22

Progress of Theoretical Physics Supplement No. 159, 2005

Neutron Scattering Investigation on Quantum Spin System
SrCu2 (BO3 )2Kazuhisa Kakurai,1,∗) Katsuyuki Nukui,2,∗∗) Naofumi Aso,2 Masakazu Nishi,2Hiroaki Kadowaki,3 Hiroshi Kageyama,4 Yutaka Ueda,5 L.-P. Regnault6 andO. Cépas7
1 Advanced Science Research Center, Japan Atomic Energy Research Institute,

Tokai, Ibaraki 319-1195, Japan
2 Neutron Science Laboratory, Institute for Solid State Physics,

The University of Tokyo, Tokai, Ibaraki 319-1106, Japan
3 Department of Physics, Tokyo Metropolitan University, Hachioji 192-0397, Japan
4 Department of Chemistry, Graduate School of Science, Kyoto University,

Kyoto 606-8502, Japan
5 Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581,

Japan
6 DRFMC-SPSMS, CEA-Grenoble, F-38402 Grenoble Cedex 9, France
7 Insitut Laue Langevin, BP 156, F-38402 Grenoble Cedex 9, France

In this paper we would like to review the zero ﬁeld inelastic neutron scattering results
obtained by using both the thermal and cold neutron triple axis instrument. The ﬁrst single
triplet state at around 3 meV is split due to the Dzyaloshinski-Moriya (DM) inter-dimer
interaction with both the out-of-plane and in-plane components. The splitting between the
Sz = ±1 states is of the order of 0.3 meV. The two triplets bound states also form a very
ﬂat bands at 4.9, 5.5 and 6.3 meV with ﬁnite line width of the order and/or twice of the
DM splitting. Finally the higher energy range up to 13 meV can be also characterized bythree rather broad, but ﬂat bands at around 8, 10 and 12 meV with the line width roughly
three times the DM splitting. These excitations can be probably assigned to three and more
triplets bound state excitations.

§1.

Introduction

After the discovery of SrCu2 (BO3 )2 as being a potential realization of the 2dimensional Shastry-Sutherland model by Kageyama et al.,1) and Miyahara andUeda,2) many experimental and theoretical investigations have been undertaken tocharacterize and understand the spin liquid state of this compound. The magnetic
properties of this compound are determined by Cu2+ ions carrying localized spins of
S = 1/2. The network of the Cu2+ ions, as will be described below, has a structure
for which strong frustration eﬀects can be expected. This system shows indeed various unique magnetic behaviours. The magnetic susceptibility shows a maximum ataround 20 K and rapidly drops toward zero with decreasing temperature and thus
suggests the existence of a spin gap. But the Curie-Weiss constant estimated fromthe high temperature susceptibility is Θ = 102.5 K and much larger than the spin∗)∗∗)

E-mail: kakurai@neutrons.tokai.jaeri.go.jp
Present address: Fuji Photo Film Co. Ltd.

Neutron Scattering Investigation on Quantum Spin System SrCu2 (BO3 )223

gap ∆ = 35 K estimated from the low T behaviour.3) High-ﬁeld magnetization measurements have demonstrated the existence of the 1/3, 1/4 and 1/8 magnetization
plateaux.1), 4) All these bulk measurements indicate that SrCu2 (BO3 )2 is an excellent
candidate of a frustrated antiferromagnetic quantum spin system. Furthermore thetheoretical investigation following these experiments indicated the possible mappingof the system on to the Shastry-Sutherland model,5) which then naturally leads tothe question of the quantum phase transition from the non-magnetic singlet groundstate for J 0 /J < 1 to magnetically ordered state for limiting J = 0 case, which isnamely S = 1/2 AF square lattice with J 0 , where J and J 0 are intra- and interdimer exchange, respectively2) (For recent experimental and theoretical review, see
Kageyama6) and Miyahara and Ueda.7) ).
It is therefore interesting and important to obtain both the characteristics of thespin dynamics of SrCu2 (BO3 )2 over the whole relevant energy scale and the detailsof the spin ﬂuctuations to determine the essential exchange coupling determining thequantum critical behaviour. We therefore have been studying the spin excitations inthis interesting 2-D system using both thermal and cold neutron inelastic scattering.
§2.

