Critical study of perturbative approaches to tunneling

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Abstract
One of the long-lasting objectives of the theory of tunneling is to express the transmission probability in terms of the wave functions of infinitely separated electrodes. This can be achieved by the application of a perturbative approach to tunneling; in this context the transfer Hamiltonian method has been developed and used. In cases such as scanning tunneling microscopy operating at small tip-sample separation, however, it becomes necessary to go beyond the original transfer Hamiltonian method. In this study we examine the modified forms of the transfer Hamiltonian method using exactly solvable one-dimensional tunneling systems. We find that it is possible to calculate the transmission probability approximately by choosing appropriate boundary conditions for the wave functions used in the transition matrix element expression. However, for low and thin barriers these modified methods still fail to give the correct results. On the other hand, Green’s-function techniques which extend the perturbation to all orders yield exact results irrespective of the boundary condition chosen at the interface.