Beyond Tersoff and Hamann: A generalized expression for the tunneling current

Abstract

We present an analytical method for treating the tunneling current between a probe tip and a sample in scanning tunneling microscopy (STM). The choice of a highly idealized tip permits a direct calculation of the tunnel current without resorting to perturbation theory (Bardeen approximation). In our model, the tip is a semi-infinite chain of spherical potential wells oriented orthogonal to the sample surface. The sample surface, however, is arbitrary. The wavefunction for the entire system is obtained by a matching procedure, from which the total current is determined. For small bias voltage the result is comparable in simplicity with that of Tersoff and Hamann [Phys. Rev. B 31, 805 (1985)]. We find that the tunnel conductance is proportional to the local density of states (LDOS) of the surface, but renormalized to include multiple reflections to all orders: σ∝ρL(r0,EF)/D, where D depends on both the tip and sample electronic structure, and the tip position r0. As the tip-surface separation increases D approaches unity, and we recover the result of TH. We find that D can be smaller or greater than unity, leading to a relative reduction or enhancement of the current. This effect which can be significant, both in the interpretation of STM images and in related spectroscopies, depends on the particular surface electronic structure.