# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 6 Probing of Thermal Nonequilibrium among Phonon Branches
Based on this, the relation between \(\psi_{\text{CW}}\) and \(\Delta\overline{T}_{\text{m}}\big{|}_{\text{CW}}\) can be expressed as:  
\[\psi_{\text{CW}}=A\Bigg{[}(0.94+2.86e^{-1.65\tau_{0}})+\frac{1}{3}\cdot\frac{P }{\pi r_{0}^{2}\tau_{\text{L}}}\cdot\frac{\delta}{G_{\text{PP}}\big{|}_{\text{CW }}}\Bigg{]}/P, \tag{12}\]  
where \(A\) is determined by Raman shift temperature coefficient and laser absorption, \(P\) is the laser power, \(r_{0}\) is the radius of laser spot. Figure 18b shows the \(\psi_{\text{CW}}\) values under three objective lenses, and Equation (12) is used to obtain \(\psi_{\text{CW}}\)-\(r_{0}\) fitting curve. Then, the energy coupling factors between OP and AP for the two Raman modes under steady state are determined as \(0.301\times 10^{15}\) W\(\cdot\)m\({}^{-3}\cdot\)K\({}^{-1}\) for E\({}_{\text{2g}}^{1}\) mode and \(0.157\times 10^{15}\) W\(\cdot\)m\({}^{-3}\cdot\)K\({}^{-1}\) for A\({}_{\text{1g}}\) mode. Afterwards, the percentages of \(\Delta T_{\text{OA}}\) and \(\Delta T_{\text{AP}}\) can be distinguished, and the temperatures of LO/TO phonon, ZO phonon, and AP are obtained, shown in Figure 18c. Specifically, the temperature difference between OP and AP takes more than 25% of the measured temperature rise under a small laser spot size. Thus, \(\Delta T_{\text{OA}}\) cannot be neglected when a small laser spot is used. However, for the FR-Raman and TD-Raman, this effect is ruled out since they only use the Raman shift change versus modulation frequency. The phonon branch temperature difference is a constant and has no effect in the physical data processing. Furthermore, in other techniques, like the TET technique, the phonon branch temperature difference is negligible. In TET, we fit the trend of the temperature change against time to determine the thermal diffusivity, th  
Figure 18: Determination of coupling factor of MoS\({}_{2}\) under steady state. (**a**) Temperature rise of acoustic phonons obtained from simulation. (**b**) Experimental values and fitting curves of \(\psi_{\text{CW}}\) against laser spot size. (**c**) Distinct temperatures of different phonon branches. Figure reproduced from [35], with permission from John Wiley and Sons. conductivity. The electrons-OP and OP-AP temperature difference will only add a constant value on the AP temperature, and does not affect the fitting results. Note in the TET technique, since the heating is over the whole sample and the inter-phonon branch heat current is significantly lower than the laser intensity in this work, the electron temperature is very close to that of the AP and their temperature difference is negligible compared with the measured temperature rise. The temperature difference between electron and AP can be calculated by \(T_{e}-T_{AP}=l^{2}RV^{-1}\bigg{[}\bigg{(}\sum G_{\text{ep}}\bigg{)}^{-1}+ \bigg{(}\sum G_{\text{PP}}\bigg{)}^{-1}\bigg{]}\), where \(I\) is the current flowing through the sample, \(R\) is its resistance at steady state, \(V\) is the volume of the sample, \(G_{\text{ep}}\) is the coupling factor between electrons and OP. As there are three phonon branches for both OP and AP, here the sums of all the corresponding coupling factors are used in the calculation. For instance, the in-plane thermal conductivity of graphene paper is obtained as 634 \(W\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\) using the TET technique. The length, width, and thickness of the sample are around 17 mm, 0.28 mm, and 28.6 \(\upmu\)m, respectively [36]. Then, based on the coupling factors, the temperature difference between electrons and AP is calculated to be around \(2.8\times 10^{-8}\) K, which is negligible compared with a measured temperature rise of 2 K [35; 37].