PHYSICAL REVIEW B 72, 064405 共2005兲

Symmetry and light coupling to phononic and collective magnetic excitations in SrCu2„BO3…2
A. Gozar,1,2,* B. S. Dennis,1 H. Kageyama,3 and G. Blumberg1,†
1Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974, USA
2University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA
3

Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
共Received 1 July 2004; revised manuscript received 6 June 2005; published 3 August 2005兲

We perform a low temperature Raman scattering study of phononic and collective spin excitations in theorthogonal dimers compound SrCu2共BO3兲2, focusing on the symmetry and the effects of external fields on themagnetic modes. The zero field symmetry and the behavior in magnetic fields of the elementary and boundmagnetic triplet states are experimentally determined. We find that a minimal 4-spin cluster forming the unit
cell is able to describe the symmetry as well as the anisotropic dispersions in external fields of the spin gap
multiplet branches around 24 cm−1. We identify two Raman coupling mechanisms responsible for the distinct
resonance behavior of these magnetic modes and we show that one of these can be ascribed to an effective
intradimer Dzyaloshinskii-Moriya spin interaction. Our data also suggest a possible explanation for the existence of a strongly bound two-triplet state in the singlet sector which has an energy below the spin gap. The
low temperature phononic spectra suggest strong spin-phonon coupling and show intriguing quasidegeneracyof modes in the context of the present crystal structure determination.
DOI: 10.1103/PhysRevB.72.064405

PACS number共s兲: 75.10.Jm, 75.50.Ee, 78.30.⫺j, 71.70.Ej

I. INTRODUCTION

Several properties make SrCu2共BO3兲2 a unique systemamong the known quantum magnets. This compound is arealization of a two-dimensional 共2D兲 spin system with adisordered ground state even at very low temperatures and aspin gap ⌬ ⬇ 24 cm−1 共3 meV兲 from the singlet 共S = 0兲ground state to the lowest excited triplet 共S = 1兲 state.1,2 The
strengths of the relevant magnetic interactions place thiscompound close to a quantum critical point 共QCP兲. Moreover, data in high magnetic fields show plateaus at commensurate 共1 / 8, 1 / 4, and 1 / 3兲 values of the saturation
magnetization.2–4 The plateau states can be thought of as
crystalline arrangements of magnetic moments separating regions of continuous rise in magnetization, the latter allowing
for an interpretation in terms of Bose-Einstein condensationof triplet excitations.5 It has also been suggested6 that dopingin this system 共regarded as a Mott-Hubbard insulator兲 maylead to a superconducting phase mediated by antiferromagnetic 共AF兲 fluctuations, a mechanism similar in spirit to oneof the scenarios proposed for the high-Tc cuprates.7
The S = 1 / 2 Cu spins are arranged in weakly coupled 2D
layers defining the 共ab兲 plane. In each of these sheets they
form orthogonal spin-dimer lattices;8,9 see Fig. 1共a兲. At Tc= 395 K the system undergoes a second order phase transition from the space group I4 / mcm to I4̄2m on cooling downfrom the high temperature side. In the I4 / mcm phase theplanes containing the Cu atoms are flat and they form mirror
symmetry elements. The transition at 395 K can be understood as the buckling of the Cu planes which lose their mirror symmetry property. Inversion symmetry is lost as well
below Tc but the number of atoms in the unit cell remains
unchanged due to the orthogonality of the spin-dimer network. Because of the same orthogonal arrangement, there are
two spin dimers in the unit cell. As a result, the spin gap
excitation, defined as the transition from the S = 0 ground1098-0121/2005/72共6兲/064405共9兲/$23.00

state to that 共usually the lowest in energy兲 S = 1 level whose
wave function contains a single spin dimer excited to thetriplet state,1,10,11 has a “fine structure” made out of six levels, three from each dimer in the unit cell.
The 2D spin-dimer lattice can be described well by taking
into account the nearest and next nearest neighbor AF superexchange interactions J1 and J2, respectively; see Fig. 1共a兲.
In this approximation and for a ratio x = J2 / J1 lower thanabout 0.7, the direct product of singlet dimers on each rung,
which is always an exact eigenstate of the Hamiltonian, isalso the ground state of the system and the spin gap is finite
although it gets renormalized down with increasing x due tomany body effects.10,12 For large x the ground state changes
and the Hamiltonian has long range AF order, other possible

FIG. 1. 共Color online兲 共a兲 The structure of the spin-dimer compound SrCu2共BO3兲2. The unit cell is shown by the short dashedsquare. The circles are the S = 1 / 2 Cu spins; the solid and long
dashed lines represent the intradimer 共J1兲 and interdimer 共J2兲 AF
superexchange interactions. The arrows correspond to the proposed
intradimer antisymmetric spin interaction dab. Below 395 K theorthogonal dimers are no longer coplanar. 共b兲 The 4-spin cluster
corresponding to the unit cell in 共a兲. As expected, the symmetry ofthis object is D2d, which is the point group associated with the
crystal space group of SrCu2共BO3兲2 below 395 K. Note that theinterdimer superexchange J2 is effectively doubled if periodic
boundary conditions are used.

