Experimental study of the dynamics of Zn2Mg Laves phase

Journal of Non-Crystalline Solids 353 (2007) 3182–3187 www.elsevier.com/locate/jnoncrysol

Experimental study of the dynamics of Zn2Mg Laves phase S. Francoual a

a,b

, M. de Boissieu a,*, R. Currat b, K. Shibata c, Y. Sidis d, B. Hennion d, A.P. Tsai e

Laboratoire de Thermodynamique et Physico-Chimie Me´tallurgique, LTPCM/ENSEEG, UMR CNRS 5614, INPG, BP 75, 38402 St Martin d’He`res cedex, France b Institut Laue Langevin, BP 156, 38042 Grenoble, cedex 9, France c Laboratoire Le´on Brillouin, CEA Saclay, 91191 Gif sur Yvette cedex, France d Neutron Science Research Center, JAERI, Ibaraki 319-1195, Japan e Materials Engineering Laboratory, NIMS, 1-1, Namiki, Tsukuba 305-0044, Japan Available online 16 July 2007

Abstract A way to address the problem of phonons in quasicrystals (QCs) is to compare their dynamical response with that of approximant crystalline structures. Although the Zn2Mg Laves phase is not a periodic approximant of a QC phase, its structure is a periodic packing of Friauf polyhedra which are basic units involved in the construction of larger icosahedral atomic clusters found in Frank–Kasper type quasicrystals. We report on the experimental study of the lattice dynamics of the Zn2Mg hexagonal phase using inelastic neutron scattering with a particular attention devoted to the behavior of transverse acoustic (TA) modes. For TA modes propagating along the (T) direction, polarized along the c axis, there is a strong bending of the dispersion curve. Whereas the broadening rate is rather slow, going like q2, a strong coupling of the acoustic mode with an higher energy optical mode takes place. For TA modes propagating along the (D) direction, polarized in the hexagonal plane, the dispersion relation reaches much higher energy: the broadening rate is however steeper, going like q4, most likely due to a mixing of several excitations in the spectral response. In any case, the width of acoustic and optical excitations is found much smaller than in quasicrystals. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.23.F; 63.20; 78.70.N Keywords: Acoustic properties and phonons; Phonons; Quasicrystals; Diffraction and scattering measurements; Phase and equilibria; Structure

1. Introduction Quasicrystals (QCs) are long-range ordered materials for which the periodicity is of higher dimensionality than the dimensionality of the physical space (see [1] for an introduction). For instance, for icosahedral (i-) phases, the arrangement of atoms in the 3D physical space is obtained by a cut of a 6D periodic space, a 6-cubic lattice, decorated by 3D objects called the atomic domains. The determination of the shape and of the chemical composition of those atomic domains is of primer importance to *

Corresponding author. Tel.: +33 4 76 82 67 03; fax: +33 4 76 82 66 44. E-mail address: [email protected] (M. de Boissieu).

0022-3093/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2007.05.052

determine the positions of atoms in the 3D quasicrystal. Alternatively, the structure can be seen in the physical space as a compact and quasiperiodic (non-periodic) packing of building blocks, the atomic clusters. Those clusters are of icosahedral symmetry (for i-phases) and present a strong chemical ordering. The dynamics of quasicrystals (see [2] for a review) has been experimentally investigated by inelastic neutron scattering (INS) in the i-ZnMgY and i-AlPdMn QC phases [3–8]. The dynamical response is shown to split into two regimes [2–4,7]: an acoustic regime with well-defined acoustic excitations originating from strong Bragg peaks and an optic regime with several broad (4 meV) dispersion-less bands. The crossover between both regimes is abrupt.

