Image charge method for electrostatic calculations in field-emission diodes

We present a method to calculate the electrostatic field between a metallic tip of arbitrary shape and a sample surface. The basic idea is to replace the electrodes by a set of ‘‘image’’ charges. These charges are adjusted in order to fit the boundary conditions on the surfaces. As an application of the method, we describe the field characteristics of a field-emission diode as a function of the gap between electrodes for different tip shapes. A comparison between numerical and analytical results is presented. The results do not depend on the overall tip geometry only for gap distances smaller than ≊1/2 the tip radius. The field enhancement factor due to the presence of small protrusions on the tip apex is calculated and their influence in near-field-emission scanning tunneling microscopy is also discussed. We show that the electron-field emission from the sample is stable against tip-shape changes due to adsorbate diffusion or atomic rearrangements.