Optical Properties of Crystalline and Amorphous Semiconductors || The Reststrahlen Region

Chapterl

THE RESTSTRAHLEN REGION

2.1 STATIC AND HIGH-FREQUENCY DIELECTRIC CONSTANTS

2.1.1 General Consideration The strong IR absorption band found in heteropolar semiconductors is essentially bonded by the TO and LO frequencies for long-wavelength vibrations. The difference between these frequencies can be related to the amount of polar character in the chemical bond in these compounds. Below the reststrahlen range in optical spectra, the real part of the dielectric constant asymptotically approaches the static (or low-frequency) dielectric constant c,. The optical constant connecting the reststrahlen-near-IR range is called the high-frequency (or optical) dielectric constant c_. The high-frequency dielectric constant c_, thus, measured for frequencies well above the long-wavelength LO phonon frequency but below the fundamental absorption edge.

The dielectric constant c, can, in principle, be written as

S. Adachi, Optical Properties of Crystalline and Amorphous Semiconductors© Kluwer Academic Publishers 1999

34 THE RESTSTRAHLEN REGION

(2.1)

where M is the reduced mass of the crystal, N is the number of unit cells per unit volume, er* is the transverse effective charge, and CUro is the long-wavelength TO phonon frequency. The second term on the right-hand side of Eq. (2.1) is the lattice contribution which arises because the LO mode in heteropolar semiconductors produces a macroscopic electric moment separating it in energy from the TO mode [1]. The dielectric constants e, and e_ are also related to the long-wavelength TO and LO phonon frequencies CUro and COLO by the Lyddane-Sachs-Teller relation:

~ = ( CO LO]2 (2.2) e_ COro

In Fig. 2.1, we show the phonon dispersion curves, together with the corresponding DOS, for (a) Si and (b) GaAs [2] . When comparing these phonon dispersion curves it becomes apparent that, while the general shapes in these figures are similar, there is one important difference. At q=O (1) the TO and LO branches are degenerate in Si, while CUro<~o in GaAs. The origin of this splitting in GaAs is the

'e 400 .2.

'" " \j 200

" cr

" ... ....

~400

~

L x w L DOS

Figure 2.1 Phonon dispersion curves along several symmetry lines, together with the corresponding DOS, for (a) Si and (b) GaAs. (From Giannozzi et ai. [2].)

2.1 STATIC AND HIGH-FREQUENCY DIELECTRIC CONSTANTS 3S

electrostatic field created by the long-wavelength (q=O) modes of lattice vibrations. Thus, from Eq. (2.2) we note that E,-E. for homopolar crystals (such as Si and Ge) and E,>E. for heteropolar crystals (such as GaAs and ZnSe).

Figure 2.2 plots the values of E, and E. as a function of the lowest-direct-gap energy Eo for some of the III-V, II-VI compounds, and alkali halides [3]. Table 2.1 also lists the values of E, and E. for some of the group-IV, III-V, II-VI, and IV-VI semiconductors. We can easily find from this figure that both E, and E.

decrease with increasing Eo values. Note that E.1fl corresponds to the long-wavelength refractive index of the material. We can then conclude that the smaller Eogap material has a larger value of the refractive index. If the refractive index in the

Will

w8

~r--r------------~---------------'

~:LA--~O--~--~--~--~2--~--~!--~~~O BAND GAP Eo ( eV )

e'nSb 15

~ elnAs

eGaAs

10 e'nP

eAlSb

·GaP ·CdTe e

ZnTe

5 ·Zn5e

o~~ __ ~ __ ~ __ +-__ ~~ __ ~ o 2 3 5 10

BAND GAP Eo (eV)

Figure 2.2 Static and high-frequency dielectric constants, £, and £.." as a function of the lowest-direct-gap energy Eo for some of the III-V, II-VI compounds, and alkali halides. (From Adachi [3].)


TABLE 2.1 Values o/f., and f.M/or the group-IV semiconductors,IIl-V,ll-VI, and IV-VI semiconductors at T=300 K.

Material C!}:stal Structure f., f.M Remark

C Diamond 5.70· Si Diamond 11.6 b

Ge Diamond 16.04 c

a-Sn Diamond 24d

3C-SiC Zincblende 9.52' 6.38 • 2H-SiC Wurtzite 6.511 El.c

6.841 Ellc 4H-SiC Hexagonal 6.56' El.c

6.78 ' EUc 6H-SiC Hexagonal 9.66 h 6.520 h El.c

10.09 h 6.742 h Ellc 15R-SiC Rhombohedral 9.66 h 6.520 h El.c

1O.09 h 6.742h Ellc c-BN Zincblende 6.80 i 4.46 i

h-BN Hexagonal 6.85 i 4.95 i El.c 5.06 i 4.lO i Ellc

BP Zincblende 11k AlN Wurtzite 9.14' 4.84' El.c

9.14' 4.84' EUc AlP Zincblende 9.8 .. 7.601 AlAs Zincblende 1O.06 ft 8.16 ft

AISb Zincblende 11.21 0 9.88 0

a-GaN Wurtzite 8.9 P 5.2p El.c 9.8 P 5.2p EUc

~-GaN Zincblende 9.40 q 5.35 Q

GaP Zincblende 11.15 ' 9.09' GaAs Zincblende 12.90' 10.86' GaSb Zincblende 15.7' 14.4' InN Wurtzite 6.6" InP Zincblende 12.50 v 9.66 v

InAs Zincblende 14.60 v 12.25 v

InSb Zincblende 17.72 w 15.68 W

MgO Rocksalt 9.9' 3.3' ZnO Wurtzite 8.1 ' 3.95' El.c

9.0' 4.05' Ellc a-ZnS Wurtzite 9.6 00 5.7 00 El.c

9.6 00 5.7·· Ellc ~-ZnS Zincblende 8.9 00 5.7 00

ZnSe Zincblende 9.3 .b 6.2 ab

ZnTe Zincblende 9.1·· 6.7 00

c-CdS Zincblende 5.3 ac

w-CdS Wurtzite 8.4 .d 5.3 .d El.c 8.9 ad 5.4 .d Ellc

w-CdSe Wurtzite 9.29·' 6.20" El.c 10.16"' 6.30" Ellc

CdTe Zincblende 10.5'" 7.3'" H~Se Zincblende 8.0·'

2.1 STATIC AND HIGH-FREQUENCY DIELECTRIC CONSTANTS

TABLE 2.1 Continued.

Material Crystal Structure E,

HgTe Zincblende 21.0 ail

PbS Rocksalt 150 ai PbSe Rocksalt 203 aj PbTe Rocksalt 400 ok

SnTe Rocksalt 380 al

aI. Fontanella et al., Appl. Opt. 16, 2949 (1977). hJ<:. Seeger, J. Appl. Phys. 63, 5439 (1988).

