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Pulsed-field magnetization, electron spin resonance, and nuclear spin-lattice relaxation
in the ˆCu3‰spin triangle
Kwang-Yong Choi and Naresh S. Dalal
Department of Chemistry and Biochemistry, Florida State University and National High Magnetic Field Laboratory,
Tallahassee, Florida 32306-4390, USA
Arneil P. Reyes and Philip L. Kuhns
National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
Yasuhiro H. Matsuda and Hiroyuki Nojiri
Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan
Sib Sankar Mal and Ulrich Kortz
School of Engineering and Science, Jacobs University Bremen, P.O. Box 750 561, 28725 Bremen, Germany
共Received 20 August 2007; published 8 January 2008兲
We report on pulsed-field magnetization, Q-band electron spin resonance 共ESR兲, and 23Na NMR measure-
ments of the S=1/2 spin triangle clusters Na9关Cu3Na3共H2O兲9共
␣
-XW9O33兲2兴共X=As and Sb兲. The pulsed-field
magnetization shows pronounced hysteresis loops and magnetization steps including the half-step magnetiza-
tion. The detailed magnetization behavior depends substantially on the diamagnetic heteroatom X. The angular
dependence of ESR parameters necessitates Dzyaloshinskii-Moriya interaction. The temperature dependence of
the 23Na spin-lattice relaxation rate, 1/T1, scales well to
共T兲T, where
共T兲is the static susceptibility. The
spin-spin relaxation rate, 1/T2, increases rapidly for temperatures below 15 K due to dipolar interactions
between the 23Na nuclei and Cu2+ spins. The two clusters exhibit a markedly different field dependence of 1/T1
at antilevel crossing points. The enhancement of 1/T1is noticeable only for X=Sb. Since the spin configura-
tions of both clusters are nearly the same, the dependence of magnetization and 1/T1on Xis ascribed to the
strong coupling of the spins to a lattice vibration, leading to an enhanced mixing of the S=1/2 chiral state.
DOI: 10.1103/PhysRevB.77.024406 PACS number共s兲: 75.50.Xx, 75.45.⫹j, 33.25.⫹k, 33.35.⫹r
I. INTRODUCTION
In recent years, magnetic molecules consisting of a small
number of exchange-coupled paramagnetic ions have been
intensively investigated.1,2This is due to the fact that they
offer an opportunity for exploring the basic principles of na-
nomagnets and have the technological potential for miniatur-
ization of electronic devices to molecular scale.
Two different classes of magnetic molecules have, in par-
ticular, acquired strong research interest. One class is the
single molecule magnets 共SMMs兲, which behave like a
single magnet with magnetic anisotropy. Their prominent
features are a magnetic bistability, quantum tunneling of
magnetization, and quantum phase interference 共Berry
phase兲.3–5The other one is antiferromagnetically coupled
spin rings. Their fundamental aspect is that the elementary
excitations are described by a quantized rotation of the Néel
vector.6In the antiferromagnetic 共AF兲spin rings, the tunnel-
ing gap can be as large as several percent of the isotropic
exchange coupling constants. In contrast, the gap from
SMMs is usually limited to the millikelvin range. Thus, the
AF spin rings have the advantage of revealing pure quantum
magnetization over the SMMs because the large tunneling
gap strongly suppresses thermal effects.
Among the AF spin rings, triangular rings are the simplest
system. Moreover, they hold a special position in the mag-
netism due to the effects of spin frustration and
spin chirality.7–11 A prototypical example is found in the
兵V6其-, 兵V15其-, 兵Cu3其-, and 兵Mn3O其-type triangular
antiferromagnets.10–14 In the 兵Cu3其spin triangle, a half-step
magnetization appears as adiabatic quantum tunneling. Very
recently, isosceles AF spin rings have been proposed for a
basic unit of molecule-based quantum computation.15 For a
realistic implementation of quantum-information processing,
all sources of decoherence are needed to be clarified. Nuclear
spins and intermolecular dipolar interactions are mostly the
well-known ones.16 In addition, spins can undergo a decoher-
ence through coupling to lattice vibrations.17 This might be
related to distinct material dependence of the magnetization
in the 兵V3其spin triangle.8However, the exact mechanism
still remains unclear. In this work, we address this issue by
studying two 兵Cu3其spin triangles having slightly different
molecular environments.
The copper共II兲-substituted polyoxotungstates
Na9关Cu3Na3共H2O兲9共
␣
-XW9O33兲2兴共abbreviated as 兵Cu3-X其
with X=As and Sb兲have a sandwich-type structure with D3h
symmetry, where 兵Cu3其resides in the central belt and is
capped by two 共
␣
-XW9O33兲Keggin subunits 关see Fig.
