nuisance defects,metrology
Detected defects that don't affect yield.
265 technical terms and definitions
Detected defects that don't affect yield.
Null-text inversion optimizes unconditional embeddings for accurate real image editing.
Find embeddings for specific images.
Sampling iterations tradeoff.
Test numerical reasoning.
Measure of lens light-gathering ability affects resolution.
# Semiconductor Manufacturing Process: Numerical Methods, Mathematics & Modeling
A comprehensive guide covering the mathematical foundations, numerical methods, and computational modeling approaches used in semiconductor fabrication processes.
## 1. Manufacturing Processes and Their Physics
Semiconductor fabrication involves sequential processes, each governed by different physics:
| Process | Governing Physics | Primary Equations |
|---------|-------------------|-------------------|
| Lithography | Electromagnetic wave propagation, photochemistry | Maxwell's equations, diffusion, reaction kinetics |
| Plasma Etching | Plasma physics, surface chemistry | Boltzmann transport, Poisson, fluid equations |
| CVD/ALD | Fluid dynamics, heat/mass transfer, kinetics | Navier-Stokes, convection-diffusion, Arrhenius |
| Ion Implantation | Atomic collisions, stopping theory | Binary collision approximation, transport |
| Diffusion/Annealing | Solid-state diffusion, defect physics | Fick's laws, reaction-diffusion systems |
| CMP | Contact mechanics, fluid-solid interaction | Preston equation, elasticity |
### 1.1 Lithography
- **Optical projection** through reduction lens system
- **Photoresist chemistry**: exposure, bake, development
- **Resolution limit**: $R = k_1 \frac{\lambda}{NA}$
- **Depth of focus**: $DOF = k_2 \frac{\lambda}{NA^2}$
### 1.2 Plasma Etching
- **Plasma generation**: RF/microwave excitation
- **Ion bombardment**: directional etching
- **Chemical reactions**: isotropic component
- **Selectivity**: differential etch rates between materials
### 1.3 Chemical Vapor Deposition (CVD)
- **Gas-phase transport**: convection and diffusion
- **Surface reactions**: adsorption, reaction, desorption
- **Film conformality**: step coverage in features
- **Temperature dependence**: Arrhenius kinetics
### 1.4 Ion Implantation
- **Ion acceleration**: keV to MeV energies
- **Stopping mechanisms**: electronic and nuclear
- **Damage formation**: vacancy-interstitial pairs
- **Channeling effects**: crystallographic orientation dependence
## 2. Core Mathematical Frameworks
### 2.1 Partial Differential Equations
Nearly every process involves PDEs of different types:
#### Parabolic (Diffusion/Heat Transport)
$$
\frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C) + R
$$
- **Application**: Dopant diffusion, thermal processing, resist chemistry
- **Characteristics**: Smoothing behavior, infinite propagation speed
- **Diffusion coefficient**: $D = D_0 \exp\left(-\frac{E_a}{k_B T}\right)$
#### Elliptic (Steady-State Fields)
$$
\nabla^2 \phi = -\frac{\rho}{\varepsilon}
$$
- **Application**: Electrostatics, plasma sheaths, device simulation
- **Boundary conditions**: Dirichlet, Neumann, or mixed
- **Properties**: Maximum principle, smoothness
#### Hyperbolic (Wave Propagation)
$$
\nabla^2 E - \mu\varepsilon \frac{\partial^2 E}{\partial t^2} = 0
$$
- **Application**: Light propagation in lithography
- **Characteristics**: Finite propagation speed
- **Dispersion**: wavelength-dependent phase velocity
### 2.2 Transport Theory
The **Boltzmann transport equation** underpins plasma modeling and carrier transport:
$$
\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_\mathbf{r} f + \frac{\mathbf{F}}{m} \cdot \nabla_\mathbf{v} f = \left(\frac{\partial f}{\partial t}\right)_{\text{coll}}
$$
Where:
- $f(\mathbf{r}, \mathbf{v}, t)$ = distribution function (6D phase space)
- $\mathbf{F}$ = external force (electric field, etc.)
