Journal of Physics and Chemistry of Solids 72 (2011) 319–323

Contents lists available at ScienceDirect

Journal of Physics and Chemistry of Solids

0022-36

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/jpcs

Pnicogen-bridged antiferromagnetic superexchange interactionsin iron pnictides

Zhong-Yi Lu a�, Fengjie Ma a,c, Tao Xiang b,c

a Department of Physics, Renmin University of China, Beijing 100872, Chinab Institute of Physics, Chinese Academy of Sciences, Beijing 100190, Chinac Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e i n f o

Available online 16 October 2010

Keywords:

D. Electronic structure

D. Superconductivity

97/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jpcs.2010.10.065

esponding author.

ail addresses: zlu@ruc.edu.cn (Z.-Y. Lu), txiang

a b s t r a c t

The first-principles electronic structure calculations made substantial contribution to study of high Tc

iron-pnictide superconductors. By the calculations, LaFeAsO was first predicted to be an antiferromag-

netic semimetal, and the novel bi-collinear antiferromagnetic order was predicted for a�FeTe. Moreover,

based on the calculations the underlying mechanism was proposed to be Arsenic-bridged antiferro-

magnetic superexchange interaction between the next-nearest neighbor Fe moments. In this article, this

physical picture is further presented and discussed in association with the elaborate first-principles

calculations on LaFePO. The further discussion of origin of the magnetism in iron-pnictides and in

connection with superconductivity is presented as well.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Since the discovery of superconductivity in LaFeAsO by partialsubstitution of O with F atoms below 26 K [1], intense studies havebeen devoted to physical properties of iron-based pnictides bothexperimentally and theoretically. Unlike study of the cuprates, thefirst-principles electronic structure calculations made substantialcontribution to study of the pnictides from the very beginning. By thecalculations, we first predicted that LaFeAsO is an antiferromagneticsemimetal [2], and then predicted that the ground state of a�FeTe isin a novel bi-collinear antiferromagnetic order [3]. These predictionswere confirmed by the later neutron scattering experiments [4,5].Moreover, based on the calculations, we proposed [6] the fluctuatingFe local moments with the As-bridged antiferromagnetic supere-xchange interaction to account for the magnetism and tetragonal-orthorhombic structural distortion in LaFeAsO.

There are, so far, four types of iron-based compounds reported,showing superconductivity after doping or under high pressures,i.e., 1111-type Re FeAsO (Re ¼ rare earth) [1], 122-type A Fe2As2

(A¼Ba, Sr, or Ca) [7], 111-type B FeAs (B ¼ alkali metal) [8], and11-type tetragonal a�FeSeðTeÞ [9]. Like cuprates, all thesecompounds have a layered structure, namely they share thesame structural feature that there exist the robust tetrahedrallayers where Fe atoms are tetragonally coordinated by pnicogen orchalcogen atoms and the superconduction pairing may happen.A universal finding is that all these compounds are in a collinear

ll rights reserved.

@aphy.iphy.ac.cn (T. Xiang).

antiferromagnetic order below a tetragonal-orthorhombicstructural transition temperature [4,10] except for a�FeTe thatis in a bi-collinear antiferromagnetic order below a tetragonal-triclinic structural transition temperature [3,11,5].

The above finding raises the question on the microscopicmechanisms underlying the structural transition and antiferro-magnetic transition, respectively, and the relationship betweenthese two transitions. There are basically two contradictive viewsupon this question. The first one [12] is based on itinerant electronpicture, which thinks that there are no local moments and thecollinear antiferromagnetic order is induced by the Fermi surfacenesting which is also responsible for the structural transition due tobreaking the four-fold rotational symmetry. On the contrary, thesecond one is based on local moment picture. The J1� J2 Heisenbergmodel was, phenomenologically [13] and from strong electroncorrelation limit [14], respectively, proposed to account for theissue. Meanwhile, we proposed [6] the fluctuating Fe localmoments with the As-bridged antiferromagnetic superexchangeinteractions as the driving force upon the two transitions,effectively described by the J1� J2 Heisenberg model as well.

