A new family of star-like icosahedral structures for small cobalt clusters F. Aguilera-Granja a , Andrés Vega b , Luis Carlos Balbás b,⇑ a Instituto de Física, Universidad Autónoma de San Luis Potosí, 7800 San Luis Potosí, Mexico b Departamento de Física Teórica, Atómica, y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain article info Article history: Received 29 October 2012 In final form 30 December 2012 Available online 5 January 2013 Keywords: Density functional theory Transition-metal clusters Electronic properties Geometrical properties abstract Using a fully unconstrained version of the density-functional method with the generalized gradient approximation to exchange and correlation, we investigate the electronic structure and related properties of free-standing Co 1326 atomic clusters with star-like geometries belonging to an unexpected icosahedral growth pattern. Our results for the binding energies, ionization potentials and electron affinities, provide compelling evidence of the preference for this new family of star-like structures of Co N clusters in the range N = 20–25 instead of the planar hcp- or fcc-based structures proposed so far as the ground state. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction Atomic nano-clusters have a wide range of possible technolog- ical applications such as the design of magnetic storage and cata- lytic devices, due to which considerable efforts are undertaken in order to characterize and to understand their geometrical and elec- tronic properties. The geometrical structure of free-standing clus- ters is difficult to be characterized since most techniques used in bulk-like systems are not suitable if the cluster is not supported on a host. Therefore, in the free-standing environment, alternative techniques have emerged to find plausible geometries. For in- stance, one can take advantage of the adsorption of light molecules on the surface of the cluster in order to get some indirect informa- tion about the geometrical shape [1]. In addition evidence was found for Co 18 , Co 19 , and Co 21 , for more than one isomer at low temperature [2]. Since this kind of experimental information is inconclusive, the combination of first-principles total energy calcu- lations with experimental techniques like infrared spectroscopy [3], trapped ion electron diffraction (TIED) [4], and with experi- mental data on ionization potential (IP) [5,6], electron affinity (EA) [7] or magnetic moment [8,9], is a good strategy to better determine the geometrical structures. Indeed, a good indicative of the plausibility of the structural and electronic configurations of pure clusters is provided by the comparison of the IP and EA with those experimentally measured. Such a comparison was made for iron clusters more than ten year ago by part of the authors in the framework of a tight-binding model [10]. In the case of Co clusters, recent DFT calculations [12,11] have shown that the planar-like structures based on fcc and/or hcp mo- tifs are the best candidates for the ground state in the small size regime. It has been also pointed out that the hybridization together with the magnetic contribution are driving forces for stabilizing those planar-like structures. In this work, we investigate a different family of structures, based on an unexpected icosahedral growth pattern. Our interest in this new family of star-like geometries comes from the fact it has been recently shown to lead to more sta- ble structures for Pd 13 and Cu 13 than any other type, including the planar-like ones [13,14]. Being Cu and Pd late transition-metal clusters with more than half-band filling, as it is Co, the new family of structures deserves to be investigated also for Co clusters, and to benchmark it against the planar-like family established so far. To this purpose, we have conducted DFT calculations of the electronic structure and related properties (total energy, ionization potential, electron affinity, magnetic moment) of Co N clusters (with N ¼ 13 to 26) with reoptimized geometries belonging to both families of structures. In addition to these planar and icosahedral like geometries, we have optimized a large number of initial struc- tures which resulted to be higher energy isomers. Specifically, we tested the ground state and first isomer resulting from: (i) a genetic algorithm search connected to a Gupta potential [15]; (ii) opti- mized structures of Pd, Rh, and Ru clusters optimized in previous calculations [16,17]; (iii) planar structures of Cu clusters grown on bridge sites preferably to hollow sites [14]. Thus, we have tested about fifteen selected initial geometries per cluster size which were already found as putative lower energy states in previous works. We think than the resulting ground state and first isomer of Co clusters given in this work are the best choice at the present computational resources for the sizes considered in this work. 0301-0104/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2012.12.037 ⇑ Corresponding author. E-mail address: [email protected](L.C. Balbás). Chemical Physics 415 (2013) 106–111 Contents lists available at SciVerse ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys
Transcript
Chemical Physics 415 (2013) 106–111
Contents lists available at SciVerse ScienceDirect
F. Aguilera-Granja a, Andrés Vega b, Luis Carlos Balbás b,⇑a Instituto de Física, Universidad Autónoma de San Luis Potosí, 7800 San Luis Potosí, Mexicob Departamento de Física Teórica, Atómica, y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain
a r t i c l e i n f o
Article history:Received 29 October 2012In final form 30 December 2012Available online 5 January 2013
Using a fully unconstrained version of the density-functional method with the generalized gradientapproximation to exchange and correlation, we investigate the electronic structure and related propertiesof free-standing Co13�26 atomic clusters with star-like geometries belonging to an unexpected icosahedralgrowth pattern. Our results for the binding energies, ionization potentials and electron affinities, providecompelling evidence of the preference for this new family of star-like structures of CoN clusters in therange N = 20–25 instead of the planar hcp- or fcc-based structures proposed so far as the ground state.
