Spin dynamics of the spin-1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2.
ABSTRACT We have performed thermodynamic and neutron scattering measurements on the S=1/2 kagomé lattice antiferromagnet ZnCu3(OH)6Cl2. The susceptibility indicates a Curie-Weiss temperature of theta CW approximately = -300 K; however, no magnetic order is observed down to 50 mK. Inelastic neutron scattering reveals a spectrum of low energy spin excitations with no observable gap down to 0.1 meV. The specific heat at low-T follows a power law temperature dependence. These results suggest that an unusual spin liquid state with essentially gapless excitations is realized in this kagomé lattice system.
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ABSTRACT: The spin wave excitations of the S=5/2 kagomé lattice antiferromagnet KFe3(OH)6(SO4)2 have been measured using high-resolution inelastic neutron scattering. We directly observe a flat mode which corresponds to a lifted "zero energy mode," verifying a fundamental prediction for the kagomé lattice. A simple Heisenberg spin Hamiltonian provides an excellent fit to our spin wave data. The antisymmetric Dzyaloshinskii-Moriya interaction is the primary source of anisotropy and explains the low-temperature magnetization and spin structure.Physical Review Letters 07/2006; 96(24):247201. · 7.94 Impact Factor
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ABSTRACT: As liquids crystallize into solids on cooling, spins in magnets generally form periodic order. However, three decades ago, it was theoretically proposed that spins on a triangular lattice form a liquidlike disordered state at low temperatures. Whether or not a spin liquid is stabilized by geometrical frustration has remained an active point of inquiry ever since. Our thermodynamic and neutron measurements on NiGa2S4, a rare example of a two-dimensional triangular lattice antiferromagnet, demonstrate that geometrical frustration stabilizes a low-temperature spin-disordered state with coherence beyond the two-spin correlation length. Spin liquid formation may be an origin of such behavior.Science 10/2005; 309(5741):1697-700. · 31.20 Impact Factor
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ABSTRACT: The oxide superconductors, particularly those recently discovered that are based on La(2)CuO(4), have a set of peculiarities that suggest a common, unique mechanism: they tend in every case to occur near a metal-insulator transition into an odd-electron insulator with peculiar magnetic properties. This insulating phase is proposed to be the long-sought "resonating-valence-bond" state or "quantum spin liquid" hypothesized in 1973. This insulating magnetic phase is favored by low spin, low dimensionality, and magnetic frustration. The preexisting magnetic singlet pairs of the insulating state become charged superconducting pairs when the insulator is doped sufficiently strongly. The mechanism for superconductivity is hence predominantly electronic and magnetic, although weak phonon interactions may favor the state. Many unusual properties are predicted, especially of the insulating state.Science 04/1987; 235(4793):1196-8. · 31.03 Impact Factor
Spin Dynamics of the Spin-1=2 Kagome Lattice Antiferromagnet ZnCu3?OH?6Cl2
J.S. Helton,1K. Matan,1M.P. Shores,2E.A. Nytko,2B.M. Bartlett,2Y. Yoshida,3Y. Takano,3A. Suslov,4Y. Qiu,5
J.-H. Chung,5D.G. Nocera,2and Y.S. Lee1
1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
3Department of Physics, University of Florida, Gainesville, Florida 32611, USA
4National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
5NIST Center for Neutron Research, Gaithersburg, Maryland 20899, USA
and Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA
(Received 19 October 2006; published 9 March 2007)
We have performed thermodynamic and neutron scattering measurements on the S ? 1=2 kagome ´
lattice antiferromagnet ZnCu3?OH?6Cl2. The susceptibility indicates a Curie-Weiss temperature of ?CW’
?300 K; however, no magnetic order is observed down to 50 mK. Inelastic neutron scattering reveals a
spectrum of low energy spin excitations with no observable gap down to 0.1 meV. The specific heat at
low-T follows a power law temperature dependence. These results suggest that an unusual spin liquid state
with essentially gapless excitations is realized in this kagome ´ lattice system.
