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Spin and charge ordering in three-leg ladders in oxyborates. E. Vallejo and M. Avignon1 1Laboratoire d’Etudes des Propriétés Electroniques des Solides (LEPES)-Centre National de la Recherche Scientifique (CNRS), BP 166, F-38042 Grenoble Cedex 9, France. (Dated: January 31, 2014) We studythe spin ordering within the3-leg ladders present in theoxyborate Fe3O2BO3 consist- ing of localized classical spins interacting with conduction electrons (one electron per rung). We also consider the competition with antiferromagnetic superexchange interactions to determine the magneticphasediagram. Besideaferromagneticphasewefind(i)aphasewithferromagneticrungs 4 ordered antiferromagnetically (ii) a zig-zag canted spin ordering along the legs. We also determine 1 the induced charge ordering within the different phases and the interplay with lattice instability. 0 Ourmodelisdiscussedinconnectionwiththelatticedimerizationtransitionobservedinthissystem, 2 emphasizing on therole of themagnetic structure. n PACSnumbers: 75.10.Hk;71.45.Lr a J 9 The ordering of the local spins interacting with con- hopping among the Fe-d5 configurations have been esti- 2 ductionelectronsremainsanimportantproblemandhas mated, leading to strongly interacting spin units of the ] become very active in the context of manganites. The Fe-triads in which all nearest-neighbor (n.n) spins are l coupling can be antiferromagnetic as in heavy-fermions antiferromagnetically coupled both above and below the e - systems or Kondo insulatorsor ferromagneticas a result structural transiton temperature10 in contradiction with tr of Hund’s coupling in manganites. This gives rise to the recentneutronresults8. Inaddition,itcanbeshownthat s general double exchange (DE) interactions1 favoring a an homogeneous magnetic phase is not compatible with . t ferromagnetic background of local spins. This ferromag- theobservedchargedistributiononthedifferentFesites. a m netic tendency is expected to be thwarted by antiferro- Inthisletter wewillshowthatthe inclusionofthe inter- magnetic superexchange (SE) interactions between the action between intinerant electrons and local spins will - d localized spins leading to interesting and unusual mag- drastically improve this picture. n netic states. Instead of the canted states conjectured by Since the local spins −→S are fairly large S = 5/2 we i o de Gennes2, spinorderingconsistingof ferromagneticis- will treat them as classical spins specified by their polar c lands coupled antiferromagnetically has been identified angles θ and ϕ (0 < θ < π,0 < ϕ < 2π) defined as [ i i i i forvariouscommensuratefillingsbothforS =1/2quan- usual with respect to a z-axis taken as the spin quanti- 1 tum spins in one dimension3 and classical spins in two zation axis of itinerant electrons. Rotating the itinerant v dimensions4. Carriers are found to be localized in the electronquantizationaxisoneachsitetomakeitparallel 7 ferromagneticislandsgivingrisetobondorderingandas 7 to −→Si, one gets the rotated electron operators with spin 6 a consequence leads to charge ordering. opposite to the local spin c+(c ) in terms of the origi- i i 1.7 vmiduTelthaeenvleiudodeuwnscisgepitoinefotaxhnyids-bcmoheraacrhtgeaensoyisrsmdteemrfionrFgetrh3eOesu2eBlxtiOisnt3genmfrcaoeymopftrhsoie-- en−aliϕeilseicnt(rθoin/2o)pde+ir.atTohrserdo+iσta(dteiσd)elaesctcr+ion=sarceosin(θdie/e2d)ds+ip↓in−- 0 less electrons. T↑he effective hopping between these elec- 4 competition between DE and SE. Fe-ludwigite contains tronsantiparalleltolocalspinsatsitesiandjistherefore :1 