chainlit,chat,interface
Chainlit builds chat interfaces for LLM apps. Streaming, memory, tools. Production-ready.
321 technical terms and definitions
Chainlit builds chat interfaces for LLM apps. Streaming, memory, tools. Production-ready.
Change point detection identifies times when statistical properties of time series change indicating regime shifts.
Channel attention mechanisms adaptively weight feature channels based on global context.
Channel shuffle permutes channels between groups enabling information flow in grouped convolutions.
Charge-induced voltage alteration uses localized charge injection to modulate circuit behavior localizing sensitive nodes.
Device itself charged then discharged.
Design ESD protection for CDM.
Chat models are optimized for multi-turn conversational interactions.
Conversational AI based on GPT models with RLHF.
Spectral GNN using Chebyshev polynomials.
ChebNet approximates spectral graph convolutions using Chebyshev polynomials reducing computational cost.
Distribute checkpoint across nodes.
Saved snapshot of model weights during training.
Checkpoints save model weights periodically. Resume training from checkpoint if interrupted. Keep best checkpoint by validation loss.
Chemical decapsulation dissolves epoxy molding compound using hot acids preserving die integrity.
Identify chemical compounds in text.
Chemical recycling recovers and purifies process chemicals for reuse reducing consumption and waste.
Chemical waste from semiconductor processing requires specialized handling neutralization and disposal due to hazardous properties.
Named entity recognition for chemistry.
Chilled water optimization adjusts setpoints and flow to minimize chiller and pump energy.
Models trained optimally.
Optimal data-to-parameter ratio.
Optimal compute allocation: scale model and data equally.
DeepMind's efficiently trained language model.
Circuit breakers halt requests to failing services allowing recovery time.
Find functional circuits in networks.
Circular economy approaches maximize material reuse refurbishment and recycling minimizing waste in semiconductor manufacturing lifecycle.
Analyze legal citation networks.
Identify factual claims in text.
System for identifying check-worthy claims.
Weight classes by inverse frequency.
AI planning using STRIPS or PDDL.
Use classifier gradients to guide generation.
Guidance without separate classifier.
Classifier-free guidance steers diffusion models using conditional and unconditional score estimates.
Control generation strength by mixing conditional and unconditional predictions.
Anthropic's vision-capable model.
Anthropic's helpful honest and harmless AI assistant.
Identify key clauses in contracts.
Poison with correctly labeled examples.
Cleanroom HVAC systems maintain temperature humidity and cleanliness requiring significant energy for air circulation and filtration.
# Semiconductor Manufacturing Cleanroom: Mathematical Modeling ## 1. Cleanroom Environment Modeling ### 1.1 Particle Dynamics The particle concentration in a cleanroom follows the **continuity equation**: $$ \frac{\partial C}{\partial t} + \nabla \cdot (C\vec{v}) = S - \lambda C $$ **Variable Definitions:** - $C$ — Particle concentration (particles/m³) - $\vec{v}$ — Air velocity vector (m/s) - $S$ — Source term / generation rate (particles/m³·s) - $\lambda$ — Removal rate coefficient (1/s) - $t$ — Time (s) **Particle Settling Velocity (Stokes' Law):** $$ v_s = \frac{\rho_p d_p^2 g C_c}{18 \mu} $$ - $\rho_p$ — Particle density (kg/m³) - $d_p$ — Particle diameter (m) - $g$ — Gravitational acceleration (9.81 m/s²) - $C_c$ — Cunningham slip correction factor - $\mu$ — Dynamic viscosity of air (Pa·s) **Cunningham Slip Correction Factor:** $$ C_c = 1 + \frac{\lambda_m}{d_p}\left[2.34 + 1.05 \exp\left(-0.39 \frac{d_p}{\lambda_m}\right)\right] $$ - $\lambda_m$ — Mean free path of air molecules (~65 nm at STP) ### 1.2 Airflow Modeling Cleanroom airflow is governed by the **Navier-Stokes equations**: $$ \rho\left(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla\vec{v}\right) = -\nabla p + \mu \nabla^2 \vec{v} + \vec{f} $$ **Variable Definitions:** - $\rho$ — Air density (kg/m³) - $\vec{v}$ — Velocity vector (m/s) - $p$ — Pressure (Pa) - $\mu$ — Dynamic viscosity (Pa·s) - $\vec{f}$ — Body forces (N/m³) **Continuity Equation (Incompressible Flow):** $$ \nabla \cdot \vec{v} = 0 $$ **Reynolds Number (Flow Regime Characterization):** $$ Re = \frac{\rho v L}{\mu} $$ - $L$ — Characteristic length (m) - $Re < 2300$ — Laminar flow (desired in cleanrooms) - $Re > 4000$ — Turbulent flow ### 1.3 Filtration Efficiency **Overall Filter Penetration:** $$ P = P_{\text{diffusion}} + P_{\text{interception}} + P_{\text{impaction}} $$ **Diffusion Mechanism (Small Particles < 0.1 µm):** $$ \eta_D = 2.7 \cdot Pe^{-2/3} $$ - $Pe = \frac{v \cdot d_f}{D}$ — Péclet number - $D = \frac{k_B T C_c}{3 \pi \mu d_p}$ — Particle diffusion coefficient - $d_f$ — Filter fiber diameter **Interception Mechanism:** $$ \eta_R = 0.6 \cdot \frac{\alpha}{Ku} \cdot \left(\frac{d_p}{d_f}\right)^2 $$ - $\alpha$ — Fiber volume fraction (solidity) - $Ku$ — Kuwabara hydrodynamic factor **HEPA/ULPA Efficiency Classification:** | Class | Efficiency | MPPS Range | |:------|:-----------|:-----------| | HEPA H13 | ≥ 99.95% | 0.1–0.3 µm | | HEPA H14 | ≥ 99.995% | 0.1–0.3 µm | | ULPA U15 | ≥ 99.9995% | 0.1–0.2 µm | | ULPA U16 | ≥ 99.99995% | 0.1–0.2 µm | ### 1.4 Temperature and Humidity Control **Heat Transfer Equation:** $$ \rho c_p \frac{\partial T}{\partial t} = k \nabla^2 T + \dot{q} $$ - $c_p$ — Specific heat capacity (J/kg·K) - $k$ — Thermal conductivity (W/m·K) - $\dot{q}$ — Volumetric heat generation (W/m³) **Psychrometric Relations (Humidity):** $$ \omega = 0.622 \cdot \frac{p_v}{p_{atm} - p_v} $$ - $\omega$ — Humidity ratio (kg water/kg dry air) - $p_v$ — Partial pressure of water vapor (Pa) - $p_{atm}$ — Atmospheric pressure (Pa) **Relative Humidity:** $$ RH = \frac{p_v}{p_{sat}(T)} \times 100\% $$ - $p_{sat}(T)$ — Saturation vapor pressure at temperature $T$ ## 2. Process Equipment Mathematics ### 2.1 Lithography #### 2.1.1 Aerial Image Formation **Hopkins Equation (Partially Coherent Imaging):** $$ I(x,y) = \left|\iint TCC(f_1, f_2; f_1', f_2') \cdot M(f_1, f_2) \cdot M^*(f_1', f_2') \, df_1 \, df_2 \, df_1' \, df_2'\right| $$ - $I(x,y)$ — Aerial image intensity - $TCC$ — Transmission Cross Coefficient - $M$ — Mask transmission function (Fourier domain) - $M^*$ — Complex conjugate of mask function #### 2.1.2 Resolution Limits **Rayleigh Criterion:** $$ R = k_1 \cdot \frac{\lambda}{NA} $$ - $R$ — Minimum resolvable feature (m) - $k_1$ — Process factor (0.25 – 0.8) - $\lambda$ — Exposure wavelength (m) - $NA$ — Numerical aperture **Depth of Focus:** $$ DOF = k_2 \cdot \frac{\lambda}{NA^2} $$ - $k_2$ — Process factor (~0.5 – 1.0) #### 2.1.3 Exposure Dose **Mack Model (Resist Response):** $$ E_{eff} = E_0 \cdot \exp\left(-\alpha z\right) \cdot \left[1 + r \cdot \exp\left(-2\alpha(D-z)\right)\right] $$ - $E_0$ — Incident dose (mJ/cm²) - $\alpha$ — Absorption coefficient (1/µm) - $z$ — Depth in resist - $r$ — Substrate reflectivity - $D$ — Resist thickness **Critical Dimension (CD) Sensitivity:** $$ \frac{\Delta CD}{CD} = \frac{1}{\gamma} \cdot \frac{\Delta E}{E} $$ - $\gamma$ — Resist contrast ### 2.2 Chemical Vapor Deposition (CVD) #### 2.2.1 Film Growth Rate **Surface Reaction Limited:** $$ R = k_s \cdot C_s $$ **Mass Transport Limited:** $$ R = h_g \cdot (C_g - C_s) $$ **Combined (Grove Model):** $$ R = \frac{k_s \cdot C_g}{1 + \frac{k_s}{h_g}} $$ - $R$ — Deposition rate (nm/min) - $k_s$ — Surface reaction rate constant (m/s) - $h_g$ — Gas-phase mass transfer coefficient (m/s) - $C_g$ — Bulk gas concentration (mol/m³) - $C_s$ — Surface concentration (mol/m³) #### 2.2.2 Arrhenius Temperature Dependence $$ k_s = A \cdot \exp\left(-\frac{E_a}{k_B T}\right) $$ - $A$ — Pre-exponential factor - $E_a$ — Activation energy (eV or J) - $k_B$ — Boltzmann constant ($1.38 \times 10^{-23}$ J/K) - $T$ — Temperature (K) #### 2.2.3 Step Coverage **Conformality Factor:** $$ SC = \frac{t_{sidewall}}{t_{top}} \times 100\% $$ **Aspect Ratio Dependence:** $$ SC \approx \frac{1}{1 + \beta \cdot AR} $$ - $AR$ — Aspect ratio (depth/width) - $\beta$ — Process-dependent constant ### 2.3 Physical Vapor Deposition (PVD) #### 2.3.1 Sputtering Yield **Sigmund Formula:** $$ Y = \frac{3\alpha}{4\pi^2} \cdot \frac{4 M_1 M_2}{(M_1 + M_2)^2} \cdot \frac{E}{U_s} $$ - $Y$ — Sputtering yield (atoms/ion) - $M_1, M_2$ — Ion and target atomic masses - $E$ — Ion energy (eV) - $U_s$ — Surface binding energy (eV) - $\alpha$ — Momentum transfer efficiency factor #### 2.3.2 Deposition Rate $$ R_{dep} = \frac{J \cdot Y \cdot M_{target}}{N_A \cdot \rho_{film} \cdot A} $$ - $J$ — Ion current density (ions/m²·s) - $M_{target}$ — Target molar mass (g/mol) - $N_A$ — Avogadro's number - $\rho_{film}$ — Film density (g/cm³) - $A$ — Deposition area (m²) ### 2.4 Plasma Etching #### 2.4.1 Etch Rate **Arrhenius Form:** $$ ER = A \cdot [F]^n \cdot \exp\left(-\frac{E_a}{k_B T}\right) $$ - $ER$ — Etch rate (nm/min) - $[F]$ — Etchant species concentration - $n$ — Reaction order - $E_a$ — Activation energy - $T$ — Wafer temperature (K) #### 2.4.2 Ion Energy Distribution **Maxwell-Boltzmann (Thermal