PHYSICAL REVIEW B 68, 104418 共2003兲

Strong anharmonicity and spin-phonon coupling in the quasi-two-dimensional quantum
spin system Sr1Àx Bax Cu2 „BO3 … 2
K.-Y. Choi,1 Yu. G. Pashkevich,2 K. V. Lamonova,2 H. Kageyama,3 Y. Ueda,3 and P. Lemmens1,41

2. Physikalisches Institut, RWTH Aachen, D-52056 Aachen, Germany
2
A.A. Galkin Donetsk Phystech NASU, 83114 Donetsk, Ukraine
3
Institute of Solid State Physics, University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan
4
Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany
共Received 7 April 2003; published 17 September 2003兲
We report Raman-scattering measurements on the quasi-two-dimensional quantum spin system
Sr1⫺x Bax Cu2 (BO3 ) 2 (x⫽0 and 0.1兲 in interlayer polarizations and temperatures between 5 K and 450 K. Upon
approaching the structural phase transition at T s ⫽395 K, a strong anharmonic softening of an interlayer mode
at 62 cm⫺1 is observed together with overtone features and a broadening of its linewidth. Moreover, an
additional 3% softening toward low temperatures points to strong spin-phonon coupling of exchange origin.
Our results suggest a significance of spin-phonon interactions, interlayer lattice dynamics, and DzyaloshinskiiMoriya interactions to understand the magnetic behavior in this compound that has earlier been discussed in
the framework of the two-dimensional Shastry-Sutherland spin system.
DOI: 10.1103/PhysRevB.68.104418

PACS number共s兲: 75.30.Gw, 78.30.⫺j, 63.20.Ls, 63.20.Ry

I. INTRODUCTION

The quasi-two-dimensional quantum antiferromagnet
SrCu2 (BO3 ) 2 has received considerable attention due to its
relevance for the two-dimensional 共2D兲 Shastry-Sutherlandmodel1,2 which is given by the Hamiltonian H⫽J 兺 NNS i S j⫹J ⬘ 兺 NNNS i S j where NN 共NNN兲 stands for nearest- 共nextnearest兲 neighbor spins. The ground state of the ShastrySutherland model is a direct product of dimer singlet states
when the ratio x⫽J ⬘ /J of the next-nearest interdimer coupling J ⬘ to the nearest-neighbor intradimer coupling J issmaller than a critical value near 0.7.3 Furthermore, the frustration between J and J ⬘ prevents a hopping of a triplet statefrom one dimer to another, leading to a flat triplet branch.4
Various experiments have confirmed that the orthogonallycoupled spin dimers in the crystallographic ab layer form adimer singlet ground state with a spin gap of ⌬⫽34 K anddispersionless elementary triplets.5,6 Multiparticle boundstates in both the singlet and triplet sectors as well as plateaus at 1/8, 1/4, and 1/3 of the saturation magnetization
were observed due to strong interactions between the welllocalized triplet excitations.5–7 In addition, SrCu2 (BO3 ) 2 (x⬇0.63–0.68) is proposed to be close to a quantum criticalpoint (x c ⫽0.691) which separates a short-range dimer phasefrom a long-range ordered Néel phase.4,8 The existence ofintermediate phases such as helical order, plaquette-singlet
order, or weak columnar-dimer order has been discussed forlarger x.9 Experiments using magnetic susceptibility underhydrostatic pressure have found evidence for a decrease of
the spin gap under pressure, i.e., the dimer ground state canat least be weakened experimentally.10
Despite the success of the 2D Shastry-Sutherland modelin understanding the ground state and spin gap of
SrCu2 (BO3 ) 2 , a lot of severe deviations show up.11 At finite
temperatures the significance of interlayer interactions is evident from the temperature dependence of the magnetic
susceptibility.12,13 Interlayer coupling destabilizes the0163-1829/2003/68共10兲/104418共9兲/$20.00

frustration-induced singlet dimer phase while enhancing
long-range antiferromagnetic correlations.14 A recent crystal
structure study has revealed a displacive, second-order structural phase transition at T s ⫽395 K. 15 The main structural
distortion was observed as a continuous corrugation of the
Cu-BO3 layer in a wide temperature range below T s . This
corrugation induces a change of the interlayer interaction
most clearly seen as a steplike jump in the magnetic susceptibility at T⫽T s . 16 Furthermore, the importance of spinphonon interactions is indicated by a substantial softening of
the sound velocity both as a function of temperature andmagnetic field.17 This implies a sensitivity of the dimer system to specific ionic displacements. The observation of amagnetic superstructure by high-field nuclear-magneticresonance measurements at the 1/8-magnetization plateau18
further supports this evidence. Finally, neutron-scattering experiments and electron spin resonance 共ESR兲 show multiparticle triplet bound states with an anomalous k dependence ofthe structure factor19,20 and zero-field splitting21 that is not inaccordance with a pure 2D Shastry-Sutherland model.22 The
splitting of the triplets may be understood by taking into
account a Dzyaloshinskii-Moriya 共DM兲 interaction which
partially lifts the magnetic frustration and suppresses the spin
gap.20
The above-mentioned results suggest that in order to develop a closer understanding of SrCu2 (BO3 ) 2 the minimalmodel should be extended to a quasi-2D J-J ⬘ model including spin-phonon coupling, lattice distortions, and DM interaction. To our present knowledge these additional terms
should lead to the observed close interplay of structure andmagnetism in this system. For further understanding especially Raman spectroscopy can provide valuable information,
as the involved degrees of freedom can be investigated simultaneously and with an unprecedented sensitivity andresolution. In previous Raman experiments a softening of alow-frequency optic mode at 62 cm⫺1 has been observed ina wide temperature interval.15 Unfortunately, these experi-

