PHYSICAL REVIEW B 69, 174420 共2004兲

X-band ESR determination of Dzyaloshinsky-Moriya interactionin the two-dimensional SrCu2 „BO3 … 2 system
Andrej Zorko* and Denis Arčon
Institute ‘‘Jožef Stefan,’’ Jamova 39, 1000 Ljubljana, Slovenia

Hans van Tol and Louis Claude Brunel
National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310

Hiroshi Kageyama
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan
共Received 4 November 2003; revised manuscript received 11 February 2004; published 20 May 2004兲
X-band ESR measurements on a single crystal of SrCu2 (BO3 ) 2 system in a temperature range between 10
K and 580 K are presented. The temperature and angular dependence of rather broad ESR spectra can be
explained by the inclusion of antisymmetric Dzyaloshinsky-Moriya 共DM兲 interaction. However, the wellaccepted picture of only out-of-plane interdimer DM vectors is not sufficient to explain the observed angulardependence. In order to account for the experimental linewidth anisotropy we additionally include sizable
in-plane components of interdimer as well as intradimer DM interaction. We also emphasize that in addition tostatic buckling of crystal planes a recently introduced dynamical mechanism allowing for instantaneous DM
interaction should be important and estimate the ‘‘average’’ size of intradimer DM vectors D⬜ ⫽3.6⫾0.5 K.
Moreover, the linewidth at an arbitrary temperature can be divided into two contributions; namely, the first part
arising from spin dynamics governed by the spin Hamiltonian of the system and the second part due tosignificant spin-phonon coupling. The nature of the latter mechanism is attributed to phonon-modulation of theantisymmetric interaction, which is responsible for the observed linear increase of the linewidth at hightemperatures.
DOI: 10.1103/PhysRevB.69.174420

PACS number共s兲: 75.30.Gw, 75.50.Mm, 76.30.⫺v

I. INTRODUCTION

Low-dimensional quantum spin systems have been intensively studied over past years, both from the experimental aswell as the theoretical point of view as they provide a reachvariety of physical phenomena. Quantum spin fluctuations aswell as magnetic anisotropy often play a crucial role in determining the ground state and low-lying magnetic excitations of these systems. In this manner a nonmagnetic singlet
ground state with a spin gap to the first magnetically excitedstate is a result of competing exchange interactions or significant frustration present in the system.
Among only a few two-dimensional 共2D兲 spin-gap systems is recently discovered SrCu2 (BO3 ) 2 with a special orthogonal network of Cu2⫹ dimers formed out of localizedS⫽ 12 spins 共see Fig. 1兲.1 Most of the magnetic properties ofthis system are well described by the Hamiltonian of the socalled 2D orthogonal dimer model,2 which takes into accountantiferromagnetic exchange interaction J of each spin with
its nearest neighbor (nn) as well as antiferromagnetic exchange coupling J ⬘ to four next-nearest neighbors (nnn),i.e., H ex ⫽J 兺 (i j) Si •S j ⫹J ⬘ 兺 兵 lm 其 Sl •Sm . Here the sums run
over pairs of spins. Although various sets of exchange parameters have been reported,2– 4 the values4 J⫽85 K and J ⬘⫽0.63J reproduce the temperature dependence of the magnetic susceptibility most accurately. At higher temperatures
the interlayer exchange coupling J ⬙ ⫽0.09J has to be included to account for the experimental susceptibility.4
The ground state of the 2D orthogonal dimer model de0163-1829/2004/69共17兲/174420共8兲/$22.50

fined by H ex is exactly solvable. It is simply a product of
singlets on each dimer up to a critical ratio of exchange
constants (J ⬘ /J) c ⫽0.68.5 The experimentally determinedvalues of isotropic exchange constants place SrCu2 (BO3 ) 2
compound to the extreme vicinity of the phase boundary
between the dimer phase and magnetically ordered phase.
The singlet state remains the ground state of the system evenwhen interlayer exchange J ⬙ is taken into account.6
Within the orthogonal dimer model the lowest-lying mag-

FIG. 1. The 2D network of Cu2⫹ ions with full lines representing the nn (J) and dashed lines the nnn (J ⬘ ) exchange coupling.
The interdimer D and intradimer D⬘ DM interactions are also presented with the thin arrows representing the orientation of corresponding bonds for DM interaction. Thick arrows are the nn DM
vectors.

