LITERATURE CITED

i. A.A. Chernov, Kristallografiya, 16, No. 4, 842 (1971). 2. L.G. Lavrent'eva, Kristallografiya, 25, No. 6, 1273 (1980); __27, No. 4, 818 (1982). 3. L.G. Lavrent'eva, L. P. Porokhovnichenko, Yu. G. Kataev, O. V. Ruzaikina, and N. N.

Bakin, in: Gallium Arsenide, No. 5 [in Russian], Tomsk (1974), p. 88. 4. L.G. Lavrent'eva, L. P. Porokhovnichenko, and P. N. Tymchishin, Izv. Vyssh. Uchebn.

Zaved., Fiz., No. 2, 143 (1974). 5. L.G. Lavrent'eva, I. V. Ivonin, and L. P. Porokhovnichenko, in: Crystal Growth, Vol. 13

[in Russian], Nauka, Moscow (1980), p. 33. 6. A.A. Chernov, in: Crystal Growth [in Russian], Vol. 3 , Nauka, Moscow (1961), p. 47. 7. A.A. Chernov, M. P. Ruzaikin, and N. S. Papkov, Poverkhnost'. Fiz., Khim. Mekh., No.

2, 94 (1982). 8. L.G. Lavrent'eva, S. E. Toropov, and L. P. Porokhovnichenko, Izv. Vyssh. Uchebn. Zaved.,

No. ii, 18 (1982).

DISPERSION OF AN ACOUSTIC POLARON IN A MAGNETIC FIELD

V. N. Gladilin and A. A. Klyukanov UDC 621.315.592:534:537.6

Interaction of conduction electrons with lattice vibrations and impurities leads (in addition to scattering) to a renormalization of the electron energy spectrum. An acousticpolaron (condenson), according to Pekar and Deigen [i], is a state that can arise in a homeopolar crystal due to the interaction of a conduction electron with the lattice deformation created by it. In recent years there have appeared many papers dealing with acoustic polarons in the absence of a magnetic field [2-6]. The theory of an acoustic polaron in a magnetic field has been considered in [7, 8] for the special case when the velocity of the polaron along the direction of the magnetic field is zero. In [9], with the help of the Bogolyubov varia- tional method, a numerical calculation of the energy and mass of the condenson resulting from its velocity of motion was done in the strong binding limit. The present paper con- siders the dispersion of an acoustic polaron in a magnetic field for arbitrary binding. In the elastic approximation, interactions with acoustic phonons and deep neutral impurities lead, as will be shown, to the same results.

The Hamiltonian of an electron in a magnetic field, interacting with impurities and phonons has the form

H : __1 (p _{_. Hph-4-, - - -izr 2m c , tvz jozje -i- ' . c.) +

1 - R i:), ( I ) (2~.)3~j'd~v.f eiz(r w h e r e r a n d Ri / a r e c o o r d i n a t e s o f t h e e l e c t r o n a n d t h e i - t h i m p u r i t y c e n t e r o f t h e f - t h k i n d ; bx] i s t h e a n n i h i l a t i o n o p e r a t o r o f a p h o n o n o f wave v e c t o r z b e l o n g i n g t o t h e j - t h b r a n c h o f t h e p h o n o n s p e c t r u m ; A=(--yH, O, O) i s t h e m a g n e t i c v e c t o r p o t e n t i a l . The v o l u m e i s n o r m a l i z e d t o u n i t y . We i g n o r e t h e e f f e c t o f t h e i m p u r i t i e s on t h e p h o n o n s p e c t r u m .

The G r e e n ' s f u n c t i o n o f an e l e c t r o n i n t h e z e r o t h L a n d a u l e v e l i s w r i t t e n i n t h e f o r m

-- i l i t

' ~ ~,~ . . ( 2 ) O ( E + i~, n~) = i% t d l e ~(s+i*)t ._ < O, n.~., nz[ ..:. e :,h,~.u,[O, n~., ~ > , O ~r

w h e r e t h e w a v e f u n c t i o n i s

/ k

~ \ : h ] 7/l ',o c h/4 x ')2 2h (Y (3)

l n o) c ,, "

V. I. Lenin Kishinev State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 50-55, June, 1983. Original article submitted December 13, 1982.