� 2013 Elsevier B.V. All rights reserved.
1. Introduction
Atomic nano-clusters have a wide range of possible technolog-ical applications such as the design of magnetic storage and cata-lytic devices, due to which considerable efforts are undertaken inorder to characterize and to understand their geometrical and elec-tronic properties. The geometrical structure of free-standing clus-ters is difficult to be characterized since most techniques used inbulk-like systems are not suitable if the cluster is not supportedon a host. Therefore, in the free-standing environment, alternativetechniques have emerged to find plausible geometries. For in-stance, one can take advantage of the adsorption of light moleculeson the surface of the cluster in order to get some indirect informa-tion about the geometrical shape [1]. In addition evidence wasfound for Co18, Co19, and Co21, for more than one isomer at lowtemperature [2]. Since this kind of experimental information isinconclusive, the combination of first-principles total energy calcu-lations with experimental techniques like infrared spectroscopy[3], trapped ion electron diffraction (TIED) [4], and with experi-mental data on ionization potential (IP) [5,6], electron affinity(EA) [7] or magnetic moment [8,9], is a good strategy to betterdetermine the geometrical structures. Indeed, a good indicativeof the plausibility of the structural and electronic configurationsof pure clusters is provided by the comparison of the IP and EAwith those experimentally measured. Such a comparison wasmade for iron clusters more than ten year ago by part of theauthors in the framework of a tight-binding model [10].
In the case of Co clusters, recent DFT calculations [12,11] haveshown that the planar-like structures based on fcc and/or hcp mo-tifs are the best candidates for the ground state in the small sizeregime. It has been also pointed out that the hybridization togetherwith the magnetic contribution are driving forces for stabilizingthose planar-like structures. In this work, we investigate a differentfamily of structures, based on an unexpected icosahedral growthpattern. Our interest in this new family of star-like geometriescomes from the fact it has been recently shown to lead to more sta-ble structures for Pd13 and Cu13 than any other type, including theplanar-like ones [13,14]. Being Cu and Pd late transition-metalclusters with more than half-band filling, as it is Co, the new familyof structures deserves to be investigated also for Co clusters, and tobenchmark it against the planar-like family established so far.
To this purpose, we have conducted DFT calculations of theelectronic structure and related properties (total energy, ionizationpotential, electron affinity, magnetic moment) of CoN clusters(with N ¼ 13 to 26) with reoptimized geometries belonging to bothfamilies of structures. In addition to these planar and icosahedrallike geometries, we have optimized a large number of initial struc-tures which resulted to be higher energy isomers. Specifically, wetested the ground state and first isomer resulting from: (i) a geneticalgorithm search connected to a Gupta potential [15]; (ii) opti-mized structures of Pd, Rh, and Ru clusters optimized in previouscalculations [16,17]; (iii) planar structures of Cu clusters grownon bridge sites preferably to hollow sites [14]. Thus, we have testedabout fifteen selected initial geometries per cluster size whichwere already found as putative lower energy states in previousworks. We think than the resulting ground state and first isomerof Co clusters given in this work are the best choice at the presentcomputational resources for the sizes considered in this work.
