NONLINEAR CONDUCTIVITY IN QUASI-ONE-DIMENSIONAL ORGANIC CRYSTALS

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NONLINEAR CONDUCTIVITY IN

QUASI-ONE-DIMENSIONAL ORGANIC CRYSTALS

E. Conwell, N. Banik

To cite this version:

E. Conwell, N. Banik. NONLINEAR CONDUCTIVITY IN QUASI-ONE-DIMENSIONAL ORGANIC CRYSTALS. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-315-C7-322.

�10.1051/jphyscol:1981738�. �jpa-00221675�

NONLINEAR CONDUCTIVITY IN QUASI-ONE-DIMENSIONAL ORGANIC CRYSTALS

E.M. Conwell and N.C. Banik

Xerox Webster Research Center, Webster, N.Y. 14580, U.S.A.

A b s t r a c t . - Large increases i n c o n d u c t i v i t y a w i t h e l e c t r i c f i e l d F, s t a r t i n g a t a few V/cm or l e s s , have been observed i n quasi-one-dimensional conductors such as TTF-TCNQ a t temperatures below the metal-to-semiconductor t r a n s i t i o n . Among the explanations t h a t have been suggested are depinned charge-or s p i n - d e n s i t y waves, conducting defects (<)>-particles) i n the charge-density waves and impact i o n i z a t i o n o f i m p u r i t i e s by hot c a r r i e r s , but none o f these i s s a t i s f a c t o r y . We show t h a t increase i n m o b i l i t y (due t o decreased phonon s c a t t e r i n g o f hot c a r r i e r s ) accounts well f o r t h e e a r l y p a r t o f the increase i n a , and speculate on mechanisms f o r e x p l a i n i n g the remainder.

I . I n t r o d u c t i o n . - The m a t e r i a l s we s h a l l be discussing are t e t r a t h i o f u l v a l e n e tetracyanoquinodimethane, TTF-TCNQ, and r e l a t e d organic s a l t s . In the c r y s t a l , as i n d i c a t e d i n F i g . 1 , the TTF molecules are l i n e d up i n stacks or chains and the TCNQ molecules i n separate s t a c k s . The TTF molecule donates approximately h an e l e c t r o n per molecule t o TCNQ, w i t h the r e s u l t t h a t both types o f stack conduct, TTF by h o l e s , TCNQ by e l e c t r o n s . Another type o f c r y s t a l we s h a l l discuss i s b i s - t e t r a m e t h y l t e t r a s e l e n a f u l v a l e n e hexafluorophosphate, (TMTSF)

PFg. The TMTSF molecule i s obtained by r e p l a c i n g t h e 4 H's i n TTF by methyl (CH,) groups and the S's by S e ' s . In t h e c r y s t a l the PFg's perform the acceptor f u n c t i o n o f TCNQ, t a k i n g e x a c t l y % e l e c t r o n from each TMTSF, but do not c a r r y any c u r r e n t . Conduction i s e n t i r e l y by holes along the TMTSF s t a c k s . These m a t e r i a l s are c a l l e d quasi-one-dimensional ( 1 - d ) because a i s much h i g h e r , by a f a c t o r 100 or more, along the stacks than perpendicular t o them.

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, oatobve 1981 page C7-315

Résumé. - Les fortes augmentations de la conductivité a avec le champ électrique F, dès qu'il est de l'ordre de quelques V/cm ont été observées dans des conducteurs quasi-unidimensionnels comme le TTF-TCNQ à des températures inférieures à celle de la transition métal-semiconducteur. Parmi les explications avancées, on trouve l'effet de la disparition de l'ancrage d'ondes de densité de charges ou de spin de défauts conducteurs (particules <(>) dans les ondes de densité de charge et de l'io- nisation par impact d'impuretés par des porteurs chauds ; mais aucune de celles-ci n'est satisfaisante.

Nous montrons que l'augmentation de mobilité due à une interaction décroissante des porteurs chauds avec les phonons rend bien compte de la première partie de l'aug- mentation de a et nous prévoyons des mécanismes pour expliquer le reste de la varia- ti on.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981738

JOURNAL DE PHYSIQUE

Fig. 1 : TTF and TCNQ molecules and t h e i r arrangement i n t h e c r y s t a l . The s t a c k o f TCNQ molecules ( d a r k c i r c l e s ) i s i n f r o n t 1 o f t h a t o f TTF molecules

TTF

(open c i r c l e s ) . From J. M i l l e r , " S y n t h e s i s and P r o p e r t i e s o f Low- Dimensional M e t a l s " eds.

