Magnetism of close packed Fe 147 clusters

Transcription

1 Phase Transitions Vol. 00, No. 00, June 2005, 1 10 Magnetism of close packed Fe 147 clusters M. E. Gruner, G. Rollmann, S. Sahoo and P. Entel a Theoretical Physics, Universität Duisburg-Essen, Campus Duisburg, Duisburg, Germany (25 June 2006) We present ab initio density functional theory calculations of the magnetic structure of perfectly cuboctahedral nanoparticles consisting of 147 iron atoms. The results are compared to geometrically relaxed cuboctahedral and icosahedral Fe 147 -clusters. It is found, that structural relaxation leads to a compression, which reduces especially the moments of the central atoms of the cluster, while surface magnetism remains practically unaffected. Keywords: Density-functional Theory, magnetic nanoparticles, FCC Fe, iron clusters 1 Introduction The magnetism of iron and its alloys has been subject of numerous research projects in the past, of particular interest is the strong interrelation between magnetism and structural and elastic properties which leads in bulk Fe to a structural phase transformation at T = 1184 K from the low temperature bcc α-phase into the fcc γ-phase and to a strongly enhanced thermal expansion in the γ-phase, the so-called anti-invar effect [1, 2]. The latter is closely related to the technologically more relevant low thermal expansion in some fcc Febased alloys, like Fe 65 Ni 35 or Fe 72 Pt 28, which is commonly referred to as the Invar-effect [3,4]. It is well established in literature that the γ-modification of bulk iron possesses a rich spectrum of magnetic structures with a layered antiferromagnetic arrangement (AFMD) with two consecutive layers with spins pointing up and down, respectively, as a candidate for the ground state spin structure [5]. Corresponding author. Markus.Gruner@uni-duisburg-essen.de Phase Transitions ISSN print/ ISSN online c 2005 Taylor & Francis Ltd DOI: / xxxxxxxxxxxx

2 2 M. E. Gruner, G. Rollmann, S. Sahoo and P. Entel There is growing interest in iron or iron-alloy nanoparticles, which could find application in biomedical area for magnetically guided transport [6] or in ultra-high density magnetic data recording [7]. Due to the tremendous computational effort needed to accurately describe these systems, comprehensive ab initio investigations of the potential energy surfaces of Fe n clusters have been limited to n 6. [8,9]. For larger clusters up to Fe 15, selected initial geometries have been optimized using various relaxation methods [10 13]. In these studies it has been revealed that small Fe-clusters prefer Jahn-Teller distorted geometries, which is also the case for the icosahedral Fe 15 cluster [14,15]. Larger Fe clusters are investigated by Duan [16], Postnikov [17] and Šipr [18]. However, they only considered high-symmetry isomers or did not allow for structural relaxations. Recently it was found [19], that the probable ground state of Fe 55 is an icosahedron, which in its inner shell is transformed towards cuboctahedral structure following the Mackay path [20]. Further calculations confirm, that this trend also holds for larger cluster sizes. Since the translational symmetry is broken in zero-dimensional systems, layered antiferromagnetic structures as observed in bulk are not necessarily to occur in clusters and nanoparticles. Therefore, it is worthwhile to investigate systematically the manifestation of antiferromagnetism in fcc (cuboctahedral) Fe-nanoparticles with more than hundred atoms in an ab initio approach as presented in this study. 2 Computational details We performed first principles calculations within the framework of density functional theory (DFT) using the Vienna Ab initio Simulation Package (VASP) [21], where the electronic wavefunctions of the valence electrons are expanded in a plane wave basis set with an energy cutoff of 268 ev and the interaction with the nuclei and the core electrons within the projector augmented wave (PAW) approach [22]. For the exchange-correlation functional the generalized gradient approximation (GGA) was used in the formulation of Perdew and Wang [23] in connection with the interpolation formula of Vosko, Wilk and Nusair [24]. Since the system is non-periodic, we restricted the integration in k-space to the Γ-point only. All clusters were placed in a cubic supercell with a fixed edge length of 22Å. Parts of the calculations include geometrical optimizations, which were carried out on the Born-Oppenheimer surface using the conjugate gradient method. The structural relaxation was stopped when the energy difference between two consecutive relaxations was less than 0.1 mev. During the geometrical optimizations, charge densities and forces were symmetrized according to

