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Calculation of magnetic and structural properties of small Co–Rh clusters
Surface Science 532–535 (2003) 334–340 www.elsevier.com/locate/susc
Calculation of magnetic and structural properties of small Co–Rh clusters S. Dennler b
a,*
, J. Morillo a, G.M. Pastor
b
a Centre d’Elaboration de Mat eriaux et d’Etudes Structurales, UPR CNRS 8011, 29 rue Jeanne Marvig, 31055 Toulouse, France Laboratoire de Physique Quantique, UMR CNRS 5626, Institut de Recherche sur les Syst emes Atomiques et Mol eculaires Complexes, 31062 Toulouse, France
Abstract Structural and magnetic properties of CoM RhN clusters with M þ N 6 4 atoms are determined by performing firstprinciples calculations in the framework of the density functional theory in a generalized gradient-corrected approximation and within the projector-augmented wave method. The role of magnetism on the most stable structures and on the energy differences among the low-lying isomers is quantified by comparing magnetic and non-magnetic solutions of the Kohn–Sham equations. As usual in small clusters, CoM RhN clusters show contracted interatomic distances with respect to the bulk. They exhibit ferromagnetic-like order with environment-dependent local magnetic moments lðiÞ. The average magnetic moments per atom lðM; NÞ and lðiÞ are generally more than a factor 2 larger than in macroscopic alloys of similar concentration. For a given cluster size M þ N, l increases with increasing number of Co atoms M. The lðiÞ at Rh atoms are remarkably enhanced by the presence of Co nearest neighbors, while the lðiÞ at Co atoms are not much affected by the presence of Rh atoms. The correlation between structure, chemical order, and environment-dependent magnetic properties is discussed. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Cobalt; Rhodium; Clusters; Density functional calculations; Magnetic phenomena (cyclotron resonance, phase transitions, etc.)
1. Introduction Experimental and theoretical investigations during the past decades have demonstrated in uncountable occasions that the reduction of system size and dimensionality gives rise to a variety of new states of matter with physical and chemical properties that are intrinsically different from those of conventional bulk crystals [1]. This has opened *
Corresponding author. Tel.: +33-5-6225-7970; fax: +33-56225-7999. E-mail address: [email protected] (S. Dennler).
numerous possibilities of generating suitably designed materials with specifically tailored properties. A remarkably intense and diverse research activity is presently developed within this field. Transition-metal (TM) clusters are a subject of particular interest in this context due to the strong environment dependence of their magnetic properties and their possible applications in magnetic recording or storage technologies. Previous experimental and theoretical studies have shown that the magnetic moments of FeN , CoN , and NiN clusters are significantly larger than the corresponding bulk magnetizations [2–6]. Moreover,
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00468-0
S. Dennler et al. / Surface Science 532–535 (2003) 334–340
non-vanishing magnetic moments have been recently observed in clusters of some 4d TMs like RhN (N 6 50) which are non-magnetic in the bulk [7–9]. This enhancement of magnetism in small particles can be qualitatively understood as a consequence of the reduction of local coordination number, which results in a stronger localization of 3d-electron states and in a reduction of the effective d-band width. The size dependence of the interatomic distances, which can be typically 20% smaller than in bulk crystals, play an important role in determining the ground-state moments. Moreover, finite-size effects and reduced symmetry are also at the origin of the very large magnetic anisotropy energies per atom found in TM clusters [10]. While many experimental and theoretical studies have already been carried out on homonuclear clusters, especially involving 3d TMs, very little is still known on mixed or alloy clusters [11,12]. This seems quite remarkable, since heterogeneous clusters are expected to show an extremely rich variety of magnetic behaviors as a function of composition and chemical order. For instance, theoretical studies of (Crx Fe1x )N have revealed interesting transitions from ferromagnetic to antiferromagnetic-like order as a function of increasing Cr concentration x, which are accompanied by important changes in the average magnetic moments per atom [11]. On the other side, very recent experimental studies [12] have shown that bimetallic Co–Rh nanoparticles present average magnetizations per atom that are remarkably enhanced with respect to macroscopic Co–Rh alloys of similar concentrations [13]. These systems are therefore very good candidates for highly stable magnetic clusters. In fact, they are expected to show particularly large magnetic anisotropies due to the stronger spin–orbit coupling at the 4d atoms. This is certainly an advantage for potential technological applications. Consequently, a systematic microscopic understanding of the properties of mixed 3d–4d TM clusters seems most worthwhile. The purpose of this paper is to investigate the structural and magnetic properties of small Co–Rh clusters. First-principles calculations based on a generalized gradient approximation (GGA) to
335
density functional theory have been performed as briefly described in Section 2. Results for small Co–Rh clusters are discussed in Section 3. Finally, Section 4 summarizes our conclusions and points to some interesting future extensions.
