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

We generalize a native Nb2O5 dissolution model [G. Ciovati, Appl. Phys. Lett. 89, 022507 (2006)] to sequential overlayer dissolutions, multilayer dissolution, and realistic temperature profiles, which may be applicable to other materials. The model is applied to secondary ion mass spectrometry depth profile measurements for varying temperature profiles and two-step oxide dissolution in Nb and found to agree well. In the context of the Meissner screening response due to impurity profiles on the length scale of the London penetration depth, the shallow diffusion of O impurities results in a substantial decrease in the peak supercurrent density near the surface. In this framework, oxide dissolution and oxygen diffusion can account for a rise in peak supportable magnetic field in SRF cavities with baking time and a suppression after the optimal baking time is reached, in good agreement with peak-field baking temperatures and times as well as recent quench field measurements.


I. INTRODUCTION
2][3][4][5][6][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. 33][14][15][16][17][18][19][20][21][22][23] In such projects, trace impurities have tremendous impact on improving superconducting cavity performance. 8,24,25Nb superconducting radio frequency cavities alloyed with nitrogen, 8,9 titanium, 26,27 or other impurities 28 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.][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][35][36][37] Elevated O concentration in Nb is associated with decreases in T c , B c , and B c1 but enhancements in κ, B c2 , and B c3 . 35,380][41] Generalizations of the Nb pentoxide dissolution and oxygen diffusion model described by Ciovati for Nb 42 could be applied to other materials like Zr 43 or Ta, 44 a material considered for superconducting qubits, to enhance the low field surface resistance and increase qubit coherence times. 45,46reservation of the Meissner state is imperative for low-loss, high accelerating field SRF cavity performance.The Bean-Livingston barrier 47 preserves the Meissner state above B c1 , where it is metastable, up to the superheating field, B sh .At B sh , the surface is absolutely unstable to vortex nucleation, allowing penetration of highly dissipative vortices that may trigger thermal instabilities. 48,49hrough surface engineering it is expected that B sh 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. 2,51A goal remains to develop impurity diffusion models to tune interstitials optimally to modify R s and E acc 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.

II. EXPERIMENTAL
SIMS measurements were made using a CAMECA 7f Geo magnetic sector SIMS instrument on Nb samples as described elsewhere. 31,52,53A Cs þ primary ion beam accelerated to 5 keV was used to impact the samples biased at À3 kV, yielding an impact energy of 8 keV.The Cs þ ion beam was rastered over an area of 150 Â 150 μm 2 with the collected data coming from a 63 Â 63 μm 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 Â 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 À in conjunction with a 93 Nb À reference signal.

