration of excess carrier density along the sample thickness, i.e.
∅𝑃𝐿 = 𝐾 ∫ ∆𝑛(𝑥, 𝑡)𝑑𝑥𝑑
0 (3.10)
where ∅𝑃𝐿 is the PL intensity, 𝑑 is the sample thickness, and 𝐾 is a constant. Therefore, the
minority carrier lifetime can be extracted from fitting the exponent PL intensity decay profile
to a single-exponential equation [46]:
∅𝑃𝐿~𝑒𝑥𝑝(−𝑡/𝜏𝑃𝐿) (3.11)
Sometimes a multi-exponential decay model is used for better fitting and describing carrier
decay mechanisms. The analytical model, however, can be very complicated depending on the
carrier transport parameters that need to be extracted from a PL decay spectrum [44].
According to the above discussion, transient PL measures the effective minority carrier
lifetime. For example, when self-absorption is considered for extra electron-hole-pair
CHAPTER 3. 37
excitation (the influence of self-absorption is more significant for direct bandgap
semiconductors such as PbS), the PL lifetime, therefore, is defined by [47]
1
𝜏𝑃𝐿=
1
𝜏𝑛𝑜𝑛−𝑟𝑎𝑑+
1
𝜏𝑆+
1
𝛾𝜏𝑟𝑎𝑑 (3.12)
where the terms 𝜏𝑛𝑜𝑛−𝑟𝑎𝑑, 𝜏𝑆, and 𝛾𝜏𝑟𝑎𝑑 are the nonradiative, surface, and radiative lifetimes,
respectively, and the term 𝛾 denotes the photon recycling factor. However, the effect of self-
absorption is not substantial for indirect bandgap semiconductors.
Figure 3.3: The mechanism for quasi-static PL imaging (a). Adapted from ref. [48]. (b) The
mechanism for low-frequency modulated PL imaging. Adapted from ref. [49].
In recent years, InGaAs camera based PL imaging of semiconductor materials and devices
is emerging with increasing frequency. However, being limited by the relatively low camera
frame rate and exposure time, PL imaging cannot be constructed at high modulation
frequencies, which, however, are essential for the characterization of high-rate carrier transport
behaviors. In other words, as most PL imaging techniques are performed at a steady state [24,
48] or low modulation frequencies [49], PL imaging cannot detect high-rate carrier transport
behaviors. Figure 3.3(a) depicts the schematic of PL imaging in a quasi-steady-state model,
where four images are taken at a low-frequency-modulated square wave excitation. The
CHAPTER 3. 38
effective minority carrier lifetime can be measured through fitting each pixel of these images
to a time domain theoretical model. Similarly, Fig.3.3(b) presents the mechanism for PL
imaging at low modulation frequencies. The low-frequency-modulated PL imaging can also
yield effective carrier lifetime images from proper frequency domain theoretical models.
Transient PL has many advantages, for example, the applied near-infrared InGaAs camera
or single detector does not need cooling, which eases the requirements for the testing
environment and improves measurement accuracy. In addition, transient PL directly measures
carrier radiative recombination without influence from thermal emission. Therefore, the
theoretical computation processes can be simplified. Currently, with the increase of the
InGaAs camera frame rate, relatively high-rate carrier transport property can be characterized
but still not high enough for low carrier lifetime CQDs. However, transient PL still has many
disadvantages as summarized below: first, without the application of a lock-in amplifier,
transient PL has a low signal-to-noise ratio (SNR), which means an extremely low signal if the
carrier lifetime is very small. In addition, system calibrations are required [24, 25]. Second,
the PL signal is a depth integration of carrier radiative recombination along the sample
thickness, which indicates that carrier transport properties at different depths cannot be
distinguished, limiting its applications in photovoltaic device characterization. Third, although
transient PL can measure multiple carrier transport parameters, this technique is still
constrained by lower camera frame rate and the requirements for high exposure time.
3.2.4 Carrier Diffusion Length Measurements
Diffusion length is the distance that photoexcited excitons or free charge carriers can
diffuse through before they recombine radiatively or non-radiatively. Given a low-level
injection approximation, the diffusion length is the square root of the product of the carrier
CHAPTER 3. 39
diffusivity and the lifetime. To measure carrier diffusion lengths in CQD thin films,
Zhitomirsky et al. [8] applied the QD bandgap tunability to develop a PL quenching method.
Specifically, in their 1D method, as shown in Fig.3.4 (a), the CQD thin film of interest is
denoted as a donor, and correspondingly, an acceptor CQD layer with smaller energy bandgap
is pre-coated atop the donor CQD layer. Laser-based photoexcitation is introduced from the
donor side, from where excitons are generated and diffuse within the donor to the acceptor
CQD layer. At steady state, the carrier concentration 𝑛(𝑥) can be approximated by
𝐷𝑑2𝑛(𝑥)
𝑑𝑥2−𝑛(𝑥)
𝜏= 𝑔(𝑥) (3.13)
in which 𝑔(𝑥) is the generation rate at x, 𝐷 is the diffusivity, and 𝜏 is the carrier lifetime.
Solving Eq. (3.13), the expression for 𝑛(𝑥) can be obtained with proper boundary conditions.
The photoluminescence intensity is proportional to 𝑛(𝑥), a theoretical fundamental that is also
applied in PCR. The PL flux is detected from the acceptor layer at different thin film
thicknesses, and the diffusion length of the donor CQD thin film can be obtained by fitting the
experimental data to the following exponential equation [8]:
𝑃𝐿𝐹𝑙𝑢𝑥 =1
𝐿𝐷(𝐴𝑒
−𝑑
𝐿𝐷 − 𝐵𝑒𝑑
𝐿𝐷) + 𝛼𝐶𝑒−𝛼𝑑 (3.14)
where 𝑃𝐿𝐹𝑙𝑢𝑥 is the PL flux from the acceptor thin film, 𝐿𝐷 is the carrier diffusion length, 𝑑 is
the thin film thickness, 𝛼 is the absorption coefficient, and A, B as well as C are constant
coefficients to fit. This 1D method is further developed into a 3D method through replacing
the boundary acceptor CQD layer with the incorporation of smaller bandgap CQDs into the
CQD thin film of interest. Through fitting Eq. (5) in the ref. [8], the carrier transport parameters
including diffusion length (LD), diffusivity (D), carrier lifetime (τ), mobility (µ), and trap
density can be obtained.
CHAPTER 3. 40
Figure 3.4: Schematic of a PL quenching method for diffusion length measurements. An
acceptor layer of CQDs with smaller bandgap should be pre-coated. Adapted from ref. [8].
Figure 3.5: Schematic of transient photoluminescence study of carrier diffusion length,
diffusivity, and lifetime. At x = L the boundary condition can be either with or without
quenching. Adapted from ref. [50].
The abovementioned methodology is based on the theoretical understanding of the charge
carrier transport mechanisms in CQD thin films. Therefore, it is not applicable for p-n junction
associated photovoltaic devices as the charge carrier transport behavior are different. Similarly,
using the non-destructive transient PL technique, Lee et al. [50] developed a methodology to
measure exciton diffusion length and diffusivity in CQD thin films. As shown in Fig.3.5, a
pulse laser is used as the photoexcitation source for CQD thin films on a substrate. The
substrate can act as either a quenching layer or a non-quenching layer. Assuming the
CHAPTER 3. 41
independence of the carrier diffusivity on the exciton density and location, i.e. D(n,x,t) = D(t),
the exciton concentration can be expressed by the one-dimensional diffusion equation,
𝜕𝑛(𝑥,𝑡)
𝜕𝑡= 𝐷(𝑡)
𝜕2𝑛(𝑥,𝑡)
𝜕𝑥2−𝑛(𝑥,𝑡)
𝜏+ 𝑔(𝑥, 𝑡) (3.15)
A time-dependent 𝐷(𝑡) does not yield an analytical solution, therefore, constant diffusivity, D
is usually applied by researchers.
It should be noted that Eq. (3.15) is also a fundamental part of the PCR theory for carrier
transport parameter measurements of CQD thin films and solar cells in this thesis. Furthermore,
a constant time-independent diffusivity is also assumed for PCR. With quenching or no
quenching boundary conditions, the exciton concentration 𝑛(𝑥, 𝑡) can be solved from Eq.
(3.15). The obtained 𝑛(𝑥, 𝑡) is a function of multiple parameters including diffusion length,
lifetime, and diffusivity. Therefore, 𝑛(𝑥, 𝑡) can be used to extract the exciton diffusivity and
diffusion length through fitting experimental transient PL data to the derived models [50]. Of
course, when converted to the frequency-domain, the 𝑛(𝑥, 𝑡) can be used in PCR methodology
that provides further advantages as discussed in Sect.3.5. Coupled with the transient PL
technique, Lee et al. [50] derived the expression for exciton diffusion length, 𝐿𝐷,
𝐿𝐷 ≈2𝐿
𝜋√2(
𝜏
𝜏𝑞− 1) (3.16)
in which 𝐿 is the QD thin film thickness (Fig. 3.5), and 𝜏𝑞 is the fitted average exciton lifetime.
Compared with PCR as discussed in Chapter 6, this methodology does not consider the
influence of trap states, moreover, this transient PL based methodology is limited by low SNR
and cannot measure very short lifetimes.
CHAPTER 3. 42
3.3 Photocarrier Carrier Radiometry (PCR)
PCR is a dynamic spectrally-gated photoluminescence to measure optoelectronic
properties in materials and devices. As will be detailed in the following sections, PCR uses
modulated lasers to excite semiconductor samples, and demodulate the near-infrared radiation
signals to quantitatively characterize carrier transport parameters. In comparison with
photothermal techniques which detect both thermal-wave and carrier-density wave, the
theoretical interpretation of PCR signal is much easier with less unknown parameters involved,
corresponding to relatively higher measurement accuracy as PCR only detect the purely
optoelectronic carrier-wave. The following sections will discuss PCR instrumentations used in
for this thesis, as well as a general theory for measuring semiconductors with continuous band
structures. The detailed analytical models developed specific CQD thin films is elaborated in
Chapter 6.
3.3.1 Photocarrier Radiometry Instrumentation
As a nondestructive, frequency-domain, and spectrally gated photoluminescence, PCR
starts with a super-bandgap frequency-modulated laser beam to create a periodic carrier-
density-wave (CDW) in semiconductor materials and devices. The periodic light emissions
from radiative recombination are detected by a single detector, and the amplitude and phase of
the periodic signal were computed using a lock-in amplifier. Consequently, the PCR spectra
are the frequency-dependent amplitudes and phases from low frequency (𝜔𝜏 ≪ 1, in which 𝜔
is the modulation angular frequency and 𝜏 is the effective carrier lifetime) to high frequency
(𝜔𝜏 ≫ 1) that can be used for following theoretical fitting process to extract useful carrier
CHAPTER 3. 43
transport parameters. As a spectrally gated frequency domain PL and the instrumentation
consists of, as shown in Fig.3.6, three key sub-systems:
1. Excitation laser generation system,
2. Dynamic photoluminescence detection system, and
3. Lock-in amplifier signal computation system.
Figure 3.6: Schematic of experimental instrumental setup for photocarrier radiometry.
The excitation laser generation system consists of a function generator, a laser, and other
optics including various mirrors as shown in Fig. 3.6. The function generator used for this
research project is from Stanford research systems, Model DS340, synthesized function
generator. Unless indicated, the PCR system for CQD materials and device characterization
for this projects uses the near-infrared diode laser at wavelength 830 nm (Melles Griot, Model
CHAPTER 3. 44
No. 56ICS115) with the highest DC power of ~ 30 mW. The laser excitation is modulated in
sine wave from 10 Hz up to 2 MHz with an accuracy of ±25ppm determined by the function
generator. From function generator the output amplitude is 10.96 Vpp, indicating the peak-to-
peak voltage of the sine wave, and the DC offset is 0.90 V. GPIB interface was used for
communication between the function generator and the computer.
The illuminating photons from samples are collected by a single detector (PDA400,
ThorLabs), prior to which a long-pass filter is used to filter out the excitation laser. The
PDA400 is an InGaAs detector with a switchable gain for detecting light signals from DC to
10 MHz. The effective diameter of the InGaAs detector is 1 mm with wavelength response
from 800 nm to 1750 nm. For modulation frequencies from 10 Hz to 100 kHz, the gain was
set to 30dB. The 1000-nm long-pass filter is mounted in front of the InGaAs detector.
Lock-in amplifier is the key component that contributes to the high signal-to-noise ratio
(SNR). As shown in Fig. 3.6, the reference signal is from the function generator, and the input
signal is from the InGaAs single detector. Time constant τ was set to 1s.
3.3.2 Theory of Lock-in Amplifier Signal Computation
Due to the critical roles that a lock-in amplifier plays in the signal processing with high
SNR for the PCR system, herein, the principle of the lock-in amplifier is briefly discussed here.
The technique used in a lock-in amplifier is called phase sensitive detection that singles out
the signal with a specific desired frequency and phase. However, noise signal at other
frequencies are rejected and have no influence on the measurement. Therefore, a lock-in
amplifier is capable of detecting AC signals as small as a few nanovolts, even when the noise
source is thousands of times larger than the signal. Without using a lock-in amplifier,
supposing a 1 µV sine wave signal at 10 MHz, in order to detect the signal, an amplifier is
CHAPTER 3. 45
required. Current good noise amplifiers have around 3nV √𝐻𝑧 of input noise. Considering an
amplfier with a bandwidth of 200 MHz and a gain of 1000, the output is expect to be 1 mV of
signal and 43 mV of broadband noise (3nV √𝐻𝑧 ×√200 MHz×1000), in which case the higher
noise than signal entails a measurement faisure. With the application of lock-in amplifier uisng
a phase sensitive detector (PSD), which enable the detection of the signal at 10 MHz while
with a bandwidth as narrow as 1 Hz (even narrower can also be achieed, such as 0.1 Hz, if
longer time contant is used). Therefore, the output noise is now only 3 µV (3nV √𝐻𝑧 ×√1
MHz×1000) which is considerablely much lower than 1 mV of signal. With a SNR as high as
300, the signal can be measured.
Suppose a reference signal is a sine wave that can be 𝑉𝑅 sin(𝜔𝑅𝑡 + 𝜑𝑅) and it can be sync
out from a function generator. The terms 𝑉𝑅 and 𝜑𝑅 are the refence signal amplitude and phase,
respectively, and 𝑡 is the time. When the sine wave output from the function generator is used
to modulate the excitaiton laser system, the response signal from samples is at the same
freuqency 𝜔𝑅 and might be 𝑉𝐼 sin(𝜔𝑅𝑡 + 𝜑𝐼), in which 𝑉𝐼 and 𝜑𝐼 are the input (i.e. output
from the samples) signal amplitude and phase, respectively. Lock-in amplifer uses a mixer to
multiply these two signals together, and it geenrates
𝑉𝑀1 = 𝑉𝑅 sin(𝜔𝑅𝑡 + 𝜑𝑅) 𝑉𝐼 sin(𝜔𝑅𝑡 + 𝜑𝐼) =1
2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼) +
1
2𝑉𝐼𝑉𝑅 sin(2𝜔𝑅𝑡 +
𝜑𝑅 + 𝜑𝐼) (3.17)
in which 𝑉𝑀1 is the signal output from the mixer 1 (mixter 2 will be introduced later). The
resultant first term is a DC component, while the second term is an AC component osilated at
higher frequency 2𝜔𝑅. The AC component can be readily removed using a low pass filter,
which generates the filtered signal 𝑉𝑀1+𝐹𝐼𝐿𝑇
CHAPTER 3. 46
𝑉𝑀1+𝐹𝐼𝐿𝑇 =1
2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼) (3.18)
Now it is evident to see the phase dependence, i.e. 𝑉𝑀1+𝐹𝐼𝐿𝑇 is proportional to the cosine of
the phase difference between the refence and the input signal (signal from samples). For the
sake of measuring 𝑉𝐼, a second mixer is used an shift the refernce signal 90o out of phase.
Therefore, reference sent to the second mixer is 𝑉𝑅 sin(𝜔𝑅𝑡 + 𝜑𝑅 − 𝜋/2). Following the
same procedure as above, the output of the second signal 𝑉𝑀2 is
𝑉𝑀2 =1
2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼 − 𝜋/2) +
1
2𝑉𝐼𝑉𝑅 sin(2𝜔𝑅𝑡 + 𝜑𝑅 + 𝜑𝐼 − 𝜋/2) (3.19)
After filtering,
𝑉𝑀2+𝐹𝐼𝐿𝑇 =1
2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼 − 𝜋/2) =
1
2𝑉𝐼𝑉𝑅 sin(𝜑𝑅 − 𝜑𝐼) (3.20)
Finally, the amplitude and phase of the input (signal from samples) can be determined, and
they are
𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 =2
𝑉𝑅√(𝑉𝑀1+𝐹𝐼𝐿𝑇)2 + (𝑉𝑀2+𝐹𝐼𝐿𝑇)2 (3.21a)
𝑃ℎ𝑎𝑠𝑒 = 𝜑𝑅 − 𝜑𝐼 = tan−1 (
𝑉𝑀2+𝐹𝐼𝐿𝑇
𝑉𝑀1+𝐹𝐼𝐿𝑇) (3.21b)
In-Phase: 𝑋 = 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 ∙ cos(𝜑𝑅 − 𝜑𝐼) (3.21c)
Quadrature: 𝑌 = 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 ∙ sin(𝜑𝑅 − 𝜑𝐼) (3.21c)
One important concept for lock-in measurements is the “time constant” τ, which can be
obtained by 𝜏 = 1/𝑓𝑐 and used to set the low-pass filer bandwidth [52]. The term 𝑓𝑐 is the
“cutoff” or “corner” frequency of the low-pass filter that has been used to remove 2𝜔𝑅
frequency components of 𝑉𝑀1 and 𝑉𝑀2, and to remove signals or noises at other frequencies
not equal to the reference frequency. The low-pass filter time constant and “rolloff” should be
selected carefully based on the nature of experiments. The filter rolloff is mostly taken one of
the four values: 6 dB/octave, 12 dB/octave, 18 dB/octave, and 24 dB/octave. Therefore, signal
CHAPTER 3. 47
with frequency 𝑓 ≪ 𝑓𝑐 will pass with unity gain, while single with frequency 𝑓 ≫ 𝑓𝑐 will be
attenuated as 𝑓−𝑛 for 6n dB/octave filter, for example, ∝ 𝑓−3 for 18 dB/octave filters. The
output is more steady and easier to be measured when the time constant increases.
3.3.3 General Theory of Photocarrier Radiometry
PCR is a full optical nondestructive dynamic spectrally gated frequency-domain PL
modality. It is also an evolution of photothermal radiometry (PTR). In comparison, PTR
detects both thermal-wave and electronic carrier-density wave (CDW) contributions, however,
lead to significant difficulties in PTR signal interpretation and computation due to a large
number of variables [53]. PCR was developed by Mandelis et al. [54] as a purely carrier-wave
laser-based detection methodology and eliminates the thermal-wave contributions. Because
PCR is only sensitive to the recombination of photoexcited carrier density waves, its signal
interpretation and computation are much simpler than that of PTR, therefore, increasing the
uniqueness and reliability of the obtained semiconductor material properties. As the detailed
theory has been elaborated in ref. [54], here, only some key conclusions that have been widely
used in this thesis will be discussed due to the limited scope of this chapter. Although the PCR
theory was developed in the scenario of Si wafers, most concepts and formulas can be directly
applied to other material systems such as GaAs [55], as well as CQD thin films and solar cells.
PCR detects photon emission from photovoltaic materials and devices. Therefore, the
discussion of its theory should start with excess carrier radiative and non-radiative
recombination mechanisms. As shown in Fig. 3.7, electrons are excited by photons with super-
bandgap energy, then the excited electrons and holes quickly thermalize and relax to the band
edge of the conduction and valence band, respectively, with the emission of phonons (or heat).
De-excitation of photoexcited carriers occurs through three recombination mechanisms:
CHAPTER 3. 48
radiative recombination, Auger recombination, and Shockley-Read-Hall (SRH) recombination.
An elaborate review of these recombination mechanisms can be found in refs. [56-58], while
will be briefly outlined here.
Radiative recombination is a direct band edge electron-hole recombination, emitting
photons with the bandgap energy of the semiconductor. As both electron and hole participate
in radiative recombination process, the recombination rate is proportional to the product of
electron and hole concentration.
Auger recombination denotes a direct nonradiative recombination of electrons and holes
via energy transfer to and emits another free carrier. It is a reverse process of MEG in QDs.
Depending on whether the energy is transferred to an electron or hole, the three-particle
interaction can be denoted by eeh or ehh. Auger recombination usually occurs under high
carrier condition with high injection levels. The recombination rate of an Auger recombination
is proportional to the product of the concentration of three particles involved.
Shockley-Read-Hall recombination is a two-step process with carriers trapped into defect
states and then followed by radiative and non-radiative recombination of these trapped states.
The photon emission from the SRH recombination is lower than the semiconductor bandgap
energy, depending on the energy level of defect states. SRH associated defect states can be
distinguished as, namely, recombination centers and traps. The capture coefficients of
electrons and holes for recombination centers are similar. However, for traps, the capture
coefficient for one carrier is higher when this carrier is trapped in, while the rate for capturing
another particle with the opposite sign for recombination is slow. It should be noted that SRH
recombination not only can happen between defects and conduction/valence band but also
CHAPTER 3. 49
occurs between defect states. The recombination rate of SRH is proportional to the product of
carrier concentrations and trap state density.
For QDs with high surface-to-volume ratios, the surface (interface) trap related surface
recombination should also be discussed. Surface recombination is defect involved. Therefore,
it is a type of SRH recombination. However, the SRH theory is derived based on a single well-
defined trap level. Interfaces or surfaces of semiconductor represent a termination of crystal
periodicity and induce a band of electronic states in the band gap. This inconsistency requires
additionally extended SRH recombination theory to deal with the continuum of surface states
across semiconductor bandgap [56].
For CQD thin films and solar cells, the radiative recombination and radiative component
of SRH recombination contribute to PCR, depending on the specific PL spectra of a sample as
shown in Fig. 6.2.
Figure 3.7: (a) Energy diagram of an n-type semiconductor with the illumination of
photoexcitation, and radiative and nonradiative recombination. Defects related states are also
depicted to carry radiative and non-radiative recombination. Adapted from ref. [54].
CHAPTER 3. 50
Figure 3.8: Schematic of one-dimensional Si wafer where an emission photon distribution is
yielded following laser excitation and carrier-wave generation. (a) A representative
semiconductor slab with thickness dz, centered at z. (b) Reflection photons from backing
support material. (c) Emissive IR photons from backing support materials at temperature Tb.
∆𝑁(𝑧, 𝜔) represents the depth- and frequency- dependent carrer-diffusion-wave, and L is the
thickness of the Si wafer. Other parameters can be found in the text. R1,2,b(λ) are reflectivity of
the front surface, back surface, and the backing support material. It should be noted that the
backing material is used to support the wafer but not necessary to be in contact with the sample.
Adapted from ref. [54].
The photoexcited carriers after ultra-fast decay to the respective band edge, accompanying
various combinations as discussed above, meanwhile diffuse within their statistical lifetime. If
the excitation laser is intensity modulated at a frequency f =ω/2π (ω is the angular frequency),
the photogenerated carrier density constitutes a spatially damped carrier-density-wave (CDW)
(or carrier-diffusion-wave). The CDW oscillates diffusely away from their generation source
due to its concentration gradient and recombines with a phase lag dependency on a delay time
that equal to the carrier statistical lifetime τ, a structure- and process-sensitive property [54,
59]. The schematic of photon excitation, absorption, and emission processes in a Si wafer is
illustrated in Fig.3.8. The one-dimensional geometry is suitable for thin semiconductor
CHAPTER 3. 51
materials or for the case of using spread laser beams of large spot size. The emission photon
power 𝑑𝑃𝑗(𝑧, 𝑡; 𝜆) at wavelength λ with a bandwidth dλ is given by [54]
𝑑𝑃𝑗(𝑧, 𝑡; 𝜆) = {𝑊𝑁𝑅[𝑇𝑇(𝑧, 𝑡); 𝜆] + 𝜂𝑅𝑊𝑒𝑅(𝜆)}𝑗𝑑𝜆; 𝑗 = 𝑟, 𝑡 (3.22)
where 𝑊𝑁𝑅[𝑇𝑇(𝑧, 𝑡); 𝜆] is the non-radiative related power per unit wavelength, 𝑊𝑒𝑅(𝜆) is the
radiative recombination generated photon power per wavelength, 𝜂𝑅 is the quantum yield for
radiative emission, 𝑇𝑇(𝑧, 𝑡) is the total temperature including the background temperature and
the temperature increase following photon absorption and heat generation and others, and the
subscript (r, t) denotes back-propagating (reflected) and forward-propagating (transmitted) as
shown in Fig. 3.8. The modulated super-band-gap laser photons impinges on the front surface
of the semiconductor and are absorbed within a short distance [α(λ)]-1 from the front surface,
where α(λ) is photon wavelength dependent absorption coefficient. The emission spectra are
within a broad wavelength range due to the various type of radiative recombination processes.
The final PCR expression was derived in one dimension by Mandelis et al. [54] as a depth
integration of excess carrier density:
𝑃(𝜔) ≈ 𝐹(𝜆1, 𝜆2) ∫ ∆𝑁(𝑧, 𝜔)𝐿
0𝑑𝑧 (3.23)
The term 𝐹(𝜆1, 𝜆2) is an instrumentation coefficient which depends on the spectral emission
bandwidth (𝜆1, 𝜆2), and the expression for 𝐹(𝜆1, 𝜆2) can be found in ref. [54].The term ∆𝑁 is
the excess free charge carrier density which depends on the material properties and carrier
transport nature. The next section discusses the derivation of ∆𝑁 with trap-mediated carrier
transport in CQD thin films.
CHAPTER 3. 52
3.4 Homodyne (HoLIC) and Heterodyne (HeLIC) Lock-in
Carrierography
HoLIC and HeLIC are the imaging evolution of PCR with the application of a CCD camera
rather than a single detector. Another difference, when compared with PCR, is that two laser
excitation systems are used for illumination. HoLIC can only construct low-frequency imaging
while HeLIC is able to image semiconductors at high frequencies. Therefore, HeLIC can
perform large-area, contactless, fast, all-optical, quantitative characterization of semiconductor
materials and devices. This section will discuss the instrumentation and signal processing
techniques of HoLIC and HeLIC and address the nonlinearity requirement for HeLIC. The as-
developed novel theoretical HeLIC signal generation models will be discussed in detail in
Chapter 8.
3.4.1 Instrumentation and Signal Processing Techniques used in
HoLIC and HeLIC Imaging
Instead of using InGaAs single detector, HeLIC uses high-speed NIR InGaAs camera for
signal collection. As shown in Fig.3.9, the InGaAs camera (Goodrich SU320 KTSW-
1.7RT/RS170) used through this study has the following features: 320×256 pixel active
elements, the spectral bandwidth of 0.9-1.7 μm, 120 fps frame rate, and exposure times tunes
between 0.13 and 16.6 ms. In comparison with PCR, two fiber-coupled diode lasers of 808 nm
wavelength were used for optical illumination. For acquiring homogeneous illumination, both
laser beams were spread and homogenized using diffusers to generate a 10 × 10 cm2 square
illumination area with small intensity variations (< 5 %). An optical long-pass filter
(Spectrogon LP-1000 nm) is mounted in front of the InGaAs camera, resulting in an effective
InGaAs camera bandwidth of 1-1.7 μm.
CHAPTER 3. 53
Figure 3.9: Experimental setup for homodyne (HoLIC) and heterodyne (HeLIC) lock-in
carrierography.
Due to the limitations of camera frame rate, a synchronous undersampling method is
employed through the application of a data acquisition module (NI USB-6259), which
produced a reference signal and an external trigger to the camera. Sixteen images per period
were scanned with a frame grabber (NI PCI-1427). The control is through homemade
LabVIEW program. To understand undersampling, the Nyquist-Shannon sampling theorem
should be reviewed:
“An analog signal with a bandwidth of fa must be sampled at a rate of fs>2fa in order to
avoid the loss of information.”
The term fs is the sampling rate. When fs = 2fa is satisfied, fs is called Nyquist rate. The
theorem is a bridge connecting continuous-time signals (analog signals) and discrete-time
signals (digital signals). In HoLIC, if the sampling method is used, for a harmonic signal with
frequency f, the sampling rate fs should be greater than 2f in order to precisely acquire the signal
CHAPTER 3. 54
information. Limited by the frame rate of 120 fps for our InGaAs camera, the highest frequency
in HoLIC can be calculated is 60 Hz. To ensure the imaging quality with sufficient exposure
time (i.e. the maximum exposure time of 16.7 ms), HoLIC imaging at 10 Hz is generally
performed.
Figure 3.10: Schematic of oversampling (a) and undersampling (b) signal processing methods.
For sampling, 16 samples are taken per one cycle (waveform), and one circle (waveform) is
skipped for undersampling.
CHAPTER 3. 55
Figure 3.11: Schematic of camera-based HeLIC imaging using an undersampling method (a)
and modulation laser frequency mixing mechanism for HeLIC imaging.
However, the low frequency (even 60 Hz) is not sufficiently high for high-frequency
imaging to generate measurable phase lags (τ ≈ 1/2πf) for materials with low carrier lifetimes.
CHAPTER 3. 56
Herein, undersampling method is used in HoLIC for higher frequency imaging. The
undersampling method is also known as harmonic sampling, bandpass sampling, IF sampling,
and IF to digital conversion. As shown in Fig.3.10, considering a 1 Hz sine wave, 16 imaging
is taken in one period, which is known as an oversampling process. The process of taking 16
images per period (or 1s, as shown in Fig. 3.10) corresponding to a frame rate of 16 Hz. If
undersampling is applied, with a skip of one waveform, the actual frame rate used is only 1
Hz. Following the same manna, through skip more waveforms (i.e. wait for more time to take
the next image), high-frequency imaging is achievable through the use of the highest frame
rate. Given the known reference frequency, the data collected using undersampling can be re-
calculated to oversampling data as shown in Fig. 3.10 (a). A complete schematic of HoLIC
using undersampling method is shown in Fig.3.11 (a) that one cycle is skipped between each
image. For example, four images are taken at 0, π/2, π, and 3π/2 phases. The main pulse train
plays the role of triggering the camera to start taking images, and the camera pulse train
initiates four images (16 in real experiments) to be taken during a fixed camera exposure period.
By shipping more cycles, higher modulation frequencies can be used while keeping the camera
frame rate unchanged. The collected reference and signal matrix are computed by lock-in
amplifier or data acquisition card shown in Fig.3.9.
With the application of the undersampling method, however, problems still arise when
high modulation of frequencies are used. They include the small timing errors and the
decreased resolution due to the limited camera exposure time. Therefore, a heterodyne
method was introduced for high-frequency HeLIC imaging through the superposition of two
modulated laser beams with a small frequency difference (the beat frequency) and the
camera measuring with a frame rate equal to the beat frequency. The experimental setup of
CHAPTER 3. 57
HeLIC is the same as that of HoLIC except that two laser excitations are modulated with a
small frequency difference as shown in Fig. 3.9. Figure 3.11(b) displays the image
generation mechanism of HeLIC imaging: two linearly combined laser irradiation
modulation with a beat frequency of 10 Hz is mixed. The interaction between the two
modulated excitation lasers and the sample is a nonlinear process that enables the generation
of an LIC image at the beat frequency as shown in Fig. 3.11(b). The camera modulation
frequency is always selected to be equal to the beat frequency, i.e., 10 Hz in this study. The
recorded image amplitudes carry information from the high modulation frequency (as
discussed in Sect. 8.2). Phase images cannot be obtained at the beat frequency due to the
close proximity of the two mixed two frequencies f1 and f2 (Fig. 3.9).
3.4.2 Requirements for HeLIC Response to Laser Excitation: Non-
linear Photoluminescence Processes
As shown in Fig. 3.11(b), HeLIC imaging requires the nonlinear combination of two carrier
density waves to create a carrier wave oscillated at the modulation frequency difference. This
section shows experimental evidence to the nonlinear response of the CQD solar cells. A
theoretical explanation to the nonlinearity is also discussed in this section with respect to
analytical models. Regarding the physics behind, the property of nonlinear response is due to
the various carrier recombination mechanisms as given in Sect. 7.3, which discusses the carrier
recombination mechanisms in CQDs.
Given the fact that the photoexcited exciton population has a linear dependence on the
incident photoexcitation intensity, for HeLIC imaging the photoexcitation laser beam was
modulated at two angular frequencies (𝜔1, 𝜔2) leading to the generation of two CDWs with a
small frequency difference Δω, mixed in a nonlinear signal processing device, a mixer, such
CHAPTER 3. 58
as a diode and a transistor [60]. The mixer creates a series of CDWs with new frequencies
including Δω. For our CQD solar cells, the sample itself acts as a mixer [61]. The nonlinear
coefficient γ of CQDs as a mixer can be obtained as shown in Fig. 3.12 (a), through fitting the
experimental DC image signal vs. laser photoexcitation intensity to I ∝ Nγ, in which I is the
average amplitude of all image pixels and N is the excitation laser power density as discussed
in Sect. 7.3. A nonlinear coefficient γ of 0.60 was extracted consistently with the requirement
for a non-linear process, γ ≠ 1, to generate a HeLIC signal [61, 62].
Figure 3.12: The nonlinear dependence of DC (a) and HeLIC (b) signals on photoexcitation
laser power density for our CQD solar cells with a typical structure: Au/PbS-EDT/PbS-
PbX2(AA)/ZnO/ITO as discussed in detail in Sect.7.3.
Theoretically, in HeLIC, as the laser excitation is modulated at two frequencies with an
angular frequency difference ∆𝜔, the excess photocarrier wave can be expressed as
∆𝑁(𝑥, 𝜔) = 2𝑛0(𝑥) + 𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑(𝑥, 𝜔1)] + 𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑(𝑥, 𝜔2)](3.24)
where 𝑛0(𝑥) is the DC component of the modulated excess CDW, 𝐴(𝑥, 𝜔𝑗) is the amplitude
of the cosinusoidally modulated CDW at 𝜔𝑗 ( 𝑗 = 1,2) and 𝜑 is the CDW phase. Equation
(3.24) is only approximate since PL emission response to photoexcitation intensity is a
fundamentally non-linear process with a non-linearity coefficient 𝛾 which was generally
CHAPTER 3. 59
measured to be between 0.5 and 2, and can be determined by plotting 𝐼 vs. 𝑁𝛾 , where 𝐼
represents the PL emission intensity and 𝑁 stands for the excitation laser power intensity.
Inserting ∆𝑁(𝑥, 𝜔) in Eq. (3.24) into Eq. (3.23) and considering the fully nonlinear response,
it can be shown that
𝑆(𝜔) = 𝐹 ∫ {2𝑛0(𝑥) + 𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑1(𝑥, 𝜔1)] + 𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 +𝑑
0
𝜑2(𝑥, 𝜔2)]}𝛾𝑑𝑥 (3.25)
Furthermore, the integrand can be expanded using the binomial theorem in the form
{2𝑛0(𝑥) + 𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑1(𝑥, 𝜔1)] + 𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑2(𝑥, 𝜔2)]}𝛾 =
∑ (𝛾𝑘)+∞
𝑘=0 ∑ (𝛾 − 𝑘𝑚
) [2𝑛0(𝑥)]𝛾−𝑘−𝑚{𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 +
+∞𝑚=0
𝜑1(𝑥, 𝜔1)]}𝑚{𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑2(𝑥, 𝜔2)]}
𝑘 (3.26)
Equation (3.25) can be further expanded using cos𝑘(𝜔𝑡) = 2−𝑘 ∑ (𝑘𝑚)𝑒𝑖(𝑘−2𝑚)𝜔𝑡∞
𝑚=0 . As only
signals modulated at the beat frequency ∆𝜔 = |𝜔2 − 𝜔1| contribute to HeLIC, the
demodulated HeLIC signal can be finally written as,
𝑆(∆𝜔) = 𝐹 ∑ ∑∏ (𝛾 − 𝑙)2𝑚+2𝑛+1𝑙=0
4𝑚+𝑛𝑚! (𝑚 + 1)! 𝑛! (𝑛 + 1)!
+∞
𝑛=0
+∞
𝑚=0
∫1
2𝑛0(𝑥)
𝛾𝑑
0
[𝐴(𝑥, 𝜔1)
𝑛0(𝑥)]
2𝑚+1
[𝐴(𝑥,𝜔2)
𝑛0(𝑥)]2𝑛+1
𝑒𝑖∆𝜑(𝑥)𝑑𝑥 (3.27)
For CQD-based thin films, ∆𝑁(𝑥, 𝜔) has been derived in Sect.6.2. In Fig.3.12(b), the
dependence of wideband (1-270 kHz) HeLIC images on the laser photoexcitation power was
investigated using a fixed 1-sun average intensity of modulated spread excitation beam while
changing the DC excitation intensity, i.e. 𝑛0(𝑥) in Eq. (3.27) changed from 0.2 to 1.1 sun. It
was found that the average amplitude of HeLIC images decreases with increasing DC
photoexcitation intensity which contrasts with its DC counterpart as shown in Fig. 3.12(b).
CHAPTER 3. 60
The decrease of the HeLIC signal with DC excitation is in agreement with Eq. (3.27) due to
the decreasing overall dependence on 𝑛0(𝑥): 𝑛0(𝑥)𝛾−2(𝑚+𝑛+2); 𝛾 = 0.6,𝑚, 𝑛 ≥ 0. As will be
discussed in detail in Sect.7.3, physically, for CQD systems featured with discrete energy
bands induced by spatial and energy disorder, the photogenerated excitons cannot dissociate
immediately into free charge carriers, in contrast to those generated in continuous energy band
semiconductors such as Si. Therefore, without the assistance of external forces including
interdot coupling and material interface effects, excitons are the dominant energy carriers in
CQD systems and act as the main radiative recombination sources [61]. Based on the foregoing
dominance of exciton dynamics in Sect.7.3, it was reported that the coefficient 𝛾 = 1 for
excitonic transitions while 𝛾 = 0.5 for carrier recombination induced by trap- or doping-
associated states. Hence, the physical meaning of Eq. (3.27) can be interpreted as an evolution
of the well-known exponential relation between photoexcitation and PL, 𝐼 ∝ 𝑁𝛾 , in the
framework of HeLIC. Therefore, the expectation of decreased HeLIC signal with increased
DC photoexcitation intensity from Eq. (3.27) results from three physical facts: unoccupied
trap-state density increases through enhanced photon absorption-mediated carrier ejection,
increased exciton density mediated carrier recombination, and the nature of radiative PL signal
collection in a heterodyne mode.
3.5 Comparison of Different Techniques and Advantages of PCR
and HeLIC
As discussed in Chapters 6 and 8, regarding charge carrier transport parameter
characterization, this thesis uses PCR and HeLIC and makes theoretical contributions to these
two novel techniques. Thereby, this section will compare general principles and features of
PCR and HeLIC with the commonly used techniques discussed above.
