GinzburgLandauSuperconductor

Homogeneous case: Landau model of a superconductor

Consider the energy density of a magnetic field {$\frac 1 2 \mathbf H\cdot\mathbf B$}, corresponding to an infinitesimal density of external work done to build it is {$\mathbf H \cdot d \mathbf B = HdH + \mathbf H\cdot d\mathbf M$} (the work done by the thermodynamic system is therefore the opposite of this). If we ignore the work done in the external coils, the first term, the second term represents the opposite of the word done by the material, per unit volume.

Assuming a sample volume {$V$} the I principle reads {$dE = TdS + V\mathbf H\cdot d\mathbf M$} and we recognize that {$E=E(S,\mathbf M)$}. Since in experiments the controllable physical parameters are temperature and H-field, it is more convenient to use the Gibbs free energy density {$g=g(T,H)=(E-TS)/V -\mathbf H \cdot\mathbf M$}, whence

{$$dg(T,\mathbf H) = -Tds - \mu_0\mathbf M \cdot d\mathbf H $$}

where {$s$} is entropy per unit volume. Because of certain analogies with superfluid ⁴He, Landau and Ginzburg postulated that the order parameter of the superconductor must be a complex function {$\psi$}, non zero below the critical temperature. Hence, following the standard Landau? approach, the Gibbs free energy density in zero magnetic field can be written

{$$ g_s(T) = g_n(T) + a(T-T_c)||\psi|^2 + \frac b 2 |\psi|^4 + \cdots$$}

Assuming {$\psi=|\psi|e^{i\theta}$} the variational condition {$\partial\delta g /\partial |\psi|^2= 0$} is {$ a(T-T_c)+ b|\psi|^2=0$}, hence

{$$\psi=(a(T_c-T)/b)^{1/2}\,e^{i\theta}$$}

whose modulus is shown in Fig. 1. Substitution in the free energy density and addition of the magnetic energy density term yields

{$$ \delta g(T,H) = \frac {a^2} {2b}(T_c -T)^2 +\frac {B^2} {2\mu_0} $$}

Index

Fig. 1 Modulus of the order parameter vs. temperature, {$|\Psi|=(a(T_c-T)/b)^{\frac 1 2}$}, valid only close to {$T_c$} (orange curve).

Condensation Energy

Following Landau's approach we first of all identify the energy difference between the superconductor and its metallic state, from the concept of critical field, {$H_c$}, above which the superconductor becomes again a normal metal (type-I).

The difference between the normal and the superconducting state is therefore due to their susceptibilities, {$M_n=\chi_P H$}, with e.g. Pauli susceptibility [1], typically {$\chi_P~10^{-4}>0$}, and {$M_s=\chi_s H=-H$} (symple cylindrical geometry, no demagnetization). The assumption of equal Gibbs free energy density at the critical field produces the plot in Fig. 2.

Fig. 2 Normalized Gibbs free energy densities, normal and superconducting, vs. normalized applied {$h=H/H_c$} field. The very small metal susceptibility is negligible on this vertical scale, be it positive or negative. The two free energy become equal at {$H=H_c$}

The free energy difference between normal metal and superconductor can be then approximated ignoring the small metal susceptibility as

{$$G_n(T,0)-G_s(T,0)\approx Gs(T,H)-Gs(T,0) = \mu_0 \frac 1 2 H_c^2$$}

This is called condensation energy density, and it will be shown that it corresponds to the condensation of composite bosons (formed by pairs of electrons) into their unique ground state.

A simple calculation for Nb, with {$B_c=0.17$} T, gives a condensation energy corresponding to 0.02 K per Nb atom, a minute energy scale compared to that set by the Fermi energy of the metal, of order 10000 K. This was a long standing conundrum in the early understanding of superconductivity.

Index

Thermodynamics

The Landau free energy density is a series expansion in {$|\psi|^2$}, valid as long as the order parameter is small, i.e for {$T\lesssim T_c$}. Introducing a second order term in free energy, at {$H=0$}

{$$\delta f = -a(T-T_c)|\psi|^2 + \frac b 2 |\psi|^4$$}

and minimizing the order parameter {$|\psi|^2$} one gets its temperature dependence

{$$|\psi|^2 = \frac a b (T_c-T)$$}

In this interval the free energy difference, obtained by putting back the order parameter in the free energy density, to obtain

{$$\delta f = - \frac 1 2 \frac {a^2} b (T_c-T)$$}

is equal to the condensation energy, therefore

{$$B_c(T) =\left(\mu_0 \frac {a^2} b (T_c-T)^2\right)^{\frac 1 2} = \sqrt{\frac {\mu_0} b} a(T_c-T), \qquad T<T_c$$}

The change in entropy and heat capacity can be also computed. The former is {$\delta s= -\frac {\partial \delta f}{\partial T} = \frac {a^2} 2b (Tc-T)$} for {$T<T_c$} and vanishes above. The latter is {$\delta c= -T\frac {\partial \delta s}{\partial T}$}, i.e.

{$$c_s = c_n + \frac {a^2} b T,\qquad T<T_c$$}

that has a jump at {$T_c$} equal to {$\frac {a^2} b T_c$}. The experimental data for Al in Fig. 3 show the jump at {$T_c$}

Fig. 3 Specific heat in Al, open squares, showing the jump at {$T_c$} K in {$H=0$}. The open circles show the same measurement in the applied field {$H = 300 Oe > H_c$}. The straight line is the specific heat of the normal metal. The behaviour at very low temperature and zero field shows the opening of a gap.

Index

Inhomogeneous case: surfaces in Van der Waals example

The inhomogeneous case was already treated by Van der Waals with his gas equation. The system undergoes a phase transition to a denser liquid. At a given pressure the square of the difference between the density and the gas density {$\rho_0$}, {$\delta \rho^2 = (\rho -\rho_0)^2$}, is indeed an order parameter and we can write the same free energy equation as for the superconductor

{$$\delta g = a(T-T_c)\delta\rho^2 + \frac b 2 \delta \rho^4$$}

with {$\delta\rho = \sqrt{\frac a b} (Tc-T)$} for {$T<T_c$}, as above. However, Van der Waals imagined a vial with condensed liquid up to a certain height {$z=0$} in equilibrium with its gas, above. Is the interface sharp, or is there a height dependent order parameter in some interval around {$z=0$}? He offered the solution considering an inhomogeneous system, {$\delta\rho(z)$} and including a gradient term

{$$\delta g = a(T-T_c)\delta\rho^2 + \frac b 2 \delta \rho^4 + c|\nabla \delta \rho|^2$$}

The total free energy is the volume integral of {$\delta g$}. Considering that the gradient, dimensionally, is an inverse length, Van der Waals assumed ta proportionality between the order parameter and its gradient, {$\frac c{\xi^2}|\nabla \delta \rho|^2= {a(T_c-T)} \delta \rho^2$}. The length

{$$\xi=\sqrt{\frac c {a(T-T_c)} }$$}

is the height over which minimized free energy is allowed to vary from liquid to gas values. We shall apply the same principle to investigate inhomogeneous superconductors.

Index

 Fritz M. London (1900 - 1954)