VOLUME 87, NUMBER 4

23 JULY 2001

PHYSICAL REVIEW LETTERS

Strong Damping of Phononic Heat Current by Magnetic Excitations in SrCu2 共BO3 兲 2
M. Hofmann,1 T. Lorenz,1 G. S. Uhrig,2 H. Kierspel,1 O. Zabara,1 A. Freimuth,1 H. Kageyama,3 and Y. Ueda3
1
II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany
Institut für Theoretische Physik, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany
3Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5 Kashiwa, Chiba 277-8581, Japan(Received 28 February 2001; published 5 July 2001)
2

Measurements of the thermal conductivity as a function of temperature and magnetic field in the 2D
dimer spin system SrCu2 共BO3 兲2 are presented. In zero magnetic field the thermal conductivity alongand perpendicular to the magnetic planes shows a pronounced double-peak structure as a function oftemperature. The low-temperature maximum is drastically suppressed with increasing magnetic field.
Our quantitative analysis reveals that the heat current is due to phonons and that the double-peak structurearises from pronounced resonant scattering of phonons by magnetic excitations.
DOI: 10.1103/PhysRevLett.87.047202

PACS numbers: 75.40.Cx, 66.70. +f, 75.10.Jm

nant scattering of phonons by magnetic excitations. With
the same parameters our model considerably improves thedescription of the magnetic field dependence of the specific heat of SCBO [16].
For our study, two samples of rectangular-bar-shaped
forms of ⬃0.6 3 1.9 3 3 mm3 with the long direction
along the a and c axes were cut from larger single crystals of SCBO grown by the traveling solvent floating zone
method [17]. The thermal conductivity was measured
with a conventional steady state method using differential chromel-Au 10.07% Fe thermocouples calibrated inmagnetic fields. Typical temperature gradients were of theorder of 0.2 K. The absolute accuracy of our measurements is of order 610% because of uncertainties in thesample geometry, whereas the relative accuracy is about
1 order of magnitude better [18].
We show in Fig. 1 the thermal conductivity ka 共T 兲(苷 kb in tetragonal SCBO) and kc 共T兲 in zero magnetic
field. For both directions pronounced double-peak structures are observed with the low-T maxima occurring at

047202-1© 2001 The American Physical Society

0031-9007兾01兾 87(4)兾047202(4)$15.00

κ (W/Km)12κa8

κ (W/Km)During the past few years the thermal conductivity (k)of low-dimensional spin systems has attracted considerable interest [1 –6]. One reason is that in these materials a large magnetic contribution k mag to the heat current
may be present as observed, e.g., for the spin ladder material Sr142x Cax Cu24 O41 [6]. Another reason is that thephononic heat current k ph probes the spectrum of magnetic
excitations as well as the spin-phonon coupling [3–5]. The
latter is very important in some of the low-dimensional
spin systems, e.g., in the spin-Peierls compound CuGeO3[4,5]. Both issues, the magnetic contribution to the heatcurrent as well as the interaction of the phonons with magnetic excitations, are to a large extent unexplored and not
well understood.
A material of particular interest in this context is
SrCu2 共BO3 兲2 [7]. The Cu21 ions form a quasi-2D
spin system which is by virtue of the crystal geometry
an experimental realization of the Shastry-Sutherlandmodel [8]. In SCBO the intradimer and interdimer
couplings are of magnitude J1 ⯝ 72 K and J2 ⯝ 43 K,
i.e., J2 兾J1 ⯝ 0.6 [9]. As expected for this ratio SCBO
has a dimerized singlet ground-state separated from theexcited triplet states by a finite gap D ⯝ 35 K, as seenin the magnetic susceptibility [10] or in inelastic neutron
scattering [11]. From the latter it is also known thatthe triplet excitations are almost dispersionless, i.e., thegroup velocity is very small, in agreement with theoretical
calculations [12 –15]. Thus, a sizable k mag is not expected
for SCBO making this material a natural candidate to
study the influence of magnetic excitations on k ph .
In this Letter we present measurements of the thermal
conductivity of SCBO along (ka ) and perpendicular (kc ) to
the 2D spin planes in a large temperature (2.5–275 K) andmagnetic field range (0–17 T). In zero field, both ka andkc show pronounced double-peak structures as a functionof T. For both directions the low-T maximum is drastically suppressed by a magnetic field. We present a model
based on a purely phononic thermal conductivity and explain the double-peak structure and its field dependence bystrong damping of the phononic heat current due to reso-