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K.-Y. CHOI et al.

ments were restricted to ab-plane polarizations of the incident and scattered electric-field vectors, neglecting important
aspects of interlayer lattice dynamics. In this paper, we reportto the best of our knowledge the first Raman-scattering dataof Sr1⫺x Bax Cu2 (BO3 ) 2 (x⫽0 and 0.1兲 in interlayer polarizations. The previously observed mode shows up as an interlayer mode with a spectral weight that is two orders of magnitude larger than that of the in-plane modes. The symmetrygroup analysis and doping dependence identify this softmode as an in-phase motion of almost all ions in the primitive cell along the z direction. Pronounced overtone features
and an additional softening by 3% below a temperature scale
given by the spin gap point to strong anharmonicities andspin-phonon coupling in SrCu2 (BO3 ) 2 . Furthermore, we
substantiate the presence of the DM interaction using ananalysis of the exact crystal symmetry.
II. EXPERIMENTAL DETAILSHigh quality single crystals of Sr1⫺x Bax Cu2 (BO3 ) 2 (x⫽0 and 0.1兲 were grown using a vertical traveling solvent
floating-zone method from a LiBO3 flux.23 Samples withtypical dimension 5⫻5⫻3 mm3 were used for Ramanscattering measurements. The measurements were done in acontinuous helium flow optical cryostat by varying temperature from 5 K to a room temperature. The high-temperature
measurements from 300 K to 450 K were carried out using aheating vacuum stage. This vacuum stage is filled with argon
gas in order to avoid a possible oxygen loss as well as acontamination of the Raman signal by rotational modes ofN2 and O2 . Note that the studied temperature interval reaching up to 177 °C can be achieved without pumping out the
sample space. Furthermore, the studied compound with amelting temperature of ⬃900 °C is stable in this temperature
range. The experiments were completely reversible. All spectra were taken in a quasibackscattering geometry with theexcitation line ␭⫽514.5 nm of an Ar⫹ laser and analyzed by
a DILOR-XY spectrometer and a nitrogen cooled chargecoupled device detector. The laser power of 5 mW was focused to a 0.1-mm-diameter spot on the interlayer surface.
The averaged laser power density amounts to ⬃70 W/cm2 .
Since the studied sample is semitransparent, the local overheating due to the incident light is expected to be less than
0.5 K. Thus, heating effects are not taken into account in theanalysis of our results.
III. RESULTS AND DISCUSSIONS
A. Symmetry analysis of phonon modes

Before discussing Raman spectra we will present a grouptheoretical analysis as a first step to understand the nature of
the soft mode and its possible effect on the magnetic properties of SrCu2 (BO3 ) 2 . Our analysis is based on the result ofa recent x-ray study15 which provides detailed information
about the temperature dependence of atomic parameters andcrystal symmetry.
In the high-temperature 共HT兲 phase SrCu2 (BO3 ) 2 hasspace group I4/mcm where Sr atoms have D 4 - (2a-兲 site
symmetry, Cu, B, and O1 atoms have C ⬘2 v - (4h-兲 site sym-

FIG. 1. 共Color online兲 共a兲 Primitive unit cell and site symmetryof the atoms in the high-temperature phase 共space group I4/mcm).
共b兲 Primitive unit cell and site symmetry of the atoms in the lowtemperature phase 共space group I4̄2m) of SrCu2 (BO3 ) 2 .

metry, and O2 atoms possess C sh - (8k-兲 site symmetry 关seeFig. 1共a兲兴. A factor group analysis gives the experimentallyobserved infrared and Raman-active modes as follows:⌫ HT ⫽5A 1g ⫹A 1u ⫹6A 2g ⫹5A 2u ⫹5B 1g ⫹4B 1u⫹5B 2g ⫹B 2u ⫹6E g ⫹11E u .

共1兲

According to this representation one expects 27 Ramanactive modes. The space group of the low-temperature 共LT兲phase is a maximal nonisomorphic subgroup of the spacegroup of high-temperature phase. This group-subgroup relation is one of the prerequisites for the application of the
Landau theory for second-order phase transitions. The LT
phase has space group I4̄2m where Sr atoms occupy D 2 (2c-兲 site symmetry, Cu, B, and O1 atoms have C s - (4i-兲 site
symmetry, and O2 atoms belong to C 1 - (8 j-兲 site symmetry
关see Fig. 1共b兲兴. The factor group analysis predicts⌫ LT ⫽9A 1 ⫹7A 2 ⫹6B 1 ⫹10B 2 ⫹17E.

