S Alarcón Villaseca

J.-M Dubois

É Gaudry

First-principles study of low-index surfaces of the Al 5 Co 2 Complex Metallic Alloy

Keywords: surfaces, first-principles calculations, electronic structure user supplied): Al5Co2, complex metallic alloy Surfaces, First-principles calculations, Electronic structure, Al 5 Co 2, Complex metallic alloy

) surfaces of the Al 5 Co 2 complex metallic alloy are studied by ab initio calculations. The relative stability of the possible surface planes built from bulk truncation is calculated and the influence of the atomic surface density, the interlayer spacing and the surface chemical composition on the plane selection is discussed. In addition, we show that the simulated images of scanning tunneling microscopy for each possible termination present a specific signature that appears to be sufficient to experimentally identify the surface plane.

Introduction

Since their discovery by Shechtman et al. in 1984 [1], quasicrystals have attracted much interest in the scientific community. Physical and chemical properties have been studied in relation to their unusual symmetries, and innovative applications are expected in various domains [START_REF] Dubois | Useful quasicrystals[END_REF], particularly in surface coating, due to their specific surface properties [START_REF] Tanabe | [END_REF][4][5][6][7][8]. Since complex metallic alloy (CMA) phases present a periodic structure which is closely related to the one of their parent quasicrystals, they are then useful, as models of the quasicrystalline phases to catch the key features of the more complex quasicrystalline phases [9][10][11].

The knowledge of the atomic and electronic surface structures of quasicrystals and/or CMAs is essential for the understanding of their attractive surface properties. However, the determination of the atomic and electronic surface structures in such complex alloys is not simple. In single metals, it is well known that the surface structure might differ from the ideal model built from truncation of the bulk structure due to interlayer relaxations, and sometimes reconstructions. In alloys, additional phenomena may occur such as chemical surface segregation or surface plane selection. These phenomena are driven by the minimization of the surface free energy, defined as the excess free energy per unit area at the surface of a material compared to the bulk [START_REF] Bechstedt | Principles of surface physics[END_REF]. Experimentally, the determination of the surface energy is difficult. The most usual methods are based on the Young equation [START_REF] Vitos | [END_REF]. A theoretical approach to determine the relative stability of the different possible surface planes is then fundamental to analyze the plane selection that occurs in quasicrystals and their related phases [14][15][16][17][18][19][20][21][22][23][24][25]. Many factors may have an influence on the surface energy. Among them, atomic densities, interlayer spacing (or gaps between planes in the case of quasicrystals) and chemical composition are key parameters [20]. To study the relative influence of these factors, we have chosen to study two different low-index surfaces of the Al 5 Co 2 alloy, which is an approximant of the decagonal phase [26]. The crystal structure of Al 5 Co 2 belongs to the space group P 6 3 /mmc (hP 28) [27,28]. Although this aluminide cannot be stricly identified as a usual sp Hume-Rothery alloy (it contains transition metal atoms), its electronic structure presents a low density of states (DOS) near the Fermi energy E F , leading to the formation of a valley in the DOS called a pseudogap that contributes to the alloy stability [29][30][31]. It is now well established that the formation of this pseudogap is due to the electron diffraction by Bragg planes combined with sp-d hybridization [29,31].

In this paper, we present a theoretical analysis of the atomic and electronic structures of the (100) and (001) surfaces. Along both crystallographic directions Al 5 Co 2 is described by a sequence of flat and puckered layers. Perpendicular to the [100] direction, two types of planes of different atomic surface densities alternate, a feature which is also found in orthorhombic Al 13 Co 4 and monoclinic Al 13 Fe 4 CMAs [28,32,33]. Perpendicular to the [001] direction, Al 5 Co 2 presents two types of planes with the same surface atomic density, but with different chemical compositions. The study of these two kinds of low-index surfaces may help to identify the key parameters responsible for the surface stability. The method used in this paper is based on density functional theory (DFT).

