# Maximum metallic resistivity: dependence of resistivity on temperature

The electrical resistivity of metals usually increases with temperature. For a metal like copper at room temperature it increases almost linearly with temperature. At melting point we see a jump in resistivity. Then it continues to increase linearly with temperature again but with a different slope. The plot shows the resistivity of copper as a function of temperature (data from Matula1979, plotted by me).

It is believed that, at least in many metals, the increase in resistivity doesn't continue unlimitedly, but saturates at high enough temperatures. This is usually explained by Mott's theories (Mott was a Nobel laureate).

I'm not an expert on this and I'm looking for an analytical formula which gives a good estimate of the saturation value of metallic resistivity, along with its derivation.

• At some point the metal vaporizes... – Jon Custer Aug 18 '19 at 16:24
• @JonCuster, I think high density metal vapor (under high pressure) is still a good conductor. What is important is the density of free electrons. Please also see my answer. – apadana Aug 18 '19 at 21:45

This formula is given for calculating saturation resistivity: $$\rho_{\mathrm{sat}} = \frac{\hbar}{0.33e^2}\frac{1}{n^{2/3}a}\cdot$$ Here, $$\hbar$$ is the reduced Planck constant, $$e$$ is electron charge, $$n$$ is electron concentration, and $$a$$ is the interatomic distance.