Electronic and magnetic structure of bulk cobalt: The

THE JOURNAL OF CHEMICAL PHYSICS 133, 024701 共2010兲

Electronic and magnetic structure of bulk cobalt: The ␣, ␤, and ε-phases from density functional theory calculations Víctor Antonio de la Peña O’Shea,1,2,a兲 Iberio de P. R. Moreira,3 Alberto Roldán,3,4 and Francesc Illas3 1

Thermochemical Processes Group, Instituto Madrileño de Estudios Avanzados en ENERGIA, C/Tulipán s/n 28933, Móstoles, Madrid, Spain 2 Departament de Química Inorgànica and Institut de Nanociència i Nanotecnologia (IN2UB), Universitat de Barcelona, 08028 Barcelona, Spain 3 Departament de Química Física and Institut de Química Teòrica i Computacional (IQTCUB), Universitat de Barcelona, C/Martí i Franquès 1, 08028 Barcelona, Spain 4 Departament de Química Física i Inorgànica, Universitat Rovira i Virgili, C/Marcel lí Domingo s/n, 43007 Tarragona, Spain

共Received 21 March 2010; accepted 9 June 2010; published online 8 July 2010兲 The geometric, electronic and magnetic properties of the three metallic cobalt phases: hcp共␣兲, fcc共␤兲, and epsilon共␧兲 have been theoretically studied using periodic density functional calculations with generalized gradient approximation 共GGA兲 and plane wave basis set. These results have been compared with those obtained with GGA+ U approach which have shown a noticeable improvement with regard to experimental data. For instance, the cohesive energy values predicted by GGA are overestimated by ⬃25%, whereas GGA+ U underestimate them by 14%–17%. On the other hand, magnetic moment values are underestimated in GGA while are overestimated for GGA+ U approach by almost the same amount. Besides, the introduction of U parameter gives rise to an electronic redistribution in the d-band structure, which leads to variations in the magnetic properties. Moreover, a higher attention has been paid in the study of the electronic and magnetic properties of the ␧-phase that has not described previously. These studies show that this phase posses special properties that could lead to an unusual behavior in magnetic or catalytic applications. © 2010 American Institute of Physics. 关doi:10.1063/1.3458691兴 I. INTRODUCTION

Cobalt based materials have a great interest in academic and industrial fields due to their chemical, magnetic, and electronic properties thank to which have potential applications as magnetic data storage1,2 and catalysis3 among others. Metallic cobalt can crystallize in three different crystal structures: Hexagonal closed-packed 共hcp兲 共␣-phase兲, facecentered cubic 共fcc兲 共␤-phase兲 and primitive cubic phase 共␧-phase兲.1,2,4 These three phases possess similar energetic stabilities and, hence, small temperatures or pressure variations give rise to changes in the crystal phase. The similar stability also renders theoretical predictions from first principles difficult either for bulk or nanoparticles. During the past years several groups have concentrated their efforts on the preparation and characterization of nanostructured systems derived from these bulk phases, mainly due to their interesting magnetic and catalytic properties. These studies have shown a strong correlation between crystal structure and magnetic properties in Co-based materials.1,2 As mentioned above, all cobalt metallic phases possess interesting magnetic properties 共in all cases the ferromagnetic structure is the most stable phase兲, mainly due to the room temperature anisotropy of the ␣-Co 共Ref. 5兲 phases, ␤-Co,6 and ␧-Co,7 whose values are: 4.2⫻ 106, 2.7⫻ 106, and 1.5⫻ 106 erg/ cm3, respectively. While the highest ana兲

