Electronic confinement by clusters in quasicrystals and approximants

MATERIALS SCBENCE & EWGIWEER1WQ ELSEVIER

Materials Science and Engineering A226-228

(1997) 986-989

A

Electronic confinement by clusters in quasicrystals and approxi nants G. Trambly de Laissardikre *, S. Roche, D, Mayou Laboyatoire d’Etz{des des PropriMs

Electronigues des Solides - CNRS, BP 166, 38042 Grelzobh, Amce

Abstract

Quasicrystalsare characterisedby a long rangequasiperiodicorder and the presenceof many icosahedral clusters arranged in a hierarchical manner. In this paper we analyse the effect of these local environments on the electronic structure by considering a model of one atomic clusterembeddedin a metallic matrix. This model leadsto ‘cltrster vittzml boz~d states’,i.e. resonances of the wavefunction by clustersat different length scales.ThesestatesSeemto be characteristicof the quasicrystallinelocal order and they are very sensitive to the geometry of the cluster. 0 1997 Elsevier Science S.A. Keywords: Quasicrystals; Scattering mechanisms; Electron localisation

1. Introduction

Quasicrystals, such as AlCuFe AlPdMn and AlPdRe, reveal unusual transport properties [I], One of the main features is the proximity of a metal-insulator transition [2-41. Indeed, they exhibit a very high resistivity (a, K = 100-200 (fi cm)- l for AlPdMn and AlCuFe; ~~ K < 1 (s2 cm) - ’ for AlPdRe) although the density of states (DOS) has a metallic character. This means that the high resistivity is due mainly to a small diffusivity of electrons. The origin of this tendency to localisation is a major question for the knowledge of quasicrystals. It is interesting to study the approximant phases, which reproduce the local order of quasicrystals, but are periodic on greater length scales. Indeed experiments indicate that approximants of quasicrystals, with unit cells of 10 to 30 A (a-AlMnSi, a-AlCuFeSi, R-AlCuFe), have similar transport properties to those of quasicrystalline phases [2-51. This suggests that t,he local atomic order on a length scale of 10 to 30 A is determinant for the particular localisation of electrons in quasicrystals and their approximants. Several bandstructure clb-initio calculations on realistic models of approximants reveal very specific properties {6,7]. The dispersion relations show flat bands corresponding to small velocities [6]. Associated * Corresponding author. 0921-5093/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PIISO921-5093(96)10832-7

with these flat bands there are fine peaks in the DOS [6]. Near the Fermi level (&) these structures of the DOS are superimposed on a wider pseudogap which is attributed to a Fermi surface/pseudo-Brillouin zone interaction [6-S]. One must note that such a fine structure is also observed in band structure calculations of elementary tight-binding model of quasiperiodic tilling [9]. Therefore, the fine peaks in the DOS of a realistic model approximant may be related to the notion of critical states [9] which is usually used to describe the particular localisation in quasiperiodic structures. A geometrical consequence of the quasiperiodicity on the local atomic order is the large number of icosahedral clusters (such as MacKay type and Bergman type) [lo,1 11. This suggests to correlate these icosahedral clusters with the fine structures of the calculated DOS which is a signature of the special tenlocalisation in quasicrystals and dency to approximants. Moreover, clusters are not isolated because the atomic distribution in quasicrystals and approximants is rather homogeneous [lo] and their DOS has a metallic character [3,4,6,7]. These remarks lead us to study the diffusion properties of clusters on various length scales in a metallic matrix within the multiple scattering theory [12,13]. We show that local environments can lead to ‘cluster virtual bound states’, i.e. resonances of the wave functions by clusters of different size.

G. Tuzmbly

2. Description

de Laissardihe

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of the model

We consider a cluster containing transition atoms (such as Mn, Fe...) and Al atoms embedded in a metallic matrix Cjellium). For electrons which have energy in the vicinity of the Fermi level, transition atoms are strong scatterers whereas Al atoms are weak scatterers. Then, following a classical approximation [14,15], we neglected the potential of Al atoms and we retain only the potential due to the d-orbitals of transition atoms. Within these approximations the studied model becomes one cluster of transition metal in a jellium (free electrons). The total DOS of this system is: n(E) = {z,(E) -IA.n(E), where no(E) and An(E) are the DOS of free electrons, and the variation of the DOS due to the cluster, respectively. A.n(E) is calculated by using the Lloyd formula [16]. The parameters of the calculation are the geometry of the cluster and the phase shift 6,(E) (I = 2) due to the potential of each transition metal. 6,(E) depends on the energy Ed of the d-orbital and on the width rd of the d-resonance [13]. Realistic parameters to simulate Mn potential in Al-Mn alloys are &%EEF M 10.88 eV and lYd z 3 eV [6,17].

