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 $E_{apex}$ has been approximated by the relation $E_{apex}=SV/\left(kr\right)$, 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 applied to a purely dielectric tip results in $k$ values far greater than the typical range for metallic tips. We put our numerical results into further context with a brief discussion of our own, separate, experimental work and the results of other publications.