Theoretical Determination of the Most Stable Structures of NimAgn

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Theoretical Determination of the Most Stable Structures of NimAgn Bimetallic Nanoalloys Mohammad Molayem,* Valeri G. Grigoryan, and Michael Springborg Physical and Theoretical Chemistry, Saarland University, D-66123, Saarbr€ucken, Germany

bS Supporting Information ABSTRACT:

The structure of the global total-energy minimum for all bimetallic NimAgn nanoalloys with m þ n = N = 2-60 atoms has been identified theoretically by combining the embedded-atom model for the total-energy evaluation with the basin-hopping algorithm for global structure optimization. All global minima structures are found to be related to icosahedra and polyicosahedra, except for some Ni- or Ag-rich clusters for N = 38. Through a careful analysis of the total energy as a function of (m,n), various particularly stable structures can be identified. The results show in most cases that Ag-rich structures are more favored and stable compared with the Ni-rich structures. By analyzing the bond-order parameter and the radial distances, we can demonstrate the existence of coreshell structures with a partial segregation of Ag to the surface of the Ni-Ag clusters.

I. INTRODUCTION The fascination of nanoscale science not only is a result of unique and extraordinary properties of systems within this size range but also comes from the possibility of tuning their properties, for instance for technological applications, through variation of their size.1 Nanoclusters show size-dependent properties that are not matched by either their bulk or by their molecular analogues.2 Among nanoclusters, bimetallic clusters (AmBn; also called nanoalloys) possess extra possibilities for varying the properties through variation both of size (N = m þ n) and of stoichiometry (m/n).3,4 The wide and diverse structural, optical, magnetic, and electronic properties of nanoalloys suggest that these can be important materials for nanodevices as well as for catalysis.3,5-9 Thus, these systems are nowadays one of the main research interests in nanoscience. Experimental and computational studies of the structures and properties of nanoalloys are highly nontrivial. In fact, theoretical prediction of their structures and properties is a complex task because not only the type of elements and the cluster size are relevant but also the composition is an important parameter.10-12 As a consequence, different structural arrangements may result, including segregation and ordered or random mixing.13 What exactly is found for a given nanoalloy is governed by a complicated r 2011 American Chemical Society

interplay between different properties of the constituent atoms like the relative strength of hetero- and homoatomic chemical bonds, bulk surface energies of the pure elements, and the differences between their Wigner-Seitz radii.3 It has been predicted that when the sizes of elements do not match, segregated core-shell structures are the favored structures.14-16 Then, the core part is occupied by elements that have higher surface energy, smaller atomic size, and stronger homoatomic chemical bonds. As a consequence, the inner part contains strained bonds when compared with the outer shell of the cluster. In this case, the icosahedron is predicted to be the dominant morphology, but for elements with close atomic sizes and stronger heteroatomic chemical bonds, an increased mixing should be found, and other structural patterns like the decahedron and fcc crystal structure will come into play.3,5 The major complexity in studying nanoalloys theoretically originates from the huge number of possible arrangements of N = m þ n atoms in an AmBn cluster, which result in different homotops. Homotops are structures that differ only in the arrangement of the A and B atoms but have the same geometrical Received: October 2, 2010 Revised: January 10, 2011 Published: March 29, 2011 7179

dx.doi.org/10.1021/jp1094678 | J. Phys. Chem. C 2011, 115, 7179–7192

The Journal of Physical Chemistry C arrangement of all atoms, and their number equals ((mþn)!)/ (m!n!) for a given (m,n).3,4 Compared with the pure clusters with only one type of atom, the existence of homotops makes the corresponding potential energy surface (PES) much more complicated. Even for the pure clusters, searching this surface for the global minimum (GM) is an NP-hard problem. As a result of these complexities, theoretical studies of structural properties of nanoalloys are fairly few. Among the nanoalloys, the nickel-silver (NimAgn) system is interesting because of possible optical applications due to the presence of Ag. For a first estimate of the structures, we consider the criteria above that predict that the stable structures of Ni-Ag nanoalloys will have a core-shell pattern. The reason is that (i) the atomic Wigner-Seitz radii of Ag and Ni have a noticeable W.S. difference (rW.S. Ag > rNi ), (ii) the surface energy of Ni is twice that of Ag, (iii) the cohesive energy of solid Ni is 50% larger than that of solid Ag, and (iv) Ni and Ag are immiscible even at high temperatures in their bulk phase. Optical analysis and low-energy ion spectroscopy confirmed the formation of core-shell structures with segregation of Ag atoms to the surface for some selected sizes and compositions.8,17 Furthermore, the absence of magic 2D structural patterns in mass spectroscopic analysis has been taken as a proof of 3D shell closures (enhanced stability) to be favored even for small Ni-Ag clusters.11 Theoretical studies using ab initio calculations are, because of the complications originating from the huge number of homotops, limited to just some few and small Ni-Ag nanoalloys. Density functional theory (DFT) has been applied to NimAgn with very small sizes to study some prechosen structures that were constructed from those of the pure clusters.18,19 The results predicted that for N e 6 planar geometries are found for Ag-rich nanoalloys and 3D geometries for Ni-rich systems. Furthermore, the calculations predicted that the optical absorption will be intermediate between the spectrum of pure Ag and Ni clusters.18 With simpler descriptions of the interatomic interactions, it is possible to study more and larger nanoalloys as well as to perform less biased structure optimizations. Therefore, the second-moment approximation to the tight-binding model (SMATB) in combination with genetic algorithms (GAs) has been used by Rossi et al. to determine several different low-energy structures for N = 34, 38, and 45.7 Subsequently, the most stable structures were reoptimized with DFT calculations. It was found that for a given size the most stable composition is the one that contains a perfect core-shell polyicosahedral (pIh) structure. This is, for example, the case for the Ni7Ag27 and Ni8Ag30 nanoalloys, which are perfect core-shell pIh, and that correspond to the most stable compositions for N = 34 and 38, respectively. At N = 45, an anti-Mackay icosahedron was found to be stable with a (m,n) = (13,32) composition. They also found that the melting point of alloyed Ag cluster with a pIh structure is higher than that of the pure Ag cluster of the same size. Ferrando et al. studied the 34-atom Ni-Ag clusters at first with an empirical potential to find the global total-energy minima.20 Subsequently, they reoptimized these structures using DFT calculations and found that the Ni7Ag27 with a five-fold pancake geometry is a particularly stable structure. A pancake contains a stack of (in some cases, slightly distorted) pentagons and single atoms. Therefore, the 11-atom pancake contains 5 þ 1 þ 5 atoms and is essentially the 13-atom icosahedron without the two capping atoms. Similarly, the 17-atom pancake contains 5 þ 1 þ 5 þ 1 þ 5 atoms and is essentially the 19-atom double-icosahedron

