Table Of ContentTypeset with jpsj2.cls <ver.1.2.2> Full Paper
Quantum Kagome antiferromagnet ZnCu (OH) Cl .
3 6 2
Philippe Mendels1∗ and Fabrice Bert1
1Laboratoire de Physique des Solides, Universit´e Paris-Sud XI Orsay, UMR 8502 CNRS, 91405
Orsay Cedex, France
0
1
0
The frustration of antiferromagnetic interactions on the loosely connected kagome lattice
2
associated to the enhancement of quantum fluctuations for S=1/2 spins was acknowledged
n
a long ago as a keypoint to stabilize novel ground states of magnetic matter. Only very re-
J
cently,themodelcompoundHerbersmithite,ZnCu (OH) Cl ,astructurallyperfectkagome
6 3 6 2
antiferromagnet,couldbesynthesizedandenablesaclosecomparisontotheories.Wereview
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l andclassify variousexperimentalresults obtainedoverthe pastyearsandunderline some of
e
- the pending issues.
r
t
s
KEYWORDS: frustration, kagome, spin liquids, quantum magnetism
.
t
a
m
-
d
n 1. Introduction
o
c After the initial papers by Anderson and Fazekas in 1973-741) and the revival of their
[
seminalmainstream proposalof aresonatingvalence bondgroundstate inthecontext of High
1
v Temperature Superconductors,2) kagome antiferromagnets have been recognized since the
1
0 early 90’s as the corner stone of the search for a quantum spin liquid state. Strong indications
8
0 for very original physics had been found in some experimental systems with a kagome based
.
1 geometry, such as magnetic freezing occuring only at very low temperatures on the scale of
0
0 the exchange energy and fluctuating ground states, two features recognized as the landmarks
1
of frustration.3,4) Only recently, a ”New Candidate emerged for a quantum spin liquid”.5)
:
v
i Namely, Herbertsmithite, a rare mineral6) which has been synthesized for the first time in
X
20057) hasbroughtatremendousexcitement tothefield.Altogether withκ(ET) Cu (CN) ,it
r 2 2 3
a
combineslowdimensionality,aquantumcharacterassociatedwithspins1/2andtheabsenceof
magneticfreezing.Thiswascelebratedas”AnEndtotheDroughtofQuantumSpinLiquids”8)
in the context of a grail quest for the synthesis of a kagome model system to compare with
theories. Although theoretical models have been developed over the last 20 years, there is still
no consensus about the expected ground state. Most of the models point at an exotic ground
state and two broad classes of spin liquids have been explored theoretically, in particular for
thekagomeantiferromagnet, (i)topologicalspinliquidswhichfeatureaspingap,atopological
orderandfractional excitations;9) (ii)algebraic spinliquidswith gapless excitations in all spin
sectors.10)
∗
E-mail: [email protected]
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Here,wefocusonthereviewoftheexperimentalpropertiesofHerbertsmithiteanddiscuss
the deviations to the ideal case and their implications with regards to the important issue of
the existence of a gap.
2. Structure, interactions and phase diagram of the generic family of parata-
camites, ZnxCu4−x(OH)6Cl2, 0 < x < 1
Herbertsmithite ZnCu (OH) Cl is the x = 1 end compound of the Zn-paratacamite fam-
3 6 2
ily ZnxCu4−x(OH)6Cl2.6,7) It can be viewed as a double variant of the parent clinoatacamite
compound (x = 0), situated at the other end of the family. Starting from the latter, a Jahn-
Teller distorted S = 1/2 pyrochlore, the symmetry first relaxes from monoclinic (P2 /n) to
1
rhombohedral (R3m) around x = 0.33, leading to a perfect kagome lattice in the a-b plane
with isotropic planar interactions; then, in the c- elongated x > 0.33 pyrochlore structure,
the magnetic bridge along c-axis between a-b kagome planes is progressively suppressed by
replacing the apical Cu2+ by a diamagnetic Zn2+. Due to a more favorable electrostatic en-
vironment, Cu2+ is expected to preferentially occupy the distorted octahedral kagome sites
only. When x = 1 the S = 1/2 ions should therefore form structurally perfect kagome layers
that are themselves well separated by diamagnetic Zn2+(Fig. 1).
