Strain coupling mechanisms and elastic relaxation associated with spin state transitions in LaCoO3

Strain coupling mechanisms and elastic relaxation associated with spin state transitions in

LaCoO3

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IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 23 (2011) 145401 (12pp) doi:10.1088/0953-8984/23/14/145401

Strain coupling mechanisms and elasticrelaxation associated with spin statetransitions in LaCoO3

Zhiying Zhang1, Johannes Koppensteiner2, Wilfried Schranz2,Dharmalingam Prabhakaran3 and Michael A Carpenter1

1 Department of Earth Sciences, University of Cambridge, Downing Street,Cambridge CB2 3EQ, UK2 Nonlinear Physics Group, Faculty of Physics, University of Vienna, Strudlhofgasse 4,A-1090, Wien, Austria3 Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road,Oxford OX1 3PU, UK

Received 14 January 2011, in final form 28 February 2011Published 23 March 2011Online at stacks.iop.org/JPhysCM/23/145401

AbstractAdvantage is taken of the wealth of experimental data relating to the evolution with temperatureof spin states of Co3+ in LaCoO3 in order to undertake a detailed investigation of themechanisms by which changes in electronic structure can influence strain, and elastic andanelastic relaxations in perovskites. The macroscopic strain accompanying changes in the spinstate in LaCoO3 is predominantly a volume strain arising simply from the change in effectiveionic radius of the Co3+ ions. This acts to renormalize the octahedral tilting transitiontemperature in a manner that is easily understood in terms of coupling between the tilt and spinorder parameters. Results from resonant ultrasound spectroscopy at high frequencies(0.1–1.5 MHz) reveal stiffening of the shear modulus which scales qualitatively with a spinorder parameter defined in terms of changing Co–O bond lengths. From this finding, incombination with results from dynamic mechanical analysis at low frequencies (0.1–50 Hz) anddata from the literature, four distinctive anelastic relaxation mechanisms are identified. Therelaxation times of these are displayed on an anelasticity map and are tentatively related tospin-spin relaxation, spin-lattice relaxation, migration of twin walls and migration of magneticpolarons. The effective activation energy for the freezing of twin wall motion below ∼590 K atlow frequencies was found to be 182 ± 21 kJ mol−1 (1.9 ± 0.2 eV) which is attributed topinning by pairs of oxygen vacancies, though the local mechanisms appear to have a spread ofrelaxation times. It seems inevitable that twin walls due to octahedral tilting must have quitedifferent characteristics from the matrix in terms of local spin configurations of Co3+. Ahysteresis in the elastic properties at high temperatures further emphasizes the importance ofoxygen content in controlling the properties of LaCoO3.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The mechanical properties of LaCoO3 have been investigatedin the context of its potential use in high temperature oxygenmembrane applications [1–4]. Wider interest over a muchlonger period has been related to spin state transitions ofCo3+ in the octahedral sites, and the changes in magneticand electronic properties which accompany these [5–15].

At low temperatures (below ∼40 K), the Co3+ ions havelow-spin (LS) configurations and LaCoO3 is a non-magneticsemiconductor. With increasing temperature, the magneticsusceptibility increases to a maximum at ∼100 K and it be-comes paramagnetic. A broader anomaly in the susceptibilitythen accompanies a semiconductor–metal transition between∼500 and ∼600 K. The nature of the spin states at thesehigher temperatures has been controversial but both transitions

0953-8984/11/145401+12$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

have generally been attributed to thermally activated changesin population of Co3+ between low-spin (LS, t6

2g, S = 0),

intermediate spin (IS, t52ge1g, S = 1) and high-spin (HS,

t42ge2

g, S = 2) configurations. Accompanying the spin statevariations are anomalies in other physical properties, includingthermal expansion [16–19], thermal conductivity [20, 21],electrical resistivity [12, 16, 22, 23], heat capacity [24], ther-mopower [9, 12] and elastic constants [25–27]. Controversialaspects of the overall behavior relate primarily to whetherthe electronic configuration between ∼100 and ∼500 K isdominated by IS states [10, 16], or a mixture of LS + HSstates [7–9, 11, 15, 28, 29], with knock-on consequences forthe nature of the high temperature transition. Experimentalevidence for the two distinct models has been reviewed inrecent papers [12–14, 21, 30–32]. A developing trend appearsto favor the LS + HS scenario, and Jirak et al [12] andKnizek et al [33] have then argued that Co3+ in the metallicphase has IS character through a charge transfer mechanismbetween LS/HS pairs. The semiconductor to homogeneousmetal transition occurs gradually via dynamic coexistence ofIS and LS/HS regions, with a percolation point for the metallicphase occurring at ∼500 K and a midpoint for metallic phaseformation at 540 K [12, 29, 33]. There is a distinct peak in theheat capacity at ∼530 K [24].

Spin state transitions of LaCoO3 are superimposed on astructural phase transition which gives a symmetry changePm3m ↔ R3c at a transition temperature reported to be∼1610 K [34] and which is assumed to be due to octahedraltilting. The transition R3c ↔ R3 had been proposedby Raccah and Goodenough [7], but subsequent studiesdisproved it [18, 35]. An important additional consequenceof this transition, which does not seem to have been givenmuch attention in discussions of the electronic transitionsand properties at much lower temperatures, is that individualgrains cooled from high temperatures contain abundanttransformation twins on a scale of ∼0.1–0.5 μm [36, 37].In other ferroelastic perovskites, individual twin wallscan have quite different properties from the bulk withintwin domains [38], implying that the changes in spinstate occur within a material which is already structurallyheterogeneous. The mobility of such ferroelastic twin wallscan also have profound implications for elastic and anelasticproperties [39–41].

An additional change in symmetry could occur as aconsequence of Jahn–Teller distortions. The IS state ofCo3+ is Jahn–Teller active but, while dynamic disorderingof individually distorted octahedra with tetragonal ororthorhombic point symmetry is possible in space group R3c,long range ordering requires a reduction in symmetry. The HSstate is only weakly Jahn–Teller active [17] and the LS state isinactive. As part of the debate about spin states, additionalpeaks in Raman spectra which are not consistent with R3csymmetry have been interpreted as being due to locally Jahn–Teller distorted CoO6 octahedra [30, 42, 43], suggesting thatthe IS state is stable. Anomalous features attributable tolocal distortions observed in infrared (IR) spectra have beendiscussed in similar terms [44]. Maris et al [45] have refinedthe structure of LaCoO3 under I 2/a symmetry (a non-standard

setting of C2/c) [21]. This depended on an interpretationof line broadening in terms of split peaks in diffractionpatterns rather than direct observation of monoclinic splitting,and is apparently contradicted by recent magnetic circulardichroism (MCD) measurements, extended x-ray absorptionfine structure (EXAFS) data and pair distribution analysisof neutron diffraction data which rule out a large staticJahn–Teller distortion for a significant fraction of the Co3+sites [11, 31, 46].

