Appl. Magn. Reson. 27, 267#278 (2004)

AppliedMagnetic Resonance© Springer-Verlag 2004
Printed in Austria

Magnetic Anisotropy of the SrCu2(BO3)2 System as Revealedby X-Band ESR
A. Zorko1, D. Arèon1, H. Kageyama2, and A. Lappas3
1
Joef Stefan Institute, Ljubljana, Slovenia
Department of Chemistry, Graduate School of Science, Kyoto University,
Kyoto, Japan
3
Institute of Electronic Structure and Laser, Foundation for Research and Technology Hellas,
Heraklion, Greece
2

Received October 3, 2003

Abstract. X-band electron paramagnetic resonance measurements on a single crystal of the highly
frustrated SrCu2(BO3)2 system are shown to provide an essential inspection of the magnetic anisotropy present in this compound. The very broad absorption lines seem to be consistent with the largest anisotropy term, namely, the antisymmetric Dzyaloshinsky-Moriya (DM) interaction allowed bysymmetry. However, the previously well-accepted model of only interdimer interaction is generalizedwith additional intradimer DM terms. Moreover, spin-phonon coupling is recognized as the cause of
the line width broadening with increasing temperature.

1 Introduction
Low-dimensional spin-gap systems with quantum-disordered singlet ground state
have been in the mainstream of scientific research in the last decade due thediversity of possible ground state phases and richness of low-lying magnetic
excitation spectra. The gap in magnetic excitation spectrum (spin gap) is a common feature in one-dimensional (1-D) antiferromagnets and is usually triggeredby competing exchange interactions or dimerization present in the system. On
the contrary, the field of spin-gap systems in two dimensions is not nearly asrich as in one dimension since the general characteristic of 2-D antiferromagnets is a tendency to exhibit 3-D order at lower temperatures due to interlayer
exchange interactions. However, sufficient frustration provided by a special exchange topology of the system can resolve the nonmagnetic singlet state as theground state of the system and is thus directly responsible for the opening of
the spin gap.
SrCu2(BO3)2 with localized S " 1/2 Cu2! spins is one of the few 2-D systems exhibiting a spin-gap behavior [1]. Already in the early stages [2] this268

A. Zorko et al.

X-Band ESR of the SrCu2(BO3)2 Systemsystem was recognized as the first chemical realization of a model proposed by
Shastry and Sutherland [3], which takes into account 2-D square lattice of antiferromagnetically coupled spins with additional selected diagonal bonds. The
Hamiltonian of the investigated system is thus given byof the system is concerned. The experimental singlet-triplet transitions are forbidden due to the fact that the Shastry-Sutherland exchange Hamiltonian commutes with a magnetic dipole operator. Such transitions can be explained only
by including spin-anisotropy terms to the original Hamiltonian, which are thenresponsible for mixing a small amount of excited states into the ground singlet
state. A feasible analysis of the observed angular-dependent fine structure was proposed by Cépas et al. [8] with the introduction of antisymmetric anisotropic exchange interaction, known as Dzyaloshinsky-Moriya (DM) interaction of the form

0ex = J å Si S j + J Så Sl S m ,( ij )(1)( lm )

where (ij) terms represent nearest-neighbor (nn) exchange coupling pairs while(lm) stands for next-nearest-neighbor (nnn) pairs of spins with exchange constants estimated to be J " 85 K and JS " 54 K [4]. The unique network of orthogonal interacting dimers composed of Cu2! ions is shown as the inset to Fig.
1. The ground state of the system remains a simple product of spin singlets oneach dimer even if interlayer exchange coupling Jì " 8 K is taken into account[5], which makes the JS-coupling extremely frustrated. As a result of the inherent geometrical frustration the lowest lying magnetic excitation  a single triplet
excitation present on one of the dimers  is extremely localized as evidenced
by almost dispersionless excitation band [6] and relatively small value of the spin
gap D " 34 K with respect to the exchange coupling constant J [6, 7].
The singlet-triplet excitation possesses another remarkable property, namely,a fine structure as revealed for the first time by a high-field electron spin resonance (ESR) experiment [7]. The interpretation of high-field ESR spectra was,in fact, quite an achievement as far as the understanding of magnetic anisotropy. +D||(-2(B-B0)fmax/f (B)∆Bpp)1/2

30
×

-D||
20

D⊥

+D||

10

-D||

×

.

