Analytical model for the radio-frequency sheath

A simple analytical model for the planar radio-frequency (rf) sheath in capacitive discharges is developed that is based on the assumptions of a step profile for the electron front, charge exchange collisions with constant cross sections, negligible ionization within the sheath, and negligible ion dynamics. The continuity, momentum conservation, and Poisson equations are combined in a single integro-differential equation for the square of the ion drift velocity, the so called sheath equation. Starting from the kinetic Boltzmann equation, special attention is paid to the derivation and the validity of the approximate fluid equation for momentum balance. The integrals in the sheath equation appear in the screening function which considers the relative contribution of the temporal mean of the electron density to the space charge in the sheath. It is shown that the screening function is quite insensitive to variations of the effective sheath parameters. The two parameters defining the solution are the ratios of the maximum sheath extension to the ion mean free path and the Debye length, respectively. A simple general analytic expression for the screening function is introduced. By means of this expression approximate analytical solutions are obtained for the collisionless as well as the highly collisional case that compare well with the exact numerical solution. A simple transition formula allows application to all degrees of collisionality. In addition, the solutions are used to calculate all static and dynamic quantities of the sheath, e.g., the ion density, fields, and currents. Further, the rf Child-Langmuir laws for the collisionless as well as the collisional case are derived. An essential part of the model is the a priori knowledge of the wave form of the sheath voltage. This wave form is derived on the basis of a cubic charge-voltage relation for individual sheaths, considering both sheaths and the self-consistent self-bias in a discharge with arbitrary symmetry. The externally applied rf voltage is assumed to be sinusoidal, although the model can be extended to arbitrary wave forms, e.g., for dual-frequency discharges. The model calculates explicitly the cubic correction parameter in the charge-voltage relation for the case of highly asymmetric discharges. It is shown that the cubic correction is generally moderate but more pronounced in the collisionless case. The analytical results are compared to experimental data from the literature obtained by laser electric field measurements of the mean and dynamic fields in the capacitive sheath for various gases and pressures. Very good agreement is found throughout.

Figure 1

Basic scheme of the density profiles in the sheath, including notations and coordinate systems used in the model. Details are explained in the text.

Figure 2

Numerical iteration of the differential equation for the rf sheath ($\beta =10,\gamma =30$) using five iteration steps (top, screening function $g$; bottom, normalized square of the ion drift velocity $W$). The initial forms $g^{\left(0\right)}$ and $W^{\left(0\right)}$ by which the iteration starts are indicated by the dashed lines. All subsequent solutions of $g^{(j>0)}$ are effectively on top of each other. Only for $W$ can the first order solution $W^{\left(1\right)}$ be identified as a separate curve but already the second and all higher orders $W^{(j>1)}$ are also all on top of each other.

Figure 3

Comparison of the approximate analytical form of the screening function $g_a$ (dashed line) and exact numerical solutions (solid lines) for combinations of two values of the density parameter ($\gamma =15\mathrm{and}50$) and three values of the collision parameter ($\beta =0,3,\mathrm{and}10$): (a) $\beta =0,\gamma =15$, (b) $\beta =0,\gamma =50$, (c) $\beta =3,\gamma =15$, (d) $\beta =3,\gamma =50$, (e) $\beta =10,\gamma =15$, (f) $\beta =10,\gamma =50.$ The order is from bottom to top: (e), (c), (a), (f), (d), (b).

Figure 4

Screening function: exact numerical solution (solid line), universal analytical approximation (dashed line). The line above the universal analytical approximation represents the collisionless case ($\beta =0,\gamma =30\sqrt{2}$) and the line below the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$).

Figure 5

Square of normalized drift velocity: exact numerical solution (solid line); analytical approximation (dashed line). The upper pair of lines represents the collisionless case ($\beta =0,\gamma =30\sqrt{2}$) and the lower pair the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$).

Figure 6

Ion density: exact numerical solution (solid line); analytical approximation (dashed line). The lower pair of lines represents the collisionless case ($\beta =0,\gamma =30\sqrt{2}$) and the upper pair the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$).

