Superconductor, critical field and critical current density

Superconductor

A superconductor is a material whose resistance becomes exactly zero below a critical temperature $T_c$ and which expels magnetic flux from its interior (Meissner effect, perfect diamagnetism). Examples: Hg ($T_c$ = 4.2 K), Pb (7.2 K), Nb$_3$Sn (18 K), YBCO (92 K).

Critical magnetic field

The critical magnetic field $H_c$ is the minimum magnetic field that destroys superconductivity at a given temperature. It is maximum ($H_0$) at 0 K and zero at $T_c$:

$$H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

  Hc
  H0|*
    |    *    normal
    | super  *
    |         *
    +----------*--- T
               Tc

Critical current density

The critical current density $J_c$ is the largest current density a superconductor can carry without becoming normal. The current's own magnetic field at the surface must stay below $H_c$. For a wire of radius $r$ (Silsbee's rule):

$$I_c = 2\pi r H_c, \qquad J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r}$$