共2兲

Subtracting the acoustic modes (B 2 ⫹E) one obtains 56
Raman-active modes. With decreasing temperature throughthe structural phase transition at T s ⫽395 K the expected
evolution of phonon modes follows a subgroup connectionof the space groups between HT and LT phases. In particular,
in a second-order phase transition the new structure is a subgroup of the old structure. Thus, there will always be a zonecenter A 1 component of the soft mode below T s regardless ofthe symmetry of the soft mode in the higher-symmetry
phase. Furthermore, the HT phase loses m z and m x mirror
planes, which are generators of the I4/mcm space group. The
translation symmetry remains the same. The 9 A 1 modes of
the LT phase are correlated with (5 A 1g ⫹4 B 1u ) modes ofthe HT phase. Thus, one of the four B 1u buckling modes atthe ⌫ point of the Brillouin zone is a soft mode in the HT

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STRONG ANHARMONICITY AND SPIN-PHONON . . .

phase approaching T s . Normal coordinates of these modes
BB
are given by Q h 1u ⫽ 21 (U 1z ⫺U 2z ⫹U 3z ⫺U 4z ) and Q k 1u⫽(1/2冑2)(U 1z ⫹U 2z ⫺U 3z ⫺U 4z ⫹U 5z ⫹U 6z ⫺U 7z ⫺U 8z )for the h and k sites, respectively. It should be noted thatthese modes are silent and include only the displacements of
the h- and k-site ions along the z direction. Such distortions
violate the m z , m x , and m y symmetry planes and preserve
the D 2d symmetry. According to the x-ray analysis in the LT
B
B
B
phase, some in-phase superposition of Q Cu1u , Q B 1u , Q O11u ,B
and Q O21u normal coordinates becomes frozen-in for temperatures below T s 关see Fig. 9共b兲 in Ref. 15兴. Precisely, this superposition represents an order parameter of the structuralphase transition. Note that Sr ions with a-site symmetry do
not participate in this order parameter.
In the LT phase four B 1u silent modes of the HT phase
II A
III A
become Raman-active A 1 modes. The Q i 1 and Q j 1 norB
mal coordinates of the A 1 mode correspond to Q h 1u andB
Q k 1u , respectively 共see the Appendix兲. As it was previously
mentioned this soft mode involves an in-phase motion ofalmost all ions in the primitive cell 共except Sr ions兲 along thez direction. The soft mode consists of a motion of atoms withheavy mass leading to a large effective mass. Thus, the energy of the soft mode is expected to be small even at low
temperatures. In addition, the A 1 normal coordinates of
IA
IA
II A
Q i 1 , Q j 1 , and Q j 1 contain xy components of a motionof the ions in the LT phase. Thus, we can expect a mixing of
the xy and z components of the normal coordinates. However, the x-ray analysis15 did not show substantial shifts ofCu, B, and O ions along x and y directions in the LT phase,
if compared to the HT phase. This implies that the A 1 modes
hold the properties of the B 1u modes of the HT phase which
consist of motions predominantly along the z direction.
Therefore, we expect a strong anisotropic Raman scatteringof the A 1 modes; that is, the dominant scattering intensity
should be seen in zz polarization.
At last, we mention that the 7 A 2 modes of the LT phaseare correlated with the (6 A 2g ⫹B 2u ) modes of the HT phase,the 6 B 1 modes with the (5 B 1g ⫹A 1u ) modes, the 10 B 2
modes with the (5 B 2g ⫹5 A 2u ) modes, and the 17 E modes
with the (6 E g ⫹11 E u ) modes, respectively. Therefore, one
expects transition-induced scattering intensity of the (4 A 1⫹B 1 ⫹4 B 2 ⫹10 E) modes in the LT phase among which(4 B 2 ⫹10 E) are infrared active in the HT phase. The actual
observability of these modes in interlayer polarizations depends on the evolution of the Raman tensor for each modethrough the structural phase transition.
B. Interlayer soft mode

Figure 2 displays Raman spectra of SrCu2 (BO3 ) 2 in (cc)and (ca) polarizations at 293 K and 410 K which lie below
and above T s ⫽395 K, respectively. The corresponding dataof Sr0.9Ba0.1Cu2 (BO3 ) 2 共not shown here兲 show only tiny frequency shifts of some phonon modes with respect to theundoped sample. Note here that the c axis is perpendicular tothe layer. Thus, these data provide mainly information about
interlayer lattice dynamics. For Raman spectra in intralayer

FIG. 2. Raman spectra of SrCu2 (BO3 ) 2 for two interlayer polarizations at 293 K and 410 K, respectively. The arrows indicate
additional modes which appear for temperatures T⬍T s ⫽395 K.
Note that the intensity of these modes in b(ca)-b polarization ispartly very weak.