2.1 Clinoatacamite
In clinoatacamite (x = 0) two successive transitions to long-range ordered phases are
observed at 18 K and 6 K. They are signalled by the development of well defined internal
fields in µSR,11) by peaks in specific heat and accidents in magnetic susceptibility.12) At
the 6 K transition a ferromagnetic component along the b-axis develops which likely relates
to some Dzyaloshinskii-Moriya anisotropy and/or weak ferromagnetic interactions between
the planes.13,14) While in the low-T phase a combination of three planar interactions, as
expected from the low symmetry of the structure, (J ∼ 53 K) and anisotropy ∼ 0.08 J
planar
are necessary to explain the inelastic neutron data,13) there are still inconsistencies between
the results from different techniques. On one hand, µSR indicates that the order parameter
fully develops at the upper transition (18 K). Below 6 K, besides some rearrangement of the
magnetic structure which might explain the change in the internal field at the muon sites,
an unexpected enhancement of fluctuations is observed.11) On the other hand, only a weak
amount of entropy is released at 18 K and a clear signature of long-range ordering appears in
neutron diffraction only below 6 K, suggesting a highly frustrated state with weak moments
between 6 and 18 K in strong contradiction with µSR results.12,14)
2.2 Phase diagram
Owingto its extreme sensitivity to static frozen phases,zero field µSR experiments enable
totrackinminutedetailsthemagnetismoftheentireparatacamitefamily.15) Forintermediate
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Cu
ZZnn//CCuu
x=0 x=0.33 x=1
P21/n R-3m
Zn/Cu
Zn/Cu II Zn
Cu III
Cu I Cu I
Cu II
Cu I Cu I Cu I Cu I
Cu I Cu I
Zn-paratacamite Herbertsmithite
Clinoatacamite
Zn Cu (OH) Cl ZnCu (OH) Cl
Cu (OH) Cl x 4-x 6 2 3 6 2
2 3
Fig. 1. (Color online) Top left: Structure of Herbertsmithite from.7) Top right: Simple sketch of the
structure where Zn and Cu sites only have been represented. Bottom: Evolution of the structure
of paratacamites when Zn content is increased. A structural transition occurs around x = 0.33
which yields well defined perfect kagome planes, assumed to be filled with Cu only.
Zn contents 0.33 < x < 0.66, the low temperature phase appears to be locally inhomogeneous
and yields two components in the µSR signal; one from a frozen magnetic environment rem-
iniscent of the ordered phase in clinoatacamite which gets more and more disordered as x
increases, and one from a dynamical environment of the muon (Fig.2). The evolution of this
frozen fraction detected in µSR yields the phase diagram plotted in Fig. 4. Interestingly for
x > 0.66, the frozen fraction gets lower than the experimental accuracy (a few percent) and
a fully ”liquid” phase appears although 1/3rd of the inter-layer 3D coupling paths are still
active. Accordingly, the small ferromagnetic component found in macroscopic magnetization
measurements disappears progressively when going from x = 0 to 1 (Fig.3).
The large domain of stability of a ”liquid” phase in the phase diagram of paratacamites
likely points at the weakness of the interplane coupling as well as at the shortness of the in
plane correlation length. No magnetic freezing has ever been observed in the x = 1 compound
downto thelowest probedtemperatureof 20 mK.15) Theupperboundvalue of ahypothetical
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x = 1.0
0.9
x =0.5
x = 0.66
n
o
zati 0.6 x = 0.33
ari
ol
P
n
o
u
M
x = 0.15
0.3
x = 0
T=1.5K
0.0 0.3 0.6
Time ( s)
Fig. 2. (Color online) Evolution of the low-T µSR asymmetry when Zn content is progressively
increased.Theoscillationsdueto theexistenceofwelldefinedstaticfields progressivelydisappear
and, for x>0.5, the frozen magnetism disappears. (Adapted from15))
frozen momentwould thenbeless than 610−4µ (µSR).Thisis completely consistent with ac
B
susceptibility results for x = 1, which behaves monotonously down to the lowest temperature
(50 mK) and with neutron scattering where no magnetic diffraction Bragg peak was detected
at variance with clinoatacamite.16)
Herbertsmithitethereforeappearsasthefirststructurallyperfectkagome antiferromagnet
notdisplayingany magnetic transition andopenswidethefield of experimental investigations
in quantum kagome antiferromagnets.
