INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 16 (2004) S911–S916

PII: S0953-8984(04)74171-5

The effects of intra-dimer Dzyaloshinsky–Moriya
interaction on the properties of SrCu2 (BO3 )2 in an
external magnetic field
S Miyahara1,6 , F Mila2 , K Kodama3, M Takigawa3 , M Horvatic4,
C Berthier4 , H Kageyama3,5 and Y Ueda3
1 Department of Physics, Aoyama Gakuin University, Sagamihara, Kanagwa 229-8558, Japan
2 Institut de Physique Théorique, Université de Lausanne, CH-1015 Lausanne, Switzerland
3 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
4

Grenoble High Magnetic Field Laboratory, CNRS and MPI-FKF, 38042 Grenoble, France

5 Department of Chemistry, Graduate School of Science, Kyoto University,

Kyoto 606-8502, Japan
E-mail: miyahara@phys.aoyama.ac.jp

Received 7 January 2004
Published 4 March 2004
Online at stacks.iop.org/JPhysCM/16/S911 (DOI: 10.1088/0953-8984/16/11/048)
Abstract
The two-dimensional spin-gap system SrCu2 (BO3 )2 shows unique physicalproperties due to the strong quantum fluctuations that follow from its lowdimensionality and its very special topology. The magnetic properties ofthis material, e.g. dimer singlet ground state, an almost localized triplet andmagnetization plateaux, are well described by a spin-1/2 antiferromagnetic
Heisenberg model on the orthogonal dimer lattice. Recently, an unusual
angular dependence of the shifts at the B sites with the external magnetic
field rotated in the (110) plane, which indicates the presence of a field-induced
staggered moment, has been observed by nuclear magnetic resonance. Such abehaviour cannot be explained by the Heisenberg model alone. We include
an intra-dimer Dzyaloshinsky–Moriya interaction as well as an anisotropic
gyromagnetic tensor in the orthogonal dimer model, and we show that theobserved field-induced staggered moment can be very well described withphysically reasonable values of the parameters.

1. Introduction
After the re-discovery of the two-dimensional spin-gap system SrCu2 (BO3 )2 [1], much
experimental and theoretical research has been done on this material [2]. The unique magnetic
properties of this material, (1) a dimer singlet ground state, (2) an almost localized triplet
6 Author to whom any correspondence should be addressed.

0953-8984/04/110911+06$30.00© 2004 IOP Publishing Ltd

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S911

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S Miyahara et al

b
3
1

y

2
a
4
3

x
J
J’

C4

D
1
4

2

D’

Figure 1. The orthogonal dimer model for SrCu2 (BO3 )2 . The unit cell contains four inequivalent
spin sites. The direction of the possible Dzyaloshinsky–Moriya interactions are also indicated.

excitation and (3) magnetization plateaux, are well described by a spin-1/2 antiferromagnetic
Heisenberg model on the orthogonal dimer lattice (figure 1). However, some features cannot
be explained by the Heisenberg model alone. For example, a splitting of the lowest excitedtriplet state, which has been observed by electron spin resonance (ESR) [3] and inelastic
neutron scattering [4] experiments, can be well described by including Dzyaloshinsky–Moriya
interactions on next nearest neighbour bonds D  [4]. Recently, an unusual angular dependenceof shifts at the B sites, with an external magnetic field rotated in the (110) plane, has been
observed by nuclear magnetic resonance (NMR) [5]. This is another property which cannot
be explained by the Heisenberg model alone. In this paper, we include a Dzyaloshinsky–
Moriya interaction on nearest neighbour bonds D and an anisotropic gyromagnetic tensor in
the orthogonal dimer Heisenberg model, and discuss their effects in the presence of an external
magnetic field.
2. NMR results
We have measured anisotropic shifts at the B sites with the external magnetic field H in theplane including the c-axis and the x direction. The angular dependence of the shift K 1 atseveral temperatures and for a magnetic field of 7 T are shown in figure 2(a). The shifts
K i (i = 1, 2, 3 and 4, see figure 1) reflect the nature of the local magnetization on Cu sites
because of the nearest neighbour hyperfine couplings between B and Cu sites (we may neglect
the other hyperfine couplings as the first approximation). If the ground state is a product of
singlets, any spin response should be very small at T = 2.5 K, which is much lower than theestimated value of the spin gap 35 K. However, the data indicate that the field induces staggeredmagnetization. The evidence for staggered magnetization can be gained most clearly from thetemperature dependence of the angular variation of (K 1 − K 2 )/2 (figure 2(b)). The amplitudeof the angular variation (K 1 + K 2 )/2 and (K 3 + K 4 )/2 approximately follow the temperaturedependence of the bulk susceptibility. However the amplitude (K 1 − K 2 )/2 does not decreaseas rapidly as other components and has a finite amplitude around T = 0 K. Note that theamplitude of the angular variation of (K 1 + K 2 )/2 and (K 3 + K 4 )/2 is strongly coupled to theuniform magnetization through the diagonal component of the hyperfine tensor. On the other
hand, (K 1 − K 2 )/2 is coupled to the staggered components.
3. Model Hamiltonian
In Cu-benzoate [6], a field-induced staggered magnetization has previously been proposed bya theory based on an effective staggered field resulting from a staggered gyromagnetic tensor

