Chemistry Reference
In-Depth Information
H
ePH
¼t
0
X
s¼";#
X
N
l¼
1
ðC
lþ
1
s
C
ls
þ
:ÞþU
X
N
n
l
"
n
l#
þ V
X
N
h.c
n
l
n
lþ
1
l¼
1
l¼
1
X
2
X
N
l¼
1
ðQ
lþ
1
Q
l
Þn
l
þ
N
K
Q
l
:
a
(8.1)
l¼
1
The naming of this hamiltonian originally comes from one of the most typical
hamiltonians in the solid-state physics, that is, the Hubbard model [
5
]. As is well
known, the Hubbard hamiltonian describes an electronic system composed by one
orbital at each site, being especially characterized by strong on-site (within-the-same-
orbital) Coulombic repulsion of the energy scale
U
. This model is widely used as a
minimum model of electron correlation typically for 3
d
electrons and is the basis for
our model construction. Here, we also assume only one orbital for each site, i.e., the
5
d
z
2
orbital for the Pt site, the 4
d
z
2
orbital for the Pd site, and the 3
d
z
2
orbital for the Ni
site, as we have already mentioned in the previous subsection. The first term and the
second term are thus a Hubbard-like part, with
C
ls
and
C
ls
being the corresponding
destruction and creation operators of the electron at the
l
th site with
spin.
However, in our MX-chain systems, only the first two terms are not sufficient.
We must include at least two key ingredients. One is the long-range Coulombic
repulsion working between different metal sites. In the above model, this effect is
represented by its shortest part, i.e., the nearest-neighboring (n. n.) term of which
the energy scale is
V
(see the third term). The word of “extended” in the model
name just comes from this term. This term gives two essential effects. One is the
stabilization of the CDW state. The other is the exciton effect that makes an
electron-hole bound state and strongly modifies the optical spectra in both the
CDW and Mott-insulator states. We will give detailed explanations about these
effects in the forthcoming sections. Regarding the part beyond the nearest
neighbors, we do not consider them explicitly, because any substantial effect
originating from it such as frustration is not observed in these materials.
One more important effect originates from the lattice. As we have already
discussed in Sect.
8.1.1
, the most important and experimentally observed lattice
effect is the halogen displacement [
6
]. We therefore include its effect as the second
line, defining
Q
l
as its displacement along the chain direction as is illustrated in
Fig.
8.2
, and
s
as its electron-lattice (e-l) coupling strength. Since this type of e-l
coupling usually induces a Peierls transition, we add “Peierls” to the name of our
hamiltonian. It will also be necessary to add an explanation to the last term. In this
model, the positions of the metal ions are assumed to be fixed because the metal
ions are connected to the backbone of the crystal through the ligands surrounding
them. For this reason, the halogen prefers the midpoint position between the two
consecutive metal ions when they have the same valencies, and so we write the
elastic term as that in Eq. (
8.1
).
a
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