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 leads to the heat transport theory which employs the idea that heat is carried by the quasiparticle modes propagating along the vortex core in a way similar to the Landauer transport theory for mesoscopic conductors. 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 transverse 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 generalize the qualitative Landauer-type picture by taking 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.

Figure 1

The spectra of quasiparticles for a vortex in the cylinder with $R∕\xi =3.5$ (a) and $R∕\xi =4.5$ (b). The gap is approximated by Eq. (20) with ${\xi }_v=\xi$, $k_F\xi =200$.

Figure 2

The temperature dependence of the effective number of modes $N_{eff}\left(T\right)$ calculated using Eq. (28) with $P_{\mu ,k_z}=1$ for two cylinders with $R∕\xi =3.5$ (a) and $R∕\xi =4.5$ (b). The gap is approximated by Eq. (20) with ${\xi }_v=\xi$, $k_F\xi =200$.

Figure 3

The effective number of modes $N_{eff}$ according to Eq. (28) with $P_{\mu ,k_z}=1$ as a function of the cylinder radius for various temperatures $T∕{\omega }_0=0.3,1.0,3.0$ (from bottom to top). The gap is approximated by Eq. (20) with ${\xi }_v=\xi$, $k_F\xi =200$.

Figure 4

Heat transport through a vortex line.

Figure 5

(a), (b) The transmission $\mid t_{uu}{\mid }^2$, $\mid t_{uv}{\mid }^2$ and (c), (d) reflection $\mid r_{uu}{\mid }^2$, $\mid r_{uv}{\mid }^2$ probabilities, respectively, as functions of momenta $k_{zu}^{\prime }$ and $k_{zv}^{\prime }$ of scattered waves for the incident electronlike quasiparticle wave with $\mu =-1∕2$, $k_z∕k_F=0.9$, and $\epsilon =0.49{\omega }_0$. Parameters are $R=4.5\xi$, $d∕\xi =16$.

Figure 6

(Color online) The transfer factor $\mathcal{T}\left(\epsilon \right)$ for different vortex line lengths $d$ (blue line is for $d∕\xi =16$ and red line is for $d∕\xi =2048$) in the cylinder with $R∕\xi =3.5$ (a) and $R∕\xi =4.5$ (b).

Figure 7

The effective number of modes $N_{eff}\left(T\right)$ calculated numerically as a function of temperature for the vortex line length $d∕{\xi }_0=16$ in cylinders with $R∕{\xi }_0=3.5$ (a) and $R∕{\xi }_0=4.5$ (b).