Microscopic mechanism of sliding charge-density-wave transport: pinning persistence, Zener-like high-field conductance, and complete mode locking in quasi-one-dimensional conductors

In quasi-one-dimensional metals such as NbSe3, an incommensurate charge-density wave (CDW) — a periodic modulation of the electronic density, coupled to a lattice distortion, with a period not commensurate with the underlying crystal lattice — depins above a threshold electric field and carries current collectively, generating narrow-band voltage noise and, under combined ac and dc fields, interference steps (mode locking) analogous to Shapiro steps in Josephson junctions. Thorne, Tucker, and Bardeen (Phys. Rev. Lett. 58, 828 (1987)) showed that complete mode locking with sharp negative wings persists at ac frequencies up to 300 MHz — as much as sixty times the CDW dielectric relaxation frequency of about 5 MHz in NbSe3 — and at ac amplitudes so large that the inverse of the time spent below the threshold field per cycle exceeds that relaxation frequency by two to three orders of magnitude, a regime in which the classical deformable-CDW model allows no time for the sub-threshold relaxation it requires; they also showed that the high-field dc CDW conductance follows the Zener form $G_{\mathrm{CDW}}=G_b\exp(-E_0/E)$ rather than the classical asymptote. They concluded that periodic impurity pinning does not vanish at large fields and frequencies and argued this as strong additional evidence that quantum tunneling underlies CDW dynamics. Later work partially reversed the theoretical interpretation (classical many-degree-of-freedom simulations reproduce complete locking) while the quantum program accumulated further support (Zener-like current-voltage curves from 9 to 474 K, h/2e magnetoconductance oscillations in CDW rings, Aharonov-Bohm effects), yet no microscopic theory exists. The open problem is a predictive microscopic theory of sliding CDW transport that decides or reconciles the quantum-tunneling and classical deformable pictures and explains the persistence of pinning at high fields and frequencies and the Zener form of the high-field conductance, including the physical origin of its parameters, while remaining consistent with the observed control of velocity coherence by defects and contacts and with normal-carrier-screening dissipation.

