Journal of Physics and Chemistry of Solids 62 (2001) 141±147

www.elsevier.nl/locate/jpcs

Neutron scattering experiments on low-dimensional quantum spin
systemsÐSpin excitations in KCuCl3 and SrCu2(BO3)2
K. Kakurai a,e,*, M. Nishi a, K. Nakajima a, K. Nukui a, N. Aso a, T. Kato b, H. Tanaka b,
H. Kageyama c, Y. Ueda c, L.P. Regnault d, B. Grenier d, B. Fùk da

Neutron Scattering Laboratory, Institute for Solid State Physics (ISSP), University of Tokyo, 106-1 Shirakata, Tokai, Ibaraki 319-1106, Japanb
Department of Physics, Faculty of Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japanc
Material Design and Characterization Laboratory, ISSP, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japand
DRFMC/SPSMS/MDN, CENG, 17, rue des Martyrs, 38054 Grenoble CeÂdex 9, Francee
CREST, Japan Science and Technology Corporation, Tsukuba, Japan

Abstract
We present the results of inelastic neutron scattering experiments on quantum spin systems KCuCl3 and SrCu2(BO3)2. Bothcompounds show non-magnetic singlet ground state behaviour at low temperature. KCuCl3 turns out to be a weakly coupleddimer system without any strong frustrating interdimer interactions. SrCu2(BO3)2, on the other hand, should be regarded as astrongly coupled dimer system with a strong frustration originating from the orthogonal arrangement of the neighbouring
dimers. The characteristics of the spin excitations in these two dimer systems are compared and discussed. q 2000 Elsevier
Science Ltd. All rights reserved.
Keywords: A. Inorganic compounds; C. Neutron scattering; D. Magnetic properties

1. Introduction
The quantum ¯uctuations leading to a ground state, whichcannot be expected in a classical spin system, continues toattract the interest of both theoretical and experimental
physicists. The interest has been revived by the discovery
of 2D S  1/2 spin correlation in high Tc superconducting
cuprates. In the course of these studies, many new lowdimensional magnetic systems with singlet ground state
have been discovered, e.g. Haldane, spin Peierls, ladder,plaquette systems [1±4]. The spin dynamics of thesesystems show a spin excitation gap from the non-magnetic
singlet ground state to the ®rst excited triplet state, which
may be regarded as a direct consequence of the quantum
spin ¯uctuations. However, little is known about the frustration effect in these extreme quantum spin systems. For
example, the magnitude of the role played by the ®nite
next-nearest neighbour interaction in the spin Peierls
compound CuGeO3 in establishing the observed singlet
ground state, remains controversial [5,6]. It is therefore,* Corresponding author. Fax: 181-29-282-8709.
E-mail address: kakurai@issp.u-tokyo.ac.jp (K. Kakurai).

interesting to study the frustration effect on spin dynamics
in similar quantum systems with and without strong frustrations.
Inelastic neutron scattering experiment on KCuCl3
revealed that this compound can be regarded as a weakly
coupled dimer system [7]. SrCu2(BO3)2, on the other hand,should be regarded as a strongly coupled dimer system witha strong frustration originating from the orthogonal arrangement of the neighbouring dimers [8,9]. A comparative
inelastic neutron scattering investigation of these two
dimer systems may give some insight into the frustration
effect on spin dynamics, of quantum magnets.
In this paper, we thus compare the characteristics of spin
excitations in two dimer systems, namely KCuCl3 as aprototype of a weakly coupled system and SrCu2(BO3)2 as
a prototype of a strongly coupled, frustrated system.
2. KCuCl3
KCuCl3 crystallises in a monoclinic structure belonging
to the space group P21/c. The lattice parameters areÊ and b  97820 0 . Thea  4.029, b  13.785, c  8.736 A
main feature of the crystal structure is the double chain of0022-3697/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0022-369 7(00)00116-5

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K. Kakurai et al. / Journal of Physics and Chemistry of Solids 62 (2001) 141±147