0038-5697/83/2606-0543507.50 �9 1983 Plenum Publishing Corporation 543

The symbol ~.--~ph, imp denotes an average over the impurity and phonon subsystems. This aver- age is performed using the method of cumulants, keeping only the first two terms in the cum-

ulant series. The average with respect to phonon coordinates is therefore done exactly since only the second cumulant survives for a harmonic oscillator. For a random distribu- tion of impurities

] ~ e~(~+~')R,1 = (2~) 3 N ~ (~ + ~'). (4 ) i

The limitation to the second cumulant is equivalent here to assuming pair interactions.

If the interaction between subsystems is switched on a t = 0 we have

.- m -g%t 1 Iv.z15 '~ s e ~..~,.,,,p= e T e x p ---~-~ . d x t d s dst e i z ( r ( s ) - r ( s O ) T ~ l ( S - - s l ) " (5)

(2~):' ~ 0

where we u se t h e g e n e r a l s u b s c r i p t ~ t o l a b e l b o t h t h e b r a n c h o f t h e phonon s p e c t r u m and t h e type of impurity. H e is the Hamiltonian of a band electron in the magnetic field. The func- tion Txl(S) for the j-th vibrational branch and f-th kind of impurity is given by

i ~ . s Tzj (s) = nxye ,1 + ( n z s 1) e - ' % / s , (6)

l ~ f ( s ) = N / , (7)

where Nf is the concentration of impurity centers of the f-th kind; nz] is the average num- ber of phonons of the j-th branch with wavevector ~ and frequency o)~/.

In order to sum the series resulting from expansion of the T-exponent in the evolution operator, we use diagrammatic techniques. If we include only diagrams with no intersecting phonon lines, it is easy to find the following equation for the Green's function:

G - l (~, x) ~ x'- '-- p : - - - - d x l G (~, x~), (8) "K d

- o o

where we introduce the notation

" ~ , X ~ K z �9

h~) C 2mo~ c ( 9 )

The e n e r g y i s m e a s u r e d f r o m t h e b o t t o n o f t h e z e r o t h L a n d a u l e v e l . We d e f i n e t h e q u a n t i t y F by t h e f o r m u l a

a~tn3litco T . 32&'N, . i V o r ~ m 3r" l' = 1 " '-l ' , ,a; P,----4V_~:h7,%//zwio, P,.~ = V~hV/2o,j 2 (10)

Here a0 i s t h e d e f o r m a t i o n p o t e n t i a l ; w i s t h e s p e e d o f s o u n d ; 9 i s t h e d e n s i t y o f t h e c r y s - t a l ; V0 and r0 p a r a m e t r i z e t h e i n t e r a c t i o n p o t e n t i a l w i t h n e u t r a l i m p u r i t i e s :

I r - - R l I ~

V,,a(r) = Vo~z~e 4~ (11) i

The c a l c u l a t i o n i s done a s s u m i n g t h a t

h w z h ti% > 1 , = <~( l, - - )i:> l. (12) 1% T tooT 2 m o y r ~

The s o l u t i o n o f (8) i s r e p r e s e n t e d in t h e f o r m

G-I(',, x) = zO) - -x~ , (13)

where z('J) is defined by the equation

iV Z ( V ) - v ) I<Z_ ~ . (14)

544

r ee %//- '/'

�9 I

2

F i g . 1. E n e r g y Re ~)0 a s a f u n c t i o n o f momentum o f t h e e l e c t r o n x . C u r v e s 1, 2, a n d 3 a r e c o n - s t r u c t e d f r o m f o r m u l a s ( 2 1 ) , ( 2 2 ) , a n d ( 2 7 ) , r e - speetively.

At the pole of the function G(~, x) we have z(~) = x 2. It follows from (14) that the pole is located at the point

v0 = x ~ - -- (15) Jxl

Equation (15) is identical to the perturbation result to second order in ~'~i and describes a level whose width diverges as x § 0; this shows the inadequacy of this approximation, and the necessity of including diagrams with intersecting phonon lines.