Ions):** $$ f(E) = \frac{2\pi}{(\pi k_B T_e)^{3/2}} \cdot \sqrt{E} \cdot \exp\left(-\frac{E}{k_B T_e}\right) $$ - $T_e$ — Electron temperature (eV or K) #### 2.4.3 Selectivity $$ S = \frac{ER_{target}}{ER_{mask}} $$ #### 2.4.4 Anisotropy $$ A_f = 1 - \frac{ER_{lateral}}{ER_{vertical}} $$ - $A_f = 1$ — Perfectly anisotropic - $A_f = 0$ — Isotropic ### 2.5 Ion Implantation #### 2.5.1 Range Distribution (Gaussian Approximation) $$ N(x) = \frac{\Phi}{\sqrt{2\pi} \Delta R_p} \cdot \exp\left[-\frac{(x - R_p)^2}{2 \Delta R_p^2}\right] $$ - $N(x)$ — Dopant concentration at depth $x$ (atoms/cm³) - $\Phi$ — Implant dose (atoms/cm²) - $R_p$ — Projected range (nm) - $\Delta R_p$ — Range straggle (nm) #### 2.5.2 Projected Range (LSS Theory) $$ R_p \approx \frac{E}{S_n(E) + S_e(E)} $$ - $S_n(E)$ — Nuclear stopping power - $S_e(E)$ — Electronic stopping power #### 2.5.3 Channeling Effect $$ \psi_c = \sqrt{\frac{2 Z_1 Z_2 e^2}{4\pi \epsilon_0 E d}} $$ - $\psi_c$ — Critical channeling angle (rad) - $Z_1, Z_2$ — Atomic numbers of ion and target - $d$ — Interplanar spacing ### 2.6 Chemical Mechanical Planarization (CMP) #### 2.6.1 Preston Equation $$ RR = K_p \cdot P \cdot V $$ - $RR$ — Removal rate (nm/min) - $K_p$ — Preston coefficient (m²/N) - $P$ — Applied pressure (Pa) - $V$ — Relative velocity (m/s) #### 2.6.2 Contact Mechanics (Hertzian) $$ P_{contact} = \frac{4E^*}{3\pi} \cdot \sqrt{\frac{a}{R}} $$ - $E^*$ — Effective elastic modulus - $a$ — Contact radius - $R$ — Particle radius #### 2.6.3 Planarization Efficiency $$ PE = \frac{Step_{initial} - Step_{final}}{Step_{initial}} \times 100\% $$ ## 3. Metrology Mathematics ### 3.1 Scatterometry (OCD) #### 3.1.1 Rigorous Coupled-Wave Analysis (RCWA) **Maxwell's Equations:** $$ \nabla \times \vec{E} = -\mu_0 \frac{\partial \vec{H}}{\partial t} $$ $$ \nabla \times \vec{H} = \epsilon \frac{\partial \vec{E}}{\partial t} $$ **Fourier Expansion of Permittivity:** $$ \epsilon(x) = \sum_{m=-\infty}^{\infty} \epsilon_m \exp\left(i \frac{2\pi m}{\Lambda} x\right) $$ - $\Lambda$ — Grating period #### 3.1.2 Diffraction Efficiency $$ DE_m = \frac{I_m}{I_0} = |r_m|^2 $$ - $DE_m$ — Diffraction efficiency of $m$-th order - $r_m$ — Complex reflection coefficient ### 3.2 Ellipsometry #### 3.2.1 Fundamental Equation $$ \rho = \tan(\Psi) \cdot e^{i\Delta} = \frac{r_p}{r_s} $$ - $\Psi$ — Amplitude ratio angle - $\Delta$ — Phase difference - $r_p, r_s$ — Complex reflection coefficients (p and s polarizations) #### 3.2.2 Film Thickness (Single Layer) $$ d = \frac{\lambda}{4\pi n_1 \cos\theta_1} \cdot \left(m\pi + \phi\right) $$ - $d$ — Film thickness (nm) - $n_1$ — Film refractive index - $\theta_1$ — Refraction angle in film - $m$ — Interference order - $\phi$ — Phase shift from interfaces #### 3.2.3 Fresnel Coefficients $$ r_p = \frac{n_2 \cos\theta_1 - n_1 \cos\theta_2}{n_2 \cos\theta_1 + n_1 \cos\theta_2} $$ $$ r_s = \frac{n_1 \cos\theta_1 - n_2 \cos\theta_2}{n_1 \cos\theta_1 + n_2 \cos\theta_2} $$ ### 3.3 Atomic Force Microscopy (AFM) #### 3.3.1 Cantilever Dynamics **Simple Harmonic Oscillator:** $$ m \frac{d^2 z}{dt^2} + \gamma \frac{dz}{dt} + k z = F_{tip-sample} $$ - $m$ — Effective mass - $\gamma$ — Damping coefficient - $k$ — Spring constant (N/m) - $F_{tip-sample}$ — Tip-sample interaction force #### 3.3.2 Resonance Frequency $$ f_0 = \frac{1}{2\pi} \sqrt{\frac{k}{m_{eff}}} $$ #### 3.3.3 Tip-Sample Forces (Lennard-Jones) $$ F(r) = \frac{A}{r^{13}} - \frac{B}{r^7} $$ - $A, B$ — Material-dependent constants - $r$ — Tip-sample separation ### 3.4 Statistical Process Control (SPC) #### 3.4.1 Process Capability Index $$ C_p = \frac{USL - LSL}{6\sigma} $$ $$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$ - $USL$ — Upper specification limit - $LSL$ — Lower specification limit - $\mu$ — Process mean - $\sigma$ — Process standard deviation #### 3.4.2 Control Limits $$ UCL = \bar{X} + 3\sigma $$ $$ LCL = \bar{X} - 3\sigma $$ #### 3.4.3 Six Sigma Metrics $$ DPMO = \frac{Number\ of\ Defects}{Number\ of\ Opportunities} \times 10^6 $$ **Sigma Level Conversion:** | Sigma Level | DPMO | Yield | |:------------|:-----|:------| | 3σ | 66,807 | 93.32% | | 4σ | 6,210 | 99.38% | | 5σ | 233 | 99.977% | | 6σ | 3.4 | 99.99966% | ## 4. Facility Modeling ### 4.1 Thermal Management #### 4.1.1 Heat Balance $$ \dot{Q}_{in} = \dot{Q}_{process} + \dot{Q}_{losses} + mc_p\frac{dT}{dt} $$ - $\dot{Q}_{in}$ — Heat input rate (W) - $\dot{Q}_{process}$ — Process heat load (W) - $\dot{Q}_{losses}$ — Heat losses (W) - $m$ — Thermal mass (kg) - $c_p$ — Specific heat (J/kg·K) #### 4.1.2 Thermal Resistance Network $$ R_{th} = \frac{\Delta T}{\dot{Q}} = \frac{L}{kA} $$ - $R_{th}$ — Thermal resistance (K/W) - $L$ — Conduction path length (m) - $k$ — Thermal conductivity (W/m·K) - $A$ — Cross-sectional area (m²) #### 4.1.3 Cooling Capacity $$ \dot{Q}_{cooling} = \dot{m} \cdot c_p \cdot \Delta T $$ - $\dot{m}$ — Mass flow rate (kg/s) - $\Delta T$ — Temperature difference (K) ### 4.2 Vibration Isolation #### 4.2.1 Transmissibility $$ T = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} $$ - $T$ — Transmissibility ratio - $r = \frac{\omega}{\omega_n}$ — Frequency ratio - $\zeta$ — Damping ratio - $\omega$ — Excitation frequency (rad/s) - $\omega_n$ — Natural frequency (rad/s) #### 4.2.2 Natural Frequency $$ \omega_n = \sqrt{\frac{k}{m}} = 2\pi f_n $$ $$ f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}} $$ #### 4.2.3 Isolation Efficiency $$ IE = \left(1 - T\right) \times 100\% $$ **Design Rule:** For effective isolation, $r > \sqrt{2}$ (frequency ratio > 1.414) ### 4.3 Ultra-Pure Water (UPW) Systems #### 4.3.1 Resistivity $$ \rho = \frac{1}{\sigma} = \frac{1}{\sum_i \lambda_i c_i} $$ - $\rho$ — Resistivity (Ω·cm) - $\sigma$ — Conductivity (S/cm) - $\lambda_i$ — Ionic equivalent conductance (S·cm²/mol) - $c_i$ — Ion concentration (mol/cm³) **Target Specification:** 18.2 MΩ·cm at 25°C (theoretical maximum for pure water) #### 4.3.2 Total Organic Carbon (TOC) $$ TOC = \frac{\Delta CO_2 \times 12}{44 \times V_{sample}} $$ - $\Delta CO_2$ — CO₂ generated from oxidation (µg) - $V_{sample}$ — Sample volume (L) - Target: < 1 ppb for advanced nodes #### 4.3.3 Particle Concentration $$ N = \frac{Counts}{V_{sampled} \times Efficiency} $$ - Specification: < 1 particle/mL at ≥ 50 nm ### 4.4 Gas Delivery Systems #### 4.4.1 Mass Flow Rate $$ \dot{m} = \rho \cdot Q = \frac{P \cdot Q \cdot M}{R \cdot T} $$ - $\dot{m}$ — Mass flow rate (kg/s) - $Q$ — Volumetric flow rate (m³/s) - $P$ — Pressure (Pa) - $M$ — Molar mass (kg/mol) - $R$ — Universal gas constant (8.314 J/mol·K) #### 4.4.2 Pressure Drop (Hagen-Poiseuille) $$ \Delta P = \frac{128 \mu L Q}{\pi d^4} $$ - $L$ — Pipe length (m) - $d$ — Pipe diameter (m) - $\mu$ — Dynamic viscosity (Pa·s) #### 4.4.3 Gas Purity $$ Purity = \left(1 - \frac{\sum Impurities}{Total}\right) \times 100\% $$ - Typical requirement: 99.9999% (6N) to 99.99999999% (10N) ## 5. Yield Modeling ### 5.1 Defect-Limited Yield #### 5.1.1 Poisson Model (Random Defects) $$ Y = e^{-D_0 \cdot A} $$ - $Y$ — Die yield (0 to 1) - $D_0$ — Defect density (defects/cm²) - $A$ — Die area (cm²) #### 5.1.2 Negative Binomial (Clustered Defects) $$ Y = \left(1 + \frac{D_0 \cdot A}{\alpha}\right)^{-\alpha} $$ - $\alpha$ — Clustering parameter (α → ∞ approaches Poisson) #### 5.1.3 Murphy's Model $$ Y = \left(\frac{1 - e^{-D_0 A}}{D_0 A}\right)^2 $$ #### 5.1.4 Seeds Model $$ Y = e^{-\sqrt{D_0 A}} $$ ### 5.2 Parametric Yield #### 5.2.1 Gaussian Distribution Model $$ Y_p = \Phi\left(\frac{USL - \mu}{\sigma}\right) - \Phi\left(\frac{LSL - \mu}{\sigma}\right) $$ - $\Phi$ — Cumulative standard normal distribution function #### 5.2.2 Combined Yield $$ Y_{total} = Y_{defect} \times Y_{parametric} \times Y_{packaging} $$ #### 5.2.3 Learning Curve $$ D_0(t) = D_{0,initial} \cdot \left(\frac{V(t)}{V_0}\right)^{-\beta} $$ - $V(t)$ — Cumulative production volume - $\beta$ — Learning rate exponent (typically 0.3–0.5) ## 6. Reference Tables ### 6.1 Process Equations Quick Reference | **Domain** | **Key Equation** | **Primary Variables** | |:-----------|:-----------------|:----------------------| | Cleanroom Particles | $\frac{\partial C}{\partial t} + \nabla \cdot (C\vec{v}) = S - \lambda C$ | $C$, $\vec{v}$, $S$, $\lambda$ | | Airflow | $\rho(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla\vec{v}) = -\nabla p + \mu \nabla^2 \vec{v}$ | $\rho$, $\vec{v}$, $p$, $\mu$ | | Lithography CD | $R = k_1 \frac{\lambda}{NA}$ | $k_1$, $\lambda$, $NA$ | | CVD Growth | $R = \frac{k_s C_g}{1 + k_s/h_g}$ | $k_s$, $C_g$, $h_g$ | | Etch Rate | $ER = A[F]^n \exp(-E_a/k_B T)$ | $[F]$, $E_a$, $T$ | | CMP | $RR = K_p \cdot P \cdot V$ | $K_p$, $P$, $V$ | | Ellipsometry | $\rho = \tan(\Psi) e^{i\Delta}$ | $\Psi$, $\Delta$, $r_p$, $r_s$ | | Process Capability | $C_{pk} = \min(\frac{USL-\mu}{3\sigma}, \frac{\mu-LSL}{3\sigma})$ | $USL$, $LSL$, $\mu$, $\sigma$ | | Yield (Poisson) | $Y = e^{-D_0 A}$ | $D_0$, $A$ | | Vibration | $T = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$ | $r$, $\zeta$ | ### 6.2 Physical Constants | **Constant** | **Symbol** | **Value** | **Units** | |:-------------|:-----------|:----------|:----------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ | J/K | | Avogadro's number | $N_A$ | $6.022 \times 10^{23}$ | mol⁻¹ | | Elementary charge | $e$ | $1.602 \times 10^{-19}$ | C | | Permittivity of vacuum | $\epsilon_0$ | $8.854 \times 10^{-12}$ | F/m | | Permeability of vacuum | $\mu_0$ | $4\pi \times 10^{-7}$ | H/m | | Gas constant | $R$ | $8.314$ | J/(mol·K) | | Planck constant | $h$ | $6.626 \times 10^{-34}$ | J·s | ### 6.3 Cleanroom Classification (ISO 14644-1) | **ISO Class** | **Max Particles ≥ 0.1 µm** | **Max Particles ≥ 0.5 µm** | **Typical Application** | |:--------------|:---------------------------|:---------------------------|:------------------------| | ISO 1 | 10 | — | Research, EUV | | ISO 2 | 100 | — | Advanced lithography | | ISO 3 | 1,000 | 35 | Leading-edge fabs | | ISO 4 | 10,000 | 352 | Advanced manufacturing | | ISO 5 | 100,000 | 3,520 | Standard IC production | | ISO 6 | 1,000,000 | 35,200 | Assembly, packaging | *Units: particles/m³* ### Math Syntax Reference | **Type** | **Syntax** | **Example** | |:---------|:-----------|:------------| | Inline math | `$...$` | `$E = mc^2$` → $E = mc^2$ | | Display math | `$$...$$` | `$$\int_0^\infty e^{-x}dx$$` | | Fractions | `\frac{a}{b}` | $\frac{a}{b}$ | | Subscript | `x_i` | $x_i$ | | Superscript | `x^2` | $x^2$ | | Greek letters | `\alpha, \beta, \gamma` | $\alpha, \beta, \gamma$ | | Partial derivative | `\frac{\partial f}{\partial x}` | $\frac{\partial f}{\partial x}$ | | Vectors | `\vec{v}` | $\vec{v}$ | | Matrices | `\begin{bmatrix}...\end{bmatrix}` | — |
# Semiconductor Manufacturing Cleanroom: Mathematical Modeling ## 1. Cleanroom Environment Modeling ### 1.1 Particle Dynamics The particle concentration in a cleanroom follows the **continuity equation**: $$ \frac{\partial C}{\partial t} + \nabla \cdot (C\vec{v}) = S - \lambda C $$ **Variable Definitions:** - $C$ — Particle concentration (particles/m³) - $\vec{v}$ — Air velocity vector (m/s) - $S$ — Source term / generation rate (particles/m³·s) - $\lambda$ — Removal rate coefficient (1/s) - $t$ — Time (s) **Particle Settling Velocity (Stokes' Law):** $$ v_s = \frac{\rho_p d_p^2 g C_c}{18 \mu} $$ - $\rho_p$ — Particle density (kg/m³) - $d_p$ — Particle diameter (m) - $g$ — Gravitational acceleration (9.81 m/s²) - $C_c$ — Cunningham slip correction factor - $\mu$ — Dynamic viscosity of air (Pa·s) **Cunningham Slip Correction Factor:** $$ C_c = 1 + \frac{\lambda_m}{d_p}\left[2.34 + 1.05 \exp\left(-0.39 \frac{d_p}{\lambda_m}\right)\right] $$ - $\lambda_m$ — Mean free path of air molecules (~65 nm at STP) ### 1.2 Airflow Modeling Cleanroom airflow is governed by the **Navier-Stokes equations**: $$ \rho\left(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla\vec{v}\right) = -\nabla p + \mu \nabla^2 \vec{v} + \vec{f} $$ **Variable Definitions:** - $\rho$ — Air density (kg/m³) - $\vec{v}$ — Velocity vector (m/s) - $p$ — Pressure (Pa) - $\mu$ — Dynamic viscosity (Pa·s) - $\vec{f}$ — Body forces (N/m³) **Continuity Equation (Incompressible Flow):** $$ \nabla \cdot \vec{v} = 0 $$ **Reynolds Number (Flow Regime Characterization):** $$ Re = \frac{\rho v L}{\mu} $$ - $L$ — Characteristic length (m) - $Re < 2300$ — Laminar flow (desired in cleanrooms) - $Re > 4000$ — Turbulent flow ### 1.3 Filtration Efficiency **Overall Filter Penetration:** $$ P = P_{\text{diffusion}} + P_{\text{interception}} + P_{\text{impaction}} $$ **Diffusion Mechanism (Small Particles < 0.1 µm):** $$ \eta_D = 2.7 \cdot Pe^{-2/3} $$ - $Pe = \frac{v \cdot d_f}{D}$ — Péclet number - $D = \frac{k_B T C_c}{3 \pi \mu d_p}$ — Particle diffusion coefficient - $d_f$ — Filter fiber diameter **Interception Mechanism:** $$ \eta_R = 0.6 \cdot \frac{\alpha}{Ku} \cdot \left(\frac{d_p}{d_f}\right)^2 $$ - $\alpha$ — Fiber volume fraction (solidity) - $Ku$ — Kuwabara hydrodynamic factor **HEPA/ULPA Efficiency Classification:** | Class | Efficiency | MPPS Range | |:------|:-----------|:-----------| | HEPA H13 | ≥ 99.95% | 0.1–0.3 µm | | HEPA H14 | ≥ 99.995% | 0.1–0.3 µm | | ULPA U15 | ≥ 99.9995% | 0.1–0.2 µm | | ULPA U16 | ≥ 99.99995% | 0.1–0.2 µm | ### 1.4 Temperature and Humidity Control **Heat Transfer Equation:** $$ \rho c_p \frac{\partial T}{\partial t} = k \nabla^2 T + \dot{q} $$ - $c_p$ — Specific heat capacity (J/kg·K) - $k$ — Thermal conductivity (W/m·K) - $\dot{q}$ — Volumetric heat generation (W/m³) **Psychrometric Relations (Humidity):** $$ \omega = 0.622 \cdot \frac{p_v}{p_{atm} - p_v} $$ - $\omega$ — Humidity ratio (kg water/kg dry air) - $p_v$ — Partial pressure of water vapor (Pa) - $p_{atm}$ — Atmospheric pressure (Pa) **Relative Humidity:** $$ RH = \frac{p_v}{p_{sat}(T)} \times 100\% $$ - $p_{sat}(T)$ — Saturation vapor pressure at temperature $T$ ## 2. Process Equipment Mathematics ### 2.1 Lithography #### 2.1.1 