Advanced Solid State Physics

16.3 Phenomenological Description

16.3.1 Thermodynamic Considerations

Since the transition between the superconducting and the normal state is reversible we can apply the
basic concepts of thermodynamics in order to obtain an expression for the free energyFas a function
of the critical field curveHcandT. The work done on a type I (completeMeissner - Ochsenfeld
effect) superconductor in an external magnetic field can be expressed by

W=

∫Ba

0

dB·M. (268)

Thus we identify the differential of the free energy as

dF=−M·dB. (269)

Since we deal with a type I superconductor we can exploit the known relationship betweenMandBa
in the superconducting state in order to obtain

dFS=

1

μ 0

B·dB, (270)

or in particular forB: 0 →Ba

FS(Ba)−FS( 0 ) =

1

2 μ 0

Ba^2. (271)

If we consider a nonmagnetic metal in the normal state and neglect the magnetic susceptibility, then
M= 0 and accordingly

FN(Ba) =FN( 0 ). (272)

Now atBa=Bacthe energies are equal in the normal and in the superconducting state:

FN(Bac) =FS(Bac) =FS( 0 ) +

1

2 μ 0

Bac^2. (273)

Hence, at zero field, we have

∆F=FN( 0 )−FS( 0 ) =

1

2 μ 0

Bac^2. (274)

Here∆F is the stabilization free energy of the superconducting state. Hence, at a temperature
T < Tc, the superconducting state is more stable up to a critical magnetic fieldBac. Above this field
the normal state is more stable. The results for the stabilization energy are in excellent agreement
with experimental results.