Journal of the Physical Society of Japan
Vol. 68, No. 6, June, 1999, pp. 1821-1823

Letters

Anomalous Magnetizations in Single Crystalline SrCu2 (BO3 )2
Hiroshi Kageyama∗ , Kenzo Onizuka, Touru Yamauchi, Yutaka Ueda,
Shingo Hane1 , Hiroyuki Mitamura1 , Tsuneaki Goto1 ,Kazuyoshi Yoshimura2 and Koji Kosuge2
Material Design and Characterization Laboratory, The Institute for Solid State Physics, The University of Tokyo,
Roppongi, Minato-ku, Tokyo 106-8666
1 Mega Gauss Laboratory, The Institute for Solid State Physics, The University of Tokyo,
Roppongi, Minato-ku, Tokyo 106-8666
2 Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502(Received February 15, 1999)

The magnetic properties of a two-dimensional spin-gap compound SrCu2 (BO3 )2 , realizing theShastry-Sutherland model, were investigated using high-quality single crystals. The analysisof the temperature-dependent magnetic susceptibility revealed a spin gap of 34 K. The magnetization curve at 0.4 K possibly has, in addition to the already established one-quarter andone-eighth plateaux, another quantized plateau at one-tenth of the full Cu moment, which is oneof the plateaux predicted by Miyahara and Ueda.
KEYWORDS: SrCu2 (BO3 )2 , Shastry-Sutherland model, quantized magnetization plateaux, exact dimer groundstateLow-dimensional materials have been the focus in thearea of condensed matter physics because the dimensionality of the structures leads to unusual and exotic magnetic/electronic phenomena such as high-Tc superconductivity.1) SrCu2 (BO3 )2 has a two-dimensional structure, which consists of alternately stacked CuBO3 - andSr-layers.2) The peculiar magnetic network of the magnetic Cu2+ ions (d9 , thus S = 1/2), in which neighboring Cu2+ –Cu2+ dimers are orthogonally connectedin the layer, has given rise to amazing aspects with respect to the ground state:3-5) SrCu2 (BO3 )2 is a highly
frustrated two-dimensional spin-gap system with a gap∆ of about 30 K. Irrespective of the interdimer interaction, the direct product of the dimer singlets is the exact
ground state, realizing the Shastry-Sutherland model6)
proposed almost 20 years ago. Moreover, a recent theory7) revealed that this compound resides in the vicinityof the critical point between the gapped exact dimer state
and the gapless Néel-ordered state.
The excited states of SrCu2 (BO3 )2 are also of greatinterest as seen in the magnetizations: we observed
quantized magnetization plateaux at 1/4 and 1/8 of thesaturation of the Cu moment Ms .3-5) The superstructures for the 1/4 and 1/8 plateaux are illustrated inFigs. 1(a) and 1(b), respectively. The origin of the observed plateaux was discussed by Miyahara and Ueda.7)
They argued that, because of the extremely localized
character of the excited triplets and also because of theirrepulsive interaction, the triplets would favor orderedrather than liquid states. Considering the tetragonal
symmetry of the original lattice, the necessary condition
for the triplets to assure ordered structures is that corresponding magnetic unit cells are square, which is ful-

