RAPID COMMUNICATIONSPHYSICAL REVIEW B

VOLUME 62, NUMBER 10

1 SEPTEMBER 2000-II

Sound-wave anomalies in SrCu2 „BO3 … 2S. Zherlitsyn,1 S. Schmidt,1 B. Wolf,1 H. Schwenk,1 B. Lüthi,1 H. Kageyama,2 K. Onizuka,2 Y. Ueda,2 and K. Ueda2
1

Physikalisches Institut, Universität Frankfurt, D-60054 Frankfurt, Germany
Institute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Tokyo 106-8666, Japan
共Received 11 July 2000兲

2The temperature and high magnetic field dependence of the longitudinal sound-wave mode c 11 in thetwo-dimensional dimer system SrCu2 (BO3 ) 2 is presented. c 11(T) shows anomalies due to strong interdimer
spin-strain coupling. We can quantitatively interpret the temperature dependence of c 11(T) together with themagnetic susceptibility ␹ m (T) with a molecular field approximation of coupled dimer triplets. The soundvelocity up to 50 T shows very sharp softening between the magnetization plateaus at low temperature. We
argue that these pronounced effects arise from a resonant interaction between the phonons and the magnetic
excitations which show softening between the plateaus.

The physics of low-dimensional spin systems is full ofsurprises. In zero magnetic field, one finds gapped groundstates for antiferromagnetic Heisenberg chains with integer
spins,1 or for dimerized spin 1/2 systems 共e.g., spin-Peierls
systems2兲, or for spin 1/2 two-leg ladders,3 to name just afew examples. Also, in the presence of a magnetic field instead of a smooth magnetization curve, one has found forsome classes of materials4 magnetization plateaus for rational fractions of the saturation magnetization. Examples given
are the quasi-one-dimensional Ni compounds,5 NH4 CuCl3 ,6and the two-dimensional SrCu2 (BO3 ) 2 .7
This latter compound has a two-dimensional structure
which consists of alternately stacked CuBO3 and Sr layers.
Within the CuBO3 plane, the Cu2⫹ -Cu2⫹ dimers are orthogonally connected, giving rise to an exact orthogonal
dimer ground state.8 This system is topologically equivalent
to the Shastry-Sutherland model.9 In the inset of Fig. 1, two
orthogonal dimers with the important exchange interactions
J,J ⬘ are shown. Numerical evaluation of the correspondingHamiltonian gives J⫽100 K, J ⬘ ⫽68 K.8,10,11 These authors also noted that the value J ⬘ /J⫽0.68 is very close to thevalue of a quantum phase transition (⬇0.7).
In this paper we report the following features for thiscompound: 共i兲 The longitudinal elastic constant c 11 exhibits astrong temperature dependence which allows us to study thestrain dependence and the elastic coupling between the Cudimers. An important result of this analysis leads to an estimate for a possible structural phase transition. 共ii兲 The soundvelocity in high magnetic fields up to 50 T exhibits characteristic features related to the magnetization plateaus, very
sharp minima before successive plateaus. A resonant interaction between the sound wave and the gap modes whichsoften between the plateaus might be responsible for theseanomalies. This evidence is direct proof for the softening of
the magnetic excitation between the plateaus. A theoreticaltreatment for these effects, especially a calculation of themagnetic excitation, is lacking so far.
In Figs. 1 and 2 we show the magnetic susceptibility andthe elastic constant c 11 as a function of temperature. Boththermodynamic quantities exhibit features typical for a lowdimensional spin gap system.
The magnetic susceptibility ␹ m , previously published,7,80163-1829/2000/62共10兲/6097共3兲/$15.00

PRB 62has a maximum at T max ⫽18 K and a strong decrease forlower temperatures indicative for a spin gap system. At hightemperatures, a Curie-Weiss behavior with ⌰⫽(J⫹4J ⬘ )/4⫽93 K is observed. For the following, it is interesting tonote that a RPA-molecular field expression12␹ m ⫽ ␹ 0 / 共 1⫺ j ␹ 0 兲共1兲gives a rather good fit for T⬍300 K. ␹ 0 is the magnetic
susceptibility of an isolated dimer, a singlet-triplet system:␹ 0 ⫽2e ⫺⌬/kT /kTZ with Z the partition function. The fit givesa singlet-triplet gap for thermodynamic quantities ⌬⫽29 K
rather close to the accepted value of 30 K,8 and j⫽⫺273 K. The interdimer coupling is j⫽4J ⬘ as expected.8
Instead of starting with the Hamiltonian with the parametersJ,J ⬘ 共Ref. 8兲 we used directly the dimer susceptibility ␹ 0with the renormalized value for ⌬⫽29 K⬍J⫽100 K.
In Fig. 2 we show the temperature dependence of thelongitudinal constant c 11 . For this mode the polarization andthe propagation is along 关100兴. With the sound velocity v
and the mass density ␳ ⫽4.1 g/cm3 one gets c 11⫽ ␳ v 2 . In
Fig. 2共a兲 we plot c 11(T) together with the temperature de0(T) taken from an empiricalpendent background c 1113evaluation. Similar to the magnetic susceptibility ␹ m , weFIG. 1. Temperature dependence of magnetic susceptibility.
Open circles: experiment from Ref. 7. Full line: fit with expressionEq. 共1兲. Inset: two neighboring dimers with exchange interactions
J,J ⬘ .
R6097

