claimbuster,nlp
System for identifying check-worthy claims.
3,145 technical terms and definitions
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}` | — |
Clebsch-Gordan tensor products combine spherical harmonic representations maintaining equivariance.
Click models estimate true relevance from biased user interactions accounting for examination probability.
Create structured clinical notes.
Summarize medical records.
Remove PHI from clinical text.
Match patients to trials using NLP.
Match patients to appropriate trials.
Draft trial protocols.
Align image and text embeddings using contrastive learning.
Use CLIP scores to guide image generation.
Optimize towards CLIP similarity.
Contrastive learning for vision-language.
Use CLIP to guide image generation.
Clock domain crossings transfer signals between different clock domains requiring synchronization.
Clock uncertainty accounts for jitter and skew in timing analysis.
Tractable continuous-time neural networks with guaranteed stability.
Financial considerations for cloud training.
# Chemical Mechanical Planarization (CMP) Modeling in Semiconductor Manufacturing ## 1. Fundamentals of CMP ### 1.1 Definition and Principle Chemical Mechanical Planarization (CMP) is a hybrid process combining: - **Chemical etching**: Reactive slurry chemistry modifies surface properties - **Mechanical abrasion**: Physical removal via abrasive particles and pad The fundamental material removal can be expressed as: $$ \text{Material Removal} = f(\text{Chemical Reaction}, \text{Mechanical Abrasion}) $$ ### 1.2 Process Components | Component | Function | Key Parameters | |-----------|----------|----------------| | **Wafer** | Substrate to be planarized | Material type, pattern density | | **Polishing Pad** | Provides mechanical action | Hardness, porosity, asperity distribution | | **Slurry** | Chemical + abrasive medium | pH, oxidizer, particle size/concentration | | **Carrier** | Holds and rotates wafer | Down force, rotation speed | | **Platen** | Rotates polishing pad | Rotation speed, temperature | ### 1.3 Key Process Parameters - **Down Force ($F$)**: Pressure applied to wafer, typically $1-7$ psi - **Platen Speed ($\omega_p$)**: Pad rotation, typically $20-100$ rpm - **Carrier Speed ($\omega_c$)**: Wafer rotation, typically $20-100$ rpm - **Slurry Flow Rate ($Q$)**: Typically $100-300$ mL/min - **Temperature ($T$)**: Typically $20-50°C$ ## 2. Classical Physical Models ### 2.1 Preston Equation (Foundational Model) The foundational model for CMP is the **Preston equation** (1927): $$ \boxed{MRR = k_p \cdot P \cdot v} $$ Where: - $MRR$ = Material Removal Rate $[\text{nm/min}]$ - $k_p$ = Preston's coefficient $[\text{m}^2/\text{N}]$ - $P$ = Applied pressure $[\text{Pa}]$ - $v$ = Relative velocity $[\text{m/s}]$ The relative velocity between wafer and pad: $$ v = \sqrt{(\omega_p r_p)^2 + (\omega_c r_c)^2 - 2\omega_p \omega_c r_p r_c \cos(\theta)} $$ Where: - $\omega_p, \omega_c$ = Angular velocities of platen and carrier - $r_p, r_c$ = Radial positions - $\theta$ = Phase angle ### 2.2 Modified Preston Models #### 2.2.1 Pressure-Velocity Product Modification $$ MRR = k_p \cdot P^a \cdot v^b $$ Where $a, b$ are empirical exponents (typically $0.5 < a, b < 1.5$) #### 2.2.2 Chemical Enhancement Factor $$ MRR = k_p \cdot P \cdot v \cdot f(C, T, pH) $$ Where $f(C, T, pH)$ represents chemical effects: - $C$ = Oxidizer concentration - $T$ = Temperature - $pH$ = Slurry pH #### 2.2.3 Arrhenius-Modified Preston Equation $$ MRR = k_0 \cdot \exp\left(-\frac{E_a}{RT}\right) \cdot P \cdot v $$ Where: - $k_0$ = Pre-exponential factor - $E_a$ = Activation energy $[\text{J/mol}]$ - $R$ = Gas constant $= 8.314$ J/(mol$\cdot$K) - $T$ = Temperature $[\text{K}]$ ### 2.3 Tribocorrosion Model For metal CMP (e.g., tungsten, copper): $$ MRR = \frac{M}{z F \rho} \cdot \left( i_{corr} + \frac{Q_{pass}}{A \cdot t_{pass}} \right) \cdot f_{mech} $$ Where: - $M$ = Molar mass of metal - $z$ = Number of electrons transferred - $F$ = Faraday constant $= 96485$ C/mol - $\rho$ = Density - $i_{corr}$ = Corrosion current density - $Q_{pass}$ = Passivation charge - $f_{mech}$ = Mechanical factor ### 2.4 Contact Mode Classification | Mode | Condition | Preston Constant | Friction Coefficient | |------|-----------|------------------|---------------------| | **Contact** | $\frac{\eta v_R}{p} < (\frac{\eta v_R}{p})_c$ | High, constant | High ($\mu > 0.3$) | | **Mixed** | $\frac{\eta v_R}{p} \approx (\frac{\eta v_R}{p})_c$ | Transitional | Medium | | **Hydroplaning** | $\frac{\eta v_R}{p} > (\frac{\eta v_R}{p})_c$ | Low, variable | Low ($\mu < 0.1$) | Where: - $\eta$ = Slurry viscosity - $v_R$ = Relative velocity - $p$ = Pressure ## 3. Pattern Density Models ### 3.1 Effective Pattern Density Model (Stine Model) The local material removal rate depends on effective pattern density: $$ \frac{dz}{dt} = -\frac{K}{\rho_{eff}(x, y)} $$ Where: - $z$ = Surface height - $K$ = Blanket removal rate $= k_p \cdot P \cdot v$ - $\rho_{eff}$ = Effective pattern density #### 3.1.1 Effective Density Calculation $$ \rho_{eff}(x, y) = \iint_{-\infty}^{\infty} \rho_0(x', y') \cdot W(x - x', y - y') \, dx' \, dy' $$ Where: - $\rho_0(x, y)$ = Local pattern density - $W(x, y)$ = Weighting function (planarization kernel) #### 3.1.2 Elliptical Weighting Function $$ W(x, y) = \frac{1}{\pi L_x L_y} \cdot \exp\left(-\frac{x^2}{L_x^2} - \frac{y^2}{L_y^2}\right) $$ Where $L_x, L_y$ are planarization lengths in x and y directions. ### 3.2 Step Height Evolution Model For oxide CMP with step height $h$: $$ \frac{dh}{dt} = -K \cdot \left(1 - \frac{h_{contact}}{h}\right) \quad \text{for } h > h_{contact} $$ $$ \frac{dh}{dt} = 0 \quad \text{for } h \leq h_{contact} $$ Where $h_{contact}$ is the pad contact threshold height. ### 3.3 Integrated Density-Step Height Model Combined model for oxide thickness evolution: $$ z(x, y, t) = z_0 - K \cdot t \cdot \frac{1}{\rho_{eff}(x, y)} \cdot g(h) $$ Where $g(h)$ is the step-height dependent function: $$ g(h) = \begin{cases} 1 & \text{if } h > h_c \\ \frac{h}{h_c} & \text{if } h \leq h_c \end{cases} $$ ## 4. Dishing and Erosion Models ### 4.1 Copper Dishing Model Dishing depth $D$ for copper lines: $$ D = K_{Cu} \cdot t_{over} \cdot f(w) $$ Where: - $K_{Cu}$ = Copper removal rate - $t_{over}$ = Overpolish time - $w$ = Line width - $f(w)$ = Width-dependent function Empirical relationship: $$ D = D_0 \cdot \left(1 - \exp\left(-\frac{w}{w_c}\right)\right) $$ Where: - $D_0$ = Maximum dishing depth - $w_c$ = Critical line width ### 4.2 Oxide Erosion Model Erosion $E$ in dense pattern regions: $$ E = K_{ox} \cdot t_{over} \cdot \rho_{metal} $$ Where: - $K_{ox}$ = Oxide removal rate - $\rho_{metal}$ = Local metal pattern density ### 4.3 Combined Dishing-Erosion Total copper thickness loss: $$ \Delta z_{Cu} = D + E \cdot \frac{\rho_{metal}}{1 - \rho_{metal}} $$ ### 4.4 Pattern Density Effects | Pattern Density | Dishing Behavior | Erosion Behavior | |-----------------|------------------|------------------| | Low ($< 20\%$) | Minimal | Minimal | | Medium ($20-50\%$) | Moderate | Increasing | | High ($> 50\%$) | Saturates | Severe | ## 5. Contact Mechanics Models ### 5.1 Pad Asperity Contact Model Assuming Gaussian asperity height distribution: $$ P(z) = \frac{1}{\sigma_s \sqrt{2\pi}} \exp\left(-\frac{(z - \bar{z})^2}{2\sigma_s^2}\right) $$ Where: - $\sigma_s$ = Standard deviation of asperity heights - $\bar{z}$ = Mean asperity height ### 5.2 Real Contact Area $$ A_r = \pi n \int_{d}^{\infty} R(z - d) \cdot P(z) \, dz $$ Where: - $n$ = Number of asperities per unit area - $R$ = Asperity tip radius - $d$ = Separation distance For Gaussian distribution: $$ A_r = \pi n R \sigma_s \cdot F_1\left(\frac{d}{\sigma_s}\right) $$ Where $F_1$ is a statistical function. ### 5.3 Hertzian Contact For elastic contact between abrasive particle and wafer: $$ a = \left(\frac{3FR}{4E^*}\right)^{1/3} $$ $$ \delta = \frac{a^2}{R} = \left(\frac{9F^2}{16RE^{*2}}\right)^{1/3} $$ Where: - $a$ = Contact radius - $F$ = Normal force - $R$ = Particle radius - $\delta$ = Indentation depth - $E^*$ = Effective elastic modulus $$ \frac{1}{E^*} = \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2} $$ ### 5.4 Material Removal by Single Abrasive Volume removed per abrasive per pass: $$ V = K_{wear} \cdot \frac{F_n \cdot L}{H} $$ Where: - $K_{wear}$ = Wear coefficient - $F_n$ = Normal force on particle - $L$ = Sliding distance - $H$ = Hardness of wafer material ### 5.5 Multi-Scale Model Framework ``` - ┌─────────────────────────────────────────────────────────────┐ │ WAFER SCALE (mm-cm) │ │ Pressure distribution, global uniformity │ ├─────────────────────────────────────────────────────────────┤ │ DIE SCALE ($\mu$m-mm) │ │ Pattern density effects, planarization │ ├─────────────────────────────────────────────────────────────┤ │ FEATURE SCALE (nm-$\mu$m) │ │ Dishing, erosion, step height evolution │ ├─────────────────────────────────────────────────────────────┤ │ PARTICLE SCALE (nm) │ │ Abrasive-surface interactions │ ├─────────────────────────────────────────────────────────────┤ │ MOLECULAR SCALE (Å) │ │ Chemical reactions, atomic removal │ └─────────────────────────────────────────────────────────────┘ ``` ## 6. Machine Learning and Neural Network Models ### 6.1 Overview of ML Approaches Machine learning methods for CMP modeling: - **Supervised Learning** - Artificial Neural Networks (ANN) - Convolutional Neural Networks (CNN) - Support Vector Machines (SVM) - Random Forests / Gradient Boosting - **Deep Learning** - Deep Belief Networks (DBN) - Long Short-Term Memory (LSTM) - Generative Adversarial Networks (GAN) - **Transfer Learning** - Pre-trained models adapted to new process conditions ### 6.2 Neural Network Architecture for CMP #### 6.2.1 Input Features $$ \mathbf{x} = [P, v, t, \rho, w, s, pH, C_{ox}, T, ...]