sheath resonance

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A slab of cold plasma is placed between two parallel plate electrodes and a potential V = V_0 e^{i \omega t}\, is established across the plates. Because of the low inertia of the electrons there exists a sheath at either electrode. Let d_p\, and d_s\, be the thicknesses of the plasma and sheath, respectively, and let A\, be the cross-sectional area of the geometry.

Contents

Uniform plasma model

Within the plasma an electric field \mathbf{E} = \mathbf{E}_0 e^{i \omega t}\, exists. First, there is a displacement current

\mathbf{J}_d = \varepsilon_0 \frac{\partial\mathbf{E}}{\partial t} = i \omega \varepsilon_0 \mathbf{E}_0 e^{i \omega t} = \omega \varepsilon_0 \mathbf{E}_0 e^{i \left(\omega t + \frac{\pi}{2} \right)}\,,

where we have assumed the ion mass is infinite. The phase shift indicates that this current is capacitive. The current is

I_d(t) = A J_d(t) = A\omega \varepsilon_0 \frac{V(t)}{d_p} e^{i \frac{\pi}{2} }\,

so the impedance is

Z_d = -i \frac{d_p}{A \varepsilon_0 \omega }\,.

There is also a conduction current due to the electrons, which, in the absence of collisions is driven directly due to the acceleration from the electric field:

\mathbf{J}_e = - n_e e \mathbf{v} = -n_e e \left( -\frac{e\mathbf{E}}{i m_e \omega} \right) = \frac{n_e e^2}{m_e \omega} \mathbf{E}_0 e^{i \left(\omega t - \frac{\pi}{2}\right)}\,.

Here the phase shift indicates that this current is inductive. The current is

I_e(t) = A J_e(t) = \frac{A n_e e^2}{m_e \omega} \frac{\mathbf{V}(t)}{d_p} e^{-i\frac{\pi}{2}}\,,

so the impedance is

Z_e = i \frac{m_e \omega d_p}{A n_e e^2 }\,.

The total current is

\mathbf{J}_p = \mathbf{J}_d + \mathbf{J}_e = i \omega \varepsilon_0 \left(1 - \frac{n_e e^2}{m_e \omega^2}\right) \mathbf{E}_0 e^{i\omega t}\,,

where we recognize the electron plasma frequency

\omega_{ep}^2 = \frac{n_e e^2}{\varepsilon_0 m_e}\,.

Thus

\mathbf{J}_p = i \omega \varepsilon_0 \left(1 - \frac{\omega_p^2}{\omega^2}\right) \mathbf{E}_0 e^{i\omega t}\,.

This is the same result we would obtain by just treating the plasma as a dielectric with permittivity

\varepsilon = \varepsilon_0 \left(1 - \frac{\omega_p^2}{\omega^2}\right)\,,

and treating the current \mathbf{J}_p\, as the dielectric displacement current \mathbf{J}_d = \tfrac{\partial \mathbf{D}}{\partial t}\,.

We can read off the plasma impedance, or calculate it from Z_e\, and Z_d\,. The result is

Z_p = - i \frac{d_p }{A \omega \varepsilon_0 \left(1 - \frac{\omega_p^2}{\omega^2}\right) }\,.

Finally, on either side is the sheath, with capacitance C_s = \frac{\varepsilon_0 A}{d_s}\,, i.e. the impedance is

Z_s = \frac{-i}{\omega C_s} = -i \frac{d_s}{\varepsilon_0 A \omega}\,.

Series resonance

The total impedance is

Z = 2 Z_s + \frac{1}{ \frac{1}{Z_d} + \frac{1}{Z_e} }\,,

or

Z = -i \frac{(d_p + 2 d_s)\omega^2 - 2 d_s \omega_{ep}^2}{A \varepsilon_0 \omega (\omega^2 - \omega_{ep}^2) }\,.

resonance occurs at Z = 0\,, or

\omega_R^2 = \frac{1}{1 + \frac{d_p}{2 d_s}} \omega_{ep}^2\,.

Parallel resonance

Reference

[1] [2] [3]

  1. ↑ L. Tonks (1931). "The High Frequency Behavior of a Plasma". Phys. Rev. 37. DOI:10.1103/PhysRev.37.1458. 
  2. ↑ L. Tonks (1931). "Plasma-Electron Resonance, Plasma Resonance and Plasma Shape". Phys. Rev. 38. DOI:10.1103/PhysRev.38.1219. 
  3. ↑ Victor P. T. Ku, Beatrice M. Annaratone (1998). "Plasma-sheath resonances and energy absorption phenomena in capacitively coupled radio frequency plasmas. Part I". J. Appl. Phys. 84 (12). DOI:10.1063/1.369025. 
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