model versioning,mlops
Track different versions of trained models.
288 technical terms and definitions
Track different versions of trained models.
Embed secret signals in model to prove ownership or detect unauthorized use.
MAML applied to RL.
Fit geometric model to optical data.
Learn environment model to improve sample efficiency.
OpenAI's content moderation.
Continuous-valued Hopfield networks equivalent to attention.
Networks built from reusable modules.
Networks composed of specialized modules.
Moisture sensitivity levels classify packages by susceptibility to moisture-induced damage during soldering.
Failures from moisture ingress.
Predict binding pose of molecules.
Simulate atomic motion over time.
Generate molecular structures.
Predict properties from molecular structure.
Design novel molecular structures.
Molecule Optimization by Learned Embeddings with Rationales generates molecules through reinforcement learning and graph grammars.
MolGAN uses domain-specific rewards like drug-likeness and synthesizability to guide molecular generation.
GAN for molecular graphs.
MolGAN uses generative adversarial networks with graph convolutional discriminators and policy gradient generators for molecular graph generation.
Moments accountant tracks privacy loss through moment-generating functions.
Features corresponding to single concepts.
Use dropout at test time for uncertainty.
Circular fingerprints for molecules.
# MOSFET: Mathematical Modeling Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) Comprehensive equations, mathematical modeling, and process-parameter relationships 1. Fundamental Device Structure 1.1 MOSFET Components A MOSFET is a four-terminal semiconductor device consisting of: - Source (S) : Heavily doped region where carriers originate - Drain (D) : Heavily doped region where carriers are collected - Gate (G) : Control electrode separated from channel by dielectric - Body/Substrate (B) : Semiconductor bulk (p-type for NMOS, n-type for PMOS) 1.2 Operating Principle The gate voltage modulates channel conductivity through field effect: $$ \text{Gate Voltage} \rightarrow \text{Electric Field} \rightarrow \text{Channel Formation} \rightarrow \text{Current Flow} $$ 1.3 Device Types | Type | Substrate | Channel Carriers | Threshold | |------|-----------|------------------|-----------| | NMOS | p-type | Electrons | $V_{th} > 0$ (enhancement) | | PMOS | n-type | Holes | $V_{th} < 0$ (enhancement) | 2. Core MOSFET Equations 2.1 Threshold Voltage The threshold voltage $V_{th}$ determines device turn-on and is highly process-dependent: $$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_{Si} \cdot q \cdot N_A \cdot 2\phi_F}}{C_{ox}} $$ Component Equations - Flat-band voltage : $$ V_{FB} = \phi_{ms} - \frac{Q_{ox}}{C_{ox}} $$ - Fermi potential : $$ \phi_F = \frac{kT}{q} \ln\left(\frac{N_A}{n_i}\right) $$ - Oxide capacitance per unit area : $$ C_{ox} = \frac{\varepsilon_{ox}}{t_{ox}} = \frac{\kappa \cdot \varepsilon_0}{t_{ox}} $$ - Work function difference : $$ \phi_{ms} = \phi_m - \phi_s = \phi_m - \left(\chi + \frac{E_g}{2q} + \phi_F\right) $$ Parameter Definitions | Symbol | Description | Typical Value/Unit | |--------|-------------|-------------------| | $V_{FB}$ | Flat-band voltage | $-0.5$ to $-1.0$ V | | $\phi_F$ | Fermi potential | $0.3$ to $0.4$ V | | $\phi_{ms}$ | Work function difference | $-0.5$ to $-1.0$ V | | $C_{ox}$ | Oxide capacitance | $\sim 10^{-2}$ F/m² | | $Q_{ox}$ | Fixed oxide charge | $\sim 10^{10}$ q/cm² | | $N_A$ | Acceptor concentration | $10^{15}$ to $10^{18}$ cm⁻³ | | $n_i$ | Intrinsic carrier concentration | $1.5 \times 10^{10}$ cm⁻³ (Si, 300K) | | $\varepsilon_{Si}$ | Silicon permittivity | $11.7 \varepsilon_0$ | | $\varepsilon_{ox}$ | SiO₂ permittivity | $3.9 \varepsilon_0$ | 2.2 Drain Current Equations 2.2.1 Linear (Triode) Region Condition : $V_{DS} < V_{GS} - V_{th}$ (channel not pinched off) $$ I_D = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_{th}) V_{DS} - \frac{V_{DS}^2}{2} \right] $$ Simplified form (for small $V_{DS}$): $$ I_D \approx \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) V_{DS} $$ Channel resistance : $$ R_{ch} = \frac{V_{DS}}{I_D} = \frac{L}{\mu_n C_{ox} W (V_{GS} - V_{th})} $$ 2.2.2 Saturation Region Condition : $V_{DS} \geq V_{GS} - V_{th}$ (channel pinched off) $$ I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 (1 + \lambda V_{DS}) $$ Without channel-length modulation ($\lambda = 0$): $$ I_{D,sat} = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 $$ Saturation voltage : $$ V_{DS,sat} = V_{GS} - V_{th} $$ 2.2.3 Channel-Length Modulation The parameter $\lambda$ captures output resistance degradation: $$ \lambda = \frac{1}{L \cdot E_{crit}} \approx \frac{1}{V_A} $$ Output resistance : $$ r_o = \frac{\partial V_{DS}}{\partial I_D} = \frac{1}{\lambda I_D} = \frac{V_A + V_{DS}}{I_D} $$ Where $V_A$ is the Early voltage (typically $5$ to $50$ V/μm × L). 