Universal threshold polarization for charge-density-wave depinning: testing $\varepsilon(0)E_T=k\,n_{\mathrm{ch}}e$ across compounds and deriving its dimensionality, Coulomb, and relaxation corrections

Many quasi-one-dimensional conductors develop a charge-density wave (CDW) below a Peierls transition temperature; the wave's phase is pinned by impurities and slides only above a threshold electric field $E_T$, while the pinned condensate exhibits a giant static dielectric constant $\varepsilon(0)$. Bardeen [Phys. Rev. Lett. 62, 2985 (1989)] argued from weak-pinning theory that the threshold polarization is universal: $\varepsilon(0)E_T=k\,n_{\mathrm{ch}}e$ per conducting chain, with $k=2/\pi$ in the Coulomb-compensated regime, reduced to $k=1/(2\pi)$ in quasi-two-dimensional blue bronze K₀.₃MoO₃ by two proposed corrections — a low-temperature doubling of the effective pinning frequency due to Coulomb-frozen density fluctuations, and a factor-of-4 reduction of chain-direction pinning energy from transverse phase coherence. He treated the abrupt low-temperature threshold $E_T^*$ (experimentally $\approx 300\,E_T$ in blue bronze) with the polarization-energy criterion $\tfrac{1}{2}\varepsilon(0)E_T^{*2}=N(0)\Delta^2$, giving $eE_T^*=\tfrac{1}{4}\lambda M_F v_p^2$ in reasonable agreement with blue bronze and (TaSe₄)₂I. The paper proposed a quantitative program: tabulate $\varepsilon(0)$, $E_T$, $E_T^*$, pinning frequency, Fröhlich mass, and relaxation times across compounds; derive the two corrections microscopically; and verify the temperature-dependent relation between $\varepsilon(0)$ and the high-frequency relaxation. Later literature measured fragments of this physics and a successor theory reformulated the threshold as a Coulomb-blockade relation with a material-dependent constant, but the systematic test and the derivations were never completed. The open problem is to determine whether the CDW threshold polarization is universal and to establish or refute the proposed corrections.

Background

In several families of quasi-one-dimensional conductors — crystals built from parallel chains of atoms along which electrons move far more easily than between chains — a Peierls instability below a critical temperature forms a charge-density wave (CDW): the electron density modulates sinusoidally with wavevector $2k_F$ (twice the Fermi wavevector $k_F$), accompanied by a periodic distortion of the ion positions. The CDW ground state is an insulating condensate whose phase $\phi$ fixes the position of the density modulation relative to the lattice; quanta of oscillation of this phase are called phasons, and because the modulation is tied to the lattice the phasons behave as if carrying an effective inertial mass $M_F$ (the Fröhlich mass), typically hundreds to tens of thousands of electron masses. A uniform phase gradient corresponds to displaced charge, so a translating ("sliding") CDW can carry an electric current — the mechanism behind the nonlinear conduction of materials such as NbSe₃, TaS₃, K₀.₃MoO₃ ("blue bronze"), and (TaSe₄)₂I, studied intensively in the 1970s–80s and reviewed by Grüner [Rev. Mod. Phys. 60, 1129 (1988)].

Random impurities pin the CDW phase: each impurity energetically prefers a particular local phase value, and the collective effect creates metastable pinned configurations. The CDW therefore remains static up to a threshold electric field $E_T$ and slides above it, with distinctive accompaniments such as narrow-band noise (a periodic voltage modulation superposed on frequency-locked current flow). The pinned but deformable condensate also shows a giant static dielectric constant $\varepsilon(0)$ along the chain direction — up to $10^7$–$10^8$ in blue bronze at low temperature, versus order $10$ in ordinary dielectrics — because the phase can distort almost freely inside a domain. In the Fukuyama–Lee–Rice weak-pinning theory [Fukuyama and Lee, Phys. Rev. B 17, 535 (1978); Lee and Rice, Phys. Rev. B 19, 3970 (1979)], many impurities within a phase-coherent domain of length $L_d$ partially cancel; the threshold field, the pinning gap $\Delta$, and $\varepsilon(0)$ are then interrelated through $M_F$ and the pinning frequency $\omega_p$ (the resonance frequency of pinned phase oscillations, accessible to microwave and THz spectroscopy): in the form used by Bardeen, $\Delta^2=8M_F\hbar^2\omega_p^2/(\pi^4 m)$ and $\varepsilon(0)=4\pi ne^2/(M_F\omega_p^2)$, with $n$ the condensed electron density.

