arXiv:cond-mat/0610540v2 [cond-mat.str-el] 28 Dec 2006
Ground state and excitation properties of the quantum kagom´ e system
ZnCu3(OH)6Cl2investigated by local probes.
Oren Ofer and Amit Keren
Physics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel
Emily A. Nytko, Matthew P. Shores, Bart M. Bartlett, and Daniel G. Nocera
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
Chris Baines and Alex Amato
Paul Scherrer Institute, CH 5232 Villigen PSI, Switzerland
We characterize the ground state and excitation spectrum of the S = 1/2, analytically pure and
perfect kagom´ e system ZnCu3(OH)6Cl2 using the following measurements: magnetization, muon
spin rotation frequency shift K, transverse relaxation time T∗
T1. We found no sign of singlet formation, no long range order or spin freezing, and no sign of spin-
Peierls transition even at temperatures as low as 60 mK. The density of states has an E1/4energy
dependence with a negligible gap to excitation.
2, and Cl nuclear spin-lattice relaxation
The study of spin 1/2 quantum magnetism on the
kagom´ e lattice is very intriguing and lively because dif-
ferent investigation strategies have led to fundamentally
different predictions regarding the ground state and ex-
citations of this system. Theories based on numerical
or approximate diagonalization of the Heisenberg Hamil-
tonian favors non-magnetic (singlet) ground state [1, 2]
with full symmetry of the Hamiltonian , negligible cor-
relation length , and an upper limit on excitation gap
of J/20 . These strategies provide conflicting reports
on the possibility of a spin-Peierls state [3, 6]. In con-
trast, semiclassical treatments based on the thermal  or
quantum  “order from disorder” approach gives rise to
a ground state with a broken symmetry of the
type, which is stable even against tunneling , with
gapless magnon excitations , and no spin-Peierls type
distortion . In these circumstances, one expects ex-
periments to provide some guidance. However, the exper-
imental situation is equally confusing since most of the
experimental studies of kagom´ e-like materials suffered
from several kinds of shortcomings, which made com-
parison with theoretical models difficult. Some material,
such as the jarosites  have spin S > 1/2. Others have
S = 1/2 but with a non perfect kagom´ e structure, such
as the volborthite Cu3V2O7(OH2)2H2O (CVO) . A
third class of materials such as SrCr9pGa12−9pO19 
and Ba2Sn2ZnGa10−7pCr7pO22  were hampered by
disorder or strong third direction interaction. In light
of these difficulties, the recent synthesis  of herbert-
smithite, ZnCu3(OH)6Cl2, with its Cu-based quantum
S = 1/2, analytically pure and perfect kagom´ e lattice,
should put the field on a new course.
In this letter we present a comprehensive study of
ZnCu3(OH)6Cl2 using local probes.
address four questions which are at the heart of the in-
vestigation of the quantum kagom´ e system: Do S = 1/2
spins on kagom´ e lattice freeze? Is the ground state mag-
netic? What is the density of excited states, and is there
a gap in the spin energy spectra? Finally, does the lat-
In our study, we
tice distort in order to accommodate spin-Peierls state?
We address these questions in the present work using
nuclear magnetic resonance (NMR) and muon spin res-
onance (µSR) local probes. We also use magnetization
measurements to calibrate the local probes.
ZnCu3(OH)6Cl2was prepared by hydrothermal meth-
ods performed at autogenous pressure.
teflon liner was charged with 16.7 g Cu2(OH)2CO3(75.5
mmol), 12.2 g of ZnCl2(89.5 mmol), and 350 mL water,
capped and placed into a custom-built steel hydrother-
mal bomb under ambient room atmosphere. The tight-
ened bomb was heated at a rate of 1◦C/min to 210◦C
and the temperature was maintained for 48 h. The oven
was cooled to room temperature at a rate of 0.1◦C/min.
