Structural, Electronic, and Magnetic Properties of Bimetallic Ni m Nb n (m + n 8) Clusters: First Principle Study

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1 J Supercond Nov Magn (2017) 30: DOI /s ORIGINAL PAPER Structural, Electronic, and Magnetic Properties of Bimetallic Ni m Nb n (m + n 8) Clusters: First Principle Study Mihai Deng 1 Zihua Xin 1,2 Xiao Yan 1 Junxian Liu 1 M. Yu 3 Received: 1 August 2016 / Accepted: 13 August 2016 / Published online: 31 August 2016 Springer Science+Business Media New York 2016 Abstract Structural, electronic, and magnetic properties of bimetallic Ni m Nb n (m + n 8) clusters have been investigated using the particle swarm optimization coupled with density functional theory. Ninety-seven stable structures of Ni m Nb n clusters were found. Among these Ni m Nb n clusters, most of the ground state of bimetallic clusters prefers compact structures when m + n > 3. The HOMO LUMO gaps of these bimetallic Ni m Nb n clusters were found in the range of ev. The exchange splitting exists in most of these binary clusters and results in non-zero magnetization in these clusters. Most of the clusters show their magnetic moment strongly depending on the size, the symmetry, the configuration, and the composition. An interesting finding is that the magnetic moment per atom in NiNb and Ni 2 Nb clusters shows a larger value (i.e., 1.5 and 1.0 μ B ) than that of the corresponding size of pure Ni clusters. Keywords Bimetallic clusters Binding energy HOMO LUMO gaps Magnetics Zihua Xin zhxin@shu.edu.cn 1 Department of Physics, Shanghai University, Shanghai , China 2 Shanghai Key Laboratory of High Temperature Superconductors, Shanghai, China 3 Department of Physics and Astronomy, University of Louisville, Louisville, KY 40292, USA 1 Introduction Nanomaterials have been widely studied for their novel properties and extensive potential applications. Among nanodimensional objects, nanoclusters occupy a very important position. The magnetic, optical, and reactive properties of clusters, which are formed by assembling a small number of atoms ranging from a few to several hundred, are peculiarly and strongly dependent on the size. Due to the unique d-shell electron structure and the potential applications in magnetic storage, catalysis, optics and biomedicine, and other fields, transition metal (TM) clusters are physically and chemically attractive for science and technology [1 6]. The studies made by Khanna et al. showed that the magnetic moments of some 3d-TM (Ni, Fe, and Co) clusters are significantly larger than the corresponding bulk magnetizations [7 9]. The investigation made by Vijay Kumar and Yoshiyuki Kawazoe [10] reveals that although Nb shows nonmagnetic in bulk state, small Nb clusters have magnetic moments due to the odd number of valence electrons. Among all the experimental and theoretical researches on TM clusters, Ni-base clusters [11 19] are one of the most significant systems. A series of density functional theory (DFT) calculations for Ni clusters [14 19] hasshownthat pure clusters prefer high coordination; the binding energy (BE) shows a smooth increase with the size of the cluster, and the calculated binding energy was generally underestimated compared to the experimental bulk value. The Jahn Teller effect plays an important role in determining the ground state of certain geometric structures [14, 15], and this makes it more difficult to determine the geometric structures of clusters.

