Chemistry Reference
In-Depth Information
Figure 1.33. Electronic densities for
0
(discontinuous lines) for an infinite linear chain of period
a
. For simplicity it has
been assumed that
ϕ
=
0 (continuous straight line) and
ϕ
=
at
at
δ
=
0 and
ψ
n
|
ψ
n
=
1.
the interaction of an occupied pair function
|
φ
(
k
)
φ
(
−
k
)
with an unoccupied pair
(
k
)
k
)
function
|
φ
φ
(
−
leadstoanenergygaprelatedto
(
k
)
k
)
φ
(
k
)
φ
(
−
k
)
|
H
per
|
φ
φ
(
−
,
where
H
per
stands for the electron-phonon interaction in the traditional BCS sce-
nario. A superconducting energy gap prevents Cooper pairs from breaking up when
there is no excitation energy greater than the gap.
7
Fermi surface
As already discussed above, bands may be partially filled and when this occurs
the energy of the highest occupied level, the Fermi energy
E
F
, lies within the
energy range of one (or more) bands. For each partially filled band there will be
a surface in
k
-space separating the occupied from the unoccupied levels. The set
of all such surfaces is known as the Fermi surface (FS) and is analytically defined
by
E
(
k
)
E
F
. The concept of FS plays a key role not only in understanding
the dimensionality of the transport properties of metals but also in explaining the
electronic instabilities of partially filled band systems.
Let us start by considering a 2D lattice for which the band dispersion is given by
E
(
k
)
=
=
E
α
+
2
E
β
a
cos
k
a
a
+
2
E
β
b
cos
k
b
b
. Figure 1.34 shows 3D band structure
7
The energy gap at
T
=
0Kisgivenby
3
.
5
k
B
T
c
within the BCS scenario and vanishes at
T
=
T
c
.