From this brief review it should be apparent that, althoughthere remains some controversy concerning details of thestructural evolution of LaCoO3, the wealth of available datafrom diverse experimental measurements makes it probably thebest characterized of all perovskites in relation to the variationof spin state of a principal cation with temperature. Thepurpose of the present study was to take advantage of thislevel of characterization to address questions concerning howelectronic structure can give rise to strain, elastic and anelasticrelaxations in perovskites, either through coupling with astructural instability or with transformation microstructures.To this end, the elastic and anelastic properties of apolycrystalline sample of LaCoO3 have been investigated indetail by dynamical mechanical analysis (DMA) at frequenciesof 0.1–50 Hz and by resonant ultrasound spectroscopy (RUS)at frequencies of ∼0.1–1.5 MHz. An essential prerequisiteto understanding the observed variations of the elastic moduliis to understand coupling of the spin order parameter withstrain and variations of the strains due to the octahedraltilting transition. Data from the literature are analyzed inthe context of a Landau model for strain/order parametercoupling therefore. It turns out also that the new datafor anelastic dissipation can be combined with relaxationtime data from the literature to distinguish four differentanelastic loss mechanisms, which are presented in the formof an anelasticity map. The results are supplemented byRUS data for a single crystal which show that, under somecircumstances, transformation twin walls can be highly mobileat high frequencies of applied stress.

2. Experimental details

A polycrystalline LaCoO3 pellet with diameter 15 mm andthickness 6 mm was purchased from PI-KEM Ltd. Usingan annular diamond saw lubricated with paraffin, rectangularparallelepiped samples were cut from the pellet. Thepolycrystalline sample for RUS measurements had dimensions3.70×2.81×2.79 mm3, mass 0.149 g, and density 5.1 g cm−3.The polycrystalline samples for DMA tests were polished tothe dimensions 0.34×2.00×8.00 mm3. An offcut of the pelletwas ground up, and investigated by powder x-ray diffraction(XRD) using a Bruker D8 diffractometer with Cu Kα radiationat 40 kV and 40 mA. The diffraction angle 2θ was scannedfrom 5◦ to 95◦ in steps of 0.02◦, and counts were collected for3 s at each step. The diffraction pattern was consistent with theexpected structure, i.e. rhombohedral R3c. A LaCoO3 singlecrystal, grown under 9 atm argon pressure using the floatingzone technique [47], was cut into a piece with mass of 1.3469 g.This had an irregular shape, with four flat surfaces and two

2

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

curved surfaces. On examination after high temperature RUSmeasurements, a large number of twin walls could be observedat the surfaces using a stereoscopic microscope.

For both polycrystalline and single crystal samples, lowtemperature RUS data were collected within the temperaturerange 10–300 K and frequency range 0.1–1.5 MHz usingdynamic resonance system (DRS) Modulus II electronics andan orange helium flow cryostat described by McKnight et al[48]. High temperature RUS data (300–1130 K) for the samefrequency range were collected using a horizontal Netzschfurnace with alumina buffer rods [49] and Stanford electronicsdescribed by Migliori and Maynard [50]. For a polycrystallinesample, Young’s modulus E is related to bulk modulus B andshear modulus G according to

E = 9G B

3B + G. (1)

Energy dissipation, i.e. the inverse quality factor Q−1, wasobtained from fitting of resonance peaks with an asymmetricLorentzian function: Q−1 = � f/ f0, where � f is full widthat half maximum (FWHM) of the resonance peak plotted asthe square of the amplitude versus frequency, and f0 is theresonance frequency.

Using a Diamond DMA from PerkinElmer, DMA data ofpolycrystalline samples were collected within the temperaturerange 125–800 K and frequency range 0.1–50 Hz. The samplewas mounted on knife edges in three point bending geometry.Static (Fs) and dynamic forces (Fd) were applied as Fs +Fd exp(iωt) in the frequency range 0.1–50 Hz using a steelrod. A phase lag δ exists between the applied force and thedeflection response of the sample. Energy dissipation tan δ,the storage modulus E ′ (real component of complex Young’smodulus), and the loss modulus E ′′ (imaginary component ofcomplex Young’s modulus) conform to

tan δ = E ′′/E ′, (2)

E = E ′ + iE ′′ = l3

4t3w

Fd

udexp(iδ), (3)

where l is the spacing between the two bottom knife edges(5.00 mm), t is the thickness of the sample (0.34 mm), w is thewidth of the sample (2.00 mm), and ud is the displacement ofthe rod. The deformation amplitude was 5 μm, and the heatingand cooling rates were 3 ◦C min−1.

3. Results

The theoretical density of LaCoO3 is 7.299 g cm−1.4LaCoO3

used in this study has 30% porosity, which gives a lowervalue of Young’s modulus, 36.7 GPa, than other reportedvalues, e.g. ∼110 GPa for 7.4% porosity [1], 76 GPa for10.2% porosity [4], 47.8 ± 7.8 GPa for 16% porosity [51].Figure 1 shows stacks of RUS scans for both polycrystallineand single crystal LaCoO3 between 10 and 800 K. Belowroom temperature, similar behaviors were observed duringheating and cooling, i.e. for both polycrystalline and single

4 JCPDS No. 25-1060.

400

300

200

100

0

T (

K)

T (

K)

T (

K)

0.500.400.300.20

f (MHz)

f (MHz)

f (MHz)

(a)

1000

800

600

400

0.220.210.200.19

(b)

800

600

400

0.300.290.280.27

(c)

Figure 1. Stacks of RUS scans for LaCoO3. (a) Polycrystallinesample at low temperatures. (b) Polycrystalline sample at hightemperatures. (c) Single crystal sample at high temperatures. Thepeak near 0.29 MHz disappears above ∼530 K during heating and nopeaks return during cooling back to room temperature. The y axis isreally amplitude in volts but spectra have been offset in proportion tothe temperature at which they were collected and the axis is thenlabeled as temperature.

crystal samples, resonance peaks gradually shift to higherfrequencies with decreasing temperature and disappear below∼50 K. During heating above room temperature, resonancefrequencies of the polycrystalline sample gradually shift tolower values and then stay unchanged. The same pattern wasobserved during cooling. On the other hand, resonance peaksdisappeared from spectra collected from the single crystalabove ∼530 K during heating and did not reappear duringcooling.

