masked region modeling, multimodal ai
Predict masked image regions.
3,145 technical terms and definitions
Predict masked image regions.
Predict masked image regions.
Models covering 100+ languages.
Material recovery extracts valuable substances from electronic waste through mechanical and chemical processes.
# Semiconductor Manufacturing Process: Materials Science & Mathematical Modeling A comprehensive guide to the physics, chemistry, and mathematics underlying modern semiconductor fabrication. ## 1. Overview Modern semiconductor manufacturing is one of the most complex and precise engineering endeavors ever undertaken. Key characteristics include: - **Feature sizes**: Leading-edge nodes at 3nm, 2nm, and research into sub-nm - **Precision requirements**: Atomic-level control (angstrom tolerances) - **Process steps**: Hundreds of sequential operations per chip - **Yield sensitivity**: Parts-per-billion defect control ### 1.1 Core Process Steps - **Crystal Growth** - Czochralski (CZ) process - Float-zone (FZ) refining - Epitaxial growth - **Pattern Definition** - Photolithography (DUV, EUV) - Electron-beam lithography - Nanoimprint lithography - **Material Addition** - Chemical Vapor Deposition (CVD) - Physical Vapor Deposition (PVD) - Atomic Layer Deposition (ALD) - Epitaxy (MBE, MOCVD) - **Material Removal** - Wet etching (isotropic) - Dry/plasma etching (anisotropic) - Chemical Mechanical Polishing (CMP) - **Doping** - Ion implantation - Thermal diffusion - Plasma doping - **Thermal Processing** - Oxidation - Annealing (RTA, spike, laser) - Silicidation ## 2. Materials Science Foundations ### 2.1 Silicon Properties - **Crystal structure**: Diamond cubic (Fd3m space group) - **Lattice constant**: $a = 5.431 \text{ Å}$ - **Bandgap**: $E_g = 1.12 \text{ eV}$ (indirect, at 300K) - **Intrinsic carrier concentration**: $$n_i = \sqrt{N_c N_v} \exp\left(-\frac{E_g}{2k_B T}\right)$$ At 300K: $n_i \approx 1.0 \times 10^{10} \text{ cm}^{-3}$ ### 2.2 Crystal Defects - **Point Defects** - **Vacancies (V)**: Missing lattice atoms - **Self-interstitials (I)**: Extra Si atoms in interstitial sites - **Substitutional impurities**: Dopants (B, P, As, Sb) - **Interstitial impurities**: Fast diffusers (Fe, Cu, Au) - **Line Defects** - **Edge dislocations**: Extra half-plane of atoms - **Screw dislocations**: Helical atomic arrangement - **Dislocation density target**: $< 100 \text{ cm}^{-2}$ for device wafers - **Planar Defects** - **Stacking faults**: ABCABC → ABCBCABC - **Twin boundaries**: Mirror symmetry planes - **Grain boundaries**: (avoided in single-crystal wafers) ### 2.3 Dielectric Materials | Material | Dielectric Constant ($\kappa$) | Bandgap (eV) | Application | |----------|-------------------------------|--------------|-------------| | SiO₂ | 3.9 | 9.0 | Traditional gate oxide | | Si₃N₄ | 7.5 | 5.3 | Spacers, hard masks | | HfO₂ | ~25 | 5.8 | High-κ gate dielectric | | Al₂O₃ | 9 | 8.8 | ALD dielectric | | ZrO₂ | ~25 | 5.8 | High-κ gate dielectric | **Equivalent Oxide Thickness (EOT)**: $$\text{EOT} = t_{\text{high-}\kappa} \cdot \frac{\kappa_{\text{SiO}_2}}{\kappa_{\text{high-}\kappa}} = t_{\text{high-}\kappa} \cdot \frac{3.9}{\kappa_{\text{high-}\kappa}}$$ ### 2.4 Interconnect Materials - **Evolution**: Al/SiO₂ → Cu/low-κ → Cu/air-gap → (future: Ru, Co) - **Electromigration** - Black's equation for mean time to failure: $$\text{MTTF} = A \cdot j^{-n} \exp\left(\frac{E_a}{k_B T}\right)$$ Where: - $j$ = current density - $n$ ≈ 1-2 (current exponent) - $E_a$ ≈ 0.7-0.9 eV for Cu ## 3. Crystal Growth Modeling ### 3.1 Czochralski Process Physics The Czochralski process involves pulling a single crystal from a melt. Key phenomena: - **Heat transfer** (conduction, convection, radiation) - **Fluid dynamics** (buoyancy-driven and forced convection) - **Mass transport** (dopant distribution) - **Phase change** (solidification at the interface) ### 3.2 Heat Transfer Equation $$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q$$ Where: - $\rho$ = density [kg/m³] - $c_p$ = specific heat capacity [J/(kg·K)] - $k$ = thermal conductivity [W/(m·K)] - $Q$ = volumetric heat source [W/m³] ### 3.3 Stefan Problem (Phase Change) At the solid-liquid interface, the Stefan condition applies: $$k_s \frac{\partial T_s}{\partial n} - k_\ell \frac{\partial T_\ell}{\partial n} = \rho L v_n$$ Where: - $k_s$, $k_\ell$ = thermal conductivity of solid and liquid - $L$ = latent heat of fusion [J/kg] - $v_n$ = interface velocity normal to the surface [m/s] ### 3.4 Melt Convection (Navier-Stokes with Boussinesq Approximation) $$\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \beta (T - T_0)$$ Dimensionless parameters: - **Grashof number**: $Gr = \frac{g \beta \Delta T L^3}{\nu^2}$ - **Prandtl number**: $Pr = \frac{\nu}{\alpha}$ - **Rayleigh number**: $Ra = Gr \cdot Pr$ ### 3.5 Dopant Segregation **Equilibrium segregation coefficient**: $$k_0 = \frac{C_s}{C_\ell}$$ **Effective segregation coefficient** (Burton-Prim-Slichter model): $$k_{\text{eff}} = \frac{k_0}{k_0 + (1 - k_0) \exp\left(-\frac{v \delta}{D}\right)}$$ Where: - $v$ = crystal pull rate [m/s] - $\delta$ = boundary layer thickness [m] - $D$ = diffusion coefficient in melt [m²/s] **Dopant concentration along crystal** (normal freezing): $$C_s(f) = k_{\text{eff}} C_0 (1 - f)^{k_{\text{eff}} - 1}$$ Where $f$ = fraction solidified. ## 4. Diffusion Modeling ### 4.1 Fick's Laws **First Law** (flux proportional to concentration gradient): $$\mathbf{J} = -D \nabla C$$ **Second Law** (conservation equation): $$\frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C)$$ For constant $D$ in 1D: $$\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$$ ### 4.2 Analytical Solutions **Constant surface concentration** (predeposition): $$C(x,t) = C_s \cdot \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$$ **Fixed total dose** (drive-in): $$C(x,t) = \frac{Q}{\sqrt{\pi D t}} \exp\left(-\frac{x^2}{4Dt}\right)$$ Where: - $C_s$ = surface concentration - $Q$ = total dose [atoms/cm²] - $\text{erfc}(z) = 1 - \text{erf}(z)$ = complementary error function ### 4.3 Temperature Dependence