CHAPTER 3. 61
Based on the discussion and review of the carrier transport characterization methodologies
(SCCD/OCVD, PCD/μ-PCD, transient PL, PL imaging, PTR, and LIT) that are commonly
used by researchers, these techniques can be divided into either transient and quasi-steady-
state (frequency-modulated) techniques, or small-spot detection and large-area imaging
techniques.
SCCD/OCVD, PCD/μ-PCD, and transient PL are transient methodologies that start with
the application of a photoexcitation laser to pump the semiconductor surface and measure the
photoexcited excess carrier decay due to various carrier recombinations. The decay of excess
carrier concentration is monitored as a function of time through short-circuit current (SCCD),
open-circuit voltage (OCVD), conductance (PCD/μ-PCD), or PL emission (transient PL). This
excess carrier decay generally follows a minority carrier lifetime dependent exponential model
as shown by Eq. (3.8). Therefore, SCCD/OCVD, PCD/μ-PCD, and transient PL usually exhibit
an exponential decay spectrum. Depending on different recombination mechanisms in specific
samples and the characterization technique, effective carrier lifetimes can be extracted through
fitting an experimental decay spectrum to theoretical models. The common features and
disadvantages of these types of techniques can be summarized as follows.
1. The carrier lifetimes measured from these techniques are generally effective carrier
lifetimes. Limited by theoretical models, these techniques measure the overall excess carrier
density decay rate that is contributed by various recombination mechanisms, which is the
definition of effective carrier lifetimes. However, it should be noted that the transient PL
technique can measure diverse types of carrier lifetimes, which are determined by different
recombination mechanisms as shown in Fig. 3.2, while precise identification of the various
slopes in a PL decay spectrum is required.
CHAPTER 3. 62
2. These techniques cannot detect depth-resolved carrier transport properties, as they are
operated only at low frequencies. Therefore, the overall properties of a sample are measured,
making them unsuitable for particular property characterizations of p-n junctions in subsurface
regions, which, however, are essential for photovoltaic device efficiency optimization.
3. Transient methodologies are restrained by their relatively low SNR, especially when
carrier lifetimes are very low (for example in ns for some types of CQDs). Compared with
techniques such as PCR, HeLIC, lock-in thermography (LIT) and photothermal radiometry
(PTR) that use a lock-in amplifier, transient techniques always have low SNR values,
indicating that the experimental signal could be too small to measure when carrier lifetimes
are very short.
PTR and LIT use frequency-modulated photoexcitation. When compared with transient
techniques, PTR and LIT have two independent amplitude and phase channels, which ensure
high measurement reliability and accuracy. In addition, these techniques have high SNR values
due to the application of lock-in amplifiers. However, PTR and LIT are limited by the high
thermal diffusion length induced low resolution. Relying on the specific theoretical models
and the depth-resolution capability, PTR and LIT can measure multiple carrier transport
parameters. Generally, carrier lifetimes measured using these techniques are effective carrier
lifetimes.
The main advantage of large-area imaging over small-spot characterization is the capability
of an overall estimation of an entire photovoltaic device. However, the common transient PL,
frequency-modulated PL, and frequency-modulated PTR are either limited by low resolution,
the lack of depth-resolved carrier transport characterization capability, low SNR, low camera
frame rate, or by the high requirements of testing environments.
CHAPTER 3. 63
Compared with transient techniques, PCR overcomes the abovementioned disadvantages
and limitations. As invented by Mandelis et al. [54] in 2003, PCR is a spectrally gated dynamic
PL that applies frequency-modulated photoexcitation and collects radiative recombination
photons through an InGaAs single detector. Through fitting experimental frequency-dependent
PCR amplitudes and phases to theoretical models, multiple carrier transport parameters
including lifetime, diffusivity, and surface recombination velocities can be extracted [54, 63,
64]. With the use of a lock-in amplifier, PCR also has high SNR values. In addition, unlike
PTR that detects both radiative and nonradiative recombination, PCR only detects carrier
radiative recombination with the application of an InGaAs detector. Therefore, the theoretical
methodology is significantly simplified, increasing the measurement accuracy and uniqueness.
With the implementation of high frequency modulated photoexcitation, PCR can measure
carrier transport behavior at high rates and shows immense potential in the depth-resolved
characterization of photovoltaic properties. With the application of heterodyne techniques,
HeLIC overcomes the limitations of low camera frame rates and high exposure time
requirements for high-quality imaging. Therefore, ultrahigh-frequency (>100 kHz) imaging is
achieved for the first time. Compared with low-frequency-modulated (30 Hz) PTR and PL
imaging, high rate carrier transport behavior can be detected through HeLIC imaging.
Compared with transient techniques, PCR and HeLIC employ frequency-modulated
photoexcitation rather than pulse excitation. Comparing their theoretical models with other
commonly used transient techniques, they all begin with solving excess carrier continuous rate
equations to generate the analytical expression of excess carrier density. Therefore, the carrier
transport mechanisms described by transient techniques and PCR as well as HeLIC are the
same. However, theoretically, PCR and HeLIC need further mathematical manipulations to
CHAPTER 3. 64
transfer the excess carrier density from the time domain to frequency domain through various
techniques such as the Fourier transform. The capability of high-frequency detection is the
main advantage of PCR and HeLIC, which leads to the determination of high-rate carrier
transport dynamic behavior, while retaining high SNR values. The minority carrier lifetimes
extracted in this thesis using PCR and HeLIC are effective carrier lifetimes with the same
definition as those for the above-mentioned commonly used techniques. However, the
difference between measured carrier transport parameters, including lifetimes, from different
techniques can still be expected even when the definition of these parameters is the same for
each technique. This is because different approximations and assumptions are made during the
theoretical model development.
Furthermore, due to the application of frequency-modulated excitation and lock-in
amplifiers, PCR and HeLIC have higher SNR values (as high as 80 dB) [9] compared with μ-
PCD. This is the most prominent advantage of frequency-domain-associated techniques over
their time-domain counterparts. Additionally, the independent amplitude and phase channels
convey more material properties when compared with the PL techniques and the μ-PCD. The
amplitude channel not only contains information on carrier transport dynamics but also reflects
sample surface optical properties. In comparison, the phase channel is only determined by the
phase lag induced during various carrier recombination processes and is not influenced by the
photon excitation or emission intensity. More importantly, for PCR and HeLIC, the modulated
frequency determines the ac carrier diffusion length. In other words, when the modulation
frequency is much smaller than 1/τ, the ac diffusion length approximately equals the dc
diffusion length. However, when the modulation frequency is equal or higher than 1/τ, the ac
diffusion length is smaller than its dc counterpart [65]. Therefore, through increasing the
CHAPTER 3. 65
modulation frequency, the ac diffusion length can be further reduced; an effect that can be
applied to perform depth-resolved material property characterization. In this way, carrier
transport properties of the surface, sub-surface, pn junction, bulk, and back surface can be
distinctively characterized. This unique advantage of PCR and HeLIC can also increase image
resolution through increasing modulation frequencies.
3.6 Conclusions
For the characterization of CQD materials and solar cells, this chapter focuses on a review
of techniques that are extensively used for characterizing carrier transport properties. These
techniques include SCCD/OPVD, PCD (μ-PCD), TRPL, PCR, HeLIC, and methodologies
developed for carrier diffusion length measurements. PCR and HeLIC show apparent
advantages over other conventional techniques with respect to the all-optical, contactless,
nondestructive, high SNR, depth-resolution, and ultrahigh-frequency characterization of high-
rate carrier transport behavior features. However, the acquired parameters retain the same
definitions as the commonly used transient techniques. HeLIC overcomes the ubiquitous
limitations of camera-based techniques such as PL and LIT imaging. Therefore, ultrahigh-
frequency imaging is realized through HeLIC for the large-area characterization of high-rate
carrier transport behavior.
For future research interest, the development of large-area, all-optical, non-destructive,
contactless, ultrahigh-frequency imaging techniques for semiconductor material and device
characterization is the trend and promising for both fundamental carrier transport mechanism
study and for industrial product quality control and estimation.
66
Chapter 4
Quantitative Carrier Transport Study through Current-voltage
Characteristics
4.1 Introduction
Generally, to analyze current-voltage characteristics of CQD solar cells, the well-known
Shockley-Queisser (S-Q) equation is often used [1], which takes on the form 𝐽𝑖𝑙𝑙𝑢(𝑉) =
𝐽0 {𝑒𝑥𝑝 [𝑞𝑉𝑎
𝑛𝑖𝑑𝑘𝐵𝑇] − 1} − 𝐽𝑝ℎ, where 𝐽𝑖𝑙𝑙𝑢 is the current density under light illumination, 𝐽0 the
saturation current density, 𝐽𝑝ℎ a photocurrent density often treated as a constant short-circuit
current density, 𝑛𝑖𝑑 the ideality factor, 𝑘𝐵 the Boltzmann constant, 𝑞 the elementary charge, T
the absolute temperature, and 𝑉𝑎 the applied voltage. This equation was derived to model diode
behavior in an electrical circuit under the assumption of infinitely large material conductivity
(or carrier mobility) which is true for most solid-state p-n junctions consisting of highly
crystalline inorganic materials with a continuous carrier transport behavior in well-defined
lattice structures such as, Ge, Si, or GaAs. With high carrier mobility on the order of 102 - 103
cm2/Vs typical of Si solar cells. However, today’s CQD-based materials and devices feature
multiple energy disorder sources due to their high surface-to-volume ratio nanostructures,
variations in confinement energy and coupling, as well as thermal broadening. All of these
point to a non-continuous hopping/tunneling transport with low mobility ranging from 10-5 to
10-1 cm2/Vs [2, 3, 4, 5-10] for PbS CQDs passivated with various ligands. This huge difference
in mobility leads to questioning the validity of applying the S-Q model to such systems.
CHAPTER 4. 67
Although many authors have developed diffusion and drift principles ad hoc [11-14] that
are usually valid for semiconductor materials with well-defined continuous energy band
structure, carrier localization within a quantum dot and the effects of disorder which also
induces e.g. Anderson localized electrons [15]. With respect to electrical transport in colloidal
quantum dots, Guyot-Sionnest [16] summarized that “Many separate sources of disorder make
it extremely unlikely that QDS based on colloidal assemblies are anywhere close to exhibiting
band-like transport behavior”. Therefore, the evidence of low dot-to-dot transmission due to
variations in confinement energy and in coupling and thermal broadening, and
electron−electron repulsion points to hopping conductivity and diffusivity [16, 17]. In CQD
ensembles, exciton hopping transport in CQD solar cells involves exciton diffusion and
dissociation distinct from charge transport mechanisms which govern conventional inorganic
silicon cells [18]. However, most studies of CQD solar cells to date are based on the classical
S-Q equation that was derived under the assumption of continuous energy band semiconductor
systems where electron-hole pairs dissociate immediately upon their generation and travel at
high charge mobility. Crystalline Si is representative of these photovoltaic materials whereas
CQD and organic solar cells are excluded from that category [19, 20]. Würfel, et al. [21]
demonstrated that the Schottky equation cannot be applied to low-mobility materials in the
way it is used for inorganic solar cells, and device parameters extracted from the Schottky
equation such as ideality factor, series resistances, and shunt resistance lack real physical
meaning and provide very limited assistance toward the optimization of device fabrication. In
comparison, material parameters including carrier mobility, diffusion length, lifetime,
diffusivity, and trap states offer more important information for solar cell device structure
optimization, starting with the selection of materials. Unfortunately, in common with organic
CHAPTER 4. 68
photovoltaic materials, CQDs constitute low carrier mobility photovoltaic materials and have
high exciton binding energy, both impediments to improving solar efficiency in a
straightforward manner. In summary, a full understanding of charge carrier transport
mechanism within CQD devices is lacking and new theoretical I-V models based on the full
understanding is demanded.
Another impediment to solar cell device efficiency is the formation of a Schottky junction
at the CQD/anode interface which forms an electric field with the direction opposite to that of
the light incidence. Schottky junctions cause holes to accumulate at the CQD/anode interface
[20, 22], thereby reducing the surface recombination velocity (SRV) as observed in organic
photovoltaic devices [23]. All of these adverse processes handicap hole extraction at the anode
and lead to low solar conversion efficiency. An experimental consequence of hole
accumulation is the formation of anomalous (including S-shaped) current-voltage (I-V) curves,
which have been reported for heterojunction CQD solar cells with increasing frequency [24,
19, 25]. Nevertheless, the origins of the formation of these anomalous I-V curves have not
been well understood, nor have they been exploited toward designing higher performance solar
cells. Wang et al. [26] attributed them to the electron accumulation effect induced by an
exciton blocking layer. Wagenpfahl et al. [23] found the reduction of hole-associated surface
recombination velocities could also give rise to S-shaped I-V curves. One approach to studying
S-shaped I-V curves is through fitting experimental data to theoretical electric circuit models.
However, although Romero et al. [27] developed equivalent electric circuit models that can
quantitatively simulate three kinds of I-V curves in terms of forward and reverse diodes, they
did not consider actual physical mechanisms.
CHAPTER 4. 69
This chapter introduces the drift-diffusion I-V model in the framework of carrier hopping
transport in CQD solar cells. Based on this model, the anomalous I-V curves that significantly
retard CQD solar cell efficiency were quantitatively analyzed using a novel double-diode
electric circuit model to elucidate their origins: imbalanced carrier mobility and Schottky
barrier. In addition, interface effects on CQD solar cell open-circuit voltage dissipation were
quantitatively discussed. The temperature-dependent carrier hopping transport in CQD solar
cells were demonstrated through the I-V model and experimental temperature-dependent I-V
characteristics of the CQD solar cells under study. At last, performance factors of our device
architecture are discussed with improvement recommendations.
4.2 Derivation of Current-voltage Model from Hopping and
Discrete Carrier Transport
4.2.1 Carrier Hopping Diffusivity and Mobility in Quantum Dot
Systems
In a QD ensemble that QDs are separated by a mean distance from its nearest neighbors
and features own size and energy manifold. The rate equation for particle population Ni(x, t)
[particles/cm3] in QD (i) can be expressed as the net rate of carriers hopping into and out of a
QD [28]:
𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑡= −
𝑁𝑖(𝑥,𝑡)
𝜏− ∑ 𝑃𝑖𝑗𝑁𝑖(𝑥, 𝑡) +𝑗 ∑ 𝑃𝑗𝑖𝑁𝑗(𝑥, 𝑡)𝑗 (4.1)
in which 𝜏 is the effective lifetime of particles including excitons and free charge carriers, 𝑃𝑖𝑗
is the probability for a particle to migrate from a QD (i) to QD (j). Considering a flow of
particles hoping in and out of a virtual volume element dV = Adx in the colloidal medium, A is
CHAPTER 4. 70
a cross sectional area. As shown in Fig.4.1, the net particle flux 𝐽𝑒(𝑥, 𝑡) (particles/cm2s) is
given by
𝐴[ 𝐽𝑒(𝑥, 𝑡) − 𝐽𝑒(𝑥 + 𝑑𝑥, 𝑡)]𝑑𝑡 = −𝜕
𝜕𝑥 𝐽𝑒(𝑥, 𝑡)𝑑𝑉𝑑𝑡 (4.2)
Combining Eqs. (4.1) and (4.2), the net particle population rate entering dV in time dt is in the
form,
𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑡= −
𝜕𝐽𝑒(𝑥,𝑡)
𝜕𝑥−𝜕𝑁𝑖(𝑥,𝑡)
𝜏 (4.3)
The first term on the right-hand side denotes trans-volume particle hopping into and/or out of
dV, and the second term represents local particle de-excitation within dV through radiative and
nonradiative recombination. It is tough to calculate the summation terms over the entire
ensemble of particles in Eq. (4.1), therefore, the nearest neighbor hopping (NNH)
approximation will be adopted as shown in Fig.4.1 (b) the 1D NNH kinetics. Under external
optical excitation induced particle population gradient, the net flux 𝐽𝑒(𝑥, 𝑡) at QD (i) is a results
of flux out of QD (i+1) and QD (i-1) into the QD (i).Therefore, these fluxes are [28]:
𝐽𝑖+1,𝑖𝑛(𝑥 + ∆𝑥, 𝑡) =1
2𝑁𝑖+1(𝑥 + ∆𝑥, 𝑡)𝑓𝑖+1𝑒
−𝛾∆𝑥 (4.4 a)
𝐽𝑖−1,𝑖𝑛(𝑥 − ∆𝑥, 𝑡) =1
2𝑁𝑖−1(𝑥 − ∆𝑥, 𝑡)𝑓𝑖−1𝑒
−𝛾∆𝑥 (4.4 a)
It should be noted that the factor 1
2 accounts for the equal probability for a particle in QDi to
hop into QDi-1 and QDi+1 in a 1D geometry. ∆𝑥 is the hopping distance (equal to interdot
distance) as shown in Fig. 4.1 (b). The factor fj denotes particle hopping frequencies, and
𝑒−𝛾∆𝑥 represents the dot-to-dot crossing probability with a hopping transmission coefficient 𝛾.
According to the flux conservation, the net flux across the virtual area A at x is given by [28]
𝐽𝑖(𝑥, 𝑡) = 𝐽𝑖−1,𝑖𝑛(𝑥 − ∆𝑥, 𝑡) − 𝐽𝑖+1,𝑖𝑛(𝑥 + ∆𝑥, 𝑡) =1
2𝑒−𝛾∆𝑥 [
𝑁𝑖−1(𝑥−∆𝑥,𝑡)
𝜏ℎ,𝑖−1−𝑁𝑖+1(𝑥+∆𝑥,𝑡)
𝜏ℎ,𝑖+1] ∆𝑥
(4.5)
CHAPTER 4. 71
Here, 𝜏ℎ,𝑗 = 1/𝑓ℎ,𝑗 is defined as the hopping time, representing particles hopping between two
nearest neighbors.
Figure 4.1: (a) A virtual volume element dV = Adx, resembling a QD, illustrates the discrete
hopping transport of excitons and charge carriers. (b) Schematic of the discrete particle flux
into and out of three adjacent virtual planes. All planes have an area of A across the thickness
direction of a CQD solar cell. Adapted from ref. [28].
The hopping velocity 𝑣ℎ,𝑗 for a particle hoping from a QD to its nearest neighbor is defined
as ∆𝑥 = 𝐿 = 𝑣ℎ,𝑗𝜏ℎ,𝑗, in which 𝐿 is the effective QD-to-QD distance and 𝑣ℎ,𝑗 is the population
gradient associated hopping velocity that includes the drift velocity 𝑣𝑑𝑟,𝑗 when external or
internal electric field E exist. Therefore, for free charge carriers, the overall velocity 𝑣𝑇,𝑗
should include both types of hopping transport:
𝑣𝑇,𝑗 = 𝑣ℎ,𝑗 ± 𝑣𝑑𝑟,𝑗 (4.6a)
The overall velocity 𝑣𝑇,𝑗 is determined by the relative direction of the particle population
gradient and the internal or external electric field. The hopping velocity, 𝑣ℎ,𝑗(𝐸) , in the
presence of electric field can be further expressed as a function of the hopping velocity 𝑣ℎ,𝑗(0)
without electric field E and the electric field:
𝑣ℎ,𝑖±1(𝐸) = 𝑣ℎ,𝑖±1(0) ±𝑞𝐸𝐿
𝑚𝑒𝑥𝑣ℎ,𝑖±1(0) (4.6b)
CHAPTER 4. 72
in which q is the free carrier charge element, 𝑚𝑒𝑥 is the effective particle mass, the ± signs
account for the motion in the direction (+) or opposite to the direction (-) of the electric field.
It should be noted that the electric filed E changes the dipole moment �� = 𝑞�� of an exciton,
in which R is an effective length between the constituent electron and hole along the direction
of the field [29]. The exciton potential energy can be expressed by ∆𝑈𝑝𝑜𝑡 = ∆�� ∙ ��. Under
electric field, the length R is stretched and shrunk through two vectors: ∆�� = ±𝑞∆��. Therefore,
the net effect of the electric field on an exciton is a drift motion with a peak defer velocity in
the direction of the electric field [28]:
𝑣𝑑𝑟,𝑗 = (2∆𝑈𝑝𝑜𝑡
𝑚𝑒𝑥)1/2
= [2𝐸(
𝑑��
𝑑𝐸)∙��
𝑚𝑒𝑥]
1/2
(4.6c)
in which 𝑑��
𝑑𝐸 is a material-medium-dependent physical property and is defined as the gradient
of the exciton dipole moment in the electric field. Implementing Eq. (4.6c), the effective
exciton mobility is derived as
𝜇𝑒𝑥 = √2𝑑��
𝑑𝐸∙��𝐸
𝑚𝑒𝑥 (4.7a)
in which ��𝐸 is a unit vector which is in the direction of the electric field. Given very close
interdot distance (~ 1 nm) for most CQD materials, 𝑁𝑖±1(𝑥 ± ∆𝑥) in Eq. (4.5) can be expanded
while only keep the first (linear) order terms to yield particle flux for both excitons and free
charge carriers [28]:
𝐽𝑖(𝑥, 𝑡) ≈ [−𝑣ℎ𝑜𝐿 (𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑥) + 𝑣𝑑𝑟,𝑖𝑁𝑖(𝑥, 𝑡)] 𝑒
−𝛾𝐿 (4.7b)
CHAPTER 4. 73
where 𝑣ℎ𝑜 is the hopping velocity when the electric field E = 0. In CQD solar cells, CQD thin
films are always assumed as the nominal p-type energy absorption layer. When compared with
conventional p-type carrier transport in continuous semiconductors:
𝐽(𝑥, 𝑡) = −𝐷 [𝜕𝑁(𝑥,𝑡)
𝜕𝑥] + 𝜇𝐸𝑁(𝑥, 𝑡) (4.8)
Comparing Eq. (4.7b) and (4.8), the hopping diffusivity can be defined by
𝐷 ≡ 𝐷ℎ(𝑇) = [𝐿2
𝜏ℎ(𝑇)] exp (−𝛾𝐿) (4.9)
Corresponding, the hopping mobility of free charge carriers can be defined as
𝜇ℎ(𝑇) = 𝜇(𝑇)exp (−𝛾𝐿) (4.10)
with 𝜇(𝑇) = 𝜇𝑒𝑥 in Eq. (4.7a) for excitons, and 𝜇(𝑇) = [𝑞𝐿
𝑚𝑒𝑓𝑣ℎ0(𝑇)] for free charge carriers
using Eq. (4.6b). As indicated in Eqs. (4.9) and (4.10), both 𝐷ℎ and 𝜇ℎ are temperature-
dependent due to the thermally activated interdot hopping process. The hopping time constant
𝜏ℎ as defined in Eq. (4.9) is associated with the hopping probability Pij [30] and can be obtained
by
1
𝜏ℎ(𝑇)=
1
𝜏0exp (−
∆𝐸𝑗𝑖
𝑘𝑇) (4.11a)
in which, ∆𝐸𝑗𝑖 is the energy difference when a particle hopping from an initial QDi to the final
QDj, i.e. ∆𝐸𝑗𝑖 = 𝐸𝑗 − 𝐸𝑖. The term 𝜏0 is a characteristic time for particles tunneling/hopping
between neighbor QDs at a distance x, and has been given by ref. [30]
1
𝜏0= 𝑓𝑒𝑥𝑝(−𝛾𝐿) (4.11b)
CHAPTER 4. 74
Therefore, 𝜏0 is determined by the tunneling attempt frequency f and the probability 𝛾 which
is dependent on the energy barrier with an effective thickness 𝐿 that separates two QDs at
location i and j, i.e. the interdot distance.
Furthermore, Guyot-Sionnest [16] proposed that the carrier hopping mobility 𝜇ℎ(𝑇) is
dependent on the sum of the energy barriers Ea that encountered by hopping particles.
𝜇ℎ(𝑇) = (𝑞𝐿2𝐸𝑎
3ℎ𝑘𝑇) exp (−𝛾𝐿 −
𝐸𝑎
𝑘𝑇) (4.12)
This expression is structurally similar to Eq. (4.10) and predicts a thermal process with an
activation energy Ea estimated to be in the range of 10-50 meV [16], which is consistent with
experimental results obtained in this work. It is obvious to see that both transport processes are
spatially limited by the nearest neighbor QD energy state differences. Clearly, when compared
with conventional continuous energy band carrier transport, in the hopping picture, a very
different physical interpretation with respect to carrier diffusivity, mobility, critical transport
length is observed. These unusual discrete hopping transport of carriers in QD ensembles
results in the breakdown of the conventional Einstein relation. In view of Eqs. (4.9), (4.10),
(4.11), and (4.12), the ratio of mobility to diffusivity in the framework of “hopping” activation
transport in QD ensembles is given by [28]:
𝜇ℎ(𝑇)
𝐷ℎ(𝑇)= (
𝑞𝜏0𝐸𝑎
3ℎ𝑘𝑇) exp (−
𝐸𝑎−∆𝐸
𝑘𝑇) ≡ (
𝑞𝜏0𝐸𝑎
3ℎ𝑘𝑇) exp (−
∆𝐸𝑎
𝑘𝑇) (4.13)
in which 𝜇ℎ is the bulk mobility as measured in the external circuit. Equation (4.13) is a special
expression of the conventional Einstein relation that can be obtained when 𝐸𝑎 = ∆𝐸 = ℎ/𝜏0.
In other words, Equation (4.13) evolves to the conventional Einstein relation when the carrier
diffusion and drift rates are limited by the same thermal activation process, which is defined
CHAPTER 4. 75
as thermal velocity in continuous band semiconductors, and when the rate limiting step is the
dot-to-dot tunneling rate 𝜏0−1.
4.2.2 Current Density J(x) across CQD Solar Cells
Figure 4.2: Schematic of one CQD solar cell energy band structure.
Disorder sources in CQD ensembles, including variations in confinement energy, electron-
electron repulsion, coupling and thermal broadening, cause CQD-based materials and devices
to exhibit discrete hopping conductivity and diffusivity [16, 17]. Considering a general CQD
solar cell structure, in which CQD layers act as a nominal p-type light absorption layer, for
example, as shown in Fig.4.2, a CQD solar cell with the energy diagram of
TiO2/CQD/MoO3/Au/Ag. At equilibrium, 𝐽(𝑥, 𝑡) = 0 across the entire CQD layer thickness,
the carrier hopping drift current density equals hopping diffusion current density
𝐷ℎ𝑑𝑁(𝑥)
𝑑𝑥− 𝜇ℎ𝐸𝑁(𝑥) = 0; 0 ≤ 𝑥 ≤ 𝑑 (4.14)
CHAPTER 4. 76
in which the subscript i, denoting the discrete spatial location, has been adequately represented
by the coordinate variable x. The electric field can be derived through integrating 𝐸(𝑥) =
− 𝑑𝑉(𝑥)
𝑑𝑥 over the thickness of the solar cell and yields [28]
𝑁(0)
𝑁(𝑑)= exp (−
𝜇ℎ𝑉0
𝐷ℎ) (4.15)
The factor 𝑉0 is defined as the potential difference at x = 0 and d, i.e. 𝑉0 = 𝑉(0) − 𝑉(𝑑).
For a non-equilibrium situation where a non-zero photovoltage 𝑉𝑎 is generated upon
optical illumination of a QD medium and lead to the injection or extraction of excitons and
free charge carriers, Eq. (4.15) becomes [28]
𝑁𝑃𝑉(0)
𝑁𝑃𝑉(𝑑)= exp {−
𝜇ℎ
𝐷ℎ[𝑉0 − 𝑉𝑎]} (4.16)
The subscript PV indicates that the solar cell is under external bias. Defining ∆𝑁𝑃𝑉(0) =
𝑁𝑃𝑉(0) − 𝑁(0) as the excess carrier population under illumination, apply a quenching
boundary condition at x = d as proposed by Zhitomirsky et al. [14] that 𝑁𝑃𝑉(𝑑) − 𝑁(𝑑) = 0,
therefore, Eqs. (4.15) and (4.16) yield
∆𝑁𝑃𝑉(0) = 𝑁𝑃𝑉(0) [𝑒𝑥𝑝 (𝜇ℎ𝑉𝑎
𝐷ℎ) − 1] (4.17)
It can be expected that, for ultrathin CQD layers in several hundred nanometers, the population
difference of photons absorbed across the thickness of the layer is relatively small, therefore,
𝑁𝑃𝑉(0) ≈ 𝑁𝑃𝑉(𝑑) sometimes can be assumed. In a one-dimensional quantum dot ensemble,
each quantum dot characterized by its own size and energy manifold is separated by a mean
distance from its neighbors. The rate equation for the net carrier (electron, hole, or exciton)
flux entering one quantum dot within a time interval 𝑑𝑡 has been given as Eq. (4.3). With an
CHAPTER 4. 77
applied dc voltage, 𝑛𝑃𝑉 becomes time-independent, therefore, Eq. (4.3) can be further written
as
𝑑2𝑁𝑃𝑉(𝑥)
𝑑𝑥2− (
𝜇ℎ𝐸
𝐷ℎ)𝑑𝑁𝑃𝑉(𝑥)
𝑑𝑥−𝑁𝑃𝑉(𝑥)
𝐷ℎ𝜏= 0 (4.18)
where 𝜏 presents the total lifetime of the charge carrier, 𝐸 = 𝑉𝑏𝑖 −𝑉𝑒𝑥𝑡
𝑑 is the electric field
across the solar cell, 𝑉𝑏𝑖 is the built-in voltage, and 𝑉𝑒𝑥𝑡 is the photovoltage. The general
solution to Eq. (4.18) takes the form as
𝑁𝑃𝑉(𝑥) = 𝐴𝑒𝑄1𝑥 + 𝐵𝑒𝑄2𝑥 (4.19)
with
𝑄1,2 =1
2(𝐶0 ±√𝐶0
2 + 4𝐶1) (4.20)
and with definitions
𝐶0 =𝜇ℎ𝐸
𝐷ℎ= (
𝜇ℎ
𝐷ℎ)𝑉𝑏𝑖−𝑉𝑎
𝑑, 𝐶1 =
1
𝐿ℎ2 , 𝐿ℎ = √𝐷ℎ𝜏 (4.21)
where 𝐿ℎ is the carrier dc hopping diffusion length. Constants A and B can be solved with the
application of the abovementioned quenching boundary conditions, therefore, highlighting the
temperature dependence, the carrier diffusion current density is given by
𝐽ℎ,𝑑𝑖𝑓𝑓(0, 𝑇) = 𝑞𝑁0𝐷ℎ(𝑇)
𝐿ℎ(𝑇)(𝑐𝑜𝑠ℎ[𝑑/𝐿ℎ(𝑇)]
𝑠𝑖𝑛ℎ[𝑑/𝐿ℎ(𝑇)]) (𝑒𝑥𝑝 [
𝜇ℎ(𝑇)𝑉ℎ
𝐷ℎ(𝑇)] − 1) (4.22)
Correspondingly, the hopping drift current density is expressed as
𝐽ℎ,𝑑𝑟𝑖𝑓(0, 𝑇) = 𝑞𝑁0𝐷ℎ(𝑇)
𝐿ℎ(𝑇)𝜇ℎ(𝑇)𝐸 (𝑒𝑥𝑝 [
𝜇ℎ(𝑇)𝑉ℎ
𝐷ℎ(𝑇)] − 1) (4.23)
CHAPTER 4. 78
The total carrier hopping density is the summation of diffusion and drift current density in the
form of
𝐽ℎ(0, 𝑇) = 𝐽ℎ,𝑑𝑖𝑓𝑓(0, 𝑇) + 𝐽ℎ,𝑑𝑟𝑖𝑓(0, 𝑇) (4.24)
4.3 Imbalanced Charge Carrier Mobility and Schottky Junction
Induced Anomalous J-V Characteristics of CQD Solar Cells
Although a sharp boost in CQD solar cell power conversion efficiency (PCE) has been
observed from 3% to 12 % in only 7 years [31], limiting factors are emerging at high frequency
to retards significant progress in CQD solar cell PCE improvement. One of the substantial
impediments to CQD solar cell device efficiency is the formation of a Schottky junction at the
CQD/anode interface which forms an electric field with the direction opposite to that of the
light incidence. This Schottky junction causes holes to accumulate at the CQD/anode interface
[20, 22], thereby reducing the surface recombination velocity (SRV) as observed in organic
photovoltaic devices [23]. These adverse processes handicap hole extraction at the anode and
lead to low solar conversion efficiency. An experimental consequence of hole accumulation is
the formation of anomalous (including S-shaped) current-voltage (I-V) curves, which have
been reported for heterojunction CQD solar cells with increasing frequency [19, 24, 25].
Nevertheless, the origins of the formation of these anomalous I-V curves have not been well
understood, nor have they been exploited toward designing higher performance solar cells.
Wang et al. [26] attributed them to the electron accumulation effect induced by an exciton
blocking layer. Wagenpfahl et al. [23] found the reduction of hole-associated surface
recombination velocities could also give rise to S-shaped I-V curves. One approach to studying
S-shaped I-V curves is through fitting experimental data to theoretical electric circuit models.
CHAPTER 4. 79
However, although Romero et al. [27] developed equivalent electric circuit models that can
quantitatively simulate three kinds of I-V curves in terms of forward and reverse diodes, they
did not consider actual physical mechanisms.
In this section, the temperature-dependent I-V characteristics are analyzed using a novel
double-diode electric circuit model and a carrier hopping transport model to interpret the
culprits of the notorious anomalous I-V characteristics of CQD solar cells. The open-circuit
deterioration induced by interface defects associated states are quantitatively analyzed. The
phonon-assisted carrier hopping transport behavior in CQD systems is addressed.
4.3.1 CQD Solar Cell Fabrication and Current-voltage
Characterization
PbS CQDs were prepared following the previous reports [32, 33]. Briefly, oleic acid (4.8
mmol), PbO (2.0 mmol) and 1-octadecene (ODE, 56.2 mmol) were mixed and heated to 95 oC
under vacuum. This was followed by the injection of bis(trimethylsilyl) sulfide and ODE at a
high temperature of 120 oC. After cooling, the PbS CQDs were successively precipitated and
re-dispersed using acetone and toluene, respectively. The products were stored in a nitrogen
glove box for further surface passivation treatments. CQD solar devices fabricated in this
manner have the structure of ITO/ZnO/PbS-TBAI CQD/PbS-EDT CQD/Au which has been
demonstrated by two groups to be stable for ca. 5 months [24], and at least 1 month [34],
respectively. S-shaped I-V curves at room temperature for a CQD solar cell with this structure
were also reported by Chuang et al. [24] but were not well-explained in terms of physical
optoelectronic processes. Devices were fabricated in air through a typical solution process [24,
32, 35]. TBAI and EDT denote tetrabutylammonium iodide and 1, 2-ethanedithiol,
respectively. They are exchange-ligands for PbS CQDs to passivate quantum dot surface trap
CHAPTER 4. 80
states and adjust the interdot distance which determines the coupling strength between two
neighboring dots. The exciton transition energy (effective band gap) of PbS CQD was
measured in a solid CQD thin film to be ca.1.4 eV. In the fabrication process, a ca.100-nm
ZnO nanoparticle layer was spin-coated onto a clean glass substrate with a pre-deposited ITO
electrode of 145 nm thickness. PbS CQD layers were deposited through a layer-by-layer spin-
coating process, after which a TBAI solution (10 mg/ml in methanol) was applied to the
substrate for 30 s, followed by successive methanol rinse-spin steps. An EDT solution (0.01
vol% in acetonitrile) and acetonitrile were used for the deposition of the PbS-EDT nanolayer.
As examined by scanning electron spectroscopy, the final thicknesses of PbS-TBAI and PbS-
EDT CQD layers were ca. 200 nm and 50 nm, respectively. In addition, a 120-nm-thick Au
anode was evaporated on top of PbS-EDT CQD layer. Fig. 4.3 shows a schematic of the
fabricated CQD solar cell. Its energy diagram is shown in Fig. 4.4 which illustrates that, under
illumination, excitons generated in PbS-TBAI dissociate into free electron and hole carriers
through interdot coupling strength and the electric field at the heterojunction interface. Free
electrons are swept onto the cathode within the depleted area. In addition, because of the
energy barrier formed at the PbS-TBAI/PbS-EDT interface, electrons are blocked from
flowing to the anode which can significantly increase Isc and Voc [24, 35]. Unfortunately, as
will be discussed later, such architecture leads to the formation of a Schottky barrier in PbS-
EDT, which prevents holes from being extracted to the external Au anode, resulting in hole
accumulation and formation of an electric field with reverse direction to the forward field in
the main heterojunction diode (PbS-TBAI/ZnO). In the following discussion, for the sake of
clarification, the subscript h refers to the heterojunction diode (ZnO/PbS-TBAI) and s refers
CHAPTER 4. 81
to the Schottky diode (PbS-EDT/Au). Fig. 4.5 shows the equivalent circuit: two diodes with
opposite electric field directions representing heterojunction and Schottky diode, respectively.
Figure 4.3: Schematic of the double-layer CQD solar cell with the structure: ITO/ZnO/PbS-
TBAI QD/PbS-EDT QD/Au.
Figure 4.4: Device energy band diagram under illumination. PbS-EDT acts as an electron
blocking layer and a Schottky barrier is formed for holes, thus prevent their extraction to the
Au anode.
Figure 4.5: Equivalent electric circuit of a double-diode model, consisting of a heterojunction
diode between ZnO and PbS-QD layers and a Schottky diode between PbS-EDT and Au.
CHAPTER 4. 82
The homojunction formed at the PbS-TBAI /PbS-EDT interface gives rise to an electric
field with forwarding direction, same as that of the heterojunction, but its strength is
diminished by the reverse Schottky barrier because of the low PbS-EDT layer thickness (50
nm). Therefore, to simplify the analysis and improve parameter fitting reliability by decreasing
the number of unknown parameters in the hopping transport theoretical model developed [28]
and used in this work, this homojunction is not considered as an independent diode in this
paper, but part of the heterojunction.
