250 N. Rolland et al.
configurations in which no atom is actually being evaporated from the center of the apex. Compared to the voronoï method, this approach is much faster. Also, the evaporated volume can now be computed all along the tip shape evolu- tion, contrary to the former approach. This will be required for the new reconstruction algorithms.
RECONSTRUCTION ALGORITHM
The proposed reconstruction algorithms proceed quite similar to the method proposed by Larson et al. (2012). In a first stage, the model is run on a given input structure representing the experimental sample. The output of the
model is a sequence of surfaces, describing the progressive field evaporation of the sample. Each of these surfaces cor- responds to a total evaporated volume (delimited by the field of view), computed as the sum of the infinitesimal volumes associated to the preceding surfaces in the sequence. In practical, these surfaces are discretized with a constant cur- vilinear abscissa step and we compute the surface normal to each point. Then, a detector radial distance is associated to each point assuming a basic linear projection law D = kθθ, where D is the radial distance on the detector, kθ the pro- jection parameter, and θ the launch angle at the tip surface. But while in standard reconstruction algorithms θ is the polar angle corresponding to a given position on the spherical apex, it is now associated to the angle between the surface normal and the revolution axis (see for instance Fig. 3a, in the particular case θ = θV). Note that the projec- tion parameter simply equates to the detector radius divided by the field of view θV. During a full reconstruction process, for a given ion of
the experimental data set, the corresponding total evapo- rated volume is computed as the sum of the atomic volumes of the preceding ions, taking into account the detection efficiency. From this volume, we determine the two surfaces of the model sequence between which the ion should be inserted. On each of these surfaces, we find the two points between which the ion should be inserted from the ion detector radial distance. Eventually, the ion position in the 3D reconstruction is calculated by mean of a linear inter- polation both in depth and lateral positions. The quality of the reconstruction will depend on the accuracy of the input structure. Here two versions of the reconstruction algorithm are proposed, depending on the way to input the structure. Those versions correspond, respectively, to the extensions of the cone angle and voltage curve standards reconstruction algorithms.
Cone Angle Version
This first version of the algorithm is a direct application of the model in its original form. The input parameters are the initial radius of curvature, the shank angle and the field of view θV on the tip surface, similarly to the standard cone angle algorithm. In addition, several interfaces can be input, with their position and field ratio. The surface shape is
computed all along the evaporation (as described in the Model section) and used to reconstruct the data set as explained above. This version is recommended if the geometry of the sample is well known (constant shank angle, initial radius of curvature, and precise interface positions), by mean of correlative microscopy for example, or if the voltage curve suffers from bias (analysis conditions have been modified during the experiment, for instance).
Voltage Curve Version
Alternatively, one can use the voltage curve as an evidence of the radius of curvature evolution at the tip apex, without further assumption of the initial sample shape. In this case, the input parameters are the Eβ0 factor of the top layer at the beginning of the evaporation and the field of view θV, simi- larly to the standard voltage curve algorithm. Like the cone angle version, several interfaces can also be input with their depth and field ratio. In the former version, the apex radius of curvature evolution was driven by the tangential con- tinuity at the base of the field evaporated area. Here, at each stage, the apex radius of curvature is found thanks to the experimental applied voltage curve. Indeed, each step cor- responds to a given total evaporated volume that can be associated to a given voltage on the voltage curve (see for instance Fig. 6a). This voltage is converted to a radius of curvature thanks to the classical relationship R = V/Eβ, where the Eβ factor is that of the current tip apex layer, at the
considered stage of the evaporation.A schematic sequence of five surfaces obtained from the algorithm during the transi- tion from a layer to another is depicted Figure 4. The two first
Figure 4. Principle of the voltage curve version of the new recon- struction algorithm. A high field red layer on top of a low field blue layer is considered. Those layers are separated by the inter- face 1, with depth z1 and field ratio f1. The hemispherical radii of the consecutive surfaces during evaporation are obtained from the classical R=V/Eβ0 relationship, taking into account the field ratio for the second layer (R=f1V/Eβ0). When both layers are contained within the field of view θV, it is necessary to model the second layer surface by a Delaunay surface. This surface is deter- mined from the field ratio and the tangential continuity at the interface (purple arrow).
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