Uniform estimates and uniqueness of stationary solutions to the drift–diffusion model for semiconductors

ABSTRACT

We study the stationary problem of the drift–diffusion model with a mixed boundary condition. For this problem, the existence of solutions was established in general settings, while the uniqueness was investigated only in some special cases which do not entirely cover situations that semiconductor devices are used in integrated circuits. In this paper, we prove the uniqueness in a physically relevant situation. The key to the proof is to derive two-sided uniform estimates for the densities of the electron and hole. We establish a new technique to show the lower bound. This together with the Moser iteration method leads to the upper bound.

KEYWORDS:

• Elliptic system
• boundary value problem
• mixed boundary condition
• lower and upper estimates

AMS Subject Classifications:

• 35A02
• 35B45
• 35J57
• 82D37

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was partially supported by JSPS KAKENHI [number 15K17569]; the second author was partially supported by JSPS KAKENHI [number 26800067].