scattering geometries we refer to Refs. 7, 11, and 15. At
room temperature two additional modes at 33 and 117 cm⫺1
appear in the (cc) polarization while five additional modes
共see the arrows in Fig. 2兲 can be seen in the (ca) polarization. Noticeably enough, four modes at 160, 221, 333, and696 cm⫺1 in the (ca) polarization turn up as a shoulder ofhigh-temperature phonon modes. The appearance of additional modes in the (ca) polarization can be ascribed to amixing of weak ab-plane motions with a strong modulationof the susceptibility tensor by a displacement along the cdirection. A strong displacement along the c direction willcause infrared-active modes of the HT phase consisting of anab-plane motion to become Raman active in the LT phase.
As listed in the Appendix, the normal coordinates of the
doubly degenerated E u mode include an xy-plane motion of
the h-site ions while that of the E g mode contains a motionof the h-site ions along the z direction. Therefore, the observed modes in the (ca) polarization can be ascribed to an
enhancement of Raman tensors of the E u modes in the LT
phase. Interestingly enough, the observed E u mode in the LT
phase is energetically very close to the frequency of the E gmode in the HT phase. The shoulder structure of the additional modes with weak intensity indicates that the symmetry
breaking is rather weak and that the crystal symmetries of the
HT and LT phases are closely related. In contrast, the 33- and117-cm⫺1 modes in the (cc) configuration and the 36-cm⫺1mode in the (ca) polarization are no longer observable in the
HT phase. Thus, these modes are attributed to a soft modethat results from the structural phase transition. This is due tothe fact that in a second-order structural phase transition an
optic soft mode is always Raman active in the lowersymmetry phase but it is not always Raman active in thehigher-symmetry phase. The 33- and 36-cm⫺1 modes in two
different polarizations have close peak energies, indicating
the same origin. The intensity of the 36-cm⫺1 mode in the(ca) polarization amounts to 10% of that of the 33-cm⫺1mode in the (cc) polarization. The corresponding mode observed in (aa) polarization has negligible intensity com-

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K.-Y. CHOI et al.

FIG. 3. 共Color online兲 Softening of a phonon mode in (cc)polarization as a function of temperature. Note that an overtone of
the 62-cm⫺1 mode is present at 124 cm⫺1 .

pared to the mode with interlayer polarization.15 This indicates an extreme anisotropy of the Raman tensor of the softmode and is consistent with the symmetry analysis showingthat the soft mode involves an in-phase motion of almost allions in a primitive cell preferably along the c axis 共interlayer
direction兲. An x-ray scattering structure analysis of
SrCu2 (BO3 ) 2 shows for T⬎T c an inversion center and completely flat CuBO3 planes. For temperatures below T⬍T c
atomic displacements are directed perpendicular to the abplane leading to a buckling of the planes.15 The in-phasemotion also explains why the observed energy of the softmode in our Raman experiments is so small. In addition, we
observe a slight increase in integrated intensity of almost allphonon modes upon cooling. This can be ascribed to achange of both dipole matrix elements and band energies
with the structural transition. The origin of the 117-cm⫺1mode will be discussed further below.
The evolution of the low-frequency modes in (cc) polarization is displayed in Fig. 3. Upon approaching the phasetransition the 60-cm⫺1 mode at T⫽5 K increases slightly up
to 62 cm⫺1 at T⫽15 K and finally disappears into the tail of
quasielastic scattering around T s while softening roughly by
44 cm⫺1 . At the same time, its linewidth broadens strongly.
Interestingly enough, an overtone of the 62-cm⫺1 mode isobserved at 124 cm⫺1 . This implies a strong anharmonicityof the soft mode. The second-order mode undergoes a small
softening by 5 cm⫺1 , suggesting that the original soft mode
has a nearly flat dispersion with a large density of states. Thiscan be explained as follows. The frequency of the soft modearound the Brillouin center will rapidly fall to zero as thetransition is approached. In contrast, phonons at the vicinityof the zone boundary will remain robust. For first-order scattering Raman spectroscopy is sensitive to the softening ofphonons near the zone center. However, the whole Brillouinzone contributes to higher-order Raman scattering. The spectral weight is proportional to the density of states if there are
no interactions. Therefore, the resulting scattering intensity
will be dominated by zone-boundary phonons with large
spectral weights. Here we draw attention to similar phonon
anomalies reported in the alternating chain system(VO) 2 P2 O7 . A strong anharmonic phonon at 123 cm⫺1 soft-

FIG. 4. 共a兲 Temperature dependence of the soft mode. The inset
shows a softening of the soft mode for temperatures T⬍15 K. 共b兲
Temperature dependence of the damping constant which is definedas the full width at half maximum. The solid lines give a fit using
Eq. 共3兲. The full square 共open triangle兲 denotes the undoped 共10%
Ba doped兲 sample.

ens in this compound by 10%.24 Most probably, this mode
modulates the exchange along the alternating chain and its
energy coincides with the zone-boundary energy of thesinglet-triplet excitation gap E ZB ⫽125 cm⫺1 of the spin
system.25 Strong spin-phonon coupling as well as the existence of a high-symmetry phase under high pressure has
been attributed to the origin for the anharmonicity of opticalphonons.26,27 In this regard, the overtone feature of the softmode in SrCu2 (BO3 ) 2 should be ascribed to the existence ofa high-symmetry phase at elevated temperatures. This relies
on an enhancement of anharmonic phonon-phonon interactions due to the strain induced by the buckling distortions of
the LT phase. Furthermore, in the LT phase, exchange paths
are strongly susceptible to ionic displacements since they
will not be mutually compensated due to the lower crystal
symmetry. As will be discussed in Sec. III C, special spin
topology together with buckling distortions induces strong
spin-phonon coupling. This suggests that the simultaneous
occurrence of anharmonic phonon interactions and strong
spin-phonon coupling in a low-symmetry phase may be a
generic feature of low-dimensional spin gap systems.
The temperature dependence of the frequency shift andthe damping constant of the soft mode is plotted in Fig. 4.
Within the experimental resolution we did not detect a difference between the undoped and the 10% Ba-doped
samples. This observation strongly confirms that the softmode does not include motions of the Sr ions, consistent
with the symmetry analysis. Both the peak position and thelinewidth change strongly over a large temperature range between 5 K and 370 K. Such a temperature dependence can be
ascribed to anharmonic terms in the vibrational potential energy. Taking into account cubic and quartic anharmonic contributions, the temperature dependence of the peak positionis given by28冉

␻ 共 T 兲 ⫽ ␻ 共 0 兲 ⫹A 1⫹

104418-42e x ⫺1

冊 冉

⫹B 1⫹

3
e y ⫺1⫹

3
共 e y ⫺1 兲 2

冊

,

共3兲

PHYSICAL REVIEW B 68, 104418 共2003兲

STRONG ANHARMONICITY AND SPIN-PHONON . . .

