Download Sensitivity Studies of Silicon Etching in Chlorine

Sensitivity studies of silicon etching in chlorineÕargon plasmas
S. Kleditzsch and U. Riedela)
Interdisciplinary Center for Scientific Computing, University of Heidelberg, D-69120 Heidelberg, Germany
共Received 19 November 1999; accepted 17 March 2000兲
In this article a well-stirred reactor model is utilized to model the etching of silicon in low-pressure
chlorine/argon plasmas. Well-stirred reactor models are increasingly common in the literature due
to their low requirements of computer resources for detailed chemical kinetics calculations. The
model predicts the spatially averaged species composition and etch rate in a plasma etch reactor by
solving conservation equations for species, mass, and the electron energy distribution function
共EEDF兲. The reactor is characterized by a chamber volume, surface area, surface area fraction of the
wafer, mass flow, pressure, power deposition, and composition of the feed gas. In such plasma etch
models, assumptions on the EEDF which are needed to determine reaction rate coefficients for
electron-impact reactions, are crucial for a prediction of steady state conditions. The model
presented in a recent article 关P. Ahlrichs, U. Riedel, and J. Warnatz, J. Vac. Sci. Technol. A 16, 1560
共1998兲兴 is extended to describe the etching of the wafer with a special set of reactions occurring on
a certain area fraction of the total reactor surface. A modified numerical procedure to solve the
species conservation equations and the EEDF is presented, which needs considerably less
computation time than the approach previously taken. Systematic sensitivity studies are presented to
identify the connection between input parameters, outflow composition, and etch rate of the process.
Such numerical studies are an important step towards fault detection and model based process
control of plasma reactors. © 2000 American Vacuum Society. 关S0734-2101共00兲00805-8兴
I. INTRODUCTION
Plasma processing is widely used in microdevice fabrication, where one of the key steps is etching of material layers
on wafer surfaces. Usually, low-pressure gas discharges are
used to dissociate and ionize a feed gas for the purpose of
reactive ion etching.1,2 Such plasma reactors show a complex
behavior due to the coupling of electrodynamic interaction,
fluid dynamics, and chemical reactions in the gas phase and
on the surface.3
Spatially averaged or zero-dimensional 共0D兲 reactor models have become common for plasma systems.1,4–6 Their major advantage lies in the small computational demands for
solving the conservation equations of the system, allowing
one to analyze the effects of model assumptions or to determine the influence of various input parameters on the plasma
composition. Besides, there are also physical reasons why
one can assume a well-stirred reactor: The low pressure of
plasma etch reactors 共1 Pa and below兲 leads to highly diffusive conditions under which the time scale for conversion of
reactants to products is much larger than that for mixing
processes.4 Thus, the outlet composition is essentially controlled by chemical kinetics.
The electron energy distribution function 共EEDF兲 in such
systems is crucial for the determination of rates of electron
impact in the plasma via cross sections.4,5,7,8 In general, in a
low-pressure partially ionized plasma the EEDF is nonMaxwellian and several approaches are reported in the literature to incorporate this effect into 0D models.4,5,7,9
In this article, the model of a low pressure plasma source
presented in Ref. 7 is extended to describe the wafer etching
a兲
Electronic mail: [email protected]
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J. Vac. Sci. Technol. A 18„5…, SepÕOct 2000
based on a special set of reactions occurring on a certain area
fraction of the reactor surface area. In an earlier article,7 the
species equations and the Boltzmann equation are solved
fully coupled. In this article, a new procedure to compute the
steady state of the model is presented, which is computationally faster than the fully coupled approach but gives the same
species concentration and EEDF. Once the steady state is
reached we focus on sensitivity studies to identify the connection between input parameters, outflow composition, and
etch rate of the process. Such numerical studies are an important step towards fault detection and model-based process
control of plasma reactors. Control strategies containing process models are needed to relate wafer and process characteristics to machine settings for improved process results.
