Tunnel current and generalized Ehrenfest theorem

Abstract

The interrelation between two different theoretical formulations of the theory of scanning tunnelling microscopy (STM) is investigated. The mathematical equivalence of the current density formulation of STM theory and the generalized Elsrenfest theorem (GET) is demonstrated. The GET is of practical importance, because it facilitates the numerical evaluation of the tunnel current and provides additional physical insight. Using formal scattering theory, the integration of the current density over a plane parallel to the sample electrode yields half the value of the total tunnel current as obtained from the generalized Ehrenfest theorem. This discrepancy is removed if an adequate normalization of the wavefunction in the current density formulation is chosen. The proof includes a generalization of Bardeen's transfer Hamiltonian expression for the transition matrix element.