VOLUME 85, NUMBER 12

PHYSICAL REVIEW LETTERS

18 SEPTEMBER 2000

Collective Singlet Excitations and Evolution of Raman Spectral Weights
in the 2D Spin Dimer Compound SrCu2 共 BO3 兲 2
P. Lemmens, M. Grove, M. Fischer, and G. Güntherodt
2. Physikalisches Institut, RWTH Aachen, 52056 Aachen, Germany

Valeri N. Kotov
Department of Physics, University of Florida, Gainesville, Florida 32611-8440

H. Kageyama, K. Onizuka, and Y. Ueda
Institute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Tokyo 106-8666, Japan(Received 7 March 2000)
Raman light scattering of the two-dimensional quantum spin system SrCu2 共BO3 兲2 shows a rich structure in the magnetic excitation spectrum, including several well-defined bound state modes at low temperature, and a scattering continuum and quasielastic light scattering contributions at high temperature.
The key to the understanding of the unique features of SrCu2 共BO3 兲2 is the presence of strong interactions between well-localized triplet excitations in the network of orthogonal spin dimers realized in thiscompound.
PACS numbers: 75.40.Gb, 75.10.Jm, 75.50.Ee, 78.30. – j

Low-dimensional quantum spin systems with a quantum disordered ground state and a finite spin gap form asubject which is of both fundamental and applied interest
to the physics community. This is due to the fascinating
and diverse physics of the spin-liquid state, allowing one
to address issues related to the nature of quasiparticle
excitations and the role of strong interaction effects. It is
known that a gapped singlet ground state is realized in one
dimension (1D) in dimerized or frustrated spin chains, andin even-leg spin ladders. Of even greater interest is theexistence of a gap in two-dimensional (2D) spin systems,
because of its potential relevance for the descriptionof high-temperature superconductivity. Unfortunately,
only very few 2D systems, such as CaV4 O9 , have been
investigated, in which the spin gap is relatively large
and not caused by anisotropies. Recently, the compound
SrCu2 共BO3 兲2 with a layered structure was identified as a
2D S 苷 1兾2 Heisenberg system with a unique exchange
topology [1,2] leading to an exact dimer ground state, thus
providing the opportunity to study the excitation spectrumof such a model quantum many-body system.
SrCu2 共BO3 兲2 has a tetragonal unit cell (D2d ) with Cu21
ions that carry a localized spin S 苷 1兾2. The spins form
dimers which consist of neighboring pairs of planar rectangular CuO4 plaquettes with a Cu-Cu distance of 2.9 Å.
The strength of the antiferromagnetic intradimer exchange
coupling is estimated to be J1 苷 100 K [3]. The spin
dimers are connected orthogonally by a triangular BO3
unit. The distance between next-nearest Cu neighbors is
5.1 Å, and the interdimer exchange J2 艐 68 K [3]. The
orthogonal arrangement of dimers thus represents a 2D
frustrated quantum spin system [1,3]. A sketch of the Cu
dimers is shown in the inset of Fig. 1(a). The ground statein this exchange topology depends critically on the ratio
J2 兾J1 [3–5]. For small J2 the system consists of isolated
0031-9007兾00兾85(12)兾2605(4)$15.00

