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Evaluation of ion bombardment profile on cylindrical cathodes

An important result in PIC-MC plasma simulations of sputter cathodes is the ion current absorption profile on the target, because this information can be subsequently used for determining the sputter erosion profile. With planar targets the extraction of the absorption data in a cut plane on the target surface is rather simple. However things become more complicated in case of curved substrates as e.g. in case of cylindrical sputter targets:

Ar+ absorption plot in whole model Ar+ absorption cells at cylindrical target

The absorption plot comprises the absorbed particle flux per second and per unit area and is stored as 3D volume plot within the cartesian volume grid structure. It can have only non-zero values in cells which intersect with meshed walls. For area normalization, the intersecting cell surfaces are used. These are always computed together with the effective cut-size cell volumina during initialization of a simulation run and stored in the file simulationcase/cell_surfaces.pos.

The right graph in the figure above shows an enlarged section of the absorption profile close to the target surface thereby illustrating the main issue in obtaining accumulated absorption profiles on curved substrates: The cartesian volume grid structure does not fit well to the cylindrical shape of the surface. Therefore, the cells intersecting with the cylinder form a step-like structure, and partially multiple intersecting cells are aligned on top of each other.

The solution is to consider the product S A between cell surface S and normalized absorption A and to take the following average along vertical direction through all cells intersecting with the cylindrical surface:

\large $<A> = \frac{\sum\limits_{i\in \rm{cells}} S_iA_i}{\sum\limits_{i \in \rm{cells}} S_i}$

This way, the total absorption cumulated over all intersecting cells in vertical direction is divided by the total intersecting cell surface, which is the proper way of averaging the absorption profile on curved surfaces. With the RIG VM postprocessing tools this can be accomplished within following script.

compiletime load "";
case = BATCH.file;
PPP obj = {
  # Create forward reference for the maximal 
  # absorption at left and right side from center line:
  float AMAX : 0.5*(A_left_max + A_right_max + abs(A_left_max - A_right_max));
T0 = 15;     # Start time for time averaging
T1 = 20;     # End time for time averaging
DT = 2.5;    # Plot interval
radius = 75; # Target radius
tilt   = 10; # Magnet tilt angle
# Compute resulting center coordinate of absorption profile
pi = 4*atan(1);
Y_center = 92.5 - radius*sin(tilt*pi/180);
# Make center coordinate also visible in obj 
obj.Y_center = Y_center;
# Read averaged absorption and cell surfaces
for(time=T0; time<=T1+1e-6; time=time+2.5) {"$case/absorption/absorption_Arplus_$(time%,2f)us.pos", "A");
}"$case/cell_surfaces.pos", "S_All");
# Consider only surfaces, where Arplus absorption is non-zero
obj.create_scalar("S", S_All*A>0 ? S_All ! 0);
# Create product A*S
obj.create_scalar("AS_product", S*A);
# Make horizontal cutplane at Target T2 averaging across upper half surface
obj.plane(-298, 0, 205,   298, 0, 205,    -298, 170, 205,   40, 0, 0, "average");
# Create averaged absorption sum A*S / sum S (if sum S > 0)
obj.create_scalar("A_avg", S>0 ? AS_product/S ! 0);
# Save plane view with projected absorption for illustration
                            "Averaged Arplus absorption %", 
# Make cumulated line profiles
# Create scalars for distinguishing absorption on left and right side from center
obj.create_scalar("A_left",  y<Y_center ? A_avg ! 0);
obj.create_scalar("A_right", y>Y_center ? A_avg ! 0);
# Convert plane profile into accumulated linescan
obj.xz_plane(Y_center, radius, "sum");
# Write relative line profiles into TXT file
             ["x", "A_left_%", "A_right_%"], 
             [x, 100*A_left/AMAX, 100*A_right/AMAX]);

The pictures below show the accumulated Ar+ flux distribution projected onto the 2D cutplane (left graph) as well as the cumulated line profiles of the racetracks on both sides (right graph).

Ar+ current density projected onto plane Cumulated line profiles of both race tracks

Projection of quantities onto cylindrical surfaces

Example 1: Magnetic field

For a cylindrical surface and a given magnetic field, it is often interesting to plot the horizontal and vertical magnetic field component. This can be obtained via the cylindric cutplane function embedded in the RIG-VM postprocessing toolkit. In the following example, a cylindrical target has the following dimensions:

  • Length: 520 mm from X=115 … 625
  • Center: Y = 167.5, Z = 247
  • Radius: 70 mm

A postprocessing script for extracting the magnetic field on the cylindrical target surface is given as follows.

compiletime load "";
case = BATCH.file;
X0 = 115;   X1 = 635;    # Cylinder elongation in X direction
Y0 = 167.5; Z0 = 247;    # Center Coordinates
R0 = 70;                 # Cylinder radius
Delta = 5;               # Search radius
NL   = 60;               # Point resolution in longitudinal direction
NPHI = 75;               # Point resolution in angular direction
PPP obj = { };"$case/field/B_0.00us.pos", "B");
obj.cylinder(X0, Y0, Z0,   X1, Y0, Z0,    R0, Delta,  NL, NPHI, "interpolate");
obj.write_pos_scalar_binary("extract/$case-B-radial.pos", "B radial [mT]", B_r*1000);
obj.write_pos_scalar_binary("extract/$case-B-parallel.pos", "B parallel [mT]", sqrt(B_x^2 + B_phi^2)*1000);

The resulting POS file can be visualized in GMSH as follows:

The point resolutions NL, NPHI in longitudinal and angular direction shall be in the same order as the resolution of the simulation output data. There may be some trials with different settings required for getting a proper visualization.

Example 2: Ion absorption profile

While some quantities such as the magnetic field exist continuously in the 3D space, other quantities such as the “absorption” exists only at the boundary between open volume and walls. For the absorption data stored in cell resolution this means, that only cells which intersect with a meshed wall can have an absorption > 0.

Therefore, taking a simple average of the absorption across several cells in the vicinity of the surface is not useful, because some of the cells will contribute zero absorption.

A workaround is to additionally load the cell surfaces S and take an average of the absorption A weighted by the cell surface

\large $<A> = \frac{\sum\limits_{cells}A\times S}{\sum\limits_{cells} S}$

This can be applied in a postprocessing script as shown below.

compiletime load "";
case = BATCH.file;
X0 = 115;   X1 = 635;    # Cylinder elongation in X direction
Y0 = 167.5; Z0 = 247;    # Center Coordinates
R0 = 70;                 # Cylinder radius
Delta = 5;               # Search radius
NL   = 60;               # Point resolution in longitudinal direction
NPHI = 75;               # Point resolution in angular direction
PPP obj = { };"$case/absorption/absorption_Arplus_10.00us.pos", "A");"$case/cell_surfaces.pos", "S");
obj.create_scalar("AxS", A*S);
obj.cylinder(X0, Y0, Z0,   X1, Y0, Z0,    R0, Delta,  NL, NPHI, "average");
                            "Ar+ absorption [1/m^2 s]", 
                            (S>0) ? AxS / S  ! 0);