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First-principles calculations for stability of atomic structures of Al-rich AlX (X=Sc–Zn) alloys, including AlMn quasicrystal: II. Medium-ranged interactions of X pairs in Al
Intermetallics 14 (2006) 913–916 www.elsevier.com/locate/intermet
First-principles calculations for stability of atomic structures of Al-rich AlX (XZSc–Zn) alloys, including AlMn quasicrystal: II. Medium-ranged interactions of X pairs in Al T. Hoshino a,1, M. Asato b,*, S. Tanaka a, F. Nakamura c,2, N. Fujima a b
a Department of Applied Physics, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan Department of General Education, Tokyo Metropolitan College of Technology, 1-10-40 Higashi-Ooi, Shinagawa, Tokyo 140-0011, Japan c Tokyo Institute of Technology, Tokyo 152-8552, Japan
Available online 28 February 2006
Abstract The AlMn quasicrystal (QC) is a metastable phase with quasiperiodic arrangement of Mackay icosahedrons (MIs) (Al42Mn12, a vacancy at the center) and is stabilized by the the addition of Si. We give qualitative discussions for the micromechanism of the stability of the atomic structures of MI in AlMn QC, as well as the Al-rich AlX (XZSc, V, Cu, Zn) alloys, by using the ab initio calculations for point defects in Al(fcc), where the minority elements in the AlX alloys are considered as impurities. We show: (1) the atomic structure of MI may be stabilized by the medium˚ , which is close to the observed interatomic ranged interaction of Mn pairs, which is strongly attractive around the interatomic distance of 4.9 A ˚ , which is close to distance for the first-neighbor Mn–Mn pairs in MI. (2) Magnetism plays an important role around the interatomic distance 7.5 A the interatomic distance of second-neighbor Mn–Mn pairs in MI. We also discuss the effect of a vacancy and Si impurities in MI. q 2006 Elsevier Ltd. All rights reserved. Keywords: B. Alloy design; B. Electronic structure of metals and alloys; D. Defects: point defects; D. Site occupancy; E. Ab-initio calculations
1. Introduction Al-rich alloys with transition metals (Sc–Ni), which are important because of the desirable technological properties of many of these alloys, form a variety of atomic structures depending on d-electron number (nd)[1]. For example, L12 for Al3Sc (ndw1.6), DO22 for Al3V (ndw3.7), and the Mackay icosahedron (MI, Al42Mn12 with a vacancy at the center) in the metastable quasicrystal (QC) phase for Al80Mn20 (ndw5.9) [2–4]; the numbers of nd are the values calculated for single X impurities in Al. It was also found that the atomic structures of small clusters at the initial stage of the Guinier–Preston (GP) zone formation of the low-concentration alloy Al1KcXc(XZ Cu, Zn; c!0.05) are correlated with high strength of alloy [5–8]. Further it is now obvious that the bulk metallic glasses (BMGs), which often contain three or more chemical elements, display the technologically desirable properties [9]. Thus, it is * Corresponding author. Tel.: C81 3 3474 4135; fax: C81 3 3471 6338. E-mail addresses: [email protected] (T. Hoshino), [email protected] (M. Asato), [email protected] (F. Nakamura). 1 Fax: C81 53 478 1276. 2 Fax: C81 3 5734 3139.
