feature engineering for materials, materials science
Create descriptive features for materials.
395 technical terms and definitions
Create descriptive features for materials.
Feature engineering creates informative features. Domain knowledge.
Method using another class's data.
Use pre-trained model as feature extractor.
Toggle features on/off without redeployment.
Feature flags enable/disable features without deploy. A/B test new models, roll back instantly if issues.
Networks significantly change during training.
Match intermediate feature distributions.
Extract multi-scale features.
Feature selection removes unimportant features. Reduce overfitting.
Feature stores manage ML features: versioning, serving, consistency between training and inference.
Centralized repository for storing and serving ML features.
Visualize what features detect.
Synthesize inputs maximizing neuron activation.
Feature visualization generates input patterns maximally activating specific neurons revealing learned features.
Simulate individual feature evolution.
Average local models in federated learning.
Federated learning at edge.
Poison in federated setting.
Federated learning trains models across decentralized devices without centralizing data.
Federated learning trains on decentralized data. Data stays on device. Privacy-preserving.
Train models on decentralized data without centralizing raw data.
Federated learning trains on distributed data without centralizing. Privacy-preserving for sensitive data.
Add proximal term to FedAvg.
Federated recommendation trains models across decentralized user devices without centralizing private interaction data.
FEDformer applies frequency-enhanced decomposed attention mixing seasonal-trend decomposition with frequency domain learning.
Normalized averaging for heterogeneous data.
Adaptive optimization for federated learning.
Personalize by separating base and head.
Use upstream data to adjust downstream.
Feedback control uses measurement results to adjust upstream process parameters.
Adjust based on measurement after processing.
Use previous layer outputs as additional input.
Collect user feedback on AI outputs. Thumbs up/down, ratings, corrections. Use to improve prompts and models.
Feedforward control compensates for known disturbances before they affect output.
Feedforward network (FFN/MLP) in transformer processes each position. Typically 4x hidden size. Most parameters here.
Front-End-Of-Line integration encompasses process steps from substrate preparation through transistor formation including wells gates and sources-drains.
# FEOL: Front End of Line in Semiconductor Manufacturing ## 1. Definition **FEOL (Front End of Line)** refers to the first portion of integrated circuit (IC) fabrication where individual transistors and other active devices are patterned directly into the silicon wafer. **Key Boundary:** The dividing line between FEOL and BEOL (Back End of Line) is typically the formation of the first metal layer—everything before that is FEOL. ## 2. Key FEOL Process Steps ### 2.1 Wafer Preparation - Starting with highly purified silicon wafer - Typical diameter: $300 \, \text{mm}$ (advanced nodes) - Crystal orientation: $\langle 100 \rangle$ for CMOS - Resistivity specification: $$ \rho = \frac{1}{q \cdot (n \cdot \mu_n + p \cdot \mu_p)} $$ where: - $q$ = electron charge ($1.6 \times 10^{-19} \, \text{C}$) - $n, p$ = carrier concentrations - $\mu_n, \mu_p$ = electron and hole mobilities ### 2.2 Well Formation - **Purpose:** Create n-wells and p-wells for PMOS and NMOS transistors - **Process:** Ion implantation - **Dopant dose calculation:** $$ D = \int_0^{t} J(t') \, dt' \quad [\text{ions/cm}^2] $$ - **Gaussian dopant profile after implantation:** $$ N(x) = \frac{D}{\sqrt{2\pi} \cdot \Delta R_p} \exp\left[-\frac{(x - R_p)^2}{2(\Delta R_p)^2}\right] $$ where: - $R_p$ = projected range - $\Delta R_p$ = straggle (standard deviation) - $D$ = dose - **Annealing requirements:** - Temperature: $900°\text{C} - 1100°\text{C}$ - Purpose: Activate dopants, repair crystal damage ### 2.3 Shallow Trench Isolation (STI) - **Process steps:** 1. Pad oxide growth ($\text{SiO}_2$) 2. Nitride deposition ($\text{Si}_3\text{N}_4$) 3. Lithography and etch 4. Trench fill with $\text{SiO}_2$ (HDP-CVD or HARP) 5. Chemical-Mechanical Polishing (CMP) - **Trench depth scaling:** $$ d_{\text{STI}} \approx 250 - 350 \, \text{nm} \quad \text{(typical)} $$ - **Aspect ratio:** $$ AR = \frac{\text{Depth}}{\text{Width}} \approx 5:1 \text{ to } 10:1 $$ ### 2.4 Gate Stack Formation #### 2.4.1 Traditional Gate Stack (Pre-45nm) - Gate dielectric: Thermal $\text{SiO}_2$ - Gate electrode: Polysilicon - **Gate oxide capacitance:** $$ C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}} = \frac{\varepsilon_0 \cdot k_{SiO_2}}{t_{ox}} $$ where $k_{SiO_2} \approx 3.9$ #### 2.4.2 High-k/Metal Gate (HKMG) — 45nm and Beyond - **High-k dielectric:** Hafnium oxide ($\text{HfO}_2$), $k \approx 22-25$ - **Equivalent Oxide Thickness (EOT):** $$ EOT = t_{\text{high-k}} \cdot \frac{k_{SiO_2}}{k_{\text{high-k}}} $$ - **Example calculation:** - If $t_{HfO_2} = 2 \, \text{nm}$ and $k_{HfO_2} = 22$: $$ EOT = 2 \, \text{nm} \cdot \frac{3.9}{22} \approx 0.35 \, \text{nm} $$ - **Metal gate work function requirements:** - NMOS: $\Phi_m \approx 4.1 - 4.3 \, \text{eV}$ (near Si conduction band) - PMOS: $\Phi_m \approx 4.9 - 5.1 \, \text{eV}$ (near Si valence band) ### 2.5 Transistor Architectures #### 2.5.1 Planar MOSFET (Legacy) - **Drain current (saturation):** $$ I_{D,sat} = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 $$ - **Threshold voltage:** $$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2 \varepsilon_{Si} q N_A (2\phi_F)}}{C_{ox}} $$ #### 2.5.2 FinFET (22nm – 5nm) - **Effective width:** $$ W_{eff} = 2 \cdot H_{fin} + W_{fin} $$ where: - $H_{fin}$ = fin height - $W_{fin}$ = fin width - **Typical dimensions at 7nm node:** - Fin pitch: $\sim 25-30 \, \text{nm}$ - Fin height: $\sim 40-50 \, \text{nm}$ - Fin width: $\sim 6-8 \, \text{nm}$ - **Drive current per fin:** $$ I_{fin} \propto \mu \cdot C_{ox} \cdot W_{eff} \cdot \frac{(V_{GS} - V_{th})^\alpha}{L} $$ where $\alpha \approx 1.0 - 1.3$ (velocity saturation effects) #### 2.5.3 Gate-All-Around (GAA) / Nanosheet FET (3nm and below) - **Channel structure:** Stacked horizontal nanosheets - **Effective width per sheet:** $$ W_{eff,sheet} = 2(W_{NS} + t_{NS}) $$ where: - $W_{NS}$ = nanosheet width - $t_{NS}$ = nanosheet thickness - **Total effective width (N sheets):** $$ W_{eff,total} = N \cdot 2(W_{NS} + t_{NS}) $$ - **Electrostatic integrity (natural length):** $$ \lambda = \sqrt{\frac{\varepsilon_{Si}}{\varepsilon_{ox}} \cdot t_{Si} \cdot t_{ox}} $$ - GAA achieves smallest $\lambda$ → best short-channel control ### 2.6 Source/Drain Engineering #### 2.6.1 Ion Implantation - **Lightly Doped Drain (LDD):** - Dose: $\sim 10^{13} - 10^{14} \, \text{cm}^{-2}$ - Energy: $\sim 1-10 \, \text{keV}$ - **Main S/D implant:** - Dose: $\sim 10^{15} - 10^{16} \, \text{cm}^{-2}$ - Energy: $\sim 10-50 \, \text{keV}$ #### 2.6.2 Epitaxial Raised Source/Drain - **PMOS strain engineering:** - Material: $\text{Si}_{1-x}\text{Ge}_x$ (typically $x = 0.3 - 0.5$) - Induces compressive strain in channel - **Lattice mismatch:** $$ \frac{\Delta a}{a} = \frac{a_{SiGe} - a_{Si}}{a_{Si}} \approx 0.042 \cdot x $$ - **Mobility enhancement:** $$ \frac{\Delta \mu_p}{\mu_p} \approx 50-100\% \text{ (with ~1-2 GPa compressive stress)} $$ - **NMOS strain engineering:** - Material: $\text{Si:C}$ or $\text{Si:P}$ - Induces tensile strain ### 2.7 Silicide Formation (Salicide Process) - **Purpose:** Reduce contact resistance at S/D and gate - **Materials evolution:** - $\text{TiSi}_2$ → $\text{CoSi}_2$ → $\text{NiSi}$ → $\text{Ni}_{1-x}\text{Pt}_x\text{Si}$ - **Sheet resistance:** $$ R_s = \frac{\rho}{t} \quad [\Omega/\square] $$ - **Typical values:** - NiSi: $\rho \approx 10-20 \, \mu\Omega\text{-cm}$ - For $t = 20 \, \text{nm}$: $R_s \approx 5-10 \, \Omega/\square$ - **Contact resistance:** $$ R_c = \rho_c / A_c $$ where $\rho_c$ is specific contact resistivity (target: $< 10^{-9} \, \Omega\text{-cm}^2$) ### 2.8 Contact Etch Stop Layer (CESL) - **Material:** Silicon nitride ($\text{Si}_3\text{N}_4$ or $\text{SiN}$) - **Stress engineering:** - Tensile CESL for NMOS: $+1.5 - 2.0 \, \text{GPa}$ - Compressive CESL for PMOS: $-2.0 - 3.0 \, \text{GPa}$ - **Strain transfer efficiency:** $$ \varepsilon_{channel} = \eta \cdot \varepsilon_{CESL} $$ where $\eta$ depends on geometry and materials ## 3. Critical FEOL Parameters and Equations ### 3.1 Lithography Resolution - **Rayleigh criterion:** $$ CD_{min} = k_1 \cdot \frac{\lambda}{NA} $$ where: - $CD_{min}$ = minimum feature size - $k_1$ = process factor ($\sim 0.25 - 0.4$) - $\lambda$ = wavelength (EUV: $13.5 \, \text{nm}$, ArF: $193 \, \text{nm}$) - $NA$ = numerical aperture - **EUV at high-NA:** - $\lambda = 13.5 \, \text{nm}$, $NA = 0.55$, $k_1 = 0.3$: $$ CD_{min} = 0.3 \cdot \frac{13.5}{0.55} \approx 7.4 \, \text{nm} $$ ### 3.2 Depth of Focus $$ DOF = k_2 \cdot \frac{\lambda}{NA^2} $$ - Trade-off: Higher NA improves resolution but reduces DOF ### 3.3 Threshold Voltage Variation - **Random Dopant Fluctuation (RDF):** $$ \sigma_{V_{th}} \propto \frac{1}{\sqrt{W \cdot L}} \cdot \sqrt{N_A \cdot t_{dep}} $$ - **Pelgrom coefficient:** $$ \sigma_{V_{th}} = \frac{A_{VT}}{\sqrt{W \cdot L}} $$ where $A_{VT} \approx 1-3 \, \text{mV} \cdot \mu\text{m}$ for advanced nodes ### 3.4 Subthreshold Swing $$ SS = \frac{k_B T}{q} \cdot \ln(10) \cdot \left(1 + \frac{C_{dep}}{C_{ox}}\right) $$ - **Ideal limit at room temperature:** $$ SS_{ideal} = \frac{k_B T}{q} \cdot \ln(10) \approx 60 \, \text{mV/decade} $$ - **GAA advantage:** Lower $C_{dep}/C_{ox}$ ratio → closer to ideal SS ### 3.5 DIBL (Drain-Induced Barrier Lowering) $$ DIBL = \frac{V_{th,lin} - V_{th,sat}}{V_{DD} - V_{D,lin}} \quad [\text{mV/V}] $$ - **Target:** $< 100 \, \text{mV/V}$ for good short-channel control ## 4. FEOL vs. BEOL Comparison | Parameter | FEOL | BEOL | |-----------|------|------| | **Focus** | Transistors, active devices | Metal interconnects | | **Temperature** | High ($\leq 1100°\text{C}$) | Low ($< 450°\text{C}$) | | **Materials** | Si, $\text{HfO}_2$, $\text{TiN}$, silicides | Cu, low-k, Ta/TaN barriers | | **Critical dimension** | Gate length, fin width | Metal pitch, via diameter | | **Dielectric** | High-k ($k > 10$) | Low-k ($k < 3$) | ## 5. Thermal Budget Considerations ### 5.1 Dopant Diffusion - **Fick's second law:** $$ \frac{\partial C}{\partial t} = D \cdot \frac{\partial^2 C}{\partial x^2} $$ - **Diffusion length:** $$ L_D = 2\sqrt{D \cdot t} $$ - **Temperature dependence:** $$ D = D_0 \exp\left(-\frac{E_a}{k_B T}\right) $$ - Example for B in Si: $D_0 \approx 0.76 \, \text{cm}^2/\text{s}$, $E_a \approx 3.46 \, \text{eV}$ ### 5.2 Thermal Budget Constraint $$ \text{Thermal Budget} = \int_0^{t_{process}} \exp\left(-\frac{E_a}{k_B T(t')}\right) dt' $$ - Must minimize while achieving required dopant activation ## 6. Yield and Defectivity ### 6.1 Poisson Yield Model $$ Y = \exp(-D_0 \cdot A) $$ where: - $D_0$ = defect density (defects/cm²) - $A$ = chip area ### 6.2 Murphy's Yield Model (more realistic) $$ Y = \left(\frac{1 - \exp(-D_0 \cdot A)}{D_0 \cdot A}\right)^2 $$ ### 