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without the two capping atoms. Throughout this work, we will repeatedly meet all of these structural motifs. Segregation and shape transitions in bimetallic nanoalloys, including NimAg3m, clusters have been studied in the N = 55309 size range using Monte Carlo simulations by Calvo et al.21 The results showed that the most stable structures at low temperatures are icosahedral. In addition, Calvo et al. found that the Ni-Ag clusters show a shape transition to a prolate geometry before the melting of the Ni core at high temperatures. In a molecular dynamics simulation that was devoted to studying the growth pattern of Ni-Ag nanoalloys, 200-300 Ag atoms were deposited on a TO core of 201 Ni atoms.22 According to the results, core-shell structure with a core of Ni is energetically most favored. In the same study, the structures of the N = 55 clusters were optimized globally by using a GA and SMATB potential. The Ni19Ag36 composition with a nonicosahedral morphology was found to be the most stable chemical ordering. SMATB was also used in combination with a GA by Rapallo et al. to study the 34- and 38-atom nanoalloys formed by size-mismatched metals.23 They found stable compositions for the 34atomic case Ni7Ag27 and for the 38-atomic cases Ni30Ag8 and Ni32Ag6. In addition, Ni13Ag32 was found to be a particularly stable structure for N = 45. In total, it can be seen that global optimizations for Ni-Ag clusters have been performed for just some selected sizes or size ranges. The most investigated cluster sizes are N = 34, 38, and 45. The purpose of the present work is to perform an exhaustive search for the GM structures for all NimAgn clusters for N = m þ n between 2 and 60. As a result, we will have a complete database of all possible GM structures for Ni-Ag nanoclusters, which can be used for more precise study of them by very accurate method like ab initio methods. It is the two-step search strategy proposed by Ferrando et al. to make the study of nanoalloys possible.20 From the enormous amount of information that results from this study, structural motifs, growths patterns, energetic properties, and so on shall be identified. We model the atomic interactions through the embedded-atom method (EAM). To search the PES for GM structures, we have used the basin-hopping (BH) algorithm that has been developed by the Wales group. One challenge is to extract physical/chemical insight from the results of a study like the present one, that is, from large listings of total energies and nuclear positions as a function of (m,n). Here we shall use the concepts of stability function, excess energy, first and second isomer energy differences, mixing energies, and mixing coefficients for the analysis of the total energies. The stability function compares binary clusters with neighboring compositions and sizes, whereas the excess energy is an indication of stability when one compares all compositions of a cluster size to each other. With the stability function, a cluster of a given size is compared with those with one atom more or less. For pure metal clusters, it has been related to the peaks in the abundance mass spectrum of clusters.24 Whereas this concept is unique for monatomic clusters, it is possible to define five different forms of the stability functions for the bimetallic clusters. Up to now, in all of the studies of the alloyed clusters, only one definition has been considered. The effect of mixing on the energy of an alloy cluster can be seen through its mixing energy. In addition, we employ the bond-order parameters and radial distances for the analysis of structural properties. In selected cases, a more detailed analysis will be presented. The outline of the Article is as follows. In Section II, we present a brief overview of our computational approach, and 7180

dx.doi.org/10.1021/jp1094678 |J. Phys. Chem. C 2011, 115, 7179–7192

The Journal of Physical Chemistry C

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Table 1. Most Stable Compositions, (m,n), for Ni-Ag Nanoalloys within the Size Range of N = 2-60a Δ2

Δ2

Δ2

Δ2(1)

Δ2(2)

EN.2 - EN.1

Δ2

Δ2

Δ2(1)

Δ2(2)

EN.2 - EN.1

n

m

(1,1)

32

(12,20)

(6,26)

(24,8)

(6,26)

(30,2)

(6,26)

(7,25)

3

(1,2)

(2,1)

(2,1)

(2,1)

(2,1)

(1,2)

33

(6,27)

(24,9)

(25,8)

(4,29)

(23,10)

(7,26)

(10,23)

4

(1,3)

(3,1)

(3,1)

(1,3)

(3,1)

(2,2)

34

(7,27)

(9,25)

(10,24)

(6,28)

(9,25)

(23,11)

(10,24)

5

(3,2)

(3,2)

(3,2)

(3,2)

(3,2)

(3,2)

35

(7,28)

(7,28)

(8,27)

(7,28)

(9,26)

(33,2)

(9,26)

6

(3,3)

(5,1)

(5,1)

(5,1)

(5,1)

(5,1)

(3,3)

