Journal of the Physical Society of Japan
Vol. 68, No. 9, September, 1999, pp. 2906-2909

Letters

Direct Observation of the Multiple Spin Gap Excitations in
Two-Dimensional Dimer System SrCu2 (BO3 )2
Hiroyuki Nojiri, Hiroshi Kageyama1 , Kenzo Onizuka1 ,Yutaka Ueda1 and Mitsuhiro Motokawa
Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577
1 Institute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Tokyo 106-8666(Received June 1, 1999)

Various spin-gap excitations have been observed in the two-dimensional dimer system
SrCu2 (BO3 )2 by ESR. The zero-field energy gap of the lowest spin-gap excitation shows a splitting into two triplet modes and the energy splitting clearly depends on the magnetic field orientation. A zero-field splitting is also found between the Sz = +1 and Sz = −1 branches of each
triplet. These behaviors are qualitatively explained by considering the anisotropic inter-dimer
and intra-dimer exchange couplings. The averaged value of the lowest spin-gap energy is determined to be 722±2 GHz (34.7 K). We have also found a second spin-gap excitation at 1140 GHz(54.7 K), which indicates that the inter-dimer coupling is strong. Besides these modes, a numberof gapped ESR absorptions are found and we propose that these excitations are caused by the
localized nature of the excited state in the present system.
KEYWORDS: SrCu2 (BO3 )2 , spin gap, ESR, magnetic excitation, high magnetic field

Low-dimensional quantum spin systems with a finite
spin gap have been the subject of intense and advanced
research in this decade. One of the factors lies in theprominent progress of solid state chemistry, enabling a
tailoring of various low-dimensional spin systems for theexperimental verification of a concept brought about by
theoretical study. Recently, the compound SrCu2 (BO3 )2
has been added to the list of two-dimensional (2D) systems with a spin-gapped ground state.1) One prominent
characteristic is that this material can be regarded asa model system with an exactly known ground state.
SrCu2 (BO3 )2 has a tetragonal unit cell and all Cu2+ions are located at equivalent sites.2) In a CuBO3 layer
sandwiched by layers of Sr ions, a dimer unit is composed of a neighboring pair of planar rectangular CuO4
and these dimers connect orthogonally by way of a triangular planar BO3 , providing a unique 2D network of
S = 1/2, as shown in the inset of Fig. 1. This 2D lattice
is topologically equivalent to a 2D square lattice with additional alternating diagonal interactions, for which thedirect product of the singlet pairs is the exact eigenstate,
as proven by Shastry and Sutherland.3) This model for
SrCu2 (BO3 )2 can thus be considered a 2D analogue ofthe Majumdar-Ghosh model with some stringent condition.4) The role of the next-nearest-neighbor interaction in the ac-plane J 2 , which brings a frustration into
the system, has also been studied theoretically by Miyahara and Ueda.5, 6) They estimated the intra-dimer coupling of J = 100 K and the next-nearest-neighbor one
of J 2 = 68 K using the experimentally obtained Weisstemperature of −92.5 K and the energy gap of 30 K. It
should be noted that the ratio J 2 /J = 0.68 is just below
the critical value of J 2 /J = 0.70 between the spin-gap
state and the Néel-ordered state. Thus, this material
resides near the border between these two states.6, 7)

Fig. 1. Examples of ESR spectra at 1.6 K for H k a. Arrows
indicate the two lowest triplets T1 and T2. Suffixes u and dindicate Sz = +1 and Sz = −1 branches, respectively. Closed
triangles and open rectangles indicate the second lowest triplet
and other multiple weak signals, respectively. The inset shows
the schematic crystal structure in the c-plane. The dotted line
shows the unit cell.

Another interesting feature appears in the magnetization curve with quantized plateaux at one-quarter,
one-eighth,1) and possibly one-tenth8) of the saturated
moment. These plateaux have been realized due tothe extremely localized nature of the low-lying excitedtriplets.6) This means that excited triplets cannot propagate freely in the 2D lattice. As a result, it is consideredthat these triplets organize a regular lattice composed of
triplets.