Temperature dependences of shear modulus G, bulkmodulus B , Young’s modulus E and inverse quality factor

3

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

0.015

0.010

0.005

0

Q-1

8006004002000

T (K)

at ~0.21 MHz at ~0.27 MHz at ~0.33 MHz at ~0.43 MHz

(b)

20

18

16

14

12

G (

GP

a)45

40

35

30

25

B, E

(GP

a)

calculated Gfitted Gfitted Bfitted E

(a)

G

Figure 2. Temperature dependences of shear modulus G, bulkmodulus B, Young’s modulus E and inverse quality factor Q−1

determined by RUS for polycrystalline LaCoO3. Changes in shearmodulus �G were determined relative to a baseline fit to data in theinterval 590–800 K.

Q−1 determined from RUS data for the polycrystalline sampleare shown in figure 2. Below room temperature, the elasticmoduli were determined from fitting 27 resonance peaks withan overall root mean square error 0.3%, an estimated error forthe shear modulus of 0.1%, and an estimated error for the bulkmodulus of 0.45%. Above room temperature, most resonancepeaks are weak and the temperature dependence of the shearmodulus was determined from the first resonance peak, whichis due to a pure shear mode, using

GT = f 2T G298

f 2298

, (4)

where f298 and G298 are the frequency of the first peak andfitted value of shear modulus at 298 K. Good agreement wasobtained between shear moduli calculated in this way and thefitted data. With increasing temperature, the shear modulusgradually decreases and then levels off at ∼590 K. Young’smodulus shows a similar trend in the temperature interval forwhich the bulk modulus could be obtained. Above ∼125 K,the bulk modulus shows a similar trend of softening withincreasing temperature, but this appears to be reversed below125 K. A peak in Q−1 was observed at ∼100 K for all theresonance peaks examined in detail. At higher temperatures,Q−1 does not change much with temperature.

After the first two heating and cooling cycles to 800 K,spectra were collected from the polycrystalline sample in aheating and cooling cycle to 1130 K and finally to 800 K.Variations in the shear modulus during the complete sequence

16

15

14

13

Gca

l (G

Pa)

12001000800600400

T (K)

1st heat 1st cool 2nd heat 2nd cool 3rd heat 3rd cool

4th heat 4th cool

Figure 3. Temperature dependences of shear modulus ofpolycrystalline LaCoO3 during repeated heating and cooling cycles.Temperatures are up to ∼800 K for the first, the second and thefourth cycles, and up to 1130 K for the third cycle.

are shown in figure 3. The first two cycles showed little or nohysteresis, but the third cycle resulted in a very large hysteresis.During heating, the shear modulus decreased with increasingtemperature up to 800 K, and then the trend reversed, i.e. theshear modulus increased with increasing temperature above800 K. During subsequent cooling, the reverse of the trendtook place at 630 K, instead of 800 K. The fourth cycle was attemperatures up to 800 K, and the hysteresis became smaller.The reverse of the trend occurred at 660 K during heating andat 620 K during cooling.

Temperature dependences of storage modulus E ′ anddissipation tan δ of polycrystalline LaCoO3 measured atfrequencies 0.1–50 Hz by DMA between 125 and 800 K areshown in figure 4. RUS data measured at ∼0.2 MHz havebeen added for comparison. Maximum tan δ and minimumE ′ values were observed at ∼590 K. The temperature Tp atthe maximum in tan δ increases with increasing frequency(∼550 K at 0.1 Hz and ∼652 K at 50 Hz), implying that itis related to some thermally activated dissipation process. Asshown in figure 5, using

ln f = ln f0 − Ea

RTp, (5)

the activation energy Ea is 182 ± 21 kJ mol−1 (1.9 ± 0.2 eV)and f0 is 1.43 × 1016 Hz.

4. Strain analysis and possible elasticsoftening/stiffening mechanisms

The most direct structural influence of changes in spinstate is the development of macroscopic strain due to theconsequential changes in ionic radius. Variations in effectiveCo–O bond lengths are reflected in the unit cell volume ofLaCoO3 [16–19, 35], but, in addition, there are changes inthe octahedral tilt angle of the R3c structure due to changesin the La/Co radius ratio [18]. With respect to the parentcubic structure, the strain variations can be described formally

4

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

0.06

0.04

0.02

0

tan(

delta

), Q

-1

8006004002000

T (K)

0.1 Hz 1 Hz 10 Hz 50 Hz 0.2 MHz

(b)

31

30

29

28

27

E '

(GP

a)30

25

20

15

G (G

Pa)

(a)

Figure 4. Temperature dependences of storage modulus E ′ anddissipation tan δ measured at different frequencies by DMA forpolycrystalline LaCoO3. The deformation amplitude is 5 μm, thedeformation is limited to be less than 100 μm, and the cooling rate is3 ◦C min−1. The data for 0.2 MHz are from RUS. Data for E ′ belowroom temperature are affected by noise, and smoother variationswere observed from other samples.

in terms of a volume strain, ea, and a shear strain, e4,which couple with two different order parameters. The orderparameter for octahedral tilting consists of three components,q1 − q3, and transforms as irreducible representation R+

4 ofspace group Pm3m. Since changes in spin state do notinvolve a change in symmetry, the spin order parameter, qspin,transforms as the identity representation. The overall freeenergy change accompanying the transition can be expressedin terms of a Landau type expansion of the form

G = 12 a(T − Tc)(q

21 + q2

2 + q23 ) + 1

4 b(q21 + q2

2 + q23 )

2

+ 14 b′(q4

1 + q42 + q4

3 ) + λ1ea(q21 + q2

2 + q23 )

+ λ2(etz(2q21 − q2

2 − q23 ) + √

3eoz(q22 − q2

3 ))

+λ3(e4q1q3 + e5q1q2 + e6q2q3) + G(qspin) + λ4eaqspin

+ λ5(e2oz + e2

tz)qspin + λ6(e24 + e2

5 + e26)qspin

+ 14 (C

o11 − Co

12)(e2oz + e2

tz) + 16 (Co

11 + 2Co12)e

2a

+ 12 Co

44(e24 + e2

5 + e26). (6)

