Collinear order and chirality-reorientation transition in the Cairo pentagonal magnet Bi$_4$Fe$_5$O$_{13}$F

1.1 MB·

by  Alexander A. Tsirlin

Collinearorderandchirality-reorientationtransition
intheCairopentagonalmagnetBi Fe O F
4 5 13
Alexander A. Tsirlin,1,∗ Ioannis Rousochatzakis,2,† Dmitry Filimonov,3
Dmitry Batuk,4 Matthias Frontzek,5 and Artem M. Abakumov3,6,‡
1ExperimentalPhysicsVI,CenterforElectronicCorrelationsandMagnetism,UniversityofAugsburg,86159Augsburg,Germany
2SchoolofPhysicsandAstronomy, UniversityofMinnesota, Minneapolis, Minnesota55455, USA
3Department of Chemistry, Lomonosov Moscow State University, 119991 Moscow, Russia
4EMAT, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium
5QuantumCondensedMatterDivision, OakRidgeNationalLaboratory, OakRidge, TN37831, USA
6Skolkovo Institute of Science and Technology, Nobel str. 3, 143026 Moscow, Russia
7 UsingacombinationofneutrondiffractionandMo¨ssbauerspectroscopy,weunravelthesequenceofmagnetic
1 ordersobservedinthefrustratedCairopentagonalmagnetBi4Fe5O13F.Theseordersincludetwoorthogonal
0 magnetic structures with opposite vector chiralities, and an intermediate, partly disordered collinear phase.
2 Stabilization of the collinear phase by quantum fluctuations was predicted theoretically, but Bi4Fe5O13F is
veryfarfromtherelevantparameterregime.Basedonabinitioband-structurecalculations,weproposethatthe
n
unusualevolutionoftheorderedstatesisdrivenbythestrongsingle-ionanisotropyontheinterlayerFe3+spins,
a
whicharelocatedbetweentheCairoplanes. Thesespinsthenplayadualrole,ononehandmediatingthe3D
J
orderandontheotherdrivingareorientationbetweenthetwoorthogonalconfigurationshavingoppositevector
3
chiralities,withthecollinearordernaturallyemergingasanintermediatephase.
1

]
l Introduction–Frustratedmagnets[1–3]hostaplethoraof lent(Fig.1),buttheclassicalphasediagramofthisextended
e
remarkable collective phenomena, ranging from topological model is qualitatively the same (Fig. 3). Furthermore, the
-
r spin liquids, long-range entanglement and fractionalized ex- calculated interactions (reported here and in [25]) place the
t
s citations [4–10], to emergent electrodynamics and magnetic twocompoundsnearlyonthesamespotinthephasediagram,
.
t monopoles [11–13], and even to spin-induced ferroelectric- and deep inside the orthogonal phase. Despite this remark-
a
m ity [14–16]. While the majority of geometrically frustrated ablesimilarity,thetwocompoundsshowqualitativelydiffer-
magnets are based on spin triangles (or tetrahedra in 3D), ent behavior. Bi Fe O orders in the anticipated orthogo-
- 2 4 9
d pentagon-based magnets, which are far more difficult to im- nal state below 238K, but Bi Fe O F, where Cairo planes
4 5 13
n plement in real materials [17], are now attracting increasing are interleaved by an additional layer of Fe3 sites, shows
o
attentionbothintheory[18–27]andexperiment[28–34]. three successive transitions at T (cid:39) 178K, T (cid:39) 71K,
c N 2
[ The main interest so far has been on the Cairo pentago- and T1 (cid:39) 62K [32], with three distinct magnetically or-
nallattice,aperiodicarrangementofirregularpentagonswith dered states that we refer to as phase I (T < T1), phase II
1
v two types of sites, one with three-fold and the other with (T1 < T < T2), and phase III (T2 < T < TN). Phase I
9 four-foldconnectivity(Fig.1). Cairo-basedmodelshostvar- is orthogonal [32], whereas the nature of phases II and III is
3 iousphasesofclassicalandquantumnature[24],magnetiza- unknowntodate.