2 252 J Supercond Nov Magn (2017) 30: On the other hand, some experimental researches on the Ni-based alloy [20 22] showed that the addition of Nb can significantly improve the performance of the alloys. Jing Li et al. [20] discovered that the Nb atom plays a crucial role in adjusting composition and improving microstructure and mechanical properties of TiNiNb alloys. All of the relative densities, tensile strength, and elastic modulus of TiNiNb alloys decrease and then increase with the addition of Nb contents. The research made by E.S. Park et al. [21]showed that a small amount of Nb addition can have a dramatic effect on plasticity enhancement in Cu Ti Zr Ni Si bulk metallic glasses. Fei Weng et al. [22] reported that with Nb addition, the hot corrosion resistance of the Ni-based superalloys is notably improved. The enhancement of mechanical properties makes Ni-based bulk metallic glasses extremely promising for potential applications. Though pure Ni [11 19] andnb[10, 23] clusters had been widely studied, few investigations on Ni and Nb bimetallic clusters are reported. So in this paper, we will pay attention to the research of Ni n Nb m bimetallic clusters. We are sure that the study of Ni n Nb m binary clusters will improve the understanding of the properties of NiNb alloy. The structures, which are fundamentally related to physical properties, are keys in the researches of clusters. However, it remains a technical difficulty to determine cluster structures experimentally. In order to get the possible geometric isomers, we used the particle swarm optimization (PSO) [24, 25] to search the available conformational space for the global minimum. Based on this, we performed a systematic study on the physical properties of obtained bimetallic Ni m Nb n (m + n 8) clusters to understand the behavior and the evolution of structural, electronic, and magnetic properties as a function of size and composition of clusters. We found that NiNb 2,Ni 2 Nb 2,Ni 4 Nb 2,Ni 5 Nb 2,and Ni 6 Nb 2 clusters have a higher relative stability compared with their neighboring binary clusters and the energy gap of the binary Ni m Nb n clusters is in the range of ev, weakly depending on the structure of the clusters, while the magnetic moment of these clusters strongly depends on the structures of the cluster. These results indicate the potential application of the bimetallic Ni m Nb n clusters in magnetic storage, catalysis, optics and biomedicine, and other fields. The article is organized as follows: the calculation methods will be briefly described in Section 2, the obtained results and corresponding discussion for Ni m Nb n (m + n 8) clusters will be presented in Section 3, and the conclusions will be given in Section 4. 2 Methods of Calculation The numbers of possible geometric configurations of clusters increase exponentially with the size of the clusters, and it will be more complex for binary clusters. So we will treat the clusters in two different routes. As the geometric structures of small clusters (m + n = 2, 3) are either line or triangle, the structural relaxation was performed using DFT directly. For larger clusters (m + n 4), we performed through the intelligent crystal structure analysis by particle swarm optimization methodology [26, 27], as implemented in the CALYPSO code [27]. This approach has been benchmarked on a variety of known systems [27] and has made several successful predictions of clusters [28]. The optimized structures obtained from PSO are taken as the initial structures for further simulation by the DFT method. The structural relaxations and electronic calculations were performed using DFT within the Perdew Burke Ernzerh (PBE) [29] type of generalized gradient approximation (GGA) and projected augmented waves (PAW) [30] as implemented in the VASP package [31, 32]. An energy Fig. 1 The optimized structures of Ni m Nb n (m + n 5) clusters. The structures of the isomer are arranged according to the ascending order of the binding energy