3 3 octahedral symmetry O h and in the case of the icosahedron with tetrahedral symmetry T h in order to speed up the time for the relaxations while preserving the stability of the structures. The calculations were performed on up to 256 processors on an IBM BlueGene/L supercomputer. 3 Results In our investigation, we found two magnetic solutions for the Fe 147 cuboctahedra. At large interatomic distances d nn a thoroughly ferromagnetic solution (labeled FM further on) with a total magnetic moment between 400 and 425µ B was determined. The FM curve collapses at d nn = Å into another solution with a smaller moment arising from reversed Fe-spins in the inner shells which is stable for smaller d nn. This solution, which we will refer to as antiferromagnetic (AF) solution (in spite of the finite total magnetic moment), forms the energetically most favorable state with its minimum at d nn = Å, around E = 18.5 mev/atom below the FM solution. The averaged total magnetic moments in both states are 2.32 µ B /atom and µ B /atom, respectively. Other ferri- or antiferromagnetic arrangements, e. g., layered AFMD structure known from bulk, appeared to be unstable and collapsed into the AF state. In comparison to bulk calculations [5], the equilibrium nearest neighbor distances for the AF-cluster structure and the AFMD-bulk structure are practically the same, while the energy difference to the FM-high-spin structure is by a factor of 2.9 larger in bulk. We also estimated a fictitious bulk modulus for the ideal cuboctahedral cluster by fitting the Murnaghan equation of state [25] to the AF-E(V ) curve presented in Fig. 1. The resulting value B = 128 GPa should be compared to the bulk value B = 127 GPa in the AFMD structure, which was found to be the lowest in energy for the fcc lattice [5]. Relaxation of the interatomic distances in the cuboctahedron further improves the total energy by 35.6 mev/atom and leads to a comparable magnetic moment of 2.37 µ B /atom. In contrast to this, an icosahedron after geometric optimization is still by E = 21.8 mev/atom more favorable than the relaxed cuboctahedron and possesses an averaged moment of 2.17 µ B /atom. The dependence of the individual magnetic moments of the AF and the FM solutions on the interatomic spacing is shown in Fig. 2. According the number of inequivalent atomic sites in the shells, the values for the local magnetic moments split up into to four different values for the magnetization are found within the shell for each d nn, although 92 and 42 atoms are present in the two outer shells. In the AF-state, the moment of the central atom has a non-monotonic dependence on the interatomic spacing, its absolute value decreases around the equilibrium neighbor distance, while it has a positive slope for smaller and

4 4 M. E. Gruner, G. Rollmann, S. Sahoo and P. Entel larger d nn. This variation, albeit damped, can also be seen in the innermost shell 1. The magnitude of the Fe-moments compares with 1.5 µ B well with the value in bulk antiferromagnetic structures [5]. In the FM state, the moment of the inner atom is with 2.2 µ B slightly lower than in the two inner shells 1 and 2, which reside uniformly at 2.4 µ B for d nn > 2.52 Å. An extended region, in which a reduced ferromagnetic low-moment solution exists, was not found, in contrast to bulk, where it is stabilized for 2.44 > d nn > 2.54 Å [5]. In the outermost shell 3 both solutions show a very similar behavior. As expected, the moment is largely enhanced here due to the reduced coordination of the surface atoms. This becomes more evident by comparing the behaviour of the magnetic moment by plotting it over the distance from the center of the particle in Fig. 3. In the upper panel the AF and the FM state are compared at a fixed value of d nn = 2.51 Å, close to the energy minimum. In the lower panel, for comparison the same plot is shown for the AF Fe 147 cuboctahedron and icosahedron after structural relaxation. In both cases, the two outer shells are polarized in the same direction with practically no significant difference in the magnetic moments of the outermost shell. Comparing the radii at which ions are found, one notices that in the relaxed clusters especially the innermost and the outermost shells are compressed, while the distances between the shells tend to increase when moving outwards. This seems to have only minor influence on the magnitude of the surface moments of the cluster; the interior moments, however, are significantly reduced. Cross sections through the magnetic structure of the relaxed icosahedron and cuboctahedron, as depicted in Fig. 4, show that the cuboctahedron forms a cuboctahedral block with reversed polarization in the center, while the icosahedron tends to exhibit shellwise alternating moments. 4 Conclusion We performed investigations on the magnetism of cuboctahedral and icosahedral Fe 147 -clusters with and without structural relaxations. The cuboctahedral clusters prefer a block-like antiferromagnetic ordering, which has comparable properties as the AFMD structure found in bulk calculations [5]. However, the moments of the icosahedron are alternating shell-wise. The two outer shells are polarized generally in the same direction, with atoms in the outermost shell having enhanced magnetic moments between 2.6 and 2.9 µ B which is at least 0.2 µ B below the moments in shell 2. Structural relaxation leads to a compression which reduces especially the magnetic moments of the inner atoms of the cluster, while surface magnetism seems to be largely unaffected.

