Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 129–132.
From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 145–148.
Original English Text Copyright © 2000 by Kageyama, Onizuka, Ueda, Nohara, Suzuki, Takagi.

SOLIDS

Low-Temperature Specific Heat Study of SrCu2(BO3)2
with an Exactly Solvable Ground State¶
H. Kageyama*1, K. Onizuka1, Y. Ueda1, M. Nohara2,
H. Suzuki2, and H. Takagi2
1

Institute for Solid State Physics, University of Tokyo,
Tokyo 106–8666, Japan2
Graduate School of Frontier Science, University of Tokyo,Tokyo 113–8656, Japan*e-mail: kage@issp.u-tokyo.ac.jp
Received July 22, 1999

Abstract—The specific heat of a two-dimensional spin gap system SrCu2(BO3)2 realizing the Shastry–Sutherland model was measured between 1.3 and 25 K under various magnetic fields up to 12 T. The analysis basedon an isolated dimer model in a low temperature region revealed that the value of the spin gap at zero field is∆ = 34.4 K. It turned out that ∆ decreases in proportion to H due to the Zeeman splitting of the excited triplet
levels. This simplest model, however, fails to reproduce the result in a high-temperature region, suggestingrather strong spin–spin correlation of the system. © 2000 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION
Exactly solvable models have been extensively
studied in the area of strongly correlated electron systems for the purpose of elucidating various exotic physical phenomena because some rigorous results can bederived from them, sometimes providing us a crucial
key to solve underlying problems of the phenomena.
Such models, even if being far from realistic, canremain tantalizing theoretical subjects owing to thebeauty of the solutions. For example, Majumdar andGhosh first proved that an exact dimer ground state forz one-dimensional spin chain imposed a stringent condition on the first and second nearest neighbor interactions [1]. Stimulated by this discovery, a number of systems with the identical exact wave function have been
explored from the theoretical point of view for one-,
two,- and three-dimensions (see, for example [2] andreferences therein). However, in spite of extensive
efforts by chemists to tailor experimental examples, nomaterial had been discovered for a long time.
Recently, we reported the magnetic properties of an
inorganic compound SrCu2(BO3)2, which consists of a
two-dimensional orthogonal dimer lattice, concludingthat this material verifies the Shastry–Sutherlandmodel, which has the exact dimer ground state [3–5].
Although an imaginary lattice Shastry and Sutherland
considered—i.e., a two-dimensional square lattice withsome additional diagonal bonds—differs from the real
one of SrCu2(BO3)2, these two are equivalent from a
topological point view. The value of the spin gap wasestimated from various measurements like measure¶This article was submitted by the authors in English.

ments of the temperature variation of the magnetic susceptibility (34 K) [6] and electron spin resonance (ESR;34.7 K) [7]. It was also found that the spin system for
SrCu2(BO3)2 is fairly frustrated, located very close to thecritical point (J'/J)c = 0.70 between the exact dimer state
and the Néel-ordered state [3, 5]: the ratio of intradimer
and interdimer interactions, respectively, J = 100 K andJ ' = 68 K, is 0.68. Furthermore, several quantized plateaux
were observed in the magnetization [3, 6, 8], which originates from the extremely localized triplet excitations [5].
In the present paper, we performed the specific heatmeasurement of SrCu2(BO3)2 under magnetic fields H
in order to obtain more information on the exchange interactions as well as the effect of the spin-gapped behavior
upon H. The data were analyzed in terms of an isolated
dimer model, and the spin gap in the absence of the field
was evaluated to be 34.4 K. Furthermore, it was found thatapplication of magnetic fields causes the Zeeman splittingof the excited triplet states, leading to a H-linear decreasein the value of the spin gap.
2. EXPERIMENTThe specific heat measurement was performed by aheat-relaxation method [9] in a temperature range
between 1.3 and 25 K under magnetic fields between 0and 12 T. A bulk single crystal of SrCu2(BO3)2 wasused, which was grown by the traveling solvent floatingzone (TSFZ) method with an image furnace using a solvent, LiBO2 under flowing O2 gas ( P O2 = 1 atm,99.99%). For a detailed procedure of the crystal
growth, see [10]. A piece of the crystal with thedimensions of 2 × 2 × 1 mm was attached to a sapphire1063-7761/00/9001-0129$20.00 © 2000 MAIK “Nauka/Interperiodica”

KAGEYAMA et al.

130
C/T, mJ/(K2 mol)
1000
SrCu2(BO3)20T
6T
9T
12 T

800

600

400

200
Photon

05

10

15

20

25
T, K

Fig. 1. C/T versus T measured at H = 0 (d), 6 (s), 9 (m), and
12 T (n). Dotted curves are the calculations based on theisolated dimer model for ∆(0) = 34.4 K. Dot-dashed curve
represents the phonon term, βT3 (β = 0.460 mJ/K4 mol).

