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Negative thermal expansion and local dynamics

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2009 J. Phys.: Conf. Ser. 190 012025

(http://iopscience.iop.org/1742-6596/190/1/012025)

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Negative thermal expansion and local dynamics

P Fornasini1, N Abd el All1, S I Ahmed2, A Sanson3 and M. Vaccari41 Dipartimento di Fisica, Universita di Trento, Via Sommarive 14, Povo (Trento), Italy2 Physics Department, Ain Shams University, Cairo, Egypt3 Dipartimento di Informatica, Universita di Verona, Verona, Italy4 ESRF, Grenoble, France

E-mail: paolo.fornasini@unitn.it

Abstract. A systematic EXAFS study has been performed on crystals affected by negativethermal expansion (NTE) and characterized by different structures: diamond-zincblende,cuprite, delafossite. Some general common features emerge from the results. The nearest-neighbours bond thermal expansion measured by EXAFS is always positive; within each familyof isostructural compounds, the stronger is the lattice NTE, the larger is the positive bondexpansion. A correlation can be established between the NTE properties and the anisotropyof relative vibrations with respect to the bond direction. EXAFS results are consistent withphenomenological models of NTE based on the competition between stretching and tensioneffects.

1. IntroductionSome physical phenomena, commonly considered as well understood, sometimes conceal subtlepeculiarities, whose experimental study and theoretical interpretation are far from trivial. Thisis the case of thermal expansion. Most substances expand when their temperature increases.One can easily understand that the asymmetry of the interaction potential energy give rise topositive expansion in two-atomic molecules. In crystals, however, where the potential energy isdefined in a many-dimensional configuration space, the situation is more complex. As a matterof fact, it is well known since long time that several solid substances undergo negative thermalexpansion (NTE) within restricted low-temperature intervals. Typical examples are tetrahedralsemiconductors such as Si, Ge, GaAs, CuCl [1].

The interest towards NTE has been renewed in the last decade by the discovery of somesubstances that undergo strong NTE over extended temperature intervals, whose prototype isZrW2O8, which contracts from 0.3 to 1500 K [2]. Materials exhibiting strong NTE are generallycharacterized by framework structures, made up by networks of corner-sharing polyhedral units[3, 4]. The applied research on NTE is focussed on the search of new substances, the optimizationof synthesis procedures and the production of composite materials tailored to meet specificthermal expansion requirements [5, 6, 7].

The discovery of strong-NTE materials has also renewed the interest for a deeper basicunderstanding of the mechanisms at the origin of NTE [8, 9]. According to a commonly acceptedphenomenological model, the net macroscopic expansion is the result of a competition betweena positive bond stretching contribution due to the anharmonicity of the effective pair potentialand a negative contribution due to tension effects [9]. When tension effects prevail over bondstretching, the solids contract upon heating.

14th International Conference on X-Ray Absorption Fine Structure (XAFS14) IOP PublishingJournal of Physics: Conference Series 190 (2009) 012025 doi:10.1088/1742-6596/190/1/012025

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Conventional techniques, such as dilatometry and Bragg diffraction, are sensitive only to thelattice thermal expansion, and cannot disentangle the bond stretching from the tension effects.As a matter of fact, the analysis of Bragg peaks gives information on the thermal variation ofthe distance between average atomic positions and on the intensity of vibrations of single atoms.

In the last years, the possibility of obtaining complementary information on the localstructure and dynamics form correlation-sensitive probes, such as EXAFS [11] and diffusescattering [10], has been experimentally demonstrated. EXAFS measures the average value ofthe instantaneous distance between neighbouring atoms and its thermal expansion, say the bondthermal expansion. Besides, EXAFS directly measures the mean square relative displacementalong the direction of the bond (parallel MSRD). The average distance 〈r〉 measured by EXAFSis larger than the distance Rc between the average positions measured by diffraction, because ofthe vibrations perpendicular to the bond. Correspondingly larger is also the thermal expansionmeasured by EXAFS. From the difference between the expansions δ〈r〉 and δRc measured byEXAFS and by diffraction, respectively, one can get information on the relative vibrations inthe plane perpendicular to the bond (perpendicular MSRD) 〈∆u2

