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

TL;DR

This study uses first-principles calculations to identify a novel pair-checkerboard antiferromagnetic order in $eta$-Fe$_4$Se$_5$, revealing its lower energy state and implications for magnetic fluctuations in FeSe-based materials.

Contribution

The paper discovers a new pair-checkerboard AFM order in $eta$-Fe$_4$Se$_5$, differing from known magnetic states, and links doping to magnetic phase transitions.

Findings

The ground state is a pair-checkerboard AFM insulator.

Electron doping induces a transition between AFM states.

The new magnetic order is energetically favored by 130 meV/Fe.

Abstract

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.

Tables

Table 1. Table 1: Total energies, averaged magnitudes of the magnetic moments of Fe spins ( | M | 𝑀 |M| ), and band gaps of the ten AFM orders shown in Fig.  2 . The energy of NM is set to zero.

AFM states	energy	$\left|M\right|$	band gap
	(meV/Fe)	(${\mu }_{\text{B}}$)	(meV)
AFM-$a$	-127.4	2.48	0.0
AFM-$b$	-77.8	2.53	82.6
AFM-$c$	-188.9	2.72	208.5
AFM-$d$	-155.5	2.66	163.6
AFM-$e$	-114.0	2.53	75.7
AFM-$f$	-245.6	2.70	290.0
AFM-$g$	-20.3	2.29	0.0
AFM-$h$	-168.4	2.47	0.0
AFM-$i$	-207.7	2.66	20.8
AFM-$j$	-115.8	2.35	0.0

Full text

I Introduction

The discovery of iron-based superconductors has opened a new path to explore unconventional mechanism of high-temperature superconductivity Kamihara-JACS . The superconducting transition temperatures of iron-based superconductors are closely related to the height of anion from the Fe-Fe square lattice and the Fe-(As/Se)-Fe angle Zhao-NatMater7 ; Okabe-PRB ; Mizuguchi-SST . By intercalating atoms between FeSe layers to form $A_{1-x}$Fe2-ySe2 compounds ($A$=K, Tl, Rb, or Cs), one can effectively modulate the height of anion and raise the superconducting transition temperature of bulk FeSe from 8 K  Hsu-PNAS to 27$\sim$48 K  Guo-PRB82 ; Maziopa-JPCM ; Fang-EPL ; Wang-EPL93 ; Sun-Nature ; Ying-Sci2 .

$A_{1-x}$Fe2-ySe2 possesses different kinds of Fe-vacancy orders. For example, the two commonly studied K2xFe2-xSe2 compounds, K2Fe3Se4 (KFe1.5Se2) and K2Fe4Se5 (K0.8Fe1.6Se2), have the $\sqrt{2}\times 2\sqrt{2}$ and $\sqrt{5}\times\sqrt{5}$ Fe-vacancy orders, respectively. These two compounds were predicted to be AFM semiconductors by the first-principles calculations Yan-PRL ; Yan-PRB83 , and later confirmed by the experimental measurements Zhao-PRL109 ; Bao-CPL28 ; Chen-PRX1 . Other kind of Fe-vacancy orders has also been observed Ding-Natcomm4 . The stoichiometric compound KFe2Se2 without Fe vacancies is unstable against mesoscopic phase separation Wang-PRB83 ; Pomjakushin-JPCM ; Ricci-PRB84 ; Ricci-PRB91 . But it can be stablized and become a high-Tc superconductor Li-PRL109 ; Zhang-PRB by proximity to K2Fe4Se5.

Recently, a series of $\beta$-Fe1-xSe materials without metal intercalations but with Fe vacancies were synthesized. Through the measurement of electron microscopy, Fe vacancies are also found to be ordered and mainly in the phases with $\sqrt{2}\times\sqrt{2}$, $\sqrt{5}\times\sqrt{5}$, and $\sqrt{10}\times\sqrt{10}$ vacancy orders Chen-PNAS . It indicates that the Fe-vacancy order is a common feature of FeSe-based superconductors. Furthermore, it was found that most of $\beta$-Fe1-xSe samples synthesized in laboratory are in the $\sqrt{5}\times\sqrt{5}$ phase, and the compound with such a vacancy order, i.e. $\beta$-Fe4Se5, is an AFM insulator Chen-PNAS ; Fang-PRB93 . As both $\beta$-Fe4Se5 and K2Fe4Se5 have the same Fe-vacancy orders, it is interesting to know whether they have the same magnetic order in the ground states.

