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)

41

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

42

( )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.

43

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

44

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

46

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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