Search for inhomogeneous Meissner screening in Nb induced by low-temperature surface treatments Open Access

Low-temperature baking (LTB) treatments are important for the surface preparation of Nb superconducting radio frequency (SRF) cavities,^1–3 a class of superconducting resonators⁴ used in particle accelerators. These procedures, performed either in vacuum^5,6 or in a low-pressure gaseous environment,^7,8 induce changes to Nb’s supercurrent density in its near-surface region through the introduction of intrinsic (e.g., dissolved oxygen from its surface oxide layer⁵) or extrinsic (e.g., infused nitrogen from the treatment atmosphere⁸) shallow impurity profiles. Chiefly, these treatments are known to mitigate the high-field Q slope (HFQS) in SRF cavities,^9–12 wherein a sudden drop in a cavity’s quality factor Q is coincident with a sharp increase in its surface resistance.³ Their mechanism of action is thought to be connected to the nanoscale alternations taking place in Nb’s near-surface region (i.e., the topmost ∼50 nm), where empirical observations^8,10,12 and model predictions^13–15 suggest that the treatment-induced defects are inhomogeneous with depth. Of particular interest is how this impacts Nb’s ability to remain in a flux-free Meissner state,^16–20 which is essential for SRF cavity operation.³ As several material properties are proportional to impurity concentration (e.g., carrier mean-free-path ℓ^21,22), they too are expected to vary non-uniformly below the surface, which should influence the element’s Meissner response;^12,15,23 however, directly probing their influence remains challenging.

Early measurements on a Nb SRF cavity cutout baked at 120 °C for 48 h (which we refer to henceforth as “120 °C bake”) showed an abrupt discontinuity in its Meissner profile,²⁴ suggesting the presence of a spatially inhomogeneous near-surface supercurrent density. Among the initial efforts to model this behavior (see, e.g., Refs. 25 and 26), one approach,²⁷ in analogy with superconducting multilayers,^28–31 approximated the inhomogeneous surface region as a superconductor–superconductor (SS) bilayer (i.e., a thin “dirty” layer atop a “clean” substrate), correctly reproducing the treatment’s abated near-surface supercurrent density.²⁷ Despite this breakthrough,³² subsequent measurements on a similarly treated Nb sample showed no such feature,³³ with true bilayer samples also producing distinct screening properties.³⁴ These differences were explained in a recent commentary,³⁵ with the discontinuity attributed to an analysis artifact. Interestingly, this work also uncovered a large non-superconducting surface “dead layer” d ≈ 35 nm.³⁵ While d ≳ 5 nm for Nb is common,^{24,33,36–38} likely due to the roughness^39,40 introduced by its surface oxidation,⁴¹ such a large value is suggestive of a surface-localized region with a diminished screening capacity. This finding is consistent with the shallow impurity profiles found in closely related LTB treatments,^15,42,43 which are predicted to “distort” the Meissner profile close to the surface (see, e.g., Ref. 15). The effect is, however, likely both gradual and subtle,^15,35 requiring scrupulous inspection.

In this work, we revisit measurements of Nb’s Meissner screening profile obtained by low-energy muon spin rotation (LE-μSR)^44,45—a sensitive, non-destructive technique for interrogating subsurface magnetic fields with nanometer depth resolution—for several LTB treatments and search for evidence of spatially inhomogeneous screening. Specifically, we examine results obtained for a Nb SRF cavity “cold spot” cutout whose surface underwent electro-polishing (EP)⁴⁶ + 120 °C baking⁵ (see Refs. 24 and 35 and Fig. 1), an SRF-grade Nb plate with a surface prepared by buffered chemical polishing (BCP)⁴⁶ + 120 °C baking⁵ (see Ref. 33 and Figs. 2 and 4), and another SRF Nb plate with a surface prepared by BCP⁴⁶ + “N₂ infusion”⁸ (see Ref. 33 and Figs. 3 and 4). By comparing the form of each sample’s screening profile against a recently proposed phenomenological model,¹² we search for signatures of inhomogeneous screening.