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GOZAR et al.

intervening states separated by QCP’s being proposed to exist around 0.7; see Ref. 1. Based on fits to magnetization
data, predictions of symmetries, relative energies, and dispersions of single and composite triplet excitations, theoreticalestimates for x range from 0.603 共Refs. 10 and 11兲 to about
0.68 共Refs. 12 and 13兲.
An analysis of the vibrational modes is of interest in
SrCu2共BO3兲2 because spin-lattice interaction has been suggested to be relevant to the magnetic dynamics at low temperatures and high magnetic fields. In particular, spin-phonon
interaction has been invoked in order to explain the selection
rules of the magnetic transitions seen in infrared 共IR兲
absorption.15 The coupling between the lattice and magnetic
degrees of freedom was also taken into account in order todescribe the spin density profile at high fields in the magnetization plateau states.4 So far the study of phononic excitations has been related mostly to the crystallographic changes
at 395 K. It has been established in Refs. 9 and 14 that thestructural transition displays soft mode behavior 共see Refs. 9
and 14 also for a phononic symmetry analysis兲. The
phononic mode which condenses at 395 K belongs to the B1u
representation of the high temperature symmetry group and,in terms of Cu atoms, it involves essentially an alternate
displacement along the c axis of the nearest neighbor dimers.
Interestingly, several phonons appearing below Tc were observed in 共ca兲 polarization as very close in energy shouldersof some of the modes found above the transition. Due to theabsence of inversion symmetry below 395 K this experimental fact suggested the existence of almost degenerate evenand odd 共transforming like the Eg and Eu representations
respectively兲 excitations in the high temperature phase.14
In terms of magnetic properties, elementary and twotriplet excitations have been studied by inelastic neutron
scattering 共INS兲,16,17 electron spin resonance 共ESR兲,18,19 and
IR absorption spectroscopy.15,20 The low temperature Raman
scattering results shown in Ref. 21 are focused on composite
magnetic excitations in the S = 0 channel and are used to
extract quantitatively the ratio x = J2 / J1. In our Raman study,
the emphasis is however on the elementary and composite
collective excitations in the S = 1 sector, the main results being related to the experimental determination of their symmetries and anisotropic behavior in magnetic fields. We arealso able to relate some of these new findings to results of a
4-spin cluster analysis as well as to propose an additional
effective spin interaction induced by spin-orbit coupling. It
has been established that due to the frustrated nature of themagnetic interactions, the one triplet excitations are local,
weakly dispersive in the reciprocal space, while two-particle
states are more mobile16 and have contributions from thewhole Brillouin zone.10,11
In spite of a lot of experimental and theoretical effort forunderstanding the magnetic properties there are several open
questions. An exact determination of the ratio of the exchange interactions x, which is important due to the proximity to the QCP, is still missing. One interesting aspect in thisregard is the observation of a magnetic state at 21.5 cm−1,
which is below the energy of the spin gap multiplet ⌬. This
could be seen in ESR,19 INS,17 and IR15 data in energy level
anticrossings in the downward dispersions with magnetic
field of some of the spin gap branches. The existence of sucha low energy mode brings into question the exact quantitative estimation of the AF exchange parameters. As we willshow, a direct comparison between our experimental findings
and theoretical predictions regarding the symmetry of thesemodes is illuminating in this respect.
A different set of questions is related to the way the external radiation field couples to the magnetic excitations. While the photon induced spin-exchange process insures the Raman coupling to S = 0 two-triplet excitations,21,22
the transitions to S = 1 states require the presence of spinorbit coupling.22 Although an effective antisymmetric
Dzyaloshinskii-Moriya 共DM兲 term originating in the spincontribution explicitly written inorbit interaction 关the HDM
cEq. 共3兲兴 has been proposed to explain the “fine structure” ofthe six levels forming the gap multiplet around 24 cm−1 seen
in a neutron scattering study, Ref. 17, this term does not mix
the singlet ground state to excited S = 1 states. As a result, thenature of the coupling leading to the observed ESR and IR
data is still to be understood. Possible candidates were suggested in the discussion of ESR data19 and dynamical spinphonon induced effective DM interactions have been invoked in order to explain the optical absorption spectra.15,20
In this article we study phononic modes and collective
magnetic excitations in SrCu2共BO3兲2 with the experimental
emphasis placed on the understanding of the magnetic Raman scattering in the S = 1 channel. In the low temperature
in-plane polarized phononic spectra we find 共in the
50–350 cm−1 energy range兲 several pairs of modes withsimilar energies. This quasidegeneracy is intriguing because
group theory analysis suggests that they are related to different atomic vibrational patterns. Regarding the magnetic Raman scattering data, our novel findings are the following.
The symmetry analysis and exact diagonalization of the
4-spin system Hamiltonian 关see Fig. 1共b兲兴 given byDM
H = H0 + HDM
c + Hab ,共1兲

which includes the main, unperturbed, Heisenberg term, H0,
the antisymmetric interdimer DM term proposed in Ref. 17,DMHDM
c , and an additional intradimer DM term, Hab , reading
H0 = J1 兺 Si · S j + J2
共i,j兲NNHDMc =兺 Si · S j ,共i,j兲NNN共2兲

dជ 共i,j兲兺c 共Si ⫻ S j兲,
共i,j兲NNN共3兲兺 dជ ab共i,j兲共Si ⫻ S j兲

共4兲

DM
=
Hab

共i,j兲NNexplains the experimentally determined symmetries of thezero field Brillouin zone center spin gap branches around
24 cm−1 共confirming the local nature of the elementary onetriplet modes兲 but fails to account for the two-triplet excitations. In the equations above i and j are nearest neighbor
共NN兲 or next nearest neighbor 共NNN兲 Cu sites and dជ c and dជ ab
are inter and intradimer antisymmetric spin interaction vectors. Remaining confined to the 4-spin cluster we find that byDM
weintroducing the effective intradimer DM interaction Hab
are also able to reproduce the observed selection rules and064405-2

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PHYSICAL REVIEW B 72, 064405 共2005兲

intensity variations of the spin gap branches in external magnetic fields. These selection rules also require that the energyof the S = 0 two-triplet bound state formed from spins confined within a unit cell is below ⌬ 共in the 4-spin cluster thisis equivalent to x 艌 0.5 in Fig. 4兲 suggesting a high binding
energy for this two particle excitation. In the last section, we
also show results of a resonance study which allows us to
identify the action of two different light coupling mechanisms to magnetic excitations.
II. EXPERIMENT

