Block Antiferromagnetism and Checkerboard Charge Ordering in Alkali-doped Iron Selenides R 1−x Fe 2−y Se 2
BBlock Antiferromagnetism and Checkerboard Charge Ordering in Alkali-doped IronSelenides R − x Fe − y Se Wei Li, Shuai Dong, Chen Fang, and Jiangping Hu
4, 3, ∗ Department of Physics, Fudan University, Shanghai 200433, China Department of Physics, Southeast University, Nanjing 211189, China Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China (Dated: October 30, 2018)By performing first-principles electronic structure calculations and analyzing effective magneticmodel of alkali-doped iron selenides, we show that the materials without iron vacancies should ap-proach a novel checkerboard phase in which each four Fe sites group together in tetragonal structure.The checkerboard phase is the ground state with a block antiferromagnetic (AFM) order and a smallcharge density wave order in the absence of superconductivity. Both of them can also coexist withsuperconductivity. The results explain mysterious 2 × PACS numbers: 74.70.-b, 74.25.Jb, 74.25.Ha, 74.20.Mn
The newly discovered alkali-doped iron selenidesuperconductors[1–3] have attracted much research at-tention because of several distinct characters that arenoticeably absent in other iron-based superconductors,such as the absence of hole pockets at Γ point of Bril-louin zone in their superconducting (SC) phases[4–6] andAFM ordered insulating phases[7–9] with very high N´eeltransition temperatures in their parental compounds[3].Due to these distinct physical characters from their pnic-tide counterparts, R − x Fe − y Se are expected to beideal grounds to test theoretical models of iron-basedsuperconductors. Models based on different mecha-nisms have suggested different pairing symmetries for R − x Fe − y Se : weak coupling approaches based onspin-excitation mediated pairing predict a d -wave pair-ing symmetry[10–13], strong coupling approaches[14–16]which emphasize the importance of next nearest neighbor(NNN) AFM local exchange coupling suggest the pair-ing symmetry is a robust s -wave, not different from the S ± -wave symmetry obtained in their pnictide counter-parts and models with orbital fluctuation mediated pair-ing suggest a S ++ -wave pairing for both iron selenide andpnictide materials[17].While the iron-selenide superconductors have gener-ated considerable excitement, there are deep confusionsregarding the delicate interplay between Fe vacancies,magnetism and superconductivity. Many latest experi-mental results in R − x Fe − y Se indicate that the insu-lating AFM and SC phases are phase separated[18–23].In particular, the recent scanning tunneling microscopy(STM) measurements on K − x Fe − y Se clearly suggestphase separation[21, 22]. The material was shown tobe phase separated into iron vacancy ordered regionsand iron vacancy free regions. The former is insu- lating and shows a √ × √ √ × √ √ × √ × √ × √ × a r X i v : . [ c ond - m a t . s up r- c on ] O c t J J’ J J’ J J X’ X (a) (b)
FIG. 1: (Color online) (a) The crystal structure of KFe Se compound in our calculation. The system consists of K(royal), Fe (dark yellow) and Se (green) atoms; (b) Schematiccheckerboard lattice structure and spin ordering pattern inthe BAF state. The lattice distortion is labeled by latticeconstants X and X (cid:48) . The magnetic exchange couplings arealso indicated. cancy ordered state and the STM and ARPES exper-imental results[6, 21]. This study clarifies the missinglink between AFM and SC phases and essentially unifiesthe understanding of various observed phases[19]. Results from LDA calculations