2. Calculational method Since we are in fine concerned with large clusters, as those experimentally observed [12], the present calculations are performed within the DFT formalism and the pseudopotential approximation, which are the only methods to deal with large TM clusters in a practically tractable way. Electronic calculations on TM elements pose a considerable theoretical challenge due to the dominant role of valence d-electrons which are characterized by a relatively strong localization, directional bonding, and high density of states around the Fermi energy. The need for an accurate treatment of exchange-correlation effects, the presence of numerous close-lying electronic configurations with different spin multiplicities, and the possibility of multiple local minima in the configurational energy surface are additional problems that render ab initio studies on TM clusters particularly difficult. Unfortunately at present time, none of the available exchange-correlation functionals are able to reproduce properly all the TM properties not even their fundamental ones in some cases [14,15]. One has to take any DFT results with some caution and compare them as far as possible with other ab initio calculations and experimental results when available. In the present study, since we are trying to infer only general tendencies in magnetic and energetic properties as a function of the structural and chemical order, this problem is not so crucial. The reported calculations have then been performed by means of the Vienna ab initio simulation program (VASP) developed at the Institut f€ ur Materialphysik of the Universit€at Wien [16] which solves the spin-polarized self-consistent Kohn– Sham equations within the projector-augmented wave (PAW) method [17]. This allows a considerably improved efficiency in terms of computation time. The exchange-correlation effects have been
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S. Dennler et al. / Surface Science 532–535 (2003) 334–340
described by means of the GGA Perdew–Wang 91 functional (PW91) [18], which is a reasonably good approximation for the calculation of fundamental state properties [19,20]. The computational parameters such as planewave energy cut-off Ec , reciprocal lattice, supercell size, etc., have been monitored in order to control the accuracy of the calculations. The results reported in Section 3 have been obtained by setting Ec ¼ 14:7 Ha. For bulk calculations, a 17 17 17 Monkhorst–Pack mesh [21] has been used to sample the Brillouin zone in the reciprocal space. For isolated clusters only the C point needs to be taken into account. In this case we consider a which has size of the supercell of about 10–15 A been checked to ensure that interactions between successive images are negligible. All structures have been fully optimized by relaxing the atomic positions until forces ~ Fi on each atom i are van). For clusishing (typically jFi j 6 0:01–0:02 eV/A ters, the geometry optimizations have been carried out for several fixed values of total spin magnetic moment Sz and several different initial configurations in order to detect nearby isomers. The structural and magnetic properties obtained in this way are in good agreement with experiments and previous DFT studies on homogeneous clusters and periodic crystals. For bulk Co the stability of the hexagonal close packed (hcp) structure is reproduced with optimized lat and c ¼ 4:023 A that tice parameters a ¼ 2:490 A compare very well with experimental results and cexp ¼ 4:069 A ). The same (aexp ¼ 2:507 A holds for the spin magnetic moment per atom: lb ðCoÞ ¼ 1:57lB (lexp b ðCoÞ ¼ 1:58lB ) [22]. In the case of Rh the face-centered cubic (fcc) structure is more stable than the hcp with a lattice parameter (aexp ¼ 3:804 A ) and a vanishing a ¼ 3:843 A magnetization. The calculated optimized geometries and magnetic properties of small homonuclear CoN and RhN (N 6 4) clusters are in good agreement with previous calculations [23,24]. In particular the Co clusters are found to be magnetic with moments per atom l ¼ 1:5–2:5lB that are enhanced with respect to bulk Co. Small Rh clusters are also magnetic with l ¼ 2lB for the dimers, and 1lB for the triangular trimer and square tetramer. As in earlier calculations [24], the
non-magnetic tetrahedron is found to be slightly more stable than the magnetic square for Rh4 . This non-magnetic state appears to be a singular case since otherwise finite moments are obtained for small N 6 50 [7–9]. The above presented results from the molecule up to the crystal show that the PW91 functional is a good approximation of the XC functional for transition-metal fundamental state properties. As a consequence one can have some confidence in the forthcoming calculations on mixed clusters. A few detailed examples of our results for pure clusters are presented in the following section for the sake of comparison with the behavior of mixed clusters.
3. Results and discussion Results for N ¼ 2 atoms are given in Table 1. First of all one can see, as usual in small clusters, that the interatomic bond lengths d are significantly smaller than in the corresponding solids. This contraction amounts to 19%, 19%, and 21% for Rh2 , CoRh, and Co2 respectively. The trends in d follow the trends observed in bulk materials (dCo < dCoRh < dRh ). The cohesive energy of the mixed dimer turns out to be larger than that of the pure dimers. This contrasts with the behavior observed for larger sizes and for the macroscopic Co0:5 Rh0:5 alloy. Concerning the magnetic moments, it is found independent of the composition ( l ¼ 2lB ). However, an interesting transfer of spinpolarized density is observed in CoRh. Indeed, the
Table 1 ), average Cohesive energy per atom Ecoh (eV), bond length d (A magnetic moment per atom l (lB ) of Co–Rh dimersa Ecoh l l(Co) l(Rh) d a
Rh–Rh
Co–Rh
Co–Co
1.74 2.00
1.83 2.00 2.29 1.43 2.08
1.52 2.00 1.93
1.82 2.21
1.96
Results are given for local moments li (lB ) as obtained by integration of the magnetization density in the bulk Wigner– and Seitz sphere around each atom i (rWS ðCoÞ ¼ 1:302 A ). rWS ðRhÞ ¼ 1:402 A
S. Dennler et al. / Surface Science 532–535 (2003) 334–340
3.00 2.50 2.00 1.50 1.00 0.50
2.50 2.00 1.50 1.00 0.50 0.00
magnetic non-magnetic 0
1
2
3
0.00 0
1
There is a regular decrease of the cohesive energy in going from the pure RhN to the pure CoN cluster in accordance with the lower cohesive energy of bulk Co compared to bulk Rh. In particular, there is no stability peak in this range of sizes. All mixed clusters are magnetic with local and average spin moments that are often a factor 2 larger than those of bulks or macroscopic alloys with similar concentrations (lðCoÞ ¼ 1:01lB and lðRhÞ ¼ 0:16lB in hcp alloy [13]). For a given cluster size, the magnetic moment per atom increases with the fraction of Co atoms. The local distribution of magnetization for some trimers is given in Fig. 3.
