Critical fields of Nb₃Sn prepared for superconducting cavities

For the LE-μSR experiments a coin-shaped sample was cut from a Nb sheet, while for the surface μSR measurements an ellipsoid sample is used which has been machined from bulk Nb. The coin is 3 mm thick and 20 mm in diameter. A hole is drilled through the center of the sample for mounting the sample during coating. The prolate ellipsoid is 9 mm in radius on the minor axis with a 22.9 mm semi-major axis. A threaded hole is drilled in one end for mounting the sample during coating and test. The muons are implanted at the opposite pole.

In the solid ellipsoid the Meissner state is supported by screening currents that augment the field at the equator and reduce the field at the poles. When the flux at the equator reaches the field of first vortex penetration ${H}_{\mathrm{vp}}$, which can be ${H}_{{\rm{c}}1}$ or in case of a surface barrier up to ${H}_{\mathrm{sh}}$, fluxoids will nucleate at the equator and redistribute uniformly inside the superconductor due to vortex repulsion. Pin-free ellipsoidal bulk samples produce a uniform vortex flux density in the mixed state since the inward directed driving force on the vortex by the surface screening currents is compensated by the vortex line length that increases for fluxoids closer to the ellipsoid axis. The magnetic field at the equator ${H}_{\mathrm{equator}}$ will be enhanced to ${H}_{\mathrm{equator}}={H}_{{\rm{a}}}/(1-N)$ where ${H}_{{\rm{a}}}$ is the applied field and N the demagnetizing factor. In our geometry the ellipsoid demagnetizing factor is N = 0.13, therefore the measured field of first penetration ${H}_{{\rm{a}}| \mathrm{entry}}=0.87{H}_{\mathrm{vp}}$. If the temperature is above 9.25 K, the critical temperature of niobium, the geometry is a superconducting Nb₃Sn shell. The field will still break in at ${H}_{{\rm{a}}| \mathrm{entry}}=0.87{H}_{\mathrm{vp}}$ but the vortex lines will directly snap to the center since the flux line length in the superconducting shell is actually less near the ellipsoid axis. The coin sample is tested in the Meissner state using the LE-μSR technique. Due to its geometry the field is enhanced by 15% at the edge and 9% in the midplane. For details refer to [14]. The samples used for μSR are shown in figure 2.

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The QPR sample chamber has a 'top hat'-like shape with its upper outside being exposed to the RF field inside the resonator, see figure 3. The actual sample is a disk of 75 mm diameter and 10 mm in height. It is electron-beam welded to a tube of 2 mm wall thickness which in turn is welded to the 'brim'. The bulk material of all parts is fine grain niobium with the Nb₃Sn coating covering all outer (and inner) surfaces. For mounting into the QPR the superconducting part of the sample chamber has to be connected to a double-sided CF100 flange. Commonly, QPR samples are brazed to this stainless steel flange. The process of coating with Nb₃Sn requires a substrate of high purity niobium, furthermore, heat related changes to the sample due to brazing or welding after coating have to be prevented. In order to enable coating with Nb₃Sn according to the process used for SRF cavities, a detachable connection of superconducting sample chamber and stainless steel bottom flange was implemented. An indium wire gasket of 1 mm diameter was used providing both thermal link to the liquid helium bath and UHV compatible sealing of the cavity vacuum. Prior to coating with Nb₃Sn the niobium substrate was characterized at HZB showing low residual rf surface resistance of about $4\,{\rm{n}}{\rm{\Omega }}$ and high RF critical field ${\mu }_{0}{H}_{{\rm{c}},\mathrm{RF}}=220\,\mathrm{mT}$ [15]. For details on the QPR refer to [16, 17].