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isotropic magnetic coercivity of the ␣-Co phase makes it preferred for magnetic recording applications, the ␤-Co phase is useful for applications as a soft magnetic material due to its symmetrical low coercivity. Although ␣-Co and ␤-Co phases can coexist at different conditions, the ␣-Co phase is more stable at room temperature and ambient pressure; while ␤-Co is a metastable phase formed at temperatures above 450 ° C.8,9 Both phases are based on a closepacking of atoms although they differ in the stacking sequence of the 关111兴 plane. The low energy required for stacking fault formation gives rise usually to the presence of both phases in the same sample. The ␧-cobalt phase possesses a more complex structure, which was reported by Dinega et al.10 The synthesis of the ␧-Co phase has been only possible by means of solution-phase chemistry processes. This is a metastable phase and its transformation to ␣-Co or ␤-Co structures can be easily achieved by annealing the sample at a suitable temperature.1,10 This ␧-structure is considered as a soft magnetic material 共as ␤-Co phase兲 and its magnetic properties favor the formation of ordered films with applications in magnetic recording. In the past few years, ␧-Co has acquired a great relevance due to the fact that it seems to be a good precursor to obtain ␣-Co nanoparticles for magnetic storage uses. The thermal-magnetic stability determined by the Curie temperature 共Tc兲 of the ␧-phase has not been reported due to the fact that ␧-Co 共as

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␣-Co兲 phase changes its structure to ␤-Co phase before achieving a paramagnetic structure. The Tc for a ␤-Co phase, under ambient pressure, is 1388 K.11,12 Furthermore, recent studies have confirmed a strong correlation between Co phase and catalytic activity. In fact, previous works reported that the behavior of cobalt-based supported catalysts in Fischer–Tropsch synthesis depends on the characteristics of the metallic phase 共i.e., fcc or hcp兲.3,13 These results are also confirmed by preliminary theoretical studies14 showing that the Fischer-Tropsch Synthesis 共FTS兲 reaction is structure sensitive. Calculations using density functional theory 共DFT兲 based methods with different exchange correlation potentials such as local spin density approximation 共LSDA兲 and generalized gradient approximation 共GGA兲 and including explicitly spin polarization, have been previously reported and addressed the study of the structural and magnetic properties of mainly ␣-Co and ␤-Co phases.15–20 It is well known that while the LSDA method provides a good description for many moderately correlated systems,17,21 it presents some shortcoming when applied to strongly correlated ones, especially in the case of Fe.22 These previous studies and the fact that cobalt systems posses large correlation effects, what makes difficult the representation of their magnetic properties. Thus, it is necessary to further explore the capability of the GGA method and to investigate alternative methods beyond the standard LDA or GGA implementations of density functional theory.23 Here, we examine the performance of methods that modify the on-site Coulomb repulsion, via ad hoc inclusion of a Hubbard effective U term, which have shown to provide quite accurate description of the electronic structure and properties of several transition metal and rareearth systems.24–28 In this work, we present an analysis of the energetic and magnetic properties associated to the different cobalt crystal phases as predicted from periodic DFT calculation, using both GGA and GGA+ U methods with plane wave basis set and focusing mainly in properties of bulk ␧-Co which have not been previously discussed in the literature. II. COMPUTATIONAL DETAILS

Spin polarized periodic DFT based calculations with two different exchange-correlation potentials have been carried out for the ␣-Co, ␤-Co, and ␧-Co phases of bulk metallic crystalline Co using the VASP program.29,30 The initial unit cells parameter were always constructed from the experimental lattice parameters, that were then fully optimized using the procedure described below. For the ␣-Co, ␤-Co, and ␧-Co the unit cell involves 4, 2, and 20 metal atoms, respectively. The total energies corresponding to the optimized geometries were calculated using the spin polarized version of the PW91 implementation of the GGA exchange correlation functional.31,32 To properly describe the magnetic behavior of all of these cobalt phases an accurate treatment of the electron correlation in the localized d orbital is crucial. Thus, in order to correct the description of the Co 3d electrons we also employed the GGA+ U approach33 in its rotational invariant form34 with an effective interaction parameter Ueff

