Coexistence of ferroelectricity and ferromagnetism in ...

THE JOURNAL OF CHEMICAL PHYSICS 125, 114305 共2006兲

Coexistence of ferroelectricity and ferromagnetism in tantalum clusters Wei Fa,a兲 Chuanfu Luo, and Jinming Dongb兲 Group of Computational Condensed Matter Physics, National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China and Department of Physics, Nanjing University, Nanjing 210093, China

共Received 20 March 2006; accepted 31 July 2006; published online 18 September 2006兲 The atomic and electronic structures of TaN 共N = 2 – 23兲 clusters have been determined in the framework of pseudopotential density-functional calculations, based upon an unbiased global search with guided simulated annealing to an empirical potential. It is found that the ground-state structures of TaN are very similar to those of NbN, showing no preference for the icosahedral growth. Also, a size- and structure-dependent ferroelectricity is found in these tantalum clusters. More importantly, it is found that the ferroelectricity and ferromagnetism can coexist in the homogeneous transition-metal cluster, offering a possibility to obtain a new type of “multiferroic” materials composed of the clusters. Finally, the far-infrared spectroscopy is suggested to be an efficient tool to distinguish the ferroelectric clusters. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2338890兴 I. INTRODUCTION

Significance of investigating the atomic and electronic structures of metal clusters was recognized a long time ago from the point of view of understanding their growth behavior and associated physical properties.1 Recently, the experimental discovery of the permanent electric dipole moments 共EDM兲 in homonuclear cold free metal clusters 共VN, NbN, and TaN兲 opened an entirely new research area due to their fundamental importance and technological applications.2 On the other hand, the spin coupling properties of these clusters are also anomalous at low temperatures, showing a magnetic moment of 1␮B only for the odd-atomic clusters.3 The occurrence of magnetism can be understood by the odd number of valence electrons in their atoms. So, tantalum is nonmagnetic in bulk while its odd-atomic clusters could be magnetic. Thus, the novel materials in which magnetic and electric orders, coexist, termed “multiferroics” or “magnetoelectrics,”4 may be synthesized from these particular clusters,5 offering another possibility to prepare new bulk materials with both ferromagnetic and ferroelectric orders. Generally speaking, physical properties of clusters are strongly correlated to their geometrical and electronic structures. For example, the EDM magnitude of smaller NbN 共N 艋 28兲 is found to show a strong size-dependent fluctuation, and the larger NbN 共N 艌 38兲 exhibits a pronounced even-odd oscillation of the EDM values.2 Two recent studies based upon density-functional theory 共DFT兲 calculations showed a geometrical origin for the ferroelectric properties of NbN.6,7 It is natural to ask whether the same phenomena exist in TaN since it belongs to the same group VB in the Periodic Table, and so it is obvious that knowledge of the geometrical structures of TaN is necessary. In the past years, the structural properties of VN and NbN a兲

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b兲

0021-9606/2006/125共11兲/114305/5/$23.00

had been intensively studied at the DFT level,8–14 and it is found that the configurations with the maximum average coordination number are preferred in their ground-state structures. The abundance spectra of autoionized NbN and TaN are similar to each other, showing the common magic number of N = 7 , 13, 15, 22, . . ..15 But they are different from other metal clusters, such as Sr, Ba, and Ni, with a maximum in the abundance intensity at N = 19, which are often grown in the icosahedral structures. As suggested in Ref. 14 the icosahedral growth is not favored in NbN; it is an interesting problem to know the growth behavior of TaN. Therefore, in this paper, we report our DFT simulations on the atomic and electronic structures of TaN up to 23 atoms, based upon which their ferroelectric and ferromagnetic properties are discussed. The following section introduces the DFT procedures adopted in our calculations. The obtained results and discussions are given in Sec. III. Some concluding remarks are offered in Sec. IV. II. COMPUTATIONAL APPROACH