Diffusion coefficient follows Arrhenius behavior: $$D = D_0 \exp\left(-\frac{E_a}{k_B T}\right)$$ | Dopant | $D_0$ (cm²/s) | $E_a$ (eV) | |--------|---------------|------------| | B | 0.76 | 3.46 | | P | 3.85 | 3.66 | | As | 0.32 | 3.56 | | Sb | 0.214 | 3.65 | ### 4.4 Point-Defect Mediated Diffusion Dopants diffuse via interactions with point defects. The total diffusivity: $$D_{\text{eff}} = D_I \frac{C_I}{C_I^*} + D_V \frac{C_V}{C_V^*}$$ Where: - $D_I$, $D_V$ = interstitial and vacancy components - $C_I^*$, $C_V^*$ = equilibrium concentrations **Coupled defect-dopant equations**: $$\frac{\partial C_I}{\partial t} = D_I \nabla^2 C_I + G_I - k_{IV} C_I C_V$$ $$\frac{\partial C_V}{\partial t} = D_V \nabla^2 C_V + G_V - k_{IV} C_I C_V$$ Where: - $G_I$, $G_V$ = generation rates - $k_{IV}$ = I-V recombination rate constant ### 4.5 Transient Enhanced Diffusion (TED) After ion implantation, excess interstitials cause enhanced diffusion: - **"+1" model**: Each implanted ion creates ~1 net interstitial - **TED factor**: Can enhance diffusion by 10-1000× - **Decay time**: τ ~ seconds at high T, hours at low T ## 5. Ion Implantation ### 5.1 Range Statistics **Gaussian approximation** (light ions, amorphous target): $$n(x) = \frac{\phi}{\sqrt{2\pi} \Delta R_p} \exp\left(-\frac{(x - R_p)^2}{2 \Delta R_p^2}\right)$$ Where: - $\phi$ = implant dose [ions/cm²] - $R_p$ = projected range [nm] - $\Delta R_p$ = range straggle (standard deviation) [nm] **Pearson IV distribution** (heavier ions, includes skewness and kurtosis): $$n(x) = \frac{\phi}{\Delta R_p} \cdot f\left(\frac{x - R_p}{\Delta R_p}; \gamma, \beta\right)$$ ### 5.2 Stopping Power **Total stopping power** (LSS theory): $$S(E) = -\frac{1}{N}\frac{dE}{dx} = S_n(E) + S_e(E)$$ Where: - $S_n(E)$ = nuclear stopping (elastic collisions with nuclei) - $S_e(E)$ = electronic stopping (inelastic interactions with electrons) - $N$ = atomic density of target **Nuclear stopping** (screened Coulomb potential): $$S_n(E) = \frac{\pi a^2 \gamma E}{1 + M_2/M_1}$$ Where: - $a$ = screening length - $\gamma = 4 M_1 M_2 / (M_1 + M_2)^2$ **Electronic stopping** (velocity-proportional regime): $$S_e(E) = k_e \sqrt{E}$$ ### 5.3 Monte Carlo Simulation (BCA) The Binary Collision Approximation treats each collision as isolated: 1. **Free flight**: Ion travels until next collision 2. **Collision**: Classical two-body scattering 3. **Energy loss**: Nuclear + electronic contributions 4. **Repeat**: Until ion stops ($E < E_{\text{threshold}}$) **Scattering angle** (center of mass frame): $$\theta_{cm} = \pi - 2 \int_{r_{min}}^{\infty} \frac{b \, dr}{r^2 \sqrt{1 - V(r)/E_{cm} - b^2/r^2}}$$ ### 5.4 Damage Accumulation **Kinchin-Pease model** for displacement damage: $$N_d = \frac{0.8 E_d}{2 E_{th}}$$ Where: - $N_d$ = number of displaced atoms - $E_d$ = damage energy deposited - $E_{th}$ = displacement threshold (~15 eV for Si) **Amorphization**: Occurs when damage density exceeds ~10% of atomic density ## 6. Thermal Oxidation ### 6.1 Deal-Grove Model The oxide thickness $x$ as a function of time $t$: $$x^2 + A x = B(t + \tau)$$ Or solved for thickness: $$x = \frac{A}{2} \left( \sqrt{1 + \frac{4B(t + \tau)}{A^2}} - 1 \right)$$ ### 6.2 Rate Constants **Parabolic rate constant** (diffusion-limited): $$B = \frac{2 D C^*}{N_1}$$ Where: - $D$ = diffusion coefficient of O₂ in SiO₂ - $C^*$ = equilibrium concentration at surface - $N_1$ = number of oxidant molecules per unit volume of oxide **Linear rate constant** (reaction-limited): $$\frac{B}{A} = \frac{k_s C^*}{N_1}$$ Where $k_s$ = surface reaction rate constant ### 6.3 Limiting Cases **Thin oxide** ($x \ll A$): Linear regime $$x \approx \frac{B}{A}(t + \tau)$$ **Thick oxide** ($x \gg A$): Parabolic regime $$x \approx \sqrt{B(t + \tau)}$$ ### 6.4 Temperature and Pressure Dependence $$B = B_0 \exp\left(-\frac{E_B}{k_B T}\right) \cdot \frac{p}{p_0}$$ $$\frac{B}{A} = \left(\frac{B}{A}\right)_0 \exp\left(-\frac{E_{B/A}}{k_B T}\right) \cdot \frac{p}{p_0}$$ | Condition | $E_B$ (eV) | $E_{B/A}$ (eV) | |-----------|------------|----------------| | Dry O₂ | 1.23 | 2.0 | | Wet O₂ (H₂O) | 0.78 | 2.05 | ## 7. Chemical Vapor Deposition (CVD) ### 7.1 Reactor Transport Equations **Continuity equation**: $$\nabla \cdot (\rho \mathbf{v}) = 0$$ **Momentum equation** (Navier-Stokes): $$\rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}$$ **Energy equation**: $$\rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T \right) = \nabla \cdot (k \nabla T) + \sum_i H_i R_i$$ **Species transport**: $$\frac{\partial (\rho Y_i)}{\partial t} + \nabla \cdot (\rho \mathbf{v} Y_i) = \nabla \cdot (\rho D_i \nabla Y_i) + M_i \sum_j \nu_{ij} r_j$$ Where: - $Y_i$ = mass fraction of species $i$ - $D_i$ = diffusion coefficient - $\nu_{ij}$ = stoichiometric coefficient - $r_j$ = reaction rate of reaction $j$ ### 7.2 Surface Reaction Kinetics **Langmuir-Hinshelwood mechanism**: $$R_s = \frac{k_s K_1 K_2 p_1 p_2}{(1 + K_1 p_1 + K_2 p_2)^2}$$ **First-order surface reaction**: $$R_s = k_s C_s = k_s \cdot h_m (C_g - C_s)$$ At steady state: $$C_s = \frac{h_m C_g}{h_m + k_s}$$ ### 7.3 Step Coverage **Thiele modulus** for feature filling: $$\Phi = L \sqrt{\frac{k_s}{D_{\text{Kn}}}}$$ Where: - $L$ = feature depth - $D_{\text{Kn}}$ = Knudsen diffusion coefficient **Step coverage behavior**: - $\Phi \ll 1$: Reaction-limited → conformal deposition - $\Phi \gg 1$: Transport-limited → poor step coverage ### 7.4 Growth Rate $$G = \frac{M_f}{\rho_f} \cdot R_s = \frac{M_f}{\rho_f} \cdot \frac{h_m k_s C_g}{h_m + k_s}$$ Where: - $M_f$ = molecular weight of film - $\rho_f$ = film density ## 8. Atomic Layer Deposition (ALD) ### 8.1 Self-Limiting Surface Reactions ALD relies on sequential, self-saturating surface reactions. **Surface site model**: $$\frac{d\theta}{dt} = k_{\text{ads}} p (1 - \theta) - k_{\text{des}} \theta$$ At steady state: $$\theta_{eq} = \frac{K p}{1 + K p}$$ Where $K = k_{\text{ads}} / k_{\text{des}}$ = equilibrium constant ### 8.2 Growth Per Cycle (GPC) $$\text{GPC} = \Gamma_{\text{max}} \cdot \theta \cdot \frac{M_f}{\rho_f N_A}$$ Where: - $\Gamma_{\text{max}}$ = maximum surface