Probing order parameter structure in iron-based superconductors using vortices

Impurities, inevitably present in all samples, induce elastic transitions between quasiparticle states on the contours of constant energy. These transitions may be seen in Fourier-transformed scanning tunneling spectroscopy experiments, sorted by their momentum transfer. In a superconductor, anomalous scattering in the pairing channel may be introduced by magnetic field. When a magnetic field is applied, vortices act as additional sources of scattering. These additional transitions may enhance or suppress the impurity-induced scattering. We find that the vortex contribution to the transitions is sensitive to the momentum-space structure of the pairing function. In the iron-based superconductors, there are both electron and hole pockets at different regions of the Brillouin zone. Scattering processes therefore represent intrapocket or interpocket transitions, depending on the momentum transfer in the process. In this work we show that while in a simple $s$-wave superconductor all transitions are enhanced by vortex scattering, in an $s_{\pm }$ superconductor only intrapocket transitions are affected. We suggest this effect as a probe for the existence of the sign change in the order parameter.

Figure 1

(Color online) Schematic plot of feature intensity vs magnetic field. The intensity of the Fourier-transformed LDOS at momenta ${\mathbf{q}}_i$ (defined as momentum transfer in the scattering processes described in Fig. 2) as a function of magnetic field. When the magnetic field is applied the vortex-induced scattering is added to the impurity scattering. For intrapocket transitions $\left({\mathbf{q}}_3,{\mathbf{q}}_4\right)$ the intensity is enhanced and for interpocket transitions $\left({\mathbf{q}}_1,{\mathbf{q}}_2\right)$ it is not.

Figure 2

(Color online) Contours of constant energy. The iron Brillouin zone with the contours of constant energy, $\omega =0.105$, right at the gap edge. The (red) contours around the $\Gamma$ points are the hole pockets and the (blue) contours around the M points are the electron pockets. The vectors ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ are interpocket transitions while ${\mathbf{q}}_3$ and ${\mathbf{q}}_4$ are intrapocket transitions in the hole and electron pockets, respectively. The nodal lines of the $s_{\pm }$ order parameter are marked by the horizontal and vertical dashed lines and the areas with a positive (negative) OP are shaded (unshaded).

Figure 3

(Color online) Quasiparticle interference cuts. The tunneling density-of-states modulations $\delta n\left(\mathbf{q}\right)$, Eqs. (10, 12, 13, 14, 15, 16, 17), plotted along $q_y=0$ for $q_x∊\left(0,\pi \right)$. The transitions ${\mathbf{q}}_i$ are the same as in Fig. 2. Both plots are for the same parameters as in Fig. 2: $\mu =1.5,\omega =0.105$. In both pallets the solid (red) line represents vortex scattering and the dashed (blue) line is for impurity scattering. Both curves of pallet (a) were generated with the $s_{\pm }$ order parameter while, for comparison, the curves in pallet (b) was generated using the absolute value $\left|{\Delta }_{\pm }\left(\mathbf{k}\right)\right|$. The presence of peaks at ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ in these plots is a confirmation that the sign change in $s_{\pm }$ plays a crucial role in their suppression in pallet (a).

Figure 4

(Color online) Quasiparticle interference patterns in the Brillouin zone. $\delta n\left(\mathbf{q}\right)$, Eqs. (10, 12, 13, 14, 15, 16, 17), for $\mu =1.5$ and $\omega =0.105$ on a $400\times 400$ lattice. Left: nonmagnetic impurity-induced interference patterns, Right: vortex-induced interference patterns. The patterns are similar except for the features close to $\left(\pm \pi ,0\right)$ and $\left(0,\pm \pi \right)$ where the interpocket transitions reside. It is clear that these transitions are missing from the modulations generated by the vortex.