Envelope function method for electrons in slowly-varying inhomogeneously deformed crystals

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.