site density [sites/cm²] - $\theta$ = surface coverage (0 to 1) - $N_A$ = Avogadro's number **Typical GPC values**: - Al₂O₃ (TMA/H₂O): ~1.1 Å/cycle - HfO₂ (HfCl₄/H₂O): ~1.0 Å/cycle - TiN (TiCl₄/NH₃): ~0.4 Å/cycle ### 8.3 Conformality in High Aspect Ratio Features **Penetration depth**: $$\Lambda = \sqrt{\frac{D_{\text{Kn}}}{k_s \Gamma_{\text{max}}}}$$ **Conformality factor**: $$\text{CF} = \frac{1}{\sqrt{1 + (L/\Lambda)^2}}$$ For 100% conformality: Require $L \ll \Lambda$ ## 9. Plasma Etching ### 9.1 Plasma Fundamentals **Electron energy balance**: $$n_e \frac{\partial}{\partial t}\left(\frac{3}{2} k_B T_e\right) = \nabla \cdot (\kappa_e \nabla T_e) + P_{\text{abs}} - P_{\text{loss}}$$ **Debye length** (shielding distance): $$\lambda_D = \sqrt{\frac{\epsilon_0 k_B T_e}{n_e e^2}}$$ **Plasma frequency**: $$\omega_{pe} = \sqrt{\frac{n_e e^2}{\epsilon_0 m_e}}$$ ### 9.2 Sheath Physics **Child-Langmuir law** (collisionless sheath): $$J_i = \frac{4 \epsilon_0}{9} \sqrt{\frac{2e}{M_i}} \frac{V_s^{3/2}}{d^2}$$ Where: - $J_i$ = ion current density - $V_s$ = sheath voltage - $d$ = sheath thickness - $M_i$ = ion mass **Bohm criterion** (ion velocity at sheath edge): $$v_B = \sqrt{\frac{k_B T_e}{M_i}}$$ ### 9.3 Etch Rate Modeling **Ion-enhanced etching**: $$R = R_{\text{chem}} + R_{\text{ion}} = k_n n_{\text{neutral}} + Y \cdot \Gamma_{\text{ion}}$$ Where: - $R_{\text{chem}}$ = chemical (isotropic) component - $R_{\text{ion}}$ = ion-enhanced (directional) component - $Y$ = sputter yield - $\Gamma_{\text{ion}}$ = ion flux **Anisotropy**: $$A = 1 - \frac{R_{\text{lateral}}}{R_{\text{vertical}}}$$ - $A = 0$: Isotropic - $A = 1$: Perfectly anisotropic ### 9.4 Feature-Scale Modeling **Level set equation** for surface evolution: $$\frac{\partial \phi}{\partial t} + F |\nabla \phi| = 0$$ Where: - $\phi(\mathbf{x}, t)$ = level set function - $F$ = local velocity (etch or deposition rate) - Surface defined by $\phi = 0$ ## 10. Lithography ### 10.1 Resolution Limits **Rayleigh criterion**: $$R = k_1 \frac{\lambda}{NA}$$ **Depth of focus**: $$DOF = k_2 \frac{\lambda}{NA^2}$$ Where: - $\lambda$ = wavelength (193 nm DUV, 13.5 nm EUV) - $NA$ = numerical aperture - $k_1$, $k_2$ = process-dependent factors | Technology | λ (nm) | NA | Minimum k₁ | Resolution (nm) | |------------|--------|-----|------------|-----------------| | DUV (ArF) | 193 | 1.35 | 0.25 | ~36 | | EUV | 13.5 | 0.33 | 0.25 | ~10 | | High-NA EUV | 13.5 | 0.55 | 0.25 | ~6 | ### 10.2 Aerial Image Formation **Coherent illumination**: $$I(x,y) = \left| \mathcal{F}^{-1} \left\{ \tilde{M}(f_x, f_y) \cdot H(f_x, f_y) \right\} \right|^2$$ Where: - $\tilde{M}$ = Fourier transform of mask transmission - $H$ = optical transfer function (pupil function) **Partially coherent illumination** (Hopkins formulation): $$I(x,y) = \iint \iint TCC(f_1, g_1, f_2, g_2) \cdot \tilde{M}(f_1, g_1) \cdot \tilde{M}^*(f_2, g_2) \cdot e^{2\pi i [(f_1 - f_2)x + (g_1 - g_2)y]} \, df_1 \, dg_1 \, df_2 \, dg_2$$ Where $TCC$ = transmission cross coefficient ### 10.3 Photoresist Chemistry **Chemically Amplified Resists (CARs)**: **Photoacid generation**: $$\frac{\partial [\text{PAG}]}{\partial t} = -C \cdot I \cdot [\text{PAG}]$$ **Acid diffusion and reaction**: $$\frac{\partial [H^+]}{\partial t} = D_H \nabla^2 [H^+] + k_{\text{gen}} - k_{\text{neut}}[H^+][Q]$$ **Deprotection kinetics**: $$\frac{\partial [M]}{\partial t} = -k_{\text{amp}} [H^+] [M]$$ Where: - $[\text{PAG}]$ = photoacid generator concentration - $[H^+]$ = acid concentration - $[Q]$ = quencher concentration - $[M]$ = protected site concentration ### 10.4 Stochastic Effects in EUV **Photon shot noise**: $$\sigma_N = \sqrt{N}$$ **Line Edge Roughness (LER)**: $$\sigma_{\text{LER}} \propto \frac{1}{\sqrt{\text{dose}}} \propto \frac{1}{\sqrt{N_{\text{photons}}}}$$ **Stochastic defect probability**: $$P_{\text{defect}} = 1 - \exp(-\lambda A)$$ Where $\lambda$ = defect density, $A$ = feature area ## 11. Chemical Mechanical Polishing (CMP) ### 11.1 Preston Equation $$\frac{dh}{dt} = K_p \cdot P \cdot v$$ Where: - $dh/dt$ = material removal rate [nm/s] - $K_p$ = Preston coefficient [nm/(Pa·m)] - $P$ = applied pressure [Pa] - $v$ = relative velocity [m/s] ### 11.2 Contact Mechanics **Greenwood-Williamson model** for asperity contact: $$A_{\text{real}} = \pi n \beta \sigma \int_{d}^{\infty} (z - d) \phi(z) \, dz$$ $$F = \frac{4}{3} n E^* \sqrt{\beta} \int_{d}^{\infty} (z - d)^{3/2} \phi(z) \, dz$$ Where: - $n$ = asperity density - $\beta$ = asperity radius - $\sigma$ = RMS roughness - $\phi(z)$ = height distribution - $E^*$ = effective elastic modulus ### 11.3 Pattern-Dependent Effects **Dishing** (in metal features): $$\Delta h_{\text{dish}} \propto w^2$$ Where $w$ = line width **Erosion** (in dielectric): $$\Delta h_{\text{erosion}} \propto \rho_{\text{metal}}$$ Where $\rho_{\text{metal}}$ = local metal pattern density ## 12. Device Simulation (TCAD) ### 12.1 Poisson Equation $$\nabla \cdot (\epsilon \nabla \psi) = -q(p - n + N_D^+ - N_A^-)$$ Where: - $\psi$ = electrostatic potential [V] - $\epsilon$ = permittivity - $n$, $p$ = electron and hole concentrations - $N_D^+$, $N_A^-$ = ionized donor and acceptor concentrations ### 12.2 Drift-Diffusion Equations **Current densities**: $$\mathbf{J}_n = q \mu_n n \mathbf{E} + q D_n \nabla n$$ $$\mathbf{J}_p = q \mu_p p \mathbf{E} - q D_p \nabla p$$ **Einstein relation**: $$D_n = \frac{k_B T}{q} \mu_n, \quad D_p = \frac{k_B T}{q} \mu_p$$ **Continuity equations**: $$\frac{\partial n}{\partial t} = \frac{1}{q} \nabla \cdot \mathbf{J}_n + G - R$$ $$\frac{\partial p}{\partial t} = -\frac{1}{q} \nabla \cdot \mathbf{J}_p + G - R$$ ### 12.3 Carrier Statistics **Boltzmann approximation**: $$n = N_c \exp\left(\frac{E_F - E_c}{k_B T}\right)$$ $$p = N_v \exp\left(\frac{E_v - E_F}{k_B T}\right)$$ **Fermi-Dirac (degenerate regime)**: $$n = N_c \mathcal{F}_{1/2}\left(\frac{E_F - E_c}{k_B T}\right)$$ Where $\mathcal{F}_{1/2}$ = Fermi-Dirac integral of order 1/2 ### 12.4 Recombination Models **Shockley-Read-Hall (SRH)**: $$R_{\text{SRH}} = \frac{pn - n_i^2}{\tau_p(n + n_1) + \tau_n(p + p_1)}$$ **Auger recombination**: $$R_{\text{Auger}} = (C_n n + C_p p)(pn - n_i^2)$$ **Radiative recombination**: $$R_{\text{rad}} = B(pn - n_i^2)$$ ## 13. Advanced Mathematical Methods ### 13.1 Level Set Methods **Evolution equation**: $$\frac{\partial \phi}{\partial t} + F |\nabla \phi| = 0$$ **Reinitialization** (maintain signed distance function): $$\frac{\partial \phi}{\partial \tau} = \text{sign}(\phi_0)(1 - |\nabla \phi|)$$ **Curvature**: $$\kappa = \nabla \cdot \left( \frac{\nabla \phi}{|\nabla \phi|} \right)$$ ### 13.2 Kinetic Monte Carlo (KMC) **Rate catalog**: $$r_i = \nu_0 \exp\left(-\frac{E_i}{k_B T}\right)$$ **Event selection** (Bortz-Kalos-Lebowitz algorithm): 1. Calculate total rate: $R_{\text{tot}} = \sum_i r_i$ 2. Generate random $u \in (0,1)$ 3. Select event $j$ where $\sum_{i=1}^{j-1} r_i < u \cdot R_{\text{tot}} \leq \sum_{i=1}^{j} r_i$ **Time advancement**: $$\Delta t = -\frac{\ln(u')}{R_{\text{tot}}}$$ ### 13.3 Phase Field Methods **Free energy functional**: $$F[\phi] = \int \left[ f(\phi) + \frac{\epsilon^2}{2} |\nabla \phi|^2 \right] dV$$ **Allen-Cahn equation** (non-conserved order parameter): $$\frac{\partial \phi}{\partial t} = -M \frac{\delta F}{\delta \phi} = M \left[ \epsilon^2 \nabla^2 \phi - f'(\phi) \right]$$ **Cahn-Hilliard equation** (conserved order parameter): $$\frac{\partial \phi}{\partial t} = \nabla \cdot \left( M \nabla \frac{\delta F}{\delta \phi} \right)$$ ### 13.4 Density Functional Theory (DFT) **Kohn-Sham equations**: $$\left[ -\frac{\hbar^2}{2m} \nabla^2 + V_{\text{eff}}(\mathbf{r}) \right] \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r})$$ **Effective potential**: $$V_{\text{eff}}(\mathbf{r}) = V_{\text{ext}}(\mathbf{r}) + V_H(\mathbf{r}) + V_{xc}(\mathbf{r})$$ Where: - $V_{\text{ext}}$ = external (ionic) potential - $V_H = e^2 \int \frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{r}'$ = Hartree potential - $V_{xc} = \frac{\delta E_{xc}[n]}{\delta n}$ = exchange-correlation potential **Electron density**: $$n(\mathbf{r}) = \sum_i f_i |\psi_i(\mathbf{r})|^2$$ ## 14. Current Frontiers ### 14.1 Extreme Ultraviolet (EUV) Lithography - **Challenges**: - Stochastic effects at low photon counts - Mask defectivity and pellicle development - Resist trade-offs (sensitivity vs. resolution vs. LER) - Source power and productivity - **High-NA EUV**: - NA = 0.55 (vs. 0.33 current) - Anamorphic optics (4× magnification in one direction) - Sub-8nm half-pitch capability ### 14.2 3D Integration - **Through-Silicon Vias (TSVs)**: - Via-first, via-middle, via-last approaches - Cu filling and barrier requirements - Thermal-mechanical stress modeling - **Hybrid Bonding**: - Cu-Cu direct bonding - Sub-micron alignment requirements - Surface preparation and activation ### 14.3 New Materials - **2D Materials**: - Graphene (zero bandgap) - Transition metal dichalcogenides (MoS₂, WS₂, WSe₂) - Hexagonal boron nitride (hBN) - **Wide Bandgap Semiconductors**: - GaN: $E_g = 3.4$ eV - SiC: $E_g = 3.3$ eV (4H-SiC) - Ga₂O₃: $E_g = 4.8$ eV ### 14.4 Novel Device Architectures - **Gate-All-Around (GAA) FETs**: - Nanosheet and nanowire channels - Superior electrostatic control - Samsung 3nm, Intel 20A/18A - **Complementary FET (CFET)**: - Vertically stacked NMOS/PMOS - Reduced footprint - Complex fabrication - **Backside Power Delivery (BSPD)**: - Power rails on wafer backside - Reduced IR drop - Intel PowerVia ### 14.5 Machine Learning in Semiconductor Manufacturing - **Virtual Metrology**: Predict wafer properties from tool sensor data - **Defect Detection**: CNN-based wafer map classification - **Process Optimization**: Bayesian optimization, reinforcement learning - **Surrogate Models**: Neural networks replacing expensive simulations - **OPC (Optical Proximity Correction)**: ML-accelerated mask design ## Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ J/K | | Elementary charge | $e$ | $1.602 \times 10^{-19}$ C | | Planck constant | $h$ | $6.626 \times 10^{-34}$ J·s | | Electron mass | $m_e$ | $9.109 \times 10^{-31}$ kg | | Permittivity of free space | $\epsilon_0$ | $8.854 \times 10^{-12}$ F/m | | Avogadro's number | $N_A$ | $6.022 \times 10^{23}$ mol⁻¹ | | Thermal voltage (300K) | $k_B T/q$ | 25.85 mV | ## Multiscale Modeling Hierarchy | Level | Method | Length Scale | Time Scale | Application | |-------|--------|--------------|------------|-------------| | 1 | Ab initio (DFT) | Å | fs | Reaction mechanisms, band structure | | 2 | Molecular Dynamics | nm | ps-ns | Defect dynamics, interfaces | | 3 | Kinetic Monte Carlo | nm-μm | ns-s | Growth, etching, diffusion | | 4 | Continuum (PDE) | μm-mm | s-hr | Process simulation (TCAD) | | 5 | Compact Models | Device | — | Circuit simulation | | 6 | Statistical | Die/Wafer | — | Yield prediction |
Math models are enhanced for mathematical reasoning and problem solving.
# Mathematics Modeling
1. Crystal Growth (Czochralski Process)
Growing single-crystal silicon ingots requires coupled models for heat transfer, fluid flow, and mass transport.
1.1 Heat Transfer Equation
$$
\rho c_p \frac{\partial T}{\partial t} + \rho c_p \mathbf{v} \cdot \nabla T = \nabla \cdot (k \nabla T) + Q
$$
Variables:
- $\rho$ — density ($\text{kg/m}^3$)
- $c_p$ — specific heat capacity ($\text{J/(kg·K)}$)
- $T$ — temperature ($\text{K}$)
- $\mathbf{v}$ — velocity vector ($\text{m/s}$)
- $k$ — thermal conductivity ($\text{W/(m·K)}$)
- $Q$ — heat source term ($\text{W/m}^3$)
1.2 Melt Convection Drivers
- Buoyancy forces — thermal and solutal gradients
- Marangoni flow — surface tension gradients
- Forced convection — crystal and crucible rotation
1.3 Dopant Segregation
Equilibrium segregation coefficient:
$$
k_0 = \frac{C_s}{C_l}
$$
Effective segregation coefficient (Burton-Prim-Slichter model):
$$
k_{eff} = \frac{k_0}{k_0 + (1 - k_0) \exp\left(-\frac{v \delta}{D}\right)}
$$
Variables:
- $C_s$ — dopant concentration in solid
- $C_l$ — dopant concentration in liquid
- $v$ — crystal growth velocity
- $\delta$ — boundary layer thickness
- $D$ — diffusion coefficient in melt
2. Thermal Oxidation (Deal-Grove Model)
The foundational model for growing $\text{SiO}_2$ on silicon.
