Global-in-time Spectral Duhamel-Picard Method for Post-Exposure Bake Simulation and Physics-Informed Neural Operators for EUV Lithography

Doyun Kim; Mircea Dusa; Werner Gilijns; Philippe Leray

physics.comp-phOct 13, 2026

Global-in-time Spectral Duhamel-Picard Method for Post-Exposure Bake Simulation and Physics-Informed Neural Operators for EUV Lithography

Authors: Doyun Kim, Mircea Dusa, Werner Gilijns, Philippe Leray

Organizations: IMEC, Kapeldreef 75, 3001 Leuven, Belgium

Abstract

We present a global-in-time spectral Duhamel-Picard method with polynomial expansions for post-exposure bake (PEB) simulation. The coupled acid-quencher reaction-diffusion equations are reformulated as nonlinear Volterra integral equations using the Duhamel principle. Full trajectories are represented by finite polynomial expansions, with temporal coefficient fields determined over the bake interval by Anderson-accelerated Picard iteration, thereby avoiding sequential time marching. In 3-D PEB simulations, we employ Chebyshev and Legendre temporal bases while evaluating the Duhamel integrals using the corresponding Gaussian quadrature rules. We demonstrate that the proposed method simulates a large domain of $1200\times1200\times100$ voxels on the order of one minute. The representation is further incorporated into a physics-informed neural operator (PINO) with a mixed-transform variant of a factorized Fourier neural operator, which is trained without precomputed solution trajectories. Together, the integral formulation and its PINO extension enable accelerated, time-marching-free prediction of concentration fields at arbitrary times within the bake interval.

Figures & tables

	Gauss–Chebyshev			Gauss–Legendre
$i$	$\xi_{i}$	$s_{i}(t_{PEB})\;[\mathrm{s}]$	$w_{i}(t_{PEB})\;[\mathrm{s}]$	$\xi_{i}$	$s_{i}(t_{PEB})\;[\mathrm{s}]$	$w_{i}(t_{PEB})\;[\mathrm{s}]$
1	$+0.980785$	$99.039$	$\phantom{0}3.831$	$+0.960290$	$98.014$	$\phantom{0}5.061$
2	$+0.831470$	$91.574$	$10.909$	$+0.796666$	$89.833$	$11.119$
3	$+0.555570$	$77.779$	$16.326$	$+0.525532$	$76.277$	$15.685$
4	$+0.195090$	$59.755$	$19.258$	$+0.183435$	$59.172$	$18.134$
5	$-0.195090$	$40.245$	$19.258$	$-0.183435$	$40.828$	$18.134$

Table 1: Comparison of

N=8

Gauss–Chebyshev and Gauss–Legendre quadrature nodes, physical times, and scaled weights over the full bake interval

[0,t_{PEB}]

for

t_{PEB}=100\,\mathrm{s}

Figure 1: Random binary mask containing horizontally oriented features, used to construct the initial acid distribution in the $800\times 800\times 200$ PEB benchmark domain.

Parameter	Description	Value
$A_{0}$	Inside / outside mask	$0.5$ / $0.0$
$Q_{0}$	Uniform initial quencher concentration	$0.05$
$P_{0}$	Uniform initial protected-group concentration	$1.0$
$D_{0}$	Uniform initial deprotected-group concentration	$0.0$
$k_{1},\,k_{4}$	Reaction rate constants	$0.01\text{--}1$
$k_{2},\,k_{3},\,k_{5}$	Reaction rate constants	$0.0001\text{--}0.01$

Table 2: Fixed initial conditions and parameter ranges used in the temporal fitting benchmark.

Figure 2: Reference time-marching trajectories for acid ( $A$ ) and protected group ( $P$ ) over $t=0$ to $100$ s, with representative acid diffusion coefficients $D_{\mathrm{acid}}=1$ and $10~\mathrm{nm}^{2}/\mathrm{s}$ .

Figure 3: Representative temporal fitting of acid trajectories at sampled voxel locations selected to cover diverse local profile shapes.

