Discuss the temperature dependence of resistivity of metals/conductors, insulators and semiconductors.
It has been observed that at low temperature, resistivity of a conductor increases at a higher power of temperature (T). Thus, over a limited range of temperature, the variation of $\rho$ with temperature (T) is expressed by the relation
$\rho = \rho_0 [1 + \alpha (T - T_0)]$
where $\rho_0$ is the resistivity at reference temperature $T_0$ (say 273 K or $0^\circ$C), $\rho$ is resistivity at temperature $T$ and $\alpha$ is the temperature coefficient of resistivity.
The temperature coefficient of resistivity is defined as :
$\alpha = \frac{(\rho - \rho_0)}{\rho_0 (T - T_0)} = \frac{\Delta \rho}{\rho_0 \Delta T}$
Thus, temperature coefficient of resistivity ($\alpha$) is defined as the change in resistivity per unit original resistivity per unit change in temperature.
The SI unit of temperature coefficient of resistivity is (kelvin)$^{-1}$ or K$^{-1}$.
Temperature coefficient of resistivity $\alpha$ is different for different materials.
(i) Metals/conductors : $\alpha$ is positive for metallic conductors i.e. their resistivity increases with increase in temperature. In most of the metallic conductors, the resistivity increases linearly with increase in temperature from room temperature to about 500 K. That is, they have linear relation upto a limited temperature range. At low temperature, the conductor like copper, aluminium etc. have non-linear dependence.
For alloys like nichrome, manganin and constantan, the value of temperature coefficient of resistivity is positive but negligible. It means, the change in the resistivity of these materials with increase in temperature is negligible. For this reason, these alloys are used for making standard resistance coils.
(ii) Semi-conductors : $\alpha$ is negative for semi-conductors. Germanium and Silicon are the examples of semi-conductors. At 0 K, semi-conductors behave as insulators but at room temperature, they behave as conductors.
The resistivity of semi-conductors depends on two factors : (i) the temperature variation and (ii) the suitable impurities added in the semi-conductors.
A small amount of specific impurities added in the semi-conductors decrease their resistivity to a large extent. Also with the increase in temperature, the resistivity of the semi-conductors decreases more rapidly as
(iii) Insulators : Resistivity of insulators (like wood, glass etc.) increases exponentially with decrease in temperature. The resistivity of insulators at absolute zero is infinitely large.
FACTS :
1. Nichrome (alloy of Ni, Cu and Cr) has high resistivity, low conductivity and a small temperature coefficient of resistivity. Another advantage of Nichrome is that it does not deteriorate mechanically or chemically even at a high temperature (e.g. when red hot) for a longer period of time. It is used for making elements of electric heaters.
2. Copper wires are used as connecting wires because the resistivity of copper is low and its electric conductivity is very high.