  Strained coherent heteroepitaxy of III-V semiconductor films such as
In$_x$Ga$_{1-x}$As/GaAs has potential for electronic and optoelectronic
applications such as high density logic, quantum computing architectures, laser
diodes, and other optoelectronic devices. Crystal symmetry can have a large
effect on the morphology of these films and their spatial order. Often the
formation of group IV strained heterostructures such as Ge deposited on Si is
analyzed using analytic models based on the Asaro-Tiller-Grinfeld instability.
However, the governing dynamics of III-V 3D heterostructure formation has
different symmetry and is more anisotropic. The additional anisotropy appears
in both the surface energy and the diffusivity. Here, the resulting anisotropic
governing dynamics are studied to linear order. The resulting possible film
morphologies are compared with experimentally observed In$_x$Ga$_{1-x}$As/GaAs
films. Notably it is found that surface-energy anisotropy plays a role at least
as important as surface diffusion anisotropy if not more so, in contrast to
previous suppositions.