36

(9,27)

(11,25)

(6,30)

(5,31)

(21,15)

(9,27)

(12,24)

7

(2,5)

(2,5)

(2,5)

(2,5)

(2,5)

(1,6)

(2,5)

37

(13,24)

(11,26)

(10,27)

(2,35)

(10,27)

(9,28)

(10,27)

8

(2,6)

(2,6)

(4,4)

(2,6)

(6,2)

(2,6)

(4,4)

38

(4,34)

(8,30)

(19,19)

(8,30)

(17,21)

(4,34)

(10,28)

9 10

(1,8) (1,9)

(1,8) (1,9)

(5,4) (6,4)

(1,8) (1,9)

(5,4) (8,2)

(1,8) (1,9)

(4,5) (4,6)

39 40

(28,11) (16,24)

(17,22) (11,29)

(28,11) (16,24)

(4,35) (16,24)

(17,22) (31,9)

(26,13) (39,1)

(14,25) (16,24)

11

(1,10)

(1,10)

(8,3)

(1,10)

(7,4)

(8,3)

(4,7)

41

(17,24)

(11,30)

(17,24)

(17,24)

(11,30)

(17,24)

(17,24)

12

(1,11)

(1,11)

(7,5)

(1,11)

(7,5)

(1,11)

(4,8)

42

(11,31)

(14,28)

(11,31)

(11,31)

(14,28)

(14,28)

(11,31)

13

(1,12)

(1,12)

(12,1)

(1,12)

(12,1)

(12,1)

(1,12)

43

(16,27)

(13,30)

(21,22)

(13,30)

(20,23)

(40,3)

(13,30)

14

(1,13)

(1,13)

(5,9)

(1,13)

(13,1)

(1,13)

(5,9)

44

(15,29)

(12,32)

(15,29)

(12,32)

(15,29)

(11,33)

(15,29)

15

(1,14)

(1,14)

(4,11)

(1,14)

(14,1)

(1,14)

(4,11)

45

(30,15)

(15,30)

(14,31)

(15,30)

(14,31)

(30,15)

(15,30)

16

(1,15)

(1,15)

(8,8)

(1,15)

(14,2)

(1,15)

(6,10)

46

(11,35)

(1,45)

(1,45)

(1,45)

(21,25)

(11,35)

(17,29)

17 18

(2,15) (2,16)

(2,15) (2,16)

(4,13) (5,13)

(1,16) (1,17)

(16,1) (9,9)

(4,13) (1,17)

(5,12) (6,12)

47 48

(14,33) (17,31)

(14,33) (17,31)

(14,33) (20,28)

(14,33) (10,38)

(14,33) (20,28)

(19,28) (44,4)

(14,33) (17,31)

19

(2,17)

(2,17)

(2,17)

(2,17)

(18,1)

(2,17)

(4,15)

49

(1,48)

(40,9)

(26,23)

(8,41)

(1,48)

(41,8)

(16,33)

20

(2,18)

(2,18)

(5,15)

(1,19)

(9,11)

(5,15)

(5,15)

50

(7,43)

(30,20)

(31,19)

(30,20)

(31,19)

(32,18)

(20,30)

21

(2,19)

(5,16)

(11,10)

(1,20)

(11,10)

(2,19)

(8,13)

51

(29,22)

(29,22)

(12,39)

(12,39)

(23,28)

(11,40)

(16,35)

22

(3,19)

(3,19)

(9,13)

(1,21)

(9,13)

(9,13)

(9,13)

52

(23,29)

(15,37)

(41,11)

(11,41)

(40,12)

(32,20)

(18,34)

23

(3,20)

(3,20)

(10,13)

(3,20)

(22,1)

(3,20)

(7,16)

53

(12,41)

(35,18)

(35,18)

(35,18)

(35,18)

(44,9)

(18,35)

24

(3,21)

(7,17)

(18,6)

(1,23)

(23,1)

(3,21)

(7,17)

54

(17,37)

(22,32)

(6,48)

(6,48)

(22,32)

(45,9)

(22,32)

25 26

(4,21) (5,21)

(3,22) (24,2)

(10,15) (17,9)

(2,23) (2,24)

(15,10) (24,2)

(4,21) (6,20)

(10,15) (7,19)

55 56

(24,31) (17,39)

(45,10) (15,41)

(44,11) (17,39)

(16,39) (14,42)

(44,11) (15,41)

(54,1) (18,38)

(18,37) (17,39)

27

(4,23)

(5,22)

(4,23)

(4,23)

(23,4)

(4,23)

(7,20)

57

(15,42)

(45,12)

(45,12)

(45,12)

(45,12)

(46,11)

(18,39)

28

(5,23)

(17,11)

(13,15)

(1,27)

(17,11)

(17,11)

(8,20)

58

(1,57)

(17,41)

(16,42)

(20,38)

(16,42)

(1,57)

(20,38)

29

(5,24)

(5,24)

(22,7)

(5,24)

(16,13)

(5,24)

(8,21)

59

(21,38)

(1,58)

(44,15)

(1,58)

(51,8)

(24,35)

(21,38)

30

(10,20)

(14,16)

(14,16)

(3,27)

(14,16)

(12,18)

(10,20)

60

(1,59)

(36,24)

(20,40)

31

(6,25)

(12,19)

(13,18)

(3,28)

(22,9)

(27,4)

(10,21)

mn

mn

2

Eexc

N

Δ2

N

N

N

n

m

mn

mn

Eexc

These compositions are defined by all of the proposed stability criteria, that is, the stability functions (eqs 4-8) as well as the first and second isomers energy difference, and the excess energy (eq 9). a

Section III is devoted to the results and the discussions. Finally, we will summarize our conclusions in Section IV.

