Charge transport in charge transfer salts by order parameter
fluctuations
Citation for published version (APA):
Kramer, G. J., Brom, H. B., & Jongh, de, L. J. (1987). Charge transport in charge transfer salts by order
parameter fluctuations. Synthetic Metals, 19(1-3), 33-38. https://doi.org/10.1016/0379-6779(87)90327-4
DOI:
10.1016/0379-6779(87)90327-4
Document status and date:
Published: 01/01/1987
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be
important differences between the submitted version and the official published version of record. People
interested in the research are advised to contact the author for the final version of the publication, or visit the
DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page
numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
Synthetic Metals,
19 (1987) 3 3 - 3 8 3 3CHARGE TRANSPORT IN CHARGE TRANSFER SALTS BY ORDER PARAMETER FLUCTUATIONS
G.J. KRAMER, H.B. BROM and L.J. DE JONGH
Kamerlingh Onnes Laboratorium, RlJksuniversiteit Leiden, Postbus 9506, 2300 RA Leiden {The Netherlands)
ABSTRACT
The conduction mechanism in Charge Transfer Salts (CTS) wlth 1:2 donor to acceptor ratio is investigated. These systems can be characterized by two dif- ferent order parameters, each corresponding to different reallsatlons of a gap at the Fermi surface. One is the usual Pelerls dlmerization, the other corre- sponds to a uniform shift of donor molecules, which induces an alternating potential on the acceptor chain. We show that small~ coherent variations of these two-fold order parameter provide a cooperative charge transport mecha- nism. We argue that this mechanism accounts for some anomalous electrical transport phenomena recently found in substituted morphollnium TCNQ2 compounds.
INTRODUCTION T h i s p a p e r d e a l s w i t h c h a r g e t r a n s f e r s a l t s w i t h 1 : 2 d o n o r t o a c c e p t o r r a t i o , c o m p l e t e c h a r g e t r a n s f e r and TCNQ a s a c c e p t o r ( I ] . I n t h i s c a s e t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n ( H u b b a r d t s U) i s much l a r g e r t h a n t h e b a n d w i d t h [ 2 ] , w h i c h J u s t i f i e s u s i n t a k i n g t h e l i m i t o f i n f i n i t e i n t e r a c t i o n . The p r o p e r Hamiltonian reads [3] i +
- Z(t + (-I) X)(CfCi+ I + h.co) + EE(-I)in I ([)
i i
where x is related to an alternation of the charge transfer integral within the unit cell due to a distance alternation between adjacent TCNQ molecules. E de- notes the alternation in local potential, which is a consequence of an asymmet- ric positioning of the donor molecule with respect to the neighbouring acceptor 0379-6779/87/$3.50 © Elsevier Sequoia/Printed in The Netherlands
3 4
molecules. Fig. I shows the generalized schematic configuration within the unit cell together with the phonon spectrum. The gap, A, in the electron spectrum is A = 2(4~ 2 + E2) ~
and t h e e l e c t r o n d i s t r i b u t i o n i n t h e ground s t a t e i s g i v e n by
Pe/o = ~ - ¥
1
(,~-F) 2)
(2)
(])
where K is a complete elliptic Integral of the first kind. Fig. 2 shows the electron distribution and its dependence on both % and E.
Consider a system in which E and • are constant over a (microscopically) large region, the initial state being (%-~8%,E-~6E) with an electron occupation of p on odd-numbered sites and l-p for even sites. Now, a change in the order parameters %-~6% ~ %+~6% and E-~6E ÷ E+~6E will result in an electron occupa- tion of p+6p, resp. l-p-6p. Since for all sites the overlap with the left-hand neighbour differs 2z from the right-hand overlap, the charge flow which estab- lishes the new equilibirum will have a preferential direction, i.e., will result in a current. This statement can be made quantitative by using Fermi's golden rule: the transition probability for an electron to hop from one site to an- other is proportional to the square of the overlap. This implies that the prob-
UNIT CELL ," . . . I L . . . J F i g . 1. Model f o r t h e 1:2 CTS l a t t i c e and i t s p h o n o n s p e c t r u m . \ \
\
'L /
D I I I I I J C,~
?