Description of the system

The crystal structure is of tetragonal symmetry with a space group of I-42m(No. 121) and a = 8.995 Å, c = 6.649 Å at room temperature. Figure 1(a) shows
the structure of SrCu2 (BO3 )2 projected along the c-axis. Layers of (BO3 )3− groups
and Cu2+ ions stack along the c-axis rotating by 90◦ , with nonmagnetic Sr2+ ionsseparating these layers. All copper ions with a localized spin S = 1/2 are crystallographycally equivalent, sitting on the 8i site. For a given CuBO3 layer, the copper
ions along with the oxygen ions of the BO3 group form dimers of planer, edgesharing CuO4 groups. The spin dimers are connected orthogonally by triangularBO3 groups. The antiferromagnetic intra-dimer (nearest neighbour, hereafter abbreviated as nn) and the inter-dimer (next nearest neighbour, hereafter abbreviatedas nnn) exchange are designated as J and J 0 , respectively (see Fig. 1(a)) and one
immediately recognizes that the antiferromagnetic inter-dimer exchange J 0 is causing a strong frustration between the neighbouring dimers because of their orthogonal
conﬁguration.
A buckling of the CuBO3 plane is observed below 395 K. The mutually orthogonal dimers form two diﬀerent planes and these two planes are slightly shifted fromeach other, as indicated in Fig. 1(b). Because the super-exchange path between theplane goes via the closed shell of the Sr2+ ions the inter-planar exchange is expected
to be much smaller than the above-mentioned intra-planar exchange. We therefore
regard the SrCu2 (BO3 )2 as a good realization of the two-dimensional orthogonal
dimer model.
There are the following coupling constants proposed theoretically for the orthogonal dimer model describing SrCu2 (BO3 )2 to explain macroscopic and microscopic
experimental ﬁndings so far. Miyahara and Ueda proposed J = 85 K, J 0 = 54 K
(J 0 /J = 0.635) and the averaged inter-planar exchange of ∼ 8 K on the basis of theexact diagonalization results explaining the low-T magnetic susceptibility, speciﬁc

24

K. Kakurai et al.

Fig. 1. (a) The structure of SrCu2 (BO3 )2 projected on ab-plane. The open circles denote Sr2+ ,black circles O2− , large grey circles Cu2+ and small grey circles B3+ ions. (b) The buckling of
the dimer plane.

heat results.7) On the other hand J = 71.5 K, J 0 /J = 0.603, and inter-planar exchange of ∼ 15 K are proposed by Knetter et al. from the triplet and bound stateenergies using the method of perturbative continuous unitary transformations.8)§3.

Wide energy range investigation using
thermal neutron triple axis instrument

We show ﬁrst the results of the thermal neutron investigation covering the fullrange of the magnetic excitation with rather coarse resolution. The experimental
results presented here are taken on IN22 spectrometer operated by CENG at ILL.
The kf was ﬁxed at 2.663 Å−1 and the horizontal collimations were 25’-40’-60’-60’.
Higher order contaminations were suppressed using the PG ﬁlter after the sample.
11 B-enriched (99.6%) bulk single crystal of 3.6 g was used. The sample was oriented
with its a- and b-axis in the scattering plane and cooled to T = 1.5 K using Orangetype pumped liquid Helium cryostat. We note that the inelastic neutron scatteringresults in the ﬁrst report9) were obtained under similar conditions using thermal
neutron triple axis instrument PONTA at JRR-3M.
Figure 2 depicts typical spectra measured along [1,−1,0] direction starting at
Q = (0,−2,0). As has been already reported in Ref. 9), the prominent peak with a 3
meV energy gap can be clearly recognized in all spectra, corresponding to the lowest
excitation from the singlet ground state to the triplet excited state (Excitation I).
The increase in the counting rate below 2 meV energy transfer is caused by the incoherent scattering and is regarded as a temperature independent background. The
horizontal bar in the ﬁgure indicates the instrumental resolution (∼ 1.2 meV at 3
meV energy transfer) and it can be readily recognized that the lowest triplet excita-Neutron Scattering Investigation on Quantum Spin System SrCu2 (BO3 )225

