Electrostatics in semiconducting devices I : The Pure Electrostatics Self Consistent Approximation

Abstract

In quantum nanoelectronics devices, the electrostatic energy is the largest energy scale at play and, to a large extend, it determines the charge distribution inside the devices. Here, we introduce the Pure Electrostatic Self consistent Approximation (PESCA) that provides a minimum model that describes how to include a semiconductor in an electrostatic calculation to properly account for both screening and partial depletion due to e.g. field effect. We show how PESCA may be used to reconstruct the charge distribution from the measurement of pinch-off phase diagrams in the gate voltages space. PESCA can also be extended to account for magnetic field and calculate the edge reconstruction in the quantum Hall regime. The validity of PESCA is controlled by a small parameter $κ= C_g/C_q$, the ratio of the geometrical capacitance to the quantum capacitance, which is, in many common situations, of the order of 1%, making PESCA a quantitative technique for the calculation of the charge distribution inside devices.

Figures (13)

• Figure 1: Side view of the split quantum wire system studied in this article. Various colors correspond to different regions. The system is infinite along the $y$ axis. The orange and yellow regions correspond to $GaAs$ and $AlGaAs$ respectively. The green region corresponds to the doped $AlGaAs$ layer. A finite density of surface charges appears at the GaAs/vacuum upper interface and is shown in violet. Metallic electrodes are shown in brown and correspond to equipotential. The electron density at special points $x_{\rm side}=-585$nm, $x_{\rm wire}=-225$nm and $x_{\rm middle}=0$nm are called $n_{\rm side}$, $n_{\rm wire}$ and $n_{\rm middle}$ respectively. We use different scales along the $x$ and $z$ directions (see text).
• Figure 2: Example of ILDOS. a): ILDOS obtained from a quantum calculation. b) ILDOS corresponding to Thomas-Fermi for a 2DEG without magnetic field. c) ILDOS in the PESCA approximation. The latter is independent of the material.
• Figure 3: Illustration of the PESCA algorithm. For each iteration, the upper panel shows the $\mathcal{D} /\mathcal{N}$ partitioning (black for $\cal D$ cells, $\cal N$ for yellow cells), the middle panel shows the potential $U(x)$ for this partitioning (blue curve) and the lower panel shows the corresponding density profile $n(x)$ (red curve). The results shown here correspond to a full Dirichlet initialization. We set $V_{mid} = -0.88V$ and $V_{gate} = -0.52 V$.
• Figure 4: fraction of Dirichlet cells ($\mathcal{D}$) in the 2DEG as a function of the number of PESCA iterations. In green all cells belonged to $\mathcal{D}$ at the first iteration. In blue all cells were set to Neumann ($\mathcal{N}$) in the initial configuration. The inset on the right corner shows the $\mathcal{D}/\mathcal{N}$ partitioning as a function of position $x$. The black regions correspond to Dirichlet cells and yellow to Neumann points.
• Figure 5: Top panel: PESCA pinch-off phase diagram for the split wire device shown in Fig.\ref{['fig:schema_dispo_2DEG']}. $n_{dop} = 1.43 10^{16} m^{-2}$, $n_s = 1.28 10^{15} m^{-2}$, $V_{off} = -0.813V$ and $V_{sc} = -0.668V$. The different lines $W_{wire}$, $W_{side}$ and $W_{middle}$ separate the different regions A-E, see text. Bottom panels: 2DEG density profile calculated for the points A-E in the phase diagram. The labels $n_{side}$, $n_{middle}$ and $n_{wire}$ indicate the 2DEG density at specific points, respectively, underneath the side gate, the middle gate and in between, see Fig.\ref{['fig:schema_dispo_2DEG']}.