Block Spin Ground State and Three-Dimensionality of $(\mathrm{K},\mathrm{Tl}{)}_y{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$

The magnetic properties and electronic structure of $(\mathrm{K},\mathrm{Tl}{)}_y{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$ is studied using first-principles calculations. The ground state is checkerboard antiferromagnetically coupled blocks of the minimal ${\mathrm{Fe}}_4$ squares, with a large block-spin moment $\sim 11.2{\mu }_B$. The magnetic interactions could be modeled with a simple spin model involving both the inter- and intrablock, as well as the nearest-neighbor and next-nearest-neighbor couplings. The calculations also suggest a metallic ground state except for $y=0.8$ where a band gap $\sim 400–550\mathrm{meV}$ opens, showing an antiferromagnetic insulator ground state for $(\mathrm{K},\mathrm{Tl}{)}_{0.8}{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$. The electronic structure of the metallic $(\mathrm{K},\mathrm{Tl}{)}_y{\mathrm{Fe}}_{1.6}{\mathrm{Se}}_2$ is highly three dimensional with unique Fermi surface structure and topology. These features indicate that the Fe-vacancy ordering is crucial to the physical properties of $(\mathrm{K},\mathrm{Tl}{)}_y{\mathrm{Fe}}_{2-x}{\mathrm{Se}}_2$.

Figure 1

1a The geometry and primitive cell from top-view. The shaded (blue) region indicates the fundamental block with the four Fe atoms at the corners. The area encircled by the dashed line for the AFM2 configuration (the ground state) is $74.75{\AA }^2$. The periodic boundary condition for such blocks extended over the whole lattice is imposed. 1b The proposed magnetic couplings. ($J_1$, $J_2$) and ($J_1^{\prime }$, $J_2^{\prime }$) represent the intrablock and interblock (n.n., n.n.n.) couplings, respectively. 1c, 1d, 1e, 1f, 1g, 1h, 1i, 1j are various magnetic configurations. The red/blue atoms indicate the Fe-atoms with positive/negative total magnetic moment, respectively. For the AFM1 configuration, two primitive cells consist a magnetic unit cell. In all figures, we show only the Fe atoms to enhance the visibility.

Figure 2

Band structure and DOS for ${\mathrm{TlFe}}_{2-x}{\mathrm{Se}}_2$: The left and right panels are for the NM and AFM2 states, respectively. Only up-spin is shown in panels 2a, 2c since the up- and down-spins are degenerate.

Figure 3

Fermi surfaces of ${\mathrm{TlFe}}_{1.6}{\mathrm{Se}}_2$ reconstructed using MLWFs at (a),(c): nonmagnetic (NM) state and (b),(d): block-spin antiferromagnetic state; (c),(d): Cross section of the Fermi surface of ${\mathrm{TlFe}}_{2-x}{\mathrm{Se}}_2$ at $k_z=0.0$ plane with $x=0.4$. Both NM and block-spin AFM Fermi surfaces are 3D-like.