Charge-density-wave depinning in NbSe3: the pinning frequency and impurity dependence, and the threshold field and effective charge near the Peierls transition (Bardeen's 1979 open questions)

In his 1979 theory of non-ohmic conduction in niobium triselenide (NbSe3), Bardeen proposed that the pinned charge-density-wave (CDW) condensate escapes pinning by Zener-type quantum tunneling, with a threshold field set by the pinning frequency and a greatly reduced effective condensate charge, and he left two issues open: how impurity pinning enters this picture — specifically the concentration dependence of the pinning frequency — and how the threshold field and effective charge behave near the Peierls transition, where a two-fluid treatment is of doubtful validity. The subsequent literature transformed rather than settled these questions: a classical framework (Fukuyama-Lee-Rice impurity pinning of a deformable elastic CDW) and a quantum framework (tunneling, later reinterpreted as Coulomb-blockade nucleation of soliton domain walls) reproduce overlapping but not identical bodies of experiment, and both impurity scalings of the threshold are observed depending on dopant. The mechanism question is still contested in publications through 2026, and both of Bardeen's issues remain open: how the impurity-concentration dependence of the pinning frequency is to be interpreted within the tunneling framework, and how the threshold field and effective condensate charge behave near the Peierls transition, where a two-fluid treatment may not be valid.

Background

NbSe3 (niobium triselenide) is a quasi-one-dimensional metal: its crystal structure contains parallel chains of niobium and selenium atoms along which electrons move far more easily than between chains. Like other quasi-one-dimensional metals it undergoes Peierls transitions — in NbSe3 there are two, at $T_{P1}=145$ K and $T_{P2}=59$ K. A Peierls transition is an instability of a one-dimensional metal in which the electron density spontaneously modulates at wave vector $2k_F$ (twice the Fermi wave vector, where $k_F$ marks the highest occupied electron momentum), opening an energy gap $\Delta$ on part of the Fermi surface (the boundary between occupied and unoccupied electron states in momentum space) and producing a charge-density wave (CDW): a collective modulation of the electron density locked to a periodic distortion of the underlying lattice. Below each transition, Monceau et al. (1976) found that electric fields of only a few volts per centimeter destroy the resistance anomalies caused by the gaps, revealing a strongly non-ohmic conduction channel: the CDW slides along the chains and carries current with it — the Frohlich mechanism, i.e. collective charge transport by a sliding incommensurate density wave (one whose wavelength bears no simple ratio to the lattice period). The observed differential conductivity follows approximately $\sigma = \sigma_n + \sigma_1\exp(-E_0/E)$, where $E$ is the applied field and $\sigma_n$ the normal (single-particle) conductivity; above threshold the sliding CDW also generates an oscillatory current whose frequency is proportional to the CDW current (narrow-band noise). In a perfect lattice an incommensurate CDW would slide freely; impurities and defects pin the CDW phase, so collective transport appears only above a finite threshold field.

Bardeen (1979) modeled the pinned CDW as a macroscopically occupied quantum state pinned to a band of pinning states separated by a pinning gap $\hbar\omega_P$, where $\omega_P$ is the frequency of small oscillations of the CDW about its pinned equilibrium. The CDW can cross this gap by tunneling, in exact analogy with Zener tunneling of electrons across the band gap of a semiconductor in a strong electric field: the probability is $P=\exp(-E_0/E)$ with $E_0 = \pi(\hbar\omega_P)^2/(4\hbar e^* v_F)$, where $v_F$ is the Fermi velocity. The electron charge $e$ is replaced by an effective charge $e^* = e\,m^*/(m^*+M_F)$, where $m^*$ is the electron band mass and $M_F \sim 10^4 m^*$ is the Frohlich mass — the extra inertia of the lattice ions carried along with the condensate. The small effective charge $e^*$ and the low pinning frequency $\omega_P$ jointly set the scale of the Zener field $E_0$ in Bardeen's estimate.

