Spin dynamics and orbital-antiphase pairing symmetry in iron-based superconductors

The symmetry of thewavefunction describing the Cooper pairs is one of the most fundamental quantities in a superconductor, but for iron-based superconductors it has proved to be problematic to determine, owing to their complex multi-band nature1–3. Here we use a first-principles many-body method, including the two-particle vertex function, to study the spin dynamics and the superconducting pairing symmetry of a large number of iron-based compounds. Our results show that these high-temperature superconductors have both dispersive high-energy and strong low-energy commensurate or nearly commensurate spin excitations, which play a dominant role in Cooper pairing. We find three closely competing types of pairing symmetries, which take a very simple form in the space of active iron3dorbitals, anddi er only in the relativequantum mechanical phase of the xz, yz and xy orbital components of the Cooper pair wavefunction. The extensively discussed s symmetry appears when contributions from all orbitals have equal sign, whereas a novel orbital-antiphase s symmetry emerges when the xy orbital has an opposite sign to the xz andyzorbitals.Thisorbital-antiphasepairingsymmetryagrees well with the angular variation of the superconducting gaps in LiFeAs (refs 4,5). The spin and the multi-orbital dynamics of iron-based superconductors are believed to play an essential role in the mechanism of superconductivity6, but a realistic modelling of magnetic excitations, and a clear physical picture for their variation across different families of iron superconductors, is currently lacking. The Cooper pairs are locked into singlets, but the orbital structure of the superconducting order parameter can be material dependent, and its connection to orbital and spin excitations is an open problem. The charge dynamics of the iron-based superconductors is controlled by the strong Hund’s coupling on the iron site7,8, which requires a theoretical approach that simultaneously treats the itinerancy of the electrons and Hund’s interaction on an equal footing. Using non-perturbative many-body method and ab initio-determined two-particle scattering amplitude (the twoparticle vertex function), we are able to accurately describe the spin dynamics and symmetry of the superconducting order parameter, and we will show that Hund’s rule coupling and orbital blocking9 play a crucial role in the superconductivity of iron superconductors. All iron-based superconductors contain the same basic motif— layers of iron atoms tetrahedrally coordinated by pnictogen or chalcogen atoms—but their spin excitation spectra varies greatly among compounds. In Fig. 1 we plot the dynamic spin structure factor S(q, ω)=χ (q,ω)/(1−exp(−h̄ω/kBT )) in the paramagnetic state for several classes of iron compounds along the high-symmetry momentum path in the first Brillouin zone of the single-iron unit cell. Here, the momentum transfer is labelled using the same convention as used in neutron scattering experiments10. We overlay the neutron scattering data10–13 for some compounds where experimental results are available, to show the good agreement between theory and experiment. We computed the magnetic excitations in the paramagnetic state, at temperatures above the spin density wave (SDW) transition (even for compounds that have a magnetically ordered ground state) and compared them with experimental results in the paramagnetic state. The bandwidth of the spin excitations, defined as the difference in the excitation energy at the two momentum points q= (1, 0) and q= (1, 1), which are the minimum and the maximum of the dispersion curve respectively, varies substantially throughout Fe compounds. In strong-coupling theories, this spin bandwidth is related to the spin-exchange constant J , and is therefore inversely proportional to the interaction strength (J ∝ t 2/U ). We find that in the Hund’s metals9, the bandwidth also increases with decreasing correlation strength, as determined by the degree ofmass renormalization. The strength of the electronic correlations is tuned by the iron 3d occupancy14 and the Fe–pnictogen distance9. The phosphorus compounds (Fig. 1a–c) exhibit the largest spin-wave bandwidth (of the order of 0.6 eV–0.45 eV), which is a consequence of their most itinerant nature among these compounds9. The mass enhancement due to correlations is increased in arsenides, and even more in the chalcogenides9, hence the spin-wave bandwidth is progressively reduced to 0.3–0.2 eV in Fig. 1d–f, and 0.15–0.10 eV in Fig. 1g–h. The intensity of the spin excitations is proportional to the size of the fluctuating moment in this energy range, which roughly correlates with the strength of correlations, hence phosphorus compounds show the weakest (Max=4) and FeTe shows the strongest (Max=20) intensity. The low-energy spin excitations are much more sensitive to the details of both the band structure and the two-particle vertex function, hence the trend across different compounds can not be guessed from either the correlation strength or from the band structure. In Fig. 2 we show S(q, ω) for the same compounds as in Fig. 1, but we take a different cut in momentum–energy space, keeping the energy fixed at ω= 5meV, and changing the momentum in the two-dimensional momentum plane (H ,K ). As is clear from Figs 1a–c and 2a–c, the low-energy spin excitations are extremely weak (Max≈ 1) in phosphorus compounds and the spin excitations at the SDW ordering vector (1, 0) is comparable to its value at the ferromagnetic ordering vector (0, 0). In strong contrast, the low-energy spin excitations are very strong in arsenides (Fig. 2d–g) and are concentrated solely at the commensurate wavevector (H , K ) = (1, 0). This is the ordering wavevector of the SDW magnetic state, which is the ground state for these parent compounds, except the superconducting LiFeAs (TC=18K).