A light blue powder was isolated from the base of the
liner by filtration, washed with water, and dried in air
to afford 21.0 g product (49.0 mmol, 97.4% yield based
on starting Cu2(OH)2CO3). Magnetic and pXRD data
were consistent with those previously reported for her-
DC magnetization m measurements were performed
on a powdered sample using a Cryogenic SQUID mag-
netometer at temperatures T ranging from 2 to 280 K
and fields H varying from 2 kG to 60 kG. In the inset
of Fig. 1 we present mT/H versus T. The data collapse
onto a single line, especially at low T, meaning that the
susceptibility is field-independent in our range of temper-
atures and fields. Also, no peak in the susceptibility is ob-
served, indicating the absence of magnetic ordering. The
only indication of interactions between spin in these mea-
surements is the fact that mT/H decreases upon cooling
whereas in an ideal paramagnetic system this quantity
should be constant. Fits of the susceptibility data at
high temperatures and lower fields than presented here
reveal a Curie-Weiss temperature of ΘCW= −314 K .
The frustration parameter TF/|ΘCW| ≈ 0.22, where TF
∼ 70 K is the temperature at which χ−1is no longer a lin-
ear function of T, indicates strong geometric frustration.
For comparison, in CVO TF/|ΘCW| ≈ 1 .
A 800 mL
FIG. 1: The muon shift K against susceptibility. In the inset,
normalized magnetization versus temperature.
Muon spin rotation and relaxation (µSR) measure-
ments were performed at the Paul Scherrer Institute,
Switzerland (PSI) in the GPS spectrometer with an He
cryostat, and in the LTF spectrometer with a dilution re-
frigerator. The measurements were carried out with the
muon spin tilted at 45◦relative to the direction of the
beam. Positron data were collected in both the forward-
backward (longitudinal) and the up-down and right when
available (transverse) detectors simultaneously.
were collected at temperatures ranging from 60 mK to
200 K with a constant field of 2 kG. In GPS we used a
resistive magnet with a calibrated field; in LTF we used
a superconducting magnet. We took data at overlapping
temperatures to calibrate the LTF field. The longitudi-
nal data, which measure muon spin T1, were found to be
useless since the T1in our field range is much longer than
the muon lifetime and showed no temperature variations.
In the lower inset of Fig. 2 we show real and imagi-
nary transverse field [TF] data taken at H = 2 kG and
T = 100 K. The data are presented in a rotating reference
frame (RRF) at a field of 1.9 kG. The TF asymmetry is
best described by ATF= A0exp?−t2/(2T∗2
2is the transverse relaxation time, and ω is the
frequency of the muon. The quality of the fit is repre-
sented by the solid line.
In Fig. 1 we depict the frequency shift, K = (ω0−
ω)/ω0 where ω0 is the free muon rotation frequency in
the RRF. The difference in frequency between free and
implanted muons is a consequence of the sample magneti-
zation; therefore, the shift is expected to be proportional
to the susceptibility. Indeed, as shown in the main panel
of Fig. 1, there is a linear relation between K and the
susceptibility χ = m/H, with the temperature as an im-
plicit parameter; some representative temperatures are
shown on the upper axis. In the upper inset of Fig. 2 we
present the field dependence of the shift at T = 10 K.
FIG. 2: A plot of the muon shift K, transverse relaxation
time σ, versus temperature. In the upper inset, the muon
shift K versus external field H. In the lower inset, real and
imaginary transverse field asymmetry for T = 100 K.
