The two-dimensional axial symmetric ICP reactor used in this model is shown in Fig. 1. In the matching box that is filled with air, a two-turn coil with the thickness of 0.6 cm is placed above the quartz window. The turns of the coil are 6.0 cm and 10.0 cm in radius, respectively. The coil is supplied with a radio frequency power source of 13.56 MHz. In the simulations, the coil current is tuned until the total deposited power in the chamber, defined as spatial integration of the electron power density, equals to the pre-designed value, i.e., 200.0 W. The working gas is delivered into the chamber from the circular open loop around the roof edge of discharging chamber and expelled away via pumping system at the chamber outlet. The total height of the chamber is 17.0 cm. The window and the matching box are both 14.0 cm in radius, and the discharging chamber beneath them is 15.0 cm in radius. The window width is 1.0 cm and the box height is 3.0 cm. A grounded substrate is seated at the bottom of the chamber, with its radius and height fixed at 13.0 cm and 4.0 cm, respectively.

Fig. 1.

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	Fig. 1. (color online) ICP chamber used in the simulations

The gas phase and surface chemistries of argon plasma used in this model are listed in Tables 1 and 2, respectively. Four types of species are considered, i.e., electrons, argon ions Ar⁺, metastable argon atoms Ar_m, and grounded atoms Ar. In Table 1, the following collision types are included, i.e., elastic, electronic excitation, de-excitation, direct ionization, multistep ionization, Penning ionization and metastable quenching. These different rate coefficients of electron collisions are calculated via the integration of the Maxwellian electron energy distribution function with their respective cross section set. The rate coefficients of all heavy species reactions are given as constants. Upon arriving at chamber surface, the argon ions are neutralized and the metastables are de-excited, and they both return to the ground state, as illustrated in Table 2. The sticking coefficients of ions and metastables at the surfaces both equal to 1.0.

Table 1.

Table 1.

Table 1. Gas phase electron impact collisions and chemical reactions.[36] . Reaction Formula Type Δε/eV Rate coefficientsa 1 e + Ar ⇒ e + Ar elastic 0 c 2 e + Ar ⇒ e + Arm excitation 11.5 c 3 e + Arm ⇒ e + Ar de-excitation −11.5 c 4 e + Ar ⇒ 2e + Ar+ direct ionization 15.8 c 5 e + Arm ⇒ 2e + Ar+ multistep ionization 4.24 c 6 Arm + Arm ⇒ e + Ar + Ar+ penning ionization 0 3.734 × 108 7 Arm + Ar ⇒ Ar + Ar metastable quenching 0 1807 a The unit of rate coefficient is m3/s·mol.

Table 1. Gas phase electron impact collisions and chemical reactions.[36] .

The space of the ICP chamber is discretized by the finite element method. The mesh set used in the simulations, i.e., mesh set I, is shown in Fig. 2(a). There are totally 17400 mesh elements that fill up the geometry, among which 9559 elements are triangle and 7841 are quadrilateral. Note that the fine mesh is used around the coil to resolve the skin effect of electromagnetic field and the boundary layer meshes are created around the plasma region border to capture the space charge separation between electrons and ions.^[37] To discuss numerical errors, the dependence of simulation results on mesh size are tested. We believe that the errors are more related to the mesh size of plasma region; hence the test is carried on the original mesh of Fig. 2(a) and a new mesh set that holds fine mesh in the plasma region, i.e., mesh set II, as shown in Fig. 2(b). The tests show that the absolute relative differences of electron densities and temperatures that are calculated by using mesh sets I and II, respectively, are about 0.05%. The difference is tiny, and the simulation results are hence independent of the mesh size.

Fig. 2.

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Fig. 2. (color online) (a) Mesh set I is used for the spatial discretization of ICP chamber in the simulations. (b) Mesh set II holds fine mesh in the plasma region as compared to mesh set I. It is used to test the dependence of simulation results on mesh size, i.e., numerical errors. The tests show that the absolute relative differences of electron densities and temperatures that are calculated by using mesh sets I and II, respectively, are about 0.05%. The difference is tiny, and the simulation results are hence independent of the mesh size.

Table 2.

Table 2.

Table 2. Surface reactions mechanism. . Reaction Formula Sticking coefficient 1 Arm ⇒ Ar 1 2 Ar+ ⇒ Ar 1

Table 2. Surface reactions mechanism. .

