TECHNーCAL REP。RT ーggN。ggg伽 - kek

ISSN 00824798
Ser. A No. 3507
September 1999
O l
'--" __ I
Direct Evidence for the Localized Single-Triplet Excitations and
the Dispersive Multiple-Triplets Excitations in SrCu,(B03)2
Hiroshi Kageyama, Masakazu Nishi, Naofumi Aso, Kenzo Onizuka,
Tomoyuki Yoshihama, Katsuyuki Nukui, Katsuaki Kodama,
Kazuhisa Kakurai and Yutaka Ueda
Technical Report of ISSP is aperiodica11y pub1ished in the
fo11owing two Series:
Series A:This series contains preprints of papers intended
for publication in a journa1or proceedings.
Series B:This series contains va1uab1e research notes (1ec−
ture notes,technical or instrument訓 information)
㎜d numerica1tab1es etc.,which are not intended
to be published e1sewhere.They must not be repro−
duced without the permission of the author(s):
SrCu2(B03)2 crystallizes in a tetragonal structure and
consists of alternately stacked Sr- and CuB03-layers [1]. Tue
Direct Evidence for the Localized Single-
magnetrc Cu2. rons m the latter laycr orgamze an S=1/2
Triplet Excitations and the Dispersive
two-dimensional (2D) Iinkage of orthogonally arranged dimers.
Figure I (b) shows a fundamental unit of the orthogonal dimers
Multiple-Triplets Excitations in
oricnted either parallcl to [1. Ij or [1 . - I], indicating in-plane
S rC u2(B03)2
interactions which act within the dimers J and between the
dimers J '. Our prcvious expcri ments [2-71 togcther with scvcral
theoretical works [8- 10] on this cuprate has revcaled following
H. Kageyamal, *, M. Nishi2. N. As02, K.
nontrivial aspects, all originating from this peculiar spin
Onizukal, T. Yoshihama2, K. Nukui2. K.
Solvable Ground Stale SrCu2(B03)2 is idcntified as a
Kodama3 K. Kakurai2 and Y. Ucdal
2D spin gap system with an exact dimcr ground state, realizing
the Shastry-Sutherland model [8-10], Measurements such as
magnetic susceptibility [2,3] , magnetization curves [2] , specif ic
l Material Design and Characterizalion Laboratory,
heat [61, nuclear magnetic resonancc (NMR) [2, 7], elcctron
Institutefor Solid State Physics, University of Totyo,
spin resonancc (ESR) [41 and Paman scattering [5] have
Roppongi, Minato-ku, Tokyo 106-8666, Japan
confirmed the spin gapped nature of this material , consistently
giving rise to A =3.0 meV as an energy gap between the
2 Neutron Scattering Laboratory, Institutefor Solid State
exact dimer ground state and the lowest triplet excited state.
Physics, University of Tokyo, 106-1 Slliralata, Toh7i,
Furthennore, ESR and Raman scattering rcvealed highcr-cncrgy
lbaraki 3]9-1]06, Japan
magnetic excitations associated with two-triplets bound states
[4, 5].
3 Division of New Materials Science, Institute for Solid
Spin Frustration: It turned out that the isolated dimer
State P/1ysics,- University of Tolyo, Roppongi, Minato-ku,
model considering nothing but J completely failcd to reproducc
Tokyo I 06-8666, Japan
our experimental data of, e,g., magnetic susccptibility [2].
Therefore J', which brings a spin frustration into the system.
PACS numbers: 75.40.Cx, 75.40.Gb
no doubt plays a crucial role in achieving the spin-singlet
(a) k
Wc pcrfomlcd inclastic ncutron scattcring on thc 2D Shastry3
Suthcrland systcm Sl(]u2( B03)2 with an exact dimcr ground
state. 'l'hrce cncrgy lcvcls at around 3, 5 and 9 mcV wcre
¥' "Jb
observed at 1.7 K. The lowest excitation at 3.0 meV is
almost dispcrsionlcss with a banctwidth of 0.2 mcV, showing
a significant constraint on a single-triplet hopping owing to
thc orthogonality of thc dimcrs. In contrast, the correlated
o IA 2 D3
two-triplets excitations at 5 meV exhibit morc dispersive
Fig. l: (a) The reciprocal latticc (h, k, O) of the nuclear unit
cell. The Brillouin 7_one used in this study is given by the
broken lines. Along thc lines A-C and D-F, the dispersion
relations of band I, and bands 11 and HI, respectively, have
been studied. (b) The fundamental unit of the orthogonal
submitted to Phys. Rev. Lett.
dimers .