dimers and the ground state is a product of singlets, while
for small J1 the model can be mapped on a 2D square lattice of spins and a Néel-ordered state is expected. It has
been shown that the critical ratio of coupling constants that
separates a gapful and a gapless state is J2 兾J1 苷 0.7 [3,4].
The experimental ratio of J2 兾J1 艐 0.68 is just below thisvalue, placing this material in the dimerized phase, close
to the Néel boundary. Thermodynamic measurements support the existence of a dimer ground state and a spin gap
of D 苷 34 K [1,6]. The magnetic specific heatand suscpx
ceptibility both show maxima (Tmax 苷 15 K, Tmax 苷 8 K)
and a rapid decrease toward lower temperatures with exponential tails at T ! 0.
We have investigated the magnetic excitation spectrumof single crystals of SrCu2 共BO3 兲2 by Raman scatteringexperiments. At low temperatures (T ø D) we observewell-defined low-energy singlet modes which can be interpreted as collective bound state excitations of two and threeelementary triplets. We demonstrate that the appearance ofthe collective modes reflects the strong triplet-triplet interactions present in the system. The spectral weight evolving
for intermediate (T 艐 D) and high (T . D) temperatures
can be described by the temperature dependence of themagnetic susceptibility and the magnetic specific heat.
The samples were investigated in quasibackscatteringgeometry with light polarizations in the ab plane of the
freshly cleaved crystal. The (a0 b 0 ) axes are rotated by 45±
within this plane. The experiments used the l 苷 488 nm
excitation line of an Ar1 ion laser and a laser fluence
below 20 W兾cm2 . The scattered light was analyzed using
a XY-Dilor Raman spectrometer and a back-illuminated
CCD detector. Measurements in a magnetic field were
performed in 90± scattering geometry. In the analysis of
our results the main emphasis has been on the role ofinteractions and their influence on the magnetic excitation© 2000 The American Physical Society

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VOLUME 85, NUMBER 12

PHYSICAL REVIEW LETTERS

200

J1
J2

(ab)
50K
295K

Intensity (arb. units)

100

7K

a)

4K
1.5K

0
(a' b'+c)
T = 2.1 K

200(a’b’)

0T
6T

1.5K
100

50K

20
40
60 -1 80
Raman shift (cm )

4K

b)

295K
0

20

40

60

80

100

-1

Raman shift (cm )FIG. 1. Raman spectra of SrCu2 共BO3 兲2 in two scattering geometries. The upper inset shows Cu dimers of 1兾4 of the unit
cell formed by J1 (full line) and J2 (dashed line). The respective polarization (arrows) of incident and scattered light is given
with respect to the crystallographic axes. The lower inset shows
spectra for B 苷 0 (full line) and 6 T (dotted line) in (a0 b 0 1 c)polarization. An additional Ag phonon is marked by an arrow.

spectrum. A detailed analysis of the phonon spectrum will
be given elsewhere.
In the low-energy region with Raman shifts comparable
to the triplet gap D 苷 24 cm21 , drastic changes and a large
shift of spectral weight occur with decreasing temperature.
This evolution leads finally to the appearance of several
new modes. In Fig. 1 Raman spectra of SrCu2 共BO3 兲2 areshown to illustrate these effects at different temperaturesand in two scattering configurations with light polarizations within the ab plane of the crystal. The spectra in
the upper panel 1(a) present data in 共ab兲:B2 and the lower
panel 1(b) shows data in 共a0 b 0 兲:B1 scattering configuration. Symmetry components are denoted with respect tothe D2d point group using b 0 苷 a 1 b.
At low temperatures (T ø D) four well-defined modes
with energies E S 苷 30, 46, 56, and 70 cm21 appear. The
dominant intensity of these modes is observed in the 共a0 b 0 兲
scattering configuration. They neither split nor shift in
an applied magnetic field up to 6 T as shown in the inset of Fig. 1(b) and therefore are assigned to spin singlets.
The only effect of the magnetic field is observed as a shiftof a weak intensity shoulder at 24 cm21 toward lower
frequencies. This signal corresponds to the elementaryspin gap. At higher temperatures (T 艐 D) all modes are
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18 SEPTEMBER 2000