15.2 ail

16.8 ai 23 aj 32.8 ok

45"'"

Remark

'M. N. Afsar et al., IEEE Trans. Micro. Theor. Tech. MTT·25, 505(1977). dR. E. Lindquist and A. W. Ewald, Phys. Rev. 135, AI91(1964). ·W. J. Moore et aI., J. Appl. Phys. 78, 7255(1995). 'Estimated from J. A. Powell [J. Opt. Soc. Am. 62, 341 (1972)]. 'Estimated from P. T. B. Shaffer [Appl. Opt. 10, 1034 (1971)]. hA. N. Pikhtin et aI., Opt. Spectrosc. 43, 420 (1977). iM. I. Eremets et al., Phys. Rev. B 52, 8854 (1995). jR. Geick et al., Phys. Rev. 146, 543 (1966). ~. Takenaka et al., Jpn. J. Appl. Phys. IS, 2021 (1976). IA. T. Collins et aI., Phys. Rev. 158,833 (1967). "'B. Monemar, Solid State Commun. 8, 1295 (1970). ftR. E. Fern and A. Onton, J. Appl. Phys. 42, 3499 (1971). OW. I. Turner and W. E. Reese, Phys. Rev. 127, 126 (1962). "H. Sobotta et al., Phys. Status Solidi B 174, K57 (1992). 'G. Mirjalili et aI., Infrared Phys. Technol. 37, 389 (1996). 'D. F. Parsons and P. D. Coleman, Appt. Opt. 10, 1683 (1971). SW. I. Moore and R. T. Holm, J. Appl. Phys. 80, 6939 (1996). 'M. Hass and B. W. Henvis, J. Phys. Chern. Solids 23, 1099 (1962). "K. L. Westra and M. J. Brett, Thin Solid Films 192, 227 (1990).

37

·S. Adachi, Physical Properties of /II-V Semiconductor Compounds: InP, lnAs, GaAs, InGaAs, and InGaAsP (Wiley-Interscience, New York, 1992).

'"R. B. Sanderson, J. Phys. Chern. Solids 26, 803 (1965). Wumericat Data and Functional Relationships in Science and Technology, edited by

K.- H. Hellwege and O. Madelung, Landolt-Bornstein, New Series, Group m, Vol. 17, Pt. b (Springer, Berlin, 1982).

YS. Cunsolo et at., Infrared Phys. 33, 539 (1992). 'E. F. Venger et al., Phys. Status Solidi 188, 823 (1995).

GlJA. Manabe et at., Jpn. J. Appl. Phys. 6, 593 (1967). abA. Deneuville et al., Phys. Rev. B 43, 6544 (1991). "Estimated from M. Cardona et at. [phys. Rev. 140, A633 (1965)]. "'M. Balkanski, in Opticat Properties of Solids, edited by F. Abeles (North-

Holland, Amsterdam, 1972), p. 529. OR. Geick et al., J. Appl. Phys. 37, 1994 (1966). "L. K. Vodop'yanov et al., Sov. Phys. Solid State 16, 912 (1974). a,s. Einfeldt et at., Phys. Rev. B 51, 4915 (1995). ~=77-K value [J. Baars and F. Sorger, Solid State Commun. 10, 875 (1972)]. aiR. Geick, Phys. Lett. 10, 51 (1964). aiR. Burkhard et al. Phys. Status Solidi B 63, 89 (1974). "0. Bauer and H. Krenn, in Handbook of Optical Constants of Solids, edited by

E. D. Palik (Academic, Orlando, 1985), p. 535. alSee, H. Lehmann et al., Appl. Phys. 25, 291 (1981). "'"R. F. Bis and I. R. Dixon, Phys. Rev. B 2, 1004 (1970).


active region of an injection laser is larger than the index of the cladding layer on both sides, the effect is like that of a waveguiding configuration which confines radiation to the neighborhood of the active region. This simple rule is known to be very useful for the design of various optical waveguiding devices.

2.1.2 Alloys To our knowledge, there have been reported no detailed experimental data on the dielectric constants of alloy semiconductors. For computational purpose, it is convenient to express the dielectric constant as a function of alloy composition. Let us consider a ternary alloy of the form AzB1_zC. We now take a linear representation for the composition dependence of the dielectric constant. The most simplest linear interpolation method is the use of the equation:

£(x) = xc(AC) + (I-x)£(BC) (2.3)

The dielectric constant £ of a medium containing a mixture of dipoles, with the ith type having polarizability uj and occurring with a density Nj , can be given by the Clausius-Mosotti relation:

£-1 =.!.E Nu (2.4) £-2 3 j I I

Based on the Clausius-Mossotti relation, we obtain another interpolation expression:

£(x) -1 = x £(AC) -1 + (I-x) £(BC) -1 (2.5) £(x)-2 £(AC)-2 £(BC)-2

In Fig. 2.3, we show the interpolated £, and £~ as a function of x for Inl_xGa.As ternary alloy. The solid and dashed lines are obtained from Eqs. (2.4) and (2.5), respectively. It is evident from the figure that these two interpolation schemes give nearly the same numeric values. It is easy to extend these expressions to more complex alloy systems, such as quaternary and pentanary systems.

2.1.3 Temperature and Pressure Effects We show in Fig. 2.4 (a), as an example, the temperature dependence of £, for GaAs measured by Samara [4] using a radio-frequency capacitance technique at different frequencies (I-100 kHz) at atmospheric pressure. Below -150 K there is no clear frequency dependence, but relatively strong frequency dependence is observed at higher temperatures. Samara considered that this is associated with dipolar contributions becoming activated at higher temperatures. The £,(1) response becomes frequency independent at 100 kHz up to -300 K and this represents the true £,(1) of the material.

Some data [5] also showed that at P-100 K the dielectric constant £, increases with increasing temperature according to the relation:

2.1 STATIC AND HIGH-FREQUENCY DIELECTRIC CONSTANTS 39

8

100~~~~~~~~~~1 0.5 x

Figure 2.3 Static and high-frequency dielectric constants, E, and E., as a function of x for Int_zGa,As ternary alloy. The solid and dashed lines are obtained from Eqs. (2.4) and (2.5), respectively.

Erl) = E.(T=O K) (1 + a.n (2.6)

The linear temperature coefficients a. determined were (2.0 1 ±O.02)x 10'4 K"' [e.(T=O K)=12.35±O.09] for GaAs, (2.27±O.02)xlO,4 K"' [E.(T=O K)=W.31±0.08] for CdTe, and (1.71±O.02)xlO,4 K"' [E.(T=O K)=8.80±O.07] for ZnSe [5]. As suggested by Samara [4], it is not likely that the quantities E. and E .. are truly linear dependence on temperature over any very wide range.

