Electron surface layer at the interface of a plasma and a dielectric wall

We study the plasma-induced modifications of the potential and charge distribution across the interface of a plasma and a dielectric wall. For this purpose, the wall-bound surplus charge arising from the plasma is modeled as a quasistationary electron surface layer in thermal equilibrium with the wall. It satisfies Poisson's equation and minimizes the grand canonical potential of wall-thermalized excess electrons. Based on an effective model for a graded interface taking into account the image potential and the offset of the conduction band to the potential just outside the dielectric, we specifically calculate the modification of the potential and the distribution of the surplus electrons for MgO, SiO${}_2$, and Al${}_2$O${}_3$ surfaces in contact with a helium discharge. Depending on the electron affinity of the surface, we find two vastly different behaviors. For negative electron affinity, electrons do not penetrate into the wall and a quasi-two-dimensional electron gas is formed in the image potential, while, for positive electron affinity, electrons penetrate into the wall and a negative space-charge layer develops in the interior of the dielectric. We also investigate how the non-neutral electron surface layer—which can be understood as the ultimate boundary of a bounded gas discharge—merges with the neutral bulk of the dielectric.

Figure 1

Qualitative sketch of an interface between a plasma and a dielectric wall. Upper panel: band structure, microscopic crystal potential merging with the image potential, and sheath potential. Lower panel: effective potential for the graded interface on which the model of an electron surface layer is based.

Figure 2

Sketch of the refined model of the interface between a plasma and a dielectric wall. In the plasma, equal densities of electrons and ions ensure quasineutrality. The positive space charge in front of the effective wall defines the plasma sheath. The ESL contains a very narrow interface-specific region (ISR), where the model of the graded interface is used, and a wide space-charge region (SCR), which allows a continuous merging with the neutral bulk of the dielectric, where intrinsic electrons and holes balance each other to guarantee charge neutrality. Note that the widths of the various regions are not to scale.

Figure 3

Plasma-supplied excess electron density $n$ (upper panel) and the potential $-\phi$ (lower panel) it gives rise to, for a MgO surface in contact with a helium discharge with $n_0={10}^7{\mathrm{cm}}^{-3}$ and $k_B{T}_e=2$ eV calculated without accounting for a SCR in the dielectric (crude ESL model). The cutoff of the interface region is $z_s=-z_0$. As can be seen, almost all of the plasma-induced wall charge is located in the well of the image potential in front of the surface.

Figure 4

Plasma-supplied excess electron density $n$ (upper panel) and the potential $-\phi$ (lower panel) it gives rise to, for an Al${}_2$O${}_3$ surface in contact with a helium discharge with $n_0={10}^7{\mathrm{cm}}^{-3}$ and $k_B{T}_e=2$ eV. The red lines show data obtained from the refined ESL model accounting for an ISR and a SCR, the boundary between the two was put at $z_s=-3z_0$, the green circles show data for a model which consists only of an ISR with cutoff $z_s=-0.9\mathrm{cm}$ (crude ESL model), and the blue triangles show data for a model consisting only of a SCR for $z<0$. Irrespective of the approximation, the plasma-induced wall charge extends deep into the bulk. Note the different scales of the axes for the left and right panels. On the spatial scale of the SCR shown in the left panels the ISR of the right panels becomes a vertical line.

Figure 5

Electron density $n$ and hole density $p$ at a GaAs surface in contact with a helium discharge with $n_0={10}^7{\mathrm{cm}}^{-3}$ and $k_B{T}_e=2$ eV calculated with the refined ESL model without ISR. The plasma-induced wall charge sits inside the GaAs wall. Deep inside the bulk charge neutrality is achieved by an equal density of electrons and holes.

Figure 6

Center of gravity $\overline{z}$ of the plasma-supplied excess electron distribution at a MgO surface (upper panel) and the $z_{90\%}$ value for the electron distribution at an Al${}_2$O${}_3$ and a SiO${}_2$ surface (lower panel), all in contact with a helium discharge with $n_0={10}^7{\mathrm{cm}}^{-3}$ and $k_B{T}_e=2$ eV, as a function of the surface temperature $T_S$. The data shown in the upper and lower panel were obtained, respectively, from the crude ESL model and the refined ESL model without an ISR.

Figure 7

Plasma-supplied surplus electron density $n$ at a SiO${}_2$ surface in contact with a helium discharge as a function of the electron temperature (upper panel, $n_0={10}^7{\mathrm{cm}}^{-3}$) and the plasma density (lower panel, $k_B{T}_e=2$ eV) for $T_S=300$ K. The refined ESL model without an ISR was employed to produce the data.

Figure 8

Density of plasma-supplied surplus electrons trapped in the ESL as well as electron and ion density in the plasma sheath (upper panel) and potential (lower panel) for a MgO surface in contact with a helium discharge ($n_0={10}^7{\mathrm{cm}}^{-3}$ and $k_B{T}_e=2$ eV). The data were obtained from the crude ESL model.

Figure 9

Density of plasma-supplied excess electrons in the ESL as well as electron and ion density in the plasma sheath (upper panel) and potential (lower panel) for a SiO${}_2$ surface in contact with a helium discharge ($n_0={10}^7{\mathrm{cm}}^{-3}$ and $k_B{T}_e=2$ eV). The refined ESL model without an ISR was employed to produce the data.