Chemistry Reference
In-Depth Information
Fig. 8.3 Illustration of the dp
model
t(l),V(l)
M d z 2 X p z
Up
U d
Q l
Q l+1
V dd
V pp
Readers may notice that the relationship between these two models is analogous
to that in cuprates. Namely, in the cuprates, both a single-band model consisting
only of Cu 3 d x 2
y 2 and a two-band model augmented by O 2 p orbitals were
discussed. In particular, the situation in the Ni complex is very similar to that in
the cuprates, from the viewpoint of the charge-transfer (CT) insulator that will be
commented later.
Another possible orbital to be added would be, for example, a molecular orbital
in the ligand. Fortunately this is not the case of ethylenediamine (en), since the
molecular orbitals in this ligand are all s orbitals and are located relatively much
higher or much lower. We must be careful that this simplification is not applied for
ligands that have
p
orbitals.
8.1.3 Phase Diagram
In this section, we discuss a ground-state phase diagram expected for the extended
Peierls-Hubbard model, because the ground state properties are essentially shared
by the dp model. Before introducing a detailed result, we present a simple analysis
in the localized limit, that is, the case of vanishing electron transfer. In the localized
limit, the picture in Fig. 8.1 becomes exact and we easily calculate the energy for
each state. In the CDW state, for example, the value of q l is optimized using the last
line in Eq. ( 8.2 ) (we rewrite the forth term as
SS l ðn l 1 n l Þq l and setting n l as
...
,0,2,0,2,
2 alternately and gives the energy of
4 S þ U per one unit cell, i.e., two pairs of “MX.” In the case of the Mott-
insulator, it has the energy of 2 V for the same segment of the chain. This easy
arithmetic gives us a simple criterion for the relative stability between the two
states; when 4 S þ
...
. As a result, q l becomes
2 V<U , the
Mott-insulator is stabilized. In other words, the e-l interaction S and the nearest-
neighbor electron mutual repulsion V collaborate to drive the CDW state, while the
on-site electron repulsion U drives the Mott-insulator state.
The validity of the above arithmetic is verified by further calculations including
the effects of electron transfer. What is interesting is that the above criterion is exact
within the framework of a Hartree-Fock (HF) calculation, as was shown by Nasu
[ 1 ]. He further applied a random-phase approximation (RPA) to this model and
considered a phase diagram even including the effects of magnon excitations. The
result is shown in Fig. 8.4 , and we see a slight intrusion of the Mott-insulator phase
into the CDW phase, which was due to the stabilization of the former by the
2 V>U , the CDW is stabilized, while, when 4 S þ
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