VOLUME 87, NUMBER 16

15 OCTOBER 2001

PHYSICAL REVIEW LETTERS

Dzyaloshinski-Moriya Interaction in the 2D Spin Gap System SrCu2共 BO3 兲 2
O. Cépas,1 K. Kakurai,2 L. P. Regnault,3 T. Ziman,1 J. P. Boucher,4 N. Aso,2 M. Nishi,2 H. Kageyama,5 and Y. Ueda 5
1
Institut Laue Langevin, BP 156, F-38042 Grenoble Cedex 9, France
Neutron Scattering Laboratory, ISSP, University of Tokyo, 106-1, Shirakata, Tokai, Ibaraki 319-1106, Japan
3
DRFMC-SPSMS, Laboratoire de Magnétisme et de Diffraction Neutronique, CEA, F-38054 Grenoble Cedex 9, France
4
Laboratoire de Spectrométrie Physique, Université J. Fourier, BP 87, F-38402 Saint Martin d’Hères Cedex, France
5
Materials Design and Characterization Laboratory, IRSP, University of Tokyo, 5-1-5 Kashiwa, Chiba 277-8581, Japan(Received 11 January 2001; published 2 October 2001)
2

The Dzyaloshinski-Moriya interaction partially lifts the magnetic frustration of the spin-1兾2 oxide
SrCu2 共BO3 兲2 . It explains the fine structure of the excited triplet state and its unusual magnetic fielddependence, as observed in previous ESR and new neutron inelastic scattering experiments. We claimthat it is mainly responsible for the dispersion. We propose also a new mechanism for the observed ESR
transitions forbidden by standard selection rules, which relies on an instantaneous Dzyaloshinski-Moriya
interaction induced by spin-phonon couplings.
DOI: 10.1103/PhysRevLett.87.167205

Strontium copper borate [SrCu2 共BO3 兲2 ] is a new example of a magnetic oxide with a spin gap [1], with aground state well described as simply a product of magnetic dimers in two dimensions on the bonds giving thestrongest magnetic exchange [2]. The weaker exchanges
are frustrated by the geometry and, as shown by Shastry
and Sutherland [3], the ground state of the isotropic Hamiltonian is independent of the value of the weaker exchange,
up to a critical value. The excitations, however, are notpurely local and cannot be explicitly given. Recent experiments by ESR [4] and neutron inelastic scattering presented here show how, in fact, there are spin anisotropies
needed for an accurate description of the dynamics. We
show the corrections to the ground state are needed and,while small, are necessary to understand many physicalproperties. For example, at finite external magnetic field
SrCu2 共BO3 兲2 appears to exhibit a number of finite magnetization plateaus [1,5], and the anisotropies determine theobservability of plateaus in different field directions. Furthermore, SrCu2 共BO3 兲2 is believed to be close in parameter
space to a quantum critical point whose nature is somewhat controversial, and while the anisotropies are small
they may be essential to its nature.
For spin 12 the leading anisotropic terms are of the
Dzyaloshinski-Moriya form [6] and exchange anisotropy.
The former is particularly relevant, since it may not be
frustrated even if the isotropic exchange is. While a small
Dzyaloshinski-Moriya interaction should not destroy a
gap generated by larger isotropic interactions it modifies
the pure locality of the ground state correlations anddelocalizes the first triplet excitation. This is because it
appears in lower order in perturbation theory than thefrustrated isotropic interactions. In this Letter we predict
the Dzyaloshinski-Moriya interactions that should be
expected in SrCu2 共BO3 兲2 from the structure and showthat they do indeed explain new features of the excitations observed with ESR and neutron inelastic scatteringexperiments.
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0031-9007兾01兾 87(16)兾167205(4)$15.00

