Numerical Analysis of RF Magnetron Discharges of
Oxygen/Argon Mixture
S. Yonemura∗ and K. Nanbu∗
∗
Institute of Fluid Science, Tohoku University, Sendai, Japan 980-8577
Abstract. The characteristics of rf planar magnetron discharges of O 2 /Ar mixture are clarified using the Particle-inCell/Monte Carlo (PIC/MC) method. The simulation is carried out for axisymmetrical magnetic fields. The spatial and temporal behavior of magnetron discharge is examined in detail. The spatial distribution of positive ion density shows a double
peak structure. The position of one peak corresponds to the peak point of electron density, while the other peak is located at
the peak point of negative ion density. The mechanism of this phenomenon is clearly explained.
INTRODUCTION
Magnetron discharges are commonly used for sputter deposition of metallic and insulating films. Much experimental
work has been reported [1, 2, 3, 4, 5, 6]. Also, considerable efforts have been made for modeling magnetron discharges
on the basis of the fluid model [7, 8], the particle tracing [9, 10], a hybrid model [11] and the self-consistent particle
model [12, 13, 14, 15]. Since gas pressure in magnetron sputtering is less than 10 mTorr, the particle approach,
i.e., the particle-in-cell/Monte Carlo (PIC/MC) method [16] is the most appropriate to examine the structure of this
discharge. However, particle modeling of magnetron discharge is really a challenging problem because of its intensive
computation. Therefore, no work was published until quite recently. Kondo and Nanbu [17, 18] developed a fast
Poisson solver for axisymmetrical electric field and performed the PIC/MC simulation of dc [17] and rf [18] planar
magnetron discharge. Although they treated the magnetron discharge of a pure argon gas, the discharge characteristics
are expected to change greatly for other gases. Especially, since oxygen is an electronegative gas, the negative ions may
largely change the discharge structure. For example, in the case when negative ions are majority of negative charges
in the plasma, a sheath structure may greatly change. This is because the sheath structure is caused by the difference
of mass between electron and positive ion, and the mass of negative ions is the same order with that of positive ions.
Using Kondo and Nanbu’s Fourier-sine-transform solver, Nanbu et al. [19] performed the PIC/MC simulation of a dc
planar magnetron discharge of O2 /Ar mixture, which is often employed for reactive sputtering in fabricating oxide
films. The rf planar magnetron discharge must be quite different from dc planar magnetron discharge. Therefore, we
examine the characteristics of a rf planar magnetron discharge of the O2 /Ar mixture in the present paper.
OUTLINE OF SIMULATION PROCEDURE
The computational domain is shown in Fig. 1. The two concentric magnets are on the back of the target. The magnetic
field is analyzed by use of the finite element method. One time step of the PIC/MC simulation consists of the particle
pushing and electric field computing. In the former stage, motion and collision of all particles are calculated. In the
latter stage, we first obtain charge density using the positions of all simulated particles at the end of the first stage, and
then we solve Poisson’s equation for the electric field.
The E-field is given by E = −∇Φ. The potential Φ can be divided into two, say, φ 1 and φ , where φ1 is the potential
due to the potential VT applied to the target and φ is that due to the charge density ρ in the discharge space. We have
φ1 = (Vdc −Vrf cos ω t)(1 − z/D), where Vdc is a dc self-bias voltage on the target, Vrf is a voltage amplitude, ω (= 2π f )
is the angular frequency of applied rf voltage, the frequency f is set at 13.56 MHz, D(=20mm) is the distance between
the electrodes. If there is a blocking capacitor in a matching network or the target material is an insulator, no dc
current flows in the external circuit. Then a dc self-bias voltage Vdc appears on the target. The self-bias voltage Vdc
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
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6 1
σ (m2)
U
6
10
–18
10
–19
10
–20
10
–21
10
–22
10
–23
10
–24
10
–25
Elastic
Ionization
Electronic excitation
10
–2
10
–1
10
0
10
1
10
2
10
3
10
4
ε (eV)
FIGURE 1. Schematic of a sputtering apparatus.
