# Iron-based superconductors

In recent years, my group has been actively involved in studying the new problems posed by the discovery in February 2008 of high-temperature Fe-based superconducting ma- terials. The enormous interest generated by the new superconductors has to do with their technological potential (they tend to be more 3D materials than the cuprates, yet with remarkably high critical fields for the corresponding Tc's); with the fact that superconductivity exists in Fe-based materials at all; and with the opportunity, now that a second class of HTS materials exists, to learn about the origins of HTS via the comparison with the cuprates. While there is some evidence that these materials are simpler to understand than the cuprates because they are less strongly correlated, they are also more complicated because several Fe bands play a role near the Fermi level. Thus to approach this problem it is important to have expertise in electronic structure theory as well as phenomenology and microscopic modelling.

### Spin fluctuation pairing theory

Weak-coupling approaches to the pairing problem in the iron pnictide superconductors have predicted a wide variety of superconducting ground states. The usual paradigm assumes that the repulsive Coulomb interaction creates an effective interaction ("spin fluctuation pairing") proportional to the magnetic susceptibility (right) strongly peaked at pi,0, the near-nesting vector of the small hole and electron Fermi surface pockets. To solve the gap equation with repulsive interaction, the gap must change sign between the pockets. In a realistic material-specific calculations, there are some modifications of this simple scheme. The Hirschfeld group performed Random Phase Approximation (RPA) calculations of the magnetic susceptibility and pairing interaction within a 5-band model accounting for Hubbard and Hund's rule interactions on each Fe site [1]. We discussed the robustness of these results for different dopings, interaction strengths, and variations in band structure. Within the parameter space explored, an anisotropic, sign-changing s-wave state and a d state were found to be nearly degenerate, due to the near nesting of Fermi surface sheets. The near degeneracy is a natural consequence of the nearly nested cylindrical Fermi sheets, and may give rise to interesting new phenomena such as order parameter collective modes.

The 5-band model was based on the Density Functional Theory (DFT) calculations of Cao et al.[2] and yielded the Fermi surface shown middle right. For undoped systems the peak in (q,w) at (pi,0) in the reduced Brillouin zone, but at finite doping and Hund's rule coupling becomes incommensurate, as shown in Fig. 1(a). The pairing interaction arises overwhelmingly from spin fluctuations near this point. Because of the near nesting of the Fermi surface sheets, a d-wave cos kx - cos ky state and s-wave cos kx cos ky state have the highest pairing eigenvalues. The s-wave state changes sign between sheets, but also exhibits nodes on the sheets. The near-degeneracy of several pairing states may account for the fact that in different materials with different dopings both indications of isotropic quasiparticle spectrum as well as nodes have been reported. Collective modes of the Bardasis-Schrieffer type may be visible in optics and Raman experiements.

More recently, these calculations have expanded to account for the 3D nature of the real materials. We have studied both KFe2Se2, an unusual material with missing central hole pocket, and a likely highly anisotropic s or d wave system. In addiiton, we explored the LiFeAs system together with the Dresden group, based on an ARPES-derived tight-binding band (right). This is a major challenge for the theory due to the very small 3D hole pockets present at the Fermi surface, and presence of incipient bands nearby.

1. "Near-degeneracy of several pairing channels in multiorbital models for the Fe-pnictides", S. Graser, P.J. Hirschfeld, T. Maier, and D.J. Scalapino, New J. Phys. 11, 025016 (2009) .
2. "Proximity of antiferromagnetism and superconductivity in LaO1-xFx FeAs: effective Hamiltonian from ab initio studies", C. Cao, P.J.Hirschfeld, and H.-P. Cheng, Phys. Rev. B 77, 220506 (2008) .
3. "Spin fluctuations and superconductivity in KxFe2-ySe2", A. Kreisel, T. Maier, Y. Wang, D.J. Scalapino and P.J. Hirschfeld, Phys. Rev. B 88, 094522 (2013) .
4. "Superconducting gap in LiFeAs from three-dimensional spin-fluctuation pairing calculations", Y. Wang, A. Kreisel, S.V. Borisenko, V. B. Zabolotnyy, B. Buechner, T. A. Maier, P . J. Hirschfeld, and D. J. Scalapino, Phys. Rev. B 88, 174516 (2013) .

RPA static magnetic susceptibility vs. q

Sign-changing s-wave state with nodes on the Fermi surfaces.

