- Email: [email protected]
Polarons in Polyacetylene
Nonlinear Problems: Present and Future
195
A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.)
0 North-Holland Publishing Company, 1982
POLARONS I N POLYACETYLENE
Alan R. Bishop and David K. Campbell Theoretical D i v i s i o n and Center f o r Nonlinear Studies Los Alamos National Laboratory Los Alamos, NM 87545 USA Recent t h e o r e t i c a l studies o f polyacetylene, (CH) have focussed on k i n k - l i k e s o l i t o n s i n the trans isomer. I t i s stown t h a t t h e same t h e o r e t i c a l models a l s o p r e m o l a r o n - l i k e s o l i t o n s i n both c i s - and W - ( C H ) . The e x p l i c i t f o r m h e s e polarons, t h e i r r e l a t i o n t o kinks, p o & i b l e i m p l i c a t i o n s f o r experiment, and open questions are discussed. INTRODUCTION During the l a s t few years the extensive t h e o r e t i c a l and experimental i n t e r e s t i n nonlinear e x c i t a t i o n s i n polyacetylene ((CH) ) has focused almost e x c l u s i v e l y on k i n k - l i k e s o l i t o n states. [l-101 This i s x h a r d l y s u r p r i s i n g , f o r a p a r t from t h e i r possible d i r e c t experimental i m p l i c a t i o n s [2,11] f o r t r a n s p o r t properties, doping mechanisms, and t h e observed semi-conductor metal t r a n s i t i o n i n (CHI , these k i n k s o l i t o n s , w i t h t h e i r b i z a r r e spin/charge assignments, [3] have s t i m b l a t e d t h e o r e t i c a l work [12,13,14] on t h e existence and r o l e o f " f r a c t i o n a l charge'' i n both s o l i d s t a t e systems and f i e l d theory models. Recently, however, i t has been discovered [15,16,17] t h a t t h e k i n k s o l i t o n s are not the only nonlinear e x c i t a t i o n s p r e d i c t e d by t h e o r e t i c a l models o f (CH)x. Indeed, i n both (numerical studies o f ) l a t t i c e models [15] and ( a n a l y t i c studies o f ) continuum theories, [16,17] a nonlinear "polaron" e x c i t a t i o n has been found. Although more conventional i n i t s p r o p e r t i e s than the k i n k s o l i t o n , the polaron, as we s h a l l see, i s the lowest energy e x c i t a t i o n a v a i l a b l e t o a s i n g l e electron. Experimentally, the polaron may thus p l a y an important r o l e i n t h e doping process o r i n e l e c t r o n i n j e c t i o n and, t h e o r e t i c a l l y , i t s presence f u r t h e r embelOne p a r t i c u l a r l y s i g n i l i s h e s the already r i c h s t r u c t u r e p r e d i c t e d f o r (CH) f i c a n t feature o f the polaron i s t h a t it, u n l i k e t h e k i n k s o l i t o n which can e x i s t only i n the trans isomer o f ( C H I , i s p r e d i c t e d t o occur i n both c i s and trans isomers. The possible i m p l i c a t i h s o f t h i s f o r comparative e x p e r x n t a l s t u d i e s o f cis and trans a r e o f great c u r r e n t i n t e r e s t . Thus i n t h i s a r t i c l e we s h a l l focus on t h i s o r e conventional" b u t nonetheless i n t e r e s t i n g nonlinear e x c i t a t i o n , describing i t s predicted nature i n d e t a i l and discussing p o t e n t i a l experimental imp1 i c a t i o n s b r i e f l y .
.
POLARONS I N m - ( C H ) , From the l a t t i c e model [3] f o r trans-(CH) , a standard sequence o f approximat i o n s [6] leads i n the c o n t i n u u m m i t 61an e f f e c t i v e a d i a b a t i c mean-field Hamiltonian i n terms o f t h e ( r e a l ) gap parameter A and e l e c t r o n f i e l d Y. The result i s
A.R. BISHOP, D.K. CAMPBELL
196
where w2/g2 i s t h e n e t e f f e c t i v e electron-phonon coupling constant, [3,6] ai i s
Q
the i t h Pauli matrix, V(y) =
(!$;),
and vF i s t h e Fermi v e l o c i t y .
[18]
For
l a t e r purposes we note t h a t i n e r i v i n g t h i s mean f i e l d Hamiltonian, t h e l a t t i c e k i n e t i c energy which would add a term p r o p o r t i o n a l t o A2(y) i n (1) has been e x p l i c i t l y ignored. Obviously, t h i s w i l l have no e f f e c t on the s t a t i c e x c i t a t i o n s we discuss below, b u t i t . w i l l prove s i g n i f i c a n t f o r on-going studies i n v o l v i n g the dynamics o f s o l i t o n s i n (CH),.
-
-
V a r i a t i o n o f H leads t o the s i n g l e p a r t i c l e e l e c t r o n wave f u n c t i o n equations
and the s e l f - c o n s i s t e n t gap equation
The summation i n (3) i s over occupied e l e c t r o n s t a t e s and s i s a s p i n l a b e l (suppressed i n (1) and (2)). Equations (2) and (3) are t h e continuum e l e c t r o n phonon equations f o r t r a n s (CH) , [5,6] and remarkably one can f i n d a n a l y t i c , closed form e x p r e s s i o f i r seveval n o n t r i v i a l solutions. [5,6,16,17] Unsurprisingly, t h i s i s r e l a t e d t o the ( p a r t i a l ) s o l i t o n features o f these equations. [16,17] Although we wish t o focus on the polaron s o l u t i o n s t o (2) and (3), f o r c l a r i t y and completeness we s h a l l b r i e f l y review t h e other solutions. F i r s t , the ground s t a t e o f t h e i n f i n i t e chain o f --(CH) e r a t e (see Figure l), and i s described by A(y) = + A. A,
= W exp (-A -1)
i s t w o - f o l d degeno r Afy) = A -,, w i t h [5,6]
(4. a)
where
The parameters i n (4) include W, t h e f u l l one-electron bandwidth (e 10 eV i s trans-(CHI 1; v , t h e Fermi v e l o c i t y (vF = aW/2, [ l a ] where a i s t h e underi n =-(CH),); and, as i n d i c a t e d p r e v i o u s l y , l a t d c e sFacing (= 1.22
POLARONS IN POL YACETYLENE
197
u2/g2, the n e t e f f e c t i v e electron-phonon coupling constant (whose value i s i n eqfect determined by (4)). Since experimentally A. 0.7 eV, eqn. (4.a) implies the dimensionless coupling A 0.4.
.. H
H
H I
H I
...c A+ /;\ /\ /\ c.. .
Figure 1: (a) two (c) gap A =
I
CI
CI
CI
I
H
H
H
H
H
-
and (b) The two schematic bond s t r u c t u r e s corresponding t o t h e degenerate ground states f o r =-(CH) A p l o t f o r =-(CH) o f the energy ber u n i t l e n g t h versus band parameter A f o r spatialTy constant A. Note the degeneracy between f A and t h e l o c a l maximum ( i n d i c a t i n g i n s t a b i l i t y ) a t A = 0.
-
.
0
the s i n g l e e l e c t r o n energy spectrum i s given by c(k) = Fig. 2) and the corresponding e l e c t r o n wave f u n c t i o n s a r e simply plan8 waves, the e x p l i c i t forms o f which are given i n r e f s . [5,6, and 171. Physically, A # 0 implies a gap (= 2A , see Fig. 2) around the fermi surface i n t h e s i n g f e e l e c t r o n spectrum; t h a t q h i s gap should e x i s t i n t h e continuum model o f ( C H I i s a d i r e c t consequence o f t h e well-known P e i e r l s i n s t a b i l i t y [19] o f one-'bimensional coupled electron-phonon systems. I n chemical terms, A, # 0 indicates a "bond a l t e r n a t i o n " between s i n g l e and double bonds (see Fig. l), and i t s value i s p r o p o r t i o n a l t o t h e lengthening (shortening) o f a s i n g l e (double) bond w i t h respect t o the i d e a l i z e d uniform bond l e n g t h which would occur were A, = 0. For
-4
f
constant A
(s8;
A. R. BISHOP, D. K. CAMPBELL
198
a) -A0
0
-2
2
Figure 2: The s p a t i a l s t r u c t u r e o f t h e gap parameter (A(y)) and t h e associated e l e c t r o n i c l e v e l s f o r --(CH) f o r (a) the ground s t a t e ; (b) the k i n k s o l i t o n ; (c) t h e polaron. foblid l i n e s i n t h e l e f t - h a n d drawings i n d i c a t e A(y); dashed l i n e s show e l e c t r o n d e n s i t i e s f o r l o c a l i z e d states. The right-hand drawings show the e l e c t r o n i c l e v e l s i n the conduction band ( s i n g l e shading), valence band (crossed shading), and gap states. Q = charge and S = spin. The two-fold degeneracy o f t h e ground s t a t e o f trans-(CH) has the very import a n t consequence t h a t a l o c a l i z e d , f i n i t e e n e r f i n k sditon corresponding can e x i s t ; t h i s k i n k (K) i n t e r p o l a t e s between t o a l o c a l i z e d l a t t i c e "defect" the degenerate ground s t a t e s (see Fig. 2) and has t h e e x p l i c i t form
-
AK(y) = Aotanh VFy/Ao
.
-
(5)
There i s a l s o an a n t i - k i n k (k) s o l i t o n with Ak(y) = -A (y). The associated s i n g l e e l e c t r o n energy l e v e l s c o n s i s t o f extended c o n d u c t i h (E(k) = + 4 and valence (E(k) = J v q band s t a t e s [5,6,17] which a r e modffied' ( e s s e n t i a l l y "phase s h i f t e d ) fr8m t h e plane waves o f t h e ground s t a t e and an The e x p l i c i t a d d i t i o n a l l o c a l i z e d "mid-gap" s t a t e ( a t E = 0, see Fig. 2). form o f the wave f u n c t i o n f o r t h i s l o c a l i z e d h a t e i s
-
POLARONS IN POL YACETYLENE
Uo(y) = No sech A0y/vF
and Vo(y) = - i N o sech Aoy/vF
w i t h No =
For e i t h e r K o r i,t h i s "mid-gap" leading t o l o c a l i z e d e x c i t a t i o n s 6,121 i n d i c a t e d i n Fig. 2. It i n t e r e s t i n " f r a c t i o n a l charge."
s t a t e can be occupied by 0, 1, o r 2 electrons, w i t h t h e b i z a r r e spin/charge assignments [3,5, i s these states t h a t have e x c i t e d the recent [12-141
One v i t a l consequence o f t h e k i n k s o l i t o n ' s i n t e r p o l a t i n g between t h e degenerate ground states i s t h a t t h e r e i s a r e s t r i c t i o n u s u a l l y c a l l e d a " t o p o l o g i c a l on t h e production o f kinks. S p e c i f i c a l l y , i n the i n f i n i t e c o n s t r a i n t " [12,17] chain polymer, kinks can be excited from the ground s t a t e o n l y i n p a i r s o f k i n k and a n t i - k i n k . The i n t e r e s t e d reader should convince himself o f t h i s by showing t h a t the production o f a s i n g l e k i n k from t h e ground s t a t e t h a t i s , going from the A(y) c o n f i g u r a t i o n i n Fig. 2a t o t h a t i n Fig. 2b requires overcoming an i n f i n i t e ( f o r an i n f i n i t e l e n g t h polymer) energy b a r r i e r .