PACS numbers: 75.10.Jm, 75.25. +z, 75.40.Gb

Miyahara and Ueda [2] have introduced the frustrated
Shastry-Sutherland model
XX
H 苷J
Si Sj 1 J 0
Si Sj
(1)
nn

nnn

for SrCu2 共BO3 兲2 , with S 苷 1兾2 and where nn stands fornearest neighbor spins and nnn for next nearest neighbors.
The lattice is shown in Fig. 1. J 苷 85 K and J 0 苷 54 K
are antiferromagnetic interactions estimated from the susceptibility and the gap [2]. The spectrum of spin excitations has several interesting features [7], in particular,
the existence of singlet bound states [8]. Figure 2a summarizes the gaps to the first excited states, calculated byexact numerical diagonalization of the finite size system(20 spins). In addition to the triplet state (solid line) calculated in [2], the energies of the two lowest singlet states(dashed lines) are given as a function of J 0 兾J. Theseroughly agree with recent calculation [9]: we find a ratio J 0 兾J ⯝ 0.62, comparing [10] the calculated ratio of
m1

m2

B
-D

-D

a

. +D

J
J’
-D

A

a

+D
J

+D .
Qb

b

c

J

ESR
(0,0)

Neutrons(1,0)

Qa

FIG. 1. Spin model including the Dzyaloshinski-Moriya interactions whose vectors are perpendicular to the plane. The arrows
show the order of the spins in the expression D ? 共Si 3 Sj 兲. At
right, the unit cell of dimers A and B with the classical groundstate with J 0 苷 0, and definition of the reciprocal space.

© 2001 The American Physical Society

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VOLUME 87, NUMBER 16

PHYSICAL REVIEW LETTERS

FIG. 2. (a) Gaps to the first excited states at q 苷 0 for D 苷 0(solid line triplets and dashed line singlets), calculated by exact
numerical diagonalization. (b) The effect of the DzyaloshinskiMoriya interaction on the first triplet for small J 0 . (c) For finite
J 0 , the exact diagonalization gives the energies of two reciprocal
points (the dots). (d) Renormalization of the splitting d and the
bandwidths W1 and W2 .

the energy of the Raman-active singlet (which is not thelowest singlet [9]) to the triplet excitation with the sameratio determined by experiment [7,8]. We find that theenergy of the lowest singlet crosses the ground state at
共J 0 兾J兲c ⯝ 0.68 (while the triplet state remains gapped), inagreement with the transition recently predicted [11], but
no evidence for the reported S 苷 1 instability [9]. Below
this value, the ground state of (1) is simply a product of
localized singlets.
Anisotropic behavior of the first triplet energy, according
to the direction of an external magnetic field, first appearedin ESR data [4], and cannot be explained by the fully
isotropic model. We show that the Dzyaloshinski-Moriya
coupling, which occurs in low-symmetry crystals, explainsthese features. In SrCu2 共BO3 兲2 , an almost perfect centerof inversion at the middle of the dimer bonds forbids the
Dzyaloshinski-Moriya interactions between the two spinsof a dimer. Each dimer is, however, separated from theneighboring dimer by a BO3 triangle for which there is no
center of inversion at the middle of the bond, allowing suchan interaction between the spins of the nearest neighbordimers. As the copper 共ab兲 plane is approximately a mirror
plane for the crystal structure the main components of D
must be perpendicular to the plane [6], thus lying along thec axis. We neglect the other components, expected to besmaller. Furthermore, using mirror planes perpendicular tothe copper plane and passing through the dimers (m1 andm2 in Fig. 1), we find an alternation of the D vector frombond to bond. Finally, the mapping of one dimer onto thenext one fixes the whole pattern of D vectors (Fig. 1). We
therefore predict an anisotropic term to the Hamiltonian:XHDM 苷
6 Dec 共Si 3 Sj 兲 ,(2)nnn

