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Charge density wave transport in linear chain compounds
400
Physica 126B (1984) 400- 408 North-Holland. Amsterdam
CHARGE DENSITY WAVE TRANSPORT IN LINEAR CHAIN COMPOUNDS George GRUNER Department o f Physics, U n i v e r s i t y
of California,
o ~ Los Angeles, CA 900L4
Recent e x p e r i m e n t s focussing on the c o h e r e n t and i n c o h e r e n t aspects o f charge d e n s i t y wave t r a n s p o r t are discussed, and r e l a t e d to the t h e o r e t i c a l developments in the f i e l d . I.
INTRODUCTION In a wide range o f metals w i t h h i g h l y a n i s o t r o p i c band s t r u c t u r e e l e c t r o n - p h o n o n i n t e r a c t i o n s lead to an e l e c t r o n - h o l e condensate, c a l l e d the charge d e n s i t y wave(CDW) ( 1 ) . The condensate is c h a r a c t e r i z e d by a comp l e x o r d e r parameter, and the e l e c t r o n i c density,-c(x) f o r a one dimensional band s t r u c t u r e is g i v e n by: .a(x)
= ~oCOS (2k F x + ;)
(I)
where ~o is the a m p l i t u d e and =! is the phase o f the CDW. The p e r i o d is r e l a t e d to the Fermi w a v e v e c t o r by , = / k F and f o r an a r b i t r a r y band f i l l i n g the p e r i o d o f the CDW is incommensurate w i t h t h a t o f the u n d e r l y i n g lattice. The p o s i t i o n and time dependent phase p ( x , t ) is r e l a t e d t o the d e n s i t y of condensed e l e c t r o n s n and to the e l e c t r i c c u r r e n t j by ( a t T = O) J = n =
e
d,~ dt
(2a)
e
d. ~ dx
(2b)
i . e . the t r a n s l a t i o n a l motion o f the condensate leads to e l e c t r i c c u r r e n t w h i l e the comp r e s s i o n o f the mode r e s u l t s in a change o f n. The above r e l a t i o n s are d i f f e r e n t from those which r e l a t e the phase o f the macroscopic wave f u n c t i o n t o the c u r r e n t and chemical p o t e n t i a l in a s u p e r c o n d u c t o r . ( 1 ) In the absence o f p i n n i n g and damping which a c t on the CDW, an a p p l i e d e l e c t r i c f i e l d would lead to s u p e r c o n d u c t i v i t y , as f i r s t suggested by F r 6 h l i c h . This has, however, not been observed in v a r i o u s m a t e r i a l s which have an incommensurate CDW because i m p u r i t i e s pin the phase ¢ o f the condensate. Thus the dc c o n d u c t i v i t y is zero f o r small a p p l i e d dc electric fields. The e l e c t r o d y n a m i c s o f the condensate was f i r s t discussed by Lee, Rice and Anderson (2) and by Rice ( 3 ) . The f l u c t u a t i o n s o f the CDW can be d e s c r i b e d by an
a m p l i t u d e mode and by a phase mode. file amp l i t u d e modehas a gap and c o n s e q u e n t l y amp l i t u d e e x c i t a t i o n s can be n e g l e c t e d at low t e m p e r a t u r e s . With no p i n n i n g and damping tile phase mode leads to a frequency d e p e n d e n c e , i . e . a gap in the s i n g l e p a r t i c l e spectrum and c o l l e c t i v e mode a t zero frequency s i m i l a r to t h a t o f a s u p e r c o n d u c t o r . Including p i n n i n g and damping in a phemomenological way the frequency dependent c o n d u c t i v i t y 2 9 r. n_e_. :.L/'T~ . , = i 2 iiTT;'/ ' T H "A"
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* This research was supported by the NSF Grants DMR 81-21394 and DMR 84-06896.
0378-4363/84/$03.00 ¢]) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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where and ~ are the (phenomenological pinn i n g frequency and damping c o n s t a n t . In E<. (3) the d i s t r i b u t i o n o f p i n n i n g e n e r g i e s which may a r i s e as the consequence of the randomly d i s t r i b u t e d i m p u r i t i e s , is n e g l e c t e d , p i n n i n g is r e p r e s e n t e d by an average frequency m* is the e f f e c t i v e mass o f the condenSate. In terms o f the s i n g l e p a r t i c l e gap band mass m el ectron-phonon c o u p l i n g constant and phonon frequency . u!*/m = 1 + 4 . . . . . . . . ~,r typical transi2 2 2 kF t i o n t e m p e r a t u r e s and c o u p l i n g c o n s t a n t m * / m 102 - 103 . For a small c o n c e n t r a t i o n of i m p u r i t i e s , the p i n n i n g energy per e l e c t r o n may be o r d e r s o f magnitude s m a l l e r than ~ the s i n g l e p a r t i c l e e n e r g i e s (bandwidth and bandgap). This leads to s t r o n g l y frequency dependent c o n d u c t i v i t y and t o a g i a n t d i e l e c t r i c cons t a n t a s s o c i a t e d w i t h the ac response o f t h ( pinned CDW mode. The l a t t e r is g i v e n by (see Eq. ( 3 ) ) .
(4)
G, Gruner
/ Charge density wave transport
When f i n i t e dc e l e c t r i c f i e l d s are applied the energy provided by the dc f i e l d to the system may exceed the pinning energy. Various evidences suggest t h a t when t h i s happens a current c a r r y i n g CDW s t a t e , represented by the t r a n s l a t i o n a l motion of the condensate develops. The threshold f i e l d ET f o r the onset o f n o n l i n e a r conduction is p r o p o r t i o n a l to the r e s t o r i n g forces associated with the pinning, and consequently ET is c l o s e l y r e l a t e d to the low frequency d i e l e c t r i c constant. Various models (4) lead to the r e l a t i o n ~ET = const x e
(6)
This leads to a displaced Fermi surface, with gaps appearing at -kF+ q and kF+q with hq = mvd. The energy d i f f e r e n c e between the r i g h t and l e f t hand side of the Fermi surface E = 2kFhvd is p r o p o r t i o n a l to the current j = nev d. The frequency associated with processes by which e l e c t r o n s are t r a n s f e r r e d from one side of the Fermi surface to the o t h e r , h~, : AE is r e l a t e d to the current density per chain JCDW by JCDW
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where the constant is of the order of I , and i t depends on the model which describes the depinning process. For a current c a r r y i n g condensate, moving with a d r i f t v e l o c i t y v d ~o(x) = Po cos((2k F x - Vd ) + ~ )
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FIGURE 1 Rea(m) and Imp(m) in TaSk. The f u l l and dotted l i n e s are f i t s to a harmonic o s c i l l a t o r response with parameters given on the f i g u r e .
