标题： 静电学在半导体器件中的应用 II：求解亥姆霍兹方程 显示英文标题

标题： Electrostatics in semiconducting devices II : Solving the Helmholtz equation

摘要： 在诸如Schrödinger-Poisson方程的平均场问题中，达到自洽性的迭代方案的收敛性以难以预测而著称。在非线性很强的区域尤其困难，例如当电子气部分耗尽或存在强磁场时。在这里，我们通过将自洽的量子静电问题映射到一个非线性Helmoltz（NLH）方程，以付出一个小误差为代价来解决这个问题。NLH方程是Thomas-Fermi近似的推广。我们证明可以通过构建一个凸泛函来建立可证明收敛的迭代方案，该泛函的最小值即为NLH问题所求解。在第二步中，该近似被提升，并通过迭代更新NLH问题直到收敛来找到初始问题的精确解。我们通过实验证明，在少量迭代中即可实现收敛，通常为一到两次。我们的算法集合为研究量子纳米电子器件中的静电效应提供了一个稳健、精确且快速的方案。 显示英文摘要

摘要： The convergence of iterative schemes to achieve self-consistency in mean field problems such as the Schr\"odinger-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.