Fig. 3: In-plane magnetic field effect on the superconducting parameters of Nb.

a In-plane magnetic field dependence of the London penetration depth \({\lambda }_{{{{{{\rm{L}}}}}}}(H)\) (red circles) is extracted from the experimental optical conductivity (see text) by using the sum-rule analysis (Methods). The white, yellow, and green shaded areas indicate the magnetic field ranges of the gapped, the gapless, and the normal state, respectively. The inset shows the normalized London penetration depth \({\lambda }_{{{{{{\rm{L}}}}}}}(H)/{\lambda }_{{{{{{\rm{L}}}}}}}(0)\) (blue circles). The green line signaling a quadratic magnetic field term is the theory of the nonlinear Meissner effect of ref. ^35,36. (Methods). The deviation from this behavior in the gapless regime is an extreme nonlinear Meissner effect. b Dependence of the London penetration depth \({\lambda }_{{{{{{\rm{L}}}}}}}(\Gamma )\) on the pair-breaking parameter Γ. The inset shows the normalized inverse squared penetration depth \({\lambda }_{{{{{{\rm{L}}}}}}}^{2}(0)/{\lambda }_{{{{{{\rm{L}}}}}}}^{2}(\Gamma )\). This is the same as the normalized superfluid density. c Order parameter ∆ (red circles) and spectroscopic gap Ω_G (blue circles) of Nb as functions of the pair-breaking parameter Γ. The experimentally accessed Γ values are indicated as purple ticks on the horizontal axis. The solid curves are the SBW theory that relates ∆ and Ω_G with Γ. The solid black line is the linear function with a slope of unity. The vertical dashed line (magenta) is used to locate Γ_g = 4.9 cm⁻¹, the pair-breaking parameter that marks the onset of the gapless regime. d Order parameter ∆ (red circles) and spectroscopic gap Ω_G (blue circles) of Nb as functions of the applied in-plane magnetic field. The magnetic field \({{\mu }_{0}H}_{{{{{{\rm{g}}}}}}}=2.4\,{{{{{\rm{T}}}}}}\) that marks the onset of the gapless regime is indicated on the horizontal axis.