Journal of the Physical Society of Japan
Vol. 76, No. 9, September, 2007, 093706#2007 The Physical Society of JapanLETTERS

Bose–Einstein Condensation of Quasi-Two-Dimensional
Frustrated Quantum Magnet (CuCl)LaNb2 O7
Atsushi K ITADA1 , Zenji H IROI2 , Yoshihiro T SUJIMOTO1 , Taro K ITANO1 ,Hiroshi K AGEYAMA1 y, Yoshitami A JIRO1;3 , and Kazuyoshi YOSHIMURA1
1

Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502
2
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581
3
CREST, Japan Science and Technology Agency (JST)(Received June 15, 2007; accepted July 25, 2007; published September 10, 2007)

A quasi-two-dimensional S ¼ 1=2 Heisenberg square-lattice antiferromagnet (CuCl)LaNb2 O7 isstudied by speciﬁc heat and magnetic susceptibility measurements in external ﬁelds up to 14 T. The
experimental results in low ﬁelds verify the absence of the order–disorder transition, as reported
previously. By further application of magnetic ﬁelds, we obtain direct evidence for the phase transition,
which can be interpreted as the Bose–Einstein condensation (BEC) of magnons. However, the criticalﬁeld of 10 T signiﬁcantly deviates from that estimated from the zero-ﬁeld gap, indicating that, unlike
known experimental examples, the one-magnon mode is not a primary source to drive the BEC.
KEYWORDS: (CuCl)LaNb2 O7 , spin gap, square lattice, ion exchange, frustration, Bose–Einstein condensation
DOI: 10.1143/JPSJ.76.093706

Field-induced transitions (FITs) occur in Heisenberg
antiferromagnetic spin systems, which, for various reasons,have a collective singlet ground state separated by an energy
gap  from the ﬁrst excited one-magnon (triplet) states. The
application of magnetic ﬁelds decreases the separation
between the ground state and the lowest branch (Mz ¼ 1)of the triplet states and ﬁnally leads to a quantum phasetransition at the critical ﬁeld Hc (corresponding to  ¼
gB Hc ). The ordered phase is described as a Bose–Einsteincondensation (BEC) of hard-core bosons for which magnetic
ﬁelds act as the chemical potential.1) This concept is
exclusively applicable to all known compounds including
TlCuCl3 ,1)
KCuCl3 ,2)Pb2 V3 O9 ,3)
PbNi2 V2 O8 ,4)
5)
6)
BaCuSi2 O6 , and NiCl2 –4SC(NH2 )2 . Theoretically, a
new route to the BEC has recently been proposed in thesquare-lattice frustrated model with modulating coupling
constants, for which the two-magnon bound states condense
above a certain magnetic ﬁeld.7)
Chimie douce, solid-state reactions at relatively low
temperatures, i.e., away from the thermodynamic equilibrium, oﬀer a rational design of the magnetic lattices innonmolecular solids.8) Recently, various exotic quantum
magnetic phenomena have emerged from a new classof S ¼ 1=2 layered transition-metal oxyhalides (CuX)[An1 Bn O3nþ1 ] (where X = Cl , Br ; A = La3þ , Ca2þ ,Naþ , . . .; B = Nb5þ , Ta5þ , Ti4þ , . . .; n ¼ 2, 3), obtained by
low-temperature ion-exchange reactions.9–12) They range
from collective singletground states in (CuCl)LaNb2 O7 ,13,14)a collinear order at 32 K with a reduced magnetic moment in(CuBr)LaNb2 O7 ,15) to successive phase transitions and 1/3
magnetization plateaus in (CuBr)Sr2 Nb3 O10 .16) Common tothis family, the CuX layer with an S ¼ 1=2 square lattice isisolated by nonmagnetic perovskite slabs, making it a quasitwo-dimensional magnet. Within the layer, the halogen atom
locates itself at the center of a copper square, so that thecompetition between ﬁrst-nearest-neighbor (1NN) and second-nearest-neighbor (2NN) exchange couplings, denotedE-mail: kitada@kuchem.kyoto-u.ac.jp
E-mail: kage@kuchem.kyoto-u.ac.jp