Figure 4: Global fitting statistics over all voxel locations for the $A$ , $Q$ , $P$ , and $D$ fields: (a) coefficient of determination $R^{2}$ , (b) trajectory mismatch relative to the time-marching reference, and (c) normalized coefficient contributions for polynomial expansions with different orders $K$ .

Figure 5: Global-in-time spectral Duhamel–Picard solutions for the acid ( $A$ ), quencher ( $Q$ ), protected-group ( $P$ ), and deprotected-group ( $D$ ) fields at $t_{PEB}=50~\mathrm{s}$ for representative diffusion settings with $D_{\mathrm{acid}}=D_{\mathrm{q}}\in\{1,10\}~\mathrm{nm}^{2}/\mathrm{s}$ . The corresponding absolute differences relative to the FD4–RK4 time-marching reference obtained using $\Delta t=0.01~\mathrm{s}$ are shown alongside the computed fields.

Figure 6: Convergence behavior of the global-in-time spectral Duhamel–Picard solver for 100 parameter combinations generated by Latin hypercube sampling over the ranges listed in Table 2 . (a) Convergence histories of the sampled cases. (b) Relationship between $k_{4}$ and the required number of Picard iterations.

Method	Device	Runtime (s)	$T/T_{\mathrm{Picard}}$
FD4–RK4	CPU	$221{,}600$	$2.8\times 10^{3}$
FD4–RK4	GPU	$2{,}700$	$33.8$
Global-in-time Spectral Duhamel-Picard	GPU	$80$	$1$

Table 3: Wall-clock runtime comparison between the global-in-time method and conventional FD4–RK4 method run on CPU and GPU.

Figure 7: Performance of the Physics-Informed Neural Operator (PINO) trained with the global-in-time PEB formulation evaluated for arbitrary validation masks at random bake times, which were not explicitly included in the training dataset. The figure shows the predicted fields and their differences relative to the reference PEB solution.

Figure 8: Mean absolute errors for (a) acid and (b) protected groups as a function of bake time.

Figure 9: Comparison of post-development resist profiles predicted by the global-in-time Duhamel–Picard method and the PINO surrogate for randomly sampled mask layouts and PEB times. (a) Three-dimensional resist profiles after development at $t_{\mathrm{PEB}}=35$ , $45$ , and $86~\mathrm{s}$ . (b) Corresponding lateral contour comparisons at $z=75$ , $50$ , and $25~\mathrm{nm}$ , with the Duhamel–Picard results shown in blue and the PINO predictions in red.

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Physics-Informed Neural Operator for Warm-Starting Background-Decomposed and Preconditioned PSFD: Enabling Scalable 3-D EUV Mask Simulation

Doyun Kim, Werner Gillijns

IMEC, Kapeldreef 75, 3001 Leuven, Belgium

Fourier Neural Operator Semiconductor Manufacturing Physics-Informed Neural Operators Physics-Based Simulation

Open paper AskPDF arXiv

We present a physics-informed neural operator (PINO) trained with pseudo-spectral frequency-domain (PSFD) equations for electromagnetic (EM) scattering problems in EUV lithography. The Fourier neural operator is factorized into a two-dimensional lateral (

xy

) branch and a one-dimensional axial (

z

) branch and is trained self-consistently with background decomposition.Thus, the full-vector coupling between the mask and the multilayer response is retained without invoking a finite-order Born approximation. In this way, the computational domain size is significantly reduced, thereby lowering the computational cost. The PINO is trained on approximately 16,000 mask designs from the LithoBench library sampled randomly at each training iteration without using precomputed EM field solutions. The PINO surrogate model yields predictions with a mean absolute error of about

7 \times 10^{-3}

for the scattered intensity of held-out mask patterns relative to the reference PSFD solution. Combined with spectral damping, the PINO warm-start initialization accelerates the background-decomposed PSFD solver on finer discretizations.

Abstract

Reaction-Diffusion Systems

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