INVESTIGATION OF MODEL SYSTEMS WITH NON-ADIABATIC ELECTRON-PHONON INTERACTION

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INVESTIGATION OF MODEL SYSTEMS WITH

NON-ADIABATIC ELECTRON-PHONON

INTERACTION

P. Reineker, A. Scheuing, C. Durst, E. Sigmund

To cite this version:

INVESTIGATION OF MODEL SYSTEMS WITH NON-ADIABATIC ELECTRON-PHONON INTERACTION

P. Reineker, A. Scheuing, C. Durst' and E. .Sigmundt

4 b t e i Z u n g T h e o r e t i s c h e P h y s i k , U n i v e r s i t a t UZm, 0-7900 UZm, F.R.G.

T h e o r e t i s c h e P h y s i k I I I , U n i v e r s i t d t S t u t t g a r t , 0-7000 S t u t t g a r t 8 0 , F.R.G.

A b s t r a c t - We c o n s i d e r t h e e i g e n v a l u e problem o f a n e l e c t r o n i c t w o s i t e s y s - tem w i t h i n t e r s i t e c o u p l i n g i n t e r a c t i n g l o c a l l y w i t h two v i b r a t i o n a l modes. By o p e r a t o r t r a n s f o r m a t i o n s t h e e i g e n v a l u e problem i s r e d u c e d t o t h a t o f a n e l e c t r o n i c two l e v e l s y s t e m c o u p l e d t o a s i n g l e v i b r a t i o n a l mode v i a d i s - p l a c i v e and t r a n s i t i v e c o u p l i n g s . The e i g e n s o l u t i o n s a r e d e t e r m i n e d numeri- c a l l y u s i n g s c a l a r and m a t r i x c o n t i n u e d f r a c t i o n methods.

I - INTRODUCTION

The i n t e r a c t i o n between e l e c t r o n i c and v i b r a t i o n a l d e g r e e s o f freedom i s o f impor- t a n c e i n many f i e l d s o f s o l i d s t a t e p h y s i c s . It p l a y s a c r u c i a l r o l e i n d e t e r m i n i n g e . g . o p t i c a l and ESR l i n e s h a p e s and t h e i r t e m p e r a t u r e dependence / 1 , 2 / , v i b r a t i o - n a l r e l a x a t i o n s and n o n - r a d i a t i v e t r a n s i t i o n s / 3 , 4 / , phonon a s s i s t e d t u n n e l i n g p r o c e s s e s / 5 , 6 / a s well a s e n e r g y t r a n s f e r mechanisms o r c h a r g e t r a n s p o r t problems 1 7 , ~ .

F o r t h e t r e a t m e n t o f t h i s k i n d o f problems many a p p r o x i m a t i o n schemes have been de- v e l o p e d 19-121 which a r e v a l i d i n l i m i t i n g c a s e s i n which a p e r t u r b a t i o n a l expan- s i o n i n powers o f a s m a l l p a r a m e t e r i s p o s s i b l e . However, i f a l l r e l e v a n t c o u p l i n g p a r a m e t e r s a r e o f t h e same o r d e r o f m a g n i t u d e , none o f t h e s e a p p r o x i m a t i o n s c a n b e u s e d .

R e c e n t l y we h a v e i n v e s t i g a t e d s u c h a s i t u a t i o n f o r an e l e c t r o n i c t w o - l e v e l s y s t e m which i s c o u p l e d t o o n e v i b r a t i o n a l mode. T h i s s i m p l e b u t n o n - t r i v i a l c o u p l i n g was u s e d my many a u t h o r s , e . g . 1 1 3 , 1 4 1 , f o r t h e d i s c u s s i o n o f d i f f e r e n t t h e o r e t i c a l a p p r o a c h e s , s u c h a s s t a t i c and a d i a b a t i c d e c o u p l i n g p r o c e d u r e s o r t r a n s f o r m a t i o n t e c h n i q u e s . R e c e n t l y , f o r s p e c i a l p a r a m e t e r c o m b i n a t i o n s i s o l a t e d e x a c t s o l u t i o n s c o u l d b e f o u n d by a t r e a t m e n t u s i n g Bargman's H i l b e r t s p a c e 1 1 5 1 . I n o u r t r e a t m e n t we have u s e d s c a l a r a s w e l l a s m a t r i x c o n t i n u e d f r a c t i o n methods and showed t h a t e x a c t n u m e r i c a l e i g e n s o l u t i o n s f o r t h i s s y s t e m may b e o b t a i n e d w i t h o u t t o o much e f f o r t and t h a t p r e v i o u s a n a l y t i c a l a p p r o x i m a t i o n s / 1 1 , 1 2 / may b e r e d e r i v e d and improved i n a s y s t e m a t i c manner.