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Transverse acoustic (TA) modes remain resolution-limited ˚ 1, after which up to a critical wavevector q equal to 0.3 A 4 they broaden rapidly with a q rate. For q values larger ˚ 1, the acoustic and optic modes cannot be disthan 0.6 A tinguished/separated anymore and an interpretation in terms of a single acoustic excitation with a well-defined wavevector is no longer possible. An additional interesting feature is the observed limit for the propagative regime of acoustic modes in those two i-QC phases. The limit is reached when the phonon mean free path is of the order ˚ and correof its wavelength: this value is of the order 10 A sponds to the diameter of atomic clusters found in those QC phases. Recent inelastic X-rays scattering measurements of TA modes in the i-CdYb quasicrystal and in its 1/1 CdYb periodic approximant lead to the same conclusion [9]. Recently, a description of the observed abrupt crossover between the acoustic and optic regimes has been given, in a phenomenological approach, using the Akhiezer mechanism generalized to QCs and the argument of a strong hybridization of sound waves and optical dispersion-less modes [10]. However a microscopic interpretation still remains to be carried out. One hypothesis is that the low energy flat branches are associated with excitations localized on atomic clusters. Because clusters are not randomly packed, but are arranged with a long range quasiperiodic order, this should give rise to a critical nature of the vibrational states in QCs i.e. states which are neither localized nor extended. This hypothesis is supported by recent X-ray charge densities studies of the a-AlReSi and aAlMnSi 1/1 cubic approximants showing a stronger covalent interatomic bonding of Al(Si) atoms with Re and Mn transition-metal elements, respectively, inside the Mackay icosahedral clusters than outside [11]. It is also supported by the characteristic wavevector for which the crossover between optical and acoustic regimes occurs in the dynamical response, together with the limit for the propagative regime, which both point to a characteristic length which is of the order of the typical cluster diameter. Recently, Duval et al. have also assigned the low lying energy modes observed in i-AlPdMn and i-ZnMgY with the characteristic frequencies of free elastic spheres embedded in a matrix, using a continuum approach [12]. However, whereas from the structural point of view, the description in terms of clusters is used, the way those clusters are precisely involved in the physical properties of QCs is still an open question (see discussion session of the ICQ9 conference [13]). In order to achieve a better understanding of the dynamics of quasicrystals, we have undertaken an experimental study of the Zn2Mg Laves phase. The Zn2Mg Laves phase belongs to the P63/mmc space group with lattice parameters ˚ , c = 8.567 A ˚ . It contains 12 atoms (4 Mg a = b = 5.221 A and 8 Zn) in the hexagonal unit cell with 3 inequivalent atoms Zn(1), Zn(2), and Mg [14]. This phase is known as the simplest Frank–Kasper phase. It can be described as periodic arrangement of space-filling Friauf polyhedra

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(Fig. 1). Those Friauf polyhedra are made of a large central Mg atom surrounded by 12 small Zn atoms at the vertices of a truncated tetrahedron. Their connectivity is shown in Fig. 1, where a one-layer arrangement is displayed. Two successive polyhedra share a common hexagonal face and have a different orientation as shown by the arrow. As can be thus noticed, the surface of this layer is a periodic tiling of hexagons and triangles units forming a Kagome lattice. Although the Zn2Mg Laves phase is not strictly speaking a crystalline approximant of any quasicrystalline phase, Friauf polyhedra can be considered as sub-units constituting larger icosahedral clusters which ones are found in Frank–Kasper type quasicrystals such as i-AlLiCu or i-ZnMgY. The ‘soccer ball’ cluster in the i-AlLiCu phase, for instance, is built of 20 Friauf polyhedra [15]. Of course, the size of the Friauf polyhedra is smal˚ , than the one of icosahedral clusters (10 A ˚ or ler, about 6 A more). Moreover, inside the whole structure, those Friauf polyhedra cannot be looked as independent/isolated units since all polyhedra are sharing faces. The Zn2Mg phase reveals thus as an interesting ‘simple’ phase with respect to the local structure of quasicrystals. It is moreover a phase with a moderate structural complexity for which we can expect a complete dynamical calculation to be carried out. The Zn2Mg lattice dynamics was investigated using inelastic neutron scattering in the mid 70s by Dorner et al. [16] at 80 K. They were searching to evidence a low energy vibrational mode associated with the particular degrees of freedom of the type 1 Zn atoms on the Kagome lattice, as predicted by Eschrig et al. [17]. In the present paper we report on measurements performed at room temperature in a same energy range up to 12 meV. In addition to the dispersion curve E(q) we also report on the q dependence of the width and of the integrated intensity of the transverse acoustic mode for different high symmetry directions. 2. Experimental details Two single grains of the Zn2Mg phase have been grown by the Bridgman method. The first one was oriented with a scattering plane defined by the reciprocal vectors (1 1 0) and (0 0 1). The second one had a scattering plane defined by the (1 0 0) and (0 1 0) reciprocal vectors, i.e. parallel to the hexagonal plane. Both samples had a volume of approximately 0.5 cm3. Their single-grain character was checked in the whole volume by hard X-ray characterisation before neutron irradiation. The sample mosaic was found to be of the order of 0.1°, and 0.2° for the first and second sample, respectively. Inelastic neutron scattering measurements have been carried out on the 2T1 triple axis spectrometer located in the Orphe´e reactor of the Laboratoire Le´on Brillouin. We used a PG002 vertically and horizontally curved monochromator and analyser. We worked with a fixed final neu˚ 1 and used a graphite tron wavelength kf equal to 2.662 A