-
-
-
-
F i n a l l y , we come t o the polaron s o l u t i o n t o equations (2) and ( 3 ) . The r e c e n t l y discovered [16,17] a n a l y t i c form f o r the gap parameter o f t h e polaron can be w r i t t e n i n t h e revealing form (c.f., eq. (5))
where tanh 2Koyo = KovF/Ao
.
(7. b)
This c o n f i g u r a t i o n f o r A(y), which corresponds t o a s p a t i a l l y l o c a l i z e d deviat i o n from t h e ground s t a t e value (here chosen t o be + A ), i s sketched i n Fig. 2c. Again t h e s i n l e e l e c t r o n states include extended s l a t e s i n t h e conduction and valence (E(k) = q-4 bands, which s t a t e s are (&(k) = + phase-shifted by th% polaron "defect"; t h e expl i c f t ?orms o f t h e corresponding wave functions are given i n r e f . 17. I n addition, two l o c a l i z e d e l e c t r o n i c states, w i t h energies symmetrically placed, form i n t h e gap a t &+ 5 f uo where
-
-
ug
-J
=
The e l e c t r o n wave functions f o r these l o c a l i z e d s t a t e s are, f o r Uo(Y) = No
and
I ( l - i ) s e c h Ko(y+yo)
+
(l+i)sech Ko(y-yo)]
E
= +wo
A. R. BISHOP, 0.K. CAMPBEL L
200
V,(Y)
= No
where No = 4&, 1
{ ( l + i ) s e c h Ko(y+yo) + (1-i)sech Ko(y-yo)] and f o r &=-wO
u-o
5
= +iVo and V-,
UIE=-,,,
(9b) :VIE=-u
0
= -iUo. 0
-
-
I t i s important t o note t h a t the polaron c o n f i g u r a t i o n f o r A(y) eq. (6) and the associated e l e c t r o n wave functions s a t i s f y the e l e c t r o n p a r t eq. (2) of the continuum equations f o r 9 KovF i n t h e allowed range 0 5 KovF 5 A., It i s
-
-
-
-
the s e l f - c o n s i s t e n t gap c o n d i t i o n eq. ( 3 ) which determines t h e s p e c i f i c value o f K v f o r an actual s o l u t i o n t o t h e coupled equations. Using some aspects o f %.o!iton theory, [17] one can, i n e f f e c t , convert t h e gap equation t o a s t r a i g h t f o r w a r d minimization problem. Apart from s i m p l i f y i n g t h e problem t e c h n i c a l l y , t h i s i s very appealing i n t u i t i v e l y , f o r the q u a n t i t y being m i n i mized i s e s s e n t i a l l y t h e ener o f t h e f u l l i n t e r a c t i n g electron-phonon system. I n p a r t i c u l a r , one f i n d s d e polaron-type c o n f i g u r a t i o n i n =-(CH), that t Ep(n++' n-, KO) = (n+
-
n- + 2)w0 +
4
KovF
-
-n4 wo t a n
-
where n are r e s p e c t i v e l y t h e occupation numbers which can be o n l y 0, 1, o r 2 o f $he l o c a l i z e d s t a t e s E+ = +UJ and e- = -w kntroducing 8 such t h a t KovF = Aosin8 and (see (8)) wo = A0co& and minimizpng E w i t h respect t o 8 g i v e P
-
.
From eq. (11) i t f o l l o w s d i r e c t l y t h a t f o r a stable, l o c a l i z e d polaron s o l u t i o n the electrons must be d i s t r i b u t e d i n one o f two c o n f i g u r a t i o n s : (1) n+ = 1, n- = 2, the " e l e c t r o n polaron" s t a t e ; or (2)
n+ = 0, n- = 1, the "hole polaron" s t a t e .
Before describing these s t a t e s i n d e t a i l , we note t h a t studying o t h e r p o s s i b l e configurations reveals some i n t e r e s t i n g phenomena. When n = n- --- f o r n+ = 0, 1, o r 2 (11) i m p l i e s 6 = n/2, so Kov = A , w = 0 fcorresponding t o t h e "mid-gap" state), and by (7.b) y + a. henceothe& p u t a t i v e trans-(CH), "bipolarons" separate i n t o an i n f i R i t e l y separated k i n k / a n t i - k w p a i r s , w i t h charges determined by the value o f nt = n-. When n+ = 0, n- = 2, 6 = 0, so w = A , K v = 0, and the "unoccupied polaron collapses t o the ground s t a t e , A'. O!herockoices o f n+ and n- lead simply t o combinations o f the above e x c i t a t?ons ( i n f i n i t e l y separated i n space); f o r example, n+ = 1, n- = 0 leads t o a c o n f i g u r a t i o n o f kink, a n t i - k i n k , and polaron.
---
Returning t o t h e t r u e , s t a b l e polaron states, we see t h a t f o r n+ = 1, n- = 2 ( e l e c t r o n polaron) o r n+ = 0, n- = 1 (hole polaron) eq. (11) y i e l d s 6 = n/4 o r
POLARONS IN POL YACETYLENE
KovF =
201
A0/G= wo
(12)
From eq. (10) the energy o f t h i s e x c i t a t i o n i s seen t o be
E =@A P n o
0.90Ao
(13)
1
From e ns. (61, (7.b), and (12) one sees t h a t the s p a t i a l e x t e n t o f t h e polaron ( = 11 f o r trans-(CH) ) i s s l i g h t l y l a r g e r than t h a t o f the k i n k (= 9 A f o r W - ( C H ) );this compirison i s made v i s u a l i n Fig. 2. Although general arguments sug&sting the existence o f polarons i n systems l i k e trans-(CH) that i s , (quasi)-one-dimensional P e i e r l s dimerized chains can b e v a n c e # , [20] , i t i s encouraging t o f i n d t h a t the continuum model, w i t h o u t any ad hoc addit i o n s , e x p l i c i t l y contains these e x c i t a t i o n s . I t i s a l s o heartening t h a t numeri c a l simulations o f the l a t t i c e model [15] have revealed a polaron e x c i t a t i o n whose s t r u c t u r e i s i n good agreement w i t h t h a t p r e d i c t e d by eq. (6), (7), and (12).
-
-
Another aspect o f the polaron f i e l d c o n f i g u r a t i o n (eq. (7)) and corresponding energy expression (eq. (10)) i s worth mentioning: namely, since t h e e l e c t r o n equations eqns. (2) are s a t i s f i e d f o r a l l K i n t h e range 0 < K 5 A /v , eq. (10) can be viewed as a s i n g l e parameter (K expression f o r tfie Rink/&!Thus, f o r example, k i n k i n t e r a c t i o n energy as a f u n c t i o n o f separ%tion, 5,. w i t h n+ = n- one f i n d s , f o r yo >> E l
-
-
9
with I K i l l where K changes w i t h y according t o eq. (7.5). T h i s expression represents a s l i g h t fhprovement on ppevious continuum estimates o f k i n k / a n t i k i n k forces and i s i n good agreement w i t h d i s c r e t e simulations. [21] Thus f a r we have t r e a t e d the polaron as a s t a t i c c o n f i g u r a t i o n , a s o l u t i o n t o the a d i a b a t i c mean f i e l d Hamiltonian, H i n eq. (1). For many experimental applications, i t i s v i t a l t o understand t h e dynamics o f k i n k / a n t i - k i n k and kinkA f i r s t step i n t h i s d i r e c t i o n i s t o r e i n s t a t e t h e polaron i n t e r a c t i o n s . l a t t i c e k i n e t i c energy term, AHw e x p l i c i t l y dropped i n d e r i v i n g H. I n t h e i t i o n a l term o f t h e form continuum l i m i t t h i s leads t o an
where M i s the mass o f a s i n g l e CH u n i t (" 13 m , w i t h m t h e p r o t o n ' s mass), a is i s the lattice,, spacing, and CL i s a constant &hose v a h e , i n trans-(CH) roughly 4.1 eV/A. [3] Unfortunately, t h e (time-dependent) e q u a t i o n s a t fd low from H z H + AH have not been solved f o r n o n - t r i v i a l l y time-dependent A, and thus dhe can asL&t say nothing d e f i n i t i v e about k i n k and polaron dynamics i n the continuum e l e c t r o n phonon models. However, some i n t r i g u i n g p a r t i a l r e s u l t s i n t h i s area4can be obtained i n two ways. F i r s t , one can r e f e r t o the phenomenological @ models [8,9] o f trans-(CH)x t o discuss dynamics. Here one f i n d s
202
A.R. BISHOP, D.K. CAMPBELL
t h a t c o l l i s i o n s between g4 kinks (and polarons [17,22]) have a f a s c i n a t i n g and complex s t r u c t u r e which suggests p o s s i b l y i n t e r e s t i n g e f f e c t s i n t r a n s p o r t and Much o f t h i s s t r u c t u r e i n o4 appears due t o a recombination processes i n ( C H I k i n k shape o s c i l l a t i o n mode. [yi3] Since arguments f o r a s i m i l a r mode i n t h e continuum (CH) theory have r e c e n t l y been advanced, [24] a s i m i l a r complex dynamics might'be expected. Second, f o r small v e l o c i t y kinks and polarons, one can crudely estimate kinematic e f f e c t s by simply making a G a l i l e a n boost o f t h e s t a t i c s o l u t i o n and r e t a i n i n g o n l y lowest order, nonvanishing terms i n the v e l o c i t y ; i n e f f e c t , t h i s gives an approximation t o the s o l i t o n ' s i n e r t i a l mass and thus gives some idea o f i t s r o l e i n t r a n s p o r t phenomena. For k i n k s o l i t o n s , the r e s u l t s are given i n r e f s . 3 and 6. For t h e polaron, r e p l a c i n g y by = y-v t i n (6) and e v a l u a t i n g the expression i n (14) gives P
y
m
J -m
4 { sech4 KO(;
4 + yo) + sech KO(;
-
yo)
a
2 2 -2 sech KO(? + yo) sech Ko(y
-
yo))
.