The I-V characteristic curve measurements were obtained under laser excitation with a
wavelength of 830 nm and excitation intensity of 100 mW/cm2. The samples were placed in
a Linkam LTS350 cryogenic chamber which can maintain a constant temperature in a range
from 77 K to 520 K. For this study, the I-V characterizations were performed at 300 K, 250 K,
200 K, 150 K, and 100 K.
4.3.2 Double-diode-equivalent Hopping Transport Model
CQD solar cell I-V curves can be quantitatively interpreted using a hopping transport
mechanism as discussed in Sect.4.2. Referring to the energy diagram in Fig. 4.4, any hole
accumulation at the PbS-EDT/Au interface due to the effective impedance presented by the
Schottky barrier can give rise to a local space charge layer (SCL) 𝑊 and an electric field with
opposite direction to the heterojunction electric field. Therefore, charge carrier hopping
transport in PbS-TBAI (heterojunction) dominated by electron transport, and in PbS-EDT
(Schottky barrier) dominated by hole transport, should be distinct. Influence of the Schottky
barrier on the net current density is expected to be strong and possibly dominant over the local
diffusion current density due to the thinness of the SCL, especially at low temperatures. To
CHAPTER 4. 83
extract carrier hopping transport parameters, the electron current across the heterojunction is
expressed as the sum of diffusion and drift currents according to Eq. (4.24):
𝐼ℎ = 𝑞𝐴𝑁0 {[𝐷ℎ(𝑇)
𝐿ℎ(𝑇)] (
𝑐𝑜𝑠ℎ[𝑑/𝐿ℎ(𝑇)]
𝑠𝑖𝑛ℎ[𝑑/𝐿ℎ(𝑇)]) + 𝜇ℎ(𝑇)𝐸ℎ} (𝑒𝑥𝑝 [
𝜇ℎ(𝑇)𝑉ℎ
𝐷ℎ(𝑇)] − 1) (4.25)
Here, 𝐴 is the CQD solar cell area exposed to the light, 𝑁0 is the electron/hole population at
equilibrium without illumination, 𝐷ℎ(𝑇) is the electron hopping diffusivity, 𝐿ℎ(𝑇) is the
electron diffusion length, 𝑑 is the CQD layer thickness, 𝜇ℎ is the electron hopping mobility in
PbS-TBAI corresponding to the heterojunction, 𝑉ℎ is the electric potential across the entire
heterojunction, and 𝐸ℎ is the electric field. According to this mechanism, the heterojunction
gives rise to an electric field across the entire CQD nanolayer, while the Schottky diode
generates a reverse electric field in the local SCL with a nominal thickness of 𝑊. Similarly, as
a combination of diffusion and drift currents the hole current flowing across the Schottky diode
can be expressed as:
𝐼𝑠 = 𝑞𝐴𝑁0 {[𝐷𝑠(𝑇)
𝐿𝑠(𝑇)] (
𝑐𝑜𝑠ℎ[(𝑑−𝑊)/𝐿𝑠(𝑇)]
𝑠𝑖𝑛ℎ[(𝑑−𝑊)/𝐿𝑠(𝑇)]) + 𝜇𝑠(𝑇)𝐸𝑠} (𝑒𝑥𝑝 [
𝜇𝑠(𝑇)𝑉𝑠
𝐷𝑠(𝑇)] − 1) (4.26)
All parameters are analogous to those of the heterojunction in Eq. (4.25), while it is noted that
𝐸𝑠 is the electric field associated with the Schottky diode. The analysis of the double-diode
electric circuit model of Fig. 4.5 yields that at point 1
𝑉 = 𝑉ℎ − 𝑉𝑠 (4.27a)
at point 2,
𝐼𝑝ℎ = 𝐼 + 𝐼ℎ (4.27b)
and
𝐼 = 𝐼𝑠 (4.27c)
CHAPTER 4. 84
therefore,
𝐼𝑝ℎ = 𝐼𝑠 + 𝐼ℎ (4.27d)
Using Eqs. (4.25) to (4.27), the external voltage 𝑉 can be written as a function of the external
current 𝐼 and the free charge carrier hopping transport parameters:
𝑉 =𝐷ℎ(𝑇)
𝜇ℎ(𝑇)𝑙𝑛
{
𝐼𝑝ℎ−𝐼
𝑞𝐴𝑁0[𝐷ℎ(𝑇)
𝐿ℎ(𝑇)
𝑐𝑜𝑠ℎ[𝑑
𝐿ℎ(𝑇)]
𝑠𝑖𝑛ℎ[𝑑
𝐿ℎ(𝑇)]+𝜇ℎ(𝑇)𝐸ℎ]
+ 1
}
−𝐷𝑠(𝑇)
𝜇𝑠(𝑇)𝑙𝑛
{
𝐼
𝑞𝐴𝑁0[𝐷𝑠(𝑇)
𝐿𝑠(𝑇)
𝑐𝑜𝑠ℎ[𝑑
𝐿𝑠(𝑇)]
𝑠𝑖𝑛ℎ[𝑑
𝐿𝑠(𝑇)]+𝜇𝑠(𝑇)𝐸𝑠]
+ 1
}
(4.28)
4.3.3 Origins of Anomalous Current-voltage Curves
To investigate the effects of imbalanced charge carrier mobilities in the PbS-TBAI and
PbS-EDT, I-V curves were simulated using Eq. (4.28) and gradually reducing the hole mobility
µs from 1 cm2/Vs to 0.005 cm2/Vs, while other hopping parameters were kept unchanged as
shown in Fig. 4.7(a). Trial values for each parameter were selected referring to reported values
for different ligand-treated PbS CQDs. Kholmicheva et al. [37] studied MOA (8-
mercaptooctanoic acid), and MPA (3-mercaptopropionic acid) treated PbS CQDs using
photoluminescence (PL) spectroscopy and measured the corresponding exciton diffusivities to
be 0.003 cm2/s and 0.012 cm2/s, respectively. Carey et al. [38] estimated free electron and hole
diffusion length in the range from 30 to 230 nm for various PbS CQDs, including EDT-treated,
pure, CdCl2 treated, bromide treated, pure fused, and solution or solid-state-iodide-treated PbS
CQDs. For electron and hole mobility in CQD nanolayers used as CQD solar cell light
absorption and charge-carrier-transport layers, Carey et al. [38], and Tang and Sargent [18]
tabulated hole (electron) mobility for three types of ligand-treated PbS CQDs. PbS CQD
CHAPTER 4. 85
nanolayers without further ligand treatments yielded hole (electron) mobility of 7.2×10-4
cm2/Vs (1×10-3 cm2/Vs for electrons), while for CdCl2-treated and butylamine-treated CQD
nanolayers, it was reported to be 1.9 ×10-3 cm2/Vs (4.2 ×10-3 cm2/Vs for electrons) and 1.5
×10-3 cm2/Vs (2 ×10-4 cm2/Vs for electrons), respectively. To our best knowledge, the strength
of the hetero/homojunction-induced electric field in PbS CQDs has not been reported.
However, the electric field should be inversely proportional to device thickness. In InAs/GaAs
QD solar cells, Kasamatsu et al. [39] reported that the electric field was 46 kV/cm when the
device thickness was ca. 300 nm. This value increased to 193 kV/cm when the device thickness
was reduced to 50-nm. Due to the thinness of our Schottky diode SLC, a relatively higher
electric field value of 1×106 V/cm was chosen for simulations. To investigate the influence of
hole mobility µs on the solar cell I-V curves, parameters common to both heterojunction and
Schottky diode were considered to have the same values.
Figure 4.6: I-V characteristic curves of a CQD solar cell measured at 300K (a), 250K (b), 230K
(c), 200K (d), 150K (e), and 100K (f).
CHAPTER 4. 86
Figure 4.7: (a) Current-voltage curves at various µs, while other parameters are kept constant,
and (b) solar cell FF as a function of µs.
Fig. 4.7(a) shows shape changes which are consistent with our experimental solar cell I-V
curves from normal exponential to S-shaped, to negative exponential when µs decreases from
1 cm/Vs to 0.005 cm/Vs. Compared with the electron mobility µh in the heterojunction, a
higher µs value for the Schottky diode facilitates the extraction of charge carriers to the external
anode. As already mentioned, for this double-diode model there are four types of currents
CHAPTER 4. 87
contributing to the final device current, i.e. both heterojunction and Schottky diode contribute
diffusion and drift currents but in opposite directions. Most excitons dissociate within the
heterojunction (ZnO/PbS-TBAI) and the homojunction (PbS-TBAI/PbS-EDT) interfaces,
where the electric fields and/or interdot coupling strength are strong enough to split bound
electron-hole pairs. In addition, the diffusion current pertaining to the Schottky diode is
negligibly small. Consequently, it is expected that high µs can enhance hole extraction
efficiency, thereby improving solar cell performance.
In contrast, when µs is very small, such as in the case of low hole mobility within PbS-
EDT, holes accumulated at the PbS-EDT/Au interface induce a local SCL and electric field. A
similar phenomenon has been reported for a depleted-heterojunction PbS CQD solar cell [28].
In the present case, the decrease of µs may be due to multi-phonon assisted carrier hopping in
CQD thin films: at high temperatures, the high phonon density results in high hole mobility.
Temperature-dependence of charge carrier mobility was reported before [40]. It was also found
[41, 42] that carrier-mobility-controlled current density increased with temperature. As shown
in Fig. 4.7(b), the solar cell FF increases with µs and saturates above 0.2 cm2/Vs. Saturation
implies that all holes that flow to the anode metal are effectively extracted when hole mobility
is high enough, however, higher hole mobility cannot improve solar cell efficiency.
For our PbS CQD solar cell the diffusion current in the Schottky diode region is negligible.
To explore the influence of diffusivity, Fig. 4.8(a) shows that high diffusivity also gives rise
to anomalous solar cell I-V curves. This simulation used the same values of all other
parameters as in Fig. 4.7(a). High diffusion current at high diffusivity Ds compromises the one-
diode heterojunction I-V behavior. Consistently, Fig. 4.8(b) shows FF decrease with increasing
hopping diffusivity.
CHAPTER 4. 88
Figure 4.8: (a) Current-voltage curves at various Ds, while other parameters are kept
constant, and (b) solar cell FF as a function of Ds.
Besides imbalanced charge carrier mobility, the Schottky barrier is another pivotal role
playing a factor in the formation of anomalous CQD solar cell I-V curves. When holes arrive
at the Schottky diode side, Fig. 4.4, they are transported to the Au anode through a phonon-
assisted hopping transport mechanism. Therefore, with decreasing temperature, the reduced
phonon population depresses this phonon-assisted hole hopping process. Consequently, more
CHAPTER 4. 89
holes accumulate at the interface, leading to anomalous I-V curves as measured. Figs. 4.6(a)
to (f) show excellent match between experimental I-V curves and best fits to the theoretical
model using Eq. (4.28).
4.3.4 Open-circuit Voltage Origin of CQD Solar Cells
The Shockley-Queisser (SQ) limit [1] was derived for continuous-band semiconductors
where photon interactions with a solar cell induce the generation and recombination of free
electron-hole pairs. This mechanism is true for inorganic solar cells when carrier binding
energy is much smaller than the thermal energy kT, in which case lattice-bound excitons
dissociate into free electrons and holes at room temperature [43]. Consequently, Voc in the case
of ideal solar cells is the result of electron and hole quasi-Fermi energy level splitting, equal
to the voltage difference between device contacts [44-46]. Carried over to CQD solar cells, the
use of the SQ limit requires that exciton binding energy should be negligible. This is not the
case with most CQD solar cells which have high exciton binding energy, especially in CQD
thin films with low coupling strength and large interdot distances. Therefore, in most practical
CQD solar cells, an exciton must encounter a heterojunction where it can dissociate into free
electrons and holes. Unfortunately, heterojunctions incur additional energy losses by enhanced
exciton recombination, which, in turn, decreases Voc and device efficiency [47-49].
Specifically, the impact of heterojunctions on Voc loss is through the formation of charge-
transfer (CT) states (which are also named bandtail states [50]) located at acceptor-donor
interfaces (the ZnO/PbS-TBAI interface in this study), a mechanism also reported for organic
solar cells [47-49, 51]. As shown in Fig. 4.4, ECT is the energy gap between Ec of ZnO and Ev
of PbS-TBAI and excitons remain electrically neutral due to their high binding energy.
Hopping excitons diffuse and induce charge transfer across the heterojunction interface to form
CHAPTER 4. 90
a bound polaron pair (BP) with binding energy EB, which is significantly smaller than the
binding energy of a bulk exciton. These types of bound pairs are prone to dissociate into free
carriers. In the case of organic photovoltaic materials, EB is typically less than 0.5 eV and thus
results in non-thermodynamically limited exciton dissociation, always be lower than the bulk
exciton binding energy (1 eV) [47]. It is known [51-53] that CT states can also absorb photons,
but due to the much lower density of these interface states than bulk states, the CT absorption
coefficient is typically two to three orders of magnitude lower than that inducing bulk exciton
transitions [52, 53]. Furthermore, because excitons are bound with smaller energy gap (ECT) at
the heterojunction interface than with bulk energy gap (Eg), bulk materials dominate absorption
whereas heterojunction interfaces dominate recombination and dissociation. Consequently, the
recombination rate increases at interfaces because it can take place via lower energy bound-
pair states. Moreover, since small ECT does not limit dissociation, electron and hole quasi-
Fermi energy level splitting is no longer the major factor in determining Voc. The maximum
possible open-circuit voltage is determined by 𝑞𝑉𝑜𝑐𝑚𝑎𝑥 = E𝐶𝑇 , which occurs in the limit of
high-incident intensity and 0 K [47]. Voc is proven to be temperature-dependent and can be
expressed by the following expression [48, 51]
𝑉𝑜𝑐(𝑇) =𝐸𝐶𝑇
𝑞+𝑘𝑇
𝑞𝑙𝑛 [
𝐽𝑠𝑐ℎ3𝑐2
𝐹𝑞2𝜋(𝐸𝐶𝑇−𝜆)] +
𝑘𝑇
𝑞𝑙𝑛(𝐸𝑄𝐸𝐸𝐿) , (4.29)
where 𝐽𝑠𝑐 is the short-circuit current density, ℎ is Plank’s constant, 𝐹 is the florescence
emission intensity, 𝜆 is the reorganization energy associated with the CT absorption process,
and 𝐸𝑄𝐸𝐸𝐿is the electron luminescence external quantum efficiency. Voc losses occur through
radiative and non-radiative CT state recombination, labeled as ∆𝑉𝑂𝐶𝑟𝑎𝑑 and ∆𝑉𝑂𝐶
𝑛𝑜𝑛, respectively.
Voc loss to radiative CT state recombination ( ∆𝑉𝑂𝐶𝑟𝑎𝑑 ) is a thermodynamically imposed
mechanism for a given material system, whereas non-radiative recombination ( ∆𝑉𝑂𝐶𝑛𝑜𝑛) can be
CHAPTER 4. 91
avoided through many approaches, for example, by removing material defects. From Eq. (4.29),
∆𝑉𝑂𝐶𝑟𝑎𝑑 and ∆𝑉𝑂𝐶
𝑛𝑜𝑛can be obtained as [48]:
∆𝑉𝑂𝐶𝑟𝑎𝑑(𝑇) = −
𝑘𝑇
𝑞𝑙𝑛 [
𝐽𝑠𝑐ℎ3𝑐2
𝑓𝑞2𝜋(𝐸𝐶𝑇−𝜆)] (4.30a)
∆𝑉𝑂𝐶𝑛𝑜𝑛(𝑇) = −
𝑘𝑇
𝑞𝑙𝑛(𝐸𝑄𝐸𝐸𝐿) (4.30b)
Table 4.1: Summary of best-fitted parameters using Eq. (4.29).
Figure 4.9(a) shows best-fits to the temperature-dependent Voc using Eq. (4.29). The best-
fitting procedure was performed 300 times, followed by statistical analysis of the fitted
parameters to generate the final parameters as summarized in Table 4.1. The detailed procedure
to assure reliability of the measured parameters is discussed in the next section. The fitted ECT
= 1.1 eV exhibits high fitting uniqueness, leading to the maximum achievable Voc = 1.1 V. For
comparison, Eg of our PbS-TBAI is ca. 1.4 eV [24, 32] since the smaller ECT does not limit
exciton dissociation [47]. As shown in Fig. 4.9(b), the decrease in Voc through radiative and
non-radiative recombination was extracted using Eqs. (4.30a) and (4.30b). The maximum Voc
exhibits insignificant change within our experimental temperature range, consistent with
results reported by Gruber et al. [51]. Therefore, the maximum Voc determined by ECT is
henceforth considered to be constant. Both radiative and non-radiative recombination Voc
losses decrease when temperature decreases.
Parameters Sample size
(fitting times)
Mean Value SD 95 % confidence
intervals
ECT , eV
300
1.1 8.6×10-9 ±9.8×10-10
f , eV2 0.0069 0.0027 ±3.1×10-4
λ, eV 0.35 0.16 ±0.018
EQEEL 5.6×10-6 2.6×10-6 ±2.9×10-7
CHAPTER 4. 92
Figure 4.9: (a) Figure of the measured open-circuit voltage (Voc) and short-circuit current (Isc)
as a function of temperature. Equation (4.29) was used for the best-fitting of Voc. (b) Voc at
various temperatures. (c) The ratio 𝛥𝑉𝑜𝑐𝑟𝑎𝑑 / 𝛥𝑉𝑜𝑐
𝑛𝑜𝑛 as a function of temperature.
CHAPTER 4. 93
Furthermore, Fig. 4.9(c) shows that ∆𝑉𝑂𝐶𝑟𝑎𝑑 and ∆𝑉𝑂𝐶
𝑛𝑜𝑛 have similar magnitudes, although
radiative recombination seems to overtake non-radiative recombination as the dominant
recombination mechanism for Voc loss at low temperatures, as expected. The maximum Voc
will be theoretically achieved at 0 K [47]. It should be noted that if CT state emission is
negligible, the maximum Voc is determined by the bulk energy band gap 𝐸𝑔. Overall, according
to the above theoretical model, to increase the CQD solar cell Voc, ECT should be enhanced in
future QD photovoltaic device design. In addition, taking our solar cells as an example, it is
also suggested to passivate ZnO/PbS-TBAI interface traps states through proper chemical
ligands to decrease the non-radiative recombination induced Voc loss.
4.3.5 Temperature-dependent Carrier Hopping Transport and
CQD Solar Cell Performance
As shown in Fig. 4.9(a), despite the enhanced non-radiative recombination, short-circuit
current increases with increasing temperature. To extract carrier hopping transport parameters
through best-fits to the six I-V curves at different temperatures, Fig. 4.6, two independent best-
fitting computation programs were used to investigate the reliability and thus the uniqueness
of the best-fitted results in a statistical analysis. These programs have been used successfully
in earlier multi-parameter fits to experimental data from an amorphous/crystalline silicon solar
cell heterojunction [54]. The ‘mean-value best fit’ minimizes the mean square variance
between the experimental data and the theoretical values. The ‘statistical best fit’ uses the
fminsearchbnd solver [55] to minimize the sum of the squares of errors between the
experimental and calculated data. This program delivers different results due to different trial
starting points generated by the program itself, thereby creating standard deviations (SD) or
variances of the theoretical curve best-fitting procedure to the experimental points. To
CHAPTER 4. 94
investigate the reliability of the fitted parameters, this procedure was repeated several hundred
times and the 100 lowest variances were selected. Based on these 100 best-fitted results, the
variance and 95 % confidence interval were calculated. The statistical mean value was used as
reliability (uniqueness) measure and the variance as a precision measurement of the associated
parameter. It is seen that the results from the two independent best-fitting programs are in very
good-to-excellent agreement. In Table 4.2, the best-fitted values of both Dh and Ds decrease
monotonically with decreasing temperature, which is consistent with the multi-phonon assisted
hopping mechanism [16, 28, 56-58] and are close to the reported values [37, 58] between 0.003
and 0.012 cm2/s for PbS CQDs which were surface passivated with different ligands.
Figure 4.10: Arrhenius plots of (a) the ratio Th(T)/Dh(T) and (b) the ratio Ts(T)/Ds(T). The
mobility and diffusivities were calculated and fitted for the PbS-TBAI and the PbS-EDT
interface, respectively.
Similar to temperature-dependent electron mobility in a two-dimensional quantum dot
superlattice [40] (electron mobility increase with temperature), the electron and hole mobilities,
𝜇ℎ and μs, decrease monotonically with decreasing temperature, which is opposite to charge
carrier mobility trends in continuous energy band structures of e.g. inorganic photovoltaic
materials. It has been found [41, 42] that the hopping conductivity of CdSe or PbSe quantum
CHAPTER 4. 95
dot arrays increases with temperature in the range from 10 K to 523 K, due to the increase of
electron mobility by means of multi-phonon assisted hopping. Fig. 4.10 shows Arrhenius plots
of [𝑇𝜇𝑗(𝑇)
𝐷𝑗(𝑇)] , j = h,s, a combination of terms identified in Eq. (4.13) which replaces the
conventional Einstein relation in the CQD hopping transport theory. These figures
experimentally prove the validity of the hopping Einstein relation and extract heterojunction,
Fig. 4.10(a), and Schottky barrier inside the SCL, Fig. 4.10(b), activation energies Ea,h = 37.2
meV and Ea,s = 29.3 meV, respectively.
Figure 4.11: (a) The CQD solar cell FFs measured at various temperatures. (b) Maximum
power of as-studied CQD solar cell measured at various temperatures.
From the fitted parameters in Table 4.2 using Eq. (4.28), the following explanation emerges:
at low temperatures the charge carrier extraction efficiency is reduced due to the reduced hole
mobility, however, the enhanced influence of the Schottky barrier on hole extraction, Fig. 4.4,
reduces the hole current. The overall effect is a decrease in short-circuit current with decreasing
temperature. Hopping transport of charge carriers in CQD thin films is a multi-phonon-assisted
process [28, 56, 58]. As a consequence, the extraction efficiency of charge carriers, including
free electrons and holes, is suppressed at low temperatures owing to the reduced thermal
CHAPTER 4. 96
energy. In Fig. 4.11 (a), the FF calculated from 𝐹𝐹 = 𝑃𝑚𝑎𝑥
𝑉𝑜𝑐𝐼𝑠𝑐, where 𝑃𝑚𝑎𝑥 is the maximum
power calculated from the I-V curves, is reduced from 0.51 mW at 310 K to 0.13 mW at 100
K as a result of reduced charge carrier extraction efficiency.
The imbalance of carrier mobility in the heterojunction (PbS-TBAI) and in the associated
Schottky diode (PbS-EDT) gives rise to S-shaped and negative exponential current-voltage
curves, as shown in Fig. 4.6. In Table 4.2, the best-fitted values of carrier mobilities 𝜇ℎ and μs,
at 300K show that 𝜇ℎ is smaller than μs, implying sufficiently high charge carrier extraction
rate in the hole extraction layer (PbS-EDT). This results in holes being able to be extracted
efficiently and, as a result, the CQD solar cell behaves like a normal one-diode device. The
best-fitted results reveal better photovoltaic material made of PbS-EDT than of PbS-TBAI.
This is also consistent with the SCL lower mobility activation energy Ea,s across the Schottky
barrier, Fig. 4.10(b), than Ea,h across the heterojunction, Fig. 4.10(a). A similar conclusion in
terms of electron mobility was reported by ref. [58] using frequency-domain photocarrier
radiometry (PCR). However, due to the complicated device architecture, each of the two fitted
mobilities leading to the Einstein plots of Figs. 4.10(a) and (b) should not be unconditionally
interpreted to be exclusively associated with the PbS-TBAI or PbS-EDT interfaces. Although
at 250 K the fitted μs > 𝜇ℎ, the 250 K I-V characteristic shapes in Figs. 4.6(b) and 4.7(a) reveal
that low mobility of charge-carrier-induced hole accumulation on the anode side has already
set in, thereby reducing hole extraction efficiency. This trend is further demonstrated at even
lower temperatures by the fitted results which show that 𝜇ℎ and μs magnitudes reverse, with
𝜇ℎ becoming larger than μs, especially at 100 K. However, it should be noted that both
unbalanced mobility and Schottky diode contribute to the formation of anomalous I-V
characteristics, although our best-fitted results are not able to provide strong evidence of the
CHAPTER 4. 97
role of the SCL electric field, the fitted values of which show high standard deviation. The
existence of the reverse Schottky diode has been established in previous reports [24, 19]. It is
important to point out from the simulated curves shown in Fig. 4.7(a) that anomalous I-V
curves can appear even without the existence of a Schottky diode. Fig. 4.11(b) shows that the
calculated maximum power decreases with decreasing temperature. This is consistent with
both mechanisms of the imbalanced charge carrier mobilities and existence of Schottky diode,
which reduce solar efficiency through hole-accumulation-induced charge carrier extraction
reduction. Apart from optimizing the work functions of the anode metal and CQD nanolayers,
for instance, a smaller anode work function can alleviate the effect of Schottky diode effect in
our solar cells, applying a high charge carrier mobility layer next to the anode so as to reduce
the interfacial activation energy measured through the Einstein relation, Fig. 4.10(a), appears
to be a potentially effective method for the improvement of CQD solar efficiency. The
suggestion of applying high charge carrier mobility layer next to the anode is supported by the
results of Zhang et al. [59] that an increased CQD solar cell PCE was achieved by employing
a hole transport interlayer between the QD film and anode metal. Furthermore, based on our
simulations, it is also recommended to apply graded hole transport layers close to the anode
which 1) enhance the intrinsic electric field to increase the hole conductivity; 2) create
additional interfaces to improve exciton dissociation; 3) alleviate the Schottky diode effect to
remove hole accumulation influence; and 4) further block electron flow to the anode. However,
this strategy may complicate the device fabrication processes.
The photogenerated current, Iph, according to the electric circuit of Fig. 4.5, is the sum of
Isc and the current flowing across the heterojunction diode. Both of these currents are controlled
by temperature-sensitive exciton and free charge carrier hopping transport. The best-fitted
CHAPTER 4. 98
values of Iph at various temperatures are shown in Table 4.2. Iph and Isc exhibit the same trend
with temperature. Iph depends on the dissociation of excitons as follows: when excitons are
generated optically, they can dissociate into free electrons and holes through two paths. They
may diffuse to the heterojunction interface where the local electric field can separate electron-
hole pairs [60, 61]; or they can be decoupled during hopping diffusion between neighboring
quantum dots [62]. Inter-dot coupling strength is determined by ligand length: short ligand
length yields strong coupling strength. However, it is possible that exciton decoupling is also
thermal energy-related, as more ambient thermal energy induces higher exciton vibration
amplitude, increasing the decoupling probability. Excitons that do not dissociate undergo
recombination through radiative and/or nonradiative processes as discussed above.
Recombination contributes to the loss of excitons and consumes photogenerated current.
Therefore, Iph decreases at lower temperatures.
Electron (hole) lifetime 𝜏ℎ(τs) is calculated from statistically fitted values of Dh (Ds) and Lh
(Ls) through the equation: τh =Lh2
Dh. As summarized in Table 4.2, both 𝜏ℎ and τs decrease with
increasing temperature which is consistent with the temperature-dependent carrier lifetime
reported by Wang et al. [17, 63] and Mandelis et al. [28]. In these cases, excitons in both
coupled and uncoupled PbS CQDs were found to possess longer lifetimes at low temperatures
due to lower radiative and non-radiative recombination. Table 4.2 shows two calculated values
of 𝜏ℎ (as well as τs) for each temperature, obtained from ‘mean-value best fit’ and ‘statistical
best fit’. They exhibit small differences demonstrating high reliability and uniqueness of the
measurements resulting from the proposed model. The non-monotonic trend of the electron
hopping diffusion length 𝐿ℎ is the result of the trade-off between increased diffusivity and
decreased lifetime with increased temperature, through 𝐿ℎ = √𝐷ℎ𝜏ℎ . Strictly speaking, the
CHAPTER 4. 99
diffusion length 𝐿ℎ is a material property, and the intrinsic lifetime affecting factors should
include trap states and dot-to-dot coupling [64, 18, 57, 62]. The best-fitted 𝐿ℎ and 𝐿𝑠 values
agree with reported values for PbS CQDs in the range between 30 nm and 230 nm [38].
The fitted SCL width 𝑊 associated with hole accumulation at 300K is larger than that at
other temperatures. Note that the fitted depletion width 𝑊 through Eq. (4.28) is an effective
value across the PbS CQD layers. In other words, the actual SCL of PbS-TBAI and PbS-EDT
is determined through a competitive process between the depletion layers of the heterojunction
and the Schottky diode. To calculate the effective SCL, the depletion extent and width of both
diodes should be considered. The increase of W at high temperatures, Table 4.2, is consistent
with changes in the accumulated hole density at the PbS-EDT CQD/Au interface which acts
as a conventional junction depletion layer. Specifically, at low temperatures, due to reduced
hole extraction efficiency as discussed above, higher density of accumulated holes results in
higher density of occupied local QD energy states, and therefore a narrow depletion layer. In
contrast, at high temperatures, hole density at the interface decreases due to higher hole
extraction efficiency, thereby, alleviating the concentration gradient and resulting in reduced
density of occupied local QD energy states, and a wider depletion layer.
CHAPTER 4. 100
Tab
le 4
.2:
Sum
mar
y o
f b
est-
fitt
ed p
aram
eter
s.
T
emper
ature
, K
Fit
ted
par
amet
ers
300
250
230
200
150
100
Mea
n-
val
ue
bes
t fi
t
Sta
tist
ical
bes
t fi
t
Mea
n-
val
ue
bes
t fi
t
Sta
tist
ical
bes
t fi
t
Mea
n-
val
ue
bes
t fi
t
Sta
tist
ical
bes
t fi
t
Mea
n-
val
ue
bes
t fi
t
Sta
tist
ical
bes
t fi
t
Mea
n-
val
ue
bes
t fi
t
Sta
tist
ical
bes
t fi
t
Mea
n-
val
ue
bes
t fi
t
Sta
tist
ical
bes
t fi
t
Ho
pp
ing
dif
fusi
vit
y
Dh,
cm
2/s
3.6
6×
10
-
4
3.5
2×
10
-4
±1
.44
×1
0-5
2.5
2×
10
-
4
2.3
6×
10
-4
±2
.74
×1
0-5
1.1
7×
10
-
4
1.0
9×
10
-4
±2
.06
×1
0-5
7.7
4×
10
-
5
6.4
8×
10
-5
±1
.28
×1
0-5
4.7
4×
10
-
5
4.1
1×
10
-5
±6
.36
×1
0-6
3.6
1×
10
-
5
3.5
3×
10
-5
±1
.49
×1
0-6
DS,
cm
2/s
3.7
7×
10
-
4
3.9
4×
10
-4
±2
.46
×1
0-5
8.4
3×
10
-
5
8.3
2×
10
-5
±2
.88
×1
0-5
6.7
6×
10
-
6
2.0
7×
10
-5
±1
.60
×1
0-5
3.5
6×
10
-
7
5.8
7×
10
-7
±4
.52
×1
0-7
2.3
0×
10
-
7
2.5
1×
10
-7
±4
.83
×1
0-8
1.2
1×
10
-
7
1.2
8×
10
-7
±9
.57
×1
0-9
Dif
fusi
on
leng
th
Lh,
cm
3
.38×
10
-
6
3.9
3×
10
-6
±8
.63
×1
0-7
4.1
3×
10
-
6
3.7
3×
10
-6
±6
.56
×1
0-7
3.6
8×
10
-
6
3.9
6×
10
-6
±2
.75
×1
0-7
3.2
2×
10
-
6
3.8
5×
10
-6
±6
.41
×1
0-7
3.9
4×
10
-
6
4.1
8×
10
-6
±2
.53
×1
0-7
4.7
5×
10
-
6
5.1
1×
10
-6
±3
.66
×1
0-7
Ls,
cm
5
.12×
10
-
6
5.7
3×
10
-6
±6
.24
×1
0-7
3.0
8×
10
-
6
3.5
3×
10
-6
±5
.63
×1
0-7
5.1
2×
10
-
6
4.5
4×
10
-6
±6
.08
×1
0-7
3.3
7×
10
-
6
2.8
0×
10
-6
±5
.93
×1
0-7
2.6
5×
10
-
6
2.4
5×
10
-6
±3
.23
×1
0-7
5.7
1×
10
-
6
5.4
9×
10
-6
±2
.25
×1
0-7
Ho
pp
ing
mo
bil
ity
μh,
cm
2/V
s
3.4
1×
10
-
3
3.2
9×
10
-3
±1
.34
×1
0-4
1.3
3×
10
-
3
1.2
5×
10
-3
±1
.45
×1
0-4
3.8
3×
10
-
4
3.2
3×
10
-4
±6
.32
×1
0-5
2.0
9×
10
-
4
1.8
5×
10
-4
±3
.50
×1
0-5
8.1
5×
10
-
5
8.6
3×
10
-5
±1
.37
×1
0-5
3.8
9×
10
-
5
3.9
2×
10
-5
±9
.20
×1
0-7
μs,
cm
2/V
s
5.5
9×
10
-
3
5.8
4×
10
-3
±3
.64
×1
0-4
1.6
1×
10
-
3
1.6
6×
10
-3
±5
.48
×1
0-5
1.0
9×
10
-
4
1.5
7×
10
-4
±4
.80
×1
0-5
5.2
7×
10
-
6
7.9
8×
10
-6
±5
.84
×1
0-6
1.2
6×
10
-
6
1.3
2×
10
-6
±2
.52
×1
0-7
6.3
2×
10
-
7
6.6
4×
10
-7
±4
.11
×1
0-8
Sp
ace
char
ge
wid
th
W,
cm
2
.33×
10
-
5
2.2
4×
10
-5
±1
.08
×1
0-6
1.8
4×
10
-
5
1.8
2×
10
-5
±1
.66
×1
0-6
1.3
9×
10
-
5
1.5
7×
10
-5
±1
.83
×1
0-6
1.6
3×
10
-
5
1.7
8×
10
-5
±1
.65
×1
0-6
1.7
7×
10
-
5
1.6
×1
0-5
±1
.84
×1
0-6
1.6
4×
10
-
5
1.7
9×
10
-5
±1
.67
×1
0-6
Pho
to-
gen
erat
ed
curr
ent
I ph,
A
8.5
6×
10
-
4
8.5
7×
10
-4
±1
.08
×1
0-1
4
6.6
9×
10
-
4
6.7
0×
10
-4
±3
.11
×1
0-1
3
4.4
9×
10
-
4
4.5
0×
10
-4
±1
.22
×1
0-6
2.8
1×
10
-
4
2.7
5×
10
-4
±8
.74
×1
0-6
1.4
1×
10
-
4
1.3
7×
10
-4
±4
.3×
10
-6
5.8
2×
10
-
5
5.7
5×
10
-5
±1
.28
×1
0-6
Lif
etim
e
τ h,
s 3
.12×
10
-
8
4.3
9×
10
-8
6.7
7×
10
-
8
5.8
8×
10
-8
1.1
6×
10
-
7
1.4
4×
10
-7
1.3
4×
10
-
7
2.2
9×
10
-7
3.2
8×
10
-
7
4.2
5×
10
-7
6.2
5×
10
-
7
7.3
9×
10
-7
τ s,
s 6
.95×
10
-
8
8.3
3×
10
-8
1.1
3×
10
-
7
1.5
0×
10
-7
3.8
8×
10
-
6
9.9
6×
10
-7
3.1
9×
10
-
5
1.3
4×
10
-5
3.0
5×
10
-
5
2.3
9×
10
-5
2.6
9×
10
-
4
2.3
6×
10
-4
CHAPTER 4. 101
4.4 Conclusions
A theoretical model of carrier discrete hopping transport in CQD materials and
photovoltaic solar cells was introduced in the framework of non-continuous energy band
structure of CQD ensembles. For further investigation of the temperature-dependent thermal
energy associated carrier hopping transport mechanism in CQD solar cells. A double-diode
electric circuit model featuring a heterojunction (PbS-TBAI/ZnO) and a Schottky diode (PbS-
EDT/Au anode) with two electric fields of opposite directions was developed and used to
quantitatively interpret experimental I-V curves obtained from a fabricated CQD solar cell
with a structure: ITO/ZnO/PbS-TBAI QD/PbS-EDT QD/Au, which exhibits anomalous I-V
characteristics at temperatures below 300K. Detailed best-fits of I-V data to the theoretical
model and simulations revealed that imbalanced charge carrier mobility is one of two factors
giving rise to S-shaped and negative exponential I-V characteristics. The other factor is the
formation of a reverse Schottky barrier for holes adjacent to the hole-extracting anode. In
addition, the existence of charge-transfer (CT) states attribute to the loss of Voc through
enhanced radiative and non-radiative recombination processes in these states which provided
quantitative insight into the nature of the Voc temperature dependence.
The presented models and quantitative I-V analysis for the discrete carrier hopping
transport, imbalanced carrier mobilities and Schottky barrier induced anomalous I-V
characteristics, and the Voc deficit at interfaces can be used to measure device transport
parameters, especially hopping mobility, aimed at minimizing Schottky barrier in order to
maximize the short-circuit current and Pmax, toward the optimization of CQD solar cell
fabrication.
102
Chapter 5
Colloidal Quantum Dot Solar Cell Efficiency Optimization:
Impact of Hopping Mobility, Bandgap Energy, and Electrode-
semiconductor Interfaces
5.1 Introduction
As discussed in Chapter 2 CQD solar cells are presently attracting immense research
interest on a global scale. However, no comprehensive device efficiency optimization
strategies have been reported aiming at achieving higher PCE, specifically for CQD solar cells.
Researchers use common sense approaches instead, trying to improve CQD solar cell
efficiency through pursuing higher carrier mobility using disparate surface passivation
materials and increasing quantum dot size for lower bandgap energy in order to harvest the
solar spectrum in a wider wavelength range. This universal strategy, however, is typically valid
for conventional solar cells of high carrier mobility such as Si solar cells, rather than for low
carrier mobility systems of a discrete carrier transport nature, such as colloidal quantum dot
(CQD) and organic solar cells. Alarmingly, however, it has been reported that state-of-the-art
high PCE solar cells are actually achieved using materials generally not exhibiting the highest
CQD carrier mobility [1]. Furthermore, researchers reverted to using smaller dots with wider
bandgap energy when they found larger dots yielded even lower PCE. This raises the crucial
question of whether higher mobility and smaller bandgap CQDs can always produce higher
PCEs in CQD solar cells. How do CQD carrier mobility and bandgap energy determine solar
cell performance? This chapter addresses these critical issues with the fabrication and study of
CQD solar cells it was discovered that the photocurrent density is voltage-dependent and is
CHAPTER 5. 103
accompanied by low carrier mobility, which indicates that the conventional constant
photocurrent assumption may be invalid for CQD solar cells. Generally, to analyze current-
voltage characteristics of CQD solar cells, the well-known Shockley-Queisser (S-Q) equation
is often used [2], which is discussed in Sect.4.1. The huge difference in mobility, 10-5 to 10-1
cm2/Vs [3-6] for PbS CQDs while 102 - 103 cm2/Vs for Si solar cell, leads to questioning the
validity of applying the S-Q model to such systems. The discovery of external-voltage-
dependent photocurrent density in PbS CQD based solar cells enabled us to revisit and relax
the constant photocurrent density assumption in the well-known S-Q equation for CQD solar
cells which has been and continues to be the prevailing assumption among researchers to-date.
A similar voltage-dependent photocurrent was also reported by Würfel et al. [7] for organic
solar cells which also have much lower carrier mobility. Therefore, this chapter, for the first
time, develops a comprehensive analysis of the dependence of CQD solar cell current-voltage
characteristics on carrier mobility and CQD bandgap energy.
Figure 5.1: Schematic of the as-fabricated CQD solar cell sandwich structure. PbX2 and AA
represent lead halide and ammonium acetate, respectively, acting as exchange-ligands for PbS
CQDs.