Within the error bars, the shift in the RRF does not de-
pend on the external field. This is in agreement with
the susceptibility results. Therefore, Fig. 2 could serve
as a conversion graph from muon spin frequency shift to
In the main panel of Fig. 2, we depict K as a function
of temperature. An additional axis is presented where
K has been converted to χ as discussed above. We find
that K (and hence χ) increases with decreasing temper-
atures and saturates below T ∼ 200 mK at a value of
χ = 15.7(5) × 10−3cm3/mol Cu; the error is from the
calibration. It should also be pointed out that the en-
ergy scale associated with spin 1/2 in a field of 2 kG is
200 mK, and the saturation could be a consequence of
the external field. The saturation of χ is a strong ev-
idence for the lack of impurities in our sample. More
importantly, it indicates the lack of singlet formation or
spin freezing. The last conclusion is also in agreement
with neutron scattering measurements  and zero field
The muon transverse relaxation rate 1/T∗
sented in Fig. 2. Roughly speaking, it has the same
temperature behavior as the shift (and as the suscepti-
2relaxation is a result of defects in the sample
causing a distribution of muons to electronic spin cou-
pling constants or a distribution of susceptibilities. It
has been shown that when the muon relaxation rate be-
haves similarly to the shift  (or susceptibility )
upon cooling, it indicates quenched distribution of either
the coupling constants or susceptibilities. In this case
the relaxation increases simply because the average mo-
ment size increases. Since the coupling constants and
susceptibility are functions of distances between muon
and electronic spin or between two electronic spins, our
results are consistent with a lack of lattice deformation
Since muons could not reveal dynamic T1information,
2is also pre-
we performed37Cl and35Cl NMR experiments on the
same sample. Using the two isotopes, we are able to
determine the origin of T1. The first step in such a mea-
surement is to find the line shape and to identify the
isotopes and transitions. This measurement was done at
a constant applied frequency of νapp = 28.28 MHz and
a varying external field H. A standard spin-echo pulse
sequence, π/2 − τ − π, was applied, and the echo sig-
nal was integrated for each H. In Fig. 3 we show a
field sweep for both Cl isotopes obtained at T = 100 K.
A rich spectrum is found and is emphasized using five
x-axis and one y-axis breakers. This rich spectrum is
a consequence of the Cl having spin 3/2 for both iso-
topes. In the case where the nuclei reside in a site with
non cubic local environment and experience an electric
field gradient, their spin Hamiltonian could be written as
H = −hνlI·(1+K)·?H+(hνQ)/6?3I2
the anisotropy parameter, K is the shift tensor, and
νl= γH/(2π). The powder spectrum of such nuclei has
two satellite peaks corresponding to the 3/2 ←→ 1/2 and
−3/2 ←→ −1/2 transitions, and a central line from the
1/2 ←→ −1/2 transition, which is split due to the pow-
der average. The transition names are presented in the
figure. The satellite peaks at T = 100 K of35Cl are at
6.52 and 7.07 T, and for37Cl at 7.91 and 8.41 T. The
lack of singularity in the satellite spectrum indicates that
the Cl resides in a site with η > 0, namely, with no xy
symmetry. These findings are consistent with a 3m site
symmetry of the Cl ion in the space group R¯3m.
In contrast to the two satellites, the spliting of the
central lines at T = 100 K is clear, and appear for the
35Cl at 6.778 and 6.801 T and for the
and 8.161 T. Under some assumptions these values could
be used to determine the parameter of the nuclear spin
Hamiltonian ; assuming that the nuclear spin opera-
tors, Ix, Iyand Izare colinear with the principal axes of
the shift tensor, and that the in-plane shift is isotropic
with K⊥ = (Kx+ Ky)/2, we find for both isotopes,
K⊥ ≃ −0.0017(5), Kz ≃ 0.035(9) and η = 0.4, and
35νQ= 3.75 MHz and37νQ= 2.55 MHz. The ratio of νQ
is as expected from the ratio of the quadrupole moments.
Due to the assumptions, the value of Kzshould only be
considered as an order of magnitude. Nevertheless, it
is not very different from that of the muon shift (in the
laboratory frame) at the same temperatures. This means
that both probes experience a similar field generated by
the Cu spins, which for muons is usually a dipolar field.
Temperature dependence field sweeps of the35Cl cen-
tral line are shown in Fig. 4. The intensities are in arbi-
trary units for clarity. The ±1/2 ←→ ∓1/2 transitions
are easily observed at T = 300 and 100 K (indicated by
the arrows in the figure) but are smeared out at lower
T. In fact, the lines become so broad that the NMR
shift cannot be followed to low temperature; hence the
importance of the µSR results.
