An atomistic model based on the spin-restricted extended Huckel theory (EHT)
is presented for simulating electronic structure and I–V characteristics of graphene
devices. The model is applied to zigzag and armchair graphene nano-ribbons
(GNR) with and without hydrogen passivation, as well as for bilayer graphene.
Further calcns. are presented for elec. fields in the nano-ribbon width direction and
in the bilayer direction to show electronic structure modification. Finally, the EHT
Hamiltonian and NEGF (Nonequil. Green’s function) formalism are used for a
paramagnetic zigzag GNR to show 2e2/h quantum conductance.
Chapter 4, Harder Questions, Answers
Q9
The ionization energies usually used to parameterize the EHM are not ordinary
atomic ionization energies, but rathervalence-state AO ionization energies, VSAO
[atomic orbital] ionization energies. What does the term “valence state” mean here?
Should the VSAO ionization energies of the orbitals of an atom depend somewhat
on the hybridization of the atom? In what way?
The term was first used by Van Vleck who explained it thus, referring to carbon
in CH 4 : “...the spins of the four electrons belonging to sp^3 were assumed paired with
those of the four atoms attached by the carbon. Such a condition of the carbon atom
we may conveniently call its valence state.” He then showed a calculation which
led to the conclusion that “The ‘valence’ state of C has about 7 or 8 more volts of
intra-atomic energy than the normal state. This is the energy required to make the C
atom acquire a chemically active condition...” [1]. Mulliken defines it saying “[it is]
a certain hypothetical state of interaction of the electrons of an atomic electron
configuration” and “A ‘valence state’ is an atom state chosen so as to have as nearly
as possible the same condition of interaction of the atom’s electronswith one
anotheras when the atom is part of a molecule.” [2].
An atom, then, is in a valence state when its electrons occupy orbitals of energies
and shapes that they would occupy if they were subject to the interactions that they
would experience in some molecule; thus one could speak of the valence state of
carbon in CH 4 (above). Clearly a valence state is an abstract concept.
The VSAO ionization energies (the terms VSIP and VSIE, valence state ioni-
zation potential or energy, can also be used) of the orbitals of an atom should
depend on the hybridization of the atom, because a carbon (say) atom, in different
hybridization states is in different structural (and thus different electronic) environ-
ments: in carbon in CH 4 (sp^3 ), H 2 C¼CH 2 (sp^2 ), and HCCH (sp), the electrons are
subject to different neighbors (four hydrogens, etc.). From CH 4 to H 2 C¼CH 2 to
HCCH, carbon is increasingly electronegative [3]. So the VSAO ionization energy
might be expected to become monotonically bigger. However Hoffmann, who
pioneered the popularization of the EHM and demonstrated its wide utility, used
the same parameters for the s and p orbitals of carbon in alkanes (sp^3 C) as in
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