Green’s function matching method for one- and zero-dimensional heterostructures

A method for the calculation of the Green's function of one-dimensional (wirelike) and zero-dimensional (boxlike) heterostructures is developed. The method is based on the formalism of surface Green's function matching (SGFM) for discrete media. The matching formulas are derived for structures consisting of an infinite ``external'' medium with an embedded array of wires (boxes), periodic in two (three) dimensions, for an arbitrary shape of the matching surface. Efficient algorithms to calculate the domain Green's functions are also proposed. Unlike the supercell methods widely used to compute the electronic structure of one- and zero-dimensional systems that scale with the supercell size, the developed approach scales with the interface size, which makes it extremely efficient to study systems such as disordered alloys, clusters of defects, and nanoscale materials. We employ it to calculate the localized and extended electronic states of a periodic array of $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{As}$ (001) nanowires using an $s{p}^{3}{s}^{*}$ tight-binding Hamiltonian with spin-orbit coupling.