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Recent theoretical progress on electronic and structural properties of clusters: Permanent electric dipoles, magnetism, novel caged structures, and their assemblies
Computational Materials Science 35 (2006) 375–381 www.elsevier.com/locate/commatsci
Recent theoretical progress on electronic and structural properties of clusters: Permanent electric dipoles, magnetism, novel caged structures, and their assemblies Vijay Kumar
*
Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan Dr. Vijay Kumar Foundation, 45 Bazaar Street, K.K. Nagar (West), Chennai 600 078, India Received 12 September 2004; received in revised form 17 October 2004; accepted 31 October 2004
Abstract We present a brief account of the recent progress in the theoretical understanding of the electronic and structural properties of clusters of metals and semiconductors from ab initio calculations. The origin of the recently observed permanent electric dipoles in Nb clusters, the occurrence of magnetism in clusters of non-magnetic elements such as Pd, Rh, and Ru, as well as the findings of the metal encapsulated clusters of Si, Ge, Sn, and Pb are discussed. Empty caged clusters of Si, Ge, and Sn have also been shown to be stable with H capping. These can be functionalized by exohedral doping for different applications while endohedral doping of the cages can be used to tailor highest occupied-lowest unoccupied molecular orbital gaps as well as the magnetic properties. Assemblies of such clusters could lead to novel nanostructures and new phases of these materials. Ó 2005 Elsevier B.V. All rights reserved. PACS: 61.46.+w; 61.48.+c; 73.22.f; 75.75.+a; 77.80.e Keywords: Permanent electric dipole moments; Magnetism; Metal clusters; Silicon cages
1. Introduction In recent years there has been a major surge in studies of clusters, nanoparticles and other novel nanostructures of matter as these are important in a variety of applications such as catalysis, semiconductor devices, sensors, optical as well as magnetic storage materials, biological systems, and so on. There is, however, a major bottleneck in the understanding of properties as structures are often difficult to obtain from experiments and one has to rely on detailed atomistic calculations.
* Address: Dr. Vijay Kumar Foundation, 45 Bazaar Street, K.K. Nagar (West), Chennai 600 078, India. E-mail address: [email protected] URL: http://www.vkf.in
0927-0256/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.10.012
Different structures as well as charge states for a given number of atoms in small systems can give rise to quite different behaviors. Ab initio calculations [1,2] are playing a key role and have come to a level of designing nanomaterials with desired properties. In bulk the chemical properties of elements, composition, and the structure play important roles in determining their behavior. For clusters size and shape become additional variables to play with and new phenomena may occur due to quantum confinement. In this brief review, we take a few examples related to the occurrence of permanent electric dipole moments (EDMs) in clusters of elements such as Nb [3], magnetism in clusters of non-magnetic elements [4–6], and metal (M) encapsulated clusters of group 14 elements [7–11]. These subjects are currently of much interest and have been attracting a lot of attention. The calculations have been done using
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the ab initio ultrasoft pseudopotential method [12] with generalized gradient approximation as well as using the gaussian program [13] with B3PW91 hybrid exchangecorrelation functional. For details, the reader is advised to see the relevant papers.
2. Results 2.1. Permanent electric dipoles in elemental clusters EDMs are known in multicomponent materials and molecules such as H2O. However, the finding of the permanent EDMs in clusters of elements such as Nb [3] is intriguing as large charge separation in homonuclear clusters is unlikely. For the occurrence of EDM an asymmetry should exist such as in a diatomic molecule NaCl. In homonuclear crystals an electric dipole layer forms on surfaces that also break the crystal symmetry. Clusters and nanoparticles have a large fraction of atoms lying on the surface and the electronic charge density decays into the vacuum as on crystal surfaces. Accordingly some charge is depleted from the cluster. This also leads to oscillations in the electronic charge density distribution [1] inside the cluster which can change significantly as a function of the size and the number of electrons in the cluster due to quantum mechanical electronic shell effects. Within the classical picture this would lead to the formation of local electric dipoles around the cluster/nanoparticle surface and further contributions arising due to the spacial variation of charge in the cluster. In the case a cluster has inversion symmetry or the vector sum of the local electric dipoles is zero, no net EDM would exist on the cluster. This is also true for a multicomponent cluster such as Ti@Si16 which has tetrahedral symmetry [7] with Ti at the center and the EDM is zero. However, distortions in the structure or low-symmetry as well as possible local variations in the bonding nature may lead to non-zero EDM. Therefore, the occurrence of EDMs should be quite common in all clusters including those of elemental clusters of metals and semiconductors. Moreover, it further follows that all supported clusters will have EDMs varying with size, shape, and orientation because of the inherent asymmetry. In general there can be three behaviors of EDMs in homonuclear clusters. (1) Zero net EDM in clusters with inversion symmetry or other high symmetries such as tetrahedron, (2) small EDM in nearly symmetric clusters with e.g., Jahn-Teller distortions, and (3) large EDMs due to the inherent lack of symmetry in a cluster as for certain sizes a symmetric structure may not be possible. Also variations in the local environments of atoms can lead to some charge transfer from one region to another even in homonuclear clusters and contribute to the formation of the EDM.
Fig. 1. Two nearly degenerate isomers of Nb13. (a) is the same as reported in Ref. [15] and (b) lies 0.21 eV lower in energy [14].