We start with the fol-lowing simple question: if the system is free of ironvacancies, what should be the ground state if it doesnot become SC? To answer this question, we performthe first principles calculation to investigate the groundstate of an iron vacancy free domain. We calculatedthe energy of a number of different possible magneti-cally ordered states, including non-magnetic (NM), ferro-magnetic (FM), collinear-AFM (CAF, the state observedin iron-pnictides[25, 26]), bicollinear-AFM (BCAF, thestate observed in FeTe)[27–30] and BAF whose pattern isshown in Fig. 1(b) where four Fe sites group to form a su-per cell. All these calculations were performed using theprojected augmented wave method[32] as implementedin the VASP code[33], and the Perdew-Burke-Ernzerhof(PBE) exchange correlation potential[34] was used. A500eV cutoff in the plane wave expansion ensures thecalculations converge to 10 − eV. For the BAF state, allatomic positions and the lattice constants were optimizeduntil the largest force on any atom was 0.005eV/˚A. Weused a 9 × × U , our LDA resultsshow that the BAF ordered state is metallic. Howeverif a finite U > . U calcula-tion shows that the state becomes insulating. There arestrong lattice distortion in the BAF state. The latticeconstant X (between two nearest sites in one supercell) TABLE I: Geometric, energetic and magnetic properties ofKFe Se . Results in the NM/FM/AFM/CAF/BCAF/BAFand optimized BAF configurations using fully optimizedstructures are all shown. ∆ E is the total energy differenceper iron atom in reference to the unoptimized experimentalstructure[1], and m Fe is the local magnetic moment on Fe.KFe Se ∆E (eV/Fe) a(˚A) c(˚A) m Fe ( µ B )NM 0 3.9136 14.0367 0FM -0.2400 3.9136 14.0367 2.781AFM -0.2384 3.9136 14.0367 2.135CAF -0.3510 3.9136 14.0367 2.446BCAF -0.3159 3.9136 14.0367 2.556BAF -0.3127 3.9136 14.0367 2.552BAF(opt.) -0.3568 3.8553 14.4099 2.635 and X (cid:48) (between two supercells) as labeled in Fig. 1(b)are X =2.6˚A, X (cid:48) =2.85˚A for U = 0eV and X =2.59˚A, X (cid:48) =3.03˚A for U = 2eV. This lattice distortion is com-parable to the lattice distortion in the √ × √ . Fe . Se phase[7].From Table I, the energies of the BCAF state and theBAF state are almost degenerate in the unoptimized lat-tice tetragonal structure. The actual ground state, infact, depends on the optimization of lattice structure.In the previous first principle calculations, the BCAFstate was shown to be the lowest energy state in the op-timized lattice which has monoclinic distortion[35, 36].The monoclinic distortion and the BCAF order can bestrongly coupled with each other because they break thesame lattice rotational symmetry. Such a strong cou-pling was observed in FeTe in which a single strong firstorder phase transition where both the BCAF orderingand the tetragonal-monoclinic distortion take place[27].However, in KFe Se , no monoclinic lattice distortion hasbeen observed. Without allowing the monoclinic latticedistortion, the BAF state becomes the ground state.From the similar symmetry analysis, the BAF statemust strongly couple to the lattice distortion shown inFig. 1(b). The lattice distortion quadroples the lat-tice unit cell to form a checkerboard pattern. In such acheckerboard lattice, the charge ordering can take place.If we calculate the electron density distribution aroundSe atoms in the Se-layer above the iron-layer, a chargeordering on the Se-layer is observed as shown in Fig. 2,which was observed in recent STM experiment[21]. Magnetic models
Now we discuss the effective magneticmodel that can interpret above calculation results. It hasbeen shown a magnetic exchange J - J - J - K model[38]where J , J and J are the nearest neighbor (NN), NNN,and the next NNN (NNNN) exchange couplings respec-tively, and K is a spin biquadratic coupling term betweentwo nearest neighbor sites is a good approximation todescribe iron-chalcogenides when the lattice distortion is FIG. 2: (Color online) Charge density distribution ( e/bohr )in the (001) plane crossing the first Se atoms layer with BAForder state within the LDA+ U ( U = 2eV) calculations. ignored[29, 30, 37, 38]. In Table I, the energies of theBCAF and BAF states in the unoptimized lattice tetrag-onal structure are almost degenerate. This degeneracyis a strong support of the model since these two statesare exactly degenerate in the J - J - J - K classical spinmodel[38]. In the magnetically ordered state, the lat-tice distortion takes place and the tetragonal symmetryis broken. The nearest neighbor exchange coupling J can take two different values J and J (cid:48) as shown in Fig.1(b). The biquadratic coupling K can be decoupled andtreated as an effective difference between J and J (cid:48) aswell[38, 39]. In general, the NNN J can also take twodifferent values, J and J (cid:48) as