J. Chem. Phys. 133, 024701 共2010兲

= U-J for Co; note that both U and J represent atomic effective parameters33 although the final results only depend on the Ueff value. The choice of Ueff is obviously an issue since it represents an external input in the calculations; a possibility is obviously to choose Ueff to fit experimental data as done to simultaneously describe CeO2 and Ce2O3.25 Nevertheless, it is important to be aware of the fact that Ueff can depend on the bulk phase studied and even in the calculated property. Thus, Anisimov et al.35 used Ueff = U-J= 7.8− 0.92 = 6.88 eV although this corresponds to LaCoO3 at the LDA+ U level and, hence, it is larger than the Ueff value for GGA+ U.25 A slightly smaller value of Ueff = 6.1 has been used by Wdowik and Parlinski in their study of CoO.36 A value of Ueff = 4.08 eV has been obtained by Wang et al. performing FLAPW calculations within the GGA functional on Co-doped SnO2.37 The same authors use Ueff = 2.4 eV in their GGA and GGA+ U study of Co under high pressure.38 Zhang et al. have used LDA+ U to study Co-doped zincblende ZnO.39 These authors explored the effect of Ueff for values between 3 and 6 eV and concluded that values larger than 3 eV do not lead to significant differences. Moreover, Vega et al.40 studied the influence of electron correlation of the temperature dependent electronic structure of ferromagnetic fcc cobalt structure using Ueff value of 3,2 eV that was explicitly obtained for these systems. In the present work, an effective UCo value of 3 eV has been used which is close to the value used for bulk Co 共Refs. 38 and 40兲 and on the range of values above described. The valence one-electron Kohn–Sham states have been expanded on a plane wave basis with a cutoff of 415 eV for the kinetic energy and the effect of the core electrons on the valence electron density, defined by the Co 3d74s2 electrons, has been taken into account through the projector augmented wave 共PAW兲 method41 as implemented by Kresse and Joubert.42 The total energy threshold defining self-consistency of the electron density was set to 10−4 eV; the convergence criterion for structural optimization was set to less than 10−3 eV for the difference of total energy from consecutive geometries ensuring that forces on all atoms are less than 0.3 eV/nm. A Gaussian smearing technique with a 0.2 eV width has been applied to enhance convergence but all energies presented in the following have been obtained by extrapolating to zero smearing 共0 K兲. Integration in the reciprocal space was carried out using the Monkhorst–Pack43 sampling of the Brillouin zone, after evaluation of several mesh of special k-points and a special 20⫻ 20⫻ 20 k-point mesh were used in other to obtain the desired accuracy in the calculated energies. Charge and spin density analysis have been performed by atomic sphere method with a atomic radius of 2.460 Å. Before ending this section, it is worth pointing out that the complex magnetic properties of Co discussed in the introduction may require to explicitly introduce relativistic effects. Although this is certainly a possibility, it is important to be aware of the fact that the complicated relativistic manybody effects are usually included in the Kohn–Sham formalism via energy functionals analogous to the exchangecorrelation functionals of the nonrelativistic formalism.44 While this is an appealing procedure, there is compelling

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Properties of ␣, ␤, and ␧-Co phases from DFT

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TABLE I. Lattice parameter 共a兲, total energy per cobalt atom 共E兲, energy per volume unit 共⌬E / V兲, cohesive energy 共Ecoh兲, average atomic spin density 共␴兲, and Fermi energy 共EF兲 as predicted by GGA and GGA+ U density functional theory based methods for the hcp, fcc, and ␧ Co bulk phases.

GGA

GGA+ U

Expt.