In this paper, we present the result for the electronic and magnetic structures of $\beta$-Fe4Se5 with the $\sqrt{5}\times\sqrt{5}$ Fe-vacancy order obtained by the first-principles density functional calculations. We find that $\beta$-Fe4Se5 is an insulator with a $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM order. In this AFM ordered state, each Fe block contains four Fe spins which are collinear AFM aligned. The band gap of $\beta$-Fe4Se5 in this $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM phase is found to be 290 meV, in agreement with the experimental observation Chen-PNAS ; Fang-PRB93 . This AFM order is stable against light electron doping. It is replaced by the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM order (which is also the AFM order observed in K2Fe4Se5) when the doping level excesses 1.7 electrons per Fe4Se5 formula.

II COMPUTATIONAL DETAILS

In our calculations the plane wave basis method is used pwscf . We adopt the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof pbe for the exchange-correlation potentials. The ultrasoft pseudopotentials vanderbilt are used to model the electron-ion interactions. After full convergence test, the kinetic energy cut-off and the charge density cut-off of the plane wave bases are taken to be 45 Ry and 480 Ry, respectively. Our calculation is done with a $\sqrt{5}\times\sqrt{5}\times 1$ tetragonal unit cell which contains one FeSe layer with 8 Fe atoms, 2 vacancies, and 10 Se atoms. A mesh of 8$\times$8$\times$10 k-points is used to sample the Brillouin zone of the $\sqrt{5}\times\sqrt{5}\times 1$ unit cell. The Gaussian broadening technique of width 0.002 Ry is used in the calculation of metallic states. The convergence is tested with a denser 16$\times$16$\times$20 k-mesh. There are two degenerate ground states, corresponding to the two degenerate chiral structures of the $\sqrt{5}\times\sqrt{5}$ lattice Yan-PRB83 ; Sabrowsky-JMMM . We choose the right-chiral structure to do the calculation.

III RESULTS AND ANALYSIS

For the vacancy ordered $\beta$-Fe4Se5, there are eight Fe atoms in each $\sqrt{5}\times\sqrt{5}\times 1$ unit cell. These eight Fe atoms can be divided into two blocks. One block contains the four Fe atoms in the middle of the unit cell enclosed by the blue box as shown in Fig. 2, and the other contains the rest four Fe atoms.

The nonmagnetic (NM) phase of $\beta$-Fe4Se5 is metallic. Figure 1 shows the band structure and the partial density of states in the NM state. There are two bands crossing the Fermi level, forming one pillar-shaped hole Fermi surface along the $\Gamma$-Z line [Figs. 1(c) and 1(e)] and one windmill-type electron Fermi surfaces around the zone center [Figs. 1(d) and 1(f)]. The low-energy physics is governed by Fe $3d$ orbitals, since the density of states (DOS) in the energy range from -2.5 eV to 1.0 eV is contributed mainly by Fe $3d$ electrons.

The ground state, or the lowest energy state, of $\beta$-Fe4Se5 in the $\sqrt{5}\times\sqrt{5}$ vacancy ordered phase is AFM ordered. In order to determine the magnetic structure of the ground state, we have calculated the electronic structures for all ten non-equivalent AFM states allowed in a $\sqrt{5}\times\sqrt{5}$ lattice. Figure 2 shows the spin configurations of these ten AFM states, labelled from AFM-$a$ to AFM-$j$. These states can be divided into three groups, depending on the number of up spins in the middle four Fe atoms, which is called an Fe block, within each unit cell enclosed by the blue square. The first group contains the spin configurations represented by (AFM-$a$)-(AFM-$c$) with just one up spin in an Fe block. The second one consists of the spin configurations represented by (AFM-$d$)-(AFM-$i$) with two up spins in an Fe block. There is one spin configuration, i.e. AFM-$j$, in the third group. It contains four up spins in an Fe block. The AFM states with one and three up Fe spins in an Fe block are energetically degenerate and physically equivalent. The states with two up spins inside an Fe block can have either a collinear AFM order [Figs. 2(d)-(g)] or a checkerboard one [Figs. 2(h) and 2(i)]. Among these states, AFM-$j$ is the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM order, which is the magnetic order observed in the ground state of K2Fe4Se5.

