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Statistical Topological Gradient and Shape Optimization for Robust Metal--Semiconductor Contact Reconstruction

Lekbir Afraites, Aissam Hadri, Mourad Hrizi, Julius Fergy Tiongson Rabago

Abstract

We develop a statistically robust framework for reconstructing metal--semiconductor contact regions using topological gradients. The inverse problem is formulated as the identification of an unknown contact region from boundary measurements governed by an elliptic model with piecewise coefficients. Deterministic stability of the topological gradient with respect to measurement noise is established, and the analysis is extended to a statistical setting with multiple independent observations. A central limit theorem in a separable Hilbert space is proved for the empirical topological gradient, yielding optimal $n^{-1/2}$ convergence and enabling the construction of confidence intervals and hypothesis tests for contact detection. To further refine the reconstruction, a shape optimization procedure is employed, where the free parameter $β$ in the CCBM formulation plays a crucial role in controlling interface sensitivity. While $β$ affects both topological and shape reconstructions, its influence is particularly pronounced in the shape optimization stage, allowing more accurate estimation of the size and geometry of the contact subregion. The proposed approach provides a rigorous criterion for distinguishing true structural features from noise-induced artifacts, and numerical experiments demonstrate the robustness, precision, and enhanced performance of the combined statistical, topological, and $β$-informed shape-based reconstruction.

Statistical Topological Gradient and Shape Optimization for Robust Metal--Semiconductor Contact Reconstruction

Abstract

We develop a statistically robust framework for reconstructing metal--semiconductor contact regions using topological gradients. The inverse problem is formulated as the identification of an unknown contact region from boundary measurements governed by an elliptic model with piecewise coefficients. Deterministic stability of the topological gradient with respect to measurement noise is established, and the analysis is extended to a statistical setting with multiple independent observations. A central limit theorem in a separable Hilbert space is proved for the empirical topological gradient, yielding optimal convergence and enabling the construction of confidence intervals and hypothesis tests for contact detection. To further refine the reconstruction, a shape optimization procedure is employed, where the free parameter in the CCBM formulation plays a crucial role in controlling interface sensitivity. While affects both topological and shape reconstructions, its influence is particularly pronounced in the shape optimization stage, allowing more accurate estimation of the size and geometry of the contact subregion. The proposed approach provides a rigorous criterion for distinguishing true structural features from noise-induced artifacts, and numerical experiments demonstrate the robustness, precision, and enhanced performance of the combined statistical, topological, and -informed shape-based reconstruction.
Paper Structure (24 sections, 13 theorems, 102 equations, 16 figures, 1 algorithm)

This paper contains 24 sections, 13 theorems, 102 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The contact resistivity problem
  3. Mathematical formulation
  4. The geometric inverse problem
  5. Notations
  6. The optimization framework
  7. Topological Sensitivity Analysis
  8. Estimate of ${J}_{1}(\varepsilon)$
  9. Estimate of ${J}_{2}(\varepsilon)$
  10. Stability of the topological gradient under noisy measurements
  11. Noisy boundary measurements and perturbed systems
  12. Lipschitz stability of the topological gradient
  13. Statistical stability under random noise
  14. Shape Sensitivity Analysis
  15. Reconstruction via topological gradient
  16. ...and 9 more sections

Key Result

Lemma 1.4

Let $\varOmega \subset \mathbb{R}^{d}$, $d \ge 2$, be a bounded Lipschitz domain, and let $\omega \Subset \varOmega$ be measurable with $\abs{\omega}>0$. Then there exists a constant $C(\varOmega) > 0$ such that for all $\varphi \in H^{1}{(\varOmega)}$,

Figures (16)

  • Figure 1: (a) A VLSI circuit; (b) simplified metal/semiconductor contact; (c) a 2D slab contact model with interface at $z=0$ and diffusion-layer thickness $z_j$. Adapted from loh1987modeling.
  • Figure 2: (a)--(i): nine representative simple cases with $g = 1$.
  • Figure 3: Effect of resistivity on topological gradient distribution
  • Figure 4: Detection of two medium-sized subregions with $g = 1$. The blue curves denote the measured boundary potentials; the positive $x$- and $y$-directions are indicated by right-angle arrows.
  • Figure 5: Comparison of the detection of two extremely small-sized subregions using $g = 1$ (top) and $g = |x|$ (bottom).
  • ...and 11 more figures

Theorems & Definitions (29)

  • Remark 1.2
  • Lemma 1.4
  • Proposition 1.5
  • proof
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 19 more