  A drastic enhancement of the thermal Hall angle in $d$-wave superconductors
was observed experimentally in a cuprate superconductor and in CeCoIn$_5$ at
low temperatures and very weak magnetic field [Phys. Rev. Lett. $\bf 86$, 890
(2001); Phys. Rev. B $\bf 72$, 214515 (2005)]. However, to the best of our
knowledge, its microscopic calculation has not been performed yet. To study
this microscopically, we derive the thermal Hall coefficient in extreme type-II
superconductors with an isolated pinned vortex based on the augmented
quasiclassical equations of superconductivity with the Lorentz force. Using it,
we can confirm that the quasiparticle relaxation time and the thermal Hall
angle are enhanced in $d$-wave superconductors without impurities of the
resonant scattering because quasiparticles around the gap nodes which become
dominant near zero temperature are restricted to the momentum in a specific
orientation. This enhancement of the thermal Hall angle may also be observed in
other nodal superconductors with large magnetic-penetration depth.
