Envelope-function approximation for nonrectangular ${\mathrm{Hg}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Cd}}_{\mathit{x}}$Te superlattices

The standard eight-band Kane model envelope-function formalism for rectangular superlattices (SL’s) is extended to compositionally graded and doping SL’s with arbitrary potential profiles. SL band structures along the growth direction are described by a set of coupled differential equations that is easily solved numerically with use of Runge-Kutta methods. A detailed application of this approach to graded ${\mathrm{Hg}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Cd}}_{\mathit{x}}$Te SL’s is presented. For a given period and a wide range of band offsets, the SL band gap increases as the composition profile changes from rectangular to sinusoidal to triangular (symmetric) to sawtooth (asymmetric). This and other effects of SL shape are discussed in terms of known quantum-well results and the inherent degree of ‘‘interdiffusion’’ in the SL. Calculated subband structures for ${\mathrm{Hg}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Cd}}_{\mathit{x}}$Te sawtooth SL’s agree well with recent tight-binding calculations. An inconsistency in the limited experimental data available for this system is identified.