Raman scattering from the ab surface of a single crystalof SrCu2共BO3兲2, grown as described in Ref. 23, was performed using an incident power density less than 1 mW focused to a 100 ␮m diameter spot. The crystallographic axes
orientation was determined by x-ray diffraction. The data inmagnetic fields, taken at a sample temperature of about 3 K,
was acquired with a continuous flow cryostat inserted in thehorizontal bore of a superconducting magnet. We used the␻L = 1.92 and 2.6 eV excitation energies of a Kr+ laser and a
triple-grating spectrometer for the analysis of the scattered
light. The data were corrected for the spectral response of thespectrometer and CCD detector.
Polarized Raman scattering can probe Brillouin zone center excitations that belong to different symmetry representations within the space group of the crystal structure. We denote by 共eineout兲 a configuration in which the incoming/
outgoing photons are polarized along the ein / eout directions
共see Fig. 1 for axes notations兲. The 共RR兲 and 共RL兲 notations
refer to circular polarizations, ein = 共â − ib̂兲 / 冑2, with eout = ein
for 共RR兲 and eout = e*in for the 共RL兲 geometry. The point groupassociated to the high temperature phase space group of the
SrCu2共BO3兲2 crystal, I4 / mcm, is D4h. The point group associated to the I4̄2m space group corresponding to the lower
temperature phase is D2d. In D2d, the 共RR兲, 共RL兲, 共aa兲, 共ab兲,
共a⬘a⬘兲, and 共a⬘b⬘兲 polarizations probe excitations which belong to the A1 + A2, B1 + B2, A1 + B1, A2 + B2, A1 + B2, andA2 + B1 irreducible representations of D2d.
III. PHONONIC SYMMETRIESIn Fig. 2 we show six low temperature Raman spectra
excited with the ␻L = 1.92 eV laser energy. The arrows point
to the observed modes above 60 cm−1 and also shown are
their energies and symmetries. Magnetic fields do not affecttheir energies which indicates the phononic nature of these
excitations. The modes below 60 cm−1 are not indexed since
they will be discussed in the next sections. The excitation at
59 cm−1 corresponds to the fully symmetric soft mode of thestructural transition at 395 K.9,14 The corresponding twophonon excitation in the A1 channel is seen at 121.8 cm−1and very close to it another sharp mode with B1 symmetry.
Similarly to the data in 共ca兲 polarization in Ref. 14, where
below 395 K several new modes appear as shoulders ofphonons existing above the transition, we observe several
pairs of modes having close energies. For example doublet
structures are observed around 155 cm−1, where we see aFIG. 2. 共Color online兲 Raman spectra at T = 3 K in six polarization configurations and zero applied field. Next to each verticalarrow we show the energies 共in cm−1兲 and symmetries of the modes
above ⬇60 cm−1. The spectra are vertically offset. The excitationenergy used is ␻L = 1.92 eV.

pair of B2 and A2 excitations, around 285 cm−1, where weobserve two modes having B1 and A1 symmetries, and also
around 320 cm−1 where we see a pair made of B2 and B1
symmetric excitations.
One way to explain this behavior, suggested in Ref. 14
is to assume that in the high temperature phase there
are phonons which are odd 共ungerade: “u”兲 and even共gerade: “g”兲 with respect to inversion but very close in energy and to try to identify them by looking at similar atomic
vibrations corresponding to “u” and “g” representations,
respectively. Following this idea in more detail and taking
into account that the set of irreducible representations
兵A1g , A2g , B1g , B2g , A1u , A2u , B1u , B2u其 of the D4h point group
becomes the set of 兵A1 , A2 , B1 , B2 , B1 , B2 , A1 , A2其 representations 共in this order兲 in D2d, we would have for example thatthe 共A1 , B1兲 pair around 284 cm−1 corresponds either to a
共A1g , A1u兲 group or to a 共B1u , B1g兲 group in the high temperature phase. This is because we chose gerade-ungerade
pairs and, as shown above, the A1 symmetric mode can originate either from a A1g or a B1u symmetric phonon while theB1 mode could be either a B1g or a A1u phonon in the hightemperature phase. Similar reasoning would suggest that the
共B1 , B2兲 group around 320 cm−1 originates either from a pairof 共B1g , A2u兲 or 共B2g , A1u兲 above 395 K and also that theorigin of the 共B2 , A2兲 group around 155 cm−1 can be pairs of
共B2g , B2u兲 or 共A2g , B1u兲 modes in the I4 / mcm phase which
has D4h as the associated point group.
We performed a symmetry analysis of the k = 0 atomic
vibrations in the high temperature phase and our conclusion
is that the approach suggested in Ref. 14 does not provide ana priori reason for the quasidegeneracy. This conclusion, as
explained in the following, is based just on a simple inspection of the character table of the D4h point group.24 One caneasily note that in D4h the even modes are symmetric withrespect to the mirror symmetry in the Cu共BO3兲 planes while
the odd vibrations are antisymmetric with respect to thissymmetry operation. This means that the “u” phonons in the064405-3

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FIG. 3. 共Color online兲 Zero field Raman data taken with the␻L = 1.92 eV excitation energy at T = 3 K in six polarization configurations. The legend shows the tetragonal symmetries probed ineach scattering geometry. See the text for notations.