tion is almost resolution limited at this moderate energy resolution. As discussed inRef. 9) its almost dispersionless nature is the consequence of the localization of thetriplet excitation due to the frustrated interdimer exchange between the orthogonalarrangement of the neighbouring dimers. We discuss the details of the lowest energy
excitation below.
On the higher energy transfer sideof this ﬁrst gapped excited state a somewhat broader peak covering the energy
regime of 4 to 7 meV can be clearlyseen (We denote this group as excitation group II.). One also notices, thatthe position of the maximum intensity
shifts roughly from 5 meV to 6 meV
when Q is changed from h = 0.1 to
0.5. With insuﬃcient counting statistics and limited number of Q points thiscan be interpreted as the evidence of themore dispersive excitation at around 5.5
meV, as in the previous thermal neutron experiment.9) But looking moreclosely at the present data sequence withimproved counting statistics (Fig. 2)
reveals an asymmetry and line widthchange not compatible with a single dispersive peak, but rather a multi-peakstructure covering this range of the energy transfer. As shown in Fig. 2 one
could indeed ﬁt the spectra in this energy range by two peaks, one with aenergy width of ∼ 0.4 meV at around
4.9 meV and the other with an energy
width of ∼ 0.8 meV at around 6.2 meV.
Fig. 2. Typical inelastic spectra as obtained
While the intensity of the peak aroundusing thermal neutron triple axis instru6.2 meV does not change by going fromment. Inset shows the phonon peak aroundQ = (0.1,−2.1,0) to (0.5,−2.5,0), the in14 meV at large Q.
tensity of the peak around 4.9 meV decreases and almost vanishes at (0.5,−2.5,0), thus creating a clear separation of thelowest excited peak at 3 meV and the next higher excitation group at this Q. We
address this point in the following section on high energy-resolution investigation ofthis energy range, where the excitations due to the bound state of two triplets are
expected.
The next excitation group, as designated group III, extend up to 13 meV andshows an even more broad feature. As indicated in Fig. 2, this group can be ﬁtted
with three broad gaussian peaks having an energy width of ∼ 1 meV. The additional
peak structure at around 14 meV is recognized as phonon contribution, as indicated26

K. Kakurai et al.

counts/mon1200(180sec)

in the large Q experiment shown in the inset. The observed energy range for themagnetic excitations is also in accord with the calculation of the dynamical structure
factor for the orthogonal dimer model by Miyahara et al.10) using exact diagonalization for Q = (2π, 2π), which shows an appreciable spectral weight extending up to
about 150 K.
The observation that the magnetic excitations extend in the energy range up to
13 meV is further substantiated by looking at the diﬀerence of the low (T = 2 K)
and high (T = 24 K) temperature spectra as shown in Fig. 3. A clear temperaturedependent feature shows up in the energy range up to 13 meV and can be attributed
to the magnetic contribution. But one should also note that there remains a broad
scattering intensity up to 13 meV above background level even at T = 24 K, which
is most probably of magnetic origin.
It is also surprising that the spec800
tral feature is totally changing although
the temperature 24 K corresponds to
Q=(0.5,-2.0,0)700
energy even less than the characteris600
T=1.5K
tic gap energy of 3 meV. If one would
T=24K
500
have an isolated dimer system with 3
400
meV gap, one would expect the inten300
sity of the lowest excitation to drop
only by roughly a factor 2 due to the200
thermally excited triplets states with100
out changing the qualitative gap char0
0
5
10
15
20
acter of the dimer excitation, as has
Energy (meV)
been observed e.g. in BaCuSi2 O6 11) and
TlCuCl3 .12) In contrast to that the dyFig. 3. The spectra for low (T = 1.5 K) andnamics of the highly frustrated dimer
high (T = 24 K) temperature at Q =system SrCu2 (BO3 )2 changes qualita(0.5,−2,0).
tively by going to temperatures above
the susceptibility maximum and seems to develop the gap only below this temperature. Whether this feature is due to the implicit quantum character of the frustrated
2-D system or the 3-D eﬀect caused by the weak inter-planar interaction is an objectof further investigation.
§4.

