Tunneling Through Asymmetric Barriers

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Abstract
Two alternative analytical approaches to the problem of tunneling through asymmetric barriers are applied to the case of the trapezoidal barrier for comparison. Stratton's method, representative of one approach, predicts somewhat smaller magnitudes and asymmetries in the current density as a function of voltage than does a modified version of Simmons' method. By using the temperature dependence, experimental data may be readily analyzed by means of Stratton's method to give the apparent barrier heights as well as the effective zero applied voltage barrier parameters. Simmons' method, representative of the other approach, when corrected also predicts asymmetry in the current density or junction resistance even for small applied voltages, i.e., V < φ/q. Excellent agreement is obtained between the modified version of Simmons' method and the numerical evaluation of the WKB approximation for the current density or junction resistance as a function of voltage. The agreement with Stratton's method, though acceptable, is not as good. Results of measurements on the Al-Al2O3-Al system show that the trapezoidal barrier shape is inadequate in accounting for the details of the observations.