SOLUTIONS TO THE SOLID PHYSICS EXAM
Ginzburg-Lindau Theory of Superconductivity
a) Explain the physical significance of the terms in the equation
The degrees of freedom in the equation vary in space. One part of the equation has values
that explain the superconducting strength. The arbitrary part determines the
superconductivity`s strength and gauge and lastly, the gradient term in the equation
determines the arbitrary motions since, they have restoring forces.
b) Show that Landau equation can be written as ζ 2 d 2/dy 2 f (x)+ f ( y )−f ( y )3=0
At zero field H = 0, J = 0 and Ψ* ∇Ψ- Ψ∇Ψ* = 0
Since, the superconducting phase is constant of position,
∇Ψ = 0 and αΨ + β|Ψ |2Ψ - h . h
2m∗¿ ∇ ¿2Ψ = 0
F = Ψ/Ψ∞ and Ψ∞ = -∞/β
1 dimension system
-h2/2m* d2/dx2f + α f + β|Ψ∞|2f3 = 0
-h2/2m* d2/dx2f = ζ2 and it can be defined as a length scale.
Since, α = 1- t and ζ2 = 1/1 – t
Hence, ζ2 d2/dx2 + f + f – f3 = 0
c) Verify that f(x) = tanh (x/ √2 ζ), sketch the Ψ and explain the significance of the
parameter.
-ζ2 d2/dx2 f + f (1 – f2) = 0
Since f = 1 – g and g is not equal to 0,
-ζ2 d2/dx2 g + (1 + g) – (1 – g) 3 = 0
-ζ2g’’ – 2g = 0

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