Advanced Solid State Physics

Hence,

(E−ε)g(E) =V

∫ 2 εm

2 εF

dE′g(E′)N(E′) =C, (321)

withCa constant. We readily obtain

g(E) =

C

E−ε

(322)

and

V

∫ 2 εm

2 εF

dE′

N(E′)

E′−ε

= 1. (323)

We now approximateN(E′)≈NFwithin this energy range and obtain the desired result

NFV

∫ 2 εm

2 εF

dE′

1

E′−ε

=NFVlog

(

2 εm−ε

2 εF−ε

)

= 1, (324)

or with the help ofε= 2εF−∆

∆ =

2 ~ωD

exp

(

1

NFV

)

− 1

. (325)

ForV > 0 (attractive interaction) the energy is lowered by excitation of aCooperpair above the
Fermilevel. Here∆is the binding energy of aCooperpair. Within the framework of the BCS
theory it is shown that this binding energy is closely related to energy gap of a superconductor.

16.4.4 Flux Quantization

In this section we aim to reproduce the result of flux quantization in a superconducting ring. Therefore
we regard a gas of charged bosons (Cooperpairs). Letψbe the pair wave function andn=ψψ∗=
const.the total number of pairs. We expressψas

ψ=

√

nexp (iθ) (326)

where the phaseθmay be a function of local space. The particle velocity operator is given by

v=

1

m

(−i~∇−qA) (327)

and, consequently, the particle flux is given by

ψ∗vψ=

n

m

(~∇θ−qA). (328)

We can write the electric current density as

j=qψ∗vψ=

qn

m

(~∇θ−qA). (329)