photoemission imaging, failure analysis advanced
Photoemission imaging captures spatial distribution of light emission from reverse-biased junctions or gate oxide breakdown.
3,145 technical terms and definitions
Photoemission imaging captures spatial distribution of light emission from reverse-biased junctions or gate oxide breakdown.
Photoemission microscopy detects light emission from hot carriers at reverse-biased junctions localizing gate oxide shorts and junction leakage.
AI-enhanced 3D reconstruction from photos.
Image emitted photons from defects.
# Semiconductor Manufacturing Process: Physics-Based Modeling and Differential Equations A comprehensive reference for the physics and mathematics governing semiconductor fabrication processes. ## 1. Thermal Oxidation of Silicon ### 1.1 Deal-Grove Model The foundational model for silicon oxidation describes oxide thickness growth through coupled transport and reaction. **Governing Equation:** $$ x^2 + Ax = B(t + \tau) $$ **Parameter Definitions:** - $x$ — oxide thickness - $A = \frac{2D_{ox}}{k_s}$ — linear rate constant parameter (related to surface reaction) - $B = \frac{2D_{ox}C^*}{N_1}$ — parabolic rate constant (related to diffusion) - $D_{ox}$ — oxidant diffusivity through oxide - $k_s$ — surface reaction rate constant - $C^*$ — equilibrium oxidant concentration at gas-oxide interface - $N_1$ — number of oxidant molecules incorporated per unit volume of oxide - $\tau$ — time shift accounting for initial oxide ### 1.2 Underlying Diffusion Physics **Steady-state diffusion through the oxide:** $$ \frac{\partial C}{\partial t} = D_{ox}\frac{\partial^2 C}{\partial x^2} $$ **Boundary Conditions:** - **Gas-oxide interface (flux from gas phase):** $$ F_1 = h_g(C^* - C_0) $$ - **Si-SiO₂ interface (surface reaction):** $$ F_2 = k_s C_i $$ **Steady-state flux through the oxide:** $$ F = \frac{D_{ox}C^*}{1 + \frac{k_s}{h_g} + \frac{k_s x}{D_{ox}}} $$ ### 1.3 Limiting Growth Regimes | Regime | Condition | Growth Law | Physical Interpretation | |--------|-----------|------------|------------------------| | **Linear** | Thin oxide ($x \ll A$) | $x \approx \frac{B}{A}(t + \tau)$ | Reaction-limited | | **Parabolic** | Thick oxide ($x \gg A$) | $x \approx \sqrt{Bt}$ | Diffusion-limited | ## 2. Dopant Diffusion ### 2.1 Fick's Laws of Diffusion **First Law (Flux Equation):** $$ \vec{J} = -D\nabla C $$ **Second Law (Mass Conservation / Continuity):** $$ \frac{\partial C}{\partial t} = \nabla \cdot (D\nabla C) $$ **For constant diffusivity in 1D:** $$ \frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2} $$ ### 2.2 Analytical Solutions #### Constant Surface Concentration (Predeposition) Initial condition: $C(x, 0) = 0$ Boundary condition: $C(0, t) = C_s$ $$ C(x,t) = C_s \cdot \text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right) $$ where the complementary error function is: $$ \text{erfc}(z) = 1 - \text{erf}(z) = 1 - \frac{2}{\sqrt{\pi}}\int_0^z e^{-u^2} du $$ #### Fixed Dose / Drive-in (Gaussian Distribution) Initial condition: Delta function at surface with dose $Q$ $$ C(x,t) = \frac{Q}{\sqrt{\pi Dt}} \exp\left(-\frac{x^2}{4Dt}\right) $$ **Key Parameters:** - $Q$ — total dose per unit area (atoms/cm²) - $\sqrt{Dt}$ — diffusion length - Peak concentration: $C_{max} = \frac{Q}{\sqrt{\pi Dt}}$ ### 2.3 Concentration-Dependent Diffusion At high doping concentrations, diffusivity becomes concentration-dependent: $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x}\left[D(C)\frac{\partial C}{\partial x}\right] $$ **Fair-Tsai Model for Diffusivity:** $$ D = D_i + D^-\frac{n}{n_i} + D^+\frac{p}{n_i} + D^{++}\left(\frac{p}{n_i}\right)^2 $$ **Parameter Definitions:** - $D_i$ — intrinsic diffusivity (via neutral defects) - $D^-$ — diffusivity via negatively charged defects - $D^+$ — diffusivity via singly positive charged defects - $D^{++}$ — diffusivity via doubly positive charged defects - $n, p$ — electron and hole concentrations - $n_i$ — intrinsic carrier concentration ### 2.4 Point Defect Coupled Diffusion Modern TCAD uses coupled equations for dopants and point defects (vacancies $V$ and interstitials $I$): **Vacancy Continuity:** $$ \frac{\partial C_V}{\partial t} = D_V\nabla^2 C_V - k_{IV}C_V C_I + G_V - \frac{C_V - C_V^*}{\tau_V} $$ **Interstitial Continuity:** $$ \frac{\partial C_I}{\partial t} = D_I\nabla^2 C_I - k_{IV}C_V C_I + G_I - \frac{C_I - C_I^*}{\tau_I} $$ **Term Definitions:** - $D_V, D_I$ — diffusion coefficients for vacancies and interstitials - $k_{IV}$ — recombination rate constant for $V$-$I$ annihilation - $G_V, G_I$ — generation rates - $C_V^*, C_I^*$ — equilibrium concentrations - $\tau_V, \tau_I$ — lifetimes at sinks (surfaces, dislocations) **Effective Dopant Diffusivity:** $$ D_{eff} = f_I D_I \frac{C_I}{C_I^*} + f_V D_V \frac{C_V}{C_V^*} $$ where $f_I$ and $f_V$ are the interstitial and vacancy fractions for the specific dopant species. ## 3. Ion Implantation ### 3.1 Range Distribution (LSS Theory) The implanted dopant profile follows approximately a Gaussian distribution: $$ C(x) = \frac{\Phi}{\sqrt{2\pi}\Delta R_p} \exp\left[-\frac{(x - R_p)^2}{2\Delta R_p^2}\right] $$ **Parameters:** - $\Phi$ — dose (ions/cm²) - $R_p$ — projected range (mean implant depth) - $\Delta R_p$ — straggle (standard deviation of range distribution) **Higher-Order Moments (Pearson IV Distribution):** - $\gamma$ — skewness (asymmetry) - $\beta$ — kurtosis (peakedness) ### 3.2 Stopping Power (Energy Loss) The rate of energy loss as ions traverse the target: $$ \frac{dE}{dx} = -N[S_n(E) + S_e(E)] $$ **Components:** - $S_n(E)$ — nuclear stopping power (elastic collisions with target nuclei) - $S_e(E)$ — electronic stopping power (inelastic interactions with electrons) - $N$ — atomic density of target material (atoms/cm³) **LSS Electronic Stopping (Low Energy):** $$ S_e \propto \sqrt{E} $$ **Nuclear Stopping:** Uses screened Coulomb potentials with Thomas-Fermi or ZBL (Ziegler-Biersack-Littmark) universal screening functions. ### 3.3 Boltzmann Transport Equation For rigorous treatment (typically solved via Monte Carlo methods): $$ \frac{\partial f}{\partial t} + \vec{v} \cdot \nabla_r f + \frac{\vec{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{coll} $$ **Variables:** - $f(\vec{r}, \vec{v}, t)$ — particle distribution function - $\vec{F}$ — external force - Right-hand side — collision integral ### 3.4 Damage Accumulation **Kinchin-Pease Model:** $$ N_d = \frac{E_{damage}}{2E_d} $$ **Parameters:** - $N_d$ — number of displaced atoms - $E_{damage}$ — energy available for displacement - $E_d$ — displacement threshold energy ($\approx 15$ eV for silicon) ## 4. Chemical Vapor Deposition (CVD) ### 4.1 Coupled Transport Equations **Species Transport (Convection-Diffusion-Reaction):** $$ \frac{\partial C_i}{\partial t} + \vec{u} \cdot \nabla C_i = D_i\nabla^2 C_i + R_i $$ **Navier-Stokes Equations (Momentum):** $$ \rho\left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla\vec{u}\right) = -\nabla p + \mu\nabla^2\vec{u} + \rho\vec{g} $$ **Continuity Equation (Incompressible Flow):** $$ \nabla \cdot \vec{u} = 0 $$ **Energy Equation:** $$ \rho c_p\left(\frac{\partial T}{\partial t} + \vec{u} \cdot \nabla T\right) = k\nabla^2 T + Q_{reaction} $$ **Variable Definitions:** - $C_i$ — concentration of species $i$ - $\vec{u}$ — velocity vector - $D_i$ — diffusion coefficient of species $i$ - $R_i$ — net reaction rate for species $i$ - $\rho$ — density - $p$ — pressure - $\mu$ — dynamic viscosity - $c_p$ — specific heat at constant pressure - $k$ — thermal conductivity - $Q_{reaction}$ — heat of reaction ### 4.2 Surface Reaction Kinetics **Flux Balance at Wafer Surface:** $$ h_m(C_b - C_s) = k_s C_s $$ **Deposition Rate:** $$ G = \frac{k_s h_m C_b}{k_s + h_m} $$ **Parameters:** - $h_m$ — mass transfer coefficient - $k_s$ — surface reaction rate constant - $C_b$ — bulk gas concentration - $C_s$ — surface concentration **Limiting Cases:** | Regime | Condition | Rate Expression | Control Mechanism | |--------|-----------|-----------------|-------------------| | **Reaction-limited** | $k_s \ll h_m$ | $G \approx k_s C_b$ | Surface chemistry | | **Transport-limited** | $k_s \gg h_m$ | $G \approx h_m C_b$ | Mass transfer | ### 4.3 Step Coverage — Knudsen Diffusion In high-aspect-ratio features, molecular (Knudsen) flow dominates: $$ D_K = \frac{d}{3}\sqrt{\frac{8k_B T}{\pi m}} $$ **Parameters:** - $d$ — characteristic feature dimension - $k_B$ — Boltzmann constant - $T$ — temperature - $m$ — molecular mass **Thiele Modulus (Reaction-Diffusion Balance):** $$ \phi = L\sqrt{\frac{k_s}{D_K}} $$ **Interpretation:** - $\phi \ll 1$ — Reaction-limited → Conformal deposition - $\phi \gg 1$ — Diffusion-limited → Poor step coverage ## 5. Atomic Layer Deposition (ALD) ### 5.1 Surface Site Model **Precursor A Adsorption Kinetics:** $$ \frac{d\theta_A}{dt} = s_0 \frac{P_A}{\sqrt{2\pi m_A k_B T}}(1 - \theta_A) - k_{des}\theta_A $$ **Parameters:** - $\theta_A$ — fractional surface coverage of precursor A - $s_0$ — sticking coefficient - $P_A$ — partial pressure of precursor A - $m_A$ — molecular mass of precursor A - $k_{des}$ — desorption rate constant ### 5.2 Growth Per Cycle (GPC) $$ GPC = n_{sites} \cdot \Omega \cdot \theta_A^{sat} $$ **Parameters:** - $n_{sites}$ — surface site density (sites/cm²) - $\Omega$ — atomic volume (volume per deposited atom) - $\theta_A^{sat}$ — saturation coverage achieved during half-cycle ## 6. Plasma Etching ### 6.1 Plasma Fluid Equations **Electron Continuity:** $$ \frac{\partial n_e}{\partial t} + \nabla \cdot \vec{\Gamma}_e = S_{ionization} - S_{recomb} $$ **Ion Continuity:** $$ \frac{\partial n_i}{\partial t} + \nabla \cdot \vec{\Gamma}_i = S_{ionization} - S_{recomb} $$ **Drift-Diffusion Flux (Electrons):** $$ \vec{\Gamma}_e = -n_e\mu_e\vec{E} - D_e\nabla n_e $$ **Drift-Diffusion Flux (Ions):** $$ \vec{\Gamma}_i = n_i\mu_i\vec{E} - D_i\nabla n_i $$ **Poisson's Equation (Self-Consistent Field):** $$ \nabla^2\phi = -\frac{e}{\varepsilon_0}(n_i - n_e) $$ **Electron Energy Balance:** $$ \frac{\partial}{\partial t}\left(\frac{3}{2}n_e k_B T_e\right) + \nabla \cdot \vec{q}_e = -e\vec{\Gamma}_e \cdot \vec{E} - \sum_j \epsilon_j R_j $$ ### 6.2 Sheath Physics **Bohm Criterion (Sheath Edge Condition):** $$ u_i \geq u_B = \sqrt{\frac{k_B T_e}{M_i}} $$ **Child-Langmuir Law (Collisionless Sheath Ion Current):** $$ J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M_i}}\frac{V_0^{3/2}}{d^2} $$ **Parameters:** - $u_i$ — ion velocity at sheath edge - $u_B$ — Bohm velocity - $T_e$ — electron temperature - $M_i$ — ion mass - $V_0$ — sheath voltage drop - $d$ — sheath thickness ### 6.3 Surface Etch Kinetics **Ion-Enhanced Etching Rate:** $$ R_{etch} = Y_i\Gamma_i + Y_n\Gamma_n(1-\theta) + Y_{syn}\Gamma_i\theta $$ **Components:** - $Y_i\Gamma_i$ — physical sputtering contribution - $Y_n\Gamma_n(1-\theta)$ — spontaneous chemical etching - $Y_{syn}\Gamma_i\theta$ — ion-enhanced (synergistic) etching **Yield Parameters:** - $Y_i$ — physical sputtering yield - $Y_n$ — spontaneous chemical etch yield - $Y_{syn}$ — synergistic yield (ion-enhanced chemistry) - $\Gamma_i, \Gamma_n$ — ion and neutral fluxes - $\theta$ — fractional surface coverage of reactive species **Surface Coverage Dynamics:** $$ \frac{d\theta}{dt} = s\Gamma_n(1-\theta) - Y_{syn}\Gamma_i\theta - k_v\theta $$ **Terms:** - $s\Gamma_n(1-\theta)$ — adsorption onto empty sites - $Y_{syn}\Gamma_i\theta$ — consumption by ion-enhanced reaction - $k_v\theta$ — thermal desorption/volatilization ## 7. Lithography ### 7.1 Aerial Image Formation **Hopkins Formulation (Partially Coherent Imaging):** $$ I(x,y) = \iint TCC(f,g;f',g') \cdot \tilde{M}(f,g) \cdot \tilde{M}^*(f',g') \, df\,dg\,df'\,dg' $$ **Parameters:** - $TCC$ — Transmission Cross Coefficient (encapsulates partial coherence) - $\tilde{M}(f,g)$ — Fourier transform of mask transmission function - $f, g$ — spatial frequencies **Rayleigh Resolution Criterion:** $$ Resolution = k_1 \frac{\lambda}{NA} $$ **Depth of Focus:** $$ DOF = k_2 \frac{\lambda}{NA^2} $$ **Parameters:** - $k_1, k_2$ — process-dependent factors - $\lambda$ — exposure wavelength - $NA$ — numerical aperture ### 7.2 Photoresist Exposure — Dill Model **Intensity Attenuation with Photobleaching:** $$ \frac{\partial I}{\partial z} = -\alpha(M)I $$ where the absorption coefficient depends on PAC concentration: $$ \alpha = AM + B $$ **Photoactive Compound (PAC) Decomposition:** $$ \frac{\partial M}{\partial t} = -CIM $$ **Dill Parameters:** | Parameter | Description | Units | |-----------|-------------|-------| | $A$ | Bleachable absorption coefficient | μm⁻¹ | | $B$ | Non-bleachable absorption coefficient | μm⁻¹ | | $C$ | Exposure rate constant | cm²/mJ | | $M$ | Relative PAC concentration | dimensionless (0-1) | ### 7.3 Chemically Amplified Resists **Photoacid Generation:** $$ \frac{\partial [H^+]}{\partial t} = C \cdot I \cdot [PAG] $$ **Post-Exposure Bake — Acid Diffusion and Reaction:** $$ \frac{\partial [H^+]}{\partial t} = D_{acid}\nabla^2[H^+] - k_{loss}[H^+] $$ **Deprotection Reaction (Catalytic Amplification):** $$ \frac{\partial [Protected]}{\partial