EFFECT OF PHONON-SCATTERING ON FREE-CARRIER ABSORPTION IN N-TYPE INDIUM-ANTIMONIDE FILMS

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ELSEVIER Physica B 205 (1995) 183-192

Effect of phonon scattering on free-carrier absorption in n-type

indium antimonide films

C h h i - C h o n g W u a' *, C h a u - J y L i n b

alnstitute of Electronics, National Chiao Tung University, Hsinchu, Taiwan bDepartment of ,4pplied Mathematics, National Chiao Tung University, Hsinchu, Taiwan

Received 15 March 1994; revised 27 July 1994

Abstract

The free-carrier absorption in n-type InSb films has been investigated for carriers confined in quasi-two-dimensional (2D) semiconductors with the nonparabolic energy band of electrons. We discuss the effect of phonon scattering on the free-carrier absorption coefficient (~) for both deformation-potential coupling and piezoelectric coupling. ~ is found to depend on the photon polarization relative to the direction normal to the quasi-2D structure, the photon frequency, the film thickness, and the temperature. ~ could be complex due to the interaction between photons, phonons, and electrons. (i) When the acoustic phonon scattering is dominant, ~ increases with decreasing the film thickness for phonons polarized parallel or perpendicular to the layer plane. It is also shown that ~t increases with decreasing photon frequency and increasing temperature for photons polarized parallel to the layer plane, while for photons polarized perpendicular to the layer plane the ct temperature-dependence is more complicated. (ii) If the piezoelectric scattering is dominant, ~ is also decreasing with increasing the film thickness for photons polarized parallel or perpendicular to the layer plane. But ct decreases with increasing temperature for photons polarized perpendicular to the layer plane. Moreover, numerical results for the parallel polarization are much smaller than those for the perpendicular polarization.

1. Introduction

There has been a growing interest in the electronic and optical properties of semiconducting layered heterojunctions [1-6], thin films I-7, 8], and inversion layers [9-11]. Due to the confinement of carriers in these quasi-two-dimensional structures, size quantization begins to play an important role in determining their electronic and optical properties. The optical absorption can take place via di- rect interband transitions, and intersubband * Corresponding author.

transitions, and indirect intraband transitions in which the carriers absorb or emit a photon while simultaneously scattering off phonons or other im- perfections. Such free-carrier absorption accounts for absorption of photons with frequency f2 lower than the band gap Eg of semiconductors, i.e., hf2 < E~. In a magnetic field, where the carrier motion is confined to a plane perpendicular to the magnetic field, the free-carrier absorption coefficient (~) depends upon the photon polarization relative to the magnetic field [12]. For carriers confined in a quasi-2D structure, it was found that depends upon the photon polarization relative to

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184 C.-C Wu, C-J. Lin / Physica B 205 (1995) 183-192

the direction normal to the quasi-2D structure [13]. In III-V semiconductors, the acoustic- phonon-carrier interaction is dominated by the deformation-potential and piezoelectric couplings [14].

In this paper, we investigate the quantum theory of the free-carrier absorption in III-V semiconductors with the nonparabolic energy band of electrons with a quasi-2D structure. It has been shown that the nonparabolicity of the energy band in semiconductors can be used to explain the longitudinal magnetoacoustic phenomenon [ 15] and the acoustic-surface-wave amplification [14]. In this calcu- lation, we consider a nondegenerate electron gas with a Maxwell-Boltzmann distribution for nondegenerate semiconductors. Thus, we make follow- ing assumptions:

(1) For a nondegenerate semiconductor, the distribution function of electrons can be represented by the Maxwell-Boltzmann distribution. The effect of Fermi energy can be neglected for the nondegenerate electron gas in a nondegenerate semiconductor [13].

(2) The interaction between electrons and acoustic phonons originates from the deformation-potential and piezoelectric couplings in III-V semiconductors.

(3) The energy band of electrons in semiconductors is assumed to be of nonparabolicity.

In Section 2, we recall the electronic states of a quasi-2D structure for the nonparabolic band structure. In Section 3, we present the quantum theory of the free-carrier absorption with confined carriers in nondegenerate semiconductors for the deformation-potential and piezoelectric couplings. We discuss two special cases: the photon is polarized parallelly or perpendicularly to the layer plane. In Section 4, some numerical results of the free- carrier absorption coefficient are presented for n- type InSb. Finally, a brief discussion about our numerical analysis is given.