The effects of intra-dimer Dzyaloshinsky–Moriya interaction on the properties of SrCu2 (BO3 )2(a) 0.1(b)(K1+K2)/2(K3+K4)/2(K1−K2)/2
H=7T

0.2

K 1 (%)

Amplitude (%)

0

-0.1

T=2.5K
T=6K
T=10K
T=20K
T=100K

-0.2
-90

-60

S913

0.1

0
-30

0
30
θ (deg.)

60

90

0

50

100

150

200

250

300

T (K)

Figure 2. (a) The anisotropic shifts at the 11 B sites with the magnetic field of 7 T rotated in the(110) plane at T = 2.5, 6, 10, 20 and 100 K. The shifts K 1 reflect a spin state at site 1. Curves
are guides to the eyes. (b) The temperature dependence of the angular variation of (K 1 + K 2 )/2,
(K 3 + K 4 )/2 and (K 1 − K 2 )/2.

and a Dzyaloshinsky–Moriya interaction [7, 8]. In the same spirit as in [7, 8], we include
a staggered gyromagnetic tensor and a Dzyaloshinsky–Moriya interaction in the orthogonal
dimer model. Note, however, that there are important differences between the two systems.
In particular, Cu-benzoate is an antiferromagnetic spin chain with a magnetic ground state,
and in which a magnetic field opens a gap, whereas SrCu2 (BO3 )2 is a two-dimensional system
which already has a spin gap in zero field.
The model Hamiltonian of the orthogonal dimer model with Dzyaloshinsky–Moriya
interactions and staggered gyromagnetic tensors is written as



 y
y
H=J
si · s j + J 
si · s j + D y(s1z s2x − s1x s2z ) + Dx
(s3 s4z − s3z s4 )n.n.
n.n.n.
A-dimer
B-dimer−((ĝ1 H) · s1 + (ĝ2 H) · s2 + (ĝ3 H) · s3 + (ĝ4 H) · s4 ).
(1)
unitcellIn the model, there are two inequivalent dimers, horizontal (A-dimer) and perpendicular(B-dimer) ones. Thus the unit cell contains four inequivalent spin sites, i.e. four inequivalentgyromagnetic tensors. If the CuBO3 planes were flat and had mirror symmetry, the J -bonds
would have an inversion centre and the Dzyaloshinsky–Moriya interaction would not beallowed. However, in the real material at low temperatures (<395 K), there is a bucklingin the plane, and the mirror symmetry with respect to the ab plane is lost [9, 10]. Thus
the centre of the J bonds is not an inversion symmetry point and the components of the
Dzyaloshinsky–Moriya interaction shown in figure 1 can exist [11].
The Dzyaloshinsky–Moriya interactions Dx and D y are connected to each other bysymmetry operations, namely the rotation C4 (see the figure) followed by the mirror I(ab-plane), leading to the relation Dx = −D y . The gyromagnetic tensor on site 1 is written as gξ cos2 ϕ + gζ sin2 ϕ 0 (gξ − gζ ) cos ϕ sin ϕg x 0 gs(2)ĝ1 == 0 gy 00gη0(gξ − gζ ) cos ϕ sin ϕ 0 gξ sin2 ϕ + gζ cos2 ϕ
gs 0 g z
based on a symmetry analysis of the crystallographic structure [9, 10]. We set the x, y, z
coordinates such that z is along the c-axis and x is along the [110] direction (see figure 1).

