flicker reduction, video generation
Minimize temporal artifacts.
395 technical terms and definitions
Minimize temporal artifacts.
Mount die face-down with solder bumps for connections.
Face-down die attachment.
Charge accumulation in SOI body.
Time allowed out of bag.
Floor markings delineate work areas pathways and storage locations improving organization.
Tiles with holes allowing air to flow from plenum below.
Estimate floating point operations.
Measure of computational cost for training or inference.
FLOPs efficiency measures actual throughput relative to theoretical peak floating-point operations.
Percentage of theoretical peak.
Floating point operations per second measure of compute capacity.
Flow controllers regulate chemical delivery rates ensuring process consistency.
Flow meters measure chemical delivery rates for process control.
Flow production creates continuous movement of work through operations minimizing batching and inventory.
Use motion to aggregate features.
Flowable CVD deposits films that flow like liquids before curing enabling void-free gap fill in extremely narrow features.
Create flowcharts from descriptions. Visualize processes.
Flowise is visual LangChain builder. Drag and drop.
CNN for optical flow estimation.
Flowtron combines Tacotron 2 with normalizing flows enabling expressive style control in speech synthesis.
Fluency assesses grammaticality and naturalness of generated language.
# Fluid Dynamics: Mathematical Modeling 1. Overview: Where Fluid Dynamics Matters Fluid dynamics plays a critical role in numerous semiconductor fabrication steps: - Chemical Vapor Deposition (CVD) — Precursor gas transport and reaction - Spin Coating — Photoresist film formation - Chemical Mechanical Planarization (CMP) — Slurry flow and material removal - Wet Etching/Cleaning — Etchant transport to surfaces - Immersion Lithography — Water flow between lens and wafer - Electrochemical Deposition — Electrolyte flow and ion transport Each process involves distinct physics, but they share a common mathematical foundation. 2. Fundamental Governing Equations 2.1 Navier-Stokes Framework The foundation is the incompressible Navier-Stokes equations. Continuity Equation $$ \nabla \cdot \mathbf{u} = 0 $$ Momentum Equation $$ \rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{F} $$ Where: - $\mathbf{u}$ — Velocity field vector - $p$ — Pressure field - $\rho$ — Fluid density - $\mu$ — Dynamic viscosity - $\mathbf{F}$ — Body forces (gravity, electromagnetic, etc.) Species Transport Equation $$ \frac{\partial C_i}{\partial t} + \mathbf{u} \cdot \nabla C_i = D_i \nabla^2 C_i + R_i $$ Where: - $C_i$ — Concentration of species $i$ - $D_i$ — Diffusion coefficient of species $i$ - $R_i$ — Reaction rate (source/sink term) Energy Equation $$ \rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T\right) = k \nabla^2 T + Q $$ Where: - $c_p$ — Specific heat capacity - $T$ — Temperature - $k$ — Thermal conductivity - $Q$ — Heat source (reaction heat, Joule heating, etc.) 3. Chemical Vapor Deposition (CVD) CVD is one of the most mathematically complex processes, coupling gas-phase transport, homogeneous reactions, and heterogeneous surface chemistry. 