Validity and microscopic basis of the tunneling theory of charge-density-wave depinning: correlation length, universal tunneling function, effective collective charge, and photon-assisted frequency response

Bardeen (1980) proposed that depinning of a charge-density wave (CDW) — the coupled electron-lattice condensate of a quasi-one-dimensional conductor such as NbSe3 or TaS3 — occurs by macroscopic quantum tunneling of the condensate through a small pinning gap, with a threshold field $E_T = \mathcal{E}_g/(e^* L)$ set by the pinning gap $\mathcal{E}_g$, the effective collective charge $e^* = e\,m/M_F$, and a correlation length $L$ over which the field must act. The theory predicted a universal reduced-field conductivity function $P(x) = (1-x^{-1})e^{-1/x}$ and, via Tucker's photon-assisted tunneling theory, a frequency-to-voltage mapping under which the ac conductivity follows the same universal function of $\omega/\omega_T$ with a threshold frequency $\omega_T = \mathcal{E}_g/\hbar$. Classical weak-pinning domain physics and the quantum tunneling picture compete across the correlation length, the impurity and finite-size dependence of the threshold, the range of validity of the reduced-field function, the effective collective charge, and the sub-threshold photon-assisted response. The surviving open core — whether CDW depinning and sliding transport is macroscopic quantum tunneling or classical nonlinear dynamics of a pinned deformable condensate, together with a quantitative determination of the collective parameters ($L$ or domain lengths, $e^*/M_F$, $\mathcal{E}_g$, $\omega_T$) — remains contested and actively pursued.

Background

In quasi-one-dimensional conductors such as the transition-metal trichalcogenides NbSe3 and orthorhombic TaS3, a Peierls instability — the one-dimensional analog of the superconducting instability, driven by the divergence of the electronic susceptibility at wavevector $2k_F$ — converts the conduction electrons below a critical temperature into a charge-density wave (CDW): the electronic density and the underlying lattice become modulated with period $\lambda = \pi/k_F$, and a gap $2\Delta$ opens at the Fermi surface. Fröhlich showed in the 1950s that such a modulated state can translate as a whole and carry current — the mechanism now called Fröhlich or sliding conductivity — because shifting the CDW phase translates the electronic states without exciting quasiparticles across the gap. In real crystals the CDW phase is pinned by impurities and, in some compounds, by commensurability with the underlying lattice, so at small dc fields only residual normal electrons respond Ohmically. Above a threshold field $E_T$ — first seen as a threshold in the sliding-mode conductivity of NbSe3 by Fleming and Grimes (1979), following the discovery of strongly nonlinear conductivity there by Monceau, Ong, Portis, Meerschaut and Rouxel (1976) — the CDW depins and slides, adding a collective conductivity whose field dependence resembles field ionization (Zener tunneling of electrons across a gap, whose rate grows as $e^{-E_0/E}$). A sliding CDW also converts part of the dc current into narrow-band current oscillations ('narrow-band noise') whose fundamental frequency $f_0$ is proportional to the CDW current — the analog of the ac Josephson relation between current and frequency in superconductors.

Two frameworks competed to explain depinning. In the classical picture, formulated for impurity pinning by Lee and Rice (1979) in what is called the Fukuyama-Lee-Rice framework (distinguishing weak pinning, where many weak impurities jointly distort the CDW into domains, from strong pinning by individual defects), the CDW is a deformable classical charge-density coupled to the lattice; depinning is classical nonlinear dynamics of an overdamped driven elastic medium, with metastable configurations, phase slip (local $2\pi$ adjustments of the CDW phase at contacts and defects), and impurity-pinning domains. In the quantum picture introduced by Bardeen (1979) and elaborated by Bardeen (1980) — building on Maki's (1977) treatment of electric-field creation of soliton pairs (kink-like $2\pi$ phase twists of the CDW) — the pinned condensate tunnels as a whole through a small pinning gap $\mathcal{E}_g$ that impurities open in the electronic spectrum near the Fermi level: a many-body analog of Zener tunneling. Because the field must do work $e^* E L$ across a coherent segment to supply the gap energy, there is a sharp threshold $E_T = \mathcal{E}_g/(e^* L)$, where $L$ is the correlation length of the tunneling segment and $e^* = e\,m/M_F$ is the effective collective charge, reduced from the bare electron charge $e$ by the ratio of the band mass $m$ to the Fröhlich mass $M_F$ — the effective inertia of the coupled electron-phonon condensate, expected from mean-field theory to exceed the band mass by several orders of magnitude. Bardeen fitted the measured nonlinear conductivity of both NbSe3 (data of Richard and Monceau, 1980) and TaS3 by the universal reduced-field function $P(x) = (1-x^{-1})e^{-1/x}$ with $x = E/E_T$ (normalized to its high-field value, $P\to1$ as $x\to\infty$), essentially parameter-free once $E_T$ was fixed, and suggested testing other linear-chain compounds.

