Fast 2D hybrid fluid-analytical simulation of inductive/capacitive discharges This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Plasma Sources Sci. Technol. 20 035009 (http://iopscience.iop.org/0963-0252/20/3/035009) Download details: IP Address: 24.7.113.151 The article was downloaded on 22/04/2011 at 15:14 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
Transcript
Fast 2D hybrid fluid-analytical simulation of inductive/capacitive discharges
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2011 Plasma Sources Sci. Technol. 20 035009
(http://iopscience.iop.org/0963-0252/20/3/035009)
Download details:
IP Address: 24.7.113.151
The article was downloaded on 22/04/2011 at 15:14
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Fast 2D hybrid fluid-analytical simulationof inductive/capacitive dischargesE Kawamura1, D B Graves2 and M A Lieberman1
1 Department of Electrical Engineering, University of California, Berkeley, CA 94720, USA2 Department of Chemical Engineering, University of California, Berkeley, CA 94720, USA
Received 31 July 2010, in final form 17 December 2010Published 19 April 2011Online at stacks.iop.org/PSST/20/035009
AbstractA fast two-dimensional (2D) hybrid fluid-analytical transform coupled plasma reactor modelwas developed using the finite elements simulation tool COMSOL. Both inductive andcapacitive coupling of the source coils to the plasma are included in the model, as well as acapacitive bias option for the wafer electrode. A bulk fluid plasma model, which solves thetime-dependent plasma fluid equations for the ion continuity and electron energy balance, iscoupled with an analytical sheath model. The vacuum sheath of variable thickness is modeledwith a fixed-width sheath of variable dielectric constant. The sheath heating is treated as anincoming heat flux at the plasma–sheath boundary, and a dissipative term is added to thesheath dielectric constant. A gas flow model solves for the steady-state pressure, temperatureand velocity of the neutrals. The simulation results, over a range of input powers, are in goodagreement with a chlorine reactor experimental study.
1. Introduction
A fast hybrid-analytical two-dimensional (2D) transformercoupled plasma (TCP) reactor model was developed usingthe finite elements simulation tool COMSOL. In additionto the availability of built-in partial differential equation(PDE) modules and diagnostics tools, the advantages ofusing a commercial PDE solver like COMSOL are improvedstandardization, transparency and portability. The simulationconsists of four basic parts: (1) an electromagnetic (EM) modelwhich includes both the inductive and capacitive coupling ofthe external coils to the plasma through a dielectric window;(2) a plasma fluid model which solves the 2D time-dependentequations for ion continuity and electron energy balance; (3) ananalytical sheath model which approximates a vacuum sheathof variable sheath thickness as a fixed-width sheath of varyingdielectric constant; (4) a gas flow model which solves for thesteady-state composition, pressure, temperature and velocityof a reactive gas. The total simulation time for a typical TCPreactor is about 70 min for a chlorine discharge and about30 min for an argon discharge on a moderate workstation witha 2.2 GHz CPU and 4 GB of memory.
Our model extends a previous 2D plasma fluid model [1]by including the capacitive coupling of the source coils to theplasma and by adding a sheath model. Our sheath modelextends a previous analytical sheath model [2] by including
a dissipative term in the sheath dielectric constant, and bytreating the sheath heating more accurately as an energy fluxentering the plasma–sheath boundary rather than a volumetricheating term. We also further extend our EM model byallowing a capacitive bias to be applied to the wafer electrode.By solving for both the capacitive and inductive fields, wecan calculate the ratio of inductive to capacitive power as wevary model parameters. By taking the capacitive fields and thesheath into account, we can calculate the sheath voltage andthus the power going into the ions.
The paper is divided into the following sections. Insection 2, we discuss the model geometry and our new EMmodel. In section 3, we summarize the bulk plasma andgas fluid models which are adapted from [1], and describehow our models are modified from [1] due to the inclusionof a sheath region and capacitive fields. In section 4, wediscuss our new analytical sheath model which is based on the‘fixed width but varying dielectric constant’ concept proposedin [2]. However, our sheath model implementation is newand differs significantly from [2], as will be described below.In section 5, we describe the TCP simulation procedure andthe TCP power balance. Then, we present the results ofchlorine TCP reactor simulations at various input powersand compare them with the experimental data of Malyshevand Donnelly [3–5]. Finally, in section 6, we present ourconclusions.
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
center of
(κ = 1)
0.1
0.1
0.2
0.3
0.20
symmetry (r = 0)
(κ = κ )
(κ = 4)
quartz window
wafer chuck
coils
inlet1
z
r (m)
conducting walls(m)
quartz spacer
bulk plasma
sheath
air
0.4
outlet
2 3 4
p
(κ = κ )s
(κ = 4)
Figure 1. The model geometry of the 2D axisymmetric TCP reactor.
2. Electromagnetic (EM) model
As shown in figure 1, the TCP model has an axisymmetriccylindrical geometry with center of symmetry at r = 0 (z-axis).The gas inlet is modeled by a 2.54 cm diameter hole at theradial center while the gas outlet is modeled by a 2.54 cmthick annular region near the radial edge at the bottom of thereactor. The outer walls are perfect conductors. The 10 cmradius wafer electrode is insulated from the outer walls by a1.905 cm wide quartz dielectric spacer. A 4 turn stove-top coilset with coils labeled 1, 2, 3 and 4 is placed 0.3175 cm abovea 1.905 cm thick quartz dielectric window. The coils have a1 cm by 1 cm cross section and are centered at r = 4, 6, 8 and10 cm. One end of the innermost coil (coil 1) is assumed to beconnected to an rf generator while one end of the outermostcoil (coil 4) is assumed to be connected to ground. The airchamber above the dielectric window has radius = 18.5 cmand height = 19 cm while the plasma chamber below thedielectric window has the same radius and height = 20 cm.The air and plasma chamber dimensions were chosen to besimilar to those described by Malyshev and Donnelly [3] inorder to facilitate comparisons with their experiments. In theplasma chamber, the bulk plasma region is surrounded by anominal 0.635 cm thick sheath region.