In LE-μSR,^44,45 positive muons μ⁺ (spin S = 1/2; gyromagnetic ratio γ/(2π) = 135.539 MHz T⁻¹; mean lifetime τ = 2.197 s) are implanted in a host material at an energy E between 0.5 and 30 keV, where they probe subsurface electromagnetic fields at depths up to ∼200 nm.⁴⁷ The technique is highly sensitive to spatial gradients in the magnetic field, making it well suited for the study of Meissner profiles.⁴⁵ In the measurement, one observes the μ⁺ spin-precession signal (detected via its radioactive decay products), which provides a measure of the field distribution p(B) sampled by the μ⁺ stopping profile ρ(z, E),⁴⁸ where z denotes the depth below the surface. Specific experimental details can be found in Refs. 24, 33, and, 34, with broader technique overviews provided elsewhere.^49–51 

To quantify the screening behavior explicitly, we fit the LE-μSR data in Figs. 1–3, using Eq. (1) and numeric solutions to Eq. (4), constraining λ(z) to follow Eq. (5) with λ_bulk = 29 nm (in accordance with “clean” Nb’s literature average^33,38) and parameterizing B(z) to contain both a finite d and a geometrically enhanced field $Bapplied/(1−Ñ)$⁠, akin to Eqs. (2) and (3). Fit results are shown in Figs. 1–3, in good agreement with the measurements. For comparison, screening profiles assuming a spatially homogeneous λ [Eqs. (1)–(3)] are also shown. A summary of these results is given in Table I, and we consider the details below.

First, we remark that the results obtained using the two screening models are quite similar; both provide the correct qualitative form of ⟨B⟩(E), yielding virtually identical values for the “experimental” parameters B_applied and $Ñ$ (see Table I). Although some quantitative differences are apparent (e.g., the E or ⟨z⟩ at which ⟨B⟩ begins to decay), both models provide a reasonable description of the Meissner profiles, as evidenced by their goodness-of-fit metric $χreduced2$⁠. This is expected, given the applicability of the London model in previous work;^24,33,35 however, we find that the inhomogeneous screening model does a better job of capturing the features of each dataset (see Table I). While this could simply be a result of the fit function’s “extra” degrees-of-freedom (i.e., six vs. four parameters), a judicious inspection of the d values suggests otherwise.

The ds identified by the inhomogeneous screening model are all systematically lower than those extracted assuming a homogeneous Meissner response (see Table I). This is most pronounced for the “EP + 120 °C bake” treatment, with the differences visible by eye in Fig. 1. Specifically, the “break in” of ⟨B⟩ from the (homogeneous) London model is much more abrupt than the data, in contrast to the smoother curvature of the generalized London equation. In essence, the London model overcompensates for this discrepancy by lengthening d, which we take as evidence for near-surface inhomogeneous screening. Similar behavior is also observed for the “BCP + 120 °C bake” sample (see Fig. 2), although the effect is less pronounced. Surprisingly, this “feature” is virtually absent from the “BCP + N₂ infusion” sample, at odds with the results in Ref. 12. It is important to stress that neither treatment^5,8 meaningfully increases the surface oxide thickness (see, e.g., Ref. 58). At this juncture, we note that the above-mentioned analysis has only considered the case of local electrodynamics governing Nb’s Meissner response; however, nonlocal effects^59,60 are also known to produce diminished screening close to the surface (see, e.g., Ref. 36). While nonlocal effects are most pronounced in the “clean” limit, they are known to be weak for Nb,³⁶ and on the basis of the impure nature of the sample surfaces (i.e., from their LTB treatments),^8,13,15 we rule out their importance. Thus, we take the differences in the ds identified by each model, in conjunction with the overall fit quality $χreduced2$⁠, as evidence supporting inhomogeneous near-surface field screening.⁶¹ 

Further evidence supporting this notion can be gleaned from the remaining fit parameters. For the two “120 °C bake”⁵ samples, despite their different surface polishings (i.e., EP vs. BCP),⁴⁶ we find that their λ_surface and L_D values are similar, suggesting the observed screening behavior is intrinsic for the treatment. Taking their weighted average, we find L_D = 61(7) nm, which is comparable to the spatial extent of the oxygen impurity profile induced by baking,^10,13,15 and λ_surface = 59(5) nm, consistent with “dirty” Nb (cf. λ ≈ 29 nm for “clean” Nb^33,38). Importantly, for both samples, we find that λ_surface > λ, implying that the screening capacity near the surface is weaker than the average over the first ∼150 nm.³³ Together, we take this consistency as further support of inhomogeneous screening caused by LTB in vacuum.