high temperature phase correspond to vibrations of the atoms
along the c axis while the “g” modes consist of in-planemovements. Due to this difference in the oscillation patterns,
one parallel and one perpendicular to the CuBO3 planes, it ishard to explain the closeness of phononic energies at thisqualitative level. This is why we find intriguing the observed
quasidegeneracy of the phononic modes in the context of thepresent crystal structure determination.
We remark that on the other hand one could find vibrations which involve similar oscillations at the “molecular”
level 共for instance groups of atoms forming the Cu-O spin
dimer structure or groups of O2 atoms bridging nearest
neighbor spin dimers兲 and which belong to different grouprepresentations because of the different “intermolecular”
phase pattern. We suggest that, remaining within the conclusions of x-ray studies9 which so far have not found evidences
for additional crystallographic changes at low temperatures
共that in turn may produce phonon splittings兲, good candidates for understanding this behavior are provided by theinterdimer BO3 molecular complexes whose rotations as awhole around the a⬘ and c axes or whose in and out of theplane translations may turn out to be similar in energies. The
data suggest that theoretical work in this respect could beinteresting.
We note that the appearance of the weak 155.9 cm−1
mode in the 共RR兲, 共ab兲, and 共a⬘b⬘兲 polarizations singles thismode out from the other excitations because it belongs to theA2 symmetry channel. The Raman coupling to this excitationis unusual compared to the other modes in the sense that it
cannot take place via two electric dipole transitions. We findseveral magnetic resonances 共see Fig. 3兲 with A2 symmetry,
but the absence of magnetic field effects suggests that thismode has a preponderant phononic character. A finite intensity of such excitation provides direct evidence for spinphonon coupling in SrCu2共BO3兲2.

FIG. 4. 共Color online兲 共a兲 The eigenvalues of the H0 term of thespin Hamiltonian of Eq. 共1兲 corresponding to the 4-spin clustershown in Fig. 1共b兲. The spin gap excitation at ⌬ = J1 is six times
degenerate in this approximation. 共b兲 Symmetry analysis of the
4-spin cluster in 共a兲 in the D2d group. The energies on the left arecalculated for a ratio x = J2 / J1 equal to 0.3. On the right we show
the energy splittings and the absolute and relative to the groundstate 共in parenthesis兲 symmetries of the 16 magnetic modes when afinite interdimer DM interaction dc 共see Ref. 17兲 is present. The
effect of a finite intradimer DM perturbation, dab, is to further split
the modes which belong to the one-dimensional representations.
IV. MAGNETIC SYMMETRIESThe irreducible representations of the D2d point groupprobed by six scattering geometries, discussed also in theexperimental section, are shown in the legend of Fig. 3. In
this figure, the six low temperature Raman spectra from Fig.
2 are shown for the frequency region below 60 cm−1. Threestrong features around 23, 29, and 38 cm−1, denoted by T1b,
S1, and T2, are observed and they transform like the A2, B1,
and A2 representations respectively. Besides these threemodes, we observe several other weaker excitations. Amongthem we see a set of three A2 symmetric modes denoted by
T3, T4, and T5. We also note the presence of the excitations
denoted by T1e and T1f which have B1 and B2 symmetries,
giving rise to a small feature seen around 25.6 cm−1 in allpolarizations except for the 共RR兲 configuration. Table I contains a summary of the observed excitations below the
60 cm−1 frequency range.
The energies of these Raman active excitations are inagreement with those where INS,16 ESR,18,19 and IR15,20 data
observed magnetic modes. The data in magnetic fields 共see
the discussion in the following section兲 confirm the magnetic
nature and the predominant S = 1 character of the “T” modes.
Therefore T1b, T1e, and T1f excitations modes seem to belong
to the spin gap multiplet while T2, T3, T4, and T5 would
correspond to multiparticle triplet channels.

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TABLE I. Collective spin excitations in zero field: notation, thepredominant spin character 共Stot兲, the z projection of the spin 共Sz兲,the energies and transition symmetries as observed experimentallyand predicted from the 4-spin cluster in Fig. 1 corresponding to k
= 0 excitations. T1a … . T1f represent elementary triplet excitations.
Generally, the modes whose energies change in external magnetic
fields are indexed by T.

Mode

Stot共Sz兲

Energy

T1a
T1b
T1c
T1d
T1e
T1f

1共±1兲
1共±1兲
1 共0兲
1 共0兲
1共±1兲
1共±1兲

22.8
22.8
24.2
24.2
25.6
25.6

S1
T2
T3
S2
T4
S3
T5

0 共0兲
1
1
0
1
0
1

28.9
37.5
40.8
43.0
44.5
45.2
50.9

Symmetry
ExperimentGroup theory

A2

B1
B2

A1
A2
E
E
B1
B2

B1
A2
A2
B1
A2
B1
A2

If the picture of real space localized elementary triplets istrue, then one expects that the analysis of the 4-spin cluster
in Fig. 1共b兲 forming the unit cell is able to predict correctly
at least some of the experimentally observed symmetries ofthese excitations. Besides symmetry analysis, numerical diagonalization of the Hamiltonian of Eq. 共1兲 allows one in
principle to also identify the energy as well as the predominant spin character of each eigenstate of this cluster. Indeed,
as can be seen by comparing Figs. 3 and 4共b兲 and looking at
the rightmost column of the Table I, the symmetries of theobserved zero field one-triplet excitations correspond to theresults of group theory analysis. The doubly degenerate E
modes, T1c and T1d, are not observed in zero field when thelight propagates parallel to the c axis because they are accessible only in 共ca兲 or 共cb兲 polarizations. The observation of
the T1e共T1f 兲 modes with B1共B2兲 symmetries at an energy of
2.8 cm−1 above the A2 symmetric mode T1b allows the determination of the magnitude of the interdimer interaction dc
关see Fig. 4共b兲兴 and also of its absolute sign.25 The fully symmetric T1a mode which, within the spin model including only
terms given in Eqs. 共2兲 and 共3兲, should be
the H0 and HDM
cdegenerate with the strong T1b 共A2 symmetric兲 excitation at
22.8 cm−1 is also not observed which is most probably due toa much weaker coupling to light in this symmetry channel.
We will discuss the coupling mechanisms in the last sectionof our paper.
We turn now to the discussion of two-triplet states. Besides the singlet ground state and six one-triplet states, as canbe seen in Fig. 4, the 4-spin cluster analysis predicts thefollowing: one S = 0 two-triplet bound state in the B1 channel, three branches with A1 and doubly degenerate E symmetries which belong to another bound S = 1 excitation and

