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TRANSPORT AND RECOMBINATION

MECHANISMS IN NANOCRYSTALLINESEMICONDUCTORS

THESISSUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY(PHYSICS)

BY

JASA RAM

under the supervision of

Prof. S.R. DHARIWAL

DEPARTMENT OF PHYSICS, JAI NARAIN VYAS UNIVERSITY

JODHPUR 342 005 INDIA2006


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Prof. S.R. Dhariwal Phone: +91-0291-2722543E-mail: [email protected]

, 2006

CERTIFICATE

This is to certify that the thesis entitled TRANSPORT AND

RECOMBINATION MECHANISMS IN NANOCRYSTALLINE

SEMICONDUCTORS is hereby submitted in full requirement for the

degree of Doctor of Philosophy (Physics) to the Jai Narain Vyas

University, Jodhpur (India). It is a record of original investigation carried

out by Mr. Jasa Ram, Scientist C, DRDO, Defence Laboratory,

Jodhpur during the period November 2000 to July 2006 under my

supervision and guidance and he has fulfilled the conditions laid down by

Jai Narain Vyas University, Jodhpur for submission of Ph.D. thesis.

The work presented in the thesis has not been submitted for any

other degree or diploma anywhere else in India or abroad.

Dr. S.R. Dhariwal

Research Supervisor

Head,

Department of Physics

J.N.V. University, Jodhpur


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TO MY PARENTS

For their faith in my education


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Acknowledgement

I express my deep sense of gratitude, profound regard and indebtness to my erudite

research supervisor Prof. S.R. Dhariwal, former Head, Department of Physics and former

Dean, Faculty of Science, J.N.V. University, Jodhpur under whose able supervision and expert

guidance, this thesis work has been carried out. His purposefulness, in-depth involvement,

passionate interest, constructive criticism, scholarly advice and constant encouragement have

abundantly helped me in this endeavor.

I avail this opportunity to express my sincere, humble and reverential gratitude to

Dr. M.P. Chacharkar, Director, Defence Laboratory, Jodhpur for resolute inspiration and keen

interest in the present work. Grateful thanks are also due to former Directors of Defence Lab

Dr. Ram Gopal and Shri R.K. Syal for their kind gesture of according permission to carry out

the work. Nevertheless, I acknowledge thanks to DRDO for providing the genial opportunity to

continue this research work.

I profoundly thanks to Shri P.K. Bhatnagar, Group Director and Dr. N. Kumar,

Additional Director of Defence Lab for their valuable suggestions, encouragements and good

wishes.

I extend my sincere thanks to Shri G.L. Baheti, Jt. Director, Head, NRMA Division of

Defence Lab for his support, suggestions and fruitful discussions. I am also thankful to

Shri D.K. Tripathi, Shri Nisheet Saxena, Mr. Ravindra, Shri K.C. Songra, Shri Gumana

Ram, Shri L.R. Meghwal, Shri V.L. Meena for their encouragement.

I hearty express my sincere gratitude to Prof. B.S. Bhandari, Dr. V.N. Ojha and

Shri Tulja Shanker Shrimali for all the cooperation and moral support during the progress of

the work.

I express grateful thanks to Madam Dr. (Mrs.) Prem Lata Dhariwal for her gratifying

compassion in sparing sirs time for this research work during holidays and even before & after

office hours.

I record my grateful thanks to Dr. R.P. Tripathi, Head, Department of Physics and

former Heads Prof. R.K. Gupta and Prof. S.R. Dhariwal for their support during the research


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work. Thanks are also due to Dr. J.K. Sharma, Dr. D.K. Sharma, Dr. G.D. Sharma,

Dr. R.J. Sengwa, Dr. S.K. Sharma, Dr. K.R. Patel and all professors of the department for

valuable technical discussions.

I desire to express grateful thanks to Dr. S.R. Vadera, Jt. Director, Defence Lab for hiskind cooperation and suggestions. Grateful thanks are also due to Dr. Desh Raj, Dr. Rashi

Mathur, Dr. K. Manzoor, Mr. Manoj Patra, Ms. Aditya, Mr. Amit Dave, Shri Hanuman

Singh, Dr. S.C. Negi, Mr. Chhagan, Shri Sukha Ram, Shri Babu Lal for all their

encouragements.

My special thanks go to colleagues Mr. Uma Shanker Mirdha and Mrs. Manu Smrity

for valuable suggestions, consistent encouragements and help in computer programming. I

appreciate the help of Mr. Dinesh Kabra and Jagdish in the literature collection. I also owe to

thank to Ashok Soni, Avdesh, Vijay Singh, Sailesh, Sailendra, Raj Kumar, S. Rajvanshi,

Neeraj and all scholars of Physics Department for all their encouragement.

Encouragements of Manak, Sidharth, Suresh, Naveen, Anand, Manish, Monu,

Abhisekh and all other friends are gratefully acknowledged.

Thanks are also due to my sister Gayatri for endeavoring help.

In the last but not the least, I offer my sincere thanks to my wife Sarita for her

endurance and patience during the course of this work. Affection for my little son Tanish, who

added a new dimension of joyfulness during concluding period of this work.

Thanks are also due to many others whose names the paper cannot accommodate but

who lives in my memory never to be forgotten.

(Jasa Ram)


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TABLE OF CONTENTS

TITLE PAGE... i

CERTIFICATE.... ii

DEDICATION..... iii

ACKNOWLEDGEMENT.. i

TABLE OF CONTENTS

LIST OF PUBLICATIONS i

ABBREVIATIONS AND SYMBOLS ii

Chapter 1 1

INTRODUCTION

1.1 : An Overview 1

1.2 : Mechanism of electron transport 4

between QWs and eigenstates of

ADQW system

1.3 : QD as a tunnel device for coherent 12

transport of electrons

1.4 : Role of nonradiative recombinations in 17

nanosized semiconductors

1.5 : Outline of the present work 20


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Chapter 2 23

ANALYTICAL EXPRESSIONS FOR

ASYMMETRIC DOUBLE QUANTUM WELLS

AND THEIR APPLICATION TO

SEMICONDUCTOR HETEROSTRUCTURES

2.1 : Introduction 23

2.2 : Transcendental Equation for ADQW 25

System

2.3 : Applications to Semiconductor 31

Heterostructures (GaAs / AlXGa1-XAs)

2.3.1 : Transport optimization using variation in 34

QW width


confining potential


material composition

2.4 : Conclusions 41

Chapter 3 44

ON THE POSSIBILITY OF DESIGNING

INTERACTING SEMICONDUCTOR QUANTUM

WELLS WITH ZERO PERTURBATION

COUPLING


3.2 : Determination of zero 45

perturbation position


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3.3 : Universality of zero perturbation 49

position

3.4 : Coupling at zero perturbation 51


Chapter 4 55

SINGLE ELECTRON CHARGING OF

QUANTUM WELLS AND QUANTUM DOTS4.1 : Introduction 55

4.2 : Mechanism of charge transfer between 59

QWs


Chapter 5 71

ELECTRON TRANSPORT THROUGH DOUBLE

BARRIER QUANTUM WELL AND

EXPLANATION FOR UNIVERSAL PHASE

BEHAVIOR5.1 : Introduction 71

5.2 : Transmission coefficient in 78

terms of coupling coefficients


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5.3 : Separation of coherent and 82

sequential tunneling coefficients and

explanation of universal phase

behavior in coherent transport


Chapter 6 91

NONRADIATIVE RECOMBINATIONS IN

NANO-SIZED SEMICONDUCTORS


6.2 : Calculation of Surface 92

recombination velocity

6.2.1 : Semi-infinite Geometry 95

6.2.2 : Semi-conductor Slab of finite 96

Thickness

6.2.3 : Cylindrical grain 103

6.3 : Discussion & Conclusions 105

Chapter 7 110

SUMMARY & CONCLUSIONS

REFERENCES 116

REPRINTS OF PUBLICATIONS


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i

LIST OF PUBLICATIONS

1. Jasa Ram and S.R. Dhariwal, Analytical expressions for asymmetricdouble quantum wells and their application to semiconductor

heterostructures, Phil. Mag. in press (2006)