A free-space magnetic permeability µ = µ0 is assumedwhile ε = κε0 depends on the relative dielectric constant κ ofthe material. For the air region above the dielectric window,κ = 1. The sheath relative dielectric constant κs is initiallyset to unity but is actually a function of the local electric field
and plasma parameters. This will be discussed in more detailin the sheath model section (section 4). For both the quartzdielectric spacer and quartz dielectric window, κ = 4. In theplasma region, the relative dielectric constant is
κp = 1 − ω2p
ω(ω − jνm), (1)
where ω is the applied rf frequency, νm is the electron–neutral momentum transfer collision frequency and ωp =(nee
2/(ε0me))1/2 is the electron plasma frequency with ne and
me the electron density and mass, respectively.In axisymmetric geometry the fields produced by the coils
separate into two modes, the inductive transverse electric (TE)fields (Eφ , Hr , Hz), and the capacitive transverse magnetic(TM) fields (Er , Ez, Hφ), which we discuss separately below.The EM model assumes that all the field components areproportional to ejωt , which eliminates the time-dependencefrom the EM equations so that time-independent Helmholtzequations can be used to solve for the TE and TM fields. Thissimplifies and speeds up the EM simulations at the cost ofignoring harmonics.
2.1. Inductive coupling of coils (TE fields)
Since the TCP coil currents flow in the azimuthal φ direction,the induced electric field must also be in the φ direction. Thus,the inductive fields are in the TE mode where the electricfield is transverse to the axis of symmetry (z-axis), but themagnetic field has components in the r and z directions, bothtransverse and parallel to the axis of symmetry. From Faradayand Ampere’s laws, the field component amplitudes Eφ , Hr
and Hz satisfy
− ∂Eφ
∂z= −jωµHr (2)
1
r
∂(rEφ)
∂r= −jωµHz (3)
∂Hr
∂z− ∂Hz
∂r= jωεEφ. (4)
Substituting (2) and (3) into (4) and introducing the dependentvariable
V ≡ 2πrEφ, (5)
the inductive loop voltage at radius r , we obtain a Helmholtzequation for V :
− ∂
∂r
(c∂V
∂r
)− ∂
∂z
(c∂V
∂z
)+ aV = 0 (6)
where
c = −1
r; a = k2
0
rκ (7)
and k20 = ω2/c2
0, with c0 the speed of light in vacuum. Theform of (6) as well as the coefficients c and a are that of theCOMSOL (v. 3.5) Helmholtz equation PDE Module in 2Drectangular geometry. We chose a 2D rectangular geometryto implement the 2D cylindrical EM equations because theCOMSOL (v. 3.5) General PDE Module we needed for theplasma and gas fluid equations is only available in Cartesian
2
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
coordinates. The COMSOL (v. 3.5) General PDE Module willbe described in more detail in the next section (section 3).
To determine the inductive fields we solve (6) in theentire domain, excluding the areas of the four coils, using thefollowing boundary conditions on V :
V = 0 on all conducting walls, spacer bottom surface
and center of symmetry (8)
V = Vn on the perimeter of the nth coil for each
of the four coils. (9)
Since V ∝ r by definition, V must be zero at the center ofsymmetry. We show below how the boundary values Vn on thefour coils are determined for a given input current Iin to the coilset. Once we solve for V (r, z), we can determine Eφ(r, z) from(5) and Hr(r, z) and Hz(r, z) from (2) and (3), respectively.
2.2. The inductance matrix L
We can define a 4 × 4 inductance matrix L which relatesthe inductive (toroidal) voltages of each coil, Vl ≡(V1, V2, V3, V4), to the inductive (toroidal) currents flowingthrough each coil, Il ≡ (I1, I2, I3, I4):
Vl ≡ jωLIl . (10)
The inverse relation is
Il = 1
jωL−1Vl . (11)
To determine the elements of L, we run four orthonormal TEsimulations with Vn = 1 for one coil while Vn = 0 for the othercoils. Then, we use Ampere’s law to calculate the resultingtoroidal coil currents Imn, where Imn is the current induced inthe mth coil from setting Vn = 1 on the nth coil:
Imn =∮
Cm
H(Vn = 1) · dl, (12)
where Cm is the perimeter of the mth coil. Note from (11) thatthe Imn are related to the matrix elements of L−1 by
L−1mn = jωImn. (13)
Finally, inverting L−1 yields the inductance matrix L. Thus,setting the inductive loop voltages of the coils is equivalent tosetting the inductive loop currents flowing through the coils.
2.3. Capacitive coupling of coils (TM fields)
The conducting coils are also capacitively coupled to theplasma. Note that the capacitive electric field lines betweenthe source coils and the plasma are in the r–z plane. Thus,the capacitive fields are in the TM mode where the magneticfield is transverse to the axis of symmetry while the electricfield has components both parallel and transverse to the axisof symmetry.
The field component amplitudes Hφ , Er and Ez satisfyAmpere and Faraday’s laws:
− ∂Hφ
∂z= jωεEr + Jr (14)
1
r
∂(rHφ)
∂r= jωεEz + Jz (15)
∂Er
∂z− ∂Ez
∂r= −jωµHφ (16)
where a specified external current density J = rJr + zJz issupplied to the coils. Substituting (14) and (15) into (16), andintroducing the dependent variable
I ≡ 2πrHφ, (17)
the poloidal rf current normal to the cross-sectional area of aloop of radius r , we obtain a Helmholtz equation for I in 2Drectangular geometry with an external source term f :
− ∂
∂r
(c∂I
∂r
)− ∂
∂z
(c∂I
∂z
)+ aI = f. (18)
Here,
c = − 1
κr; a = k2
0
r; f = 2π
(∇ × J
κ
)φ
. (19)
To determine the capacitive fields we solve (18) in theentire domain, excluding the areas of the four coils, using thefollowing boundary conditions on I :
n · ∇I = 0 on all conducting walls and spacer bottom surface
(20)
I = 0 on center of symmetry since I ∝ r by definition. (21)
Note from (14) and (15) that the first condition (20) isequivalent to setting the tangential electric field at the outerwalls to zero.