The inductively coupled plasma (ICP) and the single-phase flow (SPF) modules of the COMSOL software are taken for this gas flow study. The former module describes the plasma generation and transport, while the latter describes the background gas flow. The ICP module consists of electron, heavy species, electromagnetic field, and electrostatic field equations. In the SPF module, the Navier–Stocks equation is selected to study the incompressible gas flow, and the momentum conservation equation is selected for the compressible gas flow. The flow velocity and density of gas mixture, calculated from the SPF module, are sent to the ICP module in the compressible flow, while only the velocity variable is delivered in the incompressible flow. Reversely, the ICP module calculates the viscosity coefficient of gas mixture and transfer it to the two types of gas flow, besides for describing the plasma transport. The interacting mechanism of the two modules is shown in Fig. 3. The equations of each module will be elaborated next.

Fig. 3.

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	Fig. 3. (color online) Interactions of single-phase flow (SPF) and inductively coupled plasma (ICP) modules in this study.

2.2.1. Electron equations

The equations of electron density and energy are expressed as

The drift diffusion approximation is selected for the number and energy flux of electrons, due to the tiny mass of electrons. The fluxes of electrons density and energy are described as follows Here, n_e and n_ε are the number and energy densities of electrons. μ_e and μ_ε are the number and energy mobility coefficients. D_e and D_ε are the number and energy diffusivity coefficients. These above coefficients are related with each other, via the formulas, D_e = μ_eT_e, D_ε = μ_εT_e, and μ_ε = (5/3)μ_e. R_e and R_ε in Eq. (1) are source terms of the continuity and energy conservation equation, respectively, and they are expressed as Here, l_j is the number of electrons created or lost per collision of reactions that generate/deplete electrons. M is the number of such reactions. k_j is the rate coefficient of reaction j, taken from Table 1. n_m is the number density of reactant m of reaction j. ν is the stoichiometric coefficient of the reaction, and P is the number of reactants. ε_j is the energy loss of related electron-impact reaction. The terms of Eq. (1), (u · Δ)n_e and (u · Δ)n_ε, describe the transport of electron mass and energy via the background gas advection, and the gas flow velocity u is calculated from the SPF module. Usually, the electron directional velocity of drift and diffusion is rather higher than advective velocity caused by the gas flow; hence the influence of incompressible advection on the electron is expected to be insignificant. Nevertheless, it will be significantly affected by the mixture density in the compressible gas flow, as revealed later. P_ohm is power density deposited into the plasma through the Ohm’s heating mechanism, where σ is the electron conductivity and E_θ is the azimuthal electric field of electromagnetic effect.

Without the secondary electron emissions, the boundary conditions of the continuity and energy equations of electrons are defined as

where ν_e,th is the thermal velocity of electrons and r_e is the reflection coefficient of electrons from the chamber surface, which is set to 0.2 in this model.

2.2.2. Heavy species equations

The heavy species means all plasma species except the electrons, i.e., grounded and excited argon atoms, and argon ions. In this heavy species module, an approximation is used. If there are totally Q heavy species, only the transports of Q − 1 species are described directly via the transport equation, i.e.,

The transport of the Q^th heavy species, fixed as the feedstock molecule, is controlled via the mass conversation law, i.e., Here, ρ is the total mass density of heavy species, also named as the mixture density. The contribution of electrons to the gas mixture density is omitted due to its tiny mass weight, as compared to the heavy species. This mixture density is calculated from the SPF module. As known, it is unchangeable in the incompressible flow. Nevertheless, in the compressible flow, it can be both spatially and temporally altered via the compressibility. w_k is the mass fraction of species k, and u is the velocity of background gas flow. j_k is the diffusive and drift flux of species k and expressed as where V_k is the diffusive and drift velocity of species k, z_k is elementary charge number of charged species k, and E is the electrostatic field of spatial charge density. The multi-component mass diffusion mechanism^[38] is taken into account in this model. D_k,m is the diffusion coefficient of the gas mixture and expressed as where x_j is the molar fraction of species j and D_k,j is binary diffusion coefficient of the Chapman–Enskog theory. μ_m is the mobility of ions, given by the Einstein’s relation. In the Einstein’s relation, the ion temperature is assumed room temperature. The validity of this assumption will be discussed in the section of conclusion and further remarks. The source term of heavy species transport equation is expressed as where M_k is molecular weight, r_j is the rate of reaction j that creates/consumes species k, N is the reaction number, and l_k,j is the particle number of species k created or lost per collision of reaction j. The reaction rate, r_j, is expressed as In this formula, k is the rate coefficient, S is the number of reactants, ν is the stoichiometric coefficients, and c is the molar concentration of reactants. The boundary condition of heavy species equation considers the components of the Bohm’s velocity, species surface kinetics and advection of gas flow.