FIG. 5. 共a兲 Low-energy Raman spectra for T⬎T s after correcting with the Bose-Einstein thermal factor. The solid line is a fit witha Lorentzian line. 共b兲 The log-log plot of the normalized intensity as
a function of 兩 T⫺T s 兩 .

where x⫽ប ␻ (0)/2k B T and y⫽ប ␻ (0)/3k B T with ␻ (0), A,
and B constants. The first term represents the frequency ofsoft modes at zero temperature. The second and third terms
originate from the decay of an optic phonon into three- andfour-phonons due to anharmonicity, respectively. In the hightemperature limit, the second and third terms vary as T and
T 2 , respectively. The fit of the experimental data to Eq. 共3兲
gives ␻ (0)⫽62 cm⫺1 , A⫽⫺0.48 cm⫺1 , and B⫽⫺0.07 cm⫺1 . The smallness of the ratio B/A⫽0.15, related
to a small contribution of the four-phonon process to thesoftening, guarantees the validity of the fitting. Despite areasonable agreement between the fitted curve and the experimental data in the overall temperature range one can see
a deviation for temperatures above 350 K as well as at very
low temperature. The temperature dependence of the linewidth can also be described using a similar relation as Eq.
共3兲. The fitting values of the linewidth at zero temperature⌫(0) and the coefficients A and B are found to be 2.7, 0.24,
and 0.02 cm⫺1 , respectively. This leads to a ratio B/A⫽0.08, also suggesting a small contribution of the fourphonon process. The origin of the discrepancy in the ratioswith respect to the behavior of frequency and linewidth isnot clear. It is certainly related to the strong anharmonicity
and/or spin-phonon interaction beyond this modeling. As
shown in Fig. 4共b兲, significant deviations from the semiphenomenological model can be seen for temperatures above
310 K. The pronounced discrepancy in the high-temperature
limit arises from a stronger broadening of the linewidth thana T 2 dependence. This points to the necessity of higher-orderterms than the four-phonon processes in the model. The observation of pronounced anharmonicity 共two-phonon scattering兲 already demands a more elaborate modeling.
Figure 5 displays low-energy Raman spectra for temperatures above T s . Within the accessible low-frequency resolution we have not observed a sharp central peak around T s .
Instead, a relatively broad elastic-scattering maximum is observed which is fitted well by a Lorentzian profile. The intensity is reasonably described by 兩 T⫺T s 兩 p with the powerp⫽0.47 and 0.56 for the undoped and the 10% Ba-doped
samples, respectively 关see Fig. 5共b兲 plotted on a logarithmic
scale of 兩 T⫺T s 兩 ]. As a possible origin of the central peak the
decay of a soft mode into acoustic modes or phonon density

FIG. 6. 共Color online兲 共a兲 Displacements of copper ions withQ II A 1 symmetry on the background of the already existing buckling
distortions. 共b兲 Displacements of copper ions with Q I A 1 symmetry.

fluctuations should be discussed. We notice that the observed
behavior is in reasonable agreement to 兩 T⫺T s 兩 1/2 which is
expected within the Landau theory for structural phase transitions which are not proper ferroelastic ones.29 In this case aquadratic coupling of the order parameter ␩ to a permeability
tensor leads to only an ␩ 4 contribution to the orderparameter fluctuations in the intensity of the central peak.
Thus, this suggests that the observed central peak is related
to a fluctuation of the order parameter.
C. Spin-phonon coupling and effectof buckling distortions in SrCu2 „BO3 … 2

In dimerized spin system spin-phonon interactions play animportant role in determining the magnitude of spin gap andthe strength of dimerization.11 Optic- and/or acoustic-phonon
anomalies can accompany the onset of a dimerized state asreported in CuGeO3 共Ref. 30兲 and SrCu2 (BO3 ) 2 . 31 In
SrCu2 (BO3 ) 2 the in-plane elastic constant c 66 shows a pronounced softening of 4.5% for temperatures below 25 K
共Refs. 17 and 31兲, which is attributed to an exchange striction coupling. Furthermore, under a magnetic field appliedalong the 关100兴 direction an enormous softening of the inplane c 11 and c 66 elastic constants takes place in the vicinityof the magnetization plateaus where excited triplets become
highly mobile.17 This highlights a close relation between
spin-phonon coupling and DM interaction which promotes
hopping of triplets.20,32 One can also expect optic-phonon
anomalies around the magnetization plateaus.32 This is because the alteration of intradimer bond lengths for each site
results in the alternating magnetization.
The main aspect of the structural phase transition in
SrCu2 (BO3 ) 2 is buckling distortions due to strong anharmonicity. These lead to alternating distances between copper
ions along the z direction. As Fig. 6共a兲 displays, in the LT
phase there are two different interlayer coupling constants J 2⬙of longer bonding length and J ⬙1 of shorter bonding length.
At 100 K the relative distance between nearest-neighboring
Cu ions along the upward and downward directions of the caxis amounts to ␦ z ⫽1.02 Å, that is, 15% of the lattice
constant15 c 关as sketched in Fig. 6共a兲兴. Thus, we can safely
neglect J 2⬙ . This buckling distortion, changing the bonding
angle and distance via spin-lattice interaction, can modify
anisotropic magnetic properties as a function of temperature.
With the special structural topology in mind we will turnto the anomalous decrease of the optic-phonon frequency

