pseudo-labeling,semi-supervised learning
Generate labels for unlabeled data using current model then retrain.
9,967 technical terms and definitions
Generate labels for unlabeled data using current model then retrain.
Replace identifiers with pseudonyms.
Pseudonymization replaces identifiers with pseudonyms allowing reversal with keys.
Pixel-wise similarity.
Phase retrieval technique for imaging.
PubMedBERT is biomedical BERT. Trained on PubMed abstracts.
Biomedical QA.
Pull production produces only what downstream processes consume avoiding overproduction.
Summarize code changes.
Produce based on demand.
Measure wire bond strength.
Pump down time measures duration to achieve target vacuum level.
Pure-play foundries focus exclusively on manufacturing without competing in design.
Purpose limitation restricts data use to specified intended purposes.
Push production schedules upstream operations independently often creating excess inventory.
Mix based on saliency and complexity.
Vacuum chamber for physical vapor deposition.
# Mathematical Modeling of Metal Deposition in Semiconductor Manufacturing 1. Overview: Metal Deposition Processes Metal deposition is a critical step in semiconductor fabrication, creating interconnects, contacts, barrier layers, and various metallic structures. The primary deposition methods require distinct mathematical treatments: | Process | Physics Domain | Key Mathematics | |---------|----------------|-----------------| | **PVD (Sputtering)** | Ballistic transport, plasma physics | Boltzmann transport, Monte Carlo | | **CVD/PECVD** | Gas-phase transport, surface reactions | Navier-Stokes, reaction-diffusion | | **ALD** | Self-limiting surface chemistry | Site-balance kinetics | | **Electroplating (ECD)** | Electrochemistry, mass transport | Butler-Volmer, Nernst-Planck | 2. Transport Phenomena Models 2.1 Gas-Phase Transport (CVD/PECVD) The precursor concentration field follows the **convection-diffusion-reaction equation**: $$ \frac{\partial C}{\partial t} + \mathbf{v} \cdot \nabla C = D \nabla^2 C + R_{gas} $$ Where: - $C$ — precursor concentration (mol/m³) - $\mathbf{v}$ — velocity field vector (m/s) - $D$ — diffusion coefficient (m²/s) - $R_{gas}$ — gas-phase reaction source term (mol/m³·s) 2.2 Flow Field Equations The **incompressible Navier-Stokes equations** govern the velocity field: $$ \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} $$ With continuity equation: $$ \nabla \cdot \mathbf{v} = 0 $$ Where: - $\rho$ — gas density (kg/m³) - $p$ — pressure (Pa) - $\mu$ — dynamic viscosity (Pa·s) ### 2.3 Knudsen Number and Transport Regimes At low pressures, the **Knudsen number** determines the transport regime: $$ Kn = \frac{\lambda}{L} = \frac{k_B T}{\sqrt{2} \pi d^2 p L} $$ Where: - $\lambda$ — mean free path (m) - $L$ — characteristic length (m) - $k_B$ — Boltzmann constant ($1.38 \times 10^{-23}$ J/K) - $T$ — temperature (K) - $d$ — molecular diameter (m) - $p$ — pressure (Pa) **Transport regime classification:** - $Kn < 0.01$ — **Continuum regime** → Navier-Stokes CFD - $0.01 < Kn < 0.1$ — **Slip flow regime** → Modified NS with slip boundary conditions - $0.1 < Kn < 10$ — **Transitional regime** → DSMC, Boltzmann