Towards the theory of vibronic coupling in molecular crystals

Volume 30A, n u m b e r 6

PHYSICS

TOWARDS

THE

THEORY

LETTERS

OF

IN M O L E C U L A R

17 N o v e m b e r 1969

VIBRONIC

COUPLING

CRYSTALS

M. Z. ZGIERSKI Department of Theoretical Chemistry, Jagellonian University, Cracow, Poland Received 15 October 1969

The influence of a small frequency change of the vibrational mode in the excited electronic state of molecules forming molecular crystal on the linear vibronic coupling is investigated.

We study a model of m o l e c u l a r c r y s t a l c o n s i s t i n g f r o m m o l e c u l e s which p o s s e s s one excited elect r o n i c state well s e p a r a t e d f r o m o t h e r s . We a s s u m e that in the excited state the e q u i l i b r i u m configu r a t i o n of m o l e c u l e as well as v i b r a t i o n a l frequency of a given n o r m a l i n t r a m o l e c u l a r mode a r e changed We neglect the lattice v i b r a t i o n s and d i s p e r s i o n of the i n t r a m o l e c u l a r v i b r a t i o n which o r i g i n a t e s from i n t e r m o l e c u l a r i n t e r a c t i o n s . Our model may be d e s c r i b e d by the following Hamiltonian: H= ~Coa~a n + w

b~b n + ~

V. o

a~am

+

a+a n

E

iS

yw(b + + bn) + m

]

(1)

where y is the l i n e a r coupling constant (it is p r o p o r t i o n a l to the change of e q u i l i b r i u m configuration in n o r m a l coordinate space), a +, a n and b~, bn a r e the exciton and phonon c r e a t i o n , a n n i h i l a t i o n o p e r a t o r s , r e s p e c t i v e l y , VnOm is the e l e c t r o n i c r e s o n a n c e i n t e r a c t i o n m a t r i x e l e m e n t between the n-th and rn-th m o l e c u l e , e o is the e n e r g y of an exciton l o c a l i s e d on a given site, and Wl, w a r e the f r e q u e n c i e s of vibr a t i o n s in the excited and ground s t a t e s , r e s p e c t i v e l y . So we a s s u m e a H e i t l e r - L o n d o n type a p p r o x i m a tion and neglect effects r e s u l t i n g from configuration i n t e r a c t i o n s . The v i b r o n i c coupling is d e s c r i b e d by the l a s t t e r m of eq. (1). Using the double time t e m p e r a t u r e dependent G r e e n function method [1] we have obtained the s y s t e m of equations for the z e r o - p h o n o n - e x c i t o n and one-phonon exciton G r e e n functions, employing the standa r d method of the decoupling p r o c e d u r e for the higher o r d e r G r e e n functions. In this way we include in our a p p r o x i m a t i o n z e r o and one phonon p r o c e s s e s in the exciton t r a n s f e r f r o m one m o l e c u l e to another, and in a given m o l e c u l e we include all multiphonon p r o c e s s e s . F r o m the s o - o b t a i n e d d i s p e r s i o n r e l a t i o n of an exciton coupled both l i n e a r l y and q u a d r a t i c to i n t r a m o l e c u l a r phonons the following conclusion can be drawn: all e n e r g y changes in the F r e n k e l exciton s y s t e m r e s u l t i n g from the l i n e a r v i b r o n i c coupling, i.e., a) the uniform r e d - s h i f t of the whole band [2], b) the s y m m e t r i c a l c o m p r e s s i o n of the exciton band r e s u l t i n g from the z e r o - p h o n o n p r o c e s s e s [3], c) the a s y m m e t r i c a l c o m p r e s s i o n of the exciton band r e s u l t i n g f r o m the one-phonon p r o c e s s e s [4], d) the e x c i t o n - e x c i t o n d y n a m i c a l i n t e r a c t i o n s via phonons [5], a r e g r e a t e r for Wl w. T h e r e also a p p e a r s an additional explicit i n t e r a c t i o n between excitons r e s u l t i n g f r o m the exchange of two phonons. It is always a t t r a c t i v e for sufficiently l a r g e co. F r o m this point of view the Bose condens a t i o n of excitons s e e m s to be doubtful in the case of a s m a l l c o n c e n t r a t i o n of excitons. The t e m p e r a t u r e dependence of the i n v e r s e of the effective exciton m a s s in the z e r o - p h o n o n approxi m a t i o n is given by the following e x p r e s s i o n : 1 exp[-~2(2~t+l) ][14~2(2n+1)c(1-~ c ) + ~ y 4 ( 2 n + l ) 2 c 2 +O@3)] (2) meff w h e r e n = [exp (w/kT) - 1] -1, c = (w~ - w 2 ) / w 2, i . e . , an i n c r e a s e in the v i b r a t i o n a l frequency f o r c e s of d e c r e a s e of the effective phonon cloud which a c c o m p a n i e s the exciton, 354

