Insulating magnetism in vacancy-ordered K${}_{0.8}$Fe${}_{1.6}$Se${}_2$

We unveil the physical origin of the insulating block checkerboard antiferromagnetism in vacancy-ordered K${}_{0.8}$Fe${}_{1.6}$Se${}_2$. Our first-principles electronic structure analysis reveals its incompatibility with a simple Fermi-surface nesting or Mott insulator scenario, and suggests the picture of coexisting itinerant and localized electronic states. Consistently, we demonstrate that it can be unified with the metallic collinear or bicollinear antiferromagnetism of the vacancy-free parent compounds LaOFeAs, BaFe${}_2$As${}_2$, or FeTe in the spin-fermion model. These results indicate that the blocking effects of Hund's rule coupling and the resulting electron correlation are crucial to the electronic and magnetic structures of iron-based superconductors.

Figure 1

The in-plane patterns of the spin-up (blue balls) and spin-down (red balls) iron atoms in (a) collinear $C$-type, (b) bicollinear $E$-type, and (c) block checkerboard $X$-type AF states. Their counterparts in the presence of $\sqrt{5}\times \sqrt{5}$ Fe vacancy ordering are (d)–(f), respectively. Patterned horizontal and vertical solid lines represent $d_{yz}$ and $d_{xz}$ bonds, respectively; dashed lines denote the Fe vacancy broken bonds.

Figure 2

LDA Fermi surface at $k_z=0$ (a) and $k_z=\pi$ (b) and band structure (c) in the one-Fe unit cell notation. High intensity means high spectral weight. Shadow bands appear as shifting the main bands by $(\pi ,\pi ,\pi )$ and $\mathbf{Q}=\pm (3\pi /5,\pi /5,\pi )$, $\pm (-\pi /5,3\pi /5,\pi )$. Color coded for $d_{xz}$ (red), $d_{yz}$ (blue), and the rest $3d$ orbitals (green). Dashed lines in (a),(b) are the folded Brillouin zone boundary in the in-plane 8-Fe unit cell notation. (d) Bare spin susceptibility in the one-Fe unit cell notation, with the value at $\mathbf{Q}$ indicated by the arrow, compared with those of FeTe and a previous model (Ref. 13).

Figure 3

(a) Fe $3d$ partial DOS of the $X$-type AF state obtained from GGA+$U$ calculations with $U=0$ and 3 eV. Upper and lower panels are for spin majority and spin minority, respectively. (b) The band gap size as a function of $U$ for $X$-type (solid squares) and FM (open circles) orders, as well as for the $X$ type without TLD (open triangles). (c) Total energy difference per Fe atom induced by vanishing TLD in the $X$ type as a function of $U$.

Figure 4

(a) Itinerant energy per Fe vs $KS$ for the TLD parameter $\alpha =0$ (see text). The FM state is the energy reference. (b) Itinerant energy per Fe vs $\alpha$ for $KS=0.8$ eV. (c) Gap size vs $KS$ for $\alpha =0$. (d) Gap size vs $\alpha$ for $KS=0.8$ eV.