2. Electronic states in a thin semiconductor layer

The motion of conduction electrons parallel to the thin films may be described by plane waves, and those perpendicular to the surface will be described

by the type of standing waves depending on the structure of potential. For a square well potential along the z axis with infinitely high barriers at z = 0 and z = d, the electron field operator ~*(r) is given [16] by

~+(r) ~, ~ , b ~ n e x p ( - i k ~ [ n n z \ = .=1 k "x)sin ~---d-) '

(I)

where r = (x, z) = (x, y, z), V = dS is the film vol- ume, S being the surface area, and d the film thickness, k = (kx, k r) is the electron wave vector in the x - y plane, and bk*, is the electron creation operator. The electron energy Ekn is given [14, 17-19] by the relation

Ek,~ h 2 k 2 n 2 h 2 n 2 ek, = Ek, l + --E-~g ) = ~gm. -~- 2m.d---~,

n = 1, 2, 3, . . . , (2)

where m* is the effective mass and ek, is the effective eigenvalue due to the effective Hamiltonian [19]. The electron energy for the nonparabolic band Ek, in Eq. (2) can be written as

+ \E,] \2m* + 2-2-m--Vd~]] J

n = 1, 2, 3, ... (3)

Since (h2k2ax)/2m* + (nhn)2/2m*d 2 _~ kBT < Eg for T ~< 300 K, Eq. (3) can be expanded as

Et, n ~- - ½E~ + ½E~a, + h2k2/2m*a. (4) with a. = [1 + (2n2h2n2)/(m*d2Eg)] 1/2. (5) In the case of low photon energies, the electron energies received from the photons are also small. Thus only low quantum numbers will contribute to the system. Thus we may expand a. from Eq. (5) for the second term of Eq. (4) and obtain the expression

n 2 h 2 n 2 h 2 k 2

E~n ~- 2m.d----~ + 2m* (6)

with m* = m*a,. This expression for Ek~ is almost the same type as shown in Ref. [19]. In our present work, we consider any values of photon energies, and the energy received by electrons is not so small. Hence we use the nonparabolic band of electrons as

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C.-C. Wu, C.-J. Lin/Physica B 205 (1995) 183 192 185 shown in Eq. (4). It is seen that Eq. (6) can be

reduced to the parabolic band of electrons in semiconductors as the parameter a, is taken to a limit one. Since the quantum number n cannot be zero, thus the parabolic band of electrons is only a limit model of the nonparabolic band of electrons in III-V semiconductors.

where ~ is the photon polarization vector and p is the electron momentum. Its matrix elements for electrons in the same band are given as follows:

(i) When the photon is polarized parallelly to the layer plane,

<k'n'lH~adlkn>

3. Quantum theory of free-carrier absorption in

quasi-2D nondegenerate semiconductors

The absorption coefficient ~ for the absorption of photons can be expressed [-13] as

El/2

= - - ~ W, f~, (7)

n 0 c "~-.

where ~ is the dielectric constant of material, no is the number of photons in the radiation field,f~ is the distribution function of carriers and Wi is the transition probability. It is given with the Born approximation

2 ~

W g - ~ - f ~ [ l < f l M +

li>126(Ef - E~ - hK2 - hc%)

+ I<flM-li>12,~(Ef - E~- ht2 +

h~%)-I,

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where Ei and

E:

are the initial and final electron energies, hQ is the photon energy, h~% is the phonon energy. For interaction between electrons, photons, and phonons, the transition matrix elements

<fire+ li)

are given by

<flM+_

Ii)

V

l Ej - Ei T- hogq

< fl Vslj)__~jtHradli>

~

+ Ej -- E~ - hf2 J ' (9) w h e r e H r a d is the electron-photon interaction and V, is the scattering potential due to the elec- t r o n - p h o n o n interaction.

The electron-photon interaction is given by

e_(_( 2nhno ~ 1/2

Hrad-

**m*\**

eOV /

n'P'

(10)

e (2~hno~ 1/2

-

~ \ ~ / t

e'(hk)6"""6kx'kx6k~'kY"

(11)

(ii) When the photon is polarized perpendicularly to the layer plane,

ieh(2nhno~e/2(d )

(k'n'lHradlkn)

= m * \

ef2V ]

J'l -- cos [•(n' + n)] x n' + n 1 -- cos[n(n' -- n ) ] ] + n ' - - n

Xfk, kxfk, ky.