FIG. 5. 共Color online兲 Magnetic field dependencies of the magnetic excitations at T = 3 K using the ␻L = 1.92 eV excitation energyជ 储 ĉ; 共b兲 共RR兲Bជ ⬜ ĉ. In 共c兲 thein the following geometries: 共a兲 共RR兲B
共RL兲 polarized data is shown for 0 and 6 T magnetic fields for bothជ 储 ĉ and Bជ ⬜ ĉ. In 共a兲 and 共b兲 the vertical shift is proportional to theBmagnetic field difference with respect to the 9 T spectrum and thedashed lines are guides for the eye. Panel 共d兲 is a zoomed-in regionof panel 共b兲.

five branches of a quintuplet 共S = 2兲 state having A1, A2, B1,
and E symmetries with respect to the ground state. Theirenergies are plotted in Fig. 4共a兲 as a function of x = J2 / J1. We
note that due to symmetry reasons none of the observed A2
symmetric modes from T2 to T5, having energies higher than
30 cm−1, qualify for an interpretation as triplet bound states
generated within the 4-spin cluster. This is consistent withthe fact that larger cluster sizes are necessary in order tocapture the more delocalized nature of these excitations.
The fact that the existence of the strong A2 symmetricbound triplet state at an energy 1.55⌬ = 37.5 cm−1 has notbeen predicted by high order perturbative analysis10 共here we
refer especially to the symmetry of this excitation, not its
energy兲 suggests that other spin interactions have to be taken
into account in order to explain the excitation spectrum. Apparently the symmetry considerations would allow the28.9 cm−1 feature denoted by S1 in Fig. 3 to be interpreted as
the singlet bound state of two triplets within a unit cell. As
we show in the following section, the 28.9 cm−1 mode does

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FIG. 6. 共Color online兲 Energies 关panels 共a兲 and 共b兲兴 and intenជ 储 ĉ 共left兲 andsities 关panels 共c兲 and 共d兲兴 of the spin excitations for B
Bជ ⬜ ĉ 共right兲 from Figs. 5共a兲 and 5共b兲. Symbols represent experimental points, solid lines are the results of 4-spin cluster diagonalization as described in the text, and dashed lines are guides for theeye.

not shift in external fields, which is compatible with a collective singlet excitation as discussed in Ref. 21, but suggeststhat its internal structure is not the one derived from the4-spin cluster.
V. MAGNETIC FIELD EFFECTS AT T É 3 K

In Fig. 5, using the same mode notations, we show theinfluence of an external magnetic field applied parallel andperpendicular to the c axis on the low temperature Raman
spectra from Fig. 3. Here we summarize the relevant aspects,
noting first that energy shifts induced by magnetic fields inthis figure were observed only for the modes indexed by T in
the Table I. In Fig. 5共a兲 we observe the splitting of the T1aជ 储 ĉ, the B = 1 T spectrum
and T1b modes in magnetic fields B
showing that the A2 mode 共T1b兲 present in zero field disperses upwards with increasing magnetic field. Dashed lines
mark the dispersion of the much weaker modes T3, T4, andT5. In Fig. 5共b兲 one of the E modes at 24.2 cm−1 becomes
Raman active due to symmetry lowering for Bជ ⬜ ĉ configuration and we observe three dispersing branches of the gap
multiplet. Figure 5共c兲 shows that the B1 symmetric excitation
at 28.9 cm−1 does not change its energy with field, only avery small negative shift of the order of 0.5 cm−1 from 0 to 6
T is seen because of the crossing with the upward dispersing
gap branches seen in 共RR兲 polarization. Figure 5共d兲, which isa zoomed-in region of Fig. 5共b兲, shows that several modesជ ⬜ ĉ around 38 cm−1
become Raman active in finite fields B
where the T2 excitation lies. The internal structure of thishigher energy multiplet is composed of modes dispersing up,

down and independent of magnetic field. We remark on thesimilarity in selection rules and dynamics in magnetic fields
between the collective modes around 38 cm−1 and that of thespin gap branches around 24 cm−1. The emergence in finite
fields of several strong modes in the spin gap region precludes the observation of the dynamics of the weak T1e andT1f modes from Fig. 3.
Figure 6 summarizes the magnetic field dependencies ofthe energies and spectral weights of the most intense Raman
excitations. The symbols in Figs. 6共a兲–6共d兲 correspond to theexperimental data and the solid lines are results of calculations: the energies in panels a and b are obtained by exact
diagonalization of the Hamiltonian of Eq. 共1兲 with the parameters specified in the next paragraph; using Fermi’s
golden rule, the intensities in Fig. 6共c兲 and 6共d兲 are calculated as the square of the matrix elements between the
ground and excited states of the effective Fleury-Loudon
spin interactions describing the coupling to the external electromagnetic field.22 The form of these interaction terms, deint
noted by Hint
1 and H2 , are discussed explicitly in the next
section.
Here we discuss the parameter set used for data fitting inជ 储 ĉ and Bជ ⬜ ĉ configurations. Taking into accountboth the B
that 共i兲 the 4-spin cluster neglects many-body gap renormalization effects leading to a singlet-triplet energy independentof x = J2 / J1, see Fig. 4共a兲, and 共ii兲 the fact that when using
periodic boundary conditions there is an effective doublingof the J2 and interdimer DM interactions, we chose the following values: J1 = ⌬ = 24.2 cm−1 which is the value of thespin gap 共see the table and Refs. 16–20兲; x = 0.556 from theratio of the energies of the subgap mode at 21.5 cm−1, Refs.
15 and 19, with respect to the gap ⌬ 关see Fig. 4共a兲兴; aninterdimer DM term dc = 1.4 cm−1, which produces the splitting of the T1a,b and T1e,f branches from 24.2 cm−1 to 22.8
and 25.5 cm−1 respectively 共our value of dc is consistent withthe one proposed in Ref. 17兲; finally, using the magnetic fieldvalue where the intensities cross in Fig. 6共c兲, the value of theintradimer interaction was set to dab = 2.66 cm−1. We note
two aspects regarding the magnitude of the parameters used
above. The first is that because many-body spin interactions
are not captured within this minimal 4-spin cluster, our chosen value for x = J2 / J1 should not be taken ad litteram for thereal structure. The second aspect, discussed in more detaillater in this section in connection to the dispersion of theជ 储 ĉ configuration, is related to themagnetic modes in the B
fact that we were forced to choose x ⬎ 0.5, which is equivalent to saying that the singlet bound state has an energy
which is lower than the elementary gap excitation at
24.2 cm−1. This fact may be important regarding the experimental observation of a mode at 21.5 cm−1; see Refs. 15, 17,
and 19.
The space group symmetry of the crystal uniquely imposes for any existing static intra-dimer DM interaction 关theDM
in Eq. 共4兲兴 the configuration depicted in Fig. 1共a兲,term Habi.e., the DM vectors are perpendicular to the c-axis and dimerDMterm in the system Hamiltonian which isbonds. It is the Hab
responsible in our interpretation for the mixing of singlet andtriplet modes and allows for a finite coupling of the latter
excitations to the external photon field. As we mentioned in