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PHYSICAL REVIEW B 69, 174420 共2004兲

ZORKO, ARC̆ON, VAN TOL, BRUNEL, AND KAGEYAMA

netically excited state is a single triplet excitation present on
one of the dimers with the spin gap significantly suppressedwith respect to the intradimer exchange J. 2 Experimentally,
the first direct observation of the spin gap ⌬⫽35 K was
made by high-field ESR experiments.7 Since then its existence has been confirmed by various spectroscopic
methods.8 –10 So far, the picture of the low-lying magnetic
excitations in SrCu2 (BO3 ) 2 seems to be rather consistent.
There are, however, still missing details. For instance, theobserved anisotropic behavior of the single triplet excitationstrongly depending on the direction of applied external magnetic field could not be explained with the oversimplified
isotropic Hamiltonian H ex . To explain this fine structure oneneeds to include the magnetic anisotropy into the spin
Hamiltonian. Cépas et al.11 introduced anisotropic terms in
the form of antisymmetric Dzyaloshinsky-Moriya 共DM兲 exchange interaction. Neglecting a small buckling of the
CuBO3 plane,12 and using symmetry arguments13 the proposed form H DM ⫽ 兺 兵 i j 其 ⫾D ⬘储 ec •Si ⫻S j takes into accountonly nnn interactions D⬘ with the interdimer DzyaloshinskyMoriya vector of a magnitude D ⬘储 ⫽2.1 K pointing in the
crystal c-direction, i.e., perpendicular to 2D Cu2⫹ plane 共Fig.
1兲. On the other hand, the nn 共intradimer兲 DM interaction D
is set to zero due to the center of inversion positioned at the
middle of each dimer bond. However, the presence of a sizable intradimer DM interaction has been lately suggested byseveral experiments. Such interaction could explain recent
intriguing findings including unusual NMR shifts on B sites
and staggered magnetization,14 the relative intensities ofsinglet-triplet transitions observed in recent high-field ESR
measurements15 as well as the temperature dependence of thespecific heat at low temperatures and in high magnetic fields
共above 27 T兲.16
Moreover, though the nnn DM interaction reproduces theobserved splitting of the single triplet excitation quite nicely
and is of the expected amplitude,12 D ⬘储 ⬇⌬g/g•J ⬘ ⬇6 K, it
cannot alone satisfactorily explain the observation of singlettriplet transitions in the high-field ESR7 and far-infrared absorption experiments.9 These transitions are in principle allowed since the interdimer DM interaction mixes a finite
amount of excited states into the ground singlet state. However, due to the symmetry of the system, the abovementioned
DM terms still yield vanishing intensities of conventional
magnetic dipole transitions between the ground state and theexcited states.17 Very recently, a mechanism of electric dipole transitions between the ground and the low-lying excited states has been proposed to be responsible for the observation of these ‘‘forbidden’’ transitions in the
SrCu2 (BO3 ) 2 system.17,18 The authors explained the occurrence of infrared-active modes corresponding to magnons aswell as the dependence of the relative intensities on the external magnetic field9 by introducing a novel concept of dynamical DM interaction. In this picture, phonons induce instantaneous DM terms, which are otherwise forbidden by thesymmetry of the crystal. Nevertheless strong experimental
evidence for the dynamic DM interaction including the determination of its size is still missing.
In order to determine the major magnetic anisotropy in-teractions in SrCu2 (BO3 ) 2 we performed a detailed single
crystal X-band electron spin resonance 共ESR兲 study in thetemperature range between 10 K and 580 K, focusing inparticular on the temperature and angular dependence of the
ESR linewidth. We have chosen the ESR technique since
ESR absorption spectra are intimately related to spin anisotropy present in the system. Conducting the experiment on asingle crystal enabled us to determine the direction of theintradimer DM vectors in addition to the interdimer ones.
For the first time we also present a quantitative estimation of
the intradimer DM interaction in low magnetic fields. Our
results give strong support to the new mechanism of the
dynamical DM interaction17,18 as will be discussed in thefollowing sections and can explain a number of other experiments indicating the presence of nn DM interaction in the
SrCu2 (BO3 ) 2 system.
II. EXPERIMENTAL

Experiments were performed on a single crystal of
SrCu2 (BO3 ) 2 compound of a size 5⫻2⫻2 mm3 , which wasgrown by the traveling solvent floating zone 共TSFZ兲
method.19 Its purity was verified by bulk susceptibility measurements down to 2 K while the orientation of crystal axes
was determined from Laue x-ray back-reflection. X-band
ESR measurements were performed on the commercial
Bruker E580 FT/CW spectrometer at the Larmor frequencyof ␯ L ⬃9.5 GHz. The measurements in the temperature range
between 10 K and the room temperature were conductedusing an Oxford-cryogenics liquid-helium-flow cryostat
while measurements at temperatures between the room temperature and 580 K required a use of a high-temperature
controller with preheated-nitrogen-flow cryostat. High-field
ESR spectra on a powdered sample of SrCu2 (BO3 ) 2 at thefrequency of ␯ L ⫽93.2 GHz were recorded on a home build
spectrometer working in a transition mode. The same highpurity polycrystalline sample was used for this measurement
as in our previous X-band ESR report.20
III. THEORETICAL BACKGROUND

In the case of a large isotropic exchange interaction one
can divide the Hamiltonian of a spin system in an external
magnetic field into two parts H⫽H 0 ⫹H ⬘ , where H 0 contains only the Zeeman interaction H Z and isotropic exchange
coupling H ex , while all the spin anisotropy terms are contained in H ⬘ and can be treated as a perturbation. In general,
at temperatures well above the characteristic Zeeman splitting (k B TⰇg ␮ B B 0 ) the ESR absorption profile is formallygiven by the Fourier transform of the relaxation function
␸ (t)⫽ 具 M̃ ⫹ (t)M ⫺ (0) 典 / 具 M ⫹ M ⫺ 典 , reflecting fluctuations oftransverse magnetization operator M ⫹ ⫽g ␮ B 兺 i S ⫹i in interaction representation averaged over ensemble.21 At high temperatures (TⰇJ) the Kubo-Tomita approach21 to magnetic
resonance is justified and well established. When the isotropic exchange interaction, which determines typical spin correlation time ␶ c ⬃ប/k B J, is large compared to the observed
ESR peak-to-peak linewidth ⌬B pp (k B JⰇg ␮ B ⌬B pp ) andspin diffusion is ineffective, the ESR absorption spectrum

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X-BAND ESR DETERMINATION OF DZYALOSHINSKY- . . .