cohesive energy (eV/atom)
cohesive energy (eV/atom)
integrated local moment lðiÞ around the Co atom is enhanced to the detriment of Rh moment (see Table 1). This can be qualitative explained as a charge transfer from the 4d Co orbitals to the 5d Rh orbitals which allows the polarization of a larger number of d holes. This is consistent with the difference in the highest occupied Kohn–Sham eigenvalues calculated for the isolated atoms (EHOMO ðCoÞ EHOMO (Rh) ¼ 0.88 eV) and with their empirical Pauling electronegativities (vP (Co)¼1.88 and vP (Rh)¼2.28) [25]. Similar effects are sometimes observed in larger clusters. Results for mixed Co–Rh clusters containing up to four atoms are summarized in Figs. 1 and 2.
2
337
3.00 2.50 2.00 1.50 1.00 0.50
3.00 2.50 2.00 1.50 1.00 0.50 0.00
magnetic non-magnetic 0
1
2
3
4
0.00
3
0
M (number of Co atoms)
1
2
3
4
M (number of Co atoms)
3.00
magnetic moment (µB/atom)
magnetic moment (µB/atom)
Fig. 1. Cohesive energy per atom Ecoh (eV/atom) of CoM Rh3M and CoM Rh4M clusters as a function of M. Results are given for the optimized magnetic ground-state geometries (triangular symbols) and for the first excited magnetic isomers (square symbols). Inserts: Ecoh for the optimal magnetic (triangular symbols) and non-magnetic (diamonds) structures. The lines are a guide to the eye.
2.50 2.00 1.50 1.00 0.50 0.00 0
1
2
M (number of Co atoms)
3
3.00 2.50 2.00 1.50 1.00 0.50 0.00 0
1
2
3
4
M (number of Co atoms)
Fig. 2. Average magnetic moment per atom l in CoM Rh3M and CoM Rh4M clusters as a function of M. Results are given for the optimized ground-state geometries (triangular symbols) and for the first excited isomers (square symbols). The lines are a guide to the eye.
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S. Dennler et al. / Surface Science 532–535 (2003) 334–340
Fig. 3. Local distribution of magnetic moment lðiÞ at each atom i and nearest-neighbor bond-lengths d for fully optimized Rh3 , CoRh2 and Co3 clusters.
Contrary to one might have expected, the presence of Rh, even at concentrations above 50%, does not reduce the Co moment (typically lðCoÞ ’ 2lB ), even in some cases a slight enhancement is found. This behavior contrasts with the important reduction of Co moments observed in some macroscopic Co–Rh alloys. From the point of view of Rh, the presence of Co atoms in the cluster results in a remarkable increase of the local moments up to more than 1lB (tetrahedral Rh3 Co). It is well known that magnetism is very sensitive to the interatomic distance. In order to distinguish between the effects of the substitution itself and those resulting from the induced interatomic distance variation, homologous clusters, with different constrained geometries, have been calculated. For the Co–Rh–Rh trimer (Fig. 3) with a 2.280 A Rh–Rh distance, for example, the homologous geometry-constrained, but otherwise fully relaxed, Rh3 and Co–Rh–Rh clusters with both inter value atomic distances set to the same 2.280 A have been calculated. The comparison of the two constrained trimers shows that the Co substituted atom induces a 0.27lB increase on the central Rh; whereas the extremal Rh approximatively retains, in both of them, the same 0.85lB magnetic moment, as in the fully relaxed one. This 0.27lB increase, induced by the Co atom, leads to the total Sz observed in the fully relaxed one: the magnetism enhancement observed in mixed clusters is due to
the presence of cobalt atoms and not to the sole induced distance variation. Moreover, the Co influence seems to be limited––at least for this size range and for this geometry––to the first neighbors. The case of Co2 Rh2 clusters illustrates the influence of other effects such as disorder and segregation on the total magnetization. If we compare its two planar structures, it can be noticed in Fig. 2 that the structure where the Co atoms are separated by a strongly bonded Rh2 dimer, not only exhibits a significantly enhanced total magnetic moment (2lB enhancement) but is also more stable (0.04 eV/at), and only slightly less stable than the tetrahedral one (0.05 eV/at). The role of magnetism on the most stable structures and on the energy differences among the lowest isomers can be identified by comparing magnetic and non-magnetic calculations (inserts Fig. 1). For CoM RhN , N þ M 6 4, the relative stability of the isomers does not appear to be changed when magnetism is not taken into account in the calculations. Concerning its influence on bond lengths, we observe a very large 5% contraction in non-magnetic equilibrium structures with respect to magnetic corresponding ones. As it can be noticed in Fig. 1, the stabilization energy due to magnetism typically ranges from 0.0 to 0.5 eV/atom. It increases with the proportion of Co atoms and consequently with increasing local magnetic moment.