DOI: 10.1103/PhysRevLett.98.107204PACS numbers: 75.40.Gb, 75.25.+z, 78.70.Nx
An important challenge in condensed matter physics is
the search for quantum disordered ground states in two-
dimensional systems. Of particular interest is studying
quantum spin liquids, an example of which is the ‘‘reso-
nating valence bond’’ state proposed by Anderson .
These states are unusual in that neither translational nor
spin rotational symmetries are broken. It is believed that
the S ? 1=2 Heisenberg antiferromagnet on a kagome ´
lattice (composed of corner sharing triangles) is an ideal
system to look for spin liquid physics due to the high
degree of frustration. There is broad theoretical consensus
that the ground state of the S ? 1=2 kagome ´ antiferromag-
net is not magnetically ordered [2–8]. However, many
basic properties are still under debate, such as the magni-
tude of the gap to the first triplet state. An intriguing
possibility is the existence of deconfined S ? 1=2 spinons
as the fundamental excitations, as opposed to conventional
S ? 1 magnons.
Despite heavy theoretical interest, experimental studies
of the S ? 1=2 kagome ´ lattice have been hampered by
the difficulty in synthesizing such materials. Here, we
report thermodynamic and neutron scattering measure-
ments on powder samples of ZnCu3?OH?6Cl2, known as
herbertsmithite . As has been previously reported
, ZnxCu4?x?OH?6Cl2can be synthesized with vari-
able Zn concentration, from x ? 0 to x ? 1 (herbert-
smithite). Figure 1(a) represents the transformation from
Cu2?OH?3Cl, which has a distorted pyrochlore structure, to
ZnCu3?OH?6Cl2, which consists of Cu kagome ´ layers
separated by nonmagnetic
ZnCu3?OH?6Cl2, with space group R?3m and lattice pa-
rameters a ? b ? 6:832?A and c ? 14:049?A, appears to
be an excellent realization of the S ? 1=2 kagome ´ lattice
antiferromagnet. Initial evidence is the absence of long-
range magnetic order, as shown in the neutron diffrac-
tion scans in Fig. 1(b). In Cu2?OH?3Cl, clear magnetic
Bragg peaks are observed below ?6 K; whereas no
magnetic Bragg peaks are observable down to 1.8 K in
ZnCu3?OH?6Cl2, we performed magnetic susceptibility
measurements on powder samples. The susceptibility,
shown in Fig. 1(c), can be fit to a Curie-Weiss law at
high temperatures (T > 200 K). The resulting Curie-
Weiss temperature of ?300 ? 20 K implies an antiferro-
magnetic exchange J ’ 17 meV, calculated using the se-
ries expansion corrections for the kagome ´ lattice [11–13].
The susceptibility continually increases as the temperature
is lowered downto 1.8 K. At first glance, this behavior may
suggest the presence of several percent free spin-1=2 im-
purities yielding a Curie tail. This is certainly possible, but
is not necessarily the case. From the chemical analyses, we
calculate the stoichiometric coefficients to be 3:00 ? 0:04
on the Cu site and 1:00 ? 0:04 on the Zn site. Also, we
have measured the ac susceptibility at temperatures down
to 50 mK, as shown in the inset of Fig. 1(c). These data do
not follow the simple Brillouin function behavior expected
for free S ? 1=2 spins. In particular, the susceptibility
increase from 705 to 50 mK is much smaller than the
free spin prediction. Recently, Ofer and co-workers 
have shown that the muon Knight shift and transverse
relaxation rate have T dependences similar to the measured
susceptibility. Hence, the measured susceptibility may be
intrinsic to the Cu kagome ´ system. We note that similar
behavior is found for the frustrated S ? 1=2 nuclear mo-
ments of3He films on graphite, where the susceptibility is
found to continually increase with decreasing temperature
down to T ? J=300 . Another recent ?SR study 
emphasizes the role of defects. The roles of impurities and
exchange or Dzyaloshinskii-Moriya  anisotropies in
PRL 98, 107204 (2007)
PHYSICAL REVIEW LETTERS
9 MARCH 2007
© 2007 The American Physical Society
this system remain important topics for further investiga-
tion. We also observe a small peak in the ac susceptibility
near H ? 2 T at 50 mK which disappears upon warming to
705 mK. The overall susceptibility data indicate the ab-
sence of magnetic order or a spin gap down to 50 mK.
