ARTICLE IN PRESS

Physica B 346–347 (2004) 27–33

The 18-magnetization plateau state in the 2D quantum
antiferromagnet SrCu2ðBO3Þ2: spin superstructure, phasetransition, and spin dynamics studied by high-ﬁeld NMR
M. Takigawaa,*, K. Kodamaa, M. Horvatic! b, C. Berthierb,c, H. Kageyamaa,1,
Y. Uedaa, S. Miyaharaa,d,2, F. Beccad,3, F. Milada

Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japanb
Grenoble High Magnetic Field Laboratory, CNRS and MPI-FKF, 38042 Grenoble, FrancecLaboratorie de Spectrom!etrie Physique, Universit!e Joseph Fourier Grenoble I, St.-Martin d’H"eres 38402, FrancedInstitut de Physique Th!eorique, Universite! de Lausanne, Lausanne CH-1015, Switzerland

Abstract
We report the NMR observation of spin superstructure in the 18 magnetization plateau phase of SrCu2 ðBO3 Þ2 ; a quasi2D dimer singlet spin system, at high magnetic ﬁelds near 28 T: The 18 phase is characterized by crystalization of triplets
in a rhomboid unit cell with oscillating magnetization and separated from uniform phase by a ﬁrst-order transition.
Comparison between the Cu and B NMR spectra and theoretical calculations on the Shastry–Sutherland model
enabled us to determine the three-dimensional spin superstructure. We also found that subtle structural feature of thebuckling of magnetic layers produces anisotropic interactions, which lead to ﬁeld-induced staggered magnetization.
r 2004 Elsevier B.V. All rights reserved.
PACS: 75.25.+z; 75.30.Kz; 76.60.Jx; 76.60.Cq
Keywords: SrCu2 ðBO3 Þ2 ; Magnetization plateau; NMR; Quantum spin systems

1. Introduction
Exploration of quantum phase transitions inspin systems induced by strong magnetic ﬁeld is a
*Corresponding author. Tel./fax: +81-4-7136-3225.
E-mail address: masashi@issp.u-tokyo.ac.jp (M. Takigawa).
1
Present address: Department of Chemistry, Faculty of
Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan.
2
Present address: Department of Physics, College of Scienceand Engineering, Aoyama Gakuin University, Sagamihara,
Kanagawa 299-8558, Japan.
3
Present address: INFM-Democritos, National Simulation
Centre, and SISSA, I-34014 Trieste, Italy.

remarkably fruitful area of recent research in highmagnetic ﬁeld, see for, e.g. Refs. [1,2]. Special
attention has been paid on a class of materials
which do not order magnetically down to zerotemperature. These materials have a collective
singlet ground state with a ﬁnite energy gap to thelowest excited state.
Heisenberg chains with integer spins, spinladders, and coupled spin dimers are well-known
examples. Low lying excitations in these materials
are S ¼ 1 triplet magnons with energy dispersion
EðqÞ as a function of momentum q. The minimumof the dispersion at q ¼ Q deﬁnes the spin-gap D:

0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2004.01.014

ARTICLE IN PRESS
28

M. Takigawa et al. / Physica B 346–347 (2004) 27–33

When a magnetic ﬁeld is applied, the triplets
degeneracy is lifted by the Zeeman splitting. The
gap to the lowest Sz ¼ 1 triplet branch D  gmB H
becomes zero at the critical ﬁeld Hc ¼ D=ðgmB Þ andﬁnite density of Sz ¼ 1 triplets, i.e. ﬁnite magnetization, appears above Hc : The Bose–Einsteincondensation should then occur at sufﬁciently low
temperatures, where the order parameter of themacroscopic condensate /a0 S is equivalent to thestaggered magnetization M> ðQÞ perpendicular tothe ﬁeld. Such ﬁeld-induced antiferromagnetic
order has been indeed observed in a number ofspin-gap systems [3,4] and is well described in theframework of Bose–Einstein condensation [5,6].
Upon further increasing the ﬁeld, the magnetization does not always increase smoothly up to thesaturation. Instead, in some materials, the magnetization stays constant at fractional values of thefull saturation for a ﬁnite range of magnetic ﬁeld[7]. At magnetization plateaus the density of
triplets is kept constant against a change of thechemical potential (magnetic ﬁeld), therefore, an
excitation gap should appear again. Oshikawa
et al. [8] developed a topological theory formagnetization plateaus in one dimension basedon the Lieb-Shultz–Mattis theorem. They
P showedthat if the total magnetizationj Szj is aconserved quantity, an excitation gap exists onlywhennðS  mÞ ¼ integer;m ¼ /Sz S;