6.3 Defect Density Requirements - For 90% yield on $100 \, \text{mm}^2$ chip: $$ D_0 < \frac{-\ln(0.9)}{1 \, \text{cm}^2} \approx 0.1 \, \text{defects/cm}^2 $$ ## 7. Key Equipment in FEOL | Process | Equipment Vendor | Key Technology | |---------|------------------|----------------| | **EUV Lithography** | ASML | NXE:3600D, High-NA EUV | | **Plasma Etch** | Lam, TEL, AMAT | Atomic Layer Etch (ALE) | | **ALD** | ASM, Lam, TEL | Thermal and Plasma ALD | | **Ion Implant** | AMAT, Axcelis | High-current, precision | | **Epitaxy** | AMAT, ASM | Selective epi for S/D | | **Anneal** | AMAT, Mattson | Spike RTA, Laser anneal | | **CMP** | AMAT, Ebara | Advanced slurries | ## 8. Advanced Node Roadmap ### 8.1 Node Progression | Node | Year | Transistor | Gate Length | Key Challenge | |------|------|------------|-------------|---------------| | 22nm | 2012 | FinFET | ~25nm | First 3D transistor | | 14nm | 2014 | FinFET | ~20nm | Fin pitch scaling | | 7nm | 2018 | FinFET | ~16nm | EUV introduction | | 5nm | 2020 | FinFET | ~12nm | Multi-EUV layers | | 3nm | 2022 | GAA | ~12nm | Nanosheet release | | 2nm | 2025 | GAA | ~10nm | High-NA EUV | | A14 | 2027+ | CFET | ~8nm | Vertical stacking | ### 8.2 Transistor Density Scaling $$ \text{Density} \propto \frac{1}{(\text{CPP} \times \text{MMP})^2} $$ where: - CPP = Contacted Poly Pitch - MMP = Minimum Metal Pitch ## 9. Economic Considerations ### 9.1 Fab Cost - Leading-edge fab: USD 20-30 billion - EUV scanner cost: USD 350-400 million each (approximately) - High-NA EUV: USD 400+ million ### 9.2 Cost per Wafer $$ \text{Cost/wafer} = \frac{\text{Fab cost} + \text{Operating cost}}{\text{Wafer throughput} \times \text{Lifetime}} $$ - Typical: USD 10,000 - 20,000 per wafer at leading edge ### 9.3 Cost per Transistor $$ \text{Cost/transistor} = \frac{\text{Cost/die}}{\text{Transistors/die}} $$ - Continues to decrease despite rising fab costs (Moore's Law economics) ## 10. FEOL FEOL is the foundational phase of semiconductor manufacturing where transistors—the computational heart of integrated circuits—are created. Key aspects include: - **Process complexity:** 400+ individual process steps - **Precision:** Atomic-scale control ($< 1 \, \text{nm}$ uniformity) - **Temperature:** High thermal budgets up to $1100°\text{C}$ - **Evolution:** Planar → FinFET → GAA → CFET - **Critical metrics:** - $V_{th}$ uniformity: $\sigma < 20 \, \text{mV}$ - Defect density: $< 0.1 \, \text{defects/cm}^2$ - CD control: $< 1 \, \text{nm}$ 3-sigma
# 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·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 μm/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·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·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·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·K) - $k$ = thermal conductivity (W/m·K) - $\Delta H_j$ = enthalpy of reaction $j$ (J/mol) - $r_j$ = rate of reaction $j$ (mol/m³·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·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·K | | Planck constant | $h$ | $6.63 \times 10^{-34}$ J·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 |
Occupation statistics for fermions.
Hierarchical goal-setting.
Feudal reinforcement learning establishes manager-worker hierarchies where managers set subgoals for workers to achieve.
Dataset and benchmark for fact checking.
Fact-checking benchmark.
Few-shot learning: provide examples in prompt, model learns pattern. Zero-shot: no examples. More shots usually help.
Classify with only few examples per class.
Provide reasoning examples.
Few-shot chain-of-thought provides example reasoning chains teaching problem decomposition.
Distill with minimal data.
How performance scales with examples.