3 J Supercond Nov Magn (2017) 30: Fig. 2 The optimized structures of Ni m Nb n (m + n = 6) clusters cutoff of 550 ev for the plane-wave basis set was chosen to ensure that energy calculations are well converged to better than 1 mev/atom. We considered 10 valence electrons for Ni (3d 8 4s 2 ) and 11 valence electrons for Nb (4p 6 4d 4 5s 1 ) and preformed spin-polarized calculations. A simple cubic supercell of size up to 15.0 Å is used with periodic boundary conditions to ensure no interactions between the clusters. The structures are optimized until the total energy reaches the criterion of 0.01 mev. The stability of the different geometric structures of Ni m Nb n (m + n 8) clusters for the given composition is investigated by comparing their binding energies, defined as follows E b (Ni m Nb n ) =[me(ni)+ne(nb) E(Ni m Nb n )]/(m+n) (1) To ensure the stability of the resulting structures, Ɣ-point vibrational frequency was calculated, and the structures with imaginary frequency of phonon were abandoned. average nearest-neighbor distance (d), the binding energy (E b ), the HOMO LUMO gap (E g ), and the total magnetic moment (Mag), are presented in Tables 1, 2, 3 and 4. The results of pure Ni and Nb clusters, which are in good agreement with previous works [10 19, 23], are also given in these tables. Figure 1 and Table 1 are the results for clusters with m + n 5. From Fig. 1, we can see that the structure of the small binary cluster (m + n = 2 and 3) is similar to the structures of their corresponding pure clusters. The binding energy of NiNb (1.71 ev/atom) is larger than that of Ni 2 (1.50 ev/atom) and smaller than that of Nb 2 (2.34 ev/atom). The bond length of NiNb (2.17 Å), on the other hand, is longer than that of Ni 2 (2.09 Å) and Nb 2 (2.12 Å). The stable configuration for three-atom binary clusters is an isosceles triangle, the same as that of Nb 3, but different from that of Ni 3 (equilateral triangle). The bond lengths of Ni 2 Nb 3 Results and Discussion 3.1 Energetics and Stability of Geometric Structures Ninety-seven structurally stable Ni m Nb n (m + n 8) clusters were obtained. The geometric structures of all the stable binary Ni m Nb n clusters with different compositions and different sizes are given in Figs. 1, 2, 3 and 4. The small red and big blue spheres in these figures represent Ni and Nb atoms, respectively. For comparison, the structures of pure Ni and Nb clusters are also calculated and given in these figures. The structures with the same composition are arranged according to the ascending order of their binding energy (e.g., the energy of Ni 3 Nb-a in Fig. 1 is lower than that of Ni 3 Nb-b). The detailed information about the obtained 97 Ni m Nb n clusters, including the point-group symmetry (PGS), the average coordination number (Nc), the Fig. 3 The optimized structures of Ni m Nb n (m + n = 7) clusters

4 254 J Supercond Nov Magn (2017) 30: Fig. 4 The optimized structures of Ni m Nb n (m + n = 8) clusters are 2.21, 2.21, and 2.36 Å, slightly shorter than those of NiNb 2 (2.17, 2.42, and 2.42 Å). The binding energy is 2.21 and 2.73 ev/atom for Ni 2 Nb and NiNb 2, respectively. We also studied the linear structures for three-atom clusters, but all of them were found to exhibit the imaginary vibration frequency indicating the instability of linear structures. From Fig. 1 and Table 1, we also found that the stable structures for both Ni 4 and Nb 4 are tetrahedron. Besides the tetrahedron, the binary clusters with m + n = 4also show two new structures, rhombus and disturbed rhombus. The Ni 3 Nb-a is the only cluster which shows a planar structure among all the m + n 4 clusters, indicating that the Ni m Nb n (4 m + n 8) clusters prefer compact structures. For pure Ni and