Three-dimensional Wentzel-Kramers-Brillouin approach for the simulation of scanning tunneling microscopy and spectroscopy

Abstract

We review the recently developed three-dimensional (3D) atom-superposition approach for simulating scanning tunneling microscopy (STM) and spectroscopy (STS) based on ab initio electronic structure data. In the method, contributions from individual electron tunneling transitions between the tip apex atom and each of the sample surface atoms are summed up assuming the one-dimensional (1D) Wentzel-Kramers-Brillouin (WKB) approximation in all these transitions. This 3D WKB tunneling model is extremely suitable to simulate spin-polarized STM and STS on surfaces exhibiting a complex noncollinear magnetic structure, i.e., without a global spin quantization axis, at very low computational cost. The tip electronic structure from first principles can also be incorporated into the model, that is often assumed to be constant in energy in the vast majority of the related literature, which could lead to a misinterpretation of experimental findings. Using this approach, we highlight some of the electron tunneling features on a prototype frustrated hexagonal antiferromagnetic Cr monolayer on Ag(111) surface. We obtain useful theoretical insights into the simulated quantities that is expected to help the correct evaluation of experimental results. By extending the method to incorporate a simple orbital dependent electron tunneling transmission, we reinvestigate the bias voltage- and tip-dependent contrast inversion effect on the W(110) surface. STM images calculated using this orbital dependent model agree reasonably well with Tersoff-Hamann and Bardeen results. The computational efficiency of the model is remarkable as the k-point samplings of the surface and tip Brillouin zones do not affect the computational time, in contrast to the Bardeen method. In a certain case we obtain a relative computational time gain of 8500 compared to the Bardeen calculation, without the loss of quality. We discuss the advantages and limitations of the 3D WKB method, and show further ways to improve and extend it.