Our proposal embodies the twofold meanings shown by thecalculations [6]: (1) there are localized magnetic moments aroundFe ions and embedded in itinerant electrons in real space; (2) it isthose bands far from the Fermi energy rather than the bands nearbythe Fermi energy that determine the magnetic behavior ofpnictides, namely the hybridization of Fe with the neighbor Asatoms plays a substantial role. Here the formation of local momenton Fe ion is mainly due to the strong Hund’s rule coupling on Fe3d-orbitals [6]. In this sense, our proposal can be considered asHund’s rule correlation picture. We emphasize again that the

Z.-Y. Lu et al. / Journal of Physics and Chemistry of Solids 72 (2011) 319–323320

Arsenic atoms play a substantial role in our physical picture.Subsequently we successfully predicted from the calculations[6], based on this Hund’s rule correlation picture, that theground state of a�FeTe is in a bi-collinear antiferromagneticorder, which was later confirmed by the neutron scatteringexperiments [5,11].

Here we would like to address that our physical picture workswell on all iron-based pnictide superconductors, including LaFePO.In this article, we will show how our physical picture works onLaFePO through the elaborate first-principles electronic structurecalculations. We find that there is P-bridged next-nearest neighborFe–Fe superexchange antiferromagnetic interaction while thenearest neighbor Fe–Fe exchange interaction is very small inLaFePO. This results in the collinear antiferromagnetic order ofFe moments in the ground state, similar to the scenario in LaFeAsO.Moreover, we demonstrate that there also exits a small monocliniclattice distortion resulting from this superexchange magneticinteraction, again similar to the one in LaFeAsO [6].

2. Computational approach

In the calculations the plane wave basis method was used [15].We used the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof [16] for the exchange-correlation potentials. Theultrasoft pseudopotentials [17] were used to model the electron-ion interactions. After the full convergence test, the kinetic energycut-off and the charge density cut-off of the plane wave basis werechosen to be 600 eV and 4800 eV, respectively. The Gaussianbroadening technique was used and a mesh of 16�16�8k-points were sampled for the Brillouin-zone integration. In thecalculation, the internal atomic coordinates within the cell weredetermined by the energy minimization.

The crystal structure of LaFePO belongs to the tetragonalstructure and the space group is of P4/nmm. In the calculations,we adopted two unit cells, namely a� a� c crystal unit cell andffiffiffi

2p

a�ffiffiffi2p

a� c unit cell, respectively, as shown in Fig. 1. Here weused the experimental lattice parameters with a¼3.964 A andc¼8.512 A. We also checked the results with the optimized latticeparameters (a¼b¼3.932 A, c¼8.415 A) and there is no significantchange found.

3. Results and analysis

In order to derive the exchange interactions between the Fe–Femoments, we designed three different magnetic structures, includ-ing the nonmagnetic, the ferromagnetic, and the checkerboard(Neel) antiferromagnetic structures calculated in a� a� c crystalunit cell, and the collinear antiferromagnetic structure calculatedin

ffiffiffi2p

a�ffiffiffi2p

a� c unit cell. The calculations show that the energydifferences among the nonmagnetic, the ferromagnetic, and the

Fig. 1. Schematic top view of the FeAs or FeP layer in LaFeAsO or LaFePO. The small

dashed square is an a� a unit cell while the large dashed square is affiffiffi2p

a�ffiffiffi2p

a unit

cell. The Fe-spins are aligned in a collinear antiferromagnetic order.

checkerboard antiferromagnetic states are so small that these threestates can be considered as degenerate, similar to the ones inRef. [18]. The calculated electronic band structure and the Fermisurface are also the same as the ones in the previous calculations[19,18]. From the volumes enclosed by the Fermi surface sheets, wederive that both hole and electron carrier density is about2.75�1021/cm3. Thus LaFePO in the nonmagnetic phase behavesas a semimetal. However, the calculation upon the collinearantiferromagnetic structure in LaFePO shows that the groundstate of LaFePO is of the collinear antiferromagnetic ordering onthe Fe moments. The energy of the collinear antiferromagneticordered state is about 33 meV per formula unit cell lower than thenonmagnetic one. The magnetic moment around each Fe atom isfound to be nearly 2:0mB, smaller than the one in the calculation onLaFeAsO [6].

