Computational materials science: The emergence of predictive

S¯adhan¯a Vol. 28, Parts 3 & 4, June/August 2003, pp. 815–831. © Printed in India

Computational materials science: The emergence of predictive capabilities of material behaviour VIJAY KUMAR Dr Vijay Kumar Foundation, 45, Bazaar Street, K K Nagar (West), Chennai 600 078, India and Institute for Materials Research, Tohoku University, 2-1-1 Katahira Aoba-ku, Sendai 980-8577, Japan e-mail: [email protected] Abstract. The availability of high performance computers and development of efficient algorithms has led to the emergence of computational materials science as the third branch of materials research complementing the traditional theoretical and experimental approaches. It has created new virtual realities in materials design that are either experimentally not realizable easily or are prohibitively expensive. The possibilities of doing calculations from first principles have led to predictive capabilities that open up new avenues of discovering novel materials with desired properties, understanding material behaviour on the nano- to the macroscopic scale and helping research in new frontiers that could interface between nano-materials and drug design, as well as in understanding biological systems. Here, we describe some significant recent developments related to alloy and steel design as well as the study of matter on the nano-scale – an area that has gained much prominence in current materials research. Keywords.

Density function; materials design; nanomaterials.

1. Introduction The foundation of the quantum mechanical description of the properties of matter was laid down in the early twentieth century by the formulation of the Schrödinger equation (SE) that was considered to mark the end of chemistry, as all answers can in principle be obtained by solving it. However, the SE could be solved exactly for some atoms and for a long time it was not possible to implement this to real materials in a rigorous way as it required solving a complex many-body problem of interacting electrons and ions. Simplified approaches as well as concepts were therefore developed, together with experimental work that contributed greatly to our understanding of properties of materials and phenomena such as magnetism, semiconductor, metallic, insulator and superconducting behaviour, mechanical, thermal, optical and transport properties and cohesion. The local way of looking at the bonding in materials by chemists led to great intuition as well as simple rules that helped the development of an enormous variety of molecules and chemicals. As the understanding grew, greater attention started to be paid on developing new materials with improved properties specific to certain applications as well as to understand complex phenomena such as catalysis, corrosion, 815

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embrittlement, deformation etc. that affected the performance and production of materials and therefore had a large impact on cost. Experiments contributed greatly to this development of which steel can be described as a classic example. It is one of the most important classes of materials, where untiring experimental work has led to various special steels for specific applications even while an understanding of the properties of steel under different conditions from first principles has remained primitive, though some efforts began to appear in the late twentieth century and the term computational metallurgy was even coined (Eberhardt 1994) with the dream of designing steel from first principles. While this possibility is still far in the future as there are too many components in steel to be optimized, and the effects of multiphases, internal boundaries (microstructure), impurities and defects have to be understood, to obtain the desired behaviour, it symbolizes a new paradigm in thinking and solving materials problems as it was thought with the advent of quantum mechanics. In the past 25 years much progress has been made in the understanding of the electronic, magnetic and thermal properties of bulk and surfaces of crystals as well as simple defects and disordered materials from first principles. In recent years important developments have occurred, as we shall discuss, that show significant progress even in steel design. This became possible due to the tremendous growth in computational power and the development of density functional based approaches, efficient algorithms and their implementation to solve the quantum mechanical equations governing the motion of electrons along with the dynamics of ions. One of the areas that have gained tremendous importance in recent years, is the behaviour of matter at the nano-scale. This is important for the future devices as well as in understanding a large variety of multi-disciplinary problems such as biological systems, self-assembly of materials, chemical reactions, and possibly evolving new concepts and methods of developing materials. An understanding of structures, properties, and assembly of nano-particles is currently a very active area of research and here we shall briefly cover some of our own contributions in this direction. 2. Density functional theory: The emergence of a new era Among the many works that contributed to the development of the predictive computational materials science from a microscopic point of view, the foundation was laid by the seminal work of Walter Kohn and coworkers (Hohenberg & Kohn 1964; Kohn & Sham 1965) in the formulation of the density functional theory that reduced the complex many-body problem of solving the SE of an n-electron system with 3n variables to Kohn–Sham equations involving the electron density with only 3 variables. This enabled ab initio studies of large systems. Currently computations can be performed with good accuracy for systems of the order of 1000 atoms. Such studies also made it possible to develop as well as justify simplified approaches and improve the existing empirical approaches to understand the macroscopic behaviour of materials including static and dynamical properties as the relevant parameters could be calculated from first principles. It allowed the study of the energetics and stability of different stable and metastable phases (see Skriver 1985), effects of impurities to understand phenomena related to grain boundaries such as the effects of B and P on cohesion at grain boundaries in Fe (Wu et al 1994), stacking faults as well as defect and surface energies, adsorption on surfaces and reactions (Besenbacher et al 1998; Jacobsen et al 2001; Kroes et al 2002), phase transitions under pressure (Sluiter et al 2002), structures, bonding and associated properties of aggregates of atoms (Kumar et al 2002a), etc. and even problems such as the development of battery materials (Ceder et al 1998), matching direct-band gap materials with silicon for optical devices (Zhang et al 2001) etc.