1共a兲兴.18,19 Spin exchange couplings between Cu2+ ions occur
in an indirect way via two W and three O atoms of each
共XW9O33兲fragment. This enables us to modify the magni-
tude of spin interactions by replacing the diamagnetic het-
eroatom X. Figures 1共b兲and 1共c兲depict the spin topology of
兵Cu3-X其. For 兵Cu3-As其, the distances between the copper
ions are Cu1¯Cu2=4.696 Å and Cu2¯Cu2= 4.689 Å, while
for 兵Cu3-Sb其, the respective distances increase slightly to
PHYSICAL REVIEW B 77, 024406 共2008兲
1098-0121/2008/77共2兲/024406共8兲©2008 The American Physical Society024406-1
Cu1¯Cu2=4.871 Å and Cu2¯Cu2=4.772 Å. This is
thought to be caused by the difference of lone pair–lone pair
interactions between the two Xatoms. Since Sb has a larger
atom size than As, the lone pair interactions between Sb
become stronger, leading to a larger separation of the copper
centers.
In what follows, we report pulsed-field magnetization,
Q-band electron spin resonance 共ESR兲, and 23Na NMR spin-
lattice relaxation time 共T1兲measurements on the 兵Cu3-X其
family. We find that the detailed features of the magnetiza-
tion and nuclear spin-lattice relaxation rate depend on the
diamagnetic heteroatom Xalthough their electronic magnetic
parameters are only marginally different. This suggests that
there is a strong coupling between the spin and the lattice
vibrations which causes X-dependent enhanced T1relaxation
at electron spin energy level crossings.
This paper is organized as follows. In Sec. II, we describe
the experimental setup and conditions. In Sec. III, we present
the experimental results of magnetization and electron and
nuclear spin resonances. The discussion of the respective ex-
perimental results is given in Sec. IV. Section V provides a
summary of our findings.
II. EXPERIMENTAL DETAILS
Single crystals of Na9关Cu3Na3共H2O兲9共
␣
-AsW9O33兲2兴
·26H2O and Na9关Cu3Na3共H2O兲9共
␣
-SbW9O33兲2兴· 40H2O, re-
spectively, were prepared as described in Ref. 18. For pulsed
magnetization experiments, a dozen of small single crystals
were glued together along the field H
储
triangle plane. Mag-
netization measurements were carried out by means of a
standard inductive method using compensated pickup coils
and a nondestructive pulse magnet. Fast pulsed magnetic
fields up to 103T/s were generated by a capacitor bank of
90 kJ as described in Ref. 20. The sample was immersed in
liquid 3He to reach a temperature as low as 0.4 K. ESR
measurements of a single crystal were performed using a
commercial Bruker spectrometer operating at the Qband
共
⬇34 GHz兲.23Na NMR spectrum of powder samples was
obtained using a locally developed NMR spectrometer and a
high homogeneity 15 T sweepable magnet.21
III. EXPERIMENTAL RESULTS
A. Pulsed-field magnetization
Shown in Fig. 2is the magnetization versus magnetic
field plot for 兵Cu3-X其at 0.4 K for Horiented in the plane
comprising the spin triangle. The measurements were per-
formed in a full cycle sweep at a time scale of about 5 mil-
liseconds. A time evolution of a pulsed field is plotted in the
inset of Fig. 2. We note that the saturation magnetization is
renormalized by gS. Here, the g-factor is determined by elec-
tron spin resonance as discussed below.
In the upward sweep 共A→B兲, the magnetization of
兵Cu3-As其first increases to 1gS
B, followed by the step of
2.3gS
B, and finally approaches the saturation value of
3gS
Bin a high magnetic field of 12 T. In the down sweep
共B→C兲, the magnetization drops sharply from 3gS
Bto
1gS
Band from 1gS
Bto zero, respectively. The former
2gS
Bstep originates from the level crossing between ST
=1/2 and ST=3/2 states. The latter 1gS
Bone is related to a
splitting of the ST=1/2 state at zero field 共see below for
more details兲. The 1.3gS
Bstep seen in the upward sweep
cannot be understood within a simple energy level scheme of
a spin triangle. The contrasting magnetization between the
up and down sweeps leads to a pronounced hysteresis loop.
