On the inverse doping profile problem

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August 2012, 6(3): 465-486. doi: 10.3934/ipi.2012.6.465

On the inverse doping profile problem

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

Abstract
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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.
[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.
[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.
[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.
[5]	V. G. Cherednichenko, "Inverse Logarithmic Potential Problem,", VSP, (1996).
[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.
[7]	V. Isakov, "Inverse Source Problems,", AMS, (1990).
[8]	V. Isakov, "Inverse Problems for PDE,", Springer-Verlag, (2006).
[9]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems Imaging, 1 (2007), 95.
[10]	V. Isakov, On identification of the doping profile in semiconductors,, Contemp. Math. AMS, 494 (2009), 123. doi: 10.1090/conm/494/09647.
[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.
[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.
[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.
[14]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990).
[15]	N. I. Muskhelishvili, "Singular Integral Equations,", Nordhoff, (1953).
[16]	D. M. Olsson and L. S. Nelson, The Nelder-Mead simplex procedure for function minimization,, Technometrics, 7 (1975), 45.
[17]	E. P. Saff and V. Totik, "Logarithmic Potentials with External Fields,", Springer-Verlag, (1997).
[18]	M. T. Wolfram, Inverse dopant profiling problems from transient measurement,, J. Comp. Electronics, 22 (2007), 409. doi: 10.1007/s10825-007-0149-3.

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.
[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.
[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.
[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.
[5]	V. G. Cherednichenko, "Inverse Logarithmic Potential Problem,", VSP, (1996).
[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.
[7]	V. Isakov, "Inverse Source Problems,", AMS, (1990).
[8]	V. Isakov, "Inverse Problems for PDE,", Springer-Verlag, (2006).
[9]	V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Problems Imaging, 1 (2007), 95.
[10]	V. Isakov, On identification of the doping profile in semiconductors,, Contemp. Math. AMS, 494 (2009), 123. doi: 10.1090/conm/494/09647.
[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.
[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.
[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.
[14]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990).
[15]	N. I. Muskhelishvili, "Singular Integral Equations,", Nordhoff, (1953).
[16]	D. M. Olsson and L. S. Nelson, The Nelder-Mead simplex procedure for function minimization,, Technometrics, 7 (1975), 45.
[17]	E. P. Saff and V. Totik, "Logarithmic Potentials with External Fields,", Springer-Verlag, (1997).
[18]	M. T. Wolfram, Inverse dopant profiling problems from transient measurement,, J. Comp. Electronics, 22 (2007), 409. doi: 10.1007/s10825-007-0149-3.

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