INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 17 (2005) L61–L68

doi:10.1088/0953-8984/17/4/L02

LETTER TO THE EDITOR

Field-induced effects of anisotropic magnetic
interactions in SrCu2 (BO3 )2
K Kodama1, S Miyahara2 , M Takigawa1,7 , M Horvatić3 , C Berthier3,4 ,F Mila5 , H Kageyama6 and Y Ueda1
1 Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581,
Japan
2 Department of Physics, Aoyama Gakuin University, Sagamihara, Kanagawa 229-8558, Japan
3 Grenoble High Magnetic Field Laboratory, CNRS and MPI-FKF, BP 166-38042 Grenoble,
France4 Laboratoire de Spectrométrie Physique, Université J Fourier, BP 87-38402 St.Martin d’Hères,
France5 Institute of Theoretical Physics, Ecole Polytechnique Fédérale de Lausanne,
CH-1015 Lausanne, Switzerland
6 Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502,
Japan

E-mail: masashi@issp.u-tokyo.ac.jp

Received 22 November 2004, in final form 4 January 2005
Published 14 January 2005
Online at stacks.iop.org/JPhysCM/17/L61
Abstract
We observed a field-induced staggered magnetization in the 2D frustrated dimer
singlet spin system SrCu2 (BO3 )2 by 11 B NMR, from which the magnitudes of
the intradimer Dzyaloshinsky–Moriya interaction and the staggered g-tensor
were determined. These anisotropic interactions cause singlet–triplet mixing
and eliminate a quantum phase transition at the expected critical field Hc forgap closing. They provide a quantitative account for some puzzling phenomena
such as the onset of a uniform magnetization below Hc and the persistence ofthe excitation gap above Hc . The gap was accurately determined from theactivation energy of the nuclear relaxation rate.

Spin systems with singlet ground states exhibit a variety of quantum phase transitions inmagnetic field [1]. A generic example is the Bose–Einstein condensation of triplets when
the field exceeds the critical value at which the excitation energy vanishes. This results inan antiferromagnetic order with the staggered moment perpendicular to the field, as has been
observed, e.g., in TlCuCl3 [2]. Another possibility is the formation of a superlattice of localized
triplets due to repulsive interactions, which translates into magnetization plateaus at fractional
values of the saturated magnetization. The best known example is SrCu2 (BO3 )2 with its
7 Author to whom any correspondence should be addressed.

0953-8984/05/040061+08$30.00© 2005 IOP Publishing Ltd

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L61

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Letter to the Editor

(a)

Cu
1

3

B
O
J

y

4

2

[110]
x

J'
D
D'

(b) z (c)
1

2

3(4)

x ([110])Figure 1. The magnetic layer of SrCu2 (BO3 )2 viewed along (a) the c-(z-) and (b) the [110]-(y-)
directions. Numbers are the site indices for both Cu and B. Symbols for D and D  indicate thedirection of d in the Dzyaloshinsky–Moriya interaction d · (si × s j ) with the bond direction i → jshown by arrows.

(This figure is in colour only in the electronic version)

two-dimensional network of orthogonal dimers of Cu2+ ions (spin 1/2). This material shows
an excitation gap 0 = 35 K and plateaus at one-eighth, one-quarter, and one-third of thesaturated magnetization [3–5]. A magnetic superlattice at the 1/8-plateau has actually been
observed by NMR experiments [6].
These basic properties can be explained by a Heisenberg model on the frustrated Shastry–
Sutherland lattice,
H0 = Jsi · s j + J 
si · s j ,
(1)
nn

nnn



where J and J are the intra- and interdimer exchange interactions as shown in figure 1(a) [3, 7].
However, several aspects of low-temperature and high-field properties remain mysterious:(i) a finite magnetization appears well below the expected critical field for the gap closing
Hc = 0 /gz µB , where gz is the g-value along the field direction [5, 8];(ii) a gap seems to persist above Hc [9–11];(iii) there appears no phase transition down to T = 0 at fields below the 1/8-plateau, whichmeans that no Bose condensation occurs, while one has been observed above it [11];(iv) the magnetization shows a discontinuous jump at the lower boundary of the 1/8plateau [6, 8].
Properties (i) and (ii) suggest that triplet states are mixed into the ground state by some
anisotropic interactions. The interdimer Dzyaloshinsky–Moriya (DM) interaction,