Nb clusters, Ni 4 is not a regular tetrahedron, having two short (2.20 Å) and four long (2.33 Å) bonds. On the other hand, the ground-state configuration of Nb 4 is a regular tetrahedron (T d ) with bond length 2.52 Å. With the increase of the number of m + n, the amount of isomers obtained from PSO increases. We are interested in those with the lowest energy or a slightly higher than the lowest. After fully relaxed by DFT, the stable structures of Ni 5 are found to be a trigonal bipyramid with D 3h symmetry and a square pyramid with C 4v symmetry, and for Nb 5,the lowest energy configuration is found to be a trigonal bipyramid with C 2v symmetry. Because of the difference of the bond length, the trigonal bipyramid of Ni 5 and Nb 5 shows a different symmetry. From Fig. 1 and Table 1, one can see that the possible structures of all the binary clusters with m + n = 5 are either square pyramid or trigonal bipyramid, similar to their corresponding pure clusters. But the trigonal bipyramids for Ni 3 Nb 2 -b, Ni 3 Nb 2 -c, and NiNb 4 -b are all distorted. In these distorted trigonal bipyramids, the atoms at pole positions are close to each other forming a bond. Otherwise, the two or three bonds between the atoms at base positions disappear. The results also show that there are two isomers for Ni 5 and Ni 4 Nb clusters and three isomers for Ni 3 Nb 2 and NiNb 4 clusters, respectively. The structures and detailed results for m + n = 6clusters are given in Fig. 2 and Table 2, respectively. From Fig. 2, we found that there are two isomers for both pure Ni 6 and Nb 6 clusters; one is octahedron and the other one is trigonal bipyramid plus adatom. The energy of the octahedron structure of Ni 6 is lower than that of the trigonal bipyramid plus adatom of Ni 6, but the energy of the octahedron structure of Nb 6 is higher than that of the trigonal bipyramid plus adatom of Nb 6. This indicates that the element itself affects the stability more than the geometry of the structure. Except Ni 5 Nb-b, which shows a square pyramid plus adatom with C s symmetry, all the other binary clusters show either the octahedron structure or the trigonal bipyramid plus adatom structure, similar to their corresponding pure clusters. Figure 2 also shows that the Nirich clusters tend to take the octahedron structure (e.g., Ni 6 cluster), and the Nb-rich clusters tend to take the trigonal bipyramid plus adatom structure (e.g., Nb 6 cluster). The structures and detailed results for m + n = 7clusters are given in Fig. 3 and Table 3, respectively. We found that both pure Ni 7 and Nb 7 clusters show two kinds of geometry; one is the octahedron plus an adatom, and the other one is the pentagonal bipyramid. The binding energy

5 J Supercond Nov Magn (2017) 30: Table 1 The point-group symmetry (PGS), the average coordination number (Nc), the average nearest-neighbor distance d (in Å), the binding energy E b (in ev/atom), the HOMO LUMO gap E g (in ev), and the total magnetic moment Mag (in μ B ) for Ni m Nb n clusters with m + n 5 PGS Nc d E b E g Mag Ni 2 D h NiNb C v Nb 2 D h Ni 3 D 3h Ni 2 Nb C 2v , NiNb 2 C 2v , Nb 3 C 2v , Ni 4 D 2d Ni 3 Nb-a C s Ni 3 Nb-b C 3v Ni 2 Nb 2 C 2v NiNb 3 C Nb 4 T d Ni 5 -a D 3h Ni 5 -b C 4v Ni 4 Nb-a C s Ni 4 Nb-b C 4v Ni 3 Nb 2 -a C 2v Ni 3 Nb 2 -b C s Ni 3 Nb 2 -c C s Ni 2 Nb 3 C s NiNb 4 -a C s NiNb 4 -b C s NiNb 4 -c C 2v Nb 5 C 2v of the octahedron plus adatom structure of Ni 7 is lower than that of the pentagonal bipyramid structure of Ni 7, but the binding energy of the distorted pentagonal bipyramid structure of Nb 7 is lower than that of the octahedron plus adatom structure of Nb 7. Because of the complication of Ni m Nb n clusters with m + n = 7, we obtained 21 stable binary clusters. Twelve of them have the structures similar to the corresponding pure cluster. Moreover, there are nine binary clusters that show new