The importance of the electron mean free path for superconducting radio-frequency cavities

A well-known property of superconductors is their perfect DC conductivity and their ability to expel ambient magnetic flux, as described by Bardeen, Cooper, and Schrieffer (BCS) in 1957.^1,2 Under radio-frequency (RF) electromagnetic fields incident on a superconducting surface, the finite inertia of the Cooper pairs causes the superconductor to develop a non-zero surface resistance, $Rs$⁠. In the context of superconducting radio-frequency (SRF) accelerator physics, $Rs$ is highly relevant due to its relation to the quality factor Q₀ of an SRF structure, as shown in Eq. (1)

Here, the geometry factor G, measured in Ohms, is (as the name suggests) a scaling factor derived from the geometry of an SRF cavity; typical values for modern elliptical SRF accelerator cavities are around 270–300 Ω. U is the energy stored in the electromagnetic field in the cavity, f is the frequency of the cavity, and $Pdiss$ is the power dissipated in the cavity walls due to the surface resistance, averaged over one RF cycle. A high Q₀ is preferable for SRF cavities because it allows higher accelerating gradients for a given cryogenic cooling power budget (typically, SRF cavities are cooled between 1.8 and 4.2 K), or, alternatively, lower cooling power costs for a given accelerating field.

The surface resistance $Rs$ can be separated into a sum of the temperature-dependent “BCS resistance” $RBCS$ and the temperature-independent “residual resistance” $R0$⁠, with the relation³ in Eq. (2)

The BCS resistance typically increases with the strength of the applied field in traditionally prepared (non-doped) SRF cavities. This rising resistance manifests as a decrease in the intrinsic quality factor Q₀ of the cavities, by the relation in Eq. (1); this effect is traditionally known as the “Q-slope.” However, under certain circumstances, $RBCS$ has been observed to decrease with increasing field strength.^4,5 This field-dependent reduction in surface resistance causes an increase in Q₀; throughout the remainder of this work, we will use the term “positive Q-slope” to differentiate it from the traditional negative Q-slope in the medium- and high-field regions (the positive Q-slope effect is also commonly referred to as the “anti-Q-slope”).^4,6,7 Figure 1 shows examples of this positive Q-slope, medium-field Q-slope, and high-field Q-slope. In the field of SRF, the positive Q-slope phenomenon was discovered in 2013 by Grassellino et al. at Fermilab in niobium doped with nitrogen and by Dhakal et al. at Jefferson Laboratory.^4,5 In this context, “doping” entails treating an SRF material in a furnace with a low-pressure dopant gas atmosphere (nitrogen-doped bulk niobium is the only material considered in this work). Impurity-doping SRF cavities in this way causes both this reverse dependence of $RBCS$ on the field strength and a decrease in the low-field surface resistance, making the technology highly appealing for new accelerators requiring very efficient SRF cavities.

Unfortunately, doping cavities with nitrogen leads to an increased susceptibility to RF losses from trapped magnetic flux.⁶ Magnetic flux is usually trapped from ambient fields during the cool-down or during quench events by flux vortex pinning at impurities and defects.^9,10 Trapping flux increases the effective residual resistance $R0$ of the superconductors, which can be detrimental to accelerator operation. Further, nitrogen-doping has been shown to lower the average quench field of SRF cavities, limiting the peak available accelerating gradient.¹¹ 

In this work, we investigate the effects of impurity doping on the surface resistance to find an optimal doping level for niobium SRF accelerating structures, as measured by the electron mean free path $ℓ$ (stronger doping corresponds to shorter $ℓ$⁠). We offer new insight into how the mean free path affects the field dependence of the BCS resistance.