The Green's function can be written in the form

G -I (%', x) : %" --X2--M (%', X), (16)

where the mass operator M(v, x) satisfies the equation

M(Lx)=_Y/ ~_ "x + / / - \ \ + - . . . . + (17)

Here § corresponds to the exact Green's function (16). The order of each diagram on the right-hand side of (17) will be given by the number n of explicit phonon lines in the dia-

gram. Any of these lines and any group of lines containing m(m < n) lines forms at least one intersection with the remaining n -- 1 or n -- m lines. Contributions from phonon lines not having the above properties, i.e., not intersecting with the remaining lines, are ac- counted for implicitly by replacing electron lines corresponding to the band electron Green's function by the lines § corresponding to the exact Green's function.

The contribution from diagrams of order n is

OO I'nSl D.(v , x ) = C n \ ~ ] dxlG('v., x , ) F~-~ (',, x , x~), ( J 8 )

- - 0 0

w h e r e t h e c o n s t a n t C n i s d e t e r m i n e d b y t h e c o n f i g u r a t i o n o f p h o n o n l i n e s a n d F ( v , x , x z ) i s given by

oo

F ('4 x, xl) = J dx,:G('~ x~) O ( < .~- + x, -.~ x j . (19) - - c O

An exact summation of all diagrams with intersecting phonon lines is not possible. Therefore, we consider some limiting situations. We first calculate G(~, x) taking into ac- count only diagrams with the maximum number of intersections for a given order n. For each order there is only one such diagram and we have C~ = _91-~7 for these particular diagrams. The equation for the Green's function becomes

545

i ]1 (3-1('~, X ) = ~ - - X " - - F_ "dXlO('~, xI) 1-- dx.,O('~, x.~)O(% xq- .x ' , q-x2) . 2 r 1 6 2 "

--oo --co

We 10ok for the solution of this in the form (13). Determining the equation for z(v)and substituting z(v) = x 2 in it, we find the following for the pole of the function

(20)

~0 : x 2 - - i r 2 i r l x I

i r (21) 4X 2 L __i1' 4x "~ -t . . . . 'txl Ix[

If we neglect x in the mass operator in (20) we obtain for ~0:

V 9 : X 2 - - iI" 3iF I x I

i1' 3ilx I:' 3x~ + [x-T 4x" q- ~- T 1 F

In Fig. 1 we show the curves calculated from (21) and (22) of the energy Re ~0 of an electron interacting with the acoustic deformation potential and deep neutral impurities as a function of the electron momentum x. We see a distinct sharp minimum in the energy at x~0,41"~/% with the value of the miniumum given by Vmin~0 ,51"~ /% The fact that the differ-

ence between the two curves (21) and (22) is small indicates the validity of the solution of (20) in the form (13).

For orders n~3, diagrams with the largest number of intersecting phonon lines do not exhaust all diagrams with intersections. Thus, in the above calculations, the only diagram contributions that are accounted for exactly are those of order n~2, while the contributions of the higher order diagrams are underestimated. For z>p1/3 the results (21) and (22) give

~0 to order ~ ['3/x< In order to show the validity of our results in the case x<~F I/3, when dia- grams of order n>>1 are important, we calculate ~0 using an overestimate of the contributions of these diagrams. The total number of diagrams with n intersecting .phonon lines is obvi- ously bounded above by the number of all topologically nonequivalent irreducible diagrams with n phonon lines

n--i

K . = ( 2 n - - 1)!! - - ~ ( 2 / - - 1 ) 1 . (2n --2l - - 1)!!. (23)

I=I

For n~3

K n ~ (2n- -3) (2n--3)! ! . (24)

Out of a l l d i a g r a m s w i t h a g i v e n number n of i n t e r s e c t i n g l i n e s , t h e maximum v a l u e of t h e c o n s t a n t C n = 1 / n c o r r e s p o n d s to d i a g r a m s of t h e f o r m

, , / 2 < . . . . ~ \ \ \ 1/ 11 / \ X \

' / / / ~ ' X '\\~ J

For an u p p e r e s t i m a t e of t h e c o n t r i b u t i o n of a l l d i a g r a m s w i t h n i n t e r s e c t i n g p h o n o n l i n e s we t a k e

oo r~

KnD2~• x)~ 2F~ (2n--a)!!J: d.LG(v , x~)F"-~(v, x, x , ) .

