118 10. PLASMA MODELING
where n
0
denotes the plasma density at the sheath edge. Finally, the Poisson equation must be
fulfilled self-consistently
@
2
ˆ
@x
2
D
e
"
0
@
@x
.n
i
n
e
/: (10.18)
With x D 0 at the electrode surface (see Figs. 6.2 and 6.3), a computational grid is extended to a
sufficient depth x
p
in the plasma. e initial conditions are n
e
.x; 0/ D n
i
.x; 0/ D n
0
, where the
presheath is neglected and thereby the factor e
1=2
is omitted. With the voltage V
0
applied
at t D 0, the above system of equations is solved numerically with the boundary conditions
ˆ.0; t / D V
0
and ˆ.x
p
; t / D 0.
Are result of such a calculation is given in Fig. 10.4, which shows the velocity of the sheath
edge, which moves from the electrode into the plasma volume, vs. the position of the sheath
edge. e results relies on experimental information about the plasma density and the electron
temperature. It is in qualitative agreement with the analytical predictions of Section 6.4, with
a high initial sheath edge velocity which tends to zero when approaching the static Langmuir-
Child sheath. e prediction is in good agreement with experimental results obtained from
time-resolved laser-induced fluorescence at varying distance from the electrode.
10.4 PARTICLE-IN-CELL COMPUTER SIMULATION
e most powerful plasma simulations rely on the so-called “particle-in-cell” (PIC) model. e
plasma volume is subdivided by a two-dimensional or three-dimensional grid. All particles in
the plasma are represented by “superparticles” of each species, which represent a large number
of real particles of that species. e superparticles are allowed to move in the entire volume,
whereas the fields which govern their motion are defined on the nodes of the grid. As indicated
in Fig. 10.5, complicated configurations with superimposed electrical and magnetic fields can
be treated in this way.
e procedure of a particle-in-cell simulation is shown schematically in Fig. 10.6. A com-
putational loop is performed for sufficiently small time step t. Starting with the lower-left
edge, the forces on each particles are calculated by interpolating the fields at the grid nodes j
adjacent to the actual position x
i
of the respective particle. Subsequently, the equation of mo-
tion can be solved for each particle. Particles arriving at the walls are treated according to the
boundary conditions (loss or reemission at the walls) with a possible generation of secondary
particles (secondary electron emission, wall sputtering, surface chemistry products, etc.). Ac-
cording to the individual collision cross sections and the corresponding collision probabilities,
a Monte Carlo algorithm decides according if any particle undergoes a collision with any other
species during the time interval. In case of a collision, the velocities and flight directions of the
collision partners are revised. en, the particle densities of different species can be determined
at each grid node from the particle numbers around that grid node and the respective positions.
Finally, the electrostatic potential and the electric field are obtained from the charged parti-
10.4. PARTICLE-IN-CELL COMPUTER SIMULATION 119
2.5
2
1.5
1
0.5
0
0 1 2 3 4
Sheath Edge Position (cm)
Sheath Edge Velocity (cm/µs)
Figure 10.4: Fluid model simulation of the sheath edge dynamics after applying high-voltage
DC bias, for a nitrogen plasma at a pressure of 6:7 10
2
Pa with a plasma density of 6:25
10
9
cm
3
and an electron temperature of 0.57 eV. e applied bias voltage is 5 kV. e re-
sult (solid line) is compared to experimental data obtained by laser-induced fluorescence. e
upper and lower dotted lines correspond to a variation of the plasma density by ˙ 10%. (From
M.J. Goeckner et al. [12].)
cle densities at the grid nodes, using standard finite difference schemes for the solution of the
Poisson equation, and the definition of the forces is reentered for the subsequent time step.
Results for a DC magnetron discharge (for the geometry, see Fig. 10.5) are shown in
Fig. 10.7. Considerable inhomogeneities are obtained. Between the locations of maximum mag-
netic field, the electrostatic potential increases from the cathode potential in axial direction along
a distance of about 1 cm, which is much larger than the width of the sheath. e Ar ion density
shows a pronounced peak at the positions of maximum magnetic field and a diffusional tail. e
sharp decrease closely behind the cathode results from a strong influence of collisions at the
relatively high pressure. Neutral Cu atoms are generated by sputtering at the cathode. ey are
partly ionized in the region of high plasma density and form an ion distribution which is similar
to the Ar ion distribution, at, however, a significantly lower concentration.
120 10. PLASMA MODELING
+
–
z
r
Figure 10.5: Particle-in-cell grid (white lines) of an axisymmetric plasme volume with a su-
perimposed magnetic field (directional lines, color and diameter of points). r and z denote the
cylindrical coordinates., the grey areas indicate the electrodes of a cylindrical magnetron con-
figuration (see Section 11.2).
10.4. PARTICLE-IN-CELL COMPUTER SIMULATION 121
Integration of
equation of motion,
moving particles
F
i
→ V
i
' → X
i
Particle loss/gain at
the boundaries
(emission, absorption)
Weighting
(E,B)
j
→ F
i
Weighting
(x,v)
i
→ (ρ)
j
MC
Collision
?
Postcollision
Velocities
V
i
' → V
i
Yes
No
Integration of
Poisson’s equations
(ρ)
j
→ (E)
j
Δt
Figure 10.6: Computational scheme of a particle-in-cell plasma simulation. e index i D
1 : : : N
p
denotes the individual superparticles, where the plasma is simulated by in total N
p
su-
perparticles of different species. j D 1 : : : N
g
denotes the nodes of the discrete grid, with a total
number N
g
. x; , and denote the superparticle position, velocity, and density, respectively, F
the force acting on the superparticles, and E and B the electric and magnetic fields, respectively,
at the grid nodes.
122 10. PLASMA MODELING
0
-100
-200
-300
6E+017
4E+017
2E+017
0
1E+016
5E+015
0
20
15
10
-30
-20
-10
0
10
20
30
5
0
20
15
10
-30
-20
-10
0
10
20
30
5
0
20
15
10
-30
-20
-10
0
10
20
30
5
0
z(mm)
r(mm)
z(mm)
r(mm)
z(mm)
r(mm)
Potential (V)
Ar
+
density (m
-3
)Cu
+
density (m
-3
)
Figure 10.7: Electrostatic potential (top), Ar ion density (middle) and Cu ion density (bottom) in
an Ar DC magnetron discharge (see Section 11.2), as obtained from a PIC computer simulation.
e electrode configuration is shown in Fig. 10.5. z D 0 denotes the position of the cathode, the
center position of the anode is at z D 25 mm. e maximum of the magnetic field is 0:12T at the
position .r; z/ D (18 mm, 0). e discharge pressure is 1.3 Pa, the discharge current 300 mA.
In the top figure, the white line denotes the anode potential, V D 0. (From I. Kolev and A.
Bogaerts [13].)
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