Measurements of the first-flux-penetration field in surface-treated and coated Nb: distinguishing between near-surface pinning and an interface energy barrier

We report measurements of the first-flux-penetration field in surface-treated and coated Nb samples using (μSR) . Using thin Ag foils as energy moderators for the implanted muon spin-probes, we ‘profile’ the vortex penetration field μ0Hvp at sub-surface depths on the order of ∼10μ m to ∼100μ m. In a coated sample [Nb3Sn( 2μm )/Nb], we find that μ0Hvp is depth-independent with a value of 234.5(35) mT, consistent with Nb’s metastable superheating field and suggestive of surface energy barrier for flux penetration. Conversely, in a surface-treated sample [Nb baked in vacuum at 120 ∘ C for 48h ], vortex penetration onsets close to pure Nb’s lower critical field μ0Hc1≈170mT , but increases with increasing implantation depth, consistent with flux-pinning localized at the surface. The implication of these results for technical applications of superconducting Nb, such as superconducting radio frequency cavities, is discussed.


I. INTRODUCTION
A key technical application of the elemental type-II superconductor Nb is its use in superconducting radio frequency (SRF) cavities [1][2][3], which are utilized in particle accelerators across the globe.Crucial to their operation is maintaining Nb in its magnetic-flux-free Meissner state (i.e., to prevent dissipation caused by magnetic vortices), which generally restricts their use to surface magnetic fields up to the element's lower critical field  0  c1 ≈ 170 mT [4].Such a limitation ultimately sets a ceiling for a cavity's maximum accelerating gradient  acc (i.e., the achievable energy gain per unit length), which impacts design considerations for accelerating structures (e.g., size, operating temperature, etc.).Consequently, there is great interest in pushing SRF cavity operation up to Nb's so-called superheating field  0  sh ≈ 240 mT [5,6], where the Meissner state is preserved in a metastable configuration.Currently, the largest gradients are achieved by so-called lowtemperature baking (LTB) surface treatments, wherein a Nb cavity is baked at temperatures on the order of ∼120 • C either in vacuum [7,8] or in a low-pressure gas atmosphere [9,10].Indeed, the best performing treatments have enabled cavities to achieve surface magnetic fields beyond  0  c1 (with some even approaching  sh ) [5]; however, the underlying mechanism for this enhancement remains unclear.
Consider the typical LTB treatment, involving vacuum annealing Nb at 120 • C for a duration up to 48 h [7].Early measurements of this treatment's effect on Nb's Meissner response using low-energy muon spin rotation (LE-SR) [11,12] showed a sharp discontinuity in the screening profile [13], which led to suggestions that LTB can be used to create an "effective" superconductor-superconductor (SS) bilayer [14].Of particular interest for SRF applications is the case where a thin "dirty" layer resides atop a "clean" bulk (e.g., as a result of surface-localized inhomogeneous doping), as it offered several avenues for increasing the vortex penetration field  0  vp via a reduced current density at Nb's surface (e.g., following directly from the SS-like structure [14] or as a consequence of deformations found in the Meissner profile itself [15,16]).While both theories [14] and measurements [13] have been presented that support the interpretation of a surface barrier originating from a "dirty" layer, other measurements [17] and analyses [18] contradict such views.To resolve this discrepancy, additional measurements using alternative approaches would be highly beneficial.
One such possibility is instead using techniques capable of identifying  0  vp directly.This has been done, for example, using "bulk" muon spin rotation (SR) [19,20], which provides a local measurement of the magnetic field ∼100 µm below Nb's surface.Such studies have found  0  vp ≳  0  c1 for both LTB and coated Nb [21], the latter yielding  0  vp ≈  0  sh [6].While this provided strong evidence that a surface energy barrier [22] was preventing flux nucleation in SS samples, some ambiguity in interpreting the enhancement from LTB remained.Specifically, subsequent magnetometry measurements on identically prepared samples showed no such enhancement [4], implying an accumulation of near-surface vortices caused by pinning.The discrepancy between SR and magnetometry suggests that any pinning centers must be localized to depths less than ∼100 µm below Nb's surface.
To test these ideas, here we extend the "bulk" SR approach used in related work [6,21,23] to provide depth-resolved measurements of  0  vp in both surface-treated and coated Nb.Specifically, we make use of thin Ag foils to moderate the implantation energy of the muon spin probes, providing spatial sensitivity to depths on the order of ∼30 µm to ∼100 µm.In the presence of surface energy barrier [22],  0  vp is expected to be depth-independent, whereas surface-localized pinning is anticipated to produce larger  0  vp s deeper below the surface (see Figure 1).Using this approach, we find that  0  vp is depth-independent and close to Nb's  0  sh for a Nb sample In the presence of strong near-surface pinning in the vortex state, fluxoid penetration is localized to the sample surface, which may go undetected by the implanted  + at full beam energy.(d): Through the use of thin Ag foils as energy moderators for the  + beam, the  + probes stop closer to the surface, allowing for flux that is surfacepinned in the vortex state to be observed.coated with the A15 superconductor Nb 3 Sn [24,25], consistent with the energy barrier expected for the SS bilayer.Conversely, for Nb that has been surface-treated by LTB at 120 • C, the measured  0  vp s are comparable to Nb's  0  c1 , but increase deeper below the surface, suggesting the presence of localized pinning near the surface that prevents detection by deeper implanted muons.