7 tionplateaux[22,24,26,27],andKosterlitz-Thoulesstransi- In this Letter, we unravel the nature of these phases, and
3
tions[22]. Bynow,therearetwomainrealizationsofthislat- elucidatetheirorigin. Ourmainfindingsare:
0
tice,Bi Fe O [28–31,34]andBi Fe O F[32],althougha i) The intermediate phase II is the quantum-mechanical,
. 2 4 9 4 5 13
1 similarpentagonaltopologycanbealsoidentifiedinthemul- partially disordered collinear antiferromagnetic phase of
0
tiferroics RMn O (R=Bi, Y, or rare-earth) [35–42] that are Ref.[24](Fig.1e),despitethefactthatwearefarawayfrom
7 2 5
known for their complex interplay of commensurate and in- thecorrespondingregioninthephasediagramanddespitethe
1
: commensuratemagneticorderwithferroelectricity. large,‘classical’spinS =5/2.
v
ThesymmetricversionoftheCairoHeisenbergmodelhas ii) Phases I and III are both orthogonal, but with opposite
i
X two exchange couplings, J and J (Fig. 1a). It hosts vectorchiralitiesχ=±c,asdefinedinFig.1b.Sincethetwo
33 43
r three phases in the classical limit [24]: a coplanar orthog- statesaredegenerateontheleveloftheisotropic(Heisenberg)
a
onal phase (Fig. 1b), a collinear ferrimagnet, and a mixed model,anisotropymustplayarole.
phase in between. Quantumfluctuations convertthe latterto iii) The onset of the collinear phase II coincides with the
anon-magneticandpossiblyspin-nematicphaseforS=1/2. significantgrowingofthemomentontheinterlayerFe3sites,
Additionally,theyintroduceanothercollinearphaseforsmall whichsitbetweentheCairoplanesandnormallyacttomedi-
J /J [24]. Thisphasefeaturescollinearantiferromagnetic atethe3Dorderingbetweentheplanes,ase.g. inRef.[44].
43 33
orderonallfour-foldsitesandonhalfofthethree-foldsites, TheFe3spinsareabsentinBi Fe O (whereneighboring
2 4 9
withtheremaininghalfbeingdisordered(Fig.1c). planescoupledirectlytoeachother),sotheemergingphysical
Bi Fe O and Bi Fe O F feature dumbbells of the 4- picture is that the interlayer Fe3 spins play a vital role for
2 4 9 4 5 13
fold-sites instead of each single 4-fold site on the Cairo lat- the magnetic order within the Cairo plane. Specifically, the
tice. Additionally, the couplings J and J(cid:48) are nonequiva- III→II→I transitions can be understood as a reorientation of
43 43
2
(a)
J^
J J
43 33
c
(b) (c)
2 3 am
b
m
1 4
orthogonal collinear Fe12 Fe21
Fe11 Fe22
J33 bm
J
43
’
J43 am (d) (e) (f)
Phase I Phase II Phase III
FIG. 1. (a) The symmetric Cairo lattice with two exchange couplings, J and J . (b-c) Orthogonal and collinear phases. Note the
33 43
zero moment on half of the 3-fold sites in (c). The Fe1 sites 1-4 in (b) provide a measure of the chirality as χ=(cid:104)Γ(cid:105)/|(cid:104)Γ(cid:105)|, where Γ=
S ×S +S ×S +S ×S +S ×S .(d-f)MagneticstructuresofBi Fe O FinphasesI,II,andIII,respectively.ThetwotypesoftheJ
1 2 2 3 3 4 4 1 4 5 13 43
couplingsarealsoindicated.ThecrystalandmagneticstructuresarevisualizedusingVESTA[43].
thenominallypreferredorthogonalstate,fromoneorientation butnetmagnetizationiszero,becausesecond-neighborchains
(phase III) that satisfies the anisotropy on the in-plane spins form opposite moments. At 1.5K, the moments are about
to another orientation (phase I) that satisfies the anisotropy 4.0µ ontheFe1andFe3sitesand3.3µ ontheFe2sites.
B B
on the interlayer Fe3 spins, and in between the system must ThisdifferenceisduetothestrongerFe–Ohybridizationfor
necessarilygothroughthecollinearphaseII. the tetrahedrally coordinated Fe atoms. We note in passing
Neutron powder diffraction – All measurements were that the magnetic configuration in phase I deviates from our
performed on single-phase polycrystalline samples of previousreport,namely,allspinsintheCairoplaneareturned
Bi Fe O F prepared previously [32]. Neutron diffraction by90◦ withrespecttoFig.5inRef.[32]. Thisrevisedmodel
4 5 13
datawerecollectedatthecoldneutronpowderdiffractometer isbasedonthebetter-qualityneutrondata[47].
