Journal of the Physical Society of Japan
Vol. 74, No. 8, August, 2005, pp. 2189–2192#2005 The Physical Society of JapanLETTERSHigh Energy-Resolution Inelastic Neutron Scattering Experimentson Triplet Bound State Excitations in SrCu2 (BO3 )2
Naofumi A SO, Hiroshi K AGEYAMA1 , Katsuyuki NUKUIy, Masakazu NISHI,
Hiroaki KADOWAKI2 , Yutaka UEDA3 and Kazuhisa K AKURAI4
Neutron Science Laboratory, Institute for Solid State Physics, University of Tokyo, Tokai, Ibaraki 319-1106
1
Graduate School of Science, Kyoto University, Kyoto 606-8502
2
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397
3
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581
4
Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195(Received February 17, 2005; accepted June 13, 2005)

In this letter we report on the high energy-resolution neutron scattering studies on the dynamic
structure factor of the two-triplet bound states in the two-dimensional S ¼ 1=2 Shastry–Sutherland
system, SrCu2 (BO3 )2 , in the energy transfer range of 4 to 7 meV. In contrast to the earlier interpretationof the low energy-resolution results, these bound state excitations show hardly any dispersion, but doshow a strong wave-vector-dependent structure factor. Although these ﬁndings are in overall agreementwith the recent theoretical calculation of the 2-triplon dynamic structure factor by Knetter and Uhrig[Phys. Rev. Lett. 92 (2004) 027204], they also show distinct diﬀerences, which may be due to somesymmetry-breaking terms in the Hamiltonian that are not included in the theory, such as the interdimer
Dzyaloshinski–Moriya interactions recently discovered in this compound.
KEYWORDS: two-dimensional (2D), orthogonal-dimer system, Shastry–Sutherland, Dzyaloshinski–Moriya interaction, inelastic neutron scattering, spin gap
DOI: 10.1143/JPSJ.74.2189

After the discovery that SrCu2 (BO3 )2 is a good realizationof the two-dimensional Shastry–Sutherland model1) by
Kageyama et al.2) and Miyahara and Ueda,3) many experimental and theoretical studies have been undertaken tocharacterize and understand the spin liquid state of thisquantum antiferromagnetic system. It is now widely accepted that SrCu2 (BO3 )2 should be regarded as an S ¼ 1=2,
highly frustrated orthogonal dimer system with a singlet
ground state with a gap energy of  ¼ 34 K.3) The ratios of
the intradimer exchange J and the interdimer exchange J 0 of
J 0 =J ¼ 0:6354) and 0.6035) have been proposed, being close
to the critical value of (J 0 =Jc )  0:7, where a quantum phasetransition from a singlet dimer state to a Néel state ispredicted.3) The ﬁrst inelastic neutron scattering (INS)experiment evidenced an almost localized ﬁrst triplet
excitation resulting in an almost wave-vector-independent
spin excitation at around 3 meV.6) The localized nature of
the ﬁrst triplet excitation has been explained to be a result of
the frustration of the interdimer interaction due to theorthogonal arrangements of the dimers. This interaction,even to the ﬁfth order in perturbation, prohibits the excitedtriplet excitations from hopping from one dimer to another.7)
The numerical exact diagonalization8) results were consistent with this interpretation.
At the same time the ﬁrst INS also clearly showed theexistence of rather intense higher-energy excitations, i.e.,around 5 and 9 meV.6) These higher-energy excitations werealso observed in ESR9) and Raman scattering10) experiments
and attributed to the bound states theoretically.5,7,8,11) Hereagain the low energy-resolution experimental results with alimited number of Q-points for higher-energy excitations
seemed to be in agreement with the ﬁnite-size exact
E-mail: aso@issp.u-tokyo.ac.jp
Present address: Fuji Photo Film Co., Ltd.

