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Core restructuring for magnetic Fe55 icosahedral nanoparticles
Chemical Physics Letters 541 (2012) 101–104
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Core restructuring for magnetic Fe55 icosahedral nanoparticles Abdesslem Jedidi a,b, Alexis Markovits a,⇑, Christian Minot a, Manef Abderrabba b a b
Laboratoire de Chimie Théorique (LCT) UMR7616, UPMC Univ. Paris 06, CNRS, Case 137, 4 Place Jussieu, Paris, F-75252 Cedex 05, France Laboratoire de Matériaux, Molécules et Applications (LMMA), Université de Carthage, Boite Postale BP51, 2070 La Marsa, Tunisia
a r t i c l e
i n f o
Article history: Received 10 April 2012 In final form 25 May 2012 Available online 4 June 2012
a b s t r a c t We investigate magnetic and structural properties of Fe55 clusters from DFT calculations. Previous calculations indicate the existence of icosahedral clusters with two antiferromagnetic states with magnetic moments smaller than experimentally measured. In the present work, we find a rearrangement of the atoms of the core (the 13 saturated atoms) that tend to form a structure close to a cuboctahedron, preserving the icosahedral structure for the surface shell (the 42 surface atoms). The introduction of a Hubbard correction term leads for the most stable electronic state to a ferromagnetic state and a high magnetic moment in agreement to experiment. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The properties of metallic clusters, Mn, vary with their size, n, without clear continuity [1–3]. Some properties oscillate with the size while the geometry of the most stable aggregates does not show an obvious relationship from n to n + 1 [4,5]. The knowledge of their geometry, even though it is not easy to obtain, is a prerequisite to the study of any property. The topology of the small clusters presents a large diversity with no relation with the structure of the bulk materials [6–9]. The stability of clusters, which is size-dependant, makes some n values stand out; these magic numbers n allow a more spherical shape with a higher coordination. n = 13 allows an increase of the nearest-neighbor bonding since a central atom is then surrounded by 12 neighbors [10]. n = 55 extends this construction with another shell of 42 atoms. The magic numbers n = 13, 55 and 147 allow building cuboctahedrons as well as icosahedrons. This Letter is devoted to the icosahedral Fe55 clusters, their magnetic properties and their topology. A cluster of 55 atoms represents a relatively large size which remains accessible to first principles calculations. Previous calculations performed on Fe55 conclude that the icosahedron is the most stable structure, more stable than the cuboctahedron that could be seen as fragment of austenite, c-iron, by 0.1 eV/atom [11–13]. This comparison remains valid for small sizes (n < 200) [14]. It could be attributed to a greater flexibility for the bonding (in the icosahedron there are two nearest neighbor distances differing by 5%) and to a higher symmetry of the surface, the atom distribution from the surface shell, the envelope as opposed to the inner shells (core), being more uniform. Since the icosahedron is obtained by distorting
⇑ Corresponding author. E-mail address: [email protected] (A. Markovits). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.05.066
the square faces of the cuboctahedron into two triangular faces, the number of bonds in the icosahedral structure is larger than in the cuboctahedron [15]. The icosahedron is compact which is a requirement for having stable transition metal clusters [15]. The fragments of the bcc structures (ferrite, c-iron, is the existing phase at 0 K) are not compact enough to pertain to the class of the most stable clusters. The experimental determination of the Fe55 cluster geometry is less evident [16] suggesting a possible competition between several different structures, an icosahedral structure and a ‘jellium-like structure which could correspond to a liquid iron cluster’. For other transition metals, M = Ni, Co or Cu, the icosahedron is definitively the most stable geometry up to 1000–2000 atoms [14,16,17] even though the crystals are fcc. Au55 [18] and Pt55 [19] also have icosahedral symmetry. The determination of the structure for Pt55 is more difficult, many structures being quasi isoenergetic and clusters having symmetry reconstructing under annealing to lower energy structures, highly asymmetric and irregular [19]. The high symmetry of the icosahedron, characterizing also all the five platonic solids, has many advantages. It is generally associated with high stability. Note that this imposes a restriction on the spin–orbital occupancy. Degenerate levels should be all occupied or all vacant to avoid symmetry breaking; otherwise, this leads to a Jahn–Teller situation and pure symmetry is not the exact geometry. For Fe4 cluster, the cluster of lowest symmetry indeed has a topology close to a tetrahedron but distorted without full symmetry. It restricts the number of sites available for adsorption studies. It is associated with unusual large magnetic properties. Magnetic properties are related to the symmetry; the higher the cluster symmetry is, the larger the magnetic moment will be. The icosahedral clusters are thus expected to have high