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Calculation of the correct phase of lowest energy is crucial for successful predictive capabilities. The energy differences between different structures are generally small (∼ a tenth of an election volt) and the heat of alloy formation is similarly small though in some cases it could be about an election volt. The difference between a magnetic and a non-magnetic phase is still smaller and so are the vibrational energies. All these require higher accuracy in the calculation of the total energy of a system than is currently possible to achieve. While an exact solution of the Kohn–Sham equations is currently not possible in most cases because of the problem of treating the exchange-correlation energy, the local density approximation (LDA) (Kohn & Sham 1965) and generalized gradient method (GGA) (Perdew & Wang 1992; Perdew et al 1996) have given good results. In many cases even chemical accuracy has been achieved. In order to illustrate this, we show in figure 1, the total energies of seven crystal structures of Si with different volumes (Yin & Cohen 1982). This shows that the diamond structure has the lowest energy in agreement with experimental observation. In figure 2 we show the heat of formation for a number of alloys in the Cu3Au and CsCl structures obtained by Jóhannesson et al (2002) from first principles. Again these are in close agreement with experiments. But problems remain and need improvements. For example the band gap in semiconductors and insulators is underestimated. In the case of systems, where correlations play an important role and electrons are rather localized, again the present schemes may not be sufficient. In the case of pure iron, the non-magnetic γ phase with fcc structure is predicted (Wang et al 1985) to be the lowest in energy in LDA rather than the experimentally known ferromagnetic α phase. However, this discrepancy is corrected (Leung et al 1991) by the use

Figure 1. Total energy curves for seven phases of silicon as a function of the atomic volume normalized to the experimental volume. Dashed curve is the common tangent of the energy curves for the diamond phase and the β-tin phase (c/a = 0·552). (After Yin & Cohen 1982.)

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Figure 2. Comparison of experimental and calculated heats of formation for binary alloys that have either Cu3Au or Cu–Au type fcc or CsCl-type bcc structures. (After Jóhannesson et al 2002.)

of GGA. For the Fe–Al system both LDA and GGA fail to predict the correct ground state of Fe3Al for which L12 structure is predicted (Lechermann et al 2002) to be lower in energy than DO3 structure found in experiments. Similarly FeAl is wrongly predicted (Mohn et al 2001) to be ferromagnetic. Such discrepancies need improvements in the description of the exchange-correlation functional and are being developed. They also show the difficulties in the prediction of phases correctly due to small energy differences. However, other than some difficult cases, the successes have been many and here we shall discuss a few examples. There are several methods of solving the Kohn–Sham equations (see for example Kumar et al 1994). These include linear combination of atomic orbitals with all electrons or pseudopotentials, plane wave pseudopotential, linear muffin-tin orbital, linear augmented plane wave, and the KKR methods. In most calculations LDA or GGA functionals have been used. There have been efforts to improve upon these and hybrid methods have also been suggested which provide improved description of bonding in certain cases. On the other hand, the quantum Monte Carlo method (Foulkes et al 2001) provides almost an exact solution and is applicable to large systems (Puzder et al 2002). However, most studies are still on small molecules and clusters. Here we shall not discuss the details of these method. Rather, we focus on the physics that has been learnt. 3. The length and time scales Computational material science deals with the understanding of materials properties and behaviour at length scales encompassing atomic, nano, microscopic and also macroscopic levels. In this scale of about eight orders of magnitude, some properties of matter can be understood from first principles using the periodic boundary conditions that allow one to deal with large systems by treating a small number of atoms. There are, however, a large number of other problems related to applications of materials such as dislocation dynamics and deformation of materials under stress that are difficult to treat fully by quantum mechanics. Therefore, continuum approaches based on the finite element method or treatments using


