Velocity Distribution of Ions Incident on a Wafer in Two
Frequency Capacitively-Coupled Plasmas
G. Wakayama∗ and K. Nanbu∗
∗
Institute of Fluid Science, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai, Japan 980-8577
Abstract. The dynamic structure of two-frequency capacitively-coupled plasmas (2f-CCP) were examined using the selfconsistent Particle-in-Cell/Monte Carlo (PIC/MC) simulation. At first the dependence of the discharge structure on wafer
biasing conditions were investigated using one-dimensional (1D) computation. The results show that the plasma potential
oscillates with both frequencies. The amplitude of the high frequency oscillation is modulated by the instantaneous potential
of the low frequency biasing electrode. Furthermore, the axisymmetrical two-dimensional (2D) PIC/MC simulation was
performed to investigate the influence of a geometric configuration of the reactor on the plasma structure.
INTRODUCTION
A modern plasma etching demands the fast processing, wide plasma uniformity, damage free process etc. In order to
satisfy such requirements, plasma should be well designed and highly controllable. Inductively coupled plasma (ICP)
and surface wave plasma (SWP) have been supposed to be effective as low pressure and high plasma density sources.
However, capacitively coupled plasmas (CCPs) have been investigated most often among various plasma sources and
recent works on CCPs have revealed a new possibility for this conventional plasma source.
Colgan and Meyyappan [1] summarized many kinds of scaling laws and pointed out that very high frequency (VHF)
capacitive plasmas hold a great promise. The VHF operation realizes not only a high plasma density but also a low
electron temperature. The latter is effective in suppressing charging damages.
In reactive ion etching (RIE), etching of a wafer by radicals is enhanced by ion bombardment. When ions accelerated
in the sheath strikes the wafer perpendicularly, an anisotropic etch profile is obtained. For this purpose a low frequency
(LF) wafer biasing is also necessary to control the velocity distribution of ions incident on the wafer. Kitajima et al. [2]
reported that a functional separation of plasma sustaining and ion bombarding is achieved by use of the VHF operation
together with LF wafer biasing. An etching apparatus using 2f-CCP has been developed in industries.
In this study we first examined the dependence of the structure of 2f-CCP on biasing conditions using onedimensional (1D) PIC/MC [3] simulation. We found the plasma potential has a non-linear waveform. Next, the
axisymmetrical two-dimensional (2D) PIC/MC simulation was performed to investigate the influence of a geometric
configuration of the reactor on the plasma structure.
ONE-DIMENSIONAL ANALYSIS
Method
The 1D computational domain is shown in Fig. 1, where L represents the distance between the electrode biased
by low frequency (ELF) on the left and the electrode biased by high frequency (EHF) on the right. This domain is
discretized by equally spaced grids smaller than the Debye length. Tracing e − and Ar+ , we obtain the charge density
at the grid point. Using the Dirichlet boundary condition at z = 0 and L, we solve the Poisson equation and obtain
the electric fields at the grid point. The force on a charged particle is given by the interpolation of the electric fields
at adjacent grid points. When the time step is small enough, a motion of a charged particle and its collision can
be calculated separately [4]. A collisional event in e− –Ar collision is determined using Nanbu method [5]. These
procedure is repeated until the periodic steady state is obtained.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
1079
EHF
LF
HF
O
FIGURE 1.
Cross Section σk (10−20 m2)
ELF
z
L
PSfrag replacements
10
2
10
0
10
−2
10
−4
10
elastic
ionization
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−2
0
10
10
2
4
10
Electron Energy ε (eV)
Schematic of 1D computational model.
FIGURE 2. Cross sections for e− –Ar collision.
The discharge gas is argon, which is assumed to be spatially uniform and in equilibrium at the temperature of 300
K. According to this assumption, we have only to examine the motion of e − and Ar+ in the gas of Ar. The velocity
of Ar atom is necessary only when determining the post-collisional velocity of e − or Ar+ . As for e− –Ar collisions,
elastic collision, ionizing collision, and also 25 exciting collisions are taken into consideration based on a set of the
cross section data of Kosaki and Hayashi [6] shown in Fig. 2. Elastic collision and resonant charge-exchange are
considered for Ar+ –Ar collision; the simple and computationally efficient model proposed in our previous work [7],
the use of which was shown to reproduce the measured data of ion drift velocity and transverse diffusion coefficient,
is employed.
