Chemistry Reference
In-Depth Information
second superconductor (fig. 8.13b(iii)). Such tunnelling experiments
can be used to determine
, and further verify the existence of the
superconducting energy gap.
BCS theory makes several specific predictions concerning the supercon-
ducting transition temperature, which are in good general agreement with
experimental observation. The superconducting transition temperature,
T
c
, is predicted to depend on the energy gap at
T
=
0 K as (see e.g. Ibach
and Lüth 1995)
(
)
=
2
0
3.53
kT
c
(8.49)
More specifically, the transition temperature is given by
exp
−
1
kT
c
=
1.13
h
ν
(8.50)
g
(
E
F
)
V
ν
where
h
is a typical phonon energy in the metal (referred to as the Debye
phonon energy, see e.g. Ibach and Lüth 1995),
g
is the density of states
at the Fermi energy in the normal metal, and
V
is an interaction parameter.
g
(
E
F
)
(
)
V
is called the coupling constant, and is always less than 1, so that
kT
c
is then always much less than the Debye energy,
h
E
F
.
Equation (8.50) supports the qualitative analysis presented above, which
noted that
T
c
should increase with increasing
g
ν
. Since vibra-
tional energies varywith particle mass,
M
,as
M
−
1
/
2
, eq. (8.50) then implies
that
T
c
(
E
F
)
and
h
ν
M
−
1
/
2
. This has been well confirmed by experimental measure-
ments of the superconducting transition temperature for different isotopes
of elements such as tin and lead. It is referred to as the isotope effect, and
is one of the key pieces of evidence supporting BCS theory.
∝
8.9 BCS coherence length
The average distance,
ξ
0
, between the electrons in a Cooper pair at
T
=
0
is of order
v
F
π(
ξ
=
(8.51)
0
0
)
ξ
where
v
F
is the electron velocity at the Fermi energy, and
0
is known
as the BCS coherence length. This definition of the coherence length is
consistent with a dimensional analysis. If the electrons each gain energy
(
)
through their interaction, then they must be coherent with each other
at least over a timescale
0
(eq. (1.16)). In this time, the electrons
at the Fermi energy can travel a distance
v
F
τ
∼
/(
0
)
τ
, comparable to the coherence
