On the cyclotron resonance width of very pure Ge

Solid State Communications, Vol. 37, pp. 919—920. Pergamon Press Ltd. 1981. Printed in Great Britain. ON THE CYCLOTRON RESONANCE WIDTH OF VERY PURE Ge T.K. Srinivas, S. Chaudhury and S. Fujita Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY 14260, U.S.A. (Received 13 October 1980 by H. Kawamura) The temperature variation of the cyclotron resonance width I” for a very pure semiconductor is calculated numerically based on the earlier theory [S. Fujita & A. Lodder, Physica 83B, 117 (1976)]. The results are in excellent agreement with experiments by Kawamura et aL for very pure Ge. IN 1964 KAWAMURA et aL [1]reported on experimental measurements of the cyclotron resonance lineshape for very pure Ge with the impurity density (n8) range: 3cm3. The half-width I’ at very low temperal0”—10’ tures approximately follows the formula I’

=

0.915h314(e2/g)m*—i/4 W —1/4 0

The average power p per unit volume derived by a circularly polarized microwave of frequency w and field K is given by pamplitude = 4E2 Re {a+(~.)},

—1/2

fl~

(1)

where ~ is the dielectric constant, m* the effective mass, w0 the cyclotron frequency. This formula, which was obtained on semiclassical arguments, was found to nbe in good agreement with experiments in the 8.dependence, the T-independence below 2.6 K, the m*.dependence and again the absolute magnitude. The square-root n8ependence is quite showing the temperature variation of theunusual. width FAisgraph reproduced in Fig. 1. The outstanding features are that (a) the width F appears to be constant at the lowest temperatures [which is described by formula (1)] and (b) it decreases monotonically as temperature is raised, Many attempts [2, 3] have since been made to give foundations to the semi-classical arguments by the first principle calculations. Earlier, one of the present authors (S.F.), and his associate presented a theory [3] yielding a result which is essentially identical with equation (1) in the low temperature limit and which decreases with the temperature T, the latter being obtained with rougher approximations. This theory starts with Kubo’s formula [4] for the dynamic conductivity and utilizes the proper connected diagram expansion [5],assuming that the cause of the linewidth is entirely due to the unscreened Coulomb interaction between electrons and charged impurities. In the present communication, we report on the numerical computations of the temperature-dependent formula, equations (2)—(4). The results, which are shown in solid line in Fig. 1, are in excellent agreement with the experimental data. Since the starting formulas together with derivations and discussions were given in [3], we wifi only quote the main steps of calculations.

(2)

where a÷_(w)is the dynamic conductivity which can be expressed in terms of the current correlation integral (Kubo’s formula). After lengthy but straightforward analysis in terms of proper connected diagrams, we obtain in the extreme quantum limit, see (ref. [3], 7.3) 2n1~m*)2((3/2irm*)2 Re {a+.} = 2e I,

+

2 e -~k~), (3) dk~~ )2 + r where the Boltzmann distribution for electrons is assumed; P (kB T) ‘; and F is the energy-dependent width associated with the transition between the first and second lowest oscillator states (N = 0 and N = 1). This I’ satisfies the following equation [ref. [3], 5.9): +~ 16e4m*2fl 2 e -t (is s2(s 2)2 + 72 8 dr t = _____________

XJ

—

J

‘~‘

x

2

—

1 + (2/r~k~)t]2

(4)

‘

[s where ‘y is the reduced width ‘y(k~)

hJ”/e(k~)= 2rn*Fh_1k2,

(5)

and r 2 = (h/eB)~2the cyclotron radius 0 (h/rn *wo)l/ corresponding to the lowest oscillator states (N = 0). The t.integration in equation (4) can be carried out, yielding i 6e4m*2 + 1 = h4i~2r~Ik ~ 52(s_2)2+7254 —~

x [u2 + u3 where U

919

—

4s~r2ok~

e~’E

3+ u4)],

(6)

1(u)(2u (7)

920

CYCLOTRON RESONANCE WIDTH OF VERY PURE Ge

required for the evaluation of Re {a~}, however, has

H

‘0

Vol. 37, No. 11

the effective half-width F(T) such that at w — w0 = F(T), the function F drops to one half the ~

F(TJ

-It-

a finite maximum maximum, that is (F’)T at w — F[F(T)] = ~F(0)= ~(F’)T.

=

0. Let us define (12)

0

Using this definition we obtained the temperature-

sec I

I

I I

I

5 TEMPERATURE IN

dependent width F(T) for several points of temperature. The results are represented by the solid line in Fig. 1. Inspection of equations (4)—(5) shows that the

I

30

3

50

°K

Fig. 1. The half-width F at n n~= 1012 3 as a function of temperature T, observed by Kawamura et al. [1] are shown in circles. The solid line represents the numerical solutions from equations (2)—(5). and E

J~

1(u)

X

is the exponential integral. Equation (6) was solved with the aid of computers for a selected set of the k2-values, where the following numerical values are chosen: 16;

=

m* = 0.09m (electron mass); 2cm3

l0’

= =

=

2.97 x 10”sec.

width F(n~,Ic2) n~for large k2 and F ‘Ins for small k2. Translated in the temperature dependence, the effective0~at width OK F(T) to n~°~5 changes at T =its10K. density Ourdependence theoretical from n3for I’(T) approaches the limit region given by values equation (1) at temperatures below 0.1 K rather than below 1 —2 K, which are indicated by the experiments and analysis by Kawamura eta!. To resolve this discrepancy, it is highly desirable that experiments be carried out at very low temperature for very pure Ge and/or Si. The present theory neglects the effects of the Coulomb interaction between electrons and the electron—phonon interaction. These effects must be included, or at least correctly estimated, in the more realistic theoretical calculations.

(8)

For small k 2, the width F behaves like 1/I k2 I. This makes the integral (F)T

J dk~F

1. e’~z)/J

dk2

e~)

(9)

2.

diverges, invalidating the usual approximation: (F)T

________

K(w_w)2

+ r2)T

~ (~—~)~ +(fl~

The average F(w —w0)

2 (F[(w —w0)

+

I’2I’)~,

(10) (11)

3. 4. 5.

REFERENCES H. Kawamura, H. Saji, M. Fukai, K. Sekido & I. Imai,J. Phys. Soc. Japan 19,288 (1964). S.J.Miyake,J.Phys. Soc. Japan 20,412 (1964); A. Kawabata,J. Phys. Soc. Japan 23,999 (1967); E.E.H. Shin, P.N. Argyres & B. Lax, Phys. Rev. Lett. 28,1634 (1972);Phys. Rev. B7, 137 (1973); E.E.H. Shin, P.N. Argyres, N.J.M. Horing & B. Lax,Phys. Rev. B7, 5408 (1973). S. Fujita & A. L.odder,Physica 83B, 117 (1976). R. Kubo,J. Phys. Soc. Japan 12, 570 (1957). 59(1969). S. Fujita & C.C. Chen, mt. J. Theoret. Phys. 2,