248 N. Rolland et al.


[F = V/βR, with V the applied potential and β a geometric factor (Gomer, 1961)]. As a specificity of the model, several interfaces are now included in the sample, and each interface is assigned with a field ratio as depicted in Figure 1c.


This field ratio is defined as the ratio between the electric field required to evaporate the top layer at a given evapora-


tion rate (an experimental setup parameter), and the electric field required to evaporate the bottom layer at the same rate. The evaporation of the sample is modeled as a


progressive step by step process, each step corresponding to the removal of amaterial width dz on the tip axis (Fig. 1c). At each step, the tip shape is considered to be in an equilibrium state in which all the surface atoms are subjected to the same evaporation rate (imposed by the experimental conditions). In


addition, it is assumed that the relationship between electric field and radius of curvature holds locally on the surface, with a constant β factor [F(x,y,z) = V/βR(x,y,z)]. Note that the radius of curvature appearing in this equation is the so-called mean radius of curvature, whose inverse equates the average of the two principal radii of curvature inverses, as illustrated in Figure 2. Alternatively, a mean curvature γ can be defined as the inverse of the mean radius of curvature (γ = 1/R). Here we would like to stress on the fact that the mean radius of


curvature has two components (the principal mean radii of curvature) because we are dealing with a 3D object. Considering only the curvature linked to a two-dimensional representation of the specimen would undoubtedly leads to inaccurate prediction of the tip morphology variations related to inhomogeneous evaporation fields. From the steady state of the surface and the relationship


between electric field and mean curvature, each layer in the sample has to develop a constant mean curvature surface when field evaporated, in order to maintain equiprobability of surface atoms evaporation. For the same reason, the mean curvatures ratio between two layers is equal to their eva- poration fields ratio. Assuming that the sample has sym- metry of revolution, there is thus no other alternative than using the Delaunay surfaces to model the layers. Indeed the latter are the only constant mean curvature surfaces of revolution (Delaunay, 1841). Those surfaces, originally used to describe the shape of liquid films (Plateau, 1873), can be obtained by rotating the roulette of the conics; they fall into two groups depending on the semi-axis parameters a and b of the conic. If the conic is an ellipse (a>b), one obtains an unduloid (Fig. 1a, note that the sphere is a particular case of unduloid with b = 0). On the other hand, if the conic is a


Figure 1. Principle of the modeling of the evaporation of a multilayered specimen. (a)and (b) show the construction of the Delaunay surfaces, the unduloid (a) and the nodoid (b). They are obtained by tracing out the focus of a conic rolling without friction on an axis, and rotating the curve around the same axis. c: The black lines depict the initial model specimen. It is a hemispherical cap (radius R0) seated on a truncate cone (angle α). The emitter contains three layers, delimitated by the interfaces 1 and 2, at depth zi=1,2 and with field ratio fi=1,2. Pieces of Delaunay surfaces (red, blue and green surfaces) are used to model the emitter surface, with the constraint of a tangential contact (purple and red arrows). d:Example of a sequence of evaporation obtained for field ratios equals to 2 and 0.45 for the first and second interfaces respectively.


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