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Semiconductor Manufacturing Process Metrology: Science, Mathematics, and Modeling

A comprehensive exploration of the physics, mathematics, and computational methods underlying nanoscale measurement in semiconductor fabrication.

1. The Fundamental Challenge

Modern semiconductor manufacturing produces structures with critical dimensions of just a few nanometers. At leading-edge nodes (3nm, 2nm), we are measuring features only 10–20 atoms wide.

Key Requirements

Sub-angstrom precision in measurement
Complex 3D architectures: FinFETs, Gate-All-Around (GAA) transistors, 3D NAND (200+ layers)
High throughput: seconds per measurement in production
Multi-parameter extraction: distinguish dozens of correlated parameters

Metrology Techniques Overview

Technique	Principle	Resolution	Throughput
Spectroscopic Ellipsometry (SE)	Polarization change	~0.1 Å	High
Optical CD (OCD/Scatterometry)	Diffraction analysis	~0.1 nm	High
CD-SEM	Electron imaging	~1 nm	Medium
CD-SAXS	X-ray scattering	~0.1 nm	Low
AFM	Probe scanning	~0.1 nm	Low
TEM	Electron transmission	Atomic	Very Low

2. Physics Foundation

2.1 Maxwell's Equations

At the heart of optical metrology lies the solution to Maxwell's equations:

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}

\nabla \cdot \mathbf{D} = \rho

\nabla \cdot \mathbf{B} = 0

Where:

$\mathbf{E}$ = Electric field vector
$\mathbf{H}$ = Magnetic field vector
$\mathbf{D}$ = Electric displacement field
$\mathbf{B}$ = Magnetic flux density
$\mathbf{J}$ = Current density
$\rho$ = Charge density

2.2 Constitutive Relations

For linear, isotropic media:

\mathbf{D} = \varepsilon_0 \varepsilon_r \mathbf{E} = \varepsilon_0 (1 + \chi_e) \mathbf{E}

\mathbf{B} = \mu_0 \mu_r \mathbf{H}

The complex dielectric function:

\tilde{\varepsilon}(\omega) = \varepsilon_1(\omega) + i\varepsilon_2(\omega) = \tilde{n}^2 = (n + ik)^2

Where:

$n$ = Refractive index
$k$ = Extinction coefficient

2.3 Fresnel Equations

At an interface between media with refractive indices $\tilde{n}_1$ and $\tilde{n}_2$:

s-polarization (TE):

r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t}

t_s = \frac{2 n_1 \cos\theta_i}{n_1 \cos\theta_i + n_2 \cos\theta_t}

p-polarization (TM):

r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t}

t_p = \frac{2 n_1 \cos\theta_i}{n_2 \cos\theta_i + n_1 \cos\theta_t}

With Snell's law:

n_1 \sin\theta_i = n_2 \sin\theta_t

3. Mathematics of Inverse Problems

3.1 Problem Formulation

Metrology is fundamentally an inverse problem:

Problem Type	Description	Well-Posed?
Forward	Structure parameters → Measured signal	Yes
Inverse	Measured signal → Structure parameters	Often No

We seek parameters $\mathbf{p}$ that minimize the difference between model $M(\mathbf{p})$ and data $\mathbf{D}$:

\min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2

Or with weighted least squares:

\chi^2 = \sum_{k=1}^{N} \frac{\left( M_k(\mathbf{p}) - D_k \right)^2}{\sigma_k^2}

3.2 Levenberg-Marquardt Algorithm

The workhorse optimization algorithm interpolates between gradient descent and Gauss-Newton:

\left( \mathbf{J}^T \mathbf{J} + \lambda \mathbf{I} \right) \delta\mathbf{p} = \mathbf{J}^T \left( \mathbf{D} - M(\mathbf{p}) \right)

Where:

$\mathbf{J}$ = Jacobian matrix (sensitivity matrix)
$\lambda$ = Damping parameter
$\delta\mathbf{p}$ = Parameter update step

The Jacobian elements:

