The current density (electric current per unit area, J=I/A) can be expressed in terms of the free electron density as:

Current Density from Free Electrons

The number of atoms per unit volume (and the number free electrons for atoms like copper that have one free electron per atom) is:

Atoms per Volume

From the standard form of Ohm's law and resistance in terms of resistivity:

Electric Resistance

Electric Current

Current Density from Volts & Ohms

The next step is to relate the drift velocity to the electron speed, which can be approximated by the Fermi speed:

Fermi Speed

The drift speed can be expressed in terms of the accelerating electric field E, the electron mass, and the characteristic time between collisions.

Drift Velocity of Electron

The conductivity of the material can be expressed in terms of the Fermi speed and the mean free path of an electron in the metal.

Conductivity