0966-9795/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2006.01.009
very interesting to study systematically the micromechanism of the stability of these atomic structures which may determine the physical properties of the materials. Theoretically, it is now possible to determine the atomic and electronic structures of elemental metals, simple metallic compounds, and the point defects in those metals by the ab initio calculations, based on the density-functional theory (DFT) and the band-structure methods. For example, we have shown that the full-potential Korringa–Kohn– Rostoker (FPKKR) Green’s function method, developed by the Ju¨lich group, combined with the generalized-gradient approximation of Perdew and Wang of the DFT (PW91GGA), reproduce accurately the bulk properties of metals as well as point-defect energies [10]. We have shown that the GGA corrects the deficiency of the local-spin-density approximation (LSDA) of the DFT, such as the underestimation of the equilibrium lattice parameters and the overestimation of the bulk moduli and the monovacancy formation energies in metals. For example, the equilibrium lattice parameters, bulk moduli, and monovacancy formation energies of elemental fcc and bcc metals (Li–Au) are reproduced, respectively, within the errors of 1, 10, and 10% of the experimental values. It was also shown that the underestimation of the magnetism in Fe and Fe-compounds, due to the LSDA, is corrected by the GGA [11,12]. We can
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now use the force calculations to determine the lattice distortion around the point defects [7,13]. It is also noted that the screened FPKKR calculations have been possible for the systems of many atoms (less than w100 atoms) in an unit cell [14]. Although highly accurate for predicting alloy properties, the ab initio calculations are limited to relatively small systems. The direct application of these techniques to the determination of the atomic structures of QCs and BMGs is clearly impractical. Instead of the direct ab initio calculations, we plan to study quantitatively the micromechanism of the stability of the atomic structures of QCs and BMGs by use of the atomistic simulations [15] with the accurate embeddedatom-method potentials (EAMP), in cooperation with Shibutani’s group at Osaka University. In our project, we determine all the parameters in the EAMP by fitting to ab initio data obtained from the present ab initio calculations, following the way of Mishin et al. [16]. In the preceding paper [17], we presented the calculated results for the cohesive energies of Al and X (XZSc–Zn), Wigner-Seitz radius dependence of total energies of different atomic structures, the solution energies of X impurities in Al, the first-neighbor X–X and X–Vacancy interaction energies in Al, which may be part of the cohesive energies of AlX alloys. These are important ab initio input data to construct the accurate EAMP. Using the calculated results, we have qualitatively discussed that the sp–d interaction of X (XZSc–Ni) with the neighboring Al atoms is strong and prevents the first-neighboring of X pairs: the sp-d interaction of Al–X is stronger than the d-d interaction of X–X pairs. It was also discussed that Al atoms may easily rearrange in the Al-rich AlX alloys, because of the strong free-electron character of Al spelectrons (very small structure dependence of Al-host). In the present paper, we give the calculated results for the interatomic-distance dependence of X–X (XZSc–Zn) interaction in Al(fcc), due to the strong Al–X(sp–d) interaction, which is medium-ranged and oscillating[19,20]. We found strong attraction at some interatomic-distances, which determine the atomic structures of Al-rich AlX alloys, including the MI in the AlMn QC.
2. Pair and many-body interactions of X impurities in Al and atomic structures of Al-rich AlX (XZSc–Zn) Fig. 1 shows the atomic structures of L12 and DO22, and MI, which were found in the Al-rich AlX alloys such as Al3Sc, Al3V, Al80Mn20 [1–4]. Differently from these ordered alloys, the low concentration Al-based alloys, such as AlCu and AlZn are segregated and form different shapes of GP zones: (001)disc shape for AlCu and spherical shape for AlZn [5–8]. We show that the element (or d-electron)-dependence for these different atomic arrangements are systematically understood by use of the pair and many-body (up to the four-body) interactions of X impurities in Al, obtained by the present ab initio calculations. 2.1. First-neighbor pair and many-body interaction energies Fig. 2 shows the pair (X2, XZSc–Zn) and many-body (triangle X3 and tetrahedron X4) interaction energies among first neighbors in Al [6,8]: the magnetism, being important for Cr, Mn, and Fe, is discussed in Section 2.3. The calculated results are summarized as follows. (1) The first-neighbor X–X interactions are strongly repulsive for X (ZSc–Ni), almost zero for Cu, weakly attractive for Zn. (2) The nth many-body interactions become weaker with the increase in n. The strong X–X repulsion (XZSc–Ni) results in the ordered structures of alloys [1,3], while the attraction of Zn–Zn to the segregation phase [5]. For Cu in Al, the lattice distortion effect may be important because of a weak Cu–Cu interaction and a large misfit of atomic radii of Cu and Al [7,18]. We have already shown that the triangle interactions are very well correlated with the shapes of the small clusters at the initial stage of the GP zone formation: the triangle interaction energies of Cu and Zn are positive (0.03 eV, repulsion) and zero, corresponding to the (001) disc (Cu in Al) and the spherical shape (Zn in Al), respectively [6,8].
Fig. 1. Atomic structures of Al-rich AlX alloys (XZSc, V, Mn). The Mackay icosahedron consists of Al42Mn12 and a vacancy at the centre, and is stabilized by the substitution of a proper amount of Si atoms for Al atoms in the first shell Al12. The 1st–8th neighbours in the fcc structure are indicated by the sites 1–8 around a central O, shown in (b).
T. Hoshino et al. / Intermetallics 14 (2006) 913–916
Fig. 2. 1st-neighbour pair, triangle, and tetrahedron interaction energies of X (X=Sc–Zn) in Al.
