Simultaneous evidence for Pauli paramagnetic effects and multiband superconductivity in ${\mathrm{KFe}}_2{\mathrm{As}}_2$ by small-angle neutron scattering studies of the vortex lattice

We study the intrinsic anisotropy of the superconducting state in ${\mathrm{KFe}}_2{\mathrm{As}}_2$ by using small-angle neutron scattering to image the vortex lattice as the applied magnetic field is rotated towards the FeAs crystalline planes. The anisotropy is found to be strongly field dependent, indicating multiband superconductivity. Furthermore, the high-field anisotropy significantly exceeds that of the upper critical field, providing further support for Pauli limiting in ${\mathrm{KFe}}_2{\mathrm{As}}_2$ for fields applied in the basal plane. The effect of Pauli paramagnetism on the unpaired quasiparticles in the vortex cores is directly evident from the ratio of scattered intensities due to the longitudinal and transverse vortex lattice field modulation.

Figure 1

(a) Sample mosaic, (b) experimental geometry, and (c) vortex lattice anisotropy. (a) Photograph showing ${\mathrm{KFe}}_2{\mathrm{As}}_2$ single crystals mounted on both sides of four parallel aluminum plates. The total mass of the crystals in this mosaic is $\sim 2$ g. (b) The coordinate system is defined with $z$ along $\mathit{H}$ and $y$ vertical in the Fe-As basal plane (along $\mathit{b}$). The applied magnetic field $\mathit{H}$ is rotated away from the basal plane by an angle $\mathrm{\Omega }$. Neutron spins ($\sigma$) are parallel or antiparallel to the magnetic field. The incident neutron beam is in the $yz$-plane, at an angle $\phi$ relative to the field direction. The observed VL scattering vector is denoted $\mathit{Q}$ and the longitudinal and transverse Fourier component of the field modulation are denoted by $h_z$ and $h_x$, respectively. (c) Schematic of VL Bragg reflections lying on an ellipse in reciprocal space, with the major-to-minor axis ratio given by ${\mathrm{\Gamma }}_{\text{VL}}$ (shown here for ${\mathrm{\Gamma }}_{\text{VL}}=6$). The area of the ellipse is determined by the applied field, $\pi Q_0^2=8{\pi }^3{\mu }_{0}H/\sqrt{3}{\mathrm{\Phi }}_0$, and as a result only the filled (red) peaks are required to determine ${\mathrm{\Gamma }}_{\text{VL}}$.

Figure 2

Vortex lattice Bragg reflections. Positions in reciprocal space (horizontal and vertical axes) are normalized by the scattering vector for an isotropic triangular VL, $Q_0=1.075\times 2\pi \sqrt{{\mu }_{0}H/{\mathrm{\Phi }}_0}$. The diffraction patterns are the sum of measurements as the cryomagnet and sample mosaic are tilted about the horizontal axis perpendicular to the beam direction, which primarily satisfy the Bragg condition for reflections on the vertical axis ($Q_x=0$). However, the remaining four first-order VL reflections with $Q_x/Q_0\approx \pm 1$ and $Q_y/Q_0\approx \pm 0.5$ are visible in panel (a). (a)–(e) Measurements performed as an applied field of $0.4$ T was rotated towards the crystalline basal plane. (f)–(j) Measurements performed as a function of magnetic field at a fixed $\mathrm{\Omega }={10}^{\circ }$. Panel (f) is identical to panel (e). The VL anisotropy increases with increasing field as indicated by the white line. The color scale is adjusted separately for all nine panels to make the VL scattered intensity clearly visible. Imperfectly subtracted background scattering unrelated to the VL is masked off: In panels (a)–(d) in a circular region around $Q=0$ and in (e)–(i) in a horizontal region around $Q_y=0$ (at small $\mathrm{\Omega }$ the neutron beam is close enough to the basal plane to cause increased horizontal background scattering from crystal stacking faults and from the aluminum mounting plates).

Figure 3

Vortex lattice rocking curve, showing the scattered intensity as a function of sample tilt angle ($\phi$) relative to the rocking curve center, ${\phi }_0=0.{28}^{\circ }$. Three clear maxima are observed: A smaller center peak due to the longitudinal field modulation (non-spin flip) and a pair of stronger, Zeeman-split peaks due to the transverse field modulation (spin flip). The rocking curve is fit by three Lorentzians, each with a width of $0.{19}^{\circ }$ FWHM, and with the two Zeeman-split peaks located symmetrically around the center at $\phi -{\phi }_0=\pm 0.{40}^{\circ }$. The neutron wavelength was ${\lambda }_n=0.8$ nm.

Figure 4

Vortex lattice anisotropy as a function of the direction and magnitude of the applied magnetic field. (a) ${\mathrm{\Gamma }}_{\text{VL}}$ versus field rotation angle for different values of the applied magnetic field. The full red (${\mathrm{\Gamma }}_{ac}=5.2$) and blue ($10.8$) curves are best fits to Eq. (1) for the $0.4$ T and the combined $1.0$ and $1.4$ T data, respectively. The black lines correspond to ${\mathrm{\Gamma }}_{H_{\text{c}2}}=3.3$ [11] and $3.8$ obtained previously from low-field SANS measurements [38]. The vertical gray bars represent the $\mathrm{\Omega }\mathrm{s}$ for which ${\mathrm{\Gamma }}_{\text{VL}}$ is plotted below. (b) Vortex lattice anisotropy as a function of applied field for the two smallest values of the rotation angle. Full lines show fits to the data below the maximum possible value of ${\mathrm{\Gamma }}_{\text{VL}}={(sin\mathrm{\Omega })}^{-1}$ (dashed lines).

Figure 5

Measured intensity ratio of spin flip to non-spin flip scattering as a function of the (a) direction and (b) magnitude of the applied magnetic field. Since the two SF peaks in the rocking curve (Fig. 3) each correspond to a different incident neutron spin, the ratio is the sum of their integrated intensities divided by the integrated intensity for the NSF peak, which contains contributions from both spin directions. The line in panel (a) is calculated by using Eq. (2) and the fitted high-field value of ${\mathrm{\Gamma }}_{ac}$ from Fig. 4. Similarly, the lines in panel (b) show the expected intensity ratio from the London model, based on the fitted field dependence of ${\mathrm{\Gamma }}_{\text{VL}}$ from Fig. 4.

Figure 6

Fractional contribution of Pauli paramagnetic effects to the form factor $h_z$ parallel to the vortices obtained from Eq. (3) and the measured and calculated values of $I_{\text{SF}}/I_{\text{NSF}}$ from Fig. 5. The line is a simple exponential fit to the data.