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Structure-dependent ferroelectricity of niobium clusters (NbN, N=2-52)

The ground-state structures and ferroelectric properties of NbN (N=2-52) have been investigated by a combination of density-functional theory (DFT) in the generalized gradient approximation (GGA) and an unbiased global search with the guided simulated annealing. It is found that the electric dipole moment (EDM) exists in the most of NbN and varies considerably with their sizes. And the larger NbN (N>=25) prefer the amorphous packing. Most importantly, our numerical EDM values of NbN (N>=38) exhibit an extraordinary even-odd oscillation, which is well consistent with the experimental observation, showing a close relationship with the geometrical structures of NbN. Finally, an inverse coordination number (ICN) function is proposed to account for the structural relation of the EDM values, especially their even-odd oscillations starting from Nb38.


I. INTRODUCTION
Transition metal clusters have long been of considerable interest due to their importance in fundamental researches and tremendous potential technical applications, among which niobium cluster is one of the most thoroughly studied in both of their chemical and physical properties. Even so, the fully understanding of their dramatic size-dependent properties is still a great challenge. Recently, the ferroelectricity (FE) in free niobium clusters (Nb N , N=2-150) has been experimentally found at low temperatures 1,2 , showing the existence of electric dipole moments (EDMs), which, more importantly, exhibit a pronounced even-odd oscillation for Nb N (N ≥38). Such an interesting phenomenon has attracted immediate attention because the FE has never been found in single-element bulk materials and neither in metals. For example, a purely electronic mechanism has been proposed, considering the FE of Nb N to be caused by the electron correlations 3 . However, the large size-dependent EDMs, especially the remarkable even-odd oscillation starting from N = 38, strongly support existence of an intimate relation between the FE and structures of Nb N . Since the existing techniques cannot conclusively determine the geometrical structures of clusters, the numerical simulation on them becomes a useful method to provide more valuable information on their structures and novel physical properties.
Several first-principle calculations on Nb N up to 23 atoms had been performed 4−6 . It was concluded that the icosahedral growth is not favored for Nb N 6 . Other studies made on the smaller Nb N (N ≤10) found the similar lowest-energy structures with the high coordinated configurations. Recently, a close relationship between the asymmetrical geometrical structures of Nb N (N ≤15) and their EDMs has been revealed through a first-principle study 7 .
However, there has been no DFT study on the larger Nb N and their ferroelectric properties.
Therefore, in this paper, we will address the growth pattern of Nb N (N ≤52) and its

II. COMPUTATIONAL METHOD
Instead of the presumed symmetry constraints, an unbiased global search on the potential energy surface of clusters is preformed by the guided simulated annealing (GSA) method 8,9 , which incorporates a guiding function (GF) in the traditional simulated annealing to find the global energy minimum. In our calculations, the density of atoms and the second moment of the mass distribution 10 are adopted as the GFs. The interaction between Nb atoms is represented by an empirical many-body potential 11,12 , which has been successfully applied to predict the equilibrium geometries of Nb N (N ≤14) 13 . A number of lower-energy isomers are thereby generated, representing different local stable states in the phase space.
Then, the DMol3 package 14 based on DFT is used to further optimize the cluster structures by selecting at least five isomers with different symmetries for a given size from the above step, in which a relativistic effective core potential (RECP) 15

III. RESULTS
The calculated binding energies (BEs) per atom are shown in Fig. 1, in which the isomers and the ground-state structures of some clusters are also included. For small Nb N (N=2-12) 20 , our lowest-energy configurations and isomeric order are similar to those found in Ref. 6 except for Nb 8 , which has several competitive isomers, such as a capped distorted pentagonal bipyramid and a C 2v capped octahedron, within an energy interval of 0.1 eV.
Though the atomic packing of the small clusters is gradually changed from one size to another, there exist several structural transitions for medium-sized clusters of N=13-24, e.g. from the icosahedron to the icositetrahedron, as illustrated in Fig. 1. Nb 13 is a distorted icosahedron. By adding one atom to Nb 13 , one planar pentagon in it will be changed into the first hexagonal ring, forming the structure of Nb 14 . Again, one more atom Based upon the above structures, the second order difference ∆ 2 E(N) of BE varying with size is calculated and shown in the inset of Fig. 1 Obviously, the EDM of a niobium cluster is determined by its asymmetrical charge distribution (CD), which should be greatly influenced by its geometrical structure, such as its shape, surface atom number, and inter-atomic distances deviated from the bulk values, all of which induce a deformed CD. For example, the lowest-energy geometrical structure of Nb 6 is a distorted prism, which is more stable than the octahedron only by 0.03 eV. increasing, the oscillation amplitude becomes smaller and smaller, and finally disappears above 100 K. In our calculations, the isomer's influence on the EDM value plays also a "finite temperature" effect. The better the structure optimization is, the clearer the even-odd oscillation behavior. In addition, we also found from Table 1  Due to the un-equivalent geometrical surroundings of each atom in a cluster, the electron density should not distribute uniformly between two nearby atoms forming a chemical bond, leading to different effective charges on the atoms of a Nb N and so emergence of its EDM.
It is obvious that the effective charge on an atom decreases with increase of its coordination number, i.e. an atom with higher coordination number should possess less effective charge.
Therefore, we adopt simply an inverse coordination number (ICN) as a weight index to quantify the effective CD in a Nb N , which could be represented by a function F (N), defined as: Z i is the total bonding number with Z i the coordination number of the ith atom. R cut is the cutoff distance (3.3Å), and ⇀ R i is the position vector of the ith atom (the coordinate origin is set at the mass center of the cluster).
The F (N) values of Nb N (N= 2-52) are shown in Fig. 3 (b), which displays almost the same variation behavior as the EDMs, demonstrating it correctly describes the CD deformation in Nb N . For example, for the smaller Nb N , the ICN function reproduces again 6 the local maxima and minima of EDMs at N=6, 11,18,20,24,28,30 and at N = 4, 7, 10, 13, 15, 17, 19, 29, respectively. Specially, for the larger Nb N with N ≥38, the ICN values are enhanced for even clusters, but suppressed for odd ones, reproducing the extraordinary even-odd EDM oscillation shown in Fig. 3(a), which clearly indicates that the ICN function

IV. CONCLUSION
In summary, the equilibrium geometries, relative stabilities, and EDMs of Nb N (N =