o where n(T) is the temperature dependent condensate d e n s i t y , vo = m The main features of the frequency and f i e l d dependent response together with the current o s c i l l a t i o n s with frequencies given by Eq. (7) have been observed in a wide range of inorganic chain compounds which show phase t r a n s i t i o n s to incommensurate CDW states ( I ) . In c e r t a i n t r a n s i t i o n metal t r i c h a l c o g e n i d e s , MX3, halogenated tetrachalcogens (MX4)vH and bronzes (of which the blue bronze K0 3MoO3 is the best known example) the CDW develops along the chain d i r e c t i o n , below the phase t r a n s i t i o n the CDW modulation on the neighboring chains are c o r r e l a t e d . The main experimental features obtained in these m a t e r i a l s are summarized in several reviews, ( I ) here I give only a few examples of the observed behaviors. Fig. 1 shows Re~(~) and Im~(~) f o r TaS3, the f u l l and dotted l i n e s are f i t s to Eq. ( 3 ) , with the i n e r t i a l terms neglected. The broad overa l l agreement suggest, t h a t in terms of a c l a s s i c a l d e s c r i p t i o n , the mode displays an overdamped response. The e l e c t r i c f i e l d dependent dc c o n d u c t i v i t y ~(E) = I/V is shown in Fig. 2. Below ET, ~ is due to normal e l e c trons which are e x c i t e d above the s i n g l e par-
t i c l e gap, the excess c o n d u c t i v i t y above ET is due to the current carrying condensate. O s c i l l a t i n g currents (or e l e c t r i c f i e l d s ) depending whether constant current or constant voltage source is used have been observed both in the time and in the frequency domain and Fig. 3 shows the. r e l a t i o n between JCDW and v o in NbSe3 wlth temperature dependence of the r a t i o in the i n s e r t . (5)
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FIGURE 2 E l e c t r i c f i e l d dependent c o n d u c t i v i t y in TaS 3
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Zco~(/z~', FIGURE 3 The frequency of the c u r r e n t o s c i l l a t i o n s versus the c u r r e n t c a r r i e d by the CDW. The i n s e r t shows ICDW/f ° versus temperature. In t h i s t a l k I w i l l discuss some f u r t h e r experiments on CDW t r a n s p o r t and analyze them in terms of equations f a m i l i a r f o r an audience w i t h background on s u p e r c o n d u c t i v i t y and Josephson phenomena. A phenomenological equation of motion d e s c r i b i n g the main experimental f e a t u r e s mentioned before t r e a t s the condensate as a r i g i d o b j e c t w i t h i n t e r n a l dynamics n e g l e c t e d . Due to the i n h e r e n t p e r i o d i c i t y o f the c o l l e c t i v e mode, the pinning p o t e n t i a l is assumed to be p e r i o d i c . Consequently, the model is t h a t of a viscous motion of a p a r t i c l e in a p e r i o d i c p o t e n t i a l (6) d2x + I dx o2 e'E (~!) dt 2 dt 2k F sin2kFX = -
- -
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is given by d2 d ~ G d t + sin dt 2
~
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where e' and i~f are the e f f e c t i v e charge and mass of the condensate, and the p o t e n t i a l is assumed to be s i n u s o i d a l . The low f i e l d ac response is t h a t of a harmonic o s c i l l a t o r (see Fig. l , and Eq. ( 3 ) ) . Nonlinear d# conduction occurs f o r f i e l d s E > ET = ~ ~ L , and current o s c i l l a t i o n s r e s u l t as2the ~ r t i c l e moves down the t i l t e d s t a i r c a s e p o t e n t i a l , w i t h retion (7). Eg. (8) can be reduced to a dimens i o n l e s s form d2. 4~ E dt 2-- + " d-t + s i n = ET (9)• -l with time measured in u n i t s of and !' = ( u ~ ) - l . Equation (9) is f o r m a l l y i d e n t i c a l to the equation f o r the r e s i s t i v i t y shunted Josephson j u n c t i o n , which, in the usual n o t a t i o n
= ]
( l. }
The t h r e s h o l d f i e l d f o r CbW conduction ET cot'responds to the c r i t i c a l c u r r e n t , I i and curr e n t o s c i l l a t i o n s w i t h JCDW p r o p o r t l o n a l to correspond to the ac Josephson e f f e c t . One should emphasize, however, t h a t at t h i s p o i n ! there is only a formal correspondence between the two nonlinear" equations of motion, n the f i r s t part of the t a l k I w i l l demonstrate t h a t t h i s formal analogy can be used to analyze o t h e r phenomena, which a r i s e in the presenc~ of j o i n t ac and dc d r i v i n g f i e l d s . In the second p a r t e f f e c t s associated w i t h the i n t r i n s i c randomness, and the consequent i n c o herent phenomena w i l l be d e s c r i b e d . 2. EXPERIMENTS IN THE PRESENCE Of JOINT £r: £f~D DC DRIVING FIELDS WiUI s t r o n g l y n o n l i n e a r und frequency dependent response a broad v a r i e t y of e x p e r i ments, i n v o l v i n g the j o i n t a p p l i c a t i o n s of ac and dc f i e l d s of various amplitude can be performed to t e s t f u t h e r the dynamics of the c o l l e c t i v e mode. Several f e a t u r e s , such as the o v e r a l l m o d i f i c a t i o n of the dc and/or ac response (ac f i e l d induced dc conduction or dc e l e c t r i c f i e l d dependent ac d i e l e c t r i c constant and c o n d u c t i v i t y ) can be described using sii~iple n o n l i n e a r c i r c u i t theory by using ( ) and ( E ) as i n p u t p a r a m e t e r s . ( 7 ) Such d e s c r i p t i o n is perhaps also a p p r o p r i a t e f o r some of the mixing and r e c t i f i c a t i o n experiments performed by the I l l i n o i s grou~. (8) Here I w i l l focus on experiments where i n t e r f e r e n c e phenomena between the i n t r i n s i c o s c i l l a t i o n and the a p p l i e d ac f i e l d s are s t u d i e d . The a p p l i e d f i e l d is a combination of dc and ac e x c i t a t i o r l s V = Vdc + Vae cos ext{ and Vdc , Mac or" e x t can be v a r i e d during the experiments, w i t h the dc and ac response being detected. 2,l. I n t e r f e r e n c e phenomena Instead o f d e t e c t i n g the ac Josephson e f f e c t d i r e c t l y , the e f f e c t was f i r s t demons t r a t e d through the m o d i f i c a t i o n of the dc c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s by an a p p l i e d microwave f i e l d . Steps o c c u r r i n g due to l o c k ing between the e x t e r n a l and i n t e r n a l frequencies are c a l l e d the Shapiro steps in the Josephson l i t e r a t u r e . Corresponding e x p e r i ments were f i r s t performed on CDW systems by Monceau et a l . ( g ) . In pure specimens w i t h small dimensions, the steps can d i r e c t l y be detected by recording the I-V curves in the presence of a p p l i e d ac f i e l d . Fig. 4 shows (lO) such I-V traces f o r Vac of various a m p l i t u d e . Steps are observed in the n o n l i n e a r c o n d u c t i v i t y region~
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FIGURE 5 Step height 6V versus ac amplitude Vac f o r Shapiro steps in NbSe3. The r f frequency is 210 MHz and the step index n = 1, The s o l i d l i n e is the p r e d i c t i o n of the c l a s s i c a l model, Eq, (11), with parameters ~o2~/2~ = 80 MHz, VT = 24 mV, ~ = 0.17.
TdC(fl : IOOMNz) - 100
100
0 dc current < I >
(~A)
FIGURE 4 Dc I-V traces f o r NbSe3 in the presence of an applied r f f i e l d at frequency ~/2~ = I00 MHz and of amplitude Vac. The stop height ~V is defined in the f i g u r e . No Shapiro steps are observed f o r Vac = O, while the maximum step height is at approximately Vac = I00 mV. The arrow i n d i c a t e s the dc current which y i e l d s a fundamental noise frequency f l = I00 MHz. n is the step index. and the step height 6V is a strong f u n c t i o n of Vac. The step defined as n = 1 corresponds to an i n t e r f e r e n c e between the I00 MHz r f f i e l d and the fundamental of the i n t r i n s i c o s c i l l a t i o n . Harmonic and subharmonic steps with smaller amplitude can also be observed. Eq. (9) leads, in a high frequency (~oext > ~) l i m i t to ~V =~2VT(Vac = O) J n l V e f f l
;
(ll)
where c~ represents the volume f r a c t i o n of the specimen which responds coherently to the external f i e l d s , VT the threshold voltage and in Jn the Bessel f u n c t i o n of order n. Computer simulations in the frequency region ~e~f ~ ~o2~ also lead to o s c i l l a t i o n s of ~V as fu~i~tion of V e f f , c l o s e l y resembling Bessel f u n c t i o n behaviors. Fig. 5 shows the n=l step height as the f u n c t i o n of Vac. The Bessel func t i o n behavior, (also observed in Josephson j u n c t i o n s ) is c l e a r l y recovered, with ~o2~ values close to those observed in o(~) studies. The
c h a r a c t e r i s t i c values of ~ are between 0,1 and 1 suggesting h i g h l y coherent response. Experiments, performed as the f u n c t i o n of frequency are also in agreement with the model. I n t e r ference e f f e c t s show up also in the ac response. Experiments, performed as the f u n c t i o n of f r e quency are also in agreement with the model. I n t e r f e r e n c e e f f e c t s show up also in the ac response, and Fig. 6 shows the d i e l e c t r i c constant at 3.2 MHz as a f u n c t i o n of dc bias. The large i n d u c t i v e dips, and associated steps in Reo(;~) can also be accounted f o r (10). 2.2, D e v i l ' s s t a i r c a s e behavior The s e n s i t i v i t y of d e t e c t i o n of the i n t e r ference phenomena can be enhanced by d i s p l a y ing the d i f f e r e n t i a l resistance dV/dI. Peaks in the d e r i v a t i v e correspond to steps in the I-V curve, the width of the peak in voltage u n i t s corresponding to the height of the step. Fig. 7 shows (11) the d i f f e r e n t i a l resistance measured with and without the e x t e r n a l l y app l i e d r f voltage ~ext/2~ = 25 MHz. There appears a series of peaks at applied dc current such t h a t Pro = q
U'ext 2~
with p and q i n t e g e r s . Some of the peaks are i d e n t i f i e d on the f i g u r e . This i d e n t i f i c a t i o n was made by p l o t t i n g ICDW versus ~ext and checking t h a t the slopes were p/q times t h a t f o r the fundamental. Q u a l i t a t i v e l y , the width of the peaks decreases with increasing p and q as expected f o r mode locking between two frequencies both with slowly decaying harmonic content. The peaks consequently correspond to regions in which any harmonic of the i n t e r n a l frequency
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V (mV) FIGURE 6 The d i e l e c t r i c constant (3.2 MHz) as the funct i o n of dc b i a s . The l a r g e i n d u c t i v e dip corresponds to the i n t e r f e r e n c e between the fun ~ damental and the a p p l i e d dc f i e l d .
