Different Approaches for Multiband Transport in Semiconductors

Abstract

Most recent nanometric semiconductor devices like RITDS (Resonant interband tunneling diodes) [1, 2] exhibit a full quantum behaviour which can be exhaustively understood only starting from the Schrödinger equation. Yet, this "exact", formulation is too complicated to deal with, since the Hamiltonian should take in account the external field potential and the potential barriers due to the heterogeneous structure of the device besides the periodic potential of the crystal (we disregard in our models the field due to impurities, magnetic fields and electronic spin).In this paper we are interested to perform and compare two simplified models, deduced from the Schrödinger equation, which are able to simulate interband tunneling between conduction and the upper (not degenerate) valence band. The tool is an expansion of the full wave function in a suitable orthonormal and complete L 2 basis. The "coefficients" of this expansion are the so-called envelope functions, which represent a smoothed version of the wave function. The Kane model [3] is obtained choosing as a basis the periodic part of the Bloch functions, the so-called Wannier basis, u n,k (x) [4] evaluated at the point k = 0, where it is assumed that the minimum for the conduction band and the maximum for the valence band in the crystal momentum representation occur (n is the band index and in the case here faced we have n = c, v). The Kane model is represented by a system of two Schrödinger-like equations (here we limit ourselves to the one-dimensional formulation):where Ψ Kane c and Ψ Kane v are, respectively, the conduction and valence band envelope functions (in the Kane representation), i is the imaginary unit, is the reduced Planck constant, m is the bare mass of the carriers, E c and E v are the minimum of 1

A Hybrid Method for Computing the Schrödinger Equations with Periodic Potential with Band-Crossings in the Momentum Space

“…Different multi-band models have been developed in order to capture the inter-band transitions, e.g. [11,32,31]. In [13] the authors derived a semiclassical model for the Schrödinger equation with periodic potentials (1.1) that account for band-crossings using Wigner-Bloch theory and proposed a hybrid method for multiscale computation.…”