The origin of EDMs in NbN clusters has been studied recently [14]. It has been shown that their nonicosahedral asymmetric structures [15] are partly due to the covalent bonding that leads to the formation of permanent EDMs in these clusters. Symmetric clusters such as Nb4 have zero EDM. The calculated EDMs are in overall good agreement with experiments [3] and give support to the lowest energy structures predicted from theory as well as to the physical picture described above. However, the earlier reported structure of Nb13 (Fig. 1a) has a large EDM of 2.57 D as compared to the experimental value of 1.5 D. Further search for a lower energy structure led to another isomer (Fig. 1b) which is 0.21 eV lower in energy and has a dipole moment of 1.39 D that is in good agreement with the experimental result. For Nb10 zero or about 0.8 D EDM was obtained while the calculated EDM [14] in the lowest energy isomer is nearly zero due to the nearly symmetric structure [15], suggesting the presence of isomers in experiments. Isomers of NbN, N = 9–12 were also reported [16] from studies of reactivity of Nb clusters. Besides Nb clusters, Al14 and Si10 with capped icosahedral and tetracapped tetragonal prism structures have 1.22 and 0.82 D EDM, respectively. Therefore as discussed above there is nothing particular about EDM in Nb clusters. An interesting fallout of the occurrence of EDM is that it can be used to separate isomers and prepare size selected assemblies of neutral clusters. The EDMs on Nb clusters have been found [3] to vanish with an increase in temperature. It was speculated that this ferroelectric behavior could be a signature of the occurrence of superconductivity in bulk Nb. However, calculations [14] show that the existence of permanent EDMs is intrinsic to the atomic structure of the clusters and not related to superconductivity. At non-zero temperatures the EDM will fluctuate. It can have zero thermally averaged value if there is a transition to a symmetric structure. These thermal as well as the recently reported spin uncoupling effects [17] are yet to be understood.
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2.2. Magnetism in clusters Small particles of elements such as Fe, Ni, Ru, Rh, Pd, and Pt are good catalysts and have attracted much interest [1]. Fe and Ni are magnetic in bulk and the magnetic moments in their clusters are enhanced [18] due to the localization of electrons and the narrowing of the distribution of states because of the mean lower coordination of atoms as compared to bulk. This is also the reason that clusters of non-magnetic elements such as Ru, Rh, and Pd become magnetic [4–6]. These elements lie below Fe, Co, and Ni, respectively in the periodic table and the narrowing of the distribution of electronic states in clusters makes them behave somewhat close to the 3d bulk behavior. The magnetic moments on RhN clusters tend to vanish beyond N 100 [4] while Ru and Pd clusters have only small magnetic moments. Calculations [5,6,19] on Ru, Rh, and Pd clusters found icosahedral growth. PdN clusters with N 6 147 [19] show significant magnetic moments, though the energy cost for a lower magnetic moment isomer is quite small. Pd55 and Pd147 have 0.47 and 0.41lB/atom magnetic moments, respectively. Their cubic isomers have lower magnetic moments (0.18 and 0lB/atom, respectively) and can be present in experiments as the energy differences are small. Also other non-icosahedral lowsymmetry isomers may lie close in energy [20]. Interaction with H as well as O leads to [19] a reduction in the magnetic moment of Pd clusters. When 8 H atoms are associated with Pd13, all the magnetic moments (8lB) get quenched. For Rh clusters the calculated magnetic moments are significantly higher than the observed values [4]. The calculated magnetic moments on Rh13 is 1.62lB/atom [5] in the icosahedral structure in contrast to the experimental value [4] of 0.48 ± 0.13lB/atom. Similarly high magnetic moments have been reported by Reddy et al. and Jinlong et al. from ab initio calculations [6]. Recently a detailed study [20] of the atomic structures and magnetic properties of Rh clusters has shown non-icosahedral growth to be more favorable. These isomers also have generally lower magnetic moments as compared to the icosahedral ones and which are in closer agreement with experiments. For Ru and Rh the magnetic moments decrease [5] rapidly to zero as compared to Pd with an increase in size. Icosahedral Ru and Rh clusters with 147 atoms have nearly zero magnetic moments in agreement with experiments [4] and already approach the bulk value. Similarly in the case of icosahedral Ni147 the magnetic moment is calculated [21] to be 0.68lB/atom within GGA. This is also very close to the bulk value of 0.6lB/atom and the experimental value obtained by Billas et al. [18] at 78 K. Photoionization [22] and chemical reactivity [23] experiments suggest that Ni clusters are icosahedral upto about 120 and 800 atoms, respectively. Therefore, the calculated results agree with these
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experiments very well and show that around 100 atoms or so, the average magnetic moments in these clusters already become close to the bulk value, though the progression to bulk behavior is slow in Pd clusters. In all above studies the orbital contribution to the magnetic moments has been ignored. Guirado-Lopez et al. [24] obtained 0–0.25lB/atom orbital magnetic moments in Rh clusters using a tight binding method and fcc structures with bulk nearest neighbor bond lengths. However, Rh clusters are not fcc and the bond lengths are significantly shorter than the bulk value [5,20]. The magnetic moments and the magnetic anisotropy are sensitive to the structure as well as the size of the clusters. The loss of spherical symmetry in clusters as compared to atoms tends to quench the orbital moments. Therefore, the low-symmetry non-icosahedral structures of Rh clusters indicate that the orbital contribution to the total magnetic moment may be small. On the other hand for Ni and Pd clusters, the icosahedral structures are believed to be lowest in energy. The high symmetry and nearly spherical shape of some of these clusters may lead to significant orbital magnetic moment. A recent study [25] also indicates greater importance of orbital contribution in Ni clusters as compared to those of Rh. 2.3. Metal encapsulated cage clusters Recently novel M encapsulated cage clusters M@Xn, X = Si, Ge, and Sn and n = 10–16, have been predicted [7–10] from computer experiments based on ab initio calculations. It has been shown that one M atom changes completely the structures of these clusters and leads to fullerenelike, cubic, Frank–Kasper (FK) polyhedra, icosahedral as well as other polyhedral forms with higher stabilities as compared to the elemental Si, Ge, and Sn clusters. The relative sizes of the M and the X atoms play an important role in deciding the optimal number of X atoms that can be wraped around an M atom. This unique property can be exploited to produce mass selected clusters of silicon and other elements/ compounds with desirable properties. Indeed in recent experiments some of these predictions have come to be þ true. Al@Pbþ 12 and Al@Pb10 have been produced [26] with high abundances and little intensities of other clusters. These are isoelectronic to the predicted [9] divalent M atom doped magic clusters of X atoms. Recently high intensity of neutral Ti@Si16 clusters has also been obtained [27] with little intensities of other Ti doped clusters, in quite a similar manner as C60 [28], supporting the predicted electronic and geometric stability of this cluster [7]. The high stability of Ti@Si16 arises due to the strong covalent bonding between Ti atom and the silicon cage as well as the occurrence of a large highest occupied-lowest unoccupied molecular orbital (HOMO-LUMO) gap.
