Arbitrary tip orientation in STM simulations: 3D WKB theory and application to W(110)

We extend the orbital-dependent electron tunnelling model implemented within the three-dimensional (3D) Wentzel-Kramers-Brillouin (WKB) atom-superposition approach for simulating scanning tunnelling microscopy (STM) by including arbitrary tip orientations. The orientation of the tip is characterized by a local coordinate system centred on the tip apex atom obtained by a rotation with respect to the sample coordinate system. The rotation is described by the Euler angles. Applying our method, we highlight the role of the real-space shape of the electron orbitals involved in the tunnelling, and analyse the convergence and the orbital contributions of the tunnelling current above the W(110) surface depending on the orientation of a model tungsten tip. We also simulate STM images at constant-current condition, and find that their quality depends very much on the tip orientation. Some orientations result in protrusions on the images that do not occur above W atoms. The presence of such apparent atom positions makes it difficult to identify the exact position of surface atoms. It is suggested that this tip orientation effect should be considered in the evaluation of experimental STM images on other surfaces as well. The presented computationally efficient tunnelling model could prove to be useful for obtaining more information on the local tip geometry and orientation by comparing STM experiments to a large number of simulations with systematically varied tip orientations.