J.S. M i l l e r and A.E. E p s t e i n (NY Acad. o f Sciences, 1978), p. 26.

TCNQ

A t room t e m p e r a t u r e t h e m a t e r i a l s we a r e i n t e r e s t e d i n may be c o n s i d e r e d m e t a l l i c , w i t h band c o n d u c t i o n a l o n g t h e s t a c k s and a Fermi l e v e l *$ o f t h e way i n t o t h e band. Conduction p e r p e n d i c u l a r t o t h e s t a c k s a t room t e m p e r a t u r e i s p r o b a b l y by hopping. I n t h i s paper we s h a l l d i s c u s s o n l y c o n d u c t i o n p a r a l l e l t o t h e s t a c k s .

I t i s known t h a t as t h e t e m p e r a t u r e i s lowered, a 1-d m e t a l l i c c o n d u c t o r becomes u n s t a b l e and a t some t e m p e r a t u r e TmS undergoes a t r a n s i t i o n t o a semi- c o n d u c t i n g o r s u p e r c o n d u c t i n g ground s t a t e . The l a t t e r case has been found r e c e n t l y f o r (TMTSF)* PF6 under pressure, and f o r some o f i t s c l o s e r e l a t i v e s , b u t w i l l n o t be c o n s i d e r e d f u r t h e r here. The semiconducting s t a t e i s achieved by a gap opening u p a t t h e Fermi s u r f a c e . T h i s i s e n e r g e t i c a l l y f a v o r a b l e because i t d r i v e s e l e c t r o n s i n t o l o w e r energy s t a t e s . I n t h e case o f TTF-TCNQ t h e gap opening i s due t o t h e appearance o f a p e r i o d i c v a r i a t i o n o f t h e d e n s i t y o f conduc- t i o n e l e c t r o n s ( o r h o l e s ) , c a l l e d charge d e n s i t y waves (CDW), w i t h wave v e c t o r 2kF, where kF i s t h e Fermi wave v e c t o r . T h i s i n t u r n causes t h e i o n s t o move, d i s t o r t i n g t h e l a t t i c e ( P e i e r l s d i s t o r t i o n ) . I n t h e o t h e r t y p e o f i n s t a b i l i t y , r e c e n t l y found t o o c c u r i n (TMTSF)2 PF6 a t ambient pressure, t h e gap opening i s due t o t h e c o n d u c t i o n e l e c t r o n s ( o r h o l e s ) o f each s p i n s e p a r a t e l y f o r m i n g waves, c a l l e d s p i n - d e n s i t y waves (SDW), w i t h wave v e c t o r 2kF. The two s p i n - d e n s i t y waves a r e 180° o u t o f phase, so t h e r e i s no n e t c h a r g e d e n s i t y anywhere and no m o t i o n o f t h e l a t t i c e i o n s .

I t i s i n t h e CDW o r SDW ground s t a t e o f t h e s e m a t e r i a l s t h a t t h e n o n l i n e a r -

i t i e s t h a t a r e t h e s u b j e c t o f t h i s paper occur. B e f o r e d e s c r i b i n g them we s h a l l

d i s c u s s l o w - f i e l d c o n d u c t i o n i n t h e m a t e r i a l s s i n c e u n d e r s t a n d i n g t h a t i s a pre-

r e q u i s i t e f o r d e a l i n g w i t h h i g h - f i e l d behavior. W i t h t h e ground s t a t e a r i s i n g f r o m

c o l l e c t i v e b e h a v i o r o f t h e e l e c t r o n s , some o f t h e t h e o r i e s o f c o n d u c t i o n a r e based

on such behavior. The c u r r e n t vs. f i e l d d a t a w i l l t h e n be presented, and i t w i l l

be shown t h a t t h e c o l l e c t i v e t h e o r i e s c a n n o t e x p l a i n t h e s e data. The r e c e n t r e a l -

i z a t i o n t h a t t h e m o b i l i t i e s a r e q u i t e h i g h has l e d us t o f o r m u l a t e a h o t c a r r i e r

t h e o r y which accounts w e l l f o r t h e i n i t i a l v a r i a t i o n o f c o n d u c t i v i t y w i t h f i e l d .

A f t e r d e s c r i b i n g t h a t , we w i l l d i s c u s s t h e s i t u a t i o n a t h i g h e r f i e l d s .

