Analysis of magnetic vortex dissipation in Sn-segregated boundaries in ${\mathrm{Nb}}_3\mathrm{Sn}$ superconducting RF cavities

We study mechanisms of vortex nucleation in ${\mathrm{Nb}}_3\mathrm{Sn}$ superconducting RF (SRF) cavities using a combination of experimental, theoretical, and computational methods. Scanning transmission electron microscopy imaging and energy dispersive spectroscopy of some ${\mathrm{Nb}}_3\mathrm{Sn}$ cavities show Sn segregation at grain boundaries in ${\mathrm{Nb}}_3\mathrm{Sn}$ with Sn concentration as high as $\sim 35$ at. % and widths $\sim 3$ nm in chemical composition. Using ab initio calculations, we estimate the effect excess tin has on the local superconducting properties of the material. We model Sn segregation as a lowering of the local critical temperature. We then use time-dependent Ginzburg-Landau theory to understand the role of segregation on magnetic vortex nucleation. Our simulations indicate that the grain boundaries act as both nucleation sites for vortex penetration and pinning sites for vortices after nucleation. Depending on the magnitude of the applied field, vortices may remain pinned in the grain boundary or penetrate the grain itself. We estimate the superconducting losses due to vortices filling grain boundaries and compare with observed performance degradation with higher magnetic fields. We estimate that the quality factor may decrease by an order of magnitude $({10}^{10} \mathrm{to} {10}^9)$ at typical operating fields if 0.03% of the grain boundaries actively nucleate vortices. We additionally estimate the volume that would need to be filled with vortices to match experimental observations of cavity heating.

Figure 1

$Q$ vs $E$ curve of various ${\mathrm{Nb}}_3\mathrm{Sn}$ SRF cavities coated at Fermilab and Cornell. The GBs of a witness sample (Fermilab cavity 3) and direct cutout (Cornell cavity 2) of a high-performance cavity are characterized in STEM-EDS and showed no Sn segregation at GBs. On the other hand, witness samples of Fermilab cavities 1 and 2, which show $Q$ slope starting at 8 MV/m, were found to have Sn segregation at GBs (reprint from Ref. [27]).

Figure 2

HAADF-STEM image of a typical $\sim 2\text{−}\mu \mathrm{m}$-thick ${\mathrm{Nb}}_3\mathrm{Sn}$ coating on Nb prepared by Sn vapor diffusion.

Figure 3

The HAADF-STEM image and corresponding Nb and Sn concentration profiles across the GB between grain 1 and grain 2. Sn is segregated at the GB up to $\sim 33$ at. % Sn and the width of the Sn segregated region is $\sim 3$ nm.

Figure 4

Bright-field (BF)-STEM and corresponding Nb and Sn concentration profiles across a GB from a witness sample of high-performance ${\mathrm{Nb}}_3\mathrm{Sn}$ SRF cavity prepared at Fermilab.

Figure 5

HAADF-STEM image of the cross section of the top surface of a ${\mathrm{Nb}}_3\mathrm{Sn}$ GB.

Figure 6

Experimental $T_c$ [38] (grey squares) and calculated $T_c$ (black circles) for A15 Nb-Sn of different stoichiometries. For some stoichiometries, multiple possible defect configurations were considered. The calculated $T_c$ reaches a minimum of about 5 K in the tin-rich regime.

Figure 7

The black square interior to the film geometry marks our domain of simulation. It lies perpendicular to the applied magnetic field $H_a$. We assume there are no variations in the direction of $H_a$ so that we can simulate a 2D domain.

Figure 8

Projection of ${\alpha }^{\prime }=1-T/T_c$ onto a mesh, where ${\alpha }^{\prime }=0$ (i.e., $T\approx T_c$) in the region $\left|x\right|/\lambda \le 0.125$ and ${\alpha }^{\prime }=1$ elsewhere. The width of this region comes from experimental observations.

Figure 9

Sequence of behaviors for increasingly large applied magnetic fields. An applied magnetic field is set as a boundary condition to the top and bottom of the film, with periodic boundary conditions on the right and left sides. At moderate applied magnetic fields, vortices penetrate only into the grain boundary from the geometric defect (top). At higher fields, a critical vortex spacing is reached (second panel) above which vortices begin to penetrate the bulk from the grain boundary (third panel). Finally, having entered the bulk, vortices disperse and fill the grain with an equilibrium distribution (bottom).

Figure 10

Phase diagram of TDGL predictions for flux penetration in the presence of the grain boundary as a function of the applied field and minimum critical temperature inside the grain boundary. At small applied fields, no flux penetrates (blue). At intermediate fields, flux penetrates but is constrained to the grain boundary (yellow). At sufficiently high fields, the flux penetrates from the grain boundary into the bulk material (red).

Figure 11

Profiles of potential ${\alpha }^{\prime }$ for the transition between the segregated and nonsegregated regions. Sharp transitions (such as the blue solid curve) keep the vortices constrained to the boundary for larger applied fields. A more shallow transition (such as the orange dashed curve), however, allows vortices to escape into the grain more easily.

Figure 12

Illustrating vortex nucleation (red lines) in a superconductor (light gray region) subject to a surface magnetic field $H$. Vortex entry starts at regions where superconductivity is weakest (dark gray regions, here representing grain boundaries).

Figure 13

Illustrating a “train” of $N$ vortex lines (blue disks) moving through a grain boundary (dark gray) of a superconductor (light gray). The first vortex is subject to a surface Lorentz force from the RF field and each vortex is repelled by its nearest neighbors.

Figure 14

Estimated percentage of active surface grain boundaries (blue curve) as a function of $B_{rf}$, corresponding to the quality factor profile displayed in the yellow curve for ${\mathrm{Nb}}_3\mathrm{Sn}$ with $Q_1={10}^{10}$.

Figure 15

Sketch of our model for heat diffusion near an active grain boundary (red rectangle at the top center). Red and blue tones represent contour regions of high and low temperatures, respectively. The dashed yellow line represent the interface between Nb, whose outer surface is in contact with the He bath (bottom of the figure), and ${\mathrm{Nb}}_3\mathrm{Sn}$, whose inner surface is in contact with vacuum and the RF field (top). White arrows represent approximate directions for the heat flux. In our model, heat fronts move along one and three dimensions for distances smaller and greater than the grain size $D$, respectively. The change in the direction of heat flux at the interface is due to the wide separation of the thermal conductivities of Nb and ${\mathrm{Nb}}_3\mathrm{Sn}$.

Figure 16

Temperature as a function of distance from the grain boundary for the ${\mathrm{Nb}}_3\mathrm{Sn}$/Nb system at a bath of 2 K (blue) and 4 K (yellow). Vertical dashed lines correspond to $D=1$ $\mu \mathrm{m}$, $R_1=2$ $\mu \mathrm{m}$, and $R_2=3\mathrm{mm}$, from left to right. Bottom and top red dashed lines correspond to $T_{\text{He}}=2$ and 4 K, respectively. We have used temperature-dependent ${\kappa }_{\mathrm{Nb}3\mathrm{Sn}}$ given by Eq. (15), ${\kappa }_{\mathrm{Nb}}=10$ W/m K, and $P_{GB}=621$ nW at $B=60$ mT, according to our previous estimates.