F. Aguilera-Granja et al. / Chemical Physics 415 (2013) 106–111 107
Nowadays, it is computationally prohibitive for 3d magnetic clus-ters within this sizes range (up to 26 atoms), the use of moresophisticated search of geometries, as the basin hopping or ab initiomolecular dynamics techniques. We have first tested our DFTimplementation by comparing the results obtained for the pla-nar-like hcp structures with those previously found by Dong andGong [11] and Datta et al. [12]. Then, we have compared the elec-tronic properties of both families with available experimental re-sults to conclude that the new growth pattern of star-likeicosahedral structures are most likely to take place in the range20 6 N 6 25.
2. Details of the calculations
The electronic structure was calculated using the DFT code SIES-TA, described in detail in Ref. [18]. It employs numerical pseudo-atomic orbitals (AO) as basis sets to solve the single particleKohn-Sham equations. For the exchange and correlation potentialwe used the Perdew–Burke–Ernzerhof (PBE) form of the general-ized gradient approximation (GGA) [19]. Aditional calculationsusing the modern non-local functional proposed by Dion et al.[20] and re-parameterized by Klimes et al. [21] (KBM) were alsoperformed for the critical sizes 20 6 N 6 25. The atomic cores weredescribed by nonlocal norm-conserving Troullier–Martins pseudo-potentials (PS) [22] factorized in the Kleinman-Bylander form [23].The pseudopotential for Co was generated using the valence con-figurations 4s13d7. The s; p and d cutoff radii were 2.00, 2.00 and2.00 a.u. We have included non-linear core corrections with amatching radius of 0.75 a.u. The valence states were describedusing double-f basis with additional polarization p orbital. In thecalculations the clusters were placed in a supercell of size20� 20� 20 Å3, large enough for the interaction between the clus-ter and its replicas in neighboring cells to be negligible, and forconsidering only the C point (k ¼ 0) when integrating over theBrillouin zone. (We have checked that using a cell30� 30� 30 Å3 does not change the total energy of Co23 within±3 meV) The energy cutoff used to define the real-space grid forthe numerical calculations involving the electron density was250 Ry. Using a conjugate gradient method [24], all the structureswere fully relaxed until the interatomic forces were smaller than0.006 eV/Å. (For KBM we used forces smaller than 0.003 eV/Å,15 meV for electronic temperature, and 10�4 for energy conver-gency.) Further details about the pseudopotential and basis setsused, as well as about the pertinent tests, can be found in our pre-vious work on pure 13-atom clusters [25]. In order to crosscheckthe results and to compare with the previous works, we performedsome of the calculations using also the plane-wave code VASP[26,27], which solves the spin-polarized Kohn-Sham equationswithin the projector-augmented wave method.
3. Results and discussion
We illustrate in the panel (A) of Fig. 1 the most stable planar-like hcp structures resulting after reoptimization with SIESTA-PBE. The binding energy per atom, total spin magnetic moment,and number of bonds are also reported. We remind that in previoustheoretical works [12,11], CoN clusters up 23 atoms were investi-gated using VASP-GGA to conclude that the fcc- or hcp-planar-likestructures were the putative ground states. Therefore, in panel (A)of Fig. 1 is shown our reoptimization of the best structures re-ported in these works (for N ¼ 13—23), while for N ¼ 24, 25, 26our optimized geometries (with label hcp in Table 1) correspondto the same planar-like growth pattern.
The average nearest-neighbor coordination increases from 5.54in N ¼ 13 to 6.85 in N ¼ 26, while the increase of the average inter-
atomic distance is nearly negligible, from 2.41 Å to 2.44 Å (see be-low Table 1). These values are similar (only about 2% larger) thanprevious VASP-GGA results for these kind of structures. Our bind-ing energies are also typically 3% larger than those predicted byDatta et al. [12]. This serves as a benchmark for our theoreticalapproach.
