Three-dimensional electronic structure of the nematic and antiferromagnetic phases of NaFeAs from detwinned angle-resolved photoemission spectroscopy

We report a comprehensive angle-resolved photoemission spectroscopy study of NaFeAs, a prototypical parent compound of the Fe-based superconductors. By mechanically detwinning the samples, we show that in the nematic phase (below the structural transition at $T_s=54$ K but above the antiferromagnetic transition at $T_N=43$ K) spectral weight is detected on only the elliptical electron pocket along the longer $a_{\text{orth}}$ axis. This dramatic anisotropy is likely to arise as a result of coupling to a fluctuating antiferromagnetic order in the nematic phase. In the long-range ordered antiferromagnetic state below $T_N$, this single electron pocket is backfolded and hybridizes with the hole bands, leading to the reconstructed Fermi surface. By careful analysis of the $k_z$ variation, we show that the backfolding of spectral weight in the magnetic phase has a wave vector of ($\pi ,0,\pi$), with the $c$-axis component being in agreement with the magnetic ordering in NaFeAs observed by neutron scattering. Our results clarify the origin of the tiny Fermi surfaces of NaFeAs at low temperatures and highlight the importance of the three-dimensional aspects of the electronic and magnetic properties of Fe-based superconductors.

Figure 1

Fermi surface maps obtained at 42 eV in LH polarization of (a) unstrained and (b) strained NaFeAs samples at 60 K in the tetragonal phase (integrating spectral weight within $\pm 2$ and $\pm 4$ meV of $E_F$, respectively). Red double-headed arrows indicate the direction of the uniaxial tensile strain on the sample, which will correspond to the longer $a_{\text{orth}}$ direction in the nematic and magnetic phases. Schematic Fermi surfaces are drawn in red, with the line thickness representing the photoemission intensity in this particular measurement geometry (see also Supplemental Material [38]). The schematic Fermi surfaces also take into account data obtained in LV polarization (not shown here, but see Fig. 5 and Supplemental Material [38]), where some of the bands are better observed due to selection rules. (c)–(e) Equivalent measurements at 45 K in the nematic phase and (f)–(h) at 7 K in the magnetic phase.

Figure 2

High-symmetry cuts in the $\mathrm{\Gamma }\text{−}\overline{\mathrm{M}}$ direction, corresponding to the geometry used in Fig. 1. (a), (b) Measurements of twinned samples, in LV and LH polarizations, respectively, at (i) 60 K in the tetragonal phase, (ii) 45 K in the nematic phase, and (iii) 6 K in the magnetic phase. Inset: Band crossing, indicating a Dirac point located $\sim 5$ meV below $E_F$ on the $\mathrm{\Gamma }-{\overline{\mathrm{M}}}_{\mathrm{X}}$ line. (c), (d) Measurements of detwinned samples along $\mathrm{\Gamma }-{\overline{\mathrm{M}}}_{\mathrm{Y}}$ and $\mathrm{\Gamma }-{\overline{\mathrm{M}}}_{\mathrm{X}}$, respectively, schematically represented in (e), (i). (f) EDCs at the $\mathrm{\Gamma }$ point, showing that the inner hole band crosses $E_F$. (g), (h) Twinned and detwinned EDCs at $k=-0.87{\AA }^{-1}$, corresponding to the dashed lines. X and Y refer to the EDCs obtained from $\mathrm{\Gamma }-{\overline{\mathrm{M}}}_{\mathrm{X}}$ and $\mathrm{\Gamma }-{\overline{\mathrm{M}}}_{\mathrm{Y}}$, respectively. Features labeled with red arrows and “bf” correspond to backfolded features in the magnetic phase.

Figure 3

Twinned and detwinned measurements at the M point, using 47-eV photons. (a) Low-temperature Fermi surface map of a twinned sample. (b) High-symmetry cut in LV polarization. (c) Curvature plot of the data from (b). (d) Schematic bands, here overlaid over measurements obtained in LH polarization. Backfolded features are marked in orange color. (e)–(i) As in (a)–(d), but for the two different sample orientations of a detwinned sample. The direction of tensile strain is indicated by the red double-headed arrows. (m)–(x) Equivalent measurements at 45 K in the nematic phase and (y)–(zc) at 60 K in the tetragonal phase.

Figure 4

Analysis of $k_z$ variation of NaFeAs. (a) Plot of the EDC at normal emission as a function of photon energy. Data were obtained in LH polarization, at 60 K in the tetragonal phase. Red stars indicate the location of maximum intensity after data smoothing. (b) DFT calculation along the $\mathrm{\Gamma }$-Z-$\mathrm{\Gamma }$ direction, highlighting the $d_{z^2}$ orbital weight. (c) High-symmetry cuts at 22 eV ($\mathrm{\Gamma }$) and 31 eV (Z). The presence of the $d_{yz}$ band at the $\mathrm{\Gamma }$ point only is indicated by the green arrows. (d) DFT band structures for the M-$\mathrm{\Gamma }$-M and A-Z-A paths. (e) Fermi surface of NaFeAs according to DFT. While the experimental electronic structure of NaFeAs varies substantially, DFT correctly captures the formation of a 3D hole pocket centered at $\mathrm{\Gamma }$.

Figure 5

(a) Schematic slices of the low-temperature Fermi surface of NaFeAs in the magnetic phase. For comparison with ARPES measurements of twinned samples we show two domains, with one domain being shaded. (b) Corresponding Fermi surface maps at the $\mathrm{\Gamma }$ and Z points of the 3D Brillouin zone, in LV and LH polarization. (c) Fermi surface maps at the M and A points, and (d) corresponding high-symmetry cuts. The clear qualitative difference as a function of $k_z$ is the presence of the elliptical Fermi surface at the A point, arising from the backfolding of the hole pocket at $\mathrm{\Gamma }$. (e) Magnetic configuration of NaFeAs, based on Li et al. [3]. (f) The wave vector $Q_{\text{AFM}}$ of the magnetic backfolding in the 3D Brillouin zone.

Figure 6

Top: Schematic Fermi surfaces of NaFeAs at the $\mathrm{\Gamma }$ and $\mathrm{M}$ points, as measured by ARPES in the tetragonal phase of FeSe. Middle: Schematic for the nematic phase. Black dotted lines represent the Fermi surfaces from the tetragonal phase, showing the absence of the pocket along $b_{\text{orth}}$ and more subtle Fermi surface elongations. Bottom: The reconstructed electronic structure in the SDW phase. Black dashed lines are overlapped electron and hole bands from the nematic phase. Note that in the SDW phase the Fermi surfaces at $\mathrm{\Gamma }$ and M are not equivalent, since the $\mathrm{\Gamma }$ point actually maps to the A point.