Figure 2.4 (b) shows the pressure effect on e. and E.., for GaAs as measured by Samara [4] using a maraging steel pressure cell up to -1.2 GPa. The dielectric constants decreased almost linearly with increasing pressure. He reported logarithmic pressure derivatives for GaAs to be d(lnE.}/dp=-16.7±O.3 (at 75.6 K) and -17.3±0.3 (at 300 K), and d(lnEJ/dp=-14.0 (at 300 K) (all are in units of 10'3 GPa,t).

We can expect that the static and high-frequency dielectric constants vary in a very similar way as in the case of the band-gap energy. This comes from the fact that the dielectric constant can be simply expressed as by a sum of two different contributions [4]: intrinsic term (no temperature dependence) and lattice vibronic term (temperature-dependent term). Although the sign of the lattice vibronic contribution is opposite, it is quite similar to an expression obtained by Varshni (6]

13.7ri----~--_r----~--;r--~----~--~--~

GaA

s

~ ~ u 2 .. Ii 13

.1

i

o

:w:

o

13

.20

~ 1

3.1

'

~ .., ~ E

z 1

3.1

2

o u H

~ ~ 1

3.0

' a

13

.04

,

ES

...... ...

... T

-75

.' K

, .....

. -, .....

. (b) ", ,

12

.74

:w:

12

.70

~

on .. l e ... I Z

e 1

2.6

6 I

i; z o u g :: u 1

2.6

2 ~

C

12

.58

10

0

20

0

30

0

TEM

PER

ATU

RE

(K)

40

0

0.0

0

.1

0.2

P

RE

SS

UR

E (

GP

a)

0.3

0

.4

Fig

ure

2.4

(a)

Tem

pera

ture

dep

ende

nce

of

the

stat

ic d

iele

ctri

c co

nsta

nt £

. fo

r G

aAs

at d

iffe

rent

fre

quen

cies

(1,

10

, an

d 10

0 kH

z) a

t at

mos

pher

ic p

ress

ure.

(b)

H

ydro

stat

ic-p

ress

ure

depe

nden

ce o

f th

e st

atic

die

lect

ric

cons

tant

£

, fo

r G

aAs

at t

wo

diff

eren

t te

mpe

ratu

res

(T=7

5.6

and

300

K).

The

das

hed

lines

rep

rese

nt t

he r

espo

nse

corr

ecte

d fo

r th

e ch

ange

s in

the

dim

ensi

ons

of t

he s

ampl

e du

e to

the

fin

ite c

ompr

essi

bilit

y of

GaA

s. (

From

Sam

ara

[4].)

""" ~ ~ I ~ ~

2.2 RESTSTRAHLEN SPECTRA 41

for the' calculation of the temperature variation of the band-gap energy. The temperature and pressure effects on the long-wavelength dielectric constant (refractive index) will also be discussed in Sec. 5.2.2.

2.2 RESTSTRAHLEN SPECTRA

2.2.1 Crystalline Materials (a) Zinc-Blende-Type Crystals-As we have already shown in Fig. 1.5, the frrst BZ for the cubic semiconductors is the truncated octahedron. Phonons representing lattice vibrations occurring at a frequency OOq have an energy 11ooq, where 11 is the reduced Planck constant, and have a momentum 11q. In the primitive cell, if there are N different types of atoms either of differing mass or ordering in space, 3N vibration modes will result. In general, three of these branches, namely the acoustic branches, will disappear at the zone center (T). The remaining 3(N-l) branches will be optical branches. For the cubic crystals, GaAs or Si, e.g., N=2, and hence there are three acoustic and three optical branches, each three comprising of one longitudinal and two transverse modes.

It is noted that in homopolar crystals (diamond, Si, Ge, and u-Sn) , the fundamental vibration has no dipole moment and is IR inactive. In heteropolar crystals, as in GaAs, the frrst-order dipole moment gives rise to a very strong absorption band associated with optical modes having an of essentially zero k vector (i.e., long-wavelength optical phonons). A useful description of the theoretical and experimental aspects of this subject was given by Balkanski [7].

The complex dielectric permittivity e(oo) in the long-wavelength limit can be generally explained by a single harmonic oscillator model:

(2.7)

where 'Y is the phonon damping constant. This expression can be modified using the Lyddane-Sachs-Teller relation of Eq. (2.2) as

The reststrahlen parameters, such as e., e~, COro, COw, and 'Y, can be determined from measurements of absorption, refractive-index dispersion, and reflection at normal and oblique incidence.


We show in Fig. 2.5 spectral variations of £1(00) and Ez(oo) for GaAs obtained from a numeric modeling of Eq. (2.7) [8]. Kachare et al. [9] measured the roomtemperature IR reflectance spectra of undoped, high-resistivity GaAs. The samples used were chemically polished to remove all traces of surface damage. The reststrahlen parameters they obtained were IDro=269.2 em-I, ~o=292.77 em-I, "(=2.5 em-I, and £.=10.90. Holm et al. [10] also measured the room-temperature reflectance of semiinsulating GaAs samples. They reported IDro=268.2 em-I, ~o=291.5 em-I, "(=2.3 em-I, and £.=11.1. We averaged in Fig. 2.5 Kachare and Holm's parameters to obtain ffiro=268.7 em-I, ~o=292.1 em-I, "(=2.4 em-I, and £..=11.0.

The imaginary part of Eq. (2.7) can be modified as

£2(00) = Soo~oooy

(2.9) (00~-002)2+CJ)2y2

with

2 2

s= £.( roLO -roTo)

(2.10) 2 ro TO

200

100

,- wLO + 500 .::-! GaAs IE

I :~~; Ii 1 200 I ::wlm(-e-) ........

j:t it 100 E .--//: :: -

01--__ -__ ·_-___ -·····../ rr \ . '--'--"--'-'- 0 •

w TO!

I --e11: -------- e 2 \

-100 300 K

200 250 300 350 400 FREQUENCY (cm-1)

Figure 2.5 Spectral variation of £(00) obtained from a numeric modeling of Eq. (2.7) with the reststrahlen parameters of semiinsulating GaAs determined at room temperature (IDro=268.7 cm- I , 0\0=292.1 cm-I, "(=2.4 cm-I, and £_=11.0). (From Adachi [8].) The spectral dependence of ooIm-£(ror l is also plotted.


