Advanced Solid State Physics

Comparison with the Eq. (279) yields

λL=

√

mβ

μ 0 q^2 α

. (301)

We now introduce the dimensionless parameterκ:

κ=

λL

ξ

≡

m

~q

√

2 β

μ 0

. (302)

We now show that this parameter characterizes the transition between type I and type II supercon-
ductors. We regard a type II superconductor exposed to a magnetic field which is slightly below the
critical fieldHc 2. Accordingly, we can linearize the firstGinzburg - Landauequation:

1

2 m

(−i~∇−qA)^2 φ=αφ. (303)

The vector potential is assumed to be of the formA=Bxey, i.e.B=Bez. We obtain

−

~^2

2 m

(

∂^2

∂x^2

+

∂^2

∂y^2

)

φ+

1

2 m

(

i~

∂

∂y

+qBx

) 2

φ=αφ. (304)

We employ the ansatzφ= exp [i(kyy+kzz)]χ(x)and obtain

1

2 m

[

−~^2

d^2

dx^2

+~^2 kz^2 + (~ky−qBx)^2

]

χ=αχ. (305)

SubstitutingE=α−~

(^2) k (^2) z
2 m andX=x−
~kyqB
2 m yields
−

~^2

2 m

d^2

dX^2

χ+

q^2 B^2

2 m

X^2 χ=Eχ (306)

This is the equation of a harmonic oscillator with frequencyω=qBm and eigenvaluesEn=

(

n+^12

)
~ω.
We note that the largest value of the magnetic field is given by the lowest energy eigenvalueE 0 =
1
2 ~ω=α−

~^2 kz^2

2 m, hence:

Bmax=μ 0 Hc 2 =

2 αm

q~

. (307)

By use of Eqs. (302) and (299) we obtain the interesting result

Hc 2 =

√

2 κHc. (308)

Forκ > 1 /

√

2 we haveHc 2 > Hcand the superconductor is of type II.