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FIGURE D i f f e r e n t i a l r e s i s t a n c e vs. dc sample v o l t a g e (a) w i t h and (b) w i t h o u t a p p l i e d r f v o l t a g e at 25 MHz. The peaks in (a) correspond to steps in the d i r e c t I-V curve. A few peaks are ident i f i e d by p/g (see t e x t ) .
locks to any harmonic of thu e x t e r n a l f i e l d . Such mode l o c k i n g has been e x t e n s i v e l y studied (12) f o r the equations of motion (9) and ( ] 0 ! and both numerical and a n a l y t i c a l s o l u t i o n s are a v a i l a b l e . In g e n e r a l , a D e v i l ' s s t a i r case behavior in which i n t e r f e r e n c e occurs ! ~ any p and q i n t e g e r is recovered by such (:,~i-c u l a t i o n s for" a weakly damped system. FoY strong damping i n t e r f e r e n c e is weak and rio subharmonics are expected f o r an overdamped response. D e v i l ' s s t a i r c a s e behavior is c l e a r l y i m p l i e d by our experimental r e s u l t b~J~ due to noise e f f e c t s higher order i n t e r f e r e n c , . peaks cannot be r e s o l v e d . The e f f e c t s of two competing trequencies in d i s s i p a t i v e systems has r e c e n t l y been i n v e s t i g a t e d by means of ~ c i r c l p map. (~3) Again, frequency l o c k i n g occurs for" a l l r~t i o n a l numbers p and q, w i t h r e l a t i v e ampli tude close to what is observed e x p e r i m e n t a l l y . Furthermore, the completeness of the s t a i r ~ case can be i n v e s t i g a t e d by looking at space; unoccupied by steps l a r g e r than a d i s c r i m i n a t i o n level r . Analysis in terms o~ such des c r i p t i o n suggest t h a t the D e v i l ' s staircase. is complete, in o t h e r words a chaotic behavior is observed. Other m a n i f e s t d t i o n s of the no~l i n e a r e q u a t i o n , such as period doubling t r a n s i t i o n to chaos and i n t e r m i t t a n c y here also has been observed in some of these m a t e r i a l s , b~ the Berkeley group. (14) Although many unresolved questions remain, in p a r t i c u l a r , coHcerning the importance ot i n e r t i a e f f e c t s , the r o l e o f the normal e l e c t r o n s , the e f f e c t of noise e t c . , the n o n l i n e a r phenomena assuciated w i t h CDW t r a n s p o r t appear to be a f r u i t f u l t e s t i n g ground f o r many o{ the t h e o r e t i c a l p r e d i c t i o n s concerning chaos and t u r b u l e n c e . 3. COHERENT VERSUS INCOHERENT EFFECTS Incoherent e f f e c t s associated w i t h i n t r l n sic randomness, f i n i t e length scales over which the phase-phase o r c u r r e n t - c u r r e n t correlations p e r s i s t , or thermal noise have been completely ueglected in the previous a n a l y s i s . I t is e v i dent however t h a t randomness and the internaE deformations of the condensate play an import a n t r o l e . dS f o r ,J r i g i d condensate which int e r a c t s w i t h random i m p u r i t i e s the pinning energy vanishes in the thermodynamic l i m i t . ( ! ! A l t e r n a t i v e l y , when local deformations around i m p u r i t i e s are taken i n t o account, the pi~asephase c o r r e l a t i o n f u n c t i o n shows an exponential decay, l e a d i n g to the absence of long range order. Indeed, various experiments suggest: t h a t d i s o r d e r plays an i m p o r t a n t r o l e also in the dynamics of the c o l l e c t i v e mode,and some of the f e a t u r e s observed are s i m i l a r to those modes on random systems l i k e g l a s s e s , spin g l a s s e s , g r a n u l a r superconductors, e t c . [ wil': discuss evidences f o r such incoherence in the case o f a pinned mode, and also f o r the c u r r e n t c a r r y i n g mode and f o r phenomena which ar~
G. Gruner I Charge density wave transport
associated with the t r a n s i t i o n between the pinned and the current carrying states. 3.1. The pinned mode: frequency dependent response and metastable states The simple harmonic o s c i l l a t o r response, given by Eq. (3) gives a good overall account for the main features of the observed f r e quency dependent c o n d u c t i v i t y , see Fig. I . This description predicts that in the ~ ÷ o limit Red(~)
......
2
/
2 ~
ToS~ i20K
~ ~ "
1
/
,~
ii
%0 b -1
j.~:r
/Re~Imw
.,.#K'
Im~(~) ~
-2
Experiments conducted at low frequencies show d r a s t i c deviations from such behavior (15). Fig. 8 shows Red(o~) - Red(~ = O) and Im~ (~) for TaS3 at T = 12OK. At low frequencies both approach a power law, and the c o n d u c t i v i t y can be described by o(~) = A(i~) ~
405
(12)
over several decades in frequency. Similar behavior was found in other materials, with < 1 in a l l cases (16). While the observed frequency dependence is in clear c o n f l i c t with a single harmonic o s c i l l a t o r description, i t also is in c o n f l i c t with calculations based on the Fukuyama-Lee model (17) when both the i n ternal deformations and i n t e r a c t i o n s with the randomly d i s t r i b u t e d i m p u r i t i e s are included. Such calculations (18) lead to Reo(~)~21nw2, and Im~(~) ~ ~, the famous Mott-Bereskinskii result. Our results however are surprisingly similar to those made on glasses and disordered solids (19) where a behavior given by Eq. (12) is obtained in the low frequency l i m i t . Therefore, i t is believed that the low frequency behavior, displayed in Fig. 8 indicates the importance of disorder in the dynamics of the pinned collective mode. The amplitude A of the observed ~dependance, given by Eq. (12) is several orders of magnitude larger than that observed with other disordered systems where t r a n s i t i o n s between l o c a l i z e d single p a r t i c l e states leads to the observed ~ dependence. As the p o l a r i z a t i o n , and consequently the dielect r i c constant is proportional to the relevant length scale £ over which such p o l a r i z a t i o n can be b u i l t up, the experiments on CDW materials suggest rather large length scales involved in the ac response. There are also several observations on metastable states associated with the pinned mode, and extremely slow t r a n s i t i o n s between these metastable states (20). These have been studied by looking at the time dependent phenomena a f t e r a heat pulse or elect r i c f i e l d pulse applied to the specimens. In both cases phenomena rather s i m i l a r to those observed in spin glasses have been detected.
/
=j===- 2~
/
/ ~
/
/,~.
2
i
I_
//i
4 I
o ~
• Iwe plebe • two probe brldg
= fourp,obebridge
u
i
6
I
,
8
w
FIGURE 8 Reo(~) and Imo(~) versus frequency for TaSk. The f u l l l i n e s represent a Cole Cole l i k e ~xpression ~(~) =?=(i~T) [I + ( i ~ ) ~] -I with = 0.87 and ~-I/2~ = 25 MHz. The dashed l i n e gives the low frequency l i m i t o5 the harmonic o s c i l l a t o r response, Re~(~) ,~ ~z. 3.2.
The current carrying mode: incoherent noise phenomena Eq. (8) leads to a t o t a l l y coherent response in the current carrying mode: the spect r a l width of the current o s c i l l a t i o n s is i n f i n i t e l y sharp, the amplitude of the current o s c i l l a t i o n s is independent of the sample volume and there is no broad band noise associated with the s l i d i n g CDW mode. All of these predictions are in clear disagreement with the experimental observations. A dramatic increase of the low frequency (kHz) broad band noise (21) accompanies the onset of s l i d i n g CDW motion, as observed f i r s t in NbSe3 and subsequently in other compounds showing c o l l e c t i v e mode conduction. The f r e quency dependence can be described by an f-~ behavior, with ~ somewhat less than 1 with noise amplitudes orders of magnitude larger than those observed in random systems whose transport is due to single p a r t i c l e conduction. The broad band noise is also only weakly temperature dependent suggesting that temperature driven f l u c t u a t i o n s do not play an important r o l e . Broad band noise is also more dominant in specimens where the current o s c i l l a t i o n s are less evident, suggesting an i n t e r p l a y between coherent and incoherent e f f e c t s .
(/. (}PltHdt" / ('/ldr,~,d ddtl.~,llF ~'~',~/1'(' IrdlL~7){~Yl
406
These experiments suggest t h a t broad band noise is d i r e c t l y r e l a t e d to the dynamics of the i n t e r n a l deformations o f the CDW mode. In an o v e r s i m p l i f i e d p i c t u r e o f N domains f l u c t u a t i n g i n d e p e n d e n t l y , the c u r r e n t f l u c t u a t i o n s ' I are given by <~12>
v e l o c i t i e s is more l i k e l y , in agreement w i t h the experimental o b s e r v a t i o n s .
IOOr--
J !
12 2 where 61 represents the i n t e g r a l of c u r r e n t f l u c t u a t i o n s over a l l f r e q u e n c i e s . In the absence of r e l i a b l e experiments a v a i l a b l e in a broad frequency range and of models which c l e a r l y r e l a t e ,~I to the a v a i l a b l e degrees of freedom no q u a n t i t a t i v e a n a l y s i s is a v a i l a b l e . The l a r g e amplitude broad band noise however again suggest a small number of domains, i . e . l a r g e length scales associated w i t h these domains, in q u a l i t a t i v e agreement w i t h the r e s u l t s of the ~(:,~) s t u d i e s . The amplitude of the c u r r e n t o s c i l l a t i o n s was s t u d i e d r e c e n t l y as the f u n c t i o n of the volume of the specimens both in NbSe3 (22) and in (TaSe4)21 (23). These experiments c l e a r l y e s t a b l i s h t h a t the amplitude o f the c u r r e n t o s c i l l a t i o n s decreases w i t h i n c r e a s i n g volume, suggesting t h a t the o s c i l l a t i o n s are f i n i t e size e f f e c t s which would not be observed in the thermodynamic i m i t . Fig. 9 shows the o s c i l l a t i o n amplitude Lj I versus sample volume in NbSe3. The volume dependence o f the o s c i l l a t i o n amplitude can well be represented by a
•
•
•
.
d I%
|
!
FIGURE 9. O s c i l l a t i o n amplitude J l in NbSe3 versus sam ple v o l u m e ' The l i n e represents Ilj ] = (3 X lO-4A c m - ½ ) - ~ .
