  This paper details a complete formalism for calculating electron subband
energy and degeneracy in strained multi-valley quantum wells grown along any
orientation with explicit results for the AlAs quantum well case. A
standardized rotation matrix is defined to transform from the conventional-
cubic-cell basis to the quantum-well-transport basis whereby effective mass
tensors, valley vectors, strain matrices, anisotropic strain ratios, and
scattering vectors are all defined in their respective bases. The specific
cases of (001)-, (110)-, and (111)-oriented aluminum arsenide (AlAs) quantum
wells are examined, as is the unconventional (411) facet, which is of
particular importance in AlAs literature. Calculations of electron confinement
and strain in the (001), (110), and (411) facets determine the critical well
width for crossover from double- to single-valley degeneracy in each system.
The notation is generalized to include miscut angles, and can be adapted to
other multi-valley systems. To help classify anisotropic inter-valley
scattering events, a new primitive unit cell is defined in momentum space which
allows one to distinguish purely in-plane inter-valley scattering events from
those that requires an out-of-plane momentum scattering component.