In the negative field, the hysteresis behavior nearly disap-
pears. The magnetization of the negative field is similar to
the magnetization which is seen in the down sweep of the
positive field. We stress that the hysteresis loop is not asso-
ciated with an energy barrier since Cu2+ has no single ion
anisotropy and the Cu2+ triangle is coupled antiferromagneti-
cally. Rather, the asymmetric magnetization through the
positive and negative fields implies that the magnetization
reversal dynamics is slow on the time scale of the pulsed-
field sweep rate.
Upon switching to 兵Cu3-Sb其, the magnetization steps be-
come less sharp in comparison to 兵Cu3-As其. In addition, the
magnetization curve looks more symmetric between the
positive and negative fields. This is mainly due to the smear-
ing of the 2.3gS
Bstep in the positive field and the appear-
ance of the small step in the negative field between −5 and
−7 T, which is absent in 兵Cu3-As其. The dependence of the
magnetization on the heteroatom Xsuggests that the dynami-
cal magnetization processes are distinctly different in the two
compounds.
Cu
2
(
a)
Cu Na
H
2
O
4.696
4.689
4.696
4.871
4.772
4.871
Cu
2
Cu2Cu2
(
b)
{Cu3-As} {Cu3-Sb}
(c) Cu1
Cu1
X
FIG. 1. 共Color online兲共a兲Combined polyhedral/ball-and-stick
representation of Na9关Cu3Na3共H2O兲9共
␣
-XW9O33兲2兴. Yellow ball is
for Na atoms, cyan for Cu, green for X, and red for H2O. 关共b兲and
共c兲兴 Sketch of the 兵Cu3-X其spin triangle configuration with Cu¯Cu
distances. Cu1and Cu2represent two crystallographically inequiva-
lent Cu sites. The numbers on the solid lines are Cu¯Cu distances.
CHOI et al. PHYSICAL REVIEW B 77, 024406 共2008兲
024406-2
B. Electron spin resonance
ESR spectra of 兵Cu3-As其recorded at 8.8 K and
=34 GHz are presented in Fig. 3as a function of angle. The
angle is measured between the molecular C3axis and the
external field. The three intense peaks ranging from
1.1 to 1.2 T correspond to the electron spin transitions be-
tween the excited ST=3/2 levels. The respective transitions
are assigned according to the calculated level diagram 共refer
to Fig. 8for the numeric designations of the energy levels兲.
In addition, we also observe several weak transitions arising
from the ST=1/2 levels. We note that the ESR intensity of
the ST=1/2 group is much weaker than that of the ST=3/2
group although the ESR signals of the ST=3/2 group are
from the excited states. The weaker intensity of the ST
=1/2 group is due to the reduced magnitude of the spin
number Ssince the ESR transition probability is given by
P⬀关S共S+1兲−Sz共Sz+1兲兴.
The ESR signals of the ST=1/2 group consist of the
conventional 共⌬Mz=±1兲ESR transitions of 兩1典→兩3典and
兩2典→兩4典as well as of the symmetry-forbidden transitions of
兩1典→兩4典. The forbidden transitions show the opposite angu-
lar dependence to the allowed ones 关compare Figs. 4共b兲and
4共c兲兴. In addition, they show a strong angular dependence in
intensity 共see the left panel of Fig. 3兲. The signals are hardly
observable when the sample is rotated such that the external
field is along the triangle plane. The presence of the
symmetry-forbidden signals implies nonvanishing matrix el-
ements between the energy levels 1 and 4. This highlights the
significant role of Dzyaloshinskii-Moriya 共DM兲interactions
in understanding the magnetic behavior of 兵Cu3-X其. Here, we
note that the matrix elements can vary strongly with mag-
netic field orientation since a mixing between the energy
levels is determined by the sign and magnitude of DM inter-
actions.
Figure 4共a兲displays the temperature dependence of the
ESR intensities of the 兩7典→兩8典共full triangle兲and 兩6典→兩7典
共open rectangle兲transitions. They scale roughly with the
magnetic susceptibility
共T兲. The g-factors of the respective
transitions are plotted as a function of temperature in the
FIG. 2. 共Color online兲共a兲Magnetization curve vs pulsed magnetic field for 兵Cu3-As其at 0.4 K for H
储
triangle plane. The saturated
magnetization is normalized by gS. Arrows are a guide to sweep directions 共A→B→C→D兲.共b兲Magnetization of 兵Cu3-Sb其using a pulsed
field with the same conditions as 共a兲. The inset shows the time dependence of a pulsed magnetic field.