2. L o w - f i e l d conduction. - I n Fig. 2 we show t h e v a r i a t i o n o f t h e l o w - f i e l d o w i t h T f o r TTF-TCNQ. Below t h e t r a n s i t i o n t e m p e r a t u r e o f 54K o decreases r a p i d l y . From 25 down t o 10K t h e r e i s a r e g i o n o f c o n s t a n t a c t i v a t i o n energy, 200K, t h a t , a c c o r d i n g t o o t h e r measurements such as p h o t o c o n d u c t i v i t y , r e p r e s e n t s t h e semi- c o n d u c t i n g gap. Thus i n t h i s range TTF-TCNQ i s an i n t r i n s i c semiconductor. Below 10K t h e s l o p e o f l o g o vs. 1/T decreases. Cohen and Heeger have drawn i n a n o t h e r 1 c o n s t a n t s l o p e l i n e c o r r e s p o n d i n g t o an a c t i v a t i o n energy o f 14K. A n a t u r a l guess i s t h a t t h i s p a r t o f t h e c u r v e r e p r e s e n t s t h e r m a l e x c i t a t i o n f r o m i m p u r i t y o r d e f e c t l e v e l s w i t h i n t h e gap. That guess i s p r o b a b l y wrong. For one t h i n g ,

a1 though t h e s l o p e i n e v i t a b l y decreases a t l o w temperatures, i n o t h e r p e o p l e ' s d a t a (and perhaps i n t h o s e o f CH i f l o w e r T d a t a had been g i v e n ) i t decreases c o n t i n u - o u s l y w i t h d e c r e a s i n g T. It s h o u l d be n o t e d t h a t , a l t h o u g h t h i s m a t e r i a l and c l o s e

r e l a t i v e s have been s t u d i e d i n t e n s i v e l y f o r a number o f years, t h e o n l y s u c c e s s f u l d o p i n g experiments a r e t h o s e i n which s i m i l a r m o l e c u l e s have been s u b s t i t u t e d , say m o l e c u l e s w i t h H r e p l a c e d by D o r S b y Se. Such molecuf es do n o t change c a r r i e r c o n c e n t r a t i o n and can o n l y a f f e c t e l e c t r i c a l p r o p e r t i e s t h r o u g h t h e d i s o r d e r t h e y c r e a t e . I t has been mentioned f o r t h e case o f ( T M T S F ) ~ PF6 t h a t t h e electrochem- i c a l process b y which i t i s made appears t o be s e l f - p u r i f y i n g . When (TMTSF)~ PF6 i s prevented by p r e s s u r e f r o m under g o i n g t h e SDW t r a n s i t i o n , i t shows no s i g n o f r e s i d u a l r e s i s t a n c e down t o t h e s u p e r c o n d u c t i n g t r a n s i t i o n a t s l ~ . ~ I n l i g h t o f t h e s e f a c t s , i t i s n o t s u r p r i s i n g t h a t t h e r e i s n o t even a model f o r s h a l l o w l e v e l s due t o i m p u r i t i e s o r d e f e c t s i n t h e s e m a t e r i a l s .

What a r e t h e p o s s i b i l i t i e s f o r e x p l a i n i n g c o n d u c t i o n a t t h e s e l o w tempera-

t u r e s i n TTF-TCNQ? One mechanism t h a t m i g h t be i n v o k e d i s moving CDW's. S i n c e

t h e CDU1s a r e charged. t h e y m i g h t move i n an e l e c t r i c f i e l d , c o n s t i t u t i n g a