II. COMPUTATIONAL APPROACH Total-Energy Expression. The EAM method, which is used mostly for metallic systems, has been proven to be suitable also for metal alloys.25-28 Our experience as well as studies of others show that this method provides reliable information on the structure of metallic clusters.29-33,48 Therefore, we have chosen this approach to describe the interatomic interactions in Ni-Ag nanoalloys. Within the EAM, the many-body interactions between the atoms are modeled using embedding functions augmented by short-ranged potentials. The embedding function for each atom is the energy needed to embed that atom in a host of all other atoms of the system. In total, the total energy for an N-atomic system can be written as28 " # N N 1 Etot ¼ Fi ðFhi Þ þ Φij ðrij Þ ð1Þ 2 j ¼ 1, j6¼ i i¼1

∑

∑

In this equation, Fi(Fih) is the embedding energy of atom i, and Fih is the electron density constructed by the host atoms at the

position of atom i. Φij(rij) is the pair interaction between atoms i and j, which have an interatomic distance of rij. The embedding functions and the parameters of pair interactions are determined empirically by fitting to experimental data for the corresponding bulk systems, like the heat of solution of binary alloys, elastic constants, and sublimation energies. The EAM potential is mathematically very similar to the SMATB, although they are derived from different models of metallic bonding.34,35 The total energy of a system in SMATB potential has also a many-body term and a repulsive pair term, which describes atom-atom interactions. Parameters of this potential are also defined by fitting to empirical data, but the other motivation in choosing the EAM potential is the universality of its embedding functions. These functions depend only on the local density in the vicinity of each atom and do not depend on the source of the electron densities, that is, on the type of atoms. Therefore, the same functions can be used to calculate the energy of an atom in an alloy that are also used in the corresponding pure metals. That means we do not need to do extra fittings for heterointeractions. We extrapolated each of the pair (A-A and B-B) interaction functions for distances larger than the used cutoff distances to avoid unexpected mathematical difficulties. The values of these functions were zero on those domains out of the corresponding 7181



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Figure 1. Stability function according to NΔ2 (eq 4) for selected sizes of Ni-Ag nanoalloys versus number of Ni atoms (m).

cutoff. The continuity of functions and their first derivatives were also required at the cutoff distances. For the heteroatomic interactions, we have used a geometric mean of pure pair interactions in line with ref 27, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ ΦAB ðrÞ ¼ ΦAA ðrÞ 3 ΦBB ðrÞ Besides the usual cutoff distances beyond which the short-range interactions between atoms of the same type vanish, we have applied a cutoff distance to heteroatomic interactions that is defined as the minimum value of the corresponding short-range homoatomic interaction cutoffs. The continuity of heterointeraction function and its derivative at the cutoff distances follows from the conditions imposed in the extrapolations. Global Optimization Method. To find the GM structures of Ni-Ag nanoalloys, we used the BH method developed by Wales.36 The method is a stochastic search strategy based on Monte Carlo simulations. With this algorithm, instead of searching for the GM of the highly complex PES, one searches the GM for a modified PES with a staircase-like shape formed by basins, which are assigned to the corresponding funnels of the PES. Therefore, one considers the transformed PES surface given by37,38 ~ðX E BÞ ¼ min½EðX BÞ

ð3Þ

~ðX that is, E BÞ is the energy that is obtained after a local totalenergy optimization starting at the structure XB. In a more practical word, one starts from an initial structure (in our case a random one) and optimizes it locally to find the local minimum energy of the corresponding funnel. This funnel will be known by its minimum from now on. Subsequently, we obtain a new structure by slightly changing the Cartesian coordinates of the atoms, jump out of the current funnel, and repeat the local optimization for other funnels. A Monte Carlo loop on this procedure will

search the PES of the system under study. The new structure in each step of Monte Carlo is accepted when it has lower energy in comparison with the older one, and if not, it will be accepted with a probability given by exp((Eold - Enew)/(kBT)). This can also be interpreted as allowing the system to jump over the barriers between the funnels at a thermal energy kBT and search the funnels of the PES. Another important parameter in Monte Carlo algorithm is the acceptance ratio that defines the number of accepted trials. The most common and convenient value for the acceptance ratio that we have also used is 0.5. This means that 50% of the trial moves in Monte Carlo are accepted. Temperature, T, and the degree of perturbation in each step are adjustable parameters. We have used values obtained by requiring that the known GM structures of pure Ni and Ag clusters are identified within the shortest computation time. We set T = 0.8 and use 5000 Monte Carlo iterations to find the GM of each cluster. For a more detailed description of the BH algorithm, we refer the reader to the original work of Wales.38 So far, the BH algorithm is the only unbiased method that has been able to find the global minima based on the Marks decahedron for Lennard-Jones clusters with N = 75 atoms.38 Moreover, it has also been applied successfully to other binary clusters.39-42

III. RESULTS AND DISCUSSION As the output of our calculations, we have obtained long listings of total energies and structures as functions of (m,n). A challenge is, accordingly, to extract physical/chemical insights from those. We shall here present and discuss various quantities that we have applied to this end. Stability Function. It is well known that for monatomic clusters (i.e., m = 0 or n = 0) there exists particularly stable clusters, that is, magic numbers. These can be identified by studying the so-called stability functions, whereby one is comparing the 7182



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At first, we can compare clusters with the same size but different stoichiometries, that is,5 N

Δ 2 ¼ Eðm - 1, n þ 1Þ þ Eðm þ 1, n - 1Þ - 2Eðm, nÞ ð4Þ

Alternatively, one may fix m or n and compare each cluster with its nearby sizes, which have the same m or n. This results in n

