VOLUME 82, NUMBER 15

PHYSICAL REVIEW LETTERS

12 APRIL 1999

Exact Dimer Ground State and Quantized Magnetization Plateaus
in the Two-Dimensional Spin System SrCu2 s BO3 d2
H. Kageyama,1,2, * K. Yoshimura,1,3, † R. Stern,3 N. V. Mushnikov,2 K. Onizuka,2 M. Kato,1 K. Kosuge,1
C. P. Slichter,3 T. Goto,2 and Y. Ueda2
1

Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106-8666, Japan
3
Loomis Laboratory of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street,
Urbana, Illinois 61801-3080(Received 5 June 1998)
2

Magnetic susceptibility, Cu NQR, and high-filed magnetization have been measured in polycrystalline
SrCu2 sBO3 d2 having a two-dimensional (2D) orthogonal network of Cu dimers. This cuprate provides
a new class of 2D spin-gap system sD ­ 30 Kd in which the ground state can be solved “exactly.”
Furthermore, in the magnetization, two plateaus corresponding to 14 and 18 of the full Cu moment werefirst observed for 2D quantum spin systems. [S0031-9007(99)08878-X]
PACS numbers: 76.60. – k, 75.40.Cx

B31 ions, providing pathways for weak interdimer interaction. Each Cu21 ion has four second-nearest-neighbor(2NN) Cu21 ions (5.132 Å). In other words, each Cu
dimer is surrounded by six 2NN Cu21 ions.
The SrCu2 sBO3 d2 sample was prepared by a solid state
reaction method from SrsNO3 d2 , CuO, and B2 O3 with
99.99% purities. Powders were ground, followed by theheat treatment at 850 ±C for 2 weeks with intermediate
gridings. The powder x-ray diffracting pattern indicateda single-phase product with no impurity. The magnetic
susceptibility was measured using a superconducting quantum interference device magnetometer in the temperature
T range from 1.7 to 400 K in an applied magnetic field
H of 1.0 T. The Cu NQR spectra were measured with ahomemade phase-incoherent-type spectrometer. The frequency was scanned from 16.0 to 27.0 MHz with approximate intervals of about 0.1 MHz. The spin-lattice
relaxation rate 1yT1 at 63 Cuy65 Cu nuclear was measured

Since the so-called pseudo spin gap has been suggested
to have intimate relevance to the appearance of high-Tc
superconductivity, a considerable number of studies havebeen made on low-dimensional quantum spin systems
with a spin-singlet ground state over the past decade [1].
Experimentally, however, they are limited to quasi-one1
dimensional (quasi-1D) cases such as SrCu2 O3 (S ­ 2
ladder system) [2], Y2 BaNiO5 (S ­ 1 Haldane system)
1[3], and CusNO3 d2 2.5H2 O (S ­ 2 alternating chain) [4],except for a quasi-two-dimensional (quasi-2D) system
CaV4 O9 [5], for which a model based on the plaquette
singlets is considered to explain the origin of the spin
gap [6]. For this reason, it is important to discover otherexamples of 2D spin systems having a finite spin gap toan excited state.
In this Letter, we investigated a 2D quantum spin system SrCu2 sBO3 d2 by means of magnetic susceptibility,
nuclear quadrupole resonance (NQR), and high-field magnetization, and have found that SrCu2 sBO3 d2 is a new
spin-gap system with an exact ground state. Moreover,in the magnetization curves at 1.7 and 0.5 K, we have1
1
succeeded in observing 4 and 8 plateaus of the saturation
magnetization, which is the first observation of the quantized magnetizations in 2D spin systems.
SrCu2 sBO3 d2 has a tetragonal unit cell with the cell constants of a ­ 8.995 Å and c ­ 6.649 Å at room tempera1
tures [7]. All Cu21 ions with a localized spin S ­ 2 arelocated at crystallographically equivalent sites. The structure is characterized by the layers of interconnected rectangular planar CuO4 and triangular planar BO3 groups
as shown in Fig. 1(a). These layers extend parallel tothe c axis, and are structurally separated from each otherby planes composed of Sr21 ions. A unique 2D magnetic linkage of the Cu21 spins is formed as illustrated inFig. 1(b): The first-nearest-neighbor (1 NN) Cu pairs (thedistance of 2.905 Å) share an edge to form dimeric units,
which are connected orthogonally with each other throughFIG. 1. (a) Schematic view of the crystal structure of
SrCu2 sBO3 d2 along f001g. The closed circles, small open
circles, and large open circles denote, respectively, the Cu21 ,B31 , and O22 ions. The unit cell is represented by dottedlines. ( b) 2D coordinates of the Cu21 spins. The 1NN and
the 2NN Cu pairs are denoted by the solid and broken lines,
respectively.

