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A STRONG COUPLED ACOUSTIC DEFORMATION
POLARON IN ONE DIMENSION
E. Young, P. Shaw
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque
C6,suppZBment au n o 12, Tome
4 2 ,ddcembre 1981
page
C6-522A STRONG COUPLED ACOUSTIC DEFORMATION POLARON
I N
ONE DIMENSION
E .W. young* and P .B. ShawPennsylvania S t a t e University,
U .S.
A.A b s t r a c t .
-
A s t u d y i s undertaken of t h e system comprised of an e l e c t r o n in- t e r a c t i n g w i t h a c o u s t i c phonons v i a t h e deformation p o t e n t i a l i n one dimen- s i o n . A s t r o n g coupling a d i a b a t i c p e r t u r b a t i o n t h e o r y i s developed i n which an effective-phonon Hamiltonian i s g e n e r a t e d and determined by t h e s o l u t i o n of an i t e r a t i v e equation. Modified phonon modes a r e obtained which r e p r e s e n t e x c i t a t i o n s of t h e deformed l a t t i c e . One of t h e modes i s a t r a n s l a t i o n whichi s i n t e r p r e t e d a s a polaron. The e f f e c t i v e mass of t h e polaron i s determined. The d e n s i t y of modified phonon modes i s o b t a i n e d by a Green's f u n c t i o n method and t h e s h i f t from t h e c o n s t a n t unperturbed d e n s i t y i s computed. The energy, t o t h e next o r d e r i n i n v e r s e coupling c o n s t a n t beyond t h e s t r o n g coupling
l i m i t , i n v o l v e s t h i s s h i f t i n t h e d e n s i t y of modes due t o t h e electron-phonon i n t e r a c t i o n . I n t e r p r e t a t i o n of t h i s s h i f t i s f a c i l i t a t e d by t h e i n t r o d u c t i o n of a s c a t t e r i n g phase s h i f t . The behavior of t h i s s c a t t e r i n g s h i f t c l e a r l y i n d i c a t e s t h e presence of a bound s t a t e , t h e p o l a r o n , and a resonance.
We develop a method f o r s t u d y i n g t h e coupled system of conduction e l e c t r o n s and a c o u s t i c phonons. The method we employ h a s been used p r e v i o u s l y t o s t u d y t h e
1
coupled system of conduction e l e c t r o n s and o p t i c a l phonons
.
We make a d i s p l a c e d o s c i l l a t o r t r a n s f o r m a t i o n on t h e s t a n d a r d Hamiltonian and t r e a t t h e displacement dk' which i s a r e a l even c-number of
L,
a s a v a r i a t i o n a l parameter. The r e s u l t , upon s u b s t i t u t i n g t h e v a l u e o b t a i n e d f o r dk i n t o t h e Harniltonian, i s H = H+
H+
H 0 1 2' H ='
--d L vsech2x -+
L
1 i k x-
O dx2 3 = (z.rraa)This Hamiltonian i s t h e b a s i s of t h e s t r o n g coupling p e r t u r b a t i o n theory we pre- s e n t . The e i g e n s t a t e s of H. a r e comprised of one bound s t a t e and a continuum o r i g -
2
i n a l l y analyzed by Yukon
.
To c a r r y out t h i s p e r t u r b a t i o n t h e o r y we expand t h e e i g e n s t a t e s of H i n terms of t h e complete s e t u ( x ) and U ( X ) , I+(x)> = u ( x ) [4>+
P
1
U (X)( 4
>, where ($I> andI @
> a r e normalizable phonon s t a t e v e c t o r s . We do notP P P
$ r e s e n t t h e g e n e r a l method t o determine t h e s e s t a t e v e c t o r s t o a r b i t r a r y o r d e r i n
-1 -1 3
a
b u t r a t h e r t o one h i g h pover i~ a t h a n t h e s t r o n g coupling l i m i t.
T h i s method g e n e r a t e s an e f f e c t i v e phonon Harniltonian, H i n v o l v i n g only t h e phonon degrees ofph'
freedom, of t h e form Hph
I$>
= EI@>.
E x p l i c i t l y we c a l c u l a t e t h i s phonon Hamilton- i a n w i t h t h e r e s u l t* p r e s e n t address : Perkin-Elmer Corporation, Hain Avenue, M/S 409, Norwalk, CT 06856
R Vopt =
(F)2
A
sechE
(p-k) s e c h (p-k').kk' 2 (2)
P (P2+112
By w r i t i n g t h e phonon Hamiltonian i n t h i s form, we can i d e n t i f y V::; a s t h e m a t r i x element t h a t appears i n t h e o p t i c a l phonon model.
The l a t t i c e we c o n s i d e r has a l a r g e number of s i t e s N , t h e r e f o r e Rkk, can be
-...
thought of a s an NxN m a t r i x w i t h N r e a l eigenvalues i12 and e i g e n v e c t o r s gkn. I n t r o - n
ducing t h e Hermitean o p e r a t o r s
on
=1
gknqk,5,
=1
g& pk, we can c a s t H i n t h ek k eh
d i a g o n a l form .,
Hph =
-
$
+&[I
(5:
+
Qzo;)
-
~ l k j . (3)n
It can be shown t h a t one of t h e e i g e n f r e q u e n c i e s v a n i s h e s and t h a t t h e corresponding mode is a t r a n s l a t i o n . I f we f u r t h e r i n t r o d u c e phonon a n n i h i l a t i o n o p e r a t o r s i A = 2an-1'2
a
+
iE
7 (n#
1 ; [An.Aml =snm,
( 4 ) we can w r i t e 1n2
1 H= - - + - - + -
i 1 1 ph 6 2m* C T U [ ~ ~ ' ~ A ~ % + ~ L \ - I I ~ ~ n k (5)where II i s t h e g e n e r a t o r of t r a n s l a t i o n s . The p h y s i c a l model t h i s e f f e c t i v e Hamil- t o n i a n d e s c r i b e s i s t h a t of a f r e e p o l a r o n , t h e a c o u s t i c deformation p o l a r o n , of
2 2 2 2
mass m* given by m* = 4na
1
k d m = - ( 4 ~ ~ 1 ) m and f r e e phonons w i t h f r e q u e n c i e san.