NbSe3
JCDW (i i . e . a square r o o t dependence s t r o n g l y sugg e s t i n g random phase c o r r e l a t i o n s in the current c a r r y i n g s t a t e . The amplitude of the current o s c i l l a t i o n s can be estimated f o r comp l e t e l y coherent response, and assuming domains o f size :;tc w i t h randomly c o r r e l a t e d phases, the domain size is estimated to be of the o r d e r of l;~m3. The F o u r i e r spectrum of the c u r r e n t o s c i l l a t i o n s also has a f i n i t e s p e c t r a l w i d t h , sugg e s t i n g a d i s t r i b u t i o n of charge d e n s i t y wave v e l o c i t i e s w i t h i n the specimens. This d i s t r i bution is more i m p o r t a n t in specimens w i t h h i g h e r a n i s o t r o p y and the width also increases w i t h i n c r e a s i n g cross s e c t i o n of the specimens. Fig. lO where the F o u r i e r s p e c t r a of the c u r r e n t o s c i l l a t i o n s observed f o r NbSe3 and f o r (TaSe4)2)I is compared. The cross sections are comparable f o r the two samples, but there is a s u b s t a n t i a l d i f f e r e n c e between the a n i s o t r o p i c s of the band s t r u c t u r e : w h i l e in NbSe3 the a n i s o t r o p y of the bandwidth is a p p r o x i mately o f 3 in (TaSe4)21, i t is more than an o r d e r of magnitude l a r g e r . This d i f f e r e n c e leads to d i f f e r e n c e s in the coupling strengh between n e i g h b o r i n g chains, f o r weaker coupling and f o r l a r g e r specimens a d i s t r i b u t i o n o f CDW
o o_
0
2
4
5 z
S
(Ta Se4)2 I
I
I
I0 f (MHz)
FIGURE lO F o u r i e r t r a n s f o r m of the c u r r e n t o s c i l l a t i o n s in (TaSe4)21 and in NbSe3.
G. Gruner / Charge density wave transport
3.3. The onset of current carrying state The onset of nonlinear conduction, as predicted by Eq. (8) is smooth, divergence is obtained only in the d e r i v a t i o n dl/dV in cases of small i n e r t i a . Large i n e r t i a leads to both a discontinuous jump in the I-V curve and to hysteresis. Such hysteresis effects have been recently observed in several materials. There is however a rather long time scale involved in t h i s switching phenomena, typical values for the time T to switch to the current carrying state can be as long as several milliseconds (24). Fig. I I shows a typical I-V curve obtained in TaS3 (25) where switching from the pinned to the current carrying state (and back) occurs. No noise is observed in the pinned region, but large low frequency broad band noise appears in the state with low resistance, i . e . when the CDW state contributes to the conduction. These observations suggest that switching and hysteresis are not related to the i n e r t i a l a f f e c t s , rather they may be related to the fact that various coherent CDW regions (domains) have f i r s t to be coupled together before the current carrying state can develop (26). As the domains most probably have macroscopic dimensions, t h i s leads to long times associated with the rearrangement of the domains, which has to occur before the current carrying state can develop.
/
ToS3 #2
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1oo H
25
200
~,oo j
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20
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I
I
/ ~ i
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....
I
I
200
150
Us (mY)
FIGURE I I Direct I-V traces near to the onset of nonl i n e a r conduction in TaS3. At the high f i e l d region, the I-V plot shows i r r e g u l a r f l u c t u a tions due to the broad band noise.
407
4. CONCLUSIONS The remarkable i n t e r p l a y between coherent and incoherent e f f e c t observed in CDW transport is most probably strongly related to the macroscopic length scales which characterize the s t a t i c s and dynamics of charge density waves and to the f i n i t e dimensions of the specimens which are investigated. The r e l e vant c h a r a c t e r i s t i c length scales are the phase-phase c o r r e l a t i o n length (27) <¢(x)¢(0)> ~ exp(- ~ ) characterizing the decay of phase c o r r e l a t i o n in the p~nned ~tat~. 2Here L = h2v F /(4k F )Vo Po ni where Vo is the strength of the impurity potential and ni ~s the impurity concentration. With Vo ~ I0 -Z eV and n i ~ I0 ppm. L ~ 10-2 cm, comparable to typical sample dimensions. The currentcurrent c o r r e l a t i o n length,
< ~de( x ) ~ ( ode)
>
may be s u b s t a n t i a l l y larger than the phasephase c o r r e l a t i o n length L. This l a t t e r is relevant for observations in the current carrying state. The narrow width of the spectrum of current o s c i l l a t i o n s observed in NbSe3 (see Fig. I0) suggests a current-current c o r r e l a t i o n length comparable to the sample dimensions, in other materials such as (TaSe4)21 i t has a much shorter scale. While i t is f i r m l y established that L is f i n i t e in less than four dimensions, long range order could be established in a driven random system. Such long range order is implied by recent experiments by Zettl et a l . (28), (29). Recent theories, (30) which take the e f f e c t of disorder into account lead to marked improvement over the description where the dynamics of the internal deformations is neglected. Besides accounting for the dependence of the parameters on the impurity concentrat i o n s , such theories lead to the removal of spurious divergences in dl/dv and d~/dV at ET, and also account for metastable states, for the d i s t r i b u t i o n of pinning energies as implied by ~(~) studies etc. F i n a l l y we remark that the analysis outl i n e d here is based on a classical equation of motion with quantum aspects of CDW dynamics neglected. Explanations of a large body of experimental r e s u l t s are also possible in terms of a quantum mechanical description of CDW dynamics, (31) and the r e l a t i o n between the parameters which characterize the and E dependent response (such as given by Eq. (5) for example) is s i m i l a r w i t h i n the framework of both descriptions (4). I t is expected that a combination of classical and quantum concepts w i l l lead to a f u l l explanation ~f the b~oad variety of phenomena
408
G. Gn~nur / (71urg'u dc~Lsity wurc lt'~l~l.~'l~¢)J'l
associated with the c o l l e c t i v e response of charge density condensates.
(17)
(I8) ACKNOWLEDGEMENTS Experimental results reported in this paper were obtained in c o l l a b o r a t i o n with L. Mihaly, G. Mozurkewich and A. Z e t t l . The author acknowledges useful discussions with John Bardeen, T. Holstein and A. Janossy.
(19) (20) (21)
REFERENCES (1)
(2) (3) (4) (5) (6) (7)
(8) (9) (lO)
(ll) (12) (13) (14)
15) 16)
P. Monceau, Physica B + C I09 (1982) 1890, N.P. Ong, Can.J. Phys. 60 (1982) 757, G. Gruner, Comments in Solid State Phys. lO (1983) 183, John Bardeen, Lectures for the Internat. School of Physics "Enrico Fermi" Varenna, I t a l y , July 4-6, (1983). P.A. Lee, T.M. Rice and P.W. Anderson, Solid State Comm. 14 (1974) 703. M. Rice, in Low Dimensional Cooperative Phenomena, ed. H.J. K e l l e r , (Plenum Press, New York, 1975). Wei-Yu Wu, A. Janossy and G. Gruner, Solid State Comm. 49 (1984) I I . John Bardeen, E. Ben Jacob, A. Zettl and G. Gruner, Phys. Rev. Lett. 49 (1982) 493. G. Gruner, A. Zawadowski and P.M. Chaikin, Phys. Rev. Lett. 49 (1981) 511. G. Gruner, W.G. Clark and A.M. Portis, Phys. Rev. Lett. B24, (1981) 3641, A. Zettl and G, Gruner, Phys. Rev. B25, (1982) 2081, J.R. Tucker, J.H. M i l l e r , K. Seeger and John Bardeen, Phys. Rev. Lett. B25, (1982) 2979. J.H. M i l l e r , J. Richard, J.R. Tucker and John Bardeen, Phys. Rev. Lett. 51 (1983) 1592. P. Monceau, J. Richard, and M. Renard, Phys. Rev. Lett. 45 (1980) 43. A. Zettl and G. Gruner, Phys. Rev. B29, (1984) 755, A. Zettl and G. Gruner, Solid State Comm. 46, (1983) 501. S. Brown, G. Mozurkewich and G. Gruner, Phys. Rev. Lett 52, (1984) 2277. P. Bak and M. Hoegh Jensen, J. Phys. Al5, (1983) (1982). P. Bak, in p r i n t . R.P. Ha l l , I.M. Shervin and A. Z e t t l , Phys. Rev. B29, (1984) 7076, I.M. Shervin, R.P. Hall and A. Z e t t l , Phys. Rev. L e t t . 52, (1984) 2293. W.Y. Wu, L. Mihaly, G. Mozurkewich and G. Gruner, Phys. Rev. Lett.52 (1984) 2382. G. Gruner, Proceedings of the I n te r n a t . Conference on Nonlinear Transport in Inorganic Linear Chain Compounds, Sapporo, Oct. lO, 1983.
