Electron energy distribution in a dusty plasma: Analytical approach

Analytical expressions describing the electron energy distribution function (EEDF) in a dusty plasma are obtained from the homogeneous Boltzmann equation for electrons. The expressions are derived neglecting electron-electron collisions, as well as transformation of high-energy electrons into low-energy electrons at inelastic electron-atom collisions. At large electron energies, the quasiclassical approach for calculation of the EEDF is applied. For the moderate energies, we account for inelastic electron-atom collisions in the dust-free case and both inelastic electron-atom and electron-dust collisions in the dusty plasma case. Using these analytical expressions and the balance equation for dust charging, the electron energy distribution function, the effective electron temperature, the dust charge, and the dust surface potential are obtained for different dust radii and densities, as well as for different electron densities and radio-frequency (rf) field amplitudes and frequencies. The dusty plasma parameters are compared with those calculated numerically by a finite-difference method taking into account electron-electron collisions and the transformation of high-energy electrons at inelastic electron-neutral collisions. It is shown that the analytical expressions can be used for calculation of the EEDF and dusty plasma parameters at typical experimental conditions, in particular, in the positive column of a direct-current glow discharge and in the case of an rf plasma maintained by an electric field with frequency $f=13.56\mathrm{MHz}$.

Figure 1

(a) The electron energy distribution function calculated using the simplified model for $a_d=200\mathrm{nm}$ and different dust densities: $n_d=0$ (solid line), $3\times {10}^7\mathrm{c}{\mathrm{m}}^{-3}$ (dashed line), $5\times {10}^7\mathrm{c}{\mathrm{m}}^{-3}$ (dotted line), and $={10}^8\mathrm{c}{\mathrm{m}}^{-3}$ (dash-dotted line). (b) The same as in (a) for $n_d=3\times {10}^7\mathrm{c}{\mathrm{m}}^{-3}$ and different dust radii: $a_d=100\mathrm{nm}$ (solid line), $a_d=250\mathrm{nm}$ (dashed line), and $a_d=500\mathrm{nm}$ (dotted line). (c) The EEDF calculated using the simplified model for $n_d=0$ (dotted line) and ${10}^8\mathrm{c}{\mathrm{m}}^{-3}$ (solid line). The Maxwellian EEDF calculated at $T_{\mathrm{eff}}=3.30\mathrm{eV}$ corresponding to $n_d=0$ (curve 1) and at $T_{\mathrm{eff}}=1.24\mathrm{eV}$ corresponding to $n_d={10}^8\mathrm{c}{\mathrm{m}}^{-3}$ (curve 2). The other external conditions are the same as in (a).

Figure 2

EEDFs calculated using the more complicated numerical (solid lines) and simplified analytical (dotted lines) models for $n_d={10}^7\mathrm{c}{\mathrm{m}}^{-3}, a_d=100\mathrm{nm}$, and different rf frequencies: 0 (a), 27.12 MHz (b), 60 MHz (c), and 2.45 GHz (d). $E_p=8500\mathrm{V}/\mathrm{m}$ for $f=2.45\mathrm{GHz}$, and $E_p=300\mathrm{V}/\mathrm{m}$ for other frequencies. The other external conditions are the same as in Fig. 1. The dashed lines correspond to the Maxwellian EEDFs calculated at $T_{\mathrm{eff}}$ corresponding to the effective electron temperatures obtained from the more complicated numerical model (the temperatures are presented in Table 2).

Figure 3

The electron energy distribution functions calculated using the simplified model (dotted lines) and the more accurate model (solid lines) for different electric field amplitudes: $100\mathrm{V}/\mathrm{m}$ (a), $500\mathrm{V}/\mathrm{m}$ (b), $1000\mathrm{V}/\mathrm{m}$ (c), and $2000\mathrm{V}/\mathrm{m}$ (d). Here, $f=13.56\mathrm{MHz}, n_d=5\times {10}^7\mathrm{c}{\mathrm{m}}^{-3}, a_d=200\mathrm{nm}$, and the other external parameters are the same as in Fig. 2. The dashed lines correspond to the Maxwellian EEDFs with $T_{\mathrm{eff}}$ obtained from the more accurate model (Table 3).

Figure 4

EEDFs for different electron densities: $n_e={10}^9$ (a), ${10}^{10}$ (b), ${10}^{11}$ (c), and ${10}^{12}\mathrm{c}{\mathrm{m}}^{-3}$ (d). The solid and dotted lines correspond to the EEDFs obtained using the more accurate and simplified models, respectively. Here, $E_p=300\mathrm{V}/\mathrm{m}$ and the other parameters are the same as in Fig. 3. The dashed lines correspond to the Maxwellian EEDFs with $T_{\mathrm{eff}}$ obtained from the more accurate model (Table 4).