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Fig. 2. High symmetry M encapsulated cage clusters of X atoms. (a) Bicapped tetragonal antiprism for Pt@Sn10, (b) icosahedral Mn@Ge12, (c) hexagonal prism Si12W, (d) cubic Fe@Si14, (e) fullerene Zr@Si16, and (f) FK-Ti@Si16. M atom is inside the cage.
These developments could lead to assemblies of such clusters and the design of silicon based molecular devices. The calculated electron affinities (EAs) [7,29] are generally in good agreement with experiments. V@Siþ 16 is isoelectronic to Ti@Si16 and the FK isomer is 0.314 eV lower in energy as compared to the fullerene isomer [30]. It has a large HOMO-LUMO gap of 3.59 (2.44) eV within the hybrid B3PW91 functional (GGA) as in the case of Ti@Si16. This suggests that the charged clusters are likely to be luminescent in the visible region as it was shown [29] for Ti@Si16. Similarly Sc@Si 16 is isoelectronic to Ti@Si16. Also Si12V is isoelectronic to Si12Cr which is magic [31] similar to Si12W. It has a large HOMO-LUMO gap [30] of 2.39 (1.54) eV within B3PW91 (GGA) and should be very stable and abunþ dant as in the case of Al@Pbþ 12 and Al@Pb10 discussed above. Therefore, charging of clusters could make it possible to produce certain species exclusively because of their specific stability as well as act as a switch. Recently switching of the charged state of a single gold atom on the surface of an insulator has been demonstrated [32] by a voltage pulse in scanning tunneling microscope. The M encapsulated clusters can also change shape and the HOMO-LUMO gap significantly on charging [29]. This could also act as a switch providing novel possibilities for devices. Recently Ta@Si12 clusters with a large electron affinity of 4.06 eV have been deposited on a silicon substrate and charge transfer has been observed from the substrate to the clusters [33] and possibilities of fabricating the next generation Si-MOS transistors have been suggested. Calculations [30] on Nb and Ta doped clusters of Si show strong stability. Si12Nb and Si12Ta have large HOMO-LUMO gaps of 1.82 and 1.96 eV in the hexagonal prism struc-
ture of Si12W (Fig. 2). These are much higher than the values [8] of 0.85 and 1.34 eV for Si12M, M = Cr and W, respectively and should lead to their strong abundances. The transition M atom provides strong stability to the X clusters. When the valence of the M atom matches with the holes in the electronic shell of the X cage as well as the relative sizes of the M atom and the X cage fit well, we obtain an extraordinary stability of these clusters. This is the case for fullerene-like Zr@Si16, FKTi@Si16 (Ti, being a slightly smaller atom, prefers a different cage than Zr) [7], icosahedral Zn@Ge12 [9] and other clusters. For Ni, Ge10 is the best cage but for Sn10, Pt is best suited [34] with large HOMO-LUMO gaps. These are some of the smallest cages that can be stabilized with an M atom. For divalent M atom such as Be, X10 and X12 cages are the best [9]. X10 clusters are magic [1] but the stability of such magic clusters is further enhanced by M doping. For Fe, Ru, and Os, a 14-atom cage is best suited [7] while for Cr, Mo, and W, a 15-atom cage is the best [8]. It has also been possible to stabilize a Si20 cage [35]. Also icosahedral magnetic superatoms Mn@Ge12 and Mn@Sn12 have been predicted [9] with large magnetic moments of 5lB as well as large HOMO-LUMO gaps. These features make such clusters attractive for assembly. Selected high symmetry structures are shown in Fig. 2. The structures of these clusters can be probed using gas interaction. Experiments [36] and calculations [37] on water interaction with Ti@Sin, n = 13–16 show it to be weak. But smaller clusters have open basket-like structures in which the M atom is not fully covered with Si and can interact strongly with water. This result establishes the encapsulation of the M atom in a cage for non-reactive clusters.
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The cagelike structure of Si12W clusters was also inferred [38] from weak adsorption of hydrogen. A similar behavior was found for Si18W2 which has a double hexagonal prism structure [39] and weak interaction with hydrogen. Raman and infrared vibrational modes are also [29] useful probes for establishing the structure of these clusters. The basket or M in a cup structures are interesting for catalysis as these provide a nice way to isolate an M atom or a group of M atoms. Often in catalysis, a molecule is dissociated on a certain number of active sites. For example breaking a C–H bond may require one Ni atom while breaking a C–C bond may require more than one Ni atoms to be nearest neighbors. In order to control this and achieve selectivity, an inactive element is added to form bimetallic clusters such as CuNi. In the same spirit, it could be possible to prepare clusters with an appropriate number of catalytically active sites of atoms and partially surround them by an inactive element. Such a behavior might already be happening in bimetallic clusters of e.g., Cu–Ni in which (inactive) Cu segregates on the surface [40]. This could lead to novel design of catalysts with minimum amount of the active material. These developments have opened up new directions in finding novel Si and Ge based nanostructures. Empty cage hydrogenated fullerenes X20H20, X = Si [39], Ge and Sn [41] have also been predicted (Fig. 3). The stability of Si20 fullerene cage is particularly interesting because C20 fullerene cage is difficult to form [42] as there is significant strain in the sp2 bonding and the bonding becomes more sp3 like. In carbon fullerenes, pentagons are the places of strain, but pentagons are most favored in Si cages as it likes to be sp3 bonded. Presence of rhombii or hexagons creates strain in Si nanocage structures [10]. The fullerene isomers of some of the M encapsulated silicon clusters [7,10] have rhombii as the number of vertices is less than 20 which is the minimum number to have a fullerene structure with all pentagonal (12) faces. Therefore, in these clusters there is isolated rhombus rule [10] similar to the
Fig. 3. (a) Dodecahedral empty center cage of Si20H20. H atoms are shown by small spheres. (b) Si28W2 with M atoms inside the cages.
