380
20 Wonders of Multifield Lattice Oscillation
resonant scattering [36], strain or phonon confinement [40–42]. Information could not
be possible on the bond formation, dissociation, relaxation and vibration under given
conditions. Particularly, for the size-reduction induced phonon frequency blueshift,
redshift, and LFR creation are beyond capacity of available theories.
One often describes the size-dependent Raman shifts in the following hypothetic
relation with multiple freely adjustable parameters [35, 38],
ω(K ) = ω(∞) + A f
d 0
K
κ
,
where A f and κ are adjustable parameters used to fit the measurement. The d 0 is
the lattice constant that should contract with the solid dimension [78]. K is the
dimensionless size of the crystal. For the LFR in the THz frequencies, A f > 0, κ =
1. The LFR mode disappears when the particle size approaches infinity, ω(∞) = 0.
For the redshift, A f < 0. For Si example, ω(∞) = 520 cm
−1 , κ varies from 1.08 to =
1.44, depending on experimental conditions.
The variation of atomic number-of-layer from monolayer to bulk leads to a redshift of photon energy or band gap E G and the phonon frequency ω [79, 80]. The
photon energy E G and phonon frequency shift ωunder perturbation of size N, pressure
P, and temperature T follow the hypothetic empirical relationships with numerous
adjustable parameters [25, 35, 81, 82]:
ω(N ) − ω 0
E G (N ) − E 0
ω(T ) − ω 0
ω(P) − ω 0
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−d(a/N )
q
AN
−2
− BN
−1
− C
ω e (T ) + ω d (T )
k P + l P
2
(20.1)
where ω 0 and E 0 are the reference phonon frequency and photon energy of the bulk;
N denotes the number of atomic layers; a is the lattice constant; A, B, C, and d, q, k,
and l are adjustable parameters to match the N- and P-dependent Raman shifts. The
N
−1 and N
−2 terms stand for the potential and kinetic energies of the electron − hole
pairs in the quantum confinement scheme. These hypothetic models can reproduce
observations albeit the unclear physical meanings.
20.4.1.2 Hwang’s Scheme
Hwang and co-workers [83] presented a sophisticated theory describing the size
effect on the Raman shift of the optical and the LFR mode. The LFR is attributed to
the quadruple vibration, lattice contraction, and the size-induced optical softening
due to quantum confinement. The LFR mode at the THz regime is in acoustic category, which is specifically associated with the vibration of the nanoparticles. The
phonon energies are size dependent and vary with materials of the host matrix. The
LFR scattering from silver nanoclusters embedded in porous Al 2 O 3 [84] and SiO 2
20 Wonders of Multifield Lattice Oscillation
resonant scattering [36], strain or phonon confinement [40–42]. Information could not
be possible on the bond formation, dissociation, relaxation and vibration under given
conditions. Particularly, for the size-reduction induced phonon frequency blueshift,
redshift, and LFR creation are beyond capacity of available theories.
One often describes the size-dependent Raman shifts in the following hypothetic
relation with multiple freely adjustable parameters [35, 38],
ω(K ) = ω(∞) + A f
d 0
K
κ
,
where A f and κ are adjustable parameters used to fit the measurement. The d 0 is
the lattice constant that should contract with the solid dimension [78]. K is the
dimensionless size of the crystal. For the LFR in the THz frequencies, A f > 0, κ =
1. The LFR mode disappears when the particle size approaches infinity, ω(∞) = 0.
For the redshift, A f < 0. For Si example, ω(∞) = 520 cm
−1 , κ varies from 1.08 to =
1.44, depending on experimental conditions.
The variation of atomic number-of-layer from monolayer to bulk leads to a redshift of photon energy or band gap E G and the phonon frequency ω [79, 80]. The
photon energy E G and phonon frequency shift ωunder perturbation of size N, pressure
P, and temperature T follow the hypothetic empirical relationships with numerous
adjustable parameters [25, 35, 81, 82]:
ω(N ) − ω 0
E G (N ) − E 0
ω(T ) − ω 0
ω(P) − ω 0
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
=
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−d(a/N )
q
AN
−2
− BN
−1
− C
ω e (T ) + ω d (T )
k P + l P
2
(20.1)
where ω 0 and E 0 are the reference phonon frequency and photon energy of the bulk;
N denotes the number of atomic layers; a is the lattice constant; A, B, C, and d, q, k,
and l are adjustable parameters to match the N- and P-dependent Raman shifts. The
N
−1 and N
−2 terms stand for the potential and kinetic energies of the electron − hole
pairs in the quantum confinement scheme. These hypothetic models can reproduce
observations albeit the unclear physical meanings.
20.4.1.2 Hwang’s Scheme
Hwang and co-workers [83] presented a sophisticated theory describing the size
effect on the Raman shift of the optical and the LFR mode. The LFR is attributed to
the quadruple vibration, lattice contraction, and the size-induced optical softening
due to quantum confinement. The LFR mode at the THz regime is in acoustic category, which is specifically associated with the vibration of the nanoparticles. The
phonon energies are size dependent and vary with materials of the host matrix. The
LFR scattering from silver nanoclusters embedded in porous Al 2 O 3 [84] and SiO 2