III. OXIDE DISSOLUTION AND OXYGEN DIFFUSION
A thin overlayer decomposition and diffusion model has been proposed by Ciovati 42 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. 31We extend this model to multistep dissolutions, multilayers, and temperature profiles relevant to real furnaces.
Impurity diffusion through a multilayer system like an oxide is inherently complex and results in the need for several boundary conditions for the impurity concentration.Here, we consider irreversible decomposition in the thin layer limit where mass transport through the oxide does not delay the availability of the impurity to the substrate.This is a serviceable approximation when the diffusion lengths within the oxides are much greater than the oxide thicknesses.In the case of Nb, the migration of O through Nb 2 O 5 oxide may be slow. 54The approximation presented here may lose accuracy for short vacuum heat treatment times or low temperatures if impurity migration through the oxides is expected to be a principal component of the impurity content in the substrate.The decomposition of the native Nb oxide can be modeled by two consecutive irreversible reactions each releasing interstitial O free to migrate toward the bulk, 55 Nb 2 O 5 , NbO 2 , and NbO are represented by reactants A, B, and C, respectively, and the rate equations for these reactions are as follows: and the isothermal solutions are as follows: where X 0 is the initial concentration of the reactant.The impurity source term located at the overlayer position, x ¼ a n , is related to the rate of the production of reactant B from A's dissolution by and the rate of production of NbO is similarly proportional to production of C, These equations can be easily extended to N-sequential irreversible dissolution steps following the same procedure that only changes the form of γ N (t).
The diffusion of oxygen released by oxide dissolution is modeled using the one-dimensional Fick's second law with finite sources in a semi-infinite slab of metal existing at x !0 in the geometry shown in Fig. 1(a), where c is the concentration of the impurity and D is the diffusion coefficient.The diffusion coefficients and rate constants associated with arbitrary component, X, dissolution are assumed to exhibit Arrhenius behavior such that k X (T) ¼ A X exp( À E aX =RT) and D(T) ¼ D 0 exp( À E aD =RT).We use the method of reflection and superposition 56 to construct a solution that preserves boundary conditions for the semi-infinite slab with impurity sources located at a n , and the diffusion equation becomes where γ n (t) ¼ P X γ nX (t).The diffusion equation is solved via Fourier transform with an initial impurity concentration residing at the oxide/metal interface, c(x, 0) For a constant-temperature irreversible process, γ n is only dependent on time, which allows Eq. ( 8) to hold for any number of component dissolutions.As an example, considering only the Nb 2 O 5 dissolution parameters determined in Ref.31, Eq.( 8) is plotted in Fig. 2(a) for T ¼ 280 C. Equation ( 8) reduces to Ciovati's equation and Eq. ( 9) for a superficial oxide dissolution and diffusion process with negligible background concentration of O and negligible NbO 2 dissolution, 42 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. 31The 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. 53For 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. 57,58quation ( 6) may be applied to an engineered multi-layer surface containing an impurity impermeable layer (impurity diffusion barrier) at x ¼ d.This imposes the boundary condition @ x c(d, t) ¼ 0. This geometry is shown in Fig. 1(b).Through the method of reflection and superposition 56 to preserve the boundary conditions of the system, the source positions must be modified to and the solution, made by Fourier transform, is In Figs.2(a) and 2(b), the semi-infinite layer dissolution and finite thickness diffusion model is compared using Nb 2 O 5 decomposition and O diffusion parameters determined from a previous study 31 and are listed in Table I.It should be noted that the values of the O diffusion parameters, D 0 and E aD , are higher than that typically reported by internal friction measurements. 59It is known that the presence of other impurities can substantially increase the 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][64][65][66][67][68][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. 35,72n practice, the time heating to temperature and cooling down in furnaces can be substantial.These effects should be considered when attempting to precisely engineer impurity profiles on the length scale of the London penetration depth to engage with modern theories of superheating field enhancement. 1,2,50,73To model an arbitrary temperature profile of a real furnace for the Nb 2 O 5 oxide dissolution, we employ the one-dimensional Fick's second law with a time-dependent temperature, The initial concentration of oxygen at the oxide/metal interface is given by c(x, 0) ¼ v 0 δ(x).Starting from the rate equation of oxide dissolution, the O source is related to rate equation by The Fick's law diffusion equation is subjected to the boundary conditions c(x ¼ 1) ¼ c 1 and c 0 (x ¼ 1) ¼ 0. To test the model, three samples were vacuum heat treated with a ramp rate of 1 C/min to a maximum temperature and then allowed the cool.The temperature of the samples was measured with a thermocouple, while the furnace temperature was regulated by monitoring the furnace hot zone wall.A ramp-up/ramp-down profile was chosen to showcase the role of oxide dissolution and migration, which may be relevant for short heat treatments where these processes contribute significantly.Sample temperature measurements during vacuum heat treatment are shown in Fig. 3(a).SIMS depth profiles of those samples are shown in Fig. 3(b).To isolate the interstitial O component and remove any possible influence from the high oxygen content oxide, SIMS depth profiles start at 25 nm.The calculations in Fig. 3(b) were made using MATLAB to numerically solve Eq. ( 12) using the source term from Eq. ( 13) using the temperature profiles in Fig. 3(a).The parameters used for the calculations are presented in Table I.The largest variation in parameters for the theoretical O concentration profiles are from the oxide source term and diffusion coefficient.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 aD and D 0 are generally consistent with the previous exploration. 31Examining 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 Â 10 À18 and 7:4 + 1:6 Â 10 À16 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. 74While 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. 31e 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. 31The oxide dissolution parameters E a and A are consistent with previous estimations.