2.3 Interactions
◦
In Herbertsmithite, the 119 Cu-OH-Cu bond-angle yields a moderate planar interaction
J ≃ 180(10) K in comparison with other cuprates. Note that this value is consistent with that
observedincubaneswhereasimilarOHbridgelinksadjacentcoppers.14) FromasimpleCurie-
Weiss analysis of the variation of the high-T susceptibility with x, the interaction between
′
apical and planar Cu can be estimated to be weakly ferromagnetic, J ∼ 0.1J, in agreement
with neutron low-T data on the x = 0 compound. The paratacamite family therefore offers
an interesting possibility to explore weakly ferromagnetically coupled kagome layers7) for
0.33 < x < 1.
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Fig. 3. (Color online) Top panel: Susceptibility versus T for x = 0, 0.5, 0.66 and 1.0 in the parat-
acamite family. The corresponding symbols can be read from the bottom panel. The T = 0
ferromagnetic component decreases progressively from x = 0 (inset) to x = 1.0. Bottom panel:
The apparentCurie-Weiss temperatures are extractedfromlinear fits ofthe inverse susceptibility
upto300K.Thesevalueshavetobe correctedinordertoextracttheexactvalueoftheexchange
interaction, (see text). Adapted from.7)
3. Magnetism of the kagome planes
3.1 Static susceptibility
Probing the susceptibility is a crucial issue regarding the comparison and discrimination
between models. It has represented a major challenge since the discovery of the compound.
Various results are now reported through macroscopic susceptibility, and local techniques, by
shift NMR measurements through 35Cl17) and 17O NMR18) or µSR.19) Among all these local
probes, 17O NMR is certainly the most sensitive to the physics of the kagome planes since
oxygen from the OH group mediates the superexchange coupling between two adjacent Cu.
The hyperfine coupling of Cl to Cu is found more than one order of magnitude smaller, itself
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1.0
1.0 ction 0.8 x = 0
a
Fr 0.6 x = 0.15
n 0.8 en 0.4
o z
cti Fro 0.2 x = 0.33
a 0.6 x = 0.5
fr 0 10 20
n T (K)
e
z 0.4
o
Static
r
F
Dynamic
0.2
P2 /n R3m
1
0.0 0.2 0.4 0.6 0.8 1.0
Zn content
Fig. 4. (Color online) Phase diagram of the paratacamite family as deduced from the T →0 frozen
fractions found in zero field µSR experiments15) (inset).
much larger than the muon one. We will focus first on 17O NMR.
Typical 17O NMR spectra are displayed in Fig. 5. From a detailed analysis of the marked
singularities of the powder lineshape which positions are governed by the magnetic shift and
quadrupolar effects, two different sites can be identified. The most intense line at high T,
named main(M), is associated with O sites linked to two Cu sites in the kagome planes.
Tracking its position gives the shift, hence one can extract the hereafter called ”intrinsic”
susceptibility from 300 K down to 0.45 K. The other line (D) which is very sharp at low-T is
characteristic of O sites linked with one Cu and one Zn in the kagome planes. The existence
of this line points at the existence of defects resulting mainly from Cu-Zn site disorder. This
will be further discussed in the next sections as well as the corresponding details of the 17O
NMR spectra such as the broadening observed at low T for the (M) line.