Computational details

Structural optimizations and electronic structure calculations have been performed within the density functional theory framework, using the plane-wave Vienna ab initio simulation package VASP [34,35]. Our calculations employ the PAW-PBE pseudopotentials [36][37][38][39].

Although the present calculations are called ab initio, there are convergence parameters linked to the numerical implementation of the density functional theory framework. Two of them are the plane-wave cutoff energy E cut and the density of k-points sampling the Brillouin zone. A set of test calculations on bulk Al 5 Co 2 (28 atoms/cell) has been performed to determine the values of E cut (350 eV) and the size of the Monkhorst-Pack k-points mesh (6×6×6) to achieve a targeted precision for the total energy smaller than 5 meV/atom. A similar k-grid mesh was used for calculations using slabs (6×6×1 for the (001) surface and 1×4×6 for the (100) surface).

The optimization of the atomic coordinates and unit cell parameters was performed by minimization of the Hellmann-Feynman forces acting on the atoms via a conjugate gradient algorithm. At the end of the ab initio energy minimization, the forces sustained by atoms were less than 0.01 eV/ • A. Simulations on the surfaces of Al 5 Co 2 were achieved by building symmetric slabs, made of a stack of atomic layers, for each kind of surface terminations. Different thicknesses of slabs were used in this work: 7-layers thick slabs for the puckered (100) and (001) surface terminations, 9-layers thick slab for the flat (001) surface termination. A thinner 5-layers thick slab for the flat (100) surface was used due to the large number of atoms in the slab. A set of test calculations on the flat (001) surface has been performed to evaluate the influence of the slab thickness on the calculated energies. It appears that the surface energy difference between a 5-layers and a 9-layers thick slab is A. The surface lattice parameters were set in agreement with the lattice parameters of the bulk alloy. Atomic relaxation has been performed by allowing displacement of all atoms within the slab except those lying in the central atomic layer.

Results and discussion

The experimental lattice parameters of bulk Al 5 Co 2 are a = 7.67

• A and c = 7.61

• A [28]. Geometry optimization calculations for bulk Al 5 Co 2 lead to lattice parameters in good agreement with the experimental data: a = 7.67

• A, c = 7.59 • A.
Total energy calculations yield a formation energy equal to ∆H f = -0.47 eV/atom, which is also in good agreement with the calculated value given by Ref. [40] (-0.48 eV/atom) as well as the experimental data (-0.43 eV/atom) [START_REF] Boerde | Cohesion in metals[END_REF].

Figure 1 shows the Al 5 Co 2 atomic structure along the [001] and [100] directions. Two kinds of atomic layers are stacked when viewed along both directions: flattype (F-type) and puckered-type (P-type) atomic layers. Perpendicular to the [100] direction, the two types of planes present different atomic densities: the F-type surface unit cell contains 8 Al atoms and 4 Co atoms, while the P-type surface unit cell contains 12 Al atoms and 4 Co atoms. This leads to an atomic density of about 0.16 at/ • A 2 for the P-type surface and of about 0.12 at/ • A 2 for the F-type surface. Perpendicular to the [001] direction, the two types of planes present the same atomic density (about 0.14 at/ • A 2 ), but different chemical compositions: 7 Al atoms for the P-type surface plane and 4 Co + 3 Al atoms for the F-type surface plane. From this structural bulk model, and for each direction considered, two different kinds of surface terminations can be obtained by bulk truncation: F-type and P-type topmost layers.

Energetics

The relative energies of surfaces with different stoichiometries cannot be evaluated directly by comparing the total energies of the corresponding slabs. We thus follow the methodology described in Refs. [START_REF] Bechstedt | Principles of surface physics[END_REF][START_REF] Moll | [END_REF]43], where the surface energy γ α of a given surface α is calculated as a function of the chemical potential µ Al of the Al atom in Al 5 Co 2 :

2Aγ α = E α slab -µ Al N Al -µ Co N Co ( 1
)
where 2a shows the surface energy differences between the F-type and P-type surface planes as a function of µ Al for the two kinds of orientations considered. From Fig. 2a, it appears that the most stable (100) and (001) surface terminations are the P-type atomic planes.