DMC (LNS PSI, Villigen, Switzerland) with the wavelength As T increases toward T , the Fe1 and Fe2 moments re-
1
of 4.5082A˚ in the T range of 1.5-200K in a He-cryostat. main roughly unchanged, whereas the moment on Fe3 de-
ThemagneticstructureswererefinedbytheRietveldmethod creasessignificantlyanddropsbelow2µ at55K.Uponfur-
B
using the JANA2006 program [45]. The symmetry anal- therheating,themagneticstructurechangesabruptlyentering
ysis of possible magnetic configurations was carried out in phaseII.AllFe1sitespreservelargemomentsof3.2-3.8µ ,
B
ISODISTORT[46]. whereas the Fe2 sites split into two groups. For Fe2 , the
2
Phases I–III share the same propagation vector k = momentsincreaseto3.8µB, whereasforFe21 theydecrease
(12,12,0) resulting in a single irrep mM5− allowing the mo- to about 1.2µB, only one third of their 1.5K value. The
mentsintheabplane,similartoBi Fe O . Thisirrepcorre- magnetic structure is nearly collinear and surprisingly simi-
2 4 9
sponds to the tetragonal magnetic space group P 4 /n and lar to the collinear phase predicted for the quantum limit of
C 2
the two-fold magnetic supercell defined as: a = a−b, theCairo-latticemodel[24]. Here,thelowersymmetryofthe
m
bm=a+b,cm=c. Thesymmetryimpliesfivenonequivalent lattice renders J43 different from J4(cid:48)3, hence the moment on
Fepositions,asshowninFig.1: Fe11,Fe12,Fe21,andFe22 Fe21doesnotvanishcompletely.
within the Cairo planes and Fe3 between the planes. Fig. 2a ThenarrowregionofphaseIIisfollowedbyabroaderre-
showsthesizeofthemagneticmomentsasafunctionoftem- gionofphaseIII,whichisagainorthogonal,butwithopposite
perature[47]. vectorchiralityχ=−c, incontrasttoχ=+cinphaseI.In
PhaseI(Fig.1d)istheanticipatedorthogonalstate,where addition,themomentsonFe1 areabouttwicelargerthanon
1
spins on the Fe1 and Fe1 sites and on the Fe2 and Fe2 Fe1 , whereastheFe3momentsaretoosmalltobedetected
1 2 1 2 2
sitesaremutuallyorthogonal. Theorderingalongthecdirec- experimentally. We obtained µ = 0.29(25)µ at 90K
Fe3 B
tion is locally ferrimagnetic, Fe1↑Fe1↑Fe3↓Fe1↑Fe1↑Fe3↓, and,therefore,keptµ fixedatzerointhefinalrefinement.
Fe3
3
4.0 0
netic moments ()mB213...000 FFFFeeee2113121 FFeeF22e2112 Fe11 (a) 40621 120908 KK (c) 1.5 withcollinearB Ferrimagnetic
Mag0.0 0 FFee33202 6T0empe9r0ature1 2(K0) 150 180 bsorption (%)201 63 K (cid:144)J'J43331.0 degenerate Mixed
A
2-(10eum/mol)c 426 T1(b)T2 TN Experiment (1FTit) 2201 55 K 00..05 Bi2Fe4O(cid:242)9 Bi4FOe5Ort1h3Fogonal degeneratewithcollinearA
0.0 0.5 1.0 1.5
0 100 200 300 400 500 600 -10 -5 0 5 10
Temperature (K) Velocity (mm/s) J43(cid:144)J33
FIG.2. (a)Temperaturedependenceoforderedmomentsobtained FIG. 3. Classical phase diagram of the Cairo Heisenberg model,
fromneutrondiffraction.Errorbarsaresmallerthanthesymbolsize, withtwoFe1-Fe2couplings,J43andJ4(cid:48)3,andtwoFe1spinsoneach
linesareguide-for-they-eyeonly.(b)Fitofthemagneticsusceptibil- four-foldsiteoftheCairolattice,asobtainedin[47]usingLyonsand
ityusingexchangeparametersJ fromDFT.(c)Mo¨ssbauerspectra Kaplan’s[51]generalizationoftheLuttinger-Tiszamethod[52].The
ij
andtheirfits,asdescribedinthetext.Forfitparameters,see[47]. linesJ43=0andJ43=0correspondtodecoupledchains,andtheor-
thogonalphasebecomesdegeneratewithinfiniteothergroundstates,
includingthecollinearphasesAandBdiscussedin[47]. Thepar-
tiallydisorderedcollinearphaseof[24],whichisrelevanttophase
Mo¨ssbauer spectroscopy – The 57Fe Mo¨ssbauer spectra
II of Fig. 1(e), is stabilized by quantum fluctuations in the corner
(Fig.2c)wererecordedinthetemperaturerange55−300K around J =J(cid:48) =0. The line J =J(cid:48) maps to the model of
43 43 43 43
ina transmissionmode witha 57Co/Rh γ-ray source usinga [24] by rescaling J →J /2 and J(cid:48) →J(cid:48) /2. The two avail-
43 43 43 43
constantaccelerationspectrometerMS1104.Atroomtemper- ablecompoundsareshownbythefilledbluetriangle(basedonthe
ature, the spectrum can be decomposed into 3 doublets with parametersof[25])andthefilledreddot(basedontheparameters
the nearly 40:40:20 ratio of the intensities corresponding to reportedhere).