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K.-Y. CHOI et al.

observed in the low-temperature limit (T⬍⌬⫽34 K). As
displayed in the inset of Fig. 4共a兲, the peak energy of the softmode reaches a maximum around 15 K and then softens by
3% with further lowering temperature. The energy of the softmode 62 cm⫺1 ⫽89 K is comparable to the nearest-neighbor
intradimer coupling constant J⫽85 K. 16 Furthermore, thesoftening is observed for temperatures below the spin gap of34 K. Thus, the observed softening should be attributed to arenormalization of phonon energy due to strong spin-phonon
coupling. Further evidence for strong spin-phonon coupling
can be provided by strong quasielastic light scattering induced by fluctuations of the energy density of the spin
system.7 As already shown in the symmetry analysis, the softmode corresponds to motions of almost all ions along theinterlayer direction. Consequently, the soft mode mediates all
possible types of exchange interactions including the interlayer and the intralayer interactions. Since the soft mode has
A 1 symmetry, it suffices to consider spin-phonon coupling in
the Q II A 1 - and Q I A 1 -symmetry displacements of Cu ions.
Following Miyahara et al.32 one can derive spin-phonon coupling of the soft mode in an exchange approximation.
First, we focus on the case of the Q II A 1 normal mode
which includes motions of Cu, B, and O1 ions along the zaxis 关see Fig. 6共a兲兴. Spin-phonon interaction of the Q II A 1
symmetry reads as follows:⫺Q

II A 1

冉

␦z

冊

l⬜z

␤ 2 J ⬘ 兺 Si •S j ⫹ ␥ 2 J 1⬙ 兺 Si •S j ,l 储 NNNl⬜ NNNz

共4兲

where NNN (NNNz) stands for next-nearest neighboring Cu
ions in the ab plane 共along the z direction兲. l⬜ (l储 ) is thevector connecting them in the ab plane 共along the z direction兲. ␤ and ␥ denote the exponents of the distance dependence of J ⬘ (r) and J ⬙1 (r), respectively. It is worth mentioning that the soft mode of the Q II A 1 symmetry does not
mediate the intradimer interactions. Moreover, in the HTphase the Q II A 1 type of spin-lattice interaction completely
disappears because of the absence of buckling distortions( ␦ z ⫽0) as well as the cancellation of the interlayer contribution by each other (J 1⬙ ⫽J ⬙2 ). Assuming that ␤ ⯝ ␥ one seesthat the contribution of the interlayer and the intralayer interactions to spin-phonon coupling in the LT phase is comparable to each other with l⬜ ⫽3.48 Å, l⬜z ⫽2.81 Å and l 储⫽5.14 Å, J ⬘ ⫽0.635J, and J ⬙ ⫽0.09J. 15,2
Another contribution to spin-phonon coupling comesfrom the Q I A 1 -symmetry uniform displacements of Cu-ionswhich consist of motions of Cu, B, and O1 ions in the xy
axis 关see Fig. 6共b兲兴:⫺Q I A 1冉␣J
Si •S j ⫺ ␤
l 兺
NN

冑

l2 l
l 2储 ⫺ ⫺
4 2
l 2储

J⬘

冊

兺 Si •Sj ,

NNN

共5兲

where NN stands for nearest-neighboring Cu ions. l is thevector connecting the intradimer Cu ions. ␣ denotes the exponent of the distance dependence of J(r). Here note that
this mode participates in the soft-mode motion by an admix-

ture of the Q II A 1 normal mode due to strong anharmonicity.
Furthermore, the Q I A 1 -type displacements can modulate theintradimer and the intralayer exchange interactions. This implies that they can shift the quantum critical point x⫽J ⬘ /J
while increasing J ⬘ and decreasing J and vice versa. Becauseof the orthogonal and frustrated exchange topology and the
predominant motions of the soft mode along the z direction,
however, the spin gap and the quantum critical point will
undergo only a small change. Rather, the interlayer interactions will be strongly modulated. However, the absolutestrength of the interlayer interactions is small. All of these
explain why the observed softening of optic-phonon modes
is weak in spite of the possible contributions of the intradimer and the intralayer interactions to spin-phonon
couplings.
Finally, we will discuss another aspect of the structuralphase transition related to the magnetic behavior and therelevance of antisymmetric interaction. Pronounced buckling
distortions for temperatures below T⬍T c can lead to a modification of anisotropic magnetic properties via spin-latticeinteractions. The experimental observation of forbidden ESR
transitions,21 a splitting of the first triplet excitation into threebranches,19 and the unusual field dependence of magnetization plateaus33 are not in accordance with a ShastrySutherland model based on isotropic superexchange. To explain the anomalous behaviors Cépas et al.20 derived the DM
interactions from a simplified 2D spin model 共corresponding
to the HT phase兲, which lift partially the magnetic frustration
as well as the degeneracy of the localized triplets.
However, we highlight that the presence of the bucking
distortions in the LT phase leads to a complicated form of the
DM interactions. In contrast to the assumption of Cépas
et al.,20 the LT phase loses m z , m y , and m x mirror planes
共see Sec. III A兲. Thus, for the purpose of a more complete
description of anisotropic interactions we have performed asymmetry analysis using the exact symmetry of the LT
phase. The exchange tensor of the intradimer interaction between copper ions 1 and 3 yields

