Electrostatics in semiconducting devices II : Solving the Helmholtz equation

Abstract

The convergence of iterative schemes to achieve self-consistency in mean field problems such as the Schrödinger-Poisson equation is notoriously capricious. It is particularly difficult in regimes where the non-linearities are strong such as when an electron gas in partially depleted or in presence of a large magnetic field. Here, we address this problem by mapping the self-consistent quantum-electrostatic problem onto a Non-Linear Helmoltz (NLH) equation at the cost of a small error. The NLH equation is a generalization of the Thomas-Fermi approximation. We show that one can build iterative schemes that are provably convergent by constructing a convex functional whose minimum is the seeked solution of the NLH problem. In a second step, the approximation is lifted and the exact solution of the initial problem found by iteratively updating the NLH problem until convergence. We show empirically that convergence is achieved in a handfull, typically one or two, iterations. Our set of algorithms provide a robust, precise and fast scheme for studying the effect of electrostatics in quantum nanoelectronic devices.

Figures (10)

• Figure 1: Schematic of the different (nested) algorithms used in this work. See Section \ref{['sec:algo_solving_NLH']} for a detailed description of the algorithms. In the Non-Linear Helmholtz solver $U_i = 0$ is an initial guess used to calculate the local density of states $\rho_i$ before the iterations start.
• Figure 2: Construction of the piecewise linear ILDOS $\bar{Q}_i(E)$ (black line) from the continuous one $Q_i(E)$ (thin gray line). The red crosses correspond to the initial list of points $E_i^\alpha$. Left: the two tangents (dotted line) do not intersect inside the segment, one interpolates linearly between the two red points. Middle panel: the tangents do intersect, one uses two linear segments (the two tangents) to interpolate between the two red points. Right panel: when a new red point is inserted, the piecewise linear ILDOS $\bar{Q}_i(E)$ is updated.
• Figure 3: Solving the NLH equation using the piecewise linear Helmholtz algorithm: PESCA. a) Input ILDOS: a piecewise linear ILDOS with two branches. This corresponds to the PESCA approximation. b) Colormap and schematics of a side view of the hexagonal nanowire with top (black) and back gate (green). The color code corresponds to the charges in each cell of the nanowire. c) Convergence of $\alpha(i)$ on each site versus different iterations. $\alpha=0$ (yellow, first branch) or $\alpha=1$ (green, second branch). The initial configuration $\alpha(i)$ is random here.
• Figure 4: Solving the NLH equation using the piecewise linear Helmholtz algorithm: Thomas-Fermi. a) Input ILDOS: a piecewise linear ILDOS with two branches. This corresponds to the Thomas-Fermi approximation. b) Colormap and schematics of a side view of the hexagonal nanowire with top (black) and back gate (green). The color code corresponds to the charges in each cell of the nanowire. c) Convergence of $\alpha(i)$ on each site versus different iterations. $\alpha=0$ (yellow, first branch) or $\alpha=1$ (green, second branch). The initial configuration $\alpha(i)$ is random here.
• Figure 5: Solving the NLH equation using the piecewise linear Helmholtz algorithm: Valence-Conduction band model. a) Input ILDOS: a piecewise linear ILDOS with three branches. They correspond respectively to the valence band, the gap and the conduction band. b) Colormap and schematics of a side view of the hexagonal nanowire with top (black) and back gate (green). The color code corresponds to the charges in each cell of the nanowire. c) Convergence of $\alpha(i)$ on each site versus different iterations.$\alpha=-1$ (blue, valence band branch) $\alpha=0$ (yellow, gap branch) or $\alpha=1$ (green, conduction band branch). The initial configuration $\alpha(i)$ is random here.