We employed the DMOL3 package,16 in which a relativistic effective core potential17 and a double numerical basis, including d-polarization function, were chosen to do the electronic structure calculations. For the exchangecorrelation functional, we adopted the spin-polarized generalized gradient approximation 共GGA兲.18 A real space cutoff of 8.0 Å was used for the numerical integration. Selfconsistent field procedures were done with a convergence criterion of 10−6 a.u. on the total energy and electron density.19 Cluster geometries were determined via symmetryunrestricted structural optimizations20 through an exhaustive search among various structures, including those suggested previously for VN and NbN and new ones obtained from a robust guided simulated annealing21 共GSA兲 with an empirical many-body potential.22,23 Different spin states were also

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FIG. 1. Size evolution of 共a兲 the increase of binding energy 共BE兲 and 共b兲 the second order differences of BE in TaN. Also shown are the lowest-energy structures of tantalum clusters with odd-N TaN drawn in the upper panel, while even ones are in the lower panel.

tested in our calculations. For each TaN with N ⬎ 5, we choose at least eight lower-energy isomers with different symmetries, in which the ground-state structure of TaN was finally determined according to the calculated DFT energy. Nevertheless, it cannot strictly rule out other energetically more favorable structures due to excessive local minimum in the configuration space. Finally, the frequency check was performed and no imaginary frequencies were obtained for the ground-state TaN, verifying that they were true minima on the potential energy surfaces. The zero-point vibrational energy corrections have been included in all calculations of the bonding energies 共BEs兲. The vibrational spectra were calculated in the harmonic approximation using the finite displacement technique to obtain the force constant matrix. Positive and negative displacements with the magnitude of 0.005 Å were used in order to obtain more accurate central finite differences. The infrared intensities were obtained from the derivative of the dipole moment. III. RESULTS AND DISCUSSIONS

To compare the relative stability of different-sized tantalum clusters, we monitor the increase of binding energy 关⌬E共N兲 = E共N兲 − E共N − 1兲兴 and the second order difference of BE 关⌬2E共N兲 = E共N + 1兲 + E共N − 1兲 − 2E共N − 1兲兴, which are presented in Fig. 1, including also the ground-state structures of TaN. The BE 关E共N兲兴 is defined as the difference between the total energy of TaN and the isolated atoms composing the cluster. Both the ⌬E共N兲 and ⌬2E共N兲 exhibit magic numbers at 4, 7, 10, 15, and 22. Additionally, Ta13 shows a weak magic behavior but Ta19 is not magic, agreeing with the measured time-of-flight mass spectra.15 The ground state of Ta2 is found to have a magnetic moment of 4␮B with a bond length of 2.308 Å, which is much shorter than the experimental bulk value of 2.859 Å.

