Magnetic field dependence of cyclotron resonance linewidth for acoustic polarons in the extreme quantum limit

Solid State Communications, Vol. 37, pp. 293—294. Pergamon Press Ltd. 1981. Printed in Great Britain. MAGNETIC FIELD DEPENDENCE OF CYCLOTRON RESONANCE UNEWIDTH FOR ACOUSTIC POLARONS IN THE EXTREME QUANTUM LIMIT S. D. Choi Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, NY, 14260, U.S.A. and S. Fujita* Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada (Received 20 August 1980 by H. Kawamura) The magnetic field dependence of cyclotron resonance linewidth (CRLW) due to electron—acoustic phonon interactions in the extreme quantum limit is obtained on the basis of Kubo’s formula and Fujita’s diagram method. The 2-dimensional electron-piezoelectric phonon interaction generates a finite maximum CRLW as a function of magnetic field while CRLW for all other acoustic polaronsincrease with the magnetic field. ONE of the present authors (S.F.) and his co-workers developed a quantum theory of cyclotron resonance linewldth (CRLW) for electron-phonon interactions on the basIs of Kubo’s formula [1] and proper connected diagram expansion method [2].The theory has been applied to 3-dimensional optical polaronsat low temperatures [3] and 2- and 3-dimensional deformation potential polarons at relatively high temperatures [4,5], the results of which are In good agreement with the experimental data reported by Summers etaL [6]by Kuelbeck and Kotthaus [7], and by Miura et aL [8] respectively. In the present letter we report on the magnetic field dependence of linewidth in both 3-and 2-dimensional electron-acoustic phonon systems in the extreme quantum limit and at very low temperatures on the basis of the same theory. We make the followmg assumptions: (a) The temperature (T) isso low and the magnetic field (B) so strong that transitions between states with the two lowest oscillator quantum numbers (N =0,1) are relevant. (b) Phononsact as scatterers, and their distri-

After systematic calculations under these conditions, we obtain the linewidth functions for the transitions between the two lowest Landau levels (N 0 and 1) with the momenta k as follows:

r’(k)

~ IC~I2e_~((l+ n~)[(t2 t) —

=

h q ~ 6(~~

+ h~q)

—

t6(eq~_s~ hc~~)] + nq[(t2 t)&(Q -k 6k +

—

—

—

—

Z

—

z

Z

+ t8(eq~_a~ + lIwq)J} for 3-dimensional systems, and

r

=

-~

~ ICqI2e_t t2[(l +fl~Xh~q +ihr)’

+ nq(~hci~q+ ihr) 1] (2) for 2-dimensional systems at the resonance maximum: =

Wo.

B

L.

h~k2 ,

~,

/ F 0

bution, characterized by the Plank distribution, can be neglected against unity at very low temperatures. (c) The electron—phonon interactions are isotropic; and Bardeen—~iockley’sdeformation potential polaronsand Mahan—Hopfleld piezoelectric polarons are chosen. (d) The structures are parabolic and the electron spinsband are neglected. * Permanent address: Department of Physics and Astronomy, State University of New York at Buffalo, Amherst, New York, 14260, U.S.A.

(1)

—

m*

2m

Z

~

h

\1~’2

( \m*c~o/

+

‘~a= ~ 111 with $ (kIT)’, where m* is the electron effective mass, and ~ = sq the frequency of acoustic phonon 2with momentum q with s for the 3-dimensional being the speed of sound. ICql systems are given by r~ 1 IC~12 = q (3) 2ps ~7 —

—

293

—

294

CYCLOTRON RESONANCE UNEWIDTH FOR ACOUSTIC POLARONS

Vol. 37, No.3

Table 1. Magneticfield dependence of the width 1’ Classification

Dimensions

C~~

Deformation-potential polarons

3D 2D 3D 2D

s.,/q ..Jq i/..Jq 1/~,/q

Plezoelectric polarons for deformation potentials polarons and

(4)

2e polarons, IZq for piezoelectrlc where E 1 is the deformation potential constant, K the electromechanical coupling constant, p the mass density, e the dielectric constant, and flthe crystal volume. On the other hand, C~l2for 2-dimensional systems are hE? 1 ICq 12 = ~~—--~ q (5) for deformation potential polarons and 2e~! -~(6) I~-qI ~i’ 2 = he hp se A q for piezoelectric polarons, where e 0 is the piezoelectric constant, p’ the mass density in two dimensions, and A the area of the 2-dimensional system. At low temperatures, flq ~ 0; so only the emission terms remain. In the case of 3-dimensional polarons, we can approximate ~ as ~ siro in the extreme quantum limit and obtain

2E~I(4(2

~)

2p’s3

~

0 4 \1/2 / \5 ii F 2~ ir’1’~ rjF2\ + ‘1/ -~(~)(~ ——t—I 2 ~ exp (—~_r)erfc (~.o~.’)] —)

I

‘3/2

—

‘.,‘1/2/

2

0F \1 L~i]

(10)

S

conditions (a)—(d), and which show little impurity scattering effects. Improvements may be made by including electron transitions involving all the Landau levels with spin and nonparabolicity of the band structure. More detailed treatments will be reported in separate publications.

2:

~)

,

Iro~F~\ ‘r

for piezoelectric polarons. The B-dependence of r for these polarons cannot be put in a simple form. However, we can notice that the width F increases with B in the case of deformation potential polarons while F has a finiteTheoretical maximum results value for obtained piezoelectric shouldpolarons. be compared with experiments made on systems which satisfy the

for the deformation potential polarons and 2e2s)/(8(2ir)”2h2roe) (k~2+ ~ (8) F(k) = (m ~K for the piezoelectric polarons. For slowly moving dcctrons (1cr 0) we obtain the B-dependence of r as listed in Table 1. The results for the 2-dimensional systems can be obtained from (2), (5) and (6) as follows _______13s11r/2\51’2 ~JvIr\2/ 1 = l6irp’hs3 (ri) -~ ~i2)3~’2 —

I 4~(4 ~i~j

%,/lrI.i\ 11’

exp ~.j—~-) erfc 2

1

L-i-

1 6irhe

=

+

(7)

2~oPs)(k2+

1’ has a finite maximum

e2e~r yirl2\

—1/2

=

1’ increases with B

for deformation potential polarons and

Ic~I~ = K2e2stt!!

F(k)

B-dependence of I’ I’ B514

3. 4. 5. 6. 7.

REFERENCES R Kubo, J Phys. Soc. Japanl2, 570 (1957). Fupta, Introduction to Nonequilibrium Quantum Statistical Mechanics, Chapter 7. W. B. Saunders Philadelphia (1966); A. Lodder & S. Fujita, J.Co., Phys. Soc. Japan 25,774(1968). C. S. Raju & S. Fujita, Acta Physica Austriaca 43, 1 (1975). M. Prasad, T. K. Srinlvas & S. Fujita, Solid State Commun. 24,439 (1977). A. Suzuki, S. D. Choi & S. Fujita, J. Phys. Chem. Solids 41, 735 (1980). C. J. Summers, R. B. Dennis, B. S. Wherrett, P. G. Harper & S.D. Smith, Phys. Rev. 170,755 (1968). H. Kuelbeck & J. P. Kotthaus, Phys. Rev. Lert. 35, 1019 (1975).

8. Miura, private communication (see [5]).