2.2. Medium-ranged interactions of X–X in Al and the atomic structures of Al-rich AlX alloys We show that the tendency of atomic structures of Alrich AlX (XZSc,V, Mn) alloys, such as L12 for Al3Sc, DO22 for Al3V, MI(Al42Mn12) in the AlMn QC, are qualitatively understood by the use of the medium-range X–X interaction energies in Al (fcc). Fig. 3 shows the calculated results (without spin-polarization) for the interatomic distances (1st–8th in Fig. 1(b)) of the X–X interaction energies in Al(fcc):the magnetism for Cr, Mn, and Fe is discussed in Section 2.3. It is noted that the interactions are medium-ranged and show the oscillating behavior [19,20]. For Sc, the X–X interaction is strongly repulsive for firstneighbor pairs, strongly attractive (negative and large) for the second-neighbor pairs, and repulsive for the thirdneighbor pairs. Thus, we can easily expect that Sc atoms are arranged at the interval of the second-neighbor interatomic distance, namely the atomic structure of L12
Fig. 3. Distance dependence (1st–8th in Fig. 1(b)) of X–X interaction energies in Al. The spin-polarization effect is neglected.
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(Fig. 1(a)). For V, the interaction is strongly repulsive for the second-neighbor pairs and very weak around the thirdneighbor pairs. Thus, for Al 3V the DO 22 structure (Fig. 1(b)) may become stable compared with the L12 structure. For XZCr, Mn, and Fe, the interactions are very repulsive up to the distance being just a little bit shorter than the third-neighbor distance and become very attractive ˚ ): for Mn the around the third-neighbor distance (4.7–5.0 A ˚ strongest attraction (K0.05 eV) at 4.9 A. Thus, it is obvious that Mn impurities are arranged at the interval of the ˚ . We can easily interatomic distance between 4.7 and 5.0 A expect that the atomic structure of a high-symmetric icosahedron of 12 Mn impurities, being a sublattice of MI (see Fig. 1(c)), may be very stable because it includes a large number (30) of Mn–Mn pairs: for this atomic structure the energy gain due to the Mn–Mn interactions is as large as wK1.5 eV(ZK0.0.5!30). It is noted that this ˚ ) is very close to one of interatomic distance (4.7–5.0 A ˚ and w7.5 A ˚) the observed interatomic distances (4.7–4.9 A of Mn–Mn pairs [2], although the present calculated interatomic distance of Mn–Mn pairs may be changed a little bit by the quantitative calculations based on the atomistic simulations with the accurate EAMP, as discussed in Section 3. We also found that the Mn–Mn interaction at ˚ , corresponding to the second-neighbor of a w7.5 A sublattice icosahedron of Mn 12, may be repulsive (0.007 eV). This weak repulsion may reduce slightly the
Fig. 4. Distance dependence (1st–8th in Fig. 1(b)) of X–X (XZCr, Mn, and Fe) interaction energies in Al. The spin-polarization effect is included. See text for details.
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3. Summary and future problem
Fig. 5. Distance dependence (1st–8th in Fig. 1(b)) of Mn–Vacancy interaction energies in Al. The spin-polarization effect is included.
stability of MI. However, we show in Section 2.3 that this weak repulsion may be changed to a weak attraction (K0.005 eV) by taking into account the magnetism of Mn [21,22]. 2.3. Magnetism of Cr, Mn, and Fe Fig. 4 shows the calculated results with the spin-polarization effect for XZCr, Mn, and Fe. The calculated results for single X impurities and first-neighbor X pairs were discussed in Refs. [21–23]. For single impurities, the energy gain due to the magnetism is as large as 0.3 eV for Mn. For two impurities, we can obtain two stable solutions of ferromagnetic (FM) and antiferromagnetic (AFM) states. These magnetic energies (per impurity) of impurity pairs are always increased compared with the magnetic energies of single impurities, because of the weaker X–X (d–d) interaction compared with the Al–X (sp–d) interactions [21,22], as discussed in the preceding paper. Here we discuss only two important points. ˚ is AF. The (1) The ground state of Mn–Mn pairs at 4.9 A energy gain due to AF is almost same as that obtained by the non-magnetic Mn–Mn pairs. ˚ changes to attraction (2) The repulsive interaction at w7.5 A by the inclusion of magnetism of Mn: 0.007 eV for nonmagnetic coupling and K0.005 eV for AF coupling. Thus, the inclusion of magnetism is important for the Mn–Mn interactions which stabilize the atomic structure of MI in the AlMn QC. The detailed discussions will be given elsewhere [24]. 2.4. Roles of vacancy and Si for stability of MI The interaction between a vacancy at the center of MI and Mn of a sublattice is almost zero (or weakly repulsive) around ˚ , as shown in Fig. 5: it is noted that the distance of 4.5–4.7 A ˚ ) by the vacancy—Mn is 95% of Mn–Mn distance (4.7–5.0 A geometry of an icosahedron. We have found that this Mn– vacancy interaction may be attractive by w0.01 eV if Si atoms exist between a vacancy and Mn (for example, at sites 1 and 1 0 between sites 0 and 3, as shown in Fig. 3). The details will be discussed elsewhere [24].
