Abstract The magnetic properties of the two-dimensional dimer spin-gap system SrCu 2 (BO 3 ) 2 were investigated by the m + SR technique. The relatively slow fluctuations of spin-dimers slow down with decreasing temperature and an unusual spinfreezing process is unraveled at T f o3:75 K, well within the spin-gap temperature range (T SG E20 K). This quasi-static phase displays a Gaussian field distribution with a remarkable stability with applied longitudinal fields. In support of the criticality of the SrCu 2 (BO 3 ) 2 spin-gap ground state towards an antiferromagnetic transition, Knight-shift measurements suggest that implanted muons may liberate spin density at ToT SG that undergoes spin-freezing at very low temperatures. On the other hand, non-magnetic impurity-doping of the copper sublattice does not suppress the spin-gap ground state and does not lead to magnetic ordering effects of static nature. 

A Lappas

A Schenck

K Prassides

CiteSeerX

Physica B 326 (2003) 431–435
Spin-freezing in the two-dimensional spin-gap systems
SrCu2xMgx(BO3)2 (x=0, 0.04, 0.12)
A. Lappasa,*, A. Schenckb, K. Prassidesc
a Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, Heraklion 711 10, Greece
b Institute for Particle Physics, Swiss Federal Institute of Technology (ETH) Zurich, CH-5232 Villigen PSI, Switzerland
cSchool of Chemistry, Physics and Environmental Science, University of Sussex, Brighton BN1 9QJ, UK
Abstract
The magnetic properties of the two-dimensional dimer spin-gap system SrCu2(BO3)2 were investigated by the m
+SR
technique. The relatively slow fluctuations of spin-dimers slow down with decreasing temperature and an unusual spin-
freezing process is unraveled at Tfo3:75K, well within the spin-gap temperature range (TSGE20K). This quasi-static
phase displays a Gaussian field distribution with a remarkable stability with applied longitudinal fields. In support of
the criticality of the SrCu2(BO3)2 spin-gap ground state towards an antiferromagnetic transition, Knight-shift
measurements suggest that implanted muons may liberate spin density at ToTSG that undergoes spin-freezing at very
low temperatures. On the other hand, non-magnetic impurity-doping of the copper sublattice does not suppress the
spin-gap ground state and does not lead to magnetic ordering effects of static nature.
r 2002 Elsevier Science B.V. All rights reserved.
Keywords: Low-dimensional solids; Spin gap; Knight shift
Recent years have seen a great deal of research
exploring complicated quantum mechanical phe-
nomena associated with the mechanism of high-Tc
superconductivity in two-dimensional (2D) copper
oxides. Prominent features of these systems
include the presence of a pseudo-gap and its
evolution with doping [1]. Low-dimensional chem-
ical analogues of the cuprates with spin-gap
ground states offer opportunities to study proto-
typical systems with unconventional low-tempera-
ture behavior.
SrCu2(BO3)2 is a spin-gap (DE30K) system [2]
in which the Cu2+ ions form a 2D network of
rectangular CuO4 units with triangular BO3 group
connectivity [3]. Nearest-neighbor Cu2+ (S ¼ 1=2)
ions form dimers (dE2:90 (A), arranged orthogon-
ally to each other, while the sheets are separated by
non-magnetic Sr2+ ions. An illustration of the
tetragonal unit cell of the structure [3] is presented
in Fig. 1. The magnetic exchange pathways in
SrCu2(BO3)2 are topologically similar to the dimer
model of Shastry and Sutherland [4]. In this, a 2D
Heisenberg model allowing for nearest-neighbor
(NN; J) and next-nearest-neighbbor (NNN; J0)
magnetic exchange interactions was employed,
leading to the conclusion that the singlet dimer
state is an exact eigenstate of the spin Hamilto-
nian. At ToTSGE20K, the bulk magnetic sus-