5 5 Acknowledgements The numerical calculations of the present work have been carried out on the massively parallel IBM BlueGene/L supercomputer system of the John von Neumann Institute for Computing (NIC) at the Forschungszentrum Jülich, Germany. We thank the staff members of this institution for their continuous and substantial support. Financial support was provided from the Deutsche Forschungsgemeinschaft through the Research Training Group GRK 277, Structure and Dynamics of Heterogeneous Systems, and the Collaborative Research Center SFB 445, Nano-Particles from the Gasphase: Formation, Structure, Properties. References [1] M. Acet, H. Zähres, E. F. Wassermann, Phys. Rev. B (1994). [2] M. Acet, T. Schneider, H. Zähres, E. F. Wassermann, W. Pepperhoff, J. Appl. Phys (1994). [3] E. F. Wassermann, in: K. H. J. Buschow, E. P. Wohlfahrt (Eds.), Ferromagnetic Materials, Vol. 5, Elsevier, Amsterdam, 1990, Ch. 3, p [4] E. F. Wassermann, in: J. Wittenauer (Ed.), The Invar Effect, The Minerals, Metals & Materials Society, Warrendale, 1996, p. 51. [5] H. C. Herper, E. Hoffmann, P. Entel, Phys. Rev. B (1999). [6] A. Hütten, D. Sudfeld, I. Ennen, G. Reiss, K. Wojczykowski, P. Jutz, J. Magn. Magn. Mater (2005). [7] S. Sun, C. B. Murray, D. Weller, L. Folks, A. Moser, Science (2000). [8] G. L. Gutsev, J. Charles W. Bauschlicher, J. Phys. Chem. A (2003). [9] S. Chrétien, D. R. Salahub, Phys. Rev. B (2002). [10] P. Ballone, R. O. Jones, Chem. Phys. Lett (1995). [11] M. Castro, Int. J. Quant. Chem (1997). [12] O. Diéguez, M. M. G. Alemany, C. Rey, P. Ordejón, L. J. Gallego, Phys. Rev. B (2001). [13] Ž. Šljivančanin, A. Pasquarello, Phys. Rev. Lett (2003). [14] P. Bobadova-Parvanova, K. A. Jackson, S. Srinivas, M. Horoi, Phys. Rev. B (2002). [15] G. Rollmann, S. Sahoo, P. Entel, Phys. Stat. Solidi (a) (2004). [16] H. M. Duan, Q. Q. Zheng, Phys. Lett. A (2001). [17] A. V. Postnikov, P. Entel, J. M. Soler, Eur. Phys. J. D (2003). [18] O. Šipr, M. Košuth, H. Ebert, Surf. Sci (2004). [19] G. Rollmann, A. Hucht, M. E. Gruner, P. Entel, preprint. [20] A. L. Mackay, Acta Cryst (1962). [21] G. Kresse, J. Furthmüller, Phys. Rev. B (1996). [22] G. Kresse, J. Furthmüller, Phys. Rev. B (1999). [23] J. P. Perdew, Y. Wang, Phys. Rev. B (1992). [24] S. H. Vosko, L. Wilk, M. Nusair, Can. J. Phys (1980). [25] F. D. Murnaghan, Proc. Natl. Acad. Sci. USA (1944).

6 6 M. E. Gruner, G. Rollmann, S. Sahoo and P. Entel Figure 1: Energy and averaged magnetic moment of ideal Fe 147 cuboctahedron as a function of the nearest neighbor distance d nn. The enrgy minimum is obtained for d nn = Å. Two solutions (denoted AF and FM) emerge. The inset shows the cuboctahedral Fe 147 cluster. Figure 2: Magnetic moments of the individual atoms in AF and FM Fe 147 ideal cuboctahedra as a function of the interatomic spacing. The shells of the cuboctahedron are represented by different symbols and colors. The atoms of each shell form classes according to the O h symmetry of the system, so that only a limited number of distinguishable lines appears. Figure 3: Plot of the magnetic moment as a function of the distance from the central atom. The upper panel shows the moments of the AF and FM cuboctahedra without structural optimizations at d nn = 2.51 Å. For comparison the lower panel shows the corresponding data for the geometrically relaxed cuboctahedron and icosahedron. Figure 4: Cross section of an Fe 147 cuboctahedron (left) and icosahedron (right) after structural relaxation. The arrows denote the direction and size of the magnetic moments on the corresponding atom sites.

7 7 Moment (µ B /atom) Cubo FM Cubo AF Total energy (mev/atom) d nn (Å) Figure 1

8 8 M. E. Gruner, G. Rollmann, S. Sahoo and P. Entel Moment (µb/atom) 3 2 AF FM Shell 3 Shell 2 Shell 1 Central dnn (Å) Figure

9 9 Magnetic moment (µ B /atom) Cubo AF Cubo FM Shell 1 Shell 2 Shell 3 Cubo relaxed Ico relaxed Distance from center (Å) Figure 3

10 10 M. E. Gruner, G. Rollmann, S. Sahoo and P. Entel Figure 4