J = σE, resistivity and superconductors

Derivation of J = σE

Let n be the free-electron density and τ the average time between collisions.

1. Acceleration: a = eE/m.
2. Drift velocity: v_d = aτ = eτE/m.
3. J = n e v_d = (ne²τ/m) E.
4. Writing σ = ne²τ/m (conductivity), J = σE. The resistivity is ρ = 1/σ = m/(ne²τ).

Why resistivity rises with temperature

When temperature increases, the lattice ions vibrate more strongly. Electrons collide more often, so τ decreases. Since ρ = m/(ne²τ), ρ increases. For a metal, n stays almost constant. Near room temperature:

ρ = ρ₀[1 + α(θ − θ₀)], and R_θ = R₀(1 + αθ).

Graph of R_θ against temperature

 R
 |                              /  normal metal
 |                           /
 |                        /
 |                    _ /
 |   ____________ --          <- residual resistance at 0 K
 |
 |                        /  superconductor
 |                     /
 |            |------/
 |            |
 +------------+------------------> T
 0            Tc

For a normal metal, R falls steadily as T is lowered and levels off at a small residual value at 0 K. For a superconductor, R drops abruptly to zero at the critical temperature T_c and stays zero below it.

Superconductors

A superconductor is a material whose resistance falls to exactly zero below T_c (e.g. Hg 4.2 K, Nb 9.3 K, YBCO ~92 K).

Difference from a perfect conductor: a perfect conductor (ρ = 0) only keeps the magnetic flux it already has, so its state depends on history. A superconductor also shows the Meissner effect: it expels magnetic flux (B = 0 inside) whatever the order of cooling and field application. It is a perfect diamagnet (χ = −1).

Characteristics

1. Zero electrical resistance below T_c, so a persistent current flows without loss.
2. Meissner effect: perfect diamagnetism.
3. Critical field H_c(T) = H₀[1 − (T/T_c)²]; above it superconductivity is destroyed.
4. Critical current density: a large enough current also destroys it.
5. Type I (sharp transition) and Type II (mixed state between H_c1 and H_c2).
6. Uses: MRI magnets, maglev, SQUIDs, power cables.