For the frequency-dependent conductivity (measured in NbSe3 by Grüner, Tippie, Sanny, Clark and Ong, 1980), Bardeen imported Tucker's theory of photon-assisted tunneling (PAT), developed for superconducting tunnel-junction detectors and mixers: a tunnel junction irradiated at frequency $\omega$ responds as if quanta $\hbar\omega$ were absorbed, so ac excitation and dc bias enter through a single voltage-to-energy mapping. Replacing the electron charge by $e^*$ and the junction voltage by the voltage $V_1 = E L$ across a correlation length, and assuming small ac amplitude and a resistive-dominated response, he predicted that $\sigma(\omega)$ as a function of reduced frequency $\omega/\omega_T$ follows the same universal function as $\sigma(E)$, with a threshold frequency $\omega_T = \mathcal{E}_g/\hbar$. The most distinctive consequence is sub-threshold photon-assisted rectification: absorption of quanta $\hbar\omega$ should assist tunneling even when the combined dc and ac drive would leave a classical medium pinned, producing dc CDW current and interference features governed by the quantum energy scale $\hbar\omega$ — a signature classical sub-threshold dynamics cannot reproduce. Under combined dc and RF drives, both frameworks also predict interference: step-like mode locking of the narrow-band noise frequency (Shapiro-like steps, the analog of the current steps of Josephson junctions under microwave irradiation), but with different dependence on ac amplitude and frequency.

Five threads remained to be established: the origin and magnitude of $L$ (asserted, by analogy with superconductivity, to equal $2\ell$ with $\ell$ the electronic mean free path, but only inferred from $E_T$); the universality of $P(x)$ across other linear-chain compounds; a quantitative independent determination of $e^*$ (equivalently the Fröhlich mass $M_F$); the regime of validity of the Tucker-based frequency-to-voltage mapping and its assumptions (small ac amplitude $V_1 = E L$; resistive-dominated response); and a microscopic account of how impurity concentration $c_i$ sets both $\mathcal{E}_g$ and $L$ — the assumptions $\mathcal{E}_g \propto c_i$ and $L \propto 1/c_i$ reproducing the then-observed $E_T \propto c_i^2$ of Ta-doped NbSe3. These five threads are linked components of one overarching objective: a quantitatively tested account of what depins and carries current when a CDW slides, and of whether that account is quantum tunneling or classical dynamics.

Problem Statement

Determine the physical validity and microscopic basis of the tunneling theory of charge-density-wave (CDW) depinning, as posed by Bardeen's 1980 formulation and its five linked open threads. The overarching objective is a quantitatively tested account of what depins and carries current when a pinned CDW slides — in modern terms, a decisive discrimination between macroscopic quantum tunneling of the condensate and classical nonlinear dynamics of a deformable pinned charge-density wave — together with the collective parameters that either description requires. The specific linked questions are: (i) the origin, magnitude, and microscopic dependence of the correlation length $L$ across which the field must act — in particular whether $L$ is related to the electronic mean free path ($L \approx 2\ell$, as asserted by analogy with superconductivity) or set by impurity-domain physics, and how it depends on impurity concentration (assumed as $L \propto 1/c_i$); (ii) whether the universal reduced-field tunneling function $P(x) = (1-x^{-1})e^{-1/x}$ describes the nonlinear conductivity of linear-chain compounds beyond NbSe3 and TaS3, and what the applicability limits of the tunneling description are; (iii) a quantitative, independent (not fit-derived) determination of the effective collective charge $e^*$, equivalently the Fröhlich mass $M_F$ of the condensate; (iv) the regime of validity of the Tucker-based frequency-to-voltage mapping — whether $\sigma(\omega)$ follows the same universal function of $\omega/\omega_T$ with a threshold frequency $\omega_T = \mathcal{E}_g/\hbar$, including its assumptions of small ac amplitude $V_1 = E L$ and resistive-dominated response, and the predicted sub-threshold photon-assisted rectification; and (v) a microscopic account of how impurity concentration $c_i$ sets the pinning gap $\mathcal{E}_g$ (the assumed $\mathcal{E}_g \propto c_i$, which with the $L \propto 1/c_i$ of (i) gives $E_T \propto c_i^2$). The mechanism — quantum tunneling versus classical depinning, or a specified synthesis — must simultaneously account for the measured field-dependent and frequency-dependent response and the impurity dependence of the threshold.

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