3168© 1999 The American Physical Society

0031-9007y99y82(15)y3168(4)$15.00

VOLUME 82, NUMBER 15

PHYSICAL REVIEW LETTERS

by the saturation recovery and inversion recovery methods
on the center line of the quadrupole spectrum. The magnetization measurement was carried out up to 45 T at 4.2, 1.7,
and 0.5 K using an induction method with a wire-woundpulse magnet whose duration time is about 20 ms.
We show in Fig. 2 the T dependence of the magnetic
susceptibility xraw measured at 1.0 T. A prominent characteristic is that xraw shows a maximum at around 20 K
and it rapidly drops toward zero with reducing temperature, suggesting the existence of an energy gap in the spin
excitation spectrum. xraw was fitted well to the CurieWeiss law over the T range of 160 to 400 K with the
Weiss temperature of u ­ 292.5 K and the effective gfactor, g ­ 2.14, accompanied with a constant susceptibility x0 ­ 22.01 3 1025 emuymol Cu. As seen in theinset of Fig. 2, a small Curie-Weiss-like upturn is seenin the xraw curve below 4 K, which would appear due
to magnetic impurities and/or defects of Cu21 ions in
SrCu2 sBO3 d2 . In order to estimate the Curie-Weiss term
at low temperatures, the data below 3.5 K were fitted to
C 0 ysT 2 u 0 d. This gave u 0 ­ 22.5 K and C 0 ­ 2.7 31023 emu Kymol Cu, corresponding to 0.72% of nearly1
free S ­ 2 impurities. Spin susceptibility xspin was finally evaluated after subtracting C 0 ysT 2 u 0 d 1 x0 fromxraw . By fitting xspin at a low temperature range withxspin ~ exps2Ds yTd, we roughly evaluated the gap Ds to
be 19 6 1 K. The fitting curve is drawn in the inset of
Fig. 2.

FIG. 2. Temperature dependence of the magnetic susceptibility in SrCu2 sBO3 d2 powder. The open and closed circles represent, respectively, the measured susceptibility xraw , and spin
susceptibility xspin after subtracting the Curie-Weiss and constant terms from xraw . The solid and broken lines show thetheoretical curves based on a dimer model. The enlarged plotis shown in the inset, where the solid curve indicates the fit to
xspin ~ exps2DS yT d.

12 APRIL 1999

A microscopic investigation by means of Cu NQR
has confirmed the existence of the spin-singlet groundstate with the finite energy gap in SrCu2 sBO3 d2 . Typical spin-echo spectra of 63 Cuy65 Cu NQR lines measured
at constant t (t being the time separation between the
exciting and refocusing rf pulses) of 80 ms are shownin the inset of Fig. 3 obtained at T ­ 4.3 and 3.1 K.
The solid curves are the best calculated profile with theGaussian distribution of the electric field gradient with a
FWHM of 400 kHz (4.3 K) and 250 kMz (3.1 K). The
ratio of two resonance frequencies measured at 4.3 K
is found to be exactly the same as that of the nuclear
quadrupole moment of Cu (63 Qy65 Q ­ 1.081). Accordingly, we can conclude that these lines are classified to63
Cuy65 Cu NQR lines. The quadrupole frequency yQ for
4.3 K can be obtained to be 23.01 MHz for 63 Cu (higher
resonance branch). On the contrary, when the temperature is lowered (see the spectrum at 3.1 K), the NQR
line signal splits into two components with yQ ­ 22.85
and 23.25 MHz for 63 Cu. This split is attributable tothe nuclear spin-spin coupling in the dimer, which was
also observed for the dimer chain site in Sr14 Cu24 O41[8]. This will be treated in full detail elsewhere aswell as spin-echo oscillating behavior observed also inthis compound.
The T dependence of 1yT1 is presented in Fig. 3. The
recovery curves were fitted to the single exponential terms,
and comparison of the recovery curves for 63 Cu and 65 Cu
revealed that the relaxation process is dominantly magnetic

FIG. 3. Temperature dependence of 1yT1 measured by saturation recovery method at Kyoto University (open) and by
inversion recovery method at University of Illinois at UrbanaChampaign (closed). The solid line shows the activated T
dependence with DR , 30 K. In the inset is shown the Cu
NQR spectra measured at 4.3 and 3.1 K.