- I -
ground state. Indeed, the exchange intcractions wcrc dctcnnined
vector of Q=(2. O) . An abrupt growth in intcnsity below 2
as J= 100 K and J'=68 K [9]. The ratio of the exchange
meV is due to the elastic incoherent scattering from thc
eonstants J'/J=0.68 is, intercstingly cnough, just bclow thc
samplc. Thc I .7 K spectrum consists of tluec peaks centcred
critical boundary ((J '/J).=0.70 [9] or 0.69 1 [lO]) beyond which
at 3.0 meV, 5.0 meV and 9.7 mcV (transitions I, 11 and 111,
the system is expected to have a N6cl ordercd ground state.
respcctively), while that for 24 K no longer has any appreciablc
Localized Triplel Ercitalions: Mlyahara and Uedia
pcak, indicating that all excitations are of magnetic origin. As
theoretically showed that the wave ftmctions of thc triplct
alrcady confirmed by other measurements [2-7], transition I at
excitations are extremely localized [9]: Thc perturbation
A=3.0 meV is the excitation of a single triplet from thc
calculations of thc fifth order or less prohibit a single triplet
singlct ground state. As shown in Fig. 2(b), the pcak intcnsities
from propagating in the (a, b) plane. They argucd that this
of transitions 1-III start to decrcase with increasing temperature
localized character accounts for the crystalli7 ltion of thc triplets
and disappear at about 13 K. We can derive from this similar
observed at particular values of magnetization in high magnetic
tempcrature dependencc that transitions ll and 111 arise from
ficlds [2, 9].
multiplc-triplcts cxcitations, which will bc discusscd in dctails
In this Letter, we investigate the s pin dynarnical properties
l atcr .
of SrCu2Q303)2 by mcans of inelastic ncutron scattcring using
bulk single crystals . Almost dis persi onless magnctic excitations
at 3 meV were observcd, rcflecting thc cxtremcly localized
nature of the single-triplet hopping in the orthogonal dimer
system. Magnetic excitations at higher energies were also
investigated and discusscd i n tcnns of corrclatcd mul ti pl c-tri plcts
EI (')
E 500
1 400
Eo ('vv
Inelastic neutron scattering expcriments werc carried out
Oo 200
on ISSP-PONTA spectromcter installed at 5G bcam port of
¥tl' 100
the Japan Research Reactor 3M (JRR-3M) in Japan Atomic
I 11 Ill
Energy Research Institute, Tokai Establishment. The
o O
5 10
spectrometcr was opcrated in the unpolarizcd neutron mode
with pyrolytic graphite Q?G) monochromator and analyzer.
E (meV)
1 .5
The inelastic scans were porformed with fixed final energy
14.7 meV (kf2.67 A-1) and horizontal collimations of
(/'c I .O
open(40' )40' -sample-80'-80' . A PG filter was placed aftcr thc
sample to suppress the higher order contaminations . In ordcr
A e : E-3.15 meV
X x: AE-S.15
: E-10.65 meV
tL Al x
'N 0.5
to minimize the neutron absorption by the natural abundance
'e ¥
ee x
of roB, IIB-enriched (99.6%) bulk single crystals of
SrCu2(lIB03)2 wcre prcpared by the traveling solvent floating
zone mcthod using LilIB02 flux [1 1]. The ncutron scattering
a X*^A A
sample consisted of two single cryst als with a total volume
T (K)
of - 1.5 cm3 aligned witlrin 20' and it was orientcd with its
,_ 'L
1 5 20 25
a- and b-axes in the scattcring plane.
For convenience, we use a Brillouin zonc at (h, k. O)
Fig. 2: (a) Energy scans at Q=(2, O) obtaincd at T=1.7 K
plane as givcn in Fig. I , where we do not distinguish between
(circlcs) and 24 K (triangles). The peaks are labeled at the
thc dimers lying along [1, l] and [1, -ll・ In this casc, thc
bottom of thc figurc. The soiid curve is the fit to the data, as
space lattice vectors are transferred toa
a-b)/2 and b* a+b)/2,
describcd in the text. (b) The temperature variation of the
which correspond to the reciprocal space given by the lattice
normaliz d intensities atE=3 . 1 5 meV and Q ( I .5, 0,5) (circles),
vectors a* a -b and b
E=5.15 meV and Q (2, O) (crosses), and E=10.65 meV and
a +b . Shown in Fig. 2(a) are
typical energy scans obtained at I .7 K and 24 K for a scattering
Q=(2, O) (triangles). The broken line is a guide to the eye.
'nle Q dcpendcnce of transition I was incasured at I .7 K
along the lincs A, B and C, and that of transitions 11 and 111
along D. F. and F indicated by arrows in h ig. l(a). It was
found that, indepcndcnt of Q, the obtained profilc for tr,ansition
I at I .7 K is rcsolution limited, whereas the transitions 11 and
III show intrinsic line widths. Accordingly, the 1.7 K profile
was fitted to a combination of a delta function for transition I
and two damped hannonic oscillators for transitions 11 and 111,
the solid line in Fig. 2(a) includes the temperature independent
term of incoherent scattering around energy zero.
convoluted with the instrumental rcsolution. It is noted that
, ,
The dispersion relation of band I is shown in Fig. 3.
