Linear temperature dependence of the magnetic heat conductivity in CaCu2O3.
ABSTRACT We present experimental results for the thermal conductivity kappa of the pseudo-two-leg ladder material CaCu2O3. The strong buckling of the ladder rungs renders this material a good approximation to a S=1/2 Heisenberg chain. Despite a strong suppression of the thermal conductivity of this material in all crystal directions due to inherent disorder, we find a dominant magnetic contribution kappa mag along the chain direction. kappa mag is linear in temperature, resembling the low-temperature limit of the thermal Drude weight D th of the S=1/2 Heisenberg chain. The comparison of kappamag and Dth yields a magnetic mean-free path of l mag approximately 22+/-5 A, in good agreement with magnetic measurements.
- SourceAvailable from: Robin Steinigeweg[Show abstract] [Hide abstract]
ABSTRACT: We investigate the Heisenberg-Kitaev chain in order to uncover the interplay between two qualitatively different integrable points in the physics of heat transport in one-dimensional spin liquids. Based on linear response theory and analytical as well as numerical approaches, we explore several directions in parameter space including exchange-coupling ratios, anisotropies, and external magnetic fields. We show the emergence of purely ballistic energy transport at all integrable points, manifest in pronounced Drude weights and low-frequency suppression of regular-conductivity contributions. Moreover, off integrability, we find extended quantum chaotic regions with vanishing Drude weights and well-defined DC conductivities. In the vicinity of the Kitaev point, we observe clear signatures of the topological gap in the response function.Preprint. 12/2013;
Article: Lower Bounds for Conductivities[Show abstract] [Hide abstract]
ABSTRACT: We show how one can obtain a lower bound for the electrical, spin or heat conductivity of correlated quantum systems described by Hamiltonians of the form H = H0 + g H1. Here H0 is an interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity for g=0. The small perturbation g H1, however, renders the conductivity finite at finite temperatures. For example, H0 could be a continuum field theory, where momentum is conserved, or an integrable one-dimensional model while H1 might describe the effects of weak disorder. In the limit g to 0, we derive lower bounds for the relevant conductivities and show how they can be improved systematically using the memory matrix formalism. Furthermore, we discuss various applications and investigate under what conditions our lower bound may become exact.04/2007;
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ABSTRACT: We report a study on the heat transport of an S = 1 Haldane chain compound Ni(C3H10N2)2NO2ClO4 at low temperatures and in magnetic fields. The zero-field thermal conductivities show a remarkable anisotropy for the heat current along the spin-chain direction (κb) and the vertical direction (κc), implying a magnetic contribution to the heat transport along the spin-chain direction. The magnetic-field-induced change of the spin spectrum has obviously opposite impacts on κb and κc. In particular, κb(H) and κc(H) curves show peak-like increases and dip-like decreases, respectively, at ∼9 T, which is the critical field that minimizes the spin gap. These results indicate a large magnetic thermal transport in this material.Journal of Applied Physics 03/2013; 113(17). · 2.19 Impact Factor
arXiv:cond-mat/0612298v1 [cond-mat.str-el] 12 Dec 2006
Magnetic heat conductivity in CaCu2O3: linear temperature dependence
C. Hess,1, ∗H. ElHaes,2A. Waske,1B. B¨ uchner,1C. Sekar,1G. Krabbes,1F. Heidrich-Meisner,3and W. Brenig4
1Leibniz-Institute for Solid State and Materials Research, IFW-Dresden, 01171 Dresden, Germany
22. Physikalisches Institut, RWTH-Aachen, 52056 Aachen, Germany and
Physics Department, Faculty of Women, Ain Shams University, Cairo, Egypt
3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831, USA and
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
4Institut f¨ ur Theoretische Physik, Technische Universit¨ at Braunschweig, 38106 Braunschweig, Germany
(Dated: February 6, 2008)
We present experimental results for the thermal conductivity κ of the pseudo 2-leg ladder material
CaCu2O3. The strong buckling of the ladder rungs renders this material a good approximation to a
S = 1/2 Heisenberg-chain. Despite a strong suppression of the thermal conductivity of this material
in all crystal directions due to inherent disorder, we find a dominant magnetic contribution κmag
along the chain direction. κmag is linear in temperature, resembling the low-temperature limit of the
thermal Drude weight Dthof the S = 1/2 Heisenberg chain. The comparison of κmag and Dthyields
a magnetic mean free path of lmag ≈ 22 ± 5˚ A, in good agreement with magnetic measurements.