ceptibility of SrCu2(BO3)2 (shown in the inset of
Fig. 4) shows a characteristic thermally activated
*Corresponding author. Tel.: +30-2810-391300; fax: +30-
2810-391305.
E-mail address: lappas@iesl.forth.gr (A. Lappas).
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 6 9 - 1
behavior consistent with the opening of an
excitation gap, D: In addition, high-temperature
susceptibility measurements [2,5] find that the NN
Cu2+ interactions are strongly antiferromagnetic,
JE 100K, while the NNN interactions are
sizeable (J 0E0:68J) and with important conse-
quences for the stability of the ground state.
Theory predicts [5] that the SrCu2(BO3)2 dimer
ground state is at the borderline of the transition
from disordered spin-gap to antiferromagnetically
(AF) ordered state with the quantum critical phase
transition expected to occur at ðJ 0=JÞcE0:7:
Bearing the above in mind, we employed the
m+SR technique to authenticate the nature of the
magnetic ground state in SrCu2xAx(BO3)2
(A=Mg2+; xp0:12), search for static magnetic
order, follow the T-dependence of small moment
spin fluctuations, investigate the spatial inhomo-
geneity of the ground state, find out if the spin-gap
is modified by non-magnetic impurity dopants
(Mg2+), and answer questions regarding possible
muon-induced break-up of the dimer spin-singlets.
m+SR measurements were carried out at the
Paul Scherrer Institute (PSI), Villigen, Switzer-
land. Datasets were collected in the zero-field (ZF),
longitudinal-field (LF ¼ 10mT to 0.4 T), and
transverse-field (TF ¼ 0:6T) variants of the tech-
nique. Polycrystalline samples were pressed into
pellets (+13mm) and mounted on a silver sample
holder which was then attached on the sample
stick of a continuous flow cryostat operating
between 1.7 and 300K.
Fig. 2 shows the ZF-m+SR spectrum of
SrCu1.96Mg0.04(BO3)2 at 2K which is representa-
tive of all samples studied. This is very unusual for
systems with disordered non-magnetic singlet
ground states for which the only means of m+
spin depolarization is through the disordered
nuclear moments [6]. In the present cases, the
low-temperature ZF-m+SR data were described
well by two-component depolarization functions,
incorporating a strongly relaxing Gaussian (s1)
and a slow exponential (l2) component.
1 The
following functional forms were employed in the
analysis of ZF time-spectra for different tempera-
ture ranges:
PmðtÞ ¼ A expð2ltÞ expð212s
2t2Þ 4:5pTo10 K;
ð1Þ
PmðtÞ ¼ A expð2ltÞ 3:75pTo4:5 K; ð2Þ
PmðtÞ ¼ A1 expð212s1t
2Þ þ A2 expð2l2tÞ To3:75 K:
ð3Þ
Fig. 3 compiles the temperature dependence of
the fitted parameters. We find that at high
Fig. 1. Layered arrangement of the Cu2+ ions in the tetragonal
unit cell of SrCu2(BO3)2.
0 2 4 6 8
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
Longitudinal Field Dependence
LF 0.40 T
LF 0.20 T
LF 0.12 T
LF 0.08 T
LF 0.05 T
ZF
A
sy
m
m
et
ry
Time (µs)
∆/γ
µ~ 29 G
ZF
T= 2 K
A
sy
m
m
et
ry
Time (µs)
Fig. 2. Zero field m+SR time spectrum of SrCu1.96Mg0.04(BO3)2
at 2K (oTSG). The line is the fit to Eq. (3). Inset: Longitudinal-
field m+SR decoupling experiment (T ¼ 2K) showing the
persisting character of the Gaussian field distribution.
1Similar behavior has been observed by A. Fukaya et al. (this
conference) on a different batch of samples.
A. Lappas et al. / Physica B 326 (2003) 431–435432
temperatures the Cu nuclei of SrCu2xMgx(BO3)2
dominate the depolarization (s ¼ 0:121ð6Þ ms1)
behavior (Eq. (1)), while relatively fast spin fluc-
tuations that are already present above 4.5K
appear to slow down (Eq. (2)) as we approach
the characteristic temperature of 3.75K from
above, with the exponential relaxation rates
(Eqs. (1)–(3)) increasing smoothly from
0.168(4) ms1 at 5K to 0.23(2) ms1 at 2.1K.