I1 - MODEL HAMILTONIAN AND NUMERICAL PROCEDURE

The t o t a l H a m i l t o n i a n o f o u r t h e o r e t i c a l model s y s t e m i s g i v e n by

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JOURNAL

DE

PHYSIQUE

A+ and A a r e c r e a t i o n and 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 , f o r e l e c t r o n s a t s i t e n and B+ and

B

a r e Bose o p e r a t o r s f o r v i b r a t i o n a l q u a n t a o f a n o s c i l l a t o r l o c a l i z e d a t s i t e n . € " d e s c r i b e s t h e e n e r g y d i f f e r e n c e o f t h e e l e c t r o n between s i t e s 1 und 2 , T i t s t r a n s f e r m a t r i x e l e m e n t between t h e s e s i t e s , and g

i s

t h e c o u p l i n g s t r e n g t h between e l e c t r o n s and v i b r a t i o n s . A l l q u a n t i t i e s a r e measured i n u n i t s o f t h e v i b r a t i o n a l e n e r g y quantum. The r e l a t i o n ~A;A, = 1 h o l d s , b e c a u s e we a r e c o n s i d e r i n g o n l y o n e e l e c t r o n e i t h e r one s i t e 1 o r n o n s i t e 2 . To s i m p l i f y t h e H a m i l t o n i a n we i n t r o d u c e new o p e r a t o r s f o r t h e v i b r a t i o n s 1

-

- - - 1 2 B 1 = 2 2(bl-b2)

,

B 2 = 2 (bl+b2) ( 1 1 . 2 ) and f o r t h e e l e c t r o n s 1 It i s f u r t h e r m o r e c o n v e n i e n t t o e x p r e s s t h e H a m i l t o n i a n i n t e r m s of s p i n o p e r a - t o r s s a t i s f y i n g t h e u s u a l commutation r e l a t i o n s ( ( u X , u

]

= i u Z ; [ u X , u Y ] + = 0 .) and b e i n g d e f i n e d by Y 1

+

u _z = -(A A ₂

_{+ +}

- A+A-)

.

With t h e s e t r a n s f o r m a t i o n s t h e H a m i l t o n i a n t h e n s p l i t s i n t o two p a r t s , H = H +H 1 2 ' w i t h

H

c o n t a i n i n g o n l y v i b r a t i o n a l d e g r e e s o f f r e e d o m , H1 = b i b 2 + G(b++b ) , and 1. 1 d e s c r i b i n g a d i s p l a c e d harmonic o s c i l l a t o r . The i n t e r e s t i n g p a r t H 2 i s g l v e n by (bEb2)

The c o n n e c t i o n s between t h e p a r a m e t e r s i n ( 1 1 . 5 ) and t h o s e i n ( 1 1 . 1 ) a r e g i v e n by E = 2 ( E 2 + ~ 2 ) 1 / 2 , A = - ~ ~ T / E

,

D = ~ ~ E / E

,

G = Q ( 1 1 . 6 )

A

and t h e t r a n s f o r m a t i o n ( 1 1 . 3 ) i s d e t e r m i n e d by s i n 2 9 = 2T/€ ; c o s 2 8 =

~ E / E

.

