Calculation of tunneling current across trapezoidal potential barrier in a scanning tunneling microscope

Abstract

Accurate calculation of the tunneling currents in a scanning tunneling microscope (STM) is needed for developing image processing algorithms that convert raw data of the STM into surface topographic images. In this paper, an accurate calculation of the tunneling current for several tip–sample distances, bias voltages, and tips of a hyperboloidal shape with several radii of curvature is carried out. The main features of this calculation are the following. Non-WKB exact solutions to the trapezoidal (linear) potential in the barrier region are used to calculate the tunneling probabilities. Pauli blocking effects on both forward and reverse current densities are introduced. Finite temperature (viz. [Formula: see text]) calculation in which electrons belonging to a narrow band of energy about the Fermi level contribute to tunneling is carried out. Integration over a field line method is used to obtain tunneling currents for the nonplanar hyperboloidal shaped tips, using the expressions obtained in the paper, for planar model current densities. An estimate of the lateral resolution is introduced. Earlier works do not consider all these aspects together in a single calculation. Tunneling currents are found to increase rapidly with increasing bias voltage and decrease exponentially with increasing tip–sample distances. Airy function determined currents are a more accurate function of a tip–sample distance than the WKB determined currents. Pauli effects are found to not always reduce currents from their non-Pauli values. The lateral resolution is found to be degraded for blunter tips, larger bias voltages, and larger tip–sample distances.