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Fluid dynamic modelling and experimental diagnostics of an inductive high density plasma source (ICP)
Surface and Coatings Technology 142᎐144 Ž2001. 526᎐530
Fluid dynamic modelling and experimental diagnostics of an inductive high density plasma source ž ICP/ P. Scheubert U , P. Awakowicz, R. Schwefel, G. Wachutka Institute for Physics of Electrotechnology, Munich Uni¨ ersity of Technology, Arcisstr. 21, D-80290 Munchen, Germany ¨
Abstract A planar inductively coupled plasma source ŽICP. used for diamond deposition experiments was characterised theoretically as well as experimentally. In the typical process window Žgas pressure 400 Pa. the discharge shows strong local variations of temperature and electron density. Especially, the homogeneity of the plasma in dependence of the process pressure and coil configurations was investigated. Langmuir probe measurements for noble gas discharges in a wide pressure range were compared with theoretical data from a reactor model. A 2D-fluid plasma model, coupled self-consistently with an electrodynamic model, was used to calculate the theoretical results. Experiment and simulation show very good agreement with a wide range of parameters. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: ICP; Fluid dynamic plasma model; Langmuir probe; Transport processes
1. Introduction
Inductively coupled plasmas ŽICP. are especially used for applications which depend on a high electron density. Another typical advantage leading to a growing interest in this type of discharge is the low energy of ions leaving the discharge. For sensitive applications, like the deposition of diamond, the self bias voltage and therefore, the ion energy must not exceed a certain level w5x in order to guarantee optimal growth conditions. To enable an optimised design of discharge vessels, it
is desirable to understand basic phenomena like electron and ion generation as well as particle transport. The goal of the simulation is to gain information about particle density and temperature distributions in dependence of external parameters like neutral gas pressure and discharge power. However, the discharge geometry, as well as the geometry and location of the coils also have a crucial influence on the process. A model that supplies all these parameters can be derived from the Boltzmann equation w3,4x. The plasma is considered as a mixture of two fluids Želectrons and ions. whose behaviour is described by means of hydrodynamic equations.
2. A two fluid model U
Corresponding author. Fax: q49-89-289-23141. E-mail address: [email protected] ŽP. Scheubert..
To describe the transport processes in the discharge
0257-8972r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 7 - 8 9 7 2 Ž 0 1 . 0 1 0 5 3 - 2
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
a set of conservation equations was used. For the ions, conservation of mass Eq. Ž1. and momentum Eq. Ž2. were considered: ⭸n i q ⵜ ⭈ Ž n iª ¨ i . s n e ⭈ i z Ž Te . ⭸t
Ž1.
⭸ Ž n i m iª i . ª q ⵜ ⭈ Ž n i m iªª i i . s en i Ey n i m iªª i ¨ i ⭸t
Ž2.
Here n i , ª i denote ion number density and velocity, respectively. Particle generation is given by n e ⭈ i z ŽTe . where n e is the electron number density and i z the ionisation frequency, which is a function of the electron temperature Te . The ion mass is m i , the electric ª field is E and i is the frequency of inelastic collisions with neutral atoms. Each ion carries the elementary charge e. For electrons, conservation of mass Eq. Ž3. and energy Eq. Ž4. are considered: Ž3.
⭸n e ª ªª q ⵜ ⭈ ⌫e s yeE⭈ j e y Pc q Ph ⭸t
Ž4. ª
The electron particle flux density j e is calculated by a drift-diffusion approximation: je s
ne e ª 1 Ey ⵜn e kTe m ee m ee
where k denotes the Boltzmann constant, m e the electron mass and e the elastic electron neutral collision frequency. Energy loss caused by collisions with neutral particles and heating by an externally applied RF-field are given by Pc and Ph , respectively. The mean elecª tron energy is and the electron energy flux density ⌫e consists of a convective and thermal conductivity term: ª
⌫e s
ª 5 5 kTe kT je y ⵜkTe 2 e 2 m ee
Ž6.
