Band Theory of solids - Engineering Physics

Band Theory of solids

Introduction –

The free-electron theory of metals successfully explained the various properties of metals like heat capacity, thermal conductivity, electrical conductivity, etc. But remained some properties that were not explained by this theory. For example,

1. It could not explain the difference between conductors, insulators, and semiconductors.

2. It is noticed that divalent metals (Be, Cd, etc.) and trivalent metals (Al, In, etc.) are not good conductors even though the theory says that conductivity is proportional to electron concentration. Rather monovalent metals Cu, Ag, and Au are good conductors, even these have less concentration of electrons that divalent and trivalent metals.

3. The shape of the Fermi surface is found to be non-spherical in shape which according to the theory should be spherical.

4. some of the metals exhibit a positive hall coefficient (eg. For Be, Zn, etc.), while the free-electron theory predicts a negative hall coefficient for all the metals.

The failure of free electron theory is because of the oversimplified assumption that the electrons move in a region of zero or constant potential in the metal. However, this is not the case, the potential experienced by the electron is very complicated and to a reasonable approximation we can assume that electrons move in the periodic potential of the ion cores with the periodicity of the lattice constant.

When we consider the notion of an electron in a periodic potential we get the given ahead results –

1. There exist allowed energy bands separated by the forbidden energy bands.

2. The function E(k) are periodic in K.

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