We have presented ab initio data for the study of the micromechanism of the atomic structures of the Mackay icosahedron Al42Mn12 in the AlMn QC as well as Al-rich AlX alloys, and have shown that the interactions of X–X (XZSc– Ni) pairs are strong and medium-ranged, and show the oscillating behavior [19,20]. We have found that for XZMn the interaction is strongly repulsive up to the distance being just a little bit shorter than the third-neighbor distance, and become strongly attractive around the third neighbor distance ˚ ), which is close to one of the observed distances for (w4.9 A Mn–Mn pair [2]. This strong attraction may be a major factor for the stability of the atomic structure of MI in the AlMn QC. We presented only qualitative discussions for the stability of MI. In order to study quantitatively the present mechanism of the stability of MI, we must perform the atomistic simulations [15] with the accurate EAMP, all the parameters in which are determined by fitting to the ab initio input data, obtained from the present calculations. We believe that the inclusion of the strong interactions of Mn with the neighboring Al atoms and the medium-ranged Mn–Mn interaction, given in the present and preceding papers, is essential for the construction of EAMP of AlMn QC. References [1] Mondolf F. Aluminium alloys: structure and properties. Boston, MA: Butterworths; 1976. [2] Maret M, Pasturel A, Senillou C, Dubois JM, Chieux P. J Phys Fr 1989; 50:295. [3] Elser V, Henley CL. Phys Rev Lett 1985;55:2883. [4] Bancel PA, Heiney PA. Phys Rev B 1986;33:7917. [5] Sato T, Hirosawa S, Hirose K, Maeguchi T. Metall Mater Trans 2003; 34A:2745. [6] Hoshino T, Asato M, Zeller R, Dederichs PH. Phys Rev B 2004;70: 094118. [7] Hoshino T, Asato M, Fujima N, Papanikolaou N. Trans Mater Res Soc Jpn 2005; 30(3):877. [8] Nakamura F, Hoshino T, Tanaka S, Hirose K, Hirosawa S, Sato T. Trans Mater Res Soc Jpn 2005; 30(3):873. [9] Inoue A. Bulk amorphous alloys. Zurich: Trans Tech Publ; 1998. [10] Hoshino T, Mizuno T, Asato M, Fukushima H. Mater Trans 2001;42:2206. [11] Hoshino T, Asato M, Nakamura T, Zeller R, Dederichs PH. J Magn Magn Mater 2004;272–276:e229. [12] Hoshino T, Asato M, Nakamura T, Zeller R, Dederichs PH. J Magn Magn Mater 2004;272–276:e231. [13] Papanikolaou N, Zeller R, Dederichs PH. Phys Rev B 1997;55:4167. [14] Zeller R. Phys Rev B 1997;55:9400. [15] Ogata S, Shimizu F, Li J, Wakeda M, Shibutani Y. In this conference. [16] Mishin Y, Mehl MJ, Papaconstantpoulos DA, Voter AF, Kress ED. Phys Rev B 2001;63:224106. [17] Hoshino T, Asato M, Fujima N. In this conference. [18] Wolverton C. Philos Mag Lett 1999;79:683. [19] Friedel J. Helv Phys Acta Sci (Paris) 1987;305:171. [20] Zou J, Carlsson AE. Phys Rev Lett 1993;70:3748. [21] Hoshino T, Zeller R, Dederichs PH, Weinert M. Europhys Lett 1993;24: 495. [22] Hoshino T, Zeller R, Dederichs PH. Phys Rev B 1996;53:8971. [23] Hoshino T, Nakamura F. J. Metas Nanostat Mater 2005; 24–25:237. [24] Hoshino T. In preparation.
