Shrinking Magnetic Vortices in ${\mathrm{V}}_3\mathrm{S}\mathrm{i}$ due to Delocalized Quasiparticle Core States: Confirmation of the Microscopic Theory for Interacting Vortices

We report muon-spin rotation measurements on the conventional type-II superconductor ${\mathrm{V}}_3\mathrm{S}\mathrm{i}$ that provide clear evidence for changes to the inner structure of a vortex due to the delocalization of bound quasiparticle core states. The experimental findings described here confirm a key prediction of recent microscopic theories describing interacting vortices. The effects of vortex-vortex interactions on the magnetic and electronic structure of the vortex state are of crucial importance to the interpretation of experiments on both conventional and exotic superconductors in an applied magnetic field.

Figure 1

Fourier transform (FT) of the muon spin precession signal (green circles) in ${\mathrm{V}}_3\mathrm{S}\mathrm{i}$ at $T=3.8\mathrm{K}$ and $H=50\mathrm{k}\mathrm{O}\mathrm{e}$. The FT provides an approximate illustration of the internal magnetic field distribution but is broadened by the apodization procedure used to smooth out the ringing effects of the finite-time window ($0–3\mu \mathrm{s}$) and the noise from reduced counts at later times. Additional broadening due to VL disorder and nuclear magnetic dipoles is accounted for by a Gaussian broadening width of $\sigma \approx 1.0\mu {\mathrm{s}}^{-1}$. The solid red curve is the FT of the fit in the time domain, assuming the field profile $B\left(r\right)$ given by Eq. (1). The inset is a contour plot of the function $B\left(r\right)$ obtained from the fit.

Figure 2

The magnetic field dependence of the fit parameters at $T=3.8\mathrm{K}$. (a) The magnetic penetration depth $\lambda$. (b) The coherence length $\xi$. (c) The apex angle $\beta$ (solid circles) and the anisotropy parameter $d$ (open triangles). The dashed vertical lines indicate the field range over which the VL continuously evolves from hexagonal to square symmetry. Immediately above 7.5 kOe, $d$ and $\beta$ are poorly determined, because the VL is only slightly distorted from hexagonal symmetry.

Figure 3

(a) The magnetic field dependence of the vortex core size $r_0$ measured by $\mu \mathrm{S}\mathrm{R}$ (solid circles) at $T=3.8\mathrm{K}$, and the electronic thermal conductivity ${\kappa }_e/T$ (open triangles) from Ref. 22 extrapolated to $T\to 0$ (and normalized to the value ${\kappa }_N/T$ at $H_{c2}$). The open circles indicate the reduction of $r_0$ due to the superposition of $j\left(r\right)$ profiles from individual vortices, calculated assuming the low-field value of $\xi$. The data are plotted as a function of $1/H^{1/2}$, which is proportional to the intervortex spacing $L=({\Phi }_0/H\sin \beta {)}^{1/2}$, where ${\Phi }_0$ is the magnetic flux quantum. The dashed vertical lines indicate the field range over which the VL undergoes a continuous hexagonal-square transition. A schematic of the pair potential $\Delta \left(r\right)$ as a function of distance from the vortex core center at high (b) and low (c) magnetic field.