Electron Tunneling with Diffuse Boundary Conditions

The tunneling of electrons through potential barriers with plane boundaries involves the conservation of tangential quasimomentum (specular transmission). The boundaries between metals and insulators (e.g., oxides) in thin-film tunneling structures can be very irregular or rough, so that the specular transmission condition must be replaced by a "diffuse" transmission condition which allows nonconservation of tangential momentum. The tunnel current for diffuse boundary conditions and for arbitrary band structures and potential-barrier shapes has been calculated. Compared to the result for specular boundary conditions, the integrand in the expression for the tunnel current contains a factor $U(E, {\mathrm{p}}_t)=\Sigma \frac{\rho \left({{\mathrm{p}}_t}^{\prime }\right)\pi ({{\mathrm{p}}_t}^{\prime }, {\mathrm{p}}_t)}{\rho \left({\mathrm{p}}_t\right)}$ for each metal [$\rho \left({\mathrm{p}}_t\right)=\mathrm{linear}\mathrm{density}\mathrm{of}\mathrm{states}\mathrm{for}\mathrm{fixed}\mathrm{tangential}\mathrm{momentum}$ ${\mathrm{p}}_t, \pi ({{\mathrm{p}}_t}^{\prime }, {\mathrm{p}}_t)$ is proportional to the probability that the diffuse boundaries change ${{\mathrm{p}}_t}^{\prime }$ to ${\mathrm{p}}_t$, and the sum is over the "shadow" of the energy surface $E\left(\mathrm{p}\right)$]. Under reasonable assumptions for $\pi ({{\mathrm{p}}_t}^{\prime }, {\mathrm{p}}_t)$, no appreciable effect of the densities of state on $U$ or on the tunnel current should occur. The effects of diffuse transmission conditions on field emission from manyvalley semiconductors and on tunneling between superconductors are briefly discussed.