50
40
30
20
10

ph

κa
ph

κC

50

100

150 200
T (K)

250

κC

4

0
0

50

100

150

200

250

Temperature (K)FIG. 1. Thermal conductivity of SrCu2 共BO3 兲2 along (ka ) andperpendicular (kc ) to the magnetic planes. The inset showsthe experimental data (±) together with “conventional” phononicphphcontributions ka and kc (- -) obtained by switching off thescattering of phonons by magnetic excitations (see text).

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VOLUME 87, NUMBER 4

PHYSICAL REVIEW LETTERS

艐4.5 K. For ka , the high-T maximum lies at 艐60 K andfor kc , at 艐30 K. Above 100 K we find kc ~ T 21 , as
expected for phonon heat transport [19], whereas ka follows a T 20.6 dependence. The magnetic field dependenceof the thermal conductivity is shown in Fig. 2. The low-T
maximum is suppressed strongly by a magnetic field forboth ka and kc . The magnetic field dependence at higher
temperatures is only weak (#2%) and close to the relative
measurement accuracy. The minimum of the double-peakstructure and the low-T maximum systematically shifts tolower temperatures with increasing magnetic field.
The behavior of k observed here is reminiscent of thatfound in other low-dimensional spin systems [1,4,6]. In

κc (W/Km)10a)

H || ajh || c5

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

κa (W/Km)

0
10

H || ajh || ajh ⊥ Hb)5

H=0,6,11,13,17T
0.8

c)

d)

2

C/T ( J/(mole K ) )0
2 tripletsBbound triplet
bound singlet
1 triplet

0.4
H || c
H=0,6,9,12T
0.0

3

5

10

∆S 2B

Stot= 1
Stot= 0k30 50
Temperature (K)

100

200

FIG. 2. Thermal conductivity kc (a) and ka (b) on a logarithmic temperature scale for various magnetic fields. On increasing
field the low-T maximum is suppressed. The lines are theoretical curves calculated for the same magnetic fields via Eq. (1).
(c) Specific heat at various fields H jj c (the peak height decreases with increasing field). Data points are from Ref. [16];the lines result from our theory. (d) Level scheme used in ourmodel: The solid lines denote the energies of the ground state,
one triplet and two elementary triplets. The dashed lines indicate bound states with Stot 苷 0 or 1, which are built from twoelementary triplets. B is the binding energy and D0 is the elementary gap. Ds is the energy difference relevant to the resonance scattering according to Eq. (3).

047202-2

particular, in CuGeO3 , k also shows an almost fieldindependent maximum at ⬃20 K and a second one about
5 K, which is strongly suppressed by a magnetic field.
This has been interpreted in terms of a magnetic and aphononic contribution to the heat current with maxima at
different temperatures [4].
For several reasons, a similar interpretation is not possible in SCBO. First, due to their flat dispersion the single
triplet excitations are not expected to contribute significantly to the heat current. Second, k mag in a 2D magnetic system should be strongly anisotropic, as is observed,e.g., in Sr142x Cax Cu24 O41 [6], but not in our measurements. The lack of a strong anisotropy also excludes an
explanation in terms of multitriplet excitations observedin SCBO above ⬃5 meV [11,16,20,21]. These excitations
move much more easily than the single triplet excitations[9], but again only within the magnetic planes. Finally,
the relation k ~ cyᐉ (c is the specific heat, y is the groupvelocity, and ᐉ is the mean-free path) predicts similar T
dependencies of k mag and the magnetic specific heat cmag .
But measurements of c in fields up to 12 T [16] show thatcmag gives rise to a maximum of c兾T at temperatures close
to the minima of ka and kc for the respective magnetic
fields (see Fig. 2).
From these arguments we exclude a sizable magnetic
contribution to the heat current [22], so that k 艐 k ph . As
the cause of the double-peak structure and its magnetic
field dependence we propose scattering of phonons bymagnetic excitations. The absence of a strong anisotropyof ka and kc is then easily understood, since both theanisotropy of the spin-phonon coupling and/or of thephonon system itself are expected to be much weaker thanthat of the 2D magnetic system. In addition, this scenario
yields a straightforward explanation why the minima ofka and kc occur at the maxima of c兾T. The latter is thetemperature derivative of the magnetic entropy and thus
directly related to the number of magnetic excitations
which serve as scatterers for the phonons.
For a quantitative description we fit our data by a Debye
model for the phononic thermal conductivity [23]∆0ground state

