enzyme design,healthcare ai
Engineer enzymes for specific reactions.
3,145 technical terms and definitions
Engineer enzymes for specific reactions.
# Semiconductor Manufacturing Process: Epitaxy (Epi) Modeling ## 1. Introduction to Epitaxy Epitaxy is the controlled growth of a crystalline thin film on a crystalline substrate, where the deposited layer inherits the crystallographic orientation of the substrate. ### 1.1 Types of Epitaxy - **Homoepitaxy** - Same material deposited on substrate - Example: Silicon (Si) on Silicon (Si) - Maintains perfect lattice matching - Used for creating high-purity device layers - **Heteroepitaxy** - Different material deposited on substrate - Examples: - Gallium Arsenide (GaAs) on Silicon (Si) - Silicon Germanium (SiGe) on Silicon (Si) - Gallium Nitride (GaN) on Sapphire ($\text{Al}_2\text{O}_3$) - Introduces lattice mismatch and strain - Enables bandgap engineering ## 2. Epitaxy Methods ### 2.1 Chemical Vapor Deposition (CVD) / Vapor Phase Epitaxy (VPE) - **Characteristics:** - Most common method for silicon epitaxy - Operates at atmospheric or reduced pressure - Temperature range: $900°\text{C} - 1200°\text{C}$ - **Common Precursors:** - Silane: $\text{SiH}_4$ - Dichlorosilane: $\text{SiH}_2\text{Cl}_2$ (DCS) - Trichlorosilane: $\text{SiHCl}_3$ (TCS) - Silicon tetrachloride: $\text{SiCl}_4$ - **Key Reactions:** $$\text{SiH}_4 \xrightarrow{\Delta} \text{Si}_{(s)} + 2\text{H}_2$$ $$\text{SiH}_2\text{Cl}_2 \xrightarrow{\Delta} \text{Si}_{(s)} + 2\text{HCl}$$ ### 2.2 Molecular Beam Epitaxy (MBE) - **Characteristics:** - Ultra-high vacuum environment ($< 10^{-10}$ Torr) - Extremely precise thickness control (monolayer accuracy) - Lower growth temperatures than CVD - Slower growth rates: $\sim 1 \, \mu\text{m/hour}$ - **Applications:** - III-V compound semiconductors - Quantum well structures - Superlattices - Research and development ### 2.3 Metal-Organic CVD (MOCVD) - **Characteristics:** - Standard for compound semiconductors - Uses metal-organic precursors - Higher throughput than MBE - **Common Precursors:** - Trimethylgallium: $\text{Ga(CH}_3\text{)}_3$ (TMGa) - Trimethylaluminum: $\text{Al(CH}_3\text{)}_3$ (TMAl) - Ammonia: $\text{NH}_3$ ### 2.4 Atomic Layer Epitaxy (ALE) - **Characteristics:** - Self-limiting surface reactions - Digital control of film thickness - Excellent conformality - Growth rate: $\sim 1$ Å per cycle ## 3. Physics of Epi Modeling ### 3.1 Gas-Phase Transport The transport of precursor gases to the substrate surface involves multiple phenomena: - **Governing Equations:** - **Continuity Equation:** $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$ - **Navier-Stokes Equation:** $$\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}$$ - **Species Transport Equation:** $$\frac{\partial C_i}{\partial t} + \mathbf{v} \cdot \nabla C_i = D_i \nabla^2 C_i + R_i$$ Where: - $\rho$ = fluid density - $\mathbf{v}$ = velocity vector - $p$ = pressure - $\mu$ = dynamic viscosity - $C_i$ = concentration of species $i$ - $D_i$ = diffusion coefficient of species $i$ - $R_i$ = reaction rate term - **Boundary Layer:** - Stagnant gas layer above substrate - Thickness $\delta$ depends on flow conditions: $$\delta \propto \sqrt{\frac{\nu x}{u_\infty}}$$ Where: - $\nu$ = kinematic viscosity - $x$ = distance from leading edge - $u_\infty$ = free stream velocity ### 3.2 Surface Kinetics - **Adsorption Process:** - Physisorption (weak van der Waals forces) - Chemisorption (chemical bonding) - **Langmuir Adsorption Isotherm:** $$\theta = \frac{K \cdot P}{1 + K \cdot P}$$ Where: - $\theta$ = fractional surface coverage - $K$ = equilibrium constant - $P$ = partial pressure - **Surface Diffusion:** $$D_s = D_0 \exp\left(-\frac{E_d}{k_B T}\right)$$ Where: - $D_s$ = surface diffusion coefficient - $D_0$ = pre-exponential factor - $E_d$ = diffusion activation energy - $k_B$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K) - $T$ = absolute temperature ### 3.3 Crystal Growth Mechanisms - **Step-Flow Growth (BCF Theory):** - Atoms attach at step edges - Steps advance across terraces - Dominant at high temperatures - **2D Nucleation:** - New layers nucleate on terraces - Occurs when step density is low - Creates rougher surfaces - **Terrace-Ledge-Kink (TLK) Model:** - Terrace: flat regions between steps - Ledge: step edges - Kink: incorporation sites at step edges ## 4. Mathematical Framework ### 4.1 Growth Rate Models #### 4.1.1 Reaction-Limited Regime At lower temperatures, surface reaction kinetics dominate: $$G = k_s \cdot C_s$$ Where the rate constant follows Arrhenius behavior: $$k_s = k_0 \exp\left(-\frac{E_a}{k_B T}\right)$$ **Parameters:** - $G$ = growth rate (nm/min or μm/hr) - $k_s$ = surface reaction rate constant - $C_s$ = surface concentration - $k_0$ = pre-exponential factor - $E_a$ = activation energy #### 4.1.2 Mass-Transport Limited Regime At higher temperatures, diffusion through the boundary layer limits growth: $$G = \frac{h_g}{N_s} \cdot (C_g - C_s)$$ Where: $$h_g = \frac{D}{\delta}$$ **Parameters:** - $h_g$ = mass transfer coefficient - $N_s$ = atomic density of solid ($\sim 5 \times 10^{22}$ atoms/cm³ for Si) - $C_g$ = gas phase concentration - $D$ = gas phase diffusivity - $\delta$ = boundary layer thickness #### 4.1.3 Combined Model (Grove Model) For the general case combining both regimes: $$G = \frac{h_g \cdot k_s}{N_s (h_g + k_s)} \cdot C_g$$ Or equivalently: $$\frac{1}{G} = \frac{N_s}{k_s \cdot C_g} + \frac{N_s}{h_g \cdot C_g}$$ ### 4.2 Strain in Heteroepitaxy #### 4.2.1 Lattice Mismatch $$f = \frac{a_s - a_f}{a_f}$$ Where: - $f$ = lattice mismatch (dimensionless) - $a_s$ = substrate lattice constant - $a_f$ = film lattice constant (relaxed) **Example Values:** | System | $a_f$ (Å) | $a_s$ (Å) | Mismatch $f$ | |--------|-----------|-----------|--------------| | Si on Si | 5.431 | 5.431 | 0% | | Ge on Si | 5.658 | 5.431 | -4.2% | | GaAs on Si | 5.653 | 5.431 | -4.1% | | InAs on GaAs | 6.058 | 5.653 | -7.2% | #### 4.2.2 In-Plane Strain For a coherently strained film: $$\epsilon_{\parallel} = \frac{a_s - a_f}{a_f} = f$$ The out-of-plane strain (for cubic materials): $$\epsilon_{\perp} = -\frac{2\nu}{1-\nu} \epsilon_{\parallel}$$ Where $\nu$ = Poisson's ratio #### 4.2.3 Critical Thickness (Matthews-Blakeslee) The critical thickness above which misfit dislocations form: $$h_c = \frac{b}{8\pi f (1+\nu)} \left[ \ln\left(\frac{h_c}{b}\right) + 1 \right]$$ Where: - $h_c$ = critical thickness - $b$ = Burgers vector magnitude ($\approx \frac{a}{\sqrt{2}}$ for 60° dislocations) - $f$ = lattice mismatch - $\nu$ = Poisson's ratio **Approximate Solution:** For small mismatch: $$h_c \approx \frac{b}{8\pi |f|}$$ ### 4.3 Dopant Incorporation #### 4.3.1 Segregation Model $$C_{film} = \frac{C_{gas}}{1 + k_{seg} \cdot (G/G_0)}$$ Where: - $C_{film}$ = dopant concentration in film - $C_{gas}$ = dopant concentration in gas phase - $k_{seg}$ = segregation coefficient - $G$ = growth rate - $G_0$ = reference growth rate #### 4.3.2 Dopant Profile with Segregation The surface concentration evolves as: $$C_s(t) = C_s^{eq} + (C_s(0) - C_s^{eq}) \exp\left(-\frac{G \cdot t}{\lambda}\right)$$ Where: - $\lambda$ = segregation length - $C_s^{eq}$ = equilibrium surface concentration ## 5. Modeling Approaches ### 5.1 Continuum Models - **Scope:** - Reactor-scale simulations - Temperature and flow field prediction - Species concentration profiles - **Methods:** - Computational Fluid Dynamics (CFD) - Finite Element Method (FEM) - Finite Volume Method (FVM) - **Governing Physics:** - Coupled heat, mass, and momentum transfer - Homogeneous and heterogeneous reactions - Radiation heat transfer ### 5.2 Feature-Scale Models - **Applications:** - Selective epitaxial growth (SEG) - Trench filling - Facet evolution - **Key Phenomena:** - Local loading effects: $$G_{local} = G_0 \cdot \left(1 - \alpha \cdot \frac{A_{exposed}}{A_{total}}\right)$$ - Orientation-dependent growth rates: $$\frac{G_{(110)}}{G_{(100)}} \approx 1.5 - 2.0$$ - **Methods:** - Level set methods - String methods - Cellular automata ### 5.3 Atomistic Models #### 5.3.1 Kinetic Monte Carlo (KMC) - **Process Events:** - Adsorption: rate $\propto P \cdot \exp(-E_{ads}/k_BT)$ - Surface diffusion: rate $\propto \exp(-E_{diff}/k_BT)$ - Desorption: rate $\propto \exp(-E_{des}/k_BT)$ - Incorporation: rate $\propto \exp(-E_{inc}/k_BT)$ - **Master Equation:** $$\frac{dP_i}{dt} = \sum_j \left( W_{ji} P_j - W_{ij} P_i \right)$$ Where: - $P_i$ = probability of state $i$ - $W_{ij}$ = transition rate from state $i$ to $j$ #### 5.3.2 Molecular Dynamics (MD) - **Newton's Equations:** $$m_i \frac{d^2 \mathbf{r}_i}{dt^2} = -\nabla_i U(\mathbf{r}_1, \mathbf{r}_2, ..., \mathbf{r}_N)$$ - **Interatomic Potentials:** - Tersoff potential (Si, C, Ge) - Stillinger-Weber potential (Si) - MEAM (metals and alloys) #### 5.3.3 Ab Initio / DFT - **Kohn-Sham Equations:** $$\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{eff}(\mathbf{r}) \right] \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r})$$ - **Applications:** - Surface energies - Reaction barriers - Adsorption energies - Electronic structure ## 6. Specific Modeling Challenges ### 6.1 SiGe Epitaxy - **Composition Control:** $$x_{Ge} = \frac{R_{Ge}}{R_{Si} + R_{Ge}}$$ Where $R_{Si}$ and $R_{Ge}$ are partial growth rates - **Strain Engineering:** - Compressive strain in SiGe on Si - Enhances hole mobility - Critical thickness depends on Ge content: $$h_c(x) \approx \frac{0.5}{0.042 \cdot x} \text{ nm}$$ ### 6.2 Selective Epitaxy - **Growth Selectivity:** - Deposition only on exposed silicon - HCl addition for selectivity enhancement - **Selectivity Condition:** $$\frac{\text{Growth on Si}}{\text{Growth on SiO}_2} > 100:1$$ - **Loading Effects:** - Pattern-dependent growth rate - Faceting at mask edges ### 6.3 III-V on Silicon - **Major Challenges:** - Large lattice mismatch (4-8%) - Thermal expansion mismatch - Anti-phase domain boundaries (APDs) - High threading dislocation density - **Mitigation Strategies:** - Aspect ratio trapping (ART) - Graded buffer layers - Selective area growth - Dislocation filtering ## 7. Applications and Tools ### 7.1 Industrial Applications | Application | Material System | Key Parameters | |-------------|-----------------|----------------| | FinFET/GAA Source/Drain | Embedded SiGe, SiC | Strain, selectivity | | SiGe HBT | SiGe:C | Profile abruptness | | Power MOSFETs | SiC epitaxy | Defect density | | LEDs/Lasers | GaN, InGaN | Composition uniformity | | RF Devices | GaN on SiC | Buffer quality | ### 7.2 Simulation Software - **Reactor-Scale CFD:** - ANSYS Fluent - COMSOL Multiphysics - OpenFOAM - **TCAD Process Simulation:** - Synopsys Sentaurus Process - Silvaco Victory Process - Lumerical (for optoelectronics) - **Atomistic Simulation:** - LAMMPS (MD) - VASP, Quantum ESPRESSO (DFT) - Custom KMC codes ### 7.3 Key Metrics for Process Development - **Uniformity:** $$\text{Uniformity} = \frac{t_{max} - t_{min}}{2 \cdot t_{avg}} \times 100\%$$ - **Defect Density:** - Threading dislocations: target $< 10^6$ cm$^{-2}$ - Stacking faults: target $< 10^3$ cm$^{-2}$ - **Profile Abruptness:** - Dopant transition width $< 3$ nm/decade ## 8. Emerging Directions ### 8.1 Machine Learning Integration - **Applications:** - Surrogate models for process optimization - Real-time virtual metrology - Defect classification - Recipe optimization - **Model Types:** - Neural networks for growth rate prediction - Gaussian process regression for uncertainty quantification - Reinforcement learning for process control ### 8.2 Multi-Scale Modeling - **Hierarchical Approach:** ``` Ab Initio (DFT) ↓ Reaction rates, energies Kinetic Monte Carlo ↓ Surface kinetics, morphology Feature-Scale Models ↓ Local growth behavior Reactor-Scale CFD ↓ Process conditions Device Simulation ``` ### 8.3 Digital Twins - **Components:** - Real-time sensor data integration - Physics-based + ML hybrid models - Predictive maintenance - Closed-loop process control ### 8.4 New Material Systems - **2D Materials:** - Graphene via CVD - Transition metal dichalcogenides (TMDs) - Van der Waals epitaxy - **Ultra-Wide Bandgap:** - $\beta$-Ga$_2$O$_3$ ($E_g \approx 4.8$ eV) - Diamond ($E_g \approx 5.5$ eV) - AlN ($E_g \approx 6.2$ eV) ## Common Constants and Conversions | Constant | Symbol | Value | |----------|--------|-------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ J/K | | Planck constant | $h$ | $6.626 \times 10^{-34}$ J·s | | Avogadro number | $N_A$ | $6.022 \times 10^{23}$ mol$^{-1}$ | | Si atomic density | $N_{Si}$ | $5.0 \times 10^{22}$ atoms/cm³ | | Si lattice constant | $a_{Si}$ | 5.431 Å |
Train on sequences of few-shot tasks.
Episodic memory records specific events or experiences with temporal context.
Epistemic uncertainty arises from insufficient knowledge reducible with more data.
Uncertainty from lack of knowledge.
Epitaxial source-drain regions are grown selectively in recessed areas controlling stress and reducing resistance.
