JETP Letters, Vol. 73, No. 11, 2001, pp. 633–636. Translated from Pis’ma v Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 73, No. 11, 2001, pp. 715–718.
Original Russian Text Copyright © 2001 by Vasil’ev, Markina, Inyushkin, Kageyama.

Thermal Conductivity and Specific Heatof SrCu2(BO3)2: A Quasi-Two-Dimensional
Metal Oxide Compound with a Spin Gap
A. N. Vasil’ev1,*, M. M. Markina1, A. V. Inyushkin2, and H. Kageyama3
1 Moscow State University, Moscow, 119899 Russia* e-mail: vasil@lt.phys.msu.su
2 Institute of Molecular Physics, Russian Research Centre Kurchatov Institute,Moscow, 123182 Russia
3 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba, 277-8581 Japan

Received May 11, 2001

The thermal conductivity and specific heat of SrCu2(BO3)2, a quasi-two-dimensional metal oxide compoundwith a spin gap, were studied at low temperatures. In the temperature interval 0.4 < T < 3.2 K, the thermal conductivity of a single crystal sample in the ab plane varies according to the power law κ ∝ T 2.73. As the temperature increases further, a deep minimum is observed in the region of Tmin ≈ 9.8 K. This behavior is explainedby the scattering of phonons—the major heat carriers—on the fluctuations of the spin subsystem. © 2001 MAIK“Nauka/Interperiodica”.
PACS numbers: 66.70.+f; 65.80.+n; 75.40.Cx

In the past decade, a number of low-dimensional
quantum-mechanical systems were discovered which,when cooled down to low temperatures, exhibit theappearance of a spin gap separating the nonmagnetic
ground state from the spectrum of spin excitations. Theformation of a spin gap in such concentrated magneticsystems is manifested by their unusual low-temperaturemagnetic properties, whereby the samples behave as
nonmagnetic substances rather than exhibiting ferro- or
antiferromagnetic ordering. The ground state of thesecompounds is a spin singlet representing an isolated
configuration of spins in which the projection of their
total magnetic moment onto any direction is zero.
Whether a given system reaches a ground state with orwithout the spin gap depends on the mutual arrangement and magnitude of the spin magnetic moments, aswell as on the magnitudes and hierarchy of theexchange interaction constants.
Recently, the group of substances with a spin gap,
containing compounds such as spin-Peierls CuGeO3[1], spin ladder SrCu2O3 [2], plaquette-type system
CaV4O9 [3], and charge-ordered system NaV2O5 [4],expanded to include a low-dimensional magnetic compound SrCu2(BO3)2 [5]. The new member is close tohigh-temperature superconductors of the cuprategroup: SrCu2(BO3)2 possesses a layer structure, exhibits a pseudo-spin-gap behavior, and shows the ability toantiferromagnetic ordering upon a small change in theparameters.
SrCu2(BO3)2 has a tetragonal unit cell with theroom-temperature lattice constants a = b = 8.995 Å,c = 6.649 Å. The structure of this compound comprises
slightly corrugated ab planes formed by CuBO3, which
are separated by layers of nonmagnetic Sr2+ ions along
the c axis. As is seen in Fig. 1, the ab plane contains
rectangular planar CuO4 complexes linked by triangular BO3 groups. All Cu2+ ions possess the spin 1/2 andoccupy crystallographically equivalent positions. The
nearest-neighbor Cu2+ ions form magnetic dimers,
while triangular BO3 complexes link these dimers so as
to form an orthogonal network.
At present, SrCu2(BO3)2 is the only known substance to which the quantum-mechanical Shastry–
Sutherland model [6] can be applied. This theory
allows the ground state of an orthogonal dimer network
to be exactly calculated. Depending on the ratio ofexchange integrals in (J) and between (J ') dimers, asystem occurs either in the antiferromagnetic groundstate (for J '/J > 0.70) or in the state with localized magnetic dimers (for J '/J < 0.70). At J '/J = 0.70, the groundstate of an orthogonal dimer network is the spin fluid.
According to [7], the exchange integrals inSrCu2(BO3)2 are J ' = 68 K and J = 100 K, which corre-0021-3640/01/7311-0633$21.00 © 2001 MAIK “Nauka/Interperiodica”