Evaluating the i n t e g r a l s (using (7.b) c r u c i a l l y ) and d e f i n i n g M as the t o t a l P c o e f f i c i e n t o f v2 i n (15) leads t o
P
2
(Kl)~ ( OK Fv )
M =- ( K v ) 16a M2 o F
where
f(KoVF)
=3 8
-
2
tn
moAo3 (KoVF 1
' t VF}
($+KovF) 'O-~O'F
(16.b)
.9 m , which shows t h a t t h e polaron has an Using eq. (12), we f i n d t h a t m even smaller e f f e c t i v e mass t h a n k h e k i n t , f o r which mK 5me. [3,6] POLARONS I N =-(CH),
From the perspective o f the nonlinear e x c i t a t i o n s we are discussing, t h e most and Cis isomers o f (CH) i s that for important d i f f e r e n c e between t h e cis-(CHIx, the two chemical s t r u c t u r e s corresponding t o t h e p h a t i v e ground s t a t e do not have the same energy. Thus t h e r e i s a unique, nondegenerate ground s t a t e for-=-(CH) This s i t u a t i o n i s shown i n Figure 3. An immediate consequence i s t h a t si#ce t h e r e are n o t two degenerate ground s t a t e s f o r k i n k s o l i tons t o connect, there are no k i n k s o l i t o n s o l u t i o n s i n G - ( C H ) ! A p h y s i c a l l y i n t u i t i v e understanding o f t h i s s i t u a t i o n f o l l o w s by consideving what would happen i f one t r i e d t o create a k i n k / a n t i - k i n k p a i r from t h e ground s t a t e . Cons i d e r t h e case r e l e v a n t t o &-(CH)x where 6E/2 i s small (see Fig. 3) so t h e
trans
.
-
-
POLARONS IN POLYACETYLENE
203
Figure 3: (a) and (b) - The two schematic band s t r u c t u r e s corresponding t o t h e actual cis-(CH) ground s t a t e ((a), the c i s - t r a n s o i d c o n f i g u r a t i o n ) and t h e m e t a s A b l e s t a t e ((b), the &-'configuration.) A p l o t f o r *-(CH) o f t h e energy per u n i t l e n g t h versus band (c) gap parameter_ A f o r s p a d a l l y constant A. Note t h e unique ground s t a t e a t A = A. and the metastable s t a t e o f A = Ams.
-
metastable and ground s t a t e have almost the same energy per u n i t length, One could then imagine c o n s t r u c t i n g a c o n f i g u r a t i o n t h a t i n t e r p o l a t e d between A and A f o r a s p a t i a l distance d, andothen (an a n t i - k i n k , say), remalned i n A and A (a kin@. This c o n f i g u r a t i o n would thus l o o k i l t e r p o l a t e d between A Fo% f i n i t e d, the c o n f i g u r a t i o n would have f i n i t e l i k e eq. (6), w i t h d =mfy f o r d + 0)) i t s energy would become i n f i n i t e l i k e energy, b u t since bE/E (6E/E).d. I n t h i s sense, one can say t h a t the p u t a t i v e l i n k s i n c&-(CH) are "confined." O f course, r e f e r r i n g t o eg. (6) we see t h a t permanently confingd KK only polaronp a i r s have t h e same s t r u c t u r e as polarons: Thus, i n &-(CH),, nonlinear e x c i t a t i o n s w i l l e x i s t .
. >8,
204
A. R. BISHOP, D.K. CAMPBELL
To make t h i s precise, i t i s very useful t o study an e x p l i c i t model o f e - ( C H ) (due t o Brazovskii and Kirova [16]) which i s both a n a l y t i c a l l y solvable an8 r e l a t e d , i n a c e r t a i n l i m i t , t o the model we have examined-for trans-(CH) . One assumes [16] t h a t the gap parameter can be w r i t t e n as A(y) + Ax, w h e r e m i s s e n s i t i v e t o e l e c t r o n feedback (as f o r t r a n s ) b u t A 'is a co8s t a n t , J x t r i n s i c component.[25] This ansatz can be motivated i n t%rms o f the e f f e c t s a r i s i n g from molecular o r b i t a l s o t h e r than t h e one included e x p l i c i t l y i n eq. (11, (21, and (3). [16]
=m)
The mean f i e l d Hamiltonian f o r @-(CHIx
then becomes [16]
Although i n t h e i n t e r e s t o f s i m p l i c i t y we. have n o t i n d i c a t e d t h i s e x p l i c i t l y , i t i s very important t o note t h a t a l l the physical parameters i n cis-(CH) l a t t i c e spacing, e f f e c t i v e e l e c t r o n phonon coupling, band width, fer?iirvelocfty, band gap - can have d i f f e r e n t values from those_ i n trans. Comparing (1) and (17) we see t h a t w i t h the replacement A(y) + A(y) m y ) + A t h e e l e c t r o n equations f o r t r a n s are the same as f o r c i s . Thus, m u t a t j s mutandis, t h e s t r u c t u r e o f the e m o n spectrum and the e i g e n f u n c t i o n s b e m s a m e . The gap equation, however, i s modified t o read
-
which leads t o d i f f e r e n t c o n s t r a i n t s on s o l u i i o n s afid on bound s t a t e eigenvalues. S p e c i f i c a l l y , the ground s t a t e has the form A(y) = A. = Ai + Ae where
A:
5
-1
(19. a)
Wc exp(-Ac )
and (19. b)
Here f o r c l a r i t y we have i n d i c a t e d e x p l i c i t l y t h a t f o r c & - ( C H I t h e dimensionsee eq. (4.b) and f u l l band width, W , a#e n o t necessarl e s s coupling, A, i l y the same as f o r Z - ( C H ) It can be shown d i r e c t b t h a t t h e r e a r e no k i n k s o l u t i o n s t o the gap equatfon (la), i n c o n f i r m a t i o n o f our e a r l i e r i n t u i t i v e arguments. For polaron c o n f i g u r a t i o n s , however, we expect s o l u t i o n s , and indeed we f i n d t h a t e x a c t l y t h e same f u n c t i o n a l form given by equations (7) can s a t i s f y t h e continuum equations f o r c&-(CH) , provided t h a t K ( o r u ) i s chosen appropriately. To determine K we f o l l c h our previous pr8cedur8 and again convert t h e gap equation t o a miRimization problem for the energy o f the f u l l i n t e r a c t i n g e l e c t r o n phonon system. For fi-(CH), t h e energy expression becomes
-
.
-
-
-
POLARONS IN POL YACETYLENE
~oy[tanh-l(Ko~F~o)-KovFfio]
+
.
where y = A /A Again i n t r o d u c i n g we can w r i t 8 tfie'equation minimizing E
o
=
205
sine
(n+
-
n- + 2)
-e
such t h a t K v = A s i n 0 and wo = iocosO, w i t h respect €0 8 9s N
P
-
3.
y tane
, o 5 0 5 n/2
.
I n the l i m i t o f zero e x t r i n s i c gap, A = 0, y = 0, and as expected eq. (21) reduces t o the trans-(CH) r e s u l t o f 8s. (11). For y > 0, eq. (21) unlike equation (11) - h a s s o l u t f o n s f o r a l l combinations o f n+ and n-. These are summarized i n Table I f o r the s p e c i f i c c a s e o f y = 1.
-
Table I
-
-
n-
1
4
!
0
0
2
+2
0.71
1.43
BIPOLARON (h,h)
0
1
1
+1
0.38
0.98
POLARON (h)
0
2
0
0
0
0
GROUND STATE
1
0
3
+1
0.95
1.96
TRIPOLARON (h, h,e)
1
1
2
0
0.71
1.43
BIPOLARON (e-h)
1
2
1
-1
0.38
0.98
POLARON (e)
2
0
4
0
1.11
2.36
QUADRIPOLARON
2
1
3
-1
0.95
1.96
TRIPOLARON (e,e,h)
2
2
2
-2
0.71
1.43
BIPOLARON (e,e)
n+
g
o
INTERPRETATION
(e,e,h,h)
-
The c&-(CH) polaron states f o r various values o f N = n+ n- + 2 f o r t h e case o f y = 1. f = 2 nn+ i s the e l e c t r i c charge o f t h e state, 0 i s the angle defined i n t h e t e x t (given here i n radians), and E fi gives the f u l l energy o f the e x c i t a t i o n . The i n t e r p r e t a t i o n column gives t h g g h e r a l nature o f t h e e x c i t a t i o n (polaron, bipolaron, ...) and t h e type ( e l e c t r o n 5 e, hole 5 h).
-
-
trans-
O f p a r t i c u l a r i n t e r e s t are the "bipolaron" s o l u t i o n s i n which n+ = n-. I n ( C H I the analogous s o l u t i o n s were i n f i n i t e l y separated KK p a i r s . Indeed, one can %ee from eq. (20) t h a t y e s s e n t i a l l y plays the r o l e o f a confinement parame t e r , leading t o a term i n the energy which increases l i n e a r l y ( r e c a l l eq. (7.b)) w i t h the k i n k / a n t i - k i n k "separation," 2y0. This i s the e x p l i c i t r e a l i z a t i o n o f our e a r l i e r h e u r i s t i c argument about k i n k confinement.