Electric interaction between electrons and ions is given by Poisson’s equation: ⌬⌽ s
e Ž n y ni . 0 e
Ž7.
here, ª 0 is the permittivity of the vacuum. The electric field E is derived from a scalar electrostatic potential, ⌽, by: ª
⭸n e ª q ⵜ ⭈ j e s n e ⭈ i z Ž Te . ⭸t
ª
527
Ž5.
Es yⵜ⌽
Ž8. ª
It is assumed that the RF-field E induced by coil ª currents is orthogonal to the electrostatic field, E and therefore, it can be treated separately. 2.1. Electrodynamic model In order to calculate the heating of the plasma by the externally applied RF-field, an electrodynamic model w2x derived from Maxwell’s equations was used. Assuming time harmonic dependence for all quantities
Fig. 1. Simulation geometry and shape of induced electric fieldlines: ŽA. is the discharge vessel; ŽB. the quartz window; and ŽC. the region coil region.
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
528
of the form expŽ i t ., the differential equation describª ing the RF-induced field, E, is: ª
ª
ª
⌬ Eq k 2 Es i 0 j with k 2 s r
2 y i 0 c 02
Ž9.
Here, c 0 s Ž 0 0 .y1 r2 is the velocity of light and 0 the permeability of free space. The current density in ª the coils is given by j. Infinite electrical conductivity is assumed for the conducting walls. For the plasma, the cold plasma approximation w4x was used: s e
ne e2 ne e2 y i 2 2 m e Ž e q . m e Ž e2 q 2 .
Ž 10 .
The resulting power transferred to the plasma is then given by: Pc s ᑬ 4 < E⌰ < 2
Ž 11 .
3. Results All calculations presented here have been carried out for a cylindrical reactor vessel Žregion A in Fig. 1. with a radius R s 100 mm and a height H s 40 mm. The upper boundary of the chamber is a quartz window Žregion B, thickness 10 mm.. A set of three cylindrical coils Ž‘1’, ‘2’, ‘3’. is fixed in a distance of 10 mm above the quartz window. The coils can be powered separately with an RF-frequency of 27 MHz. Langmuir probe measurements were carried out in a radial direction. 3.1. Influence of coil configurations and neutral gas pressure Simulations and experiments with separately powered coils were performed in order to investigate the influence of different coil configurations. In Fig. 2 experimental and theoretical data are presented for the case of an argon discharge operating at a pressure of 10 Pa. Only one of the coils was powered, respectively. The maximum of the electron density is correlated with the position of the driven coil. Theory and experiment are in excellent agreement. In order to investigate the pressure dependence of this effect a pressure variation was carried out for the case, that coil ‘2’ was driven. The results are given in Fig. 3. For low pressure Ž0.5 Pa. the radial electron density profile has the shape of a Bessel function according to simple diffusion theory. For increasing pressure the discharge becomes localised in proximity to the powered coil. The off-axis maximum means that the shape of the electron density
Fig. 2. Radical electron density profiles measured and calculated for an argon discharge operating at 10 Pa. The coils ‘1’, ‘2’, ‘3’ were powered separately.
distribution forms a torus and correlates with deposition experiments which show a maximum deposition rate in this region. 3.2. Design study for optimum homogeneity The deposition process Ždiamond films. studied in
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
529
this work w1x requires a electron density in the order of 10 11 cmy3 and a neutral gas temperature in the order of 2000 K. Such process conditions can be obtained by using argon as neutral gas background at a pressure of 400 Pa. As the previously discussed simulations and measurements showed in this pressure range the electron density distribution depends directly on the posi-
Fig. 4. Calculated electron density for different coil configurations. Radial electron density profiles are shown for the case of each coil powered separately and the resulting electron density when all three coils are powered. Results are given for 0.5, 5.0 and 50 Pa in argon.