B
θ
78
14
24
13
67
56
ν=34 GHz, T=8.8 K
x10
FIG. 3. Angular dependence of ESR spectra 共derivative of the
absorption spectra versus field, vertically shifted for clarity兲mea-
sured at
=34 GHz and 8.8 K for 兵Cu3-As其. The angle is measured
between the C3axis of a triangle and the external field. The three
strong peaks of the right panel correspond to the transitions between
the ST=3/2 states. The weak peaks arise from the transitions be-
tween the ST=1/2 states. Note the different magnetic field scales of
the left and right panels, and ESR spectra of the left panel are ten
times magnified.
PULSED-FIELD MAGNETIZATION, ELECTRON SPIN…PHYSICAL REVIEW B 77, 024406 共2008兲
024406-3
inset of Fig. 4共a兲. No appreciable change is detected. This
rules out any significant structural modulations with tem-
perature change.
Figures 4共b兲and 4共c兲summarize the angular dependence
of the resonance fields. The open 共full兲symbols represent the
experimental data of 兵Cu3-As其共兵Cu3-Sb其兲. Overall, the reso-
nance fields of 兵Cu3-Sb其are lower than those of 兵Cu3-As其.
This suggests that the magnetic parameters of 兵Cu3-X其are
reduced upon replacing As with Sb. This is consistent with
the increasing distances between the copper ions in
兵Cu3-Sb其.
C. Nuclear spin resonance
To investigate the dynamics of the Cu2+ electronic spins,
we have performed NMR measurements using the 23Na nu-
clei which are coupled via dipole-dipole interactions to the
Cu2+ magnetic moments. Since three Na+nuclei are located
between the Cu2+ spins 关see Fig. 1共a兲兴, they discriminate the
spin dynamics of each Cu2+ ion. In contrast, we note that the
1H nuclei do not provide specific information since the H
atoms have spatially two different classes. The H2O mol-
ecules inside the central belt of the cluster are directly
coupled to the Cu spins, while other 1H nuclei lie far away
关see Fig. 1共a兲兴. Thus, we will focus on the 23Na NMR mea-
surements in this study.
A typical 23Na NMR spectrum is displayed in Fig. 5. The
spectrum was obtained by monitoring the fast Fourier trans-
form 共FFT兲sum of a spin echo as a function of an external
field at a fixed temperature of 1.82 K. We observe a peak,
labeled P, together with its quadrupolar wings. The ratio be-
tween the intensity of the peak P and the wings is 0.57, close
to the theoretical value of 2/3. This confirms that the peak P
originates from the +1/2↔−1/2 central transition. We note
that the sharp, intense peak next to the peak P is from the
copper coil and not from the sample since the magnetic mo-
ment sits on the Cu2+ ion.
We measured the 23Na nuclear spin-lattice relaxation rate
1/T1in the temperature range between 1.8 and 50 K at the
two fixed magnetic fields H=2 and 4.4 T. The peak P was
selectively irradiated. The recovery curve of the longitudinal
magnetization is fitted by a double exponential function,
M共t兲=M⬁关1−共0.4e−t/T1+ 0.6e−t/6T1兲兴,共1兲
characteristic for the central transition of I=3/2 nuclei.
Figure 6共a兲shows the temperature dependence of 1/T1for
兵Cu3-As其. For temperature above 50 K, the NMR signal be-
comes too weak for 1/T1measurements. 1/T1slowly de-
creases with decreasing temperatures from 50 K and then
shows a strong drop below 10 K. This is due to the depopu-
lation of the excited states into the ground state in accor-
dance with the fact that the magnetic properties at low tem-
peratures are governed by the low-lying energy states.
We recall that for even AF rings there exists a strong
enhancement of 1/T1, leading to a peak at a temperature of
the order of the exchange coupling constant J/kB.22 In the
case of the 兵V6其AF spin triangle, the temperature depen-
dence of 1/T1is well approximated by T
共T兲without show-
ing a peak. In contrast, for 兵V15其the enhanced peak of 1/T1
occurs around 50–100 K and is attributed to the interlayer
exchange couplings.23,24 In our cluster, the temperature de-
pendence is proportional to T
共T兲as is the case for 兵V6其.
This confirms that the 23Na nucleus probes the spin dynamics
FIG. 4. 共a兲Temperature dependence of ESR intensity of the
兩7典→兩8典共full triangle兲and 兩6典→兩7典共open rectangle兲transitions.
The inset displays the respective g-factor as a function of tempera-
ture. 关共b兲and 共c兲兴 Angular dependence of the resonance fields. The
open 共full兲symbols represent the experimental data of 兵Cu3-As其
共兵Cu3-Sb其兲.
