czochralski,crystal growth,silicon ingot,pure silicon
# Mathematics of the Czochralski Process
## 1. Introduction
The **Czochralski (CZ) method** is the dominant technique for growing single-crystal semiconductors, responsible for approximately 90% of silicon wafers used in integrated circuits. The mathematical modeling of this process involves:
- **Multi-phase heat transfer** with moving boundaries
- **Turbulent melt convection** driven by buoyancy and rotation
- **Capillary phenomena** controlling crystal shape
- **Mass transport** of dopants and impurities
- **Phase transition thermodynamics**
This document presents the key mathematical frameworks governing the CZ process.
## 2. Heat Transfer: The Stefan Problem
### 2.1 Overview
The CZ process is fundamentally a **moving boundary problem** where the solid-liquid interface position must be determined as part of the solution.
### 2.2 Governing Heat Equation
For both solid (crystal) and liquid (melt) regions:
$$
\rho c_p \frac{\partial T}{\partial t} + \rho c_p (\mathbf{u} \cdot \nabla T) = \nabla \cdot (k \nabla T) + Q
$$
**Where:**
- $\rho$ — density [kg/m³]
- $c_p$ — specific heat capacity [J/(kg·K)]
- $T$ — temperature [K]
- $\mathbf{u}$ — velocity field [m/s]
- $k$ — thermal conductivity [W/(m·K)]
- $Q$ — volumetric heat source [W/m³]
### 2.3 Stefan Condition at the Interface
At the **solid-liquid interface**, the Stefan condition balances latent heat release with conductive heat flux:
$$
\rho L v_n = k_s \left( \frac{\partial T}{\partial n} \right)_s - k_l \left( \frac{\partial T}{\partial n} \right)_l
$$
**Where:**
- $L$ — latent heat of fusion [J/kg]
- $v_n$ — interface velocity (normal direction) [m/s]
- $k_s$, $k_l$ — thermal conductivity of solid and liquid [W/(m·K)]
- $n$ — unit normal vector pointing into the solid
### 2.4 Boundary Conditions
| Location | Condition | Equation |
|----------|-----------|----------|
| Crystal surface | Radiation + convection | $-k\frac{\partial T}{\partial n} = h(T - T_{\infty}) + \varepsilon \sigma (T^4 - T_{amb}^4)$ |
| Melt free surface | Radiation + evaporation | $-k\frac{\partial T}{\partial n} = \varepsilon \sigma T^4 + q_{evap}$ |
| Crucible wall | Specified temperature | $T = T_{crucible}(z)$ |
| Solid-liquid interface | Melting point | $T = T_m$ |
### 2.5 Enthalpy Formulation
For numerical implementation, the **enthalpy method** avoids explicit interface tracking:
$$
\frac{\partial H}{\partial t} + \nabla \cdot (\mathbf{u} H) = \nabla \cdot (k \nabla T)
$$
**With enthalpy defined as:**
$$
H(T) = \begin{cases}
\rho_s c_{p,s} T & T < T_m \\
\rho_s c_{p,s} T_m + \rho L f_l & T = T_m \\
\rho_l c_{p,l} T + \rho L & T > T_m
\end{cases}
$$
Where $f_l \in [0,1]$ is the liquid fraction.
## 3. Fluid Dynamics: Navier-Stokes Equations
### 3.1 Governing Equations
The melt flow is governed by the **incompressible Navier-Stokes equations** with the **Boussinesq approximation** for buoyancy:
**Continuity (mass conservation):**
$$
\nabla \cdot \mathbf{u} = 0
$$
**Momentum conservation:**
$$
\rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \beta (T - T_0) \mathbf{g} + \mathbf{F}_{ext}
$$
**Where:**
- $\rho_0$ — reference density [kg/m³]
- $p$ — pressure [Pa]
- $\mu$ — dynamic viscosity [Pa·s]
- $\beta$ — thermal expansion coefficient [K⁻¹]
- $T_0$ — reference temperature [K]
- $\mathbf{g}$ — gravitational acceleration [m/s²]
- $\mathbf{F}_{ext}$ — external forces (rotation, magnetic) [N/m³]
### 3.2 Boussinesq Approximation
The density variation is linearized:
$$
\rho(T) \approx \rho_0 [1 - \beta(T - T_0)]
$$
**Validity condition:**
$$
\frac{\Delta \rho}{\rho_0} = \beta \Delta T \ll 1
$$
### 3.3 Rotational Effects
For rotating crystal (angular velocity $\Omega_c$) and crucible ($\Omega_{cr}$):
**Coriolis force:**
$$
\mathbf{F}_{Cor} = -2\rho_0 (\boldsymbol{\Omega} \times \mathbf{u})
$$
**Centrifugal force:**
$$
\mathbf{F}_{cent} = -\rho_0 \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r})
$$
### 3.4 Marangoni (Thermocapillary) Convection
At the free surface, temperature-dependent surface tension drives flow:
$$
\mu \frac{\partial u_t}{\partial n} = \frac{\partial \gamma}{\partial T} \frac{\partial T}{\partial t}
$$
**Where:**
- $\gamma$ — surface tension [N/m]
- $\frac{\partial \gamma}{\partial T}$ — temperature coefficient of surface tension [N/(m·K)]
### 3.5 Dimensionless Numbers
| Number | Definition | Physical Meaning |
|--------|------------|------------------|
| **Reynolds** | $Re = \frac{\rho U L}{\mu}$ | Inertia / Viscous forces |
| **Grashof** | $Gr = \frac{g \beta \Delta T L^3}{\nu^2}$ | Buoyancy / Viscous forces |
| **Prandtl** | $Pr = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}$ | Momentum / Thermal diffusivity |
| **Rayleigh** | $Ra = Gr \cdot Pr$ | Convection strength |
| **Marangoni** | $Ma = \frac{\left|\frac{\partial \gamma}{\partial T}\right| \Delta T L}{\mu \alpha}$ | Surface tension / Viscous forces |
**For silicon CZ growth:**
- $Ra \sim 10^8 - 10^{10}$ (turbulent regime)
- $Pr \approx 0.01$ (liquid metals)
## 4. Capillarity: Young-Laplace Equation
### 4.1 Meniscus Shape
The melt meniscus connecting the crystal to the melt surface is governed by the **Young-Laplace equation**:
$$
\Delta P = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right)
$$
**Where:**
- $\Delta P$ — pressure difference across interface [Pa]
- $\gamma$ — surface tension [N/m]
- $R_1$, $R_2$ — principal radii of curvature [m]
### 4.2 Axisymmetric Formulation
For axisymmetric geometry with meniscus profile $z = z(r)$:
$$
\gamma \left[ \frac{z''}{(1 + z'^2)^{3/2}} + \frac{z'}{r(1 + z'^2)^{1/2}} \right] = \rho g z
$$