Since the frequency COro is usually much larger than y, ~(ro) has a significant value only when ro is close to COro. For such values of ro, it is a good approximation to replace ro by COro everywhere in Eq. (2.9), except in the factor COra-ro, which gives

SroToy/4 e2(ro) = (y<roTO) (2.11)

(roro -ro )2+(y /2)2

The ~(ro) thus reaches a maximum value SCOrolY for frequency ro=COro (resonance frequency) and drops to half that peak value for frequencies ro=IDroJ:Y/2. Equation (2.11) is known as the Lorentzian line shape. The frequency region just above COro is one in which el is driven strongly negative (see Fig. 2.5). Note that in homopolar crystals (Si, Ge, and a-Sn), COrO=roLO then S=O [e,=e_, see Eq. (2.2)].

We also show that one has the relation [7]

e (ro) (e~l-e;l)ro~yro 1m - e(rotl = 2 = (2.12)

e l(ro)2+e2(ro)2 (ro~o-ro2)2+y2ro2

which becomes for ~o

( -I -I) 2 e_ -e, roLOY

ro[4(roLO _ro?+y2] (2.13)

Hence the maximum of the curve rolm-e(ro)"1 determines the frequency COr.o. This calculated result is shown in Fig. 2.5.

The nero) and k(ro) curves obtained for GaAs from Eqs. (1.19a) and (1.19b) are shown in Fig. 2.6 [8]. The absorption coefficient aero) and normal-incidence reflectivity R(ro) as calculated from Eqs. (1.21) and (1.22) are also shown in Figs. 2.7 and 2.8, respectively. Both n and k (a) dispersion curves show a strong peak at the resonance frequency ro=COro. The reflectivity R at ro close to COr.o is found to be nearly zero.

The optical properties at the reststrahlen region of semiinsulating crystals can be adequately well interpreted by the above-mentioned single harmonic oscillator model. If free carriers (electrons) are present in the medium, we must take into account plasma contribution to the dielectric permittivity (see details in Sec. 5.3.1). When the frequency of free-carrier plasma excitations (plasmons) is close to the frequency of the LO phonons (COr.o) , the two excitations interact via their macroscopic electric fields. The dielectric permittivity e(ro) in such case is given by [cf. Eq. (2.7)]

(2.14)


GaAs

10 --n

------ k

c:

300 K

0~2~0=0~~~2~5~O~'~~~~~~350

FREQUENCY (cm-1)

Figure 2.6 Numerically calculated spectral dependence of the real refractive index, nero), and extinction coefficient, k(ro), for GaAs using the same set of reststrahlen parameters as in Fig. 2.5. (From Adachi [8].)

where

ne 2 co = ( ]

112

p mee_eo (2.15)

is the plasma frequency, r is the plasmon damping constant, and n and me are, respectively, the concentration and effective mass ("optical mass") of the free electrons. The frequencies of the coupled plasmon-LO phonon modes are deduced in the limit of zero damping as the roots of the equation e(co)=O and are given by

= co~o +co~ + [( co~o +CO~]2 _ 2 2]112 co ± _ COpCOTO (2.16) 2 2

As the damping values increase, however, this equation is no longer valid. The two sharp dips in the reflectivity minima get smeared out and the coupled-mode frequencies become complex.

In the case of III-V crystals and their alloys, we must take into consideration the effects of multivalley nature of the conduction-band structure. A three-valley


5 ! WTO

4

I' GaAs E 3 ()

"I' 300 K 0 ,.... 2 C WLO

~ 1 1

0 200 300 400

FREQUENCY (cm-1)

Figure 2.7 Numerically calculated spectral dependence of the absorption coefficient, aero), for GaAs using the same set of reststrahlen parameters as in Fig. 2.5.

a:

0.5

WTO

0.0 200 250 300 350

FREQUENCY (cm-1)

Figure 2.8 Numerically calculated spectral dependence of the normal-incidence reflectivity, R(ro), for GaAs using the same set of reststrahlen parameters as in Fig. 2.5. (From Adachi [8].)


model is used to account for the r, L, and X valleys in the conduction band of such materials. Considering the r minimum as the energy reference, the electron concentration in the r minimum and those in the L (X) minima relative to the r are given by [8]

(2.17a)

nL(X) = (m~J312 [_ ilErL(X)] -- -_ exp __ _ nr m; kT

(2. 17b)

where EF is the Fermi energy and ilEn. (ilEne) is the energy gap between the r and L (X). The total free electron density can be given by n=nr+nL +nx. Then, the plasmon term in Eq. (2.14) is modified as follows:

002 2 2 2 p

-+ OOpr

+ OOpL

+ oopx

(2.18) oo(oo+iD 00(00 +irr) 00(00 +irL) oo(oo+irx)

with

( 'r ( 'r ( 'r nre nLe nxe (2.19) oor= --- , OOL= --- , 00 =

p m;£~£o p m;£~£o pX x

me £~£o

where m/, meL, and mex are the effective masses at the r, L, and X conduction bands, respectively.

In the case of p-type semiconductors, we must consider the contributions from the HH and LH subbands (with the hole effective masses mlUl* and mLH* and densities PHH and PLH, respectively). However, it should be noted that the sum of these contributions has the form similar to Eq. (2.18).

Equation (2.14) has been used as the basis for sorting out plasmon-LO phonon effects in a number of investigations [to-15]. Kukharskii [15], however, showed that this expression oversimplifies the coupling of plasmons and phonons and can lead, principally, to errors in deducing the damping constants 'Y and r. In Kukharskii's model (factorized form), £(00) is given by

(002+ir_00 -00:)(002+ir+00 -OO~) £(00) = £

~ 00(00 +iD(ro2+iyro -ro~o) (2.20)


In his model, two additional parameters are the damping constants r ± of the coupled plasmon-phonon modes. The plasmon and single-oscillator models are related to each other through the equation

(2.21)

Chandrasekhar and Ramdas [14] measured the IR reflectivity spectra of n-type GaAs doped with Te in the concentration range from 1.8x10!6 to 7.0x1018 cm-3• We reproduce in Fig. 2.9 the reflectivity spectrum as a function of carrier concentration for n-GaAs [8]. These spectra were calculated using the reststrahlen parameters as determined by Chandrasekhar and Ramdas [14]. These authors used a least-squares minimization algorithm to fit the experimental data to the theoretical expression of Eq. (2.14). They also used the factorized form of the dielectric function, Eq. (2.20), to improve the fits and found to be unnecessary.

