Semiconductor Manufacturing Process: Etch Modeling
1. Introduction
Etch modeling is one of the most complex and critical areas in semiconductor fabrication simulation. As device geometries shrink below $10\ \text{nm}$ and structures become increasingly three-dimensional, accurate prediction of etch behavior becomes essential for:
- Process Development: Predict outcomes before costly fab experiments
- Yield Optimization: Understand how variations propagate to device performance
- OPC/EPC Extension: Compensate for etch-induced pattern distortions in mask design
- Design-Technology Co-Optimization (DTCO): Feed process effects back into design rules
- Virtual Metrology: Predict wafer results from equipment sensor data in real time
2. Fundamentals of Etching
2.1 What is Etching?
Etching selectively removes material from a wafer to transfer lithographically defined patterns into underlying layers—silicon, oxides, nitrides, metals, or complex stacks.
2.2 Types of Etching
- Wet Etching
- Uses liquid chemicals (acids, bases, solvents)
- Typically isotropic (etches equally in all directions)
- Etch rate follows Arrhenius relationship:
$$
R = A \exp\left(-\frac{E_a}{k_B T}\right)
$$
where:
- $R$ = etch rate
- $A$ = pre-exponential factor
- $E_a$ = activation energy
- $k_B$ = Boltzmann constant ($1.381 \times 10^{-23}\ \text{J/K}$)
- $T$ = temperature (K)
- Dry/Plasma Etching
- Uses ionized gases (plasma)
- Anisotropic (directional)
- Dominant for modern processes ($< 100\ \text{nm}$ nodes)
2.3 Plasma Etching Mechanisms
1. Physical Sputtering
- Ion bombardment physically removes atoms
- Sputter yield $Y$ depends on ion energy $E_i$:
$$
Y(E_i) = A \left( \sqrt{E_i} - \sqrt{E_{th}} \right)
$$
where $E_{th}$ is the threshold energy
2. Chemical Etching
- Reactive species form volatile products
- Example: Silicon etching with fluorine
$$
\text{Si} + 4\text{F} \rightarrow \text{SiF}_4 \uparrow
$$
3. Ion-Enhanced Etching
- Synergy between ion bombardment and chemical reactions
- Etch yield enhancement factor:
$$
\eta = \frac{Y_{ion+chem}}{Y_{ion} + Y_{chem}}
$$
3. Hierarchy of Etch Models
3.1 Empirical Models
Data-driven, fast, used in production:
- Etch Bias Models
- Simple offset correction:
$$
CD_{final} = CD_{litho} + \Delta_{etch}
$$
- Pattern-dependent bias:
$$
\Delta_{etch} = f(\text{pitch}, \text{density}, \text{orientation})
$$
- Etch Proximity Correction (EPC)
- Kernel-based convolution:
$$
\Delta(x,y) = \iint K(x-x', y-y') \cdot I(x', y') \, dx' dy'
$$
- Where $K$ is the etch kernel and $I$ is the pattern intensity
- Machine Learning Models
- Neural networks trained on metrology data
- Gaussian process regression for uncertainty quantification
3.2 Feature-Scale Models
Semi-empirical, balance speed and physics:
- String/Segment Models
- Represent edges as connected nodes
- Each node moves according to local etch rate vector:
$$
\frac{d\vec{r}_i}{dt} = R(\theta_i, \Gamma_{ion}, \Gamma_{n}) \cdot \hat{n}_i
$$
- Where:
- $\vec{r}_i$ = position of node $i$
- $\theta_i$ = local surface angle
- $\Gamma_{ion}$, $\Gamma_n$ = ion and neutral fluxes
- $\hat{n}_i$ = surface normal
- Level-Set Methods
- Track surface as zero-contour of signed distance function $\phi$:
$$
\frac{\partial \phi}{\partial t} + R(\vec{x}) |
abla \phi| = 0
$$
- Handles topology changes naturally (merging, splitting)
- Cell-Based/Voxel Methods
- Discretize feature volume into cells
- Apply probabilistic removal rules:
$$
P_{remove} = 1 - \exp\left( -\sum_j \sigma_j \Gamma_j \Delta t \right)
$$
- Where $\sigma_j$ is the reaction cross-section for species $j$
3.3 Physics-Based Plasma Models
Capture reactor-scale phenomena:
- Plasma Bulk
- Electron energy distribution function (EEDF)
- Boltzmann equation:
$$
\frac{\partial f}{\partial t} + \vec{v} \cdot
abla f + \frac{q\vec{E}}{m} \cdot
abla_v f = \left( \frac{\partial f}{\partial t} \right)_{coll}
$$
- Sheath Physics
- Child-Langmuir law for ion flux:
$$
J_{ion} = \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{M}} \frac{V^{3/2}}{d^2}
$$
- Ion angular distribution at wafer surface
- Transport
- Species continuity:
$$
\frac{\partial n_i}{\partial t} +
abla \cdot (n_i \vec{v}_i) = S_i - L_i
$$
- Where $S_i$ and $L_i$ are source and loss terms
3.4 Atomistic Models
Fundamental understanding, computationally expensive:
- Molecular Dynamics (MD)
- Newton's equations for all atoms:
$$
m_i \frac{d^2 \vec{r}_i}{dt^2} = -
abla_i U(\{\vec{r}\})
$$
- Interatomic potentials: Tersoff, Stillinger-Weber, ReaxFF
- Monte Carlo (MC) Methods
- Statistical sampling of ion trajectories
- Binary collision approximation (BCA) for high energies
- Acceptance probability:
$$
P = \min\left(1, \exp\left(-\frac{\Delta E}{k_B T}\right)\right)
$$
- Kinetic Monte Carlo (KMC)
- Sample reactive events with rates $k_i$:
$$
k_i =
u_0 \exp\left(-\frac{E_{a,i}}{k_B T}\right)
$$
- Event selection: $\sum_{j < i} k_j < r \cdot K_{tot} \leq \sum_{j \leq i} k_j$
4. Key Physical Phenomena
4.1 Anisotropy
Ratio of vertical to lateral etch rate:
$$
A = 1 - \frac{R_{lateral}}{R_{vertical}}
$$
- $A = 1$: Perfectly anisotropic (vertical sidewalls)
- $A = 0$: Perfectly isotropic
Mechanisms for achieving anisotropy:
- Directional ion bombardment
- Sidewall passivation (polymer deposition)
- Low pressure operation (fewer collisions → more directional ions)
- Ion angular distribution characterized by:
$$
f(\theta) \propto \cos^n(\theta)
$$
where higher $n$ indicates more directional flux
4.2 Selectivity
Ratio of etch rates between materials:
$$
S_{A/B} = \frac{R_A}{R_B}
$$
- Mask selectivity: Target material vs. photoresist/hard mask
- Stop layer selectivity: Target material vs. underlying layer
Example selectivities required:
| Process | Selectivity Required |
|---------|---------------------|
| Oxide/Nitride | $> 20:1$ |
| Poly-Si/Oxide | $> 50:1$ |
| Si/SiGe (channel release) | $> 100:1$ |
4.3 Loading Effects
Microloading
Local depletion of reactive species in dense pattern regions:
$$
R_{dense} = R_0 \cdot \frac{1}{1 + \beta \cdot \rho_{local}}
$$
where:
- $R_0$ = etch rate in isolated feature
- $\beta$ = loading coefficient
- $\rho_{local}$ = local pattern density
Macroloading
Wafer-scale depletion:
$$
R = R_0 \cdot \left(1 - \alpha \cdot A_{exposed}\right)
$$
where $A_{exposed}$ is total exposed area fraction
4.4 Aspect Ratio Dependent Etching (ARDE)
Deep, narrow features etch slower due to transport limitations:
$$
R(AR) = R_0 \cdot \exp\left(-\frac{AR}{AR_0}\right)
$$
where $AR = \text{depth}/\text{width}$
Physical mechanisms:
1. Ion Shadowing
- Geometric shadowing angle:
$$
\theta_{shadow} = \arctan\left(\frac{1}{AR}\right)
$$
2. Neutral Transport
- Knudsen diffusion coefficient:
$$
D_K = \frac{d}{3} \sqrt{\frac{8 k_B T}{\pi m}}
$$
- where $d$ is feature diameter
3. Byproduct Redeposition
- Sticking probability affects escape
4.5 Profile Anomalies
| Phenomenon | Description | Cause |