4. Conclusion The present calculations have shown that small CoRh clusters are magnetic with moments that are significantly larger than those of bulk materials. Magnetic moments at Co atoms seem to be relatively independent on the Rh concentration in the cluster; in some cases even a slight enhancement with respect to pure CoN clusters is found. From the point of view of Rh, the presence of Co atoms in the cluster remarkably leads to increased local moments of Rh neighbors. Moreover it has been shown that magnetism has a significant contribution to the cluster stability. Qualitatively the results are consistent with experiments on nanometer-scale Co–Rh particles [12]. However, it is
S. Dennler et al. / Surface Science 532–535 (2003) 334–340
still unclear at present to what extent these trends are specific to the very small sizes considered here. In fact, for M þ N 6 4 atoms the local coordination numbers are so small that the finite-size effects and reduction of effective d band width should largely dominate over the Co–Rh hybridizations. In progress extended calculations for larger clusters should clarify this point. Another question of considerable interest for experiment concerns chemical ordering and the possible role of segregation on the magnetic behavior. The present calculations show that the energy differences between isomers are relatively small. Therefore, the coexistence of different chemical orderings, often showing different magnetic properties, cannot be excluded. This stresses the interest to determine the ground-state configurations as well as the first low-lying isomers, all the more that experimental synthesis conditions and matrix effects may lead preferentially to constrained structures. Calculations of larger clusters allowing a greater diversity of chemical and structural orderings are currently in progress. Moreover, the distribution of Co and Rh species within the cluster can be modified to some extent by changing the experimental growth conditions. Comparison between measured and calculated magnetic properties will then be useful to infer complementary structural information. This study has mainly focused on structural and magnetic properties. In a forthcoming paper, a more extensive presentation will be given including electronic structure considerations, which will bring more insight into the remarkable magnetic behavior of such mixed clusters and the origin of the magnetic enhancement at Rh sites induced by Co nearest neighbors. We plan also to make some specific calculations with other functionals to test the accuracy of our results, and to include spin– orbit coupling and non-linear magnetization in order to get a finer insight of the properties of these clusters.
Acknowledgements Helpful discussions with M. Respaud, D. Zitoun, M.J. Casanove and M.C. Fromen are
339
gratefully acknowledged. This work has been financed in part by the EU GROWTH project AMMARE (G5RD-CT-2001-00478). Computer resources were provided by CALMIP and IDRIS (France). The calculations have been performed using the ab initio total-energy and moleculardynamics program VASP developed at the Institut f€ ur Materialphysik of the Universit€at Wien.
References [1] See, for instance, C. Guet, P. Hobza, F. Spiegelman, F. David (Eds.), Atomic Clusters and Nanoparticles, NATO ASI, Les Houches Session LXXIII, EDP Sciences, Les Ulis, Springer, Berlin, 2001. [2] K. Lee, J. Callaway, S. Dhar, Phys. Rev. B 30 (1985) 1724; K. Lee, J. Callaway, K. Wong, R. Tang, A. Ziegler, Phys. Rev. B 31 (1985) 1796; G.M. Pastor, J. Dorantes-Davila, K.H. Bennemann, Phys. Rev. B 40 (1989) 7642. [3] J.P. Bucher, D.C. Douglass, L.A. Bloomfield, Phys. Rev. Lett. 66 (1991) 3052; I.M.L. Billas, J.A. Becker, A. Ch^atelain, W.A. de Heer, Phys. Rev. Lett. 71 (1993) 4067; I.M.L. Billas, A. Ch^atelain, W.A. de Heer, Science 265 (1994) 1682; S.E. Apsel, J.W. Emert, J. Deng, L.A. Bloomfield, Phys. Rev. Lett. 76 (1996) 1441. [4] See, for instance M. Castro, Ch. Jamorski, D. Salahub, Chem. Phys. Lett. 271 (1997) 133; B.V. Reddy et al., J. Phys. Chem. A 102 (1998) 1748; F.A. Reuse, S. Khanna, Eur. Phys. J. D 6 (1999) 77. [5] M. Respaud et al., Mat. Sci. Forum 269–272 (1998); Phys. Rev. B 57 (1998) 2925; Phys. Rev. B 59 (1999) R3934. [6] G.M. Pastor, in: C. Guet, P. Hobza, F. Spiegelman, F. David (Eds.), Atomic Clusters and Nanoparticles, NATO ASI Les Houches Session LXXIII, EDP Sciences Les Ulis, Springer, Berlin, 2001, p. 335ff. [7] A.J. Cox, J.G. Louderback, L.A. Bloomfield, Phys. Rev. Lett. 71 (1993) 923; A.J. Cox, J.G. Louderback, S.E. Apsel, L.A. Bloomfield, Phys. Rev. B 49 (1994) 12295. [8] B.V. Reddy, S.N. Khanna, B.I. Dunlap, Phys. Rev. Lett. 70 (1993) 3323. [9] P. Villase~ nor-Gonzalez, J. Dorantes-Davila, H. Dreysse, G.M. Pastor, Phys. Rev. B 55 (1997) 15084. [10] G.M. Pastor, J. Dorantes-Davila, S. Pick, H. Dreysse, Phys. Rev. Lett. 75 (1995) 326. [11] A. Vega, J. Dorantes-Davila, G.M. Pastor, L.C. Balbas, Z. Phys. D 19 (1991) 263. [12] M.C. Fromen et al., J. Mag. Mag. Mat. 242–246 (2002) 610; D. Zitoun et al., Phys. Rev. Lett. 89 (2002) 037203.