Cl NMR shift could be as well followed on partially oriented powders but due to a large
broadening when T is decreased, it becomes impossible to determine the susceptibility below
15 K.
It is worth noting that no quantitative derivation of the couplings of the probes to the
various Cu sites has been made up to now but the diversity of the observed behaviors at low
T might likely be associated to the ratios of couplings strength of each probe to the various
Cu locations including the non-kagome Zn site.20)
One can divide the behavior of the susceptibility into 3 T-ranges (Fig. 6):
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reference
1.3 K
0.47 K
5 K
10 K
15 K
20 K
M 30 K
40 K
50 K
D
60 K
85 K
175 K
6.4 6.5 6.6 6.7 6.8 6.9 7.0
H (Tesla)
Fig. 5. (Color online) 17O NMR spectra from 175 K to 0.47 K from ref.18) The shift reference is
indicated by the vertical line. (M) and (D) refer to the two oxygen sites in the kagome planes as
described in the text. Large full green circles indicate the position of the center of the line that
wouldbeexpectedfromχ .At175K,adetailedanalysiswasperformedandtwocontributions
macro
are evident from the displayed fits (see text and section 4). The hatched area represents the
contribution from (D) oxygens sitting nearby one Zn defect in the kagome plane.
3.1.1 High T: T > 150 K
In macroscopic susceptibility (χ ) experiments as well as for 17O NMR, or 35Cl NMR,
macro
an effective Curie-Weiss behavior is observed at high T. From a correct fit using high temper-
ature series expansions, one can extract a value J = 170(10) K from .21,22)
3.1.2 Intermediate T: 10 < T < 150 K
Below 150 K,macroscopic and local susceptibilities as measured from 35Cland 17O NMR,
start to markedly differ as seen directly from Fig. 5 and 6. A clear upturn is observed in
χ whereas the variation of the oxygen shift goes the opposite direction. 35Cl NMR data
macro
also show a deviation from χ . This is certainly the best evidence that the compound is
macro
not free from defects. Would it be, the only contribution to 17O and χ measurements
macro
would come from fully occupied Cu2+ kagome planes and would be identical provided that no
staggered component dominates the O signal. Through 17O shift measured from 300 K ∼ 1.7
J down to 0.45 K ∼ J/400, a broad crossover with a smooth maximum from the Curie-Weiss
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-3
4.0x10
-0.08 S
u
-0.06 s
ft (%) 4 -0.04 3.0x10-3ceptib
Lineshi 3 -00..0002 35Cl 2.0x10-3ility (cm
R 2 0 50 100 150 200 250 300 / 3
M m
N o
1.0x10-3l C
1
u
)
17
O
0 0.0
0 50 100 150 200 250 300
Temperature (K)
Fig. 6. (Coloronline)Thermalvariationofthesusceptibilityofthekagomeplanesmeasuredfromthe
shift of the main 17O NMR line (red open symbols) compared to the macroscopic susceptibility
( green solid line). The red dashed line is a guide to the eye for the low-T shift. Inset: 35Cl data
show the same trend as 17O data although the accuracy is much reduced below 50 K. Adapted
from ref.17,18)
to the low-T regime is observed between ∼ 2J/3 ∼ 120 K and ∼ J/3 ∼ 60 K, see Fig. 6.
Such maxima are also observed in other corner-sharing kagome based lattices23–26) but the T
dependence of the shift differs from one compound to another. A rapid decrease of the shift
is further observed at lower T, between J/3 and J/20 ∼ 10 K.
3.1.3 Low T: T < 5−10 K
For T < 10 K∼J/20, theshiftfinally levels off atafinite value.For instance, nodifference
can be found between the 17O spectra taken at 0.45 and 1.3 K. The non-zero value and the
T dependence of the shift lead to the conclusion that the kagome plane susceptibility of
Herbertsmithite has a finite value at T → 0 and the system is probably gapless .