The specific heat C?T? of ZnCu3?OH?6Cl2is shown in
Fig. 2(a) in various applied fields. For temperatures of a
few Kelvin and higher, the lattice contribution to the
specific heat (proportional to ?T3) is the most significant
contribution, as shown in the inset. However, this contri-
bution diminishes at low temperatures, and below ?5 K,
an additional contribution is clearly observed which arises
from the Cuspinsystem.Magnetic fieldsofa fewTesla can
significantly affect the low-T behavior, and fields of 10 T
and higher strongly suppress the specific heat below 3 K.
The difficulty in synthesizing an isostructural nonmagnetic
compound makes it hard to subtract the lattice contribution
precisely. However, the magnetic field dependence sug-
gests that the specific heat in zero applied field below1Kis
predominately magnetic in origin. As a rough measure of
the spin entropy, the field-induced change in specific heat
below 3 K, obtained by subtracting the 14 T data from the
zero field data, accounts for about 5% of the total entropy
of the spin system.
Additional specific heat measurements at zero field at
temperatures down to 106 mK were performed at the
National High Magnetic Field Laboratory (NHMFL) and
the combined data are shown in Fig. 2(b). The specific heat
at low temperatures (T < 1 K) appears to be governed by a
power law with an exponent which is less than or equal to
1. In a 2D ordered magnet, magnon excitations would give
C ? T2. The kagome ´-like compound SrCr8?xGa4?xO19
(SCGO)  and other 2D frustrated magnets  are
also observed to have C ? T2even in the absence of
long-range order [20,21]. The behavior that we observe
in ZnCu3?OH?6Cl2below 1 K stands in marked contrast.
We can fit our data to the power law C ? ?T?, though we
note that the exponent ? is sensitive to the chosen range of
temperatures that are fit. The blue line in this figure repre-
sents a linear fit with ? ? 1 over the temperature range
106 mK < T < 400 mK. The fitted value for ? is 240 ?
20 mJ=K2Cu mole. If we include higher temperatures, the
red line represents a fit with ? ? 2=3 over the temperature
range 106 mK < T < 600 mK. Extending the fitted range
to even higher temperatures can yield ? values as low as
Finally, inelastic neutron scattering measurements of the
low energy spin excitations were performed on deuterated
powder samples of ZnCu3?OD?6Cl2. High resolution mea-
ZnCu3?OH?6Cl2 in various applied fields, measured using a
Physical Property Measurement System. Inset: C?T? plotted
over a wider temperature range in applied fields of 0 T (square)
and 14 T (star). (b) C?T? in zero field measured down to 106 mK.
The lines represent power law fits as described in the text.
FIG. 1 (color online).
the pyrochlorelike lattice of Cu2?OH?3Cl to the kagome ´ layers of
ZnCu3?OH?6Cl2. (b) Magnetic diffraction scans of the two
T ? 1:4 K
Cu2?OH?3Cl data show magnetic Bragg peaks at Q ’ 0:70 and
Q ’ 0:92, which are absent for the ZnCu3?OH?6Cl2data (which
have been shifted by 2300 counts=min for clarity). (c) Magnetic
susceptibility of ZnCu3?OH?6Cl2 measured using a SQUID
magnetometer plotted as 1=?, where mole refers to a formula
unit. The line denotes a Curie-Weiss fit. Inset: ac susceptibility
(at 654 Hz) at low temperatures measured at the NHMFL in
(a) The chemical transformation from
(open) and20 K (filled).The
PRL 98, 107204 (2007)
9 MARCH 2007
surements were taken on the time-of-flight Disk Chopper
Spectrometer (DCS) at the NIST Center for Neutron
Research in Gaithersburg, MD. A sample with mass 9 g
was cooled in a dilution refrigerator and studied with
incident neutrons of wavelength 7 A˚, yielding an instru-
mental energy resolution of 0.02 meV (half-width at half-
maximum). As shown in Fig. 3(a), the spin excitations
form a broad spectrum at low energies. A notable obser-
vation is the near temperature independence of the scatter-
ing for positive energy transfers. The excitation spectrum
on the negative energy-transfer side is suppressed at low
temperatures due to detailed balance.