Aerial Image Formation **Hopkins Equation (Partially Coherent Imaging):** $$ I(x,y) = \left|\iint TCC(f_1, f_2; f_1', f_2') \cdot M(f_1, f_2) \cdot M^*(f_1', f_2') \, df_1 \, df_2 \, df_1' \, df_2'\right| $$ - $I(x,y)$ — Aerial image intensity - $TCC$ — Transmission Cross Coefficient - $M$ — Mask transmission function (Fourier domain) - $M^*$ — Complex conjugate of mask function #### 2.1.2 Resolution Limits **Rayleigh Criterion:** $$ R = k_1 \cdot \frac{\lambda}{NA} $$ - $R$ — Minimum resolvable feature (m) - $k_1$ — Process factor (0.25 – 0.8) - $\lambda$ — Exposure wavelength (m) - $NA$ — Numerical aperture **Depth of Focus:** $$ DOF = k_2 \cdot \frac{\lambda}{NA^2} $$ - $k_2$ — Process factor (~0.5 – 1.0) #### 2.1.3 Exposure Dose **Mack Model (Resist Response):** $$ E_{eff} = E_0 \cdot \exp\left(-\alpha z\right) \cdot \left[1 + r \cdot \exp\left(-2\alpha(D-z)\right)\right] $$ - $E_0$ — Incident dose (mJ/cm²) - $\alpha$ — Absorption coefficient (1/µm) - $z$ — Depth in resist - $r$ — Substrate reflectivity - $D$ — Resist thickness **Critical Dimension (CD) Sensitivity:** $$ \frac{\Delta CD}{CD} = \frac{1}{\gamma} \cdot \frac{\Delta E}{E} $$ - $\gamma$ — Resist contrast ### 2.2 Chemical Vapor Deposition (CVD) #### 2.2.1 Film Growth Rate **Surface Reaction Limited:** $$ R = k_s \cdot C_s $$ **Mass Transport Limited:** $$ R = h_g \cdot (C_g - C_s) $$ **Combined (Grove Model):** $$ R = \frac{k_s \cdot C_g}{1 + \frac{k_s}{h_g}} $$ - $R$ — Deposition rate (nm/min) - $k_s$ — Surface reaction rate constant (m/s) - $h_g$ — Gas-phase mass transfer coefficient (m/s) - $C_g$ — Bulk gas concentration (mol/m³) - $C_s$ — Surface concentration (mol/m³) #### 2.2.2 Arrhenius Temperature Dependence $$ k_s = A \cdot \exp\left(-\frac{E_a}{k_B T}\right) $$ - $A$ — Pre-exponential factor - $E_a$ — Activation energy (eV or J) - $k_B$ — Boltzmann constant ($1.38 \times 10^{-23}$ J/K) - $T$ — Temperature (K) #### 2.2.3 Step Coverage **Conformality Factor:** $$ SC = \frac{t_{sidewall}}{t_{top}} \times 100\% $$ **Aspect Ratio Dependence:** $$ SC \approx \frac{1}{1 + \beta \cdot AR} $$ - $AR$ — Aspect ratio (depth/width) - $\beta$ — Process-dependent constant ### 2.3 Physical Vapor Deposition (PVD) #### 2.3.1 Sputtering Yield **Sigmund Formula:** $$ Y = \frac{3\alpha}{4\pi^2} \cdot \frac{4 M_1 M_2}{(M_1 + M_2)^2} \cdot \frac{E}{U_s} $$ - $Y$ — Sputtering yield (atoms/ion) - $M_1, M_2$ — Ion and target atomic masses - $E$ — Ion energy (eV) - $U_s$ — Surface binding energy (eV) - $\alpha$ — Momentum transfer efficiency factor #### 2.3.2 Deposition Rate $$ R_{dep} = \frac{J \cdot Y \cdot M_{target}}{N_A \cdot \rho_{film} \cdot A} $$ - $J$ — Ion current density (ions/m²·s) - $M_{target}$ — Target molar mass (g/mol) - $N_A$ — Avogadro's number - $\rho_{film}$ — Film density (g/cm³) - $A$ — Deposition area (m²) ### 2.4 Plasma Etching #### 2.4.1 Etch Rate **Arrhenius Form:** $$ ER = A \cdot [F]^n \cdot \exp\left(-\frac{E_a}{k_B T}\right) $$ - $ER$ — Etch rate (nm/min) - $[F]$ — Etchant species concentration - $n$ — Reaction order - $E_a$ — Activation energy - $T$ — Wafer temperature (K) #### 2.4.2 Ion Energy Distribution **Maxwell-Boltzmann (Thermal Ions):** $$ f(E) = \frac{2\pi}{(\pi k_B T_e)^{3/2}} \cdot \sqrt{E} \cdot \exp\left(-\frac{E}{k_B T_e}\right) $$ - $T_e$ — Electron temperature (eV or K) #### 2.4.3 Selectivity $$ S = \frac{ER_{target}}{ER_{mask}} $$ #### 2.4.4 Anisotropy $$ A_f = 1 - \frac{ER_{lateral}}{ER_{vertical}} $$ - $A_f = 1$ — Perfectly anisotropic - $A_f = 0$ — Isotropic ### 2.5 Ion Implantation #### 2.5.1 Range Distribution (Gaussian Approximation) $$ N(x) = \frac{\Phi}{\sqrt{2\pi} \Delta R_p} \cdot \exp\left[-\frac{(x - R_p)^2}{2 \Delta R_p^2}\right] $$ - $N(x)$ — Dopant concentration at depth $x$ (atoms/cm³) - $\Phi$ — Implant dose (atoms/cm²) - $R_p$ — Projected range (nm) - $\Delta R_p$ — Range straggle (nm) #### 2.5.2 Projected Range (LSS Theory) $$ R_p \approx \frac{E}{S_n(E) + S_e(E)} $$ - $S_n(E)$ — Nuclear stopping power - $S_e(E)$ — Electronic stopping power #### 2.5.3 Channeling Effect $$ \psi_c = \sqrt{\frac{2 Z_1 Z_2 e^2}{4\pi \epsilon_0 E d}} $$ - $\psi_c$ — Critical channeling angle (rad) - $Z_1, Z_2$ — Atomic numbers of ion and target - $d$ — Interplanar spacing ### 2.6 Chemical Mechanical Planarization (CMP) #### 2.6.1 Preston Equation $$ RR = K_p \cdot P \cdot V $$ - $RR$ — Removal rate (nm/min) - $K_p$ — Preston coefficient (m²/N) - $P$ — Applied pressure (Pa) - $V$ — Relative velocity (m/s) #### 2.6.2 Contact Mechanics (Hertzian) $$ P_{contact} = \frac{4E^*}{3\pi} \cdot \sqrt{\frac{a}{R}} $$ - $E^*$ — Effective elastic modulus - $a$ — Contact radius - $R$ — Particle radius #### 2.6.3 Planarization Efficiency $$ PE = \frac{Step_{initial} - Step_{final}}{Step_{initial}} \times 100\% $$ ## 3. Metrology Mathematics ### 3.1 Scatterometry (OCD) #### 3.1.1 Rigorous Coupled-Wave Analysis (RCWA) **Maxwell's Equations:** $$ \nabla \times \vec{E} = -\mu_0 \frac{\partial \vec{H}}{\partial t} $$ $$ \nabla \times \vec{H} = \epsilon \frac{\partial \vec{E}}{\partial t} $$ **Fourier Expansion of Permittivity:** $$ \epsilon(x) = \sum_{m=-\infty}^{\infty} \epsilon_m \exp\left(i \frac{2\pi m}{\Lambda} x\right) $$ - $\Lambda$ — Grating period #### 3.1.2 Diffraction Efficiency $$ DE_m = \frac{I_m}{I_0} = |r_m|^2 $$ - $DE_m$ — Diffraction efficiency of $m$-th order - $r_m$ — Complex reflection coefficient ### 3.2 Ellipsometry #### 3.2.1 Fundamental Equation $$ \rho = \tan(\Psi) \cdot e^{i\Delta} = \frac{r_p}{r_s} $$ - $\Psi$ — Amplitude ratio angle - $\Delta$ — Phase difference - $r_p, r_s$ — Complex reflection coefficients (p and s polarizations) #### 3.2.2 Film Thickness (Single Layer) $$ d = \frac{\lambda}{4\pi n_1 \cos\theta_1} \cdot \left(m\pi + \phi\right) $$ - $d$ — Film thickness (nm) - $n_1$ — Film refractive index - $\theta_1$ — Refraction angle in film - $m$ — Interference order - $\phi$ — Phase shift from interfaces #### 3.2.3 Fresnel Coefficients $$ r_p = \frac{n_2 \cos\theta_1 - n_1 \cos\theta_2}{n_2 \cos\theta_1 + n_1 \cos\theta_2} $$ $$ r_s = \frac{n_1 \cos\theta_1 - n_2 \cos\theta_2}{n_1 \cos\theta_1 + n_2 \cos\theta_2} $$ ### 3.3 Atomic Force Microscopy (AFM) #### 3.3.1 Cantilever Dynamics **Simple Harmonic Oscillator:** $$ m \frac{d^2 z}{dt^2} + \gamma \frac{dz}{dt} + k z = F_{tip-sample} $$ - $m$ — Effective mass - $\gamma$ — Damping coefficient - $k$ — Spring constant (N/m) - $F_{tip-sample}$ — Tip-sample interaction force #### 3.3.2 Resonance Frequency $$ f_0 = \frac{1}{2\pi} \sqrt{\frac{k}{m_{eff}}} $$ #### 3.3.3 Tip-Sample Forces (Lennard-Jones) $$ F(r) = \frac{A}{r^{13}} - \frac{B}{r^7} $$ - $A, B$ — Material-dependent constants - $r$ — Tip-sample separation ### 3.4 Statistical Process Control (SPC) #### 3.4.1 Process Capability Index $$ C_p = \frac{USL - LSL}{6\sigma} $$ $$ C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right) $$ - $USL$ — Upper specification limit - $LSL$ — Lower specification limit - $\mu$ — Process mean - $\sigma$ — Process standard deviation #### 3.4.2 Control Limits $$ UCL = \bar{X} + 3\sigma $$ $$ LCL = \bar{X} - 3\sigma $$ #### 3.4.3 Six Sigma Metrics $$ DPMO = \frac{Number\ of\ Defects}{Number\ of\ Opportunities} \times 10^6 $$ **Sigma Level Conversion:** | Sigma Level | DPMO | Yield | |:------------|:-----|:------| | 3σ | 66,807 | 93.32% | | 4σ | 6,210 | 99.38% | | 5σ | 233 | 99.977% | | 6σ | 3.4 | 99.99966% | ## 4. Facility Modeling ### 4.1 Thermal Management #### 4.1.1 Heat Balance $$ \dot{Q}_{in} = \dot{Q}_{process} + \dot{Q}_{losses} + mc_p\frac{dT}{dt} $$ - $\dot{Q}_{in}$ — Heat input rate (W) - $\dot{Q}_{process}$ — Process heat load (W) - $\dot{Q}_{losses}$ — Heat losses (W) - $m$ — Thermal mass (kg) - $c_p$ — Specific heat (J/kg·K) #### 4.1.2 Thermal Resistance Network $$ R_{th} = \frac{\Delta T}{\dot{Q}} = \frac{L}{kA} $$ - $R_{th}$ — Thermal resistance (K/W) - $L$ — Conduction path length (m) - $k$ — Thermal conductivity (W/m·K) - $A$ — Cross-sectional area (m²) #### 4.1.3 Cooling Capacity $$ \dot{Q}_{cooling} = \dot{m} \cdot c_p \cdot \Delta T $$ - $\dot{m}$ — Mass flow rate (kg/s) - $\Delta T$ — Temperature difference (K) ### 4.2 Vibration Isolation #### 4.2.1 Transmissibility $$ T = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} $$ - $T$ — Transmissibility ratio - $r = \frac{\omega}{\omega_n}$ — Frequency ratio - $\zeta$ — Damping ratio - $\omega$ — Excitation frequency (rad/s) - $\omega_n$ — Natural frequency (rad/s) #### 4.2.2 Natural Frequency $$ \omega_n = \sqrt{\frac{k}{m}} = 2\pi f_n $$ $$ f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}} $$ #### 4.2.3 Isolation Efficiency $$ IE = \left(1 - T\right) \times 100\% $$ **Design Rule:** For effective isolation, $r > \sqrt{2}$ (frequency ratio > 1.414) ### 4.3 Ultra-Pure Water (UPW) Systems #### 4.3.1 Resistivity $$ \rho = \frac{1}{\sigma} = \frac{1}{\sum_i \lambda_i c_i} $$ - $\rho$ — Resistivity (Ω·cm) - $\sigma$ — Conductivity (S/cm) - $\lambda_i$ — Ionic equivalent conductance (S·cm²/mol) - $c_i$ — Ion concentration (mol/cm³) **Target Specification:** 18.2 MΩ·cm at 25°C (theoretical maximum for pure water) #### 4.3.2 Total Organic Carbon (TOC) $$ TOC = \frac{\Delta CO_2 \times 12}{44 \times V_{sample}} $$ - $\Delta CO_2$ — CO₂ generated from oxidation (µg) - $V_{sample}$ — Sample volume (L) - Target: < 1 ppb for advanced nodes #### 4.3.3 Particle Concentration $$ N = \frac{Counts}{V_{sampled} \times Efficiency} $$ - Specification: < 1 particle/mL at ≥ 50 nm ### 4.4 Gas Delivery Systems #### 4.4.1 Mass Flow Rate $$ \dot{m} = \rho \cdot Q = \frac{P \cdot Q \cdot M}{R \cdot T} $$ - $\dot{m}$ — Mass flow rate (kg/s) - $Q$ — Volumetric flow rate (m³/s) - $P$ — Pressure (Pa) - $M$ — Molar mass (kg/mol) - $R$ — Universal gas constant (8.314 J/mol·K) #### 4.4.2 Pressure Drop (Hagen-Poiseuille) $$ \Delta P = \frac{128 \mu L Q}{\pi d^4} $$ - $L$ — Pipe length (m) - $d$ — Pipe diameter (m) - $\mu$ — Dynamic viscosity (Pa·s) #### 4.4.3 Gas Purity $$ Purity = \left(1 - \frac{\sum Impurities}{Total}\right) \times 100\% $$ - Typical requirement: 99.9999% (6N) to 99.99999999% (10N) ## 5. Yield Modeling ### 5.1 Defect-Limited Yield #### 5.1.1 Poisson Model (Random Defects) $$ Y = e^{-D_0 \cdot A} $$ - $Y$ — Die yield (0 to 1) - $D_0$ — Defect density (defects/cm²) - $A$ — Die area (cm²) #### 5.1.2 Negative Binomial (Clustered Defects) $$ Y = \left(1 + \frac{D_0 \cdot A}{\alpha}\right)^{-\alpha} $$ - $\alpha$ — Clustering parameter (α → ∞ approaches Poisson) #### 5.1.3 Murphy's Model $$ Y = \left(\frac{1 - e^{-D_0 A}}{D_0 A}\right)^2 $$ #### 5.1.4 Seeds Model $$ Y = e^{-\sqrt{D_0 A}} $$ ### 5.2 Parametric Yield #### 5.2.1 Gaussian Distribution Model $$ Y_p = \Phi\left(\frac{USL - \mu}{\sigma}\right) - \Phi\left(\frac{LSL - \mu}{\sigma}\right) $$ - $\Phi$ — Cumulative standard normal distribution function #### 5.2.2 Combined Yield $$ Y_{total} = Y_{defect} \times Y_{parametric} \times Y_{packaging} $$ #### 5.2.3 Learning Curve $$ D_0(t) = D_{0,initial} \cdot \left(\frac{V(t)}{V_0}\right)^{-\beta} $$ - $V(t)$ — Cumulative production volume - $\beta$ — Learning rate exponent (typically 0.3–0.5) ## 6. Reference Tables ### 6.1 Process Equations Quick Reference | **Domain** | **Key Equation** | **Primary Variables** | |:-----------|:-----------------|:----------------------| | Cleanroom Particles | $\frac{\partial C}{\partial t} + \nabla \cdot (C\vec{v}) = S - \lambda C$ | $C$, $\vec{v}$, $S$, $\lambda$ | | Airflow | $\rho(\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla\vec{v}) = -\nabla p + \mu \nabla^2 \vec{v}$ | $\rho$, $\vec{v}$, $p$, $\mu$ | | Lithography CD | $R = k_1 \frac{\lambda}{NA}$ | $k_1$, $\lambda$, $NA$ | | CVD Growth | $R = \frac{k_s C_g}{1 + k_s/h_g}$ | $k_s$, $C_g$, $h_g$ | | Etch Rate | $ER = A[F]^n \exp(-E_a/k_B T)$ | $[F]$, $E_a$, $T$ | | CMP | $RR = K_p \cdot P \cdot V$ | $K_p$, $P$, $V$ | | Ellipsometry | $\rho = \tan(\Psi) e^{i\Delta}$ | $\Psi$, $\Delta$, $r_p$, $r_s$ | | Process Capability | $C_{pk} = \min(\frac{USL-\mu}{3\sigma}, \frac{\mu-LSL}{3\sigma})$ | $USL$, $LSL$, $\mu$, $\sigma$ | | Yield (Poisson) | $Y = e^{-D_0 A}$ | $D_0$, $A$ | | Vibration | $T = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$ | $r$, $\zeta$ | ### 6.2 Physical Constants | **Constant** | **Symbol** | **Value** | **Units** | |:-------------|:-----------|:----------|:----------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ | J/K | | Avogadro's number | $N_A$ | $6.022 \times 10^{23}$ | mol⁻¹ | | Elementary charge | $e$ | $1.602 \times 10^{-19}$ | C | | Permittivity of vacuum | $\epsilon_0$ | $8.854 \times 10^{-12}$ | F/m | | Permeability of vacuum | $\mu_0$ | $4\pi \times 10^{-7}$ | H/m | | Gas constant | $R$ | $8.314$ | J/(mol·K) | | Planck constant | $h$ | $6.626 \times 10^{-34}$ | J·s | ### 6.3 Cleanroom Classification (ISO 14644-1) | **ISO Class** | **Max Particles ≥ 0.1 µm** | **Max Particles ≥ 0.5 µm** | **Typical Application** | |:--------------|:---------------------------|:---------------------------|:------------------------| | ISO 1 | 10 | — | Research, EUV | | ISO 2 | 100 | — | Advanced lithography | | ISO 3 | 1,000 | 35 | Leading-edge fabs | | ISO 4 | 10,000 | 352 | Advanced manufacturing | | ISO 5 | 100,000 | 3,520 | Standard IC production | | ISO 6 | 1,000,000 | 35,200 | Assembly, packaging | *Units: particles/m³* ### Math Syntax Reference | **Type** | **Syntax** | **Example** | |:---------|:-----------|:------------| | Inline math | `$...$` | `$E = mc^2$` → $E = mc^2$ | | Display math | `$$...$$` | `$$\int_0^\infty e^{-x}dx$$` | | Fractions | `\frac{a}{b}` | $\frac{a}{b}$ | | Subscript | `x_i` | $x_i$ | | Superscript | `x^2` | $x^2$ | | Greek letters | `\alpha, \beta, \gamma` | $\alpha, \beta, \gamma$ | | Partial derivative | `\frac{\partial f}{\partial x}` | $\frac{\partial f}{\partial x}$ | | Vectors | `\vec{v}` | $\vec{v}$ | | Matrices | `\begin{bmatrix}...\end{bmatrix}` | — |
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