VOLUME 84, NUMBER 25

PHYSICAL REVIEW LETTERS

19 JUNE 2000

Direct Evidence for the Localized Single-Triplet Excitations
and the Dispersive Multitriplet Excitations in SrCu2 共 BO3 兲 2
H. Kageyama,1, * M. Nishi,2 N. Aso,2 K. Onizuka,1 T. Yosihama,2 K. Nukui,2 K. Kodama,3 K. Kakurai,2 and Y. Ueda1
1

Material Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo, Kashiwanoha,
Kashiwa, Chiba 277-8581, Japan2
Neutron Scattering Laboratory, Institute for Solid State Physics, University of Tokyo, 106-1 Shirakata, Tokai,
Ibaraki 319-1106, Japan
3
Division of New Materials Science, Institute for Solid State Physics, University of Tokyo, Roppongi, Kashiwanoha,
Kashiwa, Chiba 277-8581, Japan(Received 17 September 1999)
We performed inelastic neutron scattering on the 2D Shastry-Sutherland system SrCu2 共11 BO3 兲2 withan exact dimer ground state. Three energy levels at around 3, 5, and 9 meV were observed at 1.7 K. The
lowest excitation at 3.0 meV is almost dispersionless with a bandwidth of 0.2 meV at most, showing asignificant constraint on a single-triplet hopping owing to the orthogonality of the neighboring dimers.
In contrast, the correlated two-triplet excitations at 5 meV exhibit a more dispersive behavior.
PACS numbers: 75.40.Cx, 75.40.Gb

Recently, much experimental and theoretical effort has
been put into understanding the quantum spin dynamics
in low-dimensional magnetic systems. The interest has
been mainly revived by the discovery of two-dimensional(2D) S 苷 1兾2 spin correlations in high-Tc superconducting cuprates. In the course of these studies, many new lowdimensional magnetic systems with a spin-singlet groundstate have been discovered, e.g., the Haldane, spin-Peierls,
ladder, and plaquette systems, to name only a few [1]. The
spin dynamics of these systems show a spin excitation gap
from the nonmagnetic singlet ground state to the first excited triplet state, which may be regarded as a direct consequence of the quantum spin fluctuations. However, little is
known about the frustration effect in these extreme quantum spin systems. It is, e.g., still controversial as to how
big a role the finite next-nearest interaction in the spinPeierls compound CuGeO3 plays in establishing the observed singlet ground state.
SrCu2 共BO3 兲2 has a tetragonal unit cell with lattice parameters a 苷 b 苷 8.995 Å and c 苷 6.649 Å at room
temperature. It consists of alternately stacked Sr andCuBO3 planes. A neighboring pair of planar rectangular
CuO4 forms a spin dimer and the dimers are connected
orthogonally by a triangular BO3 . It was pointed out by
Miyahara and Ueda [2] that SrCu2 共BO3 兲2 can be mappedon the Shastry and Sutherland model [3], which has anexactly solvable ground state. They have also shown thata phase transition from a Néel to a singlet dimer state
occurs at the critical ratio of interdimer 共J 0 兲 and intradimer
共J兲 exchange of 共J 0 兾J兲c 苷 0.70 (or 0.691 [4]) with decreasing J 0 兾J. Based on this theoretical model, it has
been established from our previous experiments [5–10]
that SrCu2 共BO3 兲2 is an S 苷 1兾2 highly frustrated spin
gap system with an energy gap of D 苷 34 K and with aratio of J 0 兾J 苷 0.68 共J 苷 100 K, J 0 苷 68 K兲 very close
to the critical value. The exactness of the ground state
allows theorists to investigate its excited states in detail.