J. Chem. Phys. 125, 114305 共2006兲

Its BE is 4.55 eV and higher than that of the low spin state isomer 共2␮B兲 by 0.33 eV. The vibrational frequency obtained in our calculations is 319.2 cm−1, which is in reasonable agreement with the resonance Raman spectroscopy measured in an argon matrix 共300 cm−1兲.24 The lowestenergy structure of Ta3 is an acute triangle with three slightly different bond lengths 共2.399, 2.544, and 2.589 Å兲, which is about 0.02 eV lower in energy than its isosceles triangle isomer. A linear chain with a bond length of 2.335 Å, separated by a large energy gap of 2.44 eV from the lowest-energy structure, is unstable. Due to a single unpaired spin, the oddatom clusters are ferromagnetic with a magnetic moment of 1 ␮ B. Like the niobium clusters, the ground-state structures of TaN are found to undergo an early transition 共i.e., for N ⬎ 3兲 of the two-dimensional to the three-dimensional configurations. The equilibrium geometry of Ta4 is a distorted tetrahedron, which is more stable than the regular tetrahedron structure by 0.10 eV. There is no cluster with perfect symmetry in the present study, showing that Jahn-Teller distortions play an important role in the geometries of tantalum clusters.25 The lowest-energy structure we found for the pentamer is a heavily distorted trigonal bipyramid, actually it could also be viewed as a capped bent rhombus. For Ta6, a distorted octahedron with an approximate D2h symmetry is more stable than a distorted prism by 0.18 eV. In the case of Ta7, the heavily distorted pentagonal bipyramid 共PBP兲 is about 0.37 eV lower in energy than that of the symmetric PBP and 1.35 eV lower than that of the capped octahedron. The distorted bicapped octahedron with the caps located at two adjacent trigonal faces is favored for Ta8 in energy over other isomers, such as the capped PBP and the transbicapped octahedron. The bicapped PBP is the lowest-energy structure for Ta9, which is more stable than the distorted tricapped trigonal prism by 0.40 eV. The most stable configuration of Ta10 is a bicapped antiprism with an approximate D4d symmetry. The first close-lying isomer of Ta10, based on capping of a pentagonal pyramid by a near rhombus, is separated by an energy gap of 0.51 eV from the ground state. Ta11 is a four-capped PBP, which can also be viewed as two PBPs fused at a triangular face. A bulk fragment buckled by a hexagon and a rhombus, lying 0.37 eV higher in energy than the ground state of Ta11, is energetically degenerate to the three-capped hexagonal bipyramid 共HBP兲. The lowest-energy structure of Ta12 is regarded as capping of Ta11 by an apex atom, and it can be also viewed as a relaxed icosahedron missing one surface atom. Another interesting result is that a distorted five-capped HBP is found to be the most stable structure of Ta13. The relaxed icosahedron and cuboctahedron, predicted previously as the ground-state structure for most 13-atom metal clusters, are not favored for the other amorphous isomers. Two degenerate isomers compete for the global minimum of Ta14, both of which can be derived from a 15-atom icositetrahedron: one removes an apex atom and the other misses a hexagonal surface atom. The former with an approximate C6v symmetry is 0.05 eV lower in energy, which is more stable than a C4v body-centered-cubic 共bcc兲-type structure by 0.27 eV and a capped icosahedron by 1.91 eV. Ta15 possesses a fragment


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structure of the bcc bulk tantalum with a structural relaxation, which has an approximate Oh symmetry and exhibits two distinct peaks at 2.65 and 3.10 Å, respectively, in its bond length distribution. The icositetrahedron and twocapped icosahedron isomers of Ta15 are 0.72 and 2.21 eV higher in energy, respectively. Considering the instability of the icosahedron-based configurations for Ta13 – Ta15, it can be concluded that the icosahedral growth is also not favored in TaN. Ta16 and Ta17 are based on the capping of the icositetrahedron isomer of Ta15 with a dimer and a trimer, respectively, on one of the hexagonal faces, showing that TaN prefers high coordinated structures with 15 and 16 nearest neighbors of the center atom. Though the capped icosahedral isomer is separated from the ground state by a large energy gap, a capped decahedron is found as the first close-lying isomer of Ta17, lying only 0.23 eV higher in energy. Also, the ground states of the 18- and 19-atom Ta clusters are decahedral in nature. The more commonly suggested double icosahedron in other 19-atom clusters is unstable, which is 2.49 eV higher in energy, confirming again that TaN does not prefer the icosahedral growth. The ground-state structures of Ta20 – Ta23 show a trend of the hexagonal-based growth. Ta22 is a relaxed doubleinterpenetrating icositetrahedron, based upon which Ta20, Ta21, and Ta23 can be obtained by removing two surface atoms, removing one apex atom, and adding one bottom atom, respectively. However, the atomic packing with increasing cluster size is frequently changed in the size range of 2–23, making each of them to be an independent system rather than one in a particular growth sequence. It is worth noticing that the growth behavior of TaN is very similar to that of NbN. For example, both of them do not favor the icosahedral growth and show no convergence to the bulk structures up to N = 23. But this similarity may be absent when compared with the vanadium clusters, for which some bulk fragment structures exist in the smaller size range.26 For example, the distorted cuboctahedron of V13 is 1.04 eV lower in energy than the five-capped HBP, which is, however, the ground-state structure of Ta13. Based upon the above lowest-energy structures, the EDMs and magnetic moments of the tantalum clusters in the size range of 2–23 were calculated and shown in Fig. 2, both of which display a dramatic size dependence. Except some symmetric TaN at N = 2, 4–6, 10, 15, and 22 with almost zero electric moments, most of TaN have a moderate EDM value, and those with 9, 11, 12, and 20 atoms attain a large EDM. On the other hand, the odd-N TaN acquire a magnetic moment of 1␮B due to the odd number of electrons. Enhancement of ferroelectricity and ferromagnetism in TaN can be understood from its reduced atomic coordination, leading to the stronger electron localization and unequivalent interatomic distances, which easily induce the asymmetric charge distribution. Ferroelectric properties of TaN 共N = 2 – 23兲 vary considerably with size, which is very similar to the EDM curve calculated for the niobium clusters.6,7 To explore the relationship between the ferroelectricity and structures of TaN, we

