Numerical evaluation of resonant states

Abstract

The quantized states formed by semiconductor heterostructures are often resonant states, in the sense that they are not asymptotically confined and thus have a finite tunneling lifetime. These states may be readily located and characterized by finding the poles of a discretized single-particle Green's function. The denominator of the Green's function is constructed from the discrete Hamiltonian matrix, augmented by terms which describe transmitting boundary conditions and which thus render the operator non-Hermitian. The resonances are the complex-valued eigenvalues of this operator. The boundary terms are energy-dependent, and thus the eigenvalue problem is nonlinear. The eigenvalues are located using a combination of linear searching and Newton iteration in the complex energy plane. For one-dimensional problems, this technique is fast enough to be used in an interactive mode, and it has been incorporated into a general-purpose interactive heterostructure modeling program. Using such a program one may rapidly examine a variety of structures and bias conditions.