Muon-spin rotation measurements of the vortex state in ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$: Type-1.5 superconductivity, vortex clustering, and a crossover from a triangular to a square vortex lattice

Muon-spin rotation has been used to probe the vortex state in ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$. At moderate fields and temperatures a lattice of triangular symmetry is observed, crossing over to a lattice of square symmetry with increasing field and temperature. At lower fields it is found that there are large regions of the sample that are completely free from vortices which grow in volume as the temperature falls. Importantly this is accompanied by increasing vortex density and increasing disorder within the vortex-cluster-containing regions. Both effects are expected to result from the strongly temperature-dependent long-range vortex attractive forces arising from the multiband chiral-order superconductivity.

Figure 1

Example of a $\mu$SR line shape $p\left(B\right)$ (solid green line) taken at 2.8 mT, 500 mK showing a simulation of a triangular structure (dashed red line). The second (unmodeled) peak is the background peak from the sample holder, which at this temperature is large relative to the superconducting signal (see text).

Figure 2

(a) and (b) Temperature-dependent parameters for data taken at 2.8 mT. (a) The parameter $\beta$ (red line: simulation, green squares: data) indicating both the change of symmetry and the onset of stronger positional fluctuations below 400 mK. (b) The angle $\theta$ between the vortex lattice vectors ${\vec{a}}_1$, ${\vec{a}}_2$ used in our simulations of the data. (c) and (d) Parameters for field-cooled data taken at 1 K as a function of applied field. (c) The parameter $\beta$ (red line: simulation, green squares: data) indicating the crossover to a square lattice above $\sim$3 mT. (d) The angle $\theta$.

Figure 3

The fraction of the sample containing flux, derived from the measured asymmetry. (a) As a function of temperature at 2.8 mT. (b) As a function of field at 1 K, indicating the reduction of volume of flux-containing regions. The rise in the mean local flux density within these regions with decreasing temperature is also shown in (a). (c) Low-field evolution of the fraction of vortex-containing regions for $T<T_c$.

Figure 4

A phase diagram based on the $\mu$SR measurements broadly indicating the areas of vortex lattice symmetry and perfection. Symbols indicate measurement points: square lattice (squares); well-ordered triangular lattice (light, green triangles); disordered triangular lattice (dark, blue triangles); and square lattice Refs. [12, 13] (red diamonds).