Niobium's intrinsic coherence length and penetration depth revisited using low-energy muon spin spectroscopy and secondary-ion mass spectrometry

We report direct, simultaneous measurements of the London penetration depth ($λ_L$) and Bardeen-Cooper-Schrieffer (BCS) coherence length ($ξ_0$) in oxygen-doped niobium, with impurity concentrations spanning the “clean” to “dirty” limits. Two depth-resolved techniques - low-energy muon spin spectroscopy (LE-$μ$SR) and secondary-ion mass spectrometry (SIMS) - were used to quantify the element’s Meissner screening profiles, analyzed within a framework that accounts for nonlocal electrodynamics. The analysis indicates intrinsic length scales of $λ_L = 29.1(10)$ nm and $ξ_0 = 39.9(25)$ nm, corresponding to a Ginzburg-Landau (GL) parameter of $κ= 0.70(5)$. The obtained $λ_L$ and $κ$ values, accurately quantified at the nanoscale, are smaller than values commonly used in applications and modeling, and indicate that clean niobium lies at the boundary between type-I and type-II superconductivity, supporting the contemporary view that its intrinsic state may be type-I.

💡 Research Summary

The authors present a comprehensive study that directly determines the intrinsic London penetration depth (λL) and BCS coherence length (ξ0) of niobium (Nb) using a combination of low‑energy muon spin rotation (LE‑μSR) and secondary‑ion mass spectrometry (SIMS). Six Nb samples were prepared with controlled oxygen doping, spanning impurity concentrations from the clean limit (electron mean free path ℓ ≈ 380 nm) to the dirty limit (ℓ ≈ 13 nm). SIMS provided quantitative profiles of carbon, nitrogen, and oxygen, enabling calculation of ℓ via an empirical relation.

LE‑μSR experiments were performed at the Swiss Muon Source on the μE4 beamline. Positive muons (μ⁺) with energies up to 15 keV were implanted at depths of 10–150 nm. In a transverse geometry, a 25 mT magnetic field was applied perpendicular to the muon spin polarization. The time‑dependent asymmetry A(t) was recorded in both the normal state (12 K) and the Meissner state (3 K). In the superconducting state, the asymmetry shows strong damping that increases with implantation energy, reflecting a depth‑dependent internal field distribution p(B). By fitting p(B) with a sum of Gaussians, the mean internal field ⟨B⟩(E) was extracted for each implantation energy.

The relationship ⟨B⟩(E)=∫B(z)ρ(z,E)dz links the measured ⟨B⟩ to the true magnetic screening profile B(z), where ρ(z,E) is the muon stopping distribution obtained from TRIM.SP Monte‑Carlo simulations. Two analysis strategies were employed. The “staged” approach first phenomenologically fits ⟨B⟩(E) to obtain an effective field profile, then compares the result with a theoretical model that includes non‑local electrodynamics via the Pippard kernel K(q). The “direct” approach simultaneously fits the full LE‑μSR spectra using the convolution of ρ(z,E) with the non‑local B(z) model. Both methods were applied in local (London) and non‑local limits.

The non‑local model describes the current‑vector potential relationship through K(q)=ξ(T)λ(T)²/ξ(0)·(3/2)·g(x)x³, with x=qξ(T) and g(x)=(1+x²)arctan x−x. Temperature dependencies λ(T)=λL/√{1−(T/Tc)⁴} and ξ(T)⁻¹=πJ(0,T)/(2ξ0)+1/ℓ were incorporated, where J(0,T) contains the BCS gap Δ(T). Global fits across all samples yielded λL=29.1 ± 1.0 nm, ξ0=39.9 ± 2.5 nm, and a Ginzburg‑Landau parameter κ=0.70 ± 0.05. These values are about 10 nm smaller for λL than the widely quoted λL≈39 nm used in SRF cavity design, but agree with recent ab‑initio calculations and earlier β‑NMR measurements. The extracted ξ0 matches prior estimates based on critical field and specific‑heat data.

A systematic dependence of the effective penetration depth λ0 on ℓ was observed, following the Pippard relation λ0≈λL·√{1+πξ0/(2ℓ)}. This confirms that impurity scattering shifts the apparent λ, while the intrinsic λL remains constant.

The implications are significant. The intrinsic κ≈0.70 lies just below the type‑I/II boundary (1/√2≈0.707), suggesting that pure Nb is fundamentally a type‑I superconductor, contrary to the conventional classification as a weak type‑II material (κ≈0.8). This re‑evaluation impacts the modeling of superconducting radio‑frequency (SRF) cavities, where surface resistance and field penetration are sensitive to λL and κ. Moreover, the successful integration of SIMS impurity profiling with depth‑resolved LE‑μSR demonstrates a powerful methodology for correlating material purity with superconducting electrodynamics, applicable to other elemental and compound superconductors. The work thus resolves longstanding discrepancies in Nb’s fundamental length scales and provides a robust experimental framework for future investigations of clean‑limit superconductivity.