Δ 2 ¼ Eðm þ 1, nÞ þ Eðm - 1, nÞ - 2Eðm, nÞ

ð5Þ

Δ 2 ¼ Eðm, n þ 1Þ þ Eðm, n - 1Þ - 2Eðm, nÞ

ð6Þ

and m

Furthermore, we may compare clusters for which both m and n are varied, that is ð1Þ

ð7Þ

ð2Þ

ð8Þ

mn

Δ 2 ¼ Eðm, n þ 1Þ þ Eðm - 1, nÞ - 2Eðm, nÞ

mn

Δ 2 ¼ Eðm þ 1, nÞ þ Eðm, n - 1Þ - 2Eðm, nÞ

and

Figure 2. Structures of some selected sizes of Ni-Ag nanoalloys with different compositions (m,n). Dark red and gray spheres are Ni and Ag atoms, respectively.

total energy of a given cluster with those of the clusters with one atom more or less. The fact that also in our case the binding energy Eb is not a smooth function of size suggests that also here particularly stable clusters may exist. (Eb is shown in the Supporting Information.) It is, however, less obvious how to identify those. Therefore, we shall study different types of stability functions.

In all of these definitions, the clusters that have maxima in the stability function are supposed to be more stable than others (i.e., they are of magic sizes). Each of the stability functions is a function of m and n or, alternatively, of N = m þ n and either m or n. Considering the latter case and searching, for each value of N and for each of the different stability functions, the most stable cluster, we find the values that are reported in Table 1. It is seen that the different definitions often give similar, although not identical, results but also that there are cases where quite different results occur. It shall, however, be remembered that the Table reports only the most stable clusters so that small changes in the relative stability are one source for the differences in the Table. We shall now discuss a few cases in more detail. Our calculations showed that for N = 13 and 19, the GM structures are those of an icosahedron (Ih13) and double icosahedron (Ih19), respectively, for all (m,n). Using the different stability functions, we find, with one exception, that the most stable compositions for these two cluster sizes are (m,n) = (1,12) and (2,17), respectively. The exception is that for mΔ2 and mnΔ2(2) for N = 13, we find (m,n) = (12,1). For N = 19 the exception is the (18,1) cluster found by mnΔ2(2). Analyzing the structures for m = 1 or 2, we find that the Ni atoms are occupying the center of the icosahedra, cf. Figure 2. All structures formed for these sizes are more-or-less deformed icosahedra. The degree of deformation seems to depend on the composition; that is, a larger number of Ag atoms leads to an increased deformation. The existence of these deformations can be related to the differences in the bond lengths for Ni-Ni, Ag-Ag, and Ni-Ag, and also to a segregation of the Ag atoms to the surface. Figure 1 depicts the stability function of eq 4 for N = 34, 38, 55, and 60 versus the number of Ni atoms (m). We have selected the cluster sizes of 38 and 55 because they have been shown to be magic sizes for pure clusters (see, e.g., refs 29 and 43) and, together with N = 34, are among the most studied sizes of Ni-Ag nanoalloys.22,23 The case N = 60 is also shown because this size is the largest of the present study. For N = 34, Ni7Ag27 is found to be the most stable composition, followed by Ni23Ag11 and Ni21Ag13. (See Figure 1.) These clusters contain five-fold pancakes with D5h point group, based on a central Ih19, on which an outer shell of atoms is added to the T sites, that is, to the top-centers of the triangular faces 7183



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Figure 3. Stability function according to nΔ2 (eq 5) for selected sizes of Ni-Ag nanoalloys versus number of Ni atoms (m).

Figure 4. Stability function according to mΔ2 (eq 6) for three sizes of Ni-Ag nanoalloys versus number of Ni atoms (m).

formed by the atoms of the inner shell (Figure 2). This type of growth is known as TIC/polyicosahedral growth.44 This structure can also be regarded as a piece (central core) of an Ih55 icosahedron. The same structural motifs have also been found in other similar studies.7,12,20,23 With two exceptions (for m = 15 and 22), we find that the GM structures for all compositions with N = 34 atoms for m lying in the range 7-24 possess the same fivefold pancake geometry, but as m changes, due to the bond-length differences, the structures show some deviations from a perfect pancake geometry. For m = 15 and 22, some silver atoms form islands outside the Ni atoms. Our results are very similar to those of Rapallo, who found that the domain of the five-fold pancake

structures covers the whole range m = 7-27.23 In Ag- or Ni-rich clusters (i.e., m < 6 or m > 24), where no five-fold pancake is formed, the structures are polyicosahedra (pIh) composed of multi interpenetrating Ih13 icosahedra. For N = 38, the (4,34) cluster with a symmetric structure is the most stable one according to Figure 1. This cluster, with D2h symmetry, can be considered to be an ordered polyicosahedron constructed from six Ih13 icosahedra that form a perfect coreshell structure (Figure 2). Two Ag atoms are shared by all of these Ih13 icosahedra. Each icosahedron has ten common atoms with its two neighbors, and Ni atoms are at the center of four of them. This structure was also found by a tight-binding genetic-algorithm 7184



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Figure 5. Stability function according to mnΔ2(1) (eq 7) for three sizes of Ni-Ag nanoalloys versus number of Ni atoms (m).