ydiagonalization results, suggesting a more dispersive nature
for these bound states. However, since the ﬁnite-size eﬀect
for an extended bound state is expected to cause severe
restrictions and since the subsequent high energy-resolution
experiments on the ﬁrst triplet excitation clearly indicatedthe importance of the interdimer Dzyaloshinski–Moriya(DM) exchange interaction describing the Q-dependent
splitting of the triplet excitations and their ﬁeld dependence,12–14) we decided to extend the high energy-resolution
INS experiment to the higher-energy excitation range of up
to 7 meV to investigate the nature of the bound states in
SrCu2 (BO3 )2 .
We thus report in the present letter the high-resolution
INS experiment using a cold-neutron triple-axis instrument
to reveal the wave-vector-dependent excitation spectra of thebound state in the strongly frustrated two-dimensional S ¼
1=2 Shastry–Sutherland system SrCu2 (BO3 )2 .
11
B-enriched (99.6%) bulk single crystals of SrCu2 11( BO3 )2 have been prepared by the traveling solvent
ﬂoating zone method using Li11 BO2 ﬂux.15) Inelastic neutron
scattering experiments were performed on the cold-neutron
triple-axis spectrometer ISSP-HER installed at the JRR-3M
reactor at the Tokai establishment of the Japan Atomic
Energy Research Institute. The spectrometer is equippedwith a vertically focused PG (002) monochromator and ahorizontally focused PG (002) analyzer and is operated with 1 ) and colliﬁxed ﬁnal energy Ef ¼ 3:1 meV (kf ¼ 1:22 A
mations after the monochromator of open–radial collimator–
open. A combination of a PG Bragg reﬂection ﬁlter for
Ei  5:1 meV and another Be ﬁlter for Ei < 5:1 meV before
the sample and Be ﬁlter after the sample is used to reduce
higher-order contaminations. The sample consisted of fouraligned single crystals with a total mass of  9 g and wasoriented with its a - and b -axes. The notation of the unit
cell in the (HK0) plane as used by Zheng et al.16) is

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J. Phys. Soc. Jpn., Vol. 74, No. 8, August, 2005

LETTERS

SrCu2(BO3)2, JRR-3M/HER
Ef=3.1meV, T=1.5K

300
250

(a)

Q=(2,0,0)

200
150
100

Intensity / M860k (~10min)

50

I

II

III

0
300
250

(b)

Q=(2.5,0,0)
1.5K
10K
20K

200
150
100
50
I

II

III

0
300
250

(c)

Q=(2.5,0.5,0)

200
150
100
50
II

0
4.0

5.0

III

6.0

7.0

Energy (meV)Fig. 1. Examples of INS spectra at (a) Q ¼ ð2; 0; 0Þ, (b) Q ¼ ð2:5; 0; 0Þand (c) Q ¼ ð2:5; 0:5; 0Þ obtained at T ¼ 1:5 K (circles). Spectra at Q ¼
ð2:5; 0; 0Þ contain the data at T ¼ 10 K (crosses) and 20 K (triangles). The
peaks are labeled at the bottom of the ﬁgure. The solid curve is the ﬁt to
the data, as illustrated in the text.

employed. An ILL-type orange cryostat was used to cool thesample down to 1.5 K.
Constant-Q energy scans at T ¼ 1:5 K were performed at
diﬀerent Q’s in diﬀerent directions of the reciprocal plane.
Figure 1 shows typical INS spectra in the energy range of 4
to 7 meV at characteristic Q points. In Fig. 1(a) the
spectrum at Q ¼ ð2; 0; 0Þ corresponding to the ð0; 0Þ position
is depicted. In contrast to the previous thermal neutron
scattering experiment with moderate energy-resolution(1:5 meV) where one broad peak at around 5 meV with