⊥〉:

δRc = δ〈r〉 − δ〈∆u2⊥〉/2Rc . (1)

The accuracy and reliability of the bond thermal expansions and of the MSRDs, both paralleland perpendicular, obtainable from EXAFS has been checked by measuring high-quality spectraof simple model systems, such as Cu [11] and Ge [12], analyzing them by the cumulant expansionapproach and comparing the results with theoretical calculations [13].

The possibility of probing the local dynamical properties, and in particular the bond thermalexpansion and the perpendicular MSRD, suggested that EXAFS can give original information,complementary to Bragg diffraction, on the local mechanisms responsible for NTE. To evaluatethe potential of EXAFS for NTE studies, several crystals with different structures (diamond-zincblende, delafossite, cuprite) and affected by different NTE properties have been investigated[12, 14, 15, 16, 17]. In this paper, a review is attempted, updating and extending a previousreport [18] with the inclusion of the EXAFS results on delafossites and of the results recentlyobtained on cuprites by diffuse scattering. The analysis of the common features shared by thedifferent systems and of their relation to the NTE properties suggests an interpretation withinthe framework of the current models based on the competition between stretching and tensioneffects.

2. Crystal structuresThree relatively simple and representative crystal structures have been considered, wherestructural disorder due to local distortions is absent or negligible.

2.1. Diamond and zincblende cubic structuresIn tetrahedrally bonded cubic semiconductors, the lattice NTE is isotropic, limited to a smalllow-temperature interval, and its extent increases with the ionicity of the bond: it is weak in Ge,and the strongest in CuCl [19]. The tension effect responsible for NTE in diamond-zincblendecrystals has been attributed to low-frequency transverse acoustic modes along the [110] direction,with negative Gruneisen parameters, whose effect is dominant in a limited temperature interval.EXAFS measurements have been performed on Ge [12], CdTe [20] and CuCl [16], which representexamples of weak, intermediate and strong NTE, respectively.

The relatively open structure of tetrahedrally bonded crystals facilitates the movementsperpendicular to given bonds, responsible for the tension effect. However, the non-linearcharacter of the A-B-A links and the coordination of each atom to four nearest-neighboursmake the quantitative interpretation of results more complicated than for simpler linear linksA-B-A, where the B atom is only coordinated to two nearest neighbours A atoms.

14th International Conference on X-Ray Absorption Fine Structure (XAFS14) IOP PublishingJournal of Physics: Conference Series 190 (2009) 012025 doi:10.1088/1742-6596/190/1/012025

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2.2. Delafossite layered structureLinear A-B-A links are present in some crystals CuMO2 (M=Al, Sc, In, La) with the delafossitestructure, which undergo NTE along the c axis in a non-negligible temperature interval [21].The structure of these crystals is characterized by the stacking, along the c axis, of O–M–Olayers separated by sheets of Cu cations; each Cu ion is coordinated to two O atoms belongingto the two adjacent O-M-O layers, the linear O-Cu-O links being always parallel to the c axis.According to the refinement of diffraction patterns, the NTE along the c axis is due to thecontraction of the distance between the Cu and O average positions, and is accompanied by theanisotropy of the Cu thermal ellipsoids, the amplitude of vibration being stronger in the planeperpendicular to the c axis [21].

An EXAFS investigation has been performed at the Cu K edge in CuScO2 and CuLaO2 [17],the former compound being affected by a stronger NTE than the latter.