# Epitaxy (Epi) Modeling: 1. Introduction to Epitaxy Epitaxy is the controlled growth of a crystalline thin film on a crystalline substrate, where the deposited layer inherits the crystallographic orientation of the substrate. 1.1 Types of Epitaxy • Homoepitaxy • Same material deposited on substrate • Example: Silicon (Si) on Silicon (Si) • Maintains perfect lattice matching • Used for creating high-purity device layers • Heteroepitaxy • Different material deposited on substrate • Examples: • Gallium Arsenide (GaAs) on Silicon (Si) • Silicon Germanium (SiGe) on Silicon (Si) • Gallium Nitride (GaN) on Sapphire ($\text{Al}_2\text{O}_3$) • Introduces lattice mismatch and strain • Enables bandgap engineering 2. Epitaxy Methods 2.1 Chemical Vapor Deposition (CVD) / Vapor Phase Epitaxy (VPE) • Characteristics: • Most common method for silicon epitaxy • Operates at atmospheric or reduced pressure • Temperature range: $900°\text{C} - 1200°\text{C}$ • Common Precursors: • Silane: $\text{SiH}_4$ • Dichlorosilane: $\text{SiH}_2\text{Cl}_2$ (DCS) • Trichlorosilane: $\text{SiHCl}_3$ (TCS) • Silicon tetrachloride: $\text{SiCl}_4$ • Key Reactions: $$\text{SiH}_4 \xrightarrow{\Delta} \text{Si}_{(s)} + 2\text{H}_2$$ $$\text{SiH}_2\text{Cl}_2 \xrightarrow{\Delta} \text{Si}_{(s)} + 2\text{HCl}$$ 2.2 Molecular Beam Epitaxy (MBE) • Characteristics: • Ultra-high vacuum environment ($< 10^{-10}$ Torr) • Extremely precise thickness control (monolayer accuracy) • Lower growth temperatures than CVD • Slower growth rates: $\sim 1 \, \mu\text{m/hour}$ • Applications: • III-V compound semiconductors • Quantum well structures • Superlattices • Research and development 2.3 Metal-Organic CVD (MOCVD) • Characteristics: • Standard for compound semiconductors • Uses metal-organic precursors • Higher throughput than MBE • Common Precursors: • Trimethylgallium: $\text{Ga(CH}_3\text{)}_3$ (TMGa) • Trimethylaluminum: $\text{Al(CH}_3\text{)}_3$ (TMAl) • Ammonia: $\text{NH}_3$ 2.4 Atomic Layer Epitaxy (ALE) • Characteristics: • Self-limiting surface reactions • Digital control of film thickness • Excellent conformality • Growth rate: $\sim 1$ Å per cycle 3. Physics of Epi Modeling 3.1 Gas-Phase Transport The transport of precursor gases to the substrate surface involves multiple phenomena: • Governing Equations: • Continuity Equation: $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$ • Navier-Stokes Equation: $$\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}$$ • Species Transport Equation: $$\frac{\partial C_i}{\partial t} + \mathbf{v} \cdot \nabla C_i = D_i \nabla^2 C_i + R_i$$ Where: • $\rho$ = fluid density • $\mathbf{v}$ = velocity vector • $p$ = pressure • $\mu$ = dynamic viscosity • $C_i$ = concentration of species $i$ • $D_i$ = diffusion coefficient of species $i$ • $R_i$ = reaction rate term • Boundary Layer: • Stagnant gas layer above substrate • Thickness $\delta$ depends on flow conditions: $$\delta \propto \sqrt{\frac{\nu x}{u_\infty}}$$ Where: • $\nu$ = kinematic viscosity • $x$ = distance from leading edge • $u_\infty$ = free stream velocity 3.2 Surface Kinetics • Adsorption Process: • Physisorption (weak van der Waals forces) • Chemisorption (chemical bonding) • Langmuir Adsorption Isotherm: $$\theta = \frac{K \cdot P}{1 + K \cdot P}$$ Where: - $\theta$ = fractional surface coverage - $K$ = equilibrium constant - $P$ = partial pressure • Surface Diffusion: $$D_s = D_0 \exp\left(-\frac{E_d}{k_B T}\right)$$ Where: - $D_s$ = surface diffusion coefficient - $D_0$ = pre-exponential factor - $E_d$ = diffusion activation energy - $k_B$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K) - $T$ = absolute temperature 3.3 Crystal Growth Mechanisms • Step-Flow Growth (BCF Theory): • Atoms attach at step edges • Steps advance across terraces • Dominant at high temperatures • 2D Nucleation: • New layers nucleate on terraces • Occurs when step density is low • Creates rougher surfaces • Terrace-Ledge-Kink (TLK) Model: • Terrace: flat regions between steps • Ledge: step edges • Kink: incorporation sites at step edges 4. Mathematical Framework 4.1 Growth Rate Models 4.1.1 Reaction-Limited Regime At lower temperatures, surface reaction kinetics dominate: $$G = k_s \cdot C_s$$ Where the rate constant follows Arrhenius behavior: $$k_s = k_0 \exp\left(-\frac{E_a}{k_B T}\right)$$ Parameters: - $G$ = growth rate (nm/min or μm/hr) - $k_s$ = surface reaction rate constant - $C_s$ = surface concentration - $k_0$ = pre-exponential factor - $E_a$ = activation energy 4.1.2 Mass-Transport Limited Regime At higher temperatures, diffusion through the boundary layer limits growth: $$G = \frac{h_g}{N_s} \cdot (C_g - C_s)$$ Where: $$h_g = \frac{D}{\delta}$$ Parameters: - $h_g$ = mass transfer coefficient - $N_s$ = atomic density of solid ($\sim 5 \times 10^{22}$ atoms/cm³ for Si) - $C_g$ = gas phase concentration - $D$ = gas phase diffusivity - $\delta$ = boundary layer thickness 4.1.3 Combined Model (Grove Model) For the general case combining both regimes: $$G = \frac{h_g \cdot k_s}{N_s (h_g + k_s)} \cdot C_g$$ Or equivalently: $$\frac{1}{G} = \frac{N_s}{k_s \cdot C_g} + \frac{N_s}{h_g \cdot C_g}$$ 4.2 Strain in Heteroepitaxy 4.2.1 Lattice Mismatch $$f = \frac{a_s - a_f}{a_f}$$ Where: - $f$ = lattice mismatch (dimensionless) - $a_s$ = substrate lattice constant - $a_f$ = film lattice constant (relaxed) Example Values: | System | $a_f$ (Å) | $a_s$ (Å) | Mismatch $f$ | |--------|-----------|-----------|--------------| | Si on Si | 5.431 | 5.431 | 0% | | Ge on Si | 5.658 | 5.431 | -4.2% | | GaAs on Si | 5.653 | 5.431 | -4.1% | | InAs on GaAs | 6.058 | 5.653 | -7.2% | 4.2.2 In-Plane Strain For a coherently strained film: $$\epsilon_{\parallel} = \frac{a_s - a_f}{a_f} = f$$ The out-of-plane strain (for cubic materials): $$\epsilon_{\perp} = -\frac{2\nu}{1-\nu} \epsilon_{\parallel}$$ Where $\nu$ = Poisson's ratio 4.2.3 Critical Thickness (Matthews-Blakeslee) The critical thickness above which misfit dislocations form: $$h_c = \frac{b}{8\pi f (1+\nu)} \left[ \ln\left(\frac{h_c}{b}\right) + 1 \right]$$ Where: - $h_c$ = critical thickness - $b$ = Burgers vector magnitude ($\approx \frac{a}{\sqrt{2}}$ for 60° dislocations) - $f$ = lattice mismatch - $\nu$ = Poisson's ratio Approximate Solution: For small mismatch: $$h_c \approx \frac{b}{8\pi |f|}$$ 4.3 Dopant Incorporation 4.3.1 Segregation Model $$C_{film} = \frac{C_{gas}}{1 + k_{seg} \cdot (G/G_0)}$$ Where: - $C_{film}$ = dopant concentration in film - $C_{gas}$ = dopant concentration in gas phase - $k_{seg}$ = segregation coefficient - $G$ = growth rate - $G_0$ = reference growth rate 4.3.2 Dopant Profile with Segregation The surface concentration evolves as: $$C_s(t) = C_s^{eq} + (C_s(0) - C_s^{eq}) \exp\left(-\frac{G \cdot t}{\lambda}\right)$$ Where: - $\lambda$ = segregation length - $C_s^{eq}$ = equilibrium surface concentration 5. Modeling Approaches 5.1 Continuum Models • Scope: • Reactor-scale simulations • Temperature and flow field prediction • Species concentration profiles • Methods: • Computational Fluid Dynamics (CFD) • Finite Element Method (FEM) • Finite Volume Method (FVM) • Governing Physics: • Coupled heat, mass, and momentum transfer • Homogeneous and heterogeneous reactions • Radiation heat transfer 5.2 Feature-Scale Models • Applications: • Selective epitaxial growth (SEG) • Trench filling • Facet evolution • Key Phenomena: • Local loading effects: $$G_{local} = G_0 \cdot \left(1 - \alpha \cdot \frac{A_{exposed}}{A_{total}}\right)$$ • Orientation-dependent growth rates: $$\frac{G_{(110)}}{G_{(100)}} \approx 1.5 - 2.0$$ • Methods: • Level set methods • String methods • Cellular automata 5.3 Atomistic Models 5.3.1 Kinetic Monte Carlo (KMC) • Process Events: • Adsorption: rate $\propto P \cdot \exp(-E_{ads}/k_BT)$ • Surface diffusion: rate $\propto \exp(-E_{diff}/k_BT)$ • Desorption: rate $\propto \exp(-E_{des}/k_BT)$ • Incorporation: rate $\propto \exp(-E_{inc}/k_BT)$ • Master Equation: $$\frac{dP_i}{dt} = \sum_j \left( W_{ji} P_j - W_{ij} P_i \right)$$ Where: - $P_i$ = probability of state $i$ - $W_{ij}$ = transition rate from state $i$ to $j$ 5.3.2 Molecular Dynamics (MD) • Newton's Equations: $$m_i \frac{d^2 \mathbf{r}_i}{dt^2} = -\nabla_i U(\mathbf{r}_1, \mathbf{r}_2, ..., \mathbf{r}_N)$$ • Interatomic Potentials: • Tersoff potential (Si, C, Ge) • Stillinger-Weber potential (Si) • MEAM (metals and alloys) 5.3.3 Ab Initio / DFT • Kohn-Sham Equations: $$\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{eff}(\mathbf{r}) \right] \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r})$$ • Applications: • Surface energies • Reaction barriers • Adsorption energies • Electronic structure 6. Specific Modeling Challenges 6.1 SiGe Epitaxy • Composition Control: $$x_{Ge} = \frac{R_{Ge}}{R_{Si} + R_{Ge}}$$ Where $R_{Si}$ and $R_{Ge}$ are partial growth rates • Strain Engineering: • Compressive strain in SiGe on Si • Enhances hole mobility • Critical thickness depends on Ge content: $$h_c(x) \approx \frac{0.5}{0.042 \cdot x} \text{ nm}$$ 6.2 Selective Epitaxy • Growth Selectivity: • Deposition only on exposed silicon • HCl addition for selectivity enhancement • Selectivity Condition: $$\frac{\text{Growth on Si}}{\text{Growth on SiO}_2} > 100:1$$ • Loading Effects: • Pattern-dependent growth rate • Faceting at mask edges 6.3 III-V on Silicon • Major Challenges: • Large lattice mismatch (4-8%) • Thermal expansion mismatch • Anti-phase domain boundaries (APDs) • High threading dislocation density • Mitigation Strategies: • Aspect ratio trapping (ART) • Graded buffer layers • Selective area growth • Dislocation filtering 7. Applications and Tools 7.1 Industrial Applications | Application | Material System | Key Parameters | |-------------|-----------------|----------------| | FinFET/GAA Source/Drain | Embedded SiGe, SiC | Strain, selectivity | | SiGe HBT | SiGe:C | Profile abruptness | | Power MOSFETs | SiC epitaxy | Defect density | | LEDs/Lasers | GaN, InGaN | Composition uniformity | | RF Devices | GaN on SiC | Buffer quality | 7.2 Simulation Software • Reactor-Scale CFD: • ANSYS Fluent • COMSOL Multiphysics • OpenFOAM • TCAD Process Simulation: • Synopsys Sentaurus Process • Silvaco Victory Process • Lumerical (for optoelectronics) • Atomistic Simulation: • LAMMPS (MD) • VASP, Quantum ESPRESSO (DFT) • Custom KMC codes 7.3 Key Metrics for Process Development • Uniformity: $$\text{Uniformity} = \frac{t_{max} - t_{min}}{2 \cdot t_{avg}} \times 100\%$$ • Defect Density: • Threading dislocations: target $< 10^6$ cm$^{-2}$ • Stacking faults: target $< 10^3$ cm$^{-2}$ • Profile Abruptness: • Dopant transition width $< 3$ nm/decade 8. Emerging Directions 8.1 Machine Learning Integration • Applications: • Surrogate models for process optimization • Real-time virtual metrology • Defect classification • Recipe optimization • Model Types: • Neural networks for growth rate prediction • Gaussian process regression for uncertainty quantification • Reinforcement learning for process control 8.2 Multi-Scale Modeling • Hierarchical Approach: ```text ┌─────────────────────────────────────────────┐ │ Ab Initio (DFT) │ │ ↓ Reaction rates, energies │ ├─────────────────────────────────────────────┤ │ Kinetic Monte Carlo │ │ ↓ Surface kinetics, morphology │ ├─────────────────────────────────────────────┤ │ Feature-Scale Models │ │ ↓ Local growth behavior │ ├─────────────────────────────────────────────┤ │ Reactor-Scale CFD │ │ ↓ Process conditions │ ├─────────────────────────────────────────────┤ │ Device Simulation │ └─────────────────────────────────────────────┘ ``` • Applications: • Surface energies • Reaction barriers • Adsorption energies • Electronic structure 8.3 Digital Twins • Components: • Real-time sensor data integration • Physics-based + ML hybrid models • Predictive maintenance • Closed-loop process control 8.4 New Material Systems • 2D Materials: • Graphene via CVD • Transition metal dichalcogenides (TMDs) • Van der Waals epitaxy • Ultra-Wide Bandgap: • $\beta$-Ga$_2$O$_3$ ($E_g \approx 4.8$ eV) • Diamond ($E_g \approx 5.5$ eV) • AlN ($E_g \approx 6.2$ eV) Constants and Conversions | Constant | Symbol | Value | |----------|--------|-------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ J/K | | Planck constant | $h$ | $6.626 \times 10^{-34}$ J·s | | Avogadro number | $N_A$ | $6.022 \times 10^{23}$ mol$^{-1}$ | | Si atomic density | $N_{Si}$ | $5.0 \times 10^{22}$ atoms/cm³ | | Si lattice constant | $a_{Si}$ | 5.431 Å |
One complete pass through the entire training dataset.
Epsilon parameter bounds maximum privacy loss in differential privacy.
Epsilon sampling truncates distribution at absolute probability threshold.
Model has same error rates across groups.
Equipment energy efficiency standards specify maximum power consumption for process tools.
Breakdowns causing downtime.
Verify equivariance properties.
3D-aware molecular generation.
Outputs transform correctly under symmetries.
Enterprise Resource Planning integrates business processes including procurement production inventory finance and human resources.
Error detection identifies failures or suboptimal actions during execution.
Accumulate compression errors.