also shown in Fig. 1(b).However, as being proved in other iron-based supercon-ductors, the NNN coupling J is rather robust againstlattice distortion. The difference between J and J (cid:48) israther small. Therefore, the effective magnetic exchangemodel in magnetically ordered state is given by J - J (cid:48) - J - J with J being strongly FM and J , both being AFM. J (cid:48) can be weak FM or weak AFM. The similar model hasbeen shown to describe the magnetism of the √ × √ . Fe . Se phase[9, 37]. Therefore,while the exact values of the magnetic exchange couplingscan not be accurately obtained from LDA calculations,since the lattice distortions in both cases are similar, itis reasonable to believe that these values should not betoo different from those of K . Fe . Se which has beenmeasured by fitting neutron scattering experiments[9].The measured values, which are specified in Fig. 3,also give the BAF order ground state. The saved en-ergy from magnetic exchange coupling per site is givenby ( − J + 2 J + J (cid:48) ) S . This energy is sightly smallerthan the saved magnetic energy in the vacancy orderedK . Fe . Se [9, 37]. The spin wave dispersion and theimaginary part of dynamic spin susceptibility of the BAFphase are shown in Fig. 3.There is another interesting prediction if the samemagnetic model describes both K . Fe . Se and k x k y E=20meV 0.20.40.60.811.2 k x k y E=60meV 0.20.4 k x k y E=70meV 0.10.20.3k x k y E=110meV 0.20.4 k x k y E=120meV 0.10.20.30.40.5 k x k y E=130meV 0.20.4k x k y E=210meV 0.10.20.3 k x k y E=220meV 0.20.4 k x k y E=230meV 0.20.4− π − π − π − π − π − π − π − π − π πππ πππ πππππππ π ππππ FIG. 3: (Color online) Spin wave dispersion (up) and imagi-nary part of dynamic susceptibility (down) for the BAF statewith J - J (cid:48) - J - J model at J = − J (cid:48) = 15meV, J = 14meV, J = 9meV, taken from Ref. 9. The profileof the imaginary part of the dynamic susceptibility is plottedat various energies with an energy resolution of 5meV, and itis given in arbitrary unit. K x Fe Se . As mentioned before, K . Fe . Se andK x Fe Se are two phase separated regions. If the sameeffective magnetic model describes both structures, it isvery interesting to inquire into the magnetic configura-tions near the boundary of these two structures. We per-form a Monte Carlo (MC) simulation on the J - J - J - K model[38] to address this problem. A simple numericalsimulation, which includes a standard Markov Chain MCsimulation followed by a zero-temperature relaxation pro-cess, is performed to qualitatively investigate the mag-netic orders near phase boundaries. A two-dimensionalspin lattice [ L x × ( L y + L y )] is used with periodic bound-ary conditions (PBCs). Vacancies with the √ × √ L y regions for the K . Fe . Se phase. A general result we obtained as shown in Fig.4 is that the spin directions between L y and L y re- (a) MC snapshot (b) after relaxation FIG. 4: (Color online) (a) A typical MC snapshot (after 3 × MC steps) of the classical J - J - J - K magnetic model.(b) The spin pattern after the zero- T relaxation. In (a) and(b), spins in regions without vacancies and with vacanciesare in blue and red, respectively. Black circles denote thevacancies. gions are noncollinear. Since experimentally, the orderedAFM moment is along c-axis in K . Fe . Se [7], this re-sult suggests that the ordered moment in the BAF statemust be in the plane. This non-collinearity stems fromthe presence of vacancies and intrinsic magnetic frustra-tion among the magnetic exchange couplings, similar tothe study in the frustrated J - J model[40]. Recent STMresults have provided an evidence supporting this predic-tion. It was shown that the magnetic moment inducedby individual vacancy in the SC state is indeed in theplane[22].The phase separation between the vacancy orderedBAF state and the SC state has blurred the interplay be-tween magnetism and superconductivity in alkali-dopediron selenide. The above results also clarify the connec-tion. In iron-pnictides, as increasing doping suppressesCAF order, superconductivity develops. The magneticorder is able to coexist with superconductivity in a smalldoping region[41]. Even in the region where the magneticorder is