Microscopic interpretation of charge-density-wave current oscillations in NbSe$_3$: effective transport period and pinning-barrier mechanism

The 1982 Bardeen–Ben-Jacob–Zettl–Grüner experiment on the lower charge-density-wave (CDW) state of NbSe3 established that the frequency of the narrow-band current oscillations is strictly proportional to the CDW current, that the oscillation amplitude is of order the total CDW current just above threshold, that the product of amplitude and CDW current stays constant at high fields, and that a pinning velocity of about 0.4 cm/s and a pinning frequency of about 3 MHz can be extracted from an adiabatic-sliding model with a Zener-type tunneling barrier, consistent with the threshold frequency obtained from field- versus frequency-dependent conductivity scaling. The authors explicitly left open the effective spatial period of transport (the CDW wavelength of about 14 Å, or a shorter commensurate or soliton-lattice period of order the lattice constant), which fixes the effective charge transported per oscillation in the current-to-frequency ratio. They acknowledged that the maximum height of the Zener-type barrier estimate depends on its shape and that their barrier formula is also consistent with Lee–Rice impurity pinning, and they discussed in a footnote the Bak-type soliton-lattice alternative with fractional charge, while arguing that the agreement between the oscillation-derived pinning frequency and that derived from the $\sigma(E)$–$\sigma(\nu)$ scaling "lends strong support to the tunneling model of CDW depinning"; the amplitude analysis rested on the hypothesis that the CDW slides without additional dissipation, and the domain structure was concluded affirmatively (a single coherent domain, or strongly coupled segments giving a highly coherent response). Later work resolved the domain question (the amplitude scales as the inverse square root of specimen volume, from interference between velocity-coherent domains) and confirmed the linear current-frequency relation as a condensate probe, but the effective transport period and the microscopic depinning mechanism remain unsettled, with half-wavelength evidence, fractional soliton charge, classical collective pinning, and quantum tunneling pictures all still represented in the literature through 2021.

Background

NbSe$_3$ (niobium triselenide) is a quasi-one-dimensional metal: its electronic structure is dominated by chains running along the crystallographic $b$ axis, and it undergoes two Peierls transitions, at $T_{P1}\approx 145$ K and $T_{P2}\approx 59$ K. A Peierls transition is an instability of a one-dimensional metal in which the electron density spontaneously modulates with wave vector $Q = 2k_F$ ($k_F$ the Fermi wave vector); this charge-density wave (CDW) is accompanied by a periodic lattice distortion that gaps the Fermi surface and lowers the electronic energy. Because the phase of such a modulation can translate along the chain direction, Fröhlich proposed that the whole electron condensate could slide and carry current; in real crystals the phase is pinned by impurities and by boundaries, and collective conduction appears only above a threshold electric field $E_T$, first observed in NbSe$_3$ by Fleming and Grimes together with an oscillatory conduction noise. Above threshold, the current splits into a normal ohmic part and an excess collective part $I_{CDW}$, and the voltage oscillates with a sharp fundamental frequency $\nu$ — the narrow-band noise (NBN) — proportional to $I_{CDW}$; combined ac and dc drives further produce Shapiro steps (constant-current plateaus in the dc characteristic) exactly analogous to those of Josephson junctions, as shown by Monceau, Richard and Renard.

Two microscopic frameworks competed from the outset. In the classical picture initiated by Lee and Rice, the CDW phase is an elastic medium pinned by a random impurity potential; once the electric force exceeds the maximum pinning force, the phase slides, and its periodic motion through the pinning force generates the current oscillations. In the quantum picture introduced by Bardeen, the condensate depins by tunneling through a pinning gap, in close analogy with Zener tunneling (field-induced interband tunneling in a semiconductor); Bardeen, Tucker and co-workers showed that this picture predicts a scaling relation between the nonlinear field-dependent conductivity $\sigma(E)$ and the frequency-dependent conductivity $\sigma(\nu)$ measured at microwave frequencies. Weger and co-workers and Bak proposed soliton-based alternatives: in a sine-Gordon model (a standard nonlinear field model whose kink solutions, called solitons, are localized traveling wave packets that carry a topological charge), nominally incommensurate CDWs could contain large locally commensurate regions, so that current would be carried by solitons with an effective charge below $e$ — Bak analyzed existing NbSe$_3$ data to obtain $e^{*}\approx 0.3$–$0.4e$ — and the effective spatial period of transport would be the lattice constant $a\approx\lambda/4$ or the soliton-lattice spacing rather than the CDW wavelength $\lambda = 2\pi/Q \approx 14$ Å.

