Meissner effect and hard superconductors

Meissner effect

Meissner effect is the complete expulsion of magnetic flux from the interior of a material when it is cooled below its critical temperature $T_c$ in a weak magnetic field ($H < H_c$). The superconductor becomes a perfect diamagnet.

Inside the superconductor:

$$\begin{aligned} B &= \mu_0 (H + M) = 0 \\ \Rightarrow M &= -H, \quad \chi_m = \frac{M}{H} = -1 \end{aligned}$$

Screening currents flow in a thin surface layer (the London penetration depth, $\lambda_L \approx 10$ to $100$ nm) and create a field that exactly cancels the applied field inside.

   T > Tc (normal)        T < Tc (superconducting)
   | | | | | | |           \ \         / /
   | |+-----+| |            \ \ +---+ / /
   | ||     || |   cool      ) )|   |( (
   | ||     || |   ---->    / / +---+ \ \
   | |+-----+| |           / /         \ \
   | | | | | | |      flux lines bend around it

The effect is independent of history: whether the field is applied before or after cooling, the flux is expelled. This shows a superconductor is not just a perfect conductor (a perfect conductor would trap the flux present at cooling).

Type-I vs type-II superconductors

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

Why type-II is called a hard superconductor

• It stays superconducting up to a very high field $H_{c2}$, so superconductivity is hard to destroy.
• In the mixed state, flux vortices get pinned at defects, dislocations and impurities. This makes the magnetization irreversible with a large hysteresis loop, just like a hard magnetic material.
• Because of pinning, it can carry a large critical current density in strong fields.
• Most are alloys or compounds that are mechanically hard and brittle (e.g. Nb$_3$Sn).