Boltzmann Equation with Time-Varying Fields

Abstract

This chapter analyses the effects produced by a time-varying electric field in the electron kinetics. The behaviour exhibited by the electron velocity distribution function is controlled by two characteristic relaxation frequencies, one for energy and another for momentum transfer, when compared with the field frequency. The cases of high-frequency (HF) and radio-frequency (RF) fields are analysed separately, since they correspond to situations in which no time-modulation and large time-modulation exist, respectively, in the isotropic part of the electron velocity distribution. This chapter also analyses the electron kinetics under the simultaneous effects of a HF electric field and a stationary external magnetic field, with leads to electron cyclotron resonance (ECR) when the electron cyclotron frequency equals the field-frequency.

Appendices

Appendices

1.1 A.4.1 Effective Collision Frequency and Electron Density in High-Frequency Discharges

When the effective collision frequency for momentum transfer ν _m ^e, given by equation (3.139), is independent of electron velocity, the equation for electron momentum conservation (3.60) in high-frequency (HF) fields reduces to

$$\displaystyle{ n_{e}\;m\;\frac{\mathbf{dv_{ed}}} {dt} \; +\; n_{e}\;e\;\mathbf{E}\; =\; -\;n_{e}\;m\;\nu _{m}^{e}\;\mathbf{v_{ ed}}, }$$

(4.118)

with v _ed denoting the electron drift velocity. Here, we are assuming the field frequency much larger than the characteristic relaxation frequency for energy transfer ω ≫ ν _e , in order the electron density does not change appreciably during a cycle of the field oscillation. Then, for an applied field of frequency ω, we have $\mathbf{dv_{ed}}/dt = j\omega \;\mathbf{v_{ed}}$, and the drift velocity is given by the well known formula

$$\displaystyle{ \mathbf{v_{ed}}\; =\; -\; \frac{e} {m\;(\nu _{m}^{e} + j\omega )}\;\mathbf{E}\;. }$$

(4.119)

The electron current density $\mathbf{J_{e}} = -\;en_{e}\;\mathbf{v_{ed}}$, allows to obtain the complex conductivity from $\mathbf{J_{e}} = \overline{\sigma _{c}}_{e}\;\mathbf{E}$, with the form

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; \frac{e^{2}n_{e}} {m\;(\nu _{m}^{e} + j\omega )}\;, }$$

(4.120)

whereas the time-averaged power absorbed from the field (4.15) is

$$\displaystyle{ \overline{P_{E}(t)}\; =\; \frac{1} {2}\; \frac{e^{2}n_{e}\;\nu _{m}^{e}} {m\;(\nu _{m}^{e\;2} +\omega ^{2})}\;E_{0}^{\;2}\;. }$$

(4.121)

However, when the frequency ν _m ^e is velocity-dependent, we must consider the expression (4.9) for the complex anisotropic component

$$\displaystyle{ \mathbf{f_{e}^{1}}\; =\; \overline{f_{ e}^{1}}(v_{ e})\;e^{j\omega t}\;\mathbf{e_{ z}}\; =\; -\; \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{eE_{0}} {m} \,\frac{df_{e}^{0}} {dv_{e}} \;e^{j\omega t}\;\mathbf{e_{ z}}\;, }$$

(4.122)

in the collision term of equation for momentum conservation (3.55)

$$\displaystyle{ \mathbf{I_{1}}\; =\; -\int _{0}^{\infty }m\;\frac{v_{e}} {3} \;\nu _{m}^{e}\;\mathbf{f_{ e}^{1}}\;4\pi v_{ e}^{\;2}\;dv_{ e}\;, }$$

(4.123)

assuming here the interactions of collision type only and the applied field with the form $\mathbf{E} = -\;E_{0}\;e^{j\omega t}\;\mathbf{e_{z}}$. The complex drift velocity (4.10) and (4.11) is then

$$\displaystyle{ \mathbf{v_{ed}}\; =\; \overline{V _{e}}_{d}\;e^{j\omega t}\;\mathbf{e_{ z}}\; =\; -\;\frac{eE_{0}} {n_{e}m}\;\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \,dv_{e}\;e^{j\omega t}\;\mathbf{e_{ z}}\;, }$$

(4.124)

so that substituting equation (4.123) in equation (4.118), we find

$$\displaystyle{ n_{e}m\;j\omega \;\overline{V _{e}}_{d}\; -\; n_{e}\;eE_{0}\; =\; eE_{0}\int _{0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;. }$$