2. S.R. Dhariwal and Jasa Ram, On the possibility of designinginteracting semiconductor quantum wells with zero perturbation

coupling, communicated

3. S.R. Dhariwal and Jasa Ram, Mechanism of charge transport acrosssingle electron states of coupled quantum wells and quantum dots,

communicated

4. S.R. Dhariwal and Jasa Ram, Universal phase evolution in coherenttransport of electrons through a quantum dot communicated

5. S.R. Dhariwal and Jasa Ram, Suppression of nonradiativerecombination in small size semiconductors, Physica B 363, 69-75

(2005)


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ii

ABBREVIATIONS AND SYMBOLS

QWs Quantum well(s)

QDs Quantum dot(s)

QPC Quantum point contact

ADQW Asymmetric double quantum well

SDQW Symmetric double quantum well

DQD Double quantum dot

QBS Quasi bound state

DBQW Double barrier quantum well

V1, V2 Confining potentials

L, L1, L2 Quantum well widths

m, m1, m2 Electron effective masses in quantum wells

mb Electron effective mass in the barrier

k1, k2 Wave vectors

, 1, 2 Attenuation constant in the barrier

E Energy eigenvalue

p, q Ratios of effective masses

b, a1, a2 Barrier width


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ii

2, 4 Fractional occupancies in the quantum wells

t Transmission coefficient

Ef

Fermi Energy

Width of resonance

U Gate potential

C Quantum well coupling matrix

Q Inverse quantum well coupling matrix

T1, T2 Barrier transmission matrices

Decay time for quasi bound state

tC Coherent tunneling coefficient

tS Sequential tunneling coefficient

SR Surface recombination velocity

fa Sticking probability

Cluster size parameter


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Chapter 1

Introduction

1.1 An Overview

Confinement of electrons in nanosized structures known as quantum

confinement has revolutionized the science in many streams [1]. In

semiconductors, many new phenomena seem to emerge at this scale, of

which, most noteworthy are enhancement of band gap, increase in optical

efficiency, anomalous transport properties, varied dielectric properties etc.

For investigation of quantum confinement associated effects, the basic unit

is a quantum well (QW). The recent interest in electron transport between

QWs [2-8] and consequences of coupling upon eigenstates of coupled QWs

[9-12] vis--vis charging of individual QWs [2, 5, 6, 13-17] have put on

demand the necessity of quantitative understanding of such systems. Such

approach will also form the basis of understanding the mechanism

responsible for a large number of applications of tunnel coupling of QWs

like those in intersubband lasers [11, 18-24], qubits used in quantum


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Chapter 1: Introduction

computations [2, 8], memories [25], study of artificial molecules [26, 27],

tetra hertz emission and absorption devices [28-32] etc. Also, the behavior of

eigenstates in coupled QWs plays crucial role in designing of quantum point

contact (QPC) [33] for transport of electron through a quantum dot (QD)

[34]. Furthermore, this may help in understanding the fundamentals of

coherent transport of electron through a QD [35, 36], particularly the recent

reports of anomalous universal phase evolution [37-39]. In addition to

quantum confinement, the recombination mechanisms have been a subject of

wide investigation in accounting for the enhanced optical efficiency in

nanosized structures [40-48], wherein the light of practical use has become a

reality even from silicon [49]. The disordered structural features on such a

small scale, particularly in case of etching produced porous silicon [50-54]

requires a better understanding of the role played by nonradiative

recombinations in enhancing optical efficiency [55].

Simplest of the interacting QWs is an asymmetric double quantum

well (ADQW) system. The behavior of eigenstates of this system,

particularly when studied vis--vis localization and delocalization of

eigenstates in the individual QWs facilitated by means of parameters like

external electric field, tuning QW sizes or varying material composition

signifies major practical interests [3, 20, 56, 57]. Furthermore, the variation

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of these parameters so as to bring the coupled system either at the crossover

or at zero perturbation position depending upon the specific practical

requirement forms the basis of various applications of coupled QWs [2-6,

11, 18-24, 28, 33]. In addition, the interplay of eigenstates of ADQW system

and the charge transfer between QWs, specifically at and around crossover

are the studies of profound significance [2, 5, 6, 13-17].

Another important area of interest is electronics of quantum dots

(QDs). A QD can act as an artificial tunnel which is neither open nor closed

and therefore, it is a fascinating problem to study [58-61]. From basic

scientific point of view, in these systems as one tries to study the transport of

electrons, the role of quasi bound states of the QD is crucial [62, 63]. In

particular, evolution of phase of electron transmitted through the QD [37,

38] is not fully understood and remains a challenging problem [35, 37]. The

understanding of coherent transport through QDs will help in exploiting the

wave nature of electron in highly miniaturized futuristic nano-sized

electronic devices.

Besides, QDs, nanosized QWs find important applications in opto-

electronic devices [49, 64]. Efficiency of these devices depends on the

relative magnitudes of radiative and nonradiative recombination rates [43,

48]. Based on quantum confinement related effects radiative recombinations

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have been investigated in much detail [65-69]. However, little attention has

been paid to the study of reduced size on the nonradiative recombinations

[40, 70]. In the present study, we shall like to understand the size

dependence of nonradiative recombinations and its effect on the efficiency

of opto-electronic devices.

In the recent past, new and interesting problems are being posed by

researchers. This leaves a lot of scope to work on the understanding of the

sciences of transport and recombination in this new class of materials and

structures which the present thesis aims to present. Though, unabated the

pace of miniaturization has made the quantum properties of electrons crucial

in determining the design of the electronic devices. However, quantum

properties of electrons will become indispensable when electronics based on

individual molecules and single-electron effects will put back the

conventional circuits.

1.2 Mechanism of electron transport between QWs and

eigenstates of ADQW system

Determination of eigenstates of ADQW system has been a subject of

profound interest. Worth mentioning is the perspective of Man Made

Quantum Wells by Kolbas and Holonyak [10]. These authors have used

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computer simulation to illustrate interaction between coupled QWs and

compared the results with those obtained by using Kronig-Penney model.

Theoretical developments in the tunnel coupling of QWs have been

reviewed by Ferreira and Bastard [9]. As far as electronic states in

heterostructures are concerned, theoretical developments from first principle

calculations to effective mass approaches have been reviewed by Smith and

Mailhiot [71]. Microscopic calculations for the complete wave functions

include one electron calculations, pseudo potentials, tight binding and at

least in principle do not require the knowledge of the band offsets [17, 71,

72]. On the other hand, the envelop function method focuses its attention on

the modulation of the carrier wave functions which is due to the

heterostructures itself [73, 74]. In the envelop function method, variation of

effective masses is used to take properties of the host into account. The band

offsets should be the inputs to this approach. Numerical solution of

Schrdingers equation has been used by Mourokh et al [3] to explain the

effect of electric field on eigenstates of the double quantum dot (DQD)

system. These authors found a minimum separation in the energy

eigenvalues of the adjacent states upon varying the electric field as shown

the figure 1.1. Also, transition between bonding (covalent) and anti bonding

(ionic) behavior of eigenstates has been deduced. Similar numerical

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approach has been used by Neogi et al [11] to explain the intersubband

transitions in coupled DQD system. For smaller thickness of the

intermediate barrier layer, a multiple QW system becomes a supperlattice.

Such systems have been dealt by solving Schrdingers equation for a

periodic structure by Erman et al [12]. This approach is, in fact, a

Fig.1.1 Dependence of energy level separation on the applied dot to

dot electric field [3]

generalization of the Kolbas-Holonyak method [10]. Numerical method of

calculation of eigenvalues and eigenfunctions for doped intermediate barrier

[75] and for symmetric double quantum well potential enclosed within two

infinite walls [76] has also been reported recently.

