  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$_{3}$.

Strongly Correlated Electrons