22)
(23) (24)
(25) (26) (27)
(28) (29)
(30)
(31
H. Fukuyama and P.A. Lee, Phys. Rev. Bl7 (1978) 535. M.V. Eeigelman and V.M. Vinokur,Phys. Rev. Lett. 87A (1981) 53. K.L. Ngai, Comments on Solid State Phys. 9 (1979) 4. G. Mihaly and L. Mihaly, Phys. Rev. Left. 52 (1984)149. J. Richard, P. Monceau, M. Papoular and M. Renard, J. Phys. Cl5 (1982) 7157, A. Maedo, M. Naito and S. Tanaka in print, A. Zettl and G. Gruner, Solid State Comm. 46 (1983) 29. G. Mozurkewich and G. Gruner, Phys. Rev. Lett. 5l (1983) 2206. The detailed dependence of the oscillatior~ amplitude is controversial at present. I t has recently been suggested that i t is independent of the length of specimens, suggesting that current o s c i l l a t i o n s are generated at the contacts, N.P. On9, et a l . , Phys. Rev. Lett. 52 (1984) 663. G. Mozurkewich, M. Maki and G. Gruner, Solid State Comm. 49 (1983) 5. A. Zettl and G. Gruner, Phys. Rev. B26 (1982) 2298. L. Mihaly and G. Gruner, Solid State Comm. 50 (1984) 807. B. Joos and ~. Murray, Phys. Rev. B29 (1984) I094. P.A. Lee and T.M. Rice, Phys. Rev. BIg (1979) 3970. A. Z e t t l , M.B. Kaiser" and G. Gruner, Solid State Comm. in p r i n t . R. Hall and A. Z e t t l , in p r i n t . D.S. Fisher, Phys. Rev. Lett. 50 (1983 1486. R.A. Klemnl and J.R. S c h r i e f f e r , Phys. Rev. Lett. 5l (1983) 47, L. Sneddon, Phys. Rev. Lett. 52 (1984 65. John Bardeen, this conference.
Physica 126B (1984) 400- 408 North-Holland. Amsterdam
CHARGE DENSITY WAVE TRANSPORT IN LINEAR CHAIN COMPOUNDS George GRUNER Department o f Physics, U n i v e r s i t y
of California,
o ~ Los Angeles, CA 900L4
Recent e x p e r i m e n t s focussing on the c o h e r e n t and i n c o h e r e n t aspects o f charge d e n s i t y wave t r a n s p o r t are discussed, and r e l a t e d to the t h e o r e t i c a l developments in the f i e l d . I.
INTRODUCTION In a wide range o f metals w i t h h i g h l y a n i s o t r o p i c band s t r u c t u r e e l e c t r o n - p h o n o n i n t e r a c t i o n s lead to an e l e c t r o n - h o l e condensate, c a l l e d the charge d e n s i t y wave(CDW) ( 1 ) . The condensate is c h a r a c t e r i z e d by a comp l e x o r d e r parameter, and the e l e c t r o n i c density,-c(x) f o r a one dimensional band s t r u c t u r e is g i v e n by: .a(x)
= ~oCOS (2k F x + ;)
(I)
where ~o is the a m p l i t u d e and =! is the phase o f the CDW. The p e r i o d is r e l a t e d to the Fermi w a v e v e c t o r by , = / k F and f o r an a r b i t r a r y band f i l l i n g the p e r i o d o f the CDW is incommensurate w i t h t h a t o f the u n d e r l y i n g lattice. The p o s i t i o n and time dependent phase p ( x , t ) is r e l a t e d t o the d e n s i t y of condensed e l e c t r o n s n and to the e l e c t r i c c u r r e n t j by ( a t T = O) J = n =
e
d,~ dt
(2a)
e
d. ~ dx
(2b)
i . e . the t r a n s l a t i o n a l motion o f the condensate leads to e l e c t r i c c u r r e n t w h i l e the comp r e s s i o n o f the mode r e s u l t s in a change o f n. The above r e l a t i o n s are d i f f e r e n t from those which r e l a t e the phase o f the macroscopic wave f u n c t i o n t o the c u r r e n t and chemical p o t e n t i a l in a s u p e r c o n d u c t o r . ( 1 ) In the absence o f p i n n i n g and damping which a c t on the CDW, an a p p l i e d e l e c t r i c f i e l d would lead to s u p e r c o n d u c t i v i t y , as f i r s t suggested by F r 6 h l i c h . This has, however, not been observed in v a r i o u s m a t e r i a l s which have an incommensurate CDW because i m p u r i t i e s pin the phase ¢ o f the condensate. Thus the dc c o n d u c t i v i t y is zero f o r small a p p l i e d dc electric fields. The e l e c t r o d y n a m i c s o f the condensate was f i r s t discussed by Lee, Rice and Anderson (2) and by Rice ( 3 ) . The f l u c t u a t i o n s o f the CDW can be d e s c r i b e d by an
a m p l i t u d e mode and by a phase mode. file amp l i t u d e modehas a gap and c o n s e q u e n t l y amp l i t u d e e x c i t a t i o n s can be n e g l e c t e d at low t e m p e r a t u r e s . With no p i n n i n g and damping tile phase mode leads to a frequency d e p e n d e n c e , i . e . a gap in the s i n g l e p a r t i c l e spectrum and c o l l e c t i v e mode a t zero frequency s i m i l a r to t h a t o f a s u p e r c o n d u c t o r . Including p i n n i n g and damping in a phemomenological way the frequency dependent c o n d u c t i v i t y 2 9 r. n_e_. :.L/'T~ . , = i 2 iiTT;'/ ' T H "A"
.
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* This research was supported by the NSF Grants DMR 81-21394 and DMR 84-06896.
0378-4363/84/$03.00 ¢]) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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/
.
where and ~ are the (phenomenological pinn i n g frequency and damping c o n s t a n t . In E<. (3) the d i s t r i b u t i o n o f p i n n i n g e n e r g i e s which may a r i s e as the consequence of the randomly d i s t r i b u t e d i m p u r i t i e s , is n e g l e c t e d , p i n n i n g is r e p r e s e n t e d by an average frequency m* is the e f f e c t i v e mass o f the condenSate. In terms o f the s i n g l e p a r t i c l e gap band mass m el ectron-phonon c o u p l i n g constant and phonon frequency . u!*/m = 1 + 4 . . . . . . . . ~,r typical transi2 2 2 kF t i o n t e m p e r a t u r e s and c o u p l i n g c o n s t a n t m * / m 102 - 103 . For a small c o n c e n t r a t i o n of i m p u r i t i e s , the p i n n i n g energy per e l e c t r o n may be o r d e r s o f magnitude s m a l l e r than ~ the s i n g l e p a r t i c l e e n e r g i e s (bandwidth and bandgap). This leads to s t r o n g l y frequency dependent c o n d u c t i v i t y and t o a g i a n t d i e l e c t r i c cons t a n t a s s o c i a t e d w i t h the ac response o f t h ( pinned CDW mode. The l a t t e r is g i v e n by (see Eq. ( 3 ) ) .
(4)
G, Gruner
/ Charge density wave transport
When f i n i t e dc e l e c t r i c f i e l d s are applied the energy provided by the dc f i e l d to the system may exceed the pinning energy. Various evidences suggest t h a t when t h i s happens a current c a r r y i n g CDW s t a t e , represented by the t r a n s l a t i o n a l motion of the condensate develops. The threshold f i e l d ET f o r the onset o f n o n l i n e a r conduction is p r o p o r t i o n a l to the r e s t o r i n g forces associated with the pinning, and consequently ET is c l o s e l y r e l a t e d to the low frequency d i e l e c t r i c constant. Various models (4) lead to the r e l a t i o n ~ET = const x e
(6)
This leads to a displaced Fermi surface, with gaps appearing at -kF+ q and kF+q with hq = mvd. The energy d i f f e r e n c e between the r i g h t and l e f t hand side of the Fermi surface E = 2kFhvd is p r o p o r t i o n a l to the current j = nev d. The frequency associated with processes by which e l e c t r o n s are t r a n s f e r r e d from one side of the Fermi surface to the o t h e r , h~, : AE is r e l a t e d to the current density per chain JCDW by JCDW
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where the constant is of the order of I , and i t depends on the model which describes the depinning process. For a current c a r r y i n g condensate, moving with a d r i f t v e l o c i t y v d ~o(x) = Po cos((2k F x - Vd ) + ~ )
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FIGURE 1 Rea(m) and Imp(m) in TaSk. The f u l l and dotted l i n e s are f i t s to a harmonic o s c i l l a t o r response with parameters given on the f i g u r e .