saunbdupnritesseinntsthaenfoinrtmeroefst3in-lgegstlraudcdteurrsa(l3aLnLd)cohfaFrgeecaotridoenrs- given by tei,j = tν(cosθ2i cosθ2j +e−i(ϕi−ϕj)sinθ2i sinθ2j), v t = t ,t being the nearest neighbor (n.n) hopping in- ing transition at T 283K, such that long and short ν a c i c X bonds on the rungs a≈lternate along the ladder axis5. As tegrals on the rungs and along the axis of the ladder. So, to describe the magnetic structure, we represent r evidencedbyMo¨ssbauerstudies6,7 andX-raydiffraction8 a each rung can be viewed as three Fe3+ ions (triad) with the interaction between the Fe3+ localized spins −→Si and high-spin S = 5/2 local spins sharing an extra itinerant the itinerant electrons by the tight-binding Hamiltonian electron. The charge distribution among the triads is a together with SE interactions among the local spins key issue. Spin ladders have recently attracted a con- soifdaersapbilneliandtdereerstcobuuptlewdewhitahvecohnedruectainoninetleercetsrtoinnsg. cTahsee H =−X(tei,jc+i cj +h.c.)+XJij−→Si·−S→j ij ij coupling is similar to the one encountered in Fe double- h i h i perovskite systems9. In the Fe3+ d5 configurationall or- ij represents n.n sites. We further assume that this bitals being occupied in one spin channel, itinerant elec- bahndiis non-degenerate, therefore the band filling is n= trons can hop to a site i only if its spin is antiparallel to 1/3. We take the simple situation in which all the spins thelocalspin−→S . ThisisindeedequivalenttoDEwithan are in the same plane. This simplification is inspired by i effectiveantiferromagneticandinfiniteexchangeintegral. Monte Carlo simulations on 2D systems in which non- Antiferromagnetic SE interactions resulting from virtual coplanar spin configurations never seem to appear4 and 2 0,25 t = 1.2 0,20 0,15 2 JcS / tc Ia 0,10 0,05 F Ib 0,00 0,00 0,05 0,10 0,15 0,20 0,25 2 JaS / tc FIG. 2: Phase diagram as function of JaS2 and JcS2, for a typical value of t = ta = 1.2. The ditffcerent phatsces are FIG.1: (a)Magneticstructureofthe3-legladderLudwigite. tc described in thetext. The five angles α,β,γ,δ,ǫ gives the orientation of the spins on the six sites i=1−6 unit cell. (b) Magnetic structureof theIa phase. Thisstructurecanpresentazigzag modulation insulating. Forsymmetry reasonthe same occursalsoat of the angles θ1 and θ2. 2/3-filling. This gap depends on the values of the different angles and can be direct or indirect. In phase A the hopping is totally suppressed in the c- is consistent with neutron scattering results8. All copla- directionandthedipersionreducestothreeenergylevels. nar phases being degenerate, we choose the plane of the The particular phase A with fully ferromagnetic rungs I ladder, taking θ = π/2 and the hopping terms simply (β = δ = 0) is encountered at lower J . This phase is i a becomes tei,j = t2ν(1+e−i(ϕi−ϕj)). Guided by the peri- in qualitative agreement with the magnetic structure re- odicity 2c of the low temperature distorted phase5, we cently proposedfromneutronexperiments at82K8.Itis consider a unit-cell containing two rungs. We define the indeedverysimilartoaphasealreadyfoundwithMonte- magnetic structure by the five angles α,β,γ,δ,ε giving Carlocalculationsinthe2Dmodel4.AtlargerJa,canting the orientationofthe spinsonthe six sitesi=1 6unit occurswithintherungswithtwodifferentanglesβ,δ,we cellasshowninFig. 