**Where:**
- $z'$ = $\frac{dz}{dr}$
- $z''$ = $\frac{d^2z}{dr^2}$
### 4.3 Arc-Length Parameterization
Using arc-length $s$ for numerical stability:
$$
\frac{dr}{ds} = \cos\phi
$$
$$
\frac{dz}{ds} = \sin\phi
$$
$$
\frac{d\phi}{ds} = \frac{\rho g z}{\gamma} - \frac{\sin\phi}{r}
$$
**Where $\phi$ is the tangent angle to the meniscus.**
### 4.4 Growth Angle Condition
At the **triple point** (crystal-melt-gas junction):
$$
\phi_{triple} = \alpha_{gr}
$$
**Where $\alpha_{gr}$ is the characteristic growth angle:**
| Material | Growth Angle $\alpha_{gr}$ |
|----------|---------------------------|
| Silicon | 11° |
| Germanium | 13° |
| GaAs | 17° |
| Sapphire | 35° |
### 4.5 Crystal Radius Evolution
The crystal radius changes according to:
$$
\frac{dr_c}{dt} = v_p \tan(\alpha_{gr} - \theta)
$$
**Where:**
- $r_c$ — crystal radius [m]
- $v_p$ — pulling velocity [m/s]
- $\theta$ — current meniscus angle at triple point
**Stability conditions:**
- $\theta < \alpha_{gr}$ → Crystal radius **increases**
- $\theta = \alpha_{gr}$ → Crystal radius **constant** (steady state)
- $\theta > \alpha_{gr}$ → Crystal radius **decreases**
### 4.6 Capillary Constant
The capillary length scale:
$$
a = \sqrt{\frac{2\gamma}{\rho g}}
$$
**For silicon:** $a \approx 7.6$ mm
## 5. Crystal Growth Rate
### 5.1 Interface Energy Balance
The local growth velocity is determined by heat flux balance:
$$
v_g = \frac{1}{\rho L} \left[ k_s \left( \frac{\partial T}{\partial n} \right)_s - k_l \left( \frac{\partial T}{\partial n} \right)_l \right]
$$
### 5.2 Simplified Growth Rate Model
$$
G = \frac{dL}{dt} = v_p \cdot \frac{T_m - T_i}{T_m - T_s}
$$
**Where:**
- $G$ — crystal growth rate [m/s]
- $L$ — crystal length [m]
- $v_p$ — pulling rate [m/s]
- $T_m$ — melting temperature [K]
- $T_i$ — interface temperature [K]
- $T_s$ — seed temperature [K]
### 5.3 Maximum Pull Rate
The maximum pull rate is limited by heat transfer:
$$
v_{p,max} = \frac{k_s G_s}{\rho L}
$$
**Where $G_s = \left( \frac{\partial T}{\partial z} \right)_s$ is the axial temperature gradient in the crystal.**
## 6. Dopant Distribution: Scheil Equation
### 6.1 Segregation Coefficient
The **equilibrium segregation coefficient** is defined as:
$$
k_0 = \frac{C_s}{C_l}
$$
**Where:**
- $C_s$ — solute concentration in solid at interface [atoms/cm³]
- $C_l$ — solute concentration in liquid at interface [atoms/cm³]
**Typical values for silicon:**
| Dopant | $k_0$ |
|--------|-------|
| Boron (B) | 0.80 |
| Phosphorus (P) | 0.35 |
| Arsenic (As) | 0.30 |
| Antimony (Sb) | 0.023 |
| Oxygen (O) | 1.25 |
### 6.2 Scheil-Gulliver Equation
For a well-mixed melt with no solid-state diffusion:
**Liquid concentration:**
$$
C_L = C_0 (1 - f_s)^{k_0 - 1}
$$
**Solid concentration:**
$$
C_s = k_0 C_0 (1 - f_s)^{k_0 - 1}
$$
**Where:**
- $C_0$ — initial concentration in melt [atoms/cm³]
- $f_s$ — solidified fraction $(= V_s / V_{total})$
### 6.3 Effective Segregation Coefficient
The **Burton-Prim-Slichter (BPS) equation** accounts for diffusion boundary layer:
$$
k_{eff} = \frac{k_0}{k_0 + (1 - k_0) \exp\left( -\frac{v_g \delta}{D} \right)}
$$
**Where:**
- $\delta$ — boundary layer thickness [m]
- $D$ — diffusion coefficient in liquid [m²/s]
- $v_g$ — growth velocity [m/s]
### 6.4 Boundary Layer Thickness
For rotating crystal (Cochran model):
$$
\delta = 1.6 D^{1/3} \nu^{1/6} \Omega^{-1/2}
$$
**Where:**
- $\nu$ — kinematic viscosity [m²/s]
- $\Omega$ — crystal rotation rate [rad/s]
### 6.5 Lambert W Function Solution
For complex segregation problems, the Scheil equation leads to transcendental equations:
$$
\left( \frac{C_s}{C_L} \right) \ln(1 - f_s) \cdot e^{\left( \frac{C_s}{C_L} \ln(1-f_s) \right)} = \left( \frac{C_s}{C_0} \right) (1 - f_s) \ln(1 - f_s)
$$
**Solution via Lambert W function:**
$$
x \cdot e^x = y \implies x = W(y)
$$
## 7. Oxygen Transport
### 7.1 Convection-Diffusion Equation
Oxygen concentration in the melt follows:
$$
\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D_O \nabla^2 C
$$
**Where:**
- $C$ — oxygen concentration [atoms/cm³]
- $D_O$ — oxygen diffusivity in silicon melt ≈ 5×10⁻⁸ m²/s
### 7.2 Boundary Conditions
**Crucible wall (dissolution):**
$$
C_{wall} = A \exp\left( -\frac{E_a}{RT} \right)
$$
**Common models:**
- **Matsuo et al.:** $C_{wall} = 3.99 \times 10^{25} \exp\left( -\frac{1.2 \text{ eV}}{k_B T} \right)$ atoms/cm³
**Free surface (evaporation):**
$$
-D_O \frac{\partial C}{\partial n} = k_{evap} (C - C_{eq})
$$
**Crystal interface (segregation):**
$$
D_O \frac{\partial C}{\partial n}\bigg|_{melt} = D_{O,s} \frac{\partial C}{\partial n}\bigg|_{crystal} + (1 - k_O) v_g C_{interface}
$$
### 7.3 Oxygen Transport Mechanisms
```
-
┌──────────────────────────────────────────────────────────────┐
│ OXYGEN TRANSPORT │
├──────────────────────────────────────────────────────────────┤
│ │
│ SiO₂ Crucible ──→ Dissolution ──→ Si Melt │
│ │ │
│ ├──→ Evaporation │
│ │ (as SiO) │
│ │ │
│ └──→ Crystal │
│ (segregation) │
│ │
└──────────────────────────────────────────────────────────────┘
```
## 8. Magnetohydrodynamics (MHD)
### 8.1 Lorentz Force
Applied magnetic fields $\mathbf{B}$ modify melt flow through the Lorentz force:
$$
\mathbf{F}_L = \mathbf{J} \times \mathbf{B}
$$
**Where $\mathbf{J}$ is the current density:**
$$
\mathbf{J} = \sigma (\mathbf{E} + \mathbf{u} \times \mathbf{B})
$$
### 8.2 Maxwell's Equations
$$
\nabla \times \mathbf{B} = \mu_0 \mathbf{J}
$$
$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
$$
$$
\nabla \cdot \mathbf{B} = 0
$$
### 8.3 Magnetic Reynolds Number
$$
Re_m = \mu_0 \sigma U L
$$
For silicon melts: $Re_m \ll 1$, so **induced fields are negligible** (quasi-static approximation).