FIG. 5. 23Na NMR spectrum obtained by monitoring the FFT
sum of a spin echo as a function of an external field at a fixed
temperature of 1.82 K. The shaded area indicates the 23Na NMR
spectrum, which consists of the +1/2↔−1/2 central transition
共marked P兲and the quadrupolar wings. The empty solid line is 63Cu
NMR spectrum from the rf coil. The T1measurements were made
on the peak P.
CHOI et al. PHYSICAL REVIEW B 77, 024406 共2008兲
024406-4
of the Cu2+ ions, which behave like a paramagnetic ion.
In Fig. 6共b兲, we show the temperature dependence of the
spin-spin relaxation rate 1/T2together with the 23Na line-
width. The recovery of the transverse magnetization is well
described by an exponential decay with time. Both 1/T2and
the full width at half maximum 共FWHM兲are more or less
constant in the temperature range of 15–50 K. T2is of order
1 s in this temperature interval and starts to decrease steeply
for temperatures below 15 K. The temperature dependence
of 1/T2at low temperatures is opposite to that of 1/T1. Since
1/T1scales as T
共T兲, it should not be ascribed to the slowing
down of the Cu2+ spins. Instead, the low-temperature en-
hancement of 1/T2could be due to the slowing down of the
nuclear relaxation of H and/or Cu nuclei, which are coupled
to 23Na nuclei via nuclear dipolar interactions. The inset
shows the field dependence of the FWHM versus the exter-
nal field at 1.7 K. Since the linewidth increases with increas-
ing field, the broadening is not associated with the second-
order quadrupole effect on the central transition. Rather, it
originates from magnetic dipolar interactions which couple
23Na nuclei to the Cu2+ spins. The data are fitted by ⌬
=
␣
H+⌬
0.23 We obtain a local hyperfine coupling factor of
␣
=3.6 kHz T−1 and the zero-field value of dipolar interac-
tions of ⌬
0=51.1 kHz, which seems reasonable.
We now turn to the field dependence of 1/T1. The results
of 兵Cu3-X其are shown in Fig. 7. Both compounds exhibit a
strong field dependence on 1/T1as found in other molecular
magnets. This is due to the slow decay of the Cu2+ ion spin
correlation function, which is a characteristic feature of the
zero dimensionality of the system.10 As in Ref. 10, we ana-
lyze the data in terms of the simplified expression
T1
−1共T,H兲=Fzz共T兲关1/
0+
␣
0/共
e
2+
0
2兲兴,共2兲
where Fzz共T兲is a sum of the autocorrelation and nearest-
neighbor correlation function,
0is an angular frequency
measuring the Lorentz broadening,
␣
is given by the ratio
between components of the magnetic dipolar interaction ten-
sor, and
e=
␥
eHis the electronic Larmor frequency. Since
the critical enhancement is absent for the studied com-
pounds,
0is weakly Tdependent. This is evidenced by the
fact that for 兵Cu3-As其,T1
−1共H,4.2 K兲is well rescaled to
T1
−1共H,1.67 K兲. A comparison of the fit to the data reveals
that 兵Cu3-Sb其shows an enhancement of T1
−1共H,T兲at the
magnetic fields of 2 and 4.5 T, where the antilevel crossings
occur between ST=1/2 and ST=3/2共see Fig. 8兲. Noticeably,
it is present but less evident in the 兵Cu3-As其compound.
The enhancement of 1/T1at level crossings can be under-
stood in terms of a phenomenological model,25
1/T1=A⌫/共⌫2+关h
L−⌬共H兲兴2兲,共3兲
where Ais an average hyperfine coupling constant, ⌫is a
T-dependent damping factor, and ⌬共H兲is the field-dependent
energy difference between energy levels. In proximity to the
level crossing field,⌬共H兲is written by
⌫共H兲=兵关g
B共Hc−H兲兴2+⌬2其1/2,
where Hcis a critical field and ⌬is an energy gap at level
crossing fields. For the analysis of 兵Cu3-Sb其, we first subtract
from the raw data the background calculated by Eq. 共2兲. The
subtracted result is shown in the inset of Fig. 7. A reasonable
fit is obtained using Hc=4.47 T and ⌬= 0.6 K, which are
discussed below.