ð1Þ

where n is the periodicity of the ground state.
When the spin hamiltonian is mapped onto aninteracting boson system, S  m gives the numberof particles per site. Therefore, this is equivalent tothe commensurability condition that the particle
number in a unit cell of the ground state has to be
an integer. This argument was then extended toarbitrary dimensions [9]. It follows that if amagnetization plateau at a fractional value S 
m ¼ p=q (p and q are mutually prime) is observedwith q greater than the number of spins in a
crystalline unit cell, the ground state must break
the translational symmetry. Physically, such broken symmetry should result from localization of
triplets forming a superlattice due to repulsive
interactions. Magnetization plateaus are thus quite
analogous to charge-ordered insulating statesoften observed in transition metal oxides [10]. In
this paper, we describe the direct observation of amagnetic superlattice for the magnetization plateau state at 18 of the full saturation in the quasi-2D
spin-dimer system SrCu2 ðBO3 Þ2 by nuclear magnetic resonance (NMR) experiments at highmagnetic ﬁeld up to 28 T [11].
The crystal of SrCu2 ðBO3 Þ2 has a layered
structure as depicted in Fig. 1(a) where themagnetic CuBO3 layers and non-magnetic Sr
layers alternate along the c-axis. The magnetic
layers contain orthogonal dimers formed by pairsof nearest-neighbour Cu2þ ions each carrying spin
1
2: If we consider isotropic antiferromagnetic
Heisenberg exchange only between the nearest
neighbour ðJÞ and next-nearest neighbour ðJ 0 Þ
spins, we obtain the model shown in Fig. 1(b)proposed by Shastry and Sutherland two decades
ago [12,13] The ground state of this model isobvious in the two limiting cases. When J=J 0 b1; it
reduces to a collection of dimer singlets, while if
J=J 0 51; the model is equivalent to an antiferromagnet on a square lattice with a Ne! el order.
Whether there are other phases is still an open
question. It is known that for J=J 0 o0:68 thesimple product of dimer singlets is the exact
ground state. Various experiments have established that SrCu2 ðBO3 Þ2 at zero magnetic ﬁeld has
the dimer-singlet ground state with the spin-gap
D ¼ 35 K and appropriate values of the parameters are determined as J=J 0 ¼ 0:64; J ¼ 85 K:
It is also known that frustration between J andJ 0 of the Shastry–Sutherland model strongly
suppresses the kinetic energies of triplets. Indeed
neutron scattering experiments have demonstrated

Cu

J'

J

B
O
Sr
(a)

(b)

Fig. 1. (a) Schematic structure of SrCu2 ðBO3 Þ2 viewed along
the c-axis. The box shows a unit cell. (b) The Shastry–
Sutherland spin model.

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M. Takigawa et al. / Physica B 346–347 (2004) 27–33

that the dispersion of triplet magnons in
SrCu2 ðBO3 Þ2 has an extremely small width [14],even though the ratio J 0 =J is close to the criticalvalue. The most striking property of SrCu2 ðBO3 Þ2is the presence of magnetization plateaus at 18; 14;
and 13 of the full saturation [15]. According to thecommensurability condition mentioned above, theground state at these plateaus should break thetranslational symmetry. The hard-core-boson calculation indeed predicted that small kinetic energies of triplets and their repulsive interactions
lead to localization of the triplets to form stripe
superlattices at 14 and 13 plateaus. For the 18 plateau,superlattices with either square or rhomboid unit
cell were proposed [18]. However, no deﬁnite
conclusion was obtained concerning the stabilityof a speciﬁc structure [16–18].

