Weak-Field Thermal Hall Conductivity in the Mixed State of $d$-Wave Superconductors

Thermal transport in the mixed state of a $d$-wave superconductor is considered within the weak-field regime. We express the thermal conductivity, ${\kappa }_{xx}$, and the thermal Hall conductivity, ${\kappa }_{xy}$, in terms of the cross section for quasiparticle scattering from a single vortex. Solving for the cross section (neglecting the Berry phase contribution and the anisotropy of the gap nodes), we obtain ${\kappa }_{xx}(H,T)$ and ${\kappa }_{xy}(H,T)$ in surprisingly good agreement with the qualitative features of the experimental results for ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{6.99}$. In particular, we show that the simple, yet previously unexpected, weak-field behavior, ${\kappa }_{xy}(H,T)\sim T\sqrt{H}$, is that of thermally excited nodal quasiparticles, scattering primarily from impurities, with a small skew component provided by vortex scattering.

Figure 1

Calculated total, transport, and skew cross sections as a function of $kR$ from 0.5 to 15. For $\sigma$ and ${\sigma }_{\perp }$, solid lines are fits to straight lines with small negative intercepts. For ${\sigma }_{\parallel }$, the solid line is an interpolation of the numerical data in which the constant large-$kR$ behavior is extrapolated to larger $kR$. The oscillating component of the ${\sigma }_{\perp }$ data is believed to be a numerical artifact.

Figure 2

Calculated longitudinal thermal conductivity (left) and thermal Hall conductivity (right) plotted in scaling form (in units defined in the text). The inset is the measured thermal Hall conductivity (in SI units) obtained by Ong and co-workers for ${\mathrm{Y}\mathrm{B}\mathrm{a}}_2{\mathrm{C}\mathrm{u}}_3{\mathrm{O}}_{6.99}$ and reproduced here from Ref. 4.