openai sdk,python,typescript
OpenAI SDK available in Python and TypeScript. Official client. Streaming, tools.
3,145 technical terms and definitions
OpenAI SDK available in Python and TypeScript. Official client. Streaming, tools.
OpenVINO optimizes models for Intel hardware using graph optimization and kernel libraries.
Operation primitives are basic computational blocks like convolutions pooling and skip connections used in NAS.
Operational carbon emissions occur during product use phase from energy consumption.
Operator fusion combines multiple operations into single kernel reducing memory traffic and latency.
Optical emission failure analysis detects photon emission from hot carriers or breakdown events localizing defects.
Optical flow estimation computes pixel motion between frames for interpolation and stabilization.
# Semiconductor Manufacturing Process Optimization and Computational Mathematical Modeling ## 1. The Fundamental Challenge Modern semiconductor manufacturing involves **500–1000+ sequential process steps** to produce chips with billions of transistors at nanometer scales. Each step has dozens of tunable parameters, creating an optimization challenge that is: - **Extraordinarily high-dimensional** — hundreds to thousands of parameters - **Highly nonlinear** — complex interactions between process variables - **Expensive to explore experimentally** — each wafer costs thousands of dollars - **Multi-objective** — balancing yield, throughput, cost, and performance **Key Manufacturing Processes:** 1. **Lithography** — Pattern transfer using light/EUV exposure 2. **Etching** — Material removal (wet/dry plasma etching) 3. **Deposition** — Material addition (CVD, PVD, ALD) 4. **Ion Implantation** — Dopant introduction 5. **Thermal Processing** — Diffusion, annealing, oxidation 6. **Chemical-Mechanical Planarization (CMP)** — Surface planarization ## 2. The Mathematical Foundation ### 2.1 Governing Physics: Partial Differential Equations Nearly all semiconductor processes are governed by systems of coupled PDEs. #### Heat Transfer (Thermal Processing, Laser Annealing) $$ \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q $$ Where: - $\rho$ — density ($\text{kg/m}^3$) - $c_p$ — specific heat capacity ($\text{J/(kg}\cdot\text{K)}$) - $T$ — temperature ($\text{K}$) - $k$ — thermal conductivity ($\text{W/(m}\cdot\text{K)}$) - $Q$ — volumetric heat source ($\text{W/m}^3$) #### Mass Diffusion (Dopant Redistribution, Oxidation) $$ \frac{\partial C}{\partial t} = \nabla \cdot \left( D(C, T) \nabla C \right) + R(C) $$ Where: - $C$ — concentration ($\text{atoms/cm}^3$) - $D(C, T)$ — diffusion coefficient (concentration and temperature dependent) - $R(C)$ — reaction/generation term **Common Diffusion Models:** - **Constant source diffusion:** $$C(x, t) = C_s \cdot \text{erfc}\left( \frac{x}{2\sqrt{Dt}} \right)$$ - **Limited source diffusion:** $$C(x, t) = \frac{Q}{\sqrt{\pi D t}} \exp\left( -\frac{x^2}{4Dt} \right)$$ #### Fluid Dynamics (CVD, Etching Reactors) **Navier-Stokes Equations:** $$ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f} $$ **Continuity Equation:** $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 $$ **Species Transport:** $$ \frac{\partial c_i}{\partial t} + \mathbf{v} \cdot \nabla c_i = D_i \nabla^2 c_i + \sum_j R_{ij} $$ Where: - $\mathbf{v}$ — velocity field ($\text{m/s}$) - $p$ — pressure ($\text{Pa}$) - $\mu$ — dynamic viscosity ($\text{Pa}\cdot\text{s}$) - $c_i$ — species concentration - $R_{ij}$ — reaction rates between species #### Electromagnetics (Lithography, Plasma Physics) **Maxwell's Equations:** $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ **Hopkins Formulation for Partially Coherent Imaging:** $$ I(\mathbf{x}) = \iint J(\mathbf{f}_1, \mathbf{f}_2) \tilde{O}(\mathbf{f}_1) \tilde{O}^*(\mathbf{f}_2) e^{2\pi i (\mathbf{f}_1 - \mathbf{f}_2) \cdot \mathbf{x}} \, d\mathbf{f}_1 \, d\mathbf{f}_2 $$ Where: - $J(\mathbf{f}_1, \mathbf{f}_2)$ — mutual intensity (transmission cross-coefficient) - $\tilde{O}(\mathbf{f})$ — Fourier transform of mask transmission function ### 2.2 Surface Evolution and Topography Etching and deposition cause surfaces to evolve over time. The **Level Set Method** elegantly handles this: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ Where: - $\phi$ — level set function (surface defined by $\phi = 0$) - $V_n$ — normal velocity determined by local etch/deposition rates **Advantages:** - Naturally handles topological changes (void formation, surface merging) - No need for explicit surface tracking - Handles complex geometries **Etch Rate Models:** - **Ion-enhanced etching:** $$V_n = k_0 + k_1 \Gamma_{\text{ion}} + k_2 \Gamma_{\text{neutral}}$$ - **Visibility-dependent deposition:** $$V_n = V_0 \cdot \Omega(\mathbf{x})$$ where $\Omega(\mathbf{x})$ is the solid angle visible from point $\mathbf{x}$ ## 3. Computational Methods ### 3.1 Discretization Approaches #### Finite Element Methods (FEM) FEM dominates stress/strain analysis, thermal modeling, and electromagnetic simulation. The **weak formulation** transforms strong-form PDEs into integral equations: For the heat equation $-\nabla \cdot (k \nabla T) = Q$: $$ \int_\Omega \nabla w \cdot (k \nabla T) \, d\Omega = \int_\Omega w Q \, d\Omega + \int_{\Gamma_N} w q \, dS $$ Where: - $w$ — test/weight function - $\Omega$ — domain - $\Gamma_N$ — Neumann boundary **Galerkin Approximation:** $$ T(\mathbf{x}) \approx \sum_{i=1}^{N} T_i N_i(\mathbf{x}) $$ Where $N_i(\mathbf{x})$ are shape functions and $T_i$ are nodal values. #### Finite Difference Methods (FDM) Efficient for regular geometries and time-dependent problems. **Explicit Scheme (Forward Euler):** $$ \frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2} $$ **Stability Condition (CFL):** $$ \Delta t \leq \frac{\Delta x^2}{2\alpha} $$ **Implicit Scheme (Backward Euler):** $$ \frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^{n+1} - 2T_i^{n+1} + T_{i-1}^{n+1}}{\Delta x^2} $$ - Unconditionally stable but requires solving linear systems #### Monte Carlo Methods Essential for stochastic processes, particularly **ion implantation**. **Binary Collision Approximation (BCA):** 1. Sample impact parameter from screened Coulomb potential 2. Calculate scattering angle using: $$\theta = \pi - 2 \int_{r_{\min}}^{\infty} \frac{b \, dr}{r^2 \sqrt{1 - \frac{V(r)}{E_{\text{CM}}} - \frac{b^2}{r^2}}}$$ 3. Compute energy transfer: $$T = \frac{4 M_1 M_2}{(M_1 + M_2)^2} E \sin^2\left(\frac{\theta}{2}\right)$$ 4. Track recoils, vacancies, and interstitials 5. Accumulate statistics over $10^4 - 10^6$ ions ### 3.2 Multi-Scale Modeling | Scale | Length | Time | Methods | |:------|:-------|:-----|:--------| | Quantum | 0.1–1 nm | fs | DFT, ab initio MD | | Atomistic | 1–100 nm | ps–ns | Classical MD, Kinetic MC | | Mesoscale | 100 nm–10 μm | μs–ms | Phase field, Continuum MC | | Continuum | μm–mm | ms–hours | FEM, FDM, FVM | | Equipment | cm–m | seconds–hours | CFD, Thermal/Mechanical | **Information