Exact analytical solution for electrostatic field produced by biased   atomic force microscope tip dwelling above dielectric-conductor bi-layer
  An exact analytical solution based on the method of images has been obtained for the description of the electrostatic field in the system comrising of atomic force microscope (AFM)tip, dielectric, and conductor. The solution provides a step towards quantitative modelling of AFM-assisted electrostatic nanolithography in polymers. 
Sergei F. Lyuksyutov, Ruslan A. Sharipov, Grigori Sigalov, and Pavel B   Paramonov