Distorted magnetic orders and electronic structures of tetragonal FeSe from first-principles
Yong-Feng Li, Li-Fang Zhu, San-Dong Guo, Ye-Chuan Xu, Bang-Gui Liu
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Distorted magnetic orders and electronic structuresof tetragonal FeSe from first-principles
Yong-Feng Li, Li-Fang Zhu, San-Dong Guo, Ye-Chuan Xu, andBang-Gui Liu
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, ChinaBeijing National Laboratory for Condensed Matter Physics, Beijing 100190, ChinaE-mail: [email protected]
Abstract.
We use the state-of-the-arts density-functional-theory method to studyvarious magnetic orders and their effects on the electronic structures of the FeSe.Our calculated results show that, for the spins of the single Fe layer, the stripedantiferromagnetic orders with distortion are more favorable in total energy than thecheckerboard antiferromagnetic orders with tetragonal symmetry, which is consistentwith known experimental data, and the inter-layer magnetic interaction is very weak.We investigate the electronic structures and magnetic property of the distorted phases.We also present our calculated spin coupling constants and discuss the reduction ofthe Fe magnetic moment by quantum many-body effects. These results are useful tounderstand the structural, magnetic, and electronic properties of FeSe, and may havesome helpful implications to other FeAs-based materials.PACS numbers: 75.30.-m,74.10.+v,75.10.-b,71.20.-b,74.20.-z istorted magnetic orders and electronic structures of tetragonal FeSe
1. Introduction
The advent of superconducting F-doped LaFeAsO stimulates a world-wide campaignfor more and better Fe-based superconductors [1]. More superconductors wereobtained by replacing La by other lanthanides or partly substituting F for O, andhigher phase-transition temperatures ( T c ) were achieved in some of them [2, 3, 4].Furthermore, more series of Fe-based superconductors were found, including BaFe2As2series and LiFeAs series [5, 6, 7, 8]. The highest T c in these series reaches 55-56K in the case of doped SmFeAsO [4]. Various explorations have been performedto elucidate their magnetic orders, electronic electronic structures, superconductivity,and so on [9, 10, 11, 12, 13, 14, 15]. Recently, superconductivity was found even intetragonal FeSe samples under high pressure and α FeSe phases with Se vacancies[16, 17, 18, 19, 20]. Very recently, SrFeAsF was made superconducting by La andCo doping [21, 22, 23, 24, 25]. The FeSe system is interesting because its FeSe layer issimilar to the FeAs layer of the FeAs-based materials: R FeAsO series ( R : rare earthelements), A Fe2As2 series ( A : alkaline-earth elements), LiFeAs series, and SrFeAsFseries. In addition to the FeAs layers, there are R O layers in R FeAsO series, A layers in A Fe2As2 series, Li layers in LiFeAs series, and SrF layers in SrFeAsF series, but there isnothing else besides the FeSe layers for the FeSe systems. Therefore, it is highly desirableto investigate the magnetic orders, electronic structures, and magnetic properties of thetetragonal FeSe phases (or distorted phases of them).In this article we use an full-potential density-functional-theory method to studyvarious magnetic orders and their effects on the electronic structures of the FeSe.We suppose checkerboard antiferromagnetic order for the spins of the Fe layer ofthe tetragonal phase and striped antiferromagnetic orders for those of the symmetry-broken structures, and perform total-energy and force optimization to determine thestructural and magnetic parameters. Then, we investigate the corresponding electronicstructures and magnetic property of them. Our calculated result means that the stripedantiferromagnetic order is favorable for the spins of the Fe layer, which is consistentwith main known experimental data. We also discuss the reduction of the Fe magneticmoment by quantum many-body effects. More detailed results are presented in thefollowing.The paper is organized as follows. In next section, we give our computational detail.In the third section, we present our main DFT calculated results, including optimizedmagnetic structures and corresponding electronic density of states and energy bands.In the fourth section, we discuss spin interactions and many-body effects on the Femagnetic moments. Finally, we present our main conclusion in the fifth section.
