  We theoretically investigated the fundamental distinction between intrinsic
and extrinsic nonlinear thermal Hall effect in the presence of disorder at the
second-order response to the temperature gradient in terms of the
semi-classical Boltzmann equation. We found that, at low temperatures, the
intrinsic contribution of the nonlinear thermal Hall conductivity is
proportional to the square of temperature, whereas the extrinsic contributions
(side-jump and skew-scattering) are independent of temperature. This distinct
dependency on temperature provide a new approach to readily distinguish the
intrinsic and extrinsic contributions. Specifically, we analysed the nonlinear
thermal Hall effect for a tilted two-dimensional massive Dirac material. In
particular, we showed that when the Fermi energy is located at the Dirac point,
the signal is solely from the intrinsic mechanism; when the Fermi energy is
higher, the extrinsic contributions are dominant, which are uncovered to be two
to three orders of magnitude larger than the intrinsic contribution.
