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The Role of the Third Dimension in Searching Majorana Fermions in $α$-RuCl$_3$ via Phonons

Understanding phonons in $α$-RuCl$_3$ is critical to analyze the controversy around the observation of the half-integer thermal quantum Hall effect. While many studies have focused on the magnetic excitations in $α$-RuCl$_3$, its vibrational excitation spectrum has remained relatively unexplored. We investigate the phonon structure of $α$-RuCl$_3$ via inelastic neutron scattering experiments and density functional theory calculations. Our results show excellent agreement between experiment and first principles calculations. After validating our theoretical model, we extrapolate the low energy phonon properties. We find that the phonons in $α$-RuCl$_3$ that either propagate or vibrate in the out-of-plane direction have significantly reduced velocities, and therefore have the potential to dominate the observability of the elusive half integer plateaus in the thermal Hall conductance. In addition, we use low-energy interlayer phonons to resolve the low temperature stacking structure of our large crystal of $α$-RuCl$_3$, which we find to be consistent with that of the $R\bar{3}$ space group, in agreement with neutron diffraction.