### 8.4 Hartmann Number
The ratio of electromagnetic to viscous forces:
$$
Ha = B L \sqrt{\frac{\sigma}{\mu}}
$$
**Effects of magnetic field:**
- $Ha > 10$: Significant flow suppression
- $Ha > 100$: Quasi-two-dimensional flow
### 8.5 Common Magnetic Field Configurations
| Configuration | Field Direction | Primary Effect |
|---------------|-----------------|----------------|
| **Axial (VMCZ)** | Parallel to pull axis | Suppresses meridional convection |
| **Transverse (HMCZ)** | Perpendicular to axis | Creates asymmetric flow |
| **Cusp (CMCZ)** | Combined radial/axial | Controls flow at specific heights |
## 9. Integrated Thermal-Capillary Model
### 9.1 Coupled System
The complete CZ model couples multiple physics:
**Heat transfer:**
$$
\rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot (k \nabla T)
$$
**Momentum (in melt):**
$$
\rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \beta (T - T_0) \mathbf{g}
$$
**Species transport:**
$$
\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D \nabla^2 C
$$
**Interface position (Stefan condition):**
$$
\rho L v_n = [k \nabla T]_{jump}
$$
**Meniscus shape (Young-Laplace):**
$$
\gamma \kappa = \rho g z
$$
### 9.2 Radiation Heat Transfer
**Surface-to-surface radiation:**
$$
q_i = \varepsilon_i \sigma T_i^4 - \sum_{j=1}^{N} F_{ij} \varepsilon_j \sigma T_j^4
$$
**Where $F_{ij}$ is the view factor from surface $i$ to surface $j$.**
**View factor calculation:**
$$
F_{ij} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos\theta_i \cos\theta_j}{\pi r^2} dA_j dA_i
$$
### 9.3 Quasi-Steady State Assumption
For slowly varying processes, time derivatives are neglected:
$$
\frac{\partial}{\partial t} \approx 0
$$
**This is valid when:**
$$
\frac{v_p L_{thermal}}{\alpha} \ll 1
$$
## 10. Numerical Methods
### 10.1 Discretization Techniques
| Method | Application | Advantages |
|--------|-------------|------------|
| **Finite Element Method (FEM)** | Complex geometries, coupled physics | Handles irregular boundaries |
| **Finite Volume Method (FVM)** | Fluid dynamics, conservation laws | Conservative discretization |
| **Finite Difference Method (FDM)** | Simple geometries, structured grids | Computational efficiency |
### 10.2 Interface Tracking Methods
**Front-tracking:**
- Explicit interface representation
- High accuracy at interface
- Topology changes require special handling
**Phase-field:**
$$
\frac{\partial \phi}{\partial t} = M \left[ \varepsilon^2 \nabla^2 \phi - f'(\phi) + \lambda g'(\phi)(T - T_m) \right]
$$
**Level-set:**
$$
\frac{\partial \psi}{\partial t} + \mathbf{u} \cdot \nabla \psi = 0
$$
### 10.3 Turbulence Models
For high Rayleigh number flows:
**k-ε model:**
$$
\frac{\partial k}{\partial t} + \mathbf{u} \cdot \nabla k = \nabla \cdot \left( \frac{\nu_t}{\sigma_k} \nabla k \right) + P_k - \varepsilon
$$
$$
\frac{\partial \varepsilon}{\partial t} + \mathbf{u} \cdot \nabla \varepsilon = \nabla \cdot \left( \frac{\nu_t}{\sigma_\varepsilon} \nabla \varepsilon \right) + C_1 \frac{\varepsilon}{k} P_k - C_2 \frac{\varepsilon^2}{k}
$$
**Turbulent viscosity:**
$$
\nu_t = C_\mu \frac{k^2}{\varepsilon}
$$
### 10.4 Newton-Raphson Iteration
For coupled nonlinear systems:
$$
\mathbf{x}^{(n+1)} = \mathbf{x}^{(n)} - \mathbf{J}^{-1} \mathbf{F}(\mathbf{x}^{(n)})
$$
**Where $\mathbf{J}$ is the Jacobian matrix:**
$$
J_{ij} = \frac{\partial F_i}{\partial x_j}
$$
## 11. Physical
| Physical Phenomenon | Mathematical Framework | Key Equation |
|---------------------|------------------------|--------------|
| Phase change | Stefan problem | $\rho L v_n = k_s \nabla T_s - k_l \nabla T_l$ |
| Melt convection | Navier-Stokes + Boussinesq | $\rho_0 \frac{D\mathbf{u}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \beta \Delta T \mathbf{g}$ |
| Meniscus shape | Young-Laplace | $\Delta P = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right)$ |
| Dopant distribution | Scheil equation | $C_s = k_0 C_0 (1 - f_s)^{k_0 - 1}$ |
| Mass transport | Convection-diffusion | $\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D \nabla^2 C$ |
| Radiation | Stefan-Boltzmann | $q = \varepsilon \sigma T^4$ |
| MHD | Maxwell + Navier-Stokes | $\mathbf{F} = \mathbf{J} \times \mathbf{B}$ |
## Symbol Glossary
| Symbol | Description | SI Unit |
|--------|-------------|---------|
| $T$ | Temperature | K |
| $\rho$ | Density | kg/m³ |
| $c_p$ | Specific heat capacity | J/(kg*K) |
| $k$ | Thermal conductivity | W/(m*K) |
| $L$ | Latent heat of fusion | J/kg |
| $\mu$ | Dynamic viscosity | Pa*s |
| $\nu$ | Kinematic viscosity | m²/s |
| $\alpha$ | Thermal diffusivity | m²/s |
| $\beta$ | Thermal expansion coefficient | K⁻¹ |
| $\gamma$ | Surface tension | N/m |
| $\sigma$ | Electrical conductivity | S/m |
| $D$ | Diffusion coefficient | m²/s |
| $k_0$ | Segregation coefficient | — |
| $\Omega$ | Angular velocity | rad/s |
| $\mathbf{B}$ | Magnetic field | T |
czochralski,crystal growth,silicon ingot,pure silicon
# Mathematics of the Czochralski Process
## 1. Introduction
The **Czochralski (CZ) method** is the dominant technique for growing single-crystal semiconductors, responsible for approximately 90% of silicon wafers used in integrated circuits. The mathematical modeling of this process involves:
- **Multi-phase heat transfer** with moving boundaries
- **Turbulent melt convection** driven by buoyancy and rotation
- **Capillary phenomena** controlling crystal shape
- **Mass transport** of dopants and impurities
- **Phase transition thermodynamics**
This document presents the key mathematical frameworks governing the CZ process.