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semi-empirical description of interatomic interactions are still useful. This has allowed atomistic simulations on plastic deformation using multi-million atoms (Bulatov et al 1998). There are also developments of hybrid methods (multiscale simulations) which treat the atomistic details affecting the behaviour of a material in a local region and combine it with the continuum description of the rest of the system. Such an approach has been used (Diaz de la Rubia et al 2000) in modelling the microstructure of an irradiated material which evolves over a wide range of length and time scales and makes radiation damage an inherently multiscale phenomenon. In the shortest time scales (nanometre and picoseconds), recoil-induced cascades of energetic atomic displacements give rise to a highly non-equilibrium concentration of point defects and defect clusters. Over macroscopic length and longer time scales, these defects can migrate and alter the chemistry and microstructure, often degrading the mechanical and other properties of materials significantly. In general, the time scale of understanding material behaviour is much longer and ranges from femtoseconds (electronic excitations) to picoseconds (ionic vibrations and related dynamical properties) to hours, days, months and up to several years such as the life of a component in an aircraft or nuclear reactor, or even a computer which is becoming an important part of present society. These are vital problems and demand reliable solutions. For some such problems, extensive experiments are difficult to perform, take much too long a time or are economically prohibitive. Computer simulations become a very effective tool using virtual realities. Simulations based on quantum mechanical understanding of the properties of matter could make such predictions more reliable by providing more accurate parameters in models and detailed information that may not be possible to obtain from experiments. These are vast and diverse subjects to be covered in a short review. Here we discuss a few results on some recent developments in the area of alloys, steels and nano-science that are important fields affecting technological developments. Some examples are drawn from our own work on nanoparticles which have shown the strength of the predictive power of ab initio methods and the possibilities of material design from first principles. 4. Some examples There are a large variety of problems such as reactions, catalysis, structure modification upon surface creation (reconstruction and relaxation), melting etc. that need detailed information of the surface structure and dynamics for proper understanding of surface related phenomena. Artificial structures such as superlattices, phase transformations and development of new materials need proper description of interatomic interactions. While experimental advances have been crucial, much has been understood from ab initio calculations. Most importantly in such cases, empirical description is, in general, not appropriate. There are other classes of materials such as a collection of atoms forming clusters and nanoparticles that are important for the future development of materials. However, it is difficult to obtain information about structure and many other properties from experiments. In this respect, calculations from first principles have contributed greatly and are expected to even contribute more in the future with the growing importance of nano-materials. The development of ab initio molecular dynamics (Car & Parrinello 1985) by combining the classical molecular dynamics with the calculation of forces from first principles, led to the possibility of giving different treatments to materials such as by varying temperature and pressure, annealing, quenching etc. that has created virtual reality on the desktop and allowed one to perform computer experiments to monitor materials behaviour under different conditions based on first principles approaches. This has led to tremendous progress in predicting the structure of clusters, liquids, amorphous

