etch profile modeling
**Etch Profile Mathematical Modeling**
1. Introduction
Plasma etching is a critical step in semiconductor manufacturing where material is selectively removed from a wafer surface. The etch profile—the geometric shape of the etched feature—directly determines device performance, especially as feature sizes shrink below 5 nm.
1.1 Types of Etching
- Wet Etching: Uses liquid chemicals; typically isotropic; rarely used for advanced patterning
- Dry/Plasma Etching: Uses reactive gases and plasma; can be highly anisotropic; dominant in modern fabrication
1.2 Key Profile Characteristics to Model
- Sidewall angle: Ideally $90°$ for anisotropic etching
- Etch depth: Controlled by time and etch rate
- Undercut: Lateral etching beneath the mask
- Taper: Deviation from vertical sidewalls
- Bowing: Curved sidewall profile (mid-depth widening)
- Notching: Localized undercutting at material interfaces
- ARDE: Aspect Ratio Dependent Etching—etch rate variation with feature dimensions
- Loading effects: Pattern-density-dependent etch rates
2. Surface Evolution Equations
The challenge is tracking a moving boundary under spatially varying, angle-dependent removal rates.
2.1 Level Set Method
The surface is the zero level set of $\phi(\mathbf{x}, t)$:
$$
\frac{\partial \phi}{\partial t} + V_n |
abla \phi| = 0
$$
Key quantities:
- Unit normal: $\hat{n} =
abla \phi / |
abla \phi|$
- Mean curvature: $\kappa =
abla \cdot \hat{n} =
abla \cdot (
abla \phi / |
abla \phi|)$
2.2 Advantages
- Handles topology changes (merge/split)
- Well-defined normals/curvature everywhere
- Extends naturally to 3D
2.3 Numerical Notes
- Reinitialize to maintain $|
abla \phi| = 1$
- Upwind schemes (Godunov, ENO/WENO) for stability
- Fast Marching and Sparse Field are common
2.4 String/Segment Method (2D)
$$
\frac{d\mathbf{r}_i}{dt} = V_n(\mathbf{r}_i) \cdot \hat{n}_i
$$
- Advantage: simple implementation
- Disadvantage: struggles with topology changes
3. Etch Velocity Models
Velocity decomposition:
$$
V_n = V_{\text{physical}} + V_{\text{chemical}} + V_{\text{ion-enhanced}}
$$
3.1 Physical Sputtering (Yamamura-Sigmund)
$$
Y(\theta, E) = \frac{0.042\, Q(Z_2)\, S_n(E)}{U_s}\Big[1-\sqrt{E_{th}/E}\Big]^s f(\theta)
$$
Angular part:
$$
f(\theta) = \cos^{-f}(\theta)\, \exp[-\Sigma (1/\cos\theta - 1)]
$$
3.2 Ion-Enhanced Chemical Etching (RIE)
$$
R = k_1 \Gamma_F \theta_F + k_2 \Gamma_{\text{ion}} Y_{\text{phys}} + k_3 \Gamma_{\text{ion}}^a \Gamma_F^b (1 + \beta \theta_F)
$$
- Term 1: chemical
- Term 2: physical sputter
- Term 3: synergistic ion-chemical
3.3 Surface Kinetics (Langmuir-Hinshelwood)
$$
\frac{d\theta_F}{dt} = s_0 \Gamma_F (1-\theta_F) - k_d \theta_F - k_r \theta_F \Gamma_{\text{ion}}
$$
Steady state: $\theta_F = s_0 \Gamma_F / (s_0 \Gamma_F + k_d + k_r \Gamma_{\text{ion}})$
4. Transport in High-Aspect-Ratio Features
4.1 Knudsen Diffusion (neutrals)
$$
\Gamma(z) = \Gamma_0 P(AR), \quad P(AR) \approx \frac{1}{1 + 3AR/8}
$$
More exact: $P(L/R) = \tfrac{8R}{3L}(\sqrt{1+(L/R)^2} - 1)$
4.2 Ion Angular Distribution
$$
f(\theta) \propto \exp\Big(-\frac{m_i v_\perp^2}{2k_B T_i}\Big) \cos\theta
$$
Mean angle (collisionless sheath): $\langle\theta\rangle \approx \arctan\!\big(\sqrt{T_e/(eV_{\text{sheath}})}\big)$
