Semiconductor Manufacturing Process: Etch Modeling

Keywords: etch modeling, plasma etch, RIE, reactive ion etching, etch simulation, DRIE

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:

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

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:

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

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:

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

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