theFe1,Fe2,andFe3positions,respectively[47].
Upon cooling below T , the spectra reveal an additional
N
splitting indicative of the magnetic ordering. However, the Thissetofexchangeparametersreproducesthesusceptibility
spectrumat100K,inphaseIII,couldnotbeaccountedforby downto120K(Fig.2b)andpredictsTN (cid:39) 180Kinperfect
acombinationofregularsextets. Wefoundthatabout20%of agreementwiththeexperiment[47].
thespectralintensitycorrespondstoanunresolvedsextetwith Ouranalysisoftheisotropicmodel[47]placesBi Fe O F
4 5 13
a very weak hyperfine splitting. This signal corresponds to deep inside the orthogonal phase of the classical phase dia-
Fe3that,accordingtotheneutrondata,hasaverylowordered gram(Fig.3). Additionally, theinterlayerFe1–Fe3coupling
momentaboveT . BelowT ,theFe3momentsincreaseand is much weaker than the couplings within the plane. There-
2 2
awell-resolvedsextetdevelops,asexpected. fore,oneexpectsthattheFe3spinsaremoresensitivetother-
Thespectrumat55K(phaseIII)containsfivemagneticsex- mal fluctuations and decrease much faster than the spins on
tetsthatseparateintothreegroups,Fe1 andFe1 ,Fe2 and Fe1andFe2[47],inagreementwithFig.2a.
1 2 1
Fe22,andFe3,asexpectedfromthemagneticstructure. The Besidesthesetwoaspects,therestoftheexperimentalbe-
spectrum at 63K (phase II) is visually quite similar, but one havior, including the formation of phases II and III, is com-
of the Fe2 components is strongly broadened indicating the pletely unexpected on the basis of the isotropic model. First
reducedorderedmomentonFe21. of all, the two phases I and III, which have opposite vector
Microscopic description – We begin with the isotropic chiralitiesχ, aredegenerate. Second, thesystemisfaraway
(cid:80)
modeldefinedbythespinHamiltonianH= J S ·S , from any collinear phase of the model (Fig. 3), so there is a
(cid:104)ij(cid:105) ij i j
where the summation is over nearest-neighbor spins-5/2, S largeenergybarrieragainstanythermally-drivenstabilization
i
and S , and J are the exchange parameters. Their ab- ofcollinearity(wehaveinfactcheckedandexcludedthesce-
j ij
solute values are obtained from density-functional (DFT) narioofthermalstabilizationbyclassicalMonteCarlosimu-
band-structure calculations performed in the FPLO [48] and lations). Third, the scenario of quantum fluctuations driving
VASP [49, 50] codes using the DFT+U procedure [47]. The thecollinearphaseishighlyunlikelyaswell:Theonsetofthe
couplingswereextractedfromtotalenergiesofcollinearspin collinear phase is roughly taking place when the spin length
configurations. IncontrasttoRef.[32],theywerefurtherre- correction δS from quadratic spin waves approaches the full
finedbyfittingexperimentalmagneticsusceptibilitywithclas- valueS=5/2. AccordingtoFig.4ofRef.[24],thecollinear
sicalMonte-CarlosimulationsfortheHeisenbergspinmodel. phaseforS = 5/2(ifany)onsetswaybelowJ /J =0.1,
43 33
WefindJ =116K,J =38K,andJ(cid:48) =57K.TheFe1– and the actual number should be further divided by two, be-
33 43 43
Fe3 interaction that mediates the 3D ordering, is J =8K. causeherewehavetwoFe1sitesateachfour-foldsite.Atany
⊥
4
1.5 nalstructurearenotallowed,becauseorthogonalityisdefined
1.0 Fe11 48.2o byJij’s,whichareatleasttwoordersofmagnitudestronger
thantheanisotropy. Thiscompetitionbetweentheorthogonal
0.5 structure and individual single-ion anisotropies explains the
Fe1 unexpected difference in the magnetic moments on the Fe1
0.0 2 1
andFe1 sitesinphaseIII.Indeed,theFe1 momentislarger,
2 1
1.5
becauseitpointsapproximatelyalongthepreferreddirection
gy (K) 1.0 Fe21 5.0o 1(t0h0eKd)e.pOarntutrheeforothmerthheanpdr,etfheerremdodmireencttioonnFies1∆ϕis f=ar a5w◦aayt
er 0.5 Fe2 2
n 2 fromitspreferreddirection(∆ϕ=47◦)andthussmaller.