J_{ij} = \frac{\partial M_i}{\partial p_j}

Algorithm behavior:

Large $\lambda$ → Gradient descent (robust, slow)
Small $\lambda$ → Gauss-Newton (fast near minimum)

3.3 Regularization Techniques

For ill-posed problems, regularization is essential:

Tikhonov Regularization (L2):

\min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 + \alpha \left\| \mathbf{p} - \mathbf{p}_0 \right\|^2

LASSO Regularization (L1):

\min_{\mathbf{p}} \left\| M(\mathbf{p}) - \mathbf{D} \right\|^2 + \alpha \left\| \mathbf{p} \right\|_1

Bayesian Inference:

P(\mathbf{p} | \mathbf{D}) = \frac{P(\mathbf{D} | \mathbf{p}) \cdot P(\mathbf{p})}{P(\mathbf{D})}

Where:

$P(\mathbf{p} | \mathbf{D})$ = Posterior probability
$P(\mathbf{D} | \mathbf{p})$ = Likelihood
$P(\mathbf{p})$ = Prior probability

4. Thin Film Optics

4.1 Ellipsometry Fundamentals

Ellipsometry measures the change in polarization state upon reflection:

\rho = \tan(\Psi) \cdot e^{i\Delta} = \frac{r_p}{r_s}

Where:

$\Psi$ = Amplitude ratio angle
$\Delta$ = Phase difference
$r_p, r_s$ = Complex reflection coefficients

4.2 Transfer Matrix Method

For multilayer stacks, the characteristic matrix for layer $j$:

\mathbf{M}_j = \begin{pmatrix} \cos\delta_j & \frac{i \sin\delta_j}{\eta_j} \\ i\eta_j \sin\delta_j & \cos\delta_j \end{pmatrix}

Where the phase thickness:

\delta_j = \frac{2\pi}{\lambda} \tilde{n}_j d_j \cos\theta_j

And the optical admittance:

\eta_j = \begin{cases} \tilde{n}_j \cos\theta_j & \text{(s-pol)} \\ \frac{\tilde{n}_j}{\cos\theta_j} & \text{(p-pol)} \end{cases}

Total system matrix:

\mathbf{M}_{total} = \mathbf{M}_1 \cdot \mathbf{M}_2 \cdot \ldots \cdot \mathbf{M}_N = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}

Reflection coefficient:

r = \frac{\eta_0 m_{11} + \eta_0 \eta_s m_{12} - m_{21} - \eta_s m_{22}}{\eta_0 m_{11} + \eta_0 \eta_s m_{12} + m_{21} + \eta_s m_{22}}

4.3 Dispersion Models

Lorentz Oscillator Model:

\varepsilon(\omega) = \varepsilon_\infty + \sum_j \frac{A_j}{\omega_j^2 - \omega^2 - i\gamma_j \omega}

Tauc-Lorentz Model (for amorphous semiconductors):

\varepsilon_2(E) = \begin{cases} \frac{A E_0 C (E - E_g)^2}{(E^2 - E_0^2)^2 + C^2 E^2} \cdot \frac{1}{E} & E > E_g \\ 0 & E \leq E_g \end{cases}

With $\varepsilon_1$ obtained via Kramers-Kronig relations:

\varepsilon_1(E) = \varepsilon_{1,\infty} + \frac{2}{\pi} \mathcal{P} \int_{E_g}^{\infty} \frac{\xi \varepsilon_2(\xi)}{\xi^2 - E^2} d\xi

5. Scatterometry and RCWA

5.1 Rigorous Coupled-Wave Analysis

For a grating with period $\Lambda$, electromagnetic fields are expanded in Fourier orders:

E(x,z) = \sum_{m=-M}^{M} E_m(z) \exp(i k_{xm} x)

Where the diffracted wave vectors:

k_{xm} = k_{x0} + \frac{2\pi m}{\Lambda} = k_0 \left( n_1 \sin\theta_i + \frac{m\lambda}{\Lambda} \right)

5.2 Eigenvalue Problem

In each layer, the field satisfies:

\frac{d^2 \mathbf{E}}{dz^2} = \mathbf{\Omega}^2 \mathbf{E}

Where $\mathbf{\Omega}^2$ is a matrix determined by the Fourier components of the permittivity:

\varepsilon(x) = \sum_n \varepsilon_n \exp\left( i \frac{2\pi n}{\Lambda} x \right)

The eigenvalue decomposition:

\mathbf{\Omega}^2 = \mathbf{W} \mathbf{\Lambda} \mathbf{W}^{-1}

Provides propagation constants (eigenvalues $\lambda_m$) and field profiles (eigenvectors in $\mathbf{W}$).

5.3 S-Matrix Formulation

For numerical stability, use the scattering matrix formulation:

\begin{pmatrix} \mathbf{a}_1^- \\ \mathbf{a}_N^+ \end{pmatrix} = \mathbf{S} \begin{pmatrix} \mathbf{a}_1^+ \\ \mathbf{a}_N^- \end{pmatrix}

Where $\mathbf{a}^+$ and $\mathbf{a}^-$ represent forward and backward propagating waves.

The S-matrix is built recursively:

\mathbf{S}_{1 \to j+1} = \mathbf{S}_{1 \to j} \star \mathbf{S}_{j,j+1}

Using the Redheffer star product $\star$.

6. Statistical Process Control

6.1 Control Charts

$\bar{X}$ Chart (Mean):

UCL = \bar{\bar{X}} + A_2 \bar{R}

LCL = \bar{\bar{X}} - A_2 \bar{R}

R Chart (Range):

UCL_R = D_4 \bar{R}

LCL_R = D_3 \bar{R}

EWMA (Exponentially Weighted Moving Average):

Z_t = \lambda X_t + (1 - \lambda) Z_{t-1}

With control limits:

UCL = \mu_0 + L \sigma \sqrt{\frac{\lambda}{2 - \lambda} \left[ 1 - (1-\lambda)^{2t} \right]}

6.2 Process Capability Indices

$C_p$ (Process Capability):

C_p = \frac{USL - LSL}{6\sigma}

$C_{pk}$ (Centered Process Capability):

C_{pk} = \min \left( \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right)

$C_{pm}$ (Taguchi Capability):

C_{pm} = \frac{USL - LSL}{6\sqrt{\sigma^2 + (\mu - T)^2}}

Where:

$USL$ = Upper Specification Limit
$LSL$ = Lower Specification Limit
$T$ = Target value
$\mu$ = Process mean
$\sigma$ = Process standard deviation

6.3 Gauge R&R Analysis

Total measurement variance decomposition:

\sigma^2_{total} = \sigma^2_{part} + \sigma^2_{gauge}

\sigma^2_{gauge} = \sigma^2_{repeatability} + \sigma^2_{reproducibility}

Precision-to-Tolerance Ratio:

P/T = \frac{6 \sigma_{gauge}}{USL - LSL} \times 100\%

P/T Ratio	Assessment
< 10%	Excellent
10-30%	Acceptable
> 30%	Unacceptable

7. Uncertainty Quantification

7.1 Fisher Information Matrix

The Fisher Information Matrix for parameter estimation:

F_{ij} = \sum_{k=1}^{N} \frac{1}{\sigma_k^2} \frac{\partial M_k}{\partial p_i} \frac{\partial M_k}{\partial p_j}

Or equivalently:

F_{ij} = -E \left[ \frac{\partial^2 \ln L}{\partial p_i \partial p_j} \right]

Where $L$ is the likelihood function.

