L

Journal of Alloys and Compounds 317–318 (2001) 177–182

www.elsevier.com / locate / jallcom

Quantized magnetization plateaux in the orthogonal dimer system
SrCu 2 (BO 3 ) 2
a,

b,c

b,c

a

a

H. Kageyama *, Y. Narumi , K. Kindo , K. Onizuka , Y. Ueda , T. GotoaaInstitute for Solid State Physics, University of Tokyo, Roppongi, 7 -22 -1 Minato-ku, Tokyo 106 -8666, Japanb
CREST, Japan Science and Technology Corporation ( JST), Japanc
Research Center for Materials Science at Extreme Conditions, Osaka University, 1 -3, Machikaneyama, Toyonaka, Osaka 560 -8531, Japan

Abstract
We carried out high-field magnetization measurements up to 57 T and down to 0.08 K on a single crystal of a two-dimensional spin-gap
material SrCu 2 (BO 3 ) 2 . We successfully observed the predicted plateau at 1 / 3 of the total magnetization around 50 T, in which themagnetic superstructure is characterized by a novel stripe order of excited triplets. The 1 / 3 plateau is stable over much wider field region
than the previously observed 1 / 4 and 1 / 8 plateaux. A sharp jump in the magnetization between 1 / 3 and 1 / 4 plateaux may be associatedwith a first-order transition. Furthermore, several anomalous behaviors were observed in a lower-field region including field-direction
dependent magnetizations around 30 T. The obtained magnetizations are compared with theoretical calculations based on a hard-coreboson model and on a perturbation expansion method. The magnetization measurements were also performed on 1% zinc-doped single
crystal.  2001 Elsevier Science B.V. All rights reserved.
Keywords: Magnetization; SrCu 2 (BO 3 ) 2 ; Stripe; Shastry–Sutherland model

1. Introduction
There has been a resurgent interest in the phenomena ofquantized magnetization plateaux because the plateaux ofrecent interest are caused principally by quantum manybody effects [1], in contrast to classical-spin systems
previously investigated such as CoCl 2 2H 2 O [2] andCa 3 Co 2 O 6 [3]. Experimentally, several quantum-spin systems were reported to have magnetization plateaux. A 1 / 2
plateau of the total magnetization was observed in[Ni 2 (Medpt) 2 (m-ox)(m-N 3 )]Cl 4 0.5H 2 O [4], and 1 / 4 and3 / 4 plateaux in NH 4 CuCl 3 [5]. The observed plateaux forthe former and latter materials have been nicely explainedby the theoretical investigations on an S51 chain systemwith bond alternation [6] and on an S51 / 2 three-dimensionally coupled dimer system [7], respectively. It was,furthermore, shown that the necessary condition thatplateaux appear in general one-dimensional (1D) systems
is n(S2m)5integer, where n is the period of the spin state,
S the magnitude of spin and m the magnetization per site

*Corresponding author.
E-mail address: kage@issp.u-tokyo.ac.jp (H. Kageyama).

in the unit cell [8]. Recently, this condition was found tobe valid for any dimension [9].
The present paper will deal with SrCu 2 (BO 3 ) 2 for whichquantized plateaux were recently observed in the magnetization curves [10]. SrCu 2 (BO 3 ) 2 is a layered material
with a tetragonal symmetry, composed of alternatingstacks of 2CuBO 3 and Sr planes along the c-axis. In the
CuBO 3 plane, nearest-neighbor Cu 21 ions constitute an
S51 / 2 dimeric unit, and each dimer connects orthogonallywith four nearest-neighbor dimers, as shown in Fig. 1. The
intradimer and interdimer interactions, symbolized by J
and J9, respectively, lead to an exact ground state withspin gap D, thus realizing the 2D Shastry–Sutherlandmodel [11,12]. The values of J, J9 and D were estimated tobe 100, 68 [10,12] and 34 K [13–15], respectively.
The 1 / 4 and 1 / 8 plateaux in SrCu 2 (BO 3 ) 2 were firstdetected by magnetization measurements on a polycrystalline sample performed up to a magnetic field H543 T anddown to a temperature T51.5 K with a pulse magnet at
ISSP, the University of Tokyo [10], followed by thoseusing field-oriented samples for H i c and H'c [14] and asingle crystal for H'c [15]. Miyahara and Ueda explained
theoretically that the origin of the plateaux lies in thealmost localized nature of the triplet excitations [12].