IV. IMPLICATIONS FOR SUPERCONDUCTING DEVICES
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. 75The 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. 75Recent 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. 48,517][78] Recent re-analysis 79 of low-energy muon spin rotation measurements on low-temperature baked Nb indicate that strong changes in the Meissner screening profile 80 do not exist, allowing for the possibility of a smoothly varying Meissner screening profile.2][83] Recent evidence has been provided that an impurity profile may play a role in HFQS amelioration in "N-infused" cavities. 73he spatial distribution of the magnetic field and supercurrent density can be described by the one-dimensional London equation for a spatially inhomogeneous London penetration depth.This equation naturally arises from the application of Faraday's law of induction on the London equation, which describes the response of the supercurrent density to electric fields, 84,85 For O in Nb, the change in the mean free path can be estimated from the change in resistivity, 86 where σ ¼ 0:37 Â 10 À15 Ω m 237 and Δρ O (x) ¼ ac(x), where c is the percent concentration of O and a ¼ 4:5 Â 10 À8 Ω m. 87 The spatially dependent London penetration depth is assumed to be local and estimated according to the following expression: 88 Here, the clean limit of the London penetration depth λ 0 ¼ 39 nm and coherence length ξ 0 ¼ 38 nm. 89A small increase in λ by 10% from the nonlinear Meissner effect is neglected. 90ith these estimations, Eq. ( 14) 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. 79The supercurrent density was FIG. 4. SIMS depth profiles (black circles) of Nb samples first-step vacuum heat treated at 300 C for 2.5 h, reoxidized, and then second-step vacuum heat treated.(a) NL563 received a second-step 250 C vacuum heat treated for 3 h (b) NL564 received a second-step 120 C vacuum heat treatment for 72 h.The red lines are the theoretical fits using oxide dissolution and oxygen diffusion parameters as described in Table I. calculated using ∇ Â B ¼ 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 À 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 À 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.][93][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 highfield 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. 95ecently, 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 ¼ E q0 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 q0 to vary, we find E q0 ¼ 29:2 + 0:5 MV=m and v 0 ¼ 2:6 + 0:3 at.% nm, close to the value of 3.5 at.% nm extracted in Lechner et al. 31 Using E q0 ¼ 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 cavities 98 as shown in the blue dashed line in Fig. 7. FIG. 6.(a) T À t À J max 0 = max (J(x, T, t)) colormap showing the existence of a ridge of optimal baking times and temperatures to reduce the peak supercurrent compared to the clean case, J max 0 = max (J(x, T, t)).The white dashed line follows the path of minimized peak supercurrent density.
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 ¼ 3:5 at.% nm as compared with v 0 ¼ 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][102][103][104][105][106] or facilitate vortex entry through weakened superconductivity or proximity effect. 107We suggest that nano-hydride precipitation may be more involved in setting E q0 ðB p0 Þ, while the extension of E q ðB p Þ with lowtemperature 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

V. CONCLUSIONS
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 À 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][82][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 superconductivity 1 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. 6,7,109lternatively, 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.

FIG. 1 .
FIG. 1.(a) Semi-infinite slab geometry and (b) thin film on O-impermeable substrate.O(gray layers) indicates dissolvable oxides located at a n and M(blue layers) denotes the substrate absorbing the mobile species produced from overlayer dissolution.

FIG. 2 .
FIG. 2. Comparison of the semi-infinite and finite-thickness-impermeable-substrate solution to the diffusion equations using dissolution and diffusion parameters described from Ref. 31 at T ¼ 280 C. (a) Colorplot and concentration profiles of the semi-infinite slab solution and concentration profiles at selected times.(b) Colorplot and concentration profiles of the finite-thickness-impermeable-substrate solution and concentration profiles at selected times.Black vertical dashed lines indicate the location of the sources (Nb 2 O 5 layers) at x = 0, 300, and 800 nm.All thin oxide layers modeled are equal in O content.

FIG. 3 .
FIG. 3. (a) 1 C/min ramp up and ramp down temperature profiles that samples were subjected to.(b) SIMS depth profiles (black circles) of the samples and their theoretical fits (colored lines) as described in the text and using the temperature profiles in (a) within the calculation.

FIG. 5 .
FIG. 5. Normalized magnetic field, supercurrent density, and O impurity distributions predicted for various times at T ¼ 120 C.

TABLE I .
Model parameters used for theoretical O concentration profiles in Figs.3 and 4.