420
22 Layered Structures
shifts with clarification, formulation, and quantification of the bond and phonon
relaxation dynamics.
22.3 Numerical Reproduction of Phonon Relaxation
22.3.1 Number-of-Layer Dependence
22.3.1.1 Formulation
For the z dependence of the phonon frequency shift follows,
D L (z) =
ω(z)
ω(z b )
=
ω(z) − ω(1)
ω(z b ) − ω(1)
=
d b
d z
E z
E b
1/2
z
z b
(L O)
1 (T O)
where,
d b
d z
E z
E b
=
C z
C b
−(1+m/2)
(22.1)
This formulation clarifies that the direct z > 1 involvement derives the redshift
and the other blueshift of phonons. Numerical matching to the experimental result
yields the ω(1) and m with ω(z b ) and ω(z) as input for calculations, as demonstrated
in the following subsections.
22.3.1.2 Graphene and Black Phosphorus
Bond-order variation and the pattern change of sp-orbit hybridization make carbon allotropes a group of materials varying from diamond, graphite, C 60 , nanotube
(CNT), nanobud (CNB), graphene, and GNRs with properties that are amazingly
different. For instances, graphite is an opaque conductor, but diamond is an insulator
transparent to light of all wavelengths; the former shares nonbonding unpaired (or
π -bond) electrons due to sp
2 -orbital hybridization compared with the latter of the ideally sp
3 -orbital hybridization. CNT and graphene demonstrate anomalous properties
including high tensile strength, electrical conductivity, ductility and thermal conductivity. GNR performs, however, quite differently from CNT or the infinitely large
graphene because of the involvement of the two-coordinated edge atoms and the associated Dirac Fermions [83–86]. Raman phonons relax with the applied stimuli such
as the temperature and pressure as well as the allotropic coordination environment
[87, 88].
The database as a function of the number-of-layer [40, 62] of graphene and BP
[64] enabled the verification of the framework for thickness dependence of the bond
22 Layered Structures
shifts with clarification, formulation, and quantification of the bond and phonon
relaxation dynamics.
22.3 Numerical Reproduction of Phonon Relaxation
22.3.1 Number-of-Layer Dependence
22.3.1.1 Formulation
For the z dependence of the phonon frequency shift follows,
D L (z) =
ω(z)
ω(z b )
=
ω(z) − ω(1)
ω(z b ) − ω(1)
=
d b
d z
E z
E b
1/2
z
z b
(L O)
1 (T O)
where,
d b
d z
E z
E b
=
C z
C b
−(1+m/2)
(22.1)
This formulation clarifies that the direct z > 1 involvement derives the redshift
and the other blueshift of phonons. Numerical matching to the experimental result
yields the ω(1) and m with ω(z b ) and ω(z) as input for calculations, as demonstrated
in the following subsections.
22.3.1.2 Graphene and Black Phosphorus
Bond-order variation and the pattern change of sp-orbit hybridization make carbon allotropes a group of materials varying from diamond, graphite, C 60 , nanotube
(CNT), nanobud (CNB), graphene, and GNRs with properties that are amazingly
different. For instances, graphite is an opaque conductor, but diamond is an insulator
transparent to light of all wavelengths; the former shares nonbonding unpaired (or
π -bond) electrons due to sp
2 -orbital hybridization compared with the latter of the ideally sp
3 -orbital hybridization. CNT and graphene demonstrate anomalous properties
including high tensile strength, electrical conductivity, ductility and thermal conductivity. GNR performs, however, quite differently from CNT or the infinitely large
graphene because of the involvement of the two-coordinated edge atoms and the associated Dirac Fermions [83–86]. Raman phonons relax with the applied stimuli such
as the temperature and pressure as well as the allotropic coordination environment
[87, 88].
The database as a function of the number-of-layer [40, 62] of graphene and BP
[64] enabled the verification of the framework for thickness dependence of the bond