0925-8388 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S0925-8388( 00 )01316-5

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H. Kageyama et al. / Journal of Alloys and Compounds 317 – 318 (2001) 177 – 182

Fig. 1. S51 / 2 orthogonal dimer lattice realized in SrCu 2 (BO 3 ) 2 , indicating in-plane interactions which act within the dimers J and between the
dimers J9. Square unit cells containing 4, 8, 16, 20, 32 spins are
represented by dotted lines.

Moreover, in the light of the tetragonal symmetry of thecrystal structure, they initially argued the following necessary condition for the triplets to ensure ordered rather thandisordered structures: corresponding magnetic superstructures should also be of tetragonal symmetry. In fact, thiscondition satisfies the superstructures shown in Fig. 2a andb for the 1 / 8 and 1 / 4 plateaux, respectively. Following
this simple geometrical argument, one expects otherplateaux at 1 / 2, 1 / 10, 1 / 16, 1 / 32 . . . , which also havesquare unit cells as illustrated in Fig. 1.
However, recent detailed analyses carried out independently by Momoi and Totsuka [16] and by Miyahara andUeda [17] have revealed the possibility that the superstructure for the 1 / 4 plateau is of the stripe type with arectangular unit cell, as shown in Fig. 2c. According toRef. [17], the triplet–triplet interaction between the nthnearest-neighbor pair of dimers Vn does not exhibit amonotonous decrease with increasing n. Especially important is that V2 is smaller than V3 for any J9 /J. Thus, thesuperstructure in Fig. 2c, where excited triplets are arranged linearly avoiding V3 , should be more stable thanthat in Fig. 2b. Moreover, a prediction has been made thata 1 / 3 plateau with a superstructure of stripe order (see Fig.
2d) will be observable if a magnetic field higher than thatever studied is applied.
Unfortunately, in spite of intensive experimental activities [10,13–15,18–21], the existence of the stripe ordereven for the 1 / 4 plateau has not yet been confirmed. Thisis partly because the required magnetic field is beyond thecapability of the facilities currently used for microscopic

Fig. 2. Possible ordered structures between the excited triplets (open) andthe remaining singlets (solid) for the 1 / 8 plateau (a) the 1 / 4 plateau (b)and (c), and the 1 / 3 plateau (d). The magnetic unit cells are represented
by broken lines.

techniques such as neutron diffraction. In this paper, wemeasured the magnetization curves for single-crystalline
SrCu 2 (BO 3 ) 2 up to 57 T to successfully observe the 1 / 3
plateau. It should be emphasized that, unlike the case of
the 1 / 4 plateau, the stripe superstructure (Fig. 2d) is theunique candidate for the 1 / 3 plateau. It is also interesting
to investigate doping effect on the magnetization plateaux.
Since the observed plateaux in SrCu 2 (BO 3 ) 2 is closely
related to the localized character of the excited triplets, we
expect that a slight substitution of Zn for Cu, i.e. introduction of spin vacancy may significantly affect the fielddependent magnetization, possibly destroying the alreadyobserved plateaux or / and creating new plateaux.

2. Experimental
The high-field magnetization measurements were conducted using an induction method with a multilayer pulse
magnet at the High Magnetic Field Laboratory,KYOKUGEN in Osaka University. The pulse width was
about 8 ms, and the use of a plastic dilution refrigerator
allowed to measure down to 50 mK. Measurement accuracy of magnetization is less than 5% in magnitude. Highquality single crystals of SrCu 2 (BO 3 ) 2 were obtained by atravelling solvent floating-zone (TSFZ) technique using
LiBO 2 as a solvent, as described in detail in Ref. [22]. We
used one of the grown single crystals of high quality withapproximate dimensions of 23232 mm. Magnetization

H. Kageyama et al. / Journal of Alloys and Compounds 317 – 318 (2001) 177 – 182

data for H'c were collected at 0.08 and 1.5 K in magnetic
fields up to 57 T, and those for H i c were collected at 1.4
and 4.2 K up to 53 T. To investigate the impurity effect on
the plateaux, we performed the magnetization measurements at 1.3 K on 1% zinc doped crystals
Sr(Cu 0.99 Zn 0.01 ) 2 (BO 3 ) 2 grown by the TSFZ method aswell. The analysis of inductively coupled plasma–atomic
emission (ICP–AE) spectroscopy and scanning electron
microscope (SEM) for the obtained doped single crystal
ensured the successful Zn-substitution as well as homogeneity of the crystal. Single crystals growth with a zinc
ratio more than 1% has not been succeeded yet.