1-a. Jij =Ja,Jc areSEinte−ractions call this phase AII. in the two directions. Phase I presents very interesting simple structures as, for example, the phase I (α,β,γ = π,δ = β,ε = α) The kinetic energy term favors a ferromagnetic ar- a − − which can be defined in terms of only two angles θ and rangement of the local spins which competes with the 1 θ (β = θ ,α = π θ θ ). It is AF along the central SE, leading to a varietyof complex structures. After the 2 1 − 1− 2 leg so that no hopping is taking place along this leg. Fourier transformation, the dispersion of the conduction Thisstructurepresentsazig-zagmodulationoftheangles electrons is obtained from the tight-binding matrix with θ andθ and,consequentlyofthehoppingt t asshown the wave-vectork in the c-direction, π/2c<k <π/2c. 1 2 1, 2 In the general case, it consists of si−x bands ǫ (k), in Fig. 1-b. A phase called Ib tends to a ferromagnetic the values ǫ (k) are increasing from i = 1 to 6.i=F1−or6the behavior along c-direction with γ <π. i As soonasthe centralleg is AF(γ =π) the bands are band-filling n = 1/3 the two lowest bands ǫ only are 1,2 two-fold degenerate with gaps at k = π. The disper- occupied. We minimize the total energy with respect to ±2c sion of the bands are ǫ(k)=0 and the five angles α,β,γ,δ,ε . Fig. 2 shows the phase diagram as funct{ion of J S2}/t and J S2/t for a typical a c c c 1 value t = ta = 1.2 roughly estimated from the different ǫ(k) = [1+t2+ t2(cosθ1+cosθ2) cos(θ1+θ2)+ tc ± 2 − Fe-Fe distances in the triad and along the legs. Besides 1 the fully ferromagnetic (F) state characterized by the +(1 cos(θ1+θ2))cos2kc]2 . − uniform angles α = β = γ = δ = ε = 0, when J and a The lower band is filled, precisely for n = 1/3, lower- J are not too large J S2/t .0.07 and J S2/t .0.13, c c c a c ing the kinetic energy to stabilize this phase. The total wefindtwootherphases(i)atlargerJ aphaseAwhich c energy per rung E can be expressed as is antiferromagnetic in the c-direction (α = γ = ε = π) with two different angles in the rung (ii) a phase I with E 2 1 different angles (α,β,γ,δ,ε) which is the stable one in = [2+t2+ t2(cosθ1+cosθ2)+ t −π 2 a large part of the phase diagram for lower J . These c phases are further described below. Except for tche ferro- 2cos(θ1+θ2)]21 (q) 2JcS2[1 +cos(θ1+θ2)] magnetic phase, there is a gap between the two lowest − E − tc 2 bands and the middle ones. At 1/3-filling the Fermi en- J S2 a + (cosθ +cosθ ), ergy is located in this gap so that all these phases are t 1 2 c 3 3 0.50 t2=1.2 0.45 2 JcS / tc = 0.14 n1 n2 0.40 n3 1 0.35n 0.30 0 0.25 -1 0.20 00..1122 00..1144 00..1166 00..1188 00..2200 00..2222 00..2244 JaS2 / tc FIG. 4: Electronic distribution ni of the homogeneous mag- FIG. 3: Cut of the phase diagram along the line JcS2 = netic phaseas function of t. tc 0.14. showing the angles between the spins and the resulting charges. amplitude; inthis senseferromagneticandparamagnetic phasesareequivalent,onlytheeffectivehoppingsaredif- (q) being the complete Elliptic Integral of second kind E ferent in the two cases. The electronic distribution is with parameter q = 2+t2+21t2(c2o(s1−θ1c+osc(oθs1θ+2θ)−2)2)cos(θ1+θ2). shown in Fig. 4 as a function of t. We see that the The angle γ varies discontinuously between phase Ia high temperature behaviour can be reproduced only if t (γ = π) and phases F (γ = 0) and Ib (γ < π), so these is large t & 2.5 3, in particular for t 2√2 one gets transitions are first order. All other transitions are sec- n = n = 1/4 a−nd n = 1/2, but such≥t values are far 1 3 2 ond order. In the Ib phase, close to F we find a canted too large in the Fe-ludwigite ladder. But we see that ferromagnetic phase with canting within the rungs, one the same regime can be reached in the A phase as well I angleonlyβ(orequivalently δ)beingdifferentfromzero; since, in this case, the effective hopping is zero in the atthetransitionβ 0givingthesecondorderboundary c-direction which is equivalent to taking t =0 (see Fig. ltihneeIJaatpcSh2a=sethaarscceo4ssπ→(s−√ent2/t2i√al2l)y.