3. Notion of ‘cluster virtual bound states’ We now focus on the case of one icosahedron of Mn atoms, as in the MacKay icosahedron of the approximant cl-AlMnSi [18]. Fig. 1 show the calculated variation of An(E), for several diameters D of the icosahedron.

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When D --$ co, one gets the classical virtual bound states (VBS) [14,15], corresponding to an isolated Mn atom in a jellium. A.n(E) is a Lorentzian centred at Ed, with a width at half maximum equal to rd. For diameters D close to the actual value in CIAlMnSi (D = 9.2 A), the DOS in the vicinity of Ed exhibits a strong deviation from the VBS case. Indeed several peaks and shoulders appear. The width of the most narrow peaks ( - 50-100 mev> is comparable to the fine peaks of the calculated DOS in approximants [6,7]. Each peak shows a resonance due to the scattering by the cluster. They correspond to states localised by the cluster. These states are not eigenstates, but they have a finite lifetime on the order of &jr, where r is the width of the peak. Therefore, the stronger the effect of the localisation by the cluster, the narrower the peak. This particular tendency to localisation can be understood in more detail as follows. For the energies E of an incident electron such as (Ed - rd/2 < Ed + lF,/2) the potential of the transition atom is strong. In this energy range, the transition atoms can roughly be considered as hard spheres with a radius on the order of the size of the d-orbital ( - 0.5 A) [17]. By a similar effect with the Faraday cage, electrons can be confined by the cluster, provided that the wavelength 3, of the incident electron satisfies 3,k 1 (E is the distance between two hard spheres). Let us consider the simple unrealistic limit where rd is infinite. In this case, the potential of the transition atom is independent of E, and transition atoms can be considered as hard spheres for all E. The DOS (Fig: 2(a)) exhibits many peaks. If these states are confined in the region of extension D (D is the diameter of the cluster), where the potential is almost constant, one expects that their energies varies as 1/D2. In the insert of Fig. 2(a), we checked this behaviour for the 3 peaks named A, B and C. This shows that the corresponding states are localised in a region of extension D. By varying continuously rd from cc to a realistic value (Fig. 2(b)), it is clear that the fine peaks in realistic cases are the same as those in the case rd z 00. This leads us to assess that the fine peaks in An(E) are ‘cluster tktual bound states’, i.e. resonances of the wavefunction by the cluster.

4. Effect of the geometry of the cluster

10 EF 12 Energy (ev) Fig. 1. Variation of the DOS due to an Mn icosahedron (with a diameter D) in a jellium. rd = 3 eV, Ed = E, = 10.33 eV.

In the previous section, we showed that the resonance energy depends crucially on the size of the clusters. We have also observed [13] that resonances disappear quickly when D increases. Moreover the confinement by an Mn icosahedron decreases very rapidly when the icosahedron is truncated (Fig. 3). It is also interesting to consider the variation of the DOS due to other kinds of Mn cluster. For each

G. Trambly de Laissardihe et al. /Materials

El ti8 z .w

0.8

.

.peakB Y peak C

Oo l/D2 i 2

(i997) 986-989

Table 1 Size of the Mn clusters for which the DOS exhibits the narrowest peak

OpeakA

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Oil )

Cluster

No. of atoms

Tetrahedron Cube Icosahedron Dodecahedron

4 8 12 20

Diameter D (A)

Clmi” (4

8.84 8.7 12.33

3.84 4.50 4.64 3.1 4.40

Li, is the first Mn neighbour distance. We have checked the size corresponding to rl,,,>3 A.