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magnetic moments. This is the case for i-Fe13 [20]. For n = 55, the experimental magnetic moment is about 3.1 lB/atom [1,2,20]. The calculated values (2.62 and 1.93 lB/atom) are smaller [11]. Two icosahedral forms with anti-ferromagnetic states are found; in the first, the central atom has a spin opposite to that of the other atoms while in the second, the whole Fe13 core has a spin opposite to that of the surface shell [11,20]. The magnitudes of the magnetic moment of the antiferromagnetic Fe55 cluster are insufficient to explain the large experimental value that would better fit a ferromagnetic state [1,2]. The cuboctahedral structure, with a poorer stability, gives a ferromagnetic state with a larger moment (2.77 lB/atom) that is closer to the experimental determination [21] but still smaller. Icosahedral symmetry is incompatible with a periodic crystal structure and a phase transition should occur during the growth to generate a cubic structure. The most natural idea for a phase transition is a strong relaxation induced by a few additional atoms. Increasing n beyond magic numbers should imply a strong reorganization of the cluster and even if relaxation imposes differences with the ideal lattice structures, for large clusters one can expect to recognize a periodic piece of a crystal. In this Letter, we present an alternative mode, the phase transition occurring in the cluster core.
2. Computational details We performed spin–polarized periodic calculations based on density functional theory (DFT) in the generalized gradient approximation (using the PW91 exchange-correlation functional) as implemented in the VASP [22–25] code. Plane-wave basis sets (with a kinetic energy cutoff of 267.9 eV) describe the valence electrons: ten electrons for Fe, the 4s and 3d electrons. The core electrons were replaced by projector augmented wave (PAW) pseudopotentials [26,27]. The relaxation of the atomic positions in the supercell took place until the changes in the total energy are smaller than 0.001 eV. The clusters are placed in a cubic box with an edge length of 25 Å, large enough to prevent interactions between the clusters. Spin states are characterized by Na–Nb, Na and Nb being the number of alpha and beta spin electrons. For the DFT + U calculations, the Hubbard potential was taken as Ueff = U–J = 3.2 1 = 2.2 eV from Ref. [28].
3. Results and discussion 3.1. Perfect icosahedral structures The first optimization, with no constraint of spin and no initial spin distribution led to an antiferromagnetic state with Na–Nb = 94 with a random distribution. Thirteen electrons are of minority spin, one on the central atom, nine from the first shell and even 3 from the surface shell. Imposing initial spins on the core atoms leads to more regular spin distributions; with or without constraint of spin, the 13 electrons of minority spin remain in the core shell as in Ref. [11,20] but this results in an electronic state of higher energy. In Table 1 (left hand side), we report the results of calculations imposing Na–Nb that have preserved the icosahedral geometry (denoted I I). The Letter A or F indicates the magnetic state (antiferromagnetic or ferromagnetic). For the A structures, the number Nb of electrons of b spin is specified by the number following the letter A. It decreases when Na–Nb increases. The structure A13 described in Ref. [11,20] with a core with beta spin is obtained in the range Na–Nb = 94–100; the structure of lowest energy being that for Na–Nb = 100. From Na–Nb = 148 to 154, only the central atom has a beta spin as found by Köhler; Na–Nb = 150 instead of Na–Nb = 144 as found in reference [11] represents the cluster of lowest energy. The binding energy is 3.90 eV/atom, larger than 3.87 eV found by Lanzani (Na–Nb = 128) [13]. For Na–Nb 156, clusters become ferromagnetic; however their energy remains smaller by at least 0.4 eV compared to the A1 