206
A.R. BISHOP, D.K. CAMPBELL
From Table I we see t h a t both the polaron w i d t h (2yo) and depth increase w i t h n-. increasing occupation number, N = n+ + 2
-
i s largely As i n t h e case o f trans, the problem o f polaron dynamics i n &-(CHI open. One can, o f u r s e , estimate t h e i n e r t i a l mass as before, ht the more i n t e r e s t i n g possible e f f e c t s o f c o l l i s i o n dynamics - recombination, long-1 i v e d have thus f a r eluded q u a n t i t a t i v e d e s c r i p t i o n . oscillatory states
-
IMPLICATIONS AND DISCUSSION Since polarons have o n l y r e c e n t l y been discovered [13,16,17] i n the t h e o r e t i c a l the p o s s i b l e i m p l i c a t i o n s i n p a r t i c u l a r , f o r experiment - o f models f o r (CH),, t h e i r existence are n o t y e t completely understood. Obviously the existence o f polarons and kinks i n trans-(CH) , contrasted w i t h t h e existence o f ( m u l t i - ) polarons only i n provrdes a n a t u r a l d i s t i n c t i o n between these two isomers. To understand t i e q u a n t i t a t i v e i m p l i c a t i o n s o f t h i s d i s t i n c t i o n f o r experiment, however, f u r t h e r t h e o r e t i c a l studies s t r e s s i n g t h e dynamics o f k i n k and polaron i n t e r a c t i o n s are needed. Two aspects o f dynamics seem p a r t i c u l a r l y crucial. F i r s t , w i t h i n the time-de endent mean f i e l d approximation (see eqns. (1) and (14)), the nature o h n d kink-polaron i n t e r a c t i o n s must be understood. Second, one must go beyond the mean f i e l d approximation t o study the f u l l quantum dynamics o f (CH) . Although t h e r e i s , as y e t , l i t t l e quantit a t i v e progress i n e i t h e r o f the& areas, t h e r e c e n t l y noted connection between model r e l a t i v i s t i c f i e l d t h e o r i e s and the continuum theory o f (CH) o f f e r some q u a l i t a t i v e suggestions. [17] F i r s t , t h i s connection suggests that! l o n g - l i v e d o s c i l l a t o r y s t a t e s may e x i s t i n (CH) and t h a t these may be important i n (CH) dynamics. Second, known r e s u l t s on 4uantum dynamics i n f i e l d theory r a i s e t h g p o s s i b i l i t y t h a t quantum f l u c t u a t i o n s can d e s t a b i l i z e t h e polaron i n (CHI , causing i t t o disappear from the theory. Although p r e l i m i n a r y quantum Monte Carlo studies [26] do show l a r g e quantum f l u c t u a t i o n s i n (CHI , a n a l y t i c e s t i mates [24] o f the f i r s t order quantum c o r r e c t i o n s i n d i c a t e t h a t ( t o t h i s order, a t l e a s t ) the polaron i s n o t d e s t a b i l i z e d . C l e a r l y , f u r t h e r q u a n t i t a t i v e work on these problem i s required.
-
&-(m
Despite the absence o f c r i t i c a l q u a n t i t a t i v e d e t a i l , i t i s p o s s i b l e t o i n d i cate a number o f i n t e r e s t i n g , p o t e n t i a l experimental i m p l i c a t i o n s o f polarons. F i r s t , on energy grounds [16,17] one knows t h a t s i n g l e e l e c t r o n s added by should form polarons. When doping o r by i n j e c t i o n - t o c i s - o r trans-(CH) many electrons are added, i t i e n e r g e t m l y fa6orable f o r them t o form k i n k / o r multi-polarons ( i n Thus f o r --(CH) , anti-kink pairs ( i n the very l i g h t l y doped m a t e r i a l an average o f 5 one e l e c t r o n ( o r hole) pgr (CH) chain might r e f l e c t polaron t r a n s p o r t p r o p e r t i e s , whereas the more h e a v f l y q e d material would e x h i b i t (charged) k i n k t r a n s p o r t behavior. In Unfortunately t h i s simple cis-(CH),, one would n o t expect t h i s d i f f e r e n c e . p i c t u r e w i l l be complicated by, among o t h e r features, t h e presence o f n e u t r a l kinks "quenched" i n t o undoped --(CHI samples and the u n c e r t a i n t i e s over isomerization from c i s t o t r a n s upon dopifig. Nonetheless, i f polarons are present i n l i g h t l y dop=e-and=-(CH) , then t h e i r o p t i c a l absorption [27] i s s u f f i c i e n t l y d i f f e r e n t from t h a t o f d n k s [28] t h a t experiments comparing "mid-gap" and i n f r a r e d absorption should be able t o d e t e c t t h e d i f f e r e n c e . [27]
-
trans)
-
-
cis).
7
The d i s t i n c t i o n between allowed nonlinear e x c i t a t i o n s i n c i s and trans-(CH) suggests [16] a p l a u s i b l e scenario f o r t h e experimental l y o b s e r v e d m conX t r a s t s between these isomers upon p h o t o i n j e c t i o n ( e x c i t a t i o n o f an electron-hole p a i r by intense r a d i a t i o n ) . I n trans-(CH) , p h o t o i n j e c t i o n leads t o a photoc u r r e n t which has been i n t e r p r e t e n 6 , 2 9 l X a s r e s u l t i n g from a charged kink/ anti-kink pair. I n cis-(CH) , one observes instead a photoluminescence, suggesting the r e c o m b i n m o n o? t h e e-h p a i r which c o u l d f o l l o w because o f t h e e f f e c t i v e confinement o f the p u t a t i v e kinks ( t h e l o c a l i z a t i o n o f m u l t i - p o l a r o n
POLARONS IN POL YACETYLENNE
207
states i n c i s aids recombination). P r e c i s e l y the p r e v i o u s l y discussed theoreti c a l dynamical studies are what i s r e q u i r e d t o make t h i s q u a l i t a t i v e p i c t u r e more precise. I n conclusion, we r e i t e r a t e t h a t recent t h e o r e t i c a l developments [13,16,17] have shown t h a t t h e nature o f nonlinear e x c i t a t i o n s i n polyacetylene may be even more i n t r i g u i n g and complex than previously believed. The c e n t r a l remaining chald e f i n i t i v e l y t h e relevance o f t h i s elegant lenge i s t o e s t a b l i s h - o r disprove t h e o r e t i c a l s t r u c t u r e t o t h e r e a l material.
-
ACKNOWLEDGEMENTS We are g r a t e f u l t o many colleagues f o r t h e i r advice and encouragement. Special thanks are due t o J. L. Bredas, J. R. S c h r i e f f e r , and W. P. Su f o r enlightening discussions o f t h e i r numerical simulations and t o S. Etemad f o r h i s shared i n s i g h t s i n t o the experimental s i t u a t i o n . REFERENCES See f o r example, D. Bloor, Proc. Amer. Chem. S O C . , Houston, A p r i l 1980; t o appear i n J. Chem. Phys. For recent reviews see a r t i c l e s i n Proc. I n t . Conf. on Low-Dimensional Synt h e t i c Metals (Helsingor, Denmark, August 1980) i n Chemica S c r i p t a 2 (1981); Physics i n One Dimension, eds. J. Bernasconi and T. Schneider (Springer Verlag, 1981); and A. J. Heeger and A. G. MacDiarmid, p. 353-391 i n The Physics and Chemistry o f Low Dimensional Solids, ed. L. Alchcer (Reidel , 1980). Su, W. P., S c h r i e f f e r , J. R., and Heeger, A. J., Phys. Rev. L e t t . (1979); Phys. Rev. B 2 , 2099 (1980).
42,
1698
42, 408 and 416 (1977). Brazovskii, S. A., JETP L e t t e r s 28, 606 (1978) (trans. of Pisma Zh. Eksp. Teor. F i z . 28, 656 (1978)); and Soviet Phys. JETP 51, 342 (1980) (trans. o f Zh. Eksp. Teor. Fiz. 'a, 677 (1980)). Kotani, A.,
J. Phys. SOC. Japan
Takayama, H., Lin-Liu, Y. R. and Maki, K., Phys. Rev. B 3,2388 (1980); Krumhansl, J. A., Horovitz, B. , and Heeger, A. J., S o l i d State Commun. 2 , 945 (1980); Horovitz, B., S o l i d State Commun. 2 , 61 (1980). Horovitz, B.
,
Phys. Rev. L e t t .
Rice, M. J., Phys. L e t t . 7lJ,
5 , 742
(1981).
152 (1979).
Rice, M. J., and Timonen, J., Phys. L e t t .
El 368
(1979).
[lo] Mele, E. J., and Rice, M. J., i n Chemica S c r i p t a IJ, 21 (1981). [ll] See a r t i c l e s by Etemad S., and Rice, M. J., these proceedings. [12] Jackiw, R. and Rebbi, C.,
Phys. Rev. D 13, 3398 (1976); Jackiw, R., S c h r i e f f e r , J . R., Nuc. Phys. B 190, 253 (n81).
[13] Su, W. P., and S c h r i e f f e r , J. R., Phys. Rev. L e t t .
5 , 738
and
(1981).
[14] Rice, M. J., and Mele, E. J., " P o s s i b i l i t y o f S o l i t o n s w i t h charge * e / 2 i n Highly Correlated 1:2 Salts o f TCNQ," Xerox Webster p r e p r i n t , 1981.
A.R. BISHOP, D.K. CAMPBELL
208
[15] Su, W. P., and S c h r i e f f e r , J. R., Proc. Nat. Acad. Sci. 77, 5526 (Physics) (1980); Bredas, J. L. , Chance, R. R., and Silbey, R., Proceedings o f I n t e r -
national Conference on Low-Dimensional Conductors, Molecular C r y s t a l s and L i q u i d Crystals (Gordon Breach), t o be published.
[16] Brazovskii, S. , and Kirova, N.,
Pisma Zh. Eksp. Teor. Fiz.
[17] Campbell, D. K. and Bishop, A. R . , Phys. Rev. B B ( i n press).
24,
2,
6 (1981).
4859 (1981); Nuc. Phys.
[18] Following the standard conventions i n t h e continuum model, we use u n i t s w i t h h = 1. [19] P e i e r l s , R. E., Quantum Theory o f S o l i d s (Clarendon Press, Oxford, 1955) p. 108; Allender, D. Bray, J. W., and Bardeen, J . , Phys. Rev. B. 9, 119 (1974). [20] Bishop, A. R.,
S o l i d State Commun.
Y. R., unpublished.
[21] Lin-Liu,
and Maki, K.,
2 , 955
(1980).
Phys. Rev.
822
5754 (1980); Kivelson, S.
[22] The phenomenological
$4 theories, i f coupled t o a phenomenological e l e c t r o n f i e l d describing o n l y the l o c a l i z e d , "gap" s t a t e s , do c o n t a i n b o t h the k i n k s o l i t o n s ( o f a l l charges) and t h e polaron.
[23] Campbell, 0. K., and Wingate, C., [24] Nakahara, N.
, and
Maki, K.,
i n preparation.
p r e p r i n t (1981).
[25] I n another context t h i s ansatz has been discussed by Rice; see Rice, M. S., Phys. Rev. L e t t . 37, 36 (1976). [26] Su, W. P.,
unpublished.
[27] Fesser, K., Bishop, A. R . , and Campbell, 0. K.
,
i n preparation.
1281 Gammel, 3. T. and Krumhansl, J. A. Phys. Rev. B 24, 1035 (1981). Maki, K. and Nakahara, N., Phys. Rev. B 23, 5005 (1981E Horovitz, B., p r e p r i n t (1981); Kirelson, S. , Lee, T.-Y. , T i n - L i u , Y. R., Peschel, I. , and Yu, L . , p r e p r i n t (1981). [29] Etemad, S. , these Proceedings.
