Refining the spin Hamiltonian in the spin-1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2 using single crystals.
ABSTRACT We report thermodynamic measurements of the S=1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2, a promising candidate system with a spin-liquid ground state. Using single crystal samples, the magnetic susceptibility both perpendicular and parallel to the kagome plane has been measured. A small, temperature-dependent anisotropy has been observed, where χ(z)/χ(p)>1 at high temperatures and χ(z)/χ(p)<1 at low temperatures. Fits of the high-temperature data to a Curie-Weiss model also reveal an anisotropy. By comparing with theoretical calculations, the presence of a small easy-axis exchange anisotropy can be deduced as the primary perturbation to the dominant Heisenberg nearest neighbor interaction. These results have great bearing on the interpretation of theoretical calculations based on the kagome Heisenberg antiferromagnet model to the experiments on ZnCu3(OH)6Cl2.
- SourceAvailable from: Jose Abelardo Rodriguez Rivera[show abstract] [hide abstract]
ABSTRACT: The experimental realization of quantum spin liquids is a long-sought goal in physics, as they represent new states of matter. Quantum spin liquids cannot be described by the broken symmetries associated with conventional ground states. In fact, the interacting magnetic moments in these systems do not order, but are highly entangled with one another over long ranges. Spin liquids have a prominent role in theories describing high-transition-temperature superconductors, and the topological properties of these states may have applications in quantum information. A key feature of spin liquids is that they support exotic spin excitations carrying fractional quantum numbers. However, detailed measurements of these 'fractionalized excitations' have been lacking. Here we report neutron scattering measurements on single-crystal samples of the spin-1/2 kagome-lattice antiferromagnet ZnCu(3)(OD)(6)Cl(2) (also called herbertsmithite), which provide striking evidence for this characteristic feature of spin liquids. At low temperatures, we find that the spin excitations form a continuum, in contrast to the conventional spin waves expected in ordered antiferromagnets. The observation of such a continuum is noteworthy because, so far, this signature of fractional spin excitations has been observed only in one-dimensional systems. The results also serve as a hallmark of the quantum spin-liquid state in herbertsmithite.Nature 12/2012; 492(7429):406-10. · 38.60 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: At low temperatures, a spin ice enters a Coulomb phase - a state with algebraic correlations and topologically constrained spin configurations. In Ho2Ti2O7, we have observed experimentally that this process is accompanied by a non-standard temperature evolution of the wave vector dependent magnetic susceptibility, as measured by neutron scattering. Analytical and numerical approaches reveal signatures of a crossover between two Curie laws, one characterizing the high temperature paramagnetic regime, and the other the low temperature topologically constrained regime, which we call the spin liquid Curie law. The theory is shown to be in excellent agreement with neutron scattering experiments. On a more general footing, i) the existence of two Curie laws appears to be a general property of the emergent gauge field for a classical spin liquid, and ii) sheds light on the experimental difficulty of measuring a precise Curie-Weiss temperature in frustrated materials; iii) the mapping between gauge and spin degrees of freedom means that the susceptibility at finite wave vector can be used as a local probe of fluctuations among topological sectors.Physical Review X. 04/2012; 3(1).
arXiv:1202.4729v1 [cond-mat.str-el] 21 Feb 2012
Refining the Spin Hamiltonian in the Spin-1
ZnCu3(OH)6Cl2using Single Crystals
2Kagome Lattice Antiferromagnet
Tianheng Han1,‡, Shaoyan Chu2, and Young S. Lee1,‡
1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA and
2Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Dated: February 22, 2012)
Wereport thermodynamicmeasurements ofthe S=1
ZnCu3(OH)6Cl2, a promising candidate system with a spin liquid ground state. Using single crystal
samples, the magnetic susceptibility both perpendicular and parallel to the kagome plane has been
measured. A small, temperature-dependent anisotropy has been observed, where χz/χp > 1 at high
temperatures and χz/χp < 1 at low temperatures. Fits of the high-temperature data to a Curie-
Weiss model also reveal an anisotropy. By comparing with theoretical calculations, the presence of
a small easy-axis exchange anisotropy can be deduced as the primary perturbation to the dominant
Heisenberg nearest neighbor interaction. These results have great bearing on the interpretation of
theoretical calculations based on the kagome Heisenberg antiferromagnet model to the experiments
PACS numbers: 75.30.Gw 75.40.Cx 75.10.Kt 75.50.Ee
The quantum spin liquid, a fundamentally new state of
matter whose ground state that does not break conven-
tional symmetries, has generated much interest in con-
densed matter physics1,2. It has long been realized that
2Heisenberg antiferromagnet on the kagome lat-
tice (composed of corner sharing triangles) is an ideal
system in which to look for spin liquid physics due to
the high degree of frustration, small spin, and low di-
mensionality. Herbertsmithite, the x = 1 end member of
the family Zn-paratacamite (ZnxCu4−x(OH)6Cl2), is ar-
guably one of the best candidate systems to study quan-
tum spin liquids.3With weak interplane coupling, it con-
sists of kagome planes of Cu2+ions separated by layers of
non-magnetic Zn2+ions, depicted in Fig. 1(a). The cur-
rent experimental evidence is consistent with the pres-
ence of a spin liquid ground state in this material4–7.
