Chemistry Reference
In-Depth Information
the orbit. An example is given in Fig. 1.36, which shows giant Shubnikov-de
Haas oscillations in the magnetoresistance of
β
H
-(BEDT-TTF)
2
I
3
with the field
applied perpendicular to the
ab
-planes (Kang
et al.
, 1989). A single frequency of
oscillations of 3730 T is observed, which corresponds to a value of
A
of about
50% of the area of the first Brillouin zone. This result is expected from the simple
ideas of band structure described in this chapter.
β
-(BEDT-TTF)
2
I
3
(
β
-(BEDT-
TTF)
2
and I
3
) corresponds to the case
N
at
=
2 with a three-quarters-band filling or
equivalently to a half-band filling of the highest occupied band. Because of this half
filling and the shape of the band the internal area covered by the FS corresponds
to half of the total area of the first Brillouin zone. This can be clearly seen from
Fig. 1.34(f).
Beyond the one-electron approximation
So far we have been discussing the case where the one-electron picture remains
valid, that is when electronic correlations are weak. However, in many cases corre-
lations can be strong and the one-electron scenario becomes no longer applicable.
Ideal 1D systems are instrinsically electronically correlated, since a single elec-
tron cannot move freely: any movement implies the collective, and thus correlated,
motion of all neighbouring electrons.
8
This forced collectivization might be re-
laxed at higher dimensionalities. Because of this the Fermi liquid model can no
longer be applied in the case of strictly 1D interacting electron systems. Instead
the Tomonaga-Luttinger liquid theory (Tomonaga, 1950; Luttinger, 1963) is more
appropriate. This scenario is characterized by the separation of spin and charge and
the absence of a sharp edge at
E
F
. Instead of the Fermi distribution function given
by Eq. (1.13), the DOS decays as
E
F
|
υ
, where
K
−
1
1. The
Luttinger liquid exponent
K
ρ
is a dimensionless parameter controlling the decay
of all correlation functions (
K
ρ
<
|
E
−
υ
=
(
K
ρ
+
)
/
2
−
ρ
1 represent repulsive and attractive
interactions, respectively). We will discuss this point in Section 6.1. An amusing
description of quantum physics in one dimension has been recently reviewed in
Giamarchi, 2004.
As discussed earlier for linear chain systems, half-filled bands and three-quarter
filled bands lead tometallic ground states because of incomplete filling of the highest
occupied band, and some examples are TTF-TCNQ and BFS. However, when on-
site and nearest-neighbour Coulomb interactions cannot be neglected, correlation
gaps
E
corr
appear and materials exhibit semiconducting or insulating ground states.
E
corr
refers to the minimum energy required to make a charge excitation in a system
1 and
K
ρ
>
8
Concerning the effects of dimensionality I strongly recommend the topic entitled
Flatland
from E. A. Abbott, an
Anglican clergyman (1838-1926), and in particular the chapter where the main character, a square in Flatland,
meets the King of Lineland.
