Quantitative incorporation of Coulomb interactions and screening dynamics in the tunneling theory of charge-density-wave depinning

Bardeen, Lyding, Lyons, Miller, Thorne and Tucker (Synth. Met. 19, 1 (1987)) consolidated a quantum-tunneling model of charge-density-wave (CDW) depinning in quasi-one-dimensional Peierls conductors — a Zener-like tunneling probability through a pinning gap, photon-assisted-tunneling scaling of the ac response, and a non-sinusoidal pinning potential generating subharmonic steps — but stated that the theory is quantitatively reliable only where screening of CDW charge fluctuations by normal carriers is effectively complete. The paper left open: (i) quantitative incorporation of Coulomb interactions and the finite-time screening by thermally excited normal carriers into the tunneling/depining kinetics, needed to explain the low-temperature thermally activated response (threshold fields close to predictions, but current magnitudes far smaller and breakdown of photon-assisted-tunneling scaling) in $TaS_3$, $(TaSe_4)_2I$ and $K_{0.3}MoO_3$; (ii) a microscopic treatment of electron-phonon momentum relaxation beyond phenomenological rescalings; (iii) the origin of the absolute-magnitude discrepancy of subharmonic step widths; and (iv) the high-frequency validity limit and the crossover between coherent tunneling and thermally activated or incoherent regimes. The subsequent literature (classical screened-dynamics and phase-slip theories; the Miller group's time-correlated soliton-tunneling program through 2024; low-temperature creep and crossover experiments through 2023) treats these questions partially, classically, or phenomenologically, but does not resolve them: the surviving open core is a quantitative microscopic theory of CDW quantum depinning incorporating Coulomb effects and screening dynamics, with a validated temperature-frequency validity domain.