state with Na–Nb = 150. In the results for structure I I, the topology remains that of an icosahedron (see Figure 1) even though small relaxations occur. These are mainly located in the core; distances from the central atom to its first neighbors range from 2.39 to 2.46 Å and distances inside the first shell range from 2.49 to 2.65 Å. The distance variations (the values above are those for the A1 structure with Na–Nb = 150) depend on the total spin. 3.2. Core reconstructed icosahedral structures Depending on the initial geometry and spin distribution, optimization modifies or maintains the structure. Under CO adsorption, we have obtained a spontaneous reconstruction of the core while the surface shell topology remains icosahedral. We have repeated the calculations for the naked cluster, starting from this reconstructed geometry denoted C I hereafter, and we
Table 1 Energy and magnetization of two structures of the cluster Fe55. Structures of lowest energy for a given spin state are in bold character and the most stable for all is underlined. Structure II II II II II II II II II II II II II II II
E (eV)
Na–Nb
Magnetism
Structure
Na–Nb
Magnetism
389.74 388.98 389.31 389.55 389.56 389.53 392.65
94 94 96 98 100 102 148
CI
389.04
94
A13
I I I I I I
96 98 100 102 148
150 152 154 156 158 166 168 170
C C C C C C
388.30 389.54 389.59 389.85 393.25
393.01 392.87 392.71 392.62 392.56 391.60 391.26 390.51
A13 (random) A13 A13 A13 A13 A11 A1 A1
E (eV)
A1 A1 F F F F F
C C C C C C C
I I I I I I I
393.39 393.16 392.95 392.91 392.72 391.64 391.27 390.44
150 152 154 156 158 166 168 170
A13 A13 A13 A12 A1 A1 A1 F F F F F F
I I: perfect icosahedron, C I: cuboctahedron (core) icosahedron (surface shell). F: ferromagnetic, ANb: antiferromagnetic. Nb indicates the number of unpaired b electrons of minority spin which is considered as an integer while, due to delocalization, the atomic magnetism in not an integer.
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A. Jedidi et al. / Chemical Physics Letters 541 (2012) 101–104
Figure 1. Spin Distribution of Fe55 (CI–A1) with Na–Nb = 150 lB.
The two left views look like a cuboctahedron shown along a C3 axis (top and side views). In the side view, the hexagon is buckled; this does not appear in the top view. The third view is obtained by a rotation of the first along the vertical axis. It makes a distorted cube become apparent. The two superposed rhombi are not very different from squares and the distances between the superposed atoms are larger than 3.1 Å. Nevertheless the structure suggests a possible distortion toward a cube with four capped faces that forms a building block of the bcc structure. It seems important to note that the patterns of the reconstructed cores are roughly similar to topologies found in cubic crystals (the cuboctahedron is part of a fcc structure and the capped cube a part of a bcc structure), suggesting a new mode for the phase transition. Icosahedral symmetry remains for the surface, while the core reconstructs! The energies of the structures C I are lower than those of structures I I. For the magnetism, the behavior is the same. The most stable cluster is again the antiferromagnetic A1 structure with Na–Nb = 150 lB. 3.3. The influence of the Hubbard correction, DFT + U
Figure 2. Views of the Fe55 clusters. Four colors indicate the four sets of iron atoms equivalent by symmetry for the perfect icosahedral structure.
have obtained a stable aggregate (see Figure 2) which is more stable than the I I structure (see Table 1). The core topology no longer resembles an icosahedron. In Figure 3, we display three views of the core with different perspectives.
As reported by Rollmann on small iron clusters, the DFT + U allows reducing the Jahn–Teller distortion and increasing the magnetic moment [29]. We have checked the effect of U on Fe55 and the main results are reported in Table 2. The main results indeed indicate an increase of the magnetism, the ferromagnetic structures becoming those of lowest energy. Then the calculated Na–Nb value, 168 lB, is very close to the experimental value (3.08 lB/atom leads to 169.4 lB) [20]. The antiferromagnetic structures also acquire higher spin; the value of Na–Nb becomes 162 for the lowest energy of both A1 structures (I I and C I), to be compared to the results without Hubbard correction that give 150. Since the introduction of U favors symmetry, the energy difference between the two clusters vanishes. It remains noteworthy that the core reorganization remains a possibility at no thermodynamical cost for Fe55. This leaves open a possibility of preference for larger particles as Fe147.