In panel (B) of Fig. 1, are shown the optimized geometries cor-responding to the star-like icosahedral growth pattern proposed inthe present work. The structure of Co13 is similar to that predictedfor Pd13 and Cu13 [13,14]. It can be considered as a fragment of theN ¼ 19 double icosahedral structure. As increasing cluster size, theatoms tend to complete the double icosahedron with further addi-tional atoms placed on bridge sites instead of on the typical hollowsites. This leads to more open structures and therefore with lowercoordination number (icosahedrons built by growing on hollowsites were shown, in previous works, to be less stable than the pla-nar-like geometries [12,11]). A peculiar feature of this growth isthat some additional atoms do not form an umbrella but occupythe equatorial positions up to the completion of a star-like arrange-ment (as illustrated in the panel (B) of Fig. 1). For N = 20–25, thisstar-like arrangement has larger binding energy than the umbrel-la-like on-bridge icosahedron.
In Table 1 are given several properties of Co13�26 clusters for thetwo isomers with lower energy. We see that the planar-like growthis dominant for sizes N = 13–19, whereas the star-like growth ispreferred for N = 20–25. For N = 13, 20, 24, 25, and 26, both typesof isomers are very close in energy, and they will coexist at roomtemperature experiments. Except for Co13, these isomers will alsocoexist at the cryogenic temperature reached in modern experi-ments [8,9] for the magnetic moment of Co clusters. We see thatthe average coordination number and the average distance areslightly larger for the star-like than for the planar-like structures.However, the magnetic moment of planar and star-like isomersfor a given size is the same, except for Co13. When the two reportedisomers are star-like their magnetic moments are equal, and the gi-ven energy difference is only due to small geometrical deforma-tions. However, for the two planar-like isomers with 15, 17, and18 atoms, the energy difference will include also small magneticcontributions.
In Fig. 2, we show the energy difference per atom between thestar-like icosahedral structure (panel (B) of Fig. 1) and the best pla-nar-like structure (panel (A) of Fig. 1) as a function of cluster size N.As commented above, we see that the star-like icosahedral struc-tures are energetically competitive for CoN clusters with20 6 N 6 25, and particularly favorable for Co21 and Co23. In orderto provide further confidence in our assessment, we have repeatedthe calculations in the range 20 6 N 6 25 using the modern non-local KBM functional [20,21] which incorporates a description ofvan der Waals dispersion interactions. Since the KBM functionalcombines semilocal exchange and non-local correlation terms(the exchange part is chosen to resemble as much as possible theexact Hartree-Fock exchange), it provides a more physical separa-tion of exchange and correlation effects. Comparing these twoapproximations for the XC potential (PBE and KMB) allows to checkthe influence of the non-local correlation effects. The correspond-ing data, also given in Fig. 2, points to a small contribution of longrange non-local correlation effects and essentially confirm thetrends found within PBE/GGA approach. Nevertheless, it have beenshown that these non-local correlation effects make a decisive con-tribution to the magnetic moment of Feþ13 [28] allowing a coherentinterpretation of experiments [9]. In Fig. 2 are also shown the re-sults from VASP calculations, which confirm the trend alreadycommented. In particular for the sizes N = 21 and 23, the favoredgeometry is clearly star-like. Note that the differences are inmeV, and that values smaller than 23 meV will allow the coexis-tence of both isomers at room temperature.
(A)
(B)
Fig. 1. Geometrical structure of the lowest energy isomer of Co13�26 with (A) planar hcp-like, and (B) icosahedral star-like configurations, as optimized by means of PBE–PS–AO calculations. The numbers in parenthesis are the binding energy per atom (eV), the total magnetic moment (lB), and the total number of bonds, respectively.