195
A.R. Bishop, D.K. Campbell, B. Nicolaenko (4s.)
0 North-Holland Publishing Company, 1982
POLARONS I N POLYACETYLENE
Alan R. Bishop and David K. Campbell Theoretical D i v i s i o n and Center f o r Nonlinear Studies Los Alamos National Laboratory Los Alamos, NM 87545 USA Recent t h e o r e t i c a l studies o f polyacetylene, (CH) have focussed on k i n k - l i k e s o l i t o n s i n the trans isomer. I t i s stown t h a t t h e same t h e o r e t i c a l models a l s o p r e m o l a r o n - l i k e s o l i t o n s i n both c i s - and W - ( C H ) . The e x p l i c i t f o r m h e s e polarons, t h e i r r e l a t i o n t o kinks, p o & i b l e i m p l i c a t i o n s f o r experiment, and open questions are discussed. INTRODUCTION During the l a s t few years the extensive t h e o r e t i c a l and experimental i n t e r e s t i n nonlinear e x c i t a t i o n s i n polyacetylene ((CH) ) has focused almost e x c l u s i v e l y on k i n k - l i k e s o l i t o n states. [l-101 This i s x h a r d l y s u r p r i s i n g , f o r a p a r t from t h e i r possible d i r e c t experimental i m p l i c a t i o n s [2,11] f o r t r a n s p o r t properties, doping mechanisms, and t h e observed semi-conductor metal t r a n s i t i o n i n (CHI , these k i n k s o l i t o n s , w i t h t h e i r b i z a r r e spin/charge assignments, [3] have s t i m b l a t e d t h e o r e t i c a l work [12,13,14] on t h e existence and r o l e o f " f r a c t i o n a l charge'' i n both s o l i d s t a t e systems and f i e l d theory models. Recently, however, i t has been discovered [15,16,17] t h a t t h e k i n k s o l i t o n s are not the only nonlinear e x c i t a t i o n s p r e d i c t e d by t h e o r e t i c a l models o f (CH)x. Indeed, i n both (numerical studies o f ) l a t t i c e models [15] and ( a n a l y t i c studies o f ) continuum theories, [16,17] a nonlinear "polaron" e x c i t a t i o n has been found. Although more conventional i n i t s p r o p e r t i e s than the k i n k s o l i t o n , the polaron, as we s h a l l see, i s the lowest energy e x c i t a t i o n a v a i l a b l e t o a s i n g l e electron. Experimentally, the polaron may thus p l a y an important r o l e i n t h e doping process o r i n e l e c t r o n i n j e c t i o n and, t h e o r e t i c a l l y , i t s presence f u r t h e r embelOne p a r t i c u l a r l y s i g n i l i s h e s the already r i c h s t r u c t u r e p r e d i c t e d f o r (CH) f i c a n t feature o f the polaron i s t h a t it, u n l i k e t h e k i n k s o l i t o n which can e x i s t only i n the trans isomer o f ( C H I , i s p r e d i c t e d t o occur i n both c i s and trans isomers. The possible i m p l i c a t i h s o f t h i s f o r comparative e x p e r x n t a l s t u d i e s o f cis and trans a r e o f great c u r r e n t i n t e r e s t . Thus i n t h i s a r t i c l e we s h a l l focus on t h i s o r e conventional" b u t nonetheless i n t e r e s t i n g nonlinear e x c i t a t i o n , describing i t s predicted nature i n d e t a i l and discussing p o t e n t i a l experimental imp1 i c a t i o n s b r i e f l y .
.
POLARONS I N m - ( C H ) , From the l a t t i c e model [3] f o r trans-(CH) , a standard sequence o f approximat i o n s [6] leads i n the c o n t i n u u m m i t 61an e f f e c t i v e a d i a b a t i c mean-field Hamiltonian i n terms o f t h e ( r e a l ) gap parameter A and e l e c t r o n f i e l d Y. The result i s
A.R. BISHOP, D.K. CAMPBELL
196
where w2/g2 i s t h e n e t e f f e c t i v e electron-phonon coupling constant, [3,6] ai i s
Q
the i t h Pauli matrix, V(y) =
(!$;),
and vF i s t h e Fermi v e l o c i t y .
[18]
For
l a t e r purposes we note t h a t i n e r i v i n g t h i s mean f i e l d Hamiltonian, t h e l a t t i c e k i n e t i c energy which would add a term p r o p o r t i o n a l t o A2(y) i n (1) has been e x p l i c i t l y ignored. Obviously, t h i s w i l l have no e f f e c t on the s t a t i c e x c i t a t i o n s we discuss below, b u t i t . w i l l prove s i g n i f i c a n t f o r on-going studies i n v o l v i n g the dynamics o f s o l i t o n s i n (CH),.
-
-
V a r i a t i o n o f H leads t o the s i n g l e p a r t i c l e e l e c t r o n wave f u n c t i o n equations
and the s e l f - c o n s i s t e n t gap equation
The summation i n (3) i s over occupied e l e c t r o n s t a t e s and s i s a s p i n l a b e l (suppressed i n (1) and (2)). Equations (2) and (3) are t h e continuum e l e c t r o n phonon equations f o r t r a n s (CH) , [5,6] and remarkably one can f i n d a n a l y t i c , closed form e x p r e s s i o f i r seveval n o n t r i v i a l solutions. [5,6,16,17] Unsurprisingly, t h i s i s r e l a t e d t o the ( p a r t i a l ) s o l i t o n features o f these equations. [16,17] Although we wish t o focus on the polaron s o l u t i o n s t o (2) and (3), f o r c l a r i t y and completeness we s h a l l b r i e f l y review t h e other solutions. F i r s t , the ground s t a t e o f t h e i n f i n i t e chain o f --(CH) e r a t e (see Figure l), and i s described by A(y) = + A. A,
= W exp (-A -1)
i s t w o - f o l d degeno r Afy) = A -,, w i t h [5,6]
(4. a)
where
The parameters i n (4) include W, t h e f u l l one-electron bandwidth (e 10 eV i s trans-(CHI 1; v , t h e Fermi v e l o c i t y (vF = aW/2, [ l a ] where a i s t h e underi n =-(CH),); and, as i n d i c a t e d p r e v i o u s l y , l a t d c e sFacing (= 1.22
POLARONS IN POL YACETYLENE
197
u2/g2, the n e t e f f e c t i v e electron-phonon coupling constant (whose value i s i n eqfect determined by (4)). Since experimentally A. 0.7 eV, eqn. (4.a) implies the dimensionless coupling A 0.4.
.. H
H
H I
H I
...c A+ /;\ /\ /\ c.. .
Figure 1: (a) two (c) gap A =
I
CI
CI
CI
I
H
H
H
H
H
-
and (b) The two schematic bond s t r u c t u r e s corresponding t o t h e degenerate ground states f o r =-(CH) A p l o t f o r =-(CH) o f the energy ber u n i t l e n g t h versus band parameter A f o r spatialTy constant A. Note the degeneracy between f A and t h e l o c a l maximum ( i n d i c a t i n g i n s t a b i l i t y ) a t A = 0.
-
.
0
the s i n g l e e l e c t r o n energy spectrum i s given by c(k) = Fig. 2) and the corresponding e l e c t r o n wave f u n c t i o n s a r e simply plan8 waves, the e x p l i c i t forms o f which are given i n r e f s . [5,6, and 171. Physically, A # 0 implies a gap (= 2A , see Fig. 2) around the fermi surface i n t h e s i n g f e e l e c t r o n spectrum; t h a t q h i s gap should e x i s t i n t h e continuum model o f ( C H I i s a d i r e c t consequence o f t h e well-known P e i e r l s i n s t a b i l i t y [19] o f one-'bimensional coupled electron-phonon systems. I n chemical terms, A, # 0 indicates a "bond a l t e r n a t i o n " between s i n g l e and double bonds (see Fig. l), and i t s value i s p r o p o r t i o n a l t o t h e lengthening (shortening) o f a s i n g l e (double) bond w i t h respect t o the i d e a l i z e d uniform bond l e n g t h which would occur were A, = 0. For
-4
f
constant A
(s8;
A. R. BISHOP, D. K. CAMPBELL
198
a) -A0
0
-2
2
Figure 2: The s p a t i a l s t r u c t u r e o f t h e gap parameter (A(y)) and t h e associated e l e c t r o n i c l e v e l s f o r --(CH) f o r (a) the ground s t a t e ; (b) the k i n k s o l i t o n ; (c) t h e polaron. foblid l i n e s i n t h e l e f t - h a n d drawings i n d i c a t e A(y); dashed l i n e s show e l e c t r o n d e n s i t i e s f o r l o c a l i z e d states. The right-hand drawings show the e l e c t r o n i c l e v e l s i n the conduction band ( s i n g l e shading), valence band (crossed shading), and gap states. Q = charge and S = spin. The two-fold degeneracy o f t h e ground s t a t e o f trans-(CH) has the very import a n t consequence t h a t a l o c a l i z e d , f i n i t e e n e r f i n k sditon corresponding can e x i s t ; t h i s k i n k (K) i n t e r p o l a t e s between t o a l o c a l i z e d l a t t i c e "defect" the degenerate ground s t a t e s (see Fig. 2) and has t h e e x p l i c i t form
-
AK(y) = Aotanh VFy/Ao
.
-
(5)
There i s a l s o an a n t i - k i n k (k) s o l i t o n with Ak(y) = -A (y). The associated s i n g l e e l e c t r o n energy l e v e l s c o n s i s t o f extended c o n d u c t i h (E(k) = + 4 and valence (E(k) = J v q band s t a t e s [5,6,17] which a r e modffied' ( e s s e n t i a l l y "phase s h i f t e d ) fr8m t h e plane waves o f t h e ground s t a t e and an The e x p l i c i t a d d i t i o n a l l o c a l i z e d "mid-gap" s t a t e ( a t E = 0, see Fig. 2). form o f the wave f u n c t i o n f o r t h i s l o c a l i z e d h a t e i s
-
POLARONS IN POL YACETYLENE
Uo(y) = No sech A0y/vF
and Vo(y) = - i N o sech Aoy/vF
w i t h No =
For e i t h e r K o r i,t h i s "mid-gap" leading t o l o c a l i z e d e x c i t a t i o n s 6,121 i n d i c a t e d i n Fig. 2. It i n t e r e s t i n " f r a c t i o n a l charge."
s t a t e can be occupied by 0, 1, o r 2 electrons, w i t h t h e b i z a r r e spin/charge assignments [3,5, i s these states t h a t have e x c i t e d the recent [12-141
One v i t a l consequence o f t h e k i n k s o l i t o n ' s i n t e r p o l a t i n g between t h e degenerate ground states i s t h a t t h e r e i s a r e s t r i c t i o n u s u a l l y c a l l e d a " t o p o l o g i c a l on t h e production o f kinks. S p e c i f i c a l l y , i n the i n f i n i t e c o n s t r a i n t " [12,17] chain polymer, kinks can be excited from the ground s t a t e o n l y i n p a i r s o f k i n k and a n t i - k i n k . The i n t e r e s t e d reader should convince himself o f t h i s by showing t h a t the production o f a s i n g l e k i n k from t h e ground s t a t e t h a t i s , going from the A(y) c o n f i g u r a t i o n i n Fig. 2a t o t h a t i n Fig. 2b requires overcoming an i n f i n i t e ( f o r an i n f i n i t e l e n g t h polymer) energy b a r r i e r .