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The distribution function for a quasi-2D nondegenerate electron gas in nondegenerate semiconductors can be expressed as

f~" = \2n2mkaT]*

[_,=1 2--~sTJ]

I Egan h2k 2

, (13)

xexp 2kBT 2m kBTa.J where ne is the concentration of electrons.

For III-V semiconductors, there are two dominant electron-phonon scattering mechanisms:

3.1. Deformation-potential coupling

When the deformation-potential coupling is dominant, the electron-phonon potential [18, 20] reads

knT

x~ 1/2

Vs = \2pv2V ]

Edexp(iq.r),

(14)

where p is the density of material, vs is the sound velocity, q is the phonon wave vector, and Ed is the deformation potential. For wave functions given in

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186 C.-C. Wu, C.oJ. Lin /Physica B 205 (1995) 183-192

Eq. (1), one can have

I ( kBT "~ 1/2

(k' n'lVslkn) = ~ \2pv2V j

EdC~kx,kx+qx6k~.k,+q,

X [(}q:, (n/d) in' - n) + (~q:, - ~nla)~n" - .)

- - (~q:. (n/d)(n' + n ) - - (~qz. - ( n / d ) ( , '

+,)1"

(15) From Eqs. (7)-(9), (11)-(13) and (15), the free- carrier absorption coefficient for a quasi-2D electron gas can be obtained as shown in Appendix. From Eqs. (A.1) and (A.7), it can be seen that the free-carrier absorption coefficient could be complex coming from the logarithmic functions due to the interaction between electrons, photons and phonons.

3.2. Piezoelectric coupling

When the piezoelectric coupling is dominant, the electron-phonon potential is given [18] by

( kBT ~1/2 lelflPexp(iq.r)

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Vs = \2pv2 V /

eq

where tip is the appropriate piezoelectric constant. For wave functions given in Eq. (1), one can have

1 ( kaT y/2 lelfle

(~k,x,kx+q x

(k'n'lVslkn) = 2t,2pv~VJ

eq

X 6k~,, ky+qy [(~q=, (n/d)(n'-n) "~ (~qz, -(n/d)tn'-n) - ($~., (ntd~(." +.~ - - 6q=, - (n/d){.' +.)1. (17)

From Eqs. (7)-(9), (11)-(13) and (17), the free- carrier absorption coefficient for a quasi-2D electron gas can be obtained as in Appendix. From Eqs. (A.12) and (A.15), it can also be seen that the free-carrier absorption coefficient could be complex coming from the logarithmic functions in there due to the interaction between electrons, photons, and phonons.

4. N u m e r i c a l analysis and discussion

In this section, a numerical example is developed for an n-type nondegenerate InSb thin film. The

relevant values of physical parameters are taken to be [15, 211 ne = 1.75 × 1014cm -3, m* = 0.013too (too is the free electron mass), p = 5.8 gm/cm 3, e = 1 8 , E g = 0 . 2 e V , E d = 4 . 5 e V , f i e = 1.8×104 esu/cm 2 and vs = 4 x 105 cm/s.

4.1. Deformation-potential coupling

In Fig. 1 (a), the free-carrier absorption coefficient in n-type InSb films is plotted as a function of the photon frequency with d = 10 Ixm for photons polarized parallelly to the layer plane. Since the real and imaginary parts of ~ have the same order of numerical values,

I~1

appears monotonically decreasing with increasing the photon frequency [22]. It can also be seen that

I c<l

increases with temperature. We plot the free-carrier absorption coefficient in n-type InSb films as a function of the photon frequency with d = 10 txm for photons polarized perpendicularly to the layer plane as shown in Fig. l(b). Since the imaginary part of~ is quite small compared with the real part of ~, hence ~ -~ I ~1 for this case [221. It can be seen that ~ decreases with increasing the photon frequency. However, the changing of c~ with temperature appears irregularly in lower frequencies with O < 25THz at higher temperatures such as T = 77 and 300 K. In Fig. 2(a), we plot the free-carrier absorption coefficient as a function of the film thickness with the photon frequency f2 = 28 THz (or 10.6 lxm wavelength of CO2 laser) for photons polarized parallelly to the layer plane. It is shown that

I~1

decreases with increasing the film thickness and increases with temperature. In Fig. 2(b), we plot the free-carrier absorption coefficient as a function of the film thickness with 12 = 28 THz. Since the imaginary part of ~ appears some significant values only with d < 0.4 Ixm except at T = 4.2 K in which the imaginary part of ~ vanishes [221, thus it can also be approximated as ~ -

I~1

except T = 4.2K. It is shown that ~ decreases with increasing the film thickness. However, the changing of ~ with temperature does not appear in a regular trend. Some small oscillations of the free-carrier absorption coefficient with the changing of the film thickness demonstrate the confinement of electrons in these quasi-two-dimensional structures. This phenomenon is coming from the interaction between

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C.-C. Wu, C.-J. Lin / Physica B 205 (1995) 183 192 I0 -9 I~ II 187 10 -6 i: Io -B 10-10 i0-12

-....

