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ANHARMONIC EFFECTS IN ION-ELECTRON
CHAINS
J. Žmuidzinas
To cite this version:
JOURNAL DE PHYSIQUE Colloque
C6,
supplPmenl au no8 ,
Tome 39, aoPt 1978, page (3-715
ANHARMONIC EFFECTS
I N
ION-ELECTRON CHAINS?J. h u i d z i n a s
J e t Ptwpulsion Luboratcry, CaZifornia I n s t i t u t e of Technolog~l
Pasadem, California
91103, U.S.A.RQsum6.- On montre qu'un Hamiltonien de FrEhlich pour une chazne d ' i o n s e t d ' B l e c t r o n s de
conduction amSne
B
un Hamiltonien e f f e c t i f anharmonique q u a r t i q u e de phonons q u i e s t c a r a c -t 6 r i s 6 par un paramstre s a n s dimension a:
=
( 3 / 6 4 ) ( 5 / ~ ~ ) ~ ( E ~ /3
Ms2) oii 5 % E e s t un pa- mstre de couplage electron-phonon, M l a masse i o n i q u e e t s l a v r t e s s e du son. E e t t e anhar- m o n i c i t e 6 l e c t r o n i q u e e s t d ' o r d i n a i r e n 6 g l i g e a b l e (a: % 0.01) mais peut d e v e n i r grande e tdominante dans d e s systsmes 3 i o n s 16gers ayant de h a u t e s 6 n e r g i e s de Fermi, p a r exemple, dans 1'hydrogSne m 6 t a l l i q u e f i l a m e n t a i r e .
A b s t r a c t . - It i s shown t h a t a FrEhlich Hamiltonian f o r a chain of i o n s and conduction e l e c -
t r o n s l e a d s t o an e f f e c t i v e q u a r t i c anharmonic phonon Hamiltonian c h a r a c t e r i z e d by a dimen- s i o n l e s s parameter a: % ( 3 / 6 4 ) ( 5 / ~ ~ ) ~ ( c
/
1
Ms2), where €,=cF i s an electron-phonon c o u p l i n gparameter, M the ion mass, and S the speez og sound. This e l e c t r o n i c anha-nicity i s o r d i n a -
r i l y n e g l i g i b l e ( a 2 = 0.01) b u t may become l a r g e and dominant i n l i g h t - i o n systems w i t h h i g h Fermi e n e r g i e s , fo? example, i n f i l a m e n t a r y m e t a l l i c hydrogen.
I n a r e c e n t paper / I / we have shown t h a t an anharmonic I D l a t t i c e of i o n s and e l e c t r o n s may s u p p o r t s u p e r s o n i c s o l i t o n s , which under f a v o r a b l e circumstances may t r a p and t r a n s p o r t e l e c t r o n s a l o n g t h e l a t t i c e . The model Hamiltonian used t o demonstrate t h i s e f f e c t was FrEhlich w i t h added q u a r t i c a n h a r m o n i c i t i e s . No e s t i m a t e of t h e t y p i c a l s t r e n g t h of phonon a n h a m n i c i t i e s was o f f e r e d i n t h a t work because of l a c k i n g experimental d a t a and t h e o r e t i c a l d i f f i c u l t i e s . Moreover, t h e e f f e c t of conduction e l e c t r o n s on t h e phonon motion was assu- med t o be small compared t o t h e l a t t i c e anharmoni- c i t y . The purpose of t h i s paper i s t o a t t e m p t t o c o r r e c t t h e s e d e f i c i e n c i e s by f o r m u l a t i n g a theory of anharmonic phonon e f f e c t s o r i g i n a t i n g from a FrGhlich-type i n t e r a c t i o n of phonons w i t h Bloch e l e c t r o n s . We r e s t r i c t o u r a t t e n t i o n t o the zero- temperature c a s e . We b e g i n w i t h a ~ r g l i c h Hamiltonian adapted t o a I D c a s e : i n a almost s t a n d a r d n o t a t i o n w i t h
M
= 1 , E=
p2/2mb 112 P-
p ,y, =
s k , andgk
= -i€k(N/2Muk).
t
I wish t o thank P r o f e s s o rJ.R.
Vaiinys f o r exten-s i v e d i s c u s s i o n s . Zhis work was sponsored by NASA
under Contract Number NAS7-100.
