Metastability and nonlinear dynamics of sliding charge density waves

Physica 23D (1986) 45-53 North-Holland, Amsterdam

METASTABILITY AND NONLINEAR DYNAMICS OF SLIDING CHARGE DENSITY WAVES P.B. LITTLEWOOD A T& T Bell Laboratories, Murray Hill, NJ 07974, USA

Incommensurate charge-density-waves pinned by disorder show an interplay between spatial disorder and nonlinear dynamics. At low electric fields close to and below the threshold field for sliding, their behavior is dominated by many degrees of freedom and low energy metastable states with a broad distribution of relaxation times. However the response to large ac driving fields shows features apparently characteristic of a system with relatively few degrees of freedom. The resolution of this paradox will be discussed.

1. Introduction

The first proposal of collective mode conductivity from the sliding of charge density waves was made by FrShlich [1] in 1954 as a model for superconductivity. In 1973 Bardeen [2] resurrected these ideas in connection with one-dimensional conductors with incommensurate charge density waves (CDW). The first unambiguous evidence for CDW sliding was the discovery of nonlinear conductivity in NbSe 3 [3] and there are now roughly a half-dozen experimental systems demonstrating a wide range of phenomena associated with conductivity from the collective sliding mode [4]. Some of the experimental aspects have been discussed by Griiner [5] and Zettl [6] in this volume. The driving mechanism for CDW formation was first enunciated by Peierls [7] who pointed out that a quasi-lD electron gas can lower its energy by scattering from a frozen lattice distortion with a periodicity Q of twice the Fermi wavevector k F. This leads to a modulation of the charge density at wavevector Q with an accompanying periodic lattice distortion, which constitute the CDW. At the same time, a gap in the electronic density of states is opened up at the Fermi energy. Since the periodicity Q need not bear a direct relation to the lattice constant (in which case the CDW is said to be incommensurate), a rigid CDW would be free to slide. However CDW's have a finite elastic

modulus, so that the CDW will deform in order to pin to the lattice [8], or to impurities and defects in the crystal [9]. If there is strong coupling to the underlying lattice, the incommensurate CDW is best described by commensurate regions separated by phase slips ("discommensurations") and there may be a commensurate-incommensurate transition (where the Q-vector locks in to a wavevector Qc commensurate with the crystal lattice) [8]. Discommensurations will still be free to move if the lattice potential is not too strong; however they too will be pinned by defects (this even if the defect potential is arbitrarily weak) [10]. The pinning of the CDW by impurities is thus generally the dominant mechanism for removing collective mode conductivity at small electric fields in quasi-lD materials by destroying the long-range order. Nevertheless, above a small but finite threshold field E z the CDW will begin to slide, and this is the origin of the nonlinear conductivity. Some of the salient experimental phenomena which any successful theory must address are listed below. This is by no means a complete summary, and more details can be found in the references given above [4-6]. In particular, this list is restricted to electrical properties. 1) A sharp onset of nonlinear conductivity above a threshold field E T with a CDW current of the form I cc ( E - ET) ~ close to threshold, where 1G~_<2.

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

46

P.B. Littlewood/ Metastability and nonlinear dynamics of sliding CD W

2) So-called "narrow-band noise", where harmonic voltage oscillations are generated in response to a d c current [11]. The frequency of the oscillation is proportional to the CDW current, and is in fact the "washboard" frequency too = Q - v (here v is the CDW velocity). Whether the noise is generated by a bulk or contact mechanism is disputed [12]. 3) The frequency-dependent dielectric function c(to) for linear response in the pinned state is strongly enhanced at low frequencies (in the range of Hz to MHz, depending on both the material and temperature) [13]. At low frequency, the dielectric function develops a cusp of the form c(to) - Co - I t o ~ l ~ with a value of a close to, and somewhat less than one [14]. 4) Broad band noise with a roughly 1 / f character is generated while the CDW is sliding [15]. 5) There are a number of electrical hysteresis phenomena. The response to a train of square wave pulses demonstrates the "pulse sign memory effect", when the shape of the transient response depends on the polarity of the preceding pulse [16]. The low field conductivity o0 and dielectric response c 0 change in response to both thermal pulses [17] and to cycling of the electric field above threshold [18]. There is a very slow (on a time scale of hours) relaxation of o0 and c o [18, 19]. At low temperatures, an electrical polarization can be frozen into the sample [20] which undergoes slow relaxation with a stretched exponential form [21]. The low frequency I - V characteristics are often found to be hysteretic [22]. 6) The response to mixed ac and dc driving fields shows interference between the driving frequency to and the washboard frequency too, which is apparently inductive in nature (despite the fact that the system is overdamped, as we shall see). For example, while low frequency I - V characteristics are invariably capacitative, sweeps at moderate frequencies (--- MHz) with biases above threshold show inductive character [23] which is closely related to transient voltage oscillations at the washboard frequency in response to square wave pulses [24]. If a train of pulses is applied, the