Figure 7

Ratio of the two terms on the left hand side of the sheath differential equation calculated from the numerical solution for $\beta =6\pi ,\gamma =30\sqrt{2}$. The second derivative is contributing slightly only close to the maximum sheath extension.

Figure 8

Mean electric field: exact numerical solution (solid line); analytical approximation based on integration of the left hand side of Eq. (7) (dotted line) and on integration of the right hand side (dashed line). The lower set of lines represents the collisionless case ($\beta =0,\gamma =30\sqrt{2}$) and the upper set the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$). In the collisional case the dashed and dotted lines are effectively on top of each other.

Figure 9

Cubic correction parameter $a$ as a function of the density parameter γ : numerical values (dots) and analytical approximation (solid line), both based on Eq. (21). Curves start at the relevant lower limit of $\gamma \ge 15$. The dashed line represents a reasonable average value of $a=1.55$.

Figure 10

Cubic correction parameter $a$ calculated by Eq. (21) as a function of the effective parameter $\alpha =\left(12/11\right){\gamma }^2/\beta$: numerical values for $\gamma =42$ (dots) and $\gamma =84$ (triangles) and analytical solution (solid line). The dashed line represents a reasonable average value of $a=1.34$, which is close to the exact value for the parameter case investigated in this section of $a=1.36$. Some of the numerical points for the two different values of γ effectively overlap.

Figure 11

Charge-voltage relation: numerical solution (solid line); analytical solution (dashed line); and cubic form of the charge-voltage relation from Eq. (20) (dotted line). The upper set of lines belongs to the collisionless case ($\beta =0,\gamma =30\sqrt{2}$, and $a=1.56$). Here the dashed and the solid lines are effectively on top of each other but the dotted line is slightly off. The lower set of lines belongs to the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$, and $a=1.36$): Here all three lines are effectively on top of each other.

Figure 12

Dynamics of the instantaneous sheath edge position ${\xi }_s$ as function of the rf phase $\phi =\omega t$: numerical result (solid line) and analytical result (dashed line) using both a parametric presentation by $\phi \left[\mathrm{v}\left(\xi \right)\right]$, and explicit analytical approximation (dotted line). The upper set of lines represents the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$) and the lower set the collionless case ($\beta =0,\gamma =30\sqrt{2}$). For both sets all three lines are effectively on top of each other. It should be noted that for a given phase the same voltage applies in both cases of collisionality.

Figure 13

Dynamic electric field in the sheath as a function of position and for three different rf phases. Here the phase is characterized by the temporary position of the sheath edge which is set to ${\xi }_s=0,1/3,\mathrm{and}2/3$: numerical result (solid line); analytical result (dashed line). In all cases the solid and dashed lines are effectively on top of each other. The field is zero for positions $\xi <{\xi }_s$. For a given sheath edge position, the higher field strengths correspond to the collisional case ($\beta =6\pi ,\gamma =30\sqrt{2}$) and the lower field strengths to the collisionless case ($\beta =0,\gamma =30\sqrt{2}$).

Figure 14

Dynamics of the normalized electric field in the sheath as a function of position and rf phase using the fully analytical result for the collisionless case ($\beta =0,\gamma =30\sqrt{2}$).

Figure 15

Normalized displacement current as a function of the rf phase for the collisionless case ($\beta =0,\gamma =30\sqrt{2}$): numerical solution (solid line); analytical result using a parametric representation as a function of ${\xi }_s$ (dashed line); analytical result using an explicit approximate expression for the sheath edge position (dotted line); and analytical result based only on the cubic charge-voltage relation (dash-dotted line). The solid and the dashed line are effectively on top of each other.

Figure 16

Coefficient $p_1\left(\gamma \right)$ for the collisionless rf Child-Langmuir law as a function of the density parameter γ: numerical solution (dots); analytical solution using integration of the left hand side (solid line); and right hand side (dotted line) of the sheath equation [Eq. (29)]. The asymptotic valued of $133/142=0.94$ is shown by a dashed line. Curves start at the minimum value of $\gamma =15$.