yhereafter as J1 and J2 , respectively, is expected. Thissituation diﬀers markedly from the case of perovskite oxides
ABO3 with magnetic B sites that typically experience lessfrustration because the intervening superexchange anions
are located at the edges of the square (or cube), resulting in
dominating J1 bonds.
From our previous studies, it is now apparent that
(CuCl)LaNb2 O7 has a spin-singlet ground state with anenergy gap, but there remain questions to be answered. Of
particular interest is the unusual behavior under magnetic
ﬁelds. Whereas the magnetic susceptibility at 0.1 T andinelastic neutron scattering (INS) experiments at 0 T haveconsistently yielded the zero-ﬁeld gap =kB ¼ 26:7 K(corresponding to Hc ¼ 18:4 T),13) the magnetization measurements at 1.3 K using pulsed ﬁelds have revealed the onsetof magnetization at as low as 10.3 T.14) However, a problem
is that the magnetization increases slowly even in a ﬁeld
region below 10.3 T, most likely due to defects and/or
unidentiﬁed magnetic impurities; the Curietail is also
detected in the magnetic susceptibility. Therefore, it isunfortunately uncertain whether the anomaly at 10.3 T canbe truly attributed to the FIT or not. It may be a more
complicated crossover behavior or just an extrinsic eﬀectcaused by the impurities. In the present study, we have
performed measurements of the temperature variations ofspeciﬁc heat and magnetization at static magnetic ﬁelds up
to 14 T. Our data not only indicate the absence of order–
disorder transitions in the presence of a ﬁnite energy gap forlow ﬁelds but also provide evidence that (CuCl)LaNb2 O7
undergoes the FIT described by the magnon BEC, which
is probably of a new type.
The (CuCl)LaNb2 O7 sample was synthesized via an ionexchange route, as described in detail in ref. 13. To avoid
the complexity arising from the dependence of the sample on
the magnetic properties, the same sample used in previous
measurements13,14) was employed. Speciﬁc heat measurements were performed by the heatrelaxation method
for T > 2 K and in magnetic ﬁelds up to H ¼ 14 T in a
QD-PPMS system. Further, 4.43 mg of (CuCl)LaNb2 O7
powder was mixed with 2.34 mg of ﬁne gold powder in order093706-1

LETTERS

J. Phys. Soc. Jpn., Vol. 76, No. 9

A. KITADA et al.

0.9(CuCl)LaNb2O7

0.8
C (J/K mol)

2

0.7

2

C/ T (J/K mol)

0.6

0T
2T
4T
7T

1

0.5
0
2

0.4

4 T (K)

6

0.3
0.2

10 T
11 T
12 T
13 T
14 T
phonon0.1

Fig. 2. Logarithmic plot of CT 2 as a function of 1=T for H ¼ 0 T.
The solid line represents the ﬁt to eq. 1.

0.0
0

5

10

T (K)

15

20

25

Fig. 1. Speciﬁc heat divided by temperature (C=T) for (CuCl)LaNb2 O7collected with ﬁelds of H ¼ 0, 2, 4, 7, 10, 11, 12, 13, and 14 T. The dottedcurve represents the phonon contribution assumed as CL  T 3
[ ¼ 0:717 mJ/(K4 mol)]. The inset shows the C vs T plots in the lowtemperature region, where arrows denote the transition temperatures.

to improve thermal contact and pressed into a pellet. The
pellet was attached to an alumina platform with a small
amount of Apiezon N grease. The heat capacity was
obtained by ﬁtting a heat relaxation curve recorded after aheat pulse caused a temperature increase of approximately
2%. The heat capacity of the alumina platform and mixed
gold powder was measured in separate runs without the
sample and was subtracted from the data. The temperaturedependence of magnetization was measured between 2 and20 K in ﬁxed ﬁelds up to H ¼ 14 T using the QD-PPMS
system.
The total speciﬁc heat divided by the temperature (C=T)
measured in the absence of a magnetic ﬁeld is plottedby black circles in Fig. 1. The data display no trace of
anomalies associated with the phase transitions. Instead,
we observe the Schottky anomaly, i.e., a round maximum
centered at around Tmax  7 K. This is a typical behavior ofspin-singlet compounds with a ﬁnite gap to the lowest
excited state and is consistent with the magnetic susceptibility and neutron scattering experiments.13) It is notstraightforward to extract a magnetic contribution CM fromthe measured data because a nonmagnetic isomorph is notpresently available from which the lattice contribution CL 
T 3 in a low-temperature regime could be estimated.
Therefore, a rough estimation is made and the result is
indicated by the dotted line in Fig. 1, where  is assumed as
0.717 mJ/(K4 mol). The magnetic contribution is subsequently determined by subtracting the CL term from C. By
integrating CM =T between 2 and 30 K, we obtain 1.1 J/(mol K). This value is approximately 13% smaller than the
total magnetic entropy (R ln 2) for 1 mol of S ¼ 1=2 spins,
but is acceptable considering the crude assumption of thelattice contribution.
For a spin-gap compound in which the spin-singlet groundstate is separated simply from the ﬁrst excited one-magnon

