Tunneling Conductance of Asymmetrical Barriers

The voltage-dependent tunneling conductance of trapezoidal potential barriers has been calculated using two extreme models of (1) the WKB approximation and (2) perfectly sharp boundaries between the metal electrode and the insulator. We show that for both models the conductance-voltage plot is roughly parabolic at low voltages (≲0.4 V). The minimum conductance is not at zero bias unless the barrier is symmetrical and identical Fermi energies are chosen for the two metal electrodes. The inclusion of image forces does not radically alter the shape of the conductance-voltage dependence. Using reasonable barrier shapes, the asymmetry of the calculated conductance about V=0 is not as large as we frequently observe experimentally. We point out that this extreme asymmetry appears to be associated with the presence of organic impurities in the oxide layer of the junction.