NGUYEN QUOC KHANH: Dielectric Function and Plasmon Dispersion Relation 73

phys. stat. sol. (b) 197, 73 (1996)

Subject classification: 71.45 and 73.61; 77.22; S7.12

Department of Theoretical Physics, University of Ho Chi Minh City')

Dielectric Function and Plasmon Dispersion Relation of a Quasi-Two-Dimensional Electron Gas

BY NGUYEN Quoc KHANH

(Received February 22, 1996; in revised form May 6, 1996)

In this paper we generalize the method suggested by Lee and Spector to calculate the dielectric function of a quasitwo-dimensional electron gas without strict restriction on the wave vector q, the conducting layer thickness d, and the dielectric constants of the conducting and surrounding media E , E I , ~ 2 . In the case of a thin layer, the plasmon dispersion equation obtained by means of the di- electric function is solved exactly to get the dispersion relation of the collective excitations. We compare our exact solution to the approximated result of Aharonian et al. and find that the latter is valid only for very small values of qd.

1. Introduction

Recently a lot of experimental and theoretical interest has been devoted to quasi-two- dimensional (Q2D) systems such as quantum-well, heterojunctions, and superlattices [l to 31. In connection with this the dielectric function E(q , o) of these systems has at- tracted much attention because its behaviour in the small q limit can determine the dispersion relations of collective excitations and its static behaviour can determine the screening potentials of ionized impurities [4 to 61. It is well known that the image charges have considerable influence on the physical properties and the effect of image charges (EICH) has been taken into account in several works on the dielectric function and plasma dispersion relations of Q2D systems [7 to 91. Interesting conclusions of EICH on the plasma frequencies have been made by Aharonian et al. [lo] for thin semiconduc- tor films but their results are valid only for the special case ~ 1 , ~2 << E and qd << 1.

In this paper we generalize the method suggested by Lee and Spector [5] to calculate the dielectric function of a Q2D electron gas for the case of arbitrary values of q, d , ~ 1 , ~ 2 , and E . This dielectric function in the size quantum limit (qd < 1) g' ives a dispersion equation for the intraband plasmon. We will solve this equation exactly and obtain the dispersion relation of the collective excitation. We will show that in the situa- tion where the condition qd << 1 is not fulfilled, our dispersion relation differs remark- ably from that of Aharonian et al. [lo]. Finally, it will be shown that EICH is not negli- gible in real Q2D systems with significant differences between the dielectric constants of the neighbouring media. -_ -. - . ..

') Ho Chi Minh City, Vietnam.

74 NGUYEN Quoc KHANH

2. Calculation of the Dielectric Function

We consider a system where the electrons are confined in a layer of thickness d. Let the z-axis be normal to the layer which occupies a region of space 0 < z < d. Assume that the dielectric constants of regions z < 0, 0 < z < d , and d < z are €1, E , and E Z , respec- tively. The x-y motion of the electrons is considered free with the effective mass m* while the motion in the z-direction is quantized. The Hamiltonian of the electrons in our system is H = Ho + V ( r , t ) where V(r, t ) is the self-consistent potential which arises in response to the presence of an external longitudinal electric field, and HO is the unper- turbed Hamiltonian for the confined carriers. By assuming a one-dimensional potential well with infinite walls for the confining potential we obtain the following wave func- tions and energy eigenvalues for the unperturbed system:

!Pk,n(X, z ) = (l/S)1'2 exp (ik. X ) (2/d) ' '2 sin (nnz ld ) , n = 1, 2, 3 . . . (1) and

h2 k2 h 2 d 2 m 2rn*d2

Ek,n = n2Eo + 7 with Eo = __

Here k = ( k z , Icy) is the 2D wave vector of the electron, X = (z, y), S is the area of the interface, and n is the subband number.