transient oscillations become locked to the pulse [25]. In the presence of a large ac drive the dc I - V characteristics develop steps at currents such that the washboard frequency is either a harmonic or a subharmonic of the driving frequency [26]. At fixed dc bias above threshold, the ac conductivity shows an "inductive" dip at the washboard frequency [27]. A general characteristic of the response is that for low fields and frequencies the behavior is dominated by disorder, is hysteretic, and shows evidence of the importance of many degrees of freedom; at higher fields, the behavior becomes apparently more coherent and seems to be governed by relatively few degrees of freedom. The existence of a threshold field in a CDW pinned by disorder, requires that the pinned state has lost long-range order. This has recently been confirmed by X-ray diffraction in K0.3MoO3 [28] but in NbSe 3 the correlation length is beyond the range of experimental resolution [29]. The large amount of experimental data provides stringent tests of any theory. Nevertheless, a relatively simple model for the macroscopic properties was proposed some time ago [9, 30] which seems to be capable of explaining most, if not all, of the observed phenomena. A very different picture from this classical hydrodynamic model is the quantum mechanical tunnelling model of Bardeen [31]. The tunnelling model has been successful in interpreting data taken at high fields and frequencies [32] but cannot easily explain the metastability and hysteresis phenomena which are so apparent at electric fields close to and below threshold.

2.

Model

The classical model of Fukuyama, Lee and Rice (FLR) [9] treats the CDW as a deformable extended elastic medium [33]. For an incommensurate CDW of collective density Pc and a sinusoidal component

p ( r ) = P0cos ( Q . r + ep(r))

(2.1)

P.B. Littlewood/ Metastability and nonlinear dynamics of sliding CD W

the energy is given by H=

47

Firstly, let us assume that we have found a static solution ~0(r) of eq. (2.4) at field E = 0, and then expand around this solution so that ~ = ~o + ~k, leading to

f d3r-~K(w¢)2- (p~E/Q)ep +

(2.2) = WJ~b+ E + E V [cos (0i + ePo(Ri) ) sin

i

i

Here K is the elastic constant of the CDW (lengths have been scaled to make K isotropic), E the applied electric field and ~ the interaction potential of the CDW with an impurity at position R~. The phase ¢ ( r ) represents the local position of the CDW, and fluctuations in the amplitude Po are neglected. The dynamics are specified by the overdamped equation of motion [34]

h~ = - 8H/,~ep.

(2.3)

After length and time rescaling, and for a shortrange interaction potential (this requirement is not crucial, but simplifies the analysis), eq. (2.3) becomes ~ ( r ) = V2~b(r) + E + E Vsin (0, + d P ( R i ) ) 8 ( r

-

Ri).

(2.4)

i

Notice that fluctuations in the phase are threedimensional but the electric field acts to move the CDW only in the incommensurate direction of Q. In eq. (2.4) 0, = Q . R~ is a random phase from site to site, in contrast to commensurate pinning to the lattice when Q-R~ is itself periodic. In the absence of an electric field, eq. (2.2) is identical to that for long wavelength fluctuations of an X Y model in a random magnetic field, for which the ground state is known to be disordered. Experimentally, it is known that the characteristic length scale for the loss of long-range order is long, so that thermal effects (for example involving thermally activated hopping of local regions over barriers) are unimportant (at least in three dimensions). Some qualitative features of the model are easy to analyse, and provide a consistency check by comparison with some experimental scales.

+ sin (Oi + , 0 ( R , ) ) ( c o s ff - 1)].

