For Ginzburg-Landau theory, the free energy dependence of the order parameter is:
F = jdV{m|-ifA + ab^2 + 1b| a = a(T-T), bconst
A. Let us consider a bulk superconductor, with no magnetic field. The first term in the integrand above will be zero. Write your working for minimizing the free energy and find the density for the SC condensate ns = |2 as a function of temperature.
B. Then consider the same superconductor (no B field) near the surface of the sample. The order parameter / depends on the position, so this will be a case where we work to derive the nonlinear differential equation describing the position dependence of the order parameter.
C. Based on your work for part B, derive the expression for the superconducting correlation length, rather than solving for the equation. Derive it using simple dimensional arguments and explain its physical meaning and whether it depends on temperature.
D. Assume that the SC condensate is independent of position. ns = [2. For this case, the electric current density is given by:
Where is the phase of the SC condensate. Use this to prove the validity of the "London equation" V x 7 TsT m*c.
E. Lastly, show/demonstrate the theoretical Meissner effect using the above London equation and use this Maxwell equation to obtain the so-called "London penetration depth". Explain how it depends on temperature. In your working, you may need to use the following mathematical identity V V a = V(Va-a, where a is an arbitrary vector field.