The article is organized as follows: First, we discuss the
conservation equation and the numerical solution procedure
to obtain the state of the system. Then, the gas phase and
surface reaction schemes for a chlorine/argon mixture are
established. The model is applied to typical reactor conditions and the results are compared to a two-dimensional 共2D兲
study reported in Ref. 3. The central parts of the article are
the sensitivity studies with respect to bias voltage, flow rate,
power, pressure, gas temperature, and wafer temperature
variations.
II. DESCRIPTION OF THE MODEL
A. Reactor
The incoming gas mixture is characterized by the mass
fractions Y 0i of the gas phase species i, the temperature T 0 ,
and the mass flow rate ṁ 0 . Furthermore, the reactor has the
volume V and is at pressure p. The total surface area is split
0734-2101Õ2000Õ18„5…Õ2130Õ7Õ$17.00
©2000 American Vacuum Society
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S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching
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TABLE I. Electron collision processes considered, Arrhenius parameters calculated with ELENDIF and fitted over
an electron temperature range of 2.8–5.4 eV.
G1
G2
G3
G4
G5
G6
G7
G8
G9
⫺
e
e⫺
e⫺
e⫺
e⫺
e⫺
e⫺
e⫺
e⫺
⫹Cl2
⫹Cl⫺
⫹Cl
⫹Cl*
⫹Ar
⫹Cl2
⫹Cl2
⫹Cl
⫹Ar
→
→
→
→
→
→
→
→
→
⫺
e
e⫺
e⫺
e⫺
e⫺
Cl⫺
e⫺
e⫺
e⫺
⫺
⫹e
⫹e ⫺
⫹e ⫺
⫹e ⫺
⫹e ⫺
⫹Cl*
⫹Cl
⫹Cl*
⫹Ar*
into two parts, A wafer for the wafer area and A wall for the
remaining surface area representing the reactor wall.
A ‘‘three-fluid-model’’ is used to describe the different
behavior of electrons, ions, and heavy particles with their
own temperature. The 共mean兲 electron temperature and the
corresponding rates of electron collision processes are obtained from integrating the EEDF for a set of different electric fields.
Solving the species and electron energy conservation
equations gives the mass fractions, surface coverages, and
the electron temperature. The temperature of ions and heavy
particles is held constant. The speed of the positive ions at
the surface is controlled by the electric potential applied to
the wafer.
B. Electron energy distribution function
Solving the Boltzmann equation and all species equations
for every time step fully coupled is a very computing-timeconsuming process. Therefore, the following iterative approach consisting of three basic steps is used in this study:
共1兲 Keeping the gas composition fixed, the Boltzmann
equation is solved for an electric field in the range from 50
up to 250 Td 共in steps of 25 Td兲. For each value of the
electric field the EEDF and the electron temperature are obtained.
共2兲 From this information an Arrhenius fit of the reaction
rates of electron reactions as a function of the electron temperature is constructed.
共3兲 The heavy particle species continuity equations are
solved using the fit from the above step.
Typically, this iteration converges after 3–5 iterations.
The iteration is stopped if the relative change of the etch rate
is less than 10⫺5 . Using the EEDF from previous iteration
steps as an initial value for the current iteration typically
needs only seconds of cpu time. Therefore, this procedure is
at least four times faster than solving the Boltzmann equation
and species equations fully coupled 共see Ref. 7 for details兲
but results in the same species composition and EEDF for
the steady state.
In this simulation we calculate the EEDF f e ( ⑀ ) by solving
the Boltzmann equation discretized on the energy axis using
the ELENDIF code 共see Table I兲.10 The Maxwell distribution is
JVST A - Vacuum, Surfaces, and Films
⫹Cl⫹
2
⫹Cl
⫹Cl⫹
⫹Cl⫹
⫹Ar⫹
⫹Cl
A k (m3 /mol/s)
Bk
E a k (J/mol)
9.14⫻10⫺16
7.51⫻105
1.12⫻10⫺10
3.49⫻1010
8.20⫻10⫺17
5.15⫻107
3.68⫻106
4.18⫻106
8.63⫻10⫺5
5.12
1.00
4.08
0.12
5.30
0.11
0.75
0.39
0.28
7.77⫻104
3.37⫻105
4.32⫻105
5.05⫻105
7.64⫻105
5.87⫻104
4.03⫻105
8.25⫻105
6.90⫻105
perturbed by the electrical field and inelastical collisions
with heavier particles. The homogenous Boltzmann equation
is
冉 冊
⳵ f e eE
⳵fe
⫺
•ⵜ f ⫽
⳵t
m e ve e
⳵t
共1兲
.
