On the cyclotron resonance width of an electron impurity system in two dimensions

Solid State Communications, Vol. 21, PP. 1105—! 106, 1977.

Pergamon Press.

Printed in Great Britain

ON THE CYCLOTRON RESONANCE WIDTH OF AN ELECTRON IMPURITY SYSTEM IN TWO DIMENSIONS M. Prasad and S. Fujita Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, N.Y. 14260, U.S.A. (Received 21 December 1976 by H. Kawamura) Using the proper connected diagram expansion which incorporates the quasi-particle effect naturally we calculated the cyclotron resonance width F in the extreme quantum limit. 2B112 for a short range interaction, and (b) F = iT1”2 ze2 x n~ Coulomb interaction, are obtained. The field (B) and iC’ (a) h’ I’n~2for concentration (n 8)-dependence is in satisfactory agreement with experimental data. The variation F = (F~+ F~+ ~ of Matthiessen’s rule F = F1 + F2 + . . . holds when there exist scattering centers of different kinds in the system. . .

,

A SYSTEM of electrons moving in two dimensions in interaction with stationary impurities may be regarded as a simple model for the electrons in space-charge layers of a metal—oxide—semiconductor [1]. In the present letter we report on calculations of the cyclotron resonance width in the extreme quantum region based on the general theory developed earlier [2,3], the proper connected diagram expansion of Kubo’s formula for the conductivity tensor. In reference [3], the resonance width F associated with the transition between the lowest Landau oscillator quantum numbers n = 0 and n = 1 for a three-dimensional system was investigated in detail. In particular, this F was shown to satisfy the following-equation, see reference [3], equations (53) F

=

(2iT)3n 3

X

f

3q

v(q)2t2 e_t

or

2n i’ii’

[(27r)

=

2q v(q)2 t2 e_~]1/2~

$

(2a)

8 d

Here, all n

2q are redefined in two dimensions. 8, v(q) and d A significant difference from e(k~) equation (1)—is q~), the absence of the energy-difference, — e(k~ in the denominator, simply because no motion is possible in the plane geometry. This leads to the exact solution given in (2a), which prescribes the square-root concentration dependence of the resonance width for any weak potential, “weak” in the sense that the elementary scattering process is calculated to second order in the interaction strength. The field-dependence of the width can be studied by carrying out the q-integration for specific interactions: (a) Gaussian potential

d

F [e(k~)—e(k~--q~)]2 +h2F2

(1)

where n~is the concentration of impurities; e(k~)E (h2k~X2m*)_lis the kinetic energy associated with the electron motion along the magnetic field; v(q) is the Fourier transform of the impurity potential;

z4~r)

(vo/ird2) exp (— r2/d2),

v(q)

(21r)_2 =

(1

(21r)2v 1/2

2d2).

[ ~

\7T)

(—

(4)

~q

v

0

F t

$ d2r e”1~v(r) 0 exp

=

(3)

d2l —3/2 1/2 —1

(5)

ns

This is in agreement Ando’s basedresult on Green’s function with method withcalculations the selfconsistent Born approximation [4]. In a typical experimetnal condition [5], the cyclotron radius r 0 is of the perpendicular to the plane of the electron motion, the order 100 A, and is much greater than the 2r~ range x nof corresponding equation for the width F describing the potential d. In this case, from (5), F czn3” 8”~B”~. two-dimensional system, can be written down in a This behavior is in satisfactory agreement with experistraightforward manner: ments [5] on Si surface space-charge layers in the region 2n 2q v(q)2t2 et r where the surface carrier density n 2. F = (2ir) 8 d 0 1 x 1012 cm 0 + h2 F2 (2) (b) Screened Coulomb potential 1105 1”2 is the radius of the cyclotron orbit where r0 (h/eB) corresponding to the states with fl = 0. Ifwe assume that the applied magnetic field B is

$

1106 ~r)

CYCLOTRON RESONANCE WIDTH OF AN ELECTRON IMPURITY SYSTEM Vol. 21, No. 12

=

~q)

~

F

— —

h

1

=

2 +q~)~2

—~

(q

1/2 1 2 2 ~ 1/2 n~ [G(~r0q~)J

K

G(a)

=

—

ci

(6)

F

=

F 1

,

~7)

+ ci2 e Li(a)

1 ~~ 1 2/a ~ ~ 1,

rule

(8)

Here i< and ze are respectively the dielectric constant and the chargeCoulomb of the impurity. In particular for the unscreened interaction, ~ ze2 F = n~2. (9) —~~———

Unlike (5), this F does not depend on the magnetic field. Let us now assume that there are scattering centers of different kinds in the system. Let us denote by F 3 the width calculated for the j-th kind disregarding the contribution from the other kinds. The true width F for the present model is feund to be expressed by \1/2 k 21 + F2 2 + 1 “ ‘ F ‘F This is to be compared with the often-used Matthiessen’s —

+

F2 +

(11)

- - -

for the relaxation rates. Regarding neutral and charged impurities as the main sources of scattering and applying (10)Theory we have analyzed the recent experimental data [5], and -

- -

experiment are in general qualitative agreement. The scattering by charged impurities are more effective at low concentrations. The reported dip in the F—n curve may partly be accounted for by assuming that2.theIn order screening effective for n0 theory ~ l0~2must cm be to attain abecomes quantitative agreement, improved substantially, e.g. by including the electron—electron scattering. Also the contribution of transitions between Landau states with a few lowest quantum nuinbers n must be included. In conclusion, our theory developed from first principles and incorporating the quasi-particle effect [3] in a natural manner, yields simple analytical expressions (5), (7), and (9), for the resonance width in the extreme quantum limit. A further comparison between theory and as rule well(10) as the derivation of the fled experiment Matthiessen’s will be reported in amodilater

—

publication.

1.

REFERENCES LANDWEHR G., in Advances in Solid State Physics. (Edited by QUEISSER H.J.), Vol. XV., p49. Pergamon, New York (1975).

2.

LODDER A. & FUJITA S.,J. Phys. Soc. Japan 25, 774 (1968).

3. 4.

FUJITA S. & LODDER A.,Physica 83B, 117 (1976). ANDO 1., J. Phys. Soc. Japan 38, 989 (1975).

5.

ABSTREITER G., KOTTHAUS J.P., KOCH J.F. & DORDA G., Phys. Rev. B14, 2480 (1976).