II. EXPERIMENT
SR experiments were performed at TRIUMF's Centre for Molecular and Materials Science (CMMS) facility in Vancouver, Canada.Using the M20C beamline [26], a ∼100 % spin-polarized ∼4.1 MeV "surface"  + beam was extracted, spin-rotated in flight, and delivered to the high-parallel-field (i.e., "HodgePodge") spectrometer equipped with a horizontal gas-flow cryostat and a low-background (i.e., Knight shift) insert [26].A sketch of cryostat configuration is given in Figure 2.This setup is similar to that used in related experiments [6,21,23], with the external magnetic field parallel to each sample's surface (see Figure 1) and perpendicular to the implanted  + spin direction.Notably, the present work also incorporates thin Ag foils (Goodfellow, 99.95 % purity, 10 µm to 30 µm thick) as part of the cryostat assembly, acting as energy moderators for the  + beam.By varying the thickness of the foils, the range of implanted  + in the Nb samples can Schematic of the horizontal gas-flow cryostat and lowbackground (i.e., Knight shift) insert used with TRIUMF's high parallel-field (i.e., "HodgePodge") spectrometer [26].The thin Ag foils used as  + energy moderators are located between a 8 mm diameter beam collimator (also made of Ag) and the inner  + counter.be controlled on the µm scale.Simulations of the moderating effect were performed using the Stopping and Range of Ions in Matter (SRIM) Monte Carlo code [27], which incorporated all major materials along the beam's path (e.g., muon counters, moderator foils, etc. -see Figure 2), as well as compound corrections [28] to the stopping powers (where appropriate).Typical  + stopping profiles for this setup are shown in Figure 3, showing mean stopping ranges ⟨⟩ between ∼36 µm to ∼108 µm for a Nb target, along with the width (i.e., standard deviation)   of the stopping distributions [29].Note that similar simulations for Nb 3 Sn(2 µm)/Nb samples (not shown) yielded virtually identical results.In cases where a 60 µm Ag foil was used, a small fraction (∼2 %) of the implanted probes stop in the inner  + counter, located immediately upstream of the sample (see Figure 2).
In SR, the implanted  + spin probes (spin  = 1/2, gyromagnetic ratio /(2) = 135.54MHz T −1 , lifetime   = 2.197 µs) are sensitive to the local magnetic field at their stopping sites, with the temporal evolution of the ensemble's spin-polarization monitored via the anisotropic property of radioactive -decay.Specifically, each  + decay positron's emission direction is statistically correlated with the  + spin direction at the moment of decay, providing an easy means of tracking spin-reorientation.Specifically, in a two-detector setup like that used here (see Figure 2), the recorded histogram of decay events takes the form: where ± denotes each detector,  is the time (in µs) after implantation,  ±,0 and  ± denote the rate of "good" and "background" decay events,   () ∈ [−1, +1] is the muon spinpolarization, and  0 is a proportionality constant (∼0.2 here).
The most essential part of Equation ( 1) is  0   () and it may be obtained directly by taking the asymmetry of two counters: where  ≡  +,0 / −,0 accounts for imperfections in the de- Simulated stopping profiles for ∼4.1 MeV "surface"  + implanted in Nb using the SRIM Monte Carlo code [27].The profiles, represented here as histograms, were generated from 10 6  + projectiles and account for all materials in the beam's path prior to implantation (e.g., cryostat windows,  + counters, moderating foils, etc. -see Figure 2).Using Ag foils of different thicknesses (indicated in each plot's inset), mean stopping depths ⟨⟩ in the range of ∼36 µm to ∼108 µm are achieved.The width (i.e., standard deviation)   of each stopping distribution is also indicated.Note that a reduced  + implantation energy is used for panel (d), yielding a ⟨⟩ comparable to using a thicker moderating foil, as shown in panel (c).
tector pair (e.g., different efficiencies, effective solid angles, etc.).Important for this work is the temporal evolution of   (), which contains all information about the local magnetic environment below the sample's surface (i.e., at the  + stopping site).Fortunately,   () differs in each of Nb's superconducting states, allowing us to quantify their volume fraction for our set of measurement conditions.