coll
The elastical collision terms are solved for low E/n g and
higher elastic than inelastic cross sections in the Lorentz approximation
f e 共 v e , ␪ ,t 兲 ⫽ f 0 共 v e ,t 兲 ⫹ f 1 共 v e ,t 兲 cos ␪ ,
共2兲
by splitting the isotropic part f 0 from the anisotropic part f 1
of the distribution function.11 It is assumed that ac effects
can be neglected in the EEDF calculation since we are only
interested in the plasma composition averaged over an ac
cycle. The ac field is replaced by an effective dc field transferring the same power to the electrons. References 8,12, and
13 discuss the validity of this approximation in detail.
In Ref. 12 a Fourier expansion of the distribution function
is performed. The authors conclude that independent of the
degree of nonstationarity of f 1 the period average of the
isotropic distribution f 0 can be sufficiently calculated by the
lowest equation in the Fourier expansion in the case of sinusoidal electric fields, which is identical to the steady-state
limit of Eq. 共4兲, if the following two conditions for the energy relaxation frequency hold:13
␻ ⭓2
m e (m)
␯
M
and
␻⭓
兺
k⫽1
␯ (inel)
,
k
共3兲
where m e is the electron mass, M is the heavy particle mass,
␯ (m) is the mole-fraction-averaged momentum transfer colliis the energy
sion frequency for elastic collisions, and ␯ (inel)
k
transfer frequency for inelastic collisions.
The reactor is driven by a frequency of 13.56 MHz. In
this frequency regime and for the low system pressures of
interest both of the above conditions hold.7 Therefore, we
restrict the EEDF calculations to the dc case, leading to the
simplification11
f 1 共 v e ,t 兲 ⫽⫺
eE
⳵ f 0 共 v e ,t 兲
m e ␯ (m)
⳵ve
,
共4兲
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S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching
E being the electric field and e the absolute value of the
electron’s charge.
A finite-difference scheme of the electron energy ⑀
⫽ 12 m e v 2e is used to solve the partial differential equation system
冉 冊 冉 冊
⳵fe
⳵J f ⳵Je
⳵fe
⫺
⫹
⫽⫺
⳵t
⳵⑀
⳵⑀
⳵t
⳵fe
⫹
⳵t
e⫺e
2
3
冕
⬁
0
f 共 ⑀ 兲 ⑀ 3/2d⑀ .
共6兲
The rate coefficients of electron collision processes are obtained by integration over the corresponding cross section ␴ j
冉 冊冕
2e
k f k⫽
m
1/2
⬁
0
␴ j 共 ⑀ 兲 f 共 ⑀ 兲 ⑀ d⑀ .
共7兲
To cover a range of electron temperatures, the field density
E/n is varied from 50 to 250 Td. The velocity coefficients
k f k (¯⑀ r ) are fitted to Arrhenius parameters 共see Table I兲. The
chemistry is described by elementary reactions k
兺
i ⑀ 兵 gas species其
⬘ ␹ i→
␯ ik
兺
i ⑀ 兵 gas species其
⬙ ␹i ,
␯ ik
共8兲
⬘ and ␯ ik
⬙ are stoichiometric coefficients and ␹ i are
where ␯ ik
species symbols. The production rate of a species i is
␻˙ i ⫽
兺
k ⑀ 兵 gas reactions其
共 ␯ ⬙ik ⫺ ␯ ⬘ik 兲 k f k
兿
i ⑀ 兵 gas species其
␯⬘
C i ik .
