Particle Modeling of Plasma and Flow in an Inductively
Coupled Plasma Reactor
M. Shiozawa and K. Nanbu
Institute of Fluid Science, Tohoku University,
Sendai, Japan 980-8577
Abstract. For predicting the etch rate of silicon, a method of simulating flows of Cl and Cb in an inductively
coupled plasma reactor is proposed. First, electromagnetic field is calculated by assuming a spatially uniform
plasma. Next, motion and e - Ck, e - Cl collisions of electrons in the obtained electromagnetic field is
calculated using the Monte Carlo method. Elastic and all inelastic collisions are taken into consideration.
Counting the number of reactive collisions such as e + Ck —> e + 2C1 and e -f Cl2 —> Cl~ -f Cl yields
a spatial distribution of the rate of Cl production. This rate is an input data in the free motion stage of
DSMC.
INTRODUCTION
High-density plasmas have been used in etching processes for fabricating ultra large scale integrated circuits(ULSI). As the feature size of wafer is smaller than 0.25 /xm and the diameter of wafer is larger than
300 mm, the trend of plasma processing is toward low gas density, high electron density(ne > 1017 m~ 3 ), and
low electron temperature(l ~ 2 eV). Inductively coupled plasma (ICP) is one of the most promising plasma
sources because high density plasmas can be produced without an external magnetic field. Although electron
temperature of ICP is high, the cost of the reactor is cheap owing to its simple structure.
ICP reactors for etching are usually operated at gas pressure pg less than 10 mTorr. Since the mean free path
A (mm), which is approximately 50/pg(mTorr), is larger than 5 mm, the flow in the reactor is in the transition
regime in most cases. That is, the flow and plasma should be analyzed using the kinetic approach, i.e., the
particle model. However, the fluid model (continuum model) [1-3] has been used mainly in the analysis of ICP.
The governing equations for the fluid model are the electromagnetic field equations and a set of conservation
equations derived by assuming Maxwellian velocity distributions for electrons, ions, and gas molecules. It is
now well-known that electron energy distribution function (EEDF) is largely deviated from the Maxwellian.
The deviation has a great effect on the rate of reactive collisions, and this rate affects the etch rate of wafer.
Only a few study have been performed on the coupling of the plasma and flow based on the particle model.
Economou et al. [4] used the direct simulation Monte Carlo (DSMC) to examine reactive neutral and ion flow
in ICP. Nanbu et al. [5] used the Monte Carlo method for obtaining the production rate of radicals and then
the axisymmetrical DSMC method for the prediction of etch rate in ICP.
Our group is now doing an experiment on the etching of polycrystalline silicon in a chlorine ICP reactor
[6]. The objective of the present work is to treat the same problem using the particle modeling of plasma and
flow. We try to develop a reliable particle model usable in calculating the etch performance of commercial size
ICP reactors. We examine the spatial structures of the plasma and flow. Etching is governed by many plasma
parameters such as electron temperature and gas pressure. The plasma parameters obtained from measurements
are usually spatially-averaged quantities. It is much more difficult to obtain the spatial distributions of plasma
parameters from experiments, which are necessary for a better understanding of plasma etching. Plasmas have
been analyzed using PIC/MC method [7]. However, we cannot apply the PIC/MC method as it stands to the
plasma in the present ICP reactor because the reactor is too large. The PIC/MC method requires that the
number of simulated particles per cell is larger than 100. Even if we use a supercomputer, it is very difficult to
consider the whole reactor by the PIC/MC method. So we decoupled the analyses of electromagnetic filed and
plasma production in the ICP reactor. Firstly, we calculate the electromagnetic field which is produced by the
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
238
121
GasInjection
Injection
Gas
100
385
Electromagneticfield
fieldcal.
cal.
Electromagnetic
E(r, z), B(r, z)
E(r,z),B(r,z)
Quartz
Quartz
Coil
MonteCarlo
Carloelectron
electron
Monte
collisioncal.
cal.
collision
Diffusion
Chamber
Substrate
DSMCflow
flowcal.
cal.
