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8.512 Theory of Solids II
Spring 2009
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1
8.512 Theory of Solids II
Problem Set 8
Due April 13, 2009
Type II Superconductor.
We begin with the Ginzburg-Landau free energy
�
F = Δf
�
��
� �2 �
�
�
T − Tc 2 1 4
�
2eA
dr
|
ψ |
+ |
ψ
|
+ ξ 2 ��
+
ψ
��
Tc
2
i
c
(1)
and consider T slightly below Tc .
1. Calculate the free energy difference between the superconducting state and the normal
state and show that the thermodynamic critical field Hc is given by
Hc2 (T )
1
= Δf
8π
2
�
�2
Tc − T
Tc
(2)
2. Now turn on a uniform magnetic field H such that we are in the normal state (ψ = 0)
and gradually reduce H. We look for an instability towards ψ �= 0. The field at which
the instability happens is defined as Hc2 (T ). We calculate Hc2 (T ) by the following
steps.
(a) Using the condition δF = 0 under a variation of ψ, show that ψ satisfies
�
�2
� 2eA
T − Tc
ξ
+
ψ+
ψ + |
ψ |
2 ψ = 0 .
i
c
Tc
2
(3)
(b) Near the critical point, the last term in Eq. (2) can be ignored and we have a
linearized equation. Instability (ψ �= 0) occurs when the linearized equation has
a negative eigenvalue. Notice that this equation has the same form as that of
a single electron in a magnetic field, where the solution is known to be Landau
levels. Use this fact to show that
Hc2 =
where φ0 = hc/2e.
(Tc − T ) (φ0 /2π)
ξ2
Tc
(4)
2
(c) Calculate the London penetration depth λL (T ). Express Hc in Eq.(2) in terms of
�
�1/2
the temperature dependent coherence length ξ(T ) = ξ TcT−c T
and the London
penetration depth λL (T ). Show that the condition Hc2 > Hc implies
λL (T )
1
>√ .
ξ(T )
2
Equation (5) is the condition for type II superconductivity.
κ=
(5)