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Numerical Studies of the Low Pressure RF Plasma

Profile image of David VenderDavid Vender

1990, ANU

Abstract

A one-dimensional, electrostatic particle-in-cell code with non-periodic boundary conditions is used to simulate a low pressure capacitive rf plasma created between two planar electrodes. Ion and electron motion is included and ionising collisions by energetic electrons allow a steady state to be reached and maintained. Realistic values of mi/me are used but there is no attempt to model a real gas and, except for ionisation, no binary collision processes are considered. The simulation plasma is generated by driving one boundary with a sinusoidal rf voltage at a frequency of 10 MHz. The effects of scaling on the steady state, the structure and the impedance of the resulting discharge are investigated. Changes resulting from varying the amplitude of the driving voltage are examined and scaling laws for the plasma potential, electron density and power loss obtained. Sheath heating is shown to be the main electron heating process and power balance is checked. The structure of the rf sheath obtained in the simulation is compared to theoretical models of both the current driven and the voltage driven sheath. Disagreement in the maximum sheath width between the simulation and the model is ascribed to neglect of the period of sheath collapse and the use of an idealised electron density profile in the model. Sheath scaling is shown to underlie the variation of electron density and temperature with rf voltage. The electron sheath interaction is examined and found to differ considerably from current theoretical models. In the range of parameters investigated, it is essential to consider the distortion of the electron velocity distribution in the sheath. A beam-like distribution is observed when the sheath velocity changes rapidly near the time of sheath collapse and an instability develops when electrons are accelerated into the plasma as the sheath expands.

