PHYSICAL REVIEW B 70, 144417 (2004)

Far-infrared spectroscopy of spin excitations and Dzyaloshinskii-Moriya interactionsin the Shastry-Sutherland compound SrCu2„BO3…2
T. Rõõm,* D. Hüvonen, and U. Nagel
National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia

J. Hwang and T. Timusk
Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S4M1, Canada

H. Kageyama
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan(Received 30 March 2004; published 27 October 2004)
We have studied spin excitation spectra in the Shastry-Sutherland model compound SrCu2共BO3兲2 in magnetic fields using far-infrared Fourier spectroscopy. The transitions from the ground singlet state to the triplet
state at 24 cm−1 and to several bound triplet states are induced by the electric field component of the farinfrared light. To explain the light absorption in the spin system we invoke a dynamic Dzyaloshinskii-Moriya(DM) mechanism where light couples to a phonon mode, allowing the DM interaction. Two optical phonons
couple light to the singlet to triplet transition in SrCu2共BO3兲2. One is a-polarized and creates an intradimer
dynamic DM along the c axis. The other is c-polarized and creates an intradimer dynamic DM interaction, it
is in the 共ab兲 plane and perpendicular to the dimer axis. Singlet levels at 21.5 and 28.6 cm−1 anticross with thefirst triplet as is seen in far-infrared spectra. We used a cluster of two dimers with a periodic boundary
condition to perform a model calculation with scaled intra- and interdimer exchange interactions. Two static
DM interactions are sufficient to describe the observed triplet state spectra. The static interdimer DM in thec-direction d1 = 0.7 cm−1 splits the triplet state sublevels in zero field [Cépas et al., Phys. Rev. Lett. 87, 167205(2001)]. The static intradimer DM in the 共ab兲 plane (perpendicular to the dimer axis) d2 = 1.8 cm−1, allowed by
the buckling of CuBO3 planes, couples the triplet state to the 28.6 cm−1 singlet as is seen from the avoided
crossing.
DOI: 10.1103/PhysRevB.70.144417

PACS number(s): 75.10.Jm, 78.30.Hv, 71.70.Gm, 76.30.Fc

I. INTRODUCTION

In spin systems with a ground singlet state and excitedtriplet state the energy gap between the singlet and the triplet
can be tuned with an external magnetic field. In SrCu2共BO3兲2
it was discovered that in magnetic fields above 22 T, where
the spin gap is expected to close, several magnetization plateaus appear.1 At magnetization plateaus the triplets form apattern which breaks the translational symmetry of the crystal structure.2 The heavy mass of the triplet excitations arising from an almost flat dispersion of energy on momentum3
favors the build-up of magnetic superstructures. Below thecritical field SrCu2共BO3兲2 has a ground state described firstby Shastry and Sutherland.4
SrCu2共BO3兲2 consists of planes of CuBO3 and Sr atoms
between the planes. Cu2+ spins 共S = 1 / 2兲 form Cu-Cu dimers
arranged into orthogonal dimer network. SrCu2共BO3兲2 is anexperimental realization of a Shastry-Sutherland model.4 In
the model there is an antiferromagnetic intradimer exchange
coupling j1 and interdimer coupling j2 between spins on thenearest-neighbor dimers (Fig. 1). In the limit of ␣ ⬅ j2 / j1
= 0 the problem reduces to that of isolated dimers where theground state is the product of singlet states and the first excited triplet state is at energy ⌬T = j1 above the ground state,
where ⌬T is the energy per dimer. Shastry and Sutherland
showed that for 0 ⬍ ␣ 艋 0.5 singlets on all dimers is an exact
ground state too. The exactness of the ground state and the1098-0121/2004/70(14)/144417(8)/$22.50