2.3 Subthreshold Conduction 2.3.1 Weak Inversion Current Condition : $V_{GS} < V_{th}$ (exponential behavior) $$ I_D = I_0 \exp\left(\frac{V_{GS} - V_{th}}{n \cdot V_T}\right) \left[1 - \exp\left(-\frac{V_{DS}}{V_T}\right)\right] $$ Characteristic current : $$ I_0 = \mu_n C_{ox} \frac{W}{L} (n-1) V_T^2 $$ Thermal voltage : $$ V_T = \frac{kT}{q} \approx 26 \text{ mV at } T = 300\text{K} $$ 2.3.2 Subthreshold Swing The subthreshold swing $S$ quantifies turn-off sharpness: $$ S = \frac{\partial V_{GS}}{\partial (\log_{10} I_D)} = n \cdot V_T \cdot \ln(10) = 2.3 \cdot n \cdot V_T $$ Numerical values : - Ideal minimum: $S_{min} = 60$ mV/decade (at 300K, $n = 1$) - Typical range: $S = 70$ to $100$ mV/decade - $n = 1 + \frac{C_{dep}}{C_{ox}}$ (subthreshold ideality factor) 2.3.3 Depletion Capacitance $$ C_{dep} = \frac{\varepsilon_{Si}}{W_{dep}} = \sqrt{\frac{q \varepsilon_{Si} N_A}{4 \phi_F}} $$ 2.4 Body Effect When source-to-body voltage $V_{SB} \neq 0$: $$ V_{th}(V_{SB}) = V_{th0} + \gamma \left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right) $$ Body effect coefficient : $$ \gamma = \frac{\sqrt{2 q \varepsilon_{Si} N_A}}{C_{ox}} $$ Typical values : $\gamma = 0.3$ to $1.0$ V$^{1/2}$ 2.5 Transconductance and Output Conductance 2.5.1 Transconductance Saturation region : $$ g_m = \frac{\partial I_D}{\partial V_{GS}} = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th}) = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D} $$ Alternative form : $$ g_m = \frac{2 I_D}{V_{GS} - V_{th}} $$ 2.5.2 Output Conductance $$ g_{ds} = \frac{\partial I_D}{\partial V_{DS}} = \lambda I_D = \frac{I_D}{V_A} $$ 2.5.3 Intrinsic Gain $$ A_v = \frac{g_m}{g_{ds}} = \frac{2}{\lambda(V_{GS} - V_{th})} = \frac{2 V_A}{V_{GS} - V_{th}} $$ 3. Short-Channel Effects 3.1 Velocity Saturation At high lateral electric fields ($E > E_{crit} \approx 10^4$ V/cm): $$ v_d = \frac{\mu_n E}{1 + E/E_{crit}} $$ Saturation velocity : $$ v_{sat} = \mu_n E_{crit} \approx 10^7 \text{ cm/s (electrons in Si)} $$ 3.1.1 Modified Saturation Current $$ I_{D,sat} = W C_{ox} v_{sat} (V_{GS} - V_{th}) $$ Note: Linear (not quadratic) dependence on gate overdrive. 3.1.2 Critical Length Velocity saturation dominates when: $$ L < L_{crit} = \frac{\mu_n (V_{GS} - V_{th})}{2 v_{sat}} $$ 3.2 Drain-Induced Barrier Lowering (DIBL) The drain field reduces the source-side barrier: $$ V_{th} = V_{th,long} - \eta \cdot V_{DS} $$ DIBL coefficient : $$ \eta = -\frac{\partial V_{th}}{\partial V_{DS}} $$ Typical values : $\eta = 20$ to $100$ mV/V for short channels 3.2.1 Modified Threshold Equation $$ V_{th}(V_{DS}, V_{SB}) = V_{th0} + \gamma(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}) - \eta V_{DS} $$ 3.3 Mobility Degradation 3.3.1 Vertical Field Effect $$ \mu_{eff} = \frac{\mu_0}{1 + \theta (V_{GS} - V_{th})} $$ Alternative form (surface roughness scattering): $$ \mu_{eff} = \frac{\mu_0}{1 + (\theta_1 + \theta_2 V_{SB})(V_{GS} - V_{th})} $$ 3.3.2 Universal Mobility Model $$ \mu_{eff} = \frac{\mu_0}{\left[1 + \left(\frac{E_{eff}}{E_0}\right)^\nu + \left(\frac{E_{eff}}{E_1}\right)^\beta\right]} $$ Where $E_{eff}$ is the effective vertical field: $$ E_{eff} = \frac{Q_b + \eta_s Q_i}{\varepsilon_{Si}} $$ 3.4 Hot Carrier Effects 3.4.1 Impact Ionization Current $$ I_{sub} = \frac{I_D}{M - 1} $$ Multiplication factor : $$ M = \frac{1}{1 - \int_0^{L_{dep}} \alpha(E) dx} $$ 3.4.2 Ionization Rate $$ \alpha = \alpha_\infty \exp\left(-\frac{E_{crit}}{E}\right) $$ 3.5 Gate Leakage 3.5.1 Direct Tunneling Current $$ J_g = A \cdot E_{ox}^2 \exp\left(-\frac{B}{\vert E_{ox} \vert}\right) $$ Where: $$ A = \frac{q^3}{16\pi^2 \hbar \phi_b} $$ $$ B = \frac{4\sqrt{2m^* \phi_b^3}}{3\hbar q} $$ 3.5.2 Gate Oxide Field $$ E_{ox} = \frac{V_{GS} - V_{FB} - \psi_s}{t_{ox}} $$ 4. Parameters 4.1 Gate Oxide Engineering 4.1.1 Oxide Capacitance $$ C_{ox} = \frac{\varepsilon_0 \cdot \kappa}{t_{ox}} $$ | Dielectric | $\kappa$ | EOT for $t_{phys} = 3$ nm | |------------|----------|---------------------------| | SiO₂ | 3.9 | 3.0 nm | | Si₃N₄ | 7.5 | 1.56 nm | | Al₂O₃ | 9 | 1.30 nm | | HfO₂ | 20-25 | 0.47-0.59 nm | | ZrO₂ | 25 | 0.47 nm | 4.1.2 Equivalent Oxide Thickness (EOT) $$ EOT = t_{high-\kappa} \times \frac{\varepsilon_{SiO_2}}{\varepsilon_{high-\kappa}} = t_{high-\kappa} \times \frac{3.9}{\kappa} $$ 4.1.3 Capacitance Equivalent Thickness (CET) Including quantum effects and poly depletion: $$ CET = EOT + \Delta t_{QM} + \Delta t_{poly} $$ Where: - $\Delta t_{QM} \approx 0.3$ to $0.5$ nm (quantum mechanical) - $\Delta t_{poly} \approx 0.3$ to $0.5$ nm (polysilicon depletion) 4.2 Channel Doping 4.2.1 Doping Profile Impact $$ V_{th} \propto \sqrt{N_A} $$ $$ \mu \propto \frac{1}{N_A^{0.3}} \text{ (ionized impurity scattering)} $$ 4.2.2 Depletion Width $$ W_{dep} = \sqrt{\frac{2\varepsilon_{Si}(2\phi_F + V_{SB})}{qN_A}} $$ 4.2.3 Junction Capacitance $$ C_j = C_{j0}\left(1 + \frac{V_R}{\phi_{bi}}\right)^{-m} $$ Where: - $C_{j0}$ = zero-bias capacitance - $\phi_{bi}$ = built-in potential - $m = 0.5$ (abrupt junction), $m = 0.33$ (graded junction) 4.3 Gate Material Engineering 4.3.1 Work Function Values | Gate Material | Work Function $\phi_m$ (eV) | Application | |--------------|----------------------------|-------------| | n+ Polysilicon | 4.05 | Legacy NMOS | | p+ Polysilicon | 5.15 | Legacy PMOS | | TiN | 4.5-4.7 | NMOS (midgap) | | TaN | 4.0-4.4 | NMOS | | TiAl | 4.2-4.3 | NMOS | | TiAlN | 4.7-4.8 | PMOS | 4.3.2 Flat-Band Voltage Engineering For symmetric CMOS threshold voltages: $$ V_{FB,NMOS} + V_{FB,PMOS} \approx -E_g/q $$ 4.4 Channel Length Scaling 4.4.1 Characteristic Length $$ \lambda = \sqrt{\frac{\varepsilon_{Si}}{\varepsilon_{ox}} \cdot t_{ox} \cdot x_j} $$ For good short-channel control: $L > 5\lambda$ to $10\lambda$ 4.4.2 Scale Length (FinFET/GAA) $$ \lambda_{GAA} = \sqrt{\frac{\varepsilon_{Si} \cdot t_{Si}^2}{2 \varepsilon_{ox} \cdot t_{ox}}} $$ 4.5 Strain Engineering 4.5.1 Mobility Enhancement $$ \mu_{strained} = \mu_0 (1 + \Pi \cdot \sigma) $$ Where: - $\Pi$ = piezoresistive coefficient - $\sigma$ = applied stress