C 7 - 3 1 8 JOURNAL DE PHYSIQUE

c o l l e c t i v e e l e c t r o n i c c u r r e n t . U s u a l l y , however, t h e CDW's a r e p r e v e n t e d f r o m moving, t y p i c a l l y by i m p u r i t i e s o r d e f e c t s , w h i c h p r o v i d e r e g i o n s t o w h i c h e l e c - t r o n s a r e a t t r a c t e d o r r e p e l l e d . Nevertheless, CDW1s p i n n e d i n t h i s way can be depinned w i t h a v e r y s m a l l t h r e s h o l d f i e l d i n some m a t e r i a l s , n o t a b l y NbSej, and t h i s does p r o v i d e , i n t h o s e cases, a s t r o n g n o n l i n e a r i t y o f o i n an e l e c t r i c f i e l d . I n TTF-TCNQ t h e p i n n i n g must be q u i t e s t r o n g , however, because i t a r i s e s f r o m t h e e l e c t r o s t a t i c a t t r a c t i o n of t h e p o s i t i v e l y charged CDW's on t h e TTF s t a c k s f o r t h e n e g a t i v e l y charged ones on t h e TCNQ s t a c k s . To a l l o w t h e CDW's t o move i n oppo- s i t e d i r e c t i o n s , t h e e l e c t r i c f i e l d must be s t r o n g enough t o overcome t h i s e l e c - t r o s t a t i c a t t r a c t i o n . However, a mechanism has been found by which p i n n e d CDW1s can p r o v i d e c o n d u c t i v i t y . I t has been p r e d i c t e d t h e o r e t i c a l l y t h a t t h e r e can e x i s t t h e r m a l l y a c t i v a t e d d e f e c t s , i .e., compressions ( $ - p a r t i c 1 es) o r r a r e f a c t i o n s ( a n t i $ - p a r t i c l e s ) o f t h e charge d e n s i t y o s c i l l a t i o n s i n t h e CDW. These d e f e c t r e g i o n s have an excess charge, c l a c u l a t e d as f 2e,3 and can c a r r y c u r r e n t because t h e y a r e m o b i l e ( s o l i t o n s ) , a t l e a s t i n a CDW w i t h o u t d e f e c t s . I t i s t o

$ - p a r t i c l e s t h a t CH a t t r i b u t e t h e c o n d u c t i o n t h e y observe below 10K, a s s i g n i n g t o them t h e 14K a c t i v a t i o n energy. As n o t e d e a r l i e r , however, o t h e r r e s e a r c h e r s do n o t f i n d a 14K a c t i v a t i o n energy. n o r even a c o n s t a n t slope. An a l t e r n a t i v e sug- g e s t i o n t o account f o r t h e s m a l l e r s l o p e s seen a t l o w T ' s i s t h a t t h e y a r e due t o r e g i o n s o f t h e m a t e r i a l w i t h s m a l l e r gaps, t h e s e r e g i o n s d o m i n a t i n g o a t l o w temperatures because t h e l a r g e r gap r e g i o n s a r e f r o z e n out. S m a l l e r gaps c o u l d r e s u l t f r o m t h e presence o f d i s o r d e r , i m p u r i t i e s o r s t r a i n s . I t i s w e l l known f o r

( T M T s F ) ~ PF6 t h a t t h e gap decreases s t r o n g l y w i t h pressure,' d i s a p p e a r i n g a t a b o u t 10Kbar.

3. H i g h f i e l d e f f e c t s . - When t h e a p p l i e d e l e c t r i c f i e l d F i s i n c r e a s e d beyond a few V/cm a t l o w T's, c o n d u c t i o n o f TTF-TCNQ becomes h i g h l y n o n l i n e a r , as shown i n Fig. 3. A t t h e h i g h e s t f i e l d f o r which measurements were taken, ~ 4 0 0 V/cm, o had i n c r e a s e d by a f a c t o r g r e a t e r t h a n 10 4 . The e x p l a n a t i o n o f t h e s e i n c r e a s e s i n o advanced b y CH i n v o l v e d two mechanisms. Up t o

100 V/cm, where a i s n o t changing r a p i d l y , t h e y a t t r i b u t e d t h e i n c r e a s e t o i n c r e a s e d numbers of

$ - p a r t i c l e s , due t o e l e c t r i c - f i e l d - i n d u c e d decrease o f t h e i r a c t i v a t i o n energy.

The r a p i d r i s e above 100 V/cm (where t h e @ - p a r t i c l e s would a l l presumably be f r e e )

t h e y a t t r i b u t e d t o CDW's depinned b y t h e h i g h f i e l d . T h e i r e x p l a n a t i o n i n terms

o f $ - p a r t i c l e s i s i n v a l i d a t e d by t h e f i n d i n g o f C.S. ~ a c o b s e n ~ t h a t j vs. F

behaves s i m i l a r l y above 4K, t o temperatures o f 15K a t l e a s t . I f t h e s o u r c e o f

t h e n o n l i n e a r i t y were $ - p a r t i c l e s w i t h a l o w - f i e l d a c t i v a t i o n energy o f 14K, no

e f f e c t on a o f f i e l d s h o u l d be seen a t 15K. The i d e a t h a t depinned CDW's a r e

c o n t r i b u t i n g t o j a t h i g h F1s i s e l i m i n a t e d by an experiment o f CH,' demonstra-

t i n g t h a t a sample h e a v i l y i r r a d i a t e d w i t h d e u t e r o n s has t h e same b e h a v i o r a t

v e r y h i g h F's as a n u n i r r a d i a t e d one. I r r a d i a t i o n i s known t o g i v e r i s e t o

d e f e c t s t h a t p i n CDW's.

