Unravelling Finite Size Effects on Magnetic Properties of Cobalt

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C: Physical Processes in Nanomaterials and Nanostructures

Unravelling Finite Size Effects on Magnetic Properties of Cobalt Nanoparticles Jacques R. Eone, Olivier M. Bengone, and Christine Goyhenex J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11763 • Publication Date (Web): 30 Jan 2019 Downloaded from http://pubs.acs.org on February 2, 2019

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The Journal of Physical Chemistry

Unravelling Finite Size Effects on Magnetic Properties of Cobalt Nanoparticles Jacques R. Eone II, Olivier M. Bengone, and Christine Goyhenex∗ Institut de Physique et Chimie des Matériaux de Strasbourg, Université de Strasbourg, CNRS UMR 7504, 23 rue du Loess, BP 43, F-67034 Strasbourg, France E-mail: [email protected]

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Abstract Using first principle calculations, the problem of scaling magnetic properties in nanoparticles is addressed. To this aim, the local electronic structure is characterized in cobalt quasi-spherical magnetic nanoparticles, in a large size range, from 0.5 to 2 nm of diameter. First, specific patterns of the magnitude of local spin magnetic moments are evidenced depending on the shape and the size of the nanoparticles. Then, effects of local structural environment (atomic coordination, structural deformations, finite size effects, shape changes) are unravelled. In small icosahedral nanoparticles, the local spin magnetic moment is found to decrease from the surface to the center. General rules driving charge transfers are observed whereby donor atomic sites are exclusively subsurface atoms and more unexpected vertex surface atomic sites. The variation of the magnetic moment is driven by the coupling between cluster microstructure and complex hybridization effects. In larger truncated octahedral clusters, whereas some properties are still found (quasi-zero inter-atomic site charge transfer, the reversal from anti-ferromagnetic to ferromagnetic coupling of electronic sp states with d ones at vertex atomic sites), other vary such as the behavior of the local spin magnetic moment which now presents weak oscillations.

Introduction The research on metallic nanoparticles is in constant progress both experimentally and theoretically. Thanks to this continuous increase of knowledge, many potential applications have been envisaged or even realized in various fields including catalysis, magnetism and optics. 1–7 In the case of computer materials science, a common aim is still to improve the predictability of the calculations and to extend their transferability towards realistic physico-chemical environments of the considered nanoparticles. The variety of physico-chemical properties of nanoparticles are indeed strongly related to finite size effects. The main involved characteristics are the presence of inequivalent atomic sites (atomic sites will be more simply named

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sites in the rest of the text) and the variation of their number in terms of atomic coordination, the atomic bonds deformations and the shape of the particles. These statements take all their sense when considering magnetic nanoparticles. The magnetic state of a transition metal element is indeed drastically sensitive to small variations of the local electronic structure, and therefore to changes in the close atomic environment.

Advances in electronic structure descriptions together with the increasing computer accessible power has enabled the upscaling of the standard ab initio codes, mainly implementing the Density Functional Theory (DFT), towards extended sizes of systems. 2,8–11 Most interestingly, the latter fact has opened the route to new possibilities to explore or to revisit the electronic structure, and the properties related to it, of increasingly larger nanoparticles. Magnetic clusters in a size range going from a few atoms to a few tens, for the most numerous, were intensively investigated. 12–17 As magnetic nanoparticles in the range size of one hundred to a few hundreds have been much less studied within DFT calculations, one still can expect to highlight some specificities in the properties at a scale closer to experimental systems, relatively to the very small clusters. In addition, these few studies are more focused on total energy calculations and scarcely propose a detailed study at the electronic structure level. 2,18 Conversely, the studies at higher sizes could even generalize and extend to higher size some results obtained on the smaller clusters. One main aim of this article is to address this question of scaling magnetic properties in nanoparticles.