Enhancement factors : - NMOS (tensile): $+30\%$ to $+70\%$ mobility gain - PMOS (compressive): $+50\%$ to $+100\%$ mobility gain 4.5.2 Stress Impact on Threshold $$ \Delta V_{th} = \alpha_{th} \cdot \sigma $$ Where $\alpha_{th} \approx 1$ to $5$ mV/GPa 5. Advanced Compact Models 5.1 BSIM4 Model 5.1.1 Unified Current Equation $$ I_{DS} = I_{DS0} \cdot \left(1 + \frac{V_{DS} - V_{DS,eff}}{V_A}\right) \cdot \frac{1}{1 + R_S \cdot G_{DS0}} $$ 5.1.2 Effective Overdrive $$ V_{GS,eff} - V_{th} = \frac{2nV_T \cdot \ln\left[1 + \exp\left(\frac{V_{GS} - V_{th}}{2nV_T}\right)\right]}{1 + 2n\sqrt{\delta + \left(\frac{V_{GS}-V_{th}}{2nV_T} - \delta\right)^2}} $$ 5.1.3 Effective Saturation Voltage $$ V_{DS,eff} = V_{DS,sat} - \frac{V_T}{2}\ln\left(\frac{V_{DS,sat} + \sqrt{V_{DS,sat}^2 + 4V_T^2}}{V_{DS} + \sqrt{V_{DS}^2 + 4V_T^2}}\right) $$ 5.2 Surface Potential Model (PSP) 5.2.1 Implicit Surface Potential Equation $$ V_{GB} - V_{FB} = \psi_s + \gamma\sqrt{\psi_s + V_T e^{(\psi_s - 2\phi_F - V_{SB})/V_T} - V_T} $$ 5.2.2 Charge-Based Current $$ I_D = \mu W \frac{Q_i(0) - Q_i(L)}{L} \cdot \frac{V_{DS}}{V_{DS,eff}} $$ Where $Q_i$ is the inversion charge density: $$ Q_i = -C_{ox}\left[\psi_s - 2\phi_F - V_{ch} + V_T\left(e^{(\psi_s - 2\phi_F - V_{ch})/V_T} - 1\right)\right]^{1/2} $$ 5.3 FinFET Equations 5.3.1 Effective Width $$ W_{eff} = 2H_{fin} + W_{fin} $$ For multiple fins: $$ W_{total} = N_{fin} \cdot (2H_{fin} + W_{fin}) $$ 5.3.2 Multi-Gate Scale Length Double-gate : $$ \lambda_{DG} = \sqrt{\frac{\varepsilon_{Si} \cdot t_{Si} \cdot t_{ox}}{2\varepsilon_{ox}}} $$ Gate-all-around (GAA) : $$ \lambda_{GAA} = \sqrt{\frac{\varepsilon_{Si} \cdot r^2}{4\varepsilon_{ox}} \cdot \ln\left(1 + \frac{t_{ox}}{r}\right)} $$ Where $r$ = nanowire radius 5.3.3 FinFET Threshold Voltage $$ V_{th} = V_{FB} + 2\phi_F + \frac{qN_A W_{fin}}{2C_{ox}} - \Delta V_{th,SCE} $$ 6. Process-Equation Coupling 6.1 Parameter Sensitivity Analysis | Process Parameter | Primary Equations Affected | Sensitivity | |------------------|---------------------------|-------------| | $t_{ox}$ (oxide thickness) | $C_{ox}$, $V_{th}$, $I_D$, $g_m$ | High | | $N_A$ (channel doping) | $V_{th}$, $\gamma$, $\mu$, $W_{dep}$ | High | | $L$ (channel length) | $I_D$, SCE, $\lambda$ | Very High | | $W$ (channel width) | $I_D$, $g_m$ (linear) | Moderate | | Gate work function | $V_{FB}$, $V_{th}$ | High | | Junction depth $x_j$ | SCE, $R_{SD}$ | Moderate | | Strain level | $\mu$, $I_D$ | Moderate | 6.2 Variability Equations 6.2.1 Random Dopant Fluctuation (RDF) $$ \sigma_{V_{th}} = \frac{A_{VT}}{\sqrt{W \cdot L}} $$ Where $A_{VT}$ is the Pelgrom coefficient (typically $1$ to $5$ mV·μm). 6.2.2 Line Edge Roughness (LER) $$ \sigma_{V_{th,LER}} \propto \frac{\sigma_{LER}}{L} $$ 6.2.3 Oxide Thickness Variation $$ \sigma_{V_{th,tox}} = \frac{\partial V_{th}}{\partial t_{ox}} \cdot \sigma_{t_{ox}} = \frac{V_{th} - V_{FB} - 2\phi_F}{t_{ox}} \cdot \sigma_{t_{ox}} $$ 6.3 Equations: 6.3.1 Drive Current $$ I_{on} = \frac{W}{L} \cdot \mu_{eff} \cdot C_{ox} \cdot \frac{(V_{DD} - V_{th})^\alpha}{1 + (V_{DD} - V_{th})/E_{sat}L} $$ Where $\alpha = 2$ (long channel) or $\alpha \rightarrow 1$ (velocity saturated). 6.3.2 Leakage Current $$ I_{off} = I_0 \cdot \frac{W}{L} \cdot \exp\left(\frac{-V_{th}}{nV_T}\right) \cdot \left(1 - \exp\left(\frac{-V_{DD}}{V_T}\right)\right) $$ 6.3.3 CV/I Delay Metric $$ \tau = \frac{C_L \cdot V_{DD}}{I_{on}} \propto \frac{L^2}{\mu (V_{DD} - V_{th})} $$ Constants: | Constant | Symbol | Value | |----------|--------|-------| | Elementary charge | $q$ | $1.602 \times 10^{-19}$ C | | Boltzmann constant | $k$ | $1.381 \times 10^{-23}$ J/K | | Permittivity of free space | $\varepsilon_0$ | $8.854 \times 10^{-12}$ F/m | | Planck constant | $\hbar$ | $1.055 \times 10^{-34}$ J·s | | Electron mass | $m_0$ | $9.109 \times 10^{-31}$ kg | | Thermal voltage (300K) | $V_T$ | $25.9$ mV | | Silicon bandgap (300K) | $E_g$ | $1.12$ eV | | Intrinsic carrier conc. (Si) | $n_i$ | $1.5 \times 10^{10}$ cm⁻³ | Equations: Threshold Voltage $$ V_{th} = V_{FB} + 2\phi_F + \frac{\sqrt{2\varepsilon_{Si} q N_A (2\phi_F)}}{C_{ox}} $$ Linear Region Current $$ I_D = \mu C_{ox} \frac{W}{L} \left[(V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2}\right] $$ Saturation Current $$ I_D = \frac{1}{2}\mu C_{ox}\frac{W}{L}(V_{GS} - V_{th})^2(1 + \lambda V_{DS}) $$ Subthreshold Current $$ I_D = I_0 \exp\left(\frac{V_{GS} - V_{th}}{nV_T}\right) $$ Transconductance $$ g_m = \sqrt{2\mu C_{ox}\frac{W}{L}I_D} $$ Body Effect $$ V_{th} = V_{th0} + \gamma\left(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}\right) $$
Motion compensation aligns frames using estimated motion for temporal processing.
High-efficiency motors reduce electrical losses in fans pumps and compressors.
Movement pruning removes weights whose values move toward zero during training.
Message Passing Neural Network framework unifies many GNN architectures through generalized message passing operations.
Open-source commercially usable LLM.
MPT (MosaicML) is open source LLM. Now part of Databricks.
Multi-horizon Quantile Recurrent Neural Network produces probabilistic forecasts through quantile regression.
Manufacturing Resource Planning extends MRP to include capacity planning shop floor control and financial integration.
Material Requirements Planning calculates material quantities and timing based on production schedules BOMs and lead times.
Average time tool operates before failure.
Average time to fix tool after failure.
