DRIE Bosch Process Physics
Scalloping Formation Model
The characteristic scallop pattern in Bosch DRIE results from alternating etch and passivation steps:
Scallop amplitude (half-depth):
h_scallop = (R_etch × t_etch - R_pass × t_pass) / 2
Where:
- R_etch: Silicon etch rate during SF₆ step (μm/min)
- t_etch: Duration of etch step (s)
- R_pass: Passivation layer removal rate (typically minimal)
- t_pass: Duration of passivation step (s)
Lateral undercut per cycle:
Δx_lateral = R_lateral × t_etch
Total scallop wavelength:
λ_scallop ≈ h_scallop / tan(sidewall_angle)
SF₆ Silicon Etch Rate
Fluorine radical-based silicon etching with ion enhancement:
R_Si = k_Si × √(P_ion) × exp(-E_a/RT) × [F]
Where:
- k_Si: Rate constant (material dependent)
- P_ion: Ion bombardment power density (W/cm²)
- E_a: Activation energy ≈ 0.1-0.2 eV (low, ion-enhanced)
- R: Gas constant (8.314 J/mol·K)
- T: Substrate temperature (K)
- [F]: Fluorine radical concentration (∝ ICP power, SF₆ flow)
Ion flux calculation:
Γ_ion = (I_ion / e) = (J_s / e) × A
J_s ≈ 0.4 × n_e × √(k × T_e / M_ion) [Bohm current density]
Chemical vs Physical contribution:
R_total = R_chemical + Y_physical × Γ_ion
Y_physical ≈ 0.5-2.0 Si atoms/ion (angle dependent)
C₄F₈ Polymer Deposition
Fluorocarbon polymer passivation layer formation:
Polymer thickness per cycle:
d_poly = k_poly × P_C4F8 × t_pass × (1 - exp(-E_stick/kT))
Where:
- k_poly: Deposition rate constant
- P_C4F8: C₄F₈ partial pressure (mTorr)
- t_pass: Passivation time (s)
- E_stick: Sticking energy barrier
Polymer composition: (CF₂)_n chains
F/C ratio: 1.5-2.0 (tunable by adding O₂ or H₂)
Bottom vs sidewall deposition:
d_bottom / d_sidewall = 1 + β × (Γ_ion / Γ_neutral)
Ion bombardment removes polymer from horizontal surfaces
Sidewalls protected → anisotropic profile
Aspect Ratio Dependent Etching (ARDE)
Etch rate decreases with increasing trench depth due to transport limitations:
ARDE Model (Gotoh):
R(AR) = R₀ / (1 + K × AR^n)
Where:
- R₀: Etch rate for shallow features
- AR: Aspect ratio (depth/width)
- K: ARDE coefficient (depends on pressure, chemistry)
- n: Exponent ≈ 1-2 (typically 1.5)
Physical origin:
1. Neutral shadowing: Knudsen transport reduces flux at high AR
Kn = λ_mfp / D_trench
If Kn > 1: Molecular flow regime (∝ 1/AR)
2. Ion angular distribution: Finite beam divergence
Effective solid angle Ω decreases with AR
3. Reaction product removal: Re-deposition at depth
Mitigation strategies:
- Higher pressure → lower Kn → better transport
- Pulsed plasma → time for gas renewal
- Overetch compensation based on feature size
Sidewall Angle Evolution
Profile angle changes during etching:
Sidewall angle:
θ_wall = arctan(dz/dx) ≈ 90° - α_deviation
Deviation from vertical:
α_deviation = f(ion_angle, polymer_coverage, lateral_etch)
Ion trajectory deflection in sheath:
α_ion = arctan(v_lateral / v_vertical)
Collisions in sheath → angular spread ≈ 2-5°
Polymer asymmetry:
Thicker polymer on one side → bowing/tilting
Controlled by: rotation, gas injection symmetry
Microloading effect:
Dense patterns → local reactant depletion → bowing
Isolated trenches → faster etch, more vertical
Surface Roughness Evolution
Roughness increases with number of cycles:
RMS roughness:
R_rms = √[Σ(z_i - z_mean)² / N]
Scallop contribution:
R_scallop ≈ h_scallop / √3 (sinusoidal approximation)
Microscopic roughness:
R_micro from: ion damage, crystallographic effects, mask roughness transfer
Total roughness (independent sources):
R_total = √(R_scallop² + R_micro² + R_mask²)
Power spectral density:
PSD(f) = |FFT(roughness profile)|²
Low f: Scallops (controlled)
High f: Stochastic etching (material dependent)
Smoothness optimization:
- Shorter cycles → smaller scallops
- Cryogenic T → reduced lateral etching
- Post-etch smoothing (oxidation + strip)
Temperature Dependencies
Substrate temperature critically affects process:
Cryogenic regime (T < -50°C):
- Reduced F radical recombination → higher [F]
- Enhanced physisorption of reactants
- Polymer more stable → better sidewall protection
- Lower lateral etch → higher anisotropy
Room temperature (T ≈ 20°C):
- Balanced chemical/physical etching
- Moderate polymer stability
- Higher etch rates but less anisotropic
Elevated temperature (T > 50°C):
- Polymer desorption/decomposition
- Faster chemistry but poor selectivity
- Trenching, undercutting issues
Thermal activation energy:
R(T) = A × exp(-E_a / kT)
For ion-enhanced: E_a low (0.1-0.2 eV)
For thermal: E_a high (0.5-1.5 eV)
Cycle Timing Optimization
Balancing etch and passivation for optimal profile:
Cycle time ratio:
r = t_etch / t_pass
Optimal r for vertical walls:
r_opt ≈ (R_etch_lateral / R_polymer_dep) × (sidewall_area / bottom_area)
For high aspect ratio (AR > 20):
- Longer passivation → ensure sidewall coverage at depth
- Shorter etch → reduce scallop amplitude
- More cycles → finer control but slower process
Throughput consideration:
Total time = N_cycles × (t_etch + t_pass + t_transition)
t_transition ≈ 0.5-1.0 s (gas switching, stabilization)
Etch depth per cycle:
Δd = R_etch × t_etch (decreases with AR)
Target depth:
N_cycles = d_target / Δd_avg
Requires ARDE correction