10

0 .

B

F i g . 3: C u r r e n t d e n s i t y vs. e l e c t r i c f i e l d i n t e n s i t y f o r TTF-TCNQ, from Cohen and Heeger r e f e r e n c e 1.

0 IW 2 0 0 300

4w

E I V l c r n ) o o o

The p o s s i b i l i t y t h a t t h e r a p i d r i s e i n o i s due t o i m p a c t i o n i z a t i o n a c r o s s t h e gap by h o t c a r r i e r s was suggested by ~ a h l e r t , ~ who was t h e f i r s t t o f i n d t h e s e l o w T n o n l i n e a r i t i e s i n TTF-TCNQ. The i d e a o f t h e c a r r i e r s b e i n g h o t a t a11 was d i s m i s s e d by CH on t h e ground t h a t i t would r e q u i r e "a m o b i l i t y

4 2

p > 1 0 cm / V sec, a v a l u e f o u r o r d e r s o f magnitude g r e a t e r t h a n found i n even t h e p u r e s t s i n g l e - c r y s t a l o r g a n i c semiconductors."' A h o t e l e c t r o n t h e o r y was never- t h e l e s s s e t up by B l o c h e t a1,6 w h i c h l e d t o a f i t t o t h e e x p e r i m e n t a l j vs F w i t h a l o w f i e l d m o b i l i t y p o o f 40 cm 2 / V sec a t 4K. H e a t i n g o f t h e c a r r i e r s was found t o o c c u r d e s p i t e t h e l o w p because energy l o s s was k e p t v e r y s m a l l b y i n c l u d i n g o n l y l o s s e s by e m i s s i o n o f o p t i c a l phonons. The i n c r e a s e i n o on t h i s t h e o r y a r o s e m a i n l y f r o m i m p a c t i o n i z a t i o n o f s h a l l o w i m p u r i t y l e v e l s . For t h e CH sample o f Fig. 2, however, even if we assume t h a t t h e 14K s l o p e r e p r e s e n t s such l e v e l s , r e l e a s e o f a l l t h e c a r r i e r s i n v o l v e d would i n c r e a s e o a t 4K by l e s s t h a n 3 o r d e r s o f magnitude.

U n t i l perhaps a y e a r and a h a l f ago t h e p o s s i b i l i t y o f h i g h p c a r r i e r s i n t h e s e c r y s t a l s was n o t t a k e n s e r i o u s l y . I t i s w e l l known t h a t a t room T i n TTF- TCNQ and o t h e r m e t a l l i c members o f i t s f a m i l y p = 1 t o 4 cm 2 / V sec. T h i s i s t h e r e s u l t o f c a r r i e r e f f e c t i v e masses mn and m o f t h e o r d e r o f t h e f r e e e l e c t r o n

P

mass mo and t h e l a r g e number o f phonon branches, due t o t h e l a r g e number o f atoms per u n i t c e l l . The m o b i l i t y i n c r e a s e s a p p r o x i m a t e l y q u a d r a t i c a l l y w i t h d e c r e a s i n g T down t o Tms, b u t i s s t i l l o n l y a c o u p l e o f hundred cm 2 / V sec a t 60K. Below Tms, however, t h e s i t u a t i o n i s r a t h e r d i f f e r e n t , i t was d i s c o v e r e d r e c e n t l y . 4

A l t h o u g h i t i s d i f f i c u l t t o measure d i r e c t l y H a l l o r d r i f t m o b i l i t y a t l o w temper- a t u r e s i n TTF-TCNQ, one can c a l c u l a t e t h e c a r r i e r c o n c e n t r a t i o n f r o m t h e known gap and t h e d e n s i t y o f s t a t e s , w h i c h we s h a l l d i s c u s s below. Combining t h a t w i t h t h e

4 2

measured o a t 10K, we found t h a t p = x 10 cm /V sec a t t h a t temperature.4 Note t h a t t h e c a r r i e r c o n c e n t r a t i o n i s v e r y s m a l l a t 10K, < _ 1 0 ~ ~ / c m ~ . Not l o n g a f t e r we o b t a i n e d t h e s e numbers, pH was measured i n ( T M S F ) 2 PF The most r e c e n t

5 2

r e p o r t f o r t h a t m a t e r i a l i s t h a t pH = 1 0 cm / V sec a t 4 ~ 6 ' Since t h e m a t e r i a l i s i n t r i n s i c t h e r e , t h e i n d i v i d u a l m o b i l i t i e s pn and a r e even l a r g e r .