# Semiconductor Manufacturing Error Propagation Mathematics ## 1. Fundamental Error Propagation Theory For a function $f(x_1, x_2, \ldots, x_n)$ where each variable $x_i$ has uncertainty $\sigma_i$, the propagated uncertainty follows: $$ \sigma_f^2 = \sum_{i=1}^{n} \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_i^2 + 2 \sum_{i < j} \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} \, \text{cov}(x_i, x_j) $$ For **uncorrelated errors**, this simplifies to the **Root-Sum-of-Squares (RSS)** formula: $$ \sigma_f = \sqrt{\sum_{i=1}^{n} \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_i^2} $$ ### Applications in Semiconductor Manufacturing - **Critical Dimension (CD) variations**: Feature size deviations from target - **Overlay errors**: Misalignment between lithography layers - **Film thickness variations**: Deposition uniformity issues - **Doping concentration variations**: Implant dose and energy fluctuations ## 2. Process Chain Error Accumulation Semiconductor manufacturing involves hundreds of sequential process steps. Errors propagate through the chain in different modes: ### 2.1 Additive Error Accumulation Used for overlay alignment between layers: $$ E_{\text{total}} = \sum_{i=1}^{n} \varepsilon_i $$ $$ \sigma_{\text{total}}^2 = \sum_{i=1}^{n} \sigma_i^2 \quad \text{(if uncorrelated)} $$ ### 2.2 Multiplicative Error Accumulation Used for etch selectivity, deposition rates, and gain factors: $$ G_{\text{total}} = \prod_{i=1}^{n} G_i $$ $$ \frac{\sigma_G}{G} \approx \sqrt{\sum_{i=1}^{n} \left( \frac{\sigma_{G_i}}{G_i} \right)^2} $$ ### 2.3 Error Accumulation Modes - **Additive**: Errors sum directly (overlay, thickness) - **Multiplicative**: Errors compound through products (gain, selectivity) - **Compensating**: Rare cases where errors cancel - **Nonlinear interactions**: Complex dependencies requiring simulation ## 3. Hierarchical Variance Decomposition Total variation decomposes across spatial and temporal hierarchies: $$ \sigma_{\text{total}}^2 = \sigma_{\text{lot}}^2 + \sigma_{\text{wafer}}^2 + \sigma_{\text{die}}^2 + \sigma_{\text{within-die}}^2 $$ ### Variance Sources by Level | Level | Sources | |-------|---------| | **Lot-to-lot** | Incoming material, chamber conditioning, recipe drift | | **Wafer-to-wafer** | Slot position, thermal gradients, handling | | **Die-to-die** | Across-wafer uniformity, lens field distortion | | **Within-die** | Pattern density, microloading, proximity effects | ### Variance Component Analysis For $N$ measurements $y_{ijk}$ (lot $i$, wafer $j$, site $k$): $$ y_{ijk} = \mu + L_i + W_{ij} + \varepsilon_{ijk} $$ Where: - $\mu$ = grand mean - $L_i \sim N(0, \sigma_L^2)$ = lot effect - $W_{ij} \sim N(0, \sigma_W^2)$ = wafer effect - $\varepsilon_{ijk} \sim N(0, \sigma_\varepsilon^2)$ = residual ## 4. Yield Mathematics ### 4.1 Poisson Defect Model (Random Defects) $$ Y = e^{-D_0 A} $$ Where: - $D_0$ = defect density (defects/cm²) - $A$ = die area (cm²) ### 4.2 Negative Binomial Model (Clustered Defects) More realistic for actual manufacturing: $$ Y = \left( 1 + \frac{D_0 A}{\alpha} \right)^{-\alpha} $$ Where: - $\alpha$ = clustering parameter - $\alpha \to \infty$ recovers Poisson model - Smaller $\alpha$ = more clustering ### 4.3 Total Yield $$ Y_{\text{total}} = Y_{\text{defect}} \times Y_{\text{parametric}} $$ ### 4.4 Parametric Yield Integration over the multi-dimensional acceptable parameter space: $$ Y_{\text{parametric}} = \int \int \cdots \int_{\text{spec}} f(p_1, p_2, \ldots, p_n) \, dp_1 \, dp_2 \cdots dp_n $$ For Gaussian parameters with specs at $\pm k\sigma$: $$ Y_{\text{parametric}} \approx \left[ \text{erf}\left( \frac{k}{\sqrt{2}} \right) \right]^n $$ ## 5. Edge Placement Error (EPE) Critical metric at advanced nodes combining multiple error sources: $$ EPE^2 = \left( \frac{\Delta CD}{2} \right)^2 + OVL^2 + \left( \frac{LER}{2} \right)^2 $$ ### EPE Components - $\Delta CD$ = Critical dimension error - $OVL$ = Overlay error - $LER$ = Line edge roughness ### Extended EPE Model Including additional terms: $$ EPE^2 = \left( \frac{\Delta CD}{2} \right)^2 + OVL^2 + \left( \frac{LER}{2} \right)^2 + \sigma_{\text{mask}}^2 + \sigma_{\text{etch}}^2 $$ ## 6. Overlay Error Modeling Overlay at any point $(x, y)$ is modeled as: $$ OVL(x, y) = \vec{T} + R\theta + M \cdot \vec{r} + \text{HOT} $$ ### Overlay Components - $\vec{T} = (T_x, T_y)$ = Translation - $R\theta$ = Rotation - $M$ = Magnification - $\text{HOT}$ = Higher-Order Terms (lens distortions, wafer non-flatness) ### Overlay Budget (RSS) $$ OVL_{\text{budget}}^2 = OVL_{\text{tool}}^2 + OVL_{\text{process}}^2 + OVL_{\text{wafer}}^2 + OVL_{\text{mask}}^2 $$ ### 10-Parameter Overlay Model $$ \begin{aligned} dx &= T_x + R_x \cdot y + M_x \cdot x + N_x \cdot x \cdot y + \ldots \\ dy &= T_y + R_y \cdot x + M_y \cdot y + N_y \cdot x \cdot y + \ldots \end{aligned} $$ ## 7. Stochastic Effects in EUV Lithography At EUV wavelengths (13.5 nm), photon shot noise becomes fundamental. ### Photon Statistics Photons per pixel follow Poisson distribution: $$ N \sim \text{Poisson}(\bar{N}) $$ $$ \sigma_N = \sqrt{\bar{N}} $$ ### Relative Dose Fluctuation $$ \frac{\sigma_N}{\bar{N}} = \frac{1}{\sqrt{\bar{N}}} $$ ### Stochastic Failure Probability $$ P_{\text{fail}} \propto \exp\left( -\frac{E}{E_{\text{threshold}}} \right) $$ ### RLS Triangle Trade-off - **R**esolution - **L**ine edge roughness (LER) - **S**ensitivity (dose) $$ LER \propto \frac{1}{\sqrt{\text{Dose}}} \propto \frac{1}{\sqrt{N_{\text{photons}}}} $$ ## 8. Spatial Correlation Modeling Errors are spatially correlated. Modeled using variograms or correlation functions. ### Variogram $$ \gamma(h) = \frac{1}{2} E\left[ (Z(x+h) - Z(x))^2 \right] $$ ### Correlation Function $$ \rho(h) = \frac{\text{cov}(Z(x+h), Z(x))}{\text{var}(Z(x))} $$ ### Common Correlation Models | Model | Formula | |-------|---------| | **Exponential** | $\rho(h) = \exp\left( -\frac{h}{\lambda} \right)$ | | **Gaussian** | $\rho(h) = \exp\left( -\left( \frac{h}{\lambda} \right)^2 \right)$ | | **Spherical** | $\rho(h) = 1 - \frac{3h}{2\lambda} + \frac{h^3}{2\lambda^3}$ for $h \leq \lambda$ | ### Implications - Nearby devices are more correlated → better matching for analog - Correlation length $\lambda$ determines effective samples per die - Extreme values are less severe than independent variation suggests ## 9. Process Capability and Tail Statistics ### Process Capability Index $$ C_{pk} = \min \left[ \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right] $$ ### Defect Rates vs. Cpk (Gaussian) | $C_{pk}$ | PPM Outside Spec | Sigma Level | |----------|------------------|-------------| | 1.00 | ~2,700 | 3σ | | 1.33 | ~63 | 4σ | | 1.67 | ~0.6 | 5σ | | 2.00 | ~0.002 | 6σ | ### Extreme Value Statistics For $n$ independent samples from distribution $F(x)$, the maximum follows: $$ P(M_n \leq x) = [F(x)]^n $$ For large $n$, converges to Generalized Extreme Value (GEV): $$ G(x) = \exp\left\{ -\left[ 1 + \xi \left( \frac{x - \mu}{\sigma} \right) \right]^{-1/\xi} \right\} $$ ### Critical Insight For a chip with $10^{10}$ transistors: $$ P_{\text{chip fail}} = 1 - (1 - P_{\text{transistor fail}})^{10^{10}} \approx 10^{10} \cdot P_{\text{transistor fail}} $$ Even $P_{\text{transistor fail}} = 10^{-11}$ matters! ## 10. Sensitivity Analysis and Error Attribution ### Sensitivity Coefficient $$ S_i = \frac{\partial Y}{\partial \sigma_i} \times \frac{\sigma_i}{Y} $$ ### Variance Contribution $$ \text{Contribution}_i = \frac{\left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_i^2}{\sigma_f^2} \times 100\% $$ ### Bayesian Root Cause Attribution $$ P(\text{cause} \mid \text{observation}) = \frac{P(\text{observation} \mid \text{cause}) \cdot P(\text{cause})}{P(\text{observation})} $$ ### Pareto Analysis Steps 1. Compute variance contribution from each source 2. Rank sources by contribution 3. Focus improvement on top contributors 4. Verify improvement with updated measurements ## 11. Monte Carlo Simulation Methods Due to complexity and nonlinearity, Monte Carlo methods are essential. ### Algorithm ``` FOR i = 1 to N_samples: 1. Sample process parameters: p_i ~ distributions 2. Simulate device/circuit: y_i = f(p_i) 3. Store result: Y[i] = y_i END FOR Compute statistics from Y[] ``` ### Key Advantages - Captures non-Gaussian behavior - Handles nonlinear transfer functions - Reveals correlations between outputs - Provides full distribution, not just moments ### Sample Size Requirements For estimating probability $p$ of rare events: $$ N \geq \frac{1 - p}{p \cdot \varepsilon^2} $$ Where $\varepsilon$ is the desired relative error. For $p = 10^{-6}$ with 10% error: $N \approx 10^8$ samples ## 12. Design-Technology Co-Optimization (DTCO) Error propagation feeds back into design rules: $$ \text{Design Margin} = k \times \sigma_{\text{total}} $$ Where $k$ depends on required yield and number of instances. ### Margin Calculation For yield $Y$ over $N$ instances: $$ k = \Phi^{-1}\left( Y^{1/N} \right) $$ Where $\Phi^{-1}$ is the inverse normal CDF. ### Example - Target yield: 99% - Number of gates: $10^9$ - Required: $k \approx 7\sigma$ per gate ## 13. Key Mathematical Insights ### Insight 1: RSS Dominates Budgets Uncorrelated errors add in quadrature: $$ \sigma_{\text{total}} = \sqrt{\sigma_1^2 + \sigma_2^2 + \cdots + \sigma_n^2} $$ **Implication**: Reducing the largest contributor gives the most improvement. ### Insight 2: Tails Matter More Than Means High-volume manufacturing lives in the $6\sigma$ tails where: - Gaussian assumptions break down - Extreme value statistics become essential - Rare events dominate yield loss ### Insight 3: Nonlinearity Creates Surprises Even Gaussian inputs produce non-Gaussian outputs: $$ Y = f(X) \quad \text{where } X \sim N(\mu, \sigma^2) $$ If $f$ is nonlinear, $Y$ is not Gaussian. ### Insight 4: Correlations Can Help or Hurt - **Positive correlations**: Worsen tail probabilities - **Negative correlations**: Can provide compensation - **Designed-in correlations**: Can dramatically improve yield ### Insight 5: Scaling Amplifies Relative Error $$ \text{Relative Error} = \frac{\sigma}{\text{Feature Size}} $$ A 1 nm variation: - 5% of 20 nm feature - 10% of 10 nm feature - 20% of 5 nm feature ## 14. Summary Equations ### Core Error Propagation $$ \sigma_f^2 = \sum_i \left( \frac{\partial f}{\partial x_i} \right)^2 \sigma_i^2 $$ ### Yield (Negative Binomial) $$ Y = \left( 1 + \frac{D_0 A}{\alpha} \right)^{-\alpha} $$ ### Edge Placement Error $$ EPE = \sqrt{\left( \frac{\Delta CD}{2} \right)^2 + OVL^2 + \left( \frac{LER}{2} \right)^2} $$ ### Process Capability $$ C_{pk} = \min \left[ \frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma} \right] $$ ### Stochastic LER $$ LER \propto \frac{1}{\sqrt{N_{\text{photons}}}} $$
Educate personnel about ESD.
Eta sampling uses entropy-based threshold for adaptive truncation.
# Etch Film Stack Mathematical Modeling
1. Introduction and Problem Setup
A film stack in semiconductor manufacturing consists of multiple thin-film layers that must be precisely etched. Typical structures include:
- Photoresist (masking layer)
- Hard mask (SiN, SiO₂, or metal)
- Target film (material to be etched)
- Etch stop layer
- Substrate (Si wafer)
Objectives
- Remove target material at a controlled rate
- Stop precisely at interfaces (selectivity)
- Maintain profile fidelity (anisotropy, sidewall angle)
- Achieve uniformity across the wafer
2. Fundamental Etch Rate Models
2.1 Surface Reaction Kinetics
The Langmuir-Hinshelwood model captures competitive adsorption of reactive species:
$$
R = \frac{k \cdot \theta_A \cdot \theta_B}{\left(1 + K_A[A] + K_B[B]\right)^2}
$$
Where:
- $R$ = etch rate
- $k$ = reaction rate constant
- $\theta_A, \theta_B$ = fractional surface coverage of species A and B
- $K_A, K_B$ = adsorption equilibrium constants
- $[A], [B]$ = gas-phase concentrations
2.2 Temperature Dependence (Arrhenius)
$$
R = R_0 \exp\left(-\frac{E_a}{k_B T}\right)
$$
Where:
- $R_0$ = pre-exponential factor
- $E_a$ = activation energy
- $k_B$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
- $T$ = absolute temperature (K)
2.3 Ion-Enhanced Etching Model
Most plasma etching exhibits synergistic behavior—ions enhance chemical reactions:
$$
R_{total} = R_{chem} + R_{phys} + R_{synergy}
$$
The ion-enhanced component dominates in RIE/ICP:
$$
R_{ie} = Y(E, \theta) \cdot \Gamma_{ion} \cdot \Theta_{react}
$$
Where:
- $Y(E, \theta)$ = ion yield function (depends on energy $E$ and angle $\theta$)
- $\Gamma_{ion}$ = ion flux to surface (ions/cm²·s)
- $\Theta_{react}$ = fractional coverage of reactive species
3. Profile Evolution Mathematics
3.1 Level Set Method
The evolving surface is represented as the zero-contour of a level set function $\phi(\mathbf{x}, t)$:
$$
\frac{\partial \phi}{\partial t} + V(\mathbf{x}, t) \cdot |\nabla \phi| = 0
$$
Where:
- $\phi(\mathbf{x}, t)$ = level set function
- $V(\mathbf{x}, t)$ = local etch velocity (material and flux dependent)
- $\nabla \phi$ = gradient of the level set function
- $|\nabla \phi|$ = magnitude of the gradient
The surface normal is computed as:
$$
\hat{n} = \frac{\nabla \phi}{|\nabla \phi|}
$$
3.2 Visibility and Shadowing Integrals
For a point $\mathbf{p}$ inside a feature, the effective flux is:
$$
\Gamma(\mathbf{p}) = \int_{\Omega_{visible}} f(\hat{\Omega}) \cdot (\hat{\Omega} \cdot \hat{n}) \, d\Omega
$$
Where:
- $\Omega_{visible}$ = solid angle visible from point $\mathbf{p}$
- $f(\hat{\Omega})$ = ion angular distribution function (IADF)
- $\hat{n}$ = local surface normal
3.3 Ion Angular Distribution Function (IADF)
Typically modeled as a Gaussian:
$$
f(\theta) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{\theta^2}{2\sigma^2}\right)
$$
Where:
- $\theta$ = angle from surface normal
- $\sigma$ = angular spread (related to $T_i / T_e$ ratio)
4. Multi-Layer Stack Modeling
4.1 Interface Tracking
For a stack with $n$ layers at depths $z_1, z_2, \ldots, z_n$:
$$
\frac{dz_{etch}}{dt} = -R_i(t)
$$
Where $i$ indicates the current material being etched. Material transitions occur when $z_{etch}$ crosses an interface boundary.
