Extension of the homogeneous electron gas theory to first-order for semiconductors with potential gradients

The theory of Homogeneous Electron Gas (HEG), extensively used in solid-state and semiconductor physics, is extended to explicitly include the effect of the first-order potential gradient, whether the gradient be due to an external electric field or a gradual compositional variation. The resulting approximation is called the First-Order Homogeneous Electron Gas (FOHEG) in this work. Application of the first-order theory to the density of states shows that extra state density is introduced below the band edge by the potential gradient, a phenomenon called the Field-induced Band Gap Narrowing (FIBGN) in this work. To study the validity of the conventional and the first-order approximations, the carrier densities are computed by both methods and compared with the exact solution to a spherical quantum dot which is so idealized that its analytic solution is available. It is found that the first-order theory better matches the exact results than the conventional one at locations beyond the classical turning point associated with the Fermi energy.