Fig. 3. Radical electron density profiles measured and calculated for an argon discharge operating at pressures of 0.5, 2 and 5 Pa, respectively. The coil ‘2’ was powered.
tion of the coils. One might expect that a superposition of the different maxima located close to each coil leads to a more or less homogeneous density distribution over the radius. Simulations demonstrating that this approach does not lead to a improved homogeneity are shown in Fig. 4. For the pressure of 0.5 Pa the coil position has only a minor influence on the density distribution, for 5.0 Pa a
530
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
correlation of coil position and the corresponding density distribution can be observed, however, for the case that all coils are powered, the resulting distribution differs only slightly from the profile obtained by powering only the coil ‘2’. A slight improvement can be observed for the case of higher pressure Ž50 Pa. where a use of three powered coils produces an electron density profile which is significantly wider than the corresponding profile of each separately powered coil. Unfortunately the torus-like structure of the electron density profile remains leading to spatially inhomogeneous deposition conditions. This is in agreement with the experimental data.
4. Conclusion Calculated and experimental results for a planar, inductively coupled argon discharge were presented. Theoretical and measured density profiles are in excellent agreement. As shown from theory and experiment, the discharge becomes more localised in the proximity of the coils with increasing pressure. Further on, theoretical results
reveal that a superposition of several powered coils does not lead to a significantly more homogeneous discharge. For higher pressure values, differences in the shape of experimental and simulated density profiles indicate that the thermal conductivity of the plasma is overestimated by the model. A calibration of the electron-neutral collision frequency e which has crucial influence on the thermal conductivity might reduce this problem. References w1x P. Awakowicz, R. Schwefel, M. Werder, W. Kasper, Diamond deposition and plasma diagnostics in a 27 MHz inductive coupled reactor ŽICP., Diamond Relat. Mater. 6 Ž1997. 1816᎐1823. w2x J.S. Tolliver, E.F. Jaeger, L.A. Berry, D.B. Batchelor, Power deposition in high-density inductively coupled plasma tools for semiconductor processing, Phys. Plasmas 2 Ž6. Ž1995. 2597 ŽJune 1995.. w3x V.E. Golant, A.B. Zhilinsky, I.E. Sakharov, S.C. Brown, Fundamentals of plasma physics, John Wiley and Sons, 1980. w4x I.P. Shkarofsky, T.W. Johnston, M.P. Bachynski, The particle kinetics of plasmas, Addison᎐Wesley, 1996. w5x S. Uhlmann, T. Frauenheim, Diamond Relat. Mater. 5 Ž1995. 169.
Fluid dynamic modelling and experimental diagnostics of an inductive high density plasma source ž ICP/ P. Scheubert U , P. Awakowicz, R. Schwefel, G. Wachutka Institute for Physics of Electrotechnology, Munich Uni¨ ersity of Technology, Arcisstr. 21, D-80290 Munchen, Germany ¨
Abstract A planar inductively coupled plasma source ŽICP. used for diamond deposition experiments was characterised theoretically as well as experimentally. In the typical process window Žgas pressure 400 Pa. the discharge shows strong local variations of temperature and electron density. Especially, the homogeneity of the plasma in dependence of the process pressure and coil configurations was investigated. Langmuir probe measurements for noble gas discharges in a wide pressure range were compared with theoretical data from a reactor model. A 2D-fluid plasma model, coupled self-consistently with an electrodynamic model, was used to calculate the theoretical results. Experiment and simulation show very good agreement with a wide range of parameters. 䊚 2001 Elsevier Science B.V. All rights reserved. Keywords: ICP; Fluid dynamic plasma model; Langmuir probe; Transport processes
1. Introduction
Inductively coupled plasmas ŽICP. are especially used for applications which depend on a high electron density. Another typical advantage leading to a growing interest in this type of discharge is the low energy of ions leaving the discharge. For sensitive applications, like the deposition of diamond, the self bias voltage and therefore, the ion energy must not exceed a certain level w5x in order to guarantee optimal growth conditions. To enable an optimised design of discharge vessels, it
is desirable to understand basic phenomena like electron and ion generation as well as particle transport. The goal of the simulation is to gain information about particle density and temperature distributions in dependence of external parameters like neutral gas pressure and discharge power. However, the discharge geometry, as well as the geometry and location of the coils also have a crucial influence on the process. A model that supplies all these parameters can be derived from the Boltzmann equation w3,4x. The plasma is considered as a mixture of two fluids Želectrons and ions. whose behaviour is described by means of hydrodynamic equations.
2. A two fluid model U
Corresponding author. Fax: q49-89-289-23141. E-mail address: [email protected] ŽP. Scheubert..
