Physica B 312–313 (2002) 597–599

Heat transport in SrCu2 ðBO3Þ2 and CuGeO3
M. Hofmanna, T. Lorenza,*, A. Freimutha, G.S. Uhrigb, H. Kageyamac, Y. Uedac,
G. Dhalenned, A. Revcolevschid
II. Phys. Institut, Universitat
Str. 77, 50937 Koln,. zu Koln,. Zulpicher.
. Germany
Inst. furStr. 77, 50937 Koln,. Theoretische Physik, Universität zu Koln,. Zulpicher.
. GermanycInst. for Sol. State Phys., University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, JapandLab. de Physico-Chimie des Solides, Université Paris-Sud, 91405 Orsay, FranceabAbstract
In the low-dimensional spin systems SrCu2 ðBO3 Þ2 and CuGeO3 the thermal conductivities along different crystal
directions show pronounced double-peak structures and strongly depend on magnetic ﬁelds. For SrCu2 ðBO3 Þ2 theexperimental data can be described by a purely phononic heat current and resonant scattering of phonons by magnetic
excitations. A similar effect seems to be important in CuGeO3 ; too but, in addition, a magnetic contribution to the heattransport may be present. r 2002 Elsevier Science B.V. All rights reserved.
Keywords: Heat transport; Low-dimensional quantum spin systems

One topic of interest in low-dimensional (D) quantum
spin systems is the dynamics of magnetic excitations,which, e.g. are expected to move without dissipation in
an S ¼ 12 Heisenberg chain [1]. This may lead to an
unusual thermal conductivity in real, (quasi-)low-Dsystems. Experimental evidence for a large magnetic
contribution kmag to the heat transport has been foundin Sr14 Cu24 O41 containing both, spin chains and ladders[2]. Here, k shows an enormous maximum around 200 K
which is present only for heat transport parallel to theladder (and chain) direction. This has been interpreted
as a huge magnetic contribution to k; that adds to aphononic contribution kph giving rise to low-T maxima
around 20 K for all three crystal directions. In thispaper, we present a study of the anisotropic heattransport in the spin Peierls (SP) system CuGeO3 andin the 2D dimer spin system SrCu2 ðBO3 Þ2 [3], whose
magnetic planes realize the Shastry Sutherlandmodel [4].
The thermal conductivities of tetragonal SrCu2 ðBO3 Þ2parallel (kc ) and perpendicular (ka ) to the magnetic*Corresponding author. Tel.: +49-221-470-3770; fax: +49221-470-6708.
E-mail address: lorenz@ph2.uni-koeln.de (T. Lorenz).

planes show pronounced double peaks with a ﬁrst
maximum around 5 K for both directions and a second
one around 60 and 30 K for ka and kc ; respectively (seeFig. 1). In a magnetic ﬁeld, the maxima at low T aresuppressed whereas those at high T remain unchanged.
At ﬁrst sight, one might suspect that the ﬁeld-dependent
double peaks result from the superposition of aphononic and a magnetic contribution. However, thesimilar behavior of ka and kc makes any explanation
depending on a kmag very unlikely, since kmag perpendicular to the magnetic planes should be negligible.
Moreover, the triplet excitations of SrCu2 ðBO3 Þ2 arealmost dispersionless along the magnetic planes [3], sothat even within the planes kmag is expected to be verysmall. This led us to a model that is based on a purely
phononic heat current and explains the ﬁeld-dependentdouble peaks by resonant scattering of phonons bymagnetic excitations. It can be visualized as (i) theabsorption of a phonon of energy o by exciting a(magnetic) two-level system and (ii) its subsequent deexcitation by emitting another phonon of same energy
but different momentum (for more details see Ref. [5]).
Such a scattering is most active in a certain temperature
range depending on the energy splitting of the two-levelsystem. Spin conservation requires a thermally excited

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
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M. Hofmann et al. / Physica B 312–313 (2002) 597–599

H⊥ j
H = 0, 6, 11, 13, 17 TCuGeO3
10

j || c8

H || a4
H = 0, 6, 11, 14, 17 T

0351030 50
Temperature (K)100300Fig. 1. Thermal conductivity ka (top) and kc (bottom) of
tetragonal SrCu2 ðBO3 Þ2 on a logarithmic temperature scale forvarious ﬁelds. With increasing ﬁeld the low-T maximum issuppressed. Lines are theoretical curves calculated for the sameﬁelds.

