Advanced Solid State Physics

of second order phase transitions,GinzburgandLandausuggested that the free energyFSin the
superconducting state can be written in the following form

FS=FN−α|φ|^2 +

β

2

|φ|^4 +

1

2 m

∣∣

∣(−i~∇−qA)φ

∣∣

∣

2

+

1

2 μ 0

Ba^2 , (287)

whereα andβ are phenomenological constants and mis the quasiparticle’s mass. Further, FN
denotes the free energy of the normal state. The equation can be interpreted in the following way:
α|φ|^2 +β 2 |φ|^4 is a term of the typical form appearing inLandau’s theory of second order phase

transitions, 21 m

∣∣

∣(−i~∇−qA)φ

∣∣

∣

2

is an additional term which accounts for energy corrections due

to spatial variations of the order parameterφand the last term, 2 μ^10 Ba^2 represents an additional
contribution to the free energy according to Eq. (271). Minimizing the free energyFSwith respectφ
yields theGinzburg - Landauequations in terms of functional derivatives:

δFS=

δFS

δφ

δφ+

δFS

δφ∗

δφ∗. (288)

Note that|φ|=φφ∗. We obtain

δFS =

[

−αφ+β|φ|^2 φ+

1

2 m

(−i~∇−qA)φ·(i~∇−qA)

]

δφ∗+h.c.

= 0. (289)

In the kinetic term of the right hand side we have to deal with a term of the form∇φ·∇δφ∗. We
note that

∫

dr∇φ·∇δφ∗=−

∫
dr(∇^2 φ)δφ∗, where we assumed thatδφ∗vanishes at the integration
boundaries. Conclusively we obtain
∫
drδFS=

∫

dr

[

−αφ+β|φ|^2 φ+

1

2 m

(−i~∇−qA)^2 φ

]

δφ∗+h.c. (290)

This is expression is equal to zero if

−αφ+β|φ|^2 φ+

1

2 m

(−i~∇−qA)^2 φ= 0. (291)

This is the firstGinzburg - Landauequation. In order to obtain the secondGinzburg - Landau
equation, one minimizesFS, Eq. (287), with respect to the vector potentialA.

δFS

δA

=

q

2 m

[φ(−i~∇+qA)φ∗−φ∗(−i~∇−qA)φ] +

1

μ 0

∇×∇×A, (292)

where we employed some rules of functional derivatives in the sense of distributions. In particular, we
first define the functionalBa(r) =F[A] =

∫

dr′[∇×A(r′)]^2 δ(r−r′). We now write

〈

δF[A]

δA

,f

〉

=

d

dε

F[A+εf]

∣∣

∣

ε=0

= 2

∫

dr′(∇×f)·(∇×A)δ(r−r′)

= 2

∫

dr′f·∇×∇×Aδ(r−r′). (293)