First-principles calculations for stability of atomic structures of Al-rich AlX (XZSc–Zn) alloys, including AlMn quasicrystal: II. Medium-ranged interactions of X pairs in Al T. Hoshino a,1, M. Asato b,*, S. Tanaka a, F. Nakamura c,2, N. Fujima a b
a Department of Applied Physics, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561, Japan Department of General Education, Tokyo Metropolitan College of Technology, 1-10-40 Higashi-Ooi, Shinagawa, Tokyo 140-0011, Japan c Tokyo Institute of Technology, Tokyo 152-8552, Japan
Available online 28 February 2006
Abstract The AlMn quasicrystal (QC) is a metastable phase with quasiperiodic arrangement of Mackay icosahedrons (MIs) (Al42Mn12, a vacancy at the center) and is stabilized by the the addition of Si. We give qualitative discussions for the micromechanism of the stability of the atomic structures of MI in AlMn QC, as well as the Al-rich AlX (XZSc, V, Cu, Zn) alloys, by using the ab initio calculations for point defects in Al(fcc), where the minority elements in the AlX alloys are considered as impurities. We show: (1) the atomic structure of MI may be stabilized by the medium˚ , which is close to the observed interatomic ranged interaction of Mn pairs, which is strongly attractive around the interatomic distance of 4.9 A ˚ , which is close to distance for the first-neighbor Mn–Mn pairs in MI. (2) Magnetism plays an important role around the interatomic distance 7.5 A the interatomic distance of second-neighbor Mn–Mn pairs in MI. We also discuss the effect of a vacancy and Si impurities in MI. q 2006 Elsevier Ltd. All rights reserved. Keywords: B. Alloy design; B. Electronic structure of metals and alloys; D. Defects: point defects; D. Site occupancy; E. Ab-initio calculations
1. Introduction Al-rich alloys with transition metals (Sc–Ni), which are important because of the desirable technological properties of many of these alloys, form a variety of atomic structures depending on d-electron number (nd)[1]. For example, L12 for Al3Sc (ndw1.6), DO22 for Al3V (ndw3.7), and the Mackay icosahedron (MI, Al42Mn12 with a vacancy at the center) in the metastable quasicrystal (QC) phase for Al80Mn20 (ndw5.9) [2–4]; the numbers of nd are the values calculated for single X impurities in Al. It was also found that the atomic structures of small clusters at the initial stage of the Guinier–Preston (GP) zone formation of the low-concentration alloy Al1KcXc(XZ Cu, Zn; c!0.05) are correlated with high strength of alloy [5–8]. Further it is now obvious that the bulk metallic glasses (BMGs), which often contain three or more chemical elements, display the technologically desirable properties [9]. Thus, it is * Corresponding author. Tel.: C81 3 3474 4135; fax: C81 3 3471 6338. E-mail addresses: [email protected] (T. Hoshino), [email protected] (M. Asato), [email protected] (F. Nakamura). 1 Fax: C81 53 478 1276. 2 Fax: C81 3 5734 3139.
