Theory of MX Chain Compounds - Material Designs and New Physical Properties in MX-and MMX-Chain Compounds - page 120

Note that we assume real one-electron orbital functions to avoid any unnecessary
use of complex values. Such a selection is really possible, because we only need to
take the real and imaginary parts of Eq. ( 8.11 ).
Thus, we have reached a computer-friendly expression of the eigenvalue equa-
tion. The dimension of the matrix M is generally

2 by definition. However, in
a regular CDW case, it is reduced to N /2, because the states can be classified into
N /2 groups having different momenta k . In Fig. 8.8a , we show an optical conduc-
tivity spectrum thus calculated. By the way, the parameter set is different from the
previous one, but the essential features are maintained. Readers will see that the
spectrum calculated by the HF approximation (the left panel of Fig. 8.6 ) is strongly
renormalized to the new spectrum by the effect of

ðN= Þ

2

DH . In particular, the intensities
widely distributed in the former spectrum are now concentrated at the lowest
position, accompanied by a downward shift of the absorption edge. This is nothing
but an exciton effect, in which Coulombic attraction makes a pair of an electron and
a hole bound to each other and forms an exciton. We usually call this downward
shift of the absorption edge the binding energy of the exciton.
We are now closer to the understanding of the measurement shown in Fig. 8.6 .
As is seen in Fig. 8.8a , the main peak is a line spectrum, seemingly consistent with
the measured one. This is deeply related to the nature of the exciton. First, the
exciton has an energy band without losing a translational symmetry. We claim the
mutual binding of an electron and a hole, not the binding of their center of gravity.
Second, we must remember that visible or near-infra-red light has a long wave
length of about several thousand Angstroms. Therefore, the exciton state that is
accessible by such light is only K

0 state ( K is the momentum of its center of
gravity). This restriction makes the main peak a line spectrum.
So, what happened to the free electron-hole pairs that existed before the
renormalization? Obviously, a part of them have K ¼

0, and they in principle can
contribute to the optical spectrum. This calculation, on the other hand, shows that
their intensities, which are located at higher energies than that of the exciton peak,
are almost negligible. This is a feature typical to one-dimensional systems, as
reported by other authors [ 19 , 20 ]. Instead of giving mathematics, we here explain
it intuitively. In one-dimensional systems, an energy band has divergent density of
states (DOS) at both its sides, as will be seen in Fig. 8.5c . This property enhances
the renormalization, since the renormalization is accomplished gathering the states
around the band edge.
Lastly, we give a final answer to this absorption-spectrum problem. While the
treatment so far gives a line spectrum as the main peak of the optical conductivity,
we must look for one more ingredient to make the spectrum as broad as the
observation. To finish our quest, it is very important to recall that we are discussing
the strong e-l system. This strong e-l interaction is one of the two factors that give
the large CDW amplitude. Then, we think that this e-l interaction will also give a
large fluctuation to the main peak. In Fig. 8.8b-d , we show spectra in which effects
of lattice fluctuation are incorporated [ 18 ]. We here use a standard semi-classical
formula as