CHINESE JOURNAL OF PHYSICS VOL. 14. NO.

2 SUMMER, 1976

Study on the Optical Properties of

Semiconductors by Kramer+Kronig Transformation*

L . L . KI A N G (%.rat;)
Department of Physics, Tsing Hua University, Hsinchu, Taiwan

(Received 6 March 1976)

The Kramers-Kronig relations can be applied to response functions which

satisfy causality. It is easy to compute all the optical constants from the reflec-
tivity data which is measured through a portion of the frequency range by using
these relations.
A new method for optical data analysis based on expansion in Hermite func-
tions and reasonable extrapolations was employed. This method was applied to
SbzTea and Sic. The results of the computation of the optical constants for both
materials were presented and discussed. The limitations of the analysis were also
pointed out.

I. INTRODUCTION
H E electrons in crystals are arranged in energy bands separated by regions in energy for which no
“IIP electron energy states are allowed; such forbidden regions are called energy gaps or band gaps.
Photon absorption with electronic transitions from the valence band to states in the conduction band
can take place in an insulator or semiconductor, if the photon energy hw~E,; E, is the minimum
energy of the band gaps. This kind of transitions is called intraband transitions. If ho<E,, intraband
transitions can not occur. But for a degenerate semiconductor whose conduction band or valence
band is partially filled with electrons or holes, respectively, intraband transitions can occur.
Generally, there always exists a functional relation between the real and imaginary parts of any
complex optical function. Their connection is realized through the Kramers-Kronig transformationcl’3,.
These transforms connect dispersive with absorptive processes. In section II, the dispersion relation
for classical charged oscillators is introduced. In section III, there is a discussion of the numerical
method for computing Kramers-Kronig transforms. If the integral over frequency space of a function
closely related to the logarithm of the reflectivity is convergent, it is possible to expand this function
in a Her,mite series. In section IV, the results of application of this method of Kramer+Kronig trans-
forms to the material Sb,Te, and Sic are explained.

II. DISPERSION RELATIONS IN SOLID

The electric field within a solid subjected to an external field with clearly depend upon the
polarizability of the medium. It is assumed when an electromagnetic wave passes through the medium,
the positions of the electrons are altered by the induced dipole moment, and that the positions of the
nuclei are not affected by the wave. Simply, we assume that if the electrons are bound by a linear
restoring force and if there is damping proportional to the velocity, then the equation for the ath
electron is
+ This work was partially supported by the National Science Council of the Republic of China.
( I ) J. Taut, The Optical Properties of Solids, Academic Press, New York London, 1966.
(2 ) J. N. Hodgson, Optical Absorption and Dispersion in Solids, Chapman and Hall Ltd, 1970.
(3 ) J. B. Marion, Classical Electromagetic Radiation. (Academic Press, New York London, 1965).

93

I,.
I_._ _
 94 STUDY ON THE OPTICAL PROPERTIES

nzr, + .&, + rcllra = e ( E + A3TP) . (1-l)

Assume .&jm=2P,, r,/m=w:. W e g e t

P,=era(f)=

If there are N electrons per unit volume, P=xNf,P,,

(I from eq. (l-2)

(l-3)

By the dielectric medium

we know that the polarization, at a certain point and time, will not only depend on the electric field
at time t, but also on the electric fields at all times previous to it@).

4nP(t)=~~~~((l_r)E(r)dr, (I-5)

where, E(t) is a harmonic field, E(f)=E, exp( -iof)

then i(w)=1 +Smf(l)e+W. (l-6)
0
If the integral converges, this gives i(o) as a complex number
L(to)=tl(u)+icZ(ti), (l-7)
where the imaginary part Ed describes dissipation of electromagnetic energy, and the real part E~ des-
cribes dispersion. Inversion of the Fourier transform in (l-6), if the causality condition is satisfied,
then, two integral relations between Ed and E*(W) can be derived

s
e,(w)=-; IDdWre,(ol)[W/(OZ-0/2)].
0
(l-9)
Eqs. (l-8) and (l-9) are also called Kramers-Kronig relations (K-K relations). By the K-K relations,
if we know the absorption of a solid in the whole frequency range, we can determine the dispersion
in the whole frequency range. From h4axwell equations we obtain
^
n^z=e. (l-10)
I; is the complex refractive index (n^=n+ir). Also, the relation between h and j is
,. .1i;
n=
(1-11)
1-F ’
where ; is the reflected wave from the surface of a medium. For the case of normal incidence
;=r exp(iil) (1-12)
and R=r2.
The K-K relations can also be applied to other functions for which the integrals converge.
One relation which has been widely used in the analysis of experimental data gives the phase
change on reflection, d, in terms of the measured reflectance R(w)(‘y5). That relation is employed
(4) F. Borghese and E. Donato, IL Nuovo Cimenro, LIIlB, 116 (1968).
(5 ) D. L. Greenaway and G. Harbeke, J. Phys. Chem. Solids (Pergamon Press) 26, 3585 (1965).