equation - $Kn > 10$ — **Free molecular regime** → Ballistic/Monte Carlo methods 3. Surface Reaction Kinetics 3.1 Langmuir-Hinshelwood Mechanism For bimolecular surface reactions (common in CVD): $$ r = \frac{k \cdot K_A K_B \cdot p_A p_B}{(1 + K_A p_A + K_B p_B)^2} $$ Where: - $r$ — reaction rate (mol/m²·s) - $k$ — surface reaction rate constant (mol/m²·s) - $K_A, K_B$ — adsorption equilibrium constants (Pa⁻¹) - $p_A, p_B$ — partial pressures of reactants A and B (Pa) 3.2 Sticking Coefficient Model The probability that an impinging molecule adsorbs on the surface: $$ S = S_0 \exp\left( -\frac{E_a}{k_B T} \right) \cdot f(\theta) $$ Where: - $S$ — sticking coefficient (dimensionless) - $S_0$ — pre-exponential sticking factor - $E_a$ — activation energy (J) - $f(\theta) = (1 - \theta)^n$ — site blocking function - $\theta$ — surface coverage (dimensionless, 0 to 1) - $n$ — order of site blocking 3.3 Arrhenius Temperature Dependence $$ k(T) = A \exp\left( -\frac{E_a}{RT} \right) $$ Where: - $A$ — pre-exponential factor (frequency factor) - $E_a$ — activation energy (J/mol) - $R$ — universal gas constant (8.314 J/mol·K) - $T$ — absolute temperature (K) 4. Film Growth Models 4.1 Continuum Surface Evolution Edwards-Wilkinson Equation (Linear Growth) $$ \frac{\partial h}{\partial t} = \nu \nabla^2 h + F + \eta(\mathbf{x}, t) $$ Kardar-Parisi-Zhang (KPZ) Equation (Nonlinear Growth) $$ \frac{\partial h}{\partial t} = \nu \nabla^2 h + \frac{\lambda}{2} |\nabla h|^2 + F + \eta $$ Where: - $h(\mathbf{x}, t)$ — surface height at position $\mathbf{x}$ and time $t$ - $\nu$ — surface diffusion coefficient (m²/s) - $\lambda$ — nonlinear growth parameter - $F$ — mean deposition flux (m/s) - $\eta$ — stochastic noise term (Gaussian white noise) 4.2 Scaling Relations Surface roughness evolves according to: $$ W(L, t) = L^\alpha f\left( \frac{t}{L^z} \right) $$ Where: - $W$ — interface width (roughness) - $L$ — system size - $\alpha$ — roughness exponent - $z$ — dynamic exponent - $f$ — scaling function 5. Step Coverage and Conformality 5.1 Thiele Modulus For high-aspect-ratio features, the **Thiele modulus** determines conformality: $$ \phi = L \sqrt{\frac{k_s}{D_{eff}}} $$ Where: - $\phi$ — Thiele modulus (dimensionless) - $L$ — feature depth (m) - $k_s$ — surface reaction rate constant (m/s) - $D_{eff}$ — effective diffusivity (m²/s) **Step coverage regimes:** - $\phi \ll 1$ — **Reaction-limited** → Excellent conformality - $\phi \gg 1$ — **Transport-limited** → Poor step coverage (bread-loafing) 5.2 Knudsen Diffusion in Trenches $$ D_K = \frac{w}{3} \sqrt{\frac{8 R T}{\pi M}} $$ Where: - $D_K$ — Knudsen diffusion coefficient (m²/s) - $w$ — trench width (m) - $R$ — universal gas constant (J/mol·K) - $T$ — temperature (K) - $M$ — molecular weight (kg/mol) 5.3 Feature-Scale Concentration Profile Solving for concentration in a trench with reactive walls: $$ D_{eff} \frac{d^2 C}{dy^2} = \frac{2 k_s C}{w} $$ General solution: $$ C(y) = C_0 \frac{\cosh\left( \phi \frac{L - y}{L} \right)}{\cosh(\phi)} $$ 6. Atomic Layer Deposition (ALD) Models 6.1 Self-Limiting Surface Kinetics Surface site balance equation: $$ \frac{d\theta}{dt} = k_a C (1 - \theta) - k_d \theta $$ Where: - $\theta$ — fractional surface coverage - $k_a$ — adsorption rate constant (m³/mol·s) - $k_d$ — desorption rate constant (s⁻¹) - $C$ — gas-phase precursor concentration (mol/m³) At equilibrium saturation: $$ \theta_{eq} = \frac{k_a C}{k_a C + k_d} \approx 1 \quad \text{(for strong chemisorption)} $$ 6.2 Growth Per Cycle (GPC) $$ \text{GPC} = \Gamma_0 \cdot \Omega \cdot \eta $$ Where: - $\Gamma_0$ — surface site density (sites/m²) - $\Omega$ — volume per deposited atom (m³) - $\eta$ — reaction efficiency (dimensionless) 6.3 Saturation Dose-Time Relationship $$ \theta(t) = 1 - \exp\left( -\frac{S \cdot \Phi \cdot t}{\Gamma_0} \right) $$ **Impingement flux** from kinetic theory: $$ \Phi = \frac{p}{\sqrt{2 \pi m k_B T}} $$ Where: - $\Phi$ — molecular impingement flux (molecules/m²·s) - $p$ — precursor partial pressure (Pa) - $m$ — molecular mass (kg) 7. Plasma Modeling (PVD/PECVD) 7.1 Plasma Sheath Physics **Child-Langmuir law** for ion current density: $$ J_{ion} = \frac{4 \varepsilon_0}{9} \sqrt{\frac{2e}{M_i}} \frac{V_s^{3/2}}{d_s^2} $$ Where: - $J_{ion}$ — ion current density (A/m²) - $\varepsilon_0$ — vacuum permittivity ($8.85 \times 10^{-12}$ F/m) - $e$ — elementary charge ($1.6 \times 10^{-19}$ C) - $M_i$ — ion mass (kg) - $V_s$ — sheath voltage (V) - $d_s$ — sheath thickness (m) 7.2 Ion Energy at Substrate $$ \varepsilon_{ion} \approx e V_s + \frac{1}{2} M_i v_{Bohm}^2 $$ **Bohm velocity:** $$ v_{Bohm} = \sqrt{\frac{k_B T_e}{M_i}} $$ Where: - $T_e$ — electron temperature (K or eV) 7.3 Sputtering Yield (Sigmund Formula) $$ Y(E) = \frac{3 \alpha}{4 \pi^2} \cdot \frac{4 M_1 M_2}{(M_1 + M_2)^2} \cdot \frac{E}{U_0} $$ Where: - $Y$ — sputtering yield (atoms/ion) - $\alpha$ — dimensionless factor (~0.2–0.4) - $M_1$ — incident ion mass - $M_2$ — target atom mass - $E$ — incident ion energy (eV) - $U_0$ — surface binding energy (eV) 7.4 Electron Energy Distribution Function (EEDF) The Boltzmann equation in energy space: $$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + \frac{e \mathbf{E}}{m_e} \cdot \nabla_v f = C[f] $$ Where: - $f$ — electron energy distribution function - $\mathbf{E}$ — electric field - $m_e$ — electron mass - $C[f]$ — collision integral 8. MDP: Markov Decision Process for Process Control 8.1 MDP Formulation A Markov Decision Process is defined by the tuple: $$ \mathcal{M} = (S, A, P, R, \gamma) $$ **Components in semiconductor context:** - **State space $S$**: Film thickness, resistivity, uniformity, equipment state, wafer position - **Action space $A$**: Temperature, pressure, flow rates, RF power, deposition time - **Transition probability $P(s' | s, a)$**: Stochastic process model - **Reward function $R(s, a)$**: Yield, uniformity, throughput, quality metrics - **Discount factor $\gamma$**: Time preference (typically 0.9–0.99) 8.2 Bellman Optimality Equation $$ V^*(s) = \max_{a \in A} \left[ R(s, a) + \gamma \sum_{s'} P(s' | s, a) V^*(s') \right] $$ **Q-function formulation:** $$ Q^*(s, a) = R(s, a) + \gamma \sum_{s'} P(s' | s, a) \max_{a'} Q^*(s', a') $$ 8.3 Run-to-Run (R2R) Control Optimal recipe adjustment after each wafer: $$ \mathbf{u}_{k+1} = \mathbf{u}_k + \mathbf{K} (\mathbf{y}_{target} - \mathbf{y}_k) $$ Where: - $\mathbf{u}_k$ — process recipe parameters at run $k$ - $\mathbf{y}_k$ — measured output at run $k$ - $\mathbf{K}$ — controller gain matrix (from MDP policy optimization) 8.4 Reinforcement Learning Approaches | Method | Application | Characteristics | |--------|-------------|-----------------| | **Q-Learning** | Discrete parameter optimization | Model-free, tabular | | **Deep Q-Network (DQN)** | High-dimensional state spaces | Neural network approximation | | **Policy Gradient** | Continuous process control | Direct policy optimization | | **Actor-Critic (A2C/PPO)** | Complex control tasks | Combined value and policy | | **Model-Based RL** | Physics-informed control | Sample efficient | 9. Electrochemical Deposition (Copper Damascene) 9.1 Butler-Volmer Equation $$ i = i_0 \left[ \exp\left( \frac{\alpha_a F \eta}{RT} \right) - \exp\left( -\frac{\alpha_c F \eta}{RT} \right) \right] $$ Where: - $i$ — current density (A/m²) - $i_0$ — exchange current density (A/m²) - $\alpha_a, \alpha_c$ — anodic and cathodic transfer coefficients - $F$ — Faraday constant (96,485 C/mol) - $\eta = E - E_{eq}$ — overpotential (V) - $R$ — gas constant (J/mol·K) - $T$ — temperature (K) 9.2 Mass Transport Limited Current $$ i_L = \frac{n F D C_b}{\delta} $$ Where: - $i_L$ — limiting current density (A/m²) - $n$ — number of electrons transferred - $D$ — diffusion coefficient of Cu²⁺ (m²/s) - $C_b$ — bulk concentration (mol/m³) - $\delta$ — diffusion layer thickness (m) 9.3 Nernst-Planck Equation $$ \mathbf{J}_i = -D_i \nabla C_i - \frac{z_i F D_i}{RT} C_i \nabla \phi + C_i \mathbf{v} $$ Where: - $\mathbf{J}_i$ — flux of species $i$ - $z_i$ — charge number - $\phi$ — electric potential 9.4 Superfilling (Bottom-Up Fill) The curvature-enhanced accelerator mechanism: $$ v_n = v_0 (1 + \kappa \cdot \Gamma_{acc}) $$ Where: - $v_n$ — local growth velocity normal to surface - $v_0$ — baseline growth velocity - $\kappa$ — local surface curvature (1/m) - $\Gamma_{acc}$ — accelerator surface concentration 10. Multiscale Modeling Framework 10.1 Hierarchical Scale Integration ``` ┌──────────────────────────────────────────────────────────────┐ │ REACTOR SCALE │ │ CFD: Flow, temperature, concentration │ │ Time: seconds | Length: cm │ └─────────────────────────┬────────────────────────────────────┘ │ Boundary fluxes ▼ ┌──────────────────────────────────────────────────────────────┐ │ FEATURE SCALE │ │ Level-set / String method for surface evolution │ │ Time: seconds | Length: μm │ └─────────────────────────┬────────────────────────────────────┘ │ Local rates ▼ ┌──────────────────────────────────────────────────────────────┐ │ MESOSCALE (kMC) │ │ Kinetic Monte Carlo: nucleation, island growth │ │ Time: ms | Length: nm │ └─────────────────────────┬────────────────────────────────────┘ │ Rate parameters ▼ ┌──────────────────────────────────────────────────────────────┐ │ ATOMISTIC (MD/DFT) │ │ Molecular dynamics, ab initio: binding energies, │ │ diffusion barriers, reaction paths │ │ Time: ps | Length: Å │ └──────────────────────────────────────────────────────────────┘ ``` 10.2 Kinetic Monte Carlo (kMC) Event rate from transition state theory: $$ k_i = \nu_0 \exp\left( -\frac{E_{a,i}}{k_B T} \right) $$ Total rate and time step: $$ k_{total} = \sum_i k_i, \quad \Delta t = -\frac{\ln(r)}{k_{total}} $$ Where $r \in (0, 1]$ is a uniform random number. 