2.3. Cuprite cubic framework structureMost systems that exhibit NTE in large temperature intervals, such as ZrW2O8, are frameworkstructures, where polyhedral structural units centered on A atoms are connected at the cornersby linear A-B-A links. Contrary to the layered delafossite structure, however, the A–B–A linksare not parallel to each other. The vibrations perpendicular to the A–B–A links, responsible forNTE in framework structures, can often be connected to low-frequency rigid unit modes (RUM),that cause rotations of the basic polyhedral units [8].

A particularly simple structure is shared by the two NTE crystals Cu2O (cuprite) and Ag2O.The cuprite structure is made up by two interpenetrating lattices, one fcc of metal atoms (M= Ag, Cu) and one bcc of oxygen atoms (O). Each M atom is linearly coordinated to twoO atoms, while each O atom is tetrahedrally coordinated to four M atoms. Alternatively,the cuprite structure can be considered as a framework of two inter-penetrating networks ofcorner-sharing undistorted M4O tetrahedra. The crystals with the cuprite structure can thusbe considered as bridging between the diamond-zincblende structures and the more complexframework structures, such as ZrW2O8. The NTE compound Zn(CN)2 has the same frameworkstructure of cuprite, O being substituted by Zn and Cu by the group CN [22].

Silver oxide Ag2O undergoes isotropic NTE within the full explored temperature interval, likeZrW2O8. Copper oxide Cu2O, in spite of having the same structure of Ag2O, exhibits insteada behaviour similar to the zincblende CuCl, NTE prevailing only in a limited low-temperatureinterval; the low-temperature NTE is stronger in CuCl than in Cu2O [18].

EXAFS and diffraction measurements have been done on both Cu2O and Ag2O [15].

3. ResultsEXAFS measurements have been performed in temperature intervals typically varying from 20to at least 300 K; the temperature steps were of the order of 10 K at low temperatures (below100 K) and 50 K at higher temperatures.

Significant results in terms of bond expansions and MSRDs have been obtained from thecumulant analysis of the first shell, whose contribution is easily singled out and is exemptfrom multiple scattering effects. The resolution and accuracy in relative distance determinationdepend on the sample quality and on the overall performance of the synchrotron beamline.Resolutions of the order of 10−3 A in relative first-shell distances are quite standard, andin some cases 10−4 A have been achieved. Better results have been obtained for tetrahedralsemiconductors, where the backscattering atom is relatively heavy [12, 16, 20], than for cuprites[15] and delafossites [17], where the backscattering atom is oxygen.

The outer shells analysis is complicated by the difficulty of separating single-shellcontributions and by the influence of multiple scattering effects; no particularly significant resultshave been up to now obtained, with the exception of cuprites (see below).

14th International Conference on X-Ray Absorption Fine Structure (XAFS14) IOP PublishingJournal of Physics: Conference Series 190 (2009) 012025 doi:10.1088/1742-6596/190/1/012025

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3.1. First coordination shell - bond expansionIn all the seven examined compounds, the expansion of the average distance 〈r〉 measured bythe first EXAFS cumulant, say the nearest-neighbour bond expansion, is positive within thefull temperature range explored. Within each family of isostructural compounds, diamond-zincblendes (Ge, CdTe and CuCl), delafossites (CuScO2 and CuLaO2) and cuprites (Cu2O andAg2O), respectively, the stronger is the NTE measured by Bragg diffraction, the larger is thepositive thermal expansion (PTE) of the bond measured by EXAFS.

The nearest-neighbours bond expansions in Cu2O and Ag2O have been recently evaluated,by two different groups, through the analysis of the pair distribution functions (PDF) fromtotal neutron scattering [23, 24]. A positive bond expansion is found by both groups for bothcompounds. The agreement with EXAFS results is particularly good in Ref. [23]. The PDFanalysis shows a positive expansion also of the Zn–CN bond in Zn(CN)2 [22].