heavy mass of triplet excitations is the consequence of frustration originating from the special geometry of the dimer
lattice in the Shastry-Sutherland model where the bonds onneighboring dimers are orthogonal. Later on it has beenshown that singlets on all dimers is the exact ground state fora larger range of ␣ up to the quantum critical point ␣c⬇ 0.7. At the quantum critical point the spin gap vanishes
and a long-range antiferromagnetic order is established. Different theoretical approaches have been used to calculate ␣c(see Ref. 5 for review). It is possible that between the exact
singlet ground state and the anti-ferromagnetic state in certain range of ␣ other gapped spin states exist.6–11
The singlet-triplet gap in SrCu2共BO3兲2, ⌬T = 24 cm−1, hasbeen measured directly by several experimental techniques:inelastic neutron scattering,3,12 electron spin resonance13,14(ESR), Raman scattering,15 and far-infrared (FIR)
spectroscopy.16 Additional information besides ⌬T is needed
to determine the exchange parameters of SrCu2共BO3兲2. The
dispersion of the triplet excitation is not informative becauseof its flatness,3 but positions of other excited states or thetemperature dependence of thermodynamic parameters canbe used for determining the exchange parameters. Miyahara
and Ueda5 found j1 = 59 cm−1 and ␣ = 0.635. They added aninterlayer coupling j3 = 0.09j1 to the model to obtain a better
fit of the magnetization T-dependence above the critical temperature kBT ⬎ ⌬T. Based on the analysis of excitation
spectra17 j1 = 50 cm−1 and ␣ = 0.603 were proposed. Such

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PHYSICAL REVIEW B 70, 144417 (2004)RÕÕM et al.

FIG. 1. Cluster with two dimers (1,2) and (3,4). (a) Dimer (3,4)
and four nearest-neighbor dimers. The thin dashed line shows thetwo dimer cluster boundary. Thin solid lines show the distortion of
Cu-Cu superexchange bonds due to the buckling of Cu-O-B planes.
Thick solid and dashed lines are the inter- and intradimer superexchange constants j1 and j2; interdimer DM vectors (d1, solid arrow)are in the c direction and intradimer DM vectors (d2, empty arrow)in the 共ab兲 plane along a and b axis. (b) The two dimer model after
the periodic boundary condition has been applied; interdimer interactions have doubled.

scattering of parameters could be either due to the incomplete model or due to the approximations made in theoretical
calculations. SrCu2共BO3兲2 is near to the quantum criticalpoint ␣c where the energy levels of the spin system are sensitive to the choice of j1 and ␣. A singlet level in the spin gap
at 21 cm−1 found in the ESR spectra14 may help to findproper parameters for the model.
Interactions other than inter- and intradimer exchange
coupling can spoil the exactness of the ground state. This isimportant in high magnetic fields where the triplet state becomes degenerate with the ground singlet state. At this critical field even a weak interaction between the singlet and thetriplet state mixes the two states completely. The singlet andtriplet state anticrossing effects were seen in the high field
ESR experiments.14 A possible antisymmetric interaction
which couples the singlet and the triplet states is the DM
interaction. An intradimer DM is allowed by symmetry but
its strength is not known below room temperature. Aboveroom temperature d2 = 2.5 cm−1 has been estimated from the
ESR linewidth.18 The interdimer DM interaction, d1= 1.5 cm−1, perpendicular to the dimer planes12 partially lifts
the degeneracy of the triplet state but does not couple thetriplet state to the singlet state. The effect of DM interactions
on the magnetic dipole active ESR transitions in
SrCu2共BO3兲2 was investigated theoretically in Ref. 19.
Lattice distortions, static or dynamic, are important in
SrCu2共BO3兲2 since they lower the crystal symmetry and allow magnetic interactions which are otherwise forbidden in amore symmetric environment. SrCu2共BO3兲2 has a structuralphase transition at 395 K (Ref. 20) that induces a buckling of
CuBO3 planes in the low T phase. As the phase transition
point is approached from below the Raman-active 62 cm−1
optical phonon mode softens.21 Acoustic phonon modes havespin-phonon coupling at magnetization plateaus.22 It has
been proposed that a spin superstructure at 1 / 8 plateau observed by nuclear magnetic resonance at 35 mK is stabilized
by a lattice distortion.2 Instantaneous breaking of lattice sym-

FIG. 2. Differential absorption in E1 ⬜ c (two upper curves) andE1 储 c (lower curve) polarization. Spectra have been offset in thevertical direction.

metry by an optical phonon allows electric dipole active
singlet-triplet transitions23 that explains FIR polarized absorption spectra in SrCu2共BO3兲2.16
Our aim is to find out which additional interactions are
required to the Shastry-Sutherland model that add triplet corrections to the ground state. For that we do FIR absorption
measurements with polarized light in magnetic field andcompare the absorption line frequencies and intensities withvalues calculated with a two dimer model including the dynamic DM effect. The important information is in the polarization and magnetic field dependence of the FIR absorption
lines and in the avoided crossing effects.
We studied single crystals of SrCu2共BO3兲2, Ref. 24. The
first sample consisted of two pieces 0.65 mm thick in thea-direction with the total area of 12 mm2 in the 共ac兲 plane.
The second sample was 0.6 mm thick in the c-direction andhad an area of 11.5 mm2 in the 共ab兲 plane. The experimental
details are described in Refs. 16 and 25.
II. RESULTS AND DISCUSSION
A. FIR spectra and electric dipole transitions