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The eigenstates of ADQW system constitute the basis of the

mechanism of electron transfer between QWs and is a subject of great

practical interest. In a novel experiment on interacting QDs, Hatano et al [2]

have measured interdot tunnel coupling and showed that inherent asymmetry

of the capacitances of the component dots allows the determination of the

dot through which electron has passed. Schematic of the single anti crossing

reported by the authors is shown in the figure 1.2 whereas figure 1.3 shows

the electric field induced symmetry of probability density in asymmetric

system. The dynamics of the electron transfer between two QWs has also

been studied by spatially transferring electron from one quantum well to its

hole filled neighbour and detecting the near infra-red recombination

luminescence [77]. Changes in the magnetization of DQD have also been

used to investigate the crossings and anti crossings in its energy spectrum

[78]. Non invasive techniques have been employed to reveal the evidence of

electron moving between the dots [7]. In fact the coupling between the QDs

strongly influences the transport through the DQD system [5, 33, 79-81]. In

practical cases of interest, transport through QD requires it to be connected

by QPC and engineering of QPC for optimum transfer have been analyzed

by Zhang et al [33]. The QPC as charge detector has been used by Petta et al

[5] to measure the microwave driven change in occupancy of

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Fig.1.2 Schematic of single electron delocalized across both the

QDs [2]

Fig.1.3 Dependence of fractional occupancy of single electron on

the potential difference across coupled QDs [2]

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charge states and study the manipulation of single charge in DQD. Charge

delocalization in a tunable DQD has also been studied by DiCarlo et al [6] as

a function of temperature and strength of coupling.

As far as mechanism of electron transfer in double dot system is

concerned, the reference should be made to report by van der Vaart et al [82]

which shows that when elastic tunneling process are the dominant transport

mechanism, current through the dot is resonantly enhanced only when two

levels in the dots 1 and 2 align as shown in figure 1.4.

The understanding of DQD system finds direct applications in the

study of bonds in artificial molecules [26] and forms the basis of molecular

electronics. Also, it has potential applications in microwave spectroscopy

[27]. In weakly coupled QDs, a time varying potential of microwave can

induce inelastic tunnel events when electrons exchange photons of energy h

with oscillating field. This inelastic tunneling with discrete energy exchange

is known as photon assisted tunneling and has been extensively investigated

[8, 83-85]. In these applications, depending upon the strength of inter-dot

coupling, the two dots form ionic like or covalent like bonds [3]. In the first

case the electrons are localized on the individual dots while in the second

case electrons are delocalized over both the dots. The coupling between QDs

leads to bonding and anti bonding states whose energy

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Fig.1.4 Schematic potential landscape of the double quantum dot

depicting the conventional understanding of electron transport

between the QDs [82]

Fig.1.5 Schematic pumping configuration for bonding state B and

anti-bonding state A in photon assisted tunneling (PAT) [4]

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difference is proportional to the tunneling strength between the dots. For

microwave spectroscopy of the strongly coupled double dot, the pumping

configuration is shown in figure 1.5. The advantage of the pumping

configuration is that these processes can lower the amount of current but

they do not smear out the resonances [4]. Inter-sub-band lasers are another

Fig.1.6 Schematic of inter-well transition in inter-sub-band laser.

The inset shows the positions of three lowest subbands (in meV) as

a function of the narrow well width a1(in nm) for fixed values a2 = 2

nm and a3 = 10 nm [20]

fascinating application area of these systems. The DQD system has been

studied for intersubband laser right from inversion of population [20],

transport processes [19], temporal evolution [86], feedback effects [30],

efficiency [28], broadband wavelength operation [32], spectral line shape

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[87], relaxation kinetics [88], optical properties [89] to optical gain

optimization [18]. Schematic of the prevailing understanding of electron

transport in intersubband lasers achieved by optimization of thickness of the

coupled QWs so as to arrive at crossover is shown in figure 1.6. In addition,

mid to far infra-red absorption [29, 31, 90], spin qubit [91], detector of high

frequency quantum noise [92-93], negative differential conductance [94] etc.

are other potential areas of the applications in nano-devices.

1.3 QD as a tunnel device for coherent transport of

electrons

Quantum dots exemplify an important trend in condensed-matter

physics in which researchers study man-made objects rather than real atoms

or nuclei [58, 61]. As in an atom, the energy levels in a quantum dot become

quantized due to confinement of electrons allowing formation of quasi

bound states. At the same time, a single electron can tunnel through QD

resulting in coherent transport.

In a novel ingenious interference experiment in semiconductors,

Yacoby et al [39] demonstrated in 1995 the measurements of magnitude and

phase of the transmission coefficient through a quantum dot and proved

directly, for the first time, that transport through the dot has a coherent

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component. Their device consists of a GaAs/AlGaAs quantum dot with two

adjustable quantum point contacts placed inside an Aharonov-Bohm (AB)

ring which forms two arms of an electron interferometer. In a step beyond

the demonstration of coherence, an improved four terminal device version of

the experiment by Schuster et al [38] allowed to measure the phase of the

transmission amplitude through the quantum dot. The transmission phase

displayed a number of unexpected properties. Most notably, virtually the

same transmission phase was found for a whole sequence of conductance

peaks and in the conductance valley the phase displayed a sharp phase slip

as shown in figure 1.7. The field expanded substantially in 1998, when it

was realized that the coherence of quantum dot states can be controlled by

external means. Controlled decoherence was achieved [95-96] in a device

with a quantum dot that was capacitevly coupled to a quantum point contact

in close vicinity. The quantum point contact acted as a measuring apparatus

for the number of electrons on the quantum dot. Since number and phase are

conjugate variables, the measurement caused the dephasing (decoherence) of

electron states in the quantum dot. Further, the restoration of phase

coherence has also been investigated [97]. The measurement of phase and

magnitude of the reflection coefficient of a quantum dot [98] have made

further advancement in understanding of the coherence in QDs in terms of

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scattering. Recently, the interference experiments have been reported in

mesoscopic systems [37] which shows that the phase behavior depends upon

the electron occupancy of the QD. The explanations for the universal phase

evolution in the transport through a QD, particularly, the phase slip at

conductance minimum have been reviewed in 2000 by Gregor Hackenbroich

[35]. Recently, M. Avinun-Kalish et al [37] also discussed the available

explanations along with their shortcomings in explaining the experimental

results. These authors have grouped the explanations in three main classes.

The first group questions whether the measured phase is the intrinsic

transmission phase of the quantum dot or a modified phase due to multiple

paths traversing the interferometer [99, 100]. The second, considers

transport that is mediated by interplay of more than one quantum state. A

common scenario assumes an existence of a dominant level strongly coupled

to the leads, responsible for shuttling the electrons [101, 102]. After

occupation the electron is unloaded to a localized level, weakly coupled to

the leads, allowing the dominant level to be free again to transfer another

electron. Hence, the observed phase is only that of the dominant level. Based

on this idea other models examined only two levels, with one of the levels

dominant, adding spins, adding interactions, or assuming a finite

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Fig.1.7 The universal phase evolution and phase slip in the

conductance valley in electron transport through a QD [38]

temperature [103, 104]. Interaction between two levels was invoked also in a

quantum dot where the plunger gate couples with different strengths to

different energy states, leading thus to avoided level crossing and charge

shuttling between levels [35]. The third class deals with specific energies

where both the imaginary and the real parts of the transmission coefficient

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vanish [105]. These singular points, which explain the phase slips in the

valleys, might result from a deviation from a strictly zero-dimensional

system [35] or from the existence of Fano resonances in the dot [106]; but

they can not explain the in phase behaviour of all peaks. Naturally, one

would expect the breakdown of every model for some tuning parameters,

which we, thus far, have never observed. Still, some models may predict an

in phase behaviour for a very large sequence of peaks but not universal

behavior.