3169

VOLUME 82, NUMBER 15

PHYSICAL REVIEW LETTERS[63 T1 y65 T1 ­ 1.112 for 3.1 K is close to s65 gy63 gd2 ­1.1475]. Then, we fitted the data to the activated T dependence, 1yT1 ~ exps2DR yT d (DR : the dynamical spingap), and obtained DR ­ 30 K a little larger than DS ­19 K estimated from the magnetic susceptibility.
Now that we are sure that SrCu2 sBO3 d2 has the finite
energy gap between the ground and exited states, the next
step is to clarify the origin of the spin gap. From theviewpoint of the crystal structure, or rather the arrangement of magnetic Cu21 ions, one may intuitively expectthat the magnetic properties of the present material would
1
be just those of the 1NN Cu dimers. An S ­ 2 dimer
model [9] is simplest among spin-gap systems and widely
known to be applicable to many materials: CsV2 O5 [10],BaCuSi2 O6 [11], and various Cu complexes [12]. Considering the dimer bridging angle (/Cu-O-Cu) of 102.42±, itis reasonable to suppose that the intradimer exchange interaction (symbolized by J) is antiferromagnetic; J , 0.
(In general, there is an inverse correlation between thevalue of J and the angle at the bridging oxygen atom, andJ changes its sign near 97.6± [13].)
1
Our xspin data were first analyzed using an S ­ 2isolated antiferromagnetic (AF) dimer model [9], where
the spin susceptibility xd is expressed as∑∏21
1
Ng2 mB2
11.
(1)
exps22JykB T d
xd ­
3kB T
3
Here, N, mB , and kB are Avogadro number, the Bohrmagneton, and Boltzmann constant, respectively. Notethat J is a unique fitting parameter. Although we attempted to fit the xspin data in such a manner that thetemperature at maximum xd coincides with that of xspins­ 18.5 Kd, Fig. 2 clearly shows that the result was far
from satisfaction: Compared with the theoretical curvewith JykB ­ 215.0 K (solid line), the peak of xspin issignificantly suppressed. In order to resolve the great discrepancy between xspin and xd , an AF 2NN interaction
J 0 s,0d was introduced. We calculated susceptibility bya mean-field approximation given in Ref. [14]. Nevertheless, it hardly improved the fit as seen in the curve assuming J 0 ykB ­ 25 K (broken line). To obtain the best fit,
one has to assume an unreasonably large negative value ofJ 0 ykB ­ 250 K apparently beyond the validity of a meanfield approximation sjJ 0 j ø jJjd. It should be pointed out
that, if the 1NN interaction is antiferromagnetic sJ , 0d,
the 2NN Cu spins feel a spin frustration highly relevant
to a resonating-valence-bond state [15]. The effect of thespin frustration is actually seen in the fairly small spin gap
in contrast with the large Weiss temperature.
Magnetization measurement is a useful tool to clarify
the nature of a spin-singlet ground state since it can
provide crucial information on a magnetic excited state.
Figure 4 shows the magnetization M of SrCu2 sBO3 d2
as a function of H, where no hysteresis was observed
between the field increasing and decreasing processes. As
expected, the continuous transition from the singlet groundstate to the gapless magnetic state occurs at around 20 T
3170

12 APRIL 1999

FIG. 4. Magnetization curves of SrCu2 sBO3 d2 obtained
at 4.2 and 1.7 K (top); and 0.5 K ( bottom). The broken
11lines correspond to 4 and 8 of the full magnetization
Zero(­ 1y2NgmB emuymol Cu with g ­ 2.14). Inset:
temperature magnetization curve for the isolated dimer modelwith a critical field of 20 T.

