Bulk Metal Large Metal Small Metal Cluster Cluster

Figure 4.9. Illustration of how energy levels of a metal change when the number of atoms of the material is reduced: (a) valence band of bulk metal-, (b) large metal cluster of 100 atoms showing opening of a band gap; (c) small metal duster containing three atoms.

containing electrons, changes dramatically. The continuous density of states in the band is replaced by a set of discrete energy levels, which may have energy level spacings larger than the thermal energy kBT, and a gap opens up. The changes in the electronic structure during the transition of a bulk metal to a large cluster, and then down to a small cluster of less than 15 atoms, are illustrated in Fig. 4.9. The small cluster is analogous to a molecule having discrete energy levels with bonding and antibonding orbitals. Eventually a size is reached where the surfaces of the particles are separated by distances which are in the order of the wavelengths of the electrons. In this situation the energy levels can be modeled by the quantum-mechanical treatment of a particle in a box. This is referred to as the quantum size effect. The emergence of new electronic properties can be understood in terms of the Heisenberg uncertainty principle, which states that the more an electron is spatially confined the broader will be its range of momentum. The average energy will not be determined so much by the chemical nature of the atoms, but mainly by the dimension of the particle. It is interesting to note that the quantum size effect occurs in semiconductors at larger sizes because of the longer wavelength of conduction electrons and holes in semiconductors due the larger effective mass. In a semiconductor the wavelength can approach one micrometer, whereas in a metal it is in the order of 0.5 nm.

The color of a material is determined by the wavelength of light that is absorbed by it. The absorption occurs because electrons are induced by the photons of the incident light to make transitions between the lower-lying occupied levels and higher unoccupied energy levels of the materials. Clusters of different sizes will have different electronic structures, and different energy-level separations. Figure 4.10 compares the calculated energy levels of some excited states of boron clusters B6, B8, and B12 showing the difference in the energy-level separations. Light-induced transitions between these levels determines the color of the materials. This means that clusters of different sizes can have different colors, and the size of the cluster can be used to engineer the color of a material. We will come back to this when we discuss semiconducting clusters.

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