Interference phenomena in charge-density waves for monsinusoidal external drives

Solid State Cormnunications, Printed in Great Britain.

Vol.57,No.3,

INTERFERENCE PHENOMENA

pp.]65-169,

IN CHARGE-DENSITY

WAVES FOR NONSINUSOIDAL

S. E. Brown, G. Gr~ner, University of California,

0038-I098/86 $3.00 + .00 Pergamon Press Ltd.

1986.

EXTERNAL DRIVES

and L. MihAly*

Los Angeles,

CA 90024, USA

( R e c e i v e d 9 September 1985 by A.A. Maradudln) We have investigated the interference phenomena in NbSe 3 associated wlth the joint ac and dc driven charge-denslty waves (CDW). The application of nonsinusoldal periodic drives allows us to compare experiments to computer simulations, and it leads to a transparent qualitative description of the so-called "Shapiro steps" in this system. The results suggest that long time scales associated with the rearrangement of the CDW In the current carrying state are impertant for a quantitative description of the phenomena observed.

matlons of the collective modes around impurity plnnlng centers are considered. The dynamlcs of internal degrees of freedom are expected to play an important role in the dc current-voltage (I-V) characteristics, and are probably also responsible for the broad band noise associated with the current carrying CDW state. Because Eq. (2) ls formally analogous to the equation of motion whlch describes the behavior of resistively shunted Josephson junctions, many of the interference phenomena are interpreted by using results originally developed to account for Josephson phenomena. In particular, steps induced by an applied ac field in the dc I-V characteristics, called the Shaplro steps In reference to work performed in the Josephson junctions, have been interpreted in detail along those lines. ? Here recently, discrete maps have utilized to investigate the dynamics of sliding charge density waves. 8 The effect of internal degrees of freedom on the interference phenomena has not been explored. Traditionally, the studies whlch focus on these effects employ a slnusoldal ac field, given by Eq. (1), and the effect of other types of periodic excitations has not been investigated. In this Communication, we report the results of experiments where we have applied pulses rather than the slnusoldal drive to induce the step features In the I-V curves. There are clear advantages to proceeding In thls manner. We found that by using pulses for the ac drive, the damped relaxation of the C D W b e l o w threshold could be identified as the physical mechanism for phase locking. Furthermore, the results can be compared to those obtained from computer simulations based on Eq. (2), and the bllevel drive allows for exact calculations (for certain choices of the potential) that do not rely on a n a l y t i c a l approximations. A l s o , by varying the repetition rate and fill factor of the pulses, the effectiveness of the relaxation can be tested and compared to the singleparticle descrlptlon of the simulations. Our findings indicate that the qualitative features of the response can be explained by Eq. (2), but that there are distinct differences. These imply that internal degrees of freedom of the condensate are important, and they contribute

Oscillating currents I which are observed in several materials with a charge density wave (CDW) g r o u n d state a r e usually regarded as clear evidences for a highly coherent translational motion of the collective mode under the influence of an applied external electric field. Upon the application of joint ac and dc fields (1)

E = Edc + EacCOS ~t ,

the mode coupling between the I n t r i n s i c current oscillations and the externally applied field leads to a strong modification of the ac or dc response whenever the two frequencies, or their harmonics, are close to each other. These phenomena, and also the dc and ac response, can be qualitatively described In terme of a simple equation of motion, 2 which considers only the time evolution of the collective (center of mass) coordinate of the CDW m * d2--~x + r ~dt + m*~°2 sln(2kFX) dt 2

= eE

(2)

where r is the damping constant, me the pinning frequency, and m = the effective mass of the collective mode. For the overdamped case, the characteristic time is the reciprocal of the classical crossover frequency ~ = 1/~co = rlm*uo2. While the model leads to a sharp threshold electric field ET for nonlinear conduction and also to current oscillations wlth frequencies proportional to the time average current as observed by experiment, Eq. (2) fails to account for the ~ and E dependent response In detail. It is becoming increasingly evident that the dynamics of internal modes is important, and that they make important contributions to the low frequency response to small amplitude ac excitations. The frequency dependent c o n d u c t i v i t y can be analyzed by assuming a broad distribution of relaxation tlmes,3, 4 u s i n g arguments similar to those advanced for glasses, spln glasses, and other random systems. These features are borne out a l s o by computer simulations s and a n a l y t i c a l s o l u t i o n s 6 o f t h e models where t h e l o c a l d e f o r -

