Intrinsic Nature of Spontaneous Magnetic Fields in Superconductors with Time-Reversal Symmetry Breaking

We present a systematic investigation of muon-stopping states in superconductors that reportedly exhibit spontaneous magnetic fields below their transition temperatures due to time-reversal symmetry breaking. These materials include elemental rhenium, several intermetallic systems, and ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We demonstrate that the presence of the muon leads to only a limited and relatively localized perturbation to the local crystal structure, while any small changes to the electronic structure occur several electron volts below the Fermi energy, leading to only minimal changes in the charge density on ions close to the muon. Our results imply that the muon-induced perturbation alone is unlikely to lead to the observed spontaneous fields in these materials, whose origin is more likely intrinsic to the time-reversal symmetry-broken superconducting state.

Figure 1

Tetrahedral muon site in rhenium. (a) The two distinct nearest-neighbor Re environments, Re1 and Re2, in the coordination tetrahedron of the muon. (b) Radial displacements of the Re atoms as a function of their distances from the muon. PDOS for the nearest-neighbor Re atoms (c) without a muon and (d) with a muon. Energies are given with respect to the Fermi energy.

Figure 2

The lowest-energy muon site in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. (a) The local geometry of the muon site. (b) Radial displacements of atoms as a function of their distances from the muon site. PDOS for the Ru atom closest to the muon site for the structures (c) without a muon and (d) with a muon. Energies are given with respect to the Fermi energy.

Figure 3

Radial displacements of atoms as a function of their distances from the muon site for the lowest-energy muon sites in (a) ${\mathrm{LaNiC}}_2$, (b) SrPtAs, (c) ${\mathrm{Zr}}_3\mathrm{Ir}$, and (d) ${\mathrm{Re}}_6\mathrm{Zr}$.