Electron-phonon interaction, dynamical screening and collective excitations in heterostructures

Physica B 167 (1990) 101-112 North-Holland

E L E C T R O N - P H O N O N I N T E R A C T I O N , D Y N A M I C A L S C R E E N I N G AND C O L L E C T I V E E X C I T A T I O N S IN H E T E R O S T R U C T U R E S 1I. DYNAMICALLY SCREENED POTENTIAL

L. W E N D L E R , R. H A U P T and V.G. G R I G O R Y A N l Sektion Physik der Friedrich-Schiller-Universitht Jena, Max-Wien-Platz 1, Jena, DDR-6900, GDR

Received 26 September 1989 Revised 5 May 1990 In this paper we analyze the dynamically screened interaction potential of a quasi two-dimensional electron gas of a heterostructure including the effects of the optical phonons. Using the subband structure of the electronic states, we calculate the bare and the screened interaction potential of two electrons interacting via electron-electron and electronphonon interaction. The results are discussed and plotted in graphical forms.

1. I n t r o d u c t i o n

Modulation-doped heterostructures are accompanied by many-particle effects which must be included to understand the physical properties of the quasi two-dimensional electron gas ( Q 2 D E G ) in such systems. In polar semiconductors long-wave optical lattice vibrations couple to the electrons of the Q 2 D E G . Therefore, elect r o n - e l e c t r o n and e l e c t r o n - p h o n o n interactions occur and affect both the electron and the phonon system. The Q 2 D E G of a heavily doped semiconductor can support charge-density oscillations organized by long-range Coulomb fields. The polar or Fr6hlich type of the e l e c t r o n - p h o n o n interaction leads to a strong coupling between these charge-density oscillations, the plasmons, and the long-wave optical phonons, if their frequencies are comparable, forming a polaron gas. For a heavily doped bulk polar semiconductor p l a s m o n - p h o n o n coupling has been studied in great detail [1-9]. In modulation-doped semiconductor microstructures such as heterostructures this coupling also occurs. But there are two basic Permanent address: Department of General Physics, KarlMarx-Polytechnical-lnstitute, ul. Terjan 105, 375009 Erevan-9, Armenian SSR, USSR.

differences between a microstructure, which is in general a system with broken translational symmetry, and an ordinary 3D bulk crystal that one must consider: (i) Confinement of the electron motion perpendicular to the heterointerfaces leads to the occurrence of subbands and hence, the electrons form a Q2DEG. (ii) The spectrum of the optical phonons interacting with the electrons of the Q 2 D E G is altered by the interfaces of the system. Longitudinal-optical ( L O ) phonons are changed and new states, the interface phonons, occur in the spectrum of the optical phonons. In recent papers [10-15] we have studied the coupled intra- and intersuband plasmonphonons, in double heterostructures including both the Q2D character of the polaron gas and the correct spectrum of the long-wave optical phonons. In this paper, part I [16] and part III [17], we continue corresponding investigations related to a single heterostructure. In part I [16] we started with the coupling of a single electron with the long-wave optical phonons of the system. H e r e we are concerned at first with the electronic subband structure of the heterostructure followed by the calculation of the screened potential, including screening effects by both the

0921-4526/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)

IUZ

l..

~ ' ( ' # l d h ' r ('t a[.

l)Ynaltticall~, ' s('rccncd poh'nttd[

electron-electron and the e l e c t r o n - p h o n o n interaction, in this structure. The collective excitations, the coupled intra- and intersubband plasmon-phonons are calculated in part Ill [17 I.

