cyclegan,generative models
Unpaired image translation using cycle consistency.
1,005 technical terms and definitions
Unpaired image translation using cycle consistency.
Repeated stress cycles.
Measure code complexity via paths.
# 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 |
# 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 |