G(qspin) is the free energy due to changes in spin state whichcan be described by Boltzmann partitioning between discreteenergy levels [16, 18, 25, 27, 52–54]. Tetragonal (etz) andorthorhombic (eoz) shear strains, and the volume strain, ea,are defined in terms of linear strains e1, e2, e3, as etz =

1√3(2e3 − e1 − e2), eoz = e1 − e2, ea = e1 + e2 + e3 [41],

and Coik represent the elastic constants of the parent cubic

phase. For the R3c structure q1 = q2 = q3, e4 = e5 = e6,

6

4

2

0

-2

ln(f

) (

Hz)

0.00200.00180.00160.0014

1/T (1/K)

ln(f )=(37.2±4.2)-(21910±2530)/T

Figure 5. ln f versus 1/T with frequency and temperaturedetermined from maxima in tan δ for polycrystalline LaCoO3. Thestraight line fit gives Ea = 182 ± 21 kJ mol−1 (1.9 ± 0.2 eV) andf0 = 1.43 × 1016 Hz.

etz = eoz = 0, and the equilibrium condition ∂G/∂e = 0 gives

ea = −3(3λ1q21 + λ4qspin)

Co11 + 2Co

12

, (7)

e4 = − λ3q21

Co44 + 2λ6qspin

. (8)

Since ea couples with q21 and qspin, it follows that there

is an indirect coupling mechanism which will give rise toeffective coupling between the two order parameters of theform λtilt,spinq2

1 qspin, where λtilt,spin = −9λ1λ4/(Co11 + 2Co

12).If cation size was the only factor in determining the

structural evolution of LaCoO3 through the Pm3m ↔ R3coctahedral tilting transition, observed variations in strain andelastic properties should fall between two limiting cases. Acomparison with LaAlO3 and PrAlO3 is instructive in thiscontext. Using ionic radii from Shannon [55], the Goldschmidttolerance factors rA+rO√

2(rB+rO)for LaAlO3 and PrAlO3 are ∼1.01

and ∼0.94, respectively, and the transition temperatures are∼820 K [56] and ∼1860 K [57], respectively. The tolerancefactors for LaCoHSO3 (radius Co3+

HS = 0.61 A) and LaCoLSO3

(radius Co3+LS = 0.545 A) are ∼0.97 and ∼1.00, so that if

all the Co3+ was in the high-spin configuration, the transitiontemperature should also be significantly higher than if it wasall in the low-spin configuration. The change in transitiontemperature is implicit from the linear/quadratic coupling termλtilt,spinq2

1 qspin which causes the transition temperature, Tc, fora second order transition to be renormalized to T ∗

c where

T ∗c = Tc + 6λ1λ4qspin

a(Co11 + 2Co

12). (9)

This is a typical consequence of linear/quadratic orderparameter coupling [58, 59].

The symmetry breaking strain e4 is related to thelattice angle of the pseudocubic unit cell, αPC, to a goodapproximation by e4 ≈ cos αPC. For Co

44 |2λ6qspin| thestrain/order parameter coupling is expected to give e4 ∝ q2

1(equation (8), and see [41, 56] for the case of LaAlO3).Values of αPC determined from the high temperature latticeparameters [16, 18, 34] are all consistent with e4 ∝ (Tc − T )

5

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

-0.016

-0.012

-0.008

-0.004

0.000

cosα

150010005000

T (K)

[16]

[18]

[34]

Figure 6. Variation of cos α (∼e4 ∝ q21 ) as a function of temperature

for LaCoO3 using lattice parameter data from [16, 18, 34]. A linearfit to the data of [34] (dotted line) is consistent with a second ordertransition with Tc = 1740 ± 140 K. A similar fit to data of [18]between 600 and 900 K (solid line) gives Tc = 1730 ± 50 K. Thedashed line is a fit to the data of [18] between 150 and 300 K, for adifferent spin state, which would give a much lower value of Tc.

to a first approximation (figure 6) and, hence, second ordercharacter for the transition [q2

1 ∝ (Tc−T )]. The linear fit to thedata of [34] shown in figure 6 gives Tc = 1740 ± 140 K, whichis within experimental uncertainty of the value of 1610 Kbased on a nonlinear fit to the rhombohedral angle given inthe original paper. In detail, the data from [16, 18] define twotrends, however, with extrapolation to different Tc values. Alinear fit to the data of [18] between 600 and 900 K givesTc = 1730 ± 50 K. A linear fit to data between 150 and300 K, also shown in figure 6, would give a much lower valuefor Tc, implying that the effective ionic radius of Co is lessthan at temperatures above 600 K. This behavior is consistentwith more or less constant average radii (excluding the normaleffects of thermal expansion) in each temperature interval and achange in transition temperature as expressed by equation (9).

Accompanying most tilting transitions in perovskites is avolume strain. It is positive in the case of LaAlO3 [56], andthe cubic phase is stabilized by increasing pressure. Latticeparameter data for the cubic phase are not available, so a directdetermination of the volume strain for LaCoO3 is not possible.However, application of pressure up to 4 GPa causes the tiltangle to decrease and then, as pressure is increased further,this trend is reversed [60]. The first trend is due to a lowering ofspin state to S = 0, confirmed by Vanko et al [61], but the trendof increasing tilting once the lower spin state has been achievedimplies that increasing pressure stabilizes the tilted structureand, hence, that the volume strain accompanying tilting isnegative. Lowering of the spin state is also accompanied bya volume reduction (negative volume strain), as seen directlyin published lattice parameter data [16, 18].

