physical reasoning,reasoning
Understand physical world dynamics.
758 technical terms and definitions
Understand physical world dynamics.
Optimize logic during physical design.
# 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 |
Incorporate known physics into learning.
Render using physical light transport.
Neural networks constrained by physics.
Multi-finger gate structure.
Consistency between perturbed inputs.
Precision of die placement.
Place components on board.
Characterize ferroelectric materials.
Find and redact sensitive personal data.
Remove personal information.
Detect and mask PII (names, emails, SSN) in inputs and outputs. Required for privacy compliance.
Small-scale fab for developing new processes.
Small production batch for validation.
Small-scale production before HVM.
Pilot tests validate improvements on small scale before full deployment.
Pin fin heat sinks use cylindrical protrusions providing omnidirectional airflow paths.
Pins arranged in grid.
Vector database service for embeddings.
Pinecone is managed vector database. Easy scaling.
Page-locked host memory for fast transfer.
pip installs Python packages. From PyPI or requirements.
Pipeline parallelism splits model into stages across GPUs. Microbatches keep pipeline full. Reduces bubble overhead.
Pipeline parallel splits layers across GPUs. Tensor parallel splits individual layers. Used for very large models.
Split model into stages each device handles one stage in pipeline.
Physical commonsense reasoning.
Physical Interaction QA tests physical commonsense understanding.
Sulfuric acid + hydrogen peroxide highly oxidizing for organics removal.
Reduce interconnect spacing.
Pitch is takt time multiplied by pack-out quantity defining material movement rhythm.
Distance between repeating features (line + space).
Translate via intermediate language.
Pivotal tuning fine-tunes generator around inverted latent code improving reconstruction.
Paired image-to-image translation.
Upscale decoded images.
Physical layout of gates and routing of connections.
Identify locations from images.
Place and Route positions cells and connects wires. Critical for timing and power. EDA tools automate this.
Precision of component location.
Components per hour.
Efficient screening design.
Plaid connects to bank accounts. Financial data API.
Plan generation creates action sequences to achieve specified objectives.
# Travel Plan Mathematical Modeling A comprehensive optimization framework for travel planning using applied mathematics ## 1️⃣ Problem Definition ### Goal **Maximize Travel Utility:** $$ \max \; U = \sum_{i=1}^{N} \Big( V_i - C_i - T_i - R_i \Big) $$ ### Variable Definitions - $V_i$ = experience value (fun, learning, novelty) - $C_i$ = monetary cost - $T_i$ = time cost - $R_i$ = risk / uncertainty penalty - $i$ = travel components (flight, hotel, transport, food, attractions) ## 2️⃣ Core Decision Variables ### ✈️ Transportation (Inter-city) | Variable | Meaning | |----------|---------| | $x_f$ | Air ticket choice | | $C_f$ | Ticket cost | | $T_f$ | Flight time + layover | | $R_f$ | Delay / cancellation risk | **Flight Utility Function:** $$ U_{\text{flight}} = V_f - C_f - \alpha T_f - \beta R_f $$ Where: - $\alpha$ = time penalty weight - $\beta$ = risk aversion weight ### 🏨 Accommodation | Variable | Meaning | |----------|---------| | $x_h$ | Hotel / Airbnb selection | | $C_h$ | Nightly cost | | $D_h$ | Distance to city center | | $S_h$ | Safety / comfort score | **Hotel Utility Function:** $$ U_{\text{hotel}} = S_h - C_h - \gamma D_h $$ Where: - $\gamma$ = distance penalty coefficient ## 3️⃣ In-City Transportation Graph Model Model the city as a **weighted graph** $G = (V, E)$: - **Nodes** $V$: hotel, POIs, restaurants, stations - **Edges** $E$: travel routes between nodes ### Edge Weight Formula $$ w = a \cdot (\text{time}) + b \cdot (\text{cost}) + c \cdot (\text{fatigue}) $$ ### Transport Modes (Edge Types) - 🚇 **Public Transport**: subway, metro, BART, Caltrain, bus - 🚕 **Ride-hailing**: Taxi, Uber, Waymo, Robotaxi - 🚗 **Car rental**: self-drive options - 🚶 **Walking**: pedestrian routes ### Path Optimization Problem $$ \min \sum_{(i,j) \in \text{path}} w_{ij} $$ **Objective:** Find the shortest / cheapest / easiest path ### Applicable Algorithms - **Dijkstra's Algorithm** — single-source shortest path - **A-star Search** — heuristic-guided pathfinding - **Time-dependent shortest path** — dynamic edge weights ## 4️⃣ Attraction & Experience Selection (Knapsack Model) ### Candidate Set A - 🏛️ Museum - 🐠 Aquarium - 🌳 Public Park - ⛰️ Hiking area / mountain - 🏖️ Beach - 🛍️ Shopping Mall / Galleria - 🌉 Port / Bridge - 🎬 Cinema - 💻 Tech tour (e.g., Mag 7 tour) - 🍽️ Restaurant ### Constraint Optimization **Objective:** $$ \max \sum_{i \in A} V_i \cdot x_i $$ **Subject to constraints:** $$ \sum_{i \in A} T_i \cdot x_i \leq T_{\text{daily}} $$ $$ \sum_{i \in A} C_i \cdot x_i \leq B_{\text{daily}} $$ $$ x_i \in \{0, 1\} \quad \forall i \in A $$ Where: - $x_i = 1$ if attraction $i$ is selected, $0$ otherwise - $T_{\text{daily}}$ = daily time budget - $B_{\text{daily}}$ = daily monetary budget **Model Type:** 0-1 Knapsack Problem (NP-hard, solvable via dynamic programming) ## 5️⃣ Food & Restaurant Utility Model ### Utility Function $$ U_{\text{food}} = Q - C - W $$ Where: - $Q$ = quality / rating score - $C$ = price (normalized) - $W$ = waiting time penalty ### Decision Strategy by Meal Type | Meal | Strategy | Priority | |------|----------|----------| | **Breakfast** | Low-cost | Minimize $C$ | | **Lunch** | Proximity-optimized | Minimize distance | | **Dinner** | High-value | Maximize $Q$ | ## 6️⃣ Shopping & Supermarket Model ### Shopping Categories | Type | Objective | |------|-----------| | **Grocery** | Minimize cost | | **Mall / Galleria** | Maximize variety | | **Souvenir** | Maximize meaning / weight ratio | ### Shopping Utility Function $$ U_{\text{shopping}} = \frac{\text{Utility}}{\text{Weight} \times \text{Price}} $$ Or equivalently: $$ U_{\text{shopping}} = \frac{u_i}{w_i \cdot p_i} $$ Where: - $u_i$ = utility/satisfaction from item $i$ - $w_i$ = weight of item $i$ - $p_i$ = price of item $i$ ## 7️⃣ Risk & Uncertainty Modeling ### Risk-Adjusted Utility $$ U' = U - \lambda \cdot \sigma $$ Where: - $\sigma$ = uncertainty variance (standard deviation) - $\lambda$ = risk aversion coefficient ### Risk Factors - ⛈️ **Weather risk** — affects beach, hiking activities - 🚌 **Transport reliability** — delays, cancellations - 👥 **Crowd density** — peak times, tourist seasons - 🛡️ **Safety zones** — neighborhood safety scores ### Expected Utility with Uncertainty $$ E[U] = \sum_{s \in S} p_s \cdot U_s $$ Where $S$ is the set of possible states and $p_s$ is the probability of state $s$. ## 8️⃣ Popular Apps as Optimization Tools | Function | App Type | Sub-problem Solved | |----------|----------|-------------------| | **Navigation** | Google Maps, Citymapper | Shortest path $\min \sum w_{ij}$ | | **Flights** | Google Flights | $\min (C_f + \alpha T_f)$ | | **Hotels** | Booking, Airbnb | $\max S_h - C_h$ | | **Transport** | Uber, Lyft, Robotaxi | Real-time $\min w$ | | **Food** | Yelp | $\max Q - C$ | | **Planning** | Notion, TripIt | Schedule coordination | **Key Insight:** Each app ≈ partial solver for one sub-problem in the overall optimization. ## 9️⃣ Daily Schedule as a Control System ### State Variables - $E(t)$ = Energy level at time $t$ - $B(t)$ = Budget remaining at time $t$ - $T(t)$ = Time remaining at time $t$ ### State Transition Equations **Energy dynamics:** $$ E(t+1) = E(t) - k_1 \cdot a_{\text{walk}}(t) + k_2 \cdot a_{\text{rest}}(t) $$ **Budget dynamics:** $$ B(t+1) = B(t) - \sum_{i} c_i \cdot a_i(t) $$ ### Control Actions $a(t)$ - 😴 **Rest** — increases $E(t)$, consumes $T(t)$ - 🚌 **Transit** — consumes $B(t)$, $T(t)$, moderate $E(t)$ drain - 🗺️ **Explore** — consumes all resources, maximizes experience - 🍴 **Eat** — consumes $B(t)$, restores $E(t)$ ## 🔟 Master Travel Planning Equation $$ \boxed{ \max_{\text{plan}} \; \sum_{\text{days}} \Big( \text{Experience} - \text{Cost} - \text{Time} - \text{Risk} - \text{Fatigue} \Big) } $$ ### Expanded Form $$ \max_{\{x_i\}} \sum_{d=1}^{D} \left[ \sum_{i \in A_d} V_i x_i - \sum_{i} C_i x_i - \sum_{(i,j)} w_{ij} - \lambda \sigma_d - f(E_d) \right] $$ Subject to: - Daily time constraints - Daily budget constraints - Energy feasibility constraints - Logical constraints (can't be in two places at once) ## Key Insight **Travel planning is applied mathematics:** | Mathematical Domain | Application | |--------------------|-------------| | **Graph Theory** | Transportation networks, routing | | **Optimization** | Budget/time allocation | | **Probability** | Risk assessment, uncertainty | | **Control Systems** | Energy/state management | | **Game Theory** | Trade-off decisions | ### The Fundamental Trade-off $$ \text{Maximize: } \quad \frac{\text{Total Experience Value}}{\text{Total Resources Spent}} $$ ## Symbol Reference | Symbol | Description | |--------|-------------| | $U$ | Total utility | | $V_i$ | Experience value of component $i$ | | $C_i$ | Cost of component $i$ | | $T_i$ | Time cost of component $i$ | | $R_i$ | Risk penalty of component $i$ | | $\alpha, \beta, \gamma$ | Weight coefficients | | $\lambda$ | Risk aversion coefficient | | $\sigma$ | Uncertainty (standard deviation) | | $E(t)$ | Energy at time $t$ | | $B(t)$ | Budget at time $t$ | | $x_i$ | Binary decision variable | | $w_{ij}$ | Edge weight from $i$ to $j$ |
Agent first creates a plan then executes steps potentially revising the plan.
How well CMP flattens topography.
Probabilistic Latent Network is a model-based agent that learns compact latent representations of visual observations for planning in latent space.
PlanetScale is serverless MySQL. Branching workflow.