This necessitates the study of quasi bound states (QBS) of the

quantum dot in the transport perspective. On the same line of thinking, Shao

et al [59] has presented a numerical technique for open boundary quantum

transmission problem which yields, as the direct solution of appropriate

eigenvalues problem, the energies of (i) quasi bound states and transmission

poles, (ii) transmission ones and (iii) transmission zeros. The influence of

QBS on the carrier capture in the QDs has been investigated by

Magnusdottir et al [107] and Ghatak et al [63] has calculated the mean life

time of QBS of QW. The problem in the same point of view has also been

dealt in terms of capacitance of the QD by Racec at al [108] and calculation

for QBS in presence of electric field has been done by Ahn and Chuang

[109]. Transmission coefficient from QBS perspective have also been

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calculated by Kim and Lee [62] and the same AB configuration has been

calculated long back in 1984 by Gefen et al [110] but all these fails to

provide satisfactory explanation of the universal phase behavior of

transmission through a QD.

1.4 Role of nonradiative recombinations in nanosized

semiconductors

Until the advent of quantum confinement phenomenon, the study of

luminescence in materials has been mainly confined around phosphors and

the indirect band gap semiconductors materials like silicon were never

considered appropriate candidates for luminescence applications. The

quantum confinement has redefined the research in luminescence wherein

even silicon is considered as a promising lasing candidate for gaining light

[49]. The easiest route to get light emitting silicon is by means of etching to

produce porous silicon [50] which fetched much attention after the

acceptable explanation of luminescence in porous silicon by Cullis and

Canhan [64, 111] in terms of quantum confinement. The quantum

confinement effects have also been studied in amorphous silicon [112] as

well as silicon nanoclusters produced by other methods [113]. Very simple

explanations of the role of quantum confinement in silicon nanoclusters are

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available in literature [51, 114]. A number of theories to explain the optical

gap in silicon quantum dots have been reported [115].

Apart from band gap modulation, the quantum confinement has other

consequences upon the luminescent properties viz., increase in oscillator

strength [116] and lifetime [47, 117]. All such studies are aimed at the

optical gain in nanosized structures [42, 44, 45, 118]. Trwoga et al [118] has

modeled the luminescent properties of the nanoclusters in terms of

contribution of quantum confinement taking the example of silicon. The

study reveals that major contribution to luminescence is from radiative

recombinations of confined excitons. The authors have studied the variation

in oscillator strength and number of available free carriers with cluster size

and size distribution. As far as recombination mechanisms in nanosized

semiconductors are concerned, the increased efficiency is mainly attributed

to enhanced radiative recombinations in the band tail states [44, 51, 53, 66,

119-122]. In fact, it is the competition between radiative and nonradiative

recombinations which determines the luminescence efficiency [43, 48]. John

and Singh [43] have modeled the luminescent properties in terms of

competition between an activated radiative process and a Berthelot type

nonradiative process. Mehra et al [48] have also explained the luminescence

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data by considering competition between radiative and Berthelot type hoping

process.

There are a few reports which emphasises the role of nonradiative

recombinations to explain the enhanced luminescence efficiency of

nanosized semiconductor [40, 41, 55, 123-127]. Neogi et al [123] have

explained the enhanced efficiency in multi period QDs as compared to single

period QD due to reduction in nonradiative processes. Vincignerra et al [41]

have attributed enhanced luminescence efficiency in superlattices to the

absence of relevant nonradiative decay processes as evident from the

observed very long lifetime (about 0.3ms). Detailed studies of the radiative

lifetime of porous silicon using photoluminescence decay measurements

have been carried out by Hooft et al [40]. These authors have attributed high

external photoluminescence efficiency to reduction in the nonradiative

recombinations owing to low mobility, to low dimensionality and to

extremely low surface recombination rate. Surface recombination velocity

data also confirms the suppression of nonradiative recombination acting as

enhancer of optical intensity [128]. Role of nonradiative recombinations in

studies pertaining to temperature dependence of luminescence has also been

investigated [55, 124, 125]. However, little attention has been paid to

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modeling of luminescence efficiency in terms of nonradiative

recombinations to account for increased luminescence efficiency [129].

1.5 Outline of the present work

The thesis is comprised of seven chapters including the present one.

Outline of the work reported in the subsequent chapters is given below.

In Chapter 2, analytical expressions have been derived for asymmetric

double quantum well system for determination of eigenstates of coupled

QWs. Effect of coupling strength upon eigenstates has been presented.

These analytical solutions find a large number of applications, of which

applications to semiconductor heterostructures taking the example of

GaAlAs / GaxAl1-xAs have been discussed considering the variation of QW

width and confining potential. Variation in the alloy composition has also

been shown to produce effects similar to variation in QW width or depth.

The variations of energy eigenvalues and eigenfunctions give an insight of

the electron transfer in coupled QWs, details of which will be taken up in

Chapter 4.

The detailed analysis of the effect of coupling strength upon

eigenstates of coupled QWs has been carried out in Chapter 3 using the

analytical solution of the ADQW system presented in the Chapter 2. In

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particular, this chapter is devoted to the study of zero perturbation position.

The universality of the zero perturbation position has been shown

graphically as well as analytically. The explanation of the zero perturbation

position has been given in terms of transmission line theory. Further, transfer

of electron between coupled QWs at zero perturbation position has been

discussed in view of its possible applications.

In Chapter 4, a through study of mechanism of electron transfer

between coupled QWs has been presented. The analysis is based on the

formulae derived in the Chapter 2. The actual interplay of eigenstates of

coupled QWs and charging of individual QWs has been presented. The

sequence and significance of localization and delocalization positions of

eigenstates in the two coupled QWs have been discussed and the effects of

strength of coupling as well as asymmetry on these have also been

presented. The analysis is used to explain the mechanism of electron transfer

between coupled QWs.

As an attempt to explain the universal phase evolution in coherent

transport through a QD, transmission coefficient has been derived in terms

of QW coupling coefficients in the Chapter 5. The formulation clearly

separates the direct and multiple reflection components in transport through

a QD modeled as a double barrier quantum well and enable identification of

- 21 -


34/171


coherent part of the transmission coefficient, thus providing an explanation

of universal phase evolution. The role of quasi bound states of QD in the

coherent transport has also been brought into the formulation.

In Chapter 6, the first principle semi-classical calculations have been

done to derive expressions for surface recombination velocity for electrons

confined between parallel plates resembling heterostructure and in

cylindrical grain resembling quantum wire. These expressions have been

used to study the nonradiative recombination rate as a function of size of

confining structure and explain the enhanced optical efficiency in nanosized

semiconductors in terms of suppression of nonradiative recombinations.

The summary of the work carrier out and gist of the conclusions

drawn have been put together in the Chapter 7.

- 22 -


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Chapter 2

Analytical expressions for asymmetric

double quantum wells and their

application to semiconductor

heterostructures

2.1 Introduction

Increasing demand of high density nano-sized quantum wells has

resulted in extensive study of the mutual interaction of these systems. The

simplest of these is an asymmetric double quantum well (ADQW) through

which all basic features of interacting nanostructures can be easily

understood [4, 9, 130-131]. Such studies have great implications on the

electronic, optical, dielectric and other material properties. Besides this, the

problem finds direct applications in heterostructures, semiconductor devices

such as lasers [20], short wavelength light emission and absorption [28,

132], single electron transistors [133], quantum computation [2, 8, 134],


36/171

Chapter 2: Analytical expressions for ADQW and their application..

memories [25] etc and form a basis for the study of artificial molecules

[26-27].

Solution of Schrdingers equation for a symmetric double quantum

well (SDQW) with infinite outer boundaries is a text book problem which

shows that the energy states of the individual wells couple themselves in two

different ways namely symmetric and anti-symmetric interactions resulting

in a splitting of the eigenstates. The problem is solved in terms of an

inversion plane at the center of the barrier around which a transition occurs.