*On leave from Central Research Institute for Physics, Budapest, 1525 Pf 49, Hungary. 165

INTERFERENCE PHENOMENA IN CHARGE-DENSITY WAVES

166

significantly to the damping processes which are associated wlth transitions between the plnned and the current carrying state. The m a l n qualitative features of the simulation are shown in Fig. I, which demonstrates the behavior of a damped particle In a perlodlc potential under the influence of a bilevel driving field. The arguments are valid for other types of drives (e.g., sinnsoldal), but the pulsed field allows for easy lllustratlon of the motion. For clarity, we will always denote the two levels so that E 2 > El, and they are each applied for a duration t2, t I respectively. The dc Is deflned as the time average, since thls is ~x (t 2) ~x (t 1) I~

Vd{:

-

-

I

-

0 -VT ~-- 12

t~ I l

Fig . l Single-particle description of synchronization to combined dc and ac drives. The particle moves a distance 6x(t 2) during the high side of the pulse (the time dependence of the field is shown below). When the high field is switched off, the particle moves a distance 6X(tl) towards dynamical equilibrium near to the potential minimum. The time-average velocity is therefore dependent on the pulse period t I + t 2 and not the dc bias itself. wha~ Is measured in the experiments. Conslder what happens if we begin the simulation with the particle at the bottom of one of the potential wells. We first apply a constant driving field above threshold. During the time t 2 that the field is on, the particle moves an amount 6x(t2). Next, the fleld E i < E T Is turned on, wlth the result that If t I is long enough, we expect the particle to reach the bottom of the nearest potential minimum. When the sequence is repeated, the motion reproduces as before, with the period given by that of the driving field. In terms of a flux or current, the time average behaves as pk - tl + t2 , (3) where k is the period of the potential and p denotes the number of wavelengths travelled in the pulse period t I + t 2 = T. From this model, we expect that if no time is spent below thresh-

Vol. 57, No. 3

old during the rf cycle, then there wlll be no steps. Furthermore, setting t I ~ t 2 should yield an asymmetric I-V curve, since the times spent above and below threshold will be different for negative and positive biases. These features are borne out by computer simulations, which will be discussed at length in a separate publication. S u b h a r a o n l c s t e p s may o c c u r when t h e w a l t l n g t i m e b e lo w t h r e s h o l d i s s h o r t . For example, suppose t h e p a r t i c l e t r a v e l s a d i s t a n c e a p p r o x i m a t e l y (n + 1 / 2 ) X d u r i n g t h e o n - t i m e t 2. I t does n o t move f a r f r o m t h a t p o s i t i o n i f t I i s s h o r t , so when t h e h l g h p u l s e comes on a g a i n , i t a g a i n moves a p p r o x i m a t e l y t h e same d i s t a n c e , so a t t h e end o f t h e p u l s e I t a r r i v e s n e a r t o a p o t e n t i a l mlnlaum. Here, t I does n o t have t o be v e r y lo n g for the particle to relax completely. Thls twostep p r o c e s s l e a d s t o what i s c a l l e d a p / 2 s u b harmonic ( p = 2 n + l ) s i n c e t h e p a r t l c l e travels a d i s t a n c e pk i n 2 c y c l e s o f t h e d r i v i n g f i e l d . We f o u n d t h a t t h e c h o i c e o f a s l n u s o l d a l potential (which may apply to other dynamical systems llke Josephson junctions or the damped pendulum) is a special case, 9 since there are no subharmonic solutions. The subharmonlcs are generally expected as long as the period of the a~plied driving field is shorter than ~ = Flm*~o ~. To check whether the phase lock mechanism occurs In a similar fashion In CDW systems, we compared the results for sine wave and square wave drives. The r e s u l t s , which appear in Fig. 2, were obtained from experiments performed on a hlgh-purlty NbSe 3 sample at T = 48 K (threshold field E_ = 22 mV/cm). Figure 2a shows differI ential resistance vs. dc sample voltage for a low frequency sinusoidal ac drive. Many harmonIcs appear close together and, between them some subharmonlc structure is visible. No steps are seen above a cutoff voltage for both positive and negative dc biases. Thls corresponds to the situation where the time-dependent voltage no longer goes below threshold at any instance. The square wave drive (Fig. 2b) also shows the same cutoff, but there is the additional feature that the step sizes are significantly suppressed around zero dc bias. This occurs because the square wave has a peak-peak amp]itude greater than 2VT, so the field switches between levels that are both larger than E T but wlth opposite signs. On the other hand, the sine wave varies continuously in time, so there is always a nonnegllgable time spent below V TThe importance of the relaxation below threshold is also demonstrated by direct observation of the tlme-dependent voltage. Fleming I0 found that the oscillations which appear above the threshold could be synchronized to the pulse width. Under these conditions, the amplitude of the oscillations is seen to increase dramatlcal1y, as in the oscilloscope photograph of Flg.3a. We established the connection between the pulse synchronization and the steps in the I-V curve by simultaneous observation of the time average properties (differential I - V curves) and the time dependent voltage (oscilloscope). When the oscillations are synchronized, only a small decay in amplitude is observed for the time that the field is high. However, Fig. 3b shows that a detuniag of the pulse width coincides with an oscillatory response that decays significantly by the time the field is switched off, indicat-