FIG. 2. Temperature dependence of the linewidth of the X-band
ESR spectra of SrCu2 (BO3 ) 2 single crystal for B0 储 c 共circles兲 andB0⬜c 共diamonds兲. Inset shows the high-T linearly increasing behavior.

has a Lorentzian profile with a peak-to-peak linewidth22⌬B pp ⬇C冉 冊

k B M 32 1/2.
g␮B M 4

共1兲The second and the fourth moment of the resonance spectrum are given by expressions
M 2 ⫽ 具 关 H ⬘ ,S ⫹ 兴关 S ⫺ ,H ⬘ 兴 典 / 具 S ⫹ S ⫺ 典 ,共2兲
M 4 ⫽ 具 关 H⫺H Z , 关 H ⬘ ,S ⫹ 兴兴关 H⫺H Z , 关 H ⬘ ,S ⫺ 兴兴 典 / 具 S ⫹ S ⫺ 典 .
The constant C appearing in Eq. 共1兲, however, is not rigorously determined. Since the second and forth moments of apure Lorentzian line diverge, one is forced to carry out further approximations. The usually considered cut-off Lorentzian with a cut-off field at an arbitrary position yields a constant C⫽ ␲ /3. On the other hand, a more realistic lineshape
given by the Lorentzian function multiplied by the exponential function of the form22 exp关⫺兩B⫺B0兩g␮B /kBJ兴 produces aconstant of C⫽2 ␲ / 冑6. It has to be emphasized that theexchange coupling in the SrCu2 (BO3 ) 2 system is few ordersof magnitude larger than the corresponding external magnetic field in X-band ESR experiments, which means that no
deviations from Lorentzian lineshape are expected to be observable despite the multiplication with the exponential function.
On the other hand, spin diffusion may be important in
low-dimensional systems at higher temperatures. In this casesignificant deviations from Lorentzian profiles and additionalline broadening are observed.23
IV. RESULTS

Lorentzian shape of very broad X-Band ESR absorption
lines measured in a single crystal of SrCu2 (BO3 ) 2 has already been reported earlier.24 The lineshape remains Lorentzian even at 580 K, proving that spin diffusion is negligible.
The temperature dependence of the linewidth shows a broad
minimum around room temperature 共RT兲 for external magnetic field B0 parallel and perpendicular to the crystal anisotropy c-axis as shown in Fig. 2.
The increase of the linewidth below RT is significant inboth directions and reflects the evolution of spin correlations

FIG. 3. X-band ESR anisotropy of SrCu2 (BO3 ) 2 single crystal
at 525 K 共diamonds兲 and 295 K 共circles兲 with the high-T linear partsubtracted. The solid lines correspond to angular dependence predicted by Eq. 共4兲 with D ⬘储 ⫽2.4 K and D⬜ ⫽4.0 K. The inset shows‘‘raw’’ linewidth anisotropy in the crystal ac-plane 共open symbols兲
at 525 K 共diamonds兲 and 295 K 共circles兲 and in the ab-plane at 295
K 共full circles兲.

present in the investigated system. On the other hand, above
RT the linewidth exhibits at first site rather surprising linear
behavior with a slope almost independent on the angle ␪
between the external field B0 and crystal c-axis, i.e., theslope equals 0.355⫾0.01 G/K for ␪ ⫽0° and 0.325⫾0.01 G/K for ␪ ⫽90° 共inset to Fig. 2兲. The unusual high-T
increase has already been discussed on the qualitative
grounds.24 Its origin was proposed to arise from spin-orbit
coupling giving rise to an additional line-broadening mechanism. However, in the present paper also a quantitative description is given on the basis of a phonon modulation of the
Dzyaloshinsky-Moriya interaction. We shall come back tothis linear contribution to the ESR linewidth later. For the
moment, let us subtract this part from the measured linewidths and focus on the angular dependence.
Angular dependence of the linewidth at 295 K and 525 K
is shown in the inset to Fig. 3. It is significant in the crystal
ac-plane 共dependence on the polar angle ␪兲 while there isalmost no angular dependence with respect to the azimuthal
angle ␾ (ab-plane兲. The ␪-dependence of the linewidth atboth temperatures can be qualitatively described by an equation of the form A⫹B(1⫹cos2 ␪). We use such a form rather
than A ⬘ ⫹B ⬘ cos2 ␪ because of the characteristic linewidth
anisotropy expected for the nnn DM interaction as it will be
revealed in the next section. The experimentally determinedvalues of the parameters are A⫽378⫾5 G, B⫽218⫾5 G forT⫽295 K and A⫽362⫾5 G, B⫽220⫾5 G for T⫽525 K
when the high-temperature linear contribution is subtracted.
As the high-T increasing part of the linewidth has more orless the same slope for ␪ ⫽0° and ␪ ⫽90° the same set ofparameters A and B corresponds to the linewidth anisotropy
for temperatures above approximately 340 K 共see inset toFig. 2兲.
Due to the large exchange coupling one expects a strong
exchange narrowing of the ESR line. Therefore, only few
spin-anisotropy interactions are expected to account for experimental linewidths. Major anisotropy contributions to thelinewidth have already been evaluated in our previous
report,24 where the DM interaction was shown to play by far
the biggest role. In fact, among the major magnetic aniso-