Most importantly, the excitation energies arc almost Q
indepcndent. Namely, the magnitudc of the dispersion, the
diff erencebctwecn themaximurn and minimum of the excitation
energy, is AE=0.2 meV. This width is significantly small in
contrast to conventional low-dimensional quantum spin
systems. Experimentally, strong dis persions of the singl e-tri plet
excitations wcre observed in the ID spin-Peierls material
(1 t,1 ,)
CuGc03 (AE=14 meV) along the chain ,axis [12], and in the
2D plaquettc system CaV409 (AE=7 meV) parallel to thc
Fig. 3: Q-depcndence of the excitation energies of bands I, H
plaquette plane [13]. On the contrary, well-isolated clusters of
and 111 obtained at I .7 K. The arrows represcnt the encrgy
exchange-coupled paramagnetic ions, whcre intercluster
resolutions of the instrument (hWIIM). The solid curves are
interactions are rarely important, evidently exhibit flat
guides to the eye. The bars represent the intrinsic line width
dispersions in the spin excitation spectrum. Tlris has been
(FWHM) of bands 11 and 111. The thcoretical dispersion curve
experimentally shown by the neutron scattering measurements
for the single-triplet excitations [10] is given by the broken
on the isolated dimer systems Cs3Cr2Br9 (AE=1.8 meV) [14]
and BaCuSi206 (AE=0.7 meV) [15], and the four-spin system
Cu2P04 (AE=3 meV) [161・ In SrCu2(B03)2' the physical
situation is completely different because the dimers within the
Recent NMR mcasurement by Kodama et al. [7] has also
(a, b) plane arc not isolatcd but strongly intcracting and
disclosed cvidencc to support the localization of thc single
moreovcr the system is located in thc vicinity of the N6cl
tri plet.
ordcred state. As thcoretlcally shown by Miyahara and Ueda
Let us take a closer look at the Q-dependencc of band I,
[9], thc key to the dispcrsionlcss band in SrClb(B03)2 Iies not
see Fig. 3. Thc disporsion curve reachcs a maximwn (2.90
in the spatial isolation of the spin clusters but in the
meV) at the so-called (7c, O), a second maximum (3.00 meV)
orthogonality of the neighboring dimers: They proved that a
at (1c/2, 1(:12), and minima (3.10 meV) at (O. O) and the
hopping of thc single triplct from one sitc to another within
equivalent point (1c, 7c). Using the scries expansion method
each planc is possible only from the sixth order in the
up to the fifteenth order, Weihong, Hamer and Oitinan obtained
perturbation calculations, Ieading to exceedingly weak
the single-triplet excitation spectrum [10]. They argued that
dispersions of band I. It is noted that the appearancc of the
the bandwidth is quite small as anticipatcd, but increases as
quantized plateaux in the magnetization curve [2, 9] and the
J'/J approaches (J'/J).. For a comparison, thc calculated
multiple magnctic resonanccs in ESR [4] indicatc the localized
dispersion curve is shown by the brokeil line in Fig. 3, where
character of multiple-triplcts cxcitations. Our neutron scattering
we assumed J=100 K. J'=68 K and A=3.0 meV. From the
study, however, provides for the first thne a direct proof of
qualitative point of view, the calculated Q dependence of the
the significant constraint on the single-triplet excitations.
cxcitation energies nicely reproduces our experimcut 1 data
Howcver, the calculated bandwidth of 0.7 meV is (relatively
discrcte energi es. We suppose the excitations are also delocalizcd
narrow but) much widcr than thc obscrved onc. To add to tlris,
owing to the correlation of the multiple triplets as in the case
their theory yields a slightly smaller value of A=2.0 meV at
of transition 11 though, at present, thc very wcak peak intensity
(O. O) and (1 . 7c). A possiblc explanation for the quantitative
and broad nature of transition 111 do not allow a reliablc
discrepancies is that thc perturbative approach bccomes no
discussion of the dispersion relation.
longer appropriate when we discuss thc phenomena of the
F・inally, wc discuss the Q dependcnce of the intensities
of excitation I. The intensities at T= I .7 K and E=3.0 meV
system near thc critical boundary (J '/J)..
