Spin-Orbit Interactions and the Nematicity Observed in the Fe-Based Superconductors

High-resolution angle-resolved photoelectron spectroscopy is used to examine the electronic band structure of ${\mathrm{FeTe}}_{0.5}{\mathrm{Se}}_{0.5}$ near the Brillouin zone center. A consistent separation of the ${\alpha }_1$ and ${\alpha }_2$ bands is observed with little $k_z$$$ dependence of the ${\alpha }_1$ band. First-principles calculations for bulk and thin films demonstrate that the antiferromagnetic coupling between the Fe atoms and hybridization-induced spin-orbit effects lifts the degeneracy of the Fe $d_{xz}$ and $d_{yz}$ orbitals at the zone center leading to orbital ordering. These experimental and computational results provide a natural microscopic basis for the nematicity observed in the Fe-based superconductors.

Figure 1

ARPES spectrum of the $d_{xz}$ and $d_{yz}$ bands of ${\mathrm{FeTe}}_{0.5}{\mathrm{Se}}_{0.5}$. (a) Fermi function renormalized ARPES spectrum near the Brillouin zone center ($\mathrm{\Gamma }$ point). The dispersion of the ${\alpha }_1$ band (dashed red) and ${\alpha }_2$ band (solid red) are traced by the colored lines. (b) The ARPES spectral intensity at Fermi energy in the first Brillouin zone. The green line shows the location of the cut in (a), and the red circle traces the Fermi surface formed by the ${\alpha }_2$ band; the dashed red line indicates the Fermi surface formed by the ${\alpha }_3$ band.

Figure 2

Photon energy dependence of intensity of the ${\alpha }_1$ and ${\alpha }_2$ bands measured near the $\mathrm{\Gamma }$ point. (Spectra are renormalized by the Fermi function.) The inset shows second-derivative plots for the incident photon energies of 22 and 20 eV. The measurements at different photon energies correspond to $k_z$ points in units of $2\pi /c$ as indicated in the brackets: 16 eV [0.57], 18 eV [0.71], 20 eV [0.83], 22 eV [0.95], 30 eV [0.36], 50 eV [0.17], 70 eV, [0.81], 100 eV [0.61].

Figure 3

Spin-orbit bands along the high-symmetry lines of the ($1\times 1$) Brillouin zone for (a) nonmagnetic (NM) FeTe, (b) AFM FeTe, and (c) AFM ${\mathrm{FeTe}}_{0.5}{\mathrm{Se}}_{0.5}$ with the Te and Se in a $c(2\times 2)$ ordering. (d) Expanded views of the bands around $\mathrm{\Gamma }$ in boxed regions in (b) and (c); bands with (without) spin-orbit coupling in orange (blue), and the size of the dots gives the $k$-projection weight. (e) Spin-resolved local Fe density of states for (c); the gray background is the corresponding NM LDOS.

Figure 4

Calculated $k$-projected surface bands for a nine $\mathrm{Fe}-\mathrm{Te}-\mathrm{Se}$ trilayer film in a $c(2\times 2)$ configuration with equal number of Te and Se in each plane (a) without SOC and (b) with spin-orbit coupling included. (c) Spin-orbit bands for $p(2\times 2)$ eight-trilayer film with the top trilayer having a $5:3\mathrm{Te}/\mathrm{Se}$ ratio; the inset is an expanded view around $\mathrm{\Gamma }$ along the same direction as the experiments in Fig. 2. (d) Same as in (b), but with the top trilayer having only Te.

Figure 5

Effect of the electron doping by K. Comparison of the spectra of ${\mathrm{FeTe}}_{0.5}{\mathrm{Se}}_{0.5}$ (a) and $\mathrm{K}{(\mathrm{FeSe})}_2$ (b). The red line traces the dispersion of the ${\alpha }_2$ band, and the dashed red line traces the dispersion of the ${\alpha }_1$ band.