High energy-resolution results using
cold neutron triple-axis instrument

4.1. First excited triplet state
We now turn to the high energy-resolution experiments on the 3 meV ﬁrst excited triplet state and on the excitation group II. The high energy-resolution inelastic neutron scattering experiments were performed on the cold neutron triple
axis instrument ISSP-HER at JRR-3M of Japan Atomic Energy Research Institute.
The instrument is installed on a cold neutron guide and is equipped with vertically curved pyrolytic graphite PG(002) monochromator and horizontally curved

Neutron Scattering Investigation on Quantum Spin System SrCu2 (BO3 )227

Fig. 4. Series of high-resolution energy scan at T = 1.5 K for diﬀerent Q. Note, that the spectrumat Q = (1,0,0) shows a three peak, at Q = (1.5,0,0) a two peak and at Q = (1.5,0.5,0) a single
peak structure.

PG(002) analyser. A combination of PG Bragg reﬂection ﬁlter for Ei > 5.1 meV
and cooled beryllium ﬁlter for Ei < 5.1 meV before the sample is used to suppress
higher-order contamination. The experimental conditions were kf being ﬁxed at 1.22
Å−1 (Ef = 3.10 meV) and collimations after the monochromator being open-radialcollimator-open. Figure 4 shows series of energy scans at diﬀerent wave vectors
in zero ﬁeld. It can be recognized that the ﬁrst excitation around 3 meV energy
transfer indeed consists of at the maximum three excitation branches. Figure 5 summarizes the observed excitation energies of the branches in diﬀerent directions. Theenergies were determined by ﬁtting gaussian line shapes with resolution limited line
widths as indicated by the thin solid lines in the spectra of Fig. 4. From the ﬁeld
dependent neutron scattering and ESR investigations it was demonstrated that theﬁeld dependent gap at q = 0 and the anisotropic behaviour can be well accounted
for by introducing the Dzyaloshinski-Moriya (DM) inter-dimer interaction with themain component directed perpendicular to the ab-plane.13) By combining the resultof the perturbative calculation neglecting the frustrated nnn-coupling (J 0 ) valid in
the weak coupling limit and the exact numerical diagonalization for limited q valid
for the strong coupling limit, as is the case for the SrCu2 (BO3 )2 (J 0 /J ∼ 0.62), we28

K. Kakurai et al.

Fig. 5. Q dependence of the peak positions deduced from the spectra as shown in Fig. 4.
Inset: Results of exact numerical diagonalization from Ref. 12).
Fig. 6. Theoretical prediction of the splitting
due to the DM exchange with both the outof-plane and in-plane components in theJ 0 = 0 limit in (a) [110] and (b) [100] direction.