The Hamiltonian of herbertsmithite consists of a Heisen-
berg exchange term, with possible perturbations such
as a Dzyaloshinskii-Moriya (DM) interaction8–11and ex-
change anisotropy12. With a Cu-O-Cu antiferromagnetic
superexhange interaction of approximately 17 meV, no
magnetic transition or long range ordering has been ob-
served down to T = 50 mK4,5,13.
perform measurements on single crystal samples so that
comparisons can be made to theoretical calculations as-
suming different perturbations to the Hamiltonian10,11,
such as a DM interaction and exchange anisotropy, to
determine the presence and magnitude of such pertur-
bations. Resolving this issue is all the more pressing in
light of recent theoretical work which strongly points to
a spin-liquid ground state for the S=1/2 kagome lattice
with isotropic (Heisenberg) exchange.14
It is important to
Recently, large single crystal samples of the parat-
acamite family, including herbertsmithite, have been
synthesized15. A powder sample was first synthesized
inside a sealed quartz tubing and transported under a
temperature gradient in a three zone furnace for crys-
tallization.The high quality of the crystals was con-
firmed by ICP analysis, x-ray diffraction, neutron diffrac-
tion, polarized optics and thermodynamic measurements.
FIG. 1: (color online) (a) Structure of herbertsmithite with
Cu2+(big brown spheres) and Zn2+(small red spheres) dis-
played.The Cu-Cu bonds (thick black solid lines) are all
equivalent as are the Cu-Zn bonds (thin green dotted lines).
(b) The oriented single crystal sample (mass = 55.5 mg) of
herbertsmithite used in the magnetic susceptibility measure-
Anomalous synchrotron x-ray diffraction confirmed the
absence of anti-site disorder where Zn2+ions appear on
the Cu sites in the kagome layer16. Rather, the main
source of disorder is the presence of a small fraction of
excess Cu ions within the Zn interlayers. Raman spec-
troscopy provides further support of a gapless spin liquid
ground state6while µSR points to an easy axis anisotropy
parallel to c-axis for magnetization17. In this paper, the
magnetic susceptibility and specific heat have been in-
vestigated with fields applied both within and normal to
the kagome plane. The roles of an easy axis exchange
anisotropy, a Dzyaloshinskii-Moriya interaction, and an
anisotropic g-factor for the Cu magnetic moment are dis-
Magnetic susceptibility measurements were performed
on a 55.5 mg single crystal sample of herbertsmithite (2.3
mm × 2.5 mm × 2.7 mm), shown in Fig. 1(b), using a
SQUID magnetometer (Quantum Design). The nearly
cubic shape of the sample minimizes a demagnetization
corrections to the measurements, allowing for a clean
measurement of the intrinsic anisotropy of the material.
The crystalline axis and the narrow mosaic of the sample
were confirmed using an x-ray diffractometer equipped
with an area detector. A plastic holder was designed and
made for securing the crystal for susceptibility measure-
ments with magnetic field applied perpendicular (χp) or
parallel (χz) to the crystalline c-axis. The background
from the plastic holder was measured to be negligibly
small relative to the signal from the sample. In Fig. 2(a),
the quantities χzT and χpT for temperatures between
2 K and 330 K are plotted (where we assume χ = M/H
in the linear regime). The quantity χpowderT, measured
on a polycrystalline collection of several dozen random
orientated crystals from the same batch, is plotted along
with χaverageT =1
3(χz+ 2χp)T, the calculated powder
average. The latter two collapse onto the same curve as
expected, pointing to the reliability of the single crystal
In Fig. 2(b), the anisotropy ratio of the magnetic sus-
ceptibility calculated as χz/χp is plotted. As temper-
ature is increased from 2 K to 330 K, the ratio in-
creases from 0.96 to 1.12 monotonically. The presence of
anisotropy in the susceptibility agrees qualitatively with
susceptibility measurements on aligned powders18and re-
cent µSR measurement on single crystals17. In Fig. 2(c),
magnetization measurements taken at T = 5 K and 300 K
are plotted as a function of applied field. At T = 5K, the
anisotropy ratio is close to unity and the two curves over-
lap for the entire field range. At T = 300 K, there is clear
anisotropy with the c-axis being the higher susceptibility
direction. The observed magnetic anisotropy is indepen-
dent of the applied field.
