American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Green function of the interior transmission problem and its applications
  • IPI Home
  • This Issue
  • Next Article
    Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint
August  2012, 6(3): 465-486. doi: 10.3934/ipi.2012.6.465

On the inverse doping profile problem

Victor Isakov 1, and  Joseph Myers 2,

1. 

Wichita State University, 1845 Fairmount, Wichita, KS 67260-0033, United States
2. 

Friends University, 2100 W University Ave, Wichita, KS 67213, United States

Received  April 2011 Revised  February 2012 Published  September 2012

We obtain new analytic results for the problem of the recovery of a doped region $D$ in semiconductor devices from the total flux of electrons/holes through a part of the boundary for various applied potentials on some complementary part of the boundary. We consider the stationary two-dimensional case and we use the index of the gradient of solutions of the linear elliptic equation modeling a unipolar device. Under mild assumptions we prove local uniqueness of smooth $D$ and global uniqueness of polygonal $D$ satisfying some geometrical (star-shapednedness or convexity in some direction) assumptions. We design a nonlinear minimization algorithm for numerical solution and we demonstrate its effectiveness on some basic examples. An essential ingredient of this algorithm is a numerical solution of the direct problem by using single layer potentials.
Keywords: Inverse problems, inverse scattering problems..
Mathematics Subject Classification: Primary: 35R30; Secondary: 78A4.
Citation: Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465
References:
[1]

G. Alessandrini, V. Isakov and J. Powell, Local uniqueness in the inverse conductivity problem with one measurement,, Trans. of AMS, 347 (1995), 3031. doi: 10.1090/S0002-9947-1995-1303113-8. Google Scholar

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions,, SIAM J. Math. Anal., 25 (1994), 1259. doi: 10.1137/S0036141093249080. Google Scholar

[3]

M. Burger, H. W. Engl, P. A. Markowich and P. Pietra, Identification of doping profiles in semiconductor devices,, Inverse Problems, 17 (2001), 1765. doi: 10.1088/0266-5611/17/6/315. Google Scholar

[4]

M. Burger, H. W. Engl, A. Leitao and P. A. Markowich, On inverse problems for semiconductor equations,, Milan J. of Mathematics, 72 (2004), 273. doi: 10.1007/s00032-004-0025-6. Google Scholar

[5]

V. G. Cherednichenko, "Inverse Logarithmic Potential Problem,", VSP, (1996). Google Scholar

[6]

W. Fang, K. Ito and D. A. Redfern, Parameter identification for semiconductor diodes by LBIC imaging,, SIAM J. Appl. Math., 62 (2002), 2149. doi: 10.1137/S003613990139249X. Google Scholar

[7]

V. Isakov, "Inverse Source Problems,", AMS, (1990). Google Scholar

[8]

V. Isakov, "Inverse Problems for PDE,", Springer-Verlag, (2006). Google Scholar

[9]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems Imaging, 1 (2007), 95. Google Scholar

[10]

V. Isakov, On identification of the doping profile in semiconductors,, Contemp. Math. AMS, 494 (2009), 123. doi: 10.1090/conm/494/09647. Google Scholar

[11]

H. Kang, J. K. Seo and D. Sheen, Numerical identification of discontinuous conductivity coefficients,, Inverse Problems, 13 (1997), 113. doi: 10.1088/0266-5611/13/1/009. Google Scholar

[12]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions,, SIAM J. Optim., 9 (1998), 112. Google Scholar

[13]

A. Leitao, P. Markowich and J. P. Zubelli, On inverse doping profile problems for stationary voltage-current map,, Inverse Problems, 22 (2006), 1071. doi: 10.1088/0266-5611/22/3/021. Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). Google Scholar

[15]

N. I. Muskhelishvili, "Singular Integral Equations,", Nordhoff, (1953). Google Scholar

[16]

D. M. Olsson and L. S. Nelson, The Nelder-Mead simplex procedure for function minimization,, Technometrics, 7 (1975), 45. Google Scholar