In contrast, the evidence for inhomogeneous Meissner screening in the N₂ infusion sample is less compelling. Contrary to expectations from surface etching¹² and secondary ion mass spectrometry (SIMS)⁸ measurements, our identified L_D is large compared to the extent of near-surface nitrogen defects and its sizable uncertainty hinders drawing firm conclusions about this length scale. Similarly, λ_surface was found to be close to the homogeneous λ value, implying that inhomogeneities in the Meissner response are either (1) absent from this treatment, (2) beyond the resolution of the LE-μSR measurements, or (3) transpiring over depths greater than those probed in the experiments. Note that the close agreement between screening models is mirrored by their $χreduced2<1$⁠, implying that the data are overparameterized (even in the simplest case). Thus, we conclude that the current data do not support the notion of inhomogeneous screening caused by N₂ infusion; however, further experiments are required to be more conclusive.⁶² 

To further test the above-mentioned findings on LTB, we also considered an additional refinement to the Meissner profile analyses shown in Figs. 2 and 3. Noting that these data³³ originate from a common batch of samples where an independent control is available (i.e., a sample with only a “BCP” treatment—called “baseline” in Ref. 33), we re-fit the LTB data simultaneously using Eqs. (1), (4), and (5), but used the “BCP” sample’s screening profile to constrain λ_bulk [i.e., using Eqs. (1)–(3)]. As both the sample dimensions and measurement conditions are common to these Nb plates,³³ under the assumption that their “true” d is also identical, we additionally restricted the simultaneous fit so that B_applied, $Ñ$⁠, and d were treated as shared parameters. The culmination of this exercise is shown in Fig. 4, with values for the optimum parameters displayed in its inset. Clearly, these restrictions pose no impediment to the form of the inhomogeneous screening in the LTB samples, as reflected by the good agreement of their λ_surface and L_D values (see Fig. 4's inset) with those in Table I. Moreover, the fit’s common parameters (see Fig. 4's top panel) are in excellent agreement with the values reported previously for the “BCP” sample,³³ indicating that the fitting procedure has no adverse effect on describing its screening behavior. This is further confirmed by their consistency with the tabulation in Table I, apart from λ_bulk’s ∼2.6 nm difference. This difference, albeit small, suggests that the inhomogeneous screening model is not strongly sensitive to λ_bulk’s absolute value, making the literature average (29 nm)^33,38 used when fitting the data in Figs. 1–3 a reasonable choice. Overall, these results show that having an independent means of characterizing the “bulk” screening properties can greatly reduce the uncertainty in quantifying the inhomogeneous portion of the Meissner profile.

As a final remark, we note that the original LE-μSR measurements^24,33 were not optimized for the detection of Meissner profile inhomogeneities, instead focusing on their overall (i.e., average) form. In future measurements, it is apparent from Figs. 1–3 that two regions are important to focus on (1) the near-surface region (i.e., z ≲ 40 nm), where the gradual curvature of B(z) is most pronounced, and (2) deep below the surface (i.e., z ≳ 120 nm), where the decay of B(z) becomes close to its “bulk” behavior. While the former was suggested previously,³⁵ the latter is only evident upon inspection of the fit curves when extrapolated beyond the current data. Having ⟨B⟩ measurements in this region (E ≳ 25 keV) is likely crucial for independently identifying λ_bulk (i.e., in the absence of a separate “control” sample), as well as reducing the (rather large) uncertainty in the inhomogeneous screening model’s fit parameters. Similarly, it would be beneficial to characterize the impurity content in the studied samples directly (e.g., using SIMS^63,64) and parameterize λ(z) directly (i.e., from their influence on the electron mean-free-path ℓ,^21,22 see Ref. 15).

In summary, we revisited the Meissner profile data for Nb samples^24,33,35 with surfaces prepared using LTB treatments^5,8 common to SRF cavities. By comparing results obtained from fits to a (homogeneous) London equation⁵³ and the generalized London expression^53–56 with an empirical model for a depth-dependent magnetic penetration depth λ(z),¹² we identify the presence of a large non-superconducting “dead layer” d ≳ 25 nm as a likely indicator for inhomogeneities in the near-surface Meissner response. The inhomogeneities are most apparent for the “120 °C bake” treatment,⁵ with the extracted length scale between “surface” and “bulk” behavior in good agreement with both experiment¹⁰ and theory.^13,15 Conversely, they are essentially absent for the “N₂ infusion” treatment,⁸ in contrast to another report.¹² Further LE-μSR measurements, covering both shallow (z ≲ 40 nm) and deep (z ≳ 120 nm) subsurface depths, will aid in precisely quantifying this phenomenon.

We thank M. Asaduzzaman and E. M. Lechner for useful discussions and a critical reading of the manuscript. T.J. acknowledges financial support from NSERC.

The authors have no conflicts to disclose.

Ryan M. L. McFadden: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Tobias Junginger: Conceptualization (supporting); Funding acquisition (lead); Writing – review & editing (supporting).

Raw data from the LE-μSR measurements reported in Refs. 24 and 33 were generated at the Swiss Muon Source (SμS), Paul Scherrer Institute (PSI), Villigen, Switzerland. Individual data files and derived data supporting the findings of this work are available from the corresponding authors upon reasonable request.