(4.125)

In the case of a velocity-independent frequency ν _m ^e, we immediately obtain equation (4.119) from (4.125). In fact, integrating by parts

$$\displaystyle{ \int _{0}^{\infty }\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\; =\; -\int _{0}^{\infty }f_{ e}^{0}\;4\pi v_{ e}^{\;2}\;dv_{ e}\; =\; -\;n_{e}\;, }$$

(4.126)

we find

$$\displaystyle{ n_{e}m\;j\omega \;\overline{V }_{ed}\; -\; n_{e}\;eE_{0}\; =\; -\;eE_{0}\; \frac{\nu _{m}^{e}} {\nu _{m}^{e} + j\omega }\;n_{e} }$$

(4.127)

and therefore

$$\displaystyle{ \overline{V _{e}}_{d}\; =\; \frac{eE_{0}} {m\;(\nu _{m}^{e} + j\omega )}\;. }$$

(4.128)

A different situation occurs as ν _m ^e depends on the electron velocity. In this case in order to express equation (4.125) with the form (4.118) and to obtain an equivalent expression (4.120) for the complex conductivity, we should consider an effective complex collision frequency $\overline{\nu }_{eff}$ in equation (4.125), such as

$$\displaystyle{ n_{e}m\;j\omega \;\overline{V _{e}}_{d}\; -\; n_{e}\;eE_{0}\; =\; -\;n_{e}m\;\overline{\nu }_{eff}\;\overline{V _{e}}_{d}\;. }$$

(4.129)

Then, using equation (4.124) for $\overline{V _{e}}_{d}$, we obtain

$$\displaystyle{ \overline{\nu }_{eff}\; =\;\int _{ 0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;v_{e}^{\;3}\;dv_{ e}\; \times \left (\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;v_{e}^{\;3}\;dv_{ e}\right )^{\!-1}\;. }$$

(4.130)

Obviously $\overline{\nu }_{eff}$ allows to obtain ν _m ^e again as this latter frequency is time-independent. When such is not the case, equation (4.129) allows to obtain an equivalent expression to equation (4.128) but with a complex frequency in place of ν _m ^e

$$\displaystyle{ \overline{V _{e}}_{d}\; =\; \frac{eE_{0}} {m\;(\overline{\nu }_{eff} + j\omega )}\;, }$$

(4.131)

whereas the complex conductivity is

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; \frac{e^{2}n_{e}} {m\;(\overline{\nu }_{eff} + j\omega )}\;. }$$

(4.132)

However, separating $\overline{\nu }_{eff}$ in real and imaginary parts, $\overline{\nu }_{eff} =\nu _{R} + j\;\nu _{I}$, we obtain an exact equivalent expression (4.120) as follows

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; \frac{e^{2}\;n_{e}^{{\ast}}} {m\;(\nu ^{{\ast}} + j\omega )}\;, }$$

(4.133)

with n _e ^∗ and ν ^∗ denoting an effective electron density and a new effective collision frequency given by

$$\displaystyle\begin{array}{rcl} & & n_{e}^{{\ast}}\; =\; \frac{n_{e}} {1 +\nu _{I}/\omega }{}\end{array}$$

(4.134)

$$\displaystyle\begin{array}{rcl} & & \nu ^{{\ast}}\; =\; \frac{\nu _{R}} {1 +\nu _{I}/\omega }\,.{}\end{array}$$

(4.135)

Separating equation (4.130) in equations for the real and imaginary parts of complex quantities, we obtain from equation for the imaginary part

$$\displaystyle{ \nu _{R}\int _{0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\; =\; \left (1 + \frac{\nu _{I}} {\omega } \right )\int _{0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\;, }$$

(4.136)

from which we may write

$$\displaystyle{ \nu ^{{\ast}}\; =\;\int _{ 0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\; \times \left (\int _{0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;v_{e}^{\;3}\;\frac{df_{e}^{0}} {dv_{e}} \;dv_{e}\right )^{\!-1}\;. }$$

(4.137)

On the other hand, from the comparison between the imaginary part of the complex conductivity (4.12)

$$\displaystyle{ \overline{\sigma _{c}}_{e}\; =\; en_{e}\;\overline{\mu _{e}}\; =\; -\;\frac{e^{2}} {m}\int _{0}^{\infty } \frac{1} {\nu _{m}^{e} + j\omega }\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e} }$$

