Modified Schrödinger equation including nonparabolicity for the study of a two-dimensional electron gas

A modified Schrödinger equation has been obtained for calculating the energy–wave-vector dispersion relationship of the two-band Kane model and has been applied to the case of a two-dimensional electron gas. The equation is applicable when the isoenergetic surfaces are not spheres and it is expressed as an infinite series whose summation provides compact solutions. With this equation, the transverse energy levels can be obtained by using an effective mass which is independent of the transverse energy, but there is a dependence of these energy levels on the parallel energy. The boundary conditions associated with the modified Schrödinger equation have been derived by imposing the continuity of the eigenfunction and the probability current, and they have been applied to the calculation of the energy levels in an infinite and a finite square quantum well.