# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 6 Probing of Thermal Nonequilibrium among Phonon Branches  
The physical process happening inside different Raman-based methods consists of energy transfer among photons, electrons, and phonons. For phonons, three optical branches, which are the longitudinal optical (LO), transverse optical (TO), and flexural optical (ZO) branches, are included. Similarly, there are also three acoustic branches (LA, TA, and ZA). Furthermore, the temperatures of these  
Figure 16: The physical process of hot carrier diffusion. Reproduced from Yuan et al. (2018), with permission from The Royal Society of Chemistry. branches are at nonequilibrium under laser excitation. ZA phonons are the main heat carriers in the heat conduction process, while optical phonons are the ones probed by Raman spectroscopy. Thus, neglect of nonequilibrium between ZA phonons and optical phonons can induce significant underestimation of thermal conductivity. Wang et al. designed and employed a nanosecond ET-Raman technique to explore the temperature nonequilibrium among different phonon branches [35]. Figure 17a shows the energy transfer process among different energy carriers. Optical phonons (OP) receive energy from hot carriers, and will have a prominent temperature rise. Then, OP will transfer majority of the energy to acoustic phonons (AP) through energy coupling. For the temperature difference between OP and AP, we have \(\Delta T_{\text{OA}}\propto I\propto r_{0}^{-2}\), where \(I\) and \(r_{0}\) are the laser energy and radius of laser spot. Furthermore, the temperature rise of AP (\(\Delta T_{\text{AP}}\)) is related to both \(r_{0}\) and \(\kappa\), we have \(\Delta T_{\text{AP}}\propto f(\kappa)\cdot r_{0}^{-n}\) with \(n<2\). As shown in Figure 17b, with the increase of laser spot size, \(\Delta T_{\text{OA}}\) decreases to zero faster than \(\Delta T_{\text{AP}}\), which indicates that the effect of energy coupling between OP and AP is negligible under very large laser spot. In Raman-based techniques, as shown in Figure 17c the temperature rise of OP, which can be expressed as \(\Delta T_{\text{m}}=\Delta T_{\text{OA}}+\Delta T_{\text{AP}}\propto Ar_{0}^{ -2}+f(\kappa)\cdot r_{0}^{-n}\), is probed under different laser spot size. Afterwards, the percentages of \(\Delta T_{\text{OA}}\) and \(\Delta T_{\text{AP}}\) in \(\Delta T_{\text{m}}\) are determined. Figure 17: **(a)** The energy transfer process among different energy carriers in 2D materials under laser irradiation. **(b)** The temperature difference between optical phonons and acoustic phonons against laser spot size. **(c)** The determination of thermal conductivity and energy coupling coefficient between optical phonons and acoustic phonons. Figure reproduced from [35], with permission from John Wiley and Sons. In nanosecond ET-Raman experiments, the measured \(\psi\) values are linearly related to Raman intensity weighted temperature rise of the sample. The Raman intensity weighted temperature rise measured under steady state can be written as:  
\[\Delta\overline{T}_{\text{m}}\big{|}_{\text{CW}}=\Delta\overline{T}_{\text{AP}} \big{|}_{\text{CW}}+\frac{1}{3}\cdot\frac{I_{0}}{\tau_{\text{L}}}\cdot\frac{ \delta}{G_{\text{PP}}\big{|}_{\text{CW}}}, \tag{11}\]  
where \(G_{\text{PP}}\big{|}_{\text{CW}}\) is the energy coupling factor between OP and AP, \(I_{0}\) is the absorbed laser power per unit area at the center of laser spot, \(\tau_{\text{L}}\) is the laser absorption depth, and \(\delta\) (\(0<\delta<1\)) is portion of laser energy transferred from the measured Raman mode optical phonons to acoustic phonons. Figure 18a shows the variation of \(\Delta\overline{T}_{\text{AP}}\big{|}_{\text{CW}}\) against laser spot size using a 3D numerical modeling for the 55 nm thick MoS\({}_{2}\).