PACS numbers: 75.40.Gb,75.10.Pq,66.70.+f,68.65.-k
dimensional quantum spin systems has become the focus
of numerous studies [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22] since intriguing prop-
erties have been found. Firstly, a substantial magnetic
contribution to the thermal conductivity κ in addition to
a regular phononic background has been experimentally
established for spin chain and ladder compounds as well
as two-dimensional (2D) antiferromagnets such as the in-
sulating parent compounds of superconducting cuprates.
It can thus be considered a generic feature of quasi low-
dimensional magnetic materials. Secondly, the surpris-
ingly large magnetic contribution to κ of spin ladder
materials observed in the case of (Sr,Ca,La)14Cu24O41
[1, 2, 3, 4] has triggered extensive theoretical work on
possible ballistic heat transport in spin chains and lad-
ders [14, 15, 16, 17, 18, 19, 20].
The analysis of experimental data for κ can be quite
involved, in particular when phonon and magnetic en-
ergy scales do not separate as is the case for the spin
chain compounds Sr2CuO3and SrCuO2[6, 7]. The sit-
uation in the cases of spin ladder materials and the two-
dimensional cuprates is quite fortunate: here, a clear sep-
aration of phonon (κph) and magnetic (κmag) contribu-
tions allows for a robust determination of κmagitself. It
is important to note that κmag of the spin ladder com-
pounds (Sr,Ca,La)14Cu24O41[1, 2, 3, 4] shows thermally
activated behavior at low temperature T dominated by
the large spin gap of two-leg ladders [1, 2], while in the
case of La2CuO4  κmag is found to be proportional
to T2, which is the leading contribution in T to the spe-
cific heat of a 2D square lattice antiferromagnet. Thus in
these two cases, the experimentally observed κmagclearly
exhibits intrinsic properties of the underlying spin mod-
Theoretically, a consistent picture for thermal trans-
port in spin-1/2 Heisenberg chains has only recently
emerged: the integrability of this model results in a diver-
gent κmag[13, 15, 16] which is described by the so-called
thermal Drude weight Dthmultiplied by a delta function
at zero frequency. At low T this Dthdepends linearly on
T  and such behavior is more generally expected for κ
of any spin chain model with gapless excitations [16, 23].
It is the purpose of this Letter to present the first
experimental example of a quasi one-dimensional (1D)
quantum magnet (namely CaCu2O3) which exhibits a
linear T-dependence of κmagover a wide T-range. This is
an eye-catching result as it resembles the intrinsic trans-
port properties of a spin chain. The extraction of κmag
from the experimental data is very accurate since κph
of this material is strongly suppressed due to inherent
disorder. We utilize well-known expressions for Dth to
extract information on the scattering processes.
analysis yields a T-independent magnetic mean free path
lmag≈ 22±5˚ A. The mean separation of magnetic defects
along the chain direction as determined from magnetiza-
tion measurements is of the same order of magnitude.
The space group of CaCu2O3 is Pmmn with lattice
constants a = 9.946˚ A, b = 4.079˚ A, and c = 3.460˚ A
. The structure basically consists of Cu2O3structural
units arranged in the ab-plane with the geometrical form
of buckled two-leg ladders running along the b-direction.
The Cu2O3-planes are stacked along the c-direction and
are separated from each other by layers of Ca-ions. The
nearest neighbor magnetic exchange coupling of the Cu2+
spins along the 180◦Cu-O-Cu bonds which form the lad-
der legs in the b-direction is large and has been estimated
as J/kB≈ 2000 K . The intra-ladder magnetic rung-
coupling along the buckled Cu-O-Cu bonds (123◦bond-
ing angle) in the a-direction is much smaller and believed
to be in the range J⊥/kB≈ 100−300 K [24, 25, 26]. The
inter-ladder coupling along the c-axis has been estimated
FIG. 1: κa (?), κb(?) and κc (△) of CaCu2O3 as a function
of T. The dashed and solid lines represent a linear fit of the
experimental data in the range 100-300 K and the estimated
κmag in this range. Extrapolations of κmag towards low T
(assuming a T-independent lmag as extracted for T > 100 K)
corresponding to a finite (∆ = 3meV) and a vanishing spin
gap are represented by dotted and dashed-dotted lines, re-
spectively. Inset: κb from two different measurements (open
symbols depict the same curve as in the main panel).