## 2. Heat Transfer: The Stefan Problem
### 2.1 Overview
The CZ process is fundamentally a **moving boundary problem** where the solid-liquid interface position must be determined as part of the solution.
### 2.2 Governing Heat Equation
For both solid (crystal) and liquid (melt) regions:
$$
\rho c_p \frac{\partial T}{\partial t} + \rho c_p (\mathbf{u} \cdot \nabla T) = \nabla \cdot (k \nabla T) + Q
$$
**Where:**
- $\rho$ — density [kg/m³]
- $c_p$ — specific heat capacity [J/(kg·K)]
- $T$ — temperature [K]
- $\mathbf{u}$ — velocity field [m/s]
- $k$ — thermal conductivity [W/(m·K)]
- $Q$ — volumetric heat source [W/m³]
### 2.3 Stefan Condition at the Interface
At the **solid-liquid interface**, the Stefan condition balances latent heat release with conductive heat flux:
$$
\rho L v_n = k_s \left( \frac{\partial T}{\partial n} \right)_s - k_l \left( \frac{\partial T}{\partial n} \right)_l
$$
**Where:**
- $L$ — latent heat of fusion [J/kg]
- $v_n$ — interface velocity (normal direction) [m/s]
- $k_s$, $k_l$ — thermal conductivity of solid and liquid [W/(m·K)]
- $n$ — unit normal vector pointing into the solid
### 2.4 Boundary Conditions
| Location | Condition | Equation |
|----------|-----------|----------|
| Crystal surface | Radiation + convection | $-k\frac{\partial T}{\partial n} = h(T - T_{\infty}) + \varepsilon \sigma (T^4 - T_{amb}^4)$ |
| Melt free surface | Radiation + evaporation | $-k\frac{\partial T}{\partial n} = \varepsilon \sigma T^4 + q_{evap}$ |
| Crucible wall | Specified temperature | $T = T_{crucible}(z)$ |
| Solid-liquid interface | Melting point | $T = T_m$ |
### 2.5 Enthalpy Formulation
For numerical implementation, the **enthalpy method** avoids explicit interface tracking:
$$
\frac{\partial H}{\partial t} + \nabla \cdot (\mathbf{u} H) = \nabla \cdot (k \nabla T)
$$
**With enthalpy defined as:**
$$
H(T) = \begin{cases}
\rho_s c_{p,s} T & T < T_m \\
\rho_s c_{p,s} T_m + \rho L f_l & T = T_m \\
\rho_l c_{p,l} T + \rho L & T > T_m
\end{cases}
$$
Where $f_l \in [0,1]$ is the liquid fraction.
## 3. Fluid Dynamics: Navier-Stokes Equations
### 3.1 Governing Equations
The melt flow is governed by the **incompressible Navier-Stokes equations** with the **Boussinesq approximation** for buoyancy:
**Continuity (mass conservation):**
$$
\nabla \cdot \mathbf{u} = 0
$$
**Momentum conservation:**
$$
\rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \beta (T - T_0) \mathbf{g} + \mathbf{F}_{ext}
$$
**Where:**
- $\rho_0$ — reference density [kg/m³]
- $p$ — pressure [Pa]
- $\mu$ — dynamic viscosity [Pa·s]
- $\beta$ — thermal expansion coefficient [K⁻¹]
- $T_0$ — reference temperature [K]
- $\mathbf{g}$ — gravitational acceleration [m/s²]
- $\mathbf{F}_{ext}$ — external forces (rotation, magnetic) [N/m³]
### 3.2 Boussinesq Approximation
The density variation is linearized:
$$
\rho(T) \approx \rho_0 [1 - \beta(T - T_0)]
$$
**Validity condition:**
$$
\frac{\Delta \rho}{\rho_0} = \beta \Delta T \ll 1
$$
### 3.3 Rotational Effects
For rotating crystal (angular velocity $\Omega_c$) and crucible ($\Omega_{cr}$):
**Coriolis force:**
$$
\mathbf{F}_{Cor} = -2\rho_0 (\boldsymbol{\Omega} \times \mathbf{u})
$$
**Centrifugal force:**
$$
\mathbf{F}_{cent} = -\rho_0 \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r})
$$
### 3.4 Marangoni (Thermocapillary) Convection
At the free surface, temperature-dependent surface tension drives flow:
$$
\mu \frac{\partial u_t}{\partial n} = \frac{\partial \gamma}{\partial T} \frac{\partial T}{\partial t}
$$
**Where:**
- $\gamma$ — surface tension [N/m]
- $\frac{\partial \gamma}{\partial T}$ — temperature coefficient of surface tension [N/(m·K)]
### 3.5 Dimensionless Numbers
| Number | Definition | Physical Meaning |
|--------|------------|------------------|
| **Reynolds** | $Re = \frac{\rho U L}{\mu}$ | Inertia / Viscous forces |
| **Grashof** | $Gr = \frac{g \beta \Delta T L^3}{\nu^2}$ | Buoyancy / Viscous forces |
| **Prandtl** | $Pr = \frac{\nu}{\alpha} = \frac{\mu c_p}{k}$ | Momentum / Thermal diffusivity |
| **Rayleigh** | $Ra = Gr \cdot Pr$ | Convection strength |
| **Marangoni** | $Ma = \frac{\left|\frac{\partial \gamma}{\partial T}\right| \Delta T L}{\mu \alpha}$ | Surface tension / Viscous forces |
**For silicon CZ growth:**
- $Ra \sim 10^8 - 10^{10}$ (turbulent regime)
- $Pr \approx 0.01$ (liquid metals)
## 4. Capillarity: Young-Laplace Equation
### 4.1 Meniscus Shape
The melt meniscus connecting the crystal to the melt surface is governed by the **Young-Laplace equation**:
$$
\Delta P = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right)
$$
**Where:**
- $\Delta P$ — pressure difference across interface [Pa]
- $\gamma$ — surface tension [N/m]
- $R_1$, $R_2$ — principal radii of curvature [m]
### 4.2 Axisymmetric Formulation
For axisymmetric geometry with meniscus profile $z = z(r)$:
$$
\gamma \left[ \frac{z''}{(1 + z'^2)^{3/2}} + \frac{z'}{r(1 + z'^2)^{1/2}} \right] = \rho g z
$$
**Where:**
- $z'$ = $\frac{dz}{dr}$
- $z''$ = $\frac{d^2z}{dr^2}$
### 4.3 Arc-Length Parameterization
Using arc-length $s$ for numerical stability:
$$
\frac{dr}{ds} = \cos\phi
$$
$$
\frac{dz}{ds} = \sin\phi
$$
$$
\frac{d\phi}{ds} = \frac{\rho g z}{\gamma} - \frac{\sin\phi}{r}
$$
**Where $\phi$ is the tangent angle to the meniscus.**
### 4.4 Growth Angle Condition
At the **triple point** (crystal-melt-gas junction):