Within that paper Bardeen raised the two issues that define this problem. First, impurity pinning: the tunneling approach applies to impurity pinning only once the dependence of $\omega_P$ on impurity concentration $c_i$ is known. In a localized limit each impurity pins the phase independently, $\omega_P$ is independent of $c_i$, and depinning would proceed non-uniformly through the motion of phase dislocations (topological defects of the CDW phase field; the Lee-Rice mechanism); in a cooperative limit pinning is collective, $\omega_P \propto c_i$, giving $E_0 \propto c_i^2$ — the scaling reported for doped NbSe3 by Ong et al. (1979). Bardeen proposed that the two limits could be distinguished by analysis of measurements of the frequency dependence of the conductivity, $\sigma(\omega)$: a pinned collective mode contributes a characteristic resonant or relaxational feature to $\sigma(\omega)$ at frequencies of order $\omega_P$, so that feature's concentration dependence directly measures the pinning spectrum. Second, temperature dependence: experiments indicate that $E_0$ grows roughly as $(1-t)^{-1/2}$ with $t=T/T_P$ as the transition is approached. To describe this, the effective charge must carry a condensate fraction, but the static superfluid fraction $\rho_s$ (varying as $\Delta^2$, like the condensate density of a Bardeen-Cooper-Schrieffer-type ordered state) and the dynamic fraction $\rho_c$ (varying as $\Delta$; the analogue, in the theory of charge imbalance between quasiparticle branches in superconductors, of the branch that carries quasiparticle charge) lead to different divergences of $E_0$. Bardeen noted, crediting Lee, Rice, and Cross, that the two fractions apply in opposite (static and dynamic) limits and that once the various relaxation processes are considered a two-fluid separation may not be valid, so that one may have to resort to the detailed Boltzmann equations — a two-fluid model being one that splits the electrons into condensate and quasiparticle components with separately defined charges, and the Boltzmann equations being the kinetic equations governing the quasiparticle distribution.

A competing classical framework was created at nearly the same time. Fukuyama and Lee (1978) and Lee and Rice (1979) treated the pinned CDW as a deformable elastic medium: impurities pin its phase field, with strong pinning (each impurity pins locally; threshold $E_T \propto c_i$) and weak pinning (the CDW breaks into domains of a characteristic Lee-Rice length that adjust collectively; $E_T \propto c_i^2$), and depinning proceeding by the nucleation and growth of dislocation loops in the phase. In this picture the subthreshold state is a classically polarized, deformable CDW and the threshold is a depinning field rather than a tunneling threshold. Decades of subsequent experiment produced a battery of discriminating observables between the two frameworks — the colossal low-frequency dielectric constant and its correlation with the threshold field, the field dependence of the subthreshold dielectric response, the subthreshold displacement of the CDW phase measured by nuclear magnetic resonance (NMR), photon-assisted transport under microwave and combined ac/dc fields, impurity-concentration scalings, finite-size effects in thin crystals, and the temperature dependence of the threshold down to millikelvin temperatures.

Problem Statement

Within the Zener-tunneling theory of charge-density-wave conduction in NbSe3, two issues were posed and left unresolved:

(1) Impurity pinning and the pinning frequency. Determine how the pinning frequency $\omega_P$ depends on impurity concentration $c_i$ — that is, whether the localized limit ($\omega_P$ independent of $c_i$, with depinning proceeding non-uniformly via phase-dislocation motion) or the cooperative limit ($\omega_P \propto c_i$, hence threshold $E_0 \propto c_i^2$) holds.

(2) Temperature dependence near the Peierls transition. Derive the temperature dependence of the tunneling threshold $E_0(T)$, observed to grow roughly as $(1-t)^{-1/2}$ as $t = T/T_P \to 1$, and establish the correct treatment of the effective condensate charge entering $e^*$ near $T_P$: the static superfluid fraction $\rho_s \propto \Delta^2$, the dynamic fraction $\rho_c \propto \Delta$, or no two-fluid separation at all, requiring the detailed Boltzmann kinetic equations including quasiparticle relaxation and screening.

References

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