Oxide dissolution and oxygen diffusion scenarios in niobium and implications on the Bean–Livingston barrier in superconducting cavities

Reliably tuning the electron mean free path, $l$⁠, in superconducting devices via surface impurity alloying can enhance their performance.^1–7 These applications include superconducting radio frequency (SRF) cavities,^8,9 superconducting qubits,¹⁰ microwave detectors,¹¹ as well as many others. In superconducting resonators, the effect of increasing impurity content on the surface resistance, $R s$⁠, is due to the competition between the diminishing dissipative conductivity, $σ 1(l)$⁠, and the expanding London penetration depth, $λ(l)$⁠. An optimal competition between these two for $R s∝ σ 1 λ 3$ leads to a minimum at $l≈ ξ 0/2$⁠, where $ξ 0$ is the superconductor’s coherence length.³ 

Major particle accelerator projects use or plan to utilize SRF technology to facilitate fundamental research.^12–23 In such projects, trace impurities have tremendous impact on improving superconducting cavity performance.^8,24,25 Nb superconducting radio frequency cavities alloyed with nitrogen,^8,9 titanium,^26,27 or other impurities²⁸ have yielded significant reductions of the BCS surface resistance. This advantage is being exploited by the LCLS-II and LCLS-II HE projects^29,30 by using N-alloyed Nb in their continuous wave accelerators. Recently, it has been shown that a native oxide dissolution and oxygen diffusion process is responsible for the increase in quality factor in SRF cavities.^31–33 The Nb–O solid solution is composed of O dissolved in octahedral interstices,³⁴ which alter the physical and electronic properties of Nb.^34–37 Elevated O concentration in Nb is associated with decreases in $T c$⁠, $B c$⁠, and $B c 1$ but enhancements in $κ$⁠, $B c 2$⁠, and $B c 3$⁠.^35,38 The presence of O may help prevent precipitation of deleterious phases like hydrides.^39–41 Generalizations of the Nb pentoxide dissolution and oxygen diffusion model described by Ciovati for Nb⁴² could be applied to other materials like Zr⁴³ or Ta,⁴⁴ a material considered for superconducting qubits, to enhance the low field surface resistance and increase qubit coherence times.^45,46

Preservation of the Meissner state is imperative for low-loss, high accelerating field SRF cavity performance. The Bean–Livingston barrier⁴⁷ preserves the Meissner state above $B c 1$⁠, where it is metastable, up to the superheating field, $B s h$⁠. At $B s h$⁠, the surface is absolutely unstable to vortex nucleation, allowing penetration of highly dissipative vortices that may trigger thermal instabilities.^48,49 Through surface engineering it is expected that $B s h$ can be extended beyond the substrate’s intrinsic limit. Either by a controlled impurity profile^1,50 or by a dirty Nb layer directly coupled to a clean Nb substrate or separated by an insulator.^2,51 A goal remains to develop impurity diffusion models to tune interstitials optimally to modify $R s$ and $E a c c$ for an intended application. This work requires the use of secondary ion mass spectrometry (SIMS) because of its high depth resolution and ability to quantify trace impurities.

In this work, we extend the thin overlayer dissolution model to sequential overlayer dissolutions, multilayer dissolution, and realistic temperature profiles. The model is compared with SIMS depth profiles for sequential oxide dissolutions and realistic temperature profiles. We apply this model to the case of shallow migration of O in Nb and examine its expected effect on near-surface supercurrent density.

SIMS measurements were made using a CAMECA 7f Geo magnetic sector SIMS instrument on Nb samples as described elsewhere.^31,52,53 A Cs $+$ primary ion beam accelerated to 5 keV was used to impact the samples biased at $−$3 kV, yielding an impact energy of 8 keV. The Cs $+$ ion beam was rastered over an area of $150×150$  $μ$m $2$ with the collected data coming from a $63×63$  $μ$m $2$ area in the center of the larger raster. Quantitation of the SIMS oxygen depth profiles was made using an implant standard to convert the ion signal to impurity concentration. Here, we used an O implant standard dosed with O at $2× 10 15$ atoms/cm $2$ at 180 keV by Leonard Kroko Inc. to quantify the O composition of the RF penetration layer and beyond by detecting $16$O $−$ in conjunction with a $93$Nb $−$ reference signal.

