A formula is derived for the electric tunnel effect through a potential barrier of arbitrary shape existing in a thin insulating film. The formula is applied to a rectangular barrier with and without image forces. In the image force problem, the true image potential is considered and compared to the approximate parabolic solution derived by Holm and Kirschstein. The anomalies associated with Holm's expression for the intermediate voltage characteristic are resolved. The effect of the dielectric constant of the insulating film is discussed in detail, and it is shown that this constant affects the temperature dependence of the J-V characteristic of a tunnel junction.

The theory of the electric tunnel effect has been extended to asymmetric junctions—i.e., junctions having electrodes of different materials. It is found that the J—V characteristic is polarity-dependent. At lower voltages, the greater current flows when the electrode of lower work function is positively biased, in agreement with observations on junctions having one aluminum electrode. At higher voltages, there is a change in direction of rectification; i.e., greater current flows when the electrode of lower work function is negatively biased. This effect has also been experimentally observed.

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Conduction through thin dielectric films sandwiched between metal and semiconductor electrodes is reviewed.

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The thermal J-V characteristic for a tunnel junction is derived in terms of a generalized theory. The resulting functional form of the equations is similar to that of Stratton; however, in the present formulation, the physical parameters of the junction appear explicitly, and their effect upon the thermal characteristic is readily appreciated. In Stratton's work, the physical constants appear in the integrand of integral that can be solved only numerically. The theory is applied to symmetric and asymmetric junctions. For the symmetric case, it is shown that, at a given temperature, the percentage change Ĵ in the high-temperature thermal component of current from the low-temperature value increases initially with increasing voltage bias up to a maximum peak, and thereafter decreases rapidly. The voltage bias at which the component of thermal current maxima occurs is equal to the interfacial barrier height and, as such, permits what is probably the most accurate method of barrier height determination. Similar results are obtained for the asymmetric barrier; however, in this case, Ĵ depends upon the polarity of the voltage bias for V>φ1, and two Ĵ maxima occur at voltages corresponding to the two distinct interfacial barrier heights φ1 and φ2.

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This paper discusses in detail the shape of the potential barrier existing between two parallel-plane metal electrodes separated by a thin insulating film. The emission-limited current flow between the electrodes is determined for symmetric and asymmetric junctions. The asymmetric J-V (current-voltage) characteristic is of particular interest, as it can be shown that the difference in work function of the two electrodes comprising the junction can be obtained from a perfunctory study of the characteristic. The thermionic J-V characteristic is compared to the tunnel J-V characteristic. For a temperature of 300°K, and for a barrier thickness less than 40 Å, the tunnel J-V characteristic predominates. However, for barrier thickness greater than 40 Å, either the thermionic or the tunnel characteristic can predominate, depending upon the barrier height and the applied voltage. In the case of asymmetric junctions, there may be a reversal of direction of rectification with decreasing temperature. This reversal of rectification may explain the similar effect sometimes observed in tunnel junctions.