t} = -k_{cat}[H^+][Protected] $$ **Parameters:** - $[PAG]$ — photoacid generator concentration - $D_{acid}$ — acid diffusion coefficient - $k_{loss}$ — acid loss rate (neutralization, evaporation) - $k_{cat}$ — catalytic deprotection rate constant ### 7.4 Development Rate — Mack Model $$ R = R_{max}\frac{(a+1)(1-M)^n}{a + (1-M)^n} + R_{min} $$ **Parameters:** - $R_{max}$ — maximum development rate (fully exposed) - $R_{min}$ — minimum development rate (unexposed) - $a$ — selectivity parameter - $n$ — contrast parameter - $M$ — normalized PAC concentration after exposure ## 8. Epitaxy ### 8.1 Burton-Cabrera-Frank (BCF) Theory **Adatom Diffusion on Terraces:** $$ \frac{\partial n}{\partial t} = D_s\nabla^2 n + F - \frac{n}{\tau} $$ **Parameters:** - $n$ — adatom density on terrace - $D_s$ — surface diffusion coefficient - $F$ — deposition flux (atoms/cm²·s) - $\tau$ — adatom lifetime before desorption **Step Velocity:** $$ v_{step} = \Omega D_s\left[\left(\frac{\partial n}{\partial x}\right)_+ - \left(\frac{\partial n}{\partial x}\right)_-\right] $$ **Steady-State Solution for Step Flow:** $$ v_{step} = \frac{2D_s \lambda_s F}{l} \cdot \tanh\left(\frac{l}{2\lambda_s}\right) $$ **Parameters:** - $\Omega$ — atomic volume - $\lambda_s = \sqrt{D_s \tau}$ — surface diffusion length - $l$ — terrace width ### 8.2 Rate Equations for Island Nucleation **Monomer (Single Adatom) Density:** $$ \frac{dn_1}{dt} = F - 2\sigma_1 D_s n_1^2 - \sum_{j>1}\sigma_j D_s n_1 n_j - \frac{n_1}{\tau} $$ **Cluster of Size $j$:** $$ \frac{dn_j}{dt} = \sigma_{j-1}D_s n_1 n_{j-1} - \sigma_j D_s n_1 n_j $$ **Parameters:** - $n_j$ — density of clusters containing $j$ atoms - $\sigma_j$ — capture cross-section for clusters of size $j$ ## 9. Chemical Mechanical Polishing (CMP) ### 9.1 Preston Equation $$ MRR = K_p \cdot P \cdot V $$ **Parameters:** - $MRR$ — material removal rate (nm/min) - $K_p$ — Preston coefficient (material/process dependent) - $P$ — applied pressure - $V$ — relative velocity between pad and wafer ### 9.2 Contact Mechanics — Greenwood-Williamson Model **Real Contact Area:** $$ A_r = \pi \eta A_n R_p \int_d^\infty (z-d)\phi(z)dz $$ **Parameters:** - $\eta$ — asperity density - $A_n$ — nominal contact area - $R_p$ — asperity radius - $d$ — separation distance - $\phi(z)$ — asperity height distribution ### 9.3 Slurry Hydrodynamics — Reynolds Equation $$ \frac{\partial}{\partial x}\left(h^3\frac{\partial p}{\partial x}\right) + \frac{\partial}{\partial y}\left(h^3\frac{\partial p}{\partial y}\right) = 6\mu U\frac{\partial h}{\partial x} $$ **Parameters:** - $h$ — film thickness - $p$ — pressure - $\mu$ — dynamic viscosity - $U$ — sliding velocity ## 10. Thin Film Stress ### 10.1 Stoney Equation **Film Stress from Wafer Curvature:** $$ \sigma_f = \frac{E_s h_s^2}{6(1-\nu_s)h_f R} $$ **Parameters:** - $\sigma_f$ — film stress - $E_s$ — substrate Young's modulus - $\nu_s$ — substrate Poisson's ratio - $h_s$ — substrate thickness - $h_f$ — film thickness - $R$ — radius of curvature ### 10.2 Thermal Stress $$ \sigma_{th} = \frac{E_f}{1-\nu_f}(\alpha_s - \alpha_f)\Delta T $$ **Parameters:** - $E_f$ — film Young's modulus - $\nu_f$ — film Poisson's ratio - $\alpha_s, \alpha_f$ — thermal expansion coefficients (substrate, film) - $\Delta T$ — temperature change from deposition ## 11. Electromigration (Reliability) ### 11.1 Black's Equation (Empirical MTTF) $$ MTTF = A \cdot j^{-n} \cdot \exp\left(\frac{E_a}{k_B T}\right) $$ **Parameters:** - $MTTF$ — mean time to failure - $j$ — current density - $n$ — current density exponent (typically 1-2) - $E_a$ — activation energy - $A$ — material/geometry constant ### 11.2 Drift-Diffusion Model $$ \frac{\partial C}{\partial t} = \nabla \cdot \left[D\left(\nabla C - C\frac{Z^*e\rho \vec{j}}{k_B T}\right)\right] $$ **Parameters:** - $C$ — atomic concentration - $D$ — diffusion coefficient - $Z^*$ — effective charge number (wind force parameter) - $\rho$ — electrical resistivity - $\vec{j}$ — current density vector ### 11.3 Stress Evolution — Korhonen Model $$ \frac{\partial \sigma}{\partial t} = \frac{\partial}{\partial x}\left[\frac{D_a B\Omega}{k_B T}\left(\frac{\partial\sigma}{\partial x} + \frac{Z^*e\rho j}{\Omega}\right)\right] $$ **Parameters:** - $\sigma$ — hydrostatic stress - $D_a$ — atomic diffusivity - $B$ — effective bulk modulus - $\Omega$ — atomic volume ## 12. Numerical Solution Methods ### 12.1 Common Numerical Techniques | Method | Application | Strengths | |--------|-------------|-----------| | **Finite Difference (FDM)** | Regular grids, 1D/2D problems | Simple implementation, efficient | | **Finite Element (FEM)** | Complex geometries, stress analysis | Flexible meshing, boundary conditions | | **Monte Carlo** | Ion implantation, plasma kinetics | Statistical accuracy, handles randomness | | **Level Set** | Topography evolution (etch/deposition) | Handles topology changes | | **Kinetic Monte Carlo (KMC)** | Atomic-scale diffusion, nucleation | Captures rare events, atomic detail | ### 12.2 Discretization Examples **Explicit Forward Euler (1D Diffusion):** $$ C_i^{n+1} = C_i^n + \frac{D\Delta t}{(\Delta x)^2}\left(C_{i+1}^n - 2C_i^n + C_{i-1}^n\right) $$ **Stability Criterion:** $$ \frac{D\Delta t}{(\Delta x)^2} \leq \frac{1}{2} $$ **Implicit Backward Euler:** $$ C_i^{n+1} - \frac{D\Delta t}{(\Delta x)^2}\left(C_{i+1}^{n+1} - 2C_i^{n+1} + C_{i-1}^{n+1}\right) = C_i^n $$ ### 12.3 Major TCAD Software Tools - **Synopsys Sentaurus** — comprehensive process and device simulation - **Silvaco ATHENA/ATLAS** — process and device modeling - **COMSOL Multiphysics** — general multiphysics platform - **SRIM/TRIM** — ion implantation Monte Carlo - **PROLITH** — lithography simulation ## Processes and Governing Equations | Process | Primary Physics | Key Equation | |---------|-----------------|--------------| | **Oxidation** | Diffusion + Reaction | $x^2 + Ax = Bt$ | | **Diffusion** | Mass Transport | $\frac{\partial C}{\partial t} = D\nabla^2 C$ | | **Implantation** | Ballistic + Stopping | $\frac{dE}{dx} = -N(S_n + S_e)$ | | **CVD** | Transport + Kinetics | Navier-Stokes + Species | | **ALD** | Self-limiting Adsorption | Langmuir kinetics | | **Plasma Etch** | Plasma + Surface | Poisson + Drift-Diffusion | | **Lithography** | Wave Optics + Chemistry | Dill ABC model | | **Epitaxy** | Surface Diffusion | BCF theory | | **CMP** | Tribology + Chemistry | Preston equation | | **Stress** | Elasticity | Stoney equation | | **Electromigration** | Mass transport under current | Korhonen model |
Neural networks constrained by physics.