FIG. 2. Size evolution of the calculated EDM and magnetic moment of TaN 共N = 2 – 23兲, represented by the filled and open circles, respectively. The inset shows the dependence of the ICN function on cluster size.

introduce an inverse coordination number 共ICN兲 function F共N兲 to reflect the degree of asymmetry for a cluster, F共N兲 =

冨

1 B

N

兺

i,j=1 rij⬍R

cut

冨

Ri/Zi + R j/Zi , 1/Zi + 1/Z j

N Zi is the total bonding number with Zi the where B = 21 兺i=1 coordination number of the ith atom. Rcut is the cutoff distance 共2.95 Å兲,27 and Ri is the position vector of the ith atom 共the coordinate origin is set at the mass center of the cluster兲. The size evolution of the ICN function in TaN is shown in the inset of Fig. 2. Considering that the F共N兲 is decided completely by the cluster’s geometrical structure, we are satisfied with the agreement between the size variations of both the ICN function and EDM of TaN. For example, the F共N兲 values are heavily enhanced at N = 11, 12, and 20, but suppressed to zero at N = 10, 15, and 22, which are the same as those of EDMs. The physical picture behind the ICN function can be understood by the concept of effective charge on an atom as follows: the quantity of effective charge on an atom in a cluster should be inversely proportional to its coordination number, which could just be represented by the ICN function defined here. The similar size dependence of the EDM and ICN function reveals the strong correlation between the geometrical structure of cluster and its ferroelectricity, showing a same geometrical origin of the ferroelectric properties as that of NbN. It is clearly seen from Fig. 2 that our DFT calculations display a possibility of coexistence for the ferroelectricity and ferromagnetism in some TaN. As shown in Fig. 2, the odd-N TaN possesses a magnetic moment of 1␮B, confirmed by the Stern-Gerlach deflection measurement,3 due to an unpaired spin. On the other hand, most of the odd-N TaN acquire a nonzero EDM except Ta15 because of its symmetric structure, which therefore become the multiferroic or magnetoelectric clusters. Since the geometrical structure of a cluster plays an important role in determination of its properties, its structural information is very useful, which may be obtained from a comparison of the infrared resonance-enhanced multiple


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coexistence of ferroelectricity and ferromagnetism in the homogeneous transition-metal cluster, offering a possibility to obtain a new type of multiferroic materials composed of the clusters. Finally, we note that the far-infrared spectroscopy can be an efficient tool to distinguish the ferroelectric clusters. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant No. 90503012 and the state key program of China through Grant No. 2004CB619004. One of the authors 共W. F.兲 also acknowledged support from China Postdoctoral Science Foundation 共No. 2005038572兲. The DFT calculations were made on the SGI-3800 and 2000 supercomputers. 1