ij
⫽
J 13

冉

冊

xx
J 13

z
d 13

y
D 13

z
d 13

yy
J 13

x
D 13
,

y
⫺D 13

x
⫺D 13

zz
J 13

共6兲

where d i j and D i j denote the component of symmetric andantisymmetric anisotropic 共Dzyaloshinskii-Moriya兲 interactions, respectively. Here note that in the HT phase only thez
is present. An upper
symmetric anisotropic interaction of d 13z34
conventional estimation is given by d 13⫽(⌬g/g) 2 J 13
⫽1.3 K with the deviation of the g factor from 2⌬g⫽0.28
共Ref. 21兲 and J 13⫽J⫽85 K. 2 The appearance of the x and ycomponents of DM interactions should be ascribed to a corrugation of the ab plane which includes Cu(1)-Oi (2)-Cu(3)and Cu(1)-Oi (4)-Cu(3) exchange interaction paths. At 100
K the angle between the ab plane and the plane including
exchange paths amounts to ␸ ⫽3.5°. 15 This allows an estiy,xmation of the strength of the DM interactions:34 D 13
⫽sin ␸(⌬g/g)J13⫽0.63 K. Because of the symmetricity of
y
x
⫽D 13
the orientation of the DM vector lies along theD 13104418-6

PHYSICAL REVIEW B 68, 104418 共2003兲

STRONG ANHARMONICITY AND SPIN-PHONON . . .

intradimer bond. The magnitude of the DM vector has astrong dependence on in-phase motions of Oi (2) and Oi (4)ions along the z direction. In addition, under a continuous
corrugation of the Cu-BO3 layer the interdimer interactions
will also undergo an appreciable modulation. From the symmetry analysis one obtains the exchange tensor of interdimer
interactions between copper ions 1 and 2:

ij
⫽
J 12

冉

xx
J 12

z
D 12

y
D 12

z
⫺D 12

yy
J 12

x
d 12

y
⫺D 12

x
d 12

zz
J 12

冊

.

共7兲

tortions and the special topology of SrCu2 (BO3 ) 2 lead to thecoupling of spin to the soft optical phonon by exchange
modulation. Our results show that in a realistic model for
SrCu2 (BO3 ) 2 spin-phonon interactions, DM interactions,and interlayer lattice dynamics should be taken into accountto understand a variety of physical properties such as thespin gap, the proximity to a quantum critical point, andanomalous behavior of the spectroscopic data. As a result,
SrCu2 (BO3 ) 2 is closer to a quantum critical point than previously predicted.
ACKNOWLEDGMENTSz
It is worth noting that only the antisymmetric component D 12yis present in the HT phase. In contrast, in the LT phase D 12x
and d 12 components appear. An estimation of their magni-

tude was not done due to the presence of complicated exchange paths of the interdimer interactions. The analysis of
the interlayer exchange tensors has not been pursued becauseof their weak strength.
Our symmetry analysis shows the presence of the symmetric anisotropic z component of the intradimer interaction
as well as the antisymmetric anisotropic z component of theinterdimer interaction in the HT phase. The latter is suggested by the analysis of the 2D spin model by Cépas et al.20
To our best knowledge, until now the former has not been
considered in the theoretical work. It can remove partiallythe frustration because its orientation is directed antiparallelto the nearest-neighboring dimers. In the LT phase the buckling distortions induce additionally the antisymmetric anisotropic x and y components of the intradimer interaction as
well as the symmetric anisotropic x component of the interdimer interaction. The estimated value of D x,y ⬃0.6 K at 100
K is not negligible compared to D z ⯝2 K. 20 Therefore, we
can conclude that a modulation of anisotropic interactions
via spin-lattice couplings leads to an enhanced hopping of
localized triplets and a mixing of triplet and singlet states.
Thus, SrCu2 (BO3 ) 2 lies closer to the quantum critical point
共to long-range order兲 than predicted based on pure 2D spinmodels.