∆, K
40C/T 2, mJ × K/mol
108

30

107

20106

10105

051015 20H, T

104

102

SrCu2(BO3)2101

3. RESULTS AND DISCUSSION
A total specific heat divided by T, C/T, measured in
the absence of a magnetic field is plotted as a functionof T by closed circles in Fig. 1. With decreasing T from15 K, C/T rises, reaches a round maximum at 7.5 K, andthen falls rapidly, approaching naught. These behaviors, that is to say, the so-called Schottky anomalies are
typical of spin–singlet system with a finite spin gap toa lowest exited state. A gradual increase in C/T with Tabove 15 K comes from the phonon term, with is in
general known to vary as C ∝ βT3. As also shown inFig. 1, qualitatively similar features described above
appear even when magnetic fields are applied, indicating that the system still has a spin–gapped ground state
at least for H < 12 T. A prominent difference is that apeak of C/T shifts to lower temperature with rising H:
the temperature at which C/T–T curve reaches a maximum (=Tmax) for H = 6.9 and 12 T is, respectively, 7.3,
6.9 and 6.8 K, implying a reduction in the actual size ofthe spin gap ∆(H) with H. This is quantitatively discussed below.
Because of a lack of an appropriate theory for thespecific heat from the standpoint of the Shastry–Sutherland model, we will analyze the experimental data utilizing the isolated dimer model, where J ' is neglected
and only J is taken into consideration. Let us define themagnetic specific heat under a certain magnetic field Has C(H). Take the example of H = 0, C(0) is given by
the following formula,12 T
9T
6T0T

103

substrate by a small amount of Apiezon N grease. Themagnetic fields were applied perpendicular to the abplane, i.e., the Shastry–Sutherland lattice. The substrate
was weakly coupled by tungsten wires to a copper heatsink. A bare chip of Cernox resistance sensor (LakeShore) was used as a thermometer to minimize theaddenda heat capacity. The magnetic field dependenceof the thermometer was calibrated using a capacitancethermometer. The heat capacity of the sample wasobtained by subtracting the addenda heat capacity,which was determined in a separate run without thesample. No appreciable magnetic field dependence wasobserved for the addenda heat capacity. The resolutionof the measurement was about 0.5%, and the absoluteaccuracy determined from the measurement of a Cu
standard was better than 5%. The measurements wereperformed with increasing temperature.

3R ( ∆ ( 0 )/T ) exp ( ∆ ( 0 )/T )C ( 0 ) = ----------------------------------------------------------------,
2[ 1 + 3 exp ( ∆ ( 0 )/T ) ]210000.1

0.2

0.3

0.4

0.5

0.6 0.7
1/T, K–1

Fig. 2. Logarithmic plot of CT2 as a function of 1/T. Solidlines denote the fit to equation (2). Inset shows the magnetic
field variation of ∆(H).

(1)

where R is 8.30 J/(K mol) (see, for example, [11]).
Likewise, C(H) for a finite magnetic fields is easily calculated. In the low temperature limit, the magnetic specific heat the isolated dimer model can be reduced toJOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 90

No. 1

2000

LOW-TEMPERATURE SPECIFIC HEAT STUDY OF SrCu2(BO3)2

the following expression as long as the system is in a
gapful state:
C ( H ) ∝ T exp ( – ∆ ( H )/T ).
–2