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15 OCTOBER 2001

where Si and Sj are spin operators which belong to next
nearest neighbors and the sign 6 depends on the bond(see Fig. 1). ec is the unitary vector in the c direction.
The magnitude can be estimated from Moriya’s argument
Dg
to be D ⬃ gc c J 0 , where Dgc ⯝ 0.28 has been measured
by ESR [4] and J 0 ⯝ 54 K [2]. This gives roughly D ⬃0.5 meV.
The Dzyaloshinski-Moriya interaction is not frustrated
because of the alternation of its vector; classically, inthe absence of the exchange interaction J 0 , the spins of
the neighboring dimers would be perpendicular. Quantum mechanically, the product of singlets on each dimer
is no longer the exact ground state. To first order the
groundstate has corrections proportional to two triplets
D Q
sa 共tb11 tc21 2 tb21 tc11 兲. These corrections are,aﬁ共b,c兲J
however, small and we checked by exact numerical diagonalization that the position of the critical point separating the two nonmagnetic ground states is not appreciably
changed.
In order to calculate the effect of the DzyaloshinskiMoriya interaction on the first triplet excitation, we startfrom small J 0 and from the exact ground state for D 苷 0.
At J 0 苷 D 苷 0, the first excited state is a localized triplet
which is separated from the ground state by an energy gap
J. Because of the frustration, a finite dispersion should
appear only at the sixth order in J 0 兾J [2]. In contrast, theDzyaloshinski-Moriya interaction is not frustrated and thedegeneracy of the localized triplets is lifted to first orderin D. We consider the two inequivalent triplet states forthe dimers A and B (Fig. 1). To linear order in J 0 and D,
we can write the Hamiltonian for the sector of total spinS z 苷 11 (respectively, 21) as a 2 3 2 matrix projectedon the basis vectors: (i) All dimers are in singlets bar one,which is in the triplet state, spin 11 (respectively, 21).
This triplet is on the sublattice of dimers A. (ii) The samewith the triplet on the sublattice of dimers B. One obtainsµ∂J
72iDf共q兲S z 苷61H苷,(3)
62iDf共q兲Jwhere f共q兲 苷 cos共qa a兾2兲 cos共qb a兾2兲. The dispersion of
the two modes 6 (each is twice degenerate with S z 苷61) is therefore proportional to D: vq6 共S z 苷 11兲 苷
vq6 共S z 苷 21兲 苷 J 6 2D cos共qa a兾2兲 cos共qb a兾2兲. The
Dzyaloshinski-Moriya interaction has no effect on the
S z 苷 0 component of the triplet, so that its energy remains
equal to J [vq 共S z 苷 0兲 苷 J] (Fig. 2b).
In particular, at q 苷 0, we have two upper (respectively, lower) modes S z 苷 61 with v共S z 苷 61兲 苷 J 1
2D [v共S z 苷 61兲 苷 J 2 2D]. A magnetic field parallel
to z k D k c splits these modes in four branches. This isin agreement with ESR when the magnetic field is parallelto the c axis (Fig. 3a). In such an experiment, however,an S z 苷 0 state cannot be seen as the external magnetic
field is tuned to adjust the energy of the state to the frequency of the propagating wave. We have, however, a clear
prediction for the energy of the mode S z 苷 0: it remains
exactly at the middle of the four v共S z 苷 0兲 苷 J. Such
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VOLUME 87, NUMBER 16

E (meV)

ESR: open dots
NIS : solid dots

2

H // aH // c
0

(b)

0

10

20 0

H (T)

10

20

H (T)

FIG. 3. The magnetic field dependence of the triplet energies:(a) H k c and (b) H k a; ESR data from [4], the dashed line
comes from [12]. The theory of the Dzyaloshinski-Moriya interaction explains these results (solid lines).

167205-3(a)10

Q = (1,0,0)
Q = (1,0,0)
(b)
H=0
H=6T

16
8

0

2 .0

4 .0

E (meV)