o where n(T) is the temperature dependent condensate d e n s i t y , vo = m The main features of the frequency and f i e l d dependent response together with the current o s c i l l a t i o n s with frequencies given by Eq. (7) have been observed in a wide range of inorganic chain compounds which show phase t r a n s i t i o n s to incommensurate CDW states ( I ) . In c e r t a i n t r a n s i t i o n metal t r i c h a l c o g e n i d e s , MX3, halogenated tetrachalcogens (MX4)vH and bronzes (of which the blue bronze K0 3MoO3 is the best known example) the CDW develops along the chain d i r e c t i o n , below the phase t r a n s i t i o n the CDW modulation on the neighboring chains are c o r r e l a t e d . The main experimental features obtained in these m a t e r i a l s are summarized in several reviews, ( I ) here I give only a few examples of the observed behaviors. Fig. 1 shows Re~(~) and Im~(~) f o r TaS3, the f u l l and dotted l i n e s are f i t s to Eq. ( 3 ) , with the i n e r t i a l terms neglected. The broad overa l l agreement suggest, t h a t in terms of a c l a s s i c a l d e s c r i p t i o n , the mode displays an overdamped response. The e l e c t r i c f i e l d dependent dc c o n d u c t i v i t y ~(E) = I/V is shown in Fig. 2. Below ET, ~ is due to normal e l e c trons which are e x c i t e d above the s i n g l e par-
t i c l e gap, the excess c o n d u c t i v i t y above ET is due to the current carrying condensate. O s c i l l a t i n g currents (or e l e c t r i c f i e l d s ) depending whether constant current or constant voltage source is used have been observed both in the time and in the frequency domain and Fig. 3 shows the. r e l a t i o n between JCDW and v o in NbSe3 wlth temperature dependence of the r a t i o in the i n s e r t . (5)
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Zco~(/z~', FIGURE 3 The frequency of the c u r r e n t o s c i l l a t i o n s versus the c u r r e n t c a r r i e d by the CDW. The i n s e r t shows ICDW/f ° versus temperature. In t h i s t a l k I w i l l discuss some f u r t h e r experiments on CDW t r a n s p o r t and analyze them in terms of equations f a m i l i a r f o r an audience w i t h background on s u p e r c o n d u c t i v i t y and Josephson phenomena. A phenomenological equation of motion d e s c r i b i n g the main experimental f e a t u r e s mentioned before t r e a t s the condensate as a r i g i d o b j e c t w i t h i n t e r n a l dynamics n e g l e c t e d . Due to the i n h e r e n t p e r i o d i c i t y o f the c o l l e c t i v e mode, the pinning p o t e n t i a l is assumed to be p e r i o d i c . Consequently, the model is t h a t of a viscous motion of a p a r t i c l e in a p e r i o d i c p o t e n t i a l (6) d2x + I dx o2 e'E (~!) dt 2 dt 2k F sin2kFX = -
- -
+
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~
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where e' and i~f are the e f f e c t i v e charge and mass of the condensate, and the p o t e n t i a l is assumed to be s i n u s o i d a l . The low f i e l d ac response is t h a t of a harmonic o s c i l l a t o r (see Fig. l , and Eq. ( 3 ) ) . Nonlinear d# conduction occurs f o r f i e l d s E > ET = ~ ~ L , and current o s c i l l a t i o n s r e s u l t as2the ~ r t i c l e moves down the t i l t e d s t a i r c a s e p o t e n t i a l , w i t h retion (7). Eg. (8) can be reduced to a dimens i o n l e s s form d2. 4~ E dt 2-- + " d-t + s i n = ET (9)• -l with time measured in u n i t s of and !' = ( u ~ ) - l . Equation (9) is f o r m a l l y i d e n t i c a l to the equation f o r the r e s i s t i v i t y shunted Josephson j u n c t i o n , which, in the usual n o t a t i o n
= ]
( l. }
The t h r e s h o l d f i e l d f o r CbW conduction ET cot'responds to the c r i t i c a l c u r r e n t , I i and curr e n t o s c i l l a t i o n s w i t h JCDW p r o p o r t l o n a l to correspond to the ac Josephson e f f e c t . One should emphasize, however, t h a t at t h i s p o i n ! there is only a formal correspondence between the two nonlinear" equations of motion, n the f i r s t part of the t a l k I w i l l demonstrate t h a t t h i s formal analogy can be used to analyze o t h e r phenomena, which a r i s e in the presenc~ of j o i n t ac and dc d r i v i n g f i e l d s . In the second p a r t e f f e c t s associated w i t h the i n t r i n s i c randomness, and the consequent i n c o herent phenomena w i l l be d e s c r i b e d . 2. EXPERIMENTS IN THE PRESENCE Of JOINT £r: £f~D DC DRIVING FIELDS WiUI s t r o n g l y n o n l i n e a r und frequency dependent response a broad v a r i e t y of e x p e r i ments, i n v o l v i n g the j o i n t a p p l i c a t i o n s of ac and dc f i e l d s of various amplitude can be performed to t e s t f u t h e r the dynamics of the c o l l e c t i v e mode. Several f e a t u r e s , such as the o v e r a l l m o d i f i c a t i o n of the dc and/or ac response (ac f i e l d induced dc conduction or dc e l e c t r i c f i e l d dependent ac d i e l e c t r i c constant and c o n d u c t i v i t y ) can be described using sii~iple n o n l i n e a r c i r c u i t theory by using ( ) and ( E ) as i n p u t p a r a m e t e r s . ( 7 ) Such d e s c r i p t i o n is perhaps also a p p r o p r i a t e f o r some of the mixing and r e c t i f i c a t i o n experiments performed by the I l l i n o i s grou~. (8) Here I w i l l focus on experiments where i n t e r f e r e n c e phenomena between the i n t r i n s i c o s c i l l a t i o n and the a p p l i e d ac f i e l d s are s t u d i e d . The a p p l i e d f i e l d is a combination of dc and ac e x c i t a t i o r l s V = Vdc + Vae cos ext{ and Vdc , Mac or" e x t can be v a r i e d during the experiments, w i t h the dc and ac response being detected. 2,l. I n t e r f e r e n c e phenomena Instead o f d e t e c t i n g the ac Josephson e f f e c t d i r e c t l y , the e f f e c t was f i r s t demons t r a t e d through the m o d i f i c a t i o n of the dc c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s by an a p p l i e d microwave f i e l d . Steps o c c u r r i n g due to l o c k ing between the e x t e r n a l and i n t e r n a l frequencies are c a l l e d the Shapiro steps in the Josephson l i t e r a t u r e . Corresponding e x p e r i ments were f i r s t performed on CDW systems by Monceau et a l . ( g ) . In pure specimens w i t h small dimensions, the steps can d i r e c t l y be detected by recording the I-V curves in the presence of a p p l i e d ac f i e l d . Fig. 4 shows (lO) such I-V traces f o r Vac of various a m p l i t u d e . Steps are observed in the n o n l i n e a r c o n d u c t i v i t y region~
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FIGURE 5 Step height 6V versus ac amplitude Vac f o r Shapiro steps in NbSe3. The r f frequency is 210 MHz and the step index n = 1, The s o l i d l i n e is the p r e d i c t i o n of the c l a s s i c a l model, Eq, (11), with parameters ~o2~/2~ = 80 MHz, VT = 24 mV, ~ = 0.17.
TdC(fl : IOOMNz) - 100
100
0 dc current < I >
(~A)
FIGURE 4 Dc I-V traces f o r NbSe3 in the presence of an applied r f f i e l d at frequency ~/2~ = I00 MHz and of amplitude Vac. The stop height ~V is defined in the f i g u r e . No Shapiro steps are observed f o r Vac = O, while the maximum step height is at approximately Vac = I00 mV. The arrow i n d i c a t e s the dc current which y i e l d s a fundamental noise frequency f l = I00 MHz. n is the step index. and the step height 6V is a strong f u n c t i o n of Vac. The step defined as n = 1 corresponds to an i n t e r f e r e n c e between the I00 MHz r f f i e l d and the fundamental of the i n t r i n s i c o s c i l l a t i o n . Harmonic and subharmonic steps with smaller amplitude can also be observed. Eq. (9) leads, in a high frequency (~oext > ~) l i m i t to ~V =~2VT(Vac = O) J n l V e f f l
;
(ll)
where c~ represents the volume f r a c t i o n of the specimen which responds coherently to the external f i e l d s , VT the threshold voltage and in Jn the Bessel f u n c t i o n of order n. Computer simulations in the frequency region ~e~f ~ ~o2~ also lead to o s c i l l a t i o n s of ~V as fu~i~tion of V e f f , c l o s e l y resembling Bessel f u n c t i o n behaviors. Fig. 5 shows the n=l step height as the f u n c t i o n of Vac. The Bessel func t i o n behavior, (also observed in Josephson j u n c t i o n s ) is c l e a r l y recovered, with ~o2~ values close to those observed in o(~) studies. The
c h a r a c t e r i s t i c values of ~ are between 0,1 and 1 suggesting h i g h l y coherent response. Experiments, performed as the f u n c t i o n of frequency are also in agreement with the model. I n t e r ference e f f e c t s show up also in the ac response. Experiments, performed as the f u n c t i o n of f r e quency are also in agreement with the model. I n t e r f e r e n c e e f f e c t s show up also in the ac response, and Fig. 6 shows the d i e l e c t r i c constant at 3.2 MHz as a f u n c t i o n of dc bias. The large i n d u c t i v e dips, and associated steps in Reo(;~) can also be accounted f o r (10). 2.2, D e v i l ' s s t a i r c a s e behavior The s e n s i t i v i t y of d e t e c t i o n of the i n t e r ference phenomena can be enhanced by d i s p l a y ing the d i f f e r e n t i a l resistance dV/dI. Peaks in the d e r i v a t i v e correspond to steps in the I-V curve, the width of the peak in voltage u n i t s corresponding to the height of the step. Fig. 7 shows (11) the d i f f e r e n t i a l resistance measured with and without the e x t e r n a l l y app l i e d r f voltage ~ext/2~ = 25 MHz. There appears a series of peaks at applied dc current such t h a t Pro = q
U'ext 2~
with p and q i n t e g e r s . Some of the peaks are i d e n t i f i e d on the f i g u r e . This i d e n t i f i c a t i o n was made by p l o t t i n g ICDW versus ~ext and checking t h a t the slopes were p/q times t h a t f o r the fundamental. Q u a l i t a t i v e l y , the width of the peaks decreases with increasing p and q as expected f o r mode locking between two frequencies both with slowly decaying harmonic content. The peaks consequently correspond to regions in which any harmonic of the i n t e r n a l frequency
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V (mV) FIGURE 6 The d i e l e c t r i c constant (3.2 MHz) as the funct i o n of dc b i a s . The l a r g e i n d u c t i v e dip corresponds to the i n t e r f e r e n c e between the fun ~ damental and the a p p l i e d dc f i e l d .
