Enhanced vortex heat conductance in mesoscopic superconductors
  Electronic heat transport along the flux lines in a long ballistic mesoscopic superconductor cylinder with a radius of the order of several coherence lengths is investigated theoretically using both semiclassical approach and the full quantum-mechanical analysis of the Bogoliubov--de Gennes equations. The semiclassical approach is constructed analogously to the Landauer transport theory in mesoscopic conductors employing the idea that heat is carried by the quasiparticle modes propagating along the vortex core. We show that the vortex heat conductance in a mesoscopic sample is strongly enhanced as compared to its value for a bulk superconductor; it grows as the cylinder radius decreases. This unusual behavior results from a strongly increased number of single-particle transport modes due to giant mesoscopic oscillations of energy levels, which originate from the interplay between the Andreev reflection at the vortex core boundary and the normal reflection at the sample edge. We derive the exact quantum-mechanical expression for the heat conductance and solve the Bogoliubov--de Gennes equations numerically. The results of numerical computations confirm the qualitative Landauer-type picture and allow us to take into account the partial reflections of excitations. We analyze the effect of surface imperfections on the spectrum of core excitations. We show that the giant oscillations of core levels and thus the essential features of the heat transport characteristic to ideal mesoscopic samples hold for a broad class of surface imperfections as well. 
N.B. Kopnin, A.S. Mel'nikov, V.I. Pozdnyakova, D.A. Ryzhov, I.A.   Shereshevskii, and V.M. Vinokur