Anisotropic exchange within decoupled tetrahedra in the quantum breathing pyrochlore Ba3Yb2Zn5O11

4 MB·

by  J. G. Rau

Anisotropicexchangewithindecoupledtetrahedrainthequantumbreathingpyrochlore
Ba Yb Zn O
3 2 5 11∗
J. G. Rau,1, L. S. Wu,2, A. F. May,3 L. Poudel,2,4 B. Winn,2 V. O. Garlea,2 A. Huq,5 P.
† ‡
Whitfield,5 A. E. Taylor,2 M. D. Lumsden,2 M. J. P. Gingras,1,6,7 and A. D. Christianson2,4
1DepartmentofPhysicsandAstronomy, UniversityofWaterloo, Ontario, N2L3G1, Canada
2QuantumCondensedMatterDivision,OakRidgeNationalLaboratory,OakRidge,TN-37831,USA
3MaterialsScience&TechnologyDivision,OakRidgeNationalLaboratory,OakRidge,TN-37831,USA
4DepartmentofPhysics&Astronomy, UniversityofTennessee, Knoxville, TN-37966, USA
5Chemical&EngineeringMaterialsDivision,OakRidgeNationalLaboratory,OakRidge,TN37831,USA
6Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
7CanadianInstituteforAdvancedResearch,180DundasStreetWest,Suite1400,Toronto,ON,M5G1Z8,Canada
6 (Dated:January19,2016)
1
0 ThelowenergyspinexcitationspectrumofthebreathingpyrochloreBa3Yb2Zn5O11hasbeeninvestigated
2 withinelasticneutronscattering. Severalnearlyresolutionlimitedmodeswithnoobservabledispersionare
observedat250mKwhile, atelevatedtemperatures, transitionsbetweenexcitedlevelsbecomevisible. To
n
gaindeeperinsight,atheoreticalmodelofisolatedYb3+tetrahedraparametrizedbyfouranisotropicexchange
a
J constantsisconstructed.Themodelreproducestheinelasticneutronscatteringdata,specificheat,andmagnetic
susceptibilitywithhighfidelity.ThefittedexchangeparametersrevealaHeisenbergantiferromagnetwithavery
6
largeDzyaloshinskii-Moriyainteraction.Usingthismodel,wepredicttheappearanceofanunusualoctupolar
1
paramagnetatlowtemperaturesandspeculateonthedevelopmentofinter-tetrahedroncorrelations.

]
l
e
Frustratedorcompetinginteractionshavebeenrepeatedly O
- Ba
r found to be at the root of many unusual phenomena in con-
t Zn
s densed matter physics [1–5]. By destabilizing conventional Yb
.
t long-rangeorderdowntolowtemperature,frustrationinmag-
a
m neticsystemscanleadtomanyexoticphases;fromunconven-
tional multipolar [6, 7] and valence bond solid orders [1, 4]
-
d todisorderedphasessuchasclassicalandquantumspinliq-
n uids [1, 4]. Significant attention has been devoted to under-
o
standinggeometricfrustrationwhereitistheconnectivityof
c
thelatticethathinderstheformationoforder. Recently,how-
[
ever, magnets frustrated not by geometry but by competing

1
interactionshavebecomeprominentforthenovelbehaviors
v
thattheyhost.Suchcompetinginteractionsmightbeadditional
4
0 isotropicexchangeactingbeyondnearestneighbors[8–10],bi-
1 quadraticorothermultipolarinteractions[11]. Onepossibility
4 attractingeverincreasinginterestisthatcompetingstrongly
0 anisotropicinteractionsmaystabilizeawiderangeofunusual FIG.1. CrystalstructureofBa3Yb2Zn5O11(cubicspacegroupF4¯3m,
. no.216).EachYb3+ionispartofalargeandsmalltetrahedroninthe
1 phenomena.
0 breathingpyrochlorelattice.
6 Anexcitingresearchdirectioninthelattercontextconcerns
1 itselfwithso-called“quantumspinice”[12]. Thisquantum
: spinliquidcanbestabilizedbyperturbingclassicalspinice liquids[22,23]. Inmanyofthesecompounds,thephysicsis
v
i withadditionalanisotropictransverseexchangeinteractions verydelicate,showingstrongsampletosamplevariations[24]
X
that induce quantum fluctuations. Particularly interesting is orsensitivitytoverysmallamountsofdisorder[25,26]. Con-
r thepotentialrealizationofsuchphysicsintherare-earthpy- sequently,anaccuratedeterminationoftheeffectivemodelis
a
rochloresR M O [13–15],whereRisatrivalent4f rare-earth crucialinmakingdefiniteprogressinthisarea. Thisisparticu-
2 2 7
ion,andMisanon-magnetictetravalenttransitionmetalion, larlytrueincaseswheretheidealizeddisorder-freematerial
suchasM=Ti,SnorZr. Thesematerialscanbedescribedin mayfinditselfinthevicinityofatransitionbetweencompeting
termsofpseudospin-1/2degreesoffreedominteractingvia semi-classicalgroundstates[18,27,28]
anisotropicexchanges[12,15],wheretheeffectivespin-1/2 Given the critical importance played by the precise value
mapsthestatesofthecrystal-electricfieldgrounddoubletof oftheanisotropicexchanges,anumberofexperimentshave
therare-earthion. Thesematerialsdisplayawealthofinterest- beenaimedatdeterminingthosecouplings[15,16]. Thereis,
ingphenomena,fromthepossibilityofquantum[16–18]order- unfortunately,muchdifficultyinobtainingaccuratevaluesfor
by-disorderphysicsinEr Ti O [19],unconventionalordered thesecouplingsstemmingfromtwokeylimitations. First,only
2 2 7
states[20,21]aswellasseveralcandidatesforquantumspin approximatemethodsareavailabletorelatethemodeltoex-
2
periment,restrictingcomparisonstoregimeswherethetheory peratures,possiblyleadingtointerestingnewphysics[40–44]
becomescontrolled,suchasinhighmagneticfield[15,16,29] inthismaterial.
orathigh-temperature[29–32]. Second,toavoidover-fitting Experimental results: Polycrystalline samples of BYZO
theexperimentaldata,onemustworkwithareasonablenum- were synthesized by solid-state reaction in Al O cru-
2 3
beroffittingparameters;forexamplerestrictingtoasubsetof cibles[45]. Theresultingsampleswerecharacterizedbyspe-
theallowedinteractionsbyignoringinteractionsbeyondnear- cificheatandmagnetizationmeasurements[45]. Thestructure
est neighbors or possible multi-spin interactions [19]. Even wasstudiedvianeutronpowderdiffractionutilizingthePOW-
inYb Ti O ,wherethelatterconcernislargelyabsent,there GEN[47]diffractometerattheSpallationNeutronSourceat
2 2 7
currentlyremainsnoconsensusonthevaluesoftheanisotropic OakRidgeNationalLaboratory[45].Thesemeasurementscon-
exchange parameters [15, 27]. At the present time, a refer- firmthepreviouslyreportedcubicstructure[33,48,49](space
encerare-earthpyrochlore-likecompoundwithsolelybilinear groupF4¯3m,no. 216)withlatticeparametera=13.47117(3)
anisotropicinteractionsandforwhichessentiallyexactmeth- at10Kanda=13.48997(3)at300K.
ods canbe employed to compare withexperimentaldata, is Toexplorethelowenergyspectrumofmagneticexcitations
badlyneededtocementthevalidityoftheeffectivespin-1/2 in BYZO, INS data was collected using the HYSPEC spec-
descriptionofsuchmaterials. trometer[50]attheSpallationNeutronSourceatOakRidge
InthisLetter,westudyBa Yb Zn O (BYZO),aso-called NationalLaboratory. Measurementswereperformedat0.25,
3 2 5 11
breathingpyrochlore(BP)compound[33,34],whichprovides 10,and20Kutilizinga3Herefrigerator,withfixedincident
anidealplatformforunderstandingsuchanisotropicexchange neutronenergiesofEi=3.8 meV,7.5 meVand15 meV.
models. As shown in Fig. 1, BYZO consists of small tetra- INSmeasurementswithEi=3.8meVat0.25and20Kare
hedra with a short nearest-neighbor bond distance r 3Å showninFig.2. Thedataat0.25K(Fig.2(a)and(b))exhibits
<
∼
connectedbylargetetrahedrawithsizer 6Å. Becauseof severalwell-definedmodeswithnoobservabledispersion. The
>
∼
thelargeratior /r 2,theinter-tetrahedroncouplingsare Q-dependence of the inelastic scattering intensity exhibits
> < ∼ | |
expected to be small compared to the intra-tetrahedron cou- a broad peak centered near Q = 1.3Å−1 (see Fig. 2(b) and
| |
plings,leadingtoeffectivelydecoupledsmalltetrahedra. This theSupplementalMaterial[45]). Thewidthinenergyofthe
canbecomparedtotheCr-basedBPcompounds,wherethe modes is close to instrumental resolution [45]. At elevated
smallandlargetetrahedraonlydifferinsizeby 5%[35–37]. temperatures(Fig.2(b)and(d)),threenewexcitationsbecome
∼
TocharacterizeBYZOspectroscopically,wehaveinvestigated visibleresultingfromtransitionsbetweenexcitedstates.
itslowenergyspinexcitationsusinginelasticneutronscatter- Theoriginoftheobservedlowenergyexcitationsappearsto
ing(INS). Weconfirmthepictureofnearlyindependenttetra- bemodesoriginatingfromdecoupledYbtetrahedra. Several
hedra,seeingnearlyresolutionlimiteddispersion-lessmodes piecesofevidencesupportthisassertion. Lowlyingcrystal
atlowtemperatures. ThisINSdata,combinedwiththether- field levels can be excluded as the origin of these modes as
modynamicmeasurementsofRef.[33],allowsforacomplete threehigherenergycrystalfieldlevelsareexperimentallyob-
andunambiguousdeterminationofthetheeffectivemodelfor served(themaximumnumberforYb3+)withthelowestlying
BYZO.Wefindthatasingletetrahedronpseudo-spinmodel levelat 38meV[34,45]. Themagneticsusceptibilityand
∼
canquantitativelyaccountforallofthecurrentexperimental specificheatdatadonotshowanysignsoflongrangemagnetic
data on BYZO, determining the four anisotropic exchanges orderdownto0.38K[33,45]thatwouldindicatecorrelations
as well as the g-tensor. In addition to the antiferromagnetic betweenthesmalltetrahedra. Examinationoftheelasticscat-
HeisenbergexchangepostulatedinRef.[33],wefindthatsig- teringat0.25Kisconsistentwiththisconclusion,revealingno
nificantDzyaloshinskii-Moriya(DM)exchangeisneededto indicationoflongrangemagneticorder.Finally,thelackofdis-
obtainthecorrectlevelstructuredeterminedfromINS. The persionsuggeststhatthesemodesariseprimarilyfromisolated
fittedexchangeparametersarefarfromthespinicelimitre- tetrahedraandthattheinteractionsconnectingthetetrahedra
centlyconsideredinRef.[38]orthepurelyHeisenberglimits areweak. Wenotethatthereisaweakandbroadfeatureat
studiedinRef.[39]. Instead,wefindthegroundstateofeach 1meV. Wehavebeenunabletoidentifytheoriginofthis
∼
tetrahedronisdoublydegenerate,consistentwiththeresidual feature,butnotethatithasa Q-dependence[45]distinctfrom
| |
entropyobservedexperimentallyatT 300mK[33]. These thatoftheothernearlyresolutionlimitedmodes.
∼
E-doublets are nearly non-magnetic, carrying both a scalar Theoreticalmodel: Wenowusetheseexperimentalobser-
spin-chiralityaswellasoctupolar,all-in/all-outmoments. The vations,alongwiththethermodynamicdatafromRef.[33]to
stateofBYZOatcurrentlystudiedbasetemperaturesisthus constructamodelofBYZO. Giventhedispersion-lessmodes
a type of “octupolar paramagnet” without significant inter- seenintheINS,andthelargeratior /r 2betweenthelarge
> <
∼
tetrahedroncorrelations. Notwithstandingthebroadagenda andsmalltetrahedronsizes,weexpectisolatedYb tetrahedra
4
ofaccuratelydeterminingtheanisotropicexchangesinrare- toprovideaverygooddescriptionofthelowenergyphysics.
earthpyrochlorematerials,thecompletecharacterizationofthe EachofthefourYb3+ ionshasaHund’srulegroundstateof
single-tetrahedronmodelshouldprovideausefulguideforfur- 2F ,withtheJ =7/2manifoldstronglysplitbytheC (3m)
7/2 3v
therexperimentalstudiesofBYZOandotherBPs.Specifically, crystallineelectricfieldenvironment. Sincethisenergyscale
weestimatethattheinter-tetrahedroncorrelationscouldbegin isverylarge, 38meV[34],relativetotheexpectedscaleof
∼
tosetinbelow500mK,attheedgeofcurrentlyexploredtem- theintra-tetrahedroninteractions,onlythegrounddoubletis
3
FIG.2. INSdata(E =3.8meV)andcomparisontoourtheoreticalmodelat(a-c)0.25Kand(d-f)20K.Theoveralltheoreticalintensityscale
i
wasfitusingtheconstantwavevectorcut(a)at0.25K. AGaussianbroadeningwithenergydependencefollowingtheexperimentalenergy
resolutionfunction[45]wasincludedinthetheoreticalcalculation.(a,d)CutoftheINSdataaveragedoverthewindow1.25Å−1< Q <1.35Å−1
| |
at(a)0.25Kand(b)20K.ResultsformodelofEq.(2)withfittedparametersofEq.(3)arealsoshown.(Inset,a)Anillustrationofthelevel
structureofthesingle-tetrahedronmodelandthepositionsofthetransitionsfromthegrounddoubletintotheexcitedstates. (b,e)Intensity
mapofthepowderaveragedINSdataat(c)0.25Kand(d)20K.Theexcitationsarenearlydispersionfreeoverthefull Q range.(c,f)Model
| |
calculationsfortheparametersofEq.(3)areshownat(e)0.25Kand(f)20K.TheYb3+formfactorwasevaluatedinthedipoleapproximation,
asgiveninRef.[46].
relevantatlowtemperatures. Thetwostatesofthisdoublet ispartlydeterminedbytetrahedralsymmetry.Thefour-pseudo-
defineaneffectivepseudo-spinSiateachofthefourYb3+sites. spinstatesbreakintotheirreduciblerepresentationsA2⊕3E⊕
Thispseudo-spinisrelatedtothemagneticmomentµ ateach T 2T undertheactionofthetetrahedralgroup. Thisgives
i 1⊕ 2
sitethroughtheg-factors,g andg ,presentduetothelocal alevelstructureofasinglet(A ),threedoublets(E)andthree
z 2
±
C symmetry. Explicitly, triplets(T orT ). Fromtheobservedresidualentropy[33],it
3v 1 2
seemsplausiblethatthegroundstateofthetetrahedronisan
µi ≡µB g± xˆiSix+yˆiSiy +gzzˆiSiz , (1) Edoublet,whichgivesanentropyofkBln(2)/4∼0.1733kB/
where(xˆi,yˆi,zˆi)aret(cid:104)he(cid:16)localaxesof(cid:17)tetrahedr(cid:105)onsitei [45]. Yb3+.
Regardlessofthedetailedcompositionofthegrounddoublet,
Best fit parameters: The model of Eq. (2), supplemented
sinceJ =7/2,theinteractionsbetweentheYb3+areexpected
withthedefinitionofthemomentinEq.(1),isdeterminedby
tobeanisotropicand,apriori,notnecessarilyneartheIsing
the six parameters J , J , J , J , g and g . To fix these
zz z z
or the Heisenberg limit [51]. Symmetry strongly constrains ± ±± ± ±
parameters,weperformafittothespecificheatandsuscepti-
their form; each Yb3+-Yb3+ bond has symmetry C (2mm)
2v bilitydataofRef.[33]andacutoftheINSdataaveragedover
andeachsmallYb tetrahedronhasfulltetrahedralsymmetry
T (4¯3m) [33, 49].4 Assuming an effective spin-1/2 doublet therange1.25Å−1 <|Q|<1.35Å−1at0.25K. Thisisaglobal
d fit,minimizingsquareddifferencesbetweenexperimentaland
[52], there are therefore four allowed anisotropic exchange
theoreticalvaluesfromeachsetofexperimentaldatasimulta-
interactions[15],takingtheform
neously. Forthespecificheat,wefitonlythedatabelow5K
4 tominimizetheinfluenceofthesubtractionofthelatticecon-
Heff ≡ JzzSizSzj−J± Si+S−j +Si−S+j + tribution,whilethesusceptibilitydataupto30Kisused. [53].
(cid:88)i=1 (cid:88)j<i (cid:104) (cid:16) (cid:17) Threeadditionalfittingparameterswereincluded;aconstant
J γ S+S++h.c. + J ζ SzS++S+Sz +h.c. , (2) shiftofthesusceptibility,χ0,toaccountfortheVanVleckand
±± ij i j z± ij i j i j diamagneticcorecontributionsoftheYb3+ions,theintensity
where(cid:16)thebonddepen(cid:17)dentph(cid:16)ases(cid:104)γ andζ are(cid:105)define(cid:17)d(cid:105)inthe scaleoftheINScutandtheoverallscaleoftheGaussianbroad-
ij ij
SupplementalMaterial[45]. ThespectrumofthisHamiltonian eningusedinthetheoreticalINSintensity[45]. Furtherdetails
4
ofthefittingmethodologyandcomparisonstoexperimental 0.10
C Model
datacanbefoundintheSupplementalMaterial[45].
3 χ Kimuraetal.
Fromthisanalysiswefindauniquebestfitwhichprovides 0.08
excellentagreementwithalloftheknownexperimentaldata
onBYZO. Thebestfitparametersare
]K 0.06 ]ol
Jzz =−0.037meV, J± =+0.141meV, /mol 2 /mum
J =+0.158meV, Jz =+0.298meV, [J 0.04 [e
±± ± C χ
g =2.36, g =3.07. (3)
z 1
±
0.02
Comparisontothespecificheatandsusceptibilityisshownin
Fig.3. Agreementwithbothisexcellent;smalldifferencescan
beseeninthespecificheatathighertemperatures,likelydueto 0 0.00
0 2 4 6 8 10 12 14
someuncertaintyinthesubtractionofthelatticecontribution.
Comparison to a cut of the INS data at 0.25 K is shown in T [K]
Fig.2(a),alongwithanillustrationofthelevelstructureofthe
singletetrahedronmodelwiththeparametersofEq.(3). The FIG.3. Comparisonofthe(magnetic)specificheat,C,andsuscep-
tibility,χ,ofKimuraetal.[33]tothemodelofEq.(2)withfitted
levelstructurematchesverywellwiththeenergiesofthepeaks
parametersofEq.(3). Aconstantshiftχ wasincludedinthefitof
intheINScutat0.25K. Explicitlyonehasthespectrum 0
thesusceptibilitytoaccountfortheVanVleckanddiamagneticcore
contributionsoftheYb3+ions.
E 0.000meV(E), E =0.754meV(E),
0 ≡ 3(cid:48)
E =0.382meV(A ), E =1.802meV(T ),
1 2 4 2
E =0.530meV(T ), E =1.802meV(E), Discussion: The physics at very low temperatures, T
2 1 4(cid:48) E , shouldbeprimarilycontrolledbytheground E doubl(cid:28)et.
E =0.806meV(T ), (4) 1
3 2 The states of this E doublet, , are rather exotic. As in
|±(cid:105)
theHeisenberglimit,theyarelargelynon-magnetic,carrying
where the irreducible representation in T of each level is
indicated. Wenotethatthe E and E levdelsareveryclose a uniform (scalar) spin-chirality κ S˜i (S˜j S˜k) 0.4
4 4(cid:48) on each triangle of the tetrahedron≡[3(cid:104)3].·How×ever,(cid:105)d∼ue to
in energy, but not exactly equal. The model also accurately
the large DM interaction, the states additionally acquire all-
reproduces the wave vector and temperature dependence of
in/all-out(AIAO)moments. Thisisgenericallyexpectedas
the INS data as can be seen in Fig. 2(c),(d),(f). Additional
theAIAOmomentsandtheuniformspin-chiralitytransform
comparisonstomagnetizationandINSdatacanbefoundin
identicallyunderthetetrahedralsymmetry[42,43]. Explicitly,
theSupplementalMaterial[45]. Someofthefeaturesofthese
energy levels can be better understood by adopting global theprojectionofapseudo-spinSi inthelocalbasisintothe
quantizationaxesanddefiningglobalpseudo-spinoperatorsS˜i. E doublettakestheform(cid:104)±|Si|±(cid:105) = ±λzˆ withλ ∼ 0.13for
UsingthenotationofRef.[54],themodelintheglobalbasisis the parameters of Eq. (3) and Si = 0. These AIAO
(cid:104)±| |∓(cid:105)
moments are octupolar in character, with the net magnetic
parametrizedbyfouranisotropicexchangesJ ,J ,J andJ .
1 2 3 4
momentoneachtetrahedronvanishing. Duetothesmallness
ThebestfitparametersofEq.(3)correspondtothevalues[45]
oftheinter-tetrahedroninteractionswethusexpectBYZOto
J =+0.587meV, J =+0.573meV, beanoctupolarparamagnetattemperaturesmuchsmallerthan
1 2
J = 0.011meV, J = 0.117meV. (5) E1. Directsignaturesofthisunusualparamagneticstatemay
3 − 4 − appearinmoreindirectmagneticprobes,suchasthenon-linear
Since J J J and J 0toafairapproximation,these susceptibilities.
1 ∼ 2 ≡ 3 ∼
fittedparametersdescribeaHeisenbergantiferromagnetsup- Goingtoevenlowertemperaturesonecanpotentiallysee
plementedwithlarge(indirect)DMinteractionD √2J indicationsofcollectivebehaviorofthesmalltetrahedra. De-
≡ 4 ∼
0.28J [45, 55] and negligible symmetric anisotropies. We pendingonthestructureoftheinter-tetrahedroninteractions,a
−
canthusunderstandtheEdoubletgroundstateasanextension varietyofstatescouldbestabilized,suchasweakAIAOorder
of the pair of S = 0 singlets that form the ground state in or non-magnetic valence bond solid phases [42, 43]. Tanta-
theHeisenberglimit[33]. Similarly,theapproximatequintet lizinghintsoftheonsetofsuchcorrelationsmayalreadybe
E E canbemappedtothehighenergy,five-folddegen- presentintheexperimentaldata. Wenotethatthe INSdata
4 ∼ 4(cid:48)
erateS =2statesoftheantiferromagneticHeisenbergmodel. isslightlybroaderthancalculatedinstrumentalresolution(by
Indeed,whenonlyHeisenbergandDMinteractionsarepresent 0.01 meV) which may be suggestive of weak dispersion,
∼
theseremainexacteigenstatesanddegenerate,leavingonlythe while the specific heat data of Kimura et al. [33] shows a
smallsymmetricanisotropiestoprovideanysplitting. While slight upturn below 500 mK that is not explained by the
∼
thismappingisappealing,therearekeydifferences;forexam- single-tetrahedron model. We thus suspect that the current
ple,thethreeS = 1tripletspresentintheHeisenbergmodel lowest temperatures explored in BYZO are at the threshold
arestronglymixedbytheDMinteractions. ofobservingsuchinter-tetrahedroncorrelationsandpossibly
5
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UnitedStatesGovernmentpurposes.TheDepartmentofEnergy [26] H.Kadowaki,H.Takatsu,T.Taniguchi,B.Fåk, andJ.Ollivier,
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7
SupplementalMaterialfor“Anisotropicexchangewithindecoupledtetrahedrainthequantum
breathingpyrochloreBa Yb Zn O ”
3 2 5 11 ∗
J. G. Rau,1, L. S. Wu,2, A. F. May,3 L. Poudel,2,4 B. Winn,2 V. O. Garlea,2 A. Huq,5 P.
† ‡
Whitfield,5 A. E. Taylor,2 M. D. Lumsden,2 M. J. P. Gingras,1,6,7 and A. D. Christianson2,4
1DepartmentofPhysicsandAstronomy, UniversityofWaterloo, Ontario, N2L3G1, Canada
2QuantumCondensedMatterDivision, OakRidgeNationalLaboratory, OakRidge, TN-37831, USA
3MaterialsScience&TechnologyDivision,OakRidgeNationalLaboratory,OakRidge,TN-37831,USA
4Department of Physics & Astronomy, University of Tennessee, Knoxville, TN-37966, USA
5Chemical&EngineeringMaterialsDivision,OakRidgeNationalLaboratory,OakRidge,TN37831,USA
6Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
7CanadianInstituteforAdvancedResearch,180DundasStreetWest,Suite1400,Toronto,ON,M5G1Z8,Canada
(Dated:January16,2016)
I. SAMPLESYNTHESIS
4
Polycrystalline samples of Ba Yb Zn O were synthe-
3 2 5 11 Present study
sized by solid-state reaction in Al O crucibles. The high-
2 3 Kimura et al.
purityreactants(driedYb O ,BaCO ,ZnO)weregroundto-
2 3 3
getherfor5-10minutesusinganagatemillingsetinaSPEX K]
SamplePrep Mixer/Mill. The mixture was pressed into pel- Yb Cp
lets, whichwereinitiallyfiredat1150◦Cfor25-50h(inair). ol
m 2
Subsequentmilling,pelletpressing,andannealingattemper- J/
atures up to 1170◦C were utilized to promote homogeneity C [
andphasepurityinthefinalproduct. Aslightexcess(upto4
at.%)ofBaandZn-containingreactantswasutilizedtomini- Cm=Cp-Cl
mizethechanceofformingYb-containingimpurities.
0
0 5 10 15
T [K]
II. SPECIFICHEATANDMAGNETICSUSCEPTIBILITY
FIG.S1.ThetemperaturedependentspecificheatofBa Yb Zn O .
Magnetization measurements were performed upon cool- 3 2 5 11
Filled circles are from measurements of a piece taken from the
ing in an applied field of 0.1 T, and isothermal magnetiza-
samebatchasusedfortheinelasticneutronscatteringmeasurements.
tion measurements were performed at 1.9 K; Quantum De- Opencirclesarethemagneticcontributiontothespecificheat(C )
m
sign’s Magnetic Property Measurement System was utilized taken from Kimura et al. [S1], determined C by subtracting the
m
formagneticmeasurements.Specificheatmeasurementswere latticecontribution(C)estimatedfromBa Lu Zn O fromthespe-
l 3 2 5 11
performedinaQuantumDesignPhysicalPropertyMeasure- cificheatofBa3Yb2Zn5O11(Cp).
mentSystem.
The specific heat of Ba Yb Zn O is shown in Fig. S1.
3 2 5 11
Thespecificheatofasamplefromthesamebatchusedforthe The magnetic susceptibility and inverse susceptibility of a
inelastic neutron scattering measurements is compared with sampletakenfromthesamebatchasthesampleusedforthe
thatfrom Kimuraetal.[S1]. Bothmeasurementsareconsis- inelastic neutron scattering data are shown in Fig. S2 for an
tentwithamaximumat 2.4K. appliedfieldof0.1T.Amaximuminthesusceptibilityoccurs
∼
at 4K.
∼
∗This manuscript has been authored by UT-Battelle, LLC under Con- III. NEUTRONDIFFRACTION
tract No. DE-AC05-00OR22725 with the U.S. Department of Energy.
The United States Government retains and the publisher, by accepting
thearticleforpublication, acknowledgesthattheUnitedStatesGovern- Neutron powder diffraction measurements of
ment retains a non-exclusive, paid-up, irrevocable, world-wide license Ba Yb Zn O were performed with the time-of-flight
3 2 5 11
to publish or reproduce the published form of this manuscript, or al- powder diffractometer POWGEN, at the Spallation Neutron
low others to do so, for United States Government purposes. The De-
Source (SNS) at Oak Ridge National Laboratory [S2]. Data
partment of Energy will provide public access to these results of feder-
werecollectedonapowderBa Yb Zn O samplewithmass
allysponsoredresearchinaccordancewiththeDOEPublicAccessPlan 3 2 5 11
(http://energy.gov/downloads/doe-public-access-plan). 6.32 g. The data were collected for 2 hours at temperatures
†jeff[email protected] 10 K and 300 K, respectively. Structural refinement was
‡[email protected] carriedoutusingthesoftwarepackageFULLPROF[S3]
8
2
0.08 200
0.07
Yb] B = 0.1 T
mol 0.06
Yb] 0.06 [emu/0.05 150 -1 Yb]
mol 0.040 5 10 15 mol
mu/ 0.04 T (K) 100 mu/
[e [e
1/
0.02 50
0.00 0
0 100 200 300
T [K]
FIG.S2. Leftaxis: Thetemperaturedependentstaticmagneticsus-
ceptibility(χ= M/B)forBa Yb Zn O (bluecircles),measuredin
3 2 5 11
fieldB = 0.1Tfromtemperature1.9to320K. Rightaxis: Thein-
versemagneticsusceptibility(1/χ)asafunctionoftemperature.The
insetshowsanexpandedviewofthelowtemperatureregion,where
amaximuminthesusceptibilityoccursaround4K.
Atom Wyckoff x y z Biso Occ. FIG.S3. NeutronpowderdiffractiondatacollectedwithPOWGEN
Ba 24f 0.7055(3) 0.00000 0.00000 0.1313(0) 0.25000 at10K(a)and300K(b)forBa3Yb2Zn5O11. Rietveldrefinement
Yb 16e 0.8365(5) 0.8365(5) 0.8365(5) 0.1187(9) 0.16670 (redline),differencepattern(blueline)andcalculatedreflectionpo-
Zn(1) 16e 0.0832(5) 0.0832(5) 0.0832(5) 0.1155(4) 0.16670 sitions(greenticks)aresuperimposedonthedatapoints(blackcir-
Zn(2) 24g 0.25000 0.25000 0.0828(1) 0.1729(2) 0.25000 cles).Theinsetsshowanexpandedviewofthehigh Q regionofthe
O(1) 4b 0.50000 0.50000 0.50000 0.4219(9) 0.04167 data. | |
O(2) 4a 0.00000 0.00000 0.00000 0.2334(2) 0.04167
O(3) 16e 0.3419(3) 0.3419(3) 0.3419(3) 0.3313(9) 0.16670
T (K) a(Å) R R R χ2
O(4) 16e 0.6679(6) 0.6679(6) 0.6679(6) 0.3777(2) 0.16670 p wp exp
O(5) 48h 0.1660(2) 0.1660(2) 1.0000(4) 0.30015(0) 0.50000 10 13.47117(3) 9.07 8.63 1.53 31.8
300 13.48997(3) 10.7 9.23 1.99 21.5
Atom Wyckoff x y z Biso Occ. TABLES2.Comparisonoftherefinementparametersat10and300
K.
Ba 24f 0.7055(6) 0.00000 0.00000 0.7268(4) 0.25000
Yb 16e 0.8365(5) 0.8365(5) 0.8365(5) 0.4480(3) 0.16670
Zn(1) 16e 0.0832(9) 0.0832(9) 0.0832(9) 0.4184(2) 0.16670
Zn(2) 24g 0.25000 0.25000 0.0824(7) 0.5697(8) 0.25000 experimental resolution. The refined atomic parameters of
O(1) 4b 0.50000 0.50000 0.50000 0.8089(3) 0.04167 Ba Yb Zn O at10Kand300KareshowninTableS1.The
3 2 5 11
O(2) 4a 0.00000 0.00000 0.00000 0.4878(2) 0.04167 latticeconstantsandgoodnessoffitparametersaredisplayed
O(3) 16e 0.3421(4) 0.3421(4) 0.3421(4) 0.6450(5) 0.16670
inTableS2.
O(4) 16e 0.6683(0) 0.6683(0) 0.6683(0) 0.7374(4) 0.16670
O(5) 48h 0.1659(4) 0.1659(4) 1.0001(6) 0.6505(3) 0.50000
TABLE S1. Atomic parameters for Ba Yb Zn O at 10 K (top IV. INELASTICNEUTRONSCATTERING
3 2 5 11
panel)and300K(bottompanel).
Inelastic neutron scattering (INS) experiments were per-
formed on the hybrid spectrometer (HYSPEC) at the Spalla-
The neutron diffraction data at 10 and 300 K along with tionNeutronSourceatOakRidgeNationalLaboratory[S5].
the Rietveld refinement of the structural model is shown in Thedatawerecollectedat0.25K,10K,and20Kutilizinga
Fig.S3(a)and(b)respectively. Thefittedmodeldescribesthe 3Herefrigerator,withincidentenergies E =3.8,7.5,and15
i
datawelloverawidewavevectorrange(1.3< Q <21Å−1). meVandFermichopperfrequenciesof180,300,and300Hz
| |
Afewunindexedimpuritypeakswithintensitieslessthan1% respectively. To cover a large region of reciprocal space the
ofthemaindiffractionpeaksofBa Yb Zn O areobserved. center of the detector vessel, which covers 60 of scattering
3 2 5 11 ◦
Thesmallfractionofimpuritiesappearstobeconsistentwith angle,wasplacedatscatteringanglesrangingfrom33-101 .
◦
that found by Kimura et al. [S1] and indicates that the sam- ShowninFig.S4(a),andFig.S4(b)aretheintensitymaps
pleconsistsprimarilyofthecubicBa Yb Zn O phase. No oftheinelasticneutronscatteringdataofBa Yb Zn O with
3 2 5 11 3 2 5 11
site vacancies or disorder between sites was detected within E = 15 meV measured at temperatures 0.25 K and 10 K.
i
9
3
FIG.S4.IntensitymapoftheinelasticneutronscatteringfromBa Yb Zn O withE =15meVat0.25K(a)and10K(b).
3 2 5 11 i
asmallpeakaround Q =0.7Å−1.
| |
As a supplement to the data collected at 0.25 K and 20 K
12
withE =3.8meVdescribedandshowninthemaintext,data
T = 0.25 K = [1, 1.25] meV i
collectedwith E =3.8meVat10KisshowninFig.S6be-
Ei = 3.8 meV = [0.65, 0.9] meV i
low. TheHYSPECinstrumentalenergyresolutionwith E =
i
b.] 3.8meVandthe180Hzchoppersettingusedexperimentally
y [ar 8 is shown in Fig. S7 as a function of energy transfer. The |Q|
sit dependence of the inelastic spectrum is shown through a se-
en riesofcutswithanenergyrangeof[0.65,0.9]meVattemper-
Int aturesof0.25,10,and20KinFig.S8.
4
10
V. CRYSTALFIELDEXCITATIONS
0
0.0 0.5 1.0 1.5 2.0 2.5
-1 Inelastic neutron scattering data was collected with the
|Q| [A ]
ARCS[S6]time-of-flightspectrometertoprobetheexcitation
spectrumathigherenergies. Thisdatawascollectedat10K
FIG. S5. Wave vector Q dependent neutron scattering intensity
measured at 0.25 K with| in|cident energy Ei = 3.8 meV, averaged with Ei = 100 meV and shows three crystal field excitations
overtheexcitationenergywindowω=[0.65,0.9]meV(redpoints) at 38,54,and67meV.Thedatapresentedhereareconsis-
∼
andω=[1,1.25]meV(blackpoints). tentwiththeresultsofRef.[S7]whereamoredetailedanal-
ysis of the crystal field excitation spectrum and Hamiltonian
can be found. We note that for a Kramers ion such as Yb3+
Althoughthelowenergyexcitationsarenotwellresolvedwith (J = 7/2), in the absence of broken time reversal symmetry,
E =15meV,thedatashowninFig.S4confirmthatthereis theminimumdegeneracyistwosothat(2J+1)/2doubletsare
i
noadditionalexcitationsupto13.75meV,whichisconsistent expectedincludingthegroundstate. Thustheobservationof
withthemodeldescribedinthemaintext. threecrystalfieldexcitationsisstrongevidencethatthemodes
As mentioned in the main text, we noticed that there is a observedatenergieslessthan 2meVdiscussedinthemain
weakbroadfeatureintheINSspectrumnearanenergytrans- paper are due to interactions ∼between the Yb3+ within each
fer of 1 meV (main text Fig. 2(a)). However, this feature tetrahedron.
has a significantly different dependence on wave vector Q
| |
from that of the other observed modes. Fig. S5 shows the
wave vector Q dependent neutron scattering intensity mea-
| | VI. THEORETICALDETAILS
sured at 0.25 K with incident energy E = 3.8 meV, aver-
i
agedovertheexcitationenergywindowω = [0.65,0.9]meV
andω = [1,1.25]meV.Incontrasttothewelldefinedinelas- A. Model
ticmodecontainedintheenergyrangeω = [0.65,0.9]meV,
whichpeaksnear Q =1.3Å−1,thebroadfeaturecapturedby For completeness, we state our model and conventions in
| |
theintegrationrange,ω=[1,1.25]meVismuchweakerwith more detail. We consider the effective anistoropic exchange
10
4
2.0 10
Model 10K 10K Data 10K Model 9
Data
4 8
1.5
b.) 7
Intensity(ar2 [meV]ω1.0 3456
0.5
2
1
(a) (b) (c)
0 0.0 0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
ω[meV] |Q|[A˚−1] |Q|[A˚−1]
FIG.S6. Inelasticneutronscatteringdata(E=3.8meV)andcomparisontotheoreticalmodelat(a-c)10K. Theoveralltheoreticalintensity
i
scalewasfitusingtheINScutat0.25K.AGaussianbroadeningwithenergydependencefollowingtheexperimentalenergyresolutionfunction
(showninFig.S7)wasincludedinthetheoreticalcalculation.(a)CutofINSdataaveragedoverthewindow1.25Å−1< Q <1.35Å−1.Results
| |
forthebestfitsingletetrahedronmodelofthemaintextareshown.(b)IntensitymapofpowderaveragedINSdata.Theexcitationsarenearly
dispersionfreeoverthefull Q range.(c)Modelcalculationsforthebestfitsingletetrahedronmodel.TheYb3+formfactorwasevaluatedin
| |
thedipoleapproximation,asgiveninRef.[S4].
8
0.25K Model
0.15 10K Data
Ei = 3.8 meV 20K
6
Fermi Chopper Frequency 180 Hz
] )
V b.
e r
m0.10 (a
ution [ ensity 4
ol nt
s I
e
R0.05 2
0
0.00 0.0 0.5 1.0 1.5 2.0 2.5
0 1 2 3 4
[meV] Q [A˚−1]
| |
FIG. S8. The Q dependence of the inelastic neutron scattering
| |
FIG. S7. HYSPEC resolution as a function of the energy trans- dataforBa Yb Zn O , averagedovertheexcitationintheenergy
3 2 5 11
fer ∆ω, with incident energy Ei = 3.8 meV and Fermi chop- window0.65meV<ω<0.9meV.
per frequency of 180 Hz. The red line is the fit to the empiri-
cal equation y = Ae−∆ω/Γ with A = 0.12398 0.0004 meV, and
±
Γ=2.95111 0.02144meV.
±
connectandthuscanbeexpressedasamatrix
modelinthelocalbasisdefinedas 0 +1 ω ω2
+1 0 ω2 ω
γ= ω ω2 0 +1, (S2)
Heff ≡Xi=41 Xj<i hJzzSizSzj−J±(cid:16)Si+S−j +Si−S+j(cid:17)+ ω2 ω +1 0 
J±± γijSi+S+j +h.c + Jz± ζij SizS+j +Si+Szj +h.c whereω=e2πi/3.Themagneticfieldiscoupleddirectlytothe
(cid:16) 4 (cid:17) (cid:16) h i (cid:17)i effectivemomentoneachYb3+site,definedas
−µBB· g± xˆiSix+yˆiSiy +gzzˆiSiz , (S1)
Xi=1 h (cid:16) (cid:17) i µi ≡µB g± xˆiSix+yˆiSiy +gzzˆiSiz , (S3)
h (cid:16) (cid:17) i
with four symmetry allowed exchanges J , J , J and J
zz z
± ±± ±
andexternalmagneticfieldB. Thecomplexbondphasefac- whereg andg aretheg-factorsinthelocal[111]direction
z
tors γij and ζij = −γi∗j depend only on the basis sites they andintheplane±perpendiculartoit. Theselocalaxesarede-