In Fig. 2.9, the reflectance minimum at about 300 cm'! is due to the LO phonons, The position of the second minimum, called the plasma minimum, depends on the electron concentration. The plasma resonance minimum is very close to the TO-phonon line in the spectrum of the relatively lightly doped sample.

n-GaAs

-- ... - r--.. '\' ....... ~ "

-...... \\ \ , I \ \ n=7.0E18 cm-3 \ I \ \ _-------\ I, \ • ---\ ,:~ \ "' ..... n=3.3E18 em-3 \/:: \ --------

. , . ' -'

.: \ ,,---- -3 :. ../' n=8.6E17 em

I , ,'----~:1~7E;:; ~~:,; -----------

I ,

n=1.8E16 em-3

400 800 1200 FREQUENCY (cm-1)

Figure 2.9 IR reflectivity spectrum as a function of carrier concentration for n-type GaAs. These spectra were calculated using the reststrahlen parameters as determined by Chandrasekhar and Ramdas [14]. The reflectance minimum at about 300 cm'! is due to the LO phonons. The position of the second minimum, called the plasma minimum, depends on the electron concentration. (From Adachi [8].)


When the electron (carrier) concentration is larger, n=8.6x1017-7.0x1018 cm·3, the plasma minimum is shifted toward higher frequencies as seen in Fig. 2.9.

(b) Wurtzjte-Type Crystals-At normal temperature and pressure CdS crystallizes in the wurtzite lattice with two molecular units in the primitive cell which gives rise to twelve phonon branches (N=4). At the center of the BZ (1) the acoustic modes comprise the representations

(2.22a)

and the remaining nine optical modes the representations

(2.22b)

Except the silent BI modes all the other optical modes are Raman active, and the Al and EI modes are IR active for incident radiation with the polarizations E~c and El..e, respectively, c being the hexagonal axis of the crystal. We reproduce in Fig. 2.10 the phonon frequency vs. reduced wave vector results for q along the principal symmetry directions (T, L, A) of ZnO, predicted by a shell model based on elastic constants and Raman spectrum by Hewat [16]. The dashed lines at region near q~O, v~O (L) represent the initial slopes of the acoustic modes estimated from the elastic constants (sound velocities).

In the uniaxial media, two different solutions of the wave equation exist, which describe the ordinary (El..c) and extraordinary (E~c) rays [7]. Figure 2.11 shows spectral variations of EI(OO) and q(OO) for CdS obtained from a numeric modeling of Eq. (2.8). The numeric values used in the calculation are as follows: IDro=240 cm-I, "(=4.7 cm-I, e.=8.4, and E_=5.3 (El..c); IDro=232 cm-I, "(=6.3 cm-I, E,=8.9, and E_=5.4 (Ene). These values were determined from the reflectivity spectra for a crystal having its optical axis perpendicular to the surface and to the two polarizations of light at 290 K [7]. The LO phonon frequencies deduced from the curve ooIm-E(ooyl are O>r.o=302 cm-I (El..c) and 298 cm-I (Ellc).

(c) Alloys-In the classical picture, the atoms of a crystal are visualized as joined by harmonic springs and the crystal dynamics can be analyzed in terms of a linear combination of 3N normal modes of vibration. In alloy crystals, the nature of the lattice optical spectrum depends on how large is the difference between the quantities representing the lattice vibronic properties of the components (Le., atomic masses and force constants) [17]. If these quantities are similar, then the optical response of an alloy is similar to the response of a crystal with the quantities averaged over the composition ("one-mode" behavior). In one-mode system, such as most I-VII alloys, a single set of long-wavelength optical modes appears. The phonon spectrum is approximately constant in strength but varies in energy as the alloy composition is varied from one end member to the other.


18

-N Ez ¢: 12

A1 Ei ->-

() 9 C Q) ::l 0"6 Q) .....

LL

Figure 2.10 Phonon dispersion curves along the principal symmetry directions of ZnO, predicted by a shell model based on elastic constants and Raman spectrum. (From Hewat [16].) The dashed lines at region near q--+O, v--+O (1:) represent the initial slopes of the acoustic modes estimated from the elastic constants.

200 2000

CdS - Elc 150 ---- E II c

100 I' E ~

fo) 50 ~ 1000 :. I' · · · fo) · 0 · J... · · . E · . · . · . ~ -50 : .

· -100

Figure 2.11 Spectral variations of £1 and ~ for CdS obtained from a numeric modeling of Eq. (2.7). The numeric values used in the calculation are as follows: IDro=240 cm-\ "(=4.7 cm-\ £0=8.4, and £.=5.3 (E.Lc); IDro=232 cm-\ "(=6.3 cm-\ £0=8.9, and £.=5.4 (EOe). These values were determined from the reflectivity spectra for a crystal having its optical axis perpendicular to the surface and to the two polarizations of light at 290 K by Balkanski [7].


When the parameters differ strongly, the response of a system is more complex [17]; the spectrum contains a number of bands each of which corresponds to one of the components and has an intensity governed by its content in the alloy ("multimode" behavior). For example, two-mode system exhibits two distinct sets of optical modes with frequencies characteristic of each end member and strengths roughly proportional to the respective concentration. As we will see next, the longwavelength optical phonons in III-V and II-VI ternary alloys exhibit either onemode or two-mode behavior, or more rigorously, three different types of mode behavior, viz., one-, two-, and "one-two-mode" behaviors. The one-two-mode system exhibits a single mode only a part of the composition range, with two modes observed over the remaining range of composition.

An analysis of the optical spectra of multimode alloys shows that they are described satisfactorily by a set of oscillators and the number of such oscillators is equal to the number of different unit cells that occur in an alloy [17]. For example, in the case of A.BI_.C alloy there are unit cells of two types-AC and BC, so that the corresponding optical spectrum can be described satisfactorily by a twooscillator model. Tables 2.2 and 2.3 list behaviors of the long-wavelength optical modes in Si.Ge l _., some mixed III-V, II-VI, and IV-VI compounds obtained by IR optical and RS spectroscopies. Figure 2.12 shows typical examples of (a) one-mode and (b) two-mode behaviors of the long-wavelength optical modes observed in AIAsSb [18] and AIGaAs ternaries [8].

It is known that Si and Ge form a continuous range of alloys, Si.Ge l .• , from x=O (Ge) to x=1 (Si). The translational symmetry of the perfect diamond lattice forbids single phonon absorption of radiation because of a lack of a linear electric moment. Consequently, the absorption is probably due to the multiple-phonon processes which allow a second-order electric moment to be induced providing the coupling to the radiation field [19]. The loss of translational symmetry with alloying modifies the pure-crystal selection rules in IR optical spectra [20]. This also yields additional allowed RS from the entire density-of-phonon states as well as resonant or local modes [21,22]. It is found that the Si.Ge l .• alloy has three types of cell and, consequently, shows multimode behavior consisting of three main peaks due to the vibrations of Si-Si, Ge-Ge, and Si-Ge pairs [22].

Most III-V ternary compounds were found to exhibit either two- or one-twomode behavior [1]. However, it has recently been shown [18] that AlAsSb ternary displays one-mode behavior over the entire range of composition [O~I, see Fig. 2.12(a)]. GalnP is also the III-V ternary alloy for which one-mode behavior was currently assumed [23]. However, Jusserand and Slempkes [24] concluded from RS study that the alloy shows a somewhat modified two-mode behavior. Galtier et al. [25] considered from hydrostatic-pressure dependence of Raman spectra that the concept of a one-mode system is not appropriate for description of the physics of GalnP alloy.

