S-orbital continuum model accounting for the tip shape in simulated scanning tunneling microscope images

Abstract

In this paper we present a simple method accounting for tip size effects in scanning tunneling microscopy (STM) simulations. We consider the case where the tip atoms can be regarded as independent sources of $s$ orbitals and compute the tunneling current using the Bardeen formula and the approximation of incoherent scattering. By averaging over the many possible tip configurations compatible with the effective external shape of the STM probe, we show that the tunneling current is proportional, within our model, to the convolution product between the local density of states of the system and a three-dimensional step function defined by the effective tip volume. The method is tested on three systems of current scientific interest, namely, a hexabenzocoronene molecule adsorbed on Cu(111), a reconstructed Au(677) surface, and a formate molecule adsorbed on Pt(111), which we study by means of large-scale density functional theory calculations and STM experiments. An excellent agreement between experimental and simulated STM images is found. It is shown that, under typical experimental conditions, our approach recovers the results of the well-known Tersoff-Hamann modeling in the case of spherical tips, while allowing for more versatility in the choice of the shape of the STM probe. Finally we present an application of our method to one-dimensional surface models mimicking a localized defect and a surface step, thereby offering a very simple framework for the discussion of the tip-induced broadening of the surface features in the STM imaging.