states by a gap , the speciﬁc heat in the low-temperature
limit is, in general, given by the following expression:17)
2
C / T exp 
:
ð1Þ
kB T
Thus, CT 2 is plotted against 1=T on the logarithmic scale, as
shown in Fig. 2. The ﬁt with =kB ¼ 19 K is also shown.
This value is slightly smaller than that estimated by themagnetic susceptibility and neutron scattering experiment,
=kB ¼ 26:5 K. Alternatively, it is also possible to estimate
the gap using the relation kB Tmax =  0:28 in the isolated
dimer limit.17) Thus, the obtained value =kB ¼ 25 K iscloser to the values determined by magnetic susceptibility
and neutron scattering studies. Similar values of the gap
have been recently obtained by NMR.18) It is noteworthy thata deviation from the temperature-activated behavior appears
below 3 K (Fig. 2), thereby leading to the ambiguity in theevaluation of the gap size because eq. (1) is more valid atlower temperatures. In fact, such a deviation is sometimes
observed for spin-gapped materials19,20) and is ascribed tothe phonon contribution or extrinsic eﬀect involving defects
and unidentiﬁed impurities. Recent NMR experiments also
show a deviation from the temperature activation curve at
low temperatures, which are assigned as the extrinsic artifact
factors from some nonmagnetic impurities.
As soon as the magnetic ﬁelds are applied (up to 7 T), thebroad maximum of C=T shifts gradually to a lower temperature, indicating not only the absence of order-disordertransitions but also a reduction in the energy gap ðHÞ withthe ﬁeld. A deviation from the temperature activation
behavior is again appreciable below 3 K (not shown). The
concave temperature dependence of the speciﬁc heat at low
temperatures below 7 T reﬂects the spin-singlet nature of theground state. However, it changes to a linear behavior at
10 T. Finally, we observe the most spectacular feature of thepresent study: a cusp-like peak in CðTÞ at and above 11 T,
which is the ﬁrst evidence of a ﬁeld-induced phase transitionin this compound. The transition temperature TN at 11 T is
2.3 K, as indicated by the arrow in the inset of Fig. 1. Whenthe magnetic ﬁeld is further increased, the peak associatedwith the phase transition develops and shifts systematicallyto higher temperatures, and TN becomes 3.3 K at 14 T. The

093706-2

LETTERS

J. Phys. Soc. Jpn., Vol. 76, No. 9

A. KITADA et al.

2.0

5
14 T

M (T)
C (T)
M (H)
φ = 3/2

4

13 T

T (K)

12 T

1.5

10 T

3
2

8T
1.0

1

3

M (x10 emu/mol)

11 T(CuCl)LaNb2O7

6T

0

4

6

8

10

12

14

16

H (T)

0.5

Fig. 4. Phase boundary in (CuCl)LaNb2 O7 determined from the resultsof MðTÞ (triangles), CðTÞ (solid circles). The open circle represents thecritical ﬁeld determined by the magnetization curve at 1.3 K.14) The solidline represents a theoretical curve described in the text.

(CuCl)LaNb2O7
0.0
0

5

10

15

20

T (K)
Fig. 3. Low-temperature magnetizations of (CuCl)LaNb2 O7 measured
at various external magnetic ﬁelds up to 14 T.