θ1B=etwθe2e=n θF; tahnids cAanI bpehasseeesn, 3cl)osaendtothAeIpwroibthlemθ1r=eduθ2cecsotuoldtharleseosgitivees.cqTuhiteeIwaepllhtahsee forexample,closetoA ,inFig. 3for JcS2 =0.14.There- high temperature charge distribution as seen in Fig. 3 fore the transition lineIbetween I andtcA (θ , θ 0) for JaS2/tc . 0.2. However, as can be seen in Fig. 3, is also second order correspondinga to JcS2I =1√2(24−→t2)+ aisnteinncteeroesfttinhge pIoinsttrruecstuulrteinwgiftrhomθ o=urθanaalsysinisFisigth.e1e-xb-. JaS2. Forlargervaluesof JaS2 thephasetcevolvest3o2wtards This produces aazig-zagbond alter1n6atio2nwhich, in turn, 4tc tc willgiverisetoalatticeinstabilityofthesametype. Due the more general zig-zag structure θ =θ (see Fig. 3). 1 2 6 to the magnetic structure the two border sites of a rung As we mentioned the charge distribution is crucial havedifferentelectronicchargesleadingtotheformation in the Fe-ludwigite ladder so let us examine this point ofazig-zagchargeordering,n n n ,similartothe in detail. It is clear that bond ordering is linked to 2 3 1 ≈ ≫ oneobservedexperimentallybelowT . Notethataphase the spin ordering through the modulation of the hop- c of type (θ = 0,θ = π/2), on the rung, has been ping amplitudes.The ferromagneticbonds tend to local- 1 2 proposedat10K8 incontras↑t↑w−→iththeantiferromagnetic ize the extra electron. This in turn may induce differ- ordering inside the triad obtained from earlier neu- ent types of charge ordering on the non-equivalent Fe- sites in the rung. Experimentally6,7 two charge regimes tronexpe↑r↓im↑ents12. Exceptasymptoticallyi.e. Ja , →∞ wedonotfindphaseswithAFarrangementofthetriads. are identified (i) above T , the side sites 1 and 3 are c identical n = n 0.25 0.3 while the central site Finally we consider the effect of the lattice distortion 1 3 ∼ − 2 has more electrons n 0.5 (ii) below T down to of the rung with hopping t (1 δ) alternating along the 2 c a 74K the charge on site 3∼(the site which gets closer to c-directionandweintroducean±elasticenergyterm 1Bδ2 2 site 2) increases close to the charge of site 2 which re- perrung. Foranhomogeneousmagneticstatethe model mains stable, n n 0.5, and at the same time the reduces to the simple Peierls model considered by Latgé 2 3 ≈ ∼ charge of site 1 decreases to n 0.15. Of course these and Continentino13 and is unlikely to reproduce the ex- 1 values8 indicate only the tende∼ncies, since one should perimental behaviour for reasonable values of t even in have n +n +n = 1. However below 74K two con- the undimerized state as shown in Fig.4. As discussed 1 2 3 tradictory behaviours have been reported7,11. Douvalis above, the zig-zag I phase strongly favors the related a etal11 foundthatthelowtemperatureorderingbelowT rung distortion and, as expected, it occupies an impor- c persists down to T = 0, while Larrea et al7 recover the tant part of the phase diagram as shown in Fig.5 for a same charge ordering as above T . value B/t =6. Here we do not consider the more com- c c To begin with, let us look at homogeneous magnetic plicated I phase appearing at lower J . Phase I shows b c a phases i.e a phases without modulation of the hopping twodistinctregions,anundistortedonewithθ =θ and 1 2 4 hoppings (θ = 0, θ = π), one ferromagnetic and one 1 2 antiferromagnetic bond in each rung, is now stabilized 00..2255 by the distortion (Fig.5, below the dashed line). The t = 1.2 B / tc = 6 distortion δ for the F, AI and Iaa phases is shown in 00..2200 Fig.6-a as function of B/t . The existence of hopping c distortionδ =0 inA requiressmallvalues ofthe elastic I 00..1155 6 2 Ia term B/tc .