For the simplest case of Nb's normal state, where magnetic flux penetrates the sample's surface freely, the SR signal follows that of a typical transverse-field measurement [19]: where  is the time after implantation,  is the damping rate (from a Gaussian field distribution),  0  0 is magnetic field at the  + stopping site (dominated by the external applied field), and  is a phase factor.In non-superconducting Nb, the term  ≈ 0.5 µs −1 [6,21,23], causing minimal damping.
Conversely, in Nb's vortex state, where fluxoids form a periodic arrangement with a broad distribution [30,31],  is much larger and the signal is damped quickly (i.e., within the first ∼0.3µs following implantation).In the opposite limit of Nb's Meissner state, where all magnetic flux is expelled from the sample's interior, the signal follows that of a so-called dynamic zero-field Kubo-Toyabe function [32]: which is obtained from a static Kubo-Toyabe function  SGKT [33]: where the local field is fluctuating (e.g., from stochastic siteto-site "hopping" of  + [34]) at a rate , typically ∼0.7 µs −1 for SRF Nb [23,35].Note that both Equations ( 4) and ( 5) assume the local field distribution at the  + stopping site to be Gaussian, in accord with related studies [6,21,23,35].
For the present experiments, the SR signal is, in general, described by a superposition of Equations ( 3) and ( 4), which may be written as: where  ZF ∈ [0, 1] denotes the volume fraction of the zero-field component, which for superconducting Nb in an applied field corresponds to its Meissner state, and  TF, ∈ [0, 1] represents the individual transverse-field components (e.g., normal state, vortex state, etc.), subject to the constraint that   TF, ≡ 1.
Examples of this type of composite signal are shown in Figure 4.
In order to identify the  0  vp in each sample, the evolution of  ZF in (monotonically increasing) magnetic fields  0  0 up to ∼260 mT was measured at the cryostat's base temperature ( ≈ 2.7 K) following zero-field cooling.Any depth-dependence was inferred from repeat measurements using different moderator foil thicknesses (and  + beam energies).Note that, to ensure the accuracy of the applied fields reported for all superconducting states (i.e., due to its geometric enhancement from flux-expulsion), all values were derived from field calibration measurements conducted above  c (i.e., at  ≥ 10.5 K for the LTB Nb and  ≥ 20 K for the SS bilayer), where the SR signal simply follows Equation (3).Specifically, they were corrected using [36]: where  0  NS 0 is the (measured) applied field in the normal state and  ≈ 0.13 [6,21] is the demagnetization factor for our samples (see Section II A).

A. Samples
The samples used in this study are identical to those employed in previous SR measurements on the first-flux-penetration field [6,21].For completeness, we briefly restate their preparation details below.
Each Nb sample was cut from fine-grain Nb (Wah Chang Corporation) stock sheets with a residual-resistivity ratio (RRR) >150 and machined into prolate ellipsoids with a semi-major radius 22.9 mm and semi-minor radii 9.0 mm for the other axes.After machining, the samples underwent buffered chemical polishing (BCP) to remove the surface's topmost 100 µm of the material (see, e.g., [37]).For one of the samples, a typical LTB "recipe" was followed where, after degassing in vacuum at 800 • C for 4 h, the Nb ellipsoid was baked in vacuum at 120 • C for 48 h [7].A complementary magnetostatic characterization of the LTB sample can also be found elsewhere [4].Microstructure analysis was conducted on identically prepared flat "witness" samples using scanning electron microscope (SEM)/energy dispersive X-ray spectroscopy (EDX) and secondary ion mass spectrometry (SIMS).The SEM/EDX analysis revealed no unexpected features or contaminations beyond typical observations for high-RRR Nb exposed to air [38].SIMS confirmed the expected increase in near-surface oxygen concentration following LTB for depths <20 nm, in accord with models for oxygen diffusion [39][40][41].For another sample, a 2 µm Nb 3 Sn surface coating was applied using a vapour diffusion [42,43] procedure developed at Cornell University [44,45].SEM and atomic force microscopy (AFM) measurements on identically prepared "witness" samples revealed that the asgrown Nb 3 Sn exhibits surface roughness comparable to their grain size, typically in the µm range [25].Further tests on substrates prepared using different polishing techniques had no effect on the bilayer's surface roughness [46].

