  We develop a new envelope-function formalism to describe electrons in
slowly-varying inhomogeneously strained semiconductor crystals. A coordinate
transformation is used to map a deformed crystal back to geometrically
undeformed structure with deformed crystal potential. The single-particle
Schr\"{o}dinger equation is solved in the undeformed coordinates using envelope
function expansion, wherein electronic wavefunctions are written in terms of
strain-parametrized Bloch functions modulated by slowly varying envelope
functions. Adopting local approximation of electronic structure, the unknown
crystal potential in Schr\"{o}dinger equation can be replaced by the
strain-parametrized Bloch functions and the associated strain-parametrized
energy eigenvalues, which can be constructed from unit-cell level ab initio or
semi-empirical calculations of homogeneously deformed crystals at a chosen
crystal momentum. The Schr\"{o}dinger equation is then transformed into a
coupled differential equation for the envelope functions and solved as a
generalized matrix eigenvector problem. As the envelope functions are slowly
varying, coarse spatial or Fourier grid can be used to represent the envelope
functions, enabling the method to treat relatively large systems. We
demonstrate the effectiveness of this method using a one-dimensional model,
where we show that the method can achieve high accuracy in the calculation of
energy eigenstates with relatively low cost compared to direct diagonalization
of Hamiltonian. We further derive envelope function equations that allow the
method to be used empirically, in which case certain parameters in the envelope
function equations will be fitted to experimental data.