(2.5)

We shall obtain the correct scales for the problem if we replace the sums in eq. (2.5) by averages over the sample. From eq. (2.4) we have (Vcos(O+ e~0)) = 0, while (Vsin(0 + ~0)) = - E p , with Ep the pinning energy. In the case of weak pinning ( V < < I ) , we have E p - L -2, with L - V 2/(4-d) the correlation length in d < 4 dimensions [9]. In this approximation, we obtain the sine-Gordon equation = W2~ + E - Ep sin tk.

(2.6)

If spatial fluctuations in the phase are ignored equation (2.6) reduces to the single degree of freedom model of Griiner, Zawadowski and Chaikin [35]. Thus (in scaled units) the threshold field is of order Ep, and the dielectric function (for linear response) is a Lorentzian , ( w ) - (i~o + Ep) -1

(2.7)

so that the dielectric constant c o and the characteristic time scale ~" vary inversely with Ep. This simplified model allows us to estimate that c o --pJQET, 1"~ ?tQ/PcEx; a value of the high field or high frequency CDW conductivity o~ = o2c/XQ2; the "washboard" frequency o~0(E) ~ ocE/QX. The system for which the most complete set of data exists is K0.3MoO 3, for which the above relationships can be tested [36]. The agreement is excellent over a wide range of temperature, despite the fact that individual quantities vary over several orders of magnitude with temperature in this material. The approximations leading to eq. (2.6) are sufficient only to set the characteristic scales in the problem. The characteristic length scale for phase

P.B. Littlewood/ Metastability and nonlinear dynamics of sliding CD W

48

fluctuations in (2.6) is E ; 1/2, which is identical to the correlation length L; consequently fluctuations in the pinning energy cannot be neglected. While eq. (2.6) would predict that a dc field would lead to current oscillations at the washboard frequency [35], this result turns out to be valid only for systems comparable in size to the correlation length L. The inclusion of many degrees of freedom is also crucial in determining the conductivity exponent ~; the single degree of freedom model yields ~ = 1/2 [35], in disagreement with the data. In fact the crucial feature of the classical model is that there are many pinned states, because the phase is determined only modulo 2~r within a correlation length. Metastable states which are "nearby" in configuration space can be related to each other by soliton-like phase slips (see fig. 1) [37]. Since the "soliton" thickness - L, this suggests that the number of metastable states in a sample of linear dimension L 0 is of order exp ( a L / L o ) d, where a is a number of order unity. Which particular state is occupied depends on the history, and it is this which is responsible for the hysteresis and "memory" phenomena as well as

2.5

~

~

,

,

I

~

'

'

f

I

'

'

T

;

I

'

'

'

'

I

V:OA E:O

2.0

,_~ .............

/ t.s

,.

\

i

I

l

',

t.O

0.5

5O

t00

t50

200

X

Fig. 1. Two pinned configurations at zero electric field and their phase difference obtained from numerical simulation of a one-dimensional model [37]. There are 200 impurities placed at r a n d o m , with pinning strength V = 0.1. Values of @ are measured in units of 2~r.

the anomalous power law dependence of c (to) that we referred to in section 1. However, the importance of the metastable states in determining the low field and low frequency behavior leads to a paradox because it would seem to be incompatible with the observation of an apparently coherent response to large ac driving. In contrast, the single degree of freedom model will reproduce, at least qualitatively, the mode-locking between an ac driving frequency and the internal washboard frequency (although subharmonic locking can be explained only by the addition of a large, unphysical inertial term to eq. (2.6)) [27]. This model fails in other important respects, as we have discussed briefly above.

3. Solution No complete solution exists to the FLR model, even for the simplest case of a pure dc driving field in dynamic equilibrium. Following is an abbreviated list of the methods and approximations which have been applied to this problem. In high fields (E >> ET), the CDW moves uniformly and a perturbation expansion can be obtained [34, 38, 39, 40]. The perturbation series fails to any order for fields close enough to threshold and in fact no threshold field can be obtained to any finite order in perturbation theory [39]. A mean field theory (infinite-range model) has been solved by Fisher [41]. This is valid in a large number of dimensions, but many of the characteristic features of this solution appear to persist in lower dimensions. The behavior of small systems (of size comparable to the correlation length) has been studied analytically [42]. Some insight has also been gained by the use of a mode-coupling approximation [43]. There have been a number of numerical studies, principally of one-dimensional versions of the model [37, 44]. A closely-related model with many similar properties is the incommensurate (Frenkel-Kontorova) pinning model, which has been studied extensively [45]. Although

49

P.B. Littlewood/ Metastability and nonlinear dynamics of sfiding CD W

our understanding is not complete, a compilation of these various results produces a comprehensive picture of a wide variety of CDW phenomena. 3.1. D C characteristics Both the mean field theory [41] and the modecoupling approximation [43] lead to a conductivity exponent of ~ = 3/2, demonstrating the importance of large phase fluctuations above threshold. The fluctuations can be calculated to leading order in the high field perturbation theory [34] with a high field CDW conductivity of the form j / E = troo - C E -1/2.