∗ E-mail: kage@issp.u-tokyo.ac.jp

filled when M/Ms = 1/4, 1/8 (see Figs. 1(a) and 1(b)).
Based on this condition, other plateaux at 1/2, 1/10,
1/16, 1/32 · · · were predicted. Therefore, observation of
the remaining plateaux may be verifying their theory.
In order to elucidate the intrinsic magnetic properties of SrCu2 (BO3 )2 in sufficient depth, in particular tosolve the plateau problem, high-quality bulk single crystals are indispensable. Recently, we have succeeded, after extensive trial-and-error, in growing large single crystals by means of a vertical traveling solvent floating zone(TSFZ) technique.8) In this letter, we will show the results of magnetic susceptibility and high-field magnetization measurements for the single crystals, and elaborateon both the ground and the excited states.
Single crystals of SrCu2 (BO3 )2 of high purity weregrown by the TSFZ method with an image furnace using an appropriate amount of a polycrystal SrCu2 (BO3 )2and a solvent LiBO2 under flowing O2 gas (PO2 =1 atm).8) The temperature variation of the magnetic susceptibility was measured in the magnetic field H of 1.0 T
in the temperature range between 2 K and 350 K using asuperconducting quantum interference device (SQUID)
magnetometer. A piece of the obtained single crystals(0.14 g) was set on a sample holder in such a way thatthe c axis was oriented parallel (//) or perpendicular (⊥)
to the applied magnetic field. The high-field magnetization measurement was performed using an induction
method with a wire-wound pulse magnet which produces
magnetic fields up to 42 T. The duration time was about
20 ms. Two pieces of the crystals (0.19 g) were alignedin the direction, H ⊥ c and cooled to 0.4 K using a conventional 3 He-pumping device.
We first show in Fig. 2 the temperature dependenceof the parallel and perpendicular magnetic susceptibilities of SrCu2 (BO3 )2 , χraw// and χraw⊥ , respectively. For
comparison, the previous data for the powdered sample

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LettersFig. 2. Temperature variation of the magnetic susceptibilities for
SrCu2 (BO3 )2 , χraw// (square) and χraw⊥ (triangle), parallel andperpendicular to the tetragonal c axis, respectively. The previous
data for the powdered sample is also shown by the solid line.3)
The inset is the enlarged plot in the low-temperature region.
The spin susceptibilities, χspin// and χspin⊥ , are marked by thesolid symbols. The broken lines (· –·– ·, – – –) represent the fitted
curves of χspin// and χspin⊥ , respectively, to the isolated dimer
model (see text), where g// = 2.28 and g⊥ = 2.07, determinedby ESR, are used.

Fig. 1. Ordered structures between the excited triplets (open)
and the remaining singlets (solid) corresponding to 1/4 (a), 1/8(b), and 1/10 (c) plateaux. The magnetic unit cells are denotedby broken lines, and the chemical unit cell by dotted lines in (a).

are also shown by the solid line.3) At high temperatures
well above the spin gap, both χraw// and χraw⊥ obey theCurie-Weiss law χ = N g 2 µ2B S(S +1)/[3kB (T −θ)], whereN denotes Avogadro number, µB , the Bohr magneton,kB , Boltzmann constant, g, the g-factor, and θ, the Weissconstant. Using S = 1/2 for the Cu2+ ion and g// = 2.28
and g⊥ = 2.07 determined by the electron spin resonance(ESR) measurements,9) we fitted the data in the temperature range between 250 K and 350 K to the above
formula, and obtained θ = −102.5 K together with aconstant susceptibility χ0 = −2.603 × 10−5 emu/Cu mol.

The obtained Weiss constant is slightly larger in magnitude than the previous results for the polycrystal(−92.5 K)3) and for the field-oriented polycrystal along
the c axis (−97.8 K).4)
When the temperature is decreased, the susceptibilities reach a maximum at about 15 K, decrease abruptly
and then tend to approach zero at 0 K, achieving theexact dimer ground state. There is a Curie-like upturn
below 4 K, presumably due to a defect in the crystal.
It is noteworthy that the contribution of this extrinsic
term is significantly reduced compared with that for thepolycrystalline samples,3, 4) ensuring high crystal quality. Indeed, the fit to the Curie law C 0 /T in the lowtemperature region gave C 0 = 1.5 × 10−3 emu K/Cu mol,corresponding to only 0.14% of free spin-1/2. Spin susceptibilities of SrCu2 (BO3 )2 , χspin// and χspin⊥ were finally evaluated by subtracting C 0 /T +χ0 from χraw// andχraw⊥ , respectively.
The value of ∆ was re-estimated in several ways as follows: based on the fit of the spin susceptibilities below
6 K to the isolated dimer model χ = N g 2 µ2B /kB T [3 +exp(∆/T )] (see the inset of Fig. 2), where ∆ is theuniquely fitted parameter, we obtain ∆ = 34 ± 1 K,
which is in good agreement with the recent nuclear
magnetic resonance (NMR) result (34 ± 1 K),10) and isvery close to the value of 30 K evaluated from the nuclear quadrupole resonance (NQR) and the magnetization measurements.3) On the other hand, if one fits χspin//and χspin⊥ to χs ∝ exp(−∆/T ) as we did previously forthe polycrystal,3) a much smaller value of ∆ = 19 K isobtained again. Thus, the former model seems to give aLettersFig. 3. Magnetization curve of SrCu2 (BO3 )2 perpendicular tothe c direction (the solid line) measured at 0.4 K. For comparison, the parallel and perpendicular magnetizations at 0.5 K
for the field-oriented sample are shown by the dotted and thebroken lines, respectively.4)