4.2 Selectivity Definition
$$
S_{A:B} = \frac{R_A}{R_B}
$$
Design requirements:
- Mask selectivity: $S_{target:mask} < 1$ (mask erodes slowly)
- Stop layer selectivity: $S_{target:stop} \gg 1$ (typically > 10:1)
4.3 Time-to-Clear Calculation
For layer thickness $d_i$ with etch rate $R_i$:
$$
t_{clear,i} = \frac{d_i}{R_i}
$$
Total etch time through multiple layers:
$$
t_{total} = \sum_{i=1}^{n} \frac{d_i}{R_i} + t_{overetch}
$$
5. Aspect Ratio Dependent Etching (ARDE)
5.1 General ARDE Model
Etch rate decreases with aspect ratio (AR = depth/width):
$$
R(AR) = R_0 \cdot f(AR)
$$
5.2 Neutral Transport Limited (Knudsen Regime)
$$
R(AR) = \frac{R_0}{1 + \alpha \cdot AR}
$$
The Knudsen diffusivity in a cylindrical feature:
$$
D_K = \frac{d}{3}\sqrt{\frac{8 k_B T}{\pi m}}
$$
Where:
- $d$ = feature diameter
- $m$ = molecular mass of neutral species
- $T$ = gas temperature
5.3 Clausing Factor for Molecular Flow
For a tube of length $L$ and radius $r$:
$$
W = \frac{1}{1 + \frac{3L}{8r}}
$$
5.4 Ion Angular Distribution Limited
$$
R(AR) = R_0 \cdot \int_0^{\theta_{max}(AR)} f(\theta) \cos\theta \, d\theta
$$
Where $\theta_{max}$ is the maximum acceptance angle:
$$
\theta_{max} = \arctan\left(\frac{w}{2h}\right)
$$
6. Plasma and Transport Modeling
6.1 Sheath Physics
Child-Langmuir Law (Collisionless Sheath)
$$
J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M}}\frac{V_0^{3/2}}{d^2}
$$
Where:
- $J$ = ion current density
- $\varepsilon_0$ = permittivity of free space
- $e$ = electron charge
- $M$ = ion mass
- $V_0$ = sheath voltage
- $d$ = sheath thickness
Sheath Thickness (Matrix Sheath)
$$
s = \lambda_D \sqrt{\frac{2eV_0}{k_B T_e}}
$$
Where $\lambda_D$ is the Debye length:
$$
\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}
$$
6.2 Ion Flux to Surface
At the sheath edge, ions reach the Bohm velocity:
$$
u_B = \sqrt{\frac{k_B T_e}{M_i}}
$$
Ion flux:
$$
\Gamma_i = n_s \cdot u_B = n_s \sqrt{\frac{k_B T_e}{M_i}}
$$
Where $n_s \approx 0.61 \cdot n_0$ (sheath edge density).
6.3 Neutral Species Balance
Continuity equation for neutral species:
$$
\nabla \cdot (D \nabla n) + \sum_j k_j n_j n_e - k_{loss} n = 0
$$
Where:
- $D$ = diffusion coefficient
- $k_j$ = generation rate constants
- $k_{loss}$ = surface loss rate
7. Feature-Scale Monte Carlo Methods
7.1 Algorithm Overview
1. Sample particles from flux distributions at feature entrance
2. Track trajectories (ballistic for ions, random walk for neutrals)
3. Surface interactions: React, reflect, or stick with probabilities
4. Accumulate statistics for local etch rates
5. Advance surface using accumulated rates
7.2 Reflection Probability Models
Specular Reflection
$$
\theta_{out} = \theta_{in}
$$
Diffuse (Cosine) Reflection
$$
P(\theta_{out}) \propto \cos(\theta_{out})
$$
Mixed Model
$$
P_{reflect} = (1 - s) \cdot P_{specular} + s \cdot P_{diffuse}
$$
Where $s$ is the scattering coefficient.
7.3 Sticking Coefficient Model
$$
\gamma = \gamma_0 \cdot (1 - \Theta)^n
$$
Where:
- $\gamma_0$ = bare surface sticking coefficient
- $\Theta$ = surface coverage
- $n$ = reaction order
8. Loading Effects
8.1 Macroloading (Wafer Scale)
$$
R = \frac{R_0}{1 + \beta \cdot A_{exposed}}
$$
Where:
- $A_{exposed}$ = total exposed etchable area
- $\beta$ = loading coefficient
8.2 Microloading (Pattern Scale)
Local etch rate depends on pattern density $\rho$:
$$
R_{local} = R_0 \cdot \left(1 - \gamma \cdot \rho\right)
$$
Dense patterns etch slower due to local reactant depletion.
8.3 Reactive Species Depletion Model
For a feature with area $A$ in a cell of area $A_{cell}$:
$$
R = R_0 \cdot \frac{1}{1 + \frac{k_{etch} \cdot A}{k_{supply} \cdot A_{cell}}}
$$
9. Atomic Layer Etching (ALE) Models
9.1 Two-Step Process
Step 1 - Surface Modification:
$$
A_{(g)} + S_{(s)} \rightarrow A\text{-}S_{(s)}
$$
Step 2 - Removal:
$$
A\text{-}S_{(s)} + B_{(g/ion)} \rightarrow \text{volatile products}
$$
9.2 Self-Limiting Kinetics
Surface coverage during modification:
$$
\theta_{mod}(t) = 1 - \exp\left(-\Gamma_A \cdot s_A \cdot t\right)
$$
Where:
- $\Gamma_A$ = flux of modifying species
- $s_A$ = sticking probability
- $t$ = exposure time
9.3 Etch Per Cycle (EPC)
$$
EPC = \theta_{sat} \cdot \delta_{ML}
$$
Where:
- $\theta_{sat}$ = saturation coverage (ideally 1.0)
- $\delta_{ML}$ = monolayer thickness (typically 0.1–0.5 nm)
9.4 Synergy Factor
$$
S_f = \frac{EPC_{ALE}}{EPC_{step1} + EPC_{step2}}
$$
Values $S_f > 1$ indicate synergistic enhancement.
10. Process Window Modeling
10.1 Response Surface Methodology
$$
CD = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i
# Semiconductor Manufacturing Process: Etch Modeling ## 1. Introduction Etch modeling is one of the most complex and critical areas in semiconductor fabrication simulation. As device geometries shrink below $10\ \text{nm}$ and structures become increasingly three-dimensional, accurate prediction of etch behavior becomes essential for: - **Process Development**: Predict outcomes before costly fab experiments - **Yield Optimization**: Understand how variations propagate to device performance - **OPC/EPC Extension**: Compensate for etch-induced pattern distortions in mask design - **Design-Technology Co-Optimization (DTCO)**: Feed process effects back into design rules - **Virtual Metrology**: Predict wafer results from equipment sensor data in real time ## 2. Fundamentals of Etching ### 2.1 What is Etching? Etching selectively removes material from a wafer to transfer lithographically defined patterns into underlying layers—silicon, oxides, nitrides, metals, or complex stacks. ### 2.2 Types of Etching - **Wet Etching** - Uses liquid chemicals (acids, bases, solvents) - Typically isotropic (etches equally in all directions) - Etch rate follows Arrhenius relationship: $$ R = A \exp\left(-\frac{E_a}{k_B T}\right) $$ where: - $R$ = etch rate - $A$ = pre-exponential factor - $E_a$ = activation energy - $k_B$ = Boltzmann constant ($1.381 \times 10^{-23}\ \text{J/K}$) - $T$ = temperature (K) - **Dry/Plasma Etching** - Uses ionized gases (plasma) - Anisotropic (directional) - Dominant for modern processes ($< 100\ \text{nm}$ nodes) ### 2.3 Plasma Etching Mechanisms 1. **Physical Sputtering** - Ion bombardment physically removes atoms - Sputter yield $Y$ depends on ion energy $E_i$: $$ Y(E_i) = A \left( \sqrt{E_i} - \sqrt{E_{th}} \right) $$ where $E_{th}$ is the threshold energy 2. **Chemical Etching** - Reactive species form volatile products - Example: Silicon etching with fluorine $$ \text{Si} + 4\text{F} \rightarrow \text{SiF}_4 \uparrow $$ 3. **Ion-Enhanced Etching** - Synergy between ion bombardment and chemical reactions - Etch yield enhancement factor: $$ \eta = \frac{Y_{ion+chem}}{Y_{ion} + Y_{chem}} $$ ## 3. Hierarchy of Etch Models ### 3.1 Empirical Models Data-driven, fast, used in production: - **Etch Bias Models** - Simple offset correction: $$ CD_{final} = CD_{litho} + \Delta_{etch} $$ - Pattern-dependent bias: $$ \Delta_{etch} = f(\text{pitch}, \text{density}, \text{orientation}) $$ - **Etch Proximity Correction (EPC)** - Kernel-based convolution: $$ \Delta(x,y) = \iint K(x-x', y-y') \cdot I(x', y') \, dx' dy' $$ - Where $K$ is the etch kernel and $I$ is the pattern intensity - **Machine Learning Models** - Neural networks trained on metrology data - Gaussian process regression for uncertainty quantification ### 3.2 Feature-Scale Models Semi-empirical, balance speed and physics: - **String/Segment Models** - Represent edges as connected nodes - Each node moves according to local etch rate vector: $$ \frac{d\vec{r}_i}{dt} = R(\theta_i, \Gamma_{ion}, \Gamma_{n}) \cdot \hat{n}_i $$ - Where: - $\vec{r}_i$ = position of node $i$ - $\theta_i$ = local surface angle - $\Gamma_{ion}$, $\Gamma_n$ = ion and neutral fluxes - $\hat{n}_i$ = surface normal - **Level-Set Methods** - Track surface as zero-contour of signed distance function $\phi$: $$ \frac{\partial \phi}{\partial t} + R(\vec{x}) |\nabla \phi| = 0 $$ - Handles topology changes naturally (merging, splitting) - **Cell-Based/Voxel Methods** - Discretize feature volume into cells - Apply probabilistic removal rules: $$ P_{remove} = 1 - \exp\left( -\sum_j \sigma_j \Gamma_j \Delta t \right) $$ - Where $\sigma_j$ is the reaction cross-section for species $j$ ### 3.3 Physics-Based Plasma Models Capture reactor-scale phenomena: - **Plasma Bulk** - Electron energy distribution function (EEDF) - Boltzmann equation: $$ \frac{\partial f}{\partial t} + \vec{v} \cdot \nabla f + \frac{q\vec{E}}{m} \cdot \nabla_v f = \left( \frac{\partial f}{\partial t} \right)_{coll} $$ - **Sheath Physics** - Child-Langmuir law for ion flux: $$ J_{ion} = \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{M}} \frac{V^{3/2}}{d^2} $$ - Ion angular distribution at wafer surface - **Transport** - Species continuity: $$ \frac{\partial n_i}{\partial t} + \nabla \cdot (n_i \vec{v}_i) = S_i - L_i $$ - Where $S_i$ and $L_i$ are source and loss terms ### 3.4 Atomistic Models Fundamental understanding, computationally expensive: - **Molecular Dynamics (MD)** - Newton's equations for all atoms: $$ m_i \frac{d^2 \vec{r}_i}{dt^2} = -\nabla_i U(\{\vec{r}\}) $$ - Interatomic potentials: Tersoff, Stillinger-Weber, ReaxFF - **Monte Carlo (MC) Methods** - Statistical sampling of ion trajectories - Binary collision approximation (BCA) for high energies - Acceptance probability: $$ P = \min\left(1, \exp\left(-\frac{\Delta E}{k_B T}\right)\right) $$ - **Kinetic Monte Carlo (KMC)** - Sample reactive events with rates $k_i$: $$ k_i = \nu_0 \exp\left(-\frac{E_{a,i}}{k_B T}\right) $$ - Event selection: $\sum_{j < i} k_j < r \cdot K_{tot} \leq \sum_{j \leq i} k_j$ ## 4. Key Physical Phenomena ### 4.1 Anisotropy Ratio of vertical to lateral etch rate: $$ A = 1 - \frac{R_{lateral}}{R_{vertical}} $$ - $A = 1$: Perfectly anisotropic (vertical sidewalls) - $A = 0$: Perfectly isotropic **Mechanisms for achieving anisotropy:** - Directional ion bombardment - Sidewall passivation (polymer deposition) - Low pressure operation (fewer collisions → more directional ions) - Ion angular distribution characterized by: $$ f(\theta) \propto \cos^n(\theta) $$ where higher $n$ indicates more directional flux ### 4.2 Selectivity Ratio of etch rates between materials: $$ S_{A/B} = \frac{R_A}{R_B} $$ - **Mask selectivity**: Target material vs. photoresist/hard mask - **Stop layer selectivity**: Target material vs. underlying layer Example selectivities required: | Process | Selectivity Required | |---------|---------------------| | Oxide/Nitride | $> 20:1$ | | Poly-Si/Oxide | $> 50:1$ | | Si/SiGe (channel release) | $> 100:1$ | ### 4.3 Loading Effects #### Microloading Local depletion of reactive species in dense pattern regions: $$ R_{dense} = R_0 \cdot \frac{1}{1 + \beta \cdot \rho_{local}} $$ where: - $R_0$ = etch rate in isolated feature - $\beta$ = loading coefficient - $\rho_{local}$ = local pattern density #### Macroloading Wafer-scale depletion: $$ R = R_0 \cdot \left(1 - \alpha \cdot A_{exposed}\right) $$ where $A_{exposed}$ is total exposed area fraction ### 4.4 Aspect Ratio Dependent Etching (ARDE) Deep, narrow features etch slower due to transport limitations: $$ R(AR) = R_0 \cdot \exp\left(-\frac{AR}{AR_0}\right) $$ where $AR = \text{depth}/\text{width}$ **Physical mechanisms:** 1. **Ion Shadowing** - Geometric shadowing angle: $$ \theta_{shadow} = \arctan\left(\frac{1}{AR}\right) $$ 2. **Neutral Transport** - Knudsen diffusion coefficient: $$ D_K = \frac{d}{3} \sqrt{\frac{8 k_B T}{\pi m}} $$ - where $d$ is feature diameter 3. **Byproduct Redeposition** - Sticking probability affects escape ### 4.5 Profile Anomalies | Phenomenon | Description | Cause | |------------|-------------|-------| | **Bowing** | Lateral bulge in sidewall | Ion scattering off sidewalls | | **Notching** | Lateral etching at interface | Charge buildup on insulators | | **Microtrenching** | Deep spots at corners | Ion reflection at feature bottom | | **Footing** | Undercut at bottom | Isotropic chemical component | | **Tapering** | Non-vertical sidewalls | Insufficient passivation | ## 5. Mathematical Foundations ### 5.1 Surface Evolution Equation General form for surface height $h(x,y,t)$: $$ \frac{\partial h}{\partial t} = -R_0 \cdot V(\theta) \cdot \sqrt{1 + |\nabla h|^2} $$ where: - $R_0$ = baseline etch rate - $V(\theta)$ = visibility/flux function - $\theta = \arctan(|\nabla h|)$ ### 5.2 Ion Angular Distribution At wafer surface, ion flux angular