E
0.0 Asideeffectoftheseenergyconsiderationsisthatthechi-
8 rality changes from χ=+c in phase I to χ=−c in phase
Fe3
6 1 III.Thecontinuoustransformationbetweenthesetwophases
4 necessitatestheintermediatecollinearphaseIIthatexistsina
2 Fe3 narrowtemperaturerangeonly.
0 90o 2 Discussion – The main picture emerging from the experi-
0 30 60 90 120 150 180 mental data presented here is that the interlayer Fe3 spins in
j(deg) Bi Fe O F play a dual role, on one hand mediating the 3D
4 5 13
orderingandontheotherdrivingareorientationoftheorder
FIG. 4. In-plane single-ion anisotropy energies for the Fe1, Fe2,
in the planes. While details of this transition require further
andFe3sites.Thearrowsdenotepreferredspindirections.Onlythe
dedicatedtheoreticalwork,ontheexperimentalsidetheeffect
Fe3sitesarecompatiblewiththeorthogonalstructure,becausetheir
preferred directions are at 90◦ to each other. For the Fe1 and Fe2 oftheFe3spinsiscrucialforthedesignofnewCairo-lattice
sites,theanglebetweenpreferreddirectionsoftheneighboringsites magnets, because interlayer magnetic sites, which are often
is largely different from 90◦. Note that we use Fe3 and Fe3 for introducedforthesakeofstabilizingthecrystalstructure[53],
1 2
thoseFe3sitesthatarecoupledtoFe11andFe12,respectively. The arenotinnocentandinfactplaydecisiveroleforthemagnetic
angleϕismeasuredbetweenthemagneticmomentandtheam-axis, orderwithintheCairoplanes.
andallcurvesareperiodic,E(ϕ+180◦)=E(ϕ).
ThesequenceoftransitionsinBi Fe O Fiscloselyrem-
4 5 13
iniscent of the consecutive transitions in RMn O , where
2 5
an intermediate collinear phase separates two non-collinear
rate, our ab initio parameters are far away from the stability
states.However,thiscollinearphase[37]isdifferentfromour
regionofthecollinearphase. SophaseIIisnotathermody-
phaseII,becauseitdoesnotshowthecharacteristicreduction
namically stable phase of the isotropic model, but should be
in the ordered moment on part of the (Fe2) sites. Instead, it
thoughtofasanintermediatestepofareorientationtransition
mayberelatedtothecollinearphasesAandBofFig.3.
fromItoIII,drivenbyanisotropy.
Further investigation of the collinear phase II in
Effect of anisotropy – To unravel the origin of this re-
Bi Fe O F would be interesting. For example, an exter-
orientation transition, we must consider anisotropic terms. 4 5 13
nal magnetic field may influence the subtle balance between
Dzyaloshinsky-Moriya (DM) couplings are forbidden by
thetwoorthogonalstatesandleadtonewpartiallydisordered
symmetry for J bonds, but are allowed for J and J(cid:48)
33 43 43 statesresemblingphaseII.Giventhechiralnatureofthemag-
bonds. However,theanalysisofsymmetryrelationstogether
netic structures in phases I and III, detailed investigation of
withtheexperimentalmagneticstructuresinphasesIandIII
theirdielectricpropertiesisalsowarranted.
showsthattheoverallcontributionofthec-componentsofthe
DMvectorscancelsout, hencetheDMvectorsdonotdeter- Acknowledgment – We are thankful to Prof. J.-M. Perez-
minethechirality. Mato for valuable discussion on the magnetic structure. DF
andAAaregratefultotheRussianScienceFoundation(grant
Incontrast,single-ionanisotropies(Fig.4)giveustheright
14-13-00680) for support. AT was supported by the Federal
insights. WefirstnotethattheanisotropyofFe3ismorethan
Ministry for Education and Research through the Sofja Ko-
5timesstrongerthanthatofFe1andFe2.Therefore,Fe3with
itspreferreddirectionsatϕ = 0◦ and90◦ putstheFe1spins valevskaya Award of Alexander von Humboldt Foundation.