-
-
-
-
F i n a l l y , we come t o the polaron s o l u t i o n t o equations (2) and ( 3 ) . The r e c e n t l y discovered [16,17] a n a l y t i c form f o r the gap parameter o f t h e polaron can be w r i t t e n i n t h e revealing form (c.f., eq. (5))
where tanh 2Koyo = KovF/Ao
.
(7. b)
This c o n f i g u r a t i o n f o r A(y), which corresponds t o a s p a t i a l l y l o c a l i z e d deviat i o n from t h e ground s t a t e value (here chosen t o be + A ), i s sketched i n Fig. 2c. Again t h e s i n l e e l e c t r o n states include extended s l a t e s i n t h e conduction and valence (E(k) = q-4 bands, which s t a t e s are (&(k) = + phase-shifted by th% polaron "defect"; t h e expl i c f t ?orms o f t h e corresponding wave functions are given i n r e f . 17. I n addition, two l o c a l i z e d e l e c t r o n i c states, w i t h energies symmetrically placed, form i n t h e gap a t &+ 5 f uo where
-
-
ug
-J
=
The e l e c t r o n wave functions f o r these l o c a l i z e d s t a t e s are, f o r Uo(Y) = No
and
I ( l - i ) s e c h Ko(y+yo)
+
(l+i)sech Ko(y-yo)]
E
= +wo
A. R. BISHOP, 0.K. CAMPBEL L
200
V,(Y)
= No
where No = 4&, 1
{ ( l + i ) s e c h Ko(y+yo) + (1-i)sech Ko(y-yo)] and f o r &=-wO
u-o
5
= +iVo and V-,
UIE=-,,,
(9b) :VIE=-u
0
= -iUo. 0
-
-
I t i s important t o note t h a t the polaron c o n f i g u r a t i o n f o r A(y) eq. (6) and the associated e l e c t r o n wave functions s a t i s f y the e l e c t r o n p a r t eq. (2) of the continuum equations f o r 9 KovF i n t h e allowed range 0 5 KovF 5 A., It i s
-
-
-
-
the s e l f - c o n s i s t e n t gap c o n d i t i o n eq. ( 3 ) which determines t h e s p e c i f i c value o f K v f o r an actual s o l u t i o n t o t h e coupled equations. Using some aspects o f %.o!iton theory, [17] one can, i n e f f e c t , convert t h e gap equation t o a s t r a i g h t f o r w a r d minimization problem. Apart from s i m p l i f y i n g t h e problem t e c h n i c a l l y , t h i s i s very appealing i n t u i t i v e l y , f o r the q u a n t i t y being m i n i mized i s e s s e n t i a l l y t h e ener o f t h e f u l l i n t e r a c t i n g electron-phonon system. I n p a r t i c u l a r , one f i n d s d e polaron-type c o n f i g u r a t i o n i n =-(CH), that t Ep(n++' n-, KO) = (n+
-
n- + 2)w0 +
4
KovF
-
-n4 wo t a n
-
where n are r e s p e c t i v e l y t h e occupation numbers which can be o n l y 0, 1, o r 2 o f $he l o c a l i z e d s t a t e s E+ = +UJ and e- = -w kntroducing 8 such t h a t KovF = Aosin8 and (see (8)) wo = A0co& and minimizpng E w i t h respect t o 8 g i v e P
-
.
From eq. (11) i t f o l l o w s d i r e c t l y t h a t f o r a stable, l o c a l i z e d polaron s o l u t i o n the electrons must be d i s t r i b u t e d i n one o f two c o n f i g u r a t i o n s : (1) n+ = 1, n- = 2, the " e l e c t r o n polaron" s t a t e ; or (2)
n+ = 0, n- = 1, the "hole polaron" s t a t e .
Before describing these s t a t e s i n d e t a i l , we note t h a t studying o t h e r p o s s i b l e configurations reveals some i n t e r e s t i n g phenomena. When n = n- --- f o r n+ = 0, 1, o r 2 (11) i m p l i e s 6 = n/2, so Kov = A , w = 0 fcorresponding t o t h e "mid-gap" state), and by (7.b) y + a. henceothe& p u t a t i v e trans-(CH), "bipolarons" separate i n t o an i n f i R i t e l y separated k i n k / a n t i - k w p a i r s , w i t h charges determined by the value o f nt = n-. When n+ = 0, n- = 2, 6 = 0, so w = A , K v = 0, and the "unoccupied polaron collapses t o the ground s t a t e , A'. O!herockoices o f n+ and n- lead simply t o combinations o f the above e x c i t a t?ons ( i n f i n i t e l y separated i n space); f o r example, n+ = 1, n- = 0 leads t o a c o n f i g u r a t i o n o f kink, a n t i - k i n k , and polaron.
---
Returning t o t h e t r u e , s t a b l e polaron states, we see t h a t f o r n+ = 1, n- = 2 ( e l e c t r o n polaron) o r n+ = 0, n- = 1 (hole polaron) eq. (11) y i e l d s 6 = n/4 o r
POLARONS IN POL YACETYLENE
KovF =
201
A0/G= wo
(12)
From eq. (10) the energy o f t h i s e x c i t a t i o n i s seen t o be
E =@A P n o
0.90Ao
(13)
1
From e ns. (61, (7.b), and (12) one sees t h a t the s p a t i a l e x t e n t o f t h e polaron ( = 11 f o r trans-(CH) ) i s s l i g h t l y l a r g e r than t h a t o f the k i n k (= 9 A f o r W - ( C H ) );this compirison i s made v i s u a l i n Fig. 2. Although general arguments sug&sting the existence o f polarons i n systems l i k e trans-(CH) that i s , (quasi)-one-dimensional P e i e r l s dimerized chains can b e v a n c e # , [20] , i t i s encouraging t o f i n d t h a t the continuum model, w i t h o u t any ad hoc addit i o n s , e x p l i c i t l y contains these e x c i t a t i o n s . I t i s a l s o heartening t h a t numeri c a l simulations o f the l a t t i c e model [15] have revealed a polaron e x c i t a t i o n whose s t r u c t u r e i s i n good agreement w i t h t h a t p r e d i c t e d by eq. (6), (7), and (12).
-
-
Another aspect o f the polaron f i e l d c o n f i g u r a t i o n (eq. (7)) and corresponding energy expression (eq. (10)) i s worth mentioning: namely, since t h e e l e c t r o n equations eqns. (2) are s a t i s f i e d f o r a l l K i n t h e range 0 < K 5 A /v , eq. (10) can be viewed as a s i n g l e parameter (K expression f o r tfie Rink/&!Thus, f o r example, k i n k i n t e r a c t i o n energy as a f u n c t i o n o f separ%tion, 5,. w i t h n+ = n- one f i n d s , f o r yo >> E l
-
-
9
with I K i l l where K changes w i t h y according t o eq. (7.5). T h i s expression represents a s l i g h t fhprovement on ppevious continuum estimates o f k i n k / a n t i k i n k forces and i s i n good agreement w i t h d i s c r e t e simulations. [21] Thus f a r we have t r e a t e d the polaron as a s t a t i c c o n f i g u r a t i o n , a s o l u t i o n t o the a d i a b a t i c mean f i e l d Hamiltonian, H i n eq. (1). For many experimental applications, i t i s v i t a l t o understand t h e dynamics o f k i n k / a n t i - k i n k and kinkA f i r s t step i n t h i s d i r e c t i o n i s t o r e i n s t a t e t h e polaron i n t e r a c t i o n s . l a t t i c e k i n e t i c energy term, AHw e x p l i c i t l y dropped i n d e r i v i n g H. I n t h e i t i o n a l term o f t h e form continuum l i m i t t h i s leads t o an
where M i s the mass o f a s i n g l e CH u n i t (" 13 m , w i t h m t h e p r o t o n ' s mass), a is i s the lattice,, spacing, and CL i s a constant &hose v a h e , i n trans-(CH) roughly 4.1 eV/A. [3] Unfortunately, t h e (time-dependent) e q u a t i o n s a t fd low from H z H + AH have not been solved f o r n o n - t r i v i a l l y time-dependent A, and thus dhe can asL&t say nothing d e f i n i t i v e about k i n k and polaron dynamics i n the continuum e l e c t r o n phonon models. However, some i n t r i g u i n g p a r t i a l r e s u l t s i n t h i s area4can be obtained i n two ways. F i r s t , one can r e f e r t o the phenomenological @ models [8,9] o f trans-(CH)x t o discuss dynamics. Here one f i n d s
202
A.R. BISHOP, D.K. CAMPBELL
t h a t c o l l i s i o n s between g4 kinks (and polarons [17,22]) have a f a s c i n a t i n g and complex s t r u c t u r e which suggests p o s s i b l y i n t e r e s t i n g e f f e c t s i n t r a n s p o r t and Much o f t h i s s t r u c t u r e i n o4 appears due t o a recombination processes i n ( C H I k i n k shape o s c i l l a t i o n mode. [yi3] Since arguments f o r a s i m i l a r mode i n t h e continuum (CH) theory have r e c e n t l y been advanced, [24] a s i m i l a r complex dynamics might'be expected. Second, f o r small v e l o c i t y kinks and polarons, one can crudely estimate kinematic e f f e c t s by simply making a G a l i l e a n boost o f t h e s t a t i c s o l u t i o n and r e t a i n i n g o n l y lowest order, nonvanishing terms i n the v e l o c i t y ; i n e f f e c t , t h i s gives an approximation t o the s o l i t o n ' s i n e r t i a l mass and thus gives some idea o f i t s r o l e i n t r a n s p o r t phenomena. For k i n k s o l i t o n s , the r e s u l t s are given i n r e f s . 3 and 6. For t h e polaron, r e p l a c i n g y by = y-v t i n (6) and e v a l u a t i n g the expression i n (14) gives P
y
m
J -m
4 { sech4 KO(;
4 + yo) + sech KO(;
-
yo)
a
2 2 -2 sech KO(? + yo) sech Ko(y
-
yo))
.