- RHS = collision integral
**Solution approaches**:
- **Moment methods**: Fluid approximations (continuity, momentum, energy)
- **Monte Carlo sampling**: Stochastic particle tracking
- **Deterministic discretization**: Spherical harmonics expansion
### 2.3 Reaction-Diffusion Systems
Coupled species with chemical reactions:
$$
\frac{\partial C_i}{\partial t} = D_i \nabla^2 C_i + \sum_j k_{ij} C_j
$$
**Examples**:
- **Dopant-defect interactions**: Transient enhanced diffusion
- Dopants: $\frac{\partial C_D}{\partial t} = \nabla \cdot (D_D \nabla C_D) + k_{DI} C_D C_I$
- Interstitials: $\frac{\partial C_I}{\partial t} = \nabla \cdot (D_I \nabla C_I) - k_{IV} C_I C_V + G$
- Vacancies: $\frac{\partial C_V}{\partial t} = \nabla \cdot (D_V \nabla C_V) - k_{IV} C_I C_V + G$
- **Resist chemistry**:
- Photoacid generation: $\frac{\partial [PAG]}{\partial t} = -C \cdot I \cdot [PAG]$
- Acid diffusion: $\frac{\partial [H^+]}{\partial t} = D_{acid} \nabla^2 [H^+]$
- Deprotection: $\frac{\partial M}{\partial t} = -k_{amp} [H^+] M$
### 2.4 Semiconductor Device Equations
The **drift-diffusion model** for carrier transport:
$$
\nabla \cdot (\varepsilon \nabla \psi) = -q(p - n + N_D^+ - N_A^-)
$$
$$
\frac{\partial n}{\partial t} = \frac{1}{q} \nabla \cdot \mathbf{J}_n + G - R
$$
$$
\frac{\partial p}{\partial t} = -\frac{1}{q} \nabla \cdot \mathbf{J}_p + G - R
$$
**Current densities**:
$$
\mathbf{J}_n = q \mu_n n \mathbf{E} + q D_n \nabla n
$$
$$
\mathbf{J}_p = q \mu_p p \mathbf{E} - q D_p \nabla p
$$
**Einstein relation**: $D = \frac{k_B T}{q} \mu$
## 3. Numerical Methods by Category
### 3.1 Spatial Discretization
#### Finite Difference Method (FDM)
**Central difference** (second derivative):
$$
\frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta x^2}
$$
**Forward difference** (first derivative):
$$
\frac{\partial u}{\partial x} \approx \frac{u_{i+1} - u_i}{\Delta x}
$$
**Characteristics**:
- Simple implementation on regular grids
- Truncation error: $O(\Delta x^2)$ for central differences
- Challenges with complex geometries
- Stability requires careful time step selection
#### Finite Element Method (FEM)
**Variational formulation** - find $u$ minimizing:
$$
J[u] = \int_\Omega \left[ \frac{1}{2} |\nabla u|^2 - fu \right] dV
$$
**Weak form** - find $u \in V$ such that for all $v \in V$:
$$
\int_\Omega \nabla u \cdot \nabla v \, dV = \int_\Omega f v \, dV
$$
**Implementation steps**:
1. **Mesh generation**: Divide domain into elements (triangles, tetrahedra)
2. **Shape functions**: Local polynomial basis $N_i(\mathbf{x})$
3. **Assembly**: Build global stiffness matrix $\mathbf{K}$ and load vector $\mathbf{f}$
4. **Solution**: Solve $\mathbf{K} \mathbf{u} = \mathbf{f}$
**Advantages**:
- Handles complex geometries naturally
- Systematic error estimation
- Adaptive refinement possible
#### Finite Volume Method (FVM)
**Conservation form**:
$$
\frac{\partial U}{\partial t} + \nabla \cdot \mathbf{F} = S
$$