DSMC
z
250
Metal
Metal
Rdis(r, z) , Rdat(r, z)
r
o
Substrate Holder
Substrate Holder
78
f 78
174
FIGURE 1. Schematic of computational flow. Rdis
FIGURE 1. Schematic of computational flow. R^is
and R
denote the rates for dissociation and dissociaand Rdatdat
denote the rates for dissociation and dissociative attachment.
tive attachment.
FIGURE 2. Inductively-coupled plasma reactor
FIGURE 2. Inductively-coupled plasma reactor
coil current, assuming the uniform plasma. Secondly, the Monte Carlo simulation of electron−molecule and
coil current, assuming the uniform plasma. Secondly, the Monte Carlo simulation of electron—molecule and
electron−atom collisions in the obtained electromagnetic field is carried out. In this stage, a spatial distribution
electron—atom collisions in the obtained electromagnetic field is carried out. In this stage, a spatial distribution
of the production rate of radicals(atoms) is obtained. Lastly, flows of atoms and molecules in the reactor is
of the production rate of radicals(atoms) is obtained. Lastly, flows of atoms and molecules in the reactor is
calculated using DSMC method. The production rate of radicals in the second stage is the input data of the
calculated using DSMC method. The production rate of radicals in the second stage is the input data of the
flow calculation. The schematic of the computational flow is shown in Fig. 1.
flow calculation. The schematic of the computational flow is shown in Fig. 1.
METHOD OF PARTICLE SIMULATION
METHOD
OF PARTICLE SIMULATION
The schematic of the ICP reactor is shown in Fig. 2. The right side is the computational domain. We
The schematic of the ICP reactor is shown in Fig. 2. The right side is the computational domain. We
regard the electromagnetic field, plasma, and flow field as axisymmetrical. The eight-turn coil is wound on the
regard
thecylinder
electromagnetic
plasma,
The eight-turn
is wound
on the
the
quartz
with innerfield,
radius
of 100 and
mm flow
and field
heightasofaxisymmetrical.
385 mm. The quartz
cylinder iscoil
connected
with
quartz
cylinder
with
inner
radius
of
100
mm
and
height
of
385
mm.
The
quartz
cylinder
is
connected
with
diffusion chamber with inner radius of 174 mm and height of 250 mm. The radius of the substrate holder is the
78
diffusion
chamber
withgas
inner
radius ofThe
174 source
mm and
height
of 250 mm. The radius of the substrate holder is 78
mm. The
discharge
is chlorine.
gas(Cl
2 ) is uniformly fed from the shower head at the top of
mm.
discharge
is chlorine.
The source gas(Cl2) is uniformly fed from the shower head at the top of
the The
reactor
and thegas
gases(Cl
2 and Cl) are pumped out from the annular region between the substrate holder
theand
reactor
and
the
gases(Cb
and
Cl)
the wall of the diffusion chamber.are pumped out from the annular region between the substrate holder
and the wall of the diffusion chamber.
A Electromagnetic field
A Electromagnetic field
The ICP is driven by the radio frequency current to the helical coil. Each of coil loops is assumed to be
The ICP isAsdriven
the radio frequency
current
to the
helical coil.
of coil
is electric
assumedfield
to as
be
separated.
a firstby
approximation
we assume
a spatial
uniformity
of theEach
plasma.
Weloops
set the
separated. As a first approximation we assume a spatial uniformity of the plasma. We set the electric field as
(1)
E = Ê ej jωt ,
T7I
J7I „ j
Jb = Jif e
(i)
where Ê(r, z) is the amplitude and ω/2π(=f =13.56 MHz) is the frequency of the coil current. The E -field
where
E(r,z)
the amplitude
and o;/27r(=/—13.56
the frequency
of the coil
current. The E -field
can beisobtained
from Maxwell’s
equation:
has only
the is
azimuthal
component
Eθ (r, z). The Êθ MHz)
has only the azimuthal component
EQ(T,Z).