$$
\phi_{triple} = \alpha_{gr}
$$
**Where $\alpha_{gr}$ is the characteristic growth angle:**
| Material | Growth Angle $\alpha_{gr}$ |
|----------|---------------------------|
| Silicon | 11° |
| Germanium | 13° |
| GaAs | 17° |
| Sapphire | 35° |
### 4.5 Crystal Radius Evolution
The crystal radius changes according to:
$$
\frac{dr_c}{dt} = v_p \tan(\alpha_{gr} - \theta)
$$
**Where:**
- $r_c$ — crystal radius [m]
- $v_p$ — pulling velocity [m/s]
- $\theta$ — current meniscus angle at triple point
**Stability conditions:**
- $\theta < \alpha_{gr}$ → Crystal radius **increases**
- $\theta = \alpha_{gr}$ → Crystal radius **constant** (steady state)
- $\theta > \alpha_{gr}$ → Crystal radius **decreases**
### 4.6 Capillary Constant
The capillary length scale:
$$
a = \sqrt{\frac{2\gamma}{\rho g}}
$$
**For silicon:** $a \approx 7.6$ mm
## 5. Crystal Growth Rate
### 5.1 Interface Energy Balance
The local growth velocity is determined by heat flux balance:
$$
v_g = \frac{1}{\rho L} \left[ k_s \left( \frac{\partial T}{\partial n} \right)_s - k_l \left( \frac{\partial T}{\partial n} \right)_l \right]
$$
### 5.2 Simplified Growth Rate Model
$$
G = \frac{dL}{dt} = v_p \cdot \frac{T_m - T_i}{T_m - T_s}
$$
**Where:**
- $G$ — crystal growth rate [m/s]
- $L$ — crystal length [m]
- $v_p$ — pulling rate [m/s]
- $T_m$ — melting temperature [K]
- $T_i$ — interface temperature [K]
- $T_s$ — seed temperature [K]
### 5.3 Maximum Pull Rate
The maximum pull rate is limited by heat transfer:
$$
v_{p,max} = \frac{k_s G_s}{\rho L}
$$
**Where $G_s = \left( \frac{\partial T}{\partial z} \right)_s$ is the axial temperature gradient in the crystal.**
## 6. Dopant Distribution: Scheil Equation
### 6.1 Segregation Coefficient
The **equilibrium segregation coefficient** is defined as:
$$
k_0 = \frac{C_s}{C_l}
$$
**Where:**
- $C_s$ — solute concentration in solid at interface [atoms/cm³]
- $C_l$ — solute concentration in liquid at interface [atoms/cm³]
**Typical values for silicon:**
| Dopant | $k_0$ |
|--------|-------|
| Boron (B) | 0.80 |
| Phosphorus (P) | 0.35 |
| Arsenic (As) | 0.30 |
| Antimony (Sb) | 0.023 |
| Oxygen (O) | 1.25 |
### 6.2 Scheil-Gulliver Equation
For a well-mixed melt with no solid-state diffusion:
**Liquid concentration:**
$$
C_L = C_0 (1 - f_s)^{k_0 - 1}
$$
**Solid concentration:**
$$
C_s = k_0 C_0 (1 - f_s)^{k_0 - 1}
$$
**Where:**
- $C_0$ — initial concentration in melt [atoms/cm³]
- $f_s$ — solidified fraction $(= V_s / V_{total})$
### 6.3 Effective Segregation Coefficient
The **Burton-Prim-Slichter (BPS) equation** accounts for diffusion boundary layer:
$$
k_{eff} = \frac{k_0}{k_0 + (1 - k_0) \exp\left( -\frac{v_g \delta}{D} \right)}
$$
**Where:**
- $\delta$ — boundary layer thickness [m]
- $D$ — diffusion coefficient in liquid [m²/s]
- $v_g$ — growth velocity [m/s]
### 6.4 Boundary Layer Thickness
For rotating crystal (Cochran model):
$$
\delta = 1.6 D^{1/3} \nu^{1/6} \Omega^{-1/2}
$$
**Where:**
- $\nu$ — kinematic viscosity [m²/s]
- $\Omega$ — crystal rotation rate [rad/s]
### 6.5 Lambert W Function Solution
For complex segregation problems, the Scheil equation leads to transcendental equations:
$$
\left( \frac{C_s}{C_L} \right) \ln(1 - f_s) \cdot e^{\left( \frac{C_s}{C_L} \ln(1-f_s) \right)} = \left( \frac{C_s}{C_0} \right) (1 - f_s) \ln(1 - f_s)
$$
**Solution via Lambert W function:**
$$
x \cdot e^x = y \implies x = W(y)
$$
## 7. Oxygen Transport
### 7.1 Convection-Diffusion Equation
Oxygen concentration in the melt follows:
$$
\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D_O \nabla^2 C
$$
**Where:**
- $C$ — oxygen concentration [atoms/cm³]
- $D_O$ — oxygen diffusivity in silicon melt ≈ 5×10⁻⁸ m²/s
### 7.2 Boundary Conditions
**Crucible wall (dissolution):**
$$
C_{wall} = A \exp\left( -\frac{E_a}{RT} \right)
$$
**Common models:**
- **Matsuo et al.:** $C_{wall} = 3.99 \times 10^{25} \exp\left( -\frac{1.2 \text{ eV}}{k_B T} \right)$ atoms/cm³
**Free surface (evaporation):**
$$
-D_O \frac{\partial C}{\partial n} = k_{evap} (C - C_{eq})
$$
**Crystal interface (segregation):**
$$
D_O \frac{\partial C}{\partial n}\bigg|_{melt} = D_{O,s} \frac{\partial C}{\partial n}\bigg|_{crystal} + (1 - k_O) v_g C_{interface}
$$
### 7.3 Oxygen Transport Mechanisms
```
-
┌──────────────────────────────────────────────────────────────┐
│ OXYGEN TRANSPORT │
├──────────────────────────────────────────────────────────────┤
│ │
│ SiO₂ Crucible ──→ Dissolution ──→ Si Melt │
│ │ │
│ ├──→ Evaporation │
│ │ (as SiO) │
│ │ │
│ └──→ Crystal │
│ (segregation) │
│ │
└──────────────────────────────────────────────────────────────┘
```
## 8. Magnetohydrodynamics (MHD)
### 8.1 Lorentz Force
Applied magnetic fields $\mathbf{B}$ modify melt flow through the Lorentz force:
$$
\mathbf{F}_L = \mathbf{J} \times \mathbf{B}
$$
**Where $\mathbf{J}$ is the current density:**
$$
\mathbf{J} = \sigma (\mathbf{E} + \mathbf{u} \times \mathbf{B})
$$
### 8.2 Maxwell's Equations
$$
\nabla \times \mathbf{B} = \mu_0 \mathbf{J}
$$
$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
$$
$$
\nabla \cdot \mathbf{B} = 0
$$
### 8.3 Magnetic Reynolds Number
$$
Re_m = \mu_0 \sigma U L
$$
For silicon melts: $Re_m \ll 1$, so **induced fields are negligible** (quasi-static approximation).