-ii.. ._:
 L. L. KIANG 95

in this work. Since all of those optical constants are frequency dependent, we can calculate i(u) by
K-K relations. From the functions (I-11) and (1-12) the complex dielectric function $0) are easily
obtained.
The K-K relation which gives il in terms of R is

d(~)=$~“do~[-~-- In R(u~)~/(co~-O/~)]. (l-13)

0
A convenient form of this equation for calculation can be write as

(1-14)

III. METHOD FOR KRAMERS-KRONIG TRANSFORMATION

Since eq. (I-14) has a singular property at w’=w, we employ Hermite expansion for this K-K
relation to deal with the problem. Suppose we have two systems of function +((o’) and g(o) linearly
related to K(o, w’) as

ID(~)= JK( w, o’)$((w’)do’, (2-l)

if @J(W) and C&(O) have a convergent Hermite expansioncBt7’
C.W=~%X(~)~ (2-2)
tim) = I$LWL(@). (2-3)

Then the K(w, 01) can also be represented as a double Hermite expansion
K(w, w’) =Y&JL(~)~nl(~‘), (2-4)
rim
so that the integral equation (2-l) is reduced to the linear matrix equation:
aI8 = IYLLLI (2-5)
and S., =~&Ld.,, (2-6)
in which IV,, and N, are the normalization factors for the nth and mth Hermite ‘functions, and we have

unnl= - dre-“SGN(t)H,(t)H,(r), (2-7)

J-0
where SGN(r) is the signature or Signum functionc8) (- 1 for t<O, t-1 for t>O and 0 for t=O). This
Signum function is a generalized function, and has the property that

-jl- SGN(r)=2B(t). (2-g)

Using the Hermite integration relation

Ze-'"r_r(r)dr=H,_,(0)-e-~2H"_,(x), (2-9)
J0
we have
ff.,=[e-‘ZH,_~(t)SGN(t)H,(r)] I-“, +s- ~w~~H._~(~)-$ [SGN(f)H,(r)]. (2-10)
-m
As the boundary term vanishes, and I$(t)=2mH,_,(t)
(6) G. F. Koster, Solid State Phys., 5, 174 (1957).
(7) W. Procarione and C. Wood, Phys. State Sol. 42, 871 (1970).
( 8 ) M. J. Light Hill, I~froduckn lo Fourier Analysis and Generalized Furrrions, (University Press, Cambridge,
1959).
 96 STUDY ON THE OPTICAL PROPERTIES

u ,rFn = m dre-‘*~,_~(t)[2s(t)~,(t) +2mSGN(t)H,_,(~)l

J-10
= 2r~,,_,(o)r3,,,(0) -I-2r$- dte-“SGN(t)H,_,(r)lr,_,(r)
-0D
=2H,_,(O)H,(O)+2~~.-~, m-1. (2-l 1)
Obviously,
u,, =2H*(O)Hm_,(O) +2n%_,, m-1. (2-12)
Eliminating u,+ ,,,_* we get

(2-13)

Since H,.+,(O)=O, and H,,,(O)= -(3:1(2!)1~, the value of CT,,,,, depends on whether (n, m) is (even, odd)
or (odd, even).
From eq. (2-6), we have
1 1
N,,N,,, = J 2n+mnntmt ’
. .
and S,,,,= -S,,,.
.
S,, is called the matrix representation of the Hilbert transform operator in the space of Hermitc
functions. Moreover, these function possess simple properties in the complex plane.
Since the Kramers-Kronig transformation has the linearity property, we can subtract the corre-
sponding phase of a bulk portion of the K-K transformation from the data which fits the data near
the origin and also at the cut-off frequency point. Then
A@>=&,W)+&o). (2-14)

The classical charge-oscillators are the best choices for this purpose. So that for 4,(w) we can cal-
culate from eq. (l-12) and for 0(o) we can calculated by K-K transformation

(2-15)

where

g(o’) has come from the data fit by oscillators. Transforming both B(o) and 4((w’) to the space of
Hermite functions, and by eq. (2-5)
S,=Z%,&l. (2-16)
m
IV. APPLICATIONS
This method has been applied to Sb,Te, and Sic two kinds of semiconductors. The results were
quite satisfied. The reflectivities of the unpolarized light of the cleaved samples of Sb,Te,c’) and
Sic@) were shown in Fig. 1 and Fig. 2 respectively.
The data was analyzed by using two or three sets of classical electron oscillators with two or
three different resonant frequencies. The conditions of those sets of oscillators were required in con-
sistent aith the data both at low and high frequencies region. Also, &(w’) in eq. (2-15) was kept as
small as it could possible be, and rather smooth. In this studies, 4(~‘) was expanded in Hermite
series to 250 order for Sb,Te, and 100 order for Sic.
By Kramers-Kronig transformation, we calculated A(w’), then other optical constants could easily

(9) M. L. Belle, N.K. Prokofeva and M.B. Reifman, Soviet Phys. Semiconductors 1, 315 (1967).
 L. L. KI.ANG 97

80 -

2 5 4 5 CY

Fig. 1. Reflectivity of SbzTe,.