10.3 Molecular Dynamics Newton's equations of motion: $$ m_i \frac{d^2 \mathbf{r}_i}{dt^2} = -\nabla_i U(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N) $$ **Lennard-Jones potential:** $$ U_{LJ}(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right] $$ **Embedded Atom Method (EAM) for metals:** $$ U = \sum_i F_i(\rho_i) + \frac{1}{2} \sum_{i \neq j} \phi_{ij}(r_{ij}) $$ Where $\rho_i = \sum_{j \neq i} f_j(r_{ij})$ is the electron density at atom $i$. 11. Uniformity Modeling 11.1 Wafer-Scale Thickness Distribution (Sputtering) For a circular magnetron target: $$ t(r) = \int_{target} \frac{Y \cdot J_{ion} \cdot \cos\theta_t \cdot \cos\theta_w}{\pi R^2} \, dA $$ Where: - $t(r)$ — thickness at radial position $r$ - $\theta_t$ — emission angle from target - $\theta_w$ — incidence angle at wafer 11.2 Uniformity Metrics **Within-Wafer Uniformity (WIW):** $$ \sigma_{WIW} = \frac{1}{\bar{t}} \sqrt{\frac{1}{N} \sum_{i=1}^{N} (t_i - \bar{t})^2} \times 100\% $$ **Wafer-to-Wafer Uniformity (WTW):** $$ \sigma_{WTW} = \frac{1}{\bar{t}_{avg}} \sqrt{\frac{1}{M} \sum_{j=1}^{M} (\bar{t}_j - \bar{t}_{avg})^2} \times 100\% $$ **Target specifications:** - $\sigma_{WIW} < 1\%$ for advanced nodes (≤7 nm) - $\sigma_{WTW} < 0.5\%$ for high-volume manufacturing 12. Virtual Metrology and Statistical Models 12.1 Gaussian Process Regression (GPR) $$ f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}')) $$ **Squared exponential (RBF) kernel:** $$ k(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left( -\frac{|\mathbf{x} - \mathbf{x}'|^2}{2\ell^2} \right) $$ **Predictive distribution:** $$ f_* | \mathbf{X}, \mathbf{y}, \mathbf{x}_* \sim \mathcal{N}(\bar{f}_*, \text{var}(f_*)) $$ 12.2 Partial Least Squares (PLS) $$ \mathbf{Y} = \mathbf{X} \mathbf{B} + \mathbf{E} $$ Where: - $\mathbf{X}$ — process parameter matrix - $\mathbf{Y}$ — quality outcome matrix - $\mathbf{B}$ — regression coefficient matrix - $\mathbf{E}$ — residual matrix 12.3 Principal Component Analysis (PCA) $$ \mathbf{X} = \mathbf{T} \mathbf{P}^T + \mathbf{E} $$ **Hotelling's $T^2$ statistic for fault detection:** $$ T^2 = \sum_{i=1}^{k} \frac{t_i^2}{\lambda_i} $$ 13. Process Optimization 13.1 Response Surface Methodology (RSM) **Second-order polynomial model:** $$ y = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i < j} \beta_{ij} x_i x_j + \varepsilon $$ 13.2 Constrained Optimization $$ \min_{\mathbf{x}} f(\mathbf{x}) \quad \text{subject to} \quad g_i(\mathbf{x}) \leq 0, \quad h_j(\mathbf{x}) = 0 $$ **Example constraints:** - $g_1$: Non-uniformity ≤ 3% - $g_2$: Resistivity within spec - $g_3$: Throughput ≥ target - $h_1$: Total film thickness = target 13.3 Pareto Multi-Objective Optimization $$ \min_{\mathbf{x}} \left[ f_1(\mathbf{x}), f_2(\mathbf{x}), \ldots, f_m(\mathbf{x}) \right] $$ Common trade-offs: - Uniformity vs. throughput - Film quality vs. cost - Conformality vs. deposition rate 14. Summary: Mathematical Toolkit Reference | Domain | Key Equations | Application | |--------|---------------|-------------| | **Transport** | Navier-Stokes, Convection-Diffusion | Gas flow, precursor delivery | | **Kinetics** | Arrhenius, Langmuir-Hinshelwood | Reaction rates | | **Surface Evolution** | KPZ, Level-set, Edwards-Wilkinson | Film morphology | | **Plasma** | Boltzmann, Child-Langmuir | Ion/electron dynamics | | **Electrochemistry** | Butler-Volmer, Nernst-Planck | Copper plating | | **Control** | Bellman, MDP, RL algorithms | Recipe optimization | | **Statistics** | GPR, PLS, PCA | Virtual metrology | | **Multiscale** | MD, kMC, Continuum | Integrated simulation | 15. Key Physical Constants | Constant | Symbol | Value | Units | |----------|--------|-------|-------| | Boltzmann constant | $k_B$ | $1.38 \times 10^{-23}$ | J/K | | Gas constant | $R$ | $8.314$ | J/(mol·K) | | Faraday constant | $F$ | $96,485$ | C/mol | | Elementary charge | $e$ | $1.60 \times 10^{-19}$ | C | | Vacuum permittivity | $\varepsilon_0$ | $8.85 \times 10^{-12}$ | F/m | | Avogadro's number | $N_A$ | $6.02 \times 10^{23}$ | mol⁻¹ | | Electron mass | $m_e$ | $9.11 \times 10^{-31}$ | kg |
Deposit conductor films (Al Cu Ti TiN Ta TaN W etc).
Design verification at extremes.
Process voltage and temperature variations cause parameter spreads requiring robust design.
Pyramid warping correlation for flow.
PyCharm is Python IDE. Professional features, debugging.
Use Pydantic models to validate and parse LLM outputs.
pyenv manages Python versions. Switch between versions.
Pyraformer uses pyramidal attention with multi-resolution representations for efficient long-term time series forecasting.
Multi-scale vision transformer.
Pyrometers measure high temperatures through emitted infrared radiation.
pytest is Python testing framework. Simple, powerful.
Pythia is EleutherAI model suite. Different sizes for research.
Execute generated Python code for answers.
I can write or refactor Python scripts for automation, data processing, and experiments, with comments explaining each step.
PyTorch Mobile enables optimized on-device inference for PyTorch models on mobile platforms.
Performance analysis for PyTorch.
Variational algorithm for optimization.
QA on scientific papers.
Vector similarity search engine.
Qdrant is vector database in Rust. High performance.
Whether to use bias in attention projections.
Combines quantization with LoRA for memory-efficient fine-tuning on consumer GPUs.
QMIX is a value-based multi-agent RL method that learns centralized but factorized Q-functions to enable decentralized policy execution.
QPLEX factorizes joint action-values through dueling network architecture enabling scalable value decomposition in multi-agent settings.
Quantile Regression DQN approximates return distribution through fixed quantiles improving robustness in distributional reinforcement learning.
Information-seeking conversations.
Question Answering in Context evaluates information-seeking conversations.
No leads extending from body.
Four-sided package with leads.
Differences between wafer quadrants.
Quadrant patterns divide wafers into sections suggesting equipment or process asymmetry.
Quadratic loss functions penalize deviations proportional to squared error.