However, once an asymmetry is introduced either in terms of well width or

depth such an inversion plane disappears and simple explanation in terms of

symmetric and anti-symmetric coupling is not applicable. Further

complications are added when external boundaries of the wells are limited to

the same height as of the barrier between the two wells (a case of great

practical importance), for which the wave functions extend in the forbidden

regions outside the ADQW. This requires solution of Schrdingers

equations in five different regions which are to be matched at four interfaces.

The problem can be solved by a numerical iterative method as outlined by

Kolbas and Holonyak [10]. Such numerical methods have been adopted by

many worker [3, 11], most widely used of these being the Bastards envelop

function method [73, 74] which allows calculation of eigenstates and eigen

- 24 -


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function by iterative method by solving a set of equation at the boundaries.

Alternatively, the tight binding approximations and linear coupling of

eigenstates have been used by many workers [71, 135-136], validity of

which is limited to weakly interacting systems [9]. A simple picture which

appeals many is in considering two wells each having only one state and

interaction between them is introduced as a perturbation [4]. However, such

a method can be applied when barrier width is sufficiently large and

interaction is weak.

We have solved analytically the Schrdingers equation for an

asymmetric double quantum well (ADQW) system schematically shown in

figure 2.1 which results in a much simplified mathematical formulation of

the problem enabling a clear understanding of the underlying physics. The

solution finds direct applications in the design of electronic systems of

practical interest.

2.2 Transcendental Equation for ADQW System

The time independent Schrdingers equation;

[ ] 0)()()(

2 2

2

*

2

=+ xxVEdx

xd

m

h(2.1)

is solved for ADQW system shown in the figure 2.1 using the potentials and

effective masses;

- 25 -


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b + L2bx -L1 0

V=0

1 2 3 4 5

V1 V2b

V= V1L1

L2

V= V2

Figure 2.1 Schematic diagram of an asymmetric double quantumwell system

- 26 -


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- 27 -


40/171

sin)(cos00000

(cos)(sin00000

sincos000

cossin000

0000

0011100

0000sincos

0000cossin

22222

222

222

2

1

111111

1111

1

1

+

+

ee

ee

ee

bb

bb

L

L

kqkLbkqk

bkLbk

qkbkqk

bk

k

LkpkLkpk

LkLk

- 28 -



41/171


Upon solving this 8x8 determinant, we get a transcendental equation;

0)sincos()cossin( 1111111111 =++ YLkkpLkXLkkpLk (2.5)

with

,)cossin()(

)sincos()(

222222

222222

LkkqLk

LkkqLkkqX

ee

eebb

bb

++

++=

)cossin()(

)sincos()(

222221

2222221

LkqkLkpk

LkqkLkqkpkY

ee

eebb

bb

+++

+=

(2.6)

Since we are interested in eigenstate of ADQW, we limit our solutions to

(2.7)0, 21


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For - < x -L1:

,)(1x

m eKx

= (2.11)

for L1 x 0:

,cos)cossin(

sin)sincos()(

111111

111111

1

2

1

+

+=

xkLkpkLk

xkLkpkLk

pkKx e

L

m

(2.12)

for 0 x b:

{ }{ }

,sin)(

cos2sin)(

2)(

11

2

122

111112

122

1

3

1

++

+=

Lkkp

LkpkLkkp

pkKx

e

eex

xL

m

(2.13)

for b x L2:

{ }

{ },

cos

cos2

sin)()(

sin)()(

sin

cos2

sin)()(

sin)()(

2sin2

1)(

2

111

11

2

122

2

)(

11

2

122

2

)(

2

111

11

2

122

2

)(

11

2

122

2

)(

221

4

1

1

1

1

+

+

++

+

++

+

=

+

+

xk

Lkpk

Lkkpqk

Lkkpqk

xk

Lkpk

Lkkpqk

Lkkpqk

bkqkpk

Kx

e

e

e

e

bL

bL

bL

bL

m

(2.14)

- 30 -


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and for L2 x < :

eLbx

mm Lbx)(

2452)()(

+=

(2.15)

The constant K in the above equations is determined by normalization of the

wave function for the entire space extending from - to . The above

equations are most general ones and may be applied to any type of ADQW

system. When the potential barriers are formed in vacuum, the effective

masses will correspond to free electron mass and then p = q = 1. However,

in the present day electronics a great interest has developed in potential

wells formed in multi-layered semiconductor heterostructures. We choose

here one of these systems as an example of application of the above

equations.

Further, shallow quantum wells with one or two energy states in each

well are of great practical interest [5, 81]. We confine ourselves mainly to

the study of these systems, though the conclusions are of general nature and

can easily be extended to deeper wells with multiple states.

2.3 Applications to Semiconductor Heterostructures

(GaAs / AlXGa1-XAs)In the transcendental equation (2.5), we have taken constant effective

masses for various regions. This essentially assumes that E-k relations are

parabolic. Non-parabolicity of E-k relations will involve nonlinearities and

- 31 -


44/171


exact solution of Schrdingers equation is not possible. In actual

semiconductors, non-parabolicities may exist which will then make m*

a

function of E. For such cases, approximation methods are being used [138-

139] and may be helpful in getting exact characteristics of the material.

However, for simplicity of arguments, the scope of the present paper has

been limited to assuming effective masses to be energy independent.

As an illustration, we have considered GaAs / AlXGa1-XAs

heterostructure. A large number of reports are available on effective mass

and conduction band offset variations with respect to composition parameter

X [56, 140-143]. We havetaken the following dependences from ref [140];

(2.16)


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- 33 -

Figure 2.2 Probability density in GaAs / Al0.3Ga0.7As asymmetric double

quantum well (ADQW) of QW widths L1= 2.0nm and L2= 4.0nm at

different well separations: (a) 5nm, (b) 1.5nm, (c) 0.5nm and in

(d) energies of eigenstates are plotted against barrier width b.


46/171


strength of coupling. Contrary to this, the very fact of introduction of

asymmetry attaches belonging of eigenstates to individual wells and for

larger separations, charge distributions remain largely localized in individual

wells. When the barrier width is reduced, the probability distribution extends

to the other well. The electron energy eigenstates obtained here by using

analytical formula are similar in nature to those obtained by Ferreira and

Bastard [9] using numerical methods. However, mathematical formulae

presented here are much simpler and easy to use.

Next, we apply these analytical formulae to some cases of practical

interest;

2.3.1 Transport optimization using variation in QW width

Interwell transition in ADQW heterostructure comprising of one narrow

and one wide coupled QWs is being used to get lasing via intersubband

inverse population [20]. In these systems, the depopulated subband is

aligned with the higher subband of the wider QW of appropriate size so that

maximum transfer of electrons between the two becomes possible. Such an

alignment requires determination of exact well width of wider QW for a

given narrow QW such that difference of energy between eigenstates of the

two isolated QWs is minimized. The mechanism which works in the process

is shown in figure 2.3. It is a general belief that quick transfer of electron in

- 34 -


47/171


- 35 -

Figure 2.3 Process of matched interwell transfer of electron by varying the

width of larger QW coupled to a narrower one is shown for AlGaAs /

Al0.3Ga0.7As ADQW system. Probability density of the matched state is

shared equally between the two coupled QWs as shown in (b) and the

corresponding energies are shown in (d) in which the inset gives the

variation of difference of energy of the eigenstates E= E2-E1 which shows

a minimum at the same value of L2 for which probability of finding the

electron in the two QWs is equal. In these calculations, L1= 4.0nm and

b= 1.5nm have been used in fig (a) to (c) whereas dotted line infig (d) shows an additional calculation for b= 1.0nm to depict the effect of

increase in the strength of coupling.