2. Electronic

subband

structure

The schematic arrangement of thc hiycr scquence of a modulation-doped GaAs G a I , A l A s HS is given in fig. 1. The wider gap semiconductor Ga~ AI As (0.15 ~ x ~< (1.35) is doped with silicon, while the smaller gap material GaAs is nominally left undoped. Due to the conduction band discontinuity at the hetcrointerface electrons from the donors in the Ga~ , A I , A s l a y e r are transferred into the GaAs layer, where they form a Q 2 D E G in thc electron space-charge layer. The electrons of the Q 2 D E G move in a rather complex potential (fig. 1). The periodic potential of the semiconductor is represented in the effective-mass approximation for an isotropic, parabolic conduction band minimum at the Brillouin zone center (1"). On this potential the relatively slowly varying

electrostatic potential is superimposed. This mean Coulomb or Hartree potential Vii(x) is the solution of the Poisson equation with the total equilibrium charge density. The httter one is the sum of the ground-state electron densfly, which is the contribution from the induced charges in the space-charge laver ( ~ no(x): electron nuulber density in the ground-state of the unperturbed system), and of the depletion chargc density arising from the ionized accepter and donor concentrations /'V\, e~l'} , respectively. The depiction charge density is the contribution from the fixed charges m the depletion layer. Many-particle effects arc included in the exchangc-corrchttion potential V,<(x) of thc density functional method. Hence. for electrons of the Q 2 D E G of the HS the K o h n - S h a m equation ]/i:

2m

d + L~,,,(x))J.k ~(x)

/.(kll)dJ. ~(x) (1)

is valid. The effective potential

<,,(x)

V~u(x)

is given bx

(2)

V~,(x) + V,< (x) ,

~(.,,(z)'

z :-0

V,,

- .

d

E,,(z)

.l~.

<,,(x)

l

0.

(3)

g020EG C2

.

I

~gl

Eg2

I 1 2 Ev2 n -doped

d2

'

d z-

<<(z)

I. Energy band diagram in the vicinity of the h e l e n > interface and schematic a r r a n g e m e n l of layer sequence of a m o d u l a t i o n - d o p e d G a A s Ga~ A l A s H S ( N ~ > = ( 1 . 0 2 . 5 ) x

l(l'~cm ~:x

t1.15 (I.35).

+ N .\

N i~ ) .

(4)

~,E.<.[.,,(z)i 8m,,(z)

V..l,,,,(z)l.

/5)

z

t Fig.

(.,i(z

GoAs v=l

0

/-'n/-;~ I

and the exchange-correlation potential is ~iven by

undoped J

u2

.... .- VtI(:)

I E

v=2

-)~

%1

d

Gal_ x At x As v=3

~z

/L

where 14, results from the conduction-band offsct at the hetcrointerface. The Hartree potential is the solution of

where E,
103

L. Wendler et al. / Dynamically screened potential

problem the electrons move quasi-free in the x - y plane with wave vector component kll. Therefore, the single-particle electronic wave function t~kuK(X) can be written in the form

%K(x)

1 eikllx,¢~(,) '

=

(6)

where xlL is the two-dimensional position vector in the x - y plane. The confinement of the electron motion in the z-direction leads to the occurrence of Q2D subbands according to

h2k] ~K(kll)= ~'~+ 2--m- ;

K=0,1,2,3

.....

(7)

The envelope wave function ¢~:(z) for an electron in the Kth subband is given by the onedimensional Kohn-Sham equation (

h2

d2

)

2m dz 2 ÷ Veff(Z) ~K(Z) = VKCK(Z )

(8)

with

f dz ¢~:(Z)¢K,(Z ) = 6KK,.

(9)

According to the local density of states 'Trh2 K

(10)

where the unit step function O(x) = + 1 for x > 0 and 0 for x < 0, the electron number density of the ground-state no(Z ) is given by

mk u T no(z)

-

-

where NK is the sheet carrier concentration of the Kth subband. Equation (12) determines the chemical potential /x. The subband energies g~ are in principle found out by solving the K o h n Sham equation (8) self-consistently in connection with the Poisson equation (4) and the use of eqs. (5), (11) and (12) [18]. Here we are interested in an analytical solution for the envelope wave functions and subband energies, because our aim is to consider electron-phonon interactions and collective excitations. Integration of the Poisson equation for the total Hartree potential gives [19]

-

.rrh 2

VH(z ) = ~

+ n2DEPL(Z

F/2DEG

- mkBT ln[1 + expk - k--BT Ti-h~ ~K )1 = ~ NK , K

d z ' ( z ' - Z)Iq~K(Z')[ 2 0

Z2 2ZDEPL)} ,

(13)

