Short Notes K137

phys. stat. sol. (b) 154, K137 (1989) Subject classification: 71.38

Department of Physics, Neimenggu University1 ) (a) and Beotou Institute of Rare-Earth (b) The Two-Dimensional Polaron in Polyatomic Crys ta l s in a Magnetic Field

BY CHUAN-YU CHEN (a), PEI-WAN JIN (a), and JIN-LIANG GAO (b)

The ground-state energy of a polaron in polar crystals has been studied by several authors /1 to 5 / . The properties of such polaron under the influence of an external magnetic field have also been investigated I S , 7 / . The polaron in polyatomic crystals

which have many modes of the longitudinal optical (LO) phonon has been studied in recent years /8, 91. But the electron-many LO phonon branches system in a magnetic field has not been discussed.

In this note, we will discuss the electron-many LO phonon branches system in a magnetic field. Let us consider an electron located in the x-y plane which is in a polyatomic crystal. The system consists of an electron interacting with many LO phonon branches and with a magnetic field 3 = ( 0 , 0 , B ) . The Hamiltonian is written as /7 to l o /

H = G(px 1 - $2y)2 + -(p 1 + $2x)2 +c fiwiai,iai;,i + 2m Y k.i

+ g ( V k , i ~ , i exp(iifi*if) + h.c.) , k, i

where - 2 = 2 2eB/c Vk,i = , a. = e / 2 ? i E + $ ,

1

* -* T~ = (fi/2mwi)1/2 , k =(kX, ky, kz) , kl = (kx, ky, 0 ) . ( 2 )

+ p = (px, py, 0 ) and ;b = (x, y, 0 ) are the momentum and coordinate of the electron? respectively; m is the band mass of the electron; V is the volume of the crystal; wi and ai are the constant three-dimensional (3D) LO phonon frequency and the electron-phonon coupling constant of the i-th branch, respectively. a- creates a b * i-th LO mode phonon with 3D wave vector k'. E: is the dielectric constant corresponding to the ionic polarization of the i-th mode /8,9/.

+ k, i

By introducing the harmonic oscillator operators

A = [(p, - 48 1 2 y) - i(py + Tp 1 2 x)] (A@)-' , B = A+ - ie(x+iy)(2JF;)-' , ( 3 )

the Hamiltonian (1) can be rewritten as

) Huhehaote, Neimenggu, People's Republic of China.

K138 physica status solid (b) 154

H = 13, + He-ph 9 (4 )

H, = f i ~ ( A A + 2 ) / 2 m + tiwiai;,iai;,i , (4a)

(4b)

2 + 1 + k, i

+ -1 -1 He-ph = (%,iLkMk%,i + 'k,iLk Mk %,i) '

k,l

In the following, we take H, as the unperturbed Hamiltonian and He-ph as the perturbation. The unperturbed eigenstates can be written as

(6) -1 /2 + n + M (1> = (n!M!) ( A ) 10>A(B ) 10>Blnk,i> ,

where the vacuum states of A and B are defined by A1 O:'* = B 1 O I > ~ = 0 and the phonon vacuum state is given by ag,i 1 Ok, i> = 0 . The strong field limit is 17, 101

(7) , a.h. -f 0. ,

1 1

2 where we have defined hi = wc/wi with the cyclotron frequency wc = eB/mc =

B / am. In the strong field limit, the electron can only be found in the lowest Landau level n = 0. The effective Hamiltonian of the system is then

2

Heff = A<oIH!o>A = Ho,eff + H ' , (8)

+ - Lfiw + hwiai;,iai;,i , (8a)

k,i o,eff - 2 c H

(8b) 2 2 + HI = V$,i exp(-hk,/2@ )(Mk as,i + ,

k , i

2 2 where we have made use of the matrix element A<sO I Lk I 0 >A = exp( -hkL / 2f3 ) . In the Wigner-Brillouin perturbation scheme we expand the perturbation energy in

power series of the small parameters a.X.. The second-order perturbation gives the energy correction due to many LO phonon branches

1 1

AE(2) = < j l H I Il><lIH' Ij>/(E-El0) H j

=C(Ep - hwi)-laiXi(hwi) 2 , 1

Short Notes

where -1/2 + M

l j> = I M > B I O k , i > = (M!) ( B )

K139

E and E10 are the ground-state energy of the system in the state l j> and the unperturbation energy of the system in the state Il>, respectively. E is the perturbation ground-state energy. Similarly, we find after tedious calculation the fourth-order correction

