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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

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.
Citation: Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems and 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), 3031-3041. 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), 1259-1269. 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), 1765-1795. 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), 273-314. doi: 10.1007/s00032-004-0025-6. [5] V. G. Cherednichenko, "Inverse Logarithmic Potential Problem," VSP, Utrecht, 1996. [6] W. Fang, K. Ito and D. A. Redfern, Parameter identification for semiconductor diodes by LBIC imaging, SIAM J. Appl. Math., 62 (2002), 2149-2174. doi: 10.1137/S003613990139249X. [7] V. Isakov, "Inverse Source Problems," AMS, Providence, RI, 1990. [8] V. Isakov, "Inverse Problems for PDE," Springer-Verlag, New York, 2006. [9] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Problems Imaging, 1 (2007), 95-107. [10] V. Isakov, On identification of the doping profile in semiconductors, Contemp. Math. AMS, 494 (2009), 123-137. 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), 113-123. 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), 112-147. [13] A. Leitao, P. Markowich and J. P. Zubelli, On inverse doping profile problems for stationary voltage-current map, Inverse Problems, 22 (2006), 1071-1088. doi: 10.1088/0266-5611/22/3/021. [14] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990. [15] N. I. Muskhelishvili, "Singular Integral Equations," Nordhoff, Holland, 1953. [16] D. M. Olsson and L. S. Nelson, The Nelder-Mead simplex procedure for function minimization, Technometrics, 7 (1975), 45-51. [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), 409-420. 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), 3031-3041. 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), 1259-1269. 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), 1765-1795. 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), 273-314. doi: 10.1007/s00032-004-0025-6. [5] V. G. Cherednichenko, "Inverse Logarithmic Potential Problem," VSP, Utrecht, 1996. [6] W. Fang, K. Ito and D. A. Redfern, Parameter identification for semiconductor diodes by LBIC imaging, SIAM J. Appl. Math., 62 (2002), 2149-2174. doi: 10.1137/S003613990139249X. [7] V. Isakov, "Inverse Source Problems," AMS, Providence, RI, 1990. [8] V. Isakov, "Inverse Problems for PDE," Springer-Verlag, New York, 2006. [9] V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Problems Imaging, 1 (2007), 95-107. [10] V. Isakov, On identification of the doping profile in semiconductors, Contemp. Math. AMS, 494 (2009), 123-137. 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), 113-123. 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), 112-147. [13] A. Leitao, P. Markowich and J. P. Zubelli, On inverse doping profile problems for stationary voltage-current map, Inverse Problems, 22 (2006), 1071-1088. doi: 10.1088/0266-5611/22/3/021. [14] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990. [15] N. I. Muskhelishvili, "Singular Integral Equations," Nordhoff, Holland, 1953. [16] D. M. Olsson and L. S. Nelson, The Nelder-Mead simplex procedure for function minimization, Technometrics, 7 (1975), 45-51. [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), 409-420. doi: 10.1007/s10825-007-0149-3.
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