to be of the same order of magnitude [24, 25]. Along the
a-axis a weak (< 100 K) and frustrated nearest neighbor
inter-ladder coupling is expected  and an even weaker
(8−30 K) second nearest neighbor inter-ladder coupling
has been suggested to be mediated via excess Cu2+ions
on interstitial positions . Despite CaCu2O3 being a
ladder-like material, where in principle a gapped non-
magnetic ground state is expected , the material or-
ders antiferromagnetically at TN≈ 25 K [25, 26, 29] and
inelastic neutron scattering reveals a spin chain-like ex-
citation spectrum with the upper bound for a spin gap
∆ ≈ 3 meV . We therefore consider CaCu2O3as an
ensemble of weakly coupled spin chains.
We have grown single crystalline CaCu2O3 by the
traveling solvent floating zone method .
with typical dimensions of around 3.5 mm length along
the thermal current direction and 1 mm2cross section
were cut from a crystal and the thermal conductivity
of CaCu2O3 has been measured along the a, b and c-
direction (κa, κband κc) as a function of T in the range
7-300 K. We used a standard four probe technique where
errorsdue to radiation loss are minimized. Magnetization
has been measured using a superconducting quantum in-
terference device (Quantum Design MPMS XL5).
Fig. 1 shows κa, κband κcof CaCu2O3as a function of
T. Since the material is insulating, electronic heat con-
duction is negligible and we therefore expect these com-
ponents to originate from phononic heat conduction plus
a possible magnetic contribution. The thermal conduc-
tivities perpendicular to the chain direction (κaand κc)
only exhibit a weak T-dependence and share absolute val-
ues (? 4 Wm−1K−1) which are 1-2 orders of magnitude
smaller than κph of other chemically undoped cuprates
such as SrCuO2 or Sr14Cu24O41. Instead of a pro-
nounced phononic low-T peak, which is usually found in
such cases, only a small peak is present in κc while no
peak is found in κa. In fact, such strongly suppressed
κ is typical for κph with a high phonon scattering rate
. Indeed, substantial phonon-defect scattering must
be present in CaCu2O3due to inherent structural disor-
der induced by a significant Ca and oxygen deficiency be-
ing balanced by excess Cu [24, 27]. We therefore consider
κ perpendicular to the chains to be purely phononic.
A completely different behavior is observed for κ par-
allel to the chains. At low T ? 50 K κb resembles κc
although it is about three times larger.
steeply increases at T ? 50 K with this increase being
linear in T for T ? 100 K. Such a strong increase of κ
with rising T cannot be understood in terms of conven-
tional phonon heat conduction by acoustic phonons. Dis-
persive optical phonons could in principle give rise to an
increase of κphwith increasing T  and possibly play a
role in the T dependence of κa. Heat transport by opti-
cal phonons is however unable to account for the rather
large κbat 300 K since this would require unrealistically
large phonon mean free paths . In analogy with ob-
servations in other cuprate materials [1, 2, 3, 4, 6, 7, 8],
we therefore conclude that the strong increase of κband
the resulting anisotropy of the κ tensor originate from 1D
heat transport due to magnetic excitations in the weakly
coupled spin chains. While these observations and the
following analysis represent the central results of this pa-
per, we wish to point out an additional feature of the
thermal conductivity. As depicted in the inset of Fig. 1,
we observe variations of κ as measured during different
runs which are beyond the statistical error of the typi-
cal measurement. Similar has been observed in other 1D
quantum magnets as well [11, 33, 34, 35] and remains an
open issue which merits further investigation.