Volume 30A, number 6

PHYSICS LETTERS

17 November 1969

F i n a l l y , e x p r e s s i o n s for the c e n t e r of g r a v i t y E and the s q u a r e width (AE) 2 of the exciton d i p o l e - a l lowed optical t r a n s i t i o n were obtained, they read: ~2 ¢02 = Co + n ~ o Vn°° + Wl-W4¢0 (2~ + 1) ," (AE) 2 = ~2w2(2n+ 1) + ( w 2 - 80)2) ~ (2n+ 1)2 (3) 1

i . e . , the t e m p e r a t u r e of the T ~ dependence of the width in the high t e m p e r a t u r e l i m i t would give us i n f o r m a t i o n about the e x i s t e n c e of the q u a d r a t i c v i b r o n i c coupling. F o r a m o r e detailed a n a l y s i s of the p r o b l e m quoted in this l e t t e r we r e f e r to ref. 6. I a m indebted to Dr. A. Witkowski for c r i t i c a l r e m a r k s .

References 1. D.N. Zubarev, Sov. Phys., Uspekhi 3 (1960) 320. 2. M. Z. Zgierski, Aeta Phys. Polon., 36 (1969) 159,167. 3. K.Walasek, JINR preprint, Dubna 1968. S. Takeno, J. Chem. Phys., 46 (1967) 2481. 4. R. E. Merriefield, J. Chem. Phys., 40 (1964) 445. 5. A.S.Witkowski, Aeta Phys. Polon., 30 (1966) 431. 6. M. Z. Zgierski, Aeta Phys. Polon., 37 (1970) in print.

PLASMA

FREQUENCY

AND TRANSFER

INTERVALLEY IN n-GaAs

POPULATION

R. K. KAR Khaira Laboratory in Physics, University College of Science, Calcutta 9, India and M. N. MUKHERJEE Physics Department, Vidyasagar College, Calcutta 6 ,India Received 10 October 1969

Recent experimental values of current density for a sample of n-GaAs have been utilized for calculating the plasma frequency as a function of applied electric field and hence the intervalley population transfer has been calculated. F o r a s a m p l e of n - G a A s at a l a t t i c e t e m p e r a t u r e TOK (taken h e r e equal to 300OK a s most e x p e r i m e n t s have been p e r f o r m e d at this t e m p e r a t u r e ) and at an e l e c t r i c field E v o l t / c m the p l a s m a frequency is given by [1] ,COp(E). 2

/@)

nl

ml

n2

n

m2

n

-+

w h e r e coD(0) = (47rn e 2 / i n l e t ) ~t is the p l a s m a f r e quency a-t z e r o - f i e l d , n = h 1 + n 2 is the total c a r r i e r density, n 1 and n 2 the d e n s i t i e s in the lower and the upper v a l l e y r e s p e c t i v e l y , m 1 and m 2 a r e the c o r r e s p o n d i n g effective m a s s e s .

F r o m Conwell [1] m l / m 2 = 0.2. So from eq. (1) we get (Wp(E),, 2 : 02

+ 08

"1/.

(2)

P l a s m a frequency at z e r o - f i e l d Wp(0) has been e x p e r i m e n t a l l y d e t e r m i n e d by o p h c a l - r e f l e c t i vity m e a s u r e m e n t s [2-3] and R a m a n - S c a t t e r i n g m e a s u r e m e n t s [4] but ~0p(E) has not yet been e x p e r i m e n t a l l y determified. C u r r e n t d e n s i t y j is given by j = n e v d = (nlew 1 +n2ew2)E

(3) 355