Figure 6. Stability function according to mnΔ2(2) (eq 8) for three sizes of Ni-Ag nanoalloys versus number of Ni atoms (m).

study for the range of m = 3-6 composition. However, when using NΔ2, the most stable structure was the Ni13Ag25 cluster.23 The structures of the pure Ag38 and Ni38 clusters are those of a truncated octahedron (TO), which is in agreement with the results of previous studies.29,43 Also for Ni-rich nanoalloys, for this cluster size, that is, m = 36 and 37, the TO is formed. The Ag-rich clusters, that is, m = 1 to 2, and other Ni-rich nanoalloys, that is, m = 32-35, possess five-fold pancakes, resembling a part of the Ih55 icosahedron but having a vacant site. For m = 27-31, the pancake motifs are changed completely by the displacement of some of the outer-shell atoms. All other clusters with N = 38 are found to have the structure of polyicosahedra.

The stability function NΔ2 for N = 55 shows that the most stable nanoalloy of this size is the pIh Ni24Ag31 cluster with the symmetry C1 (Figure 1). The core of this nanoalloy is formed by only Ni atoms, but also some Ni atoms are found on the surface. The only other available study for this cluster size found the Ni19Ag36 to be the most stable cluster.22 The other particularly stable compositions for this size are found at m = 5, 11, 13, 16, 27, 41, and 44. The Ag-rich nanoalloys with m = 0-9 possess the five-fold symmetric structure of the Ih55 icosahedron (Figure 2). This is also true for the Ni-rich clusters with m = 36-55, except for the case of m = 7, for which we found a polyicosahedron as its GM. As is expected, all of these symmetric geometries possess 7185



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Figure 7. Energy difference between the first and second stable isomers for four selected sizes of Ni-Ag nanoalloys versus number of Ni atoms (m).

Figure 8. Excess energy per atom for NimAgn clusters as a function of (m,n) for N = m þ n from 2 to 60.

some deformation due to the difference in the size of the two types of atoms. The deformation is particularly large for the Agrich clusters, for which Ag atoms are on the surface so that the deformation can not be compensated by other bonds around them. All other compositions of this size, which were not mentioned directly, are polyicosahedra as well. As can be seen in Figure 1, the clusters (1,59), (3,57), (20,40), and (39,21) are the more stable ones for N = 60. The (1,59) cluster, which is the most stable one, is an Ih55 icosahedron with one Ni atom at the center and five extra Ag atoms which are placed on its surface. This GM has the Cs symmetry (see Figure 2).

For comparison, the stability functions for the values N = 34, 38, and 55 and as defined according to the definitions of eq 5 to 8 are shown in Figures 3-6. For N = 34 clusters, the stability functions defined by eqs 6 and 7 find different clusters, that is, (m,n) = (6,28) and (10,24) to be the most stable ones (cf. Figures 4 and 5). According to nΔ2 and mnΔ2(2), the (9,25) cluster is the most stable one (Figures 3 and 6). The (m,n) = (21,13), (23,11), and (27,7) clusters are determined as stable compositions by all definitions of stability functions. For N = 38, we find, as also was found above, that the cluster with (m,n) = (4,34) is stable compared with its neighbors, although it is no longer the most stable one. Instead, the clusters with (m,n) = (8,30) for nΔ2 and mnΔ2(1) and (m,n) = (19,19) for m Δ2 are seen to be the most stable clusters for this value of N. According to mnΔ2(2), the (17,21) cluster is magic, but the (18,2) and (19,19) clusters are very close to it. Other stable compositions that are identified by all five measures are (24,14) and (34,4). Also, the (1,37) case with a five-fold symmetric pancake structure shows a pronounced stability according to all definitions of the stability function. The case of clusters with N = 55 is interesting because different compositions are proposed to be stable by the different stability functions (Figures 3-6). In all cases, we find the (16,39) cluster to be very stable except for the mnΔ2(2). The same is the case for (m,n) = (13,42), (24,31), (41,14), and (47,8), although the (13,42) cluster is not magic again according to mnΔ2(2). Isomers Energy Difference. Stability of a cluster can also be studied by defining the energy difference between the first and second lowest energy structures, that is, isomers. High energy gap between the isomers is an indication of the thermal stability for the lower one. Figure 7 depicts this quantity for all compositions of four selected cluster sizes. When we compare the results of the isomer energy differences with those of the stability functions, one can see that for N = 34 and 38 many of the magic 7186



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Figure 9. Excess energy of the Ni-Ag nanoalloys for some selected size (N = 34, 38, 55, and 60) versus the number of Ni atoms (m).

Figure 10. Bond-order parameter as a function of composition (number of Ni atoms, m) for the global minima of four sizes of interest (N = 34, 38, 55, and 60). The inserts show the number of the three possible types of bonds versus m. Solid triangles and squares refer to the numbers of Ni-Ni and AgAg bonds, respectively, whereas open circles are for Ni-Ag bonds.

clusters found by the latter measures have also large energy gaps with respect to their second isomers (Figure 7a,b). However, some noticeable differences are also seen: for example, two new stable clusters (28,6) and (25,13) are proposed for these sizes.