7.2 Cramér-Rao Lower Bound

The covariance matrix of any unbiased estimator is bounded:

\text{Cov}(\hat{\mathbf{p}}) \geq \mathbf{F}^{-1}

For a single parameter:

\text{Var}(\hat{\theta}) \geq \frac{1}{I(\theta)}

Interpretation:

Diagonal elements of $\mathbf{F}^{-1}$ give minimum variance for each parameter
Off-diagonal elements indicate parameter correlations
Large condition number of $\mathbf{F}$ indicates ill-conditioning

7.3 Correlation Coefficient

\rho_{ij} = \frac{F^{-1}_{ij}}{\sqrt{F^{-1}_{ii} F^{-1}_{jj}}}

	$\rho$		Interpretation
< 0.3	Weak correlation
0.3 – 0.7	Moderate correlation
> 0.7	Strong correlation
> 0.95	Severe: consider fixing one parameter

7.4 GUM Framework

According to the Guide to the Expression of Uncertainty in Measurement:

Combined standard uncertainty:

u_c^2(y) = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i) + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} u(x_i, x_j)

Expanded uncertainty:

U = k \cdot u_c(y)

Where $k$ is the coverage factor (typically $k=2$ for 95% confidence).

8. Machine Learning in Metrology

8.1 Neural Network Surrogate Models

Replace expensive physics simulations with trained neural networks:

M_{NN}(\mathbf{p}; \mathbf{W}) \approx M_{physics}(\mathbf{p})

Training objective:

\mathcal{L} = \frac{1}{N} \sum_{i=1}^{N} \left\| M_{NN}(\mathbf{p}_i) - M_{physics}(\mathbf{p}_i) \right\|^2 + \lambda \left\| \mathbf{W} \right\|^2

Speedup: Typically $10^4$ – $10^6 \times$ faster than RCWA/FEM.

8.2 Physics-Informed Neural Networks (PINNs)

Incorporate physical laws into the loss function:

\mathcal{L}_{total} = \mathcal{L}_{data} + \lambda_{physics} \mathcal{L}_{physics}

Where:

\mathcal{L}_{physics} = \left\| \nabla \times \mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} \right\|^2 + \ldots

8.3 Gaussian Process Regression

A non-parametric Bayesian approach:

f(\mathbf{x}) \sim \mathcal{GP}\left( m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}') \right)

Common kernel (RBF/Squared Exponential):

k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left( -\frac{\left\| \mathbf{x} - \mathbf{x}' \right\|^2}{2\ell^2} \right)

Posterior prediction:

\mu_* = \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{y}

\sigma_*^2 = k_{**} - \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{k}_*

Advantages:

Provides uncertainty estimates naturally
Works well with limited training data
Interpretable hyperparameters

8.4 Virtual Metrology

Predict wafer properties from equipment sensor data:

\hat{y} = f(FDC_1, FDC_2, \ldots, FDC_n)

Where $FDC_i$ are Fault Detection and Classification sensor readings.

Common approaches:

Partial Least Squares (PLS) regression
Random Forests
Gradient Boosting (XGBoost, LightGBM)
Deep neural networks

9. Advanced Topics and Frontiers

9.1 3D Metrology Challenges

Modern structures require 3D measurement:

Structure	Complexity	Key Challenge
FinFET	Moderate	Fin height, sidewall angle
GAA/Nanosheet	High	Sheet thickness, spacing
3D NAND	Very High	200+ layers, bowing, tilt
DRAM HAR	Extreme	100:1 aspect ratio structures

9.2 Hybrid Metrology

Combining multiple techniques to break parameter correlations:

\chi^2_{total} = \sum_{techniques} w_t \chi^2_t

Example combination:

OCD for periodic structure parameters
Ellipsometry for film optical constants
XRR for density and interface roughness

Mathematical framework:

\mathbf{F}_{hybrid} = \sum_t \mathbf{F}_t

Reduces off-diagonal elements, improving condition number.