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SYMMETRY AND LIGHT COUPLING TO PHONONIC AND …

FIG. 7. 共Color online兲 T = 3 K Raman data in 共a⬘b⬘兲 polarizationfor ␻L = 1.92 共filled squares兲 and 2.6 eV 共empty circles兲 excitationenergies. The data points corresponding to the resonantly enhanced
mode at 29 cm−1 are multiplied by 0.15 and the corresponding line
represents a Lorentzian fit.

the Introduction, a finite intradimer interaction dab is crucial
term alone does not mix the singlet andbecause the HDM
ctriplet states. Also shown in Fig. 6 by filled squares anddashed lines are the experimental field dependencies of otherhigher energy modes observed in Fig. 4.
The agreement for this set of parameters is qualitatively
good overall and quantitatively better with regard to the enជ ⬜ ĉ case. As discussed
ergy and intensity variations for the B
DM
before, the term Hab plays a crucial role in obtaining a finiteDM
also produces
coupling to the excited S = 1 triplets. Hab
splittings of the T1a and T1b modes, of the T1e and T1f modes
as well as that of the quintuplet branch 关these splittings are
not shown explicitly in Fig. 4共b兲兴. For the spin gap branches,the magnitude of the splittings is unresolved because it is2/ ⌬ ⬇ 0.25 cm−1. The largest
very small, of the order of dab
discrepancy between the experimental data and the calculation is seen in Fig. 6共c兲. One aspect in this regard is that thevalue dab had to be chosen greater than that of dc. This is
unexpected because dc is allowed by symmetry both aboveand below the structural phase transition at 395 K while theexistence of a finite intradimer DM interaction is allowed
only below 395 K, when the mirror symmetry of the 共ab兲
plane is broken. Additional terms may be responsible for thisdisagreement, possible candidates being the in-plane components of the interdimer DM interaction, which should also beallowed below the structural phase transition.
We now discuss the existence of a magnetic mode below
the spin gap value.15,17,19 In order to reproduce the upward
dispersion with fields Bជ 储 ĉ of the T1b mode we had to choose
a value for x which is greater than 0.5, otherwise this excitation would have displayed a downward dispersion. From
Fig. 4共a兲 we observe that x = J2 / J1 艌 0.5 implies that the position of the bound singlet state is below ⌬. We suggest thatprecisely this state may be responsible for the observationsof the 21.5 cm−1 mode in Refs. 15, 17, and 19. The presenceof this excitation will also influence specific heat measurements and, in conjunction with the finite intradimer interaction dab, also the low temperature magnetization data which
is not quantitatively understood yet.1

PHYSICAL REVIEW B 72, 064405 共2005兲

The existence of this magnetic mode below the gap seems
quantitatively at odds with theoretical predictions.10,12,13
Nevertheless, perturbational calculations predict the existence of a singlet state at 25 cm−1, which is above but very
close to the spin gap.10 A quantitative reconciliation between
theory and the observed selection rules in magnetic fields
could be achieved if the coupling ratio x = J2 / J1 is slightly
increased from the value of 0.603, as determined in Ref. 10.
The experimental finding of the set of A2 modes 共T2 to T5 in
Fig. 3, all of them below the two-magnon continuum startingat 2⌬ ⬇ 48 cm−1 and whose symmetries are also not predicted by theory兲 shows that although several aspects of themagnetic bound states are understood, a complete picture of
the spin dynamics in the multitriplet sectors is still to beachieved.
VI. RESONANCE AND LIGHT COUPLING MECHANISMS

