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Unformatted text preview: Physics 927 E.Y.Tsymbal 1 Section 9: Energy bands The free electron model gives us a good insight into many properties of metals, such as the heat capacity, thermal conductivity and electrical conductivity. However, this model fails to help us other important properties. For example, it does not predict the difference between metals, semiconductors and insulators. It does not explain the occurrence of positive values of the Hall coefficient. Also the relation between conduction electrons in the metal and the number of valence electrons in free atoms is not always correct. We need a more accurate theory, which would be able to answer these questions. The problem of electrons in a solid is in general a many-electron problem. The full Hamiltoniam of the solid contains not only the one-electron potentials describing the interactions of the electrons with atomic nuclei, but also pair potentials describing the electron-electron interactions. The many-electron problem is impossible to solve exactly and therefore we need simplified assumptions. The simplest approach we have already considered, it is a free electron model. The next step in building the complexity is to consider an independent electron approximation, assuming that all the interactions are described by an effective potential. One of the most important properties of this potential is that it is periodic on a lattice ( ) ( ) U U = + r r T , (1) where T is a lattice vector. Qualitatively, a typical crystalline potential might be expected to have a form shown in Fig.1, resembling the individual atomic potentials as the ion is approached closely and flattening off in the region between ions. Fig. 1 The crystal potential seen by the electron. Within the approximation of non-interacting electrons the electronic properties of a solid can be examined by Schrdinger equation 2 2 ( ) ( ) ( ) 2 U E m + = r r r , (2) in which ( ) r is a wave function for one electron. Independent electrons, which obey a one- electron Schrdinger equation (2) with a periodic potential, are known as Bloch electrons , in contrast to "free electrons," to which Bloch electrons reduce when the periodic potential is identically zero. Now we discuss general properties of the solution of the Schrdinger equation (2) taking into account periodicity of the effective potential (1) and discuss main properties of Bloch electrons, which follow from this solution. We represent the solution as an expansion over plain waves: ( ) i c e = kr k k r . (3) This expansion in a Fourier series is a natural generalization of the free-electron solution for a zero potential. The summation in (3) is performed over all k vectors, which are permitted by the periodic Physics 927 E.Y.Tsymbal 2 boundary conditions. According to these conditions the wave function (3) should satisfy ( , , ) ( , , ) ( , , ) ( , , ) x y z x L y z x y L z x y z L = + = + = + , (4) so that 2 2 2 ; ; y...
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