Endpoint Detection Physics & Equations
Optical Emission Spectroscopy (OES)
Plasma emission intensity from excited species follows excitation-de-excitation balance:
I(λ) = (h·c/λ) · n* · A
where:
I(λ) = emission intensity at wavelength λ
n* = excited state density = (k_exc · n_e · n_gas) / (A + k_quench · n_gas)
A = Einstein coefficient (spontaneous emission rate)
k_exc = electron impact excitation rate
k_quench = quenching rate coefficient
n_e = electron density
n_gas = ground state gas density
For Si atoms etched from wafer:
Si: 288 nm (3P → 1D transition), A = 2.17×10⁷ s⁻¹
Si emission ∝ Si etch product concentration in plasma
Emission Line Ratio Method
Ratio of product emission to reactant emission indicates material transition:
R(t) = I_product(t) / I_reactant(t)
For SiO₂ → Si transition:
Product: Si emission (288 nm) - increases at endpoint
Reactant: F emission (704 nm) - decreases at endpoint
Endpoint criterion: dR/dt > threshold AND R > R_critical
R_critical typically 0.5-2.0 depending on process
Laser Interferometry
Interference from reflections at film surfaces creates periodic intensity pattern:
I_detector = I₀ · [1 + V · cos(Φ)]
where phase difference:
Φ = (4π · n · d) / λ + Φ₀
n = refractive index of film
d = film thickness
λ = laser wavelength
V = fringe visibility (0-1)
Φ₀ = phase offset from substrate reflectivity difference
As film etches: Φ(t) = (4π·n/λ) · [d₀ - R_etch·t]
One complete fringe cycle = λ/(2n) thickness change
For SiO₂ (n=1.46) at 633nm: one cycle = 217 nm
Etch Rate from Interferometry
Real-time etch rate determined from fringe frequency:
R_etch = (λ / 2n) · f_fringe
where:
f_fringe = frequency of intensity oscillations (Hz)
Example: f = 0.1 Hz, λ = 633 nm, n = 1.46
→ R_etch = 21.7 nm/s = 1300 nm/min
Endpoint detected when fringes cease (film cleared)
Mass Spectrometry
Volatile etch products detected by quadrupole mass analyzer:
Signal(m/z) ∝ partial pressure ∝ etch rate × sticking coefficient
For SiO₂ etching in CF₄/O₂:
Primary products: SiF₄ (m/z = 104), CO₂ (m/z = 44), CO (m/z = 28)
Si etching in CF₄:
Primary products: SiF₄ (m/z = 104), SiF₃ (m/z = 85)
Signal intensity:
I(m/z) = σ_ion · n(m/z) · I_electron
σ_ion = ionization cross-section
n(m/z) = species partial density
I_electron = electron beam current (70 eV standard)
Product Formation Rate
Etch product generation linked to material removal:
Γ_product = (R_etch · A_wafer · ρ_material) / M_product
where:
Γ_product = molecular flow rate (molecules/s)
R_etch = etch rate (cm/s)
A_wafer = wafer area (cm²)
ρ_material = material density (g/cm³)
M_product = molecular weight (g/mol)
For 300mm wafer, SiO₂ at 100 nm/min:
Γ_SiF₄ = 2.4×10¹⁷ molecules/s = 4×10⁻⁷ mol/s
Signal Derivative Detection
First derivative of OES or mass spec signal maximizes at endpoint:
S'(t) = dS/dt = [S(t+Δt) - S(t-Δt)] / (2Δt)
Endpoint criterion:
|S'(t)| > threshold AND S''(t) changes sign
Second derivative for noise rejection:
S''(t) = [S(t+Δt) - 2S(t) + S(t-Δt)] / Δt²
Moving average filter applied for SNR > 10
Multi-Modal Fusion
Combine multiple detection methods for robust endpoint determination:
Confidence(t) = w₁·C_OES(t) + w₂·C_interf(t) + w₃·C_MS(t)
where weights sum to 1: w₁ + w₂ + w₃ = 1
Individual confidences based on:
C_OES = sigmoid[(R(t) - R_threshold) / σ_R]
C_interf = 1 if fringes stopped, 0 otherwise
C_MS = sigmoid[(I_product(t) - I_threshold) / σ_I]
Typical weights: w₁=0.5 (OES), w₂=0.3 (interf), w₃=0.2 (MS)
Endpoint declared when Confidence > 0.8