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Fig. 1. Structure of the Zn2Mg Laves phase. The c hexagonal axis is vertical. Large and small spheres are for the Mg and Zn atoms, respectively. The Friauf polyhedron is outlined (light gray). Two Friauf polyhedra, having a different orientation, tile the hexagonal plane. For clarity, only one of the second orientation is shown (black arrow). The next layer is obtained by a mirror operation. The top Zn layer on the Fig. forms a Kagome net.

filter to suppress higher order harmonics. With this setting, we could probe excitations in a (0–4 THz) transfer energy range with an energy resolution equal to 0.22 THz, as measured on a Vanadium standard. We performed systematic constant-Q energy scans. Using the afitv software, we could extract the position, width and normalized integrated intensity of the measured excitations taking as a fitting response function a damped harmonic oscillator convoluted with the experimental resolution.

two different behaviors are observed among the four acoustic modes studied. The first one is represented by the dispersion curve of TA modes propagating along the (T) direction and polarized along the (0 0 1) one. In the dispersion curve shown in Fig. 2, full symbols correspond to the acoustic mode

3. Results

ΖΒ

( Τ ')

3

2.5

2

E [THz]

Excitations have been measured along the main high symmetry directions (D), (T, T 0 ) and (R) of the hexagonal Brillouin zone, parallel to the (0 0 1), (1 1 0) and (1 0 0) directions of the hexagonal reciprocal lattice, respectively. To enhance the acoustic part of the signal, measurements have been carried out for excitations originating from the strong (0 0 6), (2 2 0) and (3 0 0) Bragg reflections. With the two samples mountings, we could measure TA modes propagating along the (T, T 0 ) direction, polarized along the (0 0 1) or (1 1 0) axis, TA modes propagating along the (D) direction, polarized along the (1 1 0) direction and TA modes propagating along (R), polarized along the (1 2 0) direction. A particular attention has been dedicated to the broadening rate of the TA modes and to their acoustic character. The last point is achieved by computing the normalized integrated intensity which should remain constant as long as the acoustic character is the dominant one. When considering general features characterising the behavior for all the measured acoustic excitations, only

(Τ)

Γ

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

1.2

-1

q [A ] Fig. 2. Dispersion relation for modes propagating along the TT 0 direction and polarized along c*. The acoustic excitation is shown with a full circle; open symbols are for the two observed optical modes. Error bars are smaller than the symbols size. The solid line is a guide for the eyes.

S. Francoual et al. / Journal of Non-Crystalline Solids 353 (2007) 3182–3187

3

2.5

2

E [THz]