The magnetic scattering intensity is proportional to the
dynamic structure factor S?~Q;!? ? ?n?!? ? 1??00?~Q;!?,
where n?!? is the Bose occupation factor and ?00?~Q;!? is
the imaginary part of the dynamical susceptibility. We find
that part of the measured intensity for positive energy
transfers below 0.4 meVis spurious background scattering,
probably caused by multiple scattering of neutrons within
the sample environment. Therefore, to extract the intrinsic
scattering from the sample, the following procedure was
used. For negative energy transfers, ?00?!;T ? 10 K? can
be obtained by subtracting the 35 mK data (which is
essentially background) from the 10 K data and dividing
by the Bose factor. Here, ?00?!? represents the dynamical
susceptibility integrated over momentum transfers 0:25 ?
j~Qj ? 1:5?A?1and is a good measure of the local response
function. This is plotted in Fig. 3(b), where the positive !
data is obtained by using the fact that ?00?!? is an odd
function of !. Then, ?00?!;T ? 35 mK? can be ex-
tracted from the positive energy-transfer data using
S?!;T ?35 mK??S?!;T ?10 K??I?!;T ?35 mK? ?
I?!;T ?10 K?, where I?!? is the measured intensity and
the background is assumed to be temperature independent
between 35 mK and 10 K. As seen in Fig. 3(b), the data for
?00?!? at T ? 35 mK increase with decreasing !, indicat-
ing the absence of a spin gap down to 0.1 meV. Moreover,
the data may be described by a simple power law; the solid
line represents a fit to the form ?00?!? / !?with an
exponent ? ? ?0:7 ? 0:3. This apparently divergent be-
havior is unusual and again differs markedly from mea-
surements on SCGO  which yield ? ’ 0. Of course,
within the errors, we cannot rule out other functional forms
The Q dependence of the scattering is shown in
Fig. 3(c). These data were obtained by integrating over
energy transfers ?0:5 ? @! ? ?0:22 meV and subtract-
ing the 35 mK data set from the 10 K data set. We find that
the data appear to be only weakly dependent on j~Qj. Note
that due to the polycrystalline form of the sample, the data
represents the powder average of the scattering from a
crystal. The solid line represents the squared form factor
jFj2for the Cu2?ion. The deviations of the data from jFj2
suggest that the structure factor is not completely indepen-
dent of j~Qj. That is, some degree of spin correlations are
necessary to account for the relative reduction in scattering
at small j~Qj. The overall diffuse nature of the scattering
points to the absence of a well-defined length scale to
describe these correlations.
Further measurements were taken using the triple-axis
SPINS spectrometer at the NCNR with the sample
mounted inside a superconducting magnet, as shown in
Fig. 3(d). The instrument was configured in a horizontally
focusing analyzer geometry with Ef? 3:05 meV and col-
limations of guide-800-radial-open. A BeO filter was
placed in the scattered beam to reduce higher-order neu-
FIG. 3 (color online).
taken on DCS, integrated over momentum transfers 0:25 ?
j~Qj ? 1:5?A?1. (b) ?00?!?, extracted from the data as described
in the text. The line denotes a power law fit. (c) The
Q-dependence of the scattering, integrated over energy transfers
?0:5 ? @! ? ?0:22 meV. (d) Energy scans taken on SPINS at
zero field and 11.5 Tat j~Qj ? 0:6?A?1and T ? 1:2 K. The lines
are guides to the eye. Inset: Temperature dependence of the
scattering for 0:3 ? @! ? 0:5 meV and j~Qj ? 0:9?A?1. The
blue data point and line indicate the background, measured on
the energy loss side at T ? 1:5 K.