(3.1)

Narrow-band noise is not produced in an infinite system [39]; nor is there any broad-band noise generated in dynamic equilibrium [37]. In fact numerical simulations and mean field theory show

that the sliding solution is everywhere periodic in time with the washboard frequency ~0 = Qv. Locally, the phase advances in periodic jumps, but these jumps are uncorrelated at large enough distances, so that the narrow-band noise (or voltage oscillations) vanishes in an infinite system. This can be most clearly seen in fig. 2, which is a sequence of snapshots of ~(x, t) - ~(x, to) from a 1D simulation [37]. The temporal periodicity is quite evident. Broad-band noise is generated as a transient, but vanishes as the system approaches dynamical equilibrium, which can take a considerable time. The narrow-band noise vanishes on a length scale which is field-dependent and diverges as threshold is approached from above [41], so that bulk generation of narrow-band noise should become important even in long samples close enough to Ea-. We also note that in pure de, there is no evidence for hysteresis in the I - V character-

2.5

P h a s e

d i f f

1.5

e r e n c e

0.5

50

I00 Impurity p o s i t i o n

150

200

Fig. 2. A sequence of snapshots of q~(x,t) - 4~(x,to) in the equilibriumslidingstate at a fieldof E = 0.05. Other parameters as for fig. 1.

50

P.B. Littlewood/ Metastability and nonlinear dynamics of sliding CDW

istics, although the time scale to reach dynamic equilibrium becomes very long close to threshold so that hysteresis will be seen at very low frequencies.

3.3. A C response At high frequency, the ac conductivity approaches a form similar to eq. (3.1) [39, 43] a(tO)

3.2. Metastable states There are many metastable states, as we argued above. It is important to realize that the CDW is not rigidly pinned below threshold, because a small electric field can cause the CDW to roll over local barriers between different states. In fact, although it is possible to prepare (at least numerically) metastable states which have everywhere finite barriers to other states, the generic situation appears to be that a metastable state has a distribution of barriers which extends to zero [41, 43, 37]. This has a number of important consequences. Firstly, the modes of linear response depend on the local curvature of the pinning potential. Instead of a single characteristic mode frequency, one then finds a distribution of modes. This changes the behavior of the dielectric function from a Lorentzian (eq. (2.7)) to a cusp-like frequency dependence

,(o~) = '0 - (i°~'r) ~-

~---Ooo - - C'(,O - 1 / 2

(3.3)

and in the presence of a large dc bias there are strong interference features between the driving frequency and the internal washboard frequency. The perturbation theory (valid for large dc field) shows that o ( ~ ) shows a sharp dip at t~ = ~0 [39, 40], because there is enhanced dissipation of low frequency modes close to the washboard frequency. This behavior mimics an inductive response, despite the negligible CDW inertia. In a similar fashion, the derivative dc response at fixed ac bias can be seen to develop kinks ("Shapiro steps") at currents such that the washboard frequency is a harmonic of the driving frequency [34, 46]. It is clear that subharmonic structure will develop in next order of perturbation, but the calculation is too complex to carry out in detail. We stress that all of these phenomena are bulk in nature, and are to be expected in the infinite volume limit, in contrast to the narrow-band noise.

(3.2)

3.4. Mode locking Secondly, because there are always zero modes, the dc response is not the same as the low frequency limit of the ac response. At low enough frequency, the response is strongly nonlinear. Finally, we argued earlier that because the correlation length is long, the characteristic barrier heights are large in comparison to k B T , so that thermal effects would be unimportant. This is no longer true for the lowest barriers, which will be gradually overcome by thermal excitation; as the relaxation proceeds, the barriers to be overcome will increase in height and the process will slow down. This is a plausible mechanism to explain the slow relaxation of %, c o and frozen polarization at low temperatures.

