Cyclotron resonance of polarons in CdTe

Solid State Communications.

Vol. 8, pp. 1907—1909, 1970.

Pergamon Press

Printed in Great Britain

CYCLOTRON RESONANCE OF POLARONS IN CdTe K.K. Bajaj Institut de Physique, Université de Liege, Sart Tilman, Liege I, Belgium. (Received 1 September 1970 by S. Amelinckx)

Energy spectrum of a polaron in a magnetic field is calculated using Onsager’s approach. The transition energy between the two lowest Landau levels thus calculated is compared with its experimental value in CdTe at various magnetic fields. A comparison with a numerical variational calculation is also made.

RECENTLY WALDMAN’ and his collaborators have made low temperature cyclotron resonance measurements in CdTe at various far infrared frequencies below the Reststrahl band. They find an increase in the polaron mass as a function

where

V~

and

a

(~~) ( ( “4

=

=

__.

e2

—

1

—

~

—

E~

2

(4ira)’~

1)

m

—

\1/2

E 0

of the applied magnetic field which they explain in terms of electron-phonon interaction. Using

is the dimensionless coupling constant. The other symbols have their standard meaning of the usual

Fröhlich’s hamiltonian and an appropriate trial wave function they calculate numerically the

polaron theory. Y~enow calculate the energy spectrum of this system at temperature T 0°K =

transition energy between the two lowest Landau levels and find a remarkably good agreement between their calculated value and the experimental value at various magnetic fields. The purpose of this brief is toin show that the field energy spectrum of anote polaron a magnetic calculated bya much simpler method i.e. using Onsager’s theory,2 gives results which agree very well with experiment in the case of CdTe at low fields (less than bOG). The dynamical properties of our system consisting of a conduction electron moving in a parabolic band and interacting with the L—O phonon field of the lattice are specified by the 3 hamiltonian operator, H ~ [Vq aq zq.r + V~7a~ e q.r + —t

2

2

4

m2a~(2) E0(k) ahc~+ (1 a/6) 160 Th ~ak The energy spectrum of a polaron in a magnetic field is now calculated using Onsager’s theory.2 This involves evaluating the area in k-space S(E 0, k2) intercepted on the surface of constant energy =

—

—

E0(k~,kg,, k2)

—

=

E0

by a plane perpendicular to the direction of the magnetic field assumed be using along the the relation z-axis of the coordinate systemtoand

-,

S(E

=

2171

using perturbation theory. For small values of kwef md ~

0, k2) q

=

(n

+

y) 27fheB

(3)

C

Here n is a non-negative integral, y is a number that lies between zero and one and ~ is the

~q hwa~a q q (1) 1907

1908

CYCLOTRON RESONANCE OF POLARONS IN CdTe

Vol. 8, No. 22

Table 1. Comparison of observed cyclotron energies with energies calculated by present theory and with those of reference 1. Resonance field (koe)

Expt. energy (meV)

31.17 ±0.06 54.76 ±0.1

3.684 6.368

Calculated energy present theory

(meV) reference 1.

3.663 6.355

3.684 6.372

7.268 10.51 12.81

7.275 10.41 12.49

+0.3

62.9 ~ 92.5 ±0.5 114*.0 ±0.6 *

7.221 10.454 12.52 ±0.06

Energy measured at temperature T = 50°K. instead of at .T = 4.2°K. This may be partly the reason for a rather poor agreement between our theory and experiment at this field.

magnetic field. Such a calculation yields for the 4 energy levels E~(kz)

—aha

+

ha~j1

E2 kz2 (1 2,72

—

—

a/6)

a/6) (n

-~

(~:/~~)2

—

at various values of the magnetic field and compare it with the experimental values in Table 1. For fields less than 100 kG the agreement between our values and the experimental values is very good (within 1 per cent). A comparison with the results of a numerical calculation based on a variational

40f approach1 is also shown in Table 1. We have

(n

~y)2 + 2(/)hkZ

+

a

~

1112

kz’\l

(

~ (4)

\ 2rn

where ~ = eB/mc The conduction band of CdTe is known to have some non-parabolicity in the absence of the electron—phonon interaction. Using Kane’s theory; ~ we find 1 (5) E(k) = + h 22 k 1 344 1o~41 k 2rn — where E~ is the energy gap. The effect of this non-parabolicity on the Landau levels is calculated by Onsager’s theory. Using 0.0966 m~,for the polaron mass (mr), 0.090 m~for the band mass (rn), 0. 0213 eV for the optical phonon energy 0) and 0.4 for the coupling constant (a) in CdTe we calculate the transition energy (E 1 — E0) at k2 = U

used ‘y = 1/2 though strictly speaking one should use a slightly different value. Considering the assumptions involved in deriving FrOhlich’s hamiltonian and the approximations made in evaluating the energy spectrum by our method or by a variational method, an agreement between theory and experiment within a percent or so is more than one should hope for. Thus application of this particularly simple approach of Onsager to equation (2) — (this equations is valid for small k and therefore somewhat approximate at high fields) gives rather good results. The justification for using this method for4small values of n has been discussed elsewhere. Acknowledgement — Useful discussions with Prof. R. Evrard and Prof. J. Devreese are gratefully acknowledged.

Vol. 8, No. 22

1.

CYLOTRON RESONANCE OF POLARONS IN CdTe

REFERENCES WALDMAN J., LARSEN D.M. TENNENWALD P.E., BRADLEY C.C., COHN D.R. and LAX B., Phys. Rev. LeU. 23, 1033 (1969).

2.

ONSAGER L., Phil. Mag. 43, 1006 (1952)

3.

FROHLICH HO., Adv. Phys., 3, 325 (1954).

4. 5.

BAJAJ K.K., Phys. Rev. 170, 694 (1968); Nuovo Cim. 55B, 244 (1968). KANE E.O., J. Phys. Chem. Solids 1, 249 (1957).

On calcule le spectre d’énergie d’un polaron dans un champ magnétique au moyen du formalisme d’Onsager. On compare l’énergie de transition entre les deux niveaux de Landau les plus bas, calculée de cette manière, avec Ia valeur expérimentale obtenue dans CdTe pour différentes valeurs du champ magnétique. On fait également Ia comparaison avec les résultats d’un calcul numérique variationnel.

1909