2.1 General Equation
$$
x_o^2 + A x_o = B(t + \tau)
$$
Variables:
- $x_o$ — oxide thickness ($\mu\text{m}$ or $\text{nm}$)
- $A$ — linear rate constant parameter
- $B$ — parabolic rate constant
- $t$ — oxidation time
- $\tau$ — time offset for initial oxide
2.2 Growth Regimes
- Linear regime (thin oxide, surface-reaction limited):
$$
x_o \approx \frac{B}{A}(t + \tau)
$$
- Parabolic regime (thick oxide, diffusion limited):
$$
x_o \approx \sqrt{B(t + \tau)}
$$
2.3 Extended Model Considerations
- Stress-dependent oxidation rates
- Point defect injection into silicon
- 2D/3D geometries (LOCOS bird's beak)
- High-pressure oxidation kinetics
- Thin oxide regime anomalies (<20 nm)
3. Diffusion and Dopant Transport
3.1 Fick's Laws
First Law (flux equation):
$$
\mathbf{J} = -D \nabla C
$$
Second Law (continuity equation):
$$
\frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C)
$$
For constant $D$:
$$
\frac{\partial C}{\partial t} = D \nabla^2 C
$$
3.2 Concentration-Dependent Diffusivity
$$
D(C) = D_i + D^{-} \frac{n}{n_i} + D^{2-} \left(\frac{n}{n_i}\right)^2 + D^{+} \frac{p}{n_i} + D^{2+} \left(\frac{p}{n_i}\right)^2
$$
Variables:
- $D_i$ — intrinsic diffusivity
- $D^{-}, D^{2-}$ — diffusivity via negatively charged defects
- $D^{+}, D^{2+}$ — diffusivity via positively charged defects
- $n, p$ — electron and hole concentrations
- $n_i$ — intrinsic carrier concentration
3.3 Point-Defect Mediated Diffusion
Effective diffusivity:
$$
D_{eff} = D_I \frac{C_I}{C_I^*} + D_V \frac{C_V}{C_V^*}
$$
Point defect continuity equations:
$$
\frac{\partial C_I}{\partial t} = D_I \nabla^2 C_I + G_I - R_{IV}
$$
$$
\frac{\partial C_V}{\partial t} = D_V \nabla^2 C_V + G_V - R_{IV}
$$
Recombination rate:
$$
R_{IV} = k_{IV} \left( C_I C_V - C_I^* C_V^* \right)
$$
Variables:
- $C_I, C_V$ — interstitial and vacancy concentrations
- $C_I^*, C_V^*$ — equilibrium concentrations
- $G_I, G_V$ — generation rates
- $R_{IV}$ — interstitial-vacancy recombination rate
3.4 Transient Enhanced Diffusion (TED)
Ion implantation creates excess interstitials causing:
- "+1" model: each implanted ion creates one net interstitial
- Enhanced diffusion persists until excess defects anneal out
- Critical for ultra-shallow junction formation
4. Ion Implantation
4.1 Gaussian Profile Model
$$
N(x) = \frac{\phi}{\sqrt{2\pi} \Delta R_p} \exp\left[ -\frac{(x - R_p)^2}{2 (\Delta R_p)^2} \right]
$$
Variables:
- $N(x)$ — dopant concentration at depth $x$ ($\text{cm}^{-3}$)
- $\phi$ — implant dose ($\text{ions/cm}^2$)
- $R_p$ — projected range (mean depth)
- $\Delta R_p$ — straggle (standard deviation)
4.2 Pearson IV Distribution
For asymmetric profiles using four moments:
- First moment: $R_p$ (projected range)
- Second moment: $\Delta R_p$ (straggle)
- Third moment: $\gamma$ (skewness)
- Fourth moment: $\beta$ (kurtosis)
4.3 Monte Carlo Methods (TRIM/SRIM)
Stopping power:
$$
\frac{dE}{dx} = S_n(E) + S_e(E)
$$
- $S_n(E)$ — nuclear stopping power
- $S_e(E)$ — electronic stopping power
Key outputs:
- Ion trajectories via binary collision approximation (BCA)
- Damage cascade distribution
- Sputtering yield
- Vacancy and interstitial generation profiles
4.4 Channeling Effects
For crystalline targets, ions aligned with crystal axes experience:
- Reduced stopping power
- Deeper penetration
- Modified range distributions
- Requires dual-Pearson or Monte Carlo models
5. Plasma Etching
5.1 Surface Kinetics Model
$$
\frac{\partial \theta}{\partial t} = J_i s_i (1 - \theta) - k_r \theta
$$
Variables:
- $\theta$ — fractional surface coverage of reactive species
- $J_i$ — incident ion/radical flux
- $s_i$ — sticking coefficient
- $k_r$ — surface reaction rate constant
5.2 Etching Yield
$$
Y = \frac{\text{atoms removed}}{\text{incident ion}}
$$
Dependence factors:
- Ion energy ($E_{ion}$)
- Ion incidence angle ($\theta$)
- Ion-to-neutral flux ratio
- Surface chemistry and temperature
5.3 Profile Evolution (Level Set Method)
$$
\frac{\partial \phi}{\partial t} + V |\nabla \phi| = 0
$$
Variables:
- $\phi(\mathbf{x}, t)$ — level set function (surface defined by $\phi = 0$)
- $V$ — local etch rate (normal velocity)
5.4 Knudsen Transport in High Aspect Ratio Features
For molecular flow regime ($Kn > 1$):
$$
\frac{1}{\lambda} \frac{dI}{dx} = -I + \int K(x, x') I(x') dx'
$$
Key effects:
- Aspect ratio dependent etching (ARDE)
- Reactive ion angular distribution (RIAD)
- Neutral shadowing
6. Chemical Vapor Deposition (CVD)
6.1 Transport-Reaction Equation
$$
\frac{\partial C}{\partial t} + \mathbf{v} \cdot \nabla C = D \nabla^2 C - k C^n
$$
Variables:
- $C$ — reactant concentration
- $\mathbf{v}$ — gas velocity
- $D$ — gas-phase diffusivity
- $k$ — reaction rate constant
- $n$ — reaction order
6.2 Thiele Modulus
$$
\phi = L \sqrt{\frac{k}{D}}
$$
Regimes:
- $\phi \ll 1$ — reaction-limited (uniform deposition)
- $\phi \gg 1$ — transport-limited (poor step coverage)
6.3 Step Coverage
Conformality factor:
$$
S = \frac{\text{thickness at bottom}}{\text{thickness at top}}
$$
Models:
- Ballistic transport (line-of-sight)
- Knudsen diffusion
- Surface reaction probability
6.4 Atomic Layer Deposition (ALD)
Self-limiting surface coverage:
$$
\theta(t) = 1 - \exp\left( -\frac{p \cdot t}{\tau} \right)
$$
Variables:
- $\theta(t)$ — fractional surface coverage
- $p$ — precursor partial pressure
- $\tau$ — characteristic adsorption time
Growth per cycle (GPC):
$$
\text{GPC} = \theta_{sat} \cdot \Gamma_{ML}
$$
where $\Gamma_{ML}$ is the monolayer thickness.
7. Chemical Mechanical Polishing (CMP)
7.1 Preston Equation
$$
\frac{dz}{dt} = K_p \cdot P \cdot V
$$
Variables:
- $dz/dt$ — material removal rate (MRR)
- $K_p$ — Preston coefficient ($\text{m}^2/\text{N}$)
- $P$ — applied pressure
- $V$ — relative velocity
7.2 Pattern-Dependent Effects
Effective pressure:
$$
P_{eff} = \frac{P_{applied}}{\rho_{pattern}}
$$
where $\rho_{pattern}$ is local pattern density.
