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.

• Received 15 June 1964

DOI:https://doi.org/10.1103/PhysRev.136.A837