Figure 3. Three views of the core of the Fe55 C I cluster (A1). The central atom is shown in grey.
Table 2 Energy and Magnetization of two structures of the cluster Fe55 using DFT + U (U = 2.2 eV). Structures of lowest energy are in bold character. ( 1) Indicates calculations run without spin constraint. Structure I I I I I I I
I I I I I I I
E (eV) 302.85 303.81 304.35 304.92 305.13 305.33 305.04
Na–Nb
Magnetism
Structure
150 156 162 164 166 168 ( 1) 170
A1 A1 A1 F F F F
C C C C C C C
I I I I I I I
E (eV) 302.99 303.84 304.62 304.91 305.35 305.37 304.97
Na–Nb
Magnetism
150 156 162 164 166 ( 1) 168 170
A1 A1 A1 F F F F
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4. Conclusion The study of Fe55 clusters shows an opposition between the core and the shell of the nanoparticle. For this size, the core represents 13 atoms and the surface 42. In a low magnetic state, the majority spin is that of the surface atoms and the core atoms have opposite spin. Calculations including an onsite Coulombic correction lead to a ferromagnetic state with a large magnetization in good agreement with experiment. The surface shell of the most stable Fe55 nonoparticle has an icosahedral shape as for Fe13. However, the structure is more stable when the core restructures resembling a building block of a cubic crystal. This might suggest that a phase transition may not result directly from additional atoms in a crystal growth but from a rearrangement of the core, the saturated atoms of the inner shells adopting a bulk-like structure at variance with surface atoms. The icosahedral envelope appears here as a solid cauldron inside which the crystal structure could be generated. Acknowledgements We are grateful to CMCU-PHC (09G 1212) and the Institut Français de Cooperation in Tunisia (IFC) for their financial support. A.J. is grateful to the Université de Carthage. The authors thank GENCI and CCRE for computing facilities. The authors thank Prof. Michel Van-Hove for fruitful discussions. References [1] I.M.L. Billas, J.A. Becker, A. Châtelain, W.A. de Heer, Phys. Rev. Lett. 71 (1993) 4067.
[2] I.M.L. Billas, A. Châtelain, W.A. de Heer, Science 265 (1994) 1682. [3] I.V. Yudanov, A. Genest, S. Schauermann, H.-J. Freund, N. Rösch, Nano Lett. 12 (2012) 2134. [4] A.N. Andriotis, M. Menon, Phys. Rev. B: Condens. Matter 57 (1998) 10069. [5] C. Köhler, G. Seifert, T. Frauenheim, Chem. Phys. 309 (2005) 23. [6] C. Mottet, J. Goniakowski, F. Baletto, R. Ferrando, G. Treglia, Phase Transit. 77 (2004) 101. [7] B. Palpant, B. Prével, J. Lermé, E. Cottancin, M. Pellarin, M. Treilleux, A. Perez, J.L. Vialle, M. Broyer, Phys. Rev. B: Condens. Matter 57 (1998) 1963. [8] D. Reinhard, B.D. Hall, P. Berthoud, S. Valkealahti, R. Monot, Phys. Rev. Lett. 79 (1997) 1459. [9] D. Reinhard, B.D. Hall, D. Ugarte, R. Monot, Phys. Rev. B: Condens. Matter 55 (1997) 7868. [10] J.P.K. Doye, D.J. Wales, Chem. Phys. Lett. 247 (1995) 339. [11] C. Köhler, G. Seifert, T. Frauenheim, Comput. Mater. Sci. 35 (2006) 297. [12] G. Lanzani, A. Nasibulin, K. Laasonen, E. Kauppinen, Nano Res. 2 (2009) 660. [13] G. Lanzani, A.G. Nasibulin, K. Laasonen, E.I. Kauppinen, J. Phys. Chem. C 113 (2009) 12939. [14] F. Baletto, R. Ferrando, A. Fortunelli, F. Montalenti, C. Mottet, J. Chem. Phys. 116 (2002) 3856. [15] J.M. Montejano-Carrizales, J.L. Morán-López, Nanostruct. Mater. 1 (1992) 397. [16] M. Pellarin, B. Baguenard, J.L. Vialle, J. Lermé, M. Broyer, J. Miller, A. Perez, Chem. Phys. Lett. 217 (1994) 349. [17] C.L. Cleveland, U. Landman, J. Chem. Phys. 94 (1991) 7376. [18] W. Vogel, B. Rosner, B. Tesche, J. Phys. Chem. 97 (1993) 11611. [19] A. Sachdev, R.I. Masel, J.B. Adams, Catal. Lett. 15 (1992) 57. [20] H.M. Duan, Q.Q. Zheng, Phys. Lett. A 280 (2001) 333. [21] A.V. Postnikov, P. Entel, J.M. Soler, Eur. Phys. J. D 25 (2003) 261. [22] G. Kresse, J. Furthmüller, Phys. Rev. B: Condens. Matter 54 (1996) 11169. [23] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [24] G. Kresse, J. Hafner, Phys. Rev. B 49 (1994) 14251. [25] G. Kresse, J. Hafner, Mater. Sci. 6 (1996) 15. [26] G. Kresse, D. Joubert, Phys. Rev. B: Condens. Matter 59 (1999) 1758. [27] P.E. Blöchl, Phys. Rev. B: Condens. Matter 50 (1994) 17953. [28] M. Cococcioni, S. de Gironcoli, Phys. Rev. B: Condens. Matter 71 (2005) 035105. [29] P.E.G. Rollmann, Comp. Lett. 1 (2005) 288.