©2000 The American Physical Society

RAPID COMMUNICATIONS

R6098

S. ZHERLITSYN et al.

PRB 62

FIG. 3. Relative change of the elastic constant c 11 for the magnetic field sweep up 共solid circles兲 and sweep down 共open circles兲
measured at T⫽1.5 K. Field dependence of relative magnetization
m/m 0 at the same temperature is also shown 共solid line兲. Here m 0 isa saturation value of the magnetization. The magnetic field is applied along the a axis. The elastic constant minima occur between
the magnetization plateaus.
FIG. 2. 共a兲 Temperature dependence of the c 11 mode. Full line isbackground c 011 (T) as described in the text. The ultrasonic frequency was 98 MHz. 共b兲 „c 11(T)⫺c 011(T)…/c 011(T⫽0) as a functionof temperature. Open circles: experiment. Full line: fit from Eq. 共2兲.

can interpret the temperature dependence of the elastic constant c 11 with an analogous RPA expression.14 In analogy to
the magnetic susceptibility which is the response function of
the magnetization to an applied field, we can define the strain
susceptibility which is the response function to an appliedstrain.The elastic constant reads:
0c 11共 T 兲 ⫽c 11
共 T 兲 ⫺G 2 ␹ strwith共2兲

␹ str ⫽ ␹ s / 共 1⫺K ␹ s 兲 .
␹ s is the strain susceptibility of a single dimer,␹ s ⫽G ⫺2

⳵ 2F

3e ⫺⌬/kT

⳵⑀

kTZ 2

⫽
2,

共3兲

and K is the strength of the (k⫽0) dimer-dimer interaction.
The effective interaction K can be mediated, e.g., byphonons.
In Fig. 2共b兲 we compare ⌬c 11 /c 11⫽„c 11(T)
00(T)…/c 11(T⫽0) with a fit to ␹ str using the same ⌬⫺c 11⫽29 K. Here the minimum of the elastic constant is atT min ⫽13 K and the coupling constant K⫽23.8 K. The fit isalso very good. This analysis shows that with a three times
larger coupling constant K, a structural transition would takeplace. The single dimer coupling constant G⫽ ⳵ ⌬/ ⳵ ⑀ whichmeasures the strain dependence of the singlet-triplet gap
amounts to G⬇900 K 关with c 0 (T⫽0)⫽25.8⫻1011
erg/cm3 at low temperature, and the number of dimers n⫽0.74⫻1022 cm⫺3 ). So this simple RPA expressions account correctly for the T max and T min in ␹ m and ␹ str , respectively. It also gives a good overall fit for ␹ m and c 11 using thesame singlet-triplet gap. It further shows that the strain interactions are very important for this dimer compound.

Now we proceed to the magnetic field dependence. In Fig.
3 we show the relative change of the same elastic constantc 11 as a function of magnetic field up to 50 T at 1.5 K. The
ultrasonic technique recently developed for pulsed fields isdescribed elsewhere.15 In the present case the rise time of thepulse is 8 ms and the total pulse duration is 24 ms. In Fig. 3
both curves for field up and down are shown. No substantial
heating effects are discernible, we estimate it to ⌬T⬍0.5 K. For 1.5 K we observe a broad minimum at 25 T
followed by very sharp minima at 27 T, 36 T, and 42 T. In
this figure the magnetization for the same temperature is also
included. It clearly exhibits magnetization plateaus form/m 0 ⫽1/8, 1/4, and 1/3. It is seen that the sound velocity
minima are in the region where the magnetization changes
from one plateau to the next one.
The broad minimum at 25 T arises possibly from localized triplet excitations. Existence of a strong spin-phonon
coupling was pointed out recently by Ueda and Miyahara.16
If the spin-phonon coupling is strong enough there is thepossibility of the formation of a self-trapped triplet, a boundcomplex of triplet excitation, and local distortion 共tripletphonon-bound state兲. The energy of the self-trapped triplet
excitations is lower than the usual triplet branch, which mayexplain this broad minimum.
The first strong softening at 27 T is due to the interaction
between the sound wave and the soft triplet branch. Inelastic
neutron scattering determined ⌬⫽35 K.17 This leads to asoftening at 27 T with g⫽2, in agreement with our observation. We observed the same effect recently in (VO) 2 P2 O7 .18
The evolution of this sound velocity minimum with temperature is similar in both substances: Comparing ⌬c 11 /c 11 for
1.7 K and 7.4 K, we observe a small shift from 26.7 T to
27.4 T and a broadening with increasing temperature. This isseen in Fig. 4 where the field dependence of the relative
elastic constant is plotted for different temperatures. Thesharp minima between the plateaus disappear rapidly withincreasing temperature, and at 7.4 K only the first minimumsurvives.
We can estimate again the coupling constant ⳵ ⌬/ ⳵ ⑀ from