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isolated pentagon rule in carbon fullerenes. The GGA HOMO-LUMO gaps in hydrogenated empty cage X clusters are large ( 3 eV) making them interesting for optoelectronic and biological applications and for developing photo-absorption materials. The endohedral [41] and exohedral [43] derivatives of these clusters can lead to novel possibilities of functional magnetic materials as well as sensors and other applications. 2.4. Assemblies of clusters Assemblies of M encapsulated clusters of Si and other elements can be produced in different ways possibly by depositing on a surface (2D structures), growing one dimensional structures or bulk phases similar to solid C60. The binding energy between two Zr@Si16 clusters is relatively weak (1.35 eV within GGA) [7] and there is a significant reduction in the HOMO-LUMO gap from 1.58 eV for Zr@Si16 to 0.67 eV for the dimer. However, for the FK isomer of Ti@Si16, this interaction is very weak (0.05 eV) and the gap remains nearly the same as for the cluster (2.36 eV), making these species attractive for self-assembly. Molecular dimers [7] with two Z M atoms replaced with (Z 1) and (Z + 1) M atoms in the fullerene structure such as Si32ScNb, Si32ScTa, Si32YTa, and Si32YNb with 0.87, 0.89, 0.82, and 0.82 eV HOMO-LUMO gap, respectively and Si18TaRe and Si18TiFe with HOMO-LUMO gap of 1.08 and 0.84 eV, respectively in the hexagonal biprism structure [39] have been obtained [44]. Even nanocapsules such as Si28W2 in which two fullerene structures are fused (Fig. 3) have been found [44] to be stable with a HOMO-LUMO gap of 0.62 eV. The fullerene structure remains almost undistorted. It is likely that such fused empty cage structures of silicon and other group 14 elements can be formed with H termination in different forms. A nanowire of Zr@Si16 fullerenes has been predicted [44] to be semiconducting with nearly the same gap as bulk silicon. Another important development has been the finding of the silicon nanotube by the assembly of Si12Be clusters [45]. These predictions have found support from recent STM experiments [46] in which Be was deposited on a Si(111) surface and under heating conditions, a self-organized structure was obtained on a large surface area with sticks of Si24Be2 finite nanotubes and two more forms, boomrang (two such finite nanotubes fused together at an angle of 120°) and propeller (three finite nanotubes fused together symmetrically) structures. These experiments would provide further impetus for more experiments on these systems. Studies on assemblies of Si12M clusters with M a transition metal atom [47] have led to the finding of nanotubes with high magnetic moments. Similar studies have also been done on Ge nanotubes [48] which show novel piezo-magnetic behavior. These nanotubes are
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metallic and interesting for interconnects in miniature devices and magnetic applications. Elemental silicon of such a diameter tend to be distorted and has tendency to agglomerate [45]. Recently a semiconducting nanotube of Ge has also been obtained [49]. This is interesting from the point of view of devices. These studies have opened up new possibilities of developing novel structures of silicon and germanium that could have important implications for the future nanodevices.
3. Summary In summary, we have presented a brief review of some recent theoretical progress primarily derived from our own work. The permanent electric dipoles in elemental clusters arise due to asymmetry and are neither specific to Nb nor related to superconductivity. Significant progress has also been made in understanding magnetism in clusters of non-magnetic elements. This would lead to a better understanding of their catalytic properties and design of new catalysts. The M encapsulated Si, Ge, Sn, and Pb caged clusters offer exciting possibilities of size selected mass production. The shape, size and the HOMO-LUMO gap in these clusters depend upon the M and the X atoms. These results show that one M atom changes the properties of elemental Si, Ge, Sn, and Pb clusters drastically and have opened up new avenues for the development of novel elements for future nanodevices as well as new varieties of nanostructures of different materials using the idea of encapsulation. The strong stability of these clusters arises from the strong M–X interactions that also fix the size of the X cage. Many of these predictions have been confirmed by recent experiments. Assemblies of such clusters have interesting electrical, magnetic and optical properties. Similar to Si, novel core-cage structures have also been predicted for II–VI nanoparticles [50]. This has enriched this field and hopefully it will facilitate a better understanding of a large body of data on these technologically important nanoparticles.
Acknowledgements I am grateful to Y. Kawazoe for all the support and cooperation. I thank A. Kasuya, F. Pichierri, R. Belosludov, V. Sundararajan, T.M. Briere, C. Majumder, A.K. Singh, H. Kawamura, Y.-C. Bae, W.E. Pickett, and K.E. Anderson for discussions. I gratefully acknowledge the kind hospitality at the Institute for Materials Research and the support of the staff of the Center for Computational Materials Science of IMRTohoku University for the use of SR8000/H64 supercomputer facilities.