Consistency between perturbed inputs.
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Split model into stages each device handles one stage in pipeline.
Pivotal tuning fine-tunes generator around inverted latent code improving reconstruction.
Paired image-to-image translation.
Upscale decoded images.
Plaid connects to bank accounts. Financial data API.
Plan generation creates action sequences to achieve specified objectives.
Agent first creates a plan then executes steps potentially revising the plan.
Planned maintenance schedules preventive activities based on time condition or predictive indicators.
Scheduled downtime for maintenance.
Generate action sequences to achieve goals.
Plasma cleaning replaces wet chemical processes using ionized gases reducing waste and chemical usage.
Plasma decapsulation removes organic package materials using oxygen plasma with minimal die damage.
# Mathematical Modeling of Plasma Etching in Semiconductor Manufacturing ## Introduction Plasma etching is a critical process in semiconductor manufacturing where reactive gases are ionized to create a plasma, which selectively removes material from a wafer surface. The mathematical modeling of this process spans multiple physics domains: - **Electromagnetic theory** — RF power coupling and field distributions - **Statistical mechanics** — Particle distributions and kinetic theory - **Reaction kinetics** — Gas-phase and surface chemistry - **Transport phenomena** — Species diffusion and convection - **Surface science** — Etch mechanisms and selectivity ## Foundational Plasma Physics ### Boltzmann Transport Equation The most fundamental description of plasma behavior is the **Boltzmann transport equation**, governing the evolution of the particle velocity distribution function $f(\mathbf{r}, \mathbf{v}, t)$: $$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{\text{collision}} $$ **Where:** - $f(\mathbf{r}, \mathbf{v}, t)$ — Velocity distribution function - $\mathbf{v}$ — Particle velocity - $\mathbf{F}$ — External force (electromagnetic) - $m$ — Particle mass - RHS — Collision integral ### Fluid Moment Equations For computational tractability, velocity moments of the Boltzmann equation yield fluid equations: #### Continuity Equation (Mass Conservation) $$ \frac{\partial n}{\partial t} + \nabla \cdot (n\mathbf{u}) = S - L $$ **Where:** - $n$ — Species number density $[\text{m}^{-3}]$ - $\mathbf{u}$ — Drift velocity $[\text{m/s}]$ - $S$ — Source term (generation rate) - $L$ — Loss term (consumption rate) #### Momentum Conservation $$ \frac{\partial (nm\mathbf{u})}{\partial t} + \nabla \cdot (nm\mathbf{u}\mathbf{u}) + \nabla p = nq(\mathbf{E} + \mathbf{u} \times \mathbf{B}) - nm\nu_m \mathbf{u} $$ **Where:** - $p = nk_BT$ — Pressure - $q$ — Particle charge - $\mathbf{E}$, $\mathbf{B}$ — Electric and magnetic fields - $\nu_m$ — Momentum transfer collision frequency $[\text{s}^{-1}]$ #### Energy Conservation $$ \frac{\partial}{\partial t}\left(\frac{3}{2}nk_BT\right) + \nabla \cdot \mathbf{q} + p\nabla \cdot \mathbf{u} = Q_{\text{heating}} - Q_{\text{loss}} $$ **Where:** - $k_B = 1.38 \times 10^{-23}$ J/K — Boltzmann constant - $\mathbf{q}$ — Heat flux vector - $Q_{\text{heating}}$ — Power input (Joule heating, stochastic heating) - $Q_{\text{loss}}$ — Energy losses (collisions, radiation) ## Electromagnetic Field Coupling ### Maxwell's Equations For capacitively coupled plasma (CCP) and inductively coupled plasma (ICP) reactors: $$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$ $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t} $$ $$ \nabla \cdot \mathbf{D} = \rho $$ $$ \nabla \cdot \mathbf{B} = 0 $$ ### Plasma Conductivity The plasma current density couples through the complex conductivity: $$ \mathbf{J} = \sigma \mathbf{E} $$ For RF plasmas, the **complex conductivity** is: $$ \sigma = \frac{n_e e^2}{m_e(\nu_m + i\omega)} $$ **Where:** - $n_e$ — Electron density - $e = 1.6 \times 10^{-19}$ C — Elementary charge - $m_e = 9.1 \times 10^{-31}$ kg — Electron mass - $\omega$ — RF angular frequency - $\nu_m$ — Electron-neutral collision frequency ### Power Deposition Time-averaged power density deposited into the plasma: $$ P = \frac{1}{2}\text{Re}(\mathbf{J} \cdot \mathbf{E}^*) $$ **Typical values:** - CCP: $0.1 - 1$ W/cm³ - ICP: $0.5 - 5$ W/cm³ ## Plasma Sheath Physics The sheath is a thin, non-neutral region at the plasma-wafer interface that accelerates ions toward the surface, enabling anisotropic etching. ### Bohm Criterion Minimum ion velocity entering the sheath: $$ u_i \geq u_B = \sqrt{\frac{k_B T_e}{M_i}} $$ **Where:** - $u_B$ — Bohm velocity - $T_e$ — Electron temperature (typically 2–5 eV) - $M_i$ — Ion mass **Example:** For Ar⁺ ions with $T_e = 3$ eV: $$ u_B = \sqrt{\frac{3 \times 1.6 \times 10^{-19}}{40 \times 1.67 \times 10^{-27}}} \approx 2.7 \text{ km/s} $$ ### Child-Langmuir Law For a collisionless sheath, the ion current density is: $$ J = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{M_i}} \cdot \frac{V_s^{3/2}}{d^2} $$ **Where:** - $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m — Vacuum permittivity - $V_s$ — Sheath voltage drop (typically 10–500 V) - $d$ — Sheath thickness ### Sheath Thickness The sheath thickness scales as: $$ d \approx \lambda_D \left(\frac{2eV_s}{k_BT_e}\right)^{3/4} $$ **Where** the Debye length is: $$ \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} $$ ### Ion Angular Distribution Ions arrive at the wafer with an angular distribution: $$ f(\theta) \propto \exp\left(-\frac{\theta^2}{2\sigma^2}\right) $$ **Where:** $$ \sigma \approx \arctan\left(\sqrt{\frac{k_B T_i}{eV_s}}\right) $$ **Typical values:** $\sigma \approx 2°–5°$ for high-bias conditions. ## Electron Energy Distribution Function ### Non-Maxwellian Distributions In low-pressure plasmas (1–100 mTorr), the EEDF deviates from Maxwellian. #### Two-Term Approximation The EEDF is expanded as: $$ f(\varepsilon, \theta) = f_0(\varepsilon) + f_1(\varepsilon)\cos\theta $$ The isotropic part $f_0$ satisfies: $$ \frac{d}{d\varepsilon}\left[\varepsilon D \frac{df_0}{d\varepsilon} + \left(V + \frac{\varepsilon\nu_{\text{inel}}}{\nu_m}\right)f_0\right] = 0 $$ #### Common Distribution Functions | Distribution | Functional Form | Applicability | |-------------|-----------------|---------------| | **Maxwellian** | $f(\varepsilon) \propto \sqrt{\varepsilon} \exp\left(-\frac{\varepsilon}{k_BT_e}\right)$ | High pressure, collisional | | **Druyvesteyn** | $f(\varepsilon) \propto \sqrt{\varepsilon} \exp\left(-\left(\frac{\varepsilon}{k_BT_e}\right)^2\right)$ | Elastic collisions dominant | | **Bi-Maxwellian** | Sum of two Maxwellians | Hot tail population | ### Generalized Form $$ f(\varepsilon) \propto \sqrt{\varepsilon} \cdot \exp\left[-\left(\frac{\varepsilon}{k_BT_e}\right)^x\right] $$ - $x = 1$ → Maxwellian - $x = 2$ → Druyvesteyn ## Plasma Chemistry and Reaction Kinetics ### Species Balance Equation For species $i$: $$ \frac{\partial n_i}{\partial t} + \nabla \cdot \mathbf{\Gamma}_i = \sum_j R_j $$ **Where:** - $\mathbf{\Gamma}_i$ — Species flux - $R_j$ — Reaction rates ### Electron-Impact Rate Coefficients Rate coefficients are calculated by integration over the EEDF: $$ k = \int_0^\infty \sigma(\varepsilon) v(\varepsilon) f(\varepsilon) \, d\varepsilon = \langle \sigma v \rangle $$ **Where:** - $\sigma(\varepsilon)$ — Energy-dependent cross-section $[\text{m}^2]$ - $v(\varepsilon) = \sqrt{2\varepsilon/m_e}$ — Electron velocity - $f(\varepsilon)$ — Normalized EEDF ### Heavy-Particle Reactions Arrhenius kinetics for neutral reactions: $$ k = A T^n \exp\left(-\frac{E_a}{k_BT}\right) $$ **Where:** - $A$ — Pre-exponential factor - $n$ — Temperature exponent - $E_a$ — Activation energy ### Example: SF₆/O₂ Plasma Chemistry #### Electron-Impact Reactions | Reaction | Type | Threshold | |----------|------|-----------| | $e + \text{SF}_6 \rightarrow \text{SF}_5 + \text{F} + e$ | Dissociation | ~10 eV | | $e + \text{SF}_6 \rightarrow \text{SF}_6^-$ | Attachment | ~0 eV | | $e + \text{SF}_6 \rightarrow \text{SF}_5^+ + \text{F} + 2e$ | Ionization | ~16 eV | | $e + \text{O}_2 \rightarrow \text{O} + \text{O} + e$ | Dissociation | ~6 eV | #### Gas-Phase Reactions - $\text{F} + \text{O} \rightarrow \text{FO}$ (reduces F atom density) - $\text{SF}_5 + \text{F} \rightarrow \text{SF}_6$ (recombination) - $\text{O} + \text{CF}_3 \rightarrow \text{COF}_2 + \text{F}$ (polymer removal) #### Surface Reactions - $\text{F} + \text{Si}(s) \rightarrow \text{SiF}_{(\text{ads})}$ - $\text{SiF}_{(\text{ads})} + 3\text{F} \rightarrow \text{SiF}_4(g)$ (volatile product) ## Transport Phenomena ### Drift-Diffusion Model For charged species, the flux is: $$ \mathbf{\Gamma} = \pm \mu n \mathbf{E} - D \nabla n $$ **Where:** - Upper sign: positive ions - Lower sign: electrons - $\mu$ — Mobility $[\text{m}^2/(\text{V}\cdot\text{s})]$ - $D$ — Diffusion coefficient $[\text{m}^2/\text{s}]$ ### Einstein Relation Connects mobility and diffusion: $$ D = \frac{\mu k_B T}{e} $$ ### Ambipolar Diffusion When quasi-neutrality holds ($n_e \approx n_i$): $$ D_a = \frac{\mu_i D_e + \mu_e D_i}{\mu_i + \mu_e} \approx D_i\left(1 + \frac{T_e}{T_i}\right) $$ Since $T_e \gg T_i$ typically: $D_a \approx D_i (1 + T_e/T_i) \approx 100 D_i$ ### Neutral Transport For reactive neutrals (radicals), Fickian diffusion: $$ \frac{\partial n}{\partial t} = D\nabla^2 n + S - L $$ #### Surface Boundary Condition $$ -D\frac{\partial n}{\partial x}\bigg|_{\text{surface}} = \frac{1}{4}\gamma n v_{\text{th}} $$ **Where:** - $\gamma$ — Sticking/reaction coefficient (0 to 1) - $v_{\text{th}} = \sqrt{\frac{8k_BT}{\pi m}}$ — Thermal velocity ### Knudsen Number Determines the appropriate transport regime: $$ \text{Kn} = \frac{\lambda}{L} $$ **Where:** - $\lambda$ — Mean free path - $L$ — Characteristic length | Kn Range | Regime | Model | |----------|--------|-------| | $< 0.01$ | Continuum | Navier-Stokes | | $0.01–0.1$ | Slip flow | Modified N-S | | $0.1–10$ | Transition | DSMC/BGK | | $> 10$ | Free molecular | Ballistic | ## Surface Reaction Modeling ### Langmuir Adsorption Kinetics For surface coverage $\theta$: $$ \frac{d\theta}{dt} = k_{\text{ads}}(1-\theta)P - k_{\text{des}}\theta - k_{\text{react}}\theta $$ **At steady state:** $$ \theta = \frac{k_{\text{ads}}P}{k_{\text{ads}}P + k_{\text{des}} + k_{\text{react}}} $$ ### Ion-Enhanced Etching The total etch rate combines multiple mechanisms: $$ \text{ER} = Y_{\text{chem}} \Gamma_n + Y_{\text{phys}} \Gamma_i + Y_{\text{syn}} \Gamma_i f(\theta) $$ **Where:** - $Y_{\text{chem}}$ — Chemical etch yield (isotropic) - $Y_{\text{phys}}$ — Physical sputtering yield - $Y_{\text{syn}}$ — Ion-enhanced (synergistic) yield - $\Gamma_n$, $\Gamma_i$ — Neutral and ion fluxes - $f(\theta)$ — Coverage-dependent function ### Ion Sputtering Yield #### Energy Dependence $$ Y(E) = A\left(\sqrt{E} - \sqrt{E_{\text{th}}}\right) \quad \text{for } E > E_{\text{th}} $$ **Typical threshold energies:** - Si: $E_{\text{th}} \approx 20$ eV - SiO₂: $E_{\text{th}} \approx 30$ eV - Si₃N₄: $E_{\text{th}} \approx 25$ eV #### Angular Dependence $$ Y(\theta) = Y(0) \cos^{-f}(\theta) \exp\left[-b\left(\frac{1}{\cos\theta} - 1\right)\right] $$ **Behavior:** - Increases from normal incidence - Peaks at $\theta \approx 60°–70°$ - Decreases at grazing angles (reflection dominates) ## Feature-Scale Profile Evolution ### Level Set Method The surface is represented as the zero contour of $\phi(\mathbf{x}, t)$: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ **Where:** - $\phi > 0$ — Material - $\phi < 0$ — Void/vacuum - $\phi = 0$ — Surface - $V_n$ — Local normal etch velocity ### Local Etch Rate Calculation The normal velocity $V_n$ depends on: 1. **Ion flux and angular distribution** $$\Gamma_i(\mathbf{x}) = \int f(\theta, E) \, d\Omega \, dE$$ 2. **Neutral flux** (with shadowing) $$\Gamma_n(\mathbf{x}) = \Gamma_{n,0} \cdot \text{VF}(\mathbf{x})$$ where VF is the view factor 3. **Surface chemistry state** $$V_n = f(\Gamma_i, \Gamma_n, \theta_{\text{coverage}}, T)$$ ### Neutral Transport in High-Aspect-Ratio Features #### Clausing Transmission Factor For a tube of aspect ratio AR: $$ K \approx \frac{1}{1 + 0.5 \cdot \text{AR}} $$ #### View Factor Calculations For surface element $dA_1$ seeing $dA_2$: $$ F_{1 \rightarrow 2} = \frac{1}{\pi} \int \frac{\cos\theta_1 \cos\theta_2}{r^2} \, dA_2 $$ ## Monte Carlo Methods ### Test-Particle Monte Carlo Algorithm ``` 1. SAMPLE incident particle from flux distribution at feature opening - Ion: from IEDF and IADF - Neutral: from Maxwellian 2. TRACE trajectory through feature - Ion: ballistic, solve equation of motion - Neutral: random walk with wall collisions 3. DETERMINE reaction at surface impact - Sample from probability distribution - Update surface coverage if adsorption 4. UPDATE surface geometry - Remove material (etching) - Add material (deposition) 5. REPEAT for statistically significant sample ``` ### Ion Trajectory Integration Through the sheath/feature: $$ m\frac{d^2\mathbf{r}}{dt^2} = q\mathbf{E}(\mathbf{r}) $$ **Numerical integration:** Velocity-Verlet or Boris algorithm ### Collision Sampling Null-collision method for efficiency: $$ P_{\text{collision}} = 1 - \exp(-\nu_{\text{max}} \Delta t) $$ **Where** $\nu_{\text{max}}$ is the maximum possible collision frequency. ## Multi-Scale Modeling Framework ### Scale Hierarchy | Scale | Length | Time | Physics | Method | |-------|--------|------|---------|--------| | **Reactor** | cm–m | ms–s | Plasma transport, EM fields | Fluid PDE | | **Sheath** | µm–mm | µs–ms | Ion acceleration, EEDF | Kinetic/Fluid | | **Feature** | nm–µm | ns–ms | Profile evolution | Level set/MC | | **Atomic** | Å–nm | ps–ns | Reaction mechanisms | MD/DFT | ### Coupling Approaches #### Hierarchical (One-Way) ``` Atomic scale → Surface parameters ↓ Feature scale ← Fluxes from reactor scale ↓ Reactor scale → Process outputs ``` #### Concurrent (Two-Way) - Feature-scale results feed back to reactor scale - Requires iterative solution - Computationally expensive ## Numerical Methods and Challenges ### Stiff ODE Systems Plasma chemistry involves timescales spanning many orders of magnitude: | Process | Timescale | |---------|-----------| | Electron attachment | $\sim 10^{-10}$ s | | Ion-molecule reactions | $\sim 10^{-6}$ s | | Metastable decay | $\sim 10^{-3}$ s | | Surface diffusion | $\sim 10^{-1}$ s | #### Implicit Methods Required **Backward Differentiation Formula (BDF):** $$ y_{n+1} = \sum_{j=0}^{k-1} \alpha_j y_{n-j} + h\beta f(t_{n+1}, y_{n+1}) $$ ### Spatial Discretization #### Finite Volume Method Ensures mass conservation: $$ \int_V \frac{\partial n}{\partial t} dV + \oint_S \mathbf{\Gamma} \cdot d\mathbf{S} = \int_V S \, dV $$ #### Mesh Requirements - Sheath resolution: $\Delta x < \lambda_D$ - RF skin depth: $\Delta x < \delta$ - Adaptive mesh refinement (AMR) common ### EM-Plasma Coupling **Iterative scheme:** 1. Solve Maxwell's equations for $\mathbf{E}$, $\mathbf{B}$ 2. Update plasma transport (density, temperature) 3. Recalculate $\sigma$, $\varepsilon_{\text{plasma}}$ 4. Repeat until convergence ## Advanced Topics ### Atomic Layer Etching (ALE) Self-limiting reactions for atomic precision: $$ \text{EPC} = \Theta \cdot d_{\text{ML}} $$ **Where:** - EPC — Etch per cycle - $\Theta$ — Modified layer coverage fraction - $d_{\text{ML}}$ — Monolayer thickness #### ALE Cycle 1. **Modification step:** Reactive gas creates modified surface layer $$\frac{d\Theta}{dt} = k_{\text{mod}}(1-\Theta)P_{\text{gas}}$$ 2. **Removal step:** Ion bombardment removes modified layer only $$\text{ER} = Y_{\text{mod}}\Gamma_i\Theta$$ ### Pulsed Plasma Dynamics Time-modulated RF introduces: - **Active glow:** Plasma on, high ion/radical generation - **Afterglow:** Plasma off, selective chemistry #### Ion Energy Modulation By pulsing bias: $$ \langle E_i \rangle = \frac{1}{T}\left[\int_0^{t_{\text{on}}} E_{\text{high}}dt + \int_{t_{\text{on}}}^{T} E_{\text{low}}dt\right] $$ ### High-Aspect-Ratio Etching (HAR) For AR > 50 (memory, 3D NAND): **Challenges:** - Ion angular broadening → bowing - Neutral depletion at bottom - Feature charging → twisting - Mask erosion → tapering **Ion Angular Distribution Broadening:** $$ \sigma_{\text{effective}} = \sqrt{\sigma_{\text{sheath}}^2 + \sigma_{\text{scattering}}^2} $$ **Neutral Flux at Bottom:** $$ \Gamma_{\text{bottom}} \approx \Gamma_{\text{top}} \cdot K(\text{AR}) $$ ### Machine Learning Integration **Applications:** - Surrogate models for fast prediction - Process optimization (Bayesian) - Virtual metrology - Anomaly detection **Physics-Informed Neural Networks (PINNs):** $$ \mathcal{L} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{\text{physics}} $$ Where $\mathcal{L}_{\text{physics}}$ enforces governing equations. ## Validation and Experimental Techniques ### Plasma Diagnostics | Technique | Measurement | Typical Values | |-----------|-------------|----------------| | **Langmuir probe** | $n_e$, $T_e$, EEDF | $10^{9}–10^{12}$ cm⁻³, 1–5 eV | | **OES** | Relative species densities | Qualitative/semi-quantitative | | **APMS** | Ion mass, energy | 1–500 amu, 0–500 eV | | **LIF** | Absolute radical density | $10^{11}–10^{14}$ cm⁻³ | | **Microwave interferometry** | $n_e$ (line-averaged) | $10^{10}–10^{12}$ cm⁻³ | ### Etch Characterization - **Profilometry:** Etch depth, uniformity - **SEM/TEM:** Feature profiles, sidewall angle - **XPS:** Surface composition - **Ellipsometry:** Film thickness, optical properties ### Model Validation Workflow 1. **Plasma validation:** Match $n_e$, $T_e$, species densities 2. **Flux validation:** Compare ion/neutral fluxes to wafer 3. **Etch rate validation:** Blanket wafer etch rates 4. **Profile validation:** Patterned feature cross-sections ## Key Dimensionless Numbers Summary | Number | Definition | Physical Meaning | |--------|------------|------------------| | **Knudsen** | $\text{Kn} = \lambda/L$ | Continuum vs. kinetic | | **Damköhler** | $\text{Da} = \tau_{\text{transport}}/\tau_{\text{reaction}}$ | Transport vs. reaction limited | | **Sticking coefficient** | $\gamma = \text{reactions}/\text{collisions}$ | Surface reactivity | | **Aspect ratio** | $\text{AR} = \text{depth}/\text{width}$ | Feature geometry | | **Debye number** | $N_D = n\lambda_D^3$ | Plasma ideality | ## Physical Constants | Constant | Symbol | Value | |----------|--------|-------| | Elementary charge | $e$ | $1.602 \times 10^{-19}$ C | | Electron mass | $m_e$ | $9.109 \times 10^{-31}$ kg | | Proton mass | $m_p$ | $1.673 \times 10^{-27}$ kg | | Boltzmann constant | $k_B$ | $1.381 \times 10^{-23}$ J/K | | Vacuum permittivity | $\varepsilon_0$ | $8.854 \times 10^{-12}$ F/m | | Vacuum permeability | $\mu_0$ | $4\pi \times 10^{-7}$ H/m |
Plate heat exchangers transfer heat between air streams through conductive metal plates.
Calibration method using logistic regression.
Plenoxels represent scenes as voxel grids enabling fast optimization without neural networks.
Multi-step diffusion sampler.
Steer generation using attribute classifiers.
Scheduled maintenance to prevent equipment failure.
Principal Neighborhood Aggregation uses multiple aggregators and scalers enhancing GNN expressiveness on graph tasks.
Point-E generates 3D point clouds from text using diffusion models.
Point-of-use abatement treats emissions at each process tool rather than centrally.
Pointwise convolutions use 1x1 kernels for channel mixing without spatial interaction.
Score each item independently.
Corrupt training data to degrade performance.
