Physica B 326 (2003) 446–449

Spin dynamics in the two-dimensional spin
system SrCu2ðBO3 Þ2
A. Fukayaa,*, Y. Fudamotoa,1, I.M. Gata, T. Itoa,2, M.I. Larkina,A.T. Savicia, Y.J. Uemuraa, P.P. Kyriakoub, G.M. Lukeb, M.T. Roversb,H. Kageyamac, Y. Uedaca

Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027, USA
Department of Physics and Astronomy, McMaster University, Hamilton, Ont., L8S 4M1, Canadac
Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, JapanbAbstract
Muon spin relaxation measurements have been performed in the two dimensional spin system SrCu2 ðBO3 Þ2 : As thetemperature decreases, the relaxation rate increases below T ¼ 10 K; and shows a saturation below T ¼ 3 K: The line
shape at low temperatures is nearly Gaussian, yet it cannot be easily decoupled by an application of longitudinal ﬁeld.
These results indicate the existence of magnetically active spins that show slowing down of ﬂuctuations with decreasing
temperatures, yet remain dynamic even at 0:1 K: We have also examined a sample doped with small amount of nonmagnetic Zn2þ impurity, SrðCu0:99 Zn0:01 Þ2 ðBO3 Þ2 : The result was quite similar to the pure sample. Thus, anomalousbehavior of the pure sample observed in low temperatures is not due to magnetic moments induced by impurities or alattice defect.
r 2002 Elsevier Science B.V. All rights reserved.
Keywords: Spin singlet ground state; Two dimensional system; Dimer; SrCu2 ðBO3 Þ2

1. Introduction
Two dimensional (2D) quantum spin systems
are quite interesting, not only from the viewpointof magnetism but also of superconductivity. In 2D*Corresponding author. Present address: Institute for Chemical Research, Kyoto University, Gokashou, Uji 611-0011,
Japan.
1
Present address: Superconductivity Research Laboratory,
ISTEC 1-10-13 Shinonome, Koto-ku, Tokyo 135-0062, Japan.
2
Present address: Correlated Electron Research Center(CERC), National Institute of Advanced Industrial Scienceand Technology, AIST Tsukuba Central 4, Tsukuba, Ibaraki,305-8562 Japan.

Heisenberg spin systems, long range order isestablished only at zero temperature [1,2]. In mostreal materials with layered magnetic structure,
three dimensional (3D) long range order is formed
at low temperatures due to small interactions
between layers. Only a few 2D materials with asinglet ground state have been found. SrCu2 ðBO3 Þ2provides a new type of 2D spin-gap system andshow quite unique behavior [3].
SrCu2 ðBO3 Þ2 is a layered material with a 2DCuBO3 plane. The arrangement of Cu2þ spins inthat plane is topologically equivalent to theShastry–Sutherland model [4,5], which is a 2D
Heisenberg square lattice with alternating diagonal

0921-4526/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 6 5 9 - 9

A. Fukaya et al. / Physica B 326 (2003) 446–449

20 K
10 K
5 K
4 K
3 K
0.1 K

Zero field

1.0
0.8

Asymmetryinteractions. The susceptibility of SrCu2 ðBO3 Þ2shows a maximum at B15 K and decreases
toward zero below B15 K; which suggests a
non-magnetic singlet ground state [3]. The spin
gap is evaluated to be B34 K [3,6]. Theoreticaland experimental studies have been extensively
performed, and it is considered that the groundstate is an exact singlet dimer state though thespins are strongly connected in the 2D plane.
In order to study spin dynamics of SrCu2 ðBO3 Þ2 ;we performed muon spin relaxation ðmSRÞ measurements. We also examined a sample doped withsmall amount of non-magnetic Zn2þ impurity,SrðCu0:99 Zn0:01 Þ2 ðBO3 Þ2 :447

0.6
0.4
0.2
0.0
012

3

4

56t(µs)Fig. 1. Temperature dependence of time spectra in zero ﬁeld.
The solid lines are guides for the eyes.