Experiments in the 1980s noticed a striking regularity across compounds whose threshold fields differ by orders of magnitude: the product $\varepsilon(0)E_T$ — the polarization the CDW sustains at threshold — is about $0.5e$ per conducting chain [Wu, Jánossy and Grüner, Solid State Commun. 49, 1013 (1984); Mihály et al., Phys. Rev. B 37, 1047 and 38, 3602 (1988)]. Bardeen [Phys. Rev. Lett. 62, 2985 (1989)] derived this from the weak-pinning relations: the threshold polarization can be written

$$\varepsilon(0)E_T=k\,n_{\mathrm{ch}}e,$$

where $n_{\mathrm{ch}}$ is the number of conducting chains per unit transverse area; all material parameters cancel, giving a universal $k=2/\pi$ in the temperature range of Coulomb compensation — the regime in which thermally excited normal carriers screen (compensate) the long-range Coulomb cost of CDW density fluctuations. He applied the polarization-energy criterion

$$\tfrac{1}{2}\varepsilon(0)E_T^{*2}=N(0)\Delta^2,$$

with $N(0)$ the one-spin density of states at the Fermi surface, to the abrupt second threshold $E_T^*$ observed at very low temperature in blue bronze — where few screening carriers remain and $E_T^*\approx 300\,E_T$ over 20–40 K — obtaining for $k=1/(2\pi)$ the explicit prediction

$$eE_T^*=\tfrac{1}{4}\lambda M_F v_p^2,$$

with $\lambda$ the CDW wavelength and $v_p=\omega_p/2\pi$ the low-temperature pinning frequency in Hz. With $\lambda=10^{-7}$ cm, $M_F=300\,m_e$ and a low-temperature pinning frequency of 100 GHz this gives $E_T^*\approx 40$ V/cm, near the observed $\sim 30$ V/cm; for (TaSe₄)₂I ($M_F=2\times10^4\,m_e$, 35 GHz) it gives $\sim 700$ V/cm versus $\sim 1200$ V/cm observed.

The value $k=2/\pi$, however, exceeds the low-temperature values $k\approx 0.16$ (blue bronze) and $\approx 0.25$ (TaS₃ and (TaSe₄)₂I) extracted from the data of Mihály and co-workers. Bardeen proposed two corrections, each halving $k$. First, at low temperatures Coulomb effects prevent the density fluctuations that pinning requires, so phase oscillations sample the total rather than the net pinning potential — four times larger — implying a doubled effective pinning frequency and hence a halved $k$; this argument was offered as a suggestion, not derived microscopically. Second, blue bronze is quasi-two-dimensional: from resistivity anisotropy, X-ray correlation lengths [Pouget et al., J. Phys. (Paris) 46, 1731 (1985)] and preliminary transverse-domain measurements, one transverse phase-correlation length is comparable to the longitudinal one, so only about half the phase-adjustment energy lies in the chain direction, reducing the chain-direction pinning energy by a factor of 4 and $k$ by another factor of 2 — giving $k=1/(2\pi)\approx 0.16$, as observed. Finally, in the quantum tunneling picture of depinning [Bardeen, Phys. Rev. Lett. 45, 1978 (1980)], relaxation enters through a two-time-constant framework: an effective mass $M'=(\tau^*/\tau)M_F$, domain length $L'=(\tau/\tau^*)^{1/2}L_d$, and low-field threshold $eE_T=(\tau^*/\tau)\lambda M_F v_p^2$, where $\tau$ is the total and $\tau^*$ the upper relaxation time for CDW momentum dissipation; $\varepsilon(0)$ increases by the compensating factor $\tau/\tau^*$ (estimated at about 8 for Ta₁₋ₓNbₓS₃, about 60 for (TaSe₄)₂I, and 100 or more for blue bronze), leaving the product $\varepsilon(0)E_T$ unchanged. In a phenomenological resistor–capacitor model of Coulomb compensation, $\varepsilon(0)$ is independent of the RC time $\tau_0$ and proportional to the capacitance $C$, so the observed temperature dependence of $\varepsilon(0)$ in blue bronze should track the high-frequency relaxation — a relation the paper explicitly flags as needing quantitative verification.