# Semiconductor Manufacturing Process: Poisson Statistics & Mathematical Modeling ## 1. Introduction: Why Poisson Statistics? Semiconductor defects satisfy the classical **Poisson conditions**: - **Rare events** — Defects are sparse relative to the total chip area - **Independence** — Defect occurrences are approximately independent - **Homogeneity** — Within local regions, defect rates are constant - **No simultaneity** — At infinitesimal scales, simultaneous defects have zero probability ### 1.1 The Poisson Probability Mass Function The probability of observing exactly $k$ defects: $$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$ where the expected number of defects is: $$ \lambda = D_0 \cdot A $$ **Parameter definitions:** - $D_0$ — Defect density (defects per unit area, typically defects/cm²) - $A$ — Chip area (cm²) - $\lambda$ — Mean number of defects per chip ### 1.2 Key Statistical Properties | Property | Formula | |----------|---------| | Mean | $E[X] = \lambda$ | | Variance | $\text{Var}(X) = \lambda$ | | Variance-to-Mean Ratio | $\frac{\text{Var}(X)}{E[X]} = 1$ | > **Note:** The equality of mean and variance (equidispersion) is a signature property of the Poisson distribution. Real semiconductor data often shows **overdispersion** (variance > mean), motivating compound models. ## 2. Fundamental Yield Equation ### 2.1 The Seeds Model (Simple Poisson) A chip is functional if and only if it has **zero killer defects**. Under Poisson assumptions: $$ \boxed{Y = P(X = 0) = e^{-D_0 A}} $$ **Derivation:** $$ P(X = 0) = \frac{\lambda^0 e^{-\lambda}}{0!} = e^{-\lambda} = e^{-D_0 A} $$ ### 2.2 Limitations of Simple Poisson - Assumes **uniform** defect density across the wafer (unrealistic) - Does not account for **clustering** of defects - Consistently **underestimates** yield for large chips - Ignores wafer-to-wafer and lot-to-lot variation ## 3. Compound Poisson Models ### 3.1 The Negative Binomial Approach Model the defect density $D_0$ as a **random variable** with Gamma distribution: $$ D_0 \sim \text{Gamma}\left(\alpha, \frac{\alpha}{\bar{D}}\right) $$ **Gamma probability density function:** $$ f(D_0) = \frac{(\alpha/\bar{D})^\alpha}{\Gamma(\alpha)} D_0^{\alpha-1} e^{-\alpha D_0/\bar{D}} $$ where: - $\bar{D}$ — Mean defect density - $\alpha$ — Clustering parameter (shape parameter) ### 3.2 Resulting Yield Model When defect density is Gamma-distributed, the defect count follows a **Negative Binomial** distribution, yielding: $$ \boxed{Y = \left(1 + \frac{D_0 A}{\alpha}\right)^{-\alpha}} $$ ### 3.3 Physical Interpretation of Clustering Parameter $\alpha$ | $\alpha$ Value | Physical Interpretation | |----------------|------------------------| | $\alpha \to \infty$ | Uniform defects — recovers simple Poisson model | | $\alpha \approx 1-5$ | Typical semiconductor clustering | | $\alpha \to 0$ | Extreme clustering — defects occur in tight groups | ### 3.4 Overdispersion The variance-to-mean ratio for the Negative Binomial: $$ \frac{\text{Var}(X)}{E[X]} = 1 + \frac{\bar{D}A}{\alpha} > 1 $$ This **overdispersion** (ratio > 1) matches empirical observations in semiconductor manufacturing. ## 4. Classical Yield Models ### 4.1 Comparison Table | Model | Yield Formula | Assumed Density Distribution | |-------|---------------|------------------------------| | Seeds (Poisson) | $Y = e^{-D_0 A}$ | Delta function (uniform) | | Murphy | $Y = \left(\frac{1 - e^{-D_0 A}}{D_0 A}\right)^2$ | Triangular | | Negative Binomial | $Y = \left(1 + \frac{D_0 A}{\alpha}\right)^{-\alpha}$ | Gamma | | Moore | $Y = e^{-\sqrt{D_0 A}}$ | Empirical | | Bose-Einstein | $Y = \frac{1}{1 + D_0 A}$ | Exponential | ### 4.2 Murphy's Yield Model Assumes triangular distribution of defect densities: $$ Y_{\text{Murphy}} = \left(\frac{1 - e^{-D_0 A}}{D_0 A}\right)^2 $$ **Taylor expansion for small $D_0 A$:** $$ Y_{\text{Murphy}} \approx 1 - \frac{(D_0 A)^2}{12} + O((D_0 A)^4) $$ ### 4.3 Limiting Behavior As $D_0 A \to 0$ (low defect density): $$ \lim_{D_0 A \to 0} Y = 1 \quad \text{(all models)} $$ As $D_0 A \to \infty$ (high defect density): $$ \lim_{D_0 A \to \infty} Y = 0 \quad \text{(all models)} $$ ## 5. Critical Area Analysis ### 5.1 Definition Not all chip area is equally vulnerable. **Critical area** $A_c$ is the region where a defect of size $d$ causes circuit failure. $$ A_c(d) = \int_{\text{layout}} \mathbf{1}\left[\text{defect at } (x,y) \text{ with size } d \text{ causes failure}\right] \, dx \, dy $$ ### 5.2 Critical Area for Shorts For two parallel conductors with: - Length: $L$ - Spacing: $S$ $$ A_c^{\text{short}}(d) = \begin{cases} 2L(d - S) & \text{if } d > S \\ 0 & \text{if } d \leq S \end{cases} $$ ### 5.3 Critical Area for Opens For a conductor with: - Width: $W$ - Length: $L$ $$ A_c^{\text{open}}(d) = \begin{cases} L(d - W) & \text{if } d > W \\ 0 & \text{if } d \leq W \end{cases} $$ ### 5.4 Total Critical Area Integrate over the defect size distribution $f(d)$: $$ A_c = \int_0^\infty A_c(d) \cdot f(d) \, dd $$ ### 5.5 Defect Size Distribution Typically modeled as **power-law**: $$ f(d) = C \cdot d^{-p} \quad \text{for } d \geq d_{\min} $$ **Typical values:** - Exponent: $p \approx 2-4$ - Normalization constant: $C = (p-1) \cdot d_{\min}^{p-1}$ **Alternative: Log-normal distribution** (common for particle contamination): $$ f(d) = \frac{1}{d \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln d - \mu)^2}{2\sigma^2}\right) $$ ## 6. Multi-Layer Yield Modeling ### 6.1 Modern IC Structure Modern integrated circuits have **10-15+ metal layers**. Each layer $i$ has: - Defect density: $D_i$ - Critical area: $A_{c,i}$ - Clustering parameter: $\alpha_i$ (for Negative Binomial) ### 6.2 Poisson Multi-Layer Yield $$ Y_{\text{total}} = \prod_{i=1}^{n} Y_i = \prod_{i=1}^{n} e^{-D_i A_{c,i}} $$ Simplified form: $$ \boxed{Y_{\text{total}} = \exp\left(-\sum_{i=1}^{n} D_i A_{c,i}\right)} $$ ### 6.3 Negative Binomial Multi-Layer Yield $$ \boxed{Y_{\text{total}} = \prod_{i=1}^{n} \left(1 + \frac{D_i A_{c,i}}{\alpha_i}\right)^{-\alpha_i}} $$ ### 6.4 Log-Yield Decomposition Taking logarithms for analysis: $$ \ln Y_{\text{total}} = -\sum_{i=1}^{n} D_i A_{c,i} \quad \text{(Poisson)} $$ $$ \ln Y_{\text{total}} = -\sum_{i=1}^{n} \alpha_i \ln\left(1 + \frac{D_i A_{c,i}}{\alpha_i}\right) \quad \text{(Negative Binomial)} $$ ## 7. Spatial Point Process Formulation ### 7.1 Inhomogeneous Poisson Process Intensity function $\lambda(x, y)$ varies spatially across the wafer: $$ P(k \text{ defects in region } R) = \frac{\Lambda(R)^k e^{-\Lambda(R)}}{k!