FITs observed here are nearly consistent with the abrupt
increase in the magnetization at 10.3 T.14) Therefore, it isnatural to consider that the FIT should be of magnetic origin
associated with the BEC of magnons.
In order to validate this picture, the temperature dependence of magnetization was measured at various constantmagnetic ﬁelds shown in Fig. 3. While the MðTÞ curves at 6,8, and 10 T are featureless, those above 10 T signal magnetic
phase transitions as cusp-like anomalies. In Fig. 4, the ﬁeld
dependence of TN is summarized in a temperature vsmagnetic ﬁeld phase diagram. It is seen that TN ’s determinedby CðTÞ and MðTÞ are in good agreement with each other.
Furthermore, both the increase in magnetization for decreasing temperature below TN and the growth of the peak of thespeciﬁc heat around TN with increasing magnetic ﬁeld canbe attributed to the increased magnon density on the basisof the BEC scenario. From these results, we conclude that
(CuCl)LaNb2 O7 is a new experimental correspondence of
the BEC of magnons.
Figure 4 shows a theoretical curve of the Hartree–Fockapproximation that gives the power law behavior ½Hc ðTÞ 
Hc ð0Þ / T  with the critical exponent  ¼ 3=2. Here, we
assume Hc ð0Þ ¼ 9:0 T. The result appears to follow thetheory. We notice, however, that the ﬁtted results largely
depend on the choice of Hc ð0Þ. Accurate determinationof  is also hampered by rather broadened anomalies at TN ,probably due to the use of the polycrystalline specimen.
Thus, quantitative discussions on the criticality require
future experimental investigation with single crystals and atlower temperatures.
The most remarkable implication of our ﬁndings is that
(CuCl)LaNb2 O7 undergoes the magnon BEC at a surprisingly small magnetic ﬁeld around 10 T, far below the criticalﬁeld estimated from the zero-ﬁeld gap. This markedly
contrasts with other known experimental examples, inwhich the occurrence of the BEC is directly linked to thelevel crossing between the spin-singlet ground state andthe low-lying excited one-magnon state.1–6) That is, for(CuCl)LaNb2 O7 , the one-magnon states are at least notdirectly responsible for the BEC. One possibility is that
two-magnon bound states reach the ground state at 10 T anddrive the transition into a gapless spin-liquid state withouttransverse magnetic moment (spin-nematic state), as proposed theoretically for a quadrumerized two-dimensionalsquare-lattice antiferromagnet by Ueda and Totsuka.7)
Here, a key requirement for the spin-nematic order is thecompetition between ferromagnetic and antiferromagnetic
interactions. Because the Weiss temperature (9:6 K) has amuch smaller magnitude than the temperature at maximum
susceptibility (16.5 K), (CuCl)LaNb2 O7 meets this requirement and thus may be the ﬁrst experimental correspondenceof this model. Historically, the BEC scenario of the twomagnon bound states was for the ﬁrst time considered for the
Shastry–Sutherland model with antiferromagnetic intra- andinterdimer interactions.21) However, unfortunately, the additional Dzyaloshinsky–Moriya interaction in the experimental
counterpart SrCu2 (BO3 )2 causes bending of these modes inthe energy vs ﬁeld diagram and eventually drives the spinsinglet phase to the 1/8 plateau phase.22) In the case of(CuCl)LaNb2 O7 , the magnon binding may be more favoredbecause of the presence of ferromagnetic interactions.
Whether this scenario or some other explains the BEC in
this compound is an interesting issue that will be analyzed
in the future.
In conclusion, we have measured the temperature
variations of the speciﬁc heat and magnetization for(CuCl)LaNb2 O7 . The results provide evidence that the
BEC occurs at as low as 10 T. On the basis of a comparison
with previous zero- and low-ﬁeld experiments, we strongly
believe that the one-magnons do not play the main role inthe ﬁeld-induced transition, thus providing a new mechanism that leads to the BEC. It seems that frustrated quantum
spin systems with antiferromagnetic and ferromagnetic
interactions have hardly received adequate attention, with
only limited theoretical studies.7,23) Therefore, the present

