Let us consider an electron of mass m and charge '-e' characterized by wave vector k such that it is related with velocity v of electron by the relation

mv = ℏ k ....1 where ℏ= reduced plank's constant

In presence of electric field E, the equation of motion for electric field written as

Let us consider a fermi sphere of radius k_f. Initially when no electric field is applied, we suppose that its centre is situated at origin i.e at k(0). When electric field is applied, the fermi sphere so that its centre is displaced to a new position k(t) after time t. With this physical background, integrating equation 2,

Because of collision of electron with phonon, lattice impurities and imperfections, the displaced sphere may be brought into steady state in electric field. With collision time t = 𝜏 in equation 3, the displacement of sphere in steady state can be written as

From equation 1,

If n be the electronic concentration, then current density is

This is ohm's law

Electrical conductivity (𝜎 ) = ne²𝜏 / m .........7

The reciprocal of electrical conductivity is electrical resistivity

Electrical resistivity (𝜌 ) = 1 / 𝜎 = m / ne²𝜏 ......8

Electronic thermal conductivity

The thermal conductivity of the particle of velocity 'v' heat capacity per unit volume 'c' and mean free path λ is given by the expression

k = 1/3 cv λ .....1

For free electrons,

Here the subscript 'f' stands for fermi level

where n = N/v = electronic concentration

KB = Boltzmann constant

T= absolute temperature

E_f = fermi energy

E_f = - 1/2 mv_f ² ....4

λ_f = v_f 𝜏 ....5

Substituting equation 3,4 and 5 in equation 2, we get

This is the expression of electronic thermal conductivity.

In the context of lattice impurities, the electrical conductivity of a material can be affected by the presence of impurities or defects in its crystal lattice structure. These impurities can scatter electrons as they move through the material, leading to an increase in the material's resistance.

However, Ohm's law still applies to materials with lattice impurities, as long as the temperature and other environmental factors remain constant. In such cases, the voltage applied across the material will still be proportional to the current flowing through it, and the resistance of the material can still be calculated using Ohm's law.

It's worth noting that the presence of impurities or defects can also affect the behaviour of the material at higher voltages or currents, leading to non-linear behaviour or even breakdown of the material. In such cases, more complex models are needed to describe the material's behaviour, beyond the simple linear relationship given by Ohm's law.

This note is a part of the Physics Repository.

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