In this context, we will take the case of cobalt. Cobalt or cobalt-based nanoparticles are extensively investigated for their wide range of technical applications in fields like high density magnetic storage, magneto-optics, catalysis and nanomedicine. 3,7,19–22 For these fields of study, rationalizing the link between the structure and the magnetic properties is of prime importance to enable the design of nanoparticles with targeted properties. In the case of magnetic systems, one tricky point is to unravel the effects of the local atomic environment

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by attributing the change in local properties to charge transfer and/or to the deformations of local densities of states (LDOS) and finally to relate these changes to local distance deformations which are enhanced in nanoparticles relatively to extended surfaces and large particles. In this aim, the present work is devoted to accurate electronic structure calculations, using first-principle methods, of the local electronic properties in cobalt nanoparticles containing 13 to 405 atoms. Two quasi-spherical shapes are considered namely the icosahedron (Ih) and the truncated octahedron (TO) or Wulff shape. Careful mappings of the local spin magnetic moments, the charge distributions per site and orbital and the inter-atomic distances are presented and related to each other in order to give a general picture of the finite size effects in local properties of magnetic nanoparticles. In the following, we choose e (charge of one electron) equal to 1 so that, in the following, the term charge will refer to a number of electrons.

Methods The SIESTA method The widely used code SIESTA 8 is an implementation of the solution of the Kohn-Sham equations (density functional theory) using numerical basis orbitals sets and pseudopotentials to describe the core electrons and their nucleus. The practical implementation of DFT calculations with SIESTA has been performed by using norm-conserving, improved Troullier-Martins pseudopotentials 23 with core corrections, a double-ζ polarized basis set (DZP) and the GGA-PBE exchange-correlation functional. 24 The latter is well known to be a relevant choice for magnetic metallic 3d elements like Co, Fe or Ni. 25 The convergence of the self-consistency and all the results have been carefully checked upon the numerical parameters of the calculations, including the size of the basis set and the so-called mesh cut-off, which is an energy term defining the fineness of the real-space grid on which the Poisson equation is solved. A sufficiently high value of this mesh-cutoff term, here between 4


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200 and 300 Ry (greater than the usual default value of 150 Ry), gave a good compromise between accuracy and computation time. Between these two values, the total energy variation |∆Etot | < 0.02eV /atom and the variation of the local spin magnetic moment |∆µ| < 0.005µB /atom. Reference calculations were performed on surface systems within the supercell scheme using a slab of twelve layers, each plane containing four atoms. The calculations with nanoparticles were performed like for isolated molecules, i.e. a cluster of atoms is placed in a vacuum of sufficiently large spans in all directions in order to effectively isolate it from its images introduced by the application of the periodic boundary conditions inherent to most DFT calculations. In the present work, a cubic supercell was used with at least 2.0 nm of spacing between the periodic cluster images in each direction. For all the systems studied in the present work, the relaxation procedure is performed by using the conjugate gradients method where the minimization of the forces is stopped if the maximum residual atomic force is smaller than 0.04 eV /˚ A.

The local charge determination is performed within a Mulliken analysis. 26 It is worth noticing that the accuracy of the obtained Mulliken populations is very sensitive to the choice and optimization of the basis. However we have previously shown it to be accurate enough for metallic (Co,Pt) based systems, further Bader analysis giving the same results in this case. 27

The tight-binding linear muffin-tin orbital (TB-LMTO) approach For comparison, we also performed TB-LMTO-ASA calculations. The use of TB-LMTOASA method to supplement SIESTA calculations, has been motivated by the fact that this approach, based on Green function formalism, can model accurately true two-dimensional systems as surfaces with open boundary conditions, i.e. without resorting to periodic conditions. The use of this method also enables a direct comparison with results of Ref. 28 for Co(111) surfaces. 5



The tight-binding linear muffin-tin orbital method, 29 within the atomic sphere approximation (ASA) has been used. TB-LMTO-ASA results have been obtained using a surface Green’s function technique. 30 This approach allows to simulate on an ab initio level the open boundaries of a surface with a true semi-infinite stacking of layers, from the substrate to the vacuum (without resorting to periodic conditions). This is realized by using a combination of delocalized basis functions in reciprocal space within the two-dimensional plane of the surface (xy), and tight-binding real space basis functions in the direction perpendicular to the surface. The electronic structure was determined self-consistently within the local spin density approximation, with Vosko-Wilk-Nusair parameterization for the exchange-correlation potential. 29 To model the surfaces, we used ten layers of metals (Co or Fe), with four layers of so-called empty spheres to simulate the vacuum on top of the metal surfaces. For the TBLMTO method we have not considered structural relaxation. The TB-LMTO-ASA method is particularly accurate and efficient for the closed-packed structures considered here.

