114
S. Yanagisawa and I. Hamada
0
0.5
1
1.5
2
-20
-15
-10
-5
0
5
10
E-E
F
(eV)
M
K
E vac
1
+
1
−
Γ
Γ
Fig. 4.10 Band structure of graphene obtained by using the rev-vdW-DF2 functional. The origin
of energy is the Fermi level (E F ). The symmetry points are = (0, 0), M = (1/2, 0), and K =
(1/3, 1/3) in the unit of the basic reciprocal lattice vectors. The horizontal dashed line indicates
the vacuum level (E vac ). The low-lying IPSs corresponding the symmetric (1 + ) and antisymmetric
(1 − ) are indicated by the arrows. (Reprinted from [134], with the permission of AIP Publishing)
Table 4.6 IPS levels of graphene with respect to the Fermi level calculated with rev-dW-DF2
along with those with LDA, PBE, and LDA+image tail (z o = 3.0 Bohr). For comparison those
obtained with B86Rx+PBEc and B86Rx+PBEsolc are also shown. IPS levels with respect to the
vacuum level are shown in parentheses. The rev-vdW-DF2 lattice constant was used for all the
calculations. The unit of the energy level is eV. (Reprinted from [134], with the permission of AIP
Publishing)
Method
1 +
1 −
2 +
2 −
rev-vdW-DF2
2.85
3.70
4.31
4.35
(−1.54)
(−0.69)
(−0.07)
(−0.04)
LDA
3.31
4.25
4.41
4.46
(−1.16)
(−0.23)
(−0.07)
(−0.02)
PBE
3.27
4.03
4.24
4.29
(−0.98)
(−0.22)
(−0.01)
(+0.04)
B86Rx+PBEc
3.42
4.00
4.18
4.23
(−0.82)
(−0.24)
(−0.06)
(−0.01)
B86Rx+PBEsolc
3.42
4.05
4.25
4.30
(−0.89)
(−0.25)
(−0.05)
(+0.02)
LDA+image tail a
2.94
3.69
4.16
4.22
(−1.47)
(−0.72)
(−0.25)
(−0.19)
a Ref. [128]
more attractive and longer-range potential than PBE one. It was also found that
the exchange potentials used in the improved vdW-DFs such as optB86b-vdW,
vdW-DF-cx, and rev-vdW-DF2 are more attractive and longer ranged than PBE