y
y
H =
Di j (six s j − si s xj ),(2)
nnn

where |Di j | = D  and the sign alternates as shown in figure 1(a), was required to explain the
splitting of triplet energy levels [12, 9]. However, it does not have matrix elements between
singlet and one-triplet states. Although the intradimer DM interaction is a candidate, noestimate for its magnitude has been made so far.

Letter to the Editor

L63

In this letter, we report observation of a field-induced staggered magnetization by 11 B
NMR experiments, which is also caused by singlet–triplet mixing. Quantitative estimates of
the intradimer DM interaction and the staggered g-tensor are obtained. Furthermore, theseanisotropic interactions provide quantitative accounts for the above points (i) and (ii). We also
report the behaviour of the gap obtained from the nuclear relaxation rate upon entering into
the 1/8-plateau.
A single crystal of SrCu2 (BO3 )2 was grown by the travelling-solvent-floating-zone method
using LiBO2 solvent [13]. The NMR measurements below 18 T were performed at ISSP,
University of Tokyo, while data at higher fields were obtained using a 20 MW resistive magnet
at the Grenoble High-Magnetic-Field Laboratory.
We first describe the anisotropic interactions compatible with the crystal structure (spacegroup I 42m [14]). A crucial feature is the non-coplanar buckling of the magnetic CuBO3
layers as depicted in figure 1(b). This allows additional anisotropic interactions,




4


 y
z x
z
z y
x z
H1 = −µB H ·gi · si + D(s1 s2 − s1 s2 ) −(s3 s4 − s3 s4 ) ,
(3)
i=1

A

B

where A (B) denotes dimers along the x- (y−) direction. The second term, the intradimer DMinteraction, is allowed since the buckling breaks inversion symmetry at the centre of the dimer
bonds. The first, Zeeman term involves anisotropic g-tensors, and for site 1 is given by


0 −gs
gx
g1 =
,
0
gy
0
gz
−gs 0
while g2 , g3 , and g4 are obtained from g1 by symmetry operations of the crystal (figure 1).
The sign of the x z-component is opposite for g2 , i.e., gs represents the staggered component.
The diagonal components were determined as gx = g y = 2.05 and gz = 2.28 from ESR
measurements [9]. We estimate gs = 0.023 by assuming that the principal axis of g1 coincides
with that of the electric field gradient at Cu1 nuclei (5.6◦ from the c-axis [15]), which is nearly
perpendicular to the plane containing the four oxygen atoms surrounding one Cu1.
In magnetic fields, the anisotropic Hamiltonian H1 results in a non-collinear magnetizationconsisting of four sublattices mi = gi · si µB (i = 1–4). We define the uniform andAstaggered moments on A and B dimers as mA
u = (m1 + m2 )/2, ms = (m1 − m2 )/2 andBB
mu = (m3 + m4 )/2, ms = (m3 − m4 )/2. The crystal symmetry tells us that an external
BAfield Hext  z (Hext  x) induces staggered moments mAs  x and ms  y (ms  z).
Such non-collinear moments, unobservable from bulk magnetization measurements, can be
accurately detected by 11 B
NMR. The magnetic hyperfine field acting on a nuclear spin at site
i is expressed as Hihf = 4j =1 Ai j · m j , where Ai j is the hyperfine coupling tensor between
the i th nuclear spin and m j . The projection of Hihf along Hext , denoted as Hi , is obtained
from the shift of the NMR frequency.
We first discuss the results for a small field (6.948 T) rotated in the (110)- (x z-) plane.
The NMR spectra consist of several sets of quadrupole-split lines. Each set was assigned toa particular site based on the known anisotropy of the quadrupole splitting [16]. In figure 2,
H1 is plotted against the angle θ between Hext and the c-axis at several temperatures. By
symmetry, H2 (θ ) = H1(−θ ) and H3 (θ ) = H4 (θ ) (not shown). The θ -dependence can be wellfitted by a simple sinusoidal curve (curves in figure 2),
H1 + H2 = Hu cos 2θ + const,
H1 − H2 = Hs sin 2θ.
T -dependences of Hu and Hs are plotted in figure 3.