0966-9795/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2006.01.009
very interesting to study systematically the micromechanism of the stability of these atomic structures which may determine the physical properties of the materials. Theoretically, it is now possible to determine the atomic and electronic structures of elemental metals, simple metallic compounds, and the point defects in those metals by the ab initio calculations, based on the density-functional theory (DFT) and the band-structure methods. For example, we have shown that the full-potential Korringa–Kohn– Rostoker (FPKKR) Green’s function method, developed by the Ju¨lich group, combined with the generalized-gradient approximation of Perdew and Wang of the DFT (PW91GGA), reproduce accurately the bulk properties of metals as well as point-defect energies [10]. We have shown that the GGA corrects the deficiency of the local-spin-density approximation (LSDA) of the DFT, such as the underestimation of the equilibrium lattice parameters and the overestimation of the bulk moduli and the monovacancy formation energies in metals. For example, the equilibrium lattice parameters, bulk moduli, and monovacancy formation energies of elemental fcc and bcc metals (Li–Au) are reproduced, respectively, within the errors of 1, 10, and 10% of the experimental values. It was also shown that the underestimation of the magnetism in Fe and Fe-compounds, due to the LSDA, is corrected by the GGA [11,12]. We can
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now use the force calculations to determine the lattice distortion around the point defects [7,13]. It is also noted that the screened FPKKR calculations have been possible for the systems of many atoms (less than w100 atoms) in an unit cell [14]. Although highly accurate for predicting alloy properties, the ab initio calculations are limited to relatively small systems. The direct application of these techniques to the determination of the atomic structures of QCs and BMGs is clearly impractical. Instead of the direct ab initio calculations, we plan to study quantitatively the micromechanism of the stability of the atomic structures of QCs and BMGs by use of the atomistic simulations [15] with the accurate embeddedatom-method potentials (EAMP), in cooperation with Shibutani’s group at Osaka University. In our project, we determine all the parameters in the EAMP by fitting to ab initio data obtained from the present ab initio calculations, following the way of Mishin et al. [16]. In the preceding paper [17], we presented the calculated results for the cohesive energies of Al and X (XZSc–Zn), Wigner-Seitz radius dependence of total energies of different atomic structures, the solution energies of X impurities in Al, the first-neighbor X–X and X–Vacancy interaction energies in Al, which may be part of the cohesive energies of AlX alloys. These are important ab initio input data to construct the accurate EAMP. Using the calculated results, we have qualitatively discussed that the sp–d interaction of X (XZSc–Ni) with the neighboring Al atoms is strong and prevents the first-neighboring of X pairs: the sp-d interaction of Al–X is stronger than the d-d interaction of X–X pairs. It was also discussed that Al atoms may easily rearrange in the Al-rich AlX alloys, because of the strong free-electron character of Al spelectrons (very small structure dependence of Al-host). In the present paper, we give the calculated results for the interatomic-distance dependence of X–X (XZSc–Zn) interaction in Al(fcc), due to the strong Al–X(sp–d) interaction, which is medium-ranged and oscillating[19,20]. We found strong attraction at some interatomic-distances, which determine the atomic structures of Al-rich AlX alloys, including the MI in the AlMn QC.
2. Pair and many-body interactions of X impurities in Al and atomic structures of Al-rich AlX (XZSc–Zn) Fig. 1 shows the atomic structures of L12 and DO22, and MI, which were found in the Al-rich AlX alloys such as Al3Sc, Al3V, Al80Mn20 [1–4]. Differently from these ordered alloys, the low concentration Al-based alloys, such as AlCu and AlZn are segregated and form different shapes of GP zones: (001)disc shape for AlCu and spherical shape for AlZn [5–8]. We show that the element (or d-electron)-dependence for these different atomic arrangements are systematically understood by use of the pair and many-body (up to the four-body) interactions of X impurities in Al, obtained by the present ab initio calculations. 2.1. First-neighbor pair and many-body interaction energies Fig. 2 shows the pair (X2, XZSc–Zn) and many-body (triangle X3 and tetrahedron X4) interaction energies among first neighbors in Al [6,8]: the magnetism, being important for Cr, Mn, and Fe, is discussed in Section 2.3. The calculated results are summarized as follows. (1) The first-neighbor X–X interactions are strongly repulsive for X (ZSc–Ni), almost zero for Cu, weakly attractive for Zn. (2) The nth many-body interactions become weaker with the increase in n. The strong X–X repulsion (XZSc–Ni) results in the ordered structures of alloys [1,3], while the attraction of Zn–Zn to the segregation phase [5]. For Cu in Al, the lattice distortion effect may be important because of a weak Cu–Cu interaction and a large misfit of atomic radii of Cu and Al [7,18]. We have already shown that the triangle interactions are very well correlated with the shapes of the small clusters at the initial stage of the GP zone formation: the triangle interaction energies of Cu and Zn are positive (0.03 eV, repulsion) and zero, corresponding to the (001) disc (Cu in Al) and the spherical shape (Zn in Al), respectively [6,8].
Fig. 1. Atomic structures of Al-rich AlX alloys (XZSc, V, Mn). The Mackay icosahedron consists of Al42Mn12 and a vacancy at the centre, and is stabilized by the substitution of a proper amount of Si atoms for Al atoms in the first shell Al12. The 1st–8th neighbours in the fcc structure are indicated by the sites 1–8 around a central O, shown in (b).
T. Hoshino et al. / Intermetallics 14 (2006) 913–916
Fig. 2. 1st-neighbour pair, triangle, and tetrahedron interaction energies of X (X=Sc–Zn) in Al.