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Hence, it will be very interesting to study the fielddependence of the magnetic excitation of this system.
However, since the critical field is above 20 T, only a
few methods can be applied to investigate the magnetic
excitation in such high fields. Among the many different methods, ESR has established its unique status asa probe of magnetic excitation particularly to study its
field variation.9-11) Thus, we have performed a systematic study of the magnetic excitations of SrCu2 (BO3 )2
by using a submillimeter-wave ESR.
Submillimeter-wave ESR measurements have been
performed in pulsed magnetic fields up to 30 T. Highpurity bulk single crystals of SrCu2 (BO3 )2 were grownby the travelling solvent floating zone (TSFZ) method.12)
The grown materials were checked and the crystal axes
were determined by the Laue X-ray photograph.
ESR spectra at 1.6 K are depicted in Fig. 1 for H k a.
Among the many different kinds of ESR signals, a setof two resonance peaks marked by arrows shows characteristic frequency dependence. As the frequency is
increased, the resonance field first decreases and thenincreases again at higher frequencies. The resonancefields are summarized as the frequency field diagram in
Figs. 2(a) and 2(b) for H k c and H k a, respectively.
As shown in these figures, the above-mentioned ESR signals, marked by closed circles (T1u and T1d ) and closed
rectangles (T2u and T2d ), are associated with a zerofield energy gap of about 720 GHz. This value is close
to that estimated by different methods.1, 8, 13-15) Thesefindings indicate clearly that the two modes correspondto the transitions between the ground singlet state andthe first excited triplet state (we call these modes singlettriplet transition hereafter). As shown in Figs. 2(a) and2(b), we have observed two branches of Sz = +1 and
Sz = −1 for each triplet mode T1 or T2 (we use notations such as T1 when we treat T1u and T1d as a set of
triplet). The Sz = 0 branch cannot be detected by ourfield-scanning ESR.
The lowest spin-gap excitations T1 and T2 exhibit acomplicated structure. For H k a, a zero-field splitting
exists between two branches of Sz = +1 and Sz = −1.
The modes T1u and T2u or T1d and T2d are degenerated
at zero field and the centers of Sz = +1 and Sz = −1
branches are identical between T1 and T2 modes. A
small splitting observed at high fields between T1u andT2u or between T1d and T2d cannot be explained by theanisotropy of the g-value in the c-plane because all Cu
sites are equivalent with respect to magnetic fields for
H k a.2) Thus, we infer that a small lattice distortion
occurs at low temperatures, though detailed structural
analysis at low temperatures is required to clarify thispoint. On the other hand, for H k c, no zero-field splitting exists between the two branches of Sz = +1 and
Sz = −1, while two triplets T1 and T2 exhibit a splitting at zero field. These characteristic behaviors will
be explained below by considering the anisotropic interdimer and intra-dimer exchange couplings. The slope ofthese modes for each field direction is nearly identicalto that of the paramagnetic resonance signals indicatedby dashed lines in Figs. 2(a) and 2(b). The g-values are
determined as g a = 2.05 and g c = 2.28 by the paramag-

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Fig. 2. The frequency-field diagrams at 1.6 K for H k c (a) andfor H k a (b). Dashed lines indicate the position of paramagnetic
resonance. Closed circles and closed rectangles indicate the two
triplet modes T1 and T2. Closed triangles indicate the second
lowest triplet. Closed diamonds and open rectangles are otherstrong and weak signals, respectively, as mentioned in the text.
The magnetization curve of the single crystal measured at 0.5 K
taken from ref. 8 is also plotted for H k a.

netic resonance peak positions at 25 K. From Figs. 2(a)and 2(b), we determine the average value of the lowest
spin gap as 722±2 GHz (34.7 K).
This assignment of T1 and T2 to the singlet-triplet
transition is further confirmed by the temperature dependence of ESR spectra, as shown in Fig. 3. The intensity of the two peaks marked by arrows decreases as thetemperature is increased. This behavior indicates thatthese signals are the transitions from the ground stateof the system. As is well known, the ESR transition
between the ground singlet state and the excited triplet
state due to the mechanism of magnetic dipole transition
is usually forbidden. However, the presence of a nonsecular term such as the Dzyaloshinsky-Moriya (DM)interaction, non-equivalent g-tensors or the anisotropic
exchange interaction (AE) makes it possible to observe
this transition by means of ESR. We speculate that the
AE is the origin of the break down of the selection rulein SrCu2 (BO3 )2 and the reason is discussed below.
Since each Cu is bridged by a planar CuO4 unit whichlies parallel in the c-plane, we can expect that the crystal
field on each Cu2+ ion exhibits a tetragonal symmetry
and the principal axis is along the c-axis. One of theevidences of the tetragonal crystal field is the anisotropyof the g-value. The anisotropy of the g-value in the cplane is less than 5% of that in the ac-plane. This angular dependence is consistent with the tetragonal crystal
field expected from the symmetry of the CuO4 unit. Ina tetragonal crystal field, we can also expect that J cc