Figure 7 shows two low temperature Raman spectra taken
in 共a⬘b⬘兲 polarization with two incoming laser frequencies,
␻L = 1.92 and 2.6 eV. The point we want to make in thisparagraph is that we observe two qualitatively different behaviors. First we notice that the area under the peak corresponding to the T1b mode in the spectrum taken with theexcitation energy ␻L = 1.92 is of the same order of magnitude
as the one in the spectrum taken with the 2.6 eV laser excitation frequency. The same observation is also true for themode denoted by T2. From this perspective, a different behavior is observed for other modes and an example is thegroup of modes denoted by T1e, S1, S2, and S3. In this lattercase for instance, it is clearly seen in our spectra that themode S1 in the spectrum taken with the ␻L = 2.6 eV excitation is several orders of magnitude stronger than when using
the ␻L = 1.92 eV laser line. Also, while the modes T1e, S2,
and S3 are barely visible when ␻L = 1.92 eV 共see also Fig. 3兲,they become quite strong for ␻L = 2.6 eV. Figure 5 shows inaddition that other modes in the 50 to 70 cm−1 energy range
also become visible only in the ␻L = 2.6 eV spectrum. We
conclude this paragraph by saying that the group 兵T1b , T2其
has a different resonance behavior than the group
兵T1e , S1 , S2 , S3其 in the sense that their relative intensities arequite different for the two laser excitation frequencies used.
Note that in the paragraph above we commented only on
the relative intensities of the Raman modes when the ␻L= 1.92 eV excitation was used compared to the 2.6 eV line.
Although the absolute intensities of each the Raman excitations can be affected by optical corrections of the data, thiseffect cancels out when relative intensity ratios are considered. In other words, it is possible in principle that opticalcorrections can change the ratio of the absolute values of theRaman intensities of the T1b or T2 modes when taken with
␻L = 1.92 and 2.6 eV laser energies, respectively. The sameobservation applies when we relatively compare the intensities of the T1e, S1, S2, and S3 for these two excitations. However, the important point is that the ratio of intensities shouldbe affected in exactly the same way if the same resonancemechanism was responsible for both the 兵T1b , T2其 group of
excitations on one hand and for the group of modes
兵T1e , S1 , S2 , S3其 on the other hand. The conclusion following

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GOZAR et al.

from the above discussion is that the data in Fig. 7 proves theexistence of two light coupling mechanisms to the observed
low energy excitations. In fact, further support for our statement comes from preliminary data which show that the 1.92
and 2.6 eV laser excitations energies correspond to regions in
the visible range where distinct features of the reflectivity
spectrum are observed.26 While certainly interesting in thisrespect, a complete determination of the optical parametersof SrCu2共BO3兲2 in the visible range and the identification of
the specific high energy electronic intermediate states involved in the coupling mechanisms goes beyond the scope of
our article and, as explained above, cannot change our conclusions.
The resonance of the B1 symmetric Te magnetic mode,enhanced for ␻L = 2.6 eV, is similar to the one corresponding
to S1, S2, and S3 excitations as well as to the behavior of the
new modes seen around 55, 59, and 68 cm−1. The results ofperturbational analysis regarding energy scales andsymmetries,10 in particular of a B1 symmetric S = 0 states at
45 cm−1, argues for the magnetic nature of the mode S3, Ref.
21. Just on the account of the similar resonance behavior of
the S3 excitation with respect to the modes found above
50 cm−1 one cannot unambiguously identify the latter as
magnetic bound states as well. The lack of energy shift inmagnetic fields is consistent with such an interpretation, but
the same would be true if they had a phononic origin.
We discuss below the nature of the two light couplingmechanisms to magnetic excitations. For the set of modes
discussed in Figs. 6共c兲 and 6共d兲 we propose that the couplingtakes place via the spin-orbit interaction which can be writz
ten in an effective form as Hint
1 ⬀ 共ein ⫻ eout兲Stot 共Ref. 22兲. As
it is found experimentally, in zero field this interaction
Hamiltonian probes indeed excitations with A2 symmetry
共the T1b mode兲 and the calculated magnetic field dependent
intensities for this and other modes in Figs. 6共c兲 and 6共d兲 isalso in agreement with the experimental results. The coupling to the T1e and T1f modes from Fig. 3 can be possibly
understood if we invoke the usual effective spin interaction
corresponding to the photon induced spin exchange process
Hint
2 ⬀ 兺具i,j典共einrij兲共eoutrij兲Si · S j. Here the sum runs over pairsof lattice sites, Si and S j are the exchanged spins on sites iand j, respectively, while rij is the vector connecting thesesites.22 The explicit expression of this interaction for several
polarizations in the 4-spin cluster approximation contains finite coupling in B1 and B2 channels for the triplet T1e andT1f states. This explains the presence of the 25.6 cm−1 magnetic modes in all polarizations except 共RR兲.
The difference in the coupling strengths seen in Fig. 7 isthus understandable because these two light coupling mechaint
nisms given by Hint
1 and H2 need not be simultaneously inresonance with the same high energy excited electronic
states. The photon induced spin exchange Hamiltonian, Hint
2 ,is usually invoked in order to explain Raman active S = 0
two-magnon type excitations in various magnetic systems.22
SrCu2共BO3兲2 is an example where this Hamiltonian, in thepresence of singlet-triplet mixing DM interactions, can beused to account for coupling to S = 1 states. We also note thatin principle the photon induced spin exchange could provide
coupling to the 21.5 cm−1S = 0 bound state below the spin
gap. The reason this mode is not directly observed in our
spectra for any of the two excitation energies used in Fig. 6
is an open question. However, one possible explanation isthat the Raman form factor for exciting pairs of k = 0 magnons is vanishing as opposed to the case zone boundarymodes. An example is the case of the Raman vertex calculated for the 2D square lattice within the spin-wave
approximation27 and using the Fleury-Loudon22 interaction.
Consequently, both the 21.5 and the 28.9 cm−1 excitations
could be attributed to S = 0 bound states but originating fromdifferent parts of the Brillouin zone and having substantiallydifferent binding energies.
VII. SUMMARY