2.2. Medium-ranged interactions of X–X in Al and the atomic structures of Al-rich AlX alloys We show that the tendency of atomic structures of Alrich AlX (XZSc,V, Mn) alloys, such as L12 for Al3Sc, DO22 for Al3V, MI(Al42Mn12) in the AlMn QC, are qualitatively understood by the use of the medium-range X–X interaction energies in Al (fcc). Fig. 3 shows the calculated results (without spin-polarization) for the interatomic distances (1st–8th in Fig. 1(b)) of the X–X interaction energies in Al(fcc):the magnetism for Cr, Mn, and Fe is discussed in Section 2.3. It is noted that the interactions are medium-ranged and show the oscillating behavior [19,20]. For Sc, the X–X interaction is strongly repulsive for firstneighbor pairs, strongly attractive (negative and large) for the second-neighbor pairs, and repulsive for the thirdneighbor pairs. Thus, we can easily expect that Sc atoms are arranged at the interval of the second-neighbor interatomic distance, namely the atomic structure of L12
Fig. 3. Distance dependence (1st–8th in Fig. 1(b)) of X–X interaction energies in Al. The spin-polarization effect is neglected.
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(Fig. 1(a)). For V, the interaction is strongly repulsive for the second-neighbor pairs and very weak around the thirdneighbor pairs. Thus, for Al 3V the DO 22 structure (Fig. 1(b)) may become stable compared with the L12 structure. For XZCr, Mn, and Fe, the interactions are very repulsive up to the distance being just a little bit shorter than the third-neighbor distance and become very attractive ˚ ): for Mn the around the third-neighbor distance (4.7–5.0 A ˚ strongest attraction (K0.05 eV) at 4.9 A. Thus, it is obvious that Mn impurities are arranged at the interval of the ˚ . We can easily interatomic distance between 4.7 and 5.0 A expect that the atomic structure of a high-symmetric icosahedron of 12 Mn impurities, being a sublattice of MI (see Fig. 1(c)), may be very stable because it includes a large number (30) of Mn–Mn pairs: for this atomic structure the energy gain due to the Mn–Mn interactions is as large as wK1.5 eV(ZK0.0.5!30). It is noted that this ˚ ) is very close to one of interatomic distance (4.7–5.0 A ˚ and w7.5 A ˚) the observed interatomic distances (4.7–4.9 A of Mn–Mn pairs [2], although the present calculated interatomic distance of Mn–Mn pairs may be changed a little bit by the quantitative calculations based on the atomistic simulations with the accurate EAMP, as discussed in Section 3. We also found that the Mn–Mn interaction at ˚ , corresponding to the second-neighbor of a w7.5 A sublattice icosahedron of Mn 12, may be repulsive (0.007 eV). This weak repulsion may reduce slightly the
Fig. 4. Distance dependence (1st–8th in Fig. 1(b)) of X–X (XZCr, Mn, and Fe) interaction energies in Al. The spin-polarization effect is included. See text for details.
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3. Summary and future problem
Fig. 5. Distance dependence (1st–8th in Fig. 1(b)) of Mn–Vacancy interaction energies in Al. The spin-polarization effect is included.
stability of MI. However, we show in Section 2.3 that this weak repulsion may be changed to a weak attraction (K0.005 eV) by taking into account the magnetism of Mn [21,22]. 2.3. Magnetism of Cr, Mn, and Fe Fig. 4 shows the calculated results with the spin-polarization effect for XZCr, Mn, and Fe. The calculated results for single X impurities and first-neighbor X pairs were discussed in Refs. [21–23]. For single impurities, the energy gain due to the magnetism is as large as 0.3 eV for Mn. For two impurities, we can obtain two stable solutions of ferromagnetic (FM) and antiferromagnetic (AFM) states. These magnetic energies (per impurity) of impurity pairs are always increased compared with the magnetic energies of single impurities, because of the weaker X–X (d–d) interaction compared with the Al–X (sp–d) interactions [21,22], as discussed in the preceding paper. Here we discuss only two important points. ˚ is AF. The (1) The ground state of Mn–Mn pairs at 4.9 A energy gain due to AF is almost same as that obtained by the non-magnetic Mn–Mn pairs. ˚ changes to attraction (2) The repulsive interaction at w7.5 A by the inclusion of magnetism of Mn: 0.007 eV for nonmagnetic coupling and K0.005 eV for AF coupling. Thus, the inclusion of magnetism is important for the Mn–Mn interactions which stabilize the atomic structure of MI in the AlMn QC. The detailed discussions will be given elsewhere [24]. 2.4. Roles of vacancy and Si for stability of MI The interaction between a vacancy at the center of MI and Mn of a sublattice is almost zero (or weakly repulsive) around ˚ , as shown in Fig. 5: it is noted that the distance of 4.5–4.7 A ˚ ) by the vacancy—Mn is 95% of Mn–Mn distance (4.7–5.0 A geometry of an icosahedron. We have found that this Mn– vacancy interaction may be attractive by w0.01 eV if Si atoms exist between a vacancy and Mn (for example, at sites 1 and 1 0 between sites 0 and 3, as shown in Fig. 3). The details will be discussed elsewhere [24].