Equation (6) provides a basis for predicting the likelyform of elastic constant variations. In the case of LaAlO3, thePm3m ↔ R3c transition is accompanied by a discontinuoussoftening of the single crystal elastic constants at Tc and,

hence, softening of the bulk and shear moduli, due tolinear/quadratic strain/order parameter coupling [41]. Changesin the bulk modulus due to a spin transition arise from bilinearcoupling of the volume strain with the spin order parametertogether with Boltzmann partitioning of spin states, whichleads to a characteristic dip in values through the temperatureinterval over which the transition occurs [27]. This is seenbelow ∼100 K, for example, in variations of moduli derivedfrom the velocities of longitudinal ultrasonic waves in singlecrystals [25, 27] and in polycrystalline samples [26]. RUS datafrom the present experiment show the onset of this dip in bulkmodulus values below ∼125 K (figure 2). The shear modulus,on the other hand, depends on the linear/quadratic terms withcoefficients λ5 and λ6 in equation (6). The influence of these isto renormalize C44 and (C11 − C12) according to

C44 = Co44 + 2λ6qspin, (10)

12 (C11 − C12) = 1

2 (Co11 − Co

12) + 2λ5qspin. (11)

Taking C44 and (C11 − C12) as representing the dominantcontributions to the shear modulus, and bearing in mind thatthe shear modulus is approximately a linear combination ofthese, it follows that changes in spin state will give rise to anincrease or decrease in the shear modulus which should displaythe same temperature dependence as qspin.

Spin state configurations are generally expressed in termsof the proportion of atoms in a high-spin or intermediate spinconfiguration with respect to the number in the low-spin state.Here it is convenient to use a spin order parameter whichwould be linear with the standard definitions but has a differentreference state. For the present purposes, qspin is taken torepresent a change in mean spin state with respect to theequilibrium structure at temperatures above ∼700 K which isdefined to have qspin = 0. This allows qspin variations to becharacterized in terms of the average Co–O bond length, dCoO.Figure 7(a) contains data from [18], and a function of the form

d = d1 + d2 coth

(s

T

), (12)

has been fitted to data at the highest available temperaturesto represent the reference structure (qspin = 0) together withnormal effects of thermal expansion [62, 63]. The saturationtemperature, s, was fixed at 150 K which seems adequate forsaturation of thermal expansion in other perovskites [64]. Thedifference between this fit and the observed values of dCoO,�dCoO in figure 7, then represents qspin as the order parameterfor the semiconductor phase with respect to the metallic phase.Variations of �dCoO with temperature (figure 7(b)) mimicthe pattern of variations of spin order parameter determinedusing spectroscopic techniques (e.g. figure 3 of [11], figure 10of [61]). The plateau of constant �dCoO and, hence, of qspin

matches the model result for a mixed 50% LS + 50% HS stategiven for the same temperature interval in figure 2 of [33].Changes in shear modulus defined in the same manner asan excess with respect to a fit to data at high temperatures(figure 7) follow the variation of �dCoO qualitatively, thusmore or less conforming to the pattern expected on the basisof allowed strain/order parameter coupling. Similarly, fitting

6

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

0.008

0.006

0.004

0.002

0

Δ

10008006004002000T (K)

4

3

2

1

0

ΔG

(GP

a)

ΔdCoO1

ΔdCoO2

ΔG

(b)

1.96

1.95

1.94

1.93

dC

oO(Å

)d

CoO

(Å)

Δ

ΔdCoO1

dCoO2

(a)

Figure 7. Changes in the Co–O bond length, dCoO, provide a measureof the degree of spin state order in LaCoO3 (data from [18]). In (a) abaseline has been fit to data in the temperature range 900–1000 K,using equation (12) with s = 150 K. The difference between this fitand observed values, �dCoO1, is shown in (b) and represents an orderparameter defined with respect to the metallic phase of LaCoO3. Anequivalent procedure has produced the variation of �dCoO2, whichrepresents an order parameter for the LS phase with respect to thespin state stable between 150 and 350 K. Changes in shear modulus�G were determined relative to a baseline fit to data at hightemperatures 590–800 K, as shown in figure 2.

of equation (12) to data between 200 and 350 K defines thereference state for the transition from paramagnetic insulatorto non-magnetic states. The difference between the fit andobserved values should scale with the order parameter for thistransition (figure 7(b)) and, indeed shows the same pattern ofvariation as obtained in the model of [33].

A quite separate cause of elastic stiffening could be thedevelopment of Jahn–Teller distortions of the CoO6 octahedra,though these would have to be dynamically averaged in therhombohedral structure. Evidence for such distortions mightappear in the extent of deviations of the octahedra fromcubic geometry. The magnitude of e4 for LaCoO3 at 5 Kis ∼1.5%, whereas the equivalent value for PrAlO3 is only∼0.5% [57]. This is in spite of fact that they both have similarPm3m ↔ R3c transition temperatures (∼1860 K for PrAlO3)and a similar maximum octahedral tilt angle of ∼10◦ [57, 65].Distortions of the CoO6 octahedra must provide a significantcontribution to the total strain in addition to the effect of simplerotations, therefore. This can be quantified using the octahedraldistortion parameter [66]

η = cT cos φ

aT

√6

, (13)

where aT, cT are lattice parameters for the trigonal unit cell andφ is the octahedral tilt angle. The octahedral tilt angle is given

12x103

8

4

04 (°

4 )0.040.030.020.010

1-

(b)

100

80

60

40

20

0

2 (°2 )

-0.015-0.010-0.0050cosα

0.04

0.03

0.02

0.01

0

1-

(a)

Figure 8. Correlations between tilt and strain parameters: φ is theoctahedral tilt angle, cos α is the macroscopic shear strain and 1 − ηis a measure of average octahedral distortion. The data wereextracted from refined structural parameters given by Radaelli andCheong [18]. (a) Neither of φ2 and 1 − η show the expected linearrelationship with cos α. (b) Distortions of the octahedra scale closelywith φ4 over most of the temperature interval for which data areavailable (5–1000 K).

by [66]tan φ = 2(x − 0.5)

√3, (14)

where x is related to the refined oxygen position. The valueof (1 − η) is zero in the cubic structure, but is 0.03 forLaCoO3 at low temperatures [65], which is much larger thanthe value of ∼0.006 for the low temperature rhombohedralstructure of PrAlO3. The normal expectation is for (1 − η) ∝φ2 ∝ q2

1 [56] but the data of [18] show that neither scalesin this manner (figure 8(a)). Rather (1 − η) varies linearlywith φ4 (figure 8(b)) showing that although the octahedraldistortion forms an essential part of the driving mechanism forthe R3c ↔ Pm3m transition, it still depends on the tilting ina regular manner. In spite of the large octahedral distortions,there appear to be no overt or abrupt changes in strain behaviorwhich might be attributable to the development of Jahn–Tellerdistortions.

