Energy Levels

The prediction of discontinuous energy levels was a breakthrough in the area of limited systems. The limited number of electrons results in different characters of electrons in the systems with a limited size. Two basic assumptions of Kubo's theory are [29, 55-57]: (1) the Poisson distribution of the energy levels close the Fermi surface is held, while the Fermi statistics is invalid in the nanoparticles; (2) the electronic neutral is valid, because it is extremely difficult to take (or put) one electron out from (or in) a nano-particle because it is necessary to overcome the extremely large Coulomb energy. Considering the effects of the discreteness of the energy levels on the physical properties of metal particles, Kubo and co-workers [29, 55-57] proposed a famous relation,

4Ef 3N

for the gap between the nearest neighboring energy levels, where EF is Fermi energy, N is the number of the conduction electrons in the particles, and V is the volume of the particles. If the particles were in the shape of balls, from Eq. (1), one would have a relation of 8 a d-3. Therefore, the energy gap increases rapidly with decreasing particle size. Because the average energy gap 8 close to the Fermi energy could be much larger than the thermal energy kBT at low temperatures, the discreteness of the energy levels would affect obviously the thermodynamic properties of the materials, and thus the specific heat; the susceptibility of the particles would differ evidently from those of bulk materials. It was found that the specific heat and the susceptibility of the particles depend on the parity (i.e., odd-even) of the number of the electrons in the systems. It is evident that this kind of the quantum size effect becomes much more pronounced only if the following condition is satisfied: the gap of the discreteness of the energy levels is larger than the thermal energy kBT, the magnetostatic energy, the magne-toelectric energy, the photon energy, etc. It was estimated that the temperature for a metallic particle of 10 nm showing the quantum size effect is about 2 K; that for a particle of 2 nm could be room temperature. Several groups tried to modify Kubo's theory in order to detect the effects of different distributions of the energy levels and to reach a better agreement between the theoretical and experimental results [58-62]. For details of this topic, readers are referred to the review article by Halperin [63].

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