1.9　Critical magnetic fields

A superconductor in a sufficiently strong magnetic field will return from the super conducting state to the normal state.This field is named as the critical magnetic field Hc .For type II superconductors, the upper critical magnetic field Hc2 depends on the vortex structure.For type I superconductors, the superconducting state can be destroyed by the thermodynamic critical magnetic field Hc .The critical mag netic field increases as temperature is lowered and reaches the maximum value at T=0 K.Generally, the thermodynamic critical field Hc is lower than Hc2 for type II.

The superconducting state can also be destroyed by the self-field from a large transport current through the material. This maximum transport current is the so-called critical current density Jc （see section 1.8）.It depends on the properties and the geometry of the superconductor.Both the critical currents and the critical magnetic fields are directly related to the temperature.With lower temperature, the better property is achieved.This critical magnetic field is also a very important parameter for industrial applications.

Type I superconductors have a single critical field Hc , above which superconduc tivity vanishes. Type II superconductors have two critical fields, the lower critical magnetic field Hc1 ，and the upper critical magnetic field Hc2 ，between which they allow partial penetration of the magnetic field in the form of magnetic vortices.

The splitting energy level of the electron ground state in a magnetic field is called as the Zeeman splitting energy level. When the Zeeman splitting energy levelΔE of electrons becomes comparable to the energy gap 2Δ，Cooper pairs are broken and superconductivity vanishes.This magnetic field is named as the Pauli limiting field, and is the theoretical upper critical field Hc2 of superconductors.

Type I superconductors have only one critical field Hc , and Eq.（1. 4.7）has shown the difference Gn −Gs in the Gibbs free energy between the normal and the superconducting states, which is proportional to the critical magnetic field H2c ，

Since this is a thermodynamic expression, Hc is named as the thermodynamic critical field. Both type I and type II superconductors have thermodynamic critical fields.In addition, a type II superconductor has both a lower and upper critical field, Hc1 and Hc2 ，respectively.The lower critical field Hc1 is given by20

The upper critical field Hc2 occurs when the flux density is so dense that the cores of vortices overlap. It can also be described by the Ginzburg-Landau coherence length ξ GL and the quantum of magnetic flux Φ0 ，

whereμ0 is the permeability of free space and Φ0 is the quantized flux which is expressed as Φ0 =hc/2e=（2.07×10−15Wb）.

Combining Eq.（1. 7.5），（1.9.2）and（1.9.3），we have

The value of Hc at T＜Tc varies with temperature, however the critical mag netic field at absolute zero Hc （0）is constant. The properties and functions depend on the material itself.The relation between the critical magnetic field Hc and temperature T is15

If the applied magnetic field is larger than Hc （0）at 0 K, then even at abso lute zero, the superconducting state will be destroyed. The magnetic field which destroys superconductivity is a very important parameter, as it determines the maximum current density（which generates a strong magnetic field）in practice applications.

The upper critical field Hc2 （0）of a type II superconductor is very high so that direct measurement is not realistic, however it can be estimated using the slope of Hc2 with increasing temperature, the approximate formula is16

For HTSC with planar structure, the critical magnetic fields are larger when the external field applied is perpendicular to the c-axis than when it is applied parallel to this axis.