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ZORKO, ARC̆ON, VAN TOL, BRUNEL, AND KAGEYAMA

tropy interactions expected for a spin S⫽ 21 system, namely,
the dipolar interaction, the hyperfine coupling, the symmetricanisotropic exchange and the antisymmetric DM interaction,
the latter interaction is the only one giving linewidths of thecorrect order of magnitude.
V. ANALYSIS AND DISCUSSION
A. Linewidth anisotropy

Exploring the originally proposed picture of only interdimer DM interactions with corresponding DM vectors parallel to crystal c-axis proposed by Cépas et al.,11 the angulardependence of the linewidth is expected to be of the form
共see the Appendix兲
⌬B pp ⬇

D ⬘储 22␲ kB
共 1⫹cos2 ␪ 兲 .
冑6 g ␮ B 冑6 共 3J 2 ⫹3J ⬘ 2 ⫺2JJ ⬘ 兲

共3兲

The above equation possesses no ␾-dependence, which is inexcellent agreement with the experiment 共Fig. 3兲. On theother hand, the ␪-dependence of the form B(1⫹cos 2 ␪ ) is
predicted. However, the experimental results presented inFig. 3 differ from this prediction in one important detail.
Namely, there is an additional constant parameter A, which
has to be included to yield a satisfactory fit, even though thehigh-temperature linearly increasing part is subtracted fromthe ‘‘raw’’ data. We emphasize once again that the parameter
A is of the same order of magnitude as parameter B. Therefore, the magnetic anisotropy introducing the additional A
term should be substantial and cannot be simply ignored.
This raises a fundamental question about the origin of this
supplementary contribution to the linewidth. Symmetric anisotropy interaction, which should give the second largest
contribution to the linewidth, can be safely dismissed from

⌬B pp ⬇

the possible causes of the observed discrepancy for at least
two reasons. It is more than an order of magnitude too small
and it yet again yields angular dependence of the form (1⫹cos2 ␪).24
We have already proposed in our initial report24 that theobserved angular dependence of the linewidth originates
fromadditional
magnetic-anisotropy
termsof
Dzyaloshinsky-Moriya type that should be included in thespin Hamiltonian. As already emphasized, when treating the
CuBO3 crystal planesas ideally planar only the interdimer
DM interactions with vectors perpendicular to this plane areallowed for symmetry reasons.11 On the other hand, a finite
buckling of these planes has been reported below the displacive, second order structural phase transition occurring at
T s ⫽395 K. 12 Bending of neighboring CuO4 plaquettes allows for finite in-plane components of interdimer DM interaction D⬜⬘ as well as for in-plane intradimer coupling D⬜ .
Moreover, the direction of DM vectors of the latter interaction is well defined, due to the fact that a mirror plane perpendicular to crystal ab-plane containing a dimer still exists.
The nn DM vectors thus have to point perpendicularly to thedirection of the dimer 共Fig. 1兲. The interdimer D⬜⬘ coupling
has recently been proposed to be responsible for the finestructure of the single triplet excitation at wave vector Q⫽(1.5,0,0) observed in high-resolution inelastic neutron
scattering experiments.25 It has been estimated to be of theorder of 35% of D ⬘储 in the J ⬘ ⫽0 limit. Since the ESR linewidth is proportional to the square of the amplitude of DM
vector, the in-plane component of the nnn DM interaction
yields smaller corrections to the linewidth. For this reason it
can be neglected in the first approximation.
Taking into account the additional intradimer DM interaction D⬜ , the angular dependence of the ESR linewidth isgiven by the following expression 共see the Appendix兲,