+ with /'/2DEPL = ( N A - ND)ZDEPL the n u m b e r of fixed charges per unit area in the depletion layer of thickness ZDEpc, in which all donors and acceptors are ionized. In eq. (13) we have neglected the image potential. This potential arises by solving the Poisson equation for a layered system due to the different polarizabilities of the different semiconductor materials. The neglect of the image potential is possible because for the system GaAs-Ga~_xAlxAs its effect is very small [18]. In the Hartree approximation Vxc(Z) is neglected and each electron moves in the Hartree potential produced by all the other electrons and the depletion charges. Within the Hartree approximation one can apply the triangular well potential for VH(Z) as a useful approximation. Using a linear dependence in z for VH(Z ) eq. (13) is approximated by

(11) The sheet carrier concentration F/2DEG (electron concentration per unit area) is obtained from (11) to be

Ux z +

80 8s I

Vn(z) =

eFz ; z > 0 , . 0 > z,

(14)

where F is the effective electric field, F =

e(Y/2DEPL÷f?'/2DEG)

,

0~
(15)

EOEsl (12)

The numerical coefficient f allows the fitting of

L. W e n d l e r et al. / D y n a m i c a l @ screened potential

104

the slope of Vu(z ) in (14) to the exact Hartree potential (see fig. 1) in the interesting region of the HS. The coefficient gives for j = 1 the field at the heterointerface, for f - 0 . 5 the average field in the electron space-charge layer, and for f - 0 the field produced by the depletion-charges alone. The solutions of (8) with (14) in the region z > 0 are Airy functions of the first kind [20]

¢~,,(z) : C~Ai( ~

),

(16)

between the constants Ko and K I on the one hand and the constants K~ and K, on the other hand:

12Kt

)

2

(23)

2

KI~ -- K o K I Jc K I "

and

K~-

(K.+K~)

(24)

The usual way to determine the constants K<+and K, is the variational approach which gives [23]

with

K(I

:

i( 12m,, ( tl2DI t, 1 4

2

fT(iFsl~-

,,

))

~2 ll2llE(;

( - 1 ) ~'

Cx = ~d IAi'(

"

(17)

and [221 KI = 1.(t3x<, .

zt" = eF " and d

( h-' \ ~ /

(26)

(18)

it.3

(19)

Hcrc Ai'(-x~,) denotes the derivative of Ai( z) at the zeros x,, of Ai(-z). The eigenvalues d~. are given by the condition C g ( 0 ) = 0. It follows

'~. = eFdxt,. .

(20)

Another possibility to obtain constants K,, and K~ is to fit directly the Airy functions. From the position of the maximum of the Airy function of the 0th subband one obtains 1

K'i --

"

d

¢,,(z) = 2,¢;+/ez e

~"=

(21)

and [221 ¢,(z) = K~ 2Z(1 - K3Z) e ~'=

(22)

The orthonormalization (9) of the envelope wave function results in the following relations

(27)

and from the zero of the Airy function of the 1st subband follows

(Z(j The first zeros of A i ( - z ) are given by &, 2.33810741, x~ = 4.08794944, x~ = 5.52055983, x 3 6.78670809 and x 4 = 7.94413359. For a discussion of the dynamically screened potential and the collective excitations [17] it is useful to discuss some properties of this simple model. To keep all the analytic calculations feasible it is necessary to approximate the Airy functions. A suitable form is [21]

"

Zt)KII

K z --

~ 3 (2S)

~-(I -

Z I

The advantage of this latter method is that the numerical coefficient f will not be fixed and can be chosen, for instance, to fit the eigenvalues with respect to self-consistent results [18]. In fig. 2 we have plotted the spatial dependence of the envelope wave function in the different approximations. It is apparent that the shapes of the Airy function are well reproduced by the used approximates. But for large values of z the Airy function goes down more rapidly to zero than the approximates do. Hence. the localization of the electron gas at the heterointerface is weaker if using eqs. (21) and (22) instead of eq. (16) for the envelope wave function. In fig. 3 the electronic subband structure of the considered HS is shown. In tables 1-3 some

L. Wendler et al. / Dynamically screened potential

15

0.2?