Results
The computational conditions are: L = 20 mm, gas pressure p = 25 mTorr, high biasing frequency f H = 60 MHz,
amplitude of HF Vrf,H = 150 V, low biasing frequency fL = 2 MHz, and amplitude of LF Vrf,L = 100 V, 300 V,
and 500 V. The electric potentials on both electrodes are φ(0) = V dc,L + Vrf,L cos ωL t, φ(L) = Vdc,H + Vrf,H cosωH t,
respectively, where t is time, ω = 2πf , and Vdc is the dc self-bias voltage. In this study Vdc,H is chosen to be 0 V as
the reference potential of the system.
In order to discuss the plasma structure changing in a cycle of low or high frequency, let us introduce ϕ L = ωL t and
ϕH = ωH t as a phase for each frequency. Figure 3 shows the dependence of a spatial distribution of electric potential
on the LF phase. The interval of the sampled phase is not exactly π/2, since ϕ H = π/2 is fixed. Thus the sampled
phases are ϕL = π/60(' 0), 29π/60(' π/2), 61π/60(' π), and 89π/60(' 3π/2).
In this study we adjusted the self-bias voltage so as to satisfy the condition that the net current flowing to the
electrode vanishes in one cycle of LF. This model is equivalent to the case including the external circuit when a
blocking capacitor is large enough. In the case of the computational model with a general external circuit, the model
suggested by Verboncoeur et al. [8] should be employed. We obtained the self-bias voltage of −0.2 V for ELF. The
voltage is close to zero since the geometric symmetry leads to the plasma structure symmetry in 1D simulation.
Figure 4 shows the modulation of the electron density near ELF in one cycle of LF. At ϕ L ' 0 the LF bias voltage
is high as shown in Fig. 3. The potential drop between the ELF and the plasma bulk becomes small. Therefore the
electrons come nearest to the ELF. On the contrary, at ϕL ' π the electrons are repelled by the large potential drop.
Thus the electron density is zero in the sheath region.
Next the changes of electron density distributions in one cycle of HF are shown in Figs. 5 and 6. The LF phase ϕ L
is around 0 for Fig. 5 and π for Fig. 6. The results that the electron density distributions are modulated by HF bias are
observed. The modulated region is from 0 mm to 3.6 mm for Fig. 5 and from 4.4 mm to 5.6 mm for Fig. 6. On the other
hand, in Fig. 4, which shows the effect of the LF bias on the electron density distribution, the modulated region is from
0.8 mm to 5.2 mm. The discrepancy of the width of the modulated region between Fig. 4 and Fig. 5 or Fig. 4 and Fig. 6
indicates the difference of the contribution to the sheath structure. In the case that V rf,H = 150 V and Vrf,L = 500 V,
the LF bias has a greater contribution to the sheath modulation than the HF bias.
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FIGURE 3. Dependence of spatial distribution of electric
potential on the LF phase (ϕH = π/2).
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FIGURE 5. Dependence of electron density distribution near
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FIGURE 4. Dependence of electron density distribution near
the ELF on the LF phase (ϕH = π/2).
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FIGURE 6. Dependence of electron density distribution near
the ELF on the HF phase (ϕL ' π).
The effects of two-frequency biasing are clearly observed in Fig. 7. This result shows the time variation of the
plasma potential Vp in one cycle of LF for Vrf,L = 100 V, 300 V, and 500 V, where the plasma potential Vp is the
potential at the middle point of two electrodes (z = L/2). The plasma potential is modulated by not only LF but also
HF. It is found that the amplitude of the HF oscillation depends on a phase of LF. In 0 ≤ ϕ L ≤ π/2 or 3π/2 ≤ ϕL ≤ 2π
the amplitude of the HF oscillation is smaller than that in the other phase for all cases. This result is explained as
follows. In no magnetic field and electropositive plasma the plasma potential is higher than the electrode potential.