This work is based on experiments performed at the Swiss
along a and b . This anisotropy does not choose the chi-
m m
spallationneutron sourceSINQ,Paul ScherrerInstitut, Villi-
rality,buttheanisotropyofFe2prefersthestructureofphase
gen,Switzerland.
Iwithχ=+c,asconfirmedbyadirectenergyminimization.
AboveT ,inphaseIII,thepreferreddirectionofFe3plays
2
norole,andtheFe1andFe2spinsarelefttoformanorthogo-
nalstructure,eventhoughtheirpreferreddirectionsdonotfa-
vorit: forexample,thepreferreddirectionsofFe11 andFe12 ∗ [email protected]
differ by 48.2◦ only, whereas Fe21 and Fe22 spins prefer to † [email protected]
be nearly collinear. However, deviations from the orthogo- ‡ [email protected]
5
[1] IntroductiontoFrustratedMagnetism:Materials,Experiments, theS =1/2HeisenbergantiferromagnetontheCairopentagon
Theory(SpringerSeriesinSolid-StateSciences,Berlin,2011). lattice,”J.Phys.Soc.Jpn.83,053702(2014).
[2] FrustratedSpinSystems,2nded.(WorldScientific,2013). [27] M.Isoda,H.Nakano, andT.Sakai,“Frustration-inducedmag-
[3] L. Balents, “Spin liquids in frustrated magnets,” Nature 464, netic properties of the spin-1/2 Heisenberg antiferromagnet
199–208(2010). on the Cairo pentagon lattice,” J. Phys. Soc. Jpn. 83, 084710
[4] P.W.Anderson,“Resonatingvalencebonds: Anewkindofin- (2014).
sulator?”Mat.Res.Bull8,153–160(1973). [28] N.Shamir,E.Gurewitz, andH.Shaked,“Themagneticstruc-
[5] S.Yan,D.A.Huse, andS.R.White,“Spin-liquidgroundstate ture of Bi Fe O – analysis of neutron diffraction measure-
2 4 9
ofthes = 1/2kagomeHeisenbergantiferromagnet,”Science ments,”ActaCryst.A34,662–666(1978).
332,1173–1176(2011). [29] A.K.Singh,S.D.Kaushik,B.Kumar,P.K.Mishra,A.Veni-
[6] S.Depenbrock,I.P.McCulloch, andU.Schollwo¨ck,“Nature madhav,V.Siruguri, andS.Patnaik,“Substantialmagnetoelec-
of the spin-liquid ground state of the S = 1/2 Heisenberg triccouplingnearroomtemperatureinBi Fe O ,”Appl.Phys.
2 4 9
model on the kagome lattice,” Phys. Rev. Lett. 109, 067201 Lett.92,132910(2008).
(2012). [30] E. Ressouche, V. Simonet, B. Canals, M. Gospodinov, and
[7] T.-H.Han,J.S.Helton,S.Chu,D.G.Nocera,J.A.Rodriguez- V.Skumryev,“Magneticfrustrationinaniron-basedCairopen-
Rivera,C.Broholm, andY.S.Lee,“Fractionalizedexcitations tagonallattice,”Phys.Rev.Lett.103,267204(2009).
in the spin-liquid state of a kagome-lattice antiferromagnet,” [31] M.N.Iliev,A.P.Litvinchuk,V.G.Hadjiev,M.M.Gospodinov,
Nature(London)492(2012). V.Skumryev, andE.Ressouche,“Phononandmagnonscatter-
[8] A. Kitaev, “Anyons in an exactly solved model and beyond,” ingofantiferromagneticBi Fe O ,”Phys.Rev.B81,024302
2 4 9
AnnalsofPhysics321,2–111(2006). (2010).
[9] L. Savary and L. Balents, “Quantum spin liquids: a review,” [32] A. M. Abakumov, D. Batuk, A. A. Tsirlin, C. Prescher,
Rep.Prog.Phys.80,016502(2017). L.Dubrovinsky,D.V.Sheptyakov,W.Schnelle,J.Hadermann,
[10] M. J. P. Gingras and P. A. McClarty, “Quantum spin ice: a and G. Van Tendeloo, “Frustrated pentagonal Cairo lattice in
searchforgaplessquantumspinliquidsinpyrochloremagnets,” the non-collinear antiferromagnet Bi Fe O F,” Phys. Rev. B
4 5 13
Rep.Prog.Phys.77,056501(2014). 87,024423(2013).
