As the resistivity of a material is given as

ρ=m/ne²t

This shows that the resistivity is related to the number density n of free electrons in the material and relaxation time t. The variation of resistivity of material with temperature is different in different materials and it is discussed below:

(a)    Metals: In most metals, number density n of free electrons does not change with temperature but an increase in temperature increases the amplitude of vibration of lattice ions of the metal. Therefore, the Collision of free electrons with ions or atoms while drifting towards the positive end of the conductor becomes more frequent, resulting in a decrease in relaxation time. Thus resistivity of conductor increases with increase in temperature. At low temperature, resistivity increases at a higher power of T.

It is found that the temperature dependence of resistivity of a metal is given by the relation

ρ = ρ ₀[1+α_t(T-T­₀)]

Where ρ and ρ₀ are the resistivity at temperature T and T₀ respectively and α_t is called temperature coefficient of resistivity.

Or                              α_r= (ρ – ρ ₀)/ ρ ₀(T-T₀)=d ρ / ρ₀ (1/dT)

Thus, α_ris defined as the fractional change in resistivity (dρ / ρ₀) per unit change in temperature (dT)

For Conductors. The value of α_r is positive, showing that their resistivity increase with increase in temperature. For most metals the resistivity increases linearly with increase in temperature over a temperature range of about 500 k, above the room temperature.

(b)   Semi conductors: In case of semi- conductors, the value of α_r is negative. It means the resistivity of semi- conductors decreases as temperature increases.

(c) Insulators: The resistivity increases exponentially with decrease in temperature in case of semiconductors . It becomes infinitely large at temperature near absolute zero i.e. the conductivity is almost zero at o k.

The temperature dependence of resistivity of semi-conductors and insulators is given by:

ρ = ρ ₀e^E_g/2kT

Where K= Boltzmann constant

(1.381*10^-23 j mole ^-1k^-1)

T= absolute temperature

E_g=Energy band gap between conduction band and valence band or activation    energy for conduction

The classification of non-conduction materials into insulators and semiconductors depends upon the E_g.

(i)                  If E_g= 1eV, the value of resistivity is not very high therefore, the materials are called semi-conductors.

(ii)                If E_g≥1eV, the value of resistivity is very high and the materials are called insulators.