One of the most remarkable results is that a single triplet
can propagate only from the sixth order in the perturbation
calculations, indicating the significant constraint on the
hopping as shown by Miyahara and Ueda [2]. In addition,
the fifteenth-order calculation of the dispersion relation of
the single triplet by Weihong et al. revealed a prominent
J 0 兾J dependence of the bandwidth [4].
So far no direct experimental investigation has been
carried out to address the questions as to whether the
single-triplet excitation is actually localized and, if so, howdifficult is its hopping process. The appearance of thefractional magnetization plateaus of the total magnetization [5] is associated with the localization of the multitriplets and also with the commensurability to the lattice,which is a kind of Mott transition in the broad sense of
the word. But, the magnetization measurement provides
only the static information of the excited triplets, and theapplied magnetic field splits the triplet state into three levels: Sz 苷 21, 0, 11 (Zeeman splitting), making the situation different from that considered by Miyahara and Ueda.
Likewise, the dynamical properties of the two-triplet mode
has not been investigated yet—although this mode has
been detected by ESR [7] and Raman scattering [8]; bothmethods can probe the excitation only at Q 苷 共0, 0兲.
It is thus interesting and important to probe directly the wave vector dependence of spin excitationsin SrCu2 共BO3 兲2 , the Shastry-Sutherland model withJ 0 兾J 苷 0.68 close to the critical boundary, as the prototype of a strongly frustrated quantum spin system. In
this Letter, we report the first inelastic neutron scatteringexperiments in bulk single crystal to investigate thedynamical properties of the triplet excitations.
Inelastic neutron scattering experiments were carriedout on the ISSP-PONTA spectrometer installed at a 5 G
beam port of the Japan Research Reactor 3M (JRR-3M)
in the Japan Atomic Energy Research Institute, Tokai
Establishment. The spectrometer was operated in the5876© 2000 The American Physical Society

0031-9007兾00兾84(25)兾5876(4)$15.00

VOLUME 84, NUMBER 25

PHYSICAL REVIEW LETTERS

unpolarized neutron mode with a pyrolytic graphite(PG) monochromator and analyzer. The inelastic scans
were performed with fixed final energy Ef 苷 14.7 meV
共kf 苷 2.67 Å21 兲 and horizontal collimations of open
共400 兲-400 -sample-800 -800 . A PG filter was placed after the
sample to suppress the higher order contaminations. In
order to minimize the neutron absorption by the naturalabundance of 10 B, 11 B-enriched (99.6%) bulk single
crystals of SrCu2 共11 BO3 兲2 were prepared by the travelingsolvent floating zone method using Li11 BO2 flux [11].
The neutron scattering sample consisted of two single
crystals with a total volume of ⬃1.5 cm3 aligned within
200 and it was oriented with its a and b axes in the
scattering plane.
For convenience, we use a Brillouin zone at the 共h, k, 0兲
plane as given in Fig. 1, where we do not distinguish between the dimers lying along 关1, 1兴 and 关1, 21兴. In thiscase, the space lattice vectors are transferred to am 苷 共a 2
b兲兾2 and bm 苷 共a 1 b兲兾2, which correspond to the reⴱ
ciprocal space given by the lattice vectors am
苷 aⴱ 2 b ⴱⴱⴱⴱand bm 苷 a 1 b . Shown in Fig. 2(a) are typical energyscans obtained at 1.7 and 24 K for a scattering vector of
Q 苷 共2, 0兲. An abrupt growth in intensity below 2 meVis due to the elastic incoherent scattering from the sample.
The 1.7 K spectrum consists of three peaks centered at 3.0,
5.0, and 9.7 meV (transitions I, II, and III, respectively),while that for 24 K no longer has any appreciable peak,indicating that all excitations are of magnetic origin. As
already confirmed by other measurements [5–10], transition I at D 苷 3.0 meV is the excitation of a single triplet
from the singlet ground state. As shown in Fig. 2(b), thepeak intensities of transitions I–III start to decrease withincreasing temperature and disappear at about 13 K. We
can derive from this fact that transitions II and III arise
from multitriplet excitations, which will be discussed indetail later. It would be important to point out that the
3 meV mode loses all its intensity at 13 K, only one-thirdof spin gap energy (34 K), which is not characteristic ofmost gapped systems [12–16]. We suppose this anomalous

FIG. 1. (a) The reciprocal lattice 共h, k, 0兲 of the nuclear unit
cell. The Brillouin zone used in this study is given by the brokenlines. Along the lines A C and D F, the dispersion relations of
the band I, and bands II and III, respectively, have been studied.
(b) The fundamental unit of the orthogonal dimers.

