Impurity density dependence of cyclotron-resonance line width

Solid State Communications Vol. 16, pp. 117—118, 1975.

Pergamon Press.

Printed in Great Britain

IMPURITY DENSITY DEPENDENCE OF CYCLOTRON-RESONANCE LINE WIDTH A. Lodder Natuurkundig Laboratorium, Vrije Universiteit, Amsterdam, The Netherlands (Received 16 September 1974 by A.R. Miedema)

It is shown that the by now commonly accepted expression for the cyclotron-resonance line width leads to a n~’~-dependence for InSb, where n8 is the impurity concentration. The order of magnitude is in agreement with experiment.

RECENTLY’ some confusion arose with respect to the description of the cyclotron-resonance line width 2 pointed out that the (CRLW). In 1967 Kawabata CRLW is a typical transport property: in the expression for the CRLW “gain and loss” factors must be included. These factors are comparable withthe [1 cos (p, p’)] factor in the conventional expression for the relaxation time of the electron-impurity system in the static case without magnetic field, in which factor the scattering of electrons with momentum p’ into the reference state characterized by the momentum p and vice versa are accounted for. —

in accordance with the result of Kawabata. This expression is n = Re w(q) 2 ~

-

X

f

K(O, n

[(

—

—

K(o, n; p/2)

+

Wq~].+ 7nq~00

1

i(nw~+ wq~)+ 710,nqzj

()

Here n

In 1968 Kawabata’s result wasexpression confirmedfor by the a 3 The lowest order general theory. CRLW derived from this theory was identical to the expression ofKawabata.

8 stands for the concentration of the impurities, ~2is the volume of the system and w(q) is the Fourier transform of the) potential of aLandau single scatterer. n(= 0, 1, 2,.. denotes the oscillator The quanturn number, w~= hq~/2m,c~,= eH/mc and p = X2( 2 + q~).The functionK(n, n’;t) = _e_ttn~n!/n’! 9 L~~’~(t) X L~+~”’~(r), where ~ ~(t) are associate Laguerre polynomials. The expression to be evaluated for the conditions of the experiment,5 ~c ~ follows from (1) 2 (p12)2 ~ 710 = h2(2~)3 d3q Iw(q) I 2 2 (2) + Tio assuming a weak q 2-dependence of the width. The point to be made in this communication is, that both Shin et aL4 and Lodder and Fujita’ wiongly neglected ~ with respect to 7io, which has a consequence for the impurity density dependence of the width. The reasoning was, that a potential w(q) with a large screening length a would favour small q-contributions .

Recently Shin et aL4 showed that for charged impurities with screening length a, for a X (hc/ell)~, it is not allowed to neglect the line width in the energy denominators [see(1), below] which occur in the expression for the CRLW 7io~However, in their ex~‘

pression for 710 no factors are included corresponding to the gain and loss factor mentioned above, 3 an expression for the From the general theory the features of the CRLW follows which combines expressions given by Kawabata and by Shin et al On the one hand the line width occurs in the energy denominators in agreement with the result of Shin eta!. On the other hand the gain and loss factors are present 117

5

118

DEPENDENCE OF CYCLOTRON-RESONANCE UNE WIDTH

in the integral. It is clear that this neglect generally leads to a ni-dependence of 7io. For InSb5 however 7io 1.4 X 1011 sec~while, for q~= 1/a, w~ 6.3 X 1012 sec~ so that expression (2) has to be evaluated fully. For a Gaussian potential w(q) = w 2~2, it is found that 0e~ ,

3/2

7io

— —

73~’2 10

+00

l)~

=

-00

2x2)/h}

=

$

(h/m)~

(5)

n$~(ze2I~L)2

2e~ dt (tt+ )2 _____

00

p +1

=

—

(p2

+ 2p)e” E 1(p),

while

x4+1 ~ exp(—(4m710a

_____

and F(p)

____________________

<

c~x2 F(x \ — —

where

fl 8W0 2~2h2(t~2m)~X2(4a2/X2 +

Vol. 16, No. 1

E i(P)

S~— 00

=

p

The following conclusions can be drawn: (i) Since the integral in (3) is wealdy dependent Ofl Yio for the experimental conditions, this expression predicts a n 2”3-dependence of the width. 8 (ii) For very low H-fields (A ~ a) an increase of Yio is expected with H, while in the experimental region (A a) a decrease of ho with increasing H follows. The calculation of Yio with expression (2) has been done also for a delta function potential w(q) = wo~and an estimate is made for a screened Coulomb potential w(q) = 47rZe2/(q2 + a2)KL. For the delta function potential the result is 3/2

—

The function F(X2/2a2) behaves such that in the experimental region (A a) the width decreases with increasing H, while for low fields an increase with the field follows. We conclude that all potentials result in a n 213-de8 pendence, instead of a ni-dependence predicted earlier, for the CRLW, while the value increases when the magnetic field decreases. It is worthwhile to settle these points by an experiment independent of the one of Kaplan et a!. the more so as the value of Yio calculated from (5) for H = 10 kG, n 4 cm3, 8 = 2.8 X iO’ ,~

n

8w~ 2 (4) 2irh2(h/m)~A which has the same features (i) and (ii) as the result for the Gaussian potential. Tio

is the exponential integral.

—

The estimate for the screened Coulomb potential, which is valid for the experimental conditions Yio ~ h/2ma2, is

afor = A, KL = 17 and men = 0.014 m, which is typical InSb, is Yio = 1.1 X l0’~sec~’ which is compared with the experimental value5 1.4 X 1011 sec~,quite reasonable in view of the approximations involved and the model character of the calculation. ,

REFERENCES 1.

LODDER A. and FUJITA S.,Phys. Lett. 46A, 381 (1974).

2. 3.

KAWABATA A., J. Phys. Soc. Japan 23, 999 (1967). LODDER A. and FUJITA S.,J. Phys. Soc. Japan 25, 774 (1968).

4.

SHIN E.E.H., ARGYRES P.N. and LAX B., Phys. Rev. B7, 3572 (1973).

5.

KAPLAN R., McCOMBE B.D. and WAGNER RJ., Solid State Commun. 12, 967 (1973).