A
In single metals, the anisotropy of the surface energy as a function of the ori- A 2 ) for both P-type and F-type terminations although the densities of both terminations differ. The formation of the (001) surface leads to the break of 0.59 bonds per • A 2 30 bonds in the surface unit cell for the P-type termination and only 0.53 bonds per • A 2 27 bonds in the surface unit cell for the F-type termination, altough these two terminations present the same surface atomic density number of atoms in the surface unit cell. For both (100) and (001) low-index surfaces, a simple relation between the number of broken bonds in the first environment of each atom and the surface energy seems then difficult to obtain. Indeed, in such a complex metallic alloy, some of the Al-Co bonds may have enhanced covalency and therefore all broken bonds may not contribute to the surface energy in the same way [46].

F
Another parameter that influences the surface energy is the chemical composition of the surface layer. When comparing the stabilities of the two (001) terminations it appears that the most stable termination, among the two topmost surfaces with the same atomic density considered, is the Al-rich atomic plane. An interpretation of this result is the presence of cobalt atoms in the (001) F-type surface plane: the elementary surface energy of cobalt (2.522 J.m -2 [47], 2.550 J.m -2 [START_REF] Boerde | Cohesion in metals[END_REF]) is higher than that of aluminium (1.143 J.m -2 [47], 1.160 J.m -2 [START_REF] Boerde | Cohesion in metals[END_REF]). The presence of cobalt atoms in the topmost layer is unfavoured, the low-density (100) surface with a surface cobalt atomic composition of 33% being more stable than the denser F-type (001) surface (surface cobalt atomic composition of 57%).

It is interesting then to compare the surface energies of the Al 5 Co 2 CMA to the more simple AlCo alloy. In the case of the (110) surface of AlCo, all planes are equivalent and the surface energy can be deduced from the equation used to calculate the surface energy in single metals: γ = 1

2A (E slab -N E bulk ) where E slab is the total energy of the symmetric slab considered, E bulk is the total energy of the corresponding bulk and N is the number of layers considered in the slab. In this approach, we consider a perfect (110)AlCo surface, we do not consider the deviation from the bulk stoichiometry in the second layer as highlighted by Ref. [48]. However, our results for the interlayer distances and the surface rippling are in good agreement with the experimental data: we found a rippling for the topmost layer of 0.19

• A (0.18

• A from Ref. [48]) and an interlayer distance d 12 = 2.04 [48]). The surface energy calculated for (110)AlCo (2.07 J.m -2 ) is higher than the (100) surface energies calculated for the two considered terminations. This confirms that the presence of Co atoms at the surface is unfavoured: (100) F-type Al 5 Co 2 presents a surface atomic density much lower than (110)AlCo, but the (110)AlCo contains much more Co surface atoms. for the bulk calculations are in good agreement with previous studies [29][30][31].

• A (2.04 • A from Ref.
The effect of the surface is mainly confined to the two topmost surface layers. Concerning the F-type termination, the Co-3d band width of the surface layer is slightly reduced by about 0.2 eV compared to the bulk. The Co d-band is also shifted by about 0.4 eV towards lower binding energies compared to the corresponding bulk "F-type" Co d-band. These effects are attributed to the reduction in the coordination number of surface atoms compared to the bulk. Similar shifts of the calculated transition metal d-band center of mass of the surface DOS have also been reported for the tenfold surface of the d-Al-Ni-Co quasicrystal (≃ 0.8 eV) and for the fivefold surface of the i-Al-Pd-Mn quasicrystal (≃ 0.5 eV) [9,10].