Key phenomena:
- Dishing: over-polishing of soft materials (e.g., Cu)
- Erosion: oxide loss in high-density regions
- Within-die non-uniformity (WIDNU)
7.3 Contact Mechanics
Hertzian contact pressure:
$$
P(r) = P_0 \sqrt{1 - \left(\frac{r}{a}\right)^2}
$$
Pad asperity models:
- Greenwood-Williamson for rough surfaces
- Viscoelastic pad behavior
8. Lithography
8.1 Aerial Image Formation
Hopkins formulation (partially coherent):
$$
I(\mathbf{x}) = \iint TCC(\mathbf{f}, \mathbf{f}') \, M(\mathbf{f}) \, M^*(\mathbf{f}') \, e^{2\pi i (\mathbf{f} - \mathbf{f}') \cdot \mathbf{x}} \, d\mathbf{f} \, d\mathbf{f}'
$$
Variables:
- $I(\mathbf{x})$ — intensity at image plane position $\mathbf{x}$
- $TCC$ — transmission cross-coefficient
- $M(\mathbf{f})$ — mask spectrum at spatial frequency $\mathbf{f}$
8.2 Resolution and Depth of Focus
Rayleigh resolution criterion:
$$
R = k_1 \frac{\lambda}{NA}
$$
Depth of focus:
$$
DOF = k_2 \frac{\lambda}{NA^2}
$$
Variables:
- $\lambda$ — exposure wavelength (e.g., 193 nm for DUV, 13.5 nm for EUV)
- $NA$ — numerical aperture
- $k_1, k_2$ — process-dependent factors
8.3 Photoresist Exposure (Dill Model)
Photoactive compound (PAC) decomposition:
$$
\frac{\partial m}{\partial t} = -I(z, t) \cdot m \cdot C
$$
Intensity attenuation:
$$
I(z, t) = I_0 \exp\left( -\int_0^z [A \cdot m(z', t) + B] \, dz' \right)
$$
Dill parameters:
- $A$ — bleachable absorption coefficient
- $B$ — non-bleachable absorption coefficient
- $C$ — exposure rate constant
- $m$ — normalized PAC concentration
8.4 Development Rate (Mack Model)
$$
r = r_{max} \frac{(a + 1)(1 - m)^n}{a + (1 - m)^n}
$$
Variables:
- $r$ — development rate
- $r_{max}$ — maximum development rate
- $m$ — normalized PAC concentration
- $a, n$ — resist contrast parameters
8.5 Computational Lithography
- Optical Proximity Correction (OPC): inverse problem to find mask patterns
- Source-Mask Optimization (SMO): co-optimize illumination and mask
- Inverse Lithography Technology (ILT): pixel-based mask optimization
9. Device Simulation (TCAD)
9.1 Poisson's Equation
$$
\nabla \cdot (\epsilon \nabla \psi) = -q(p - n + N_D^+ - N_A^-)
$$
Variables:
- $\psi$ — electrostatic potential
- $\epsilon$ — permittivity
- $q$ — elementary charge
- $n, p$ — electron and hole concentrations
- $N_D^+, N_A^-$ — ionized donor and acceptor concentrations
9.2 Carrier Continuity Equations
Electrons:
$$
\frac{\partial n}{\partial t} = \frac{1}{q} \nabla \cdot \mathbf{J}_n + G - R
$$
Holes:
$$
\frac{\partial p}{\partial t} = -\frac{1}{q} \nabla \cdot \mathbf{J}_p + G - R
$$
Variables:
- $\mathbf{J}_n, \mathbf{J}_p$ — electron and hole current densities
- $G$ — carrier generation rate
- $R$ — carrier recombination rate
9.3 Drift-Diffusion Current Equations
Electron current:
$$
\mathbf{J}_n = q n \mu_n \mathbf{E} + q D_n \nabla n
$$
Hole current:
$$
\mathbf{J}_p = q p \mu_p \mathbf{E} - q D_p \nabla p
$$
Einstein relation:
$$
D = \frac{k_B T}{q} \mu
$$
9.4 Advanced Transport Models
- Hydrodynamic model: includes carrier temperature
- Monte Carlo: tracks individual carrier scattering events
- Quantum corrections: density gradient, NEGF for tunneling
10. Yield Modeling
10.1 Poisson Yield Model
$$
Y = e^{-A D_0}
$$
Variables:
- $Y$ — chip yield
- $A$ — chip area
- $D_0$ — defect density ($\text{defects/cm}^2$)
10.2 Negative Binomial Model (Clustered Defects)
$$
Y = \left(1 + \frac{A D_0}{\alpha}\right)^{-\alpha}
$$
Variables:
- $\alpha$ — clustering parameter
- As $\alpha \to \infty$, reduces to Poisson model
10.3 Critical Area Analysis
$$
Y = \exp\left( -\sum_i D_i \cdot A_{c,i} \right)
$$
Variables:
- $D_i$ — defect density for defect type $i$
- $A_{c,i}$ — critical area sensitive to defect type $i$
Critical area depends on:
- Defect size distribution
- Layout geometry
- Defect type (shorts, opens, particles)
11. Statistical and Machine Learning Methods
11.1 Response Surface Methodology (RSM)
Second-order model:
$$
y = \beta_0 + \sum_{i=1}^{k} \beta_i x_i + \sum_{i=1}^{k} \beta_{ii} x_i^2 + \sum_{i
Matrix profile is an efficient data structure storing nearest neighbor distances for all subsequences enabling motif discovery and anomaly detection.
Maximum iterations limit agent loops preventing infinite execution.
Max tokens parameter limits total generation length.
Measure distribution difference.
Learn piecewise linear activation.
Average time until failure.
Average time to failure.
Means-ends analysis reduces differences between current and goal states through operator selection.