Gaussian

Lorentzian
θ=0°
θ=90°

0
0

20

40
(2(B-B0)/∆Bpp)260

0 DM = å ± D|| ec Si ´ S j .
( ij )

269(2)

Such antisymmetric interaction is quite common in highly anisotropic, i.e., lowdimensional systems, with large exchange coupling and no center of inversion
between interacting spins. It is the first-order perturbation effect of spin-orbit
coupling and can often be by far the largest anisotopic term of the Hamiltonian.
The magnitude of the DM interaction can be estimated from the g-shift andexchange constant, D ; Dg/gJS. As the Cu-B-O planes are almost flat [9], onlyinterdimer DM interaction was employed with DM vectors D " 2.1 K [8] pointing in the crystal c-direction (inset to Fig. 1), due to symmetry arguments originally proposed by Moriya [10].
Magnetic anisotropy determines the spin dynamics in the system and is thus
reflected in ESR spectra through the observed line widths and lineshifts. To establish the nature of the largest anisotropic terms, we previously conducted Xband ESR measurements on a powdered sample of SrCu2(BO3)2 and concluded
that symmetric anisotropic exchange and/or antisymmetric DM interaction had tobe considered in order to justify the observed broad spectra [11]. However, performing experiments on single crystals offers an additional insight into the anisotropy of the system. Namely, the angular dependence of the line width can yield
supplementary information on the direction of anisotropy axes in addition to themagnitude of anisotropic contributions. Moreover, the presence of diffusive motionof spins can be verified through the enhanced contributions of secular parts ofanisotropic terms. Measurements on single crystals thus often allow one to ruleout particular anisotropy terms in favor of other ones. Performing X-band ESR
on a single crystal of SrCu2(BO3)2 hence seems a reasonable way to verify thevalidity of the picture of spin dynamics proposed for this particular system.
Furthermore, it can yield information on additional anisotropic terms which have
not yet been evaluated or even observed by other spectroscopic techniques.

80

Fig. 1. Line shape analysis for X-band absorption lines of SrCu2(BO3)3 single crystal for external
magnetic field applied parallel (open circles) and perpendicular (full diamonds) to crystal c-axis at
525 K. Solid lines correspond to absorption profiles of Lorentzian and Gaussian type. The inset shows
the 2-D network of Cu2! ions, with thick lines representing the nn exchange coupling J and thinlines related to the nnn exchange JS. The direction of interdimer D as well as intradimer D^ DM
interaction is also presented.

2 Material and Methods
A single crystal of SrCu2(BO3)2 was grown by the travelling solvent floating zone
(TSFZ) method from a polycrystalline SrCu2(BO3)2 powder with LiBO2 as a solvent [12]. The direction of crystal axes was determined by means of X-ray scattering and the high purity was verified by almost no low-temperature Curie-like270

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X-Band ESR of the SrCu2(BO3)2 System

upturn of the magnetic susceptibility down to 2 K. A crystal of a size of 5 by
2 by 2 mm was used for ESR experiments.
X-band ESR measurements were performed on a commercially available
Bruker E580 FT/CW spectrometer working at a Larmor frequency of nL " 9.5
GHz. The temperature of 525 K was reached using high-temperature controller
with preheated-nitrogen-flow cryostat.

ternal magnetic field B0, with respect to the crystal anisotropy c-axis, is shownin Fig. 1 for T " 525 K. In the plot (#(B # B0)f max/f S(B)DBpp)1/2 vs. (2(B # B0)/
DBpp)2 Lorentzian profiles yield a linear dependence, while Gaussian lines produce a much faster (exponentially) increasing function due to the rapid decreaseof the signal in the tails of resonant absorption spectra. Here, f S(B) represents
the derivative absorption spectrum with peaks )f max
S . Spin diffusion results in afunction somewhere in-between the two extreme cases [16]. As it can be seenfrom Fig. 1 the presence of spin diffusion can be excluded from the SrCu2(BO3)2
system as the absorption profiles obey Lorentzian dependence.