### 8.4 Hartmann Number
The ratio of electromagnetic to viscous forces:
$$
Ha = B L \sqrt{\frac{\sigma}{\mu}}
$$
**Effects of magnetic field:**
- $Ha > 10$: Significant flow suppression
- $Ha > 100$: Quasi-two-dimensional flow
### 8.5 Common Magnetic Field Configurations
| Configuration | Field Direction | Primary Effect |
|---------------|-----------------|----------------|
| **Axial (VMCZ)** | Parallel to pull axis | Suppresses meridional convection |
| **Transverse (HMCZ)** | Perpendicular to axis | Creates asymmetric flow |
| **Cusp (CMCZ)** | Combined radial/axial | Controls flow at specific heights |
## 9. Integrated Thermal-Capillary Model
### 9.1 Coupled System
The complete CZ model couples multiple physics:
**Heat transfer:**
$$
\rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot (k \nabla T)
$$
**Momentum (in melt):**
$$
\rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \beta (T - T_0) \mathbf{g}
$$
**Species transport:**
$$
\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D \nabla^2 C
$$
**Interface position (Stefan condition):**
$$
\rho L v_n = [k \nabla T]_{jump}
$$
**Meniscus shape (Young-Laplace):**
$$
\gamma \kappa = \rho g z
$$
### 9.2 Radiation Heat Transfer
**Surface-to-surface radiation:**
$$
q_i = \varepsilon_i \sigma T_i^4 - \sum_{j=1}^{N} F_{ij} \varepsilon_j \sigma T_j^4
$$
**Where $F_{ij}$ is the view factor from surface $i$ to surface $j$.**
**View factor calculation:**
$$
F_{ij} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos\theta_i \cos\theta_j}{\pi r^2} dA_j dA_i
$$
### 9.3 Quasi-Steady State Assumption
For slowly varying processes, time derivatives are neglected:
$$
\frac{\partial}{\partial t} \approx 0
$$
**This is valid when:**
$$
\frac{v_p L_{thermal}}{\alpha} \ll 1
$$
## 10. Numerical Methods
### 10.1 Discretization Techniques
| Method | Application | Advantages |
|--------|-------------|------------|
| **Finite Element Method (FEM)** | Complex geometries, coupled physics | Handles irregular boundaries |
| **Finite Volume Method (FVM)** | Fluid dynamics, conservation laws | Conservative discretization |
| **Finite Difference Method (FDM)** | Simple geometries, structured grids | Computational efficiency |
### 10.2 Interface Tracking Methods
**Front-tracking:**
- Explicit interface representation
- High accuracy at interface
- Topology changes require special handling
**Phase-field:**
$$
\frac{\partial \phi}{\partial t} = M \left[ \varepsilon^2 \nabla^2 \phi - f'(\phi) + \lambda g'(\phi)(T - T_m) \right]
$$
**Level-set:**
$$
\frac{\partial \psi}{\partial t} + \mathbf{u} \cdot \nabla \psi = 0
$$
### 10.3 Turbulence Models
For high Rayleigh number flows:
**k-ε model:**
$$
\frac{\partial k}{\partial t} + \mathbf{u} \cdot \nabla k = \nabla \cdot \left( \frac{\nu_t}{\sigma_k} \nabla k \right) + P_k - \varepsilon
$$
$$
\frac{\partial \varepsilon}{\partial t} + \mathbf{u} \cdot \nabla \varepsilon = \nabla \cdot \left( \frac{\nu_t}{\sigma_\varepsilon} \nabla \varepsilon \right) + C_1 \frac{\varepsilon}{k} P_k - C_2 \frac{\varepsilon^2}{k}
$$
**Turbulent viscosity:**
$$
\nu_t = C_\mu \frac{k^2}{\varepsilon}
$$
### 10.4 Newton-Raphson Iteration
For coupled nonlinear systems:
$$
\mathbf{x}^{(n+1)} = \mathbf{x}^{(n)} - \mathbf{J}^{-1} \mathbf{F}(\mathbf{x}^{(n)})
$$
**Where $\mathbf{J}$ is the Jacobian matrix:**
$$
J_{ij} = \frac{\partial F_i}{\partial x_j}
$$
## 11. Physical
| Physical Phenomenon | Mathematical Framework | Key Equation |
|---------------------|------------------------|--------------|
| Phase change | Stefan problem | $\rho L v_n = k_s \nabla T_s - k_l \nabla T_l$ |
| Melt convection | Navier-Stokes + Boussinesq | $\rho_0 \frac{D\mathbf{u}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho_0 \beta \Delta T \mathbf{g}$ |
| Meniscus shape | Young-Laplace | $\Delta P = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right)$ |
| Dopant distribution | Scheil equation | $C_s = k_0 C_0 (1 - f_s)^{k_0 - 1}$ |
| Mass transport | Convection-diffusion | $\frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D \nabla^2 C$ |
| Radiation | Stefan-Boltzmann | $q = \varepsilon \sigma T^4$ |
| MHD | Maxwell + Navier-Stokes | $\mathbf{F} = \mathbf{J} \times \mathbf{B}$ |
## Symbol Glossary
| Symbol | Description | SI Unit |
|--------|-------------|---------|
| $T$ | Temperature | K |
| $\rho$ | Density | kg/m³ |
| $c_p$ | Specific heat capacity | J/(kg*K) |
| $k$ | Thermal conductivity | W/(m*K) |
| $L$ | Latent heat of fusion | J/kg |
| $\mu$ | Dynamic viscosity | Pa*s |
| $\nu$ | Kinematic viscosity | m²/s |
| $\alpha$ | Thermal diffusivity | m²/s |
| $\beta$ | Thermal expansion coefficient | K⁻¹ |
| $\gamma$ | Surface tension | N/m |
| $\sigma$ | Electrical conductivity | S/m |
| $D$ | Diffusion coefficient | m²/s |
| $k_0$ | Segregation coefficient | — |
| $\Omega$ | Angular velocity | rad/s |
| $\mathbf{B}$ | Magnetic field | T |
d-nerf, 3d vision
NeRF with deformation field.
