Effect of interstitial impurities on the field dependent microwave surface resistance of niobium Open Access

Nitrogen-doping is a surface treatment which allows nitrogen atoms to be absorbed as interstitial impurities in the niobium lattice. This treatment has shown a dramatic improvement of superconducting radio-frequency (SRF) properties, even at relatively low concentrations of nitrogen (lower than 200 ppm). In particular, the cavity quality factor, or Q-factor (Q₀), which is inversely proportional to the power dissipated on the cavity walls, can increase by a factor of three at medium values of accelerating field (⁠$Eacc=16 MV/m$⁠).¹ The Q-factor is determined by the cavity RF surface resistance R_s: $Q0=G/Rs$⁠, where G = 270 Ω is the geometrical factor which is independent on material properties. The RF surface resistance can be decomposed in two contributions, one temperature dependent called BCS surface resistance (⁠$RBCS$⁠) and one temperature independent called residual resistance (⁠$Rres$⁠).

In this letter, we analyze these two surface resistance contributions for bulk niobium resonators, looking at both the mean free path and the RF field dependencies. The findings here reported allow a much better understanding of which surface treatment is required to maximize the Q-factor for a certain RF field, taking into account the external DC magnetic field trapped during the cooldown through the superconducting (SC) transition.

Q-factor maximization is extremely beneficial in order to decrease the cryogenic cost of continuous-wave accelerators. For this reason, the Linear Coherent Light Source (LCLS-II) at SLAC embraced the nitrogen-doping technology as treatment for the SRF cavities of the superconducting linear accelerator.² 

The BCS surface resistance was defined by Mattis and Bardeen.³ Based on the Bardeen-Cooper-Schrieffer theory of superconductivity,⁴ $RBCS$ decays exponentially with the temperature and depends on several material parameters, such as London penetration depth $λL$⁠, coherence length ξ₀, energy gap Δ, critical temperature $Tc$⁠, and mean free path $ℓ$⁠. Of most interest here is that from the Mattis-Bardeen calculation, $RBCS$ as a function of the mean free path shows a minimum around $ξ0/2$⁠.⁵ 

It is well known that nitrogen-doping affects the BCS contribution¹ which, in contrary of what happens with standard treatments, decreases with accelerating field. This results in an increasing of Q-factor with accelerating field called anti-Q-slope. The mechanisms that govern the anti-Q-slope are not well understood yet, even though some theories have been proposed.^6,7

This letter adds important insight on the BCS surface resistance field dependence, suggesting that the decreasing of $RBCS$ may be due to an increasing of the energy gap Δ with the RF field.

The introduction of interstitial impurities and the subsequent change in mean free path affects also $Rres$⁠. Principal sources of residual losses are condensed gasses, material inclusions, hydrides, and trapped magnetic flux.⁸ This last contribution defines the trapped flux sensitivity which in turn strongly depends on the mean free path.⁹ Here, we show a complete and detailed study which gives the most clear picture of trapped flux dissipation in SRF niobium cavities, from its dependence on the mean free path to its dependence on the RF field. This part of the study is of crucial importance in order to understand both the surface treatment of SRF cavities and the level of magnetic field shielding needed in cryomodules.

The amount of trapped flux depends on both the amount of external magnetic field which surrounds the cavity during the SC transition, and on the cooldown details, which affects the magnetic flux trapping efficiency. In particular, fast cooldowns with large thermal gradients along the cavity length help to obtain efficient magnetic flux expulsion, while slow and homogeneous cooling through transition leads to full flux trapping.^10–13 

The cavities analyzed are single cell 1.3 GHz Tesla-type bulk niobium cavities.¹⁴ After the fabrication, about 120 μm is removed from the inner surface via electro-polishing (EP) and then UHV baked at 800 °C for 3 h. If no further treatment is done, such cavities are called EP. Other treatments may be done after these steps, such as another run of EP, buffer chemical polishing (BCP), 120 °C baking and N-doping followed by EP removal. Details on the nitrogen-doping process can be found elsewhere.^1,15 The surface treatment of the analyzed cavities is summarized in Table SI of the supplementary material.

A schematic of the instrumentation used to characterize the trapped flux surface resistance may be found in Ref. 9 (Fig. 1). Helmholtz coils adjust the magnetic field around the cavity, three Bartington single axis fluxgate magnetometers monitor the external magnetic field at the cavity equator and thermometers monitor the cooldown details.

In order to estimate the trapped flux surface resistance, each cavity was measured after two different cooldowns: (i) compensating the magnetic field outside the cavity during the SC transition and (ii) cooling slowly the cavity with about 10–20 mG of external magnetic field. After each cooldown, the cavities were tested at the vertical test facility at Fermilab.