.
.a
. 2
F i g . 2. ( E , ~ ) - p a r a m e t e r s p a c e . The d r a w n l i n e s a r e c o n t o u r s o f e q u a l p f o r t h e g r o u n d s t a t e . R e v o l u t i o n a l o n g r p r o d u c e s a n e t c h a r g e f l o w a s s e e n f r o m t h e i n s e r t . F a t ( t h i n ) a r r o w s i n d i c a t e l a r g e ( s m a l l ) t r a n s f e r i n t e g r a l s .ability difference for charge flow from an even-numbered site to the left or right equals
p(+)_p(÷) . (t + ~)2 _ (t - .~)2 = . 4t~ = --2~ for • < t (t + ~)2 + (t - ~)2 2t 2 + 2~ 2 t
Since in the above-descrlbed change in the parameter space an amount of charge dp is redistributed we have for the amount of charge (6Q) that passes along all even sites:
6Q = & p ' ( P ( + ) - P C * ) ) = 6p • 2-2-~ ( 4 ) t
Or, more general
AQ = 2 I dp • -- ( 5 )
t parameter
space
where the integral is taken over parameter space and its value will depend not only on the end and starting points but also on the path in between. Note that for odd sites b o t h dp and ( P ( + ) - P ( ÷ ) ) c h a n g e sign, which implies that 6Q i s site-lndependent and an overall shift of the electron gas is accomplished. It is important that generally Q is non-zero if one integrates over a closed loop in parameter space, as the reader may verify by considering fig. 2. For a small closed trajectory one has
q e f f = ~ do 2~ = ~ 2~ ~-- ~-- [( ~ )*dE + ( ~ ) * d ~ ] = , ~ j ~ _ ~p (Dp~2A (6) where qeff is the transported charge and A the area of the closed loop. Since charge is transported by going along any closed trajectory, an electric field will manifest itself as a force in (E,~)-space.
We conclude that coherent variations of the two order parameters • and E can give rise to the cooperative motion of the electron gas, which is impossible for a commensurate system with just a single order parameter.
OPTICAL PHONONS
So far, we have dealt with a hypothetical chain with a charge density dis- tortion w h i c h was correlated over the entire chain. In the actual situation of a CTS, the ground state will be characterized by a set of order parameters E* and %*, and at finite temperature lattice vibrations will cause small, local fluctuations of these order parameters. Since • and E can be defined at unit c e l l l e v e l a n d a r e d e t e r m i n e d by i n t r a u n i t c e l l d i s t a n c e s , v a r i a t i o n s o f • a n d E s h o u l d be a t t r i b u t e d t o o p t i c a l p h o n o n s . As a c o n s e q u e n c e o f t h e a b o v e d i s - c u s s e d c h a r g e t r a n s p o r t a r i s i n g f r o m v a r i a t i o n s i n E and ~, t h e o p t i c a l p h o n o n modes w i t h m- a n d E - c h a r a c t e r r e s p e c t i v e l y w i l l be c o u p l e d by t h e e l e c t r i c
36
field. If we treat the phonons c l a s s i c a l l y , we can write the local, time-de-
p e n d e n t o r d e r p a r a m e t e r s i n t e r m s o f phonon c o n t r i b u t i o n s :
1:(X, tR) = E ~k sln(kx - bJ kt R + ~ , k ) (7a) k
E(X,tR) ffi r E k sln(kx - o#E,kt R + ~E,k) (Tb) k The phonons t h e m s e l v e s a r e - f o r t h e u n p e r t u r b e d l a t t i c e - s o l u t i o n s o f t h e equations of motion MxX(x) - K Xxx(X ) + M ~2 _~(X) " 0 (ga) %,opt MEE(x) - ~ E x x ( X ) + H E ~ , o p t E ( x ) - 0 (gb) bl x, M E denote effective masses; K , k g are spring constants. ~ denotes the two- vector of place (x) and time (tR) , while Xxx and Exx are the second derivatives
