Spontaneous $π$ flux trapping in granular rings of unconventional superconductors

TL;DR

The work analyzes spontaneous flux trapping in granular rings of unconventional superconductors through symmetry-guided derivations of first-order Josephson couplings. It establishes a no-go theorem: single-band chiral superconductors cannot trap spontaneous $\pi$ flux in granular rings, while a time-reversal-invariant helical equal-spin pairing state can, even with interface disorder. Incorporating full crystalline symmetry and local inversion-symmetry–induced spin–orbit coupling, the authors propose a minimal two-band model for $\beta$-Bi$_2$Pd that stabilizes HES pairing and naturally explains the observed half-quantum flux in Little–Parks experiments. They further show that $\pi$-flux trapping in HES rings is robust to interface disorder and predict phase diagrams and $\pi$-ring probabilities for three- and multi-grain configurations, suggesting experimental tests with controlled grain orientations. Overall, the results constrain the pairing symmetry in $\beta$-Bi$_2$Pd and point to a spin-orbit–driven mechanism for robust $\pi$-flux physics in granular unconventional superconductors.

Abstract

We study Josephson couplings in unconventional superconductors and generalize the Sigrist-Rice formula by incorporating symmetry constraints and interface orientation disorder. Applying this framework to granular superconducting rings, we establish a no-go result that single-band chiral superconductors cannot spontaneously trap a magnetic flux. This rules out chiral $p$-wave pairing in $β$-Bi$_2$Pd, in light of the half-quantum flux observed in recent Little-Parks experiments. Incorporating the full crystalline and time-reversal symmetries, we show that a fully gapped helical equal-spin pairing state, naturally stabilized by spin-orbit coupling arising from local inversion-symmetry breaking, is instead favored. We further find that granular rings of such superconductors can trap a spontaneous $π$ flux in a manner robust against interface disorder.

Figures (6)

• Figure 1: (a) Josephson junction between two superconducting grains. The pairing order parameters of the grains are $\hat{\Delta}_{L,R}\mathrm{e}^{\mathrm{i} \phi_{L,R}}$ on the left and right, respectively. The crystallographic $a$-axis of each grain is rotated by an angle $\alpha_{L(R)}$ relative to the normal direction of the junction interface $\mathbf{n}_{L\rightarrow R}$. (b) A zoomed-in view of a local segment of the junction interface, highlighted by the blue box in (a). In the presence of interface orientation disorder, the local normal direction $\tilde{\mathbf{n}}_{L \rightarrow R}$ (blue) deviates from the average normal direction $\mathbf{n}_{L \rightarrow R}$ (gray) by an angle $\beta$. (c) A superconducting ring composed of $N_g$ grains, enclosing a magnetic flux $\Phi_B$ through its central hole. The $n$-th grain has its crystallographic $a$-axis oriented at an angle $\alpha_n$ with respect to the $x$-axis, and its pairing order parameter is $\hat{\Delta}_n \mathrm{e}^{\mathrm{i} \phi_n}$. Without loss of generality, we set $\alpha_1 = 0$ in the first grain.
• Figure 2: Three-grain superconducting ring. (a) Geometric configuration of the ring, with grain orientations labeled by $\alpha_{1,2,3}$. (b) Phase diagram for three-grain rings of HESP superconductor plotted in the $(\alpha_2,\alpha_3)$ plane with $\alpha_{1}=0$. The red-shaded region indicates configurations supporting spontaneous $\pi$ flux trapping.
• Figure 3: Four-grain superconducting ring. (a) Geometric configuration of the ring, with grain orientations labeled by $\alpha_{1,2,3,4}$. (b) Phase diagram for four-grain rings of HESP superconductor plotted in the $(\alpha_2,\alpha_3,\alpha_4)$ space with $\alpha_{1}=0$. Colored planes mark the boundaries between $0$- and $\pi$-ring configurations. (c-e) $2$D slices of the phase diagram at $\alpha_4=0$, $\alpha_4\in(0,\pi/2)$ and $\alpha_4\in(\pi/2,\pi)$, respectively. The yellow-shaded region indicates configurations supporting spontaneous $\pi$ flux trapping. The colored lines represent the intersections of these slices with the corresponding colored phase boundary planes in (b).
• Figure S1: Geometric configuration and phase diagrams of a three-grain ring composed of$d$-wave superconductors. (a). Schematic of a three-grain ring, showing grain orientations $\{\alpha_{1,2,3}\}$, and interface orientations $\{\gamma_{12,23,31}\}$. For simplicity, $\alpha_3 =0$. (b). Phase diagram in the dirty junction limit, plotted as a function of $\alpha_1$ and $\alpha_2$. The red-shaded region denotes grain configurations that support spontaneous $\pi$-flux trapping. (c). Phase diagram for clean Josephson junctions, plotted as a function of $\alpha_{12}$ and $\alpha_2$ under the constraints $\gamma_{23} = \pi$, $\gamma_{31} = 2\pi$, and the relation $\alpha_1+\alpha_2 = \pi/2$. The red-shaded region again indicates $\pi$-ring configurations.
• Figure S2: Probability of forming a $\pi$-ring as a function of number of grains, $N_g$. $P(N_g)$ are presented for $d$-wave superconductors with clean (blue circles) and dirty (green triangles) Josephson junctions, as well as for helical equal-spin pairing superconductors (red crosses). The dashed (resp. dotted) line indicates the exact value of $P(N_g=3)=1/4$ (resp. $P(N_g=4)=1/4$) for helical equal-spin pairing superconductors. The solid line indicates $P=1/2$ in the large $N_g$ limit.