CHAPTER 5. 104
Furthermore, as will be discussed in detail in Chapter 8, most researchers have reported
solar cell efficiencies based on small-spot (<0.1 cm2) testing, including professional
certification characterizations of solar cells towards an entry in the Solar Cell Efficiency Tables.
This, however, raises questions about the overall solar cell performance and stability
estimations. Furthermore, no imaging studies of CQD solar cells have been reported in efforts
to acquire an insightful physical picture of defect or contact effects on key solar cell
performance parameters. In order to study the contact/CQD interface influence on the
performance of CQD solar cells to further develop this device efficiency optimization strategy,
this chapter uses a large-area photovoltaic device non-destructive imaging (NDI) carrier-
diffusion-wave characterization technique (HoLIC), as discussed in details in Sect.3.4, to
obtain open-circuit voltage distribution and carrier collection efficiency images and were thus
able to elucidate the effects of the CQD/electrode interfaces on solar cell performance within
the framework of our drift-diffusion J-V model.
The presented theoretical model and large-area characterization technique can be of
significance for guiding CQD solar cell optimization with respect to CQD surface passivation
ligand selection and the determination of CQD energy bandgap (or quantum dot size), as well
as for solar cell fabrication quality control.
5.2 Derivation of Carrier Hopping Drift-diffusion J-V Model for
CQD Solar Cells
As it has already been discussed in Chapter 4, Disorder sources in CQD ensembles,
including variations in confinement energy, electron-electron repulsion, coupling and thermal
broadening, cause CQD-based materials and devices to exhibit discrete hopping conductivity
and diffusivity [8, 9]. Using intensity modulated illumination, the distribution and hopping
CHAPTER 5. 105
transport of excitons and charge carriers follow a diffusion-wave behavior. The theory of
particle-population-gradient-induced diffusive transport through spatial profiles of discrete
hopping into and out of a quantum dot was developed in detail in ref. [10]. In this chapter, the
prevailing assumption of constant photocurrent [10] was relaxed. Based on the photocarrier
hopping diffusion-wave theory to solve the carrier population rate equation, quenching and
surface recombination velocity (SRV) associated boundary conditions were assumed with
respect to both hopping diffusion and drift current densities for high-efficiency CQD solar cell
structures, thus overcoming the voltage limitation [10] which leads to a decreased J with V
when the applied voltage is higher than the built-in potential. Specifically, in a one-
dimensional quantum dot ensemble, each quantum dot characterized by its own size and
energy manifold is separated by a mean distance from its neighbors. Therefore, the rate
equation for the net carrier flux entering one quantum dot within a time interval 𝑑𝑡 can be
written as [10]:
𝜕𝑛𝑃𝑉(𝑥,𝑡)
𝜕𝑡= −
𝜕𝐽𝑒(𝑥,𝑡)
𝜕𝑥−𝑛𝑃𝑉(𝑥,𝑡)
𝜏 (5.1)
where 𝑛𝑃𝑉(𝑥) is the carrier concentration under illumination, 𝜏 is the lifetime, and 𝐽𝑒(𝑥, 𝑡) is
the carrier hopping flux in units of s-1cm-2. The subscript PV indicates that the solar cell is
under external bias. With an applied dc voltage, 𝑛𝑃𝑉 becomes time-independent, therefore, Eq.
(5.1) can be further written as
𝑑2𝑛𝑃𝑉(𝑥)
𝑑𝑥2− (
𝜇𝑒𝐸
𝐷𝑒)𝑑𝑛𝑃𝑉(𝑥)
𝑑𝑥−𝑛𝑃𝑉(𝑥)
𝐷𝑒𝜏= 0 (5.2)
where 𝐸 = 𝑉𝑏𝑖 −𝑉𝑎
𝑑 is the electric field across the solar cell, d is the CQD thin film thickness as
shown in Fig. 5.2, 𝐷𝑒 is the carrier hopping diffusivity, 𝜇𝑒is the hopping mobility, 𝑉𝑏𝑖 is the
built-in voltage, and 𝑉𝑎 is the photovoltage. Equation (5.2) is subject to a surface boundary
CHAPTER 5. 106
condition at x = 0 as shown in Fig. 5.2: 𝑛𝑃𝑉(0) − 𝑛(0) = ∆𝑁0, where ∆𝑁0 is the excess carrier
population generated by the photovoltaic effect. A second (quenching) boundary condition for
the CQD thin films at x = d is 𝑛𝑃𝑉(𝑑) − 𝑛(𝑑) = 0 , indicating an infinite SRV and the
immediate recombination of electrons and holes when they drift or diffuse to the interface.
Therefore, solving Eq. (5.2) with ∆𝑛𝑃𝑉(𝑥) = 𝑛𝑃𝑉(𝑥) − 𝑛(𝑥), it can be found that
∆𝑛𝑃𝑉(𝑥) =∆𝑁0(𝑒
𝑄2𝑑+𝑄1𝑥−𝑒𝑄1𝑑+𝑄2𝑥)
𝑒𝑄2𝑑−𝑒𝑄1𝑑 (5.3)
The excess carrier population ∆𝑁0 = 𝑛(0) (𝑒𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) can be derived through integrating
the electric field over the thickness of a solar cell under both dark and illumination conditions
[10]. Accordingly, the hopping drift and diffusion current densities [A/cm2] can be expressed
as
𝐽𝑒,𝑑𝑖𝑓𝑓 = −𝑞𝐷𝑒𝑑∆𝑛𝑃𝑉(𝑥)
𝑑𝑥|𝑥=0
= −𝑞𝐷𝑒𝑛(0) (𝑄1𝑒
𝑄2𝑑−𝑄2𝑒𝑄1𝑑
𝑒𝑄2𝑑−𝑒𝑄1𝑑) (𝑒
𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) (5.4)
𝐽𝑒,𝑑𝑟𝑖𝑓𝑡 = 𝑞𝜇𝑒𝐸𝑛(0) (𝑒𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) = 𝑞𝜇𝑒 (
𝑉𝑏𝑖−𝑉𝑎
𝑑) 𝑛(0) (𝑒
𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) (5.5)
with
𝑄1,2 =1
2(𝐶0 ±√𝐶0
2 + 4𝐶1) (5.6)
and with the definitions
𝐶0 =𝜇𝑒𝐸
𝐷𝑒= (
𝜇𝑒
𝐷𝑒)𝑉𝑏𝑖−𝑉𝑎
𝑑, 𝐶1 =
1
𝐿𝑒2 (5.7)
where 𝐿𝑒 is the dc hopping diffusion length. Eventually, the total dark current density 𝐽𝑒
(A/cm2) including drift and diffusion components can be expressed as
𝐽𝑒 = 𝐽𝑒,𝑑𝑖𝑓𝑓 + 𝐽𝑒,𝑑𝑟𝑖𝑓𝑡 = 𝑞𝑛(0) [−𝐷𝑒 (𝑄1𝑒
𝑄2𝑑−𝑄2𝑒𝑄1𝑑
𝑒𝑄2𝑑−𝑒𝑄1𝑑) + 𝜇𝑒 (
𝑉𝑏𝑖−𝑉𝑎
𝑑)] (𝑒
𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) (5.8)
CHAPTER 5. 107
Therefore, through adding electron and hole dark current densities and considering the
photocurrent density, the total carrier hopping current density under illumination can be
expressed as usual by
𝐽𝑖𝑙𝑙𝑢 = 𝐽𝑒 + 𝐽ℎ − 𝐽𝑝ℎ (5.9)
It should be noted that the hopping drift current density 𝐽𝑑𝑟𝑖𝑓𝑡 is 𝑉𝑎 dependent. Specifically,
the electric field (𝑉𝑏𝑖−𝑉𝑎
𝑑) leads to 𝐽𝑑𝑟𝑖𝑓𝑡 decreasing with 𝑉𝑎 , while the excess carrier
population 𝑛(0)𝑒𝜇𝑒𝑉𝑎𝐷𝑒 yields an exponential dependence of 𝐽𝑑𝑟𝑖𝑓𝑡 on 𝑉𝑎.
Figure 5.2: Schematic of the CQD solar cell energy band structure. PbX2 and AA represent
lead halide and ammonium acetate, respectively, acting as exchange-ligands for PbS CQDs.
In general, the boundary condition at x=d is not always an infinite SRV. There are four
types of surface recombination, namely, minority carrier hole recombination at the cathode,
majority carrier hole recombination at the anode, minority carrier electron recombination at
the anode, and majority carrier electron recombination at the cathode. Here, however, for the
CHAPTER 5. 108
sake of simplification, all types of surface recombination are treated in the same manner. The
surface recombination rate at d can be defined as
𝐽(𝑑) = 𝑆(𝑑)𝑛𝑃𝑉(𝑑) (5.10)
𝐽(𝑑) is a carrier flux in units of s-1cm-2. Solving Eq. (5.2) subject to boundary condition Eq.
(5.10), the total current density 𝐽𝑒 (A/cm2) is given by
𝐽𝑒 = 𝑞𝑛(0){−𝐷𝑒[𝑄1 − (𝑄1 − 𝑄2)𝑓] + 𝜇𝑒𝐸} (𝑒𝜇𝑒𝑉𝑎𝐷𝑒 − 1) (5.11)
with the definition
𝑓 ≡𝑒𝑄1𝑑[𝑆(𝑑)−𝜇𝑒𝐸+𝐷𝑒𝑄1]
[𝑆(𝑑)−𝜇𝑒𝐸](𝑒𝑄1𝑑−𝑒𝑄2𝑑)+𝐷𝑒(𝑄1𝑒𝑄1𝑑−𝑄2𝑒
𝑄2𝑑) (5.12)
Smaller surface recombination velocity S means more excess charge carriers across the
interface which should result in better solar cell performance and S = 0 corresponds to Ohmic
contact behavior as reported by Kirchartz et al. [11]. Also they found that surface
recombination played the role of a carrier recombination source in the way that a zero surface
recombination rate corresponds to preclude carrier recombination at the contacts (interface),
while an infinite recombination rate adds a new recombination pathway, leading to strongly
decreased Voc and PCE at high carrier mobility when carrier recombination is significantly
increased.
5.3 Experimental CQD Solar Cell Efficiency Optimization
The synthesis of oleic-acid-capped CQDs, ZnO nanoparticles, and CQD solar cells follows
the methods in Sect.4.3.1 and ref. [12] with the exception that PbS-TBAI CQD layer in Fig.4.3
was replaced by PbX2/AA-exchanged (PbX2: lead halide, AA: ammonium acetate) PbS inks
[13]. Figure 5.1 shows the sandwich structure of the CQD solar cells under study.
CHAPTER 5. 109
Following the device efficiency optimization strategy (See Sect. 5.5) based on the fact that
there are optimized bandgap energy and carrier mobility for a given type of CQD solar cells,
this section describes how the CQD solar cell efficiency was improved experimentally through
varying the CQD bandgap energy (dot size) and altering the CQD carrier mobility using
various surface passivation ligands and different ligand exchange methods[13]. Specifically,
instead of using PbX2/AA in solution for ligand exchange, two types of CQD solar cells using
solid-state layer-by-layer exchange with tetrabutylammonium iodide (TBAI) and solution
exchange with methylammonium lead iodide (MAPbI3) as ligands [13] were fabricated and
labeled PbS-TBAI and PbS- MAPbI3, respectively. The structures of these samples are shown
in Fig. 5.1 except that PbS- PbX2/AA is replaced by PbS-TBAI and PbS-MAPbI3 for our
control samples. As already reported [13], consistent with the efficiency optimization strategy,
the dependencies of CQD solar cell external quantum efficiency and current density on CQD
bandgap energy Eg were initially found to increase with the reduction of Eg, then decrease
when Eg became smaller than a threshold value between 1.28 eV and 1.38 eV. Limited by the
scope of the experiments, an exception was the current density dependence of PbS-PbX2/AA
which exhibited a monotonic increase with decreasing Eg without attaining a threshold value
yet. In comparison, Voc exhibited a positive linear dependence on the CQD bandgap energy,
and it was also exchange-ligand-dependent with PbS- PbX2/AA and PbS- MAPbI3 possessing
the highest and lowest Voc, respectively, at all bandgap energies. All of these experimental
results are in good agreement with our theoretical model predictions (Sect.5.5). Although
mobility measurements are not discussed here, the PbS- PbX2/AA CQD solar cells were
characterized with photothermal deflection spectroscopy and found to have fewer bandtail
states and higher CQD packing density compared with the controls, as well as higher
CHAPTER 5. 110
uniformity characterized by grazing-incidence small-angle X-ray scattering measurements.
Therefore, PbS- PbX2/AA CQD solar cells were found to have the highest PCE of all CQD
solar cells. Regardless of the fact that PbS- MAPbI3 CQD thin films had higher carrier mobility
than PbS-TBAI [14], PbS-TBAI-based CQD solar cells exhibited higher PCE values than that
of PbS- MAPbI3 which goes against the common sense that higher mobility corresponds to
better device performance. This perceived anomaly is, however, consistent with the theoretical
model of Sect. 5.5. Ultimately, the CQD solar cells were optimized with a certified efficiency
of 11.28 %, a bandgap energy of 1.32 eV and a PbS- PbX2/AA thickness of 350 nm.
J-V characteristics were obtained using a Keithley 2400 source measuring unit under
simulated AM1.5 illumination (Sciencetech class A) in a continuous nitrogen flow
environment. Furthermore, the calibration for spectral mismatch was carried out using a
reference solar cell (Newport). Finally, following experimental methods described in Sect.3.4,
LIC imaging of the CQD solar cells was performed in a room-temperature nitrogen
environment under 10 Hz modulation frequency.
5.4 Non-constant Photocurrent in CQD Solar Cells
The photocurrent density generated with illumination at short circuit can be expressed as
[15]
𝐽𝑝ℎ = 𝑞 ∫𝑏𝑠(𝐸)𝐸𝑄𝐸(𝐸)𝑑𝐸 (5.13)
where 𝑏𝑠(𝐸) is the incident spectral photon flux and EQE is the solar cell external quantum
efficiency which depends on the material absorption coefficient, charge separation efficiency,
and carrier collection ability in the device, but is independent of the incident optical spectral
distribution. It should be noted that 𝐽𝑝ℎ in Eq. (5.13) corresponds to the maximum photocurrent
density that can be collected. However, as discussed above, the assumption of a constant 𝐽𝑝ℎ
CHAPTER 5. 111
is not always true for CQD solar cells. Contrary to intuition, Fig.5.3(a) demonstrates that the
experimental current density Jillu under illumination is not equal to Jdark - Jsc; instead, the
current density difference (Jph, according to the S-Q equation discussed in the introduction)
between Jillu and Jdark is obviously voltage-dependent with a shape resembling Jillu. The
amplified dark current density Jdark is shown in Fig.5.3(b) that exhibits a typical exponential
J-V curve. This deviation can be attributed to the low carrier mobility in CQD and organic
solar cells [7, 16]. Solar cell efficiency is dominated by three main loss factors, namely, the
non-radiative recombination at the heterojunction interfaces or thin film/contact interfaces, the
inefficient collection of photogenerated excitons and charge carriers, and the parasitic
absorption of the contact layers [16]. Compared with high-mobility systems such as Si solar
cells, the much lower carrier hopping mobility significantly reduces the charge carrier
extraction rate. As a consequence, charge carriers or excitons recombine substantially at or
near the location where they are created [16]. In addition, low carrier hopping mobility causes
an almost open-circuit condition within the solar cell device which occurs even at short circuit,
leading to approx. 95% of the photogenerated carriers becoming lost to recombination as
reported by Würfel et al. [7] for organic solar cells with a carrier mobility equal to 10-6 cm2/Vs.
This phenomenon is well pronounced also in CQD solar cells with increased photoactive layer
thickness and/or under high illumination intensity conditions. Jph is constant only when the
carrier mobility is adequately high, comparable to that of commercial Si solar cells, so that the
driving forces for the transport of electrons and holes can be neglected. For CQD solar cells,
however, carrier hopping mobility and diffusivity are very small [3-6]. Furthermore, the higher
carrier concentration under illumination than in the dark additionally increases the conductivity
of the material [𝜎𝑒,ℎ = 𝑒𝜇𝑒,ℎ𝑛𝑒,ℎ, with 𝑛𝑒,ℎ being the carrier density of electron (e) or holes
CHAPTER 5. 112
(h)] for a given hopping mobility, so the influence of driving forces for electron and hole
extraction begins to emerge [7]. With this consideration in mind, we set out to develop an
analytical expression for the voltage-dependent Jph considering a carrier hopping drift and
diffusion transport mechanism.
Figure 5.3: (a) Experimental data and theoretical best-fits of current density vs. voltage under
illumination and in the dark; (b) The dark current density in (a) amplified. Comparison between
Jdark - Jsc, and Jdark, as well as Jillu, as a function of voltage, is also shown in (a). Equations (5.9),
(5.11) and (5.20) were used for the best-fits of the J-V characteristics. The best-fitted Jph at Va
= 0 (representing Jsc) is 24.9 mA and 7.9×10-7 mA under illumination and in the dark,
respectively.
For our solar cell sample ZnO/PbS-PbX2(AA)/PbS-EDT in Figs. 5.1 and 5.2,
photogenerated excitons dissociate into free electrons and holes when generated in CQD layers,
resulting in electric-field-dependent photocurrent with a fractional contribution, 𝜂′, a function
of hopping drift lengths. At voltage Va, the mean carrier hopping drift length ��𝑑𝑟𝑖𝑓𝑡 can be
expressed as:
��𝑑𝑟𝑖𝑓𝑡 = (𝜇ℎ𝜏ℎ + 𝜇𝑒𝜏𝑒)(𝑉𝑏𝑖−𝑉𝑎)
𝑑 (5.14)
Therefore, the fraction 𝜂′ can be extracted as the ratio of ��𝑑𝑟𝑖𝑓𝑡 to the total carrier hopping drift
transport length (the CQD solar cell thickness), i.e.
CHAPTER 5. 113
𝜂′ =��𝑑𝑟𝑖𝑓𝑡
𝐿𝐶𝑄𝐷+𝐿𝑍𝑛𝑂 (5.15)
According to the well-known Shockley-Queisser equation, Jph reverses the direction of Jdark
which is the net current density comprising drift and diffusion current densities and has a
direction same as that of a diffusion current density under forward bias. Therefore, the
direction of the built-in electric field (Ei) under equilibrium conditions is the positive direction
of the photocurrent. Based on the assumption that photocarriers will be fully extracted if their
hopping drift length is larger than the solar cell thickness, a case also addressed in refs. [13,
17], the hopping drift photocurrent density can be obtained through
𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 = {
𝐽𝑝ℎ′ × 𝜂′, 𝑤ℎ𝑒𝑛 − 1 < 𝜂′ < 1
𝐽𝑝ℎ′ , 𝑤ℎ𝑒𝑛 𝜂′ ≥ 1
−𝐽𝑝ℎ′ , 𝑤ℎ𝑒𝑛 𝜂′ ≤ −1
(5.16)
where 𝐽𝑝ℎ′ is the maximum hopping drift photocurrent density that is constant at a given
illumination condition and is independent of the external voltage. Furthermore, CQD solar cell
efficiencies are deteriorated due to carrier transport toward wrong electrodes. Therefore,
researchers have tried to add additional energy barriers, for example, the extra PbS-EDT CQD
layer in Fig.5.2 was deposited to prevent electron diffusion to the Au electrode [13, 18].
Although the influence of carrier diffusion induced carrier loss is not significant in our CQD
solar cells because of the extra energy barrier introduced by PbS-EDT, for most other CQD
solar cell architectures these effects are substantial and better understanding is required. Hence,
in comparison with electric field induced 𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 , the diffusion-associated photocurrent
𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 is also studied and found to be constant across the entire external voltage range.
Specifically, hole diffusion to the ZnO is negligible due to the built-in energy barrier, while
CHAPTER 5. 114
there are no (or there are much smaller) energy barriers for electron diffusion in the CQD layer.
Thus, the mean diffusion distance is given by
��𝑑𝑖𝑓𝑓 = √𝐷𝑒𝜏𝑒 (5.17)
and
𝜂′′ =��𝑑𝑖𝑓𝑓
𝐿𝐶𝑄𝐷 (5.18)
Analogous to the derivation of hoping drift photocurrent density in Eq. (5.16), the hopping
diffusion photocurrent density can be given by:
𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 = {−𝐽𝑝ℎ
′′ , 𝑤ℎ𝑒𝑛 �� ≥ 𝐿𝐶𝑄𝐷
−𝐽𝑝ℎ′′ × 𝜂′′, 𝑤ℎ𝑒𝑛 �� < 𝐿𝐶𝑄𝐷
(5.19)
Similarly, 𝐽𝑝ℎ′′ is the maximum diffusion photocurrent density. Equation (5.19) reveals the
negative hopping diffusion photocurrent, implying that 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 will diminish the active total
photocurrent. The negative 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓is due to that carrier transport in a wrong direction toward
the incorrect electrodes, which offsets the drift photocurrent as shown in Fig.5.4(a) and (b).
Therefore, the total photocurrent density 𝐽𝑝ℎ can be obtained from adding 𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 and 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓.
Figure 5.4(a) shows the dependence of 𝐽𝑝ℎ on the external voltage using Eq. (5.16)
considering only 𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 , while Fig. 5.4(b) is simulated using Eqs. (5.16) and (5.19) with the
addition of 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 which leads to lower 𝐽𝑝ℎ at the same mobility when compared with Fig.
5.4(a). It should be noted that 𝐽𝑝ℎ′ and 𝐽𝑝ℎ
′′ equal 35mA/cm2 for the simulation in Figs. 5.4(a)
and (b). The simulated 𝐽𝑝ℎ decreases with voltage except at high carrier mobilities (1 cm2/Vs
or higher) where constant 𝐽𝑝ℎ values across the entire voltage range are obtained as shown in
Figs. 5.4(a) and (b). Under reverse bias in Fig. 5.4(a), the external applied electric field has
the same direction as Ei, thereby helping to extract charge carriers and increase Jph until it
CHAPTER 5. 115
saturates to the maximum 𝐽𝑝ℎ′ − 𝐽𝑝ℎ
′′ . Therefore, high reverse voltage assists the extraction of
charge carriers, contributing to the overall 𝐽𝑝ℎ[7]. This is consistent with the saturated Jdark-
Jillu under reverse bias as shown in Fig. 5.3(a). The experimental Jdark-Jillu in Fig. 5.3(a) is
found to reduce under a forward bias, a behavior demonstrated by the simulated results in Figs.
5.4(a) and (b). The reduction in 𝐽𝑝ℎ in Fig. 5.4(a) is expected, as the net electric filed Enet
reduces with increasing Va. The resultant negative Jph results from insufficient drift
photocurrent to offset the negative hopping diffusion photocurrent density which is mirrored
by the negative Jdark-Jillu values when the forward voltage is larger than Voc. Therefore, some
special situations can be expected to arise for extremely small mobilities, such as negative Jph
for a mobility of 0.001 cm2/Vs (which should be much smaller for actual CQD solar cells) in
the entire Va range when the mobility is very small or the diffusion photocurrent is sufficiently
high. However, due to the restrictive assumptions as discussed below behind Eqs. (5.16) and
(5.19) and the carrier transport parameter values used for this simulation, the negative Jph for
the mobility of 0.001 cm2/Vs in Fig. 5.4(b) does not mean that CQD solar cells with such a
low mobility cannot materialize. In other words, Jph at 0.001 cm2/Vs can be simulated to be
positive simply by adjusting the simulation parameters, for example, considering a much
smaller 𝐽𝑝ℎ′′ in Eq. (5.19) or 𝐽𝑝ℎ
′′ = 0 as shown in Fig. 5.4(a). A very small 𝐽𝑝ℎ′′ is true for our
CQD solar cells as indicated by Fig.5.2 due to the energy barriers introduced by the PbS-EDT
CQD layer. Figures 5.4(a) and (b) also show that all Jph curves converge at a voltage
corresponding to the intrinsic voltage Vbi used in Eq. (5.14). Furthermore, lower carrier
hopping mobility 𝜇∗ lead to a lower voltage at which 𝐽𝑝ℎ starts to drop due to a lower carrier
hopping drift extraction efficiency (i.e. the reduced drift length) resulting from the reduced net
electrical field strength, in other words, lower mobility requires higher net electric field to
CHAPTER 5. 116
extract the photocarriers. These simulated results validate the fact that when carrier hopping
mobility is low, the driving forces (electric field and diffusion gradient) for carrier transport
start to impact 𝐽𝑝ℎ.
Figure 5.4: Simulated photocurrent density Jph (a) using Eq.(5.16) without Jph,diff, and (b) using
Eqs.(5.16) and (5.19) with Jph,diff at different effective carrier hopping mobilities; and (c) (Jillu
-Jdark)/𝐽𝑝ℎ𝑚𝑎𝑥 as a function of the external voltage at various effective mobilities using Eq. (5.20).
However, the experimental J-V characteristics in Fig. 5.3(a) illustrate that the difference
between Jdark and Jillu exhibits a nonlinear (exponential-like) dependence on the applied voltage,
contrary to the linear dependence shown in Fig. 5.4(a) and (b) using Eqs. (5.16) and (5.19).
According to the hopping drift-diffusion J-V model, 𝐽𝑝ℎ can be obtained through solving Eq.
(5.2) by implementing a voltage- and position-dependent carrier generation rate. For instance,
CHAPTER 5. 117
the photocurrent density 𝐽𝑝ℎ at 𝑥 = 0 can be obtained when 𝐽𝑝ℎ(0) = −𝑆(0)[𝑛𝑃𝑉(0) − 𝑛(0)]
that yields an exponential dependence of 𝐽𝑝ℎ on the applied voltage. Furthermore, free
electrons and holes dissociated from photogenerated excitons generated in CQD layers
contribute to electric-field-dependent and diffusion-related photocurrents with fractional
contribution 𝜂′ and 𝜂′′, respectively, as discussed above. However, the diffusion photocurrent
has been reported with negligible influence by Schilinsky et al. [19], especially for our CQD
solar cells with the additional PbS-EDT layer as shown in Fig.5.2. Therefore, considering the
drift photocurrent extraction efficiency and that the CQD layers are the main carrier transport
layer, an empirical ad hoc exponential dependence of photocurrent density on the applied
voltage to represent the aforementioned hopping drift-related photocurrent densities in Eq.
(5.16) is given according to refs.[7, 17]:
𝐽𝑝ℎ = 𝐽𝑝ℎ𝑚𝑎𝑥 [1 − 𝑒𝑥𝑝 (
(𝑉𝑎−𝑉𝑏𝑖)𝜇∗𝜏∗
𝑑2)] (5.20)
where 𝐽𝑝ℎ𝑚𝑎𝑥 is the maximum photocurrent density that can be extracted at a given illumination
level from Eq. (5.13). For the sake of simplification, electrons and holes are considered to have
the same transport parameters as defined by 𝜏∗ = 𝜏𝑒 = 𝜏ℎ, 𝐿∗ = 𝐿𝑒 = 𝐿ℎ, 𝐷∗ = 𝐷𝑒 = 𝐷ℎ, and
𝜇∗ = 𝜇𝑒 = 𝜇ℎ . Figure 5.4(c) shows the ratio (Jillu - Jdark) / 𝐽𝑝ℎ𝑚𝑎𝑥 as a function of 𝑉𝑎 using
Eq.(5.20), in agreement with the results reported for organic solar cells using a semiconductor
device simulation tool TCAD Sentaurus from Synopsys Inc. [7]. Furthermore, the excellent
fitting of the experimental J-V characteristics in Fig. 5.3 to the empirical expression of Eq.
(5.20) corroborates the validity of the ad hoc hopping drift-diffusion model proposed under
the assumption of nonlinear exponential dependence of Jph on Va. For a high carrier hopping
mobility such as 1 cm2/Vs, all photogenerated carriers can be extracted, resulting in a typical
inorganic solar cell behavior with a constant Jph across the entire voltage range of interest
CHAPTER 5. 118
except for the case when the external voltage becomes larger than the built-in voltage, Vbi,
Fig.5.2, and the photocurrent density drops to 0 mA/cm2. Similar to Figs. 5.4(a) and (b) under
linear-dependent photocurrent density, the photocurrent changes direction if the external
voltage increases beyond the value of the built-in potential as shown in Fig. 5.4(c). It should
be noted that due to low hopping mobilities, sometimes dark and illuminated J-V curves cross,
for example, the crossover point of Jdark and Jillu in Fig.5.3(a), because of the higher-carrier-
population-induced higher conductivity upon illumination [7]. This effect, however, is not
important in materials and devices with high carrier hopping mobility.
5.5 Impact of Hopping Mobility and Bandgap Energy
Using experimental parameter values, Figure 5.5(a) exhibits the simulated solar cell J-V
characteristics and their dependence on the carrier hopping mobility ranging from 0.001
cm2/Vs to 10 cm2/Vs. For the sake of better comparison with our experimental CQD solar cells,
the bandgap used for simulation is 1.32 eV, same as our experimentally optimized CQD energy
bandgap. The mobility of CQD thin films used in our CQD solar cells has been measured to
be ca. 2×10-2 cm2/Vs [20], which is also in agreement with the experimental results reported
by Yazdani et al. [21] for CQDs. As shown in Fig. 5.5(b), the simulated Voc and Jsc precisely
match with those measured from our CQD solar cell in Fig.5.3(a). Furthermore, with the
increase in mobility, Fig. 5.5(b), Voc decreases while Jsc increases then saturates at high
effective mobilities μ*. The reduced Voc with carrier mobility μ* has been intensively
investigated for organic solar cells, however, such studies are insufficient for CQD solar cells.
Wang et al. [22] and Shieh et al. [23] attributed the Voc loss to enhanced recombination with
dark charge carriers injected from contacts at high mobilities. While it also occurs in our model,
CHAPTER 5. 119
the fast extraction of charge carriers at high μ* is another reason for low Voc as predicted by
Mandoc et al. [24] and Deibel et al. [25].
Figure 5.5: (a) Simulated carrier-mobility-dependent J-V characteristics; (b) open-circuit
voltage Voc, and short-circuit current density Jsc; (c) fill factor FF; and (d) power conversion
efficiency PCE (d). The CQD thin film bandgap used in 1.32 eV same as our experimentally
optimized bandgap for the CQD solar cell in Fig. 5.3. The CQD solar cell carrier hopping
mobility was estimated from our previous study [20]. Equations (5.9), (5.11) and (5.20) were
used for the simulations.
Furthermore, Tress et al. [26] successfully simulated this trend of Voc decline with mobility
through various recombination mechanisms including Langevin recombination,
recombination via charge transfer states, and trap-assisted recombination. The essential
principle is the interplay between the high-mobility-boosted carrier extraction from high drift
current and the enhanced carrier loss resulting from increased carrier recombination rates at
CHAPTER 5. 120
high carrier mobilities. The developed drift-diffusion current-voltage model operates as a self-
consistent system considering the carrier concentration, surface recombination, and carrier
hopping mobility to interpret the dependence of Voc, Jsc, FF, and PCE on mobility. Specifically,
Voc is carrier concentration dependent and is determined by the energy difference between the
electron and hole quasi-Fermi levels [27]. At low carrier hopping mobilities, low
recombination rates yield high carrier concentrations at open circuit according to
𝑒𝑉𝑜𝑐 = ∆𝐸𝐹 = 𝑘𝑇𝑙𝑛 (𝑛𝑝
𝑛𝑖2) (5.21)
in which 𝑛 , 𝑝 , 𝑛𝑖 are the electron, hole, and intrinsic carrier concentrations, respectively.
Furthermore, within the framework of direct electron-hole recombination, the recombination
is given by
𝑅0 = 𝛽(𝑛𝑝 − 𝑛𝑖2) (5.22)
where 𝛽 is the recombination constant. Langevin theory gives a description connecting the
carrier mobility with recombination rate through 𝛽 =𝑒(𝜇𝑒+𝜇ℎ)
𝜖0𝜖𝑟 with 𝜖0𝜖𝑟 the permittivity of the
materials [28] and 𝜖𝑟 ≈ 20 for our CQD thin films [29]. Therefore, Tress et al. [26] derived
the open-circuit voltage 𝑉𝑜𝑐 as a function of mobility through 𝛽,
𝑉𝑜𝑐 =1
𝑒[𝐸𝑔 − 𝑘𝐵𝑇𝑙𝑛 (
𝛽𝑁𝑐𝑁𝑉
𝐺)] (5.23)
in which 𝑁𝑐 and 𝑁𝑉 are effective density of states on the order of 1019 cm-3 for CQDs [29] in
conduction and valence bands, respectively. The term 𝐺 is a carrier generation rate which
equals ca. 1×1022cm-3s-1 from ref. [26]. Therefore, 𝑉𝑜𝑐 decreases from 0.64 V to 0.49 V with
increasing carrier hopping mobility from 0.001 cm2/Vs to 10 cm2/Vs as simulated in Figs.
5.5(a) and (b). Taking 0.023 cm2/Vs as our solar cell’s mobility, the simulated Voc is found to
be close to the experimental value of 0.63 V. Although current CQD fabrication techniques
CHAPTER 5. 121
cannot enable mobilities in a wide range, in agreement with Fig. 5.5(b) for CQD solar cells
our previous study [30] found that Voc was reduced at higher temperatures which corresponds
to higher carrier mobility due to the nature of phonon-assisted carrier hopping transport in
these materials [31, 32]. Similar simulation results for Voc have also been reported for other
low-mobility solar cell systems [11, 25, 26] using an implicit solar cell simulator. As estimated
from fitting in Fig.5.3(a), the SRV used for Fig. 5.5 is 1×10-3 cm/s through Eq. (5.11). However,
when an infinite SRV is used, i.e. Eq. (5.8) with a quenching boundary condition indicating
immediately recombination of all carriers arriving at the contact, Voc decreases much more
dramatically at high mobilities, and this effect was demonstrated in refs.[24, 25] for organic
solar cells. However, Deibel et al. [25] found that a dramatic reduction of Voc can be avoided
if a finite surface recombination rate is considered. Infinite SRV is not reasonable, of course,
as discussed in ref. [22], also considering the significantly improved CQD surface quality of
our CQD solar cells through solution-ligand exchanges that leave few unsatisfied dangling
bonds on the CQD surface [13]. Furthermore, also from Fig. 5.5(b), Voc is directly proportional
to the bandgap energy Eg of the active photovoltaic CQD thin films, while the simulated eVoc
values are almost half the corresponding Eg, in agreement with the experimentally reported
results for CQD solar cells [33, 34], indicating that only half of the photon energy is harvested.
This is due to the energy loss of excitons and charge carriers to bandgap trap states and/or
bandtail states [13].The enhanced carrier collection efficiency of the photogenerated current at
higher carrier hopping mobility facilitates the increase of short-circuit current density. Short-
circuit current density was obtained at zero voltage using Eqs. (5.9), (5.11), and (5.20). From
Fig. 5.4, it is expected that Jsc rises with mobility due to the enhanced drift photocurrent density
resulting from increased mobility according to Eqs. (5.14) - (5.16). Theoretically, based on our
CHAPTER 5. 122
model, photocurrent saturation at high mobilities occurs because all photoexcited carriers are
extracted at short circuit as shown in Fig. 5.5(b), mirrored by the maximum Jph values at 0 V
for the high mobilities in Fig. 5.4. In practice, further enhancement of Jsc can be achieved
through enhancing photoexcitation intensity or absorption with small bandgap CQDs or
thicker CQD layers.
Furthermore, the trade-off between Jsc and Voc results in peaked FF and PCE with respect
to carrier hopping mobilities, Figs. 5.5(c) and (d). The simulated FF of 0.62 at ca. 0.023 cm2/Vs
in Fig. 5.5(c) is in agreement with the value of 0.63 estimated in our CQD solar cells
characterized in Fig. 5.3(a). Fill factor is a measure of the current-voltage characteristic shape
of solar cells. Compared with other parameters, FF can markedly elucidate carrier
recombination strength [35]. Before reaching the optimized mobility as shown in Fig. 5.5(c),
FF improves dramatically from the significantly enhanced carrier drift current and the
marginally decreased Voc as discussed above. The steep rise of FF in the low mobility regime
is attributed to the increased charge carrier extraction outside the device with mobility increase
while carrier recombination still remains at a relatively low level as per the drift-diffusion J-V
model and Fig. 5.5(b). After attaining the optimal carrier mobility, the decline in FF is an
indication that carrier recombination starts to overtake extraction. As for Figs. 5.5(c) and (d),
similar results of mobility-dependent FF as well as PCE are also found in low mobility organic
solar cells [26]. Consequently, the study of the competition between carrier extraction and
recombination, as well as the tradeoff between Voc and Jsc with mobility, is helpful for CQD
solar cell fabrication. Additionally, the simulated PCE of 9.3 % is comparable to the
experimental result of 10 % shown in Fig. 5.3(a). Through literature review, despite the fact
that relatively high field-effect [36-38] and terahertz radiation [39, 40] mobilities have been
CHAPTER 5. 123
reported on the order of 1-30 cm2/Vs for CQDs, the highest reported solar cell PCE to-date
was achieved by employing active materials with relative lower field-effect mobilities in a
range from 10-3 to 10-2 cm2/Vs [3, 41, 42]. Similar to our model, Zhitomirsky et al. [1]
attributed this to trap-state-limited carrier diffusion lengths, in other words, the low PCE values
in higher carrier mobility materials and devices are results of increased trap-state-assisted
recombination. Before the maximum PCE is attained, increasing the carrier mobility improves
the PCE, however, beyond this regime, increasing the hopping mobility simply enables a
higher rate of carrier recombination. Therefore, instead of intuitively pursuing higher carrier
mobility, a more effective suggestion for solar cell performance optimization should be to
reduce the trap states, which also reduce bimolecular recombination due to strengthened
interdot coupling and enhance the diffusion length, then to further increase carrier mobility. In
conclusion, the simulated results in Fig. 5.5 are in agreement with the finding that “low
mobility might help mitigate a particular loss mechanism in a certain material…” as reported
by Street et al. [28].
Table 5.1: Parameters used for the CQD solar cell simulations.
Parameter
Symbol Value Unit References
Carrier concentration at equilibrium n(0) 1×1016 cm-3 29, 43, 44
Solar cell thickness d 4.1×10-5 cm 13
Effective hopping diffusion length L* 350 nm 13, 45
Effective carrier hopping diffusivity D* Varied, 1×10-4-
0.01
cm2/
s
14, 43
Effective carrier hopping mobility μ* Varied, 1×10-3-10,
0.023 for our
CQDs
cm2/
Vs
29, 20, 46
Built-in voltage Vbi Varied, 0.1-1.5 V
Surface recombination velocity S 1×10-3 cm/s Estimated
from fitting
CQD bandgap energy Eg Varied, 0.5-3.6 eV
CHAPTER 5. 124
Figure 5.6: Theoretical simulations of CQD solar cell electrical parameters: (a) Voc and Jsc, (b)
PCE, and (c) FF as functions of CQD bandgap energy (Eg) for five different carrier hopping
mobilities. The maximum photocurrent 𝐽𝑝ℎ𝑚𝑎𝑥 is the same as Jsc at the mobility of 0.1 cm2/Vs.