FIG. 6. 共a兲23Na spin-lattice relaxation rate 共1/T1兲of 兵Cu3-As其
at 2 T 共full square兲and 4.4 T 共open triangle兲as a function of tem-
perature. The solid line is the temperature dependence of
共T兲Tat
4.4 T. 共b兲Temperature dependence of spin-spin relaxation rate,
1/T2,共full triangle兲and 23Na FWHM 共open square兲for 兵Cu3-As其at
4.4 T.
FIG. 7. Field dependence of spin-lattice relaxation rate 1/T1at
1.67 K for 兵Cu3-As其compound and at 1.72 K for 兵Cu3-Sb其com-
pound, respectively. The solid line is a fit of the data to Eq. 共2兲.
Inset: Field dependence of 1/T1subtracted by the background,
which is obtained by Eq. 共3兲. The solid line is a calculated curve
with ⌬=0.6 K.
PULSED-FIELD MAGNETIZATION, ELECTRON SPIN…PHYSICAL REVIEW B 77, 024406 共2008兲
024406-5
IV. DISCUSSION
To figure out the underlying physics, we first determine
the energy level diagram by starting from a general spin
Hamiltonian of a spin triangle ring,
H=兺
l=1
3
兺
␣
=x,y,z
Jll+1
␣
Sl·Sl+1 +兺
l=1
3
Dll+1 ·关Sl⫻Sl+1兴
+
B兺
l=1
3
Sl·g
˜
ll ·Hl,共4兲
where the exchange coupling constants Jll+1
␣
, the DM vectors
Dll+1, and the g-tensors g
˜
ll are defined as site-dependent
quantities with a periodic boundary condition.
The magnetic parameters are obtained by diagonalizing
the 8⫻8 matrix numerically. We stress that all parameters
are uniquely fixed by considering 共i兲a crystal symmetry, 共ii兲
the positions of the magnetization steps 共see Fig. 2兲, and 共iii兲
angular dependence of the ESR signals 关see Figs. 4共b兲and
4共c兲兴. The resulting values are listed in Table I, and the cor-
responding energy level diagram is depicted in Fig. 8. Over-
all, the magnetic parameters of 兵Cu3-Sb其are slightly smaller
than those of 兵Cu3-As其. This is consistent with an increase of
Cu¯Cu distance upon replacing As with Sb.
The determined energy levels show that the excited state
of ST=3/2 lies about 6 K above the ST=1/2 ground state. In
addition, the two ST=1/2 states are split with a sizable en-
ergy gap of about 1 K. With the aid of numerical simula-
tions, we find that both the strong DM interactions, amount-
ing to 12% of Jij, and a small isosceles distortion contribute
equally to the opening of the gap. This also leads to a lifting
of two different spin chiral states, which are defined as
1=共兩↑↓↓典+e2
i/3兩↓↑↓典+e4
i/3兩↓↓↑典兲/冑3,
2=共兩↑↓↓典+e4
i/3兩↓↑↓典+e2
i/3兩↓↓↑典兲/冑3.
On applying an external field, a ST=1/2 state crosses with
aST=3/2 state at 2 T and then the ground state changes to a
ST=3/2 state at 4.47 T. We note that the two different spin
chiral states show different level crossings with the ST=3/2
states; energy level 1 has an antilevel crossing with the ST
=3/2 state, while energy level 2 shows a tiny admixture to it
in the field interval of 4–4.47 T.
Based on this fact, we can provide an explanation for the
observed magnetization features. At 0.4 K, the spins mostly
occupy levels 1 and 2 since they are separated by about 1 K
from the higher energy levels. When the field is swept up-
wards, the spins of level 1 transit to the lowest ST=3/2 state
at HC2=4.47 T. The respective magnetization 共thin dotted
line兲is sketched in Fig. 9. The magnetization jump by 1gS
B
at zero field is attributed to the opening of the tiny gap be-
tween the ST=1/2 states. As possible origins, dipolar inter-
actions and/or intramolecule hyperfine couplings are dis-
cussed, but there is yet no consensus on the exact
mechanism.26 The magnetization jump of 2gS
Bat HC2
=4.47 T arises from the level crossing between the ST=1/2
and ST=3/2 states. Level 2 undergoes successive transitions
of ST=1/2→ST=3/2→ST=1/2共dashed line兲. Thus, the
magnetization by 2gS
Bwill occur in the field regime be-
tween HC1and HC2. If we assume that the field sweeping rate
is faster than the relaxation rate between levels 1 and 2, the
TABLE I. A comparison of the magnetic parameters of the
兵Cu3-X其compounds.