o f t and E w i t h r e s p e c t t o p o s i t i o n . I f an e l e c t r i c f i e l d i s p r e s e n t , an e x t e r - n a l f o r c e Fx (FE) w i l l a p p e a r on t h e r . h . s , o f e q . ( 8 a , b ) as a c o n s e q u e n c e o f e l e c t r o n - p h o n o n c o u p l i n g . For t h e e x t e r n a l f o r c e ( e x c e r t e d by an e l e c t r i c f l e l d ) on a g e n e r a l i z e d c o o r d i n a t e q, one h a s Fq = -dW/dq, where dW i s t h e work done by t h e e l e c t r i c f i e l d ( + ) , which e q u a l s +-6Q and 6Q i s g i v e n by e q . ( 5 ) . T h i s y i e l d s x - AN F ( x ) " li_m - ~ =
Ax(x)+0
t
,Z(~)
and likewise FE(~) 2+p(~) Sp " - t(~)~(~),E(~)
(E,x+Ax) - l~m 2+ f d p ( x ) . t ( x )A~Cx)÷0 t.A~(~) (z,~)
( 9 a ) (9b)Combining (9) and (8) and t r a n s f o r m i n g t o k - s p a c e we f i n d t h a t t h e phonon modes obey t h e r e l a t i o n s :
+ (o2 kXk(tR ) . -2¢ +k(tR )
~ k ( t R ) + w2 k E k ( t R ) = - 2 g ( g p ) ~ , v , . ~ ( t ~ ) g E L ~ (lOb) L e t us r e s t r i c t o u r s e l v e s t o t h e c a s e where x*-E*-O, f o r which ( d p / d x ) - O . Then
(1On,b) s i m p l i f y t o
+ ~2,kX k = 0 (lla)
~k
+ " 2 +
37 where we have introduced a dissipative term in (llb) to avoid unphysical re-
suits (7 is inversely proportional to the phonon llfe time). Note that the asymmetry in (ii) with respect to E and ~, could already be anticipated by examining figure 2: the fleld-lnduced force will always be perpendicular to contours of equal p as a consequence of (5). The situation is similar to oscil- lators, of which one (g) serves as the driving force for the other (E). The solution of (ii) is Zk(tR) = A.sin(~ ,ktR) ER(tR) = Ek s i n ( ~ , k t R + ~k ) where Ek " 2eA (12a) (12b)
~,k]
The physical interpretation of the above is the following: due to an electric field, part of the E-mode oscillation amplitude is transferred to frequency while being phase-coupled to the ~-mode oscillation. We see that the coupling
o f o p t i c a l p h o n o n s i n t h e p r e s e n c e o f a n e l e c t r i c f i e l d g i v e s i n d e e d r i s e t o c o h e r e n t v a r i a t i o n s o f • and E, w h i c h was shown e a r l i e r t o be t h e c o n d i t i o n f o r c h a r g e t r a n s p o r t . U s i n g e q s . ( 5 ) and ( 7 ) one f i n d s f o r t h e c u r r e n t ~ 1 ~ ( 1 3 ) I = Z ' (~)E,~,~kEk~ , k E sln(~E,k) k DISCUSSION I t i s p o s s i b l e t o g i v e a n o r d e r o f m a g n i t u d e e s t i m a t e f o r t h e c u r r e n t r e s u l t i n g f r o m a p a i r o f c o u p l e d p h o n o n s . L e t u s l o o k more c l o s e l y a t two p h o n o n s , e a c h o f a d i f f e r e n t b r a n c h and i d e a l l y c o u p l e d i n p h a s e . We r e c a l l t h a t t h i s p h o n o n p a i r w i l l c a u s e t h e s y s t e m t o make t i n y r e v o l u t i o n s a r o u n d t h e
ground state configuration (E*,x*). Using eq.(13) we can write
m m t ~ p ~ r ~ x / t ~
I = ~kU kUE,kl~-~E J k~-~'-) (14)
w h e r e u m a n d u~ a r e t h e v i b r a t i o n a l a m p l i t u d e s o f b o t h modes f o r a s i n g l e