Evidence of the approach towards a cooperative Jahn–Teller transition to a monoclinic structure could be provided,quite separately, by elastic softening with falling temperature,following the normal pattern for a pseudoproper ferroelastictransition. A perovskite structure with space group C2/c couldoccur by combining R-point tilting with a Jahn–Teller orderparameter that transforms as the irreducible representation�+

3 [67]. Softening in the R3c structure occurs becausethe Jahn–Teller order parameter couples bilinearly with a

7

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

30

25

20

15

10

5

0

-5

ln(1

/)

( in

sec

onds

)

0.0300.0200.0100

1/T (K-1

)

[13][27][26], longitudinalDMA, this study[9][25][26], transverseRUS, this study

spin-spin?

spin-spin/lattice

polarons?twin walls

Figure 9. Arrhenius plot for thermally activated relaxationalprocesses in LaCoO3. Straight lines represent equation (5) withvalues of τ0, Ea obtained from fits to experimental data collectedbetween temperature limits indicated by the filled symbols.

tetragonal strain, et, as has been observed in single crystalsbelonging to the LaAlO3–PrAlO3 solid solution, where theJahn–Teller active cation is Pr3+ [41, 68]. Related softening ofshear elastic moduli is observed also in (Pr, Ca)MnO3 [69–72],and (La, Ca)MnO3 [73–77] but has not been observed inLaCoO3 (this study). On the other hand, the C2/c structurecan also develop by combining R-point tilting with a Jahn–Teller order parameter which transforms as the irreduciblerepresentation R+

3 , and which corresponds to the a-typeordering of KCuF3 [78]. In this case, the strain/orderparameter coupling is linear/quadratic and the transition wouldbe improper, without the same shear mode softening expectedof a pseudoproper ferroelastic material.

5. Dissipation behavior

Acoustic dissipation requires relaxation of some process in acrystal which, firstly, is coupled with a strain and, secondly,has a relaxation time concordant with the frequency ofthe applied stress. Such a relaxation may depend on themobility of defects, including grain boundaries, dislocations,twin walls and point defects, or it might be intrinsic anddepend, for example, on correlated changes of hydrogenbonds in a structure. Together with data from the presentstudy collected at frequencies of ∼0.1–50 Hz (DMA) and∼0.1–1.5 MHz (RUS), there are data from the literature for∼10–100 MHz (pulse-echo ultrasonics) [25–27] which revealrelaxation phenomena that appear to be associated specificallywith twin wall motion, changes in spin states and magneticpolarons. These are considered as being thermally activatedand conforming to equation (5). The effect of changing τ0 andEa is most easily displayed on an Arrhenius plot (figure 9)which might be regarded as the basis of an anelasticitymap [79] for LaCoO3.

5.1. Twin wall relaxation

Ferroelastic switching by displacement of twin walls occursunder near static loading conditions in LaCoO3 [1–4]. Debye-

type acoustic dissipation peaks at frequencies of a few Hz andat temperatures in the vicinity of ∼400–500 K from DMAmeasurements are understood to arise from effective pinningof transformation twin walls in other R3c (LaAlO3) andI 4/mcm [(Ca, Sr)TiO3] perovskites with octahedral tiltingonly [39, 40, 80–82]. The twin walls are mobile at hightemperatures but become pinned at low temperatures, with anactivation energy for depinning of ∼90 kJ mol−1 (∼0.9 eV)which fits with oxygen vacancies as providing the pinningmechanism [39, 40]. A higher temperature freezing intervalat ∼600–650 K with a higher activation energy of 182 ±21 kJ mol−1 (1.9 ± 0.2 eV) has been found for Sr(Zr, Ti)O3

in the Imma structure [83], and a similar result has beenfound for twin wall related dissipation in orthorhombic(Sr, Ba)SnO3 [82, 84]. This value of Ea is a factor oftwo greater than the expected activation energy for oxygendiffusion. The pinning process responsible for the higheractivation energy is not known, but one possibility is pinningby pairs of oxygen vacancies with additivity of the activationenergy for pinning by individual vacancies. A similar resulthas been found for acoustic dissipation in PbZr0.3Ti0.7O3 byHe et al [85], who observed a broadened Debye peak in torsionpendulum data collected at 0.1–5 Hz. The activation energythey obtained was also a factor of ∼2 greater than expectedfor oxygen diffusion and the relaxation time of ∼10−16 s wastwo orders shorter than expected for point defect relaxation.They proposed that this could be understood in terms of therelaxation of clusters of oxygen vacancies near the twin walls.As discussed below, relatively high concentrations of oxygenvacancies can develop in LaCoO3 and a similar explanationof the high activation energy for depinning of twin walls isproposed here.

Harrison et al [40, 81] showed that the overall relaxationprocess involves a spread of activation energies rather thana single, discrete thermally activated step. This can alsobe expressed as a spread of relaxation times and modeledusing data collected as a function of temperature at constantfrequency [86–88] with

tan δ(T ) = tan δm

[cosh

{Ea

kBr2(β)

(1

T− 1

Tm

)}]−1

. (15)

The value tan δm is the maximum of tan δ obtained at thetemperature, Tm, of the maximum of the dissipation peak,and r2(β) is a width parameter which describes a Gaussiandistribution spread in relaxation times for the dissipationprocess. Here a linear baseline has been fit to the DMA datacollected at 0.1–50 Hz and equation (15) fit to data in figure 4between ∼450 and ∼750 K, with Ea = 1.91 eV, yielding r2(β)

values of 3–3.5. If the dissipation process involves a singlerelaxation time, the value of β is 0 and the value of r2(β)

is 1. Figure 1.2.23 of [87] gives β = 4–5 for r2(β) = 3–3.5.The spread in relaxation time which this result implies showsthat, in detail, the interaction of twin walls with defects mustinvolve a variety of local mechanisms with different values ofEa and/or f0.

Extrapolation of the Arrhenius fit of the DMA data tohigher frequencies (figure 5) would give an expected twin wallfreezing temperature of ∼900 K at ∼0.25 MHz. Dissipation

8

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

does not obviously increase in the RUS spectra from thepolycrystalline sample in the vicinity of 900 K, and there isno real evidence of twin wall related dissipation. This impliesstronger pinning of twin walls in LaCoO3 than in LaAlO3. Onthe other hand, superattenuation of the resonances occurs inthe single crystal sample on heating up to temperature above∼530 K. This is comparable to the onset of twin wall relatedsuperattenuation in single crystal of LaAlO3 (∼600 K) [89].It appears that grain boundaries may play a role in reducingtwin wall mobility, though the fact that the single crystal ofLaCoO3 remains superattenuating on cooling back down toroom temperature implies that the pinning processes have notbeen fully characterized.