Magnetic parameters 兵Cu3-As其兵Cu3-Sb其
J12
x=J12
y4.50 K 4.49 K
J12
z4.56 K 4.54 K
J23
x=J23
y=J31
x=J31
y4.03 K 3.91 K
J23
z=J31
z4.06 K 3.96 K
D12
z=D23
z=D31
z0.529 K 0.517 K
D12
x=D12
y0.529 K 0.517 K
g11
xx =g11
yy 2.25 2.24
g22
xx =g22
yy 2.10 2.11
g33
xx =g33
yy 2.40 2.40
gii
zz共i=1–3兲2.06 2.07
FIG. 8. 共Color online兲Energy level diagram for 兵Cu3-As其. Field
dependence of the eight eigenvalues is obtained by solving the
Hamiltonian 共4兲numerically.
level 1
level 2
(level 1 + level 2)/2
Magnetization (µB/gS)
1
2
3
Hc1(
4T
)Hc2(
4.47T
)
Magnetic field (T)
FIG. 9. 共Color online兲A schematic of magnetization following
energy levels 1 and 2 in an upward sweep. See the text for details.
CHOI et al. PHYSICAL REVIEW B 77, 024406 共2008兲
024406-6
resulting magnetization will be given by the average of the
two processes, yielding the half-step magnetization of 1gS
B
共see the solid thick line in Fig. 9兲. The observed step of
1.3gS
Bis somewhat bigger. A rough estimate suggests that
15% of spins relax from level 2 to level 1 in the course of the
field sweeping.
Above 8 T, the magnetization increases continually from
2.3gS
Bto 3gS
B. Upon approaching point Bof 12 T, the
field sweep speed, dH/dt, goes to zero 关see a cosine type
time evolution of the pulsed field in the middle inset of Fig.
1兴. Thus, the spins in level 1 undergo a thermal relaxation to
the ground state on approaching 12 T. In the down sweep
after the saturation, the spins are confined in level 1. As a
result, the magnetization will be governed by level 1: ST
=3/2→ST=1/2→ST=1/2→ST=3/2共thin dotted line兲.
This result suggests that in the down sweep, the thermal re-
laxations can be quenched and, thus, spins can be decoupled
from environments. This is an important result since the de-
coupling leads to enhanced coherence times for quantum
computation. We note that the time interval of the decoupled
spin state amounts to several milliseconds 共see the inset of
Fig. 1兲. This implies that a dephasing time of the electron
spins is estimated to be the same order of magnitude, which
is much longer than the switching time of a modern elec-
tronic device.
Compared to 兵Cu3-As其, the magnetization of 兵Cu3-Sb其be-
comes more symmetric. At the same time, the difference be-
tween the up and down sweeps becomes smaller. This im-
plies fast repopulation of the spins between levels 1 and 2.
This fact is surprising considering that the spin triangle con-
figuration changes marginally by replacing X. The field de-
pendence of the nuclear spin-lattice relaxation rate 1/T1pro-
vides additional insight. At an antilevel crossing point, a
strong enhancement of 1/T1is expected due to cross relax-
ation effects or magnetization fluctuations.27 For 兵Cu3-As其,
we find no pronounced peaks around the antilevel crossing
points. In contrast, 兵Cu3-Sb其shows the peaks at 2 and
4.47 T. Since the magnetic parameters and the energy level
structures of the two compounds are nearly the same, the
different magnetization and 1/T1behavior indicate that there
exists an additional relaxation channel. Since the static lattice
structure of 兵Cu3-X其hardly depends on X, a lattice vibration
might be the responsible factor.17
The 兵Cu3-X其cluster contains internal vibration modes of
the copper triangle subunit. In a regular triangle, a breathing
vibration will conserve symmetry. As a result, there will be
no mixing between the two ST=1/2 ground states which are
orthogonal to each other. However, an out-of-phase stretch-
ing vibration will break the local symmetry. Then, the dy-
namical lattice distortion will cause an admixture of the two
chiral states 1 and 2, just as the static lattice distortions do.
Actually, such a dynamical effect is inferred theoretically in
the quantum tunneling of 兵Fe10其ring.17 Due to the larger
atom size of Sb, the deviation from a regular triangle is
slightly bigger in 兵Cu3-X其. In this case, we expect more en-
hanced effect of a dynamical lattice distortion on spin relax-
ation in 兵Cu3-Sb其than in 兵Cu3-As其. This explains qualita-
tively the faster spin relaxation and the more symmetric
magnetization observed in 兵Cu3-Sb其. We recall that the ma-
terial dependence of magnetization is also found in the tri-
angles made of V4+ ions.8Our study indicates that a spin-
lattice coupling might be a significant channel of
decoherence in the spin triangle.