The high quality of the susceptibility data allows for
further analysis to better understand the intrinsic be-
havior of the interacting spins on the kagome layers.
The primary results of this paper are shown in Fig. 3.
For the susceptibility data at high temperatures, Curie-
Weiss fits were performed for each data set taken at var-
ious fields. The Curie-Weiss temperatures and g-factors
determined from the fits (which take into account the
corrections based on high temperature series expansion
calculations19,20) are plotted in Fig. 3(a). For both field
orientations, the Curie-Weiss temperatures and g-factors
increase slightly upon lowering the applied field below
≃ 0.2 T. At µ0H= 1 T, the anisotropy ratio
for the g-factor is gz/gp= 1.10 at T = 330 K. A simi-
lar g-factor anisotropy, though slightly smaller, has been
deduced from ESR work21on powders.
It is important to separate out the anisotropy of the
Cu moments intrinsic to the kagome planes from that
related to the impurity spins. The experimentally mea-
sured magnetic susceptibility originates from both the
kagome plane and the weakly interacting Cu2+impuri-
ties on the interlayer sites. Assuming that the intrinsic
kagome susceptibility becomes much smaller than the im-
purity contribution as T → 0 K, consistent with recent
NMR measurement on single crystal samples22, we model
the impurity susceptibility with a Curie-Weiss law where
ΘCW ≃ 1.3 K23. The best fit gives an estimated 10%
Cu ions which occupy the interlayer sites for this non-
deuterated sample. Then, by assuming a temperature
independent anisotropy for the impurities, the impurity
FIG. 2: (color online) (a) Magnetic susceptibility, plotted as
MT/H = χT, measured under an applied field of µ0H = 1 T
which was oriented both perpendicular to and along the c-
axis.The data from a powder sample is also plotted and
compared to the “average” of the single crystal results, as de-
scribed in the text. (b) The anisotropy χz/χpof the measured
susceptibility plotted as a function of temperature. (c) Mag-
netization versus field measurements at T = 5 K and 300 K.
The vertical scale for each temperature is indicated by the
contribution to the susceptibility can be subtracted re-
vealing the anisotropy of the intrinsic kagome spins. The
only remaining free parameter is the anisotropy ratio for
the impurities (χz/χp)imp, and in our analysis, we use
the value (χz/χp)imp= 1.
The deduced anisotropy of the susceptibility for the
intrinsic kagome spins is plotted in Fig. 3(b). The main
observation, which is relatively insensitive to the model
parameters, is that χz/χpfor the intrinsic susceptibility
is a monotonically increasing function of temperature for
T > 150 K. This provides useful information on the im-
portance of additional terms in the spin Hamiltonian, as
we discuss further below. Moreover, since we have de-
duced the anisotropy of the g-factor resulting from the
Curie-Weiss analysis, we can correct for this in deter-
mining χz/χpratio for the intrinsic kagome spins. The
g-factor corrected data is also plotted in Fig. 3(b). At low
temperatures (below ∼ 5 K) where the impurity contri-
bution dominates the susceptibility, the measured ratio
for χz/χp is actually less than 1. If we assume a value
FIG. 3: (color online) (a) Curie-Weiss temperatures and g-
factors calculated from fits of the magnetic susceptibilities
between T = 150 K and 330 K, as described in the text.
The proper vertical scale for each data set is indicated by the
arrow. (b) The susceptibility anisotropy ratio of the intrin-
sic kagome spins after subtracting the impurity contribution,
with and without a correction for the g-factor anisotropy, as
described in the text. In the model for subtracting the impu-
rity contribution, temperature independent anisotropy ratios
(χz/χp)imp were assumed. The three curves represent exact
diagonalization calculations11for the anisotropy ratio consid-
ering the effects of an easy-axis exchange anisotropy and a
DM interaction separately.
of (χz/χp)imp = 0.95, the deduced anisotropy ratio for
the intrinsic kagome spins exhibits a slight upturn as the
temperature is cooled below T ≈ 100 K. However, the
main conclusions of our analysis based on the data for
temperature above T = 150 K are not quantitatively
Our experimental results shed light on the roles played
by various perturbations to the spin Hamiltonian beyond
the nearest neighbor Heisenberg model. The observed
anisotropy of the intrinsic susceptibility can be compared
with theoretical calculations using 15-site exact diagonal-
ization (ED) by Rigol and coworkers11. If an easy-axis
exchange anisotropy HEA=∆Σi,j(Sx
∆ < 0 is considered, the shape of the anisotropy ver-
sus temperature curve matches our measurements over a
wide temperature range, as shown in Fig.3(b). In fact,
comparing our g-factor corrected data with the calcula-
tion with ∆ = −0.1 J give a good match for the slope for
T > 150 K as well as the magnitude for χz/χp. The pres-
FIG. 4: (color online) (a) Low temperature specific heat data
on a single crystal sample of herbertsmithite measured un-
der various applied fields with two orientations.
