Features of Electromagnetic Field Excitation in a Capacitive HF Discharge. I. General Aspects. A Simple Model of Symmetric Discharge

Abstract—

Electrodynamic properties of a low-pressure (electron collision frequency much lower than the field frequency) capacitive high-frequency (HF) discharge with large-area electrodes maintained by an electromagnetic field with frequency higher than 13 MHz is studied analytically. The discharge is sustained by surface waves propagating along the plasma–space-charge sheath–metal interface. When describing the sheath, dispersion curves for even and odd surface waves in a three-layer structure (sheath–plasma–sheath) surrounded by metal boundaries are calculated within the framework of the matrix model. It is demonstrated that the field of the fundamental mode (the quasi-TEM mode) must be taken into account for correct discharge impedance calculation in the case of low electron densities, while the field of the surface wave must be taken into account in the case of high electron densities. An approximate expression governing the discharge impedance that is based on accounting only for one mode is derived. The expression is valid at low electron densities when the surface waves are absent, as well as high electron densities exceeding the value giving rise to a geometric resonance in the plasma–sheath system.

Appendices

ANALYSIS OF DISPERSION EQUATION FOR WAVES IN THE THREE-LAYER SYSTEM

Propagation constants of the waves, appearing in the above equations, calculation can be simplified by considering a plane problem, representing a structure infinite in the X0Z plane, that consists of three dielectric layers with dielectric permittivities ε₁, ε_P, ε₂, and thicknesses d₁, 2L–d₁–d₂, d₂, respectively, instead of a cylindrical one. The 0X axis is parallel to the direction of wave propagation, the 0Z axis is perpendicular to it, and the 0Y axis is perpendicular to the structure surface. The equivalence of these two approaches follows from the possibility of expanding a plane wave in cylindrical waves, as well as from the possibility of solving the above problem by the separation of variables method [93, 94] both in cylindrical and plane coordinate systems. The wave prorogation constant along the structure h is defined by the determinant of the system of equations describing field distribution along the 0Z axis, that depend only on this coordinate and are identical in both coordinate systems.

Electrodes made of metal are located at the boundaries of the structure of thickness 2L. It is convenient to choose the coordinate system in such a way that zero on the 0Z axis would be located at the medium point of the positive column, i.e., so that the plasma and sheath boundaries have coordinates –L₂ and L₂ (d₁ is the thickness of the upper sheath, d₂ is the thickness of the lower sheath, and 2L = 2L₂ + d₁ + d₂). For an E-wave, equations (1), (3), and (4), with zero boundary conditions (2) for the radial electric field imposed at the walls in the discharge chamber, yield the following expressions governing the electromagnetic field:

$$\begin{gathered} \left\{ {\begin{array}{*{20}{c}} {{{e}_{z}}\left( z \right)} \\ {{{e}_{x}}\left( z \right)} \\ {{{b}_{y}}\left( z \right)} \end{array}} \right\} = {{A}_{1}} \\ \times \,\left( {\begin{array}{*{20}{c}} { - \frac{{ih}}{{\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{1}}} }}\coth \left( {\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{1}}} \left( {{{L}_{2}}\, + \,{{d}_{1}}\, - \,z} \right)} \right)} \\ { - \sinh \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{1}}} \left( {{{L}_{2}} + {{d}_{1}} - z} \right)} \right)} \\ {\sqrt {\frac{{{{\varepsilon }_{0}}}}{{{{\mu }_{0}}}}} \frac{{ik{{\varepsilon }_{1}}}}{{\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{1}}} }}\coth \left( {\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{1}}} \left( {{{L}_{2}}\, + \,{{d}_{1}}\, - \,z} \right)} \right)} \end{array}} \right) \\ \times \;\frac{{\exp \left( { - i\omega t + ihx} \right)}}{{\coth \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{1}}} {{d}_{1}}} \right)}} \\ \end{gathered} $$

(A.1)

in the upper sheath,

$$\left\{ {\begin{array}{*{20}{c}} {{{e}_{z}}\left( z \right)} \\ {{{e}_{x}}\left( z \right)} \\ {{{b}_{y}}\left( z \right)} \end{array}} \right\} = \frac{{\exp \left( { - i\omega t + ihx} \right)}}{{\coth \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} {{L}_{P}}} \right)}}$$

$$\, \times \left\{ {{{A}_{2}}\left( {\begin{array}{*{20}{c}} { - \frac{{ih}}{{\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} }}\coth \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{1}}} z} \right)} \\ {\sinh \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{1}}} z} \right)} \\ {\sqrt {\frac{{{{\varepsilon }_{0}}}}{{{{\mu }_{0}}}}} \frac{{ik{{\varepsilon }_{P}}}}{{\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} }}\coth \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{2}}} z} \right)} \end{array}} \right)} \right.$$

(A.2)

$$\, + \left. {{{B}_{2}}\left( {\begin{array}{*{20}{c}} { - \frac{{ih}}{{\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} }}\sinh \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{1}}} z} \right)} \\ {\coth \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} z} \right)} \\ {\sqrt {\frac{{{{\varepsilon }_{0}}}}{{{{\mu }_{0}}}}} \frac{{ik{{\varepsilon }_{P}}}}{{\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} }}\sinh \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} z} \right)} \end{array}} \right)} \right\}$$

in plasma, and

$$\begin{gathered} \left\{ {\begin{array}{*{20}{c}} {{{e}_{z}}\left( z \right)} \\ {{{e}_{x}}\left( z \right)} \\ {{{b}_{y}}\left( z \right)} \end{array}} \right\} = {{A}_{3}} \\ \times \;\left( {\begin{array}{*{20}{c}} { - \frac{{ih}}{{\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{2}}} }}\coth \left( {\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{2}}} \left( {z\, + \,{{L}_{2}}\, + \,{{d}_{2}}} \right)} \right)} \\ {\sinh \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{2}}} \left( {z + {{L}_{2}} + {{d}_{2}}} \right)} \right)} \\ {\sqrt {\frac{{{{\varepsilon }_{0}}}}{{{{\mu }_{0}}}}} \frac{{ik{{\varepsilon }_{2}}}}{{\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{2}}} }}\coth \left( {\sqrt {{{h}^{2}}\, - \,{{k}^{2}}{{\varepsilon }_{2}}} \left( {z\, + \,{{L}_{2}}\, + \,{{d}_{2}}} \right)} \right)} \end{array}} \right) \\ \times \;\frac{{\exp \left( { - i\omega t + ihx} \right)}}{{\coth \left( {\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{2}}} {{d}_{2}}} \right)}} \\ \end{gathered} $$

