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Parameter-interval estimation for cooperative reactive sputtering processes

Fabian Schneider, Christian Wölfel

Abstract

Reactive sputtering is a plasma-based technique to deposit a thin film on a substrate. This contribution presents a novel parameter-interval estimation method for a well-established model that describes the uncertain and nonlinear reactive sputtering process behaviour. Building on a proposed monotonicity-based model classification, the method guarantees that all parameterizations within the parameter interval yield output trajectories and static characteristics consistent with the enclosure induced by the parameter interval. Correctness and practical applicability of the new method are demonstrated by an experimental validation, which also reveals inherent structural limitations of the well-established process model for state-estimation tasks.

Parameter-interval estimation for cooperative reactive sputtering processes

Abstract

Reactive sputtering is a plasma-based technique to deposit a thin film on a substrate. This contribution presents a novel parameter-interval estimation method for a well-established model that describes the uncertain and nonlinear reactive sputtering process behaviour. Building on a proposed monotonicity-based model classification, the method guarantees that all parameterizations within the parameter interval yield output trajectories and static characteristics consistent with the enclosure induced by the parameter interval. Correctness and practical applicability of the new method are demonstrated by an experimental validation, which also reveals inherent structural limitations of the well-established process model for state-estimation tasks.

Paper Structure

This paper contains 16 sections, 8 theorems, 27 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Uncertain nonlinear reactive sputter processes
  3. Literature survey
  4. Structure of the contribution
  5. State-space process model
  6. Problem formulation
  7. Notation and preliminaries on monotonicity
  8. Objectives
  9. Monotonicity analysis
  10. Section overview
  11. Monotonicity under fixed parameterization
  12. Monotonicity with respect to parameter variations
  13. Main theorem
  14. Parameter-interval estimation method
  15. Experimental validation
  16. ...and 1 more sections

Key Result

Lemma 1

Consider a continuously differentiable function $f \colon \mathbb{D}_{\mathrm{x}} \subseteq \mathbb{R}^n \to \mathbb{R}$. If $\mathop{\mathrm{sign}}\nolimits \left( \frac{\partial f}{\partial \bm x}(\bm x) \right) = \bm r \quad \forall\, \bm x \in \mathbb{D}_{\mathrm{x}},$ then $f$ is monotone

Figures (7)

  • Figure 1: Material flows in reactive sputtering described by the well-established process model $\varSigma(\bm p)$ (cf. \ref{['eq:processModel:berg2014ss:system']}).
  • Figure 2: Ordered quantities of a (parameter-) monotone system lead to ordered output trajectories \ref{['eq:performanceSpecification:enclose:trajectory']}.
  • Figure 3: Structure graph of $\varSigma(\bm p)$ indicating monotonically increasing (+) and monotonically decreasing (-) couplings between $u(t)$ and $\bm x(t)$ (Lemmas \ref{['lem:processModell:parameterMonotone', 'lem:processModell:cooperative']}) and between $\bm x(t)$ and $\bm y(t)$ (Lemmas \ref{['lem:processModell:monotone:p:to:x:y', 'lem:processModell:monotoneOutputFunction:xy']}).
  • Figure 4: $\widetilde{\bm r}_{\bm \pi}$, $\bm r_\mathrm{y}$ represent monotonically increasing (+) and monotonically decreasing (-) couplings in the structure graph of the static characteristic \ref{['eq:performanceSpecification:enclose:static']}.
  • Figure 5: $\bm r_{\bm \pi}$ and $\bm r_\mathrm{y}$ represent monotonically increasing (+) and monotonically decreasing (-) couplings in the structure graph of the static characteristic \ref{['eq:performanceSpecification:enclose:static']} for an increase in $\bm p$ in direction $\bm r_\mathrm{p}$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Lemma 1: Monotone function
  • Lemma 2: Monotone output function
  • Lemma 3: Cooperative class of systems $\varSigma(\bm p)$
  • Lemma 4: (Non-)Monotone static characteristic
  • Lemma 5: Parameter-monotone output function
  • Lemma 6: Parameter-monotone class of systems $\varSigma(\bm p)$
  • Lemma 7: Parameter-monotone static characteristics
  • Theorem 1: Monotonicity of the class of systems $\varSigma(\bm p)$
  • Remark 1