Scattering theory of subsurface impurities observed in scanning tunneling microscopy

Subsurface impurities observed in scanning tunneling microscopy (STM) are investigated theoretically. As a scattering potential of the impurities, the screened Coulomb potential is used. Wave functions are solved exactly by numerical calculations. Qualitative behavior of STM images is discussed in terms of analytical expressions derived perturbatively, and the origin of the Friedel oscillation observed in STM images of subsurface impurities is clarified. It is found that to produce the Friedel oscillation, the spatial range of the impurity potential must be shorter than the Fermi wavelength. In the short-range case, calculated STM images show oscillating behavior also as a function of the depth of impurities. When the depth is small, the corrugation amplitude increases with the increase of the depth and reaches the maximum at the depth of about the Fermi wavelength. Ring structures observed in STM of transition-metal-dichalcogenide surfaces are also explained by the subsurface impurities.