We thank C. Baumann for helpful discussions. This work
was supported by DFG/SPP 1073 and NATO Collaborative
Linkage Grant No. PST.CLG.977766.
APPENDIX

Here we list some normal coordinates which are relevant
for the analysis of the structural and magnetic properties of
SrCu2 (BO3 ) 2 .
Low-temperature phase,
4i—positions 共Cu, B, and O1 ions兲,A 1 —irreducible representation,
I A1

Qi

⫽

2 冑2

II A 1Qi

共 ⫺U 1xy ⫹U 2x̄y ⫹U 3xy ⫺U 4x̄y 兲 ,1⫽ 共 U 1z ⫺U 2z ⫹U 3z ⫺U 4z 兲 ,2

A 2 —irreducible representation,A

Q i 2⫽

1
2 冑2

共 ⫺U 1x̄y ⫺U 2xy ⫹U 3x̄y ⫹U 4xy 兲 ,B 1 —irreducible representation,
B

Q i 1⫽
IV. CONCLUSIONS

We have presented a Raman-scattering study ofSr1⫺x Bax Cu2 (BO3 ) 2 (x⫽0 and 0.1兲 in interlayer polarizations. With decreasing temperatures through the structuraltransition, a pronounced interlayer soft mode is observed. A
symmetry analysis shows that this soft mode has an A 1 symmetry corresponding to in-phase motions of almost all ionsalong the interlayer direction. This explains the extreme anisotropy of the Raman tensor of the soft mode as well as its
low energy. The doping dependence further confirms this. In
addition, a displacement of atomic motions along the z direction leads to a small modulation of the intralayer and theinterlayer interactions as well as the appearance of additional
symmetric and antisymmetric anisotropic interactions. For
temperatures below the spin gap such a modulation results inan additional softening of the soft mode by 3%, indicatingstrong spin-phonon coupling. In particular, the buckling dis-

1

1
2 冑2

共 ⫺U 1x̄y ⫹U 2xy ⫹U 3x̄y ⫺U 4xy 兲 ,B 2 —irreducible representation,
IB

Q i 2⫽

1
2 冑2

II B 2Qi

共 ⫺U 1xy ⫺U 2x̄y ⫹U 3xy ⫹U 4x̄y 兲 ,1⫽ 共 U 1z ⫹U 2z ⫹U 3z ⫹U 4z 兲 ,2

E—irreducible representation;

104418-7

1IE⫽ 共 U 1x ⫹U 2x ⫹U 3x ⫹U 4x 兲 ,
Q 1,i
2
1
IE
⫽⫺ 共 U 1y ⫹U 2y ⫹U 3y ⫹U 4y 兲 ,Q 2,i2

PHYSICAL REVIEW B 68, 104418 共2003兲

K.-Y. CHOI et al.

1
II E
Q 1,i
⫽ 共 U 1y ⫺U 2y ⫹U 3y ⫺U 4y 兲 ,
2

1
B
II A
Q h 1u ⫽ 共 U 1z ⫺U 2z ⫹U 3z ⫺U 4z 兲 ⫽Q i 1
2
E g —irreducible representation for 4h positions,

1
II E⫽ 共 ⫺U 1x ⫹U 2x ⫺U 3x ⫹U 4x 兲 ,
Q 2,i
2

1
E
III E
,
Q 1,hg ⫽ 共 U 1z ⫺U 2z ⫺U 3z ⫹U 4z 兲 ⫽Q 1,i
2

1
III E
⫽ 共 U 1z ⫺U 2z ⫺U 3z ⫹U 4z 兲 ,
Q 1,i
2

1
E
III E
,
Q 2,hg ⫽ 共 ⫺U 1z ⫺U 2z ⫹U 3z ⫹U 4z 兲 ⫽Q 2,i
2

1
III E
⫽ 共 ⫺U 1z ⫺U 2z ⫹U 3z ⫹U 4z 兲 ,Q 2,i2

E u —irreducible representation for 4h positions,A 1 —irreducible representation for 8 j positions,

1
IE
IE
,
Q 1,hu ⫽ 共 U 1x ⫹U 2x ⫹U 3x ⫹U 4x 兲 ⫽Q 1,i2

1
IA
Q j 1 ⫽ 共 U 1xy ⫹U 2xy ⫺U 3x̄y ⫺U 4x̄y
4

II A 1Qj

III A 1Qj

⫺U 5xy ⫹U 6xy ⫹U 7x̄y ⫹U 8x̄y 兲 ,

1
IE
IE
,
Q 2,hu ⫽⫺ 共 U 1y ⫹U 2y ⫹U 3y ⫹U 4y 兲 ⫽Q 2,i2

1⫽ 共 ⫺U 1x̄y ⫹U 2x̄y ⫺U 3xy ⫹U 4xy
4

1
II E
IIE
,
Q 1,h u ⫽ 共 U 1y ⫺U 2y ⫹U 3y ⫺U 4y 兲 ⫽Q 1,i2⫹U 5x̄y ⫺U 6x̄y ⫹U 7xy ⫺U 8xy 兲 ,

1
II E
II E
,
Q 2,h u ⫽ 共 ⫺U 1x ⫹U 2x ⫺U 3x ⫹U 4x 兲 ⫽Q 2,i
2

⫽

1
2 冑2

共 U 1z ⫹U 2z ⫺U 3z

B 1u —irreducible representation for 8k positions,⫺U 4z ⫹U 5z ⫹U 6z ⫺U 7z ⫺U 8z 兲 ,B

Q k 1u ⫽High-temperature phase,B 1u —irreducible representation for 4h positions,1
2 冑2

共 U 1z ⫹U 2z ⫺U 3z⫺U 4z ⫹U 5z ⫹U 6z ⫺U 7z ⫺U 8z 兲 .

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