fault tolerance in training, infrastructure
Continue despite failures.
3,145 technical terms and definitions
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Quantum computing with error correction.
FBNet uses differentiable NAS with hardware-aware latency loss to discover efficient architectures optimized for specific mobile devices.
Fast Causal Inference algorithm discovers causal graphs allowing for hidden confounders.
Identify important features.
Method using another class's data.
Match intermediate feature distributions.
Visualize what features detect.
Synthesize inputs maximizing neuron activation.
Federated learning at edge.
Poison in federated setting.
Federated learning trains models across decentralized devices without centralizing data.
FEDformer applies frequency-enhanced decomposed attention mixing seasonal-trend decomposition with frequency domain learning.
Use previous layer outputs as additional input.
# 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 |
Distill with minimal data.
Generate with minimal steps.
FFT convolutions compute long kernels efficiently in frequency domain.
FFT convolution performs spatial convolution through frequency domain multiplication for large kernels.
Single-step gradient-based attack.
Failures occurring at customer.
Thick oxide for isolation (older technology).
Fill rate measures order fulfillment completeness as percentage of requested items delivered.
Generate code for middle section given surrounding context.
Normalization without batch statistics.
Fine-tuning APIs (OpenAI, Anthropic) let you customize models without infrastructure. Easy but less control.
Classify entities into detailed types.
Detailed emotional states.
Classify on scale (very negative to very positive).
# Comprehensive Guide: LLM Training, AI Architecture & the Modern AI Stack
## 1. Foundational Hierarchy from Math to Intelligence
### The Conceptual Stack
The AI field builds upon layers of abstraction, each enabling the next:
- **Mathematics & Statistics**
- Linear algebra (vectors, matrices, tensors)
- Calculus (gradients, optimization)
- Probability theory (distributions, Bayesian inference)
- Information theory (entropy, mutual information)
- **Machine Learning**
- Statistical learning from data
- Pattern recognition without explicit programming
- Core paradigm: learn from examples, not rules
- **Deep Learning**
- Neural networks with multiple layers
- Hierarchical feature learning
- End-to-end differentiable systems
- **Artificial Intelligence**
- The overarching goal
- Systems exhibiting intelligent behavior
- ML/DL are currently the most successful approaches
### Mathematical Foundation
The fundamental learning objective in supervised learning:
$$\min_{\theta} \mathcal{L}(\theta) = \min_{\theta} \frac{1}{N} \sum_{i=1}^{N} \ell(f_\theta(x_i), y_i)$$
Where:
- $\theta$ = model parameters
- $f_\theta$ = model function
- $\ell$ = loss function
- $(x_i, y_i)$ = training examples
## 2. The Transformer Revolution
### Why Transformers Changed Everything
Before 2017, sequence modeling relied on recurrent architectures:
- **RNNs (Recurrent Neural Networks)**
- Sequential processing (cannot parallelize)
- Vanishing gradient problem
- Limited long-range dependencies
- **LSTMs (Long Short-Term Memory)**
- Gating mechanisms for memory
- Better long-range, but still sequential
- Computationally expensive for long sequences
### The Attention Mechanism
The breakthrough paper "Attention Is All You Need" (Vaswani et al., 2017) introduced:
**Self-Attention Formula:**
$$\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V$$
Where:
- $Q$ = Query matrix $\in \mathbb{R}^{n \times d_k}$
- $K$ = Key matrix $\in \mathbb{R}^{n \times d_k}$
- $V$ = Value matrix $\in \mathbb{R}^{n \times d_v}$
- $d_k$ = dimension of keys (scaling factor)
- $n$ = sequence length
**Multi-Head Attention:**
$$\text{MultiHead}(Q, K, V) = \text{Concat}(\text{head}_1, ..., \text{head}_h)W^O$$
$$\text{head}_i = \text{Attention}(QW_i^Q, KW_i^K, VW_i^V)$$
### Key Architectural Components
| Component | Function | Mathematical Operation |
|-----------|----------|----------------------|
| Multi-Head Attention | Multiple parallel attention patterns | $h$ parallel attention functions |
| Positional Encoding | Injects sequence order | $PE_{(pos,2i)} = \sin(pos/10000^{2i/d})$ |
| Feed-Forward Networks | Non-linear transformations | $\text{FFN}(x) = \max(0, xW_1 + b_1)W_2 + b_2$ |
| Layer Normalization | Training stability | $\text{LN}(x) = \gamma \odot \frac{x - \mu}{\sigma} + \beta$ |
| Residual Connections | Gradient flow | $\text{output} = x + \text{SubLayer}(x)$ |
### BERT vs GPT: Two Paradigms
**BERT (Bidirectional Encoder Representations from Transformers)**
- Architecture: Encoder-only
- Context: Bidirectional (sees left and right)
- Pre-training objectives:
- **MLM (Masked Language Modeling):**
$$\mathcal{L}_{\text{MLM}} = -\mathbb{E}\left[\log P(x_{\text{mask}} | x_{\text{context}})\right]$$
- **NSP (Next Sentence Prediction):**
$$\mathcal{L}_{\text{NSP}} = -\mathbb{E}\left[\log P(\text{IsNext} | A, B)\right]$$
- Best for: Classification, NER, extractive QA
**GPT (Generative Pre-trained Transformer)**
- Architecture: Decoder-only
- Context: Autoregressive (left-to-right only)
- Pre-training objective:
$$\mathcal{L}_{\text{GPT}} = -\sum_{t=1}^{T} \log P(x_t | x_{
Unique identifiers in model behavior.
Finite capacity scheduling respects resource constraints ensuring realistic achievable schedules.
Advanced position encoding method.
Fisher information pruning estimates parameter importance using Fisher information matrix.
Weight averaging by Fisher information.
Pre-defined sparse attention structure.
FixMatch combines consistency regularization with pseudo-labeling using strong augmentation and confidence thresholding for semi-supervised learning.
Create test fixtures automatically.
Few-shot vision-language model.
T5 model fine-tuned on many instruction tasks.
Finetuned Language Net scales instruction tuning across many tasks and templates.
Optimized attention algorithm that reduces memory usage and speeds up training/inference.
Flax and Haiku are neural network libraries for JAX. Functional style. Google ecosystem.
Flex testing applies board bending to evaluate solder joint robustness under mechanical strain.
Measure of computational cost for training or inference.
FLOPs efficiency measures actual throughput relative to theoretical peak floating-point operations.
Flowise is visual LangChain builder. Drag and drop.
Temperature mapping using fluorescence.
Flying probe testers use movable probe heads for board-level testing without fixed fixtures enabling flexible low-volume testing.