Fig. 2. Typical constant-Q inelastic spectra from KCuCl3 at T  7
and 57 K. (From Ref. [7].)Fig. 1. Double chain structure in KCuCl3 and the con®guration of
the hole orbital dx 22y 2) of the Cu 21 ion.

edge-sharing octahedra of CuCl6 along the a-axis as shownin Fig. 1. All the octahedra are distorted due to the Jahn±
Teller effect, and their elongated axes are aligned in thesame direction. Hence, all the Cu 21 ions in a double chain
are magnetically equivalent. The susceptibility result by
Tanaka et al.[10] shows no anomaly pointing to a magnetic
phase transition, rather a broad maximum around 30 K. The
susceptibility then vanishes exponentially towards zerotemperature.
To clarify the nature of this singlet ground state behaviour
we performed the inelastic neutron scattering experiment ona large single crystal of KCuCl3, grown from a melt by the
Bridgeman technique. The experiment was performed using
the thermal neutron triple axis spectrometer ISSP-PONTA
installed at the research reactor JRR-3M, JAERI, Tokai. The
scattering plane was selected to be the a p ±c p plane of thecrystal. Fig. 2 shows a typical energy spectrum taken at Q 0; 0; 21 reciprocal point. A resolution limited single excitation peak is observed at around 3 meV. The magnetic

origin of this excitation is veri®ed by looking at the temperature dependence. As can be seen from Fig. 2, at T  57 K
there is no appreciable peak above the background level.
The energies of the magnetic excitations at different Q are
deduced by ®tting a Gaussian peak, as indicated by the solidline in Fig. 2. Fig. 3 shows the summary of the excitationenergies at different wave vectors. The lowest excitationenergy for the reciprocal points in the a p ±c p plane is evaluated to be 2.7 meV by averaging the values for Q 0; 0; 21 and the equivalent points at Q  1; 0; ^1: Thisvalue is very close to the gap energy of 2.74 meV deduced
from the previous magnetisation measurements [11].
Furthermore, we see that the dispersion, parallel and perpendicular to the chemical double chains, i.e. in a p- and c pdirection, show band widths of the same order of magnitude,
and that the excitation gap is almost as large as the dispersion band width. Thus, KCuCl3 should be described as aweakly coupled antiferromagnetic dimer system rather
than a quasi one-dimensional double spin chain, as initially
proposed based on chemical intuition [10].
The dispersion relation of such a weakly coupled dimer
system can be modelled within the random phase approximation [12].
Assuming the exchange constant J2 as the dominant dimer
forming antiferromagnetic interaction Jd, the exchange
constants as shown in Table 1 can be introduced to describe
the observed excitation energies [13]. The solid lines in Fig.

Table 1
Values of the intradimer exchange constant Jd and the effective interdimer exchange constants Ji with the path direction vector ri
Exchange constants

Path direction vector ri (latticeunit vector)

Absolute valuesof exchange
constants (meV)

Jd
Ja2c
Jabc
Ja
Jbc

(0 0 0)^ (2 0 1)^ (1 1/2 1/2), ^(1 21/2 1/2)
^ (1 0 0)^ (0 1/2 1/2), ^(0 21/2 1/2)

4:11 ^ 0:02
0:37 ^ 0:02
0:32 ^ 0:02
0:26 ^ 0:01
0:02 ^ 0:01

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143

Fig. 3. Summary of the observed excitation peak positions for different Q  (h,0,l). The solid lines indicate calculated dispersion relations, as
explained in the text.