**Discrete form** (cell $i$):
$$
\frac{dU_i}{dt} = -\frac{1}{V_i} \sum_{\text{faces}} F_f A_f + S_i
$$
**Characteristics**:
- Conserves quantities exactly by construction
- Natural for fluid dynamics
- Upwinding for convection-dominated problems
### 3.2 Time Integration
#### Explicit Methods
**Forward Euler**:
$$
u^{n+1} = u^n + \Delta t \cdot f(u^n, t^n)
$$
**Runge-Kutta 4th order (RK4)**:
$$
u^{n+1} = u^n + \frac{\Delta t}{6}(k_1 + 2k_2 + 2k_3 + k_4)
$$
Where:
- $k_1 = f(t^n, u^n)$
- $k_2 = f(t^n + \frac{\Delta t}{2}, u^n + \frac{\Delta t}{2} k_1)$
- $k_3 = f(t^n + \frac{\Delta t}{2}, u^n + \frac{\Delta t}{2} k_2)$
- $k_4 = f(t^n + \Delta t, u^n + \Delta t \cdot k_3)$
**Stability constraint** (CFL condition for diffusion):
$$
\Delta t < \frac{\Delta x^2}{2D}
$$
#### Implicit Methods
**Backward Euler**:
$$
u^{n+1} = u^n + \Delta t \cdot f(u^{n+1}, t^{n+1})
$$
**Crank-Nicolson** (second-order accurate):
$$
u^{n+1} = u^n + \frac{\Delta t}{2} \left[ f(u^n, t^n) + f(u^{n+1}, t^{n+1}) \right]
$$
**BDF Methods** (Backward Differentiation Formulas):
$$
\sum_{k=0}^{s} \alpha_k u^{n+1-k} = \Delta t \cdot f(u^{n+1}, t^{n+1})
$$
- BDF1: Backward Euler (1st order)
- BDF2: $\frac{3}{2}u^{n+1} - 2u^n + \frac{1}{2}u^{n-1} = \Delta t \cdot f^{n+1}$ (2nd order)
**Characteristics**:
- Unconditionally stable (A-stable)
- Requires nonlinear solver per time step
- Essential for stiff systems
#### Operator Splitting
**Strang splitting** for $\frac{\partial u}{\partial t} = Lu + Nu$ (linear + nonlinear):
$$
u^{n+1} = e^{\frac{\Delta t}{2} L} e^{\Delta t N} e^{\frac{\Delta t}{2} L} u^n
$$
**Applications**:
- Separate diffusion and reaction
- Different time scales for different physics
- Preserves second-order accuracy
### 3.3 Linear Algebra
#### Direct Methods
**LU Factorization**: $\mathbf{A} = \mathbf{L}\mathbf{U}$
**Sparse direct solvers**:
- PARDISO (Intel MKL)
- SuperLU
- MUMPS
- UMFPACK
**Complexity**: $O(N^\alpha)$ where $\alpha \approx 1.5-2$ for 3D problems
#### Iterative Methods
**Conjugate Gradient (CG)** for symmetric positive definite:
```text
┌─────────────────────────────────────────────────────┐
│ r_0 = b - Ax_0 │
│ p_0 = r_0 │
│ for k = 0, 1, 2, ... │
│ α_k = (r_k^T r_k) / (p_k^T A p_k) │
│ x_{k+1} = x_k + α_k p_k │
│ r_{k+1} = r_k - α_k A p_k │
│ β_k = (r_{k+1}^T r_{k+1}) / (r_k^T r_k) │
│ p_{k+1} = r_{k+1} + β_k p_k │
└─────────────────────────────────────────────────────┘
```
**GMRES** (Generalized Minimal Residual) for non-symmetric systems
**BiCGSTAB** (Bi-Conjugate Gradient Stabilized)
#### Preconditioning
**Purpose**: Transform $\mathbf{A}\mathbf{x} = \mathbf{b}$ to $\mathbf{M}^{-1}\mathbf{A}\mathbf{x} = \mathbf{M}^{-1}\mathbf{b}$
**Common preconditioners**:
- **ILU** (Incomplete LU): Approximate factorization
- **Multigrid**: Hierarchical coarse-grid correction
- **Domain decomposition**: Parallel-friendly
**Multigrid V-cycle**:
$$
\text{Solution} \leftarrow \text{Smooth} + \text{Coarse-grid correction}