In a quaternary alloy of the A.BI .• CyD1•y type, there are four kinds of unit cells: AC, AD, BC, and BD. On the other hand, in the A.ByC,D (AB.CyDz) type there


TABLE 2.2 Behavior of the long-wavelength optical modes in mixed Si-Ge and l/l-V alloy crystals.

System

Si-Ge

AIGaN AIGaP AIGaAs AIGaSb AlInP AlInAs AlInSb GaloN GalnP GalnAs GalnSb AIAsSb GaNAs GaPAs GaPSb GaAsSb InPAs InPSb InAsSb AIGaAsSb GalnPSb

Behavior

three

one two two two one-two two two one two two one-two one two two two two two two one-two three four

References

a

b c d e f g h

j k I m n 0

p q r p s t u

GalnPAs/InP four v GalnP As/GaAs four w GaInAsSb three x AIGaInP three f AIGaInAs three y InPAsSb three z

QT. Ishida et aI., J. Phys. Soc. Jpn 53, 2584 (1984). b K. Hayashi et al., Solid State Commun. 77, 115 (1991). C G. Armelles et al., Solid State Commun. 65, 779 (1988). dB. Jusserand and J. Sapriel, Phys. Rev. B 24, 7194 (1981) . • G. Lucovsky et al., Phys. Rev. B 12, 4135 (1975). tH. Asahi et ai., J. Appl. Phys. 65, 5007 (1989). g S. Emura et ai., J. Appl. Phys. 62, 4632 (1987). h v. P. Gnezdilov et al., J. Appt. Phys. 74, 6883 (1993). j K. Osamune et at., J. Appt. Phys. 46, 3432 (1975). j B. Jusserand and S. Slempkes, Solid State Commun. 49, 95 (1984). kG. Landa et at., Solid State Commun. 86, 351 (1993). I Z. C. Feng et at., Can. J. Phys. 69, 386 (1991). mI. Sela et al., Phys. Rev. B 46, 16142 (1992). "T. Prokofyeva et al., Appl. Phys. Lett. 73, 1409 (1998).

51

oN. D. Strahm and A. L. McWhorter, in Light Scattering Spectra of Solids, edited by G. B. Wright (Springer, Berlin, 1969), p. 455.

P Y. T. Cherng et at., J. Appt. Phys. 65, 3285 (1989). qT. C.McGlinn et al., Phys. Rev. B 33, 8396 (1986). 'N. P. Kekelidze et al., J. Phys. Chem. Solids 34, 2117 (1973).


• G. Lucovsky and M. F. Cohen, Solid State Commun. 8, 1397 (1970). I D. H. Jaw et at., J. Appl. Phys. 69, 2552 (1991). "D. H. Jaw and G. B. Stringfellow, J. Appl. Phys. 72, 4265 (1992). YR. K. Soni et at., J. Appl. Phys. 59, 2184 (1986). W T. Sugiura et al., Jpn. J. Appl. Phys. 32, 2718 (1993) . • C. Pickering, J. Electron. Mater. IS, 51 (1986). YR. Borroff et at., Appl. Phys. Lett. 53, 1652 (1988). , A. G. Belov et al., SOy. Phys. Solid State 22, 839 (1980).

are three kinds of unit cells: AD, BD, and CD (AB, AC, and AD). We can, thus, once expect ''four-'' or "three-mode" behavior of the long-wavelength optical modes in such quaternary alloys. Indeed, the quaternary compounds GaInPSb and GalnPAs are reported to show four-mode behavior. Also, GalnAsSb, AIGaInP, AIGalnAs, and InPAsSb exhibit three-mode behavior (see Table 2.2). However, the GaInAsSb system showed a three-phonon mode with GaAs, InSb, and mixed InAs/GaAs characteristics [26]. It was considered that the behavior of this alloy system may be termed "three-mode" because of the degeneracy of the GaAs- and InAs-like modes.

Mixed II-VI ternaries grow either in the zinc-blende or wurtzite structure and exhibit one- or two-mode behavior (see Table 2.3). However, optical phonons in ZnMn-based diluted magnetic semiconductors (such as ZnMnSe and ZnMnTe) exhibit a variation with composition that is intermediate between the one- and twomode situation. The II-VI quaternaries of the A)3yCp or ABxCP, type show a three-mode behavior, as can be expected from the number of different cells.

IR optical spectra of bulk alloy semiconductors consist of lattice absorption bands, the number of which is usually equal to the number of different unit cells. For example, reflectivity spectra of AlxGa1_xAs alloy consist of two lattice bands [two-mode behavior, see Table 2.2 and also Fig. 2.12 (b)]. The dielectric permittivity in such material can be successfully interpreted by the following additive multi oscillator form:

(2.23)

or by the factorized form:

(2.24)

where Sj [=£_(IDr.o/-roro/)] is the oscillator strength, rorOj (IDr.o) is the TO (LO) phonon frequency, and 'Ii is the damping constant of the jth lattice oscillator.

Figure 2.13 shows the calculated £1(0)) and ~(o» spectra for AlxGa1_xAs alloy


TABLE 2.3 Behavior of the long-wavelength optical modes in mixed II-VI and IV-VI crystals.

S~stem Behavior References

MgZnS one a MgZnSe two b MgCdTe two c ZnCdS one d ZnCdSe intennediate e ZnCdTe two f ZnMnS one g ZnMnSe intennediate h ZnMnTe intennediate CdMnSe two j CdMnTe two k HgZnTe two I HgCdTe two m ZnSSe two n ZnSTe two 0

ZnSeTe one p CdSSe two n CdSTe two q CdSeTe two r MgZnSSe three a MgCdMnTe three c ZnCdMnTe three s ZnCdHgTe three t CdSSeTe three u