At ϕL ' π the plasma potential is smaller than the amplitude 150 V of HF in Fig. 3. Therefore the potential of EHF
directly affects the plasma potential in the plasma bulk.
Figure 8 shows the time variation of the sheath voltage Vsh in one cycle of LF where Vsh is the potential drop from
the plasma bulk to the ELF. Hence Vsh = VLF − Vp where VLF = φ(0) and Vp = φ(L/2). The ions are accelerated to
the electrodes by the sheath voltage and strikes the electrodes with the widely differing energies mainly determined by
collisions and the value of ϕL or ϕH .
The ion energy distributions (IEDs) at LF and HF electrodes are shown in Fig. 9. These two IEDs show a good
agreement since the plasma structure is almost symmetric. Generally ions are too heavy to follow the instantaneous
change of the electric field when the bias frequency is high enough such as 60 MHz. Therefore the IED can be predicted
by the Vsh excluding the HF bias component. We see from the IEDs that the number of ions with small energy is large
and that the maximum energy of ions is almost 400 eV, which is less than V rf,L (= 500 V). The reason is that the sheath
1081
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25 mTorr
−400
LF 2 MHz
−600
100 V
300 V
500 V
HF 60 MHz
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FIGURE 7. Time variation of plasma potential φ(L/2) in
bulk.
PSfrag replacements
0
LF 2 MHz
2π
FIGURE 8. Time variation of sheath potential (potential
drop from bulk plasma to ELF).
25 mTorr
HF 60 MHz
150 V
0.008
LF 2 MHz
500 V
0.004
ELF (z = 0)
EHF (z = L)
0
0
200
400
Ion Energy ( eV )
FIGURE 9. Ion energy distribution at ELF and EHF.
width is so thick that ions lose energy by Ar+ –Ar collisions in the sheath region.
Figure 10 shows the time-averaged ion density distributions for V rf,L = 100 V, 300 V, and 500 V. The maximum
value of the ion density is at the middle point of two electrodes for all cases and is close to each other. However the
sheath width decreases as the amplitude of LF increases.
Figure 11 shows the velocity distributions of ions incident on the ELF, where the velocity is the component normal
to the electrode. Since the range of variation of |Vsh | becomes wider as Vrf,L increases, the velocity distributions
becomes broader for a larger amplitude of LF. Another feature of ion velocity distribution is smooth distributions for
Vrf,L = 300 V and 500 V. The large amplitude of LF leads to a thicker sheath width as shown in Fig. 10 and hence,
more collisions in the sheath region. Thus the sawtoothed distribution for V rf,L = 100 V is smoothened at Vrf,L = 300 V
or 500 V.
AXISYMMETRICAL TWO-DIMENSIONAL ANALYSIS
Method
The 2D computational domain is shown in Fig. 12. The radius of chamber is 150 mm, the radius of EHF is 110 mm,
the radius of ELF is 100 mm, and the distance between two electrodes is 20 mm as in case of 1D analysis. The origin
1082
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150 V
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100 V
300 V
500 V
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HF 60 MHz
150 V
0.05
0
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FIGURE 10. Dependence of time-averaged electron density
distribution on the amplitude of LF.
LF 2 MHz
100 V
300 V
500 V
−40
−20
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0
FIGURE 11. Dependence of ion velocity distribution at ELF
on the amplitude of LF.
is at the center of the ELF.
The Poisson equation in the cylindrical coordinate system is
∂ 2 φ 1 ∂φ ∂ 2 φ
ρ
+
+
=− ,
∂r2 r ∂r ∂z 2
0
(1)
where φ is the electric potential, ρ is the charge density, and 0 is the dielectric constant of vacuum.