2. Computational detail
Our calculations are performed by using a full-potential linearized augmented plane wave(FLAPW) method within the density functional theory (DFT)[26], as implemented in istorted magnetic orders and electronic structures of tetragonal FeSe ❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵ ✘✘✘✘✘✘✘✘✘✘✘✘ ❵❵❵❵❵❵❵❵❵❵❵❵❵❵❵ .............................. .................................................. ❵❵❵❵❵❵❵❵ ............... ............... ✘✘✘✘✘✘❵❵❵❵❵❵❵❵✘✘✘✘✘✘ .......... r rrrr rrr X YZ Γ U S TR
NNJ JJN NNJ NNN JJJ NNN (a) (b) (e)(c) (d)
Figure 1. (color online). Schematic structures of tetragonal FeSe (a)and orthorhombic FeSe (b); and the spin configurations of the checkerboardantiferromagnetic order (c) and the striped antiferromagnetic order (d). The big ballswith Fe1 and Fe2 indicates the two kinds of Fe atoms with different spin orientations,and the small one Se atom. The sign ⊗ means that the Fe1 spin orients upward withrespect to the paper plane, and ⊙ the Fe2 spin downward. The first Brillouin Zone ofthe orthorhombic FeSe is shown in (e) with the high-symmetry points labelled. the Vienna package WIEN2k [27]. The generalized gradient approximation (GGA) isused for the exchange and correlation potentials [28]. We take the 3 d and 4 s statesof Fe and the 4 s and 4 p states of Se as valence states, and the 3 p states of Fe and3 d states of Se are treated as semicore states. The core states include all the lowerstates. The core states are treated in terms of radial Dirac equation and thus the fullrelativistic effect is included. For the valence and semicore states, the relativistic effectis calculated under the scalar approximation, with spin-orbit interaction being neglected[29]. The radii of Fe and Se muffin-tin spheres are 2.05 and 2.00 a.u., respectively. Toget more accurate results, we take R mt × K max =9.0 and make the angular expansionup to l=10 in the muffin-tin spheres. We use 1000 k points in the calculations. Fordifferent magnetic orders the k points in the first Brillouin zone are chosen differentlybecause of the different symmetry. The self-consistent calculations are controlled bythe charge density, and the convergence standard is that the difference between inputcharge density and output one is less than 0.00005 per unit cell.We show the unit cell of the tetragonal FeSe (PbO structure) in Fig. 1a. It isa layered structure, in which Fe atom occupies 2a position and Se atom 2c position.There are six possible magnetic configurations for the Fe ions if the tetrahedral FeSe istorted magnetic orders and electronic structures of tetragonal FeSe { } planes,the Fe moments can couple ferromagnetically or antiferromagnetically. As a result, wehave four different antiferromagnets, namely (a) checkerboard-FM, (b) checkerboard-AF, (c) stripe-FM, and (d) stripe-AF. In addition, when we force Fe moments in theFe plane have FM order, the self-consistent calculations yield zero moments for the Femoments, which means that FM order is unstable for FeSe, independent of the interlayerspin arrangements. Therefore, we can actually construct five magnetic orders for thissystem.
3. Main calculated results
The parameters in our calculations are taken from the experimental values. We use a = 3 . c = 5 . µ B for the two checkerboard orders, and about 2.0 µ B for the two striped orders.The total moment for one Fe atom is estimated to be a little larger for the striped AForders. For the striped AF orders, the Se position parameter remains almost the same Table 1.
The magnetic order, the relative total energy per formula unit (∆ E inmeV, with the lowest stripe-FM set as reference), the magnetic moment in the Femuffin-tin sphere ( M in µ B ), and the internal Se position parameter u Se of the twostriped antiferromagnetic orders and the two checkerboard ones. The results of thenonmagnetic order are presented for comparison. Magnetic order ∆ E (meV) Moment M ( µ B ) u Se checkerboard-FM 72 1.82 0.2570checkerboard-AF 72 1.81 0.2426stripe-FM 0 1.98 0.2590stripe-AF 5 2.00 0.2592nonmagnetic 154 0.00 0.2471 istorted magnetic orders and electronic structures of tetragonal FeSe total Fe1 Fe2 Se inter total Fe1 Fe2 Se inter -6 -5 -4 -3 -2 -1 0 1 2630 Energy (eV) -6 -5 -4 -3 -2 -1 0 1 2630 D en s i t y o f S t a t e s D en s i t y o f S t a t e s Energy (eV) (b)(a)
Figure 2. (color online). Spin-dependent density of states (DOS, in units of states/eVper formula unit) of the stripe-FM (a) and stripe-AF (b). The upper part of each panelis the majority-spin DOS and the lower part the minority-spin DOS. The Fe1 DOS isemphasized by thick red (or gray) solid lines, and the Fe2 DOS by thick blue (or gray)dotted lines. The black thin solid line indicates the spin-dependent total DOS, andthe others are projected DOS in the muffin tin sphere of Se atom (dot-dash) and theinterstitial region (pink or gray thin solid). when the interlayer spin coupling is changed from FM to AF. Because it is impossible todistinguish between FM and AF spin alignment in the z direction by density-functional-theory calculation, we present calculated results for both FM and AF arrangement inthe z direction in the following.We present the spin-dependent density-of states (DOS) of the FeSe in the twostriped AF orders in Fig. 2. There is no energy gap near the Fermi levels and thereforethe FeSe for each AF order shows a metal feature. The Fe atom has different crystallineenvironment for different magnetic orders, and thus its states are reformed in differentways. It can be seen that the main peaks occupy the states in the energy window from-2.2 eV to -1.5 eV. Almost all the partial DOS of Fe atom comes from the 3d sates, andthe DOS of Se atom consists mainly of the p states.We present the electronic energy bands of the FeSe in the two striped AF orders in istorted magnetic orders and electronic structures of tetragonal FeSe Γ S X Γ Y T Γ Z R U Z T E n e r gy ( e V ) Γ S X Γ Y T Γ Z R U Z T E F Γ S X Γ Y T Γ Z R U Z T E n e r gy ( e V ) Γ S X Γ Y T Γ Z R U Z T E F (a)(b) Figure 3.