共9兲
The rate coefficient k f k is modeled by an Arrhenius law
k f k ⫽A k T
Bk
exp
冉 冊
⫺E a k
RT
1 1
vi
␶
⌫ k4
共11兲
j
or for positive charged ions by
k f k⫽ ␬ S k
inel.
where J f is the contribution of electrical field and J e is the
elastic collision term. The last two terms on the right-hand
side are contributions from electron–electron and inelastic
collisions.7
The cross sections used are the same as in Ref. 7. The
reduced averaged electron temperature in eV is calculated
from
¯⑀ r ⫽
k f k ⫽S k
共5兲
,
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1
␶
⌫jk
共12兲
v Bohm,mod ,
with the modified Bohm velocity (m i is the mass of the
considered positive ion, and n ⫺ is the sum of densities of all
negative ions multiplied by their charge兲5,15
v Bohm, mod⫽
冉 冊冉
kT e
mi
1/2
冊
n ⫺ 共 T ion /T e 兲 1/2⫹n e
,
n e ⫹n ⫺
共13兲
the mean species velocity is
¯
v i⫽
冉 冊
8kT
␲mi
1/2
共14兲
,
and a correction term for etching reactions
␬⫽
冉
E ion
⫺1
E0
冊
1/2
共15兲
,
which describes the dependence of etching reaction rates
from the energy of positive ions at the surface E ion having a
minimum energy E 0 共see Table II兲. This factor has been
found in experimental studies.16
The sticking coefficients for the recombinations of positive ions on the wall contains a transport-limitation factor of
0.4 共Ref. 4兲 and the probability of these reactions is 1.0 共see
Table III兲.
D. Conservation equations
Solving the continuity equation for species mass and electron energy for species mass fractions Y i , surface coverages
⌰ i , and T e gives
N
.
共10兲
Reaction rates for heavy particle reactions are taken from
Ref. 14.
C. Surface chemistry
Two different types of surfaces and two different sets of
reactions are considered in the model. The reactor wall consists of alumina and contains two species, W(s) and WCl
⫻(s). The silicon wafer has four surface species Si(s),
SiCl(s), SiCl2 (s), SiCl3 (s) and the bulk species Si(b). The
recombination reactions of ions and electrons, the recombination of chlorine radicals at the wall, and the deexcitation of
metastable species are taken from Ref. 7. The sticking coefficient S k , the number of surface species involved in the
reaction ␶ k , and the surface density ⌫ j are used to calculate
the reaction rate coefficient k f k for neutral gas species by
J. Vac. Sci. Technol. A, Vol. 18, No. 5, SepÕOct 2000
␳V
g
⳵Y i
˙ i M i V,
ṡ k M k A⫹ṡ i M i A⫹ ␻
⫽ṁ 0 共 Y 0i ⫺Y i 兲 ⫺Y i
⳵t
k⫽1
兺
⳵ ⌰ i ṡ i
⫽ ,
⳵t
⌫
共16兲
⳵Te
3 kB
␳ VY e
2 me
⳵t
⫽
5 kB 0 0 0
kB
ṁ Y e 共 T e ⫺T e 兲 ⫹
T ṁ 0 共 Y 0e ⫺Y e 兲
2 me
me e
N
⫺
g
kB
kB
˙ eM eV 兲
T eY e
ṡ k M k A⫹
T 共 ṡ M A⫹ ␻
me
me e e e
k⫽1
⫹
5 kB
␻˙ M V 共 T⫺T e 兲 ⫺ ␩˙ e V⫺ ␨˙ e V⫹ P e .
2 me e e
兺
˙ i is the formaṡ i is the surface formation rate of species i, ␻
˙
˙
tion rate in the gas phase. ␩ e and ␨ e are energy losses by
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S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching
2133
TABLE II. Sticking coefficients S k of reactions at the wafer surface at T⫽500 K 共see Ref. 3兲. Etching reactions
are depended on the energy of the positive ions. The threshold energy E 0 is assumed to be 10 eV.
