Polaron with the anisotropic mass

Solid State Communications,

Vol. 14, pp. 533—536, 1974. Pergamon Press.

Printed in Great Britain

POLARON WITH THE ANISOTROPIC MASS~

M. Matsuura and S. Wang Department of Physics, University of Waterloo,Waterloo, Ontario, Canada (Received 26 October 1973 by A.G. Chynoweth)

Mass correction fonnulae for the polaron with the anisotropic mass are derived and used to calculate the bare masses from the observed anisotropic polaion masses in T1C1 and T1Br.

THE POLARON, i.e., the electron interacting with the phonons (the acoustic or optical phonons), has been studied for many years.’ In polar ciystals the electron interacts strongly with the longitudinal optical (LO) phonons and the mass observed in experiments such as cyclotron resonance is affected very much by this (optical) polaron effect. The mass so observed is usually called the (optical) polaron mass. Its relation with the bare mass of the electron has been of considerable interest and has been studied in some detail, but only for the case of isotropic polaron mass.1~Recently Hodby eta!.6 have observed the anisotropic polaron masses in

the mass and the momentum in the /.direction, ~ a~Xand akx denote, respectively, the energy, the creation and the annihilation operators for X.type phonons with the wave vector k. nk~= Vk,x is the electron—phonon coupling constant. We and Vk,x = assume the properties t~k,X = ~ V_~.7 First of all, let us apply the second order perturbation theory to the problem in question. Starting with the vacuum state of the phonons, the energy correction for the electron state p, i.e. (ps, p~ 1Pz), is given by

TLCI and TlBr, which are attributed to the hole resonances. Their observed results imply this anisotropic effect is significant. In this work we first derive the formulae, which enable us to calculate the bare masses from the observed anisotropic masses and then apply the derived results to hole resonances in T1CI and TLBr.

/

~(2)

=

—

~

~,

—

~z&

m~/

\2nij

(2)

AE(2)

—

—

~‘

u2

~

~

2

E hk~.Pj/mj)+ x~f

1+12

~.e~1~+E ~k~X~k,X ~ 2nij kX

~

hkjpj/mj) I

where V~.{e’~a~+ h.c.}

/

I

For small p we obtain

We start with the following Hamiltonian,

+ ~

~

‘

—

H=

+

(1)

(3)

i

X

=

(1~.~k.x +Z h2kf/2m~.) j

k.A

wherej takes x,y and z. ni,~and P, are, respectively,

+...

/

Thus the total energy of the electron can be written

~ Work supported by the National Research Council

+

533

=

6E +

~

+...,

(4)

POLARON WITH THE ANISOTROPIC MASS

534 where

Vol. 14, No.7

which is the mass correction formula in the inter= — ~

V~X

k

(5)

intermediate coupling theory8 We may also take

and rn 1’~/rn~= 11(1 —2~).

Here

(6) =

3(h2kJ/rnj). =

A

V.,xX

~

AIO>+ ~Ak,x4,xIO>

Equations (5) and (6) are, respectively, the self~ energy and mass correction formula in second order

as in the Haga approximation.49 Choosing Fkx = — XVk,x, we obtain the following equations for A and Ak,x

perturbation theory.

(E~_oE_~)

Now we come to consider the problem in question by a variational method.2’4 We may take the following ~‘TR = ei k,A)Ul~> (7)

=

A

and (E~)_~E_t~k.x__Z~ j

)

as a trial function. Here

U

=

=~

expI~Fk,X(a~,x—ak,A)

j

and I~,>issome phonon function. The form of Fk,?. and I ~> will be determined later. Then the polaron energy with the momentum p may be determined from the variation of E(p) = (WTR I H I ‘4!TR>. in the intermediate coupling theory2 we take 10,>= tO>, O>being the vacuum state of the

=

,

(13)

~p)p,.

=

For small p equations (12) and (13) yield (E(~)

—

6E

1—ii~(pfl—~ kV~Y (8)

2m~

(i~(p)—A)Fk,x,

~

Setting p

—

~ Pj) ~(~~)(1 +

~)

~a

= —

=

0 (i * j) in this equation gives E(pj)

=

s5E+pJ/2rnff*,

(14) (15)

where

where 2k~/2rn~)+ ~~((p) l)/rnij~-I (h and i~~,(j)is the quantity satisfying the equation y

where

~ mj

~

1rn,•

phonons. Then the variational condition for Fk, i.e., 6E(J~)/~Fk = 0 yields E(p)

~ i~(p) (12)

—~

Now letmi~/rn~ us discuss = the (1 +~)/(1 derived results —~). in relation (16)to

[~k~x+~

flj(p) Pj

~

= k, A

2. V~,~hk1 Y

(9)

Solving this equation for small p we obtain ~j(P) = 2~/(1+ 2’~)+... . Substituting this solution into equation (8) yields

E(p)

=

p~/~rnf* + ...

öE + ~

those obtained for the usual optical polaron problem. In the usual optical polaron problem rn~= rn, 1 = rn2 = rn, has the only interaction the longitudinal with themode electron, of the andoptical the electron— phonon phonon interaction constant Vk,, is given by

(1

+

2’ij),

(17)

where U= IL.(41ra)”2 (h,I2mw)”4and the dimensionless electron—phonon coupling constant ~ =

(11)

e2(l/e,.. 1/es) (rn/2h3w)”2. Here ~2is the crystal volume, t~,is the dispersioniess LO phonon energy, e~,and ~ are, respectively, the hig~iand the low —

=

~

(10)

where n4*/mj

I

U

V~ = ~

Vol. 14, No.7

POLARON WITH THE ANISOTROPIC MASS

535

Table 1. Experimental polaron masses in reference 6 and the calculated bare masses. m1 = m~= my and m2 = m~.SP, ICTand HA are the results of the second order perturbation theory, the Intermediate coupling theory and the Haga approximation, respectively. Values In the brackets we the results of the calculation using the formulaefor the isotropic polaron. The units for the mass are the free electron mass m = 1.