A thin overlayer decomposition and diffusion model has been proposed by Ciovati⁴² to describe the decomposition of Nb $2$O $5$ and O diffusion in the Nb $2$O $5$/Nb system to describe the migration of O in Nb during heat treatments between $100$ and $200 °$C. Ciovati’s model considers one-dimensional O diffusion from an initial interstitial oxygen concentration present at the oxide/metal interface and diffusion of oxygen from thermal decomposition of an oxide overlayer. Recently, vacuum heat-treated Nb measured by secondary ion mass spectrometry found oxygen diffusion profiles consistent with Ciovati’s model.³¹ We extend this model to multistep dissolutions, multilayers, and temperature profiles relevant to real furnaces.

Using Eq. (9) with the contributions from Nb $2$O $5$ and NbO $2$ dissolution source terms, we have attempted to fit the SIMS measurements in our previous work.³¹ The fit to that data did not yield NbO $2$ dissolution parameters to any reasonable confidence, and results are not presented here. We suspect that a reasonable extraction of dissolution parameters will require a deeper exploration of the heat treatment time and temperature parameter space and hinge on high-quality SIMS measurements on single crystals to combat the effects of ion channeling.⁵³ For higher temperatures and longer vacuum heat treatment times, the effects of NbO $2$ dissolution will become relevant depending on the nature of the decomposition,⁵⁵ where the NbO $2$ layer has been shown to dissolve after Nb $2$O $5$ dissolution.^57,58

The finite-thickness-impermeable-substrate model may be realized by multilayer Nb films deposited on an oxygen diffusion barrier material like MgO, Si, Al $2$O $3$⁠, or AlN^63–69 at temperatures low enough to preserve the native Nb oxide and then allow the newly deposited Nb surface to oxidize. In context of SIMS measurements, an impermeable substrate geometry could allow for precise determinations of oxide decomposition and O diffusion in Nb for native oxide or anodized Nb^70,71 dissolution without lengthy secondary ion mass spectrometry measurements by only needing to depth profile to the impermeable substrate. The applicability of this structure to superconducting RF structures may allow thorough explorations of preservation of the Bean–Livingston barrier by introducing a dirty layer on top of a clean Nb substrate as described by Kubo,² but care must be taken not to introduce too much oxygen during dissolution as many superconducting properties can be degraded.^35,72

A two-step impurity alloying process may be advantageous for SRF cavity performance to first establish a background concentration and then introduce a shallow impurity profile for enhancing peak magnetic field. Sequential heat treatments after reoxidation can be modeled by evaluating the first alloying step using Eq. (8), then using that as the initial condition for the following step evolved numerically under Eq. (7) using a replenished source. Figure 4 shows modeling of samples vacuum heat treated, oxidized, and vacuum heat treated again. The parameters used for the modeling are listed in Table I.

As shown in Table I, there is some spread in oxide dissolution and O diffusion parameters of fits to the diffusion model in limits not previously explored. The O diffusion parameters $E a D$ and $D 0$ are generally consistent with the previous exploration.³¹ Examining $D(T)$ extracted from these single curve fits at the extremes, at $100 °$C, $D$ has an average diffusion coefficient of $1.6±0.5× 10 − 18$ and $7.4±1.6× 10 − 16$ cm $2$/s at $350 °$C. For, $u 0$⁠, the parameter quantifying the amount of dissolvable Nb $2$O $5$⁠, we find $190±40$ at. % nm suggesting that the magnitude of O content can be determined to be $±20%$⁠. There may be a decrease in $u 0$ in the two-step baked samples due to a thinner Nb $2$O $5$ in the mid-T baked type samples.⁷⁴ While these parameters are generally within the deviations expected from RSF variation on polycrystalline Nb, real differences in oxide thickness and composition before dissolution could be a compounding factor to the spread. Determination of $v 0$ is only possible for baking temperatures where oxide dissolution is not dominant. Determination of $v 0$ was only possible for NL564. It was found that $v 0$ was much larger than previously observed.³¹ We speculate that the enhanced interstitial O concentration may be due to the fact that this oxide was grown in air, in contrast to previously investigated samples.³¹ The oxide dissolution parameters $E a$ and $A$ are consistent with previous estimations.