III. RESULTS
Figure 4 depicts typical time-differential SR data in one of our samples (LTB Nb [7]), showing the contrast in signals for different material states.In the normal state, where the local field at the  + stopping sites is dominated by the (transverse) applied magnetic field  0  0 , coherent spin-precession is observed with minimal damping, consistent with a narrow field distribution.By contrast, in the Meissner state,  0  0 is completely screened [47] and the local field is dominated, in part, by Nb's 100 % abundant 93 Nb nuclear spins, resulting in the characteristic (dynamic) "zero-field" signal [32].At applied fields just above the vortex penetration field  0  vp , a mixed signal with both zero-and transverse-field components is observed, the latter being strongly damped.As the field is further increased beyond  0  vp , the sample fully enters a vortex state, where the signal resembles that of the normal state, but the broad field distribution from the vortex lattice causes strong damping of the spin-polarization.Similar behavior is observed at other implantation ranges, as well as in the Nb 3 Sn(2 µm)/Nb sample (not shown).In line with the experiment's description in Section II, these observations can be quantified through fits to Equation ( 6), yielding good agreement with the data in all cases (typical reduced- 2 ≈ 1.07).A subset of the fit results are shown in Figure 4.
To aid in identifying  0  vp and its depth-dependence, we plot Typical SR data in surface-treated Nb, illustrating the evolution with applied magnetic field  0  0 .Both panels (a) and (b) display the same dataset; however, (b) specifically represents the initial 0.5 µs.The temperature  and  0  0 for each superconducting state are detailed in the panel's inset.The solid lines are fits to the data points using Equation ( 6).In the normal state, coherent spin-precession is observed with minimal damping, consistent with a narrow local field distribution.In the Meissner state, the applied field is completely screened and the local field is dominated by the host's 93 Nb nuclear spin, resulting in the characteristic (dynamic) "zero-field" signal.At fields just above the vortex penetration field, a mixed signal with both zero-and transverse-field components is observed, the latter being strongly damped.Finally, in the vortex state, the broad field distribution causes strong damping of the transverse-field response.
the measured  ZF s vs.  0  0 for each Nb sample in Figure 5.The resulting "curves" all have a sigmoid-like shape, where  ZF = 1 for  0  0 <  0  vp , with  ZF decreasing rapidly towards zero once  0  0 ≥  0  vp .This behavior is phenomenologically captured by a logistic function: where  0 is the curve's height,  denotes the "steepness" of the transition, and  0  m represents its midpoint.Fits of the measured volume fractions to Equation ( 8) are shown in Figure 5 as guides to the eye.It is clear from Figure 5 that  0  vp 's depth-dependence is quite different for LTB and coated Nb.In the LTB sample, fluxpenetration is detected just above Nb's  0  c1 , with the onset pushed to higher  0  0 s for larger  + ranges.Similarly, the field span of this "transition" (i.e., going from zero-to full-fluxpenetration) also increases slightly deeper below the surface.By contrast, in the Nb 3 Sn(2 µm)/Nb, first-flux-penetration onsets close to Nb's  0  sh , with this value and the transition's width both virtually unaffected by  + 's proximity to the sample's surface.Qualitatively, this disparity between the Nb treatments indicates that different mechanisms are likely responsible for determining each sample's  0  vp .To quantify these differences explicitly, we use a nonparametric approach to identify  0  vp for each "curve" in Figure 5. Noting that each  ZF has a statistical uncertainty of ∼4 % (determined from fitting), we define  0  vp to correspond to the applied field where where  ZF ≤ 0.96 (i.e., the very onset of flux-penetration).Due to the finite "sampling" of our measurements, this field is estimated as midpoint between the pair of  ZF s on either side of the threshold criteria [48].These values are marked graphically in Figure 5 by vertical dotted colored lines.Similarly, we take the width of the zeroto full-flux-penetration "transition" Δ vp to be the field range where 4 % ≤  ZF ≤ 96 %.To correct for any influence from the finite span of the  + stopping profile we additionally normalize the Δ vp s by the width   of the implantation distribution, which varies for different beam implantation conditions (see Figure 3).For each of our samples, the dependence of both  0  vp and Δ vp /  on the implanted  + range ⟨⟩ is shown in Figure 6.For comparison, measured values for additional surface-treated samples [6,21] have also been included [49].Lastly, to facilitate comparison between the measured  0  vp s, we correct for  minor temperature differences and extrapolate our values to 0 K using the empirical relation [50]: where  is the absolute temperature,  c ≈ 9.25 K is Nb's critical temperature [51], and  0  vp (0 K) is the value at absolute zero.