3 indicate the calculated dispersion relations with theseparameters, in good agreement with the experimental ®ndings.
One should note that the measured susceptibility of
KCuCl3 is well reproduced by the isolated dimer susceptibility with the above mentioned intradimer exchange
energy, consistent with the conclusions derived from theinelastic neutron scattering results.
These results clearly show that KCuCl3 is a weakly
coupled dimer system with the dominant antiferromagnetic
intradimer exchange of Jd  24:11 meV and weak interdimer exchanges, which when compared to the intradimer
exchange are smaller roughly by an order of magnitude. We
note that, an independent inelastic neutron scattering investigation on the same compound has been performed by
Cavadini et al. [14] and their results are consistent withthose presented here. Fig. 4 shows the temperature dependence of the excitation peak intensity at Q  0; 0; 21: It
can be seen that the peak structure disappears at around T 

30 K; corresponding roughly to the observed energy gap of
2.7 meV.

3. SrCu2(BO3)2
SrCu2(BO3)2 has a tetragonal unit cell (D2d) with lattice at roomparameters a  b  8:995 and c  6:649 A
temperature. It consists of alternately stacked Sr- andCuBO3-planes. A neighbouring pair of planar rectangular
CuO4 forms a spin dimer and the dimers are connected
orthogonally by a triangular BO3 (Fig. 5). The solid andbroken lines connecting the Cu-ions in Fig. 5 denote theintradimer Jd and interdimer J 0 interactions, respectively.
Miyahara and Ueda [9] showed that this 2D lattice in thea±b plane of the compound is topologically equivalent to
the Shastry±Sutherland lattice studied earlier [15], andpointed out that this system has an exact dimer groundstate even when J 0 ± 0:

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K. Kakurai et al. / Journal of Physics and Chemistry of Solids 62 (2001) 141±147

Fig. 4. Temperature dependence of the peak intensity of the excitation at Q  0; 0; 21:

As can be seen from Fig. 5, the orthogonal arrangement of
the dimers with J 0 -exchanges introduce spin frustration. It
turned out that, the experimental data of the magnetic
susceptibility [8] cannot be reproduced by an isolated
dimer model in which J 0 is neglected. Fig. 6 shows thetemperature dependence of the magnetic susceptibility in
SrCu2(BO3)2. The solid line indicates the isolated dimer
susceptibility, in which the intradimer exchange is chosensuch that the maximum is located at the same temperature as
the experimental result. The failure of the model is evident.
Miyahara and Ueda showed, by the numerical diagonalization of a ®nite size Shastry±Sutherland model, that theobserved susceptibility can be quite well reproduced by

Fig. 5. Schematic drawing of the CuBO3 plane in SrCu2(BO3)2.

choosing Jd  100 and J 0  68 K [9]. The ratio of theexchange constants, J 0 =Jd  0:68; is just below the criticalboundary of J 0 =Jd c  0:70 [9] (or 0.69 [16]), beyond
which the system is expected to exhibit a NeÂel ordered state.
The ®rst inelastic neutron scattering experiments were
performed using the thermal neutron triple-axis spectrometer ISSP-PONTA [17]. Vertically curved pyrolytic
graphite (PG) monochromator and analyser were used andthe ®nal neutron energy was ®xed at Ef  14:7 meV kf  21 : A PG ®lter was placed after the sample to2:67 A

Fig. 6. Magnetic susceptibility of SrCu2(BO3)2 depicted by open
points. The solid line indicates the calculated susceptibility for an
isolated dimer model displaying the maximum at the same temperature as the experimental result.

K. Kakurai et al. / Journal of Physics and Chemistry of Solids 62 (2001) 141±1470

145

Fig. 7. Typical energy scans at Q  (2,0,0) for T  1.7 (open points)
and T  24 K (triangles). (From Ref. [13]).

suppress the higher-order contamination. The horizontal
collimations were chosen as open-40 0 -sample-80 0 -80 0 . In
order to minimise the neutron absorption by 10B, weprepared a 11B-enriched sample SrCu2( 11BO3)2. Bulk single
crystals were prepared from the travelling solvent ¯oatingzone method using the ¯ux Li 11BO2 [18]. Two pieces of thesingle crystals with a total volume of ,1.5 cm 3 were aligned

Fig. 8. Q-dependence of the excitation energies of mode I, II and III
obtained at T  1.7 K. The upper and lower Q-values indicate thescattering vectors, at which the lowest mode I and the other modes II
and III are observed. The arrows represent the energy resolutions of
the instrument (FWHM), at these energy transfers. The solid lines
are only guides to the eyes. The vertical bars represent the intrinsic
line widths (FWHM) of mode II and III.