Note t h a t we a r e u s i n g screened phonon f r e q u e n c i e s wk and screened electron-phonon coupling c o n s t a n t s
gk. The parameter
E
i s of o r d e r cF; f o r example,from ( 1 ) and t h e e f f e c t i v e small-k electron-phonon m a t r i x element v e f f = - i ( 4 n 2e2k/kFT2L) X
k
( N / M ) ' / ~ ( L = l e n g t h of I D c h a i n ) given by Pines
121, one can i n f e r t h a t
6
=
2 F/3. The use ofvk 4ne2/k2L, i m p l i c i t i n v k e f f , i s j u s t i f i e d by Williams and Bloch 131 f o r t h e I D c a s e . We n e g l e c t a l l Umklapp p r o c e s s e s , s o t h a t [ b k , b k , + 7
= 6kk,,
e t c . Considering t h e Green's f u n c t i o n a l c o r r e s - ponding t o ( 1 ) and i n t e g r a t i n g o u t t h e e l e c t r o n v a r i a b l e s , we g e t 7, = /d$d$' exp i { i Tr I n G(@)+
l
'
(2%)-l (k.2- wk2 ) 1+k12+ source terms), ( 2 ) 2 kwhere G - ~ (U) = (po
-
cp) 6ppl-
U i s t h e in-PP' P-P
'
v e r s e e l e c t r o n Green's f u n c t i o n f o r an a r b i t r a r y p o t e n t i a l U . Expanding t h e e l e c t r o n loop term Vzi Tr I n G t o f o u r t h o r d e r i n $, we f i n d , i n t h e long
wavelength limit and f o r k
=
s k and s 2 << vp2,t h a t V
=
V 2+
V 3+
V4, wherew i t h a 2 = 2, ag = aq = 1 . The c a l c u l a t i o n of (3)
i s
q u i t e t e d i o u s a l g e b r a i c a l l y and was done w i t h t h e h e l p of MACSYMA 141. We d i s c a r d t h e V2t e r m b e c a u s e i t r e p r e s e n t s a r e n o r m a l i z a t i o n e f f e c t o f t h e phonon f r e q u e n c i e s , and s u c h r e n o r m a l i z a t i o n was presumably a l r e a d y c a r r i e d o u t t o o b t a i n t h e s c r e e n e d phonon f r e q u e n c i e s I+ i n t h e ( e f f e c t i v e )
Fr6hlich H a m i l t o n i a n ( I ) . We a r e mainly i n t e r e s t e d i n t h e a n h a m n i c phonon terms
V g
and V,,. I t s h o u l d be s t r e s s e d t h a t o u r p r o c e d u r e o f d e r i v i n g t h e anharmonic terms i s n o t e n t i r e l y s a t i s f a c t o r y i n a s - much a s we a r e n o t t r e a t i n g t h e s c r e e n i n g and anhar- monic e f f e c t s c o n s i s t e n t l y ; It would b e o f i n t e r e s tt o do t h e more d i f f i c u l t , c o n s i s t e n t c a l c u l a t i o n of t h e s e e f f e c t s .
By comparing t h e e f f e c t i v e anharmonic pho- non H a m i l t o n i a n , which can b e r e a d o f f from ( 2 ) and ( 3 ) , w i t h ( 2 . 2 ~ ) and (2.4b) of / I / and u s i n g N = 2
6
I = 2 L k /n and cF = kF2/2mb, we o b t a i n a n F e x p r e s s i o n f o r t h e d i m e n s i o n l e s s q u a r t i c anharmoni- c i t y p a r a m e t e r a 2 i n t r o d u c e d i n / I / : 1 a 2 = (3164) ( ~ / E ~ ) ~ ( E ~ /7
M S ~ ) . ( 4 ) A s i m i l a r e x p r e s s i o n c a n b e o b t a i n e d f o r t h e s t r e n g t h p a r a m e t e r o f t h e c u b i c phonon a n h a m o n i c i t y . However, a c u b i c phonon term c a n b e d i s p o s e d o f , a s f a r a s i t s c o n t r i b u t i o n t o t h e c u b i c Boussinesq e q u a t i o n d e r i v e d i n / l / i s c o n c e r n e d , by s i m p l y s h i f - t i n g t h e m a c r o s c o p i c s t r a i n f i e l d by a c o n s t a n t and r e d e f i n i n g t h e p a r a m e t e r s a , S , and d (= l a t t i c e c o n s t a n t ) . For t h i s r e a s o n we h e n c e f o r t h d i s r e g a r d t h e c u b i c term.The p a r a m e t e r ( 4 ) measures t h e phonon a n h a r - m o n i c i t y induced o r mediated by c o n d u c t i o n e l e c t r o n s ; t h i s e f f e c t i s i n a d d i t i o n t o any i n t r i n s i c anharmo- n i c i t : ~ t h a t t h e l a t t i c e may p o s s e s s ( a s a r e s u l t of anharmonic i o n - i o n f o r c e s ) . We s h a l l d i s t i n g u i s h t h e s e two d i f f e r e n t c o n t r i b u t i o n s by t h e s u b s c r i p t s e and i ; a 2 = a