19 JUNE 2000

temperature dependence indicates the strong spin frustration as well as the quantum effect of some kind in thisorthogonal dimer spin system. A similar behavior has
also been seen in Raman scattering study, in which several well-resolved modes for the multitriplets excitations
below 4.2 K are considerably damped already at 7 K [8].
The Q dependence of transition I was measured at 1.7 Kalong the lines A, B, and C, and that of transitions II and
III along D, E, and F indicated by arrows in Fig. 1(a). It
was found that, independent of Q, the obtained profile fortransition I at 1.7 K is resolution limited, whereas the transitions II and III show intrinsic linewidths. Accordingly,
the 1.7 K profile was fitted to a combination of a delta
function for transition I and two damped harmonic oscillators for transitions II and III, convoluted with the instrumental Gaussian resolution. In spite of large instrumental
resolution about 1.3 meV for transition I, the excitationenergies can be determined to an accuracy of 0.1 meV,less than 10% of the instrumental resolution since thescattering intensity is strong enough. It is noted that thesolid line in Fig. 2(a) includes the temperature independent
term of incoherent scattering around energy zero.
The dispersion relation of band I is shown in Fig. 3.
Most importantly, the excitation energies are almost Q
independent. Namely, the magnitude of the dispersion,
the difference between the maximum and minimum of
the excitation energy, is DE 苷 0.2 meV at most. This

FIG. 2. (a) Energy scans at Q 苷 共2, 0兲 obtained at T 苷 1.7 K(circles) and 24 K (triangles). The peaks are labeled at thebottom of the figure. The solid curve is the fit to the data, as
described in the text. (b) The temperature variation of thenormalized intensities at E 苷 3.15 meV and Q 苷 共1.5, 0.5兲(circles), E 苷 5.15 meV and Q 苷 共2, 0兲 (crosses), and E 苷
10.65 meV and Q 苷 共2, 0兲 (triangles). The broken line is a
guide to the eye.

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FIG. 3. Q dependence of the excitation energies of bands I,
II, and III obtained at 1.7 K. The arrows represent the energy
resolutions of the instrument (FWHM). The solid curves are
guides to the eye. The bars represent the intrinsic linewidth(FWHM) of bands II and III. The theoretical dispersion curvefor the single-triplet excitations by Weihong et al. [4] with J 苷
100 K and J 0 苷 68 K is given by the broken line.

width is extremely small in contrast to conventional
low-dimensional quantum spin systems. Experimentally, strong dispersions of the single-triplet excitations
were observed in the 1D spin-Peierls material CuGeO3
共DE 苷 14 meV兲 along the chain axis [12], and in the2D plaquette system CaV4 O9 共DE 苷 7 meV兲 parallel tothe plaquette plane [13]. On the contrary, well-isolated
clusters of exchange-coupled paramagnetic ions, whereintercluster interactions are rarely important, evidently exhibit flat dispersions in the spin excitation spectrum. Thishas been experimentally shown by the neutron scattering
measurements on the isolated dimer systems Cs3 Cr2 Br9
共DE 苷 1.8 meV兲 [14] and BaCuSi2 O6 共DE 苷 0.7 meV兲[15], and the four-spin system Cu2 PO4 共DE 苷 3 meV兲[16]. In SrCu2 共BO3 兲2 , the physical situation is completely
different because the dimers within the 共a, b兲 plane are
not isolated but strongly interacting, and moreover thesystem is located in the vicinity of the Néel ordered state.
As theoretically shown by Miyahara and Ueda [2], thekey to the dispersionless band in SrCu2 共BO3 兲2 lies not in
the spatial isolation of the spin clusters but in the orthogonality of the neighboring dimers: They proved that a hopping of the single triplet from one site to another withineach plane is possible only from the sixth order in the perturbation calculations, leading to an exceedingly weak dispersion of band I. As mentioned above, the appearance ofthe quantized plateaus in the magnetization curve [2,5] andthe multiple magnetic resonances in ESR [7] indicate the5878