completely suppressed, orthorhombic lattice dis-tortion which couples the fluctuating short range CAForder[42, 43], can survive and coexist with superconduc-tivity. Our result suggests that the similar physics cantake place in R − x Fe − y Se . The absence of iron vacan-cies in the SC state suggests that the true SC materialhas a chemical formula R − x Fe Se . The parent state ofthis material should be a BAF state. Increasing dopingsuppresses the BAF state and leads to SC. While it is stilldifficult to determine whether the BAF and SC can co-exist, we can safely argue that, similar to iron-pnictides,a lattice distortion as shown in Fig. 1(b) that couples tothe short range BAF fluctuation should be able to coexistwith SC.This picture provide explanations to many puz- zling phenomena observed in alkali-doped iron selenides R − x Fe − y Se . First, in ARPES measurements, a weakbut large electron pocket at Γ point was observed[6]. Thispocket is almost identical to the electron pockets at M point, suggesting the electron pocket is a folded pocketdue to translational symmetry breaking in the SC state.Moreover, in recent STM experiment[21], a 2 × × √ × √ R − x Fe − y Se has a checkerboard phase in which eachfour Fe sites group together in a tetragonal structure.The checkerboard phase approaches a BAF order in theabsence of superconductivity. The phase also exhibitssmall charge density modulation on Se sites. Magneticproperties related to this state are calculated. Combiningwith the strong experimental evidence of phase separa-tion between vacancy ordered and vacancy free phases,we suggest the checkerboard phase is the parent state ofthe superconductor. Acknowledgement:
We thank H. Ding, D. L. Feng, P.C. Dai, N. L. Wang, H. H. Wen, X. Chen, Q. K. Xue, T.Xiang and Y. Y. Wang for useful discussion. S.D. wassupported by the 973 Projects of China (2011CB922101),NSFC (11004027) and NCET (10-0325) ∗ Electronic address: [email protected][1] J. Guo et al. , Phy. Rev. B. , 180520(R) (2010).[2] M. Fang et al. , Europhys. Lett. , 27009 (2011).[3] R. H. Liu et al. , Europhys. Lett. , 27008 (2011).[4] Y. Zhang et al. , Nature Materials , 273 (2011).[5] X. Wang et al. , Europhys. Lett. , 57001 (2011).[6] D. Mou et al. , Phys. Rev. Lett. , 107001 (2011).[7] W. Bao et al. , Chinese Phys. Lett. , 086104 (2011).[8] W. Bao et al. , arXiv:1102.3674 (2011).[9] M. Wang et al. , arXiv:1105.4675 (2011).[10] C. Platt, R. Thomale, and W. Hanke, Ann. Phys. (Berlin) , 638 (2011).[11] S. Maiti et al. , arXiv:1104.1814 (2011).[12] Y.-Z. You, H. Yao, and D.-H. Lee, Phys. Rev. B ,020406(R) (2011).[13] T. A. Maier et al. , Phys. Rev. B , 100515(R) (2011).[14] C. Fang et al. , Phys. Rev. X , 011009 (2011).[15] J. Hu and H. Ding, arXiv:1107.1334 (2011).[16] R. Yu et al. , arXiv:1103.3259 (2011).[17] T. Saito et al. , Phys. Rev. B , 140512(R) (2011).[18] C. H. Li et al. , Phys. Rev. B , 184521 (2011).[19] Y. J. Yan et al. , arXiv:1104.4941 (2011).[20] F. Chen et al. , arXiv:1106.3026 (2011).[21] P. Cai et al. , arXiv:1108.2798 (2011).[22] W. Li et al. , arXiv:1108.0069 (2011).[23] R. H. Yuan et al. , arXiv:1102.1381 (2011).[24] Z. Wang et al. , Phys. Rev. B , 140505 (2011).[25] J. Zhao et al. , Nature Physics , 55 (2009).[26] J. Zhao et al. , Phys. Rev. Lett. , 167203 (2008).[27] W. Bao et al. , Phys. Rev. Lett. , 247001 (2009).[28] S. Li et al. , Phys. Rev. B , 054503 (2009).[29] F. Ma et al. , Phys. Rev. Lett , 177003 (2009). [30] C. Fang, B. Andrei Bernevig, and J. Hu, Europhys. Lett. , 67005 (2009).[31] O. J. Lipscombe et al. , Phys Rev Lett , 057004(2011).[32] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[33] G. Kresse and J. Furthmuller, Phys. Rev. B , 11169(1996).[34] J. P. Perdew, K. Burke, and M. Erznerhof, Phys. Rev.Lett. , 3865 (1996).[35] X. Yan, et al , Phys. Rev. B 84, 054502 (2011).[36] X. Yan, et al , Phys. Rev. B 83, 233205 (2011).[37] C. Fang et al , arXiv:1103.4599 (2011).[38] J. Hu et al , arXiv:1106.5169 (2011).[39] A.L. Wysocki, K.D. Belashchenko and V. P. Antropov,Nature Physics, 7 485 (2011).[40] B. Xu et al , arXiv:1104.1848 (2011).[41] D. C. Johnston, Adv. Phys. , 803 (2010).[42] C. Fang et al. , Phys Rev B , 224509 (2008).[43] C. Xu et al , Phys. Rev. B,78