As the result of the polarization sensitive measurement of
FIR spectra we have identified that the main resonances inthe spectra are electric dipole transitions, rather than being
magnetic dipole transitions. In Fig. 2 differential absorption
spectra at 4.4 K relative to 15 K, are displayed. The strong
absorption lines at 52.3 and 53.5 cm−1 were identified16 as
electric dipole transitions, that are active in E1 储 a polarization. We see the same for the 43.0 cm−1 singlet and T0 and T1
triplets (see Table I) at 24.2 and 37.5 cm−1, respectively,
which are present in the spectra measured with E1 储 a regardless of H1 being perpendicular to the c axis or parallel to it.
The lines are missing in E1 储 c polarization,26 instead a newline appears at 25.5 cm−1, which is identified as anothercomponent of the triplet T0.
The triplets are split by the magnetic field B0. Differential
absorption spectra in E1 储 a polarization for one magnetic
field direction, B0 储 a, measured relative to the zero field, are

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TABLE I. Singlet and triplet excitations observed in the FIR spectra at 4.4 K in the order of increasing
zero field energies ប␻0 (in cm−1 units). When a line is visible in two E1 polarizations, both are indicated. The
corresponding H1 polarizations are also indicated. S and T label the singlet and triplet states; ⫹ (⫺) denotes
levels which energy increases (decreases) with B0 and 0 indicates levels where the energy stays constant; ga
and gc are the g-factors with B0 储 a and B0 储 c, respectively. The labeling of T0 levels is shown in Fig. 5. The
zero field intensities A0 (in cm−2 units) of T0 (쐓) are described in the text and in Figs. 4–6. High energy
excitations are labeled by their energies.
Label

E1

H1

S1
T0m共±兲
T0p,m共0兲
T0p共±兲
S2
T1共±兲
T1共±兲
T1共0兲
T38.7共±兲
T38.7共0兲
T39.1共±兲
S39.7
T40.5共±兲T40.7共±兲T40.7共0兲
T41.1共 兲
T42.7共+兲S43
T43.5共±兲
S44.7
S47.0
T48.2共±兲
S52.3
S53.5

a
a,c
a,c
a,c
a,c
c
a
a
a
c
c
a
a
c
a,c
c
a
a
c
c
c
c
a
a

c
c,a
c,a
c,a
c,a
a
b,c
c
c
a
a
c
c
a
c,a
a
c
b,c
a
a
a
a
b,c
b,c

ប␻0

A0

21.50± 0.03
22.72± 0.05
24.11± 0.05
25.51± 0.05
28.57± 0.03
37.49± 0.03
37.51± 0.04
37.69± 0.09
38.74± 0.03
38.70± 0.15
39.08± 0.15
39.71± 0.04
40.45± 0.03
40.67± 0.03
40.70± 0.16
41.11± 0.1342.7± 0.2
43.00± 0.16
43.54± 0.03
44.7± 0.4
47.04± 0.04
48.21± 0.09
52.24± 0.08
53.44± 0.07쐓쐓
쐓
0.3± 0.2
0.9± 0.2
0.9± 0.2

0.19± 0.05
0.2± 0.1
0.2± 0.1
0.4± 0.1
0.2± 0.1
2.6± 0.3
0.2± 0.1

0.04± 0.02
86± 14
24± 3

ga

gc

1.988

2.219

1.988

2.219

1.996
2.001

2.264
2.23

2.026
2.067

2.29

1.97
2.243
2.10
2.25
2.31

2.27

displayed in Fig. 3. We see an anticrossing of the T0m共−兲
level with the singlet S1 at 21.5 cm−1 and an anticrossing of
the T0p共+兲 level with the singlet S2 at 28.6 cm−1. All the
peaks in the measured spectra in different light polarizations
and B0 directions were fitted with Lorentzians. The results
are summarized in Table I and displayed in Figs. 4–6. The
states above 38 cm−1 are labeled by their zero field frequencies. The magnetic field independent energy levels are labeled as singlets with the exception of those in the middle ofthe triplet levels T共±兲.
B. Dynamic Dzyaloshinskii-Moriya mechanismand optical transitions: Two dimers

FIG. 3. Differential absorption spectra in magnetic field B0 储 a at
4.4 K. Vertical offset equals the magnetic field value in Tesla.

