VOLUME 86, NUMBER 21

PHYSICAL REVIEW LETTERS

21 MAY 2001

Soft Acoustic Modes in the Two-Dimensional Spin System SrCu2 共 BO3 兲 2
B. Wolf,1 S. Zherlitsyn,1 S. Schmidt,1 B. Lüthi,1 H. Kageyama,2 and Y. Ueda 2
2

1
Physikalisches Institut, Universität Frankfurt, D-60054 Frankfurt, Germany
Institute for Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan(Received 18 January 2001)

SrCu2 共BO3 兲2 is a two-dimensional dimerized quantum spin system which is close to a quantum criticalpoint. The sound velocity for the longitudinal and transverse acoustic modes shows strong spin-latticeeffects. The shear c66 mode exhibits a pronounced softening of 4.5% as a function of temperature andsoftens more than 25% in fields up to 50 T. This huge effect occurs in the vicinity of the magnetization plateaus m兾m0 苷 1兾4 and 1兾3. We can analyze quantitatively the temperature dependence of all
measured elastic modes c11 , c44 , and c66 with an exchange striction mechanism. The soft c66 mode with
B2g symmetry enables us to predict the possible symmetry of the condensed triplets in some plateaus.
DOI: 10.1103/PhysRevLett.86.4847

The physics of low-dimensional spin systems has a long
history beginning with an analysis of the antiferromagnetically coupled spin chain by Bethe in 1931. In recent
years many new effects were found especially for onedimensional spin systems. Many studies deal with the energy gap between the nonmagnetic ground state and thefirst excited state, e.g., in antiferromagnetic Heisenberg
chains with integer spin or dimerized spin 1兾2 systems.
An external magnetic field can also produce spin gaps,
which are seen as quantization steps in the magnetization
curve. Recently Oshikawa et al. [1] gave the condition forthe occurrence of magnetization plateaus in a spin chain:n共S 2 m兲 苷 integer, where n is the period of the spin
ground state, S is the spin, and m is the magnetization
per magnetic ion. But also in the two-dimensional system SrCu2 共BO3 兲2 magnetization plateaus were observed.
Of particular importance is the study of two-dimensional
spin systems, mainly because of its relevance to high temperature superconductivity. Very few examples of 2D spin
systems are known up to now.
SrCu2 共BO3 兲2 is a two-dimensional coupled dimer system with remarkable properties. This tetragonal compoundconsists of alternately stacked CuBO3 and Sr layers.
Within the CuBO3 plane, the Cu21 -Cu21 dimers are
orthogonally connected, giving rise to an exact orthogonal
dimer ground state [2] (see Fig. 1) according to the
Shastry-Sutherland model [3]. In the magnetic field
SrCu2 共BO3 兲2 exhibits plateaus in the magnetization at
1兾8, 1兾4, and 1兾3 of the saturation magnetization [4].
Furthermore, it is close to a quantum critical point (QCP)
for the exchange parameters J 苷 100 K, J 0 苷 68 K [2,5](for a definition of J, J 0 , see Fig. 1). In zero magnetic
field the singlet-triplet splitting is D ⬵ 35 K deduced
from neutron scattering (with negligible dispersion) [6],from ESR [7], and from Raman scattering [8].
Recently there have been several theoretical papers investigating the quantum critical point for such a system.
Apart from the dimer singlet ground state for approximately J 0 兾J , 0.7 and the possible magnetic ground state
0031-9007兾01兾86(21)兾4847(4)$15.00

PACS numbers: 62.65.+k, 75.45.+j

for J 0 兾J . 0.7 various papers discussed the possibility ofintermediate phases such as helical spin arrangements [9],plaquette singlet phase [10], or second order phase transitions with nontrivial critical exponents [11]. Further developments give quantitative calculations for excited states[12] in zero magnetic field.
We want to make several important points in this Letter.
As we show below very dramatic sound velocity phenomena have been discovered both as a function of temperatureand high magnetic field indicating that the dimer system
as shown in Fig. 1 is very sensitive to specific ionic displacements. Therefore, care has to be taken in applying thetheoretical results for the QCP to the case of SrCu2 共BO3 兲2because the lattice degrees of freedom have not been taken
into account up to now. Other important features which we
1

3

J
2

4

J'

q
e(c66)e(c11)yxe(c44)

FIG. 1. Geometric arrangement of the dimers and of the different elastic modes. Here q is the ultrasound wave vector; e共cii 兲 isthe polarization for the different acoustic cii modes. In the caseof the c44 mode the polarization is parallel to the z direction.

