ensemble methods,machine learning
Combine multiple models for better predictions.
422 technical terms and definitions
Combine multiple models for better predictions.
Ensembling combines multiple models. Bagging, boosting, stacking.
Ensembles combine multiple models. Diversity improves robustness. Disagreement indicates uncertainty.
Enthalpy wheels transfer both sensible and latent heat recovering moisture and thermal energy.
Resolve which entity is meant.
Entity embeddings in knowledge graphs learn representations of items and attributes for enhanced recommendations.
Named Entity Recognition extracts entities (person, org, location). SpaCy, transformers. Information extraction.
NER extracts named entities (people, places, orgs) from text. Useful for knowledge base population and structured data.
Connect mentions to knowledge bases.
Connect mentions to knowledge base entities.
Mask named entities during pre-training.
Predict masked entity types.
Track mentioned entities.
Encourage diverse predictions.
Control software dependencies.
Maintain stable temperature and humidity.
Protect from environment.
Environmental monitoring tracks cleanroom conditions ensuring specification compliance.
Apply stress to find weak units.
Combined environmental stresses.
TEM in controlled gas environment.
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 |
# Mathematical Modeling of Epitaxy in Semiconductor Front-End Processing (FEP) ## 1. Overview Epitaxy is a critical **Front-End Process (FEP)** step where crystalline films are grown on crystalline substrates with precise control of: - Thickness - Composition - Doping concentration - Defect density Mathematical modeling enables: - Process optimization - Defect prediction - Virtual fabrication - Equipment design ### 1.1 Types of Epitaxy - **Homoepitaxy**: Same material as substrate (e.g., Si on Si) - **Heteroepitaxy**: Different material from substrate (e.g., GaAs on Si, SiGe on Si) ### 1.2 Epitaxy Methods - **Vapor Phase Epitaxy (VPE)** / Chemical Vapor Deposition (CVD) - Atmospheric Pressure CVD (APCVD) - Low Pressure CVD (LPCVD) - Metal-Organic CVD (MOCVD) - **Molecular Beam Epitaxy (MBE)** - **Liquid Phase Epitaxy (LPE)** - **Solid Phase Epitaxy (SPE)** ## 2. Fundamental Thermodynamic Framework ### 2.1 Driving Force for Growth The supersaturation provides the thermodynamic driving force: $$ \Delta \mu = k_B T \ln\left(\frac{P}{P_{eq}}\right) $$ Where: - $\Delta \mu$ = chemical potential difference (driving force) - $k_B$ = Boltzmann's constant ($1.38 \times 10^{-23}$ J/K) - $T$ = absolute temperature (K) - $P$ = actual partial pressure of precursor - $P_{eq}$ = equilibrium vapor pressure ### 2.2 Free Energy of Mixing (Multi-component Systems) For systems like SiGe alloys: $$ \Delta G_{mix} = RT\left(x \ln x + (1-x) \ln(1-x)\right) + \Omega x(1-x) $$ Where: - $R$ = universal gas constant (8.314 J/mol$\cdot$K) - $x$ = mole fraction of component - $\Omega$ = interaction parameter (regular solution model) ### 2.3 Gibbs Free Energy of Formation $$ \Delta G = \Delta H - T\Delta S $$ For spontaneous growth: $\Delta G < 0$ ## 3. Growth Rate Kinetics ### 3.1 The Two-Regime Model Epitaxial growth rate is governed by two competing mechanisms: **Overall growth rate equation:** $$ G = \frac{k_s \cdot h_g \cdot C_g}{k_s + h_g} $$ Where: - $G$ = growth rate (nm/min or $\mu$m/min) - $k_s$ = surface reaction rate