Flow Between Scales:** - **Upscaling:** Parameters computed at lower scales inform higher-scale models - Reaction barriers from DFT → Kinetic Monte Carlo rates - Surface mobilities from MD → Continuum deposition models - **Downscaling:** Boundary conditions and fields from higher scales - Temperature fields → Local reaction rates - Stress fields → Defect migration barriers ## 4. Optimization Frameworks ### 4.1 The General Problem Structure Semiconductor process optimization typically takes the form: $$ \min_{\mathbf{x} \in \mathcal{X}} f(\mathbf{x}) \quad \text{subject to} \quad g_i(\mathbf{x}) \leq 0, \quad h_j(\mathbf{x}) = 0 $$ Where: - $\mathbf{x} \in \mathbb{R}^n$ — process parameters (temperatures, pressures, times, flows, powers) - $f(\mathbf{x})$ — objective function (often negative yield or weighted combination) - $g_i(\mathbf{x}) \leq 0$ — inequality constraints (equipment limits, process windows) - $h_j(\mathbf{x}) = 0$ — equality constraints (design requirements) **Typical Parameter Vector:** $$ \mathbf{x} = \begin{bmatrix} T_1 \\ T_2 \\ P_{\text{chamber}} \\ t_{\text{process}} \\ \text{Flow}_{\text{gas1}} \\ \text{Flow}_{\text{gas2}} \\ \text{RF Power} \\ \vdots \end{bmatrix} $$ ### 4.2 Response Surface Methodology (RSM) Classical RSM builds polynomial surrogate models from designed experiments: **Second-Order Model:** $$ \hat{y} = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \sum_{j>i}^{k} \beta_{ij} x_i x_j + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \epsilon $$ **Matrix Form:** $$ \hat{y} = \beta_0 + \mathbf{x}^T \mathbf{b} + \mathbf{x}^T \mathbf{B} \mathbf{x} $$ Where: - $\mathbf{b}$ — vector of linear coefficients - $\mathbf{B}$ — matrix of quadratic and interaction coefficients **Design of Experiments (DOE) Types:** | Design Type | Runs for k Factors | Best For | |:------------|:-------------------|:---------| | Full Factorial | $2^k$ | Small k, all interactions | | Fractional Factorial | $2^{k-p}$ | Screening, main effects | | Central Composite | $2^k + 2k + n_c$ | Response surfaces | | Box-Behnken | Varies | Quadratic models, efficient | **Optimal Point (for quadratic model):** $$ \mathbf{x}^* = -\frac{1}{2} \mathbf{B}^{-1} \mathbf{b} $$ ### 4.3 Bayesian Optimization For expensive black-box functions, Bayesian optimization is remarkably efficient. **Gaussian Process Prior:** $$ f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')) $$ **Common Kernels:** - **Squared Exponential (RBF):** $$k(\mathbf{x}, \mathbf{x}') = \sigma^2 \exp\left( -\frac{\|\mathbf{x} - \mathbf{x}'\|^2}{2\ell^2} \right)$$ - **Matérn 5/2:** $$k(\mathbf{x}, \mathbf{x}') = \sigma^2 \left(1 + \frac{\sqrt{5}r}{\ell} + \frac{5r^2}{3\ell^2}\right) \exp\left(-\frac{\sqrt{5}r}{\ell}\right)$$ where $r = \|\mathbf{x} - \mathbf{x}'\|$ **Posterior Distribution:** Given observations $\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^{n}$: $$ \mu(\mathbf{x}^*) = \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{y} $$ $$ \sigma^2(\mathbf{x}^*) = k(\mathbf{x}^*, \mathbf{x}^*) - \mathbf{k}_*^T (\mathbf{K} + \sigma_n^2 \mathbf{I})^{-1} \mathbf{k}_* $$ **Acquisition Functions:** - **Expected Improvement (EI):** $$\text{EI}(\mathbf{x}) = \mathbb{E}\left[\max(f(\mathbf{x}) - f^+, 0)\right]$$ Closed form: $$\text{EI}(\mathbf{x}) = (\mu(\mathbf{x}) - f^+ - \xi) \Phi(Z) + \sigma(\mathbf{x}) \phi(Z)$$ where $Z = \frac{\mu(\mathbf{x}) - f^+ - \xi}{\sigma(\mathbf{x})}$ - **Upper Confidence Bound (UCB):** $$\text{UCB}(\mathbf{x}) = \mu(\mathbf{x}) + \kappa \sigma(\mathbf{x})$$ - **Probability of Improvement (PI):** $$\text{PI}(\mathbf{x}) = \Phi\left(\frac{\mu(\mathbf{x}) - f^+ - \xi}{\sigma(\mathbf{x})}\right)$$ ### 4.4 Metaheuristic Methods For highly non-convex, multimodal optimization landscapes. #### Genetic Algorithms (GA) **Algorithmic Steps:** 1. **Initialize** population of $N$ candidate solutions 2. **Evaluate** fitness $f(\mathbf{x}_i)$ for each individual 3. **Select** parents using tournament/roulette wheel selection 4. **Crossover** to create offspring: - Single-point: $\mathbf{x}_{\text{child}} = [\mathbf{x}_1(1:c), \mathbf{x}_2(c+1:n)]$ - Blend: $\mathbf{x}_{\text{child}} = \alpha \mathbf{x}_1 + (1-\alpha) \mathbf{x}_2$ 5. **Mutate** with probability $p_m$: $$x_i' = x_i + \mathcal{N}(0, \sigma^2)$$ 6. **Replace** population and repeat #### Particle Swarm Optimization (PSO) **Update Equations:** $$ \mathbf{v}_i^{t+1} = \omega \mathbf{v}_i^t + c_1 r_1 (\mathbf{p}_i - \mathbf{x}_i^t) + c_2 r_2 (\mathbf{g} - \mathbf{x}_i^t) $$ $$ \mathbf{x}_i^{t+1} = \mathbf{x}_i^t + \mathbf{v}_i^{t+1} $$ Where: - $\omega$ — inertia weight (typically 0.4–0.9) - $c_1, c_2$ — cognitive and social parameters (typically ~2.0) - $\mathbf{p}_i$ — personal best position - $\mathbf{g}$ — global best position - $r_1, r_2$ — random numbers in $[0, 1]$ #### Simulated Annealing (SA) **Acceptance Probability:** $$ P(\text{accept}) = \begin{cases} 1 & \text{if } \Delta E < 0 \\ \exp\left(-\frac{\Delta E}{k_B T}\right) & \text{if } \Delta E \geq 0 \end{cases} $$ **Cooling Schedule:** $$ T_{k+1} = \alpha T_k \quad \text{(geometric, } \alpha \approx 0.95\text{)} $$ ### 4.5 Multi-Objective Optimization Real optimization involves trade-offs between competing objectives. **Multi-Objective Problem:** $$ \min_{\mathbf{x}} \mathbf{F}(\mathbf{x}) = \begin{bmatrix} f_1(\mathbf{x}) \\ f_2(\mathbf{x}) \\ \vdots \\ f_m(\mathbf{x}) \end{bmatrix} $$ **Pareto Dominance:** Solution $\mathbf{x}_1$ dominates $\mathbf{x}_2$ (written $\mathbf{x}_1 \prec \mathbf{x}_2$) if: - $f_i(\mathbf{x}_1) \leq f_i(\mathbf{x}_2)$ for all $i$ - $f_j(\mathbf{x}_1) < f_j(\mathbf{x}_2)$ for at least one $j$ **NSGA-II Algorithm:** 1. Non-dominated sorting to assign ranks 2. Crowding distance calculation: $$d_i = \sum_{m=1}^{M} \frac{f_m^{i+1} - f_m^{i-1}}{f_m^{\max} - f_m^{\min}}$$ 3. Selection based on rank and crowding distance 4. Standard crossover and mutation ### 4.6 Robust Optimization Manufacturing variability is inevitable. Robust optimization explicitly accounts for it. **Mean-Variance Formulation:** $$ \min_{\mathbf{x}} \mathbb{E}_\xi[f(\mathbf{x}, \xi)] + \lambda \cdot \text{Var}_\xi[f(\mathbf{x}, \xi)] $$ **Minimax (Worst-Case) Formulation:** $$ \min_{\mathbf{x}} \max_{\xi \in \mathcal{U}} f(\mathbf{x}, \xi) $$ **Chance-Constrained Formulation:** $$ \min_{\mathbf{x}} f(\mathbf{x}) \quad \text{s.t.} \quad P(g(\mathbf{x}, \xi) \leq 0) \geq 1 - \alpha $$ **Taguchi Signal-to-Noise Ratios:** - **Smaller-is-better:** $\text{SNR} = -10 \log_{10}\left(\frac{1}{n}\sum_{i=1}^{n} y_i^2\right)$ - **Larger-is-better:** $\text{SNR} = -10 \log_{10}\left(\frac{1}{n}\sum_{i=1}^{n} \frac{1}{y_i^2}\right)$ - **Nominal-is-best:** $\text{SNR} = 10 \log_{10}\left(\frac{\bar{y}^2}{s^2}\right)$ ## 5. Advanced Topics and Modern Approaches ### 5.1 Physics-Informed Neural Networks (PINNs) PINNs embed physical laws directly into neural network training. **Loss Function:** $$ \mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}} + \gamma \mathcal{L}_{\text{BC}} $$ Where: $$ \mathcal{L}_{\text{data}} = \frac{1}{N_d} \sum_{i=1}^{N_d} |u_\theta(\mathbf{x}_i) - u_i|^2 $$ $$ \mathcal{L}_{\text{physics}} = \frac{1}{N_p} \sum_{j=1}^{N_p} |\mathcal{N}[u_\theta(\mathbf{x}_j)]|^2 $$ $$ \mathcal{L}_{\text{BC}} = \frac{1}{N_b} \sum_{k=1}^{N_b} |\mathcal{B}[u_\theta(\mathbf{x}_k)] - g_k|^2 $$ **Example: Heat Equation PINN** For $\frac{\partial T}{\partial t} = \alpha \nabla^2 T$: $$ \mathcal{L}_{\text{physics}} = \frac{1}{N_p} \sum_{j=1}^{N_p} \left| \frac{\partial T_\theta}{\partial t} - \alpha \nabla^2 T_\theta \right|^2_{\mathbf{x}_j, t_j} $$ **Advantages:** - Dramatically reduced data requirements - Physical consistency guaranteed - Effective for inverse problems ### 5.2 Digital Twins and Real-Time Optimization A digital twin is a continuously updated simulation model of the physical process. **Kalman Filter for State Estimation:** **Prediction Step:** $$ \hat{\mathbf{x}}_{k|k-1} = \mathbf{F}_k \hat{\mathbf{x}}_{k-1|k-1} + \mathbf{B}_k \mathbf{u}_k $$ $$ \mathbf{P}_{k|k-1} = \mathbf{F}_k \mathbf{P}_{k-1|k-1} \mathbf{F}_k^T + \mathbf{Q}_k $$ **Update Step:** $$ \mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{H}_k^T (\mathbf{H}_k \mathbf{P}_{k|k-1} \mathbf{H}_k^T + \mathbf{R}_k)^{-1} $$ $$ \hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k (\mathbf{z}_k - \mathbf{H}_k \hat{\mathbf{x}}_{k|k-1}) $$ $$ \mathbf{P}_{k|k} = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \mathbf{P}_{k|k-1} $$ **Run-to-Run Control:** $$ \mathbf{u}_{k+1} = \mathbf{u}_k + \mathbf{G} (\mathbf{y}_{\text{target}} - \hat{\mathbf{y}}_k) $$ Where $\mathbf{G}$ is the controller gain matrix. ### 5.3 Machine Learning for Virtual Metrology **Virtual Metrology Model:** $$ \hat{y} = f_{\text{ML}}(\mathbf{x}_{\text{sensor}}, \mathbf{x}_{\text{recipe}}, \mathbf{x}_{\text{context}}) $$ Where: - $\mathbf{x}_{\text{sensor}}$ — in-situ sensor data (OES, RF impedance, etc.) - $\mathbf{x}_{\text{recipe}}$ — process recipe parameters - $\mathbf{x}_{\text{context}}$ — chamber state, maintenance history **Domain Adaptation Challenge:** $$ \mathcal{L}_{\text{total}} = \mathcal{L}_{\text{task}} + \lambda \mathcal{L}_{\text{domain}} $$ Using adversarial training to minimize distribution shift between chambers. ### 5.4 Reinforcement Learning for Sequential Decisions **Markov Decision Process (MDP) Formulation:** - **State** $s$: Current wafer/chamber conditions - **Action** $a$: Recipe adjustments - **Reward** $r$: Yield, throughput, quality metrics - **Transition** $P(s'|s, a)$: Process dynamics **Policy Gradient (REINFORCE):** $$ \nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta} \left[ \sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t|s_t) \cdot G_t \right] $$ Where $G_t = \sum_{k=t}^{T} \gamma^{k-t} r_k$ is the return. ## 6. Specific Process Case Studies ### 6.1 Lithography: Computational Imaging and OPC **Optical Proximity Correction Optimization:** $$ \mathbf{m}^* = \arg\min_{\mathbf{m}} \|\mathbf{T}_{\text{target}} - \mathbf{I}(\mathbf{m})\|^2 + R(\mathbf{m}) $$ Where: - $\mathbf{m}$ — mask transmission function - $\mathbf{I}(\mathbf{m})$ — forward imaging model - $R(\mathbf{m})$ — regularization (manufacturability, minimum features) **Aerial Image Formation (Scalar Model):** $$ I(x, y) = \left| \int_{-\text{NA}}^{\text{NA}} \tilde{M}(f_x) H(f_x) e^{2\pi i f_x x} df_x \right|^2 $$ **Source-Mask Optimization (SMO):** $$ \min_{\mathbf{m}, \mathbf{s}} \sum_{p} \|I_p(\mathbf{m}, \mathbf{s}) - T_p\|^2 + \lambda_m R_m(\mathbf{m}) + \lambda_s R_s(\mathbf{s}) $$ Jointly optimizing mask pattern and illumination source. ### 6.2 CMP: Pattern-Dependent Modeling **Preston Equation:** $$ \frac{dz}{dt} = K_p \cdot p \cdot V $$ Where: - $K_p$ — Preston coefficient (material-dependent) - $p$ — local pressure - $V$ — relative velocity **Pattern-Dependent Pressure Model:** $$ p_{\text{eff}}(x, y) = p_{\text{applied}} \cdot \frac{1}{\rho(x, y) * K(x, y)} $$ Where $\rho(x, y)$ is the local pattern density and $*$ denotes convolution with a planarization kernel $K$. **Step Height Evolution:** $$ \frac{d(\Delta z)}{dt} = K_p V (p_{\text{high}} - p_{\text{low}}) $$ ### 6.3 Plasma Etching: Plasma-Surface Interactions **Species Balance in Plasma:** $$ \frac{dn_i}{dt} = \sum_j k_{ji} n_j n_e - \sum_k k_{ik} n_i n_e - \frac{n_i}{\tau_{\text{res}}} + S_i $$ Where: - $n_i$ — density of species $i$ - $k_{ji}$ — rate coefficients (Arrhenius form) - $\tau_{\text{res}}$ — residence time - $S_i$ — source terms **Ion Energy Distribution Function:** $$ f(E) = \frac{1}{\sqrt{2\pi}\sigma_E} \exp\left(-\frac{(E - \bar{E})^2}{2\sigma_E^2}\right) $$ **Etch Yield:** $$ Y(E, \theta) = Y_0 \cdot \sqrt{E - E_{\text{th}}} \cdot f(\theta) $$ Where $f(\theta)$ is the angular dependence. ## 7. The Mathematics of Yield **Poisson Defect Model:** $$ Y = e^{-D \cdot A} $$ Where: - $D$ — defect density ($\text{defects/cm}^2$) - $A$ — chip area ($\text{cm}^2$) **Negative Binomial (Clustered Defects):** $$ Y = \left(1 + \frac{DA}{\alpha}\right)^{-\alpha} $$ Where $\alpha$ is the clustering parameter (smaller = more clustered). **Parametric Yield:** For a parameter with distribution $p(\theta)$ and specification $[\theta_{\min}, \theta_{\max}]$: $$ Y_{\text{param}} = \int_{\theta_{\min}}^{\theta_{\max}} p(\theta) \, d\theta $$ For Gaussian distribution: $$ Y_{\text{param}} = \Phi\left(\frac{\theta_{\max} - \mu}{\sigma}\right) - \Phi\left(\frac{\theta_{\min} - \mu}{\sigma}\right) $$ **Process Capability Index:** $$ C_{pk} = \min\left(\frac{\mu - \text{LSL}}{3\sigma}, \frac{\text{USL} - \mu}{3\sigma}\right) $$ **Total Yield:** $$ Y_{\text{total}} = Y_{\text{defect}} \times Y_{\text{parametric}} \times Y_{\text{test}} $$ ## 8. Open Challenges 1. **High-Dimensional Optimization** - Hundreds to thousands of interacting parameters - Curse of dimensionality in sampling-based methods - Need for effective dimensionality reduction 2. **Uncertainty Quantification** - Error propagation across model hierarchies - Aleatory vs. epistemic uncertainty separation - Confidence bounds on predictions 3. **Data Scarcity** - Each experimental data point costs \$1000+ - Models must learn from small datasets - Transfer learning between processes/tools 4. **Interpretability** - Black-box models limit root cause analysis - Need for physics-informed feature engineering - Explainable AI for process engineering 5. **Real-Time Constraints** - Run-to-run control requires millisecond decisions - Reduced-order models needed - Edge computing for in-situ optimization 6. **Integration Complexity** - Multiple physics domains coupled - Full-flow optimization across 500+ steps - Design-technology co-optimization ## 9. Optimization summary Semiconductor manufacturing process optimization represents one of the most sophisticated applications of computational mathematics in industry. It integrates: - **Classical numerical methods** (FEM, FDM, Monte Carlo) - **Statistical modeling** (DOE, RSM, uncertainty