Numerical computation of tunneling fluxes

Abstract

The computation of tunneling probabilities in three dimensions is a numerical challenge, because the small transition probabilities associated with the overlap of exponentially vanishing wave function-tails require large computational accuracy. In scattering situations arising, e.g., in electron tunneling in metal-molecule-metal junctions, this is compounded by the need to provide a proper truncation procedure at the numerical boundaries of the computed system and by the need to account for electrostatic fields and image interactions. This paper describes a numerical methodology to deal with these problems. A pseudopotential that describes the underlying system is assumed given. Electrostatic fields and image interactions are evaluated for the given boundary conditions from numerically solving Laplace and Poisson equations. Tunneling probabilities are computed using a grid-based absorbing boundary conditions Green’s function method. An efficient and exact way to implement the absorbing boundary conditions by using the exact self-energy associated with separating the scattering system from the rest of the infinite space is described. This makes it possible to substantially reduce the size of the grid used in such calculations. Two applications, an examination of the possibility to resolve the spatial structure of an electron wave function in an electron cavity by scanning tunneling microscopy, and a calculation of electron tunneling probabilities through water, are presented.