Although the dependence of the low-field surface resistance on the mean free path observed in superconductors¹² with minimum near $ℓ≈ξ0/2$ (where ξ₀ is the BCS coherence length) can be calculated numerically from the BCS theory,^3,13–15 the dependence of this resistance on the strength of the RF field is not yet fully understood. Recent theoretical work by Gurevich offers a promising explanation for the decrease of $RBCS$ with increasing RF field.¹⁶ Here, the RF field changes the density of states of the Bogoliubov quasiparticles (i.e., electron-like quasiparticle excitations above the superconducting ground state^17,18) in a way that tends to reduce their number density and thus reduce the Ohmic losses due to their movement. An important effect in this theory is the overheating of the quasiparticles, which offers an opposing force to the reduction of the surface resistance by increasing the effective temperature of the quasiparticles. This overheating is controlled by the “overheating parameter” α, encapsulating material parameters such as the Kapitza interface conductance $hK$⁠, the thermal conductivity κ, and Y, the energy transfer rate between quasiparticles and phonons, by the relations in Eqs. (3) and (4) (Eqs. (13) and (14), respectively, in Gurevich's paper¹⁶)

Here, T is the quasiparticle temperature, T₀ is the experimental bath temperature, $RBCS,0$ is the low-field BCS resistance, $μ0Ha=Ba$ is the RF magnetic field at the surface, $μ0Hc=Bc$ is the thermodynamic critical field, $RBCS$ is the BCS resistance at the given applied field and temperature, and d is the thickness of the SRF cavity wall.

In order to study the effects of nitrogen doping in niobium SRF cavities, we prepared many cavities using nitrogen-doping techniques to achieve a range of values of the mean free path $ℓ$ in the RF penetration layer, ranging from $ℓ=$ 4 nm to over 200 nm. It is necessary to mention here that the doped layer is typically 5–50 μm thick and does not extend through the bulk; $ℓ$ is essentially uniform over the RF penetration depth, which is on the order of 100 nm. This is not true for 120 °C-baked cavities, which are not discussed in this work.

These preparations were done on single-cell 1.3 GHz TESLA-shape cavities.¹⁹ Table I summarizes their properties and preparation techniques. We performed vertical RF tests of these cavities, measuring the quality factor Q₀ as a function of field at many temperatures, as well as the low-field Q₀ and resonant frequency f₀ as a function of bath temperature T₀.^7,11,20,21 We used the change in resonant frequency f₀ to determine the change in RF penetration depth λ using methods described previously by Ciovati and others.^22,23 We used the quality factor Q₀ to determine total surface resistance $Rs$ using the relation in Eq. (1). We performed a combined fit of $Rs(T0)$ and $Δλ(T0)$ to the BCS theory using SRIMP;¹³ fits from cavity C4(P1) are given in Fig. 2 and serve as representative examples of the procedure followed for all cavities. This BCS fitting yielded the mean free path $ℓ$⁠, the energy gap Δ, the coherence length ξ, and the residual resistance R₀. Fitting to both $Rs(T0)$ and $Δλ(T0)$ was very helpful for determining precise values for the fit coefficients as the two fits are more sensitive to energy gap and mean free path, respectively. More information on these fitting techniques is available in previous reports of work at Cornell by Meyers, Valles, and others.^6,21,24–26 Equipped with R₀, we were then able to calculate the BCS resistance $RBCS$ as a function of field and temperature using the relation in Eq. (2).

We then used these material parameters and Gurevich's theory¹⁶ to calculate theoretical predictions of the BCS surface resistance as a function of surface field and temperature, using the overheating parameter α as a free fitting parameter in Eq. (3), with no additional constraints. We found it necessary to use an additional field-independent fitting parameter s as a scaling factor, with the relation $s=RBCS,meas/Rthy$⁠. Here, $RBCS,meas$ is the BCS surface resistance extracted experimentally and $Rthy$ is the predicted surface resistance from Gurevich's theory as described above. For a given cavity, s was fixed for all temperatures. The scaling factor s was typically near unity; we believe that this accounted for systematic experimental errors that vary from cavity to cavity (e.g., uncertainty in G, the geometry factor), which we usually cite as 10%. We discuss this further below. It is important to note here that the theory does not include any explicit dependence of the field-dependent resistance on the mean free path; in our analysis, we sought to investigate any possible dependence of the overheating parameter on $ℓ$⁠.