2.2.3. Electromagnetic equations

The electromagnetic field in this discharge is calculated via the following equation,

which is deduced from the Maxwell’s equation (details about this deduction can be found in the appendix). Here, j is the imaginary unit and ω is the angular frequency, ε₀ and ε_r are the vacuum permittivity and the relative permittivity of specific media, μ₀ and μ_r are the vacuum permeability and the relative permeability of specific media, A is the magnetic vector potential and J_a is the applied coil current. σ is the electron conductivity and calculated as, where n_e is the electron number density; m_e and q are the electron mass and charge, respectively; and ν_e is the collision frequency of electrons with neutrals. The magnetic insulation condition, n × A = 0, is taken as boundary for this electromagnetic equation. Note that the general inductive discharges operate at two modes, i.e., E and H modes. In our present study, only the H mode is discussed. The above electromagnetic vector equation presented actually can describe both the E and H mode discharges. We therefore in the appendix illustrate explicitly how to use this equation to describe the H mode discharge.

2.2.4. Electrostatic equations

The Poisson’s equation is used to calculate the electrostatic field,

where ρ_V is the charge density of the space. The boundary condition of zero potential is used at chamber surfaces that are grounded, such as the sidewall and the bottom, while the condition of surface charge accumulation, i.e., is used at the surface of dielectric window. Here, ρ_s is the surface charge density, accumulated by the fluxes of arriving ion and electron. ρ_n is the normal unit vector of dielectric window surface. D is the electric displacement vector and related to the electrostatic field via the formula, D = ε_rε₀E.

2.2.5. The Navier–Stocks equation for gas flow

When the temperature variation in the gas flow is insignificant and the flow velocity is low, i.e., with the Mach number less than 0.3, the flow can be assumed incompressible.^[39] The Navier–Stocks equation can therefore be used, i.e.,

Here μ is the gas dynamic viscosity and u is the mass weighted flow velocity. In the incompressible flow, the pressure, p, has actually no physical meaning, while the pressure gradient formed is useful for it drives the gas flow. ρ is the total mass density of gas mixture and unaltered in the incompressible flow.

Slip velocity at the walls is negligible in this work due to the selected high-pressure range and no-slip boundary condition is therefore used. At the gas inlet, fixed inlet velocities are taken as boundary condition. At the chamber outlet, fixed gas pressure boundary condition is used. In the incompressible flow, the gas mixture density is fixed in the whole space, whose value is given via the state equation of ideal gas with the outlet pressure and the room gas temperature. Here, the validity of assuming room gas temperature will be discussed in the section of discussion and further remarks.

2.2.6. Momentum conservation equation of gas flow

When the compressibility is taken into account, the continuity and momentum conservation equations are used for the gas flow, i.e.,

No-slip velocity boundary condition is used at the walls and fixed pressure boundary condition is used at the outlet. At the inlet, the boundary condition of gas flow rate, in a unit of sccm, is taken. In the compressible flow, the pressure itself is meaningful and related to the gas mixture density via the state equation of ideal gas, again at the condition of room gas temperature.

In Fig. 4, the velocity profiles of incompressible flow are plotted at the inlet velocity of 200 m/s, the power of 200 W, and the pressure of 20 mTorr. Two peaks occur in the radial velocity profile, which are located under the inlet nozzle and at the substrate corner, respectively. The axial velocities are maximal under the inlet nozzle and second largest in the annular passage between the substrate and the sidewall. As the inlet velocity is axial oriented, the axial velocity magnitude is about one factor larger than the radial velocity. In Fig. 5, the advective mass flux profiles of metastables at the same discharge condition are plotted. Two maxima occur in the radial advective flux, which is similar to the radial velocity. Nevertheless, a tilted “bulb” shape is seen from the axial advective flux profile that is quite different with the morphology of axial velocity. The radial fluxes above the substrate edge are distributed similar to the radial velocity the most. In a small area just beneath the nozzle, where the radial and axial velocities are both peaked, the two components of advective flux, nevertheless, are both minimal. This is because the metastables densities at the locations where the velocities of the two components peak, are quite low. Instead, a blue peak of radial flux and the head of an axial flux bulb occur more in the bulk chamber due to the high metastables densities therein. Of interest is in the outlet passage, where the metastables densities are supposed to be low, prominent axial fluxes are observed. This is related to the novel metastables density profile formed by the gas flow process.

