32
2 Theory: Bond-Electron-Energy Correlation
1.5
1.0
0.5
0.0
12
10
8
6
4
2
0
ΔE ν (12)
Effective atomic CN (z)
ΔE ν (z)=ΔE ν (12)[1+Δ i ]
DE n (z)
Δ Δ
E ν
ν
Κ
E ν (1)
B
S 2
S 1
S defect
K increase
(a)
(b)
Fig. 2.3 a Illustration of the correspondence between the cluster size (K, upper part) to the atomic
CN dependence of the CLS (lower part). b The number-of-atom dependence of the Cr n 2p energy
shifts. Th energy shifts from 537.5 eV for an isolated Cr atom to E 2p (n = 13) = 574.4 eV for z =
2 [20], which follows the BOLS prediction in the z ≤ 2 regime of (a). Reprinted with permission
from [20, 47]. Copyright 2005 and 2009 American Physical Society
induced strong localization and strong correlation” premise of Anderson [18] to the
irregularly-coordinated systems including amorphous glasses [19].
Figure 2.3a illustrates the cluster-size or the effective atomic-CN dependency of
the CLS. If the solid grows from an isolated atom to a solid of infinitely large, the
CL peak shifts along the line in the lower part of (a). Measurements [12, 20, 21]
confirmed that when the solid grows from an isolated atom the BE of the νth level
deepens from the initial E ν (0) level sharply to a maximum at z = 2, and then restores
gradually in a K
−1 fashion to the bulk E ν (12) [12], as the atomic CN increases [20].
The K is the characteristic size of a nanocrystal. The z = 2 corresponds to the effective
CN for an atom in an fcc unit cell or in the monatomic chain. The size resolved 2p
level shift of Cr n (n = 2 − 13) clusters (b) does follow the z ≤ 2 prediction [20] and
the number-of-layer resolved C 1s level shift of graphene follows the z > 2 size trend
as well [22].
2.3.2.3 Point Defect and Monolayer Skin
For a point defect or a monolayer solid skin, one only needs to consider the relaxation of one representative bond for all under the same coordination environment.
Incorporating Eq. (2.5) into Eq. (2.4) yields:
ν (z)
ν (12)
= (1 + H z ) =
E z
E b
= C
−m
z
ν (z ≥ 2) = ν (12)(1 + H z ) = [E ν (12) − E ν (0)]C
−m
z
(2.6)
2 Theory: Bond-Electron-Energy Correlation
1.5
1.0
0.5
0.0
12
10
8
6
4
2
0
ΔE ν (12)
Effective atomic CN (z)
ΔE ν (z)=ΔE ν (12)[1+Δ i ]
DE n (z)
Δ Δ
E ν
ν
Κ
E ν (1)
B
S 2
S 1
S defect
K increase
(a)
(b)
Fig. 2.3 a Illustration of the correspondence between the cluster size (K, upper part) to the atomic
CN dependence of the CLS (lower part). b The number-of-atom dependence of the Cr n 2p energy
shifts. Th energy shifts from 537.5 eV for an isolated Cr atom to E 2p (n = 13) = 574.4 eV for z =
2 [20], which follows the BOLS prediction in the z ≤ 2 regime of (a). Reprinted with permission
from [20, 47]. Copyright 2005 and 2009 American Physical Society
induced strong localization and strong correlation” premise of Anderson [18] to the
irregularly-coordinated systems including amorphous glasses [19].
Figure 2.3a illustrates the cluster-size or the effective atomic-CN dependency of
the CLS. If the solid grows from an isolated atom to a solid of infinitely large, the
CL peak shifts along the line in the lower part of (a). Measurements [12, 20, 21]
confirmed that when the solid grows from an isolated atom the BE of the νth level
deepens from the initial E ν (0) level sharply to a maximum at z = 2, and then restores
gradually in a K
−1 fashion to the bulk E ν (12) [12], as the atomic CN increases [20].
The K is the characteristic size of a nanocrystal. The z = 2 corresponds to the effective
CN for an atom in an fcc unit cell or in the monatomic chain. The size resolved 2p
level shift of Cr n (n = 2 − 13) clusters (b) does follow the z ≤ 2 prediction [20] and
the number-of-layer resolved C 1s level shift of graphene follows the z > 2 size trend
as well [22].
2.3.2.3 Point Defect and Monolayer Skin
For a point defect or a monolayer solid skin, one only needs to consider the relaxation of one representative bond for all under the same coordination environment.
Incorporating Eq. (2.5) into Eq. (2.4) yields:
ν (z)
ν (12)
= (1 + H z ) =
E z
E b
= C
−m
z
ν (z ≥ 2) = ν (12)(1 + H z ) = [E ν (12) − E ν (0)]C
−m
z
(2.6)