Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg-Landau theory approach

Vortex motion in type-II superconductors is studied starting from a variant of the time-dependent Ginzburg-Landau equations, in which the order-parameter relaxation time is taken to be complex. Using a method due to Gor’kov and Kopnin, we derive an equation of motion for a single vortex (B≪${\mathit{H}}_{\mathit{c}2}$) in the presence of an applied transport current. The imaginary part of the relaxation time and the normal-state Hall effect both break ‘‘particle-hole symmetry,’’ and produce a component of the vortex velocity parallel to the transport current, and consequently a Hall field due to the vortex motion. Various models for the relaxation time are considered, allowing for a comparison to some phenomenological models of vortex motion in superconductors, such as the Bardeen-Stephen and Nozières-Vinen models, as well as to models of vortex motion in neutral superfluids. In addition, the transport energy, Nernst effect, and thermopower are calculated for a single vortex. Vortex bending and fluctuations can also be included within this description, resulting in a Langevin-equation description of the vortex motion. The Langevin equation is used to discuss the propagation of helicon waves and the diffusional motion of a vortex line. The results are discussed in light of the rather puzzling sign change of the Hall effect which has been observed in the mixed state of the high-temperature superconductors.