Phase

aa 共Å兲

E 共eV/ Coatom兲

⌬E / V b 共KJ/ mol A3兲

Ecoh 共eV兲

hcp 共␣兲 fcc 共␤兲 ␧共FM兲 ␧共NM兲 hcp 共␣兲 fcc 共␤兲 ␧共FM兲 ␧共NM兲 hcp 共␣兲 fcc 共␤兲 ␧

1.642 3.518 6.057 6.057 1.642 3.548 6.128 6.128 1.633 3.545 6.097

⫺7.01 ⫺6.99 ⫺6.96 ⫺6.69 ⫺4.55 ⫺4.67 ⫺4.58 ⫺3.73 ¯ ¯ ¯

0.00 0.57 2.02 4.38 0.00 0.20 2.13 9.28 ¯ ¯ ¯

⫺5.49 ⫺5.48 ⫺5.45 ⫺4.99 ⫺3.66 ⫺3.79 ⫺3.70 ⫺2.74 ⫺4.39d ¯ ¯

␴ 共e− / atom兲

c

1.61 1.64 1.65 ¯ 1.83 1.87 1.91 ¯ 1.72d 1.75e 1.70f

EF 共eV兲 5.43 5.45 5.30 5.07 5.49 5.29 5.15 4.31 ¯ ¯ ¯

Correspond to a cell parameter for fcc and epsilon phase and b = c / a for hcp phase. ⌬E / V = 共E␣ / V␣−Ex / Vx兲, where x is the Co phase. c Spin density assigned to each atom can be consider an estimation of the spin only magnetic moment for this atom. The relation between atomic spin density and the magnetic moment of a given atom 共or ion兲 is not straightforward. d Reference 48. e Reference 52. f Reference 10. a

b

evidence that we are still far from having an accurate universal exchange-correlation functional and that it is likely that the error introduced by this functional is larger than the one originated by the neglect of relativistic effects. Consequently, these studies are based on the application of standard spin polarized DFT and the only additional relativistic corrections are those entering into the PAW potential to describe the effect of the core electrons on the valence shells without including noncollinear magnetism effect even if this is possible through the PAW formalism.45 Consequently, only the spin contribution to the magnetic moment has been included. For systems such as bulk Co with an almost filled d shell this is a reasonable choice, whereas noncollinear magnetism is more likely to occur in small magnetic clusters.45 Note also that in the absence of spin orbit interaction, where there is no contribution from orbital to magnetic moment, the spin directions are not linked to the crystalline structure and, therefore, the system is invariant under a general common rotation of all spins. III. RESULTS AND DISCUSSION

First, a brief description of the optimized crystal structure and properties for each of the three Co phases 共␣, ␤, and A

B

␧兲 studied as described by GGA and GGA+ U methods is presented. Not surprisingly, the GGA values for the ␣ and ␤ phases are in good agreement with previous works15–19 and have been discussed recently.20 Therefore, we will omit a direct comparison with these previous studies and will focus only on the results for the ␧-phase and on the new GGA + U set of data. The whole set of calculated lattice constants, stability energy per Co atom and per volume unit 共⌬E / V兲, cohesion energy 共Ecoh兲, total magnetic moments 共␮B兲 estimated from the spin density and Fermi energy 共EF兲 are summarized in Table I. These quantities are useful to examine the relative stability of the Co phases although one must bear in mind that the results obtained correspond to 0 K. Temperature effects could be taken into account by appropriate molecular dynamics simulations. However, these have not been considered in the present work. To carry out the comparison of the optimized results obtained for ␣-Co phase the axial ratio c/a has been used. This value is the same for both GGA and GGA+ U methods 共1.642 Å兲 and is slightly larger than the values found in the literature 共1.622 Å兲.46 This increase in the c / a value could be TABLE II. Local coordination and nearest-neighbor distances for the two type of Co atoms in ␧-cobalt phase. Atom

Near neighbors’

GGA

GGA+ U

Expt.a

Type I

Type I 共3兲 Type II 共3兲 Type II 共3兲 Type II 共3兲 Type I 共2兲 Type I 共2兲 Type I 共2兲 Type II 共4兲 Type II 共2兲

2.295 2.432 2.557 2.590 2.432 2.557 2.559 2.518 2.613

2.307 2.469 2.590 2.599 2.469 2.590 2.599 2.572 2.582

2.281 2.487 2.543 2.587 2.487 2.543 2.587 2.544 2.580

C

Type II

FIG. 1. Sketch view of different bulk cobalt structures: 共a兲 hcp or ␣, 共b兲 fcc or ␤, and 共c兲 ␧ 共including a detail of the different cobalt position兲.

a

Reference 10.