distribution: $$ \Gamma(\theta, \phi) = \Gamma_0 \cdot f(\theta) \cdot g(E) $$ Common models: - **Gaussian distribution:** $$ f(\theta) = \frac{1}{\sqrt{2\pi}\sigma_\theta} \exp\left(-\frac{\theta^2}{2\sigma_\theta^2}\right) $$ - **Thompson distribution** (for sputtered neutrals): $$ f(E) \propto \frac{E}{(E + E_b)^3} $$ ### 5.3 Visibility Calculation For a point on the surface, visibility to incoming flux: $$ V(\vec{r}) = \frac{1}{2\pi} \int_0^{2\pi} \int_0^{\theta_{max}(\phi)} f(\theta) \sin\theta \cos\theta \, d\theta \, d\phi $$ where $\theta_{max}(\phi)$ is determined by local geometry (shadowing) ### 5.4 Surface Reaction Kinetics Langmuir-Hinshelwood mechanism: $$ R = k \cdot \theta_A \cdot \theta_B $$ where surface coverages follow: $$ \frac{d\theta_i}{dt} = s_i \Gamma_i (1 - \theta_{total}) - k_d \theta_i - k_r \theta_i $$ - $s_i$ = sticking coefficient - $k_d$ = desorption rate - $k_r$ = reaction rate ### 5.5 Plasma-Surface Interaction Yield Ion-enhanced etch yield: $$ Y_{etch} = Y_0 + Y_1 \cdot \sqrt{E_{ion} - E_{th}} + Y_{chem} \cdot \frac{\Gamma_n}{\Gamma_{ion}} $$ where: - $Y_0$ = chemical baseline yield - $Y_1$ = ion enhancement coefficient - $E_{th}$ = threshold energy (~15-50 eV typically) - $Y_{chem}$ = chemical enhancement factor ## 6. Modern Modeling Approaches ### 6.1 Hybrid Multi-Scale Frameworks Coupling different scales: ``` - ┌─────────────────────────────────────────────────────────────┐ │ REACTOR SCALE │ │ Plasma simulation (fluid or PIC) │ │ Output: Ion/neutral fluxes, energies, angular dist. │ └────────────────────────┬────────────────────────────────────┘ │ Boundary conditions ▼ ┌─────────────────────────────────────────────────────────────┐ │ FEATURE SCALE │ │ Level-set or Monte Carlo │ │ Output: Profile evolution, etch rates │ └────────────────────────┬────────────────────────────────────┘ │ Parameter extraction ▼ ┌─────────────────────────────────────────────────────────────┐ │ ATOMISTIC SCALE │ │ MD/KMC simulations │ │ Output: Sticking coefficients, sputter yields │ └─────────────────────────────────────────────────────────────┘ ``` ### 6.2 Machine Learning Integration - **Surrogate Models** - Train neural network on physics simulation outputs: $$ \hat{y} = f_{NN}(\vec{x}; \vec{w}) $$ - Loss function: $$ \mathcal{L} = \frac{1}{N} \sum_{i=1}^{N} \|y_i - \hat{y}_i\|^2 + \lambda \|\vec{w}\|^2 $$ - **Physics-Informed Neural Networks (PINNs)** - Embed physics constraints in loss: $$ \mathcal{L}_{total} = \mathcal{L}_{data} + \alpha \mathcal{L}_{physics} $$ - Where $\mathcal{L}_{physics}$ enforces governing equations - **Virtual Metrology** - Predict CD, profile from chamber sensors: $$ CD_{predicted} = g(P, T, V_{bias}, \text{OES}, ...) $$ ### 6.3 Computational Lithography Integration Major EDA tools couple lithography + etch: 1. Litho simulation → Resist profile $h_R(x,y)$ 2. Etch simulation → Final pattern $h_F(x,y)$ 3. Combined model: $$ CD_{final} = CD_{design} + \Delta_{OPC} + \Delta_{litho} + \Delta_{etch} $$ ## 7. Challenges at Advanced Nodes ### 7.1 FinFET / Gate-All-Around (GAA) - **Fin Etch** - Sidewall angle uniformity: $90° \pm 1°$ - Width control: $\pm 1\ \text{nm}$ at $W_{fin} < 10\ \text{nm}$ - **Channel Release** - Selective SiGe vs. Si etching - Required selectivity: $> 100:1$ - Etch rate: $$ R_{SiGe} \gg R_{Si} $$ - **Inner Spacer Formation** - Isotropic lateral etch in confined geometry - Depth control: $\pm 0.5\ \text{nm}$ ### 7.2 3D NAND Extreme aspect ratio challenges: | Generation | Layers | Aspect Ratio | |------------|--------|--------------| | 96L | 96 | ~60:1 | | 128L | 128 | ~80:1 | | 176L | 176 | ~100:1 | | 232L+ | 232+ | ~150:1 | Critical issues: - ARDE variation across depth - Bowing control - Twisting in elliptical holes ### 7.3 EUV Patterning - Very thin resists: $< 40\ \text{nm}$ - Hard mask stacks with multiple layers - LER/LWR amplification: $$ LER_{final} = \sqrt{LER_{litho}^2 + LER_{etch}^2} $$ - Target: $LER < 1.2\ \text{nm}$ ($3\sigma$) ### 7.4 Stochastic Effects At small dimensions, statistical fluctuations dominate: $$ \sigma_{CD} \propto \frac{1}{\sqrt{N_{events}}} $$ where $N_{events}$ = number of etching events per feature ## 8. Industry Tools ### 8.1 Commercial Software | Category | Tools | |----------|-------| | **TCAD/Process** | Synopsys Sentaurus Process, Silvaco Victory Process | | **Virtual Fab** | Coventor SEMulator3D | | **Equipment Vendor** | Lam Research, Applied Materials (proprietary) | | **Computational Litho** | Synopsys S-Litho, Siemens Calibre | ### 8.2 Research Tools - **MCFPM** (Monte Carlo Feature Profile Model) - University of Illinois - **LAMMPS** - Molecular dynamics - **SPARTA** - Direct Simulation Monte Carlo - **OpenFOAM** - Plasma fluid modeling ## 9. Future Directions ### 9.1 Digital Twins Real-time chamber models for closed-loop process control: $$ \vec{u}_{control}(t) = \mathcal{K} \left[ y_{target} - y_{model}(t) \right] $$ ### 9.2 Atomistic-Continuum Coupling Seamless multi-scale simulation using: - Adaptive mesh refinement - Concurrent coupling methods - Machine-learned interscale bridging ### 9.3 New Materials Modeling requirements for: - 2D materials (graphene, MoS$_2$, WS$_2$) - High-$\kappa$ dielectrics - Ferroelectrics (HfZrO) - High-mobility channels (InGaAs, Ge) ### 9.4 Uncertainty Quantification Predicting distributions, not just means: $$ P(CD) = \int P(CD | \vec{\theta}) P(\vec{\theta}) d\vec{\theta} $$ Key metrics: - Process capability: $C_{pk} = \frac{\min(USL - \mu, \mu - LSL)}{3\sigma}$ - Target: $C_{pk} > 1.67$ for production ## Summary Etch modeling spans from atomic-scale surface reactions to reactor-scale plasma physics to fab-level empirical correlations. The art lies in choosing the right abstraction level: | Application | Model Type | Speed | Accuracy | |-------------|------------|-------|----------| | Production OPC/EPC | Empirical/ML | ★★★★★ | ★★☆☆☆ | | Process Development | Feature-scale | ★★★☆☆ | ★★★★☆ | | Mechanism Research | Atomistic MD/MC | ★☆☆☆☆ | ★★★★★ | | Equipment Design | Plasma + Feature | ★★☆☆☆ | ★★★★☆ | As geometries shrink and structures become more 3D, accurate etch modeling becomes essential for first-time-right process development and continued yield improvement.
# Mathematical Modeling of Plasma Etching in Semiconductor Manufacturing ## Introduction Plasma etching is a critical process in semiconductor manufacturing where reactive gases are ionized to create a plasma, which selectively removes material from a wafer surface. The mathematical modeling of this process spans multiple physics domains: - **Electromagnetic theory** — RF power coupling and field distributions - **Statistical mechanics** — Particle distributions and kinetic theory - **Reaction kinetics** — Gas-phase and surface chemistry - **Transport phenomena** — Species diffusion and convection - **Surface science** — Etch mechanisms and selectivity ## Foundational Plasma Physics ### Boltzmann Transport Equation The most fundamental description of plasma behavior is the **Boltzmann transport equation**, governing the evolution of the particle velocity distribution function $f(\mathbf{r}, \mathbf{v}, t)$: $$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{collision}} $$ **Where:** - $f(\mathbf{r}, \mathbf{v}, t)$ — Velocity distribution function - $\mathbf{v}$ — Particle velocity - $\mathbf{F}$ — External force (electromagnetic) - $m$ — Particle mass - RHS — Collision integral ### Fluid Moment Equations For computational tractability, velocity moments of the Boltzmann equation yield fluid equations: #### Continuity Equation (Mass Conservation) $$ \frac{\partial n}{\partial t} + \nabla \cdot (n\mathbf{u}) = S - L $$ **Where:** - $n$ — Species number density $[\text{m}^{-3}]$ - $\mathbf{u}$ — Drift velocity $[\text{m/s}]$ - $S$ — Source term (generation rate) - $L$ — Loss term (consumption rate) #### Momentum Conservation $$ \frac{\partial (nm\mathbf{u})}{\partial t} + \nabla \cdot (nm\mathbf{u}\mathbf{u}) + \nabla p = nq(\mathbf{E} + \mathbf{u} \times \mathbf{B}) - nm\nu_m \mathbf{u} $$ **Where:** - $p = nk_BT$ — Pressure - $q$ — Particle charge - $\mathbf{E}$, $\mathbf{B}$ — Electric and magnetic fields - $\nu_m$ — Momentum transfer collision frequency $[\text{s}^{-1}]$ #### Energy Conservation $$ \frac{\partial}{\partial t}\left(\frac{3}{2}nk_BT\right) + \nabla \cdot \mathbf{q} + p\nabla \cdot \mathbf{u} = Q_{\text{heating}} - Q_{\text{loss}} $$ **Where:** - $k_B = 1.38 \times 10^{-23}$ J/K — Boltzmann constant - $\mathbf{q}$ — Heat flux vector - $Q_{\text{heating}}$ — Power input (Joule heating, stochastic heating) - $Q_{\text{loss}}$ — Energy losses (collisions, radiation) ## Electromagnetic Field Coupling ### Maxwell's Equations For capacitively coupled plasma (CCP) and inductively coupled plasma (ICP) reactors: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ $$ \nabla \cdot \mathbf{D} = \rho $$ $$ \nabla \cdot \mathbf{B} = 0 $$ ### Plasma Conductivity The plasma current density couples through the complex conductivity: $$ \mathbf{J} = \sigma \mathbf{E} $$ For RF plasmas, the **complex conductivity** is: $$ \sigma = \frac{n_e e^2}{m_e(\nu_m + i\omega)} $$ **Where:** - $n_e$ — Electron density - $e = 1.6 \times 10^{-19}$ C — Elementary charge - $m_e = 9.1 \times 10^{-31}$ kg — Electron mass - $\omega$ — RF angular frequency - $\nu_m$ — Electron-neutral collision frequency ### Power Deposition Time-averaged power density deposited into the plasma: $$ P = \frac{1}{2}\text{Re}(\mathbf{J} \cdot \mathbf{E}^*) $$ **Typical values:** - CCP: $0.1 - 1$ W/cm³ - ICP: $0.5 - 5$ W/cm³ ## Plasma Sheath Physics The sheath is a thin, non-neutral region at the plasma-wafer interface that accelerates ions toward the surface, enabling anisotropic etching. ### Bohm Criterion Minimum ion velocity entering the sheath: $$ u_i \geq u_B = \sqrt{\frac{k_B T_e}{M_i}} $$ **Where:** - $u_B$ — Bohm velocity - $T_e$ — Electron temperature (typically 2–5 eV) - $M_i$ — Ion mass **Example:** For Ar⁺ ions with $T_e = 3$ eV: $$ u_B = \sqrt{\frac{3 \times 1.6 \times 10^{-19}}{40 \times 1.67 \times 10^{-27}}} \approx 2.7 \text{ km/s} $$ ### Child-Langmuir Law For a collisionless sheath, the ion current density is: $$ J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M_i}} \cdot \frac{V_s^{3/2}}{d^2} $$ **Where:** - $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m — Vacuum permittivity - $V_s$ — Sheath voltage drop (typically 10–500 V) - $d$ — Sheath thickness ### Sheath Thickness The sheath thickness scales as: $$ d \approx \lambda_D \left(\frac{2eV_s}{k_BT_e}\right)^{3/4} $$ **Where** the Debye length is: $$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$ ### Ion Angular Distribution Ions arrive at the wafer with an angular distribution: $$ f(\theta) \propto \exp\left(-\frac{\theta^2}{2\sigma^2}\right) $$ **Where:** $$ \sigma \approx \arctan\left(\sqrt{\frac{k_B T_i}{eV_s}}\right) $$ **Typical values:** $\sigma \approx 2°–5°$ for high-bias conditions. ## Electron Energy Distribution Function ### Non-Maxwellian Distributions In low-pressure plasmas (1–100 mTorr), the EEDF deviates from Maxwellian. #### Two-Term Approximation The EEDF is expanded as: $$ f(\varepsilon, \theta) = f_0(\varepsilon) + f_1(\varepsilon)\cos\theta $$ The isotropic part $f_0$ satisfies: $$ \frac{d}{d\varepsilon}\left[\varepsilon D \frac{df_0}{d\varepsilon} + \left(V + \frac{\varepsilon\nu_{\text{inel}}}{\nu_m}\right)f_0\right] = 0 $$ #### Common Distribution Functions | Distribution | Functional Form | Applicability | |-------------|-----------------|---------------| | **Maxwellian** | $f(\varepsilon) \propto \sqrt{\varepsilon} \exp\left(-\frac{\varepsilon}{k_BT_e}\right)$ | High pressure, collisional | | **Druyvesteyn** | $f(\varepsilon) \propto \sqrt{\varepsilon} \exp\left(-\left(\frac{\varepsilon}{k_BT_e}\right)^2\right)$ | Elastic collisions dominant | | **Bi-Maxwellian** | Sum of two Maxwellians | Hot tail population | ### Generalized Form $$ f(\varepsilon) \propto \sqrt{\varepsilon} \cdot \exp\left[-\left(\frac{\varepsilon}{k_BT_e}\right)^x\right] $$ - $x = 1$ → Maxwellian - $x = 2$ → Druyvesteyn ## Plasma Chemistry and Reaction Kinetics ### Species Balance Equation For species $i$: $$ \frac{\partial n_i}{\partial t} + \nabla \cdot \mathbf{\Gamma}_i = \sum_j R_j $$ **Where:** - $\mathbf{\Gamma}_i$ — Species flux - $R_j$ — Reaction rates ### Electron-Impact Rate Coefficients Rate coefficients are calculated by