John Bardeen's 1985 theory of charge-density-wave (CDW) transport in quasi-one-dimensional metals explains nonlinear conduction above a threshold field by Zener-like macroscopic quantum tunneling of CDW electrons, but it is explicitly restricted to an idealized regime: the polarization field is neglected (validity only for applied fields Eappl≳2ETE_{appl}\gtrsim 2E_{T}), pinning is modeled by periodic domains, and the relaxation time τ∗\tau^{*} that sets the phase-correlation length L=2vFτ∗∼c0/ωpL=2v_{F}\tau^{*}\sim c_{0}/\omega_{p} is not derived; in addition, the key numerical estimates used to fix the pinning gap (κa=(π/4)κm\kappa_{a}=(\pi/4)\kappa_{m}; pinning energy =−13=-\tfrac{1}{3} of the kinetic-energy increase) are uncontrolled. No complete microscopic theory of near-threshold CDW tunneling transport exists; the problem remains open at its full source scope.

Background

Below a Peierls transition temperature, linear-chain metals such as NbSe3_3, TaS3_3, and K0.3_{0.3}MoO3_3 develop a charge-density wave (CDW): a modulation of the electron density, ρ=ρ0+ρ1cos⁡(2kFx+ϕ(x,t))\rho=\rho_{0}+\rho_{1}\cos(2k_{F}x+\phi(x,t)), formed jointly by electrons and ions with wave vector 2kF2k_{F} (kFk_{F} is the Fermi wave vector), which opens an energy gap at the Fermi surface. Because the condensate can translate, it can carry current (Fröhlich conduction), but impurities pin the wave to the lattice until a threshold electric field ETE_{T} is exceeded; above threshold these materials show strongly nonlinear conduction, hysteresis, and narrow-band noise (periodic voltage oscillations whose frequency is proportional to the CDW current). In the classical description, built on the Fukuyama-Lee-Rice model (the CDW phase ϕ\phi interacting with random impurity pinning), the CDW is a deformable classical object that slides once the field overcomes the pinning force, and the phase breaks into Lee-Rice domains of length LdL_{d} within which it remains coherent.

John Bardeen proposed in 1979-1980, and developed through a close interaction with experiment, that depinning is instead macroscopic quantum tunneling: a phase-coherent domain has only one thermal degree of freedom, so its total pinning energy can far exceed kBTk_{B}T, and current above threshold arises from Zener-like tunneling of CDW electrons across a tiny impurity-induced pinning gap, with tunneling probability P(E)=exp⁡(−E0/E)P(E)=\exp(-E_{0}/E). Maki had earlier derived a similar soliton-pair creation probability for a single sine-Gordon chain, with the phason velocity c0c_{0} (the sound-like velocity at which phase oscillations of the condensate propagate) in the denominator. In the 1985 Letter, Bardeen derived, in an idealized model with periodic pinning domains, the tunneling activation field E0=πEg2/(4ℏe∗vF)E_{0}=\pi E_{g}^{2}/(4\hbar e^{*}v_{F}) with Eg=ℏωpE_{g}=\hbar\omega_{p}, where ωp\omega_{p} is the pinning frequency (the oscillation frequency of the pinned condensate in a pinning well), vFv_{F} is the Fermi velocity, and e∗=e m/(m+MF)e^{*}=e\,m/(m+M_{F}) is the effective electronic charge with MFM_{F} the Fröhlich mass, the ionic contribution to the condensate inertia, which is much larger than the bare electron mass mm. He further derived the scaling frequency of the ac conductivity, ωs=(c0/vF)ωp\omega_{s}=(c_{0}/v_{F})\omega_{p}, which controls the observed law σ(ω)=σbexp⁡(−ωs/ω)\sigma(\omega)=\sigma_{b}\exp(-\omega_{s}/\omega) and connects the dc and ac responses through photon-assisted tunneling (a theory developed for superconducting tunnel junctions, in which microwave quanta assist CDW tunneling); the domain length Ld=πc0/ωpL_{d}=\pi c_{0}/\omega_{p}; and the CDW current ICDW=ncNchee∗EL P(E)/(πℏ)I_{\mathrm{CDW}}=n_{c}N_{ch}ee^{*}EL\,P(E)/(\pi\hbar), where LL is the phase-correlation (free) path, ncn_{c} the condensate fraction, and NchN_{ch} the number of chains adding coherently. The elementary tunneling event creates a pair of π\pi solitons (phase kinks across which the CDW phase advances by half a wavelength, each carrying charge ee per chain), and the single-particle gap Δ\Delta entering the Zener exponent was estimated by equating the impurity pinning energy to −13-\tfrac{1}{3} of the kinetic-energy increase of the phase deformation and by using an averaged inverse decay length κa=(π/4)κm\kappa_{a}=(\pi/4)\kappa_{m}. The companion experiment by Thorne, Miller, Lyons, Lyding, and Tucker found these relations in quantitative agreement with NbSe3_3 and TaS3_3 data well above threshold.