P

C7-320 JOURNAL DE PHYSIQUE

What i s i t t h a t makes u so h i g h below Tms? P a r t o f t h e answer t o t h i s q u e s t i o n l i e s i n t h e r e l a t i o n s h i p between energy and t h e wave v e c t o r k. Above TmS s i n c e t h e s e m a t e r i a l s have r e l a t i v e l y narrow c o n d u c t i o n bands, t h e w i d t h

E

= 0.5 t o 1 e v, t h e energy

t a k e s t h e t i g h t - b i n d i n g f o r m

E~

= ( ~ ~ / 2 ) ( 1 - ^COS k b ) ( 1 1

where b i s t h e l a t t i c e spacing. Expanding t h e cosine, we f i n d t h e u s u a l q u a d r a t i c r e l a t i o n between

and k f o r s t a t e s near t h e bottom o f t h e band, w i t h an e f f e c -

2 2

t i v e mass m = 2 ).( / s o b . For TTF-TCNQ, m - ^{2 mo,} The gap a r i s e s from m i x i n g o f s t a t e s a t k and k - ^{2 kF.} P e r t u r b a t i o n t h e o r y g i v e s t h e energy o f s t a t e s below

a s

where 2 A i s t h e gap. Again, f o r s t a t e s c l o s e t o t h e gap

and t h e r a d i c a l can be expanded t o g i v e a q u a d r a t i c r e l a t i o n between Ek and (k-kF), k F b e i n g t h e wave v e c t o r a t t h e band edge, w i t h t h e e f f e c t i v e mass

a = (2 2 / s o b 2 ) [ ( E o / ~ ~ ) s i n 2 kFb i cos t F b I-' ^{( 3 )}

The f a c t o r i n t h e f i r s t p a r e n t h e s i s i s r e c o g n i z e d as t h e e f f e c t i v e mass a t t h e bottom o f t h e u n s p l i t band. For a q u a r t e r - f i l l e d band s i n kFb and cos kFb a r e 2-', and t h e t e r m 2 cos k b ( w h i c h g i v e s t h e d i f f e r e n c e between mn and m ) i s q u i t e

F P

s m a l l compared t o t h e o t h e r t e r m i n t h e a n g u l a r b r a c k e t . N e g l e c t i n g cos kFb, we see t h a t m below Tms i s s m a l l e r t h a n t h a t above by a f a c t o r

~ A / E ~ . T h i s f a c t o r i s q u i t e s m a l l . For TTF-TCNQ, where A s 200K, i t i s c 0.1 , w h i l e f o r (TMTSF)2 PF6, where A s 24K, i t i s l e s s t h a n 0.01. Thus a s m a l l mass i s a s s o c i a t e d w i t h a s m a l l gap, j u s t as i n InSb, f o r example. These s m a l l masses, p l u s t h e f a c t t h a t t h e v e r y many o p t i c a l branches o f t h e phonon spectrum a r e f r o z e n o u t , a r e what g i v e r i s e t o t h e h i g h m o b i l i t i e s a t l o w temperatures i n t h e s e m a t e r i a l s .

The h i g h m o b i l i t i e s suggest t h a t t h e dominant s c a t t e r e r s a r e a c o u s t i c pho- nons. We have c a l c u l a t e d LI due t o a c o u s t i c phonons, pat, and f i n d t h i s t o be t h e case.4 The c a l c u l a t i o n i s based on t h e usual p e r t u r b a t i o n t h e o r y . For t h e r - mal e l e c t r o n s , w h i c h a r e c l o s e t o t h e band edge, t h e square of t h e m a t r i x element v a r i e s as ( k - ^kF) 2 ^{/w, where w} i s t h e a c o u s t i c phonon frequency, i t s e l f p r o p o r - t i o n a l t o ( k - ^kF). These e l e c t r o n s i n t e r a c t w i t h l o n g wavelength phonons, f o r which n + 4 = kB T/