The
EQ
can
be
obtained
from
Maxwell's
equation:
2
∂
1 ∂
1
∂2
2
2
+
µ
+
ω
−
+
(2)
0
p
d 2\ Êθ = 0,
∂r2 I r d∂r
r2
∂z
(2)
~7T
r or
)1
where µ0 is the permeability of vacuum and p is the dielectric constant of plasma given by
where /^o is the permeability of vacuum and ep is the dielectric constant of plasma given by
239
,Qv
(3)
Here CQ is the permittivity of vacuum, a;pe is the plasma angular frequency for electron, and f m is the momentum
transfer collision frequency between electron and gas molecule. We solve eq. (2) for the boundary condition
Eg = 0 on the metal surfaces. The boundary condition on the inner surface of the quartz cylinder is determined
by an iterative procedure [3] . Let us write the field at the point (r, z) on the quartz surface as
z) = (Ee)c + (Ee)p.
(4)
Here the first term represents the field due to n coils. It is given by
(5)
1=1
where 7C is the coil current, rc is the coil radius, and
G(mi} =
(2 - mi)lC(mi) - 2S(mt)
——
——
4rcr
i = 7—;——\9 , /
m
Here z\ is the axial position of the /th coil, and /C( ) and £( ) are the complete elliptic integrals of the first and
the second kind, respectively. The second term in eq. (4) represents the field due to the azimuthal current in
the plasma. The current in divided into K x L sub-currents at r — rk(k=\, 2, • • •, K) and z = z/(/=l, 2, • • •,
L). We have
(7)
k=l 1=1
where
The amplitude of the sub-current at (r^z/) is
Ip = (TpEe(rk,zi)A
(8)
where crp ~ juep is the conductivity of plasma and A(~ ZirrkArAz) is the cross sectional area of the subcurrent. We see from eq. (8) that eq. (7) depends on the unknown EQ. That is, an iterative procedure is
necessary to determine the boundary condition.
The coil current is required to calculate the electromagnetic field. However, measurement of the rf current is
very difficult in experiments. The rf power input into the coil can be much easily measured. The time-averaged
power deposition per unit volume is
(9)
The total power deposition into the plasma reactor can be obtained by integrating eq. (9). Since the power
deposition depends on the coil current, we chose the coil current Ic in a way that the total power deposition
agrees with the measured rf power.
Once EQ is obtained, the magnetic field components Br and Bz can be easily obtained. The components Br
and Bz are given by
Br = -
(10)
240
B
Motion and collision of electron
Next, we consider the motion and collision of electrons. Ions are not taken into consideration. The background gas is Cl2 and Cl which is assumed be spatially uniform. We solve the equation of motion for electron:
m^ = -e(E + vxB),
(12)
du
where v is the electron velocity, ra is the mass of electron, t is the time, and —e is the charge of electron. Since
E and B are calculated in advance irrespective of electron motion, our simulation is not self-consistent. The
equation of motion is integrated using the leap-frog scheme. In the calculation of E- and B- fields, we did
not consider the electrostatic field, which is important in the sheaths near walls. We replaced the sheath by
the simplest model as follows. The walls are assumed to be electrostatically floating, and hence the potential
difference <I>f between the plasma bulk and the wall is given by [8]
where /CB is the Boltzmann constant and M is the mass of ClJ. We have e<3>f = 21.8 eV for &Te = 4.0 eV.
The electrons whose kinetic energy £_L are larger than e$f are let to pass through the sheath and be absorbed
on the wall. Here the kinetic energy e± is £j_ = rat^/2, where vj_ is the component of electron velocity
perpendicular to the wall. Electrons that are incident on the wall are removed if e± > e$f and specularly
reflected if £j_ < e<I>f . Electrons arriving at the annular exit at z — 0 are let to cross the region with probability
Pout and be reflected with probability (1 — P ou t)- The magnitude of P0ut is determined in such a way that the
electron density averaged over the computational domain agrees with a prescribed value.