We define $Rres$ as the sum of the two terms: the trapped flux residual resistance, $Rfl$⁠, and the “intrinsic” residual resistance, R₀, in order to distinguish the effect of trapped flux from other contributions; therefore,

where T is the temperature and $Btrap$ is the trapped field. Since at very low temperatures, $RBCS$ becomes negligible, the Q-factor is measured at 1.5 K, and the residual resistance is calculated as R_res = G/Q(1.5 K). If during the cooldown, the amount of trapped flux is minimized, then $Rfl≃0$ and $Rres≃R0$⁠. In order to obtain $Rfl≃0$⁠, the magnetic field outside the cavity was compensated during the cooldown through the SC transition. The average value of magnetic field measured at the cavity equator was always lower than 1 mG. Alternatively, when possible, the measurement was done after a complete magnetic flux expulsion (⁠$BSC/BNC∼1.77$ at the equator).¹⁶ We have observed that these two methods gave the same results within the measurements uncertainties. On the other hand, after the cavity trapped some external field: $Rres(Btrap)=Rfl(Btrap)+R0$⁠.

$RBCS$ and $Rfl$ are therefore estimated as follows:

$Rres(Btrap)$ was always calculated from the RF measurements after slow cooldowns so that the amount of trapped flux tends to the amount of external field: $Btrap≃BNC$⁠.

The trapped flux sensitivity S describes the amount of cavity losses per unit of trapped field and can be defined as

The values of sensitivity estimated for the cavities analyzed are listed in Table SI of the supplementary material and are shown as a function of the mean free path in Fig. 1. The mean free path of the majority of the cavities analyzed is estimated by means of a C++ translated version of SRIMP¹⁷ implemented in the OriginLab data analysis program.

The cavity resonance frequency as a function of temperature during the cavity warm up is acquired in order to obtain the variation of the penetration depth, $Δλ$⁠, as a function of T close to $Tc$⁠.⁵ 

The SRIMP code¹⁷ is used to interpolate $Δλ$ versus temperature to estimate the mean free path and the reduced energy gap (⁠$Δ/kTc$⁠). The fixed parameters are: critical temperature, coherence length (ξ₀ = 38 nm) and London penetration depth (λ_L = 39 nm). The critical temperature, $Tc$⁠, was estimated for each cavity directly from the measurement of the resonance frequency as a function of temperature during the warm-up through the normal-conducting state. The resonance frequency drops at transition, and stabilizes when the cavity is normal-conducting, allowing the visualization of the $Tc$ value.

The 120 °C baked cavities mean free path is estimated from low energy muon spin rotation (LE-μSR)¹⁸ measurements performed on a representative cavity cut-out,¹¹ since the fit with SRIMP would have introduced larger error. Indeed, the 120 °C baking treatment modifies the mean free path of about the first 60 nm, and for temperatures close to T_c the penetration depth becomes larger than the modified layer, about thousands of nanometers, probing a region which is not representative of the mean free path in the RF layer at low temperature.

Figure 1 shows that the sensitivity has a bell-shaped trend as a function of the mean free path. The sensitivity is minimized for both very small (120 °C bake cavities) and very large (EP and BCP cavities) mean free paths, and it is maximized around ℓ ≃ 70 nm. Taking into account the optimal N-doped cavities, when over-doped they show the highest sensitivity (ℓ between 70 and 100 nm), while the 2/6 recipe¹⁵ gives the lowest sensitivity (ℓ around 120–180 nm). From this trend, we can also infer that going toward even lighter doping recipes, it should be possible to further decrease the trapped flux sensitivity. We are not taking into account doped cavities with ℓ < 70 nm since they show low sensitivity but large intrinsic residual resistance, since “non-optimally” doped,¹ which nullifies the beneficial effect of interstitial nitrogen.

The values of sensitivity obtained for EP and BCP cavities are in agreement with the previous studies,¹⁹ in which sensitivity of 0.35 nΩ/mG was measured for a 1.5 GHz cavity made out high purity niobium sheet. Trapped flux sensitivity of 1.3 GHz EP niobium cavities was studied also in Ref. 20 in which larger values were found compared to the values of both ours and Ref. 19.

The experimental data in Fig. 1 shows some scatter that may be due to a larger uncertainty on the mean free path values than the error bars since the large number of fit parameters. Differences in terms of pinning force and dimension or position of pinning centers between the analyzed cavities may also increase the data scattering.

The vortex dissipation may be considered as coming from two contributions: (i) static due to the normal-conducting core of the vortex⁸ and (ii) dynamic due to the vortex oscillation driven by the Lorentz force in the presence of the RF field.^21–23 The bell-shaped trend of sensitivity as a function of mean free path (Fig. 1) may be obtained even from the static contribution alone;²⁴ however, our preliminary theoretical results suggest that also the dynamic dissipation may lead to the same trend.

In Fig. 2, it can be seen that the trapped flux surface resistance, and therefore the sensitivity, increases with the RF field. A field dependence of $Rfl$ was also found studying large grain cavities²⁵ and niobium on copper thin film cavities.²⁶ As hypothesized also in Ref. 26, the possible explanation to this phenomenon might be the progressive depinning of vortexes from their pinning center, driven by the increasing of the RF field.