Etching reactions with chlorine radicals
S1
S2
S3
S4
Sk
⫹Si共s兲
⫹ SiCl(s)
⫹SiCl2 (s)
⫹SiCl3 (s)
Cl
Cl
Cl
Cl
→
→
→
→
SiCl(s)
SiCl2 (s)
SiCl3 (s)
SiCl4
0.99
0.20
0.15
0.0001
⫹SiCl2 (s)⫹Si(b)
⫹SiCl2 (s)⫹Si(b)
⫹SiCl2 (s)⫹Si(b)
⫹SiCl2 (s)⫹Si(b)
0.80
0.50
0.30
0.10
Deposition of SiCl2 on the wafer surface
S5
S6
S7
S8
SiCl2 ⫹Si(s)
SiCl2 ⫹SiCl(s)
SiCl2 ⫹SiCl2 (s)
SiCl2 ⫹SiCl3 (s)
⫹Si(s)
⫹Si(s)
⫹Si(s)
⫹Si(s)
→
→
→
→
Si(s)
SiCl(s)
SiCl2 (s)
SiCl3 (s)
Ion etching reactions
S9
S10
S11
S12
S13
S14
S15
S16
S17
e ⫺ ⫹Ar⫹
e ⫺ ⫹Ar⫹
e ⫺ ⫹Cl⫹
e ⫺ ⫹Cl⫹
e ⫺ ⫹Cl⫹
e ⫺ ⫹Cl⫹
2
e ⫺ ⫹Cl⫹
2
e ⫺ ⫹Cl⫹
2
⫺
e ⫹Cl⫹
2
⫹SiCl2 (s)
⫹SiCl3 (s)
⫹SiCl(s)
⫹SiCl2 (s)
⫹SiCl3 (s)
⫹Si(s)
⫹SiCl(s)
⫹SiCl2 (s)
⫹SiCl3 (s)
⫹Si(b)
⫹Si(b)
⫹Si(b)
⫹Si(b)
⫹Si(b)
⫹Si(b)
⫹Si(b)
⫹Si(b)
⫹Si(b)
elastic and inelastic collisions of electrons with heavy particles, and ṁ is the total mass flux into the reactor. For the
pressure the ideal gas law is used,
Ng
p⫽R ␳
兺
i⫽1
Y iT i
.
Mi
共17兲
The power deposited to the reactor is split into three parts.
The first one is the heating of ions from their formation
temperature T to the predefined ion temperature 关set to a
typical value of T i ⫽5800 K 共Ref. 5兲兴
P i⫽
兺 ␻˙ i M i V„h i共 T i 兲 ⫺h i共 T 兲 ….
i⫽ ⑀ 兵 ion其
共18兲
Furthermore, positive ions are accelerated in the plasma
sheath
TABLE III. Sticking coefficients at T⫽500 K for relaxation and recombination reactions 共see Ref. 7兲.
Recombination of Cl
S18
S19
Cl
Cl
⫹W(s)
⫹WCl共s兲
Sk
→
→
WCl(s)
W(s)
→
→
Cl
Ar
1.0
1.0
→
→
→
Cl
Cl2
Ar
0.40
0.40
0.40
⫹Cl2
1.0
0.15
Relaxation of Cl* and Ar*
S20
S21
Cl*
Ar*
⫹ M
⫹ M
Recombination of positive ions
S22
S23
S24
e⫺
e⫺
e⫺
⫹Cl⫹
⫹Cl⫹
2
⫹Ar⫹
JVST A - Vacuum, Surfaces, and Films
SiCl2 ⫹Si(s)⫹Ar
SiCl3 ⫹Si(s)⫹Ar
SiCl2 ⫹Si(s)
SiCl2 ⫹SiCl(s)
SiCl4 ⫹Si(s)
SiCl2 ⫹Si(s)
SiCl2 ⫹SiCl(s)
SiCl2 ⫹SiCl2 (s)
SiCl4 ⫹SiCl(s)
→
→
→
→
→
→
→
→
→
P s⫽
兺
i⫽ ⑀ 兵 ion其
0.16
0.16
0.13
0.16
0.19
0.13
0.16
0.16
0.19
ṡ i M i A ⑀ i RT e .