冉

冊

1/2
共 8D ⬘储 2 ⫹3D⬜2 ⫹ 共 8D ⬘储 2 ⫺D⬜2 兲 cos2 ␪ 兲 3
2␲ kB,
冑6 g ␮ B 96共 32D ⬘储 2 J 21 ⫹3D⬜2 J 22 ⫹ 共 32D ⬘储 2 J 21 ⫺D⬜2 J 22 兲 cos2 ␪ 兲with J 21 ⫽3J 2 ⫹3J ⬘ 2 ⫺2JJ ⬘ and J 22 ⫽13J 2 ⫹6J ⬘ 2 . Due to thefact that dimers form an orthogonal network, the linewidth
remains ␾-independent, which is, as already stressed, in accordance with the experiment. The derived equation fits theobserved linewidth anisotropy rather well, as shown in Fig.
3. The parameters obtained for 525 K are D ⬘储 ⫽2.4⫾0.1 K
and D⬜ ⫽4.0⫾0.1 K. At 295 K the parameters are only
slightly bigger, i.e., by approximately 0.1 K. Moreover, thehigh-T parameters seem to be valid for all temperatures
above approximately 340 K as the linewidth simply increases with similar slope for all directions above this temperature. The D ⬘储 ⫽2.4 K term is in good agreement with theprevious estimation of the interdimer DM interaction determining the fine splitting of triplet excitations.11
We stress that the inclusion of a finite in-plane interdimer

共4兲

DM interaction D⬜⬘ 共Fig. 1兲 slightly modifies the value ofD⬜ . If the DM interaction between nnn spins 1 and 3 is of
the form D13⫽(D ␰⬘ ,D ␩⬘ ,D ⬘储 ), the four symmetry operators of
the SrCu2 (BO3 ) 2 space group (I4̄2m) below T s ⫽395 K,
namely, the mirror planes (m 1 x,x,z and m 2 x,x̄,z) and thefourfold rotoinversion axes (4̄ ⫺ 0,1/2,z;0,1/2,1/4 and4̄ ⫹ 1/2,0,z;1/2,0,1/4), determine all the remaining nnn DM
vectors as indicated in Fig. 1. The resulting linewidth isqualitatively unchanged as well as the size of D ⬘储 term. However, the in-plane components of nn and nnn DM interaction
are coupled. Taking the latter from high-resolution inelastic
neutron scattering results25 to amount for D ⬘␰ 2 ⫹D ␩⬘ 2 ⫽D⬜⬘ 2⬇0.1D ⬘储 2 , our fit gives a smaller value of the former interaction at 525 K, i.e., D⬜ ⫽3.6 K. However, since there is a174420-4

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X-BAND ESR DETERMINATION OF DZYALOSHINSKY- . . .

certain ambiguity about the size of D⬜⬘ as it has been estimated only in the J ⬘ ⫽0 limit, at this stage we can only makean estimation that the intradimer DM interaction is of theorder of D⬜ ⫽3.6⫾0.5 K.
B. Nature of the in-plane DM interaction

The value of the nn DM interaction D⬜ ⫽3.6 K seems to
be unexpectedly high at first glance. This interaction has
been disregarded until very recently,14 –16 although it should
be considerable due to finite buckling of CuBO3 planes.
At room temperature where the bending angle12 of CuO4
plaquettes with respect to c-axis is ␣ ⫽4° it is of the order ofD⬜ ⫽& sin ␣(⌬g/g)J⬇1 K. 26 We note, however, that this interaction is expected to vanish at temperatures above T s⫽395 K, i.e., at temperatures where the buckling disappears.
Surprisingly, the ESR linewidth anisotropy is not affected bythis structural phase transition at all.
A credible explanation of this puzzle is obtained if dynamical effects are taken into account.17,18 In this picture
lattice vibrations instantaneously break local symmetry, allowing for additional in-plane terms of DM interaction.
However, there are requirements to be fulfilled before including this mechanism into the interpretation of our X-band
ESR results. First of all, dynamical effects can be observed
when the characteristic phonon frequency is small compared
to characteristic exchange frequency ␻ e ⬃k B J/ប determining
the spin correlation time ␶ c . Since optical phonons areneeded to produce required lattice distortions, significant
softening of a particular normal mode should be present.
Second, the mean square displacements of ions, participatingin the aforementioned lattice motion, from their equilibrium
positions should be large enough to break the local symmetry
significantly. It seems that both conditions are fulfilled in the
SrCu2 (BO3 ) 2 system.
Of our particular interest is the observation of a soft mode
detected by the Raman scattering experiment.26 The Raman
shift of this optical mode amounts to 62 cm⫺1 共89 K兲 at T
⫽15 K and progressively softens by 44 cm⫺1 just below T s⫽395 K. It causes in-phase motion of almost all ions within
the primitive cell 共with the exception of Sr2⫹ ions兲, preferably along the crystal c-direction. The symmetry of this softmode is also consistent with the observed anomalous anharmonicity of lattice properties, namely, a flattening of the local potentials of ions and a significant enhancement of theirmean square displacements in the crystal c-direction, bothphenomena progressively getting larger when approaching
the transition to high-T phase.12 For instance, the U 33 principal components of the displacement tensor of Cu ions andO1 ions connecting them yield enhanced vibration amplitudes in the crystal c-direction in the high-T phase, which arevirtually the same as static displacements 共0.25 Å for Cu ionsand 0.33 Å for O1 ions兲 of these ions at 100 K.12 Such lattice
vibrations break the local symmetry in a very similar way as
it is broken in the low-T phase by finite buckling since thedistortions are quasi-static on the time-scale of spin correlation time ␶ c . Therefore, it is not surprising that the ESR
linewidth anisotropy does not change qualitatively above RT
despite crossing the structural phase transition. We proposethat the origin of nn DM interaction D⬜ ⫽3.6 K is a combination of static, i.e., buckling of CuBO3 planes, and dynamical mechanism and evolves progressively into a dynamicalsource with increasing temperature. The obtained parameteris thus an ‘‘averaged’’ value since the displacements are timedependent. Our experimental findings strongly support the
new picture of dynamical DM interaction.17,18
By introducing additional anisotropic terms of DM type tothe spin Hamiltonian of the system, one has to consider carefully whether the observed fine structure of triplet excitations
can be theoretically reproduced. In fact, it has been shownusing the finite temperature Lanczos 共FTL兲 method on a
finite-size Shastry-Sutherland lattice that the splitting remains primarily given by the out-of-plane interdimer DMinteraction.27
C. High-temperature linewidth behavior