RAPID COMMUNICATIONSPRB 62SOUND-WAVE ANOMALIES IN SrCu2 BO3 ) 2FIG. 4. Relative change of the elastic constant c 11 versus magnetic field at three different temperatures. Only field sweeps up areshown. The curves were shifted arbitrarily for the sake of clarity.

the softening of the first triplet at 27 T where ⌬c 11 /c 11⬇2.8%. Since the first magnetic excitation shows nodispersion17 we can again use the free energy of weakly
coupled dimers. We get ⳵ ⌬/ ⳵ ⑀ ⬇700 K in rough agreementwith the estimate from the c 11 temperature dependence givenabove. This indicates that both effects are due to an exchange
striction coupling.
Now we discuss the additional steep minima effects at 36
T and 42 T, the steep minima before each plateau (m/m 0F. D. M. Haldane, Phys. Lett. 93A, 464 共1983兲; Phys. Rev. Lett.
50, 1153 共1983兲.
2
For a review, see J. P. Boucher and L. P. Regnault, J. Phys. I 6,1939 共1996兲.
3
For a review of ladder systems, see E. Dagotto and T. M. Rice,Science 271, 618 共1996兲.
4
M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev. Lett. 78,
1984 共1997兲; K. Totsuka, Eur. Phys. J. B 5, 705 共1998兲.
5
R. Sato et al., J. Magn. Magn. Mater. 177-181, 645 共1998兲.
6
W. Shiramura et al., J. Phys. Soc. Jpn. 67, 1548 共1998兲.
7H. Kageyama et al., Phys. Rev. Lett. 82, 3168 共1999兲.
8S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 共1999兲.
9B. S. Shastry and B. Sutherland, Physica B & C 108B, 1069
1R6099⫽1/4,1/3). We argue that these are likewise resonant effects
between the sound wave and the corresponding soft mode
before the plateau is reached. Within the plateau the excitation modes have a gap, thus keeping the magnetization constant. While we are not aware of any theory describing theseexcitation modes and their interaction with the elastic modes,
we can give the field dependence of our sound velocities
between the plateaus: ⌬c 11 /c 11⬀(B⫺B c ) ␥ with ␥ ⬇0.5 forthe resonances before the plateaus for B⬎B c at m/m 0⫽1/4,1/3.
In summary, we have given temperature and field dependencies of a longitudinal elastic constant in the twodimensional compound SrCu2 (BO3 ) 2 . We find considerable
phonon induced dimer-dimer coupling and a strain dependence of the singlet-triplet gap of 700–900 K in rough agreement with exchange striction coupling constants ( ⳵ J/ ⳵ ⑀⬇⫺9J).14 The sharp softening of c 11 between the plateaus
indicates a resonant interaction with the magnetic excitation
spectrum which softens between the plateaus. Further investigations of other elastic modes can help in elucidating thestrain-magnetic excitation interaction in this material. It isclear that phonon effects are very important in this compound. A more detailed theoretical treatment of this interaction and of the magnetic excitations is needed.
This research was supported in part by a grant fromBMBF 13N6581A/3 and by a Grant-in-Aid for Encouragement of Young Scientists from the Ministry of Education,
Science, Sports and Culture, Japan. A helpful discussion
with G. S. Uhrig is gratefully acknowledged.

共1981兲.
S. Miyahara and K. Ueda, Phys. Rev. B 61, 3417 共2000兲.
11
Z. Weihong et al., Phys. Rev. B 60, 6608 共1999兲.
12
R. M. White, Quantum Theory of Magnetism 共Springer, Berlin,
1983兲.
13
Y. P. Varshni, Phys. Rev. B 2, 3952 共1970兲.
14
B. Lüthi, in Dynamical Properties of Solids, edited by G. K.
Horton and A. A. Maradudin 共North-Holland, Amsterdam,
1980兲.
15B. Wolf et al., Europhys. Lett. 48, 182 共1999兲.
16K. Ueda and S. Miyahara 共unpublished兲.
17H. Kageyama et al. 共unpublished兲.
18
B. Wolf et al., J. Appl. Phys. 87, 7055 共2000兲.
10