To describe the transport processes in the discharge
0257-8972r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 7 - 8 9 7 2 Ž 0 1 . 0 1 0 5 3 - 2
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
a set of conservation equations was used. For the ions, conservation of mass Eq. Ž1. and momentum Eq. Ž2. were considered: ⭸n i q ⵜ ⭈ Ž n iª ¨ i . s n e ⭈ i z Ž Te . ⭸t
Ž1.
⭸ Ž n i m iª i . ª q ⵜ ⭈ Ž n i m iªª i i . s en i Ey n i m iªª i ¨ i ⭸t
Ž2.
Here n i , ª i denote ion number density and velocity, respectively. Particle generation is given by n e ⭈ i z ŽTe . where n e is the electron number density and i z the ionisation frequency, which is a function of the electron temperature Te . The ion mass is m i , the electric ª field is E and i is the frequency of inelastic collisions with neutral atoms. Each ion carries the elementary charge e. For electrons, conservation of mass Eq. Ž3. and energy Eq. Ž4. are considered: Ž3.
⭸n e ª ªª q ⵜ ⭈ ⌫e s yeE⭈ j e y Pc q Ph ⭸t
Ž4. ª
The electron particle flux density j e is calculated by a drift-diffusion approximation: je s
ne e ª 1 Ey ⵜn e kTe m ee m ee
where k denotes the Boltzmann constant, m e the electron mass and e the elastic electron neutral collision frequency. Energy loss caused by collisions with neutral particles and heating by an externally applied RF-field are given by Pc and Ph , respectively. The mean elecª tron energy is and the electron energy flux density ⌫e consists of a convective and thermal conductivity term: ª
⌫e s
ª 5 5 kTe kT je y ⵜkTe 2 e 2 m ee
Ž6.
Electric interaction between electrons and ions is given by Poisson’s equation: ⌬⌽ s
e Ž n y ni . 0 e
Ž7.
here, ª 0 is the permittivity of the vacuum. The electric field E is derived from a scalar electrostatic potential, ⌽, by: ª
⭸n e ª q ⵜ ⭈ j e s n e ⭈ i z Ž Te . ⭸t
ª
527
Ž5.
Es yⵜ⌽
Ž8. ª
It is assumed that the RF-field E induced by coil ª currents is orthogonal to the electrostatic field, E and therefore, it can be treated separately. 2.1. Electrodynamic model In order to calculate the heating of the plasma by the externally applied RF-field, an electrodynamic model w2x derived from Maxwell’s equations was used. Assuming time harmonic dependence for all quantities
Fig. 1. Simulation geometry and shape of induced electric fieldlines: ŽA. is the discharge vessel; ŽB. the quartz window; and ŽC. the region coil region.
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
528
of the form expŽ i t ., the differential equation describª ing the RF-induced field, E, is: ª
ª
ª
⌬ Eq k 2 Es i 0 j with k 2 s r
2 y i 0 c 02
Ž9.
Here, c 0 s Ž 0 0 .y1 r2 is the velocity of light and 0 the permeability of free space. The current density in ª the coils is given by j. Infinite electrical conductivity is assumed for the conducting walls. For the plasma, the cold plasma approximation w4x was used: s e
ne e2 ne e2 y i 2 2 m e Ž e q . m e Ž e2 q 2 .
Ž 10 .
The resulting power transferred to the plasma is then given by: Pc s ᑬ 4 < E⌰ < 2
Ž 11 .
