Microscopic origin of subthreshold ac-dc decoupling and photon-assisted-tunneling scaling in charge-density-wave conductors (NbSe3)

Below the depinning threshold field, the charge-density wave (CDW) in NbSe3 stores charge in metastable polarized states, yet its small-signal dynamics at megahertz frequencies behaves as if no dc bias were applied: mixing experiments show an effective dc voltage of zero in the high-frequency response for all subthreshold biases, direct mixing vanishes below threshold over applied frequencies of 1-500 MHz, and rectification measured at 5 MHz loses the threshold singularity that is sharp at 1 kHz. A photon-assisted tunneling (PAT) description fits these data once derivatives of the dc current-voltage curve are replaced by finite differences with a quantum step proportional to frequency, with a scaling parameter of about 1 mV/MHz, roughly 10^5 times the single-electron value h/e; the zero-bias harmonic mixing signal saturates above about 12 MHz. The 1984 experiments that established this ac-dc decoupling left its mechanism unresolved, and four decades of subsequent work have produced phenomenological descriptions (capacitive coupling between an intrinsic tunneling conductance and a polarization capacitance; screening of the CDW polarization by normal carriers near strong-pinning centers) and classical counter-analyses of comparable explanatory power, but no microscopic first-principles theory. The open problem is to derive, from a stated microscopic model of the pinned CDW, the collapsed subthreshold high-frequency response and the voltage-frequency scaling with its saturation scale that these experiments established. The wider dispute over whether CDW depinning is quantum tunneling of the condensate or classical dynamics of a deformable elastic medium, developed in work after 1984, is background context for this mechanism problem rather than part of it.

Background

Charge-density waves (CDWs) are collective electronic states of quasi-one-dimensional metals. Below a Peierls transition temperature, the electron density modulates with period twice the Fermi wavelength (NbSe3, with transitions near 144 K and 59 K, is the archetype), and the phase of the modulation becomes a collective coordinate: translating it carries the condensed electrons as a current, while static impurities pin it. A finite threshold field $E_T$ (about 0.01-1 V/cm in the classic materials) must be exceeded before the CDW slides; below threshold the condensate polarizes without transporting charge and can remain in metastable polarization states with extremely long relaxation times. Above threshold, the sliding CDW current is accompanied by periodic current oscillations (narrow-band noise) whose frequency is proportional to the CDW current, so applied ac signals can interfere with the internal oscillation.

Two competing frameworks were developed in the early 1980s. In the classical picture, the pinned CDW is a deformable elastic medium (the Fukuyama-Lee-Rice model, in which the CDW phase is an elastic field pinned by impurity potentials) or, in its simplest form, a single overdamped coordinate in a periodic pinning potential that a dc bias progressively tilts until depinning occurs. This picture predicts a divergent differential conductance at threshold and a strong dependence of the low-frequency ac conductivity on dc bias near threshold. In the quantum picture, Bardeen's tunneling theory treats depinning as Zener-type tunneling of condensed electrons through a small pinning gap, the measured dc current being fit by $I(E) = G_n E + G_b (E - E_T)\exp(-E_0/E)$ for $E \gtrsim E_T$, with $G_n$ and $G_b$ the normal and CDW conductances and $E_0$ a Zener activation field. Zettl and Grüner (Phys. Rev. B 29, 755 (1984)) had shown that a large-amplitude ac signal lowers the dc threshold of NbSe3 but leaves it sharply defined until it is completely suppressed.