We assume that each coil is supplied from above by axialfeed currents. Only the regions directly above the coils have anon-zero external current density Jz(r) and a non-zero externalsource term f = (2π/κ)∂Jz/∂r . In our case, we choose asinusoidal current feed so that
Jzn(r) = −in
2πwrn
[cos
(2π(r − rn)
w
)+ 1
], (22)
and
f = 2πin
κrnw2sin
(2π(r − rn)
w
). (23)
Here, rn is the r-coordinate of the center of the nth coil and w
is the width of the coil (coil extends from rn−w/2 to rn +w/2).Note that Jzn(r) was chosen so that the capacitive (poloidal)current through the nth coil equals −in:
2π
∫ rn+w/2
rn−w/2rJzn(r) dr = −in. (24)
We show below how to determine the axial feed currents inof the four coils for a given input current Iin from the rfgenerator. Once we solve for the poloidal current I (r, z), wecan determine Hφ from (17) and Er and Ez from (14) and (15),respectively.
3
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
Figure 2. Coil circuit driven by a specified input current Iin from the rf generator.
2.4. The capacitance matrix C
Analogous to the inductance matrix L in the TE simulations,we can define a 4 × 4 capacitance matrix C which relates thefour capacitive (poloidal) coil currents ic ≡ (i1, i2, i3, i4) to thefour capacitive (poloidal) coil voltages vc ≡ (v1, v2, v3, v4):
ic = jωCvc. (25)
The inverse relation is
C−1ic = jωvc. (26)
To determine the elements of C, we run four orthonormal TMsimulations with in = 1 for one coil while in = 0 for the otherthree coils. Then, we calculate the resulting capacitive coilvoltages vmn, where vmn is the capacitive voltage induced inthe mth coil from setting in = 1 on the nth coil:
vmn =∫
Pm
E(in = 1) · dl. (27)
Here, Pm is any path from coil m to ground. In our case,we chose each Pm to start from the bottom surface of coil m
and go to the right in a horizontal straight line to a point onthe grounded outer wall. In general, E is not the gradient ofa scalar potential (E �= −∇�) so that voltages depend onthe path. However, for relatively low frequencies (e.g. 13.56to 60 MHz), the voltages are approximately path independent.From (26), the vmn are related to the matrix elements of C−1 by
C−1mn = jωvmn. (28)
Finally, inverting C−1 yields C. Thus, setting the capacitivefeed currents of the coils is equivalent to setting the capacitivevoltages of the coils.
2.5. Coupling of TE and TM coil fields
The coil circuit is shown in figure 2 for a specified inputcurrent Iin from the rf generator. Note that the inductive loopvoltages Vl = (V1, V2, V3, V4) and inductive loop currentsIl = (I1, I2, I3, I4) are coupled with the capacitive coilvoltages vc = (v1, v2, v3, v4) and capacitive coil currentsic = (i1, i2, i3, i4).
From Kirchoff’s voltage law applied to the circuit infigure 2, we obtain the relation
Vl = Mvc (29)
where
M =
1 −1 0 00 1 0 00 0 1 −10 0 0 1
. (30)
From Kirchoff’s current law, we find similarly
ic = −MTIl + Iin (31)
where Iin ≡ (Iin, 0, 0, 0). The inductance matrix L andcapacitance matrix C obtained in the previous sections canbe used with (29) and (31) to determine all the voltages andcurrents in figure 2 in terms of the input current Iin to the coilset. Substituting (25) into (31) yields
jωCvc = −MTIl + Iin. (32)
Next substituting (29) into (32), we obtain
jωCM−1Vl = −MTIl + Iin. (33)
Finally, substituting (10) into (33), we obtain
SIl = Iin, (34)
whereS = MT − ω2CM−1L. (35)
Inverting (34) yields
Il = S−1Iin. (36)
Then, Vl = jωLIl , vc = M−Vl , and ic is obtained from(31). Thus, for a specified value of the input current Iin, we candetermine the inductive voltages Vl on the coil perimeters andthe capacitive feed currents ic for each coil. Recall that Vl givesthe boundary conditions needed for the TE field solution whileic gives the external excitation terms needed for the TM fieldsolution. Thus, solving the circuit determines the inductiveand capacitive fields of the TCP reactor for any given inputcurrent Iin to the coil set.
2.6. Capacitive bias option for the wafer electrode
As with the fields due to the capacitive coupling of the TCPcoils to the plasma, the fields due to an rf capacitive bias currentIB applied to the wafer electrode also have TM symmetry. Wesolve (18) with the boundary conditions:
n · ∇I = 0 on all conducting walls (37)
I = IB = const on spacer bottom surface (38)
I = 0 on center of symmetry. (39)
4
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
The TCP coils are floating in this simulation so that the axialfeed current for each coil is set to zero.
Recall from (17) that I (r) gives the total current flowingnormal to the cross-sectional area of a loop of radius r . Hence,the I (r) = IB = const condition on the bottom surface of thedielectric spacer means that there is no current within the spacerthat is perpendicular to its bottom surface, (i.e. no z-directioncurrent within the spacer). All the z-direction current fromthe current source is contained within the wafer electrode.When a current source is applied at the wafer electrode, thedielectric spacer prevents conduction current from traveling inthe r direction, so all the applied conduction current goes in thez-direction and is contained within the wafer electrode. Thedielectric spacer acts as a blocking capacitor with a potentialdifference and displacement current in the r-direction, but notthe z-direction. The potential difference across the dielectricspacer gives the value of the applied rf bias voltage at the waferelectrode:
Vrf =∫
Pds
Er dr, (40)
where Pds refers to the path along the bottom surface of thedielectric spacer. Then, the rf power supplied by the capacitivebias is given by
Prf = 12 Re(IBV ∗
rf ), (41)
where V ∗rf is the complex conjugate of Vrf .