We assume an external perturbing potential of the form

V , ( X , z , t ) = V , ( X , z ) exp (iot) (3)

and use the Ehrenreich-Cohen self-consistent field prescription ill] to calculate the in- duced electron density,

where a = (k, n) is a composite index and

in which f0(Ea) is the Fermi function. The self-consistent potential is composed of V, and an induced potential V' which is related to the induced charge density n ' ( X , z ) by the Poisson equation

AV' = -(4ne2/c) n ' ( X , z ) . (6) By taking the 2D Fourier transform

q ( z ) = 1 dX V ' ( X , z ) exp (-ZqX) S

S

we obtain

(7)

0; otherwise. (8)

Dielectric Function and Plasmon Dispersion Relation of an Electron Gas 75

The complete solution of ( 8 ) which satisfy the boundary conditions at z = 0, d are [5]

A e-Qz; d < z < c m , I ( z ) + B eq’ + C e-Qz; D eq‘ ;

0 < z < d , -m < z < 0 ,

(9)

where

7rz cos (n - nl) - cos (72 + n’) -

d - 2 ‘ 2 ) d k ‘ , k + q , q 2 + (n - n/)2 (f) 42 + (n + n’)2 (f)

(10)

(11)

B = [I(O) e-qd (1 ~ ~ r 2 ) - I ( d ) (1 + ~ r 1 ) ] / G , (12)

C = [ I ( d ) (1 - Er1) - I ( 0 ) eqd (1 + ~ r a ) ] / G , (13)

D = Er1 {I(()) [(I + E,Z) eqd + (1 - ~ , 1 ) e-“] - 21(d)}/G, (14) G = eqd (1 + Er1 + Er2 + ~ r 1 ~ r 2 ) - e-qd (1 - (15)

E,1 = & / E l , Er2 = & / & a . (16)

8ne2 I ( 4 = 2 @&la

A = ~ r 2 eqd { I ( d ) [(I - Er1) epqd + (1 + e+qd] - 21(O)}/G,

- + ~ , 1 ~ , 2 ) , and

Equations (9) to (16) give the following matrix elements of the induced potentials:

where

and

1 4 1 - I , *(n’ - n) - 611 + I , i (n’ - n) 6’1 - I , *(n’ + n) - + 1, *(d + n) 4 - ( q 2 + (n’ - ny (3)’ 42 + (nl + n)2 (3) Rn‘rL,1/1 = 2

1 1.

.i

.( q 2 + (n - n’)2 (3)’ q 2 + (n + n y (;)2

1 -

1

q2 + (1 - Z ’ Y (3)’ q 2 + (1 + q2 ( $ ) 2

1 -

1

76 NGUYEN Quoc KHANH

Note that when the conditions I’ = I and n = n’ hold, 6 l g ~ l , h ( ~ t - ~ ) gives two but 6p+l,i(n,+n) gives unity 1. From (17) and V = V, + Vs we get

(k+q, 1’1 VO Ik, E)=(l’\ 11) =c dnldn’/’-C 8nxk+q,nr;k,n;l‘1 (n‘i vq(z) I n ) > (20)

n‘n ( k ) where the subscript q in the potentials denotes the 2D Fourier components. The quan- tity in the bracket of (20) is just the dielectric function

&n’,n,L’,l(q> w , = (Snldn’lf - c 8nxk+q,nI,k,n;lfl) ; Of ? (21) k

where x is the quantity defined in (18).

3. Dispersion Relations of Collective Excitations of a Q2D System in the Size Quantum Limit

In the size quantum limit all the electrons are confined to the ground state of the system where n = 1 since the layer is so thin that the energy differences between the different subbands are very large. In this limit, intersubband transitions cannot take place at low temperatures and the dielectric function reduces to

We note that in the case E~ = ~2 = &b, Rl(q) reduces to the result obtained by Lee and Spector,

where sr = E/eb . For a pure 2D system, d goes to zero, Rl(q) in (24) reduces to ~,d/2q, and the two-dimensional dielectric function is given by

For the general case, by denoting y(q) = q R l ( q ) ( ~ 1 + Q ) / E ~ , we can write the dielectric function in the form

El l , l l (q l 0) = 1 - y(q) &tD(4), (26)

Dielectric Function and Plasmon Dispersion Relation of an Electron Gas 77

t . B 3

0 0 . 2 0.4 0.6 0.8

4d +

Fig. 1. The plasma collective modes (28) (curve 1) and (29) (curve 2) in a GaAs layer embedded in media with ~1 = 11.6 and ~2 = 1. The frequency is given in units of wo and the layer parameters are: E = 13.1, rn* = 0.067rn,, n, = 10’O cm-’, d =

where QiD(q) is the polarizability of a two-dimensional non-interacting electron gas gi- ven by

cm, and T = OK

For the case of a 2D degenerate electron gas, the explicit expression for QiD(q) and the zeros of the dielectric function give the following dispersion relation of plasmon excitation [12] :

where qF = ( 2 ~ r n , ) l / ~ and B = h2q~&d/4rn*e2qR1(q). Here n, and qF are, respectively, the 2D electron density and the Fermi wave vector. If E 1 E 2 / E 2 and qd go to zero (28) gives, in the small q limit, the result given by (17) of [lo],