P

AE(4) = T ( E p - hwi)-2(Ep - 2hwil )-l(hwihwii ) 2 aiXiaii Xii P$K(O.S)/&] , (11) i,i

2 n / 2 where K(z) = J’ (1-zsin e)-’”d63 is the complete elliptic integral of the first kind. To obtain the ground-state energy correction to the order (aihi)2 and (aiAi)(aiiki), we first expand the denominator in (9), and set E = 0 in ( l l ) , and then

combine (9) with (11). This gives

0

P

AE = - Jii - C aiAihwi - +C[. t K ( 0 . 5 ) / ~ ] a i X i a i ~ h i ~ f i w i ~ + ~ l ~ a i h i ~ i , qhw. S 2 i i,i‘ 4 i,i 1

(12’)

Thus, we find finally the ground-state energy of the two-dimensional (2D) polaron in polyatomic crystals within a strong magnetic field limit

E = < j ( H o l j > + AEs g

2 = $ hwc - C aiXifiwi + 0.0649419 C (aiAi) hwi t 2 i i

+ 0.0649419 7, aihia i l A ~ I hwi . (13) i,i’ z i

The first term in (13) is the Landau level energy in strong magnetic field. The second and third terms are the coupling energies of electron, mabetic field, and separate LO phonon branches. And the last term is the coupling energy of the electron with the magnetic field and the interaction among different LO phonon branches, via the electron recoil.

In an arbitrary magnetic field, the electron is no longer confined to the lowest Landau level. The Hamiltbnian is still given by ( 4 ) but the unperturbed eigenstates are

The second-order perturbation energy from He-ph is

K140 physica status solid (b) 154

= - >; V i , i (hwi)-' i;,i

+ where we have made use of the relations A A I = n I n>A and 15, 71

m

(nfiwc + fiwi)-l = @ai)-' 0 $exp[-(nhFl)t]dt . (16)

Substituting (2) in (15) , and replacing the discrete sum over

obtain after some calculation

by an integral, we

Therefore the ground-state energy of the system is

1 i 2 1 2 na E = < i l H li> + AE = (n + z)tiwC - x fiwi r ( l / x i ) / r ( 2 + l / h i ) . (18)

g 0 i Ti

The meaning of various terms in (18) is self-evident. Now we discuss these results as follows:

2 1. In the strong magnetic field limit, Xi! + and exp(-kit) + 0.Equation (17)

becomes

which Leads to the same result as (12) in second-order correction. 2 2 2. In the weak field limit, hi! + 0 and exp(-Xit)*l-hit. Thus (17) becomes

m

AE = - Z 3 d- a i X i f i w i i e-t(Xft)-112dt = - X aifiwi , i i

which is the usual result that the ground-state energy correction up to the second- order perturbation is the sum of the contributions of the different phonon branches.

3. In the simplest case of diatomic cubic crystals, only one mode of the LO

phonon is present. In this case, (12) and (17) become

Short Notes K141

and

A E = - > ~ ~ M . (22)

Equation (21) is the same result as that in 171 concerning the g round-s t a t e energy up to f o u r t h - o r d e r correct ion of the 2D polaron within the strong magnetic f ie ld limit. A n d (22) is identical t o the result f o r the usual 2D polaron in the absence of a magnet ic field 14, 51. I t shows that the (12) and (17) are respectively general izat ions of the results obtained by Larsen 171 and by Devreese and coworkers

14, 51.

References

111 J. SAK, Phys. Rev . B S , 3981 (1972). 121 E. EVANS and D.L. MILLS, Solid S t a t e Commun. 11, 1093 (1972); Phys. Rev .

B 8, 4004 (1973). 131 W.J. HUYBRECHTS, Solid State Commun. 28, 95 (1978). / 4 / XIAOGUANG WU, F.M.PEETERS, and J.T. DEVREESE, Phys. Rev . B 3 l ,

3420 (1985). 151 F.M. PEETERS, XIAOGUANG WU, and J.T. DEVREESE, Phys. Rev. B 33,

3926 (1986). 161 F.M. PEETERS and J.T. DEVREESE, Phys. Rev. B 3 l , 3689 (1985). 171 D.M. LARSEN, Phys. Rev. B 33, 799 (1986).

I81 M. MATSUURA, J. Phys. C lo, 3345 (1977).

191 Y. LfiPINE, Solid State Commun. 40, 367 (1981). 1101 C.Y. CHEN, T.Z. DING, and D.L. LIN, Phys. Rev. B 2, 4398 (1987);

- 36, 9816 (1987).

(Received July 8, 1988)

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