Class 12 Physics Semiconductor Electronics Band theory of solids

Band theory of solids

In a substance, as many atoms are close to each other, the energy levels of the atom form a continuous band, where in the electrons move.  This is called band theory of solids.

  • We know that in an atom, the protons and the neutrons constitute the central part called the nucleus
  • The electrons revolve around the nucleus in defined orbits
  • The orbits are named as 1s, 2s, 2p, 3s, 3p, 3d etc. each of which has a discrete energy level
  • All electrons in the same orbit have the same energy
  • The electrons in the innermost orbits which are completely filled constitute the valence electrons whereas the electrons in the outermost orbit which do not completely fill that shell are called conduction electrons
  • As seen in the diagram below, both Si and Ge have 4 electrons in the outermost shell

  • When in the crystal, the atoms are close to each other and hence they may be flow of electrons from one atom to another in the conduction band
  • Let us discuss in detail by considering interatomic distance in the X-axis and energy in the Y-axis:
  • As seen in the diagram below, the graph is divided into 4 regions – Region A, B, C and D
  • In the region A, the interatomic distance is large between atoms and in region D, the interatomic distance is small

 

       Region A

  • Consider that the Si or Ge crystal contains N atoms. Electrons of each atom will have discrete energies in different orbits
  • If the atoms are isolated, that is, separated from each other by a large distance, the electron energy will be the same
  • However, in a crystal, the atoms are close to each other separated by a distance of 2-3 Ao. Hence, electrons interact with each other and also with the neighbouring atoms
  • The overlap or the interaction will be felt more by the electrons in the outermost orbit while the inner electron energies will remain unaffected
  • Hence, in the case of Si and Ge crystals, we need to consider the changes in energies of electron in the outer most orbit only
  • For Si, the outermost orbit is the third orbit (n = 3) while for Ge, the outermost orbit is fourth orbit (n = 4)
  • The number of electrons in both cases is 4 – namely 2s and 2p. Hence, the outer electrons in the crystal is 4
  • The maximum possible number of outer electrons in the orbit is 8 (2s + 6p electrons)
  • This is the case of well separated or isolated atoms as shown in region A

Region B

  • Suppose the atoms start coming nearer to each other to form a solid.
  • The energies of the electrons in the outermost orbit may increase or decrease, due to the interaction between electrons of different atoms
  • The 6N states for l=1, which originally had identical energies in the isolated atoms, spread out and form an energy band as shown in the region B
  • Similarly, the 2N states for l = 0 split into a second band separated from the first one

Region C

  • At still smaller spacing, however, there comes a region in which the bands merge with each other
  • The lowest energy state that is a split from the upper atomic level appears to drop below the upper state that has come from the lower atomic level
  • In this region, no energy gap exists where the upper and the lower energy states gets mixed

Region D

  • If the distance between the atom further decreases, the energy bands again split apart and are separated by an energy gap Eg
  • The total number of available energy states 8N has been re-apportioned between the two bands (4N states each in the lower and upper energy bands)
  • Here there are exactly as many states in the lower band (4N) as there are available valence electrons from the atoms (4N)
  • This lower band called the valence band is completely filled while the upper band is completely empty. The upper band is called the conduction band

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