The Hamiltonian for a spin pair with exchange coupling j
and DM interaction d on the bond connecting spins k and lreads

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FIG. 4. (Color online) Magnetic field dependence of line positions and line areas in E1 储 c polarization at 4.4 K; (a), (b) B0 储 c;(c), (d) B0 储 a. Solid lines are theresults of the calculation based on
the two dimer model: j1= 24 cm−1, 2j2 = 9.8 cm−1, 2d1= 1.4 cm−1, and d2 = 1.8 cm−1.
Dashed lines in panels (a) and (c)are fits with parameters given in
Table I.

冉

kl
Hstat
= j−

冊

兩d兩2
1
Sk · Sl + Sk · dd · Sl + d · 关Sk ⫻ Sl兴
4j
2j+ g␮BB0 · 共Sk + Sl兲.

V = eQE1 = eq共a† + a兲E1 ,共1兲

Here we included Shekhtman corrections27,28 which are quadratic in d (see also Ref. 25). The last term is the Zeeman
energy of spins in the magnetic field B0 where g is the electron spin g-factor and ␮B is the Bohr magneton.
The formalism to introduce the spin-phonon coupling issimilar to one used in Refs. 23 and 25. We are interested insinglet to triplet transitions. Therefore the relevant term is theantisymmetric DM interaction d共Q兲 · 关Sk ⫻ Sl兴 which couples
the singlet to the triplet state. We expand the DM vector
d共Q兲 into a power series of the lattice normal coordinate Q
d共Q兲 = d共0兲 +冏 冏⳵dQ + ... ,⳵Q Q=0

共2兲

where d共0兲 ⬅ d is the static DM interaction in (1). We keep
terms linear in Q. The full Hamiltonian for a spin pair including the phonons iskl+ ប␻ pa†a + q共a† + a兲dQ · 关Sk ⫻ Sl兴,
Hkl = Hstat

共3兲

where dQ ⬅ 兩⳵d / ⳵Q兩Q=0. The lattice normal coordinate Q ispresented in terms of phonon creation and annihilation operators a† and a, Q = q共a† + a兲, where q is the transformation
coefficient and ␻ p is the phonon frequency. The spin-phonon
coupling term in (3) is linear in a† and a. Therefore thephonon states with the occupation numbers n and n⬘ arecoupled where n⬘ = n ± 1. We will consider only two phononstates 兩0典 and 兩1典, which is justified when kBT Ⰶ ប␻ p.
The normal coordinate Q in the dynamic DM singlet to
triplet optical transition mechanism belongs to an opticalphonon. Electric dipole coupling between a phonon and lightin the long wavelength limit is

共4兲

where e is the effective charge associated with the latticenormal coordinate Q. Here we assumed E1 储 Q and dropped
the time dependence of V. Once the eigenstates of (3) are
known the optical transition probability between the groundstate 兩␾典 and the excited state 兩␾⬘典 is calculated as I= 兩具␾⬘兩V兩␾典兩2.
To calculate optical transitions in SrCu2共BO3兲2 we use a
two dimer model depicted in Fig. 1. In this model intradimer
and interdimer superexchange interactions j1 and j2 are considered. The interdimer static DM vector d1 is along the caxis and alternates from bond to bond. The intradimer static
DM vector d2 exists due to the buckling of Cu-O-B planes.20
The direction of DM vectors is defined by the right-hand rule
where the path is along the Cu-O-Cu bond (for d1 Cu-O-BO-Cu) in the direction of increasing spin index k. In thevector product Sk ⫻ Sl the spin with a smaller index is on theleft, k ⬍ l. When a periodic boundary condition is applied tothe two dimer cluster, bounded by a box drawn with a thindashed line in Fig. 1(a), an effective spin model is obtained
where the interdimer interactions are doubled, Fig. 1(b). The
doubling is necessary to conserve the number of nextnearest-neighbor bonds, which is four.
The Hamiltonian for the two dimer cluster is the sum of
pairwise interactions (3) where the sum runs over all thebonds in the cluster. We will use a basis 兩ABn典, where A runs
over the singlet S and three triplet components T−, T0, and T+
on the j1 bond of the dimer (1,2) and B over the singlet andtriplet states of the dimer (3,4). n is the number of phonons,
0 or 1. The basis has 32 components. Below we consider aand c-axis phonons, shown in Fig. 7, named by the directionof their electric dipole moment.
1. Energy levels