obtained the dispersion relation for the ﬁrst triplet excitation as indicated in theinset of Fig. 5. Notable features are the q-dependent splitting due to the anisotropic
exchange and the ﬁnite bandwidth due to the nnn-coupling. The agreement of thedispersion as determined by the high energy-resolution investigation in [1,1,0] direction and the calculation is excellent and can be regarded as strong evidence for theimportance of the anisotropic DM inter-dimer interaction in the spin dynamics ofthis quantum system. The splitting δ is renormalized by the ﬁnite J 0 and from theexact diagonalization for J 0 /J = 0.62, one obtains δ ∼ 2.0D. From the zero-ﬁled
splitting δ = 0.352 ± 0.008 meV at q = 0 (Q = (2,0,0)), the z-component of the
DM interaction is estimated to be D ∼ 2.1 K. This ﬁnding is in complete agreementwith the recent chopper spectrometer investigation by Gaulin et al.14) employing aslightly better energy resolution.
An anisotropic DM interaction with only z-component would predict a single
peak behaviour both at qa = qb = π/2, e.g. (1.5,0.5,0) and qa = π/2, qb = 0, e.g.
(1.5,0,0). The experiment however clearly shows at least a two peak structure at Q =(1.5,0,0). This suggests that the degeneracy at this point is lifted by another component of the DM interaction. Recalling that the 2-D dimer plane in SrCu2 (BO3 )2
below room temperature is not totally ﬂat, but shows some buckling (Fig. 1(b)), theNeutron Scattering Investigation on Quantum Spin System SrCu2 (BO3 )229

symmetry allows the introduction of the in-plane DM component, which may lift
the degeneracy at (1.5,0,0), but leaves the result for [1,1,0] direction unchanged, as
indicated by the weak coupling limit results depicted in Fig. 6. As shown in Ref. 15)
the Zeeman splitting of the two peaks into the three peak structure observed at thisQ can be also well accounted for by introducing the in-plane component of DM.
The high energy-resolution investigation of the lowest triplet excitation revealed
that most of the observed band width is attributed to the anisotropic exchange interdimer DM interaction and the bandwidth due to the frustrated nnn-interaction J 0 is
indeed very small and is of the order of 0.2 meV. Thus the strongly localized character of the triplet excitation in this orthogonal dimer system is again impressively
demonstrated.
4.2. Two triplets bound states
We now turn to the high energy-resolution investigation of the excitation group
II in the energy range of 4 to 7 meV. As already pointed out in the ﬁrst section
the ﬁrst inelastic neutron scattering results on this energy range was interpreted asindicating a more dispersive nature of this higher energy mode and thus suggesting acorrelated hopping of the two-triplets excited states. However the above-mentioned
data with better statistic already indicated that the Q-dependence in (1,1,0) direction
could be explained by number of ﬂat modes displaying diﬀerent structure factors.
To substantiate this interpretation we performed high resolution experiment on thebound states in the energy range of 4 to 7 meV.16) The experimental conditions were
the same as for the previous high resolution experiment looking at the ﬁrst triplet
excitation.
Figure 7 depicts typical inelastic spectra in the energy range of 4 to 7 meV.
With this high resolution it is obvious that the excitation group in this energy range
consists at least of three peaks with ﬁnite widths. The solid lines in Fig. 7 show
the results of ﬁtting procedure assuming three gaussian peaks convoluted with theresolution function (in the ﬁtting procedure another higher energy mode at around 8
meV is assumed, which we know from the coarse resolution experiment, see Fig. 2.).
Figure 8 summarizes the ﬁtting results. Figure 8(a) shows the Q-dependence ofthe energy positions and the bars represent the intrinsic line widths of the individual excitations. Figure 8(b) indicates the Q-dependent integrated intensity of theindividual excitations. Following features clearly emerge from this analysis.
1) The excitations in the energy range of 4 to 7 meV should be described ascollection of almost ﬂat modes (bandwidth of at most 0.2 meV) at 4.8, 5.5 and 6.25
meV energy transfer with intrinsic line widths of 0.3, 0.5 and 0.5 meV, respectively.
2) The integrated intensities of the upper two modes show only small modulation
with Q, but the intensity of the 4.8 meV mode shows a strong Q dependence, notably
disappears at Q = (2.5,0.5,0).
We remark that these ﬁndings are consistent with the coarse resolution measurement, where we approximated the observed spectra in this energy range by two
ﬂat modes at 4.9 and 6.2 meV energy transfer with line widths of 0.4 meV and 0.7
meV respectively (see Fig. 1).

