Complete and incomplete Meissner effect

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).

Complete Meissner effect in type-I

• In a type-I superconductor, $B = 0$ inside for all fields up to $H_c$. Magnetization rises linearly, $M = -H$.
• At $H = H_c$, flux enters all at once and the material becomes normal; $M$ drops to zero.
• There is no intermediate state in a long sample, so the expulsion is complete right up to $H_c$. Examples: Pb, Hg, Sn.

Incomplete Meissner effect in type-II

• Below the lower critical field $H_{c1}$, flux is fully expelled (complete Meissner effect).
• Between $H_{c1}$ and $H_{c2}$ it is energetically favourable for flux to enter as thin normal cores called vortices (fluxons), each carrying one flux quantum $\Phi_0 = h/2e = 2.07\times10^{-15}$ Wb, surrounded by circulating supercurrents.
• The material between vortices stays superconducting (zero resistance), but $B \ne 0$ inside on average. Hence the Meissner effect is incomplete in this mixed (vortex) state.
• As $H$ increases, vortex density rises; at $H_{c2}$ the cores overlap and the material becomes normal. Examples: NbTi, Nb$_3$Sn, YBCO.

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

 Mixed state (top view of type-II slab)
   +-------------------+
   |  o    o    o    o |   o = vortex (normal
   |    o    o    o    |       core with flux)
   |  o    o    o    o |   rest = superconducting
   +-------------------+