Absorption of an Electron by a Dielectric Wall

We introduce a method for calculating the probability with which a low-energy electron hitting the wall of a bounded plasma gets stuck in it and apply the method to a dielectric wall with positive electron affinity smaller than the band gap using MgO as an example. In accordance with electron beam scattering data we obtain energy-dependent sticking probabilities significantly less than unity and question thereby for electrons the perfect absorber assumption used in plasma modeling.

Figure 1

(a) Potential energy step of height $\chi$ modeling the wall and a scattering trajectory bringing an electron entering the wall back to the plasma. Half circles are the moduli of the electron momenta outside and inside the wall at the entrance ($E$) and exit ($E^{\prime }$) point connected by phonon emission events. Also shown is the Coulomb barrier $U_w$ the electron has to overcome to reach the wall, the energy distribution $f(E)$ it may have, and a realistic surface potential (red dashed curve). The tilt of the potential step due to the wall charge responsible for $U_w$ is not shown. (b) Illustration of Eqs. (4)–(5) defining $S(E,\xi )$. The potential step leads to a transmission probability $\mathcal{T}$, while the emission of phonons to a conditional backscattering probability $\mathcal{E}$ proportional to a quantity $Q$, obtained from the invariant embedding approach, that is, by adding an infinitesimally thin layer to the wall and requiring $Q$ to be invariant against it [12]. Notice the definition of the angles and energies characterizing the electron. Inside (outside) the wall the angles, measured with respect to the surface normal $\widehat{n}$, are denoted $\theta$ ($\beta$) and ${\theta }^{\prime }$ (${\beta }^{\prime }$).

Figure 2

Electron sticking probability for MgO obtained from Eq. (4). On the left is shown $S(E,\xi )$ for all direction cosines $\xi$ and energies $E$ up to the band gap $E_g$. Total reflection occurs in the white region. Below the yellow dotted line inelastic backscattering has no effect on $S(E,\xi )$. On the right, $S(E,\xi )$ (solid curves) and $\mathcal{T}(E,\xi )$ (dashed curves) are plotted as a function of $E$ for representative $\xi$. The material parameters are ${\overline{m}}_e=0.4$, $\chi =1\mathrm{eV}$, $E_g=7.8\mathrm{eV}$ [41], and $\omega =0.1\mathrm{eV}$ [42]. The value for ${\overline{m}}_e$ is strictly applicable only at the band edge. Away from it ${\overline{m}}_e$ depends on momentum. From the band structure [44] we expect our results are reliable up to 5 eV, enough for a comparison with experiment (see Fig. 3).

Figure 3

Electron sticking probability $\overline{S}(E;1)$ for normal incident onto a MgO wall obtained from Eq. (12) (solid curves). We show data for $C=0$ (black), 1 (red), 2 (green), and $\infty$ (blue) with $\overline{\mathcal{T}}(E;1)$ also included for $C=0$ and $C=\infty$ (dashed curves). Symbols are data from electron beam scattering experiments, where two methods, denoted potential adjusted method (pam) and potential subtraction method (psm), have been employed to determine the energy of the electron [37]. Material parameters as in Fig. 2.