  Thermal properties are of great interest in modern electronic devices and
nanostructures. Calculating these properties is straightforward when the device
is made from a pure material, but problems arise when alloys are used.
Specifically, only approximate bandstructures can be computed for random alloys
and most often the Virtual Crystal Approximation (VCA) is used. Unfolding
methods [T. B. Boykin, N. Kharche, G. Klimeck, and M. Korkusinski, J. Phys.:
Condens. Matt. 19, 036203 (2007).] have proven very useful for tight-binding
calculations of alloy electronic structure without the problems in the VCA, and
the mathematical analogy between tight-binding and valence-force-field
approaches to the phonon problem suggest they be employed here as well.
However, there are some differences in the physics of the two problems
requiring modifications to the electronic structure approach. We therefore
derive a phonon alloy bandstructure (vibrational mode) approach based on our
tight-binding electronic structure method, modifying the band-determination
method to accommodate the different physical situation. Using the method, we
study In$_x$Ga$_{1-x}$As alloys and find very good agreement with available
experiments.