These magnetic states can be modeled by the followingfrustrated Heisenberg model with the nearest and next-nearestneighbor couplings J1 and J2

H¼ J1

X

/ijS

~Si �~Sjþ J2

X

5 ijb

~Si �~Sj, ð1Þ

whereas /ijS and 5 ijb denote the summation over the nearestand next-nearest neighbors, respectively. From the calculatedenergy data, we find that J1 � 0:5 meV=S2 and J2 � 7:5 meV=S2: Ifthe spin of each Fe ion S¼1, then J1 � 0:5 meV and J2 � 7:5 meV (thedetailed calculation is referred to Ref. [6, Appendix]). In comparisonwith LaFeAsO [6], the J1 and J2 are smaller by 7 times for LaFePO.

Fig. 2(a) and (b) show the calculated charge distributions on the(0 0 1) plane crossing four Fe atoms and the plane (1 1 0) crossing Fe–P–Fe atoms, respectively. We see that there is a strong bondingbetween Fe and P ions. This indicates that the next-nearest neighborcoupling J2 is induced by the superexchange bridged by P 3p-orbitals.This superexchange is antiferromagnetic because the intermediatedstate associated with the virtual hopping bridged by P ions is a spinsinglet. The charge distribution between the two nearest Fe ions isfinite but almost none between the next-nearest Fe ions. Thus there isa direct exchange interaction between the two nearest neighboring Fespins. Because of the exists of the strong Hund’s coupling induceddirect ferromagnetic exchange interaction and the indirect antiferro-magnetic superexchange interaction bridged by P 3p-orbitals, com-bined with the degeneration of different magnetic states in a� a� c,the nearest neighbor coupling J1 is nearly zero.

For the collinear antiferromagnetic structure, our calculationsshow that there is further small structural distortion that shrinksalong the spin-parallel alignment (x-axis in Fig. 1) and expandsalong the spin-antiparallel alignment (y-axis in Fig. 1), causingstructural transition from tetragonal structure to monoclinicstructure. This is easily understandable that the direct exchangefavors a shorter Fe–Fe separation while the superexchange favors alarger Fe–P–Fe angle. It turns out that the angle g in ab-plane(see Fig. 1) is no longer rectangular and optimized to be 90.571 witha small energy gain of about 3 meV, similar to the finding in

Fig. 2. Calculated charge density distribution of LaFePO in the (0 0 1) plane crossing

Fe–Fe atoms (a) and in the (1 1 0) plane crossing Fe–P–Fe atoms (b).

Z.-Y. Lu et al. / Journal of Physics and Chemistry of Solids 72 (2011) 319–323 321

LaFeAsO [6]. Such a small distortion is found to have a small effectupon the electronic structure and the Fe moments. Here weemphasize that our calculations show that the driving forceupon this structural distortion is nothing but the magneticinteraction. More specifically it is the superexchange interactionJ2 that breaks the rotational symmetry to induce both the structuraldistortion and the Fe-spin collinear antiferromagnetic ordering.

Fig. 3 shows the total and projected density of states of LaFePO inthe collinear antiferromagnetic state. The most of the states aroundthe Fermi level are of Fe 3d characters. The corresponding electronicdensity of states is 1.45 states per eV per formula unit cell, which isless than the one of nomagnetic state (¼5.78 states per eV performula unit cell). The specific heats are evaluated as 0.84 mJ/(K2 mol)and 6.86 mJ/(K2 mol) for the collinear antiferromagnetic stateand nonmagnetic state, respectively, consistent with the specificheat measurement [20]. The paramagnetic susceptibility in thenonmagnetic state is about 1:18� 10�9 m3=mol.

Fig. 4 shows the collinear antiferromagnetic electronic bandstructure and the Fermi surface. Unlike the ones in thenonmagnetic state [19], there are now only two bands crossingthe Fermi level, one small hole cylinder along GZ and two smallelectron cylinders formed between G and X . From the volumesenclosed by these Fermi surface sheets, we find that the hole carrierdensity is about 1.34�1020/cm3 and the electron carrier density isabout 1.21�1020/cm3.