# Semiconductor Manufacturing Process: Multi-Physics Coupling & Mathematical Modeling ## 1. Overview: Why Multi-Physics Coupling Matters Semiconductor fabrication involves hundreds of process steps where multiple physical phenomena occur simultaneously and interact nonlinearly. At the 3nm node and below, these couplings become critical—small perturbations propagate across physics domains, affecting yield, uniformity, and device performance. ## 2. Key Processes and Their Coupled Physics ### 2.1 Plasma Etching (RIE, ICP, CCP) **Coupled domains:** - Electromagnetics (RF field, power deposition) - Plasma kinetics (electron/ion transport, sheath dynamics) - Neutral gas fluid dynamics - Gas-phase and surface chemistry - Heat transfer - Feature-scale transport and profile evolution **Coupling chain:** ``` RF Power → EM Fields → Electron Heating → Plasma Density → Sheath Voltage ↓ ↓ Ion Energy Distribution ← ─────────────────────────┘ ↓ Surface Bombardment + Radical Flux → Etch Rate & Profile ↓ Feature Geometry Evolution → Local Field Modification (feedback) ``` ### 2.2 Chemical Vapor Deposition (CVD/ALD) **Coupled domains:** - Fluid dynamics (often rarefied/transitional flow) - Heat transfer (convection, conduction, radiation) - Multi-component mass transfer - Gas-phase and surface reaction kinetics - Film stress evolution ### 2.3 Thermal Processing (RTP, Annealing) **Coupled domains:** - Radiation heat transfer - Solid-state diffusion (dopants) - Defect kinetics - Thermo-mechanical stress (slip, warpage) ### 2.4 EUV Lithography **Coupled domains:** - Wave optics and diffraction - Photochemistry in resist - Stochastic photon/electron effects - Mask/wafer thermal-mechanical deformation ## 3. Mathematical Framework: Governing Equations ### 3.1 Electromagnetics (Plasma Systems) For RF-driven plasma, the **time-harmonic Maxwell's equations**: $$ \nabla \times \left(\mu_r^{-1} \nabla \times \mathbf{E}\right) - k_0^2 \epsilon_r \mathbf{E} = -j\omega\mu_0 \mathbf{J}_{ext} $$ The **plasma permittivity** encodes the coupling to electron density: $$ \epsilon_r = 1 - \frac{\omega_{pe}^2}{\omega(\omega + j\nu_m)} $$ Where the **plasma frequency** is: $$ \omega_{pe} = \sqrt{\frac{n_e e^2}{m_e \epsilon_0}} $$ **Key parameters:** - $n_e$ — electron density - $e$ — electron charge - $m_e$ — electron mass - $\epsilon_0$ — permittivity of free space - $\nu_m$ — electron-neutral collision frequency - $\omega$ — angular frequency of RF excitation > **Note:** This creates a **strong nonlinear coupling**: the EM field depends on plasma density, which in turn depends on power absorption from the EM field. ### 3.2 Plasma Transport (Drift-Diffusion Approximation) **Electron continuity equation:** $$ \frac{\partial n_e}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_e = S_e $$ **Electron flux:** $$ \boldsymbol{\Gamma}_e = -\mu_e n_e \mathbf{E} - D_e \nabla n_e $$ **Electron energy density equation:** $$ \frac{\partial n_\epsilon}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_\epsilon + \mathbf{E} \cdot \boldsymbol{\Gamma}_e = S_\epsilon - \sum_j \varepsilon_j R_j $$ **Where:** - $n_e$ — electron density - $\boldsymbol{\Gamma}_e$ — electron flux vector - $\mu_e$ — electron mobility - $D_e$ — electron diffusion coefficient - $S_e$ — electron source term (ionization, attachment, recombination) - $n_\epsilon$ — electron energy density - $\varepsilon_j$ — energy loss per reaction $j$ - $R_j$ — reaction rate for process $j$ **Ion transport** (for multiple species $i$): $$ \frac{\partial n_i}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_i = S_i $$ ### 3.3 Neutral Gas Flow (Navier-Stokes Equations) **Continuity equation:** $$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$ **Momentum equation:** $$ \rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{body} $$ **Where:** - $\rho$ — gas density - $\mathbf{u}$ — velocity vector - $p$ — pressure - $\boldsymbol{\tau}$ — viscous stress tensor - $\mathbf{F}_{body}$ — body forces **Low-pressure corrections (Knudsen effects):** At low pressures where Knudsen number $Kn = \lambda/L > 0.01$, slip boundary conditions are required: $$ u_{slip} = \frac{2-\sigma}{\sigma} \lambda \left.\frac{\partial u}{\partial n}\right|_{wall} $$ Where: - $\lambda$ — mean free path - $L$ — characteristic length - $\sigma$ — tangential momentum accommodation coefficient ### 3.4 Species Transport and Chemistry **Convection-diffusion-reaction equation:** $$ \frac{\partial c_k}{\partial t} + \nabla \cdot (c_k \mathbf{u}) = \nabla \cdot (D_k \nabla c_k) + R_k $$ **Gas-phase reaction rates:** $$ R_k = \sum_j \nu_{kj} \, k_j(T) \prod_l c_l^{a_{lj}} $$ **Where:** - $c_k$ — concentration of species $k$ - $D_k$ — diffusion coefficient - $R_k$ — net production rate - $\nu_{kj}$ — stoichiometric coefficient - $k_j(T)$ — temperature-dependent rate constant - $a_{lj}$ — reaction order **Surface reactions (Langmuir-Hinshelwood kinetics):** $$ r_s = k_s \theta_A \theta_B $$ **Surface coverage:** $$ \theta_i = \frac{K_i c_i}{1 + \sum_j K_j c_j} $$ ### 3.5 Heat Transfer **Energy equation:** $$ \rho c_p \frac{\partial T}{\partial t} + \rho c_p \mathbf{u} \cdot \nabla T = \nabla \cdot (k \nabla T) + Q $$ **Heat sources in plasma systems:** $$ Q = Q_{Joule} + Q_{ion} + Q_{reaction} + Q_{radiation} $$ **Joule heating (time-averaged):** $$ Q_{Joule} = \frac{1}{2} \text{Re}(\mathbf{J}^* \cdot \mathbf{E}) $$ **Where:** - $\rho$ — density - $c_p$ — specific heat capacity - $k$ — thermal conductivity - $Q$ — volumetric heat source - $\mathbf{J}^*$ — complex conjugate of current density ### 3.6 Solid Mechanics (Film Stress) **Equilibrium equation:** $$ \nabla \cdot \boldsymbol{\sigma} = 0 $$ **Constitutive relation with thermal strain:** $$ \boldsymbol{\sigma} = \mathbf{C} : (\boldsymbol{\epsilon} - \boldsymbol{\epsilon}_{th} - \boldsymbol{\epsilon}_{intrinsic}) $$ **Thermal strain tensor:** $$ \boldsymbol{\epsilon}_{th} = \alpha(T - T_0)\mathbf{I} $$ **Where:** - $\boldsymbol{\sigma}$ — stress tensor - $\mathbf{C}$ — stiffness tensor - $\boldsymbol{\epsilon}$ — total strain tensor - $\alpha$ — coefficient of thermal expansion - $T_0$ — reference temperature - $\mathbf{I}$ — identity tensor **Stoney equation** (wafer curvature from film stress): $$ \sigma_f = \frac{E_s h_s^2}{6(1-\nu_s)h_f}\kappa $$ **Where:** - $\sigma_f$ — film stress - $E_s$ — substrate Young's modulus - $\nu_s$ — substrate Poisson's ratio - $h_s$ — substrate thickness - $h_f$ — film thickness - $\kappa$ — wafer curvature ## 4. Feature-Scale Modeling At the nanometer scale within etched features, continuum assumptions break down. ### 4.1 Profile Evolution (Level Set Method) The etch front $\phi(\mathbf{x},t) = 0$ evolves according to: $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ **Local etch rate** depends on coupled physics: $$ V_n = \Gamma_{ion}(E,\theta) \cdot Y_{phys}(E,\theta) + \Gamma_{rad} \cdot Y_{chem}(T) + \Gamma_{ion} \cdot \Gamma_{rad} \cdot Y_{synergy} $$ **Where:** - $\phi$ — level set function (zero at interface) - $V_n$ — normal velocity of interface - $\Gamma_{ion}$ — ion flux (from sheath model) - $\Gamma_{rad}$ — radical flux (from feature-scale transport) - $Y_{phys}$ — physical sputtering yield - $Y_{chem}$ — chemical etch yield - $Y_{synergy}$ — ion-enhanced chemical yield - $\theta$ — local incidence angle - $E$ — ion energy ### 4.2 Feature-Scale Transport Within high-aspect-ratio features, **Knudsen diffusion** dominates: $$ D_{Kn} = \frac{d}{3}\sqrt{\frac{8k_BT}{\pi m}} $$ **Where:** - $d$ — feature diameter/width - $k_B$ — Boltzmann constant - $T$ — temperature - $m$ — molecular mass **View factor calculations** for flux at the bottom of features: $$ \Gamma_{bottom} = \Gamma_{top} \cdot \int_{\Omega} f(\theta) \cos\theta \, d\Omega $$ ### 4.3 Ion Angular and Energy Distribution At the sheath-feature interface: $$ f(E, \theta) = f_E(E) \cdot f_\theta(\theta) $$ **Angular distribution** (from sheath collisionality): $$ f_\theta(\theta) \propto \cos^n(\theta) \exp\left(-\frac{\theta^2}{2\sigma_\theta^2}\right) $$ **Where:** - $f_E(E)$ — ion energy distribution function - $f_\theta(\theta)$ — ion angular distribution function - $n$ — exponent (depends on sheath collisionality) - $\sigma_\theta$ — angular spread parameter ## 5. Multi-Scale Coupling Strategy ``` ┌─────────────────────────────────────────────────────────────┐ │ REACTOR SCALE (cm–m) │ │ Continuum: Navier-Stokes, Maxwell, Drift-Diffusion │ │ Methods: FEM, FVM │ └─────────────────────┬───────────────────────────────────────┘ │ Boundary fluxes, plasma parameters ▼ ┌─────────────────────────────────────────────────────────────┐ │ FEATURE SCALE (nm–μm) │ │ Kinetic transport: DSMC, Angular distribution │ │ Profile evolution: Level set, Cell-based methods │ └─────────────────────┬───────────────────────────────────────┘ │ Sticking coefficients, reaction rates ▼ ┌─────────────────────────────────────────────────────────────┐ │ ATOMIC SCALE (Å–nm) │ │ DFT: Reaction barriers, surface energies │ │ MD: Sputtering yields, sticking probabilities │ │ KMC: Surface evolution, roughness │ └─────────────────────────────────────────────────────────────┘ ``` **Scale hierarchy:** 1. **Reactor scale (cm–m)** - Continuum fluid dynamics - Maxwell's equations for EM fields - Drift-diffusion for charged species - Numerical methods: FEM, FVM 2. **Feature scale (nm–μm)** - Knudsen transport in high-aspect-ratio structures - Direct Simulation Monte Carlo (DSMC) - Level set methods for profile evolution 3. **Atomic scale (Å–nm)** - Density Functional Theory (DFT) for reaction barriers - Molecular Dynamics (MD) for sputtering yields - Kinetic Monte Carlo (KMC) for surface evolution ## 6. Coupled System Structure The full system can be written abstractly as: $$ \mathbf{M}(\mathbf{u})\frac{\partial \mathbf{u}}{\partial t} = \mathbf{F}(\mathbf{u}, \nabla\mathbf{u}, \nabla^2\mathbf{u}, t) $$ **State vector:** $$ \mathbf{u} = \begin{bmatrix} n_e \\ n_\epsilon \\ n_{i,k} \\ c_j \\ T \\ \mathbf{E} \\ \mathbf{u}_{gas} \\ p \\ \boldsymbol{\sigma} \\ \phi_{profile} \\ \vdots \end{bmatrix} $$ **Jacobian structure reveals coupling:** $$ \mathbf{J} = \frac{\partial \mathbf{F}}{\partial \mathbf{u}} = \begin{pmatrix} J_{ee} & J_{e\epsilon} & J_{ei} & J_{ec} & \cdots \\ J_{\epsilon e} & J_{\epsilon\epsilon} & J_{\epsilon i} & & \\ J_{ie} & J_{i\epsilon} & J_{ii} & & \\ J_{ce} & & & J_{cc} & \\ \vdots & & & & \ddots \end{pmatrix} $$ **Off-diagonal blocks** represent inter-physics coupling strengths. ## 7. Numerical Solution Strategies ### 7.1 Coupling Approaches **Monolithic (fully coupled):** - Solve all physics simultaneously - Newton iteration on full Jacobian - Robust but computationally expensive - Required for strongly coupled physics (plasma + EM) **Partitioned (sequential):** - Solve each physics domain separately - Iterate between domains until convergence - More efficient for weakly coupled physics - Risk of convergence issues **Hybrid approach:** - Group strongly coupled physics into blocks - Sequential coupling between blocks ### 7.2 Spatial Discretization **Finite Element Method (FEM)** — weak form for species transport: $$ \int_\Omega w \frac{\partial c}{\partial t} \, d\Omega + \int_\Omega w (\mathbf{u} \cdot \nabla c) \, d\Omega + \int_\Omega \nabla w \cdot (D\nabla c) \, d\Omega = \int_\Omega w R \, d\Omega $$ **SUPG Stabilization** for convection-dominated problems: $$ w \rightarrow w + \tau_{SUPG} \, \mathbf{u} \cdot \nabla w $$ **Where:** - $w$ — test function - $c$ — concentration field - $\tau_{SUPG}$ — stabilization parameter ### 7.3 Time Integration **Stiff systems** require implicit methods: - **BDF** (Backward Differentiation Formulas) - **ESDIRK** (Explicit Singly Diagonally Implicit Runge-Kutta) **Operator splitting** for multi-physics: $$ \mathbf{u}^{n+1} = \mathcal{L}_1(\Delta t) \circ \mathcal{L}_2(\Delta t) \circ \mathcal{L}_3(\Delta t) \, \mathbf{u}^n $$ **Where:** - $\mathcal{L}_i$ — solution operator for physics domain $i$ - $\Delta t$ — time step - $\circ$ — composition of operators ## 8. Specific Application: ICP Etch Model **Complete coupled system summary:** | Physics Domain | Governing Equations | Key Coupling Variables | |----------------|---------------------|------------------------| | EM (inductive) | $\nabla \times (\nabla \times \mathbf{E}) + k^2\epsilon_p \mathbf{E} = 0$ | $n_e \rightarrow \epsilon_p$ | | Electron transport | $\nabla \cdot \Gamma_e = S_e$ | $\mathbf{E}_{dc}, n_e, T_e$ | | Electron energy | $\nabla \cdot \Gamma_\epsilon = Q_{EM} - Q_{loss}$ | $T_e \rightarrow$ rate coefficients | | Ion transport | $\nabla \cdot \Gamma_i = S_i$ | $n_e, \mathbf{E}_{dc}$ | | Neutral chemistry | $\nabla \cdot (c_k \mathbf{u} - D_k\nabla c_k) = R_k$ | $T_e \rightarrow k_{diss}$ | | Gas flow | Navier-Stokes | $T_{gas}$ | | Heat transfer | $\nabla \cdot (k\nabla T) + Q = 0$ | $Q_{plasma}$ | | Sheath | Child-Langmuir / PIC | $n_e, T_e, V_{dc}$ | | Feature transport | Knudsen + angular | $\Gamma_{ion}, \Gamma_{rad}$ from reactor | | Profile evolution | Level set | $V_n$ from surface kinetics | ## 9. EUV Lithography: Stochastic Multi-Physics At EUV wavelength (13.5 nm), photon shot noise becomes significant. ### 9.1 Aerial Image Formation $$ I(\mathbf{r}) = \left|\mathcal{F}^{-1}\left[\tilde{M}(\mathbf{f}) \cdot H(\mathbf{f})\right]\right|^2 $$ **Where:** - $I(\mathbf{r})$ — intensity at position $\mathbf{r}$ - $\tilde{M}(\mathbf{f})$ — mask spectrum (Fourier transform of mask pattern) - $H(\mathbf{f})$ — pupil function (includes aberrations, partial coherence) - $\mathcal{F}^{-1}$ — inverse Fourier transform ### 9.2 Photon Statistics $$ N \sim \text{Poisson}(\bar{N}) $$ $$ \sigma_N = \sqrt{\bar{N}} $$ **Where:** - $N$ — number of photons absorbed - $\bar{N}$ — expected number of photons - $\sigma_N$ — standard deviation (shot noise) ### 9.3 Resist Exposure (Stochastic Dill Model) $$ \frac{\partial [PAG]}{\partial t} = -C \cdot I \cdot [PAG] + \xi(t) $$ **Where:** - $[PAG]$ — photoactive compound concentration - $C$ — exposure rate constant - $I$ — local intensity - $\xi(t)$ — stochastic noise term ### 9.4 Line Edge Roughness (LER) $$ \sigma_{LER} \propto \sqrt{\frac{1}{\text{dose}}} \cdot \frac{1}{\text{image contrast}} $$ > **Note:** This requires **Kinetic Monte Carlo** or **Gillespie algorithm** rather than continuum PDEs. ## 10. Process Optimization (Inverse Problem) ### 10.1 Problem Formulation **Objective:** Minimize profile deviation from target $$ \min_{\mathbf{p}} J = \int_\Gamma \left|\phi(\mathbf{x}; \mathbf{p}) - \phi_{target}\right|^2 \, d\Gamma $$ **Subject to physics constraints:** $$ \mathbf{F}(\mathbf{u}, \mathbf{p}) = 0 $$ **Control parameters** $\mathbf{p}$: - RF power - Chamber pressure - Gas flow rates - Substrate temperature - Process time ### 10.2 Adjoint Method for Efficient Gradients **Gradient computation:** $$ \frac{dJ}{d\mathbf{p}} = \frac{\partial J}{\partial \mathbf{p}} - \boldsymbol{\lambda}^T \frac{\partial \mathbf{F}}{\partial \mathbf{p}} $$ **Adjoint equation:** $$ \left(\frac{\partial \mathbf{F}}{\partial \mathbf{u}}\right)^T \boldsymbol{\lambda} = \left(\frac{\partial J}{\partial \mathbf{u}}\right)^T $$ **Where:** - $\boldsymbol{\lambda}$ — adjoint variable (Lagrange multiplier) - $\mathbf{u}$ — state variables - $\mathbf{p}$ — control parameters ## 11. Emerging Approaches ### 11.1 Physics-Informed Neural Networks (PINNs) **Loss function:** $$ \mathcal{L} = \mathcal{L}_{data} + \lambda \mathcal{L}_{PDE} $$ **Where:** - $\mathcal{L}_{data}$ — data fitting loss - $\mathcal{L}_{PDE}$ — PDE residual loss at collocation points - $\lambda$ — regularization parameter ### 11.2 Digital Twins **Key features:** - Real-time reduced-order models calibrated to equipment sensors - Combine physics-based models with ML for fast prediction - Enable predictive maintenance and process control ### 11.3 Uncertainty Quantification **Methods:** - **Polynomial Chaos Expansion (PCE)** — for parametric uncertainty propagation - **Bayesian Inference** — for model calibration with experimental data - **Monte Carlo Sampling** — for statistical analysis of outputs ## 12. Mathematical Structure The semiconductor manufacturing multi-physics problem has a characteristic mathematical structure: 1. **Hierarchy of scales** (atomic → feature → reactor) - Requires multi-scale methods - Information passing between scales via homogenization 2. **Nonlinear coupling** between physics domains - Varying coupling strengths - Both explicit and implicit dependencies 3. **Stiff ODEs/DAEs** - Disparate time scales (electron dynamics ~ ns, thermal ~ s) - Requires implicit time integration 4. **Moving boundaries** - Etch/deposition fronts - Requires interface tracking (level set, phase field) 5. **Rarefied gas effects** - At low pressures ($Kn > 0.01$) - Requires kinetic corrections or DSMC 6. **Stochastic effects** - At nanometer scales (EUV, atomic-scale roughness) - Requires Monte Carlo methods ## Key Physical Constants | Symbol | Value | Description | |--------|-------|-------------| | $e$ | $1.602 \times 10^{-19}$ C | Elementary charge | | $m_e$ | $9.109 \times 10^{-31}$ kg | Electron mass | | $\epsilon_0$ | $8.854 \times 10^{-12}$ F/m | Permittivity of free space | | $\mu_0$ | $4\pi \times 10^{-7}$ H/m | Permeability of free space | | $k_B$ | $1.381 \times 10^{-23}$ J/K | Boltzmann constant | | $N_A$ | $6.022 \times 10^{23}$ mol$^{-1}$ | Avogadro's number | ## Common Dimensionless Numbers | Number | Definition | Physical Meaning | |--------|------------|------------------| | Knudsen ($Kn$) | $\lambda / L$ | Mean free path / characteristic length | | Reynolds ($Re$) | $\rho u L / \mu$ | Inertia / viscous forces | | Péclet ($Pe$) | $u L / D$ | Convection / diffusion | | Damköhler ($Da$) | $k L / u$ | Reaction / convection rate | | Biot ($Bi$) | $h L / k$ | Surface / bulk heat transfer |
Use multiple LLM providers for redundancy. Failover if one is down. Route based on cost or capability.
# Semiconductor Manufacturing: Multi-Scale Problems and Mathematical Modeling ## 1. The Multi-Scale Hierarchy Semiconductor manufacturing spans roughly **12 orders of magnitude** in length scale, each with distinct physics: | Scale | Range | Phenomena | Mathematical Approach | |-------|-------|-----------|----------------------| | **Quantum/Atomic** | 0.1–1 nm | Bond formation, electron tunneling, reaction barriers | DFT, quantum chemistry | | **Molecular** | 1–10 nm | Surface reactions, nucleation, atomic diffusion | Kinetic Monte Carlo, MD | | **Feature** | 10 nm – 1 μm | Line edge roughness, profile evolution, grain structure | Level set, phase field | | **Device** | 1–100 μm | Transistor variability, local stress | Continuum FEM | | **Die** | 1–10 mm | Pattern density effects, thermal gradients | PDE-based continuum | | **Wafer** | 300 mm | Global uniformity, edge effects | Equipment-scale models | | **Reactor** | ~1 m | Plasma distribution, gas flow | CFD, plasma fluid models | ### Fundamental Challenge **Physics at each scale influences adjacent scales, creating coupled nonlinear systems with vastly different characteristic times and lengths.** ## 2. Key Processes and Mathematical Structure ### 2.1 Plasma Etching — The Most Complex Multi-Scale Problem #### 2.1.1 Reactor Scale (Continuum) **Electron density evolution:** $$ \frac{\partial n_e}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_e = S_e - L_e $$ **Ion density evolution:** $$ \frac{\partial n_i}{\partial t} + \nabla \cdot \boldsymbol{\Gamma}_i = S_i - L_i $$ **Poisson equation for electric potential:** $$ \nabla^2 \phi = -\frac{e}{\epsilon_0}(n_i - n_e) $$ Where: - $n_e$, $n_i$ = electron and ion densities - $\boldsymbol{\Gamma}_e$, $\boldsymbol{\Gamma}_i$ = electron and ion fluxes - $S_e$, $S_i$ = source terms (ionization) - $L_e$, $L_i$ = loss terms (recombination) - $\phi$ = electric potential - $e$ = elementary charge - $\epsilon_0$ = permittivity of free space #### 2.1.2 Feature Scale — Profile Evolution via Level Set **Level set equation:** $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ Where: - $\phi(x,t) = 0$ defines the evolving surface - $V_n$ = local etch rate (normal velocity) **The local etch rate $V_n$ depends on:** - Ion flux and angle distribution (from sheath physics) - Neutral species flux (from transport) - Surface chemistry (from atomic-scale kinetics) #### 2.1.3 The Coupling Problem The feature-scale etch rate $V_n$ requires: - Ion angular/energy distributions → from sheath models - Sheath models → depend on plasma conditions - Plasma conditions → affected by loading (total surface area being etched) **This creates a global-to-local-to-global feedback loop.** ### 2.2 Chemical Vapor Deposition (CVD) / Atomic Layer Deposition (ALD) #### 2.2.1 Gas-Phase Transport (Continuum) **Navier-Stokes momentum equation:** $$ \rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} $$ **Species transport equation:** $$ \frac{\partial C_k}{\partial t} + \mathbf{u} \cdot \nabla C_k = D_k \nabla^2 C_k + R_k $$ Where: - $\rho$ = gas density - $\mathbf{u}$ = velocity field - $p$ = pressure - $\mu$ = dynamic viscosity - $C_k$ = concentration of species $k$ - $D_k$ = diffusion coefficient - $R_k$ = reaction rate #### 2.2.2 Surface Kinetics (Stochastic/Molecular) **Adsorption rate:** $$ r_{ads} = s_0 \cdot f(\theta) \cdot F $$ Where: - $s_0$ = sticking coefficient - $f(\theta)$ = coverage-dependent function - $F$ = incident flux **Surface diffusion hopping rate:** $$ \nu = \nu_0 \exp\left(-\frac{E_a}{k_B T}\right) $$ Where: - $\nu_0$ = attempt frequency - $E_a$ = activation energy - $k_B$ = Boltzmann constant - $T$ = temperature #### 2.2.3 Mathematical Tension **Gas-phase transport is deterministic continuum; surface evolution involves discrete stochastic events. The boundary condition for the continuum problem depends on atomistic surface dynamics.