[11] C. Castelnovo, R. Moessner, and S. L. Sondhi, “Magnetic [33] M.G.Rozova,V.V.Grigoriev,I.A.Bobrikov,D.S.Filimonov,
monopolesinspinice,”Nature451,42–45(2008). K.V.Zakharov, O.S.Volkova, A.N.Vasiliev, E.V.Antipov,
[12] C.L.Henley,“TheCoulombphaseinfrustratedsystems,”Ann. A.A.Tsirlin, andA.M.Abakumov,“Synthesis,structureand
Rev.Condens.MatterPhys.1,179–210(2010). magneticorderingofthemullite-typeBi Fe Cr O solidso-
2 4−x x 9
[13] J. Rehn and R. Moessner, “Maxwell electromagnetism as an lutionswithafrustratedpentagonalCairolattice,”DaltonTrans.
emergentphenomenonincondensedmatter,”Phil.Trans.Royal 45,1192–1200(2016).
Soc.LondonA374(2016),10.1098/rsta.2016.0093. [34] S.R.Mohapatra, A.Swain, C.S.Yadav, S.D.Kaushik, and
[14] S.-W. Cheong and M. Mostovoy, “Multiferroics: a magnetic A. K. Singh, “Unequivocal evidence of enhanced magnetodi-
twistforferroelectricity,”NatureMaterials6,13–20(2007). electriccouplinginGd3+ substitutedmultiferroicBi Fe O ,”
2 4 9
[15] T.Arima,“Spin-drivenferroelectricityandmagneto-electricef- RSCAdv.6,112282–112291(2016).
fects in frustrated magnetic systems,” J. Phys. Soc. Jpn. 80, [35] K. Saito and K. Kohn, “Magnetoelectric effect and low-
052001(2011). temperaturephasetransitionsofTbMn O ,”J.Phys.:Condens.
2 5
[16] Y.Tokura,S.Seki, andN.Nagaosa,“Multiferroicsofspinori- Matter7,2855(1995).
gin,”Rep.Prog.Phys.77,076501(2014). [36] N.Hur, S.Park, P.A.Sharma, J.S.Ahn, S.Guha, andS-W.
[17] Thisismostlybecauseofthewell-knownincompatibilityofthe Cheong,“Electricpolarizationreversalandmemoryinamul-
five-foldsymmetrywithlatticeperiodicity. tiferroicmaterialinducedbymagneticfields,”Nature429,392
[18] R.MoessnerandS.L.Sondhi,“Isingmodelsofquantumfrus- (2004).
tration,”Phys.Rev.B63,224401(2001). [37] G.R.Blake,L.C.Chapon,P.G.Radaelli,S.Park,N.Hur,S-W.
[19] V.Urumov,“ExactsolutionoftheIsingmodelonapentagonal Cheong, andJ.Rodr´ıguez-Carvajal,“Spinstructureandmag-
lattice,”J.Phys.A35,7317(2002). neticfrustrationinmultiferroicRMn O (R =Tb, Ho, Dy),”
2 5
[20] K.S.Raman,R.Moessner, andS.L.Sondhi,“SU(2)-invariant Phys.Rev.B71,214402(2005).
spin-1 Hamiltonians with resonating and other valence bond [38] C.Vecchini, L.C.Chapon, P.J.Brown, T.Chatterji, S.Park,
2
phases,”Phys.Rev.B72,064413(2005). S-W. Cheong, and P. G. Radaelli, “Commensurate magnetic
[21] A.Jagannathan, B.Motz, andE.Vedmedenko,“Novelprop- structuresofRMn O (R=Y,Ho,Bi)determinedbysingle-
2 5
erties of frustrated low-dimensional magnets with pentagonal crystalneutrondiffraction,”Phys.Rev.B77,134434(2008).
symmetry,”Phil.Mag.91,2765–2772(2011). [39] A.B.Harris,A.Aharony, andO.Entin-Wohlman,“Orderpa-
[22] A.Ralko,“PhasediagramoftheCairopentagonalXXZspin- rametersandphasediagramofmultiferroicRMn O ,”Phys.