0 ll”N”““l”l

3

5

7

’ ‘1

9

Energy (eV) Fig. 2. Variation of the DOS due to one Mn icosahedron in a jellium: {a) r, = co and D = 9.21 A. (Insert: energies of the peaks A, B and C versus the diameter D of the icosahedron. The crosses, the points and the circles are the calculated energies.) (b) rd = 3 eV, Ed = 9.30 eV and D = 9.21 A.

than 3 A. This assumes that we consider only alloys in which the Mn atoms are not first neighbours. The size values for which the DOS exhibits the narrowest peaks are reported in Table 1, and the corresponding DOS are drawn on Fig. 4. These results show that confinement effects are present in all cases. However, the strongest confinement (i.e. the narrowest peaks) are observed in the case of the icosahedron and the dodecahedron. This suggests that the local orders, which are characteristic of the quasiperiodicity, have a more pronounced effect on the electronic confinement.

cluster, given in Table 1, we calculated An(E) for various sizes of the cluster. We consider only the sizes for which the first Mn neighbour distances are larger

(1Mnatom) =: g 3

12 : -

% ‘G 9‘2 l! 2 =:6 % % d Z I-

9.5

1

I

10

I

,/1

I(1

I

I1

10.5 11 Energy (eV)

1-t

: -

( 1 Mn cube 1

_ _ - ( 1 Mnicosahedron)

z 3. 4

I

11.5

_ ( 1 Mn tetrahedron

12

Fig. 3. Variation of the DOS due to one truncated Mn icosahedron in a jellium: rd = 3 eV, Ed = E, = 10.88 eV and D = 9.21 I\. (a) Complete Mn icosahedron (12 Mn) (same as Fig. l), (b) Mn icosahedron with 1 Mn vacancy (11 Mn), (c) Mn icosahedron with 2 Mn vacancies sited in opposite positions (10 Mn), (d) Mn icosahedron with 2 neighbouring Mn vacancies (10 Mn), (e) 1 Mn atom in a jellium.

- ( 1 Mn dodecahedro 0 ~“1”~‘1~1’1”“1”11”‘1 0 2 4 6 8 lo EF I2 Energy (eV) Fig. 4. Variation of the DOS due to several small clusters in a jellium, Their diameters (Table 1) are the diameters for which An(E) exhibits the narrowest peak (for the dodecahedron it is the case D = 12.33 A).

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Moreover for the two large clusters, Fig. 5(a) and 5(b), a wide pseudogap (width -0.5 eV) appears at E% 10.7 eV and E z 11.3 eV, respectively. This is attributed to the diffraction of the electrons by the network of Mn atoms (Hume-Rothery mechanism [8,17]).

6. Conclusion

Fig. 5. Variation of the DOS due to large clusters in a jellium. (a) For one icosahedron of Mn icosahedra (i.e. 12 Mn icosahedra) obtained after an i&ation by a factor rz of an initial Mn icosahedron. The diameter of each Mn icosahedron is D = 9.21 A, so the diameter of the icosahedron of icosahedra is about Dz2 + D z 35 d, (F, = 3 eV, Ed = 10.88 ev). (b) For 192 Mn atoms from the Mn network of the actual AI,Mn phase (orthorhombic Cmcm). The Mn positions are taken from 48 ( = 4 x 4 x 3) neighbouring unit cells of A1,Mn (F, = 3 eV, Ed = 10.88 eV).

The main result of this work is evidence of the existence of cluster virtual bound states at different length scales: clusters, clusters of clusters... Their lifetime is larger for clusters with icosahedral symmetry. Moreover those states are very sensitive to the geometry of the cluster. This tendency to localisation should have strong effects on the transport properties of quasicrystals and their related approximants. However, we have shown elsewhere [13] that this tendency to localisation has a small effect on the total energy. This is in good agreement with the Hume-Rothery picture of quasicrystals and approximants.

References PI C. Janet and R. Mosseri (Eds.),

5. Confinement by large clusters Quasicrystals are characterised by infiation symmetry, therefore their atomic structure may exhibit a hierarchy of icosahedral clusters [ll]. Fig. 5(a) gives An(E) for twelve Mn icosahedra (i.e. 144 Mn atoms) obtained after inflation by a factor r* from an initial Mn icosahedron (r is the golden mean) [l l]. This DOS exhibits new fine peaks with respect to the case of one Mn icosahedron (Fig. 1). This shows the existence of cluster virtual bound states at the length scales of the icosahedron of icosahedra [13]. For comparison, Fig. 5(b) shows Aln(E) for a cluster containing 192 Mn atoms located on the Mn network of the actual crystalline phase AI,Mn. Al,Mn is not an approximant of a quasicrystal and it does not have similar transport properties to quasicrystals. An(E) exhibit some deviations from the VBS, however no fine peaks are observed.

Proc.

5th lilt.

ConJ: Quasicrys-

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