Pair-checkerboard antiferromagnetic order in $\beta$-Fe$_4$Se$_5$ with $\sqrt{5}\times\sqrt{5}$ ordered Fe vacancies

The electronic structure of recently discovered $\beta$-Fe$_4$Se$_5$ with $\sqrt{5} \times \sqrt{5}$ ordered Fe vacancies is calculated using first-principles density functional theory. We find that the ground state is an antiferromagnetic (AFM) insulator in agreement with the experimental observation. In K$_2$Fe$_4$Se$_5$, it is known that the ground state is $\sqrt{5} \times \sqrt{5}$-blocked-checkerboard AFM ordered. But for this material, we find that the ground state is $\sqrt{5} \times \sqrt{5}$-pair-checkerboard AFM ordered, in which the intrablock four Fe spins exhibit the collinear AFM order and the interblock spins on any two nearest-neighboring sites are antiparallel aligned. This state is about 130 meV/Fe lower in energy than the $\sqrt{5} \times \sqrt{5}$-blocked-checkerboard AFM one. Electron doping can lower the energy of the $\sqrt{5} \times \sqrt{5}$-blocked-checkerboard AFM state and introduce a transition between these two states, suggesting that there is strong AFM fluctuation in FeSe-based materials upon doping. This provides a unified picture to understand the AFM orders in $\beta$-Fe$_4$Se$_5$ and in alkali-metal intercalated FeSe materials.