# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 3 Time-Domain Differential and Frequency-Resolved Raman
### Time-Domain Differential Raman (TD-Raman)
Furthermore, the accumulative Raman emission for one excitation cycle (from 0 to \(t_{\rm e}\)) is calculated based on the equation below:  
\[E_{\omega}(\omega,t_{\rm e})=I_{0}\int_{0}^{t_{\rm e}}\big{(}1-A\overline{ \Delta T}^{*}\big{)}\exp\bigg{[}-\frac{4\ln 2\cdot\big{(}\omega-\omega_{0}+B \overline{\Delta T}^{*}\big{)}^{2}}{\big{(}\Gamma_{0}+C\overline{\Delta T}^{* }\big{)}^{2}}\bigg{]}dt, \tag{4}\]  
where \(I_{0}\), \(\omega_{0}\), \(\Gamma_{0}\) are the corresponding Raman properties at the beginning of laser heating, \(A\), \(B\), \(C\) are the changing rate of Raman intensity, Raman shift, and linewidth against the normalized temperature. Then, a parameter named Fourier number \(Fo\) (\(Fo=\alpha t/L^{2}\)) is substituted into Equation (4) to get the following equation:  
\[E_{\omega}(\omega,Fo_{\rm e})=I_{0}\int_{0}^{Fo_{\rm e}}\big{(}1-A\overline{ \Delta T}^{*}\big{)}\exp\bigg{[}-\frac{4\ln 2\cdot\big{(}\omega-\omega_{0}+B \overline{\Delta T}^{*}\big{)}^{2}}{\big{(}\Gamma_{0}+C\overline{\Delta T}^{* }\big{)}^{2}}\bigg{]}dFo, \tag{5}\]where \(F_{0\text{e}}=\alpha t_{\text{e}}/L^{2}\). In order to eliminate the integration time's effect, a normalized intensity is used and the expression is:  
\[E_{\omega}^{*}(\omega,F_{0\text{e}})=\frac{I_{0}}{F_{0\text{e}}}\int_{0}^{F_{0 \text{e}}}(1-A\overline{\Delta T}^{*})\exp\left[-\frac{4\ln 2\cdot(\omega- \omega_{0}+B\overline{\Delta T}^{*})^{2}}{\left(\Gamma_{0}+C\overline{\Delta T }^{*}\right)^{2}}\right]dFo. \tag{6}\]  
Based on Equation (5), the reconstructed Raman spectra per cycle at different \(F_{0\text{e}}\) is shown in Figure 5a. With the increase of \(F_{0\text{e}}\), the Raman peak is redshifted (softening), the linewidth becomes slightly broader, and the Raman intensity increases significantly. The variations of normalized Raman intensity and Raman shift against \(F_{0\text{e}}\) are shown in Figure 5b,c. As the temperature increases rapidly at the beginning of laser heating, the normalized Raman intensity and Raman shift decrease quickly correspondingly. With the increase of heating time, these two parameters will reach a constant when the sample reaches steady state. Based on Equation (6), different trial values of thermal diffusivity are used to find the best fitting curve for the experimental data. The thermal diffusivity of the silicon cantilever is determined at \(9.17\times 10^{-5}\) m\({}^{2}\)/s, which is very close to the reference value of \(8.66\times 10^{-5}\) m\({}^{2}\)/s [23]. Similarly, the thermal diffusivity of carbon nanotube fiber is also determined, which is around \(1.74\times 10^{-5}\) m\({}^{2}\)/s [24]. Although this technique has only been used in 1D materials for concept design and testing, it can also be used to characterize the thermal diffusivity of 2D materials, either supported or suspended. For the TD-Raman technique, neither temperature rise nor laser absorption information are needed. Therefore, it provides higher level physics understanding. However, a technical issue faced in experiment is that, when the heating time is too short, the overall laser on time is very short, and it takes extremely long time to collect the Raman signal. Thus, stage shift or environment noise will increase the uncertainty, which indicates that it is extremely challenging to study very fast thermal transport phenomena. This issue can be resolved using the frequency-resolved Raman (FR-Raman) technique without sacrificing the measurement accuracy.