Tunneling theory without the transfer-Hamiltonian formalism. I.

Abstract

Tunneling between two semi-infinite systems across an abrupt junction (of zero thickness) is discussed without invoking the transfer-Hamiltonian formalism. A lucid and novel interpretation of the significance of the transfer Hamiltonian is obtained. A rigorous and easily applied separation of the junction into its two component subsystems is introduced. The current is expressed in terms of characteristics of these two subsystems. Keldysh's perturbation theory for nonequilibrium processes is used to include the effects of the external potential $V$ to all orders. In this, the analysis is similar to that of Caroli et al. The extreme-tight-binding approximation of these authors is avoided; this less-restricted formulation differs in important details from the preceding work. It is shown that tunneling current (density) within a differential energy interval can generally be expressed as proportional to the product of appropriately defined "left" and "right" local densities of states (in energy). This confirms the form of the extensions of the consequence of the transfer-Hamiltonian formalism, proposed by Appelbaum and Brinkman and by Caroli et al. The new formulation of tunneling manifestly applies in the zero-thickness extreme-strong-coupling limit, where the current formulation of the transfer Hamiltonian fails. The extension of the formalism to junctions of finite width has been worked out and is reported in a second paper, in which it is demonstrated that the results for the abrupt junction follow from those of the finite junction in the zero-width limit, and agree with them qualitatively. The formalism is specifically suited to include many-body interactions.