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FIGURE D i f f e r e n t i a l r e s i s t a n c e vs. dc sample v o l t a g e (a) w i t h and (b) w i t h o u t a p p l i e d r f v o l t a g e at 25 MHz. The peaks in (a) correspond to steps in the d i r e c t I-V curve. A few peaks are ident i f i e d by p/g (see t e x t ) .
locks to any harmonic of thu e x t e r n a l f i e l d . Such mode l o c k i n g has been e x t e n s i v e l y studied (12) f o r the equations of motion (9) and ( ] 0 ! and both numerical and a n a l y t i c a l s o l u t i o n s are a v a i l a b l e . In g e n e r a l , a D e v i l ' s s t a i r case behavior in which i n t e r f e r e n c e occurs ! ~ any p and q i n t e g e r is recovered by such (:,~i-c u l a t i o n s for" a weakly damped system. FoY strong damping i n t e r f e r e n c e is weak and rio subharmonics are expected f o r an overdamped response. D e v i l ' s s t a i r c a s e behavior is c l e a r l y i m p l i e d by our experimental r e s u l t b~J~ due to noise e f f e c t s higher order i n t e r f e r e n c , . peaks cannot be r e s o l v e d . The e f f e c t s of two competing trequencies in d i s s i p a t i v e systems has r e c e n t l y been i n v e s t i g a t e d by means of ~ c i r c l p map. (~3) Again, frequency l o c k i n g occurs for" a l l r~t i o n a l numbers p and q, w i t h r e l a t i v e ampli tude close to what is observed e x p e r i m e n t a l l y . Furthermore, the completeness of the s t a i r ~ case can be i n v e s t i g a t e d by looking at space; unoccupied by steps l a r g e r than a d i s c r i m i n a t i o n level r . Analysis in terms o~ such des c r i p t i o n suggest t h a t the D e v i l ' s staircase. is complete, in o t h e r words a chaotic behavior is observed. Other m a n i f e s t d t i o n s of the no~l i n e a r e q u a t i o n , such as period doubling t r a n s i t i o n to chaos and i n t e r m i t t a n c y here also has been observed in some of these m a t e r i a l s , b~ the Berkeley group. (14) Although many unresolved questions remain, in p a r t i c u l a r , coHcerning the importance ot i n e r t i a e f f e c t s , the r o l e o f the normal e l e c t r o n s , the e f f e c t of noise e t c . , the n o n l i n e a r phenomena assuciated w i t h CDW t r a n s p o r t appear to be a f r u i t f u l t e s t i n g ground f o r many o{ the t h e o r e t i c a l p r e d i c t i o n s concerning chaos and t u r b u l e n c e . 3. COHERENT VERSUS INCOHERENT EFFECTS Incoherent e f f e c t s associated w i t h i n t r l n sic randomness, f i n i t e length scales over which the phase-phase o r c u r r e n t - c u r r e n t correlations p e r s i s t , or thermal noise have been completely ueglected in the previous a n a l y s i s . I t is e v i dent however t h a t randomness and the internaE deformations of the condensate play an import a n t r o l e . dS f o r ,J r i g i d condensate which int e r a c t s w i t h random i m p u r i t i e s the pinning energy vanishes in the thermodynamic l i m i t . ( ! ! A l t e r n a t i v e l y , when local deformations around i m p u r i t i e s are taken i n t o account, the pi~asephase c o r r e l a t i o n f u n c t i o n shows an exponential decay, l e a d i n g to the absence of long range order. Indeed, various experiments suggest: t h a t d i s o r d e r plays an i m p o r t a n t r o l e also in the dynamics of the c o l l e c t i v e mode,and some of the f e a t u r e s observed are s i m i l a r to those modes on random systems l i k e g l a s s e s , spin g l a s s e s , g r a n u l a r superconductors, e t c . [ wil': discuss evidences f o r such incoherence in the case o f a pinned mode, and also f o r the c u r r e n t c a r r y i n g mode and f o r phenomena which ar~
G. Gruner I Charge density wave transport
associated with the t r a n s i t i o n between the pinned and the current carrying states. 3.1. The pinned mode: frequency dependent response and metastable states The simple harmonic o s c i l l a t o r response, given by Eq. (3) gives a good overall account for the main features of the observed f r e quency dependent c o n d u c t i v i t y , see Fig. I . This description predicts that in the ~ ÷ o limit Red(~)
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Experiments conducted at low frequencies show d r a s t i c deviations from such behavior (15). Fig. 8 shows Red(o~) - Red(~ = O) and Im~ (~) for TaS3 at T = 12OK. At low frequencies both approach a power law, and the c o n d u c t i v i t y can be described by o(~) = A(i~) ~
405
(12)
over several decades in frequency. Similar behavior was found in other materials, with < 1 in a l l cases (16). While the observed frequency dependence is in clear c o n f l i c t with a single harmonic o s c i l l a t o r description, i t also is in c o n f l i c t with calculations based on the Fukuyama-Lee model (17) when both the i n ternal deformations and i n t e r a c t i o n s with the randomly d i s t r i b u t e d i m p u r i t i e s are included. Such calculations (18) lead to Reo(~)~21nw2, and Im~(~) ~ ~, the famous Mott-Bereskinskii result. Our results however are surprisingly similar to those made on glasses and disordered solids (19) where a behavior given by Eq. (12) is obtained in the low frequency l i m i t . Therefore, i t is believed that the low frequency behavior, displayed in Fig. 8 indicates the importance of disorder in the dynamics of the pinned collective mode. The amplitude A of the observed ~dependance, given by Eq. (12) is several orders of magnitude larger than that observed with other disordered systems where t r a n s i t i o n s between l o c a l i z e d single p a r t i c l e states leads to the observed ~ dependence. As the p o l a r i z a t i o n , and consequently the dielect r i c constant is proportional to the relevant length scale £ over which such p o l a r i z a t i o n can be b u i l t up, the experiments on CDW materials suggest rather large length scales involved in the ac response. There are also several observations on metastable states associated with the pinned mode, and extremely slow t r a n s i t i o n s between these metastable states (20). These have been studied by looking at the time dependent phenomena a f t e r a heat pulse or elect r i c f i e l d pulse applied to the specimens. In both cases phenomena rather s i m i l a r to those observed in spin glasses have been detected.
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w
FIGURE 8 Reo(~) and Imo(~) versus frequency for TaSk. The f u l l l i n e s represent a Cole Cole l i k e ~xpression ~(~) =?=(i~T) [I + ( i ~ ) ~] -I with = 0.87 and ~-I/2~ = 25 MHz. The dashed l i n e gives the low frequency l i m i t o5 the harmonic o s c i l l a t o r response, Re~(~) ,~ ~z. 3.2.
The current carrying mode: incoherent noise phenomena Eq. (8) leads to a t o t a l l y coherent response in the current carrying mode: the spect r a l width of the current o s c i l l a t i o n s is i n f i n i t e l y sharp, the amplitude of the current o s c i l l a t i o n s is independent of the sample volume and there is no broad band noise associated with the s l i d i n g CDW mode. All of these predictions are in clear disagreement with the experimental observations. A dramatic increase of the low frequency (kHz) broad band noise (21) accompanies the onset of s l i d i n g CDW motion, as observed f i r s t in NbSe3 and subsequently in other compounds showing c o l l e c t i v e mode conduction. The f r e quency dependence can be described by an f-~ behavior, with ~ somewhat less than 1 with noise amplitudes orders of magnitude larger than those observed in random systems whose transport is due to single p a r t i c l e conduction. The broad band noise is also only weakly temperature dependent suggesting that temperature driven f l u c t u a t i o n s do not play an important r o l e . Broad band noise is also more dominant in specimens where the current o s c i l l a t i o n s are less evident, suggesting an i n t e r p l a y between coherent and incoherent e f f e c t s .
(/. (}PltHdt" / ('/ldr,~,d ddtl.~,llF ~'~',~/1'(' IrdlL~7){~Yl
406
These experiments suggest t h a t broad band noise is d i r e c t l y r e l a t e d to the dynamics of the i n t e r n a l deformations o f the CDW mode. In an o v e r s i m p l i f i e d p i c t u r e o f N domains f l u c t u a t i n g i n d e p e n d e n t l y , the c u r r e n t f l u c t u a t i o n s ' I are given by <~12>
v e l o c i t i e s is more l i k e l y , in agreement w i t h the experimental o b s e r v a t i o n s .
IOOr--
J !
