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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

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