5.2. Spin-spin and spin-lattice relaxation

Hoch et al [13] gave fit parameters of 1/τ0 = f0 = 7(±1) ×1011 s−1 and Ea = 7.1 ± 0.5 meV for thermally activatedspin-spin relaxation in the temperature interval 30–70 Kfrom electron paramagnetic resonance (EPR) spectroscopy.At higher temperatures, there is a significant dispersionwith frequency of longitudinal sound waves at ultrasonicfrequencies which can also be described in terms of a thermallyactivated process with 1/τ0 = 1.87 × 108 s−1 and Ea =11 meV [27]. Both fits are shown as straight lines in figure 9and, while the attempt frequencies are quite different, thesimilar activation energies suggest a closely related thermalbarrier in each case. On the basis of Mossbauer data,the lifetime of a HS state was determined to be 10−8 sat 200 K [9]. This point sits exactly on the straight linefrom [27] for the ultrasonic data (figure 9), indicating thatthe acoustic dispersion is also related to spin-spin transitiondynamics. Furthermore, since there is no frequency dispersionof transverse waves [27], the strain coupling mechanismcan only involve a volume strain without any shear straincomponent. This is consistent with the strain analysispresented above, in which it was concluded that direct couplingbetween qspin and macroscopic strain is exclusively with ea.Finally, Murata et al [25] observed strong attenuation of10 MHz longitudinal sound waves with a distinct peak at∼50 K. This point lies close to the extrapolation of theArrhenius fit from [27] shown in figure 9, and appears to bedue to the same loss mechanism, therefore. A similar peak inattenuation of 10 MHz longitudinal ultrasonic waves througha polycrystalline sample was observed by Zhang et al [26] at∼90 K, and this also plots close to the fit from [27] (figure 9).This combination of data seems to describe two separate, spin-related relaxation processes which differ only in the attemptfrequency.

Knizek et al [33] and Jirak et al [12] have argued that thestable structure between ∼150 and 350 K contains an intimatemixture of HS and LS cobalt cations. An externally appliedvolume strain would convert some proportion of these fromLS to HS states or vice versa, depending on the sign of thestress (compression or expansion). Relaxations of the localstructure in response to this on the time scale of the appliedstress would then account for the frequency dispersion andattenuation of acoustic waves. The same effect could alsooccur if the LS and HS cations formed local clusters. An

externally induced compression or expansion would then giverise to a volume strain by displacements of the boundariesbetween clusters from changes of spin state across them.The difference in attempt frequency from EPR measurementscould be reconciled with the acoustic dissipation data if thespin-spin relaxation process being sampled does not includethe significantly slower coupling with lattice strain whichgives rise to the acoustic dispersion and attenuation, and thelines for short relaxation times in figure 9 have been labeledspeculatively as ‘spin-spin’ and ‘spin-spin/lattice’ on this basis.This would imply that some spin transitions can be sampledin a time scale which excludes the lattice relaxation, however,and the two lines might more simply represent two differentrelaxation processes.

5.3. Relaxation related to magnetic polarons

The steep increase in dissipation observed at 100 K at∼0.30 MHz in the RUS experiment (figure 2) does not matchup with the spin-spin or twin wall related dynamics whenshown as a single point on figure 9. The point marks the onsetof a temperature interval of significant attenuation rather thanthe peak of dissipation but it falls close to the extrapolationof an Arrhenius fit to data from a different set of ultrasonicmeasurements. Zhang et al [26] obtained 1/τ0 = 1.2×1010 s−1

and Ea = 0.11 eV from the temperature dependence of aDebye-like attenuation peak in longitudinal ultrasonic dataat 10 and 15 MHz. A similar peak appears to be presentalso in data for the transverse mode. This activation energy isindistinguishable from the value reported by Yamaguchi et al[22] and is close to the values of 0.14 and 0.146 eV reported bySenaris-Rodriguez and Goodenough [9] and English et al [23].It also matches the activation energy for polaron migration inmanganite perovskites (∼0.1 eV) [90–92]. The existence ofmagnetic polarons or excitons at low temperatures has beendiscussed by many authors [12, 22, 29, 32, 93–95]. Polaronscan be bound up with spin state changes because of thepossibility that defects next to Co3+ stabilize the HS state evenwhen the matrix contains LS states [93]. Giblin et al [96] haveargued that oxygen non-stoichiometry is the dominant sourceof such defects in LaCoO3. The fact that an attenuation peakoccurs also for shear waves at about the same temperature asfor longitudinal waves [26] suggests that the defect responsibleis coupled to both volume and shear strains. This contrastswith the peak in attenuation at lower temperatures in their datawhich exists only for longitudinal waves and is presumablydue to spin-spin relaxations coupled to a pure volume strain,as described above.

6. Hysteresis and oxidation state

Experience from other studies of the properties of LaCoO3

when measured in air at high temperatures suggests that somevariation in oxygen content occurs. According to Knizek et al[52], sintered polycrystalline samples of various lanthanidecobaltates lose oxygen when heated up to 1073 K, but thechange in oxygen content is less than from 3.0 to 2.997 performula unit. This is apparently sufficient to give a small but

9

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

measurable hysteresis in thermal expansion above ∼700 K [97]and more extensive changes in shear modulus (this study). Thesingle crystal sample used here had been grown in an argonatmosphere, and had an initial oxygen content of 2.99% [47].This may have changed during heating to high temperatures butunless oxygen diffusion is sufficiently rapid to allow completeequilibration, it is possible that the main influence will occurat grain surfaces in a single crystal and at grain boundaries ina polycrystalline sample. The increase in shear modulus foundabove ∼800 K (figure 3) seems most likely to be due to changesin the elastic properties of a surface layer with increasingproportions of Co2+ and the possible changes in spin statethat this would also entail, but further elasticity measurementswith variable oxygen fugacity would be needed to test this ideafurther. High concentrations of oxygen vacancies in samplesproduced by synthesis at high temperatures in air could alsoaccount for the different pinning characteristics of twin wallsin polycrystalline LaCoO3, in comparison with polycrystalline(Ca, Sr)TiO3. Variations in oxygen content might also be insome way responsible for the high attenuation of single crystalLaCoO3 on cooling from high temperatures.

