Electron Transport, Ionization and Attachment

Abstract

This chapter is devoted to the analysis of electron transport by using the Boltzmann equation. Contrary to the previous chapters where only an electric field exists, here the electron drift also results from a density gradient. The chapter initiates by considering the situation where only the density gradient exists, which leads to free diffusion of electrons. Since the diffusion to the walls leads to the disappearing of electrons from the swarm, the reintroduction of secondary electrons produced by ionization into the distribution needs to be properly taken into account to obtain the breakdown self-sustaining field. Here, only the breakdown produced by a high-frequency (HF) field is considered, since in the direct-current (DC) case the situation is much more complicated because the breakdown also depends on the nature of the cathode.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-09253-9_12

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-09253-9_12

Appendices

Appendices

1.1 A.5.1 Expansion of the Boltzmann Equation in Spherical Harmonics

In this appendix we present the derivation of the term with the gradient in the space of positions of the Boltzmann equation (5.1) using the expansion in spherical harmonics (Cherrington 1980; Delcroix 1963, 1966). In this case using the expansion (5.2) we write the Boltzmann equation as

$$\displaystyle{ \sum _{l}\frac{\partial f_{e}^{l}} {\partial t} \;P_{l}\; +\;\sum _{l}v_{ez}\;\frac{\partial f_{e}^{l}} {\partial z} \;P_{l}\; =\;\sum _{l}I^{l}(f_{ e}^{l})\;P_{ l}, }$$

(5.129)

having assumed the anisotropy directed along the z axis. Since $v_{ez} = v_{e}\;\cos \theta$ and making use of the recursion relation (3.209), we may write

$$\displaystyle{ \sum _{l}\frac{\partial f_{e}^{l}} {\partial t} \;P_{l}\; +\;\sum _{l}v_{e}\;\frac{\partial f_{e}^{l}} {\partial z} \left ( \frac{l + 1} {2l + 1}\,P_{l+1} + \frac{l} {2l + 1}\,P_{l-1}\right ) =\;\sum _{l}I^{l}(f_{ e}^{l})\;P_{ l}. }$$

(5.130)

The second left-hand side term can be rewritten in terms of the polynomial P _l as follows

$$\displaystyle{ \sum _{l}\frac{\partial f_{e}^{l}} {\partial t} \;P_{l}\; +\;\sum _{l}v_{e}\left ( \frac{l} {2l - 1}\;\frac{\partial f_{e}^{l-1}} {\partial z} + \frac{l + 1} {2l + 3}\;\frac{\partial f_{e}^{l+1}} {\partial z} \right )P_{l}\; =\;\sum _{l}I^{l}(f_{ e}^{l})\;P_{ l}, }$$

(5.131)

so that the equation of order l is

$$\displaystyle{ \frac{\partial f_{e}^{l}} {\partial t} \; +\; v_{e}\left ( \frac{l} {2l - 1}\;\frac{\partial f_{e}^{l-1}} {\partial z} + \frac{l + 1} {2l + 3}\;\frac{\partial f_{e}^{l+1}} {\partial z} \right ) =\; I^{l}(f_{ e}^{l}), }$$

(5.132)

from which the equations (5.7) and (5.8) are immediately obtained.

1.2 A.5.2 Longitudinal Diffusion

In the time of flight experiments for determining the electron mobility μ _e , the presence of an axial electric field makes the diffusion to be different in directions perpendicular and parallel to the field (Parker and Lowke 1969; Wedding et al. 1985; Nakamura 1987). The continuity equation, in the assumption of null electron source or sink, is

$$\displaystyle{ \frac{\partial n_{e}} {\partial t} \; +\; (\triangledown.\;\boldsymbol{\Gamma }_{\mathbf{e}})\; =\; 0, }$$

(5.133)

in which the electron particle current density is

$$\displaystyle{ \boldsymbol{\Gamma }_{\mathbf{e}}\; =\; -\;D_{eT}\;\triangledown _{T}n_{e}\; -\; D_{eL}\;\triangledown _{z}n_{e}\; -\; n_{e}\;\mu _{e}\;\mathbf{E}\;, }$$