After solving the equation of motion by time step At, we consider electron— Cl and electron— Cl2 collisions
in the same time step by the Monte Carlo method. We used a set of collision cross sections shown in Figs.
3 and 4 [9] . The number of electron— C12 collisional events is 9, i.e., elastic collision, vibrational excitation,
four electronic excitations, dissociation, dissociative attachment, and ionization. The number of electron— Cl
collisional events is 11, i.e., elastic collision, nine electronic excitations, and ionization. The total number of
events for electron— C12 and electron— Cl collisions is 20. Whether a collision occurs in At or not and which
event occurs can be determined by use of one random number [10]. The collision probability Pk for event k is
2£\ 1 / 2
At,
ra/
(14)
where a^ is the collision cross section for event A: and ng is the number density of gas C12 or Cl. The densities
of C12 and Cl are necessary in starting simulation. However, the densities of C12 and Cl are unknown until
the flow fields calculation of C12 and Cl are completed. In the ICP, the rate of the dissociation is large and
the density of Cl is often larger than the density of C12 [11]. The ratio of C12 and Cl affects the electron
temperature because of the difference of inelastic collision cross sections between C12 and Cl. As an example,
the C12 to Cl density ratio is set to 1 from a simple estimation that the electron temperature does not depend
strongly on the ratio of C12 to Cl. The temperatures of C12 and Cl are set to 400 K. The velocities of C12
and Cl are randomly sampled from the Maxwellian distributions. The post-collision velocity of electron is
determined using the conservation equations for momentum and energy, taking into account the energy loss in
case of inelastic collisions.
The E- and B- fields change in time with a frequency of 13.56 MHz. After an initial transcient, a system
of simulated electrons turns into a periodic steady state. Then we obtain the electron temperature, electron
density, and rates of ionization, dissociation, and dissociative attachment.
Note that the input data for the above simulation are the spatially averaged electron density, the C1/C12
density ratio, the total gas pressure, and the gas temperature in addition to the E- and B- fields. Also we
need a rough estimate of the electron temperature for determining the floating potential.
C
Flow fields of C12 and Cl
In the previous section, the flow fields of C12 and Cl are assumed to be spatially uniform and at rest. Really,
the number densities and gas temperature are not uniform in the reactor, and these quantities affect the etch
241
U
10'
10'
0.01
0.1
1
0.01
10
0.1
1
10
Electron Energy 8 (eV)
Electron Energy E (eV)
FIGURE 4. Electron-Cl collision cross sections,
lielastic, 2:ionization, 3:excitation(4s), 4:excitation(5s),
5:excitation(6s), 6:excitation(3d), 7:excitation(4d), 8:excitation(5d), 9:excitation(4p), 10:excitation(5p), ll:excitation (metastable)
FIGURE
3.
Electron-Cl2
collision
cross
sections. Irelastic, 2:ionization, 3:vibrational excitation,
4:excitation(B3Il), 5:excitation(C1II), 6:excitation(B1II
-f C1!!), 7:excitation(1n + 1 E + ), 8:dissociation, 9:dissociative attachment
rate. Therefore, let us now examine the flows of Cl2 and Cl using DSMC method [12]. Owing to the principle
of decoupling, the motion and collision of particle can be treated separately. In the stage of motion calculation,
we first consider the injection of C12 from the shower head and pumping of molecules(Cl2 and Cl) at the
annular exit at z — 0. Cl is produced by such reactions as dissociation(e + C12 —> e -f 2 Cl) and dissociative
attachment( e -f C12 —» Cl~ + Cl), accompanied by consumption of C12. The reaction rates were obtained
in section B. The number of C12 increases not only by the injection but also by the recombination of Cl on
the walls of the reactor and the surface of the substrate holder. The treatment of pumping and recombination
is important in the present simulation. The molecule arrived at the annular region at z = 0 is removed with
probability PpUmP and reflected with probability (1 — P pU m P )- The probability Ppump is adjusted in such a
way that the spatially-averaged total gas pressure agrees with a given computational condition. Similarly, the
recombination probability of Cl, Prcmbr is adjusted in such a way that the C12/C1 ratio of the spatially-aver aged
densities obtained agrees with the given value of 1.
