Chen's derivative rule revisited: Role of tip-orbital interference in STM

Abstract

On the occasion of its 25th anniversary, we revise Chen's derivative rule for electron tunneling [C. J. Chen, Phys. Rev. B 42, 8841 (1990)] for the purpose of computationally efficient simulations of scanning tunneling microscopy (STM) based on first-principles electronic structure data. The revised model allows the weighting of tunneling matrix elements of different tip-orbital characters by an arbitrary energy-independent choice or based on energy-dependent weighting coefficients obtained by an expansion of the tip single-electron wave functions/density of states projected onto the tip-apex atom. Tip-orbital interference in the STM junction is included in the model by construction and can be analyzed quantitatively. As a further advantage, arbitrary tip geometrical orientations are included in the revised model by rotating the coordinate system of the tip apex using Euler angles and redefining the weighting coefficients of the tunneling matrix elements. We demonstrate the reliability of the model by applying it to two functionalized surfaces of recent interest where quantum interference effects play an important role in the STM imaging process: N-doped graphene and a magnetic ${\mathrm{Mn}}_{2}\mathrm{H}$ complex on the Ag(111) surface. We find that the proposed tunneling model is 25 times faster than the Bardeen method concerning computational time, while maintaining good agreement. Our results show that the electronic structure of the tip has a considerable effect on STM images, and the Tersoff-Hamann model does not always provide sufficient results in view of quantum interference effects. For both studied surfaces, we highlight the importance of interference between $s$ and ${p}_{z}$ tip orbitals that can cause a significant contrast change in the STM images. Our method, thus, provides a fast and reliable tool for calculating STM images based on Chen's derivative rule, taking into account the electronic structure and local geometry of the tip apex.