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C(T) of pressed powder samples was studied in therange from 1.6 to 300 K with the aid of a quasi-adiabatic microcalorimeter. The errors of determination ofthe κ and C values at low temperatures were about 10%.
Figure 2 shows the temperature dependence of thespecific heat of SrCu2(BO3)2 . Against a monotonic
buildup of the specific heat with increasing temperature, the curve exhibits a clearly pronounced maximumat T ≈ 8.3 K related to the formation of a spin excitation
spectrum [11, 12]. The inset in Fig. 2 shows a temperature dependence of the longitudinal sound velocity v inthe [100] direction (reproduced from [13]). As is seen,
there is a minimum at T ≈ 13.3 K on the background ofv(T) decreasing with the increase in the sample temperature.

Fig. 1. Schematic diagrams showing the (a) crystal and(b) magnetic structures of SrCu2(BO3)2 in the ab plane:(䊉) Cu2+ ions; (䊊) B3+ ions; (䊊) O2– ions.

Fig. 2. The temperature dependence of specific heat inSrCu2(BO3)2 . The bottom inset shows the temperaturedependence of the longitudinal sound velocity in the [100]direction (data from [13]).

sponds to the case of localized dimers in the groundstate at an exchange integral ratio J '/J = 0.68 close tothe critical value. The spin gap width បω0 /kB in
SrCu2(BO3)2 amounts to 34 K [5, 7–9].
A sample of SrCu2(BO3)2 studied in our experiments was cut to size 6 × 1.6 × 0.4 mm from a single
crystal ingot grown by the floating zone technique [10].
The thermal conductivity κ(T) of the single crystal sample was studied in a broad range of temperatures from0.4 to 300 K by the method of stationary thermal flux inthe [100] direction (in the ab plane). The specific heatFigure 3 shows the temperature variation of the thermal conductivity of SrCu2(BO3)2 . At the lowest temperatures, the curve obeys the power law κ(T) ∝ T 2.73; asthe temperature grows, the κ value passes through amaximum at T ≈ 3.7 K, then drops to show a minimum
at T ≈ 9.8 K, and increases again with a maximum at T ≈47 K. A maximum value of the thermal conductivity ofSrCu2(BO3)2 at T ≈ 3.7 K amounts to κ = 28 W/(m K).
A difference in the temperatures at which the features
were observed in the curves of κ(T), C(T), and v(T) canbe related to the fact that the ground state inSrCu2(BO3)2 is attained without a clearly pronounced
phase transition at a certain temperature.
The involved temperature variation of the thermal
conductivity exhibiting the alternation of maxima andminima is probably a general characteristic of the systems with spin gaps. The κ(T) curves with two peaks
were previously reported for many low-dimensional
compounds, although different interpretations weregiven in various particular cases. In CuGeO3, the hightemperature peak in the thermal conductivity wasexplained by the magnon mechanism, and the low-temperature peak, by the phonon mechanism of the heattransfer [14]. The double peak observed in the thermal
conductivity of Sr14 – xCaxCu24O41 was interpreted interms of the magnon propagation over spin chains andladders [15]. In NaV2O5 , both peaks in κ(T) wereattributed to the phonon heat transfer [16].
In our opinion, a more justified interpretation of theκ(T) behavior in the case of SrCu2(BO3)2 is based on
the phonon heat transfer mechanism. According to theneutron scattering data [9], the spin excitations hardlyspread over the lattice and, hence, cannot participate in
the heat transfer. The minimum of κ(T) at Tmin ≈ 9.8 K
is probably indicative of a strong phonon coupling tothe spin subsystem.
We have analyzed the temperature variation of thethermal conductivity within the framework of a simple
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THERMAL CONDUCTIVITY AND SPECIFIC HEAT

