Chemistry Reference
In-Depth Information
Figure 1.35. Nesting vectors for: (a)
E
β
a
<
E
β
b
=
0 and (b)
E
β
a
<
E
β
b
<
0.
quasi-1D case of
E
β
a
<
0 (Fig. 1.34(d)) nesting is only possible through the
vector
q
shown in Fig. 1.35(b). Metallic systems with nested FS are electronically
unstable and therefore are likely to undergo a metal-insulator phase transition. In
the insulating phase a gap opens at
E
F
thus destroying the FS.
Finally, let us consider the effect of applying an external magnetic field
B
(Kartsovnik, 2004). In a magnetic field, the motion of quasi-particles becomes
partially quantized according to the expression:
E
β
b
<
l
¯he
|
B
|
1
2
E
(
B
,
k
z
,
l
)
=
+
+
E
(
k
z
)
,
(1.60)
m
∗
where
E
(
k
z
) is the energy at zero magnetic field of the motion parallel to
B
,
l
is a
quantum number (
l
∈ Z
) and
m
∗
is an averaged effective mass.
Themagnetic field quantizes themotion of the quasi-particles in the plane perpen-
dicular to
B
: the resulting levels are known as Landau levels, and the phenomenon is
called Landau quantization. The Landau-level energy separation is given by
mul-
m
∗
, known as the cyclotron frequency,
which corresponds to the semiclassical frequency at which the quasi-particle orbits
the FS. Magnetic quantum oscillations are caused by the Landau levels passing
through
E
F
. This results in an oscillation of the electronic properties of the system,
periodic in 1
tiplied by the angular frequency
ω
c
=
e
|
B
|
/
. Experimentally, the oscillations are usually measured in the mag-
netization or the resistivity, effects known as de Haas-van Alphen or Shubnikov-de
Haas, respectively.
Landau quantization only occurs for sections of FS corresponding to closed
k
-space orbits in the plane perpendicular to
B
. The frequency of the oscillation
(in tesla) is given by (
h
/
/
|
B
|
2
π
e
)
A
, where
A
is the cross-sectional
k
-space area of
