  Stacking variations in quasi-2D materials can have an important influence on
material properties, such as changing the topology of the band structure.
Unfortunately, the weakness of van der Waals interactions makes it difficult to
compute the stacking dependence of properties, and even in a material as simple
as graphite the stacking energetics remain unclear. Mo$_{1-x}$W$_{x}$Te$_{2}$
is a material in which three differently-stacked phases are conveniently
accessible by temperature changes: $1T^{\prime}$, $T^*_d$, and the reported
Weyl semimetal phase $T_d$. The transitions proceed via layer sliding, and the
corresponding interlayer shear mode (ISM) is relevant not just for the stacking
energetics, but for understanding the relationship between the Weyl physics and
structural changes. However, the interlayer interactions of
Mo$_{1-x}$W$_{x}$Te$_{2}$ are not well understood, with wide variation in
computed properties. We report inelastic neutron scattering of the ISM in a
Mo$_{0.91}$W$_{0.09}$Te$_{2}$ crystal. The ISM energies are generally
consistent with the linear chain model (LCM), as expected given the weak
interlayer interaction, though there are some discrepancies from predicted
intensities. However, the interlayer force constants $K_x$ in the $T^*_d$ and
$1T^{\prime}$ phases are substantially weaker than that of $T_d$, at 76(3)% and
83(3)%, respectively. Considering that the relative positioning of atoms in
neighboring layers is approximately the same regardless of overall stacking,
our results suggest that longer-range influences, such as stacking-induced band
structure changes, may be responsible for the substantial change in the
interlayer vibrational coupling. These findings should elucidate the stacking
energetics of Mo$_{1-x}$W$_{x}$Te$_{2}$ and other van der Waals layered
materials.