and open symbols to the optical excitations. The acoustic dispersion curve rapidly bends over and departs significantly from linearity becoming almost flat at the Brillouin zone boundary. In the same time, the intensity of the optical excitation located around 2.7 THz increases, while the intensity of the TA mode vanishes progressively. This is exemplified in Fig. 3(b) which shows the evolution of the normalized integrated intensity for both excitations. When considering the width of the acoustic mode as a function of the wavevector, we observe a slight increase, going like q2 (Fig. 3(a)). The width of the TA excitation remains however small when compared to what is observed in quasicrystals. The second behavior is represented by the dispersion of TA modes propagating along the (D) direction and polarized along the (1 1 0) direction. We present the results of the measurement in an extended zone scheme in order to allow full comparison with the dispersion curves usually shown for quasicrystals. Unlike the previous case, and as shown in Fig. 4, the dispersion of the acoustic mode only slightly departs from linearity up to relatively high energy values, of the order of 2.6 THz, and for wavevectors values ˚ 1. The vertical dashed lines indicate the posiup to 0.8 A tion of the first and second Brillouin zone boundaries along the (D) direction. For q values beyond the first Brillouin zone, the measured transverse excitation is no longer an acoustic mode but an optical mode. However, the excitation remains of acoustic character, as shown by the evolution of its normalized integrated intensity in Fig. 5(b): the ˚ 1. The evolution of norm is almost constant up to 0.6 A

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1.5

1

0.5

0

0

0.2

0.4

q [A

0.6 -1

0.8

]

Fig. 4. Dispersion relation for modes propagating along the (D) direction and polarized along (1 1 0). The acoustic excitation is shown with a full circle; open symbols are for the two observed optical modes. Error bars are smaller than the symbols size. The solid line is a guide for the eyes.

the width of the acoustic mode as a function of the wavevector shows a broadening rate going like q4 (Fig. 5(a)). Two optical excitations have also been measured to which correspond the open circles in Fig. 4.

0.5

0.5 0.4

0.3

[THz]

[THz]

0.4

0.2 0.1

0.2 0.1

0 110

0.3

4

0 7000 6000

6000

Norm [a.u]

Norm [a.u]

8000

4000 2000

5000 4000 3000

0 0

0.2

0.4

0.6

0.8

-1

q [A ] Fig. 3. (a) Evolution of the width of the acoustic excitation (full circle) and of the optical one (open circle) of Fig. 2. The acoustic excitation width increases as q2 (solid line). (b) Same for the evolution of the normalized integrated intensity. There is a clear intensity exchange between the acoustic and optical mode.

2000 0

0.2

0.4

0.6

0.8

-1

q [A ] Fig. 5. (a) Evolution of the width of the acoustic excitation of Fig. 4. The width increases with a q4 rate (solid line). (b) Evolution of the normalized integrated intensity for the acoustic mode of Fig. 4. The intensity is ˚ 1. constant and only slightly diminishes for q values larger than 0.6 A

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4. Discussion The dynamics of the Zn2Mg phase was previously studied experimentally and theoretically in the mid 70s. Our experimental results are in agreement with the experiment results of Dorner et al. obtained at low temperature [16]. However, whereas Dorner et al. only measured acoustic excitations, we could evidence a few additional optical modes in the low-energy spectral response up to 4 THz. Moreover, they did not report on the evolution of the width and integrated intensity of the excitations. Given the number of atoms in the unit cell of the Zn2Mg Laves phase, the maximum number of branches expected for each direction is of 36, number which can be already reduced after considerations of the symmetries of that hexagonal phase. However, the number of branches remains already quite large and a full analysis of our data would require their identification and labelling. This can only be achieved through a modeling of the measured dispersion curves using interatomic interactions. Such a work was first initiated in the Zn2Mg Laves phase by Eschrig et al. [17], who used pseudo-potentials to calculate the dynamical response. Within their calculations, they could predict the existence of low-lying optical vibrations corresponding to the vibrations of those Zn atoms located at the vertices of the triangle tiles in the Kagome layers. Successive layers lead to a double tetrahedron decorated with Zn atoms. Kagome modes correspond to a rotational vibration of this double tetrahedron around the central c axis. Similarly to Dorner et al., we did not find any low-lying optical vibration along the (D) direction. All optical excitations were found to have an energy of at least 1.6 THz (about 7 meV) which means that if such Kagome vibrations are to be observed, they must be searched for higher energy values than the ones predicted by Eshrig et al. New calculations are thus necessary to interpret further our measurements. This is underway. It is also interesting to compare our present results with those obtained in icosahedral phases. The dynamical response obtained in the Zn2Mg is markedly different from the one observed in quasicrystals. The response function is much more structured. The present measured optical excitations are well-defined and can be distinguished one from the other. Their width is moreover less or at most equal to 0.3 THz, and some of them show a clear dispersion relation. In QCs, we usually have to face with rather broad optical excitations showing almost no dispersion and with an average width of 1 THz. Acoustic modes are also much better defined in the Zn2Mg crystalline phase. Up to ˚ 1, all transverse acoustic modes are found q = 0.6 A almost resolution-limited, and have a width smaller than 0.1 THz (0.4 meV). At that q value, the broadening of TA modes is already quite important in QCs: the broadening is indeed abrupt in those materials and starts for a ˚ 1. In fact, in the dynamical wavevector equal to 0.3 A response of the i-AlPdMn and i-ZnMgY quasicrystals, ˚ 1 is the experimental limit beyond which single q = 0.6 A