3.2 Dynamical susceptibility
On the dynamical side, inelastic neutron scattering (INS) data taken at 35 mK (Fig. 7),
′′
indicate an increase of χ (ω) with decreasing ω and clearly supportsthe absence of a spin gap
at least larger than 0.1 meV. The apparently divergent power law behavior χ′′(ω) ∼ ω−0.7(3)
′′
is also quite unusual. Moreover χ is only weakly Q dependent and points at the absence of
a characteristic length to describe the correlations.16)
T relaxation measurements complement the insight into the structure of the excitation
1
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)
s
it T = 35 mK
n
4
u
T = 10 K
.
b
r
a
(
)
2
(
"
0
0.00 0.25 0.50 0.75
(meV)
Fig. 7. (Color on line) Imaginary part of the dynamical susceptibility integrated over momentum
transfer.The apparently divergentbehavior is indicated by a power law fit, as representedby the
full line on the 35 mK data. The exponent -0.5(3)is in line with the -0.7(3)obtained from earlier
data, in ref.16)
spectrum.Thethermaldependenceoftherelaxationrateobtainedfor17O,18) 35Cland63Cu17)
NMR are plotted in Fig. 8. Quite strikingly, in the low T regime (T < 30 K∼ J/6), the high
field relaxation rate of O, Cl and Cu nuclei follow the same rather unusual variation, namely
a power law with the sublinear exponent T−1 ∼ T0.71(5).17,18) Given the filtering form factors
1
which are quite different for all probes, this indicates that excitations are likely weakly Q-
dependent. The T-dependence clearly demonstrates the absence of a gap in the excitations
andconfirmtheneutronresults.Note thatavery differentT-behavior,namelyamoderateyet
observable increase of the relaxation followed by a saturation is observed in µSR experiments
for T < 1 K but this might be related to weak interactions between magnetic defects.15)
4. Magnetic defects and their contribution to macroscopic measurements
In addition to the previously reported discrepancy between 17O shift and macroscopic
susceptibility, there are several types of arguments which consistently fit with a scenario
of dilution of the kagome magnetic lattice of Herbertsmithite. This comes out from inter-site
mixingdefectsofCuandZnspeciesevenforcompoundswiththeidealstoechiometry.Namely,
part of the non-magnetic Zn2+ substitutes copper on the kagome plane, and respectively, to
preserve stoichiometry, the same amount of copper substitutes zinc on the inter-plane site.
One should note that samples of different origins yield very similar results in macroscopic
measurements, 35Cl NMR and µSR, so that the various interpretations which have been
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)
1
10 - s
(
)
1
) T
-1 s 1/
m 5 (
3
) ( 1 0.8 ),
1 1
T - s
1/ 1 m
17 ( 17 0.4 ) (1
T
O
63 -1 0.71(5) 1/
T ~T (
Cu 1 3
6
0.1 35 0.0
Cl 0 5 10 15
0.1
0 15 30 45 60
Temperature (K)
Fig. 8. (Color on line) Semi-log plot of the relaxation rates obtained for all probe nuclei, 17O, 63Cu
and 35Cl. Inset: Linear plot of T−1 with a power law fit for the low T 17O data. Adapted from
1
ref.17,18)
successively proposed relate rather with progress in the understanding of Herbertsmithite
physics than with differences between samples.
It is fair to point that a different picture of perfectly filled Cu2+ kagome planes was
initially favored based on a quite funded Jahn-Teller argument.7) Indeed, since Cl− are quite
far from Cu2+ located in the kagome planes, the quasi-planar configuration is thought to
be most favorable to Cu occupation through Jahn-Teller stabilization. This statement was
reinforced by the interpretation of early shift measurements through µSR which was not then
confirmedbyother local probes.Yet, as pointedabove, itistoday quitehardtoexplain allthe
experimental findings within the initial framework of what was initially hailed as a ”perfect”
compound. The Jahn-Teller energy stabilization might not be large enough to completely
prevent inter-site mixing and it would be interesting to know more about the Cu Jahn-Teller
energy in this compound.
4.1 Direct evidence of intersite mixing defects
Although Zn and Cu have close neutron scattering cross-sections, structural refinements
are sensitive enough, as compared to x-ray ones, to suggest a large amount 7-10% of such
inter-site mixing defects,27,28) depending on the origin of the samples.
NMR spectra taken at high T, in the uncorrelated regime, see Fig. 5, give a local evi-
dence for this. Indeed, the NMR powder average spectrum displays many singularities which
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