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V. Kumar, Y. Kawazoe, Phys. Rev. Lett. 90 (2003) 055502. V. Kumar, Phys. Rev. B 23 (1981) 3756. V. Kumar, Y. Kawazoe, to be published. H. Prinzbach, A. Weiler, P. Landenberger, F. Wahl, J. Wo¨rth, L.T. Scott, M. Gelmont, D. Olevano, B.v. Issendorff, Nature 407 (2000) 60. F. Pichierri, V. Kumar, Y. Kawazoe, Chem. Phys. Lett. 383 (2004) 544. V. Kumar, Y. Kawazoe, to be published. A.K. Singh, V. Kumar, T.M. Briere, Y. Kawazoe, Nano Lett. 2 (2002) 1243; A.K. Singh, V. Kumar, Y. Kawazoe, J. Mater. Chem. 14 (2004) 555. A.A. Saranin, A.V. Zotov, V.G. Kotlyar, T.V. Kasyanova, O.A. Utas, H. Okado, M. Katayama, K. Oura, Nano. Lett. 4 (2004) 1469. A.K. Singh, T.M. Briere, V. Kumar, Y. Kawazoe, Phys. Rev. Lett. 91 (2003) 146802. A.K. Singh, V. Kumar, Y. Kawazoe, Phys. Rev. B 69 (2004) 233406. A.K. Singh, V. Kumar, Y. Kawazoe, Phys. Rev. B 71 (2005) 075312. A. Kasuya, R. Sivamohan, Y.A. Barnakov, I.M. Dmitrik, T. Nirasawa, V.R. Romanyuk, V. Kumar, S.V. Mamykin, K. Tohji, B. Jeyadevan, K. Shinoda, T. Kudo, O. Terasaki, Z. Liu, R.V. Belosludov, V. Sundararajan, Y. Kawazoe, Nature Mater. 3 (2004) 99; V. Kumar, R. Belosludov, Y. Kawazoe, A. Kasuya, to be published.
Recent theoretical progress on electronic and structural properties of clusters: Permanent electric dipoles, magnetism, novel caged structures, and their assemblies Vijay Kumar
*
Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan Dr. Vijay Kumar Foundation, 45 Bazaar Street, K.K. Nagar (West), Chennai 600 078, India Received 12 September 2004; received in revised form 17 October 2004; accepted 31 October 2004
Abstract We present a brief account of the recent progress in the theoretical understanding of the electronic and structural properties of clusters of metals and semiconductors from ab initio calculations. The origin of the recently observed permanent electric dipoles in Nb clusters, the occurrence of magnetism in clusters of non-magnetic elements such as Pd, Rh, and Ru, as well as the findings of the metal encapsulated clusters of Si, Ge, Sn, and Pb are discussed. Empty caged clusters of Si, Ge, and Sn have also been shown to be stable with H capping. These can be functionalized by exohedral doping for different applications while endohedral doping of the cages can be used to tailor highest occupied-lowest unoccupied molecular orbital gaps as well as the magnetic properties. Assemblies of such clusters could lead to novel nanostructures and new phases of these materials. Ó 2005 Elsevier B.V. All rights reserved. PACS: 61.46.+w; 61.48.+c; 73.22.f; 75.75.+a; 77.80.e Keywords: Permanent electric dipole moments; Magnetism; Metal clusters; Silicon cages
1. Introduction In recent years there has been a major surge in studies of clusters, nanoparticles and other novel nanostructures of matter as these are important in a variety of applications such as catalysis, semiconductor devices, sensors, optical as well as magnetic storage materials, biological systems, and so on. There is, however, a major bottleneck in the understanding of properties as structures are often difficult to obtain from experiments and one has to rely on detailed atomistic calculations.
* Address: Dr. Vijay Kumar Foundation, 45 Bazaar Street, K.K. Nagar (West), Chennai 600 078, India. E-mail address: [email protected] URL: http://www.vkf.in
0927-0256/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.10.012
Different structures as well as charge states for a given number of atoms in small systems can give rise to quite different behaviors. Ab initio calculations [1,2] are playing a key role and have come to a level of designing nanomaterials with desired properties. In bulk the chemical properties of elements, composition, and the structure play important roles in determining their behavior. For clusters size and shape become additional variables to play with and new phenomena may occur due to quantum confinement. In this brief review, we take a few examples related to the occurrence of permanent electric dipole moments (EDMs) in clusters of elements such as Nb [3], magnetism in clusters of non-magnetic elements [4–6], and metal (M) encapsulated clusters of group 14 elements [7–11]. These subjects are currently of much interest and have been attracting a lot of attention. The calculations have been done using
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the ab initio ultrasoft pseudopotential method [12] with generalized gradient approximation as well as using the gaussian program [13] with B3PW91 hybrid exchangecorrelation functional. For details, the reader is advised to see the relevant papers.
2. Results 2.1. Permanent electric dipoles in elemental clusters EDMs are known in multicomponent materials and molecules such as H2O. However, the finding of the permanent EDMs in clusters of elements such as Nb [3] is intriguing as large charge separation in homonuclear clusters is unlikely. For the occurrence of EDM an asymmetry should exist such as in a diatomic molecule NaCl. In homonuclear crystals an electric dipole layer forms on surfaces that also break the crystal symmetry. Clusters and nanoparticles have a large fraction of atoms lying on the surface and the electronic charge density decays into the vacuum as on crystal surfaces. Accordingly some charge is depleted from the cluster. This also leads to oscillations in the electronic charge density distribution [1] inside the cluster which can change significantly as a function of the size and the number of electrons in the cluster due to quantum mechanical electronic shell effects. Within the classical picture this would lead to the formation of local electric dipoles around the cluster/nanoparticle surface and further contributions arising due to the spacial variation of charge in the cluster. In the case a cluster has inversion symmetry or the vector sum of the local electric dipoles is zero, no net EDM would exist on the cluster. This is also true for a multicomponent cluster such as Ti@Si16 which has tetrahedral symmetry [7] with Ti at the center and the EDM is zero. However, distortions in the structure or low-symmetry as well as possible local variations in the bonding nature may lead to non-zero EDM. Therefore, the occurrence of EDMs should be quite common in all clusters including those of elemental clusters of metals and semiconductors. Moreover, it further follows that all supported clusters will have EDMs varying with size, shape, and orientation because of the inherent asymmetry. In general there can be three behaviors of EDMs in homonuclear clusters. (1) Zero net EDM in clusters with inversion symmetry or other high symmetries such as tetrahedron, (2) small EDM in nearly symmetric clusters with e.g., Jahn-Teller distortions, and (3) large EDMs due to the inherent lack of symmetry in a cluster as for certain sizes a symmetric structure may not be possible. Also variations in the local environments of atoms can lead to some charge transfer from one region to another even in homonuclear clusters and contribute to the formation of the EDM.