structures, in which Ni 6 Nb-b, Ni 4 Nb 3 -b, Ni 3 Nb 4 -b, Ni 3 Nb 4 -c, Ni 2 Nb 5 - c, and NiNb 6 -c show tricapped tetrahedron structures, and Ni 5 Nb 2 -c Ni 4 Nb 3 -c, and Ni 2 Nb 5 -d show bicapped trigonal bipyramid structures. From Table 3, we found that there are several isomers for each composition of Ni m Nb n clusters with m + n = 7. The composition Ni 5 Nb 2,Ni 4 Nb 3, Ni2Nb 5, and NiNb 6 all have four isomers. The other compositions of this binary system have two or three isomers. The structures and detailed results for m + n = 8clusters are given in Fig. 4 and Table 4. From Fig. 4 and Table 4, Table 2 The point-group symmetry (PGS), the average coordination number (Nc), the average nearest-neighbor distance d (in Å), the binding energy E b (in ev/atom), the HOMO LUMO gap E g (in ev), and the total magnetic moment Mag (in μ B ) for Ni m Nb n clusters with m + n = 6 PGS Nc d E b E g Mag Ni 6 -a D 3d Ni 6 -b C 2v Ni 5 Nb-a C 4v Ni 5 Nb-b C s Ni 4 Nb 2 -a C 2v Ni 4 Nb 2 -b D 4h Ni 4 Nb 2 -c C 2v Ni 4 Nb 2 -d C Ni 3 Nb 3 -a C s Ni 3 Nb 3 -b C s Ni 3 Nb 3 -c C Ni 2 Nb 4 -a C Ni 2 Nb 4 -b C s Ni 2 Nb 4 -c C 2v Ni 2 Nb 4 -d C s NiNb 5 C Nb 6 -a C 2v Nb 6 -b D 2h one can see that four isomers exist for the pure Ni 8 clusters, and there is a little difference in binding energy among these four structures. For pure Nb 8 clusters, there are two isomers; one is the bicapped octahedron II, the other is the hexagonal bipyramid. There are 23 possible structures of the binary clusters. Eighteen of them have similar structures as their corresponding pure clusters, and five of them show new structures. The Ni 6 Nb 2 -c shows bicapped octahedron III, in which the two capped atoms are in different positions compared to those of bicapped octahedron II. Ni 6 Nb 2 -e, Ni 5 Nb 3 -a, Ni 4 Nb 4 -a, and Ni 3 Nb 5 -a show a tricapped trigonal bipyramid structure. Same as the m + n = 7 clusters, all the possible compositions of m + n = 8 clusters have their isomers. The Ni 6 Nb 2 composition has five isomers, and the others have four, three, or two isomers. In summary, most of the Ni m Nb n (m + n 8) clusters prefer compact structures when m + n > 3. Ni-rich binary clusters tend to adopt the geometries of their corresponding pure Ni clusters, and Nb-rich clusters tend to adopt the geometries of their corresponding pure Nb clusters. When the total number of atoms (m + n) islarger, more stable structures will have a larger nearest-neighbor distance. For the same size of clusters, the average nearestneighbor distance increases with an increase of the number of Nb atoms, except for two atom clusters. From the tables

6 256 J Supercond Nov Magn (2017) 30: Table 3 The point-group symmetry (PGS), the average coordination number (Nc), the average nearest-neighbor distance d (in Å), the binding energy E b (in ev/atom), the HOMO LUMO gap E g (in ev), and the total magnetic moment Mag (in μ B ) for Ni m Nb n clusters with m + n = 7 PGS Nc d E b E g Mag Ni 7 -a C 3v Ni 7 -b D 5h Ni 6 Nb-a C s Ni 6 Nb-b C Ni 5 Nb 2 -a D 5h Ni 5 Nb 2 -b C s Ni 5 Nb 2 -c C Ni 5 Nb 2 -d C s Ni 4 Nb 3 -a C 2v Ni 4 Nb 3 -b C s Ni 4 Nb 3 -c C Ni 4 Nb 3 -d C Ni 3 Nb 4 -a C 2v Ni 3 Nb 4 -b C 3v Ni 3 Nb 4 -c C Ni 2 Nb 5 -a C Ni 2 Nb 5 -b C Ni 2 Nb 5 -c C Ni 2 Nb 5 -d C NiNb 6 -a C NiNb 6 -b C 2v NiNb 6 -c C s NiNb 6 -d C s Nb 7 -a C Nb 7 -b C s given above, we can also see that the average coordination number increases with the size of clusters. The ground state of Ni m Nb n tends to have a maximum average coordination number and shows more compact comparing with the other isomers. For example, the coordination number of the ground state of Ni 7 Nb is 4.75, named with Ni 7 Nb-a in Fig. 4, which is larger than that of the other isomers. This results in Ni 7 Nb-a being more compact than the others. The second-order difference of energy, E, isalsoan important indicator that