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Although the star-like icosahedral clusters have about 10%higher average coordination number and 2% larger average nearestneighbor distance than the hcp- and fcc-planar clusters, the totalmagnetic moment is practically the same in both structures, exceptfor N ¼ 25 (slightly larger moment in the planar geometry). In or-der to quantify the role of magnetism on stabilizing any of the twostructural arrangements, we optimized both of them also in theparamagnetic solution for all the clusters. In Fig. 3 we also plotthe energy difference per atom between the two structural growth
patterns as a function of cluster size N in the paramagnetic solu-tions. The results show that magnetism is a driving mechanismleading the clusters to adopt the star-like icosahedral structuresfor N ¼ 20 to 25, and the hcp- or fcc-like planar structures forN ¼ 13 to 16.
A good indicative of the plausibility of the structural and elec-tronic configurations of pure clusters is provided by the compari-son of the IP and EA with those experimentally measured. Aslong as the values of IP and EA are calculated as the difference of
Table 1Electronic properties of the two lowest energy isomers of Co13�26 clusters with planar-fcc, or planar-hcp, or icosahedral star-like (Star and Star-2) structures, optimized in thiswork by means of PBE–PS–AO calculations. The binding energy per atom Eb (eV/atom), magnetic moment per atom l/atom (lB), average coordination number (ACN), averagedistance (Å), and binding energy per atom difference, Delta (meV), are given. Those isomers marked with ⁄ correspond to the icosahedral star-like structure, and are reported forthe first time in this paper.
Fig. 2. Energy difference, in meV per atom, of the star like structure and the planarlike. Differences smaller than 23 meV will allow coexistence of both isomers atambient temperature. In the range N = 20–25 the results of three differentcalculations are represented.
Fig. 3. Energy difference of the magnetic configuration and the paramagnetic onefor Co13�26 clusters in the corresponding lowest energy state calculated within thePBE–PS–AO approach.
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the total energy of the neutral and charged clusters (cation and an-ion, respectively, for IP and EA), these properties are well definedwithin DFT. Here, we calculate the vertical IPs and EAs, whichmeans to use for the charged cluster the optimized geometry ofits neutral counterpart. In Fig. 4, we provide such a comparisonfor the two structural families. Although a quantitative descriptionof these electronic properties is out of the scope of the DFT, a clearimprovement of the the qualitative agreement is obtained for CoN
clusters with 20 6 N 6 25 with star-like icosahedral structures,while in this size range, the values obtained for the metastable
hcp- or fcc-like planar structures clearly depart from the experi-mental trend. Again here, a better treatment of the non-local cor-relation effects, using the KBM functional, does not modify theoverall results.
Among the clusters for which the star-like structures are morestable, the largest energy difference with respect to the planar-likeones is found for N ¼ 23. Some structural isomers of Co23 are illus-trated in Fig. 5. The corresponding electronic and structural prop-erties are shown in Table 2. These results show that although theplanar like structures (hcp and fcc) are low-lying isomers, thereis a poly-icosahedral structure that is closer to the putative ground
Fig. 4. Ionization and electronic affinity of CoN clusters in the ground state as afunction of the size N. The calculations were performed within the PBE–PS–AOapproach. Comparison with the experimental values [5,7] is fairly good. Thecalculated values for planar-like structures with N = 20–25 are also given. In theinset is represented the second difference of energy for the lowest energystructures. The step of IP and EA at N ¼ 19, and the peak of the second differenceof energy, indicate enhanced stability of Co19, which is related to the clustergeometry (see text).
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state. It is important to notice that the average coordination num-ber of the putative ground state (star-like icosahedron) is amongthe largest ones. The second isomer is a structural reorganizationof the ground state which moves a basal Co atom to the free bridgesite in the intermediate plane, forming a pentagonal external ringwithout change of magnetism.