Results and discussion Reference study of close-packed extended surfaces of cobalt As we will focus more on icosahedral nanoparticles presenting only facets of (111) orientation, we consider, in this benchmark study, only the extended close-packed Co(111) surface. We think it important to perform such preliminary work in the aim to ensure the correctness of the analyses, as well of the local magnetic moments as of the charge analysis, knowing that these quantities are very sensitive to the method of calculation and the one for analyzing the charge. We have used the SIESTA-DFT 8 and the TB-LMTO 29 to extract the number of electrons per atom and the spin magnetic moment per site. The results for local 6


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magnetic moments and of total number of valence electrons per site are presented in Figure 1 where N (N=1-5) denotes the number of plane from the deepest one in the bulk (N=1) to the surface (N=5). It is worth mentioning that, in LMTO-ASA calculation, with so-called Empty spheres (ES) to simulate the vacuum of the surface, there is a non negligible charge transfer from Co surface atom to the first ES layer (typically 0.22 electron). This expresses the delocalization and extension of electron in dangling bond due to the surface. For the sake of comparison with Siesta calculations, the LMTO results for the last Co atom (N=5) in Figure 1 and Figure 2, accounts for the sum of electrons of the Co surface atom, and the electron of the first ES atom above the surface. It is worth noticing that we choose e (charge of one electron) equal to 1 so that, in the following, charge will refer to a number of valence electrons.

FCC - Co(111)

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Figure 1: Co(111) surface. Upper panel: magnetic moment per site; lower panel: total charge per site. The layers (abscissa) are numbered from bulk (N=1) to surface (N=5). The horizontal lines mark the bulk values of respectively the local spin magnetic moment (1.61µB and the total charge on a bulk atomic site (9.0 e).

Both unrelaxed and relaxed slabs are considered with SIESTA, while TB-LMTO is performed on unrelaxed slabs only. The LMTO results are consistent with previous studies on surfaces. 28 Then all the results are consistent with each other whatever the method which gives confidence in the present performed analyses and in their transferability to the studies of nanoparticles. In summary, the local spin magnetic moment is (as expected) strongly 7



Nd (e/atom)

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Figure 2: Co(111) surface. From top to bottom : charge per orbital d, charge per orbitals sp, total charge per site (spd). The layers (abscissa) are numbered from bulk (N=1) to surface (N=5). The horizontal lines mark the bulk values. enhanced at the surface layer, due to the decreasing of the coordination at the surface. The presence of the surface still have an influence on the subsurface plane where the local magnetic moment, even if it decreases, is still higher than in the next deeper planes where the bulk value is rapidly reached, anyway, already from the third plane from the surface (N=3 in Figure 1). The relaxation (SIESTA-DFT results) does not affect the results in a noticeable way. Both charge and local spin magnetic moment variations are tiny and definitely below the accuracy of standard DFT calculations. A closer analysis to up and down population separately, represented in Figure 2, shows that the effects are also limited to the two first planes from the surface. The relaxations at the surface do not change the trends in the spin up - spin down distribution. Additionally one can see an anti-ferromagnetic coupling between sp and the d electronic distribution, which is found with both used methods and

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which is preserved at the surface. It is worth noticing that, despite the very good agreement on the total distribution, a noticeable difference is visible between SIESTA and LMTO regarding the electronic distribution between sp spin up and down. It however does not affect the general conclusions in particular in the case of the anti-ferromagnetic coupling between sp and the d electronic distribution.

In summary, the variations of both charges and magnetic moments are very small and the main variations in the local magnetic properties are essentially due to the change in atomic coordination. The observed main effect is an anti-ferromagnetic coupling between ”sp” and ”d” electrons. However, in nanoparticles, where deformations are more important, the role of relaxation could take a larger part in the change of local magnetic properties. Moreover, changes of charge distributions at the surface may alter, if not the magnetic properties, the surface reactivity towards adsorbed gases. Based on these considerations, the next section is devoted to the nanoparticles and the effects of finite size and deformations on the local electronic structure and magnetic properties, that we intend to quantify in the aim of a better rationalization of magnetic properties in a computer material design perspective.

