Microscopic theory of charge-density-wave conduction under combined ac and dc fields in NbSe3: quantum tunneling versus classical depinning of the collective phase

The source paper (G. Grüner, A. Zettl, W. G. Clark, and J. Bardeen, Phys. Rev. B 24, 7247 (1981)) reported that the semiconductor (Zener-tunneling) model of charge-density-wave (CDW) depinning, combined with Tucker's photon-assisted tunneling formalism, describes the dc and ac conductivity of NbSe3 for separate drives but fails decisively for simultaneously applied ac and dc fields: no photon-assisted tunneling is observed, the joint-response threshold is smoothed rather than sharpened, and the ac-dc mixing response is classical only at low frequency, falling to a few percent of the classical prediction near the pinning frequency. The paper posed, as an open problem, a microscopic theory based on the collective phase model that includes damping and pinned-mode excitations, explains the absence of photon-assisted tunneling, reproduces the frequency crossover, grounds the phenomenological parameters ($E_0\approx 2E_T$, effective charge $e^*$, correlation length $L$), and extends soliton tunneling to whole-condensate motion. Subsequent work is mixed: phase-sensitive microwave harmonic-mixing experiments reported photon-assisted-tunneling signals (1983-84); classical Fukuyama-Lee-Rice simulations reproduced much of the mixing and mode-locking response (1986-87); dedicated interference experiments contradicted classical deformable-CDW predictions (1987); Bardeen's 1989 critique found 'no consensus' on the correct approach; and the time-correlated soliton-tunneling program (2000-2023) constructed a phase-based quantum theory but remains contested, its authors stating in 2023 that the CDW transport mechanism 'remains poorly understood at the microscopic level'. The problem - a consensus microscopic theory of CDW conduction under combined ac and dc fields, deciding quantum tunneling versus classical depinning - remains open.

Background

NbSe$_3$ (niobium triselenide) is a quasi-one-dimensional metal - a linear-chain compound whose electronic conduction is primarily along one crystal axis - which undergoes two Peierls transitions, at about 145 K and 59 K. A Peierls transition is an instability of a one-dimensional metal in which the electron density and the lattice distort together into a charge-density wave (CDW), a state whose charge density is modulated, $\rho(x)=\rho_0+\rho_1\cos(2k_F x+\phi)$, and in which an energy gap opens at the Fermi surface. The phase $\phi$ of the modulation is a collective variable: an unpinned, sliding CDW carries an electric current (the collective conduction mechanism envisaged in Fröhlich's original theory of the sliding charge-density wave). In real crystals, impurities pin this phase, so a finite threshold electric field $E_T$ (of order 0.1-1 V/cm in NbSe$_3$) is required before the CDW slides, producing strongly nonlinear dc conduction; while sliding, the CDW generates narrow-band noise, an oscillatory current at a frequency proportional to the CDW drift velocity.

Two frameworks were developed to describe the field- and frequency-dependent conductivity of pinned CDWs. In the classical pinning model of Grüner, Zawadowski, and Chaikin, built on the impurity-pinning theory of Fukuyama and Lee and of Lee and Rice, the CDW phase obeys a damped equation of motion in a periodic pinning potential (Eq. (2.2) of the source paper); small oscillations of the pinned CDW near the pinning frequency $\omega_q$ (the pinned mode) give a broad low-frequency contribution to the ac conductivity with no sharp threshold frequency. In the tunneling model of Bardeen (1979, 1980), the pinned CDW is mapped onto a gapped semiconductor: the pinning energy over a correlation length $L$ acts as a gap, and the collective depinning current is computed as a Zener tunneling probability (electric-field-assisted quantum tunneling across the gap) $P(E)=[1-(E_T/E)]\exp(-E_0/E)$, where the effective charge $e^*=e/(1+M_F/m)$ (with $M_F$ the Fröhlich mass, the effective inertia of the coupled electron-phonon condensate, and $m$ the electron band mass) rescales the field. The tunneling model predicts an exact scaling between the dc field response and the ac frequency response, $\hbar\omega=e^*EL$, and a threshold frequency tied to the threshold field by $\hbar\omega_T=e^*LE_T$. To describe simultaneous ac and dc drives, the formalism Tucker developed for superconductor-insulator-superconductor (SIS) tunnel-junction mixers was adopted; in that photon-assisted tunneling (PAT) formalism, in the lineage of Tien and Gordon's treatment of microwave-assisted tunneling, a voltage $V_0+V_1\cos\omega t$ produces current components depending on $V_0\pm\hbar\omega/e^*$, so microwave photons of energy $\hbar\omega$ should assist the tunneling and raise the dc current even when the classical field amplitude is below threshold.