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[13] G. Mora€ıtis, H. Dreysse, M.A. Khan, Phys. Rev. B 54 (1996) 10. [14] R. Besson, A. Legris, J. Morillo, Phys. Rev. Lett. 89 (22) (2002) 225502/1-4, and references therein. [15] M. F€ ahnle, M. Komelj, R.Q. Wu, G.Y. Guo, Phys. Rev. B 65 (2002) 144436/1-5. [16] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) RC558; G. Kresse, J. Furthm€ uller, Phys. Rev. B 54 (1996) 11169. [17] P.E. Bl€ ochl, Phys. Rev. B 50 (1994) 24; G. Kresse, J. Joubert, Phys. Rev. B 59 (1999) 1758. [18] P. Ziesche, H. Eschrig (Eds.), Electronic Structure of SolidsÕ91, Akademie Verlag, Berlin, 1991; J.P. Perdew et al., Phys. Rev. B 46 (1992) 6671. [19] N.A. Lambropoulos, J.R. Reimers, J. Chem. Phys. 116 (23) (2002) 10277.
[20] S. Yanagisawa, T. Tsuneda, K. Hirao, J. Chem. Phys. 112 (2) (2000) 545. [21] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [22] C. Kittel, Introduction to Solid State Physics, seventh ed., Wiley, New York, 1996. [23] See, for instance C. Jamorski, A. Martinez, M. Castro, D.R. Salahub, Phys. Rev. B 55 (1997) 16; M. Pereiro et al., Int. J. Quant. Chem. 81 (2001) 422–430. [24] See, for instance Y. Jinlong, F. Toigo, W. Kelin, Phys. Rev. B 50 (1994) 11; C.H. Chien, E. Blaisten-Barojas, M.R. Pederson, Phys. Rev. A 58 (1998) 3. [25] M.C. Ball, A.H. Norbury, Physical Data for Inorganic Chemists, Longman, London, 1974.
Calculation of magnetic and structural properties of small Co–Rh clusters S. Dennler b
a,*
, J. Morillo a, G.M. Pastor
b
a Centre d’Elaboration de Mat eriaux et d’Etudes Structurales, UPR CNRS 8011, 29 rue Jeanne Marvig, 31055 Toulouse, France Laboratoire de Physique Quantique, UMR CNRS 5626, Institut de Recherche sur les Syst emes Atomiques et Mol eculaires Complexes, 31062 Toulouse, France
Abstract Structural and magnetic properties of CoM RhN clusters with M þ N 6 4 atoms are determined by performing firstprinciples calculations in the framework of the density functional theory in a generalized gradient-corrected approximation and within the projector-augmented wave method. The role of magnetism on the most stable structures and on the energy differences among the low-lying isomers is quantified by comparing magnetic and non-magnetic solutions of the Kohn–Sham equations. As usual in small clusters, CoM RhN clusters show contracted interatomic distances with respect to the bulk. They exhibit ferromagnetic-like order with environment-dependent local magnetic moments lðiÞ. The average magnetic moments per atom lðM; NÞ and lðiÞ are generally more than a factor 2 larger than in macroscopic alloys of similar concentration. For a given cluster size M þ N, l increases with increasing number of Co atoms M. The lðiÞ at Rh atoms are remarkably enhanced by the presence of Co nearest neighbors, while the lðiÞ at Co atoms are not much affected by the presence of Rh atoms. The correlation between structure, chemical order, and environment-dependent magnetic properties is discussed. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Cobalt; Rhodium; Clusters; Density functional calculations; Magnetic phenomena (cyclotron resonance, phase transitions, etc.)
1. Introduction Experimental and theoretical investigations during the past decades have demonstrated in uncountable occasions that the reduction of system size and dimensionality gives rise to a variety of new states of matter with physical and chemical properties that are intrinsically different from those of conventional bulk crystals [1]. This has opened *
Corresponding author. Tel.: +33-5-6225-7970; fax: +33-56225-7999. E-mail address: [email protected] (S. Dennler).