Figure 3 shows typical results of the BCS surface resistance as a function of applied RF magnetic field, as well as the theoretical predictions based on Gurevich's work.¹⁶ Figure 3(a) shows the measurements of the BCS resistance for a heavily doped cavity, with a mean free path of 4.5 ± 1.3 nm, achieved with a 990 °C nitrogen bake followed by chemical removal (etching) of 5 μm of material. Here, the theoretical predictions match very well with experimental results, with no quasiparticle overheating (⁠$α≈0)$ (by the theory, α should never get to zero due to the contributions from κ and $hK$⁠, but the effects of these on $RBCS$ in this case are minimal¹⁶). Figure 3(b) shows similar measurements for a cavity with a medium doping level, with a mean free path of 34 ± 10 nm after an 800 °C nitrogen bake and a 12 μm etch. Again, there is good agreement with theory and experiment. Here, an overheating parameter of $α=0.44$ was found to best fit the results of the RF measurements at 2.1 K, with α generally decreasing with decreasing temperature. Figure 3(c) shows the results of a lightly doped cavity, with a mean free path of 213 ± 64 nm after receiving an 800 °C nitrogen bake followed by 40 μm of material removal. Here, with $ℓ$ approaching the clean limit, experimental results and theoretical predictions diverge; though the theoretical predictions are well within the quoted uncertainty interval, we believe that this uncertainty accounts mainly for systematic errors that are absorbed into the scaling parameter s. The discrepancy between the overall trends of the theoretical predictions and the experimental data suggests that the thermal effects become characteristically different at higher mean free path, as the positive Q-slope transitions to the more commonly observed medium-field Q-slope, suggesting additional effects not considered in the theory.

Figure 4 shows the low-field 2 K BCS resistance of the cavities listed in Table I, measured at $∼20$ mT (corresponding to an accelerating gradient of $∼5$ MV/m in a TESLA cavity), as a function of the mean free path $ℓ$⁠. The BCS resistance here, and later at 16 MV/m, has been normalized to average critical temperature and energy gap, by the formula $RBCS,norm=RBCS exp (−Δavg/kBTc,avg+Δ/kBTc)$⁠. We do not see any strong dependence of the energy gap on doping level, suggesting that this is a valid normalization technique. This resistance corresponds very well to the theoretical 2 K BCS resistance, also shown in Fig. 4, which has its minimum at $ℓ=25±1$ nm for average material parameters. This is a good confirmation of the accuracy of our measurement techniques.

We next investigated the two fitting parameters, s and α, for dependences on any material or superconductor parameters. The scaling parameter s, which was fixed for each cavity at all temperatures, shows little dependence on any physical parameters. The dependence on mean free path, or lack thereof, is shown in Fig. 5(a). For α, we saw a general positive slope with increasing temperature, as mentioned above and as shown in Figure 5(b). This trend is not surprising given the dependence of α on the BCS surface resistance (see Eq. (4)), which increases rapidly with temperature, but it serves as evidence supporting the derivation of α by Gurevich.¹⁶ In an effort to account for the dependence of the quasiparticle overheating on the experimental temperature and focus our study on the dependence on actual thermal parameters, we derive from the overheating parameter α a normalized overheating parameter $α′$ by Eq. (5)

This reduces Eqs. (3) and (4) to Eqs. (6) and (7)

When plotting this $α′$ as a function of temperature, we see that the dependence has been removed to first order, as shown in Fig. 5(c). This suggests that the quasiparticle overheating is independent of temperature to first order, at least within this band of temperatures between 1.8 K and 2.1 K for the cases where $α′$ is large enough compared to its uncertainty to extract relative trends with T₀.