d-optimal design, doe
Minimize variance of parameter estimates.
d-vector, audio & speech
D-vector is a deep neural network embedding that represents speaker characteristics for verification and multi-speaker speech synthesis.
d2d (die-to-die variation),d2d,die-to-die variation,manufacturing
Variation between dies on same wafer.
dac, dac, reinforcement learning advanced
Discriminator-Actor-Critic combines adversarial imitation learning with off-policy RL improving sample efficiency.
dagger, imitation learning
Iterative imitation learning.
dagster,data assets,orchestration
Dagster is data orchestration with software-defined assets. Type-safe pipelines.
dall-e 3, dall-e, multimodal ai
DALL-E 3 improves text-to-image through enhanced caption generation and refined training.
dall-e tokenizer, dall-e, multimodal ai
DALL-E tokenizer uses discrete visual codes for autoregressive image generation from text.
daly city,colma,serramonte
# Daly City, California
## Overview
Daly City is a densely populated city located in **San Mateo County, California**, immediately south of San Francisco. It serves as a critical residential and commercial hub in the San Francisco Bay Area.
## Basic Information
- **Official Name:** City of Daly City
- **County:** San Mateo County
- **State:** California, USA
- **Incorporated:** March 22, 1911
- **Named After:** John Donald Daly (1841–1923), a dairy farmer and landowner
## Geographic Data
### Location Coordinates
$$
\text{Latitude: } 37.6879° \, N
$$
$$
\text{Longitude: } 122.4702° \, W
$$
### Area Calculations
- **Total Area:** $7.7 \, \text{mi}^2$ (approximately $19.9 \, \text{km}^2$)
- **Land Area:** $7.6 \, \text{mi}^2$
- **Water Area:** $0.1 \, \text{mi}^2$
Area conversion formula:
$$
A_{\text{km}^2} = A_{\text{mi}^2} \times 2.58999
$$
Example:
$$
7.7 \, \text{mi}^2 \times 2.58999 = 19.94 \, \text{km}^2
$$
### Elevation
- **Average Elevation:** $\approx 520 \, \text{ft}$ ($158 \, \text{m}$)
- **Highest Point:** $\approx 925 \, \text{ft}$ ($282 \, \text{m}$)
Elevation conversion:
$$
h_{\text{meters}} = h_{\text{feet}} \times 0.3048
$$
## Population Statistics
### Census Data
| Year | Population | Change |
|------|------------|--------|
| 2000 | 103,621 | — |
| 2010 | 101,123 | $-2.4\%$ |
| 2020 | 104,901 | $+3.7\%$ |
### Population Density
$$
\rho = \frac{P}{A}
$$
Where:
- $\rho$ = population density
- $P$ = population
- $A$ = area
Calculation:
$$
\rho = \frac{104,901}{7.7 \, \text{mi}^2} \approx 13,623 \, \text{people/mi}^2
$$
In metric:
$$
\rho_{\text{metric}} = \frac{104,901}{19.94 \, \text{km}^2} \approx 5,261 \, \text{people/km}^2
$$
## Demographics
### Ethnic Composition (Approximate)
- **Asian:** $\approx 56\%$
- Filipino: $\approx 33\%$ (largest subgroup)
- Chinese: $\approx 15\%$
- Other Asian: $\approx 8\%$
- **White:** $\approx 18\%$
- **Hispanic/Latino:** $\approx 20\%$
- **Black/African American:** $\approx 3\%$
- **Other/Mixed:** $\approx 3\%$
### Diversity Index
The Simpson's Diversity Index can be calculated as:
$$
D = 1 - \sum_{i=1}^{n} p_i^2
$$
Where $p_i$ is the proportion of each ethnic group.
For Daly City:
$$
D = 1 - (0.56^2 + 0.18^2 + 0.20^2 + 0.03^2 + 0.03^2)
$$
$$
D = 1 - (0.3136 + 0.0324 + 0.04 + 0.0009 + 0.0009)
$$
$$
D = 1 - 0.3878 = 0.6122
$$
A diversity index of $D \approx 0.61$ indicates **moderate-to-high diversity**.
## Climate Data
### Climate Classification
- **Köppen Classification:** $Csb$ (Mediterranean Climate / Warm-summer Mediterranean)
### Temperature Statistics
| Metric | Value (°F) | Value (°C) |
|--------|------------|------------|
| Average High | $63°F$ | $17.2°C$ |
| Average Low | $49°F$ | $9.4°C$ |
| Annual Mean | $56°F$ | $13.3°C$ |
Temperature conversion formula:
$$
T_C = \frac{5}{9}(T_F - 32)
$$
### Precipitation
- **Annual Rainfall:** $\approx 20 \, \text{inches}$ ($508 \, \text{mm}$)
Conversion:
$$
P_{\text{mm}} = P_{\text{in}} \times 25.4
$$
### Fog Frequency
Daly City is known for its frequent fog due to:
$$
\text{Fog Formation} \propto \frac{\Delta T \times H}{W}
$$
Where:
- $\Delta T$ = temperature differential (ocean vs. land)
- $H$ = humidity
- $W$ = wind speed
The city experiences fog approximately **100+ days per year**.