numerous possibilities of generating suitably designed materials with specifically tailored properties. A remarkably intense and diverse research activity is presently developed within this field. Transition-metal (TM) clusters are a subject of particular interest in this context due to the strong environment dependence of their magnetic properties and their possible applications in magnetic recording or storage technologies. Previous experimental and theoretical studies have shown that the magnetic moments of FeN , CoN , and NiN clusters are significantly larger than the corresponding bulk magnetizations [2–6]. Moreover,
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00468-0
S. Dennler et al. / Surface Science 532–535 (2003) 334–340
non-vanishing magnetic moments have been recently observed in clusters of some 4d TMs like RhN (N 6 50) which are non-magnetic in the bulk [7–9]. This enhancement of magnetism in small particles can be qualitatively understood as a consequence of the reduction of local coordination number, which results in a stronger localization of 3d-electron states and in a reduction of the effective d-band width. The size dependence of the interatomic distances, which can be typically 20% smaller than in bulk crystals, play an important role in determining the ground-state moments. Moreover, finite-size effects and reduced symmetry are also at the origin of the very large magnetic anisotropy energies per atom found in TM clusters [10]. While many experimental and theoretical studies have already been carried out on homonuclear clusters, especially involving 3d TMs, very little is still known on mixed or alloy clusters [11,12]. This seems quite remarkable, since heterogeneous clusters are expected to show an extremely rich variety of magnetic behaviors as a function of composition and chemical order. For instance, theoretical studies of (Crx Fe1x )N have revealed interesting transitions from ferromagnetic to antiferromagnetic-like order as a function of increasing Cr concentration x, which are accompanied by important changes in the average magnetic moments per atom [11]. On the other side, very recent experimental studies [12] have shown that bimetallic Co–Rh nanoparticles present average magnetizations per atom that are remarkably enhanced with respect to macroscopic Co–Rh alloys of similar concentrations [13]. These systems are therefore very good candidates for highly stable magnetic clusters. In fact, they are expected to show particularly large magnetic anisotropies due to the stronger spin–orbit coupling at the 4d atoms. This is certainly an advantage for potential technological applications. Consequently, a systematic microscopic understanding of the properties of mixed 3d–4d TM clusters seems most worthwhile. The purpose of this paper is to investigate the structural and magnetic properties of small Co–Rh clusters. First-principles calculations based on a generalized gradient approximation (GGA) to
335
density functional theory have been performed as briefly described in Section 2. Results for small Co–Rh clusters are discussed in Section 3. Finally, Section 4 summarizes our conclusions and points to some interesting future extensions.
2. Calculational method Since we are in fine concerned with large clusters, as those experimentally observed [12], the present calculations are performed within the DFT formalism and the pseudopotential approximation, which are the only methods to deal with large TM clusters in a practically tractable way. Electronic calculations on TM elements pose a considerable theoretical challenge due to the dominant role of valence d-electrons which are characterized by a relatively strong localization, directional bonding, and high density of states around the Fermi energy. The need for an accurate treatment of exchange-correlation effects, the presence of numerous close-lying electronic configurations with different spin multiplicities, and the possibility of multiple local minima in the configurational energy surface are additional problems that render ab initio studies on TM clusters particularly difficult. Unfortunately at present time, none of the available exchange-correlation functionals are able to reproduce properly all the TM properties not even their fundamental ones in some cases [14,15]. One has to take any DFT results with some caution and compare them as far as possible with other ab initio calculations and experimental results when available. In the present study, since we are trying to infer only general tendencies in magnetic and energetic properties as a function of the structural and chemical order, this problem is not so crucial. The reported calculations have then been performed by means of the Vienna ab initio simulation program (VASP) developed at the Institut f€ ur Materialphysik of the Universit€at Wien [16] which solves the spin-polarized self-consistent Kohn– Sham equations within the projector-augmented wave (PAW) method [17]. This allows a considerably improved efficiency in terms of computation time. The exchange-correlation effects have been
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described by means of the GGA Perdew–Wang 91 functional (PW91) [18], which is a reasonably good approximation for the calculation of fundamental state properties [19,20]. The computational parameters such as planewave energy cut-off Ec , reciprocal lattice, supercell size, etc., have been monitored in order to control the accuracy of the calculations. The results reported in Section 3 have been obtained by setting Ec ¼ 14:7 Ha. For bulk calculations, a 17 17 17 Monkhorst–Pack mesh [21] has been used to sample the Brillouin zone in the reciprocal space. For isolated clusters only the C point needs to be taken into account. In this case we consider a which has size of the supercell of about 10–15 A been checked to ensure that interactions between successive images are negligible. All structures have been fully optimized by relaxing the atomic positions until forces ~ Fi on each atom i are van). For clusishing (typically jFi j 6 0:01–0:02 eV/A ters, the geometry optimizations have been carried out for several fixed values of total spin magnetic moment Sz and several different initial configurations in order to detect nearby isomers. The structural and magnetic properties obtained in this way are in good agreement with experiments and previous DFT studies on homogeneous clusters and periodic crystals. For bulk Co the stability of the hexagonal close packed (hcp) structure is reproduced with optimized lat and c ¼ 4:023 A that tice parameters a ¼ 2:490 A compare very well with experimental results and cexp ¼ 4:069 A ). The same (aexp ¼ 2:507 A holds for the spin magnetic moment per atom: lb ðCoÞ ¼ 1:57lB (lexp b ðCoÞ ¼ 1:58lB ) [22]. In the case of Rh the face-centered cubic (fcc) structure is more stable than the hcp with a lattice parameter (aexp ¼ 3:804 A ) and a vanishing a ¼ 3:843 A magnetization. The calculated optimized geometries and magnetic properties of small homonuclear CoN and RhN (N 6 4) clusters are in good agreement with previous calculations [23,24]. In particular the Co clusters are found to be magnetic with moments per atom l ¼ 1:5–2:5lB that are enhanced with respect to bulk Co. Small Rh clusters are also magnetic with l ¼ 2lB for the dimers, and 1lB for the triangular trimer and square tetramer. As in earlier calculations [24], the
non-magnetic tetrahedron is found to be slightly more stable than the magnetic square for Rh4 . This non-magnetic state appears to be a singular case since otherwise finite moments are obtained for small N 6 50 [7–9]. The above presented results from the molecule up to the crystal show that the PW91 functional is a good approximation of the XC functional for transition-metal fundamental state properties. As a consequence one can have some confidence in the forthcoming calculations on mixed clusters. A few detailed examples of our results for pure clusters are presented in the following section for the sake of comparison with the behavior of mixed clusters.