Figure 17

Coefficient $p_2$ for the collisional rf Child-Langmuir law as a function of the effective parameter $\alpha =\left(12/11\right){\gamma }^2/\beta$: numerical values for $\gamma =42$ (dots) and $\gamma =84$ (triangles) and analytical approximation (solid line). The dashed line represents the asymptotic analytical value $p_2=1.22$ for very large α. Some of the numerical points for the two different values of γ effectively overlap.

Figure 18

Square of normalized drift velocity: exact numerical solution (solid line); analytical approximation (dashed line). The different cases are all calculated for the same density parameter $\gamma =30\sqrt{2}$ and various values of the collision parameter $\beta =k\pi$ with $k=0.1,0.5,1.0,2.0$ (from top to bottom) representing the transition region between the collisionless and the collisional regime.

Figure 19

Mean sheath electric field as a function of the distance from the powered electrode $x$ for three different cases: krypton at $p=1\mathrm{Pa}$ (dots), helium at $p=50\mathrm{Pa}$ (squares), and hydrogen at $p=80\mathrm{Pa}$ (triangles). The solid lines represent the fields calculated by integrating the left hand side of Eq. (7) using the analytical solutions: helium and hydrogen collisional solution, krypton collisionless solution. In case of krypton, the dashed line represents alternative integration of the right hand side of Eq. (7).

Figure 20

Absolute displacement current density for the helium discharge at 50 Pa as determined from the experimental data (dots) and the analytical sheath model using the same input parameters as in Fig. 19 in the cubic charge-voltage relation. The cases $\varepsilon =0$ (dashed line) and $\varepsilon =0.23$ from the experiment (solid line) are compared. The experimental data points are inferred from a direct differentiation of the electric field at the electrode.

Figure 21

Charge-voltage characteristic for the helium discharge at $p=50\mathrm{Pa}$: dots (measurement); analytically calculated characteristic using the same data as in Figs. 26 and 27 (solid line); cubic charge-voltage relation using the calculated cubic correction parameter $a=1.30$ (dashed line). The solid and the dashed lines are effectively on top of each other.

Figure 22

Comparison of spatially and temporally resolved electric fields in a capacitive rf discharge in krypton at $p=10\mathrm{Pa}$. The points are experimental values obtained by laser electric field measurements at different phases of the rf cycle: $\phi =0.639$ (squares), $\phi =1.83$ (triangles), $\phi =3.37$ (stars), $\phi =4.90$ (dots). The aspect ratio and the intervals of the axis are set identical to the original figure taken from Ref. 60, as shown in Fig. 23 for comparison. Here $x$ is the distance to the electrode.

Figure 23

Original figure of the experimental data shown in Fig. 22 and comparison to the computational results using the Brinkmann sheath model [60].

Figure 24

The function $h_1$ (upper solid curve) and $h_2$ (lower solid curve) for a linear electric field as a function of $W^{\prime }/\beta W$. The dashed line is at 1 to indicate the value used in the fluid equation.

Figure 25

Error in the drift velocity by use of the simplified fluid equation for a linear electric field ($u$, exact solution; $u_a$, solution of the simplified fluid equation) as function of the spatial coordinate.

Figure 26

Voltage wave form over the sheath at the powered electrode in case of $a=1$ (top figure) and $a=2$ (bottom figure) for various values of the symmetry parameter ε (top to bottom): ε = 1, 0.75, 0.5, 0.25, 0. The solid lines represent the exact form given Eq. (B16) and the dashed lines the approximation given by Eq. (B17).

Figure 27

Screening function $g$ as determined numerically for $\gamma =60$ and $\beta =15$ (solid line) and the approximate analytical form $g_a={\xi }^{\nu }$ with an exponent of $\nu =0.391$, optimized according to Eq. (C3) (dashed line). Both curves are effectively on top of each other.

Figure 28

Exponent ν for the simple power scaling approximation to the screening function $g$ as function of the density parameter γ and the collision parameter β. The surface representing the values of ν is semitransparent in order to allow comparison with the proposed universal exponent of $\nu =3/8$, which is represented by a grid. The surface of the optimized ν values intercepts with the grid at about the diagonal going from the left to the right hand corner.