Computation of CEX-Ion Flux and Sputtering Rate
4.1 Obtaining the Electrostatic Potential
Referring to Figure 4.2, let us assume that an ion scattered fromP at the angleθ, with respect to the primary ion velocity vector,u, has a trajectory passing throughˆ S, in the absence of any external forces (Path 1). Under the influence of external forces, this same ion will, in general, follow a path that does not pass throughS (Path 2). In order for an ion to scatter onto a path passing through S (Path 3), it must scatter at an angleθ∗.
The contribution to the density and energy distribution atS from ions scattering atθ∗ will be different than the contribution from those scattering atθ. First, the number of ions scattered into a solid angleδΩ will be different for each angle, due to the different values of the differential cross- section dσ/dΩ. Second, the area,δA, through which ions pass that are scattered into a solid angle˜ δΩ will be different for each scattering angle, due to the effect the external forces have on distorting the streamtube formed by these ions. Third, the kinetic energy of the ions scattered into the two different angles will be different at the pointS, due to the dependence of the initial scattered energy
S
P ˆ u
θ
θ∗ 1
2 3
Figure 4.2: Example of scattering solutions. In the absence of any external forces, a CEX ion scattered at an angle,θ, with respect to the primary-ion velocity vector, will pass through the point S (Path 1). If external forces are present, the same ion will travel on a trajectory that does not pass throughS (Path 2). The CEX ion must scatter at a different angle, θ∗, in order to travel on a trajectory that passes through S, if external forces are present (Path 3). Due to the different scattering angles, the energy of an ion that travels to S along Path 1 will be different than the energy of an ion that travels along Path 3.
on the scattering angle (Equation 2.18). In order to compute an accurate count of the CEX ions passing through S and their corresponding energies, the electric field (obtained from the electric potential Φ) influencing their trajectories must be known. In this section we present a model for determining the electric potential, Φ, from the ion density,ni.
The small mass and resulting high mobility of electrons leads to the situation known as Debye shielding, in which the electrons move to nearly neutralize the electric field at distances larger than the Debye length from an ion [44]. We thus assume that the plasma is in a state of quasi-neutrality for length scales greater than the Debye length. Under the assertion of quasi-neutrality, the electron and ion charge densities are very nearly equal;neni [12].
In molecular dynamics models it is typical to assume that the neutralizing electrons behave as a fluid [45], for which the conservation of momentum equation is
me
Due
Dt ≡me
∂ue
∂t +ue· ∇ue
=
i
Fi. (4.1)
We assume that the only significant forces acting on the electrons are pressure and electrostatic
forces. Let us also assume that the electrons have reached a steady state in thermal equilibrium as an isolated population, and that they behave as an ideal gas [12]. With these assumptions, the momentum equation for the electrons yields
meue· ∇ue=−∇p ne
+qe∇Φ =−kBTe∇ne
ne
+qe∇Φ, (4.2)
whereneis the electron number density,Teis the electron equilibrium temperature,qeis the electron charge, andkB is Boltzmann’s constant. It is common to assume that the inertial term is negligible, due to the small mass of the electrons, especially in comparison with the ions [14, 46]. Thus
∇Φ =kBTe
qe
∇ne
ne
, (4.3)
for which integration yields the Boltzmann relation
Φ−Φ0=kBTe
qe
ln ne
ne,0
, (4.4)
where Φ0 is a reference potential assigned to the reference density, ne,0. The relation between the electrostatic potential and the electron density in Equation 4.4 is commonly referred to as the barometric potential law [12, 14, 47]. From the condition of quasi-neutrality, the electrostatic potential can be inferred if the density of either species is known. Assuming that the electric potential is the only source of any significant force acting on the ions [14], the equation of motion for any ion is
mi
dui
dt =−qe∇Φ. (4.5)
It is immediately evident that a problem arises if we attempt to use Equation 4.4 to compute the electric potential. In order to do so we must have a prioriinformation about the ion density which is the quantity we are attempting to compute in the first place. Computational methods such as Particle-in-Cell (PIC) models approach this problem by solving for the ion trajectories and electrostatic potential self-consistently, through sequential time-integration of the equations of
motion [48]. At each time step the total charge density is computed, from which the potential is updated by Poisson’s equation or the barometric potential law [46] for the next step. The equations of motion (Equation 4.5) are then integrated through another time step, using the new potential, from which the total charge density can be updated. Self-consistent electric potentials and ion trajectories through the domain of interest are obtained by integrating through a sufficiently large number of time steps.
In this work we need to compute the electric field downstream of multiple grid shapes. The time required for a complete self-consistent computation of the electric field using time-integration methods, such as PIC, for multiple grids makes using these methods impractical. In Section 4.2 we present an alternate method to compute the electric field, which, though not strictly self-consistent, may be adequate for our purposes.