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materials, reconstruction on surfaces, phase transitions etc., stability of various known phases and the prediction as well as stability of new materials. Here we present results on three problems that demonstrate the level of progress achieved. 4.1 Designing superalloys The calculation from first principles of the total energy of a system with a few to a few tens of atoms in a unit cell has become routine and requires little time on present day workstations. This can be seen from recent work (Jóhannesson et al 2002) where 192016 combinations of possible fcc and bcc alloys having up to four components have been studied with 32 different metals using the density functional theory and an evolutionary algorithm set up that enables to identification of 20 most stable known and new superalloys. These most stable alloys prefer bcc structure and typically involve an equal number of early and late transition metals so that the average number of d electrons per atom is five and the d band is half-filled making only the occupied bonding states analogous to the strongest cohesion found in Mo and W that also prefer bcc structure and have half-filled d bands. The bcc structure is, however, not preferred for structural alloys as these are less ductile. Adding the requirement that fcc structure should be more stable than other structures, it was found that addition of Pd or Pt causes alloys of early and late transition metals to possess f cc structure. However, these metals are expensive. Excluding such near-noble metals, a new set of alloys was obtained that was dominated by transition metal silicides. However, transition metal silicides are generally brittle. Here it is interesting to note that alloys based on Ni3 Si (L12 -type structure) such as Ni3 (Si,Ti) are known to be candidates for high temperature structural materials and chemically resistant parts. Therefore, such calculations provided quantum mechanical foundation to some of the known superalloys. Excluding Si and expensive alloys, Ni3Al was found to be the best superalloy which is also known. The next favoured alloy was found to be based on Ti. Ni3 Ti has hexagonal DO24 structure and as such it is not used as structural material. However, Ti is the main alloying component in Ni3 Si and Ni3Al. TiAl is a well-known intermetallic compound and has been extensively developed together with TiAl3 for commercial applications as high temperature superalloys. Several new alloys such as Al3 Sc, Al2 ZnZr, Al2 ZnTi, Al2 ZnSc, Al2 ZnHf, Al2 CuTi, and Al2 CuZr having L12 structure were found to be interesting cases. The development of new functional and structural materials is largely based on a trial-and-error method and it has been realized that it is impossible to screen all combinations experimentally. These studies illustrate that computer experiments could help to identify materials with e.g. required structure or without certain costly elements but having the desired properties. This could then be the basis for further experimental development to optimize large number of considerations for practical usage. 4.2 Designing alloy steel Much of the development of steel alloys has been by guess-work and empirical correlations with composition, and the obtained microstructure, as well as with properties. Such experimental methods are expensive. Combinatorial approaches (Cahn 2001) are being used to speed up the development of new alloys with desired properties. An approach (Olsen 1997) based on a hierarchy of design models characterized by specific empirical and ab initio methods, has been used to design materials with properties combining strength, toughness and resistance to impurity embrittlement. The first principles method is used to predict the alloying elements that enhance cohesion between alloy grains and make alloys stronger, whereas the micromechanic level design employs semi-empirical thermodynamic databases.


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Recently composition-elastic property map has been used (Vitos et al 2002) to predict basic steel compositions that have improved and controllable properties as compared to the available commercial grade. The elastic data can give fundamental information about mechanical deformation and structural stability of alloys as well as performance at the macroscopic level. There are empirical correlations between the polycrystalline elastic moduli and technologically important properties such as strength, hardness, and wear. The shear modulus, G for a large set of materials is a general indicator of mechanical hardness and resistance to plastic deformation. On the other hand, the bulk modulus, B is a measure of average bond strength and represents opposition to fracture. It is strongly correlated with the cohesive energy, a large value of which is also an indicator of corrosion resistance. The ratio B/G is a measure of the ductility (ability to change without fracture) versus brittleness (fracture without appreciable plastic deformation). Ductile alloys are characterized by high B/G ratio (> 1·75) while low B/G is representative of brittle systems. Elastic property maps have been recently calculated (Vitos et al 2002a) from first principles for austenitic stainless steels that are Fe-based with low carbon and have a minimum of 12 at.% Cr as well as Ni and a few other elements such as Mo, Si, Al etc., distributed in compositionally disordered manner on an fcc lattice. In this study the disorder has been treated using the coherent potential approximation (see, for recent developments, Vitos et al 2001) by considering Fe–Cr–Ni alloys in the concentration range of 13·5–25·5 at.% of Cr and 8–24 at.% of Ni. This range includes the compositions of the commercial grades from the AISI (American Iron and Steel Institute) 300 austenitic stainless steel series. These are paramagnetic at room temperature. Assuming that the chemistry of protective chromiumoxide surface film is not altered by changing Cr and Ni concentrations, an increase in the cohesive energy and thus the bulk modulus has been used to indicate an improved resistance of steels to various forms of localized corrosion. Calculated values of B and G of 304, 316, and 310 austenitic steels were obtained in very good agreement with experimental results within 5·1 and 1% respectively. Also the effects of addition of Mo were reproduced correctly in that it increases B of the alloys. As shown in figure 3, alloys with large G have low and intermediate Cr (< 20 at.%) and low Ni (< 15 at.%) concentrations and are indicated as region I. Within this group G decreases monotonically with both Cr and Ni from a maximum of 81 GPa near 14 at.% Cr and 8 at.% Ni to about 77 GPa. The high Cr containing alloys (family II) have lowest G values (< 75 GPa) with a minimum at around Fe25 Cr20 Ni while the intermediate G values (region III) are for alloys with moderate Cr (< 20 at.%) and high Ni (> 15 at.%) contents. The Ni contents play an important role in the stabilization of the austenitic phase against the ferrite phase which has the bcc structure. On the other hand Ti is a strong ferrite stabilizer. So there is a delicate balance between Ti and Ni. Both Ni and Ti enhance B conferring excellent corrosion-resistance quality on austenitic steels. Further improvements are obtained by addition of Mo and Si but these elements decrease G. Also, as discussed earlier, Si has a tendency for brittleness. The B/G ratio shows that addition of Cr and Ni as compared to the commercial grades of 8–13 at.% Ni and 17–19 at.% Cr, increases ductility. Thus 310 steel has about 10% higher B/G ratio as compared to the 316 steel. These maps can be used to obtain alloys with improved properties. Stainless steel with high G and high hardness can be obtained from low Cr and Ni contents (family I), but these have low B values and therefore, less corrosion resistance. This can be improved by addition of Mo such as 2 at.% substitution of Fe in Fe13·5 Cr8 Ni leads to an increase of 4% in B. On the other hand, steels in family III have intermediate hardness. Steels with the composition of approximately 18 at.% Cr and 24 at.% Ni are more ductile as compared to the commercial grades low Ni austenites and due to large B these should have excellent corrosion resistance. Also high Ni