3 Results and Discussion
Angular-dependent X-band ESR spectra on a single crystal of SrCu2(BO3)2 wererecorded at 295 and 525 K, i.e., at two temperatures well above the characteristic intradimer exchange temperature J " 85 K. Bearing in mind that the 00 "0z ! 0ex part of the total spin Hamiltonian 0 " 00 ! 0 S is dominant, theanisotropic part 0 S can be treated as a perturbation and the Kubo-Tomita approach to magnetic resonance should be justified at such high temperatures [13].
Here, the 0z and 0ex terms represent the Zeeman and isotropic exchange (Eq.
(1)) parts of the Hamiltonian, respectively. As we are dealing with S " 1/2 spins,
no single-spin anisotropy terms are present. So the magnetic anisotropy is inprinciple driven by the dipolar interactions 0dd between neighboring magnetic
moments, the hyperfine coupling 0hf of electron spins with nuclear spins, thesymmetric anisotropic exchange interaction 0ae (pseudo-dipolar interaction) andthe antisymmetric DM interaction 0DM, i.e.,
0 S " 0dd ! 0hf ! 0ae ! 0DM.

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3.1 g-Factor Anisotropy
Next we focus on the anisotropy of the g-factor presented in Fig. 2 for T " 295
K. As expected from the crystal structure [9], the g-tensor is axially symmetricwith a pronounced angular dependence in the ac-plane (q-dependence) and only
a minor dependence with respect to the azimuthal angle f (ab-plane). Fitting the
q-dependence to the equation
g =

(4)

where w eff = kB J / D is the characteristic frequency of the exchange modulationand M2 is the second moment given by

M 2 = [ H S, S+ ][ S- , H S]

S+ S- .

The line shape is expected to be Lorentzian. However, line broadening anddeviations from Lorentzian profiles are often detected in low-dimensional systems at higher temperatures due to spin diffusion [15].
The standard analysis of the line shape of ESR absorption spectra for
SrCu 2(BO 3) 2 for both the parallel and the perpendicular direction of the ex-2.30
θ-dependence
2.25

g-factor

2 kB2 M 2
2 kB M 2»,
3 Dg m B w eff
3 g mB J(5)

g||2 cos 2 q + g ^2 sin 2 q

yields parallel and perpendicular g-factors with respect to the c-axis: g "2.269 ) 0.002 and g^ " 2.057 ) 0.002. Such g-shifts are common for Cu2! 3d9
configuration where the crystal field is stronger than the spin-orbit coupling [17,The absorption lines should be extremely exchange narrowed as the isotropic exchange is by far the superior interaction present in the investigated system (kBJ @ gmBB0 ; gmBDBpp). In this regime, the peak-to-peak line width isgiven by the expression [14]DBpp »

271

φ-dependence

2.20

2.15

2.10

2.05
0

30

60

90
φ, θ (°)

120

150

180

Fig. 2. Anisotropy of the g-tensor of the X-band ESR spectra for the SrCu2(BO3)3 single crystal in
the crystal ac-plane (q-dependence; full circles) and in the ab-plane (f-dependence; open circles) at
295 K. The fit (solid line) corresponds to Eq. (5).

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X-Band ESR of the SrCu2(BO3)2 System

p. 48]. Crystal field thus effectively quenches orbital motion producing an effective spin of 1/2 for the ion. Orbital moments are partially restored by spinorbit coupling leading to g-shifts. The anisotropy of the g-shifts in the ab-plane
is much smaller. It can be limited by experimental accuracy to a value smaller
than 0.002. At 525 K, the g-factor shows virtually the same angular dependencewith unchanged eigenvalues.
3.2 Spin-Anisotropy Contributions to Line Widths
Line widths of the X-band ESR spectra at 295 and 525 K are presented in Fig.
3. As already emphasized [11] all the anisotropic terms included in the Hamiltonian Eq. (3) have to be considered in determination of the contributions to
the ESR line width. Furthermore, the Lorentzian line shape of absorption spectra confirms the assumption of the presence of strong exchange narrowing, sothat the use of Eq. (4) is well justified. Following the previous estimates ofthe dipolar and hyperfine contribution to the second moment M2dd < (gmB/kB1000
G)2 and M2hf < (gmB/kB200 G)2 [11], line widths of the order of DBdd
pp < 2 G andhf< 0.1 G are expected, which are several orders of magnitude smaller thanDBppthe experimental values. Employing symmetric anisotropic exchange, which isa result of second-order perturbation of the spin-orbit coupling (its amplitude
can be evaluated as AE ; (Dg/g)2J " 1.0 K), and antisymmetric DM interaction
yields the second moment of the form [18]1050