© 2001 The American Physical Society

4847

PHYSICAL REVIEW LETTERS

Eexs 苷

X

关J共d 1 ui 2 uj 兲 2 J共d兲兴Si ? Sj .

(1)

ij

Here d 苷 Ri 2 Rj measures the distance between two
magnetic ions and ui is the displacement vector for the ion
Ri . With a sound wave given by u 苷 Ue exp关i共q ? r 2
vt兲兴, where e is the polarization vector, U is the amplitude, v is the frequency of the sound wave, the exchange
4848

21 MAY 2001
c44

0

c11
-5

-2

∆ c/c (10 )outline below deal with a quantitative description of thetemperature dependence of the elastic constants. Finallythe dramatic softening of some symmetry elastic constantsin high magnetic fields enables us to predict the symmetryof the ground state for the various plateaus.
Recently we have established with ultrasonic experiments a new spectroscopic tool suitable for pulsed highmagnetic fields. In a series of papers [13–15] we haveshown that with ultrasonic velocity and attenuation we caninvestigate soft magnetic modes, magnetoacoustic quantum oscillations, and different phase transitions in pulsed
fields up to 50 T. This technique can be complementary
to ESR in pulsed fields [16] because it covers the low
frequency region. The longitudinal mode c11 was usedpreviously to investigate the softening of the magnetic excitations in SrCu2 共BO3 兲2 [15]. This was possible throughthe resonant coupling of the sound wave and the soft magnetic excitations between the magnetic plateaus.
The sample has a size of about 3.9 3 4 3 1 mm3 . Two
opposite surfaces normal to the [100] crystallographic
direction were polished for the ultrasonic experiments.
We used resonance quartz transducers with the fundamental frequency of 10 MHz glued by liquid polymer(Thiokol-32) in the case of a transverse sound excitation.
A broad frequency band piezoelectric film glued on thesample with a two component epoxy was used for theexcitation of the longitudinal sound waves. The experimental setup is described in Ref. [14].
In Fig. 2 we show our main results obtained at 1.5 K,
the relative elastic constant changes as a function of applied pulsed fields B k a up to 50 T for the elastic modes
c11 , c44 , and c66 . All three modes propagate in the [100]direction with the polarization vectors e along [100] forthe longitudinal c11 mode, along [001] out of the tetragonal plane for the shear c44 mode, and along [010] for the
shear c66 mode as indicated in Fig. 1. Clearly the in-planemode c66 shows a dramatic softening with increasing field
between the plateaus of more than 25% compared to 2.5%
for c11 and no softening at all for c44 . For c66 and c11
we notice sharp spikes at the corresponding field values
where the magnetization changes from one plateau to thenext one, i.e., at 27, 36, and 43 T for B k a.
What is the reason for the dramatic softening of c66 andwhat makes the big differences in the different modes?
As for the c11 investigated before in detail [15], we showthat exchange striction effects are responsible for these
phenomena. This exchange striction coupling arises fromthe strain dependence of the exchange interaction [18]:Sr Cu2(B O3)2

-10

c66

-15
(a )

-20
-25
0.3

m /m 0VOLUME 86, NUMBER 21

(b )

0.0
0

10

20

30
B(T)

40

50

FIG. 2. Relative change of the elastic constant cij (a) as a function of high magnetic field in SrCu2 共BO3 兲2 at T 苷 1.5 K. Magnetic field direction was B k 关100兴. The ultrasonic frequency
was 10 MHz (c44 and c66 modes) and 86 MHz (c11 mode). Also
shown is the magnetization (b) for B k 关100兴 from Ref. [17].

striction term reads
X
i共q?R2vt兲
Eexs 苷
U共dJ兾dd ? e兲 共q ? d兲 共Si ? Si1d 兲ei.
i(2)Here we have expanded the exponential to first orderbecause qd ø 1 for sound waves of 100 MHz. Usually Eq. (2) tells us [共dJ兾dd ? e兲 苷 0 or 共q ? d兲 苷 0] thatshear waves propagating along symmetry directions do notcouple with this mechanism whereas longitudinal waves
do. However, with Eq. (2) and the geometry of Fig. 1 it isevident that the c66 mode can couple to the spin system via
Eq. (2), whereas c44 can couple only in higher order. Explicit calculations give for example the distance between
the magnetic ions 12 in Fig. 1. It is given by the following
expressions:pdc11 and c66 modes: 12 ⬵ 2 a关1 1 2a 兴 ,pd2
c44 mode: 12 ⬵ 2 a关1 1 4a2 兴 ,pwhere 2 a is the undisturbed distance and d is proportional to the amplitude U.
It is seen that the c11 and c66 modes couple linearly inthe strain or amplitude U to the spin system whereas c44
couples only quadratically in the strain. These considerations explain the absence of any essential field dependent
effect for c44 . They explain also the temperature dependence of the various elastic modes shown in Fig. 3.