Fig. 4.

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	Fig. 4. (color online) Profiles of radial (a) and axial (b) velocity components for the inlet velocity of 200 m/s, the power of 200 W and the pressure of 20 mTorr, at the incompressible gas flow.

Fig. 5.

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	Fig. 5. (color online) Profiles of radial (a) and axial (b) advective fluxes of metastables for the inlet velocity of 200 m/s, the power of 200 W and the pressure of 20 mTorr, at the incompressible gas flow.

In Fig. 6, the metastables density profiles, when the gas flow, i.e., advection, is not included and at the inlet velocity of 200 m/s, are plotted at the discharge condition of 20 mTorr and 200 W. The density profile without the advection is symmetric with respect to the half height plane of the chamber. At the 200 m/s inlet velocity, this symmetry is destroyed by the advection. In particular, the density profile in the upstream chamber is horizontally compressed inward for the radial diffusion in this region is counteracted by the advective component. In the downstream, the profile is horizontally prolonged outward, which, we believe, is a synergistic effect of radial and axial advective fluxes. The densities of metastables, as well as its axial advective fluxes, near the outlet passage are increased, along with the prolonged density profile in the downstream. The density profile border is in total shifted away from the chamber wall. As a result, the atoms loss on the wall is reduced and the metastables density magnitude is slightly increased.

Fig. 6.

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	Fig. 6. (color online) Metastables density profiles when the gas flow is not included (a) and at 200 m/s inlet velocity, at the discharge condition of 20 mTorr and 200 W.

In Fig. 7, the metastables density profiles when the gas flow is not included and at the inlet velocity of 200 m/s are plotted at the discharge condition of 50 mTorr and 200 W. The metastables density is peaked under the dielectric window, different with the 20 mTorr, because the multistep and Penning ionizations that consume metastables happen more frequently at this pressure. At high pressure, the deformations of density profile, i.e., upstream compression and downstream prolong, via the advection are more obvious. The strong influence of advection on metastables is because the peak of its density profile is close to the gas flow path and the interaction of diffusion and advection is stronger. Moreover, prominent high metastables densities are seen in the passage, as illustrated by a red dashed line circle, formed by strong axial advective fluxes therein. As noted, these high metastables densities of passage could cause diffusions to both the substrate and the sidewall, forming novel double-way diffusion scheme.

Fig. 7.

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	Fig. 7. (color online) Metastable density profiles when the gas flow is not included (a) and at 200 m/s inlet velocity, at the discharge condition of 50 mTorr and 200 W.

In Figs. 8(a) and 8(b), the metastable density profiles without advection and at the inlet flow rate of 2000 sccm in the scope of compressible flow are plotted, at the discharge condition of 20 mTorr and 200 W. The density profile without advection is symmetric and center peaked. At 2000-sccm flow rate, all distortions of neutral density profile occurred in the incompressible gas flow, i.e., inward shift of upstream border, outward extension of downstream border, and novel double-way diffusion in the outlet passage, are observed in the compressible flow as well. Besides, the metastable density profile becomes localized and the density magnitude decreases obviously. To explain this new change, in Fig. 8(c) the metastable density profile without advection while at the high pressure of 40 mTorr is plotted, still at the power of 200 W. It is seen that except for the peripheral density profile distortions, the bulk density profile and the peak magnitude of metastables at 40 mTorr when excluding advection are similar to the case of 2000 sccm. It implies that the compressible gas flow seemingly has the same effect as the pressure. To assess this guess, we plot the electronic excitation rate profiles of the above three cases in Fig. 9. It is seen that the peripheral excitation rate profile shifts toward the dielectric window and the peak magnitude of excitation rate increases whenever increasing the flow rate or the pressure, as compared to the 20 mTorr without advection. The electronic excitations are known to occur at high threshold, 11.56 eV and the electron temperature decreases with the pressure due to in-elastic electron collisions. These two effects cooperatively result in the shift of its rate profile toward the coil.

Fig. 8.

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	Fig. 8. (color online) Metastable density profiles without advection (a) and at 2000 sccm inlet flow rate (b), at the discharge condition of 20 mTorr and 200 W. Also plotted is the metastable density profile of 40 mTorr and 200 W without advection (c) for comparing its profile with the metastable density profile of 2000 sccm (b).

Fig. 9.