strongly damped. They are replaced by a continuum of
scattering with a center of gravity near 50 cm21 [Fig. 1(a),T 苷 7 K], corresponding roughly to the energy 2D. For
even higher temperatures (T . D) quasielastic scatteringwith a Lorentzian spectral function is detected. The lattertwo scattering intensities are observed in the 共ab兲 scattering configuration.
To understand these dramatic changes of the Raman
spectra with temperature we compare them with observations related to the triplet excitation spectrum and then map
the temperature dependence of the scattering intensity onto
the corresponding thermodynamic data, i.e., the magnetic
susceptibility and the magnetic part of the specific heat. To
gain a deeper insight into the structure of the spectrum wewill also identify the interactions leading to the formationof singlet bound states in the relevant Heisenberg model at
T 苷 0, and present estimates for their binding energies.
Recent ESR [7] and neutron scattering investigations [8]on SrCu2 共BO3 兲2 at low temperatures observed triplet excitations with a spin gap of D 苷 34 K. This triplet branch
has a very small dispersion of only 2 K pointing to extremely localized excitations [4]. In addition, a second
triplet branch D0 苷 55 K with a larger dispersion of 17 K
was observed. This branch can be interpreted as a triplet
bound state of two elementary triplets (see discussion below). Frustration due to the interdimer coupling J2 canlead to the reduction of the ratio D0 兾D 苷 1.62 below 2(corresponding to noninteracting magnons) [8].
The four modes that we observe in Raman scatteringin SrCu2 共BO3 兲2 for T ø D are clearly related in energy
and temperature scale to the spin gap of the compound. A
phonon origin of these modes can be ruled out due to theirtemperature dependence and the nonexistence of a structural phase transition below room temperature. On theother hand, in low-dimensional spin systems with strong
triplet-triplet interactions, well-defined modes can appear
below the scattering continuum [9–11]. In this case thelight scattering process is better described as spin conserving (DS 苷 0) scattering on singlet bound states. Suchstates composed of two elementary triplets have been observed, e.g., in the low-temperature dimerized phases of
CuGeO3 and NaV2 O5 [12–14].
To illustrate the mechanism of bound state formation
at T 苷 0 in the 2D Heisenberg model H 苷 Si,j Jij Si ? Sj
with Jij . 0 defined in the inset of Fig. 1(a) (the ShastrySutherland model [15]), we have derived the effective
ybosonic representation in terms of triplets (ti ), excited
above the singlets, formed on the stronger (J1 ) bonds:
X y
X y
y
共ti 3 ti 兲 ? 共tj 3 tj 兲
ti ? ti 1 V
H 苷 J1
i

1

X

具i,j典
y
y
y
兵iG共ti 3 ti 兲 ? 共ti1x̂ 2 ti2x̂ 1 H.c.兲其

i[A

1

X

i[B

兵x̂ ! ŷ其 ,(1)where V 苷 2G 苷 2J2 兾2. The site indices run overthe square lattice formed by the dimers (singlet pairs of

VOLUME 85, NUMBER 12

PHYSICAL REVIEW LETTERS

spins connected by J1 bonds), and 具i, j典 stands for nearest
neighbors, while sublattice A共B兲 is formed by the vertical(horizontal) dimers. An important feature of Eq. (1) isyy
the absence of quantum fluctuations (i.e., ti ? tj terms),
reflecting the fact that the ground state is an exact productof singlets. The triplet excitations would have been completely localized in the absence of the G term in Eq. (1),and hopping appears only in order 共G兾J1 兲6 ⬃ 共J2 兾J1 兲6 ,leading to a small bandwidth compared to the gap [3,4].
Next we present estimates for the energies of the collective, two-particle excitations. The zero-momentum twomagnon bound state with S 苷 0, constructed by excitingPy
S y
two triplets, has the form jC S 典 苷 i,j ci,jti ? tj j0典.
Starting from the limit J2 兾J1 ø 1, the dominant contribution to binding comes from the two-particle interaction[V term in Eq. (1)], with corrections of order 共J2 兾J1 兲2and higher. This interaction provides an effective attraction between two triplets. Assuming the triplets are
localized, the binding problem can be solved exactly, andthe singlet binding energy defined as e S 苷 2D 2 E2S ise S 苷 Z 2 J2 . Here E2S is the energy of the singlet boundstate, and we have estimated the renormalization factor Zby evaluating the appropriate lowest order diagrams to be
Z 艐 0.75 at J2 兾J1 苷 0.65. We have found, by comparing
neutron scattering data for the one-particle dispersion
from [8] with the dispersion obtained by high-order
perturbative expansions [4] that the ratio J2 兾J1 苷 0.65
is consistent with the experimental results. Indeed, forthis ratio the theoretical bandwidth [BW 苷 difference
between the energy at k 苷 共p, 0兲 and 共0, 0兲] and gap areBW 艐 0.04J1 , D 艐 0.36J1 [4], consistent with the measured values BW 艐 0.3 meV and D 艐 3 meV [8]. Fromthe same analysis we estimate J2 艐 61 K 艐 42 cm21 .
For the higher value of J2 兾J1 苷 0.678 the bandwidth isapproximately twice as large [4] which would make thisratio inconsistent with the neutron scattering data. Putting
everything together we have the estimate for the singlet
binding energy e S 苷 Z 2 J2 艐 24 cm21 . Our experimental value for the lowest singlet mode in Fig. 1(b) withS
E2S 苷 30 cm21 is eRaman艐 18 cm21 .
The analysis based on the localized picture somewhat
overestimates the binding since corrections of order
共J2 兾J1 兲2 and higher have been neglected. Such high-ordercorrections lead to the development of a strong dispersion
as well as create several two-particle singlet modes
with different energies, classified according to the group
D2d . Detailed calculations have recently been performed[16,17], providing quantitatively accurate results for theenergies in the different symmetry sectors. Only themodes with G3 共xy兲 symmetry contribute to the Raman
intensity, and, as T ! 0, their contribution grows for the(a0 b 0 ) and vanishes for the (ab) geometry [17], consistent
with our observations. In addition, the calculated energies[17] are in excellent agreement with the two lowest (30and 46 cm21 ) singlet modes observed in our experiment,
which we therefore interpret as two-particle bound states.