Background

In many quasi-one-dimensional metals — $NbSe_3$, orthorhombic $TaS_3$, $(TaSe_4)_2I$, and the blue bronze $K_{0.3}MoO_3$ — the Fermi surface is unstable at low temperature to a charge-density wave (CDW), a periodic modulation of the electronic charge density with wave vector $2k_F$, accompanied by a periodic lattice distortion (the Peierls transition) that opens an energy gap (the Peierls gap) over most of the Fermi surface. The phase of the modulation is a collective degree of freedom whose sliding would carry current (Fröhlich conduction), but impurities pin it, so a finite threshold electric field $E_T$ is required for collective motion. In the classical picture — the Fukuyama-Lee-Rice model of impurity pinning, in which the CDW is an elastically deformable medium in a random impurity pinning potential — the sliding condensate is overdamped and depins classically; the condensate carries a very large effective (Fröhlich) mass $M_F$ because the electrons drag the lattice distortion with them, and the gapless collective phase excitations are called phasons. In 1979-1980 John Bardeen proposed instead that depinning in these materials is a quantum-coherent process: electrons of the condensate tunnel through a small pinning gap $E_g$ at the Fermi surface (the energy needed to advance the electronic wave vector by $2k_F$), giving a Zener-type tunneling probability $P(E)=\exp(-E_0/E)$ with characteristic field $E_0=\pi E_g^2/(4\hbar v_F e)$, where $v_F$ is the Fermi velocity and $e$ the electronic charge. The 1987 paper audited here — a conference summary by Bardeen, Lyding, Lyons, Miller, Thorne and Tucker — consolidates this 'tunneling model'. It relates the pinning gap to the pinning frequency $\omega_p$ through $E_g=(2/\pi)\sqrt{M_F/m}\,\hbar\omega_p$; derives a tunneling-weighted drift frequency $\hbar\omega_d=(m/M_F)eE(2v_F\tau^*)P(E)$, where $\tau^*$ is the condensate's phase-correlation time for tunneling; defines a scaling frequency $\omega_s$ (in the revised theory of Bardeen, Phys. Rev. Lett. 55, 1010 (1985), $\omega_s=(c_0/v_F)\,\omega_p$, with $c_0$ the phason velocity); proposes that the ac conductivity follows photon-assisted tunneling (PAT) — the standard theory of the ac response of tunnel junctions, in which absorbed photons assist single-particle tunneling, here applied to the condensate — giving $\sigma(\omega)=\sigma_b\exp(-\omega_s/\omega)$, where $\sigma_b$ is the high-frequency conducting limit of the condensate's ac response, from which detection, mixing and harmonic generation follow from the dc current-voltage curve; models the pinning energy as the non-sinusoidal potential $V(\theta)=-V_0-V|\cos\theta|$ whose Fourier content explains the strong harmonic content of the narrow-band noise (the current oscillations at frequency $\omega_d=2k_F v_d$, with $v_d$ the drift velocity, that accompany sliding); and predicts subharmonic (rational $p/q$) constant-current steps under combined dc and ac drive. Earlier evidence for tunneling of CDWs in $TaS_3$ was reported in Miller et al. (1983, 1985), and the accompanying detailed experiments in Thorne et al. (1985, 1987); the microscopic basis was elaborated in Bardeen (1987, 1989). Against experiment, the model gives the observed exponential field dependence of the CDW current, the correct scale of threshold fields, and the correct relative subharmonic step widths. The same paper states the model's quantitative limitations, which define this problem. (i) At low temperatures in the semiconducting CDW materials ($TaS_3$, $(TaSe_4)_2I$, $K_{0.3}MoO_3$) the response becomes thermally activated: threshold fields remain comparable to the predictions, but the CDW current magnitudes are much smaller than predicted, and the PAT scaling of $\sigma(\omega)$ breaks down. The authors attribute this to the finite time required for thermally excited normal carriers (electrons and holes excited across the Peierls gap) to screen the charge fluctuations that a moving CDW polarization produces; the theory omits the Coulomb forces from these density fluctuations and is valid only when screening is effectively complete — as it is at higher temperature or in $NbSe_3$ with its incomplete Peierls gap. (ii) Electron-phonon momentum relaxation is included only phenomenologically, through modified scales $E_0'=(\tau/\tau^*)E_0$ and $\omega_s'=\sqrt{\tau^*/\tau}\,\omega_s$ with $\tau=\tau_{el-ph}+\tau^*$, where $\tau_{el-ph}$ is the electron-phonon momentum-relaxation time. (iii) The calculated absolute magnitudes of the subharmonic step widths come out smaller than the idealized estimate, although the relative widths agree. (iv) The upper-frequency validity of the theory ($\omega_d$ up to order $\sqrt{M_F/m}\,\omega_p$ or $1/\tau$) is identified only heuristically, leaving the breakdown regimes and the crossover between coherent tunneling and thermally activated (or incoherent) response unmapped.

Problem Statement

The problem is to complete the tunneling theory of CDW depinning into a fully quantitative microscopic theory with a validated validity domain. The common objective, encompassing the four components below, is to make the tunneling model of charge-density-wave depinning quantitatively correct wherever it is applied, by incorporating the physics it currently omits. Concretely: (1) Incorporate quantitatively, within the tunneling/depining kinetics, the Coulomb interactions arising from CDW density fluctuations and the finite-time screening of those fluctuations by thermally excited normal carriers (electrons and holes excited across the Peierls gap): derive the resulting low-temperature, partially screened response — including why threshold fields stay comparable to predictions while CDW current magnitudes are strongly suppressed and the photon-assisted-tunneling scaling $\sigma(\omega)=\sigma_b\exp(-\omega_s/\omega)$ breaks down — in materials such as $TaS_3$, $(TaSe_4)_2I$ and $K_{0.3}MoO_3$. (2) Replace the phenomenological electron-phonon momentum-relaxation rescalings $E_0'=(\tau/\tau^*)E_0$ and $\omega_s'=\sqrt{\tau^*/\tau}\,\omega_s$ by a microscopic treatment of electron-phonon relaxation and its temperature dependence, as needed for materials with very high pinning frequencies. (3) Identify the physical origin of the quantitative discrepancy in the absolute magnitudes (as opposed to the relative widths) of the subharmonic steps — the calculated absolute step widths come out smaller than the idealized estimate, while the relative widths agree. (4) Establish systematically the theory's validity limits: the high-frequency breakdown and the crossover boundaries between coherent quantum tunneling, thermally activated response, and incoherent transport in the temperature-frequency plane.

References

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