3.1 Reactor-Scale Transport In a typical showerhead reactor, gas enters through distributed holes and flows toward a heated wafer. The classic stagnation-point flow solution applies. Similarity Solution For axisymmetric flow toward a disk: $$ u_r = r f'(\eta), \quad u_z = -\sqrt{\nu a} \cdot f(\eta) $$ Where: - $\eta = z\sqrt{a/\nu}$ — Similarity variable - $a$ — Strain rate parameter - $\nu$ — Kinematic viscosity This yields the Hiemenz equation : $$ f''' + ff'' - (f')^2 + 1 = 0 $$ With boundary conditions: - $f(0) = f'(0) = 0$ (no-slip at surface) - $f'(\infty) = 1$ (far-field condition) 3.2 Key Dimensionless Groups Damköhler Number $$ \text{Da} = \frac{k_s L}{D} $$ Physical meaning: Ratio of surface reaction rate to diffusive transport rate. | Regime | Condition | Implication | |--------|-----------|-------------| | Transport-limited | $\text{Da} \gg 1$ | Uniformity controlled by flow | | Reaction-limited | $\text{Da} \ll 1$ | Uniformity controlled by temperature | Péclet Number $$ \text{Pe} = \frac{UL}{D} $$ Physical meaning: Ratio of convective to diffusive transport. Grashof Number $$ \text{Gr} = \frac{g\beta \Delta T L^3}{\nu^2} $$ Physical meaning: Ratio of buoyancy to viscous forces (important in horizontal reactors). Where: - $g$ — Gravitational acceleration - $\beta$ — Thermal expansion coefficient - $\Delta T$ — Temperature difference 3.3 Surface Boundary Conditions The critical coupling between transport and chemistry at the wafer surface: $$ -D \left.\frac{\partial C}{\partial n}\right|_{\text{surface}} = k_s \cdot f(C, T, \theta) $$ This is a Robin boundary condition linking diffusive flux to surface kinetics. Langmuir-Hinshelwood Kinetics $$ R = \frac{k C}{1 + KC} $$ Features: - First-order at low concentration ($C \ll 1/K$) - Zero-order (saturated) at high concentration ($C \gg 1/K$) Sticking Coefficient Model $$ s = s_0 \cdot f(T) \cdot (1 - \theta) $$ Where: - $s_0$ — Base sticking coefficient - $\theta$ — Surface coverage fraction 3.4 Multi-Scale Challenge CVD spans enormous length scales: | Scale | Dimension | Physics | |-------|-----------|---------| | Reactor chamber | 0.1–1 m | Continuum CFD | | Boundary layer | 1–10 mm | Convection-diffusion | | Surface features | 10–100 nm | Ballistic/Knudsen transport | | Molecular mean free path | 0.1–10 μm | Molecular dynamics | Knudsen Number $$ \text{Kn} = \frac{\lambda}{L} $$ Where $\lambda$ is the molecular mean free path. | Regime | Condition | Modeling Approach | |--------|-----------|-------------------| | Continuum | $\text{Kn} < 0.01$ | Navier-Stokes | | Slip flow | $0.01 < \text{Kn} < 0.1$ | Navier-Stokes + slip BC | | Transition | $0.1 < \text{Kn} < 10$ | DSMC, Boltzmann | | Free molecular | $\text{Kn} > 10$ | Ballistic transport | 4. Spin Coating Spin coating deposits thin photoresist films through centrifugal spreading and solvent evaporation. 4.1 Thin Film Lubrication Theory For a thin viscous layer ($h \ll R$) on a rotating disk, the lubrication approximation applies: $$ \frac{\partial h}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r}(r h \bar{u}_r) = -E $$ Where: - $h(r,t)$ — Film thickness - $\bar{u}_r$ — Depth-averaged radial velocity - $E$ — Evaporation rate 4.2 Velocity Profile Integrating the momentum equation with: - No-slip at substrate ($u_r = 0$ at $z = 0$) - Zero shear at free surface ($\partial u_r / \partial z = 0$ at $z = h$) Yields: $$ u_r(z) = \frac{\rho \omega^2 r}{2\mu}(2hz - z^2) $$ Depth-averaged velocity: $$ \bar{u}_r = \frac{\rho \omega^2 r h^2}{3\mu} $$ 4.3 Emslie-Bonner-Peck Solution For a Newtonian fluid without evaporation: $$ \frac{\partial h}{\partial t} = -\frac{\rho \omega^2}{3\mu} \cdot \frac{1}{r}\frac{\partial (r h^3)}{\partial r} $$ For uniform initial thickness $h_0$: $$ h(t) = \frac{h_0}{\sqrt{1 + \dfrac{4\rho \omega^2 h_0^2 t}{3\mu}}} $$ Asymptotic behavior: - Short time: $h \approx h_0$ - Long time: $h \propto t^{-1/2}$ 4.4 Non-Newtonian Photoresists Real photoresists are shear-thinning. Using a power-law model : $$ \tau = K\left(\frac{\partial u}{\partial z}\right)^n $$ Where: - $K$ — Consistency index - $n$ — Power-law index ($n < 1$ for shear-thinning) The governing equation becomes: $$ \frac{\partial h}{\partial t} = -\frac{n}{2n+1}\left(\frac{\rho \omega^2}{K}\right)^{1/n} \frac{1}{r}\frac{\partial}{\partial r}\left(r