18 SEPTEMBER 2000

In SrCu2 共BO3 兲2 the four modes in Fig. 1(b) show ratios
E S 兾D 苷 1.25 2.9. Two of the well-pronounced peaks (at56 and 70 cm21 ) are above the two-particle threshold (2D).
The smallness of the one-particle bandwidth makes it possible to resolve such higher energy peaks, since the weightof the scattering background is expected to be small. The
pronounced sharpness of the peaks indicates that they canbe interpreted as three-particle excitations, whose existence is indeed possible due to the localized nature of
the states. A three-particle
singlet bound state can be
P
y
y
y
S
constructed as jF S 典 苷 i,j,k fi,j,k
共ti 3 tj 兲 ? tk j0典, andits energy on the same level of approximation as discussed above (localized triplets) is E3S 苷 3D 2 Z 3 J2 艐
54 cm21 , in very good agreement with the energy of themode at 56 cm21 . The appearance of additional levels(as well as finite dispersion) at higher order is also expected. The total Raman scattering intensity, reflecting thepresence of two- and three-particle singlets with spectral
weights I2 and I3 , is IB 共v兲 苷 I2 d共v 2 E2S 兲 1 I3 d共v 2
E3S 兲. In order to estimate the ratio of the two intensities wenotice that while light couples directly to the singlet twomagnon state, the three-magnon state has to be created via
the action of the G (magnon number nonconserving) termin Eq. (1) (i.e., this term provides a vertex correction to theyyRaman operator R ⬃ ti ? tj ). Consequently, we estimate
2
at T 苷 0, I3 兾I2 ~ 共J2 兾J1 兲 艐 0.4, in good agreement withthe experimental ratio of 0.32 (at T 苷 1.5 K). Notice thatin the hypothetical case G 苷 0, when the excitations arecompletely localized, the three-particle states do not contribute to the Raman intensity. Thus the G term plays animportant role providing both a finite bandwidth and coupling to higher bound state modes. Let us also mention
that three-particle bound states have been predicted theoretically in quasi-1D systems [11]. Our Letter presents thefirst experimental evidence for their existence in a (2D)
dimerized spin system.
The bound state modes experience strong damping withincreasing temperature [see Fig. 1(b)], due to scattering on
thermally excited triplet states. The decrease of the scattering intensity IB with temperature is governed by the density of excitations: IB 共T 兲 ~ 共1 2 Ae2D兾kB T 兲, where A is aconstant. In Fig. 2(a) the integrated intensity of the boundstates is shown together with a fit based on the equation
above (A 苷 215.7) and fixing D 苷 34 K from experimental data. The macroscopic temperature-induced populationof triplet states has a direct effect on the excitation spectrum. For intermediate temperatures [5–100 K, typical
spectrum shown at 7 K in Fig. 1(a)] a broad continuumof scattering is observed that is of magnetic origin [18].
However, in comparison with a 2D antiferromagnet with amaximum of the two-magnon continuum at Emax 苷 2.8 J[19] its energy is very small. The reason for the maximum
appearing at only 50 cm21 艐 2D is the smallness of the
one-particle bandwidth in SrCu2 共BO3 兲2 . The temperaturedependence of the two-magnon scattering intensity shows
a pronounced increase in the temperature range where theintensity of the bound states is dropping [see the open
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VOLUME 85, NUMBER 12