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Core restructuring for magnetic Fe55 icosahedral nanoparticles Abdesslem Jedidi a,b, Alexis Markovits a,⇑, Christian Minot a, Manef Abderrabba b a b
Laboratoire de Chimie Théorique (LCT) UMR7616, UPMC Univ. Paris 06, CNRS, Case 137, 4 Place Jussieu, Paris, F-75252 Cedex 05, France Laboratoire de Matériaux, Molécules et Applications (LMMA), Université de Carthage, Boite Postale BP51, 2070 La Marsa, Tunisia
a r t i c l e
i n f o
Article history: Received 10 April 2012 In final form 25 May 2012 Available online 4 June 2012
a b s t r a c t We investigate magnetic and structural properties of Fe55 clusters from DFT calculations. Previous calculations indicate the existence of icosahedral clusters with two antiferromagnetic states with magnetic moments smaller than experimentally measured. In the present work, we find a rearrangement of the atoms of the core (the 13 saturated atoms) that tend to form a structure close to a cuboctahedron, preserving the icosahedral structure for the surface shell (the 42 surface atoms). The introduction of a Hubbard correction term leads for the most stable electronic state to a ferromagnetic state and a high magnetic moment in agreement to experiment. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction The properties of metallic clusters, Mn, vary with their size, n, without clear continuity [1–3]. Some properties oscillate with the size while the geometry of the most stable aggregates does not show an obvious relationship from n to n + 1 [4,5]. The knowledge of their geometry, even though it is not easy to obtain, is a prerequisite to the study of any property. The topology of the small clusters presents a large diversity with no relation with the structure of the bulk materials [6–9]. The stability of clusters, which is size-dependant, makes some n values stand out; these magic numbers n allow a more spherical shape with a higher coordination. n = 13 allows an increase of the nearest-neighbor bonding since a central atom is then surrounded by 12 neighbors [10]. n = 55 extends this construction with another shell of 42 atoms. The magic numbers n = 13, 55 and 147 allow building cuboctahedrons as well as icosahedrons. This Letter is devoted to the icosahedral Fe55 clusters, their magnetic properties and their topology. A cluster of 55 atoms represents a relatively large size which remains accessible to first principles calculations. Previous calculations performed on Fe55 conclude that the icosahedron is the most stable structure, more stable than the cuboctahedron that could be seen as fragment of austenite, c-iron, by 0.1 eV/atom [11–13]. This comparison remains valid for small sizes (n < 200) [14]. It could be attributed to a greater flexibility for the bonding (in the icosahedron there are two nearest neighbor distances differing by 5%) and to a higher symmetry of the surface, the atom distribution from the surface shell, the envelope as opposed to the inner shells (core), being more uniform. Since the icosahedron is obtained by distorting
⇑ Corresponding author. E-mail address: [email protected] (A. Markovits). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.05.066
