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# 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}` | — |