Fig. 1. Two nearly degenerate isomers of Nb13. (a) is the same as reported in Ref. [15] and (b) lies 0.21 eV lower in energy [14].
The origin of EDMs in NbN clusters has been studied recently [14]. It has been shown that their nonicosahedral asymmetric structures [15] are partly due to the covalent bonding that leads to the formation of permanent EDMs in these clusters. Symmetric clusters such as Nb4 have zero EDM. The calculated EDMs are in overall good agreement with experiments [3] and give support to the lowest energy structures predicted from theory as well as to the physical picture described above. However, the earlier reported structure of Nb13 (Fig. 1a) has a large EDM of 2.57 D as compared to the experimental value of 1.5 D. Further search for a lower energy structure led to another isomer (Fig. 1b) which is 0.21 eV lower in energy and has a dipole moment of 1.39 D that is in good agreement with the experimental result. For Nb10 zero or about 0.8 D EDM was obtained while the calculated EDM [14] in the lowest energy isomer is nearly zero due to the nearly symmetric structure [15], suggesting the presence of isomers in experiments. Isomers of NbN, N = 9–12 were also reported [16] from studies of reactivity of Nb clusters. Besides Nb clusters, Al14 and Si10 with capped icosahedral and tetracapped tetragonal prism structures have 1.22 and 0.82 D EDM, respectively. Therefore as discussed above there is nothing particular about EDM in Nb clusters. An interesting fallout of the occurrence of EDM is that it can be used to separate isomers and prepare size selected assemblies of neutral clusters. The EDMs on Nb clusters have been found [3] to vanish with an increase in temperature. It was speculated that this ferroelectric behavior could be a signature of the occurrence of superconductivity in bulk Nb. However, calculations [14] show that the existence of permanent EDMs is intrinsic to the atomic structure of the clusters and not related to superconductivity. At non-zero temperatures the EDM will fluctuate. It can have zero thermally averaged value if there is a transition to a symmetric structure. These thermal as well as the recently reported spin uncoupling effects [17] are yet to be understood.
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2.2. Magnetism in clusters Small particles of elements such as Fe, Ni, Ru, Rh, Pd, and Pt are good catalysts and have attracted much interest [1]. Fe and Ni are magnetic in bulk and the magnetic moments in their clusters are enhanced [18] due to the localization of electrons and the narrowing of the distribution of states because of the mean lower coordination of atoms as compared to bulk. This is also the reason that clusters of non-magnetic elements such as Ru, Rh, and Pd become magnetic [4–6]. These elements lie below Fe, Co, and Ni, respectively in the periodic table and the narrowing of the distribution of electronic states in clusters makes them behave somewhat close to the 3d bulk behavior. The magnetic moments on RhN clusters tend to vanish beyond N 100 [4] while Ru and Pd clusters have only small magnetic moments. Calculations [5,6,19] on Ru, Rh, and Pd clusters found icosahedral growth. PdN clusters with N 6 147 [19] show significant magnetic moments, though the energy cost for a lower magnetic moment isomer is quite small. Pd55 and Pd147 have 0.47 and 0.41lB/atom magnetic moments, respectively. Their cubic isomers have lower magnetic moments (0.18 and 0lB/atom, respectively) and can be present in experiments as the energy differences are small. Also other non-icosahedral lowsymmetry isomers may lie close in energy [20]. Interaction with H as well as O leads to [19] a reduction in the magnetic moment of Pd clusters. When 8 H atoms are associated with Pd13, all the magnetic moments (8lB) get quenched. For Rh clusters the calculated magnetic moments are significantly higher than the observed values [4]. The calculated magnetic moments on Rh13 is 1.62lB/atom [5] in the icosahedral structure in contrast to the experimental value [4] of 0.48 ± 0.13lB/atom. Similarly high magnetic moments have been reported by Reddy et al. and Jinlong et al. from ab initio calculations [6]. Recently a detailed study [20] of the atomic structures and magnetic properties of Rh clusters has shown non-icosahedral growth to be more favorable. These isomers also have generally lower magnetic moments as compared to the icosahedral ones and which are in closer agreement with experiments. For Ru and Rh the magnetic moments decrease [5] rapidly to zero as compared to Pd with an increase in size. Icosahedral Ru and Rh clusters with 147 atoms have nearly zero magnetic moments in agreement with experiments [4] and already approach the bulk value. Similarly in the case of icosahedral Ni147 the magnetic moment is calculated [21] to be 0.68lB/atom within GGA. This is also very close to the bulk value of 0.6lB/atom and the experimental value obtained by Billas et al. [18] at 78 K. Photoionization [22] and chemical reactivity [23] experiments suggest that Ni clusters are icosahedral upto about 120 and 800 atoms, respectively. Therefore, the calculated results agree with these
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experiments very well and show that around 100 atoms or so, the average magnetic moments in these clusters already become close to the bulk value, though the progression to bulk behavior is slow in Pd clusters. In all above studies the orbital contribution to the magnetic moments has been ignored. Guirado-Lopez et al. [24] obtained 0–0.25lB/atom orbital magnetic moments in Rh clusters using a tight binding method and fcc structures with bulk nearest neighbor bond lengths. However, Rh clusters are not fcc and the bond lengths are significantly shorter than the bulk value [5,20]. The magnetic moments and the magnetic anisotropy are sensitive to the structure as well as the size of the clusters. The loss of spherical symmetry in clusters as compared to atoms tends to quench the orbital moments. Therefore, the low-symmetry non-icosahedral structures of Rh clusters indicate that the orbital contribution to the total magnetic moment may be small. On the other hand for Ni and Pd clusters, the icosahedral structures are believed to be lowest in energy. The high symmetry and nearly spherical shape of some of these clusters may lead to significant orbital magnetic moment. A recent study [25] also indicates greater importance of orbital contribution in Ni clusters as compared to those of Rh. 2.3. Metal encapsulated cage clusters Recently novel M encapsulated cage clusters M@Xn, X = Si, Ge, and Sn and n = 10–16, have been predicted [7–10] from computer experiments based on ab initio calculations. It has been shown that one M atom changes completely the structures of these clusters and leads to fullerenelike, cubic, Frank–Kasper (FK) polyhedra, icosahedral as well as other polyhedral forms with higher stabilities as compared to the elemental Si, Ge, and Sn clusters. The relative sizes of the M and the X atoms play an important role in deciding the optimal number of X atoms that can be wraped around an M atom. This unique property can be exploited to produce mass selected clusters of silicon and other elements/ compounds with desirable properties. Indeed in recent experiments some of these predictions have come to be þ true. Al@Pbþ 12 and Al@Pb10 have been produced [26] with high abundances and little intensities of other clusters. These are isoelectronic to the predicted [9] divalent M atom doped magic clusters of X atoms. Recently high intensity of neutral Ti@Si16 clusters has also been obtained [27] with little intensities of other Ti doped clusters, in quite a similar manner as C60 [28], supporting the predicted electronic and geometric stability of this cluster [7]. The high stability of Ti@Si16 arises due to the strong covalent bonding between Ti atom and the silicon cage as well as the occurrence of a large highest occupied-lowest unoccupied molecular orbital (HOMO-LUMO) gap.
