# The self-consistent quantum-electrostatic problem in strongly non-linear regime

## TL;DR

This paper introduces a stable, convergent algorithm for solving the challenging self-consistent quantum-electrostatic problem in highly non-linear regimes, enabling accurate modeling of quantum nanoelectronics devices.

## Contribution

A novel, stable, and intrinsically convergent algorithm for solving the Poisson-Schrödinger problem in strongly non-linear conditions.

## Key findings

- Successfully modeled quantum Hall stripe formations.
- Accurately calculated differential conductance in quantum point contacts.
- Demonstrated stability and convergence in highly non-linear regimes.

## Abstract

The self-consistent quantum-electrostatic (also known as Poisson-Schr\"odinger) problem is notoriously difficult in situations where the density of states varies rapidly with energy. At low temperatures, these fluctuations make the problem highly non-linear which renders iterative schemes deeply unstable. We present a stable algorithm that provides a solution to this problem with controlled accuracy. The technique is intrinsically convergent including in highly non-linear regimes. We illustrate our approach with (i) a calculation of the compressible and incompressible stripes in the integer quantum Hall regime and (ii) a calculation of the differential conductance of a quantum point contact geometry. Our technique provides a viable route for the predictive modeling of the transport properties of quantum nanoelectronics devices.