The illumination intensity used for the simulation is AM1.5 spectrum at 1 sun intensity.
Equations (5.9), (5.11) and (5.20) were used for the simulations.
CQDs are promising in solar cell fabrication due to their dot-size-tunable bandgap energy,
thereby making the structural design for harvesting more solar energy much easier. Using the
parameter values in Table 5.1, a simulation of CQD bandgap-energy-dependent PCE, Fig.5.6,
was carried out using Eq.(5.13), for the sake of simplification, with an approximated average
EQE = 0.76 when the incident light energy is higher than the CQD thin film bandgap energy
according to our fitting shown in Fig.5.3; otherwise, EQE = 0. However, more precise
CHAPTER 5. 125
experimental EQE values as a function of wavelength can be found in ref. [13] for further
research investigations. Therefore, considering AM1.5 excitation, 𝐽𝑝ℎ𝑚𝑎𝑥
was obtained by
integrating the product of EQE, 𝑏𝑠(𝐸) , and the photon energy according to Eq. (5.13).
Subsequently, Jsc was calculated by combining Eqs. (5.9), (5.11), (5.13) and (5.20) at 0 V
external voltage. Smaller Eg facilitates absorption of photons with lower energy, leading to an
increased maximum photocurrent density 𝐽𝑝ℎ𝑚𝑎𝑥 , Fig. 5.6(a). However, due to the hopping
mobility-dependent Jsc, which equals Jph at short circuit, 𝐽𝑝ℎ𝑚𝑎𝑥 converges to Jsc only at 0.1
cm2/Vs across the whole simulated Eg range. The slight drop in Jsc at small Eg for all mobilities
is due to the scarcity of significant low-wavelength solar energy according to the nature of the
AM1.5 solar spectrum, i.e. 𝐽𝑝ℎ𝑚𝑎𝑥 starts to saturate as shown in Fig. 5.6(a). In addition, it is also
due to the decreased built-in voltage Vbi which is reduced when the CQD photovoltaic material
Eg decreases according to Eq. (5.20). Therefore, compared with higher mobilities, Jsc is
expected to start to decrease at higher Eg for low carrier mobilities as shown in Fig. 5.6(a). A
linear dependence of Voc on Eg was found and extracted by linear best fits of the experimental
data [13] from our CQD solar cells, Fig. 5.6(a), and it could be expressed as
𝑉𝑜𝑐 = 0.387𝐸𝑔 + 0.095 (5.24)
Here, the units of 𝑉𝑜𝑐 and 𝐸𝑔 are V and eV, respectively. A similar linear dependence of Voc on
Eg has also been reported by Bozyigit et al. [33] in the form of 0.27Eg+0.09 for ligand EDT
capped PbS CQD solar cells. Insofar as the trade-off between Jsc and Voc, an optimized CQD
bandgap energy for a maximized PCE value can be expected. In other words, although small-
bandgap CQDs facilitate solar energy absorption in a wider wavelength range, the reduced Eg
compromises Voc according to Eqs. (5.23) and (5.24). The simulated PCE in Fig. 5.6(b) are
through Eqs. (5.9), (5.11), (5.13), (5.20), and (5.24) and the calculated Jsc and Voc as discussed
CHAPTER 5. 126
above. Therefore, applying the given carrier hopping transport parameters as tabulated in Table
5.1, the simulation of PCE in Fig. 5.6(b) yields an optimized Eg of ~1.12 eV for a mobility of
0.02 cm2/Vs, a mobility estimated for our CQD systems [20]. Due to the nature of AM1.5
spectrum, multiple PCE sub-peaks are also observed, a feature consistent with the Jsc in Fig.
5.6(a) and the well-known Shockley-Queisser limit simulation. As discussed in Sect. 5.3, the
experimentally optimized Eg is 1.32 eV, which is close to the sub-peak labeled in Fig. 5.6(b).
It should be noted that, experimentally, only CQD bandgap values in a range between 1.28-
1.48 eV were tried [13]. The PCE simulation implies there is still room for further PCE
improvement using CQD materials at this mobility. The bandgap-dependent PCEs for other
mobilities in Fig. 5.6(b) reveal a blue-shift of the optimized Eg with mobility increase, which
is in agreement with ref. [7]. The shift to small bandgap is a result of relatively high carrier
drift current at high mobilities. Specifically, the reduced Eg diminishes the intrinsic electric
field and the drift current starts to decrease at smaller Eg for higher carrier mobilities when
compared with lower-mobility CQD solar cells. This overall non-monotonic behavior in Fig.
5.6(b) implies an increased PCE with the simulated carrier hopping mobility in a range where
the carrier extraction rate still surpasses the recombination rate. Furthermore, possible
simulation deviation of this model is expected due to the use of linear Eg –dependent Voc
through Eq. (5.24) which, however, is derived based on experimental data in a narrow Eg range,
probably not sufficiently accurate for a wide-range Eg simulation in this study. In addition, for
small Eg with high photoexcited carrier densities Voc should be further reduced as there will be
an exponential dependence of Voc on carrier concentration according to Eqs. (5.21) and (5.23).
The latter will lead to even lower Voc at smaller Eg than Eq. (5.24) predicts, resulting in a shift
of the optimized Eg to large Eg CQDs. Figure 5.6(c) shows a monotonic increase of FF with Ea
CHAPTER 5. 127
as well as with μ* in the range between 0.001 cm2/Vs and 0.1 cm2/V. As discussed above, the
increased FF values indicate that with increased bandgap, carrier extraction plays an
increasingly important role in carrier transport over recombination processes according to Fig.
5.6(c).
Figure 5.7: (a) Simulated CQD solar cell (a) Voc and Jsc, as well as FF and PEC (b), as functions
of the illumination intensity. Equations (5.9), (5.11) and (5.20) were used for the simulations.
The foregoing hopping drift-diffusion J-V model was further examined by studying the
effects of illumination intensity on the solar cell Voc, Jsc, FF, and PCE. Figure 5.7(a) validates
the enhanced Jsc with illumination intensity due to boosted photoexcited carrier densities. The
simulated Voc shows an exponential correlation with the excitation intensity as shown in Fig.
5.7(a) that is mirrored in the well-known relation 𝑉𝑜𝑐 = 𝐴 +𝑛𝑖𝑑𝑘𝑇
𝑞𝑙𝑛𝑋, in which A is a constant
and X represents the illumination intensity. According to recombination mechanisms including
the Langevin and trap-state-assisted Shockley-Read-Hall (SRH) recombination theories, the
carrier recombination rate changes proportional to the carrier concentration. For example,
through direct bimolecular recombination, Eq. (5.22), the carrier recombination rate grows
with carrier concentration proportional to the photoexcitation intensity.
With carrier recombination increase, Fig. 5.7(b) shows decreasing FF at high excitation
intensities, in good agreement with the experimental findings reported by [47, 48]. Because of
CHAPTER 5. 128
the increased carrier recombination rate at high photocarrier injection levels, PCE increases
only slightly from ~8.0% at 0.001 sun to ~10.1% at 1 sun illumination in agreement with our
CQD solar cells in Fig. 5.3(a), above which the PCE increase slows down and even saturates
at high illumination intensities. This implies that with the consideration of various carrier
recombination pathways, carrier radiative and nonradiative recombination through different
mechanisms such as direct biomolecule or SRH approaches can degrade CQD solar cell
performance significantly at high mobilities and/or high photocarrier injection levels. However,
one should be aware that at sufficiently high mobilities comparable to conventional Si solar
cells, this model should not be applied as the effects of electric and diffusion forces become
trivial and negligible. Approaches to reduce carrier recombination in CQD systems can be
through reducing exciton binding energy and/or through removing material trap states. High
exciton binding energy facilitates the probability for electrons and holes to recombine [49]. An
effective approach is to dissociate excitons through strengthening interdot coupling and/or
increase interface energy barriers through a heterojunction architecture. Strong interdot
coupling can be realized with the use of high-quality and monodispersed CQDs that remove
defects and trap states in CQDs. Therefore, improving CQD quality through various methods
as discussed in the introduction always contributes to improved CQD solar cell performance.
5.6 Impact of Electrode-semiconductor Interfaces Using
Homodyne Lock-in Carrierography
Equation (5.10) of the hopping drift-diffusion model above reveals the dependence of CQD
solar cell J-V characteristics on surface recombination velocity 𝑆(𝑑), a parameter determined
by the CQD thin film surface passivation or trap states at CQD semiconductor/Au electrode
interfaces, Figs.5.1 and 5.2. A better electrode coating with lower interface states leads to low
CHAPTER 5. 129
𝑆(𝑑) that results in higher CQD solar cell performance. To investigate the CQD/Au interface
influence on CQD solar cell performance, non-destructive imaging (NDI) of carrier population
distributions and key photovoltaic parameters was carried out using homodyne lock-in
carrierography (HoLIC) [50], a spectrally gated dynamic frequency-domain
photoluminescence imaging method as reported by Hu et al. [51] and also discussed in Sec.7.2.
The HoLIC images show the complexity and inhomogeneity of the electrode-coating-
associated surface recombination in our experimental CQD solar cells. As shown in Fig.5.8,
the J-V and P-V characteristics of one solar cell with lower PCE when compared with Fig.5.3
exhibited similar Voc values but much lower Jsc, and were best-fitted to the combination of Eqs.
(5.9), (5.11), and (5.20) with the consideration of Au-electrode-modified surface
recombination velocity.
Figure 5.8: (a) Experimental J-V characteristics; and (b) output power curves as a function of
photovoltage. Continuous lines are best fits to J-V characteristics and output power using Eqs.
(5.9), (5.11) and (5.20).
CHAPTER 5. 130
Figure 5.9: (a) A photograph of a CQD solar cell sample; and (b) its LIC image at open circuit
after the cell was flipped over. The excitation laser was frequency-modulated at 10 Hz at a
mean intensity of 1 sun. The eight Au-coated thin film electrodes on the top in (a) are electrical
contacts while dark brown regions are without Au contact layers. Both regions have an energy
structure as shown in Fig. 5.2. The Au electrode circumscribed with a dashed rectangle in (a)
and also shown in the flipped over orientation in (b) is further studied in Figs. 5.10 and 5.11.
For excitons and dissociated free carrier radiative recombination, the voltage-dependent
optical carrier flux corresponding to its electrical counterpart Eq. (5.9) was introduced by
Mandelis et al. [52] and subsequently used by Liu et al. [53] for mc solar cells as their Eq.(5.18);
it was further adapted here with a different optoelectronic coefficient 𝑚′ [= (𝜇∗
𝐷∗)′
] for hopping
transport in CQD solar cells:
𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑟 = 𝐽𝑅 − 𝐽𝑅0[𝑒𝑚′𝑉(ℏ𝜔) − 1] (5.25)
𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑅 can also be obtained experimentally through
𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑟 = 𝑞𝐶𝐿𝐼𝐶[𝐿𝐼𝐶(𝑉𝑜𝑐) − 𝐿𝐼𝐶(𝑉(ℏ𝜔))] (5.26)
where 𝑞 is the elementary electron charge, 𝐿𝐼𝐶 is the HoLIC signal at photovoltage 𝑉(ℏ𝜔),
𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑟 is the non-equilibrium radiative recombination current density, 𝐽𝑅 and 𝐽𝑅0 are
the relevant current-density-like quantities, and 𝐶𝐿𝐼𝐶 is a coefficient defined as [53]
𝐶𝐿𝐼𝐶 =|𝐼𝑖|(1−𝑅)𝜂
ℏ𝜔𝑖𝑛 [𝐿𝐼𝐶(𝑉𝑜𝑐)−𝐿𝐼𝐶(0))][1−𝜂𝑐𝑒(ℏ𝜔,𝑉=0,𝑇)
1−𝜆𝑖𝑛𝜆𝑒𝑚−1 ] (5.27)
CHAPTER 5. 131
where |𝐼𝑖| is the peak value of the incident modulated illumination intensity, 𝑅 is the surface
reflectance, 𝜂 and 𝜂𝑐𝑒(ℏ𝜔, 𝑉 = 0, 𝑇) are the quantum efficiency for exciton and charge carrier
photo-generation and photocarrier-to-current collection efficiency, ℏ𝜔𝑖𝑛 is the incident photon
energy, and 𝜆𝑖𝑛 and 𝜆𝑒𝑚 are, respectively, the incident and emitted photon wavelength. In a
manner similar to the electrical Eqs. (5.9) and (5.11), Eq. (5.25) links the exciton and free
carrier radiative recombination flux which is an optically measurable quantity to its electrical
parameter counterparts. The expressions for Jph, J0, Voc, and m’ have been derived by Liu et al.
[53] using the optical parameters in Eq. (5.25) and a photocarrier-to-current collection
efficiency which is the ratio of the photocarrier flux collected by the solar cell electrodes
(giving rise to the photocurrent) to the incident photocarrier flux. A photograph of our solar
cell is presented in Fig. 5.9(a), and the corresponding HoLIC image shown in Fig. 5.9(b)
reveals the inhomogeneity of the Au contact regions which are distinguishable from those
without Au layers. A dashed rectangle in Fig. 5.9(a) is circumscribed around the perimeter of
an Au layer, same as the one circumscribed in Fig. 5.9(b), is further studied in detail in Figs.
5.9-5.11. The LIC image-generating laser was introduced from the ITO/ZnO side, Figs.5.1 and
5.2. The image contrast originates in inhomogeneous exciton and free charge carrier
population distributions due to mechanical or electrical defect-induced photocarrier lifetime
variations. Defect-induced trap states act as thermal capture and emission centers of a
nonradiative recombination nature that diminish exciton and free charge carrier hopping
lifetimes, resulting in HoLIC image signal decreases. According to Fig. 5.9(b), regions (solar
cell pixel images) with electrodes on the right-hand side appear to have higher defect densities
or worse contacts than those on the left-hand side. These observations are consistent with our
experimental results that low solar cell efficiency is associated with J-V measurements of these
CHAPTER 5. 132
particular solar cell units. These results show that carrier diffusion-wave-based HoLIC
imaging has excellent potential for non-destructive inspection of CQD solar cells.
Figure 5.10. HoLIC images of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a) at
open-circuit 0.64 V (a), 0.60 V (b), 0.56 V (c), 0.35 V (d), 0.20 V (e), and short-circuit (f). The
excitation laser was frequency-modulated at 10 Hz at a mean intensity of 1 sun.
For solar cell PCE optimization, dynamic carrier distribution visualization as a function of
applied external voltage, Fig. 5.10, is crucial for optimizing device fabrication with respect to
materials and nanoparticle deposition techniques. With decreasing external voltage, the HoLIC
image exhibits different trends in different regions within the entire solar cell unit image,
indicating highly inhomogeneous carrier hopping transport. These variations in performance
CHAPTER 5. 133
of each device lead to a poor overall solar cell behavior and should be considered seriously as
optimization issues for commercial CQD solar cell fabrication. Specifically, points A, B, C,
and the area inside the dashed rectangle were studied as shown in Fig. 5.10(a). Excitons and
charge carriers extracted into the external circuit through the Au and ITO contacts in Figs.5.1
and 5.2 can be probed through HoLIC signal differences between open-circuit and short-circuit
conditions, Figs.5.11(a) and (b). It can be observed that more photogenerated exciton and free
charge carriers are collected at points A and B than at point C and at regions close to the edge
of the contacts.
Table 5.2: Optical counterparts of CQD solar cell electrical parameters, obtained through
best-fitting of the experimental data in Figs.5.11(b) and (d) to Eq. (5.25).
Sample JR/ CLIC
(C·mV)
JR0/ CLIC
(C·mV)
m’
(V-1)
Point A 1.78×10-18 2.34×10-19 3.43
Point B 1.66×10-18 1.71×10-19 3.76
Point C 1.56×10-18 1.02×10-18 1.49
Selected Region 1.45×10-18 6.45×10-19 1.89
Excellent best-fits to our theoretical model Eq. (5.25) have been achieved as shown in
Figs.5.11(b) and (d), in addition to the best-fitted optical parameters shown in Table 5.2. The
results reveal that high-amplitude regions in Figs.5.11(a) and (c) yield higher optical
counterparts of Jph and along with lower optical saturation current densities. A conclusion can
be reached from Fig. 5.11(b) that defects compromise the optical Isc more significantly than
Voc which remains almost constant at all three selected points and inside the dashed rectangle
region. The HoLIC images thus suggest that material surface and interface treatments for
eliminating CQD surface defects may benefit Voc enhancement only in a limited manner
although they can raise Jsc significantly. Therefore, smaller Jsc values arising from higher
defect state density result in reduced maximum-output-powers Pmax as shown in Figs.5.11(c)
CHAPTER 5. 134
and (d), which present the maximum-output-power mapping and its voltage dependence,
respectively.
Figure 5.11: LIC of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a): (a) LIC (Voc)
- LIC(Vsc); (b) [LIC (Voc) - LIC (V)] vs. V; (c) [LIC (Voc) - LIC(VPM)]VPM ; and (d) [LIC(Voc) -
LIC(V)]V vs. V characteristics. The excitation laser was frequency-modulated at 10 Hz at a
mean intensity of 1 sun. (b) and (d) are best-fitted to Eq. (5.25). Points A, B, C, and the dashed
rectangle region are shown in (a) and (c). It should be noted that values calculated for the
dashed rectangle region are based on averaging the LIC amplitudes over all pixels in this region.
CHAPTER 5. 135
Figure 5.12: Open-circuit voltage Voc LIC contour mapping of the circumscribed CQD solar
cell Au electrode in Fig. 5.9.
In comparison, the solar cell pixel Voc image, Fig. 5.12, is more homogeneous within the
Au electrode region as Voc is determined primarily by the work function difference between its
corresponding electrodes rather than device defects or defect-affected carrier hopping transport
behavior. Figures 5.11 and 5.12 clearly demonstrate the critical importance of electrode
influence on CQD solar cell performance. The respective maximum power and carrier
collection HoLIC images are critical to the evaluation of the CQD solar cell quality due to the
low carrier hopping mobility and diffusivity which result in low collection efficiency of
carriers generated far away (compared to a diffusion length) from the carrier extraction
electrodes.
5.7 Conclusions
A comprehensive study of CQD solar cell efficiency dependence on carrier mobility, CQD
energy bandgap energy, and electrode interface was performed. Optimized carrier hopping
CHAPTER 5. 136
mobilities and bandgap energies can be determined from studying their impact on CQD solar
cell efficiencies. Furthermore, the universally applied assumption of constant photocurrent was
relaxed and its variation with voltage and mobility was analyzed for CQD solar cells. This
voltage- and mobility dependent photocurrent density was demonstrated to originate from the
competition between carrier extraction rate and recombination rate (for example, through trap
states) in CQD solar cells. Large-area inspection of CQD solar cell carrier population
distribution, collection efficiency, and Voc using LIC NDT revealed a strong correlation
between Au electrode/CQD interface associated surface recombination effects and solar cell
performance, overcoming limits of conventional small-dot characterization methodologies.
The developed self-consistent hopping drift-diffusion model, together with large-area HoLIC
NDT pave the way for a comprehensive quantitative strategy for device fabrication toward
high-efficiency solar cells that can be of keen interest to the CQD solar cell community. The
presented efficiency optimization strategy is summarized below.
1. Attempts to enhance the carrier diffusion length by reducing trap state density either
through selecting proper interdot linking ligands that have lower lattice mismatch with the
CQD crystal lattice or through the use of solution-based ligand exchanges rather than the solid
state layer-by-layer method [13] should always be the first priority.
2. According to the achieved diffusion length, the proper CQD thin film thickness for
solar cells should be determined, and the present new drift-diffusion transport model that
introduces voltage-dependent Jph should be used to find optimized μ and Eg at a maximized
PCE for the given estimated parameters as shown in Table 5.1.
CHAPTER 5. 137
3. Only after the diffusion length improves, efforts to reduce CQD polydispersity or
strengthen interdot coupling to reach an optimized μ at a given CQD Eg should be implemented
guided by the parameter relationships shown in Fig.5.5.
4. Since μ cannot be characterized in as straightforward a manner as Eg, the preferred
procedure should be to use dot-size-tunable Eg, instead of varying μ, toward achieving
maximum PCE according to Fig.5.6.
5. Eliminating material surface and interface defects benefit Voc enhancement only in a
limited manner but can raise Jsc significantly. Therefore, selection of electrode metals and
contact procedures yielding optimal carrier density distributions, extraction efficiency, and Voc
as visualized by LIC imaging is an effective tool for further CQD solar cell efficiency
improvement.
138
Chapter 6
Photocarrier Radiometry Study of Quantitative Carrier
Transport in CQD Thin Films
6.1 Introduction
Understanding carrier transport dynamics and shedding light on energy dissipation
mechanisms in optoelectronics is essential to devise efficiency optimization. QD disorder in
the form of energy and/or geometry, originating in dot shape, size, composition, surface
chemistry, and capping ligands, as well as the degree of polydispersity and superlattice order
in thin films, disrupts the formation of continuous energy band structures in CQD ensembles.
Depending on the level of QD disorder, there are four possible carrier transport mechanisms
[1] as discussed in Sect.2.2.1. Phonon-assisted hopping is the most prevailing mechanism that
has been widely applied in studying various QD systems [2-8] in which carriers hop from one
dot to the next depending on the interdot distance, coupling strength, temperature, and the type
of carrier. Moreover, strengthening the capping-ligand-controlled interdot coupling has been
reported in PbSe QDs as originating in the Coulomb blockade dominated insulating regime
and into the hopping conduction dominated semiconductor regime [2], and has also been found
to assist exciton dissociation into free electrons and holes [9, 10, 11]. Furthermore, Lee et al.
[11] and Liu et al. [4] have observed a monotonic increase in hole mobility with increasing
QD size, while electron mobility exhibits a peak at a QD diameter of 6 nm, which can be
ascribed to the compromise between reduced activation energy (lower hopping energy barrier)
and weakened interdot coupling strength amongst larger QDs. The phonon-assisted hopping
CHAPTER 6. 139
transport mechanism predicts a temperature-dependent carrier mobility, diffusivity, lifetime,
conductivity, and conductance of QD devices [2, 3, 5, 7, 8, 12-14]. However, due to the large
specific surfaces of QDs, even with the application of capping ligands, QD trap states still
hinder the efficiency of CQD based electronic devices through acting as undesired radiative
and nonradiative recombination centers [4, 15-21]. Important as these effects are, a systematic
study of trap-state-modified carrier transport is still lacking.
Despite the importance of carrier dynamics to QD optoelectronic and electronic device
efficiency optimization, current characterization techniques are still not able to provide
sufficient feedback information about carrier transport kinetics in QD substrates and devices.
As discussed in Chapter 3, carrier mobility can be characterized by linearly increasing voltage
(CELIV) [20, 22, 23], time of flight (TOF) [22,24], transient photovoltage [20, 25], and by
using field effect transistors (FET) [20, 26, 27]. Nonetheless, these methods require thick QD
films and a completed device. Although Zhitomirsky et al. [28] introduced photoluminescence
(PL) quenching for carrier diffusion length measurement in CQD thin films, additional coating
and/or embedding different types of CQDs are compulsory. Nowadays, carrier lifetime is
measured mostly by Voc (open-circuit voltage) transient decay [22] and transient PL [20, 29,
30] for devices and substrates, respectively. However, due to the fragile nature of materials
comprising photovoltaic devices, especially organic and QD-based solar cells, and due to their
disadvantages as elaborated in Sect.3.5, most of these conventional techniques are suitable
neither for industrial in-line mass manufacturing of electronic devices at any and all fabrication
stages nor for optoelectronic process analysis involving light-carrier interactions.
Therefore, this chapter introduces the all-optical, fast, and non-destructive technique,
photocarrier radiometry (PCR) into CQD thin film characterization with its instrumentation,
CHAPTER 6. 140
general principles, and advantages as discussed in Sect. 3.5. A trap-state-mediated carrier
hopping transport model was developed and applied for the extraction of multiple carrier
transport parameters of different ligand-capped PbS CQD thin films. The temperature-
dependent carrier transport dynamics was investigated in perovskite-passivated PbS CQD thin
films. Combined with a carrier hopping transport model, PCR was shown to exhibit great
potential in QD materials characterization for fundamental physics research of carrier transport
dynamics, in addition to being an all-optical, nondestructive and promising technique for
industrial device quality control as discussed in Sect.3.5.
6.2 PCR Theory for CQDs: Trap-state-mediated Carrier
Transport Model
Figure 6.1: (a) Schematic of carrier hopping transport in PbS CQD thin films embedded in a
surface-passivation ligand matrix when excited by a frequency-modulated laser source. (b)
Illustration of carrier generation, dissociation, hopping transport, and trapping processes in a
CQD assembly. Se and Sh are the ground states for electrons and holes, respectively. Ea,1 and
Ea,2 are the activation energies associated with exciton binding energy (Eb) and trap-mediated
transition process, respectively. Eg and Eg, opt are, respectively, the electronic and optical band
gap energy.
Figures 6.1(a) and (b) exhibit the schematic of surface-passivated and laser-illuminated
PbS CQDs in a ligand matrix. Upon laser excitation, excitons will firstly be generated within
the CQDs and diffuse away through a carrier hopping mechanism [Fig.6.1(a)], during which
process, excitons may dissociate into free charge carriers. All these particles, including
CHAPTER 6. 141
excitons and their dissociated charge carriers, can recombine radiatively, or be bound to or
trapped in trap states and recombine radiatively or non-radiatively [Fig. 6.1(b)].
Therefore, the rate equation for the population 𝑁𝑖(𝑥, 𝑡) of charge carriers in quantum dot 𝑖
[31] must include the presence of trap states acting as thermal emission and capture centers.
Such trap states have been reported in thiol-capped PbS QDs [32], and in glass-encapsulated
PbS QDs [33], also several trap-related emission bands have been reported for PbS QDs in
polyvinyl alcohol [34]. Taking into consideration that those trap states acting as thermal
emission and capture centers, the carrier rate equation can be expressed as:
𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑡= −∑ 𝑃𝑖𝑗𝑗 𝑁𝑖(𝑥, 𝑡) + ∑ 𝑃𝑗𝑖𝑗 𝑁𝑗(𝑥, 𝑡) + ∑ {𝑒𝑖𝑘(𝑇)𝑛𝑇𝑘(𝑥, 𝑡) − 𝐶𝑖𝑘𝑁𝑖(𝑥, 𝑡)[𝑁𝑇𝑘 −
𝑚𝑘=1
𝑛𝑇𝑘(𝑥, 𝑡)]} −𝑁𝑖(𝑥,𝑡)
𝜏+ 𝐺0(𝑥, 𝑡) (6.1)
where 𝑘 denotes trap level, 𝑒𝑖𝑘 is the thermal emission rate of charge carriers from the trap
level 𝑘, 𝐶𝑖𝑘 is the charge carrier capture coefficient, 𝜏 is the carrier lifetime, 𝑁𝑇𝑘 is the trap
density of level 𝑘, 𝑛𝑇𝑘 is the trapped carrier density, 𝑃𝑖𝑗 (𝑃𝑗𝑖) is the hopping probability from
the 𝑖𝑡ℎ (𝑗𝑡ℎ) QD to the 𝑗𝑡ℎ (𝑖𝑡ℎ) QD. Here, 𝐶𝑖𝑘𝑁𝑇𝑘 is defined as the carrier-trapping rate 𝑅𝑇𝑘.
Go is the photocarrier generation rate. In the PbS CQD system under consideration, all the trap
states at distinct levels are considered to have the same effects on carrier transport behavior,
i.e. trapping and detrapping carriers. Although distinguishing them is possible using photo-
thermal deep level transient spectroscopy at various temperatures and can yield a more detailed
structure of trap levels, however, it is not necessary for the present optoelectronic transport
property characterization. Therefore, the rate equation for carrier population can be further
developed to
CHAPTER 6. 142
𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑡= −
𝜕𝐽𝑒(𝑥,𝑡)
𝜕𝑥+ 𝑒𝑖(𝑇)𝑛𝑇(𝑥, 𝑡) − 𝐶𝑖𝑁𝑖(𝑥, 𝑡)[𝑁𝑇 − 𝑛𝑇(𝑥, 𝑡)] −
𝑁𝑖(𝑥,𝑡)
𝜏+ 𝐺0(𝑥, 𝑡) (6.2)
where the charge carrier current density 𝐽𝑒(𝑥, 𝑡) is a function of the hopping diffusivity 𝐷ℎ and
can be written as
𝐽𝑒(𝑥, 𝑡) = −𝐷ℎ(𝑇)𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑥 (6.3)
Hopping diffusivity 𝐷ℎ is a fundamental photovoltaic electronic property, which depends on
the interdot distance 𝐿, charge carrier hopping probability 𝛾, and temperature 𝑇, through the
following relationship:
𝐷ℎ(𝑇) =𝐿2
𝜏0𝑒−𝛾𝐿−∆𝐸𝑗𝑖/𝑘𝐵𝑇 (6.4)
where 𝜏0 is the hopping time of a carrier from one QD to another, 𝛾 is the hopping
transmission coefficient, 𝐿 is the effective interdot distance, 𝑇 is the temperature, ∆𝐸𝑗𝑖 is the
energy difference of a hopping particle (exciton or dissociated carrier) between QD states (i)
and (j) [35], and 𝑘𝐵 is the Boltzmann constant. Since trapped charge carriers (𝑛𝑇) can be
emitted from trap states or re-captured, the kinetic equation for 𝑛𝑇 is given by
𝜕𝑛𝑇(𝑥,𝑡)
𝜕𝑡= −𝑒𝑖(𝑇)𝑛𝑇(𝑥, 𝑡) + 𝐶𝑖𝑁𝑖(𝑥, 𝑡)[𝑁𝑇 − 𝑛𝑇(𝑥 , 𝑡)] (6.5)
Combining Eqs. (6.2) - (6.5) yields an expression for the kinetics of the carrier population in
a QD ensemble involving the charge carrier generation, capture, and release from trap states,
as well as the carrier hopping diffusion:
𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑡+ {𝐶𝑖[𝑁𝑇 − 𝑛𝑇(𝑥, 𝑡)] +
1
𝜏(𝑇)}𝑁𝑖(𝑥, 𝑡) = 𝐺0(𝑥, 𝑡; 𝜔) + 𝑒𝑖(𝑇)𝑛𝑇(𝑥, 𝑡) + 𝐷ℎ(𝑇)
𝜕2𝑁𝑖(𝑥,𝑡)
𝜕𝑥2
(6.6)
CHAPTER 6. 143
There is much evidence for the existence of bright (or singlet) and dark (or triplet) states in
PbS QDs [33, 36-38]. Non-radiative recombination processes arise from charge carriers
trapped in both singlet and triplet states:
𝜕𝑁𝑖(𝑥,𝑡)
𝜕𝑡=
𝜕𝑁𝑠(𝑥,𝑡)
𝜕𝑡+𝜕𝑁𝑡(𝑥,𝑡)
𝜕𝑡 (6.7)
𝑁𝑠 and 𝑁𝑡 denote the carrier population in singlet and triplet states, respectively. An energy-
level relation between singlet and triplet states has been proposed [33]
𝑁𝑠(𝑥, 𝑡) = 𝑅𝑠𝑡𝑒−∆𝐸
𝑘𝐵𝑇𝑁𝑡(𝑥, 𝑡) (6.8)
where ∆𝐸 is the energy difference between the two split energy levels, and 𝑅𝑠𝑡 is an energy-
level degeneracy constant equal to 1/3. To simplify the notation, let
𝐴(𝑇) = 𝑅𝑠𝑡𝑒−∆𝐸
𝑘𝐵𝑇 (6.9)
Furthermore, for the harmonic laser excitation at frequency 𝑓𝑟𝑒 = 𝜔/2𝜋, 𝑁𝑡(𝑥, 𝑡), 𝑛𝑇(𝑥, 𝑡)
and 𝐺0(𝑥, 𝑡; 𝜔) can be written as,
𝑁𝑡(𝑥, 𝑡) =1
2𝑁𝑡(𝑥; 𝜔)(1 + 𝑒
𝑖𝜔𝑡) (6.10a)
𝑛𝑇(𝑥, 𝑡) =1
2𝑛𝑇(𝑥; 𝜔)(1 + 𝑒
𝑖𝜔𝑡) (6.10b)
𝐺0(𝑥, 𝑡) =1
2𝐺0(𝑥; 𝜔)𝛽𝑒
−𝛽𝑥(1 + 𝑒𝑖𝜔𝑡) (6.10c)
where 𝜔 is the modulation angular frequency and 𝛽 is the optical absorption coefficient.
The kinetics of the trapping rate Eq. (6.5) can be modified in the frequency domain to yield an
expression for the trapped carrier density 𝑛𝑇(𝑥;𝜔)
CHAPTER 6. 144
𝑛𝑇(𝑥; 𝜔) ≈ (𝐶𝑖𝑁𝑇𝜏𝑖
1+𝑖𝜔𝜏𝑖)𝑁𝑡(𝑥, 𝜔) (6.11)
where 𝜏𝑖 is defined by the carrier emission rate
1
𝜏𝑖(𝑇)= 𝑒𝑖(𝑇) (6.12)
Solving Eqs. (6.6)- (6.9) subject to frequency domain Eq. (6.10), and taking only the
modulated components gives
𝑑2𝑁𝑡(𝑥,𝜔)
𝑑𝑥2−
1
𝐷ℎ(𝑇)[𝑖𝜔 +
1
𝜏𝐸(𝑇)−
𝑅𝑇
[1+𝐴(𝑇)][1+𝑖𝜔𝜏𝑖(𝑇;𝑥,𝜔)]]𝑁𝑡(𝑥, 𝜔) = −
𝐺0𝛽𝑒−𝛽𝑥
𝐷ℎ(𝑇)[1+𝐴(𝑇)] (6.13)
Here, 𝜏𝐸(𝑇) is the effective carrier lifetime, defined as
1
𝜏𝐸(𝑇)≡
1
1+𝐴(𝑇)[
1
𝜏𝑡(𝑇)+
𝐴(𝑇)
𝜏𝑠(𝑇)] (6.14)
and 𝜏𝑡(𝜏𝑠) is the triplet (singlet) lifetime.
For CQD thin films with a thickness d (200 nm for CQD solar cell devices [13, 15]), charge
carriers at the boundaries should be quenched due to the high density of trap states. Equation
(6.13), therefore, can be solved with the boundary conditions: 𝑁𝑡(𝑥, 𝜔) = 0; 𝑥 = 0, 𝑑, viz.
𝑁𝑡(𝑥, 𝜔) = 𝐵1(𝜔, 𝑇)𝑒𝐾1𝑥 − 𝐵2(𝜔, 𝑇)𝑒
−𝐾1𝑥 + [𝐾2(𝑇,𝛽)
𝐾12(𝑇,𝜔)−𝛽
] 𝑒−𝛽𝑥 (6.15)
where the parameters are defined as
𝐾12(𝑇; 𝜔) =
1
𝐷ℎ(𝑇){𝑖𝜔 +
1
𝜏𝐸(𝑇)−
𝑅𝑇
[1+𝐴(𝑇)][1+𝑖𝜔𝜏𝑖(𝑇)]} (6.16a)
𝐾2(𝑇, 𝛽) =𝐺0𝛽
𝐷ℎ(𝑇)[1+𝐴(𝑇)] (6.16b)
𝐵1(𝜔, 𝑇) = [𝐾2(𝑇,𝛽)
𝐾12(𝑇,𝜔)−𝛽2
] (𝑒−𝐾1𝑑−𝑒−𝛽𝑑
𝑒𝐾1𝑑−𝑒−𝐾1𝑑) (6.16c)
CHAPTER 6. 145
𝐵2(𝜔, 𝑇) = [𝐾2(𝑇,𝛽)
𝐾12(𝑇,𝜔)−𝛽2
] (𝑒𝐾1𝑑−𝑒−𝛽𝑑
𝑒𝐾1𝑑−𝑒−𝐾1′ 𝑑) (6.16d)
The radiative emission (i.e., PCR) signal can be expressed as an integral of the charge carrier
population over the thickness of the active layer [39]:
𝑆(𝜔) = 𝐹(𝜆1, 𝜆2) ∫ 𝑁𝑖(𝑥,𝑑
0𝜔)𝑑𝑥 (6.17)
Here, 𝐹(𝜆1, 𝜆2) is an instrumentation coefficient which depends on the spectral emission
bandwidth [𝜆1, 𝜆2 ] of the near-infrared detector. From Eqs. (6.16) and (6.17), the final
expression for the PCR signal can be obtained
𝑆(𝜔)
𝐹(𝜆1,𝜆2)= [
𝐾2(𝑇,𝛽)
𝛽2−𝐾12(𝑇,𝜔)
] {(1+𝑒−𝛽𝑑)(1−𝑒−𝐾1𝑑)
2
𝐾1(1−𝑒−2𝐾1𝑑)−
1
𝛽(1 − 𝑒−𝛽𝑑)} (6.18)
It should be noted that when the trap state density 𝑁𝑇 = 0,
𝐾1(𝑇;𝜔) = √1+𝑖𝜔𝜏𝐸(𝑇)
𝐷ℎ(𝑇)𝜏𝐸(𝑇) =
1
𝐿ℎ(𝑇;𝜔) (6.19)
which is the conventional carrier diffusion wavenumber [7], and 𝐿ℎ(𝑇; 𝜔) is the effective
charge carrier hopping diffusion length.
6.3 CQD Thin Film Homogeneity and Optical Properties
The PbS CQDs were synthesized and purified using the same method as described in
Sect.4.3.1 and ref. [40]. These CQD thin films have a thickness of 200 nm as characterized by
scanning electron spectroscopy. As shown in the room temperature PL spectra, Fig.6.2, three
CQD thin film samples capped with the abovementioned three ligands have the same band-to-
band energy gap of 1.21 eV, while, for further investigation, perovskite MAPbI3 was also
applied to passivate CQD thin films with larger QD size, implying a smaller energy band gap
CHAPTER 6. 146
of 1.09 eV. To clarify, the perovskite MAPbI3 passivated PbS CQD thin film with larger dot
size is labeled PbS-MAPbI3-B throughout this paper, while the one with smaller dot size is
labeled PbS-MAPbI3. It is also shown in Fig. 6.2 that the PL peaks for each type of CQD thin
films depend on the QD size, as well as on surface capping ligands. In addition, secondary PL
emission peaks are also characterized, such as those at 0.81 eV (PbS-MAPbI3) and 0.83 eV
(PbS-TBAI), as well as the PL shoulder at 0.99 eV (PbS-EDT). These secondary PL emission
peaks originate from recombinations that occur through defect-induced donors/acceptors
arising from unpassivated surface states, structural defects, or other changes induced during
ligand exchange processes. Similar types of defect-induced donor/acceptor radiative emission
have also been reported in other materials, such as ZnO nanowires [41], MoS2[42], and InP
[43].
Figure 6.2: Photoluminescence (PL) spectra of four PbS CQD thin films surface passivated
with MAPbI3, EDT, and TBAI.