IV. DISCUSSION
Consistent with our main observations in Figure 5 for the vortex penetration field, Figure 6(a) shows that  0  vp s in Nb 3 Sn(2 µm)/Nb (extrapolated to 0 K) remain depthindependent with an average value of 234.5 (35) mT, in excellent agreement with Nb's  0  sh (0 K) ≈ 240 mT [5,6].That the flux penetration occurs in such close proximity to the superheating field is strong evidence for the presence of an interface barrier at the SS boundary, similar to a Bean-Livingston (BL) surface energy barrier [22], as anticipated by the theoretical framework for superconducting multilayers [14,52].Conversely, in LTB Nb the  0  vp s are much lower, remaining close to (but slightly above)  0  c1 for all measurements.In this case, however, a modest depth-dependence is observed, with  0  vp increasing gradually with increasing ⟨⟩.That these details coincide is significant, as it sets LTB apart from other surface treatments, where flux-penetration occurs at Nb's lower critical field.We note that, though small, the observation of such a depth-dependence is inconsistent with a surface energy barrier being solely responsible for pushing  0  vp >  0  c1 [53].An alternative possibility is the presence of pinnning centers localized near LTB Nb's surface, which has been suggested previously [4,21].In such a case,  0  vp closely approximates  0  c1 due to the presence of pinning, which diminishes towards the sample's center, resulting in delayed flux penetration (i.e., the pinning centers act as supplemental flux "blockades," providing resistance to the free motion of the fluxoids, which would otherwise uniformly distribute throughout the sample) [21].Independent of the mechanistic details, from our data we identify the length scale over which flux-penetration is retarded.Using a simple linear fit to the measured values, we find that  0  vp ≈  0  c1 at a mean depth of ∼14 µm, characterizing the distance in which it is delayed for our sample geometry.Note that, there is a significant proportion of muons that stop significantly closer to the surface at this average depth.We shall return the implications of this quantity later on.
Further insight into the flux-penetration mechanism for the LTB and SS samples can be gleaned from the (normalized) flux-entry "transition" widths Δ vp /  , which are shown in Figure 6(b).The span from first-to full-flux penetration provides a measure for the "haste" in which vortices nucleate through the probe  + stopping depths where, in the presence of near-surface pinning, we expect that the Meissner-vortex transition becomes "extended" to a larger range of applied fields (i.e., the presence of pinning centers delays full-flux penetration).Thus, we suggest that Δ vp /  serves as a proxy for each treatment's pinning strength.Indeed, we observe that the values for LTB Nb are all similar, which also suggests that the pinning strength appears to be depth-independent and exceeds the values of all other surface preparations shown in Figure 6.However, given the relatively large uncertainty in each measurement, drawing firm conclusions about treatment-specific differences is challenging.This is clear from their average values, which turned out to be 0.56 (15) mT/µm (for LTB) and 0.37 (18) mT/µm (for Nb 3 Sn(2 µm)/Nb), respectively.Interestingly, both quantities are comparable to that of Nb in the absence of any treatment (0.46 (14) mT/µm), which is larger than both the values ob-tained for 1400 • C annealing (0.14 (13) mT/µm), as well as 1400 • C annealing + 120 • C LTB (0.22 (14) mT/µm).We note that, in line with our expectations for Δ vp /  , its value is minimized for the 1400 • C treatment, which is known to release virtually all pinning [21].Thus, although not as conclusive as the  0  vp measurements, the large values for LTB insinuate that pinning is strongest for this treatment.With these results in mind, we will explore their implications in the remaining discussion.
First, we consider the SS bilayer Nb 3 Sn(2 µm)/Nb, whose high  0  vp is favorable for technical applications requiring operation in a flux-free state (e.g., SRF cavities).In fact, direct current (DC) measurements of first-flux-penetration using a Hall probe magnetometer on a 1.3 GHz single-cell SRF cavity of similar composition are in good agreement with our result [5], with similar  0  vp values reported for surface coatings other than Nb 3 Sn, such as hybrid physical chemical vapour deposition (HPCVD) MgB 2 [6].The combination of our work and related studies [5,6,21] provides compelling evidence for the use of SS bilayers as an empirical means for increasing  0  vp , consistent with earlier observations using SR [6,21].For further insight into why such a treatment is so effective at enhancing  0  vp , we must consider the theory of superconducting multilayers (see, e.g., [14]), which we shall briefly outline below.
In bilayer superconductors, by analogy with the BL barrier at the surface of "bulk" superconductors [22], the discontinuity in each material's (coupled) electromagnetic response at the SS interface is responsible for creating a second (sub-surface) barrier that impedes flux penetration [52].Specifically, electromagnetic continuity across the SS boundary creates a "coupling" between the layer's properties, leading to marked deviations from the lone material's native behavior when the surface layer penetration depth is larger than the substrate.Microscopically, this is predicted to manifest in the heterostructure's Meissner screening profile with a distinct bipartite form [52], which was recently confirmed experimentally for Nb 1−x Ti x N(50 nm to 160 nm)/Nb samples [54].A weaker Meissner screening current is observed, which, as a corollary, provides enhanced protection against flux nucleation (see, e.g., [14]).While the  0  vp ≈  0  sh we observe is in good agreement with this prediction, the theory also suggests that  0  vp can be improved further still through optimizing the Nb 3 Sn coating's thickness.This enhancement is achievable by ensuring a flux-free surface layer and enabling superheating in both the surface and substrate layers.Achieving this involves precise adjustments to the thickness of the surface layer and the presence of an interface barrier at the SS boundary for the substrate layer.For an SS structure, the optimum thickness of the surface layer  opt s is defined by [14]: where   and   sh denote the magnetic penetration depth and superheating field for the surface ( = s) and substrate ( = sub) layers.Using literature values for these quantities ( Nb 3 Sn = 173(32) nm [5,55];  Nb 3 Sn sh = 430(110) mT [5,55];  Nb = 29.01(10)nm [17]; and  Nb sh = 237(29) mT [5,6]), Equation (10) yields  opt s = 210(60) nm or, equivalently, ∼1.2 Nb 3 Sn .Similar predictions have been made for Nb 1−x Ti x N/Nb [54].It would be interesting to test these explicitly, for example, using the experimental formalism employed in this work.Investigating this phenomena in closely related superconductor-insulatorsuperconductor (SIS) heterostructures would also be fruitful, as they offer similar means of enhancing  0  vp [14].