2 5
1 magnet under a magnetic field,” Phys. Rev. B 84, 184434 Rev.Lett.100,217202(2008).
2
(2011). [40] A.B.Sushkov,M.Mostovoy,R.Valde´sAguilar,S.-W.Cheong,
[23] M. Rojas, O. Rojas, and S. M. de Souza, “Frustrated Ising and H. D. Drew, “Electromagnons in multiferroic RMn O
2 5
modelontheCairopentagonallattice,”Phys.Rev.E86,051116 compounds and their microscopic origin,” J. Phys.: Condens.
(2012). Matter20,434210(2008).
[24] I.Rousochatzakis,A.M.La¨uchli, andR.Moessner,“Quantum [41] J.W.Kim,S.Y.Haam,Y.S.Oh,S.Park,S.-W.Cheong,P.A.
magnetism on the Cairo pentagonal lattice,” Phys. Rev. B 85, Sharma, M. Jaime, N. Harrison, Jung Hoon Han, G.-S. Jeon,
104415(2012). P. Coleman, and K. H. Kim, “Observation of a multiferroic
[25] Z.V.PchelkinaandS.V.Streltsov,“Abinitioinvestigationof criticalendpoint,”PNAS106,15573–15576(2009).
theexchangeinteractionsinBi Fe O : TheCairopentagonal [42] K.Cao,G.-C.Guo,D.Vanderbilt, andL.He,“First-principles
2 4 9
latticecompound,”Phys.Rev.B88,054424(2013). modeling of multiferroic RMn O ,” Phys. Rev. Lett. 103,
2 5
[26] H.Nakano,M.Isoda, andT.Sakai,“Magnetizationprocessof 257201(2009).
6
[43] K.MommaandF.Izumi,“VESTA3forthree-dimensionalvisu- nallocal-orbitalminimum-basisband-structurescheme,”Phys.
alizationofcrystal,volumetricandmorphologydata,”J.Appl. Rev.B59,1743–1757(1999).
Crystallogr.44,1272–1276(2011). [49] G.KresseandJ.Furthmu¨ller,“Efficiencyofab-initiototalen-
[44] R. Nath, K. M. Ranjith, J. Sichelschmidt, M. Baenitz, Y. Sk- ergycalculationsformetalsandsemiconductorsusingaplane-
ourski,F.Alet,I.Rousochatzakis, andA.A.Tsirlin,“Hindered wave basis set,” Computational Materials Science 6, 15 – 50
magneticorderfrommixeddimensionalitiesinCuP O ,”Phys. (1996).
2 6
Rev.B89,014407(2014). [50] G. Kresse and J. Furthmu¨ller, “Efficient iterative schemes for
[45] V. Pet˘r´ıc˘ek, M. Dus˘ek, and L. Palatinus, “Crystallographic abinitiototal-energycalculationsusingaplane-wavebasisset,”
computingsystemJANA2006:Generalfeatures,”Z.Krist.229, Phys.Rev.B54,11169–11186(1996).
345–352(2014). [51] D. H. Lyons and T. A. Kaplan, “Method for determining
[46] B.J.Campbell,H.T.Stokes,D.E.Tanner, andD.M.Hatch, ground-statespinconfigurations,”Phys.Rev.120,1580–1585
“ISODISPLACE:aweb-basedtoolforexploringstructuraldis- (1960); T.A.KaplanandN.Menyuk,“Spinorderinginthree-
tortions,”J.Appl.Cryst.39,607–614(2006). dimensionalcrystalswithstrongcompetingexchangeinterac-
[47] See Supplemental material at [...] for: i) Neutron diffraction tions,”Phil.Mag.87,3711–2785(2007).
patternsanddetailsoftherefinement;ii)fittingparametersfor [52] J.M.LuttingerandL.Tisza,“Theoryofdipoleinteractionin
theMo¨ssbauerspectra;iii)detailsofelectronicstructurecalcu- crystals,”Phys.Rev.70,954–964(1946).
lations;iv)Detailsonthederivationofthephasediagrampre- [53] J. Cumby, R. D. Bayliss, F. J. Berry, and C. Greaves, “Syn-
sented in Fig. 3; v) Theoretical evidence for the qualitatively theticanaloguesofFe(II)–Fe(III)mineralscontainingapentag-
differentT-dependenceoftheFe3andFe1/Fe2moments. onal’Cairo’magneticlattice,”DaltonTrans.45,11801–11806
[48] K. Koepernik and H. Eschrig, “Full-potential nonorthogo- (2016).