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At. -% Cr

(a)

At. -% Cr

(b)

At. -% Cr

(c)

At. -% Ni

Figure 3. The calculated shear (G) and bulk (B) modulus of austenitic stainless steels and the ratio B/G as a function of the Cr and Ni concentrations. The balance is Fe. (a) Shear modulus, (b) bulk modulus, and (c) B/G ratio. (After Vitos et al 2002a.)


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concentration that makes Fe–Cr austenitic, allows reduction in carbon content, lowering the susceptibility of these steels to intergranular corrosion. These studies have thus opened up new perspectives in steel design. 4.3 Design of nano-materials : Nano-particles of metals and semiconductors Properties of matter with dimensions of a few to a few tens of ångstroms, also referred to as nano-particles, are often very different from those of then bulk material and are size- and shape-dependent. An outstanding example is the bright luminescence from nanostructures of silicon in the visible range due to quantum confinement (see e.g. Belomoin et al 2002) while bulk silicon is an inefficient emitter of light in the near-infrared range because of its indirect band gap. The colour of the emitted light can be changed by changing the particle size. This is because properties of semiconductors depend on the band gap that varies significantly with the size of the nano-particle as the band edge states arise from infinitely large systems. It gives rise to different properties of semiconductor nano-particles in a large size range. The finding of visible luminescence in silicon nanostructures has tremendous implications for future opto-electronic devices and has potential (Polman 2002) for the development of silicon-based lasers as well as optical connections in microelectronics. Magnetic properties of nano-materials are important for storage devices. Magnetic moments and magnetic ordering in nano-particles could be quite different from that in the corresponding bulk and this still lacks proper understanding. On the other hand, nanoparticles of metals are important for catalysis. Metallic nano-particles tend to attain bulk-like properties for smaller sizes as the Fermienergy lies in a band. An example of how different a material could be at the nanoscale is the observation of different colours of gold ranging from blue to red as the size is varied (Mulvaney 2001). A difficult experimental problem is the determination of the structure of nano-particles particularly in the small size range. As the properties are generally structure dependent, understanding the structure-property relation is an important aspect of nano-science research. Also, often there is a size distribution in experiments and the properties may be significantly affected due to impurities. Quantification of the size distribution as well as the impurities and defects are major difficulties from an experimental point of view. Also, for bulk material, a lot of thermodynamic data are available on alloys and phase diagrams, as well as much theoretical work having been done. However, little knowledge has been accumulated on phases that may exist in the nano-form. The role of computer simulations, therefore, becomes even more important as quantitative experimental information is difficult to obtain. Further, for such systems empirical methods will in general be non-applicable as the interatomic interactions are likely to be very different because of the different structures that are themselves needed to be found. First principles calculations are making it possible to have quantitative understanding of the properties of small nano-particles and helping to develop a better interpretation of the experimental results and making predictions of new possibilities to design materials with desired properties. As the current interest in nano-materials is expanding, such studies are likely to grow in the future. Nano-particles have a large fraction of atoms lying on the surface and hence often their structures differ from the bulk. Metal particles often have icosahedral structures that are forbidden for bulk crystals. Such higher symmetry in nano-particles and lower mean coordination could give rise to interesting electronic and magnetic properties such as the occurrence of magnetism in nanoparticles of non-magnetic metals (Kumar & Kawazoe 2002a). On the other hand in semiconductor nanoparticles such as those of carbon (fullerenes) and III-V compound semiconductors, there are novel caged structures that have led to completely new ideas for the development of materials using clusters as the building blocks (Krätschmer et al