acoustic excitations can not be distinguished from the broad optic bands, and their dispersion curves followed anymore. At that q wavevector value, the observed acoustic modes have a width which is much larger, 0.9 and 0.5 THz for the i-AlPdMn [3,4] and i-ZnMgY [7] phases, respectively, to be compared to 0.1 THz in the Zn2Mg phase. Another interesting point is that we found two different broadening rates of the acoustic signal in the Zn2Mg phase. A q2 broadening is what is expected for the interaction of phonons with defects in the elasticity theory: it is thus not surprising to find such behavior. However, the q4 broadening is more unexpected. It occurs for TA modes propagating along the (D) direction, far after the dispersion relation has crossed the first Brillouin zone boundary. In the q range where the broadening is observed, lower energy optical excitations are already present (see Fig. 4). The broadening can thus be due to an hybridization with those optical excitations, as proposed for quasicrystals, and/or to a mixing with other optical modes. Since a similar q4 fitting law is obtained for the broadening of transverse acoustic modes in the i-ZnMgY and i-AlPdMn QC phases, although with a much steeper/higher coefficient, such an abrupt broadening might have the same origin. Detailed calculations are necessary to clarify that point. Finally, the experimentally measured response function, with both, the dispersion relation and the intensity distribution, is a very strong constraint for any tentative calculation and for its validation. It will permit, in particular, to question the validity of interatomic pair potentials in the calculations of the dynamical response of the Zn2Mg phase but also of larger structures as 1/1 (or higher order) cubic approximants aimed at model quasicrystals. 5. Conclusion We have studied the lattice dynamics of the Zn2Mg Laves phase. Although the Zn2Mg phase can not be considered as a periodic approximant of any QC phase, it can be described as a periodic packing of Friauf polyhedra which are basic units involved in the construction of larger icosahedral atomic clusters in the Frank–Kasper type complex cristalline and quasicristalline metallic. We find that the response function is much more structured than what is observed in quasicrystals. In particular, all excitations, optical and acoustic ones, are well-defined with a width at most equal to 0.3 THz, i.e. 3–4 times smaller than in quasicrystals. Only two distinct behaviors are observed for the measured transverse acoustic modes. For TA modes propagating along the (T) direction, polarized along the c axis, there is a strong bending of the dispersion curve. Whereas the broadening rate of these TA modes is rather slow and goes like q2, we observed a strong coupling of the acoustic mode with an higher energy optical mode as indicated by the intensity transfer in the response function. For TA modes propagating along the (D) direction, polarized in the hexagonal plane, the dispersion reaches much higher energy: the broadening rate of the acoustic excita-

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tion is however steeper, going like q4, most likely due to a mixing of several excitations in the spectral response. A complete interpretation of the data, with a labeling of the different measured branches, requires further model calculation with appropriate interatomic interactions. Those experimental data will strongly constraint this modeling. This is underway. Acknowledgments We thank C. Antion for her help in the sample preparation. We thank P. Bastie for his help in the hard X-ray characterisation of the single grains. References [1] C. Janot, Quasicrystals: A Primer, Oxford Science, Oxford, 1992. [2] M. Quilichini, T. Janssen, Rev. Mod. Phys. 69 (1997) 277. [3] M. de Boissieu, M. Boudard, R. Bellissent, M. Quilichini, B. Hennion, R. Currat, A.I. Goldman, C. Janot, J. Phys.: Condens. Mat. 5 (1993) 4945.

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