9.3 Atomic-Scale Considerations

At the 2nm node and beyond:

Line Edge Roughness (LER):

\sigma_{LER} = \sqrt{\frac{1}{L} \int_0^L \left[ x(z) - \bar{x} \right]^2 dz}

Power Spectral Density:

PSD(f) = \frac{\sigma^2 \xi}{1 + (2\pi f \xi)^{2(1+H)}}

Where:

$\xi$ = Correlation length
$H$ = Hurst exponent (roughness character)

Quantum Effects:

Tunneling through thin barriers
Discrete dopant effects
Wave function penetration

9.4 Model-Measurement Circularity

A fundamental epistemological challenge:

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Why it matters</text><rect x="524" y="104" width="94" height="28" rx="6" fill="#161b22" stroke="#34d399"/><text x="571" y="122" fill="#34d399" font-size="9" text-anchor="middle">measure</text><rect x="636" y="104" width="94" height="28" rx="6" fill="#161b22" stroke="#38bdf8"/><text x="683" y="122" fill="#38bdf8" font-size="9" text-anchor="middle">vs target</text><rect x="636" y="164" width="94" height="28" rx="6" fill="#161b22" stroke="#a99cf0"/><text x="683" y="182" fill="#a99cf0" font-size="9" text-anchor="middle">APC feedback</text><rect x="524" y="164" width="94" height="28" rx="6" fill="#161b22" stroke="#e0b13a"/><text x="571" y="182" fill="#e0b13a" font-size="9" text-anchor="middle">tune litho/etch</text><line x1="618" y1="118" x2="632" y2="118" stroke="#8b949e" stroke-width="1.5"/><path d="M636 118 l-8 -4 l0 8 Z" fill="#8b949e"/><line x1="683" y1="132" x2="683" y2="160" stroke="#8b949e" stroke-width="1.5"/><path d="M683 164 l-4 -8 l8 0 Z" fill="#8b949e"/><line x1="632" y1="178" x2="620" y2="178" stroke="#8b949e" stroke-width="1.5"/><path d="M618 178 l8 -4 l0 8 Z" fill="#8b949e"/><line x1="571" y1="160" x2="571" y2="134" stroke="#8b949e" stroke-width="1.5"/><path d="M571 132 l-4 8 l8 0 Z" fill="#8b949e"/><text x="526" y="214" fill="#adb5bd" font-size="8.5">Every step is measured, compared to</text><text x="526" y="226" fill="#adb5bd" font-size="8.5">target, and fed back &#8212; advanced</text><text x="526" y="238" fill="#adb5bd" font-size="8.5">process control (APC).</text><line x1="526" y1="252" x2="728" y2="252" stroke="#30363d" stroke-width="1"/><text x="526" y="271" fill="#a99cf0" font-size="10.5" font-weight="700">AI twist</text><text x="526" y="287" fill="#adb5bd" font-size="8.5">At 2 nm, GAA &amp; 3D-NAND, parameters</text><text x="526" y="299" fill="#adb5bd" font-size="8.5">are correlated and throughput is brutal.</text><text x="526" y="319" fill="#34d399" font-size="8.5" font-weight="600">ML inverse models + virtual metrology</text><text x="526" y="331" fill="#34d399" font-size="8.5" font-weight="600">predict results from tool sensor data.</text><!-- ============ BOTTOM CARDS ============ --><rect x="20" y="384" width="226" height="70" rx="7" fill="#111a24" stroke="#30363d"/><text x="32" y="404" fill="#34d399" font-size="10.5" font-weight="700">Metrology &#8212; dimensions &amp; films</text><text x="32" y="421" fill="#adb5bd" font-size="8.7">Measures CD, film thickness and</text><text x="32" y="434" fill="#adb5bd" font-size="8.7">overlay to sub-nm &#8212; the numbers</text><text x="32" y="447" fill="#adb5bd" font-size="8.7">that keep every layer on target.</text><rect x="267" y="384" width="226" height="70" rx="7" fill="#111a24" stroke="#30363d"/><text x="279" y="404" fill="#38bdf8" font-size="10.5" font-weight="700">Inspection &#8212; defects</text><text x="279" y="421" fill="#adb5bd" font-size="8.7">Scans for particles and pattern</text><text x="279" y="434" fill="#adb5bd" font-size="8.7">faults (bright/dark-field, e-beam)</text><text x="279" y="447" fill="#adb5bd" font-size="8.7">and maps their coordinates.</text><rect x="514" y="384" width="226" height="70" rx="7" fill="#111a24" stroke="#30363d"/><text x="526" y="404" fill="#a99cf0" font-size="10.5" font-weight="700">AI twist: virtual metrology</text><text x="526" y="421" fill="#adb5bd" font-size="8.7">ML inverse models predict results</text><text x="526" y="434" fill="#adb5bd" font-size="8.7">from tool data &#8212; keeping up at</text><text x="526" y="447" fill="#adb5bd" font-size="8.7">2 nm, GAA &amp; 3D-NAND.</text></svg>