Fig. 9. High resolution spectra at Q  (1,0,0), (1.5,0,0) and(1.5,0.5,0). Inset: the peak positions of excitation mode I as determined by Gaussian ®ts. The vertical bars represent the line widths(FWHM) as obtained by Gaussian ®ts.

within 20 min. The sample was mounted with its a- and baxis in the scattering plane.
Fig. 7 shows a typical pro®le of the energy scan for the
scattering vector of Q  (2,0,0) measured at 1.7 K (openpoints). The 1.7 K spectrum is characterised by threemodes of excitations; namely a sharp peak at 3 meV, asomewhat broader peak at around 5.0 meV and a verybroad signal in the energy range of 6±12 meV, which weshall designate as excitations I, II and III, respectively. The
spectrum at 24 K, in contrast, has no appreciable peak structures, indicating that excitations I, II and III are of magnetic
origin.
As already con®rmed, for example, by ESR [19], theexcitation mode I is the lowest triplet excitation from thesinglet ground state, and the gap energy D  3:0 meV isconsistent with the values estimated from different measurements [19±22].
Fig. 8 depicts the Q-dependence of the energy centers ofthese excitation modes. The vertical bars do not represent
error bars, but indicate the observed intrinsic line widths ofthe excitations. Besides the small gap energy compared withthe estimated intradimer exchange energy of 100 K, it isremarkable that the ®rst excited states show an almost ¯atdispersion despite the large interdimer exchange energy
value.
To measure the band width of the lowest excitation
mode more reliably, we performed a cold neutron triple
axis experiment on IN12 at Institut Laue Langevin. The
chosen spectrometer condition was Ef ®xed at 5 meV withhorizontal collimations after the vertically focused PG

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K. Kakurai et al. / Journal of Physics and Chemistry of Solids 62 (2001) 141±147

Fig. 10. Temperature dependence of the peak intensities at
DE  3.15, 5.15 and 10.65 meV, at Q  (1.5,0.5,0) for the ®rst
and (2,0,0)for the next two. The intensities are normalised at low
temperature. The broken line is a guide to the eyes only. (From Ref.
[13]).

monochromator 40-sample-open-open, to allow the use of
the horizontally curved PG analyser. A cooled Be-®lter wasinserted in the scattered beam, to eliminate higher ordercontamination. Fig. 9 shows three typical spectra at different
Q vectors, which include the extrema suggested by the thermal neutron experiment. The inset of Fig. 9 shows the peak
positions at these wave vectors, and it can be clearly seenfrom this high resolution measurement that the 3 meV mode
has indeed a very small band width of 0.2 meV. It should be
noted that the 3 meV excitation peak at Q  1; 0; 0 has asmall intrinsic line width of the order of 0.2 meV, while thepeak at Q  1:5; 0:5; 0 is resolution limited even with thishigh energy resolution. (Note, that the vertical bars in theinset represent the line widths (FWHM) of Gaussian ®ts.)
Furthermore, it is clear from the spectrum at Q  1; 0; 0;that there is a clear energy gap between the 3 and 5 meV
excitations.
Fig. 10 shows the temperature dependence of thepeak intensities at 3.15, 5.15 and 10.65 meV energy
transfer, as obtained by the thermal neutron experiment.
All peak structures disappear at about 13 K, corresponding to an energy much smaller than even the lowest gap
energy of 34 K.

