Magnetic frustration and iron-vacancy ordering in iron chalcogenide

We study the spin- and vacancy-ordered states in 122 iron chalcogenides ($A_{1-y}$Fe${}_{2-x}$Se${}_2$) by inspecting the magnetic ground states of a $J_1$-$J_2$-$J_3$ model on different vacancy-ordered lattices observed/conjectured in these compounds. A highly frustrated $J_1$-$J_2$-$J_3$ model was first applied to the study of magnetism in FeTe and was reported to explain the inelastic neutron-scattering data qualitatively. We find that the vacancy-ordered states are generally energetically favored for their reduction of magnetic frustration inherent to the spin-exchange model and, especially, that the 245 vacancy-spin-ordered state minimizes the magnetic exchange energy among all known vacancy-ordered states, in line with the fact that it has the highest vacancy-ordering phase transition temperature and the largest ordered moment in all iron-based superconductors. Thus, our study provides an electronic perspective for understanding the various vacancy orderings in these compounds. Then we focus on the experimentally well-studied 245 state and calculate the spin-wave spectrum and dynamic spin susceptibility. Finding that the key features of these calculated quantities are consistent with a recent inelastic neutron-scattering experiment, we conclude that we have obtained a qualitative local spin model for the 245 state. We also discuss the possibility of a unified local-moment description for all iron chalcogenides based on our result.

Figure 1

The bicollinear antiferromagnetic spin structure of Fe spins observed in the 11 (FeTe/Se) systems. The figure does not reflect a small lattice distortion in the magnetic phase. The distortion can be viewed as an elongation of the unit cell along the diagonal direction, which is 45 deg away from the lattice distortion observed in iron pnictides. From the figure, it is easy to see that in this ground state $J_1$ and $J_2$ are highly frustrated.

Figure 2

Schematic of the $J_1$-$J_2$-$J_3$ model in the 245 state. $J_1$ and $J_1^{\prime }$ are NN couplings when the two spins belong to the same and different four-spin blocks, respectively; $J_2$ and $J_2^{\prime }$ are NNN couplings when the two spins belong to the same and different four-spin blocks, respectively; $J_3$ and $J_3^{\prime }$ are TNN couplings when the two spins belong to the same and different four-spin blocks, respectively. Couplings not plotted in the figure can be obtained by translating the plotted couplings by the lattice period of the vacancy-ordered phase.

Figure 3

Possible vacancy orderings other than the experimental 245 state and their ground-state spin configurations calculated assuming a $J_1$-$J_2$-$J_3$ model with ferromagnetic $J_1$. (a) The armchair dimer crystal pattern. (b) The square dimer crystal pattern. These two patterns correspond to $(x=1,y=0.5)$ or $A$Fe${}_2$Se${}_4$. (c) and (d) The two different possible vacancy patterns for $(x=0.5,y=0)$ or $A_2$Fe${}_3$Se${}_4$. (e) and (f) Two different possible patterns for $A_4$Fe${}_4$Se${}_6$: (e) the dimer vacancy ordering similar to the CAFM magnetic phase and (f) the diagonal stripe pattern where the magnetic ordering is incommensurate.

Figure 4

The spin-wave spectra of the lowest three branches (the other one is too high to be drawn in the same plot) using the parameters (a) $(J_1,J_1^{\prime },J_2,J_2^{\prime },J_3,J_3^{\prime })=(-30,-10,20,20,9,0)$ meV and (c) $(J_1,J_1^{\prime },J_2,J_2^{\prime },J_3,J_3^{\prime })=(-30,-30,20,20,9,0)$ meV. The corresponding band structures are given in (b) and (d), in which the red dashed lines represent the highest branch and the black solid lines represent the lower three branches. $M_s$, $X_s$, and $Y_s$ are high-symmetry points in the BZ that are defined in the text.

Figure 5

The dynamic factor $S\left(\omega \right)$ contributed by the spin wave. The black solid line corresponds to $(J_1,J_1^{\prime },J_2,J_2^{\prime },J_3,J_3^{\prime })=(-30,-10,20,20,9,0)$ meV; the blue dashed line corresponds to $(J_1,J_1^{\prime },J_2,J_2^{\prime },J_3,J_3^{\prime })=(-30,-30,20,20,9,0)$ meV; and the red dotted line corresponds to $(J_1,J_1^{\prime },J_2,J_2^{\prime },J_3,J_3^{\prime })=(-12,-4,9,9,0,0)$ meV. The main panel only includes the contribution from the lowest three branches, as the highest branch is too high to be plotted in the same energy range. Inset: the contribution by the fourth and highest branch.

Figure 6

The phase diagrams against (a) $J_1$ and $J_1^{\prime }$ taking $J_3=9/20J_2$ and $J_3^{\prime }=0$ and (b) $J_1$ and $J_2$ taking $J_1^{\prime }=J_1$, $J_2^{\prime }=J_2$, and $J_3^{\prime }=0$. Solid lines mark second-order transitions calculated from the spin wave, and dotted lines mark possible first-order transitions. FM is the ferromagnet, AFM1 is the block spin, and AFM2 is the commensurate state plotted in Fig. 2(b) of Ref. 38.