Finally, we address the potential of a spin triangle as a
molecular switch or quantum computation. We start with a
triangle with a linear chain consisting of three S=1/2 spins.
If the system has a sizable uniaxial anisotropy and ferromag-
netic exchange coupling, it can be regarded as a SMM,
where two Sz=±3/2 spin states comprise a memory unit. A
quantum spin gate operation may be possible by using pulsed
electron paramagnetic resonance technique if the three spins
have different gvalues. In such a scheme, all possible spin
states of ST=3/2 are utilized, and thus, it is more effective
than other SMMs. However, much work is needed for its
realization because of a fast relaxation of ST=1/2 spin.
In the case of AF coupled triangle spin rings, we can
adopt a different scheme by exploiting an internal degree of
freedom: spin chirality. The right and/or left spin chiralities
are well-defined and distinguishable quantum states. Above
all, the relaxation time of a chiral state can be as slow as a
few milliseconds as shown in the magnetization study of
兵Cu3-X其. This is because a spin chirality is defined for the
whole triangle and, thus, the fluctuations of individual spin
are not crucial. The most remarkable feature is that a tunnel-
ing gap, a mixing between two spin chiralities, and a result-
ing decoherence time can be tuned by controlling molecular
symmetry. As shown in Fig. 8, the tunneling gap originates
from the DM interaction, which is closely related to a struc-
tural chirality. Further, the present NMR study uncovers that
the mixing between the two chiral states is determined by the
structural distortion from a regular triangle, that is, deviation
from C3symmetry. Thus, we see that there are two control
knobs of a quantum state in a spin triangle: 共i兲a structural
chirality and 共ii兲a structural distortion. If we attach a photo-
reactive molecule to the triangle, an induced structural
change might control the tunneling gap. This also indicates
that an excitation of resonant phonons pertaining to dynami-
cal distortion modes might be possible.
Next we discuss other sources of decoherence: dipolar
interaction and nuclear hyperfine interaction. Compared to
other SMMs, the dipolar interaction is less serious in a spin
triangle because the total spin is as small as S=3/2. Such
weak interactions can be suppressed by means of a fast
sweeping magnetic field as shown in the present work. In
this context, an effect of nuclear spins 共e.g., hydrogen兲on
nonmagnetic ions can be quenched as well. In addition, the
application of high magnetic field leads to the suppression of
the relaxation rate as indicated by the steady decrease of
1/T1with increasing magnetic fields. Regarding the on-site
hyperfine coupling, it cannot be avoided by the dilution of
material. This problem can be overcome with the help of the
chemical engineering of nuclear spins. A considerably long
decoherence time is expected for a spin triangle made of
nuclear-spin-less S=1/2 ions. We note that many well estab-
lished procedures are currently available for a coherent ma-
nipulation of nuclear spin polarization. Thus, it may be pos-
sible to manipulate the flip of electron spin via the nuclear
spin.26
PULSED-FIELD MAGNETIZATION, ELECTRON SPIN…PHYSICAL REVIEW B 77, 024406 共2008兲
024406-7
V. CONCLUSIONS
We have presented a comparative study of the S=1/2
兵Cu3-X其共X= As and Sb兲spin triangle systems using pulsed-
field magnetization, Q-band ESR, and 23Na NMR T1mea-
surements. The analysis of ESR measurements needed the
introduction of the Dzyaloshinskii-Moriya interaction. The
hysteresis loops were found to be quite pronounced and in-
dicative of adiabatic quantum tunneling. We find that the
magnetization and 1/T1behavior is sensitive to the type of
heteroatom X.1/T1shows a clear peak at the antilevel cross-
ing when X=As, but less so when X=Sb. This is attributed to
the different dynamical mixing of a S=1/2 chiral state be-
tween the two compounds, caused by a coupling of the spins
to the lattice vibration. Also, it is seen that a field sweep rate
can be adjusted to decouple the spins from the environment.
This should lead to enhanced decoherence times, a prerequi-
site for efficient quantum computation. The present work,
thus, indicates that a spin triangle is a promising candidate
for molecular implementation of the quantum spin informa-
tion process. Its extension to 兵Cu4其and 兵Cu5其rings28,29 seems
worthwhile.
ACKNOWLEDGMENTS
This work was partly supported by Grant-in-Aid for Sci-
entific Research on priority Areas “High Field Spin Science
in 100 T” 共No. 451兲from MEXT, Japan, NSF Grant No.
DMR-0506946, and DFG Grant No. KO-2288/6-1.
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