# American Institute of Mathematical Sciences

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 and Optimization, doi: 10.3934/naco.2021034
##### References:
 [1] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002. doi: 10.1137/1.9780898719208. [2] M. Dashti Ardakani and M. Khodadad, Shape estimation of a cavity by inverse application of the 2D elastostatics problem, International Journal of Computational Methods, 10 (2013), 1350042. doi: 10.1142/S0219876213500424. [3] Y. El Yazidi and A. Ellabib, Reconstruction of the depletion layer in MOSFET by genetic algorithms, Mathematical Modeling and Computing, 7 (2020), 96-103.  doi: 10.23939/mmc2020.01.096. [4] Y. El Yazidi and A. Ellabib, An iterative method for optimal control of bilateral free boundaries problem, Mathematical Methodes in Applied Science, (2021), 1–20. doi: 10.1002/mma.7527. [5] Y. El Yazidi and A. Ellabib, Augmented Lagrangian approach for a bilateral free boundary problem, Journal of Applied Mathematics and Computing, 2021. doi: 10.1007/s12190-020-01472-y. [6] A. Ellabib and A. Nachaoui, On the numerical solution of a free boundary identification problem, Inverse Problems in Engineering, 9 (2001), 235-260.  doi: 10.1080/174159701088027764. [7] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003. doi: 10.1137/1.9780898718690. [8] M. Hinze and R. Pinnau, Second-order approach to optimal semiconductor design, Journal of Optimization Theory and Applications, 133 (2007), 179-199.  doi: 10.1007/s10957-007-9203-3. [9] M. Hinze, B. Kaltenbacher and T. N. T. Quyen, Identifying conductivity in electrical impedance tomography with total variation regularization, Numerische Mathematik, 138 (2018), 723-765.  doi: 10.1007/s00211-017-0920-8. [10] C.-H. Huang and C.-C. Shih, A shape identification problem in estimating simultaneously two interfacial configurations in a multiple region domain, Applied Thermal Engineering, 26 (2006), 77-88.  doi: 10.1016/j.applthermaleng.2005.04.019. [11] V. G. Korneev and U. Langer, Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic Problems, World Scientific, 2013. doi: 10.1142/9035. [12] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer Vienna, 2013. doi: 10.1007/978-3-7091-3678-2. [13] M. H. Mozaffari, M. Khodadad and M. Dashti Ardakani, Simultaneous identification of multi-irregular interfacial boundary configurations in non-homogeneous body using surface displacement measurements, Journal of Mechanical Engineering Science, 231 (2016), 1-12.  doi: 10.1177/0954406216636166. [14] A. Novruzi and M. Pierre, Structure of shape derivatives, Journal of Evolution Equations, 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y. [15] O. Pironneau, Optimal Shape Design For Elliptic Systems, Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-87722-3. [16] H. Prautzsch, W. Boehm and M. Paluszny, Bézier and B-Spline Techniques, Springer, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-662-04919-8. [17] K. V. Price, Differential evolution: a fast and simple numerical optimizer, in Proceedings of North American Fuzzy Information Processing, (1996), 524–527. doi: 10.1109/NAFIPS.1996.534790. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] S. Salhi, Heuristic Search The Emerging Science of Problem Solving, Palgrave Macmillan, 2017. doi: 10.1007/978-3-319-49355-8. [20] R. Storn and K. Price, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328. [21] S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley & Sons Singapore Pte. Limited, 2012. [22] J. P. Zolesio and J. Sokolowski, Introduction to Shape Optimization Shape Sensitivity Analysis, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9.

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##### References:
 [1] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002. doi: 10.1137/1.9780898719208. [2] M. Dashti Ardakani and M. Khodadad, Shape estimation of a cavity by inverse application of the 2D elastostatics problem, International Journal of Computational Methods, 10 (2013), 1350042. doi: 10.1142/S0219876213500424. [3] Y. El Yazidi and A. Ellabib, Reconstruction of the depletion layer in MOSFET by genetic algorithms, Mathematical Modeling and Computing, 7 (2020), 96-103.  doi: 10.23939/mmc2020.01.096. [4] Y. El Yazidi and A. Ellabib, An iterative method for optimal control of bilateral free boundaries problem, Mathematical Methodes in Applied Science, (2021), 1–20. doi: 10.1002/mma.7527. [5] Y. El Yazidi and A. Ellabib, Augmented Lagrangian approach for a bilateral free boundary problem, Journal of Applied Mathematics and Computing, 2021. doi: 10.1007/s12190-020-01472-y. [6] A. Ellabib and A. Nachaoui, On the numerical solution of a free boundary identification problem, Inverse Problems in Engineering, 9 (2001), 235-260.  doi: 10.1080/174159701088027764. [7] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003. doi: 10.1137/1.9780898718690. [8] M. Hinze and R. Pinnau, Second-order approach to optimal semiconductor design, Journal of Optimization Theory and Applications, 133 (2007), 179-199.  doi: 10.1007/s10957-007-9203-3. [9] M. Hinze, B. Kaltenbacher and T. N. T. Quyen, Identifying conductivity in electrical impedance tomography with total variation regularization, Numerische Mathematik, 138 (2018), 723-765.  doi: 10.1007/s00211-017-0920-8. [10] C.-H. Huang and C.-C. Shih, A shape identification problem in estimating simultaneously two interfacial configurations in a multiple region domain, Applied Thermal Engineering, 26 (2006), 77-88.  doi: 10.1016/j.applthermaleng.2005.04.019. [11] V. G. Korneev and U. Langer, Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic Problems, World Scientific, 2013. doi: 10.1142/9035. [12] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer Vienna, 2013. doi: 10.1007/978-3-7091-3678-2. [13] M. H. Mozaffari, M. Khodadad and M. Dashti Ardakani, Simultaneous identification of multi-irregular interfacial boundary configurations in non-homogeneous body using surface displacement measurements, Journal of Mechanical Engineering Science, 231 (2016), 1-12.  doi: 10.1177/0954406216636166. [14] A. Novruzi and M. Pierre, Structure of shape derivatives, Journal of Evolution Equations, 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y. [15] O. Pironneau, Optimal Shape Design For Elliptic Systems, Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-87722-3. [16] H. Prautzsch, W. Boehm and M. Paluszny, Bézier and B-Spline Techniques, Springer, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-662-04919-8. [17] K. V. Price, Differential evolution: a fast and simple numerical optimizer, in Proceedings of North American Fuzzy Information Processing, (1996), 524–527. doi: 10.1109/NAFIPS.1996.534790. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] S. Salhi, Heuristic Search The Emerging Science of Problem Solving, Palgrave Macmillan, 2017. doi: 10.1007/978-3-319-49355-8. [20] R. Storn and K. Price, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328. [21] S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley & Sons Singapore Pte. Limited, 2012. [22] J. P. Zolesio and J. Sokolowski, Introduction to Shape Optimization Shape Sensitivity Analysis, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9.
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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