3. Results and discussion
Fig. 3a shows the obtained magnetization curves of
SrCu 2 (BO 3 ) 2 for Hi c and for H'c, where within theexperimental accuracy, no hysteresis was observed upon

Fig. 3. (a) Magnetization curves for SrCu 2 (BO 3 ) 2 measured at 0.08 K for
H'c and at 1.4 K for H i c (solid), and for 1% Zn doped single crystal
measured at 1.3 K (dotted). The broken lines correspond to 1 / 3, 1 / 4 and1 / 8 of the total magnetization. (b) Differential magnetization curves
DM /DH for SrCu 2 (BO 3 ) 2 .

179

increasing or decreasing H. We also show the differential
magnetization curves DM /DH in Fig. 3b. It appears thatthe difference between the two curves arises only from theg-factor. As demonstrated in Fig. 4, the parallel andperpendicular magnetizations can be normalized using gi 52.28 and g' 52.05 determined by electron spin resonance(ESR) [13]. Strictly speaking, there is a slight difference
around 30 T, which will be discussed later. Note that thisis the first measurement of parallel magnetization using thesingle crystal. The obtained magnetization curve for Hi c
has more distinct 1 / 4 and 1 / 8 plateaux compared with thatobtained using the field-oriented polycrystal [14] with poor
orientation, as evaluated by nuclear magnetic resonance.
On the other hand, the result for H'c below 42 T is
almost identical to that previously reported using the single
crystal [15]. As shown in Fig. 3a, the critical fields are
determined to be H1 529.6 T, H2 531.2 T, H3 536.7 T and
H4 543.6 T for H'c, and H1 526.9 T, H2 528.2 T, H3 534.1 T and H4 539.1 T for H i c, which is almost consistent with the previous results [14,15].
A glance at Fig. 3a reveals a wide plateau at 1 / 3 of thefull magnetization. As mentioned earlier, the 1 / 3 plateau
cannot accommodate any magnetic superstructure with asquare unit cell. Therefore, this observation confirms forthe first time the existence of the novel stripe order, where
the magnetic unit cell is composed of four singlets and two
triplets. The onset field H5 of the 1 / 3 plateau is estimatedto be 43.6 T for H'c and 40.2 T for H i c. As evident in
Fig. 3a and b, the 1 / 3-plateau phase is stable over a muchwider field range than the 1 / 8 and 1 / 4 plateaux.
The common feature of the observed three plateau
phases is the gradual increase of the magnetization with H,
indicating nonstoichiometry of the ordered triplets. It isnoted that this effect does not originate from thermal
fluctuation because, as will discussed later, the contribution

Fig. 4. Normalized parallel and perpendicular magnetizations for H'c(solid) and for H i c (broken), where gi 52.28 and g' 52.05 are used.