Results on Cobalt nanoparticles In the case of transition metals, many various shapes can be found depending on the size of the nanoparticles, in particular in the subnanometer range. 31 For instance, for 13 atoms metallic clusters, geometries like HCP, FCC, cubic and mixed icosahedral, decahedral and octahedral shapes have been listed. 32 From 55 atoms, the icosahedral shape becomes more frequent for some metals including cobalt. 33 We therefore take this archetypal shape to perform our study of the influence of structural changes onto local electronic structure and magnetism. Such particle is based on the Mackay icosahedron, 34 a shape deriving from the regular cuboctahedron. It is obtained therefore starting from a regular cuboctahedron where 9



the constituting atoms are separated by the interatomic distances of the bulk Co in an FCC stacking. The intra-layer distances are then expanded by about 5% relatively to the interlayer distances, which leads to a distortion of the (100) facets into triangular (111) facets. It is worth noticing that starting in the DFT calculations with a cuboctahedron leads to its deformation towards an icosahedron which is more stable within the same number of atoms. To our knowledge, the question of the accurate transition to another morphology at higher sizes is not clearly established. From recent DFT calculations it is clear that up to 309 atoms the icosahedral shape is favored for Co nanoparticles. 18 On the other hand, at much higher size (diameter ≈ 10 nm), the experiments show clearly the Wulff shape (truncated octahedron or TO in the following) of Co nanoparticles, 35 as expected for close-packed transition metals. In this context, we will compare the main results for the icosahedra to the case of a TO morphology for a higher size (405 atoms, ≈ 2 nm of diameter). This choice is also justified by total energy calculations that we performed for these closed shells structures. The icosahedral structures are the most stable structures relatively to truncated octahedra from the lowest sizes (13 atoms, ≈ 0.5 nm to around ≈ 1.5 nm (309 atoms) of diameter where both structures are almost degenerated on an energy point of view. Based on these calculations and on the fact that our main purpose is to relate finite size effects to magnetic properties, we selected the icosahedron and the truncated octahedron for representing respectively the smallest sizes of nanoparticles and the largest ones. For the present study they can be considered as relevant model systems and are anyway directly related to actual shapes.

The relaxed icosahedral nanoparticles are represented in Figure 3 where the represented atoms are colored following their local spin magnetic moment. At first glance, a specific pattern is observed for the variation of the local spin magnetic moment that is a shell behavior where the magnetic moment is constant in each concentric shell of atoms of the quasi-spherical nanoparticle and decreases from the surface to the central atom shell by shell. The interatomic distances inside these particles are then analyzed in order to consider

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the structural effects (relaxation, deformation) and their interplay with the local electronic structure.

Figure 3: Images of icosahedral nanoparticles (Ih) colored following the local spin magnetic moment per site. 36 Top images: three-dimensional view. Bottom images: two-dimensional cross-section of the upper represented nanoparticle. The color scale ranges from the minimal to the maximal value for each nanoparticle independently. Number of atoms in the nanoparticles, from left to right: 13, 55, 147 and 309. The minimal value, 1.34 µB is obtained in the central atomic site of the 309 atoms-Ih. The maximal value, 2.42 µB is obtained at the vertex atomic sites of the 13 atoms-Ih.

The local distribution of interatomic distances, 1st (d1 ) and 2nd (d2 ) nearest neighbors for each atom, is represented in Figure 4 for the relaxed icosahedral nanoparticles. A general characteristic shown by these distributions is the decreasing mean interatomic distance from one concentric shell to the other, going from the surface shell to the central atom, meaning that the core of the icosahedron is in the strongest compressive state which is a known fact for icosahedral nanoparticles. 37 In more details, when looking closer to the local interatomic distance variations, some other specificities are noticeable. First, there is a dilation of the second nearest distance (d2 ) implying the vertex atoms of the surface, which is a consequence 11



of the intra-layer dilation. Looking at the particular case of vertex sites, their first neighbors distances are contracted relatively to the other surface atoms, but in return their second nearest neighbors distances are clearly longer. For the two largest nanoparticles (containing 147 and 309 atoms), the distribution of the second nearest distances is very inhomogeneous with a particular behavior identified along the axis joining the central atom to the surface vertex atom. Along this direction, there is a connection between the enhanced contraction of first nearest neighbors and the enhanced dilation of second nearest neighbors distances. This configuration is expected to play a part in the evolution of the electronic structure through variable hybridizations of the orbitals and charge distributions.