h^{(2n+1)/n}\right) $$ 4.5 Evaporation and Marangoni Effects Coupled Concentration Equation $$ \frac{\partial(h x_s)}{\partial t} + \frac{1}{r}\frac{\partial}{\partial r}(r h x_s \bar{u}_r) = -\frac{e}{\rho_s} $$ Where: - $x_s$ — Solvent mass fraction - $e$ — Evaporation mass flux - $\rho_s$ — Solvent density Marangoni Stress Surface tension gradients drive Marangoni flows: $$ \tau_{\text{surface}} = \frac{\partial \sigma}{\partial r} = \frac{d\sigma}{dC}\frac{\partial C}{\partial r} $$ Marangoni Number $$ \text{Ma} = \frac{\Delta\sigma \cdot L}{\mu \alpha} $$ Where $\alpha$ is thermal diffusivity. 5. Chemical Mechanical Planarization (CMP) CMP combines chemical etching with mechanical abrasion, mediated by slurry flow between pad and wafer. 5.1 Reynolds Lubrication Equation For the thin fluid film: $$ \frac{\partial}{\partial x}\left(h^3 \frac{\partial p}{\partial x}\right) + \frac{\partial}{\partial y}\left(h^3 \frac{\partial p}{\partial y}\right) = 6\mu U \frac{\partial h}{\partial x} + 12\mu \frac{\partial h}{\partial t} $$ Terms: - Left side: Pressure-driven (Poiseuille) flow - First term on right: Shear-driven (Couette) flow (wedge effect) - Second term on right: Squeeze film effect 5.2 Slurry as Suspension CMP slurries contain abrasive particles exhibiting complex rheology. Shear-Induced Migration (Leighton-Acrivos) $$ \mathbf{J}_{\text{shear}} = -K_c a^2 \phi \nabla(\dot{\gamma} \phi) - K_\eta a^2 \dot{\gamma} \phi^2 \nabla(\ln \eta) $$ Where: - $a$ — Particle radius - $\phi$ — Particle volume fraction - $\dot{\gamma}$ — Shear rate - $K_c, K_\eta$ — Empirical constants Physical effect: Particles migrate from high-shear to low-shear regions. Effective Viscosity (Krieger-Dougherty) $$ \eta_{\text{eff}} = \eta_0 \left(1 - \frac{\phi}{\phi_m}\right)^{-[\eta]\phi_m} $$ Where: - $\phi_m$ — Maximum packing fraction (~0.64) - $[\eta]$ — Intrinsic viscosity (~2.5 for spheres) 5.3 Material Removal Models Classical Preston Equation $$ \text{MRR} = K_p \cdot p \cdot V $$ Where: - MRR — Material removal rate - $K_p$ — Preston coefficient - $p$ — Applied pressure - $V$ — Relative velocity Enhanced Models $$ \text{MRR} = f(\tau_{\text{shear}}, \phi_{\text{particle}}, k_{\text{chem}}, T) $$ Incorporating: - Fluid shear stress: $\tau = \mu \left.\dfrac{\partial u}{\partial z}\right|_{\text{surface}}$ - Local particle flux - Chemical reaction rate - Temperature-dependent kinetics 5.4 Contact Mechanics When pad asperities contact wafer: Greenwood-Williamson Model $$ p_{\text{contact}} = \frac{4}{3} E^* n \int_d^\infty (z-d)^{3/2} \phi(z) \, dz $$ Where: - $E^*$ — Effective elastic modulus - $n$ — Asperity density - $\phi(z)$ — Asperity height distribution - $d$ — Separation distance Force Balance $$ p_{\text{fluid}} + p_{\text{contact}} = P_{\text{applied}} $$ 6. Wet Etching: Mass Transfer Limited Processes 6.1 Convective-Diffusion Equation $$ \frac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D \nabla^2 C $$ At the reactive surface (fast reaction limit): $$ C|_{\text{surface}} = 0 $$ Etch rate: $$ \text{ER} \propto D \left.\frac{\partial C}{\partial n}\right|_{\text{surface}} $$ 6.2 Rotating Disk Solution (Levich) For a wafer rotating in etchant: Velocity Components $$ u_r = r\omega F(\zeta), \quad u_\theta = r\omega G(\zeta), \quad u_z = \sqrt{\nu\omega} H(\zeta) $$ Where $\zeta = z\sqrt{\omega/\nu}$. Boundary Layer Thickness $$ \delta = 1.61 D^{1/3} \nu^{1/6} \omega^{-1/2} $$ Mass Flux (Levich Equation) $$ j = 0.62 D^{2/3} \nu^{-1/6} \omega^{1/2} C_\infty $$ Key insight : The etch rate is uniform across an infinite disk . This explains why rotating processes achieve excellent uniformity. 