10 - / ~ 5 E

%

105

n2DEG=5×1011cm-2

!1././~- ~

0

n2DEPL: 8 × 1010cm-2

0.~5

\.

5

>

•

10

\ . /

0.10

>~

./

15

~~

0.05

-%--n2DEs : 5.1011cm-2 n2DEPL--8.1010cm-2

10 2

5

m'-" 0 'E z, O

9._~

o

0 10

20

30

2

c°0

Z (nm) Fig. 2. Envelope wave functions %(z) and %(z) for electrons

10

20

30

in the two lowest subbands for n2DEG = 5 X 10 H cm -2 and n2DvPL = 8 × 10~°cm 2. The solid lines correspond to (21) and (22) using the fitted values of (27) and (28) with f = 0.5. The dashed lines correspond to (21) and (22) using the variational approach (25) and (26). The dashed-dotted lines correspond to the Airy function (16).

Fig. 3. Electron n u m b e r density, conduction-band edge and subband energy of the triangular well potential approximation of a GaAs-GaovsA1025As HS at T = 0 K , n2DEC, = 5 X 1 0 H c m 2, n2DEpL = 8 x 1 0 I°cm 2 and f = 0 . 5 (corresponding to a field strength at the interface F = 8 . 1 5 x 10 ~ Vm ').

Table 1 Subband energies and chemical potential of a G a A s - G a I xAl~As HS for n2ujp I = 8 X 10 m cm 2 and f=

II ~2.

n2DCG

40

z (nm)

"~0

~d,

T%

At (meV)

(10 ~2 cm 2)

(meV)

(meV)

(meV)

0K

77 K

300 K

(/.2 0.3536 0.5 (1.8 1.0

31.78 38.91 45.14 56.75 63.85

55.56 68.03 78.92 99.22 111.64

75./13 91.87 106.59 133.99 150.77

39.00 51.69 63.21 85.66 99.99

35.75 50.12 62.20 84.84 99.03

- 14.44 12.30 31.51 62.59 80.03

300 K

Table 2 See table 1; f = 0 . 5 .

n2DE, i

~o

~

~2

0K

77 K

0.2 0.3536 //.5 0.8 I. 0

36.08 45.73 54.05 69.39 78.72

63.09 79.95 94.50 121.32 137.63

85.20 107.97 127.62 163.83 185.86

43.31 58.51 72.12 98.30 114.86

40.26 57.21 71.45 98.01 114.62

7.83 22.02 43.64 78.87 98.79

L. Wendler el al. / Dynamically screened potential

106

Tablc 3 See table 1: f = l.O. F

n:m ,~

g~,

(',

,~.

0.2 0.3536 0.5 0.8 1.0

48.44 64.84 78.72 103.94 119.14

84.70 113.37 137.63 181.73 208.31

114.38 153.10 185.86 245.41 281.31

0K 55.67 77.62

96.79 132.85 155.28

77 K

3()(} K

52.87 76.54 t;6.32 132.76 155.25

%46 46.5l 73.76 118.97 144.SS

results are given for the s u b b a n d energies and the chemical potential of the G a A s - G a l ,AI,As HS.

an electron from the s u b b a n d () to K by a n o t h e r electron which b e c o m e s scattered from 0 to K'. T h e direct C o u l o m b part of the bare e l e c t r o n electron interaction potential is given bv

3. The d y n a m i c a l l y screened potential

e .f~t,'(q ), V,,,,,,.(qll ) - 2 F , , e , l q

2

In part 1 [16] the interaction of a single electron with the different kinds of long-wave o p t i c a l p h o n o n s of the HS was discussed. But in the case of a HS the electrons form a Q 2 D E G in the electron space-charge layer. Therefore, a fermion-boson many-particle problem for layered structures must be investigated. H e r e we use our formalism derived in ref. [10] and apply it to a HS. In this p a p e r we have derived the Dyson e q u a t i o n for the s c r e e n e d interaction potential in the " s u b b a n d s p a c e " . In the electric q u a n t u m limit it reads