^T $$ Where: - $P$ = Pressure - $v$ = Velocity - $t$ = Polish time - $\rho$ = Pattern density - $w$ = Feature width - $s$ = Feature spacing - $pH$ = Slurry pH - $C_{ox}$ = Oxidizer concentration - $T$ = Temperature #### 6.2.2 Multi-Layer Perceptron (MLP) $$ \mathbf{h}^{(1)} = \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) $$ $$ \mathbf{h}^{(2)} = \sigma(\mathbf{W}^{(2)} \mathbf{h}^{(1)} + \mathbf{b}^{(2)}) $$ $$ \hat{y} = \mathbf{W}^{(out)} \mathbf{h}^{(2)} + \mathbf{b}^{(out)} $$ Where: - $\sigma$ = Activation function (ReLU, tanh, sigmoid) - $\mathbf{W}^{(i)}$ = Weight matrices - $\mathbf{b}^{(i)}$ = Bias vectors #### 6.2.3 Activation Functions | Function | Formula | Use Case | |----------|---------|----------| | **ReLU** | $\sigma(x) = \max(0, x)$ | Hidden layers | | **Sigmoid** | $\sigma(x) = \frac{1}{1 + e^{-x}}$ | Output (binary) | | **Tanh** | $\sigma(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ | Hidden layers | | **Softmax** | $\sigma(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}}$ | Classification | ### 6.3 CNN-Based CMP Modeling (CmpCNN) #### 6.3.1 Architecture ``` Input: Layout Image (Binary) + Density Map ↓ Conv2D Layer (3×3 kernel, 32 filters) ↓ MaxPooling2D (2×2) ↓ Conv2D Layer (3×3 kernel, 64 filters) ↓ MaxPooling2D (2×2) ↓ Flatten ↓ Dense Layer (256 units) ↓ Dense Layer (128 units) ↓ Output: Post-CMP Height Map ``` #### 6.3.2 Convolution Operation $$ (I * K)(i, j) = \sum_m \sum_n I(i+m, j+n) \cdot K(m, n) $$ Where: - $I$ = Input image (layout) - $K$ = Convolution kernel - $(i, j)$ = Output position ### 6.4 Loss Functions #### 6.4.1 Mean Squared Error (MSE) $$ \mathcal{L}_{MSE} = \frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2 $$ #### 6.4.2 Root Mean Square Error (RMSE) $$ RMSE = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2} $$ #### 6.4.3 Mean Absolute Percentage Error (MAPE) $$ MAPE = \frac{100\%}{N} \sum_{i=1}^{N} \left| \frac{y_i - \hat{y}_i}{y_i} \right| $$ ### 6.5 Transfer Learning Framework For adapting models across process nodes: $$ \mathcal{L}_{transfer} = \mathcal{L}_{target} + \lambda \cdot \mathcal{L}_{domain} $$ Where: - $\mathcal{L}_{target}$ = Target domain loss - $\mathcal{L}_{domain}$ = Domain adaptation loss - $\lambda$ = Regularization parameter ### 6.6 Performance Metrics | Metric | Formula | Target | |--------|---------|--------| | $R^2$ | $1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}$ | $> 0.95$ | | RMSE | $\sqrt{\frac{1}{N}\sum(y_i - \hat{y}_i)^2}$ | $< 5$ Å | | MAE | $\frac{1}{N}\sum|y_i - \hat{y}_i|$ | $< 3$ Å | ## 7. Slurry Chemistry Modeling ### 7.1 Kaufman Mechanism Cyclic passivation-depassivation process: $$ \text{Metal} \xrightarrow{\text{Oxidizer}} \text{Metal Oxide} \xrightarrow{\text{Abrasion}} \text{Removal} $$ ### 7.2 Electrochemical Reactions #### 7.2.1 Copper CMP **Oxidation:** $$ \text{Cu} \rightarrow \text{Cu}^{2+} + 2e^- $$ **Passivation (with BTA):** $$ \text{Cu} + \text{BTA} \rightarrow \text{Cu-BTA}_{film} $$ **Complexation:** $$ \text{Cu}^{2+} + n\text{L} \rightarrow [\text{CuL}_n]^{2+} $$ Where L = chelating agent (e.g., glycine, citrate) #### 7.2.2 Tungsten CMP **Oxidation:** $$ \text{W} + 3\text{H}_2\text{O} \rightarrow \text{WO}_3 + 6\text{H}^+ + 6e^- $$ **With hydrogen peroxide:** $$ \text{W} + 3\text{H}_2\text{O}_2 \rightarrow \text{WO}_3 + 3\text{H}_2\text{O} $$ ### 7.3 Pourbaix Diagram Integration Stability regions defined by: $$ E = E^0 - \frac{RT}{nF} \ln Q - \frac{RT}{F} \cdot m \cdot pH $$ Where: - $E$ = Electrode potential - $E^0$ = Standard potential - $Q$ = Reaction quotient - $m$ = Number of H⁺ in reaction ### 7.4 Abrasive Particle Effects #### 7.4.1 Particle Size Distribution (PSD) Log-normal distribution: $$ f(d) = \frac{1}{d \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln d - \mu)^2}{2\sigma^2}\right) $$ Where: - $d$ = Particle diameter - $\mu$ = Mean of $\ln(d)$ - $\sigma$ = Standard deviation of $\ln(d)$ #### 7.4.2 Zeta Potential $$ \zeta = \frac{4\pi \eta \mu_e}{\varepsilon} $$ Where: - $\eta$ = Viscosity - $\mu_e$ = Electrophoretic mobility - $\varepsilon$ = Dielectric constant ### 7.5 Slurry Components Summary | Component | Function | Typical Materials | |-----------|----------|-------------------| | **Abrasive** | Mechanical removal | SiO₂, CeO₂, Al₂O₃ | | **Oxidizer** | Surface modification | H₂O₂, KIO₃, Fe(NO₃)₃ | | **Complexant** | Metal dissolution | Glycine, citric acid | | **Inhibitor** | Corrosion protection | BTA, BBI | | **Surfactant** | Particle dispersion | CTAB, SDS | | **Buffer** | pH control | Phosphate, citrate | ## 8. Chip-Scale and Full-Chip Models ### 8.1 Within-Wafer Non-Uniformity (WIWNU) $$ WIWNU = \frac{\sigma_{thickness}}{\bar{thickness}} \times 100\% $$ Where: - $\sigma_{thickness}$ = Standard deviation of thickness - $\bar{thickness}$ = Mean thickness ### 8.2 Pressure Distribution Model For a flexible carrier: $$ P(r) = P_0 + \sum_{i=1}^{n} P_i \cdot J_0\left(\frac{\alpha_i r}{R}\right) $$ Where: - $P_0$ = Base pressure - $J_0$ = Bessel function of first kind - $\alpha_i$ = Bessel zeros - $R$ = Wafer radius ### 8.3 Multi-Zone Pressure Control For zone $i$: $$ MRR_i = k_p \cdot P_i \cdot v_i $$ Target uniformity achieved when: $$ MRR_1 = MRR_2 = ... = MRR_n $$ ### 8.4 Full-Chip Simulation Flow ``` - ┌─────────────────────┐ │ Design Layout (GDS)│ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Density Extraction │ │ ρ(x,y) for each │ │ metal/dielectric │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Effective Density │ │ ρ_eff = ρ * W │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ CMP Simulation │ │ z(t) evolution │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Post-CMP Topography │ │ Dishing/Erosion Map │ └──────────┬──────────┘ ↓ ┌─────────────────────┐ │ Hotspot Detection │ │ Design Rule Check │ └─────────────────────┘ ``` ## 9. Process Control Applications ### 9.1 Run-to-Run (R2R) Control #### 9.1.1 EWMA Controller $$ \hat{y}_{k+1} = \lambda y_k + (1 - \lambda) \hat{y}_k $$ Where: - $\hat{y}_{k+1}$ = Predicted output for next run - $y_k$ = Current measured output - $\lambda$ = Smoothing factor $(0 < \lambda < 1)$ #### 9.1.2 Recipe Adjustment $$ u_{k+1} = u_k + G^{-1} (y_{target} - \hat{y}_{k+1}) $$ Where: - $u$ = Process recipe (time, pressure, etc.) - $G$ = Process gain matrix - $y_{target}$ = Target output ### 9.2 Virtual Metrology $$ \hat{y} = f_{VM}(\mathbf{x}_{FDC}) $$ Where: - $\hat{y}$ = Predicted wafer quality - $\mathbf{x}_{FDC}$ = Fault Detection and Classification sensor data ### 9.3 Endpoint Detection #### 9.3.1 Motor Current Monitoring $$ I(t) = I_0 + \Delta I \cdot H(t - t_{endpoint}) $$ Where $H$ is the Heaviside step function. #### 9.3.2 Optical Endpoint $$ R(\lambda, t) = R_{film}(\lambda, d(t)) $$ Where reflectance $R$ changes as film thickness $d$ decreases. ## 10. Current Challenges and Future Directions ### 10.1 Key Challenges - **Sub-5nm nodes**: Atomic-scale precision required - Thickness variation target: $< 5$ Å (3σ) - Defect density target: $< 0.01$ defects/cm² - **New materials integration**: - Low-κ dielectrics ($\kappa < 2.5$) - Cobalt interconnects - Ruthenium barrier layers - **3D integration**: - Through-Silicon Via (TSV) CMP - Hybrid bonding surface preparation - Wafer-level packaging ### 10.2 Future Model Development - **Physics-informed neural networks (PINNs)**: $$ \mathcal{L} = \mathcal{L}_{data} + \lambda_{physics} \cdot \mathcal{L}_{physics} $$ Where: $$ \mathcal{L}_{physics} = \left\| \frac{\partial z}{\partial t} + \frac{K}{\rho_{eff}} \right\|^2 $$ - **Digital twins** for real-time process optimization - **Federated learning** across multiple fabs ### 10.3 Industry Requirements | Node | Thickness Uniformity | Defect Density | Dishing Limit | |------|---------------------|----------------|---------------| | 7nm | $< 10$ Å | $< 0.05$/cm² | $< 200$ Å | | 5nm | $< 7$ Å | $< 0.03$/cm² | $< 150$ Å | | 3nm | $< 5$ Å | $< 0.01$/cm² | $< 100$ Å | | 2nm | $< 3$ Å | $< 0.005$/cm² | $< 50$ Å | ## Symbol Glossary | Symbol | Description | Units | |--------|-------------|-------| | $MRR$ | Material Removal Rate | nm/min | | $k_p$ | Preston coefficient | m²/N | | $P$ | Pressure | Pa, psi | | $v$ | Relative velocity | m/s | | $\rho$ | Pattern density | dimensionless | | $\rho_{eff}$ | Effective pattern density | dimensionless | | $L$ | Planarization length | $\mu$m | | $D$ | Dishing depth | Å, nm | | $E$ | Erosion depth | Å, nm | | $w$ | Feature width | nm, $\mu$m | | $h$ | Step height | nm | | $t$ | Polish time | s, min | | $T$ | Temperature | K, °C | | $\eta$ | Viscosity | Pa$\cdot$s | | $\mu$ | Friction coefficient | dimensionless | ## Key Equations ### Preston Equation $$ MRR = k_p \cdot P \cdot v $$ ### Effective Density $$ \rho_{eff}(x,y) = \iint \rho_0(x',y') \cdot W(x-x', y-y') \, dx' dy' $$ ### Material Removal (Density Model) $$ \frac{dz}{dt} = -\frac{K}{\rho_{eff}(x,y)} $$ ### Dishing Model $$ D = D_0 \cdot \left(1 - e^{-w/w_c}\right) $$ ### Erosion Model $$ E = K_{ox} \cdot t_{over} \cdot \rho_{metal} $$ ### Neural Network $$ \hat{y} = \sigma(\mathbf{W}^{(n)} \cdot ... \cdot \sigma(\mathbf{W}^{(1)} \mathbf{x} + \mathbf{b}^{(1)}) + \mathbf{b}^{(n)}) $$
Jointly attend to multiple modalities.
Train multiple views of data.
Co-training trains multiple models on different views of data where each model labels examples for others' training sets.
Train multiple models on different views of data and share predictions.
Simplified representation for larger scales.
Train from coarse to fine details.
Multi-scale attention.
Cocktail party problem refers to separating individual speakers from mixed audio in multi-talker scenarios.
Amount of code change over time.
Find similar code segments.
Complete code understanding context.
Autocomplete code based on context and comments.
Measure code complexity.
Describe what code does in natural language.
LLMs that write code from natural language descriptions.
Code models specialize in programming tasks through code-focused training.
Improve code performance automatically.
Various code quality measures.