Influence of Spin-Orbit Coupling in Iron-Based Superconductors

We report on the influence of spin-orbit coupling (SOC) in Fe-based superconductors via application of circularly polarized spin and angle-resolved photoemission spectroscopy. We combine this technique in representative members of both the Fe-pnictides (LiFeAs) and Fe-chalcogenides (FeSe) with tight-binding calculations to establish an ubiquitous modification of the electronic structure in these materials imbued by SOC. At low energy, the influence of SOC is found to be concentrated on the hole pockets, where the largest superconducting gaps are typically found. This effect varies substantively with the $k_z$ dispersion, and in FeSe we find SOC to be comparable to the energy scale of orbital order. These results contest descriptions of superconductivity in these materials in terms of pure spin-singlet eigenstates, raising questions regarding the possible pairing mechanisms and role of SOC therein.

Figure 1

Electronic structure with (black) and without (red) SOC for (a) LiFeAs and (b) FeSe. Orbitally projected eigenstates illustrate the substantial departure from cubic harmonics near the zone center in both materials. The $k_z$ dispersion for FeSe, not shown, is not markedly different from $\mathrm{\Gamma }$. The color scale indicates possible values of $⟨S_z⟩$. The red curves are labeled by their primary character—the $d_{xy}$ states play a critical role, as seen for the upper state in LiFeAs, and for both $d_{xz}$ and $d_{yz}$ states in FeSe, where the states can no longer be factorized into orbital and spin sectors. We note that each state is twofold Kramers’ degenerate with a net spin of zero: the degenerate state of the opposite spin is not shown for clarity.

Figure 2

(a) ARPES near normal emission for LiFeAs at $h\nu =26\mathrm{eV}$, corresponding to $0.1\overline{\mathrm{\Gamma }Z}$ with the TB model overlain in orange. (b) Tight-binding model for LiFeAs along the high-symmetry direction. The color scale indicates the expectation value of $⟨L_z{S}_z⟩$.

Figure 3

(a) Measurement of out of plane spin polarization asymmetry at normal emission (orange), $k_{||}=-0.06{\AA }^{-1}$ (red), and $k_{||}=-0.14{\AA }^{-1}$ (purple). The inset shows the Fermi surface for this region of $\overrightarrow{k}$, with symbols indicating the momenta for the three curves. (b) The calculated map of $P_z$ (red minimum, blue maximum) near normal emission; vertical lines correspond to curves in (a). (c) Spin polarization asymmetry along $\overline{\mathrm{\Gamma }Z}$. Spin polarization asymmetry was measured at $h\nu =26$ (black), 31 (red), 36 (yellow), 41 (orange), and 46 (purple) eV, corresponding to $k_z=0.1Z$, $0.5Z$, $0.9Z$, $0.7Z$, and $0.35Z$, respectively. Inset: ARPES at normal emission as a function of photon energy—vertical lines indicate the photon energies (and $k_z$ values) for the spin measurements.

Figure 4

(a) ARPES image of FeSe at $T=110\mathrm{K}$ (sum of linear vertical and horizontal polarization maps at $h\nu =37\mathrm{eV}$). (b) VLEED energy distribution curves (EDCs) for $I_P$ (purple) and $I_A$ (red), as defined above, measured near $k=0\AA$. The grey curve is the computed spin polarization asymmetry [$P_z$ of Eq. (1)] for these EDCs. (c) Energy splitting between hole bands for the shaded region in (a) above (orange) and below (blue) the orthorhombic transition. (d) EDCs at $k_F$ above (orange) and below (blue) the orthorhombic transition temperature. Arrows illustrate the spread in peak positions across the transition.