19 JUNE 2000

localized character of multitriplets excitations. Our neutron scattering study, however, provides for the first timea direct proof of the significant constraint on the singletriplet excitations. Recent NMR measurement by Kodama
et al. [10] has also disclosed evidence to support the localization of the single triplet.
Let us take a closer look at the Q dependence of band I
in Fig. 3. The dispersion curve reaches a maximum(3.10 meV) at the so-called 共p, 0兲, a second maximum(3.00 meV) at 共p兾2, p兾2兲, and a minimum (2.90 meV)at (0,0) and the equivalent point 共p, p兲. Using the series
expansion method up to the fifteenth order, Weihong,Hamer, and Oitmaa obtained the single-triplet excitation
spectrum [4]. They argued that the bandwidth is quite
small as anticipated, but increases as J 0 兾J approaches
共J 0 兾J兲c . For a comparison, the calculated dispersion curveis shown by the broken line in Fig. 3, where the exchange
energies J 苷 100 K and J 0 苷 68 K are assumed. From
the qualitative point of view, the calculated Q dependence of the excitation energies nicely reproduces our
experimental data. However, the calculated bandwidth of
0.7 meV is (relatively narrow but) much wider than theobserved one. In addition to this discrepancy, their theory
yields a slightly smaller value of D 苷 2.0 meV at (0, 0)
and 共p, p兲. A possible explanation for the quantitative
discrepancies is that the perturbative approach becomesno longer appropriate when we discuss the phenomena of
the system near the critical boundary 共J 0 兾J兲c . Anotherpossible explanation could be the three-dimensional effect,i.e., the existence of the interlayer coupling, which also
could lead to a slightly different estimation of J and J 0 .
We now discuss the transition II which occurs at energy transfer of about 5 meV. This excitation has alreadybeen observed by ESR [7] and Raman scattering [8] (at4.7 meV), and the magnetic field dependence of the ESR
frequency has identified this mode as the second triplet
state [7]. Transition II is understood based on two triplets
coupled by Jt 共.0兲: If Jt 苷 0, the corresponding transition
occurs only at 2D 苷 6.0 meV. But when Jt is finite, one
expects a separation of the transition into three energy levels at E0 苷 2D 2 2Jt for the singlet state, E1 苷 2D 2 Jt
for the triplet state, and E2 苷 2D 1 Jt for the quintet
state. Using E1 苷 4.7 meV [7,8] and E0 苷 3.7 meV [8],Jt is determined to be 1.1–1.3 meV.
Figure 3 shows that the Q dependence of band II
is qualitatively identical to that of band I. The striking
difference between bands I and II is that the latter shows
more dispersive behavior with a bandwidth of 1.5 meV.
This observation may indicate that the propagation ofcorrelated two triplets is much easier than that of thesingle triplet. Very recent preliminary considerationof the two-triplet excitations by Miyahara and Ueda
indicates that two triplets sitting on the nearest neighborsites can propagate within the fourth-order perturbation
calculations, leading to stronger dispersions of band II[17]. Considering the instrumental energy resolution, thefull width at half maximum (FWHM) of band II for any

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Q points is obtained to be approximately 0.7 meV. The
finite width possibly indicates a spin continuum and/or
a short lifetime of the coupled two triplets. One of theremaining problems regarding this mode is to determine
the distance between a particular pair of triplets. Thismay be possible if one analyzes the Q dependent intensity
variation of this mode, which will be our future work.
The newly observed transition III which appears at
8–12 meV may be interpreted within the frameworkof correlated three triplets or more. Compared withtransition II, the observed profile of transition III is considerably broader [see Figs. 2(a) and 3], which may result
from a much wider spin continuum and/or a much shorter
lifetime of the bound state. Or one might think of the caseof several excited states lying at nearly degenerate discreteenergies. We suppose the excitations are also delocalized
owing to the correlation of the multitriplets as in the caseof transition II, though, at present, the very weak peakintensity and broad nature of transition III do not allow a
reliable discussion of the dispersion relation.
Finally, we discuss the Q dependence of the intensities of excitation I. The intensities at T 苷 1.7 K
and E 苷 3.0 meV were measured over various Q
points. The data taken along three directions in thereciprocal lattice are shown in Fig. 4. The observed
periodic intensity modulation was compared with the
dynamical structure factor which can be obtained by
calculating the transition probability from the singlet
ground state to the lowest triplet excited state. For
simplicity, let us consider a noninteracting pair of orthogonal dimers 共J 0 苷 0兲 with the intradimer distance
2.905 Å. Then, the intensity at Q 苷 共h, k兲 is given by
the incoherent superposition of the two single dimer