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Fig. 3. Temperature dependence of ESR spectra at 716.7 GHz.
Arrows indicate T1 and T2 modes.

deviates from J aa , where J cc and J aa are the components parallel and perpendicular to the c-axis of theintra-dimer exchange coupling J, respectively. As is wellknown, the order of AE can be estimated approximatelyby the relation AE ∼ (∆g/g)2 J i , where ∆g/g is theanisotropy of the g-value normalized by the average ofg, and J i is the isotropic part of exchange constant. By
using g a = 2.05, g c = 2.28 and J i ≈ J = 100 K, weevaluate AE∼1 K. This value is comparable to the zerofield splitting of about 30 GHz between Sz = +1 and
Sz = −1 branches in Fig. 2(b). To understand the difference in the zero-field splitting between H k c andH k a configurations, let us consider an isolated dimerof S = 1/2 coupled by an antiferromagnetic exchange interaction with exchange anisotropy. By diagonalizing the
Hamiltonian of a dimer with such anisotropic exchange
coupling, we can demonstrate that the splitting at zero
field exists between Sz = +1 and Sz = −1 branches forH ⊥ c and that no splitting exists for H k c between
these two branches. This field-orientation dependence
is consistent with the experimental results of the splitting between Sz = 1 and Sz = −1 branches shown in
Figs. 2(a) and 2(b). Thus, we conclude that the AE isthe origin of the above-mentioned splitting and this term
makes the singlet-triplet transition allowable for ESR.
Next, let us discuss the parallel splitting of the lowest energy spin gap excitation for H k c into the T1
and T2 modes. This splitting cannot be attributed tothe difference of the g-value because the splitting exists
even at zero fields and both modes are parallel to eachother, as shown in Fig. 2(a). Such a splitting can be expected when an inter-dimer interaction J 0 exists. Since
the splitting depends on the field orientation, we speculate that the splitting is caused by the anisotropy of J 0
and is not related to the absolute total value of J 0 . To
confirm this point, we measured the angular dependenceof two peaks T1d and T2d , as shown in Fig. 4. By using
the angular dependence of the g-value, the zero-field energy of the two triplets is evaluated as a function of fieldorientation, as shown in the inset of Fig. 4. It should
be noted that the splitting exhibits a sinusoidal angulardependence. Thus, we speculated that this parallel split-

Fig. 4. A angular dependence of T1 and T2 modes at 1.6 K andat 428.9 GHz in the ac-plane. The number for each spectrum
is the angle θ between magnetic field and the a-axis. The inset
shows the angular dependence of the zero-field energy gaps ofT1 (closed circles) mode and T2 (closed rectangles) mode.

ting is caused by the anisotropic inter-dimer exchangeinteraction.
When a finite J 0 exists, we can expect that the second
excited state splits from the lowest triplet state due to J 0 .
In practice, the second lowest singlet-triplet transition isfound and the energy gap is about 1140 GHz (see closed
triangles in Figs. 2(a) and 2(b)). Since J 2 = 68 K is comparable to the intra-dimer coupling J in the present system, a theoretical calculation of the energy levels of theexcited states is very difficult and a perturbative treatment may not be successful. Although it cannot be applied directly to the present case, it is useful to consider
the energy level of the following simple S = 1 dimer
model to understand why the second excited state is thetriplet. When J 0 < J , we can treat two dimers coupled
by J 0 as a dimer of two S = 1 spins. In this case, theground state (E = 0), the first excited state (E = J 0 )and the second excited state (E = 3J 0 ) are S = 0 singlet, S = 1 triplet and S = 2 quintet, respectively, where
E is the energy of each level. More correctly, we haveto include all four nearest-neighbored dimers. In bothcases, however, the first excited state is the triplet as
long as the condition J 0 < J holds. Thus, we speculatethat the second triplet branch is the transition between
the ground state and the excited state in which the two
triplets are coupled by J 0 . The fact that the intensity ofthis transition increases monotonically with decreasing
temperature indicates that this mode is the transition
from the ground state. The weak intensity of the second
triplet branch compared to that of the lowest triplet at722±2 GHz can be understood because the former transition is caused by the simultaneous excitation of two
dimers. It should be noted that the S = 1 dimer model
is not realistic and that J 0 cannot be related to J 2 . Itis essential to include all of the many-body interactions
to evaluate the correct energy levels of the excited states
theoretically. Experimentally, however, the second energy gap was also observed by neutron scattering experiments,16) and the value of the energy gap is consistent2909