In the collision stage, we calculate C12-C12, C12-C1, and C1-C1 collisions in time step At. We used the maximal
collision number method [13]. Molecules are regarded as hard sphere. Let Vc be the cell volume and W be the
weight. The maximal collision number in At is, for s-s(Cl2-Cl2, C1-C1) and s-r(C!2-Cl) collisions
= -WNs(Ns-l)(9s
Nsr =
(15)
(16)
where Ns is the number of simulated molecules in a cell, (#sr)max is the maximal relative speed between unlike
molecules, and asr is the total collision cross section. Since the molecular model is hard sphere, scattering
is isotropic. As is mentioned before, we use the dissociation rate averaged over the azimuthal angle in the
flow simulation. This results in an axisymmetrical flow. Collisions in the axisymmetrical flow are treated by
Riechelmann and Nanbu's method [14].
Even after determining the values of the adjustable probabilities Ppump and Prcmb? we should continue the
calculation to reduce the statistical fluctuations of time-averaged data. Sampled flow properties are the flow
velocities, densities, temperatures, and pressures of C12 and Cl.
242
FIGURE 5. Electric field amplitude \E0\
FIGURE 6. Power deposition Pdep
FIGURE 7. Electron number density
FIGURE 8. Electron temperature
R
0.6
0.6
22
(10 m-V)
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
ill (i 2
ill °
ill
I
nil
(1022m-V)
(1022m-V)
2
1.6
°0
0.1
r (m)
(a)
0.2
n
y €„ 1
Illlllilil
iiiiiiitiiif , i
0.1
r (m)
0.2
0.1
r (m)
0.2
0.1
r (m)
0.2
(b)
(c)
(d)
FIGURE 9. Reaction rates: (a) Ionization(e + C12 -+ e 4- ClJ), (b) Ionization(e + Cl -»• e + Cl+), (c) Dissociation(e
+ C12 -* e + Cl + Cl), (d) Dissociative attachment(e + C12 -> Cl~ + Cl).
243
(a)
YC12
(b) yCl
FIGURE 10. Number densities of C12 and Cl
(a)
(b) yCl
YC12
FIGURE 11. Pressures of C12 and Cl
0.6
0.6: ;;; ' ioom/s
100 m/s
0.4
0.2
0.1
0.1
0.2
r (m)
0.2
r (m)
(a) yCl2
(b) yCl
FIGURE 12. Flow velocities of C12 and Cl
244
RESULTS AND DISCUSSION
Electromagnetic field is first calculated for the coil current Ic —8.0 A, electron density ne = 5.0 x 1016 m~ 3 ,
momentum-transfer collision frequency vm = 1.5 x 107 s"1, and total gas pressure pg = 1 Pa(7.5 mTorr). The
power deposition to the plasma was found to be 492 W. Figures 5 and 6 show the electric field amplitude \E0\
and the power deposition. The electric field penetrates into the region near the wall. Let us define the skin
depth S by \Ee(R - 5,z)/Ee(R,z)\ = e"1, where R = 0.1 m and z = 0.448 m. Then we have 5 = 27 mm.
This is comparable to the theoretical value of 8 = c/u;pe=24 mm, where c is the speed of light. The region
of the power deposition is much narrower than the region of the electric field penetration because the power
deposition is proportional to \Eo\2.
Motion and collision of electrons are calculated for the C^/Cl density ratio of 1 using the E- field in Fig.
5. Electron density and electron temperature are shown in Figs. 7 and 8, respectively. The electron density
decreases in the downward direction of the reactor. It is almost uniform in the diffusion chamber. The decrease
of the electron density near the wall of the quartz cylinder is due to high electron temperature. In fact, Fig.
