Novel Magnetism and Local Symmetry Breaking in a Mott Insulator with Strong Spin Orbit Interactions

0.82 MB·

by  L. Lu

Magnetism and Local Symmetry Breaking in a Mott Insulator with Strong Spin Orbit
Interactions
L. Lu1, M. Song1, W. Liu1, A. P. Reyes2, P. Kuhns2, H. O. Lee3,4, I. R. Fisher3,4, and V. F. Mitrovi´c1,†
1Department of Physics, Brown University, Providence, RI 02912, U.S.A.
2National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA
3Department of Applied Physics and Geballe Laboratory for Advanced Materials, Stanford University, California 94305, USA
4Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,
2575 Sand Hill Road, Menlo Park, California 94025, USA
(Dated: January 24, 2017)
7
Study of the combined effects of strong electronic correlations with spin-orbit coupling (SOC)
1
0 representsacentralissueinquantummaterialsresearch. Predictingemergentpropertiesrepresents
2 a huge theoretical problem since the presence of SOC implies that thespin is not a good quantum
number. Existing theories propose the emergence of a multitude of exotic quantum phases, distin-
n
guishablebyeitherlocalpointsymmetrybreakingorlocalspinexpectationvalues,eveninmaterials
a
withsimplecubiccrystalstructuresuchasBa2NaOsO6. Experimentaltestsofsuchtheoriesbylocal
J
probesarehighlysoughtfor. Here,wereportonlocalmeasurementsdesignedtoconcurrentlyprobe
2
spin and orbital/lattice degrees of freedom of Ba2NaOsO6. We find that a novel canted ferromag-
2
netic phase which is preceded by local point symmetry breaking is stabilized at low temperatures,

as predicted by quantumtheories involvingmultipolar spin interactions.
]
l
e
-
r Introduction has a face-centered-cubic lattice, as shown in Fig. 1c.
t Thermodynamic and reflectivity measurements charac-
s Magnetic Mott insulators with strong spin-orbit cou-
. terize this material as a 5d1 ferromagnetic (FM) Mott
t pling (SOC) represent an intriguing class of materials
a insulator with a moderate ordered moment ∼0.2µ per
m where various exotic quantum phases, that include spin B
formula unit and T ∼6.3 K [12, 17]. This relatively
liquid,multipolarchargeorder,topologicalinsulatorand c
- smallvalueoftheorderedmomentwasconfirmedinµSR
d semimetal,Weylsemimetal,andAxioninsulator,arepre-
n dicted to emerge [1–10]. SOC is a relativistic effect that measurements [18]. Taking SOC into account, the antic-
o ipated ground state for a perfectly cubic point symme-
in the strong regime leads to local entanglement of spin
c try is J = 3. Yet, the magnetic entropy removed at
[ and orbital degrees of freedom. This entanglement re- eff 2
sultsindrasticallydifferentphysicsthanincasesofweak FM transition is only Rln2 [12]. Though, the most un-
1 usualobservationisthatthe FM stateeasyaxisis inthe
SOC. The associated emergent phenomena are particu-
v [110] direction, as this does not occur in standard Lan-
larly rich in metal oxides containing 5d transition-metal
7
dau theory for ferromagnetism in a cubic symmetry [3].
1 ions owing to the comparable magnitude of both strong
This uncommon magnetism can either be explained by
1 electroncorrelationsandSOC.Particularlyinterestingis
6 the case ofmaterials with a double perovskitesstructure the density functional theory (DFT) electronic structure
0 calculations, that include effects of electron correlation,
[11–15],forwhichithasbeenproposedthatpartiallifting
. a strongSOC, andanisotropicexchangeinteraction[19];
1 ofdegeneracyofthetotalangularmomentumeigenstates
0 or, by quantum models including multipolar exchange
induces a highly nontrivial multipolar exchange interac-
7 interactions arising from strong SOC [3, 20]. Moreover,
tions [3, 4]. These peculiar interactions promote quan-
1 quantum models identify the quadrupolar/orbitalorder-
: tumfluctuationsandthusgeneratenovelquantumstates
v ing, as a driving mechanism for the FM phase that de-
impossiblewithoutstrongSOC[3,4,16]. Suchstatesin-
i velops atop it [3, 20]. This quadrupolar order is charac-
X clude an unconventional antiferromagnet with the dom-
terized by the orbital polarizationthat is distinct on the
r inant magnetic octupole and quadrupole moments, an
a two sublattices. As such polarizations cannot be time-
unusual noncollinear ferromagnet with a doubled unit
reversalconjugates,when magnetismonsets, a net ferro-
cell and magnetization along the [110] axis, and biax-
magneticmomentresults. Moreover,becausethis antici-
ial spin nematic phase with quadrupolar order with pre-
patedquadrupolarordering,thatmanifestsinabreaking
servedtime-reversalsymmetrystabilizedinabroadinter-
ofthelocalcubicsymmetry,onsetsathighertemperature
mediate temperaturerangeaboveanymagneticordering
thanmagneticorder,many-bodyquantumbasedmodels
temperature. A key feature of these many-body quan-
alsoaccountforthemissingentropyattheFMtransition
tummodelsisthatsignificantinteractionsarefourthand
[12,20](see SupplementaryDiscussion3). Experimental
sixthorderin the effective spins, due to stronglyorbital-
confirmationofthemicroscopicquantummodelsrequires
dependent exchange.
the observation of two effects. These comprise a struc-
ArepresentativematerialinthisclassisBa2NaOsO6,a tural change, that precedes magnetic order, associated
double perovskite with Na and Os ions inhabiting alter- with the quadrupolar ordering and local spin expecta-
nate cation B sites, which for an undistorted structure
2
a
cal electron spin susceptibility and electric field gradient
{
H = 9 T 20 K (EFG).Thatis,wededucethatthismodificationreflects
15 K asplittingofNaintotwositessensingdifferenthyperfine
13 K fields due to electronic spins and breaking of the local
{ PM
12 K cubic point symmetry with both sites sensing the same
e (a. u.) { 10 KBLPS 10.5 K 11 K quTadoreusptaoblelifsrheqthueensceyn,sirteilvaitteydotfoouthremEeaFsGur.ementstopu-
ud 9 K tative lattice distortions, orbital order, and magnetism,
gnit 8 K we firstinspect the temperature (T)dependence of23Na
Ma cBFLMPS + 76 KK NMR spectra in Ba2NaOsO6. These 23Na spectra re-
veal the distribution of the hyperfine fields and the elec-
5 K
4.2 K tronic charge and are thus a sensitive probe of both the
II I
electronicspinpolarization(localmagnetism)andcharge
4.2 K δq H = 15 T distribution (orbital order and lattice symmetry). Tem-

perature evolution of 23Na NMR spectra is plotted in
-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0
Fig. 1a. For T > 12K, the spectrum consists of a sin-
ω − ω0 (MHz) gle narrow NMR line, evidencing the PM state. Below
13 K, the NMR line broadens and splits into multiple
b c
20 c peaks indicating onset of significant changes in the the
BLPS
local symmetry, thereby producing EFG, that is asym-
15
metric (non-cubic) charge distribution. Below 10 K, the
H (T)10 BcFLMPS + PM b a 23Naspectraclearlysplitinto6peaks,thatistwosetsof
triplet lines, labeled as I and II in Fig. 1a, that are well
5 EFG = 0 separated in frequency.
Os
0 BLCS?
O Two magnetic sites. Broadly speaking, the emer-
0 5 10 15 20 25
T (K) EFG = 0 Na gence of these two sets of triplets indicates appearance
FIG.1: Phasediagram of Ba2NaOsO6 deducedfrom of two distinct magnetic sites, i.e. two nuclear sites that
the NMR spectra. sensetwodifferentlocalfields,inthe lattice. Inafieldof
9T,thetransitiontotheLROstateonsetsinthevicinity
tion values, that differ from the average ones. Here, we of 10 K, which is significantly higher than the transition
report the first observation of such effects. Namely, we temperature observed in low field thermodynamic mea-
observe exotic local spin expectation values along with surements in [12]. Our data measured at the different
the structural changes, and infer the exact microscopic applied magnetic fields indicate that the transition tem-
nature of the FM state and lattice distortions. We show perature increases with the increasing field, confirming
that the FM state is in fact a type of canted ferromag- the magnetic nature of the transition (see Supplemen-
net with two sub-lattice magnetization, and that cubic tary Note 2). The line shape and the fact that the two
symmetry breaking occurs at a temperature above the sets of triplet lines, I and II, are well separated in fre-
N´eel temperature and it involves deformation of oxygen quency implies that the low temperature LRO magnetic
octahedra presumably reflecting a complicated pattern state is commensurate. Furthermore, both sets of lines
of staggered orbital order. In light of our NMR find- are shifted to frequencies below that of spectra in the
ings,wecompiledthephasediagramsketchedinFig. 1b. PMstate. Thisdemonstratesthatthenetlocalmagnetic
Ourresultsareinstartlinglygoodagreementwithrecent fieldsonbothNasitesareofthesamesignindicatingthat
theoreticalpredictions basedonquantummodels [3,20]. the LRO order is likely ferromagnetic.
Thus, our findings establish that such quantum models
Point symmetry breaking and EFG. Besides the
represent an appropriate theoretical framework for pre-
splittingintosetsIandII,thatreflectstheappearanceof
dictingemergentpropertiesinmaterialswithbothstrong
twodistinctmagneticsitesinthelowT phase,weobserve
correlations and SOC, in general.
additional splitting of each set of the spectral lines into
Results three peaks. Moreover, as visible in Fig. 1a, the addi-
NMR Spectra. In a lattice with cubic symmetry tionalsplittingisdiscernibleattemperatureshigherthan
only one Na NMR site is present, which results in a nar- that for the onset of LRO state, apparent in emergence
rowsinglepeak spectruminthe insulatingparamagnetic of sets I and II. This splitting, labeled as δ in Fig. 1a,
q
(PM)state. Ourmaindiscoveryisthat,onreducingtem- originatesfromquadrupoleinteraction,implyingchanges
perature, the Na NMR line undergoes a complex mod- in local charge distribution induced by modifications of
ification, as evident in Fig. 1a, reflecting changes in lo- electronic orbitals and/or local lattice symmetry.
3
a δq H = 15 T b 200 c c d H = 0 H > 0
nitude (a. u.) 409o876500000ooooo T = 8 K δ (KHz)q11846200000 [001] [111] [110] b a I = 3/2 ωωω ωωω +−δδqq
Mag 2300oo 0Ang2l0e fr4o0m [60001] 8(0de1g0.)0 EFG ≠ 0 EFG = 0 { EFG ≠ 0
100oo δq Vzz H OOs Δω ω δq ∼ Δω
-2.0 -1.ωI6I −- 1ω.20(M-0H.8z)I-0.4 0.0 +--+θ μ EFG ≠ 0 Na ω ω −δq ω ω +δδqq >>Δω
FIG. 2: Local cubic symmetry breaking in the ordered phase.
For nuclear sites with spin I >1/2,such as 23Na with 15 T at4 K,deepinthe LROphase(see Fig. 1a). Com-
I = 3/2, and non-zero EFG, quadrupole interaction be- parison reveals that δ varies by less than 2 %, which is
q
tweennuclearspinandEFGsplitsotherwisesingleNMR oftheorderoftheerrorbars. Furthermore,wemeasured
line to 2I lines, as illustrated in Fig. 2d. Thus, 23Na the spectra at 15 T and 8 K as a function of the of the
spectral line splits in three in the presence of non-zero angle(θ) betweenH and [001]crystallineaxis as plotted
EFG. Nonetheless, for small finite values of the EFG in Fig. 2a. The angle dependence of the splitting δ is
q
three peaks arenotnecessarilydiscernible, inwhichcase displayed in Fig. 2b. It indeed follows the exact func-
significant line broadening can only be observed, as de- tional dependence expected for the δ originating from
q
pictedinFig. 2d. Atsiteswithcubicpointsymmetrythe quadrupole interactions. Both the observedinsensitivity
EFGiszero(seeSupplementaryNote5),asisthecasefor of δ to the strength of the magnetic field and its de-
q
Na nucleiinthe hightemperature PMphase. Therefore, pendence on θ indicate that δ splitting originates from
q
the observedline broadening and subsequent splitting of structural/orbitaldistortionsforwhichtheprincipalaxes
the Na spectra into triplets, in the magnetically ordered of the EFG coincide with those of the crystal(see Meth-
phase, indicates breaking of the cubic point symmetry, ods). Moreover,the insensitivity of δ to the strength of
q
caused by local distortions of electronic charge distribu- the magnetic field rules out the possibility that the de-
tion. These distortions, marking the broken local point tecteddistortionsoriginatefromtrivialmagnetostriction
symmetry(BLPS)phase,occurabovethetransitioninto effects on the crystal. Our finding, that structural dis-
the magnetic state, as depicted in Fig. 1b. To confirm tortionis presentin the LRO phase,is in contrastto the
this finding, we measured low T spectra as a function of predictions made by the first-principles DFT calculation
strengthandorientationoftheappliedmagneticfield,as [19, 22]. This is important in so far that it clearly shows
we describe next. thatquantummodelsbasedoncomplexmultipolarinter-
Specifically,weexamine δ ,the averageseparationbe- action generating high-order spin exchange is consistent
q
tweentwoadjacentquadrupolarsatellitelines,asafunc- with the observednature ofemergent phases in Mott in-
tion of strength (H) and orientation (H) of the applied sulators with the strong SOC [3, 20].
magneticfield. Thesizeofδq isproportionaltothemag- Lattice distortions and orbital order. To resolve
nitude of the EFG and the square of the spin operator themicroscopicnatureoftheobservedlatticedistortions
(Iˆz) projected along the principal axis of the EFG (Vzz) and determine their magnitude, we performed detailed
(seeSupplementaryNote5). Instrongappliedfield,asis numericalcalculationsoftheEFG,andthusδ ,basedon
q
the caseinourexperiment,the sizeofδq iscontrolledby the point charge approximation [21] (see Supplementary
the projectionofIˆ2 alongH, proportionalto the cos2(θ) Note 5). We found that our observation, revealing equal
z
of the angle θ between H and V , as shown in Fig. 2b. δ ontwomagneticallyinequivalentNasites,canbebest
zz q
Evidently, δ is at its maximum for H applied in the di- explained by a scenario involving distortions of the O2−
q
rection of V , which in our experiment corresponds to octahedra, surrounding Na+ ions as depicted in Fig. 2c.
zz
[001] direction. Therefore, if the splitting δ originates In this scenario,one structurally distinct Na site in non-
q
fromquadrupole interactions, δ shouldremainconstant cubic environment is generated. As it results from our
q
asstrengthofthetheappliedfieldischangedandshould calculations,distortionsthatgenerateorthorhombiclocal
follow |(3cos2θ−1)/2| function of θ as its orientation is symmetryattheNasitearerequiredtoaccountforboth
varied [21]. We compare δ in fields ranging from 7 T to the amplitude of the detected splitting (δ ≈ 190kHz)
q q
4
andits dependence onthe field orientation. Thus,in the the same order of magnitude. Presence of H implies
stag
ordered phase, our observations are explained by the or- that the LRO state contains two-sublattice magnetiza-
thorhombic distortionsthat comprise ofdominantdefor- tionwithsignificantantiparallelcomponents. Suchmag-
mations along[001]andone ofthe crystallineaxis inthe netization naturally accounts for the appearance of two
(110)plane. Eventhoughwefindseveralpossibledistor- magnetically inequivalent Na sites, that is the appear-
tionswhichcaninducetheobservedsplitting,weempha- ance of distinct local fields hH i and hH i. In Fig. 4a,
I II
size that they allinvolve a symmetry-loweringtransition the internal field at the Na site in one plane consists of
to an orthorhombic point symmetry in the LRO phase. a sum of the four Os moments on the same layer, and
Consideringtheobservedamplitudeofδ ,wededucethat thus equal magnetic moments from one sub-lattice la-
q
a typical magnitude of the distortion along any particu- beled A, and two Os above/below in neighboring layers
lar direction in the LRO phase does not exceed 0.8 % of with magnetic moment pointing in a different direction
the respective lattice constant. Above T , in the BLPS (sub-lattice labeled B), and thus producing different lo-
c
phase, the width of the NMR spectra allows us to place cal field than A moments at the Na site. Na nuclei in
an upper limit on distortions. We infer that the limit the next plane will then sense four type B and two type
equalsto 0.02%of the respectivelattice constant,asany A Os moments. This generates two sets of inequivalent
deformations that exceed this value would cause visible Na sites and causes the magnetic splitting in spectrum
splitting of the NMR spectra in the PM state. betweentripletI andII, astwotypes ofmoments induce
In the PM state, BLPS phase is characterized by sig- different local fields at the Na site.
nificant broadening of the NMR spectra. This broaden- Next, we inspect the local fields as the applied mag-
ing growsrapidlyondecreasingtemperaturetowardsTc. neticfieldwasrotatedinthe(1¯10)planeofthecrystal,as
The angle dependence of the broadening does not coin- illustrated in Fig. 4a. This is essential for ensuring that
cidewitheitherthatoftheinternaluniformorstaggered weunderstandwhatcomponentsofanisotropicmagnetic
fields,indicatingthatthebroadeningpredominantlyorig- susceptibilityarebeingmeasured. ForHk[001],H ,as
stag
inates from lattice distortions. Because in the BLPS wellas hH i, reachesits maximumvalue, while both H
II u
phasewedonotobservewelldefinedsplitting,butrather and hH i are at their minimum. In principle, H should
I u
convolutedbroadening,theexactdependenceofδqonthe scale as bulk magnetization (M). As evident in Fig. 4c,
field orientation is unknown. Consequently, dominant thisisnotthecasehere. Thisfindingrevealsthatdespite
tetragonal distortions along [001] direction can in prin- the fact that the net magnetization is aligned with the
ciple account for the line broadening in the PM phase. [110] axis, the local fields, i.e. spin expectation values,
Thus, the BLPS phase can be viewed as the PM phase are not. This very fact was predicted to arise as a di-
in which the cubic point symmetry is broken by either rectconsequenceoflattice distortionsdrivenby complex
dominanttetragonaldeformationsoftheoxygentetrahe- interactions in this class of materials [3].
dra along [001] direction or orthorhombic distortions, as
The angular dependence of the internal fields is used
is the case in the LRO phase (see Supplementary Note
to deduce the exact spin orientation in the LRO phase
5 and Supplementary Discussion 1). Hence, it is possi-
bycalculatingthelocalH andH attheNasitefora
u stag
ble that the solid line, indicating T into LRO magnetic
c given spin orientation. The local field consists of a con-
phase, in Fig. 1b denotes symmetry lowering transition
tributions from electronic spins at six nearest-neighbor
from tetragonal-to-orthorhombicphase as well.
Os ions mediated via anisotropic hyperfine interaction
LRO magnetic state. We emphasize that detected (Methods). By performing full lattice sum, we calculate
lattice distortions lead to only one magnetically distinct the local fields at the Na site as the direction of the ap-
Na site. Thus, we conclude that the observed magnet- pliedfieldisrotatedin(1¯10)planeofthecrystal. Wefind
ically distinct Na sites (labeled as I and II in Fig. 1) that the model that best describes our observations, as
must originate from the novel type of magnetism and illustratedbythesolidlinesinFig. 4a,isatwo-sublattice
not from lattice distortions. In order to deduce the mi- cantedFMmodel,recentlyproposedin[20]anddepicted
croscopic nature of the LRO magnetism, we next dis-
cuss the temperature and field evolution of the local
a b
fields. We used the NMR shift data to infer the local
t1au21hsna[ehindfHtoderrItimi2per−l)em(.tHhiWHInueaIedI=nind]b)o21IytIfi[e,hetHtalhhdsIeasid,tfi+ewHnrshohutHteecrmIdoeIiro(]tr)msheesaeeepnnaSodtvnueosdprtfspaalgttgehoegmeetierhseenendttaatloi(krrHcyeeansNltsafiopogveteelec=dsr- |H| (mT)u19876543000000000 H || 179[50 TT 0T 1] H (mT)stag5432100000 79 TT
15 T
20
tra. Temperature evolution of these local fields is dis- 0
10
played in Fig. 3. Well below Tc, Hu increases with in- 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1.0
creasingH,whileHstag (observableonlybylocalprobes) T/Tc T/Tc
remainsconstant. Interestingly,bothHuandHstagareof FIG.3: Temperaturedependenceoftheuniformand
staggered fields.
5
a b
dicatesthatspin-spininteractionsarehighlyanisotropic.
y
140 HI 50 [110]
130 HII 45 Discussion
mT)111200 4305 H A φ We have performed microscopic measurements on
|H| (u1890000 H1sstt aMgoment 32120055stag (mT) φ B x aBam2NoadOelsOsy6s.teOmurosftaMticotNtMinRsumlaetaosrurwemithentsstrroenvgealStOhCat,
70 [001] H = 15 T [111] [110] 10 d c the local cubic symmetry breaking, induced by defor-
60 T = 8 K 5 mation of the oxygen octahedra, precedes the formation
0 10 20 30 40 50 60 70 80 90 of the LRO magnetism. We establish that LRO state is
Angle (θ) b
the exotic canted two-sublattice FM state, believed to
c
FU) 00..2210 MHu [111] 112105 | H B A a bisetdhreivfiernstbdyirtehcetsdteatgegcetrioend qofuasdurcuhpaolcaormoprdleexr [q3u].anTtuhmis
/M (µb00..1198 [001] 111005u (mT| sotbasteervwaittihonthoefdbisottihnctthleoclaolcsaplincuebxipcescytmatmioentrvyaluberse.akOinugr
[110] 100 ) Os and appearance of two-sublattice exotic FM phase is
0.17 O
0 50 100 Na in line with theoretical predictions based on quantum
Angle (θ(
models with multipolar magnetic interactions. The
FIG. 4: Resolution of the spin orientation in the or- fact that spin-spin interactions are indeed mediated by
dered phase. complex multipolar interactions, as suggested in [3], is
further confirmed by our finding that the symmetry of
theinferredAtensordoesnotreflectanylocalsymmetry
in Fig. 4b. This model consists of two inequivalent sub-
of the distorted O octahedra. Moreover, it is proposed
latticeswithmomentsineachlayerintheXYplane par-
that two-sublattice magnetic structure is the very man-
allel to each other, forming FM order, while moments
ifestation of staggered quadrupolar order and that this
in the neighboring layers point to a different direction.
ordering drives the formation of LRO magnetism [3, 9].
Specifically,momentsintwoadjacentlayersaresymmet-
Thus, our finding that LRO phase is a two-sublattice
ric about [110]axis, that is they form an angle ±φ, with
canted FM implies that broken cubic symmetry phase is
[110] axis, as depicted in Fig. 4b and d. As direction
a staggered quadrupolarly ordered phase with distinct
of the applied field is varied spin-plane follows the di-
orbital polarization on two-subattices. In summary, our
rection of H while spins remain staggered about [110]
findings clearly demonstrate that microscopic quantum
axis. Moments arranged in this fashion induce an uni-
models with multipolar magnetic interactions are an ap-
form field in [110]direction, providing an overallshift to
propriate theoretical framework for predicting emergent
the NMR spectrum, and forma staggeredpatternin the
quantumphasesinMottinsulatorswiththestrongSOC.
direction perpendicular to [110]. Thus, for H||[001] such
spin arrangementgenerates lowestH andlargestH ,
u stag Lastly, presented unique direct observation of both
as observed, when the canting angle φ exceeds 45◦. The
local cubic symmetry breaking and exotic long range
exactvalue ofthe canting angle is determined by the fit-
ordered (LRO) magnetic state is the confirmation of
ting procedure described below. The curves in Fig. 4A,
the theoretical proposal that the combination of the un-
are fits to the data using the simulated local fields with
usual multipolar interactions, generic for spin-orbitally
relative strength of the off-diagonal terms of the hyper-
entangled effective spins, and/or structural transitions
finecouplingtensor,A,(A /A ),magnitudeofthelocal
ij ii or quadrupolar order can lead to a highly frustrated
Os moments, and the canting angle φ between the spins
quantum regime evenfor systems with spin greaterthan
on two different sub-lattices, as fitting parameters. Con-
S = 1/2 [20]. Thus, our work illustrates that such com-
straining the diagonal terms of A to be close to those
plex quantum states might be found in other frustrated
foundinthePMstate,wefindthemomentofµ≈0.6µ
B materialswithbothstrongcorrelationsandSOC[23,24].
andφ≈67◦. Usingthededucedvalueofφandformalism
in [20], we estimate the ratio of in-plane to intra-plane
METHODS
coupling constant to be ≈ 4 (see Supplementary Note
4). The value of the moment is in agreement with the NMR methods. The measurements were done at
effective moment deduced from the fit to a Curie-Weiss Brown University for magnetic field up to 9 T and at
law in the PM state in [12]. Large canting angle φ ex- the NHMFL in Tallahassee, FL at higher fields. In both
plains the smaller moment detected in the FM state in laboratorieshigh homogeneity superconducting magnets
bulk measurements [12, 18] due to partial cancelation of were used. The temperature control was provided by
nonparallel magnetic moments. We emphasize that the 4He variable temperature insert. The NMR data were
symmetry of the inferred A tensor reflects neither local recorded using a state-of-the-art laboratory-made NMR
tetragonal nor orthorhombic symmetry of the distorted spectrometer. The spectra were obtained, at each given
Ooctahedra(seeSupplementaryNotes3and4). Thisin- value of the applied field, from the sum of spin-echo
6
Fourier transforms recorded at constant frequency
intervals. We used a standard spin echo sequence Quadrupolar Interaction and EFG: In the sim-
(π/2 − τ − π). Shape of the spectra presented in the plest case of a field with axial symmetry, interaction
manuscript are independent of the duration of time between eq, the electric field gradient (EFG), and the
interval τ. Since nuclear spin I of 23Na equals to 3/2 nucleus, with spin I and the quadrupole moment Q,
and at low temperatures, both Na sites (I and II) are is described by the Quadrupole Hamiltonian, H =
Q
in non-cubic environments, three distinct quadrupolar (eQ)(eq)[3I2−I(I +1)]. For nuclear spin I = 3/2, as is
4I(2I−1) z
satellite lines are observed per site [21]. The shift
the case of 23Na, the energy eigenstates of H are given
Q
was obtained from the frequency of the first moment
by, E = (eQ)(eq) [3m2−I(I+1)]. Than, the frequencies
of spectral distribution of set of triplet lines using a 4I(2I−1)
between different quadrupole satellite transitions equal,
gyromagnetic ratio of 23γ = 11.2625 MHz/T. The same
gyromagnetic ratio was used for all frequency to field
(eQ)(eq)
scale conversions. ωm→m−1 = [3(2m−1)]=
h4I(2I−1)
1
Sample. High quality single crystal of Ba2NaOsO6 2, for |+3/2i→|+1/2i (1)
with a truncated octahedral morphology were grown
0, for |+1/2i→|−1/2i
from a molten hydroxide flux, as described elsewhere
[12, 17]. Crystal quality was checked by x-ray diffrac- 1
− , for |−1/2i→|−3/2i.
tion, using a Bruker Smart Apex CCD diffractometer, 2
whichindicatedthattheroomtemperaturestructurebe-
Therefore, in a magnetic field applied along the prin-
longs to the Fm¯3m space group [12]. NMR measure-
cipal axis of the EFG only 3 NMR lines (transitions)
ments were performed for a single crystal with a volume
will be observed with equal splitting δ . In this case,
of approximately 1 mm3. The quality of the sample was q
the quadrupolesplittingδ betweendifferentquadrupole
confirmedbythesharpnessof23NaNMRspectrabothin q
satellites is simply given by δ = 1 (eQ)(eq) =
the high temperature paramagnetic state and low tem- q 2h
1 (Quadrupole moment) × (EFG). In our experiment
perature quadrupolar split spectra. 2h
equal splitting is observed between quadrupole satellites
The sample was both zero-field and field-cooled. We
lines plotted in Fig. 1a for Hk[001] indicating that the
did not detect any influence of the samples cooling his-
principalaxisofthe EFGmustbe alongH,thatisalong
tory on the NMR spectra. Nevertheless for consistency,
theaxisofthecrystal. Further,wecanestimatethevalue
all results presented in the paper were obtain in field-
oftheEFGusingexperimentallydeterminedvalueofthe
cooled conditions. The sample was mounted to one of
splitting.
the crystal faces and rotated with respect to the applied
For anisotropic charge distributions, quadrupole
field about an axis using a single axis goniometer. The
Hamiltonian expressed in the coordinate system define
rotation angle, for applied fields below 9 T, was inferred
by the principal axes of the EFG, is given by
from the signal of two perpendicularly positioned Hall
sensors. In addition, to ensure that data was taken eQV
H (x,y)= zz (3Iˆ2−Iˆ2)+η(Iˆ2−Iˆ2) , (2)
with no external pressure applied, the mounted sample Q 4I(2I−1)h z x y i
was placed in a solenoid coil with cross sectional area
significantly larger than that of the sample. In this way, where η ≡|V −V |/V is asymmetry parameter and
xx yy zz
no pressure is exerted on the sample as coil contracts on V , V , and V are diagonalcomponents of the EFG.
xx yy zz
cooling. Here, V is defined as the principle component of the
zz
EFG and |V | < |V | < |V |, by convention. In this
xx yy zz
Transition Temperature: Transition temperature case,thesplittingisgivenby,δ = (eQ)(Vzz) 1+ η2 1/2.
(T ) from paramagnetic (PM) to low temperature q 2h (cid:16) 3 (cid:17)
c
Thus, the value of δ is dictated by both V and
ferromagnetic (FM) state was determined by examining q zz
anisotropy parameter. In the high field limit, when H
the NMR shift and spectral line shapes, as described Q
is a perturbation to the dominant Zeeman term, the an-
in detail in Supplementary Note 2. Onset temperature
gular dependence of the splitting is given by
for breaking of local cubic symmetry, shown in Fig. 1b,
was identified as temperature below which the second
∆
moment of the NMR spectral line, measuring the δq = q(3cos2θ−1+ηsin2θcos2φ), (3)
2
spectral width, increases notably as compared to that
in high temperature PM phase. We point out that this whereθ istheanglebetweenthe appliedfieldHandV .
zz
temperature does not necessarily correspond to the true As in the case of axially symmetric EFG, in the coordi-
onset temperature for orbital ordering, which could be natesystemdefinedbytheprincipalaxesoftheEFGonly
undetectable in our experiment due to the subtlety of 3 NMR lines (transitions) will be observed with equal
the effect. splitting δ between them. When H is rotated in such
q
7
coordinate system only 3 NMR lines are observed and
the magnitude of the splitting between these lines de-
pendsonlyonangleθ. Thefactthatweobservenomore
[1] Gang Chen, Leon Balents, and Andreas P. Schnyder,
than 3 lines per set (I or II) regardless of the angle be-
Spin-OrbitalSinglet and QuantumCritical Point on the
tween H and [001] crystalline axis, as shown in Fig. 2B, DiamondLattice: FeSc2S4, Phys.Rev.Lett.102,096406
indicates that H was rotated in the coordinate system (2009).
defined by the principal axes of the EFG. Therefore, the [2] G. Jackeli and G. Khaliullin, Mott Insulators in the
principal axes of the EFG must coincide with those of StrongSpin-OrbitCouplingLimit: FromHeisenbergtoa
the crystal. QuantumCompass andKitaevModels, Phys. Rev. Lett.
102, 017205 (2009).
[3] Gang Chen, Rodrigo Pereira, and Leon Balents, Exotic
phases induced by strong spin-orbit coupling in ordered
In a material with cubic symmetry, it is thus possible double perovskites, Phys. Rev. B 82, 174440 (2010).
to stabilize three different domains, each with the prin- [4] Gang Chen and Leon Balents, Spin-orbit coupling in d2
ordered double perovskites, Phys. Rev. B, 84, 094420
ciple axis of the EFG, V , pointing along any of the 3
zz (2011).
equivalentcrystalaxes. Further,localmagnetic field has
[5] DmytroPesin andLeonBalents, Mottphysicsandband
to be parallel to V in each domain. The facts that the
zz topology in materials with strong spin-orbit interaction,
splitting is the largest for Hk[001] (Fig. 2B), and that Nat Phys., 6(5), 376-381 (2010).
only 3 peaks per setare observedfor Hk[110]imply that [6] J. Radi´c, A. Di Ciolo, K. Sun, and V. Galitski, Exotic
two domains are plausible in the crystal. One domain is Quantum Spin Models in Spin-Orbit-Coupled Mott In-
characterizedbypureuniaxial3z2−r2 distortionswhere sulators, Phys. Rev. Lett., 109, 085303 (2012).
[7] William S. Cole, Shizhong Zhang, Arun Paramekanti,
V is in [001] direction, while the other is distinguished
zz and Nandini Trivedi, Bose-Hubbard Models with Syn-
byx2−y2distortionswhereV istheninthe(110)plane.
zz thetic Spin-Orbit Coupling: Mott Insulators, Spin Tex-
Inthe simplestcaseVzz isparallelto[001]directionwith tures, and Superfluidity, Phys. Rev. Lett., 109, 085302
η = 0 indicating tetragonal local symmetry. In the sec- (2012).
ond case V is aligned along [100] direction with η is of [8] Johannes Reuther, Ronny Thomale, and Simon Trebst,
zz
the order of 1, implying orthorhombic local symmetry. Finite-temperature phase diagram of the Heisenberg-
To determine the exact local symmetry, splittings δ ob- Kitaev model, Phys. Rev. B, 84, 100406 (2011).
q
[9] William Witczak-Krempa,Gang Chen,YongBaek Kim,
tainedforHrotatedaboutoneofthecrystallineaxisand
andLeonBalents,Correlatedquantumphenomenainthe
about [110] direction have to be analyzed.
strong spin-orbit regime, Annual Review of Condensed
Matter Physics, 5(1), 57-82 (2014).
[10] Zohar Nussinov and Jeroen van den Brink, Compass
models: Theory and physical motivations, Rev. Mod.
Calculation of Internal Fields: In the LRO phase,
Phys., 87, 1-59 (2015).
the component of the internal hyperfine field parallel
[11] TomokoAharenetal.,Structureandmagneticproperties
to H, at an Na site, is given by Hiknt = hˆ ·PhiiAi ·µi, of the S =1 geometrically frustrated double perovskites
where hˆ is a unit vector in the applied field direction, La2LiReO6 and Ba2YReO6, Phys. Rev. B, 81, 064436
A is the symmetric 3×3 hyperfine coupling tensor with (2010).
i [12] A. S. Erickson et al., Ferromagnetism in the Mott insu-
the ith nearest-neighbor Os atom and µ is its magnetic
i lator Ba2NaOsO6, Phys. Rev. Lett., 99, 016404 (2007).
moment (see Supplementary Note 4). In the PM phase, [13] KazuhiroYamamura,MakotoWakeshima,andYukioHi-
hyperfinecouplingtensorisdiagonal. Due tocomplexity natsu, Structural phase transition and magnetic prop-
of the orbitals mediating the exchange paths, which can erties of double perovskites Ba2CaMO6 (M=W, Re,
induce multipolar exchange interactions [3, 9] between Os), Journal of Solid State Chemistry, 179(3), 605-612
(2006).
neighboring Os spins and broken cubic symmetry, the
[14] C. R. Wiebe et al., Frustration-driven spin freezing in
onffon-dzieargooninaltheelemLRenOtspohfasteh.eWhyeppeorfiinnteotuetnstohratAeivjenarief theS = 12 fccperovskiteSr2MgReO6, Phys. Rev. B,68,
134410 (2003).
moment is not exclusively localized on Os site but the [15] C.R.Wiebe,J.E.Greedan,G.M.Luke,andJ.S.Gard-
spin density is distributed to O [19], our modeling of ner, Spin-glass behavior in the S =1/2 fcc ordered per-
Hi is valid. This is because the complexity of the spin ovskite Sr2CaReO6, Phys. Rev. B, 65, 144413 (2002).
int
density is accounted for in A. For simplicity, we treat [16] Tyler Dodds,Ting-Pong Choy,and YongBaek Kim, In-
terplaybetweenlatticedistortionandspin-orbitcoupling
moment as S = 1/2 localized on Os as was done in
in double perovskites, Phys. Rev. B,84, 104439 (2011).
[20]. By performing full lattice sum, we calculate the
[17] Katharine E Stitzer, Mark D Smith, and Hans-Conrad
local H and H at the Na site as well as Na NMR
u stag zurLoye,CrystalgrowthofBa2MOsO6(M=Li,Na)from
spectra, which is a histogram of the local field compo- reactivehydroxidefluxes, Solid State Sciences,4(3),311
nent projected along the applied field, as the direction - 316 (2002).
oftheappliedfieldisrotatedin(1¯10)planeofthecrystal. [18] Andrew J Steele et al., Low-moment magnetism in the
double perovskitesBa2MOsO6 (M=Li, Na), Phys. Rev.
8
B, 84(14), 144416 (2011). ordered phase. (a) 23Na spectra in low temperature
[19] ShrubaGangopadhyayandWarrenE.Pickett,Spin-orbit ordered state as a function of the angle between the
coupling, strong correlation, and insulator-metal transi- applied magnetic field and [001] crystalline axis. In this
tions: The Jeff = 32 ferromagnetic Dirac-Mott insulator case, H was rotated in the (1¯10) plane of the crystal
Ba2NaOsO6, Phys. Rev. B, 91, 045133 (2015).
which contains three high symmetry directions: [001],
[20] Hiroaki Ishizuka and Leon Balents, Magnetism in S =
1 doubleperovskiteswith strongspin-orbit interactions, [111], and [110]. (b) The mean peak-to-peak splitting
2
Phys. Rev. B, 90, 184422 (2014). (δq) between any two adjacent peaks of the triplets I
[21] A.Abragam, Principles of Nuclear Magnetism. Number and II. Error bars reflect the scattering of deduced (δq)
216-263. Oxford University Press, 1996. values. Solid line is the fit to |(3cos2θ−1)/2|, where θ
[22] H. J. Xiang and M.-H. Whangbo, Cooperative effect of denotes the angle between the principal axis of the EFG
electron correlation and spin-orbit coupling on the elec-
(V ) and the applied magnetic field (H) as depicted
tronic and magnetic properties of Ba2NaOsO6, Phys. zz
for nuclei spins I = 3/2 with magnetic moment µ (see
Rev. B, 75, 052407 (2007).
Supplementary Note 5). (c) Schematic of a proposed
[23] A. M. Cook, S. Matern, C. Hickey, A. A. Aczel, A.
Paramekanti, Spin-orbit coupled jeff = 1/2 iridium mo- lattice distortion involving the O2− ions. The distortion
ments on the geometrically frustrated fcc lattice, Phys. breakscubicsymmetryattheNasitegivingrisetofinite
Rev. B, 92, 020417(R) (2015). electric field gradient (EFG). Different types and colors
[24] Judit Romha`nyi, Leon Balents, and George Jack- of double arrows illustrate unequal magnitude of oxygen
eli, Spin-Orbit Dimers and Non-Collinear Phases
displacements. Displacements along different axes can
in d1 Cubic Double Perovskites, Preprint at
comprise either from compression or elongation of the
http://arxiv.org/abs/1611.00646 (2016).
oxygen bonds (see Supplementary Note 5). Depicted
distortion magnitude is not to scale with the distance
Figure Legends:
between nearest neighbor Os atoms, and is amplified for
1. Phase diagram of Ba NaOsO deduced from clarity. (d) Schematic of the energy levels of a spin-3/2
2 6
the NMR spectra. (a) Temperatureevolutionof23Na nucleus in a finite magnetic field and in the presence
spectra at 9 T (and at 15 T, shaded trace) magnetic of quadrupolar interaction with EFG, generated by
field applied parallel to [001] crystalline axis. Narrow surrounding electronic charges, and resulting NMR
single peak spectra characterize high temperature para- spectra. In the absence of quadrupolar interaction
magnetic (PM) state. At intermediate temperatures, spectrum consists of a single narrow line at frequency
broaderandmorecomplexspectrarevealtheappearance ω and of width ∆ω. In the presence of quadrupolar
of electric field gradient (EFG) induced by breaking of interaction the centre transition remains at frequency
local cubic symmetry. Splitting into 2 sets of triplet ω, while the satellite transitions appear at frequencies
lines (labeled as I and II), reflecting the existence of shift by ± δq, proportional to the magnitude of the
two distinct magnetic sites in the lattice, is evident EFG. For small values of the EFG, satellite transition
at lower temperatures. Zero of frequency is defined cannotbe resolvedand only line broadening is observed.
as ω = 23γH. Abbreviation: PM, paramagnetic; Strictly speaking, there is also a broadening due to the
0
BLPS, broken local cubic symmetry; and, cFM, canted distribution of magnitude of the EFG itself, but this is
ferromagnetic. The charge density in the theoretically manifested only on the satellites and not on the central
predicted quadrupolar phase [9] is sketched in the inset. transition. Inourcase,thisdistributioncanbeneglected
(b) Sketch of the phase diagram based on our NMR as all the lines show the same width.
measurements. Squares indicate onset temperature for
the local cubic symmetry breaking, determined from 3. Temperature dependence of the uniform
our NMR data as explained in the text, in the PM and staggered fields. Absolute value of the uniform
phase. Circles denote T , transition temperature into internal field, H , (a) and the staggered internal field,
c u
canted ferromagnetic (cFM) phase, as deduced from the H , (b) as a function of reduced temperature for
stag
NMR data,while diamondmarks T as determined from various Hk[001]. Typical error bars are on the order of
c
thermodynamic measurements in Ref. [12]. The solid a few per centand notshownfor clarity. Lines areguide
line indicates phase transition into cFM state and also to the eyes.
possible tetragonal-to-orthorhombic phase transition.
The dashed line denotes cross over to the broken local 4. Resolution of the spin orientation in the
point symmetry (BLPS) phase, as detected by NMR. ( ordered phase. (a) The magnitude of the internal
c) The high-temperature undistorted crystal structure field associated with triplet I (H ) and II (H ) (solid
I II
of Ba NaOsO . In this case, point symmetry at the symbols), and the first moment of the entire spectrum
2 6
Na site is cubic leading to zero electric field gradient (uniform field, H ) as a function of the angle between
u
(EFG).Principalcrystallographicaxesareshownaswell. the applied magnetic field and [001] crystalline axis
at 8 K and 15 T. Open symbols depict the angular
2. Local cubic symmetry breaking in the dependence of the staggered field, H . Typical error
stag
9
bars are on the order of a few per cent and not shown Os site. Planes containing moments from sub-lattice A
for clarity. Dotted lines are guide to the eyes. Solid and B are shown. Lines are guide to the eyes.
lines are calculated fields from the spin model sketched
in part D, as described in the text. (b) Schematic of the
net magnetization in the XY plane in the spin model, AcknowledgementsWe wouldliketothankL.Balents
consistent with our data and proposed in [20]. Arrows and M. Horvati´c for illuminating discussions. The study
of different shades depict spins from two sub-lattices, was supported in part by the the National Science
labeled as A and B. These spins are canted by angle Foundation (DMR-0547938 and DMR-1608760). The
±φ with respect to [110] direction. (c) Comparison study at the NHMFL was supported by the National
of the angular dependence of bulk magnetization per Science Foundation under Cooperative Agreement
FU (formula unit) from Ref. [12] (open symbols) and no. DMR95-27035 and the State of Florida. Work
H determined from our NMR measurements. (d) at Stanford University was supported by the DOE,
u
Schematic of the spin model consistent with our data Office of Basic Energy Sciences, under Contract No.
and proposed in [20]. Different shades and orientation DE-AC02-76SF00515.
of arrows indicate distinct ionic (spin) environments on