Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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] 2130 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 2130 2131 S. Kleditzsch and U. Riedel: Sensitivity studies of silicon etching 2131 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 2132 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 , 2132 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 2133 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- 2134 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. 2135 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.