# Semiconductor Manufacturing Process Measurement Uncertainty: Mathematical Modeling ## 1. The Fundamental Challenge At modern nodes (3nm, 2nm), we face a profound problem: **measurement uncertainty can consume 30–50% of the tolerance budget**. Consider typical values: - Feature dimension: ~15nm - Tolerance: ±1nm (≈7% variation allowed) - Measurement repeatability: ~0.3–0.5nm - Reproducibility (tool-to-tool): additional 0.3–0.5nm This means we cannot naively interpret measured variation as process variation—a significant portion is measurement noise. ## 2. Variance Decomposition Framework The foundational mathematical structure is the decomposition of total observed variance: $$ \sigma^2_{\text{observed}} = \sigma^2_{\text{process}} + \sigma^2_{\text{measurement}} $$ ### 2.1 Hierarchical Decomposition For a full fab model: $$ Y_{ijklm} = \mu + L_i + W_{j(i)} + D_{k(ij)} + T_l + (LT)_{il} + \eta_{lm} + \epsilon_{ijklm} $$ Where: | Term | Meaning | Type | |------|---------|------| | $L_i$ | Lot effect | Random | | $W_{j(i)}$ | Wafer nested in lot | Random | | $D_{k(ij)}$ | Die/site within wafer | Random or systematic | | $T_l$ | Measurement tool | Random or fixed | | $(LT)_{il}$ | Lot × tool interaction | Random | | $\eta_{lm}$ | Tool drift/bias | Systematic | | $\epsilon_{ijklm}$ | Pure repeatability | Random | The variance components: $$ \text{Var}(Y) = \sigma^2_L + \sigma^2_W + \sigma^2_D + \sigma^2_T + \sigma^2_{LT} + \sigma^2_\eta + \sigma^2_\epsilon $$ **Measurement system variance:** $$ \sigma^2_{\text{meas}} = \sigma^2_T + \sigma^2_\eta + \sigma^2_\epsilon $$ ## 3. Gauge R&R Mathematics The standard Gauge Repeatability and Reproducibility analysis partitions measurement variance: $$ \sigma^2_{\text{meas}} = \sigma^2_{\text{repeatability}} + \sigma^2_{\text{reproducibility}} $$ ### 3.1 Key Metrics **Precision-to-Tolerance Ratio:** $$ \text{P/T} = \frac{k \cdot \sigma_{\text{meas}}}{\text{USL} - \text{LSL}} $$ where $k = 5.15$ (99% coverage) or $k = 6$ (99.73% coverage) **Discrimination Ratio:** $$ \text{ndc} = 1.41 \times \frac{\sigma_{\text{process}}}{\sigma_{\text{meas}}} $$ This gives the number of distinct categories the measurement system can reliably distinguish. - Industry standard requires: $\text{ndc} \geq 5$ **Signal-to-Noise Ratio:** $$ \text{SNR} = \frac{\sigma_{\text{process}}}{\sigma_{\text{meas}}} $$ ## 4. GUM-Based Uncertainty Propagation Following the Guide to the Expression of Uncertainty in Measurement (GUM): ### 4.1 Combined Standard Uncertainty For a measurand $y = f(x_1, x_2, \ldots, x_n)$: $$ u_c(y) = \sqrt{\sum_{i=1}^{n} \left(\frac{\partial f}{\partial x_i}\right)^2 u^2(x_i) + 2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n} \frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_j} u(x_i, x_j)} $$ ### 4.2 Type A vs. Type B Uncertainties **Type A** (statistical): $$ u_A(\bar{x}) = \frac{s}{\sqrt{n}} = \sqrt{\frac{1}{n(n-1)}\sum_{i=1}^{n}(x_i - \bar{x})^2} $$ **Type B** (other sources): - Calibration certificates: $u_B = \frac{U}{k}$ where $U$ is expanded uncertainty - Rectangular distribution (tolerance): $u_B = \frac{a}{\sqrt{3}}$ - Triangular distribution: $u_B = \frac{a}{\sqrt{6}}$ ## 5. Spatial Modeling of Within-Wafer Variation Within-wafer variation often has systematic spatial structure that must be separated from random measurement error. ### 5.1 Polynomial Surface Model (Zernike Polynomials) $$ z(r, \theta) = \sum_{n=0}^{N}\sum_{m=-n}^{n} a_{nm} Z_n^m(r, \theta) $$ Using Zernike polynomials—natural for circular wafer geometry: - $Z_0^0$: piston (mean) - $Z_1^1$: tilt - $Z_2^0$: defocus (bowl shape) - Higher orders: astigmatism, coma, spherical aberration analogs ### 5.2 Gaussian Process Model For flexible, non-parametric spatial modeling: $$ z(\mathbf{s}) \sim \mathcal{GP}(m(\mathbf{s}), k(\mathbf{s}, \mathbf{s}')) $$ With squared exponential covariance: $$ k(\mathbf{s}_i, \mathbf{s}_j) = \sigma^2_f \exp\left(-\frac{\|\mathbf{s}_i - \mathbf{s}_j\|^2}{2\ell^2}\right) + \sigma^2_n \delta_{ij} $$ Where: - $\sigma^2_f$: process variance (spatial signal) - $\ell$: length scale (spatial correlation distance) - $\sigma^2_n$: measurement noise (nugget effect) **This naturally separates spatial process variation from measurement noise.** ## 6. Bayesian Hierarchical Modeling Bayesian approaches provide natural uncertainty quantification and handle small samples common in expensive semiconductor metrology. ### 6.1 Basic Hierarchical Model **Level 1** (within-wafer measurements): $$ y_{ij} \mid \theta_i, \sigma^2_{\text{meas}} \sim \mathcal{N}(\theta_i, \sigma^2_{\text{meas}}) $$ **Level 2** (wafer-to-wafer variation): $$ \theta_i \mid \mu, \sigma^2_{\text{proc}} \sim \mathcal{N}(\mu, \sigma^2_{\text{proc}}) $$ **Level 3** (hyperpriors): $$ \begin{aligned} \mu &\sim \mathcal{N}(\mu_0, \tau^2_0) \\ \sigma^2_{\text{meas}} &\sim \text{Inv-Gamma}(\alpha_m, \beta_m) \\ \sigma^2_{\text{proc}} &\sim \text{Inv-Gamma}(\alpha_p, \beta_p) \end{aligned} $$ ### 6.2 Posterior Inference The posterior distribution: $$ p(\mu, \sigma^2_{\text{proc}}, \sigma^2_{\text{meas}} \mid \mathbf{y}) \propto p(\mathbf{y} \mid \boldsymbol{\theta}, \sigma^2_{\text{meas}}) \cdot p(\boldsymbol{\theta} \mid \mu, \sigma^2_{\text{proc}}) \cdot p(\mu, \sigma^2_{\text{proc}}, \sigma^2_{\text{meas}}) $$ Solved via MCMC methods: - Gibbs sampling - Hamiltonian Monte Carlo (HMC) - No-U-Turn Sampler (NUTS) ## 7. Monte Carlo Uncertainty Propagation For complex, non-linear measurement models where analytical propagation fails: ### 7.1 Algorithm (GUM Supplement 1) 1. **Define** probability distributions for all input quantities $X_i$ 2. **Sample** $M$ realizations: $\{x_1^{(k)}, x_2^{(k)}, \ldots, x_n^{(k)}\}$ for $k = 1, \ldots, M$ 3. **Propagate** each sample: $y^{(k)} = f(x_1^{(k)}, \ldots, x_n^{(k)})$ 4. **Analyze** output distribution to obtain uncertainty Typically $M \geq 10^6$ for reliable coverage interval estimation. ### 7.2 Application: OCD (Optical CD) Metrology Scatterometry fits measured spectra to electromagnetic models with parameters: - CD (critical dimension) - Sidewall angle - Height - Layer thicknesses - Optical constants The measurement equation is highly non-linear: $$ \mathbf{R}_{\text{meas}} = \mathbf{R}_{\text{model}}(\text{CD}, \theta_{\text{swa}}, h, \mathbf{t}, \mathbf{n}, \mathbf{k}) + \boldsymbol{\epsilon} $$ Monte Carlo propagation captures correlations and non-linearities that linearized GUM misses. ## 8. The Deconvolution Problem Given observed data that is a convolution of true process variation and measurement noise: $$ f_{\text{obs}}(x) = (f_{\text{true}} * f_{\text{meas}})(x) = \int f_{\text{true}}(t) \cdot f_{\text{meas}}(x-t) \, dt $$ **Goal:** Recover $f_{\text{true}}$ given $f_{\text{obs}}$ and knowledge of $f_{\text{meas}}$. ### 8.1 Fourier Approach In frequency domain: $$ \hat{f}_{\text{obs}}(\omega) = \hat{f}_{\text{true}}(\omega) \cdot \hat{f}_{\text{meas}}(\omega) $$ Naively: $$ \hat{f}_{\text{true}}(\omega) = \frac{\hat{f}_{\text{obs}}(\omega)}{\hat{f}_{\text{meas}}(\omega)} $$ **Problem:** Ill-posed—small errors in $\hat{f}_{\text{obs}}$ amplified where $\hat{f}_{\text{meas}}$ is small. ### 8.2 Regularization Techniques **Tikhonov regularization:** $$ \hat{f}_{\text{true}} = \arg\min_f \left\{ \|f_{\text{obs}} - f * f_{\text{meas}}\|^2 + \lambda \|Lf\|^2 \right\} $$ **Bayesian