θ-dependence, 525 K

1000

θ-dependence, 295 K

950

φ-dependence, 295 K∆Bpp(G)öS ( S + 1) æyçç 2å AE 2 + å éë( Dlmx )2 + ( Dlm )2 + 2( Dlmz )2 ùû ÷÷ , (6)3N è (ij )( lm )
ø

where the components of the DM vector are given in the laboratory frame withthe z-axis pointing in the direction of the external magnetic field B0. With Eqs.
ae<(4) and (6) the contribution of the symmetric part can be evaluated as DBpp
k B AE 2 /2 3 g m B J < 30 G, while the antisymmetric DM interaction gives a valueDM< 2kB D 2 / 3 g mB J < 460 G taking into account the interdimer DM inof DBppteraction D " 2.1 K as proposed in ref. 8.
The above analysis favors the DM interaction as the only candidate capableof inducing sufficiently large line widths, as observed in the current experiment.
This conclusion could have been already reached by conducting the experimenton the powdered sample, where the room temperature peak-to-peak line width
was around 760 G. However, the fundamental advantage of the present singlecrystal experiment is the power to test the proposed picture of magnetic anisotropy in the SrCu2(BO3)2 system given by Eq. (2) through the angular dependenceof the observed line widths. This can in fact yield additional information about
the direction of DM vectors as discussed below.
3.3 Line Width Anisotropy Determined by DM Interaction
Let the external magnetic field B0 have an arbitrary orientation described by thepolar angle q with respect to the crystal c-axis and the azimuthal angle f in theab-plane. Applying the transformation of the DM vector from laboratory frameto crystal frame gives the expression of the angular dependence of the DM second moment
M 2DM =900

S ( S + 1)
3N

a 2
b 2
c 2
) (1 + sin 2 q cos 2 f ) + ( Dlm) (1 + sin 2 q sin 2 f ) + ( Dlm) (1 + cos 2 q )´ å {( Dlm

850( lm )abac+ 2 DlmDlm
sin 2 q sin f cos f + 2 Dlm
Dlm
sin q cos q cos f

800

b
c
+ 2 Dlm
sin q cos q sin f }.
Dlm

750
700
650

M 2ae,DM =

273

0

30

60

90
θ , φ (°)

120

150

180

Fig. 3. Line width anisotropy of the X-band ESR absorption lines for the SrCu2(BO3)3 single crystal in the crystal ac-plane (full symbols) at 295 K (circles) and 525 K (diamonds) and in the abplane at 295 K (open circles). Experimental data is fitted to equation of the form A ! Bcos2q (solidlines).

(7)

Entering only the interdimer DM interaction D " (0,0,D) into Eq. (7) yields
the angular dependence of the second moment of the form M2DM R 1 ! cos2q.
However, the angular dependence of the observed line width at 295 and 525 K
is significantly different from this prediction (Fig. 3). q-dependence can be welldescribed by an equation of the form A ! B(1 ! cos2q) with A and B contributions of the same order of magnitude. The parameters thus obtained are A "478 ) 5 G, B " 218 ) 5 G for room temperature and A " 541 ) 5 G, B " 220 ) 5
G for the higher temperature. The angular-dependent part B is temperature inde-

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X-Band ESR of the SrCu2(BO3)2 System275