resulting in ﬁne structure in the NMR spectra.
NMR experiments were done on a single crystal
grown by the travelling solvent-ﬂoating zone
method using a 20 MW resistive magnet at theGrenoble High Magnetic Field Laboratory.
First, we brieﬂy review the Cu NMR results,
which have been reported in Ref. [11]. We show in
Fig. 2 the Cu NMR spectrum at the magnetic ﬁeldof 26 T (below the onset of the 18-plateau) and27:6 T (inside the 18-plateau) obtained near
TB50 mK: The spectrum at 26 T can be reproduced as a superposition of six lines coming fromsingle Cu site, three lines split by quadrupole
interaction for the two isotopes 65 Cu and 63 Cu; asindicated by the line in the inset. This conﬁrms
that the magnetization is largely uniform. The
NMR spectrum shows a drastic change when theﬁeld is increased up to 27:6 T: There are many
sharp peaks distributed over a extremely wide
frequency range from 100 to over 400 MHz;
pointing to many distinct Cu sites with different
magnetization. This gives a direct evidence for acommensurate magnetic superstructure.
In spite of the apparent complexity of thespectrum, a large number of sharp peaks allowed
us to determine the distribution of hyperﬁne ﬁelds
quite accurately. We found that in order toreproduce all the peak positions of the spectrum,at least 11 distinct hyperﬁne ﬁelds have to be
assumed [11]. The distribution of the hyperﬁneﬁelds is shown in the middle panel of Fig. 3(b),2. Spin superstructure in the 18 magnetization
plateau phase
We have used nuclear magnetic resonance(NMR) at the Cu and B sites to detect superstructure at the 18 plateau phase that occurs in theﬁeld range from 27 to 28:5 T when the ﬁeld isapplied parallel to the c-axis [11]. Since the local
hyperﬁne ﬁeld at the nuclei is determined by theinteraction with neighbouring electron spins,
magnetic superstructure should give rise to distinct
types of nuclei with different hyperﬁne ﬁelds,
H=26 T

63 Cu

Intensity

63 Cu

Intensity

0

29

65 Cu

H=27.6 T

65 Cu

260 280 300 320

Frequency (MHz)

100

150

200

250

300

350

400

450

Frequency (MHz)Fig. 2. The Cu NMR spectrum at 26 T (inset) and 27:6 T (main panel). The arrows in the main panel shows the reference NMR
frequency in diamagnetic materials.

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30

2
3

4

5
1

B1

3

B3
6

B4

Experiment

7
8

4

2

site <Sz >1 0.406
2 0.00604
3 -0.00264
4 -0.210

(a)

1

5

6

8
7

B2

site <Sz >5 0.00306
6 0.294
7 0.0301
8 -0.0265

-20

-10

0

10

<Hn > (T)(b)

Fig. 3. (a) Calculated magnetization proﬁle. Hatched circles
show the B sites shown for later discussion. Solid (open) circles
indicate the Cu sites with magnetization parallel (antiparallel)to the external ﬁeld with the circle size representing themagnitude. There are eight distinct Cu sites in one layer. The
values of local magnetization are listed below. (b) Histogram of
the hyperﬁne ﬁeld distribution. The middle panel shows theexperimental results, while the upper (lower) panel indicates thecalculated results without (with) the transferred hyperﬁne
coupling. See Ref. [11] for details.

where long (short) lines indicate that the popula1
tion of the site is 18 (16). The NMR spectrum
generated from this distribution is shown by the
thick line in Fig. 2. By symmetry consideration, it
is concluded that only the rhomboid-type unit cellis compatible with the NMR observation of morethan 11 sites [11]. The middle panel of Fig. 3(b)shows that two sites have large negative hyperﬁneﬁelds, which through a negative hyperﬁne couplingmeans large magnetization along the ﬁeld. Thus
the triplets must be spatially extended, instead ofbeing conﬁned on one dimer as envisaged by theapproximate treatment of hard-core-boson model[16–18]. Note also that some sites have positive
hyperﬁne ﬁeld, i.e. local magnetization opposite to
the magnetic ﬁeld. Therefore, the magnetization
oscillates within a unit cell.
In order to gain insight into the detailed spatial
magnetization proﬁle, the Shastry–Sutherlandmodel with spin-phonon coupling was solved fora 16-sites cluster with the periodic boundary
condition appropriate for the rhomboid unit cell
[11,19]. Typical result of the calculated magnetization proﬁle is shown in Fig. 3(a). The spin-phonon
coupling was included in order to select a uniqueground state out of the degenerate ground state
manifold [11,19]. This is physically justiﬁed since
sizeable spin-phonon coupling was indeed observed by sound velocity measurements [20]. We
found that the magnetization-proﬁle depends only
weakly on the spin-phonon coupling parameters
and the resultant lattice displacements is less than
1%. Therefore, we speculate that the superstructure will be stable in the thermodynamic limit
without the spin-phonon coupling.
The magnetization proﬁle in Fig. 3(a) demonstrates that one ‘‘triplet’’ unit basically consists ofthree dimers; a central dimer with large parallel
moments and neighbouring dimers on both sidesof the central dimer with antiparallel moments.
The large negative moments next to the central
dimer are due to the unfrustrated exchange ﬁeld
from the central dimer. Similar calculations showthat this unit will survive in the 14 and 13 plateau
phases [19]. The magnetization decays rapidly as
the distance from the central dimer increases.
However, the oscillation persists over the entire
unit cell. This is analogous to Friedel oscillations
around impurities in metals or staggered magnetization near impurities in quantum spin chains[21,22] and two-dimensional superconducting cuprates [23]. The distribution of the hyperﬁne ﬁeld
obtained from the calculated magnetization-proﬁleshows good agreement with the experimental
results as shown in Fig. 3(b), in particular, when
the transferred hyperﬁne interactions with neighbouring sites are considered.
While the Cu NMR spectrum combined withthe theoretical results yielded detailed knowledge
about the magnetization proﬁle within one layer,
the 11 B (nuclear spin 32) NMR spectrum turned out
to provide important information about the threedimensional stacking pattern. This is because thehyperﬁne interactions between B nuclei andneighbouring Cu2þ spins are rather weak and thedipolar ﬁelds from adjacent layers are relatively
important. The B NMR spectra at the ﬁeld of
27:5 T are shown in Fig. 4 at three different
temperatures. The spectrum at 550 mK consists ofthree sharp quadrupole-split lines indicating auniform hyperﬁne ﬁeld. On the other hand, thespectrum at 150 mK showed a large number ofpeaks corresponding to the superstructure in the 18

