Unresolved issues in the quantum-tunneling theory of charge-density-wave depinning: sliding mechanism, below-threshold response, and coupling to narrow-band noise

John Bardeen's 1985 chapter "Depinning of Charge-Density Waves by Quantum Tunneling" proposed that the nonlinear conduction of quasi-one-dimensional metals above a depinning threshold is macroscopic quantum tunneling of the charge-density-wave (CDW) phase over a pinning gap within phase-coherent domains, and showed that this model, together with photon-assisted tunneling theory, accounts for a wide range of dc, ac, and radio-frequency mixing experiments. The chapter itself closed with unresolved issues: no direct evidence for quantum effects (such as photon-assisted tunneling steps) existed; the below-threshold response was not understood; the revised ac-dc scaling had an undetermined lower frequency limit; a microscopic derivation of the equations of motion incorporating quantum tunneling, large-amplitude ac effects, and coupling to narrow-band noise was lacking; and the predicted zero-bias harmonic-mixing signal was not observed unless a finite difference frequency was used. Four decades of subsequent literature have not resolved this: classical models and quantum-tunneling proposals each account for parts of the measured transport phenomenology. The consolidated problem - determine the microscopic mechanism of CDW depinning and construct a theory consistently describing quantum tunneling, below-threshold response, and narrow-band noise, with a direct experimental discriminator between quantum and classical depinning - remains open.

Background

The physical setting. In certain highly anisotropic ("quasi-one-dimensional") metals - such as niobium triselenide (NbSe$_3$), orthorhombic tantalum trisulfide (o-TaS$_3$), and the blue bronze K$_{0.3}$MoO$_3$ - electrons can lower their energy by forming a charge-density wave (CDW): a periodic modulation of the electron density, with wavelength set by twice the Fermi wavelength, accompanied by a matching distortion of the underlying lattice (a Peierls transition, the one-dimensional analog of a superconducting gap opening over an entire Fermi surface). Because the condensed electrons carry a current when the density wave translates rigidly over the lattice periodicity - so-called Fröhlich conduction - a CDW is in principle a frictionless collective conductor. In practice, impurities and defects pin the phase of the wave to the lattice, so a minimum threshold field $E_{th}$ must be exceeded before the CDW slides; above threshold the current-voltage curve is strongly nonlinear, and the excess current oscillates coherently in time, producing narrow-band noise (NBN) - a sharp peak in the current-noise spectrum at a frequency proportional to the CDW current, with harmonics that in the simplest pinning-potential model decay as $1/n^2$.

The classical description. The standard classical theory treats the CDW phase as a classical, overdamped elastic medium (the Fukuyama-Lee-Rice model: the CDW phase as an elastic variable pinned by random impurity potentials), with pinning weak enough that the phase remains coherent over domains of a characteristic length (Lee-Rice domains). Depinning is then classical depinning of an elastic medium, and radio-frequency irradiation of a sliding CDW produces current steps at dc biases where the internal oscillation frequency locks to the applied frequency. These steps are universally called Shapiro steps, by analogy with the current steps that radio-frequency irradiation produces in superconducting Josephson junctions, but in CDW conductors they are conventionally understood as classical interference or mode-locking (frequency locking) effects rather than as quantum signatures.

Bardeen's tunneling model. Bardeen instead proposed that CDW depinning is macroscopic quantum tunneling: the pinned CDW sits in a false vacuum, and the phase advances by quantum tunneling across a small pinning gap $\Delta$ that impurity fluctuations open at the Fermi level within a Lee-Rice domain of length $L$. Because the current is carried by the electron-phonon condensate, the condensate's inertia is characterized by the Fröhlich mass $M_F$, which greatly exceeds the electron band mass because the entire lattice distortion moves with the condensate, and the condensate responds to fields with an effective charge $e^* = (m/M_F)e$, where $m$ is the electron band mass and $e$ the elementary charge. With $\omega_P$ the frequency associated with the pinning gap, the model predicts a threshold relation $e^* E_{th} L = \hbar \omega_P$ and a dc current of Zener-tunneling form (Zener tunneling: field-assisted quantum tunneling through a barrier, as in electrical breakdown of insulators)

