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Far Infrared Studies of Silicon using Terahertz Spectroscopy
Amartya Sengupta, Aparajita Bandyopadhyay, John F Federici and Nuggehalli M Ravindra
Department of Physics, New Jersey Institute of Technology, Newark, NJ 07103, USA
Keywords: Terahertz Spectroscopy, Electronic Materials, Characterization
Abstract
In this work, the optical properties of p- type silicon wafers, of various thicknesses, have been studied in
the frequency range of 0.2 – 1.2 THz. It is seen that, for low resistivity silicon, the optical properties are
dominated by the presence of dopants. The analysis technique deployed in the present work explores a
general iterative procedure to determine the real and imaginary parts of the complex dielectric constant
without utilizing Kramers-Kronig relationships. This study not only holds scientific relevance in
material science but also opens up rich avenues for novel applications of terahertz spectroscopy to
semiconductors.
Introduction
In recent years, Terahertz (1 THz = 1012
cycles/sec and 300 µm in wavelength) spectroscopy has
become a standard technique for non-contact, non-invasive, real-time characterization technique for the
measurement of material parameters. Measurement of optical properties, in general, is important for
studying energy band-structure, impurity levels, excitons, localized defects, lattice vibrations, and
certain magnetic excitations. The far-infrared or the THz region of the electromagnetic spectrum is of
critical importance in the spectroscopy of condensed matter systems as the optical and electronic
properties of semiconductors and metals are greatly influenced by excitons and Cooper pairs whose
energies are resonant with THz photons. The THz regime also coincides with certain inelastic processes
like tunneling and quasi-particle scattering in solids. At the same time, THz time domain spectroscopy
(THz-TDS) is a reliable tool for studying confinement energies in artificial dielectrics (ADs) such as
artificially synthesized nanostructures and for non-contact estimation of interface traps in high dielectric
constant (K) materials [1-3]. It has also been used to study the effects of grain size dependent scattering
in various materials [4, 5]. Most recently, the possibility of interferometric imaging for security
screening applications is also being considered using THz radiation [6, 7].
During the past several years, researchers have exploited the THz range of frequencies for material
identification and characterization, which has been possible due to the availability of a variety of sources
and detectors [8-10]. The allure of THz-TDS can be attributed to the facts that (a) coherent detection
enhances the signal to noise ratio (SNR) of the measurement; (b) time resolved studies with sub
picosecond time resolution is possible in the far-infrared component of the electromagnetic spectrum;
and (c) compared to other methods such as millimeter wave spectroscopy, THz-TDS has more spatial
resolution and it can record both the amplitude and phase of the THz waves simultaneously.
In the field of material characterization, THz spectroscopy bears special significance as it can serve the
purpose of an in – situ, non – contact measurement tool during device fabrication. This allows the
The Physics and Materials Challenges for Integrated Optics - A Step in the Future for Photonic DevicesOrganized by Animesh Jha, Andrew Bell, Nuggehalli M. Ravindra, and Andy R. Harvey
Materials Science & Technology 2005
39
semiconductor device production line to achieve a dynamic control over the device properties even
before the final quality control check of the packaged products. Even though similar techniques exist at
other wavelengths, THz-TDS offers a unique opportunity to estimate material parameters in the interval
between the high frequency limit of modern electronics and the low frequency limit of most practical
lasers and other incoherent sources [12-14].
Previous studies involving silicon have shown that it is an exceptional optical material in the infrared
range of frequencies [15-19]. In this work, a general method has been discussed to determine the
complex index of refraction of semiconductors which could lead to estimation of other related
parameters of the material such as electrical conductivity. Different types of silicon wafers have been
used to validate the analysis which can be divided into three different classes on the following basis:
resistivity, polishing and presence of oxide layer.
Experimental Arrangement
THz Set - up
The experimental arrangement consists of a Ti: Sapphire laser emitting 125 fs pulses at 800 nm, part of
which pumps an Auston switch consisting of a semi insulating GaAs wafer with a gold transmission line
structure microlithographically imprinted on it [20]. This acts as a coplanar stripline (CPS) antenna
when an AC bias is applied to it and becomes the source of THz radiation with a center frequency of
about 0.5 THz [21]. A schematic of the THz generator is shown in Fig 1.
Figure 1: Configuration of the Auston switch used in the present set-up; A
is the LTG-GaAs substrate, B is the transmission line structure, C is the
pump laser beam, D is the source of ± 5 V bias at 12 KHz. The values of
the switch are L = 1mm, b = 60 µm, d = 10 µm and w = 20 µm
A silicon ball lens mounted above the antenna collects the emitted THz beam and guides it through a set
of gold plated off axis parabolic mirrors to the detector. The detection scheme is just the reverse of the
generation process, where the incoming THz electric field provides the bias for the antenna which is
optically gated by the other part of the laser pulse. The sample being studied is placed at the focus of the
THz beam between two parabolic mirrors. The experimental layout for THz – TDS is shown in Fig. 2.