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Figure 4: Local distribution of interatomic distances in the relaxed icosahedral nanoparticles (Ih) containing from 13 to 309 atoms: 1rst (d1 ) and 2nd (d2 ) nearest neighbors for each atom. The x axis indicates the atom number starting from the central atom and following the shell by shell structure of the nanoparticle towards the surface. The vertical lines delimit each shell of atoms.

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Shell analysis of the electronic structure in Co icosahedral nanoparticles The variations of the magnetic moment and of the distances (1st nearest neighbors) are occurring from one shell to another in the nanoparticles so that the local electronic properties, in relation to the atomic local environment, can be analyzed by considering values per shell. From Figure 3, it can indeed be clearly seen that, in relation to the symmetric, quasi-spherical icosahedral configuration, the spin magnetic moment presents a rather constant value per shell except for the surface where sites with different coordination numbers coexist. It increases steadily, starting from a much reduced magnetic moment at the central site to an enhanced magnetic moment at the surface sites. This can be related to the variation of the compressive state which is itself related to the reduction of the next nearest distances when going towards the center of the nanoparticles. At the surface, as expected, the local spin magnetic moment increases when the coordination reduces and reaches a maximum at the least coordinated vertex sites. Pursuing this analysis per shell, in a more quantitative way, we have plotted both the average magnetic moment per shell (Figure 5) and the total charge variation per shell (Figure 6). In each case, the initial (not relaxed) configuration has also been added. As already mentioned and well visible in Figure 5, for each size the average magnetic moment increases from one shell of atoms to another, going from the central atom to the surface. At the surface the reduced coordination dominates the enhancement of the magnetic moment. Then one general effect of the relaxation is to reduce the overall interatomic distances relatively to the initial configuration, leading to a reduced local spin magnetic moment, in particular in the core of the two larger nanoparticles which becomes in a strong compressive state. As a matter of fact, from the size of N ≥ 147 atoms, and inside the nanoparticles from the first subsurface shell, the magnetic moment is decreasing well below the bulk value, so that the curve of the inner part of the particle is always under the bulk value which is reached or overcome only from the subsurface. This effect is a particular finite size effect which leads to a very different behavior in a nanoparticle from what could 13



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Figure 5: Icosahedra : average magnetic moment (µB /atom) per shell. The concentric shells (abscissa) are numbered from the central atom (N=1) to the surface (N=2,3,4,5). The horizontal line marks the bulk value of the local spin magnetic moment: (1.62 µB ).

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Figure 6: Icosahedra : total charge variation per shell. The concentric shells (abscissa) are numbered from the central atom (N=1) to the surface (N=2,3,4,5).

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be inferred from a rationale purely based on coordination numbers. It is worth noticing that the central atomic site is in particular subject to a drop of the magnetic moment which is even more marked after relaxation. The central atom is indeed in a particular state of compression (shortened first nearest neighbors) and certainly subject also to the perturbation originating from the surface. This perturbation, owing to its symmetry produces strong deformations in the local density of states which will be presented in the last section of the article. As expected, its influence decreases when the size increases but even so, it acts on a long range distance and the bulk magnetic moment is probably a limit which is reachable only in the core of very large particles. This behavior, already mentioned for very small clusters of a few atoms, 38 is found here to persist in larger nanoparticles of a few hundreds of atoms. Remaining in the shell analysis, a first insight in the electronic state is to quantify the total charge transfers between layers which is done in Figure 6. While in the initial structure the total charge transfers are very limited, in the relaxed structure a very specific behavior appears. The subsurface layer acts as a single donor of electrons transferred to the close layers mainly at the surface and with a less extent to its close inner shells. Exactly the same behavior is already mentioned in the literature for magnetic Pt nanoparticles 39 so that it can be described as a general phenomenon. In addition, as reported in Ref.