$$
### 3.4 Monte Carlo Methods
#### Particle-in-Cell (PIC) for Plasmas
**Algorithm**:
1. **Push particles**: $\mathbf{x}^{n+1} = \mathbf{x}^n + \mathbf{v}^n \Delta t$
2. **Weight to grid**: $\rho_j = \sum_p q_p W(\mathbf{x}_p - \mathbf{x}_j)$
3. **Solve fields**: $\nabla^2 \phi = -\rho/\varepsilon_0$
4. **Interpolate to particles**: $\mathbf{E}_p = \sum_j \mathbf{E}_j W(\mathbf{x}_p - \mathbf{x}_j)$
5. **Accelerate**: $\mathbf{v}^{n+1} = \mathbf{v}^n + (q/m)\mathbf{E}_p \Delta t$
**Monte Carlo Collisions**: Null-collision method for efficiency
#### Direct Simulation Monte Carlo (DSMC)
**For rarefied gas dynamics** (high Knudsen number):
$$
Kn = \frac{\lambda}{L} > 0.1
$$
**Algorithm**:
1. Move particles (ballistic)
2. Index/sort particles into cells
3. Select collision pairs probabilistically
4. Perform collisions (conserve momentum, energy)
5. Sample macroscopic properties
#### Kinetic Monte Carlo (KMC)
**For atomic-scale processes**:
**Rate calculation**: $k_i = \nu_0 \exp\left(-\frac{E_a}{k_B T}\right)$
**Event selection** (BKL algorithm):
1. Calculate total rate: $R_{tot} = \sum_i k_i$
2. Select event $j$ with probability $k_j / R_{tot}$
3. Advance time: $\Delta t = -\ln(r) / R_{tot}$ where $r \in (0,1)$
4. Execute event
5. Update rates
### 3.5 Interface Tracking
#### Level Set Methods
**Interface** = zero contour of $\phi(\mathbf{x}, t)$
**Evolution equation**:
$$
\frac{\partial \phi}{\partial t} + v_n |\nabla \phi| = 0
$$
**Signed distance property**: $|\nabla \phi| = 1$
**Reinitialization** (maintain distance property):
$$
\frac{\partial \phi}{\partial \tau} = \text{sign}(\phi_0)(1 - |\nabla \phi|)
$$
**Advantages**:
- Handles topological changes naturally
- Curvature: $\kappa = \nabla \cdot \left( \frac{\nabla \phi}{|\nabla \phi|} \right)$
- Normal: $\mathbf{n} = \frac{\nabla \phi}{|\nabla \phi|}$
#### Fast Marching Method
**For static Hamilton-Jacobi equations**:
$$
|\nabla T| = \frac{1}{F}
$$
**Complexity**: $O(N \log N)$ using heap data structure
**Application**: Arrival time problems, distance computation
## 4. Key Application Areas
### 4.1 Lithography Simulation
#### Simulation Chain
```text
┌─────────────────────────────────────────────────────┐
│ Mask (GDS) → Optical Simulation → Aerial Image → │
│ → Resist Exposure → PEB Diffusion → Development → │
│ → Final Profile │
└─────────────────────────────────────────────────────┘
```
#### Hopkins Formulation (Partially Coherent Imaging)
$$
I(x,y) = \iint\iint J(f,g) H(f,g) H^*(f',g') O(f,g) O^*(f',g') \times
$$
$$
\exp[2\pi i((f-f')x + (g-g')y)] \, df \, dg \, df' \, dg'
$$
Where:
- $J(f,g)$ = source intensity distribution
- $H(f,g)$ = pupil function
- $O(f,g)$ = mask spectrum
#### SOCS Decomposition
**Sum of Coherent Systems**:
$$
I(x,y) \approx \sum_{k=1}^{N} \lambda_k |h_k * m|^2
$$
- $\lambda_k$ = eigenvalues (decreasing)
- $h_k$ = eigenkernels
- Typically $N \sim 10-30$ sufficient