(4.138)

and the imaginary part of equation (4.133), we obtain

$$\displaystyle{ n_{e}^{{\ast}}\; =\; -\;(\nu ^{{\ast}\;2} +\omega ^{2})\int _{ 0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;. }$$

(4.139)

When ν _m ^e is velocity-independent, equations (4.137) and (4.139) reduce to ν ^∗ = ν _m ^e and n _e ^∗ = n _e . Finally, in terms of the EEDF normalized through equation (3.166), ν ^∗ and n _e ^∗ are written as follows

$$\displaystyle\begin{array}{rcl} & & \nu ^{{\ast}}\; =\;\int _{ 0}^{\infty } \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;u^{3/2}\; \frac{df} {du}\;du\; \times \left (\int _{0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;u^{3/2}\; \frac{df} {du}\;du\right )^{\!-1}{}\end{array}$$

(4.140)

$$\displaystyle\begin{array}{rcl} & & n_{e}^{{\ast}}\; =\; -\;\frac{2} {3}\;n_{e}\;(\nu ^{{\ast}\;2} +\omega ^{2})\int _{ 0}^{\infty } \frac{1} {\nu _{m}^{e\;2} +\omega ^{2}}\;u^{3/2}\; \frac{df} {du}\;du\;.{}\end{array}$$

(4.141)

As firstly pointed out in Whitmer and Herrmann (1966), the two effective parameters ν ^∗ and n _e ^∗ have the advantage to permit the writing of a relation of the type (4.120) for the electron conductivity in HF fields, for gases in which the collision frequency for momentum transfer depends on the electron energy.

Exercises

Exercise 4.1:

Determine the expression of the velocity-averaged collision frequency for momentum transfer valid as ω ≫ ν _m ^e, that allows to keep the time-averaged power absorbed from the field $\overline{P_{E}(t)}$ with the same form as the effective collision frequency for momentum transfer is independent of energy.

Resolution:

When ω ≫ ν _m ^e and ν _m ^e(v _e ) = const, we obtain the following expression for the time-averaged power absorbed from the field from equations (4.15) and (4.16)

$$\displaystyle{\overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m} \;\frac{\nu _{m}^{e}} {\omega ^{2}} \;E_{rms}^{\;2}\;,}$$

so that as ν _m ^e(v _e ) ≠ const, we must consider the energy-averaged frequency

$$\displaystyle{\nu ^{{\ast}}\; =\; -\; \frac{1} {n_{e}}\int _{0}^{\infty }\nu _{ m}^{e}\;\frac{df_{e}^{0}} {dv_{e}} \;\;\frac{4\pi v_{e}^{3}} {3} \;dv_{e}\;,}$$

in order an equivalent expression for $\overline{P_{E}(t)}$ may be used

$$\displaystyle{\overline{P_{E}(t)}\; =\; \frac{e^{2}n_{e}} {m} \;\frac{\nu ^{{\ast}}} {\omega ^{2}}\;E_{rms}^{\;2}\;.}$$

Integrating by parts the frequency ν ^∗ can also be written as

$$\displaystyle{\nu ^{{\ast}}\; =\;< \frac{1} {v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{\nu _{m}^{e}\;v_{e}^{\;3}} {3} \right ) >\;.}$$

Using the normalization (3.166) the first expression for ν ^∗ writes as follows

$$\displaystyle{\nu ^{{\ast}}\; =\; -\;\frac{2} {3}\int _{0}^{\infty }\nu _{ m}^{e}\;u^{3/2}\; \frac{df} {du}\;du\;.}$$

This frequency corresponds to the effective collision frequency introduced in Appendix A.4.1, in the limit ω ≫ ν _m ^e, and it differs from the effective collision frequency $\nu _{m}^{{\prime}}$ used with a DC field to write the electron mobility as $\mu _{e} = e/(m\;\nu _{m}^{{\prime}})$ (see Exercise 3.6). Obviously, both result in ν _m ^e when the frequency ν _m ^e is velocity-independent.

Exercise 4.2:

Obtain the time-averaged power absorbed from the field by the electrons, for the case of a high-frequency field ω ≫ ν _m ^e, directly from equation (4.22) for the isotropic component of the electron velocity distribution f _e ⁰.

