Theory of resonant tunneling of charge carriers within the framework of the Green's function method and the biorthogonal formalism

A general approach to the analysis of resonant tunneling and scattering effects is developed within the framework of the Green's functions method. The proposed mathematical apparatus is based on the biorthogonal formalism of quantum theory that allows one to describe systems with non-Hermitian Hamiltonians. Such operators are typical for problems wherein particle is supposed to leave the physical system and go to infinity. The stationary Schrodinger equation with a particle source is considered in abstract operator form. The solution of that problem is expressed in a general form in terms of the corresponding Green's operator. Transmission coefficients and their dependences on the particle energy are determined for a single rectangular potential barrier and a double Gaussian barrier. The result of numerical calculation for a single rectangular barrier is compared with the well-known analytical solution of this problem. The proposed method makes it possible to analyze the particles tunneling and scattering in structures with an arbitrary number and shape of potential barriers with high accuracy.