A noteworthy feature in the density of states is the existence of a pseudogap at the Fermi level E F of the bulk material: n(E F ) = 0.11 states/eV/atom, in agreement with the value of Trambly et al. (0.13 states/eV/atom [29]). Concerning the F-type (001) surface, the pseudogap is hardly noticeable on the DOS of the topmost atomic layer, while it appears in the next subsurface atomic layers. This behaviour is in agreement with the shallower surface pseudogap calculated on the fivefold surface of i-Al-Pd-Mn quasicrystal [11]. However, the decrease of the pseudogap depth is much more pronounced in the latter case.

Simulated STM images

STM images are simulated within the Tersoff-Hamann approximation [49,50]. The calculated images are shown in Fig. 3 for each type of surface termination (V bias = 0.7 V, d tip-surf ace = 3 • A).

The Al 5 Co 2 (001) surface

The STM image corresponding to the Al-rich termination shows lines of bright triangles. This contrast is due to the presence of three Al atoms slightly above (≃ 0.44

The Al 5 Co 2 (100) surface

The STM image of the F-type termination shows very bright spots aligned along the [001] direction and a rather bright zig-zag feature. The contrast of the brightest spots are due to Al atoms at the mean position of the atomic plane. The contrast of the zig-zag feature is due to the six remaining Al atoms slightly above the mean position of the atomic plane and to the two Co atoms slightly below the mean position of the atomic plane. The two remaining Co atoms are about 0.28

•

A below the mean position of the atomic plane, they do not contribute to the bright contrast of the STM image.

In the case of the P-type termination, we can observe sequences of zig-zag rows of bright pentagons. The pentagonal features are due to the Al atoms slightly above the mean position of the topmost atomic plane (about 0.26-0.29 arranged in pentagonal features. The center of the pentagons is occupied by a Co atom underneath the mean position of the plane (about 0.51

• A). The simulated STM image shows also lines of grey contrast, which is due to a Co atom slightly above the mean position of the plane (about 0.22 • A).

Conclusion

The atomic and electronic structures of the (100) and (001) surfaces of the Al 5 Co 2 CMA have been studied by ab initio calculations.

Among the three main factors influencing the surface energy and the selection of the surface plane in complex metallic alloys [20], the interlayer spacing does not seem to play a major role here since it is the same for the two energetically different terminations considered for a given low-index surface. The comparison of the stability of the two surface structures with the same number of atoms in the surface unit cell -the (001) surface -shows that the Al-richer atomic layer is preferred as surface termination. The comparison of the stability of the two surface structures with close atomic compositions (25% or 33% at. Co for the two considered terminations of the (100) surface) shows that the denser atomic layer is preferred as surface termination. However, the influence of the atomic density, which is not always directly related to the number of broken bonds in such a complex alloy, is difficult to highlight rigorously, since in the examples considered here, one cannot set stricly apart its influence from the influence of the atomic composition. The comparative study of the four considered surface models shows that the surface energy and the percentage of Co surface atoms increase in the following sequence: (100) P-type termination, (100) F-type termination, (110)AlCo surface, (001) F-type termination. The atomic composition seems then to have a significant impact on the surface plane selection, the presence of Co surface atoms being unfavoured.

In this paper, we have also shown that the determination of the electronic structure of each surface leads to a specific STM image: the differences between the two considered topmost layers appear to be sufficient to identify the surface plane by scanning tunneling microscopy. Experimentally, studies of the Al 5 Co 2 (100) surface would be interesting, but the synthesis of a millimeter-sized sample is difficult, since (i) this phase is formed during a peritetic reaction at a temperature higher than 1400 K, (ii) it requires a narrow temperature range for the crystal growth (about 50 o C), combined with a narrow composition range for the Al 5 Co 2 phase (± 3%at Al composition). However, it might be worth: since the stable (100) termination built from bulk truncation (P-type surface) contains Co surface atoms, one may expect an attractive behaviour for catalysis applications.