As a close to our discussion of the Nb 3 Sn/Nb bilayer, it is interesting to consider its synthesis.We note that in our sample, as is common for heterostructures prepared by thermal diffusion, the composition of the Nb 3 Sn/Nb interface deviates appreciably from each layer's respective "bulk" [25,56].Specifically, within the first few hundred nanometers from the heterojunction, a localized Sn deficiency (enhancement) is present in the Nb 3 Sn (Nb) layers, with the former known to lower Nb 3 Sn's  c , making the region a poor superconductor [25,56].The presence of such inhomogeneities, however, do not meaningfully impact  0  vp , as indicated by it's agreement with Nb's  0  sh .Testing the extent in which this remains true, for example, on samples with extended defect regions, would be interesting.Similarly, these nanoscale inhomogeneities at the SS interface could be examined directly using a depth-resolved technique such as LE-SR [11,12], with the caveat that the Nb 3 Sn layer thickness must be compatible with the technique's spatial sensitivity (i.e., subsurface depths typically <150 nm).Finally, given the presence of a secondary energy barrier at the SS interface, we suggest that incorporating SS bilayers into SRF cavity structures holds great promise for surpassing the inherent limitations of current Nb cavity technology (i.e., enabling higher accelerating gradients and enhanced performance in particle accelerators).
We now turn our attention to the 120 • C LTB treatment [7], which is well-known in SRF applications for its ability to alleviate the so-called high-field  slope (HFQS) "problem" [2,57], where a rapid decrease in a cavity's quality factor () occurs as the peak surface magnetic field exceeds ∼100 mT [58].True to this fashion, our finding of  0  vp s in excess of Nb's  0  c1 underscores its utility in this domain; however, the treatment's depth-dependent  0  vp and relatively large Δ vp /  make it unique among the comparison treatments reported here.As mentioned above, near-surface pinning of the fluxlines provides the most likely explanation for these facts.The observed delay in flux-penetration would then arise from the pinning centers acting as "supplementary barriers," impeding the movement of vortices from the edges of the sample to the center [21].The relatively large Meissner-vortex transition "widths" observed here also support this interpretation.It has been suggested by others that material inhomogeneities, such as interstitial oxygen or hydrogen precipitates, may dominate the pinning mechanism [59].For further insight into the matter, it is instructive to consider some of the treatment's finer details, which we do below.
During LTB, the heat treatment induces changes to Nb's superfluid density in its outermost nanoscale region through the dissolution and diffusion of oxygen originating from the metal's native surface oxide [7].This alteration is believed to result in a "dirty" region localized near Nb's surface (i.e., the first ∼50 nm), as explained by an oxygen diffusion model [39].This length scale aligns well other work, including an experiment that used repeat HF "rinses" to remove the topmost ∼50 nm and (essentially) restore the HFQS following LTB [60].Similarly, an increase of the ratio of the upper and surface critical fields after baking was explained by the presence of an impurity layer of thickness smaller than Nb's coherence length [61], which is of similar magnitude.Other work on related treatments have also found similar results [41,62], and the first ∼10 nm may be particularly enriched with interstitial oxygen [63].It has been suggested that the LTB effect (i.e., HFQS mitigation) is due to the strong suppression of hydride precipitation [64], as oxygen efficiently traps interstitial (or "free") hydrogen that has accumulated during standard chemical treatments, such as BCP or electropolishing (EP) [62,64,65].Indeed, LTB has been linked to changes in the vacancy structure in Nb's near-surface region [66,67], supporting the prevailing idea that nanoscale niobium hydrides cause the HFQS [64].These works all point to the importance of surface defects, especially those closest to the surface.