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31

H=27.5T

550mK

Intensity

(a)

B2

B1 B4

B3

Intensity

480mK

0
150mK

372

374
376
Frequency (MHz)

-0.2

378

Fig. 4. The B NMR spectrum for the magnetic ﬁeld of 27:5 Talong the c-axis at different temperatures.

phase. These two types of spectra coexist at
480 mK: Close inspection of the spectrum at
150 mK allows us to generate the distribution of
the hyperﬁne ﬁeld shown in Fig. 5(b), whichfaithfully reproduces the observed NMR spectrum
when convoluted with the quadrupole structure.
There are three peaks with large negative
hyperﬁne ﬁelds, in contrast to the case of Cu
NMR where two sites have a large negative
hyperﬁne ﬁeld. The difference should be ascribedto the dipolar ﬁelds at the B sites from neighbouring layers. We found that the B NMR spectrum
can be best explained by the stacking patternshown in Fig. 5(a). The central dimer of the triplet
units in the adjacent layers are located above andbelow the dimer containing sites 7 and 8 shown in
Fig. 3(a). Although the B1 and B2 sites in Fig. 3(a)are equivalent within one layer, the B1 site has asizable positive dipolar ﬁeld from the large
moments on the neighbouring layers while thedipolar ﬁeld at the B2 site is very small. Therefore,
the total hyperﬁne ﬁeld at the B1 site is muchlarger than the ﬁeld at the B2 site. A similar
situation occurs for the B3 and B4 sites. The

(b)

-0.1

0

0.1

<Hn> (T)Fig. 5. (a) Three-dimensional stacking pattern of the superlattice compatible with the B NMR spectrum. The triplet units
on adjacent layers are distinguished by black and grey circles.
(b) Hyperﬁne ﬁeld distribution at the B sites in the 18 phase.
Convolution of the distribution and the quadrupole structure
reproduces the NMR spectrum at 150 mK: The vertical lines
are the calculated results including the dipolar ﬁelds fromneighbouring layers.

difference in dipolar ﬁeld is apparent in thecalculated hyperﬁne ﬁeld distribution shown by
the lines in Fig. 5(b), which is the sum of thetransferred hyperﬁne ﬁelds from the nearest andnext nearest Cu spins and the long-ranged dipolarﬁelds. The values of the transferred hyperﬁne ﬁelds
are adjusted to best reproduce the experimental
spectrum. This stacking pattern is consistent withthe results of the hard-core boson calculation [19],which concludes that triplets on adjacent layers
tend to sit as far as possible due to repulsive
interaction.

3. Nature of the phase transition
We discussed the nature of the quantum phasetransition into the 18 phase in Ref. [11] based on theevolution of the B NMR spectrum as a function ofmagnetic ﬁeld. The ﬁrst-order phase transition is