Ft
%
30I’

2 ct-

IC )-

, ! / I !
C )- 3 5 7 9 II 13 (ov 1

Fig. 2. Reflectivity of @ic.

A--- ”
98 STUDY ON THE OPTICAL PROPERTIES

O-

-10 -

-20 -

-50 .

I ,
1 2 5 4 m 5 cl’

Fig. 3. Dielectric constants <I and ~2 of Sb2Tes from Kramers-Kr6nig analysis.

1.1

0.

0.

-0

ev-
Fig. 4. Dielectric constants EL and EZ o f PSiC from Kramers-Krijnig analysis.

.__.
 L.L. KIXNG - 99

Fig. 5. Optical constants n and 6 of SbzTes from Kramers-Kranig analysis.

0.5
t

-0.5-
/

/ I
3 5 7 9 II I3 (ev)
*Fig. 6. Optical constant n and d of FSiC from Kramers-KrGnig analysis.

be computed as i by eq. (I-11) and E^ by eq. (l-10). Fig. 3 and Fig. 5 show the real and imaginary
parts of the complex dielectric constant. Figs. 4 and 6 show the real and imaginary parts of the
index of refraction.
Compared the spectra of the complex dielectric constant from Figs. 3 and 5, this method had
better results in low energies range than the OPW method( It was obviously that the spectrum
of the imaginary refraction index (Figs. 4 and 6) was similar to the spectrum of reflectivity. If we
use the values of d(w) (aSiC’s) to calculate the absorption coefficients, the results are quite acceptable
by comparison with the experimental results(*lvlz’.
From Fig. 4, the plasma frequency of Sb,Te, was calculated as wpz0.14ev which was approxi-
mate in agreement with the results of Hurch et uZ.(‘~‘. They estimated o,-0.129 ev from thermal
(10) Frank Herman. J.P. Van Dyke and R.L. Kortum, Math. Res. Bull. 4, S167 (1969).
(11) W.G. Spitzer, D.A. Kleiman, C. J. Frosch and D. J. Walsh, Silicon Carbide, 1960, eds. by O’Connor,
J. R. & Smiltens, J., Pergamon.
(12) H.R. Philipp and E.A. Taft, Silicon Carbide, 1960, eds. by O’Connor,’ J.R. & Smiltens, J., Pergamon.
(13) Z. Hurych and C. Wood, Solid State Comm. 9, 181 (1971).

-u..-, ..-. ,.. ._

 100 STUDY ON THE OPTICAL PROPERTIES

modulation, and o,=O.169 ev by the method of extrapolations. Also we calculated the D. C. electrical
conductivity a,=2.06x 10*3sec-1 and the electron relaxation time z=O.19 x IO-‘*sec. The plasma
frequency deduced from these two values was 0,~0.152 ev, this value was in good agreement with.
The plasma frequency of SIC was calculated from Fig. 6 as ~~~0.2 ev. The obtained value is a
little higherc”) (they indicated op- 1.8 rad/s=0.12 ev). By extrapolation values of eI, we calculated
G,,= 1.1 x lO%ec+, which was one order less than J. H. Racettes’ results(15’ (u,=8.0 x 103(ohm-cm)-* =
7.2x 10*W1), also the relaxation time 2=0.4x 10~“sec. This two values deduced the plasma frequency
as op 2: 0.24 ev.
Since we used the extrapolation in both ends of the low and the high frequencies in this method,
the procession of the analyzed values has a limitation. In our calculation, since the measured re-
flectivity of Sb,Te, was in the range of 0.2 ev-6.3 ev, the calculated values of the plasma frequency
tip, the DC electrical conductivity 6, and the relaxation time T were acceptable. However, the results
obtained from the energy loss function Im-!- were obviously incorrected for which frequencies
( s)
higher than 0=4.5 ev. As for the sample Sic, the calculated values of tip, 0, and r were not so
close to the experimental values that calculated as the values for Sb,Te,. However, in the range of
2-11 ev, the results were still good, that was helpful for studying the band structure of Sic.

V. ACKNOWLEDGEMENT
I would like to express my appreciation to Dr. John C. Shaffer, Northern Illinois University,
.
for his kindness advice and guidence for this work.

(14) T. S. Moss, G. J. Burrell, B. Ellis, Semiconductor Oplo-Electronics, (Butterworth, 1973).

(15) J.H. Racette, Phys. Rev. 107, 1542 (1957).