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1990). For this, it is important to understand the atomic, electronic, thermal and other properties of nano-particles from first principles. The density functional theory within the framework of the LDA, GGA or hybrid methods for the exchange-correlation energy has been very successful in this direction. The calculated results are often in good agreement with the available experimental results obtained from spectroscopic measurements such as photoemission, abundance spectra, ionization potentials (IPs), electron affinities (EAs), polarizabilities, magnetic moments, optical absorption, Raman or infrared measurements, fragmentation as well as Coulomb explosion. These have helped to develop a good understanding of the properties at least in the small size range and establish certain trends as a function of the size of clusters that could be helpful in the understanding of large clusters (Kumar et al 2002a). Clusters of simple metals exhibit electronic shell structure similar to the shell model of nucleus and this leads to magic behaviour of clusters with 8, 20, 40, 58, 92, . . . valence electrons (Kumar et al 2002a). This has been understood and verified from ab initio calculations on clusters of alkali metals, aluminum and Ga. On the other hand clusters of divalent metals show non-metal–metal transition as the size grows due to the closed electronic shell structure of atoms. With increasing size, there is delocalization of electrons and hybridization of the occupied s states occurs with the unoccupied atomic p and d states. Such a transition was predicted (Kumar & Car 1991) for clusters of Mg having about 20 atoms and has recently been confirmed from experiments (Diederich et al 2001; Thomas et al 2002). Another novel aspect is the atom-like behaviour of some clusters. As an example, Al13 has 39 valence electrons that is one electron short of the electronic shell closure at 40 valence electrons. It has large electron affinity of about 3·7 eV very similar to the electron affinity of Cl. So it behaves like a halogen atom. Its interaction with an alkali atom has been shown (Kumar 1998) to lead to large gain in energy and a charge transfer to the Al13 cluster similar to NaCl molecule. On the other hand Al7 with 21 valence electrons behaves like a Na atom as it has one electron more than the electron shell closing at 20. This illustrates how differently matter could behave by just changing the size. This has been confirmed from the measurement of IPs of clusters which often varies with size (Schriver et al 1990). A detailed study of hydrogen interaction has been carried on Al clusters (Kawamura et al 2002) and even formation of a polymer form AlN H3N with high concentration of H and linear or cyclic structures has been predicted (Kawamura et al, unpublished manuscript). This could be interesting as a hydrogen rich material. The electronic structure of atoms obeys Hund’s rules due to the spherical symmetry of the potential. In the case of clusters with open electronic shells, often Jahn–Teller distortions are energetically preferred over symmetric structures. However, in the case of the Al12 Cu cluster that has an odd number of valence electrons (37 besides the ten 3d electrons of Cu), the electronic structure has been predicted (Kumar & Kawazoe 2001a) to follow Hund’s rule such that it has 3 µB magnetic moment with perfect icosahedral symmetry and Cu atom at the centre of the icosahedron as shown in figure 4. The highest occupied up-spin state has 3-fold symmetry (2p states in the spherical jellium model) and is fully occupied while the lowest unoccupied down-spin state (2p), also a 3-fold degenerate state, is fully unoccupied. This is followed by a significant gap. Therefore, this cluster behaves like an atom with half-filled p state in an atom. Accordingly it was suggested (Kumar & Kawazoe 2001a) that it should interact strongly with a trivalent atom. Studies on Al13 Cu indeed showed strong stability that has also been observed (Thomas et al 2001). In this case the added Al atom is, however, incorporated within the shell of Al atoms. Copper atom being slightly smaller in size occupies the central position. However, doping of aluminum clusters with a bigger atom such as Sn or Pb has been shown to lead to the segregation of the impurity atom at the surface (Kumar & Sundararajan 1998) as one expects on the basis of the surface segregation theory of alloys


825 down spin

up spin 1g

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Energy [eV]