7. Discussion

Variations in spontaneous strains and elastic properties ofLaCoO3 as a function of temperature are consistent with themain influence of Co3+ spin state arising from the associatedchange in ionic radius. The influence on the volume straingives rise to changes in the effective transition temperatureof the octahedral tilting transition and, because of the linearcoupling with spin order parameters, to a characteristic dip inthe bulk modulus. On the other hand, the influence of spinstate on shear strains is seen mainly through linear/quadraticterms which do not give rise to significant changes in strain butstill renormalize the shear elastic constants. In this context,changes in the shear modulus might be used as an indirectmeasure of the spin order parameters and the pattern obtainedis at least consistent with other measures of these. For asystem with three possible spin states, there will be twospin order parameters to describe partitioning between them.Elasticity data do not provide direct evidence of which spinstates will be stable at each temperature. However, abruptchanges in shear modulus which might signify a transitionto a monoclinic structure have not been observed and thelack of any elastic softening with falling temperature rulesout Jahn–Teller ordering driven by an order parameter with�+

3 symmetry. Local or dynamic Jahn–Teller order with R+3

symmetry would still be permitted.As with the equilibrium elastic properties, the anelastic

loss behavior is consistent with spin state variations beingcoupled almost exclusively to volume strain. Dispersionand attenuation of longitudinal ultrasonic waves at MHzfrequencies define a distinct trend with thermal activationbetween spin states that includes a lattice relaxation. Shorterrelaxation times from EPR measurements (GHz frequencies)have then been interpreted as spin transitions without couplingto a lattice relaxation. A third trend in attenuation behaviorappears in longitudinal and transverse ultrasonic data as well

as in RUS data which are determined mainly by shear modes.This has been related to possible magnetic polarons, making alink between strain relaxation effects (shear and volume) andtransport properties through the spin coupling.

The ferroelastic twin walls of LaCoO3 have potentiallyinteresting properties in their own right. As argued on thebasis of the effect of pressure, octahedral tilting is accompaniedby a negative volume strain. On this basis the structureof the twin walls, with tilt angles which pass through zero,will occupy a larger volume than the structure within thetwin domains. A larger volume will stabilize high-spinstates of Co3+ located on the twin walls and it is inevitablethat the average spin state must be different in the vicinityof the twin walls in comparison with the spin state withinthe twin domains. In addition, evidence from the twinrelaxation behavior implies relatively high concentrations ofoxygen vacancies on the twin walls, perhaps even as clustersof vacancies. These will further modify the local spinstates, and there must be an expectation, also, of relativelyhigh concentrations of magnetic polarons at the twin walls.Rhombohedral LaCoO3 containing transformation twins isthus a heterogeneous material at all temperatures below theR3c ↔ Pm3m transition temperature. The known spinstate transitions will affect the properties and structure ofthe twin walls and matrix to different extents, leading tolocal heterogeneities in physical properties. For example,the walls could have higher electrical conductivity than thematrix, they could preserve locally high-spin configurationsof Co3+ into the stability field of the LS state and they couldact as nuclei for formation of percolating metallic regions atthe metal–semiconductor transition. In this respect, LaCoO3

should be added to a growing list of perovskites with ferroicand multiferroic properties in which twin wall propertiescould have a direct bearing on potential nanoscale deviceapplications [38, 98].

8. Conclusions

Observations of the spin state transitions in LaCoO3 from theperspective of elastic and anelastic relaxations have providedthe following insights. Although these are listed here forthe specific case of LaCoO3, they are likely to apply moregenerally to other perovskites with multiple instabilities anddefects.

(1) The principal influence on macroscopic strain arises fromthe change in ionic radius of Co3+ with spin state. Thisis seen overtly as a change in transition temperaturefor the octahedral tilting transition. Variations of bulkand shear moduli can be understood in terms of linearcoupling of the spin order parameter(s) with volume strain,and linear/quadratic coupling with shear strain. Theshear modulus then mirrors changes in the spin orderparameters.

(2) The lack of pseudoproper ferroelastic softening of theshear modulus with falling temperature probably rules outthe possibility that zone center (�+

3 ) cooperative Jahn–Teller distortions of the CoO6 octahedra might develop.

10

J. Phys.: Condens. Matter 23 (2011) 145401 Z Zhang et al

This does not rule out the possibility of local Jahn–Tellerordering with R+

3 symmetry, however.(3) Intrinsic relaxation processes occur through coupling

of changes in spin configuration of Co3+ with volumestrain. These are frequency dependent such that, at GHzfrequencies, spin-spin relaxations can occur but probablywithout time for lattice relaxations. At MHz frequenciesspin-lattice relaxations cause frequency dispersion in thevelocities of longitudinal ultrasonic waves.

(4) Extrinsic relaxation processes occur by coupling also withshear strain. Dissipation of ultrasonic waves at MHzfrequencies near 200 K has been interpreted as arisingfrom relaxation of magnetic polarons, which are likelyto be associated with oxygen vacancies. Attenuationof acoustic resonances at frequencies near 1 MHz andtemperatures below ∼100 K has been attributed to thesame relaxation process.

(5) Evidence of twin wall mobility in response to an appliedshear stress is provided by variations of the phase anglein three point bending measurements at Hz frequencies.The twin walls become pinned below ∼ 590 K with aspread of relaxation time and an activation energy whichhas been tentatively attributed to oxygen vacancy clusters.The absence of attenuation of acoustic resonances near1 MHz at higher temperatures in polycrystalline samplesimplies some role for grain boundaries in the pinningprocess given that strong attenuation attributed to twinwall mobility is observed in a single crystal sample above∼530 K.

(6) Twin walls due to the octahedral tilting transition inLaCoO3 are likely to have structural and physicalproperties that are quite different from the matrix in whichthey lie. In particular, they are expected to have higherspin states, higher concentrations of oxygen vacancies andmagnetic polarons, and higher electrical conductivity.

Acknowledgments

The RUS facilities in Cambridge were established with fundingfrom the Natural Environment Research Council (NERC)(Grant No. NE/B505738/1). Timothy W Darling at Universityof Nevada, Andy Buckley and Paul A Taylor at Universityof Cambridge are thanked for ongoing assistance with RUSfacilities. Financial support from NERC under Grant No.NE/F017081/1, from the Austrian FWF under Grant No.P19284-N20, and from the University of Vienna within theIC ‘Experimental Materials Science—Bulk NanostructuredMaterials’, is also gratefully acknowledged.

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