In this context Bardeen, Ben-Jacob, Zettl and Grüner (1982) reported, for the lower CDW of NbSe$_3$: (i) $\nu$ strictly linear in $I_{CDW}$, with $I_{CDW}/\nu = n_c e \lambda A$ ($n_c$ the density of electrons condensed into the CDW, $A$ the cross-sectional area), the temperature dependence of the ratio following that of the Bardeen–Cooper–Schrieffer (BCS) order parameter (the characteristic temperature dependence of the superconducting energy gap); (ii) an oscillation amplitude $\Delta I$ of order the total CDW current just above threshold; and (iii) a product $\Delta I \times I_{CDW}$ approximately constant within experimental scatter for applied voltages from roughly $2V_T$ to $6V_T$. They interpreted this with a model in which the CDW slides adiabatically (without additional dissipation) over a periodic pinning potential $V(\varphi)$ in the phase $\varphi$: energy conservation for the drift velocity $V_d$ and the periodic velocity component $V_1(\varphi)$ gives $V(\varphi) = -m^{*} V_d V_1(\varphi)$ ($m^{*}$ the electron band mass), and matching to a Zener-type barrier $V_{max} = h^2\nu_p^2/(16E_F) = (1/2)m^{*}V_p^2$ ($E_F$ the Fermi energy) yields a pinning velocity $V_p \approx 0.39$ cm/s and pinning frequency $\nu_p \approx 2.8$ MHz, in approximate agreement with the threshold frequency $\nu_T \approx 3.5$ MHz independently extracted from the $\sigma(E)$–$\sigma(\nu)$ scaling on crystals of the same batch. Requiring the pinning barrier per coherent domain (a region that slides as a single degree of freedom) to exceed the thermal energy, with a Fröhlich mass ratio $\alpha$ (the ratio of the total ion-plus-electron inertia to the electron inertia, of order $10^3$), gives a minimal domain of $N_c \gtrsim 4\times 10^{10}$ condensed electrons. The analysis left open the effective spatial period of transport, acknowledged the shape dependence of the barrier estimate together with its consistency with the Lee–Rice impurity-pinning picture, and rested on the hypothesis of dissipationless adiabatic sliding; these caveats, and the subsequent literature summarized in the introductory paragraph, frame the present problem.

Problem Statement

The problem, as left open by Bardeen, Ben-Jacob, Zettl and Grüner (1982) and not closed by the subsequent literature, is to give a quantitative, model-independent microscopic account of the sliding-CDW current oscillations in NbSe$_3$ — model-independent in that it fixes the effective transport period, the charge per cycle, and the barrier height and prefactors from the measured quantities without presupposing a particular depinning mechanism, and then states which of the candidate mechanisms the data favor and which they exclude. Its two inseparable determinations are: (1) The effective transport period and charge per cycle. Determine which spatial period governs transport — the CDW wavelength $\lambda \approx 14$ Å of the incommensurate picture, a shorter lattice-based period $a \sim \lambda/4$ associated with commensurate or soliton-lattice regions (Bak; Weger and co-workers), or an effective half-wavelength as later evidence suggested — and hence the effective charge transported per oscillation, which fixes the quantitative interpretation of $I_{CDW}/\nu$ and the absolute calibration of condensed-carrier densities extracted from narrow-band noise. (2) The mechanism and magnitude of the pinning barrier. Establish whether depinning is quantum (Zener-type) tunneling of the condensate — the picture favored in 1982 on the basis of the $\sigma(E)$–$\sigma(\nu)$ scaling agreement — classical collective motion over an impurity-generated pinning potential of the Lee–Rice type, with which the 1982 barrier formula $V_{max} = h^2\nu_p^2/(16E_F) = (1/2)m^{*}V_p^2$ ($E_F$ the Fermi energy) is also consistent, or a soliton-lattice picture with fractional charge; and determine the barrier height and its prefactors model-independently, given that $V_{max}$ depends on the shape of the barrier. The two determinations are entangled: the 1982 extraction of the pinning parameters $V_p \approx 0.39$ cm/s and $\nu_p \approx 2.8$ MHz (versus $\nu_T \approx 3.5$ MHz from the $\sigma(E)$–$\sigma(\nu)$ scaling) assumes both $\lambda = 14$ Å and dissipationless adiabatic sliding — the relation $V(\varphi) = -m^{*} V_d V_1(\varphi)$ ($m^{*}$ the electron band mass, $V_d$ the drift velocity, $V_1(\varphi)$ the periodic velocity component) requires that the CDW slide without additional dissipation over the peaks and valleys of the pinning potential — and the associated stability estimate (a Fröhlich mass ratio $\alpha$ of order $10^3$, giving a minimal coherent domain of $N_c \gtrsim 4\times 10^{10}$ condensed electrons) rests on the same extraction, so the period choice and any dissipative or nonadiabatic corrections enter the barrier calibration. The 1982 single-domain interpretation of the oscillation amplitude (amplitude $\Delta I$ of order the total CDW current just above threshold; product $\Delta I \times I_{CDW}$ approximately constant within experimental scatter, $\approx 7\times 10^{-11}$ A$^2$ for applied voltages from roughly $2V_T$ to $6V_T$), and its later supersession by the many-domain interference phenomenology, are settled background and progress context rather than open subgoals; any proposed microscopic account must of course be consistent with these measured amplitude laws and with the other measured quantities in which the period and the mechanism jointly enter (the $I_{CDW}/\nu$ slope, $\Delta I$ versus $I_{CDW}$, the $\sigma(E)$–$\sigma(\nu)$ scaling, and the interference and mode-locking spectra).

References

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