On the other hand, below 3.75K the Gaussian
depolarization component begins to grow
quickly at the expense of the exponential
one. The corresponding relaxation rate, s1 ap-
proaches saturation at the lowest temperatures
(B2.7(1) ms1) and gives rise to a sizeable field
width which varies little among the different
compositions, i.e. /DB2S1=2 ¼ 31x¼0; 29x=0.04,
and 26.5x=0.12G at 2K. The rapid growth of the
relaxation rate and the Gaussian field spread are
consistent with component #1 of Eq. (3) reflecting
a quasi-static volume fraction (A1) which di-
minishes with increasing Mg-content from 44%
in the parent compound to E26% for x ¼ 0:12:
Component #1 cannot arise from paramagnetic
S=1/2 impurities present in the samples, as
magnetic susceptibility measurements put an
upper limit of 0.1% to such impurities. By fitting
a power-law expression to the T-dependence of the
Gaussian (Eq. (3)) depolarization rate, s1 ¼
s0½1 ðT=Tf Þb; a freezing temperature, Tf ¼
3:75ð2ÞK (bE0:2220:3) for the electronic mag-
netic moments can be extracted. In order to
explore further the nature of the Gaussian
component of Eq. (3), we performed additional
LF-m+SR experiments at ToTf : The time-depen-
dence of the muon spin depolarization at applied
LFs is similar for all compositions. A good
description of the LF spectra at To3:75K
was achieved with Eq. (3), while at higher-Ts
Eq. (2) was more appropriate. The inset of Fig. 2
shows a typical decoupling experiment. The
exponential component (A2) represents the domi-
nant volume fraction in the presence of an applied
LF and the corresponding relaxation rates appear
to diverge when approaching Tf : The maximum
depolarization rate is reached at about 3.5K (e.g.
for x ¼ 0: lB0:23 ms1 in ZF and lB0:08 ms1 in
0.2 T LF) and then diminishes at lower tempera-
tures (e.g. at 2K: lZFB0:22 ms
1, l0:1 TB0:06 ms
1,
l0:2 TB0:02 ms
1). Very surprisingly though, the
Gaussian component, s; survives even after the
application of HLFB0:05T (cD=gm), while its
volume fraction appears to shrink somewhat. Only
when fields of the order ofHLFB0:2T are reached,
component #1 of Eq. (3) is completely decoupled.
Such a very unusual, persisting Gaussian relaxa-
tion has been seen before in other systems with
spin-singlet ground states. For example, in the
frustrated Kagom!e lattice system SrCr8Ga4O19, a
similar behavior was observed and was attributed
to a dilute source of a magnetic local field, which
migrates spatially through the lattice [7].
In an attempt to understand the origin of the
slowing down and eventual freezing of the
electronic magnetic moments at ToTSG; we
0
20
40
60
80
100
A
total
# 1
# 2
quasi-static fraction
26 %
35 %
44 %
A
m
pl
itu
de
s:
A
1,
 A
2 
(%
)
0 1 2 3 4 5 6 7 8 9 10 11
0.0
0.1
0.2
0.3
1.0
2.0
3.0
T
f
 Temperature (K)
λ
2
σ
1
R
el
ax
at
io
n 
R
at
es
  (
µs
-1
)
x=0.0
x=0.4
x=0.12
(a)
(b)
Fig. 3. Temperature dependence of the fitted parameters
(Eqs. (1)–(3)) for SrCu2xMgx(BO3)2 (x ¼ 0; 0.04, 0.12) in zero
field: (a) the sample volume fractions (A1 þ A2 ¼ Atotal)
involved in the m+ spin-depolarization and (b) the correspond-
ing relaxation rates. At T > 3:75K, s is on averageB0.12ms1,
whereas l becomes smaller with increasing temperature—the
line through s1 is a power-law fit to determine Tf (E3.75K).
A. Lappas et al. / Physica B 326 (2003) 431–435 433
performed TF-m+SR (at 0.6 T) measurements
between 5 and 90K. We find that at Tp20K,
while entering the spin-gap regime, the local field
distribution, initially centered at n0B81:3MHz,
becomes broader and two additional lines (#2, #3)
gradually separate out while approaching 5K. We
calculated the Knight shift (Fig. 4) for the three
components according to the formula:
Ks;i ¼
gmBs;i
n0
¼
niðTÞ  n0
n0
 4p
1
3
 Nxx
 