FIG. 4. Observed and calculated Q dependence of the scattering intensity for excitation I.

19 JUNE 2000

៬ ? d៬ 2 兲 苷៬ ? d៬ 1 兲 1 sin2 共Q
contributions as I共Q兲 ~ sin2 共Q22sin 关0.717共h 1 k兲兴 1 sin 关0.717共h 2 k兲兴, where 2d៬ 1
and 2d៬ 2 are the vectors connecting individual spins of
the two orthogonal dimers. The Q dependence of themagnetic form factor is not included, because it does
not affect the result considerably in the limited Q range
studied. Included in Fig. 4 are these theoretical curves,in qualitative agreement with the experimental findingsin spite of the unrealistic assumption about J 0 . This
fact reflects the frustrating arrangement of the J 0 bonds
between the dimers.
In summary, we performed inelastic neutron scatteringexperiments on SrCu2 共11 BO3 兲2 to find three magnetic excitations. The lowest excitation from the ground state wasobserved at 3.0 meV, in agreement with previous measurements. We observed for the first time the almost
dispersionless excitation in strongly correlated spin systems, originating not from the spatial isolation of the spinclusters but from the peculiar orthogonal dimer network.
The second triplet mode at around 5 meV can be interpreted as correlated two-triplet excitation, but it is moredispersive than that of the 3 meV excitation. Likewise, thehigh-energy excitation in the energy range of 8–12 meVwould arise from the correlated three-triplet or multitriplet
excitations.
The authors are grateful to H. Nojiri, P. Lemmens, M.
Takigawa, S. Miyahara and K. Ueda for stimulating discussion. This work was supported by a Grant-in-Aid for
Encouragement of Young Scientists from The Ministry of
Education, Science, Sports and Culture.

*Electronic address: kage@issp.u-tokyo.ac.jp[1] See, e.g., M. Hase et al., Phys. Rev. Lett. 70, 3651 (1993);
J. Darriet and J. P. Regnault, Solid State Commun. 86, 409(1993); M. Azuma et al., Phys. Rev. Lett. 73, 2626 (1994);
S. Taniguchi et al., J. Phys. Soc. Jpn. 64, 2758 (1995).
[2] S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 (1999).
[3] B. S. Shastry and B. Sutherland, Physica (Amsterdam)
108B, 1069 (1981).
[4] Z. Weihong, C. J. Hamer, and J. Oitmaa, Phys. Rev. B 60,
6608 (1999).
[5] H. Kageyama et al., Phys. Rev. Lett. 82, 3168 (1999).
[6] H. Kageyama et al., J. Phys. Soc. Jpn. 68, 1821 (1999).
[7] H. Nojiri et al., J. Phys. Soc. Jpn. 68, 2906 (1999).
[8] P. Lemmens et al., cond-mat/0003094.
[9] H. Kageyama et al., Physica (Amsterdam) 281 & 282B,
667(2000).
[10] K. Kodama et al. (unpublished).
[11] H. Kageyama et al., J. Cryst. Growth 206, 65 (1999).
[12] M. Nishi, O. Fujita, and J. Akimitsu, Phys. Rev. B 50, 6508
(1994).
[13] K. Kodama et al., J. Phys. Soc. Jpn. 66, 793 (1997).
[14] B. Leuenberger et al., Phys. Rev. B 30, 6300 (1984).
[15] Y. Sasago et al., Phys. Rev. B 55, 8357 (1997).
[16] M. Hase et al., Phys. Rev. B 56, 3231 (1997).
[17] S. Miyahara and K. Ueda (private communication).

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