4. Discussion
The experimental ®ndings presented above indicate thatthe spin dynamics in both systems can be interpreted as thatof exchange connected dimers. In the case of KCuCl3, thedimers are weakly coupled and the interdimer exchanges are
not strongly frustrating. Hence, the excited triplet state on a
dimer can propagate due to the interdimer exchange andexhibits a dispersive behaviour with a band width of thep
order of 2 Jd ´J 0 ; where Jd and J 0 are characteristic intra
and interdimer exchange energies, respectively. The sizeof Jd is re¯ected in the averaged gap energy of the ®rst triplet
excitation.
In the case of SrCu2(BO3)2, however, the dimers arestrongly coupled but the two interdimer exchange paths
form a triangular lattice together with the intradimer
exchange due to the orthogonal arrangement of the dimers,
thus causing a strong frustration. This frustration then gives
the ®rst triplet excitation a very strong localised nature,
which is re¯ected in the dispersion with a narrow band
width of 0.2 meV at most. This fact is consistent with therecent theoretical consideration by Miyahara and Ueda,
where they prove that a hopping of the single-triplet excitation from one site to the neighbouring site within the plane,
is possible only from the sixth order in the perturbation
calculation. The gap energy is much smaller than the intradimer exchange energy, because the system is located close
to the critical ratio of interdimer and intradimer exchange ofJ 0 =Jd  0:7; where, according to Miyahara and Ueda [9], aphase transition from the singlet ground state to a NeÂel stateis expected. One should mention here that Weihong et al.
[17] calculated the single-triplet excitation using the series
expansion method and they predicted a prominent J 0 /Jddependence of the bandwidth near the critical ratio, which
is not in accord with our experimentally observed small
band width.
The next striking difference between the spectrum of
KCuCl3 and SrCu2(BO3)2 is the visibility of many higher
energy excitation branches or continuum in the latter,
while in the former the spectrum is dominated by the single
excitation branch associated with the ®rst excited triplet
state. The excitation mode II in SrCu2(BO3)2 at around
5 meV can be interpreted as the second triplet state associated with the coupled two-triplets excitation. The continuum like excitations at even higher energies up to 12 meVthen can be understood as multi-triplets excitations. One
may therefore conclude, that the spectral weight of these
multi-triplets excitations is considerably enhanced in afrustrated system, rather than in a weakly coupled, hence
unfrustrated, system. In the light of this interpretation, the`second triplet state' at 5 meV in SrCu2(BO3)2 may berelated to the observation of the `second gap' in CuGeO3,
where the frustration due to the next-nearest neighbour
exchange coupling is thought to play an important role toform the singlet ground state.
The temperature dependence of the excitation spectrum in
SrCu2(BO3)2 is also different from that of KCuCl3. In
KCuCl3, the excitation peak structure disappears roughly
at a temperature corresponding to the smallest gap energy,
as one would commonly expect; while in SrCu2(BO3)2, theexcitation peak structure including the multi-triplet excitations disappears at a much lower temperature than the smallest gap energy given by the single-triplet excitation.
A quantitative calculation of the dynamical structure
factor and its temperature dependence for SrCu2(BO3)2 is

K. Kakurai et al. / Journal of Physics and Chemistry of Solids 62 (2001) 141±1470

highly desirable, to verify whether the large spectral weightof the multi-triplets excitations and the peculiar temperaturedependence are indeed consequences of the strong frustration.
5. Summary
To summarise, we reported inelastic neutron scattering
investigations on KCuCl3 and SrCu2(BO3)2. The twosystems can be regarded as examples of quantum systems
with singlet ground state. Both systems can be described as
coming from interconnected dimers, but while in KCuCl3
the dimers are only very weakly coupled and unfrustrated,in SrCu2(BO3)2 they are strongly coupled and frustrated. The
characteristics in the excitation spectrum and its temperaturedependence of the latter may be attributed to the frustration
effect and the near criticality, which are absent in the former
system.
Acknowledgements
One of the authors (K.K.) would like to thank Prof J.
Flouquet and the Neutron Scattering Group of CENG for
their hospitality during his stay in Grenoble.
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