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32

demonstrated by a discontinuous jump of themagnetization deduced from the averaged hyperﬁne ﬁelds at the Cu sites. The order parameters of
the 18 phase is given by the spatial variation of thelocal magnetization, for which the overall width of
the NMR spectrum provides a good measure. It
was found that the spectral shape does not change
in the 18 phase at low temperatures when the ﬁeld ischanged, again consistent with the discontinuoustransition.
The results in Fig. 4 demonstrate that the phasetransition into the 18 state as a function oftemperature occurs near 500 mK for the ﬁeld of
27:5 T: There is a ﬁnite range of temperature
where the uniform phase and the 18 phase coexist
similar to the case of ﬁeld-induced transition [11].
The width of the entire spectrum, however,decreases slightly with increasing temperature asshown by the arrows in Fig. 4. Thus the orderparameter is reduced with increasing temperature,in contrast to the case of ﬁeld-induced transition.
The plots in Fig. 6 show the temperaturedependence of the intensity of the spectrum fromthe uniform phase (solid circles) and the 18 phase(open circles). The peak height of the centre line istaken as a measure for the intensity of the uniform
phase and the peak height at 372:5 MHz (at150 mK) is plotted to present the intensity of the1
8 phase. These intensities represent the volume
fraction of each phase, which varies continuously

15

H=27.5T

Intensity

10

5plateau phase
uniform phase
0
0.2

0.3

0.4
T (K)

0.5

0.6

Fig. 6. Spectrae intensity of the uniform and the 18-plateau
phases plotted as a function of temperatures.

over a rather wide temperature range. The overall
width of the spectrum from the 18 phase, however,develop discontinuously, implying that the temperature induced transition is also ﬁrst order.

4. The Dzyaloshinski–Moriya interaction and theﬁeld-induced staggered magnetization
Although the spin superstructure in theplateau phase is basically understood by the
Shastry–Sutherland model, some properties of
SrCu2 ðBO3 Þ2 cannot be explained by an isotropic
Heisenberg model alone. For example, the anisotropic Dzyaloshinski–Moriya (DM) interaction on
the second nearest neighbour (interdimer) bonds
have to be considered in order to explain the
splitting of the triplet excitation energies observedby neutron scattering [24] and electron spinresonance [25,26] experiments.
Another feature is the non-coplanar structure of
the CuBO3 layers. There is a structural phasetransition at 395 K; below which buckling occurs
in the CuBO3 layers and the intradimer J bonds
lose inversion symmetry [27]. This buckling resultsin the alternate tilting of the principal axis of the gtensor form the c-axis toward four equivalent/1 1 0S directions on the one hand. On the other
hand, it results in the intradimer DM interaction.
Both of these perturbations produce an effective
staggered ﬁeld in the presence of a uniform
magnetic ﬁeld. Recently such ﬁeld-induced staggered magnetization has been indeed observed by
NMR experiments. We found that the orientation
dependence of the B NMR frequency shift at low
ﬁeld ð7 TÞ shows unusual variation with temperature. The direction at which the NMR shift isextremal moves signiﬁcantly below 20 K [28]. This
indicates rapid growth of the ﬁeld-induced staggered magnetization at low temperatures, where
the uniform magnetization becomes exponentiallysmall.
Miyahara et al. [29] have shown that such abehaviour can be well described by considering
the ground state of an isolated dimer with boththe DM interaction (D-vector perpendicular to themagnetic ﬁeld) and the alternating g-tensor. The
key observation is that both interactions mix

1
8

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M. Takigawa et al. / Physica B 346–347 (2004) 27–33

the singlet ground state with the lowest triplet
branch so that staggered magnetization linear inﬁeld appears in the lowest-order perturbation. Itshould be noted that the interdimer DM interaction discussed earlier [24] affects the dispersion of
the triplets but does not have matrix elements
between singlet and one triplet states. At sufﬁciently high ﬁelds, the mixing causes ﬁnite magnetization to appear below Hc ¼ D=ðgmB Þ andanticrossing of the two levels at Hc ; both of whichappear to be relevant to experimental results. It
has been indeed puzzling that the magnetization
rises rather gradually below the expected Hc andthere is no indication of Bose condensation ofmagnons, instead, the spin-gap persists above Hc[30]. It remains to be seen to what extent the simple
picture derived from these anisotropic interactions
captures the essential physics at high magnetic
ﬁelds.

5. Summary
The spin superstructure in magnetization plateaus breaking translational symmetry was experimentally observed for the ﬁrst time in the 18 phaseof SrCu2 ðBO3 Þ2 : The three-dimensional spatial
distribution of the magnetization was determinedby combining the NMR spectrum and numericalcalculations. The plateau phase is separated fromthe uniform phase by a ﬁrst-order transition. A
subtle structural feature of this compound brings
anisotropic perturbations not included in the
Shastry–Sutherland model, which lead to ﬁeldinduced staggered magnetization.
The work is supported by Grant-in Aid for
Scientiﬁc Research on Priority Area (B) on ‘‘Fieldinduced new quantum phenomena in magnetic
systems’’ from Ministry of Education, Culture,

33

Sports, Science , and Technology of Japan and the
Swiss National Fund.

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