Shadowing: $\theta_{\max}(z) = \arctan(w/2z)$
4.3 Sheath Potential
$$
V_s \approx \frac{k_B T_e}{2e} \ln\Big(\frac{m_i}{2\pi m_e}\Big)
$$
5. Profile Phenomena
5.1 Bowing (sidewall widening)
$$
V_{\text{lateral}}(z) = \int_0^{\theta_{\max}} Y(\theta')\, \Gamma_{\text{reflected}}(\theta', z)\, d\theta'
$$
5.2 Microtrenching (corner enhancement)
$$
\Gamma_{\text{corner}} = \Gamma_{\text{direct}} + \int \Gamma_{\text{incident}} R(\theta) G(\text{geometry})\, d\theta
$$
5.3 Notching (charging)
Poisson: $
abla^2 V = -\rho/(\epsilon_0 \epsilon_r)$
Charge balance: $\partial \sigma/\partial t = J_{\text{ion}} - J_{\text{electron}} - J_{\text{secondary}}$
Deflection: $\theta_{\text{deflection}} \approx \arctan\big(q E_{\text{surface}} L / (2 E_{\text{ion}})\big)$
5.4 ARDE (RIE lag)
$$
\frac{ER(AR)}{ER_0} = \frac{1}{1 + \alpha AR^\beta}
$$
6. Computational Approaches
- Monte Carlo (feature scale): launch particles, track, reflect/react, accumulate rates
- Flux-based / view-factor: $V_n(\mathbf{x}) = \sum_j R_j \Gamma_j(\mathbf{x}) Y_j(\theta(\mathbf{x}))$
- Cellular automata: $P_{\text{etch}}(\text{cell}) = f(\Gamma_{\text{local}}, \text{neighbors}, \text{material})$
- DSMC (gas transport): molecule tracing with probabilistic collisions
7. Multi-Scale Integration
| Scale | Range | Physics | Method |
|---------|----------|-------------------------------|-------------------------|
| Reactor | cm–m | Plasma generation, gas flow | Fluid / hybrid PIC-MCC |
| Sheath | μm–mm | Ion acceleration, angles | Kinetic / fluid |
| Feature | nm–μm | Transport, surface evolution | Monte Carlo + level set |
| Atomic | Å | Reaction mechanisms, yields | MD, DFT |
7.1 Coupling
- Reactor → species densities/temps/fluxes to sheath
- Sheath → ion/neutral energy-angle distributions to feature
- Atomic → yield functions $Y(\theta, E)$ to feature scale
7.2 Governing Equations Summary
- Surface evolution: $\partial S/\partial t = V_n \hat{n}$
- Neutral transport: $\mathbf{v}\cdot
abla f + (\mathbf{F}/m)\cdot
abla_v f = (\partial f/\partial t)_{\text{coll}}$
- Ion trajectory: $m\, d^2\vec{r}/dt^2 = q(\vec{E} + \vec{v}\times\vec{B})$
8. Advanced Topics
8.1 Stochastic roughness (LER)
$$
\sigma_{LER}^2 = \frac{2}{\pi^2 n_s} \int \frac{PSD(f)}{f^2} \, df
$$
8.2 Pattern-dependent effects (loading)
$$
\frac{\partial n}{\partial t} = D
abla^2 n - k_{\text{etch}} A_{\text{exposed}} n
$$
8.3 Machine Learning Surrogates
$$
\text{Profile}(t) = \mathcal{NN}(\text{Process conditions}, \text{Initial geometry}, t)
$$
Uses: rapid exploration, inverse optimization, real-time control.
9. Summary and Diagrams
9.1 Complete Flow
```svg
```
9.2 Equations
| Phenomenon | Equation |
|----------------------|-------------------------------------------------|
| Level set evolution | $\partial \phi/\partial t + V_n \|
abla \phi\| = 0$ |
| Angular yield | $Y(\theta) = Y_0 \cos^{-f}(\theta) \exp[-\Sigma(1/\cos\theta - 1)]$ |
| ARDE | $ER(AR)/ER_0 = 1/(1 + \alpha AR^\beta)$ |
| Transmission prob. | $P(AR) = 1/(1 + 3AR/8)$ |
| Surface coverage | $\theta_F = s_0\Gamma_F / (s_0\Gamma_F + k_d + k_r\Gamma_{\text{ion}})$ |
9.3 Mathematical Elegance
- Geometry via $\phi$ evolution
- Physics via $V_n$ models
Modular structure enables independent improvement of geometry and physics.