constant - $h_g$ = gas-phase mass transfer coefficient - $C_g$ = gas-phase reactant concentration ### 3.2 Temperature Dependence The surface reaction rate follows Arrhenius behavior: $$ k_s = A \exp\left(-\frac{E_a}{k_B T}\right) $$ Where: - $A$ = pre-exponential factor (frequency factor) - $E_a$ = activation energy (eV or J/mol) ### 3.3 Growth Rate Regimes | Temperature Regime | Limiting Factor | Growth Rate Expression | Temperature Dependence | |:-------------------|:----------------|:-----------------------|:-----------------------| | **Low T** | Surface reaction | $G \approx k_s \cdot C_g$ | Strong (exponential) | | **High T** | Mass transport | $G \approx h_g \cdot C_g$ | Weak (~$T^{1.5-2}$) | ### 3.4 Boundary Layer Analysis For horizontal CVD reactors, the boundary layer thickness evolves as: $$ \delta(x) = \sqrt{\frac{\nu \cdot x}{v_{\infty}}} $$ Where: - $\delta(x)$ = boundary layer thickness at position $x$ - $\nu$ = kinematic viscosity (m²/s) - $x$ = distance from gas inlet (m) - $v_{\infty}$ = free stream gas velocity (m/s) The mass transfer coefficient: $$ h_g = \frac{D_{gas}}{\delta} $$ Where $D_{gas}$ is the gas-phase diffusion coefficient. ## 4. Surface Kinetics: BCF Theory The **Burton-Cabrera-Frank (BCF) model** describes atomic-scale growth mechanisms. ### 4.1 Surface Diffusion Equation $$ D_s \nabla^2 n_s - \frac{n_s - n_{eq}}{\tau_s} + J_{ads} = 0 $$ Where: - $n_s$ = adatom surface density (atoms/cm²) - $D_s$ = surface diffusion coefficient (cm²/s) - $n_{eq}$ = equilibrium adatom density - $\tau_s$ = mean adatom lifetime before desorption (s) - $J_{ads}$ = adsorption flux (atoms/cm²$\cdot$s) ### 4.2 Characteristic Diffusion Length $$ \lambda_s = \sqrt{D_s \tau_s} $$ This parameter determines the growth mode: - **Step-flow growth**: $\lambda_s > L$ (terrace width) - **2D nucleation growth**: $\lambda_s < L$ ### 4.3 Surface Diffusion Coefficient $$ D_s = D_0 \exp\left(-\frac{E_m}{k_B T}\right) $$ Where: - $D_0$ = pre-exponential factor (~$10^{-3}$ cm²/s) - $E_m$ = migration energy barrier (eV) ### 4.4 Step Velocity $$ v_{step} = \frac{2 D_s (n_s - n_{eq})}{\lambda_s} \tanh\left(\frac{L}{2\lambda_s}\right) $$ Where $L$ is the inter-step spacing (terrace width). ### 4.5 Growth Rate from Step Flow $$ G = \frac{v_{step} \cdot h_{step}}{L} $$ Where $h_{step}$ is the step height (monolayer thickness). ## 5. Heteroepitaxy and Strain Modeling ### 5.1 Lattice Mismatch $$ f = \frac{a_{film} - a_{substrate}}{a_{substrate}} $$ Where: - $f$ = lattice mismatch (dimensionless, often expressed as %) - $a_{film}$ = lattice constant of film material - $a_{substrate}$ = lattice constant of substrate **Example values:** | System | Lattice Mismatch | |:-------|:-----------------| | Si₀.₇Ge₀.₃ on Si | ~1.2% | | Ge on Si | ~4.2% | | GaAs on Si | ~4.0% | | InAs on GaAs | ~7.2% | | GaN on Sapphire | ~16% | ### 5.2 Strain Components For biaxial strain in (001) films: $$ \varepsilon_{xx} = \varepsilon_{yy} = \varepsilon_{\parallel} = \frac{a_s - a_f}{a_f} \approx -f $$ $$ \varepsilon_{zz} = \varepsilon_{\perp} = -\frac{2C_{12}}{C_{11}} \varepsilon_{\parallel} $$ Where $C_{11}$ and $C_{12}$ are elastic constants. ### 5.3 Elastic Energy For a coherently strained film: $$ E_{elastic} = \frac{2G(1+\nu)}{1-\nu} f^2 h = M f^2 h $$ Where: - $G$ = shear modulus (Pa) - $\nu$ = Poisson's ratio - $h$ = film thickness - $M$ = biaxial modulus = $\frac{2G(1+\nu)}{1-\nu}$ ### 5.4 Critical Thickness (Matthews-Blakeslee) $$ h_c = \frac{b}{8\pi f(1+\nu)} \left[\ln\left(\frac{h_c}{b}\right) + 1\right] $$ Where: - $h_c$ = critical thickness for dislocation formation - $b$ = Burgers vector magnitude - $f$ = lattice mismatch - $\nu$ = Poisson's ratio ### 5.5 People-Bean Approximation (for SiGe) Empirical formula: $$ h_c \approx \frac{0.55}{f^2} \text{ (nm, with } f \text{ as a decimal)} $$ Or equivalently: $$ h_c \approx \frac{5500}{x^2} \text{ (nm, for Si}_{1-x}\text{Ge}_x\text{)} $$ ### 5.6 Threading Dislocation Density Above critical thickness, dislocation density evolves: $$ \rho_{TD}(h) = \rho_0 \exp\left(-\frac{h}{h_0}\right) + \rho_{\infty} $$ Where: - $\rho_{TD}$ = threading dislocation density (cm⁻²) - $\rho_0$ = initial density - $h_0$ = characteristic decay length - $\rho_{\infty}$ = residual density ## 6. Reactor-Scale Modeling ### 6.1 Coupled Transport Equations #### 6.1.1 Momentum Conservation (Navier-Stokes) $$ \rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} $$ Where: - $\rho$ = gas density (kg/m³) - $\mathbf{v}$ = velocity vector (m/s) - $p$ = pressure (Pa) - $\mu$ = dynamic viscosity (Pa$\cdot$s) - $\mathbf{g}$ = gravitational acceleration #### 6.1.2 Continuity Equation $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 $$ #### 6.1.3 Species Transport $$ \frac{\partial C_i}{\partial t} + \mathbf{v} \cdot \nabla C_i = D_i \nabla^2 C_i + R_i $$ Where: - $C_i$ = concentration of species $i$ (mol/m³) - $D_i$ = diffusion coefficient of species $i$ (m²/s) - $R_i$ = net reaction rate (mol/m³$\cdot$s) #### 6.1.4 Energy Conservation $$ \rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T\right) = k \nabla^2 T + \sum_j \Delta H_j r_j $$ Where: - $c_p$ = specific heat capacity (J/kg$\cdot$K) - $k$ = thermal conductivity (W/m$\cdot$K) - $\Delta H_j$ = enthalpy of reaction $j$ (J/mol) - $r_j$ = rate of reaction $j$ (mol/m³$\cdot$s) ### 6.2 Silicon CVD Chemistry #### 6.2.1 From Silane (SiH₄) **Gas phase decomposition:** $$ \text{SiH}_4 \xrightarrow{k_1} \text{SiH}_2 + \text{H}_2 $$ **Surface reaction:** $$ \text{SiH}_2(g) + * \xrightarrow{k_2} \text{Si}(s) + \text{H}_2(g) $$ Where $*$ denotes a surface site. #### 6.2.2 From Dichlorosilane (DCS) $$ \text{SiH}_2\text{Cl}_2 \rightarrow \text{SiCl}_2 + \text{H}_2 $$ $$ \text{SiCl}_2 + \text{H}_2 \rightarrow \text{Si}(s) + 2\text{HCl} $$ #### 6.2.3 Rate Law $$ r_{dep} = k_2 P_{SiH_2} (1 - \theta) $$ Where: - $P_{SiH_2}$ = partial pressure of SiH₂ - $\theta$ = surface site coverage ### 6.3 Dimensionless Numbers | Number | Definition | Physical Meaning | |:-------|:-----------|:-----------------| | Reynolds | $Re = \frac{\rho v L}{\mu}$ | Inertia vs. viscous forces | | Prandtl | $Pr = \frac{\mu c_p}{k}$ | Momentum vs. thermal diffusivity | | Schmidt | $Sc = \frac{\mu}{\rho D}$ | Momentum vs. mass diffusivity | | Damköhler | $Da = \frac{k_s L}{D}$ | Reaction rate vs. diffusion rate | | Grashof | $Gr = \frac{g \beta \Delta T L^3}{\nu^2}$ | Buoyancy vs. viscous forces | ## 7. Selective Epitaxial Growth (SEG) Modeling ### 7.1 Overview In SEG, growth occurs on exposed Si but **not** on dielectric (SiO₂/Si₃N₄). ### 7.2 Loading Effect Model $$ G_{local} = G_0 \left(1 + \alpha \cdot \frac{A_{mask}}{A_{Si}}\right) $$ Where: - $G_{local}$ = local growth rate - $G_0$ = baseline growth rate - $\alpha$ = pattern sensitivity factor - $A_{mask}$ = dielectric (mask) area - $A_{Si}$ = exposed silicon area ### 7.3 Pattern-Dependent Growth Sources of non-uniformity: - Local depletion of reactants over Si regions - Species reflected/desorbed from mask contribute to nearby Si - Gas-phase diffusion length effects ### 7.4 Selectivity Condition For selective growth on Si vs. oxide: $$ r_{deposition,Si} > 0 \quad \text{and} \quad r_{deposition,oxide} < r_{etching,oxide} $$ **Achieved by adding HCl:** $$ \text{Si}(nuclei) + 2\text{HCl} \rightarrow \text{SiCl}_2 + \text{H}_2 $$ Nuclei on oxide are etched before they can grow, maintaining selectivity. ### 7.5 Faceting Model Growth rate depends on crystallographic orientation: $$ G_{(hkl)} = G_0 \cdot f(hkl) \cdot \exp\left(-\frac{E_{a,(hkl)}}{k_B T}\right) $$ Typical growth rate hierarchy: $$ G_{(100)} > G_{(110)} > G_{(111)} $$ ## 8. Dopant Incorporation ### 8.1 Segregation Coefficient **Equilibrium segregation coefficient:** $$ k_0 = \frac{C_{solid}}{C_{liquid/gas}} $$ **Effective segregation coefficient:** $$ k_{eff} = \frac{k_0}{k_0 + (1-k_0)\exp\left(-\frac{G\delta}{D_l}\right)} $$ Where: - $k_0$ = equilibrium segregation coefficient - $G$ = growth rate - $\delta$ = boundary layer thickness - $D_l$ = diffusivity in liquid/gas phase ### 8.2 Dopant Concentration in Film $$ C_{film} = k_{eff} \cdot C_{gas} $$ ### 8.3 Dopant Profile Abruptness The transition width is limited by: - **Surface segregation length**: $\lambda_{seg}$ - **Diffusion during growth**: $L_D = \sqrt{D \cdot t}$ - **Autodoping** from substrate $$ \Delta z_{transition} \approx \sqrt{\lambda_{seg}^2 + L_D^2} $$ ### 8.4 Common Dopants for Si Epitaxy | Dopant | Type | Precursor | Segregation Behavior | |:-------|:-----|:----------|:---------------------| | B | p-type | B₂H₆, BCl₃ | Low segregation | | P | n-type | PH₃, PCl₃ | Moderate segregation | | As | n-type | AsH₃ | Strong segregation | | Sb | n-type | SbH₃ | Very strong segregation | ## 9. Atomistic Simulation Methods ### 9.1 Kinetic Monte Carlo (KMC) #### 9.1.1 Event Rates Each atomic event has a rate following Arrhenius: $$ \Gamma_i = \nu_0 \exp\left(-\frac{E_i}{k_B T}\right) $$ Where: - $\Gamma_i$ = rate of event $i$ (s⁻¹) - $\nu_0$ = attempt frequency (~10¹²-10¹³ s⁻¹) - $E_i$ = activation energy for event $i$ #### 9.1.2 Events Modeled - **Adsorption**: $\Gamma_{ads} = \frac{P}{\sqrt{2\pi m k_B T}} \cdot s$ - **Desorption**: $\Gamma_{des} = \nu_0 \exp(-E_{des}/k_B T)$ - **Surface diffusion**: $\Gamma_{diff} = \nu_0 \exp(-E_m/k_B T)$ - **Step attachment**: $\Gamma_{attach}$ - **Step detachment**: $\Gamma_{detach}$ #### 9.1.3 Time Advancement $$ \Delta t = -\frac{\ln(r)}{\Gamma_{total}} = -\frac{\ln(r)}{\sum_i \Gamma_i} $$ Where $r$ is a uniform random number in $(0,1]$. ### 9.2 Density Functional Theory (DFT) Provides input parameters for KMC: - Adsorption energies - Migration barriers - Surface reconstruction energetics - Reaction pathways **Kohn-Sham equation:** $$ \left[-\frac{\hbar^2}{2m}\nabla^2 + V_{eff}(\mathbf{r})\right]\psi_i(\mathbf{r}) = \varepsilon_i \psi_i(\mathbf{r}) $$ ### 9.3 Molecular Dynamics (MD) **Newton's equations:** $$ m_i \frac{d^2 \mathbf{r}_i}{dt^2} = -\nabla_i U(\mathbf{r}_1, \mathbf{r}_2, ..., \mathbf{r}_N) $$ Where $U$ is the interatomic potential (e.g., Stillinger-Weber, Tersoff for Si). ## 10. Nucleation Theory ### 10.1 Classical Nucleation Theory (CNT) #### 10.1.1 Gibbs Free Energy Change $$ \Delta G(r) = -\frac{4}{3}\pi r^3 \cdot \frac{\Delta \mu}{\Omega} + 4\pi r^2 \gamma $$ Where: - $r$ = nucleus radius - $\Delta \mu$ = supersaturation (driving force) - $\Omega$ = atomic volume - $\gamma$ = surface energy #### 10.1.2 Critical Nucleus Radius Setting $\frac{d(\Delta G)}{dr} = 0$: $$ r^* = \frac{2\gamma \Omega}{\Delta \mu} $$ #### 10.1.3 Free Energy Barrier $$ \Delta G^* = \frac{16 \pi \gamma^3 \Omega^2}{3 (\Delta \mu)^2} $$ #### 10.1.4 Nucleation Rate $$ J = Z \beta^* N_s \exp\left(-\frac{\Delta G^*}{k_B T}\right) $$ Where: - $J$ = nucleation rate (nuclei/cm²$\cdot$s) - $Z$ = Zeldovich factor (~0.01-0.1) - $\beta^*$ = attachment rate to critical nucleus - $N_s$ = surface site density ### 10.2 Growth Modes | Mode | Surface Energy Condition | Growth Behavior | Example | |:-----|:-------------------------|:----------------|:--------| | **Frank-van der Merwe** | $\gamma_s \geq \gamma_f + \gamma_{int}$ | Layer-by-layer (2D) | Si on Si | | **Volmer-Weber** | $\gamma_s < \gamma_f + \gamma_{int}$ | Island (3D) | Metals on oxides | | **Stranski-Krastanov** | Intermediate | 2D then 3D islands | InAs/GaAs QDs | ### 10.3 2D Nucleation Critical island size (atoms): $$ i^* = \frac{\pi \gamma_{step}^2 \Omega}{(\Delta \mu)^2 k_B T} $$ ## 11. TCAD Process Simulation ### 11.1 Overview Tools: Synopsys Sentaurus Process, Silvaco Victory Process ### 11.2 Diffusion-Reaction System $$ \frac{\partial C_i}{\partial t} = \nabla \cdot (D_i \nabla C_i - \mu_i C_i \nabla \phi) + G_i - R_i $$ Where: - First term: Fickian diffusion - Second term: Drift in electric field (for charged species) - $G_i$ = generation rate - $R_i$ = recombination rate ### 11.3 Point Defect Dynamics **Vacancy concentration:** $$ \frac{\partial C_V}{\partial t} = D_V \nabla^2 C_V + G_V - k_{IV} C_I C_V $$ **Interstitial concentration:** $$ \frac{\partial C_I}{\partial t} = D_I \nabla^2 C_I + G_I - k_{IV} C_I C_V $$ Where $k_{IV}$ is the recombination rate constant. ### 11.4 Stress Evolution **Equilibrium equation:** $$ \nabla \cdot \boldsymbol{\sigma} = 0 $$ **Constitutive relation:** $$ \boldsymbol{\sigma} = \mathbf{C} : (\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{thermal} - \boldsymbol{\varepsilon}^{intrinsic}) $$ Where: - $\boldsymbol{\sigma}$ = stress tensor - $\mathbf{C}$ = elastic stiffness tensor - $\boldsymbol{\varepsilon}$ = total strain - $\boldsymbol{\varepsilon}^{thermal}$ = thermal strain = $\alpha \Delta T$ - $\boldsymbol{\varepsilon}^{intrinsic}$ = intrinsic strain (lattice mismatch) ### 11.5 Level Set Method for Interface Tracking $$ \frac{\partial \phi}{\partial t} + v_n |\nabla \phi| = 0 $$ Where: - $\phi$ = level set function (interface at $\phi = 0$) - $v_n$ = interface normal velocity ## 12. Advanced Topics ### 12.1 Atomic Layer Epitaxy (ALE) / Atomic Layer Deposition (ALD) Self-limiting surface reactions modeled as Langmuir kinetics: $$ \theta = \frac{K \cdot P \cdot t}{1 + K \cdot P \cdot t} \rightarrow 1 \quad \text{as } t \rightarrow \infty $$ **Growth per cycle (GPC):** $$ GPC = \theta_{sat} \cdot d_{monolayer} $$ Typical GPC values: 0.5-1.5 Å/cycle ### 12.2 III-V on Silicon Integration Challenges and models: - **Anti-phase boundaries (APBs)**: Form at single-step terraces - **Threading dislocations**: $\rho_{TD} \propto f^2$ initially - **Thermal mismatch stress**: $\sigma_{thermal} = \frac{E \Delta \alpha \Delta T}{1-\nu}$ ### 12.3 