(4)

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Letter to the Editor

5

H1(mT)

0

2.5 K

-5

6K
20 K

-10

100 K

-90

-60

-x

θ (deg.)

-30

0

30

60

90

z

x

Figure 2. Angular variation of H1 at different temperatures in the external field of 6.948 T.

Internal Fields (mT)

10

5

Hs

H su

Hu x 0.53

H ss

0
0

20

40

60

80

T (K)
Figure 3. Temperature dependences of Hu , Hs , (4) Hsu , and Hss (6) at Hext = 6.948 T.

The θ -dependence of H1 is not symmetric around θ = 0, leading to a finite Hs. Both mA,Bs
and mA,B
contribute to Hs through the uniform (diagonal) and the staggered (off-diagonal)uand mA,B
had the same T -dependence, thisparts of the coupling Ai j , respectively. If mA,Busshould stand for Hs and Hu . This is indeed not the case, since Hs extrapolates to a large finite
value at T = 0, while Hu almost vanishes similarly to the magnetic susceptibility [4]. This
is induced at T = 0 at field values Hext  Hc ,proves that a sizable staggered moment mA,Bs
while the uniform moment is nearly zero. This can be qualitatively explained by considering
H1 as a perturbation for an isolated dimer [17]. Since H1 has matrix elements between singlet
and triplets on the same dimer, triplets are mixed into the ground state, resulting in finite
mA,B
to first order in perturbation. In contrast, mA,B
is much smaller than mA,Bsus , since onlyA,Bhigher order terms contribute to mu . Although field-induced staggered moments have beenreported in the Haldane chain compound NENP [18, 19],this is the first observation in quasi-2D
systems.
For a quantitative analysis, we writeA
H1 − H2 = 2 (Bxz + C xz ) m A
uz sin θ + m ux cos θA+ 2 (Bxx − C xx ) m A
sx sin θ + 2 (Bzz − C zz ) m sz cos θ B
+ 4Dxz m uz sin θ + m Bux cos θ + 4Dxy m Bsy sin θ,(5)

Letter to the Editor

L65

where we have defined B = A11 , C = A12 , and D = A13 and used symmetry properties toobtain other Ai j . The sin 2θ dependence of H1 − H2 in (4) is reproduced if the moments vary
B
A,B
A
A,B
as m Asx , m sy , m uz ∝ cos θ , and m sz , m ux ∝ sin θ . This is approximately confirmed by thenumerical calculations presented below. We defineBσx ≡ m A
sx (θ = 0) = m sy (θ = 0),

Mz ≡ m A,Buz (θ = 0),σz ≡ −m A
sz (θ = π/2),

MxA(B) ≡ m A(B)ux (θ = π/2).

Combining these relations with (5), we obtain
Hs = Hsu + Hss(6)

Hsu = (Bxz + C xz + 2Dxz ) Mz + (Bxz + C xz ) MxA + 2Dxz MxB

Hss = Bxx − C xx + 2Dxy σx − (Bzz − C zz ) σz .

(7)
(8)