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In a μSR experiment, 100% spin polarized muons are implanted one at a time into the sample, stop in interstitial sites and precess with characteristic Larmor frequencies determined by the local magnetic field ${H}_{\mathrm{int}}$. When the muon decays it emits a fast positron which has a propensity to emerge along the direction of the muon spin. Two positron detectors (symmetrical placed around the sample) detect the positron signal. The time evolution of the muon spin polarization P(t) gives information about the local field experienced by the muon. For a general introduction to μSR refer to [18, 19]. If the field has fully penetrated, or the measurement is performed in the London layer using the LE-μSR technique, an oscillating depolarization will be observed

$$\begin{eqnarray}&&P(t)=\exp \left(-\displaystyle \frac{1}{2}{\sigma }^{2}{t}^{2}\right)\cdot \cos \left(\omega t+\displaystyle \frac{\pi \phi }{180}\right)\end{eqnarray} \tag{ 2 }$$

with

$$\begin{eqnarray}&&\omega ={\gamma }_{\mu }{H}_{\mathrm{int}},\end{eqnarray} \tag{ 3 }$$

where ${\gamma }_{\mu }=2\pi$ 13.55 kHz G⁻¹ is the gyromagnetic ratio of the muon and ϕ a phase which can depend on the external field. The leading term of equation (2) describes the damping of the oscillation assuming a Gaussian distribution of the local field around ${H}_{\mathrm{int}}$ with a full width at half maximum σ.

If the sample is in the Meissner state and the muon is implanted in the bulk with the surface muon beam, the muon does not sense the external magnetic field. The local magnetic field sensed in this case is from the nuclear dipolar fields. In case of significant muon diffusion, which is the case for high RRR niobium [14], the polarization is described by the dynamic zero field Kubo–Tuyabe function ${P}_{\mathrm{ZF}}^{\mathrm{dyn}}$ [20]. Therefore, in the field of first vortex penetration measurements ${P}_{\mathrm{ZF}}^{\mathrm{dyn}}$ will be sensed for low applied field. When the field has fully penetrated the sample equation (2) describes the depolarization. If the field penetrates gradually, which is the case if there is pinning, there can be coexisting field-free and penetrated areas. In this case, the total polarization function is comprised of two terms as

$$\begin{eqnarray}\begin{array}{rcl}P(t) & = & {f}_{0}\cdot {P}_{\mathrm{ZF}}^{\mathrm{dyn}}(t)\\ & & +{f}_{1}\cdot \exp \left(-\displaystyle \frac{1}{2}{\sigma }^{2}{t}^{2}\right)\cdot \cos \left(\omega t+\displaystyle \frac{\pi \phi }{180}\right).\end{array}\end{eqnarray} \tag{ 4 }$$

The first term is due to the volume fraction being in the Meissner state and the second term describes the volume fraction containing magnetic field. These fractions are represented by the coefficients f₀ and f₁, which can take values from 0 to 1. At zero magnetic field inside the sample f₀ equals 1 and f₁ equals 0. A significant decrease of f₀ from its low field value equal to 1 signifies vortex penetration. For details refer to [14].

Two different muon beams are used for these studies. At TRIUMF so called surface muons (kinetic energy 4.1 MeV) are used with an implantation depth of about 130 μm deep in the bulk of a niobium sample. If measured above the critical temperature of niobium, the geometry of the samples can therefore be considered as a superconducting shell and the screening of the Nb₃Sn coating alone can be determined. This allows detection of the transition from the Meissner to a vortex state. For details refer to [14].

At PSI, a low energy muon beam is available [21, 22]. By varying its energy, it can be used as a probe for local magnetism at various depths between a few and up to about 300 nm depending on the material density. Applying a field below ${H}_{{\rm{c}}1}$ this allows to measure the local field ${H}_{\mathrm{int}}$ as a function of depth. This experiment therefore allows for a direct measurement of the London penetration depth.

The QPR [23, 24] is a dedicated sample test cavity which can be used to measure the RF quench field as a function of temperature. It is operated in a bath cryostat filled with superfluid helium which is pressure stabilized providing stable RF conditions. Regarding the sample chamber, only the stainless steel flange at its bottom is in contact with liquid helium. The RF sample disk is cooled via heat conduction through the sidewalls of the sample chamber. The bottom side of the sample disk is equipped with a resistive DC heater in its center and a calibrated Cernox temperature sensor at the radial position of highest RF field. Using the heater, arbitrary sample temperatures between 1.8 and approx. $25\,{\rm{K}}$ can be achieved. The heater power is PID controlled, providing temperature stability better than $\pm 2\,\mathrm{mK}$ at 18 K. Due to the weak thermal link to the liquid helium bath and the low thermal conductivity of the stainless steel flange, the maximum temperature difference along the RF surface is about 0.1 K at 18 K. Note that only the sample temperature is changed, whereas the helium bath temperature is kept at 1.8 K throughout the entire measurement.

