Electronic and magnetic structures of the ternary iron selenides $A$Fe${}_2$Se${}_2$ ($A=\mathit{Cs}$, Rb, K, or Tl)

We have studied the electronic and magnetic structures of the ternary iron selenides $A$Fe${}_2$Se${}_2$ ($A$ $=$ Cs, Rb, K, or Tl) using first-principles electronic structure calculations. We find that the ground state of these compounds is bicollinearly antiferromagnetically ordered, with the Fe moments having collinear antiferromagnetic order in each bipartite sublattice. This bicollinear antiferromagnetic order results from the superexchange interactions of Fe moments mediated by the Se $4p$ orbitals. We have also determined the density of states at the Fermi level, the specific heat coefficient, the Pauli susceptibility, and other related physical properties in both the nonmagnetic and bicollinear antiferromagnetic states for these compounds. The underlying mechanism is discussed according to the electronic structure analysis.

Figure 1

$A$Fe${}_2$Se${}_2$ ($A=\mathit{Cs}$, Rb, K, or Tl) with the ThCr${}_2$Si${}_2$-type structure: (a) a tetragonal unit cell containing two formula units; (b) a primitive unit cell containing one formula unit.

Figure 2

CsFe${}_2$Se${}_2$ in the nonmagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 3

Total and orbital-resolved partial densities of states (DOS) of $A$Fe${}_2$Se${}_2$ ($A=\mathit{Cs}$, Rb, K, or Tl) per formula unit in the nonmagnetic state: (a) CsFe${}_2$Se${}_2$; (b) RbFe${}_2$Se${}_2$; (c) KFe${}_2$Se${}_2$; (d) TlFe${}_2$Se${}_2$.

Figure 4

RbFe${}_2$Se${}_2$ in the nonmagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 5

KFe${}_2$Se${}_2$ in the nonmagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 6

TlFe${}_2$Se${}_2$ in the nonmagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 7

CsFe${}_2$Se${}_2$ in the metastable checkerboard antiferromagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 8

RbFe${}_2$Se${}_2$ in the metastable checkerboard antiferromagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 9

KFe${}_2$Se${}_2$ in the metastable checkerboard antiferromagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 10

TlFe${}_2$Se${}_2$ in the metastable checkerboard antiferromagnetic state with a primitive crystal unit cell: (a) the energy band structure; (b) and (c) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1 and 2 in (a), respectively.

Figure 11

(a) Schematic top view of the FeSe layer in the ground state with bicollinear antiferromagnetic order, in which the dashed rectangle is an $a\times 2a$ unit cell and the Fe spins are shown by arrows. (b) Brillouin zone of $A$Fe${}_2$Se${}_2$ ($A=\mathit{Cs}$, Rb, K, or Tl) in the bicollinear antiferromagnetic state.

Figure 12

CsFe${}_2$Se${}_2$ in the ground state with bicollinear antiferromagnetic order: (a) the energy band structure; (b), (c), and (d) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1, 2, and 3 in (a), respectively. The corresponding Brillouin zone is shown in Fig. 11b.

Figure 13

RbFe${}_2$Se${}_2$ in the ground state with bicollinear antiferromagnetic order: (a) the energy band structure; (b), (c), and (d) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1, 2, and 3 in (a), respectively. The corresponding Brillouin zone is shown in Fig. 11b.

Figure 14

KFe${}_2$Se${}_2$ in the ground state with bicollinear antiferromagnetic order: (a) the energy band structure; (b), (c), and (d) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1, 2, and 3 in (a), respectively. The corresponding Brillouin zone is shown in Fig. 11b.

Figure 15

TlFe${}_2$Se${}_2$ in the ground state with bicollinear antiferromagnetic order: (a) the energy band structure; (b), (c), (d), and (e) show the Fermi surface sheets due to the bands crossing the Fermi energy denoted by lines 1, 2, 3, and 4 in (a), respectively. The corresponding Brillouin zone is shown in Fig. 11b.

Figure 16

Projected density of states of the five Fe $3d$ orbitals around one of the four Fe atoms in the bicollinear antiferromagnetic state unit cell of CsFe${}_2$Se${}_2$. The Fermi energy is set to zero.

Figure 17

Total and orbital-resolved partial densities of states (spin-up part) per four formula cells of CsFe${}_2$Se${}_2$ in the bicollinear antiferromagnetic state. The Fermi energy is set to zero.