multi physics coupling, multiphysics modeling, coupled simulation, process simulation, transport phenomena, heat transfer plasma coupling, electromagnetic plasma
# 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 |
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# 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 |
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