pendent and coincides very well with the order-of-magnitude prediction of 460
G at q " 0°, which should be asymptotically approached when going to infinite
temperature. However, unexpectedly, there is an additional term A, which is of
the same amplitude as the B term. In contrast to B, A is T-dependent. The magnetic anisotropy introducing such contribution to the line width should be quite
strong as the amplitudes of A and B terms are virtually the same. The symmetae
can be excluded for at least two rearic-anisotropy-interaction contribution DBppsons. The first one is pointed out by the above order-of-magnitude calculationand the second one is due to the characteristic angular dependence of this interaction [19]. Namely, symmetric anisotropic exchange interaction can be presented
by a traceless tensor whose principal axes coincide with those of the g-tensor.
As the SrCu2(BO3)2 system is axially symmetric to a very good approximation,with the anisotropy axis pointing in the crystal c-direction, the contribution of
Hae to the line width is expected to vary with the orientation of external magnetic field according to M2ae R 1 ! cos2q. This is the same angular dependence
as provided by DM interdimer interaction with DM vectors perpendicular to theab-plane and as such cannot provide an adequate explanation of the additional
A term observed in the experiment.
We propose that the origin of the experimental angular dependence of theline width lies in additional magnetic-anisotropy terms of the DM type. Although,in the first approximation, treating the crystal ab-plane as strictly planar, DM
vectors are obliged to point into the c-direction and no intradimer interaction isallowed due to symmetry reasons, one can loosen these constraints if a buckling of the ab-planes is taken into account [20]. Bending of neighboring CuO4
plaquettes forming dimers suppresses inversion symmetry within dimers thus
allowing finite intradimer DM interaction D^. In addition, also a finite in-planecomponent of interdimer interactions Dab is endorsed. In fact, it has been recently
reported that the latter interaction is responsible for the observed anomalies inhigh-resolution inelastic neutron scattering experiments [21]. A sizable in-planeinterdimer component of the order of 30% of D was suggested. The ESR line
width is proportional to the square of the amplitude of DM vector, which would
imply that the corrections of the Dab term amount in a correction of only about
10% of the contribution to the line width due to D interaction. However, theobserved effect in our experiment is much larger.
The behavior of the line width in the ab-plane is less exciting as there isvirtually no f-dependence (Fig. 3). The line widths seem to be randomly scattered as a function of the azimuthal angle with the amplitude of 5 G around thevalue of 686 G. Such behavior corresponds to the addition of intradimer DMinteraction since in this case the DM vectors are forced to lie in the ab-plane ina direction perpendicular to dimers due to symmetry; there is a mirror plane including a dimer and a mirror plane perpendicular to a dimer (see inset to Fig.
1). A detailed analysis of the effect of intradimer DM interaction on the ESR
line width [22]. Preliminary results suggest that this interaction is of about thesame magnitude as the well-excepted interdimer interaction perpendicular to theplane. However, it has to be stressed that in addition to the calculation of thesecond moment, a detailed analysis of the exchange modulation frequency weffis required since there is a high degree of frustration present in the system.
Furthermore, the effect of additional DM interaction on the fine structure of single
triplet excitations has to be considered carefully.
At the moment, the origin of the intradimer DM exchange interaction seems
to some extent unclear. The buckling of the ab-planes is rather small as the bending angle of CuO4 plaquettes amounts to 8° at room temperature [20]. This angle
continuously drops towards 0° approaching the structural phase transition at 395
K so that in the high-temperature phase only the D interdimer component isagain allowed in the static-lattice picture. However, as this interaction alone still
does not allow an observation of singlet-triplet transitions in high-field ESR experiments, a DM interaction of a dynamical origin was proposed [23]. In thispicture, lattice vibrations instantaneously break local symmetry allowing for the
DM anisotropy. It seems that our ESR results would favor the dynamical mechanism, as there is only a small difference in the angular dependence of the line
width at 295 and 525 K, i.e., at temperatures well below and well above thestructural phase transition, respectively.
The effect of the buckling of the ab-planes on the anisotropy of the g-factor
within the planes has also been considered. Since dimers are tilted by half thebending angle with respect to the c-axis (4° at room temperature), one can perform a rotation of the g-tensor of an individual dimer, whose principal axes aretilted due to buckling, into the crystal frame. This transformation yields two different components of the in-plane g-factor when the in-plane external magnetic
field is applied, i.e., ga^ " (g2^cos2q ! g2sin2q)1/2 for the case when the magnetic
field is parallel to the dimer and gb^ " g^ for the magnetic field perpendicular
to the dimer, where g^ and g|| are the g-factor components in the eigenframe of
the g-tensor. However, due to rather the small corrugation of the ab-planes, theeffect is unobservably small as ga^ # g^ < 0.001. This finding is consistent withalmost no f-dependence of the g-factor shown in Fig. 2.
3.4 Temperature Dependence of Line Widths
Finally, let us comment on the temperature dependence of the observed line width
anisotropy (Fig. 3). The increase of the line width at 525 K with respect to room
temperature is reverse to the behavior below room temperature, previously reported for a powdered sample [11]. However, a similar increase was also observed in a powdered sample when going beyond room temperature. The increaseof the line width between 295 and 525 K in q " 0° direction is virtually thesame as in the case when the external field is applied in q " 90° direction. Bothdirections yield 65 ) 5 G broader lines at 525 than at 295 K. Trying to give an
explanation for such angular-independent behavior, or the observed line broadening in the first place, one can immediately exclude spin diffusion as the origin of the increase since the line shape remains perfectly Lorentzian even at
525 K. The next step would be the incorporation of static spin correlations, which