6.3 Feature-Scale Transport In high-aspect-ratio trenches: Knudsen Diffusion $$ D_{\text{Kn}} = \frac{d}{3}\sqrt{\frac{8RT}{\pi M}} $$ Where: - $d$ — Trench width - $M$ — Molecular weight Concentration Profile in Trench For a trench of depth $L$ with reactive bottom: $$ \frac{d^2 C}{dz^2} = 0 \quad \text{(diffusion only)} $$ With boundary conditions: - $C(0) = C_{\text{top}}$ (top of trench) - $-D\dfrac{dC}{dz}\big|_{z=L} = k_s C(L)$ (reactive bottom) Solution: $$ \frac{C(z)}{C_{\text{top}}} = 1 - \frac{z}{L} \cdot \frac{1}{1 + D/(k_s L)} $$ Thiele Modulus $$ \phi = L\sqrt{\frac{k_s}{D}} $$ - $\phi \ll 1$: Reaction-limited (uniform etch in feature) - $\phi \gg 1$: Transport-limited (RIE lag) 7. Immersion Lithography At 193 nm wavelength, water ($n \approx 1.44$) fills the gap between lens and wafer, increasing numerical aperture. 7.1 Free Surface Dynamics Capillary Number $$ \text{Ca} = \frac{\mu U}{\sigma} $$ Where $\sigma$ is surface tension. - $\text{Ca} < \text{Ca}_{\text{crit}} \approx 0.1$: Stable meniscus - $\text{Ca} > \text{Ca}_{\text{crit}}$: Bubble entrainment risk Young-Laplace Equation $$ \Delta p = \sigma \kappa = \sigma \left(\frac{1}{R_1} + \frac{1}{R_2}\right) $$ Where $\kappa$ is the interface curvature. 7.2 Interface Tracking Methods Level Set Method $$ \frac{\partial \phi}{\partial t} + \mathbf{u} \cdot \nabla \phi = 0 $$ Where: - $\phi > 0$: Liquid phase - $\phi < 0$: Gas phase - $\phi = 0$: Interface Volume of Fluid (VOF) $$ \frac{\partial \alpha}{\partial t} + \nabla \cdot (\alpha \mathbf{u}) = 0 $$ Where $\alpha$ is the volume fraction. 7.3 Thermal Management Light absorption heats the water: $$ \rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T\right) = k\nabla^2 T + Q_{\text{abs}} $$ Refractive Index Sensitivity $$ \frac{dn}{dT} \approx -1 \times 10^{-4} \text{ K}^{-1} $$ Temperature variations cause refractive index changes, introducing imaging errors (aberrations). 8. Numerical Methods 8.1 Finite Volume Method (FVM) The workhorse for semiconductor CFD. Starting from integral form: $$ \frac{\partial}{\partial t}\int_V \rho \phi \, dV + \oint_S \rho \phi \mathbf{u} \cdot \mathbf{n} \, dS = \oint_S \Gamma \nabla \phi \cdot \mathbf{n} \, dS + \int_V S_\phi \, dV $$ Discretization $$ \frac{(\rho \phi)_P^{n+1} - (\rho \phi)_P^n}{\Delta t} V_P + \sum_f F_f \phi_f = \sum_f \Gamma_f (\nabla \phi)_f \cdot \mathbf{A}_f + S_\phi V_P $$ Where: - $P$ — Cell center - $f$ — Face index - $F_f = \rho \mathbf{u}_f \cdot \mathbf{A}_f$ — Face flux 8.2 Advection Schemes | Scheme | Order | Properties | |--------|-------|------------| | Upwind | 1st | Stable, diffusive | | Central | 2nd | Unstable for high Pe | | QUICK | 3rd | Good accuracy, bounded | | MUSCL | 2nd | TVD, shock-capturing | 8.3 Pressure-Velocity Coupling SIMPLE Algorithm 1. Guess pressure field $p^*$ 2. Solve momentum for $\mathbf{u}^*$ 3. Solve pressure correction: $\nabla \cdot (D \nabla p') = \nabla \cdot \mathbf{u}^*$ 4. Correct: $p = p^* + \alpha_p p'$, $\mathbf{u} = \mathbf{u}^* - D \nabla p'$ 5. Iterate until convergence 8.4 Moving Boundary Problems For etching/deposition where geometry evolves: Arbitrary Lagrangian-Eulerian (ALE) $$ \left.\frac{\partial \phi}{\partial t}\right|_{\chi} + (\mathbf{u} - \mathbf{u}_{\text{mesh}}) \cdot \nabla \phi = \text{RHS} $$ Where $\mathbf{u}_{\text{mesh}}$ is mesh velocity. Level Set Velocity Extension $$ \frac{\partial d}{\partial \tau} + \text{sign}(\phi)(|\nabla d| - 1) = 0 $$ Reinitializes the level set to a signed distance function. 8.5 Stiff Chemistry CVD with multiple reactions has time scales from ns (gas reactions) to s (deposition). Operator Splitting 1. Solve transport: $\dfrac{\partial C}{\partial t} + \mathbf{u} \cdot \nabla C = D\nabla^2 C$ 2. Solve chemistry: $\dfrac{dC}{dt} = R(C)$ (using stiff ODE solver) Implicit Methods For stiff systems: $$ \mathbf{C}^{n+1} = \mathbf{C}^n + \Delta t \cdot \mathbf{R}(\mathbf{C}^{n+1}) $$ Requires Newton iteration with Jacobian $\partial R_i / \partial C_j$. 