Tunneling from an Independent-Particle Point of View

A method is developed for calculating wave functions through regions of varying band structure. This method is applied to tunneling problems using the transition-probability approach of Bardeen. It is found that the experiments of Giaever involving tunneling into superconductors cannot be understood strictly in terms of an independent quasi-particle model of the superconductor. The observed proportionality of the tunneling probability to the density of states depends upon the matrix elements being constant which, in turn, depends upon a many-particle feature of the problem. This feature does not carry over to fluctuations in the density of states arising from band structure, and contributions to the current are not expected to be proportional to the density of states in that case. Instead, a projection in wave-number space of the appropriate constant-energy surface enters. Tunneling systems are discussed which involve semiconductors, semimetals, and transition metals as well as simple metals. Finally, alterations in the properties arising from alterations in the nature of the boundary regions are discussed.