Detailed Model Results at 60 mTorr

4.4 RF Plasma Transport Models

4.4.1 Isothermal Diffusion Solution

4.4.2.3 Detailed Model Results at 60 mTorr

0 50 100 150 200 250 0

0.5 1 1.5 2 2.5

3x 10¹⁸

Time (µs) nm(m−3)

z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm

60 mTorr

0 50 100 150 200 250

0 0.5 1 1.5 2 2.5

3x 10¹⁸

Time (µs) nr(m−3)

z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm

60 mTorr

0 50 100 150 200 250

0 0.5 1 1.5 2 2.5

3x 10¹⁸

Time (µs) np(m−3)

z = −7 cm z = −6 cm z = −5 cm z = −4 cm z = −3 cm

60 mTorr

Figure 4.23: Ar I excited state population densities vs. time calculated by the model with pAr = 60 mTorr. The same simulation parameters were used as in Fig. 4.19. The left panel shows the 4s metastable state population density, the center panel shows the 4sresonant state population density, and the right panel shows the 4pstate population density.

Te(z, t) into a predicted ion saturation current.

−20 −15 −10 −5 0 0

1 2 3 4x 10¹⁹

z (cm) Electron Density (m−3)

−20 −15 −10 −5 00

1 2 3 4

Electron Temperature (eV)

Antenna edges

n_e T_e

t = 23.5 µs

−200 −15 −10 −5 0

4 8 12x 10¹⁹

z (cm) Electron Density (m−3)

−20 −15 −10 −5 00

1 2 3

Electron Temperature (eV)

Antenna edges

T_e

n_e

t = 118.5 µs

−20 −15 −10 −5 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

5x 10¹⁸

z (cm) Population Density (m−3)

n_m n_r n_p Antenna edges

t = 23.5 µs

−20 −15 −10 −5 0

0 2 4 6 8 10 12 14x 10¹⁷

z (cm) Population Density (m−3)

n_m n_r n_p Antenna edges

t = 118.5 µs

Figure 4.24: Spatial dependence of ne, Te,nm,nr, andnp calculated by the 1D numerical discharge model at p_Ar = 60 mTorr. The same simulation parameters were used as in Fig.

4.19. The density and temperature profiles are plotted at t = 23.5 µs (i.e., 5µs after the start of the simulation) and at t= 118.5µs, during the quasi-steady state phase.

saturation current curves shown in Fig. 4.19). The excited state populations fall moving away from the power deposition region. As in Fig. 3.11, the 4smetastable and 4sresonant manifolds have similar population densities because the collisional transition rate between these states is high (see Table 3.1) and most of the resonant line emission is re-absorbed. In the quasi-steady state, the 4plevel has a higher population density than the 4slevels inside the antenna, butnp decreases more rapidly moving downstream. It is interesting that even though three-body recombination was assumed to occur exclusively into the 4p level, it is the 4s levels whose population densities peak in the afterglow at around t≈ 225µs: this is because the effective radiative decay rate for the 4p state (A_{ps,ef f.} ∼ 3×10⁷ s⁻¹) far exceeds the three-body recombination rate (νr ∼10²–10⁴ s⁻¹). Balancing the population

180 190 200 210 220 230 240 250 0

0.5 1 1.5 2 2.5 3 3.5

4x 10¹⁹

Time (µs) Electron Density (m−3 )

With recombination No recombination z = −7 cm

z = −6 cm

z = −5 cm

Figure 4.25: Electron density decay in the afterglow from 1D discharge model calculations with and without three-body recombination included. The calculation parameters used were otherwise the same as in Fig. 4.19.

and depopulation rates per unit volume for the 4p state yields np/ne ≈νr/A_{ps,ef f.}, so the 4p population density in the afterglow is lower than the electron density by 3–5 orders of magnitude. On the other hand, A_{rg,ef f.} ≈ 5×10⁵ s⁻¹, so the resonant and metastable state population densities can rise to n_r ≈ n_m ≈ (ν_r/A_{rg,ef f.})n_e ∼ 0.01n_e when these energy levels are populated from above by spontaneous decays from the 4p state following recombination.

The spatial profiles of the plasma density, electron temperature, and excited state pop- ulation densities are plotted for two representative times in Fig. 4.24. After the initial transient phase of the discharge, Te is approximately uniform inside the power deposition region and approximately uniform far downstream, but there is a relatively steep temper- ature gradient in the transition regions around z ∼ −15.5 cm and z ∼ −7 cm that drives heat flux away from the antenna. Interestingly, nm, nr, and np peak near the edges of the power deposition region and are lower at the center of the antenna due to the complex balance of populating and de-populating processes included in Eqs. 4.32, 4.33, and 4.34. It is also notable that there is a substantial population density of metastable excited atoms far from the antenna region.

0 50 100 150 200 250

−1.5

−1

−0.5 0 0.5 1 1.5

Time (µs)

Dimensionless Power Density

Heat flux Recombination

Elastic collisions

Inelastic collisions Losses to walls

z = −5 cm

180 190 200 210 220 230 240 250

−0.02

−0.015

−0.01

−0.005 0 0.005 0.01 0.015 0.02

Time (µs)

Dimensionless Power Density

z = −5 cm Heat flux

Losses to walls Recombination

Elastic collisions

Inelastic collisions

Figure 4.26: Relative contributions of the processes included in Eq. 4.40 to the overall change in the dimensionless electron pressure at z=−5 cm, just downstream of the power deposition region. The right panel zooms in on the afterglow period (t > 200 µs). The same simulation parameters were used as in Fig. 4.19.

A useful feature of numerical modeling is the ability to artificially turn off physical processes in order to explicitly see their impact on the plasma evolution. For example, Fig. 4.25 illustrates the effect of turning off three-body recombination by setting ν_r = 0.

The simulation results are unchanged for t≤215µs, indicating that recombination was of negligible importance during the main power-on period (t≤200µs), as expected. However, later in the afterglow, the density decreases too slowly when recombination is not included, particularly near the antenna where n_e was high initially (the recombination rate per unit volume is approximately proportional to n³_e—see Eq. 4.27).

We can also plot the magnitude of each term in the model equations as a function of time or position in order to monitor their relative importance. One example is given in Fig.

4.26, which illustrates the contributions of the various terms in Eq. 4.40 to the overall rate of change of the dimensionless electron pressure at z =−5 cm. The left panel shows that heat flux from the antenna region is approximately balanced by the sum of diffusive energy losses to the walls (the terms involving D and E in Eq. 4.40) and electron energy losses due to inelastic collisions (the ionization and excitation/de-excitation terms in Eq. 4.40) during the main discharge. In the afterglow, diffusive losses to the walls (i.e., evaporative cooling) are the dominant energy loss mechanism, and three-body recombination provides some reheating of the electrons. The effect of elastic electron-neutral collisions is negligible

during both periods.