FIGURE 2.
e−Ar cross sections.
is determined in such a way that the net dc current per one rf cycle becomes zero. The potential φ is governed by
Poisson’s equation
∂ 2φ 1 ∂ φ ∂ 2φ
ρ
+
(1)
+ 2 =− ,
∂ r2 r ∂ r
∂z
ε0
where ε0 is the permittivity of free space. The boundary conditions are φ (r, 0) = φ (r, D) = φ (R, z) = 0 and
(∂ φ /∂ r)r=0 = 0, where R(=90mm) is the radius of the computational domain. The detailed treatment to solve
Poisson’s equation by using the Fast Fourier-sine transform is given in Refs. 17, 18, and 19.
The leap-frog scheme is used to integrate the equation of motion
m
dv
= q(E + v × B)
dt
(2)
+
where m is the mass, q is the charge and v is the velocity. The charged species considered are electron, O+
2, O ,
+
−
n
n
n
n−1/2
O and Ar . Let (x , v ) be the position and velocity at the time point
) we have
n∆t. For a given data (x , v
(xn+1 , vn+1/2 ). Whether a collision occurs or not is judged at n + 12 ∆t. If it occurs, vn+1/2 is replaced by the postcollision velocity v n+1/2
. The Runge-Kutta-Gill scheme is used to determine x n+1 from (xn+1/2 , vn+1/2
). The data for
∗
∗
the next leap-frog step are (x n+1 , vn+1/2
).
∗
The e−Ar collision is treated using the model developed by Surendra et al. [20]. The ionizing collision, elastic
collision, and 25 exciting collisions are taken into consideration on the basis of Kosaki and Hayashi’s [21] cross
section data shown in Fig. 2.
The e−O2 collision is treated using the cross section data of Itikawa et al. [22]. Figure 3 shows a set of the
cross sections. The vibrational excitation consists of 6 nonresonant and 13 resonant collisions. Seven types of
electronic excitation are considered. The direct ionization is e + O 2 → 2e + O+
2 , the dissociative ionization is e + O 2 →
2e + O+ + O, the dissociative attachment is e + O2 → O− + O, and the ion pair formation is e + O 2 → e + O+ + O− .
The cross sections for the last two reactions, by which O − is produced, are small.
The numbers of e−Ar and e−O2 collisional events are 27 and 31, respectively. The total number of collisional
events is 58. The probability that the kth event occurs in ∆t e is
Pe (k) = ng σk (ε )
2ε
m
1/2
∆te
(3)
where ng is the number density of Ar for e−Ar collisions or the number density of O 2 for e−O2 collisions, σ k is the
integrated cross section for the kth event, ε (= mv2 /2) is the electron energy, and m is the mass of electron. Sampling
one out of 58 events can easily be done using Nanbu’s method [23]. For e−O 2 collision isotropic scattering is assumed
866
–18
10
–19
10
–20
10
–21
10
–22
10
–23
10
–24
10
–25
20
z (mm)
10
Elastic
Direct ionization
Dissociative ionization
Electronic excitation
Vibrational excitation
Dissociative attachment
Ion pair formation
σ (m2)
10
10
0
0
10
20
30
40
50
60
70
80
90
r (mm)
FIGURE 4. Magnetic flux density.
–2
10
–1
10
0
10
1
10
2
10
3
10
4
ε (eV)
FIGURE 3. e−O2 cross sections.
TABLE 1. Ion − neutral-species collisions.
No.
Collisional event
Type of reaction
1
2
3
4
5
6
7
8
9
10
11
Elastic
Charge exchange
Elastic
Elastic
Charge exchange
Elastic
Elastic
Charge exchange
Elastic
Elastic
Elastic
Ar+ + Ar → Ar+ + Ar
Ar + + Ar → Ar + Ar+
Ar+ + O2 → Ar+ + O2
+
O+
2 + O2 → O2 + O2
+
O+
+
O
→
O
2
2 + O2
2
+ + Ar
O+
+
Ar
→
O
2
2
O+ + O2 → O+ + O2
O + + O2 → O + O+
2
O+ + Ar → O+ + Ar
O− + O2 → O− + O2
O− + Ar → O− + Ar
for all collisional events. The post-collision velocities of electron and ion are determined by use of the conservation
equations for momentum and energy.