The BCS surface resistance at 2 K is extrapolated after the cooldown with no flux trapped, as the difference between the $Rs$ measured at 2 K and at 1.5 K (Eq. (2)). $RBCS$ measured at low field is shown in Fig. 3 (upper graph) as a function of the mean free path. The upper graph shows the results obtained at low field (5 MV/m), while the lower at medium field (16 MV/m). For EP cavities, the mean free path is calculated for one cavity (AES018) and the other one is fixed at the same value, assuming that they should show very similar values. For one N-doped cavity (AES005), the mean free path is directly measured on a cavity cut-out with LE-μSR.¹⁵ 

The green diamonds represent doped cavities, while the pink circles are Niobium cavities with different standard treatments (120 °C bake, BCP, and EP). The black curves are theoretical curves of $RBCS$ versus mean free path estimated using SRIMP¹⁷ for different reduced energy gap values.

In both field regimes, doped cavities show lower values of $RBCS$ than the non-doped cavities, proving that $RBCS$ is lowered with the introduction of interstitial impurities. At medium field, the difference in $RBCS$ between doped and non-doped cavities is maximized due to the opposite trend of this surface resistance contribution as a function of the accelerating field.

The values of $RBCS$ obtained for all the cavities analyzed cannot be described by one single theoretical curve, both at low and medium fields, suggesting that the mean free path is not the only parameter changing with the introduction of impurities. Following this hypothesis, one of the other parameters on which the BCS surface resistance depends on (⁠$λL$⁠, ξ₀, Δ, $Tc$⁠) is changing as well. In the low field case, fixing all the other parameters and changing the reduced energy gap $Δ/kTc$⁠, the 120 °C baked, BCP, and EP cavities are interpolated with $Δ/kTc=1.95$⁠, while doped cavities are better interpolated setting $Δ/kTc=2$⁠. At medium field, the difference is even larger being $Δ/kTc=1.85$ for 120 °C baked cavities and $Δ/kTc=2.05$ for doped cavities. For BCP and EP cavities, $Δ/kTc$ is probably slightly larger than the value assumed for 120 °C baked cavities.

In addition, we observe a possible field dependence of the gap. Comparing the upper and lower graphs of Fig. 3, for doped cavities $Δ/kTc$ increases passing from 5 to 16 MV/m. This variation may be the reason why $RBCS$ decreases with the RF field for doped cavities. Increasing of the energy gap with the RF field has been measured in the past,²⁷ and in that case the enhancement of superconductivity was attributed to non-equilibrium effects.^28,29 In the Eliashberg theory, the minimum frequency at which non-equilibrium effects may be visible depends on the inelastic collision time of quasi-particles scattering with phonons $τE$ and for niobium this minimum frequency is around 15 GHz.²⁸ Future work will be focused on the measurement of $τE$ of nitrogen-doped samples in order to understand how N-doping modify this parameter.

Adding together the measured values of $RBCS$ and sensitivity, we illustrate which treatment shows the highest Q-factors depending on the amount of trapped flux.

In order to visualize that we calculate for each treatments among EP, 120 °C baking and N-doping the Q-factor are as follows:

where the values of $Btrap$ ranges from 0 to 20 mG.

Among the N-doped cavities, we chose the 2/6 N-doping treatment, which is one of the greatest interests for high-Q application, since it shows the best compromise between $RBCS$ and sensitivity values exploited so far.

The Q-factor as a function of $Btrap$ is shown in Fig. 4. From this graph, it is clear that the 2/6 N-doped cavity shows the highest values of Q-factor as long as the trapped field is lower than 10 mG, i.e., within the range of realistic values of magnetic field achievable in modern cryomodules.

The intrinsic residual resistance depends on many parameters, some related to the surface treatments and others related to the bulk itself.³⁰ 120 °C baked cavities usually show value of R₀ greater than both EP and optimized N-doped cavities.^1,31 Because of that, the calculation was carried out assuming as intrinsic residual resistance: R₀ = 4 nΩ for the 120 °C baked cavity and R₀ = 2 nΩ for both EP and N-doped cavities, which are common values found for these treatments.

Using these findings as guidance, our future work will investigate the performance of N-doped cavities with even larger mean free path values. The treatment that produces best compromise between BCS surface resistance and sensitivity is probably still unexplored.

This letter provides insight on the physics behind the lowering of the BCS surface resistance, suggesting that with the introduction of impurities another material parameter must change other than the mean free path. The most likely candidate is the energy gap. In addition, the anti Q-slope of N-doped cavities may be explained as the decreasing of the energy gap with the field caused by microwave-driven non-equilibrium effects.

In summary, from a practical point of view, these results are of crucial importance in order to identify the best surface treatment that allows to reach the highest Q-factors, taking into account all the surface resistance contributions and their dependencies on mean free path, RF field, and DC external magnetic field.

See supplementary material for mean free path, sensitivity and BCS surface resistance data and for more information on the cavities surface treatments.

This work was supported by the United States Department of Energy, Offices of High Energy and Nuclear Physics and by the DOE HEP Early Career grant of A. Grassellino. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.