23 JULY 2001phµ ∂3 Z QD 兾T 4 x
kBx e t共v, T 兲
kB
苷
T3
dx ,
2
2p y h̄
共ex 2 1兲20(1)where y is the sound velocity, QD the Debye temperature,v is the phonon frequency, x 苷 h̄v兾kB T , and t共v, T 兲 isthe mean lifetime of a phonon. The total scattering ratet 21 is given by
21
21
21
21
1 tpt
1 tum
1 tres.
t 21 苷 tbd(2)

21
21 , and t 21 refer to the standard relaxation, tpt
Here tbd
um
processes for conventional phonon heat transport, i.e., toboundary scattering, scattering by point defects, and umk21
lapp scattering, respectively [24]. tres
describes the scattering of phonons on magnetic excitations, which will beconsidered in the following.

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VOLUME 87, NUMBER 4

PHYSICAL REVIEW LETTERS

For the double-peak structure of the phononic thermal
conductivity we need a scattering mechanism that is mostactive in a certain temperature range. We therefore consider resonant scattering [23,25 –27]. The physical picture is the following: A phonon of energy v is absorbed,
thereby exciting the magnetic system (under the constraintof spin conservation). Then another phonon of the sameenergy is emitted and the magnetic system is deexcited.
If excitation (or deexcitation) starts from a localized state,such as the dispersionless triplet excitations of SCBO there
is no directional correlation between the absorbed and theemitted phonon. The net effect is therefore the same asthat of pointlike defects, except for the frequency dependence of the matrix element Mk 0 ,k 共v兲, and the scattering
21
is of the form
rate tres21
tres共v兲 苷 Cv 2 Mk20,k 共v兲F共T 兲 苷 C

v 4 Ds2
F共T兲 .
共v 2 2 Ds2 兲2(3)Here, Ds is the energy splitting of the two levels involved,
C is the overall coupling strength between phonons and thetwo-level system, and F共T兲 contains information on thestate of the two-level system [23,25,26]. Notably, resonant
scattering starting from the magnetic ground state is noteffective. The reason is that the ground state is a coherent,
translational invariant state with zero momentum so thatthe emitted and absorbed phonons must have the same
momentum and energy.
With this scenario in mind we consider resonant scattering within the magnetic excitation spectrum of SCBO
(Fig. 2d). The lowest excited state is a single triplet
with excitation energy D0 . It is thermally excited withprobability c1 苷 1 2 关1 1 m1 exp共2bD0 兲兴21 .
Hereb 苷 共kB T兲21 , and m1 苷 1 1 2 cosh共bh兲 contains themagnetic field dependence through h 苷 gmB H. In theresonance process a second triplet is excited on one of
the two adjacent dimers, and it combines with the first,
thermally excited triplet to a bound triplet state so that the
total spin is conserved. The energy of the bound triplet isgiven by 2D0 2 B, where B is the binding energy [28].
The resonance energy to be used in Eq. (3) is therefore
Ds 苷 共2D0 2 B兲 2 D0 苷 D0 2 B. A complete description of the thermal factors of the resonance process takes
account of (i) the fact that a thermally excited triplet has
to be present and (ii) all resonance processes involving
the two adjacent dimers. This yields [18]F共T兲 苷 2c1 关N0 1 N1 1 共N0 2 N1 兲2 兴

