Weak pinning of the charge-density wave revisited

Synthetic Metals 86 (1997) 2225-2226

ELSEVIER

Weak pinning of the charge-density Ioan Bildea,’ Institut

fir

Theoretische

Chemie,

wave revisited

Horst Koppel and Lorenz S. Cederbaum

UniversitGt

Heidelberg, Im Neuenheimer

Feld 253, D-69120

Heidelberg,

Germany

Abstract

An overview of numerical simulation and experimental works that contradict the Fukuyama-Lee-Rice (FLR) theory of weak pinning is presented. Within an alternative, analytic approach, we find a salomonic answer to the problem of three-dimensional weak pinning: short- and large-scale phase deformations coexist, but their role is different. The present results provide a unified picture of the CDW weakly pinned by impurities, and explain puzzling aspects of numerical simulations, disagreement between previous theories and experiments, e. g. in the case of NbSes, and the white line effect. Keywords:

models of non-linear

1. Inconsistencies

phenomena;

of Fukuyama-Lee-Rice

other phase transitions

theory

The charge density wave (CDW) containing a random distribution of quenched impurities is usually described as a frustrated system, characterized by a complex order parameter. The frustration is due to two antagonisticconditions: the intrinsic CDW periodicity tends to impose a uniform order parameter, while the random impuritiesdetermine a nonuniform CDW order parameter, deformed to take advantage from the CDW-impurity interaction. The magnitude of this deformation depends on the value of the Fukuyama-Lee-Rice (FLR) parameter E, defined as the ratio between the energy gained by optimizing the phase around each impurity to the value preferred by the CDW-impurity interaction and the accompanying elastic energy cost. As briefly summarized below, the most delicate problem turned out to be the three-dimensional pinning in the limit E << 1, to which the present work is exclusively devoted. While large phase distortions (- ?r) at individual impurity sites yield an overall energy increase with respect to the state with uniform phase (whose energy is set zero, as usual), one can show that a state with negative energy can be obtained if the system breaks up into phase coherent (Fukuyama-Lee) domains comprising a large number of impurities [l, 2, 31. The total energy FFLR of this collectively pinned CDW is proportional to E* and the ratio of the energy gained from the CDW-impurity interaction to the corresponding elastic energy cost r G -Fi/Fel has a value independent of E, PFLR = 4/3 [2, 41. The average * On leave from Institute Space

Sciences,

of Atomic

76900 Bucharest-Msgurele,

0379-6779lP7/S17.00 Q 1997 Elsekr PII SO379-6779(96)04813-8

Physics,

Institute

Romania

Scienw S.A All rights reserved

for

phase gradient (inversely proportional to the domain length LFLR) was predicted to be proportional to E’ [2, 51. None of these results was confirmed by numerical simulations [5, 41. In the state with lowest energy found by numerical simulation FNS O( &2 (I FFLR 1 < )FNS 1for E < 1) and the average phase gradient is proportional to E [5], while at small E the values of the parameter r tend to saturate to 2 [4]. From experimental side, we mention two pieces of work. First, the case of doped NbSea, appeared controversial. The concentration dependence of the threshold field for non-linear conduction [6], along with the large characteristic length of phase-coherent domains revealed by high-resolution x-ray experiments [7] agree with the FLR prediction. On the other hand, except for very fast variations (over lengths shorter than the BCS correlation length), phase deformations over short distances were also identified [8]. Second, the “white-line” effect, detected in a variety of materials, clearly demonstrates that CDW phases and impurity positions are correlated [9]. 1. Coexistence variations

of short- and large-scale in weak-coupling limit

phase

Recently, we reinvestigated analytically the problem of CDW pinning by impurities and found a very important role played by phase deformations around individual impurities; details are to be found in Ref. [1’2, 131. Here we shall only give a brief description and insist on the implications of those results. In a first stage [12] of investigation, a simple variational ansatz was employed to minimize the GinsburgLandau (GL) functional. This approach demonstrated