** ### 2.3 Lithography #### 2.3.1 Aerial Image Formation (Wave Optics) **Hopkins formulation for partially coherent imaging:** $$ I(\mathbf{r}) = \sum_j w_j \left| \iint M(f_x, f_y) H_j(f_x, f_y) e^{2\pi i(f_x x + f_y y)} \, df_x \, df_y \right|^2 $$ Where: - $I(\mathbf{r})$ = image intensity at position $\mathbf{r}$ - $M(f_x, f_y)$ = mask spectrum (Fourier transform of mask pattern) - $H_j(f_x, f_y)$ = pupil function for source point $j$ - $w_j$ = weight for source point $j$ #### 2.3.2 Photoresist Chemistry **Exposure (photoactive compound destruction):** $$ \frac{\partial m}{\partial t} = -C \cdot I \cdot m $$ **Post-exposure bake diffusion (acid diffusion):** $$ \frac{\partial h}{\partial t} = D_h \nabla^2 h $$ **Development rate (Mack model):** $$ R = R_0 \frac{(1-m)^n + \epsilon}{(1-m)^n + 1} $$ Where: - $m$ = normalized photoactive compound concentration - $C$ = exposure rate constant - $I$ = intensity - $h$ = acid concentration - $D_h$ = acid diffusion coefficient - $R_0$ = maximum development rate - $n$ = dissolution selectivity parameter - $\epsilon$ = dissolution rate ratio #### 2.3.3 Stochastic Challenge at Advanced Nodes At EUV wavelength (13.5 nm), photon shot noise becomes significant: $$ \text{Fluctuation} \sim \frac{1}{\sqrt{N}} $$ Where $N$ = number of photons per feature area. **This translates to line edge roughness (LER) of ~2-3 nm — comparable to feature dimensions.** ### 2.4 Diffusion and Annealing Classical Fick's law fails because: - Diffusion is mediated by point defects (vacancies, interstitials) - Defect concentrations depend on dopant concentration - Stress affects diffusion - Transient enhanced diffusion during implant damage annealing #### Five-Stream Model $$ \frac{\partial C_s}{\partial t} = \nabla \cdot (D_s \nabla C_s) + \text{reactions with } C_I, C_V, C_{As}, C_{AV}, \ldots $$ Where: - $C_s$ = substitutional dopant concentration - $C_I$ = interstitial concentration - $C_V$ = vacancy concentration - $C_{As}$ = dopant-interstitial pair concentration - $C_{AV}$ = dopant-vacancy pair concentration **This creates a coupled nonlinear system of 5+ PDEs with concentration-dependent coefficients spanning time scales from picoseconds to hours.** ## 3. Mathematical Frameworks for Multi-Scale Coupling ### 3.1 Homogenization Theory For problems with periodic microstructure at scale $\epsilon$: $$ -\nabla \cdot \left( A^\epsilon(x) \nabla u^\epsilon \right) = f $$ Where $A^\epsilon(x) = A(x/\epsilon)$ oscillates rapidly. #### Two-Scale Expansion $$ u^\epsilon(x) = u_0\left(x, \frac{x}{\epsilon}\right) + \epsilon \, u_1\left(x, \frac{x}{\epsilon}\right) + \epsilon^2 \, u_2\left(x, \frac{x}{\epsilon}\right) + \ldots $$ This yields an **effective coefficient** $A^*$ that captures microscale physics in a macroscale equation. **Rigorous for linear elliptic problems; much harder for nonlinear, time-dependent cases in manufacturing.** ### 3.2 Heterogeneous Multiscale Method (HMM) **Key Idea:** Run microscale simulations only where/when needed to extract effective properties for the macroscale solver. ``` ┌────────────────────────────────────────┐ │ MACRO SOLVER (continuum PDE) │ │ Uses effective coefficients D*, k* │ └──────────────────┬─────────────────────┘ │ Query at macro points ▼ ┌────────────────────────────────────────┐ │ MICRO SIMULATIONS (MD, KMC, etc.) │ │ Constrained by local macro state │ │ Returns averaged properties │ └────────────────────────────────────────┘ ``` #### Mathematical Formulation **Macro equation:** $$ \frac{\partial U}{\partial t} = F\left(U, D^*(U)\right) $$ **Micro-to-macro coupling:** $$ D^*(U) = \langle d(u) \rangle_{\text{micro}} $$ Where the micro simulation is constrained by the macroscopic state $U$. ### 3.3 Kinetic-Continuum Transition #### Boltzmann Equation $$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_x f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = Q(f,f) $$ Where: - $f(\mathbf{x}, \mathbf{v}, t)$ = distribution function - $\mathbf{v}$ = velocity - $\mathbf{F}$ = external force - $m$ = particle mass - $Q(f,f)$ = collision operator #### Chapman-Enskog Expansion Derives Navier-Stokes equations in the limit: $$ Kn \to 0 $$ Where the **Knudsen number** is defined as: $$ Kn = \frac{\lambda}{L} $$ - $\lambda$ = mean free path - $L$ = characteristic length #### Spatial Variation of Knudsen Number | Region | Knudsen Number | Valid Model | |--------|---------------|-------------| | Bulk reactor | $Kn \ll 1$ | Continuum (Navier-Stokes) | | Feature trenches | $Kn \sim 1$ | Transitional regime | | Surfaces, small features | $Kn \gg 1$ | Kinetic (Boltzmann) | ### 3.4 Level Set and Phase Field Methods #### 3.4.1 Level Set Method **Interface definition:** $\{\mathbf{x} : \phi(\mathbf{x},t) = 0\}$ **Evolution equation:** $$ \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ **Advantages:** - Handles topology changes naturally (merging, splitting) - Implicit representation avoids mesh issues **Challenges:** - Maintaining $|\nabla \phi| = 1$ (signed distance property) - Velocity extension from interface to entire domain #### 3.4.2 Phase Field Method **Diffuse interface evolution:** $$ \frac{\partial \phi}{\partial t} = M\left[\epsilon^2 \nabla^2 \phi - f'(\phi) + \lambda g'(\phi)\right] $$ Where: - $M$ = mobility - $\epsilon$ = interface width parameter - $f(\phi)$ = double-well potential - $g(\phi)$ = driving force - $\lambda$ = coupling constant **Advantages:** - No explicit interface tracking required - Natural handling of complex morphologies **Challenges:** - Resolving thin interface requires fine mesh - Selecting appropriate interface width $\epsilon$ ## 4. Fundamental Mathematical Challenges ### 4.1 Stiffness and Time-Scale Separation | Process | Characteristic Time | |---------|-------------------| | Electron dynamics | $10^{-12}$ s | | Surface reactions | $10^{-9}$ – $10^{-6}$ s | | Gas transport | $10^{-3}$ s | | Feature evolution | $1$ – $10^{2}$ s | | Wafer processing | $10^{2}$ – $10^{4}$ s | **Time scale ratio:** $\sim 10^{16}$ between fastest and slowest processes. **Direct simulation is impossible.