Microscopic foundation and experimental test of quantum-tunneling depinning of charge-density waves

A pinned charge-density wave (CDW) — the Peierls-condensed, modulated ground state of a quasi-one-dimensional metal — carries collective current only above a threshold electric field, and whether it depins and moves as a coherently tunneling macroscopic quantum object or as a classically deforming elastic medium has been contested since Bardeen's tunneling theory (1979-1985). J. Bardeen (Physica B+C 126, 342 (1984)) recast that theory in soliton form and left open threads: the microscopic derivation of the soliton energy and Fröhlich mass, the treatment of memory and hysteresis below threshold, dissipative corrections to the tunneling rate, the justification of the photon-assisted-tunneling description of the ac response, and the experimental demonstration of macroscopic quantum coherence. The question remains open through 2024, though unevenly: single-chain soliton building blocks and adjacent Caldeira-Leggett-type treatments of dissipation exist but were never assembled into a controlled theory of the pinned, domain-scale collective coordinate; memory and hysteresis were reinterpreted within classical metastability rather than resolved inside the tunneling framework; the photon-assisted-tunneling substitution remains heuristic; and an active experimental program (low-temperature creep, 'quantized' microcrystal states, h/2e conductance oscillations in rings) reports quantum signatures whose interpretation is still contested by the phase-slip and metastability mainstream.

Background

A charge-density wave (CDW) is a modulation of both the electronic density and the lattice, $\rho(x)=\rho_0\cos(2k_F x+\phi)$, that forms in quasi-one-dimensional metals below a Peierls transition (the electron-phonon-coupling-driven instability of a one-dimensional metal), opening a gap $E_g$ at the Fermi surface ($k_F$ is the Fermi wave vector and $\phi$ the collective phase of the condensate). Because the phase $\phi$ is free to slide, the condensed electrons can in principle carry current with little dissipation (Fröhlich conduction), but with an effective inertial mass $M$ per electron far larger than the band mass $m$ ($M/m$ typically $10^2$-$10^3$); this mass ratio enters all tunneling formulas below. In real crystals, randomly distributed impurities pin the phase: a finite threshold field $E_{th}$ must be exceeded before the CDW moves, above which the current-voltage relation is strongly nonlinear and history-dependent. Small-amplitude phase oscillations of the pinned condensate are phasons, characterized over a Fukuyama-Lee-Rice (FLR) weak-pinning domain of length $L_d$ (in weak pinning, many individually weak impurities pin the phase collectively over such a domain, rather than each impurity pinning alone) by a pinning-gapped mode of frequency $\omega_p$; narrow-band noise (NBN) is the periodic current signal emitted by the sliding CDW. Two competing microscopic pictures of depinning exist. (i) The classical picture: the CDW is an elastic medium in a random pinning potential that slides as an overdamped, deformable charge among many metastable configurations, with phase slips (localized $2\pi$ jumps of $\phi$, at contacts or defects) required to sustain dc flow; memory and hysteresis reflect the glassy metastable configuration. (ii) The quantum picture (Bardeen): the pinned charge must tunnel through pinning barriers; in the 1984 soliton version, each phase-coherent domain tunnels by nucleating a pair of $\pi$-solitons (kinks of the phase) with energy $2E_\phi=\sqrt{M/m}\,\hbar\omega_p$ and effective charge $e_s=\sqrt{m/M}\,e$, where $e$ is the electron charge, giving a Zener-type tunneling probability $P(E)=\exp(-E_0/E)$ in a field $E$ (Zener tunneling: field-driven tunneling through a band gap, as in semiconductor breakdown), with activation field $E_0$ of order $\pi E_g^2/(4\hbar e v_F)$ ($v_F$ is the Fermi velocity), and a predicted doubling of the narrow-band-noise frequency relative to the classical sliding value. Photon-assisted tunneling (PAT) is the extension of single-particle tunneling theory to ac-driven junctions (Tucker, 1979): currents follow from photon-sideband weights, implemented for CDWs as a finite-difference replacement of dc-bias derivatives, and used heuristically to explain CDW ac detection and mixing. Macroscopic quantum tunneling (MQT) is the escape of a collective coordinate through a barrier at temperatures too low for thermal activation, familiar from Josephson junctions; for CDWs it implies temperature-independent subthreshold creep, size-dependent thresholds, and, in ring geometries, conductance oscillations with magnetic-flux period h/2e (an Aharonov-Bohm-type effect). The open problem is whether the quantum picture can be given a microscopic foundation and experimentally verified — or whether the classical picture suffices.

Problem Statement

Construct and experimentally test a microscopic theory of the depinning and nonlinear transport of a pinned charge-density wave (CDW) in a quasi-one-dimensional Peierls conductor, treating the condensate as a quantum object, or establishing that the classical picture suffices. The open threads of this program include: deriving, from the electron-phonon-plus-impurity Hamiltonian, the collective tunneling coordinate and its parameters - the $\pi$-soliton (domain-wall) energy $E_\phi$, the effective soliton charge $e_s$, and the Fröhlich mass ratio $M/m$ - together with the pinned domain length $L_d$ and plasma frequency $\omega_p$ that set the activation field $E_0$ in the tunneling probability $P(E)=\exp(-E_0/E)$; determining how the glassy metastable configuration of the pinned state, the origin of hysteresis and pulse-duration memory below threshold, enters the distribution of local depinning fields and hence the tunneling rates; computing the dissipative corrections to the tunneling rate from the coupling to normal quasiparticles and phonons, as explicit functions of material parameters; establishing the conditions of field, frequency, and amplitude under which the photon-assisted-tunneling (PAT) finite-difference description of the coupled ac-dc response — the replacement, borrowed from single-particle tunnel-junction theory, of dc-bias derivatives by finite differences — is a valid approximation; and experimentally demonstrating macroscopic quantum coherence of the pinned condensate through observable, size- and temperature-dependent signatures (such as temperature-independent subthreshold creep, i.e. slow current flow below the depinning threshold) in regimes where they can be distinguished from classical deformable-charge sliding and phase-slip creep. These threads are open aspects of a single target - a microscopically founded and experimentally verified quantum theory of depinning - rather than a conjunctive checklist of separately mandated deliverables.

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