The analysis of the 11 B NMR spectrum in the 1/8-plateau phase [20] indicates that all
hyperfine couplings except B are approximately given by the classical dipolar fields that we cancalculate. Since the diagonal components of B+C+2D were determined previously [16], all the
coupling parameters in the above equations are known except for Bxz . At high temperatures,the staggered moments σx , σz are much smaller than the uniform moments Mz , MxA,B , and
MxA ≈ MxB . Assuming σx = σz = 0 above T = 70 K, which is also supported by preliminary
numerical calculations, we determined Bxz from the data of Hs and the magnetization at
70 K, using (6). The parameters in (7) and (8) are determined as Bxz + C xz + 2Dxz = 0.10,
Bxz + C xz = 0.09, 2Dxz = 0.01, Bxx − C xx + 2Dxy = −0.06, and Bzz − C zz = −0.32(T µ−1B ). We then obtained Hsu at other temperatures (figure 3) from the magnetization data.
Finally, subtracting this from Hs, Hss is determined in the whole temperature range as shownin figure 3. This demonstrates that the staggered magnetization increases very steeply below
10 K and saturates at lower temperatures.
Measurements of Hi were extended to higher fields up to 26 T. The magnetization datain [8] was used to calculate Hsu ,8 which was then subtracted from the data of Hs to obtain Hss.
In the main panel of figure 4, we plot Hsu and Hss in the low-temperature limit.
In order to see if these results are explained by anisotropic interactions, we have calculated
mi by exact diagonalization (ED) of the Hamiltonian H0 + H1 + H at T = 0 and Hext = 6.948 T
in the (110)-plane for clusters with up to 24 sites for various values of D. Values of otherparameters are fixed: J = 85 K, J  = 54 K [3], D  /J = −0.02 [12], and gs = 0.023. Byrequiring that the calculated value of Hs from (6) to (8) agrees with the experimental data at thelowest temperature (2.5 K), we obtained D/J = 0.034. Taking into account the uncertainty in
gs leads to possible values of D/J between 0.030 (for gs = 0.030) and 0.038 (for gs = 0.014).
The choice of D  has little effect on the calculated values of mi .
Using the same parameter values, we have also calculated mi at higher fields, as shownin the inset of figure 4. The agreement between the calculated results of Hsu and Hss shownby the lines in the main panel of figure 4 and the experimental data is satisfactory up to
about 20 T. Thus we now have a quantitative explanation for the development of a uniform
magnetization at fields as low as 18 T. Note that the staggered moment is sizable even at low
fields (∼0.03 µB /Cu at 15 T, figure 4 inset), where the uniform moment is negligible. The
experimental and theoretical results of Hss deviate at higher fields, probably due to finite-size
effects as discussed below.
Let us now discuss the dynamics. The nuclear spin–lattice relaxation rate (1/T1 ) was
measured for Hext  c up to 27.5 T. It shows an activated behaviour at low temperatures
We estimated MxA and MxB separately by using the magnetization data of (MxA + MxB )/2 and assuming the ratio
MxA /MxB = 2.7 for the field range 18–26 T as indicated by the ED calculations (figure 4 inset).

8

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Letter to the Editor

Internal Fields (mT)

80

60

(µB )

MxA

σx

0.2

MxB

σz

Mz

0.1

40
0

0

10

Hsu

20

20

Hext (T)

H ss
0

5

10

15

20

25

Hext(T)
Figure 4. Field dependences of Hsu and Hss measured at low temperatures, 2.5 K (for Hext = 7
and 12 T), 1.5 K (15 and 17.9 T), 0.64 K (20 T), 0.46 K (24 T), and 0.21 K (26 T). The curves are
obtained from the ED results of σx , σz , Mz , and MxA,B shown in the inset by using (7) and (8).

40

100

1/T1 (s -1)

30

∆ (K)

26.5 T

10

26 T
1

24 T

20

10

22 T

20 T
0

1

27.5 T2

3

4

5

1/T (K-1)

6

7

observed
calculated

0
0

10

20

30

Hext(T)
Figure 5. Field dependence of the excitation gap (solid circles in the main panel) determined fromthe activated behaviour of 1/ T1 (inset) compared with the results of ED calculation (open circles).
The data at 27.5 T, where the NMR spectrum shows many lines in the 1/8-plateau phase, wereobtained for the line with the largest (negative) hyperfine field.