The effect of the dynamic DM interaction on the positionof energy levels is small because we take ប␻ p = 100 cm−1

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FIG. 5. (Color online) Magnetic field dependence of line positions and line areas in E1 储 a polarization at 4.4 K; (a), (b) B0 储 c;(c), (d) B0 储 a. Solid lines are theresults of the calculation based on
the two dimer model: j1= 24 cm−1, 2j2 = 9.8 cm−1, 2d1= 1.4 cm−1, and d2 = 1.8 cm−1.
Dashed lines in panels (a) and (c)are fits with parameters given in
Table I. The solid line in panel (b)is the sum of two theoretical line
areas of S0 to T0m共0兲 and to T0p共0兲transitions shown by dashed lines.
Dashed lines in (d) are eye guides(see text). The 18 T point a panels(a) and (b) was measured at 1.8 K.

that is substantially larger than the singlet-triplet gap. We use
this value since there are no optical phonons with substantial
spectral weight below 100 cm−1 as our transmission measurements show. The energy spectrum can be analyzed separately from the dynamic DM effect because of the high phonon energy. The calculated energy levels are the same in
Figs. 4 and 5. In these figures only the zero phonon levels of
the triplet T0 and S2 are shown. The levels with one excitedphonon are offset by ប␻ p to higher energies and are notshown.
In a two dimer system two singlets, two triplets, and aquintet are present. The ground state is a product of singlets
兩SS典. The first triplet is a linear combination of 兩ST典 and 兩TS典.
In the two dimer model the singlet-triplet splitting is notrenormalized by the interdimer coupling j2 and the energy of
the triplet excitation is ET0 = j1. The second singlet, a boundstate of two triplets, is at ES1 = 2j1 − 2共2j2兲. To stress the factthat in the two dimer model with a periodic boundary condition the interdimer bonds are effectively doubled, we write
2j2 explicitly. There are two other bound states of two triplets, a triplet at ET1 = 2j1 − 共2j2兲 and a quintet at EQ = 2j1+ 共2j2兲. These energies and the ground state wave function
are slightly changed by the static DM interactions d1 and d2.
The spin states 兩STi典 and 兩TiS典 are strongly mixed by theinterdimer d1 since they are degenerate in any field.
The states are labeled in Fig. 5. The following parameters
were used to fit the energy spectra plotted in Figs. 4 and 5.
The energy of one-triplet sublevels T0m共0兲 and T0p共0兲 gives
us j1 = 24.0 cm−1. To get the singlet S2 at 28.6 cm−1 we use
2j2 = 9.8 cm−1. Triplet levels are split in zero field by 2d1= 1.4 cm−1. The intradimer d2 = 1.8 cm−1 induces an avoided
crossing of T0p共+兲 and S2. In a simplified picture the onetriplet excitation is the 兩ST典 (or 兩TS典) state and the excited
singlet is 兩TT典. d2 “flips” the singlet to the triplet state on oneof the dimers and thus couples T0p共+兲 to S2.

2. c-axis phonon

The optical c-axis phonon bends the Cu-O-Cu bond in thec-direction. We assume that the bending action of the phononis the same on both dimers, Fig. 7. As a result the dynamic
DM interaction on the dimer (1,2) is qcdQc ⬅ d3c= 共−d3c , 0 , 0兲 and on the dimer (3,4) d3c = 共0 , d3c , 0兲; the orientation of the Cartesian coordinates is the same as in Fig.
1(b). The calculated and the measured transition probabilities
as a function of magnetic field are plotted in Figs. 4(b) and4(d) for two field orientations. In zero field a line at
25.5 cm−1 is present. The area of this line is the only scaling
parameter between the theory and the experiment. Note thatthe transition to the triplet level, which anticrosses with S2, is
optically active when B0 储 c. When B0 储 a there is no anticrossing for the optically active triplet level.
The overall agreement between the theory and the experiment is good. There is a disagreement between the intensitiesof the middle and lower triplet components in the theory andin the experiment, Fig. 4(d). In the theory the intensity of the
middle component is approximately three times as strong as
the lower component while in the experiment they are equal.
We tried several changes in our model to make the intensitiesof the two triplet components more equal and none of themhelped. These unfruitful changes were the shift of the phonon
frequency, a small out-of-plane component of B0 and an inplane component of the interdimer DM vector d1.
3. a-axis phonon

The optical a-axis phonon bends the Cu-O-Cu bond in thea-direction and creates a dynamic DM interaction in thec-direction, Fig. 7. If we choose E1 储 a the dynamic DM interaction is created on dimer (1,2), qadQa ⬅ d3a = 共0 , 0 , d3a兲.
In general, for an arbitrary orientation of E1 in the 共ab兲
plane, both dimers will acquire a certain d3a. For the time
being we assume E1 储 a.