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Fig. 2. High symmetry M encapsulated cage clusters of X atoms. (a) Bicapped tetragonal antiprism for Pt@Sn10, (b) icosahedral Mn@Ge12, (c) hexagonal prism Si12W, (d) cubic Fe@Si14, (e) fullerene Zr@Si16, and (f) FK-Ti@Si16. M atom is inside the cage.
These developments could lead to assemblies of such clusters and the design of silicon based molecular devices. The calculated electron affinities (EAs) [7,29] are generally in good agreement with experiments. V@Siþ 16 is isoelectronic to Ti@Si16 and the FK isomer is 0.314 eV lower in energy as compared to the fullerene isomer [30]. It has a large HOMO-LUMO gap of 3.59 (2.44) eV within the hybrid B3PW91 functional (GGA) as in the case of Ti@Si16. This suggests that the charged clusters are likely to be luminescent in the visible region as it was shown [29] for Ti@Si16. Similarly Sc@Si 16 is isoelectronic to Ti@Si16. Also Si12V is isoelectronic to Si12Cr which is magic [31] similar to Si12W. It has a large HOMO-LUMO gap [30] of 2.39 (1.54) eV within B3PW91 (GGA) and should be very stable and abunþ dant as in the case of Al@Pbþ 12 and Al@Pb10 discussed above. Therefore, charging of clusters could make it possible to produce certain species exclusively because of their specific stability as well as act as a switch. Recently switching of the charged state of a single gold atom on the surface of an insulator has been demonstrated [32] by a voltage pulse in scanning tunneling microscope. The M encapsulated clusters can also change shape and the HOMO-LUMO gap significantly on charging [29]. This could also act as a switch providing novel possibilities for devices. Recently Ta@Si12 clusters with a large electron affinity of 4.06 eV have been deposited on a silicon substrate and charge transfer has been observed from the substrate to the clusters [33] and possibilities of fabricating the next generation Si-MOS transistors have been suggested. Calculations [30] on Nb and Ta doped clusters of Si show strong stability. Si12Nb and Si12Ta have large HOMO-LUMO gaps of 1.82 and 1.96 eV in the hexagonal prism struc-
ture of Si12W (Fig. 2). These are much higher than the values [8] of 0.85 and 1.34 eV for Si12M, M = Cr and W, respectively and should lead to their strong abundances. The transition M atom provides strong stability to the X clusters. When the valence of the M atom matches with the holes in the electronic shell of the X cage as well as the relative sizes of the M atom and the X cage fit well, we obtain an extraordinary stability of these clusters. This is the case for fullerene-like Zr@Si16, FKTi@Si16 (Ti, being a slightly smaller atom, prefers a different cage than Zr) [7], icosahedral Zn@Ge12 [9] and other clusters. For Ni, Ge10 is the best cage but for Sn10, Pt is best suited [34] with large HOMO-LUMO gaps. These are some of the smallest cages that can be stabilized with an M atom. For divalent M atom such as Be, X10 and X12 cages are the best [9]. X10 clusters are magic [1] but the stability of such magic clusters is further enhanced by M doping. For Fe, Ru, and Os, a 14-atom cage is best suited [7] while for Cr, Mo, and W, a 15-atom cage is the best [8]. It has also been possible to stabilize a Si20 cage [35]. Also icosahedral magnetic superatoms Mn@Ge12 and Mn@Sn12 have been predicted [9] with large magnetic moments of 5lB as well as large HOMO-LUMO gaps. These features make such clusters attractive for assembly. Selected high symmetry structures are shown in Fig. 2. The structures of these clusters can be probed using gas interaction. Experiments [36] and calculations [37] on water interaction with Ti@Sin, n = 13–16 show it to be weak. But smaller clusters have open basket-like structures in which the M atom is not fully covered with Si and can interact strongly with water. This result establishes the encapsulation of the M atom in a cage for non-reactive clusters.
