  The material $\alpha$-RuCl$_3$ has been the subject of intense scrutiny as a
potential Kitaev quantum spin liquid, predicted to display Majorana fermions as
low energy excitations. In practice, $\alpha$-RuCl$_3$ undergoes a transition
to a state with antiferromagnetic order below a temperature $T_{\rm N}$
$\approx$ 7 K, but this order can be suppressed by applying an external
in-plane magnetic field of $H_\parallel$ = 7 T. Whether a quantum spin liquid
phase exists just above that field is still an open question, but the reported
observation of a quantized thermal Hall conductivity at $H_\parallel$ $>$ 7 T
by Kasahara and co-workers $\big[$Kasahara ${\it et \ al}$., Nature ${\bf
559}$, 227 (2018)$\big]$ has been interpreted as evidence of itinerant Majorana
fermions in the Kitaev quantum spin liquid state. In this study, we re-examine
the origin of the thermal Hall conductivity $\kappa_{\rm xy}$ in
$\alpha$-RuCl$_3$. Our measurements of $\kappa_{\rm xy}$($T$) on several
different crystals yield a temperature dependence very similar to that of the
phonon-dominated longitudinal thermal conductivity $\kappa_{\rm xx}$($T$), for
which the natural explanation is that $\kappa_{\rm xy}$ is also mostly carried
by phonons. Upon cooling, $\kappa_{\rm xx}$ peaks at $T \simeq$ 20 K, then
drops until $T_{\rm N}$, whereupon it suddenly increases again. The abrupt
increase below $T_{\rm N}$ is attributed to a sudden reduction in the
scattering of phonons by low-energy spin fluctuations as these become partially
gapped when the system orders. The fact that $\kappa_{\rm xy}$ also increases
suddenly below $T_{\rm N}$ is strong evidence that the thermal Hall effect in
$\alpha$-RuCl$_3$ is also carried predominantly by phonons. This implies that
any quantized signal from Majorana edge modes would have to come on top of a
sizable -- and sample-dependent -- phonon background.
