Planar thermal Hall effect from phonons in a Kitaev candidate material

Kitaev materials are a promising platform for the realization of quantum spin liquid states. The thermal Hall effect has emerged as a potential probe of exotic excitations within such states. In the Kitaev candidate material $\alpha$-RuCl$_{3}$, however, the thermal Hall conductivity $\kappa_{xy}$ has been attributed not only to exotic Majorana fermions or chiral magnons, but also to phonons. It has been shown theoretically that the former two types of heat carriers can generate a "planar" thermal Hall effect, whereby the magnetic field is parallel to the heat current, as observed experimentally, but it is unknown whether phonons also could. Here we show that a planar thermal Hall effect is present in another Kitaev candidate material, Na$_{2}$Co$_{2}$TeO$_{6}$. On the basis of a striking similarity between the temperature and field dependence of $\kappa_{xy}$ and that of the phonon-dominated thermal conductivity $\kappa_{xx}$, we argue that the planar thermal Hall effect in Na$_{2}$Co$_{2}$TeO$_{6}$ is generated by phonons. The phonon contributed planar $\kappa_{xy}$ also shows a strong sample dependence, which indicates an extrinsic origin of the mechanism. By conducting a complete study with different in-plane configurations of heat current $J$ and magnetic field $H$, $i.e.$ $H$ $\|$ $J$ and $H$ $\bot$ $J$, we observe a large difference in $\kappa_{xy}$ between these two configurations, which reveals that the direction of the heat current $J$ may play an important role in determining the planar thermal Hall effect. Our observation calls for a re-evaluation of the planar thermal Hall effect observed in $\alpha$-RuCl$_{3}$.