(5.134)

assuming the applied electric field directed along the z axis, and being D _eT and D _eL the transverse and longitudinal electron diffusion coefficients. Inserting (5.134) into (5.133) and considering $\mathbf{E} = -\;E\;\mathbf{e_{z}}$, we obtain

$$\displaystyle{ \frac{\partial n_{e}} {\partial t} \; -\; D_{eT}\left (\frac{\partial ^{2}n_{e}} {\partial x^{2}} \; +\; \frac{\partial ^{2}n_{e}} {\partial y^{2}} \right ) - D_{eL}\;\frac{\partial ^{2}n_{e}} {\partial z^{2}} \; +\;\mu _{e}\;E\;\frac{\partial n_{e}} {\partial z} \; =\; 0, }$$

(5.135)

whose solution for motion and spread of a pulse, starting from the origin at t = 0, is

$$\displaystyle{ n_{e}(x,y,z,t)\; =\; \frac{N_{0}} {4\pi D_{eT}t\sqrt{4\pi D_{eL } t}}\;\exp \left (-\;\frac{x^{2}\! +\! y^{2}} {4D_{eT}t} \right )\;\exp \left (-\;\frac{(z -\mu _{e}Et)^{2}} {4D_{eL}t} \right ). }$$

(5.136)

This solution is normalized as follows

$$\displaystyle{ \int _{-\infty }^{\infty }\int _{ -\infty }^{\infty }\int _{ -\infty }^{\infty }n_{ e}(x,y,z,t)\;dx\;dy\;dz\; =\; N_{0}. }$$

(5.137)

If a short pulse of N ₀ electrons starts from z = 0 at t = 0, in a transverse infinite medium, the density n _e (z, t) is

$$\displaystyle{ n_{e}(z,t)\; =\; \frac{N_{0}} {\sqrt{4\pi D_{eL } t}}\;\exp \left (-\;\frac{(z - v_{ed}\;t)^{2}} {4D_{eL}t} \right )\;, }$$

(5.138)

being v _ed  = μ _e E the electron drift velocity. This density passes through a maximum at

$$\displaystyle{ t_{M}\; =\; \frac{z} {v_{ed}}\;\sqrt{1 + \left (\frac{D_{eL } } {v_{ed}\;z}\right )^{\!\!2}} -\;\frac{D_{eL}} {v_{ed}^{\;2}}. }$$

(5.139)

In Pitchford and Phelps (1982) and Phelps and Pitchford (1985) we may find the calculation of D _eL from an expansion in powers of the spatial gradients of the electron distribution function f _e (r, v _e, t) in cylindrical geometry

$$\displaystyle{ f_{e}(\mathbf{r},\mathbf{v_{e}},t)\; =\; f(\mathbf{v_{e}})\;n_{e}(\mathbf{r},t)\; -\;\frac{g_{r}(\mathbf{v_{e}})} {n_{o}} \;\frac{\partial n_{e}} {\partial r} \; -\;\frac{g_{z}(\mathbf{v_{e}})} {n_{o}} \;\frac{\partial n_{e}} {\partial z}, }$$

(5.140)

where f(v _e), g _r (v _e), and g _z (v _e) are velocity distributions normalized such that

$$\displaystyle{ \int _{\mathbf{v_{e}}}f(\mathbf{v_{e}})\;\mathbf{dv_{e}}\; =\; 1\;;\;\;\;\int _{\mathbf{v_{e}}}g_{r}(\mathbf{v_{e}})\;\mathbf{dv_{e}}\; =\;\int _{\mathbf{v_{e}}}g_{z}(\mathbf{v_{e}})\;\mathbf{dv_{e}}\; =\; 0, }$$

