Abstract
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.
Highlights
In so-called Slotboom variables, after scaling, a typical semiconductor device is described by the elliptic quasilinear system λ2∆V = δ2(eV u − e−V v) − C, divJn = δ4Q(, V, u, v)(uv − 1), Jn = μnqnieV ∇u, divJp = −δ4Q(, V, u, v)(uv − 1), Jp = −μpqnie−V ∇v in Ω and the boundary conditions
In order to avoid “inverse crimes,” different grid sizes and discretizations were used, and the position of point sources and interpolation points varied from production of data to solution of the inverse problem, thereby solving the inverse problem in a different finite-dimensional computational space than that which was used for solving the direct problem
It would be useful to find singular value decomposition for the linearized inverse problem to have better understanding of ill-conditioning and of how many parameters one can expect to reconstruct in a stable way
Summary
Let u∗1, v1∗ solve the boundary value problem div(au∇u∗1) = C1Qu∗1 − C1Qv1∗, div(av∇v1∗) = −C2Qu∗1 + C2Qv1∗ on Ω, ∂ν u∗1 = ∂ν v1∗ = 0 on ΓN , u∗1 = v1∗ = 0 on Γ0, u∗1 = v1∗ = 1 on Γ1 It was shown in [9] (by using Green’s type identities) that the data of the Inverse Problem 2 uniquely determine g1∗ = C1−1au∂ν u∗1 + C2−1av∂ν v1∗ on Γ0. Uniqueness in this linear problem is controlled by indices of gradient of u which are topological (not geometrical) tools. Using continuity of the index with respect to deformation of curves and approximating Γ∗ by curves Γ(1) we complete the proof
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