# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 3 Time-Domain Differential and Frequency-Resolved Raman
### Frequency-Domain Energy Transport State-Resolved Raman (FET-Raman)  
For FR-Raman, the data fitting is for the Raman shift against the modulation frequency, and it takes quite tremendous measurements. An alternative, named FET-Raman, is to fix the frequency, but vary the laser power and study the Raman shift change against laser power [27; 28]. Here, we take the work on MoSe\({}_{2}\) to introduce this technique. The physical principle of this technique is shown in Figure 10. During each heating period, since the thermal diffusion length of MoSe\({}_{2}\) in the cross-plane direction is much longer than the sample thickness, the thermal transport in the cross-plane direction can be neglected. The first energy transport state is the steady state heating constructed by a CW laser, shown in Figure 10b. By using different laser powers (\(P\)), a parameter named Raman shift power coefficient (RSC) is obtained: \(\psi_{\text{CW}}=\partial\omega/\partial P=\alpha\cdot(\partial\omega/ \partial T)\cdot f_{1}(\kappa)\), where \(\alpha\) is laser absorption coefficient, \(\partial\omega/\partial T\) is Raman shift temperature coefficient, and \(\kappa\) is in-plane thermal conductivity of MoSe\({}_{2}\). The second energy transport state is a transient state heating constructed by a square wave modulated CW laser, shown in Figure 10c,d. After a sufficient number of heating cycles, the sample temperature will vary periodically with time. As illustrated in Section 3.2, the energy transport state changes from quasi-steady state to steady state with the decrease of frequency. Therefore, an appropriate frequency should be selected to construct a transient state with good sensitivity. Based on the curves shown in Figure 9a, this frequency should be around the middle of quasi-steady state to steady state range. Then, a similar RSC value is also obtained: \(\psi_{\text{FR}}=\partial\omega/\partial P=\alpha\cdot(\partial\omega/\partial T )\cdot f_{2}(\kappa,\rho c_{p})\), where \(\rho c_{p}\) is volumetric heat capacity of the sample.  
As the thermal diffusion lengths in the two states are different, a dimensionless normalized RSC \(\Theta=\psi_{\text{FR}}/\psi_{\text{CW}}=f_{3}(\kappa,\rho c_{p})\) is used to completely eliminate the effects of \(\alpha\) and \(\partial\omega/\partial T\). In the experiment, very low laser powers are employed to ensure a moderate temperature rise of the sample. This ensures the effect of volumetric heat capacity change with temperature can also be ignored. That is, the normalized RSC is only related to the in-plane thermal conductivity of MoSe\({}_{2}\). A three-dimensional (3D) numerical modeling is then conducted to characterize the temperature profile under these two states. A theoretical relation between the ratio of temperature rises in the two states and in-plane thermal conductivity is obtained. The in-pane thermal conductivity of the sample can be determined by interpolating the experimental data into the curve. As shown in Figure 11a, \(\kappa\) of a 36 nm-thick MoSe\({}_{2}\) is determined as \(10.8\pm 1.6\) W-m\({}^{-1}\).K\({}^{-1}\). Figure 11b shows the \(\kappa\) values of MoSe\({}_{2}\) with different  
Figure 9: **(a)** The various theoretical fitting curves of experimental data. **(b)** The standard deviation \(\delta\) of experimental data to theoretical curves as a function of \(\kappa_{\text{AC}}\). Reprinted from [26], with the permission of AIP Publishing.  
thickness from different studies, and the feasibility of FET-Raman technique firmly verified. Figure 11c shows the blue shift of Raman spectra with the increased thickness, which means the interlayer Van der Waals force in MoSe\({}_{2}\) is increasing. Additionally, combining with the TET technique, the FET-Raman can also be used to characterize the anisotropic thermal conductivity of carbon fibers [28].