Finally, we measured the37Cl spin-lattice relaxation
to determine spin gap and excitation spec-
z− I2+ η?I2
where νQ is the quadrupole frequency, 0 ≤ η ≤ 1 is
37Cl at 8.148
FIG. 3: A field sweep of35Cl and37Cl.
trum. The data were taken at a field of 8.15 T which
corresponds to the low field peak of the central line.
We use a saturation recovery pulse sequence.
5 we depict T−1
normalized by γ2where37γ = 3.476
MHz/T on a semi-log scale. T−1
ing down to 50 K and then sharply decreases. We also
present35Cl (T1γ2)−1where35γ = 4.172 MHz/T below
50 K in order to determine the origin of the dynamic
fluctuations. These measurements were done under the
same conditions as37Cl. When considering all temper-
atures we find that T35
netic relaxation mechanism we expect this ratio to equal
(37γ/35γ)2= 0.69. From a quadrupole based mechanism
we anticipate (37Q/35Q)2= 0.62 where Q is the nuclear
quadrupole moment. Our finding is in favor of relaxation
mediated by a magnetic mechanism as indicated by the
increases upon cool-
= 0.75(10). From a mag-
peratures. The arrows indicate the central line singularities
observed at high-T but smeared out at low T.
35Cl field sweep (ν = 28.28MHz) at different tem-
FIG. 5: A semi-log plot of the Cl inverse spin-lattice relax-
ation, (γ2T1)−1, versus temperature. Inset, a linear plot of
the low-temperature region. The black line is a fit to Eq. 1.
The dashed line is straight.
overlapping (T1γ2)−1data points in Fig. 5.
In the inset of Fig. 5 we zoom in on the low T data us-
ing a linear scale. A first glance suggests that at low tem-
perature 1/T1is a linear function of T as indicated by the
dashed line. A remanent relaxation at zero temperature
1could be due to magnetic fluctuations form other
nuclear moments such as the protons or copper, since
they continue to fluctuate even when the electronic mo-
ments stop. Thus the spin lattice relaxation due to elec-
tronic contribution only 1/Te
1T) = const. This relation is expected in the case
of free fermions and might be related to recent theories
A different approach to T1interpretation is in terms of
magnon Raman scattering where
1≡ 1/T1(T) − 1/Tn
ρ2(E)·n(E)·[n(E) + 1]dE (1)
with ρ being the density of states, ∆ the gap, A is a
constant derived from the hyperfine coupling, and n(E)
the Bose-Einstein occupation factor . This expres-
sion is constructed from the population of magnons be-
fore and after the scattering, with the associated density
of states and the assumption that they exchanged negli-
gible amount of energy with the nuclei since its Zeeman
splitting is much less than a typical magnon energy. How-
ever, in frustrated magnets the magnon might not be the
proper description of the excitations [13, 14]. Neverthe-
less, we use Eq. 1 since it is expected for any kind of
bosonic excitations, and since there is no other available
theory. We assume ρ(E) ∼ Eα, with α and ∆ as fit pa-
rameters. The fit of Eq. 1 to the data is presented as the
solid line in Fig. 5, and in its inset. We find α = 0.23(1)
and ∆ = 0.5(2) K. Comparing to J = 209 K , this is
a negligibly small gap. It indicates that most likely there
is no gap in the spin energy spectra, in agreement with
Ref. , and ρ(E) ∼ E1/4.
To conclude, susceptibility measurements down to
60 mK suggest that there is no freezing and only a sat-
uration of susceptibility, namely, no singlet formation.
The data also do not support the presence of lattice de-
formation.Finally, Cl NMR T1 measurements find a
negligibly small magnetic gap and the density of states
ρ ∼ E1/4. Thus, ZnCu3(OH)6Cl2 is an exotic magnet
with no broken continuous symmetry but gapless exci-
tations. It might be an example of algebraic spin liquid
We would like to thank the PSI facility for support-
ing the µSR experiments and for continuous high quality
beam, and the NATO CollaborativeLinkage Grant, refer-
ence number PST.CLG.978705. We acknowledge helpful
discussions with Young. S. Lee and Philippe. Mendels,
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