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doi: 10.3934/naco.2021034
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## A new hybrid method for shape optimization with application to semiconductor equations

 1 Université Cadi Ayyad, laboratoire de Mathématiques Appliquées et Informatique 2 Faculté des Sciences et Techniques, Avenue Abdelkrim El khttabi B. P. 549, Marrakech, Maroc

* Corresponding author: Youness El Yazidi

Received  May 2021 Revised  July 2021 Early access August 2021

The aim of this work is to reconstruct the depletion region in pn junction. Starting with famous drift diffusion model, we establish the simplified equation for the considered semiconductor. There we call the shape optimization technique to formulate a minimization problem from the inverse problem at hand. The existence of an optimal solution of the optimization problem is proved. The proposed numerical algorithm is a combined Domain Decomposition method with an efficient hybrid conjugate gradient guided by differential evolution heuristic algorithm, the finite element method is used to discretize the state equation. At the end we establish several numerical examples, to prove the validity of theoretical results using the proposed algorithm, in addition we show some simulation of the depletion region approximation under two different functioning modes.

Citation: Youness El Yazidi, Abdellatif ELLABIB. A new hybrid method for shape optimization with application to semiconductor equations. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021034
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##### References:
the geometry of the PN-Junction
Example 1: the optimal interfaces versus the exact ones
Example 2: the optimal interfaces versus the exact ones
Example 1: the optimal interfaces versus the exact ones
Example 2: the optimal interfaces versus the exact ones
Example 1: the obtained results with noisy data
Example 2: the obtained results with noisy data
Forward mode
Reverse mode
The used parameters in DE
 Population size Max generations Mutation scale $\varrho$ Crossover ratio $\sigma$ 25 10 0.1 0.75
 Population size Max generations Mutation scale $\varrho$ Crossover ratio $\sigma$ 25 10 0.1 0.75
Comparison of errors
 Noise level 0% 1% 5% 10% Example 1 0.0095 0.0184 0.0676 0.1779 Example 2 0.0213 0.0243 0.0458 0.0589
 Noise level 0% 1% 5% 10% Example 1 0.0095 0.0184 0.0676 0.1779 Example 2 0.0213 0.0243 0.0458 0.0589
Optimal cost for each functioning mode
 Depletion mode Enhancement mode $V^+$ $V^-$ cost $V^+$ $V^-$ cost $+0.1V$ $-0.0V$ $0.018$ $-0.3V$ $+0.3V$ $0.029$ $+0.3V$ $-0.3V$ $0.047$ $-0.6V$ $+0.6V$ $0.063$ $+0.6V$ $-0.6V$ $0.079$ $-0.6V$ $+1.2V$ $0.085$
 Depletion mode Enhancement mode $V^+$ $V^-$ cost $V^+$ $V^-$ cost $+0.1V$ $-0.0V$ $0.018$ $-0.3V$ $+0.3V$ $0.029$ $+0.3V$ $-0.3V$ $0.047$ $-0.6V$ $+0.6V$ $0.063$ $+0.6V$ $-0.6V$ $0.079$ $-0.6V$ $+1.2V$ $0.085$
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