DC conductivity of metal

σ = nq²τ/m

where n is the density of carrier, q is the charge of the carriers, tau is the τ time, m is the effective mass.

The temperature dependence of σ comes from the temperature dependence of τ.

τ can be found from scattering theory. It turns out that

1/τ ~ A², where A is the amplitude of the vibration.

A² ~ n, where n is the number of phonons.

As we know that <n>=1/(exp(hω/k_BT)-1).

At hight T, <n> ~ T, which is the linear relation found for 1/τ or σ.

At low T, we have to consider not only the number of vibrational modes excited, but also the scattering effect of these mode.

The former gives T^-3 relation and the latter gives T². So the total relation will be T^-5.