# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 4 Energy Transport State-Resolved Raman
(**a**) Physical processes happening in MoS\({}_{2}\) upon laser irradiating. (**b**,**c**) Two transient states in picosecond laser heating under 50\(\times\) and 100\(\times\) objective lenses. (**d**–**f**) Three steady states under a CW laser with 20\(\times\), 50\(\times\), and 100\(\times\) objective lenses. Reproduced from [32], with permission of the PCCP Owner Societies. \(\Theta_{3}=\chi_{\text{CW}_{3}}/(\chi_{\text{PS2}}-\chi_{\text{PS}_{1}})\) are obtained. Furthermore, all these coefficients, which are related to the temperature rise of the sample, are functions of \(\rho c_{p}\), \(R\), \(D\), and \(\kappa\). A 3D numerical modeling is employed to calculate the temperature rise to determine \(R\), \(D\), and \(\kappa\) simultaneously. To normalize the (\(\kappa\), \(D\), \(R\)) space data, a normalized probability distribution function \(\Omega_{i}=\exp\left[-\left(\Theta_{i}-\Theta_{\exp\_i}\right)^{2}/\left(2 \sigma_{i}^{2}\right)\right]\) (\(i\) = 1, 2, and 3) is employed. Furthermore, \(\Theta_{i}\) and \(\Theta_{\exp\_i}\) are normalized RSCs from 3D modeling and experiment, respectively. \(\sigma_{i}\) is the experimental uncertainty. Then, \((\kappa,D,R)\) of the sample can be determined when a composite probability distribution function \(\Omega(\kappa,D,R)=\Omega_{1}\cdot\Omega_{2}\cdot\Omega_{3}\) is equal to 1. Figure 13 shows the determination of the three parameters for 2.4 nm-thick MoS\({}_{2}\). With the increase of probability level from 0.65 to 1.0, the (\(\kappa\), \(D\), \(R\)) space range is decreased to have only one point in the space that could give \(\Omega(\kappa,D,R)=1\). Based on this, there three parameters can be obtained as \(\kappa_{0}=60.3\) W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\), \(D_{0}=7.92\) cm\({}^{2}\cdot\)s\({}^{-1}\), and \(R_{0}=1.82\times 10^{-6}\) K\(\cdot\)m\({}^{2}\cdot\)K\({}^{-1}\), respectively. In addition to supported 2D materials, the ET-Raman technique can also be used for suspended 2D materials. However, a strong heat accumulation will happen in suspended samples because of the very short pulse interval for the picosecond laser, a nanosecond laser is used instead [33; 34]. Wang et al. used one CW laser and one nanosecond laser with the same wavelength to construct the steady state heating and transient state heating [33]. The in-plane thermal conductivities of suspended MoS\({}_{2}\) and MoSe\({}_{2}\) with different thickness are measured. However, the hot carrier effect is not considered in this work. Zobeiri et al. developed a three-state nanosecond ET-Raman technique to measure \(\kappa\) and \(D\) of nm-thick suspended WS\({}_{2}\) films [34]. In the three-state nanosecond ET-Raman technique, the three heat transport states are constructed with two lasers and two objective lenses. As shown in Figure 14a, a CW laser with a 20\(\times\) objective lens is used to construct the steady state. Figure 14b shows that two transient states are constructed using a nanosecond pulsed laser and two different objective lenses (20\(\times\) and 100\(\times\)). Since the thickness is very thin, the temperature distribution in the thickness direction is assumed to be uniform. Similarly, the three RSCs under the three states are obtained as \(\psi_{CW}\), \(\psi_{ns20}\), and \(\psi_{ns100}\). As shown in Figure 14c, \(\psi_{CW}\) is a function of \(\alpha\), \(\kappa\), \(D\), and Raman temperature coefficient (\(\partial\omega/\partial T\)). Meanwhile, both \(\psi_{ns20}\) and \(\psi_{ns100}\) are a function of \(\alpha\), \(\kappa\), \(D\), \(\rho c_{p}\), and \(\partial\omega/\partial T\), shown in Figure 14d. Furthermore, the effects of \(\kappa\) and \(D\) can be distinguished by using the two objective lenses to vary the local heating size. Considering the moderate temperature rise in the experiment, \(\rho c_{p}\) can be taken as a constant. Figure 13: The normalized probability distribution function \(\Omega(\kappa,D,R)\) with the probability of 0.65 in (**a**), 0.80 in (**b**), 0.95 in (**c**), and 1.0 in (**d**).