Electronic thermal resistivity and quasi-particle collision cross-section in semi-metals

Electron-electron collisions lead to a T-square component in the electrical resistivity of Fermi liquids. The case of liquid ${}^3$He illustrates that the $\mathit{\text{thermal}}$ resitivity of a Fermi liquid has a T-square term, expressed in m$\cdot$W${}^{-1}$. Its natural units are $\mathrm{\hslash }\mathrm{/}{k}_F{E}_F^2$. Here, we present a high-resolution study of the thermal conductivity in bismuth, employing magnetic field to extract the tiny electronic component of the total thermal conductivity and resolving signals as small as $\approx 60\mu$K. We find that the electronic thermal resistivity follows a T-square temperature dependence with a prefactor twice larger than the electric T-square prefactor. Adding this information to what has been known for other semi-metals, we find that the prefactor of the T-square thermal resistivity scales with the square of the inverse of the Fermi temperature, implying that the dimensionless fermion-fermion collision cross-section is roughly proportional to the Fermi wavelength, indicating that it is not simply set by the strength of the Coulomb interaction.