Kinetic modeling of the electric double layer at a dielectric plasma-solid interface

For a collisionless plasma in contact with a dielectric surface, where with unit probability electrons and ions are, respectively, absorbed and neutralized, thereby injecting electrons and holes into the conduction and valence bands, we study the kinetics of plasma loss by nonradiative electron-hole recombination inside the dielectric. We obtain a self-consistently embedded electric double layer, merging with the quasineutral, field-free regions inside the plasma and the solid. After a description of the numerical scheme for solving the two sets of Boltzmann equations, one for the electrons and ions of the plasma and one for the electrons and holes of the solid, to which this transport problem gives rise to, we present numerical results for a $p$-doped dielectric. Besides potential, density, and flux profiles, plasma-induced changes in the electron and hole distribution functions are discussed, from which a microscopic view on plasma loss inside the dielectric emerges.

Figure 1

Illustration of the potential energy profiles for an electric double layer at a floating dielectric plasma-wall interface (not to scale). Shown are the edges of the conduction ($U_{*}$) and valence ($U_{vb}$) bands, the edge for the motion of valence band holes ($U_h$), the position of the trap levels ($E_t$), and the potential energy for electrons $(U_e$) and ions ($U_i$) on the plasma side. The origin of the energy axis is the potential just outside the solid, $\chi$ is the electron affinity, and $E_g$ the energy gap. The positions $z_1$ and $z_p$ are the end points of the double layer and $z_w$ is the location of the plasma source. Also shown are the potential energies at these positions, playing an important role in the modeling. As explained in the main text, we cannot resolve the energy space up to the energies where carriers are actually injected from the plasma into the solid. In the simplified model for the solid side, shown in the shaded area, we inject carriers at artificially low energies by source functions included in the matching conditions at $z=0$.

Figure 2

Electric field multiplied by the dielectric function across the interface for parameter sets A and B, plotted respectively in black and orange (grey), showing the matching (2) to be satisfied. The maximum at $U_c\approx 6.8$ eV comes from the Schwager-Birdsall boundary condition. Since $U_p\approx 5$ eV effectively corresponds already to $z\approx \infty$ the maximum has no physical meaning. It is an artifact of implementing in a collisionless plasma a field-free, quasineutral bulk region.

Figure 3

Distribution functions for the injected electrons (upper panels) and holes (lower panels) for parameter set A at $U_c=0$, that is, at the interface and at $U_c=21.4\mathrm{meV}$. Instead of $E$ and $T$ the variables $E-U_s$ and $T_z=E-U_s-T$ are used, where we attached to the latter a sign to denote the distributions of left- ($T_z<0$) and right-moving ($T_z>0$) particles. The injection peaks at $E-U_{*}=I_{*}^{\mathrm{in}}=0.2$ eV and at $E-U_h=I_h^{\mathrm{in}}=0.15$ eV are clearly visible in the data for $U_c=0$. Away from the interface, the peak gradually vanishes. Replicas due to phonon emission and absorption can be also seen, as well as the drop at $T_z=0$, signaling right-moving states to be less populated than left-moving ones. Colors (shading) are used only for visibility reasons and the triangular shape is due to energy conservation.

Figure 4

Total (solid lines) and directional (long and short dashed lines) densities of the injected carriers for the parameter set A. For plotting purposes we introduced reference densities $n_{*}^{\text{ref}}=6\times {10}^{12}{\text{cm}}^3$ and $n_h^{\text{ref}}={10}^{12}{\text{cm}}^3$. The positive difference of left- and right-moving densities $\mathrm{\Delta }{n}_s$, shown in the inset, is on the same order of magnitude for electrons [blue (dark grey)] and holes [red (grey)], indicating that for both polarities surplus carriers move more likely to the left than to the right.

Figure 5

Potential energy (upper panel) and net charge density (lower panel) for parameter set A as a function of $z$ in units of the Debye lengths, ${\lambda }_D^w=1.821\mu \text{m}$ and ${\lambda }_D^p=9.107\mu \text{m}$. The kink in $U_c$ at $z=0$ signals the matching condition (2). To fit the density profiles into a single plot, we scaled them on the solid side by $n_0^w={10}^{13}{\text{cm}}^{-3}$ and on the plasma side by $n_0^p=-{10}^{12}{\text{cm}}^{-3}$.

Figure 6

Mobile charge carriers (upper panel) and the fluxes (lower panel) as a function of $U_c$ for parameter set A. Data for electrons (holes, ions) are plotted in blue (red) [dark grey (grey)]. Inside the solid, electrons prevail for $U_1/2<U_c<0$, while for $U_c<U_1/2$ holes dominate, indicating the $p$-doped bulk. The quasineutral bulk plasma emerges on the other side of the interface for $U_c\approx U_p$. Also seen is the negative sheath in front of the plasma source at $U_c=U_w$. The electron and ion fluxes of the plasma merge at $U_c=0$ with the electron and hole fluxes of the solid, which then decay nonconcurrently due to the variation of the occupancy of the traps shown in the inset, placing empty (occupied) traps, required for electron (hole) recombination, further away from (closer to) the interface.

Figure 7

Schematic illustration of the three-dimensional integration domain for conduction band electrons spanned by the variables $U_c, E$, and $T$. The domain is cut by the surface $v_{*}(U_c,E,T)=0$ leading to turning points at $T=E-U_{*}\left(U_c\right)$ separating the region where electrons are allowed to move shown in dark [$T<E-U_{*}\left(U_c\right)$] from the forbidden region [$T>E-U_{*}\left(U_c\right)$]. For the valence band holes the domain is divided by the function $T=E-U_h\left(U_c\right)$ giving rise to a different shape. The elementary discretization step, to be the same in all three dimensions, $\mathrm{\Delta }=\hslash {\omega }_0/n$ with $\hslash {\omega }_0$ the phonon energy. The energy cutoff $E_{\mathbf{k}}^{\mathrm{cutoff}}=N\hslash {\omega }_0$ is used for the total kinetic energy $T+T_z$ of the carriers measured from the bottom of the bands, limiting thereby the variable $E$. For the results presented in this work we typically used $n=8, N=14$ yielding around ${100}^3$ discretization points.