integration over the EEDF: $$ k = \int_0^\infty \sigma(\varepsilon) v(\varepsilon) f(\varepsilon) \, d\varepsilon = \langle \sigma v \rangle $$ **Where:** - $\sigma(\varepsilon)$ — Energy-dependent cross-section $[\text{m}^2]$ - $v(\varepsilon) = \sqrt{2\varepsilon/m_e}$ — Electron velocity - $f(\varepsilon)$ — Normalized EEDF ### Heavy-Particle Reactions Arrhenius kinetics for neutral reactions: $$ k = A T^n \exp\left(-\frac{E_a}{k_BT}\right) $$ **Where:** - $A$ — Pre-exponential factor - $n$ — Temperature exponent - $E_a$ — Activation energy ### Example: SF₆/O₂ Plasma Chemistry #### Electron-Impact Reactions | Reaction | Type | Threshold | |----------|------|-----------| | $e + \text{SF}_6 \rightarrow \text{SF}_5 + \text{F} + e$ | Dissociation | ~10 eV | | $e + \text{SF}_6 \rightarrow \text{SF}_6^-$ | Attachment | ~0 eV | | $e + \text{SF}_6 \rightarrow \text{SF}_5^+ + \text{F} + 2e$ | Ionization | ~16 eV | | $e + \text{O}_2 \rightarrow \text{O} + \text{O} + e$ | Dissociation | ~6 eV | #### Gas-Phase Reactions - $\text{F} + \text{O} \rightarrow \text{FO}$ (reduces F atom density) - $\text{SF}_5 + \text{F} \rightarrow \text{SF}_6$ (recombination) - $\text{O} + \text{CF}_3 \rightarrow \text{COF}_2 + \text{F}$ (polymer removal) #### Surface Reactions - $\text{F} + \text{Si}(s) \rightarrow \text{SiF}_{(\text{ads})}$ - $\text{SiF}_{(\text{ads})} + 3\text{F} \rightarrow \text{SiF}_4(g)$ (volatile product) ## Transport Phenomena ### Drift-Diffusion Model For charged species, the flux is: $$ \mathbf{\Gamma} = \pm \mu n \mathbf{E} - D \nabla n $$ **Where:** - Upper sign: positive ions - Lower sign: electrons - $\mu$ — Mobility $[\text{m}^2/(\text{V}\cdot\text{s})]$ - $D$ — Diffusion coefficient $[\text{m}^2/\text{s}]$ ### Einstein Relation Connects mobility and diffusion: $$ D = \frac{\mu k_B T}{e} $$ ### Ambipolar Diffusion When quasi-neutrality holds ($n_e \approx n_i$): $$ D_a = \frac{\mu_i D_e + \mu_e D_i}{\mu_i + \mu_e} \approx D_i\left(1 + \frac{T_e}{T_i}\right) $$ Since $T_e \gg T_i$ typically: $D_a \approx D_i (1 + T_e/T_i) \approx 100 D_i$ ### Neutral Transport For reactive neutrals (radicals), Fickian diffusion: $$ \frac{\partial n}{\partial t} = D\nabla^2 n + S - L $$ #### Surface Boundary Condition $$ -D\frac{\partial n}{\partial x}\bigg|_{\text{surface}} = \frac{1}{4}\gamma n v_{\text{th}} $$ **Where:** - $\gamma$ — Sticking/reaction coefficient (0 to 1) - $v_{\text{th}} = \sqrt{\frac{8k_BT}{\pi m}}$ — Thermal velocity ### Knudsen Number Determines the appropriate transport regime: $$ \text{Kn} = \frac{\lambda}{L} $$ **Where:** - $\lambda$ — Mean free path - $L$ — Characteristic length | Kn Range | Regime | Model | |----------|--------|-------| | $< 0.01$ | Continuum | Navier-Stokes | | $0.01–0.1$ | Slip flow | Modified N-S | | $0.1–10$ | Transition | DSMC/BGK | | $> 10$ | Free molecular | Ballistic | ## Surface Reaction Modeling ### Langmuir Adsorption Kinetics For surface coverage $\theta$: $$ \frac{d\theta}{dt} = k_{\text{ads}}(1-\theta)P - k_{\text{des}}\theta - k_{\text{react}}\theta $$ **At steady state:** $$ \theta = \frac{k_{\text{ads}}P}{k_{\text{ads}}P + k_{\text{des}} + k_{\text{react}}} $$ ### Ion-Enhanced Etching The total etch rate combines multiple mechanisms: $$ \text{ER} = Y_{\text{chem}} \Gamma_n + Y_{\text{phys}} \Gamma_i + Y_{\text{syn}} \Gamma_i f(\theta) $$ **Where:** - $Y_{\text{chem}}$ — Chemical etch yield (isotropic) - $Y_{\text{phys}}$ — Physical sputtering yield - $Y_{\text{syn}}$ — Ion-enhanced (synergistic) yield - $\Gamma_n$, $\Gamma_i$ — Neutral and ion fluxes - $f(\theta)$ — Coverage-dependent function ### Ion Sputtering Yield #### Energy Dependence $$ Y(E) = A\left(\sqrt{E} - \sqrt{E_{\text{th}}}\right) \quad \text{for } E > E_{\text{th}} $$ **Typical threshold energies:** - Si: $E_{\text{th}} \approx 20$ eV - SiO₂: $E_{\text{th}} \approx 30$ eV - Si₃N₄: $E_{\text{th}} \approx 25$ eV #### Angular Dependence $$ Y(\theta) = Y(0) \cos^{-f}(\theta) \exp\left[-b\left(\frac{1}{\cos\theta} - 1\right)\right] $$ **Behavior:** - Increases from normal incidence - Peaks at $\theta \approx 60°–70°$ - Decreases at grazing angles (reflection dominates) ## Feature-Scale Profile Evolution ### Level Set Method The surface is represented as the zero contour of $\phi(\mathbf{x}, t)$: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ **Where:** - $\phi > 0$ — Material - $\phi < 0$ — Void/vacuum - $\phi = 0$ — Surface - $V_n$ — Local normal etch velocity ### Local Etch Rate Calculation The normal velocity $V_n$ depends on: 1. **Ion flux and angular distribution** $$\Gamma_i(\mathbf{x}) = \int f(\theta, E) \, d\Omega \, dE$$ 2. **Neutral flux** (with shadowing) $$\Gamma_n(\mathbf{x}) = \Gamma_{n,0} \cdot \text{VF}(\mathbf{x})$$ where VF is the view factor 3. **Surface chemistry state** $$V_n = f(\Gamma_i, \Gamma_n, \theta_{\text{coverage}}, T)$$ ### Neutral Transport in High-Aspect-Ratio Features #### Clausing Transmission Factor For a tube of aspect ratio AR: $$ K \approx \frac{1}{1 + 0.5 \cdot \text{AR}} $$ #### View Factor Calculations For surface element $dA_1$ seeing $dA_2$: $$ F_{1 \rightarrow 2} = \frac{1}{\pi} \int \frac{\cos\theta_1 \cos\theta_2}{r^2} \, dA_2 $$ ## Monte Carlo Methods ### Test-Particle Monte Carlo Algorithm ``` 1. SAMPLE incident particle from flux distribution at feature opening - Ion: from IEDF and IADF - Neutral: from Maxwellian 2. TRACE trajectory through feature - Ion: ballistic, solve equation of motion - Neutral: random walk with wall collisions 3. DETERMINE reaction at surface impact - Sample from probability distribution - Update surface coverage if adsorption 4. UPDATE surface geometry - Remove material (etching) - Add material (deposition) 5. REPEAT for statistically significant sample ``` ### Ion Trajectory Integration Through the sheath/feature: $$ m\frac{d^2\mathbf{r}}{dt^2} = q\mathbf{E}(\mathbf{r}) $$ **Numerical integration:** Velocity-Verlet or Boris algorithm ### Collision Sampling Null-collision method for efficiency: $$ P_{\text{collision}} = 1 - \exp(-\nu_{\text{max}} \Delta t) $$ **Where** $\nu_{\text{max}}$ is the maximum possible collision frequency. ## Multi-Scale Modeling Framework ### Scale Hierarchy | Scale | Length | Time | Physics | Method | |-------|--------|------|---------|--------| | **Reactor** | cm–m | ms–s | Plasma transport, EM fields | Fluid PDE | | **Sheath** | µm–mm | µs–ms | Ion acceleration, EEDF | Kinetic/Fluid | | **Feature** | nm–µm | ns–ms | Profile evolution | Level set/MC | | **Atomic** | Å–nm | ps–ns | Reaction mechanisms | MD/DFT | ### Coupling Approaches #### Hierarchical (One-Way) ``` Atomic scale → Surface parameters ↓ Feature scale ← Fluxes from reactor scale ↓ Reactor scale → Process outputs ``` #### Concurrent (Two-Way) - Feature-scale results feed back to reactor scale - Requires iterative solution - Computationally expensive ## Numerical Methods and Challenges ### Stiff ODE Systems Plasma chemistry involves timescales spanning many orders of magnitude: | Process | Timescale | |---------|-----------| | Electron attachment | $\sim 10^{-10}$ s | | Ion-molecule reactions | $\sim 10^{-6}$ s | | Metastable decay | $\sim 10^{-3}$ s | | Surface diffusion | $\sim 10^{-1}$ s | #### Implicit Methods Required **Backward Differentiation Formula (BDF):** $$ y_{n+1} = \sum_{j=0}^{k-1} \alpha_j y_{n-j} + h\beta f(t_{n+1}, y_{n+1}) $$ ### Spatial Discretization #### Finite Volume Method Ensures mass conservation: $$ \int_V \frac{\partial n}{\partial t} dV + \oint_S \mathbf{\Gamma} \cdot d\mathbf{S} = \int_V S \, dV $$ #### Mesh Requirements - Sheath resolution: $\Delta x < \lambda_D$ - RF skin depth: $\Delta x < \delta$ - Adaptive mesh refinement (AMR) common ### EM-Plasma Coupling **Iterative scheme:** 1. Solve Maxwell's equations for $\mathbf{E}$, $\mathbf{B}$ 2. Update plasma transport (density, temperature) 3. Recalculate $\sigma$, $\varepsilon_{\text{plasma}}$ 4. Repeat until convergence ## Advanced Topics ### Atomic Layer Etching (ALE) Self-limiting reactions for atomic precision: $$ \text{EPC} = \Theta \cdot d_{\text{ML}} $$ **Where:** - EPC — Etch per cycle - $\Theta$ — Modified layer coverage fraction - $d_{\text{ML}}$ — Monolayer thickness #### ALE Cycle 1. **Modification step:** Reactive gas creates modified surface layer $$\frac{d\Theta}{dt} = k_{\text{mod}}(1-\Theta)P_{\text{gas}}$$ 2. **Removal step:** Ion bombardment removes modified layer only $$\text{ER} = Y_{\text{mod}}\Gamma_i\Theta$$ ### Pulsed Plasma Dynamics Time-modulated RF introduces: - **Active glow:** Plasma on, high ion/radical generation - **Afterglow:** Plasma off, selective chemistry #### Ion Energy Modulation By pulsing bias: $$ \langle E_i \rangle = \frac{1}{T}\left[\int_0^{t_{\text{on}}} E_{\text{high}}dt + \int_{t_{\text{on}}}^{T} E_{\text{low}}dt\right] $$ ### High-Aspect-Ratio Etching (HAR) For AR > 50 (memory, 3D NAND): **Challenges:** - Ion angular broadening → bowing - Neutral depletion at bottom - Feature charging → twisting - Mask erosion → tapering **Ion Angular Distribution Broadening:** $$ \sigma_{\text{effective}} = \sqrt{\sigma_{\text{sheath}}^2 + \sigma_{\text{scattering}}^2} $$ **Neutral Flux at Bottom:** $$ \Gamma_{\text{bottom}} \approx \Gamma_{\text{top}} \cdot K(\text{AR}) $$ ### Machine Learning Integration **Applications:** - Surrogate models for fast prediction - Process optimization (Bayesian) - Virtual metrology - Anomaly detection **Physics-Informed Neural Networks (PINNs):** $$ \mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}} $$ Where $\mathcal{L}_{\text{physics}}$ enforces governing equations. ## Validation and Experimental Techniques ### Plasma Diagnostics | Technique | Measurement | Typical Values | |-----------|-------------|----------------| | **Langmuir probe** | $n_e$, $T_e$, EEDF | $10^{9}–10^{12}$ cm⁻³, 1–5 eV | | **OES** | Relative species densities | Qualitative/semi-quantitative | | **APMS** | Ion mass, energy | 1–500 amu, 0–500 eV | | **LIF** | Absolute radical density | $10^{11}–10^{14}$ cm⁻³ | | **Microwave interferometry** | $n_e$ (line-averaged) | $10^{10}–10^{12}$ cm⁻³ | ### Etch Characterization - **Profilometry:** Etch depth, uniformity - **SEM/TEM:** Feature profiles, sidewall angle - **XPS:** Surface composition - **Ellipsometry:** Film thickness, optical properties ### Model Validation Workflow 1. **Plasma validation:** Match $n_e$, $T_e$, species densities 2. **Flux validation:** Compare ion/neutral fluxes to wafer 3. **Etch rate validation:** Blanket wafer etch rates 4. **Profile validation:** Patterned feature cross-sections ## Key Dimensionless Numbers Summary | Number | Definition | Physical Meaning | |--------|------------|------------------| | **Knudsen** | $\text{Kn} = \lambda/L$ | Continuum vs. kinetic | | **Damköhler** | $\text{Da} = \tau_{\text{transport}}/\tau_{\text{reaction}}$ | Transport vs. reaction limited | | **Sticking coefficient** | $\gamma = \text{reactions}/\text{collisions}$ | Surface reactivity | | **Aspect ratio** | $\text{AR} = \text{depth}/\text{width}$ | Feature geometry | | **Debye number** | $N_D = n\lambda_D^3$ | Plasma ideality | ## Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Elementary charge | $e$ | $1.602 \times 10^{-19}$ C | | Electron mass | $m_e$ | $9.109 \times 10^{-31}$ kg | | Proton mass | $m_p$ | $1.673 \times 10^{-27}$ kg | | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ J/K | | Vacuum permittivity | $\varepsilon_0$ | $8.854 \times 10^{-12}$ F/m | | Vacuum permeability | $\mu_0$ | $4\pi \times 10^{-7}$ H/m |
# Etch Profile Mathematical Modeling 1. Introduction Plasma etching is a critical step in semiconductor manufacturing where material is selectively removed from a wafer surface. The etch profile—the geometric shape of the etched feature—directly determines device performance, especially as feature sizes shrink below 5 nm. 1.1 Types of Etching - Wet Etching: Uses liquid chemicals; typically isotropic; rarely used for advanced patterning - Dry/Plasma Etching: Uses reactive gases and plasma; can be highly anisotropic; dominant in modern fabrication 1.2 Key Profile Characteristics to Model - Sidewall angle: Ideally $90°$ for anisotropic etching - Etch depth: Controlled by time and etch rate - Undercut: Lateral etching beneath the mask - Taper: Deviation from vertical sidewalls - Bowing: Curved sidewall profile (mid-depth widening) - Notching: Localized undercutting at material interfaces - ARDE: Aspect Ratio Dependent Etching—etch rate variation with feature dimensions - Loading effects: Pattern-density-dependent etch rates 2. Surface Evolution Equations The challenge is tracking a moving boundary under spatially varying, angle-dependent removal rates. 