Beyond the local Bardeen–Stephen model: the microscopic, nonlocal theory of vortex motion in type-II superconductors in the clean, low-temperature, low-field regime, and the experimental verification of its distinctive predictions

The 1965 Bardeen–Stephen theory explained the motion of Abrikosov vortices and the flux-flow resistivity of type-II superconductors with a local, clean-limit ($l > \xi_0$), near-zero-temperature model in which a vortex carries an essentially normal core of radius near the coherence length, crossed directly by the transport current. The authors themselves identified the theory's open ends: the nonlocal corrections to the flow pattern and dissipation (including the surface-charge artifact of the sharp core boundary), the extension to the dirty limit ($l \ll \xi_0$) and finite temperature through time-dependent Ginzburg–Landau (TDGL) dynamics, the first-principles origin of the core radius and of the empirical flux-flow resistivity law, the unexplained mixed-state Hall-angle anomaly in alloys, the experimentally unconfirmed dispersion of helical waves on vortex lines, and whether transport current really flows through the core in an exact nonlocal theory. The open objective, in the clean, low-temperature, low-field regime, is to establish the exact microscopic flux-flow conductivity, current, and charge distribution around moving vortices and the dispersion of helical waves on a vortex line, and to determine what measurement would confirm or refute the experimentally unconfirmed dispersion; departures from the Bardeen–Stephen law were still being identified in and after 2016.

Background

In a type-II superconductor in a magnetic field between its lower and upper critical fields ($H_{c1}$ and $H_{c2}$), magnetic flux penetrates as Abrikosov vortices: singular lines, each carrying one flux quantum $\Phi_0 = hc/2e$ and surrounded by circulating supercurrents. A transport current of density $\mathbf{J}$ exerts on each line a Lorentz force per unit length $\Phi_0\,\mathbf{J}\times\hat{\mathbf{k}}$ (with $\hat{\mathbf{k}}$ along the applied field), so unless vortices are pinned by material defects they move steadily, and the moving flux generates an electric field by Faraday's law. The mixed state therefore has a finite resistivity, the flux-flow resistivity, measured by Kim, Hempstead, and Strnad, who fit it by an empirical corresponding-states law in which the flux-flow resistivity $\rho_f$ relative to the normal-state resistivity $\rho_n$ rises roughly linearly with the mean induction $B$ toward $H_{c2}$.

Bardeen and Stephen (1965) gave the first dynamical theory of this state that reproduced the empirical laws. In their local model, a vortex is a line of essentially normal metal of radius $a$ of order the coherence length $\xi$ (the scale on which the superconducting order parameter recovers from zero at the vortex center), the electric field of the moving line drives the transport current directly through this core, and the Lorentz force is balanced by viscous drag proportional to the vortex velocity. The model predicts a vortex viscosity, the flux-flow resistivity

$\rho_f/\rho_n \approx B/H_{c2},$

a mixed-state Hall angle equal to the normal-state Hall angle evaluated at the core's magnetic field, a vortex effective mass, and a dispersion relation for helical waves — bending waves of a single vortex line that are circularly polarized in the weakly damped limit (elliptically polarized when the Hall angle is finite), the superconducting analogue of Kelvin waves on a vortex in a superfluid — which differs from the earlier collective-mode result of de Gennes and Matricon by a factor $H/H_{c2}$.

