Etch Modeling

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:

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 $E_i$:

where $E_{th}$ is the threshold energy

2. Chemical Etching
- Reactive species form volatile products
- Example: Silicon etching with fluorine

3. Ion-Enhanced Etching
- Synergy between ion bombardment and chemical reactions
- Etch yield enhancement factor:

3. Hierarchy of Etch Models

3.1 Empirical Models

Data-driven, fast, used in production:

• Etch Bias Models
• Simple offset correction:

• Pattern-dependent bias:

• Etch Proximity Correction (EPC)
• Kernel-based convolution:

• 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:

• 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 $\phi$:

• Handles topology changes naturally (merging, splitting)

• Cell-Based/Voxel Methods
• Discretize feature volume into cells
• Apply probabilistic removal rules:

• 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:

• Sheath Physics
• Child-Langmuir law for ion flux:

• Ion angular distribution at wafer surface

• Transport
• Species continuity:

• 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:

• Interatomic potentials: Tersoff, Stillinger-Weber, ReaxFF

• Monte Carlo (MC) Methods
• Statistical sampling of ion trajectories
• Binary collision approximation (BCA) for high energies
• Acceptance probability:

• Kinetic Monte Carlo (KMC)
• Sample reactive events with rates $k_i$:

• 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:

• $A = 1$: 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:

where higher $n$ indicates more directional flux

4.2 Selectivity

Ratio of etch rates between materials:

• Mask selectivity: Target material vs. photoresist/hard mask
• Stop layer selectivity: Target material vs. underlying layer

Example selectivities required:

4.3 Loading Effects

Microloading

Local depletion of reactive species in dense pattern regions:

where:
- $R_0$ = etch rate in isolated feature
- $\beta$ = loading coefficient
- $\rho_{local}$ = local pattern density

Macroloading

Wafer-scale depletion:

where $A_{exposed}$ is total exposed area fraction

4.4 Aspect Ratio Dependent Etching (ARDE)

Deep, narrow features etch slower due to transport limitations:

where $AR = \text{depth}/\text{width}$

Physical mechanisms:

1. Ion Shadowing
- Geometric shadowing angle:

2. Neutral Transport
- Knudsen diffusion coefficient:

• where $d$ is feature diameter

3. Byproduct Redeposition
- Sticking probability affects escape

4.5 Profile Anomalies

5. Mathematical Foundations

5.1 Surface Evolution Equation

General form for surface height $h(x,y,t)$:

where:
- $R_0$ = baseline etch rate
- $V(\theta)$ = visibility/flux function
- $\theta = \arctan(|\nabla h|)$

5.2 Ion Angular Distribution

At wafer surface, ion flux angular distribution:

Common models:

• Gaussian distribution:

• Thompson distribution (for sputtered neutrals):

5.3 Visibility Calculation

For a point on the surface, visibility to incoming flux:

where $\theta_{max}(\phi)$ is determined by local geometry (shadowing)

5.4 Surface Reaction Kinetics

Langmuir-Hinshelwood mechanism:

where surface coverages follow:

• $s_i$ = sticking coefficient
• $k_d$ = desorption rate
• $k_r$ = reaction rate

5.5 Plasma-Surface Interaction Yield

Ion-enhanced etch yield:

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:

6.2 Machine Learning Integration

• Surrogate Models
• Train neural network on physics simulation outputs:

• Loss function:

• Physics-Informed Neural Networks (PINNs)
• Embed physics constraints in loss:

• Where $\mathcal{L}_{physics}$ enforces governing equations

• Virtual Metrology
• Predict CD, profile from chamber sensors:

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:

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:

• Inner Spacer Formation
• Isotropic lateral etch in confined geometry
• Depth control: $\pm 0.5\ \text{nm}$

7.2 3D NAND

Extreme aspect ratio challenges:

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:

• Target: $LER < 1.2\ \text{nm}$ ($3\sigma$)

7.4 Stochastic Effects

At small dimensions, statistical fluctuations dominate:

where $N_{events}$ = number of etching events per feature

8. Industry Tools

8.1 Commercial Software

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:

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:

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:

As geometries shrink and structures become more 3D, accurate etch modeling becomes essential for first-time-right process development and continued yield improvement.