Robust Simulation of Poisson's Equation in a P-N Diode Down to 1 {\mu}K

TL;DR

This paper introduces a reformulation of Poisson's equation that enables stable and convergent simulation of a p-n diode at ultra-low temperatures down to 1 microkelvin, surpassing previous temperature limits.

Contribution

The authors develop a new formulation of Poisson's equation that improves convergence in deep-cryogenic simulations, allowing stable modeling at microkelvin temperatures.

Findings

Achieved convergence of Poisson's equation at 1 microkelvin.

Demonstrated potential and carrier density profiles at ultra-low temperatures.

Provided Python implementation details for reproducibility.

Abstract

Semiconductor devices are notoriously difficult to simulate at deep-cryogenic temperatures. The lowest temperature that can be simulated today in commercial TCAD is around 4.2 K, possibly 100 mK, while most experimental quantum science is performed at 10 mK or lower. Besides the challenges in transport solvers, one of the main bottlenecks is the non-convergence in the electrostatics due to the extreme sensitivity to small variations in the potential. This article proposes to reformulate Poisson's equation to take out this extreme sensitivity and improve convergence. We solve the reformulated Poisson equation for a p-n diode using an iterative Newton-Raphson scheme, demonstrating convergence for the first time down to a record low temperature of one microkelvin using the standard IEEE-754 arithmetic with double precision. We plot the potential diagrams and resolve the rapid variation of the carrier densities near the edges of the depletion layer. The main Python functions are presented in the Appendix.