9. Dimensionless | Group | Definition | Physical Meaning | |-------|------------|------------------| | Reynolds (Re) | $\dfrac{\rho UL}{\mu}$ | Inertia / Viscosity | | Péclet (Pe) | $\dfrac{UL}{D}$ | Convection / Diffusion | | Damköhler (Da) | $\dfrac{k_s L}{D}$ | Reaction / Transport | | Knudsen (Kn) | $\dfrac{\lambda}{L}$ | Mean free path / Length | | Capillary (Ca) | $\dfrac{\mu U}{\sigma}$ | Viscous / Surface tension | | Marangoni (Ma) | $\dfrac{\Delta\sigma \cdot L}{\mu \alpha}$ | Marangoni / Viscous | | Grashof (Gr) | $\dfrac{g\beta \Delta T L^3}{\nu^2}$ | Buoyancy / Viscous | | Schmidt (Sc) | $\dfrac{\nu}{D}$ | Momentum / Mass diffusivity | | Sherwood (Sh) | $\dfrac{k_m L}{D}$ | Convective / Diffusive mass transfer | | Thiele ($\phi$) | $L\sqrt{\dfrac{k_s}{D}}$ | Reaction / Diffusion in pores | 10. Current Research Frontiers 10.1 Machine Learning Integration - Surrogate models replacing expensive CFD for real-time process control - Physics-informed neural networks (PINNs) for solving PDEs - Digital twins for predictive maintenance and optimization 10.2 Atomic Layer Processes (ALD/ALE) - Highly transient, surface-reaction-dominated - Requires time-dependent modeling of pulse/purge cycles - Surface coverage evolution: $$ \frac{d\theta}{dt} = k_{\text{ads}} C (1-\theta) - k_{\text{des}} \theta $$ 10.3 Extreme Aspect Ratios - 3D NAND with aspect ratios > 100 - Transition to molecular flow ($\text{Kn} > 0.1$) - Transmission probability methods : $$ P = \frac{1}{1 + 3L/(8r)} $$ 10.4 EUV-Related Flows - Hydrogen buffer gas flow for debris mitigation - Tin droplet dynamics in source - Molecular outgassing and mask contamination 10.5 Plasma-Flow Coupling Low-pressure plasma processes require multi-physics: $$ \nabla \cdot \mathbf{J}_e = S_e - R_e \quad \text{(electron continuity)} $$ $$ \nabla \cdot \mathbf{J}_i = S_i - R_i \quad \text{(ion continuity)} $$ $$ \nabla \cdot (\epsilon \nabla \phi) = -e(n_i - n_e) \quad \text{(Poisson)} $$ Coupled to neutral gas Navier-Stokes equations.
Temperature mapping using fluorescence.
SiO2 with fluorine for lower k.
Use F radicals to etch Si SiO2 (CF4 SF6 NF3).
Remaining flux after reflow.
Flying probe testers use movable probe heads for board-level testing without fixed fixtures enabling flexible low-volume testing.
Factorization Machines model feature interactions through factorized parameters generalizing matrix factorization to arbitrary features.
Failure Mode and Effects Analysis systematically identifies potential failures and their impacts.
Mix with random masks from Fourier space.
FNet replaces attention with Fourier transforms for efficiency.
Replace attention with Fourier transform.
Focal loss down-weights easy examples by adding a modulating factor to cross-entropy addressing class imbalance and hard example mining.
Down-weight easy examples.
Focal loss focuses on hard examples. Handles class imbalance.
Test different focus and dose combinations.
Combine FIB sample prep with APT.
Use ion beam to mill and image samples.
Use ion beam to repair mask defects.
Create parameters for MD simulations.
Force field learning predicts molecular forces and energies using equivariant graph neural networks.
Forced convection uses fans or blowers to increase airflow over heat sinks dramatically improving cooling capacity.
Constrain generation to match pattern.
Forecast error variance decomposition attributes forecast uncertainty to shocks in different variables.
Segment moving objects.
Loss of earlier learned information.
Share gate between nFET and pFET.
Prove correctness using formal methods.
Adjust formal vs informal tone.