Electrostatic potential for a hyperbolic probe tip near a semiconductor

The electrostatic potential resulting from a metallic probe tip near a semiconductor is examined. A solution is formulated assuming circular symmetry and using prolate spheroidal coordinates in the vacuum and Cartesian coordinates in the semiconductor. The result is most directly applied to the case of a hyperbolic probe tip, but other shapes (for example, a small hemispherical protrusion on the tip apex) can also be handled. Numerical results are given for representative cases that might be encountered in scanning probe microscopy.