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5(d) the B0 储 E1 case is shown. Here are optically active thetriplet levels which anticross with the singlet states. In
B0 ⬜ E1 field orientation, Fig. 6, the optically active triplet
levels do not anticross with the singlet states. The mutualorientation of B0 and E1 is important because E1 储 a createsd3a on the dimer (1,2) and not on (3,4). Which set of the
twofold degenerate triplet levels is optically active depends
on the relative orientation of B0 and d2 on the dimer whered3a ⫽ 0. In Fig. 5, B0 储 d2 and in Fig. 6, B0 ⬜ d2. An additional
splitting of T0m共±兲 and T0p共±兲 by 0.6 cm−1 seen in Fig. 6 isbecause B0 is out of 共ab兲 plane by 9°.
C. Static and dynamic DM in SrCu2„BO3…2

FIG. 6. (Color online) Line positions (a) and line areas (b) in
E1 储 a and B0 储 b configuration at 4.4 K. The lines are results of thecalculation based on the two dimer model and dynamic DM interaction. The additional splitting of triplet components (triangles) iscaused by the magnetic field B0 being misaligned by 9° out of the
共ab兲 plane (Ref. 26). In panel (b) the line area (triangles up or
triangles down) is a sum of line areas of split components.

In zero magnetic field the transition to the central triplet
component is observed, Fig. 2. As the B0 储 c field is turnedon, Fig. 5(b), the central line, being a sum of two overlapping transitions, conserves its intensity. The experimentallyobserved drop in intensity with increasing field is a T effect.
At 1.8 K (18 T field) the intensity is recovered. Besides thestrong central line there are in zero field two sidepeaks ten
times weaker at 22.7 and 25.5 cm−1 corresponding to transitions to the twice degenerate states T0m共±兲 and T0p共±兲. The
dynamic DM interactions due to the a- and c-axis phonons inthis B0 orientation give zero intensity for the sidepeaks. The
detailed analysis of the mechanism causing these weak transitions is difficult because in other polarizations and fieldorientations stronger mechanisms are prevailing. The sidepeaks split in the magnetic field and an avoided crossing
with S1 and S2 is seen in the experiment.
When the magnetic field is in the 共ab兲 plane two cases
must be considered, B0 储 E1 and B0 ⬜ E1. In Figs. 5(c) and

FIG. 7. Intradimer dynamic DM interactions. A lattice distortion
with the normal coordinate Q (solid arrow) creates an intradimer
DM interaction d3 (empty arrow). The c-axis phonon creates a dynamic DM interaction on both dimers while the a-axis phonon affects the dimer (1,2) only.

We have shown that the first triplet state energy spectra
are well described with two static DM interactions, d1 andd2. The information about d1 and d2 is contained in the position of energy levels and in the FIR absorption line intensities. The interdimer d1 determines the magnetic field dependence of intensities and the triplet state level energy
splitting. The intradimer d2 determines the extent of the
avoided crossing with S2 and the magnetic field dependence
of intensities near the avoided crossing points. Over the magnetic field range of our experiment the intensities of thesinglet-triplet absorption lines do not depend on the dynamic
part of the DM interaction, because the phonon energies arelarge compared to the triplet state energy.
Other inter- and intradimer DM interaction components
besides d1 and d2 have been considered to describe experimental data.18,29 These are the in-plane component of theinterdimer DM dxy and the symmetry-forbidden intradimer
DM dz in the c-direction. We included dxy and dz in the two
dimer model and found that calculations with nonzero dxy
and dz give results contradicting with the experiment. Our
argument, which is independent of whether a particular infrared transition is allowed or forbidden, relies on the observed and calculated crossing-anticrossing effects between
the triplet and the singlet states.
If B0 储 c and dxy ⫽ 0 then T0m共+兲 would have an avoided
crossing with S2 contradicting the experiment, whereT0p共+兲 anticrosses with the singlet [Fig. 5(a)]. Also dz does
not give any anticrossing between S2 and T0m共+兲 or T0p共+兲.
In high field nonzero d2 creates an avoided crossing between
the ground state S0 and the triplet branch T0m共−兲 as observedin the experiment14 while nonzero dxy or dz do not create an
avoided crossing between S0 and T0m共−兲 or T0p共−兲. However,
the two dimer model does not predict the experimentallyobserved14 avoided crossing between S0 and T0p共−兲.
In B0 储 a field orientation both dxy and dz add, in addition
to d2, to the avoided crossing of one of the triplet components with S2. The experimental data can be fitted with asingle value d2 = 1.8 cm−1 in both field orientations, B0 储 aand B0 储 c. If dxy and dz were comparable in magnitude to d2,then the extent of avoided crossing would be different in
B0 储 a and B0 储 c field orientations.
Our conclusion is that the dominant DM interactions are
d1 = 0.7 cm−1 and d2 = 1.8 cm−1. In the magnetization plateau
state the lattice parameters of SrCu2共BO3兲2 may change due
to spin-phonon coupling.22 Our calculation of energy levels