2. Experimental
2.0
SrC u2(BO3)2Sr(C u0.99Zn0.01)2(BO3)2
1.5

1/T1/2 (µs-1)

We used powder samples of pure SrCu2 ðBO3 Þ2and a sample doped with Zn, SrðCu0:99 Zn0:01 Þ2
ðBO3 Þ2 ; for mSR measurements. The measurements
were carried out using a surface muon beam at
TRIUMF in Canada. We used a He-ﬂow type
cryostat and a dilution refrigerator. Time spectra
were measured down to 100 mK:

1.00.5

3. Results and discussion
We show the temperature dependence of thetime spectrum in zero ﬁeld (ZF) in Fig. 1. As thetemperature decreases below B10 K; the relaxation becomes fast. Below B3 K; the time spectrum
is almost independent of temperature.
The shape of the time spectra and the decoupling behavior are quite anomalous. The details
are described later. At present, we have not founda reasonable function explaining the relaxation inboth of ZF and longitudinal ﬁelds. Thus, we deﬁnea half-width of the half-maximum (HWHM) of thetime spectra, T1=2 ; whose inverse reﬂects therelaxation rate. We plotted 1=T1=2 as a functionof temperature in Fig. 2. As the temperature
decreases, 1=T1=2 starts to decrease at B10 K: It
shows an anomaly at B3 K and then remains
nearly constant down to 0:1 K:
Thus, we see the existence of some magnetically
active spins in a nominally pure SrCu2 ðBO3 Þ2 at

0.0051015

2025T(K)Fig. 2. Temperature dependence of 1=T1=2 in zero ﬁeld.

low temperatures. This enhancement of relaxation
rate shows the slowing down of ﬂuctuations ofelectronic spins. Since the ground state is considered to be non-magnetic, the muon spin should
not relax, at least, at low temperatures. Since themSR method is quite sensitive to small magnetic
moments, mSR might detect intrinsic magnetic
properties of samples which cannot be detected byother methods. Impurity and probe effects are alsosuggested as a possibility of origin of anomalousenhancement [7,8]. The origin of the anomalous
spin freezing has not yet been clariﬁed.
To study whether a small amount of impurity
affects the behavior, we examined a sample dopedA. Fukaya et al. / Physica B 326 (2003) 446–449

ZF
50G
500G
2kG
1kG

1.00.8

Asymmetrywith 1% Zn2þ : The susceptibility is almost
identical to that in the pure sample, though theincrease of the susceptibility at low temperatures isslightly larger than that in the pure sample. Thissuggests that the ground state is still a spin singletstate, but there are quasi-free spins induced near
the doped Zn2þ ions. The results of the mSR
measurements were almost the same as for thepure sample. We have plotted 1=T1=2 of the timespectra for the doped sample in Fig. 2. This result
suggests that the origin of the enhancement of therelaxation rate observed in the pure sample is notdue to Cu2þ moments induced by impurities orlattice defects.
The temperature dependence of 1=T1=2 in Fig. 2
shows an anomaly at TB3 K: It seems to suggest
some phase transition or spin freezing. However,the behavior is quite different from generally
observed phase transition or spin freezing. In
general phase transitions or spin freezing, the time
spectrum relaxes exponentially above transition orfreezing temperatures. In SrCu2 ðBO3 Þ2 ; the shapeof the time spectrum at early times in ZF is
Gaussian-like over the whole temperature region
examined. Gaussian-like line shapes generally
result from quasi-static local ﬁelds.
The most noticeable point is the decoupling
behavior by a longitudinal ﬁeld (LF). We show theLF dependence of time spectra measured at 0:1 Kin Fig. 3. The long time tail shows only slight
relaxation and shifts upward (decouples) withincreasing LF. The time spectrum at early time isindependent of LF. All of these facts are characteristic of decoupling behavior of quasi-static
local ﬁelds. Assuming that the local ﬁeld is staticand the distribution of the magnitude is Gaussian(not Maxwellian), the value estimated from thetime spectrum at to1 ms is B30 G: This value ofthe static ﬁeld should be decoupled by a small LF.
However, even in a LF of 2 kG; the time spectrum
shows relaxation. Thus, the value of the local ﬁeld
estimated from the LF dependence is much larger
than the one estimated from the shape of the time
spectra at early times, it is an ‘undecouplable
Gaussian’. This tendency is observed in the timespectra even at TB4 K:
Such anomalous ‘undecouplable Gaussian’ relaxation is observed in several systems with