corresponding to the gap of 30 K in agreement with the
T121 result. With decreasing temperature, this transition
becomes more distinct. For comparison, we show thetheoretical magnetization at zero temperature based on theisolated dimer model with a critical field of 20 T [4,14],
again in disagreement with the experimental data.
Our experimental study has triggered a theoreticalsearch for this 2D spin system. Recently, Miyahara andUeda showed that SrCu2 sBO3 d2 is close to the criticalpoint between the spin-singlet state and the Néel orderedstate [16]. The unusual behavior of the susceptibility
is a consequence of the closeness to the criticality, andtheir theoretical calculations reproduced the x-T curvequite satisfactorily. Furthermore, SrCu2 sBO3 d2 having
the 2D orthogonal dimer network [17] has been found to
be a clean system with the “exact” dimer ground state.
SrCu2 sBO3 d2 is the first example where this exact groundstate is achieved.
The most important finding in the present high-field1
study is that the two plateaus corresponding to 4 and1
8 of the saturation moment appear in the magnetizations at 1.7 and 0.5 K. The critical fields are evaluated by extrapolating the M vs H curves in the slope
1
1
regions to the zero, 8 and 4 plateau lines in Fig. 4.
As a result, Hc1 ­ 20.9 T, Hc2 ­ 27.9 T, Hc3 ­ 29.8 T,
Hc4 ­ 37.0 T, and Hc5 ­ 41.0 T are obtained. With
increasing H, we have the gapped and gapless groundstates by turns: In the gapped regions H , Hc1 , Hc2 ,H , Hc3 , and Hc4 , H , Hc5 , SrCu2 sBO3 d2 has theVOLUME 82, NUMBER 15

PHYSICAL REVIEW LETTERS

finite energy gap between the ground and the lowest
exited states, and, in the gapless regions Hc1 , H , Hc2 ,
Hc3 , H , Hc4 , and Hc5 , H, the system has no excitation gap and the magnetization increases continuously.
The observed plateaus are not flat although the data at
0.5 K have clearer plateaus than in the case of 1.7 K. Thisis partly because of thermal fluctuation and also becauseof the fact that the measurement using powder sample
averages the field-direction-dependent critical fields dueto the anisotropy of the g factor.
Theoretically, several 1D quantum spin models havebeen shown to have magnetization plateaus [18–20]. For
11
example, the 2 and the 3 plateaus appear, respectively,
in an S ­ 1 alternating AF Heisenberg chain and in an1
S ­ 2 AF Heisenberg chain with period-3 exchange coupling. Recently, these quantization conditions have been
generalized by Oshikawa, Yamanaka, and Affleck [21].
They investigated the general Heisenberg spin chains atzero temperature in the presence of H and showed thatthe magnetization per site m is topologically quantized as
nsS 2 md ­ integer, where n is the period of the groundstate, and S is the magnitude of spin. From the experimental point of view, the quantized plateaus have been indeed
found in the magnetization curves of 1D compounds
1
fNi2 smedptd2 sm-oxd sm-N3 dgClO4 ? 0.5H2 O ( 2 plateau)13[22] and NH4 CuCl3 ( 4 and 4 plateaus) [23], both of which
are considered to satisfy the quantization conditions.
In contrast to the 1D spin systems mentioned above,
the problem of the quantized magnetization plateaus
in 2D spin systems has not been addressed yet. It
should be stressed that SrCu2 sBO3 d2 is the first example
having quantized magnetization plateaus in the 2D spin
systems. The recent theory by Miyahara and Ueda
suggests that triplet excitations from the ground state are11
almost localized, which accounts for the 4 and 8 plateaus
in the 2D orthogonal dimer lattice [16]. Moreover, other1 1 1plateaus at 2 , 10 , 16 , . . . have been predicted.
In summary, we have investigated both the static andthe dynamical magnetic properties of SrCu2 sBO3 d2 bymeans of magnetic susceptibility, Cu NQR, and highfield magnetization measurements, and have found theexistence of the spin-singlet ground state with the energy
gap of 30 K. It is concluded that SrCu2 sBO3 d2 is a new
spin-gap system with the exact dimer ground state [17].
Furthermore, the quantized magnetization plateaus have11been observed at 4 and 8 of the full magnetization, whichis the first case in the 2D quantum spin systems. The
recent theoretical investigation has succeeded to explainthese interesting magnetic behaviors of SrCu2 sBO3 d2 [16].
Investigations using a single crystal are necessary in orderto understand fully the nature of the spin-singlet groundstate and to observe the predicted magnetization plateaus.
It is also interesting to examine the effect of hole/electron
doping, which may induce superconductivity.
We are grateful to Professor A. N. Vasil’ev andProfessor M. Takigawa for helpful discussions. The

12 APRIL 1999

work in Urbana was supported by the Science and Technology Center for Superconductivity under NSF Grant
No. DMR91-20000 and the U.S. DOE Division of Materials Research under Grant No. DEFG02-91ER45439.