Next we focus on the high-T linearly increasing part of
the ESR linewidth. For a pure spin Hamiltonian ESR linewidths are normally expected to approach a constant value
for temperatures TⰇJ. However, several mechanisms canlead to a temperature dependent ESR linewidth at such temperatures including spin diffusion, static spin correlations
and spin-phonon coupling. Although the spin diffusion
mechanism may become significant at higher temperatures,the measured Lorentzian ESR lineshape speaks strongly
against it.24 Second, the effect of static spin correlations 共i.e.,
short-range order兲 on the linewidth can be observed in lowdimensional magnetic systems even up to temperatures ofthe order of T⬃10J. 28 Following the original paper of Sooset al.,29 the temperature-dependent second moment due tothe DM interaction can be calculated
M DM
2 ⫽

S 共 S⫹1 兲
3N
⫹

再兺

(i j)

共共 D xi j 兲 2 ⫹ 共 D iyj 兲 2 ⫹2 共 D zi j 兲 2 兲共 1⫺C i j 兲y⫹2D zi j D zik 兲 C jk
共 D xi j D xik ⫹D iyj D ik
兺(i j)k⫽i, j⫹兺兺 共 D xi j D xkl ⫹D iyj D kly ⫹2D zi j D zkl 兲(i j) (kl)k,l⫽i, j⫻ 共 C ik C jl ⫺C il C jk 兲冎␹C.
␹共 T 兲

共5兲

The temperature dependence due to static spin correlations is
hidden in the ratio between Curie susceptibility ␹ c and themeasured susceptibility ␹ (T) as well as in functions C i j reflecting two-site static spin correlation between spins at sites
i and j. Each of the three sums in Eq. 共5兲 yields its own
characteristic temperature behavior, with the first one approaching the infinite-temperature second moment and thelast two going to zero when increasing the temperature. It ishighly unlikely that the two-site correlation function C i jwould evolve with temperature in such a way to produce
angular-independent line broadening as observed in current
investigation. Another reason allowing us to dismiss static

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ZORKO, ARC̆ON, VAN TOL, BRUNEL, AND KAGEYAMA

nate over phonon-induced Zeeman transition within S⫽1
multiplet. In a simple picture of uncorrelated dimers thefinite-lifetime contribution to the linewidth is given by theequation22⌬B linpp ⬇

FIG. 4. Log-to-log plot of the linewidth of X-band ESR spectrafrom SrCu2 (BO3 ) 2 single crystal with the high-T contribution subtracted for B0 储 c 共circles兲 and B0⬜c 共diamonds兲. Full lines represent
fits to an equation of the form ⌬B⫽A/(T⫺T * ) p . The inset showsa high-field ESR spectrum of powdered SrCu2 (BO3 ) 2 sample at RT
共circles兲 and a fit to anisotropic Lorentzian powder distribution 共fullline兲.

spin correlations as the origin of the observed broadening isalmost ideal linear dependence in a rather broad temperature
range, i.e., between 3.5J and 7J. In such a broad region one
would expect significant bending of the linewidth curves towards the infinite temperature value if the broadening wasdue to short-range order effects.28
Therefore, the spin-phonon coupling should be the main
broadening mechanism at high temperatures. The significance of such coupling has been highlighted from the dramatic softening of elastic constants both with temperature
and applied magnetic field.30 Spin-phonon interaction causes
lifetime broadening effects. As the increase of the linewidthis linear, normal direct phonon processes should be involved.
In particular, the observed line broadening can be either due
to the usual spin-lattice relaxation between Zeeman split
low-lying excited states or a reflection of transitions between
the ground state and these states induced by a phonon modulation of antisymmetric Dzyaloshinsky-Moriya interaction.31
For the former mechanism a strong field-dependent behavior
is typical while no field dependence is expected in the lattercase.31 In the inset to Fig. 4 a high-field spectrum of a powdered sample, recorded at a frequency of 93.2 GHz at RT, isshown. As it can be seen, it is nicely fitted with a Lorentzianfunction for powder spectra yielding anisotropic linewidthsof ⌬B app ⫽⌬B bpp ⫽699⫾10 G and ⌬B cpp ⫽890⫾10 G. These
parameters are virtually the same as the single-crystal
X-band values ⌬B ␪pp⫽90° ⫽690⫾5 G and ⌬B ␪pp⫽0° ⫽907
⫾5 G. Since the linear contribution to the linewidth at RT isof the order of 100 G and is the same at two resonance fields
differing by almost a factor of 10, we propose that the spinlattice contribution to the linewidth is due to fluctuating DM
interaction.
Making a rather crude approximation while neglecting
correlation effects between a pair of interacting spins andtheir neighbors, the linewidth is determined by phononinduced transitions between a singlet state and triplet statesof a two spin system. Since J is two orders of magnitudelarger than characteristic Zeeman energy and the phonon
density scales as ␻ 2 in the Debye approximation, it is not
surprising for the modulated-DM interaction effect to domi-冓k B4 共 ␭R 兲 2 D 2 J 2 1 2 1⫹
␳ប3
c 5t 3 c 5l9) g ␮ B
8z

