34
2 Theory: Bond-Electron-Energy Correlation
Generally, the experimentally observed size (K), shape (τ ), and bond nature (m)
dependence of the E ν (τ , K, m), of a nanosolid follows a scaling relation based on
the core-shell configuration [12],
E ν (K ) − E ν (12)
E ν (12)
=
bK
−1
(Experiment)
H (Theory)
with
⎧
⎨
⎩
H =
i≤3
γ i
E zi
E zb
=
i≤3
γ i
C
−m
zi − 1
(Skin resolved perturbation)
γ i = N i
N = V i
V = τ K
−1 C i ≤ 1
(Fraction of skin atoms)
(2.7)
As the dimensionless form of size, the K is the number of atoms lined along the
radius of a spherical dot (τ = 3), or a cylindrical rod (τ = 2), or cross the thickness of
a thin plate (τ = 1) where τ is the shape factor. N i is the number of atoms and V i the
volume of the ith atomic layer, respectively. E ν (K) is the peak energy of the νth band
for a K-sized solid. E zi is the energy of a bond in the ith layer between z-coordinated
atoms. The weighting factor, γ i , represents the fraction of undercoordinated atoms
in the ith shell of a K-sized and τ -shaped nanosolid. Subscript i counts from the
outermost layer inward up to three as no bond order loss happens at i > 3.
Generally, the size dependent CN for a spherical dot follows empirically [12],
⎧
⎨
⎩
z 1 = 4
1 − 0.75K
−1
z 2 = z 1 + 2
z 3 = z 2 + 4
(2.8)
K > 0 corresponds to a solid and K < 0 to a cavity. K = 0 to a flat skin and K =
∞ to an atom inside the ideal bulk. The z 1 changes with the curvature in the order:
1 = z dimer < z cluster < z nanocrystal < z flat-skin < z cavity < z bluk = 12. At K ≤ 0.75, the solid
degenerates into an isolated atom. For a spherical dot at the lower end of the size
limit, K = 1.5 (Kd = 0.43 nm for an Au spherical dot example, or an fcc unit cell),
z 1 = 2, which is equivalent to an atomic chain, the edge of a monolayer graphene,
and to the primary fcc unit cell having 13 atoms. Bonds between atoms of the same z
values perform identically. This LBA expression covers all sizes and shapes varying
from a dimer, to a monatomic chain, a monolayer atomic sheet, a hollow cavity, a
flat skin and bulk solid. The BOLS-NEP notion applies to all the undercoordinated
systems without discrimination of the stricture phase or bond nature.
2.3.2.5 BOLS-LBA Versus Quantum Confinement
Figure 2.2b illustrates the BOLS-NEP notion for the sized matter [24] compared
with the theory of Quantum Confinement (QC) that was firstly proposed by chemists
in 1982 for the light emission and band gap modulation by crystal size reduction.
Electron-hole pairs, or excitons, moving in the confinement box in the potential
2 Theory: Bond-Electron-Energy Correlation
Generally, the experimentally observed size (K), shape (τ ), and bond nature (m)
dependence of the E ν (τ , K, m), of a nanosolid follows a scaling relation based on
the core-shell configuration [12],
E ν (K ) − E ν (12)
E ν (12)
=
bK
−1
(Experiment)
H (Theory)
with
⎧
⎨
⎩
H =
i≤3
γ i
E zi
E zb
=
i≤3
γ i
C
−m
zi − 1
(Skin resolved perturbation)
γ i = N i
N = V i
V = τ K
−1 C i ≤ 1
(Fraction of skin atoms)
(2.7)
As the dimensionless form of size, the K is the number of atoms lined along the
radius of a spherical dot (τ = 3), or a cylindrical rod (τ = 2), or cross the thickness of
a thin plate (τ = 1) where τ is the shape factor. N i is the number of atoms and V i the
volume of the ith atomic layer, respectively. E ν (K) is the peak energy of the νth band
for a K-sized solid. E zi is the energy of a bond in the ith layer between z-coordinated
atoms. The weighting factor, γ i , represents the fraction of undercoordinated atoms
in the ith shell of a K-sized and τ -shaped nanosolid. Subscript i counts from the
outermost layer inward up to three as no bond order loss happens at i > 3.
Generally, the size dependent CN for a spherical dot follows empirically [12],
⎧
⎨
⎩
z 1 = 4
1 − 0.75K
−1
z 2 = z 1 + 2
z 3 = z 2 + 4
(2.8)
K > 0 corresponds to a solid and K < 0 to a cavity. K = 0 to a flat skin and K =
∞ to an atom inside the ideal bulk. The z 1 changes with the curvature in the order:
1 = z dimer < z cluster < z nanocrystal < z flat-skin < z cavity < z bluk = 12. At K ≤ 0.75, the solid
degenerates into an isolated atom. For a spherical dot at the lower end of the size
limit, K = 1.5 (Kd = 0.43 nm for an Au spherical dot example, or an fcc unit cell),
z 1 = 2, which is equivalent to an atomic chain, the edge of a monolayer graphene,
and to the primary fcc unit cell having 13 atoms. Bonds between atoms of the same z
values perform identically. This LBA expression covers all sizes and shapes varying
from a dimer, to a monatomic chain, a monolayer atomic sheet, a hollow cavity, a
flat skin and bulk solid. The BOLS-NEP notion applies to all the undercoordinated
systems without discrimination of the stricture phase or bond nature.
2.3.2.5 BOLS-LBA Versus Quantum Confinement
Figure 2.2b illustrates the BOLS-NEP notion for the sized matter [24] compared
with the theory of Quantum Confinement (QC) that was firstly proposed by chemists
in 1982 for the light emission and band gap modulation by crystal size reduction.
Electron-hole pairs, or excitons, moving in the confinement box in the potential