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A

B

-10

GGA

Density of states

Density of states

GGA

-5

0

E-EF (eV)

5

-10

-5

E-EF (eV)

0

GGA+U

GGA+U Density of states

Density of states -10

5

-5

0

E-EF (eV)

5

-10

-5

C

E-EF (eV)

0

5

Density of states

GGA

-10

-5

E-EF (eV)

0

5

Density of states

GGA+U

-10

-5

E-EF (eV)

0

5

FIG. 2. Total 共straight line兲, Co-4s 共gray pattern兲, and Co-3d 共dashed pattern兲 for Density of states calculated using GGA and GGA+ U methods for ␣ 共a兲, ␤ 共b兲, and ␧ 共c兲 phases. Dashed line indicates the Fermi level. All curves are represented for spin up and down contributions

due to the hcp domain that could be deformed to better match the fcc staking. In an ideal hcp structure the b value is 1.633 Å and coincides with an fcc structure observed along the 关111兴 direction. In the case of ␤-Co phase GGA and GGA+ U calculations predict lattice parameter values of 3.518 and 3.548 Å, respectively. Both values are in agreement with the experimental one 共3.545 Å兲 共Ref. 47兲 being the GGA+ U value closer to experiment. In the case of ␧-Co phase, 6.057 Å 共GGA兲 and 6.128 Å 共GGA+ U兲 values were

obtained for the lattice parameter, which are close to the experimental value of 6.097 Å.46 The small differences are considered within the method error 共⬃2%兲. For the ␣-Co phase, the predicted nearest neighbors distances are 2.478 Å 共GGA兲 and 2.487 Å 共GGA+ U兲 that are closer to the experimental value of 2.497 Å. Whereas for the for ␤-Co phase we find 2.488 Å 共GGA兲 and 2.509 Å 共GGA+ U兲, being the experimental value 2.506 Å. The ␧-Co phase is a very special

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Properties of ␣, ␤, and ␧-Co phases from DFT

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TABLE III. Analysis of the atomic spin density for each atom in the ␣-Co structure as predicted from GGA and GGA+ U calculation. The decomposition of the total spin density in atomic components is obtained by proper projection.