3. Results All calculations presented here have been carried out for a cylindrical reactor vessel Žregion A in Fig. 1. with a radius R s 100 mm and a height H s 40 mm. The upper boundary of the chamber is a quartz window Žregion B, thickness 10 mm.. A set of three cylindrical coils Ž‘1’, ‘2’, ‘3’. is fixed in a distance of 10 mm above the quartz window. The coils can be powered separately with an RF-frequency of 27 MHz. Langmuir probe measurements were carried out in a radial direction. 3.1. Influence of coil configurations and neutral gas pressure Simulations and experiments with separately powered coils were performed in order to investigate the influence of different coil configurations. In Fig. 2 experimental and theoretical data are presented for the case of an argon discharge operating at a pressure of 10 Pa. Only one of the coils was powered, respectively. The maximum of the electron density is correlated with the position of the driven coil. Theory and experiment are in excellent agreement. In order to investigate the pressure dependence of this effect a pressure variation was carried out for the case, that coil ‘2’ was driven. The results are given in Fig. 3. For low pressure Ž0.5 Pa. the radial electron density profile has the shape of a Bessel function according to simple diffusion theory. For increasing pressure the discharge becomes localised in proximity to the powered coil. The off-axis maximum means that the shape of the electron density
Fig. 2. Radical electron density profiles measured and calculated for an argon discharge operating at 10 Pa. The coils ‘1’, ‘2’, ‘3’ were powered separately.
distribution forms a torus and correlates with deposition experiments which show a maximum deposition rate in this region. 3.2. Design study for optimum homogeneity The deposition process Ždiamond films. studied in
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
529
this work w1x requires a electron density in the order of 10 11 cmy3 and a neutral gas temperature in the order of 2000 K. Such process conditions can be obtained by using argon as neutral gas background at a pressure of 400 Pa. As the previously discussed simulations and measurements showed in this pressure range the electron density distribution depends directly on the posi-
Fig. 4. Calculated electron density for different coil configurations. Radial electron density profiles are shown for the case of each coil powered separately and the resulting electron density when all three coils are powered. Results are given for 0.5, 5.0 and 50 Pa in argon.
Fig. 3. Radical electron density profiles measured and calculated for an argon discharge operating at pressures of 0.5, 2 and 5 Pa, respectively. The coil ‘2’ was powered.
tion of the coils. One might expect that a superposition of the different maxima located close to each coil leads to a more or less homogeneous density distribution over the radius. Simulations demonstrating that this approach does not lead to a improved homogeneity are shown in Fig. 4. For the pressure of 0.5 Pa the coil position has only a minor influence on the density distribution, for 5.0 Pa a
530
P. Scheubert et al. r Surface and Coatings Technology 142᎐144 (2001) 526᎐530
correlation of coil position and the corresponding density distribution can be observed, however, for the case that all coils are powered, the resulting distribution differs only slightly from the profile obtained by powering only the coil ‘2’. A slight improvement can be observed for the case of higher pressure Ž50 Pa. where a use of three powered coils produces an electron density profile which is significantly wider than the corresponding profile of each separately powered coil. Unfortunately the torus-like structure of the electron density profile remains leading to spatially inhomogeneous deposition conditions. This is in agreement with the experimental data.
4. Conclusion Calculated and experimental results for a planar, inductively coupled argon discharge were presented. Theoretical and measured density profiles are in excellent agreement. As shown from theory and experiment, the discharge becomes more localised in the proximity of the coils with increasing pressure. Further on, theoretical results
reveal that a superposition of several powered coils does not lead to a significantly more homogeneous discharge. For higher pressure values, differences in the shape of experimental and simulated density profiles indicate that the thermal conductivity of the plasma is overestimated by the model. A calibration of the electron-neutral collision frequency e which has crucial influence on the thermal conductivity might reduce this problem. References w1x P. Awakowicz, R. Schwefel, M. Werder, W. Kasper, Diamond deposition and plasma diagnostics in a 27 MHz inductive coupled reactor ŽICP., Diamond Relat. Mater. 6 Ž1997. 1816᎐1823. w2x J.S. Tolliver, E.F. Jaeger, L.A. Berry, D.B. Batchelor, Power deposition in high-density inductively coupled plasma tools for semiconductor processing, Phys. Plasmas 2 Ž6. Ž1995. 2597 ŽJune 1995.. w3x V.E. Golant, A.B. Zhilinsky, I.E. Sakharov, S.C. Brown, Fundamentals of plasma physics, John Wiley and Sons, 1980. w4x I.P. Shkarofsky, T.W. Johnston, M.P. Bachynski, The particle kinetics of plasmas, Addison᎐Wesley, 1996. w5x S. Uhlmann, T. Frauenheim, Diamond Relat. Mater. 5 Ž1995. 169.