Vol. 57, No. 3

INTERFERENCE PHENOMENA IN CHARGE-DENSITY

300

I

I

I

167

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A

200 E O

~ IO0

20d "0

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I

I

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sample voltage (mY) Fig. 2 Demonstration that the synchronization mechanism for CDWs also requires a relaxation time below the threshold field. The ac drive in (a) is a low frequency sine wave. The recording of differential resistance vs. sample voltage shows many harmonics plus subharmonics in between them. The steps do not appear at dc biases above a cutoff (see text). The drive in (b) is a square wave, which shows the same cutoff and the additional feature of suppressed steps near zero bias occur because the field switches between two levels, both above threshold but of the opposite polarity.

(a:

(b]

Fig. 3 Oscilloscope photographs exhibiting synchronization and also decay effects due to dephasing. The drive in each case is a square wave of peak-peak amplitude ~ 2Em.z a) the sample is driven from ground to above threshold with E > E T photographed. The pulse width has been set so the response is synchronized. b)same as (a) but with a slightly different period, so the response is not synchronized and the observed oscillation amplitude decays significantly by the end of the pulse, c) same peak-peak pulse amplitude as before,but the other side of the pulse is not below threshold. No oscillations are observed.

INTERFERENCE

1 68

PHENOMENA IN CHARGE-DENSITY

together on the negative side, where the system spends a longer time above threshold). For short t I (Eq. 4b), the step width increases linearly in tl, and the relative slopes behave approximately as I/t 2. When the proper value for the threshold voltage Is in- serted, however, the absolute widths are approx- imately one order of magnitude smaller than that suggested by Eq. 4b. Serious shortcomings also arise when we consider the results for long t i. The step widths approach a constant value, but they do not vary as I/t 2. In fact, it is ex- pected that the harmonic steps are "dense", that is, they should touch each other as in the inset. This result was not obtained for short times t2, In clear disagreement with the computer model. A detalled comparison shows that the single particle model does not work for the t 2 = 2.0 ps elther. We obtain a value ~ = 0.25 ps by fitting the saturated step height through Eq. (4a). The simulation then gives the step height as a function of tl, wlth the basic result that for any waiting time larger than ~, the harmonic steps are "dense", a n d no subharmonlcs appear at all. This is contrary to the experimental findings, which show the step size to be too small until t i is greater than ~(* = 2.0 lJs. Subharmonlcs then occur in a region o f parameter space (t I > ~) where there is no possibility for them in a single-partlcle description. We do not believe that the above characteristic times • and ~* have any specific signlficance. They point, however, to a spectrtm of relaxation times which extends to frequencies

Ing a dephaslng of sources. When the field is switched to the same level but from above threshold, no oscillatory structure is seen on the pulse (Fig. 3c). This implies that the oscillations can only he phase correlated to the drive if a significant amount of time is spent below threshold. Computer simulations based on the deformable Fukuyama-Lee Hamiltonian result in decaying oscillations u n d e r conditions similar to that in Fig. 3b, II pointing to the importance of internal modes for a full description. Since the internal modes influence the time dependent response, and this in turn has been related to the steps in the I-V curves, it is expected that the deviations due to multiple degrees of freedom show up here also. The observations are suggestive of a convenient way to test the effectiveness of the relaxation of the center of mass coordinate with regard to the size of the steps. By fixing the time t 2 above threshold and varying the waiting time t I spent below, the width of the steps (single particle simulation) for two limiting cases can be found analytically: &E = E T ~

=

,

t i >> ~

(qa)

t--i , t i << z (4b) t2 Figure 4 shows the width of the steps plotted for three different values of t 2. A typical recording of the differential resistance vs. sample voltage appears in the inset (in this case, the steps are closer &E

2ET

I

I

I

Vol. 57, No. 3

WAVES

I

1

I

1

NbSe 3 T=48K

0,8

VT = 2.6mY O SO

•

+

0.6

A

> E

+ aa I

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c~ o~

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I00

+

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I

I

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2,0

I

I

4,0

r = 0,25/.~s I

I

6,0

woiting time T l(Fs) Fig.