48/171


the system under consideration is because of minimization of energy

difference so that the two QWs resonate. What we find here is that for a

finite coupling of QWs there always exists a finite energy difference

between eigenstates. Stronger is the coupling, higher is the energy

difference, E. As one tries to minimize E by changing width of the QW,

one shifts the system slowly from an asymmetric to a symmetric one in

which for the eigenstate under consideration, the probability of finding the

electron in each QW become nearly equal. It is this equalization of

probability that makes the transfer efficient under an elastic tunneling at the

same energy rather than transfer of electron from one eigenstate to another.

The analytical formulae presented here can be used to calculate the exact

width of the complementary QW which corresponds to minimum energy

difference between nearest eigenstates in such a way that probability density

of a particular eigenstate is shared equally between the two QWs as shown

in fig. 2.3(b). In addition, energy for photon or phonon assisted interwell

transfer of electron can also be calculated using these analytical formulae.

2.3.2 Transport optimization using variation in confining potential

The effect of variation of the confining potential by varying it for one of

the wells while keeping the other as constant is shown in figure 2.4 for a

typically chosen GaAs / Al0.3Ga0.7As ADQW with QW sizes L1= 2.0nm and

- 36 -


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- 35 -

Figure 2.3 Process of matched interwell transfer of electron by varying the

width of larger QW coupled to a narrower one is shown for AlGaAs /

Al0.3Ga0.7As ADQW system. Probability density of the matched state is

shared equally between the two coupled QWs as shown in (b) and the

corresponding energies are shown in (d) in which the inset gives the

variation of difference of energy of the eigenstates E= E2-E1 which shows

a minimum at the same value of L2 for which probability of finding the

electron in the two QWs is equal. In these calculations, L1= 4.0nm and

b= 1.5nm have been used in fig (a) to (c) whereas dotted line infig (d) shows an additional calculation for b= 1.0nm to depict the effect of

increase in the strength of coupling.


50/171


L2= 4.0nm. We note that the ground state of narrower QW reaches a

minimum energy difference with respect to second eigenstate of the wider

QW at V2 725meV making the behavior of ADQW system resembling a

SDQW. This confirms the proposition that the size asymmetry can be

neutralized by introducing confining potential asymmetry in an opposite

direction. The energy level separation at this point is similar to energy level

splitting in SDQW. Further, we note that the separation between the two

eigenstates increases as one move away from the matched condition. As far

as probability distribution is concerned, we note that size asymmetry in

ADQW results in the preferential distribution of charge in the individual

wells to which the particular states belong. Here, we find that by varying

confining potentials V2 (or V1) the charge can be redistributed as shown in

figure 2.4(b), ADQW behaves like SDQW for this eigenstate.

The effect described above is similar to that obtained by applying an

electric field. A large number of evidences exists in the literature that the

asymmetry of size of ADQW can be neutralized by applying an external

electric field [3, 26]. The effect of change in V2 with respect to V1 as a

matter of fact is equivalent to applying an electric field, since the purpose of

electric field is also to create a potential difference between the two wells. In

the simplest term, the electric field is equivalent to (V2-V1)/b in the present

- 38 -


51/171


formulations. Thus, our calculations provide an easy means of calculating

the effect of electric field on the eigenstates of ADQW. To elaborate this

point further, we have plotted in the inset of figure 2.4(d) the separation of

energy levels E = E2-E1 against V2, in which a minima is obtained similar

to that reported by Lev G. Mourokh et al [3] by applying an electric field

and at this point the probability of finding the electron in the two wells, for

eigenstate under consideration is equal.

2.3.3 Transport optimization using variation in material composition

Results similar to those obtained by variation in V2 can also be obtained

by variation in material composition of the alloy. In figure 2.5, we have

plotted energy eigenstates and probability functions for typically chosen

Al0.45Ga0.55As / AlAs / AlXGa1-XAs system with size parameters L1= 4.0nm

and L2= 8.0nm. As is clear from figure 2.5(b) and (d) the size asymmetry

can be compensated by varying the material mole fraction of the coupled

QW, an effect similar to that obtained by varying QW width or confining

potential. Thus, we find that minimization of energy difference E as well as

probability redistribution can also be accomplished by suitably selecting the

alloy composition. In addition, the estimation of Al mole fraction in the

AlGaAs alloy itself is a very interesting problem and has been attempted

- 39 -


52/171


- 37 -

Figure 2.4 Probability density in GaAs / Al0.3Ga0.7As ADQW of well

widths L1= 2.0nm and L2= 4.0nm, QW separation b= 2.5nm and for

confining potentials V1= 330meV and V2 equal to (a) 550meV, (b) 725meV

and (c) 850meV are plotted. These show shift of charge density with

variation of V2 and neutralization of size asymmetry for the given set of

parameters for ground state at V2= 725meV. Variation of energy of the

eigenstates with confining potential V2 is shown in (d) and inset shows the

variation of difference of energy of the eigenstates of the two wells similar

to that reported in the literature [11] for ADQW system by applying anexternal electric field. An additional dotted line in fig (d) shows calculation

for b= 1.0nm to depict the effect of increase in the strength of coupling.


53/171


rigorously by many researchers [56-57]. We here provide simple and easy to

use analytical formulae for calculating Al mole fraction by measuring the

band offset data of coupled heterostructure layer. In addition, certain

properties of these heterostructures depends on the Al mole fraction [89] and

mechanism underlying such effects can be better understood in terms of

variation of energy states and probability redistribution in QWs using the

analysis presented here.

2.4 Conclusions

In the present chapter, based on the first principles, we have derived

analytical expressions for an asymmetric double quantum well system. This

analytical approach provides the energy eigenvalues as roots of a simple

transcendental equation and wave functions for corresponding eigenstates

can be calculated using standard formulae provided by this method. In our

calculation, we have provision to accommodate all necessary ADQW

parameters such as individual quantum well widths and confining potentials,

separation of wells, effective masses in different regions etc.

Further, we have discussed some of the applications of the these

analytical expressions for GaAs / AlXGa1-XAs semiconductor

heterostructures which reveal their simplicity in use and effectiveness in

- 41 -


54/171


accommodating various material parameters as well as in understanding the

underlying basic physics of ADQW.

(i) Effect of variation of separation of the two wells upon the energy

eigenvalues and corresponding wave functions has been studied for an

ADQW having one eigenstate for each isolated well. The main feature of

ADQW is localization of eigenstates in individual QWs as compared to

SDQW in which a complete delocalization exists. In ADQW, delocalization

of probability density increases as a result of coupling when the barrier

width is reduced. Results thus obtained are in agreement with those reported

in the literature using numerical methods.

(ii) The effect of asymmetry can be minimized for a particular

eigenstate by varying width of one of the QWs so that probability density in

two coupled QWs becomes nearly equal for that energy level which in turn

results in an efficient transfer of electrons. This occurs at the minimum

energy difference between the eigenstate under consideration and its nearest

state, corresponding to minimization of asymmetry and ADQW behaving

like a SDQW for this state.

(iii) The biasing of ADQW result in interesting properties and its

variation can be used in tuning the QWs in a way similar to that obtained by

- 42 -


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varying QW width. Effects are similar to those obtained by applying an

electric filed and are in agreement with those reported in literature.

(iv) The consequences of varying Al mole fraction on the properties of

these heterostructures and estimation of Al mole fraction itself is of great

practical interest and can be better understood in terms of band alignment

and probability redistribution which in turn can be calculated using the

analytical formulae presented here.

Thus a simple mathematical formulation provides an important method

for understanding, investigating and optimization of properties of ADQW.

