Path-integral approach to the electron density of states at the interface of a single modulation-doped heterojunction

Abstract

The electronic density of states (DOS) at the interface of a single modulation-doped heterojunction is calculated both in the fluctuation band tail and in the semiclassical limit using the path-integral method. Due to charge density inhomogeneities in the heavily doped barrier region, random potential fluctuations are generated in whose minima carriers are localized, resulting in a band-tail density spectrum. The screening of the long-range potential fluctuations, which are important for the problem considered, is accounted for by using the two-dimensional Thomas-Fermi model. The statistical properties of the random impurity charge distribution are taken into account using the binary correlation function of the random potential for the specific geometry of the problem in two limiting cases of the general correlation function. Analytical expressions for the dimensionless functions of the exponential and preexponential of the band-tail DOS, as a function of the energy, are obtained in the weak and strong screening limit for the quasi-2D case under consideration and are compared to the respective functions for the general d-dimensional case. The dependence of the band-tail DOS behavior on the relevant parameters of the system, namely, spacer layer thickness, doping layer thickness, 2D EG thickness, and 3D-impurity concentration is studied numerically for ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}\ensuremath{-}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ modulation doped heterostructures and numerical results for the 2D-DOS are presented. The band-tail results for the 2D-DOS are compared with the Kane approximation derived by taking the limit $\stackrel{\ensuremath{\rightarrow}}{t}0$ in order to determine $\ensuremath{\rho}(E)$ in the whole energy range. We compare the results from the two computed cases of the path-integral expression for the DOS corresponding to the two limits of the general correlation function with each other. On the other hand, we compare the semiclassical limit and the white Gaussian noise limit of the above results with many other theoretical methods such as the generalized semiclassical method, multiple scattering method and the simulations resulting from the tight-binding model.