√2t. This is easily obtained from the total JcS 00/ ..t11c00 Ia energy per rung E which reduces to tEc = −tp2(1+δ2) Iaa +21tBcδ2 inthe3-siteproblem. TheIaa phasepresentsthe 00..0055 largest distortion among these phases, showing clearly F the bond order related to the ferromagnetic character of 00..0000 the bonds. The corresponding charges on the Fe sites in 00..0000 00..0055 00..1100 00..1155 00..2200 00..2255 2 phases A and I are shown on Fig. 6-b. We see that JaS / tc I aa the phase A represents better than others the experi- I FIG.5: F−IaphasediagramfortheelasticparameterB/tc = mental charges both above (δ = 0) and below (δ 6= 0) p6aarstiacufluanrcctaiosnesooffJIatacS.2TahneddJiscttcSo2r.teNdotpehathseatIaAaIoacncudrAsIbIelaorwe ttohe1/st2r,uwcthuilrealnt1ra=nsni3tio=n1T/c4ii.nenth2eruemndaiisntsorctoendstpahnatseeqaunadl n approaches 1/2 whereas n decreases in the distorted thedashed line. 3 1 phase. In the I phase it is site-3 which has the largest aa electronic chargen =0.5 contraryto experimental esti- 3 mate both above T and below for 74K <T <T . c c Our results are consistent with the existence of a A- typephaseasproposedat82K8butimplythatitpersists above T . On the other hand, the I-type structure pro- c posedat10K8shouldpresentchargeorderingandlattice distortion,incontradictionwiththerecentMo¨ssbauerre- sultsofLarreaetal.7. We haveshownthatsimultaneous spin and charge ordering in qualitative agreement with the experimental behaviour for T > 74K occurs from the competition between DE and SE interactions. The bonding is strongly reinforced by the ferromagnetic cor- relations, therefore this may induce a lattice instability as observed. Below 74K, the experimental results7,8,11 are contradicting and further experiments are required to clarify the low temperature situation. Our approach hasemphasizedtheimportanceofthemagneticstructure FIG. 6: (a) Lattice distortion of the rung among the F, AI and bring to light the interplay between spin ordering, and Iaa phases as function of B/tc. (b) The corresponding charge ordering and lattice distortion. charges on theFe sites in phases AI and Iaa. We are grateful to J. Dumas, M. A. Continentino and P. Bordet for helpful discussions. E. V. acknowledge a wide distorted one. A phase I with fully dimerized CONACyT for financial support. aa 1 P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675 8 P. Bordet et al.,(unpublished). (1955). 9 E. Carvajal, O. Navarro, R. Allub, M. Avignon and B. 2 P-G. de Gennes, Phys.Rev. 118, 141 (1960). Alascio, Eur. Phys. J. B 48, 179 (2005). 3 D. J. Garcia, K.Hallberg, C. D.Batista, M. Avignon and 10 M.-H. Wangbo, H.-J. Koo, J. Dumas and M. A. Conti- B. Alascio, Phys.Rev. Lett. 85, 3720 (2000). nentino, Inorg. Chem. 41, 2193 (2002). 4 H. Aliaga, B. Normand, K. Hallberg, M. Avignon and B. 11 A. P. Douvalis, A. Moukarika, T. Bakas, G. Kallias and Alascio, Phys.Rev.B 64, 024422 (2001). V. Papaefthymiou, J. Phys.: Condens. Matter 14, 3302 5 M. Mir et al.,Phys. Rev.Lett. 87, 147201 (2001). (2002). 6 J. Larrea J., D. R. Sanchez, F. J. Litterst, E. M. Baggio- 12 J. P. Attfield, J. F. Clarke and D. A. Perkins, Physica B Saitovitch, J. Phys.: Condens. Matter 13, L949 (2001). 180-181, 581 (1992). 7 J. Larrea J., D. R. Sanchez, F. J. Litterst, E. M. Baggio- 13 A.LatgéandM.A.Continentino,Phys.Rev.B66,094113 Saitovitch, J. C. Fernandes, R. B. Guimaraes and M. A. (2002). Continentino, Phys. Rev.B 70, 174452 (2004).