--.. --..._

% %%. ,.. IIII1"~1"1~1 "~1 "'1~'111 I! t1111 I IIII

'E

U v

i0-~o

10-~1

10-12

i0-13

10-+4

(b)

NNN

-'.

~. -,

-...

%°% "~,+~ "% %-°++ "'% %, , I I , I I , 0 20 40 60 B0 100 0 20 40 60 80 100

(THz)

Fig. 1. Free-carrier absorption coefficient in n-InSb films due to acoustic phonon coupling as a function of photon frequency with d = 10 lam for photons polarized (a) parallel and (b) perpendicular to the layer plane for various temperatures: T = 300 K (----), T = 7 7 K ( ), T = 19.7K( . . . . ), and T = 4.2K ( ... ).

photons, phonons, and electrons in nondegenerate semiconductors.

4.2. Piezoelectric coupling

We plot the free-carrier absorption coefficient in n-type InSb films as a function of the photon frequency with d = 10 ~tm for photons polarized parallel to the layer plane as shown in Fig. 3(a). Since the values of the imaginary part of ~ are quite small compared to those of the real part of ~ [22], thus ~1~1. It can be seen that ~ decreases monotonically with increasing the p h o t o n frequency and increases with temperature. While for photons polarized perpendicularly to the layer plane as shown in Fig. 3(b), it is shown that ~ decreases monotonically with increasing the photon frequency, but decreases with increasing temperature. This is

different from that in Fig. 3(a) for photons polarized parallel to the layer plane. F o r this case, the imaginary part of ~ appears quite small compared with the real part of ct in the order of 10- 6 [-22]. In Fig. 4(a), the flee-carrier absorption coefficient is plotted as a function of the film thickness with t2 = 2 8 T H z (or 10.6 Mm wavelength of CO2 laser) for photons polarized parallel to the layer plane. It can be seen that ~ decreases monotonically with increasing the film thickness and also increases with temperature. In Fig. 4(b), we plot the flee- carrier absorption coefficient as a function of the film thickness for photons polarized perpendicularly to the layer plane with f2 = 28 T H z (or 10.6 ~tm wavelength of CO2 laser). It can be seen that in low temperatures ~ decreases monotonically with increasing the film thickness. But ct decreases with increasing temperature after the film thickness

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188 C.-C Wu, C.-J. Lin / Physica B 205 (1995) 183-192 E 10 -1 1 0 - 3 10-5 10 - 7 1 0 - 9 10 -11

-",..?'.,',X

".... " . \ \ \

• -.. • .. "\ \

"...

"-:..

".. " \ k - \ -.. "; i . . . I . . . .,J 10 -1 1 I0 1 0 - 3 10-5 10-7 "7 E - 10- 9 1 0 -11 10 -I 3 ( b ) • ". ",,k

%

... 10 -1 1 10 d Clam) d (lam)

Fig. 2. Free-carrier absorption coefficient in n-InSb films due to acoustic phonon coupling as a function of the film thickness with Q = 28 THz for photons polarized (a) parallel and (b) perpendicular to the layer plane for various temperatures: T = 300 K ( ), T = 7 7 K ( -), T = 19.7K ( .. . . . ), and T = 4.2K ( ... ).

d = 3 x I0-1 lam. This is different from that Fig. 4(a). However, at higher temperatures such as T = 77 and 300 K, ~ oscillates with the film thickness in a region of small thickness of films, d < 3 x 10-1 ~tm, and then decreases monotonically with increasing the film thickness. This is coming from the fact that there exists a cusp min- imum point in the real part of ~ and there are some oscillations in the imaginary part of ~ in the region d < 3 x 10 -1 lam [22]. Moreover, the real and imaginary parts of ~ in this region appear the same order of numerical values. After passing this region, the imaginary part of ~ drops abruptly and discontinuously to a quite small value compared with the real part of ~ as T = 77 and 300 K for the effect of phonons becoming more considerably.

