1.3　Meissner-Ochsenfeld effects

The Meissner-Ochsenfeld effect8（often abbreviated as the Meissner effect）is one important signature of the superconductivity.Perfect diamagnetism is observed due to expulsion of the magnetic flux from inside the superconducting material forfields below a critical field when the temperature is below the critical temperature Tc .This important property allows us to distinguish a superconducting material from a perfect conductor.

According to Lenz's law, when a changing magnetic field is applied to a con ductor, it will induce an electric current in the conductor that creates an opposing magnetic field. However, the Meissner-Ochsenfeld effect is the spontaneous expul sion which occurs during the transition to the superconducting state.Suppose there is a superconductor in the normal state, placed in a constant external magnetic field.When it is cooled below the critical temperature Tc , the abrupt expulsion of the internal magnetic field would occur, which is not expected based on Lenz’s law.

The dimensionless magnetic susceptibility is defined asχ=M/H, and the magnetic flux density can be expressed as

whileμ0 is the permeability of free space.The property of perfect diamagnetism means that the susceptibility χ=−1，i.e.the magnetization M=−H, so there can be no B field inside a superconductor.

If a superconductor in the normal state is cooled below Tc by zero field cooled（ZFC）method, the normal state to the superconducting state transition happens without the presence of any magnetic field. Thereafter the cooled superconductor is placed in an external magnetic field, the magnetic field will be expelled from the superconductor.If this superconductor in normal state is placed in an external mag netic field, the field will penetrate and have the same value inside and outside.When a superconductor in the normal state is cooled below its Tc by field cooled（FC）method, the normal state to the superconducting state transition happens with the presence of a magnetic field, thereafter the field will be expelled from the supercon ductor.This phenomenon of superconductors is named as the Meissner-Ochsenfeld effect, and a similar phenomenon cannot be found in conventional conductors and demonstrates that superconductivity is a new thermodynamic phase.

The superconductors that totally exclude applied magnetic flux are known as type I superconductors. The Meissner-Ochsenfeld effect exists in type I supercon ductors below the critical field Hc , but the superconductivity will be destroyedabove Hc .In type I superconductors whose thicknesses are much greater than the penetration depth λ，internal magnetic fields and transport currents are merely able to exist only in a surface layer of thicknessλ.

The magnetic properties of the type II superconductor are more complex. type II superconductors are similar to type I superconductors but the presence of a Meissner-Ochsenfeld effect only happens when the magnetic applied field is smaller than a lower critical field Hc1 .When the applied magnetic field is higher than Hc1 but lower than its upper critical field Hc2 ，magnetic vortices begin to penetrate into the superconductor and the material enters the mixed state.The magnetic field can partially penetrate into the superconductor, and the field is only partially excluded, i.e.diamagnetism is not perfect.

Movement of vortices in a type II superconductor can cause heat dissipation, i. e.a finite resistivity, called flux flow resistivity, the superconductor is no longer in the superconducting state of type I superconductor.If the current is sufficiently small, the vortices are stationary, and the resistivity is zero.The resistance below the critical current density Jc is negligible compared with that of normal conductors.Indeed, there is no complete Meissner-Ochsenfeld effect in type II superconduc tors above Hc1 ，since in the mixed state, the magnetic field penetrates into the superconductor in the shape of the vortices.

Interaction between the magnetic field and the type I superconductor in Meis sner-Ochsenfeld state produces a magnetic levitation force, but this force is much smaller than that of a type II superconductor in the mixed state. This is a very important matter, but also confusing.Magnetic levitation in the type II super conductor looks similar to the Meissner-Ochsenfeld effect, but there is a critical difference.The Meissner-Ochsenfeld effect completely excludes the magnetic fields below Hc1 to the field penetrating depthλon the superconductor surface, and it gives rise to a small levitation forces.In the mixed state of a type II superconduc tors, the pinning forces hold the superconductor in a place thereby allowing it to levitate stably.The levitation forces of the type II superconductors in the mixed state are much larger than the levitation force in a pure Meissner-Ochsenfeld state in the type I superconductors.

Svedlindh et al.13reported the observation of a paramagnetic Meissner-Ochsen-feld effect（PME）14 in a BiSrCaCuO superconductor.The Meissner-Ochsenfeld effect is characterized by a diamagnetic response of the superconducting mate rial when a magnetic field is applied, but in several cases this magnetic response in FC experiments can be paramagnetic, challenging the conventional Meissner Ochsenfeld effect.