The Letter itself marks what the theory omits. It works with the net field E=Eappl−EpolE=E_{appl}-E_{pol}, where EpolE_{pol} is the polarization field generated by the CDW's own displaced charge, but neglects polarization, so its results hold "without correction only for Eappl≳2ETE_{appl}\gtrsim 2E_{T}". Pinning is treated as periodic, whereas in real samples the domains are disordered, their lengths are random variables, and many metastable phase configurations coexist and dominate near-threshold behavior. The rapid relaxation that transfers the momentum and energy gained by tunneling electrons to the macroscopically occupied phonon modes is characterized only qualitatively: the length L=2vFτ∗L=2v_{F}\tau^{*} (τ∗\tau^{*} a relaxation time) is identified with c0/ωpc_{0}/\omega_{p}, equivalently L/Ld∼π−1L/L_{d}\sim\pi^{-1}, as an order of magnitude, with no microscopic derivation of τ∗\tau^{*}. And the numerical estimates behind Δ\Delta carry no error bounds.

Problem Statement

Can the microscopic quantum-tunneling theory of charge-density-wave depinning in quasi-one-dimensional metals be completed, removing the three idealizations explicitly flagged in the source Letter, so that it holds in the near-threshold regime the original treatment does not cover? Specifically:

  1. Polarization: derive the CDW polarization field EpolE_{pol}, and hence the relation between the applied field and the net driving field E=Eappl−EpolE=E_{appl}-E_{pol}, self-consistently, incorporating its back-action on the tunneling rate, so that the theory is valid below Eappl≈2ETE_{appl}\approx 2E_{T}, down to and through the threshold region.

  2. Disorder and metastability: replace the periodic-domain pinning model with disordered phase-coherent domains of distributed sizes and treat the many possible metastable configurations, so that all of the near-threshold phenomena (hysteresis, memory, metastable switching, and the rounding and position of the measured threshold) are accounted for within the tunneling framework.

  3. Relaxation dynamics: identify the microscopic channels by which momentum and energy are transferred from the tunneling electrons to the macroscopically occupied phonon modes, and derive or rigorously bound the relaxation time τ∗\tau^{*}, thereby establishing, or correcting, the identification of the phase-correlation length L=2vFτ∗L=2v_{F}\tau^{*} with c0/ωpc_{0}/\omega_{p} (equivalently L/Ld∼π−1L/L_{d}\sim\pi^{-1}) across materials and temperature regimes.

The Letter's uncontrolled numerical estimates used to fix the pinning gap Δ\Delta — the averaged inverse decay length κa=(π/4)κm\kappa_{a}=(\pi/4)\kappa_{m} and the identification of the impurity pinning energy with −13-\tfrac{1}{3} of the kinetic-energy increase — are part of the theory, and clarifying their footing is part of completing it; deriving formal validity ranges for them in disordered, finite-temperature, multi-chain samples, and demarcating the quantum and classical depinning frameworks, lie outside the objective.

The completed theory would reduce to the published expressions (E0=πEg2/(4ℏe∗vF)E_{0}=\pi E_{g}^{2}/(4\hbar e^{*}v_{F}) with Eg=ℏωpE_{g}=\hbar\omega_{p}, ωs=(c0/vF)ωp\omega_{s}=(c_{0}/v_{F})\omega_{p}, Ld=πc0/ωpL_{d}=\pi c_{0}/\omega_{p}, and the Zener-form current) in the ideal periodic, high-field limit, and its near-threshold predictions would be quantitatively testable against experiments on NbSe3_3 and TaS3_3. The three idealizations are to be removed as a single objective, and a quantitative demonstration that the microscopic tunneling framework cannot be continued into the threshold region would also resolve the question.

References

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