Wx~,"( qll' w) = ~ (6~..,. WKL( qll" co)X{t-' )( q!l' co)) l

× W;[K,(qlI,

co),

with the form fact()] / dz 1J d z ' J'(,v,v'( qll) = j" x

T h e e l e c t r o n - e l e c t r o n interaction potential contains the optical dielectric constant ~,~, which includes the dielectric screening of the semiconductor b a c k g r o u n d arising from the high-energy electronic excitations across the band gap. Using the e n v e l o p e wave functions eqs. (21) and (22) it follows that

3(• 2K()/t++

,(. .[ ,)()(qll)

~

4,+ "-' )' 2K(, ' (33)

(

'(

,1 ll(qll) WKK.(qlI, co) = V ~.~.,(ql) ) + V~,Cqll, co),

¢~.(z)¢,,(z) e ,,'Hi= =i ¢ ~ . ( z , )¢,,(z , ).

(32)

(29)

where WKK,(qil, co) is the bare and W~?t..(qll, co) is the s c r e e n e d interaction potential. T h e p r o p e r polarization function X ~)" K ( qil' 02) was calculated in rcf. [10] in a full R P A t r e a t m e n t in the z e r o - t e m p e r a t u r e limit. T h e bare interaction potential is given by

(31

KoK2 ) (<,+ K~)~

(30)

w h e r e V&.K,(qll ) r e p r e s e n t s the bare e l e c t r o n electron interaction and V~,.)~,,(q!l , co) is the bare electron-phonon interaction potential. The potential W~<~<,(qu, co) signifies the scattering of

+ X 1 + ql~) Ko +

~ K t .

)

',,, ~

KI) ~

K I

(34)

L. Wendler et al. / Dynamically screened potential

and

107

with .,, 9 3.~1/2 ZtKoK2)

fC0(qll) - (~7~Ko+ ~-7)g {[10(K o - K,)(3K,, + K,

)3

fK~,(qll)=S(qll) ,m

f fdz

e ql[(Z+Z')a~

×

- 10(3K()- 2K,)(3K o + K,)2(2K0 -- qll)

dz' (Z t$/~ (Z t$, .

(37)

+ 3(9K o -- 5K1)(3K o 4- K~)(2Ko -- qll) 2 H e r e S(qll) is a structure factor, which for the half-space g e o m e t r y is

-- 4 ( 2 K o -- K~ ) ( 2 K o -- q l l ) 3 ] ( K 0 + K, + qll) -4

+ [4(2K0 -- K1)(ZKo + qll) 2

S ( q l l ) - e~l - e~2 E~ 1 ~

+ (5K0 -- K')(3K0 + K')(ZK0 + qll) + 2K0(3K o + K,)Z](zK0 + qrl)-3}.

(35)

and for the g e o m e t r y where m e d i u m 2 forms a layer for 0 > z > - a 2 it follows

C

In lag. 4 the form factors fKK'(qll) are depicted. It can be seen that for the e l e c t r o n - e l e c t r o n interaction the ( 0 - 0 ) intrasubband scattering is the dominant process for all wave vectors. Besides of the direct C o u l o m b potential V~K,(qll ) and the bare e l e c t r o n - p h o n o n interph action potential WgK,(qll , ~) there is still a further potenial, the classical image potential im VKK,(qlI), which contributes to the bare interaction potential WKK,(qlI, W). This potential, neglected in eq. (30), arises by solving the Poisson equation for a layered system if the semiconductor materials have different polarizabilities. T h e classical image potential is given by im g~f¢'(qr[) -

e

2

im 2eoe~ql I fK~'(qll),

(36)

e~, sinh(ql a : ) + E~ cosh(qlla2) -- ~'~2 e ~ cosh(q41a2) + e'~ sinh(qlla2)

~1

S(qll ) =

e_, sinh(ql,a:) + e 3 cosh(qlla2) " - e~, c o s h ( % a . , ) + e~ sinh(qtla_~ )

