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Bayesian inversion for the identification of the doping profile in unipolar semiconductor devices

Leila Taghizadeh, Ansgar Jüngel

TL;DR

The paper develops a rigorous Bayesian framework in infinite dimensions to identify the doping profile in unipolar semiconductor devices from voltage–current data, using a stationary linearized drift–diffusion forward model. It combines a Gaussian Matérn–Whittle prior with a preconditioned Crank–Nicolson MCMC sampler to estimate the posterior distribution of the exponential potential V_e and, from it, the doping profile via C(x)=γ(x)−λ^2Δ(ln γ)(x) with γ=e^{V_e}. The study proves posterior well-posedness under standard assumptions and demonstrates, in a 2D pn-diode, that the method yields accurate posterior means with quantified uncertainties, while illuminating the trade-offs between data richness, prior regularization, KL truncation, and mesh resolution. The approach offers a principled, uncertainty-aware alternative to deterministic methods for semiconductor doping profiling and can be extended to bipolar devices and memristive systems, with implications for noninvasive diagnostics and device design.

Abstract

A rigorous Bayesian formulation of the inverse doping profile problem in infinite dimensions for a stationary linearized unipolar drift-diffusion model for semiconductor devices is given. The goal is to estimate the posterior probability distribution of the doping profile and to compute its posterior mean. This allows for the reconstruction of the doping profile from voltage-current measurements. The well-posedness of the Bayesian inverse problem is shown by proving boundedness and continuity properties of the semiconductor model with respect to the unknown parameter. A preconditioned Crank-Nicolson Markov chain Monte-Carlo method for the Bayesian estimation of the doping profile, using a physics-informed prior model, is proposed. The numerical results for a two-dimensional diode illustrate the efficiency of the proposed approach.

Bayesian inversion for the identification of the doping profile in unipolar semiconductor devices

TL;DR

The paper develops a rigorous Bayesian framework in infinite dimensions to identify the doping profile in unipolar semiconductor devices from voltage–current data, using a stationary linearized drift–diffusion forward model. It combines a Gaussian Matérn–Whittle prior with a preconditioned Crank–Nicolson MCMC sampler to estimate the posterior distribution of the exponential potential V_e and, from it, the doping profile via C(x)=γ(x)−λ^2Δ(ln γ)(x) with γ=e^{V_e}. The study proves posterior well-posedness under standard assumptions and demonstrates, in a 2D pn-diode, that the method yields accurate posterior means with quantified uncertainties, while illuminating the trade-offs between data richness, prior regularization, KL truncation, and mesh resolution. The approach offers a principled, uncertainty-aware alternative to deterministic methods for semiconductor doping profiling and can be extended to bipolar devices and memristive systems, with implications for noninvasive diagnostics and device design.

Abstract

A rigorous Bayesian formulation of the inverse doping profile problem in infinite dimensions for a stationary linearized unipolar drift-diffusion model for semiconductor devices is given. The goal is to estimate the posterior probability distribution of the doping profile and to compute its posterior mean. This allows for the reconstruction of the doping profile from voltage-current measurements. The well-posedness of the Bayesian inverse problem is shown by proving boundedness and continuity properties of the semiconductor model with respect to the unknown parameter. A preconditioned Crank-Nicolson Markov chain Monte-Carlo method for the Bayesian estimation of the doping profile, using a physics-informed prior model, is proposed. The numerical results for a two-dimensional diode illustrate the efficiency of the proposed approach.
Paper Structure (12 sections, 3 theorems, 37 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 12 sections, 3 theorems, 37 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Forward model
  3. Drift-diffusion equations
  4. Inverse problem
  5. Semiconductor Bayesian inverse problem
  6. Likelihood, prior, and posterior distribution
  7. Well-posedness of the Bayesian inverse problem
  8. Preconditioned Crank--Nicolson Markov chain Monte--Carlo method
  9. Markov chain Monte--Carlo methods
  10. Infinite-dimensional setting: prior sampling
  11. Numerical results
  12. Conclusions

Key Result

Theorem 3

Let $X$ and $Y$ be Banach spaces and let $\mu_0$ be a Gaussian measure with $\mu_0(X)=1$. If $\Phi$ satisfies Assumption assump, then there exists a unique posterior measure $\mu^y$ for 3.bayes and $\mu^y$ is Lipschitz continuous with respect to $y$ in the Hellinger distance.

Figures (8)

  • Figure 1: Cross-section of a $pn$-diode with two Ohmic contacts $\Gamma_N$ and $\Gamma_P$. The junction $\Gamma$ separates the $n$- and $p$-regions.
  • Figure 2: Schematic diagrams of the forward (top) and inverse (bottom) problems for semiconductors with $I=\widehat{J}_n\cdot\nu|_{\Gamma_N}$.
  • Figure 3: True doping profile (left), solution $V_e$ to the Poisson equation (middle), and solution $\widehat{u}$ to the continuity equation (right).
  • Figure 4: True parameter field $V_e$ (synthetic truth) with the locations of the measurement points as black dots (left), a sample from the prior field (middle), and the reconstructed $V_e$ (right).
  • Figure 5: The trace plots (left) and histograms (right) show the posterior Markov chains for $V_e$ for the first, 200th, and 400th parameters of the discretized parameter domain (from top to bottom).
  • ...and 3 more figures

Theorems & Definitions (5)

  • Definition 1: Well-posedness in the Hellinger distance
  • Theorem 3: Well-posedness and stability
  • Theorem 4: Reduced assumptions for well-posedness
  • Theorem 5
  • proof