the square faces of the cuboctahedron into two triangular faces, the number of bonds in the icosahedral structure is larger than in the cuboctahedron [15]. The icosahedron is compact which is a requirement for having stable transition metal clusters [15]. The fragments of the bcc structures (ferrite, c-iron, is the existing phase at 0 K) are not compact enough to pertain to the class of the most stable clusters. The experimental determination of the Fe55 cluster geometry is less evident [16] suggesting a possible competition between several different structures, an icosahedral structure and a ‘jellium-like structure which could correspond to a liquid iron cluster’. For other transition metals, M = Ni, Co or Cu, the icosahedron is definitively the most stable geometry up to 1000–2000 atoms [14,16,17] even though the crystals are fcc. Au55 [18] and Pt55 [19] also have icosahedral symmetry. The determination of the structure for Pt55 is more difficult, many structures being quasi isoenergetic and clusters having symmetry reconstructing under annealing to lower energy structures, highly asymmetric and irregular [19]. The high symmetry of the icosahedron, characterizing also all the five platonic solids, has many advantages. It is generally associated with high stability. Note that this imposes a restriction on the spin–orbital occupancy. Degenerate levels should be all occupied or all vacant to avoid symmetry breaking; otherwise, this leads to a Jahn–Teller situation and pure symmetry is not the exact geometry. For Fe4 cluster, the cluster of lowest symmetry indeed has a topology close to a tetrahedron but distorted without full symmetry. It restricts the number of sites available for adsorption studies. It is associated with unusual large magnetic properties. Magnetic properties are related to the symmetry; the higher the cluster symmetry is, the larger the magnetic moment will be. The icosahedral clusters are thus expected to have high
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magnetic moments. This is the case for i-Fe13 [20]. For n = 55, the experimental magnetic moment is about 3.1 lB/atom [1,2,20]. The calculated values (2.62 and 1.93 lB/atom) are smaller [11]. Two icosahedral forms with anti-ferromagnetic states are found; in the first, the central atom has a spin opposite to that of the other atoms while in the second, the whole Fe13 core has a spin opposite to that of the surface shell [11,20]. The magnitudes of the magnetic moment of the antiferromagnetic Fe55 cluster are insufficient to explain the large experimental value that would better fit a ferromagnetic state [1,2]. The cuboctahedral structure, with a poorer stability, gives a ferromagnetic state with a larger moment (2.77 lB/atom) that is closer to the experimental determination [21] but still smaller. Icosahedral symmetry is incompatible with a periodic crystal structure and a phase transition should occur during the growth to generate a cubic structure. The most natural idea for a phase transition is a strong relaxation induced by a few additional atoms. Increasing n beyond magic numbers should imply a strong reorganization of the cluster and even if relaxation imposes differences with the ideal lattice structures, for large clusters one can expect to recognize a periodic piece of a crystal. In this Letter, we present an alternative mode, the phase transition occurring in the cluster core.
2. Computational details We performed spin–polarized periodic calculations based on density functional theory (DFT) in the generalized gradient approximation (using the PW91 exchange-correlation functional) as implemented in the VASP [22–25] code. Plane-wave basis sets (with a kinetic energy cutoff of 267.9 eV) describe the valence electrons: ten electrons for Fe, the 4s and 3d electrons. The core electrons were replaced by projector augmented wave (PAW) pseudopotentials [26,27]. The relaxation of the atomic positions in the supercell took place until the changes in the total energy are smaller than 0.001 eV. The clusters are placed in a cubic box with an edge length of 25 Å, large enough to prevent interactions between the clusters. Spin states are characterized by Na–Nb, Na and Nb being the number of alpha and beta spin electrons. For the DFT + U calculations, the Hubbard potential was taken as Ueff = U–J = 3.2 1 = 2.2 eV from Ref. [28].
