Electronic Structures and Magnetic Order of Ordered-Fe-Vacancy Ternary Iron Selenides ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$ and $A{\mathrm{Fe}}_{1.5}{\mathrm{Se}}_2$ ($A=\mathrm{K}$, Rb, or Cs)

By the first-principles electronic structure calculations, we find that the ground state of the Fe-vacancies ordered ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$ is a quasi-two-dimensional collinear antiferromagnetic semiconductor with an energy gap of 94 meV, in agreement with experimental measurements. This antiferromagnetic order is driven by the Se-bridged antiferromagnetic superexchange interactions between Fe moments. Similarly, we find that crystals $A{\mathrm{Fe}}_{1.5}{\mathrm{Se}}_2$ ($A=\mathrm{K}$, Rb, or Cs) are also antiferromagnetic semiconductors but with a zero-gap semiconducting state or semimetallic state nearly degenerated with the ground states. Thus, rich physical properties and phase diagrams are expected.

Figure 1

${\mathrm{TlFe}}_x{\mathrm{Se}}_2$ with the ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$-type structure: (a) a tetragonal unit cell containing two formula units without any vacancy; (b) schematic top view of the Fe-Fe square layer with one-quarter Fe vacancies ordered in rhombus ($x=1.5$), in which there are two inequivalent Fe atoms, namely, 2-Fe-neighbored and 3-Fe-neighbored Fe atoms. (c) Schematic top view of the Fe-Fe square layer with one-quarter Fe vacancies ordered in square ($x=1.5$). The filled circles denote the Fe atoms while the empty circles denote the Fe vacancies. The square and rectangle enclosed by the solid lines denote the unit cells.

Figure 2

Electronic structure of ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$ with the Fe vacancies ordered in rhombus [Fig. 1b] in the nonmagnetic state: (a) the band structure, (b) the Brillouin zone, and (c) the Fermi surface. The Fermi energy sets to zero.

Figure 3

Electronic structure of ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$ with the Fe vacancies ordered in square [Fig. 1c] in the nonmagnetic sate: (a) the band structure, (b) the Brillouin zone, and (c) the Fermi surface. The Fermi energy sets to zero.

Figure 4

Schematic top view of four possible magnetic orders in the Fe-Fe square layer with one-quarter Fe vacancies ordered in rhombus: (a) checkerboard Néel order in which the nearest-neighboring Fe moments are antiparallel ordered; (b) $A$-collinear AFM order in which the Fe moments are antiferromangtic ordered along the line without vacancies; (c) P-collinear AFM order in Fe moments are antiferromagnetic ordered along the lines with vacancies; (d) bicollinear AFM order. The squares or rectangles enclosed by the solid lines denote the magnetic unit cells.

Figure 5

Schematic top view of two possible magnetic orders in the Fe-Fe square layer: (a) $c$-AFM order in which the next-nearest Fe moments are antiparallel ordered; (b) $h$-AFM order in which half pairs of the next-nearest Fe moments are in antiparallel while the other half pairs are in parallel. The squares enclosed by the solid lines denote the magnetic unit cells.

Figure 6

Electronic band structure of ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$ or $A{\mathrm{Fe}}_{1.5}{\mathrm{Se}}_2$ ($A=\mathrm{K}$, Rb, or Cs) with one-quarter Fe vacancies ordered in rhombus [see Fig. 1b] and with an $A$-collinear AFM order [see Fig. 4b] in the ground state, which is an antiferromagnetic semiconductor. (a) ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$, (b) ${\mathrm{KFe}}_{1.5}{\mathrm{Se}}_2$, (c) ${\mathrm{RbFe}}_{1.5}{\mathrm{Se}}_2$, (d) ${\mathrm{CsFe}}_{1.5}{\mathrm{Se}}_2$. The Brillouin zone is shown in Fig. 2b. Here the top of the valence band sets to zero. Note that $\Gamma \mathrm{\text{−}}N(\Gamma \mathrm{\text{−}}{N}^{\prime })$ corresponds to the antiparallel- (parallel-)aligned moment line.

Figure 7

Projected density of states (PDOS) at the five Fe-$3d$ orbitals around (a) 2-Fe-neighbored Fe atom and (b) 3-Fe-neighbored atom in the $A$-collinear AFM ordered ground state of ${\mathrm{TlFe}}_{1.5}{\mathrm{Se}}_2$. Here the top of the valence band sets to zero and $x$ axis is along the antiferromagnetic direction while $y$ axis is along the ferromagnetic direction in Fig. 4b.