HOMOGENEOUS BROADENING OF ZERO-PHONON LINES FOR A MULTILEVEL SYSTEM IN A CRYSTAL : THE ROLE OF THE ELECTRON-PHONON INTERACTIONS

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HOMOGENEOUS BROADENING OF

ZERO-PHONON LINES FOR A MULTILEVEL SYSTEM IN A CRYSTAL : THE ROLE OF THE

ELECTRON-PHONON INTERACTIONS

A. Barchielli, E. Mulazzi

To cite this version:

A. Barchielli, E. Mulazzi. HOMOGENEOUS BROADENING OF ZERO-PHONON LINES FOR A MULTILEVEL SYSTEM IN A CRYSTAL : THE ROLE OF THE ELECTRON- PHONON INTERACTIONS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-475-C6-477.

�10.1051/jphyscol:19816138�. �jpa-00221202�

JOURNAL DE PHYSIQUE

CoZZoque C6, suppldment au n o 12, Tome 42, ddeembre

1981

page

C6-475

HOMOGENEOUS BROADENING OF ZERO-PHONON L I N E S FOR A M U L T I L E V E L SYSTEM I N

A CRYSTAL

:

THE ROLE OF THE ELECTRON-PHONON INTERACTIONS

A. Barchielli and E. ~ u l a z z i *

I s t i t u t o d i F i s i c a deZllUniversitd, v i a Celoria 16, Milano, I t a l y and I s t i t u t o Nazionale Fisiea NucZeare, Sezione d i Milano, I t a l y .

* I s t i t u t o d i Fisica d e l l ' U n i v e r s i t d , v i a Celoria

16,

Milano, I t a l y and Gruppo NazionaZe S t r u t t u r a Materia deZ CNR, Milano, I t a l y .

Abstract.- We present t h e e v a l u a t i o n o f the homogeneous broadening f o r a Jahn T e l l e r c e n t r e i n a c r y s t a l when i n t e r a c t i o n s which cause i n t e r l e v e l m i x i n g a r e present. C o n t r i b u t i o n s coming o u t from one and two-phonon processes a r e taken i n t o account. The s p e c i a l case i n which t h e m i x i n g i s due t o t h e presence o f s p i n - o r b i t i n t e r a c t i o n i s a l s o t r e a t e d .

We consider a m u l t i l e v e l system i n i n t e r a c t i o n w i t h t h e l a t t i c e phonons, e.g. an o p t i c a l e l e c t r o n of an i m p u r i t y i n a c r y s t a l . T h i s i s t h e t y p i c a l problem o f d i s c r e t e l e v e l s embedded i n a continuum. I n a f i r s t approximation, these d i s c r e t e l e v e l s have an i n t r i n s i c w i d t h when t h e i n t e r a c t i o n w i t h t h e continuum mixes them. Anyhow, t h e problem i s n o t t r i v i a l if one i s i n t e r e s t e d i n o b t a i n i n g t h e expressions f o r t h e widths o f t h e zero phonon l i n e s and t h e multi-phonon processes when the l e v e l s a r e degenerate and Jahn-Teller i n t e r a c t i o n s a r e present. We study t h i s problem i n t h e framework o f t h e open system theory, by u s i n g a method r e c e n t l y developed i n Ref.1.

As i t has a l r e a d y been shown i n Ref.2, t h e o p t i c a l response f u n c t i o n can be w r i t t e n as ( l + M ( t ) ) exp Gt, where exp G t represents t h e "lrlarkovian" behaviour which dominates i n the l o n g time l i m i t , and M ( t ) represents the "memory" c o r r e c t i o n s t o t h e Markovian behaviour which dominate i n t h e s h o r t and i n t e r m e d i a t e times. M ( t ) i s determined by t h e multi-phonon processes, w h i l e the r e a l p a r t o f G(R~G=-Y/E) g i v e s t h e w i d t h y,and therefore, t h e homogeneous broadening o f the zero-phonon l i n e . I n t h i s s h o r t r e p o r t we 1 im i t ourselves t o show t h e expressions o f such widths f o r a simple model : o n l y t h r e e e l e c t r o n i c l e v e l s a r e considered, t h e ground and two e x c i t e d

l e v e l s ; t h e i n t e r a c t i o n w i t h t h e l a t t i c e phonons i s t r e a t e d i n t h e l i n e a r approxima- t i o n . The electron-phonon i n t e r a c t i o n H can be w r i t t e n as

eP

where

r

l a b e l s the symmetry group r e p r e s e n t a t i o n s according t o which t h e various operators transform, v i s t h e p a r t n e r index; t h e u(r,v) a r e the (r,v)-symmetric d i s - placements and t h e h. .(T,v) a r e t h e electron-phonon i n t e r a c t i o n m a t r i c e s (i,j=1,2

1

J

r e f e r t o the two e x c i t e d l e v e l s ) ; f o r i = j they r e p r e s e n t t h e i n t e r a c t i o n s on t h e two

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816138

C6-4 7 6 JOURNAL DE PHYSIQUE

excited e l e c t r o n i c levels (Jahn-Tel l e r i n t e r a c t i o n s ) , while f o r i # j they represent the interactions which mix the two excited levels (mixing i n t e r a c t i o n s ) . No i n t e r a c t - ion involves the e l e c t r o n i c ground level.