VOLUME 86, NUMBER 21

PHYSICAL REVIEW LETTERS

the c11 mode (see Fig. 3). xs for c11 was given in Ref. [15]and xs for c66 readsΩµ ∂2 æ
d2Z
1 d2F
kT
dZ
xs 苷 2 2
, (4)
苷 2 Z
2
2
2
Z
d´
d´
G1 d´

Sr Cu2(B O3)2

c44
0

-2

∆ c/c (10 )

c11

-2

-4

c66
0

20

40

60

80

100

T(K )
FIG. 3. Temperature dependences of the elastic constants cij
in SrCu2 共BO3 兲2 (symbols) and fit (solid line) made in assumption of exchange striction coupling (see text for details). The
ultrasonic frequencies were 10 MHz (c44 mode), 47 MHz (c66mode), and 98 MHz (c11 mode).

Figure 3 gives cij 共T 兲 for c11 , c44 , and c66 acoustic
modes. Plotted is the temperature dependence of relative change normalized by the cij value at T 苷 1.5 K:
关cij 共T 兲 2 cij 共1.5 K兲兴兾cij 共1.5 K兲. The absolute values of
cij 共1.5 K兲 are given in Table I. The temperature dependence of the c11 mode was already discussed together
with the magnetic susceptibility xm before [15]. A RPAmolecular field expression for xm and c11 could explain
the temperature dependence quantitatively. For the c11 andc66 modes the expressions readcii 共T 兲 苷 ciio 共T 兲 2 G12 Nxstr ,(3)

where xstr 苷 xs 兾共1 2 Kxs 兲. K is the strength of the(q 苷 0) dimer-dimer interaction, xs is the strain susceptibility of a single dimer, N is the density of dimers, andG1 苷 j≠D1 兾≠´j is a single dimer coupling constant as explained in Ref. [15]. For the c11 mode we have to consider
only the singlet and the first excited triplet state we takeD1 ⬵ 30 K [15]. For the c66 mode we have to include
the next higher triplet state at D2 苷 58 K [6] because c66
couples very strongly to this level. This explains the higher
minimum position compared to c11 . In addition the softening of the c66 mode begins at temperatures higher as for
TABLE I. Values of some elastic constants cij in SrCu2 共BO3 兲2[15] at T 苷 1.5 K and some exchange striction coupling
constants.
cij

共1011 erg兾cm3 兲

K (K)

G1 (K)

G2 (K)

c11
c66
c44

25.8
8.2
0.96

23
219
···

970
1300
···

···
3770
···

≠2 D1
≠´2

21 MAY 2001

(K)

···
···
1680

where Z 苷 1 1 3 exp共2D1 兾kT 兲 1 3 exp共2D2 兾kT 兲. The
fit of Eq. (3) to the experimental results in Fig. 3 is rather
good for c11 and c66 . In order to show our original experimental results we did not subtract temperature background from our experimental data. Instead we found thisbackground, using empirical equations [19], and added it
to the exchange striction fit [see Eqs. (3) and (4)]. No
background was added for the c66 mode fit because of thevery large (in comparison to the background) softeningof this mode. It turns out that the effective interaction
K is ferrodistortive for c11 (K 苷 23 K) and antiferrodistortive for c66 (K 苷 219 K). The coupling constant G2 苷j≠D2 兾≠´j is 2.9 times larger than G1 (see Table I).
For c44 we know from above that the coupling is ofhigher order in the strain. Therefore instead of a coupling
共1兾J兲≠J兾≠´ 苷 共1兾D兲≠D兾≠´ we have a higher ordercoupling constant 共1兾J兲≠2 J兾≠´2 苷 共1兾D兲≠2 D兾≠´2 andwe get∂µ3N ≠2 D1
2D1
0
c44 苷 c44 1
.
(5)
exp
Z ≠´2
kT
This expression gives again a good fit to the experimental results as seen from Fig. 3 with a value of the coupling
constant ≠2 D1 兾≠´2 苷 1680 K. In summary, we conclude
that the anomalous temperature dependence of the elastic
constants is adequately described with this exchange striction coupling. Now we return to Fig. 2, the high field effects on cij . Again we can argue that c44 should not couple
to the magnetic excitation in lowest order as discussed
above. Experimentally it does not show any magnetic fielddependence except for a small hardening between eachplateau of 0.1%. On the other hand Eq. (2) shows that
one can have a resonant interaction between phonons andmagnetic excitations because of the special geometry ofq and B. In the presence of high magnetic fields B 苷y y
共Bx , 0, 0兲 we get Si Sj 苷 Six Sjx 1 Si Sj 1 Siz Sjz 苷 共具S典 1yydSix 兲 共具S典 1 dSjx 兲 1 dSi dSj 1 dSiz dSjz . Therefore we
can have bilinear phonon-magnon coupling for the c66 andc11 modes such as dS x ´ii or H ⬃ g共aq1 bq 1 aq bq1 兲 witha, a1 , b, b 1 magnon and phonon operators. Such a resonant interaction close to the crossover leads to coupledphonon-magnon modes of the form [18]共v 2 vm 兲 共v 2 yq兲 苷 g2 ,(6)