We have presented ab initio data for the study of the micromechanism of the atomic structures of the Mackay icosahedron Al42Mn12 in the AlMn QC as well as Al-rich AlX alloys, and have shown that the interactions of X–X (XZSc– Ni) pairs are strong and medium-ranged, and show the oscillating behavior [19,20]. We have found that for XZMn the interaction is strongly repulsive up to the distance being just a little bit shorter than the third-neighbor distance, and become strongly attractive around the third neighbor distance ˚ ), which is close to one of the observed distances for (w4.9 A Mn–Mn pair [2]. This strong attraction may be a major factor for the stability of the atomic structure of MI in the AlMn QC. We presented only qualitative discussions for the stability of MI. In order to study quantitatively the present mechanism of the stability of MI, we must perform the atomistic simulations [15] with the accurate EAMP, all the parameters in which are determined by fitting to the ab initio input data, obtained from the present calculations. We believe that the inclusion of the strong interactions of Mn with the neighboring Al atoms and the medium-ranged Mn–Mn interaction, given in the present and preceding papers, is essential for the construction of EAMP of AlMn QC. References [1] Mondolf F. Aluminium alloys: structure and properties. Boston, MA: Butterworths; 1976. [2] Maret M, Pasturel A, Senillou C, Dubois JM, Chieux P. J Phys Fr 1989; 50:295. [3] Elser V, Henley CL. Phys Rev Lett 1985;55:2883. [4] Bancel PA, Heiney PA. Phys Rev B 1986;33:7917. [5] Sato T, Hirosawa S, Hirose K, Maeguchi T. Metall Mater Trans 2003; 34A:2745. [6] Hoshino T, Asato M, Zeller R, Dederichs PH. Phys Rev B 2004;70: 094118. [7] Hoshino T, Asato M, Fujima N, Papanikolaou N. Trans Mater Res Soc Jpn 2005; 30(3):877. [8] Nakamura F, Hoshino T, Tanaka S, Hirose K, Hirosawa S, Sato T. Trans Mater Res Soc Jpn 2005; 30(3):873. [9] Inoue A. Bulk amorphous alloys. Zurich: Trans Tech Publ; 1998. [10] Hoshino T, Mizuno T, Asato M, Fukushima H. Mater Trans 2001;42:2206. [11] Hoshino T, Asato M, Nakamura T, Zeller R, Dederichs PH. J Magn Magn Mater 2004;272–276:e229. [12] Hoshino T, Asato M, Nakamura T, Zeller R, Dederichs PH. J Magn Magn Mater 2004;272–276:e231. [13] Papanikolaou N, Zeller R, Dederichs PH. Phys Rev B 1997;55:4167. [14] Zeller R. Phys Rev B 1997;55:9400. [15] Ogata S, Shimizu F, Li J, Wakeda M, Shibutani Y. In this conference. [16] Mishin Y, Mehl MJ, Papaconstantpoulos DA, Voter AF, Kress ED. Phys Rev B 2001;63:224106. [17] Hoshino T, Asato M, Fujima N. In this conference. [18] Wolverton C. Philos Mag Lett 1999;79:683. [19] Friedel J. Helv Phys Acta Sci (Paris) 1987;305:171. [20] Zou J, Carlsson AE. Phys Rev Lett 1993;70:3748. [21] Hoshino T, Zeller R, Dederichs PH, Weinert M. Europhys Lett 1993;24: 495. [22] Hoshino T, Zeller R, Dederichs PH. Phys Rev B 1996;53:8971. [23] Hoshino T, Nakamura F. J. Metas Nanostat Mater 2005; 24–25:237. [24] Hoshino T. In preparation.