12 2 where 61 represents the i n t e g r a l of c u r r e n t f l u c t u a t i o n s over a l l f r e q u e n c i e s . In the absence of r e l i a b l e experiments a v a i l a b l e in a broad frequency range and of models which c l e a r l y r e l a t e ,~I to the a v a i l a b l e degrees of freedom no q u a n t i t a t i v e a n a l y s i s is a v a i l a b l e . The l a r g e amplitude broad band noise however again suggest a small number of domains, i . e . l a r g e length scales associated w i t h these domains, in q u a l i t a t i v e agreement w i t h the r e s u l t s of the ~(:,~) s t u d i e s . The amplitude of the c u r r e n t o s c i l l a t i o n s was s t u d i e d r e c e n t l y as the f u n c t i o n of the volume of the specimens both in NbSe3 (22) and in (TaSe4)21 (23). These experiments c l e a r l y e s t a b l i s h t h a t the amplitude o f the c u r r e n t o s c i l l a t i o n s decreases w i t h i n c r e a s i n g volume, suggesting t h a t the o s c i l l a t i o n s are f i n i t e size e f f e c t s which would not be observed in the thermodynamic i m i t . Fig. 9 shows the o s c i l l a t i o n amplitude Lj I versus sample volume in NbSe3. The volume dependence o f the o s c i l l a t i o n amplitude can well be represented by a
•
•
•
.
d I%
|
!
FIGURE 9. O s c i l l a t i o n amplitude J l in NbSe3 versus sam ple v o l u m e ' The l i n e represents Ilj ] = (3 X lO-4A c m - ½ ) - ~ .
NbSe3
JCDW (i i . e . a square r o o t dependence s t r o n g l y sugg e s t i n g random phase c o r r e l a t i o n s in the current c a r r y i n g s t a t e . The amplitude of the current o s c i l l a t i o n s can be estimated f o r comp l e t e l y coherent response, and assuming domains o f size :;tc w i t h randomly c o r r e l a t e d phases, the domain size is estimated to be of the o r d e r of l;~m3. The F o u r i e r spectrum of the c u r r e n t o s c i l l a t i o n s also has a f i n i t e s p e c t r a l w i d t h , sugg e s t i n g a d i s t r i b u t i o n of charge d e n s i t y wave v e l o c i t i e s w i t h i n the specimens. This d i s t r i bution is more i m p o r t a n t in specimens w i t h h i g h e r a n i s o t r o p y and the width also increases w i t h i n c r e a s i n g cross s e c t i o n of the specimens. Fig. lO where the F o u r i e r s p e c t r a of the c u r r e n t o s c i l l a t i o n s observed f o r NbSe3 and f o r (TaSe4)2)I is compared. The cross sections are comparable f o r the two samples, but there is a s u b s t a n t i a l d i f f e r e n c e between the a n i s o t r o p i c s of the band s t r u c t u r e : w h i l e in NbSe3 the a n i s o t r o p y of the bandwidth is a p p r o x i mately o f 3 in (TaSe4)21, i t is more than an o r d e r of magnitude l a r g e r . This d i f f e r e n c e leads to d i f f e r e n c e s in the coupling strengh between n e i g h b o r i n g chains, f o r weaker coupling and f o r l a r g e r specimens a d i s t r i b u t i o n o f CDW
o o_
0
2
4
5 z
S
(Ta Se4)2 I
I
I
I0 f (MHz)
FIGURE lO F o u r i e r t r a n s f o r m of the c u r r e n t o s c i l l a t i o n s in (TaSe4)21 and in NbSe3.
G. Gruner / Charge density wave transport
3.3. The onset of current carrying state The onset of nonlinear conduction, as predicted by Eq. (8) is smooth, divergence is obtained only in the d e r i v a t i o n dl/dV in cases of small i n e r t i a . Large i n e r t i a leads to both a discontinuous jump in the I-V curve and to hysteresis. Such hysteresis effects have been recently observed in several materials. There is however a rather long time scale involved in t h i s switching phenomena, typical values for the time T to switch to the current carrying state can be as long as several milliseconds (24). Fig. I I shows a typical I-V curve obtained in TaS3 (25) where switching from the pinned to the current carrying state (and back) occurs. No noise is observed in the pinned region, but large low frequency broad band noise appears in the state with low resistance, i . e . when the CDW state contributes to the conduction. These observations suggest that switching and hysteresis are not related to the i n e r t i a l a f f e c t s , rather they may be related to the fact that various coherent CDW regions (domains) have f i r s t to be coupled together before the current carrying state can develop (26). As the domains most probably have macroscopic dimensions, t h i s leads to long times associated with the rearrangement of the domains, which has to occur before the current carrying state can develop.
/
ToS3 #2
3C
1oo H
25
200
~,oo j
/
~
20
__I
I
I
/ ~ i
I
~
I
'
I
....
I
I
200
150
Us (mY)
FIGURE I I Direct I-V traces near to the onset of nonl i n e a r conduction in TaS3. At the high f i e l d region, the I-V plot shows i r r e g u l a r f l u c t u a tions due to the broad band noise.
407
4. CONCLUSIONS The remarkable i n t e r p l a y between coherent and incoherent e f f e c t observed in CDW transport is most probably strongly related to the macroscopic length scales which characterize the s t a t i c s and dynamics of charge density waves and to the f i n i t e dimensions of the specimens which are investigated. The r e l e vant c h a r a c t e r i s t i c length scales are the phase-phase c o r r e l a t i o n length (27) <¢(x)¢(0)> ~ exp(- ~ ) characterizing the decay of phase c o r r e l a t i o n in the p~nned ~tat~. 2Here L = h2v F /(4k F )Vo Po ni where Vo is the strength of the impurity potential and ni ~s the impurity concentration. With Vo ~ I0 -Z eV and n i ~ I0 ppm. L ~ 10-2 cm, comparable to typical sample dimensions. The currentcurrent c o r r e l a t i o n length,
< ~de( x ) ~ ( ode)
>
may be s u b s t a n t i a l l y larger than the phasephase c o r r e l a t i o n length L. This l a t t e r is relevant for observations in the current carrying state. The narrow width of the spectrum of current o s c i l l a t i o n s observed in NbSe3 (see Fig. I0) suggests a current-current c o r r e l a t i o n length comparable to the sample dimensions, in other materials such as (TaSe4)21 i t has a much shorter scale. While i t is f i r m l y established that L is f i n i t e in less than four dimensions, long range order could be established in a driven random system. Such long range order is implied by recent experiments by Zettl et a l . (28), (29). Recent theories, (30) which take the e f f e c t of disorder into account lead to marked improvement over the description where the dynamics of the internal deformations is neglected. Besides accounting for the dependence of the parameters on the impurity concentrat i o n s , such theories lead to the removal of spurious divergences in dl/dv and d~/dV at ET, and also account for metastable states, for the d i s t r i b u t i o n of pinning energies as implied by ~(~) studies etc. F i n a l l y we remark that the analysis outl i n e d here is based on a classical equation of motion with quantum aspects of CDW dynamics neglected. Explanations of a large body of experimental r e s u l t s are also possible in terms of a quantum mechanical description of CDW dynamics, (31) and the r e l a t i o n between the parameters which characterize the and E dependent response (such as given by Eq. (5) for example) is s i m i l a r w i t h i n the framework of both descriptions (4). I t is expected that a combination of classical and quantum concepts w i l l lead to a f u l l explanation ~f the b~oad variety of phenomena
408
G. Gn~nur / (71urg'u dc~Lsity wurc lt'~l~l.~'l~¢)J'l
associated with the c o l l e c t i v e response of charge density condensates.
(17)
(I8) ACKNOWLEDGEMENTS Experimental results reported in this paper were obtained in c o l l a b o r a t i o n with L. Mihaly, G. Mozurkewich and A. Z e t t l . The author acknowledges useful discussions with John Bardeen, T. Holstein and A. Janossy.
(19) (20) (21)
REFERENCES (1)
(2) (3) (4) (5) (6) (7)
(8) (9) (lO)
(ll) (12) (13) (14)
15) 16)
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22)
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(31
H. Fukuyama and P.A. Lee, Phys. Rev. Bl7 (1978) 535. M.V. Eeigelman and V.M. Vinokur,Phys. Rev. Lett. 87A (1981) 53. K.L. Ngai, Comments on Solid State Phys. 9 (1979) 4. G. Mihaly and L. Mihaly, Phys. Rev. Left. 52 (1984)149. J. Richard, P. Monceau, M. Papoular and M. Renard, J. Phys. Cl5 (1982) 7157, A. Maedo, M. Naito and S. Tanaka in print, A. Zettl and G. Gruner, Solid State Comm. 46 (1983) 29. G. Mozurkewich and G. Gruner, Phys. Rev. Lett. 5l (1983) 2206. The detailed dependence of the oscillatior~ amplitude is controversial at present. I t has recently been suggested that i t is independent of the length of specimens, suggesting that current o s c i l l a t i o n s are generated at the contacts, N.P. On9, et a l . , Phys. Rev. Lett. 52 (1984) 663. G. Mozurkewich, M. Maki and G. Gruner, Solid State Comm. 49 (1983) 5. A. Zettl and G. Gruner, Phys. Rev. B26 (1982) 2298. L. Mihaly and G. Gruner, Solid State Comm. 50 (1984) 807. B. Joos and ~. Murray, Phys. Rev. B29 (1984) I094. P.A. Lee and T.M. Rice, Phys. Rev. BIg (1979) 3970. A. Z e t t l , M.B. Kaiser" and G. Gruner, Solid State Comm. in p r i n t . R. Hall and A. Z e t t l , in p r i n t . D.S. Fisher, Phys. Rev. Lett. 50 (1983 1486. R.A. Klemnl and J.R. S c h r i e f f e r , Phys. Rev. Lett. 5l (1983) 47, L. Sneddon, Phys. Rev. Lett. 52 (1984 65. John Bardeen, this conference.