Evaluating the i n t e g r a l s (using (7.b) c r u c i a l l y ) and d e f i n i n g M as the t o t a l P c o e f f i c i e n t o f v2 i n (15) leads t o
P
2
(Kl)~ ( OK Fv )
M =- ( K v ) 16a M2 o F
where
f(KoVF)
=3 8
-
2
tn
moAo3 (KoVF 1
' t VF}
($+KovF) 'O-~O'F
(16.b)
.9 m , which shows t h a t t h e polaron has an Using eq. (12), we f i n d t h a t m even smaller e f f e c t i v e mass t h a n k h e k i n t , f o r which mK 5me. [3,6] POLARONS I N =-(CH),
From the perspective o f the nonlinear e x c i t a t i o n s we are discussing, t h e most and Cis isomers o f (CH) i s that for important d i f f e r e n c e between t h e cis-(CHIx, the two chemical s t r u c t u r e s corresponding t o t h e p h a t i v e ground s t a t e do not have the same energy. Thus t h e r e i s a unique, nondegenerate ground s t a t e for-=-(CH) This s i t u a t i o n i s shown i n Figure 3. An immediate consequence i s t h a t si#ce t h e r e are n o t two degenerate ground s t a t e s f o r k i n k s o l i tons t o connect, there are no k i n k s o l i t o n s o l u t i o n s i n G - ( C H ) ! A p h y s i c a l l y i n t u i t i v e understanding o f t h i s s i t u a t i o n f o l l o w s by consideving what would happen i f one t r i e d t o create a k i n k / a n t i - k i n k p a i r from t h e ground s t a t e . Cons i d e r t h e case r e l e v a n t t o &-(CH)x where 6E/2 i s small (see Fig. 3) so t h e
trans
.
-
-
POLARONS IN POLYACETYLENE
203
Figure 3: (a) and (b) - The two schematic band s t r u c t u r e s corresponding t o t h e actual cis-(CH) ground s t a t e ((a), the c i s - t r a n s o i d c o n f i g u r a t i o n ) and t h e m e t a s A b l e s t a t e ((b), the &-'configuration.) A p l o t f o r *-(CH) o f t h e energy per u n i t l e n g t h versus band (c) gap parameter_ A f o r s p a d a l l y constant A. Note t h e unique ground s t a t e a t A = A. and the metastable s t a t e o f A = Ams.
-
metastable and ground s t a t e have almost the same energy per u n i t length, One could then imagine c o n s t r u c t i n g a c o n f i g u r a t i o n t h a t i n t e r p o l a t e d between A and A f o r a s p a t i a l distance d, andothen (an a n t i - k i n k , say), remalned i n A and A (a kin@. This c o n f i g u r a t i o n would thus l o o k i l t e r p o l a t e d between A Fo% f i n i t e d, the c o n f i g u r a t i o n would have f i n i t e l i k e eq. (6), w i t h d =mfy f o r d + 0)) i t s energy would become i n f i n i t e l i k e energy, b u t since bE/E (6E/E).d. I n t h i s sense, one can say t h a t the p u t a t i v e l i n k s i n c&-(CH) are "confined." O f course, r e f e r r i n g t o eg. (6) we see t h a t permanently confingd KK only polaronp a i r s have t h e same s t r u c t u r e as polarons: Thus, i n &-(CH),, nonlinear e x c i t a t i o n s w i l l e x i s t .
. >8,
204
A. R. BISHOP, D.K. CAMPBELL
To make t h i s precise, i t i s very useful t o study an e x p l i c i t model o f e - ( C H ) (due t o Brazovskii and Kirova [16]) which i s both a n a l y t i c a l l y solvable an8 r e l a t e d , i n a c e r t a i n l i m i t , t o the model we have examined-for trans-(CH) . One assumes [16] t h a t the gap parameter can be w r i t t e n as A(y) + Ax, w h e r e m i s s e n s i t i v e t o e l e c t r o n feedback (as f o r t r a n s ) b u t A 'is a co8s t a n t , J x t r i n s i c component.[25] This ansatz can be motivated i n t%rms o f the e f f e c t s a r i s i n g from molecular o r b i t a l s o t h e r than t h e one included e x p l i c i t l y i n eq. (11, (21, and (3). [16]
=m)
The mean f i e l d Hamiltonian f o r @-(CHIx
then becomes [16]
Although i n t h e i n t e r e s t o f s i m p l i c i t y we. have n o t i n d i c a t e d t h i s e x p l i c i t l y , i t i s very important t o note t h a t a l l the physical parameters i n cis-(CH) l a t t i c e spacing, e f f e c t i v e e l e c t r o n phonon coupling, band width, fer?iirvelocfty, band gap - can have d i f f e r e n t values from those_ i n trans. Comparing (1) and (17) we see t h a t w i t h the replacement A(y) + A(y) m y ) + A t h e e l e c t r o n equations f o r t r a n s are the same as f o r c i s . Thus, m u t a t j s mutandis, t h e s t r u c t u r e o f the e m o n spectrum and the e i g e n f u n c t i o n s b e m s a m e . The gap equation, however, i s modified t o read
-
which leads t o d i f f e r e n t c o n s t r a i n t s on s o l u i i o n s afid on bound s t a t e eigenvalues. S p e c i f i c a l l y , the ground s t a t e has the form A(y) = A. = Ai + Ae where
A:
5
-1
(19. a)
Wc exp(-Ac )
and (19. b)
Here f o r c l a r i t y we have i n d i c a t e d e x p l i c i t l y t h a t f o r c & - ( C H I t h e dimensionsee eq. (4.b) and f u l l band width, W , a#e n o t necessarl e s s coupling, A, i l y the same as f o r Z - ( C H ) It can be shown d i r e c t b t h a t t h e r e a r e no k i n k s o l u t i o n s t o the gap equatfon (la), i n c o n f i r m a t i o n o f our e a r l i e r i n t u i t i v e arguments. For polaron c o n f i g u r a t i o n s , however, we expect s o l u t i o n s , and indeed we f i n d t h a t e x a c t l y t h e same f u n c t i o n a l form given by equations (7) can s a t i s f y t h e continuum equations f o r c&-(CH) , provided t h a t K ( o r u ) i s chosen appropriately. To determine K we f o l l c h our previous pr8cedur8 and again convert t h e gap equation t o a miRimization problem for the energy o f the f u l l i n t e r a c t i n g e l e c t r o n phonon system. For fi-(CH), t h e energy expression becomes
-
.
-
-
-
POLARONS IN POL YACETYLENE
~oy[tanh-l(Ko~F~o)-KovFfio]
+
.
where y = A /A Again i n t r o d u c i n g we can w r i t 8 tfie'equation minimizing E
o
=
205
sine
(n+
-
n- + 2)
-e
such t h a t K v = A s i n 0 and wo = iocosO, w i t h respect €0 8 9s N
P
-
3.
y tane
, o 5 0 5 n/2
.
I n the l i m i t o f zero e x t r i n s i c gap, A = 0, y = 0, and as expected eq. (21) reduces t o the trans-(CH) r e s u l t o f 8s. (11). For y > 0, eq. (21) unlike equation (11) - h a s s o l u t f o n s f o r a l l combinations o f n+ and n-. These are summarized i n Table I f o r the s p e c i f i c c a s e o f y = 1.
-
Table I
-
-
n-
1
4
!
0
0
2
+2
0.71
1.43
BIPOLARON (h,h)
0
1
1
+1
0.38
0.98
POLARON (h)
0
2
0
0
0
0
GROUND STATE
1
0
3
+1
0.95
1.96
TRIPOLARON (h, h,e)
1
1
2
0
0.71
1.43
BIPOLARON (e-h)
1
2
1
-1
0.38
0.98
POLARON (e)
2
0
4
0
1.11
2.36
QUADRIPOLARON
2
1
3
-1
0.95
1.96
TRIPOLARON (e,e,h)
2
2
2
-2
0.71
1.43
BIPOLARON (e,e)
n+
g
o
INTERPRETATION
(e,e,h,h)
-
The c&-(CH) polaron states f o r various values o f N = n+ n- + 2 f o r t h e case o f y = 1. f = 2 nn+ i s the e l e c t r i c charge o f t h e state, 0 i s the angle defined i n t h e t e x t (given here i n radians), and E fi gives the f u l l energy o f the e x c i t a t i o n . The i n t e r p r e t a t i o n column gives t h g g h e r a l nature o f t h e e x c i t a t i o n (polaron, bipolaron, ...) and t h e type ( e l e c t r o n 5 e, hole 5 h).
-
-
trans-
O f p a r t i c u l a r i n t e r e s t are the "bipolaron" s o l u t i o n s i n which n+ = n-. I n ( C H I the analogous s o l u t i o n s were i n f i n i t e l y separated KK p a i r s . Indeed, one can %ee from eq. (20) t h a t y e s s e n t i a l l y plays the r o l e o f a confinement parame t e r , leading t o a term i n the energy which increases l i n e a r l y ( r e c a l l eq. (7.b)) w i t h the k i n k / a n t i - k i n k "separation," 2y0. This i s the e x p l i c i t r e a l i z a t i o n o f our e a r l i e r h e u r i s t i c argument about k i n k confinement.
