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

Abstract

We report 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. Depth-resolved low-energy muon spin spectroscopy (LE-$μ$SR) was 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 $κ$ value are smaller than values commonly used in applications and modeling, indicating 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.

Figures (3)

• Figure 1: Typical data in Nb (sample Nb-SR18). In the normal state, the $\mu^{+}$ asymmetry $A(t)$ is weakly damped with a spin-precession rate that is independent of implantation energy $E$. By contrast, the damping of $A(t)$ in the Meissner state is strong, increasing with increasing $E$, which is accompanied by a decrease in the rate of spin-precession. The solid colored lines denote fits to a model approximating the field distribution as a sum of Gaussians (described in the text and Supporting Material sm). The $\mu^{+}$ implantation profile $\rho(z, E)$ and mean stopping depth $\langle z \rangle$, simulated using the TRIM.SP Monte Carlo code 1991-Eckstein-SSMS-10trimsp, are shown in the inset for each $E$.
• Figure 2: Meissner screening profiles in each Nb sample, derived from a "staged" analysis of the data. Here, the mean magnetic field $\langle B \rangle$ is plotted as a function of $\mu^{+}$ implantation energy $E$ (with the corresponding mean stopping depth $\langle z \rangle$ also indicated) in both the normal and Meissner states. Differences in screening capacity are visually evident, with each sample's oxygen impurity concentration $[\ch{O}]$ and electron mean-free-path $\ell$ (derived from ) indicated in each panel. The solid and dotted colored lines represent a global fit of the "staged" data to the nonlocal model for the screening profile $B(z)$ convolved with the $\mu^{+}$ stopping distribution $\rho(z,E)$ (described in the text). For comparison, the dashed colored lines show similar fits assuming $B(z)$ is governed by local electrodynamics.
• Figure 3: Dependence of the (effective) magnetic penetration depth at 0, determined the "staged" analysis approach in the local limit, on the electron mean-free-path $\ell$. The solid coloured line denotes a best fit to \ref{['eq:lambda-impurity']}, with corresponding values for the London penetration depth $\lambda_{L}$ and the Pippard/ coherence length indicated in the plot.