2 .0

4 .0

E (meV)0count s/ mon600000

a crucial test can be made by neutron inelastic scatteringexperiments: we come back to this point next.
A transverse magnetic field leads to the diagonalization of a 6 3 6 matrix which reduces to two equal
3 3 3 matrices in the basis of the zero-field eigenstates.
Therefore, the energies remain twice degenerate in atransverse magnetic field and are given by vq6 苷 J 6
p
4D 2 f 2共q兲 1 共g⬜ mB H⬜ 兲2 , vq0 苷 J. This form also fits
the ESR results in transverse magnetic field (Fig. 3b),apart from a small splitting in the high field regime which
may be accounted for by the differences in the g tensors
from the dimer A and B.
These expressions are first order in J 0 兾J (although corrections like DJ 0 兾J 2 may appear), which is not accurate
for SrCu2 共BO3 兲2 (J 0 兾J ⯝ 0.62). We performed exact numerical diagonalization to see how these results survive inthe strong coupling limit. We know, for instance, that the
gap between the ground state and the first triplet state isrenormalized to order 共J 0 兾J兲2 [2]. In addition, we find,for D 苷 0, a finite bandwidth W for the triplet dispersion. This bandwidth is smaller than that found previously
from a perturbative treatment [13]. We interpret this difference as a breakdown of the perturbative series close to
the quantum critical point. When D ﬁ 0, we calculated
again the splitting d between lower and upper modes atq 苷 0. The result d共J 0 兾J ! 0兲 苷 4D is renormalized byfinite J 0 (Fig. 2d). Taking the relevant parameters, we findd共J 0 兾J 苷 0.62兲 ⯝ 2.0D. On the other hand, the dispersion is changed from that calculated in the limit of small
J 0 : the energy of the S z 苷 0 mode acquires a small dispersion W 共D 苷 0兲 (Figs. 2c and 2d), though smaller thanthe S z 苷 61 dispersions.
In order to test the prediction of the energy of theS z 苷 0 mode, we compare it to neutron inelastic scattering measurements in the presence of a magnetic field,
up to H 苷 12 T. They have been realized on the threeaxes spectrometer CRG/CEA-IN12 installed on a cold
neutron guide at the Institut Laue Langevin. The final
wave vector was kept fixed at kf 苷 1.55A21 . A vertical curved pyrolytic graphite PG(002) monochromator anda horizontally curved PG(002) analyzer were used anda cooled beryllium filter was placed after the sample toeliminate higher-order contamination. The collimations
were 400 -open-open. A single crystal of about 15 3 6 3
6 mm3 was installed in a cryomagnet with the c axis
aligned along the vertical field. In the following, we referto the 共aⴱ bⴱ 兲 reciprocal plane defined in [7] (Fig. 1). In
reciprocal lattice units, the wave vector components are defined as Qa,b 苷 qa,b a兾共2p兲, where a is the lattice parameter. Examples of energy scans obtained at Q 苷 共1, 0, 0兲 forH 苷 0 and H 苷 6T are shown in Fig. 4. In Fig. 4a, weobserve a broad single peak at E ⯝ 3 meV, corresponding to the lowest triplet excitation branch (the signal above⯝4.4 meV corresponds to a higher triplet [7] as well as
the dashed line in Fig. 4b). For H 苷 0, it is of note thatthe 3 meV mode is appreciably broader than the energy
resolution (⯝0.2 meV at full width at half maximum). In
Fig. 4b, the application of H shows a Zeeman splitting of
this mode in five distinct lines. The five peaks become
resolution limited. Their energies are shown in Fig. 3a.
The energies of the S z 苷 61 excitations at Q 苷
共1, 0, 0兲 (solid dots) agree with the ESR field-dependent
branches [Q 苷 共0, 0, 0兲], confirming the translational
symmetry of the model. In addition, we find the energyof the S z 苷 0 mode of the triplet (solid squares) whichcannot be seen by ESR. It lies exactly at the middle of thefour other branches, as predicted above. Furthermore, theobserved intensities of the S z 苷 61 peaks are all equaland roughly 1兾4 of that of the S z 苷 0 peak (Fig. 4b).
We calculate the spin correlation function transverse to
the scattering wave vector which gives the intensities.
Because of the purely imaginary off-diagonal elements inthe matrix (3), the upper (S z 苷 61) and lower (S z 苷 61)
have the same intensity at zero field. Both are split inlongitudinal field into two modes of half the zero-field
intensity. Similarly, for the S z 苷 0 sector, there are, infact, two degenerate modes at zero field which remain
degenerate at finite field with the original weight. Eachweight is twice that of S z 苷 11 since only spin correlation functions transverse to the momentum transfer
contribute to the intensities. The intensity factor 1兾4 thenfollows. The agreement with experiment suggests that thecount s/ mon300000(a)

4

15 OCTOBER 2001

PHYSICAL REVIEW LETTERS

FIG. 4. Scans in the neutron experiment at Q 苷 共1, 0, 0兲:(a) the broad signal comes from three superimposed modes(dashed lines) and (b) Zeeman splitting of the modes.