3. Results and discussion Results for N ¼ 2 atoms are given in Table 1. First of all one can see, as usual in small clusters, that the interatomic bond lengths d are significantly smaller than in the corresponding solids. This contraction amounts to 19%, 19%, and 21% for Rh2 , CoRh, and Co2 respectively. The trends in d follow the trends observed in bulk materials (dCo < dCoRh < dRh ). The cohesive energy of the mixed dimer turns out to be larger than that of the pure dimers. This contrasts with the behavior observed for larger sizes and for the macroscopic Co0:5 Rh0:5 alloy. Concerning the magnetic moments, it is found independent of the composition ( l ¼ 2lB ). However, an interesting transfer of spinpolarized density is observed in CoRh. Indeed, the
Table 1 ), average Cohesive energy per atom Ecoh (eV), bond length d (A magnetic moment per atom l (lB ) of Co–Rh dimersa Ecoh l l(Co) l(Rh) d a
Rh–Rh
Co–Rh
Co–Co
1.74 2.00
1.83 2.00 2.29 1.43 2.08
1.52 2.00 1.93
1.82 2.21
1.96
Results are given for local moments li (lB ) as obtained by integration of the magnetization density in the bulk Wigner– and Seitz sphere around each atom i (rWS ðCoÞ ¼ 1:302 A ). rWS ðRhÞ ¼ 1:402 A
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3.00 2.50 2.00 1.50 1.00 0.50
2.50 2.00 1.50 1.00 0.50 0.00
magnetic non-magnetic 0
1
2
3
0.00 0
1
There is a regular decrease of the cohesive energy in going from the pure RhN to the pure CoN cluster in accordance with the lower cohesive energy of bulk Co compared to bulk Rh. In particular, there is no stability peak in this range of sizes. All mixed clusters are magnetic with local and average spin moments that are often a factor 2 larger than those of bulks or macroscopic alloys with similar concentrations (lðCoÞ ¼ 1:01lB and lðRhÞ ¼ 0:16lB in hcp alloy [13]). For a given cluster size, the magnetic moment per atom increases with the fraction of Co atoms. The local distribution of magnetization for some trimers is given in Fig. 3.
cohesive energy (eV/atom)
cohesive energy (eV/atom)
integrated local moment lðiÞ around the Co atom is enhanced to the detriment of Rh moment (see Table 1). This can be qualitative explained as a charge transfer from the 4d Co orbitals to the 5d Rh orbitals which allows the polarization of a larger number of d holes. This is consistent with the difference in the highest occupied Kohn–Sham eigenvalues calculated for the isolated atoms (EHOMO ðCoÞ EHOMO (Rh) ¼ 0.88 eV) and with their empirical Pauling electronegativities (vP (Co)¼1.88 and vP (Rh)¼2.28) [25]. Similar effects are sometimes observed in larger clusters. Results for mixed Co–Rh clusters containing up to four atoms are summarized in Figs. 1 and 2.
2
337
3.00 2.50 2.00 1.50 1.00 0.50
3.00 2.50 2.00 1.50 1.00 0.50 0.00
magnetic non-magnetic 0
1
2
3
4
0.00
3
0
M (number of Co atoms)
1
2
3
4
M (number of Co atoms)
3.00
magnetic moment (µB/atom)
magnetic moment (µB/atom)
Fig. 1. Cohesive energy per atom Ecoh (eV/atom) of CoM Rh3M and CoM Rh4M clusters as a function of M. Results are given for the optimized magnetic ground-state geometries (triangular symbols) and for the first excited magnetic isomers (square symbols). Inserts: Ecoh for the optimal magnetic (triangular symbols) and non-magnetic (diamonds) structures. The lines are a guide to the eye.
2.50 2.00 1.50 1.00 0.50 0.00 0
1
2
M (number of Co atoms)
3
3.00 2.50 2.00 1.50 1.00 0.50 0.00 0
1
2
3
4
M (number of Co atoms)
Fig. 2. Average magnetic moment per atom l in CoM Rh3M and CoM Rh4M clusters as a function of M. Results are given for the optimized ground-state geometries (triangular symbols) and for the first excited isomers (square symbols). The lines are a guide to the eye.
