Vortex Flow in Superconductors
Authors: Yntema G. B. 
Published: February 2004

Electrical resistance observed1 in superconductors in the mixed state is interpreted as a measure of the motion of Abrikosov vortices in a direction transverse to the imposed net current. Additional evidence of flow of vortices has been provided by dc transformer action2,3 and by heat transport4 in the direction of vortex flow. The connection between the resistive voltage drop and the flow of vortices is understood5 in terms of the superconducting order parameter, which is a complex number varying in space and time. A vortex, which is formed at one edge of the sample, moves across it, and is destroyed at the other edge, has a ``zipper'' effect on the phase of the order parameter. On one side of the path of the vortex, the phase is raised by π (for a single quantum vortex); on the other side it is lowered by the same amount. This process makes no net change in the physical state of the sample; yet it requires a pulse of voltage difference between the ends of the sample, because the time derivative of phase of the order parameter is proportional6 to electrostatic potential (more generally, to the chemical potential for electrons). A state of steady flow of vortices thus involves a steady difference of potential between the ends of the sample. A voltmeter registers this difference. There is no net induced emf to be registered. The dissipation associated with the electrical resistance of a sample in which there is vortex flow occurs in the form of Joule heating produced by normal (i.e., nonsuper) currents.7 Most of this dissipation is in the cores of the vortices, where the material is at least approximately normal and where the electric field is strongest. The electric field in a moving core is partly induced magnetically but is mostly the gradient of electrostatic potential which is associated with the rapid changes of order parameter on opposite sides of the core. A moving vortex not only produces heat but also carries heat along with it, transversely to the electric current and to the magnetic field. A plausible model for the mechanism of this heat transport is based on the available excited states of the superconducting system of electrons as described by BCS. Each available level has a thermal probability of being occupied. The spectrum of levels available varies from place to place in the material according to the local value of the energy gap, which practically vanishes in the core of each vortex, but is significantly large between cores. A particular excitation can migrate only in regions where the energy gap is less than the excitation energy. Each low-energy excitation is therefore trapped, rattling about within a definite core. When a core moves, the trapped excitations are carried along. When a vortex is eventually destroyed at the edge of the sample, its trapped excitations are stranded at the last position of the core. As the gap there goes up, so does the energy of each excitation. The excitation probability which corresponded to thermal equilibrium at the orignal energy is excessive at higher energies. Until the energy becomes so great that the excitation is no longer trapped, the excitation probability can readjust only by a net probability of conversion of energy from the electronic excitation into lattice heat. Similarly, when a vortex is formed, its core absorbs heat from the lattice. The net result is transportation of heat from the location of formation to the location of destruction. The detailed mechanisms by which forces are applied to vortices remain obscure. But by thermodynamic arguments8 we find a force in the direction of j×B due to net electrical current and a thermal force in the direction of − ▿T. In a superconductor in which the pinning of vortices is slight, we should be able (at least as laboratory curiosities) to use vortex flow as the basis of an electrically driven low-temperature refrigerator and of a thermally driven source of electrical energy.