CHAPTER 6. 147
Figure 6.3: Homodyne lock-in carrierography images of PbS-MAPbI3 (a), PbS-MAPbI3-B (b),
PbS-EDT (c), and PbS-TBAI (d) measured at 10 Hz. Note all the samples were placed on an
aluminum platform for imaging.
As shown in Fig.3.6 the experimental PCR setup for CQD thin films. Constant
characterization temperatures in a range from 100 K to 300 K were maintained using a Linkam
LTS350 cryogenic chamber same as that described in Sect.4.3.1. Photothermal spectroscopy
was performed with the same PCR system at a fixed laser modulation frequency and scanning
temperature. CQD thin films are promising candidates for QD photovoltaic devices; however,
their efficiency is considerably limited by mechanical and electrical defects in CQD thin film
materials. Therefore, we performed homogeneity examination through homodyne lock-in
carrierography imaging (details are shown in Sects.3.4 and 8.2) [44], as shown in Fig.6.3. The
image contrast arises from the modulated carrier wave (including free charge carriers and
excitons) density distribution. Regions with high amplitude values originate from high carrier
CHAPTER 6. 148
density, consistent with high carrier transport parameters including carrier lifetime, diffusivity,
and low trap state density.
6.4 Carrier Transport Kinetics in Various CQD Thin Films
Excitons and free charge carrier transport in CQD thin films is a multi-fact-determined
process as discussed in Chapter 2. The temperature dependence of carrier transport kinetics is
discovered in Chapter 4 using hopping drift-diffusion current-voltage model for high-
efficiency CQD solar cells, herein, using PCR and the trap states involved carrier transport
PCR model, the temperature-dependent transport kinetics will be further investigated in CQD
thin films that surface-passivated with various ligands or consisting of various size of CQDs.
Furthermore, as the ligand-controlled carrier transport kinetics are also explored to reveal more
fundamental physics behind this energy transport behavior. In addition, carrier transport
parameters are obtained through fitting experimental data to theoretical models, therefore, the
fitting uniqueness and measurement reliability are analyzed at the end of this section.
6.4.1 Temperature-dependent Carrier Transport Kinetics
PCR can generate independent carrier diffusion-wave amplitude and phase channels
simultaneously from a single frequency scan, both of which can be used for data analysis
through a best fitting to increase the accuracy and reliability of the best-fitted parameters.
Detailed derivation of PCR amplitude and phase and the Matlab-based computational fitting
for parameter extraction are discussed in Sect. 6.5. Furthermore, a parametric theory as
discussed in Sect. 6.5 is used to examine the uniqueness and reliability of the best-fitted
parameters and demonstrates that all six parameters can be resolved in the framework of our
equations and experimental data sets. For example, Sect.6.5 shows the determinants and
sensitivity coefficients for parameters Dh and τE. Therefore, this validated methodology was
CHAPTER 6. 149
employed for parameter extraction through this study. The experimental and best-fitted PCR
amplitude and phase frequency scans of PbS-MAPbI3 at various temperatures (300 K-100 K),
using Eqs. (6.18), (6.21) and (6.22), are presented in Fig.6.4. Due to the reduced carrier-phonon
interactions at low temperatures and a concomitant increase in the radiative emission rate
accomplished by a decrease in the nonradiative decay rate, the PCR amplitude increases at low
temperatures. In addition, at low temperatures, the increased carrier lifetime yields an
increased PCR phase lag when compared with that at higher temperatures.
Figure 6.4: PCR amplitude (a) and phase (b) of MAPbI3-passivated CQD thin films (PbS-
MAPbI3) measured at various frequencies ranging from 10 Hz to 100 kHz and temperatures
between 100 K and 300 K.
CHAPTER 6. 150
Table 6.1: Best-fitted parameters for PbS-MAPbI3 CQD thin films at different temperatures.
Parameters Temperature (K)
300 250 200 150 100
Dh (cm2/s) 2.41×10-3
±2.75×10-4
5.41×10-4
±1.76×10-6
2.47×10-4
±1.72×10-5
3.62×10-5
±1.44×10-6
1.04×10-6
±3.93×10-13
τE (µs) 0.45 ±0.15 0.66 ±0.21 1.35 ±0.39 4.75 ±0.65 5.37±3.68×10-8
RT (s-1) 2.40×1013
±4.55×1012
2.02×109
±2.37×108
5.54×106
±6.93×105
1.08×105
±3.70×102
2.36×104
±8.99×10-4
𝑒𝑖 (s-1)
3.41×108
±1.22×108
2.97×107
±9.38×106
2.18×107
±3.45×105
4.55×105
±5.33×102
6.70×104
±0.00059
Lh (μm) 0.33 0.19 0.18 0.13 0.023
Figures 6.5(a), (c)-(f) and Table 6.1 show the measurements of five temperature-dependent
carrier hopping transport parameters: Dh, τE, RT, 𝑒𝑖, and Lh of PbS-MAPbI3.With the increase
in temperature from 100 K to 300 K, the best-fitted hopping diffusivity Dh increases
dramatically from 1.04×10-6 cm2/s to 2.41×10-3 cm2/s. The latter value is comparable to the
previously reported values of 0.012 cm2/s and 0.003 cm2/s [9] measured at room temperature
by transient PL spectroscopy for 3-mercaptopropionic acid (MPA) and 8-mercaptooctanoic
acid (MOA) passivated CQD thin films, respectively. The temperature-dependent behavior of
the carrier-wave diffusivity Dh is consistent with the phonon-assisted carrier hopping transport
mechanism. It should be noted that a tunneling transport mechanism is not taken into
consideration due to its non-phonon-assisted transport nature [1, 4, 45-48]. Hopping transport
of carriers within the CQD assembly is carried out through the temperature-dependent nearest
neighbor hopping (NNH) or Efros-Shklovskii variable-range-hopping (ES-VRH) [1, 3, 5, 14].
NNH does not occur at extremely low temperatures because, on average, hopping between
nearest neighbor states requires higher activation energy [5]. Hopping distance is always
optimized spontaneously to yield the highest carrier mobility, and the optimized distance
decreases with increasing temperature [5, 49]. With the temperature rising above a threshold
CHAPTER 6. 151
value, carrier hopping behavior switches from ES-VRH to NNH which has the same hopping
distance as the interdot spacing determined by thermal energy. Kang et al.[5] found that the
hopping distance was longer than the interdot spacing at lower temperatures, and the optimized
distance was equal to the nearest neighbor distance in a temperature range from 40 K to 75 K,
indicating a threshold temperature lower than the minimum temperature of this study. Eq. (6.4),
which does not assume a conventional Einstein relation, predicts an exponential increase in 𝐷ℎ
with increasing temperature, and an exponential decrease with increasing average barrier width
(ligand length).
Figure 6.5: Best-fitted hopping diffusivity Dh (a), and Arrhenius plot of Dh for the extraction
of the carrier hopping transport activation energy (b) of the MAPbI3-passivated (PbS-MAPbI3)
CQD thin film. For the same sample, (c)-(e) are the best-fitted effective exciton lifetime 𝜏𝐸,
carrier trapping rate RT , and Arrhenius plot of thermal emission rate 𝑒𝑖, respectively. (f) Carrier
hopping diffusion length Lh calculated from the best-fitted 𝜏𝐸 and Dh values.
Furthermore, Fig. 6.5(b) shows the activation energy (96.2 meV) obtained from the
Arrhenius plot of the trap-state-mediated hopping diffusivity. It should be reiterated that the
calculated activation energy is trap-state-mediated, i.e., an average energy barrier must be
CHAPTER 6. 152
overcome when carriers hop over the interdot energy barrier and hop out of trap states, as
shown in Fig. 6.1(b). The activation energy extracted from the Dh Arrhenius plot is consistent
with those obtained from the thermal emission rate 𝑒𝑖 as well as those from the photo-thermal
spectra which will be discussed later, corresponding to shallow trap states with an energy depth
much smaller than that of defect-related states measured by PL (Fig.6.2). These activation
energy values are mirrored by the shallow trap states of ca. 0.1 eV (from the conduction band)
obtained from photocurrent quenching [50, 51].
Figure 6.5(c) shows the same trends of carrier lifetime dependence on temperature as our
earlier reported results [7, 8]: carrier lifetime increases with decreasing temperature which is
due to the reduced non-radiative decay rate at low temperatures, a result of decreased phonon-
carrier interactions. Similar values of carrier lifetimes at room temperature can also be found
elsewhere [7, 8, 52-55] in a range from 0.01 µs to 5 µs. Carrier lifetime can be influenced by
many intrinsic QD properties including size, surface ligands, and QD composition. Figures
6.5(d) and (e) show the temperature-dependent carrier trapping rate RT and the thermal
emission rate 𝑒𝑖, respectively. Table 6.1 shows that ei increases from ~104 s-1 at 100 K to ~108
s-1 at 300 K. The activation energy of 106.3 meV, originating from shallow trap states, were
extracted from the Arrhenius plot of 𝑒𝑖 as shown in Fig.6.5(e) and agrees with the activation
energy measured from the hopping diffusivity Dh. At lower temperatures, more carriers are
localized at the excitation sites and the smaller population of phonons freezes these
photogenerated carriers in trap states, which is mirrored by the much lower hopping diffusivity
Dh when compared with the values obtained at room temperature. Therefore, it is reasonable
to conclude that, with the help of phonons at high temperatures, the more widely distributed
carriers are subject to a relatively higher carrier trapping rate, i.e., at high temperature more
CHAPTER 6. 153
trap states are empty which results in an increased RT. Furthermore, due to the higher ambient
thermal energy, the thermal emission rate from the trap states is higher as shown in Fig.6.5(e).
Uing the best-fitted Dh and 𝜏𝐸 values, the hopping diffusion lengths were calculated
through 𝐿ℎ = (𝜏𝐸𝐷ℎ)1/2 , Fig.6.5(f). Although 𝜏𝐸 decreases with temperature, the hopping
diffusion length still increases dramatically from 23 nm to 0.33 µm when the temperature rises
from 100 K to 300 K, because the diffusivity increase is stronger than the lifetime decrease.
Diffusion length is capping-ligand-dependent, for example, at room temperature, diffusion
lengths of PbS CQD thin films treated with different ligands vary widely: with partially fused
PbS CQDs (230 nm) [56], with CdCl2 (80 nm) [24], with ethanethiol (140 nm) [57], and with
3-mercaptopropionic acid (MPA, 100 nm-1000 nm) [58]. Notwithstanding the fact that the
hopping diffusion length can vary as a function of probe method, the Dh values at room
temperature obtained in this study additionally indicate the high photocarrier diffusion ability
of the perovskite photovoltaic material MAPbI3.
6.4.2 Carrier Hopping Activation Energy and Exciton Binding
Energy
40 50 60 70 80 90 100 110 120 130 140
0.1
1
10
100
Best fits
Am
pli
tud
e (m
V)
1/kT (eV-1)
PbS-TBAI
PbS-EDT
PbS-MAPbI3
PbS-MAPbI3-B
Figure 6.6: Temperature scans of the PCR amplitude for different ligands passivated PbS CQD
thin films. The continuous lines are the best fits to each set of data using Eq. (6.20).
CHAPTER 6. 154
Following the temperature-dependent carrier transport dynamics, this section discusses the
trap-state-mediated carrier hoping activation energies and the exciton binding energies. These
properties were studied using PCR photothermal spectra with Arrhenius plot analysis. Figure
6.6 shows the Arrhenius plots of the temperature-dependent PCR amplitude at temperatures
ranging from 90 K to 300 K. Taking into the consideration of both radiative and non-radiative
recombination pathways in PbS CQD thin films, the temperature-dependent dynamic PL (PCR
amplitude) intensity I(T) can be described by the following expression [59-61]
𝐼(𝑇) = 𝐼0
1+∑ 𝐴𝑖𝑒𝑥𝑝(−𝐸𝑖𝑘𝐵𝑇
)𝑖
(6.20)
where 𝐼0 is a normalizing factor, 𝐸𝑖 is the activation energy of the process (i), and 𝐴𝑖 is the
carrier transition rate for process (i). The activation energy is the energy difference between
the original and the final energy states within a carrier transition process. Here, we assume that
our exciton complexes are in the ground state with energy E0 and at least two carrier transition
channels with higher energy states E1 and E2, which should be overcome for the transition
process of excitons to occur. It should be noted that the distribution of excitons in these three
levels is govern by Boltzmann statistics featuring an equilibrium temperature behavior, which
leads to the derivation of Eq. (6.20) with i = 2. Therefore, the activation energy can be
expressed as: Ea,1 = E1- E0 and Ea,2 = E2- E0. The PL emission of PbS CQD thin films in the
entire experimental temperature range cannot be fitted using only one activation energy level
as different carrier dynamic transport processes dominate in different temperature ranges. The
best-fitted curves to the photo-thermal spectra of the four samples using Eq. (6.20), Fig. 6.6,
are the results of two strategies applied for the fitting: first, the entire thermal spectrum was
fitted across the entire temperature range, while the number of activation energy levels was
increased until a satisfactory fit was achieved. For PbS-EDT, PbS-MAPbI3, and PbS-MAPbI3-
CHAPTER 6. 155
B, two activation levels were found to be adequate. When three levels were attempted for these
samples, the third activation energy was identical to one of the first two activation energies.
Compared with other samples, PbS-TBAI exhibits two distinguishable trends in the entire
temperature range, which cannot be accounted for by Eq. (6.20). Therefore, the PbS-TBAI
data were split into two regimes using the dashed line boundary in Fig. 6.6. The two sub-ranges
were fitted separately, using two energy levels (high temperature end) and one energy level
(low temperature end). Second, to investigate the temperature-dependent trap effects on carrier
transport, each spectrum was divided into 5 parts with 100 K, 150 K, 200 K, 250 K, and 300
K being the average temperature (central temperature) in each range, and then each range was
fitted using only one energy level through Eq. (6.20). All the best-fitted activation energies
through these two strategies are summarized in Table 6.2.
Table 6.2: Activation energies at different temperatures for PbS CQD thin films passivated
with various ligands.
Samples
Activation energy fitted in separate temperature
range (one level fitting) (meV)
Activation energy fitted
across the whole
temperature range (meV)
Ea Ea,1 Ea,2 Ea,3
300 K 250 K 200 K 150 K 100 K
PbS-
MAPbI3 275.90 147.51 110.87 98.77 59.55 53.20 147.41
PbS-
MAPbI3-B 190.94 170.23 92.25 95.98 69.96 35.21 107.24
PbS-TBAI 233.81 236.27 77.92 46.45 52.25 51.55 273.94 25.91
PbS-EDT 172.94 148.36 113.63 101.29 59.35 45.21 129.94
CHAPTER 6. 156
Generally, a higher activation energy, accounting for the trap-state-related thermally
activated carrier transition process, dominates in the high-temperature range. In comparison,
at relatively low temperatures, lower activation energies are usually observed, which can be
ascribed to phonon energy [62], exciton binding energy [62-65], and exciton dislocation
binding energy [64]. As shown in Table 6.2, thermal quenching was observed across the entire
temperature range and the activation energy increases with temperature for all samples. The
temperature-dependent activation energy of PbS CQD thin films using the same method was
also reported by Wang et al. [66]. Therefore, as shown in Fig. 6.1(b), Ea,2 is associated with
shallow trap states, and the best-fitted values are close to the activation energies measured
from the carrier hopping diffusivity [Fig. 6.5(b)], and from the thermal emission rate [Fig.
6.5(e)]. As shown in Fig. 6.1 (b), contrary to deep level trap states, which operate as
recombination centers, carriers that are trapped in these shallow trap states do not recombine
but escape from these traps quickly. Nevertheless, it is difficult to identify the source of Ea,1
based on this study. In Table 6.2, with the exception of PbS-TBAI, activation energies Ea at
300 K are higher than either Ea,1 or Ea,2. Moreover, Ea at 100 K is almost equal to Ea,1 (except
for PbS-MAPbI3-B with bigger dot size), which is consistent with the elimination of shallow-
trap-state-related carrier transport processes as contributors to the activation energy Ea,2.
Nonetheless, the Ea,1 process is active throughout the entire temperature range regardless of
the type of surface capping ligands and QD size. Also, the value of Ea,1 is consistent with the
exciton binding energy in a range from 50 to 200 meV for QDs with a diameter of 1-2 nm [67].
In addition, the exciton dissociation occurs in the course of all carrier transport kinetics in our
experimental temperature range. Therefore, it is reasonable to assign Ea,1 to be the exciton
binding energy (Eb) as depicted in Fig. 6.1 (b). It should also be noted from Fig. 6.1 (b) that
CHAPTER 6. 157
the exciton binding energy in quantum confined systems is the energy difference between
exciton transition (optical gap, Eg, opt) and electronic bandgap (Eg), i.e., Eb = Eg - Eg,opt, which
can be approximated through the electron-hole Coulomb interaction [67] and affected by the
material dielectric constant. Electron-hole Coulomb interaction predicts that exciton binding
energy is proportional to 1/R [67], where R is the radius of the QDs. This is consistent with
the much smaller Ea,1 (i.e. Eb, 35.21 meV) of PbS-MAPbI3-B than that of other CQD thin films,
as exciton binding energies for smaller CQDs (PbS-EDT, PbS-TBAI, and PbS- MAPbI3) have
higher values ranging from 45.21 meV to 53.20 meV. This measured exciton binding energy
is similar to the activation energy of ca. 40 meV for exciton dissociation in the PbS CQD solar
cell as reported by Gao et al. [10] and in PbSe QD films measured by Mentzel et al. [68]. As
for PbS-TBAI, the activation energy does not exhibit a monotonic increase with temperature.
The additional activation energy Ea,3 of 25.91 meV might result from many possible
mechanisms, such as exciton delocalization energy [64] from donors or acceptors resulting
from the capping ligand TBAI. The identification of Ea,3 needs further investigation.
The foregoing discussion summarizes that activation energies Ea for CQD thin films arise
from two carrier transition channels except that for PbS-TBAI which has three channels, i.e.
exciton dissociation (Ea,1) and shallow-trap-related thermal activation (Ea,2). Consequently, the
extraction of Ea using Eq.(6.20) at only one activation energy level is subject to the assumption
that, in each temperature range (with a central temperature of 300 K, 250 K, 200 K, 150 K, or
100 K), only one carrier transition process (Ea,1 or Ea,2) is dominant. It should be noted that
deeper lying trap states require higher Ea for carrier transitions, as expected. Therefore,
comparing Ea values at different central temperatures (Table 6.2) with the corresponding Ea,1
and Ea,2 as discussed above, it must be kept in mind that the exciton dissociation process occurs
CHAPTER 6. 158
across the entire temperature range, while the activation energies (Ea,2) for trap-mediated
carrier transitions decrease with decreasing temperature. Furthermore, at the low temperature
of 100 K, Ea for all samples is approximately equal to Ea,1, indicating a negligible contribution
of Ea,2 to the overall activation energy at this temperature. All of these facts point to the
following conclusion: deep-lying trap states dominate carrier transport at higher temperatures,
while shallow trap states control carrier transport at low temperatures, in agreement with [44].
These effects may arise because carrier distributions are localized near their generation sites
at low temperatures due to the low values of Dh.
6.4.3 Ligand- and Size-dependent Carrier Transport Kinetics
Equation (6.18) describes the PCR signals generated by the carrier transport in CQD thin
films. Besides temperature, surface passivation ligands and the QDs geometry are also
substantial factors to carrier transport properties in CQD thin films [1, 3-6, 11, 14]. In addition
to passivating QD unsaturated surface bonds to minimize or eliminate surface trap states,
solution exchange ligands reduce the interdot spacing and enhance the coupling strength
between neighboring QDs. When trap states are not the dominant factors for carrier hopping
transport, smaller interdot spacing, according to Eq. (6.4), results in increased diffusivity.
Figure 6.7 shows the PCR amplitude and phase frequency scans at 100 K for CQD thin films
passivated with four different ligands and the best fits of Eq. (6.18) to each curve. The best-
fitted parameters of carrier transport properties in these CQD thin films are tabulated in Table
6.3. The interdot spacing values of PbS-TBAI and PbS-MAPbI3 CQD thin films measured by
grazing-incidence small-angle scattering (GISAXS) are 3.50 nm and 3.30 nm, respectively
[69]. In addition, Liu et al. [4] calculated the nominal EDT length to be ca. 0.43 nm which
should result in a smaller interdot spacing than the other ligands due to its smaller molecule
CHAPTER 6. 159
size. The small interdot spacing of PbS-EDT allows the increase of hopping diffusivity above
that of MAPbI3 passivated CQD thin films, consistent with Eq. (6.4). The slightly higher
carrier diffusivity of PbS-MAPbI3 than PbS-MAPbI3-B originates in the lower hopping
activation energy of PbS-MAPbI3 (59.55 meV) at 100 K than PbS-MAPbI3-B (69.96 meV)
with shallow, yet deeper lying, trap states, as shown in Table 6.2. On the contrary, at 300 K,
when trap states start to play key roles in carrier hopping transport, PbS-MAPbI3 carriers face
higher transport activation energy (275.90 meV) than PbS-MAPbI3-B carriers (190.94 meV),
Table 6.2, resulting in a smaller Dh (2.41×10-3cm2/s) than PbS-MAPbI3-B (1.80×10-2 cm2/s),
Table 6.4. In addition, the values of activation energies obtained from the photothermal spectra
also explain the slightly higher Dh of PbS-TBAI (Ea = 233.81 meV) than that of PbS-MAPbI3
CQD thin films, as both samples feature significant defect-related states, Fig.6.2, which seem
to become limiting factors of the carrier hopping diffusion at room temperature.
0.01 0.1 1 10 100
0.4
0.6
0.8
1
PbS-MAPbI3-B
PbS-MAPbI3
PbS-TBAI
PbS-EDT
Continuous lines are best fits
No
rmal
ized
Am
pli
tud
e (m
V)
Frequency (kHz)
(a)
0.01 0.1 1 10 100-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
(b)
PbS-MAPbI3-B
PbS-MAPbI3
PbS-TBAI
PbS-EDT
Continuous lines are best fitsPh
ase
(deg
ree)
Frequency (kHz)
Figure 6.7:100 K PCR amplitudes (a) and phases (b) of CQD thin films passivated with four
different ligands, and the best fits to each curve using Eq. (6.18).
However, the PCR signal resolution (i.e., phase lag and amplitude decrease) for the PbS-
EDT samples is too low to be resolved and fitted for the extraction of these parameters even
when the modulation frequency is increased to 1 MHz. This is not a limitation inherent in the
CHAPTER 6. 160
PCR technique, but rather a consequence of the inability of our equipment to measure samples
with too poor transport properties.
Table 6.3: Summary of the best-fitted parameters for CQD thin films surface passivated with
various ligands. These parameters were evaluated for 100 K measurement.
Samples PbS-EDT PbS-TBAI PbS-MAPbI3-B PbS-
MAPbI3
Hopping
diffusivity
Dh (cm2/s)
1.62×10-6
±2.83×10-8
8.58×10-7
±3.74×10-13
8.82×10-7
±6.07×10-14
1.04×10-6
±3.93×10-13
Effective carrier
lifetime τE (µs)
2.78
±7.13×10-6
3.79
±1.42×10-8
7.66
±3.53×10-8
5.37
±3.68×10-8
Thermal emission
rate
ei (s-1)
6.53×104
±1.16
6.80×104
±0.00086
4.96×104
±0.0010
6.70×104
±0.00059
Trapping rate RT
(s-1)
4.72×104
±0.80
5.24 ×104
±3.71×10-4
2.04×104
±3.43×10-4
2.36×104
±8.99×10-4
Absorption
coefficient
𝛽 (cm-1)
8.57×107
±2.83×106
6.22×106
±1.43
7.80×106
±0.34
2.79×106
±0.56
Generation rate
G0 (cm-3s-1)
1.84×109
±7.22×107
1.81×108
±3.12
1.44×108
±2.50
3.04×107
±0.31
Diffusion length
Lh (μm) 0.017 0.018 0.026 0.024
Interdot spacing
(nm)
0.43 (nominal
ligand length) [4] 3.50 [69] 3.30 [69] 3.30 [69]
Table 6.4: Summary of the best-fitted parameters for CQD thin films surface passivated with
various ligands. These parameters were evaluated for 300 K measurement.
Samples PbS-TBAI PbS-MAPbI3-B PbS-MAPbI3
Hopping diffusivity
Dh (cm2/s)
5.09×10-3
±6.96×10-4
1.80×10-2
±2.30×10-3
2.41×10-3
±2.75×10-4
Effective carrier lifetime τE (µs) 0.16
±0.06
0.51
±0.15
0.45
±0.15
Thermal emission rate
𝑒𝑖 (s-1)
7.35×109
±4.93×109
8.33×1010
±5.78×1010
6.993×109
±1.35×109
Diffusion length Lh (μm) 0.29 0.96 0.33
CHAPTER 6. 161
The effective carrier lifetime , 𝜏𝐸 , and the thermal emission rate, ei, at 100 K, are
summarized for all ligands in Table 6.3. Consistently with the foregoing mechanism of the
temperature-dependent carrier hopping lifetime as shown in Table 6.1, values of 𝜏𝐸 for PbS-
TBAI (0.16 µs) and PbS- MAPbI3-B (0.51 µs) at 300 K greatly increase when the temperature
decreases to 100 K (3.79 s and 7.66 s, respectively) as shown in Table 6.3. On the contrary,
the thermal emission rate 𝑒𝑖 was found to significantly increase from ~104 s-1 at 100 K (Table
6.3) for all samples to ~1010 s-1 at 300 K (Table 6.4). Comparing all three samples, PbS-
MAPbI3-B at 300 K exhibits the highest thermal emission rate 𝑒𝑖. Returning to the effective
carrier lifetime 𝜏𝐸, MAPbI3-passivated PbS CQD thin films exhibit longer lifetime at 100 K
than PbS-EDT and PbS-TBAI, while PbS-MAPbI3-B lifetime remains the highest amongst all
the tested samples at both 100 K (Table 6.3) and 300 K (Table 6.4). This is not unexpected
because PbS-MAPbI3-B does not exhibit any defect states induced secondary PL emission
peak as shown in Fig.6.2.
Regarding the calculated diffusion length Lh, both MAPbI3-passivated samples have
similar Lh of ca. 24-26 nm at 100 K, a temperature at which the influence of trap states is not
significant. These values are higher than those of TBAI and EDT treated samples, Table 6.3.
Considering the QD size (~2 nm) and interdot spacing (~3 nm), the short diffusion length
indicates that carriers hop across only a few QDs before statistically recombining. At 300 K,
trap states limit carrier transport which is, nevertheless, assisted by phonon interactions in
overcoming the hopping activation energy. This trade-off between trap-state limitations and
phonon assistance results in longer diffusion lengths Lh at high temperatures. Tables 6.3 and
6.4 show that PbS-MAPbI3-B possesses the longest Lh at both high and low temperatures,
indicating that this material is optimal for solar cell performance improvement.
CHAPTER 6. 162
The carrier trapping rate RT as defined in Sect. 6.2 is proportional to the trap state density
and was measured to be on the order of 104 s-1 at 100 K (Table 6.3). Consistently with the PL
spectra in Fig. 6.2, PbS-MAPbI3, as shown in Table 6.3, was fitted with higher RT than PbS-
MAPbI3-B. In addition, the strong trap peak of the PbS-TBAI spectrum is also consistent with
its higher RT than that of PbS-EDT. MAPbI3-capped PbS CQD thin films exhibit relatively
lower carrier trapping rates than PbS-EDT and PbS-TBAI (Table 6.3) due to the smaller lattice
mismatch between PbS and the MAPbI3 perovskite material [30] which results in better surface
passivation. The absorption coefficient 𝛽 at 100 K (Table 6.3), is on the order of 106 cm-1 for
all samples except PbS-EDT with a slightly higher value of 8.57×107 (± 2.83×106). From the
same table, the exciton generation rate G0 at 100 K is between 3×107 cm-3s-1 and 2×109 cm-3s-
1 for all our PbS CQD thin films.
6.5 Fitting Uniqueness and Reliability – Parameter Extraction
from PCR
PCR can generate independent carrier diffusion-wave amplitude and phase channels
simultaneously from a single frequency scan, both of which can be used for data best fitting to
increase the accuracy and reliability of best-fitted parameters. For the PCR measurements
carried out in this Chapter, at each temperature, 120 points (60 points each for amplitude and
phase) were used in one computational fitting process to extract six to-be-measured parameters:
𝛽, 𝐷ℎ, 𝜏𝐸 , 𝑅𝑇 , 𝑒𝑖 and 𝐺0, from Eq.( 6.18). The expressions for the PCR amplitude and phase can
be derived from Eq. (6.18):
𝐴(𝜔) = √𝑆𝑅2(𝜔) + 𝑆𝐼
2(𝜔) (6.21)
𝜙(𝜔) = 𝑡𝑎𝑛−1 [𝑆𝐼(𝜔)
𝑆𝑅(𝜔)] (6.22)
CHAPTER 6. 163
where 𝑆𝑅(𝜔) and 𝑆𝐼(𝜔) are the real and imaginary parts of the PCR signal 𝑆(𝜔), respectively,
in Eq. (6.18).
Figure 6.8: The determinant of diffusivity Dh (a) and effective carrier lifetime 𝜏𝐸 (b) in the
PCR phase channel. Diamonds indicate frequencies at which linear dependence occurs; no
such linear dependencies were found for the amplitude channel of all parameters. (c) and (d)
are the sensitivity coefficients of 𝜏𝐸 in the amplitude and phase channel, respectively. Besides
the measured parameters in this figure, other parameters were also treated similarly to yield
the best-fitted values for all samples as shown in Tables 6.1, 6.3, and 6.4.
To extract these carrier transport parameters, the energy difference between singlet and
triplet states was first found to be 37.24 meV for PbS CQDs as reported earlier [27]. The best-
fitted parameters for charge carrier dynamics in MAPbI3-passivated CQD thin films with small
dots are tabulated in Table 6.1 as shown in Sect.6.4.1. The Matlab-based computational
CHAPTER 6. 164
program employs the fminsearchbnd solver [70] which minimizes the sum of the squares of
errors between the experimental and calculated data [7, 51].
Furthermore, a parametric theory was also applied to test the uniqueness of our best-fitted
parameters. First, the sensitivity coefficient 𝑓 was defined as the first derivative of function 𝑆
with respect to a specific parameter [71]. According to the measurement theory, the sensitivity
coefficients of the parameters to-be-measured (𝛽, 𝐷ℎ, 𝜏𝐸 , 𝑅𝑇 , 𝑒𝑖 and 𝐺0) can be obtained for
amplitude and phase channel as follows:
𝑓(𝐴) = [𝜕𝐴(𝜔)
𝜕𝛽 𝜕𝐴(𝜔)
𝜕𝐷ℎ 𝜕𝐴(𝜔)
𝜕𝜏𝐸 𝜕𝐴(𝜔)
𝜕𝑅𝑇 𝜕𝐴(𝜔)
𝜕𝑒𝑖 𝜕𝐴(𝜔)
𝜕𝐺0] (6.23)
𝑓(𝜙) = [𝜕𝜙(𝜔)
𝜕𝛽 𝜕𝜙(𝜔)
𝜕𝐷ℎ 𝜕𝜙(𝜔)
𝜕𝜏𝐸 𝜕𝜙(𝜔)
𝜕𝑅𝑇 𝜕𝜙(𝜔)
𝜕𝑒𝑖 𝜕𝜙(𝜔)
𝜕𝐺0] (6.24)
Mathematically, all these parameters should be linearly independent over the range of
observations (frequencies), the number of which should be larger than the volume of unknown
parameters. Taking the amplitude channel as an example, the following relation should be
valid:
𝐶1𝜕𝐴(𝜔)
𝜕𝛽+ 𝐶2
𝜕𝐴(𝜔)
𝜕𝐷ℎ+ 𝐶3
𝜕𝐴(𝜔)
𝜕𝜏𝐸+ 𝐶4
𝜕𝐴(𝜔)
𝜕𝑅𝑇+ 𝐶5
𝜕𝐴(𝜔)
𝜕𝑒𝑖+ 𝐶6
𝜕𝐴(𝜔)
𝜕𝐺0≠ 0 (6.25)
where 𝐶1 − 𝐶6 are coefficients, not all equal to zero. Eq. (6.25) is satisfied if, and only if, the
determinant of the 6×6 matrix of the sensitivity coefficient is not equal to zero, i.e.,
𝑑𝑒𝑡𝐴 =
[ 𝜕𝐴(𝜔1)
𝜕𝛽⋯
𝜕𝐴(𝜔1)
𝜕𝐺0
⋮ ⋱ ⋮𝜕𝐴(𝜔6)
𝜕𝛽⋯
𝜕𝐴(𝜔6)
𝜕𝐺0 ]
≠ 0 (6.26)
Although mathematically, if these six unknown parameters can be determined, measurements
at only six frequency points are sufficient, however, for better computational fitting reliability,
CHAPTER 6. 165
60 frequency points in both amplitude and phase channels were measured in this study. Eqs.
(6.23), (6.24) and (6.26), as well as the expression corresponding to Eq.( 6.26) for the PCR
phase channel, were used and the results showed that when the parameters at each temperature
were assigned the values in Tables 6.1, 6.3, and 6.4 as presented in the previous sections, the
determinants at all experimental frequencies were not zero. For example, as shown in Figs.
6.8(a) and (b) for Dh and 𝜏𝐸, when the range of an estimated parameter value was extended,
the PCR phase channel exhibited some values that do not satisfy linear independence. In
comparison, no such values were found for the PCR amplitude channel. Theoretically, these
results indicate that all six parameters can be resolved in the framework of our equations and
experimental data sets.
In addition to linear independence which indicates whether a parameter can be determined
reliably, the signal response sensitivity to each parameter is another important factor which
shows to what extent the PCR signal will change when the specific parameter value is varied.
To analyze the sensitivities of the measured carrier transport parameters, best-fits for PbS-
MAPbI3 at 100 K were used as the database. For, example, Figs. 6.8(c) and (d) show the
sensitivity coefficients of 𝜏𝐸 in the two channels. In the amplitude channel, larger sensitivity
is found when 𝜏𝐸 is smaller than 4 ns. In the phase channel, the PCR signal is more sensitive
to 𝜏𝐸 at high frequencies. The frequency-dependent parameter sensitivity provides important
clues for optimal experimental measurements, i.e., the proper frequency window can be chosen
through the analysis of parameter sensitivity coefficients for the calculation of specific
parameters with optimal accuracy. However, it should be noted that the sensitivity values of
different parameters are not comparable due to the difference in their units. Furthermore, the
values of other parameters influence the sensitivity coefficient of a specific parameter. From
CHAPTER 6. 166
the sensitivity coefficients in Fig. 6.8 and that of other parameters, it is interesting to see that
some of the parameters have a positive correlation with the PCR signal while others, a negative
correlation.
6.6 Conclusions
This chapter, first, introduces the Instrumentation of PCR system and discusses concepts
of modulated laser excitation, detector signal collection, and lock-in amplifier based signal
computation. The general theory of PCR was also discussed. Second, coupled with the fully
optical non-destructive PCR technique, a novel quantitative methodology was developed to
characterize carrier transport dynamics for QD systems by deriving a trap-state-mediated
carrier hopping transport model. Multiple materials and carrier transport parameters for PbS-
EDT, PbS-TBAI, PbS-MAPbI3, and PbS-MAPbI3-B CQD thin films were measured at
different temperatures. The observed monotonic dependence of effective carrier lifetime 𝜏𝐸,
hopping diffusivity Dh, carrier trapping rate RT, and hopping diffusion length Lh on the
temperature in the range from 100 K to 300 K is consistent with a phonon-assisted carrier
hopping transport mechanism in PbS CQD thin films. For all samples, trap-state-mediated
activation energies were found to be in a range between 100 meV and 280 meV. Photothermal
spectroscopy modified from the PCR system was also used to measure exciton binding
energies as a function of dot size. From PL spectroscopy, it was shown that perovskite
(MAPbI3) passivated thin films with larger dot size (bandgap energy: 1.09 eV) are free of
obvious defect states induced secondary PL emission. These thin films exhibited the highest
carrier lifetime and hoping diffusivity at 300 K, thus proving to be better photovoltaic materials
than PbS-MAPbI3, as well as TBAI, or EDT treated CQD thin films.
CHAPTER 6. 167
The PCR technique sheds lights on the temperature- and ligand-dependent carrier transport
dynamics in photovoltaic CQD thin films, hence, benefiting CQD solar cell efficiency
optimization through a better understanding of device energy dissipation physics and through
quantitative recombination process analysis in CQD surface trap states with a goal to
minimizing their effects through ligand passivation and bandgap energy engineering. The
results of this study can be further applied in directing high-efficiency CQD solar cell
fabrication in conjunction with the development of an improved PCR theory for photovoltaic
devices.
168
Chapter 7
Carrier Recombination Mechanism, Energy Band Structure, and
Inhomogeneity-affected Carrier Transport in Perovskite Shelled
PbS CQD Thin Films Using PCR and HoLIC
7.1 Introduction
Chapter 6 introduces the PCR technique through discussing trap-state-mediated carrier
transport PCR theory, and the dependence of carrier transport on CQD dimensions and
temperatures. However, the recombination mechanisms and band energy structure of CQDs
are not provided. Therefore, this chapter applies the theoretical models and findings discussed
in Chapter 6 to investigate the trap-state-mediated carrier transport mechanisms further.
In specific, charge carrier recombination processes and sub-bandgap energy states in
perovskite passivated CQD thin films for photovoltaic applications are discussed using PL
spectroscopy, excitation power dependent PL intensity, and photocarrier radiometry (PCR)-
based photothermal spectroscopy. Quantitative analysis of carrier transport properties was
carried out through PCR frequency scans. An energy band structure is proposed based on the
above energy states study. It should be noted that the as discussed recombination mechanism
leads to the theoretical explanation of the nonlinear response of radiative recombination to
laser excitation intensity, which is fundamental for HeLIC technique as discussed in Chapter
8. Furthermore, the sample inhomogeneity-associated variation of carrier transport in CQD
thin films is also discussed through the combination of PCR, HoLIC, and HeLIC.
Interpretation of HoLIC and HeLIC imaging contrast is addressed that image amplitudes can
CHAPTER 7. 169
reflect carrier density distribution, which is proportional to defects or trap states associated
effective carrier lifetime.
7.2 Experimental Details and CQD Thin Film Synthesis
The schematic of the LIC imaging setup is shown by Fig.3.9 and discussed in detail in
Sect.3.4. To characterize charge carrier recombination processes and sub-bandgap states
within CQD thin films, excitation laser intensity scans, PCR temperature (photothermal
spectroscopy) scans and laser modulation frequency scans were carried out with a conventional
PCR system as discussed in Sect.3.3. Specifically, incorporated within the PCR system,
photothermal spectra were obtained through linear temperature scans by measuring PCR
signals from the sample at a fixed laser modulation frequency while the temperature was
reduced from 300 K to 100 K at a sufficiently slow (quasi-equilibrium) rate of 5 oC/min using
the Linkam LTS350 cryogenic stage. An average 1 sun laser excitation intensity was used for
the measurements. The PL emission from the sample was collected by a single detector
connected to a lock-in amplifier. Different modulation frequencies were used for the
temperature scans.
