Correlated Quantum Transport of Density Wave Electrons

Recently observed Aharonov-Bohm quantum interference of the period $h/2e$ in charge density wave rings strongly suggests that correlated density wave electron transport is a cooperative quantum phenomenon. The picture discussed here posits that quantum solitons nucleate and transport current above a Coulomb blockade threshold field. We propose a field-dependent tunneling matrix element and use the Schrödinger equation, viewed as an emergent classical equation as in Feynman’s treatment of Josephson tunneling, to compute the evolving macrostate amplitudes, finding excellent quantitative agreement with voltage oscillations and current-voltage characteristics in ${\mathrm{NbSe}}_3$. A proposed phase diagram shows the conditions favoring soliton nucleation versus classical depinning.

Figure 1

(a) Potential energy vs $\theta$ for $\varphi \sim 2\pi n$. (b) $u(\varphi )$ when $\theta =2\pi E/E^{*}>\pi$ as the phases ${\varphi }_k(x)$ tunnel coherently into the next well. (c) Time-correlated soliton tunneling model.

Figure 2

(a) Theoretical (solid lines) vs experimental (dashed lines [21]) voltage oscillations of an ${\mathrm{NbSe}}_3$ crystal at 52 K for current pulse amplitudes (bottom to top, offset by 0, 0.25, 0.5, and 0.75 V for clarity): $9.90$, $10.89$, $11.49$, and $11.88\mu \mathrm{A}$. (b) Simulated DW current vs field for $\gamma =1.5$ and several $q_0$. Dotted lines (Table ): Bardeen’s modified Zener function [22]. (c) Simulated $R=dV/dI$ vs $I/I^{*}$, where $I^{*}\equiv E^{*}\ell /R_n$, where $R_n$ is the normal resistance at zero bias, for several $q_0$ and $\gamma =1.5$. (d) Theoretical (solid lines, Table ) vs experimental (dotted lines) $dV/dI$ vs $I$ for an ${\mathrm{NbSe}}_3$ crystal.

Figure 3

(a) Blue bronze $I-V$ curves [25]. (b) (left): $u$ vs $\varphi$, showing $\theta \ge \pi$ quantum instability and $\theta \ge {\theta }_c$ classical depinning. (right): $u(\varphi )$ at $\theta =\pi$ for several $\alpha =u_E/u_0$. (c) Phase diagram showing pinned, soliton nucleation, and classically depinned states. Solid arrow: $u_E/u_0\ll 1$, for which soliton pair creation dominates. The dotted arrow shows both soliton nucleation and classical depinning. Since $u_E\propto 1/ϵ$, the path curves to the left (right) if $ϵ$ increases (decreases). Dashed arrows (a,c): classical depinning dominates.