We give in the following the perturbative expressions (up to the fourth order) of r f o r the higher excited level (level 2) by considering two d i f f e r e n t cases:

where

wZ1

i s the frequency difference between the two excited levels and

umax

i s the maximum frequency of the one-phonon spectrum.

a ) In t h i s case the electron can decay form the level

2

t o the 1 through the emiss- ion of one phonon. This process gives -yL1), the second order contribution t o We neglect a l l the fourth order terms but the one containing the Jahn-Teller i n t e r - actions only (yJT). This term can be comparable t o y h l ) i f the Jahn-Teller i n t e r a c t - ion couplings

h J T

a r e stronger than the mixing ones aM(say lJT=$). Under these as- sumptions we have

y =

y i ' ) +

y J T a

where N i s t h e degeneracy of the level 2, n(w) i s the phonon population and the

~ ~ ( ~ 2 ) a r e the d e n s i t i e s of one-phonon s t a t e s of r s y m e t r y . r h l ) i s due t o the emission process of one phonon whose frequency i s equal to

wz1.

These processes a r e more important a s the pr(w2) f o r

W = W ~ ,

a r e more intense. In Eq.(3) we have excluded the exceptional case t h a t a l l P,(W:~) vanish. The expression of y d l l i s well known i n the framework of open system theory.3

yJT

i s determined by phonon absorption and re- emission virtual processes (note t h a t

y J T

too i s positive). I t i s worthwhile t o note t h a t Y(:) i s the only one contribution to t h e homogeneous broadening

y

a t T=O°K.

b) In t h i s second case the principal contribution Y(;) t o

y

i s due t o the decay of the electron from level 2 to 1 through the emission of two phonons. Moreover, a l s o phonon absorption and re-emission processes contribute t o

Y

. Now the expression of

y

i s :

where

y J T

i s given by Eq.(4) and

+W max

^$9'

f r ( u ) f r l (wZ1-w)

I ^dm

- W

max

^{W 2}

1

ym=

$ A

^{p V #}

^A Tr(h12(~~v)h21(r.v)h12(r:~L)h21 ( r l , v l ) ) j'nax f r ( w ) f r (-w)

-

^W

^dw

max (w-uZ1

^)L

The function f,(w) i s given by

f r ( ~ )

= (8)

Note t h a t yh2) i s determined by both the Jahn-Teller interactions ( h i i ( r , v ) ) and the mixing ones ( h . . ( r , v ) , i # j ) . In f a c t , the two phonon emission can be determined only

1

J

through a two steps process, where the two d i f f e r e n t kinds of i n t e r a c t i o n s a r e invol- ved.

A

two phonon emission through a one s t e p process could happen only i f we had con- sidered quadratic interactions.

A t

T=O°K the expression of i s given by

pr(W2)prl

(

( W ~ ~ - W ) ~ )

u L

.

. ^1~ 1 Tr{(h22(r.v)h21(r' ,"I)-h2,(r8 , ~ ' ) h , , ( r , v ) ) . ( h . c . ) }

(9)

Note t h a t

yJT

and

ym

do not contribute a t T=O°K, because they involve phonon absorpt- ion and re-emission processes.

The former expressions f o r the homogeneous broadening

y

hold a l s o i n the case where the mixing interaction i s determined only by the spin-orbit interaction. Let the Hamiltonian we consider have the expression

=

e

⁺

Hep

⁺ HSO

+ H~~ (10)

He i s now the electron Hamiltonian where the spin-orbit interaction acting on the de- generate levels has been already taken i n t o account; HSO i s the p a r t of the electro- nic spin-orbit interaction which mixes the electron levels and H i s t h e e-p i n t e r - action Hamiltonian on the degenerate levels. Let us consider now the case of spin-or- ep b i t coupling comparable with the e-p one. Taking i n t o account two excited levels as before, when

w ~ ~

(case a ) , the homogeneous broadening i s given by Eq.(2) where

< w ~ ~ ~ yJT

i s given by Eq.(4) and r(A)by

The 1

k ; k >

a r e the electronic s t a t e s ; k=1,2 numbers t h e two levels and k i s the dege-

neracy index; a l l the other symbols have been given before. Note t h a t

y

i s determined by the combined action of

H

which does not mix the e l e c t r o n i c l e v e l s and

HSO

which mixes them. eP

1.

G.W.

Ford,

J.T.

Lewis and J.R. McConnell, Phys. Rev.A 19,

907

(1979).

2.

A.

Barchielli, E . Mulazzi and

G.

Parravicini, to appear in Phys. Rev.

B.

3. H.Spohn, Rev. Mod. Physics

-

53,

569

(1980).