(5.141)

and n _o is the gas number density. The continuity equation is then

$$\displaystyle{ \frac{\partial n_{e}} {\partial t} \; -\; D_{eT}\;\frac{1} {r}\; \frac{\partial } {\partial r}\left (r\;\frac{\partial n_{e}} {\partial r} \right ) - D_{eL}\;\frac{\partial ^{2}n_{e}} {\partial z^{2}} \; +\; v_{ed}\;\frac{\partial n_{e}} {\partial z} \; =\; 0. }$$

(5.142)

Using the two-term expansion in Legendre polynomials for the velocity dependent functions (5.141), we obtain the following expression for the electron drift velocity in accordance with (3.171)

$$\displaystyle{ v_{ed}\; =\;\int _{\mathbf{v_{e}}}v_{e}\;\cos \theta \;\;f(\mathbf{v_{e}})\;\;\mathbf{dv_{e}}\; =\; -\;\frac{E} {n_{o}}\;\frac{e} {3}\;\sqrt{ \frac{2} {m}}\int _{0}^{\infty } \frac{u} {\sigma _{m}^{e}(u)}\;\frac{df^{0}} {du} \;du, }$$

(5.143)

whereas the transverse and longitudinal electron diffusion coefficients, in two-term approximation, are given by (see Pitchford and Phelps 1982; Phelps and Pitchford 1985)

$$\displaystyle\begin{array}{rcl} n_{o}\;D_{eT}& =& \int _{\mathbf{v_{e}}}v_{e}\;\sin \theta \;\;g_{r}(\mathbf{v_{e}})\;\;\mathbf{dv_{e}}\; =\; \frac{1} {3}\;\sqrt{ \frac{2} {m}}\int _{0}^{\infty } \frac{u} {\sigma _{m}^{e}(u)}\;f^{0}(u)\;du{}\end{array}$$

(5.144)

$$\displaystyle\begin{array}{rcl} n_{o}\;D_{eL}& =& \int _{\mathbf{v_{e}}}v_{e}\;\cos \theta \;\;g_{z}(\mathbf{v_{e}})\;\;\mathbf{dv_{e}}\; =\; \frac{1} {3}\;\sqrt{ \frac{2} {m}}\int _{0}^{\infty }u\;g_{ z}^{1}(u)\;du \\ & =& n_{o}\;D_{eT}\; -\; \frac{E} {n_{o}}\;\frac{e} {3}\;\sqrt{ \frac{2} {m}}\int _{0}^{\infty } \frac{u} {\sigma _{m}^{e}(u)}\;\frac{dg_{z}^{0}} {du} \;du\; -\; v_{ed}\int _{0}^{\infty }g_{ z}^{0}(u)\;\sqrt{u}\;du\;. \\ & & {}\end{array}$$

(5.145)

The functions f ⁰(u) and g _z ⁰(u) are normalized such that

$$\displaystyle{ \int _{0}^{\infty }f^{0}(u)\;\sqrt{u}\,du\; =\; 1\;;\;\;\;\int _{ 0}^{\infty }g_{ z}^{0}(u)\;\sqrt{u}\;du\; =\; 0\;. }$$

(5.146)

The transverse diffusion coefficient agrees with the free diffusion coefficient previously reported through equation (3.193) and derived later on in (5.12), whereas D _eL  ≡ D _eT as $g_{z}^{0}(u) \rightarrow 0$ throughout (i.e. for null axial density gradients).

Exercises

Exercise 5.1:

Determine the power lost in ionization from equation (5.75).