~rn* qd + (&I + & I ) / &

w = [- 4nn,e2

In the special cases qd << ( ~ 1 + ~ a ) / & << 1 and ( ~ 1 + ~ a ) / & << qd << 1, (29) reduces to

(29)

78 NGUYEN Quoc KHANH

1.5

1.25 t

0.75

0 .5

0 .25

0

0 0.2 0.4 0.6 0.8

Qd +

Fig. 2. The plasma collective mode (28) in a GaAs layer for the two cases: ~2 = 1 (curve 1) and ~2 = 11.6 (curve 2). The frequency is given in units of wo and the other parameters are: E = 13.1, = 11.6, m* = 0.067m,, n, = l O l o cm-', d = cm, and T = 0 K

and

respectively. We note that our condition E~EZIE' << 1 for getting (29) is less strict than the condition &I/&, E Z / E << 1 used in [lo]. Now let us consider a numerical example, namely, the case of GaAs layers with parameters taken from [5 ] . In Fig. 1 the dispersion relations (28) (curve 1) and (29) (curve 2) are plotted. These curves show that in the region where the condition qd << 1 breaks down, the approximated expression (29) dif- fers remarkably from the exact solution (28). This means that the dispersion relation (29), derived by Aharonian et al., is valid only in the long wavelength limit (qd < 0.1 in Fig. 1). The effect of image charges is illustrated in Fig. 2, where the dispersion relation (28) is plotted for the two cases ~1 = 11.6, ~2 = 1 (curve 1) and ~1 = ~2 = 11.6 (curve 2). It is seen that the difference between the two cases is considerable. In other words, EICH is very important, especially in systems with significant differences between the dielectric constants of the neighbouring media. Note that we also find a similar beha- viour in the case of a PbSe layer with parameters given in [lo]. Therefore our dielectric function (22) and exact dispersion relation (28) may be very useful because they allow us to calculate the spectrum of collective modes and other properties of the Q2D elec- tron gas, including EICH for large ranges of q and d. Finally, we note that the plasmon mode is well-defined only if the dispersion curve lies outside the electron-hole pair con-

Dielectric Function and Plasmon Dispersion Relation of an Electron Gas 79

tinuum. This condition is expressed by the inequality [la]

The inequality (32) is satisfied for small q up to a certain qc, at which

w = o + . (33) Using (28) we can solve this equation numerically and find that undamped plasma exci- tations in the systems described in Fig. 1 and 2 exist only for qd 5 0.75. Because of this reason, the curves in Fig. 1 and 2 were plotted only for qd 5 0.8.

4. Conclusions

In this paper, by making use of the method developed by Lee and Spector, we have calculated the dielectric function and derived the plasmon dispersion relations of a Q2D electron gas without strict restrictions on q, d, E, ~ 1 , and E2. We have shown that our dielectric function reduces to that of Lee and Spector and our plasma dispersion rela- tions reduce to those of Aharonian et al. in special cases. By comparing our exact disper- sion relation to that derived by Aharonian et al. we have found that the latter is valid only for very small qd even in the special cases EIIE, EZ/E << 1. Therefore we hope that our results will be of help in future theoretical and experimental investigations on the physical properties of Q2D systems [13].

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[lo] K. H. AHARONIAN, H. L. ERKNAPETIAN, and D. R. TILLEY, phys. stat. sol. (b) 150, 133

[ll] H. EHRENREICH and M. H. COHEN, Phys. Rev. 115, 786 (1959). [12] A. CZACHOR, A. HOLAS, S. R. SHARMA, and K. S. SINGWI, Phys. Rev. B 25, 2144 (1982). [13] J. YOSHINO, H. MANEKATA, and L. CHONG, J. Vacuum Sci. Technol. B 5, 683 (1987).

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