** #### Solution Strategies - **Implicit time integration** with adaptive stepping - **Quasi-steady state approximations** for fast variables - **Operator splitting:** Treat different physics on different time scales - **Averaging/homogenization** to eliminate fast oscillations ### 4.2 High Dimensionality The kinetic description $f(\mathbf{x}, \mathbf{v}, t)$ lives in **6D phase space**. Adding internal energy states and multiple species → intractable. #### Reduction Strategies - **Moment methods:** Track $\langle 1, v, v^2, \ldots \rangle_v$ rather than full $f$ - **Monte Carlo:** Sample from distribution rather than discretizing - **Proper Orthogonal Decomposition (POD):** Find low-dimensional subspace - **Neural network surrogates:** Learn mapping from inputs to outputs ### 4.3 Stochastic Effects at Nanoscale At sub-10nm, continuum assumptions fail due to: - **Discreteness of atoms:** Can't average over enough atoms - **Shot noise:** Finite number of photons, ions, molecules - **Line edge roughness:** Atomic-scale randomness in edge positions #### Mathematical Treatment **Stochastic PDEs (Langevin form):** $$ du = \mathcal{L}u \, dt + \sigma \, dW $$ Where $dW$ is a Wiener process increment. **Master equation:** $$ \frac{dP_n}{dt} = \sum_m \left( W_{nm} P_m - W_{mn} P_n \right) $$ Where: - $P_n$ = probability of state $n$ - $W_{nm}$ = transition rate from state $m$ to state $n$ **Kinetic Monte Carlo:** Direct simulation of discrete events with proper time advancement. ### 4.4 Inverse Problems and Control **Forward problem:** Given process parameters → predict outcome **Inverse problem:** Given desired outcome → find parameters #### Manufacturing Requirements - Recipe optimization - Run-to-run control - Fault detection/classification #### Mathematical Challenges - **Ill-posedness:** Multiple solutions, sensitivity to noise - **High dimensionality** of parameter space - **Real-time constraints** for feedback control #### Approaches - **Regularization:** Tikhonov, sparse methods - **Bayesian inference:** Uncertainty quantification - **Optimal control theory:** Adjoint methods - **Surrogate-based optimization:** Using ML models ## 5. Current Frontiers ### 5.1 Physics-Informed Machine Learning #### Loss Function Structure $$ \mathcal{L} = \mathcal{L}_{\text{data}} + \lambda_{\text{physics}} \mathcal{L}_{\text{PDE}} + \lambda_{\text{BC}} \mathcal{L}_{\text{boundary}} $$ Where: - $\mathcal{L}_{\text{data}}$ = data fitting loss - $\mathcal{L}_{\text{PDE}}$ = physics constraint (PDE residual) - $\mathcal{L}_{\text{boundary}}$ = boundary condition constraint - $\lambda$ = weighting hyperparameters #### Methods - **Physics-Informed Neural Networks (PINNs):** Embed governing equations as soft constraints - **Neural operators (DeepONet, FNO):** Learn mappings between function spaces - **Hybrid models:** Combine physics-based and data-driven components #### Challenges Specific to Semiconductor Manufacturing - Sparse experimental data (wafers are expensive) - Extrapolation to new process conditions - Interpretability requirements for process understanding - Certification for high-reliability applications ### 5.2 Uncertainty Quantification at Scale Manufacturing requires predicting **distributions**, not just means: - What is $P(\text{yield} > 0.95)$? - What is the 99th percentile of line width variation? #### Polynomial Chaos Expansion $$ u(\mathbf{x}, \boldsymbol{\xi}) = \sum_{k} u_k(\mathbf{x}) \Psi_k(\boldsymbol{\xi}) $$ Where: - $\boldsymbol{\xi}$ = random input parameters - $\Psi_k$ = orthogonal polynomial basis functions - $u_k(\mathbf{x})$ = deterministic coefficient functions #### Challenge: Curse of Dimensionality 50+ random input parameters is common in semiconductor manufacturing. #### Solutions - Sparse polynomial chaos - Active subspaces (dimension reduction) - Multi-fidelity methods (combine cheap/accurate models) ### 5.3 Quantum Effects at Sub-Nanometer Scale As features approach ~1 nm: - **Quantum tunneling** through gate oxides - **Quantum confinement** affects electron states - **Atomistic variability** in dopant positions → device-to-device variation #### Non-Equilibrium Green's Function (NEGF) Method For quantum transport: $$ G^R(E) = \left[ (E + i\eta)I - H - \Sigma^R \right]^{-1} $$ Where: - $G^R$ = retarded Green's function - $E$ = energy - $H$ = Hamiltonian - $\Sigma^R$ = self-energy (contact + scattering) - $\eta$ = infinitesimal positive number ## 6. Conceptual Framework ### Unified View of Multi-Scale Modeling ``` ATOMISTIC MESOSCALE CONTINUUM EQUIPMENT (QM/MD/KMC) (Phase field, (CFD, FEM, (Reactor-scale Level set) Drift-diff) transport) │ │ │ │ │ Coarse │ Averaging │ Lumped │ ├───graining────►├──────────────────►├───parameters───►│ │ │ │ │ │◄──Boundary ────┤◄──Effective ──────┤◄──Boundary──────┤ │ conditions │ coefficients │ conditions │ │ │ │ │ ─────┴────────────────┴───────────────────┴─────────────────┴───── Information flow (bidirectional coupling) ``` ### Key Mathematical Requirements - **Consistency:** Coarse-grained models recover fine-scale physics in appropriate limits - **Conservation:** Mass, momentum, energy preserved across scales - **Efficiency:** Computational cost scales with information content, not raw degrees of freedom - **Adaptivity:** Automatically refine where and when needed ## 7. Open Mathematical Problems | Problem | Current State | Mathematical Need | |---------|--------------|-------------------| | **Stochastic feature-scale modeling** | KMC possible but expensive | Fast stochastic PDE methods | | **Plasma-surface coupling** | Often one-way coupling | Consistent two-way coupling with rigorous error bounds | | **Real-time model-predictive control** | Simplified ROMs | Fast surrogates with guaranteed accuracy | | **Variability prediction** | Expensive Monte Carlo | Efficient UQ for high-dimensional inputs | | **Atomic-to-device coupling** | Sequential handoff | Concurrent adaptive methods | | **Inverse design** | Local optimization | Global optimization in high dimensions | ## Key Equations Summary ### Transport Equations $$ \text{Continuity:} \quad \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 $$ $$ \text{Momentum:} \quad \rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f} $$ $$ \text{Energy:} \quad \rho c_p \frac{DT}{Dt} = k \nabla^2 T + \dot{q} $$ $$ \text{Species:} \quad \frac{\partial C_k}{\partial t} + \nabla \cdot (C_k \mathbf{u}) = D_k \nabla^2 C_k + R_k $$ ### Interface Evolution $$ \text{Level Set:} \quad \frac{\partial \phi}{\partial t} + V_n |\nabla \phi| = 0 $$ $$ \text{Phase Field:} \quad \tau \frac{\partial \phi}{\partial t} = \epsilon^2 \nabla^2 \phi - f'(\phi) $$ ### Kinetic Theory $$ \text{Boltzmann:} \quad \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_x f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = Q(f,f) $$ $$ \text{Knudsen Number:} \quad Kn = \frac{\lambda}{L} $$ ### Stochastic Modeling $$ \text{Langevin SDE:} \quad dX = a(X,t) \, dt + b(X,t) \, dW $$ $$ \text{Fokker-Planck:} \quad \frac{\partial p}{\partial t} = -\nabla \cdot (a \, p) + \frac{1}{2} \nabla^2 (b^2 p) $$ ## Nomenclature | Symbol | Description | Units | |--------|-------------|-------| | $\rho$ | Density | kg/m³ | | $\mathbf{u}$ | Velocity vector | m/s | | $p$ | Pressure | Pa | | $T$ | Temperature | K | | $C_k$ | Concentration of species $k$ | mol/m³ | | $D_k$ | Diffusion coefficient | m²/s | | $\phi$ | Level set function or phase field | — | | $V_n$ | Normal interface velocity | m/s | | $f$ | Distribution function | — | | $Kn$ | Knudsen number | — | | $\lambda$ | Mean free path | m | | $E_a$ | Activation energy | J/mol | | $k_B$ | Boltzmann constant | J/K |
Multi-agent systems coordinate multiple agents to solve complex problems through collaboration.
Train across cloud providers.
Use multiple control inputs.
Train on crops of different sizes.
Use multiple crops of different scales.
Generate large images in parts.
Multi-domain recommendation jointly models user preferences across multiple domains leveraging shared patterns.
Multiple exits at different depths.
Multi-fidelity NAS evaluates architectures at different training lengths resolutions or data subsets for efficiency.
Approaches to use multiple GPUs.
Multi-horizon forecasting predicts multiple future time steps simultaneously or autoregressively.