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FAR-INFRARED SPECTROSCOPY OF SPIN…

did not account for static lattice distortions and therefore wecannot make any conclusions about dxy and dz and thestrength of d1 and d2 in high magnetic fields.
The intensity of the FIR singlet-triplet transitions depends
on the strength of the dynamic DM and on the frequency andthe oscillator strength of the phonon. Since the particularphonons involved in the dynamic DM effect in SrCu2共BO3兲2
are not known we can give only the relative strength of dynamic DM interactions. The a- and c-polarized singlet-triplet
transitions have similar oscillator strengths. These are
2.0 cm−2 共E1 储 a兲 and 1.7 cm−2 共E1 储 c兲 if we compare the twolower spectra in Fig. 2 which have been measured on thesame sample by changing the direction of the light polarization. The ratio of the dynamic DM interactions for the two
mechanisms is d3a / d3c = 冑2 ⫻ 2.0/ 1.7= 1.5 if we assume thata- and c-axis phonons have equal frequencies and oscillator
strengths. The factor 2 accounts for the a-axis phonon creating a dynamic DM only on the dimer with its axis perpendicular to E1.
D. Staggered g-tensor

The importance of the staggered g-tensor in SrCu2共BO3兲2
was pointed out by Miyahara et al.30 The staggered g-tensor
exists in SrCu2共BO3兲2 because of the buckling of Cu-O-B
planes below 395 K. It mixes singlet and triplet states similar
to the static DM interaction d2. The strength of the staggered
g-tensor interaction can be estimated and we show that its
effect on the energy of the spin levels is small compared withthe effect of d2. The Zeeman term HZs couples singlet andtriplet states on a single dimer and is proportional to gs␮BB0,
where gs = 共gx̄ − gz̄兲sin ␾ cos ␾ (Ref. 30). The angle ␾ ⬇ 6° is
the buckling angle of the Cu-O-B plane.20 The components
gx̄ and gz̄ of the Cu ion g-tensor are not known but we takegx̄ ⬇ ga = 1.998 and gz̄ ⬇ gc = 2.219 (Table I) and get gs= 0.023. The staggered term HZs increases linearly with magnetic field. The largest field where the anticrossing between
T0 and S2 takes place is 5 T. In this field the magnitude of
the staggered g-tensor term in the Hamiltonian is 0.05 cm−1,
which is much smaller than the static intradimer DM term
d2 = 1.8 cm−1. We conclude that the dominant coupling between the singlet and the triplet is due to the static DM
interaction d2.
E. States of bound triplets