Within the ∼50 nm "dirty" region, it is expected that quantities sensitive to the density of (nonmagnetic) scattering centers (e.g., the carrier mean-free-path, the magnetic penetration depth, etc.) be altered from their (clean-limit) "bulk" values.As the doping is likely inhomogeneous over this length scale, a similar character may be imparted on dependent quantities.Early experimental results seemed to favor this possibility [13], with other authors suggesting a strong likeness of LTB Nb to an "effective" SS bilayer [14].Subsequent experiments, however, have shown that such a distinction is far from clear, with both a recent a commentary [18] and a separate LE-SR experiment [17] showing that the effects are homogeneous over subsurface depths spanning ∼10 nm to ∼160 nm.Such a finding was rather surprising, given the aforementioned related work [39,60,61] and that doping from the closely related nitrogen infusion treatment [10] yields inhomogeneous superconducting properties over the same length scale [16].We point out, however, that the observed electromagnetic response for LTB [17,18] is consistent with the absence of an interface energy barrier preventing flux-penetration [15,16], in line with our present findings.
As alluded above, the "dirty" nature of LTB provides an ample environment for pinning centers, which can serve as seeds for flux penetration.While other experiments are clear on their surface proximity, their ∼50 nm localization is quite different from the micrometer depth-dependence we observe for  0  vp , which warrants further consideration.As is shown in Figure 6(a), LTB's  0  vp varies approximately linearly in ⟨⟩.From a fit to a function of the form  0  vp (⟨⟩) ≈  + ⟨⟩, we parameterize this trend, but postulate that Nb's  0  c1 acts as the floor for  0  vp .Upon equating the two relations, we find that  0  vp (⟨⟩) ≈  0  c1 when ⟨⟩ ≈ 14 µm.Clearly, this scale is considerably larger than the "dirty" region's extent [39,60,61].We argue that, despite the LTB effect being confined to the very near surface, this "layer" could introduce pinning over a µm length scale in an ellipsoidal geometry.This is a consequence of the fact that, even if flux lines penetrate further into the material, they must both enter and exit through the "dirty" region.The pinning strength is directly influenced by the flux line's path length through this volume, which in the case of an ellipsoid is minimized for the straight path along its equator, but maximized for a (curved) trajectory close to the surface (see Figure 1).Indeed, magnetometry studies demonstrate that LTB can significantly alter the pinning characteristics in this geometry [4].We emphasize that an ellipsoidal geometry is the ideal means for probing intrinsic pinning effects, as opposed to other sample forms (e.g., rectangular prisms) where additional geometric effects are present [36].In line with the generally accepted view that LTB changes the concentration of pinning centers (i.e., from the redistribution of near-surface defects) [41,60,68,69], our identification of a length scale associated with flux pinning may prove useful in further refining their microscopic distribution.In the future, it would be interesting to use this finding as a constraint for simulations of flux-entry in ellipsoidal geometries (see, e.g., Ref. 70).
In terms of SRF cavity performance, it is important to highlight that our investigation on the LTB "dirty" layer cannot explain situations where cavities in radio frequency (RF) operation exhibit  0  vp s above  0  c1 , reaching values as high as ∼190 mT [59,71], equivalent to accelerating gradients of ∼45 MV m −1 [2].Recall that, if the Meissner state of any (type-II) material persists above  0  c1 , it must do so in a metastable state.For DC fields, flux penetration can only be prevented by an energy barrier [22], generally anticipated for defect-free surfaces.From the delineations above, it is clear that LTB results in surfaces that are anything but defect-free and any prospect for achieving such high  0  vp fields during RF operation depends on the interplay between the time needed for the vortex core formation and the RF period.In such cases, maintaining a flux-free state above  0  c1 necessitates the time required for vortex penetration to exceed the operating RF period (i.e., the inverse RF frequency) of the cavity [72].Comparing this study with  0  vp ∼ 190 mT [59,71], we suggest that LTB cavities need a longer flux nucleation time than the RF period to sustain the Meissner state.Alternatively, SS bilayers can maintain that up to its  0  vp ∼ 235 mT, even in the DC flux penetration case, offering a more robust approach for achieving higher accelerating gradients than LTB.
Considering the above details, it is apparent that LTB Nb differs fundamentally from that of an SS bilayer in both its composition and mechanism for impeding flux entry.Concerning the latter, our present findings, in conjunction with related work [4,21], support the notion that LTB does not create a supplemental energy barrier, but instead postpones vortex penetration above  0  c1 due to pinning.While it remains an open question as to if this behavior could be further engineered to benefit SRF applications, it is apparent that careful control over the near-surface doping is crucial.Advances in this area are already apparent [41].In future work, it would be interesting to test these ideas on related LTB treatments [8][9][10], as well as the recently discovered "mid-" treatments that are known to produce very small surface resistances [73,74].

