22.3 Numerical Reproduction of Phonon Relaxation
429
22.3.3.2 MX 2 Debye Thermal Decay
Figure 22.8 shows the theoretical reproduction of the thermally softened Raman shifts
for the MX 2 with derived E coh , and θ D given in Table 22.1. The ω x (1) and ω x can
also be obtained from matching the number-of-layer effect on the phonon frequency
shift. Theoretical match is realized by adjusting the θ D and the E coh without needing
any hypothetic parameters [100], as the Raman shift features the (E/d
2 )
1/2 that relaxes
functionally with the integration of Debye specific heat. The slow decrease of the
Raman shift at very low temperatures arises from the small
T
0 ηdt values as the
specific heat η(t) is proportional to T
τ for the τ-dimensional system at very low
temperatures. These results imply that the θ D determines the width of the shoulder,
the 1/E coh and the thermal expansion coefficient determine the slope of the curve at
high temperatures.
Results derived from the matching to measurements agree with the BOLS-LBA
expectation of the bond thermal relaxation premise. The atomic cohesive energy for
the MX 2 follows the E coh-MoS 2 (1.35 eV) < E coh-WS 2 (2.15 eV) < E coh-MoSe 2 (2.30 eV)
< E coh-WSe 2 (3.21 eV) order. The MoS 2 shows higher rates of thermal expansion and
bond energy attenuation than those of others, due to its intrinsically weaker atomic
cohesive energy. In addition, the E coh also ranks the Debye temperature θ D and the
frequency offset ω x . Although the E
1
2g mode of the WS 2 deviates slightly, the
entire statistical trend holds. Theoretical production of the experimental observation
suggests that the bond nature determines the thermal relaxation dynamics.
One may refer the atomic cohesive energy of the compounds to their electronegativity difference between the constituent M and X elements. The electronegativity
values are W(1.7), Mo(1.8), S(2.5), and Se(2.4) [101]. Differences in electronegativity values of the constituent elements M and X of the compounds represent the
polarity of chemical bonds, which are in the order: MoSe 2 (0.6) < WSe 2 (0.7) =
MoS 2 (0.7) < WS 2 (0.8).
22.3.3.3 BP and (W, Mo)S 2
Figure 22.9 shows the BOLS-LBA reproduction of the measured pressure and temperature dependent Raman shifts for the layered BP [70, 106, 107]. The reproduction
of the P-stiffened and T-softened Raman shifts results in the compressibility β, energy
density E den , elastic modulus B, and the Debye temperature featured in Table 22.1
[90]. The compressibility β = 0.32 GPa
−1 and energy density E den = 9.46 eV nm
−3 .
The θ D = 466 K determines the width of the curve shoulder and the mode cohesive
energy E coh-m = 2.11 eV determines the slope of the curve at high temperature.
The database as a function of temperature [108] and pressure [46] for graphene
and the data for the pressure dependence for (W, Mo)S 2 [109] phonon frequency
shifts allow for information derived from the theoretical matching to measurements.
Figure 22.10a shows the BOLS-LBA reproduction of the pressure-resolved phonon
relaxation for (W, Mo)S 2 with derived information summarized in Table 22.1.
429
22.3.3.2 MX 2 Debye Thermal Decay
Figure 22.8 shows the theoretical reproduction of the thermally softened Raman shifts
for the MX 2 with derived E coh , and θ D given in Table 22.1. The ω x (1) and ω x can
also be obtained from matching the number-of-layer effect on the phonon frequency
shift. Theoretical match is realized by adjusting the θ D and the E coh without needing
any hypothetic parameters [100], as the Raman shift features the (E/d
2 )
1/2 that relaxes
functionally with the integration of Debye specific heat. The slow decrease of the
Raman shift at very low temperatures arises from the small
T
0 ηdt values as the
specific heat η(t) is proportional to T
τ for the τ-dimensional system at very low
temperatures. These results imply that the θ D determines the width of the shoulder,
the 1/E coh and the thermal expansion coefficient determine the slope of the curve at
high temperatures.
Results derived from the matching to measurements agree with the BOLS-LBA
expectation of the bond thermal relaxation premise. The atomic cohesive energy for
the MX 2 follows the E coh-MoS 2 (1.35 eV) < E coh-WS 2 (2.15 eV) < E coh-MoSe 2 (2.30 eV)
< E coh-WSe 2 (3.21 eV) order. The MoS 2 shows higher rates of thermal expansion and
bond energy attenuation than those of others, due to its intrinsically weaker atomic
cohesive energy. In addition, the E coh also ranks the Debye temperature θ D and the
frequency offset ω x . Although the E
1
2g mode of the WS 2 deviates slightly, the
entire statistical trend holds. Theoretical production of the experimental observation
suggests that the bond nature determines the thermal relaxation dynamics.
One may refer the atomic cohesive energy of the compounds to their electronegativity difference between the constituent M and X elements. The electronegativity
values are W(1.7), Mo(1.8), S(2.5), and Se(2.4) [101]. Differences in electronegativity values of the constituent elements M and X of the compounds represent the
polarity of chemical bonds, which are in the order: MoSe 2 (0.6) < WSe 2 (0.7) =
MoS 2 (0.7) < WS 2 (0.8).
22.3.3.3 BP and (W, Mo)S 2
Figure 22.9 shows the BOLS-LBA reproduction of the measured pressure and temperature dependent Raman shifts for the layered BP [70, 106, 107]. The reproduction
of the P-stiffened and T-softened Raman shifts results in the compressibility β, energy
density E den , elastic modulus B, and the Debye temperature featured in Table 22.1
[90]. The compressibility β = 0.32 GPa
−1 and energy density E den = 9.46 eV nm
−3 .
The θ D = 466 K determines the width of the curve shoulder and the mode cohesive
energy E coh-m = 2.11 eV determines the slope of the curve at high temperature.
The database as a function of temperature [108] and pressure [46] for graphene
and the data for the pressure dependence for (W, Mo)S 2 [109] phonon frequency
shifts allow for information derived from the theoretical matching to measurements.
Figure 22.10a shows the BOLS-LBA reproduction of the pressure-resolved phonon
relaxation for (W, Mo)S 2 with derived information summarized in Table 22.1.