Using the finite volume method (FVM) we discretize eq. (1). To solve eq. (1) we impose the Dirichlet condition for
all boundaries. The sidewall is assumed to be grounded (φ = 0). On the two electrodes the potential is the same as the
corresponding bias voltage. On the dielectric the potential is assumed to be
φ(r) = φ(R0 ) +
ln(r/R0 )
[φ(R1 ) − φ(R0 )] ,
ln(R1 /R0 )
(2)
where R0 is the radius of upper or lower electrode and R1 is the radius of chamber. This equation is derived based on
the assumption that the potential between R0 and R1 is equal to that between infinitely long concentric cylinders with
radii R0 and R1 filled with vacuum.
Collisions and self-bias voltages are treated as were done in the 1D calculations.
Results
The computational conditions are the same as the 1D model, that is, p = 25 mTorr, V rf,H = 150 V, and so on. Now
we show the results for Vrf,L = 500 V as a typical case of the axisymmetrical 2D simulation.
Figure 13 shows the spatial distribution of the time-averaged ion density. Excepting the region near the sidewall, the
ion density at constant r is analogous to that in Fig. 10. The ion density takes a maximum near the center of the two
electrodes. The maximum ion density decreases in the radial direction. Figure 14 shows the spatial distribution of the
time-averaged potential. Unlike the 1D simulation, the self-bias voltages are not zero since the electrode has a finite
radius, in other words, the plasma structure is asymmetric in 2D model. The self-bias voltages are −328 V and −4.8 V
for ELF and EHF, respectively.
Figure 15 shows the spatial distribution of the time-averaged electric field in the radial direction. The radial
component of the electric field Er has a negative value near the dielectric since the self-bias voltage is negative.
On the other hand, Er is positive near the sidewall due to ion sheath formation. At all events the field strength of |E r |
is smaller than that of the axial component |Ez | of the electric field. Figure 16 shows the spatial distribution of the
time-averaged electric field Ez in the axial direction. The axial component of the electric field has a similar structure
to the 1D simulation except for the attenuation near the sidewall. Figure 17 shows the spatial distributions of the ion
1083
Sidewall (Metal)
Dielectric
Electrode of HF (EHF)
HF
z
110
150
20
100
O
r
Electrode of LF (ELF)
LF
Schematic of axisymmetrical 2D computational model.
0
0.02
0.04
0
0.06
r
0.08
-250
0.04
0.06
r
0.08
-500
0.1
0.12
1
FIGURE 14.
Time-averaged ion density.
2
0.14
0
z
0.0
2
1
0
FIGURE 13.
0.0
0.14
0.0
0.12
0.02
0.0
0.1
-2
0
φ
2
0
16
-3
4
Ion Density [ 10 m ]
FIGURE 12.
Dielectric
z
Time-averaged electric potential.
flux to the ELF, EHF, and sidewall. These distributions correspond to the distribution of the ion density in Fig. 13. The
maximum value of the ion flux to the sidewall is about the half of the flux to the electrode. Figure 18 shows the IEDs
for the ions incident on the ELF, EHF, and sidewall. The maximum energy of the incident ions is about 200 eV for the
EHF and sidewall. On the other hand the energy of the ions incident on the ELF is widely distributed up to 850 eV.
Figure 19 shows the dependence of the radial distributions of the ion flux onto the ELF on the electrode gap which
is the distance between two electrodes. The ion flux at the edge of the electrode becomes large as the electrode gap
decreases. Consequently the uniformity of the ion flux is improved. On the other hand the IED is almost unchanged as
shown in Fig. 20. These results indicate the possibility that the spatial distribution of the ion flux can be controlled by
changing the electrode gap.
1084
1
5
0.04
0.06
r
0.08
-5
r
Ion Flux (1020 m−2s−1)
ELF (z = 0)
EHF (z = L)
Sidewall
0
25 mTorr
HF 60 MHz LF 2 MHz
500 V
150 V
0
0.01
PSfrag replacements
0.02
Surface distribution of ion fluxes at ELF, EHF,
z
Axial component of time-averaged electric
2D-Axisymmetrical
0.015
0.01
25 mTorr
HF 60 MHz
150 V
LF 2 MHz
500 V
ELF (z = 0)
EHF (z = L)
Sidewall
0.005
0
z (m)
FIGURE 17.
and sidewall.