N. ASO et al.

an intrinsic line width of  0:7 meV was observed,6) one
clearly recognizes three excitation peaks centered at about
4.8, 5.5 and 6.2 meV in the present experiment with higher
energy-resolution. The solid line in the spectrum is atentative ﬁt using three Gaussian peaks convoluted with the
instrumental resolution. (Note that we have assumed higherenergy excitations above 7 meV, as has been shown in theprevious low energy-resolution experiment, to account for
the increase in the intensities towards 7 meV.) The three
excitations, which will be referred to as excitations I, II and
III, all have ﬁnite intrinsic energy widths (full width at half
maximum (FWHM)) of 0.42, 0.51 and 0.77 meV, respectively. The ﬁnite intrinsic energy widths may arise from theﬁnite lifetime of the excitations or the existence of numerous
discrete excitations. A closer look at excitation I in Fig. 1(a)suggests an intrinsic peak structure pointing to a two-peakstructure centered at 4.7 and 4.9 meV as marked by thearrows. The fact that the recent ESR measurement by Nojiri
et al.17) shows at least ﬁve lines between 4.7 and 5.6 meVstrongly suggests that the latter is the case at this low
temperature. However, to clarify how many peaks the
spectrum in this energy range is composed of, we would
need an even higher energy-resolution that is not yet
available in neutron scattering. We therefore continue to ﬁt
the spectra at diﬀerent Q points using the groupings of (I),(II) and (III) as introduced above for Q ¼ ð2; 0; 0Þ.
Figure 1(b) depicts the spectrum for Q ¼ ð2:5; 0; 0Þ. The
overall features of this spectrum are similar to those of Q ¼
ð2; 0; 0Þ. In this graph the spectra taken at 10 and 20 K are
also shown. The temperature dependence of the intensities
shows that the well-structured spectrum at 1.5 K becomes
ﬂat and featureless when increasing the temperature up to 10
and 20 K. This clearly shows that the excitations in thisenergy range are of magnetic origin.
Figure 1(c) shows the spectrum at Q ¼ ð2:5; 0:5; 0Þ. In
contrast to the spectra at the previous two wave vectors, it isobvious that excitation (I) has no spectral weight. The two
broad peaks associated with (II) and (III) excitations areobserved to be centered at 5.5 and 6.2 meV. These results are
consistent with the previous low energy-resolution results
assigning one broad peak at around 6 meV with an intrinsic
energy width of 0.7 meV.6)
Figure 2 summarizes the ﬁt results, as described above, of
all the spectra in diﬀerent directions. Figure 2(a) displays
the energies and widths, indicated by the vertical bars, of thethree excitation groups and Fig. 2(b) depicts their integrated
intensities. It is clear from these results that all the excitationgroups have almost constant Q-independent energies andthat the spectral weights of the groups strongly vary with Q.
In particular, the intensity of excitation group (I) acquires its
maximum at Q ¼ ð2; 0; 0Þ and decreases gradually bymoving away from this point. At around Q ¼ ð2:5; 0:5; 0Þ
the spectral weight of (I) vanishes and the intensities of (II)and (III) dominate the spectra. These characteristic structure
factors of the dispersionless excitation groups led to theenergy points at around 5 and 6 meV in the previous coarse
energy-resolution results (Fig. 3 of ref. 6), which at that time
were explained by the strongly dispersing character of thebound triplets excitations. These high energy-resolution
results clearly demonstrate that there are many isolated
energy levels of bound states in the energy range of 5 to

J. Phys. Soc. Jpn., Vol. 74, No. 8, August, 2005

LETTERS

7.0

Energy (meV)

6.5
6.0
5.5
5.0
4.5

(a)

Integrated Intensity (arb. units)

4.0
30
I
II
III

25
20
15
10
5

(b)
0
(2,0)

(2.5,0)(2.5,0.5)(2,0)

Wave Vector (r.l.u.)
Fig. 2. (a) Q dependence of the energy of excitations I (circles), II(triangles) and III (squares), obtained at 1.5 K. The bars represent the
intrinsic linewidth (FWHM) of excitations I, II and III. The arrows
represent the energy resolution of the instrument (FWHM) at 5.0 and
6.0 meV. (b) Q dependence of the integrated intensity of excitations I, II
and III.

300

6.5

250

6.0

200

5.5

150

5.0

100

4.5

50

4.0

Intensity / M860k (~10min)

Energy (meV)

SrCu2(BO3)2, (HK0)-zone, T=1.5K
7.0

0
(2,0)

(2.5,0)(2.5,0.5)(2,0)

Wave Vector (r.l.u.)
Fig. 3. Image plot of the scattering intensities in the energy–wave vector
plane. Intensities are indicated by the scale on the right hand side.

