Within-Wafer Non-Uniformity (WIWNU) Characterization
In capacitively coupled plasma (CCP) reactors, the RF electric field creates standing wave patterns that vary with radial position. The electric field distribution follows Bessel function behavior:
where J₀ is the zeroth-order Bessel function, λ is the wavelength at 13.56 MHz (~22m in vacuum, but modified by plasma permittivity), and m represents azimuthal mode number. The radial variation in field intensity directly affects local ionization rates and ion flux to the wafer surface.
The voltage drop across the plasma sheath varies radially due to edge effects and non-uniform plasma density. This affects ion bombardment energy:
where V₀ is the center sheath voltage, R is wafer radius (150mm for 300mm wafers), and α is the edge enhancement factor (typically 0.1-0.3). Higher sheath voltage at the edge increases ion energy and etch rate, contributing to center-fast or edge-fast profiles depending on chemistry.
Ion flux to the wafer surface exhibits radial non-uniformity due to diffusion and drift processes:
where Φ₀ is the peak flux and β controls the width of the distribution. Edge effects cause flux depletion near the wafer periphery due to recombination at chamber walls and reduced confinement.
The residence time of reactive species varies with radial position, affecting local etch chemistry:
where V_chamber is the chamber volume, Q_flow is the gas flow rate, and f(r) is a radial distribution function. Longer residence times at the center lead to greater depletion of reactive species, while the edge receives fresher gas supply. This creates center-slow etch profiles in reaction-rate-limited regimes.
Wafer temperature varies radially due to non-uniform heating from plasma and cooling from the backside:
where ΔT is the center-to-edge temperature difference (typically 5-20°C) and n is typically 2-4. Temperature affects etch rate through Arrhenius-type surface reaction kinetics. A 10°C increase can change etch rate by 5-15% depending on activation energy.
Within-wafer non-uniformity is quantified as the relative standard deviation:
where σ is the standard deviation of etch rates across all measurement zones and μ is the mean. Advanced nodes require WIWNU < 2% for critical layers. Alternative metrics include range (max-min)/2×mean and site-to-site uniformity for specific measurement patterns (49-point, 121-point).
The outer 3-5mm of the wafer (edge exclusion zone) is typically excluded from uniformity calculations due to extreme edge effects. This region experiences:
The penetration depth of RF fields into the plasma is limited by skin effect:
where c is speed of light, ω is angular frequency (2π × 13.56 MHz), and ε_r is the relative permittivity of the plasma (negative for frequencies below plasma frequency). Skin depth of 1-10 cm means that large-area reactors can have significant field attenuation across the wafer, contributing to non-uniformity.
In regions with high plasma density and etch rate, reactive species can be depleted faster than they are replenished by gas flow. This creates a feedback loop where high-rate regions become reaction-limited. The depletion factor D is given by:
where k_etch is the etch rate constant, n_wafer is wafer site density, A(r) is the local area element, and k_flow accounts for gas mixing efficiency.
Wafer bow (curvature) affects gap spacing to the upper electrode, creating non-uniform sheath width:
where h_bow is the bow height at center (can be 50-200 μm). Thermal expansion during processing can increase bow by 20-50%, significantly impacting uniformity. Sheath width scales with gap^0.5 to gap^0.75, so even small gap variations translate to measurable etch rate differences.