Breathing Pyrochlore Lattice Realized in A-Site Ordered Spinel Oxides LiGaCr4O8 and LiInCr4O8

0.37 MB·English

by  Yoshihiko Okamoto

Breathing Pyrochlore Lattice Realized in A-Site Ordered Spinel Oxides LiGaCr O
4 8
and LiInCr O
4 8
Yoshihiko Okamoto1,∗, Gøran J. Nilsen1,†, J. Paul Attfield2, and Zenji Hiroi1
1Institute for Solid State Physics,
University of Tokyo, Kashiwa 277-8581, Japan
2Centre for Science at Extreme Conditions and School of Chemistry,
University of Edinburgh,
Edinburgh EH9 3JZ, United Kingdom
3
(Dated: January 30, 2013)
1
0 A unique type of frustrated lattice is found in two A-site ordered spinel oxides, LiGaCr4O8 and
2 LiInCr4O8. Because of the large size mismatch between Li+ and Ga3+/In3+ ions at the A site,
n the pyrochlore lattice, made up of Cr3+ ions carrying spin 3/2, becomes an alternating array of
a small and large tetrahedra, i.e., a “breathing” pyrochlore lattice. We introduce a parameter, the
J breathingfactorBf,whichquantifiesthedegreeoffrustrationinthepyrochlorelattice: Bf isdefined
9 asJ′/J,whereJ′andJarenearest-neighbormagneticinteractionsinthelargeandsmalltetrahedra,
2 respectively. LiGaCr4O8 withBf ∼0.6showsmagneticsusceptibilitysimilartothatofconventional
CrspineloxidessuchasZnCr2O4. Incontrast,LiInCr4O8 withasmallBf ∼0.1exhibitsaspin-gap
] behavior in its magnetic susceptibility, suggesting a proximity to an exotic singlet ground state.
i
c Magnetic long-range order occurs at 13.8 and 15.9 K for LiGaCr4O8 and LiInCr4O8, respectively,
s in both cases likely owing to thecoupling to structural distortions.
-
l
r PACSnumbers: ValidPACSappear here
t
m
t. Transition metal oxides AB2O4 crystallizing in the (cid:12)(cid:4)(cid:13) (cid:12)(cid:14)(cid:13)
a
spinel structure provide us with a rich playground for
m (cid:8)(cid:9) (cid:16)
studying the physics of geometrical frustration. Transi- (cid:11)
(cid:16)(cid:15)
-
d tion metal B atoms, which are octahedrally coordinated
n byoxideions,formathree-dimensionalnetworkoftetra- (cid:1)(cid:2) (cid:10)
(cid:1)(cid:2)
o hedra, i.e., the pyrochlore lattice. Various interesting (cid:3)(cid:4)(cid:5)(cid:6)(cid:7) (cid:10)(cid:15)
c phenomena have been observed arising from geometri-
[
calfrustrationconcerning the spin andchargedegreesof

1 freedom on this lattice. Typical examples are the Ver-
v
wey transition in Fe O [1, 2], a heavy-Fermion state in
3 4
6 LiV O [3], and a heptamer formation in AlV O [4]. FIG.1: (coloronline)(a)CrystalstructureofLiGaCr4O8and
3 2 4 2 4 LiInCr4O8. Coordination polyhedra made of oxide ions are
9 ACr2O4 with a nonmagnetic A2+ ion, such as Zn2+, depicted. (b)Breathing pyrochlorelatticemadeofCr3+ ions
6 Mg2+, Cd2+, or Hg2+ at the tetrahedral site, and Cr3+ embedded in the two compounds. Cr-Cr bonds on the small
1. ions at the octahedral site is of particular interest as a (filled sticks) and large tetrahedra (open sticks) have bond
0 frustrated spin system [5]. It is a Mott insulator with lengthsdandd′ andantiferromagneticinteractionsJ andJ′,
3 three 3d electrons localized at Cr3+, yielding localized S respectively.
1
= 3/2 Heisenberg spin. Various magnitudes of antifer-
:
v romagnetic interactions occur between nearest-neighbor
Xi spins, as evidenced by a range of negative Weiss tem- conventionalspinel oxides;an inversioncenter at the oc-
peratures of −390, −370, −70, and −32 K for A = Zn, tahedral site present in Fd¯3m is missing in F¯43m. A
r
a Mg, Cd, and Hg, respectively [6, 7]. ACr O undergoes structural model was proposed in which Li and Ga/In
2 4
antiferromagnetic long-range order at 12, 12.4, 7.8, and atoms alternately occupy the tetrahedral sites, resulting
5.8 K, respectively [6–8], which is accompanied by a lat- inthezinc-blendetypearrangement,althoughstructural
tice distortionwhichlowersthe crystalsymmetry [8–10]. refinements were not performed [11]. This type of A-site
Plausibly, there is an inherent structural instability in order is likely, because it minimizes electrostatic energy
the spinelstructure thatcancouple with the spindegree arisingfromthe largedifference in the valence states be-
of freedom so as to lift the magnetic frustration. tween Li+ and Ga3+/In3+.
In this Letter, we study two spinel oxides,LiGaCr O WeareinterestedintheCrpyrochlorelatticesofthese
4 8
andLiInCr O ,whichbothcontaintwometalionsatthe compounds, because the local chemical pressure caused
4 8
Asite. JoubertandDurifpreparedthemin1966[11]and by the difference in ionic radii of Li+ and Ga3+/In3+
found that they crystallize ina modified spinel structure should result in the Cr tetrahedra expanding and con-
4
with space group F¯43m, a subgroup of Fd¯3m for the tracting alternately, whilst keeping their shapes regular,
2
unit) (a) LiGaCr4O8 TLtriAoInnBCLdrEi4ffOrIa8:c(Ctbirooyntsh.taFTll¯4oh3gemral)aptdhteiiccteeprmcaorinanmsetdaetnbetyrsimsfeoaarn=LsioG8f.a2pC5o5wr14d(Oe7r8)naaennudd-
b. 8.4205(5) ˚A, respectively. The range of R factors obtained
sity (ar (b) LiInCr4O8 atecnrotswsiatlhlbaaLneksBoafildfiattaoafroenglyivtehne,amnadinarpehfaosuen.dBtoisbtehceotnhseisr--
en mal displacement parameter. The following constraints are
nt assumed: B(Ga1) = B(Ga2), B(In1) = B(In2), and B(Li1)
I
= B(Li2).
0.8 1.0 1.2 1.4 1.6
d-spacing (Å) x y z Occ. B (˚A2)
LiGaCr4O8 (Rp = 3.72-4.77, Rwp = 4.79-6.49)
Li1 4a 0 0 0 0.994(7) 1.96(24)
FIG.2: (coloronline)Powderneutrondiffractionpatternsfor
Ga1 4a 0 0 0 0.006(7) 0.53(4)
LiGaCr4O8 (a)andLiInCr4O8 (b)measuredonbank6(2θ= Li2 4d 3/4 3/4 3/4 0.006(7) 1.96(24)
154◦)oftheGEMdiffractometeratroomtemperature. Filled
Ga2 4d 3/4 3/4 3/4 0.994(7) 0.53(4)
circles are experimental data, and vertical bars indicate the
Cr 16e 0.3728(3) x x 1 0.33(3)
positionsofBraggreflections. Thecurveonthedatashowsa
O1 16e 0.13649(14) x x 1 0.44(3)
calculated pattern, and the bottom curve shows a difference
O2 16e 0.61889(13) x x 1 0.37(3)
plot between the experimental and calculated intensities.
LiInCr4O8 (Rp = 4.12-5.84, Rwp = 5.92-7.11)
Li1 4a 0 0 0 0.992(11) 1.09(28)
In1 4a 0 0 0 0.008(11) 0.35(11)
as shown in Fig. 1(b). We call this type of lattice Li2 4d 3/4 3/4 3/4 0.008(11) 1.09(28)
the “breathing” pyrochlore lattice. The resulting mod- In2 4d 3/4 3/4 3/4 0.992(11) 0.35(11)
Cr 16e 0.3719(3) x x 1 0.14(3)
ulation in bond lengths produces two kinds of nearest-
O1 16e 0.1377(2) x x 1 0.38(4)
neighbormagneticinteractions,J andJ′,withoutreliev-
O2 16e 0.61069(14) x x 1 0.18(4)
ing frustration. The spin Hamiltonian of the breathing
pyrochlorelatticecanthusbewrittenasH =JΣ S ·S
ij i j
+ J′Σ S · S , where the summations over ij in the
ij i j
firstandsecondtermsrunoverthe Cr-Crbondsofsmall mentwasperformedusingthe Fullprofprogramon4out
and large tetrahedra, respectively; a uniform pyrochlore ofthe 6 constantanglebanks ofexperimentaldata using
antiferromagnet is yielded when J = J′, and isolated common structural parameters. Magnetic susceptibility
tetramers are realized for J′ = 0. and heat capacity measurements were performed in an
A theoretically predicted groundstate for the uniform MPMSandPPMS(bothQuantumDesign),respectively.
pyrochlore antiferromagnet is a spin liquid with a finite Powder neutron diffraction patterns taken at room
spin gap [12–14]. This state has not yet been evidenced temperature for polycrystalline samples of LiGaCr O
4 8
experimentally because actual compounds always suffer and LiInCr O are shown in Fig. 2. In addition to re-
4 8
fromvariousperturbationssuchaslatticedeformationor flections expected for Fd¯3m, forbidden reflections that
defects. In the weak coupling limit, J′ = 0, the ground are allowed for F¯43m, such as 002, are observed. Ri-
state is also a gapped state, but with a considerably etveld refinements using data from 4 banks were at-
larger gap due to singlet formation on isolated tetrahe- tempted starting from either case of full occupation of
dra. Itwouldthereforebe intriguingtoexaminehowthe Li at the 4a site and Ga/In at the 4d site, as in the case
twostatesareconnectedasafunctionofB =J′/J inthe of LiFe Rh O [15], or vice versa. Irrespective of the
f 2 3 8
breathing pyrochlorelattice. LiGaCr O and LiInCr O initial conditions, the refinements converged in a model
4 8 4 8
are apparently the right compounds to study this issue. in which the 4a and 4d sites are fully occupied by Li
The study on the breathing pyrochlore lattice would en- and Ga/In, respectively, within error bars. This is con-
able us to get a novel insight on the ground state of the sistent with the model by Joubert and Durif [11]. De-
uniform pyrochlore lattice as well as to explore a new tails of the refinement are given in Table I. The lattice
phenomenon caused by the breathing. parameters obtained are 8.2551(7) and 8.4205(5) ˚A, re-
Polycrystalline samples of LiGaCr O and LiInCr O spectively, which are similar to 8.243(3) and 8.411(3) ˚A
4 8 4 8
were prepared by the solid state reaction method. reported previously [11]. Thus, we have confirmed that
Li CO , Cr O , and Ga O /In O powders were mixed the F¯43mmodelwithperfectA-site orderisappropriate
2 3 2 3 2 3 2 3
in 1:4:1 molar ratio, then the mixture was sintered at forthetwocompoundsandsuccessfullyobtainedreliable
1000◦Cforonedayandat1100◦Cforanotherday. En- atomic coordinates.
ergy dispersive powder neutron diffraction experiments The zinc blende type order of small Li+ and large
were carried out at room temperature on the General Ga3+/In3+ ions on the 4a and 4d sites gives rise to a
Materials (GEM) diffractometer at the ISIS pulsed neu- local chemical pressure on the Cr pyrochlore lattice, re-
tron source. Rietveld analysis for the structural refine- sulting in an alternation of the size of the Cr tetrahe-
4
3
5 (b)22..86 LiGaCr4O8 −1ol-Cr) 860000 TN TN
(a) 2.4 µH = 5 T m
−-331χ (10cm mol-Cr) 4321 0ZnC1T0r20 O(IKn4)2L00iGGaa3C0r04LOi5432I8n0000C0000rcm(mol-Cr) χ4O31−−8 -3−-33110χ (10cm mol-Cr)(µc / Cr))B222222......642020210 LL0iiµIGn05HaCC0rT =40r. O04( .051O1K18.0 0 8TT)T11 0TTT.11N1T5 TNT FiorsteafyIsmGLednpi.Gitl4ve:aid,dCa(re−c2terdC / T (mJ K oa4splpObomeyr842ce00otata000ines0nvmluedinrlpyeeL.de)irITaantte1Cum0mrr4epaOegCrn8apT.et/ t2u(FTi0KLcriel)iffilIoendeLrdCelidprpGa4se3Oonan0olCd8ydfrce4o0rOnpya8cesnetn4dao0lsfl9yihnmTeeabtfsooaclrmsaeprpaealcpechs--
00 100 T (K2)00 300 -2M (10 00 2 LiInCr44O8
µH (T)
0 ZnCr O , as indicated in Fig. 3(a). It shows a broad
2 4
peak at ∼45 K, which represents a development of anti-
FIG. 3: (color online) (a) Temperature dependence of mag- ferromagneticshort-rangeorder. InthecaseofZnCr O ,
2 4
netic susceptibility χ measured in a magnetic field of 1 T for the correspondingbroadpeak appearsat∼30 K,reflect-
polycrystalline samples of LiGaCr4O8 and LiInCr4O8. The ing stronger magnetic interactions in LiGaCr O . On
4 8
data for a polycrystalline sample of ZnCr2O4 are also shown furthercooling,χdecreasessteeplyat∼14K,wherelong-
forcomparison[6]. Solidlinesarecalculatedcurvesproduced
by classical Monte Carlo simulations with J′ = 0.5J (J = range order sets in, as also evidenced by a sharp peak in
53.3 K) for the Ga data and by the exact result for an iso- heat capacity at TN = 13.8 K (Fig. 4), close to that
lated tetrahedron (J = 56.8 K) for the In data. The inset of ZnCr2O4. Therefore, the breathing of the pyrochlore
shows inversesusceptibilities with Curie-Weiss fits. (b) Tem- latticeinLiGaCr O haslittle influenceonthe magnetic
4 8
peraturedependencesoffield-cooledandzero-field-cooledχ’s properties.
measured at various magnetic fields for LiGaCr4O8 (upper) In contrast, the χ of LiInCr O shows a temperature
and LiInCr4O8 (lower). (c) M-H curves for LiGaCr4O8 and 4 8
dependence which is obviously distinguishable from that
LiInCr4O8 measuredat5K.Thebrokenlineshowsaninitial
straight line. of the Ga analogue: it rapidly decreases with decreas-
ing temperature below 65 K, which is reminiscent of the
opening of a spingap. The gapsize is roughlyestimated
dra. The Cr-Cr distances of the small and large tetrahe- to be ∆ = 56.8(2) K by a fit to the exact result for an
dra, denoted as d and d′, are found to be 2.867(4) and isolated tetrahedron of S = 3/2, where the diamagnetic
2.970(4) ˚A for LiGaCr O , and 2.903(4) and 3.052(4) contribution of core electrons of −4.0 × 10−5 cm3 mol-
4 8
˚A for LiInCr O , respectively. The differences between Cr−1 is included [16], as shown in Fig. 3(a). At yet
4 8
d and d′ are 3.5% and 4.9%, respectively, meaning a lower temperature, a sharp peak in C /T (Fig. 4) indi-
p
stronger alternation or breathing in LiInCr O than in cateslong-rangeorderatT =15.9K.Basedonthis,we
4 8 N
LiGaCr O . believethe groundstateofLiInCr O tobe inproximity
4 8 4 8
Figure3(a)showsthetemperaturedependenceofmag- to a spin-gapped state, despite the long-range order at
netic susceptibility χ for LiGaCr O and LiInCr O . low temperature. As shown in Fig. 3(b), the χ of each
4 8 4 8
The inverse of χ, shown in the inset, exhibits a linear compound is nearly insensitive to magnetic fields below
temperature dependence above ∼100 K, following the 1 T, beyond which it suddenly increases at 5 T below
Curie-Weiss law χ = C/(T −θW), where C and θW are TN. Thisisclearlyobservedinthe M-H curvesshownin
the Curie constant and the Weiss temperature; a fit to Fig. 3(c), where an upturn from the initial straight line
thedatabetween200and350KyieldsC =2.025(3)cm3 appears at approximately 2 T in each compound. This
K mol-Cr−1 and θ = −658.8(4)K for LiGaCr O , and maybe dueto aspinfloptransitioninthe orderedstate.
W 4 8
C = 1.899(4) cm3 K mol-Cr−1 and θW = −331.9(4) K WenowconsidermagneticinteractionsintheCrspinel
for LiInCr4O8. The values of C correspond to effective oxides. Generally, it is well established that the shorter
moments of µeff = 4.024 µB and 3.897 µB per Cr atom, the Cr-Cr distance, the stronger the antiferromagnetic
or Lande g-factors of g = 2.078 and 2.012 for S = 3/2, interaction,asshowninFig. 5,becauseexchangeinterac-
respectively. The large and negative θW indicates that tionsaredominantlymediatedbydirectoverlapbetween
averagemagnetic interactionsare stronglyantiferromag- Cr t orbitals along the Cr-Cr bond [17, 18]. For exam-
2g
netic. ple,theJ ofZnCr O is33-45K[19–21],almosttentimes
2 4
The χ versus T curve of LiGaCr O resembles that of larger than 4 K for HgCr O [7, 22], which comes from
4 8 2 4
4
before,whichliesclosetotheuniversallineinFig. 5. We
) d A Cr O also tried to fit the data by taking account of J′ in the
K 60 2 4 8
n ( mean-fieldapproximation. However,the improvementof
o
cti LiIn fitting was limited, and it resulted in an unreasonably
era 40 d Zn2 large ferromagnetic J′. Thus, we assume J′ ∼ 6 K from
nt the universal relation in Fig. 5, which gives B ∼ 0.1 for
etic i 20 A2 = LiGa d´ LiInCr4O8. The breathing factor can be anfimportant
n Hg parameter to tune the ground state of pyrochlore lattice
Mag d´ Cd 2 antiferromagnets.
2
0 Finally, we describe the characteristicsof the ordering
2.80 2.90 3.00 3.10
transitions in the two compounds. Their C /T curves,
Cr-Cr distance (Å) p
shownin Fig. 4, exhibit sharppeaks at 13.8 and 15.9K,
respectively, clear evidence for magnetic transitions. In-
FIG. 5: (color online) Nearest-neighbor magnetic interaction
terestingly, the peak shape for LiGaCr O is apparently
versus the Cr-Cr distance for various Cr spinel oxides. The 4 8
different from that of a conventional second-order mag-
iJntveanlsuietyof[1Z9n–C21r]2.OT4haereJe’sstoimf CatdeCdrb2yO4fitatnindgHχg,CCrp2,Oa4ndarEePeRs- netictransition: theCp/T showsabroadshoulderat∼16
timated byCurie-Weiss fitstoχ[6,7,22]. Thedottedlineis KaboveTN andgraduallydecreaseswithincreasingtem-
a guide for the eyes. Plotted for LiGaCr4O8 or LiInCr4O8 is perature,indicating that a large spin entropy is retained
a pair of points corresponding to the two Cr-Cr distances in above T due to the development of antiferromagnetic
N
thesmall and large tetrahedra. short-rangeorder. InLiInCr O ,thereisalsoalargeen-
4 8
tropyreleaseaboveT ,associatedwithaspin-gapforma-
N
tion. Thesemagnetictransitionsarequiterobustagainst
a 4% difference in the Cr-Cr distance between ZnCr2O4 magnetic fields: the two Cp/T curves measuredat 0 and
(2.944 ˚A) and HgCr2O4 (3.062 ˚A). Interestingly, there 9 T nearly overlap to each other in each compound.
is an approximate linear relationship between J and the
The originoflong-rangeorderinthe presenttwocom-
bond length for ACr O , as indicated by the dotted line
2 4 pounds is not perfectly confirmed. In the absence of
in Fig. 5.
structural distortions, geometrical frustration should re-
A remarkable feature of the present Ga and In com- main and favor a spin-liquid ground state. In ACr O ,
2 4
pounds is the extremely short Cr-Cr distance within the astructuraltransitionalwaystakesplacesimultaneously
small Cr4 tetrahedra compared with that of ACr2O4: d with the magnetic transition. The similar TN’s of 5-15
= 2.867 ˚A (Ga) and 2.903 ˚A (In) are much shorter than K,inspiteofthebroadvariationofJ,suggestacommon
2.944˚AinZnCr2O4[23],suggestingthatJ intheGaand structural instability that is coupled with spins through
In compounds are larger than J = 33-45 K in ZnCr2O4. strong spin-lattice interactions. This is also likely the
On the other hand, the distance d′ = 2.970 ˚A within case for the present compounds. We plan to investi-
the large tetrahedra of LiGaCr4O8 is comparable to the gate the crystal and magnetic structures across TN by
Cr-Cr distance of ZnCr2O4, while the d′ = 3.052 ˚A for means of NMR, X-ray, and neutron diffraction exper-
LiInCr4O8 liesbetween3.041˚AinCdCr2O4 and3.062˚A iments. Moreover, inelastic neutron scattering experi-
in HgCr2O4. Assuming that the linear relationship ob- mentswillaidinestablishingthepresence(orabsence)of
served in ACr2O4 holds in LiGaCr4O8 and LiInCr4O8, a gapin either material,as wellas allowingfor quantita-
we arrive at estimates of J ∼ 60 and 50 K and J′ ∼ 30 tivedeterminationofJ andJ′. Althoughthecompounds
and 6 K, which give a breathing factor Bf = J′/J ∼ 0.5 assume long-range ordered ground states, fingerprints of
and 0.12, respectively. neighboring spin liquid states or the effect of frustration
To more accurately estimate J and J′, we carried out may be observable as in Ref. 24 and 25.
fitting ofthe χ datashowninFig. 3(a). ClassicalMonte In summary, two spinel oxides LiGaCr O and
4 8
Carlo simulations were performed using the spinmc pro- LiInCr O withthetetrahedralsitesalternatelyoccupied
4 8
gram of the ALPS package (1024 sites, periodic bound- by Li+ and Ga3+/In3+ ions are found to be unique frus-
ary conditions). Although it was difficult to determine trated antiferromagnets with breathing pyrochlore lat-
the two values uniquely, we could obtain J for various tices. LiGaCr O , with a lesser degree of breathing,
4 8
Bf’s, e.g. J = 62.9(5), 53.3(1), and 51.0(3) K for Bf = shows similar magnetic properties to the conventional
0.3, 0.5, and 0.6, respectively; a typical fitting curve for Cr spinel oxides with a uniform pyrochlore lattice, while
Bf = 0.5 is shown in Fig. 3(a). Taking into account of LiInCr4O8 shows a spin-gap behavior caused by a large
J ∼ 50 K from the universalline in Fig. 5, we decide Bf alternation of magnetic interactions in the more breath-
= 0.6 for LiGaCr4O8. ing pyrochlore lattice. The breathing of the pyrochlore
In contrast to LiGaCr O , the fit to the Monte Carlo lattice appears to be an important parameter to explore
4 8
results was poor for LiInCr O . Thus, we adopt J = interesting phenomena in frustrated magnets.
4 8
56.8(2) K from the fitting to the exact result mentioned We thank H. Tsunetsugu, M. Isobe, H. Ueda, and T.
5
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