triplet and the phonon excites a second triplet thatcombines with the ﬁrst one to total spin Stot ¼ 1: Thus,
the scattering rate depends on the thermal population of
excited triplets. It also depends on the magnetic ﬁeld dueto the Zeeman splitting. Within this model our experimental data for zero ﬁeld can be ﬁtted very well. Since
these ﬁts ﬁx all parameters the data for higher ﬁelds arecalculated from the Zeeman splitting without further
parameter adjustment. As shown by the solid lines inFig. 1, our model describes the general features of ka
and kc very well; kc is even quantitatively reproduced up
to 17 T; whereas the ﬁeld dependence of ka is slightly
overestimated.
The thermal conductivities along the a; b; and c axesof orthorhombic CuGeO3 are shown in Fig. 2. Our
results conﬁrm those previously measured for kc (jj tothe spin chains) [6] showing one maximum below and asecond one above the SP transition at TSP B14 K: Withincreasing ﬁeld the low T maximum is continuously
suppressed up to 14 T and then slightly increases again.
This non-monotonic ﬁeld dependence is related to theoccurrence of the so-called incommensurate phase above
B12:5 T [7]. Although the data of CuGeO3 qualitatively
resemble those of SrCu2 ðBO3 Þ2 the situation in the twocompounds is rather different. A sizeable kmag Fthough
not necessary for an explanation of a ﬁeld-dependent
double peakFmay be present in CuGeO3 since thetriplet excitations have considerable dispersion [10]. Inaddition, there is the SP transition. Above TSP theenergy gap closes and strong structural ﬂuctuations areobserved up to B35 K [8]. All these effects mayinﬂuence kph and/or kmag making a reliable descriptionof k difﬁcult.
In order to clarify whether a magnetic contribution tokc is present in CuGeO3 we have also measured kb andka (> to the spin chains). As shown in Fig. 2, kb behaves

Thermal conductivity

κa (W/Km)40

κc (W/Km)

j || aH || aSrCu2(BO3)28(W/Km)

598

0T8T

j || a020

j || b0T
12 T16 T14 T

j || c0T4, 6, 8,10, 12, 14 T

1006030

16 T0351030 50
Temperature (K)100300Fig. 2. Thermal conductivity ka ; kb and kc (from top tobottom) of orthorhombic CuGeO3 on a logarithmic temperature scale for various ﬁelds. ka is ﬁeld-independent whereas kband kc decrease with increasing ﬁeld up to 14 T and thenincrease again.

very similar to kc apart from the absolute value that is afactor of B3 smaller for kb than for kc : An anisotropy
ratio of B3 is present between the sound velocities alongc and b (vc B7600 m=s and vb B2400 m=s [9]), but also
for the triplet dispersions along c and b [10]. Therefore,
one can neither exclude nor conﬁrm the presence of a
sizeable magnetic contribution to the heat current fromthis anisotropy. Finally, we consider the a-axis. The
sound velocity amounts to va B3600 m=s [9] whereas
the magnetic dispersion is essentially zero [10], i.e. kmag
ais negligible. At 300 K the anisotropy between kc ; kb and(the purely phononic) ka reﬂects that of the vi(i ¼ a; b; c). Therefore kmagand kmag
can also becb
neglected at room temperature. This is not surprisingbecause the magnetic coupling constants in CuGeO3 aremuch smaller; Jc B120–180 K and Jb B0:1Jc dependingon the model [10,11]. Unfortunately, ka cannot help toseparate kph and kmag along b and c at low T because thelayered structure of CuGeO3 along a limits the mean
free path by the so-called sheet-like faults. This prevents
a strong increase of ka at low T and already explains thecompletely different behavior of ka compared to kb andkc ; regardless their ﬁeld-dependent double peaks.
In conclusion, the thermal conductivities of CuGeO3and SrCu2 ðBO3 Þ2 show pronounced, magnetic-ﬁelddependent double peaks. For SrCu2 ðBO3 Þ2 this arises
from resonant scattering of a phononic heat current bymagnetic excitations. In CuGeO3 a similar effect seems
to play a role but further studies are necessary in orderto clarify the dynamics of the magnetic excitations.

M. Hofmann et al. / Physica B 312–313 (2002) 597–599

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