#### Rigorous Electromagnetic Methods
**RCWA** (Rigorous Coupled Wave Analysis):
- Fourier expansion of fields and permittivity
- Matrix eigenvalue problem per layer
- S-matrix or T-matrix propagation
**FDTD** (Finite Difference Time Domain):
$$
\frac{\partial \mathbf{E}}{\partial t} = \frac{1}{\varepsilon} \nabla \times \mathbf{H}
$$
$$
\frac{\partial \mathbf{H}}{\partial t} = -\frac{1}{\mu} \nabla \times \mathbf{E}
$$
- Yee grid staggering
- PML absorbing boundaries
- Handles arbitrary 3D structures
#### Resist Models
**Dill exposure model**:
$$
\frac{\partial M}{\partial t} = -I(z,t) M C
$$
$$
I(z,t) = I_0 \exp\left[ -\int_0^z (AM(\zeta,t) + B) d\zeta \right]
$$
**Enhanced Fujita-Doolittle development**:
$$
r = r_{\max} \frac{(1-M)^n + r_{min}/r_{max}}{(1-M)^n + 1}
$$
### 4.2 Plasma Process Modeling
#### Multi-Scale Framework
```text
┌─────────────────────────────────────────────────────┐
│ Reactor Scale (cm) Feature Scale (nm) │
│ ↓ ↑ │
│ Plasma Model → Flux/Distributions │
│ ↓ ↑ │
│ Surface Fluxes → Profile Evolution │
└─────────────────────────────────────────────────────┘
```
#### Fluid Plasma Model
**Continuity**:
$$
\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = S_s
$$
**Momentum** (drift-diffusion):
$$
n_s \mathbf{u}_s = \pm \mu_s n_s \mathbf{E} - D_s \nabla n_s
$$
**Energy**:
$$
\frac{\partial}{\partial t}\left(\frac{3}{2} n_e k_B T_e\right) + \nabla \cdot \mathbf{q}_e = \mathbf{J}_e \cdot \mathbf{E} - P_{loss}
$$
**Poisson**:
$$
\nabla \cdot (\varepsilon \nabla \phi) = -e(n_i - n_e)
$$
#### Feature-Scale Model
**Surface advancement**:
$$
v_n = \Gamma_{ion} Y_{ion}(\theta, E) + \Gamma_{neutral} S_{chem}(\theta) - \Gamma_{dep}
$$
Where:
- $\Gamma_{ion}$ = ion flux
- $Y_{ion}$ = ion-enhanced yield (angle, energy dependent)
- $S_{chem}$ = chemical sticking coefficient
- $\Gamma_{dep}$ = deposition flux
### 4.3 TCAD Device Simulation
#### Scharfetter-Gummel Discretization
**Current between nodes** $i$ and $j$:
$$
J_{ij} = \frac{q D}{\Delta x} \left[ n_j B\left(\frac{\psi_j - \psi_i}{V_T}\right) - n_i B\left(\frac{\psi_i - \psi_j}{V_T}\right) \right]
$$
**Bernoulli function**:
$$
B(x) = \frac{x}{e^x - 1}
$$
**Properties**:
- Exact for constant field
- Numerically stable for large bias
- Preserves current continuity
#### Quantum Corrections
**Density gradient model**:
$$
n = N_c \exp\left(\frac{E_F - E_c - \Lambda}{k_B T}\right)
$$
$$
\Lambda = -\frac{\gamma \hbar^2}{6 m^*} \frac{\nabla^2 \sqrt{n}}{\sqrt{n}}
$$
**Schrödinger-Poisson** (1D slice):
$$
-\frac{\hbar^2}{2m^*} \frac{d^2 \psi_i}{dz^2} + V(z) \psi_i = E_i \psi_i
$$
$$
n(z) = \sum_i |\psi_i(z)|^2 f(E_F - E_i)
$$
## 5. Multi-Scale and Multi-Physics Coupling
### 5.1 Length Scale Hierarchy
```text
┌─────────────────────────────────────────────────────┐
│ Atomic Feature Device Die Wafer │
│ (0.1 nm) (10 nm) (100 nm) (1 mm) (300 mm) │
│ │ │ │ │ │ │
│ └────┬─────┴────┬─────┴────┬────┴────┬────┘ │
│ │ │ │ │ │
│ Ab initio KMC Continuum Pattern │
│ DFT MD PDE Effects │