|------------|-------------|-------|
| Bowing | Lateral bulge in sidewall | Ion scattering off sidewalls |
| Notching | Lateral etching at interface | Charge buildup on insulators |
| Microtrenching | Deep spots at corners | Ion reflection at feature bottom |
| Footing | Undercut at bottom | Isotropic chemical component |
| Tapering | Non-vertical sidewalls | Insufficient passivation |
5. Mathematical Foundations
5.1 Surface Evolution Equation
General form for surface height $h(x,y,t)$:
$$
\frac{\partial h}{\partial t} = -R_0 \cdot V(\theta) \cdot \sqrt{1 + |
abla h|^2}
$$
where:
- $R_0$ = baseline etch rate
- $V(\theta)$ = visibility/flux function
- $\theta = \arctan(|
abla h|)$
5.2 Ion Angular Distribution
At wafer surface, ion flux angular distribution:
$$
\Gamma(\theta, \phi) = \Gamma_0 \cdot f(\theta) \cdot g(E)
$$
Common models:
- Gaussian distribution:
$$
f(\theta) = \frac{1}{\sqrt{2\pi}\sigma_\theta} \exp\left(-\frac{\theta^2}{2\sigma_\theta^2}\right)
$$
- Thompson distribution (for sputtered neutrals):
$$
f(E) \propto \frac{E}{(E + E_b)^3}
$$
5.3 Visibility Calculation
For a point on the surface, visibility to incoming flux:
$$
V(\vec{r}) = \frac{1}{2\pi} \int_0^{2\pi} \int_0^{\theta_{max}(\phi)} f(\theta) \sin\theta \cos\theta \, d\theta \, d\phi
$$
where $\theta_{max}(\phi)$ is determined by local geometry (shadowing)
5.4 Surface Reaction Kinetics
Langmuir-Hinshelwood mechanism:
$$
R = k \cdot \theta_A \cdot \theta_B
$$
where surface coverages follow:
$$
\frac{d\theta_i}{dt} = s_i \Gamma_i (1 - \theta_{total}) - k_d \theta_i - k_r \theta_i
$$
- $s_i$ = sticking coefficient
- $k_d$ = desorption rate
- $k_r$ = reaction rate
5.5 Plasma-Surface Interaction Yield
Ion-enhanced etch yield:
$$
Y_{etch} = Y_0 + Y_1 \cdot \sqrt{E_{ion} - E_{th}} + Y_{chem} \cdot \frac{\Gamma_n}{\Gamma_{ion}}
$$
where:
- $Y_0$ = chemical baseline yield
- $Y_1$ = ion enhancement coefficient
- $E_{th}$ = threshold energy (~15-50 eV typically)
- $Y_{chem}$ = chemical enhancement factor
6. Modern Modeling Approaches
6.1 Hybrid Multi-Scale Frameworks
Coupling different scales:
``
-
┌─────────────────────────────────────────────────────────────┐
│ REACTOR SCALE │
│ Plasma simulation (fluid or PIC) │
│ Output: Ion/neutral fluxes, energies, angular dist. │
└────────────────────────┬────────────────────────────────────┘
│ Boundary conditions
▼
┌─────────────────────────────────────────────────────────────┐
│ FEATURE SCALE │
│ Level-set or Monte Carlo │
│ Output: Profile evolution, etch rates │
└────────────────────────┬────────────────────────────────────┘
│ Parameter extraction
▼
┌─────────────────────────────────────────────────────────────┐
│ ATOMISTIC SCALE │
│ MD/KMC simulations │
│ Output: Sticking coefficients, sputter yields │
└─────────────────────────────────────────────────────────────┘
6.2 Machine Learning Integration
- Surrogate Models
- Train neural network on physics simulation outputs:
$$
\hat{y} = f_{NN}(\vec{x}; \vec{w})
$$
- Loss function:
$$
\mathcal{L} = \frac{1}{N} \sum_{i=1}^{N} \|y_i - \hat{y}_i\|^2 + \lambda \|\vec{w}\|^2
$$
- Physics-Informed Neural Networks (PINNs)
- Embed physics constraints in loss:
$$
\mathcal{L}_{total} = \mathcal{L}_{data} + \alpha \mathcal{L}_{physics}
$$
- Where $\mathcal{L}_{physics}$ enforces governing equations
- Virtual Metrology
- Predict CD, profile from chamber sensors:
$$
CD_{predicted} = g(P, T, V_{bias}, \text{OES}, ...)