Resolution:

Multiplying the symmetric of the left-hand side member of equation (4.22) by the electron velocity $u = \frac{1} {2}mv_{e}^{\;2}$ and integrating in all velocity space, we obtain

$$\displaystyle{\overline{P_{E}(t)}\; =\;\int _{ 0}^{\infty }u\;\;\frac{1} {2}\left (\frac{eE_{0}} {m} \right )^{2} \frac{1} {v_{e}^{\;2}}\; \frac{d} {dv_{e}}\left (\frac{v_{e}^{\;2}} {3} \; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \right )\;4\pi v_{e}^{\;2}\;dv_{ e}\;.}$$

Integrating now by parts, we find

$$\displaystyle\begin{array}{rcl} \overline{P_{E}(t)}& =& -\;\frac{1} {2}\left (\frac{eE_{0}} {m} \right )^{2}\int _{ 0}^{\infty } \frac{d} {dv_{e}}\left (\frac{1} {2}\;mv_{e}^{\;2}\right )\;\frac{v_{e}^{\;2}} {3} \; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\;\frac{df_{e}^{0}} {dv_{e}} \;4\pi \;dv_{e} {}\\ & =& -\;\frac{1} {2}\;\frac{(eE_{0})^{2}} {m} \int _{0}^{\infty }\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e} {}\\ \end{array}$$

and identifying Re$\{\overline{\sigma _{c}}_{e}\}$ given by equation (4.16) in this expression, we can write $\overline{P_{E}(t)}$ under the form

$$\displaystyle{\overline{P_{E}(t)}\; =\; \frac{1} {2}\;\mbox{ Re}\{\overline{\sigma _{c}}_{e}\}\;E_{0}^{\;2}\;,}$$

in accordance with equation (4.15).

Exercise 4.3:

Write the expression for the mean power absorbed from the field per electron, $\overline{P_{E}(t)}/n_{e}$, in terms of the mean energy absorbed per collision u _c (u) by an electron of energy u given by equation (4.25).

Resolution:

Replacing equation (4.25) into equations (4.15) and (4.16), we find

$$\displaystyle{\frac{\overline{P_{E}(t)}} {n_{e}} \; =\; -\; \frac{1} {n_{e}}\int _{0}^{\infty }\nu _{ m}^{e}\;u_{ c}\;\;\frac{df_{e}^{0}} {dv_{e}} \;\;\frac{4\pi v_{e}^{\;3}} {3} \;dv_{e}\;.}$$

Making the replacement of v _e with u and using the EEDF normalized such as

$$\displaystyle{\int _{0}^{\infty }f\;\sqrt{u}\;du\; =\; 1\;,}$$

with both distributions linked each other through equation (3.167), we obtain

$$\displaystyle{\frac{\overline{P_{E}(t)}} {n_{e}} \; =\; -\;\frac{2} {3}\int _{0}^{\infty }u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\;\; \frac{df} {du}\;\;du\;.}$$

On the other hand, starting from the first expression in the answer to Exercise 4.2, we also obtain

$$\displaystyle{\frac{\overline{P_{E}(t)}} {n_{e}} \; =\; \frac{2} {3}\int _{0}^{\infty }u\; \frac{d} {du}\left (u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\;\; \frac{df} {du}\right )\;du}$$

and integrating by parts we find the same expression.

Exercise 4.4:

Write the expression for the complex mobility (4.12) and for the time-averaged absorbed power from the field per volume unit (4.15) and (4.16) in the case of an HF field, using the EEDF normalized through equation (3.166).

Resolution:

The two expressions asked are:

$$\displaystyle\begin{array}{rcl} \overline{\mu _{e}}& =& -\;\frac{2} {3}\; \frac{e} {m}\int _{0}^{\infty } \frac{u^{3/2}} {\nu _{m}^{e} + j\omega }\;\frac{df} {du}\;du {}\\ \overline{P_{E}(t)}& =& -\;\frac{2} {3}\;\frac{e^{2}n_{e}} {m} \;E_{rms}^{\;2}\int _{ 0}^{\infty }u^{3/2}\; \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\; \frac{df} {du}\;du\;. {}\\ \end{array}$$

Although the notation seems to be identical the first expression represents a complex quantity and the second a time-averaged value of a real quantity.

Exercise 4.5:

Write the expressions of the time-averaged gain produced by the field in velocity space, $\overline{G_{E}(t)}$, and of the power absorbed by the electrons in terms of this gain, for the case of a HF frequency electric field.