+
−
+
Ions in the discharge space are O+
2 , O , O , and Ar . The ion−neutral-species collisions considered are in Table 1.
The collision probability and the post-collision velocity of ion are determined using Nanbu and Kitatani’s model [24].
The potential U(r) is assumed to be that for induced dipole, i.e., U(r) = −a/r 4 , where the constant a is determined
from the polarizability of neutral species. Then the collision probability of ion is given by
Pi =
8a
µ
1/2
πβ∞2 ng ∆ti
(4)
where µ is the reduced mass, ng is the number density of ion’s collision partner, ∆t i is the ion time step and β ∞ (= 9)
is the cutoff of the dimensionless impact parameter β . Equation (4) is the sum of the probabilities for elastic collision
and charge exchange collision. If the charge exchange does not occur as in numbers 3, 6, 9, 10, 11 in Table 1, Eq. (4)
is the probability for elastic collision. For number 2 in Table 1 the charge-exchange higher limit β ex of the parameter
β is Aεr1/4, where A = 2.6 and εr (= µ g2 /2) is the kinetic energy of relative motion in units of eV and g is the relative
velocity between a colliding pair. For number 5, the value of A is set 1.9. This value is determined in such a way that
drift velocities of O +
2 resulting from Nanbu and Kitatani’s model agree with the measured data [25, 26]. The treatment
of number 8 in Table 1 is given in Ref. 19.
867
TABLE 2. Recombinations and electron detachment.
No. Type of reaction
Rate constant
Cross section
O− + O+
2 → 3O
O− + O+
→ O + O2
2
O− + O+ → 2O
e + O+
2 → 2O
e + O− → 2e + O
1
2
3
4
5
kc g−1
kc (π kTc /2µ ) 1/2 g−2
kc (π kTc /2µ ) 1/2 g−2
(k c /2)(m/2e) 1/2 εc ε −3/2
kc (m/2e) 1/2 (ε − εc )1/2 /ε
kc
kc(Tc /T )1/2
kc(Tc /T )1/2
kc(εc /Te )
kc exp(−εc /Te)
ne(m−3)
20
4e+15
3e+15
2e+15
1e+15
15
z(mm)
15
10
20
4e+15
3e+15
2e+15
1e+15
10
5
5
0
0
10
20
30
40
50
60
70
80
0
90
0
10
20
30
40
50
60
r(mm)
r(mm)
(a) Ar/O2=100/0
(b) Ar/O2=90/10
FIGURE 5.
z(mm)
ne(m−3)
70
80
90
Electron density. (a) Ar/O2 =100/0. (b) Ar/O2 =90/10.
We also consider the reactions in Table 2 [27]. The first four are recombination and the last is electron detachment.
These reactions influence the number densities of charged particles. The values of kc , Tc and εc are in Ref. 27. For
numbers 4 and 5, Te is the electron temperature, m is the mass of electron, e (J/eV) is the charge of electron, and ε is
the energy of electron. The units of Te , εc and ε are eV. The cross section for number 5 has the threshold energy εc .
Each of reactions can be treated as was done for the eighth collision in Table 1. See Ref 19 again.
RESULTS AND DISCUSSION
The time step ∆te for electron motion is chosen as f −1 /1800 = 4.10 × 10 −11 s. The ion time step is chosen to be
∆ti = 10∆te . The total gas pressure pg and gas temperature Tg are 5 mTorr and 323 K, respectively. The voltage
amplitude is Vrf =200 V. The magnetization M in the magnets is 0.25T. The magnetic flux density is shown in Fig. 4.
+
The secondary electron emission coefficient γ is chosen at 0.12 for all positive ions Ar + , O+
2 , and O considered here.