0.6
0.4
0.20.0(a)
ZF
50G
500G
1kG
2kG

500G 1kG 2kG

1.0
0.8

Asymmetry

448

0.6
0.4
0.20(b)

50G

ZF

0.0
12

3

4

56t(µs)Fig. 3. Longitudinal ﬁeld dependence of time spectra measured
at T ¼ 0:1 K: (a) The solid lines are the result of ﬁtting byEq. (3) with fLF ¼ 0:034: (b) The solid lines are the results ofﬁtting by Eq. (3) without multiplying HLF by fLF ðfLF ¼ 1Þ:geometrically-frustrated exchange interactions,such as kagom!e-lattice SrCr8 Ga4 O19 [9], the heavy
fermion compound CePt2 Sn2 [10], and the 1D spin
system Y2 BaNiO5 with charge doping [11]. The
details of this line shape have not yet been
understood. In these materials, the decoupling
behaves as if the efﬁciency of the applied ﬁeld weremultiplied by a numerical factor fLF ð0ofLF o1Þ;
which is independent of the applied ﬁeld [10].
We analyzed the data by a similar method toRef. [10]. In SrCu2 ðBO3 Þ2 ; we assumed that thedistribution of the magnitudes of the static local
ﬁelds Hs is Gaussian,pﬃﬃﬃ

2H2PðHs Þ ¼ pﬃﬃﬃ exp  s2 ;
ð1Þ
2sps

A. Fukaya et al. / Physica B 326 (2003) 446–449

where s is the width of the Gaussian distribution.
Thisdistribution
is different from PðHx;y;z Þ ¼
pﬃﬃﬃﬃﬃ
ﬃ2
ð1= 2psÞ expðHx;y;z=2s2 Þ; which is assumed inthe Kubo-Toyabe function [12]. The time spectrum of powder samples with static local ﬁelds isrepresented byGstat ðt; HLF Þ ¼12Z pZ Ndy0449

ﬂuctuating ﬁeld or ﬂuctuating time is assumed, theGaussian-like line shape cannot be reproduced.
One of the anomalous points of the experimental
results in SrCu2 ðBO3 Þ2 is that the relaxation rate
remains constant below B3 K down to 0:1 K:
These results may suggest that the ﬂuctuations aredHPðHs Þ0ðHs cos y þ HLF Þ2 þ Hs2 sin2 y cosðgmqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 tÞHs2 þ 2Hs cos yHLF þ HLF

2 þ 2H H
Hs2 þ HLFs LF cos y

We approximate the relaxation function by aproduct of a static function Gstat ðt; HLF Þ and adynamical function expðlðHLF ÞtÞ: Since decoupling behaves as if fLF HLF is applied instead of
HLF ; the ﬁtting function isGz ðt; fLF HLF Þ ¼ expðlðHLF ÞtÞ Gstat ðt; fLF HLF Þ:

ð3Þ

The solid lines in Fig. 3(a) are the results of ﬁtting.
The value of s was common in all LF. From theﬁtting, we obtained s ¼ 28 G and fLF ¼ 0:034; andthe LF dependence is successfully reproduced. Forcomparison, we also ﬁt the time spectra by Eq. (3)without multiplying HLF by fLF ðfLF ¼ 1Þ: The
decoupling behavior cannot be reproduced at all
as shown in Fig. 3(b).
At present, we consider that the ‘undecouplable
Gaussian’ results from dynamically ﬂuctuatingﬁelds. If the ﬂuctuations of SrCu2 ðBO3 Þ2 are
general thermal ﬂuctuations that result in spins
ﬂuctuating randomly in time, the shape of the time
spectrum should be exponential-like. Even if someconventional distribution of the magnitude of theð2Þ

not general thermal ﬂuctuations but rather have aquantum origin.
Acknowledgements
One of the authors (A.F.) was supported by
JPSJ Postdoctoral Fellowships for Research
Abroad. Work at Columbia was supported by
National Science Foundation, DMR 01-02752 andDMR 98-02000.
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