Calculating electrostatic interactions in atomic force microscopy with semiconductor samples

Electrostatic interactions are important in non-contact atomic force microscopy (AFM) measurement. Previous reports had focused on the calculation of electrostatic interactions in AFM with metal and dielectric samples, and the present work extended the discussion to semiconductor samples based on Green’s function theory and Debye-Hückel theory, considering sample dielectric polarization and free carriers at the same time. In order to enhance the calculation efficiency, an equivalent charge method was implemented and developed with a linear algebra-based algorithm. The calculation results of two limiting cases, metal and dielectric limit with infinite and zero carrier concentrations respectively, were in good agreement with the boundary element method. For a finite carrier concentration, it is found that the electrostatic force on the tip cone is quickly saturated whereas that on the tip apex slowly increases as the carrier concentration increases. On the other hand, the interaction radius on the sample surface is found independent of the sample free carriers, but it linearly increases as the tip-sample distance. Our work can be useful for the carrier concentration detection of semiconductor samples using non-contact electrical AFM modes such as Kelvin probe force microscopy and electrostatic force microscopy.