However, plotting as a function of mean free path, we see a clear dependence that is well approximated by a linear relationship with an offset at zero. Figure 5(d) shows this result, with an offset linear (affine) fit for $α′$ averaged over temperature. Averaging the values over all temperatures, given the temperature-independence indicated in Fig. 5(c), we found an offset of $γ=0.02±0.21×10−3$ K m²/W and a slope $β=2.1±0.8×104$ K m/W, according to the relation in Eq. (8)

Quite important to emphasize here is that this approximation is only valid in the short-mean-free-path limit, as the overheating is not adequately described by the theory for $ℓ>50$ nm. This may be due to the linear approximation of heat transfer for weak overheating in Gurevich's theory.¹⁶ 

The relation in Eq. (8) makes a compelling suggestion that the mean free path plays a strong, fundamental role in the overheating of the quasiparticles, resulting in the observed variation of the field-dependent surface resistance. The shorter mean free path correlates with smaller overheating and therefore stronger reduction in $RBCS$ with increasing RF field.

This correlation invites a comparison between the empirically determined form of $α′$ in Eq. (8) and the theoretical derivation of $α′$ in Eq. (7); in particular, we are motivated to investigate the theoretical form for factors that may depend on the mean free path of the RF penetration layer and for factors that we expect to remain constant for all cavities studied here. Since the Kapitza interface conductance $hK$ and thermal conductivity κ are properties of the substrate cavity, not of the nitrogen-doped RF surface, we do not expect these parameters to change with doping level. Moreover, because all five TESLA cavities considered in this study were manufactured from the same niobium stock, we are further logically inclined to $hK$ and κ as constants across the set of cavities. Indeed, for typical values of these properties for niobium,^3,16,28 this contribution is given by $(d/κ+1/hK)≈0.07$ to $0.50×10−3$ K m²/W, on the same order of magnitude as the confidence interval of the linear fit for γ (as seen in Fig. 5(d)). This suggests that $(βℓ)−1$ is approximately equal to the parameter Y in Eq. (7), representing the electron-phonon energy transfer rate, in this short-mean-free-path limit. In this case, the longer mean free path indicates a lower energy transfer rate from quasiparticles to phonons. A potential explanation for this is elastic scattering of the quasiparticles on impurities: quasiparticles moving on lattices with a longer mean free path (i.e., longer spacing between impurity scattering sites) have longer characteristic elastic scattering times and thus lower energy transfer rates, leading to the stronger overheating observed for long mean free path $ℓ$⁠.

Using this dependence of $α′$ on $ℓ$⁠, we generated theoretical predictions of the field dependence of the surface resistance for a range of mean free paths. We calculated the surface resistance at 68 mT (⁠$Eacc≈16$ MV/m for a TESLA cavity), the operating accelerating gradient spec of the Linac Coherent Light Source II (LCLS-II), to compare to experimental data at the 16 MV/m operating gradient and at low field; LCLS-II is a relevant choice for this comparison because it is an SRF-powered continuous-wave free-electron laser under construction at the SLAC National Accelerator Laboratory, using nitrogen-doped 1.3 GHz niobium TESLA cavities.^29,30 Figure 6 shows the low field BCS resistance (normalized in Δ and $Tc$⁠, as described above) at 21 mT (∼5 MV/m), the theoretical BCS resistance for average material parameters, the theoretical BCS resistance at 68 mT calculated from Gurevich's theory¹⁶ and the $α′(ℓ)$ model presented here, and the experimental data at 68 mT. We see agreement with the high-field experimental data, and in particular, with the shifted minimum in $RBCS$ as a function of $ℓ$⁠, to about 17 nm at 68 mT from the low-field minimum of 25 nm for these material parameters.

As mentioned in the Introduction, nitrogen-doped niobium SRF cavities are susceptible to large increases in the residual resistance R₀ due to trapped magnetic flux. While the reduction in BCS resistance described above is very enticing, any increase in R₀ can counteract this, leading to decreased performance in an SRF accelerating structure.

To investigate this effect, we tested many 1.3 GHz single-cell TESLA cavities at varying doping levels, measured by the mean free path $ℓ$⁠, with varying cool-down rates and externally applied uniform DC magnetic fields in order to study the residual resistance in each cavity as a function of trapped magnetic flux.⁶ These cavities and preparations were the same as those used for the study on the field-dependent BCS surface resistance. Trapped flux was measured by flux gate magnetometers placed on the outer surface of the cavity at the iris, the connection between the beam tubes and the cavity cell. Figure 7(a) shows representative results for the residual resistance as a function of accelerating field for several amounts of trapped flux.

