7 First-Principles Investigations of Electronically …
159
Diagonalization of a single-electron Hamiltonian provides delocalized orbitals,
ψ
M
HB =
I C MI
ψ
I
H , where M indexes a HOMO-derived state.
In a periodic system comprising molecules of equivalent MO energies, an eigenstate from the Hamiltonian is equally delocalized over all the constituent molecules.
In this limit, the one-electron orbital is described as a Bloch orbital labeled by a
wavevector in the first Brillouin zone. By contrast, in a realistic molecular aggregate with energy variations among MO energies, delocalized and localized orbitals
can coexist. A convenient measure for quantifying the extent of delocalization
is the inverse participation ratio (IPR) [57]. For example, the IPR of the Mth
HOMO-derived state is given by
IPR(M ) =
1
I |C MI | 4
(7.5)
This molecular IPR quantifies the number of molecules over which the wave
function is delocalized.
As compared to the polarization effect, an experimental characterization of the
spatial extent of electronic states is more challenging. The observation of bandlike transport [105] possibly indicates the existence of delocalized Bloch orbitals,
although the extent of the delocalization still remains to be determined. Matsui et al.
have used the electron spin resonant technique to obtain the spatial extent of charge
carriers [74], characterizing the distribution of trapped states in a PEN thin-film
transistor. They found that at 20 K the major trap states comprise localized states
spanning around 1.5 and 5 molecules, with a broad distribution of states extending
over 6–16 molecules. Motivated by their work, Hoshi et al. [57] have presented the
large-scale electronic structure calculations for the disordered PEN thin film and
analyzed the IPRs of the HOMO-derived states. The roles of disorder in the electronic states, as well as its influence on the charge-transport mechanism, have been
intensively discussed [33, 110].
7.2.2 Polarization Energies of Excited States
Here, we briefly discuss the polarization energy of an excited state. The HOMO or
LUMO energy corresponds to the energy level of a single charge (electron or hole)
carrier. By contrast, an electronic excitation simultaneously creates an electron and a
hole; thus, the excitation energy corresponds to the energy of an e–h pair. Therefore,
we consider the polarization energy of an excited state as the interaction between the
e–h pair and the environment.
Similar to the polarization energies of an anion or cation state, the polarization
energy of an excited state, P
ex , is introduced as
E
(s)
= E
(g)
− P
ex
,
(7.6)
159
Diagonalization of a single-electron Hamiltonian provides delocalized orbitals,
ψ
M
HB =
I C MI
ψ
I
H , where M indexes a HOMO-derived state.
In a periodic system comprising molecules of equivalent MO energies, an eigenstate from the Hamiltonian is equally delocalized over all the constituent molecules.
In this limit, the one-electron orbital is described as a Bloch orbital labeled by a
wavevector in the first Brillouin zone. By contrast, in a realistic molecular aggregate with energy variations among MO energies, delocalized and localized orbitals
can coexist. A convenient measure for quantifying the extent of delocalization
is the inverse participation ratio (IPR) [57]. For example, the IPR of the Mth
HOMO-derived state is given by
IPR(M ) =
1
I |C MI | 4
(7.5)
This molecular IPR quantifies the number of molecules over which the wave
function is delocalized.
As compared to the polarization effect, an experimental characterization of the
spatial extent of electronic states is more challenging. The observation of bandlike transport [105] possibly indicates the existence of delocalized Bloch orbitals,
although the extent of the delocalization still remains to be determined. Matsui et al.
have used the electron spin resonant technique to obtain the spatial extent of charge
carriers [74], characterizing the distribution of trapped states in a PEN thin-film
transistor. They found that at 20 K the major trap states comprise localized states
spanning around 1.5 and 5 molecules, with a broad distribution of states extending
over 6–16 molecules. Motivated by their work, Hoshi et al. [57] have presented the
large-scale electronic structure calculations for the disordered PEN thin film and
analyzed the IPRs of the HOMO-derived states. The roles of disorder in the electronic states, as well as its influence on the charge-transport mechanism, have been
intensively discussed [33, 110].
7.2.2 Polarization Energies of Excited States
Here, we briefly discuss the polarization energy of an excited state. The HOMO or
LUMO energy corresponds to the energy level of a single charge (electron or hole)
carrier. By contrast, an electronic excitation simultaneously creates an electron and a
hole; thus, the excitation energy corresponds to the energy of an e–h pair. Therefore,
we consider the polarization energy of an excited state as the interaction between the
e–h pair and the environment.
Similar to the polarization energies of an anion or cation state, the polarization
energy of an excited state, P
ex , is introduced as
E
(s)
= E
(g)
− P
ex
,
(7.6)