Resolution:

The gain produced by a HF field in velocity space is obtained by inserting (4.20) into equations (3.92) and (3.95)

$$\displaystyle{\overline{G_{E}(t)}\; =\; -\;\frac{1} {6}\left (\frac{eE_{0}} {m} \right )^{2} \frac{\nu _{m}^{e}} {\nu _{m}^{e\;2} +\omega ^{2}}\;\frac{df_{e}^{0}} {dv_{e}} \;4\pi v_{e}^{\;2}\;,}$$

so that using equation (4.25), we still have

$$\displaystyle{\overline{G_{E}(t)}\; =\; -\;\frac{1} {3}\;\frac{\nu _{m}^{e}\;u_{c}} {m} \;\frac{df_{e}^{0}} {dv_{e}} \;4\pi v_{e}^{\;2}\; >\; 0\;.}$$

On the other hand using equation (3.119), the time-averaged power absorbed by the electrons is given by

$$\displaystyle\begin{array}{rcl} \overline{P_{E}(t)}& =& -\int _{0}^{\infty }u\;\frac{d\overline{G_{E}}} {dv_{e}} \;dv_{e}\; =\;\int _{ 0}^{\infty }\overline{G_{ E}}\;mv_{e}\;dv_{e} {}\\ & =& -\int _{0}^{\infty }\nu _{ m}^{e}\;u_{ c}\;\frac{df_{e}^{0}} {dv_{e}} \;\frac{4\pi v_{e}^{3}} {3} \;dv_{e}\;. {}\\ \end{array}$$

In terms of the EEDF normalized through equation (3.166), we find

$$\displaystyle{\overline{G_{E}(t)}\; =\; -\;n_{e}\;\frac{2} {3}\;u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\; \frac{df} {du}}$$

and

$$\displaystyle{\overline{P_{E}(t)}\; =\int _{ 0}^{\infty }\overline{G_{ E}}\;du\; =\; -\;n_{e}\;\frac{2} {3}\int _{0}^{\infty }u^{3/2}\;\nu _{ m}^{e}\;u_{ c}\; \frac{df} {du}\;du\;.}$$

Exercise 4.6:

By writing the drift velocity of electrons under a RF field as $v_{ed}(t) = V _{d0}\;\cos (\omega t + \Phi )$ show that the power absorbed from the field (4.116) can be expressed as

$$\displaystyle{P_{E}(t)\; =\; (P_{E})_{0} +\; \frac{en_{e}E_{0}} {2} \;V _{d0}\;\cos (2\omega t + \Phi )\;.}$$

Obtain the limit values of P _E (t) at low and high field frequencies.

Resolution:

The power absorbed is given by

$$\displaystyle\begin{array}{rcl} P_{E}(t)& =& (\mathbf{J_{e}}\;.\;\mathbf{E})\; =\; en_{e}\;V _{d0}\;\mbox{ Re}\{e^{j(\omega t+\Phi )}\}\;E_{ 0}\;\mbox{ Re}\{e^{j\omega t}\} {}\\ & =& \frac{en_{e}E_{0}} {2} \;V _{d0}\;[\cos (\Phi ) +\cos (2\omega t + \Phi )]\;, {}\\ \end{array}$$

so that using equation (4.111) we obtain

$$\displaystyle{P_{E}(t)\; =\; (P_{E})_{0} +\; \frac{en_{e}E_{0}} {2} \;V _{d0}\;\cos (2\omega t + \Phi )\;.}$$

When ω ≪ ν _m ^e, v _ed (t) and E(t) are in phase so that

$$\displaystyle{P_{E}(t)\; =\; en_{e}E_{0}V _{d0}\;\cos ^{2}(\omega t)\;,}$$

with $V _{d0} = eE_{0}/(m\nu _{m}^{e})$ and $\overline{P_{E}(t)} = en_{e}E_{0}V _{d0}/2$. On the other hand, when ω ≫ ν _m ^e, the phase shift of the drift velocity approaches $-\;\pi /2$ and we obtain

$$\displaystyle{P_{E}(t)\; =\; \frac{en_{e}E_{0}} {2} \;V _{d0}\;\sin (2\omega t)\;,}$$

with $V _{d0} = eE_{0}/(m\omega )$ in accordance with equation (4.18), and therefore $\overline{P_{E}(t)} = 0$.