etch plasma 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 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 | 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(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 | Phenomenon | Equation | | Level set evolution | ∂φ/∂t + V_n |∇φ| = 0 | | Angular yield | Y(θ) = Y_0 cos^(-f)(θ) exp[-Σ(1/cos θ - 1)] | | ARDE | ER(AR)/ER_0 = 1/(1 + α AR^β) | | Transmission prob. | P(AR) = 1/(1 + 3AR/8) | | Surface coverage | θ_F = s_0Γ_F / (s_0Γ_F + k_d + k_rΓ_ion) | 9.3 Mathematical Elegance - Geometry via φ evolution - Physics via V_n models Modular structure enables independent improvement of geometry and physics.