Generalized scattering matrix method for electrostatic calculations of nanoscale systems

We present a theoretical formalism specially suited for the simulation of electrostatic force microscopy (EFM) magnitudes. The method allows for an exact three-dimensional description of the electrostatic interaction. Furthermore, any other calculation that involves different interactions may also be considered since the theory is developed specifically from the Sturm-Liouville equation. The main advantage of this technique is that the influence of the sample, included in the scattering matrix, must be calculated only once at the beginning of the simulation. Once the scattering matrix has been obtained, sources can be moved easily without calculating the whole system again. This characteristic allows us to simulate full scans of magnitudes directly measurable by an electrostatic force microscope such as the capacitance between a metallic tip and a dielectric nanowire. Additionally, the formalism goes beyond standard tip-sample geometries by considering the presence of point charges inside the sample.