(4)with the probabilities N0 苷 1 2 c1 of having a dimer in
its singlet state and N1 苷 c1 兾3 of having a dimer in the
triplet state with a particular S z component. Note that in
the resonance process the total S z component is conserved
due to spin conservation. Therefore Ds does not dependon the magnetic field. It is only c1 that is field dependent.
In zero field, Eqs. (3) and (4) yield an excellent fit(not shown) for D0 艐 35 K and Ds 艐 20 K, indicating
047202-3

23 JULY 2001

strong binding effects (B 艐 15 K). This is in good agreement with inelastic neutron scattering and ESR experiments showing the lowest bound triplets at 艐55 K and anelementary gap of D0 艐 35 K (see Fig. 2d) [11,20]. However, the decrease of the thermal conductivity in magnetic
fields computed by Eqs. (3) and (4) is much stronger thanthat found in our experiments.
We can account for this discrepancy by considering asecond, magnetic field independent, contribution to theresonance scattering. It arises from the presence of boundsinglet excitations with energy 2D0 2 2B [9,28] (seeFig. 2d). Since 2B 艐 D0 in SCBO, the energy of thesebound singlets is close to that of single triplet excitations
yielding comparable thermal populations of these states.
Therefore bound singlets have to be treated as scattering
centers for resonance scattering also. In this case a dimer
adjacent to a bound singlet is excited and the corresponding three triplets combine to Stot 苷 0. Since Stot 苷 0, thiscontribution to the resonance scattering does not depend
on a magnetic field and lowers the sensitivity of the totalresonance scattering rate to a magnetic field. The contribution of the bound singlets to the scattering rate is treatedin analogy to that of the single triplets. The probability
that a pair of adjacent dimers is in a state made of two
triplets is given by c2 苷 1 2 关1 1 2m1 exp共2bD0 兲兴兾Z,
where Z is the complete partition sum of the systemof two triplets. In this way we achieve a theoreticaldescription comprising single triplets with density per
dimer n1 and two-triplet states with density per dimer n2 .
Exclusion effects — a dimer cannot be in a single-triplet
state and involved in one of the bound states simultaneously — are accounted for by n1 苷 c1 共1 2 4n1 2 2n2 兲
and n2 苷 2c2 共1 2 2n1 2 3n2 兲.
A complication arises from a repulsive interaction y
between neighboring triplets. The energy D0 of a single
triplet is smaller than J1 due to virtual processes between
neighboring dimers resulting from the dimer-dimer coupling. If a triplet is excited on a neighboring dimer theseprocesses are (in part) blocked and D0 increases. Accounting for this on a mean-field level yields y 苷 R共n1 1
3n2 兾2兲 with R 苷 共J1 2 D0 兲兾2 [18]. In the calculation ofk, one has to replace D0 ! D0 1 py in all equations (i.e.,in c1 , c2 , and the energies given in [28]), where p 苷 2
refers to single triplets and p 苷 3兾2 refers to bound statesfrom two triplets.
To illustrate the validity of our model we compute thespecific heat for various magnetic fields. For the realistic
parameters D0 苷 36 K, R 苷 18 K (from J1 苷 72 K [9]),and B 苷 17 K we obtain the results depicted in Fig. 2c. At
low T the agreement is very good; position and height ofthe peak are considerably improved relative to the isolated
dimer model [16]. At higher T we presume that the local
model developed above is too simplistic since the dispersion [9,11] of the bound states is neglected.
The resonant scattering, decisive for the thermal conductivity, is calculated in the same model with the same21 苷 Cv 4 共A 1 A 兲 withparameters. We obtain tres
1
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VOLUME 87, NUMBER 4

PHYSICAL REVIEW LETTERS

A1 苷

2n1 关N0 1 N1 1 共N0 2 N1 兲2 兴 共D0 2 B兲2,
关共D0 2 B兲2 2 v 2 兴2

(5a)

