Simplified calculations of the lateral distribution for the current in tunnelling junctions having general shapes

A simplified method for calculations of the current in a tunnelling junction with its orientation and spatial distribution is described. The current at test points on either electrode is calculated for the shortest distance with expressions for point-to-point tunnelling by Simmons (1963) and then summed. The calculated distribution of the current in a spherical-tip/planar-sample model of a scanning tunnelling microscope (STM) agrees with that from finite-element simulations. This method may be applied to models having arbitrary shapes, and results are presented for a paraboloidal-tip/corrugated-sample STM model as the first approximation to the atomic structure.