8 shows that the temperature is high near the quartz wall. The electron temperature is low and constant in
the diffusion chamber because the power is not deposited there. The distribution of the electron temperature
agrees with that of the power deposition. The reaction rates of ionization(e 4- Cl2 —> 2e 4- ClJ and e 4- Cl
—» 2e 4- Cl + ), dissociation, and dissociative attachment are shown in Fig. 9. The ionizations and dissociation
occurs in the region of high electron temperature. The dissociative attachment occurs in the region of low
electron temperature because the cross section for this reaction is large in the low energy region. See Fig. 3.
Therefore, the rate of dissociative attachment is low in the region of high electron temperature near the quartz
cylinder. See Fig. 8.
Flow fields of Cl2 and Cl are calculated for Cl2 mass flow rate of Q = 200 seem and wall temperature of
Tw = 400 K. The gas Cl2 is fed at the sonic condition from the shower head. Number densities of Cl2 and
Cl and pressures of Ck and Cl are shown in Figs. 10 and 11, respectively. We found that the choice of
Prcmb — 0.042 and PpUmP = 0.095 satisfies the given conditions that the C^/Cl ratio of the spatially averaged
densities is unity and the spatially averaged total pressure is 1 Pa, really the latter being 1.098 Pa. The density
nci2 decreases with z in the quartz cylinder because the dissociation rate is high. The increase of nci2 near the
cylindrical wall of the diffusion chamber is due to the recombination(2Cl —> C^) on the wall. The density nci
is high in the upper part of the quartz cylinder. Since the production and consumption of Cl are balanced near
the wall, the density nci is comparatively constant in the radial direction in the quartz cylinder. The pressure
of Cl2 is low in the quartz cylinder and high in the diffusion chamber. On the other hand, the pressure of Cl
is high in the quartz cylinder because nci is high. The flow velocities of C^ and Cl are shown in Fig. 12. The
main flow of C^ from the shower head to the annular exit is caused by the gas injection and pumping. Owing
to the production of Cl2 on the wall, there occurs a cross flow of Cl2 from the wall to the center of the reactor.
However, the cross flow is immediately sucked by the main flow. Since the pressure of the Cl gas is high in
the quartz cylinder, the gas is pushed out into the diffusion chamber. The cross flow of Cl is directed to the
surfaces of the quartz, metal, and substrate because they are the sink of Cl due to the recombination.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
J. D. Bukowski and D. B. Graves, J. Appl. Phys. 80(5), 2614-2623 (1996).
R. A. Stewart, P. Vitello, D. B. Graves, E. J. Jaeger and L. A. Berry, Plasma Sources Sci. Technol. 4, 36-46 (1995).
P. L. G. Ventzek, R. J. Hoekstra, and M. J. Kushner, J. Vac. Sci. Technol. B 12(1), 461-477 (1994).
D. J. Economou, T. J. Bartel, R. S. Wise, and D. P. Lymberopoulos, IEEE Trans. Plasma Sci. 23, 581-590(1995).
K. Nanbu, T. Morimoto, and M. Suetani, IEEE Trans. Plasma Sci. 27, 1379-1388 (1999).
H. Sasaki, K. Nanbu and M. Takahashi, Proc. 22nd Int. Symp. Rarefied Gas Dynamics, ed. G. A. Bird, Sydney,
(2000).
C. K. Birdsall, IEEE Trans. Plasma Sci. 19, 65-85 (1991).
M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, Wiley, New
York, (1994).
W. L. Morgan (unpublished).
K. Nanbu, Jpn. J. Appl. Phys. 33, 4752. (1994).
M. V. Malyshev, V. M. Donnelly, and S. Samukawa, J. Appl. Phys. 84, 1222-1230 (1998).
Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, UK, (1994).
K. Nanbu, Report of the Institute of Fluid Science, Tohoku University, Sendai, Japan, 8 77-125 (1996).
D. Riechelmann and K. Nanbu, Phys. Fluids A 5, 2585. (1993).
245