PHYSICAL REVIEW LETTERS

triangles in Fig. 2(b)]. At higher temperatures (not shownhere) it decreases again and is then superimposed on theintensity of quasielastic scattering. In Fig. 2(b) we compare the integrated scattering intensity with the behaviorof the magnetic susceptibility x共T 兲. Calculations of thelatter quantity exist, but their agreement with experimentis not fully satisfactory [1,4], and consequently we takex共T 兲 directly from experiment [1]. Notice that both thetwo-magnon continuum intensity and x共T 兲 exhibit strong
variations on the same temperature scale, even though weare not aware of a rigorous sum rule relating the twoquantities.
Quasielastic scattering connected with fluctuations of
the magnetic energy density is found at high temperatures (T . D) in our Raman scattering experiments. It
has the expected Lorentzian spectral function [20]. To determine the evolution as a function of temperature we use
the hydrodynamic form of the respective correlation function [21], which includes the magnetic specific heat cm 共T 兲and the thermal diffusion constant DT . In the high temperature approximation (h̄v兾kB T ø 1) the result is [20]k c T 2D k2
I共v兲 ~ h̄B v 2m1共DTTk 2 兲2 , where k is the scattering wave vector.
A fit to this equation can then be used to estimate cm 共T 兲from the integrated intensity, scaled by T 2 . In Fig. 2(c) the
result of this procedure is found in good agreement withthe measured specific heat.
1,2

a)

0,8

Intensity (arb. units)

0,4

IB(T)
bound state intensity

0,0

0,8

b)
0,4

χ(T)
continuum intensity

0,0

0,8

c)

0,4

cm(T)
2
quasielastic intensity I / T

0,0

2

4

6

8

10

12

14

16

Temperature (K)FIG. 2. Mapping of Raman scattering intensities (open symbols) on thermodynamic quantities (lines) as a function of temperature. The scattering intensities and the thermodynamic datafrom Refs. [1,6] are normalized to their maxima.

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18 SEPTEMBER 2000

In conclusion, we have shown that the low-energy excitation spectrum of the 2D compound SrCu2 共BO3 兲2 has arich and complex structure. Our main result is that at low
temperature (T ø D) the spectrum contains several wellpronounced singlet modes which are interpreted as collective two-particle and novel three-particle singlet boundstates of strongly localized triplets. We have demonstrated
that since quantum fluctuations are absent in the groundstate (due to the unique dimer arrangement), the triplettriplet interactions can lead to large binding energies andscattering intensities, and consequently make the collective modes observable. At intermediate (T ⬃ D) and high(T ¿ D) temperatures, where the quantum (interaction)effects are not important, the light scattering is dominatedby a two-magnon continuum and quasielastic contributions. We relate the spectral weight evolution in theseregimes to thermodynamic quantities.
We acknowledge fruitful discussions with C. Pinettes,
P. H. M. van Loosdrecht, G. S. Uhrig, C. Gros, R. Valentí, W. Brenig, W. Atkinson, P. Hirschfeld, and D. Tanner. Financial support by DFG/SFB 341 and NSF Grant
No. DMR-9357474 (V. N. K.) is gratefully acknowledged.

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