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The cagelike structure of Si12W clusters was also inferred [38] from weak adsorption of hydrogen. A similar behavior was found for Si18W2 which has a double hexagonal prism structure [39] and weak interaction with hydrogen. Raman and infrared vibrational modes are also [29] useful probes for establishing the structure of these clusters. The basket or M in a cup structures are interesting for catalysis as these provide a nice way to isolate an M atom or a group of M atoms. Often in catalysis, a molecule is dissociated on a certain number of active sites. For example breaking a C–H bond may require one Ni atom while breaking a C–C bond may require more than one Ni atoms to be nearest neighbors. In order to control this and achieve selectivity, an inactive element is added to form bimetallic clusters such as CuNi. In the same spirit, it could be possible to prepare clusters with an appropriate number of catalytically active sites of atoms and partially surround them by an inactive element. Such a behavior might already be happening in bimetallic clusters of e.g., Cu–Ni in which (inactive) Cu segregates on the surface [40]. This could lead to novel design of catalysts with minimum amount of the active material. These developments have opened up new directions in finding novel Si and Ge based nanostructures. Empty cage hydrogenated fullerenes X20H20, X = Si [39], Ge and Sn [41] have also been predicted (Fig. 3). The stability of Si20 fullerene cage is particularly interesting because C20 fullerene cage is difficult to form [42] as there is significant strain in the sp2 bonding and the bonding becomes more sp3 like. In carbon fullerenes, pentagons are the places of strain, but pentagons are most favored in Si cages as it likes to be sp3 bonded. Presence of rhombii or hexagons creates strain in Si nanocage structures [10]. The fullerene isomers of some of the M encapsulated silicon clusters [7,10] have rhombii as the number of vertices is less than 20 which is the minimum number to have a fullerene structure with all pentagonal (12) faces. Therefore, in these clusters there is isolated rhombus rule [10] similar to the
Fig. 3. (a) Dodecahedral empty center cage of Si20H20. H atoms are shown by small spheres. (b) Si28W2 with M atoms inside the cages.
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isolated pentagon rule in carbon fullerenes. The GGA HOMO-LUMO gaps in hydrogenated empty cage X clusters are large ( 3 eV) making them interesting for optoelectronic and biological applications and for developing photo-absorption materials. The endohedral [41] and exohedral [43] derivatives of these clusters can lead to novel possibilities of functional magnetic materials as well as sensors and other applications. 2.4. Assemblies of clusters Assemblies of M encapsulated clusters of Si and other elements can be produced in different ways possibly by depositing on a surface (2D structures), growing one dimensional structures or bulk phases similar to solid C60. The binding energy between two Zr@Si16 clusters is relatively weak (1.35 eV within GGA) [7] and there is a significant reduction in the HOMO-LUMO gap from 1.58 eV for Zr@Si16 to 0.67 eV for the dimer. However, for the FK isomer of Ti@Si16, this interaction is very weak (0.05 eV) and the gap remains nearly the same as for the cluster (2.36 eV), making these species attractive for self-assembly. Molecular dimers [7] with two Z M atoms replaced with (Z 1) and (Z + 1) M atoms in the fullerene structure such as Si32ScNb, Si32ScTa, Si32YTa, and Si32YNb with 0.87, 0.89, 0.82, and 0.82 eV HOMO-LUMO gap, respectively and Si18TaRe and Si18TiFe with HOMO-LUMO gap of 1.08 and 0.84 eV, respectively in the hexagonal biprism structure [39] have been obtained [44]. Even nanocapsules such as Si28W2 in which two fullerene structures are fused (Fig. 3) have been found [44] to be stable with a HOMO-LUMO gap of 0.62 eV. The fullerene structure remains almost undistorted. It is likely that such fused empty cage structures of silicon and other group 14 elements can be formed with H termination in different forms. A nanowire of Zr@Si16 fullerenes has been predicted [44] to be semiconducting with nearly the same gap as bulk silicon. Another important development has been the finding of the silicon nanotube by the assembly of Si12Be clusters [45]. These predictions have found support from recent STM experiments [46] in which Be was deposited on a Si(111) surface and under heating conditions, a self-organized structure was obtained on a large surface area with sticks of Si24Be2 finite nanotubes and two more forms, boomrang (two such finite nanotubes fused together at an angle of 120°) and propeller (three finite nanotubes fused together symmetrically) structures. These experiments would provide further impetus for more experiments on these systems. Studies on assemblies of Si12M clusters with M a transition metal atom [47] have led to the finding of nanotubes with high magnetic moments. Similar studies have also been done on Ge nanotubes [48] which show novel piezo-magnetic behavior. These nanotubes are
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metallic and interesting for interconnects in miniature devices and magnetic applications. Elemental silicon of such a diameter tend to be distorted and has tendency to agglomerate [45]. Recently a semiconducting nanotube of Ge has also been obtained [49]. This is interesting from the point of view of devices. These studies have opened up new possibilities of developing novel structures of silicon and germanium that could have important implications for the future nanodevices.
3. Summary In summary, we have presented a brief review of some recent theoretical progress primarily derived from our own work. The permanent electric dipoles in elemental clusters arise due to asymmetry and are neither specific to Nb nor related to superconductivity. Significant progress has also been made in understanding magnetism in clusters of non-magnetic elements. This would lead to a better understanding of their catalytic properties and design of new catalysts. The M encapsulated Si, Ge, Sn, and Pb caged clusters offer exciting possibilities of size selected mass production. The shape, size and the HOMO-LUMO gap in these clusters depend upon the M and the X atoms. These results show that one M atom changes the properties of elemental Si, Ge, Sn, and Pb clusters drastically and have opened up new avenues for the development of novel elements for future nanodevices as well as new varieties of nanostructures of different materials using the idea of encapsulation. The strong stability of these clusters arises from the strong M–X interactions that also fix the size of the X cage. Many of these predictions have been confirmed by recent experiments. Assemblies of such clusters have interesting electrical, magnetic and optical properties. Similar to Si, novel core-cage structures have also been predicted for II–VI nanoparticles [50]. This has enriched this field and hopefully it will facilitate a better understanding of a large body of data on these technologically important nanoparticles.
Acknowledgements I am grateful to Y. Kawazoe for all the support and cooperation. I thank A. Kasuya, F. Pichierri, R. Belosludov, V. Sundararajan, T.M. Briere, C. Majumder, A.K. Singh, H. Kawamura, Y.-C. Bae, W.E. Pickett, and K.E. Anderson for discussions. I gratefully acknowledge the kind hospitality at the Institute for Materials Research and the support of the staff of the Center for Computational Materials Science of IMRTohoku University for the use of SR8000/H64 supercomputer facilities.
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