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S. Dennler et al. / Surface Science 532–535 (2003) 334–340
Fig. 3. Local distribution of magnetic moment lðiÞ at each atom i and nearest-neighbor bond-lengths d for fully optimized Rh3 , CoRh2 and Co3 clusters.
Contrary to one might have expected, the presence of Rh, even at concentrations above 50%, does not reduce the Co moment (typically lðCoÞ ’ 2lB ), even in some cases a slight enhancement is found. This behavior contrasts with the important reduction of Co moments observed in some macroscopic Co–Rh alloys. From the point of view of Rh, the presence of Co atoms in the cluster results in a remarkable increase of the local moments up to more than 1lB (tetrahedral Rh3 Co). It is well known that magnetism is very sensitive to the interatomic distance. In order to distinguish between the effects of the substitution itself and those resulting from the induced interatomic distance variation, homologous clusters, with different constrained geometries, have been calculated. For the Co–Rh–Rh trimer (Fig. 3) with a 2.280 A Rh–Rh distance, for example, the homologous geometry-constrained, but otherwise fully relaxed, Rh3 and Co–Rh–Rh clusters with both inter value atomic distances set to the same 2.280 A have been calculated. The comparison of the two constrained trimers shows that the Co substituted atom induces a 0.27lB increase on the central Rh; whereas the extremal Rh approximatively retains, in both of them, the same 0.85lB magnetic moment, as in the fully relaxed one. This 0.27lB increase, induced by the Co atom, leads to the total Sz observed in the fully relaxed one: the magnetism enhancement observed in mixed clusters is due to
the presence of cobalt atoms and not to the sole induced distance variation. Moreover, the Co influence seems to be limited––at least for this size range and for this geometry––to the first neighbors. The case of Co2 Rh2 clusters illustrates the influence of other effects such as disorder and segregation on the total magnetization. If we compare its two planar structures, it can be noticed in Fig. 2 that the structure where the Co atoms are separated by a strongly bonded Rh2 dimer, not only exhibits a significantly enhanced total magnetic moment (2lB enhancement) but is also more stable (0.04 eV/at), and only slightly less stable than the tetrahedral one (0.05 eV/at). The role of magnetism on the most stable structures and on the energy differences among the lowest isomers can be identified by comparing magnetic and non-magnetic calculations (inserts Fig. 1). For CoM RhN , N þ M 6 4, the relative stability of the isomers does not appear to be changed when magnetism is not taken into account in the calculations. Concerning its influence on bond lengths, we observe a very large 5% contraction in non-magnetic equilibrium structures with respect to magnetic corresponding ones. As it can be noticed in Fig. 1, the stabilization energy due to magnetism typically ranges from 0.0 to 0.5 eV/atom. It increases with the proportion of Co atoms and consequently with increasing local magnetic moment.
4. Conclusion The present calculations have shown that small CoRh clusters are magnetic with moments that are significantly larger than those of bulk materials. Magnetic moments at Co atoms seem to be relatively independent on the Rh concentration in the cluster; in some cases even a slight enhancement with respect to pure CoN clusters is found. From the point of view of Rh, the presence of Co atoms in the cluster remarkably leads to increased local moments of Rh neighbors. Moreover it has been shown that magnetism has a significant contribution to the cluster stability. Qualitatively the results are consistent with experiments on nanometer-scale Co–Rh particles [12]. However, it is
S. Dennler et al. / Surface Science 532–535 (2003) 334–340
still unclear at present to what extent these trends are specific to the very small sizes considered here. In fact, for M þ N 6 4 atoms the local coordination numbers are so small that the finite-size effects and reduction of effective d band width should largely dominate over the Co–Rh hybridizations. In progress extended calculations for larger clusters should clarify this point. Another question of considerable interest for experiment concerns chemical ordering and the possible role of segregation on the magnetic behavior. The present calculations show that the energy differences between isomers are relatively small. Therefore, the coexistence of different chemical orderings, often showing different magnetic properties, cannot be excluded. This stresses the interest to determine the ground-state configurations as well as the first low-lying isomers, all the more that experimental synthesis conditions and matrix effects may lead preferentially to constrained structures. Calculations of larger clusters allowing a greater diversity of chemical and structural orderings are currently in progress. Moreover, the distribution of Co and Rh species within the cluster can be modified to some extent by changing the experimental growth conditions. Comparison between measured and calculated magnetic properties will then be useful to infer complementary structural information. This study has mainly focused on structural and magnetic properties. In a forthcoming paper, a more extensive presentation will be given including electronic structure considerations, which will bring more insight into the remarkable magnetic behavior of such mixed clusters and the origin of the magnetic enhancement at Rh sites induced by Co nearest neighbors. We plan also to make some specific calculations with other functionals to test the accuracy of our results, and to include spin– orbit coupling and non-linear magnetization in order to get a finer insight of the properties of these clusters.
Acknowledgements Helpful discussions with M. Respaud, D. Zitoun, M.J. Casanove and M.C. Fromen are
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gratefully acknowledged. This work has been financed in part by the EU GROWTH project AMMARE (G5RD-CT-2001-00478). Computer resources were provided by CALMIP and IDRIS (France). The calculations have been performed using the ab initio total-energy and moleculardynamics program VASP developed at the Institut f€ ur Materialphysik of the Universit€at Wien.
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