} $$ where the integrated intensity is: $$ \Lambda(R) = \iint_R \lambda(x,y) \, dx \, dy $$ ### 7.2 Cox Process (Doubly Stochastic) The intensity $\lambda(x,y)$ is itself a **random field**: $$ \lambda(x,y) = \exp\left(\mu + Z(x,y)\right) $$ where: - $\mu$ — Baseline log-intensity - $Z(x,y)$ — Gaussian random field with spatial correlation function $\rho(h)$ **Correlation structure:** $$ \text{Cov}(Z(x_1, y_1), Z(x_2, y_2)) = \sigma^2 \rho(h) $$ where $h = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ ### 7.3 Neyman Type A (Cluster Process) Models defects occurring in clusters: 1. **Cluster centers:** Poisson process with intensity $\lambda_c$ 2. **Defects per cluster:** Poisson with mean $\mu$ 3. **Defect positions:** Scattered around cluster center (e.g., isotropic Gaussian) **Probability generating function:** $$ G(s) = \exp\left[\lambda_c A \left(e^{\mu(s-1)} - 1\right)\right] $$ **Mean and variance:** $$ E[N] = \lambda_c A \mu $$ $$ \text{Var}(N) = \lambda_c A \mu (1 + \mu) $$ ## 8. Statistical Estimation Methods ### 8.1 Maximum Likelihood Estimation #### 8.1.1 Data Structure Given: - $n$ chips with areas $A_1, A_2, \ldots, A_n$ - Binary outcomes $y_i \in \{0, 1\}$ (pass/fail) #### 8.1.2 Likelihood Function $$ \mathcal{L}(D_0, \alpha) = \prod_{i=1}^n Y_i^{y_i} (1 - Y_i)^{1-y_i} $$ where $Y_i = \left(1 + \frac{D_0 A_i}{\alpha}\right)^{-\alpha}$ #### 8.1.3 Log-Likelihood $$ \ell(D_0, \alpha) = \sum_{i=1}^n \left[y_i \ln Y_i + (1-y_i) \ln(1-Y_i)\right] $$ #### 8.1.4 Score Equations $$ \frac{\partial \ell}{\partial D_0} = 0, \quad \frac{\partial \ell}{\partial \alpha} = 0 $$ > **Note:** Requires numerical optimization (Newton-Raphson, BFGS, or EM algorithm). ### 8.2 Bayesian Estimation #### 8.2.1 Prior Distribution $$ D_0 \sim \text{Gamma}(a, b) $$ $$ \pi(D_0) = \frac{b^a}{\Gamma(a)} D_0^{a-1} e^{-b D_0} $$ #### 8.2.2 Posterior Distribution Given defect count $k$ on area $A$: $$ D_0 \mid k \sim \text{Gamma}(a + k, b + A) $$ **Posterior mean:** $$ \hat{D}_0 = \frac{a + k}{b + A} $$ **Posterior variance:** $$ \text{Var}(D_0 \mid k) = \frac{a + k}{(b + A)^2} $$ #### 8.2.3 Sequential Updating Bayesian framework enables sequential learning: $$ \text{Prior}_n \xrightarrow{\text{data } k_n} \text{Posterior}_n = \text{Prior}_{n+1} $$ ## 9. Statistical Process Control ### 9.1 c-Chart (Defect Counts) For **constant inspection area**: - **Center line:** $\bar{c}$ (average defect count) - **Upper Control Limit (UCL):** $\bar{c} + 3\sqrt{\bar{c}}$ - **Lower Control Limit (LCL):** $\max(0, \bar{c} - 3\sqrt{\bar{c}})$ ### 9.2 u-Chart (Defects per Unit Area) For **variable inspection area** $n_i$: $$ u_i = \frac{c_i}{n_i} $$ - **Center line:** $\bar{u}$ - **Control limits:** $\bar{u} \pm 3\sqrt{\frac{\bar{u}}{n_i}}$ ### 9.3 Overdispersion-Adjusted Charts For clustered defects (Negative Binomial), inflate the variance: $$ \text{UCL} = \bar{c} + 3\sqrt{\bar{c}\left(1 + \frac{\bar{c}}{\alpha}\right)} $$ $$ \text{LCL} = \max\left(0, \bar{c} - 3\sqrt{\bar{c}\left(1 + \frac{\bar{c}}{\alpha}\right)}\right) $$ ### 9.4 CUSUM Chart Cumulative sum for detecting small persistent shifts: $$ C_t^+ = \max(0, C_{t-1}^+ + (x_t - \mu_0 - K)) $$ $$ C_t^- = \max(0, C_{t-1}^- - (x_t - \mu_0 + K)) $$ where: - $K$ — Slack value (typically $0.5\sigma$) - Signal when $C_t^+$ or $C_t^-$ exceeds threshold $H$ ## 10. EUV Lithography Stochastic Effects ### 10.1 Photon Shot Noise At extreme ultraviolet wavelength (13.5 nm), **photon shot noise** becomes critical. Number of photons absorbed in resist volume $V$: $$ N \sim \text{Poisson}(\Phi \cdot \sigma \cdot V) $$ where: - $\Phi$ — Photon fluence (photons/area) - $\sigma$ — Absorption cross-section - $V$ — Resist volume ### 10.2 Line Edge Roughness (LER) Stochastic photon absorption causes spatial variation in resist exposure: $$ \sigma_{\text{LER}} \propto \frac{1}{\sqrt{\Phi \cdot V}} $$ **Critical Design Rule:** $$ \text{LER}_{3\sigma} < 0.1 \times \text{CD} $$ where CD = Critical Dimension (feature size) ### 10.3 Stochastic Printing Failures Probability of insufficient photons in a critical volume: $$ P(\text{failure}) = P(N < N_{\text{threshold}}) = \sum_{k=0}^{N_{\text{threshold}}-1} \frac{\lambda^k e^{-\lambda}}{k!} $$ where $\lambda = \Phi \sigma V$ ## 11. Reliability and Latent Defects ### 11.1 Defect Classification Not all defects cause immediate failure: - **Killer defects:** Cause immediate functional failure - **Latent defects:** May cause reliability failures over time $$ \lambda_{\text{total}} = \lambda_{\text{killer}} + \lambda_{\text{latent}} $$ ### 11.2 Yield vs. Reliability **Initial Yield:** $$ Y = e^{-\lambda_{\text{killer}} \cdot A} $$ **Reliability Function:** $$ R(t) = e^{-\lambda_{\text{latent}} \cdot A \cdot H(t)} $$ where $H(t)$ is the cumulative hazard function for latent defect activation. ### 11.3 Weibull Activation Model $$ H(t) = \left(\frac{t}{\eta}\right)^\beta $$ **Parameters:** - $\eta$ — Scale parameter (characteristic life) - $\beta$ — Shape parameter - $\beta < 1$: Decreasing failure rate (infant mortality) - $\beta = 1$: Constant failure rate (exponential) - $\beta > 1$: Increasing failure rate (wear-out) ## 12. Complete Mathematical Framework ### 12.1 Hierarchical Model Structure ``` - ┌─────────────────────────────────────────────────────────────┐ │ SEMICONDUCTOR YIELD MODEL HIERARCHY │ ├─────────────────────────────────────────────────────────────┤ │ │ │ Layer 1: DEFECT PHYSICS │ │ • Particle contamination │ │ • Process variation │ │ • Stochastic effects (EUV) │ │ ↓ │ │ Layer 2: SPATIAL POINT PROCESS │ │ • Inhomogeneous Poisson / Cox process │ │ • Defect size distribution: f(d) ∝ d^(-p) │ │ ↓ │ │ Layer 3: CRITICAL AREA CALCULATION │ │ • Layout-dependent geometry │ │ • Ac = ∫ Ac(d)$\cdot$f(d) dd │ │ ↓ │ │ Layer 4: YIELD MODEL │ │ • Y = (1 + D₀Ac/α)^(-α) │ │ • Multi-layer: Y = ∏ Yᵢ │ │ ↓ │ │ Layer 5: STATISTICAL INFERENCE │ │ • MLE / Bayesian estimation │ │ • SPC monitoring │ │ │ └─────────────────────────────────────────────────────────────┘ ``` ### 12.2 Summary of Key Equations | Concept | Equation | |---------|----------| | Poisson PMF | $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$ | | Simple Yield | $Y = e^{-D_0 A}$ | | Negative Binomial Yield | $Y = \left(1 + \frac{D_0 A}{\alpha}\right)^{-\alpha}$ | | Multi-Layer Yield | $Y = \prod_i \left(1 + \frac{D_i A_{c,i}}{\alpha_i}\right)^{-\alpha_i}$ | | Critical Area (shorts) | $A_c^{\text{short}}(d) = 2L(d-S)$ for $d > S$ | | Defect Size Distribution | $f(d) \propto d^{-p}$, $p \approx 2-4$ | | Bayesian Posterior | $D_0 \mid k \sim \text{Gamma}(a+k, b+A)$ | | Control Limits | $\bar{c} \pm 3\sqrt{\bar{c}(1 + \bar{c}/\alpha)}$ | | LER Scaling | $\sigma_{\text{LER}} \propto (\Phi V)^{-1/2}$ | ### 12.3 Typical Parameter Values | Parameter | Typical Range | Units | |-----------|---------------|-------| | Defect density $D_0$ | 0.01 - 1.0 | defects/cm² | | Clustering parameter $\alpha$ | 0.5 - 5 | dimensionless | | Defect size exponent $p$ | 2 - 4 | dimensionless | | Chip area $A$ | 1 - 800 | mm² |
Poisson yield model assumes random defects with constant density predicting yield as exponential of negative defect area product.
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