FIG. 3. The DFT results for the vibrational and far-infrared 共FIR兲 spectra of Ta10, shown by the solid and dashed lines, respectively, in different geometrical structures: 共a兲 the ground state and 共b兲 the first close-lying isomer. The spectra are folded with a 3 cm−1 Lorentzian broadening. Constant-value images of the charge deformation density, defined as the total charge density minus the density of the isolated atoms, in these two isomers are also depicted in each panel, in which the direction of the EDM is denoted by the orange arrow.

photon dissociation spectroscopy with the DFT calculations, as those done successfully for the vanadium and niobium clusters.28–30 So, we make DFT calculations of the vibrational spectra and corresponding far-infrared spectra for the ground-state structure and the first close-lying isomer of Ta10, shown in Figs. 3共a兲 and 3共b兲, respectively, from which it can be clearly seen that these spectra depend very sensitively on the geometric and electronic structures of the cluster. The ground-state structure of Ta10 has a near zero EDM due to its approximate D4d symmetry, whose vibrational modes at low frequencies are infrared inactive. However, the first close-lying isomer of Ta10 is a three-capped PBP with a large EDM value of 1.076 D, showing the distinctly differently charge distribution from top to bottom as illustrated in the inset of Fig. 3共b兲. Almost all of the vibrational modes of this isomer are infrared active except for one mode at 124.8 cm−1, thus inducing the much richer spectral structure. Other ferroelectric TaN also display rich infrared-active spectra, compared with their nonferroelectric isomers, because of the existence of the electric dipole moment in the ferroelectric material. IV. SUMMARY

In conclusion, we have performed the DFT calculations on the geometrical and electronic structures of TaN up to 23 atoms, based upon which their ferroelectric and ferromagnetic properties are also studied. The geometrical features of TaN show a similarity to those of NbN, both of which favor nonicosahedral growth and have only a bcc fragment structure at N = 15. It is found that the ferroelectricity of TaN up to 23 atoms shows a close relationship with their geometrical structures. More importantly, our calculations have found the

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Lindsay, J. Chem. Phys. 103, 3289 共1995兲兴. R. Fletcher, Practical Methods of Optimization 共Wiley, New York, 1980兲, Vol. 1. The total energy was converged to 10−6 a.u. in the self-consistent loop and the force on each atom was less than 10−3 a.u. 21 C. I. Chou and T. K. Lee, Acta Crystallogr., Sect. A: Found. Crystallogr. A58, 42 共2001兲; C. I. Chou, R. S. Han, S. P. Li, and T. K. Lee, Phys. Rev. E 67, 066704 共2003兲. 22 M. W. Finnis and J. E. Sinclair, Philos. Mag. A 50, 45 共1984兲. 23 G. J. Ackland and R. Thetford, Philos. Mag. A 56, 15 共1987兲. 24 Z. Hu, B. Shen, J. R. Lombardi, and D. M. Lindsay, J. Chem. Phys. 96, 8757 共1992兲. 20

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J. E. Fowler, A. García, and J. M. Ugalde, Phys. Rev. A 60, 3058 共1999兲. H. Wu, S. R. Desai, and L. Wang, Phys. Rev. Lett. 77, 2436 共1996兲. 27 The cutoff distance was verified for all the tantalum clusters by calculating the distribution of atom pair-correlation function. 28 A. Fielicke, A. Kirilyuk, C. Ratsch, J. Behler, M. Scheffler, G. von Helden, and G. Meijer, Phys. Rev. Lett. 93, 023401 共2004兲. 29 C. Ratsch, A. Fielicke, A. Kirilyuk, J. Behler, G. Helden, G. Meijer, and M. Scheffler, J. Chem. Phys. 122, 124302 共2005兲. 30 A. Fielicke, C. Ratsch, G. Helden, and G. Meijer, J. Chem. Phys. 122, 091105 共2005兲. 25 26