2.7. Power deposition in bulk plasma
Once we determine the inductive TE and capacitive TM fieldsof the TCP reactor, we can calculate the power deposition to thebulk plasma, i.e. the bulk Ohmic heating. The time-averagedpower density profile in the bulk plasma is given by
pdep = 12 Re(JT · E∗) = 1
2 Re(σp)( |Eφ|2︸ ︷︷ ︸inductive
+ |Er |2 + |Ez|2︸ ︷︷ ︸capacitive
),
(42)where JT = (σp + jωε0)E is the total plasma current densityand σp = ε0ω
2p/(νm + jω) is the plasma conductivity. Note that
the power deposition in the bulk plasma has both capacitive andinductive components. The power deposition in the sheath willbe discussed below in the sheath section (section 4).
3. Bulk plasma and gas flow models
The bulk plasma and gas flow models used in our simulationsare adapted from those of Hsu et al [1]. The major differencesbetween our model and that of Hsu et al are the inclusion of(i) a sheath and sheath heating model, and (ii) the capacitivecoupling of the source coils to the plasma.
3.1. Bulk plasma fluid model
The plasma fluid model consists of solving the time-dependentconservation equations for each ion species simultaneouslywith the electron power balance equation. The assumptionsof the model are:
(1) Constant and uniform ion temperature Ti = 0.052 V foreach ion species.
(2) Maxwellian electron energy distribution.(3) Quasi-neutral and ambipolar plasma: ne = ∑
i n+i −∑i n−i and Γe = ∑
i Γ+i − ∑i Γ−i . Here the variables
n and Γ refer to the particle densities and fluxes while thesubscripts e, +i and −i refer to the electron, positive andnegative ion species, respectively.
(4) Ambipolar electric field neglecting electron inertia: Ea =−Te∇ne/ne, where Te is the electron temperature in unitsof volts.
(5) Collisionally dominated ion transport: ion flux for eachion species ±i is given by Γ±i = −D±i[∇n±i ±(n±i/ne)(Te/T±i)∇ne] with the ion diffusion coefficientD±i = eT±i/(M±iν±i ), where M±i and ν±i are the ionmass and ion–neutral collision frequency, respectively.Note that the ‘drift’ term proportional to ∇ne has oppositesigns for positive and negative ions.
(6) The electron heat flux Qe is given by
Qe = 5
2ΓeeTe − 5
2
nee2Te
meνm∇Te. (43)
The first term is the energy transported by electron netmotion while the second term is the electron thermalconduction.
The ion continuity equation for each ion species ±i isgiven by
∂n±i
∂t+ ∇ · Γ±i = R±i , (44)
where R±i is the net rate of creation of the ion species ±i. TheHsu et al model has no sheath and uses the chamber walls asthe outer boundary for the plasma fluid equations. Our modelincludes a sheath and uses the plasma–sheath boundary as theouter boundary for the plasma fluid equations. The boundaryconditions for positive ions are zero flux at the radial centerline(r = 0) and Bohm flux ni(eTe/Mi)
1/2 at the plasma–sheathboundary. The boundary conditions for negative ions are zeroflux at the radial centerline and zero density at the plasma–sheath boundary.
The electron power balance equation is given by
3
2
∂
∂t(neeTe) + ∇ · Qe = −eEa · e + pdep − pcoll, (45)
where the first term on the right-hand side (RHS) is the Jouleheating from the ambipolar field, the second term pdep, givenby (42), is the Joule heating from the inductive and capacitivefields, and the last term pcoll is the power lost to electron–neutral collisions. In a typical fluid model without a sheath, theboundary conditions are zero energy flux at the radial centerlineand an outward (from plasma) Maxwellian flux
Qe · n = (2eTe)Γe · n, (46)
at the chamber walls. Here n is the unit normal vector to theboundary surface and points away from the bulk plasma.
We modified boundary condition (46) to take the sheathinto account. In our model, the boundary conditions are zeroenergy flux at the radial centerline, and an energy flux
Qe · n = (2eTe + eV shMin)Γe · n − Sstoc − SohmSh (47)
5
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
at the plasma–sheath boundary. Here V shMin, Sstoc and SohmSh
are the minimum dc sheath voltage, the stochastic heating fluxand the sheath Ohmic heating flux, respectively. Assumingthat electrons enter the sheath when the dc sheath voltageis at a minimum, an electron requires an additional energyeV shMin at the plasma–sheath boundary to overcome the sheathpotential barrier and arrive at the walls with energy 2eTe.Also, in addition to an outgoing (from plasma) Maxwellianelectron energy flux due to electron wall losses, there is also anincoming (to plasma) electron energy flux due to the stochasticand sheath Ohmic heating. We will discuss how to calculateV shMin, Sstoc and SohmSh below in the sheath model section(section 4).
For both the ion conservation equations and the electronpower balance equation we use the COMSOL (v. 3.5) GeneralPDE Module which has the form
da∂X
∂t+ ∇ · Γ = F. (48)
In this equation, X is the conserved quantity (e.g. n±i or Te),and the coefficient da can be a constant or a variable. Thevector Γ is the flux term (e.g. Γ±i , Qe), and F is the source termon the RHS. The General PDE Module in COMSOL (v. 3.5)provides the required generality for our set of equations, butit is available only in Cartesian coordinates. Thus, all thedivergences must be written as
∇ · Γ = ∂z
∂z+
1
r
∂(rr)
∂r= ∂z
∂z+
∂r
∂r+
r
r. (49)
The ‘extra’ third term r/r has to be added to the RHS of (48)in the source term F .
3.2. Gas flow model
The gas flow model is mostly adapted from Hsu et al’smodel [1] and will only be briefly summarized below. Themodel consists of five basic parts:
(1) A neutral mass continuity equation which solves for thegas pressure p.
(2) A neutral energy balance equation which solves for gastemperature Tg.
(3) A neutral r-momentum balance equation which solves forthe r-component of the gas velocity vr .