␴Co 共e− / atom兲 GGA Atom

4s

4p

3d

Tot

4s

1 2

⫺0.01 ⫺0.01

⫺0.05 ⫺0.05

1.67 1.67

1.61 1.61

⫺0.02 ⫺0.02

case since, compared to the other cobalt phases, exhibits different structural properties. The ␧-Co, described by Dinega et al.10 posses a cubic structure 共space group P4132兲 with a unit cell similar to that of ␤-manganese.46 This structure contains 20 cobalt atoms, per unit cell, divided in two types: Eight atoms of type I and twelve atoms of type II 关Fig. 1共c兲兴. In Table II, the nearest neighbor’s distances for each cobalt atom type are reported. The calculated total energy per cobalt atom shows that, for GGA calculations, the ␣-Co phase is more stable than the ␤-Co and ␧-Co phases. However, the GGA+ U calculations show that the ␤-Co phase has a lower energy per Co atom than ␣-Co and ␧-Co phases. However, the structural stability should be expressed as energy per cell volume 共Ex / Vx, where x is Co-phase兲 and, in both GGA and GGA+ U, the calculated ⌬E / V 共E␣ / V␣ − Ex / Vx, where x is Co-phase兲 values show that the ␣-Co structure is the most stable, as expected from experimental studies 共Table I兲. The cohesive energy 共Ecoh兲 provides an alternative important way to evaluate the stability of the cobalt phases and the accuracy of the computational method. The Ecoh calculated values in Table I are similar to the experimental value.48 In fact, for all cobalt phases the GGA calculations furnish values of the cohesive energy that overestimate the experimental value by ⬃25%, while the GGA+ U method underestimate this property by 14%–17%. These data illustrate that GGA and GGA+ U methods, within the form of one of the most widely used functional such as PW91 provides a rather approximate description of the binding between heavy-element atoms,49 a feature exhibited also by other GGA functionals.50 Next, let us discuss the main features of the electronic structure of the different Co phases. Upon inclusion of the on-site Hubbard electron correlation correction to GGA parameter 共UCo兲, the calculated value of the Fermi level 共EF兲 shifts and the variation depends on the cobalt phase. For the fcc and epsilon phases EF increases by 0.16 and 0.15 eV, respectively, whereas for the hcp phase it decreases by 0.06 eV. This behavior could be explained thanks to the fact that the inclusion of U parameter leads to an expansion of the filled and semifilled subbands, which could also explain the energy stabilization previously observed in these phases. Additional information about the electronic structure of these Co bulk phases can be obtained from the total 共spin up and down兲, Co 共4s兲, and Co 共3d兲 density of states 共DOS兲 obtained for GGA and GGA+ U solutions shown in Fig. 2.

GGA+ U 4p 3d ⫺0.08 ⫺0.08

1.93 1.93

Tot 1.83 1.83

Previous theoretical studies have revealed that the 3d-band has a complex structure divided in five nondegenerate subbands.40 The complete or quasicomplete filled bands are magnetically inactive, while upper bands provide the magnetic behavior to this system. Present study shows that all three phases exhibit related features with concomitant similarities in their electronic properties, although a detailed analysis of the DOS profiles also reveals some differences in the electronic structure of the different cobalt phases. Thus, in the GGA calculations, the density of states for the d-band in epsilon phase is broader than in fcc and hcp structures. Besides, the electron distribution density around Fermi level is slightly higher for epsilon phase than for fcc phase and lower for hcp. These data corroborate the magnetic values obtained. On the other hand, the calculations performed using the Hubbard correction lead to similar DOS profiles, although a shift in the energy of the up spin states toward more negatives values is observed. Obviously, the inclusion of the U parameter leads to a variation in the 3d band structure in all cases, giving rise to a redistribution of the density of states in the subbands, leading to an increase in the spin down density that corroborates the enhancement observed in the average atomic spin density 共␴兲. These energy gaps are different for each structures being higher for epsilon phase. One of the main properties of these systems is their magnetic behavior. Experimental studies have shown that all metallic cobalt phases have a stable structure which is strongly ferromagnetic at ambient pressure. As previously commented, the magnetic properties of ␣ and ␤ phases has been studied, corroborating that the hcp and fcc nonmagnetic 共NM兲 phases are less stable than the ferromagnetic 共FM兲 one.15,20 In the case of epsilon phase this kind of calculations has not been previously evaluated. Our studies show that, as found for the fcc and hcp structures, the magnetic phase is more stable than the NM phase 共Table I兲. Moreover, the stability and cohesion energy differences in both NM and FM are higher in the GGA+ U calculation, which indicates that the use of the Hubbard algorithm leads to a destabilization of the NM phase. On the other hand, the magnetovolume effect studies in the ␧-phase show a stabilization of the NM phase with the volume decrease. Thus, our calculations show that these phases lose their magnetic properties into a nonmagnetic structure. Two self-consistent stable structures, with a volume of 214.17 Å3 共GGA兲 and 204.99 Å3 共GGA+ U兲 were obtained. Experimental data that corroborate this behavior

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TABLE IV. Analysis of the atomic spin density for each atom in the ␤-Co structure as predicted from GGA and GGA+ U calculation. The decomposition of the total spin density in atomic components is obtained by proper projection.