Figure 2: Experimental arrangement of the THz Spectroscopy system
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Samples Studied
Table I summarizes the different types of wafers that were used in this study. All the wafers were of 4”
diameter and were supplied by Virginia Semiconductors Inc and Silicon Sense Inc.
Table I. Parameters of wafers that were used in the experiments
Wafer Thickness Resistivity Polish Exposed Side
p – type Silicon 50 µm 10 – 100 Ω-cm Double side <100>
p – type Silicon 250 µm 10 – 100 Ω-cm Double side <100>
p – type Silicon 475 µm 1 – 2 kΩ-cm Single side <100>
SiO2 on Silicon SiO2: 0.5 µm
Silicon: 700 µm18 – 22 Ω-cm Double side <100>
Theoretical Analysis
Time resolved THz spectroscopy measurements provide simultaneous information about the amplitude
and phase of the samples under study. One reference waveform Eref (t) is measured without the sample
or with a sample of known dielectric properties, and a second measurement Esample (t) is performed, in
which the THz radiation interacts with the sample. The transmission spectrum is calculated using the
Discrete Fourier Transform (DFT) of the sample and reference measurements:
exp
( )( )
( )
sample
ref
ET
E
νν
ν= (1)
In the case of optically thin samples (the transit time of the THz pulse through the sample is comparable
to the width of the pulse itself), the overlap between successive echoes limits the ability to break up the
transmitted THz signal through the sample into individual echoes and hence we have to consider the
effects of multiple reflections through the sample.
The following conditions are assumed in the analysis:
• the electromagnetic response of all the media is linear;
• the sample is homogeneous with two optically flat and parallel sides;
• the sample and the overlayers are isotropic without surface charges or the presence of an
interface.
The transmitted electric field through the sample Esample (ν), is given by,
, ,2
0, 1 , 1
( ) ( , ) ( , ) ( )a b m a b m
k
sample ab m ab m
ka b a ba b a b
E T P d R P d Eν ν ν ν= =∞
== =≠ ≠
= ∏ ∏ (2)
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The second term is the Fabry – Perot term arising out of multiple reflections within the thin samples. In
the above equation, E(ν) is the electric field of the emitted THz signal, Rab, Tab are the Fresnel reflection
and transmission coefficients at the a-b interface [20], and Pm is the propagation coefficient in medium
m over a distance d and is given by,
( , ) expa
a
n dP d i
c
νν = − (3)
with ( ) ( ) ( )a a an n iν ν κ ν= + being the complex refractive index of medium a. Hence the complex
transmission coefficient ( )T ν taking into account Fabry – Perot effects is given by [17, 23],
2
2
4 2 ( 1)exp
( 1)( , , )
1 41 exp
1
n n li
cnT n l
n n li
cn
π ν
νπ ν
−−+
=−− −+
(4)
( , , ) ( , , ) ( , , ) exp ( ( , , ) ( , , ))FP FPT n l n l n l i n l n lν ρ ν ρ ν θ ν θ ν= +single single (5)
To evaluate the Fabry – Perot contribution, the samples were assumed dispersionless and to have κ << 1,
so that the total complex transmission coefficient can be expressed in terms of the following functions,
2 2
2 2
1
2
2
2
1
2
4 2( , , ) exp
( 1)
2 ( 1)( , , ) tan
( 1)
1( , , )
1 42 1 cos
1
1 4sin
1( , , ) tan
1
1
FP
FP
n ln l
n c
n ln l
c n n
n l
n nl
n c
n nl
n cn l
n
n
κ πκνρ νκ
π κθ νκ
ρ νπν
πν
θ ν
−
−
+= −+ +
−= ++ +
=−−+
−+=
−+
single
single
4cos 1
nl
c
πν −
(6)
The transmission spectrum deconvolution obtained from the Fourier transform of the measured signals
as shown in Eq. (1) is compared with the modeled transfer function of Eq. (6) using a minimization
algorithm which evaluates the sum square error ε2, defined as,
( ) ( )2 2 2ε ρ θ= ∆ + ∆ (7)
where
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( )expsingle
expsingle
( ) ( ) ~ ( )
( ) ( ) ~ arg ( )
FP
FP
T
T
ρ ρ ν ρ ν ν
θ θ ν θ ν ν
∆ =
∆ = + (8)
Minimization of the error gives a set of (n,κ, l) values for the sample which are the effective optical
quantities of the assumed dispersionless medium. Using these values in the single pass model, i.e.
without Fabry – Perot effects, the actual optical quantities of the dispersive medium are obtained as
functions of frequency as,
( )single
2
single
( ) 1 arg ( )2
( ( ) 1)( ) ln ( )
2 4 ( )
cn T
l
c nT
l n
ν νπν
νκ ν νπν ν
= −
+= (9)
From the above set of n(ν) and κ(ν), the real and imaginary parts of the dielectric constant of the
material are calculated as functions of frequency [24].