3. Results and discussion 3.1. Perfect icosahedral structures The first optimization, with no constraint of spin and no initial spin distribution led to an antiferromagnetic state with Na–Nb = 94 with a random distribution. Thirteen electrons are of minority spin, one on the central atom, nine from the first shell and even 3 from the surface shell. Imposing initial spins on the core atoms leads to more regular spin distributions; with or without constraint of spin, the 13 electrons of minority spin remain in the core shell as in Ref. [11,20] but this results in an electronic state of higher energy. In Table 1 (left hand side), we report the results of calculations imposing Na–Nb that have preserved the icosahedral geometry (denoted I I). The Letter A or F indicates the magnetic state (antiferromagnetic or ferromagnetic). For the A structures, the number Nb of electrons of b spin is specified by the number following the letter A. It decreases when Na–Nb increases. The structure A13 described in Ref. [11,20] with a core with beta spin is obtained in the range Na–Nb = 94–100; the structure of lowest energy being that for Na–Nb = 100. From Na–Nb = 148 to 154, only the central atom has a beta spin as found by Köhler; Na–Nb = 150 instead of Na–Nb = 144 as found in reference [11] represents the cluster of lowest energy. The binding energy is 3.90 eV/atom, larger than 3.87 eV found by Lanzani (Na–Nb = 128) [13]. For Na–Nb 156, clusters become ferromagnetic; however their energy remains smaller by at least 0.4 eV compared to the A1 state with Na–Nb = 150. In the results for structure I I, the topology remains that of an icosahedron (see Figure 1) even though small relaxations occur. These are mainly located in the core; distances from the central atom to its first neighbors range from 2.39 to 2.46 Å and distances inside the first shell range from 2.49 to 2.65 Å. The distance variations (the values above are those for the A1 structure with Na–Nb = 150) depend on the total spin. 3.2. Core reconstructed icosahedral structures Depending on the initial geometry and spin distribution, optimization modifies or maintains the structure. Under CO adsorption, we have obtained a spontaneous reconstruction of the core while the surface shell topology remains icosahedral. We have repeated the calculations for the naked cluster, starting from this reconstructed geometry denoted C I hereafter, and we
Table 1 Energy and magnetization of two structures of the cluster Fe55. Structures of lowest energy for a given spin state are in bold character and the most stable for all is underlined. Structure II II II II II II II II II II II II II II II
E (eV)
Na–Nb
Magnetism
Structure
Na–Nb
Magnetism
389.74 388.98 389.31 389.55 389.56 389.53 392.65
94 94 96 98 100 102 148
CI
389.04
94
A13
I I I I I I
96 98 100 102 148
150 152 154 156 158 166 168 170
C C C C C C
388.30 389.54 389.59 389.85 393.25
393.01 392.87 392.71 392.62 392.56 391.60 391.26 390.51
A13 (random) A13 A13 A13 A13 A11 A1 A1
E (eV)
A1 A1 F F F F F
C C C C C C C
I I I I I I I
393.39 393.16 392.95 392.91 392.72 391.64 391.27 390.44
150 152 154 156 158 166 168 170
A13 A13 A13 A12 A1 A1 A1 F F F F F F
I I: perfect icosahedron, C I: cuboctahedron (core) icosahedron (surface shell). F: ferromagnetic, ANb: antiferromagnetic. Nb indicates the number of unpaired b electrons of minority spin which is considered as an integer while, due to delocalization, the atomic magnetism in not an integer.
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Figure 1. Spin Distribution of Fe55 (CI–A1) with Na–Nb = 150 lB.