measures the relative stability of a cluster with respect to its immediate neighbors. The definition of E is given with (2). E(Ni m Nb n ) = E b (Ni m Nb n ) 1/2[E b (Ni m+1 Nb n 1 ) +E b (Ni m 1 Nb n+1 )] (2) We calculated the second-order difference of energy of the ground state according to (2), and the results are given in Fig. 5. From this figure, we can see that though the second- Table 4 The point-group symmetry (PGS), the average coordination number (Nc), the average nearest-neighbor distance d (in Å), the binding energy E b (in ev/atom), the HOMO LUMO gap E g (in ev), and the total magnetic moment Mag (in μ B ) for Ni m Nb n clusters with m + n = 8 PGS Nc d E b E g Mag Ni 8 -a C 2v Ni 8 -b C s Ni 8 -c C 2v Ni 8 -d C s Ni 7 Nb-a C s Ni 7 Nb-b C Ni 7 Nb-c C s Ni 6 Nb 2 -a C 2v Ni 6 Nb 2 -b C s Ni 6 Nb 2 -c C s Ni 6 Nb 2 -d C 2v Ni 6 Nb 2 -e C 2v Ni 5 Nb 3 -a C s Ni 5 Nb 3 -b C s Ni 5 Nb 3 -c C Ni 4 Nb 4 -a D 2d Ni 4 Nb 4 -b C 2v Ni 4 Nb 4 -c C s Ni 3 Nb 5 -a C s Ni 3 Nb 5 -b C Ni 3 Nb 5 -c C Ni 2 Nb 6 -a C Ni 2 Nb 6 -b C s Ni 2 Nb 6 -c C NiNb 7 -a C s NiNb 7 -b C s NiNb 7 -c C s Nb 8 -a C 2v Nb 8 -b D 3d order difference of energies mainly shows an oscillatory behavior with respect to n, the E still has a highest peak when the amount of Nb atoms is 2. This behavior implies that the corresponding clusters, such as NiNb 2,Ni 2 Nb 2, Ni 4 Nb 2,Ni 5 Nb 2,andNi 6 Nb 2, have a higher relative stability compared with the neighboring binary clusters, and this indicates that it is difficult for the structural evolution from the clusters containing two Nb atoms to the clusters containing one or three Nb atoms. 3.2 Electronic Structure and Magnetic Properties The electronic structures and magnetic properties of the obtained clusters are calculated. The HOMO LUMO gaps and magnetic moment of obtained Ni m Nb n clusters are also

7 J Supercond Nov Magn (2017) 30: Fig. 5 Second-order difference of energy as the function of the number of Nb atoms of the ground state of Ni m Nb n (m + n 8) clusters. The horizontal axis n is the number of Nb atoms in the cluster given in Tables 1, 2, 3 and 4. Generally, the HOMO LUMO gap of clusters is a good indicator for the optical spectrum of the material formed from clusters. Regarding the data in Tables 1, 2, 3 and 4, the average gap of all the binary clusters is 0.28 ev, which is between that of pure Ni n (0.15 ev) and pure Nb n (0.61 ev) clusters. The HOMO LUMO gap (E g ) of the ground state as the function of the composition isgiveninfig.6. It is found that pure Ni m (2 m 8) clusters have small gaps (between 0.1 and 0.2 ev). While the gap in the pure Nb n (2 n 8) clusters strongly depends on the geometry of the cluster, the two isomers Nb 7 have a small gap ( ev), while Nb 2,Nb 4,andNb 8 isomers show a large gap ( ev); other Nb m clusters show their gaps between 0.39 and 0.56 ev, respectively, consistent with previous results [10 19]. The HOMO-LUMO Fig. 6 Average HOMO LUMO gaps of Ni m Nb n (m + n 8) clusters gaps in the bimetallic Ni m Nb n clusters, on the other hand, do not strongly depend on the geometry of the clusters. For a given composition, the gap is fluctuated within a range of 0.2 ev, and for a given size of the cluster, the gap mainly tends to slowly increase with the increasing number of Nb atoms. The exception is found in some Ni m Nb n clusters, for example, Ni 2 Nb 2 has a large gap of 0.52 ev but Ni 1 Nb 3 has asmallgapof0.21ev. Through the analysis of the Bader charge [33] distribution of Ni m Nb n (m + n = 8) clusters, the charge transfer between the Ni and Nb atoms was obtained and are given in Table 5. The results show that in pure Ni or Nb clusters, there is almost no charge transfer. But in the binary clusters, the Nb atoms lost charge while the Ni atoms gain charge. The Nb atom in Ni 7 Nb loses the most electrons ( e), and the Ni atom in NiNb 7 gain the most