$$I_{CDW} = \sigma_b (E - E_{th}) \exp(-E_0/E)$$

for $E > E_{th}$, where $\sigma_b$ is a constant and the characteristic field is $E_0 = \pi\hbar\omega_P^2/(4 v_F e^*)$ in the simple Zener model; accounting for the damping of the condensate (dissipative quantum tunneling theory) replaces the Fermi velocity $v_F$ by the phason velocity $u_P = (m/M_F)^{1/2} v_F$ (phasons: the collective phase oscillation modes of the CDW). Applying photon-assisted tunneling (PAT) theory - the framework by which photons of energy $\hbar\omega$ assist quantum tunneling, as in superconducting junctions - yields a scaling between the frequency-dependent ac conductivity and the field-dependent dc conductivity, $\sigma(\omega) \sim \sigma_{dc}(\hbar\omega/(e^* L))$, valid for angular frequencies $\omega > 2\omega_T$ with $\omega_T = e^* E_{th} L/\hbar$. A revised version of the scaling, motivated by the experimental observation that dc biases below threshold barely affect any ac response, scales with the voltage above threshold, $V' = V - V_T$, and expresses the ac conductivity, rectification (the dc response to an ac drive), and harmonic mixing (generation of a current at the difference frequency $2\omega_1 - \omega_2$ of two applied ac signals; direct mixing analogously uses the difference $\omega_1 - \omega_2$ of two frequencies) through finite differences of the dc current $I'(V' \pm n\hbar\omega/e^*)$.

What the chapter left open. The chapter's concluding assessment states that despite the model's success in explaining many experiments, several issues remained unresolved: (i) no direct evidence for quantum effects, such as photon-assisted tunneling steps, had been found - noting in particular that the rf-induced steps observed in CDW systems are equally well accounted for as classical interference, so they do not discriminate; (ii) the response of CDWs biased below threshold was not fully understood; (iii) the revised scaling required a lower frequency limit of applicability that was not precisely determined; (iv) a more basic microscopic derivation of the equations of motion for CDWs, incorporating quantum tunneling, large-amplitude ac effects, and their coupling to narrow-band noise, was needed; (v) the interplay between the oscillatory current (narrow-band noise) and quantum tunneling had not been fully integrated; and (vi) the theory of harmonic mixing predicted a zero-bias signal that was not observed - the signal was recovered only when a finite difference (output) frequency was used, indicating an incomplete understanding of the below-threshold response.

Problem Statement

Determine the microscopic mechanism of charge-density-wave depinning and transport in quasi-one-dimensional conductors, and settle the specific issues that Bardeen's quantum-tunneling model left open. Concretely, the problem comprises:

(1) Mechanism discrimination. Establish whether CDW depinning above threshold is quantum tunneling of the phase-coherent condensate over the pinning gap (as in Bardeen's model, with effective charge $e^* = (m/M_F)e$ where $m$ is the electron band mass and $M_F$ the Fröhlich mass, threshold relation $e^* E_{th} L = \hbar \omega_P$ where $L$ is the phase-coherent domain length and $\omega_P$ the frequency associated with the pinning gap, and Zener-form current $I_{CDW} = \sigma_b (E - E_{th}) \exp(-E_0/E)$ above threshold, with $\sigma_b$ a constant and $E_0$ the characteristic tunneling field), or classical depinning of an overdamped elastic medium, or a hybrid (for example soliton nucleation, solitons being localized kink-like distortions of the density-wave phase); and supply a direct experimental discriminator between the quantum-tunneling and classical (or hybrid) mechanisms.

(2) Microscopic equations of motion and response. Derive, from a basic microscopic starting point, the equations of motion of a pinned CDW underlying the tunneling model and its ac/dc response, incorporating quantum tunneling, large-amplitude ac effects, and the coupling of the oscillatory narrow-band noise to the tunneling process; determine the below-threshold regime, in which small-signal ac dynamics are decoupled from dc bias and the predicted zero-bias harmonic-mixing signal is recovered only at a finite difference frequency; and fix the lower frequency limit of validity of the revised ac-dc scaling.

Elements (1) and (2) are both parts of this target. Element (1) asks whether depinning is quantum tunneling, classical, or hybrid, together with a concrete, falsifiable experiment that would distinguish the quantum-tunneling mechanism from the classical or hybrid ones. Element (2) asks for a derivation of the microscopic equations of motion of the tunneling model and its ac/dc response that begins from a stated microscopic model, is internally consistent, incorporates quantum tunneling, large-amplitude ac effects, and the coupling of the oscillatory narrow-band noise to the tunneling process, reproduces the ac conductivity, rectification, and harmonic-mixing responses including the below-threshold regime, and fixes the lower frequency limit to within an order of magnitude. These equations of motion are those of the tunneling model whatever the outcome of the determination in element (1).

References

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15. John H. Miller, Jr., Martha Y. Suárez-Villagrán, and Johnathan O. Sanderson, "Quantum transport of charge density wave electrons in layered materials", Materials Today Physics 41, 101326 (2024). DOI 10.1016/j.mtphys.2024.101326, https://doi.org/10.1016/j.mtphys.2024.101326 (also arXiv:2310.10512, https://arxiv.org/abs/2310.10512; the quantum school's most recent review, whose abstract states that the CDW "is known to carry electric current en masse, but the transport mechanism remains poorly understood at the microscopic level")
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