└─────────────────────────────────────────────────────┘
```
### 5.2 Coupling Approaches
#### Sequential (Parameter Passing)
```text
┌─────────────────────────────────────────────────────┐
│ Lower Scale → Parameters → Higher Scale │
└─────────────────────────────────────────────────────┘
```
**Examples**:
- DFT → activation energies → KMC rates
- MD → surface diffusion coefficients → continuum
- Feature-scale → pattern density → wafer-scale
#### Concurrent (Domain Decomposition)
Different physics in different regions, coupled at interfaces:
**Handshaking region**:
$$
u_{atomic} = u_{continuum} \quad \text{in overlap zone}
$$
**Force matching** or **energy-based** coupling
#### Homogenization
**Effective properties** from microstructure:
$$
\langle \sigma \rangle = \mathbf{C}^{eff} : \langle \varepsilon \rangle
$$
**Application**: Pattern-density effects in CMP
### 5.3 Multi-Physics Coupling
#### Monolithic vs. Partitioned
**Monolithic**: Solve all physics simultaneously
$$
\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}
\begin{pmatrix} u_1 \\ u_2 \end{pmatrix} =
\begin{pmatrix} f_1 \\ f_2 \end{pmatrix}
$$
- Strong coupling
- Large, often ill-conditioned systems
**Partitioned**: Iterate between physics
```
while not converged:
Solve Physics 1 with fixed Physics 2 variables
Solve Physics 2 with fixed Physics 1 variables
Check convergence
```
- Reuse existing solvers
- May have stability issues
## 6. Uncertainty Quantification
### 6.1 Sources of Uncertainty
- **Process variations**: Dose, focus, temperature, pressure
- **Material variations**: Film thickness, composition, defect density
- **Model uncertainty**: Parameter calibration, structural assumptions
- **Measurement noise**: Metrology errors
### 6.2 Polynomial Chaos Expansion
**Expansion**:
$$
u(\mathbf{x}, \boldsymbol{\xi}) \approx \sum_{k=0}^{P} u_k(\mathbf{x}) \Psi_k(\boldsymbol{\xi})
$$
Where:
- $\boldsymbol{\xi}$ = random variables (inputs)
- $\Psi_k$ = orthogonal polynomial basis
- $u_k$ = deterministic coefficients
**Basis selection**:
| Distribution | Polynomial Basis |
|--------------|------------------|
| Gaussian | Hermite |
| Uniform | Legendre |
| Beta | Jacobi |
| Exponential | Laguerre |
**Statistics from coefficients**:
- Mean: $\mathbb{E}[u] = u_0$
- Variance: $\text{Var}[u] = \sum_{k=1}^{P} u_k^2 \langle \Psi_k^2 \rangle$
### 6.3 Stochastic Collocation
**Algorithm**:
1. Select collocation points $\boldsymbol{\xi}^{(q)}$ (Gauss quadrature, sparse grids)
2. Solve deterministic problem at each point
3. Construct interpolant/response surface
4. Compute statistics by integration
**Advantages**:
- Non-intrusive (uses existing solvers)
- Flexible basis
- Good for smooth dependence on parameters
### 6.4 Sensitivity Analysis
**Sobol indices** (variance decomposition):
$$
\text{Var}[u] = \sum_i V_i + \sum_{i
Numerical understanding.
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Approximate attention using Nyström method.