New Atom Probe Tomography Reconstruction Algorithm for Multilayered Samples 249
Figure 2. a: Mean radius of curvature of a saddle surface. The 3D mean curvature inverse is defined as the average of the two principal radii of curvature inverses. On one hand, on a cross-section corresponding to the minimal radius of curvature, one would measure a negative radius. On the other hand, the measured radius would be positive on a cross- section corresponding to the maximal radius of curvature. Eventually, the mean radius of curvature is zero where the surface normal vector is drawn. b: Mean radius of curvature on a field emitter surface containing two layers (high field red layer and low field blue layer). The green arrow is the surface normal, and the two planes are orthogonal planes containing this normal. The vertical plane also contains the revolution axis (large black arrow). Due to the cylindrical symmetry of the emitter, the principal radii of curvature correspond to the so-called meridian (radius of the blue circle, in the plane containing the revolution axis) and parallel radius (radius of the purple arc of circle in the second plane, equals to the small black arrow length).
hyperbola (b>a), one obtains a nodoid (Fig. 1b). Thus, each layer surface is modeled by a piece of a Delaunay surface; in Figure 1c, the field evaporated surfaces of layer 2 and layer 3 are the piece of the unduloid filled in blue in Figure 1a and the piece of the nodoid filled in green Figure 1b, respectively. As the mean radius of curvature of those surfaces is directly related to the semi-axis length a (2a = R), the ratio of the a parameters between two successive layers equates to the inverse of the field ratio of the corresponding interface. In addition, from the tangential continuity condition (depicted by purple arrows in Fig. 1c), one can infer a direct relationship between the b parameters of successive layers. Eventually, the tangential continuity at the base of the field evaporated area (red arrow in Fig. 1c) unambiguously determines the set of surfaces representing the layer surfaces. We refer the reader to Rolland et al. (2015a) for the calculation
details.Acomplete description of the tip morphology evolution during the evaporation can be obtain by repeating the operation for successive dz steps as illustrated in Figure 1d. Originally, a voronoï partition of the output (set of
points describing the successive emitter surfaces during the evaporation) was used to associate an evaporated volume to the successive steps. Here, we propose to compute this volume in a simpler manner. At each stage, corresponding to the removal of a material width dz on the tip axis is asso- ciated an evaporated volume dv: dv = πrV2dz, with rV the radial distance associated to the field of view θV seen by the detector (Fig. 3a). The formula may seem quite unusual as it is often assumed that the infinitesimal volume dv associated to the evaporation of a small material width dz is propor- tional to the sample surface analyzed area SA so that dv = SAdz. However, the latter is mathematically incorrect as
Figure 3. a:Definition of the evaporated volume associated to each step of the modeling. n is the surface normal corresponding to the field of view θV, and rV the associated radial distance. dz is the in depth increment between two successive surfaces computed in the model. The evaporated volume corresponding to this sur- face is assimilated to the volume 2.π
.rV.dz of the orange cylinder. b: Example showing the inaccuracy of common evaporated volume evaluation. The evaporated volume (considering full field of view) corresponding to the depth z in the blue tip equates the red cylinder volume (2.π.R.z).
highlighted in the simple example Figure 3b. Indeed, the analyzed volume (filled in blue) is clearly given by πR2z (the volume of the cross-hatched cylinder), whereas SAz = 2πR2z. Obviously, the step dz has to be as small as possible to ensure an accurate evaporated volume evaluation. In prac- tical, dz has to be even smaller than the characteristic size of an atom to get the correct evolution of the tip morphology; indeed, as pointed out in Larson et al. (2012), there are
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