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H. Kageyama et al. / Journal of Alloys and Compounds 317 – 318 (2001) 177 – 182of thermal fluctuation to the magnetizations can be neglected at least below 1.5 K. This nonstoichiometric naturein SrCu 2 (BO 3 ) 2 is strikingly in contrast to the cases of[Ni 2 (Medpt) 2 (m-ox)(m-N 3 )Cl 4 0.5H 2 O, [4] and NH 4 CuCl 3[5], where the nearly flat plateaux were observed.
It is interesting to note the nature of the transitions
between the plateaux. A sudden jump was observed
between the 1 / 4 and 1 / 3 plateaux. This may indicate afirst-order phase transition between the two states, although there is no hysteresis between the field increase anddecrease processes. The first-order phase transition between the plateaux would be a consequence of the extremely localized triplet excitations, as discussed theoretically [12,16]. On the contrary, a complex behavior is seen
between the 1 / 8 and 1 / 4 plateaux. The magnetization
grows gradually above H2 until H approaches to H3 whenthe magnetization suddenly rises to the 1 / 4 plateau. A
possible explanation is that the system is in a liquid(disordered) state for H2 ,H,H3 and its transition to the
1 / 8-plateau phase at H2 is of a second order, while that to
the 1 / 4-plateau phase at H3 is of a first order.
In Ref. [15], we pointed out the existence of the 1 / 10
plateau, judging from the perpendicular magnetization that
displays a bump at about 1 / 10 of the total magnetization,
as observed in this experiment as well. The behavior of theparallel magnetization, however, is not indicative of the1 / 10 plateau, as seen in Fig. 4. It is possible that theappearance of the 1 / 10 plateau depends on the field
direction, i.e. the orientation of the excited triplets. Alternatively, this phenomenon could be a quantum effect ofsome kind.
Let us focus our attention on the temperature dependence of the magnetization curves. Shown in Fig. 5 are thenormalized magnetizations at 4.2, 1.5 and 0.08 K. Compared with the 1.5 K curve, the 4.2 K curve is considerably
obscured due to the effect of thermal fluctuation. However,the 0.08 K curve is very similar to the 1.5 K curve (apartfrom the anisotropy around 30 T as mentioned earlier),
indicating that thermal fluctuation is negligible at least
below 1.5 K and thus these two curves are almost
equivalent to those for zero-temperature limit. Nevertheless, the magnetizations for both 1.5 and 0.08 K begin tohave a finite value at about 16 T. This value is considerably lower than the expected critical field of Hc 524.5
T determined from k B D5g' mB Hc . Here, k B and mB denotethe Boltzmann constant and the Bohr magneton, respectively, and D534 K and g' 52.05 are used. The ESR
measurement [13] has also revealed anomalous behaviors
in this field region, which include multiple magnetic
excitations and the deviation of field-dependent spin gap
D H from the relation D H 5gmB H /k B . We infer from thesefacts that the mixing occurs between the initial spin-singlet
ground state and the excited triplet states mediated by, e.g.
the Dzyaloshinsly–Moriya interaction and as a result the
ground state becomes partially magnetic below Hc .

Fig. 5. Temperature dependence of magnetization curves. Hc denotes thecritical field expected from the relation described in the text. Dotted lines
are the guides to the eye.

There are several theoretical calculations of the magnetizations on the Shastry–Sutherland model with varying
J9 /J ([16,17,23,24] and G. Misguich, private communication). In Fig. 6a and b, we compare the experimental datawith the calculations in terms of a hard-core boson modelby Momoi and Totsuka [16] (theory I) and Miyahara andUeda [17] (theory II), where J5100 K and J9568 K areused. Shown in Fig. 6c is the result of Fukumoto andOguchi [23] based on the perturbation expansion method
up to the third order, where J585 K and J9555 K are
assumed. It is noteworthy that these three models quantitatively reproduce the experimental data particularly aroundthe 1 / 3 plateau. The main difference between theories I
and II, in short, is that the former considers correlated
hopping terms and neglects long-range interactions, whereas the case is reversed in the latter. In connection with thisdifference, the 1 / 3 plateau ends at 80 T according totheory I, while theory II predicts that it ends at 110 T.
Moreover, theory II (and theory III) concluded that a 1 / 2
plateau exists, but theory I did not. Thus, the future
direction of this area of study will be one that requires
even higher magnetic fields.
Finally, we shortly discuss the impurity effect on themagnetizations. It is naively expected that the substitutionof a Zn ion for the Cu site should break a spin-singlet paircreating a localized S51 / 2. Thus created localized moment might affect the already observed plateaux and / orinduce new plateaux. Unlike such an expectation, no
appreciable change was observed in the magnetizations

H. Kageyama et al. / Journal of Alloys and Compounds 317 – 318 (2001) 177 – 182

181

the expected critical field may indicate that the groundstate is no longer the perfect dimer singlet state. The 1%
zinc substitution for the Cu site does not affect theobserved plateaux nor spin-singlet ground state. We consider the number of spin vacancies created by the Zn
substitution may be too small to induce some dramaticchanges. To further understand the role of spin vacancies
on the Shastry–Sutherland system, preparation as well as
magnetization measurements of samples with higher Zn
concentration is desirable and is in progress.

Acknowledgements
It is our great pleasure to thank S. Miyahara, K. Ueda,
T. Momoi, K. Totsuka, Y. Fukumoto, A. Oguchi for kindly
providing us with their theoretical results and also for
motivating us to carry out this experiment. H.K. wishes tothank Ritsuko Ezumi for her continuous encouragement.
This work was partially supported by a Grant-in-Aid for
Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan.

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Fig. 6. Theoretical magnetizations (solid curves) calculated by (a) Momoi
and Totsuka [16], (b) Miyahara and Ueda [17], and (c) Fukumoto andOguchi [23]. The dotted curve is the experimental result for H i cnormalized by gi 52.28.

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Fig. 3.

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