共19兲
The energy available to the electrons P e is the difference of
the total power P coupled into the reactor and the power used
for heating the ions
P e ⫽ P⫺ P i ⫺ P s .
共20兲
III. RESULTS
A. Base case and comparison to two-dimensional
simulations
To get an estimate for the accuracy of the well-mixed
approach, we compare our results to the numerical study of
Hoekstra, Grapperhaus, and Kushner reported in Ref. 3. In
this simulation, Hoekstra and co-workers examine an argon/
chlorine discharge with a ratio of argon to chlorine of 70:30
at a flow rate of 150 sccm. A 1 kW inductively coupled
plasma reactor is operated at 13.56 MHz with a bias of 150
V 共see Table IV兲. They report an etch rate of 76–180
nm min⫺1 depending on the distance from the center of the
11
⫺3
wafer. The peak of the Cl⫹ ⫹Cl⫹
2 density is 6.36⫻10 cm
13
⫺3
and the peak of the Cl radical density is 8.12⫻10 cm
共see Table V兲.
In our 0D simulation we find a Cl⫹ ⫹Cl⫹
2 density of
1.65⫻1011 cm⫺3 and a Cl radical density of 2.23⫻1013
cm⫺3 which is of the same order but a lower than the results
of the 2D simulations. However, as our model can predict
spatially averaged values only this is the expected result. The
electron temperature is not reported for the two-dimensional
study, but the value of 4.1 eV we obtain is in good agree-
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S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching
FIG. 1. Sensitivity with respect to the ion energy over the surface.
ment with the value of 4 eV given in Ref. 17 for a pure argon
plasma at 13 mTorr and 50–250 W power input. The etch
rate of 265 nm min⫺1 of our simulation is too high compared
to what is reported in Ref. 3. The reason for this might be a
modeled channel in the 2D simulation. A certain fraction of
particles hits one of the sidewalls of the channel and, therefore, does not contribute to the surface etch rate. In a
volume-averaged approach the total particle flux towards the
surface contributes to the etching.
2134
FIG. 2. Sensitivity with respect to the gas flow through the reactor.
concentration in the exhaust gas mixture, while reactive ion
etching 共reactions S9–S17兲 gives higher ion concentrations
and a lower chlorine coverage.
In the following discussion we report sensitivities with
respect to a variation of the bias voltage, the gas flow into the
reactor, the power coupled into the gas phase, the pressure,
the temperature of the neutral gas species, and the temperature of the wafer surface. The standard values of these input
parameters are listed in Table IV.
共1兲 Bias voltage 共see Fig. 1兲: A higher bias voltage increases the etch rate only by a higher ion etch rate. Therefore, the coverage of chlorine is reduced, which can be de-
B. Sensitivity studies and interpretation of results
The relative sensitivity E (rel)
i, j of an output variable x i with
respect to an input parameter f j can be calculated by varying
this parameter and dividing the relative change of the result
by the relative change of the parameter18
E (rel)
i, j ⫽
f j ⌬x i
.
x i⌬ f j
共21兲
The results of Figs. 1–6 are obtained from a variation of
⫺2%, ⫺1%, 1%, and 2% of the input parameter and subsequent averaging of the sensitivities obtained. As an example, the etch rate shows a sensitivity of ⫹0.4 with respect
to the ion energy 共see Fig. 1兲. This means that an increase of
10% in the ion energy results in a 4% increase in the etch
rate.
There are two different paths of etching included in the
model, which are radicalic etching and reactive ion etching.
The following sensitivity studies show that the variation of
the input parameters can change the contribution of these
two mechanisms to the total etch rate.
The radicalic path 共reactions S1–S4兲 promotes a higher
coverage of chlorine on the wafer surface and a higher SiCl4
J. Vac. Sci. Technol. A, Vol. 18, No. 5, SepÕOct 2000
FIG. 3. Sensitivity with respect to the power coupled into the reactor.