( 1 1 . 7 ) The e i g e n v e c t o r s o f t h e t o t a l H a m i l t o n i a n f a c t o r i z e i n t o a p r o d u c t o f s t a t e vec- t o r s c o r r e s p o n d i n g t o HI and H2 and t h o s e o f H2 a r e r e p r e s e n t e d i n t e r m s o f e i g e n - s t a t e s o f o Z by

The e x p a n s i o n c o e f f i c i e n t s

x

and

X-

depend on t h e v i b r a t i o n a l d e g r e e s o f freedom and a r e e x p r e s s e d by h a r m o n i l o s c i l l a t o r f u n c t i o n s

E i s t h e eigenvalue, I t h e 2-dimensional u n i t m a t r i x , and we have used t h e notation 1161 :

Q, =

6

( ~

+

o

sox)

~ , Q, = n I

+

c u Z , Q; = ( 0 0 ~

+

sox)

.

(11.12)

Introducing t h e t r a n s f e r matrices S; and Sn by

"'n+l = S: "'n

*

"'n-1 =

s,$Jn

, (11.13)

we end up with t h e following homogeneous s e t of equations whose determinant has t o vanish i n o r d e r t o guarantee n o n - t r i v i a l e i g e n s o l u t i o n s :

To evaluate t h e determinant t h e t r a n s f e r matrices must be known. From ( I I . 1 1 , 1 3 ) we obtain r e c u r r e n t e r e l a t i o n s which allow t h e following continued f r a c t i o n r e - p r e s e n t a t i o n f o r Sm, S i :

Because of t h e physical s t r u c t u r e of our problem ( e x i s t e g c e of a lowest energy eigenvalue) S i s a f i n i t e continued f r a c t i o n whereas Sm i s i n f i n i t e and has t o be approximat!d by breaking o f f (11.15) i n an a p p r o p r i a t e manner. The eigenvalues a r e then c a l c u l a t e d from (11.14) and t h e eigenvectors from (11.13).

I11 - RESULTS

To solve t h e eigenvalue problem of H (11.5) numerically 1171 we s t a r t e d from t h e homogeneous system of equations (11.g4) with a f i x e d index n = M . Via (11.12) M d e f i n e s t h e matrices Q QM and Q; of dimension

2 ,

an$ it a l s o g i v e s t h e s t a r t i n g point f o r t h e it-ratio!' procedure which determines S and S,,,. As mentioned i n s e c . 11.. S i i s a f i n i t e matrix continued f r a c t i o n

which

can be c a l c u l a t e d whithout approximation (we confined M t o values ( 2 0 ) . S 1s an i n f i n i t e MCF which i s approximated by a f i n i t e one of l e n g t h K (by s e t t i n g

s;+~

= 0 ) . The two i n t e - g e r s K and M can be chosen f r e e l y . The value of K determines t h e accuracy of t h e c a l c u l a t i o n and t h e M value d e f i n e s t h e "window" i n which t h i s numerical procedure allows t o determine t h e eigenvalues most e a s i l y . The l a t t e r a r e c a l c u l a t e d from t h e condition

+ +

F(E) = Det B ( E ) = Det[Qi S i

+

(QM - E I )

+

QNSw ) = 0 (111.1) I n evaluating (11111) one has t o t a k e c a r e of some numerical f a c t s . For t h e calcu- l a t i o n of S+ and SM numerous matrix summations, m u l t i p l i c a t i o n s and i n v e r s i o n s a r e

C7-276 JOURNAL DE PHYSIQUE

necessary. The multiplications and summations can be treated very accurately and rapidly by a computer. Matrix inversions, however, are more difficult to handle,be- cause from a mathematical point of view a non-singular behaviour of the matrices is required. Nevertheless, if the values of these determinants are f 0 but very small, the numerical treatment must be controlled carefully to avoid computer overflow.

Fig. 1

-

F(E) for E = 1, A = 2, D = 0 , Fig. 2 - F(E) for E = 1, A = 2, D = 0 ,

K = 20, M = 1. K = 20, M = 10.

Fig.

3

- Energy eigenvalues E for different approximations: (a) MCF: dots (b) approxi- mation of 1121:--- (c) numerical diagona- lization of a 100by100 matrix: ---

.