␴Co 共e− / atom兲 GGA Atom

4s

4p

3d

Tot

4s

1 2 3 4

⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01

⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05

1.70 1.70 1.70 1.70

1.64 1.64 1.64 1.64

⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01

GGA+ U 4p 3d ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06

1.95 1.95 1.95 1.95

Tot 1.87 1.87 1.87 1.87

DOS analysis which illustrates that the use of GGA+ U methodology leads to an increase of the number of states around of Fermi level 共Fig. 2兲. The obtained spin density per atom 共␴Co兲 for the different phases and its decomposition in atomic symmetries is summarized in Tables III–V. For the hcp and fcc phases the ␴Co values for all atoms are identical with values of 1.61 for GGA and 1.87 for GGA+ U for the fcc phase and slightly higher values for hcp which are of about 1.64 for GGA and 1.87 for GGA+ U. These values are quite close to those reported in previous studies using similar approaches.18,19 In the case of epsilon phase different ␴Co values are obtained in function of the atom type. The eight atoms of type I exhibit a lower ␴Co value while for the twelve atoms of type II the atomic spin density is larger 共Table V兲. Unfortunately, for the ␧ phase comparison with previous studies is not possible because they are inexistent. Note also that in the case of Co

have not been found. However, a similar behavior, for the other Co phases, was previously described from experimental51 and theoretical20 studies. The calculated average atomic spin density 共␴兲 obtained for each phase from GGA and GGA+ U are also summarized in Table I and are in agreement with experimental magnetic moments.52 Nevertheless, the ␴ values obtained from the GGA method are underestimated with respect to experimental results; however, when the Coulomb correction is used these values are overestimated. Moreover, the average atomic spin density of the ␧-Co phase presents the highest value in both GGA and GGA+ U calculation. These results are corroborated by the examination of spin density values over each atom and projected on the 4s, 4p, and 3d states 共Tables III–V兲, where an occupation increase in the d band is observed when the GGA+ U method is used. These results are in agreement with the electronic population observed in

TABLE V. Analysis of the atomic spin density for each atom in the ␧-Co structure as predicted from GGA and GGA+ U calculation. The decomposition of the total spin density in atomic components is obtained by proper projection.

␴Co 共e− / atom兲 GGA Atom

4s

4p

3d

Tot

4s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

⫺0.02 ⫺0.02 ⫺0.01 ⫺0.01 ⫺0.02 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01

⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05 ⫺0.05

1.67 1.66 1.66 1.64 1.66 1.65 1.66 1.64 1.72 1.71 1.65 1.66 1.76 1.76 1.73 1.73 1.84 1.84 1.77 1.77

1.59 1.59 1.59 1.57 1.59 1.58 1.58 1.57 1.66 1.65 1.59 1.60 1.70 1.70 1.68 1.67 1.78 1.78 1.72 1.71

⫺0.02 ⫺0.01 ⫺0.02 ⫺0.02 ⫺0.02 ⫺0.02 ⫺0.02 ⫺0.02 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01 ⫺0.01

GGA+ U 4p 3d ⫺0.08 ⫺0.08 ⫺0.08 ⫺0.08 ⫺0.08 ⫺0.08 ⫺0.08 ⫺0.08 ⫺0.07 ⫺0.07 ⫺0.07 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06

1.97 1.97 1.97 1.96 1.97 1.96 1.96 1.95 2.03 2.00 2.01 2.01 2.03 2.02 2.01 2.01 2.02 2.02 2.01 2.01

Tot 1.87 1.88 1.88 1.86 1.88 1.87 1.87 1.86 1.95 1.92 1.93 1.94 1.95 1.94 1.94 1.94 1.95 1.94 1.93 1.93

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Properties of ␣, ␤, and ␧-Co phases from DFT