The CQD thin films studied in this work are of the lead sulfide (PbS) kind and were surface
passivated with methylammonium lead triiodide perovskite (MAPbI3) to remove CQD surface
defect states and adjust interdot distances. The detailed fabrication process is available in
Sect.4.3.1, and the scanning electron microscopy (SEM) and transmission electron microscopy
(TEM) images of these perovskite (MAPbI3)-passivated CQD thin films can be found in ref.[1].
With excellent surface passivation of perovskite thin shells onto CQD surfaces, the CQD solar
cells exhibited improved open-circuit voltage and power conversion efficiency compared with
those without MAPbI3 treatment or those treated with other ligands [1]. Two CQD thin films
CHAPTER 7. 170
(samples A and B) were fabricated with different free-exciton PL emission peaks for LIC
imaging analysis towards qualitative thin film homogeneity and mechanical defect
characterization for the purpose of CQD solar cell efficiency optimization in lab fabrication
processes.
7.3 Charge Carrier Recombination Mechanism for PbS CQDs:
Nonlinear Response
Figure 7.1: Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs. Due
to the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between QD
and MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles.
To a great extent, semiconductor quality is determined by structural defects arising from
dangling bonds on QD surfaces, such as point defects including vacancies and interstitials
performing as very efficient trap states for electrons, holes, and excitons. They exhibit a strong
influence on optical and electrical properties of the host semiconductor materials. In PbS QDs,
potential point defects include Pb-vacancies (VPb), S-vacancies (Vs), Pb-interstitials (Pbi), S-
interstitials (Si) and antisites (PbS and SPb). These possible defects can be introduced during
the material fabrication process, acting as recombination centers. MAPbI3 has been reported
CHAPTER 7. 171
to be an excellent candidate for passivating PbS CQD surface defects because of the minimal
lattice mismatch between the MAPbI3 and PbS, a schematic shown in Fig. 7.1. As a result, no
defect-related peaks were found in PL spectra of MAPbI3-PbS CQDs in solution [1].
Figure 7.2: Near-band-edge photoluminescence via variable radiative and nonradiative
transitions. (a) Free-exciton recombination, (b) and (c) recombination of donor (D)- and
acceptor (A)-bound excitons (DX, AX), (d) donor-acceptor pair recombination (DA), (e)
recombination of a free electron with a neutral acceptor (eA), (f) recombination of a free hole
with a neutral donor (hD).
However, a better understanding of charge carrier recombination processes in CQD thin
films is necessary. As shown in Fig. 7.2, photogenerated excitons, and free electrons and holes
may undergo the following transitions in PbS CQDs: radiative free-exciton recombination
(FE), radiative acceptor- and donor-bound exciton (AX, DX) recombination, nonradiative
donor-acceptor pair (DA) recombination, radiative recombination of a free electron and a
neutral acceptor (eA), and radiative recombination of a free hole and a neutral donor (hD).
These recombination processes have also been encountered in other low-dimensional nano-
systems including ZnO nanocrystals [2], MoS2, MoSe2 and WSe2 monolayers [3], ZnSe
nanowires [4], InGaN/GaN multiple quantum wells [5], InAs/GaAsSb quantum dots [6], and
CHAPTER 7. 172
PbS quantum dots [7]. Unlike semiconductor materials with continuous energy bands, CQDs
exhibit excitonic behavior with photogenerated excitons having much higher binding energy
than that of excitons in semiconductor materials with continuous energy bands, and are
incapable of forming continuous band structures due to the dot size polydispersity and the
energy band disorder. Therefore, excitons are bound together and are not able to separate into
free electrons and holes without the help of external forces including interdot coupling effects
and electric fields [8]. Physical descriptions of processes (I) to (VI) in Fig. 7.2 have been
detailed by Schmidt et al.[9], while for MAPbI3-PbS CQDs, four unique conditions are
considered: first, their n-type conducting property (only for calculation simplification), which,
although not measured in this study, hinges on previous evidence of the n-type conducting
property of CQD-perovskite LEDs [10]; second, exciton recombination dominates the
transitions; third, excitons bound to ionized donors and acceptors can be neglected due to their
weak transition probabilities; and fourth, the excitation laser energy is higher than the CQD
bandgap Eg. Therefore, the rate formulas, adapted from Schmidt et al. [9], can be expressed by
𝑑𝑛
𝑑𝑡= 𝑔𝑃 − 𝑎𝑛2 (7.1)
𝑑𝑛𝐹𝐸
𝑑𝑡= 𝑎𝑛2 + 𝑃 − (
1
𝜏𝐹𝐸+
1
𝜏𝐹𝐸𝑛𝑟) 𝑛𝐹𝐸 − 𝑏𝑛𝐹𝐸𝑁𝐷0 (7.2)
𝑑𝑛𝐷𝑋
𝑑𝑡= 𝑏𝑛𝐹𝐸𝑁𝐷0 − (
1
𝜏𝐷𝑋+
1
𝜏𝐷𝑋𝑛𝑟 ) 𝑛𝐷𝑋 (7.3)
𝑑𝑁𝐷0
𝑑𝑡= 𝑔(𝑁𝐷 − 𝑁𝐷0)𝑛 − 𝑚𝑁𝐷0𝑃 − 𝑏𝑛𝐹𝐸𝑁𝐷0 + (
1
𝜏𝐷𝑋+
1
𝜏𝐷𝑋𝑛𝑟 ) 𝑛𝐷𝑋 − 𝑓𝑁𝐷0𝑛 (7.4)
where 𝑃 is the excitation laser intensity, 𝜏𝐹𝐸 and 𝜏𝐹𝐸𝑛𝑟 are the radiative and nonradiative
lifetimes of free excitons, respectively, and 𝜏𝐷𝑋 and 𝜏𝐷𝑋𝑛𝑟 are the respective radiative and
nonradiative lifetimes of donor bound excitons undergoing transition DX. n, 𝑛𝐹𝐸 , and 𝑛𝐷𝑋 are
the concentrations of free electrons, free excitons, and donor bound excitons, respectively.
Furthermore, 𝑁𝐷 and 𝑁𝐷𝑜 are the concentrations of donors and neutral donors, respectively.
CHAPTER 7. 173
𝑎, 𝑏, … , 𝑓 are the coefficients associated with the processes shown in Fig. 7.2, while 𝑔 and 𝑚
are coefficients for exciton generation and electron excitation, respectively. Solving Eqs. (7.2)
and (7.3) in the steady state, yields the luminescence intensities of free and bound excitons,
IFE, and IDX defined as
𝐼𝐹𝐸 ∝𝑛𝐹𝐸
𝜏𝐹𝐸=
𝐵
𝜏𝐹𝐸𝑛2 (7.5)
𝐼𝐷𝑋 ∝𝑏𝑁𝐷𝐵
1+𝜏𝐷𝑋𝜏𝐷𝑋𝑛𝑟𝑛2 (7.6)
where
𝐵 =𝑎
(1
𝜏𝐹𝐸+
1
𝜏𝐹𝐸𝑛𝑟)+𝑏𝑁𝐷
(7.7)
Provided the probabilities of free-to-bound transitions be proportional to the respective
transition rates, luminescence intensities of free-to-bound (donors) transitions, 𝐼ℎ𝐷 can be
expressed by
𝐼ℎ𝐷 ∝ 𝑛 𝑁𝐷0 (7.8)
Solving Eq. (7.1) in the steady state, yields 𝑛 ∝ 𝑃0.5. Using this relationship in Eqs. (7.5) to
(7.8), it is found that 𝐼 ∝ 𝑃 for excitonic transitions and 𝐼 ∝ 𝑃0.5 for free-to-bound transitions.
Therefore, the γ value can be used to characterize the type of a radiative charge carrier
recombination process. Experimentally, γ is generally measured to be between 1 and 2 for
excitonic emissions including transitions (a)-(c) in Fig. 7.2, and less than 1 for free-to-bound
(acceptors/donors) emissions as shown by transitions (d)-(f). For example, γ = 0.69 for sub-
bandgap recombinations and γ = 1.48 for band-edge associated recombinations for PbS-TBAI
QD/PbS-EDT QD devices [7], in which TBAI and EDT denote tetrabutylammonium iodide
CHAPTER 7. 174
and 1,2-ethanedithiol, respectively. Therefore, the value of γ can be used to physically interpret
the linear or nonlinear behavior of a charge carrier recombination process with respect to the
laser excitation and reveals the underlying radiative recombination process. It is an important
parameter for the HeLIC process, because nonlinear PL responses to the laser excitation, acting
as a nonlinear frequency mixer, are essential for PL based heterodyne LIC imaging [11-14].
As opposed to CQD thin films, excitons in polycrystalline and amorphous Si wafers, upon
their formation, immediately dissociate into free electrons and holes, with the dominant
recombination process being via defect states. In this situation, the nonlinear recombination
term 𝑎𝑛2 in the rate equation 𝑑𝑛
𝑑𝑡= 𝑔𝑃 − 𝑎𝑛2 − ℎ𝑛(𝑁𝐷 −𝑁𝐷0) − 𝑒𝑛𝑁𝐴0 (ℎ is the coefficient
for nonradiative transitions of free electrons to ionized donors, and 𝑁𝐴0 is the concentration of
natural acceptors) [9], can be neglected, resulting in 𝑛 ∝ 𝑃. Solving Eqs. (7.5) - (7.8) using the
relationship 𝑛 ∝ 𝑃, it follows that 𝐼 ∝ 𝑃2 for excitonic transitions and 𝐼 ∝ 𝑃 for free-to-bound
transitions. This conclusion is consistent with experimental LIC results for Si wafers with
exponent γ >1 [13].
Figure 7.3: PCR amplitude vs. excitation power at three different temperatures for sample A
(a) and sample B (b), at 10 kHz laser modulation frequency.
CHAPTER 7. 175
To identify charge carrier transition types of various radiative recombination processes for
the study of non-linear PCR responses, Fig. 7.3 shows the excitation power dependent PCR
signals of sample A and B at three temperatures (300 K, 200 K, and 100 K). Based on the
foregoing discussion of radiative recombination processes, experimental data were fitted
to 𝐼 ∝ 𝑃𝛾. At room temperature (300 K), the γ value of sample A was 0.86 which is close to,
but less than, unity and is indicative of donor/acceptor-related free-to-bound recombinations.
However, free-exciton-like transitions also exist. By comparison, the only dominant emission
peak (i.e. the excitonic recombination) of sample B results in a γ value of 0.94, and the small
deviation from unity is probably due to the very few sub-bandgap trap states involved in non-
radiative exciton trapping which compromises only slightly the strength of the PCR amplitude,
much less so than that of trap rich sample A. Both samples exhibited reduced γ values when
temperature decreased. The values of γ at 100 K for both samples are close to 0.7 which is
likely indicative of the onset of free-to-bound transitions (γ=0.5) starting to dominate the
radiative emission processes as opposed to free-exciton-like emissions (γ=1). This can be
explained by the following typical behavior of exchange coupled excitons. The presence of
bright and dark states can be attributed to the nonstoichiometry of QD surfaces [15], lattice
mismatch during ligand exchange processes, and incomplete surface bond termination or
chemical changes [16]. Although the presence of these exciton bright and dark states has not
been proven directly, evidence of the presence of these states is accumulating [17-20]. De
Lamaestre et al. [17] studied the temperature dependent PL intensity and decay rates of PbS
nanocrystals in a silicate glass and found a large energy splitting (ca. 30 meV) of the exciton
ground state fine structure which showed evidence for the existence of a triplet state. Nordin
et al. [18] has observed the PL emission from two active states with an energy separation of
CHAPTER 7. 176
ca. 6 meV that is close to the theoretically reported energy difference between triplet and
singlet states, ca. 10 meV. Gaponenko et al. [19] provided another piece of evidence using
steady-state and time-resolved PL, and developed a theoretical energy-level model considering
the lowest 1S-1S exciton state splitting that can present a consistent quantitative description of
experimental results. Gao et al. [20] discussed charge trapping in bright and dark states of
coupled PbS quantum dot films with analysis of temperature-dependent PL from dots of
different sizes or different surface passivation. Considering the state splitting, although the
physics behind the γ exponent decrease with decreasing temperature is not well understood, it
is well known that triplet exciton states with spin 1 (parallel spins) are energetically more
stable than singlet exciton states with spin 0 (antiparallel spins). Furthermore, triplet states
have a higher statistical weight of 3 (allowed values of spin components:-1, 0 and 1) than that
of singlet states (statistical weight = 1, spin component 0). Consequently, at low temperatures,
most excitons condense into triplet states, from which they cannot decay radiatively to the S =
0 ground state. However, triplet excitons have longer lifetimes and higher probability for
nonradiative recombination than deeper dark states. As a result, the contribution of excitonic
recombinations to dynamic PL (measured by PCR) decreases with reduced temperature in
favor of nonradiative triplet recombinations. It is concluded that the presence of large densities
of trap states can reverse the non-radiative recombination suppression rate at low temperatures
previously observed due to decreased phonon populations in PbS CQD thin films [21]. Instead,
enhanced nonradiative recombinations reduce the radiative emission, an effect which may
severely compromise the solar efficiency of photovoltaic solar cells fabricated using this type
of CQDs.
7.4 Energy Band Structure
CHAPTER 7. 177
7.4.1 Photoluminescence of CQD Thin Films
Figure 7.4 (a) shows PL spectra of perovskite MAPbI3-passivated PbS CQD thin films,
samples A and B with different dot sizes. The measurements were performed at room
temperature using an excitation laser wavelength of 375 nm, and two dominant radiative PL
emission peaks located at 1169 nm and 1553 nm were observed for Sample A. The 1169 nm
PL emission peak corresponds to the band-edge excitonic recombination (PbS is a direct
bandgap semiconductor) including transitions a and b/c, Fig. 7.2, but is dominated by the free
exciton recombination (transition a) at room temperature [22]. The 1553 nm emission peak
originates from recombinations that occur through donors/acceptors (transitions e/f, Fig. 7.2)
arising from unpassivated surface states, structural defects or other chemical changes induced
during ligand exchange processes. In contrast, only the band-edge excitonic recombination
peak (1232 nm) is observed for sample B. When compared with PL spectra of other CQD thin
films with different QD sizes, Fig. 7.4 (b), although the CQD and thin film synthesis processes
are the same, it is found that these PL peaks are quantum dot size dependent, indicating a
complexity of surface passivation mechanism which requires further investigation.
Figure 7.4: (a) Photoluminescence (PL) spectra of MAPbI3-passivated PbS (MAPbI3−PbS)
thin films (samples A and B) spin-coated on glass substrates. (b) PL spectra of MAPbI3−PbS
thin films fabricated through the same process as that of samples A and B but with different
QD sizes.
CHAPTER 7. 178
The surfaces of CQDs typically contain a lot of recombination centers because of the abrupt
termination of the semiconductor crystal periodicity even within a QD. The absence of sub-
bandgap-state-related emissions implies better surface passivation of sample B than A,
revealing the inevitable diversity of the lab fabrication processes. It is also reasonable that
sample A, with a smaller dot size (due to its wider bandgap from PL characterization) that
leads to higher surface-to-volume ratio, incorporates more trap states, making itself a more
difficult candidate for surface passivation. From Fig. 7.4 (a), energy bandgaps of samples A
and B were calculated to be ca. 1.1 eV and 1.0 eV, respectively. The emission from sub-
bandgap states in sample A is ca. 0.3 eV lower than the band-edge emissions. In Fig. 7.4 (a),
sample A has a smaller FWHM (full width at half maximum) than sample B, reflecting a
narrower quantum dot size distribution, as the broadening of PL peaks arises from the quantum
dot size polydispersity with specific spectral components originating from dots of specific
sizes. It should be noted that with the use of a long-pass filter, PL emission detected by the
InGaAs camera is in a wavelength range from 1000 nm to 1700 nm.
7.4.2 PCR Photothermal Spectra of CQD Thin Films
Further investigation of sub-bandgap states using PCR photothermal spectroscopy is
presented in Fig.7.5. From Figs. 7.5 (a) and (b), corresponding to samples A and B,
respectively, it is seen that sample A exhibits more and deeper troughs at all frequencies.
Furthermore, these troughs shift toward higher temperature with increasing modulation
frequency of the excitation laser which can be attributed to three dominant sub-bandgap (trap
state) levels: (I), (II) and (III). Thermal emission rates of carriers from trap states are
temperature dependent processes. When the emission rate matches (or is resonant with) the
modulation frequency (a process called “rate window”), a dynamic photoluminescence (PCR
CHAPTER 7. 179
signal) enhancement occurs due to the increased number of free de-trapped carriers
recombining radiatively, exhibiting a peak in the amplitude of the photo-thermal spectrum at
the resonance temperature. Amplitude peaks and phase troughs follow opposite trends, i.e.
when there is an amplitude peak in the amplitude spectrum, a trough forms in a corresponding
phase spectrum. The phase signal is more sensitive than the amplitude channel which can be
influenced easily by, for example, sample surface reflections and shallow surface states. The
phase locks on the rate of the photo-thermal emissions only and is little or not sensitive to
factors complicating the amplitude spectrum. For an optoelectronic sample with one or more
traps like the one in Fig. 7.5(a), the phase trough associated with a trap state characterized by
a fixed carrier de-trapping energy Ea, shifts to higher temperatures with increasing modulation
frequency. This is caused by the change in the rate-window resonance condition between the
faster thermal ejection rate of trapped carriers at the higher temperature [a Boltzmann factor,
Eq. (7.9)] and the modulation frequency, which is now satisfied at a higher frequency [23, 24].
The activation energy of sub-bandgap trap levels in sample A can be calculated through
Arrhenius-plot fitting of the photo-thermal emission rate, 𝑒𝑛 [23]:
𝑒𝑛(𝑇) = 𝑅𝑛𝜎𝑛𝑒𝑥𝑝 (−𝐸𝑎
𝑘𝑇) (7.9)
Here 𝑅𝑛 is a material constant and 𝜎𝑛 is the exciton capture cross section. At each trough,
𝑒𝑛(𝑇𝑡𝑟𝑜𝑢𝑔ℎ) = 2.869𝑓, in which 𝑓 is the pulse repetition frequency [24]. Figure 7.5(c) shows
the best-fitted activation energies for three dominant sub-bandgap levels in sample A. These
sub-bandgap trap levels exhibit similar activation energies ranging from 33.8 meV to 40.7
meV. Associated with the colloidal environment and surface states in PbS QDs, these shallow
level multi-energetic traps can capture excitons which undergo nonradiative recombinations
CHAPTER 7. 180
or phonon-mediated radiative emissions [21]. Activation energies for trap states in EDT
passivated PbS QDs have also been measured by Bozyigit et al. [25] using thermal admittance
spectroscopy to be around 100 meV for QDs with an energy bandgap of 1.1 eV. Activation
energy differences mostly originate from different exchange-ligands. Chuang et al. [7] also
reported a ~230 meV energy difference (activation energy) between band-edge and sub-
bandgap state emission using PL, which is close to ~ 260 meV following the model proposed
by reference [25] for an EDT treated PbS CQDs with an energy bandgap of 1.1 eV.
Figure 7.5: Photocarrier radiometry (PCR) photothermal spectra of MAPbI3-passivated PbS
(MAPbI3-PbS) thin films spin-coated on glass substrates, samples A (a) and B (b). (c)
Arrhenius plots of the PCR phase troughs, I, II, and III, as shown in (a), and best-fitted to Eq.
(7.9) for the extraction of activation energies for each sub-bandgap trap level.
As shown in Fig. 7.6, which incorporates information from PL spectroscopy and photo-
thermal temperature scans, there are two trap levels located in the PbS QD bandgap: a deep
level (𝐸𝑎 = 0.3 eV) and shallow levels (𝐸𝑎 in a range from 33.8 meV to 40.7 meV). As shown
in Fig. 7.6, when excitons are free roaming they may experience radiative recombination,
become captured in trap states or diffuse to the next QD through nearest-neighbor-hopping
(NNH). The coupling strength between two QDs induces exciton dissociation into free charge
carriers. Trapped excitons require overcoming an activation energy barrier to become de-
trapped and undergo radiative recombinations. For shallow states, the activation energy can be
provided by thermal energy, therefore, PCR photothermal spectroscopy with a lock-in rate
CHAPTER 7. 181
window can reveal shallow trap states. With regard to sample B, Fig. 7.5(b) shows that the
PCR thermal spectra are much smoother at all frequencies, indicative of very few sub-bandgap
states in this sample, and consistent with the PL spectroscopy results, Fig. 7.4 (a).
Figure 7.6: (a) Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs.
Due to the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between
QD and MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles
(a); (b) energy band structure (assumed n-type) of a PbS- MAPbI3 nanolayer, sample A,
featuring shallow and deep level trap states. Excitons are excited in the right QD and diffuse
through nearest-neighbor-hopping (NNH) to the next QD, during which process the coupling
strength between two QDs dissociate excitons into free charge carriers. Carriers may
experience radiative recombination or captured by different types of trap states, where non-
radiative recombination or de-trapping may occur.
7.5 Large-area Imaging and Carrier Transport of CQD Thin Films
7.5.1 Qualitative Large-area Imaging
Figure 7.7 (a) and (b) presents photos of MAPbI3-passivated PbS thin films. The thin films
were spin-coated on glass substrates with an area of 25×25 mm2 and stored together in a
nitrogen environment for further study. It is observed that these thin films are visually
homogeneous with few visible imperfections. From a series of CQD thin films with different
QD size, we have experimentally observed that the CQD thin film color changes slightly with
QD size. This is consistent with the expected change in quantum confinement which affects
the optical absorption coefficient of these quantum dots, thereby accounting for the slight color
CHAPTER 7. 182
difference between Figs. 7.7 (a) and (b). In contrast, homodyne and heterodyne LIC images
shown in Fig.7.7 (c) - (f) illustrates significant degrees of inhomogeneity.
Figure 7.7: Photos of MAPbI3 -PbS thin films, (a) sample A and (b) sample B. 1 kHz homodyne
LIC amplitude images of MAPbI3- PbS thin films, (c) sample A and (d) sample B. 20 kHz
heterodyne LIC amplitude images of MAPbI3-PbS thin films, (e) sample A and (f) sample B.
Note the very different signal strength scales associated with the two samples.
The physical origins of the LIC spatial contrast are due to the free photocarrier density
diffusion-wave distributions, which depend on charge carrier transport parameters, mainly the
effective exciton lifetime as well as the hopping diffusivity, de-trapping time, and trap state
density as shown in Sect.6.4. Specifically, large amplitudes and phase lags correspond to high
photocarrier density, a result of long local carrier lifetimes; however, for regions associated
with mechanical damages or intrinsic material defects, the lower amplitude is generally
expected because defects lead to a significant increase of nonradiative recombination rates
resulting in a reduction of carrier lifetimes. Comparison between regions 2 and 3 in Fig.7.7(d)
CHAPTER 7. 183
provides a direct example of the LIC image contrast arising from different photocarrier
diffusion-wave distributions, with region 3 being indicative of longer carrier lifetimes. This
may be due to different spin-coating and ligand passivation processes or unexpected surface
chemical reactions upon exposure to ambient air.
Regarding mechanical-damage-induced defects, as shown in Fig.7.7 (f), the area of a
scratched letter B on the front surface of sample B exhibits lower PCR amplitude values
compared with its neighboring regions. This is attributed to the damage-induced lower
photocarrier density diffusion-wave, leading to the enhanced probability of nonradiative
recombinations into defect states, thus shorter lifetimes. It should be noted that the scratch was
produced after the homodyne image of sample B was obtained and shown in Fig. 7.7(d). For
both A and B, homodyne and heterodyne LIC amplitude images show prominent
inhomogeneities in the charge carrier density distributions.
7.5.2 PCR Characterization of Carrier Transport Parameters
The spatial resolution of LIC images is determined by the ac hopping diffusion length
through
𝐿(𝜔) = √ 𝐷ℎ𝜏𝐸
1+𝑖𝜔𝜏𝐸 (7.10)
Shorter 𝐿(𝜔) yields higher spatial resolution that can be achieved through increasing the
angular modulation frequency 𝜔. Limited by the camera frame rate and exposure time, the
highest frequency achieved in this study with homodyne lock-in carrierography is 1 kHz for
high-quality images. However, better image resolution relies on higher laser modulation
frequencies [12-14, 26, 27], so heterodyne imaging was performed at 20 kHz. Higher
frequency imaging is limited by the low radiative emissions of our samples but can be attained
by using laser sources with higher power. Compared with Fig.7.7 (c), Fig.7.7 (e) exhibits
CHAPTER 7. 184
higher spatial resolution with more detailed features of charge carrier density wave
distributions. As for sample B, Fig.7.7 (f) with a laser frequency of 20 kHz exhibits limited
image resolution improvement when compared with Fig.7.7 (d). This is probably due to the
longer effective exciton lifetimes of sample B than A, as shown in Table 7.1 through PCR
frequency characterization and computational best-fits, leading to a much smaller change of
the longer hopping diffusion length 𝐿(𝜔), Eq. (7.10).
Figure 7.8: Phase diagram of PCR frequency scans in three different regions 1-3 as shown in
Fig.7.7 (c)-(d) and the best-fits of experimental data to Eq. (6.18) in Sect. 6.2.
Table 7.1: Summary of best-fitted parameters using Eq. (6.18) in Sect. 6.2.
Parameters Region 1
(Sample A)
Region 2
(Sample B)
Region 3
(Sample B)
Dh (cm2/s) 0.0181± 0.0061 0.0106 ± 0.0049 0.00870± 0.00437
τE (μs) 0.43 ± 0.10 1.56 ± 0.48 2.05 ± 0.20
ΔE (meV) 25.3 ± 3.1 25.4 ± 3.6 25.5± 3.2
τi (ns) 1.33 ± 0.62 0.75 ± 0.45 2.92 ± 0.78
NT (×1013 s-1) 20.05±2.30 2.99 ± 0.35 3.18 ± 0.38
(×107 cm-1) 4.55± 0.87 1.58 ± 0.49 1.30 ± 0.53
Go (×1025 cm-2/s) 4.85± 0.69 4.94 ± 0.72 5.03 ± 0.70
CHAPTER 7. 185
To investigate exciton transport parameters and material properties of MAPbI3- passivated
PbS CQD thin films, PCR frequency scans, and theoretical best-fits were performed. Sample
surface spots were selected as shown in Fig.7.7 (c)-(d) from regions 1 to 3. Equation (6.18) in
Sect.6.2 was used to fit the experimental PCR data for the extraction of parameters involved.
Figure 7.8 presents the PCR frequency scans of the three regions as well as the best-fits of each
curve. Region 1 exhibits a smaller phase lag than regions 2 and 3, indicating shorter lifetime
[21, 28], while region 3 exhibits slightly larger phase lag than region 2, consistent with the
slightly longer lifetime of region 3.
The best fitting program was implemented through a fminsearch solver minimizing the
square sum of errors between the experimental and calculated data. To establish the uniqueness
and reliability of our measurements, different starting points were generated automatically by
the solver for each fitting process, so that the best-fitted results fluctuated about their mean
values. The fitting procedure was repeated several hundred times and 100 sets of results with
the smallest variance were selected for statistical calculation as tabulated in Table 7.1. The
table summarizes the best-fitted carrier transport and material property parameters including
Dh, 𝜏𝐸, ΔE, 𝜏𝑖, 𝑁𝑇, , and Go for the three regions, providing room temperature measurements.
The CQD thickness of 200 nm was measured by scanning electron microscopy. Region 3
exhibits the highest effective exciton lifetime of 2.05 ± 0.20 μs, consistent with the LIC image
results. The relatively long lifetime could be the effect of dielectric screening similar to that
observed in other IV-VI semiconductor nanocrystals [29]. Similar theoretical results of exciton
lifetimes have also been reported to be in a range between 1 and 3 μs for PbS CQDs and PbSe
CQDs [30]. Additional same range experimental data have been reported in the literature from
time-resolved PL spectroscopy: between 4.97 and 2.74 μs for small PbSe CQDs with diameter
CHAPTER 7. 186
from 2.7 to 4.7 nm) [31]; 1 μs for oleic-acid-capped PbS CQDs [29]; and 2.5 – 0.9 μs for PbSe
CQDs, 3.2 -4.3 nm in diameter [32]. Furthermore, both our A and B samples exhibit Dh values
on the order of 10-2 cm2/s. Comparable results using transient PL spectroscopy for 3-
mercaptopropionic acid (MPA) ligands or 8-mercaptooctanoic acid (MOA) linked PbS CQDs
were also reported elsewhere [33]. Those PbS QDs were doped with metal nanoparticles to
introduce fixed exciton dissociation distances away from the location of their formation. The
best-fitted separation energy ΔE between singlet and triplet states is ca. 25 meV, which
indirectly provides another piece of evidence for the existence of dark and bright states in PbS
CQDs. Manifestations of the existence of these states in PbS CQD thin films with similar
separation energies were also reported elsewhere [21]. Consistent with the PL spectra, Fig. 7.4
(a), Table 7.1 exhibits that region 1 has the highest carrier trapping rate 𝑁𝑇, indicating the
highest trap state density amongst all three regions. Table 7.1 further presents the de-trapping
lifetimes, i, reflecting the time an exciton resides in a trap state before being released. The de-
trapping activation energies obtained from Fig. 7.5 (c) are tentatively attributed to phonon-
mediated photo-thermal interactions and the associated interface trap states appear in the
energy diagram of Fig. 7.6 (b). Room temperature NNH and the carrier diffusivity Dh are
determined by the interdot coupling strength. The absorption coefficient 𝛽 was evaluated
through best-fitting to be on the order of 107 cm-1 for the 800 nm excitation. For comparison,
an absorption coefficient on the order of 105 cm-1 for excited oleic acid-capped PbS CQDs
suspended in tetrachloroethylene (C2Cl4) [30] was measured using a UV-vis-NIR
spectrophotometer. Furthermore, using the same method, the PbSe CQDs bandgap absorption
coefficient was found to be ~ 106 cm-1 and decreased with the dot size [34]. It is hypothesized
that these large differences may arise from different surface passivation ligands, dot
CHAPTER 7. 187
dimensions, and sample state (solvent/solid). Finally, the exciton generation rate Go is an
excitation source determined parameter which essentially remains constant across the three
regions 1-3, as expected.
Apart from fitting uniqueness, sample stability is also essential for measurement accuracy,
hence, sample stability was examined under laser excitation for the duration of one complete
PCR frequency scan. Figure 7.9 illustrates the sample A PCR phase time dependence measured
at 100 kHz over 25 minutes. It exhibits a standard deviation of only 0.12o, which allows
concluding that the PCR phase is not time dependent and the excitation laser has negligible
influence on CQD thin films.
Figure 7.9: PCR phase dependence on time over 25 minutes, the duration of a PCR frequency
scan. Sample A at 100 kHz laser modulation frequency.
Coupled with the theoretical best-fits, a conclusion can be reached that in Fig.7.6, excitons,
undergo recombination, or dissociate into free charge carriers which can recombine radiatively
CHAPTER 7. 188
through donors/acceptors, or through trap state-induced nonradiative transitions, giving rise to
non-linear heterodyne LIC image responses to laser excitation. The combined PCR, LIC
imaging and photothermal temperature scans of perovskite-shelled PbS CQD thin films were
shown to yield quantitative information about key exciton transport processes like effective
lifetimes and other hopping transport parameters extracted from the theoretical exciton
diffusion-wave density trap model as discussed in Sect.6.2.
7.6 Conclusions
High-frequency InGaAs-camera-based HoLIC and HeLIC images of CQD thin films, as
well as temperature scanned photothermal emission rates, activation energies, and trap
densities were obtained to qualitatively characterize CQD nanolayer properties. It was
demonstrated that a MAPbI3-shelled PbS CQD thin film exhibits non-linear PCR signal
response that acts as an effective frequency mixer giving rise to heterodyne LIC images,
originating from free-to-bound and trap-state associated recombination. Furthermore,
quantitative analysis of exciton transport processes using PCR frequency scans yielded carrier
transport parameters including effective exciton lifetimes and diffusivities of MAPbI3-
passivated CQD. Combined with LIC imaging, PCR frequency scans, and photothermal
temperature scans can provide fast, quantitative, contactless, nondestructive evaluation of
charge carrier transport as well as material properties of CQD materials and electronic devices.
This combined analytical methodology can be used for improved control of PbS CQD solar
cell fabrication and performance/efficiency optimization.
189
Chapter 8
Heterodyne and Homodyne Lock-in Carrierography Imaging of
Carrier Transport in CQD Solar Cells
8.1 Introduction
With intensive researches on device architecture engineering [1-3], surface materials
chemistry [4-6], synthesis methodologies [7, 8], charge carrier dynamics [9-11], and
theoretical modeling [12-14], CQD solar cell power to electricity conversion efficiency has
boosted from only 3 % to today’s 13.4 % in a short 7-year period [15]. However, as discussed
in Sects. 5.1 and 5.6, conventional small-spot (< 0.1 cm2) testing, arises seriously questionable
overall CQD solar PCE and stability on a large-scale. Therefore, nowadays, large-area
photovoltaic solar cells prevail, the characterization of which fulfills various purposes
including shading effects, fundamental carrier transport dynamics, and mechanical and
electrical defect evaluation. Therefore, large-area characterization methodologies are needed
for CQD solar cell efficiency optimization. Spatially resolved photoluminescence (PL) and
electroluminescence (EL) constitute powerful methodologies for the characterization of silicon
wafers [16-18] and solar cells [18-21]. They yield measurements of minority carrier hopping
lifetime [16, 17, 20], open-circuit voltage [18, 20, 21], current density [21], series resistance
[21], fill factor [18, 21], and quality monitoring in different device fabrication steps [19].
Furthermore, due to the high signal-to-noise ratio (SNR), synchronous frequency-domain
imaging methodologies are emerging, such as the lock-in thermography (LIT) [22] that has
been used for determining series resistance and recombination current at a frequency of 20 Hz.
However, static (dc) PL and EL, as well as low-frequency LIT that limited by the low camera
CHAPTER 8. 190
frame rate, cannot monitor electronic transport kinetics and recombination dynamics. The
latter, however, are key parameters for the determination of photovoltaic energy conversion
and dissipation.
To address these critical issues, this chapter discusses large-area HoLIC qualitative
carrier distribution imaging of CQD solar cells, which was mostly used for material or device
homogeneity and quality estimation. Furthermore, what is addressed is the development of
large-area HeLIC quantitative imaging of carrier transport parameters in these CQD solar cells.
HoLIC imaging and PCR (discussed in Chapters 3 and 6) are the same type of dynamic
spectrally gated frequency-domain photoluminescence modality and can yield quantitative
information about carrier transport dynamics with accuracy and precision superior to the time-
resolved PL due to their intrinsically high signal-to-noise ratio (SNR) by virtue of lock-in
demodulation [23, 24]. HoLIC has advantages as an all-optical non-destructive imaging
technique for large-area photovoltaic device imaging, yet it is limited to the low modulation
frequency range (<1 kHz) due to the low frame rates of even the state-of-the-art conventional
cameras [9]. Using a single InGaAs detector, PCR can attain high-frequency characterization
(> 100 kHz), however, the fast large-area imaging capability is compromised. Through creating
a slow enough beat frequency component, HeLIC overcomes the high-frequency limitation of
conventional camera-based optical characterization techniques and the poor SNR at short
exposure times associated with high frame rates [25]. Therefore, with higher SNR than dc
photoluminescence (PL) imaging, HeLIC can attain a wide range of frequency-dependent ac
carrier diffusion lengths to generate depth-selective/resolved high-frequency imaging of
carrier transport parameters in large-scale devices.
CHAPTER 8. 191
Thereby, to study carrier transport dynamics in CQD solar cells and the effects of CQD
layer inhomogeneity (for example, induced at various fabrication stages) and contact/film
interface effects on solar cell performance, This chapter, for the first time, develops a large-
area quantitative characterization methodology through combining a J-V model with PCR and
HeLIC to quantitatively produce carrier lifetime, diffusivity, and drift and diffusion length
images for a high-efficiency CQD solar cell under frequency modulated excitation. The
proposed methodology overcomes the limitations of small-dot testing, providing a fast,
contactless, and kinetic property characterization technique that is also suitable for in-line solar
cell quality monitoring in the industrial photovoltaic manufacturing process.
8.2 Theories of Homodyne and Heterodyne Lock-in
Carrierography
Section 3.4.1 discusses the instrumental setups and various signal processing techniques
that used in HoLIC and HeLIC. With the demonstration of nonlinear response from CQD solar
cells (Sect.3.4.2), this section will quantitatively describe the working principles of HoLIC and
HeLIC.
