  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.

Mesoscale and Nanoscale Physics;Applied Physics;Computational Physics