Bardeen’s tunnelling theory as applied to scanning tunnelling microscopy: a technical guide to the traditional interpretation

This article is a thorough tutorial on the traditional approach to Bardeen's tunnelling theory in the context of scanning tunnelling microscopy. We follow Duke's interpretation of Bardeen's theory as the tunnelling analogue of Oppenheimer's perturbation theory for field ionization of atomic hydrogen. We explain Oppenheimer's ideas, Fermi's golden rule, and Bardeen's formulae; and we derive the celebrated Tersoff–Hamann formula using an argument due to Chen. Finally, we discuss the application of Bardeen's theory to scanning tunnelling microscopy beyond the Tersoff–Hamann approximation.