(4) A neutral z-momentum balance equation which solves forthe z-component of the gas velocity vz.
(5) A set of species continuity equations which solve for theneutral species mass fractions wj where j iterates over allthe neutral species.
The total flow (in sccm) is set at the inlet surface while thepressure is set to a reference value (p = pref ) at the outletsurface. The equations are all solved for the steady state (i.e.all ∂/∂t terms set to zero), and the COMSOL General PDEModule described above is used for each equation.
We used Hsu et al’s gas flow model with only a fewmodifications. First, we used a newer and more up todate chlorine reaction set and rate coefficients compiled byThorsteinsson and Gudmundsson [6]. These rate data are
substantially improved over those used in prior models forCl2 plasmas. Second, we changed the Cl recombinationcoefficient at the walls from γrec = 0.6 to a much lowervalue of γrec = 0.02. Third, we used COMSOL’s extrusioncoupling variable tool to map the ion fluxes at the plasma–sheath boundary to the chamber walls.
When comparing theoretical models with experimentaldata, both Malyshev and Donnelly [4] and Corr et al [8] foundthat the Cl recombination rate at stainless steel reactor wallswas much lower than the value of 0.6 reported by Kota et al [9].The explanation is that the stainless steel walls are passivatedby a large flux of chlorine neutrals and ions to the walls, thusreducing Cl atom recombination. Corr et al found the bestagreement between model and experiment was obtained forγrec = 0.02. A more accurate method requiring no adjustableparameters is to use a γrec which is a function of the Cl to Cl2ratio, as was done by Thorsteinsson and Gudmundsson [6].Since our simulations were mostly done in a regime where γrec
is a slowly varying function of the Cl to Cl2 ratio with a valuebetween 0.01 to 0.05, we decided to just use a constant valueof 0.02.
In our model, the gas fluid equations are solved in boththe bulk plasma and the sheath, but the plasma fluid equationsare solved only in the bulk plasma. Hence, the ion fluxes areknown at the plasma–sheath boundary but not at the chamberwalls. Thus, to take the ion surface reactions into account, wemap the values of the ion fluxes at the plasma–sheath boundaryto the chamber walls.
4. Analytical sheath model
Due to the quasi-neutrality assumption, Poisson’s equationis not solved, and the sheaths are not resolved in the bulkplasma fluid model. The local time-averaged sheath thicknesss actually depends on the local E field as well as local plasmaparameters such as ne and Te. However, since the plasma–sheath boundary is also the boundary used in the bulk plasmafluid equations, it is computationally inconvenient to adjust theposition of the plasma–sheath boundary whenever the localfields and plasma parameters change. Instead, as proposed byLee et al [2], we assume a sheath with constant thickness s0
and varying relative dielectric constant κs to mimic an actualvacuum sheath of varying thickness.
The major differences between our sheath model and Leeet al’s model is that (i) we treat the stochastic and sheath Ohmicheating as incoming heat fluxes at the plasma–sheath boundaryrather than a volumetric heating term in order to capture thelocal nature of the sheath heating, and (ii) we add a dissipative(imaginary) term to the sheath dielectric constant in order toaccount for the electron and ion sheath heating. Without thisdissipative term, we do not get a balance between the powerabsorbed by the plasma and the power supplied by the coilcircuit.
4.1. Fixed-width sheath model
For typical inductively coupled plasmas, the plasma density ishigh and the sheath is thin and collisionless. In this case, the
6
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
time-averaged sheath width is given by [7, chapter 11],
s = 21/4
1.23
(0.82ε0
e
)1/2V
3/4sh
n1/2e T
1/4e
, (50)
where the dc sheath voltage V sh is related to the rf sheathvoltage magnitude Vsh by
V sh = 0.83Vsh + V shMin = 0.83Vsh + 3.26Te. (51)
The second term V shMin = 3.26Te is added to ensure aminimum width smin = 2.61λD in the case the local Vsh = 0.The rf sheath voltage magnitude Vsh at every point along theplasma–sheath boundary is given by
Vsh = |E · n|s0, (52)
where n is the unit vector perpendicular to the plasma–sheathboundary surface. Note that only the capacitive fields Er andEz contribute to the sheath voltages. If we assume a sheath offixed width s0 and varying relative dielectric constant
κs = s0
s, (53)
we will be able to mimic an actual vacuum sheath with varyingsheath thickness.
For example, the sheath voltage Vvac in a vacuum sheathwith width s and vacuum electric field Evac is Vvac = |Evac·n|s.In a sheath of fixed width s0 and relative dielectric constant κs,the electric field would be E = Evac/κs. This results in asheath voltage of Vsh = |Evac · n|(s0/κs) = |Evac · n|s = Vvac.Thus, the fixed-width sheath of varying dielectric constant hasthe same sheath voltage as the vacuum sheath of varying sheaththickness.
4.2. Complex sheath dielectric constant
The sheath capacitive electric fields Er and Ez can heat boththe ions and electrons. The electrons are heated via stochasticand sheath Ohmic heating while the positive ions are heated asthey accelerate across the dc sheath potential drop of 0.83Vs.The stochastic heating flux Sstoc and the sheath Ohmic heatingflux SohmSh are given by [7, chapter 11]:
Sstoc = 0.45
√me
eε0ω
2VsT1/2
e (54)
and
SohmSh = 0.5
π
(me
2e
)ε0ω
2νm sVs. (55)
Both Sstoc and SohmSh are included in the bulk plasma modelas incoming heat fluxes at the plasma–sheath boundary in theelectron power balance equation. The ion heating flux Sion isthe total positive ion flux multiplied by the dc sheath voltagedrop:
Sion = 0.83eVsh
∑i
(ni
√eTe/Mi), (56)
where the sum is over all the positive ion species i.