In order to separate the phononic (κph,b) and magnetic
(κmag) parts of κb we assume that κph,b is of a similar
magnitude and exhibits a similar T-dependence to the
purely phononic κa and κc. Possible errors due to the
crudeness of this estimation become small at T ? 100 K
where the strongly increasing magnetic part of κb be-
comes clearly larger than any possible phononic ther-
mal conductivity. Since κa and κc are only weakly T-
dependent and κbincreases linearly in this T-regime, it
is natural to conjecture κph,b ≈ const and κmag ∝ T
at T ? 100 K. Indeed, a linear fit in the range 100-
300 K (dashed line in Fig. 1) describes the data al-
most perfectly and yields κph,b = 1.2 Wm−1K−1and
κmag = 0.055 Wm−1K−2× T. We plot the extracted
κmag for T > 100 K as a solid line in Fig. 1.
at lower T cannot be inferred from our data. Return-
ing to the inset we note that the alternative curve for
κbalso increases linearly with T for T ? 100 K yielding
κph,b= 2.2 Wm−1K−1and κmag= 0.041 Wm−1K−2×T.
We therefore assume that the difference in the two results
for κbrepresents the error in determining the intrinsic κ
of our sample. κph,b could certainly exhibit a slight T-
dependence, either increasing (as κa) or decreasing (as
normally expected for κph) with rising T. These uncer-
tainties are irrelevant to our further analysis.
Qualitatively, the linear increase of κmagwith T is a re-
markable result as it directly reflects the low-temperature
behavior of the thermal Drude weight of a Heisenberg
chain [15, 16], where (in SI units)
vT . (1)
As the exchange integral J/kB along the chain is of the
order of 2000 K, we may safely neglect any deviations
from the linear behavior that become relevant at tem-
peratures T ? 0.15J/kB[15, 16], i.e., T ∼ 300 K.
As mentioned earlier, the structure of CaCu2O3 sug-
gests a model of weakly coupled chains. We have checked
that any deviations of κmagfrom a linear T-dependence
due to the presence of a small spin gap can only occur
in the low temperature regime which is dominated by
phonons and thus experimentally hardly observable. Us-
ing a Boltzmann-type expression for κmag(see Eq.(1) in
Ref. ) to estimate the effect of a small gap, we fur-
ther find that at T > 100 K a pure spin chain and a
weakly coupled ladder for J⊥/J ? 0.05 both result in
the same linear T-dependence (cf. Fig. 1). For the lat-
ter, ∆ ? 3 meV since ∆ ≈ 0.4J⊥. It is thus justified
to use Eq. (1) to analyze our data at high T.
In the presence of external scattering, the thermal con-
ductivity of a single chain ˜ κmag is rendered finite with
a width ∼ 1/τ and may be approximated by ˜ κmag =
Dthτ/π. Combining this with Eq. (1) to calculate the
magnetic mean free path lmag= vτ from our experimen-
tal data for κmagyields
where N = 4/ac is the number of spin chains per unit
area in the crystal. A kinetic approach as described in
Ref. 7 yields the same result in the low temperature limit.
Note further that in our model, τ is an energy indepen-
dent quantity which seems to be a reasonable approxi-
mation, since even at T = 300 K ≪ J/kB only spinons
in a small region of the Brillouin zone can contribute to
κmag. In this region of the Brillouin zone v ≈ Jaπ/2? is
constant and hence lmagcan also be regarded as indepen-
dent of energy. From Eq. 2 and the experimental data
we obtain lmag= 22±5˚ A for the entire range 100-300 K
, corresponding to about 5 − 6 lattice spacings.
M (10-3µB per Cu)
(M-M0)-1 (104µB per Cu)-1
FIG. 2: Magnetization of CaCu2O3 as a function of tempera-
ture with a magnetic field µ0H = 1 Tesla parallel to the three
crystallographic axes. The solid lines represent Curie-Weiss-
type fits to the data in the range 100-360 K which yield ∼ 3%
free spins with respect to Cu. Inset: Inverse magnetization
after subtracting a constant M0 which roughly accounts for
van Vleck and chain magnetism .
Within the framework of Boltzmann-type kinetic mod-
els, T-independent mean free paths for magnetic excita-
tions have already been observed in the spin ladder mate-
rial (Sr,Ca,La)14Cu24O41[1, 2, 4, 5] and the 2D antifer-
romagnet La2CuO4 where in the low-T regime scat-
tering off defects dominates over other possible scattering
mechanisms. In analogy with these cases it seems rea-
sonable to explain our result of a T-independent lmagin
CaCu2O3by dominant spinon-defect scattering as well.
