Field of first magnetic flux entry and pinning strength of superconductors for rf application measured with muon spin rotation

The performance of superconducting radiofrequency (SRF) cavities used for particle accelerators depends on two characteristic material parameters: field of first flux entry $H_{\text{entry}}$ and pinning strength. The former sets the limit for the maximum achievable accelerating gradient, while the latter determines how efficiently flux can be expelled related to the maximum achievable quality factor. In this paper, a method based on muon spin rotation ($\mu \mathrm{SR}$) is developed to probe these parameters on samples. It combines measurements from two different spectrometers, one being specifically built for these studies and samples of different geometries. It is found that annealing at $1400°\mathrm{C}$ virtually eliminates all pinning. Such an annealed substrate is ideally suited to measure $H_{\text{entry}}$ of layered superconductors, which might enable accelerating gradients beyond bulk niobium technology.

Figure 1

Generic plot of the quality factor Q as a function of the accelerating gradient.

Figure 2

(a) Schematic displaying the components of the HPF spectrometer and the beam trajectory (b) 3D render of the spectrometer. In the LAMPF spectrometer (not shown) the orientation of the magnetic field is parallel to muon momentum, therefore a steering magnet is not required.

Figure 3

Muon stopping distance in Nb as simulated by TRIM. The simulation takes into account all obstacles the muons encounter in their path such as beamline windows and scintillators.

Figure 4

Polarization function in different states. (a) In the Meissner state the depolarization is caused by randomly orientated nuclear dipole fields resulting in the characteristic dynamic Kubo-Toyabe polarization function. For comparison a fit to the static Kubo-Toyabe polarization function is plotted as well. This fit does not give a good representation for longer times. (b) In the mixed state, which is defined by coexisting macroscopic areas in the Meissner and vortex state, the Kubo-Toyabe polarization function is combined with a fast decaying oscillating function. (c) In the vortex state all muons sense the external field and there is no signature of the Kubo-Toyabe polarization function left. Here the strong damping is caused by the non uniformity of the vortex field structure. (d) In the normal state all muons probe the same field yielding weaker damping. Note the different time scale in this subplot.

Figure 5

Phase $\varphi$ and $\alpha$ as a function of applied field $H_{\mathrm{a}}$ above $T_{\mathrm{c}}$. Triangles/Squares are for the HPF/LAMPF spectrometer.

Figure 6

Fit parameter $f_0$ as a function of applied field for a coin sample in transverse geometry (LAMPF). Fixing $\alpha$ from the normal state calibration yields a smother curve, while the phase correction eliminates the effect of reduced initial polarization due to spin rotation outside of the sample.

Figure 7

Four generic arrangements of sample, muon and field direction using the LAMPF and HPF spectrometers. Red/blue color indicates areas of high/low magnetic field (compare to flux line density). The direction of the muon spin is always perpendicular to the beam direction, the sample surface and the applied field. (a) Note the high flux line density at the edges of the coin in the transverse geometry (LAMPF). (b) In the parallel geometry (HPF) the field enhancement at the edges is much weaker. (c,d) For the ellipsoid there is no edge boundary. Here the field will first nucleate at the equator, where the muons are implanted in the HPF setup. Simulations have been performed with Comsol Multiphysics assuming the samples to be in a perfect Meissner state (relative magnetic permeability ${\mu }_r=0$).

Figure 8

Example of samples used in the experiment. Top, from left to right: Coins after BCP, after deposition of Nb3Sn (the whole in the middle was used to hang the sample in the furnace) and cut from a 1.3 GHz cavity half cell. Bottom: Ellipsoids used in LAMPF (left) and HPF (right).

Figure 9

Flux applied to a thin circular disk transverse to an applied field where $H_{\mathrm{a}}>H_{\mathrm{a}|\text{entry}}$. The field breaks in at the edges first at $H_{\mathrm{edge}}<H_{\mathrm{a}|\text{entry}}$. Above $H_{\mathrm{a}|\text{entry}}$ the flux lines will move to the center of the sample, which is the position with lowest line tension. Redrawn after [15].

Figure 10

Magnetization curves for a disk sample with different pinning and geometry. Solid lines correspond to the sample with $\mathrm{b}/\mathrm{a}=0.3$ while the dash line correspond to pin-free sample with $\mathrm{b}/\mathrm{a}=0.15$ and 0.50. The critical current density $J_{\mathrm{c}}$ is a measure of the pinning strength. The larger $J_{\mathrm{c}}$ the stronger the pinning. Adapted from [15].

Figure 11

Fit parameter $f_0$ signifying the volume fraction probed by the muons which is in the field free Meissner state as a function of applied field in four geometries. (a) Chemically etched samples with no heat treatment: The apparent differences in $H/H_0$ are correlated to the different sensitivity to pinning of the four geometries. (b) Annealing at $1400°\mathrm{C}$ virtually releases all pinning.

Figure 12

Fit parameter $f_0$ signifying the volume fraction probed by the muons which is in the field free Meissner state as a function of applied field for coins in transverse geometry (LAMPF) at 2.5 K. (a) BCP removes the outer damaged layer and releases pinning. Additional surface treatments show no effect on the pinning strength. (b) Different masks, probing different areas on the sample visualize how flux breaks in from the corner of the sample in the transverse coin geometry. (c) A smaller coin of 0.8 mm thickness and 18 mm diameter. The sample was first etched (BCP), followed by annealing ($1400°\mathrm{C}$) and another BCP. The BCP treatment after annealing does not alter the pinning strength. (d) Formed and flat geometries. Formed samples are denoted with dashed lines. This plot shows that forming increases the pinning strength, which is virtually eliminated by a subsequent annealing at $1400°\mathrm{C}$.

Figure 13

Fit parameter $f_0$ signifying the volume fraction probed by the muons which is in the field free Meissner state as a function of applied field for ellipsoid samples with different heat treatments measured in (a) the LAMPF spectrometer and (b) the HPF spectrometer.

Figure 14

Fit parameter $f_0$ signifying the volume fraction probed by the muons which is in the field free Meissner state as a function of applied field for $2\mu \mathrm{m}$ ${\mathrm{Nb}}_3\mathrm{Sn}$ on Nb (a) in different geometries (b) of an ellipsoid in transverse field measured at different temperatures in the LAMPF spectrometer.

Figure 15

Measured field of first flux entry of a ${\mathrm{Nb}}_3\mathrm{Sn}$ coated Nb ellipsoid. The lines are predictions for the superheating field $H_{\mathrm{sh}}$ of Nb and the lower critical field $H_{\mathrm{c}1}$ of ${\mathrm{Nb}}_3\mathrm{Sn}$ and Nb taking into account the demagnetization factor of this geometry $N=0.13$.