Atomic view of resistivity, ρ = m/ne²τ

Current density

Current density J is the current per unit area, taken perpendicular to the flow. J = I/A (unit A/m²). It is a vector along the direction of positive charge flow. For carriers of density n, charge q and drift velocity v_d, J = nq v_d.

Mobility

Mobility μ is the drift velocity per unit electric field: μ = v_d/E (unit m²V⁻¹s⁻¹). It tells how easily carriers move through the material.

Atomic view of resistivity

In a metal the free electrons move randomly at high speed (about 10⁶ m/s) and collide with the lattice ions and impurities. With no field the average velocity is zero. A field E gives each electron an acceleration a = eE/m between collisions. After a collision the electron loses its memory of the earlier motion, so the average time between collisions is the relaxation time τ.

 field E: ----------------------------->

 electron:  <--.     .--.     .--o (start)
                \   /    \   /
                 . .      . .        . = collision
 net drift: <--------  (opposite to E, as the electron is negative)

Derivation of ρ = m/(ne²τ)

1. Acceleration of an electron: a = eE/m.
2. Average (drift) velocity gained in time τ: v_d = aτ = eEτ/m.
3. Current density: J = n e v_d = n e (eEτ/m) = (ne²τ/m) E.
4. Ohm's law in vector form is J = σE, so σ = ne²τ/m.
5. Resistivity is the reciprocal: ρ = 1/σ = m/(ne²τ).

Also μ = v_d/E = eτ/m, so σ = neμ. Resistivity rises with temperature because lattice vibrations increase, which reduces τ.