30

K. Kakurai et al.

Fig. 8. (a) Q dependence of the peak positions. The vertical bars indicate the intrinsic line widths. (b) Integrated intensities of
the individual peaks versus Q.
Fig. 7. Typical inelastic spectra at T = 1.5 K
in the energy range of 4 to 7 meV. The
solid lines are the results of ﬁt procedure,as described in the text.

Fig. 9. Two-triplon contribution as calculatedby Knetter and Uhrig using J 0 /J = 0.603
and J = 71.5 K. (by courtesy of Drs. Ch.
Knetter and G. S. Uhrig)

Recently Knetter and Uhrig calculated the dynamic structure factor of the two-triplon sector of thetwo-dimensional spin 1/2 ShastrySutherland model using the method of
perturbative continuous unitary transformations.17) Their main conclusion isthat the rather ﬂat bands of the twotriplon state are distributed over the energy range of 4.5 to 7 meV. This energy
range corresponds to the previously predicted range due to the correlated hopping. And since many bound states occur the strong level repulsion ﬂattens

Neutron Scattering Investigation on Quantum Spin System SrCu2 (BO3 )231

the bands so that no signiﬁcant dispersion arises. Figure 9 reproduces their calculated spectra for Q = (2,0,0) and (2.5,0.5,0) with J 0 /J = 0.603 and J = 71.5
K. The qualitative agreement with the experimental spectra presented in Fig. 7 isstriking. It can therefore be concluded that the rather ﬂat bands of excitations in theenergy transfer range of 4.9 to 6.3 meV should be associated with the number of ﬂat
two-triplon bound states distributed in this energy range showing very characteristic
Q-dependent spectral weight.
§5.

Conclusions

To summarize the inelastic neutron scattering investigations utilizing both thermal and cold neutron spectroscopy revealed the characteristic spin dynamics of theorthogonal dimer system SrCu2 (BO3 )2 . The negligible kinetic energy of the single
triplet excitation due to the frustrated inter-dimer exchange renders each individual
band a very small band width of at most 0.2 meV. The ﬁrst single triplet state ataround 3 meV is split due to the Dzyaloshinski-Moriya inter-dimer interaction withboth the out-of-plane and in-plane components. The splitting between the Sz = ±1
states is of the order of 0.3 meV. The two triplets bound states also form a very
ﬂat bands at 4.9, 5.5 and 6.3 meV with ﬁnite line width of the order and/or twiceof the Dzyaloshinski-Moriya splitting. Finally the higher energy range up to 13
meV can be also characterized by three rather broad, but ﬂat bands at around 8,
10 and 12 meV with the line width three times the Dzyaloshinski-Moriya splitting.
These excitations can be probably assigned to higher bound state excitations. The
ﬁnite widths observed in the two triplets and higher bound states excitations may
arise from the existence of numerous discrete excitations, as has been indicated by
the ESR experiment by Nojiri et al.18) Unfortunately the energy resolution of thepresent experiments does not allow to address this issue. We probably have to wait
for the operation of a cold neutron chopper instrument at the MW-class spallation
source to clarify this issue.
The series of the inelastic neutron investigations revealed the entire spin dynamics of SrCu2 (BO3 )2 and should contribute to the deeper understanding of this2-dimensional quantum system. Some of the prominent features of spin dynamics
expected for the Shastry-Sutherland model are found in SrCu2 (BO3 )2 , but it is ofgreat interest how the novel anisotropic Dzyaloshinski-Moriya exchange terms modify the quantum critical behaviour in this real system. And whether the origin of theunexpected temperature dependence of the spin dynamics is intrinsic to the 2-D system or a 3-D eﬀect due to the ﬁnite inter-plane exchange awaits further experimental
and theoretical investigations.
Acknowledgements
We would like to thank Drs. J. P. Boucher, S. Miyahara, H. Nojiri, K. Ueda, G.
Uhrig and T. Ziman for stimulating and illuminating discussions. Special thanks are
due to Drs. Ch. Knetter and G. S. Uhrig to allow us to cite their results in Fig. 9.

32

K. Kakurai et al.
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