electrons ( e). Magnetism is an important property of TM clusters. From Tables 1, 2, 3 and 4, we found that all the pure Ni clusters have net magnetization, but only some of the pure Nb clusters have net magnetization. The total magnetic moment of the bimetallic Ni m Nb n clusters was found strongly depending on the symmetry and configuration of the clusters, even for the same composition. For example, the isomers of Ni 3 Nb-a (rhombus symmetry) and Ni 3 Nb-b (tetrahedron symmetry) have different symmetries and different total magnetic moments (i.e., 1.0 and 3.0 μ B );the isomers of Ni 4 Nb 2 -b (octahedron symmetry) and Ni 4 Nb 2 -c (octahedron symmetry) have the same symmetry but different total magnetic moments (i.e., 2.0 and 4.0 μ B );and the isomers of Ni 2 Nb 5 x (x = a, b, c, d) have different symmetries and configurations but have the same total magnetic moment of 1.0 μ B. For further comparison, we plotted the magnetic moment per atom (μ atom ) of the ground state structures for each Ni m Nb n cluster in Fig. 7. We found that for a given size of the Ni m Nb n clusters (i.e., for fixed n + m), there is always some particular cluster having a larger magnetic moment than others, such as NiNb in Ni m Nb n clusters with n + m = 2, Ni 2 Nb in Ni m Nb n clusters with n + m = 3, NiNb 3 in Ni m Nb n clusters with n + m = 4, Ni 3 Nb 2 -a and NiNb 4 -a in Ni m Nb n clusters with n + m = 5, Ni 2 Nb 4 -a in Ni m Nb n clusters with n + m = 6, Ni 4 Nb 3 - ainni m Nb n clusters with n + m = 7, and Ni 4 Nb 4 -a in Ni m Nb n clusters with n + m = 8, respectively. The most interesting finding is that the magnetic moments of NiNb and Ni 2 Nb are even larger than those of the pure Ni 2 and Ni 3 clusters. For understanding the magnetic properties, we also calculated the partial density of states (PDOS) for the most stable structures of m + n = 8clustersasshowninFig.8. The green short dash, blue dash, and red solid curves represent the density of the s, p, andd states, respectively. Gaussian broadening has been used while drawing the PDOS

8 258 J Supercond Nov Magn (2017) 30: Table 5 Net charge (e) of each atom at the ground state of Ni m Nb n (m + n = 8) clusters. The numbers in the first line represent the atom number. The data above the dashed line are for Ni atoms, while the others are for Nb atoms 1st 2nd 3rd 4th 5th 6th 7th 8th Ni Ni 7Nb Ni 6Nb Ni 5Nb Ni 4Nb Ni 3Nb Ni 2Nb NiNb Nb curves and the Fermi level was shifted to zero. We can see from this figure that the DOS near the Fermi energy levels are mainly dominated by delectrons; sand pelectrons make Fig. 7 Magnetic moment per atom of the ground state of Ni m Nb n (m+ n 8) clusters a small contribution. This is consistent with the electronic structure of the pure Ni or Nb bulk system. An obvious shift between the DOS of up-spin and down-spin states can be seen in Fig. 8, except for Nb 8. It exhibits different degrees of exchange splitting for different clusters. The magnitude of the magnetic moments of a cluster is correlated with the degrees of exchange splitting between up-spin and downspin bands. The larger the exchange splitting is, the larger the magnetic moment is. In these Ni m Nb n (m + n = 8) clusters, the Ni 8 cluster has the largest exchange splitting, and the total magnetic moment of Ni 8 is the largest one among all the clusters with eight atoms. While the Nb 8 cluster has no exchange splitting, there is no magnetic moment in this cluster. For the other seven binary clusters, the exchange splitting also exists. But it is much smaller than that of the Ni 8 cluster; therefore, the magnetization is smaller than Ni 8 cluster. With the increase of the number of Nb atoms, the exchange splitting decreases. This behavior is completely consistent with the variation of magnetic moments, which indicates the magnetic moments decrease with respect to the number of Nb atoms.