 -Co

 -Co

 -Co

Type II

GGA

GGA+U

Type I

FIG. 3. Schematic representation of the 共ELF兲 for the ␣-Co hcp; ␤-Co fcc and ␧-Co epsilon phases as obtained from GGA 共left兲 and GGA+ U 共right兲.

nanoparticles different types of Co atoms will be present at the surface depending on the crystalline structure of the particle. This fact can lead to different magnetic behavior as a function of the particle shape. The different behavior of magnetic and catalytic properties of Co nanoparticles with relation to its shape and phase are actually under study. Figure 3 shows plots of the electron localization function 共ELF兲 for the different phases as obtained by the GGA and GGA+ U methods. In the case of hcp and fcc structures, the ELF shows an homogenous electron distribution. The main difference between these two structures is that hcp phase shows a radial electron distribution while the fcc phase shows an evident polarization. However, in the case of ␧-Co phase a different ELF distribution is observed depending on the atom type, which one can tentatively relate to the magnetic behavior of this phase. It is also important to point out that, for the three phases, the ELF plots in Fig. 3 corresponding to calculations with the GGA+ U potential, show a more localized character than in the case of the GGA calculations, as expected from the well known trend of GGA+ U to localize the electron density of the levels where the correction is applied. IV. CONCLUSIONS

The lattice parameter and structural parameters of the fcc, hcp, and ␧ phases of bulk Co have been obtained at the GGA and GGA+ U levels of DFT. For the hcp and fcc phases, the calculated values are in good agreement with

J. Chem. Phys. 133, 024701 共2010兲

experimental data and previous GGA calculations. Nevertheless, the GGA+ U values reported in the present work seem to be more accurate and closer to the experimental ones. The cohesive energy values predicted by these density functional methods are also near to the experimental data. However, GGA overestimates this property by ⬃25% and GGA+ U underestimate it by 14%–17%. The density of states for the different phases presents some marked differences. In the GGA calculation, the electron distribution density around Fermi level is slightly higher for epsilon than for the fcc phase; and lower for the hcp one. On the other hand, the GGA+ U calculation shows a shift in the energy for the up spin states toward more negatives values. This energy shift is higher for epsilon phase and leads to an increase in the average atomic spin density 共␴兲. The study of the magnetic properties shows that the three cobalt phases exhibit a stable ferromagnetic state. However, the ␴Co values obtained from GGA method are smaller than the experimental values while the use of the Coulomb correction overestimates these values. Interestingly enough, the calculated atomic spin density values for the ␧-Co phase are the highest. Besides, in the ␧-Co phase, the ␴Co values depend on the cobalt atom type, a feature which for cobalt nanoparticles can lead to different magnetic behavior as a function of the particle shape. Moreover, the present results show that each cobalt phase, in particular the ␧ one, has inherent structural and electronic properties that could help to understand the different behavior in magnetic storage or catalytic application as H2 production or Fischer–Tropsch synthesis. In summary, density functional calculations within the GGA method allows reaching a rather detailed description of the atomic and electronic structure of the ␣, ␤ phases of bulk Co with experimental values and with previous theoretical studies. For these two phases, comparison with experimental data indicates that the GGA+ U values are more accurate. The precise description of the hcp and fcc Co bulk phases strongly suggest that the present description of the ␧ phase; which has not been previously described, is also equally accurate. ACKNOWLEDGMENTS

V.A.P.O. acknowledges financial support from the MICINN in the Ramon y Cajal Program. A.R. thanks Universitat Rovira i Virgili, for supporting his pre-doctoral research. Financial support has been provided by Spanish MICINN 共Grant Nos. FIS2008-02238, CTQ2008-06549-C02-01, and ENE2009-09432兲 and in part by Generalitat de Catalunya 共Grant Nos. 2009SGR1041, 2009SGR462, and XRQTC兲. Computational time has been provided by the Centre de Supercomputació de Catalunya 共CESCA兲 by generous grants from Universitat de Barcelona and CIRIT. 1

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