quantification) - **Optimization theory** (convex/non-convex, single/multi-objective, deterministic/robust) - **Machine learning** (neural networks, Gaussian processes, reinforcement learning) - **Control theory** (Kalman filtering, run-to-run control, MPC) The field continues to evolve as feature sizes shrink toward atomic scales, process complexity grows, and computational capabilities expand. Success requires not just mathematical sophistication but deep physical intuition about the processes being modeled—the best work reflects genuine synthesis across disciplines.
Optimization-based inversion iteratively refines latent codes minimizing reconstruction error.
Optimize considering variability.
Optimize latent to match image.
An orchestrator routes each request to the best model or tool (cheap vs. expensive, code vs. chat) and can chain multiple steps into a workflow.
Constrain convolutions to be orthogonal.
Multi-modal instruction tuning.
Out-of-distribution inputs differ from training data potentially causing failures.
Outbound logistics handles product distribution from factories to customers.
Extend image beyond boundaries.
Outpainting extends images beyond original boundaries generating consistent extrapolations.
Extend image beyond its original boundaries.
Output constraints limit response characteristics like length style or content.
Output filters block unsafe responses before delivery to users.
Post-process model outputs to remove harmful sensitive or off-policy content.
Check generations before returning.
Decline benign requests unnecessarily.
Duplicate rare examples.
Overconfidence occurs when models express higher certainty than warranted by accuracy.
Train beyond optimal point.
Furnace to grow thermal oxide (dry or wet oxidation).
Ozone treatment oxidizes organic contaminants and disinfects water without residual chemicals.
Probably Approximately Correct learning provides theoretical bounds on sample complexity for learning with high probability and accuracy.
Package decapsulation removes molding compound and heat spreaders exposing die for failure analysis.
Package failure analysis investigates defects in packaging materials interconnects and interfaces using cross-sectioning acoustic microscopy and X-ray.
Package thermal modeling predicts temperature distributions using finite element analysis and computational fluid dynamics.
Packed sequences concatenate examples without padding for memory efficiency.
Padding masks prevent attention to padded positions during processing.
Padding extends shorter sequences to batch length enabling parallel processing.
Paged attention manages KV cache in non-contiguous memory blocks like virtual memory.
PagedAttention (vLLM) manages KV cache like virtual memory. Reduces fragmentation. Enables larger batches.
Equivariant molecular network.
Polarizable Atom Interaction Neural Network uses equivariant message passing for molecular property prediction.
Paired t-tests compare related measurements on same units.
Pairwise comparisons judge relative quality between two responses.
Compare two outputs and pick better one.
Pairwise ranking learns relative preferences between item pairs for recommendation ordering.
Learn from item comparisons.
Google's large-scale language model.
Create wide panoramic images.
Duplicate inheritance structures.
Parallel sampling generates multiple sequence candidates simultaneously.
Parameter binding fills function arguments with appropriate values from context.