Meissner effect types and critical current

Meissner effect: type-I vs type-II

The Meissner effect is the expulsion of flux ($B = 0$, $M = -H$) below $T_c$. The way it ends as $H$ increases separates the two types:

• Type-I: complete diamagnetism up to $H_c$, then an abrupt drop to $M = 0$ (normal). Single, low $H_c$. Examples: Pb, Hg.
• Type-II: complete only up to $H_{c1}$; between $H_{c1}$ and $H_{c2}$ flux partly enters as vortices (mixed state), $-M$ falls gradually; normal above $H_{c2}$. Examples: NbTi, Nb$_3$Sn.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

Critical current

The critical current $I_c$ is the maximum current a superconductor can carry without becoming normal. The self-field of the current must remain below $H_c$ (Silsbee's rule).

For a wire of radius $r$, by Ampere's law the surface field is $H = I/(2\pi r)$. Setting $H = H_c$:

$$\begin{aligned} I_c &= 2\pi r H_c = 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \\ J_c &= \frac{I_c}{\pi r^2} = \frac{2H_c}{r} \end{aligned}$$

        H = I/(2πr)
      .-~~~~~-.
     / .-----. \
    | (  I ⊙  ) |  wire radius r
     \ '-----' /
      '-~~~~~-'

With an applied field $H_a$, $I_c = 2\pi r (H_c - H_a)$, so $I_c$ falls with field and with temperature.