Experimental polaron mass

si

ICT

HA

m1

0.74

0.412 (0377)

0.491 (0.476)

0.453 (0.431)

1772

0.55

0.256 (0.306)

0.386 (0.370)

0.310 (0.341)

0.577 (0.493)

0.663 (0.627)

0.619 (0.566)

0.252

0.362

(0.340)

(0.400)

0.318 (0.371)

TlBr

0.98 .nCl in2

0.58

frequency dielectric constants. For this optical polaron problem we obtain = a/12 and equations (6), (II) and (16) become

the present anisotropic polaron.

‘~

=

ni,~

(1

—

~

m1

I ~ a/6)’ in1

=

(1 + a/6), and

(1 +a/l2)

(I

—

(18)

a/l 2)’

respectively. Here j = x, y and z. The first, second, and third results in equation (18) are the well known results for the optical polaron in second order per. 2 and turbation theory,intermediate coupling Intheory the Haga approximation,4 respectively. the optical polaron problem more reliable expressions for the mass correction have been derived by Feynman3 using the path integral method and by Larsen4 using a variational method which includes the two phonon correlation term. Actually, in the region a ~ 3.5 Feynman’s results are very close to those of second order perturbation theory, whereas Larsen’s results are very close to those of Haga approximation. The differences between the results of Feynman’s and Larsen’s theories are significant and the Larsen’s results seem to be more reliable than Feynman’s in this region of a.4’10 We note that for a ~ 3.5 the intermediate coupling theory does not yield better results for the effective mass than the second order perturbation theory.5 l’his may be also applied to

We now apply the above derived results to the cyclotron resonances of holes in TIC1 and T1Br. These hole resonances are anisotropic with (1, 0, 0) or a-type symmetry and the corresponding observed polaron masses are summarized in Table 1,6 There m1 =m~=my,m, =m~andz axisischosento lie along the line joining the center of the Brillouin zone to the point I, where the maximum of the valence band exists. In TiC and T1Br there is no anisotropy of the dielectric constants and thus we can use the usual electron—LO phonon coupling constant of equation (17). Crystal parameters are taken from reference 11. In Table I we also list the bare masses (i.e., masses without the optical phonon effect) calculated from equations (6), (11) and (16) and those calculated from expressions in equation(18) which neglect the anisotropy of the masses. We see that the amsotropy of the masses affects the bare masses very much (by 10—20%). The magnitude of this effect depends on the theory used. From the discussion following equation (18) we may expect that the present results for the bare masses obtained by Hap approximation are more reliable than others because a is around 3 for the materials considered here.

Other materials for which the anisotropic masses have been observed are I1—IV compounds such as CdS and ZnS, which have the longitudinal hole masses

536

POLARON WITh THE ANISOTROPIC MASS

0.04—0.8 and the transverse hole masses 1_5.~2 However in these materials there are also the anisotropy of the dielectric constants12 and the piezoelectric polaron effect (which arises from the lack of the inversion symmetry and has the amsotropic nature).”8 Accordingly the situation in these materials is very complicated and the application of the above derived results to these materials is not considered in thJ5 work.

Vol. 14, No.7

In summary we have derived the mass correction formulae for the polaron with the anisotropic mass and calculated the bare masses from the anisotropic polaron masses of hole cyclotron resonances in TICI

and T1Br. Acknowledgements We are grateful to Professor G.D. Whitfield for sending us reference 8 before the publication. —

REFERENCES 1.

Polarons in Ionic Crystals and Polar Semiconductors, (edited by J.T. DEVREESE) North-Holland, London (1972).

2. 3.

LEE, T.D., LOW F.E. and PINES D., Phys. Rev. 90, 297 FEYNMAN R.P., Phys. Rev. 97, 660 (1955).

4.

LARSEN D.M., Phys. Rev. 144,697 (1966); 174, 1046(1968).

5.

SHENGP.andDOWJ.D.,Phys.Rev. B4, 1343 (1971).

6. 7.

HODBY J.W., JENKiN G.T., KOBAYASHJ K. and TAMURA H., Solid Stare Commun. 10, 1017 (1972). Without this natural assumption and the assumption that there is no term like pjpj/mu (1*/) in the Hamiltonian (I), we have the contribution proportional top, which cannot be treated as a mass correction.

8.

LICARI J.J. and WHITFIELD G.D. (to be published); they have discussed the anisotropic piezoelectric polaron. Their equation (42) for the velocity v/~p)(= p 1/rnf), derived with use of the intermediate coupling theory, yields the present result of equation (11) with Wk,X = V~Ik I and the electron—phonon coupling constant of the piezoelectric polaron when p is small and rn~’= 0 (i * /). HAGAE.,frog. Theor.Phys. 13,355 (1955).

9. 10. 11.

12.

(1953).

LARSEN D.M.,Phys. Rev. 172, 967 (1968). BACHRACH R.Z. and BROWN F.C.,Phys. Rev. BI, 818 (1970). SEGALL B.,Phys. Rev. 163, 769 (1967).

Die Masse Korrektur Formeln fir da~Polaron mit der anisotropen Masse

werden hergeleitet und benutzet, urn die baren Massen aus der beobachteten Polaronmasse von TlCl und TIBr zu berechnen.