A2 苷

3n2 关N0 1 N1 1 2共N0 2 N1 兲2 兴D20,
共D20 2 v 2 兲2(5b)where N1 苷 共n1 1 3n2 兾2兲兾3 and N0 苷 1 2 3N1 . Using
21
together with the usual scattering rates [Eq. (2)],this tres
we fit the experimental ka and kc by Eq. (1). The soundvelocity (y 苷 7600 m兾s) along the a direction and QD 苷
453 K are calculated from the measured elastic constantc11 [29]. Because of the lack of experimental data for c33
we take the same velocity for the fit of kc . The relaxation
21
21
rate tbd
is then obtained by tbd
苷 y兾L with the characteristic sample lengths L 苷 0.75 mm (ka ) and 0.5 mm(kc ) of the two samples. The point defect scattering rate is
21
given by tpt
苷 Pv 4 and the phonon umklapp scattering
21
rate is approximated by tum
苷 UTv 3 exp共QD 兾uT兲 [24].
Since the gap D0 , the binding energy B, and the interaction
R are determined from the specific heat, there remain only
four adjustable parameters (P, U, u, and C).
As shown in Fig. 2 our model yields an almost perfect
fit of the measured kc (for H 苷 0) over the entire T range.
Above 10 K the fit of ka is also almost perfect, whereas
below 10 K a slight T shift (#1 K) between fit and experimental data occurs. We use P 苷 7.7 3 10243 s3 共7.9 310243 s3 兲 , U 苷 2.6 3 10231 s2 兾K (5.7 3 10231 s2 兾K),u 苷 5 (10), and C 苷 3.35 3 106 s3 共1.15 3 106 s3 兲 forthe calculation of ka 共kc 兲. These values are reasonable
for an insulator and are comparable in magnitude to thosefound, e.g., in the spin ladders [6]. Different values for ka
and kc are expected in an anisotropic crystal.
Accounting for the Zeeman splitting in c1 and c2 , wecalculate ka and kc in magnetic fields without further parameter adjustment. Using g 艐 2.07 [10], our model reproduces the overall influence of the magnetic field very
well. The high-T maxima remain unchanged, whereas the
low-T maxima are continuously suppressed. Again theagreement between calculated and experimental values isbetter for kc than for ka . Whereas the field influence on
ka is slightly overestimated by our model the experimental
values of kc are even quantitatively reproduced up to thehighest field.
In the calculations, one can switch off the resonant scattering on magnetic excitations by setting C 苷 0. k ph 共C 苷
0兲 obtained in this way is much larger than the measured
thermal conductivity, as shown in the inset of Fig. 1. As
expected for resonant scattering the damping of the phonon
heat transport is most pronounced for T 艐 Ds . Note, however, that even at room temperature the suppression is sizable. This strong damping gives further evidence for alarge spin-phonon coupling in SCBO as has been inferred
before from sound wave anomalies [29].
In summary, the thermal conductivity of the lowdimensional quantum spin system SrCu2 共BO3 兲2 has acharacteristic double-peak structure with two maxima at
low temperatures and a pronounced magnetic field de047202-4

23 JULY 2001

pendence. A significant magnetic contribution to the heatcurrent can be excluded, since the double-peak structure
is not anisotropic and because the magnetic excitationsare (almost) localized. Our quantitative analysis in termsof resonant scattering of phonons on magnetic excitations
explains the double-peak structures and their magnetic
field dependence very well and gives evidence for strong
spin-phonon coupling. Therefore, we call for further
investigations whether the same mechanism is at work inother low-dimensional spin systems.
We acknowledge useful discussions with E. MüllerHartmann, M. Grüninger, and A. P. Kampf. This work
was supported by the Deutsche Forschungsgemeinschaft
in SFB 341 and SP 1073.

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[27] The more conventional scattering mechanisms where, e.g.,a phonon decays into two magnetic excitations are farless effective, basically because of the discreteness of themagnetic excitation spectrum.
[28] The coupling BSa Sb between two triplets on adjacentdimers leads to the total energies 2D0 2 2B, 2D0 2 B,
and 2D0 1 B for Stot 苷 0, 1, and 2, respectively.
[29] S. Zherlitsyn et al., Phys. Rev. B 62, 6097 (2000).

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