phonon, which is (quantum mechanically [4])
I n c a l c u l a t i n g t h e a c t u a l c o n t r i b u t i o n t o t h e c o n d u c t i v i t y we t a k e a c h a i n s y s - tem w i t h 107 u n i t c e l l s p e r cm and 1014 c h a i n s p e r cm 2. The r e d u c e d m a s s M, I n v o l v e d i n t h e o s c i l l a t i o n s w i l l be i 0 - I 0 0 a . m . u , d p / d u E and d ( % / t ) / d u a r e
38
expected to be of the order of 1 A -I, which implies that a change in the molec- ular positions of 1 A would reverse the charge distribution or dlmerizatlon. Using these numerical data, we estimate the current for a system with one cou- pled pair or phonons per chain to be 0.I-I A/cm 2. In actual systems for which the model may be applicable (like substituted morphollnlum TCNQ2 compounds, the room temperature conductivity is of the order of I-I0 (R.cm) -I. At this temper- ature, optical phonons are rather abundant and one finds that only 1 in every 105 phonons need to be coupled by an electric field of I V/cm. Another, perhaps more physical way of expressing the power of the mechanism is to say that the field-lnduced admixture of one phonon mode into the other at i V/cm needs to be only 10 -5 to account for the observed conductlvltles.
In conclusion we state that the model presented in this paper is able t o account for the conductlvlties found in the class of morphollnlum TCNQ 2 salts [5]. Moreover, it predicts temperature-actlvated conductivity, Just as the single electron model does, but here the activation energy is not dependent on the electron gap but rather on the optical phonon energy. It should therefore be expected to contribute whenever single particle conductivity fails because of a large gap-value or low carrier mobility. This different origin of the activation energy would explain the absence of any correlation between the cal- culated gap (using eq.(2)) and the observed activation energy [6]. Although at present it is not possible to give an explanation for the anomalous field and frequency dependence of the conductivity observed in a number of substituted morphollnium TCNQ 2 salts, examination of the data indicate that the origin is most likely to be found in lattlce-coupled charge transport [7] for which our model is certainly a good candidate.
REFERENCES
1 For an o v e r v i e w o f r e c e n t work i n 1-D c o n d u c t o r s s e e e . g . , P r o c e e d i n g s o f t h e ICSM'84, Mol C r y s t . L i q . C r ~ s t . , 119 (1985) 117-121.
2 S. Mazumdar and Z.G. Soos, P h y s . Rev. B, 23 (1981) 2810.
3 A s i m i l a r s y s t e m has been c o n s i d e r e d w i t h r e s p e c t t o t h e p r e s e n c e o f s o l i t o n s by S. K i v e l s o n , P h y s . Rev. B~ 28 (1983) 2653, and W.P. Su, S o l i d S t a t e Connu., 48 (1983) 479.
4 H. Bottger, Principles of the Theory of Lattice Dynamics, Academie-Verlag, Berlin, 1983.
5 R.J.J. Visser, Th.W.L. van Heemstra and J.L. de Boer, M0 I. Cryst. Liq. Cryst. 85 (1982) 2 6 5 .
6 S. van Smaalen and J. Kommandeur, Phys. Rev. B, 31 (1985) 8056.
7 G.J. Kramar, H.B. Brom, L.J. de Jongh and J.L. de Boer, Festk~rperprobleme Advances in Solld State Physics XXV,(1985) 167; and G.J. Kramer, J.L. Joppe, H.B. Brom, L.J. de Jongh and J.L. de Boer, Synth. Met.,19 (1987)439(these Proceedings)