[17]

E. P. Saff and V. Totik, "Logarithmic Potentials with External Fields,", Springer-Verlag, (1997). Google Scholar

[18]

M. T. Wolfram, Inverse dopant profiling problems from transient measurement,, J. Comp. Electronics, 22 (2007), 409. doi: 10.1007/s10825-007-0149-3. Google Scholar

show all references

References:
[1]

G. Alessandrini, V. Isakov and J. Powell, Local uniqueness in the inverse conductivity problem with one measurement,, Trans. of AMS, 347 (1995), 3031. doi: 10.1090/S0002-9947-1995-1303113-8. Google Scholar

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions,, SIAM J. Math. Anal., 25 (1994), 1259. doi: 10.1137/S0036141093249080. Google Scholar

[3]

M. Burger, H. W. Engl, P. A. Markowich and P. Pietra, Identification of doping profiles in semiconductor devices,, Inverse Problems, 17 (2001), 1765. doi: 10.1088/0266-5611/17/6/315. Google Scholar

[4]

M. Burger, H. W. Engl, A. Leitao and P. A. Markowich, On inverse problems for semiconductor equations,, Milan J. of Mathematics, 72 (2004), 273. doi: 10.1007/s00032-004-0025-6. Google Scholar

[5]

V. G. Cherednichenko, "Inverse Logarithmic Potential Problem,", VSP, (1996). Google Scholar

[6]

W. Fang, K. Ito and D. A. Redfern, Parameter identification for semiconductor diodes by LBIC imaging,, SIAM J. Appl. Math., 62 (2002), 2149. doi: 10.1137/S003613990139249X. Google Scholar

[7]

V. Isakov, "Inverse Source Problems,", AMS, (1990). Google Scholar

[8]

V. Isakov, "Inverse Problems for PDE,", Springer-Verlag, (2006). Google Scholar

[9]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems Imaging, 1 (2007), 95. Google Scholar

[10]

V. Isakov, On identification of the doping profile in semiconductors,, Contemp. Math. AMS, 494 (2009), 123. doi: 10.1090/conm/494/09647. Google Scholar

[11]

H. Kang, J. K. Seo and D. Sheen, Numerical identification of discontinuous conductivity coefficients,, Inverse Problems, 13 (1997), 113. doi: 10.1088/0266-5611/13/1/009. Google Scholar

[12]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions,, SIAM J. Optim., 9 (1998), 112. Google Scholar

[13]

A. Leitao, P. Markowich and J. P. Zubelli, On inverse doping profile problems for stationary voltage-current map,, Inverse Problems, 22 (2006), 1071. doi: 10.1088/0266-5611/22/3/021. Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). Google Scholar

[15]

N. I. Muskhelishvili, "Singular Integral Equations,", Nordhoff, (1953). Google Scholar

[16]

D. M. Olsson and L. S. Nelson, The Nelder-Mead simplex procedure for function minimization,, Technometrics, 7 (1975), 45. Google Scholar

[17]

E. P. Saff and V. Totik, "Logarithmic Potentials with External Fields,", Springer-Verlag, (1997). Google Scholar

[18]

M. T. Wolfram, Inverse dopant profiling problems from transient measurement,, J. Comp. Electronics, 22 (2007), 409. doi: 10.1007/s10825-007-0149-3. Google Scholar

[1]

Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139
[2]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[3]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[4]

Gabriel Katz. Causal holography in application to the inverse scattering problems. Inverse Problems & Imaging, 2019, 13 (3) : 597-633. doi: 10.3934/ipi.2019028
[5]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267
[6]

Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215
[7]

Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749
[8]

Laurent Bourgeois, Houssem Haddar. Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems & Imaging, 2010, 4 (1) : 19-38. doi: 10.3934/ipi.2010.4.19
[9]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[10]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
[11]

Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems & Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[12]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225
[13]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[14]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[15]

Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239
[16]

Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[17]

Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems & Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[18]

Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449
[19]

Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[20]

François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems & Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences