High-field Phase Diagram and Spin Structure of Volborthite Cu3V2O7(OH)2/2H2O

0.48 MB·English

by  Makoto Yoshida

Typeset with jpsj2.cls <ver.1.2> Full Paper
High-field Phase Diagram and Spin Structure of Volborthite Cu V O (OH) 2H O
3 2 7 2 2
·
Makoto Yoshida∗, Masashi Takigawa†, Steffen Kra¨mer1, Sutirtha Mukhopadhyay1, Mladen Horvatic´1,
Claude Berthier1, Hiroyuki Yoshida2, Yoshihiko Okamoto, and Zenji Hiroi
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
1Laboratoire National des Champs Magnetique Intenses, LNCMI - CNRS (UPR3228), UJF, UPS and INSA, BP 166, 38042
Grenoble Cedex 9, France
2National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan
2
1
0 We report results of 51V NMR experiments on a high-quality powder sample of volborthite
2 Cu3V2O7(OH)2·2H2O, a spin-1/2 Heisenberg antiferromagnet on a distorted kagome lattice.
Following the previous experimentsin magnetic fields B below 12 T, theNMR measurements
n
havebeen extendedto higher fieldsup to 31 T. In addition to thetwo already known ordered
a
J phases (phases I and II), we found a new high-field phase (phase III) above 25 T, at which
a second magnetization step has been observed. The transition from the paramagnetic phase
0
1 to the antiferromagnetic phase III occurs at 26 K, which is much higher than the transition
temperatures from the paramagnetic to the lower field phases I (B < 4.5 T) and II (4.5 < B
] < 25 T). At low temperatures, two types of the V sites are observed with different relaxation
l
e rates and line shapes in phaseIII as well as in phase II.Ourresults indicate that both phases
- II and III exhibit a heterogeneous spin state consisting of two spatially alternating Cu spin
r
t systems,oneofwhichexhibitsanomalous spinfluctuationscontrastingwiththeothershowing
s
a conventional static order. Themagnetization of thelatter system exhibitsa suddenincrease
.
t upon entering into phase III, resulting in the second magnetization step at 26 T. We discuss
a
m the possible spin structurein phase III.
d- KEYWORDS: kagomelattice,frustration,volborthite,Cu3V2O7(OH)2·2H2O,NMR,spinstructure,phase
n diagram, high-field
o
c
[ 1. Introduction significantlydisturbs the intrinsicpropertiesatlowtem-

2 Thekagomelattice,atwo-dimensional(2D)networkof peratures.17,18)
v corner-sharing equilateral triangles, is known for strong Volborthite Cu3V2O7(OH)2 2H2O has distorted
·
0 geometrical frustration. In particular, the ground state kagome layers formed by isosceles triangles. Conse-
6
of the quantum S = 1/2 Heisenberg model with a quently, it has two inequivalent Cu sites and two kinds
2
0 nearest-neighbor antiferromagnetic (AF) interaction on of exchange interactions (J and J′ ) as shown in Fig. 1.
. the kagome lattice is believed to show no long-range The Cu2 sites form linear chains, which are connected
0
1 magnetic order. Theories have proposed various ground through the Cu1 sites. The magnetic susceptibility χ
1 states such as spin liquids with no broken symme- obeys the Curie-Weiss law χ = C/(T + θW) above
1 try with1) or without2) a spin-gap or symmetry break- 200 K with θW = 115 K, exhibits a broad maximum
v: ing valence-bond-crystal states.3) Real materials, how- at 20 K, and approaches a finite value at the lowest
i ever, have secondary interactions, which may stabilize temperatures.11,19) An unusual magnetic transition
X a magnetic order. Effects of the Dzyaloshinsky-Moriya has been observed near 1 K,20–23) which is much
r (DM) interaction,4) spatially anisotropicexchangeinter-
a
actions,5–7) and longer range Heisenberg interactions8)
havebeen theoreticallyinvestigated.Inthe classicalsys-
tem, it is known that order-by-disorder effect9) or DM
interaction10) favors long-range order with the √3 √3
×
pattern or the Q = 0 propagationvector, respectively.
On the experimental side, intensive efforts have been
devotedtosynthesizematerialscontainingS =1/2spins
onakagomelattice.11–14) Candidatematerialsknownto
date,however,departfromtheidealkagomemodelinone
way or another, such as disorder, structural distortion,
anisotropy, or longer range interactions. For example,
herbertsmithite ZnCu (OH) Cl realizes a structurally
3 6 2
ideal kagome lattice12) and exhibits no magnetic order
down to 50 mK.15) However, chemical disorder replac- Fig. 1. (Color online) Schematic structure of volborthite pro-
ing 10% of Cu2+ spins with nonmagnetic Zn2+ ions16) jected onto the a-b plane. The H and O sites are not shown.
The V sites are located both below and above the Cu kagome
layers.TheupperandlowerVsitesarerelatedbyinversionwith
∗E-mailaddress:[email protected] respecttotheCusites,henceallVsitesareequivalent.
†E-mailaddress:[email protected]
2 J.Phys.Soc.Jpn. FullPaper AuthorName
lower than θ , consistent with strong frustration in a
W
kagome lattice. However, a recent density functional
(cid:6)(cid:7)(cid:8)(cid:7)(cid:9)(cid:7)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)
calculation24) proposedthatthefrustrationisattributed
to the competition between a ferromagnetic J and an (cid:3)(cid:1) (cid:16)(cid:17)(cid:7)(cid:18)(cid:12)
antiferromagnetic J′′ between second neighbors along (cid:30)
(cid:29)
the chain(Fig. 1) rather than the geometry of a kagome (cid:28)
lattice. Thus the appropriate spin model for volbor- (cid:24)(cid:21)
thite has not been settled yet. Recently, anomalous (cid:26)(cid:27) (cid:3)
(cid:25)
sequential magnetization steps were reported in a high (cid:24)
(cid:21)
quality polycrystalline sample at 4.3, 25.5, and 46 T.19) (cid:23)
Furthermore, a magnetization plateau was observed at (cid:22)
(cid:21)
2/5 of the saturation magnetization above 60 T,25) in (cid:20)
spite of the theoretical prediction for a plateau26) or a (cid:1)(cid:0)(cid:3)
ramp27) at the 1/3 of the saturation magnetization in (cid:19) (cid:19)(cid:19) (cid:19)(cid:19)(cid:19)
an isotropic kagome lattice.
The magnetic transition at T∗ 1 K has been de- (cid:1) (cid:2) (cid:3)(cid:1) (cid:3)(cid:2) (cid:4)(cid:1) (cid:4)(cid:2) (cid:5)(cid:1)
∼
tected by 51V NMR,21,22) muon spin relaxation,20) and
heat capacity23) experiments. The NMR measurements (cid:31)(cid:7)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) (cid:14)(cid:12)!" #$%
revealedasharppeakinthenuclearrelaxationrate1/T
1 Fig. 2. Thephasediagramofvolborthiteproposedfromthepre-
and broadening of the NMR spectrum due to develop- vious19,22)andpresentstudies.Thesquares(triangles)represent
ment of spontaneous moments for the magnetic field B thephaseboundariesdeterminedfromthevariationoftheNMR
below 4.5 T.22) A kink was also observed in the heat spectra as a function of temperature (magnetic field). The cir-
capacity.23) Recently, development of short range spin cles and stars indicate the peak of 1/T122) and magnetization
steps,19) respectively.Thelinesareguidetotheeyes.Thereisa
correlation was detected by neutron inelastic scattering
finiterangeoffield,wherephaseIIandIIIcoexist.
experiments.28) However,thelowT phase,whichwecall
phase I, shows various anomalies incompatible with a
conventional magnetic order.22) The NMR line shape
served in phase III in a similar way as in phase II, indi-
is not rectangular but can be fit to a Lorentzian, sug-
catingthatthehigh-fieldphaseIIIexhibitsalsoahetero-
gesting a state in which the amplitude of the static mo-
geneousspinstate.OurresultsshowthatthoseCuspins
menthasspatialmodulation,suchasaspindensitywave
which are dominantly coupled to the V sites with small
(SDW) state or a spatially disordered state. The behav-
1/T areresponsibleforthe increaseofmagnetizationat
2
ior 1/T T provides evidence for dense low-energy ex-
1 the second step. We propose that the second magneti-
∝
citations. The spin-echo decay rate 1/T is anomalously
2 zation step is ascribed to the ferromagnetic alignment
large, pointing to unusually slow fluctuations.
of these Cu spins and discuss possible spin structure in
Above 4.5 T, at which the first magnetization step
phaseIII,withparticularreferencetothetheoriesonthe
occurs, another magnetic phase appears with a different anisotropic kagome lattice in the limit J J′.5,6)
spin structure and dynamics.22,29) In this phase, which ≫
wecallphaseII,theNMRresultsshowcoexistenceoftwo 2. Experiment
typesoftheVsiteswithdifferentvaluesof1/T2 andline The 51VNMR measurementshavebeen performedon
shapes.29) Thesiteswithlarge1/T havesimilarfeatures
2 a high-quality powder sample prepared by the method
to the V sites in phase I. They are strongly coupled to
described in ref. 19. The data at high magnetic fields
the Cu spins with a spatially modulated structure and
above 18 T were obtained by using a 20 MW resistive
largetemporalfluctuations.Onthe otherhand,thesites
magnet at LNCMI Grenoble. The NMR spectra were
withsmall1/T showarectangularspectralshape,which
2 recorded at fixed resonance frequency (ν ) by sweeping
0
is compatible with a conventional order having a fixed
magneticfield(B)inequidistantstepsandsummingthe
magnitude of the ordered moment, such as a N´eel state
Fouriertransformsofrecordedspin-echos,whichwereob-
or a spiral order. The numbers of these two sites are tainedbyusingthepulsesequenceπ/2 τ π/2.30)They
nearlyequalina wide rangeof B andT.Basedonthese − −
areplottedagainsttheinternalfieldB =ν /γ B.The
int 0
observations, a heterogeneous spin state, in which two −
NMR spectra arelabeledby the referencemagnetic field
typesofCumomentsformaperiodicstructure,hasbeen
B =ν /γ. We determined 1/T by fitting the spin-echo
0 0 1
proposed.29)
intensity M(t) as a function of the time t after a comb
In this paper, we report results of the NMR experi-
of several saturating pulses to the stretched exponential
ments in higher magnetic fields up to 31 T. The phase function M(t)=M M exp (t/T )β ,where M is
eq 0 1 eq
diagram determined from the previous22,29) and the − {− }
the intensity atthe thermalequilibrium.This functional
present experiments is shown in Fig. 2. A new high-
form was used to quantify the inhomogeneous distribu-
field phase defined as phase III is found above 25 T, at
tion of 1/T . When the relaxation rate is homogeneous,
1
whichthe secondmagnetizationstepoccurs.The transi-
the value of β is close to one.
tion from the paramagnetic phase to the phase III takes
place at 26 K, which is much higher than the transition
temperatures from the paramagnetic to phases I and II.
At low temperatures, two types of the V sites are ob-
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OKK KQS
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7 .+,04 Fig. 4. (Coloronline)Temperaturedependences of1/T1 andthe
strechexponent β at30.1T.
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powderpattern broadened by a non-axial anisotropic magnetic
xyz{
shift.
Fig. 5. (Color online) Recovery curves M(t) measured at posi-
tions A, B, and C in Fig. 3 at 1.3 K. The solid lines show
3. Results and Discussion the fitting to the stretched exponential function M(t)=Meq−
3.1 Temperature dependence of NMR spectra in high M0exp{−(t/T1)β}.
magnetic fields
Figure 3 shows the temperature dependence of the
and the strech exponent β at 30.1 T. In the tempera-
51V NMR spectra at B = 30.1 T. The spectra above
0 ture range between 15 and 50 K, 1/T is independent of
30 K are compatible with a powder pattern due to 1
temperature. In spite of the clear change in the spectral
anisotropic magnetic shifts (inset of Fig. 3). The width
shape,thereisnosignificantanomalyin1/T near26K.
due to the quadrupolar interactions is about 0.02 T,22) 1
Let us recall that while the transition from the param-
muchsmallerthantheobservedwidthinthisfieldrange.
agnetic phase to phase I is marked by a clear peak of
As temperature decreases, the spectral shape begins to
1/T , only a small anomaly in 1/T was observed at the
change near 26 K. Below 20 K, the spectra show two 1 1
transitionfromtheparamagneticphasetophaseII.22,29)
peaks. Such a structure, which cannot be explained by
Thus the dynamic signature at the magnetic transition
the anisotropic paramagnetic shift, indicates an inter-
becomesgraduallyobscuredforhigher-fieldphases.How-
nalfieldduetospontaneousantiferromagnetic(AF)mo-
ever, it is still puzzling that, unlike in phase II, 1/T no
ments. Therefore, our result indicates an AF transition 1
longer decreases below the magnetic ordering tempera-
near 26 K at 30 T. Similar measurements at 25.6 T in-
ture in phase III.
dicate the transition near 22 K. As shown in Fig. 3, the
As shown in Fig. 4, β decreases with decreasing tem-
spectralwidthrapidlyincreaseswithdecreasingtemper-
perature below 10 K, indicating more inhomogeneous
aturebelow15K.Thespectralshapebecomesinsensitive
distributions in 1/T . With further decreasing temper-
to the temperature below 4 K, where three broad peaks 1
ature, the distribution of 1/T becomes too large to de-
labeled as A, B, and C are observed. 1
terminetherepresentativevalueof1/T .Inaddition,the
The transition temperature of 26 K is much higher 1
relaxation rates depend on the spectral position at low
than those between the paramagneticphase and phase I
temperatures. Figure 5 shows the recovery curves M(t)
or II below 12 T.22) Therefore,this suggestsa new mag-
measured at positions A, B, and C in Fig. 3 at 1.3 K.
netic phase at high fields which is distinct from phases I
TheintensityatAisrecoveredwithin0.1s,whilethein-
andII.Indeed,the magnetizationdatashowsthe second
tensityatC doesnotsaturateevenat10s.Thesecurves
stepnear26T,indicatingafield-inducedmagnetictran-
can be fit to the stretched exponential function, and we
sition.Thisnew high-fieldphasewillbe calledphaseIII,
obtain 1/T = 311 s−1 and β = 0.49 for A and 1/T
as shown in Fig. 2. 1 1
= 2.9 s−1 and β = 0.33 for C. The recovery curve at
Figure 4 shows the temperature dependences of 1/T
1
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(cid:136) (cid:133)(cid:135)(cid:131)(cid:132)(cid:143) Fig. 7. (Coloronline)Magneticfielddependence ofthecenterof
gravity of the spectrum M1. The experimental temperature is
(cid:133)(cid:134)(cid:131)(cid:132)(cid:143) 0.3 K for the data below 15 T and 0.4 K for the data above
15T. The open and solidcircles represent M1 forthe slow and
(cid:133)(cid:133)(cid:143) fastcomponents, respectively(seethetext).
(cid:133)(cid:132)(cid:143)
broad peak indicated by the dashed line is present in all
(cid:158) (cid:149)(cid:132)(cid:144)(cid:131)(cid:134)(cid:143)
(cid:159)
the spectra, shifting slightly to higher B with increas-
int
ing magnetic field. In phase II, an additional peak and
(cid:129)(cid:130)(cid:131)(cid:132) (cid:130)(cid:131)(cid:130) (cid:130)(cid:131)(cid:132) (cid:130)(cid:131)(cid:133) (cid:130)(cid:131)(cid:134) (cid:130)(cid:131)(cid:135) a shoulder indicated by the dotted lines are observed at
lower and higher B , respectively. They gradually dis-
(cid:151) (cid:155)(cid:156)(cid:157) int
(cid:152)(cid:153)(cid:154) appears above 22 T. Instead, two additional peaks indi-
cated by the solid lines are observed in phase III. These
Fig. 6. (Color online) Magnetic field dependence of the NMR
twopeaksgraduallydevelopabove24T.Asthemagnetic
spectra at 0.4 K. Straight lines are guide to the eye to follow
differentcomponents (peaks) ofthespectra. fieldincreasesfrom22to28T,thereisagradualtransfer
ofthespectralweightofthepeaksdenotedbythedotted
lines to those denoted by the solid lines, indicating the
B shows an intermediate behavior between the curves coexistence of phases II and III.
measured at A and C. Consequently, M(t) at B does In order to investigate the phase boundary in more
not fit to the stretched exponential function. These re- detail, we examine the first and second moments of the
sults show that the spectra at low temperatures consist spectra defined as,
at least of two components. This is quite similar to the
behavior observed in phase II, where the previous NMR M1 = Z BintI(Bint)dBint
measurements show the coexistence of two components
characterized by the large and small values of 1/T2.29) M = (B M )2I(B )dB (1)
The two component behavior will be discussed in detail 2 Z int− 1 int int
in section 3.3. The onset of rapid spectral broadening
where I(B ) is the NMR spectrum normalized as
int
(Fig. 3) and the strongly inhomogeneous distribution of
I(B )dB = 1. Figure 7 shows the magnetic field
int int
1/T (Fig. 4) indicate that there may be another tran-
1 Rdependence of M . The data below (above) 15 T were
1
sition or crossover at 10 - 15 K. This is shown by the
taken at 0.3 K (0.4 K). At these sufficiently low tem-
dotted line in Fig. 2.
peratures, difference of 0.1 K makes no change in the
spectral shape. As shown in Fig. 7, the data below 22 T
3.2 Magnetic field dependence of NMR spectra
lie onastraightlinethroughthe origin.Above22T,M
1
Figure 6 shows the field dependence of the 51V NMR departs from the straight line. The magnetic field de-
spectra at 0.4 K. The center of gravity of the spectra pendence of M in Fig. 7 reproduces the magnetization
1
shifts to larger values of Bint with increasing magnetic curve19) including the second magnetization step at 26
field. This shift corresponds to the increase of the mag- T.
netization. On the other hand, the width and shape of Figure 8 shows the magnetic field dependence of the
the spectrum are related to the magnetic structure. As line width, i.e. square root of the second moment M ,
2
shown in Fig 6, the spectral shapes are almost indepen- at various temperatures. At 0.3 K, √M decreases with
2
dent of the magnetic field in the ranges of 18 - 21 T and increasingB below4.5T(phaseI).Fortherangeofmag-
above28T.However,the spectra above28T havea dif- netic field 5 - 22 T and at 0.3 - 0.4 K, √M is almost
2
ferentshape fromthe spectrabelow 21T.Thisindicates independentofB,indicatingthatmagneticstructurere-
that the transition from phase II to phase III occurs in mains unchanged over the entire field range of phase II.
the magnetic field region between 21 and 28 T, and the Above28T, √M becomes independentofthe magnetic
2
magnetic structure in phase III is significantly different fieldagain.Thusthe regionabove28T shouldbelongto
from that in phase II. In Fig. 6, we observe that the
J.Phys.Soc.Jpn. FullPaper AuthorName 5
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(cid:204)˝˛ˇ (cid:218)(cid:219)(cid:220)(cid:221)
Fig. 8. (Color online) Magnetic field dependence of square root Fig. 9. (Color online) Temperature dependences of FWHM at
of thesecond momentM2. Theopen andsolidcirclesrepresent various magnetic fields. The data below 11.5 T are from in ref.
the data at 0.3 and 0.4 K, respectively. The solidtriangles and 22.
squaresrepresentthedataat1.35and4.2K,respectively.
on the powder-pattern NMR spectra for various mag-
phaseIII.FortherangeofB between22and28T,√M2 netic states. Figure 10 shows the schematic illustration
increases monotonically with increasing magnetic field. of the typical powder-pattern for three different antifer-
Thisfieldregionislikelytocorrespondtothe broadened romagneticstates.Theoriginofthehorizontalaxisisset
transition due to anisotropy in the powder sample. The atthecenterofgravityofthespectrum.Fig.10(a)shows
similar broadened transition between phases I and II is aGaussianspectrumwhichisexpectedifthe magnitude
alsoobservedaround4.5T withthe width ofthe coexis- of the internal field is randomly distributed. Fig. 10(b)
tence regionof about 1 T.22) In both cases,the width of showsaspectrumforaspindensitywave(SDW)state,32)
the transitionis about20- 25%ofthe center fieldof the wherethemagnitudeoftheinternalfieldhasasinusoidal
transition. A likely origin of the broadened transition is modulation. Fig. 10(c) shows a rectangular spectrum,
theanisotropyofgfactor,whichisestimatedtobeabout whichisexpectedwhentheinternalfieldhasafixedmag-
17% from the electron spin resonance measurements.31) nitude.Examplesofthis caseincludeasimpleN´eelstate
As shownin Fig. 8, √M2 at 0.4 and 1.35 K show sim- withatwo-sublatticestructureandaspiralorderwithan
ilar behavior above 18 T, indicating that the transition isotropic hyperfine coupling. For both cases (b) and (c),
field is unchanged at these temperatures. On the other we assume that the direction of the internal field is ran-
hand, √M2 at 4.2 K lie on a straight line through the domly distributed with respect to the external field, i.e.
origin below 18 T. This behavior indicates a paramag- spin-flop transition does not occur. As shownin Fig. 10,
netic state, where the internal field is induced by the both the SDW and Gaussian distribution give a central
externalmagneticfield.However,√M2 departsfromthe peak. They differ only in the way the intensity vanishes
straight line above 22 T, indicating the appearance of onbothsides.SinceNMRspectraofpowdersamplesare
a spontaneous internalfield. Above 28 T, √M2 becomes determinedbythe distributionofthe magnitude ofBint,
independentofthemagneticfield.Therefore,forthefield it may be difficult to distinguish from the NMR spectra
above 28 T, there must be an AF order even at 4.2 K. alone between anSDW order,where B has a coherent
int
Thus obtainedphase boundariesareplottedby the solid spatial dependence as cos(Q r), and a random distri-
triangles in Fig. 2. bution of B , in particular i·f the distribution is not a
int
In order to establish the phase boundary between Gaussian. On the other hand, it is easy to distinguish
phase II and the paramagnetic phase completely, the thoseorderedstateswithspatialmodulationinthemag-
temperature dependence of the full width at half maxi- nitude of moments such as SDW from the ones with a
mum(FWHM) ofthe spectra wasmeasuredas shownin fixed magnitude of moments. The spectra observed in
Fig. 9 at various magnetic fields. Here, the data below phase I belongsto the formercase asreportedin ref.22,
11.5 T are the same as those presented in ref. 22. The i.e. the magnitude of the internal field is spatially mod-
transition temperatures are determined by the onset of ulated.
line broadening. Above 8.5 T, the onset is insensitive to In phase II, the previous NMR results show coexis-
the magnetic field as indicated by the arrow in Fig. 9. tence of two types of V sites, V and V sites, which
f s
ThetransitionsobservedbyourNMRmeasurementsare are characterizedby the large(fast) and small(slow)re-
summarized in Fig. 2. Phases I and II are characterized laxationrates,respectively.29) Thespin-echodecaycurve
by a weak field dependences of the transition tempera- in phase II was well fit to the two-component function
ture.PhaseIIIischaracterizedbymuchhighertransition I(τ) = A exp( 2τ/T )+A exp( 2τ/T ). The coeffi-
f 2f s 2s
temperatures compared with phases I and II. cients A and A− are proportional−to the number of V
f s f
and V sites within the frequency window covered by
s
3.3 Spin structure the exciting rf-pulse. Figure 11(a) shows the spectra for
To consider spin structures in the low-temperature the V and V sites at 6 T and 0.3 K reported in ref.
f s
phases of volborthite, we begin with general discussion 29.Thesespectraareobtainedby plottingA andA at
f s
6 J.Phys.Soc.Jpn. FullPaper AuthorName
(cid:10)(cid:11)(cid:12) (cid:13)(cid:11)(cid:14)(cid:15)(cid:15)(cid:16)(cid:11)(cid:17) )R+ (cid:22)(cid:23)(cid:18)(cid:24) !"#$%
&!$#’(
(cid:31)
(cid:5)(cid:9) (cid:30)(cid:27)
(cid:8) (cid:29)
(cid:4)(cid:7) (cid:26)
(cid:5)(cid:6) (cid:27)(cid:28) (cid:18)(cid:19)(cid:20)(cid:21)
(cid:4) (cid:26)
(cid:3) (cid:25)
(cid:246)(cid:247)ł øœß
)*+ <=/->?
(cid:2) :
(cid:254)
(cid:0)
(cid:1) ;
(cid:253) ?NOP <=//->?
(cid:254)(cid:255) BF
(cid:253) E
(cid:252) D KL?M
A Q
C
B
A
@
3 =1-./8
G
æ(cid:242)(cid:243) (cid:244)ı
H=0.IJ
Ł(cid:236) ,-./ -.- -./ -.0 -.1 -.2
º
Œ
(cid:231) 3456789
Ø
Ł
(cid:231)
(cid:230) Fig. 11. (Coloronline)(a)NMRspectraforthefast(solidcircles)
and slow (open circles) components at 0.3 K in the magnetic
fieldof6.0T.(b)NMRspectraatτ =10(thicksolidline)and
110 µs (thick dashed line) in the magnetic field of 30.1 T. The
(cid:229) red (blue) dotted line represents the NMR spectra for the fast
(cid:237) (slow)component.
(cid:238)(cid:239)(cid:240)
Fig. 10. (Color online)Typical powder-patternspectra forthree
cases,(a)Gaussiandistributioninthemagnitudeoftheinternal are labeled as A, B, and C. The intensity of the peak
field,(b) aSDWstate wheretheinternal fieldhas asinusoidal A markedly decreases with increasing τ from 10 to 110
modulation, and (c) an N´eel state with a simple two-sublattice µs. That is, the A component has large 1/T as well as
2
structure.
large 1/T . The left half of the peak A of the spectrum
1
for τ = 10 µs can be fit to a Gaussian labeled as “fast”,
whichdisappearsinthespectrumforτ =110µs.Bysub-
variousfrequenciesν withthefixedB =6.0T,afterthe
0
tracting the Gaussiancomponent from the spectrum for
two-component fitting has been done. The spectrum of
τ =10µs,weobtainarectangularcomponentshownby
theV sitesresemblesaGaussian.Therefore,theV sites
f f
the dottedline labeledas“slow”.Then,the “slow”com-
in phase II have a similar feature to the V sites in phase
ponent is quite similar to the spectrum for τ = 110 µs,
I. These V sites should be surrounded by Cu moments
indicating that the two-component description is valid
withspatiallyvaryingmagnitude.Ontheotherhand,the
also in phase III.
spectrum of the V sites shows a rectangular-likeshape.
s
ThecommonfeatureofphasesIIandIIIaretwotypes
indicating that the internal field at the V sites is more
s
ofVsites,the V andV sites,ofsimilarcharacteristics.
homogeneous, compared with that at the V sites. The f s
f
AttheV sites,thespectralshapesresemblesaGaussian
spin structure of the V sites can be a two-sublattice f
s
andtherelaxationratesarelarge.Ontheotherhand,the
N´eel order or a spiral order with an isotropic hyperfine
V siteshavearectangular-likespectralshapewith rela-
coupling. When the hyperfine coupling tensor is slightly s
tivelysmallrelaxationrates.However,theaveragevalues
anisotropic,aspiralorderleadstoasmallmodulationof
of B for the two V sites are quite different for phases
the internalfield,resulting ina trapezoidalspectrum.In int
II andIII as showninFigs. 11(a)and(b). Ingoing from
reality,thismaybedifficulttodistinguishfromarectan-
phase II to phase III, the center of gravity of the whole
gularone.Thuswecannotselectauniquespinstructure.
spectrumshiftstohighervaluesofB duetoincreaseof
The multi-component behavior is observed also in int
the magnetization. What is more interesting is that the
phaseIIIbythe1/T measurementsatlowtemperatures
1
relativepositionsofthetwocomponentschangesubstan-
asshowninFig.5.Thesecomponentsalsoexhibitdiffer-
tially between phases II and III. In phase II, the centers
ent spin-echo decay rates. In Fig. 11(b), two spectra at
of gravity of the fast and slow components are located
30.1 T and 2.5 K are displayed, one for τ = 10 µs and
almostatthesameposition,asshowninFig.11(a).This
the other for τ = 110 µs. The spectrum for τ = 10 µs
shows that the two types of Cu spins, each of which is
is the same as the one shown in Fig. 3, and the peaks
dominantly coupled to either V or V sites, have al-
f s
J.Phys.Soc.Jpn. FullPaper AuthorName 7
mostthesamefield-inducedaveragedmagnetization.On by a symmetry breaking superstructure.
the other hand, in phase III, the center of gravity of the SinceaVsiteislocatedapproximatelyaboveorbelow
slowcomponentis locatedatamuchlargervalueofB thecenterofCuhexagonofthekagomelattice,thedom-
int
thanthatofthefastcomponent.Therefore,theaveraged inant source of the hyperfine field at V nuclei should be
magnetizationcoupledtotheV sitesismuchlargerthan confinedwithinthesixCuspinsonahexagon.Becauseof
s
that coupled to the V sites in phase III. The centers of thedistortedstructureofvolborthite,therearethreedis-
f
gravity of the fast and slow components above 28 T are tincthyperfinecouplingtensorsA ,A ,andA asshown
a b c
showninFig.7.Theformerisonthestraightlineextrap- in Fig. 12(a). The coupling tensors to the other three
olated from phase II, while the latter is located at much spinsA′,A′,andA′ aregeneratedbythemirrorreflec-
a b c
larger B . Therefore, we conclude that the Cu spins tion perpendicular to the b axis at the V site. A and A′
int
coupled to the V sites are responsible for the increase are the same hyperfine tensors but oriented differently,
s
of magnetization at the second step. and they will be effectively different only if the tensor
The center of gravity M is related to the magneti- is anisotropic. The coupling constant A determined in
1 hf
zation M by the relation M = M /A , where A is theparamagneticstateshouldbeequaltotheaverageof
1 hf hf
the coupling constant determined in the paramagnetic the diagonalcomponentsofthe sumofthese sixtensors.
phaseA =0.77T/µ .21,22) Indeed,the centerofgrav- ThehyperfinefieldsattheV andV sitesarewrittenas
hf B α β
ity of the whole spectrum at 31 T, M = 0.165 T, gives B =(A s +A′s′)+(A s +A′s′)+(A s +A′s′)and
1 α a κ a κ b ρ b ρ c λ c λ
the magnetization of 0.21 µ in good agreement with B =(A s +A′s′)+(A s +A′s′)+(A s +A′s′),re-
B β a λ a λ b σ b σ c κ c κ
the measuredmagnetizationof0.22µ .19) Byusing this spectively,where s ands′ denote twoneighboringspins
B ǫ ǫ
relation, the averaged magnetization coupled to the V on the same type of sites (i.e., ǫ stands for Cu1 , Cu1 ,
f ρ σ
(V ) sites, M (M ), at 31 T are estimated to be Cu2 , or Cu2 ).
s fast slow κ λ
0.16 (0.32 µ ). The integrated intensities of the spectra Both1/T and1/T aredeterminedbythetimecorre-
B 1 2
of the fast and slow components are almost the same. lationfunction ofthe hyperfine field.33) Since the hyper-
This means that numbers of the two sites N and N finecouplingtotheuniformmagnetizationA islargely
f s hf
arenearlyequal.Moreprecisedeterminationoftheratio isotropic,22)weexpectthisisalsothecasefortheindivid-
N :N wouldrequirepreciseknowledgeofthefrequency ualcouplingtensorsA ,A ,andA .Iftheyhavesimilar
f s a b c
dependence of 1/T and 1/T . magnitudes,the couplingtotheCu2sites isidenticalfor
1 2
In phase II, a heterogeneous spin state was pro- the two V sites, and the different relaxation rates must
posed,29) in which the V and V sites spatially alter- beascribedtothedifferenceinthetimecorrelationfunc-
f s
nate, because of magnetic superstructure on Cu sites. tions of the Cu1 spins s and s . The model proposed
ρ σ
Arguments were given in ref. 29 why this is more likely in ref. 29 is shownin Fig. 12(b). The Cu1 sites develop
σ
than the macroscopic phase separation. First, the ratio a long range magnetic order with a uniform magnitude
N : N should vary as a function of B and T, if the of moments. The spin fluctuations at these sites should
f s
V and V sites belong to distinct phases. Experimen- have a small amplitude, leading to the slow relaxation
f s
tally, however, N and N are nearly equal irrespective at the V sites. On the other hand, the Cu1 sites show
f s s ρ
of B and T. Second, the V and V sites show similar a modulation in the magnitude of the orderedmoments.
f s
T-dependenceof1/T .29) Thismeans thatthe twotypes The absenceoffully developedstatic momentswouldal-
2
ofCu spins coupled to the V andV sites, respectively, low unusually slow fluctuations responsible for the large
f s
are interacting in a certain way to establish a common 1/T at the V sites.
2 f
T-dependence. Again, this is not possible if they belong In phase II, the spectra of the V and V sites have
f s
to different phases. almost the same M , which is proportional to the field.
1
Althoughitisdifficulttodeterminethesuperstructure Therefore, all Cu sites have both an AF component and
fromtheNMRdataonapowdersamplealone,plausible a field-induced uniform component. In phase III, M is
1
structures were discussed in the previous paper as fol- largely shifted only at the V sites. This shift can be
s
lows.29) ThemostsimplewaytogeneratetwotypesofV explained by ferromagnetic alignment of the Cu1 sites
σ
siteswiththesameabundanceistoformasuperstructure as shown in the right panel of Fig. 12(b). As shown in
doubling the size of the unit cell. On the ideal kagome Fig.11,thewidthofthespectrumoftheV sitesinphase
s
lattice, the three directions k , k ,and k + k (Fig. 12 IIIisnarrowerthanthatinphaseII.Thisdecreaseofthe
1 2 1 2
a) are all equivalent. However, in volborthite, the struc- widthisalsounderstoodbytheferromagneticsaturation
ture is uniform along k and inequivalent Cu1 and Cu2 of the Cu1 sites. Because the ferromagnetic moment is
2 σ
sites alternate along k . Then it is more likely that the alwaysparalleltothemagneticfield,thespectralwidthis
1
superstructure develops along k . Consequently, each of determined by the anisotropy of the hyperfine coupling.
1
the Cu1 and Cu2 sites are divided into two inequivalent Indeed, the spectral shape of the V sites in phase III is
s
sitesCu1 ,Cu1 andCu2 ,Cu2 ,asshowninFig.12(a). similar to a paramagnetic powder pattern. In contrast,
ρ σ κ λ
ItfollowsthattherearetwotypesoftheVsites,V and the width in phase II should be dominantly determined
α
V , as shown in Fig. 12(a). They should correspond to by the AF component. On the other hand, the position
β
the V and V sites. It should be emphasized that dis- and the width of the spectrum of the V sites do not
f s f
tinct spin states on the crystallographicallyinequivalent change substantially at the transition between phases II
Cu1 andCu2 sites do not accountfor our observationof and III. This indicates that the Cu1 spins keep spatial
ρ
two different V sites that are otherwise crystallographi- modulationofthemagnitudeofmomentinphaseIII.We
cally equivalent. Our results can be accounted for only should stress that the width of the V spectrum is much
s
8 J.Phys.Soc.Jpn. FullPaper AuthorName
The similaritybetween our modelshownin Fig.12(b)
andthetheoreticalpredictionontheanisotropickagome
modelbecomesmoreapparentinhighmagneticfields.A
moderate magnetic field should easily drive the small-Q
spiral order of the Cu1 sites into a ferromagnetic state.
This situation is consistent with our observation of in-
duced ferromagnetism of Cu1 spins in phase III. How-
σ
ever, our results indicate that only a half of the Cu1
sites undergoes ferromagnetic saturation across the sec-
ond magnetization step at 26 T. It appears most likely
that the other half of the Cu1 sites (the Cu1 sites) get
ρ
fully polarized across the third step at 46 T. Stouden-
mire and Balents investigated the spin structure of the
Cu2sites forthe sameanisotropickagomemodelinhigh
magneticfields,whereCu1momentsarefully alignedby
the field.6) They found that the Cu2 sites show a quan-
tum phase transition from a ferrimagnetic order along
the field direction to anantiferromagnetic orderperpen-
dicular to the field with increasing field. A remarkable
aspect of the theory is the prediction for a magnetiza-
tion plateau at 2/5 of the saturation magnetization in
the ferrimagnetic region immediately below the transi-
tion field.
The 2/5 plateau has been actually observed in the re-
centmagnetizationmeasurementsabove60 T.25) There-
fore, we further examine if such a scenariois compatible
with our experimental results. When the Cu1 sites ex-
hibit the full moment, s = 1/2,in the plateau region,
z
h i
the Cu2 sites should have the averaged moment s =
z
h i
1/20 in order to account for the total magnetization of
2/5ofthesaturation.Ifweassumethatallthehyperfine
Fig. 12. (Color online) Possible magnetic structures in volbor-
coupling tensors A , A , and A are isotropic and have
thiteprojectedonto thea-bplane.(a)Superstructurealongthe a b c
k1 directionprovidestwoinequivalentVsites,Vα andVβ,and the same magnitude, Aa = Ab = Ac = Ahf/6 = 0.13
fourinequivalent Cusites,Cu1ρ,Cu1σ,Cu2κ,andCu2λ.(b)If T/µB, the contribution from the four Cu2 sites to Bint
the coupling tensors Aa, Ab, and Ac are largely isotropic and is 0.055 T at all V sites in the plateau region. Here, we
havesimilarmagnitudes,differentrelaxationbehaviorattheVf used the averaged g value of 2.15.31,34) This contribu-
andVssitesshouldbeascribedtothedistinctspinstatesatthe
Cu1ρ andCu1σ sites.(c)IfAa isthedominantcoupling,differ- tiontoBint willbereducedto0.028Tat30T,assuming
ent fluctuations at the Cu2κ and Cu2λ sites lead to distinction that the magnetization at the Cu2 sites is proportional
between Vf andVs. to B. In addition, the full moments at the Cu1σ sites
produce B = 0.28 T at the V sites. The sum of these
int s
gives B = 0.30 T, which is reasonably close to the
int
smallerthantheshiftM1inphaseIII,consistentwiththe observed M1 (= 0.24 T) at the Vs sites at 30 T (Fig.
largely isotropic hyperfine coupling in the paramagnetic 7). On the other hand, M = 0.12 T at the V sites at
1 f
phase.22) 30 T corresponds to s = 0.17 at the Cu1 sites. The
z ρ
h i
Our results show interesting correspondence with the summationofthe magnetizationsatthe Cu2,Cu1 ,and
σ
theory on the anisotropic kagome model in the limit Cu1 sites amounts to the total magnetization of 0.28
ρ
J J′.5,6) Schnyder, Starykh, and Balents studied the µ at 30 T, which is in reasonable agreement with the
≫ B
ground state of this model in zero magnetic field. They experimental magnetization of 0.22 µ . Therefore, this
B
found that the Cu1 sites develop either a ferromagnetic model is semiquantitatively compatible with the NMR
order or a spiral order with a long wave length with the and magnetization measurements.
full moment, while the Cu2 sites have only a small or- Let us remark that the V and V sites show ex-
s f
dered moment.5) The spin fluctuations at the Cu2 sites tremely different relaxationrates in phase III. As shown
associated with the large energy scale J should also inFig.5,theV sites(lineC)showtwoordersofmagni-
∼ s
lead to a small contributions to 1/T1 and 1/T2 at the tudesmaller1/T1thantheVf sites(lineA).Thespectra
V sites. The irrelevance of the Cu2 sites for both the in Fig. 11(b) for different values of τ indicate orders of
static internal field and the dynamic relaxation rates at magnitude difference in 1/T . In contrast, the difference
2
the V sites is consistent with our model that the differ- in 1/T between the V and V sites is only a factor of
2 f s
encebetweentheVf andVs sitesareascribedtothetwo fourinphaseII.29) Infacttheextremelysmallrelaxation
types of the Cu1 sites.It should be noted,however,that rates at the V sites in phase III is what should be ex-
s
thetheorydoesnotaccountforthesuperstructurewhich pected if the moments at the Cu1 sites are completely
σ
produces the different Vf and Vs sites. saturated by magnetic fields and the contribution from
J.Phys.Soc.Jpn. FullPaper AuthorName 9
the Cu2 sites is heavily suppressed for J J′. On the 2) M.Hermele,Y.Ran,P.A.Lee,andX.-G.Wen:Phys.Rev.B
other hand, in phase II, the spins at the Cu≫1 sites have 77(2008) 224413.
σ
finite fluctuations eventhough they show a conventional 3) R. R. P. Singh and D. A. Huse: Phys. Rev. B 76 (2007)
magneticorder.The anomalousfluctuations ofthe Cu1ρ 180407(R).
spins can then couple to the fluctuations of the Cu1 4) O. C´epas, C. M.Fong, P. W. Leung, and C. Lhuillier: Phys.
σ
Rev.B78(2008) 140405(R).
spins, leading to the similar temperature dependence of
5) A.P.Schnyder, O.A.Starykh,andL.Balents:Phys.Rev.B
1/T2 at the Vf and Vs sites.29) 78(2008) 174420.
The above discussion is based on the assumption that 6) E. M. Stoudenmire and L. Balents: Phys. Rev. B 77 (2008)
all the hyperfine coupling tensors are largely isotropic 174414.
and have similar magnitudes. However, the low symme- 7) F. Wang, A. Vishwanath, and Y. B. Kim: Phys. Rev. B 76
(2007) 094421.
try of the volborthite structure may result in large dif-
8) J.-C. Domenge, P. Sindzingre, C. Lhuillier, L. Pierre: Phys.
ference in the hyperfine coupling. In particular, Aa and Rev.B72(2005) 024433.
Ac involve significantly different V-O-Cu hybridization 9) J. N. Reimers and A. J. Berlinsky: Phys. Rev. B 48 (1993)
paths. If one of them, say Aa, is the dominant coupling, 9539.
the differentrelaxationratesfor the two Vsites must be 10) M.Elhajal,B.Canals,andC.Lacroix:Phys.Rev.B66(2002)
ascribed to the different dynamics of s and s . An ex- 014422.
λ κ
11) Z.Hiroi,M.Hanawa,N.Kobayashi,M.Nohara,H.Takagi,Y.
ample of this case is showninFig. 12(c).However,there Kato,andM.Takigawa: J.Phys.Soc.Jpn.70(2001) 3377.
has been no microscopic theory which supports such a 12) M.P.Shores,E.A.Nytko,B.M.Bartlett,andD.G.Nocera:
scenario. J.Am.Chem.Soc.127(2005)13462.
13) Y.Okamoto,H.Yoshida,andZ.Hiroi:J.Phys.Soc.Jpn.78
4. Summary (2009) 033701.
We have presented the 51V NMR results of the S = 14) K.Morita,M.Yano,T.Ono,H.Tanaka,K.Fujii,H.Uekusa,
Y.Narumi,andK.Kindo:J.Phys.Soc.Jpn.77(2008)043707.
1/2 distorted kagome lattice volborthite. Following the
15) J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M.
previous experiments in magnetic fields below 12 T, the Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H.
NMR measurements have been extended to higher fields Chung,D.G.Nocera,andY.S.Lee:Phys.Rev.Lett.98(2007)
upto31T.Inadditiontothealreadyknowntwoordered 107204.
16) S.-H.Lee,H.Kikuchi,Y.Qiu,B.Lake,Q.Huang,K.Habicht,
phases(phasesIandII),wefoundanewhigh-fieldphase
andK.Kiefer:Nat.Mater.6(2007) 853.
above 25 T, at which the second magnetization step oc-
17) T.Imai,E.A.Nytko,B.M.Bartlett,M.P.Shores,andD.G.
curs. This high-field phase (phase III) has the transition Nocera:Phys.Rev.Lett. 100(2008) 077203.
temperature of about 26 K, which is much higher than 18) A.Olariu,P.Mendels,F.Bert,F.Duc,J.C.Trombe,M.A.de
those of phase I (1 K) and phase II (1 - 2 K). At low Vries,andA.Harrison:Phys.Rev.Lett. 100(2008) 087202.
19) H.Yoshida,Y.Okamoto,T.Tayama,T.Sakakibara,M.Toku-
temperatures,twotypes ofthe Vsites areobservedwith
naga,A.Matsuo,Y.Narumi,K.Kindo,M.Yoshida,M.Taki-
different relaxation rates and line shapes in phases II gawa,andZ.Hiroi:J.Phys.Soc.Jpn.78(2009) 043704.
and III. Our results indicate that both phases exhibit a 20) A. Fukaya, Y.Fudamoto, I. M.Gat, T.Ito, M.I.Larkin, A.
heterogeneous spin state, where an ordered state with T. Savici, Y. J. Uemura, P. P. Kyriakou, G. M. Luke, M. T.
non-uniformmagnitudeofmomentwithanomalousfluc- Rovers, K. M. Kojima, A. Keren, M. Hanawa, and Z. Hiroi:
Phys.Rev.Lett.91(2003)207603.
tuationsalternateswithamoreconventionalorderwitha
21) F.Bert,D.Bono,P.Mendels,F.Ladieu,F.Duc,J.-C.Trombe,
fixedmagnitudeoftheorderedmoment.Thelattergroup andP.Millet:Phys.Rev.Lett. 95(2005)087203.
ofspinsareresponsibleforthe increaseofmagnetization 22) M. Yoshida, M. Takigawa, H. Yoshida, Y. Okamoto, and Z.
at the second step between the phase II and phase III. Hiroi:Phys.Rev.Lett.103(2009)077207.
We proposed a possible spin structure in phase III com- 23) S. Yamashita, T. Moriura, Y. Nakazawa, H. Yoshida, Y.
Okamoto, andZ.Hiroi:J.Phys.Soc.Jpn.79(2010)083710.
patiblewiththeNMRandmagnetizationmeasurements.
24) O.Janson,J.Richter,P.Sindzingre,andH.Rosner:Phys.Rev.
Our model has common features with the theories on B82(2010)104434.
the anisotropic kagome lattice in the limit J J′,5,6) 25) Y.Okamoto,M.Tokunaga,H.Yoshida,A.Matsuo,K.Kindo,
although the formation of heterogeneous super≫structure andZ.Hiroi:Phys.Rev.B83(2011) 180407(R).
was not predicted by the theories. 26) K.Hida:J.Phys.Soc.Jpn.70(2001)3673.
27) H.NakanoandT.Sakai:J.Phys.Soc.Jpn.79(2010)053707.
Acknowledgment 28) G.J.Nilsen,F.C.Coomer,M.A.deVries,J.R.Stewart,P.
P.Deen,A.Harrison,H.M.Ronnow:arXiv:1001.2462v2.
We thank F. Mila for stimulating discussions. This
29) M. Yoshida, M. Takigawa, H. Yoshida, Y. Okamoto, and Z.
work was supported by MEXT KAKENHI on Priority Hiroi:Phys.RevB84(2011) 020410(R).
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the MEXT-GCOE program, and EuroMagNET under
H.Yoshida,Y.Okamoto,Z.Hiroi:J.Phys.:Conf.Series145
the EU contract NO. 228043.
(2009) 012010.
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