Spin-dependent energy bands of the stripe-FM (a), and stripe-AF (b). Theleft panel for each magnetic order shows the energy bands of Fe1 spin-up channel withthe Fe1 atomic character emphasized by circles, and the right one those of Fe1 spin-down channel. The spin of Fe1 orients in antiparallel to that of Fe2. The emphasizedenergy bands consist of circles with different diameters, and the larger the diameter,the more the atomic character. There are so many circles (or dots) for a given bandthat they seams to be connected forming a line for the band.
Fig. 3. The plots look like lines, but consist of hollow circles. We choose the k pointsuniformly for convenient comparison. The circle diameter is proportional to the Fe1spin-up or Fe1 spin-down atomic character of the band at that k point. The dispersionalong the z direction is much stronger than those in other similar Fe-based materials.This can be attributed to the short distance between the successive Fe layers. It canbe seen that the d states of Fe play a key role for all the magnetic structures. For istorted magnetic orders and electronic structures of tetragonal FeSe
4. Spin interactions and many-body effects on the magnetic moments
In order to further investigate the magnetic moment, we use the following AFMHeisenberg spin model to describe the spin properties of the Fe atoms in the stripedAFM phase. H = X ij J ij ~S i · ~S j (1)where ~S i is quantum spin operator for site i , and J ij is the exchange coupling constantsbetween the two spins at sites i and j . For the striped AFM phase, the nearest couplingconstant in the x direction is J x , and that in the y direction J y . For the tetrahedralphase, we should have J x = J y , but for the striped phase we have J x = J y because thecrystalline distortion in the xy plane. We limit non-zero exchange coupling constantsup to the next nearest neighboring spins, J ′ . For convenience, we define J = ( J x + J y ) / δ = J x − J y . Our DFT calculation yields J x = 10meV, J y = 8meV, and J ′ = 5meV.These means J = 9meV and δ = 2meV. The J x and J y comes from the superexchangethrough the two nearest Se atoms and J ′ from that through the one nearest Se atom,and as a result, we should have J ≈ J ′ , which supports our DFT results. A v e r a g e s p i n J'/J ∆ S J'/J
Figure 4.
Zero-temperature average spin h S z i vs. the coupling constant ratio J ′ /J for the two-dimensional spin model. The inset shows ∆ S (defined in text) vs. J ′ /J .The ⊙ signs indicate calculated points. We treat the spin Hamiltonian (1) with the above parameters with spin wavetheory[30]. As usual, the average spin for zero temperature, h S z i , can be given by istorted magnetic orders and electronic structures of tetragonal FeSe h S z i = S − ∆ S , where S is the spin value of the Fe atom in the FeSe and ∆ S thecorrection due to quantum many-body effects. Presented in Fig. 4 are our calculatedresults for h S z i and ∆ S as functions of the parameter J ′ /J . It is clear that h S z i decreaseswith decreasing J ′ /J , getting to zero at J ′ = J y /
2. In fact, the striped AFM structureis not the magnetic ground state of the FeSe any more if J ′ is smaller than J y /
2. Forreal samples of the FeSe, one should have the parameter relations J ′ ≈ J/ δ issmall but finite, and therefore the experimental spin value is small compared to theDFT value S .
5. Conclusion
In summary, we have used the full-potential density-functional-theory method to studyvarious magnetic orders and their effects on the electronic structures of the FeSe. Wefind that, for the spins of the single Fe layer, the striped antiferromagnetic orderswith the broken symmetry are more favorable in total energy than the checkerboardantiferromagnetic orders with tetragonal symmetry, and the inter-layer magneticinteraction is very weak. Then, we investigate the corresponding electronic structuresand magnetic property of the distorted phases with the striped antiferromagnetic orders.Our calculated result that the striped antiferromagnetic order is favorable for the spinsof the Fe layer is consistent with main known experimental data. We also present ourcalculated spin coupling constants and conclude that the reduction of the Fe magneticmoment is caused by quantum many-body effects. These results are useful to understandthe structural, magnetic, and electronic properties of FeSe, and may have some helpfulimplications to other FeAs-based materials.
Acknowledgements
This work is supported by Nature Science Foundation of China (Grant Nos. 10774180,10874232, and 60621091), by Chinese Department of Science and Technology (Grant No.2005CB623602), and by the Chinese Academy of Sciences (Grant No. KJCX2.YW.W09-5).
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