Comparative Apex Electrostatics of Atom Probe Tomography Specimens

Rigorous electrostatic modeling of the specimen electrode environment is required to better understand the fundamental processes of atom probe tomography (APT) and guide the analysis of APT data. We have developed a simulation tool that self-consistently solves the nonlinear electrostatic Poisson equation along with the mobile charge carrier concentrations and provides a detailed picture of the electrostatic environment of APT specimen tips. We consider cases of metals, semiconductors, and dielectrics. Traditionally in APT, and regardless of specimen composition, the apex electric field \(E_\mathrm{apex}\) has been approximated by the relation \(E_\mathrm{apex} = SV / (kr)\), which was originally derived for sharp, metallic conductors; we refer to this equation as the “K-factor approximation”. Here, SV is tip-electrode bias, r is the radius of curvature of the tip apex, and k is a dimensionless fitting parameter with \(1.5< k < 8.5\). As expected, our Poisson solver agrees well with the k-factor approximation for metal tips; it also agrees remarkably well for semiconductor tips-regardless of the semiconductor doping level. We ascribe this finding to the fact that even if a semiconductor tip is fully depleted of majority carriers under the typical SV conditions used in APT, an inversion layer will appear at the apex surface. The inversion forms a thin, conducting layer that screens the interior of the tip, thus mimicking metallic behavior at the apex surface. By contrast, we find that the k-factor approximation yields a very poor representation of the electrostatics of a purely dielectric tip. We put our numerical results into further context with a brief discussion of our own separate work and the results of other publications.