0.14
2
0.3
0.1
0.12
1
PSfrag replacements
0.1
2D-Axisymmetrical
0.2
-3
0
FIGURE 16.
field.
r (m)
0.05
0.1
z
Radial component of time-averaged electric
0
0.08
0.0
2
1
0
FIGURE 15.
field.
0.0
0.14
-2
0.06
0.0
0.12
0.04
0.0
0.1
-1
0.02
Ez
0
0
Ion Energy Distribution ( eV−1 )
0.02
Er
0
0
FIGURE 18.
and sidewall.
0
400
800
Ion Energy ( eV )
Ion energy distribution incident on ELF, EHF,
Conclusion
In the case of the 1D model, the dynamic structure of 2f-CCP was clarified using PIC/MC simulation. In the 1D
simulation the self-bias voltage is almost zero for both electrodes since the plasma structure is symmetric.
The electron density is modulated by not only LF but also HF. In the phase when the plasma potential becomes high
due to the potential of the ELF, the amplitude of the HF oscillation becomes small. This trend is striking for a larger
amplitude of LF. As the amplitude of LF increases, the flux of ions incident on the electrode with high energy increase
and the IED becomes wider.
For the 2D model, the effects of the geometric configuration of the reactor on the plasma structure were examined.
Although the radii of the electrode only differs 10 %, a large difference of self-bias voltage between two electrodes
arises. Consequently the IED for the ELF differs noticeably from that for the EHF.
As the electrode gap becomes narrower, the ion flux at the edge of the electrode increases and hence, the ion flux
uniformity on the ELF is improved. The size of the electrode gap has little effect on the IED.
1085
at ELF
LF 2 MHz
0.2
Electrode Gap
15 mm
18 mm
20 mm
22 mm
25 mTorr
HF 60 MHz
150 V
0
0
PSfrag replacements
0.05
0.1
Radial Position r ( m )
FIGURE 19. Surface distribution of ion fluxes at ELF for
several distance between two electrodes.
Ion Energy Distribution ( eV−1 )
Ion Flux (1020 m−2s−1)
PSfrag replacements
0.4
HF 60 MHz LF 2 MHz 25 mTorr
150 V
500 V
0.003
Electrode Gap
15 mm
18 mm
20 mm
22 mm
0.002
0.001
at ELF
0
0
400
800
Ion Energy ( eV )
FIGURE 20. Ion energy distribution incident on ELF for
several distance between two electrodes.
ACKNOWLEDGMENTS
The computations have been carried out using the SGI Origin 2000 and NEC SX-5 at the Institute of Fluid Science,
Tohoku University.
REFERENCES
1. Colgan, M. J., and Meyyappan, M., “Very High Frequency Capacitive Plasma Sources,” in High Density Plasma Sources
Design, Physics and Performance, edited by O. A. Popov, Noyes Publications, Park Ridge, 1995.
2. Kitajima, T., Takeo, Y., Petrovic, Z. L., and Makabe, T., Appl. Phys. Letters, 77, 489 (2000).
3. Birdsall, C. K., and Langdon, B., Plasma Physics via Computer Simulation, McGrawHill, New York, 1985.
4. Nanbu, K., IEEE Trans. Plasma Sci., 28, 971 (2000).
5. Nanbu, K., Jpn. J. Appl. Phys., 33, 4752 (1994).
6. Kosaki, K., and Hayashi, M., Denkigakkai zenkokutaikai yokoshu, Tech. rep., Inst. Electr. Eng. Jpn., Tokyo (1992), [in
Japanese].
7. Nanbu, K., and Wakayama, G., Jpn. J. Appl. Phys., 38, 6097 (1999).
8. Verboncoeur, J. P., Alves, M. V., Vahedi, V., and Birdsall, C. K., J. Comput. Phys., 104, 321 (1993).
1086