1.9 Critical magnetic fields Ë 13electrons becomes comparable to the energy gap 2훥, Cooper pairs are broken and
superconductivity vanishes. This magnetic field is named as the Pauli-limiting field
and is the theoretical upper critical fieldHc2of superconductors.
Type I superconductors have only one critical fieldHc, and Eq. (1.8) has shown
the differenceGn−Gsin the Gibbs free energy between the normal and the supercon-
ducting states, which is proportional to the critical magnetic fieldH^2 c,
Gn−Gs=^1
2휇 0 H^2 c. (1.37)Since this is a thermodynamic expression,Hcis 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 an upper critical fields,Hc1
andHc2, respectively. The lower critical fieldHc1is given by [20]
Hc1=훷^0 ln휅
4 휋휇 0 휆^2. (1.38)
The upper critical fieldHc2occurs when the flux density is so dense that the cores
of vortices overlap. It can also be described by the GL coherence length휉GLand the
quantum of magnetic flux훷 0 ,
Hc2= 훷^0
2 휋휇 0 휉^2 GL, (1.39)
where휇 0 is the permeability of free space and훷 0 is the quantized flux which is
expressed as훷 0 =hc/ 2 e=( 2. 07 × 10 −^15 Wb).
Combining Eqs. (1.30), (1.38) and (1.39), we have
Hc2/Hc1=^2 휆2
휉^2 GLln휅= 2 휅^2 /ln휅. (1.40)The value ofHcatT<Tcvaries with temperature; however, the critical magnetic
field at absolute zeroHc( 0 )is constant. The properties and functions depend on the
material itself. The relation between the critical magnetic fieldHcand temperatureT
is [15]
Hc=Hc( 0 ) 1 −T
Tc
2
¡. (1.41)If the applied magnetic field is larger thanHc( 0 )at 0 K, then even at absolute zero, the
superconducting state will be destroyed. The magnetic field which destroys supercon-
ductivity is a very important parameter, as it determines the maximum current density
(which generates a strong magnetic field) in practice applications.