where g is proportional to the coupling constant, y is
the sound velocity, and q is the wave number. vm 苷
gjB 2 Bc j is the magnetic excitation mode which follows
a linear field dependence up to 27 T as shown by ESR
measurements [5]. Magnetic excitations in higher fields
have not been observed yet. From Eq. (6) we get for the4849

VOLUME 86, NUMBER 21

PHYSICAL REVIEW LETTERS

relative velocity changeDyvm 兾v0 1 1
苷
y
s 2
µ ∂2
共vm 兾v0 2 1兲2
g
2
1
2 1.
4
v0

(7)

Here v0 is the acoustic frequency. Equation (7) canonly qualitatively describe the sound velocity as a functionof the magnetic field for the c11 and c66 modes.
The difference between the effects of the c11 and c66
modes which amount to an order of magnitude must be explained by a detailed microscopic consideration of the exchange striction coupling constant in the two cases. Simple
length change effects give the same magnitude for the twomodes as outlined above. Angular dependences of the exchange and other factors have to be taken into account fora full description of such a coupling. Such a calculation
has not yet been performed.
We can try to describe the huge difference between c11
and c66 in another way in analogy to the soft mode concepts. We explain it similarly to the recently discussed
charge ordering phenomena and the sound wave-charge
fluctuation coupling [20,21]. In these cases the symmetry of the corresponding elastic constant (c44 for Yb4 As3
and c66 for NaV2 O5 ) predicts the symmetry of the frozen
charge order (body diagonal for Yb4 As3 and zigzag structure for NaV2 O5 ). Likewise in our case of SrCu2 共BO3 兲2
we can argue that the soft c66 mode predicts the symmetryof the resonantly coupled magnetic excitation and consequently of the triplet condensation for the corresponding
plateaus as calculated theoretically [2]. Taking the data ofFig. 2 we predict especially for the plateaus 1兾4 (aboveB 苷 36 T) and 1兾3 (above 43 T) a B2g -type symmetry arrangement. Calculations of such an arrangement are given
in Ref. [2] for the square unit cell. The hard core bosonmodel gives rather an arrangement of B1g symmetry forthe 1兾4 plateau. The plateau for 1兾8 should give a mixtureof B1g and B2g symmetry.
We have shown that the exchange striction coupling canaccount quantitatively for the temperature dependence ofthe three acoustic modes; it can also explain the absence ofa magnetic field effect for c44 and for the resonant couplingof the sound wave with the magnetic excitations for thec11 and c66 modes. There are other possibilities such as asingle ion magnetoelastic coupling. But for S 苷 1兾2 spin
the quadrupolar matrix elements are zero and therefore thiscoupling should give only small effects.
In summary, we have given evidence for a very dramatic spin-lattice coupling in the two-dimensional dimer
substance SrCu2 共BO3 兲2 . The in-plane shear mode exhibitsa very strong softening at magnetic field values between
the magnetization plateaus, but also as a function of temperature. This means that the Cu21 -Cu21 dimers are verystrain dependent especially for the ´xy strain. Any theory
describing the magnetism of this system has to include thespin phonon coupling. This has to be done for describing
4850

21 MAY 2001

the magnetic excitations, the general phase diagram, andespecially for parameter choices which include the quantum critical point.
We gratefully acknowledge helpful discussions with
G. S. Uhrig. This research was supported in part by a grantfrom BMBF 13N6581A/3, DFG Schwerpunktprogramm
1073, and a Grant-in-Aid for Encouragement of Young
Scientists from the Japanese Ministry of Education,
Science, Sports, and Culture.

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