Fig. 3. Calculated total and orbital-resolved partial density of states (spin-up part) of

Fig. 4. (a) The electronic band structure of LaFePO in the collinear antiferromagnetic state

Fermi surface sheets: a hole-type cylinder alongGZ and two electron-type pockets betwe

to the antiparallel-aligned moment line.

4. Origin of the magnetism

By projecting the density of states onto the five 3d orbitalsof Fe in the collinear antiferromagnetic state of LaFePO (Fig. 5), wefind that the five up-spin orbitals are almost filled and the fivedown-spin orbitals are only partially filled. Moreover, the down-spin electrons are nearly uniformly distributed in these five 3dorbitals. This indicates that the crystal field splitting imposedby P atoms is relatively small and the Fe 3d-orbitals hybridizestrongly with each other. As the Hund rule coupling is strong,this would lead to a magnetic moment formed around each Featom, as found in our calculation. Actually, this is a universalfeature for all iron pnictides, as we first found in LaFeAsO [6]. Thepolarization of Fe magnetic moments is thus due to Hund’s rulecoupling.

In low temperatures, the Fe moments will interact with each otherto form an antiferromagnetic ordered state. These ordered magneticmoments have been observed by elastic neutron scattering and otherexperiments in LaFeAsO and other pnictides [4,11]. However, they arenot exactly the moments obtained by the density functional theory(DFT) calculations, as we indicated first for LaFeAsO [6]. This isbecause the DFT calculation is done based on a small magneticunit cell and the low-energy quantum spin fluctuations as well astheir interactions with itinerant electrons are frozen by the finiteexcitation gap due to the finite-size effect. Thus the moment obtained

LaFePO in the collinear antiferromagnetic state. The Fermi energy is set to zero.

. The Fermi energy is set to zero. (b) The correspondent labels in Brillouin zone. (c) The

enG and X .GX corresponds to the parallel-aligned moment line andGX corresponds

Fig. 5. Calculated total and projected density of states at the five Fe-3d orbitals around one of the four Fe atoms in the collinear antiferromagnetic state unit cell. The Fermi

energy is set to zero.

Z.-Y. Lu et al. / Journal of Physics and Chemistry of Solids 72 (2011) 319–323322

by the DFT is the bare moment of each Fe ion. It should be larger thanthe ordered moment measured by the neutron scattering and otherexperiments. Our calculations show that the bare magnetic momentaround each Fe atom is about 2mB. Moreover, the frustration betweenthe J1 and J2 terms will further suppress the antiferromagneticordering at the two Fe-sublattices, each connected only by the J2terms. All of these together will reduce strongly the average magneticordered moment around each Fe measured by experiments, like theneutron scattering.

In high temperatures, there is no net static moment in theparamagnetic phase due to the thermal fluctuation, but the baremoment of each Fe ion can still be measured by a fast local probe likeESR (electron spin resonance). Recently the bare moment of Fe hasbeen observed in the paramagnetic phase of BaFe2As2 by the ESRmeasurement [21]. The value of the moment detected by ESR is about2:222:8mB in good agreement with our DFT result [6], which is butsignificantly larger than the ordered moment in the antiferromagneticphase.

Fig. 6. Various critical superconductivity temperature Tc versus antiferromagnetic

superexchange interaction J2 for iron-based pnictides or chalcogenides with doping

or under high pressures. Tc are taken from Refs. [1,7–9,22–30].

5. Discussion in connection with superconductivity

As we see above, the next nearest neighbor antiferromagneticsuperexchange interaction affects strongly the magnetic structure ofthe ground state. The magnetic fluctuation induced by the antiferro-magnetic superexchange interactions should be responsible for thesuperconducting pairing. Upon doping, the long range ordering will besuppressed. However, we believe that the remanent antiferromagneticfluctuation will survive, similar as in cuprate superconductors. We plotthe superconducting critical temperatures Tc versus J2 for the ironpnictide compounds with doping carriers or by applying high pressuresin Fig. 6. As we see, the maximum critical temperatures Tc are inproportion to the next-nearest neighbor antiferromagnetic supere-xchange interaction J2. This suggests that there would exist an intrinsicrelationship between the superconductivity and the Pnicogen-bridgedantiferromagnetic superexchange interactions.