The residual resistance was found to increase linearly with trapped flux, leading to the simple scaling relation given in Eq. (9), where $R0,B$ is the portion of the residual resistance due to trapped flux losses

Here, S provides a convenient measure of the sensitivity of the residual resistance to trapped flux for a given cavity surface. When analyzing S as a function of mean free path, we found that our results compared agreeably with theoretical predictions.^6,31 The theory involves RF losses due to the oscillation of magnetic flux vortices in the superconductor, with the sensitivity varying with mean free path, and with a fit parameter relating the mean free path to the mean spacing between flux pinning sites. From our cavity data, we fitted a mean pinning site spacing $ℓp=75ℓ$⁠, as shown in Fig. 7(b).⁶ 

Combining this theoretical prediction of trapped flux sensitivity with the prediction of $RBCS$ at a typical operating gradient of 16 MV/m and operating temperature of 2 K, we can derive a model to find the optimal mean free path for a given amount of trapped magnetic flux. Adding $RBCS(ℓ,2 K,16 MV/m)+R0(ℓ,Btrapped)$ and plotting over mean free path and trapped flux gives the result seen in Fig. 8. Minimizing $Rs$ by optimizing mean free path for a given amount of trapped flux results in the dashed line shown in the figure. It is important to note here that there exists a second minimum at lower mean free path, below 5 nm; however, in this heavily doped region, cavities exhibit a very low average quench field.¹¹ This gradient limitation makes cavities in this regime currently unusable for accelerator operations.

In the context of accelerator design, this means that one can choose a minimum achievable trapped flux for the accelerating structure and from this determine the ideal mean free path for doping. Any cavities with smaller values of trapped flux will have a lower overall surface resistance, since $R0(ℓ,Btrapped)$ increases monotonically as a function of trapped flux. The amount of trapped flux depends on ambient fields, the quality of the magnetic shielding, and cooldown procedures. With good shielding, typical ambient fields in SRF cryomodules are near 0.5 μT (5 mG); anywhere from 0% to 100% of this ambient field may be trapped during cooldown, depending on cooldown parameters, most notable of which is the spatial temperature gradient as the cavity crosses $Tc$⁠.^10,32–34 For an operating gradient of 16 MV/m at 2 K, the results in Fig. 8 show that, unless more than ∼50% of such an ambient field can be expelled during cooldown, light doping is preferable over heavy doping. On the other hand, if the trapped field can be limited to <0.2 μT (2 mG), then the stronger positive Q-slope of heavy doping makes it a preferable option.

We have found strong evidence that the electron mean free path plays an important role in the surface resistance of SRF materials. Beyond the well-known minimization of the low-field BCS resistance near $ℓ=ξ0/2$⁠, this role is twofold: (1) decreasing the mean free path into the dirty limit causes an increasingly strong positive Q-slope, resulting in very low BCS resistances at technologically interesting accelerating gradients, but it also (2) increases susceptibility to increases in the effective residual resistance due to trapped magnetic flux. We have drawn a theoretical connection between the observed dependence on mean free path of the field-dependent BCS resistance and the overheating of quasiparticles, which increases linearly with mean free path in the dirty limit. This is linked either to a decreasing energy transfer rate from quasiparticles to phonons or to an additional energy loss channel from quasiparticles due to elastic scattering on impurities which decreases as the mean free path increases. We have used this model to calculate the BCS resistance for short-mean-free-path cavities at operating accelerator gradients, and combined this with theoretical results for the flux-trapping sensitivity to find an optimal mean free path for a given technologically achievable trapped flux.

This work was supported in part by the U.S. National Science Foundation under Award No. PHY-1549132, the Center for Bright Beams, and by the U.S. DOE LCLS-II High Q Project and NSF Grant No. PHY-1416318. We would like to thank A. Gurevich for helpful discussions on his work.^16,31 We would also like to thank A. Grassellino for bringing nitrogen doping to the field of SRF.