3.2. First coordination shell - MSRDsEinstein models have been fitted to the temperature dependence of both parallel andperpendicular MSRDs, 〈∆u2

||〉 and 〈∆u2⊥〉, respectively. The effective force constants k|| and

k⊥ obtained from the Einstein frequencies (k = 4π2ν2µ, where µ is the reduced mass) allow adirect comparison for different systems. The force constant k|| measures the effective stiffness ofthe nearest-neighbour bond. Within each family of isostructural compounds, a stronger bondPTE and a stronger lattice NTE are accompanied by a smaller value of k||, corresponding toa looser bond. The difference between bond PTE and lattice NTE is due to the perpendicularMSRD. Within each family of isostructural compounds, a stronger bond PTE and a strongerlattice NTE are accompanied by a smaller value of k⊥, suggesting a direct correlation betweenperpendicular MSRD and tension effects.

The perpendicular MSRDs are significantly larger than twice the parallel MSRDs forall studied compounds, reflecting a perpendicular to parallel anisotropy of relative atomicvibrations. Within each family of isostructural compounds, a correlation can be establishedbetween the NTE properties and the anisotropy of relative vibrations, measured by the ratioξ = k||/k⊥. Disk-shaped thermal ellipsoids have been measured also by diffraction for thecentral B atoms of A–B–A links in cuprites [15] and in delafossites [21]. The anisotropy ofrelative vibrations seems, however, to be more relevant than the anisotropy of absolute atomicvibrations: in the diamond-zincblende systems, the MSRDs are strongly anisotropic, while theMSDs are isotropic for symmetry reasons; in delafossites, the MSRD anisotropy of the Cu–Opair is larger in CuScO2 than in CuLaO2, say in the system affected by the larger NTE, whilethe anisotropy of the Cu ellipsoid is larger in CuLaO2 than in CuScO2 [17]. For Cu2O and Ag2O,the first-shell MSRD anisotropy has been recently reproduced by lattice dynamical calculationsperformed using an interatomic potential [25].

Although the actual extent of correlation cannot be obtained solely from Bragg diffractionmeasurements, the upper and lower bounds to correlation can be evaluated from the uncorrelatedMSDs [26]. The comparison of the MSRDs from EXAFS with the lower and upper boundsdetermined from the MSDs is useful to evaluate the phase relations between the two atoms. Bythis procedure, one has shown that in cuprites the metal and oxygen atoms move in phase alongthe bond direction, and that the perpendicular correlation is much weaker than the parallelcorrelation [15].

3.3. Second coordination shells in cupritesIn most framework structures, the cations are at the center and the oxgen atoms at the cornersof the polyhedra, respectively. In cuprites, the situation is reversed; besides, each M atom has12 M next-nearest-neighbours at the same crystallographic distance, which can be grouped intotwo sets: 6 atoms (Type A) belong to the same network of the central M atom, to which they are

14th International Conference on X-Ray Absorption Fine Structure (XAFS14) IOP PublishingJournal of Physics: Conference Series 190 (2009) 012025 doi:10.1088/1742-6596/190/1/012025

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connected via a bridging nearest-neighbour oxygen atom, while the remaining 6 atoms (Type B)belong to the other network, and do not share a nearest-neighbour oxygen atom with the centralM atom. A thorough second-shell analysis of the EXAFS spectra, taking into account multiplescattering contributions, allowed us to distinguish the different behaviours of Type A and TypeB atomic pairs [15]. M–M pairs belonging to the same network, and corresponding to an edge ofa M4O tetrahedron (Type A), reduce their average distance when temperature increases, whileM–M pairs belonging to different networks (Type B) increase their average distance. Besides,the parallel MSRD is larger for Type B pairs than for Type A.

The contraction of Type A distances has recently been qualitatively explained in terms ofdynamic distortions of the M4O tetrahedra [24]. A more complete quantitative model, basedon rigid M–O rods, has reproduced both the contraction of Type A and the expansion of TypeB distances, as well as the different MSRDs behaviours of Type A and Type B pairs [27]. Thedifferent MSRDs behaviours have been confirmed by lattice dynamics calculations [25].