The lowest energy isomer of the (28,6) cluster is a fragment of a six-fold pancake structure, whereas the second isomer is an unsymmetrical pIh. Both lowest isomers of the (25,13) cluster are pIh but with no similarity. Some of the stable clusters found 7187



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Figure 11. Ratio of the average radial distances of the Ni atoms to that of the Ag atoms in NimAgn clusters as a function of (m,n) for N = m þ n from 2 to 60.

by the stability functions are not thermally stable because the energies of their first and second isomers are very close to each other. The (6,28) cluster, which is stable according to the mΔ2 and mnΔ(1) 2 , turns out to have two isomers with close energies. The first isomer is an unsymmetrical pIh but with perfect coreshell structure, whereas the second one is a five-fold pancake. Figure 7b indicates that many of the N = 38 clusters are not thermally stable, although we find them among the magic clusters in Figures 1 and 3-6. These are the (9,29), (12,26), (18,20), (24,14), and (26,12) clusters. All of these clusters, except the (24,14), have different homotops as their first and second isomers with very close values of the energy, but for the m = 24 cluster, we find two different structures for the two lowest isomers. According to Figure 7c, the (35,20) and (42,13) clusters are stable, whereas they do not yield peaks in the stability function graphs. Both isomers of the (35,20) cluster possess two different pIh structures. Both isomers of the (42,13) cluster are Ih55, but in the second isomer, an Ag atom from a vertex has left its site to sit on a T site of the last icosahedron shell. We find a set of clusters that are stable according to the stability functions, but they are not thermally stable. These clusters are those with m = 3, 20, 33, and 52. Despite these differences, many of the stable clusters defined by the stability functions are also thermally stable. These clusters are (7,48), (13,42), (16,39), (24,31), (45,10), and (54,1). Although we can just calculate the NΔ2 for N = 60, by considering the common stable clusters found for other sizes by the stability functions and the isomers energy differences, we can use the latter measure to predict other possible stable clusters of this size. These clusters are singled out by their high peaks in Figure 7d. Here the (36,24) cluster shows to be very stable. The first and second isomers of it are both pIh’s formed on a base of the Ih55 with high degrees of deformations. Other stable clusters of these size are those with m = 4, 16, 21, 27, 47, and 52. Excess Energy. Instead of using the stability functions in identifying particularly stable clusters, one may also consider the so-called excess energy12,45 Eexc ¼ Eðm, nÞ - m

EðAgN Þ EðNiN Þ -n N N

ð9Þ

Figure 12. Mixing coefficients M (upper part) and mixing energy per atom (lower part) for NimAgn clusters as a function of (m,n) for N = m þ n from 2 to 60.

Here E(NiN) and E(AgN) are the energies of the pure Ni and Ag clusters with N = m þ n atoms, respectively. The excess energy is, per construction, zero for pure clusters. Negative values imply that the mixing is favored, and the smallest value for a given N corresponds to the most stable cluster for this size when comparing with all possible compositions.3 Figure 8 shows Eexc/N for all of the clusters we have considered. For almost all sizes and stoichiometries, Eexc/N is negative, implying that some mixing is (almost) always favored by Ni-Ag nanoalloys. It is moreover interesting to notice that there is a certain size range, that is, m = 10 and n = 22, for which Eexc/N is particularly negative, suggesting that these clusters are the most stable ones. Another possible explanation, that is, that the corresponding pure Ag and Ni clusters are particularly unstable, does not hold here. To obtain more detailed information, Figure 9 shows Eexc for four cluster sizes versus the number of Ni atoms. In general, the excess energy starts to decrease monotonically from zero and after arriving at a minimum increases again roughly monotonically to zero. On top of these general trends, small N-specific 7188



Figure 13. Mixing coefficient for different sizes of Ni-Ag nanoalloys versus total energy, sorted according to the number of Ag atoms (n) that increases from left to right for each curve. The insert shows the results for N = 55 and 38.

deviations are seen, especially for N = 55 and 60. These can be caused by sudden changes in the structure compared with their neighboring sizes. For N = 34, (m,n) = (10,24) is seen to correspond to the lowest value of Eexc with the clusters (9,25) and (7,27) having very close values of the excess energy (Figure 9a). The fact that these clusters are particularly stable was also found with the help of the stability functions (Figure 3-6). The structures of all of these can be described as being five-fold pancakes, but the clusters (9,25) and (10,24) have lower symmetries, that is, C2 in contrast with D5h for the (7,27) cluster. A structural difference is seen for (22,12) which has a five-fold pancake geometry, although two of the Ag atoms are placed at other positions outside Ni atoms. The neighboring compositions, (21,13) and (23,11), are complete five-fold pancakes. For N = 38, the minimum of Eexc is found for the (10,28) cluster. Also the (9,29) cluster has a low value of the excess energy, in accord with the high stability of this cluster as predicted by the mnΔ2(2) stability function. Both of them can be considered to be polyicosahedra with a broken five-fold pancake structure in which some of the Ag atoms are placed outside some Ni atoms near the surface. As seen in Figure 9, the excess energy of clusters with N = 34 and 38 shows a plateau for m = 7-14 and 8-13, respectively. The reason is that all of these structures are structurally very similar. For N = 34, the structures are all five-fold pancakes, and for N = 38, they are all polyicosahedra, in which a part of the fivefold pancake is still formed although containing some deformations, and having extra atoms attached to it. The excess energy for N = 55 is very low but shows oscillations for the clusters with m = 15-25. As mentioned above, in this range, all structures are polyicosahedra. As a result of the higher surface energy of Ni, the Ni atoms tend to occupy the inner sites of the clusters. The excess energy minimum of this size is found for the (18,37) cluster and is only 0.011 eV higher than that of the (16,39) cluster. The corresponding structures for these compositions are the 55-atomic icosahedrons. The peak shown at (23,32) cannot be easily explained because all neighboring clusters have the polyicosahedron structure. If we neglect some deformations