Resolution:

The equation for electron energy conservation (5.125) is obtained multiplying both sides of the equation for evolution of the isotropic component of the velocity distribution function (5.113) by the electron energy $u = \frac{1} {2}\;mv_{e}^{\;2}$ and integrating over the whole velocity space

$$\displaystyle{ \frac{\partial } {\partial t}\left (n_{e} < u >\right )\; +\;\ldots \ldots \ldots \ldots \ldots.\; =\;\int _{ 0}^{\infty }u\;(I^{0}\; +\; J^{0})\;4\pi v_{ e}^{\;2}\;dv_{ e}\;.}$$

Then, the term for P _ion in equation (5.125) is given by

$$\displaystyle{P_{ion}\; =\; -\int _{0}^{\infty }u\;J_{ ion}^{0}\;4\pi v_{ e}^{\;2}\;dv_{ e}\;,}$$

with J _ion ⁰ expressed by equation (5.75). Thus, we may write

$$\displaystyle\begin{array}{rcl} P_{ion}& =& -\int _{0}^{\infty }u\; \frac{2} {mv_{e}}\;n_{o}\;\Bigg\{\left ( \frac{u} {\Delta } + u_{I}\right )\;\sigma _{ion}\left ( \frac{u} {\Delta }\; +\; u_{I}\right )\;f_{e}^{0}\left ( \frac{u} {\Delta } + u_{I}\right )\; \frac{1} {\Delta } {}\\ & & \qquad + \left ( \frac{u} {1 - \Delta } + u_{I}\right )\;\sigma _{ion}\left ( \frac{u} {1 - \Delta } + u_{I}\right )\;f_{e}^{0}\left ( \frac{u} {1 - \Delta } + u_{I}\right )\; \frac{1} {1 - \Delta } {}\\ & & \qquad - u\;\sigma _{ion}(u)\;f_{e}^{0}(u)\Bigg\}\;\;4\pi v_{ e}^{\;2}\;dv_{ e}\;. {}\\ \end{array}$$

Replacing v _e with the electron energy u and using for simplification the electron energy distribution function (EEDF) normalized such that

$$\displaystyle{n_{e}\; =\;\int _{ 0}^{\infty }F(u)\;\sqrt{u}\;du\;,}$$

in which

$$\displaystyle{F(u)\; =\; \frac{4\pi } {m}\;\sqrt{ \frac{2} {m}}\;\;f_{e}^{0}(v_{ e})\;,}$$

we find

$$\displaystyle\begin{array}{rcl} P_{ion}& =& -\;\sqrt{ \frac{2} {m}}\;\;n_{o}\Bigg\{\int _{0}^{\infty }u^{{\prime}}\;\left (\frac{u^{{\prime}}} {\Delta } + u_{I}\right )\;\sigma _{ion}\left (\frac{u^{{\prime}}} {\Delta }\; +\; u_{I}\right )\;F\left (\frac{u^{{\prime}}} {\Delta } + u_{I}\right )\; \frac{1} {\Delta }\;du^{{\prime}} {}\\ & & \qquad +\int _{ 0}^{\infty }u^{{\prime\prime}}\;\left ( \frac{u^{{\prime\prime}}} {1 - \Delta } + u_{I}\right )\;\sigma _{ion}\left ( \frac{u^{{\prime\prime}}} {1 - \Delta } + u_{I}\right )\;F\left ( \frac{u^{{\prime\prime}}} {1 - \Delta } + u_{I}\right )\; \frac{1} {1 - \Delta }\;du^{{\prime\prime}} {}\\ & & \qquad -\int _{u_{I}}^{\infty }u^{2}\;\sigma _{ ion}(u)\;F(u)\;du\Bigg\}\;. {}\\ \end{array}$$

Making the replacements $u = u^{{\prime}}/\Delta + u_{I}$, $u^{{\prime}} = \Delta \;(u - u_{I})$, and $du^{{\prime}} = \Delta \;du$ in the first term, and $u = u^{{\prime\prime}}/(1 - \Delta ) + u_{I}$, $u^{{\prime\prime}} = (1 - \Delta )\;(u - u_{I})$, and $du^{{\prime\prime}} = (1 - \Delta )\;du$ in the second term, we obtain