k k 3
The spectrum of modes i n t h e presence of t h e electron-phonon i n t e r a c t i o n i s t h e same a s t h e unperturbed spectrum. The q u a n t i t y of i n t e r e s t i s t h e s h i f t i n t h e den- s i t y of v i b r a t i o n a l modes A(w) given by A(w) =
I
6(w-
Q ) , il>
0 , i n terms of t h enZ1 n n -
unknown modes of t h e deformed l a t t i c e . To c a l c u l a t e t h e d e n s i t y of modes we d e f i n e a r e t a r d e d Green's f u n c t i o n D k k , ( t ) and i t s F o u r i e r t r a n s f o r m G k k l ( t ) . The d e n s i t y
2w
A(@) is g i v e n by A(w) = -
I
I m Gkk(w), i n terms of t h e n o n - i n t e r a c t i n g wave v e c t o r s kk. We f u r t h e r i n t r o d u c e a continuum s c a t t e r i n g amplitude ~ ( k , k ' ,U) i n terms of which
-
t h e s h i f t generated by t h e electron-phonon i n t e r a c t i o n a(w) can be w r i t t e n
-
.t
a
A(w)= l
$'dk I T(k,k;w) R -CO l4.
(6) k2-
(*io)I n t h e Born approximation i n t h e continuum l i m i t , T(k,klw) i s r e p l a c e d V ( k , k 7 ) and Eq. (6) becomes
- d
A(w) = A (W) - V(W7W)
.
B dw
w
(7)
The r e s u l t i s p l o t t e d i n Fig. 1. The s h i f t i n t h e d e n s i t y of modes i n t h i s approxi- mation h a s t h e behavior m
-
2AB(w) + O(o ) ,
w
+ 0; iiB(w) + ~ ( o - ~ ) ,w
+ m;1
do $(W) = 0. (8)0-
A s a r e s u l t of t h e h i g h frequency behavior of AB(@), t h e s h i f t i n z e r o p o i n t energy 2 d i v e r g e s . This d i v e r g e n t c o n t r i b u t i o n i s given by Ediv = -8nms2 !?m(* w a x / m s )
which i s t h e r e s u l t of Reference ( l ) .
To f a c i l i t a t e c a l c u l a t i o n of h(w) without ap2roximation we f i r s t w r i t e i n m a t r i x T
n o t a t i o n V = BB (T = t r a n s p o s e ) . I f we d e f i n e a m a t r i x R(w) by t h e r e l a t i o n R(W) =
C6-524 JOURNAL DE PHYSIQUE
- 1
a
A(*) =
;
I m Tr ~-'(w) R(w). F u r t h e r we can o b t a i n an even s i m p l e r e x p r e s s i o n f o r 1 dt h e s h i f t a(w) =
- -
& ( a ) , where &(W) and R(w) a r e r e l a t e d by d e t R(w) = r ( w ) e i 6 (W) IT dw(r(w) r e a l ) . Gaussian q u a d r a t u r e h a s been used t o compute 6(w), which i s p l o t t e d i n Fig. 2. The f u l l s h i f t i n t h e d e n s i t y of modes e x h i b i t s t h e behavior
which i s analgous t o Levinson's theorem i n p o t e n t i a l t h e o r y i n t h e presence of a bound s t a t e . The absence of one v i b r a t i o n a l mode ( t h e t r a n s l a t i o n a l mode) r e s u l t s i n a d e p l e t i o n of modes c o n c e n t r a t e d a t z e r o frequency, w i t h l e s s e r d e p l e t i o n a t h i g h e r frequency.
The phase s h i f t does n o t f a l l monotonically t o z e r o a s i t would i n t h e presence of a bound s t a t e alone. I n f a c t , t h e s h i f t r i s e s a b r u p t l y a t
w
Q, 0.5, r e a c h e s amaximum a t
w
?,
l, and f a l l s monontonically t o z e r o t h e r e a f t e r . T h i s behavior sug- g e s t s t h e presence of a resonance a s w e l l a s a bound s t a t e . The enhancement o c c u r s n a t u r a l l y a t wavelengths comparable t o t h e s i z e of t h e polaron. These "resonating" phonons i n t e r a c t most s t r o n g l y ; hence t h e y a r e lowered i n frequency t h e most. A s a r e s u l t . we observe a bunching phonon modes s l i g h t l y beloww
= 1.l.lol
Fig. 1: The s h i f t i n d e n s i t y of modes F i g . 2: The phase s h i f t 6 ( w ) / 1 ~ v e r s u s
w.
Dashed curve i s Born approx., v e r s u s W.s o l i d w i t h o u t approx.
F i n a l l y we c a l c u l a t e n u m e r i c a l l y t h e s h i f t i n z e r o p o i n t energy r e s u l t i n g from t h e s h i f t i n t h e phonon d e n s i t y of modes. Using t h i s r e s u l t , we can w r i t e t h e ground s t a t e energy momentum r e l a t i o n a t low p i n dimensional u n i t s i n t h e form
-
(1) G. W h i t f i e l d and P.B. Shaw, Phys. Rev.