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in the bonds, especially in the surface regions, then for both cases of (22,33) and (24,31) a five-fold pancake core is formed. However, the atoms from the caps have been moved to other sites to form a stable 13-atomic icosahedral core, together with the atoms of the last shell. The exception mentioned above for the (7,48) cluster manifests itself also in Eexc as a peak. For N = 60, Eexc has a minimum for the (20,40) cluster, in agreement with its maximum for NΔ2. The structure of this cluster is a truncated Ih55 with five additional atoms on the side sites. Magic Numbers and Growth Patterns. As mentioned above, in Table 1, we have collected the most stable clusters as a function of the total number of atoms in the clusters and when using the various descriptors (i.e., stability function or excess energy). At first, we observe the interesting and fairly general finding that in many of the cases Ag-rich clusters are found to be more stable than Ni-rich ones. Next, we shall bring a brief overview of the structures of some magic clusters. The (3,2) cluster with a triangular bipyramid structure is found to be the most stable cluster with five atoms in all cases. For N = 6, two particularly stable clusters are found, that is, the (3,3) and (5,1) clusters, which both are octahedra. The pentagonal bipyramid with D5h symmetry group, which has two Ni atoms on the top vertices of the pyramids, is the GM for N = 7. The cluster with (m,n) = (2,6) consists of a six atomic Ni octahedron to which two Ag atoms are attached to the edge (E) positions. The (4,4) cluster contains a pentagonal bipyramid to which one Ag atom is attached at a T site (triangular face) above the Ni atoms. Increasing the size of the clusters, we find the same growth pattern, that is, the addition of atoms at the T sites, until the 13atomic icosahedral Ih13 cluster is obtained. For larger clusters, this Ih13 icosahedron remains a dominating structural motif. At first, atoms one by one are added to the surfaces of this motif (in some cases connected to some structural distortions) until the Ih19 double icosahedra are reached for N = 19. Subsequently, the Ih19 double icosahedra with various stoichiometries are the central part of all magic clusters found for N = 19-23. Starting at N = 23, a third icosahedron is being formed that, as is the case for the second one, shares several atoms with the already formed icosahedra. As a consequence of the formation of more interpenetrating Ih13 icosahedra outside the central Ih19, the first five-fold pancake structures are constructed for N = 33 for all stable clusters, with the (4,29) and (6,27) clusters being the only exceptions. The two cases of (4,29) and (6,27) have no five-fold structures but are instead formed from several Ih13 icosahedra. For N > 34, the five-fold pancake is not recovered anymore. Instead, all structures are polyicosahedra up to N = 38. On the basis of the symmetric structure of (4,34), the cluster (4,35) has one extra Ag atom that is placed on the top of this structure along the central axis. (See Figure 2.) For the (28,11) cluster, a six-fold pancake is formed that has a vacant site on one side. (See Figure 2.) The next interesting magic structure is found for N = 46 and has the (1,45) stoichiometry. This structure is a part of the Ih55 icosahedron. The same structural motif is also seen for some of the slightly larger clusters, that is, for the (1,48), (40,9), (41,11), (35,18), and (6,48) clusters (Figure 2). The clusters for N = 4054 are polyicosahedra. This is also the case for the (14,33) cluster, which is found to be of particularly high stability independent of the measure we use. At N = 57, the (45,12) cluster is composed of an Ih55 with two more atoms on its surface. The same structure is also formed for (m,n) = (1,57) and (1,58), but these two have 7189



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Figure 14. Mixing coefficients of the Ni-Ag clusters versus the number of Ni atoms m for clusters with N = 13, 19, 34, 38, 55, and 60 atoms.

three and four extra atoms, respectively, outside the outermost shell of the Ih55 core. Bond-Order Parameter And Radial Distances. The mixing versus segregation of the nanoalloys can be studied through the bond-order parameter, σ.46 For an AmBn nanoalloy, it is defined as σ ¼

NA - A þ NB - B - NA - B NA - A þ NB - B þ NA - B

ð10Þ

Here Ni-j (i,j = A, B) is the number of nearest-neighbor bonds between atoms of type i and j. σ is positive for segregated, almost zero for disorderly mixed, and negative for mixed and onion-like phases of nanoalloys. Figure 10 shows the bond-order parameter versus number of Ni atoms for all compositions of four selected sizes, N = 34, 38, 55, and 60. In addition, in each case, we also show the corresponding number of Ni-Ni, Ag-Ag, and Ni-Ag bonds. Having only positive values of σ implies some degree of segregation, and as we shall argue below, this segregation is mainly due to the formation of core-shell-like structures. For the clusters with comparable numbers of Ni and Ag atoms, there is a relatively large number of Ni-Ag bonds so that σ obtains lower values. Therefore, the lowest value of σ for N = 34 is found for the (16,18) and (17,17) clusters. For N = 38, 55, and 60, the lowest values of the bond-order parameter are found for more asymmetric clusters, that is, (m,n) = (16,22), (23,32), and (34,26), respectively. We see that none of these is among those we have identified as particularly stable ones. From the absolute values of the numbers of different types of bonds (Figure 10), we can see that by starting from pure Ag clusters the number of Ag-Ag bonds decreases monotonically to a value close to 0 that is obtained not only for the pure Ni clusters. This implies that for the Ni-rich clusters, the Ag atoms

are well-separated. The number of Ni-Ni bonds is nonzero already for clusters with just a few Ni atoms. As we shall argue below, this difference is a consequence of a spatial separation of the Ag and Ni atoms: the former are mainly found in the outer parts of the clusters, and the latter are mainly found in the inner parts. Finally, the largest number of Ni-Ag bonds is found for the clusters with an equal number of Ni and Ag atoms, which may not be a surprise. The spatial separation of the Ni and Ag atoms can be (partially) identified through the so-called radial distances. We show in Figure 11 the ratio of average radial distances of Ni and Ag atoms for all of the considered clusters. Distances are considered from the center of clusters. The segregation of Ag atoms to the sites with larger distances from the center of the clusters is clearly recognized because the ratio of distances is mostly

Theoretical Determination of the Most Stable Structures of NimAgn

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