Investigation of a new numerical method for the exact calculation of one-dimensional transmission coefficients: Application to the study of limitations of the WKB approximation

A new technique for an exact, numerical calculation of one-dimensional transmission coefficients has been recently derived by Vigneron and Lambin. This has been extended to allow different right and left asymptotic values of the potential, V(+∞) ≠ V(−∞). The algorithm is assessed by using it to calculate the J- V characteristics for a variety of model tungsten (W-W) metal-vacuum-metal junctions with different interelectrode separations. Comparison of these results with the corresponding J- v characteristics previously calculated using the WKB approximation reveals some unexpected limitations on the accuracy of the WKB method. These are consistent with well known inadequacies of the WKB approximation. Namely, the WKB barrier transmission coefficient T WKB always exceeds the exact transmission coefficient T for energies close to the bottom of the barrier. Furthermore, at the top of the barrier, T WKB assumes the infinite energy value of unity. This suggests that either T WKB> T for all energies or there exist one or more energy intervals between the bottom and the top of the barrier in which T WKB ⩽ T. For all barriers investigated here and by Vigneron and Lambin, exactly one such interval was found. This result can also be proven analytically for the square barrier. It is also found that for a given barrier shape the location of the intersection of T WKB and T may depend on the barrier width. Furthermore, the energies for which T WKB = T determine whether J WKB approaches the exact current density J as the barrier width increases. For some barriers, with widths larger than some optimum value, J WKB diverges from J with increasing barrier thickness. It is concluded that the Vigneron-Lambin algorithm is generally faster and more accurate than the WKB approximation.