Problem Statement

Determine, with measured parameters and microscopic theory, whether the threshold polarization for charge-density-wave depinning is universal, and establish or refute the correction physics proposed for it. The relations under test are $\varepsilon(0)E_T=k\,n_{\mathrm{ch}}e$ (threshold polarization per conducting chain; $k=2/\pi$ predicted for quasi-one-dimensional compounds in the Coulomb-compensated regime, $k=1/\pi$ predicted for quasi-one-dimensional compounds at low temperature where only the Coulomb-freezing correction applies, and $k=1/(2\pi)$ predicted for quasi-two-dimensional blue bronze K0.3MoO3 where both corrections apply), the abrupt-threshold criterion $\tfrac{1}{2}\varepsilon(0)E_T^{*2}=N(0)\Delta^2$, and the explicit prediction $eE_T^*=\tfrac{1}{4}\lambda M_F v_p^2$ for $k=1/(2\pi)$, doubled to $eE_T^*=\tfrac{1}{2}\lambda M_F v_p^2$ for $k=1/\pi$ (with $\lambda$ the CDW wavelength, $M_F$ the Fröhlich mass, $v_p=\omega_p/2\pi$ the low-temperature pinning frequency). Specifically:

(a) Cross-compound test: assemble or measure, for CDW compounds with independently determined high-frequency parameters (pinning frequency $\omega_p$, Fröhlich mass $M_F$, relaxation times $\tau$ and $\tau^*$), the static dielectric constant $\varepsilon(0)$ (with its frequency window specified), the gradual threshold $E_T$, and where observable the abrupt low-temperature threshold $E_T^*$, and determine whether $k=\varepsilon(0)E_T/(n_{\mathrm{ch}}e)$ takes the predicted value in each regime within experimental uncertainties — $2/\pi$ for quasi-one-dimensional compounds in the Coulomb-compensated regime, $1/\pi$ for quasi-one-dimensional compounds at low temperature where only the Coulomb-freezing correction applies, and $1/(2\pi)$ for quasi-two-dimensional compounds where both corrections apply. A single well-characterized compound in a regime where the screening state and dimensionality are established, whose $k$ deviates beyond uncertainties from the corresponding predicted value, would refute the universality claim.

(b) Coulomb-freezing correction: derive microscopically — or refute — the proposed low-temperature doubling of the effective pinning frequency, in which oscillations about the pinning potential sample the total rather than the net pinning energy because Coulomb effects prevent the density fluctuations, and the consequent factor-of-2 reduction of $k$.

(c) Dielectric–relaxation relation: establish quantitatively whether the temperature-dependent $\varepsilon(0)$ tracks the high-frequency relaxation as the two-time-constant framework requires ($M'=(\tau^*/\tau)M_F$, $L'=(\tau/\tau^*)^{1/2}L_d$, $eE_T=(\tau^*/\tau)\lambda M_F v_p^2$, and $\varepsilon(0)$ proportional to the capacitance $C$ in the resistor–capacitor model), including the implied constancy of $\varepsilon(0)E_T$ as $\varepsilon(0)$ varies with temperature.

(d) Dimensionality correction: test and microscopically justify — or refute — the quasi-two-dimensional factor-of-4 reduction of chain-direction pinning energy (a further factor of 2 in $k$), through measurements of transverse phase-correlation lengths and their effect on $E_T$, $E_T^*$ and $\varepsilon(0)$.

These four questions are linked aspects of one objective: a quantitative account of the CDW threshold polarization across compounds and temperature regimes, deciding whether the universality claim and its proposed corrections survive experimental test and microscopic derivation.

References

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