*Present address: Institute for Solid State Physics,
University of Tokyo, Tokyo, Japan.
†
Permanent address: Department of Chemistry, Graduate
School of Science, Kyoto University, Kyoto, Japan.
[1] See, for example, J. R. Schrieffer, X. G. Wen, andS. C. Zhang, Phys. Rev. B 39, 11 663 (1989); E. Dagotto,
J. Riera, and D. Scalapino, Phys. Rev. B 45, 5744 (1992);
T. Tanimoto, H. Kohno, and H. Fukuyama, J. Phys. Soc.
Jpn. 63, 2739 (1994); M. Troyer, H. Tsunetsugu, andD. Würts, Phys. Rev. B 50, 13 515 (1994).
[2] M. Azuma, Z. Hiroi, M. Takano, K. Ishida, andY. Kitaoka, Phys. Rev. Lett. 73, 2626 (1994).
[3] J. Darriet and J. P. Regnault, Solid State Commun. 86, 409(1993).
[4] J. C. Bonner, S. A. Friedberg, H. Kobayashi, and
D. L. Meier, Phys. Rev. B 27, 248 (1983).
[5] S. Taniguchi, T. Nishikawa, Y. Yasui, Y. Kobayashi,
M. Sato, T. Nishioka, M. Kontani, and K. Sano, J. Phys.
Soc. Jpn. 64, 2758 (1995).
[6] N. Kato and M. Imada, J. Phys. Soc. Jpn 64, 4105 (1995).
[7] R. W. Smith and D. A. Keszler, J. Solid State Chem. 93,
430 (1991).
[8] M. Takigawa, N. Motoyama, H. Eisaki, and S. Uchida,Phys. Rev. B 57, 1124 (1998).
[9] R. L. Carlin, Magnetochemistry (Springer-Verlag, Berlin,1986), Chap. 5, p. 75.
[10] M. Isobe and Y. Ueda, J. Phys. Soc. Jpn. 65, 3142 (1996).
[11] Y. Sasago, K. Uchinokura, A. Zheludev, and G. Shirane,
Phys. Rev. B 55, 8357 (1997).
[12] D. J. Hodgson, Prog. Inorg. Chem. 19, 173 (1975).
[13] V. H. Crawford, H. W. Richardson, J. R. Wasson,
D. J. Hodgson, and W. E. Hatfield, Inorg. Chem. 15, 2107
(1976).
[14] Y. Sassago, M. Hase, K. Uchinokura, M. Tokunaga, and
N. Miura, Phys. Rev. B 52, 3533 (1995).
[15] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973).
[16] S. Miyahara and K. Ueda, cond-mat / 9807075.
[17] Theoretically, the 2D spin network of SrCu2 sBO3 d2
is equivalent to that for which B. S. Shastry andB. Sutherland solved the ground state exactly [ Physica(Amsterdam) 108B, 1069 (1981)].
[18] T. Tonegawa, T. Nakao, and M. Kaburagi, J. Phys. Soc.
Jpn. 65, 3317 (1996).
[19] K. Hida, J. Phys. Soc. Jpn. 63, 2359 (1994).
[20] K. Okamoto, Solid State Commun. 98, 245 (1996).
[21] M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev.
Lett. 78, 1984 (1997).
[22] Y. Narumi, M. Hagiwara, R. Sato, K. Kindo, H. Nakano,and M. Takahashi, Physica (Amsterdam) 246B – 247B,
509 (1998). medpt ­ methyl-biss3-aminopropyldamine.
[23] W. Shiramura, K. Takatsu, B. Kurniawan, H. Tanaka,
H. Uekusa, Y. Ohashi, K. Takizawa, H. Mitamura, and
T. Goto, J. Phys. Soc. Jpn. 67, 1548 (1998).

3171