However, a constant lmag over such a large T-range is
very surprising because spinon-phonon scattering should
become increasingly important at higher T and eventu-
ally lead to a T-dependent mean free path. Despite this
intuition the data suggest that at 300 K the probability of
a spinon scattering off a phonon is still much lower than
that of scattering off a defect.
to conclude that the density of relevant defects is very
high in the material, resulting in scattering off defects
being much more important than other scattering pro-
cesses. This is consistent with our quantitative result for
lmagwhich is about 1-2 orders of magnitude smaller than
in spin chain systems with similar magnetic exchange,
e.g. SrCuO2 and Sr2CuO3, where much larger and T-
dependent mean free paths have been found [6, 7].
It is obvious to search for the origin of the high den-
sity of static scattering centers as suggested by our data
in the strong off-stoichiometry of CaCu2O3. The oxygen
It is therefore natural
deficiency is clearly most relevant for creating vacancies
within the Cu2O3chain structures which must create lo-
cal scattering sites. In order to estimate the density of
such scattering sites we consider the T-dependence of the
magnetization M of our sample, shown in Fig. 2, which
is in good agreement with previous results for crystals
from a different sample growth [25, 26]. M(T) exhibits a
sharp anomaly at the N´ eel temperature TN≈ 25 K and
a Curie-like behavior at T > TN indicating ∼ 3% of free
spins (with respect to Cu) in the material. Kiryukhin et
al. suggested  that these free moments are located
directly in the chains and originate from chain interrup-
tions. According to their analysis the high-T Curie tail
in the magnetization originates from chain segments with
a length between ∼ 40 and 80˚ A. In contrast, a recent
electron spin resonance study by Goiran et al.  sug-
gests that the free moments are more likely to arise from
excess Cu2+ions where each of these ions resides on an
interstitial site in the vicinity of an oxygen vacancy in a
neighboring chain structure. Within the latter scenario
a lower limit for the density (per unit length along the
b-direction) of oxygen vacancies n ? 0.03/b can be in-
ferred from the M(T) data. This yields an upper limit
for the mean distance d between the induced local scat-
tering sites within a chain of d =
estimate we take into account that each vacancy creates
a local structural distortion which affects at least two
chains since the individual chain structures are strongly
interwoven. In either case the extracted mean length of
non-distorted chain segments d should be regarded as an
upper limit for the magnetic mean free path since not ev-
ery chain distortion affecting κmagnecessarily contributes
to the magnetization. This agrees reasonably well with
the analysis of the thermal conductivity as lmagis found
to be somewhat smaller (by a factor 2-3) than d.
We mention that the T-independent scattering rate as
found in our experiment is in conflict with recent calcula-
tions of a scattering rate τ−1
induce slight disorder in the magnetic exchange coupling
. A possible reason for this discrepancy could be re-
lated to the nature of the defects in our sample. Very
likely the degree of the distortion of the chain at the
actual scattering sites is too large for this model to be
applicable. Detailed experimental and theoretical inves-
tigations are underway to clarify this issue.
To conclude, we have studied the thermal conductivity
of CaCu2O3as a function of temperature T. The thermal
conductivity parallel to the chains exhibits a pronounced
linear increase with increasing T which we attribute to
magnetic heat transport within the chains of the mate-
rial. The linear increase resembles the intrinsic thermal
conductivity of a spin chain with a constant scattering
rate. We extract a value for the magnetic mean free path
lmag≈ 22 ± 5˚ A which is in reasonably good agreement
with the mean distance between magnetic defects in the
material as determined from magnetization data.
2n? 68˚ A. In this
imp∝ T−1for impurities which
It is a pleasure to thank A.V. Sologubenko, X. Zo-
tos, A.L. Chernyshev, S.-L. Drechsler, V. Kataev and
R. Klingeler for stimulating discussions and A.P. Petro-
vic for proofreading the manuscript. F.H-M. is supported
by NSF grant DMR-0443144.
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