9 J Supercond Nov Magn (2017) 30: Fig. 8 Partial density of states of the ground state of Ni m Nb n (m + n = 8) clusters. The green dotted, the blue dash, and the red solid curves represent the density of the s, p, and d states, respectively

10 260 J Supercond Nov Magn (2017) 30: Conclusion We obtained 97 stable structures of Ni m Nb n (m + n 8) clusters by the particle swarm optimization and density functional theory. We found that most of Ni m Nb n (m + n 8) clusters prefer compact structures when m + n > 3. Generally, the structures of bimetallic clusters are similar to those of their corresponding pure Ni or Nb clusters. For a giving composition, the ground state of the cluster tends to have a maximum average coordination number and shows more compact compared with its isomers. The results reveal that the stabilities of clusters rely not only on the elements but also on the size, the structure, and the composition. The study of second-order difference of energy revealed that the binary clusters have a higher relative stability when the number of the Nb atom is 2. The HOMO LUMO gaps of the obtained bimetallic Ni m Nb n structures range from 0.07 to 0.52 ev and mainly tend to slowly increase with the increasing number of Nb atoms. The total magnetic moment of the bimetallic Ni m Nb n clusters was found strongly depending on the size, the symmetry, the configuration, and the composition of the clusters. The exchange splitting of PDOS of the clusters is the reason for the appearance of magnetization. The PDOS of eight-atom clusters also show that these clusters are mainly dominated by d electrons. Acknowledgments This work was partly supported by the Shanghai Key Laboratory of high temperature superconductors (No. 14DZ ) and high-performance computing platform of Shanghai University. References 1. Billas, I.M., Châtelain, A., de Heer, W.A.: Science 265, 1682 (1994) 2. Alonso, J.A.: Chem. Rev. 100, 637 (2000) 3. Parida, P., Kundu, A., Pati, S.K.: J. Clust. Sci. 20, 355 (2009) 4. Aguilera-Granja, F., García-Fuente, A., Vega, A.: Phys. Rev. B 78, 1 (2008) 5. Piotrowski, M.J., Piquini, P.: Phys. Rev. B. 81(2008), 1 (2010) 6. Soltani, A., Bouderbala, W., Boudjahem, A.: J. Clust. Sci. 27, 715 (2016) 7. Louderback, J.G., Cox, A.J., et al.: Z. Phys. D. 26, 301 (1993) 8. Billas, I.M., Châtelain, A., de Heer, W.A.: Science 265, 1682 (1994) 9. Khanna, S.N., Linderoth, S.: Phys. Rev. Lett. 67, 742 (1991) 10. Kumar, V., Kawazoe, Y.: Phys. Rev. B. 65, 1 (2002) 11. Parks, E.K., Winter, B.J., et al.: J. Chem. Phys. 94, 1882 (1991) 12. Pinegar, J.C., Langenberg, J.D., et al.: J. Chem. Phys. 102, 666 (1995) 13. Ganteför, G., Eberhardt, W.: Phys. Rev. Lett. 76, 4975 (1996) 14. Desmarais, N., Jamorski, C., et al.: Phys. Lett. 294, 480 (1998) 15. Xie, Z., Ma, Q.M., et al.: Phys. Lett. A. 342, 459 (2005) 16. Futschek, T., Hafner, J., Marsman, M.: J. Phys. Condens. Matter. 18, 9703 (2006) 17. Reuse, F.A., Khana, S.N.: Chem. Phys. Lett. 234, 77 (1995) 18. Castro, M., Jamorski, C., Salahub, D.R.: Chem. Phys. Lett. 271, 133 (1997) 19. Michelini, M.C., Diez, R.P., Jubert, A.H.: Int. J. Quantum Chem. 85, 22 (1999) 20. Li, J., Wang, H., Liu, J., Ruan, J.: Mater.: Sci. Eng. A. 609, 235 (2014) 21. Park, E.S., Kim, D.H., et al.: J. Non. Cryst. Solids 351, 1232 (2005) 22. Weng, F., Yu, H., et al.: J. Mater. Res. 29, 2596 (2014) 23. Hales, D.A., Lian, L., Armentrout, P.B.: Int. J. Mass Spectrom. Ion Process. 102, 269 (1990) 24. Eberhart, R., Kennedy, J.: In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, p. 39 (1995) 25. Kennedy, J., Eberhart, R.: In: Proceedings, IEEE International Conference, p (1995) 26. Wang, Y., Lv, J., Zhu, L., Ma, Y.: Comput. Phys. Commun. 183, 2063 (2012) 27. Wang, Y., Lv, J., et al.: Phys. Rev. B. 82, (2010) 28. Lv, J., Wang, Y., et al.: J. Chem. Phys. 137, (2012) 29. Perdew, J., Burke, K., Ernzerhof, M.: Phys. Rev. Lett. 77, 3865 (1996) 30. Kresse, G.: Phys. Rev. B. 59, 1758 (1999) 31. Kresse, G.: Phys. Rev. B. 54, (1996) 32. Kresse, G., Furthmüller, J.: Comput. Mater. Sci. 6, 15 (1996) 33. Bader, R.F.W., Matta, C.F.: J. Phys. Chem. A. 108, 8385 (2004)