$$\displaystyle\begin{array}{rcl} P_{ion}& =& -\;\sqrt{ \frac{2} {m}}\;\;n_{o}\Bigg\{\int _{u_{I}}^{\infty }(u - u_{ I})\;u\;\sigma _{ion}(u)\;F(u)\;\Delta \;du {}\\ & & \qquad +\int _{ u_{I}}^{\infty }(u - u_{ I})\;u\;\sigma _{ion}(u)\;F(u)\;(1 - \Delta )\;du {}\\ & & \qquad -\int _{u_{I}}^{\infty }u^{2}\;\sigma _{ ion}(u)\;F(u)\;du\Bigg\} {}\\ & \!\!=\!\!& \sqrt{ \frac{2} {m}}\;\;n_{o}\int _{u_{I}}^{\infty }u_{ I}\;u\;\sigma _{ion}(u)\;F(u)\;du {}\\ & \!\!=\!\!& n_{o}\;n_{e}\;u_{I}\;C_{ion}\;, {}\\ \end{array}$$

in which

$$\displaystyle{C_{ion}\; =\; \frac{1} {n_{e}}\;\;\sqrt{ \frac{2} {m}}\int _{u_{I}}^{\infty }u\;\sigma _{ ion}(u)\;F(u)\;du}$$

is the electron ionization rate coefficient.

Exercise 5.2:

Determine the power lost in electron attachment.

Resolution:

The power lost in electron attachment is obtained from

$$\displaystyle{P_{att}\; =\; -\int _{0}^{\infty }u\;J_{ att}^{0}\;4\pi v_{ e}^{\;2}\;dv_{ e}\;,}$$

where J _att ⁰ has the same form as the third term of equation (5.75) but for attachment

$$\displaystyle{J_{att}^{0}\; =\; -\; \frac{2} {mv_{e}}\;n_{o}\;u\;\sigma _{att}\;f_{e}^{0}\;.}$$

Substituting and replacing v _e with the electron energy u, we obtain

$$\displaystyle{P_{att}\; =\; \frac{8\pi } {m^{2}}\;n_{o}\int _{0}^{\infty }u^{2}\;\sigma _{ att}(u)\;f_{e}^{0}(u)\;du\;.}$$

Using now the EEDF F(u) as defined in Exercise 5.1, we may write

$$\displaystyle{P_{att}\; =\;\int _{ 0}^{\infty }u^{3/2}\;\nu _{ att}(u)\;F(u)\;du\;,}$$

with ν _att (u) denoting the electron attachment frequency given by

$$\displaystyle{\nu _{att}(u)\; =\; n_{o}\;v_{e}\;\sigma _{att}(v_{e})\; =\; n_{o}\;\sqrt{\frac{2u} {m}} \;\;\sigma _{att}(u)\;.}$$

Replacing ν _att (u) in P _att , we obtain at the end

$$\displaystyle{P_{att}\; =\; n_{e} <\!\! u\;\nu _{att}\!\! >\;.}$$

Exercise 5.3:

Write the expressions for the electron drift velocity (5.119) and for the power brought to the electron drift by the ionization-attachment balance (5.126) in terms of the EEDF F(u) used in the previous exercises.

Resolution:

From equation (5.119) we obtain

$$\displaystyle{v_{ed}\; =\; -\; \frac{1} {n_{e}n_{o}}\;\;\sqrt{ \frac{2} {m}}\int _{0}^{\infty } \frac{u} {3\sigma _{m}^{e}}\left (eE\;\frac{dF} {du} \; +\; (\alpha -\eta )\;F\right )du\;,}$$

with $\sigma _{m}^{e}$ denoting the effective electron cross-section for momentum transfer, while from (5.126) we find

$$\displaystyle{P_{flow}\; =\; -\;\sqrt{ \frac{2} {m}}\; \frac{\alpha -\eta } {n_{o}}\int _{0}^{\infty } \frac{u^{2}} {3\sigma _{m}^{e}}\left (eE\;\frac{dF} {du} \; +\; (\alpha -\eta )\;F\right )\;du\;.}$$

In the case of pure SF₆, P _flow is < 0 at E∕n _o  < 370 Td and P _flow  > 0 at E∕n _o  > 370 Td, so that in the first situation P _flow represents a gain energy term.