Background

A charge-density wave (CDW) is a periodic modulation of the electronic density, coupled to a lattice distortion, that forms below a Peierls transition — the metal-insulator instability of a quasi-one-dimensional metal, in which the lattice distorts at twice the Fermi wave vector and opens an energy gap at the Fermi level — in many quasi-one-dimensional metals, most prominently the trichalcogenide NbSe3. The CDW is normally pinned to the lattice by its phase-dependent interaction with randomly distributed impurities; when the applied electric field exceeds a threshold $E_T$, the CDW depins and slides, carrying current collectively and producing nonlinear dc conduction. Because the CDW-impurity interaction is periodic under rigid displacement of the CDW by one wavelength, a sliding CDW generates coherent voltage oscillations called narrow-band noise (NBN), whose frequency is proportional to the CDW current. Under a combined dc and ac voltage, the internal NBN frequency locks to rational multiples of the applied frequency, producing steps of nearly constant CDW current on the dc current-voltage characteristic (harmonic steps for $\omega_{\mathrm{NBN}}/\omega=p$, subharmonic for $\omega_{\mathrm{NBN}}/\omega=p/q$), directly analogous to the Shapiro steps (constant-voltage plateaus) of a resistively shunted Josephson junction driven by combined ac and dc currents. A step is completely locked when the dc CDW current is exactly constant over a finite dc-voltage interval. Two competing theories were developed in the late 1970s and 1980s. The classical deformable-CDW model, built on the Fukuyama-Lee-Rice description of an overdamped elastic medium (a charged 'rubber sheet') with impurity pinning, treats the CDW phase as a many-degree-of-freedom classical field; within it, the moving CDW's internal deformations relax toward lower-energy configurations, and when the CDW moves so fast that significant relaxation cannot occur, the periodic pinning force is expected to vanish. Bardeen's alternative treats the CDW metal as a macroscopic quantum system in which the condensate depins by quantum tunneling through a small pinning gap over distances of order a micrometer; it predicts a Zener-like nonlinear conductance of the collective mode, $G_{\mathrm{CDW}}=G_b\exp(-E_0/E)$, analogous to Zener tunneling of electron-hole pairs across a band gap in a strong electric field. In this context, Thorne, Tucker, and Bardeen (1987) reported high-coherence NbSe3 interference experiments designed to test specific predictions of the classical model, in the form given by Coppersmith and Littlewood (Phys. Rev. Lett. 57, 1927 (1986)): complete mode locking should require damped sub-threshold relaxation, and should disappear at large ac amplitudes and frequencies, as should the sharp negative 'wings' adjacent to the interference peaks. They observed complete harmonic and subharmonic locking with sharp negative wings at ac frequencies up to 300 MHz — as much as sixty times the CDW dielectric relaxation frequency (about 5 MHz in NbSe3), the characteristic rate at which the pinned internal deformations relax — and at such large ac amplitudes that the time $\delta t = 2\pi E_T/(\omega E_{\mathrm{ac}})$ spent below the threshold field per cycle (with $\omega$ the ac angular frequency and $E_{\mathrm{ac}}$ the ac field amplitude) corresponds to a rate up to three orders of magnitude above that relaxation frequency, far too short for any relaxation to occur; they also measured the high-field dc CDW conductance out to 150 times threshold, finding that the classical asymptotic form (conductance linear in $E^{-1/2}$) derived by Sneddon, Cross, and Fisher (Phys. Rev. Lett. 49, 292 (1982)) fits only between about 3 and 15 times threshold, with significant deviations at higher fields, while the Zener form with $E_0=215$ mV/cm and $G_b=1.554$ mS fits the data from 2 to 150 times threshold. The paper also identified extrinsic velocity-distribution broadening (from defects, contacts, and nonuniformities, which fragment the crystal into regions carrying different CDW currents) as the factor controlling whether complete locking is observed at all, and cited evidence that in K0.3MoO3 (blue bronze) and (TaSe4)2I the high-field collective conductance is thermally activated and limited by dissipation from screening of the CDW charge fluctuations by normal (uncondensed) carriers. The authors concluded that periodic impurity pinning does not vanish at large fields and frequencies, that the classical deformable model is fundamentally deficient, and that the data provide strong additional evidence for quantum tunneling in CDW dynamics, while leaving the microscopic mechanism unquantified.

Problem Statement

Establish a microscopic, predictive theory of sliding charge-density-wave transport in quasi-one-dimensional conductors that decides between, or validly reconciles, the quantum-tunneling picture and the classical deformable-medium picture, and within a single framework accounts for the interference and high-field phenomenology of Thorne, Tucker, and Bardeen (1987). The theory must explain: (i) why the periodic impurity pinning of the CDW persists essentially undiminished at large electric fields and ac frequencies — specifically, why complete mode locking with sharp negative wings is observed at ac frequencies up to about 300 MHz in NbSe3, as much as sixty times the CDW dielectric relaxation frequency (about 5 MHz, the characteristic rate at which the pinned internal deformations relax), and at ac amplitudes so large that the inverse of the time spent below the threshold field per cycle exceeds that relaxation frequency by two to three orders of magnitude, a regime in which the relaxation-time argument of the classical deformable model predicts that the pinning should vanish; and (ii) the high-field dc CDW conductance — that it follows the Zener form $G_{\mathrm{CDW}}=G_b\exp(-E_0/E)$ rather than the classical asymptote, and what physically fixes the parameters $E_0$ and $G_b$ in the material studied. The theory must also be consistent with the recorded phenomenology that sample preparation, defect populations, and contact geometries determine the internal CDW velocity distributions that broaden the interference steps and control whether complete locking is observed, and that in (TaSe4)2I and K0.3MoO3 the high-field collective conductance is thermally activated and limited by dissipation from normal-carrier screening of the CDW charge fluctuations. The objective is a microscopic mechanism for how the CDW depins and transports charge — deciding between the two pictures or reconciling them within one framework that accounts for both (i) and (ii) — rather than merely reproducing or discriminating the phenomenology.

References

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