39, it does not depend on the shape, at least considering icosahedra,

cuboctahedra and decahedra in the same size range from 13 to 309 atoms, which supports its general character. Concerning the used methods, the work of Ref. 39 has been performed with the DFT code Quantum ESPRESSO. With the DFT SIESTA code we reach the same conclusions for the shell behavior of charge distributions and spin magnetic moments, so that on the methodological point of view, both works are well consistent with each other. However, reduced to the charge transferred per atom, the involved values do not exceed a few hundredths of electron per atom and it is therefore not obvious how the magnetic properties can be accordingly modified in Co systems. As these features have not been studied into detail, they deserve a closer analysis which will be further described at the

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local electronic level, in the section devoted to the local charge analysis in icosahedral Co nanoparticles. It is worth mentioning that, before this, the local spin magnetic moment has been also extracted for each site, the only local particularity being the slightly enhanced magnetic moment at under-vertex sites. The latter are visibly under the influence of the surface, as mentioned before for the surfaces, and this effect is more marked in the case of vertices relatively to facets. In addition, at this stage, no other particular correlation could be found between local magnetic moments and the local structure apart from the decrease of the local magnetic moment with the reduced 1st next neighbors distances and its enhancing with the decreasing atomic coordination at the surface.

Mapping of the magnitude of the local spin magnetic moment in truncated octahedron shaped nanoparticles (TO, Wulff shape) Although the shape transition is not well defined in Co, one can attempt to extend the study to the TO shapes which should become more stable than the icosahedral ones at sufficiently higher size. 35 We take here the case of a TO of 405 atoms which should be enough representative. Anyway, the DFT calculations start to be difficult to handle at larger sizes, in particular relaxations can hardly be carried out in a reliable way. The relaxed nanoparticles are visualized in Figure 7 where the represented atoms are colored following the magnitude of local spin magnetic moment. The highest spin magnetic moment is 1.96 µB at vertex sites and the lowest one is 1.56 µB at the center of the nanoparticle. Concentric weak oscillations, dictated by the presence of the more open (100) facets, 28,40 are visible in the core of the nanoparticle. In the TO studied here, the amplitude of these oscillations is anyway very small (≤ 0.04 µB ) and the average magnetic moment in the core of the nanoparticle is 1.59 µB which is very close to the bulk value (1.61 µB ).

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Figure 7: Nanoparticle with the shape of a truncated octahedron (TO, Wulff shape) containing 405 atoms. The color scale ranges from the minimal to the maximal value of spin magnetic moment (µmin = 1.57 µB and µmax = 1.96 µB ). Local charge analysis in icosahedral Co nanoparticles For a detailed and comprehensive local analysis, we will consider the charge per site and orbital (up, down and total respectively) and finally some projected densities of states (PDOS). The latter will be used to better clarify the relative importance of pure charge transfer and deformations of the PDOS on the modification of the local magnetism. The local charge distributions indeed deserve to be examined in order to detect singularities apart from the previously identified average behavior per shell. The charges per site (spd), decomposed in spin up and spin down contributions, are reported in Figure 8 (the total charge (Nup + Ndown ) is also shown). For the sake of clarity and simplicity in the presentation of the data, only the results for the two larger icosahedra are presented. Particularities in the case of the two lowest sizes, if any, will be presented in the text.

On the total charge curves of Figure 8, it is clear that the subsurface shell presents a lack of charge, mainly a minority spin charge loss (spin down), and the other shells a slight excess of charge equally distributed between minority (spin down) and majority (spin up) populations. One single exception is at the vertex sites which also undergo a charge loss. This effect appears already from the smallest icosahedron with at least 3 shells (55 atoms). 17



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Figure 8: Charge per site (spd) decomposed in spin up (majority spin, red) and spin down (minority spin, blue) contributions for the icosahedra of 147 and 309 atoms. The total charge (sum, right y axis) is represented in green. The x axis indicates the atom number starting from the central atom and following the shell by shell structure of the nanoparticle towards the surface. The vertical lines delimits each shell of atoms. The loss is rather undergone by the minority spin bands since there is clearly an asymmetry between the increase of the majority spin population and the larger decrease of the minority spin one. It therefore contributes, in addition to the coordination effect, to the increase of the magnetic moment while probably to a lesser extent. The other noteworthy fact is that the redistribution between majority and minority populations is very important at the center, in particular for the icosahedron of 147 atoms. On the total population, at this site, an excess of 0.04 e appears for the icosahedron of 147 atoms and of 0.02 e for the icosahedron of 309 atoms. These value are too small to relate them to a significant change in the spin local magnetic moment which is more likely due to the deformations of the LDOS in response to local structural changes and the persistent effect due to the finite size of the particles. The latter is logically more pronounced for the smaller nanoparticle of 147 atoms than for the larger one of 309 atoms. However, it is interesting to have a closer look to the charge redistribution effects by also separating d(up, down) and sp(up, down) contribution per site. 18