wðTÞ; ð4Þ
where n0 is the frequency of the external field, ni
(i ¼ 1; 2; 3) is the frequency of each component,
Nxx is the demagnetization factor, and wðTÞ is the
volume susceptibility. The T-variation of the
corresponding sample volume fractions and re-
laxation rates are similar to those shown in Fig. 3
for the ZF measurements, i.e. TF #1 is associated
with the ZF exponential component, whereas TF
#2 and #3 are related to the ZF Gaussian
component. With ZF/LF and TF experiments
probing effects of the same nature, we note the
similarity of K1(T) to wðTÞ; as shown in the inset of
Fig. 4. Assuming that component #1 of the TF
data reflects roughly the bulk susceptibility around
TSG; then the Curie-like behavior (ni ¼ n0;i þ C=T)
exhibited by the spins associated with components
#2 and #3 may be ascribed to some spin-density,
which is liberated by the muon itself.
In summary, m+SR investigations of the
SrCu2xMgx(BO3)2 ground state indicate that the
spins associated with the dimer singlet state are
fluctuating relatively slowly (nB1=/lSLF¼0:1 TB
66MHz) at 7.5K before slowing down (B9MHz)
at 3.5K close to a characteristic temperature, Tf
that is 5 times smaller than TSG: Interestingly, this
point is also marked by the appearance of a
secondary process, which sets in abruptly and is
consistent with the freezing of spins liberated from
the dimer state. This transition is marked by a
persisting (non-decouplable upon the application
of HLF) Gaussian field distribution and points to a
m+-induced effect suggested before for other low-
dimensional spin-gap systems [8]. It is presumably
an indication of how close SrCu2(BO3)2 is to a
quantum critical phase transition from a spin-gap
to an AF ordered state. In addition, Mg-dilution
of the Cu-sublattice does not induce static
magnetic order; instead, it decreases the volume
fraction associated with the free-spins in accord
with the picture of a m+-associated perturbation of
the SrCu2(BO3)2 quantum critical ground state.
0 20 40 60 80 100
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
10 100
-0.02
-0.01
0.00
0.01
# 1
T
SG
K
ni
gh
t S
hi
ft,
 K
1 
(%
)
T (K)
0
1
2
3
4
5
χ (x 10 -3 em
u/m
ol)
SrCu
2
(BO
3
)
2
 TF-µSR
H
ext
= 0.6 T # 3
# 2
# 1
K
ni
gh
t S
hi
ft 
(%
)
Temperature (K)
Fig. 4. Knight shifts of the muon precession frequencies observed in SrCu2(BO3)2 as a function of temperature. The lines through the
data for components #2 and #3 are Curie-law fits. Inset: Temperature dependence of the Knight shift, K1; of component #1 and of the
bulk molar dc susceptibility (H=1T) in SrCu2(BO3)2.
A. Lappas et al. / Physica B 326 (2003) 431–435434
Acknowledgements
We thank A. Amato for technical assistance
with the m+SR experiments, I. Mastoraki for help
with the measurements and Paul Scherrer Institute
for the provision of muon beamtime. We acknowl-
edge support by NATO (KP, AL), the General
Secretariat for Research & Technology (Greece)
through a ‘‘Greece-Slovenia Joint Research &
Technology Program’’ (AL) and the Marie-Curie
Fellowship program of the European Union
(Contract no. HPMD-CT-2000-00050) (AL).
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A. Lappas et al. / Physica B 326 (2003) 431–435 435


Spin-freezing in the two-dimensional spin-gap systems SrCu 2Àx Mg x (BO 3 ) 2 (x=0, 0.04, 0.12)

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Spin-freezing in the two-dimensional spin-gap systems SrCu 2Àx Mg x (BO 3 ) 2 (x=0, 0.04, 0.12)

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