Role of magnetic ions in the thermal Hall effect of the paramagnetic insulator TmVO${}_4$

In a growing number of materials, phonons have been found to generate a thermal Hall effect, but the underlying mechanism remains unclear. Inspired by previous studies that revealed the importance of Tb${}^{3+}$ ions in generating the thermal Hall effect in a family of pyrochlores, we investigated the role of Tm${}^{3+}$ ions in TmVO${}_4$, a paramagnetic insulator with a different crystal structure. We observe a negative thermal Hall conductivity in TmVO${}_4$ with a magnitude such that the Hall angle, $\mathrm{\mid }{\kappa }_{xy}$/${\kappa }_{xx}\mathrm{\mid }$, is approximately 1 x 10${}^{-3}$ at $H$ = 15 T and $T$ = 20 K, typical for a phonon-generated thermal Hall effect. In contrast to the negligible ${\kappa }_{xy}$ found in the nonmagnetic pyrochlore analog (where the Tb${}^{3+}$ ions are replaced with Y${}^{3+}$), we observe a negative ${\kappa }_{xy}$ in YVO${}_4$ with a Hall angle of magnitude comparable to that of TmVO${}_4$. This shows that the Tm${}^{3+}$ ions are not essential for the thermal Hall effect in this family of materials. Interestingly, at an intermediate Y concentration of $x$ = 0.3 in Tm${}_{1-x}$Y${}_x$VO${}_4$, ${\kappa }_{xy}$ was found to have a positive sign, pointing to the importance of impurities in the thermal Hall effect of phonons.