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	Fig. 9. (color online) Electronic excitation rates without advection (a) and at 2000 sccm inlet flow rate (b), at the discharge condition of 20 mTorr and 200 W. Also plotted is the excitation rate at 40 mTorr and 200 W without advection (c) for comparing its profile with the excitation profile of 2000 sccm (b).

In Fig. 10, the peaked values of both electron density and temperature profiles versus the flow rate are plotted, at the discharge condition of 20 mTorr and 200 W. The electron density linearly increases while the temperature monotonically deceases with the flow rate. These trends of electron density and temperature against the flow rate are quite similar to their variations with the pressure,^[40] which again testifies that the compressible gas flow and the pressure have the same effects on the plasma, except for the neutral density profile distortions caused by incompressible gas flow. Besides, the variations of electron density and temperature with flow rate, this model predicted, qualitatively agree well with the experimental work of Doh and Horiike,^[24] where the electron density decreased and electron temperature increased with the residence time in argon inductive coupled plasma. Here, we believe that the inverse relation between the residence time and the gas flow rate still works.

Fig. 10.

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	Fig. 10. (color online) Peaked values of electron density and temperature profiles versus the gas flow rate at 20 mTorr and 200 W.

To explain the above discovering, in Fig. 11 the calculated gas mixture density profiles of 2000 sccm, with widely and finely resolved legends, respectively, are plotted, at the discharge condition of 20 mTorr and 200 W. At the wide and complete legend resolution, i.e., in Fig. 11(a), the mixture density peaks at the inlet nozzle and sinks at the pumping port. The mixture density minimum at the port, 4.276 × 10⁻⁵ kg/m³, is calculated via the outlet pressure value, i.e., 20 mTorr, which serves as the boundary condition of the flow equation, based on the state relation of ideal gas. As seen, the most prominent variations of gas mixture density occur at the inlet nozzle and outlet port while it is more or less unchanged in the chamber bulk. At the fine but incomplete color legend resolution, i.e., in Fig. 11(b), the mixture density of bulk chamber lies in a small range of 8.32 kg/m³ ∼ 8.4 × 10⁻⁵ kg/m³. The three yellow circles in this figure enclose the mixture densities that that are out of the limits of the fine resolution legend. In the experiments, the pressure is always measured at the locations near the outlet port.^[40] The simulations reveal that this measurement can severely underestimate the bulk pressure when the pressure is measured at the locations near the outlet. This observation is qualitatively agreeable well with the work of Ref. [41], where the neutral argon density profile was determined in a helicon source by using the spectroscopic measurements and the collisional-radiative model, and the meaning of the measured neutral profile on the ionization degree of bulk plasma was discussed.

Fig. 11.

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Fig. 11. (color online) Calculated gas mixture density profiles with wide (a) and fine (b) resolution legends, at the discharge condition of 20 mTorr, 200 W, and 2000 sccm. In Fig. 11(b), a special way of demonstrating this fine data structure around the chamber periphery is used, i.e., a small range of data, 8.32 × 10−5 ∼ 8.4 × 10−5, is used to define the color bar while a wide range of actual data, 4.3 × 10−5 ∼ 8.4 × 10−5, is represented by this color bar.

In Fig. 12(a), the center mixture density versus the flow rate is plotted, at the discharge condition of 20 mTorr and 200 W, together with the outlet mixture density given by the pre-set outlet pressure of 20 mTorr, 4.276 × 10⁻⁵ kg/m³, that is unvaried with flow. In Fig. 12(b), the mixture density versus pressure without gas flow is plotted. The center mixture density significantly increases with the flow rate, as shown in Fig. 12(a). It resembles the trend of mixture density with pressure in Fig. 12(b). This figure interprets why the compressible flow has the same effect on plasmas as the pressure, i.e., it actually increases bulk pressure. Note that the present study is based on the boundary condition that fixes pressure at the outlet position, which is frequently used in relevant studies.^[22,23] In the real industry or academia lab, the complicated pumping system is installed and the outlet boundary condition is therefore more complex than the fixed pressure. Application of this new outlet boundary condition, i.e., pumping speed and configuration, into the gas flow model should amend the above conclusions to some extent. This is certainly a new interesting work for future.

Fig. 12.

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	Fig. 12. (color online) (a) Center mixture density versus the flow rate and the outlet mixture density that is determined by the outlet pressure of 20 mTorr and unvaried with flow rate, at 20 mTorr and 200 W. The mixture density versus pressure without gas flow at the power of 200 W is plotted(b).