$$
6.3 Computational Lithography Integration
Major EDA tools couple lithography + etch:
1. Litho simulation → Resist profile $h_R(x,y)$
2. Etch simulation → Final pattern $h_F(x,y)$
3. Combined model:
$$
CD_{final} = CD_{design} + \Delta_{OPC} + \Delta_{litho} + \Delta_{etch}
$$
7. Challenges at Advanced Nodes
7.1 FinFET / Gate-All-Around (GAA)
- Fin Etch
- Sidewall angle uniformity: $90° \pm 1°$
- Width control: $\pm 1\ \text{nm}$ at $W_{fin} < 10\ \text{nm}$
- Channel Release
- Selective SiGe vs. Si etching
- Required selectivity: $> 100:1$
- Etch rate:
$$
R_{SiGe} \gg R_{Si}
$$
- Inner Spacer Formation
- Isotropic lateral etch in confined geometry
- Depth control: $\pm 0.5\ \text{nm}$
7.2 3D NAND
Extreme aspect ratio challenges:
| Generation | Layers | Aspect Ratio |
|------------|--------|--------------|
| 96L | 96 | ~60:1 |
| 128L | 128 | ~80:1 |
| 176L | 176 | ~100:1 |
| 232L+ | 232+ | ~150:1 |
Critical issues:
- ARDE variation across depth
- Bowing control
- Twisting in elliptical holes
7.3 EUV Patterning
- Very thin resists: $< 40\ \text{nm}$
- Hard mask stacks with multiple layers
- LER/LWR amplification:
$$
LER_{final} = \sqrt{LER_{litho}^2 + LER_{etch}^2}
$$
- Target: $LER < 1.2\ \text{nm}$ ($3\sigma$)
7.4 Stochastic Effects
At small dimensions, statistical fluctuations dominate:
$$
\sigma_{CD} \propto \frac{1}{\sqrt{N_{events}}}
$$
where $N_{events}$ = number of etching events per feature
8. Industry Tools
8.1 Commercial Software
| Category | Tools |
|----------|-------|
| TCAD/Process | Synopsys Sentaurus Process, Silvaco Victory Process |
| Virtual Fab | Coventor SEMulator3D |
| Equipment Vendor | Lam Research, Applied Materials (proprietary) |
| Computational Litho | Synopsys S-Litho, Siemens Calibre |
8.2 Research Tools
- MCFPM (Monte Carlo Feature Profile Model) - University of Illinois
- LAMMPS - Molecular dynamics
- SPARTA - Direct Simulation Monte Carlo
- OpenFOAM - Plasma fluid modeling
9. Future Directions
9.1 Digital Twins
Real-time chamber models for closed-loop process control:
$$
\vec{u}_{control}(t) = \mathcal{K} \left[ y_{target} - y_{model}(t) \right]
$$
9.2 Atomistic-Continuum Coupling
Seamless multi-scale simulation using:
- Adaptive mesh refinement
- Concurrent coupling methods
- Machine-learned interscale bridging
9.3 New Materials
Modeling requirements for:
- 2D materials (graphene, MoS$_2$, WS$_2$)
- High-$\kappa$ dielectrics
- Ferroelectrics (HfZrO)
- High-mobility channels (InGaAs, Ge)
9.4 Uncertainty Quantification
Predicting distributions, not just means:
$$
P(CD) = \int P(CD | \vec{\theta}) P(\vec{\theta}) d\vec{\theta}
$$
Key metrics:
- Process capability: $C_{pk} = \frac{\min(USL - \mu, \mu - LSL)}{3\sigma}$
- Target: $C_{pk} > 1.67$ for production
Summary
Etch modeling spans from atomic-scale surface reactions to reactor-scale plasma physics to fab-level empirical correlations. The art lies in choosing the right abstraction level:
| Application | Model Type | Speed | Accuracy |
|-------------|------------|-------|----------|
| Production OPC/EPC | Empirical/ML | ★★★★★ | ★★☆☆☆ |
| Process Development | Feature-scale | ★★★☆☆ | ★★★★☆ |
| Mechanism Research | Atomistic MD/MC | ★☆☆☆☆ | ★★★★★ |
| Equipment Design | Plasma + Feature | ★★☆☆☆ | ★★★★☆ |
As geometries shrink and structures become more 3D, accurate etch modeling becomes essential for first-time-right process development and continued yield improvement.