Finally we address a few remarks about the critical number Co19
at which the transition from planar-like to star-like growth occurs.At that size there is a big drop of the calculated ionization potential(IP), according to the experiments [5], only if the structure changesfrom planar-like to star-like growth motif (see Fig. 4). That drop inthe IP, indicates a special stability of Co19, which correlates withthe peak in the second energy difference, E(N � 1) +E(N + 1) � 2E(N), as shown in the inset of Fig. 4. Thus, the stabiliza-tion of Co19 is a geometry effect related to the completion of atomic
Fig. 5. Isomers of Co23 in a decreasing order of the binding energy, with labels as in Tablethe free bridge site in the intermediate plane.
fcc shells. In addition, the magnetism plays a roll, as shown inFig. 3. However, the magnetism stabilize that fcc structure insteadof the hcp one as claimed by Datta and coworkers [29]. In Table 3are collected our results for the late 3d atomic clusters with 19atoms. We see that the Co19 structure is fcc-like instead of thehcp-like obtained by Datta and coworkers [29] using VASP codewithin the PBE/GGA for the exchange and correlation potential.The energy difference between the paramagnetic and the magneticsolutions, however, is smaller for fcc than for hcp-like geometry.Nevertheless, the magnetism seems to be the stabilizing factorfor Fe19 and Ni19, as in the Datta et al calculations. For Cu19 ourgeometry is the same than that of Datta et al. [29], but the Cu clus-ters are not magnetic.
4. Summary
As a summary, our DFT-GGA calculations of the electronic struc-ture with geometrical optimization for small CoN clusters point toan unexpected structural growth pattern, at least in the range20 6 N 6 25. This growth pattern is based on icosahedral motifswith additional atoms placed on bridge sites instead of on the typ-ical hollow sites, and with some on-bridge atoms occupying theequatorial positions up to the completion of a star-like arrange-ment, instead of forming an umbrella. The special stability ofCo19 shown in the steeper drop of experimental IP and EA trends,as well as in the theoretical second energy difference, is mainlydue to the completion of fcc atomic shells. Magnetism turns outto be a driving mechanism leading the clusters to adopt the star-like icosahedral structures, particularly for Co21 and Co23, and thehcp-like or fcc-like planar ones for N ¼ 13 to 16. The better quali-tative agreement with the experimental IP and EA in the range20 6 N 6 25 obtained for these structures as compared with thehcp- of fcc-like planar ones proposed so far as the ground state,gives further support to the plausibility of the new growth pattern.Modern experiments based in infrared spectroscopy [3] or trappedion electron diffraction techniques [4] will provide, when available,definitive insights in the growth pattern of cobalt clusters. Itshould be noted that a difference smaller than 23 meV between
2. The isomer 2 results from isomer 1 by moving a Co atom from the basal plane to
Table 3Electronic properties of some X19 atomic clusters: Binding energy, Eb (eV), spin, l (lB), and energy difference between the paramagnetic solution and the magnetic one, M (eV).The larger the value of M the larger the contribution from the magnetic energy. The fcc is an octahedral structure, hcp is a piece of bulk hcp, Ico is the doble icosahedral structure,and Ico-l is the star-like icosahedral structure. Bold numbers indicate the ground state.
Table 2Electronic properties of the Co23 isomers represented in Fig. 5: binding energy per atom, Eb, magnetic moment, l, atomic coordinationnumber, ACN, average bond length, lav , and total energy difference with the ground state, Delta.
Isomer Type Eb/atom (eV) l (lB) ACN lav (Å) Delta (eV)
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the total energy per atom of both types of structure, as predictedby the different theoretical approaches used in this work, will al-low coexistence of both type of growth pattern in experiments atroom temperature. Preliminar calculations for cobalt clusters of se-lected larger sizes (N P 30) indicate that the star-like icosahedralstructure can be also present.
Acknowledgements
We acknowledge the financial support from the Spanish Minis-try of Science and Innovation (Project FIS2011-22957) in conjunc-tion with the European Regional Development Fund and by theJunta de Castilla y León (Project VA104A11-2). F.A-G, acknowledgethe financial support from PROMEP-SEP-CA230 and to the Ministe-rio de Educación, Cultura y Deporte of Spain (Ref. SAB2011-0024).
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