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X-Band ESR of the SrCu2(BO3)2 System

can play an important role in the temperature evolution of the line width in lowdimensional systems and can be sometimes seen in a rather broad temperature
range between T 8 J and T ; 10J [14]. Soos et al. [14] showed that including
static spin correlations into an ordinary nn square lattice leads to the temperature-dependent second moment

4 Conclusions

M 2 ( J , T ) = ( M 2S F S ( J , T ) + M 2A F A ( J , T ))cC,
c (T )(9)

where indexes S and A stand for spin symmetric and spin antisymmetric contributions, respectively. The infinite-temperature second moments M2S,A are multiplied by functions describing temperature evolutions of static spin correlations
FS,A(J,T) and by the ratio between Curie-law susceptibility cC and the observed
one c(T). If the increase of the line width is due to static spin correlations, onewill expect to observe an angular-dependent increase as the angular dependence
is contained in the infinite-temperature second moments and is thus simply multiplied by corrections originating from static spin correlations. The calculationof M2(J,T) in the SrCu2(BO3)2 system is more complicated due to the form of
its Hamiltonian and a simple factorization into high-T second moment and a Tdependent function is not possible. However, one still expects to detect an angular-dependent increase, which is not experimentally observed [22]. Excluding
also short-range order effects consequently indicates temperature-independent Linewidths with the assumption of the pure spin Hamiltonian determining the spin
dynamics.
Such temperature-independent behavior is often observed in magnetic crystals with significant exchange narrowing. It is then plausible that spin-phonon
coupling has to be taken into consideration in the SrCu2(BO3)2 system in orderto account for the increase in the line width with the temperature. In fact, normal modes can modulate anisotropic spin interactions via spin-phonon interaction, which then induces transitions between energy levels of the system. Theline width, regulated by the finite life-time of a spin in a particular energy level,
is in this scenario expected to rise with temperature due to the increasing number of phonons. However, the A-term cannot entirely be of the dynamical origin
as one might expect. Its increase between 295 and 525 K is too small with respect to the increase of the convex function describing the number of phonons
that characterizes the direct phonon process [17, p. 112] of relaxation. Higherorder (Raman) phonon processes are expected to yield even faster temperaturedependent broadening of the line width with increasing temperature. Therefore
we propose that the temperature-dependent part of the A-term is due to spinphonon coupling, while its major (constant) part is due to magnetic anisotropyof DM origin. Spin-lattice coupling has indeed been reported to play a crucialrole in a dramatic softening of the elastic constants in SrCu2(BO3)2 [24]. It wassuggested to be very strong and to be taken into consideration in any future
model considering this particular system.

277

In summary, angular dependence of X-band ESR spectra of a SrCu2(BO3)2 single
crystal have been presented at two different temperatures. The magnetic anisotropy governing the spin dynamics has been shown to be quite substantial as theobserved line widths are very broad in spite of the presence of strong isotropic
exchange coupling. Among the possible anisotropy mechanisms the antisymmetric anisotropic DM interaction was determined as the only one capable of producing sufficiently large line widths. However, previously proposed interdimer
DM exchange coupling, with DM vectors perpendicular to the crystal ab-plane,
does not suffice for an adequate description of the experimentally observed angular dependence. In fact, sizable intradimer DM interaction has to be employed
to provide a satisfactory explanation. Its origin, though, still remains unclear. It
can be due to dynamical effects and/or is a fingerprint of buckling of the crystal ab-planes. In addition, the angular-independent increase of the line width withincreasing temperature above room temperature suggests the presence of strong
spin-phonon coupling in the system.
Acknowledgements
We thank the General Secretariat for Science & Technology (Greece) for theprovision of financial support through a Greece-Slovenia Joint Research & Technology Program. A.L. and H.K. thank the Yamada Science Foundation (Japan)
for supporting their work in low-dimensional materials.
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Authors address: Andrej Zorko, Joef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
E-mail: andrej.zorko@ijs.si