Feature Profile Evolution Simulator

Advanced level-set and string-model based simulation of trench and via profile evolution during plasma etching. Model anisotropic vs isotropic components, sidewall angle dynamics, bowing, undercutting, and critical dimension control for sub-10nm features.

Key Physics

Profile evolution combines directional ion bombardment with isotropic radical etching. The level-set method tracks the moving boundary while accounting for angular-dependent yields, redeposition, and shadowing effects critical for high aspect ratio features.

Running
1.0×

Etch Profile Parameters

Etch Depth

285
nm

Bottom CD

48
nm

Sidewall Angle

89.2
degrees

Aspect Ratio

5.9
:1

CD Loss (Total)

4.2
nm

Bowing (Max)

1.8
nm

3D Trench Profile Evolution

Theoretical Background

Level-Set Method

The evolution of the etch front is described by the level-set equation:

∂φ/∂t + F(x, y, z, n)·|∇φ| = 0 where φ is the level-set function and F is the etch rate normal to the surface

String Model for 2D Profiles

Discretized profile evolution using connected nodes:

x_i(t+Δt) = x_i(t) + v_x(θ_i)·Δt y_i(t+Δt) = y_i(t) + v_y(θ_i)·Δt where θ_i is the local surface angle

Angular-Dependent Etch Rate

Combined ion and radical contributions:

R(θ) = R_ion·Y(θ)·cos^n(θ) + R_radical·(1 - α·cos(θ)) Y(θ) = Y_0·[1 + β·cos(θ)] (angular yield function) n ≈ 1-2 for physical etching

Anisotropy Factor Definition

Quantifies directional vs isotropic etching:

A = 1 - (R_lateral / R_vertical) A = 0: purely isotropic A = 1: perfectly anisotropic (vertical only)

Sidewall Angle Calculation

Local slope determines sidewall angle:

θ_wall = arctan(dx/dy) For vertical profile: θ_wall = 90° Reentrant profile: θ_wall > 90°

Critical Dimension Evolution

CD changes due to lateral etching and redeposition:

dCD/dt = -2·R_lateral + 2·Γ_redep·s_redep·V_atom Redeposition flux: Γ_redep = ∫ Y(θ)·Γ_ion·(1-s_top)·dθ

Bowing and Undercutting

Profile distortions from differential etching:

Bowing: occurs when mid-trench etches faster than top/bottom ΔCD_bow = ∫(R_lateral(z) - R_lateral_avg) dt Undercutting: mask erosion causing CD > nominal

Aspect Ratio Dependent Effects

Transport limitations at high AR:

R_eff(AR) = R_0 / [1 + (AR/AR_crit)^p] where AR_crit depends on sticking probability and angular spread

Advanced Profile Simulation Methods

Monte Carlo Trajectory Simulation

Stochastic ion and neutral transport:

Probability of collision in dx: P_coll = 1 - exp(-dx/λ_mfp) Mean free path: λ_mfp = k_B T / (√2 π d² p) Scattering angle: cos(θ_scatter) = 1 - 2ξ/(1 + α - ξ(1-α)) where α = [(m1-m2)/(m1+m2)]² and ξ = random[0,1]

Redeposition and Sputtering Effects

Material removed from sidewalls can redeposit:

Γ_redep(x,y) = ∫∫ Y(x',y',θ')·Γ_ion(x',y')·G(x,y|x',y') dx' dy' where G is geometric view factor accounting for line-of-sight Sticking probability determines capture vs bounce

Microtrench Formation

Enhanced etching at trench bottom corners:

Mechanism: Ion reflection from sidewalls causes focused bombardment Depth: h_microtrench ∝ Γ_reflected · Y_corner · t Mitigation: Reduce pressure, increase directionality, optimize bias

Faceting and Roughness Evolution

Development of crystallographic planes and surface texture:

Facet growth rate depends on Miller indices (hkl): R_facet(hkl) = R_0 · [1 + β_facet · cos(θ_hkl)] Roughness amplitude growth: σ(t) = σ_0 · exp(λ_roughness · t) where λ depends on angular distribution and sticking

Notching at Dielectric Interfaces

Charging-induced ion deflection:

Electric field at interface: E_lateral ∝ (σ_charge / ε_r) Ion deflection angle: Δθ = arctan(eE_lateral·s / (2E_ion)) Notch depth increases with: AR, ion dose, dielectric thickness

Grass and Residue Formation

Incomplete etch clearing mechanisms:

Causes: 1. Micromasking from contamination particles 2. Redeposition of non-volatile etch products 3. Polymer-rich regions with low etch rate Clearing strategies: O₂ descum, over-etch with tuned chemistry

Time-Multiplexed Etching

Pulsed plasma for improved profile control:

Duty cycle: D = t_on / (t_on + t_off) Benefits: - Reduced ion energy spread (narrow IEDF) - Lower surface temperature (less resist damage) - Enhanced radical-to-ion ratio tunability Effective etch rate: R_eff = D · R_continuous · f(t_off)

Optical Emission Endpoint Detection During Profiling

Real-time monitoring of profile evolution:

Intensity change at breakthrough: I(λ) = I_substrate(λ) - I_masklayer(λ) For SiO₂: Monitor 480nm (SiO*), 777nm (O*) For Si: Monitor 288nm (Si*), 390nm (Si⁺) Derivative signal: dI/dt shows sharp peak at endpoint

Process Window and Optimization

Design of Experiments (DOE) for Profile Tuning

Multi-parameter optimization:

Response surface: Profile_metric = β₀ + Σβᵢxᵢ + Σβᵢⱼxᵢxⱼ + ε Key factors: Pressure, Power, Gas ratio, Bias, Temperature Responses: CD, Angle, Depth, Uniformity, Selectivity Use central composite or Box-Behnken designs

Sensitivity Analysis

Parameter tolerance windows:

Sensitivity: S_parameter = ∂(Output) / ∂(Parameter) Critical parameters (high S): Bias voltage, Pressure Robust parameters (low S): Electrode gap, Flow rate 3σ process window: |Δ Output| < Spec when |Δ Param| < 3σ_drift

Multi-Objective Optimization

Balancing competing requirements:

Objectives: 1. Maximize etch rate 2. Minimize CD bias 3. Maximize selectivity 4. Minimize LWR Pareto frontier: Trade-off curve where no objective can improve without degrading another Weighted sum: F = Σwᵢ·fᵢ(x) subject to constraints

Machine Learning for Profile Prediction

Neural network models trained on simulation/experimental data:

Input layer: [Pressure, Power, Time, Gas_flows, Temperature] Hidden layers: Dense networks with ReLU activation Output layer: [Depth, CD, Angle, Roughness, Selectivity] Training: 1000+ simulation runs or wafer measurements Accuracy: R² > 0.95 for interpolation within training space