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VOLUME 87, NUMBER 16

PHYSICAL REVIEW LETTERS

ratio of intensities is a good test of a mechanism based oncouplings of Dzyaloshinski-Moriya symmetry, even in theregime of large J 0 兾J.
Since the spectra are well fitted for both directions, we
can extract the coupling D from the zero-field splittingd 苷 0.352 6 0.008 meV. Using the numerical result d ⯝2.0D (Fig. 2d), this gives D ⯝ 0.18 meV, consistent withthe approximate value from Moriya’s formula, as given
earlier. We obtain the gap (which is no longer simply J
but the function shown in Fig. 2a) and the g factors in both
directions: gk 苷 2.24 6 0.09 and g⬜ 苷 2.00 6 0.09.
We then discuss the origin of the finite intensities in
ESR [4] and infrared absorption [12]. No magnetic transition can be induced from a purely singlet ground state.
Spin-orbit couplings generate anisotropies that may permit
transitions from the ground state to the original triplet excited states. First, note that Dzyaloshinski-Moriya interactions are not sufficient to explain the observed
transitions.
P
Indeed, the magnetic operator HM 苷 h i Si1 is odd under
the symmetry transformation: the reflection using the mirror plane m1 and the rotation of p around the dimer bond.
The ground state and the triplet state are, however, evenunder this transformation. The matrix element therefore
vanishes, leading to zero intensity. Second, the exchange
anisotropy only (both in J and J 0 ) leaves the ground stateas the product of singlets, leading again to zero intensity.
With both Dzyaloshinski-Moriya and exchange anisotropy,
the transition can exist but should be very small, of the order of l6 , where l is the spin-orbit coupling. Note alsothat an exchange anisotropy of order D 2 兾J always occurs
with a Dzyaloshinski-Moriya coupling [6], but unlike theunfrustrated case [14], here it does not eliminate the splitting which is linear in D. Third, we have mentioned thatthe anisotropy of the g tensors may be responsible for thevery small splitting of the modes in high field. A staggered field between dimers (due to perpendicular orientations of oxygen
copper ions) is writtenP
P ions surroundingHM 苷 hDg共 i[A Si1 2 i[B Si1 兲. This term is odd in
the transformation introduced above and transitions arethus forbidden for the same reasons. An intradimer staggered field should be very small giving an intensity weestimate as l2 a 2 , where a — a few degrees — is the tilting
angle of the crystal structure [15].
Consider now electric dipole transitions between
magnetic states. We restrict ourselves here to a phononassisted mechanism. A purely electronic mechanism may,in fact, also apply [16]. Treating the spin-phonon interactions perturbatively leads to an effective matrix element
between pure magnetic states j具 fjHE ji典j2 , where theeffective operator HP
E has now a correction of spin-orbit
origin [17]: HE 苷 nn gSi ? Sj 1 h ? 共Si 3 Sj 兲. The
first term comes from a transition via a virtual phonon

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15 OCTOBER 2001

as an intermediate state. The second term, which comesfrom the spin-orbit coupling should be jhj ⬃ lg, witha direction that depends on details of the phonons. Thismechanism gives an intensity proportional to h 2 , suggesting that the observed transitions may be electric dipole.
Further polarized experiments can test this: if the intensityof the transition changes in rotating the crystal aroundthe magnetic (respectively, electric) field of the wave, thetransition would be demonstrated explicitly to be electric
dipole (respectively, magnetic).
We have shown that the static Dzyaloshinski-Moriya interaction explains the magnetic field behavior of the first
triplet excitation (Fig. 3) and gives a dispersion (Fig. 2),
because it is not frustrated. We extracted its value fromthe zero-field splitting, taking into account a finite J 0 :D 苷 0.18 meV. Since the direction of the main anisotropy
is given by D k c, we expect clear magnetization plateaus
only when the magnetic field is applied along the c axis.
This is consistent with current results [5]. The static interaction does not, however, explain the finite intensity for the
ESR transitions. We presented an alternative mechanism
using a dynamical Dzyaloshinski-Moriya interaction.
We thank Professor H. Nojiri for providing us with his
ESR data and for interesting discussions.

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[4] H. Nojiri et al., J. Phys. Soc. Jpn. 68, 2906 (1999).
[5] K. Onizuka et al., J. Phys. Soc. Jpn. 69, 1016 (2000).
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(2000).
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