High-field Q-slope (HFQS) is a phenomenon characterized by a rapid decay of the quality factor, $Q 0$⁠, above accelerating gradients between 20 and 30 MV/m, which is preceded by a slower decay in $Q 0$ below the onset of HFQS.⁷⁵ The amelioration of this effect on electropolished Nb surfaces has involved “low-temperature baking” consisting of vacuum heat treatment at temperatures between 110 and $160 °$C for several hours, e.g., $120 °$C for 48 h.⁷⁵ Recent theories of extending the peak magnetic fields in SRF cavities have focused around reducing the peak current density near the surface to preserve the Bean–Livingston barrier. Guarding the Bean–Livingston barrier with impurity profiles or a “dirty” surface bilayer^1,2,50,73 extends the peak magnetic field before the metastable Meissner state becomes absolutely unstable to vortex penetration.^48,51

Here, the clean limit of the London penetration depth $λ 0=39$ nm and coherence length $ξ 0=38$ nm.⁸⁹ A small increase in $λ$ by $∼10%$ from the nonlinear Meissner effect is neglected.⁹⁰ With these estimations, Eq. (14) can be calculated for different temperature and times for the heat treatment process using the oxide dissolution and O diffusion model described in Ref. 31. Figure 5 shows the normalized magnetic field, supercurrent density, and oxygen impurity profile at different temperatures and times of heat treatment. There is a small widening of the Meissner screening profile, $B/ B 0$⁠. Such a small change in Meissner screening profile may be observable in muon spin rotation experiments but may be difficult to distinguish.⁷⁹ The supercurrent density was calculated using $∇× B= μ 0 J$⁠. Starting with short baking times at $120 °$C, the supercurrent density is deformed from its exponential form and shifted away from the surface. With longer baking times, the peak supercurrent density extends further from the surface and is suppressed until the spatial distribution of the impurity profile is uniform enough to start returning to its exponential nature. The ratio, $J max 0/max(J(x,T,t))$⁠, of peak supercurrent density of clean Nb, $J max 0$⁠, to the maximum spatial supercurrent density, $max(J(x,T,t))$⁠, in the O-alloyed Nb represents the factor of peak field enhancement before the depairing current density of the clean limit is reached in O-alloyed Nb. This ratio can be mapped out in the O-alloying $T−t$ space as shown in Fig. 6. There exists a ridge of heat treatment times and temperatures that minimize the peak spatial supercurrent density with respect to the clean case. Despite slightly enhanced O concentration near the surface for higher temperature processes, this ridge in $T−t$ space shows no significant difference as a function of temperature. This invariance is due to the form of Eq. (16) where the penetration depth would change negligibly with a small increase in O from oxide dissolution. Furthermore, the ridge is consistent with the temperatures and times where efficacy for HFQS amelioration has been shown in previous works for low temperature baking at different temperatures.^91–94 Notably, the model predicts that the $120 °$C for 48 h recipe may not ideally minimize the spatial distribution of the supercurrent density and an ideal baking time may be more consistent with the $120 °$C for 24 h. It should be noted that the proposed mechanism of high-field Q-slope amelioration described here is much different from that of Ref. 42, which proposed that the effect is due to a minimization of O near the surface upon heat treatment for some time. This scenario was explicitly ruled out.⁹⁵ 

Recently, insightful measurements by Bafia et al.⁹⁶ clearly showed a rise and gentle decay of the quench field, $E q$⁠, with baking time and that cavities baked at different temperatures tend to fall on a universal line when scaled with oxygen diffusion depth using the model of Ref. 97. These observations are consistent with the model of high-field Q-slope amelioration proposed here. Since the quench fields (peak magnetic fields, B_p) are essentially unaffected by baking temperature, we can rescale the O-diffusion depth in Fig. 5 of Ref. 96 to effective baking times at $120 °$C, as shown in Fig. 7. The peak field should scale as $E q= E q 0 J max 0/max(J(x,T,t))$⁠. A best fit of this model, plotted in red in Fig. 7, using the oxide dissolution and oxygen diffusion parameters of Lechner et al.³¹ and allowing the interstitial O content $v 0$ and $E q 0$ to vary, we find $E q 0=29.2±0.5 MV / m$ and $v 0=2.6±0.3$ at. % nm, close to the value of 3.5 at. % nm extracted in Lechner et al.³¹ Using $E q 0=29.2$ MV/m (B_p = 124 mT) and the parameters extracted in Lechner et al., an optimal treatment is estimated to produce $∼50$ MV/m (B_p = 212 mT) TESLA-shaped cavities⁹⁸ as shown in the blue dashed line in Fig. 7. Between $∼100$ and $200 °$C, O diffusion is dominated by the migration of interstitial O located near the oxide/metal interface, quantified by $v 0$⁠. The peak supercurrent density can be suppressed by loading the surface with a greater concentration of impurities, which enlarges the London penetration depth. This enhanced O concentration must be spread further by diffusion. The enhancement of peak field in the blue dashed line of Fig. 7 as compared with the red line is due to the larger O concentration of $v 0=3.5$ at. % nm as compared with $v 0=2.6$ at. % nm. Naturally, optimal bake times will scale with the diffusion coefficient, $D$⁠.