The model's assumptions are severe, and the authors stated the resulting open ends. First, the current–momentum relation is local: superconducting electrodynamics is actually nonlocal over the Pippard coherence length $\xi_0 = \hbar v_F/\pi\Delta_0$, the intrinsic size of a Cooper pair ($v_F$ is the Fermi velocity and $\Delta_0$ the zero-temperature energy gap); in the local treatment the boundary of the core acquires an unphysical surface charge, which a correct nonlocal theory would smear out over a distance of order $\xi_0$, and the resulting corrections to the flow pattern and to the dissipation were not worked out. Second, the theory is restricted to relatively clean material, mean free path $l > \xi_0$, near $T = 0$: the dirty limit $l \ll \xi_0$ and finite temperature require genuine nonequilibrium dynamics of the order parameter — a time-dependent Ginzburg–Landau (TDGL) equation, which governs the relaxation of the complex superconducting order parameter $\Delta(\mathbf{r},t)$ — and Tinkham had proposed that the time rate of change of the superconducting energy gap is itself a dissipation channel. Third, the core radius $a(B,T)$ and its field and temperature dependence enter through an interpolating ansatz rather than a derivation, so the empirical resistivity law was reproduced only approximately. Fourth, Hall-effect measurements on alloys by Niessen and Staas, contemporaneous with the observations of Reed, Fawcett, and Kim, showed a mixed-state Hall angle that grows as the field drops below $H_{c2}$ and can exceed the normal-state value — unexplained by the model, which predicts the normal-state angle. Fifth, the helical-wave dispersion was derived but awaited experimental confirmation. Sixth, the conclusion that the transport current flows through the core depended on the sharp-core and specific-relaxation assumptions and awaited proof in an exact nonlocal theory. A key microscopic ingredient for everything beyond the local model was supplied by Caroli, de Gennes, and Matricon: the bound quasiparticle states localized in the vortex core, whose dynamics ultimately govern clean-limit vortex dissipation. Modern treatments of these questions use the TDGL equation and nonequilibrium quasiclassical (Eilenberger–Keldysh) superconductivity — a Green's-function formalism that reduces the microscopic Bardeen–Cooper–Schrieffer theory of superconductivity to equations for quasiparticles moving along the Fermi surface — in which the motion of the core bound states and their spectral flow (the continuous transfer of quasiparticles between bound-state levels as the vortex moves, which converts the line's motion into dissipation) replace the Ohmic normal-core picture.

Problem Statement

The problem is to replace the local, clean-limit, zero-temperature Bardeen–Stephen model with the correct microscopic theory of vortex motion in a type-II superconductor carrying a transport current, in the clean, low-temperature, low-field regime, and to verify that theory's distinctive predictions. The local model leaves the following open directions: (i) the flow pattern, charge distribution, and dissipation of a steadily moving vortex with nonlocality over the Pippard coherence length $\xi_0$ included, removing the surface-charge artifact of the local model's sharp core boundary; (ii) the nonequilibrium order-parameter dynamics (the time-dependent Ginzburg–Landau equation) for impure superconductors, including the correct relaxation rate of the gap and any additional dissipation from its time rate of change; (iii) a first-principles account of the flux-flow resistivity and of the effective core size and their dependence on magnetic field and temperature, in place of the empirical law and interpolating ansatz; (iv) an explanation of the measured mixed-state Hall angle, including its growth as the field decreases below $H_{c2}$ and its excess over the normal-state value in alloys; (v) a determination of the dispersion relation of helical waves on a vortex line and its experimental confirmation or refutation, including the predicted modification by the factor $H/H_{c2}$ relative to the de Gennes–Matricon collective-mode form; and (vi) the current distribution in and around the moving core in the exact nonlocal theory, settling whether the transport current crosses the core. The required scope is the clean, low-temperature, low-field regime: (a) the nonlocal flow pattern, charge distribution, and dissipation of a steadily moving vortex of (i), taken together with the current distribution in and around its core of (vi) that settles whether the transport current crosses the core, with the smeared charge and current distribution replacing the local model's surface-charge artifact; (b) the exact microscopic flux-flow conductivity and effective core size of (iii); and (c) the helical-wave dispersion of a vortex line of (v), split into (c1) its dispersion relation, including the predicted $H/H_{c2}$ modification relative to the de Gennes–Matricon form, and (c2) a measurement that would confirm or refute it, including whether it is quantitatively sensitive to the predicted $H/H_{c2}$ factor, together with an account of why the dispersion remains unobserved. Directions (ii) (the dirty-limit TDGL near $T_c$) and (iv) (the mixed-state Hall anomaly) are addressed by prior work and lie outside the required scope.

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