(39) In our earlier treatments we have always omitted the classical image potential because of its small value. This is true for the system G a A s Ga~_~AI,.As, for which in this paper we also neglect the image potential in the ground state as well as in the dynamical properties. But in farinfrared experiments (see section 7 of part III) the vacuum in - a , > z is replaced by a highly conducting metal for which e3---~-~ is valid. In this case the contribution of the image potential to screen the bare e l e c t r o n - e l e c t r o n interaction is not small and hence, must be included. T h e structure factor follows from (39) for e3---~-~

1.0

eC~l -- em2 0.8

(38)

8~2

S(qll)

=

WCzl

coth(qrla 2)

4- e~2 coth(qlla2 ) .

(40)

0.6

Using the envelope wave functions (21) and (22) one finds

0.t.

fc /'f~o

0.2

im

0

t

I

[

5

10

15

V,,,,(qji) 20

e: -

16K~

2eoe~lql I (2Ko + qlt)6

S(

)

qlJ ,

(41)

q. (107m-1 ) Fig. 4. Form factors f%x'(qH) of the direct electron-electron interaction of a GaAs-Ga(~.75Alo.:sAs HS for nmEc, = 5 × 10It cm 2, n2DEm = 8 × 10~Ocm 2 and f = 0 . 5 .

im

V'I(qll)-

e-

3

3 2

4KoK 2qll

2e0e~qll (K04-K, 4-qrr) s S ( q l l ) ' (42)

1.. ~4"mth'r el al.

10g

l)vmmucalh" s'creem'd potemial

and inl

Vl,,(q

.'~/,',,:(q,,.

e-h coil 1 ( ~ ~:,, ( . . . .

q ) -

)

)

8,,,',/12./', c

( K~,

•

\,,,q:

qll

2~-',F, lq!l (2K,, 4 q )~(K, + K l - q:l )~ 5 ; ( q ). (43)

V ph K~""( qll"

(co + i~): x l~,.(qll)

I

( I. F H

X: ((K(,

with (45)

q'l)

Throughout this paper we only use retarded functions ( a - - , ( l ' ) . The basic electron-phonon interaction vertex is

l))

' F,I

/-'~I

q. 12K~(KI, ÷ KI )%

q~)

((K,, -P KI) 2 -- (/?)I

co~(qll) (44)

ii I

• 4(Ko

".. (3(K,, + K,):

"

M~',(qll)M~"'(

,:~

+

( ' fl (OI i

~o/(qll)

--

(O)

,:~ lq: (K~ , ,/~) 3

/~:IIl:( ql" [I-)

The bare clcctron-phonon interaction potential is given by

./[~,(q!l)=

1

q])l"

~ KI )2

(>))

I.. iN According to eelS. (45) and (47), (48),/~,~. ~iven by the product of the interaction vertex functions. For the form factor of the L() p h o n o n s w e delinc • E I

I

I

/i,,,,(q=l)=~:l/IK,,(qi,

q

[1

q:)M1,,,( q

,

q ) (5])

which gives M~,.,,,(qll)-'v:A

-/

d z % , ( z ) l ; ( q , i, z)¢~,..(z).

(46)

,, J'"'(q:')

.-t,.o., ( , 4<,q: ;:,i

__
<

SK,; ' /',,'tq)

qll ( 2 K , + q ! l ) ' "

<,

.IZ( q )

For the matrix elements of the interface phonons of the HSit follows (using eq. (15) of part l 16])

. loo ~ q! )

,)

1tq,,)

~21

{1.