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I. Bdldea et al. /Synthetic Metals 86 (1997) 22252226

that, contrary to the Lee-Rice claim [2], phase adjustments at individual impurities are possible even for small E; the energy gain they cause is much larger than that obtained within the FLR mechanism. Actually, the approach of Ref. [12] succeeded to explain all findings of numerical simulations listed above. This gave the puzzling results of numerical simulations [5, 41 a clear physical content. Neither the finite-size effects, invoked to motivate the E-dependence found in Ref. [5], nor the unusual dependence on the concentrat.ion in the artificial lattice of Ref. [4] were responsible for the deviations from the FLR theory. The source of this disagreement is the physical fact that, rather than phase adjustments within domains containing many impurities, those occurring at individual impurities are essential for the behavior of aforementioned quantities. In addition, they are responsible for the experimentally observed “white-line” effect [9]. In a second stage, the GL functional was minimized directly [13], without resorting to any variational ansatz. This was done by means of a diagrammatical technique [lo, 111. This more elaborate method reconfirmed the essential role played by the short-scale phase variations, but adjustments at large scale were also found. In understanding the physics of impurity pinning, the threshold field for non-linear conduction plays a crucial part. As previously analysed (see Table 1 of Ref. [12]), the two numerical simulations [5, 43 led to results for the threshold field in quantitative and qualitative disagreement among themselves. However, the methods used in the two studies were different. Using the canonical conjugation of the average phase and the applied field to determine the threshold field &h, the value obtained analytically [12] a g rees with that of numerical simulation [5]. It was shown [12] that this value of ,?&, oc raig2 (ni-impurity concentration) corresponds to the energy required to dislodge the pinned CDW and to convert it in a CDW with uniform phase. Alternatively, one can compute the threshold field directly, by means of the motion equation for the CDW phase. Applying this method, numerical simulation [4] and analytical approach [ll, 131 led to values compatible to each other, but the result of this method, &, oc rafc4 is different from that deduced by means of the other method. As a matter of fact, the value Eth or nfE4 does not correspond to the total energy of the pinned CDW, but rather to the much smaller energy gain due to large-scale phase deformations. The result &, oc npc4 was initially found [2] by simply balancing the electric field energy to that gained by the formation of FLR domains; nevertheless, the role of short-scale phase deformations was overlooked in Ref. [2]. The experimental data in Ta- and T&doped NbSes [6] agree with the functional dependence &, o( nf&4. In spite of a crude theoretical estimation [14], indicating that the energy gained by strong pinning is larger than that via the FLR weak-pinning mechanism, this fact and the observation of large-scale phase correlations [7] were taken as strong experimental support for the FLR weak-pinning mechanism. Nevertheless, as demonstrated both by subsequent experiments [8] and by our theoretical study [12, 131, the phase deformations in the

weak-coupling limit are more complex than initially imagined by Lee and Rice [2]. The results presented above have important implications for the CDW physics, in particular for the non-linear conduction. Their direct consequence is that, at least for fields not very far from the onset of non-linear conduction, the phase of CDW is not uniform. Within the employed analytical method [ll, 131, estimates for both the threshold field and the energy gained by large-scale phase correlation have uncertainity factors of order of unity. Still open is therefore the question whether the two energies are equal or only of the same order of magnitude, without a simple, direct relationship between them [15]. In the latter case, not only phase deformations at individual impurities are expected in the regime of non-linear conduction: also a certain large-scale phase correlation could exist. A preliminary experimental report [16] seems to give support to this alternative. Although the theoretical results presently available are not sufficient, the dependence of the threshold field on the concentration in the artificial lattice used in Ref. [4] also makes plausible this hypothesis. Because, as already discussed [12], such a dependence would be excluded if the contributions of the local phase deformations were absent. Presumably, only further numerical simulations could offer a definite answer to this problem. The authors thank the Sonderforschungsbereich 247 at Universitit Heidelberg for financial support. In the early stage, I. B. was supported by Alexander van Humboldt Foundation. References PI H. Fukuyama and P. A. Lee, Phys. Rev. B 1’7 535 (1978). PI P. A. Lee and T. M. Rice, Phys. Rev. B 19, 3970 (1979).

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