Figures (53)
Figure 1.1: Profiles obtained with (a) isotropic etching and (b) anisotropic etching.
Figure 1.1: Profiles obtained with (a) isotropic etching and (b) anisotropic etching.
Figure 1.2: Etch reactor configurations. (a) Symmetric system, (b) Reactive ion etchine conficuration. so that higher densities of charged particles and chemically active species result from
Figure 1.2: Etch reactor configurations. (a) Symmetric system, (b) Reactive ion etchine conficuration. so that higher densities of charged particles and chemically active species result from
pressure of 0.3 mTorr. At this pressure there is a range of input power at which the
pressure of 0.3 mTorr. At this pressure there is a range of input power at which the
sure. The traces were obtained in the above-mentioned etching reactor (Figure 1.3) by
sure. The traces were obtained in the above-mentioned etching reactor (Figure 1.3) by
Figure 2.2: The evaluation of velocity ‘leaps’ over the evaluation of position. is integrated using the leap-frog method schematically illustrated in Figure 2.2. The After the force acting on each particle has been calculated, the equation of motion velocity and position of each particle are determined at alternate half-integral time steps | Time Integration
Figure 2.2: The evaluation of velocity ‘leaps’ over the evaluation of position. is integrated using the leap-frog method schematically illustrated in Figure 2.2. The After the force acting on each particle has been calculated, the equation of motion velocity and position of each particle are determined at alternate half-integral time steps | Time Integration
Figure 3.1: The total number of electron sheets, the average plasma potential nd the mean electron energy shown during the development to the steady 1836 and effective neutral pressure 20 mTorr).
Figure 3.1: The total number of electron sheets, the average plasma potential nd the mean electron energy shown during the development to the steady 1836 and effective neutral pressure 20 mTorr).
Table 3.2: The changes in plasma parameters resulting from individually hoosing diagnostics, in particular n., €, and P,, and reducing the scaling parameters the changes in the plasma parameters resulting from individually halving each scaling
Table 3.2: The changes in plasma parameters resulting from individually hoosing diagnostics, in particular n., €, and P,, and reducing the scaling parameters the changes in the plasma parameters resulting from individually halving each scaling
Table 3.1: Fluctuation amplitudes of the most important plasma parameters
Table 3.1: Fluctuation amplitudes of the most important plasma parameters
Figure 3.2: The spatio-temporal evolution of the electric field averaged over ten rf cycles.
Figure 3.2: The spatio-temporal evolution of the electric field averaged over ten rf cycles.
Figure 3.4: The distribution of ions in phase space. sound speed is indicated in the figure showing that 3.3.1 is satisfied in the equality.
Figure 3.4: The distribution of ions in phase space. sound speed is indicated in the figure showing that 3.3.1 is satisfied in the equality.
Figure 3.5: The average electron energy (squares) and the average speed of the ions arriving at the sheath (circles) as a function of applied rf voltage. Figure 3.5: The average electron energy (squares) and the average speed of The arrow indicates the ion sound speed.
Figure 3.5: The average electron energy (squares) and the average speed of the ions arriving at the sheath (circles) as a function of applied rf voltage. Figure 3.5: The average electron energy (squares) and the average speed of The arrow indicates the ion sound speed.
Figure 3.6: The electron velocity distribution averaged over twenty rf cycles.
Figure 3.6: The electron velocity distribution averaged over twenty rf cycles.
Figure 3.7: The central electron density as a function of applied voltage.
Figure 3.7: The central electron density as a function of applied voltage.
shown together with the applied voltage V.
shown together with the applied voltage V.
mplies that the total capacitance is independent of the applied voltage (when the higher
mplies that the total capacitance is independent of the applied voltage (when the higher
This can be seen in Figure 3.12 where the field energy is shown with the total kinetic oscillates at double the rf frequency. This periodic transfer of power between the plasma
This can be seen in Figure 3.12 where the field energy is shown with the total kinetic oscillates at double the rf frequency. This periodic transfer of power between the plasma
power lost in acceleration of electrons in the sheath P,,, (0).
power lost in acceleration of electrons in the sheath P,,, (0).
Figure 3.14: The spatio-temporal distribution of ionising collisions collected over 20 rf cycles. Figure 3.14: The spatio-temporal distribution of ionising collisions collected
Figure 3.14: The spatio-temporal distribution of ionising collisions collected over 20 rf cycles. Figure 3.14: The spatio-temporal distribution of ionising collisions collected
low voltage V,z S 200 V. This can be seen in Figure 3.15 where the maximum sheath which accompanies the increase in V,;. While a higher voltage leads to an expansion
low voltage V,z S 200 V. This can be seen in Figure 3.15 where the maximum sheath which accompanies the increase in V,;. While a higher voltage leads to an expansion
limit of validity of the analytic solutions.
limit of validity of the analytic solutions.
were obtained by averaging over 100 rf cycles.
were obtained by averaging over 100 rf cycles.
motion to estimate the components of the current. To get an equation for the sheath to values of H, V,, and J, that match the simulation values for an excitation voltage of A solution can then be obtained by truncating the sum 4.3.2.18 at the desired level of
motion to estimate the components of the current. To get an equation for the sheath to values of H, V,, and J, that match the simulation values for an excitation voltage of A solution can then be obtained by truncating the sum 4.3.2.18 at the desired level of
Figure 4.5: The instantaneous position of the sheath at the live electrode un- der standard conditions. The circles are scaled experimental measurements civen in Reference [1771]. Figure 4.5: The instantaneous position of the sheath at the live electrode un-
Figure 4.5: The instantaneous position of the sheath at the live electrode un- der standard conditions. The circles are scaled experimental measurements civen in Reference [1771]. Figure 4.5: The instantaneous position of the sheath at the live electrode un-
the ion and electron densities in the sheath are shown at two times during the sheath for standard conditions. The densities have been averaged over 50 rf cycles. transition layer at the sheath ‘edge’ is not small and there are both a finite electric
the ion and electron densities in the sheath are shown at two times during the sheath for standard conditions. The densities have been averaged over 50 rf cycles. transition layer at the sheath ‘edge’ is not small and there are both a finite electric
Figure 4.7: The plasma potential (solid line) and the potential across the sheath at the grounded electrode (dashed) shown during one rf cycle for FF: 4 Mtr "wi. 2... ... 1... kL... 222k BA wR LLY © ....34..—
Figure 4.7: The plasma potential (solid line) and the potential across the sheath at the grounded electrode (dashed) shown during one rf cycle for FF: 4 Mtr "wi. 2... ... 1... kL... 222k BA wR LLY © ....34..—
plasma obtained with standard conditions. The individual points correspond to phase a chosen number of particles and four such trajectories are shown in Figure 5.1 for a
plasma obtained with standard conditions. The individual points correspond to phase a chosen number of particles and four such trajectories are shown in Figure 5.1 for a
Figure 5.2: Electron velocity distribution at three phases in the rf cycle 6 = 0 (Solid line), 6 = x/2 (dashed) and 0 = m (dash-dotted). The date + a
Figure 5.2: Electron velocity distribution at three phases in the rf cycle 6 = 0 (Solid line), 6 = x/2 (dashed) and 0 = m (dash-dotted). The date + a
potential shown in Figure 5.3a. The spectrum was derived from the potential in the of plasma oscillations so these can be clearly seen in the frequency spectrum of the
potential shown in Figure 5.3a. The spectrum was derived from the potential in the of plasma oscillations so these can be clearly seen in the frequency spectrum of the
vanishes. Between the centre and the edge of the ion sheath, the ion distributions have
vanishes. Between the centre and the edge of the ion sheath, the ion distributions have
to be reasonably close to thermal although the wings are slightly hotter than the bulk. In Figure 5.5b, the distribution at the centre is shown for an ion mass m; = 20m,. The
to be reasonably close to thermal although the wings are slightly hotter than the bulk. In Figure 5.5b, the distribution at the centre is shown for an ion mass m; = 20m,. The
by the power density j;-E shown in Figure 5.6a. The dark areas in the figure represent the initial phase during the rf cycle at which it enters the sheath.
by the power density j;-E shown in Figure 5.6a. The dark areas in the figure represent the initial phase during the rf cycle at which it enters the sheath.
plasmas.
plasmas.
shown in Figure 5.8 that power deposition in the sheath is the only significant electro
shown in Figure 5.8 that power deposition in the sheath is the only significant electro
Table 5.2: Comparison of sheath heating power loss with estimates from the theoretical model for a sinusoidal current. The values of power are given
Table 5.2: Comparison of sheath heating power loss with estimates from the theoretical model for a sinusoidal current. The values of power are given
Figure 5.9: Electron velocity distributions in the sheath. The grey areas
Figure 5.9: Electron velocity distributions in the sheath. The grey areas
standard conditions. It is clearly complex although the general form of a broad plateau
standard conditions. It is clearly complex although the general form of a broad plateau

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