Temperature dependence of resistivity: The resistivity of a material is dependent on temperature. It is experimentally found that for a wide range of temperatures, the resistivity of a conductor increases with increase in temperature according to the expression,
`rho_(T)=rho_(0)[1+alpha(T-T_(0))] " " ...(1)`
where `rho_(T)` is the resistivity of a conductor at `T_(0)C, rho_(0)` is the resistivity of the conductor at some reference temperature To (usually at `20^(@)C`) and `alpha` is the temperature coefficient of resistivity. It is defined as the ratio of increase in resistivity per degree rise in temperature to its reisstivity at `T_(0)`.
From the equation (1), we can write
`rho_(T)-rho_(0)=alpha rho_(0)(T-T_(0))`
`therefore alpha=(rho_(T)-rho_(0))/(rho_(0)(T-T_(0)))=(Delta rho)/(rho_(0)DeltaT)`
where `Deltarho=rho_(T)-rho_(0)` is change in resistivity for a change in temperature `DeltaT=T-T_(0)`. Its unit is per `""^(@)C`.
`alpha ` of conductors: For conductors `alpha` is positive. It the temperature of a conductor increases, the average kinetic energy of electrons in the conductor increases. This results in more frequent collisions and hence the resistivity increases.
Even though, the resistivity of conductors like metals varies linearly for wide range of temperatures, there also exists a nonlinear region at very low temperatures. The resistivity approaches some finite value as the temperature approaches absolute zero.
As the resistance is directly proportional to resistivity of the material, we can also write the resistance of a conductor at temperature `T^(@)C` as
`R_(T)-R_(0) = [1+alpha(T-T_(0))]" " ...(2)`
The temperature coefficient can be also be obtained from the equation (2),
`R_(T)-R_(0)=alpha R_(0)(T-T_(0))`
`therefore alpha=(R_(T)-R_(0))/(R_(0)(T-T_(0)))=(DeltaR)/(R_(0)DeltaT)`
`alpha= (DeltaR)/(R_(0)DeltaT)" " ...(3)`
where `DeltaR=R_(T)-R_(0)` is change in resistance during the change in temperature `DeltaT=T-T_(0)`
`alpha` of semiconductors: For semiconductors, the resistivity decreases with increase in temperature. As the temperature increases, more electrons will be liberated from their atoms (Refer unit 9 for conduction in semi conductors). Hence the current increases and therefore the resistivity decreases. a semiconductor with a negative temperature coefficient of resistance is called a thermistor.
We can understand the temperature dependence of resistivity in the following way. The electrical conductivity, `sigma = (n e^(2)tau)/(m) (m)/(n e^(2)tau)`. As the resistivity is inverse of `sigma`, it can be written as,
`sigma=(n e^(2)tau)/(m) (m)/(n e^(2)tau) " " ...(4)`
The resistivity of materials is
(i) inversely proportional to the number density (n) of the electrons
(ii) inversely proportional to the average time between the collisions `(tau)`.
In metals, if the temperature increases, the average time between the collision `(tau)` decreases and n is independent of temperature. In semiconductors when temperature increases, n increases and `tau` decreases, but increase in n is dominant than decreasing `tau`, so that overall resisitivity decreases.