(figure 5 inset). The activation gap () shown in figure 5 follows a linear H -dependence
below 20 T,  = 0 − gz µB H with 0 = 34.8 K and gz = 2.28 as expected. At higher fields,
however, it deviates from this relation and remains finite even above the expected critical field
Hc = 0 /gz µB = 22.7 T. The gap reaches a minimum at 26.5 T, which is at the boundary
to the 1/8-plateau, and increases again inside the 1/8-plateau (27.5 T). The finite gap above
Hc , which naturally explains the absence of Bose condensation, is consistent with earlier
reports. For example, the gap was estimated from the specific heat data [11] as  = 4.6(3.2) K at Hext = 22 (24) T. The ESR data [10] also show a similar deviation from the linear
H -dependence of the lowest triplet energy near Hc .

Letter to the Editor

L67

One expects that the singlet–triplet mixing due to H1 will prevent the gap from closing,as has been discussed for NENP [21]. We have confirmed this by numerical calculations.
The energy of the lowest excited state obtained by ED calculations using the same parametersis plotted in figure 5. The constant value at low fields is due to the singlet bound state of
two triplets having lower energy than one triplet [22]. However, they do not contribute to
the nuclear relaxation. Near 22 T, the results clearly indicate level repulsion due to mixing,
reproducing the experimental behaviour. Let us note one subtle point. The interdimer DMinteraction H splits the one-triplet excitations into two branches [12]. Only one of them ismixed with the singlet by H1 and this branch must have a lower energy for the level repulsion
to occur. This determines the sign of D  to be negative. Note that the sign of D  was not known
before.
Finally, the theory is not applicable when the field is too close to the 1/8-plateau. The
cluster size of our calculation is not large enough to take proper account of interaction between
triplets, which should become important near the 1/8-plateau.
In conclusion, we have shown that a large staggered magnetization is induced by themagnetic field in the presence of the intradimer DM interaction and staggered g-tensor, whose
magnitudes are obtained as 0.03  D/J  0.038 for 0.014  gs  0.03. The singlet–
triplet mixing and level repulsion caused by these interactions account for the finite uniform
magnetization below Hc and the persistence of the excitation gap above Hc. They thus turn thequantum phase transition at Hc into a crossover. This is the first example of such field-induced
phenomena in quasi-2D spin systems. Our determination of the magnitudes and sign of the
DM interactions should set the stage for further investigation of other remarkable properties
of SrCu2 (BO3 )2 . In particular, the signature of a phase transition in the specific heat observed
above the 1/8-plateau by Tsujii et al [11] calls for a precise understanding of the interplay
between the DM interaction and other bosonic interaction such as bound state formation [22]and correlated hopping [23].
We thank M Oshikawa, K Ueda, and T Ziman for stimulating discussions. This work is
supported by Grants-in-Aid for Scientific Research from the MEXT Japan and by the Swiss
National Fund.
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[4] Kageyama H et al 1999 Phys. Rev. Lett. 82 3168
[5] Onizuka K et al 2000 J. Phys. Soc. Japan 69 1016
[6] Kodama K et al 2002 Science 298 395[7] Miyahara S and Ueda K 1999 Phys. Rev. Lett. 82 3701
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[8] Kageyama H et al 2001 J. Alloys Compounds 317/318 177[9] Nojiri H et al 1999 J. Phys. Soc. Japan 68 2906
[10] Nojiri H et al 2003 J. Phys. Soc. Japan 72 3243
[11] Tsujii H et al 2003 Preprint cond-mat/0301509[12] Cépas O et al 2001 Phys. Rev. Lett. 87 167205[13] Kageyama H et al 1999 J. Cryst. Growth 206 65[14] Smith R W and Keszler D A 1991 J. Solid State Chem. 93 430[15] Kodama K et al, unpublished[16] Kodama K et al 2002 J. Phys.: Condens. Matter 14 L319
[17] Miyahara S et al 2004 J. Phys.: Condens. Matter 16 S911
[18] Chiba M et al 1991 Phys. Rev. B 44 2838

L68[19] Fujiwara N et al 1993 Phys. Rev. B 47 11860[20] Kodama K et al, unpublished[21] Mitra P P and Halperin B I 1994 Phys. Rev. Lett. 72 912[22] Knetter C et al 2000 Phys. Rev. Lett. 85 3958
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Letter to the Editor