PbSeTe two v

«D. Wang et at., J. Appt. Phys. 80, 1248 (1996). b D. Huang et at., Appl. Phys. Lett. 67, 3611 (1995). c E. Oh et al., Phys. Rev. B 48, 15040 (1993). tiM. Ichimura et al., Phys. Rev. B 46, 4273 (1992). • R. G. Alonso et at., Phys. Rev. B 40, 3720 (1989). 'D. N. Talwar et al., Phys. Rev. B 48, 17064 (1993). , A. Anastassiadou et al., Solid State Commun. 69, 137 (1989). ·P. D. Lao et at., Phys. Rev. B 48, 11701 (1993). j D. L. Peterson et at., Phys. Rev. B 33, 1160 (1986). iR. G. Alonso et at., Phys. Rev. B 43, 9610 (1991). l S. Venugopalan et al., Phys. Rev. B 25, 2681 (1982). 1M. P. Volz et at., Solid State Commun. 75, 943 (1990). m S. P. Kozyrev et at., Sov. Phys. Solid State 34, 1984 (1992). A I. F. Chang and S. S. Mitra, Phys. Rev. 172, 924 (1968). o P. Gupta et at., Thin Solid Films 227, 66 (1993). P S. Nakashima et al., J. Phys. Soc. Jpn 30, 1508 (1971). q R. Pal et at., J. Phys. D: Appl. Phys. 26, 704 (1993). 'Z. C. Feng et al., J. Cryst. Growth 138, 239 (1994). • E. Oh et al. Phys. Rev. B 45, 10934 (1992). I J. Polit et al. Phys. Status Solidi B 208, 21 (1998). N V. M. Burlakov et al., Phys. Status Solidi B 128, 389 (1985). • H. Finkenrath et at., Phys. Status Solidi B 95, 179 (1979).

S3

S4 THE RESTSTRAHLEN REGION

31001.---1.--0 ...... 25--'--0.1....50--'----=0-=. 7=-5 -'--~1.00

X

&Or-----------------~

(b)

't 400 AlAs-like

> U Z UJ => @ a: LL300

TO

x Figure 2.12 Long-wavelength LO and TO phonon frequencies in (a) AlAsSb (one-mode, from Sela et at. [18]) and (b) AlGaAs alloy (two-mode behavior, from Adachi [8]).

(x=0, 0.18, 0.36, and 0.54) [8]. Kim and Spitzer [27] measured the IR reflectivity spectra of LPE-grown AlxGa!_..As in the frequency range 240-700 cm-! at close to normal incidence (within approximately a 10° angle of incidence). The reflectivity data were analyzed by using the KK dispersion relations and by curve fitting to the two forms for the classical dielectric function, one the additive form of Eq. (2.23) and the other the factorized form of Eq. (2.24). In Fig. 2.13, the reststrahlen parameters used were taken from the factorized-form results of Kim and Spitzer. The numeric values of these parameters are listed in Table 2.4. The high-frequency


dielectric constant, t~, is simply obtained from the known values of GaAs and AlAs using the linear variation assumption of Eq. (2.3). The t(co) spectrum of AlAs is also calculated using the reststrahlen parameters of the additive form reported by Perkowitz et al. [28] and is shown in Fig. 2.13.

The corresponding n(co) and k(co) spectra are shown in Fig. 2.14 [8]. The two phonon bands, GaAs-like and AlAs-like, are clear and distinct in Figs. 2.13 and 2.14. No additional structure is seen in the reflectivity spectra in the frequency range 50-700 cm- I .

100

50

-50

-100

(\/

co

200

150

100

50

250

; )\ , \

0 250 ,\

(a)

- x=O(GaAs) ----- x=O.18 --- x=O.36 --- x=O.54 - x=1.0 (AlAs)

300 350 400 FREQUENCY (cm-1)

(b) - x=O(GaAs) ----- x=O.18 --- x=O.36 ___ xcO.54

- x=1.0(AIAs)

G~ike

300 350 400 FREQUENCY (cm-1)

Figure 2.13 Numerically calculated spectral dependence of the dielectric function, e(ro)=el(ro)+iez(ro), for AlxGal_..t\s alloy using the reststrahlen parameters as determined by Kim and Spitzer (x=O-O.S4, see Table 2.4) [27] and by Perkowitz et al. (x=l.O) [28]. (From Adachi [8].)


TABLE 2.4 Reststrahlen parameters for AIPaJ . .As alloy obtained from the factorized form [0. K. Kim and W. G. Spitzer, J. Appl. Phys. 50, 4362 (1979)].

GaAs-Like Mode AlAs-Like Mode

x IDro row X (TO) X (LO) IDro 0\.9 y(TO) y(LO) e. 0 268.8 292.8 2.65 2.85 10.90 0.14 267.1 285.7 5.67 4.85 358.8 369.0 10.56 11.31 10.57 0.18 266.9 283.4 8.76 4.24 360.1 372.4 12.20 10.24 10.47 0.30 265.2 278.3 8.64 6.15 360.2 379.1 12.10 9.42 10.16 0.36 264.5 276.5 10.69 5.58 360.4 381.3 12.23 8.08 10.04 0.44 262.9 273.7 10.05 6.44 360.2 385.4 9.55 7.90 9.84 0.54 261.8 269.8 12.43 7.97 361.5 390.1 8.75 8.68 9.60

(d) Temperature Effect-It is noted that the phonon frequency usually decreases with increasing temperature. The static and high-frequency dielectric constants are functions of temperature; they are usually increase with increasing temperature (see Sec. 2.1.3). The damping constant y in Eqs. (2.7)-(2.14) may also be strongly dependent on temperature (that will increase with increasing temperature). We can thus easily expect that the optical constants in the reststrahlen range of semiconductors are dependent on temperature.

Figure 2.15 shows, as examples, the reststrahlen refractive-index [(a)] and transmission spectra [(b)] for high-resistivity GaAs measured at 300 and 8 K by Johnson et at. [29]. These results show that the ionic contribution to the far-IR refractive index is different at 300 and 8 K, while the near-IR refractive index remains unchanged. The two transmission spectra shown in Fig. 2.15 (b) also suggest that cooling of this material has a marked effect on the optical absorption process.

2.2.2 Amorphous Materials The simplest way to describe the structure of an amorphous material consists of considering that the short-range order is preserved whereas the extended atomic network is random (i.e., the lack of long-range order) [30]. If the short-range order in the amorphous form is similar to that in the corresponding crystal, one can expect the Raman intensity and IR absorption to show some similarity to the phonon DOS of the crystal: essentially, the DOS in the amorphous form should be a broadened version of the crystalline state [31,32].

The main features of the phonon DOS in amorphous solids can be extracted from neutron scattering, IR absorption, and RS spectroscopies. In IR absorption spectroscopy, the absorption coefficient a(ro) is related to the phonon DOS g(ro) as

a(ro) - IM(ro) 12 g(ro) (2.25)

where M(ro) is a frequency-dependent dipole matrix element. In a single crystal


c

10

5

(a) x=O(GaAs) x=0.18 x=O.36 x=O.54 x=1.0 (AlAs)

o 200 250 300 350 400

(b)

10

5

o 200

FREQUENCY (cm-1)

x.O(GaAs) x .. O.18 x-o.36 x.O.54 x .. 1.0 (AlAs)

57

Figure 2.14 Numerically calculated spectral dependence of the real refractive index, n(ro), and extinction coefficient, k(ro), for AI%Gal'~s alloy using the same sets of reststrahlen parameters as in Fig. 2.12. (From Adachi [8].)

with one optical active phonon at COro(I), IM(roW is the Lorentzian-type function centered at COro(I), whose width is given by the damping constant resulting from the anharmonicity [see Eq. (2.11)]. In this case, IM(ro) 12 projects a narrow line out of the phonon density of the crystal.