We performed PIC/MC simulation for the cases of pure argon gas and Ar/O 2 mixture gas with partial pressure ratio of
90/10.
Figure 5 shows the time-averaged electron number density for two cases. Since the field is axisymmetrical and the
electron densities show contours like them of a mountain in the r − z plane, plasmas have the shape of a torus. The
contour lines for pure argon discharge look like an ellipse, while that of a mixture gas discharge shows a distortion.
The electron density for the mixture discharge is slightly higher than that for the pure argon discharge. In both cases,
most of the electrons are confined in the region between the two magnetic poles where the magnetic field has a large
radial (parallel to the target) component, so that almost all ions are generated by the electron impact in this region.
Figure 6 shows the time-averaged number density of Ar + ion. The argon ion density for the pure argon discharge
shows almost the same distribution as the electron density, while that for the mixture gas discharge is very different
+
from the electron density and shows double peaks. Other positive ions O +
2 and O (not shown) also show similar
868
nAr+(m−3)
20
4e+15
3e+15
2e+15
1e+15
10
z(mm)
15
5
0
0
10
20
30
40
50
60
70
80
90
20
1.5e+16
1.4e+16
1.3e+16
1.2e+16
1.1e+16
1e+16
9e+15
8e+15
7e+15
6e+15
5e+15
4e+15
3e+15
2e+15
1e+15
15
10
z(mm)
nAr+(m−3)
5
0
0
10
20
30
40
50
60
r(mm)
r(mm)
(a) Ar/O2=100/0
(b) Ar/O2=90/10
70
80
90
FIGURE 6. Ar+ ion density. (a) Ar/O2 =100/0. (b) Ar/O2 =90/10.
nO−(m−3)
20
1.4e+16
1.3e+16
1.2e+16
1.1e+16
1e+16
9e+15
8e+15
7e+15
6e+15
5e+15
4e+15
3e+15
2e+15
1e+15
10
z(mm)
15
5
0
0
10
20
30
40
50
60
70
80
90
r(mm)
FIGURE 7. O− ion density. (Ar/O2 =90/10).
+
14
13 −3
double-peak distributions. The maximum densities of O +
2 and O ions are 6×10 , 6×10 m , respectively.
This phenomenon can be explained from the negative ion density. Figure 7 shows the time-averaged number density
of O− ion. We see that the higher (upper) peak of Ar + density is located at the peak location of O − density. Ion
motions are hardly affected by the magnetic field applied here and also cannot follow the oscillating electric field of
13.56 MHz. Therefore, the ion motions are governed by the time-averaged electric fields. Figure 8 shows the timeaveraged potential for two cases. Both distributions show the shape of a plateau. The flat space is much larger for
the mixture gas discharge. The dc self-bias voltages Vdc for the pure argon and mixture gas discharges are 15.1 V
and 12.7 V, respectively. The negative ions concentrate to the plateau of the time-averaged potential. Figure 9 shows
the generation and loss rates of O − ion. O− ions are generated where there are many electrons, and are lost where
time-averaged potential is high. From these, we can say that O− ions are generated, lost, and trapped in the plasma
bulk. Therefore O− ion density in steady state is governed only by a balance of generation and loss of O − ions. The
similar trapping of negative ions in capacitive discharges was reported [28]. The positive ions are distributed in such a
way that the sum of charges of negative ions and electrons are neutralized. Consequently, their distributions have two
869
φ(V)
20
98
90
80
70
60
50
40
30
20
10
10
z(mm)
15
20
98
90
80
70
60
50
40
30
20
10
15
10
5
5
0
0
10
20
30
40
50
60
70
80
0
90
0
10
20
30
40
50
60
r(mm)
r(mm)
(a) Ar/O2=100/0
(b) Ar/O2=90/10
FIGURE 8.
z(mm)
φ(V)
70
80
90
Potential. (a) Ar/O2 =100/0. (b) Ar/O2 =90/10.