Quantum Dot Formation (Stranski-Krastanov) **Critical thickness for islanding:** $$ h_{SK} \approx \frac{\gamma}{M f^2} $$ **Island density:** $$ n_{island} \propto \exp\left(-\frac{E_{island}}{k_B T}\right) \cdot F^{1/3} $$ Where $F$ is the deposition flux. ### 12.4 Machine Learning in Epitaxy Modeling **Physics-Informed Neural Networks (PINNs):** $$ \mathcal{L}_{total} = \mathcal{L}_{data} + \lambda_{PDE}\mathcal{L}_{physics} + \lambda_{BC}\mathcal{L}_{boundary} $$ Where: - $\mathcal{L}_{data}$ = data fitting loss - $\mathcal{L}_{physics}$ = PDE residual loss - $\mathcal{L}_{boundary}$ = boundary condition loss - $\lambda$ = weighting parameters **Applications:** - Surrogate models for reactor optimization - Inverse problems (parameter extraction) - Process window optimization - Defect prediction ## 13. Key Equations | Phenomenon | Key Equation | Primary Parameters | |:-----------|:-------------|:-------------------| | Growth rate (dual regime) | $G = \frac{k_s h_g C_g}{k_s + h_g}$ | Temperature, pressure, flow | | Surface diffusion length | $\lambda_s = \sqrt{D_s \tau_s}$ | Temperature | | Lattice mismatch | $f = \frac{a_f - a_s}{a_s}$ | Material system | | Critical thickness | $h_c = \frac{b}{8\pi f(1+\nu)}\left[\ln\frac{h_c}{b}+1\right]$ | Mismatch, Burgers vector | | Elastic strain energy | $E = M f^2 h$ | Mismatch, thickness, modulus | | Nucleation rate | $J \propto \exp(-\Delta G^*/k_BT)$ | Supersaturation, surface energy | | Species transport | $\frac{\partial C}{\partial t} + \mathbf{v}\cdot\nabla C = D\nabla^2 C + R$ | Diffusivity, velocity, reactions | | KMC event rate | $\Gamma = \nu_0 \exp(-E_a/k_BT)$ | Activation energy, temperature | ## Physical Constants | Constant | Symbol | Value | |:---------|:-------|:------| | Boltzmann constant | $k_B$ | $1.38 \times 10^{-23}$ J/K | | Gas constant | $R$ | 8.314 J/mol$\cdot$K | | Planck constant | $h$ | $6.63 \times 10^{-34}$ J$\cdot$s | | Electron charge | $e$ | $1.60 \times 10^{-19}$ C | | Si lattice constant | $a_{Si}$ | 5.431 Å | | Ge lattice constant | $a_{Ge}$ | 5.658 Å | | GaAs lattice constant | $a_{GaAs}$ | 5.653 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 |
Epoch = one pass through entire dataset. Iteration = one batch update. More epochs can improve learning but risk overfitting.
One complete pass through the entire training dataset.
Standard molding material.
Privacy parameter smaller means more private.
Epsilon parameter bounds maximum privacy loss in differential privacy.
Epsilon sampling truncates distribution at absolute probability threshold.
Epsilon-greedy strategies in recommendation exploration randomly recommend items with probability epsilon for exploration.
Simple exploration: random action with probability ε.
Sample tasks uniformly.
Equalization compensates for frequency-dependent channel loss restoring signal integrity.
Equalized odds requires equal true positive and false positive rates across groups.
Model has same error rates across groups.
Equalized odds: equal error rates across groups. Balances false positives and negatives.
Find solutions to mathematical equations.
Formal acceptance of new tool.
Reference performance level.
What tool can achieve.
Model individual tool behavior.
Equipment effectiveness quantifies how well assets achieve intended function.
Equipment energy efficiency standards specify maximum power consumption for process tools.