Free Energy of the Superconducting State

4 Thermodynamics of Superconductivity 26.4.1 Free Energy of the Superconducting State 

Let G]v(H,T) and Gsc(Tf/T) be the free energies of the normal and superconductive states, respectively, as a fimction of applied magnetic field and temperature. The work required to destroy the superconducting state in a type-1 superconductor is obtained from 

The free energy of the superconducting state as a function of applied field can be written as 

Since the free energy of the normal state is imaffected by the presence of a magnetic field 

The difference in free energy between the normal and superconductive states is given by subtracting Equation 26.9 from Equation 26.10 

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Free energies of the normal and the superconducting state as a function of T/Tc- The slopes of the two free energies are the same at Tq indicating a second-order transformation. The free energy of the superconductivity state is found by integrating the entropy Equation 26.17. 

We now start with the exponential form of the heat capacity equation (Equation 26.12) and work backward to find the free energy of the superconducting state and eventually obtain the critical magnetic field as a function of temperature. 

Taking the free energy of the normal state to be zero at T = 0, the free energy of the superconducting state at T = 0 is obtained by integrating the entropy from Tc to 0 and adding this to the Gcs(Tc) which is the same as Gn(Tc). 

At first, the superconducting state was not thought to be a thermodynamic equilibrium state. But, as we know from Chapter 2, any equilibrium state must be the result of a free energy minimization. It can be shown that the superconducting and normal states will be in equilibrium when their free energies are equal, and that the free energy difference between the normal state at zero field, G H = 0), and the superconducting state at zero field, Gs H = 0), is... 

The Meissner effect is a very important characteristic of superconductors. Among the consequences of its linkage to the free energy are the following (a) The superconducting state is more ordered than the normal state (b) only a small fraction of the electrons in a solid need participate in superconductivity (c) the phase transition must be of second order that is, there is no latent heat of transition in the absence of any applied magnetic field and (d) superconductivity involves excitations across an energy gap. 

As is done in the GL-theory for a single even-parity order parameter, we write the free energy density difference between the superconducting state and the norma state as an expansion in even powers of the complex gap function A(k), which is related to the anomalous thermal average of the microscopic theory [28] where c is the electron annihilation operator with wave vector k and spin t. However, for the multiple-order parameter case we must expand A(k) as a linear combination of the angular momentum basis functions Yj(k)),... 

There is, however, another type of transition possible in two dimensions, a transition between states without LRO. This is the Kosterlitz-Thouless transition [8] mentioned in Sections II and V.B.l. It is relevant to superconductivity, commensurate-incommensurate transitions [61], planar magnetism, the electron gas system, and to many other systems in two dimensions. It involves vortices (thus the requirement of a two-component order parameter) characterized by a winding number q = (1/2-rr) dr V0, in which 0 is the phase of the order parameter (see also Ref. 4), the amplitude being fixed. These free vortices have an energy [see Eq. (28)] given by... 

In order to understand how the crystal-field splitting of RE-impurities influences the jump in the specific heat at i.e. AC(T ), one needs a more detailed theory than for the determination of T (ni) or HaiT). This can be seen by expanding the free energy difference between the superconducting and normal state Fs. near Tc in terms of the order parameter A (Ginzburg-Landau expansion). 

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