The source experiment (Miller, Richard, Thorne, Lyons, Tucker, and Bardeen, Phys. Rev. B 29, 2328 (1984)) probed the nonlinear ac response of the NbSe3 CDW at 125 K by mixing: with applied voltage $V(t) = V_0 + V_1\cos(2\pi\nu_1 t) + V_2\cos(2\pi\nu_2 t + \phi)$, the current generated at the difference frequency (direct mixing) or, with the two applied tones arranged as $\nu_2 = 2\nu_1 + \nu_0$, at the harmonic-mixing output frequency $\nu_0$ has, for small amplitudes, an amplitude proportional to the second or third derivative of the dc current-voltage ($I$-$V$) curve. Photon-assisted tunneling theory, the quantum theory of mixers (Tucker, IEEE J. Quantum Electron. 15, 1234 (1979)) adapted to CDWs, replaces these derivatives by finite differences evaluated at voltage offsets proportional to the photon energy, e.g. $d^2I/dV_0^2$ becomes the second difference of $I(V)$ at $V_0 \pm \alpha\nu_1$. The parameter $\alpha$ is fixed by a scaling relation between the frequency-dependent ac admittance at zero bias and the voltage-dependent dc conductance; the experiments give $\alpha \approx 1$ mV/MHz, of order $10^5$ times the single-electron value $h/e$, interpreted as collective tunneling over a distance of order 10 micrometers in a sample of length about 1 mm. Later work related $\alpha$ to independently measurable microscopic quantities: the Fröhlich mass $M_F$ (the effective inertial mass of the condensed electrons moving collectively, much larger than the electron mass, which scales down the condensate's effective charge per carrier) and the Lee-Rice domain length (the distance over which the pinned CDW phase remains coherent, set by the balance between the CDW's elastic stiffness and random impurity pinning).

Crucially, agreement with all prior ac data required treating every subthreshold dc bias as equivalent to zero bias -- a collapsed effective $I$-$V$ curve with effective bias $V_0' = 0$ for $|V_0| < V_T$ and $V_0' = \mathrm{sgn}(V_0)(|V_0| - V_T)$ above. The 1984 measurements established three striking facts: (i) a bias-independent harmonic mixing signal below threshold, whose zero-bias value grows with output frequency until saturating for $\nu_0 \gtrsim 12$ MHz (also observed in TaS3); (ii) vanishing direct mixing below threshold for applied frequencies from 1 to 500 MHz; and (iii) rectification measured at 1 kHz under a large 10-MHz drive shows a sharp peak at threshold while the same rectification measured at 5 MHz does not -- direct evidence that the MHz dynamics is decoupled from the dc polarization. The authors concluded that a subthreshold dc field "apparently distorts the CDW into a different metastable configuration, without affecting its high-frequency behavior." Under large ac drive, $d^2I/dV^2$ beyond the initial peaks exhibits interference effects associated with the coherent current oscillations, which the paper did not analyze. What is missing -- then and now -- is a microscopic derivation of this behavior rather than a phenomenological description of it.

Problem Statement

Construct and validate a microscopic theory of the subthreshold dynamics of the pinned charge-density wave in NbSe3 and the related quasi-one-dimensional CDW conductors (TaS3, K0.3MoO3, (TaSe4)2I), which within one framework derives, rather than assumes or fits: (1) the ac-dc decoupling -- an effective dc bias $V_0' = 0$ in the high-frequency response for all $|V_0| < V_T$ while the condensate is statically polarized into metastable configurations -- together with the source paper's own evidence that the ac and dc responses recouple at low probe frequency, namely the sharp threshold peak in rectification measured at 1 kHz and its absence in the rectification measured at 5 MHz under the same large 10-MHz drive; and (2) the voltage-frequency scaling $\Delta V = \alpha\nu$ with $\alpha \approx 1$ mV/MHz (of order $10^5 h/e$), with the collective-tunneling interpretation over a distance of order 10 micrometers in a sample of length about 1 mm, the vanishing of direct mixing below threshold over applied frequencies from 1 to 500 MHz, and the saturation of the zero-bias harmonic mixing signal for output frequencies $\nu_0 \gtrsim 12$ MHz. The interference features in $d^2I/dV^2$ under large ac drive, which the source paper reported but did not analyze, and the later phenomenological relations of $\alpha$ to the Fröhlich mass and the Lee-Rice domain length remain part of the background described above. The behaviors listed in (1) and (2) are to follow within a single framework for NbSe3, with the TaS3 harmonic-mixing saturation reproduced as the corresponding check; the other conductors named above are related context.

References

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