The single detector based PCR, and the InGaAs camera based HoLIC and HeLIC methods
collect photons from radiative recombination of charge CDWs or excitons in CQD
photovoltaic materials and devices. The recombination rate (RR) is proportional to the product
of the concentration of electrons n and holes p in the form of RR = knp [26], in which k is a
material-dependent constant which can be obtained from the semiconductor’s absorption
coefficient. Using the CDW ∆𝑁(𝜔, 𝑥) in the frequency domain, the homodyne and heterodyne
CHAPTER 8. 192
signals can be modeled through a depth integral of the radiative recombining free photocarrier
densities:
𝑆(𝑡) = ∫ 𝑑𝑥 ∫ ∆𝑁(𝜔, 𝑥)[∆𝑃(𝜔, 𝑥) + 𝑁𝐴] 𝐹(𝜆)𝑑𝜆𝜆2
𝜆1
𝑑
0 (8.1)
where ∆𝑃(𝜔, 𝑥) is the excess hole CDW and is equal to ∆𝑁(𝜔, 𝑥) according to the quasi-
neutrality approximation, i.e. the photogenerated excess electron and hole concentrations are
identical across the thin film thickness. 𝐹(𝜆) is an instrumental coefficient that depends on the
spectral detection bandwidth (𝜆1, 𝜆2) of the near-infrared detector. 𝑁𝐴 is the equilibrium
majority carrier concentration determined by material doping resulting from in-air oxidation
of our CQD thin films and solar cells. For single frequency modulation, the excess carrier
density waves can be expressed as
∆𝑁(𝑥, 𝜔) = ∆𝑛0(𝑥) + 𝐴(𝑥, 𝜔)𝑐𝑜𝑠[𝜔𝑡 + 𝜑(𝑥, 𝜔)] (8.2)
Here ∆𝑛0(𝑥) , 𝐴(𝑥, 𝜔) and 𝜑(𝑥, 𝜔) are the dc component, ac amplitude and phase of the
photogenerated excess electron CDW ∆𝑁(𝑥, 𝜔), respectively. Therefore, considering lock-in
detection at only the fundamental frequency term, the homodyne lock-in carrierography signal
from Eq. (8.1) can be given by
𝑆ℎ𝑜(𝜔) = ∫ {𝐴2(𝑥, 𝜔)𝑐𝑜𝑠2[𝜔𝑡 + 𝜑(𝑥, 𝜔)] + [𝑁𝐴 + 2∆𝑛0(𝑥)]𝐴(𝑥, 𝜔)𝑐𝑜𝑠[𝜔𝑡 +𝑑
0
𝜑(𝑥, 𝜔)]}𝑑𝑥 (8.3)
In comparison, in HeLIC the incident laser excitation is modulated at two different angular
frequencies 𝜔1 and 𝜔2. The excess electron CDW is
∆𝑁(𝑥, 𝜔) = 2∆𝑛0(𝑥) + 𝐴(𝜔1, 𝑥)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑(𝜔1, 𝑥)] + 𝐴(𝜔2, 𝑥)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑(𝜔2, 𝑥)](8.4)
The reference beat frequency is∆𝜔 = |𝜔2 − 𝜔1| , indicating that radiative recombination
modulated at other frequencies will be filtered, and hence the corresponding signal can be
expressed as
CHAPTER 8. 193
𝑆ℎ𝑒(∆𝜔) = ∫ 𝐴(𝑥, 𝜔1)𝐴(𝑥, 𝜔2)cos [∆𝜔𝑡 + ∆𝑑
0𝜑(𝑥, 𝜔1, 𝜔2 )]𝑑𝑥 (8.5)
where ∆𝜑 = 𝜑(𝑥, 𝜔2) − 𝜑(𝑥, 𝜔1). Using the Fourier transformation, ∆𝑁 in frequency domain
is given by
∆𝑁(𝑥, 𝜔) = 4𝜋∆𝑛0(𝑥)𝛿(𝜔) + 2𝜋𝐴(𝑥, 𝜔1)𝑒𝑖𝜑(𝑥,𝜔1)𝛿(𝜔 − 𝜔1) +
2𝜋𝐴(𝑥, 𝜔2)𝑒𝑖𝜑(𝑥,𝜔2)𝛿(𝜔 − 𝜔2) (8.6)
The demodulated signal for PCR and HoLIC becomes,
𝑆ℎ𝑜(𝜔) = ∫ (2∆𝑛0(𝑥) + 𝑁𝐴)∆𝑁(𝑥, 𝜔)𝑑𝑥𝑑
0 (8.7)
whereas that for HeLIC becomes
𝑆ℎ𝑒(∆𝜔) = ∫ ∆𝑁∗(𝑥, 𝜔1)∆𝑁(𝑥, 𝜔2)𝑑𝑥𝑑
0 (8.8)
where * denotes complex conjugation. It should be noted that the HeLIC phase is very small
on the order of (10-3)o due to the small frequency difference ∆𝜔.
8.3 Carrier Transport Theory of CQD Solar Cells under
Modulated Photoexcitation
The nature of photocarrier generation, discrete hopping transport, and recombination in
CQD-based thin films was found to follow a hopping diffusion transport behavior [14, 27]
under frequency-modulated laser excitation that reveals details of carrier hopping transport
dynamics in these photovoltaic materials. Here, in order to extract charge carrier hopping
transport dynamics in CQD-based solar cells, light-matter interaction under modulated-
frequency excitation is investigated as an extension of conventional current-voltage
characterization of CQD solar cells under DC laser excitation [12, 14, 28]. Reviewing the CQD
solar cell structure in Fig.5.1 in Chapter 5 and Fig.8.1, due to the larger bandgap energy of
ZnO than the incident excitation photon energy as well as the thicker CQD layers than ZnO,
CHAPTER 8. 194
charge carriers and excitons are considered to be generated only in CQD layers, thus
contributing to the primary current within this type of solar cell.
Figure 8.1: Schematic of CQD solar cell sandwich structure (a), and the corresponding band
energy structure (b) also shows the illumination depth profile, the photocarrier density wave
distribution and the intrinsic and external electric fields.
Therefore, in a manner similar to the carrier hopping transport model under static excitation
[14], the rate equation for electrons in the nominal p-type CQD layers under dynamic
illumination can be written as
𝜕∆𝑛(𝑥,𝑡)
𝜕𝑡=
𝜕𝐽𝑒(𝑥,𝑡)
𝜕𝑥−∆𝑛(𝑥,𝑡)
𝜏+ 𝑔(𝑥, 𝑡) (8.9)
∆𝑛(𝑥, 𝑡) is the excess electron density and 𝐽𝑒(𝑥, 𝑡) is the electron current flux; 𝜏 is the nominal
minority carrier electron lifetime, and 𝑔(𝑥, 𝑡) is the carrier generation rate. Considering that
the ambipolar diffusion coefficient and mobility, 𝐽𝑒(𝑥, 𝑡) can be further defined by [29, 30]
𝐽𝑒(𝑥, 𝑡) = 𝐷𝑒𝜕∆𝑛(𝑥,𝑡)
𝜕𝑥+ 𝜇𝑒𝐸∆𝑛(𝑥, 𝑡) (8.10)
where 𝐷𝑒 is the diffusivity, 𝜇𝑒 is the mobility, and 𝐸 is the electric field (a constant value given
as the difference between the external and intrinsic electric fields), Eq. (8.9) is reduced to a
diffusion equation which can be solved using the Green function method and transferred to the
CHAPTER 8. 195
frequency domain through a Fourier transformation [14]. Upon harmonic optical excitation,
the photoexcited excess carrier distribution follows the Beer-Lambert Law:
𝑔(𝑥, 𝜔) =𝛽𝜂𝐼0
2ℎ𝜈𝑒−𝛽𝑥(1 + 𝑒𝑖𝜔𝑡) (8.11)
where 𝛽 is the optical absorption coefficient, 𝜂 is the quantum yield of the photogenerated
carriers, and ℎ and 𝜈 are the Plank constant and the frequency of incident photons, respectively.
𝐼0 denotes the incident photon intensity. In the one-dimensional geometry, the boundary
conditions at 𝑥 = 0 and 𝑑 , Fig. 8.1, can be written as functions of surface recombination
velocities (𝑆1and 𝑆2 at 𝑥 = 0 and 𝑥 = 𝑑, respectively,) and the excess carrier density at the
corresponding boundaries, Fig. 8.1.
𝐷𝑒𝜕∆𝑁(𝑥,𝜔)
𝜕𝑥|𝑥=0
= 𝑆1∆𝑁(0,𝜔) (8.12a)
−𝐷𝑒𝜕∆𝑁(𝑥,𝜔)
𝜕𝑥|𝑥=𝑑
= 𝑆2∆𝑁(𝑑, 𝜔) (8.12b)
where ∆𝑁(𝑥, 𝜔) is the Fourier transformed counterpart of ∆𝑛(𝑥, 𝑡), a carrier-density-wave
(CDW). Therefore, the final expression of excess carriers ∆𝑁(𝑥, 𝜔) can be obtained as follows
∆𝑁(𝑥, 𝜔) =𝜂𝐼0𝛽
4ℎ𝜈𝐷𝑒(1−𝑅𝑒1𝑅𝑒2𝑒−2𝐾𝑒𝑑)[𝛽2−(𝑄02+𝜎𝑒
2)]{([(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] −
𝑅𝑒1[(𝜌𝑒 − 1) + 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝐾𝑒+𝛽)𝑑)𝑒−𝐾𝑒𝑥 + (𝑅𝑒2[(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] +
[(1 − 𝜌𝑒) − 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝛽−𝐾𝑒)𝑑)𝑒−𝐾𝑒(2𝑑−𝑥) − 2(1 − 𝑅𝑒1𝑅𝑒2𝑒
−2𝐾𝑒𝑑)𝑒−𝛽𝑥} (8.13)
with the following definitions,
𝑄0 =𝜇𝑒
2𝐷𝑒�� [cm-1]; 𝜎𝑒 = √
1+𝑖𝜔𝜏
𝐷𝑒𝜏 [cm-1] (8.14a)
𝑅𝑒𝑗 =𝐷𝑒√𝑄0
2+𝜎𝑒2−𝑆𝑗
𝐷𝑒√𝑄02+𝜎𝑒
2+𝑆𝑗
, 𝑗 = 1, 2 (8.14b)
𝐾𝑒 = √𝑄02 + 𝜎𝑒2 − 𝑄0 [cm-1]; 𝜌𝑒 =
𝛽
𝐾𝑒 (8.14c)
CHAPTER 8. 196
As shown in Fig.8.1, the electric field �� = ��𝑖 + ��𝑒𝑥𝑡 (Ei > Eext) prevents the spreading of the
excess CDW density towards the 𝑥 = 𝑑 terminal of the solar cell. On the contrary, when the
net electric field switches its direction it facilitates the spreading of the CDW concentration
gradient, resulting in reduced energy barriers. Here, all discussion will be based on the
condition that �� prevents the spreading of the excess electron CDW density, i.e. �� and ��𝑖 have
the same direction, which is a general working condition for traditional solar cells.
Ultimately, when substitute Eq. (8.13) into Eqs. (8.7) and (8.8) under low injection levels,
i.e. 2∆𝑛0(𝑥) ≪ 𝑁𝐴 , the final expression of Eq. (8.7) after the necessary mathematical
manipulations becomes
𝑆ℎ𝑜(𝜔) ≈ 𝐴𝑁𝐴 [𝐵(1−𝑒−𝑑𝐾𝑒)+𝐶(𝑒−𝑑𝐾𝑒−𝑒−2𝑑𝐾𝑒)
𝐾𝑒+𝐷(𝑒−𝑑𝛽−1)
𝛽] (8.15)
Coefficients A, B, C, and D are defined according to Eq. (8.13), as follows,
𝐴 =𝜂𝐼0𝛽
4ℎ𝜈𝐷𝑒(1−𝑅𝑒1𝑅𝑒2𝑒−2𝐾𝑒𝑑)[𝛽2−(𝑄02+𝜎𝑒
2)] (8.16a)
𝐵 = [(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] − 𝑅𝑒1[(𝜌𝑒 − 1) + 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝐾𝑒+𝛽)𝑑 (8.16b)
𝐶 = 𝑅𝑒2[(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] + [(1 − 𝜌𝑒) − 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝛽−𝐾𝑒)𝑑 (8.16c)
𝐷 = 2(1 − 𝑅𝑒1𝑅𝑒2𝑒−2𝐾𝑒𝑑) (8.16d)
Correspondingly, the final expression of Eq. (8.8) is derived as
𝑆ℎ𝑒(∆𝜔) =1
2𝜋𝐴∗(𝜔1)𝐴(𝜔2) {−
𝐵∗(𝜔1)𝐵(𝜔2)[𝑒−𝑑[𝐾𝑒
∗(𝜔1)+𝐾𝑒(𝜔2)]−1]
𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)
+
𝐵∗(𝜔1)𝐶(𝜔2)[𝑒
−𝑑[𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)]−𝑒−2𝑑𝐾𝑒(𝜔2)]
−𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)
+𝐵∗(𝜔1)𝐷(𝜔2)[𝑒
−𝑑[𝐾𝑒∗(𝜔1)+𝛽]−1]
𝐾𝑒∗(𝜔1)+𝛽
+
𝐶∗(𝜔1)𝐵(𝜔2)[𝑒−𝑑[𝐾𝑒
∗(𝜔1)+𝐾𝑒(𝜔2)]−𝑒−2𝑑𝐾𝑒∗(𝜔1)]
𝐾𝑒∗(𝜔1)−𝐾𝑒(𝜔2)
+
𝐶∗(𝜔1)𝐶(𝜔2)[𝑒−𝑑[𝐾𝑒
∗(𝜔1)+𝐾𝑒(𝜔2)]−𝑒−2𝑑[𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)]]
𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)
−
CHAPTER 8. 197
𝐶∗(𝜔1)𝐷(𝜔2)[𝑒−𝑑[𝐾𝑒
∗(𝜔1)+𝛽]−𝑒−2𝑑𝐾𝑒∗(𝜔1)]
𝐾𝑒∗(𝜔1)−𝛽
+𝐷∗(𝜔1)𝐵(𝜔2)[𝑒
−𝑑[𝛽+𝐾𝑒(𝜔2)]−1]
𝛽+𝐾𝑒(𝜔2)−
𝐷∗(𝜔1)𝐶(𝜔2)[𝑒−𝑑[𝛽+𝐾𝑒(𝜔2)]−𝑒−2𝑑𝐾𝑒(𝜔2)]
−𝛽+𝐾𝑒(𝜔2)−𝐷∗(𝜔1)𝐷(𝜔2)[𝑒
−2𝛽𝑑−1]
2𝛽} (8.17)
where 𝐴∗, 𝐵∗, 𝐶∗, and 𝐷∗are the complex conjugates of A, B, C, and D, respectively, in Eq.
(8.16).
8.4 Quantitative Colloidal Quantum Dot Solar Cell Imaging
8.4.1 Device Fabrication and Characterization Details
Oleic-acid-capped CQDs and ZnO nanoparticles were synthesized following the previously
published method as discussed in Sect.4.3.1 and [31]. As shown in Fig.5.1 and 8.1, the CQD
solar cells have a sandwich structure of PbS CQDs surface-capped with two different ligands,
i.e. PbX2/XX (PbX2: lead halide, AA: ammonium acetate) and EDT (1, 2-ethanedithiol). The
oxygen plasma etching of CQD solar cell surface to add additional surface trap states for
further carrier lifetime study was performed through our plasma etch system (PE-50) at room
temperature with an oxygen flow rate of 10 cc/min in a vacuum environment at a pressure of
100 mTorr. J-V characteristics were obtained using a Keithley 2400 source measuring
instrument under simulated AM1.5 illumination in a nitrogen environment. The experimental
setups for camera-based HoLIC and HeLIC, as well as for single detector based PCR, are
identical to that discussed in Chapter 3.
8.4.2 Quantitative HeLIC Imaging of Carrier Transport in CQD
Solar Cells
Figure 8.2 (a) is a photograph of the as-synthesized multilayer CQD solar cell sample
(structure shown in Figs. 5.1 and 8.1) with metallic Au contacts on the top. The bottom Au
CHAPTER 8. 198
contact so labeled in Fig. 8.2 (a) is connected to an indium tin oxide (ITO) conducting layer.
Each golden circle-rectangle-shaped region, for example, the one circumscribed in a dashed
rectangle, represents a complete solar cell structure with top Au contact, while other regions
with brown color are without top contact Au deposition. The highest PCE of this type of solar
cell has been certified to be as high as 11.28% [31], while device performances vary from
sample to sample due to factors such as electrical and mechanical defects introduced during
various device fabrication processes, as well as film-inhomogeneity-associated short-circuit
effects. As shown in Fig. 8.2 (b), a HoLIC image of the corresponding solar cell sample in Fig.
8.2 (a) reveals the charge carrier distribution within an entire solar device, thereby elucidating
the influence of the Au electrode on charge carrier transport, and depicting the CQD solar cell
inhomogeneity that originates from various defects. The solar cell contact electrode A
circumscribed within the dashed rectangle in Figs. 8.2 (a) and (b) has a PCE of ca. 9%, Fig.8.3,
and has been further investigated using HeLIC and PCR.
Figure 8.2: (a) A photograph of the CQD solar cell sample under study, and (b) the
corresponding HoLIC image of this solar cell. The dashed-rectangle-circumscribed solar cell
A is selected for further studies as shown in Figs. 8.3 and 8.4. The HoLIC characterization was
carried out at 10 Hz. It should be noted that for carrierographic imaging, the sample was flipped
over with the top Au contact on the bottom, resulting in mirror image positions being assumed
in (a) and (b) by the dashed rectangles and inscribed solar cells.
CHAPTER 8. 199
Figure 8.3: Current-density-voltage characteristic of the CQD solar cell shown circumscribed
by a dashed rectangle in Fig.8.2(a).
For the investigation of contact effects on carrier lifetime, single-detector based PCR as
discussed in Chapter 7 was used to study the carrier lifetimes within regions with and without
Au contact deposition and to compare with the HeLIC imaging methodology. The PCR single-
element InGaAs detector can detect a spot area equal to the size of the circular tip of a solar
cell electrode unit (e.g. A and B in Figs. 8.2) and can measure the average carrier lifetime in a
spot region. The large phase lag in PCR phase with increased frequency indicates longer carrier
lifetimes. Therefore, as directly shown in Fig.8.4, according to the larger phase lag at high
frequencies, area C without contact Au deposition exhibits a longer carrier lifetime than its
counterparts in regions A and B, Figs. 8.2. Similarly, region A presents a slightly longer
lifetime than B. The lifetime difference between regions A, B, and the area C is manifested by
the quantitative fitting of experimental frequency-dependent PCR phases to Eq.8.15 and are
consistent with the results from the HeLIC high-frequency imaging method.
CHAPTER 8. 200
Figure 8.4: Frequency-dependent PCR phase spectra of the solar cell electrode units A, B, and
area C without Au contact (Figs. 8.2) at 200 K. Equation (8.15) is used for the best fitting of
each curve. The characterization spot area of the single-detector based PCR is the same as the
area of the circular Au contact tip.
Figure 8.5: High-frequency HeLIC images at 1 kHz (a) and 100 kHz (b) for the CQD solar cell
shown in Fig. 8.2.
The frequency-dependent AC diffusion length enables the characterization of photovoltaic
device properties at different depths using HeLIC. A comparison of HeLIC images at 1 kHz
and 100 kHz is presented in Fig. 8.5, which qualitatively reveals depth-resolved HeLIC images
through the evolution of CQD solar cell image patterns with increased frequency. In other
words, at the low frequency of 1 kHz, the longer AC diffusion length is expected to yield
CHAPTER 8. 201
different image contrast from HeLIC images at the higher frequency, 100 kHz, which
corresponds to a shorter AC diffusion length. Importantly, the HeLIC images reveal the
inhomogeneity of the CQD solar cell and the quality of each CQD solar cell unit.
Figure 8.6: The frequency-dependent average HeLIC image amplitudes of the CQD solar cell
shown in Fig. 8.2. The HeLIC images in Fig.8.5 are also included.
In comparison, as a large-area imaging technique, HeLIC can image an entire solar cell
sample. The dependence of HeLIC amplitudes on modulation frequency is shown in Fig. 8.6
in which each point is the average amplitude of an entire HeLIC image at the corresponding
frequency. The best fitting of the data in Fig. 8.6 into Eq.8.17 yields the overall carrier transport
parameters for the CQD solar cell measured at 200 K. For the best-fitting, as tabulated in Table
8.1, other parameters involved in Eq. (8.17) were taken from the literature or measured
experimentally. Specifically, carrier lifetime τ = 2.98 ± 0.06 µs, diffusivity De = 3.60 × 10-5 ±
4.00 × 10-6 cm2/s, diffusion length Ldiff = 99.10 ± 5.42 nm, and drift length Ldrif = 47.01 ± 6.04
nm. Compared with their room temperature (293 K) counterparts, except for τ, other carrier
CHAPTER 8. 202
transport parameters are smaller at 200 K: at 293 K, De is on the order of 10-3 cm2/s, and Ldiff
and Ldiff are around 400 nm. The carrier lifetime τ decreases to around 500 ns when the
temperature increases from 200 K to 293 K, a phenomenon attributed to increased nonradiative
recombination and consistent with phonon-assisted carrier hopping transport within spatial and
energy disordered CQD systems [11, 12, 14, 32].
Table 8.1: Summary of the parameters used for heterodyne lock-in carrierography best-fits to
Eq. (8.17).
Parameters E (Vcm-1) Sj (cm/s) 𝜂 𝐼0(Js-1cm-2) 𝛽(cm-1) d (cm)
Parameter
values used for
fitting
1.2×104
0 1 0.1 (1 sun) 107 360×10-7
References and
experimentally
obtained
parameters
𝑉𝑜𝑐𝑑𝑠𝑜𝑙𝑎𝑟 𝑐𝑒𝑙𝑙
ideal
situation
ideal
situation
experimentall
y measured
[27] [31]
The carrier transport parameter images for CQD solar cell region E, Fig. 8.7, were
reconstructed using the same method mentioned above. These HeLIC images were taken at
various frequencies between 400 Hz and 270 kHz. Therefore, the carrier lifetime image of
electrode E at 200 K was constructed as shown in Fig. 8.7(b). Regions with Au contacts exhibit
shorter τ of ca. 2.3 µs than the surrounding regions. This can be ascribed to the enhanced
interface-induced defects and traps that decrease carrier lifetime through increased non-
radiative recombination. This finding is consistent with the PCR phase study of carrier
lifetimes in regions with and without Au coating as shown in Fig. 8.4. For comparison, at 293
K the carrier lifetime τ image of the same electrode also yielded a shorter τ of ca. 0.5 µs in Au
regions than the lifetime outside the Au/CQD interfaces, Fig. 8.7(b). Comparison between the
lifetime images at 293 K and 200 K revealed that the increased carrier lifetime at the low
CHAPTER 8. 203
temperature is due to the reduced carrier-phonon interactions which act as necessary phonon-
mediation pathways for trap state related non-radiative recombination [9, 11, 32].
Figure 8.7: (a) 400 Hz HeLIC image of the CQD solar cell region E, Fig. 8.2, and its carrier
lifetime τ image (b) at 200 K. (c) For comparison, the carrier lifetime image of the same
electrode E at 293 K. (d)-(f) are images of carrier diffusivity, diffusion, and drift lengths,
respectively, at temperature 200 K.
The measured carrier lifetimes from HeLIC images are comparable to those measured from
transient photovoltage for PbS CQD solar cells (3-6 μs) [33], impedance spectroscopy for PbS
CQD thin films (3 μs) [34], time-resolved PL spectroscopy for PbS-capped CQDs (1 μs) and
for CdSe nanocrystals (0.88 μs) [35], and from absorbance spectroscopy for PbS CQDs (1-1.8
CHAPTER 8. 204
μs) [36]. Additionally, for a similar type of solar cells to ours except for PbS-PbX2/AA (Fig.5.1)
replacement by PbS-TBAI (tetrabutylammonium iodide surface passivated PbS CQDs), Wang
et al. [37] reported a carrier lifetime of ca. 0.5 μs using impedance spectroscopy. Compared
with these literature carrier lifetime data, the slightly lower carrier lifetime measured in this
paper may be attributed to the different types of samples, also it should be noted that the CQD
solar cells characterized in this study are not of our highest efficiency which is expected to
have higher carrier lifetimes. In addition, the carrier lifetimes characterized by various
transient methodologies were obtained through fitting the time-dependent PL (or other
electrical parameters such as photovoltage) decay spectrum to a simplified exponential decay
model which is also commonly used for lifetime extraction for thin films. It is apparent that
the complexity of carrier transport behavior via various pathways in CQD solar cell devices
was ignored, inevitably leading to deviations from the actual carrier lifetime in solar cell
devices.
Furthermore, for electrode E (Fig. 8.2) at 200 K, carrier diffusivity, and diffusion and drift
lengths were also obtained as shown in Figs. 8.7(d)-(f). Specifically, the carrier diffusivity, Fig.
8.7(d), was imaged to be on the order of 10-5 cm2/s which is much smaller than its room
temperature counterpart (ca. 10-3 cm2/s). Fig. 8.7(d) also shows the effects of the Au/CQD
interfaces on the carrier diffusivity with a lower average De in the Au region. Interface-induced
trap states can trap, de-trap, or recombine carriers, a process that inhibits carrier hopping
diffusion transport. Therefore, with the extraction of τ and De, Ldiff = De was also reconstructed
to be approx. 120 nm, which is much shorter than ca. 400 nm at room temperature. The reduced
Ldiff at low temperature is attributed to the decreased availability of thermal energy for the
phonon-assisted carrier hopping transport within the CQD thin film [12, 14, 32]. With the
CHAPTER 8. 205
presence of interface trap states or defects, carriers in pure CQD layers can be transported
about 30 nm longer than those in Au regions, hopping across ~ 10 more QDs. As shown in Fig.
8.7(f), the effects of interface traps are also substantiated through carrier drift length Ldrif
images using HeLIC, i.e., lower carrier drift lengths of ca. 50 nm in Au regions are obtained
than in other regions.
8.5 Further HeLIC Carrier Lifetime Imaging of CQD Solar Cells
8.5.1 HeLIC Imaging at Various Frequencies
Depth-selective/resolved high-frequency (1 kHz-100 kHz) HeLIC images of CQD solar
cells with a structure shown in Fig.5.1 without Au electrode deposition were obtained as shown
in Fig.8.8. For comparison, HeLIC images taken at different frequencies are presented with all
ranges (the differences between two neighboring values on the color scales of Fig. 8.8) equal
to one-fifth of the difference between the maximum and minimum pixel amplitudes of each
image. Therefore, similar to Fig.8.5, it was found that with increasing modulation frequency
in the HeLIC images the low amplitude pattern A fades while the high amplitude pattern B
spreads out. Different HeLIC images at various modulation frequencies can exhibit different
image contrast emerging from electronic property variations with depth. Another phenomenon
that should be noted is that HeLIC image amplitudes decrease with the modulation frequency,
therefore, degrading the image quality as shown in the Fig.8.9 the HeLIC image at 270 kHz
for the same solar cell in Fig. 8.8. To obtain an optimized contrast, the image range of 270 kHz
image is not set in the same way for images in Fig. 8.8 because of the low-amplitude of 270
kHz image. Hence, the contrast of Fig. 8.9 should not be compared with those in Fig. 8.8.
Despite the low amplitude, benefiting from the high signal-to-noise ratio of HeLIC as
CHAPTER 8. 206
discussed in Sect.3.4, 270 kHz image can still be used for carrier transport parameter imaging
construction as shown in Fig.8.10.
Figure 8.8: HeLIC images of a CQD solar cell at different modulation frequencies 1 kHz (a),
10 kHz (b), 50 kHz (c), and 100 kHz (d) as labeled.
Figure 8.9: 270 kHz HeLIC images of the same CQD solar cell shown in Fig. 8.8.
CHAPTER 8. 207
8.5.2 HeLIC Lifetime Imaging of CQD Solar Cells with/without
Plasma Etching
103
104
105
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24 (a)
Region 3
Region 4
HeL
IC (
mV
)
Frequency (Hz)
: 0.56 s
: 0.63 s
0.3 0.4 0.5 0.6 0.7
0.0
0.2
0.4
0.6
0.8
1.0(d)
Lifetime (s)
With 15s plasma etching Without plasma etchingN
orm
aliz
ed C
ou
nts
Figure 8.10: (a) Frequency-dependent HeLIC image average amplitude for regions 3 and 4 of
the CQD solar cell shown in Fig. 8.8(a) without plasma etching; and (b) lifetime imaging of
the same CQD solar cell. (c) Furthermore, bar-plotted lifetime statistical distribution for the
above CQD solar cell without plasma etching and another CQD solar cell of the same type
except with 15 s plasma etching.
The mechanisms of carrier generation, hopping transport, and recombination under
frequency-modulated excitation for CQD solar cells have been developed as Eq. (8.17).
Following the same procedure, the lifetimes of CQD solar cells with and without 15 s plasma
etching were obtained, Fig. 8.10. Furthermore, the average lifetimes in regions 3 and 4,
rectangle-circumscribed in Fig. 8.8(a), were best-fitted to Eq. (8.17) through taking the average
CHAPTER 8. 208
HeLIC amplitudes of pixels in each region at each frequency (ranging from 1 kHz to 270 kHz).
As shown in Fig. 8.10(a), frequency-dependent HeLIC amplitudes and their best-fits yielded
average lifetime measurements for regions 3 and 4 equal to 0.56 μs and 0.63 μs, respectively.
The slightly higher value in region 4 is in agreement with the HeLIC lifetime image as shown
in Fig. 8.10(b) for the same CQD solar cell through best-fitting of 27 HeLIC images to Eq.
(8.17) including those shown in Figs. 8.8 and 8.9. It is interesting to see that the lifetime image
in Fig. 8.10(b) resembles the HeLIC images in Fig. 8.8. The dependence of HeLIC amplitude
on modulation frequency can directly reflect carrier lifetime through a simplified model fk~
1/(2πτ), in which fk is the knee frequency at which the amplitude starts to drop as shown in Fig.
8.10(a).
Furthermore, oxygen plasma dry etching created more trap states in our CQD solar cells,
paving an additional way to investigate the influence of defects and trap states on carrier
lifetimes and to demonstrate the validity of this HeLIC methodology. Specifically, oxygen gas
plasma is efficient in breaking chemical bonds of PbS CQDs and the ligands, creating extra
dangling chemical bonds on the surface of quantum dots and the ligand agents. Moreover, the
oxygen species including ionized oxygen can be added into CQDs acting as oxygen interstitial-
associated defects. These effects on the CQD solar cell from oxygen plasma dry etching are
expected to reduce carrier lifetime through enhanced non-radiative recombination induced by
the increased material trap states and defects. The lifetime distribution image of an oxygen
plasma etched CQD solar cell is shown Fig. 8.10(c), and is plotted as a histogram in Fig.
8.10(d) along with the carrier lifetime distribution of the intact CQD solar cell. The two
dominant lifetime peak distributions as shown in Fig. 8.10(d) for both CQD solar cells with
and without plasma etching may originate from the nature of the sample itself as regions with
CHAPTER 8. 209
distinct image patterns were deliberately chosen for characterization. This pattern of CQD
solar cells is in agreement with the HeLIC amplitude image contrast (Fig. 8.8), where two
different prominent regions can be observed. The difference in FWHM of each lifetime
distribution can be attributed to the homogenization nature of the plasma-etched CQD solar
cell or the plasma etching effects as manifested by the significant narrowing of the FWHM
after plasma etching.
8.5.3 HeLIC Lifetime Imaging for Interface Effects on CQD Solar
Cells
Furthermore, to study the influence of surface recombination velocity on CQD solar cell
performance, CQD/electrode interface effects were also investigated. Fig. 8.2 (a) is a
photograph of the CQD solar cell (structure shown in Fig.5.1 and 8.1) under study with the Au
contacts on top. In addition to J-V characteristics of the solar cell unit A as shown in Fig.8.3,
CQD solar cell unit B was also characterized with a PCE as high as 8.38 %. Using the same
lifetime extraction methodology, the lifetime images of two adjacent CQD solar cells were
obtained as shown Fig. 8.11 (a) and the histogram of the imaged lifetimes is presented in Fig.
8.11 (b). Apparently, regions with Au electrodes exhibit slightly shorter lifetime that can be
ascribed to increased trap state densities formed at the Au/PbS-EDT interfaces, which act as
nonradiative recombination centers and compromise carrier hopping lifetimes. As shown in
Fig. 8.11 (a), the CQD solar cell region A corresponds to region A in Fig. 8.2 which has a high
PCE value of ca. 9.0 %, Fig.8.3. In comparison, the J-V characteristic demonstrated a poor
performance of region D in Fig. 8.2 with a very low efficiency less than 1%. These results are
in agreement with the lifetime images in Fig. 8.11 (a) that region A is more homogeneous than
region D with fewer fabrication-associated defects.
CHAPTER 8. 210
Figure 8.11: (a) Lifetime image of two adjacent CQD solar cell units A and D which reveals
the device homogeneity and influence of electrode contacts on carrier hopping transport, and
(b) the barplot of carrier hopping lifetime image in (a). It should be noted that the top Au
contacts as shown in Fig. 8.2 (a) were on the bottom through flipping the sample over for all
the HeLIC imaging.
8.6 Conclusions
This chapter introduces large-area imaging techniques HoLIC and HeLIC for CQD solar
cell imaging. The Instrumentation and signal processing principles including sampling,
undersampling, and heterodyne are reviewed with details. The nonlinear response of PL
emission to laser excitation intensity which is a prerequisite for HeLIC was further investigated
experimentally. The theories for HoLIC and HeLIC were discussed with great efforts.
Combined with the as-developed carrier transport model under dynamic excitation, the
mathematical expressions for HoLIC, HeLIC, and PCR were derived. HeLIC as a large-area
imaging technique that can perform ultra-high frequency (270 kHz for the as-studied CQD
solar cells) is the state-of-the-art advanced imaging technique for photovoltaics to generate
qualitative imaging of carrier transport parameters that are essential parameters for CQD solar
cell efficiency optimization. The combination of HoLIC, HeLIC, and PCR as emerging
dynamic quantitative interface non-destructive imaging methodologies shows great potential
CHAPTER 8. 211
for fundamental photovoltaic optoelectronic transport studies and industrial in-line or off-line
manufactured solar cell device characterization.
Specifically, it was demonstrated that, compared with regions without Au contact,
enhanced trap state density at the Au/CQD interface results in lower minority carrier lifetime
(ca. 0.5 μs and 2.3 μs at 293 K and 200 K, respectively, in agreement with literature transient
photovoltaic results). The dependence of HeLIC images on modulation frequencies manifests
the potent applications of HeLIC for probing solar cell surface and sub-surface (including p-n
junctions) properties. In addition to the elucidation of large-area defect-related device
homogeneity that originates from various fabrication stages, the carrier hopping lifetime
imaging using HeLIC shows the dominant effective carrier lifetime of ca. 0.60 μs, which
reduces to ca. 0.36 μs after 15 s plasma etching that created more surface trap states.
212
Chapter 9
Conclusions and Outlook
9.1 Conclusions
The present research work adds to the versatility of carrier transport dynamics and current-
voltage mechanisms in CQD systems, to the ultrahigh-frequency testing of carrier transport,
and to the quantitative large-area ultrahigh-frequency imaging via miscellaneous theoretical,
conceptual, and experimental advances. All these scientific and engineering efforts contribute
to reaching the final goal of CQD solar cell efficiency optimization.
9.1.1 Advances in Carrier Transport and J-V Mechanisms of
CQD Systems
Due to the significant electronic, electrical, and optical difference between CQD systems
(low carrier mobility and discrete energy band system) and conventional inorganic Si systems
(high carrier mobility and continuous energy band system), this thesis revisited the
conventional Si solar cell working principles and introduced carrier discrete hopping transport
in CQD systems. Based on the nature of hopping transport behavior, novel carrier drift-
diffusion current-voltage models were developed for different CQD solar cell working
conditions. Therefore, the phonon-assisted carrier hopping transport property was
demonstrated from temperature-dependent CQD solar cell current-voltage characteristics, the
study of which led to the quantitative transport mechanism derivations of the double-diode
mechanism, imbalanced carrier mobilities, and Schottky barrier effects. These novel
mechanisms interpret the S-shaped current-voltage characteristics which have been found to
CHAPTER 9. 213
deteriorate CQD solar cell efficiency considerably. Meanwhile, open-circuit voltage deficit
was further quantitatively studied with the finding of the existence of charge transfer states at
p-n junction interfaces that were quantitatively modeled and ascribed to the high radiative and
nonradiative recombination probability at interfaces. Furthermore, carrier recombination
processes in CQD thin films was also quantitatively described, which led to a complete CQD
energy band structure. Moreover, rather than being constant, voltage- and mobility-dependent
photocurrent was found experimentally in high-efficiency CQD solar cells and was modeled
quantitatively, which demonstrated the invalidity of constant photocurrent assumption. The
study of CQD energy bandgap and carrier mobility demonstrated optimized energy bandgaps
and mobilities for higher solar cell efficiency.
9.1.2 Advances in Ultrahigh-frequency Diagnostics of Carrier
Transport
Carrier transport dynamics in photovoltaic solar cells are essential for fundamental
physical understating of energy transport and loss in solar cells, and for solar cell carrier
efficiency optimization. Based on the understanding of carrier transport mechanisms and
working principles of CQD solar cells. This thesis introduces a trap-state-mediated carrier
transport model for single detector based PCR high-frequency characterization of carrier
lifetime, hopping diffusivity, mobility, and diffusion length in CQD systems. The highest
frequency applied is 1 MHz. However, higher frequency can be achieved depending on
experimental Instrumentation.
High-frequency PCR characterization of carrier transport parameters further demonstrated
the phonon-assisted carrier hopping transport in CQD systems. The carrier transport
dependencies on dot size, ligand, temperature, and hopping activation energy were
CHAPTER 9. 214
investigated. It was found that large CQDs tend to have fewer trap states, while perovskite-
passivated CQDs exhibit long lifetime and high diffusivity due to a low trap state density.
These transport behavior are reflected by their trap-state-mediated hopping transport activation
energies.
9.1.3 Advances in Quantitative Large-area Ultrahigh-frequency
Imaging
Traditional small-spot testing techniques arise doubtful overall photovoltaic materials and
device quality and stability estimation. This thesis developed analytical methodologies for
HoLIC and HeLIC, which are all-optical, large-area, fast, and non-destructive imaging
techniques. Overcoming the limitations of low frame rate and long exposure time even for the
state-of-the-art IR cameras, ultrahigh-frequency imaging of CQD solar cells was achieved for
the first time through heterodyne technologies and an excess carrier-diffusion-wave model.
Therefore, large-area imaging of CQD solar cells at high modulation frequencies can be
obtained now, and the quantitative imaging of carrier transport parameters including carrier
lifetime, diffusivity, and diffusion and drift lengths was obtained through the development of
a novel HeLIC signal generation methodology. The acquired high-frequency images and the
carrier transport parameter imaging are essential for CQD solar cell homogeneity and quality
estimation, for fundamental physical carrier transport study, and for studying CQD/contact
interface influence. Due to the enhanced trap states at CQD/contact interfaces, carrier lifetime
was found to reduce.
Qualitative HoLIC large-area imaging was also achieved for CQD thin films and solar cells.
HoLIC can estimate large-area device homogeneity and quality, which reveals a preliminary
CHAPTER 9. 215
assessment of the carrier lifetime, photocarrier collection efficiency, and output power of an
entire photovoltaic device.
9.2 Outlook
This thesis has shown the advances in high-efficiency CQD solar cells, novel CQD solar
cell working principles, and high-frequency, large-area, and nondestructive characterization
techniques. For future investigation in this area, the following fields are of great interest and
necessity.
Further study of the energy band evolution from localized energy states to extended
energy bands with the increase of interdot coupling strength. This needs to surface
passivate CQDs with short exchange ligands to achieve closely packed CQD
ensembles. This study can be conducted through theoretical carrier transport study and
PCR and HeLIC carrier transport parameter characterization.
Study of hot carriers in CQD materials and devices. Develop or adopt PCR system
through combining it with photothermal techniques for the testing of radiative and
nonradiative (thermal) emission of hot carriers. The applications of these hot carriers
can be very effective in increasing both current density and open-circuit voltage.
In this thesis, the lifetime discussed and measured is identified as effective lifetime,
however, for future investigation, if necessary, different lifetimes including bulk
lifetime and surface lifetime should be distinguished from further development of
theoretical carrier transport models.
To realize complete depth-selective/resolved imaging of HeLIC, a wider modulation
frequency range can be tried for the precise investigation of p-n junction or interface
properties that are essential for high-efficiency CQD solar cell optimization.
CHAPTER 9. 216
All the as-developed theoretical models can be helpful as references for solar cells of
other types. Therefore, using these mechanisms and models with the proper adoption
of PCR and HeLIC signal generation models for other types of solar cells is also a
promising research field.
217
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