The o n l y s i g n i f i c a n t d i f f e r e n c e f r o m t h e 3-d case i s i n t h e d e n s i t y o f s t a t e s 9 ( =1/2 n dEk/dk). T h i s v a r i e s as ( k - k F ) - l , i.e., i n v e r s e l y as t h e square r o o t o f t h e energy measured f r o m t h e band edge, t h u s d e c r e a s i n g w i t h i n c r e a s i n g (k - kF) o r Ek f r o m t h e - v a l u e a t t h e band edge c h a r a c t e r i s t i c o f a I - d c r y s t a l . When t h e r e l a x a t i o n t i m e T i s c a l c u l a t e d f r o m t h e Golden Rule, t h e p r o d u c t o f t h e square o f t h e m a t r i x element, n + % and t h e

9

d e n s i t y of s t a t e s l e a d s t o T = ( k - kF), i.e., T = E', E r e p r e s e n t i n g t h e energy measured from t h e band edge. T h i s d i f f e r s f r o m t h e r e s u l t f o r t h e 3-d case, T E-%, p r e c i s e l y by t h e f a c t o r t h a t g i v e s t h e d i f f e r e n c e between t h e 3-d and 1-d d e n s i t i e s o f s t a t e s . P u t t i n g numbers i n t o t h e t h e o r e t i c a l e x p r e s s i o n

4 2

o b t a i n e d f o r pac,4 we f i n d uaCi 10 cm / V sec f o r TTF-TCNQ, i n good agreement w i t h t h e v a l u e deduced e a r l i e r from t h e measured a and c a l c u l a t e d c a r r i e r c o n c e n t r a - t i o n .

Knowing t h e s c a t t e r i n g mechanism and i t s energy dependence, we a r e i n a p o s i - t i o n t o c a l c u l a t e a as a f u n c t i o n o f F. We have so f a r done o n l y t h e s i m p l e s t t h e o r y , assuming an e l e c t r o n t e m p e r a t u r e Te which i s o b t a i n e d by e q u a t i n g t h e r a t e o f energy g a i n f r o m t h e f i e l d , ep ( T e ) F', t o t h e average r a t e o f l o s s t o a c o u s t i c phonons. From t h i s we o b t a i n f o r t h e change i n u,A p, a t a f i e l d F i n t h e warm e l e c t r o n r e g i o n

where v S i s t h e sound v e l o c i t y . W i t h vs = 3 x 1 0 cm/sec and p a l i t t l e o v e r 10 5 4 cm / V sec a t 4K ( o b t a i n e d by c o r r e c t i n g t h e v a l u e deduced a t 10K f o r v 2

T-%)

a c t h i s l e a d s t o a&uo

a few % a t 10V/cm, i n agreement w i t h t h e d a t a of Fig. 3.

For (TMTSFl2 PF6 au/uo o f a few % was found a t a much l o w e r f i e l d , 20 mV/cm, a t 4 ~ . I t was suggested t h a t t h e n o n l i n e a r i t y i n t h i s case i s due t o depinned ~ SDW's.' S t r o n g arguments a g a i n s t t h i s s u g g e s t i o n were p u t f o r w a r d by C h a i k i n e t a ~ We f i n d good agreement w i t h warm e l e c t r o n t h e o r y f o r t h i s case a l s o . . ~

5 2

The v a l u e o f pn o r u p deduced f r o m t h e measured pH o f 1 0 cm /V sec a t 4 ~ , * w i t h t h e d i f f e r e n c e i n mn and m t a k e n f r o m Eq. ( 3 ) , i s 6 x l o 6 cm 2 /V sec. When t h i s

P 5

i s i n s e r t e d i n Eq. ( 4 ) , w i t h vS = 3 x 10 cm/sec, t h e r e s u l t i s indeed Av/uo o f a few % a t 20 mV/cm.