The relative energies of the ten AFM states with respect to the energy of the NM state are given in Table I. The ground state of $\beta$-Fe4Se5 is that shown in Fig. 2(f), namely the state labelled by AFM-$f$. It is a $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM state. A similar AFM state, called the pair-checkerboard AFM state, was found to be the ground state of bulk $\beta$-FeSe based on the first principles

electronic structure calculations Cao-arXiv1407 . AFM-$f$ can be regarded as a realization of this pair-checkerboard AFM order on the $\sqrt{5}\times\sqrt{5}$ Fe-vacancy ordered lattice.

In the ground state, the lattice undergoes a tetragonal to monoclinic structure transition induced by the spin-lattice interaction. This changes the angle, $\varphi$, defined in Fig. 2(f) from 90∘ to 89.2∘ with an energy gain of 0.8 meV/Fe. Consequently, there is a tiny difference between the lattice parameters along the $a$ and $b$ axes, which are equal to 8.4519 Å and 8.4855 Å, respectively. This kind of structure transition is similar to the tetragonal-orthorhombic phase transition observed in other parent compounds of iron-based superconductors Cruz-Nature ; Rotter-PRB78 ; McQueen-PRL103 ; Ma-PRL102 .

Figure 3 shows the band structure of $\beta$-Fe4Se5 in the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM state obtained with a tetragonal unit cell by ignoring the monoclinic distortion. It indicates clearly that $\beta$-Fe4Se5 is an AFM insulator with a direct energy gap of about 290 meV, in agreement with the experimental observation Chen-PNAS ; Fang-PRB93 .

Figure 4 shows the orbital-resolved partial DOS in the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM state with tetragonal symmetry. This AFM state is invariant under the following two kinds of transformations: One is to combine an $ab$-plane reflection with a fractional translation of (-1/2, 1/2, 0), the other is to take a spin inversion and then a 180∘ rotation around the $c$-axis. This symmetry property ensures the spin-up/down DOS at an Fe atom to equal the spin-down/up DOS at the diagonal Fe atom inside the same block. Thus we need only to show the orbital-resolved partial DOS for two Fe atoms, labelled by Fe1 and Fe2 in the figure. For Fe1, the five up-spin orbitals are almost completely filled, but the five down-spin orbitals are only partially filled. The DOS at Fe2 behaves similarly, but the up and down spins are reversed. The magnetic moment of Fe2 is slightly larger than that of Fe1. This leads to the difference in the DOSs between the up/down-spin part of Fe1 and the down/up-spin one of Fe2. The unoccupied states are mainly contributed by $d_{z^{2}}$ and $d_{xy}$ orbitals.

We also find that the AFM states defined by AFM-$b$, AFM-$c$, AFM-$d$, AFM-$e$, and AFM-$i$ are insulating. The excitation energy gaps are equal to 82.6 meV, 208.5 meV, 163.6 meV, 75.7 meV, and 20.8 meV, respectively. For AFM-$d$, the band gaps for the up- and down-spin electrons are not equal to each other, since the corresponding up- and down-spin configurations are not equivalent. They equal 163.6 meV and 233.1 meV, respectively. The other AFM states, including the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM state, i.e. AFM-$j$, are metallic. This differs from the case in K2Fe4Se5 Yan-PRL ; Bao-CPL28 ; Chen-PRX1 where the ground state is blocked-checkerboard AFM ordered but insulating. AFM-$a$ is in fact not a truly AFM state, although we initially set the up- and down-spin moments the same in the amplitude in our calculation. But after full optimization, we find that it is actually a ferrimagnetic state with a net magnetization about 1.97 $\mu_{\text{B}}$ per unit cell.

In $\beta$-Fe4Se5, the energy of the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM state is 129.9 meV/Fe lower than the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM one. In contrast, the energy of the former is about 95.8 meV/Fe higher than the latter in K2Fe4Se5. Apparently, the difference results from the intercalation of K atoms in Fe4Se5: First, the intercalated K atoms impose a tensile strain on each Fe4Se5 layer, which enlarges the $a$-axis lattice constant from 8.41 Å in $\beta$-Fe4Se5 to 8.69 Å in K2Fe4Se5. Second, the intercalated atoms dope electrons into Fe4Se5 layers.

To understand the effect of tensile strain introduced by K atoms, we calculate the energies of $\sqrt{5}\times\sqrt{5}$-pair-checkerboard and $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM states as a function of the $a$-axis lattice constant for $\beta$-Fe4Se5 with tetragonal unit cell. The lattice constant along the $c$-axis and internal atomic positions are relaxed to minimize the total energy. In a wide range of the $a$-axis lattice constant, as shown in Fig. 5(a), we find that the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM order is more stable than the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM one. It suggests that the difference between $\beta$-Fe4Se5 and K2Fe4Se5 is not due to the tensile strain effect.

To understand the charge doping effect, we introduce extra electrons to $\beta$-Fe4Se5 and to K2Fe4Se5 with all K atoms removed (namely, K2Fe4Se5-2K). A jellium background with positive charge is introduced to compensate the doped electrons and to avoid numerical divergence in the calculation. Note that different AFM states have different optimized lattice constants. For example, the optimized lattice constants are $a$=8.4682 Å and $c$=6.5241 Å in the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM state, and $a$=8.1092 Å and $c$=6.2119 Å in the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM order of $\beta$-Fe4Se5. In order to avoid the complication introduced by the lattice parameters, we do the calculation by taking the experimental lattice constants, i.e., $a$=8.41 Å and $c$=5.47 Å for $\beta$-Fe4Se5 Chen-PNAS , $a$=8.6929 Å and $c$=14.0168 Å for K2Fe4Se5-2K Bao-CPL28 .

For both materials without electron doping, we find that the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM state is lower in energy than the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard one [Figs. 5(b) and 5(c)]. Upon electron doping, the energy difference between these two states gradually decreases. A reversion in the total energy happens at a critical doping roughly equal to 1.7 electrons per formula for both $\beta$-Fe4Se5 and K2Fe4Se5-2K. The $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM state is 57.5 meV/Fe lower than the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard one by adding 2 electrons per formula to K2Fe4Se5-2K. Here K2Fe4Se5-2K+2$e$ is isoelectronic to K2Fe4Se5. Thus adding electron to Fe4Se5 layers reproduces correctly the ground-state AFM order of K2Fe4Se5. It suggests that the difference in the magnetic orders between $\beta$-Fe4Se5 and K2Fe4Se5 results mainly from the doping effect introduced by the intercalated K atoms. Furthermore, by doping 2 electrons per formula to $\beta$-Fe4Se5, we find that the metallic $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM state becomes insulating, with an energy gap of 324 meV [Fig. 5(d)]. This is reminiscent of the insulating AFM state of K2Fe4Se5 Yan-PRB83 .

With the increase of electron doping, we find that the nearest-neighboring Fe-Fe bond length inside the inner Fe block decreases. This leads to a tetramer lattice distortion of Fe atoms, in both the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard and the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM states. The tetramer distortion also exists in the ground state of K2Fe4Se5 Yan-PRB83 . It is in fact this tetramer distortion that stablizes the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM order in K2Fe4Se5. To check how important this effect is in $\beta$-Fe4Se5, we carry out a total-energy calculation for this material without any tetramer distortion by using the lattice parameters obtained from the bulk $\beta$-FeSe. We find that the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM state is about 27.8 meV/Fe higher in energy than the $\sqrt{5}\times\sqrt{5}$-pair-checkerboard one at the doping of 2 electrons per Fe4Se5 formula. This shows that it is the tetramer lattice distortion enhanced by electron doping that lowers the energy of the $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM state.

The above results are obtained without considering the van der Waals (vdW) interaction between FeSe layers. This overestimates the lattice constant along the $c$-axis. If the vdW correctionGrimme-JCC ; Barone-JCC is taken into account, we find that the ground state is still $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM insulator. The band gap is 172 meV, and the $a$- and $c$-axis lattice constants are respectively 8.2440 Å and 5.5619 Å, in agreement with the experimental results Chen-PNAS .

IV SUMMARY

In conclusion, we have studied the electronic and magnetic structures of $\beta$-Fe4Se5. Similar to K2Fe4Se5, we find that the ground state of $\beta$-Fe4Se5 is an AFM insulator. But the ground-state is $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM ordered, unlike in K2Fe4Se5 where the ground state is $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM ordered. The band excitation gap of $\beta$-Fe4Se5 is 290 meV, in agreement with the experimental result. The $\sqrt{5}\times\sqrt{5}$-pair-checkerboard AFM order is robust against the lattice tensile strain. But electron doping, introduced by extra Fe or intercalated alkali-metal atoms, can turn it into a $\sqrt{5}\times\sqrt{5}$-blocked-checkerboard AFM state above a critical doping level.

Acknowledgements.

This work is supported by National Natural Science Foundation of China (Grant Nos. 11404383, 91421304, 11474331, and 11474004). M.G. is also supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY17A040005 and K.C.Wong Magna Fund in Ningbo University.