635

Debye lattice thermal conductivity model [17], inwhich
T D /T4 x
k B  k B 3 3x e τ ( x, T )-------------------------- d x.
---Tκ ( T ) = -------------22x2π v s  ប (–1)
e0∫Here, x = បω/kBT, ω is the phonon frequency, τ–1 =

∑τ

–1
i

is the total phonon relaxation rate in various
scattering processes, TD ≈ 450 K is the Debye temperature [13], and vs is the polarization-averaged soundvelocity (vs ≈ 4.5 × 105 cm/s [13]).
i

A satisfactory description of the low-temperaturethermal conductivity is obtained with an allowance for–1
the phonon scattering at the sample boundaries ( τ b =vs /lC), dislocations ( τ d–1= Adω), and point defects–1( τ pd= Apdω4) in the three-phonon processes ( τ 3 ph =A3phω2T 3) and on the spin subsystem fluctuations–1( τ flu = Afluω4T 2Cs) [18, 19]. The Casimir length lC =1.12S 1/2 (S is the sample cross section area) determinedfrom the sample geometry was 0.84 mm. In the above
relaxation terms, Ai are the fitting parameters and Cs isthe heat capacity of the spin subsystem determined bysubtracting the phonon contribution from the experimental profile (see the inset in Fig. 3).
–1

The parameters of scattering on dislocations andpoint defects were determined from a power-law
approximation of the low-temperature (0.4 < T < 3.2 K)
branch of the thermal conductivity, where κ ∝ T 2.73.
Deviation of the κ(T) value from the cubic law expectedin the boundary scattering regime can be related to asignificant phonon scattering on dislocations. Since thethermal conductivity at high temperatures is determined by the phonon scattering on the spin density
fluctuations, which hinders reliable estimation of theparameters of the three-phonon scattering process, ourselection for A3ph is rather arbitrary.
The solid curve in Fig. 3 shows the approximation of
the temperature dependence of the thermal conductivity in SrCu2(BO3)2 calculated with an allowance of all
the aforementioned scattering processes. The dashedcurve represents the κ(T) variation determined withneglect of the phonon scattering on the spin subsystem
fluctuations. As is seen, the phonon scattering on thespin density fluctuations decreases the thermal conductivity by almost two orders of magnitude at a temperature in the region of the minimum in the κ(T) curve.
Such a large depression of the phonon thermal conductivity probably indicates that the entire spectrum ofthermal phonons is involved in interaction with the spinsubsystem. This conclusion is confirmed by the resultsof ultrasonic measurements showing that the low-freJETP LETTERS

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Fig. 3. The temperature dependence of the specific heat of aSrCu2(BO3)2 single crystal in the [100] direction. Open circles represent the experimental data, solid curve shows theresults of calculations with an allowance of the phonon scattering on the spin subsystem fluctuations, and dashed curve
corresponds to the calculation ignoring the spin–phononinteraction. The inset illustrates determination of the heat
capacity Cs of the spin subsystem (see the text).

quency acoustic phonons actually participate in theinteraction [13].
Recently, the minimum observed in the temperaturedependence of the thermal conductivity in SrCu2(BO3)2
was given an alternative interpretation based on the resonance scattering of phonons in a two-level system ofmagnetic excitations [12]. In our opinion, this explanation should be taken with care since the resonance scattering can be effective only within a certain narrow partof the phonon spectrum.
The authors are grateful to S. Zherlitsin for kindly
providing the results of ultrasonic measurements. This
study was supported by the Russian Foundation for
Basic Research (project nos. 02-99-17828 and 00-0216255), INTAS (grant no. 99-0155), and NWO (grantno. 047-008-012).
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Translated by P. Pozdeev

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