Page 1

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

Recently,

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

magnetic

heattransportoflow-

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 [9] 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-

els.

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 [15] 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.

Our

The space group of CaCu2O3 is Pmmn with lattice

constants a = 9.946˚ A, b = 4.079˚ A, and c = 3.460˚ A

[24]. 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 [25]. 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

Page 2

2

0100200 300

T (K)

0

5

10

15

κ (Wm-1K-1)

0 100

κb

200 300

5

10

15

κmag

κa

κc

CaCu2O3

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 [25] 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 [26]. Despite CaCu2O3 being a

ladder-like material, where in principle a gapped non-

magnetic ground state is expected [28], 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 [37]. We therefore consider CaCu2O3as an

ensemble of weakly coupled spin chains.

We have grown single crystalline CaCu2O3 by the

traveling solvent floating zone method [30].

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

Samples

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[7] or Sr14Cu24O41[2]. 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

[31]. 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 [32] 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 [38]. 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.

However, κb

κmag

Page 3

3

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)

Dth=(πkB)2

3?

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. [2]) 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⊥[36]. 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

lmag=3

π

?

k2

BN

κmag

T

,(2)

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

[39], corresponding to about 5 − 6 lattice spacings.

100200300

T (K)

0.2

0.4

0.6

0.8

1.0

1.2

M (10-3µB per Cu)

H||a

H||b

H||c

0100200300

T (K)

0.0

0.5

1.0

1.5

2.0

(M-M0)-1 (104µB per Cu)-1

µ0H=1 Tesla

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 [25].

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[9] 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

Page 4

4

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 [25] 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. [26] 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

[22]. 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.

1

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.

∗c.hess@ifw-dresden.de

[1] A. V. Sologubenko et al., Phys. Rev. Lett. 84, 2714

(2000).

[2] C. Hess et al., Phys. Rev. B 64, 184305 (2001).

[3] K. Kudo et al., J. Phys. Soc. Jpn. 70, 437 (2001).

[4] C. Hess et al., Phys. Rev. Lett. 93, 027005 (2004); Phys-

ica B 312-313, 612 (2002).

[5] C. Hess et al., Phys. Rev. B 73, 104407 (2006).

[6] A. V. Sologubenko et al., Phys. Rev. B 62, R6108 (2000).

[7] A. V. Sologubenko et al., Phys. Rev. B 64, 054412 (2001).

[8] P. Ribeiro et al., J. Mag. Mag. Mater. 290-291, 334

(2005).

[9] C. Hess et al., Phys. Rev. Lett. 90, 197002 (2003).

[10] M. Hofmann et al., Phys. Rev. B 67, 184502 (2003).

[11] A. V. Sologubenko et al., Phys. Rev. B 68, 094432 (2003).

[12] A. V. Sologubenko et al., Europhys. Lett. 62, 540 (2003).

[13] X. Zotos, F. Naef, and P. Prelovsek, Phys. Rev. B 55,

11029 (1997).

[14] J. V. Alvarez and C. Gros, Phys. Rev. Lett. 89, 156603

(2002).

[15] A. Kl¨ umper and K. Sakai, J. Phys. A: Math. Gen. 35,

2173 (2002).

[16] F. Heidrich-Meisner et al., Phys. Rev. B 66, 140406 (R)

(2002); ibid. 68, 134436 (2003); Phys. Rev. Lett. 92,

069703 (2004).

[17] X. Zotos, Phys. Rev. Lett. 92, 067202 (2004).

[18] K. Saito, Phys. Rev. B 67, 064410 (2003).

[19] E. Orignac, R. Chitra, and R. Citro, Phys. Rev. B 67,

134426 (2003).

[20] P. Jung, R.W. Helmes, and A. Rosch, Phys. Rev. Lett.

96, 067202 (2006).

[21] E. Shimshoni, N. Andrei, and A. Rosch, Phys. Rev. B

68, 104401 (2003).

[22] A. V. Rozhkov and A. L. Chernyshev, Phys. Rev. Lett.

94, 087201 (2005); A. L. Chernyshev and A. V. Rozhkov,

Phys. Rev. B 72, 104423 (2005).

[23] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 76,

3192 (1996).

[24] T. K. Kim et al., Phys. Rev. B 67, 024516 (2003).

[25] V. Kiryukhin et al., Phys. Rev. B 63, 144418 (2001).

[26] M. Goiran et al., New J. Phys. 8, 74 (2006).

[27] K. Ruck et al., Mater. Res. Bull. 36, 1995 (2001).

[28] T. Barnes et al., Phys. Rev. B 47, 3196 (1993).

[29] P. Sengupta, W. Zheng, and R. R. P. Singh, Phys. Rev.

B 69, 064428 (2004).

[30] C. Sekar and G. Krabbes and A. Teresiak, J. Cryst.

Growth 275, 403 (2005).

[31] R. Berman, P. G. Klemens, and F. E. Simon, Nature

166, 864 (1950).

[32] C. Hess and B. B¨ uchner, Eur. Phys. B 38, 37 (2004).

[33] A. V. Sologubenko, private communication.

[34] S. Uchida, private communication.

[35] C. Hess et al., unpublished results.

[36] D. C. Johnston et al., cond-mat/0001147 (unpublished).

Page 5

5

[37] B. Lake et al., to be published.

[38] A comparison with the structurally closely related doped

La2CuO4 [32] would suggest lph≈ 270˚ A.

[39] The error in lmag arises from the difference between the

two curves in the inset of Fig. 1.