冔k B T. 共6兲
⍀Taking the number of independent pairs as z⫽1, ␭R⫽10
with the nearest-neighbor distance R and dJ/dr⫽⫺␭J, 32 ␳
⫽4.1 kg/dm3 , 33 an approximate average velocity30 c⫽4600 m/s in place of the complicated angular average andthe above estimated intradimer DM interaction D⫽D⬜
⫽3.6 K, we can calculate the slope of the linearly increasing
part to be of the order of 0.14 G/K. This result is in a reasonable agreement with the experimentally observed slope
0.34 G/K, bearing in mind that the mean velocity, which isburdened with the biggest uncertainty, is taken to the powerof 5. Moreover, Eq. 共6兲 is strictly valid only in a crude approximation of independent dimers. Since interactions withother neighbors in general shorten the lifetime of a spin in acertain energy level, the slope is expected to be larger in thereal system than the value our estimation produces. The fairagreement of this line-broadening mechanism with the experiment once again justifies the introduction of intradimer
DM interaction.
D. Low-temperature linewidth behavior

At the end let us give a brief discussion on the temperature dependence of the ESR linewidth below RT. Figure 4
shows a log-log plot of the linewidth without the hightemperature linear part. The spin-phonon contribution to thelinewidth at low temperatures becomes insignificant, which
justifies the subtraction of the linear term instead of a term
proportional to the actual number of phonons.
The evolution of the low-temperature part of the linewidthwith temperature yields a linear dependence in the log-log
plot to a reasonably good approximation in a rather broad
temperature range between 15 K and 150 K. Solid lines correspond to a fit to the function ⌬B⫽A/(T⫺T * ) p , characteristic of critical broadening effects. The obtained values forthe exponent are p⫽0.41⫾0.01 for ␪ ⫽0° and p⫽0.47⫾0.01 for ␪ ⫽90°, while the value of the characteristic temperature T * is within the experimental error very close to
zero. We have reported similar observations on the linewidth
behavior in powder samples.20 Such critical behavior typically occurs near phase-transition temperatures with induced
long-range order in the low-temperature phase. In the case of
the investigated system it is probably due to critical enhancement of antiferromagnetic fluctuations and thus a fingerprintof the importance of quantum criticality effects in the
SrCu2 (BO3 ) 2 system, as the system is believed to be situatedin the extreme vicinity of the quantum borderline between
the nonmagnetic and magnetically ordered ground state.
However, the observed temperature interval 共15 K to 150 K兲,in which this theory seems to be consistent, is surprisingly
broad. In this respect, the results of our investigation call for
a comprehensive theoretical evaluation of the temperature

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PHYSICAL REVIEW B 69, 174420 共2004兲

X-BAND ESR DETERMINATION OF DZYALOSHINSKY- . . .

evolution of the ESR spectra in the SrCu2 (BO3 ) 2 system atlower temperatures. Such theory would also have to accountfor the observed small deviations from ‘‘linear’’ dependence
below 15 K.

thanks the Yamada Science Foundation 共Japan兲 for supporting his work in low-dimensional systems and the MECSST
共Japan兲 through Grant-in-Aid for Scientific Research No.
40302640.