## Transportation
### BART Stations
- **Daly City Station** (opened 1973)
- **Colma Station** (opened 1996)
### Commute Statistics
Average commute time:
$$
\bar{t} \approx 32 \, \text{minutes}
$$
Modal split (approximate):
- **Drive Alone:** $\approx 55\%$
- **Public Transit:** $\approx 25\%$
- **Carpool:** $\approx 12\%$
- **Walk/Bike/Other:** $\approx 8\%$
## Economic Data
### Median Household Income
$$
\tilde{Y} \approx 95{,}000\,\text{ USD (2023 estimate)}
$$
### Housing Statistics
- **Median Home Price:** approx USD 950,000
- **Median Rent:** approx USD 2,500/month
Housing affordability ratio:
$$
\text{Affordability Ratio} = \frac{\text{Median Home Price}}{\text{Median Income}} = \frac{950,000}{95,000} \approx 10
$$
A ratio $> 5$ indicates a **severely unaffordable** housing market.
## Cultural Significance
### "Little Manila"
Daly City has one of the **highest concentrations of Filipino Americans** in the United States.
Filipino population percentage:
$$
P_{\text{Filipino}} = \frac{\text{Filipino Residents}}{\text{Total Population}} \times 100 \approx 33\%
$$
### Historical Reference: "Little Boxes"
The song *"Little Boxes"* (1962) by **Malvina Reynolds** was inspired by the **Westlake neighborhood** tract housing.
Verse structure analysis (syllables per line):
$$
\text{Meter Pattern} = \{8, 8, 8, 6\} \quad \text{(approximate)}
$$
## Key Landmarks
- **Westlake Shopping Center** — One of the first planned shopping centers in the U.S.
- **Serramonte Center** — Major regional shopping mall
- **Top of the Hill Park** — Scenic overlook
- **Thornton State Beach** — Coastal access point
## Government
### City Council Structure
- **Council Members:** 5
- **Mayor:** Elected by council (rotates annually)
### Budget Formula (Simplified)
$$
B_{\text{total}} = R_{\text{tax}} + R_{\text{fees}} + G_{\text{state}} + G_{\text{federal}}
$$
Where:
- $B_{\text{total}}$ = total budget
- $R_{\text{tax}}$ = tax revenue
- $R_{\text{fees}}$ = fees and permits
- $G_{\text{state}}$ = state grants
- $G_{\text{federal}}$ = federal grants
## Distance Calculations
### Distance from Key Locations
| Destination | Distance (mi) | Distance (km) |
|-------------|---------------|---------------|
| San Francisco (Downtown) | $8$ | $12.9$ |
| San Jose | $40$ | $64.4$ |
| Oakland | $20$ | $32.2$ |
| SFO Airport | $5$ | $8.0$ |
Haversine formula for great-circle distance:
$$
d = 2r \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right) + \cos(\phi_1)\cos(\phi_2)\sin^2\left(\frac{\Delta\lambda}{2}\right)}\right)
$$
Where:
- $r$ = Earth's radius ($\approx 3,959 \, \text{mi}$ or $6,371 \, \text{km}$)
- $\phi_1, \phi_2$ = latitudes
- $\Delta\phi$ = difference in latitude
- $\Delta\lambda$ = difference in longitude
## Reference
| Category | Value |
|----------|-------|
| **Population** | $\approx 104,901$ |
| **Area** | $7.7 \, \text{mi}^2$ |
| **Density** | $\approx 13,623 \, \text{/mi}^2$ |
| **Elevation** | $\approx 520 \, \text{ft}$ |
| **Founded** | 1911 |
| **Filipino %** | $\approx 33\%$ |
| **Median Income** | approx USD 95,000 |
| **Climate** | $Csb$ (Mediterranean) |
damascene process,cmp
Pattern trenches fill with metal CMP to remove excess.
dan (do anything now),dan,do anything now,ai safety
Type of jailbreak prompt claiming to remove restrictions.
dan prompts, dan, ai safety
Jailbreak technique.
dann, dann, domain adaptation
Adversarial training for domain adaptation.
dare, dare, model merging
Sparsify then merge models.
dark knowledge, model compression
Information in soft targets beyond hard labels.
dark knowledge, model optimization
Dark knowledge is information in teacher's soft predictions beyond hard labels aiding distillation.
darkfield inspection,metrology
Scatter light off defects for enhanced detection.
darts, darts, neural architecture search
Differentiable Architecture Search enables gradient-based NAS by relaxing the discrete architecture search space into continuous representations.
dask,parallel,distributed
Dask parallelizes pandas/numpy across cores or cluster. Lazy evaluation. Scale out analysis.
data annotation,data
Process of labeling data for training or evaluation.
data anonymization, training techniques
Data anonymization removes identifying information preventing re-identification.
data anonymization,privacy
Remove or obscure identifying information.
data augmentation privacy, training techniques
Privacy-preserving augmentation creates variations without exposing original data.
data augmentation,model training
Transform existing data (paraphrase translate corrupt) to increase training diversity.
data card, evaluation
Data cards describe dataset characteristics collection and potential biases.
data card,documentation
Documentation of dataset characteristics and collection.
data clumps, code ai
Groups of data appearing together.
data collection,automation
Automatically gather process data and metrology results.
data contamination detection,evaluation
Check if test data leaked into training set.
data contamination,evaluation
When test data appears in training data inflating scores.
data deduplication, data quality
Remove duplicate training examples.
data deduplication,data quality
Remove duplicate examples from dataset.
data drift,mlops
When input data distribution changes over time.
data efficiency of vit, computer vision
How much data ViT needs.
data extraction,parsing,scraping
LLMs extract structured data from unstructured text: invoices, resumes, contracts. Output JSON for downstream use.
data filtering strategies, data quality
Remove low-quality data.
data filtering,data quality
Remove low-quality or irrelevant examples.
data labeling,annotation,gt,quality
I can explain data labeling strategies, guidelines, QC loops, and how to design good annotation instructions.
data leakage,ai safety
When model exposes training data or sensitive information.
data loading pipeline, infrastructure
Efficient data feeding to GPUs.
data minimization, training techniques
Data minimization collects only necessary information reducing privacy risks.
data mix,domain,proportion
Data mix balances domains (web, books, code). Proportions affect model capabilities. Code improves reasoning.
data mixing strategies, training
Combine different data sources.
data ordering effects, training
Impact of data sequence.
data parallel,model parallel,hybrid
Data parallel: same model, different data. Model parallel: split model across GPUs. Hybrid for largest models.
data parallelism,model training
Replicate model on each device process different data batches.
data pipeline,etl,orchestration
Data pipelines (Airflow, Dagster) orchestrate ETL for training data. Reliable, versioned, scheduled.