Letters

with the present result. Thus, we conclude that the existence of the second spin gap for both H k c and H k a,
is attributable to the large J 0 in the present system.
When we examine the frequency-field diagram shownin Figs. 2(a) and 2(b), we note that a number of ESR
signals are observed besides the lowest and the second
lowest energy modes of the singlet-triplet transition. We
categorize these signals into the following two types: (1)strong ESR absorption appearing just below the criticalfield H c marked by closed diamond, and (2) other weak
ESR signals marked by open rectangles. To investigate
the origin of the strong ESR mode (1), we measured thetemperature dependence of the ESR spectra, as shownin Fig. 3. The signal located at 22 T is the transition in
the excited states. The intensity of this peak is almost
constant between 10–25 K and it decreases rapidly below
10 K. This behavior is completely different from that expected for a simple isolated dimer model and the details
will be given in a separate paper. Since the intensityof all ESR peaks appearing around 15–17 T decreases as
the temperature is increased, these signals are considered
to be the excitations from the ground state. It should
also be noted that the intensity of these peaks is muchstronger than that of the singlet-triplet transition and iscomparable to the signal observed at 22 T and at 7 K.
Moreover, these signals are not related to the magnetic
excitations at a plateau because the fields are well below
H c , where the lowest triplet state intersects the singlet
ground state. These findings indicate that the strong
signals are related to the ground state but are not a
forbidden transition, such as a singlet-triplet transition.
Hence, we infer that the ground state of the system near
the H c is not a perfect nonmagnetic singlet state; i.e.,part of the singlet is broken and, as a consequence, amagnetic ground state is recovered locally for the finite
mixing between the ground singlet state and the excitedstates. In practice, the bending of modes T1d and T2d
is observed above 15 T, as shown in Fig. 2(a) and themagnetization is non-zero even at 0.5 K in the field well
below H c .8)
Finally, we discuss the origin of the weak signals
marked by open rectangles. It appears that many ofthem lie almost parallel to the transition T1 or T2. This
indicates that these ESR signals have a finite zero-fieldenergy gap and thus, they are the transitions between
the ground state and the excited state. We speculatethat these weak absorptions are the localized multiple
spin-gap excitations caused by the extremely localized
nature of the excited state proposed theoretically.6) Usually, an excited triplet in the dimerized spin-gap system
can propagate for the finite inter-dimer coupling andthus, a dispersion is observed. In the present system,a triplet excitation is extremely localized and its propagation is considerably suppressed. In this case, we expectthat a dispersive mode may be replaced with a numberof excitations which consist of nearly degenerated discrete energy levels. Namely, we can expect a state where
triplets are coupled as dimer, trimer, quadramer, etc. In

this case, the excitation energy of each level may be givenby the sum of the formation energy of the isolated triplet,the interactions with other triplets located nearby and
the Zeeman energy. This idea can explain why most of
the weak ESR peaks appear above the Sz = −1 branchof the second lowest singlet-triplet transition marked byclosed triangles. It also explains why the number ofpeaks increases when a field approaches to H c . It isbecause the number of surrounding triplets is increasedaround H c . As is well known, for the Ising system, such
localized excitations are realized for their discrete energy
levels. In practice, localized modes have been observed
as a spin-cluster excitations.17, 18) However, such a localized excitation has not been observed for the Heisenberg
system. We propose that the present observation of themultiple spin-gap excitations is the manifestation of theextremely localized nature of the excited state in the
Shastry and Sutherland model.
To summarize, we have directly observed singlettriplet transition of SrCu2 (BO3 )2 . Two sets of welldefined triplet modes are found and the energy gaps
are evaluated to be 722±2 GHz (34.7 K) and 1140 GHz(54.7 K). In addition to these transitions, we have also
observed multiple spin-gap excitations, which may berealized by the extremely localized nature of the excitedstate.
Acknowledgements
This work was partly supported by a Grant-in-Aid
from the Ministry of Education, Science, Sports and Culture, Japan.
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