We study by Raman scattering collective magnetic excitations in the spin-dimer compound SrCu2共BO3兲2. Regarding
the one-triplet sector, we showed that by using a 4-spin cluster approximation and by including an additional intradimer
DM interaction we are able to explain the observed zero fieldsymmetry selection rules and the rich behavior in magnetic
fields. We are also able to experimentally demonstrate theexistence of two effective magnetic light scattering Hamiltonians which are responsible for their resonance behavior. The
4-spin approximation fails to account for the excitations seenin the multiparticle magnetic sectors. In particular, the existence of a set of four modes below the onset of the twotriplet continuum 共at 37.5, 40.8, 44.5, and 50.9 cm−1 in theA2 symmetry channel兲 shows that further theoretical analysis
is required in order to understand the nature of these composite excitations. We suggest a possible explanation for the
existence of a subgap collective mode in terms of a strongly
bound singlet state which can be generated within the spaceof 4 nearest neighbor spin dimers.
Experimental data in the energy range below 350 cm−1
shows the existence of several quasidegenerate phonons.
General symmetry arguments suggest that these excitations,which are very close in energy, involve different vibrational
patterns 共in plane and c-axis atomic motions, respectively兲.
The failure of group theory to provide an understanding of
this interesting behavior at a qualitative level calls for a detailed theoretical investigation of the vibrational modes. We
also find a weak A2 symmetric phonon at 155.9 cm−1. Theenergy and symmetry of this excitation suggest that this is a
coupled spin-phonon excitation and can be evidence for the
existence of magnetoelastic interactions in SrCu2共BO3兲2.
ACKNOWLEDGMENTS

We thank T. Rõõm, G. S. Uhrig, T. Siegrist, and R. Sternfor discussions. A.G. also gratefully acknowledges the collaboration with Sasa Dordevic.

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SYMMETRY AND LIGHT COUPLING TO PHONONIC AND …

*Permanent address: Brookhaven National Laboratory, Upton, NY
11973; Email address: agozar@bnl.gov†
Email address: girsh@bell-labs.com
1 For a recent theoretical review, see S. Miyahara and K. Ueda, J.
Phys.: Condens. Matter 15, R327 共2003兲.
2 H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto, and Y. Ueda,
Phys. Rev. Lett. 82, 3168 共1999兲.
3
K. Onizuka, H. Kageyama, Y. Narumi, K. Kindo, Y. Ueda, and T.
Goto, J. Phys. Soc. Jpn. 69, 1016 共2000兲.
4 K. Kodama, M. Takigawa, M. Horvati, C. Berthier, H. Kageyama,
Y. Ueda, S. Miyahara, F. Becca, and F. Mila, Science 298, 395
共2002兲.
5
T. M. Rice, Science 298, 760 共2002兲.
6
B. S. Shastry and B. Kumar, Prog. Theor. Phys. Suppl. 145, 1
共2002兲.
7 P. W. Anderson, Science 235, 1196 共1987兲.
8 R. W. Smith and D. A. Keszler, J. Solid State Chem. 93, 430
共1991兲.
9
K. Sparta, G. J. Redhammer, P. Roussel, G. Heger, G. Roth, P.
Lemmens, A. Ionescu, M. Grove, G. Güntherodt, F. Hüning, H.
Lueken, H. Kageyama, K. Onizuka, and Y. Ueda, Eur. Phys. J. B
19, 507 共2001兲.
10 C. Knetter, A. Bühler, E. Müller-Hartmann, and G. S. Uhrig,
Phys. Rev. Lett. 85, 3958 共2000兲.
11 C. Knetter and G. S. Uhrig, Phys. Rev. Lett. 92, 027204 共2004兲.
12
S. Miyahara and K. Ueda, J. Phys. Soc. Jpn. 共Suppl.兲, 69, 72
共2000兲.
13
S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 3701 共1999兲.
14 K.-Y. Choi, Yu. G. Pashkevich, K. V. Lamonova, H. Kageyama,

PHYSICAL REVIEW B 72, 064405 共2005兲
Y. Ueda, and P. Lemmens, Phys. Rev. B 68, 104418 共2003兲.
T. Rõõm, D. Hüvonen, U. Nagel, J. Hwang, T. Timusk, and H.
Kageyama, Phys. Rev. B 70, 144417 共2004兲.
16 H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama, K.
Nukui, K. Kodama, K. Kakurai, and Y. Ueda, Phys. Rev. Lett.
84, 5876 共2000兲.
17
O. Cépas, K. Kakurai, L. P. Regnault, T. Ziman, J. P. Boucher, N.
Aso, M. Nishi, H. Kageyama, and Y. Ueda, Phys. Rev. Lett. 87,
167205 共2001兲.
18 H. Nojiri, H. Kageyama, K. Onizuka, Y. Ueda, and M. Motokawa, J. Phys. Soc. Jpn. 68, 2906 共1999兲.
19 H. Nojiri, H. Kageyama, Y. Ueda, and M. Motokawa, J. Phys.
Soc. Jpn. 72, 3243 共2003兲.
20
T. Rõõm, U. Nagel, E. Lippmaa, H. Kageyama, K. Onizuka, andY. Ueda, Phys. Rev. B 61, 14342 共2000兲.
21 P. Lemmens, M. Grove, M. Fischer, G. Güntherodt, V. N. Kotov,
H. Kageyama, K. Onizuka, and Y. Ueda, Phys. Rev. Lett. 85,2605 共2000兲.
22 P. A. Fleury and R. Loudon, Phys. Rev. 166, 514 共1968兲.
23 H. Kageyama, K. Onizuka, T. Yamauchi, and Y. Ueda, J. Cryst.
Growth 206, 65 共1999兲.
24
See, for instance, G. F. Koster, Properties of the Thirty-Two Point
Groups 共M.I.T press, 1963兲.
25 A sign change will interchange the position of these modes
around the gap at energy ⌬.
26 S. V. Dordevic 共private communication兲.
27
A. W. Sandvik, S. Capponi, D. Poilblanc, and E. Dagotto, Phys.
Rev. B 57, 8478 共1998兲.
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