VI. CONCLUSIONSAPPENDIX

In summary, a comprehensive X-band ESR study on asingle crystal of a highly frustrated SrCu2 (BO3 ) 2 system has
been given in a broad temperature range between 10 K and
580 K. The anisotropy of the rather broad linewidth shedsome important additional light on the spin anisotropy of theinvestigated systems. Previously proposed out-of-plane interdimer DM interaction as the dominant anisotropic contribution cannot adequately account for the experimental angulardependence. In addition to this interaction D ⬘储 ⫽2.4⫾0.1 K
one has to incorporate also sizable in-plane DM components.
Taking the value of the in-plane nnn DM vector D⬜⬘⬇0.35D ⬘储 , as suggested by recent inelastic neutron scattering experiments, allowed us to evaluate for the first time also
the intradimer DM interaction, D⬜ ⫽3.6⫾0.5 K, which isbelieved to be partially responsible for many anomalous
findings including staggered magnetization, high-field ESRactive transitions, and temperature dependence of the specific heat. Moreover, as in the high-T phase 共above 395 K兲
the in-plane components of DM interaction are prohibited bysymmetry; an explanation of the nature of these terms basedon a novel dynamical DM interaction has been given. Taking
into account the presence of a soft mode with a particularsymmetry we emphasized that lattice distortions are quasistatic on the time-scale of the spin correlation time and that
dominant ion displacements take place along crystal
c-direction. Second, the observed anisotropic linewidthshows a peculiar temperature dependence, which can be decomposed into two contributions. The high-temperature linear contribution is arising from the interplay between spin
and lattice degrees of freedom and has been attributed to the
induced transitions between energy levels due to phononmodulation of the nn Dzyaloshinsky-Moriya interaction. On
the other hand, the low-temperature part exhibits criticalbroadening-like behavior, which could be a consequence ofenhanced antiferromagnetic correlations.

In low-dimensional spin systems with significant g-shifts
and large isotropic exchange coupling the antisymmetric
Dzyaloshinsky-Moriya interaction is often the dominant spin
anisotropy contribution. In this case the second and the forth
moments of the absorption line are given in the hightemperature limit 共neglecting static spin correlations兲 bycomponents of DM vectors in the laboratory frame as22

ACKNOWLEDGMENTS

It is a pleasure to thank Dr. A. Lappas for stimulating
discussions on the magnetic properties of SrCu2 (BO3 ) 2 . HK

M DM
2 ⫽
M DM
4 ⫽

S 共 S⫹1 兲„共 D xi j 兲 2 ⫹ 共 D iyj 兲 2 ⫹2 共 D zi j 兲 2 …,
3N (i j)兺再兺

S 2 共 S⫹1 兲 2 2
J 2 „共 D xi j 兲 2 ⫹ 共 D iyj 兲 2 ⫹2 共 D zi j 兲 2 …
N
3 (i j) i j⫹

1„共 F xi jk 兲 2 ⫹ 共 F iyjk 兲 2 ⫹2 共 F zi jk 兲 2 ⫹ 共 F xjki 兲 2
18 (i jk)

共A1兲

兺

冎

y 2
z
2
⫹ 共 F yjki 兲 2 ⫹2 共 F zjki 兲 2 ⫹ 共 F xki j 兲 2 ⫹ 共 F kij 兲 ⫹2 共 F ki j 兲 … ,where the three-site term sums over functions F ␣i jk ⫽J i j (D ␣ik⫺D ␣jk )⫹J ik (D ␣i j ⫹D ␣jk ). The expected ESR linewidth anisotropy is then obtained by transforming the DM vectors fromthe laboratory frame to the crystal frame
D xi j ⫽D ai j cos ␪ cos ␸ ⫹D bi j cos ␪ sin ␸ ⫺D ci j sin ␪ ,D iyj ⫽⫺D ai j sin ␸ ⫹D bi j cos ␸ ,共A2兲

D xi j ⫽D ai j sin ␪ cos ␸ ⫹D bi j sin ␪ sin ␸ ⫹D ci j cos ␪ .
We denote an arbitrary orientation of applied constant magnetic field (z-direction in the laboratory frame兲 with the polar
angle ␪ and the azimuthal angle ␾ in the crystal frame of
SrCu2 (BO3 ) 2 system with c-axis as the polar axis. Using theequations 共1兲, 共A1兲 and 共A2兲 with the DM-vectors patternshown in Fig. 1 we derive the expression for the angulardependence of the ESR linewidth in the SrCu2 (BO3 ) 2 system for the case when only out-of-plane nnn DM interactionD⬘ is present and the case which also takes into account thein-plane nn DM coupling D:

共1兲 D⬘ ⫽(0,0,D ⬘储 ),D⫽0:
D ⬘储 2
2␲ kB⌬B pp ⬇
共 1⫹cos2 ␪ 兲 ,冑6 g ␮ B 冑6 共 3J 2 ⫹3J ⬘ 2 ⫺2JJ ⬘ 兲
共2兲 D⬘ ⫽(0,0,D ⬘储 ),D⫽(D⬜ ,0,0) or D⫽(0,D⬜ ,0):
⌬B pp ⬇

冉

共A3兲

冊

1/2
共 8D ⬘储 2 ⫹3D⬜2 ⫹ 共 8D ⬘储 2 ⫺D⬜2 兲 cos2 ␪ 兲 3
2␲ kB,
冑6 g ␮ B 96共 32D ⬘储 2 J 21 ⫹3D⬜2 J 22 ⫹ 共 32D ⬘储 2 J 21 ⫺D⬜2 J 22 兲 cos2 ␪ 兲

where J 21 ⫽3J 2 ⫹3J ⬘ 2 ⫺2JJ ⬘ and J 22 ⫽13J 2 ⫹6J ⬘ 2 .
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共A4兲

PHYSICAL REVIEW B 69, 174420 共2004兲

ZORKO, ARC̆ON, VAN TOL, BRUNEL, AND KAGEYAMA*Author to whom correspondence should be addressed. Email ad-

19

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