Usadel Bilayer Calculator

Estimates T_c and the superconducting gap of a proximitized S/N (or S/S) bilayer using the Cooper-limit approximation.

Material 1 (superconductor)

Material: Al — T_c = 1.20 K, D = 5×10−³ m²/s Ti — T_c = 0.45 K, D = 2×10−&sup4; m²/s Nb — T_c = 9.25 K, D = 2×10−³ m²/s Ta — T_c = 4.40 K, D = 5×10−&sup4; m²/s Au — T_c = 0 K, D = 2×10−² m²/s W — T_c = 0.015 K, D = 3×10⁻⁵ m²/s Custom… Thickness d₁ [nm]:

Material 2 (proximitized normal metal or weaker SC)

Material: Ti — T_c = 0.45 K, D = 2×10−&sup4; m²/s Al — T_c = 1.20 K, D = 5×10−³ m²/s Nb — T_c = 9.25 K, D = 2×10−³ m²/s Ta — T_c = 4.40 K, D = 5×10−&sup4; m²/s Au — T_c = 0 K, D = 2×10−² m²/s W — T_c = 0.015 K, D = 3×10⁻⁵ m²/s Custom… Thickness d₂ [nm]: Calculate

Formulas & model

Quantity	Formula / Note
Coherence length ξ	ξ = √(ℏD / (2πkBTc))
Effective Tc (Cooper limit)	Tc,eff = exp[(d1 ln Tc,1 + d2 ln Tc,2) / (d1 + d2)]
Gap estimate	Δ(0) ≈ 1.76 kB Tc,eff  (BCS)
Regime (Cooper limit)	d < 0.3 ξ for both layers → uniform proximitized bilayer
Regime (coupled)	d < ξ for both layers → coupled but non-uniform gap
Thick film regime	d ≥ ξ for either layer → strong spatial variation, Cooper limit invalid

Note: The Cooper-limit formula assumes equal density of states (thickness-weighted average). It is most accurate when both layers are thin compared to their coherence lengths. For thick films the result is an overestimate and a full Usadel numerical solution is required.