+
a. 2 . F o r t y p i c a l I D s y s t e m s con- s i d e r e d i n ( l ) , one h a s5
% cF % 0.05 eV, M % 100 Xp r o t o n mass, and S % lo5 cm/s, s o t h a t a e 2 I O - ~
by ( 4 ) . I n ( 1 ) we have examined c a s e s where
a << a 2 = a m O ( 1 ) . The c a s e o f s m a l l t o t a l a 2 ,
e . i
i f i t c a n b e r e a l i z e d , i s of c o n s i d e r a b l e i n t e r e s t b e c a u s e i t l e a d s t o l a r g e - a m p l i t u d e l a t t i c e s o l i t o n s . L e t us t r y t o e s t i m a t e a i 2 . One c a n e a s i l y show from ( 2 . 2 ~ ) .of / l / and e q u a t i o n ( 3 4 ) o f / 5 / t h a t
a . 2 = cd2/4y, where y and c a r e t h e harmonic and q u a r t i c f o r c e c o n s t a n t s . F o r t h e l i n e a r c h a i n w i t h Morse model p o t e n t i a l c o n s i d e r e d b y ~ l a k i d a and S i k l b s / 6 / one h a s y = ~ a ~ and c / y 2 = 7/D, where D and a-l a r e t h e d e p t h and r a n g e o f t h e p o t e n t i a l . T h i s g i v e s a; ( 7 / 4 ) ( d a ) 2 . To o b t a i n a 2, a % I O - ~ one t h e r e f o r e n e e d s v e r y i long-range e f f e c t i v e i o n - i o n p o t e n t i a l s : a-' = 13d Because o f e l e c t r o n s c r e e n i n g e f f e c t s , one o r d i n a - r i l y w i l l have a-'
<
d , i m p l y i n g a . 2 = 0 ( 1 ) , a s assumed i n / I / . I n view o f t h i s d i s c u s s i o n i t t h e r e f o r e seems u n l i k e l y t h a t t h e e l e c tron-induced a n h a r m o n i c i t i e s a r e g o i n g t o p l a y a n i m p o r t a n t r o l e i n o r d i n a r y m e t a l l i c I D s y s t e m s ( e . g . , c o n d u c t i n g o r g a n i c c h a r g e t r a n s f e r s a l t s ).
The s i t u a t i o n might b e q u i t e d i f f e r e n t i n s y s t e m s w i t h l i g h t i o n s and h i g h Fermi e n e r g i e s ( e . g . , f i l a m e n t a r y m e t a l l i c hydrogen / 7 / , where a e 2 c o u l d c o n c e i v a b l y be 2-4 o r d e r s of magnitude l a r g e r , a s one may e s t i m a t e from ( 4 ) . However, i n s u c h s y s t e m s one must be c a u t i o u s a b o u t s e p a r a t i n g c o r e and c o n d u c t i o n e l e c - t r o n s and c o n s e q u e n t l y decomposing a 2 i n t o i t s i o n i c and e l e c t r o n i c p a r t s .R e f e r e n c e s
/ I / Zrnuidzinas, J . S . , Phys. Rev.
B E ,
i n p r e s s . / 2 / P i n e s , D., Elementary E x c i t a t i o n s i n S o l i d s(W.A. Benjamin, I n c . (1963)) 243.
1 3 1 W i l l i a m s , P.F. and Bloch, A.N., Phys. Rev.
B=
(1974) 1097./ 4 / M i t Mathlab Group, Macsyma R e f e r e n c e Manual, V e r s i o n Nine ( J u l y 1977). The work o f t h e Mathlab Group a t M i t i s s u p p o r t e d , i n p a r t , by Dod u n d e r C o n t r a t Number E(ll-1)-3070 and by NASA under Grant NSG 1323.
1 5 1 P a t h a k , K . N . , Phys. Rev.
139
(1965) A1569. 1 6 1 P l a k i d a , N.M. and S i k l b s , T., Phys. S t a t . S o l .33 (1969) 113.