The velocity magnitude profiles at the inlet velocity of 100 m/s and at the inlet flow rates of 200 sccm and 2000 sccm are sequentially plotted in Fig. 13, at 20 mTorr and 200 W. When the flow is incompressible, and the inlet velocity is 100 m/s, the velocity magnitude peaks at the inlet and the peaked value is 112.8 m/s. The model predicts that the velocity magnitude profile is not changed, but the peaked value linearly increases on increasing the inlet velocity (see Fig. 14(b)). When the flow is compressible and the inlet flow rates are low, e.g., at 200 sccm, the velocity magnitude profile (with a small peaked value of 17.6 m/s) in Fig. 13(b) is similar to that of the incompressible flow. When increasing the flow rate, e.g., to 2000 sccm, the velocity magnitude profile is changed, with its peak shifting from the inlet to the pump port, as shown in Fig. 13(c). This trend is more clear in Fig. 14(a), where the velocity magnitude of pump port increases faster with flow rate than at the inlet.

Fig. 13.

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	Fig. 13. (color online) Velocity magnitude profiles at 100 m/s inlet velocity and incompressible flow (a), and at 200 sccm (b) and 2000 sccm (c) inlet flow rates and compressible flow, at 20 mTorr and 200 W.

The different characteristics of velocity magnitude profiles at the two flows are analyzed. At the incompressible flow with fixed mixture density, the gas flows freely across the chamber. Nevertheless, in the compressible flow the mixture density is varied, and the fluid becomes elastic. It can efficiently respond to the external disturbance via its pressure alteration, proportional to the mixture density alteration, as shown in Fig. 11. At the inlet, the direction of mixture density gradient is along the velocity direction, i.e., from zero to the maxima. So, the fluid elasticity resists the flow, and the velocity magnitude is reduced. At the pump port, the direction of mixture density gradient is against the velocity direction, from the minimum to zero. So, the fluid elasticity accelerates the flow, and the velocity magnitude is increased. This mechanism can be described by the definition of compressibility, ∇ · u = −∇ ln ρ · u, analogous to the Poisson’s equation that describes the variation of active field with the source term. As seen, at the inlet, an equivalent negative source is formed for the velocity field, while at the port an equivalent positive source term is formed. In addition, at 2000 sccm, obvious gradient of axial velocity is observed in the pumping port channel and meanwhile a maxima of velocity magnitude is formed at the port outlet, as compared to the case of 200 sccm. This is because the gradient of mixture density in the channel holds the same direction as that of the port outlet, both increasing the velocity field; but the increasing rate at the port outlet is stronger, due to the strong mixture density there. Accordingly, the mixture density gradient in the channel at 2000 sccm is stronger than the 200 sccm, as the model predicted. This is a positive feedback process, as the above compressibility equation predicted.

Fig. 14.

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	Fig. 14. (color online) (a) Velocity magnitudes at the inlet and the pump port of the chamber versus the flow rate at compressible flow and (b) velocity magnitude that is peaked under the inlet nozzle in the profile versus the inlet velocity at incompressible flow, at 20 mTorr and 200 W.

The influence of the compressibility on velocity magnitude profile becomes less important at high pressure, like 50 mTorr in Fig. 15 where the velocity magnitudes at the inlet and pump port both significantly increase with the flow rate. This is because it is more difficult to compress dense gases. As checked, the maximal pressure variation caused by the inlet flow rate of 2000 sccm at 20 mTorr is indeed larger than that of 50 mTorr, i.e., 4.0 Pa versus 3.5 Pa. In addition, the velocity magnitudes at 50 mTorr is totally decreased, as compared with the 20 mTorr, due to the high viscosity at high pressures.

The flow velocity profile is tightly related with the residence time calculations and depicts the picture of plasma species residing in the chamber. Moreover, the velocity magnitude profile evolution implies that the residence time of background gas in the chamber might not be proportional to the gas flow rate, which is contradictory to the present recognition that based on one simple formula, τ_r = pV/(p_atmQ), where p is the gas pressure, V is the plasma volume, Q is the gas flow rate in units of standard cubic centimeter per minute (sccm), and Q_atm is defined as atmosphere pressure.^[42] The literature review revealed the importance of residence time in both the plasma dynamics and etching and deposition process.

Fig. 15.

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	Fig. 15. (color online) Velocity magnitudes at the inlet and the pump port of the chamber versus the flow rate at compressible flow and 50 mTorr and 200 W.