Physica B 329–333 (2003) 1020–1023

Quantum phase transitions in the orthogonal dimer
system SrCu2ðBO3 Þ2
Hiroshi Kageyama*, Nikoray V. Mushnikov, Masaki Yamada,
Tsuneaki Goto, Yutaka Ueda
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

Abstract
Inspired by recent enthusiastic theoretical works on the phase diagram of the (generalized) Shastry–Sutherlandmodel, we measured the temperature dependence of magnetic susceptibility of SrCu2 ðBO3 Þ2 at high pressures up to
7 kbar: We employed a clumping type of a pressure cell made by CuTi alloys, which allows us to obtain the sample
magnetization with a high sensitivity. We found that the temperature at maximum susceptibility decreases withincreasing pressure, implying the reduction of the spin gap. This result strongly suggests that the system nears the phaseboundary to an antiferromagnetically ordered or a plaquette singlet state.
r 2003 Elsevier Science B.V. All rights reserved.
PACS: 75.10.m; 75.30.Kz
Keywords: SrCu2 ðBO3 Þ2 ; Shastry–Sutherland model; Spin gap; High pressure

1. Introduction
Over the last three decades, geometrical spin frustration has been one of the most interesting and challenging
issues in the condensed matter physics. Three spins
sitting on the corners of one rigid triangle, coupled
antiferromagnetically with respect to each bond, is themost fundamental unit for conventional frustrated
models. By arranging rigid triangles in some space,
one can tailor various lattices with geometrical frustration, as exempliﬁed in Fig. 1. Figs. 1(a) and (b) show the
one-dimensional (1D) cases, a delta chain [1] and azigzag chain [2], respectively. If the rigid triangles arepacked completely on a plane, one obtains a 2D
triangular lattice as shown in Fig. 1(c) [3], while one
quarter of periodic vacancies in the triangular lattice
results in the so-called kagom!e lattice (Fig. 1(d)) [4].
Shown in Fig. 1(e) is one of the 3D versions, called the
pyrochlore lattice, consisting of the alternate stack of the*Corresponding author.
E-mail address: kage@issp.u-tokyo.ac.jp (H. Kageyama).

dilute triangular and dense kagome! layers along the
cubic ½1 1 1 [5].
In contrast, antiferromagnetically coupled spins on a
2D square lattice are not at all frustrated because of theabsence of such triangles. Nevertheless, geometricalfrustration can be induced into this lattice by introducing some regular depletion (Fig. 2(a)) [6] or by adding
some diagonal interactions (Figs. 2(b)–(d)) [7–9]. Compared with the former models based on the rigid
triangles, only little attention had been paid for thelatter models so far presumably due to the lack ofexperimental correspondences. But recent discovery ofmaterials such as CaV4 O9 [10], Li2 VOSiO4 [11],SrCu2 ðBO3 Þ2 [12] sparked experimental and theoreticalinvestigations on the respective models of square latticeorigin.
The Shastry–Sutherland lattice with the intradimer ðJÞ
and interdimer ðJ 0 Þ bonds is characterized by theorthogonality of the neighboring dimers. Because ofthis peculiar topology, many intriguing phenomena havebeen found for the Shastry–Sutherland model and thecorresponding material SrCu2 ðBO3 Þ2 ; including an exact

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0921-4526(02)02222-6

H. Kageyama et al. / Physica B 329–333 (2003) 1020–1023

1021

J
(a)

J'

(b)

(c)

J''

(d)
c

b

a
Fig. 3. The 3D lattice of SrCu2 ðBO3 Þ2 :(e)Fig. 1. Geometrically spin frustrated models consisting only of
rigid triangles. (a) delta chain, (b) zigzag chain, (c) triangularlattice, (d) 14 regularly depleted triangular lattice (kagom!elattice), (e) pyrochlore lattice.