The parameter values are E =

4,

0 < A < 3, D = 2 J 2

,

K = 10,

M

= 0.

r e g i o n s can be d i s t i n g u i s h e d . I n one r e g i o n , a p a r t from some p o l e s , F(E) i s a c o n t i - nous f u n c t i o n and t h e energy eigenvalues can be found e a s i l y from F(E) = 0. I n o t h e r r e g i o n s F(E) behaves i n an i r r e g u l a r manner, p o l e s and o r d i n a r y zeros cannot be separated. Comparing b o t h f i g u r e s one sees t h a t t h e r e g i o n which i s f a v o u r a b l e f o r t h e search o f energy eigenvalues i s s h i f t i n g w i t h M. W i t h i n c r e a s i n g values o f M a l l eigenvalues o f t h e system can be c a l c u l a t e d . The accuracy of t h e method i n dependence o f K can be seen i n t a b l e 1 where t h e g r und s t a t e energy i s given. The energy v a l u e o b t a i n e d f o r K = 10 c o i n c i d e s up t o

lo-'

w i t h t h e one determined f o r K = 20. F i g . 3 gives energy elgenvalues as a f u n c t i o n o f A u s i n g v a r i o u s approxi- mation schemes.

I V - CONCLUDING REMARKS

We have shown t h a t c o n t i n u e d f r a c t i o n s a l l o w t o determine n u m e r i c a l l y exact eigen- s o l u t i o n s i n a convenient manner f o r systems, which cannot be s o l v e d a n a l y t i c a l l y . A d d i t i o n a l r e s u l t s , e s p e c i a l l y t h e comparison w i t h a n a l y t i c approximations and e i g e n s o l u t i o n s f o r an e l e c t r o n i c t w o - l e v e l system coupled t o two v i b r a t i o n a l modes, have been o b t a i n e d and a r e contained i n more d e t a i l e d p u b l i c a t i o n s 1181.

REFERENCES

/I/ Kohler, J. and Reineker, P., Chem.Phys. 39 (1985) 209 /2/ Schmid, U. and Reineker, P., Molecular P q s . , i n p r i n t

/3/ Stoneham, A.M., Theory o f D e f e c t s i n S o l i d s (Oxford: Clarendon Press, 1975) /4/ Benk, S. and Sigmund, E., J. Phys. C 18 (1985) 533

/ 5 / Dick, B.G., Phys. Rev. B 16 (1977) 33% / 6 / Wagner, M., Z. Phys. B 3 2 7 1 9 7 9 ) 225

/ 7 / Reineker, P. i n : E x c i t o ~ D y n a m i c s i n Molecular C r y s t a l s and Aggregates: S t o c h a s t i c L i o u v i l l e Equation Approach, Springer Tracts i n Modern Physics Vol. 94 ( B e r l i n : S p r i n g e r , 1982)

/ 8 / Kenkre, V.M. i n : E x c i t o n Dynamics i n Molecular C r y s t a l s and Aggregates: The Master Equation Approach, S p r i n g e r T r a c t s i n Modern Physics, Vol. 94 ( B e r l i n : S p r i n g e r , 1982)

/ 9 / M e r r i f i e l d , R.E., Rad. Research 20 (1963) 154 _-

/ l o /

F u l t o n , R.L. and Gouterman, M., J. Chem. Phys. 4 1 (1964) 2280 /11/ F r i e s n e r , R. and S i l b e y , R., J. Chem. Phys. 74 (1981) 1166 1121 F r i e s n e r , R. and S i l b e y , R., J. Chem. Phys. (1981) 3925 1131 Wagner, M., J. Phys. C 15 (1982) 5077

1141 Gutsche, E., Phys. S t a t T S o l . (b) 109 (1982) 583

1151 Reik, H.G., Nusser, H. and R i b e i r c ~ . ~ . ~ . , J. Phys. A 15 (1982) 3491 1161 Risken, H., The Fokker Planck Equation: Methods o f S o l u t i o n and A p p l i c a t i o n s ,

S p r i n g e r S e r i e s i n S y n e r g e t i c s ( B e r l i n : Springer, 1984) 1171 Durst, C., Diploma Thesis, U n i v e r s i t a t S t u t t g a r t , 1984