more appropriate description of the χ-T , at least in thistemperature region.
Let us now move onto the result of the magnetization measurements. The solid line in Fig. 3 denotes
the perpendicular magnetization M⊥ at 0.4 K as a function of H, which is a major improvement over previous data (also shown in Fig. 3 by the broken lines),4)
possibly adding a new plateau as described in the nextparagraph. We see from Fig. 3 that the gap closes
at about H1 = 22.5 T, corresponding to ∆ = 31.3 K(where we assumed g⊥ = 2.079) ). As observed previously, there are two quantized plateaux at M/Ms = 1/4,1/8, which evidently become more distinct. The fieldinduced gapped state11) for the 1/4 plateau exists between H5 = 39.1 T and H6 = 41.6 T, and that for the1/8 plateau between H3 = 30.1 T and H4 = 31.7 T. The
fact that the 1/4 plateau is wider than the 1/8 plateau,i.e., (H6 − H5 )/(H4 − H3 ) ∼ 1.5, suggests that the former state has a bigger field-induced gap than the latter, in contrast to the case of NH4 CuCl3 ,12) in which
the observed plateaux at 1/4 and 3/4 have almost thesame width. The estimated critical fields H3 ∼ H6 aresomewhat higher than those for the oriented polycrystal(H ⊥ c).4) This means that the orientation of the polycrystal was poor as checked by NMR measurement9) and,as a consequence, the M⊥ vs H curve for the oriented
polycrystal must involve the contribution of the parallel
component M// .
Of greater significance here is that the magnetization
curve has an inflection at H2 = 27.5 T, which, broadly
speaking, appears at 1/10 of the saturation moment.
This anomaly is possibly attributed to the 1/10 plateau,
which is, as mentioned earlier, one of the plateaux predicted by Miyahara and Ueda.7) The superstructure cor-1823

responding to the 1/10 plateau is illustrated in Fig. 1(c),where the magnetic unit cell is square and is five times
as large as the chemical unit cell. According to their
theory, with descending plateaux as 1/2, 1/4, 1/8, 1/10,1/16, 1/32 · · · , the field-induced gap decreases, resulting
in narrower plateau width. This is the reason for therather obscure 1/10 plateau in contrast to the clear 1/4
and 1/8 plateaux and for the unsuccessful observation ofother predicted plateaux such as 1/16 and 1/32 plateaux.
Therefore, lower temperature measurement is probably
necessary in order to ensure the existence of the 1/10
plateau and to detect the remaining plateaux.
Let us now look at the detailed curvature of the magnetization in the gapless regions. As is well known for
one-dimensional spin systems, the pmagnetization near
the plateau edge is proportional to |H − Hc | (Hc : thecritical field).13) Similar behavior is seen for the edges
at H3 , H4 and H6 , while those at H1 and H5 are somewhat round. Another point to note is the tendency of themagnetization curve to shoot up at 27 T or 37 T toward
the next plateau edge. Presently, there exists no theory
to describe such characteristics of SrCu2 (BO3 )2 or moregenerally of two-dimensional spin systems.
In summary, we have measured the temperaturedependent magnetic susceptibilities and the high-fieldmagnetization using single crystals of SrCu2 (BO3 )2 .
From the analysis of the susceptibility data, we evaluatedthe spin gap to be 34 K, in good accordance with the results of NMR and the magnetization measurements. Inthe magnetization, in addition to the previously observed
1/4 and 1/8 plateaux of the saturation Cu moment, another plateau at 1/10 is probably present, which strongly
supports the theory by Miyahara and Ueda.
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