S, J/(K mol)
122Rln2(2)Thus CT2 is plotted against 1/T in a logarithmic scale asshown in Fig. 2. One can see all data roughly follows alinear reversal-temperature dependency. Using thereduced expression of equation (2), we obtained ∆(0) =35.9 K, ∆(6 T) = 27.5 K, ∆(9 T) = 22.5 K, and ∆(12 T) =16.8 K. The deviation from the calculations (the solidlines in Fig. 2) in lower temperature region, more prominent in case of lower field, is for a most part due to thephonon contribution which is neglected here and will
be included later. The obtained values of ∆(H) are plotted in the inset of Fig. 2 as a function of H. It is clearthat ∆(H) decreases nearly in proportion to H. The origin of the decrease should be the Zeeman splitting of
the excited states. Namely, a three-fold degeneracy ofthe lowest excited triplet states (S = 1) in the absence ofthe magnetic field is lifted up by applied magnetic field.
∆(0) was estimated to be 35.0 K using the following
relation: ∆(H) = ∆(0) – gµBH, where g is the g-factor of
the Cu2+ electron spin and µB is the Bohr magneton. An
isotropic g-value, i.e., g = 2.0 was assumed. The obtained
value of ∆(0) is consistent with that obtained in other measurements using a single crystalline SrCu2(BO3)2 such as
the magnetic susceptibility (34 K) [6], ESR (34.7 K) [7]and Boron nuclear magnetic resonance (B-NMR; 36 K)[12], Cu-NMR (35 K) [12], and neutron scattering(34 K) [13].
Next, let us take a phonon term into consideration.
Then, the total specific heat is given by the sum of themagnetic and phonon terms, C = C(H) + βT3. Dottedcurves in Fig. 1 denote the results of the global leastsquare fit in the T range well below the spin-gap size,
namely, 2.6 K < T < 4.8 K for 0 T, 2.4 K < T < 4.1 K for
6 T, 2.1 K < T < 3.5 K for 9 T, and 1.5 K < T < 2.8 K for
12 T, from which we obtained once again a reasonablevalue of ∆(0) = 34.4 K together with β = 0.460 mJ/(K4 mol)and g =2.03. The phonon contribution is independently
shown by the dot-dashed curve in Fig. 1, which also
seems to reproduce the temperature dependence of theexperiment above 15 K.
As demonstrated above, it seems that the isolated
dimer model nicely reproduces the experimental data,
providing a consistent value of ∆(0). In a higher temperature region, however, the deviation between the experiment and the theory is appreciable. One can noticefrom Fig. 1 that experimental Tmax is lower as comparedwith the theoretical one in any magnetic field, andabove Tmax the value of experimental C/T is much suppressed. In Fig. 3, we show the T variation of the magnetic entrope of the system for H = 0, which shouldreach 2Rln2 ideally in the high-T limit. For comparison, a theoretical curve for the isolated dimer model for∆(0) = 34.4 K is shown by the solid line. The experimental entropy starts to deviate largely from the theo-13110864SrCu2(BO3)2200

102030T, KFig. 3. Magnetic entropy of SrCu2(BO3)2 at H = 0 (circles).
Solid curve represents the magnetic entrope for the isolated
dimer model for ∆(0) = 34.4 K.

retical one at around 10 K. For example, the magnetic
entrope at 25 K is still about and 74% of that for the isolated dimer model and 62% of the full entropy. Thisindicated that the spin system of SrCu2(BO3)2 is effectively correlated over much higher temperatures, andthus consistent with the estimation of exchange constants by Miyahara and Ueda; J = 100 K and J' = 68 K
[5]. It is noteworthy that the value of J is identical withthat of ∆(0) for the isolated dimer model, and J = 35 K(=∆(0)) derived from the isolated dimer model is toomuch smaller.
To summarize, we have measured the specific heatof SrCu2(BO3)2 under various magnetic fields. From
the fitting based on the isolated dimer model, the gap
was estimated to be 34.4 K, which is in good agreementwith the values determined from other physical measurements. With increasing H, the gap decreases in proportion with H. The simple dimer model, however, cannot explain the data at all in higher-temperature, region,
suggesting rather stronger correlation of the spin system. We are looking forward to a theory based on theShastry–Sutherland model with J and J' to reproduce
our specific heat data over the whole T range.
REFERENCES
1. C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10,1399 (1969).
2. M. P. Gelfand, Phys. Rev. B 43, 8644 (1991); T. Oguchi
and H. Kitatani, J. Phys. Soc. Jpn. 64, 1403 (1994).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 90

No. 1

2000

132

KAGEYAMA et al.

3. H. Kageyama, K. Yashimura, R. Stern, et al., Phys. Rev.
Lett. 82, 3168 (1999).
4. B. S. Shastry and B. Sutherland, Physica B 108, 1069
(1981).
5. S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701
(1999).
6. H. Kageyama, K. Onizuka, T. Yamauchi, et al., J. Phys.
Soc. Jpn. 68, 1821 (1999).
7. H. Nojiri, H. Kageyama, K. Onizuka, et al. (submitted toJ. Phys. Soc. Jpn.); cond-mat/9906072.

8. H. Kageyama, K. Onizuka, Y. Ueda, et al., J. Phys. Soc.
Jpn. 67, 4304 (1998).
9. R. Bachmann, E. J. DiSalvo, Jr., T. H. Geballe, et al.,Rev. Sci. Instrum. 43, 205 (1972).
10. H. Kageyama, K. Onizuka, T. Yamauchi, et al. (submitted to J. Crystal Growth).
11. R. L. Carlin, Magnetochemistry (Springer-Verlag, Berlin, 1986).
12. M. Takigawa, private communication.
13. H. Kageyama et al. (in preparation).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 90

No. 12000