Sample dependence of the half-integer quantized thermal Hall effect in the Kitaev candidate $\alpha$-RuCl$_3$

We have investigated the sample dependence of the half-integer thermal Hall effect in $\alpha$-RuCl$_3$ under a magnetic field tilted 45 degree from the $c$ axis to the $a$ axis. We find that the sample with the largest longitudinal thermal conductivity ($\kappa_{xx}$) shows the half-integer quantized thermal Hall effect expected in the Kitaev model. On the other hand, the quantized thermal Hall effect was not observed in the samples with smaller $\kappa_{xx}$. We suggest that suppressing the magnetic scattering effects on the phonon thermal conduction, which broaden the field-induced gap protecting the chiral edge current of the Majorana fermions, is important to observe the quantized thermal Hall effect.