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The results are gathered in Figs. 9 and 10. On the overall, the charge variation per site, is the most important for sp populations. At the subsurface sites, the charge variation, obtained by comparison with bulk, reaches on the average −0.16 e/atom for the cluster of 13 atoms (not shown, limit case where the subsurface site is also the center) but it decreases progressively when the size of the nanoparticles increases (it is about −0.04e/atom for the larger size of 309 atoms). For the d population, the charge variations are lower, from −0.02 to +0.01 e/atom, the latter value being reached for the size of 55 atoms (not shown) and does not change afterwards when the size increases. Putting sp and d together, the maximal charge loss per atom at the subsurface of the nanoparticles is therefore evolving rather rapidly, as a function of the size, towards the limit value observed at the surface which is −0.02e/atom. Then, it is worth noticing that some deviations to the mean value, per atom and per shell, are occurring at particular sites like the sites situated along the axis joining the vertices to the center. It is indeed the place where the distance deformations are the most important which favors certainly some charge transfers. However, again, the involved quantities are very small. The electronic distribution presents however a particularity at the least coordinated vertex sites (and at any size of nanoparticle). There the anti-ferromagnetic coupling, observed in extended systems (bulk and surfaces) between sp and and the d electronic distribution, becomes strongly ferromagnetic with a total reversal between the minority spin and majority spin electronic fillings. As the contribution to the magnetic moment becomes now positive, this effect also contributes to enhance the magnetic moment and this, moreover, exclusively at these vertex sites. The effect, at least in the size range of this study, seems independent sp of the size of the nanoparticles, with a value of 0.12 ≤ ∆Nup−down ≤ 0.3, the tendency sp being a decreasing of ∆Nup−down when the size increases. Finally, we go back to the case of

the drop of the magnetic moment at the center of the nanoparticle. After finding that the charge transfers are too small at this place to affect noticeably the local magnetic moment, it remains to take a closer look at the deformation of the PDOS as a consequence to local structural changes. The PDOS at the center of the two largest relaxed nanoparticles (147

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and 309 atoms) are represented in the graph of Figure 11 together with the PDOS in the pure bulk cobalt system. It can be seen that the PDOS undergo strong deformation in the center of the nanoparticles and are rather different from the bulk one. As found previously, the charge transfer to the central site being very small (≤ 0.04 e), only the PDOS deformations should be at the origin of the drop in the magnetic moment at the center of the considered nanoparticles.

5.0 4.8

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Figure 9: Icosahedra : d charge per site decomposed in spin up (majority spin, red) and spin down (minority spin, blue) contributions for the icosahedra of 147 and 309 atoms. The corresponding charge in bulk is represented by an horizontal line: red for spin up and blue for spin down. The x axis indicates the atom number starting from the central atom and following the shell by shell structure of the nanoparticle towards the surface. The vertical lines delimits each shell of atoms.

Local charge analysis in truncated octahedron shaped nanoparticles (TO, Wulff shape) It is now interesting to check how these behaviors scale with larger sizes and a different shape, taking again the case of a TO of 405 atoms. On the overall, charge transfers are never exceeding 0.02 e confirming the low impact of charge transfers in the magnetic properties 20


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0.8

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0.6

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Figure 10: Icosahedra : sp charge per site decomposed in spin up (majority spin, red) and spin down (minority spin, blue) contributions for the icosahedra of 147 and 309 atoms. The corresponding charge in bulk is represented by an horizontal line: red for spin up and blue for spin down. The x axis indicates the atom number starting from the central atom and following the shell by shell structure of the nanoparticle towards the surface. The vertical lines delimits each shell of atoms.

Figure 11: Spin up and down contributions to the local (projected) densities of states at the central site of the relaxed icosahedra of 147 (dotted black line) and 309 (dashed blue line) atoms respectively. The energies refer to the Fermi energy EF . The red curve corresponds to the bulk material reference.