### Role of disorder in superconducting state

Schematic depiction of intra- and interband scattering of a quasiparticle from k to k' in isotropic 2-band superconductor.

a) Nodal s+/- and s++ states. b) Sequences of gap structure (top) and density of states (bottom) as disorder increases in an s+/- state. Only the last step in the sequence (bound state formation due to interband scattering) is unique to this state.

At present experiments on the superconducting state present many apparent paradoxes, with some interpreted in terms of a fully gapped superconducting state, and others indicating low-energy excitations[1]. While some of these differences may arise due to differences in electronic structure and local interactions, it is also possible that disorder may account for some. If intraband scattering processes (left) are assumed to dominate, they simply average the angular structure of the order parameter on each Fermi surface sheet, as in the conventional s-wave case. This has the effect of eventually "lifting" the nodes of the superconducting order parameter. Thus if intraband scattering dominates, clean systems will display the low-energy excitations characteristics of nodes, while dirty systems will be gapped. If interband dominates, the gap nodes will simply be smeared.

In the presence of several bands and both intra- and interband scattering the resultant behavior can be quite complex. For example, a number of experiments on the suppression of Tc by disorder have declared that Tc suppression is "too slow" for an s+/-, since it is slower than the Abrikosov-Gor'kov like rate expected for a symmetric 2-band system with opposite sign gaps and inter- and intraband scattering equal (maximum pairbreaking rate). In Ref. 1, Wang et al. showed that essentially any Tc suppression rate is possible when plotted against an obsevable measure of disorder, e.g. residual resistivity.

Thus it is difficult to use controlled disorder to identify s+/- states. One exception is the remakable behavior in a nodal system P-doped Ba122 identified by Mizukami et al. [2], who measured a nonmonotonic exponent in the low-T penetration depth (see left for a schematic description of what happens in a nodal s-wave system with intra and inter-band scattering). Another possiblity is discussed in this paper on quasiparticle interference .

1. "Using controlled disorder to to distinguish $s_\pm$ and $s_{++}$ gap structure in Fe-based superconductors", Y. Wang, V. Mishra, A. Kreisel, and P.J. Hirschfeld, Phys. Rev. B 87, 094504 (2013) .
2. "Disorder-induced topological change of the superconducting gap structure in iron pnictides", Y. Mizukami,, M. Konczykowski, Y. Kawamoto, S. Kurata, S. Kasahara, K. Hashimoto, V. Mishra, A. Kreisel, Y. Wang, P. J. Hirschfeld, Y. Matsuda, and T. Shibauchi, Nat. Comm. 5, 5657 (2014).

### Disorder, electronic nematicity and transport

Fe-based superconducting materials are known to exhibit strong "electronic nematic" tendencies, i.e. the electronic system has a large susceptibility to deform, breaking C4 symmetry, when a weak tetragonal symmetry breaking perturbation is applied. Impurities are known to be able to "freeze" fluctuations of these ordered phase around themselves, creating an emergent defect state. In a phase in proximity to magnetic stripe order characteristic of Fe-based superconductors, these "nematogens" take the form shown on the right: they elongate and grow above the transition, then freeze below into remarkably long defect structures which have been seen in STM. Recently, the PI and group calculated the scattering rate anisotropy to be expected from such objects, and were able to explain a number of puzzling aspects of transport experiments on BaFe2As2.

1. "Origin of electronic dimers in the spin-density wave phase of Fe-based superconductors", M. N. Gastiasoro, P. J. Hirschfeld, and B. M. Andersen, Phys. Rev. B 89, 100502(R) (2014).
2. "Emergent defect states as a source of resistivity anisotropy in the nematic phase of iron pnictides", Maria N. Gastiasoro, I. Paul, Y. Wang, P. J. Hirschfeld, and Brian M. Andersen, Phys. Rev. Lett. 113, 127001 (2014) .
3. "Anomalous effect of Lifshitz transitions on DC transport in magnetic phases of Fe-based superconductors", Y. Wang, M. N. Gastiasoro, B. M. Andersen, M. Tomic, H.O. Jeschke, Roser Valenti, I. Paul and P.J. Hirschfeld, Phys. Rev. Lett. 114, 097003 (2015) .

a) Anisotropic resistivity in Ba(Co,Fe)2As2 [1] and schematic picture of calculated result for defect in each case. b) Transport scattering rate from nematogen above TN calculated over Fermi surface.