- 43 -


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Chapter 3

On the possibility of designing interacting

semiconductor quantum wells with zero

perturbation coupling

3.1 Introduction

A large number of efforts are being made to transmit electrons between

quantum wells (QWs) elastically [3-6, 11, 20, 28, 33]. For example in lasers,

attempts have been made to couple the QW of the host material with another

of larger width so that second eigenstate of the later coincides the ground

state of the former to provide a means of depopulation [4, 20, 28]. However

in doing so, generally the interaction between the two QWs will perturb the

original eigenstates [4]. While trying to understand these systems, we have

come across an important property which shows that whenever a QW is

coupled with another to form an asymmetric double quantum well system


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Chapter 3: On the possibility quantum wells with zero perturbation coupling

- 45 -

and if the phase traversed in the second QW is an integral multiple of, the

eigenstate of the first QW remains unperturbed. This condition of zero

perturbation coupling is universal and is independent of the strength of

interaction. The results may find important applications in interacting

heterostructures like those used in designing inter-sub-band lasers [18-24]

and laterally coupled quantum dots (QDs) [2-3, 33].

3.2 Determination of zero perturbation position

The two coupled QWs of arbitrary size and depth form an asymmetric

double quantum well (ADQW) system shown in the figure 2.1. Analytical

expressions for eigenstates of this system have been discussed in Chapter 2,

which when expressed in terms of phase angles a free wave would travel in

the two QWs; 1 = k1L1 and 2 = k2L2 shows that the eigenstates of this

system obey the transcendental equation;

0)sincos()cossin( 111111 =++ YkpXkp

with

( ) ( ){ } 222

2222

22

222

sin

cos2

ee

ebb

b

kqkq

qkX

+

+=

( ) ( ){ } 2222222221221

sin

cos2

ee

ebb

b

kqkqpk

qkpkY

++

+=(3.1)


58/171


- 46 -

where effective masses in the two QWs and the barrier are m1, m2 and mb

respectively and their ratios are represented by p = mb / m1 , q = mb / m2.

Since we are interested in eigenstates of ADQW, we limit our solutions to

0EV,V 21


59/171


- 47 -

Figure 3.1 Energy eigenstates of two coupled QWs forming asymmetric

double quantum well system are plotted by keeping the width L1 of the

first QW fixed and varying the width L2 of the second QW, where

strength of coupling is varied by changing the barrier width b. The values

are calculated by taking numerical example of GaAs / Al0.3Ga0.7As system

using m1 = m2 = 0.067mo, mb = 0.0919mo, L1 = 2.0nm and

V1 = V2 = 330meV. All eigenstates irrespective of the strength of couplinge

- bpass through a common point Q which shows zero perturbation

coupling at these points.


60/171


- 48 -

Figure 3.2 Energy eigenstates of two coupled QWs forming asymmetric

double quantum well system are plotted by keeping the widths L1 and L2of the QWs fixed and varying the potential V2 of the second QW, where

strength of coupling is varied by changing the barrier width b. The values

are calculated by taking numerical example of GaAs / Al0.3Ga0.7As system

using m1 = m2 = 0.067mo, mb = 0.0919mo, L1 = 2.0nm and V1 = 330meV. All

eigenstates irrespective of the strength of coupling e- b

pass through acommon point Q which shows zero perturbation coupling at these

points.


61/171


- 49 -

Similar effects have been obtained when k2 is varied by varying V2

instead of L2 as shown in figure 3.2. In order to make our result further

explicit, we have plotted eigenstates as a function of phase angle 2

= k2L

2in

figure 3.3(a) by varying L2 and in figure 3.3(b) by varying k2 (in terms of

V2). These two graphs clearly shows the universality of zero perturbation

position at 2 = k2L2 = n irrespective of the variation of k2 or L2. Further, in

experimental conditions both L2 and V2 can be varied to achieve a fine

tuning.

3.3 Universality of zero perturbation position

Universality of this property of the point Q in Fig. 3.1 can be proved

by using the analytical result given by equation (1). By putting 2 = k2L2 =

n where n is an integer, the equation becomes independent of the barrier

width b and reduces to

0cossin)( 11121

22 =+ pkkp (3.2)

Roots of this equation give eigenstates of isolated host QW of width L1.

Thus, for an eigenvalue E1 of the host QW, a corresponding k2 can be

calculated and L2 can then be determined to satisfy the above condition. For

this value of L2 even after coupling the QWs, the energy level E1 remains

unchanged whereas the wave function extends to the other QW.


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- 50 -

Figure 3.3 Graphical representation of zero perturbation position at

= k2L2 = n obtained by (a) variation in L2 and (b) variation in V2 using

the parameters used in figures 3.1 and 3.2 respectively. Energy eigenstates

of two coupled QWs forming asymmetric double quantum well system are

plotted as functions of2 = k2L2. All states passes through 2 = k2L2 = n.


63/171


- 51 -

3.4 Coupling at zero perturbation

To further our understanding of the system around this point, we have

plotted in figure 3.4 the wave functions and find that the form of

distribution function in the first QW remains invariant at all strengths of

coupling. Whereas the probability density * in figure 3.5 and variations in

the fractional occupancies in the two QWs as a function of barrier width in

figure 3.6 show that its magnitude reduces as the strength of the coupling is

increased. This shows that coupling with the second QW does not reflect the

wave backward and a perfectly matched coupling can be achieved. The

absence of reflection can be understood by noting that for an isolated QW a

solution xeK = exists in the barrier region and at the boundary of the

QW we have,

=

pdx

d(3.3)

If the function of a large QW formed by combining widths L1+L2*

where

L2*

is the width of the matched QW corresponding to the energy E1 the

condition (3) can be found to exist at x = L1. Since, the solutionx

eK

=

is a valid solution throughout the barrier, if a barrier is introduced at this

place, it will always find continuity. This shows that in a QW of width

L1+L2*, introduction of barrier of any width at x = L1 leaves the eigenstate


64/171


- 52 -

Figure 3.4 (a) The wave functions , (b) the probability densities * and(c) the fractional occupancies in the two QWs are plotted for coupled QW

system with the host QW width of L1 = 2nm matched with a second QW of

width L2 = 5.3nm so that energy eigenvalue of the state remains constant

at E = 127.3 meV. As the strength of the coupling is increased, the

occupancy in the coupled QW increases without any change in the energy

eigenvalue as well as the wave form of the host QW.


65/171


- 53 -

unperturbed. On the other hand, when the phase elapsed in the second QW is

an integer multiple of , the condition at the two boundaries remains

essentially unchanged because of symmetry (or anti-symmetry). The

situation here resembles that of a transmission line terminated with

characteristic impedance which acts effectively as an infinite line with no

reflection and any half wavelength line added in between acts as 1:1

transformer.

3.4 Conclusions

For a given eigenstate of a quantum well (QW), it is always possible

to find another QW in such a way that the coupling leaves the original

eigenstate of the host QW unperturbed irrespective of the strength of

interaction. The condition is universal and is met with whenever the second

QW has appropriate width and depth so that phase traveled by an electron

wave through it is an integral multiple of. Thus, for every QW for a given

eigenstate another QW can always be found which is completely matched

with the host and electron can tunnel from the host to the guest without

perturbing the former. This new type of coupling may find important

applications in heterostructures in which the barrier width may vary because

of disorders at the interfaces and in coupled quantum dots where separation


66/171


- 54 -

between them may not be controllable to a fine precision. An elastic

coupling between QWs reduces the possibility of energy loss in electron

transfer and may find important applications in nano-electronic devices. This

may also find useful applications in heterostructures and quantum dots in

which monochromaticity is an essential requirement. It may be further

pointed out that though we have analyzed only a system of two QWs, the

matched coupling described above is applicable for multiple QWs where a

host QW can be coupled to as many QWs as required and none of them will

disturb the original eigenstate.