The total power deposition profile psh in the sheath is then
psh = Sstoc + SohmSh + Sion
s0. (57)
Analogous to the power density profile pdep for the bulk plasmagiven in (42), the power density profile for the sheath is givenin terms of the capacitive electric fields as
psh = 12σsh(|Er |2 + |Ez|2), (58)
where σsh is the sheath conductivity which we assume to bereal. Using (57) in (58), and solving for σsh, we obtain
σsh = 2
s0
Sstoc + SohmSh + Sion
|Er |2 + |Ez|2 . (59)
Recall that for any material, the conductivity σ is related to therelative dielectric constant κ by κ = 1 − jσ/(ωε0). Thus, thedissipative or imaginary part of κs is given by
Im(κs) = − 2
ωε0s0
Sstoc + SohmSh + Sion
|Er |2 + |Ez|2 . (60)
Typically, this dissipative term is small and represents aperturbation to the mostly capacitive vacuum sheath.
For the fixed-width sheath of varying dielectric constant,the real part of κs is again set equal to the ratio s0/s. Thus, thecomplex sheath dielectric constant is
κs = s
s0− j
2
ωε0s0
Sstoc + SohmSh + Sion
|Er |2 + |Ez|2 . (61)
The real part of κs allows us to take the varying sheath thicknessinto account while the imaginary part of κs allows us to takethe electron and ion sheath heating into account. Note thatsince the dissipative or imaginary part of κs is only due to thecapacitive fields, only the capacitive TM equation (18) usesthe complex κs given in (61). The inductive TE equation (6)just uses κ = Re(κs) in the sheath region.
5. TCP reactor simulations
A flow chart depicting the TCP simulation procedure is shownin figure 3. The simulation steps are as follows:
(1) Load the initial profiles for densities, Te, p and Tg (e.g.uniform). Calculate the initial plasma dielectric constantκp from the initial profiles, and set the initial sheathdielectric constant κs to unity.
(2) Calculate the inductance L and capacitance C matricesby conducting four orthonormal TE and TM simulations.Then, for a given input current Iin to the coil set, solve thecoil circuit to obtain the four inductive coil voltages (Vl)and the four capacitive coil currents (ic). Recall that theformer give the boundary terms for the TE field solutionwhile the latter give the external excitation terms for theTM field solution.
(3) Solve simultaneously for the steady-state gas pressure p,gas temperature Tg and gas velocity components vr and vz.If the gas is reactive (e.g. chlorine), solve for the neutralspecies mass fractions wj .
7
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
I
iSolve coil circuit
for and c VL(source andboundary terms)
4 orthonormalTE & TM simulationsto get & matricesL C
L, C
Gas Flow Model to get p & T
ci , VL
input current ILoad initial profiles and
in
in
sκ , κp
p & fields
TE & TM to get p & fieldsdep
p0
Plasma Fluid Modelto get n , n , & Ti e e
p & T
g
g
dep
κ ,κs0
Sheath Model to getsheath parameters
Figure 3. Flow chart of TCP simulation procedure.
(4) Solve the TE and TM field equations simultaneously toobtain the inductive and capacitive fields. Then, calculatethe time-averaged power density profile pdep from (42).
(5) Solve the bulk plasma fluid equations for the particledensities and Te simultaneously, using pdep from the EMsolution in the electron power balance equation. Thestochastic (Sstoc) and sheath Ohmic heating (SohmSh) fluxescalculated from the analytical sheath model are treated asincoming heat fluxes at the plasma–sheath boundary. Usethe updated plasma parameters to calculate κp.
(6) Use the analytical sheath model to calculate the sheathvoltage V sh, sheath thickness s and sheath heatingcomponents Sstoc, SohmSh and Sion. Use the updated sheathparameters to calculate the complex sheath dielectricconstant κs.
(7) Repeat the cycle again using the updated κp and κs in theEM model.
The above cycle is repeated for 15 iterations of 640 rf cycleseach for a total of 9600 rf periods (or about 0.71 ms forf = 13.56 MHz). It should be noted that L and C can berecomputed every two iterations rather than every iterationto save simulation time. The total simulation time wasabout 70 min for chlorine and 30 min for argon when usinga moderate workstation with a 2.2 GHz CPU and 4 GB ofmemory.
5.1. TCP power balance
The total power absorbed by the plasma Pabs is due to both bulkOhmic heating and sheath heating. The bulk plasma powerPbulk is the volume integral of the bulk power deposition profilepdep over the volume of the bulk plasma Vbulk:
Pbulk =∫
Vbulk
pdep dV = PbInd + PbCap, (62)
where
PbInd = 1
2
∫Vbulk
Re(σp)|Eφ|2 dV, (63)
and
PbCap = 1
2
∫Vbulk
Re(σp)(|Er |2 + |Ez|2) dV (64)
are the inductive and capacitive components of Pbulk
respectively. The sheath heating Psh is the surface integralof the sum of the stochastic heating flux Sstoc, sheath Ohmicheating flux SohmSh and the ion heating flux Sion over the surfacearea of the sheath Ash:
Psh =∫
Ash
(Sstoc + SohmSh + Sion)dA. (65)
Since Sstoc, SohmSh and Sion are all due to capacitive fields, Psh
is purely capacitive. Thus, the total power absorbed by theplasma is
Pabs = PbInd︸︷︷︸inductive
+ (PbCap + Psh)︸ ︷︷ ︸capacitive
= PabsInd + PabsCap, (66)
where PabsInd and PabsCap are the inductive and capacitivecomponents of Pabs, respectively.
The total power lost by the plasma Ploss is equal to thesum of the power lost to electron–neutral collisions (Pcoll), theambipolar field (Pa) and electron and ion wall losses (Pewall
and Piwall):
Ploss = Pcoll + Pa + Pewall + Piwall. (67)
From figure 2, the total rf power input to the coil circuit is
Prf = 12 Re(Iinv
∗1). (68)
Iin is the input current to the coil circuit while v1 (the capacitivevoltage of coil 1) is the input voltage of the coil circuit. Notethat Prf is also equal to the sum of the powers dissipated by theplasma-loaded inductors and capacitors of the coil circuit:
Prf = PrfCap + PrfInd, (69)
where
PrfCap = 1
2
4∑n=1
Re(inv∗n), (70)
and
PrfInd = 1
2
4∑n=1
Re(InV∗n ). (71)
Power balance implies that
Pabs = Ploss = Prf . (72)
In addition, the inductive power input to the coil circuit mustequal the inductive power absorbed by the plasma:
PrfInd = PabsInd = PbInd (73)
while the capacitive power input to the coil circuit must equalthe capacitive power absorbed by the plasma:
PrfCap = PabsCap = PbCap + Psh. (74)
8
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
Table 1. Power balance results of TCP reactor model for Iin = 16 Aand 67 A.