In order to probe the RF critical field of the sample, single rectangular RF pulses are used. A quench decreases the loaded quality factor of the resonator, leading to a drop of transmitted RF power. The transmitted power is recorded with a time resolution of $50\,\mu {\rm{s}}$, allowing for a precise determination of the peak RF field. Figure 4 shows four examples of transmitted RF pulse traces, converted to peak RF field on the sample. The temperature measurement is not fast enough to capture the temporal evolution of the sample but it is used to differentiate between a quench occurring on the sample or somewhere else inside the resonator.

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The field configuration inside the QPR provides an advantage over tests using elliptical cavities: in case of a 1.3 GHz TESLA-shaped single-cell cavity the ratio of peak magnetic field and square root of stored energy is given by $\tfrac{{B}_{\mathrm{pk}}}{\sqrt{U}}=36.5\,\tfrac{\mathrm{mT}}{\sqrt{{\rm{J}}}}$. Hence, a stored energy of $U=7.5\,{\rm{J}}$ is required to achieve ${B}_{\mathrm{pk}}=100\,\mathrm{mT}$. For the QPR this is reduced significantly:

$$\begin{eqnarray}&&{U}_{\mathrm{QPR}}={\left(\displaystyle \frac{100\mathrm{mT}}{319\mathrm{mT}/\sqrt{{\rm{J}}}}\right)}^{2}=0.098\,{\rm{J}}.\end{eqnarray} \tag{ 5 }$$

In order to suppress pre-quench heating the RF time constant $\tau ={Q}_{{\rm{L}}}/\omega$ has to be small, but reducing the loaded quality factor ${Q}_{{\rm{L}}}$ increases the required RF drive power. Assuming the cavity to be strongly overcoupled, the required forward power is given by ${P}_{{\rm{f}}}=\tfrac{U}{4\tau }$. This yields for the case of equal rise time and magnetic field:

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{{\rm{f}},\mathrm{TESLA}}}{{P}_{{\rm{f}},\mathrm{QPR}}}=\displaystyle \frac{{U}_{\mathrm{TESLA}}}{{U}_{\mathrm{QPR}}}=\displaystyle \frac{7.5}{0.098}=76.\end{eqnarray} \tag{ 6 }$$

Clearly, nearly two orders of magnitude more RF power must be applied to the TESLA cavity to achieve the quench field. The maximum achievable field level inside the QPR is limited by the cavity quench field at about 120 mT. Hence, the critical field can only be measured at elevated temperatures with ${R}_{{\rm{s}}}$ being dominated by BCS losses. Using ${P}_{\mathrm{diss}}\propto {R}_{{\rm{s}}}\propto {\omega }^{2}$ yields for localized heating

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{\mathrm{diss},\mathrm{TESLA}}}{{P}_{\mathrm{diss},\mathrm{QPR}}}=\displaystyle \frac{{\omega }_{\mathrm{TESLA}}^{2}}{{\omega }_{\mathrm{QPR}}^{2}}={\left(\displaystyle \frac{1.3\mathrm{GHz}}{414\mathrm{MHz}}\right)}^{2}\approx 10.\end{eqnarray} \tag{ 7 }$$

For this reason, the requirement of equal rise time in QPR and cavity measurements can be relaxed which in turn reduces the necessary RF power. Furthermore, time-resolved numerical simulations of the QPR sample chamber show a significant impact of the sample thickness on the temperature dynamics [17]. Due to the thermal diffusivity of niobium, the sample disk being 10 mm in thickness reduces the non-equilibrium temperature rise during the RF rise time compared to cavity-grade material of 2–3 mm thickness. Note that the scenario of a thin sample disk is approximately comparable to the situation of a single-cell cavity test at temperatures above 4.2 K with inefficient cooling by gaseous helium only. Hence, at HZB a solid-state amplifier with a few hundred watts is sufficient, while measurements with TESLA cavities had to be performed with MW-class klystrons [11, 17, 25].

