Motion of electrons in semiconductors under inhomogeneous strain with application to laterally confined quantum wells

A general treatment for finding the energy levels of electrons and holes in a system with slowly varying inhomogeneous strain is given in the envelope-function approximation. An eight-band model is derived, then block diagonalized to 2×2 and 6×6 for the conduction and valence bands, respectively. To first order, i.e., in the so-called effective-mass approximation, the gradient of the strain tensor does not appear in the Hamiltonian; in the second-order approximation, both the strain variation of the effective masses and the gradients of the strain tensor appear. The second-order effect can be significant within the range of applicability of the theory when the strain is not sufficiently small and its variation is not sufficiently slow. The general theory is first applied to the case of homogeneous strain (for the conduction band), which gives the strain dependence of the band structure, with comparison to previous work. The theory is then applied to laterally strain-confined quantum wells, with the first-order approximation. In the strain-confined system, the valence band is treated by four simultaneous envelope-function equations. Under certain conditions, they can be reduced to a pair of independent equations. Numerical results for the energy levels in a specific quantum wire are given. In the strain-confined system, the conditions under which the second-order effect could be significant are discussed.