The TCP reactor model was used to simulate a chlorine reactorwith applied rf frequency f = 13.56 MHz, total flow set to100 sccm at the inlet surface, and pressure set to 10 mTorr atthe outlet surface. The rf input current Iin was varied from 15to 70 A, resulting in an absorbed power Pabs of 5.3 to 815 W.
Table 1 shows the power balance results of the TCP reactormodel for input current Iin = 16 A and 67 A. For low values ofIin, both Pabs and ne are low, and capacitive coupling dominatesover inductive coupling while the opposite is true for highvalues of Iin. As required for simulations that are run untilsteady state, the power absorbed by the plasma is equal to thepower lost by the plasma: Pabs = Ploss. The power absorbedby the plasma Pabs and the power supplied by the coil circuitPrf agree to within 3%. We also see similar good agreementbetween PrfInd and PabsInd, and between PrfCap and PabsCap. Theagreement between Prf and Pabs depends on the refinement ofthe computational mesh. (For example, recall that in order tocalculate the L matrix needed to solve the coil circuit, it isnecessary to take the line integrals of the H field around thetiny source coils.) The more refined the mesh, the better theagreement at the cost of more memory and simulation time.
The agreement between Prf and Pabs is also due to thedissipative (imaginary) term added to the sheath dielectricconstant κs. Recall from the flow chart in figure 3 that thebulk plasma and sheath models calculate κp and κs and send theresults back to the EM model. The EM model then recalculatesthe fields based on these updated dielectric constants. Thedissipative (imaginary) components of κp and κs provide thebulk and sheath heating, respectively, to the plasma. Thus,if we do not add a dissipative term to κs to account for Psh,the power supplied by the coil circuit Prf will only equal Pbulk
rather than Pabs = Pbulk + Psh. We confirmed this by doingother simulations in which we did not add a dissipative termto κs.
Figure 4 shows (a) the electron density ne (diamonds)in units of 1015 m−3 at the discharge center, and the ratioof inductive to capacitive power absorbed by the plasmaPabsInd/PabsCap (stars) as a function of the rf input current Iin
(A). As Iin increases from 15 to 70 A, the TCP reactor goes from
10 100I (A)
0.1
1
10
100 e
P /PabsInd absCap
n (10 m )-3
in
15
Figure 4. Model results for ne (1015 m−3, diamonds) at thedischarge center, as well as PabsInd/PabsCap (stars) versus Iin (A).
a low density capacitive mode to a high density inductive mode.The E to H transition appears to occur around Iin = 19 A.
Simulation results comparing a low power case withPabs = 6.0 W and a high power case with Pabs = 763 W areshown in figures 5 and 6. Figure 5 shows the Cl molar fractionxCl of the discharge for (a) Pabs = 6.0 W and (b) Pabs = 763 W.For the high power case, most of the chlorine gas is dissociated(xCl ≈ 0.8 at discharge center) while the opposite is true forthe low power case (xCl ≈ 0.02 at discharge center). Figure 6shows the gas temperature Tg (K) for (a) Pabs = 6.0 W and (b)Pabs = 763 W. For the high power case, there is significant gasheating with Tg varying from about 320 K at the chamber wallsto about 800 K at the discharge center. For the low power case,the gas heating is much less significant.
For the high power case of Pabs = 763 W, figure 7shows the (a) inductive and (b) capacitive components of thepower density profile (W m−3) in the bulk plasma. Comparingfigure 7(a) with figure 7(b), we see that at high power, thereactor is in an inductive mode where inductive couplingdominates over capacitive coupling.
The model reactor geometry is similar to that describedin Malyshev and Donnelly (2/2000) [3], having the samedimensions for both the air and plasma chambers. This allowedus to compare the model results with the experimental data ofMalyshev and Donnelly [3–5]. Figure 8 presents the modelresults (circles) and Malyshev and Donnelly data (triangles) fora 10 mTorr, 100 sccm chlorine TCP reactor. The four plots inthe figure show (a) ne (m−3) (b) Te (V), (c) Cl density nCl (m−3)and (d) Cl2 density nCl2 (m−3) at the discharge center versusPabs (W). The power coupling efficiency of the Malyshev andDonnelly reactor was assumed to be about 75% [10]. In otherwords, we assumed that Pabs/Prf = 0.75 for the Malyshev andDonnelly reactor. This is a good estimate when the reactor isin inductive mode, but in capacitive mode, the actual couplingefficiency may be much smaller. There is good agreementbetween the model results and Malyshev and Donnelly’s datafor ne, Te, nCl and nCl2 at the discharge center.
The Malyshev and Donnelly measurements for ne versusPabs show a gap in the region between Pabs = 45 and
9
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
Figure 5. Model results of the 10 mTorr, 100 sccm chlorine reactor, showing Cl molar fraction xCl for (a) Pabs = 6.0 W and (b) Pabs = 763 W.
Figure 6. Model results of the 10 mTorr, 100 sccm chlorine reactor, showing gas temperature Tg (K) for (a) Pabs = 6.0 W and(b) Pabs = 763 W.
110 W (see figure 8(a)). Malyshev and Donnelly sawan abrupt transition from capacitive to inductive couplingwhich somewhat depended on pressure and matching networktuning [5]. In our simulations, we see a smooth E to Htransition with no gaps (see figure 4). One explanation forthe discrepancy is that our external circuit is a simple currentsource with no matching network. Another is that since wedid not know the exact coil configuration of the Malyshevand Donnelly reactor, we used a somewhat different coilconfiguration, resulting in a lower capacitive coupling.