- - o o

Then t h e f o l l o w i n g e q u a t i o n r e s u l t s f o r G(v , x ) :

(25)

�9 _]

0 .1 (v, x) : : ~ - - x: 2I' dxiG ('q xl) d~ [/~e -~ 1 --2I' ~F(~, x, X I (26) ~3, 'k

�9 - - c o ~ )

I t c a n be shown t h a t t h e s o l u t i o n of (26) i s r e p r e s e n t a b l e i n t h e f o r m ( 1 3 ) . A f t e r i n t e g r a - t i o n w i t h r e s p e c t t o x z , x2 , and r a n d s u b s t i t u t i o n of z ( ~ ) = x 2 i n t h e mass o p e r a t o r , we f i n d t h e f o l l o w i n g e x p r e s s i o n f o r t h e p o l e :

546

�9 '~o = -- 1 / / iPlx[_ ~ e -7 K, - K o '~- x~ |/~-7~en [1 - err t ] /7 / ) ] . 2

(27)

Here

I x I ~, (28) iF

Kn(z) is the MacDonald function, and erf(z) is the probability integral [error function].

Curve 3 of Fig. 1 shows the dependence of the energy Re v0 on the momentum x calculated from (27). Comparison of curves i, 2, and 3 shows that the estimate (27) gives the same qualitative dispersion law as (21) and (22).

At x = O, when the electron is at rest, the correction to the energy goes to zero. In

the limit of large velocities one can ignore the polaron effect because the lattice deforma- tions cannot follow the electron. The minimum polaron energy is attained~a~ .r~0,4Pl/3. For velocitiea of this order, interaction of the electron with lattice deformations is at amaximum.

We note that the entire calculation is done assuming perfectly elastic electron scatter- ing. Actually there is always some inelasticity and for x2~e/h o~,~ (~ is the energy chara- terizing the inelasticity of the scattering) the results (21), (22), and (27) are not correct. If we account for the inelasticity, the correction to the energy at x = 0 is nonzero. Natu- rally the minimum in the acoustic polaron energy described above is possible for x:#0 only when the condition ]~/3>>~/h0~, is satisfied.

LITERATURE CITED

i. M.F. Deigen and S. I. Pekar, Zh. Eksp. Teor. Fiz., 21, 803 (1951). 2. G. Whitfield and P. B. Shaw, Phys. Rev., 21, 4349 (1980). 3. N. Tokuda, Sol. State Commun., 35, 1025 (1980). 4. J. Thomchick, Phys. Lett., 64A, 71 (1977). 5. G.B. Norris and G. Whitfield, Phys. Rev. B, 21, 3658 (1980). 6. P.B. Shaw and E. W. Young, Phys. Rev. B, 24, 714 (1981). 7. L.S. Kukushkin, Pis'ma Zh. Eksp. Teor. Fiz., 7, 251 (1968). 8. L.S. Kukushkin, Zh. Eksp. Teor. Fiz., 57, 1224 (1969). 9. K.S. Kabisov and E. P. Pokatilov, Ukr. Fiz. Zh., 17, 671 (1972).

CALCULATION OF THE LINEAR PARAMETERS OF A SURFACE-HELICAL INSTABILITY

IN SEMICONDUCTOR PLATES

G. F. Karavaev, N. L. Chuprikov, and B. A. Uspenskii

UDC 621.315.592

We have constructed in [i] a nonlinear theory of a surface-helical instability (SHI) of a semiconductor plasma in plates. The dispersion equation and an equation for the ampli- tude and the nonlinear frequency shift of a wave established as a result of the development of an instability were derived. The dispersion equation was previously obtained in [2] and analyzed for some particular cases permitting an analytic investigation.

A discussion is performed in this paper of the results of the numerical solution of the dispersion equation without any constraints on the parameters of the equation. The notation of the different physical quantities corresponds to [i].

It is assumed within the framework of the SHI theory constructed that there exists in the sample a perturbation wave of the plasma density and the potential of the electric field

V. D. Kuznetsov Siberian Physicotechnical Institute at Tomsk State University. Trans- lated from Izvestiya Vysshikh Uchebnykh Zavedenii, Eiziki, No. 6, pp. 55-59, June, 1983. Original article submitted December 16, 1982.

0038-5697/83/2606-0547507.50 �9 1983 Plenum Publishing Corporation 547