Etch Profile Mathematical Modeling

Keywords: etch profile modeling, etch profile, plasma etching, level set, arde, rie, profile evolution

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

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

                  Plasma Parameters
                          ↓
              Ion/Neutral Energy-Angle Distributions
                          ↓
    ┌─────────────────────┴─────────────────────┐
    ↓                                           ↓
Transport in Feature                    Surface Chemistry
(Knudsen, charging)                   (coverage, reactions)
    ↓                                           ↓
    └─────────────────────┬─────────────────────┘
                          ↓
                  Local Etch Velocity
                    Vn(x, θ, Γ, T)
                          ↓
              Surface Evolution Equation
              ∂φ/∂t + Vn|∇φ| = 0
                          ↓
                   Etch Profile

9.2 Equations

abla \phi\| = 0$ |

9.3 Mathematical Elegance

• Geometry via $\phi$ evolution
• Physics via $V_n$ models

Modular structure enables independent improvement of geometry and physics.

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