20.4 Overview on Theoretical Progress
381
[85] was attributed to being arisen from the quadrupolar vibration modes that are
enhanced by the excitation of the surface plasmas of the encapsulated Ag particles.
The mode choice by LFR scattering is due to the stronger plasmon-phonon coupling. For an Ag particle smaller than 4 nm, the size dependence of the LFR peak
frequency approximates Lamb’s theory [86], which gives vibrational frequencies of
a homogeneous elastic body in a spherical form.
The mechanism for the LFR mode enhancement is analogous to the case of
surface-plasma enhanced Raman scattering from molecules adsorbed on rough metal
surfaces. The surface acoustic phonons are eigen frequencies of a homogeneous elastic sphere under stress-free boundary conditions, which gives rise to a low-frequency
ω that is in the range of THz regime. The LFR mode corresponds to the spheroidal and
the torsional mode of vibrations of a spherical or an ellipsoidal particle. Spheroidal
motions are associated with dilation and it depends strongly on the cluster material
through the transverse and the longitudinal sound velocities, v t and v l , respectively.
The sound velocity in a medium depends functionally on the Young’s modulus and
the mass density, i.e., v ~ (Y/ρ)
0.5 ~
√
E b , where E b is the bond energy [87].
One can also ascribe the polarized LFR peaks as the confined LA-like and
the depolarized LFR as the TA-like acoustic phonons [88]. Interface between the
nanoparticle and the support matrix leads to a redshift for both the polarized and the
depolarized LFR peaks. This approach improves the fit to the measurement compared with Lamb’s model. Atomistic simulations [89] suggested that the morphology of nanoscopic Ag grains (twined icosahedra, Mark’s decahedra, and irregular
nanograins) introduces a high degree of complexity into the phonon spectra with the
total and the partial vibrational density of states and phonon localization.
Hwang et al. also suggested that the size-reduction induced lattice contraction
stems the LFR blueshift. For example, CdS x Se 1−x nanocrystals embedded in a
borosilicate (B 2 O 3 –SiO 2 ) glass matrix undergoes the size-dependent compressive
strain [90]. The lattice strain raises the surface tension when the crystal size is
reduced. It is suggested that the compressive stress overcomes the redshift of the
confined phonon due to negative dispersion and thus drives the LFR blueshift. The
LFR blueshift is also related to the bond length and energy that are functionally
dependent on the entropy, latent heat of fusion, and the melting point in a classical
thermodynamic manner [91].
The high-frequency optical modes shift in opposite directions. The Raman
blueshift is usually suggested to be activated by surface disorder [92], surface stress
[93, 94], phonon quantum confinement, and surface chemical effect [95, 96]. The
Raman shifts of TiO 2 particles are attributed to the effects of decreasing particle
size on the force constants and vibrational amplitudes of the nearest neighbor bonds
[97]. However, the effect of stress can usually be ignored for hydrogenated silicon,
in which hydrogen atoms passivate the surface dangling bonds to reduce the bond
strains and the residual stress [98, 99].
The phonon confinement model [95] ascribes the asymmetric Raman redshift
to relaxation of the q-vector selection-rule for the excitation of the Raman active
phonons due to their localization. The relaxation of the momentum conservation rule
arises from the finite crystalline size and the diameter distribution of the nanosolid
381
[85] was attributed to being arisen from the quadrupolar vibration modes that are
enhanced by the excitation of the surface plasmas of the encapsulated Ag particles.
The mode choice by LFR scattering is due to the stronger plasmon-phonon coupling. For an Ag particle smaller than 4 nm, the size dependence of the LFR peak
frequency approximates Lamb’s theory [86], which gives vibrational frequencies of
a homogeneous elastic body in a spherical form.
The mechanism for the LFR mode enhancement is analogous to the case of
surface-plasma enhanced Raman scattering from molecules adsorbed on rough metal
surfaces. The surface acoustic phonons are eigen frequencies of a homogeneous elastic sphere under stress-free boundary conditions, which gives rise to a low-frequency
ω that is in the range of THz regime. The LFR mode corresponds to the spheroidal and
the torsional mode of vibrations of a spherical or an ellipsoidal particle. Spheroidal
motions are associated with dilation and it depends strongly on the cluster material
through the transverse and the longitudinal sound velocities, v t and v l , respectively.
The sound velocity in a medium depends functionally on the Young’s modulus and
the mass density, i.e., v ~ (Y/ρ)
0.5 ~
√
E b , where E b is the bond energy [87].
One can also ascribe the polarized LFR peaks as the confined LA-like and
the depolarized LFR as the TA-like acoustic phonons [88]. Interface between the
nanoparticle and the support matrix leads to a redshift for both the polarized and the
depolarized LFR peaks. This approach improves the fit to the measurement compared with Lamb’s model. Atomistic simulations [89] suggested that the morphology of nanoscopic Ag grains (twined icosahedra, Mark’s decahedra, and irregular
nanograins) introduces a high degree of complexity into the phonon spectra with the
total and the partial vibrational density of states and phonon localization.
Hwang et al. also suggested that the size-reduction induced lattice contraction
stems the LFR blueshift. For example, CdS x Se 1−x nanocrystals embedded in a
borosilicate (B 2 O 3 –SiO 2 ) glass matrix undergoes the size-dependent compressive
strain [90]. The lattice strain raises the surface tension when the crystal size is
reduced. It is suggested that the compressive stress overcomes the redshift of the
confined phonon due to negative dispersion and thus drives the LFR blueshift. The
LFR blueshift is also related to the bond length and energy that are functionally
dependent on the entropy, latent heat of fusion, and the melting point in a classical
thermodynamic manner [91].
The high-frequency optical modes shift in opposite directions. The Raman
blueshift is usually suggested to be activated by surface disorder [92], surface stress
[93, 94], phonon quantum confinement, and surface chemical effect [95, 96]. The
Raman shifts of TiO 2 particles are attributed to the effects of decreasing particle
size on the force constants and vibrational amplitudes of the nearest neighbor bonds
[97]. However, the effect of stress can usually be ignored for hydrogenated silicon,
in which hydrogen atoms passivate the surface dangling bonds to reduce the bond
strains and the residual stress [98, 99].
The phonon confinement model [95] ascribes the asymmetric Raman redshift
to relaxation of the q-vector selection-rule for the excitation of the Raman active
phonons due to their localization. The relaxation of the momentum conservation rule
arises from the finite crystalline size and the diameter distribution of the nanosolid