Several states besides the one-triplet excitation are
infrared-active (Table I). We showed that the two dimer
model explains well the energies of the one-triplet states andtransitions to them. In the two dimer model with j1= 24 cm−1 and 2j2 = 9.8 cm−1 we get several two triplet
states: a singlet, a triplet, and a quintet of two bound tripletsat 28.4, 38.2, and 57.8 cm−1, respectively.
SrCu2共BO3兲2 has two low energy singlet states S1 and S2
which both anticross with triplet state levels (Figs. 4 and 5).
In the two dimer model only one singlet of bound triplets ispossible and the anticrossing occurs only with T0p共±兲 states.
In the experiment an anticrossing is observed between S2 andT0p共+兲, Fig. 5(a). The observed anticrossing between S1 andT0m共−兲 cannot be explained by the two dimer model. In Sec.
II C we show that other DM interactions besides d2 are weakor absent in SrCu2共BO3兲2 in the studied B0 range, although
they may have proper symmetry to couple S1 and T0m共−兲.
The energy of the 38.2 cm−1 triplet in the two dimer
model is in the range where triplets are present in
SrCu2共BO3兲2. There is a triplet at 37.5 cm−1 labeled as T1(Table I). FIR transitions to this state are active in E1 储 apolarization, Fig. 2. In E1 储 c the transitions are weaker (TableI). The T1共0兲 level is FIR active when B0 储 c and T1共±兲 are
active when B0 ⬜ c. All this, polarization and magnetic fielddependence, is consistent with the dynamic DM mechanismof the FIR absorption where the dynamic DM is along the caxis. The a-axis phonon creates a dynamic DM in the direction parallel to the c axis. The intradimer dynamic DM interaction d3a does not give any transitions to bound states of
triplets. We considered a possibility that the a-axis phonon
modulates the static interdimer d1. We found that the patternof dynamic interdimer DM vectors with the same symmetry
as d1 [Fig. 1(b)] gives selection rules that apply to the37.5 cm−1 T1 triplet. Transitions to other states are forbiddenin the first order of this dynamic DM interaction. The lattice
deformation that creates such a pattern of dynamic DM vectors is of A1 symmetry and is not an optical phonon; in the A1
symmetry mode Cu atoms on j1 bond move along the bondin antiphase. We conclude that the two dimer model is not
sufficient to account for transitions to states of bound triplets,
except to S2.
Quintet states were observed by high field ESR.14 Their
extrapolated zero field energies are in the range 46– 58 cm−1.
There are two E1 储 a singlet excitations at 52.3 and 53.5 cm−1in this range (Table I). The quintet 共S = 2兲 has a mS = 0 spinlevel which has the same magnetic field dependence of energy as the S = 0 state. However, the observed singlets at 52.3
and 53.5 cm−1 are not the mS = 0 components of the quintet.
If in one B0 field orientation the mS = 0 level is infraredactive then in the 90° rotated field orientation other levels,mS = ± 1 or mS = ± 2, become active. We studied all possible
B0, E1 orientations relative to crystal axes and did not findthe splitting of the 52.3 and 53.5 cm−1 excitations in themagnetic field although they are one to two orders of magnitude more intensive than other magnetic excitations in FIR
spectra.
We assigned the a-axis polarized 共E1 储 a兲 43.0, 52.3, and53.5 cm−1 singlet excitations to magnetic excitations becauseof the magnetic field and temperature dependence of theirenergy and intensity.16 Whether they could be phonons activated by magnetic interactions needs a further study.
III. CONCLUSIONSIn SrCu2共BO3兲2 the ground state is not exactly a productof singlets on dimers as in the Shastry-Sutherland model,
because the intradimer Dzyaloshinskii-Moriya interaction d2
mixes the ground singlet state with the triplet. From the observed anticrossing between T0p共+兲 and S2 we get d2= 1.8 cm−1. This is comparable to the interdimer DM, d1= 0.7 cm−1, which determines the triplet state energy levelzero field splitting. Both d1 and d2 determine the magnetic

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field dependence of the absorption line intensities.
Although magnetic dipole singlet-triplet transitions are allowed by d2, the experimentally observed polarization andmagnetic field dependencies of absorption line intensities are
not described by this interaction. Instead, singlet-triplet transitions are allowed by the dynamic DM mechanism where
the electric field component of FIR light couples to a nonsymmetric phonon, which creates the DM interaction. There
are two dynamic DM mechanisms in SrCu2共BO3兲2. In one
case the FIR light couples to an a-axis phonon and in theother case to a c-axis phonon. This is consistent with thecalculations of Cépas and Ziman23 who used a two dimer
model in the j2 = 0 limit.
The experiment also yielded information about higher
triplet and singlet excitations. Several of these absorption

lines are identified as electric dipole transitions. The two
dimer cluster is too small to describe these transitions. Also,
we had to use renormalized values of j1 and j2 to calculate
the energy levels because the actual spin excitations are delocalized over a larger cluster. Obviously a bigger cluster isneeded for proper calculation of magnetic excitations in
SrCu2共BO3兲2. Nevertheless, the two dimer model gives us agood description of the one-triplet excitation.

*Electronic address: roomtom@kbfi.ee

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ACKNOWLEDGMENTS

We thank G. Blumberg and O. Cépas for fruitful discussions. This work was supported by the Estonian Science
Foundation Grant Nos. 4926, 4927, and 5553.

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