Figure 159: Illustration of the dependence of the magnetization on the applied magnetic field for type
I and type II superconductors. In case of type I superconductors (completeMeissner -
Ochsenfeldeffect) the magnetization above the critical field (i.e. in the normal state)
is too small to be resolved in this figure. In case of a type II superconductor (incomplete
Meissner - Ochsenfeldeffect) the magnetic fieldBstarts to penetrate the supercon-
ductor at a critical valueHc 1 which is lower than the thermodynamic critical valueHc.
Above the second critical pointHc 2 the compound is normal conducting, the state between
Hc 1 andHc 2 is called vortex state orShubnikovphase. Note that the area under both
curves is the same.
known as type I superconductors. Type I superconductors are typically pure materials with low values
of electrical resistance. For other materials, typically alloys or transition metals with a high electric
resistance in the normal state, a different behavior is observed. These materials are referred to as type
II superconductors since they undergo a second order phase transition at the critical magnetic field.
Such a transition is depicted in Fig. 159(b). Type II superconductors posses two critical magnetic
fields,Hc 1 andHc 2. BelowHc 1 theMeissner - Ochsenfeldeffect is complete, while betweenHc 1
andHc 2 it is said to be incomplete. However, the sample is superconductive up to the second critical
fieldHc 2. BetweenHc 1 andHc 2 the magnetic field within the superconductor is not equal to zero,
B 6 = 0. I.e., the sample is threaded by flux lines. This state is referred to as the vortex state or
Shubnikovphase. Note thatHc 2 can be larger thanHcby orders of magnitude (which is in case of
type II superconductors defined by the thermodynamics of the system).
16.2.3 Heat Capacity
Measurements show that for temperatures below the critical temperature,T < Tc, the entropy of the
system is decreased, see Fig. 160. This contains the information that the superconducting state is
more ordered than the normal state. This observation was of crucial importance for the development
of microscopic theories. Since the change in entropy is very small, this must mean that only a small
fraction of all conduction electrons participate in the transition to the superconducting state.
Furthermore, the heat capacity is discontinuous at the critical temperature forBa= 0, see Fig. 161
and the observed form suggests an exponential form with argument proportional to−^1 T, while in the
normal state, the specific heat increases linearly with temperature. This allows the interpretation of
excitations of electrons over an energy gap, see Fig. 161 and 162. The energy gap is well described