Key questions:

How do we validate models when "truth" requires modeling?
Reference metrology (TEM) also requires interpretation
What does it mean to "know" a dimension at atomic scale?

Key Symbols and Notation

Symbol	Description	Units
$\lambda$	Wavelength	nm
$\theta$	Angle of incidence	degrees
$n$	Refractive index	dimensionless
$k$	Extinction coefficient	dimensionless
$d$	Film thickness	nm
$\Lambda$	Grating period	nm
$\Psi, \Delta$	Ellipsometric angles	degrees
$\sigma$	Standard deviation	varies
$\mathbf{J}$	Jacobian matrix	varies
$\mathbf{F}$	Fisher Information Matrix	varies

Computational Complexity

Method	Complexity	Typical Time
Transfer Matrix	$O(N)$	$\mu$s
RCWA	$O(M^3 \cdot L)$	ms – s
FEM	$O(N^{1.5})$	s – min
FDTD	$O(N \cdot T)$	s – min
Monte Carlo (SEM)	$O(N_{electrons})$	min – hr
Neural Network (inference)	$O(1)$	$\mu$s

Where:

$N$ = Number of layers / mesh elements
$M$ = Number of Fourier orders
$L$ = Number of layers
$T$ = Number of time steps

Semiconductor Metrology Equipment Landscape (2024). Metrology in semiconductor manufacturing measures the physical dimensions, compositions, and electrical properties of features ranging from 300 mm wafer-scale uniformity down to sub-nanometer atomic layers — in-line at production throughput without destroying the wafer. The equipment market exceeds 8 billion USD (2023), dominated by four companies: KLA Corporation ($\sim$55% of process control revenue), Applied Materials (electron-beam review, Uniprocessor), Hitachi High-Tech (CD-SEM), and Onto Innovation (optical film/overlay). Every critical etch, deposition, lithography, and CMP step requires a paired metrology measurement to close the process control loop.

Critical Metrology Measurements at Each Process Step. After lithography: overlay alignment ($<$1.5 nm 3$\sigma$) by optical or e-beam, CD measurement ($\pm$0.3 nm) by OCD scatterometry or CD-SEM. After etch: CD uniformity, profile angle (88–90$^\circ$), remaining film thickness, and sidewall roughness (LER $<$ 2 nm). After deposition: film thickness ($\pm$0.5%), refractive index (composition), stress, and particle count. After CMP: remaining thickness, dishing, erosion, and surface roughness. After implant: dose ($\pm$1%), energy (keV), and junction depth. The metrology budget at advanced nodes consumes 5–10% of total wafer cycle time — a non-trivial throughput penalty that drives demand for faster optical techniques that can replace slow electron-beam methods.

In-Line vs Offline vs In-Situ Metrology. In-line metrology measures every lot (25 wafers) at dedicated metrology stations between process steps — OCD, ellipsometry, and CD-SEM operate here at 50–200 WPH. Offline metrology (TEM, AFM, SIMS) requires sample preparation and runs at 1–5 measurements per day — used only for process development and failure analysis. In-situ metrology (optical emission for etch endpoint, reflectometry for deposition thickness, pyrometry for temperature) operates inside the process chamber in real-time, enabling closed-loop control without removing the wafer. The trend at advanced nodes is migrating all possible measurements from in-line to in-situ — eliminating the 15–30 minute queue time between process and measurement that degrades APC (advanced process control) responsiveness.

metrologysemiconductor metrologymeasurementcharacterizationellipsometryscatterometry

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