Etude par Sonde Atomique Tomographique de la formation de nano ...

tel-00751814, version 1 - 14 Nov 2012
Appendix 4. Field evaporation model
Appendixes
A model, which simulates the field evaporation of atoms and their trajectories from the tip
apex to a detector was developed in the laboratory GPM [8–10]. The field emitter is
represented as a stack of atoms in the form of a cylinder terminated by a hemispherical cap of
radius R. The shape of the atomic volume is defined by the Wigner–Seitz cell of the chosen
crystalline structure [11].
This tip is submitted to an electric potential, V0. Far from the tip, the potential is fixed at 0.
Between the sample and null voltage electrode the partial distribution of potential can be
calculated by solving numerically the Laplace equation:
ΔV=0 (1)
The space between the tip and the electrode is divided into a regular cubic grid of discrete
points. Each site is represented by three coordinates i, j, k. The potential in each site is noted
Vi,j,k (Figure 12). In each site, local Laplace equation (1) can be applied. By this way, the value
of potential V in each point of the volume can be estimated. This numerical solution can be
simplified to local mean value in each point. Thus, the value in each point is determined by the
six first neighbours of this point:
1
6
�V �V
�V
�V
�V
�V
�
V (2)
� ijk
i�1,
j,
k i�1,
j,
k i,
j�1,
k i,
j�1,
k i,
j,
k�1
i,
j,
k�1
Figure 12. Each cell is given by its coordinates i,j,k [12].
This condition has to be satisfied in each point of the tip. Just the limits of the volume are
not affected by the potential resolution.
V i-1,j,k
V i,j+1,k
V i,j,k
V i,j-1,k
V i+1,j,k
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