I

e:fi%~ (

....... 4~,,q~ , ~- a

I

~-. )

(47) and 1,>

M10(qll)

;4 FItC i~ 1

t l 2 ("';

2co (qll) :

4(Ki:K'~_

'

q

2(]:, K, ~ ,~

q~ (K ° + K1 + q,)4 48)

Under the assumption that L_ is much grcater than the spatial extension of the space-chargc layer for the LO phonons of the HS (modes of thc GaAs half-space) (eq. (20)in part I 116]) the following matrix elements are obtained:

/(53 )

Lind

/.IE(q!l)

e:h% 4~-',,q! I

( 1

I

~"

F, I

32(K,,K))

)]-/],,(~| q!l ) qll

t

(2K,, + qll)~(K" + K~ + ql!)4 [ •

(54)

L. Wendler et al. / Dynamically screened potential

In fig. 5 the form factors f~o(ql)), f{l(qll) and f~0(qll) are plotted for the HS modelled by the half-space configuration. The plotted form factors are normalized to that of the interaction of strict 2D electrons with bulk LO phonons given by f L O ( q,,)

e21iOOLl ( 1

4eoql~I

e.,

= (w

{[I

(1 -

0.3

o

0.2

,,.I--

4,--

5

10

15

20

q,, (107m-1) (a) 0.07 0.06 0.05

o

0.04

.,%o 0.03

0.O2 0.01 0 0

10

5

15

20

q,,(107m-1 ) (b) 0.15

0~0

2 7,.._~

-L(qll, o.) +

0.05

I*

,1,))

WL,L,(qII, W)XL'(" (qfl, W))

L '=0

-- Wl0(qll , o))W0l(qll

}' , x X(,')(qll. ~o)

with K' # K.

o

(55)

.

- w,.,(q H, ×

0.4

0.1

It can be seen that the dominant contribution results from the scattering with the LO phonons of the GaAs half-space. Only at low values of qll the dominant interaction for the (0-0) intersubband scattering is that with the two interface phonon modes. If one uses the Airy function to calculate the form factors, the magnitude of the form factor f~K(qll) would be higher and that of LI fKK'(qll) would be lower than that calculated with the approximates. This is true because of the stronger localization of the electrons if one uses their exact envelope wave functions. Noticeable is that the form factors f~0(qll) are all different from zero. This result for the HS differs from that obtained for a double heterostructure (DHS). In the latter case all form factors f~0 vanish because of the symmetry of the DHS [14]. Now we are able to calculate the dynamically screened interaction potential W~K'(qlI, w). Considering the Dyson equation for W~K, it can be seen that a further approximation is necessary to calculate W~5~.,. One possible approximation is to include only a few subbands. In the twosubband model the dynamically screened potential has the following form

w'L(qrt,

0.5

1)

e7'

109

, o))xl)l)(qll

I

I

I

I

5

10

15

20

q. (107 m-1 )

, o3)

(c)

(56)

Fig. 5. Form factors f~x(qlO (normalized to the form factor f ' °(qll)) of a G a A s - G a . 75Alo_,sAs HS modelled by a halfspace geometry for n m ~ o = 5 × l0 I1 cm : nzDEpt. = 8 X 1010 cm--" and f = 0.5. (a) f [)./f '°. (b) f {,/f ' ° and (c) f ',[/f L°.

L. Wendler et al.

11()

Dvnanficallv screened potential

If one further neglects the off-diagonal elements, the screened potential has a very simple form. From eq. (56) it follows

W~,~,(qll,W)=lW~,.~(q!l , ¢ o ) ~

x"'(,- qll" c°)l '

(57)

The screened potential then is related to the total longitudinal dielectric function of the system (eq. (42) in ref. [10]) by the following equation

F~.~.(q,l,co) ...... ,

01

02

o-Oi I,

0.3

,

10_7J ~ ~~J u ~ " ' k \\

0.4

i

0.5

,

b) ,L0=~./+×1013S-1 /' C) OO='7'l . 101'51 /'/

1o -°

\V _

0

qll kF

_

v; J_

L

!0

5 q,, (10 ? m-1 )

Fig. 6. Screened potential W~,~,(qi , to) (normalized to u ' (4=~,)) o f a G a A s - G a , ~,AI, .~As HS modelled by a haltspace geometry for /121)1 t , = I x 1(1~: cm ' H2DI I'1 = ~ "/ 101" cm : and f = 0.5 for three different frequencies in comparison to V(,,( q,~ ). A negative potential is shown by' dashed

curx,'cs,

~×lO~:~s" b! vo:5×10~3s '• ,

11! to~

o/ co:

oc o oo

c) w : 7 . 1 0 ~S'

/

1

/

i b

10 8 _ [

10~°

101!