(a) Inelastic neutron scattering data
PRL 98, 107204 (2007)
9 MARCH 2007
tron contamination. The resulting instrumental energy
resolution was about 0.06 meV. An applied field of
11.5 T transfers spectral weight from lower to higher
energies. This demonstrates that a significant fraction of
the low energy scattering is magnetic in origin, since the
incoherent and phonon background would not respond to
the applied field in this manner. The magnetic signal in
zero field extends down to below 0.2 meV, consistent with
the analysis of the above DCS data. In 11.5 T, the magnetic
signal becomes peaked around @! ’ 1:4 meV, which is
close to the Zeeman energy g?BH. However, the half-
width of this peak of about 0.21 meV is significantly
broader than the resolution. Therefore, the peak does not
simply originate from Zeeman excitations of noninteract-
ing spins, which would result in a narrow energy peak, but
involve spins which are part of the interacting system. The
integrated spectral weight of the zero field magnetic signal
for @! < 1 meV accounts for at most 20% of the total
scattering expected from a S ? 1=2 spin system (an esti-
mate made by normalization to the incoherent elastic
scattering from the sample and also to a vanadium stan-
dard). The inset of Fig. 3(d) shows the temperature depen-
dence of the inelastic signal with energy transfers
integrated over the range 0:3 ? @! ? 0:5 meV. There is
a small increase in the signal when the sample is cooled
below ?5 K, though, for the most part, the intensity is
largely independent of temperature in this range.
Our experimental results suggest an intriguing picture
for the ground state properties of the S ? 1=2 kagome ´
lattice antiferromagnet. A hallmark of the quantum spin
liquid in two dimensions is the existence of deconfined
S ? 1=2 spinons as the fundamental magnetic excitation.
A rich variety of spin liquid states have been theoretically
proposed in which the spinons can be described as bosonic
[6,8,23], fermionic [24,25], or even as Dirac fermions .
We note that several of these theories are based on trian-
gular lattice Hamiltonians, and they may not have clear
extensions to the kagome ´ lattice antiferromagnet. Using a
naive comparison to a generic model of fermionic spinons
with a Fermi surface, one would expect C ? ?T. From our
linear fit below 400 mK, the value of ? indicates a Fermi
temperature of TF? 110 K. However, other forms for the
specific heat (such as C / T2) may hold at higher
temperatures where the lattice contribution prevents us
from clearly identifying the magnetic contribution.
The neutron scattering measurements of the excitation
spectrum at low temperatures are also consistent with
expectations of deconfined spinons in a spin liquid. We
find no evidence of a spin gap down to ?J=170, much
lower than the prediction from exact diagonalization stud-
ies for a spin gap of ?J=20 . The power law behavior of
?00?!? is interesting and may indicate a spin liquid with
critical spin correlations . Our observation of a diffuse
Q dependence for the inelastic scattering suggests that if a
singlet spin liquid picture is correct, then the singlets are
not restricted to nearest neighbor dimers, since no well-
defined length scale is indicated by the data. The near
temperature independence of S?~Q;!?, similar to observa-
tions in f-electron systems , may indicate the proxim-
ity to a quantum critical point. Many of the current theories
for 2D spin liquids were formulated to describe experi-
mental results [29,30] for S ? 1=2 triangular lattice sys-
tems. More theoretical studies based explicitly on the
S ? 1=2 kagome ´ Heisenberg antiferromagnet (including
the possible effects of impurities and exchange or
Dzyaloshinskii-Moriya anisotropies) are certainly impor-
tant for further comparisons with experimental results.
We thank P.A. Lee, A. Keren, J.W. Lynn, Q. Huang,
T. Senthil, and X.-G. Wen for useful discussions and
E. Palm and T. Murphy for help with the measurements
at the NHMFL. The work at MIT was supported by the
NSF under Grant No. DMR 0239377, and in part by the
MRSEC program under Grant No. DMR 02-13282. This
work used facilities supported in part by the NSF under
Agreement No. DMR-0454672. A portion of this work was
performed at the NHMFL, which is supported by NSF
Cooperative Agreement No. DMR-0084173, by the State
of Florida, and by the DOE.
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