F r o m our numerical results presented here, it is shown that the free-carrier absorption coefficient in n-type InSb films depends upon the photon polar-

ization, the p h o t o n frequency, the film thickness, and the temperature. When the acoustic p h o n o n scattering is dominant, the free-carrier absorption coefficient appears irregular dependence on temperature for photons polarized perpendicularly to the layer plane. However, when the piezoelectric scattering is dominant, the free-carrier absorption coefficient increases with decreasing temperature for photons polarized perpendicularly to the layer plane. Thus the dependence of temperature on the free-carrier absorption coefficient for the piezoelectric scattering is quite different from that for the deformation-potential scattering in this kind of polarization. We have also shown that the effect of the nonparabolicity of electrons in semiconductors will be relevant at high electron energy and thus a high photon energies. At low frequencies the parabolic band of electrons is a rea- sonable approximation. The deviation for the approximation at low frequencies will be from

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C.-C. Wu, C.-J. Lin / Physica B 205 (1995) 1 8 3 - 1 9 2 189 o t-- t~ 0 -? i1) e.,- ff) E o v 10 - 4 i 0 - o 10 -a 10 -10 10 2 10 . 4 . ~ ( a ) \ \ \ • -. \ % . ".... \ \ . °%.° ~ _ ~ , °°%"° % ~ , , 7'5 25 50 f~ (THz) 10-o 10 -a ¢.) "o "o "o "5 0~ v i E o 104 102 E

~"..

(b)

~\ "..

\ - . - . X \ "....

-...

\ , , , - , . , .

.. ...

_ 10 - 1 2 10 -1° I 0 -2 I I I 0 1 0 0 0 2 5 5 0 7 5 1 0 0 f~ ( T H z )

Fig. 3. Free-carrier absorption coefficient in n-InSb films due to piezoelectric coupling as a function of photon frequency with d = 10 ~tm for p h o t o n s polarized (a) parallel and (b) perpendicular to the layer plane for various temperatures: T = 3 0 0 K ( - - ) , T = 7 7 K ( - - - ) , T = 19.7K ( . . . ), a n d T = 4 . 2 K ( ... ).

Eqs. (4)-(6) as

l ~ 2 h 4 k 2 n 2

A E k , - 2m,E d2Eg .

This is very small if we take a limit n -- 1.

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Acknowledgement

This study was supported by N a t i o n a l Science Council, Republic of China, under contract number: NSC-81-0404-E009-539.

Appendix

Deformation-potential coupling

(i) When the photon is polarized parallel to the layer plane, 1= l nI ~ L g(U,i)- + 47nin I + nz7 -

LT ~ \ ~ )

- ~7 2(n I + 2n/)21n U 2 = - 1 +~ 1 1}

"}-~2=-11 ~ , ).(U/~f)_ 1 + 5 7ninf , (A.1)

where

2n4 eEn~kB T E 2

C~D = f2aellE p v 2 m , c d a h 2 , (A.2)

(nv,y

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190 C.-C. Wu, C.-J. Lin / Physica B 205 (1995) 183 192

'E

U v 10 3 10 10 -1 10-3 10-5

10-7

"..\

- \

(a)

" X

_-,. \

- \

',,

~ "% \% 10 5 10 4 ---_ 10 3 0

10 2

10 A

II

rl

( b )

°eooe Q°oeo e Qe % °. ~e • o

",.,,

"\%

..

• %%1

" \

%. I ~ I I 10 10 2 I0 -I I 10 10 2

d (I..tm)

d (larn)

Fig. 4. Free-carrier absorption coefficient in n-InSb films due to piezoelectric coupling as a function of the film thickness with D = 28 THz for photons polarized (a) parallel and (b) perpendicular to the layer plane for various temperatures: T = 300 K ( ), T = 7 7 K ( - - - ) , T = 19.7K ( . . . ), and T = 4.2K ( ... ).

hEk 2

kilT = zEg -

2m* '

(A.4)

1

U ~ = ~,(n: +_

n,) 2 - ~

(A.5)

and

N :

can be determined from

ni < N f <~ ni + df2/ltvs.