(a)

(b)

(c)

(d)

neighboring dimers along the c-axis are also connected
orthogonally, thus making the grand state of the 3Dmodel still solvable exactly [27]. Considering theestimated coupling constants ðJ ¼ 84 K; J 0 ¼ 54 K;
J 00 ¼ 8 K [28]; J ¼ 70 K; J 0 ¼ 42 K; J 00 ¼ 15 K [21]),
this material must be located in the vicinity of the phaseboundaries to an antiferromagnetically ordered state aswell as a resonating plaquette singlet state.
In spite of these vigorous theoretical activities,
however, there has been no experimental observationof the quantum phase transitions. Although the solid
solution Sr1x Ax Cu2 ðBO3 Þ2 ðA ¼ Ca; Ba; 0oxo0:3Þ
was prepared to investigate the chemical pressure effect,
the obtained susceptibility did not exhibit systematicchange with x and A due to marked difference of sample
quality [29]. This makes it difﬁcult to estimate theexchange couplings reliably in this solid solution. On theother hand, application of physical pressure is considered to be a clean tool to tune exchange couplings. In
this paper, we will report on the susceptibility measurement at various pressures up to 7 kbar:

2. Experimental
Fig. 2. Geometrically frustrated models derived from the 2D
square lattice. (a) 15 regularly depleted square lattice realized by
CaV4 O9 ; (b) J1  J2 model realized by Li2 VOSiO4 ; (c) tilted
triangular lattice, (d) Shastry–Sutherland lattice realized by
SrCu2 ðBO3 Þ2 :dimer ground state [9,12–15], fractional plateaux in themagnetization curve [12,16–19], and multiple triplets
excitations [20–23]. Theoretically, the possibility ofquantum phase transitions from the dimer singlet state
to other states has been discussed intensively[9,15,21,24–26]. For instance, Koga considered thegeneralized Shastry–Sutherland model adding the coupling between the layers ðJ 00 Þ: As illustrated in Fig. 3,

We recently developed a high-pressure clamp cellconsisting of CuTi alloys with 3 wt% Ti [30]. The
magnetization of the CuTi pressure cell is extremely
small, namely, of the order of 107 emu: Furthermore, it
is almost independent of temperature ðTÞ and pressure
ðPÞ: These characteristics are based on the fact that theparamagnetic susceptibility of Ti almost compensates
the diamagnetic susceptibility of Cu. Therefore, we canprecisely measure even weak magnetizations of materials
like the present compound. Sample magnetizations were
measured under pressure up to 7 kbar using the CuTi
cell with the outer diameter of 8:8 mm; which was
installed in a SQUID magnetometer. Single crystals of
SrCu2 ðBO3 Þ2 of high quality were grown by thetravelling solvent ﬂoating zone method using a solvent

H. Kageyama et al. / Physica B 329–333 (2003) 1020–1023

LiBO2 [31]. One piece of grown crystals with anappropriate dimensions was compressed in a Teﬂon
capsule 3 mm in diameter and 10 mm in length,
ﬁlled with a liquid pressure-medium, Fluorinert
ðCF 70: CF77 ¼ 1:1Þ: The magnetic ﬁeld ðHÞ of 1 T
was applied perpendicular to the c-axis and temperature
was scanned between 3 and 30 K:

3. Results and discussion
For magnetization measurements under high pressures, the pressure cell made by Co-contaminated CuBe
alloys has been usually employed [32]. However, thispressure cell itself has ﬁnite magnetization because of the
Co impurities in the commercially available alloys. To
make matters worse, the magnetization of the pressure
cell varies as a function of both P and T: The
susceptibility measurement for SrCu2 ðBO3 Þ2 using thistype of pressure cell was previously performed between
5oTo30 K [33]. As anticipated, the total magnetization (SrCu2 ðBO3 Þ2 þ pressure cell) even at ambientpressure does not approach to zero as T-0: Moreover,at a given T; it remarkably increases with P:
The use of the new type of pressure cell has
dramatically changed the situation into better. Fig. 4
demonstrates the result of temperature dependence ofthe magnetic susceptibilities for 3oTo30 K at 0, 2.0,
4.0, 6.0, and 7:0 kbar; using the CuTi pressure cell. At
higher temperatures, the obtained susceptibilities arealmost independent of pressure, signiﬁcantly in contrast
to the result of previous magnetization as mentioned
above. With decreasing T; however, marked pressure
dependence appears. The temperature at the maximum
susceptibility TMax and the temperature at the halfmaximum susceptibility THM decrease with the applica-tion of pressure. With further decreasing T; all thesusceptibilities decrease to zero as T-0: This meansthat the ground state remains spin singlet at any pressure
applied in this study. It should be pointed out that theplaquette singlet state also has nonmagnetic groundstate. However, we believe that the dimer singlet phasestill survives at least until 7 kbar since the susceptibilitychanges quite systematically with P:
We plotted TMax and THM as a function of P in Fig. 5.
Both values decrease in proportion to P; which would be
attributable to the reduction of the energy gap D fromthe singlet ground state to the ﬁrst (triplet) excited state.
It is noted, however, that in spin-gap materials it is inprinciple possible to evaluate the value of D by ﬁtting thesusceptibility to a function assuming an activation-typeof T dependence. In SrCu2 ðBO3 Þ2 ; however, such aﬁtting using the data even without a pressure cell (i.e., at
the ambient pressure) gave much smaller value of 19 K
[12] than the actual value ðD ¼ 34 KÞ obtained by e.g.,
ESR [13]. Therefore, the pressure variation of D cannot
be estimated from the present experiment. Even then, as
seen in Fig. 5, extrapolation of TMax and THM to higher
P may predict the critical pressure Pc to be 25B35 kbar:
Possibly, Pc can be smaller because the transitions either
to the ordered state or to the plaquette singlet state areconsidered to be of ﬁrst order [25]. This result has
further motivated us to investigate the pressure effect.
Actually, various experiments at high(er) pressure(magnetization, nuclear magnetic resonance, neutron

20

(a)
15

TMax (K)

1022

5

3.0
0
10

2.5
2.0
1.5
1.0
0.5
0.0

6
4
2

SrCu2(BO3)2

1T

0

0

5

10

15

20

25

(b)

8

0 kbar
2.0
4.0
6.0
7.0

THM (K)χ (x10 -3 emu/mol Cu)

3.5

10

30

T (K)
Fig. 4. Temperature dependence of the magnetic susceptibilityof SrCu2 ðBO3 Þ2 at high pressures up to 7:0 kbar:

0

5

10

15

20

25

30

P (kbar)Fig. 5. Pressure dependence of (a) TMax and (b) THM : Brokenlines are the guides to the eyes.

H. Kageyama et al. / Physica B 329–333 (2003) 1020–1023

scattering/diffraction, Raman scattering, speciﬁc heat)
are in progress.
Finally, we would like to discuss the origin of thepressure effect. Since the structure of SrCu2 ðBO3 Þ2consists of the CuBO3 (Shastry–Sutherland) layers,intercalated by nonmagnetic Sr2þ ions, the applicationof P would affect only interlayer distance, thus changing
J 00 while keeping J and J 0 unchanged. However, this is
not the case. Theoretical considerations show that D is
independent of J 00 for a given set of J 0 =J; while it largely
depends on J 0 =J [15,25]. In fact, the theoreticalsusceptibility as a function of J 0 =J is quite similar to
the experimental susceptibility as a function of P [15].
Considering these facts, it is natural to think that withincreasing P both interlayer and intralayer coupling
constants change toward the direction of the phaseboundary.

4. Conclusion
We performed the magnetic susceptibility experiments
for SrCu2 ðBO3 Þ2 at high pressures using a new type ofpressure cell made by CuTi alloys with almost nonmagnetic susceptibility. Although the pressure in ourhand is not enough to induce the quantum phasetransition, we obtain a strong piece of evidence that thesystem gets closer to the phase boundary. Higher P
experiments will be conducted soon.

Acknowledgements
This work is supported by Grant-in-Aid for ScientiﬁcResearch (C) (No. 40302640) from Ministry of Education, Science, Technology, Sports and Culture, Japan.
We thank S. Miyahara, A. Koga, K. Totsuka, P.
Lemmens, K. Kodama, M. Takigawa for helpful
discussions. H.K. thanks Yoto Kageyama for illuminating discussions.

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