4 Nanoscale First-Principles Electronic Structure Simulations of Materials. . .
113
rigorous theoretical background, have been limited to the abovementioned works
[112–117]. The computational or numerical difficulty involved, as mentioned above,
seems to be a critical limiting factor. Furthermore, determination of the quasiparticle
wave functions in a self-consistent manner is necessary in case of a hybridized
interface involving appreciable charge transfer at the interface. In terms of the
modification of the electronic charge density at a hybridized organic-metal interface
in a self-consistent manner, a theoretical methodology within the framework of DFT,
such as the optimally tuned range-separated hybrid (OT-RSH), might be a method
of choice [118–120].
Unoccupied electronic states at surfaces and interfaces are important as they
are relevant to the charge carrier transport in the electronic devices. As such, the
image potential state (IPS) is a fundamental electronic state emerging at metal
surfaces, which is characterized by a set of Rydberg-like series induced by the
Coulombic tail of the potential [121, 122]. Image potential has a form of V im =
−e 2 /4(z − z 0 ), where z 0 is the position of the so-called image plane, the effective
position of the surface plane, and the energy levels in the image potential is given
by E n = −0.85 eV/(n + a) 2 with n and a being the quantum number and the socalled quantum defect, respectively. The IPSs have been shown to exist even for
graphitic materials including single-crystal graphite [123], highly oriented pyrolytic
graphite [124], carbon nanotube [125], fullerene [126], and fullerite [127]. Double
Rydberg states of IPSs are also predicted for freestanding graphene [128] by using
the LDA augmented by the image potential tail (“LDA+image tail”), and it was
suggested that the IPS is the origin of the interlayer state [nearly free electron
(NFE) state] of graphite [129–131]. The double Rydberg states were confirmed for
graphene and bilayer graphene on SiC by using the scanning tunneling spectroscopy
[132]. Computationally however, it is well-known that the semilocal approximation
to the exchange-correlation functional fails to reproduce the image potential, and
the dynamic and nonlocal correlation is necessary to describe it accurately. Indeed
it has been shown that by using the GW method to evaluate the energy-dependent
electron self-energy, an image potential for a metal surface is reproduced [133].
As an alternative, vdW-DF has been used to study graphene and some graphite
materials [134], as it contains a dynamical and nonlocal piece of correlation in an
approximate manner. In the following, electronic structures of graphitic materials
obtained by using vdW-DF are discussed with the emphasis on the IPSs.
In Fig. 4.10, the band structure of graphene obtained by using the rev-vdW-DF2
[10] functional is shown [134]. It is in good agreement with that obtained with the
LDA+image tail potential of Ref. [128]. In particular, the low-lying IPS levels agree
well (Table 4.6), suggesting that vdW-DF improves the description of graphene’s
IPS. In order to clarify the role of the nonlocal correlation, calculations without
the nonlocal correlation [B86R exchange plus PBE correlation (B86Rx+PBEc) and
B86R exchange plus PBEsol correlation (B86Rx+PBEsolc)] were performed. It
was found that the IPS levels obtained with B86Rx+PBEc and B86Rx+PBEsolc
are similar to those with PBE and underestimated, suggesting that the nonlocal
correlation plays an important role in describing IPSs accurately. By further analyzing the exchange-correlation potential, it was found that the vdW-DF generates
113
rigorous theoretical background, have been limited to the abovementioned works
[112–117]. The computational or numerical difficulty involved, as mentioned above,
seems to be a critical limiting factor. Furthermore, determination of the quasiparticle
wave functions in a self-consistent manner is necessary in case of a hybridized
interface involving appreciable charge transfer at the interface. In terms of the
modification of the electronic charge density at a hybridized organic-metal interface
in a self-consistent manner, a theoretical methodology within the framework of DFT,
such as the optimally tuned range-separated hybrid (OT-RSH), might be a method
of choice [118–120].
Unoccupied electronic states at surfaces and interfaces are important as they
are relevant to the charge carrier transport in the electronic devices. As such, the
image potential state (IPS) is a fundamental electronic state emerging at metal
surfaces, which is characterized by a set of Rydberg-like series induced by the
Coulombic tail of the potential [121, 122]. Image potential has a form of V im =
−e 2 /4(z − z 0 ), where z 0 is the position of the so-called image plane, the effective
position of the surface plane, and the energy levels in the image potential is given
by E n = −0.85 eV/(n + a) 2 with n and a being the quantum number and the socalled quantum defect, respectively. The IPSs have been shown to exist even for
graphitic materials including single-crystal graphite [123], highly oriented pyrolytic
graphite [124], carbon nanotube [125], fullerene [126], and fullerite [127]. Double
Rydberg states of IPSs are also predicted for freestanding graphene [128] by using
the LDA augmented by the image potential tail (“LDA+image tail”), and it was
suggested that the IPS is the origin of the interlayer state [nearly free electron
(NFE) state] of graphite [129–131]. The double Rydberg states were confirmed for
graphene and bilayer graphene on SiC by using the scanning tunneling spectroscopy
[132]. Computationally however, it is well-known that the semilocal approximation
to the exchange-correlation functional fails to reproduce the image potential, and
the dynamic and nonlocal correlation is necessary to describe it accurately. Indeed
it has been shown that by using the GW method to evaluate the energy-dependent
electron self-energy, an image potential for a metal surface is reproduced [133].
As an alternative, vdW-DF has been used to study graphene and some graphite
materials [134], as it contains a dynamical and nonlocal piece of correlation in an
approximate manner. In the following, electronic structures of graphitic materials
obtained by using vdW-DF are discussed with the emphasis on the IPSs.
In Fig. 4.10, the band structure of graphene obtained by using the rev-vdW-DF2
[10] functional is shown [134]. It is in good agreement with that obtained with the
LDA+image tail potential of Ref. [128]. In particular, the low-lying IPS levels agree
well (Table 4.6), suggesting that vdW-DF improves the description of graphene’s
IPS. In order to clarify the role of the nonlocal correlation, calculations without
the nonlocal correlation [B86R exchange plus PBE correlation (B86Rx+PBEc) and
B86R exchange plus PBEsol correlation (B86Rx+PBEsolc)] were performed. It
was found that the IPS levels obtained with B86Rx+PBEc and B86Rx+PBEsolc
are similar to those with PBE and underestimated, suggesting that the nonlocal
correlation plays an important role in describing IPSs accurately. By further analyzing the exchange-correlation potential, it was found that the vdW-DF generates