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BAND THEORY OF SOLIDS

According to quantum free electron theory of metals, a electron in a metal experiences constant(or zero) potential and free to move inside the crystal but will not come out of the metal because an infinite potential exists at the surface.

Bloch Theorem: According to this theorem, the periodic potentials due to the positive ions in metal have been considered. (i.e. the electrons moves in a periodic potential provided by lattice). If the electron moves through these ions, it experiences varying potentials. The potential of an electron at the positive ion site zero and is maximum in between two ions. i.e. the potential experienced by an electron varies periodically ...view middle of the document...

The potential of electron varies periodically with periodicity of ion core ‘a’ which is nothing but inter-atomic spacing. It is assumed that the potential energy of the electrons is zero near nucleus of the positive ion core and maximum when it is lying between the adjacent nuclei as shown fig.

Fig. One dimensional periodic potential

The width of the potential well and barrier are ‘a’ and ‘b’ respectively. The potential energy of an electron in the well is zero and in the barrier is V0. The periodicity of the potential is a+b. This model is an highly artificial, but it illustrates many of the characteristics features of the behavior of electrons in periodic lattice. The energies and wave functions of electrons associated with model can be calculated by solving the Schrödinger’s wave equation for two regions I and II.

The Schrödinger’s equations are

for region-I 0 < x < a

and for region-II -b < x < a

Since E less than Vo, define two +ve quantities, α2 = 2mE/ ħ2 and β2 = 2m(V0– E )/ħ2 for region 0 < x < a

for region -b < x < a

According to Bloch Theorem, the solutions of above equations can be written as

Ψ(x) = uK (x) eiKx

The consists of a plane wave eiKx modulated by the periodic function uK(x), where this UK(x) is periodic with the periodicity of the lattice.

uK(x) = uK(x+a)

where K is propagating constant along x-direction and is given by K =2π/λ is a Propagation wave vector. In order to simplify the computations, an assumption made regarding the potential barrier. As Vo increases the width of the barrier ‘b’ decreases so that the product Vob remains constant. After tedious calculations, the possible solutions for energies are obtain from the relation

Psin∝aαa+cos∝a=cosKa ------------ (1)

where P = mVo ab /ħ2 is scattering power of the potential barrier. It is a measure with which electrons in a crystal are attracted to the ions on the crystal sites.

The left hand side of the equation (1) is plotted as a function of αa for the value of P=3π/2. The right side of the equation imposes a limitations on the values of left side function, i.e., between -1 to +1 as indicated by the horizontal lines in fig. Hence only certain range of α are allowed.

Fig. a) For large value of P b) P⟶∞ c) P⟶0

The permitted values of energy are shown as solid lines between white portions. This gives rise to the concept of ranges of permitted values of for a given ion lattice spacing a.

Conclusions:

1. The energy spectrum of electron consists of alternate regions of allowed energy and un allowed regions i.e. The motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions.

2. As the value of a increases, the width of allowed energy bands also increases and the...

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