3. 2 .

 2 Figure2bshows the calculated local density of states (LDOS) on surface and subsurface planes for the Al 5 Co 2 (001) slab with F-type termination. The results

• 12 • 41 •

 1241 A) the mean position of the plane. The other Al atoms, at ≃ 0.A underneath the mean position of the atomic plane contribute to the dark contrast in the STM image.The STM image corresponding to the F-type termination shows similar triangles although connected to each other by a bright spot. The bright constrast of the triangle is due to the 3 Al atoms slightly (≃ 0.15 • A) above the mean position of the atomic plane. The bright spot connecting the triangles is due to the Co atom situated at about the same z-position as the 3 Al atoms. The other three Co atoms slightly underneath the mean position of the topmost atomic plane, do not appear in the STM image.

Figure 1 .

 1 Figure 1. Atomic structure of Al 5 Co 2 along the [100] (a) and [001] (b) directions.

Figure 2 .

 2 Figure 2. (a) Calculated surface energies of the different topmost layers considered in each orientation. The chemical potential µ bulkAl corresponding to f cc aluminium is taken as reference. (b) Densities of states of the surface and sub-surface planes compared to the corresponding bulk DOS for the F-type (001) topmost layer. The label S refers to the surface plane, S-1 to the plane immediately below, etc... The DOS of each surface atomic plane are compared with the DOS of the corresponding atomic plane in the bulk structure.

Figure 3 . 3 •

 33 Figure 3. Simulated STM images corresponding to the (100) and (001) surfaces (V bias = 0.7 V, d tip-surf ace = 3 • A).

Figure 1 .Figure 2 .

 12 Figure 1. Atomic structure of Al 5 Co 2 along the [100] (a) and [001] (b) directions.

Figure 3 . 3 •

 33 Figure 3. Simulated STM images corresponding to the (100) and (001) surfaces (V bias = 0.7 V, d tip-surf ace = 3 • A).

  is the surface area, µ Al (respectively µ Co ) is the chemical potential of the Al (respectively Co) atom and N Al (respectively N Co ) is the number of Al (respectively Co) atoms. Since Al 5 Co 2 is a stable alloy (∆H f = µ bulk Al5Co2 -5µ bulk Al -2µ bulk Co < 0), the chemical potentials µ Al and µ Co in Al 5 Co 2 have to be smaller than the corresponding µ bulk Al and µ bulk Co in the elemental bulk system, leading to the range of variation of µ Al : ∆Hf 5 < µ Al -µ bulk Al < 0. Figure

  entation is explained by the "broken bonds" model, that states that the surface energy is proportional to the product of the number of broken bonds times the bonding energy[44,45]: the densest atomic planes, whose atoms are surrounded by the most complete environment, presents the lowest surface energy. In Al 5 Co 2 , the situation is more complex, since all atoms in bulk Al 5 Co 2 do not have the same kind of environment, contrary to what happens in simpler metals. The atoms in bulk Al 5 Co 2 occupy five types of inequivalent sites. One can then define five types of clusters surrounding each atom in the structure[27] (see Tab. 1). The creation of the (100) surface induces the break of the same number of bonds (0.61 bonds per

	July 19, 2010	10:45	Philosophical Magazine	gaudry
			•
			o r
			P
				e
				e r
				R
				e
				v i e
				w
				O n l
				y

Taylor & Francis and I.T. Consultant

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

July 19, 2010 10:45 Philosophical Magazine gaudry

Acknowledgements

The European Network of Excellence on "Complex Metallic Alloys" (NMP3-CT-2005-500145), and the "Agence Nationale de la Recherche" (ANR-08-BLAN-0041-01) are acknowledged for their financial support. Computing time was partially supplied by the "Institut Français Développement et de Recherche en Informatique Scientifique" in Orsay (project 99642). One of us (SAV) thanks INC-CNRS and the "Région Lorraine" for financial support during his PhD thesis.