206
A.R. BISHOP, D.K. CAMPBELL
From Table I we see t h a t both the polaron w i d t h (2yo) and depth increase w i t h n-. increasing occupation number, N = n+ + 2
-
i s largely As i n t h e case o f trans, the problem o f polaron dynamics i n &-(CHI open. One can, o f u r s e , estimate t h e i n e r t i a l mass as before, ht the more i n t e r e s t i n g possible e f f e c t s o f c o l l i s i o n dynamics - recombination, long-1 i v e d have thus f a r eluded q u a n t i t a t i v e d e s c r i p t i o n . oscillatory states
-
IMPLICATIONS AND DISCUSSION Since polarons have o n l y r e c e n t l y been discovered [13,16,17] i n the t h e o r e t i c a l the p o s s i b l e i m p l i c a t i o n s i n p a r t i c u l a r , f o r experiment - o f models f o r (CH),, t h e i r existence are n o t y e t completely understood. Obviously the existence o f polarons and kinks i n trans-(CH) , contrasted w i t h t h e existence o f ( m u l t i - ) polarons only i n provrdes a n a t u r a l d i s t i n c t i o n between these two isomers. To understand t i e q u a n t i t a t i v e i m p l i c a t i o n s o f t h i s d i s t i n c t i o n f o r experiment, however, f u r t h e r t h e o r e t i c a l studies s t r e s s i n g t h e dynamics o f k i n k and polaron i n t e r a c t i o n s are needed. Two aspects o f dynamics seem p a r t i c u l a r l y crucial. F i r s t , w i t h i n the time-de endent mean f i e l d approximation (see eqns. (1) and (14)), the nature o h n d kink-polaron i n t e r a c t i o n s must be understood. Second, one must go beyond the mean f i e l d approximation t o study the f u l l quantum dynamics o f (CH) . Although t h e r e i s , as y e t , l i t t l e quantit a t i v e progress i n e i t h e r o f the& areas, t h e r e c e n t l y noted connection between model r e l a t i v i s t i c f i e l d t h e o r i e s and the continuum theory o f (CH) o f f e r some q u a l i t a t i v e suggestions. [17] F i r s t , t h i s connection suggests that! l o n g - l i v e d o s c i l l a t o r y s t a t e s may e x i s t i n (CH) and t h a t these may be important i n (CH) dynamics. Second, known r e s u l t s on 4uantum dynamics i n f i e l d theory r a i s e t h g p o s s i b i l i t y t h a t quantum f l u c t u a t i o n s can d e s t a b i l i z e t h e polaron i n (CHI , causing i t t o disappear from the theory. Although p r e l i m i n a r y quantum Monte Carlo studies [26] do show l a r g e quantum f l u c t u a t i o n s i n (CHI , a n a l y t i c e s t i mates [24] o f the f i r s t order quantum c o r r e c t i o n s i n d i c a t e t h a t ( t o t h i s order, a t l e a s t ) the polaron i s n o t d e s t a b i l i z e d . C l e a r l y , f u r t h e r q u a n t i t a t i v e work on these problem i s required.
-
&-(m
Despite the absence o f c r i t i c a l q u a n t i t a t i v e d e t a i l , i t i s p o s s i b l e t o i n d i cate a number o f i n t e r e s t i n g , p o t e n t i a l experimental i m p l i c a t i o n s o f polarons. F i r s t , on energy grounds [16,17] one knows t h a t s i n g l e e l e c t r o n s added by should form polarons. When doping o r by i n j e c t i o n - t o c i s - o r trans-(CH) many electrons are added, i t i e n e r g e t m l y fa6orable f o r them t o form k i n k / o r multi-polarons ( i n Thus f o r --(CH) , anti-kink pairs ( i n the very l i g h t l y doped m a t e r i a l an average o f 5 one e l e c t r o n ( o r hole) pgr (CH) chain might r e f l e c t polaron t r a n s p o r t p r o p e r t i e s , whereas the more h e a v f l y q e d material would e x h i b i t (charged) k i n k t r a n s p o r t behavior. In Unfortunately t h i s simple cis-(CH),, one would n o t expect t h i s d i f f e r e n c e . p i c t u r e w i l l be complicated by, among o t h e r features, t h e presence o f n e u t r a l kinks "quenched" i n t o undoped --(CHI samples and the u n c e r t a i n t i e s over isomerization from c i s t o t r a n s upon dopifig. Nonetheless, i f polarons are present i n l i g h t l y dop=e-and=-(CH) , then t h e i r o p t i c a l absorption [27] i s s u f f i c i e n t l y d i f f e r e n t from t h a t o f d n k s [28] t h a t experiments comparing "mid-gap" and i n f r a r e d absorption should be able t o d e t e c t t h e d i f f e r e n c e . [27]
-
trans)
-
-
cis).
7
The d i s t i n c t i o n between allowed nonlinear e x c i t a t i o n s i n c i s and trans-(CH) suggests [16] a p l a u s i b l e scenario f o r t h e experimental l y o b s e r v e d m conX t r a s t s between these isomers upon p h o t o i n j e c t i o n ( e x c i t a t i o n o f an electron-hole p a i r by intense r a d i a t i o n ) . I n trans-(CH) , p h o t o i n j e c t i o n leads t o a photoc u r r e n t which has been i n t e r p r e t e n 6 , 2 9 l X a s r e s u l t i n g from a charged kink/ anti-kink pair. I n cis-(CH) , one observes instead a photoluminescence, suggesting the r e c o m b i n m o n o? t h e e-h p a i r which c o u l d f o l l o w because o f t h e e f f e c t i v e confinement o f the p u t a t i v e kinks ( t h e l o c a l i z a t i o n o f m u l t i - p o l a r o n
POLARONS IN POL YACETYLENNE
207
states i n c i s aids recombination). P r e c i s e l y the p r e v i o u s l y discussed theoreti c a l dynamical studies are what i s r e q u i r e d t o make t h i s q u a l i t a t i v e p i c t u r e more precise. I n conclusion, we r e i t e r a t e t h a t recent t h e o r e t i c a l developments [13,16,17] have shown t h a t t h e nature o f nonlinear e x c i t a t i o n s i n polyacetylene may be even more i n t r i g u i n g and complex than previously believed. The c e n t r a l remaining chald e f i n i t i v e l y t h e relevance o f t h i s elegant lenge i s t o e s t a b l i s h - o r disprove t h e o r e t i c a l s t r u c t u r e t o t h e r e a l material.
-
ACKNOWLEDGEMENTS We are g r a t e f u l t o many colleagues f o r t h e i r advice and encouragement. Special thanks are due t o J. L. Bredas, J. R. S c h r i e f f e r , and W. P. Su f o r enlightening discussions o f t h e i r numerical simulations and t o S. Etemad f o r h i s shared i n s i g h t s i n t o the experimental s i t u a t i o n . REFERENCES See f o r example, D. Bloor, Proc. Amer. Chem. S O C . , Houston, A p r i l 1980; t o appear i n J. Chem. Phys. For recent reviews see a r t i c l e s i n Proc. I n t . Conf. on Low-Dimensional Synt h e t i c Metals (Helsingor, Denmark, August 1980) i n Chemica S c r i p t a 2 (1981); Physics i n One Dimension, eds. J. Bernasconi and T. Schneider (Springer Verlag, 1981); and A. J. Heeger and A. G. MacDiarmid, p. 353-391 i n The Physics and Chemistry o f Low Dimensional Solids, ed. L. Alchcer (Reidel , 1980). Su, W. P., S c h r i e f f e r , J. R., and Heeger, A. J., Phys. Rev. L e t t . (1979); Phys. Rev. B 2 , 2099 (1980).
42,
1698
42, 408 and 416 (1977). Brazovskii, S. A., JETP L e t t e r s 28, 606 (1978) (trans. of Pisma Zh. Eksp. Teor. F i z . 28, 656 (1978)); and Soviet Phys. JETP 51, 342 (1980) (trans. o f Zh. Eksp. Teor. Fiz. 'a, 677 (1980)). Kotani, A.,
J. Phys. SOC. Japan
Takayama, H., Lin-Liu, Y. R. and Maki, K., Phys. Rev. B 3,2388 (1980); Krumhansl, J. A., Horovitz, B. , and Heeger, A. J., S o l i d State Commun. 2 , 945 (1980); Horovitz, B., S o l i d State Commun. 2 , 61 (1980). Horovitz, B.
,
Phys. Rev. L e t t .
Rice, M. J., Phys. L e t t . 7lJ,
5 , 742
(1981).
152 (1979).
Rice, M. J., and Timonen, J., Phys. L e t t .
El 368
(1979).
[lo] Mele, E. J., and Rice, M. J., i n Chemica S c r i p t a IJ, 21 (1981). [ll] See a r t i c l e s by Etemad S., and Rice, M. J., these proceedings. [12] Jackiw, R. and Rebbi, C.,
Phys. Rev. D 13, 3398 (1976); Jackiw, R., S c h r i e f f e r , J . R., Nuc. Phys. B 190, 253 (n81).
[13] Su, W. P., and S c h r i e f f e r , J. R., Phys. Rev. L e t t .
5 , 738
and
(1981).
[14] Rice, M. J., and Mele, E. J., " P o s s i b i l i t y o f S o l i t o n s w i t h charge * e / 2 i n Highly Correlated 1:2 Salts o f TCNQ," Xerox Webster p r e p r i n t , 1981.
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[15] Su, W. P., and S c h r i e f f e r , J. R., Proc. Nat. Acad. Sci. 77, 5526 (Physics) (1980); Bredas, J. L. , Chance, R. R., and Silbey, R., Proceedings o f I n t e r -
national Conference on Low-Dimensional Conductors, Molecular C r y s t a l s and L i q u i d Crystals (Gordon Breach), t o be published.
[16] Brazovskii, S. , and Kirova, N.,
Pisma Zh. Eksp. Teor. Fiz.
[17] Campbell, D. K. and Bishop, A. R . , Phys. Rev. B B ( i n press).
24,
2,
6 (1981).
4859 (1981); Nuc. Phys.
[18] Following the standard conventions i n t h e continuum model, we use u n i t s w i t h h = 1. [19] P e i e r l s , R. E., Quantum Theory o f S o l i d s (Clarendon Press, Oxford, 1955) p. 108; Allender, D. Bray, J. W., and Bardeen, J . , Phys. Rev. B. 9, 119 (1974). [20] Bishop, A. R.,
S o l i d State Commun.
Y. R., unpublished.
[21] Lin-Liu,
and Maki, K.,
2 , 955
(1980).
Phys. Rev.
822
5754 (1980); Kivelson, S.
[22] The phenomenological
$4 theories, i f coupled t o a phenomenological e l e c t r o n f i e l d describing o n l y the l o c a l i z e d , "gap" s t a t e s , do c o n t a i n b o t h the k i n k s o l i t o n s ( o f a l l charges) and t h e polaron.
[23] Campbell, 0. K., and Wingate, C., [24] Nakahara, N.
, and
Maki, K.,
i n preparation.
p r e p r i n t (1981).
[25] I n another context t h i s ansatz has been discussed by Rice; see Rice, M. S., Phys. Rev. L e t t . 37, 36 (1976). [26] Su, W. P.,
unpublished.
[27] Fesser, K., Bishop, A. R . , and Campbell, 0. K.
,
i n preparation.
1281 Gammel, 3. T. and Krumhansl, J. A. Phys. Rev. B 24, 1035 (1981). Maki, K. and Nakahara, N., Phys. Rev. B 23, 5005 (1981E Horovitz, B., p r e p r i n t (1981); Kirelson, S. , Lee, T.-Y. , T i n - L i u , Y. R., Peschel, I. , and Yu, L . , p r e p r i n t (1981). [29] Etemad, S. , these Proceedings.