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S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching
FIG. 4. Sensitivity with respect to the gas pressure.
duced from the positive sensitivities of the less chlorined
surface species. This parameter of the calculation is important, because the bias voltage is not constant over the wafer
surface, but varies from 48 eV on the edge to 69 eV in the
center of the wafer.
共2兲 Gas flow 共see Fig. 2兲: The significantly negative sensitivities of the etch products SiClx show, that with increased
flow rate, their concentration is reduced due to a lower
production–dilution ratio. For the same reason the shorter
residence time reduces the total concentration of the ions and
excited species.
2135
FIG. 6. Sensitivity with respect to the wafer surface temperature.
共3兲 Power 共see Fig. 3兲: The sensitivity of nearly zero indicates that changing the power in the 1000 W region leaves
the electron temperature constant, while the electron density
is proportional to power. This trend is also found in pure
chlorine plasmas.7 The contribution of reactive ion etching
becomes larger with increasing power, which is expressed in
the shifting of etch products to SiCl2 and reduced chlorine
coverage of the wafer.
共4兲 Pressure 共see Fig. 4兲: Increasing pressure leads to a
shorter average time between collisions of particles. Therefore, the electron temperature decreases. Furthermore, the
recombination of ions and relaxation of excited species is
favored, as can be seen in the negative sensitivities of these
species. A higher contribution of radicalic etching leads to
higher SiCl4 concentration and chlorine coverage on the wafer.
TABLE IV. Input parameters.
Inflow mixture Cl2 /Ar
Gas temperature T
Inflow temperature T 0
Wafer temperature T wafer
Wall temperature T wall
Ion temperature T ion
Wafer surface density ⌫ wafer
Wall surface density ⌫ wall
Reactor volume V
Wafer area A wafer
Wall area A wall
Pressure p
Power P
Mass flux into the reactor ṁ
Ion energy at the wafer E ion
Wafer density ␳ wafer
FIG. 5. Sensitivity with respect to the gas temperature.
JVST A - Vacuum, Surfaces, and Films
30/70 vol %
500 K
500 K
500 K
500 K
5800 K
1⫻1015 cm⫺2
1.5⫻1015 cm⫺2
8024 cm3
314 cm2
3522 cm2
10 mTorr
1000 W
5⫻10⫺6 kg s⫺1
68 eV
2120 g cm⫺3
2136
S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching
TABLE V. Densities of gas-phase species.
a
Species
Density (cm⫺3 )
e⫺
Cl⫹
Cl⫹
2
Ar⫹
Cl⫺
Cl
Cl2
Ar
Cl*
Ar*
1.82⫻1011
7.98⫻1010
8.49⫻1010
3.88⫻1010
2.20⫻1010
2.23⫻1013
1.32⫻1013
1.20⫻1014
1.86⫻1011
1.52⫻1011
Literaturea
冎
6.36⫻1011 共max.兲
8.12⫻1013 共max.兲
Reference 3.
共5兲 Gas temperature 共see Fig. 5兲: A higher gas temperature generally leads to a lower gas-phase density. The concentration sums of the neutral and charged gas-phase species,
on which the rates of the etch reactions are depended, are
lower. This explains the small negative sensitivity of the etch
rate. The lower density is the reason for the positive sensitivity of the electron temperature, since the mean-free path of
electrons is larger.
共6兲 Wafer temperature 共see Fig. 6兲: Increasing the wafer
temperature, the importance of radicalic etching is increased,
because its rate is determined by the wafer temperature,
while ion etching depends only on electron and ion temperature. The negative sensitivity of the etch rate can be explained by the small contribution of the radicalic path to the
total etch rate (⬍1%).
In the context of fault detection and model-based process
control these sensitivity studies are useful to identify the
probable cause of a reactor malfunction. If, for example, the
etch rate observed in a real process is too low the sensitivity
studies for this chlorine/argon etch reactor would indicate
that the gas temperature or the wafer temperature is not set
correctly for this process. These two input quantities have a
negative correlation to the etch rate 共see Figs. 5 and 6兲 in
contrast to ion energy, flow rate, power, or pressure which all
have a positive correlation 共Figs. 1–4兲. Or, as a second example, if there is a large amount of unused molecular chlorine measured at the reactor exit, this might be due to an
error in flow rate, gas temperature, or wafer temperature 共all
three input parameters have a positive correlation兲, but not,
for example, due to an error in power input which has a
negative correlation.