(A.3)

in the lower sheath.

The relation between the amplitudes of symmetric and anti-symmetric modes in the surface wave, and the dispersion relation, follow from equalities of tangent components of the electromagnetic field:

$$\begin{gathered} \frac{{{{B}_{{2 \pm }}}}}{{{{A}_{{2 \pm }}}}} = - \frac{{\left( {\frac{{{{\varepsilon }_{P}}\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{1}}} }}{{{{\varepsilon }_{1}}\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} }}\tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{1}}} {{d}_{1}}} \right) + \tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} {{L}_{2}}} \right)} \right)}}{{\left( {1 + \frac{{{{\varepsilon }_{P}}\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{1}}} }}{{{{\varepsilon }_{1}}\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} }}\tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} {{L}_{2}}} \right)\tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{1}}} {{d}_{1}}} \right)} \right)}} \\ = \frac{{\left( {\frac{{{{\varepsilon }_{P}}\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{2}}} }}{{{{\varepsilon }_{2}}\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} }}\tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{2}}} {{d}_{2}}} \right) + \tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} {{L}_{2}}} \right)} \right)}}{{\left( {1 + \frac{{{{\varepsilon }_{P}}\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{2}}} }}{{{{\varepsilon }_{2}}\sqrt {{{h}^{2}} - {{k}^{2}}{{\varepsilon }_{P}}} }}\tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}} {{L}_{2}}} \right)\tanh \left( {\sqrt {h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{2}}} {{d}_{2}}} \right)} \right)}}. \\ \end{gathered} $$

(A.4)

The minus sign denotes the surface wave with a shorter wavelength, while the plus sign denotes the surface wave with a longer wavelength. Introducing variables ${{\tilde {p}}^{2}} = h_{ \pm }^{2} - {{k}^{2}}{{\varepsilon }_{P}}$, ${{p}^{2}} = {{k}^{2}}{{\varepsilon }_{P}} - h_{ \pm }^{2}$, $a_{{1,2}}^{2} = h_{ \pm }^{2} -$ ${{k}^{2}}{{\varepsilon }_{{1,2}}}$, $\tilde {a}_{{1,2}}^{2} = {{k}^{2}}{{\varepsilon }_{{1,2}}} - h_{ \pm }^{2}$, we have

$$\frac{{{{B}_{{2 \pm }}}}}{{{{A}_{{2 \pm }}}}} = - \frac{{\left( {\frac{{{{\varepsilon }_{P}}{{a}_{1}}}}{{{{\varepsilon }_{1}}p}}\tanh \left( {{{a}_{1}}{{d}_{1}}} \right) + \tanh \left( {p{{L}_{2}}} \right)} \right)}}{{\left( {1 + \frac{{{{\varepsilon }_{P}}{{a}_{1}}}}{{{{\varepsilon }_{1}}p}}\tanh \left( {p{{L}_{2}}} \right)\tanh \left( {{{a}_{1}}{{d}_{1}}} \right)} \right)}} = \frac{{\left( {\frac{{{{\varepsilon }_{P}}{{a}_{2}}}}{{{{\varepsilon }_{2}}p}}\tanh \left( {{{a}_{2}}{{d}_{2}}} \right) + \tanh \left( {p{{L}_{2}}} \right)} \right)}}{{\left( {1 + \frac{{{{\varepsilon }_{P}}{{a}_{2}}}}{{{{\varepsilon }_{2}}p}}\tanh \left( {{{a}_{2}}{{L}_{2}}} \right)\tanh \left( {p{{d}_{2}}} \right)} \right)}}.$$

(A.5)

Dispersion relation (12) follows from (A.4) and (A.5).

APPENDIX B

FIELD STRUCTURE OF THE SURFACE WAVE, HIGHER-ORDER MODES, AND WAVEGUIDE MODES

The following notations for cylindrical functions were used in Tables 1–3: ${{P}_{0}}(x) = {{J}_{0}}(x)$, $H_{0}^{{(1)}}(x),H_{0}^{{(2)}}(x)$, ${{Q}_{0}}(x) = {{I}_{0}}(x)$, ${{K}_{0}}(x)$, ${{P}_{1}}(x) = {{J}_{1}}(x)$, $H_{1}^{{(1)}}(x)$, $H_{1}^{{(2)}}(x)$, and ${{Q}_{1}}(x) = {{I}_{1}}(x)$, $- {{K}_{1}}(x)$. The notations for the propagation constant are ${{\tilde {h}}_{{n \pm }}} = i{{h}_{{n \pm }}}$, ${{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} }_{{n + }}}\, = \,\pi (1{\text{/}}2\, + \,n){\text{/}}L$, ${{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} }_{{n - }}} = (\pi n){\text{/}}L$, and ${{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{h} }_{{n \pm }}} = \sqrt {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{a} _{{n \pm }}^{2} - {{k}^{2}}}$.

Table 3. Field distribution for TEM wave and decaying waves outside of plasma