Muons have been implanted with energies between 3.3 and 17.3 keV corresponding to mean stopping depths between 15 and 55 nm, see figure 5. The applied field was ${\mu }_{0}{H}_{a}=10\,\mathrm{mT}$ and the temperature 5 K. For each depth the local magnetic field sensed by the muons is derived from the time evolution of the muon spin polarization function P(t) using equation (2). The data has been analyzed using the musrfit software [26]. The whole data was also collectively fitted to the one-dimensional London model using the BMW libraries for musrfit⁹ taking into account the stopping distribution of the muons in the material. For an isotropic local superconductor a solution of the form

$$\begin{eqnarray}&&B={B}_{0}\exp \left(-x/\lambda \right)\end{eqnarray} \tag{ 8 }$$

is obtained, where x is the distance from the surface. The fit parameters found are  B₀ = 11.4 mT and $\lambda (5K)\,=160.5(3.9)\,\mathrm{nm}$. λ is expected to change according to

$$\begin{eqnarray}&&\lambda (T)=\displaystyle \frac{\lambda (0K)}{\sqrt{1-{\left(T/{T}_{{\rm{c}}}\right)}^{4}}},\end{eqnarray} \tag{ 9 }$$

from which $\lambda (0K)$ = 160.0(3.9) nm can be derived being only 0.3% shorter than $\lambda (5K)$. The lower critical field ${H}_{{\rm{c}}1}$ can be expressed as [27]

$$\begin{eqnarray}&&{\mu }_{0}{H}_{{\rm{c}}1}=\displaystyle \frac{{{\rm{\Phi }}}_{0}}{4\pi {\lambda }^{2}}\mathrm{ln}(\kappa +0.5),\end{eqnarray} \tag{ 10 }$$

where ${{\rm{\Phi }}}_{0}$ is the magnetic flux quantum. To obtain the Ginzburg–Landau parameter $\kappa =\lambda /{\xi }_{{\rm{GL}}}$ from the measured value of λ, literature values of the London penetration depth ${\lambda }_{L}$ and the BCS coherence length ${\xi }_{0}$ are required. A formula is derived using the fact that the BCS and the Ginzburg–Landau coherence length are both correlated to the magnetic flux quantum ${{\rm{\Phi }}}_{0}$ [27]

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{\lambda (T,l)}{{\xi }_{\mathrm{GL}}}=\displaystyle \frac{2\sqrt{3}}{\pi }\displaystyle \frac{{\lambda }^{2}}{{\xi }_{0}{\lambda }_{{L}}}.\end{eqnarray} \tag{ 11 }$$

Note that ${\xi }_{0}$ and ${\lambda }_{L}$ are both fundamental material properties defined for the clean stoichiometric material. The values taken from literature can be found in table 1. From $\lambda (0K)$ =160 nm, κ = 60(15) and ${\mu }_{0}{H}_{{\rm{c}}1}$ = 28(2) mT are derived. The thermodynamic critical field can be calculated using [27]

$$\begin{eqnarray}&&{H}_{{\rm{c}}}=\displaystyle \frac{\sqrt{2}\kappa {H}_{{\rm{c}}1}}{\mathrm{ln}\kappa }.\end{eqnarray} \tag{ 12 }$$

The value of 600(100) mT is consistent with 520 mT reported in [1], see table 1. Finally, the superheating field is calculated using equation (1) to be 500(120) mT. On a side note the value of B₀ is 14% larger than the applied field. This field enhancement is close to what is expected at the edge of this coin-shaped sample (N = 0.15)¹⁰ .

