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Reconstruction of the Doping Profile in Vlasov-Poisson

Ru-Yu Lai, Qin Li, Weiran Sun

Abstract

We study the inverse problem of recovering the doping profile in the stationary Vlasov-Poisson equation, given the knowledge of the incoming and outgoing measurements at the boundary of the domain. This problem arises from identifying impurities in the semiconductor manufacturing. Our result states that, under suitable assumptions, the doping profile can be uniquely determined through an asymptotic formula of the electric field that it generates.

Reconstruction of the Doping Profile in Vlasov-Poisson

Abstract

We study the inverse problem of recovering the doping profile in the stationary Vlasov-Poisson equation, given the knowledge of the incoming and outgoing measurements at the boundary of the domain. This problem arises from identifying impurities in the semiconductor manufacturing. Our result states that, under suitable assumptions, the doping profile can be uniquely determined through an asymptotic formula of the electric field that it generates.
Paper Structure (8 sections, 9 theorems, 130 equations)

This paper contains 8 sections, 9 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Mathematical problem setup and main result
  3. The forward problem
  4. Notation
  5. Properties of the characteristics
  6. Proof of the well-posedness
  7. Reconstruction of the doping profile
  8. Proof of Theorem \ref{['thm:uniqueness']} (Uniqueness result)

Key Result

Theorem 1.1

Let $\Omega \subseteq {\mathbb R}^d$ with $d\geq 2$ be an open, bounded, and strictly convex domain with a $C^2$ boundary. Let $\Lambda_{N_1}$ and $\Lambda_{N_2}$ be the Albedo operators associated with $N_1, N_2\in W^{1,\infty}(\Omega)$ in eq:main respectively. Suppose that for any $(x_0, p_0) \in Then $N_1 = N_2 \hbox{in}\Omega$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • ...and 9 more