Microscopic Theory of Tunneling: General Theory and Application to the Static Impurity

Abstract

A microscopic theory of impurity-assisted tunneling is constructed in which the current-carrying (extended) eigenstates of the average one-electron potential in a tunnel junction are utilized as the basis functions which are mixed by the Hamiltonian associated with the presence of a static or dynamic impurity anywhere in the system. In such a system, the one-electron propagator and its concomitant current across the junction can be calculated at zero bias by standard techniques for manipulating temperature Green's functions. The presence of a finite bias across the junction is incorporated into the theory via the principle of rigid occupancy; i.e., the equilibrium occupation of the (current-carrying) many-body eigenstates of the system is taken to be unaffected by the presence of the bias. Therefore the rigid-occupancy hypothesis relates the nonequilibrium current flow to the equilibrium (zero-bias) properties of the junction system. Thus, we obtain a theory of nonequilibrium current flow which is not based on linear-response theory. This hypothesis is incorporated into the Matsubara perturbation theory by treating the chemical potentials of the left- and right-hand electrodes as separate Lagrangian mulitpliers determined after the completion of all Matsubara sums to be related by ${\mu }_R={\mu }_L-eV$. Therefore, for purposes of constructing and solving Dyson's equations for the renormalized one-electron propagators, the theory reduces to the conventional equilibrium theory defined using distorted-wave (i.e., non-plane-wave) states. The perturbation theory is shown to yield the conventional one-electron-theory results for the case of a static impurity potential in the barrier. Resonant elastic tunneling through impurity states of energy $E_r$ near the zero-bias Fermi energy $\zeta$ causes conductance minima (maxima) for $E_r\mathrm{}\left(0\right)>\zeta$ [$E_r\mathrm{}\left(0\right)<\zeta$]. The transfer-Hamiltonian results are recovered by expanding the transmission probability.