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2p 1f

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1s Al 13 Cu

Figure 4. Perfect icosahedral structure of Al12 Cu cluster with Cu at the centre. It has an odd number of valence electrons and the electronic structure obeys Hund’s rule with the up-spin 2p states fully occupied and the down-spin 2p states fully empty. The Al13 Cu cluster has an electronically closed shell with a large gap and is magic. (After Kumar & Kawazoe 2001a)

(Kumar 1981). Evidence for such a behaviour has been obtained recently (Li & Wang 2002). On the other hand ordering has been predicted (Kumar 1999) in Al10 Li8 cluster such that a Li atom is at the centre of a decahedral cluster of Al10 . This cluster has 38 valence electrons and therefore, the behaviour is different from the known magic clusters of simple metals. The effects of d electrons have been studied (Kumar & Kawazoe 2001b) on the growth behaviour and the electronic structure of Sr clusters. In the case of Sr, the 4d states just start to be occupied. This has profound effect on the growth and leads to icosahedral structures in contrast to the findings on small Mg clusters. Clusters with upto 147 atoms have been shown to prefer icosahedral structures. It was also shown from extrapolation of results that clusters with several thousand atoms could have structures different from the bulk. An important suggestion was made that clusters of elements with large compressibility are likely to have icosahedral growth. This is generally true as one finds in large clusters of alkali metals, alkaline earth metals as well as in clusters of rare gases, all of which have high compressibility. It was further confirmed from the behaviour of Nb clusters that did not show icosahedral growth (Kumar & Kawazoe 2002b). Transition metals in the middle of a d series have low compressibilities due to the strong d bonding. On the other hand, palladium clusters again show icosahedral growth (Kumar & Kawazoe 2002a) as the contribution of d electrons to bonding decreases significantly towards the end of a d series. An important result, however, is the occurrence of magnetism in Pd clusters. Figure 5 shows the variation of the magnetic moments as a function of size. Large moments can be seen even on clusters with 147 atoms. This is in contrast to the non-magnetic behaviour in the bulk as well as of atoms. The lower coordination, high symmetry and elongation of the surface bonds (see variation of the bond lengths in figure 5) are responsible for the occurrence of magnetism. The magnetic moments show an oscillatory behaviour as a function of the size (figure 5) and clusters with 147 atoms have a moment of 60 µB . The difference in the energies of the magnetic and non-magnetic states is, however, quite small. Magnetic moments are seen to be distributed over the whole cluster and at least in the first 3–4 shells near the surface. The binding energy increases monotonically towards the bulk value as the size increases and is in good agreement with the experimental result. Similar

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Magnetic moment [µB/atom]

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Figure 5. (a) Magnetic moment per atom, (b) binding energy per atom, and (c) mean nearest neighbour bond length in PdN clusters. The numbers in (b) show the size of the clusters. The points are connected to aid the eyes. The inset in (a) shows the mean magnetic moment per atom in successive atomic shells of icosahedral 13-, 55-, 147- atom clusters. The inset in (b) shows the energies of the spin isomers of icosahedral Pd55 with respect to the ground state of 26 µB . (After Kumar & Kawazoe 2002a.)