(A.6)

(ii) When the photon is polarized perpendicular-

ly to the layer plane,

(

at

~

(n,)2 = ~D t a, exp -- ~ : J ni=l n.f=l n'=l

[1-- cos ,(n: + n')

1--cosTt(n:--n'!] 2

X ani n f -t- n" '~ n f - - ~'

(_a4{ £', :.V,',_+

+*.:4

x exp

2~J

In

~ = _ 1

\v,; +-£~,~)

+ 1 + 1 + ~,' ~ ' I V ; I - - 2]exp[2V,'~- E;f] # = - 1 ) . = - 1

x E,[ -

).v;# +

E~:]},

where

(A.7)

f ~tv,h \ ,

v:::~d-~)~,

_+,,,)~,,.,+a.,)~,,.,-a.,), tA.8)

h Q a . f

(A.9)

- - t kBT(a~: -

a,,)

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c.-c. Wu, c.-J. Lin / Physica B 205 (1995) 183 192

191 and the exponential-integral function

Edx)

is de-

fined [23, 24] as

;;e-'

fx e'

E~(x)

= --

- - dt =

- dr.

(A.IO)

t _ ~ t N '

and

N f

are determined from

ni < N' <

N z <~

ni + df2/nvs.

(A.11)

Piezoelectric coupling H' -~- n i - 1 ,,,i:~ K[f + 1 + 1

- ( K I i ) - ' + ~'

2 '

2P[K'if, l~Wi')] 2 = - 1 / t = 1

+1

}

+ (a. s - a.,) ~'

2Q[KI:, W;)] ,

(A.15)

2 = - 1

where

(i) When the photon is polarized parallel to the

layer plane,

= ~p

atexp

-- ~ ) 3

E

ani 1 n i = l n s = l

× exp(

- a " ' ~ , 2 m * d Z y k " T a n ' [ 8 1 n { n: + n,~

t

L

\ns - n,)

- 31n(Ui'~'~-

2 ' (U~:) -1

p

11

+ 2?n~n:ln[16U~U~] - 87n~n: ~

In(7) + ~

+ 1

- 27 ~'

2(n: +

2nl) 21n(n: +

2ni) 2 = - 1 ~ u ~ --4(ln3)?n~nf+~[1

+ 47(n} + n~ )] In (~--~))}

(A.12)

where

2n6 e 4 nekB T fl 2 O~p ---- ~3gSl2pv2m,cd3h 2

(A.13)

and

N:

can be determined from

ni < N: <~ ni + dO/nVs.

(A.14)

(ii) When the photon is polarized perpendicular-

ly to the layer plane,

NS ~ (nl)2ania2'[

1-cOs"(nf'~",)

x _{n i = l} _ny=l~ _{n ' = l} (a.s -

a.,) 2

n: + n'

7 (

__

a.,']

1 - cosn(n: + n')

2ex p

n : - - n ' 2 z )

K ~: = hf2a.s(a.: - a., )- 1

- ½E,(a.,a., + a2: - a 2, - a.,a.s)(a, l - a.,)- 1,

(A.16)

Wi~ + = (nhvs/d)tn' +_ ni)(a.: + a.,)(a.s - a.,) 1,

(A.17)

[

/2m*a.,d

2V'~ 7 - 1 P(K, V) = - kBT

5 - 2 1 n t - - 2 - ~ h 2

. ) J K I /4m* 2aZ.fl4K V \

x ( K + V ) - ' +

2--21n t-

~

)

(2m*a. dZK) {2m*a.,daV'~](K +

V) -2

+In\-

)ln\-

J3

[-'

{2m*a" d2V'X

_.)-

1]

x In L-

n2h---5

[4m,2a2 d2KV\-[

+ 1 - 1 n t

-~-h-; ) J ( K + V ) - I

-1

V

+ / k . r ) - expt

)

_ V

{ (4m*2a2'd4KV'] -

3K 1 + V-~_(k~T)-Xln\.

~

-)

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192 C - C Wu, C-J. Lin / Physica B 205 (1995) 183-192

_ 2kBTK_2}exp(K + V'~Ei(

K+__ V'~

\ kaT /

kBT J

(A.18)

and

Q(K, V ) =

V -1

exp

Ei kaT

J

_ exp (Kk_~TV)Ei (K _ - _V'~,

_{kBT J.~}

_(A.19)

N' a n d Ny can be d e t e r m i n e d from

nl < N' < N 1. <<. ni + df2/nvs.

(A.20)

T h e s u m s over 2 or # with 27 indicate 2, # = - 1 to + 1 w i t h o u t 0. F o r the n o t a t i o n for 2 o r / ~ , + 1 m e a n s " + ", and - 1 m e a n s " - "

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EFFECT OF PHONON-SCATTERING ON FREE-CARRIER ABSORPTION IN N-TYPE INDIUM-ANTIMONIDE FILMS