Charge kinetics across a negatively biased semiconducting plasma-solid interface

An investigation of the self-consistent ambipolar charge kinetics across a negatively biased semiconducting plasma-solid interface is presented. For the specific case of a thin germanium layer with nonpolar electron-phonon scattering, sandwiched between an Ohmic contact and a collisionless argon plasma, we calculate the current-voltage characteristic and show that it is affected by the electron microphysics of the semiconductor. We also obtain the spatially and energetically resolved fluxes and charge distributions inside the layer, visualizing thereby the behavior of the charge carriers responsible for the charge transport. Albeit not quantitative, because of the crude model for the germanium band structure and the neglect of particle-nonconserving scattering processes, such as impact ionization and electron-hole recombination, which at the energies involved cannot be neglected, our results clearly indicate (i) the current through the interface is carried by rather hot carriers and (ii) the perfect absorber model, often used for the description of charge transport across plasma-solid interfaces, cannot be maintained for semiconducting interfaces.

Figure 1

Illustration of the electric potential across a negatively biased semiconducting plasma-solid interface (not to scale). The bias voltage $U_B$ is applied between the bulk plasma and the Ohmic contact to the left of the semiconducting layer, as sketched in the bottom right. Measuring the potential with respect to its value in the plasma bulk, $U_B=U_I-U_{\text{SC}}$, with $U_I$ the sheath (interface) potential at $z=0$ and $U_{\text{SC}}$ the potential drop across the semiconducting layer. In the Schwager-Birdsall approach [14] to the collisionless plasma, the presheath potential $U_W$ stretches over the entire region $z>z_p$, where $z_p$ is the location of the bulk plasma, which is effectively infinitely far away from the interface.

Figure 2

Energy-resolved source functions $S_{*}^{<}$ (left) and $S_h^{<}$ (right) in arbitrary units for the floating interface, that is, the situation where electron and ion fluxes are equal. Under bias, with a net flux, the functions look qualitatively similar, only the numerical values and the scales are different. The material parameters are taken from Table 1, the thickness of the germanium layer is $1\mu \mathrm{m}, U_W=0.97\mathrm{eV}$, and $U_B=-5.125\mathrm{eV}$. Note, $T^{\prime }>E-U_{*}\left(0\right)$ and $T>E-U_h\left(0\right)$ are energetically not allowed. Since the injection of holes spreads over a larger energy range, the absolute values of $S_h^{<}$ are smaller than the values of $S_{*}^{<}$. Integrated over energy, however, the source functions ensure flux equality, as required for the floating interface.

Figure 3

Current-voltage characteristic across a device consisting of a germanium layer sandwiched between an argon plasma and an Ohmic contact. The orange lines belong to the reflecting plasma-germanium interface whereas the blue lines show data for an interface which perfectly absorbs electrons from the plasma and lets them never return to it. Different layer thicknesses are considered. For comparison, we also plot the characteristic of a perfect absorber without a germanium layer, that is, for $U_{\text{SC}}=0$ (solid black line, hardly seen in the main panel). The inset shows for the perfectly absorbing interfaces the zero crossings of the net flux. The germanium layer shifts the crossings to lower voltages because of the potential drops $U_{\text{SC}}$.

Figure 4

As a function of $U_B$ and for the reflecting plasma-solid interface, the electric field at the interface (top panel), the potential drop $U_{\text{SC}}$ across the germanium layer (center panel), and the electron and ion and hole fluxes flowing from the plasma towards the Ohmic contact (bottom panel). Reflected and emitted electrons refer, respectively, to electrons quantum mechanically reflected at the potential step connected with the plasma-solid interface and electrons emitted back to the plasma due to collisions inside the semiconductor. While the electric field is identical for the layer thicknesses considered, $U_{SC}$ changes with the thickness. Electric field and $U_{\text{SC}}$ are effectively identical to what one obtains for absorbing matching conditions. Fluxes are given only for the case of a $1\text{−}\mu \mathrm{m}$-thick germanium layer.

Figure 5

Macroscopic properties for the reflecting $1\text{−}\mu \text{m}$-thick interface at the floating point, for which $U_I=5.1\mathrm{eV}, U_{\text{SC}}=0.025\mathrm{eV}$, and $U_W=0.97\mathrm{eV}$. Top to bottom: electric field, electric potential [shifted so that $U\left(z_0\right)=0$], fluxes, and densities. The last two are only given for the injected surplus electrons and holes, the Maxwellian contributions due to the intrinsic carriers, giving rise to the double layer shown in the inset of the top panel, are subtracted. In addition to the net fluxes and densities per species, we also plot the direction-resolved quantities.

Figure 6

In arbitrary units, the energy-resolved fluxes across the interface with a $1\text{−}\mu \mathrm{m}$-thick germanium layer. The interface is at the floating point with potential drops given in the caption of Fig. 5. On the left (right), the net electron and hole (electron and ion) fluxes inside the germanium layer (plasma) are shown, with high- and low-energy contributions plotted in separate panels. The ion flux, located at $U_I+U_W+U_{\text{SC}}-\chi -E_g=1.39\mathrm{eV}$, is shown by the red line (not belonging to the color scale). Its energetic spread is negligible on the scale of the other fluxes. The energy scale is chosen such that within the germanium layer $E=0$ occurs at the minimum of $U_{*}$ and $U_h$, respectively.

Figure 7

Distribution functions in arbitrary units for electrons (top two rows) and holes (bottom two rows) at different spatial locations of a $1\text{−}\mu \mathrm{m}$-thick nearly floating germanium layer. Regions of high and low energy are shown in separate panels. On the horizontal axis, positive values of $|T_z|=E-U_s-T$ show $F_s^{>}$ and negative values depict $F_s^{<}$. The triangular shape is due to the energy restriction $0<T<E-U_s$. Note the logarithmic (linear) scale for low (high) energies. At $z=-0.9$ and $-0.5\mu \mathrm{m}$, the high-energy distributions are multiplied by the factors given in the panels to utilize the same scale. The potential drops are as in Figs. 5 and 6.