Tunneling theory without the transfer-Hamiltonian formalism. IV. The abrupt (zero-width) three-dimensional junction

A general three-dimensional many-body theory of tunneling across an abrupt junction was developed independently of the transfer-Hamiltonian formalism. The theory is based on Keldysh's perturbation theory for nonequilibrium processes. The extension of our previously published one-dimensional theory to three dimensions produced significant new insight. It was demonstrated that a "transfer-Hamiltonian-like" expression for the (energy density of the) tunneling current can be derived. However, this expression depends on a "transmissivity" weighted-average product of the spectral densities of the "uncoupled" subsystems (electrodes) over the interface, rather than on the product of their local energy densities of state at the interface. This qualitative agreement with Appelbaum and Brinkman's theory was shown to reflect the fact that Appelbaum and Brinkman's tunneling theory may be derived from our theory if it is linearized in the (pseudo) transfer Hamiltonian ${\mathcal{H}}^{\prime }\left(\overrightarrow{\mathrm{r}}\right)$ defined by us. Thus, several conjectures concerning the transfer-Hamiltonian formalism were confirmed. Namely, it is a "thick-barrier approximation" involving a (pseudo) perturbing potential which is correct to first order in this pseudo-operator. The general theory is complemented by a brief examination of the translationally invariant and of the ordered planar junctions. The important consequences of the convention used in defining the spectral densities, and Green's functions for the uncoupled electrodes were further elucidated. An examination of the "surface Green's function" introduced by Garcia-Moliner and Rubio indicates that our formalism explicitly accounts for the contribution of interfacial states to the tunneling. Though cast into a many-body formalism, the present results were derived for a noninteracting system. The effects of a nontrivial interaction are currently investigated and will be reported in a subsequent publication.