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 φ(x, t):

∂φ/∂t + Vn |∇φ| = 0

Key quantities:

• Unit normal: n̂ = ∇φ / |∇φ|
• Mean curvature: κ = ∇ · n̂ = ∇ · (∇φ / |∇φ|)

2.2 Advantages

• Handles topology changes (merge/split)
• Well-defined normals/curvature everywhere
• Extends naturally to 3D

2.3 Numerical Notes

• Reinitialize to maintain |∇φ| = 1
• Upwind schemes (Godunov, ENO/WENO) for stability
• Fast Marching and Sparse Field are common

2.4 String/Segment Method (2D)

dr_i/dt = V_n(r_i) · n̂_i

• Advantage: simple implementation
• Disadvantage: struggles with topology changes

3. Etch Velocity Models

Velocity decomposition:

V_n = V_physical + V_chemical + V_ion-enhanced

3.1 Physical Sputtering (Yamamura-Sigmund)

Y(θ, E) = (0.042 Q(Z_2) S_n(E) / U_s) [1-√(E_th/E)]^s f(θ)

Angular part: f(θ) = cos^(-f)(θ) exp[-Σ (1/cos θ - 1)]

3.2 Ion-Enhanced Chemical Etching (RIE)

R = k_1 Γ_F θ_F + k_2 Γ_ion Y_phys + k_3 Γ_ion^a Γ_F^b (1 + β θ_F)

• Term 1: chemical
• Term 2: physical sputter
• Term 3: synergistic ion-chemical

3.3 Surface Kinetics (Langmuir-Hinshelwood)

dθ_F/dt = s_0 Γ_F (1-θ_F) - k_d θ_F - k_r θ_F Γ_ion

Steady state: θ_F = s_0 Γ_F / (s_0 Γ_F + k_d + k_r Γ_ion)

4. Transport in High-Aspect-Ratio Features

4.1 Knudsen Diffusion (neutrals)

Γ(z) = Γ_0 P(AR), where P(AR) ≈ 1/(1 + 3AR/8)

More exact: P(L/R) = (8R/3L)(√(1+(L/R)²) - 1)

4.2 Ion Angular Distribution

f(θ) ∝ exp(-m_i v_⊥²/2k_B T_i) cos θ

Mean angle (collisionless sheath): ⟨θ⟩ ≈ arctan(√(T_e/(eV_sheath))) Shadowing: θ_max(z) = arctan(w/2z)

4.3 Sheath Potential

V_s ≈ (k_B T_e / 2e) ln(m_i / 2π m_e)

5. Profile Phenomena

5.1 Bowing (sidewall widening)

V_lateral(z) = ∫₀^θ_max Y(θ') Γ_reflected(θ', z) dθ'

5.2 Microtrenching (corner enhancement)

Γ_corner = Γ_direct + ∫ Γ_incident R(θ) G(geometry) dθ

5.3 Notching (charging)

Poisson: ∇²V = -ρ/(ε₀ ε_r) Charge balance: ∂σ/∂t = J_ion - J_electron - J_secondary Deflection: θ_deflection ≈ arctan(q E_surface L / (2 E_ion))

5.4 ARDE (RIE lag)

ER(AR)/ER_0 = 1/(1 + α AR^β)

6. Computational Approaches

• Monte Carlo (feature scale): launch particles, track, reflect/react, accumulate rates
• Flux-based / view-factor: V_n(x) = Σ_j R_j Γ_j(x) Y_j(θ(x))
• Cellular automata: P_etch(cell) = f(Γ_local, neighbors, 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(θ, E) to feature scale

7.2 Governing Equations Summary

• Surface evolution: ∂S/∂t = V_n n̂
• Neutral transport: v·∇f + (F/m)·∇_v f = (∂f/∂t)_coll
• Ion trajectory: m d²r/dt² = q(E + v×B)

8. Advanced Topics

8.1 Stochastic roughness (LER)

σ²_LER = (2/π² n_s) ∫ PSD(f)/f² df

8.2 Pattern-dependent effects (loading)

∂n/∂t = D∇²n - k_etch A_exposed n

8.3 Machine Learning Surrogates

Profile(t) = NN(Process conditions, Initial geometry, t)

Uses: rapid exploration, inverse optimization, real-time control.

9. Summary and Process Flow

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 Key Equations

9.3 Mathematical Elegance

• Geometry via φ evolution
• Physics via V_n models

Modular structure enables independent improvement of geometry and physics.