For f i e l d s beyond t h e warm e l e c t r o n r a n g e t h e p a r a b o l i c a p p r o x i m a t i o n we have made f o r Ek and t h e d e n s i t y - o f - s t a t e s , a s w e l l as e q u i p a r t i t i o n f o r t h e phonons, a r e no l o n g e r v a l i d and c a l c u l a t i o n s must be done n u m e r i c a l l y . I t i s found t h a t uac i n c r e a s e s w i t h i n c r e a s i n g Te o n l y by a b o u t a f a c t o r 3 f o r TTF-TCNQ, and t h e n decreases w i t h f u r t h e r i n c r e a s e i n Te. How t h e n do we a c c o u n t f o r a n i n c r e a s e i n

a by a f a c t o r > 1 0 ? As n o t e d e a r l i e r , impact i o n i z a t i o n o f i m p u r i t i e s c o u l d n o t 4 a c c o u n t f o r such a l a r g e f a c t o r , a l t h o u g h we c a n n o t e l i m i n a t e t h e p o s s i b i l i t y t h a t i t accounts f o r some o f t h e i n c r e a s e . We suggest t h a t a good p a r t o f t h i s

i n c r e a s e i s due t o t h e b a r r i e r s t h a t must e x i s t . between r e q i o n s w i t h d i f f e r e n t

gaps, p o s t u l a t e d e a r l i e r t o e x p l a i n t h e c o n t i n u o u s l y v a r y i n g s l o p e o f t h e l o w

t e m p e r a t u r e l o g a vs. 1/T. A d d i t i o n a l evidence f o r a r a n g e o f gaps i s t h e c o n t i n -

uous background o f p h o t o c o n d u c t i v i t y and i n f r a r e d a b s o r p t i o n observed a t e n e r g i e s

below t h e gap. A1 so, f o r T < 20K t h e microwave a i s o r d e r s o f magnitude l a r q e r

t h a n dc a. A l t h o u g h t h e b a r r i e r s a r e n o t expected t o be l a r g e ( < ZOK), h e a t i n g

o f t h e c a r r i e r s would i n c r e a s e a o v e r a range o f f i e l d s by p e r m i t t i n g more

C7-322 JOURNAL DE PHYSIQUE

c a r r i e r s t o surmount t h e b a r r i e r s . W i t h f u r t h e r i n c r e a s e i n Te, c r e a t i o n o f e l e c - t r o n - h o l e p a i r s s h o u l d o c c u r , f i r s t across s m a l l e r gaps and t h e n a c r o s s p r o g r e s - s i v e l y l a r g e r gaps.

I n summary, we have d i s c u s s e d some mechanisms f o r n o n l i n e a r c o n d u c t i v i t y t h a t a r e u n i q u e t o quasi 1-d conductors, n o t a b l y depinned CDW's o r SDW's and + - p a r t i - c l e s . These e f f e c t s do n o t o c c u r i n TTF-TCNQ and r e l a t e d o r g a n i c compounds. Due t o t h e s m a l l e f f e c t i v e masses i n t h e s e m i c o n d u c t i n q s t a t e , h i g h m o b i l i t i e s and h o t c a r r i e r s a t f a i r l y l o w f i e l d s s h o u l d be a common phenomenon i n t h i s f a m i l y o f m a t e r i a l s . C a r r i e r h e a t i n g accounts w e l l f o r t h e i n i t i a l d e p a r t u r e s f r o m Ohm's l a w i n TTF-TCNQ, TSeF-TCNQ and (TMTSF)2 PF6, t h e o n l y compounds i n which n o n l i n e a r c o n d u c t i v i t y has been i n v e s t i g a t e d a t l o w temperatures. To o b t a i n a q u a n t i t a t i v e u n d e r s t a n d i n g o f l a r g e r d e p a r t u r e s r e q u i r e s more knowledge o f l o w - f i e l d c o n d u c t i o n a t l o w temperatures and of t h e phonon spectrum t h a n i s now a v a i l a b l e . Among t h e mechanisms t h a t must p l a y an i m p o r t a n t r o l e a r e surmounting o f b a r r i e r s between r e g i o n s of d i f f e r e n t gap and c r e a t i o n o f e l e c t r o n - h o l e p a i r s by i m p a c t i o n i z a t i o n .

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(1 980).

M.J. Rice, A.R. Bishop, J.A. Krumhansl and S.E. T r u l l i n g e r , Phys. Rev. L e t t . 36, 432 (1976).

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N.C. Banik, E.M. Conwell and C.S. Jacobsen, S o l i d S t a t e Commun. 3, 267 ( 1 981 1,.

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A.N. Bloch, T.F. C a r r u t h e r s , T.O. P o e h l e r and D.O. Cowan, i n "Chemistry and P h y s i c s o f One-Dimensional M e t a l s " , ed. H.J. K e l l e r (Plenum, N.Y. 1976) pp. 47-86.

P.M. Chaikin, G. Gruner, E.M. E n g l e r and R.L. Greene, Phys. Rev. L e t t . 45,

1874 (1980).

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