2.1 Level Set Method The surface is the zero level set of $\phi(\mathbf{x}, t)$: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ Key quantities: - Unit normal: $\hat{n} = \nabla \phi / |\nabla \phi|$ - Mean curvature: $\kappa = \nabla \cdot \hat{n} = \nabla \cdot (\nabla \phi / |\nabla \phi|)$ 2.2 Advantages - Handles topology changes (merge/split) - Well-defined normals/curvature everywhere - Extends naturally to 3D 2.3 Numerical Notes - Reinitialize to maintain $|\nabla \phi| = 1$ - Upwind schemes (Godunov, ENO/WENO) for stability - Fast Marching and Sparse Field are common 2.4 String/Segment Method (2D) $$ \frac{d\mathbf{r}_i}{dt} = V_n(\mathbf{r}_i) \cdot \hat{n}_i $$ - Advantage: simple implementation - Disadvantage: struggles with topology changes 3. Etch Velocity Models Velocity decomposition: $$ V_n = V_{\text{physical}} + V_{\text{chemical}} + V_{\text{ion-enhanced}} $$ 3.1 Physical Sputtering (Yamamura-Sigmund) $$ Y(\theta, E) = \frac{0.042\, Q(Z_2)\, S_n(E)}{U_s}\Big[1-\sqrt{E_{th}/E}\Big]^s f(\theta) $$ Angular part: $$ f(\theta) = \cos^{-f}(\theta)\, \exp[-\Sigma (1/\cos\theta - 1)] $$ 3.2 Ion-Enhanced Chemical Etching (RIE) $$ R = k_1 \Gamma_F \theta_F + k_2 \Gamma_{\text{ion}} Y_{\text{phys}} + k_3 \Gamma_{\text{ion}}^a \Gamma_F^b (1 + \beta \theta_F) $$ - Term 1: chemical - Term 2: physical sputter - Term 3: synergistic ion-chemical 3.3 Surface Kinetics (Langmuir-Hinshelwood) $$ \frac{d\theta_F}{dt} = s_0 \Gamma_F (1-\theta_F) - k_d \theta_F - k_r \theta_F \Gamma_{\text{ion}} $$ Steady state: $\theta_F = s_0 \Gamma_F / (s_0 \Gamma_F + k_d + k_r \Gamma_{\text{ion}})$ 4. Transport in High-Aspect-Ratio Features 4.1 Knudsen Diffusion (neutrals) $$ \Gamma(z) = \Gamma_0 P(AR), \quad P(AR) \approx \frac{1}{1 + 3AR/8} $$ More exact: $P(L/R) = \tfrac{8R}{3L}(\sqrt{1+(L/R)^2} - 1)$ 4.2 Ion Angular Distribution $$ f(\theta) \propto \exp\Big(-\frac{m_i v_\perp^2}{2k_B T_i}\Big) \cos\theta $$ Mean angle (collisionless sheath): $\langle\theta\rangle \approx \arctan\!\big(\sqrt{T_e/(eV_{\text{sheath}})}\big)$ Shadowing: $\theta_{\max}(z) = \arctan(w/2z)$ 4.3 Sheath Potential $$ V_s \approx \frac{k_B T_e}{2e} \ln\Big(\frac{m_i}{2\pi m_e}\Big) $$ 5. Profile Phenomena 5.1 Bowing (sidewall widening) $$ V_{\text{lateral}}(z) = \int_0^{\theta_{\max}} Y(\theta')\, \Gamma_{\text{reflected}}(\theta', z)\, d\theta' $$ 5.2 Microtrenching (corner enhancement) $$ \Gamma_{\text{corner}} = \Gamma_{\text{direct}} + \int \Gamma_{\text{incident}} R(\theta) G(\text{geometry})\, d\theta $$ 5.3 Notching (charging) Poisson: $\nabla^2 V = -\rho/(\epsilon_0 \epsilon_r)$ Charge balance: $\partial \sigma/\partial t = J_{\text{ion}} - J_{\text{electron}} - J_{\text{secondary}}$ Deflection: $\theta_{\text{deflection}} \approx \arctan\big(q E_{\text{surface}} L / (2 E_{\text{ion}})\big)$ 5.4 ARDE (RIE lag) $$ \frac{ER(AR)}{ER_0} = \frac{1}{1 + \alpha AR^\beta} $$ 6. Computational Approaches - Monte Carlo (feature scale): launch particles, track, reflect/react, accumulate rates - Flux-based / view-factor: $V_n(\mathbf{x}) = \sum_j R_j \Gamma_j(\mathbf{x}) Y_j(\theta(\mathbf{x}))$ - Cellular automata: $P_{\text{etch}}(\text{cell}) = f(\Gamma_{\text{local}}, \text{neighbors}, \text{material})$ - DSMC (gas transport): molecule tracing with probabilistic collisions 7. Multi-Scale Integration | Scale | Range | Physics | Method | |---------|----------|-------------------------------|-------------------------| | Reactor | cm–m | Plasma generation, gas flow | Fluid / hybrid PIC-MCC | | Sheath | μm–mm | Ion acceleration, angles | Kinetic / fluid | | Feature | nm–μm | Transport, surface evolution | Monte Carlo + level set | | Atomic | Å | Reaction mechanisms, yields | MD, DFT | 7.1 Coupling - Reactor → species densities/temps/fluxes to sheath - Sheath → ion/neutral energy-angle distributions to feature - Atomic → yield functions $Y(\theta, E)$ to feature scale 7.2 Governing Equations Summary - Surface evolution: $\partial S/\partial t = V_n \hat{n}$ - Neutral transport: $\mathbf{v}\cdot\nabla f + (\mathbf{F}/m)\cdot\nabla_v f = (\partial f/\partial t)_{\text{coll}}$ - Ion trajectory: $m\, d^2\vec{r}/dt^2 = q(\vec{E} + \vec{v}\times\vec{B})$ 8. Advanced Topics 8.1 Stochastic roughness (LER) $$ \sigma_{LER}^2 = \frac{2}{\pi^2 n_s} \int \frac{PSD(f)}{f^2} \, df $$ 8.2 Pattern-dependent effects (loading) $$ \frac{\partial n}{\partial t} = D\nabla^2 n - k_{\text{etch}} A_{\text{exposed}} n $$ 8.3 Machine Learning Surrogates $$ \text{Profile}(t) = \mathcal{NN}(\text{Process conditions}, \text{Initial geometry}, t) $$ Uses: rapid exploration, inverse optimization, real-time control. 9. Summary and Diagrams 9.1 Complete Flow ```text Plasma Parameters ↓ Ion/Neutral Energy-Angle Distributions ↓ ┌─────────────────────┴─────────────────────┐ ↓ ↓ Transport in Feature Surface Chemistry (Knudsen, charging) (coverage, reactions) ↓ ↓ └─────────────────────┬─────────────────────┘ ↓ Local Etch Velocity Vn(x, θ, Γ, T) ↓ Surface Evolution Equation ∂φ/∂t + Vn|∇φ| = 0 ↓ Etch Profile ``` 9.2 Equations | Phenomenon | Equation | |----------------------|-------------------------------------------------| | Level set evolution | $\partial \phi/\partial t + V_n \|\nabla \phi\| = 0$ | | Angular yield | $Y(\theta) = Y_0 \cos^{-f}(\theta) \exp[-\Sigma(1/\cos\theta - 1)]$ | | ARDE | $ER(AR)/ER_0 = 1/(1 + \alpha AR^\beta)$ | | Transmission prob. | $P(AR) = 1/(1 + 3AR/8)$ | | Surface coverage | $\theta_F = s_0\Gamma_F / (s_0\Gamma_F + k_d + k_r\Gamma_{\text{ion}})$ | 9.3 Mathematical Elegance - Geometry via $\phi$ evolution - Physics via $V_n$ models Modular structure enables independent improvement of geometry and physics.
I can discuss high-level AI ethics, bias risks, and mitigations, and help you think about responsible uses of your system.
Simple ODE solver for diffusion.
# EUV Lithography: Mathematical Modeling Framework **Extreme Ultraviolet (EUV) lithography** uses 13.5 nm wavelength light to pattern semiconductor features at 7nm, 5nm, 3nm nodes and beyond. This document provides a comprehensive mathematical modeling reference. ## 1. Optical Image Formation ### 1.1 Partially Coherent Imaging (Hopkins Formulation) The aerial image intensity at wafer level is governed by the **Hopkins imaging equation**: $$ I(x,y) = \iint TCC(f_1, f_2) \cdot \tilde{M}(f_1) \cdot \tilde{M}^*(f_2) \cdot e^{i2\pi(f_1-f_2)\cdot\mathbf{r}} \, df_1 \, df_2 $$ Where the **Transmission Cross-Coefficient (TCC)** captures the optical system behavior: $$ TCC(f_1, f_2) = \iint J(f) \cdot H(f + f_1) \cdot H^*(f + f_2) \, df $$ **Key parameters:** - $J(f)$ — Effective source (mutual intensity function) - $H(f)$ — Pupil function including aberrations - $\tilde{M}(f)$ — Fourier transform of mask reflectance - $f_1, f_2$ — Spatial frequency coordinates ### 1.2 Aberration Modeling with Zernike Polynomials Pupil aberrations are expanded in **Zernike polynomials**: $$ H(\rho, \theta) = P(\rho) \cdot \exp\left[i \sum_{n,m} Z_{nm} R_n^m(\rho) \cos(m\theta)\right] $$ **Zernike terms and their physical meaning:** - $Z_4$ — Defocus - $Z_5, Z_6$ — Astigmatism - $Z_7, Z_8$ — Coma - $Z_9$ — Spherical aberration - $Z_{nm}$ — Higher-order aberrations > **Note:** For EUV at $\lambda = 13.5$ nm, even sub-nanometer wavefront errors cause significant image degradation. ### 1.3 Key Optical Parameters | Parameter | Symbol | EUV Value | High-NA EUV | |-----------|--------|-----------|-------------| | Wavelength | $\lambda$ | 13.5 nm | 13.5 nm | | Numerical Aperture | $NA$ | 0.33 | 0.55 | | Partial Coherence | $\sigma$ | 0.2–0.9 | 0.2–0.9 | | k1 Factor | $k_1$ | 0.3–0.4 | 0.3–0.4 | **Resolution limit (Rayleigh criterion):** $$ R_{min} = k_1 \frac{\lambda}{NA} $$ ## 2. EUV Mask (Reticle) Modeling ### 2.1 Multilayer Reflectance — Transfer Matrix Method EUV masks use **Mo/Si multilayer mirrors** (40–50 bilayer pairs). Reflectance is computed via the **transfer matrix method**: $$ \begin{pmatrix} E_0^+ \\ E_0^- \end{pmatrix} = \prod_{j=1}^{N} M_j \begin{pmatrix} E_N^+ \\ 0 \end{pmatrix} $$ Each layer contributes a **characteristic matrix**: $$ M_j = \begin{pmatrix} \cos\beta_j & \frac{i\sin\beta_j}{\eta_j} \\ i\eta_j\sin\beta_j & \cos\beta_j \end{pmatrix} $$ **Where:** - $\beta_j = \frac{2\pi}{\lambda} n_j d_j \cos\theta_j$ — Phase thickness of layer $j$ - $\eta_j = n_j \cos\theta_j$ — Effective admittance (TE polarization) - $n_j$ — Complex refractive index of layer $j$ - $d_j$ — Physical thickness of layer $j$ ### 2.2 Complex Refractive Index at EUV At 13.5 nm wavelength, materials have refractive indices very close to vacuum: $$ \tilde{n} = 1 - \delta + i\beta $$ **Typical values:** | Material | $\delta$ | $\beta$ | |----------|----------|---------| | Si | 0.00183 | 0.00184 | | Mo | 0.0076 | 0.0064 | | Ru | 0.0104 | 0.0171 | | TaN (absorber) | 0.041 | 0.038 | ### 2.3 3D Mask Effects (M3D) — Rigorous EM Simulation Because feature sizes approach the wavelength, **rigorous electromagnetic simulation** is required. **Maxwell's Equations (Time-Harmonic Form):** $$ \nabla \times \mathbf{E} = -i\omega\mu_0 \mathbf{H} $$ $$ \nabla \times \mathbf{H} = i\omega\epsilon \mathbf{E} $$ **Numerical Methods:** - **RCWA** (Rigorous Coupled-Wave Analysis) - Expands fields in Fourier series - Efficient for periodic structures - Computational complexity: $O(N^3)$ where $N$ = number of harmonics - **FDTD** (Finite-Difference Time-Domain) - Direct time-stepping of Maxwell's equations - Yee grid discretization: $$ \frac{\partial E_x}{\partial t} = \frac{1}{\epsilon}\left(\frac{\partial H_z}{\partial y} - \frac{\partial H_y}{\partial z}\right) $$ - **FEM** (Finite Element Method) - Variational/weak formulation - Adaptive meshing for complex geometries **Shadowing Effect:** The 6° chief ray angle in EUV creates pattern-dependent asymmetries: $$ \Delta CD_{shadow} \approx 2h \cdot \tan(6°) \approx 0.21h $$ where $h$ is the absorber height. ## 3. Stochastic Effects ### 3.1 Photon Shot Noise — The Critical EUV Challenge At $\lambda = 13.5$ nm, photon energy is **~92 eV** (compared to 6.4 eV for ArF at 193 nm): $$ E_{photon} = \frac{hc}{\lambda} = \frac{1240 \text{ eV} \cdot \text{nm}}{13.5 \text{ nm}} \approx 92 \text{ eV} $$ **Number of photons per pixel:** $$ N_{photons} = \frac{E \cdot A_{pixel}}{h\nu} = \frac{E \cdot A_{pixel} \cdot \lambda}{hc} $$ **Example calculation:** - Dose: $E = 30$ mJ/cm² - Pixel area: $A = (10 \text{ nm})^2 = 10^{-12}$ cm² - Result: $N \approx 200$ photons **Relative fluctuation (Poisson statistics):** $$ \frac{\sigma_N}{N} = \frac{1}{\sqrt{N}} \approx 7\% $$ ### 3.2 Stochastic Line Edge Roughness (LER) Total LER combines multiple independent noise sources: $$ \sigma_{LER}^2 = \sigma_{photon}^2 + \sigma_{PAG}^2 + \sigma_{acid}^2 + \sigma_{develop}^2 $$ **Component models:** - **Photon shot noise:** $$\sigma_{photon} \propto \frac{1}{\sqrt{D}}$$ where $D$ is dose - **PAG (Photo-Acid Generator) distribution:** - Poisson statistics on molecular positions - Typical density: $\rho_{PAG} \sim 1\text{–}2 \text{ nm}^{-3}$ - Fluctuation: $\sigma_{PAG} \propto \frac{1}{\sqrt{\rho_{PAG} \cdot V}}$ - **Acid diffusion:** - Random walk during post-exposure bake - Diffusion length: $L_d = \sqrt{2D_a t_{PEB}}$ - **Development roughness:** - Percolation-type dissolution - Depends on polymer molecular weight distribution ### 3.3 Resolution-LER-Sensitivity (RLS) Trade-off The fundamental **RLS triangle** constrains lithographic performance: $$ Z = R^3 \cdot L^2 \cdot S^{-1} $$ **Where:** - $R$ — Resolution (half-pitch) - $L$ — Line edge roughness (3σ) - $S$ — Sensitivity (dose to clear) > **Physical limit:** This product has a quantum mechanical minimum that cannot be overcome without fundamental changes to imaging physics. **Practical targets for advanced nodes:** | Node | Resolution | LER (3σ) | Dose | |------|------------|----------|------| | 7 nm | 18 nm HP | < 3.6 nm | ~20 mJ/cm² | | 5 nm | 14 nm HP | < 2.8 nm | ~30 mJ/cm² | | 3 nm | 10 nm HP | < 2.0 nm | ~40 mJ/cm² | ## 4. Resist Chemistry Modeling ### 4.1 Chemically Amplified Resist (CAR) Kinetics **Step 1: Exposure — Acid Generation** Photo-acid generator (PAG) decomposition follows first-order kinetics: $$ [H^+] = [PAG]_0 \left(1 - e^{-\sigma \Phi}\right) $$ **Where:** - $[PAG]_0$ — Initial PAG concentration - $\sigma$ — Absorption cross-section (cm²) - $\Phi$ — Photon fluence (photons/cm²) **Quantum yield consideration:** $$ [H^+] = \phi_a \cdot [PAG]_0 \left(1 - e^{-\sigma \Phi}\right) $$ where $\phi_a$ is the acid generation quantum yield. ### 4.2 Post-Exposure Bake — Reaction-Diffusion System **Acid diffusion with quenching:** $$ \frac{\partial [H^+]}{\partial t} = D_a \nabla^2[H^+] - k_q[H^+][Q] $$ **Deprotection reaction (catalytic):** $$ \frac{\partial [P]}{\partial t} = -k_{amp}[H^+][P] $$ **Parameter definitions:** - $[P]$ — Protected polymer concentration - $[Q]$ — Quencher concentration - $D_a$ — Acid diffusion coefficient (~10–50 nm²/s) - $k_q$ — Quenching rate constant - $k_{amp}$ — Catalytic amplification rate **Acid blur (diffusion length):** $$ \sigma_{acid} = \sqrt{2 D_a t_{PEB}} $$ ### 4.3 Development Rate Models **Mack Model (Enhanced Notch Model):** $$ R = R_{max} \frac{(a+1)(1-m)^n}{a + (1-m)^n} + R_{min} $$ **Where:** - $R$ — Development rate - $m$ — Normalized inhibitor concentration (0 to 1) - $R_{max}$ — Maximum development rate (fully deprotected) - $R_{min}$ — Minimum development rate (unexposed) - $n$ — Dissolution selectivity parameter - $a$ — Notch parameter **Original Dill Parameters:** $$ \frac{dm}{dE} = -C \cdot I \cdot m $$ **Dissolution contrast:** $$ \gamma = \frac{d \ln R}{d \ln E}\bigg|_{E=E_{th}} $$ ## 5. Computational Lithography / OPC ### 5.1 Inverse Lithography Problem **Objective function:** $$ M_{opt} = \arg\min_M \mathcal{L}(M) $$ **Loss function with regularization:** $$ \mathcal{L}(M) = \underbrace{\left\| I_{target} - \mathcal{F}(M) \right\|^2}_{\text{Pattern fidelity}} + \underbrace{\lambda_1 R_{MRC}(M)}_{\text{Mask rules}} + \underbrace{\lambda_2 R_{MEEF}(M)}_{\text{Error sensitivity}} $$ **Where:** - $\mathcal{F}$ — Forward optical model (mask → aerial image) - $R_{MRC}$ — Mask rule check penalty (manufacturability) - $R_{MEEF}$ — Mask error enhancement factor penalty - $\lambda_1, \lambda_2$ — Regularization weights ### 5.2 Gradient Computation — Adjoint Method **Gradient of loss with respect to mask:** $$ \frac{\partial \mathcal{L}}{\partial M} = 2 \cdot \text{Re}\left[ TCC^{\dagger} \cdot (I_{simulated} - I_{target}) \right] $$ **Iterative update (gradient descent):** $$ M^{(k+1)} = M^{(k)} - \alpha \frac{\partial \mathcal{L}}{\partial M}\bigg|_{M^{(k)}} $$ ### 5.3 Source-Mask Optimization (SMO) **Joint optimization problem:** $$ \min_{S,M} \sum_{i=1}^{N_{patterns}} w_i \left\| I_i^{target} - I(S, M_i) \right\|^2 $$ **Subject to constraints:** - Source intensity: $\int\int S(x,y) \, dx \, dy = 1$ - Source non-negativity: $S(x,y) \geq 0$ - Mask manufacturability: $M \in \mathcal{M}_{feasible}$ **Source parameterization (multipole illumination):** $$ S(\sigma_x, \sigma_y) = \sum_{k=1}^{N_{poles}} A_k \cdot G(\sigma_x - \sigma_{x,k}, \sigma_y - \sigma_{y,k}; \sigma_0) $$ ### 5.4 Mask Error Enhancement Factor (MEEF) $$ MEEF = \frac{\partial CD_{wafer}}{\partial CD_{mask}} \cdot M $$ where $M$ is the reduction ratio (typically 4×). **For ideal imaging:** MEEF = 1 **For sub-resolution features:** MEEF can exceed 2–5 ## 6. Flare and Scattered Light ### 6.1 Mirror Roughness Contribution **Flare PSF from mid-spatial-frequency roughness:** $$ PSF_{flare}(r) = \frac{1}{(2\pi)^2} \int_0^\infty PSD(f) \cdot |H_{scatter}(f)|^2 \cdot J_0(2\pi f r) \cdot 2\pi f \, df $$ **Where:** - $PSD(f)$ — Power spectral density of surface roughness - $H_{scatter}(f)$ — Scattering transfer function - $J_0$ — Bessel function of the first kind ### 6.2 Total Flare Level **Definition:** $$ Flare = \frac{\int\int PSF_{flare}(r) \, dA}{\int\int PSF_{total}(r) \, dA} $$ **Typical values:** - DUV (193 nm): 1–2% - EUV (13.5 nm): 3–10% ### 6.3 Impact on Image Contrast **Modified aerial image:** $$ I_{with flare}(x,y) = (1 - F) \cdot I_{ideal}(x,y) + F \cdot \bar{I} $$ where $F$ is the flare fraction and $\bar{I}$ is the average intensity. **Contrast degradation:** $$ NILS_{with flare} = NILS_{ideal} \cdot (1 - F) $$ ## 7. Overlay and Distortion Modeling ### 7.1 Wafer Distortion — Polynomial Model **Displacement fields:** $$ \delta x(x,y) = \sum_{i+j \leq n} a_{ij} x^i y^j $$ $$ \delta y(x,y) = \sum_{i+j \leq n} b_{ij} x^i y^j $$ **Common terms and physical meaning:** | Coefficient | Physical Effect | |-------------|-----------------| | $a_{10}, b_{01}$ | Translation | | $a_{10}, b_{01}$ (scaled) | Magnification | | $a_{01}, b_{10}$ | Rotation | | $a_{20}, b_{02}$ | Asymmetric magnification | | Higher order | Field curvature, distortion | ### 7.2 Overlay Budget **Total overlay (RSS):** $$ OVL_{total} = \sqrt{OVL_{tool}^2 + OVL_{process}^2 + OVL_{wafer}^2 + OVL_{mask}^2} $$ **Typical requirements:** | Node | Overlay Spec | |------|-------------| | 7 nm | < 3.0 nm | | 5 nm | < 2.0 nm | | 3 nm | < 1.5 nm | ### 7.3 Machine Learning for Overlay Correction **Neural network predictor:** $$ (\delta x, \delta y)_{predicted} = f_{NN}(\mathbf{x}_{sensors}, \mathbf{x}_{context}, \mathbf{x}_{history}) $$ **Features include:** - Wafer stage sensor data - Temperature measurements - Previous layer metrology - Process history ## 8. Monte Carlo Stochastic Simulation ### 8.1 Full Stochastic Model Chain **Sequential simulation steps:** 1. **Photon absorption** - Poisson-distributed absorption events - Position: $(x_i, y_i, z_i)$ random in resist volume - Number: $N \sim \text{Poisson}(\bar{N})$ 2. **Secondary electron cascade** - Each 92 eV photon generates ~3–4 secondary electrons - Electron trajectory: Monte Carlo scattering simulation - Energy deposition: Bethe stopping power 3. **Acid generation** - Binomial sampling at PAG molecule locations - $P(\text{activation}) = 1 - e^{-\sigma E_{local}}$ 4. **Acid diffusion (PEB)** - Random walk simulation - Step size: $\Delta r = \sqrt{2D_a \Delta t}$ 5. **Deprotection** - Probabilistic reaction at each polymer site - $P(\text{deprotect}) = 1 - e^{-k_{amp}[H^+]\Delta t}$ 6. **Development** - Kinetic Monte Carlo (KMC) dissolution - Site-dependent removal rates ### 8.2 Secondary Electron Blur **Point spread function:** $$ PSF_{e}(r) = \frac{1}{2\pi\sigma_e^2} \exp\left(-\frac{r^2}{2\sigma_e^2}\right) $$ **Typical blur:** $\sigma_e \approx 2\text{–}5$ nm for EUV resists **Energy-dependent blur:** $$ \sigma_e(E) = a \cdot E^b $$ ### 8.3 Kinetic Monte Carlo (KMC) Development **Transition rate for site removal:** $$ k_i = k_0 \exp\left(-\frac{E_a - \gamma \cdot n_{neighbors}}{k_B T}\right) $$ **Where:** - $E_a$ — Base activation energy - $\gamma$ — Neighbor interaction energy - $n_{neighbors}$ — Number of connected neighbors - $k_B T$ — Thermal energy **Algorithm:** 1. Calculate all site rates $\{k_i\}$ 2. Total rate: $K = \sum_i k_i$ 3. Time step: $\Delta t = -\frac{\ln(u_1)}{K}$, $u_1 \sim U(0,1)$ 4. Select site with probability $P_i = k_i/K$ 5. Remove site, update neighbors 6. Repeat until target depth reached ## 9. Process Window Analysis ### 9.1 Bossung Curves **CD as a function of focus (parabolic approximation):** $$ CD(F) = CD_0 + \alpha(F - F_{best})^2 $$ **Where:** - $CD_0$ — Best focus CD - $F_{best}$ — Best focus position - $\alpha$ — Curvature coefficient (nm/nm²) ### 9.2 Depth of Focus (DOF) **Rayleigh DOF:** $$ DOF = k_2 \frac{\lambda}{NA^2} $$ **Typical values:** | System | NA | DOF | |--------|-----|-----| | EUV | 0.33 | ~80–100 nm | | High-NA EUV | 0.55 | ~35–45 nm | ### 9.3 Exposure Latitude **Definition:** $$ EL = \frac{E_{max} - E_{min}}{E_{nominal}} \times 100\% $$ **Normalized Image Log-Slope (NILS):** $$ NILS = w \cdot \frac{1}{I} \frac{dI}{dx}\bigg|_{edge} $$ where $w$ is the feature width. **Correlation:** Higher NILS → larger exposure latitude ### 9.4 Process Variability Band (PVB) **Total CD variation from all sources:** $$ PVB = \sqrt{\sum_i \left(\frac{\partial CD}{\partial p_i}\right)^2 \sigma_{p_i}^2} $$ **Contributing parameters $(p_i)$:** - Dose variation - Focus variation - Mask CD variation - Aberrations - Resist thickness - PEB temperature ## 10. Machine Learning Integration ### 10.1 Applications in EUV Modeling | Application | ML Method | Input | Output | |-------------|-----------|-------|--------| | Aerial image prediction | CNN | Mask layout | Image intensity | | Hotspot detection | Deep learning | Pattern clips | Pass/fail | | Resist model calibration | Gaussian process | Process params | CD | | Virtual metrology | Random forest | Sensor data | Wafer quality | | OPC acceleration | GNN | Layout graph | OPC corrections | ### 10.2 Convolutional Neural Networks for Imaging **Architecture for mask → aerial image:** $$ I_{predicted} = CNN_\theta(M_{input}) $$ **Loss function:** $$ \mathcal{L}(\theta) = \frac{1}{N}\sum_{i=1}^{N} \|I_i^{rigorous} - CNN_\theta(M_i)\|^2 $$ ### 10.3 Physics-Informed Neural Networks (PINNs) **Combined loss function:** $$ \mathcal{L} = \underbrace{\mathcal{L}_{data}}_{\text{Measurement fit}} + \lambda \cdot \underbrace{\mathcal{L}_{physics}}_{\text{Maxwell residual}} $$ **Physics loss (Maxwell's equations):** $$ \mathcal{L}_{physics} = \|\nabla \times \mathbf{E} + i\omega\mu_0\mathbf{H}\|^2 + \|\nabla \times \mathbf{H} - i\omega\epsilon\mathbf{E}\|^2 $$ ### 10.4 Transfer Learning for New Processes **Pre-train on simulation data:** $$ \theta^{(0)} = \arg\min_\theta \mathcal{L}_{simulation}(\theta) $$ **Fine-tune on experimental data:** $$ \theta^{*} = \arg\min_\theta \mathcal{L}_{experimental}(\theta; \theta^{(0)}) $$ ## 11. Summary ### 11.1 Mathematical Tools by Domain | Domain | Mathematical Framework | |--------|----------------------| | Source modeling | Plasma physics, MHD, radiation transport | | Optical imaging | Fourier optics, Hopkins TCC, Zernike polynomials | | Mask modeling | Transfer matrices, RCWA/FDTD, Maxwell solvers | | Aerial image | Convolution, partially coherent imaging | | Stochastics | Poisson statistics, random walks, percolation | | Resist chemistry | Reaction-diffusion PDEs, Arrhenius kinetics | | OPC/ILT | Inverse problems, gradient optimization, regularization | | Process control | Statistical process control, Kalman filtering | | ML augmentation | CNNs, PINNs, surrogate models, transfer learning | ### 11.2 Key Equations Summary **Resolution:** $$R = k_1 \frac{\lambda}{NA}$$ **Depth of Focus:** $$DOF = k_2 \frac{\lambda}{NA^2}$$ **Photon count:** $$N = \frac{E \cdot A \cdot \lambda}{hc}$$ **Shot noise:** $$\sigma_N / N = 1/\sqrt{N}$$ **RLS Trade-off:** $$Z = R^3 \cdot L^2 \cdot S^{-1}$$ **Hopkins Imaging:** $$I = \iint TCC \cdot \tilde{M}(f_1) \cdot \tilde{M}^*(f_2) \cdot e^{i2\pi(f_1-f_2)\cdot\mathbf{r}} df_1 df_2$$ ### 11.3 Emerging Challenges - **High-NA EUV (0.55 NA)** - Anamorphic optics (4× in x, 8× in y) - Polarization effects become critical - DOF reduced to ~35 nm - **Stochastic limits** - Approaching quantum limits of imaging - New resist chemistries needed - Multi-trigger and dry resist concepts - **Computational complexity** - Full chip rigorous simulation infeasible - ML surrogate models essential - Real-time OPC requirements
Event-based temporal graphs represent edge additions and deletions as continuous-time events.
Evol-Instruct iteratively evolves instructions making them more complex and diverse.
Use evolutionary algorithms for NAS.
Evolutionary neural architecture search uses genetic algorithms with mutation and crossover operations to explore discrete architecture spaces.
EvolveGCN learns temporal graph representations by evolving GCN parameters over time using RNNs rather than maintaining separate models per snapshot.
Evolved normalization and activation.
Order within batches.
Run generated code get errors and iterate to fix.
Execution traces document complete agent behavior sequences for analysis.
Uncertainty with confidence interval.
Expanding window forecasting retrains on all historical data including newest observations.
Evaluate robustness over transformations.
Expediting accelerates delivery of critical materials through special handling or premium shipping.
Replay old examples.
Expert capacity limits tokens assigned to each expert preventing overload.
Expert parallelism distributes experts across devices for scalable MoE training.
Distribute MoE experts across different GPUs.
Expert routing assigns tokens to appropriate specialized sub-networks.
Mechanism to select which experts process each input.
Interpret ML model decisions.