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enhancement of moments has been found in clusters of Ni and the occurrence of magnetism in clusters of Ru and Rh has been predicted (Kumar & Kawazoe, unpublished data). These are important catalysts and the occurrence of magnetism can have important consequences for reactions. Accordingly the changes in the magnetic properties have been studied due to H adsorption (Kumar & Kawazoe 2002a, and to be published). An interesting finding is the metal-non-metal transition due to H adsorption. Such behaviour has important implications for the catalytic properties of Pd particles. However, rhodium clusters could have enhancement of magnetic moments with H adsorption. It is to be noted that experimentally it has been difficult to measure magnetic moments on these clusters and conflicting results were obtained. The theoretical results have clarified this. Bonding nature in clusters could be different from bulk and this is particularly so in the case of semiconductors. Studies on tin clusters (Majumder et al 2001) showed strong enhancement of bonding in small clusters that could be responsible for the higher melting temperature observed in these clusters (Shvartsburg & Jarrold 2000). Recently extensive computer experiments have been performed on metal encapsulation of silicon (Kumar & Kawazoe 2001c, 2002c, Kumar et al 2002b; Sun et al 2002) and germanium (Kumar & Kawazoe 2002d,e) clusters. These have led to the discovery of silicon fullerene M@Si16 , M = Zr and Hf in which 16 Si atoms surround the metal atom and other new forms such as the Frank–Kasper polyhedron of M@Si16 with M = Ti and Hf (figure 6). These have higher stabilities as well as higher symmetries as compared to the elemental Si clusters (Ho et al 1998). Surprisingly, the band gap in these metal doped clusters is even higher (up to 2·35 eV within GGA) than known for the pure Si clusters. This property makes such clusters interesting for photoluminescence. Also depending upon the size of the metal atom, the size and structure of the cage is different. This gives strong size selectivity to these clusters. Therefore, it is possible that one can prepare clusters with desired properties by changing the metal atom. Recently these clusters have been produced in the laboratory (Bergeron & Castleman 2002; Ohara et al 2002) confirming our predictions. The interaction between the clusters is found to be weak (Kumar & Kawazoe 2001c) and this makes it possible to assemble such clusters to develop new materials. A nanowire of silicon fullerene has been studied (Kumar & Kawazoe, unpublished) and it is a semiconductor similar to silicon. On the other hand assembly of a Si12 Be cluster (Kumar & Kawazoe 2002e) has been found to lead to the formation of silicon nanotube which is metallic (Singh et al 2002). These findings have opened up new avenues in research on nano-particles

(a)

(b)

Figure 6. (a) Fullerene and (b) Frank–Kasper polyhedral isomers of M@Si16 clusters with M = Ti, Zr, and Hf. The metal atom is at the centre. (After Kumar & Kawazoe 2001c.)

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where metal encapsulation can be used to produce novel structures. Recently, hydrogenated silicon fullerenes have been predicted (Kumar & Kawazoe 2002f) that have empty cages. It has been difficult to produce silicon with empty cages as there is a preference for sp 3 bonding in contrast to carbon. Our research has shown that silicon can be stabilized with empty cage structures also. The optical gap of Zr@Si16 fullerene and Ti@Si16 Frank–Kasper polyhedron lie in the red and violet regions, respectively (Briere et al unpublished). Therefore, these are new forms of silicon that are photoluminescent. These developments have opened up new possibilities of silicon nanoforms. Currently much efforts are going on molecular electronics and the findings of such clusters make silicon promising even at such small scales. 5. Summary The above mentioned developments are outstanding and fulfill the dream that quantum mechanics alone predicts the behaviour of matter. However, independent developments have taken place such as those using the classical molecular dynamics with empirical potentials that are even today important because the problems of practical materials are far too complicated to be understood in a fully ab initio manner at least for the present and perhaps it may not be necessary as one can work with appropriate models to suit the time and length scales involved in the concerned problem. However, one important difference has occurred that some of the parameters needed in such studies could be obtained from ab initio calculations so that such calculations could have improved predictability. We have covered in this brief review a few developments that indicate the strengths and potentials of such approaches. As computer power on the desktop grows, the utility of such approaches will become more and more apparent so that much exploration may become possible to find suitable materials for certain applications before an actual experiment is done. Such materials could then be refined by experiments. This will hopefully save much cost as well as time and possibly lead to better solutions. Significant progress is already taking place in this direction, though the number of cases where simulations have preceded experiments are still few. The past century has been dominated by many experimental discoveries and findings of suitable materials, but we hope that this century will see much more predictions and design of materials based on simulations.

I am grateful to Prof Y Kawazoe for all the support and to all my collaborators for the fruitful cooperation. I would like to acknowledge hospitality at the Institute for Materials Research as well as the support of the staff of the Center for Computational Materials Science of the IMR-Tohoku University for the use of the supercomputer facilities as well as their support. References Belomoin G, Therrien J, Smith A, Rao S, Twesten R, Chaieb S, Nayfeh M H, Wagner L, Mitas L 2002 Observation of a magic discrete family of ultrabright Si nanoparticles. Appl. Phys. Lett. 80: 841–843 Bergeron D E, Castleman A W Jr 2002 Insights into the stability of silicon and metal-doped silicon cluster ions: Reactive etching with O2 . In Int. Symp. on Small Particles and Inorganic Clusters 11, Strasbourg. Besenbacher F, Chorkendorff I, Clausen B S, Hammer B, Molenbroek A M, Norskov J K, Stensgaard I 1998 Design of a surface alloy catalyst for steam reforming. Science 279: 1913–1915


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