Electron Scattering and Matthiessen’s Rule
The resistivity ρ of a metal containing small amount of impurities may be written as:
ρ = ρ0i+ ρ(T) (1)
Where ρ 0 is a constant which increases with increasing impurity content and ρ (T) is the temperature –dependent part of the resistivity.
Equation (1) is known as the Matthiessen’s rule
Explanation. If a metal contains impurities, the electrons are scattered by impurity atoms. This scattering is in addition to that caused by the lattice vibrations. The reason for this scattering is that the field in the vicinity of the impurities is different from that near the host atoms and deviations from the periodicity of the potential thus produced, act as scattering centres for electrons. As the scattering cross-section for the two types of scattering are additive and that the relaxation times are proportional to the reciprocals of the scattering cross-sections, we can write for the total relaxation time as
Where ti and tl are the relaxations times associated with scattering by impurity atoms and by lattice vibrations respectively.
Now, as the resistivity is inversely proportional to the relaxation time of the electrons, we have a temperature dependent term, arising from 1/ti contributing to the total resistivity and a temperature independent term, arising from 1/ti
Thus ρ = ρ 0i+ ρ(T)
The latter contribution to the resistivity will be independent of temperature because, for not too high impurity concentrations, 1/tiis proportional only to the impurity concentration. Sometimes ti itself is found to be slightly temperature dependent, but in general the temperature independent part predominates strongly.
The increased resistivity due to the introduction of impurity atoms does not disappear at absolute zero. The resistivity that remains at T=0 K is usually called the residual resistivity.