The above analysis suggests that there are some similaritiesbetween pnictide and cuprate superconductors. Here we wouldalso like to further address the difference between them is thatthe undoped cuprate is an antiferromagnetic Mott insulatorwhile the undoped pnictide is an antiferromagnetic semimetal.In particular, in undoped pnictide there already coexist bothlocal magnetic moments and itinerant electrons. This is similaras in the doped cuprate superconductors. In cuprate superconduc-tors, the low-energy physics can be effectively described by the t� J

model, in which the coupling between local spins is the

antiferromagnetic interaction and the doped holes or electronscan hop on the lattice.

For iron-based pnictides, the low energy spin dynamicscan be approximately described by an antiferromagnetic Heisen-berg model (Eq. (1) with the nearest and the next-nearestneighbor exchange interactions. However, the Fe spin (ormagnetic moment) is not quantized since the electronsconstituting the moment can propagate on the lattice, like inhole or electron doped high Tc superconductivity cuprates. Besidesthese superexchange interactions, the on-site Hund’s rule couplingamong different Fe 3d orbitals is important. This is because thecrystal splitting of Fe 3d levels is very small and the spins of Fe 3delectrons are polarized mainly by this interaction. Thus we believethat the low-energy physical properties of these iron-basedpnictides can be approximately described by the followingeffective Hamiltonian

H¼X

/ijS,abtabij cyiacjb�JH

X

i,aab

~Sia �~Sibþ J1

X

/ijS,ab

~Sia �~Sjbþ J2

X

0ijT,ab

~Sia �~Sjb,

ð2Þ

where /ijS and 0ijT represent the summation over the nearestand the next-nearest neighbors, respectively. a and b are the

Z.-Y. Lu et al. / Journal of Physics and Chemistry of Solids 72 (2011) 319–323 323

indices of Fe 3d orbitals. cyiaðciaÞ is the electron creation(annihilation) operator.

~Sia ¼ cyia~s2

cia ð3Þ

is the spin operator of thea orbital at site i. The total spin operator atsite i is defined by ~Si ¼

Pa~Sia. In Eq. (2), JH is the on-site Hund’s

coupling among the five Fe 3d orbitals. The value of JH is generallybelieved to be about 1 eV. tabij are the effective hopping integralsthat can be determined from the electronic band structure in thenonmagnetic state [31]. It has been shown that the nearest and thenext-nearest neighbor exchange coupling constants J1 and J2 inEq. (2) can be calculated from the relative energies ofthe ferromagnetic, square antiferromagnetic, and collinearantiferromagnetic states with respect to the nonmagnetic state.For the corresponding detailed calculations, please refer to theappendix in our paper in Ref. [6]. Here we have assumed that thecontribution of itinerant electrons to the energy is almostunchanged in different magnetically ordered states. Since thebare tabij and JH can be considering independent of magneticstructures, the relative energies between different magneticstates are not affected by itinerant electrons.

6. Conclusion

In conclusion, we have presented the first-principles calculationof the electronic structure of LaFePO. We find that like the situationin LaFeAsO, there are the next-nearest neighbor antiferromagneticsuperexchange interactions bridged by P 3p-orbitals, which aremuch larger than the nearest neighbor ones. This gives rise to thecollinear antiferromagnetic ordering of Fe spins in the ground stateas observed in LaFeAsO by the neutron scattering. However,because of much smaller exchange interactions J2 and J1 incomparison with the ones in LaFeAsO, the corresponding transitiontemperatures should be very low in LaFePO, very likely less than10 K scaled according to LaFeAsO. This needs further experiment totest and verify. Based on the analysis of electronic and magneticstructures, we proposed that the low-energy physics of ironpnictides can be effectively described by the t� JH� J1� J2 model,defined by Eq. (2).

Acknowledgements

This work is supported by National Natural Science Foundation ofChina and by National Program for Basic Research of MOST, China.

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