Tunneling theory without the transfer Hamiltonian formalism. V. A theory of inelastic-electron-tunneling spectroscopy

The currently accepted interpretation of inelastic-electron-tunneling spectra (IETS) is based on frankly phenomenological formulations of tunneling. The most commonly applied theories invoke either the golden rule or the linear-response formulation of the transfer Hamiltonian formalism. The equivalence of the theories has been asserted, but never fully established. A generalization of the golden rule analysis of inelastic tunneling to finite temperatures is presented. It is shown that this generalization is equivalent (at all temperatures) to the linear-response theory for inelastic tunneling involving single-phonon excitations in the barrier. In agreement with experiment, both theories predict an asymmetry of the differential tunneling conductivity, $\frac{d^{2}I}{d{\mathrm{}\left(\mathrm{eV}\right)\mathrm{}}^2}$, under bias reversal. A new many-body formulation of IETS is developed by applying Feuchtwang's tunneling theory to a one-dimensional junction. The tunneling electrons are assumed to interact with the vibrational modes of an impurity in the barrier via a nonlocal electron-phonon interaction, which is supposed to be totally screened out in the metal electrodes. The interaction is assumed to give rise to electronic self-energy operator, which is treated in the Migdal approximation. A formally exact expression of the tunneling current is obtained, and the first-order terms in the self-energy are explicitly presented. It is shown that these lead to three distinct inelastic contributions to the current. The first, or ordinary term corresponds to the one-phonon contribution to the generalized golden rule expression for the tunneling current. The other two represent more subtle impurity effects that can be interpreted as inelastic internal field emission (in the barrier) and resonant inelastic tunneling, respectively. The latter can also be interpreted as tunneling via an effective density of states induced by the electron-impurity-phonon interaction. This dynamic or resonant density of states is spatially localized in the barrier. It is suggested that a resonant inelastic channel may be misinterpreted as a nonresonant channel associated with an incorrect value of energy transfer between the tunneling electron and the vibrational impurity mode. The phenomenological parameters in the golden rule formulation of tunneling are interpreted in terms of the new and more complete theory. The extension of the analysis to a full three-dimensional theory is considered and the consequent simple modification of our conclusions is discussed.