The ncxt argument concems transition 11 which occurs
were mcasured over various Q-points. As typical examples,
at energy transfer of about 5 meV. This excitation has bccn
the data taken along three directions in the rcciprocal latticc
already observcd by ESR [5] and Raman scattering [6] (at 4.7
arc shown in Fig. 4. The observed periodic intensity modulation
meV), and thc magnetic field depcndencc of the ESR frequcncy
was compared with the dynamical structure factor which can
has identified this mode as the second triplct state [5]. As
be obtained by calculating the transition probability from the
discusscd in Rcfs. [51 and [6], transition 11 is undcrstood basod
singlct ground state to the lowcst triplct cxcited statc. For
on two triplets coupled by J,(>0): If J,=0, the corrcsponding
simplicity, Ict us considcr a non-interacting pair of orthogonal
transition occurs on]y at 2A=6.0 meV. But when J* is finitc,
dimers (J'=0) with the intradimer dist,ancc 2.905 A. Thcn, the
one expects a separation of the transition into tlucc energy
intcnsity at Q ll, k) is given by the superposed form, sin[a(h-
levels at E0=2A-2J, for the singlet state. El=2A-J, for the
k)]2+sin[a(h+k)]2 (a=0.717), where the Q dependence of thc
triplet state, and E2=2A+J, for the quintct statc. Using /"I=4'7
magnetic fonn factor is not includcd, bccausc it docs not
meV [5. 6] and Eo=-3'7 mcV [6]. J* is detcrntincd to be I . I -
affcct the rcsult considcrably in the limited Q-range studied.
l.3 mcV.
Includcd in Fig. 4 are thc theorctical curvcs, which achicve
Figure 3 shows that the Q dcpendence of band 11 is
close conforrnity with the experimental ones in spitc of the
qualitatively idcntical to that of band I. The striking difference
unrealistic assumphon about J'. This fact reflects thc spin
betwecn bands I and 11 is that the latter shows more dispersive
frustration bctween the dimers. A slight deviation bctween the
behavior w:ith a bandwidth of 1.5 meV. This observation may
theory and experiment may bc corrected by including the
indicate that the propagation of correlated two Uiplets is much
interdimer interactions.
casier than that of the single triplet. Very recently, preliminary
consideration of the two-triplets excitations by Miyahara and
Ueda indicates that two triplets sitting on the nearest neighbor
sites can propagate within the fourth-order perturbation
calculations [17J. Considering the instrumental energy
resolution, the full width at half maximurn (IWHM) of band
II for any Q points is obtained to be approximately 0.7 meV.
The finite width possibly indicates a spin continuum and/or a
short lifetime of the coupled two triplets. One of the remaining
problems regarding this mode is to detennine thc distance
between a particular pair of the triplets. This may be possible
if one analyses the Q and intensity variations of this mode,
1 OO
interpreted within the framework of correlated three triplets or
1 OO
the transition 111 is considerably broad (see Figs. 2(a) and 3).
Transition 111 wlrich appears at 8-12 meV may be
1 oo
more. Compared with the transition II. the observed profile of
Q-(h,h) (b)
which will be our future work.
which may result from a much wider spin continuum and/or a
much shorter lifetime of the bound state. Or one might think
Fig. 4: Observed and calculated Q iependcnce of the scattering
the case of several excited states lying at nearly degenerated
intensity for excitation I.
Mod,㎜d N.Nis阯,J.Phys.S㏄.Jpn.“,793(1997)。
Fei1e,‘㎜d Jl K.Kj’㎝1s,Phys.Rev.B,30.6300(1984)一
血e a1most dispersio血ess cu〃e fo“he firs“me in s血ong1y
ooπc1atω1−3D spin systems,ohgimting fmm伽e p㏄u1i町
○汕10gon刮dimcr1letwork mt from血e iso1adon or血e sp㎞
口5]Y.Sasago,K.Uc肚mk山ra,A.Zheludev,‘㎜d G.S㎞r三㎜e,
to the nrst approxim汕on by血e localized model o〔he
coπe1atωユwo trip1cts,but its exci胞don sp㏄血㎜n is more
dispersive血al1血atof血e3meV mωc.Likewise,血ehighest−
en町gy mode煎9meV wo血d㎞serrom血e ooπe1ated此㏄or
The au血ors胴帥tcr山o H.N句iH,P.レmn㎝s,M、
T11is work w㎜ by a Grmレin−Aid forEhoourag㎝1e1lt
Yo㎜1g Sciendsts from The Mi㎡s岬of Educadon,Scien㏄,
Spo111s∼md Cu1111肥.
*Elec血o皿ic address:kage@issp.u−tokyo.acjp
凹R,W.Smith md D.A,Kes加r,J.So1id State Chem.
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Hl㎜e,H.Miレ㎜㎜a,T.G伽o,Kl Yos肚mura,ε㎜d K.
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Motokawa,to aP畔町in J.Phys.Soc.Jpn.
Kageyama,K.Onizuka,md Y.Ueda,to be published。
Takagi,alld Y.Ucda,to app㎝r in Physi㎝B.
0此㎜ka,md Y.U地,㎜published.
エ8】B.S.Shas甘y md B.Su血eH㎜d,Physica108B,1069
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to apP㎝r in J.C町sωGmw血.
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