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Chapter 4

Single electron charging of quantum wells

and quantum dots

4.1 Introduction

The recent impetus in research pertaining to charging in coupled QWs

and QDs from various aspects and in variety of systems has put forward the

requirement of a detailed quantitative study of interplay between single

electron charging and quantized energy states of the coupled system [3-6,

13-17]. The novel application like spin entanglement using charging in DQD

by Hatano et al [2] has increased the significance of such studies. These

authors have presented charging diagram around the crossover position and

explained the delocalization of states in terms of symmetry and asymmetry

of coulomb diamonds. Change in fractional occupancy as the coupled

system is varied around energy crossovers has been presented by these

authors. Furthermore, the photon / phonon assisted tunneling based

applications require a detailed understanding of mechanism of charging of


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Chapter 4: Single electron charging of quantum wells and quantum dots

- 56 -

QWs / QDs vis--vis energy eigenstates of the system. Even the

manipulation of single charge in coupled QDs has been shown by Petta et al

[5] using microwave excitation. These authors conclude that the

understanding of charge manipulation across DQD is crucial for spin

manipulation experiments and the same has been shown by Hatano et al [2].

Di Carlo et al [6] has used QPC for charge readout to investigate charge

delocalization. These authors have accurately determined the inter dot

coupling using the charge readout. The formation as well as control of

covalent / ionic bonds in DQD system also requires the detailed

understanding of charge transport. The demand of quantitative description of

charging of quantum structures is further enhanced by experimental studies

for imaging of single electron charge states in QDs using various techniques

[13, 144-147]. In addition, for the study of transport through a QD, it is

required to be connected with some QPC. The system acts as if a bigger QD

(QPC) is coupled with smaller one for transfer of charge and therefore, such

studies as well as the designing of QPC [33] have put on demand the

understanding of charge transfer in QDs. Moreover, the explanation of

anomalous universal phase evolution in electron transport through a QD by

Hackbroich et al [60] in terms of avoided crossings can be better understood

if charging and discharging profiles are studied around the crossovers. It is


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to be noted that Silvestrov and Imry [101] have given an explanation of

universal phase evolution using a model for charging of a QD.

The single electron charging effects in small area metal tunnel

junctions have been reviewed by Averin and Likharev [148]. Later, Averin

et al [14] accommodated the energy quantization in the formulations to

explain the charging of QWs and QDs by taking into account the change in

electrostatic energy of the capacitance of QWs and QDs. The authors have

emphasized the importance of discreteness of the charge accumulated but

have given only the extension of the conventional description [148] of the

single electron charging effects. Though, discreteness of charge

accumulation in the QWs has been considered but actual interplay between

single electron charging and eigenstates of the system is not obvious in the

formulation. The charge re-distribution in the tunneling process has been

studied by Fong et al [149]. These authors have accounted for the symmetry

properties of the resonantly coupled QD states to govern the charging of QD.

The analysis is based on the scaled version of density functional theory and

has been presented in terms of bonding and anti-bonding molecular resonant

tunneling states between coupled QDs. In this analysis, Fermi level of the

emitter is supposed to align with discrete levels of the QD to resonantly

transfer charge to it. In fact, such an alignment of states is never achieved for


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any finite coupling of emitter with QD. Moreover, the coupling itself greatly

influences the charging characteristics. The effect of strong shift in the QDs

energy states upon addition of single electron has been considered by

Johnson et al [16] to explain their experimental findings. These authors have

analyzed the single electron charging in QDs in the transport experiments

taking into consideration 0D states of QDs. All these investigations lead to a

common theme of charging of QDs and QWs in exact quantization

framework while taking into account the effect of finite coupling through

which charge is actually transferred. An ADQW may form the most

appropriate system to bring the interplay of actual quantization of energy

states and charging effects in one frame work so as to resolve finer features

of charging of QWs and QDs together with exact quantitative analysis of the

mechanism of charge transport. An experimental study of tunneling time in

GaAs / AlxGa1-xAs ADQW system has been presented by Roussignol et al

[17]. These authors have observed reduction in tunneling time for fixed

barrier width as the system is moved from minimum energy difference to be

greater then LO phonon energy. The dispersion relations for ADQW system

calculated using finite difference approximation along with eigen functions

have been given by these authors to explain the reduction of tunneling time.

Although, a qualitative picture of charge transfer in terms of inter band


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relaxation has been discussed in this paper, the authors have concluded that

charge transfer mechanism should not be sensitive to the effect of band

banding. Such a conclusion may be a result of the qualitative picture of

charging. Quantitative analysis presented in this chapter clearly brings out

the finer details of charge transfer at the cross over and satisfactorily explain

the recently reported experimental results of Hatano et al [2].

4.2 Mechanism of charge transfer between QWs

A quantitative analysis of the transport of electron through coupled

QWs is crucial in designing and implementation of nano electronic devices.

The simplest method for understanding the mechanism is to investigate the

electron states of an ADQW system in which electron states can be

manipulated by applying gate voltages. Also, the laterally coupled QDs can

be modeled as two QWs coupled through a barrier. In general, these

quantum wells have asymmetry of size as well as potential energy of which

the later can be varied by applying gate voltages. Thus, asymmetric double

quantum well (ADQW) system forms a basis for understanding the

mechanism of electron transport between QWs and QDs. The conventional

understanding of the charge transfer is that each QW has discrete energy

states and as the potential energy of the QWs is varied, the states align


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themselves to allow a resonant tunneling [82]. However, a closer look at the

one electron states of the coupled system shows that the states never align

but slip around the crossover positions as the QW depth is changed [4, 9]

(here by changing the gate voltage). As pointed out by Hatano et al [2], the

mechanism of electron transfer is delocalization of electron states around the

crossover. In Chapter 2, we have given an analytical solution for an ADQW

system, in which we have also arrived at a similar conclusion. When we go

through further details of calculations, we find that the charge transport takes

place in three steps, in which delocalization of energy states of the two QWs

occurs one after the other separated by a potential energy gap. For this, we

first calculate the fractional probabilities of electron occupancy in the two

QWs defined by 1 and 2 respectively, where

=11

*0

2*21 dxdx

L

(4.4)

and

=1

2

*

0

4*42 dxdx

L

(4.5)

For understanding the mechanism of charging of QWs and QDs, we

have considered example of GaAs / Al0.3Ga0.7As ADQW system, in which

one QW is treated as host from which an electron is to be transferred to


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second QW to which a gate voltage is applied. We start our argument with

states around a single crossover. By crossover, we here mean that the two

isolated QWs will have energy eigenstates E10

and E20

which meet at a

common point in the E-k2 plane. When coupled together, these states interact

in such a way that energy states slip around this crossover and the crossover

is avoided. This is shown in figure 4.1(a). Now, these coupled energy states

become E1 and E2 shown by solid lines in the figure. The energy difference

E = E2E1 goes through a minimum as one changes k2 (by changing the

potential energy V2 of the second QW). This minimum ofE is shown in

figure 4.1(b). It is around this minimum rather than exact crossover position

that the entire charging process takes place. In order to understand charging

process, we have plotted in figure 4.1(c), the fractional occupancies for the

two states E1 and E2 as function of wave vector k2. At the extreme left, the

first eigenstate is mostly localized in the first QW. As k2 is increased, at the

point A there is a delocalization in which probability of finding the electron

in his state becomes equal in two QWs (1 = 2). As one tunes the system

towards the E minimum position, at the point B, the states get almost

completely localized in the second QW with next higher state E2 having

same fractional occupancy also localized in the second QW.


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Figure 4.1 (a) Eigenstates of GaAs / Al0.3Ga0.7As system using typical

ADQW parameters with L1 = 200nm and L2 = 50nm. Solid lines are for

b = 5nm whereas dotted lines show the energy crossover; (b) Energy

difference between the eigenstates of coupled QWs; (c) Fractional

occupancy 1 in the first QW is shown by dashed lines whereas 2 in the

second QW by solid lines. Charging diagram clearly shows exact interplay

between eigenstates of coupled QWs and their charging wherein point A

corresponds to delocalization of state E1, point B corresponds to

localization of both states in the second QW signifying the actual chargetransfer position and point C to delocalization of state E2 in k2-E space.

Thus, electron remains in second QW between A and C and in firs

transport and re combination mechanism in nano crystalline semiconductors

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