N. ASO et al.

2191

6.5 meV. This energy range corresponds to the previously
discussed bandwidth of enhanced correlated hopping basedon the coarse energy-resolution results. At the same time theresults clearly show that the dispersion of the individual
bound-state mode in this energy range is at most 0.5 meVand hence the bandwidth due to the correlated hopping of thebound triplets excitations is experimentally determined to beof the order of 0.5 meV in SrCu2 (BO3 )2 . One should notethat the ﬁnite-size exact diagonalization results8) at limited
wave-vector positions indicated a bound-state excitationlevel at around 5 meV with a bandwidth of 0.5 meV.
Figure 3 is an image plot of the scattering intensities inthe energy–wave vector plane. The above conclusions drawn
from the ﬁt results can be reconciled and the almost
dispersionless nature and the Q-dependent structure factorof the excitations are clearly visible.
Very recently Knetter and Uhrig calculated the dynamical
structure factor of two-triplet bound-state excitation, which
they call 2-triplon, in SrCu2 (BO3 )2 by means of perturbative
continuous unitary transformations.18) In contrast to thenumerical exact diagonalization calculation, this perturbative method can provide continuous distribution in thehigher-order bound states’ spectra and hence can be directlycompared with the INS results. They could well reproduce
the spectra at Q ¼ ð2; 0; 0Þ and ð2:5; 0:5; 0Þ [Figs. 2(a) and2(c)] using J 0 =J ¼ 0:603 and J ¼ 6:16 meV (see Fig. 2 ofref. 18). The spectral density of the 2-triplon channel (Fig. 3of ref. 18) calculated with the same set of the parameters
also well reproduces the essential features of our results
represented in Fig. 3. They interpret the nonsigniﬁcantmomentum dependence of the 2-triplon eigenenergies asevidence for level repulsion. Due to the negligible kinetic
energy of the single-triplet state, there are a relatively large
number of individual bound states involved and theirenergetic repulsion causes each individual band to be very
ﬂat.
Although the overall agreement seems to be satisfactory,
particularly the momentum dependence of the spectral
weight at an energy transfer of 4.8 meV and the rather
strong intensity at around 6 meV at Q ¼ ð2:5; 0:5; 0Þ, there
are some features in the observed spectra which are notaccounted for by the calculation. First of all the spectral
weight of excitation group (II) at 5.5 meV is much lesspronounced in the calculation, although the theory predicts
eigenenergy states at that energy. Secondly the observed
intensities at around 6.5 meV are absent in the calculation.
Furthermore, experimentally we do see a ﬁnite width in the
excitation groups, which probably indicates that each modeconsists of numerous split states, as discussed above. These
discrepancies may be due to the interdimer DM interactionsin SrCu2 (BO3 )2 not included in the calculation. The highenergy-resolution INS measurements of the ﬁrst triplet
excitation at around 3 meV revealed the Q-dependent
splitting of the modes in zero ﬁeld and Zeeman splitting ina ﬁeld, which can be only explained by assuming theinterdimer DM interactions.12,13) The qualitative changes in
the kinetic energies of the ﬁrst triplet excitation due to thisterm may show up in the subtle modiﬁcations of the boundstate spectra, as has been observed in the experiment.
To summarize, we have performed high energy-resolution
INS experiments on the two-triplet excitations in the

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J. Phys. Soc. Jpn., Vol. 74, No. 8, August, 2005

LETTERS

Shastry–Sutherland model substance SrCu2 (BO3 )2 . In contrast to the earlier interpretation of the low energy-resolution
results, there are a relatively large number of isolated energy
levels of two-triplet bound states, which show hardly any
dispersion, but show a strong wave-vector-dependent structure factor. These ﬁndings are in overall agreement with therecent theoretical calculation of the ‘‘2-triplon’’, i.e., the twotriplet bound state, dynamic structure factor obtained by
Knetter and Uhrig,18) but show also distinct diﬀerences,which may be due to some symmetry-breaking terms in the
Hamiltonian that are not included in the theory, such as theinterdimer Dzyaloshinski–Moriya interactions recently discovered in this compound.
Acknowledgment
The authors would like to thank Drs. C. Knetter and G. S.
Uhrig for sending us their theoretical calculations prior to
publication and for illuminating discussions and suggestions.
They are also grateful to Drs. H. Nojiri, S. Miyahara andK. Ueda for stimulating discussions. This work has been
partly supported by a Grant-in-Aid from the Ministry of
Education, Culture, Sports, Science and Technology, Japan.

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