IV. CONCLUSIONS AND FUTURE WORK
By comparing different parameter variations, necessary
changes of parameters for an optimal operation can be derived. Comparing the variations of pressure and power shows
J. Vac. Sci. Technol. A, Vol. 18, No. 5, SepÕOct 2000
2136
that the pressure has only little influence on unused chlorine,
while an increase of power by 10% reduces chlorine by
14%. On the other hand, a lower pressure leads to less
SiCl4 , which gives a higher etching yield (Cl/Si ratio兲. A
lower Cl2 /Ar ratio increases the etching yield, but at the cost
of more Ar consumption.
With regard to the numerical model a significant reduction of the computation time is achieved by an iterative solution of the Boltzmann equation and conservation equation
for species, surface coverage, and electron temperature without loss of any accuracy compared to a fully coupled solution of all conservation equations. The spacially averaged
model is able to predict not only the outlet composition of
the reactor but also the sensitivity of the outlet state with
respect to various input parameters with very little computational resources.
In future studies we want to address the issue of uncertainties in the electron-impact cross sections. Here, to estimate the influence of the cross sections on the results, the
threshold energy of a given cross section and its magnitude
have to be varied. First studies seem to indicate that these
variations need to be defined very carefully to obtain meaningful results.
Furthermore, the influence of the etch products SiClx will
be investigated as soon as electronic cross sections are available. We expect a lower electron temperature due to an increased collision probability in the gas phase.
1
M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges
and Materials Processing 共Wiley, New York, 1994兲.
2
M. Meyyappan, in Computational Modeling in Semiconductor Processing, edited by M. Meyyappan 共Artech House, Boston, MD, 1995兲,
Chap. 5.
3
R. J. Hoekstra, M. J. Grapperhaus, and M. J. Kushner, J. Vac. Sci. Technol. A 15, 1913 共1997兲.
4
E. Meeks and J. W. Shon, IEEE Trans. Plasma Sci. 23, 539 共1995兲.
5
R. S. Wise, D. P. Lymberopoulos, and D. J. Economou, Plasma Sources
Sci. Technol. 4, 317 共1995兲.
6
M. Meyyappan, Vacuum 47, 215 共1996兲.
7
P. Ahlrichs, U. Riedel, and J. Warnatz, J. Vac. Sci. Technol. A 16, 1560
共1998兲.
8
P. Jiang and D. J. Economou, J. Appl. Phys. 73, 8151 共1993兲.
9
S. C. Deshmukh and D. J. Economou, J. Appl. Phys. 72, 4597 共1992兲.
10
L. W. Morgan, Comput. Phys. Commun. 58, 127 共1990兲.
11
Y. Raizer, Gas Discharge Physics 共Springer, New York, 1991兲.
12
R. Winkler, H. Deutsch, J. Wilhelm, and C. Wilke, Beitr. Plasmaphys. 24,
285 共1984兲.
13
R. Winkler, H. Deutsch, J. Wilhelm, and C. Wilke, Beitr. Plasmaphys. 24,
303 共1984兲.
14
P. L. G. Ventzek, M. Grapperhaus, and M. J. Kushner, J. Vac. Sci. Technol. B 12, 3118 共1994兲.
15
N. S. J. Braithwaite and J. E. Allen, J. Phys. D: Appl. Phys. 21, 1733
共1988兲.
16
J. P. Chang et al., J. Vac. Sci. Technol. A 15, 1853 共1997兲.
17
M. J. Grapperhaus and M. J. Kushner, J. Appl. Phys. 81, 569 共1997兲.
18
J. Warnatz, U. Maas, and R. W. Dibble, Combustion 共Springer, Berlin,
1996兲.