The thin-film Al-${\mathrm{Al}}_2$${\mathrm{O}}_3$-Al system is studied. The current density as a function of temperature and voltage through aluminum oxide less than 30 Å thick is analyzed using the theory of Stratton. The charge transport mechanism is established as tunneling by the functional form of the small temperature dependence. By fitting the current density data to the temperature dependence for $79°\mathrm{K}<T<\mathrm{}300\mathrm{}°\mathrm{K}$ and $0\le V\le 1.0$ V, the integrals arising from the WKB approximation are evaluated for an arbitrary barrier shape. The region of applied voltage for which quadratic power series expansions in $V$ of these integrals are valid is established as $0.0\le V\le 0.3$ V. Assuming a trapezoidal barrier shape, approximate values for the barrier heights of 0.89 and 0.78 eV and an electron effective mass of $\frac{m^{*}}{m}=1\mathrm{}$ are deduced. Comparisons are drawn between the more general forms of Stratton's and Simmons' theories.

• Received 13 December 1963

DOI:https://doi.org/10.1103/PhysRev.134.A1094