The two left views look like a cuboctahedron shown along a C3 axis (top and side views). In the side view, the hexagon is buckled; this does not appear in the top view. The third view is obtained by a rotation of the first along the vertical axis. It makes a distorted cube become apparent. The two superposed rhombi are not very different from squares and the distances between the superposed atoms are larger than 3.1 Å. Nevertheless the structure suggests a possible distortion toward a cube with four capped faces that forms a building block of the bcc structure. It seems important to note that the patterns of the reconstructed cores are roughly similar to topologies found in cubic crystals (the cuboctahedron is part of a fcc structure and the capped cube a part of a bcc structure), suggesting a new mode for the phase transition. Icosahedral symmetry remains for the surface, while the core reconstructs! The energies of the structures C I are lower than those of structures I I. For the magnetism, the behavior is the same. The most stable cluster is again the antiferromagnetic A1 structure with Na–Nb = 150 lB. 3.3. The influence of the Hubbard correction, DFT + U
Figure 2. Views of the Fe55 clusters. Four colors indicate the four sets of iron atoms equivalent by symmetry for the perfect icosahedral structure.
have obtained a stable aggregate (see Figure 2) which is more stable than the I I structure (see Table 1). The core topology no longer resembles an icosahedron. In Figure 3, we display three views of the core with different perspectives.
As reported by Rollmann on small iron clusters, the DFT + U allows reducing the Jahn–Teller distortion and increasing the magnetic moment [29]. We have checked the effect of U on Fe55 and the main results are reported in Table 2. The main results indeed indicate an increase of the magnetism, the ferromagnetic structures becoming those of lowest energy. Then the calculated Na–Nb value, 168 lB, is very close to the experimental value (3.08 lB/atom leads to 169.4 lB) [20]. The antiferromagnetic structures also acquire higher spin; the value of Na–Nb becomes 162 for the lowest energy of both A1 structures (I I and C I), to be compared to the results without Hubbard correction that give 150. Since the introduction of U favors symmetry, the energy difference between the two clusters vanishes. It remains noteworthy that the core reorganization remains a possibility at no thermodynamical cost for Fe55. This leaves open a possibility of preference for larger particles as Fe147.
Figure 3. Three views of the core of the Fe55 C I cluster (A1). The central atom is shown in grey.
Table 2 Energy and Magnetization of two structures of the cluster Fe55 using DFT + U (U = 2.2 eV). Structures of lowest energy are in bold character. ( 1) Indicates calculations run without spin constraint. Structure I I I I I I I
I I I I I I I
E (eV) 302.85 303.81 304.35 304.92 305.13 305.33 305.04
Na–Nb
Magnetism
Structure
150 156 162 164 166 168 ( 1) 170
A1 A1 A1 F F F F
C C C C C C C
I I I I I I I
E (eV) 302.99 303.84 304.62 304.91 305.35 305.37 304.97
Na–Nb
Magnetism
150 156 162 164 166 ( 1) 168 170
A1 A1 A1 F F F F
104
A. Jedidi et al. / Chemical Physics Letters 541 (2012) 101–104
4. Conclusion The study of Fe55 clusters shows an opposition between the core and the shell of the nanoparticle. For this size, the core represents 13 atoms and the surface 42. In a low magnetic state, the majority spin is that of the surface atoms and the core atoms have opposite spin. Calculations including an onsite Coulombic correction lead to a ferromagnetic state with a large magnetization in good agreement with experiment. The surface shell of the most stable Fe55 nonoparticle has an icosahedral shape as for Fe13. However, the structure is more stable when the core restructures resembling a building block of a cubic crystal. This might suggest that a phase transition may not result directly from additional atoms in a crystal growth but from a rearrangement of the core, the saturated atoms of the inner shells adopting a bulk-like structure at variance with surface atoms. The icosahedral envelope appears here as a solid cauldron inside which the crystal structure could be generated. Acknowledgements We are grateful to CMCU-PHC (09G 1212) and the Institut Français de Cooperation in Tunisia (IFC) for their financial support. A.J. is grateful to the Université de Carthage. The authors thank GENCI and CCRE for computing facilities. The authors thank Prof. Michel Van-Hove for fruitful discussions. References [1] I.M.L. Billas, J.A. Becker, A. Châtelain, W.A. de Heer, Phys. Rev. Lett. 71 (1993) 4067.
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