approach:** $$ p(f_{\text{true}} \mid f_{\text{obs}}) \propto p(f_{\text{obs}} \mid f_{\text{true}}) \cdot p(f_{\text{true}}) $$ With appropriate priors (smoothness, non-negativity) to regularize the solution. ## 9. Virtual Metrology with Uncertainty Quantification Virtual metrology predicts measurements from process tool data, reducing physical sampling requirements. ### 9.1 Model Structure $$ \hat{y} = f(\mathbf{x}_{\text{FDC}}) + \epsilon $$ Where $\mathbf{x}_{\text{FDC}}$ = fault detection and classification data (temperatures, pressures, flows, RF power, etc.) ### 9.2 Uncertainty-Aware ML Approaches **Gaussian Process Regression:** Provides natural predictive uncertainty: $$ p(y^* \mid \mathbf{x}^*, \mathcal{D}) = \mathcal{N}(\mu^*, \sigma^{*2}) $$ $$ \mu^* = \mathbf{k}^{*T}(\mathbf{K} + \sigma^2_n\mathbf{I})^{-1}\mathbf{y} $$ $$ \sigma^{*2} = k(\mathbf{x}^*, \mathbf{x}^*) - \mathbf{k}^{*T}(\mathbf{K} + \sigma^2_n\mathbf{I})^{-1}\mathbf{k}^* $$ **Conformal Prediction:** Distribution-free prediction intervals: $$ \hat{C}(x) = \left[\hat{y}(x) - \hat{q}, \hat{y}(x) + \hat{q}\right] $$ Where $\hat{q}$ is calibrated on held-out data to guarantee coverage probability. ## 10. Control Chart Implications Measurement uncertainty affects statistical process control profoundly. ### 10.1 Inflated Control Limits Standard control chart limits: $$ \text{UCL} = \bar{\bar{x}} + 3\sigma_{\bar{x}} $$ But $\sigma_{\bar{x}}$ includes measurement variance: $$ \sigma^2_{\bar{x}} = \frac{\sigma^2_{\text{proc}} + \sigma^2_{\text{meas}}/n_{\text{rep}}}{n_{\text{sample}}} $$ ### 10.2 Adjusted Process Capability True process capability: $$ \hat{C}_p = \frac{\text{USL} - \text{LSL}}{6\hat{\sigma}_{\text{proc}}} $$ Must correct observed variance: $$ \hat{\sigma}^2_{\text{proc}} = \hat{\sigma}^2_{\text{obs}} - \hat{\sigma}^2_{\text{meas}} $$ > **Warning:** This can yield negative estimates if measurement variance dominates—indicating the measurement system is inadequate. ## 11. Multi-Tool Matching and Reference Frame ### 11.1 Tool-to-Tool Bias Model $$ y_{\text{tool}_k} = y_{\text{true}} + \beta_k + \epsilon_k $$ Where $\beta_k$ is systematic bias for tool $k$. ### 11.2 Mixed-Effects Formulation $$ Y_{ij} = \mu + \tau_i + t_j + \epsilon_{ij} $$ - $\tau_i$: true sample value (random) - $t_j$: tool effect (random or fixed) - $\epsilon_{ij}$: residual **REML (Restricted Maximum Likelihood)** estimation separates these components. ### 11.3 Traceability Chain $$ \text{SI unit} \xrightarrow{u_1} \text{NMI reference} \xrightarrow{u_2} \text{Fab golden tool} \xrightarrow{u_3} \text{Production tools} $$ Total reference uncertainty: $$ u_{\text{ref}} = \sqrt{u_1^2 + u_2^2 + u_3^2} $$ ## 12. Practical Uncertainty Budget Example For CD-SEM measurement of a 20nm line: | Source | Type | $u_i$ (nm) | Sensitivity | Contribution (nm²) | |--------|------|-----------|-------------|-------------------| | Repeatability | A | 0.25 | 1 | 0.0625 | | Tool matching | B | 0.30 | 1 | 0.0900 | | SEM calibration | B | 0.15 | 1 | 0.0225 | | Algorithm uncertainty | B | 0.20 | 1 | 0.0400 | | Edge definition model | B | 0.35 | 1 | 0.1225 | | Charging effects | B | 0.10 | 1 | 0.0100 | **Combined standard uncertainty:** $$ u_c = \sqrt{\sum u_i^2} = \sqrt{0.3475} \approx 0.59 \text{ nm} $$ **Expanded uncertainty** ($k=2$, 95% confidence): $$ U = k \cdot u_c = 2 \times 0.59 = 1.18 \text{ nm} $$ For a ±1nm tolerance, this means **P/T ≈ 60%**—marginally acceptable. ## 13. Key Takeaways The mathematical modeling of measurement uncertainty in semiconductor manufacturing requires: 1. **Hierarchical variance decomposition** (ANOVA, mixed models) to separate process from measurement variation 2. **Spatial statistics** (Gaussian processes, Zernike decomposition) for within-wafer systematic patterns 3. **Bayesian inference** for rigorous uncertainty quantification with limited samples 4. **Monte Carlo methods** for non-linear measurement models (OCD, model-based metrology) 5. **Deconvolution techniques** to recover true process distributions 6. **Machine learning with uncertainty** for virtual metrology ### The Fundamental Insight At nanometer scales, measurement uncertainty is not a nuisance to be ignored—it is a **primary object of study** that directly determines our ability to control and optimize semiconductor processes. ## Key Equations Quick Reference ### Variance Decomposition $$ \sigma^2_{\text{total}} = \sigma^2_{\text{process}} + \sigma^2_{\text{measurement}} $$ ### GUM Combined Uncertainty $$ u_c(y) = \sqrt{\sum_{i=1}^{n} c_i^2 u^2(x_i)} $$ where $c_i = \frac{\partial f}{\partial x_i}$ are sensitivity coefficients. ### Precision-to-Tolerance Ratio $$ \text{P/T} = \frac{6\sigma_{\text{meas}}}{\text{USL} - \text{LSL}} \times 100\% $$ ### Process Capability (Corrected) $$ C_{p,\text{true}} = \frac{\text{USL} - \text{LSL}}{6\sqrt{\sigma^2_{\text{obs}} - \sigma^2_{\text{meas}}}} $$ ## Notation Reference | Symbol | Description | |--------|-------------| | $\sigma^2$ | Variance | | $u$ | Standard uncertainty | | $U$ | Expanded uncertainty | | $k$ | Coverage factor | | $\mu$ | Population mean | | $\bar{x}$ | Sample mean | | $s$ | Sample standard deviation | | $n$ | Sample size | | $\mathcal{N}(\mu, \sigma^2)$ | Normal distribution | | $\mathcal{GP}$ | Gaussian Process | | $\text{USL}$, $\text{LSL}$ | Upper/Lower Specification Limits | | $C_p$, $C_{pk}$ | Process capability indices |
Measurement uncertainty quantifies doubt about measurement results from systematic and random errors.
Reverse engineer neural network algorithms.
Reverse-engineer neural network internals to understand computation at neuron/circuit level.
50th percentile lifetime.
Resolve medical acronyms.
Generate doctor-patient conversations.
Extract medical terms.
Analyze X-rays MRI CT scans.
Extract knowledge from papers.
Answer medical questions.
Draft clinical reports.
Identify medications mentioned.
Medusa adds multiple prediction heads generating several tokens per forward pass.
NVIDIA's framework for large language models.
Determine if specific data was in the training set.
Membrane filtration removes particles and microorganisms through size-exclusion using porous membranes.
Edit multiple facts simultaneously.
Retrieve from large external memory.
Memory consolidation transfers important information from working to long-term storage.
How models store information.
Networks with explicit external memory for facts.
Memory pools preallocate buffers reducing allocation overhead during inference.
Memory retrieval selects relevant past information based on current context.
Give agents long-term persistent memory across sessions using vector stores or databases.
Cache previous segments for longer context.
Memory update mechanisms in temporal GNNs maintain node states updated by events and read for predictions.
Use external memory for long videos.
Memory-bound operations are limited by data transfer rather than computation affecting actual deployment speed.
Methods to reduce memory usage.
Model merging combines weights from multiple fine-tuned models. Can get benefits of each without retraining.
Mesh generation creates explicit polygonal surface representations of 3D objects.
Long chain of method calls.
Message passing allows agents to send structured information to collaborators.
GNN framework using message passing.