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of ferromagnetic nanoparticles. Then, the characteristics of the subsurface layer, acting as a single donor of electrons transferred to the close layers, is not found anymore. The only two sites presenting a charge loss are eventually the vertex and their underlying neighbors in the subsurface with a charge loss reaching 0.06 e for under-vertex sites. Then, as for the smaller icosahedral nanoparticles, at the single least coordinated vertex sites, the antiferromagnetic coupling, observed in extended systems (bulk or surfaces) between sp and d electronic distribution, becomes strongly ferromagnetic with a total reversal between the minority and majority spin electronic fillings.

Conclusions Within both a shell and a local fine analyses of DFT calculations, we have unraveled the effects of local structural environment including atomic coordination, structural deformations and finite size effects in cobalt ferromagnetic nanoparticles. In particular different patterns of the local spin magnetic moment are evidenced. They depend strongly on the shape of the nanoparticle and the relaxation effects, while charge transfers between sites are very limited.

Co icosahedra containing 13 to 309 have first been considered. For each size, the average magnetic moment decreases in each shell of atoms, from the surface to the center. Therefore a shell analysis of the quantities obtained from SIESTA-DFT calculations has been performed. Concerning the charge analysis, a particular effect of donor-acceptor behavior between the sublayer and its close environment, in particular to the surface layer, is observed in ferromagnetic Co icosahedra. It can be considered as a general effect, since it has already been obtained in weakly magnetic nanoparticles of Pt. 39 However, while in Ref. 39 the magnetic behavior is dictated by these charge transfers, it is definitely not the case for particles made of initial magnetic elements like Co where inter-sites charge transfers can be considered negligible to describe the magnetic behavior. Then a particular magnetic behavior is observed

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exclusively at vertex sites with the reversal from an anti-ferromagnetic coupling (bulk behavior) to a ferromagnetic one for sp electrons. The particular compressive state present in icosahedral nanoparticles leads to a redistribution between up and down electronic populations leading to a drop of the magnetic moment. Since the electronic transfer is too weak, the drop of the local magnetic moment is the consequence of important deformations of the local densities of states through hybridizations due to the important structural deformations.

In the attempt to extend the rationale on the scaling of magnetic properties with both the increasing size of nanoparticles and shape transition, we have given an overview of the same type of analysis in the case of a TO nanoparticle containing 405 atoms. The cartography of the local spin magnetic moments is different from the one of the icosahedron, since in the core, oscillations of weak amplitude around the bulk value (< 0.04 µB ) are observed. Then some characteristics are well preserved on the whole range of sizes and shapes, like : the weak charge transfer between atoms in the particles, the maximal charge loss at vertex and under-vertex sites and finally the reversal from an anti-ferromagnetic coupling to a ferromagnetic one for sp electrons of vertex sites exclusively. The latter effect leads to an increased local magnetic moment at these sites. Then conversely to icosahedral nanoparticles, the particular donor-acceptor shell behavior observed in icosahedra between subsurface and surface shells is, in TO nanoparticles, limited to the under-vertex and vertex sites.

Connecting, in this way, the different contributions between structural changes and electronic structure in magnetic nanoparticles enables to extend the transferability of DFT calculations towards realistic physico-chemical environments of magnetic nanoparticles and therefore to contribute to a better choice of systems with targeted properties either magnetic or catalytic. In the aim of a further multi-scale approach, the important result on the local neutrality per site and orbital (no charge transfer) is fundamental for the implementation of self-consistent semi-empirical algorithms 41 for the optimization of the structure of larger

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particles having sizes inaccessible within first principle calculations.

Acknowledgement The authors thank the High Performance Computing center of the University of Strasbourg for supporting this work by providing scientific support and access to computing resources. Part of the computing resources were funded by the Equipex Equip@Meso project (Programme Investissements d’Avenir) and the CPER Alsacalcul/Big Data. Part of the calculations were also performed thanks to the ”French Institut du Développement et des Ressources en Informatique Scientifique” through a grant of computer time (Project No. 100525).

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Graphical TOC Entry

Magnetic nanoparticles show specific patterns of the magnitude of the local spin magnetic moment, depending on their size and shape.

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Unravelling Finite Size Effects on Magnetic Properties of Cobalt

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