4. DiscussionThe possibility offered by EXAFS of measuring the true nearest-neighbours thermal expansionrepresents the solution of an old crystallographic problem, which was previously faced byapproximate models, such as the riding model or the TLS model [28]. Similar information can inprinciple be obtained from diffuse scattering [10]. The main strength and limitation of EXAFSwith respect to diffuse scattering are the selectivity of atomic species and the insensitivity tolong-range distances, respectively. Investigations on the same materials, such as on cuprites[15, 23, 24], can contribute to assess the relative merits of the two techniques, for instanceclarifying the relevance of the asymmetry of distributions.

It is appealing to relate equation (1) to the current phenomenological NTE models [9], andto connect the bond expansion δ〈r〉 and the variation of the perpendicular MSRD of the right-hand side of (1) with the bond stretching and the tension effects, respectively. Such a connectioncould give a stronger quantitative foundation to the phenomenological models, clarifying the verymeaning of the effective pair potential responsible for the stretching effect and its relations tothe bare interaction potential and the full crystal potential.

In principle, the effective EXAFS pair potential depends on the statistically averagedbehaviour of all the atoms in the crystal. It has been recently shown, by taking into accountonly the nearest-neighbours interactions in copper and nickel, that the effective potential issignificantly stronger and less asymmetric than the single-bond potential [29]. However, theanharmonicity of the effective potential seems not sufficient to account for the bond expansion,directly measured by the first EXAFS cumulant. Experiments [12, 11] and simulations [30, 13]agree in indicating that, while the shape of the effective potential can be considered to agood approximation independent of temperature, its position cannot. The shift of the effectivepotential can give a non negligible contribution to the bond expansion.

The comparative study of NTE materials shows that bond expansion and perpendicularvibrations (say stretching and tension effects) are strongly correlated for a pair of atomsembedded in a crystal. The increase of the NTE strength corresponds to larger perpendicularvibrations and stronger perpendicular to parallel anisotropy, accompanied by a loosening and ashift of the effective potential that favors the positive bond expansion. Otherwise stated, whenthe anisotropy of relative vibrations increases, the tension effect giving rise to NTE increases, butmore room is also at disposal to the nearest-neighbours bond for expanding under the influenceof the stretching effect.

Further work is under way, to better understand the relations between effective potentialand bond expansion, to evaluate the anharmonicity contribution to the perpendicular MSRD,and to find a connection between the local EXAFS picture and the Gruneisen lattice dynamicalapproach, along the guidelines recently proposed for the RUM models [31].

14th International Conference on X-Ray Absorption Fine Structure (XAFS14) IOP PublishingJournal of Physics: Conference Series 190 (2009) 012025 doi:10.1088/1742-6596/190/1/012025

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5. ConclusionsThe results obtained up to now confirm the potential of EXAFS as a probe of local dynamicsin crystals and suggest that EXAFS can give unique insights on the mechanisms responsible forNTE. EXAFS results represent new experimental constraints for theoretical calculations of thedynamical properties of matter, in particular concerning anharmonicity and correlation effects,and should stimulate the development of more effective phenomenological models of NTE. Adeeper understanding of the local dynamical behaviour in bulk crystalline systems is preliminaryto careful studies of thermal effects in non-crystalline materials, as well as in nano-structuredmaterials and at interfaces.

AcknowledgmentsThis work is based on measurements performed with synchrotron radiation on different materialsat different times. The authors acknowledge ESRF for provision of its facilities and the staffs ofthe BM08 and BM29 beamline for technical assistance. The authors are grateful to G. Dalba,R. Grisenti, F. Rocca, G. Artioli, M. Dapiaggi, J. Purans, A. Kuzmin, D. Diop and A. Sleightfor cooperation in experimental activity and/or for useful advices.

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14th International Conference on X-Ray Absorption Fine Structure (XAFS14) IOP PublishingJournal of Physics: Conference Series 190 (2009) 012025 doi:10.1088/1742-6596/190/1/012025

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