anisotropy ratio of the specific heat measured in the two field
ence of an anisotropic exchange is consistent with recent
µSR measurements on single crystal herbertsmithite17
and work on partially aligned powders18. The difference
in the Curie-Weiss temperatures for the two field orienta-
tions are also consistent with the deduced magnitude of
the easy-axis exchange anisotropy. That is, in Fig.3 (a),
ΘCW for the field along the c-axis is larger than that
for the field within the kagome plane by about 10%, as
one would expect for an easy-axis exchange anisotropy of
∆ ≈ −0.1 J.
The DM interaction, HDM=Σi,jDz(?Si×?Sj)z+?Dp·
(?Si ×?Sj), has a much smaller effect on the anisotropy
ratio11. For a wide range of Dzand Dpvalues (the out-
of-plane and the in-plane components of the DM vector,
respectively), the primary effect is to slightly increase the
anisotropy ratio from unity, where χz/χpmonotonically
decreases as temperature increases. The results of two
model calculations which only include a DM term are
plotted (one with Dz= 0.2J and one with Dp= 0.2J).
Our data appear to rule out such scenarios where only
a DM term is present, as a small easy-axis exchange
anisotropy is needed to give the observed χz/χp< 1 as
well as the temperature dependence.
The specific heat was measured on a 4.10 mg single
crystal sample of herbertsmithite using a Quantum De-
sign Physical Property Measurement System (PPMS).
The sample was prepared so that its orientation could
be changed in-situ without re-measuring the background.
The specific heat was measured in two field orientations:
with the field oriented in the kagome plane Cpand per-
pendicular to the plane Cz. The data for a wide range of
applied fields up to µ0H = 14 T are plotted in Fig. 4(a).
The ratio Cz/Cpis plotted in Fig. 4(b) which reveals a
small magnetocaloric anisotropy. As a check of system-
atic errors in the measurement, the data measured in
zero field, taken with both crystal orientations, collapse
onto the same line. At temperatures higher than 15 K
(not shown), the specific heat under all applied magnetic
fields are identical for both orientations, within error.
At the lowest temperatures (below ∼ 10 K), it is likely
that the magnetocaloric anisotropy derives from the im-
purities. The observed Cz/Cp> 1 for kBT ≪ µ0H indi-
cates that the impurities are easier to polarize with an in-
plane field. This idea is further supported by the observa-
tion in Fig. 2(b) that the ratio χz/χpbegins to decrease
very rapidly upon cooling below T = 5 K. This indicates
the impurity moments develop a g-factor anisotropy with
(χz/χp)imp< 1 at low temperatures. As another possi-
bility, it has been shown that the presence of a DM inter-
action can mix the triplet and singlet states10so that the
total spin is not a good quantum number. The observed
field independence of the anisotropy for the susceptibil-
ity coupled with the field dependence of anisotropy of
the specific heat point to the possibility that the singlet
states may be coupled to the applied field. Further the-
oretical calculations would be useful to determine how
the thermodynamic quantities should behave under the
application of in-plane and out-of-plane fields.
In summary, we have measured the anisotropy of
the magnetic susceptibility in a single crystal sample
of herbertsmithite. The temperature dependence of the
anisotropy allows one to deduce the important additional
terms in the spin Hamiltonian beyond nearest neighbor
Heisenberg exchange. A comparison with previous ex-
act diagonalization calculations indicates the presence of
an easy-axis exchange anisotropy with ∆ ≈ −0.1 J. This
small value for the anisotropy indicates that the Heisen-
berg model is a reasonable approximation to define the
physics of herbertsmithite. However, calculations start-
ing from the Ising limit and approaching the Heisenberg
limit for the S=1/2 kagome antiferromagnet may provide
useful insight into the behavior of herbertsmithite. A
field- and temperature- dependent anisotropy in the spe-
cific heat measured in different field orientations is also
observed. Further theoretical calculations which include
an anisotropic exchange interaction as well as a DM in-
teraction would be most useful for a detailed comparison
with the data.
We thank D. G. Nocera, A. Keren, J. S. Helton, M.
Rigol, S. Todadri, and P. Lee for useful discussions. This
work was supported by the Department of Energy (DOE)
under Grant No. DE-FG02-07ER46134.
‡email: firstname.lastname@example.org, email@example.com
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