Theory of the Tunneling Spectroscopy of Collective Excitations

The transfer-Hamiltonian model of a tunnel junction is extended to incorporate the interaction of a tunneling electron with impurities and collective excitations in the barrier region. The associated elastic and inelastic contributions to the tunnel current are evaluated, and are compared with the consequences of (bulk) electron-collective-mode coupling in the conducting electrode. Expressions for incoherent electronphonon coupling via potential fluctuations in the barrier and coherent coupling via constituent atoms of the barrier are derived. Various contributions to both the elastic and inelastic currents due to these coupling mechanisms are classified and evaluated. A prototype square-barrier model, which simulates the potential metal-semiconductor, metal-oxide-metal, or direct $p-n$ tunnel junctions, is used to described the "one-electron" aspects of the tunneling characteristics. For a narrow, high barrier at $T=0\mathrm{}$, the one-phonon impurity-induced conductance is proportional to ${\mathrm{}\left(\mathrm{eV}\right)\mathrm{}}^2\ominus (\hslash {\omega }_D-eV)+{\left(\hslash {\omega }_D\right)}^2\ominus (eV-\hslash {\omega }_D)$, where $\hslash {\omega }_D$ is the highest phonon energy. The one-phonon constituent-atom-induced conductance under similar conditions is proportional to ${\mathrm{}\left(\mathrm{eV}\right)\mathrm{}}^4\ominus (\hslash {\omega }_D-eV)+{\left(\hslash {\omega }_D\right)}^4\ominus (eV-\hslash {\omega }_D)$. The calculated line shape, temperature dependence, carrier-mass dependence, and impurity-density dependence of the impurity-induced inelastic conductance are shown to agree reasonably well with existing experimental data.