In an amorphous material, where the translational symmetry is relaxed, at least in principle, all phonons are optically active and M(ro) may be different from zero in the whole frequency range of the phonons [33].

Figure 2.16 (a) shows the room-temperature IR absorption coefficient a(ro) for

58

4.5

4.0

3.5

c

3.0

2.5

80

o! 60 z 0 ill 40 If)

:i If)

~ 20 a: t-

O

(0)

, -o-3OOK -e- 8K

10

(b)

100 400 WAVELENGTH (pm)

, .

- 300K 8K

WAVELENGTH ( pm )

THE RESTSTRAHLEN REGION

Figure 2.15 Far-IR real refractive-index [(a)] and transmission spectra [(b)] for high-resistivity GaAs at 300 and 8 K. (From Johnson et at. [29].)

a-Ge [34]. The lower part of the figure shows the phonon DOS for c-Ge as obtained by neutron inelastic scattering (dashed line) and a Gaussian-broadened version of the crystalline DOS (solid line). We represent in Fig. 2.17 the equivalent results for GaAs [33,35,36].

All first-order IR absorption is forbidden in c-Ge and the absorption band in the amorphous material [Fig. 2.16 (a)] arises from the lack of crystallinity. If the transition is IR active in the crystal, it is relatively strong in the spectrum of the amorphous counterpart but is broadened considerably [Fig. 2.17 (b)]. It is also apparent from Figs. 2.16 and 2.17 that there is some relations between the IR absorption and the vibrational DOS. As mentioned before, this is in contrast to the typical crystalline case where only a few special vibrations of suitable momenta and symmetry are IR active (see Fig. 2.7). All structures are also shifted to lower frequencies in the amorphous material relative to the crystal.


200

Ge

150 - (a) ,... IE 0 :::: 100 a ~

50

-a Oi

\ 1',

" I' 1\ I , I\,.. I '

LO(r) -l TO(r) I

S9

Figure 2.16 (a) Room-temperature IR absorption coefficient aero) vs. wave number for aGe. Also shown for comparison is the reduced Raman spectrum. (b) Phonon density of states of c-Ge (dashed line) and a-Ge (solid line). (From Brodsky and Lurio [34].)

If the matrix elements M(ro) were weakly dependent on ro, the absorption curve would very resemble the amorphous phonon DOS. A comparison in Figs. 2.16 and 2.17 of the broadened DOS with the experimental absorption spectra clearly shows that this is not the case: while M(ro) is nearly constant in the higher-frequency range, it must decrease rapidly at lower frequencies. A consequence of this fact is that the TA branch does not appear in the IR absorption (a-GaAs) or its strength is very weak (a-Ge). In the elastic quasi-continuous limit, Pretti et al. [33] have argued that the frequency dependence of M(ro) can be given by

(2.26)

where ~ is the frequency at the maximum of the phonon DOS. Above roM they


6

'E u

M g ~

'3 ~

2

o 100 200 300 ENERGY (cm-1 )

Figure 2.17 (a) Room-temperature IR absorption coefficient aero) vs. wave number for aGaAs. Also shown for comparison is the reduced Raman spectrum. (From Prettl et al. [33].) (b) Phonon density of states of c-GaAs (dashed line) and a-GaAs (solid line) taken from Dolling and Cowley [35] and Smith et al. [36], respectively.

assumed the IM(ro) 12 to be independent of 0). The agreement with experiment was found to be excellent.

Also shown by the dashed lines in Figs. 2.16 (a) and 2.17 (a) are the roomtemperature Raman spectra. The RS cross section OR of an amorphous material is given by

OR oc ~ lel 2 f (11ro ) g(O) 0) IJ kT

(2.27)

where f(x) is the Bose-Einstein distribution factor and Cjj measures the coupling of the vibrational modes of frequency 0) with the incident and scattered light. Considering the quasi-continuous limit, PrettI et al. [33] also obtained the dependence of ICj/ro) 12 on ro (O)<O)M):

REFERENCES 61

(2.28)

Above COM it was assumed to be independent of co. They found fair agreement between the IR absorption, RS, and neutron scattering DOS. For small co Connell [37] simply assumed that M(co)oc co2 and CJco)occo so that

a(co) 0"' g( co) ex: __ oc ~ (2.29)

co 4 co2

where O"R'ECOj·IO"R is the reduced Raman cross section (representing the density of phonon states).

Felty et al. [38] studied the IR absorption oands in charcogenide glass alloy As2SxSe3-x- They observed two clear bands at frequencies approximately 300 and 215 cm·1 which did not vary in position but only in strength. The frequencies corresponded to the reststrahlen bands in pure ASzS3 (-300 cm·l) and As2Se3 (-215 cm· I ). On the other hand, Taylor et al. [39] observed a qualitatively different behavior in a T12SexTel.xAs2SeyTe3.y glass system. The intensity of the two observed IR bands did not change much with composition but their frequencies shifted. To our knowledge, however, no detailed experimental or theoretical data on the reststrahlen bands in tetrahedrally-coordinated amorphous semiconductor alloys has been reported so far.

REFERENCES

1. G. P. Srivastava, The Physics oj Phonons (Adam Hilger, Bristol, 1990). 2. P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, Phys. Rev. B 43,

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11. A. Mooradian and G. B. Wright, Phys. Rev. Lett. 16, 999 (1966). 12. J. D. Axe and R. Hammer, Phys. Rev. 162, 700 (1967). 13. c. G. Olson and D. W. Lynch, Phys. Rev. 177, 1231 (1969). 14. H. R. Chandrasekhar and A. K. Ramdas, Phys. Rev. B 21, 1511 (1980). 15. A. A. Kukharskii, Solid State Commun. 13, 1761 (1973). 16. A. W. Hewat, Solid State Commun. 8, 187 (1970). 17. G. M. Zinger, 1. P. Ipatova, and A. 1. Ryskin, Sov. Phys. Semicond. 18, 13

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(1984). 26. C. Pickering, 1. Electron. Mater. IS, 51 (1986). 27. O. K. Kim and W. G. Spitzer, J. Appl. Phys. 50, 4362 (1979). 28. S. Perkowitz, R. Sudharsanan, S. S. Yom, and T. 1. Drummond, Solid State

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Optical Properties of Crystalline and Amorphous Semiconductors || The Reststrahlen Region