093706-3

J. Phys. Soc. Jpn., Vol. 76, No. 9

LETTERS

compound together with its family15,16) would provide
suitable opportunities to investigate novel quantum phenomena including the BEC for systems with mixed interactions.
The authors thank Y. Baba, N. Oba, H. Ueda, and K.
Totsuka for fruitful discussions. This study has been
supported by Grants-in-Aid for Young Scientists (A) (No.
17684018) and Science Research on Priority Area (Nos.
19052004 and 16076210) from the Ministry of Education,
Culture, Sports, Science and Technology of Japan.
1) T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka: Phys. Rev. Lett.
84 (2000) 5868.
2) A. Oosawa, T. Takamatsu, K. Tatani, H. Abe, N. Tsujii, O. Suzuki,
H. Tanaka, G. Kido, and K. Kindo: Phys. Rev. B 66 (2002) 104405.
3) T. Waki, Y. Morimoto, C. Michioka, M. Kato, H. Kageyama, K.
Yoshimura, S. Nakatsuji, O. Sakai, Y. Maeno, H. Mitamura, and
T. Goto: J. Phys. Soc. Jpn. 73 (2004) 3435.
4) N. Tsujii, O. Suzuki, H. Suzuki, H. Kitazawa, and G. Kido: Phys. Rev.
B 72 (2005) 104402.
5) S. E. Sebastian, N. Harrison, C. D. Batista, L. Balicas, M. Jaime, P. A.
Sharma, N. Kawashima, and I. R. Fisher: Nature 441 (2006) 617.
6) V. S. Zapf, D. Zocco, B. R. Hansen, M. Jaime, N. Harrison,
C. D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, and A.
Paduan-Filho: Phys. Rev. Lett. 96 (2006) 077204.
7) H. Ueda and K. Totsuka: private communication.
8) A. Stein, S. W. Keller, and T. E. Mallouk: Science 259 (1993) 1558.
9) T. A. Kodenkandath, J. N. Lalena, W. L. Zhou, E. E. Carpenter,
C. Sangregorio, A. U. Falster, W. B. Simmons, Jr., C. J. O’Connor,
and J. B. Wiley: J. Am. Chem. Soc. 121 (1999) 10743.

A. KITADA et al.

10) T. A. Kodenkandath, A. S. Kumbhar, W. L. Zhou, and J. B. Wiley:
Inorg. Chem. 40 (2001) 710.
11) H. Kageyama, L. Viciu, G. Caruntu, Y. Ueda, and J. B. Wiley:
J. Phys.: Condens. Matter 16 (2004) S585.
12) H. Kageyama, T. Kitano, R. Nakanishi, J. Yasuda, N. Oba, Y. Baba,
M. Nishi, Y. Ueda, Y. Ajiro, and K. Yoshimura: Prog. Theor. Phys.
Suppl. 159 (2005) 39.
13) H. Kageyama, T. Kitano, N. Oba, M. Nishi, S. Nagai, K. Hirota,
L. Viciu, J. B. Wiley, J. Yasuda, Y. Baba, Y. Ajiro, and K. Yoshimura:
J. Phys. Soc. Jpn. 74 (2005) 1702.
14) H. Kageyama, J. Yasuda, T. Kitano, K. Totsuka, Y. Narumi,
M. Hagiwara, K. Kindo, Y. Baba, N. Oba, Y. Ajiro, and K.
Yoshimura: J. Phys. Soc. Jpn. 74 (2005) 3155.
15) N. Oba, H. Kageyama, T. Kitano, J. Yasuda, Y. Baba, M. Nishi,
K. Hirota, Y. Narumi, M. Hagiwara, K. Kindo, T. Saito, Y. Ajiro, andK. Yoshimura: J. Phys. Soc. Jpn. 75 (2006) 113601.
16) Y. Tsujimoto, Y. Baba, N. Oba, H. Kageyama, T. Fukui, Y. Narumi,
K. Kindo, T. Saito, M. Takano, Y. Ajiro, and K. Yoshimura: J. Phys.
Soc. Jpn. 76 (2007) 063711.
17) R. L. Carlin: Magnetochemistry (Springer-Verlag, Berlin, 1986).
18) M. Yoshida, N. Ogata, M. Takigawa, J. Yamaura, N. Oba, T. Kitano,
H. Kageyama, Y. Ajiro, and K. Yoshimura: to be published in J. Phys.
Soc. Jpn. 76 (2007) No. 10; cond-mat/0706.3559.
19) H. Kageyama, K. Onizuka, Y. Ueda, M. Nohara, H. Suzuki, and
H. Takagi: J. Exp. Theor. Phys. 90 (2000) 129.
20) P. R. Hammar, D. H. Reich, C. Broholm, and F. Trouw: Phys. Rev. B
57 (1998) 7846.
21) T. Momoi and K. Totsuka: Phys. Rev. B 62 (2000) 15067.
22) H. Nojiri, H. Kageyama, Y. Ueda, and M. Motokawa: J. Phys. Soc.
Jpn. 72 (2003) 3243.
23) N. Shannon, T. Momoi, and P. Sindzingre: Phys. Rev. Lett. 96 (2006)
027213.

093706-4