101:'

n2DEG (cm - 2 )

In fig. 6 the 0-0 element of the screened potential W~,~ is plotted. Comparing to the bare Q2D Coulomb potential VI{.( ql ) a large enhancement of the screening is visible especially near the poles of the screened potential which are equal to the collective excitations of the system (see part Ill [17] and eq. (58) with Fi;]'"(qll, w ) = ( ) ) . Figure 7 shows the dependence of the sc on the density n:m~; of screened potential W,.~ Q 2 D E G for different frequencies.

¢~

10-7 ~_ E m

~ , V~,~.(qll )[W~?n,(q, ,, o,)] (SS)

0

i

Fig. 7. Screened potential W~,;,(q.w) (normalized to t" (4rr,",)) of a G a A s G a , , - , A I , > A s HS modelled by a half space geometry for n:,~ ,, S ~ 1(11' on1 " and [ 0.5 t'or q 5 :< I0 r m t and three different frequencies. A negative potential is shown by dashed curves.

References [1] B.13. Varga, Phys. Rev. 137 (1965) A 1896. [2] A. Mooradian and G.B. Wright. Phys. Re~. l,ctt. In (1966) 999. [3] A. Mooradian and A.I,. McWhortcr, Phys. Rc'~. Letl. 19 (1967) 849. [4] P,. Tell and R.J. Martin, Phys. Rex'. 167 (1968) 381. 15] (i.D. Mahan. in: Polarons in Ionic Crystals and Pohu Scmiconductors, ,I.T. Devreese, cd. (North-Holhmd. A m s t e r d a m , 1972) p. 553. 161 I,.F. L c m m c n s and J.T. Devrcese. Solid State ('o[l/nlun. 14 (1974) 1339, [71 1,.[:. L e m m c n s . ;:. Broscns and ,I.T. Devreesc, S¢llid State (k~mmun. 17 (1975) 337. ISl M.E, Kim, A. l)as and S.D. Senturia, Phys. Re~. B IS (1~)78) 6890. I,)l w Richter. in: Polarons and Excitons in Polar Semiconductors and hmic Crystals, ,I.F. Devreese and F. Pec lers. cds. N A T O ASI Series, Series B Physics, Vol. 1()8 ( H e n u m . New York and l x m d o n , 1984) p. 209. [101 [.. Wendler and R. Pechstedt. Phvs, Star. Sol. (b) 13S ( 1 9 8 6 ) 197. [11] L. Wendler and P,. Pechstedt, Phys. Star. Sol. (b) 141 (1987) 129. [12] [_. Wcndlcr and P,. Pechstedt, Phys. Rcv. B 35 (1987) 5887. [13] 1.. Wendlcr. Solid State ( ' o m n l u n . 65 (1988) 1197. [14] l_. Wcndlcr, R. Haupt and V.G. Grigoryan, Phys. Star. Sol. (b) I49 (19~4) K 123. [151 L. Wendler and k4(i. (}rigoryan. Solid Stale ( ' o m m u n . 71 (1989) 527.

L. Wendler et al. / Dynamically screened potential

[16] L. Wendler, R. Haupt and V.G. Grigoryan, Physica B 167 (1990) 91 part I, this issue). [17] L. Wendler, R. Haupt and V.G. Grigoryan, Physica B 167 (1990) 113 (part III, this issue). [18] F. Stern and S. Das Sarma, Phys. Rev. B 30 (1984) 840. [19] T. Ando, A. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437.

111

[20] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards. Applied Mathematics, Ser. 55 (1964). [21] F.F. Fang and W.E. Howard, Phys. Rev. Lett. 16 (1966) 797. [22] E. Yamaguchi, J. Appl. Phys. 56 (1984) 1722. [23] F. Stern and W.E. Howard, Phys. Rev. 163 (1967) 816.