RO−,loss(m−3s−1)
20
2e+18
1.5e+18
1e+18
5e+17
10
z(mm)
15
20
4e+18
3.5e+18
3e+18
2.5e+18
2e+18
1.5e+18
1e+18
5e+17
15
10
5
5
0
0
10
20
30
40
50
60
70
80
0
90
0
r(mm)
(a) Generation rate of O
z(mm)
RO−,gen(m−3s−1)
10
20
30
40
50
60
70
80
90
r(mm)
−
(b) Loss rate of O
−
FIGURE 9. Generation and loss rates of O − ion. (Ar/O2 =90/10).
peaks.
Figure 10 shows flux of charged particles to the target and substrate. The peak value of Ar + ion flux to the substrate
is 1/3 of that to the target. In Fig. 10, O − flux is not included because O − ions never reach either electrodes. This
means that negative ion resputtering does not occur in the present case. Negative ion resputtering is the phenomenon
that negative ions, generated in the plasma or emitted from the surface of the target by positive ion bombardment,
are accelerated toward the substrate. In the present case we assume that the target is pure metal. Therefore there is
no emission of negative ion from the target. Since the self-bias voltage of target is positive in this condition, even if
we used the oxide target, negative ion resputtering might not occur. Figure 11 shows the energy distributions of ions
incident on two electrodes. Ions incident on the grounded substrate are more energetic than those on the target due to
the positive self-bias voltage of the target. From these findings, positive ion resputtering of the substrate is problematic
in the present sputtering apparatus.
870
Flux to substrate (m-2s-1)
Flux to target (m-2s-1)
5 101 9
electron
O2 + (x10)
O+ (x100)
A r+
4 101 9
3 101 9
2 101 9
1 101 9
0
0
20
40
60
80
1 101 9
electron
O2 + (x10)
O+ (x100)
A r+
8 101 8
6 101 8
4 101 8
2 101 8
0
0
20
r (mm)
40
60
80
r (mm)
(a) Target
(b) Substrate
FIGURE 10. Flux to electrodes. (a) Target. (b) Substrate. (Ar/O 2 =90/10).
0.1
0.07
0.06
O2 +
O+
A r+
0.06
f(εi) (eV-1)
f(εi) (eV-1)
0.08
0.04
O2 +
O+
A r+
0.05
0.04
0.03
0.02
0.02
0.01
0
0
0
20
40
60
εi (eV)
80
100
120
0
(a) Target
20
40
60
εi (eV)
80
100
120
(b) Substrate
FIGURE 11. Energy distribution of ions incident on electrodes. (a) Target. (b) Substrate. (Ar/O 2 =90/10).
CONCLUSION
We have simulated the radio-frequency magnetron discharges of pure argon and Ar/O 2 (=90/10) mixture by using the
particle-in-cell/Monte Carlo method. We conclude as follows.
1. The argon ion density in Ar/O 2 mixture discharge is much higher than that in pure argon discharge.
2. The positive ion density in Ar/O 2 mixture gas discharge shows a double peak structure. The position of one peak
corresponds to the peak point of electron density, while the other peak is located at the peak point of negative ion
(O− ) density.
3. The O− ions are trapped in the plasma bulk by the time-averaged potential. Since O − ions are generated and lost
only in the plasma bulk, the density distribution of them is governed by a balance of generation and loss of O −
ions.
4. The positive ions are distributed in such a way that the sum of charges of negative ions and electrons are
neutralized. Consequently, their distributions have two peaks.
5. The positive ion flux to the substrate is 1/3 of that to the target. Moreover, positive ions incident on the substrate
have higher energy than those incident on the target. These suggest that the substrate may be sputtered by positive
ions.
871
6. The cause of item 5 is the appearance of a positive self-bias on the target.
ACKNOWLEDGMENTS
The numerical simulations were carried out using the supercomputer SX-5 at the Institute of Fluid Science of Tohoku
University. We would like to express our thanks to Dr. S. Kondo and Mr. K. Mitsui in the graduate course of Tohoku
University for allowing us to refer to their program.
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