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S Miyahara et al

The diagonal elements of the g-tensor with respect to its principal axes are denoted by gξ , gηand gζ . One of the principal axes is normal to the mirror plane (the (110) plane) and one otherprincipal axis is tilted slightly from the z-axis. Here the angle ϕ is the angle between the ζ -axis
and the z-axis. The other three tensors are related to the tensor of equation (2) by symmetry
operations, and the results are






0 −gs
0
0
gx
gy
gy 0 0ĝ2 =,ĝ3 = 0ĝ4 = 0 gx gs .
gx −gs ,0gy0gz
0 −gs gz
0 gs g z−gs 0(3)
From ESR experiments, the diagonal terms can be estimated to be gx = g y = 2.05 andgz = 2.28 [12]. The off-diagonal elements gs are staggered, i.e. they are opposite on sites
1 and 2, respectively 3 and 4. Since it is difficult to estimate the value for gs directly fromexperiments, we treat it as a parameter in the model.
4. Ground state
Without Dzyaloshinsky–Moriya interactions and staggered gyromagnetic tensors, the groundstate of the orthogonal dimer Heisenberg model is the product of the dimer singlets [13, 14],

 1
|s =
|si =(4)√ (|↑↓i − |↓↑i ).
2iiHere i denotes nearest-neighbour bonds.
To clarify the effects of Dzyaloshinsky–Moriya interactions and staggered gyromagnetic
tensors, we performed a perturbation calculation in the limit Dx /J, gs H /J  g x H /J  1.
Since the main magnetic properties can be well described by the Heisenberg model, it is natural
to assume that Dx and gs H are much smaller than J . In the presence of an external magneticyalong the [110] direction H = (H sin θ, 0, H cos θ ), the expectation values Six , Si  andzSi  are given to first order in D y /J and gs H /J by:
gs H cos θ D y gz H cos θ+S1x  = −
(5)

22
x
x
(6)
S2  = −S1 
g
D
H
sin
θ
g
H
sin
θ
s
y
x
S1z  =
(7)
−

22
z
z
(8)
S2  = −S1 
gs H cos θ D y gz H cos θyS3  = −
+
(9)

22
y
y
S4  = −S3 .
(10)
It is clear that the staggered magnetization is induced by the external fields. The expectationyyvalues of other components, S1 , S2 , S3x , S3z , S4x  and S4z , vanish.
Finally, let us consider the case H ∼ (0, 0, Hc ). Here Hc is the critical field at which
the spin gap would collapse in the absence of the Dzyaloshinsky–Moriya interactions andstaggered gyromagnetic tensor. In this field region, we may take the lowest energy states: |s(equation (4)) and |e1 (i ), where|s j .
(11)|e1 (i ) = |t1 ij =i

Although one triplet states (equation (11)) are degenerate, only the matrix element between the
ground state and a Fourier transformation of equation (11) with q = 0 can be non-zero, since

The effects of intra-dimer Dzyaloshinsky–Moriya interaction on the properties of SrCu2 (BO3 )2

S915

the ground state for the Heisenberg model |s is uniform. Thus in the perturbation calculationsin the third order for J  /J and first order for D y /J and gs H /J , the ground state on the A-dimer
bond is given by solving the 2 × 2 matrix:Dy+ √gs2 H
02  3   2
.
(12)
Dy
gs
√
− gz HJ 1 − JJ − 12 JJ
2 + 2HThe state on the A-dimer bond is nothing but the ground state for an isolated dimer with arenormalized spin gap. Note that a state on the B-dimer is also obtained through a similarcalculation. Thus the gap persists even at Hc and the magnetization can be finite below Hc .
This picture can be a good approximation for the fields around Hc, where we can neglect the
mixing of the bound state of two triplets to the ground state. These features are qualitatively
consistent with the results observed in the experiments [3, 12, 15, 16].
5. Conclusion
Including an intra-dimer Dzyaloshinsky–Moriya interaction D and staggered gyromagnetic
tensors, one can qualitatively explain the staggered magnetization which was observed in
NMR experiments. Let us comment on the effects of Dzyaloshinsky–Moriya interactions on
next-nearest-neighbour bond D  , which was included to explain the magnetic behaviours forlowest triplet excitations [4]. If we apply the D  term to the ground state, two of the singlet
states are promoted to triplets. Thus a non-zero expectation value for Siα  (α = x, y, z) only
appears at second order in D  /J , and it would be much smaller than that deduced from NMR
experiments since D  /J ∼ 0.02. In contrast, Siα  is proportional to D, and if we assume that
D ∼ D  , the magnitude of the calculated expectation values is consistent with the observed
one. On the other hand, in the lowest branches of excitations the effects of D  are more
important than those of D. If we apply the D  term to the state (11), the triplet can hop to
the neighbouring bonds, and the magnitude of the splitting observed in the lowest branch of
the excitation is proportional to D  , whereas the D term would lead to a splitting proportional
to D 2 . These results indicate that, although D and D  have in principle similar effects since
they both break SU (2) symmetry, D dominates the behaviour of the ground state while D  is
crucial to understanding the lowest branch of the triplet excitations.
Acknowledgments
We would like to express our sincere thanks to K Ueda for continuous discussions and manysuggestions. We thank N Furukawa, and F Becca for many helpful discussions.
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