Results
Demonstration of Reciprocity Principle
The time domain transmission measurements and the corresponding frequency domain spectra, for all
the double side polished (DSP) wafers under study of thickness 50 µm, 250 µm and 700 µm,
demonstrate the reciprocity principle in the sense that the two opposite faces of the wafers yield identical
transmission spectra under THz illumination. Figures 1 (a) and 1 (b) show the experimental time domain
and corresponding frequency domain plot for the 250 µm thick silicon wafer. This is particularly
interesting as it reflects the ability of THz radiation to characterize the substrate property even in the
presence of native oxides on the surface. Corresponding plots obtained for other DSP wafers are not
shown here.
Figure 1 (c) shows the frequency domain plot for the single side polished 475 µm thick wafer. The
difference in the spectra for two opposite faces suggests that, THz is sensitive to surface roughness and
appropriate analysis of the measurement can lead to the determination of roughness.
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Figure 1: Plots of the THz signal in (a) time domain and (b), (c) DFT of the same.
Determination of Optical Parameters
Using the analysis described in the previous section, both the real and the imaginary parts of the
complex refractive indices as a function of frequency are determined for the wafers from Eq. (9). Figure
2 (a) shows a comparison of the experimentally obtained and numerically extracted frequency dependent
refractive indices for the 250 µm thick DSP silicon wafer. The first one shows the characteristic Fabry –
Perot oscillations while the numerically corrected value of refractive indices has monotonic variation of
about 10% over the frequency range of 0.2 to 1.2 THz. Figure 2 (b) shows the corresponding
comparison of extinction coefficient for the same wafer. The absorption is mainly due to the presence of
the free carriers in the doped wafers. In fact, for a very lightly doped wafer, or conversely, for a
semiconductor of high resistivity, THz absorption would be almost negligible. Thus the determination of
refractive indices would be more precise which opens up the possibility of non – contact
characterization of high resistivity semiconductors using THz radiation.
In previously published reports in the literature [15, 16, 23], the variations observed in the refractive
indices have been much larger compared to the results in the present study as shown in Figure 2 (c). The
apparent discrepancies cannot be attributed to the difference in resistivity of the silicon wafers. Small
variations in the optical parameters are anticipated in the THz regions as the dispersion becomes smaller
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at longer wavelengths. Furthermore, the refractive index, by definition, does not depend on wafer
thickness and it is observed in this study for wafers of different thickness.
Figure 2: Comparison of (a) refractive index, (b) extinction coefficients
and (c) data from Ref [16].
Calculation of the Complex Dielectric Constant and AC Conductivity
Using the frequency dependent complex refractive index, real and imaginary parts of the complex
dielectric constants are calculated. The results of these calculations are plotted in Figures 3 (a) and 3 (b)
respectively for the 250 µm thick DSP silicon wafer.
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Figure 3: Plots of (a) the real part and (b) the imaginary part of the dielectric constant
The imaginary part of the complex dielectric constant is directly related to the AC conductivity [24].
The frequency dependent conductivity of the sample is shown in Figure 4. The value of the conductivity
at the lowest frequency, that is, at 0.2 THz gives a fair estimation of the DC conductivity, which is
equivalent to a resistivity of 2 – cm. This value of the resistivity is in good agreement with an
independent four probe measurement on the same wafer which yielded 4.5 – cm.
Figure 4: Conductivity as a function of frequency
Conclusions
A non-contact, non-destructive and real-time characterization technique for semiconductors using THz-
TDS has been introduced. By measuring the optical transmission spectrum, the analysis yields optical
parameters such as complex index of refraction by minimizing the errors between the calculated and the
theoretical values from the time domain experimental data and extends it to determine the electrical
parameters like the dielectric constant and conductivity of semiconductors. Also, since the effect of
multiple reflections has been included in the model, it opens up new vistas for non invasive material
characterization techniques for both optically thin and thick materials. Efforts to determine the mobility,
density of free carriers and the effect of sample roughness on the THz transmission spectra are currently
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ongoing. The future scope of expanding this study will include materials having multiple interfaces such
as the presence of overlayers on the substrate.
Acknowledgements
We acknowledge the assistance of Vishal R. Mehta in performing the four probe resistivity
measurements.
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