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The muon polarization function was measured as a function of applied field at three temperatures above and one below ${T}_{{\rm{c}}}[{\rm{Nb}}]$ using surface muons implanted ≈130 μm in the bulk. Figure 6 displays the fit parameter f₀ as derived from equation (4) as a function of applied field divided by 1 − N, where N is the demagnetization factor of the ellipsoid. From this plot the intrinsic DC field of first vortex penetration ${H}_{\mathrm{vp},\mathrm{DC}}$ can be derived for each temperature as the field for which f₀ significantly deviates from 1. For temperatures above ${T}_{{\rm{c}}}[{\rm{Nb}}]$ the geometry is a 2 μm superconducting shell, while below ${T}_{{\rm{c}}}[{\rm{Nb}}]$ both materials can potentially provide shielding and it is not possible to directly derive the field of first vortex penetration for the Nb₃Sn layer alone. The field of first vortex penetration for the Nb₃Sn shell at 0 K is derived by extrapolation taking only data above 9.25 K into account. A value of $\mu {H}_{\mathrm{vp}}=28(12)\,\mathrm{mT}$ is found, see section 5.

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The RF quench field was measured at various temperatures $T\gt 8\,{\rm{K}}$, see figure 7. This temperature limitation was due to the fact that RF instabilities and the quenching resonator ruled out measurements for ${B}_{\mathrm{pk}}\gt 100\,\mathrm{mT}$.

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The measured data can be described by the empirical relation for the temperature dependence of the critical thermodynamic field

$$\begin{eqnarray}&&{H}_{{\rm{c}}}(T)={H}_{{\rm{c}}}(0)\cdot \left(1-{\left(\displaystyle \frac{T}{{T}_{{\rm{c}}}}\right)}^{2}\right)\end{eqnarray} \tag{ 13 }$$

only in the temperature range $T\gt 16.3\,{\rm{K}}$. This choice is somewhat arbitrary, but expanding the fit data range to lower temperature reduces the fit quality significantly. Extrapolation to low temperature yields ${\mu }_{0}{H}_{\mathrm{vp},\mathrm{RF}}=200(5)\,\mathrm{mT}$ and ${T}_{{\rm{c}}}\,=18.5(1)\,{\rm{K}}$. Note that the predicted temperature dependence for ${H}_{\mathrm{sh}}$ is almost identical to that of ${H}_{{\rm{c}}}$.

In order to estimate a possible impact of pre-quench RF heating, which would falsify the assumed sample temperature, the acquired pulse shapes are investigated. After a quench, the loaded quality factor of the QPR is dominated by the partially normal conducting sample. The corresponding decay time ${\tau }_{\mathrm{quench}}$ is determined by fitting the drop of transmitted power (see figure 8). After the RF drive power is switched off, the sample becomes superconducting within few $100\,\mu {\rm{s}}$ and the following unperturbed decay time ${\tau }_{{\rm{L}}}$ is again obtained from an exponential fit (see figure 8).

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From ${\tau }_{\mathrm{quench}}$ and the electrical resistivity of Nb₃Sn in the normal conducting state ($8-20\,\mu {\rm{\Omega }}\,\mathrm{cm}$ [31]), the quenched sample area is estimated yielding $2-3\,{\mathrm{mm}}^{2}$. Note that for $T\gt 16\,{\rm{K}}$ the increasing temperature dependent surface resistance of the sample begins to reduce the decay time ${\tau }_{{L}}$ of the superconducting state (see figure 9). For $T\gt 17\,{\rm{K}}$, the change of loaded quality factor decreases such, that no steep decrease of transmitted power is observed anymore and fits of ${\tau }_{\mathrm{quench}}$ are meaningless to deduce the quenched sample area. The increase observed for the 17 K point in figure 9 hence is considered to reflect this measurement difficulty. This is also visible in the pulse traces shown in figure 4.

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However, the fact that for $T\lt 17\,{\rm{K}}$ ${\tau }_{\mathrm{quench}}$, and hence the quenched area, do not depend on temperature (see figure 9) is a strong indication of a localized quench spot without excessive pre-quench heating affecting the measurement accuracy. This is in agreement with a quench limitation set by vortex line nucleation (see the following discussion). In comparison, measurements of a bulk niobium sample with RF quench fields consistent with the global superheating limit showed a quench area of about $100\,{\mathrm{mm}}^{2}$ [17].