There is no analytic method presently available to calculate the behavior for large ac biases when the voltage is driven close to or below threshold, and numerical simulations provide most of the information on the behavior of the model. Rather than considering a pure ac driving, the most appropriate study turns out to be a sequence of square-wave field pulses from below to above threshold. Remarkably, one find that the CDW configuration rapidly "locks" into a repeating configuration. If the time between pulses is long enough that the CDW relaxes close to a static configuration between pulses, the configurations repeat periodically so that after n pulses the CDW has advanced by precisely m wavelengths, and is

P.B. Littlewood/ Metastability and nonlinear dynamics of sliding CD 14/ I

'

I

f

51

I

/~,\

/

C U P

e n

t

I

Y

O

~

~

m

- -

. . . .

0

]

. . . .

5

J 10 time

I

I

l

I

I 15

. . . .

20

Fig. 3. The spatially-averaged C D W velocity plotted for a sequence of twelve identical square wave pulses from zero field to E = 2. For convenience in visualizing the results, subsequent pulses to the first have been displaced vertically by Av = 0.4. These results were obtained by a simulation of a 1D model (eq. (2.4)) with V = 1 and 50 impurities. After the sixth pulse, the system has locked into a state with n = 3, m = 11.

in an i d e n t i c a l pinned configuration [47]. At the same time, the current exhibits well-defined oscillations (at a frequency close to the washboard o~o = Q v ) locked to the pulse (fig. 3). While this is precisely the effect needed in order to explain the experiments [27], it should not be understood in terms of competing periodicities (i.e. ~/0~0), because the locked states are determined by the height and length of the pulses; the time between pulses is irrelevant. Experimental evidence for this point of view has been given recently [48]. This result is all the more surprising because we have argued in favor of the existence of large numbers of metastable pinned states; the response to a sequence of pulses shows that the CDW configuration is only able to explore an infinitesimal fraction of those states. Some understanding can be gained by remembering that the sliding

state in dynamic equilibrium is u n i q u e . Although this state is never reached in the course of the numerical simulations, as the pulse length is increased it is approached more and more closely. The number of pinned configurations that are reached from "quenching" the equilibrium sliding state at different times is a small set of the total. We find that the intermediate configurations never differ by more than one wavelength (A@ < 2~r) locally from each other, which supports this picture. However, there is hysteresis, in that there exists more than one sequence of locked configurations for a given set of parameters. These different sequences can be obtained from differing starting configurations of the CDW (but identical arrangements of impurities) before the initial pulse. The behavior is similar to that described by a circle map but careful study has shown that it is

52

P.B. Littlewood / Metastability and nonlinear dynamics of sliding CD W

different, and cannot be explained with a single degree of freedom [47]. Furthermore, the system always locks when the time between pulses is long, and we have not observed chaotic trajectories.

[4]

4. Conclusion We believe that there is strong evidence that the F L R model is both qualitatively and quantitatively in agreement with a large number of experiments probing very different regimes. This model, with its combination of quenched disorder and strong nonlinearity, exhibits a variety of phenomena. At low electric fields and low frequencies, the effects of disorder are apparent in the observed hysteresis and "memory", and in an anomalous behavior of the low frequency conductivity. Here the behavior is similar to that of a frozen glass. The apparently coherent nature of the response to large ac fields turns out to be also directly related to the phase fluctuations induced by disorder. The sharp structure in the frequency-dependent response at fields above threshold is produced by enhanced damping by the impurities of fluctuations driven by the external field. Finally, and perhaps most remarkably, the CDW can be "trained" by a sequence of pulses to explore only a minute fraction of the metastable states, leading to an apparent mode-locking to a large ac drive.

[5] [6] [7] [8] [9] [10] [11] [12]

[13] [14]

[15]

[16]

[17]

Acknowledgements [18]

The author has benefitted from continuing discussions with R.J. Cava, S.N. Coppersmith, D.S. Fisher, R.M. Fleming, L.F. Schneemeyer and C.M. Varma. The author is also grateful to S.E. Brown, G. Griiner and L. Mihaly for communicating both ideas and data prior to publication.

[19] [20] [21] [22] [23]

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P.B. Littlewood/ Metastability and nonlinear dynamics of sliding CDW

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53

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