4

Waiting time dependence of the step height for various values of t 2 (see Fig. I). The dashed line is the simulation result with the characteristic time T = G.25 ~s. The inset shows a typical recording of differential resistance vs. sample voltage when t I~ t 2.

Vol. 57, No. 3

INTERFERENCE

PHENOMENA

significantly lower than the characteristic frequencies which appear in Eq. (2). Because the system is overdamped, the fundamental characteristic tlme is the reciprocal of the socalled crossover frequency, on the order of 2~ x 108 sec -I. The parameter ls associated wlth the center of mass motion as discussed in reference to Flg. I. The characterlstle tlmes ~ and ~* are instead associated with the relaxation of the condensate Into a pinned state after the current Is turned off, i.e., wlth the buildup of local deformations around pinning centers. It is expected, however, that the response cannot be characterized by a slngle relaxation but by a spectrum of relaxation times, Just llke the stretched exponential or logarlthmlc decay of the polarization which follows an electric field pulse Is suggestive of a relaxation dlstrlbutlon. Moreover, the presence of subharmenlc steps indicates that the descrlptlon by a set of independent oscillators having different parameters

IN CHARGE-DENSITY

WAVES

may not be complete (thls becomes apparent when the computer-slmulated subharmenic steps are compared to the experiments= a dlstrlbutlon of relaxation times, deduced from the width of the flrst harmonic response, completely smears out the subharmonlc steps). Therefore, we expect that the results for <* to be mere relevant to the long time relaxation of internal degrees of freedom, observed also in the pulse-memory experlments. Our results call for a complete treatment of the Fukuyama-Lee Hamiltonlan of deformable CDWs; the application of a hllevel ac drive (lnstead of sine waves) slgnificantly simplifies the calculations In the nonlinear regime.

We would like to thank R. Brulnsma and G. Mozurkewich for helpful discussions. Thls work was supported in part by the National Science Foundation under Grant No. DHR 84-06896.

References I. R. M. Fleming and C. c. Grimes, Phys. Rev. Lett. 42, 1423 (1979). For a review, see G. Gr~ner and A. Zettl, Phys. Reports 119, 117 (1985). 2. G. Gr~ner, A. Zawadowskl, and P. M. C h a l k l n , Phys. Rev. L e t t . 4_66, 511 (1981). 3. W. Wu, L. H l h ~ l y , G. Mozurkewlch, and G. GrUner, Phys. Rev. L e t t . 52, 2382 (1984). 4. R. J. Cava, R. M. Fleming, P. B. L l t t l e w o o d , E. A. Rletman, L.F. Schneemeyer, and R. G. Dunn, Phys. Rev. B30, 3228 (1984). S. P. B. L l t t l e w o o d , Proceedings o f the I n t e r n a t i o n a l Conference on Charge-density Waves In S o l i d s , Lecture Notes i n Physics 217, S p r l n g e r - V e r l a g , 1985. 6. D. S. F i s h e r , Phys. Rev. B31, 1396 (1985). 7. A. Z e t t l and G. G r 0 n e r , P h y s . Rev. B29, 755 (1984).

169

8. M.Ya. A z b e l and P. Bak, P h y s . Rev. B3_O0, 3722 ( 1 9 8 4 ) . 9. B o t h n u m e r i c a l and a n a l y t i c a l c a l c u l a t i o n s have been done previously f o r a sinusoldal drive [see for example P. F. Tua and J. Ruvalds, Solid State Comm. 54, 471 (1985), John Bardeen (to be published)]. Our simulation shows that the same result is obtained for a pulsed driving fleld. 10. R. M. Fleming, Solld State Comm. 43, ~67 (1982). ii. H. Matsukawa and H. Takayama, Solid State Comm. SO, 283 (1984), and S. N. Coppersmith and P. B. Littlewood, Proceedings of the International Conference on Charge Density Waves In Sollds, Lecture Notes In Physics 217, Springer Yerlag 198S.