The analysis presented here shows that Bean–Livingston barrier preservation via oxygen diffusion is capable of capturing the increase and decrease in quench field with vacuum heat treatment time. In this framework, what could be the role of nano-hydride precipitates? It is known that severe plastic deformation occurs in Nb affected by Q-disease, which may facilitate Bean–Livingston breakdown well below its theoretical maximum^99,100 topographically^101–106 or facilitate vortex entry through weakened superconductivity or proximity effect.¹⁰⁷ We suggest that nano-hydride precipitation may be more involved in setting $E q 0( B p 0)$⁠, while the extension of $E q( B p)$ with low-temperature vacuum annealing is related to prevention of vortex entry of the imperfect surface. Recent calculations show that vortex nucleation and dissipation through hydrides may be especially lossy and bring about high-field Q-slope.¹⁰⁸ 

We further developed a model of overlayer dissolution and diffusion potentially relevant for a variety of materials with a dissolving overlayer. The model has been extended to consecutive irreversible reactions, multilayers, varying temperature, sequential dissolution, and films on impermeable substrates. These methods may be useful in guiding heat treatments toward tailored O content deep in to the substrate where a single native oxide dissolution would be insufficient. In particular, varying O diffusion depths at a constant elevated O content near the surface may provide insight into the effectiveness of hydride blocking by O.¹⁰⁸ The model was compared to SIMS O concentration depth profiles and shows promise but will require a dataset spanning larger portions of the $T−t$ space to determine parameters related to NbO $2$ dissolution. This in-depth investigation of the native oxide dissolution and O diffusion process using single crystals is a topic of future study.

It was shown that optimal low-temperature baking times and temperatures scale with oxygen diffusion parameters^81–83 because the oxygen impurities bend the supercurrent. An optimal nanostructuring of the surface with O requires that the O impurities extend over a length scale set by the London penetration depth. The models of oxygen diffusion and supercurrent density deformation described here also provide insight into other optimal baking temperatures and times that have been previously explored in literature. This reduction in the peak supercurrent density also explains the rise and reduction of the onset of quench fields with greater baking times observed by Bafia et al.⁹⁶ Beside superconducting RF cavities, this process should be applicable to devices. While the modified London equation is oversimplified for O-alloyed Nb, quasi-classical theories of superconductivity¹ that consider the modification of superconducting parameters with O content^35,36 will provide more accurate predictions.

Beyond the impurity alloying described here, the oxide dissolution process may provide an opportunity to modify the metallic oxide on the surface. Tuning the thickness of the overlayer and its contact resistance may have a considerable impact on deforming the quasi-particle density of states toward enhancing quality factors that improve energy efficiency of particle accelerator applications.^6,7,109 Alternatively, a metallic oxide could be formed on the surface by dissolving both the NbO $2$ and Nb $2$O $5$ and then preserved by capping the surface with a robust insulating O diffusion barrier.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177, including supplemental funding by the Office of Science, Office of Nuclear Physics Early Career Award to A. D. Palczewski. J. W. Angle was supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Grant No. DE-SC-0018918 to Virginia Polytechnic Institute and State University. We gratefully acknowledge C. Baxley, R. Overton, and T. Harris for support in sample preparation.

The authors have no conflicts to disclose.

E. M. Lechner: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (lead). J. W. Angle: Data curation (equal); Formal analysis (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). A. D. Palczewski: Funding acquisition (equal); Supervision (supporting); Writing – review & editing (supporting). F. A. Stevie: Supervision (supporting); Writing – review & editing (supporting). M. J. Kelley: Conceptualization (supporting); Funding acquisition (equal); Supervision (equal); Writing – review & editing (supporting). C. E. Reece: Conceptualization (supporting); Funding acquisition (equal); Supervision (equal); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.