The source paper measured the conductivity of both CDW phases of NbSe$_3$ as a function of dc field $E$ and ac frequency $\omega$ and found: (i) the field-frequency scaling predicted by the tunneling model holds well for $E\gtrsim 2E_T$, and the tunneling functional form fits the nonlinear dc conductivity; (ii) in contrast to the sharp dc threshold there is no threshold frequency, the low-frequency ac response instead requiring the pinned-mode contribution; (iii) under joint dc and rf excitation (up to 150 MHz), no photon-assisted tunneling is observed, and only the low-frequency classical limit of the ac-dc mixing response is seen - at higher frequencies the mixing signal falls to a few percent of the classical prediction near the pinning frequency. The semiconductor mapping, while successful for separate ac or dc drives, thus fails for simultaneous drives: the paper notes that the single-particle quantum frequency $\hbar\omega_0=e^*V_0$ survives in the semiconductor model's tunneling-electron wave functions but is not a frequency of an actual CDW, whose only natural frequencies are those of its motion. The authors also noted that Maki's theory of soliton-pair creation by tunneling - solitons being localized, particle-like kink distortions of the CDW phase - treats soliton formation but not the motion of the CDW as a whole, and that the phenomenological parameters entering the comparison - the ratio $E_0\approx 2E_T$ in NbSe$_3$ (about $5E_T$ in TaS$_3$), the effective charge $e^*$, and the correlation length $L$ - lacked microscopic justification. The paper therefore posed the problem of constructing a theory of CDW conduction under combined ac and dc fields based on the collective phase model, going beyond the single-particle semiconductor mapping.

Problem Statement

Construct - and test against the experimental data on NbSe$_3$ reported in the source paper - a microscopic theory of charge-density-wave conduction based on the collective phase model of the CDW (rather than on the single-particle semiconductor mapping), describing the response of NbSe$_3$ to simultaneously applied dc and ac electric fields. The theory must:

(1) incorporate damping and the internal pinned-mode excitations (oscillations of the pinned CDW near the pinning frequency $\omega_q$) alongside the translational collective mode, so as to reproduce the absence of a threshold frequency in the ac response and the observed field-frequency scaling for $E\gtrsim 2E_T$;

(2) account for the observed absence of photon-assisted tunneling in joint ac-dc experiments of the source geometry (a dc bias with a superposed radio-frequency amplitude, at frequencies up to 150 MHz) - specifically, why the single-particle quantum frequency $\hbar\omega_0=e^*V_0$ of the semiconductor mapping, where $V_0$ is the applied dc voltage, produces no photon-assisted effects in the real CDW;

(3) reproduce the crossover from the classical low-frequency limit of ac-dc mixing to the high-frequency regime in which the measured mixing response drops to a few percent of the classical prediction for $\omega\gtrsim\omega_q$ (the pinning frequency of the pinned mode; in the source's data at 32 K, $\omega_q/2\pi=113$ MHz, numerically coincident with the threshold frequency $\omega_T$ assigned from the field-frequency scaling), in agreement with the observed absence of photon-assisted tunneling in the source's joint-drive geometry;

(4) provide a microscopic basis for the phenomenological parameters of the ac and dc response - the empirical ratio $E_0\approx 2E_T$ in NbSe$_3$ (the source notes $E_0\approx 5E_T$ in TaS$_3$ as a comparison), the magnitude and justification of the effective charge $e^*$, and the correlation length $L$ - and extend soliton- or phase-slip-based tunneling mechanisms (of the type Maki developed for soliton-pair creation, a soliton being a localized kink-like distortion of the CDW phase) to the motion of the CDW as a whole.

These deliverables jointly serve one overarching objective: deciding, at the microscopic level, whether CDW depinning and the combined ac-dc response are governed by quantum tunneling of the condensate or by classical overdamped dynamics of the pinned phase, and demonstrating quantitative agreement with the source paper's own observations on NbSe$_3$ (field-frequency scaling for $E\gtrsim 2E_T$, absence of a threshold frequency with the pinned-mode contribution near $E_T$, absence of photon-assisted tunneling under the joint drive, and the low-frequency classical mixing limit with its drop to a few percent of the classical prediction for $\omega\gtrsim\omega_q$).

References

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