V. CONCLUSION
Using SR, we measured the depth-dependence (on the µm scale) of the vortex penetration field in Nb ellipsoids that received either a LTB surface-treatment or a 2 µm coating of Nb 3 Sn.In each sample, the measured field of first-flux-entry is greater than Nb's lower critical field of ∼170 mT, suggesting their applicability for SRF cavities.In the coated sample, we find a depth-independent  0  vp = 234.5 (35) mT, consistent with Nb's superheating field and the presence of interface energy barrier preventing flux penetration.Conversely, in LTB Nb, its  0  vp is only moderately larger than Nb's  0  c1 , increasing slightly with increasing depths below the surface.The latter observation, in conjunction with the increased span of the Meissner-vortex transition, suggests pinning from surfacelocalized defects.Our findings confirm that the introduction of a thin superconducting overlayer on Nb can effectively push the onset of vortex penetration up the superheating field, but rules our LTB as a means of achieving this.We suggest that its success is rather due to effects specific to the operation under RF fields, such as the time required for vortex nucleation.These findings validate the potential of employing superconducting bilayers to achieve a flux-free Meissner state up to the superheating field of the substrate.

FIG. 1 .
FIG.1.Sketch of the present SR experiment on superconducting Nb samples with ellipsoidal shape in an applied magnetic field parallel to the ellipsoid major axis (---), with the magnetic flux lines (-) also indicated.(a): In the Meissner state, complete flux expulsion from the ellipsoid's interior is achieved.Without any energy moderation for the  + beam, the magnetic probes stop well-below the sample surface and experience no external contribution to their local field.(b): In the vortex state, some magnetic flux penetrates the sample as quantized fluxoids with a field-depended lattice arrangement, leading to a broad local field distribution samples by the  + beamspot.(c): In the presence of strong near-surface pinning in the vortex state, fluxoid penetration is localized to the sample surface, which may go undetected by the implanted  + at full beam energy.(d): Through the use of thin Ag foils as energy moderators for the  + beam, the  + probes stop closer to the surface, allowing for flux that is surfacepinned in the vortex state to be observed.
FIG. 3.Simulated stopping profiles for ∼4.1 MeV "surface"  + implanted in Nb using the SRIM Monte Carlo code[27].The profiles, represented here as histograms, were generated from 10 6  + projectiles and account for all materials in the beam's path prior to implantation (e.g., cryostat windows,  + counters, moderating foils, etc. -see Figure2).Using Ag foils of different thicknesses (indicated in each plot's inset), mean stopping depths ⟨⟩ in the range of ∼36 µm to ∼108 µm are achieved.The width (i.e., standard deviation)   of each stopping distribution is also indicated.Note that a reduced  + implantation energy is used for panel (d), yielding a ⟨⟩ comparable to using a thicker moderating foil, as shown in panel (c).

FIG. 5 .
FIG. 5.Volume fraction of the zero-field SR signal  ZF at different applied magnetic field  0  0 and mean  + implantation depths ⟨⟩.The ⟨⟩ and measurements temperature  are mentioned in the figure's inset.The solid colored lines denotes fits to a logistic function, intended to guide the eye.The  0  vp at each ⟨⟩ are shown using colored dotted vertical lines.Vertical dotted-dashed and dashed brown lines are included to mark Nb's lower critical field  0  c1 and superheating field  0  sh at 2.7 K. Note that Nb 3 Sn has a considerably smaller  0  c1 = 25.0(14)mT[5] compared to the fields shown here.(a): In LTB Nb, the vortex penetration field  0  vp is comparable to  0  c1 , but shows a strong ⟨⟩-dependence, increasing with increasing ⟨⟩.(b): In Nb 3 Sn(2 µm)/Nb,  0  vp is ⟨⟩-independent and close to  0  sh .
Δ 0  vp , divided by   , confirm that the stopping distributions are not significantly different.The dotted and dash-dotted horizontal lines are the average values of Δ vp /  , ⟨Δ vp /  ⟩ for LTB and Nb 3 Sn(2 µm)/Nb samples, respectively.
[6,21]Summary of the first-flux penetration measurements at different mean  + implantation depths ⟨⟩ for LTB Nb and Nb 3 Sn(2 µm)/Nb.For comparison, we include re-analyzed results for additional SRF Nb treatments (originally reported elsewhere[6,21]).(a): Measured vortex penetration fields  0  vp .The horizontal dashed and dotted-dashed brown lines denote Nb's superheating field  0  sh and lower critical field  0  c1 , respectively.In Nb 3 Sn(2 µm)/Nb,  0  vp is ⟨⟩-independent and close to  0  sh , whereas  0  vp ≈  0  c1 in LTB Nb, increasing modestly with increasing ⟨⟩.The other surface-treatments have  0  vp s that are similarly close to  0  c1 .The cyan color solid line represents the "straight line" fit applied to the LTB data, providing an estimate of the depth where  0  vp =  0  c1 .(b): Measured Meissner-vortex transition "widths"