Note that the model results of Te in figure 8(b) lie betweenthe LP and TRG-OES experimental data. However, althoughthe measurements show Te increasing by about 13% between
Pabs = 375 to 700 W, the simulation Te is fairly constant. Aspower goes up, Tg rises and ng falls, and we might expectTe to increase in the simulations. However, as power goesup, Cl2 dissociation also increases, leading to a decline in theCl− ion to electron density ratio α since Cl− ions are createdfrom dissociative electron attachment of Cl2. This changein α could be significant. Referring to chapter 10 of [7],dropping the recombination term in (10.4.1) and using (10.4.4)for the positive ion flux +s , where n+ = (1 + α)ne0 with ne0
the assumed uniform electron density, one gets the positiveion balance equation to determine Te in an electronegativedischarge with small recombination loss:
KizngV = 2D+(1 + α)A/deff . (75)
10
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
Figure 7. Model results of the 10 mTorr, 100 sccm chlorine reactor for the high power case Pabs = 763 W, showing (a) the inductive and(b) the capacitive components of the power density profile (W m−3) in the bulk plasma.
Figure 8. Model results (circles) and experimental data from Malyshev and Donnelly [3–5] (triangles) for a 10 mTorr, 100 sccm, chlorineTCP reactor, showing (a) ne (m−3), (b) Te (V), (c) nCl (m−3) and (d) nCl2 (m−3) at the discharge center versus Pabs (W).
Here Kiz is the ionization rate coefficient, V and A are thevolume and surface area of the plasma, and deff is an effectivediffusion length in the bulk plasma. Increasing the powerincreases Tg and reduces ng, but also reduces (1 + α). If
ng and (1 + α) both go down, then from equation (75),Te may not change much. Equation (75) also explainswhy the Thorsteinsson and Gudmundsson’s global model(see figure 8 in [6]) shows that Te is fairly constant with
11
Plasma Sources Sci. Technol. 20 (2011) 035009 E Kawamura et al
pressure (although the measurements show Te decreasing withincreasing pressure). As pressure rises, so does ng, and wemight expect Te to decrease. However, α and (1 + α) also goup with pressure, keeping Te fairly constant.
6. Conclusion
Our TCP reactor model couples a bulk plasma fluid model withan analytical sheath model and includes both the inductive andcapacitive coupling of the source coils to the plasma. Thishybrid fluid-analytical approach allows fast computation ofchemically active plasmas with flow as well as predictionsof inductive and capacitive plasma sustaining mechanisms.Solving for the capacitive fields and including a sheath modelallows us to calculate the sheath voltage and thus the sheathheating of the electrons and ions. For collisionless sheaths, thepower absorbed by the ions in the sheath allows us to predictthe ion energy impacting surfaces (and, in principle, the ionenergy distribution). Treating the stochastic and sheath Ohmicheating as incoming heat fluxes at the plasma–sheath boundarycorrectly captures the local nature of the sheath heating.A dissipative (imaginary) term is added to the sheath dielectricconstant to account for the electron and ion sheath heating,enabling a correct power balance between the power suppliedby the coils and the power absorbed by the plasma. Codes canbe shared quickly among COMSOL users with the ability tochange reactor parameters fairly easily.
We conducted chlorine TCP reactor simulations for rfinput current Iin from 15 to 70 A, resulting in a range ofPabs from 5.3 to 815 W. Solving for both the capacitive andinductive fields, allowed us to calculate separately and thencompare the capacitive and inductive heating of the plasma.The capacitive fields were also used by the sheath modelto calculate the sheath parameters (i.e. sheath voltage andthickness), allowing us to calculate the power absorbed bythe ions as well as the sheath heating of the electrons. At highabsorbed powers, the inductive coupling dominates over thecapacitive coupling, the plasma density is high, the chlorine
gas is mostly dissociated, and significant gas heating occursin the discharge. The chlorine TCP reactor model showedgood agreement with Malyshev and Donnelly’s experimentaldata for ne, Te, nCl and nCl2 at the discharge center. Thenext steps include incorporating matching network/powergenerator models into the coil/circuit model, developing amulti-frequency sheath model, and adding more reaction sets(currently, Ar/O2/Cl2). We also plan to couple a particle codeto the hybrid fluid-analytical model to obtain the ion energyand angular distributions at the wafer electrode.
Acknowledgments
We thank A J Lichtenberg at the University of California atBerkeley, J T Gudmundsson at the University of Iceland andNeil Benjamin at Lam Research Corporation for many usefulsuggestions. This work was supported by gifts from LamResearch Corporation and Micron Corporation, in part by theUC Discovery Grant ele07-10283 under the IMPACT programand in part by the Department of Energy Office of FusionEnergy Science Contract DE-SC0001939.
References
[1] Hsu C C, Nierode M A, Coburn J W and Graves D B 2006J. Phys. D: Appl. Phys. 39 3272
[2] Lee I, Lieberman M A and Graves D B 2008 Plasma SourcesSci. Technol. 17 015018
[3] Malyshev M V and Donnelly V M 2000 J. Appl. Phys. 87 1642[4] Malyshev M V and Donnelly V M 2000 J. Appl. Phys. 88 6207[5] Malyshev M V and Donnelly V M 2001 J. Appl. Phys. 90 1130[6] Thorsteinsson E G and Gudmundsson J T 2010 Plasma
Sources Sci. Technol. 19 015001[7] Lieberman M A and Lichtenberg A J 2005 Principles of
Plasma Discharges and Materials Processing 2nd edn(New York: Wiley-Interscience)
[8] Corr C S, Despiau-Pujo E, Chabert P, Graham W G, Marro F Gand Graves D B 2008 J. Phys. D: Appl. Phys. 41 185202
[9] Kota G P, Coburn J W and Graves D B 1998 J. Vac. Sci.Technol. A 16 270
