# 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)  
The TD-Raman technique is inspired by the transient electro-thermal (TET) technique, which is developed for effective thermal characterization of one-dimensional (1D) materials [23,24]. In this technique, a single modulated laser beam is used for both sample heating and thermal probing. The concept of this technique is shown in Figure 4a. Both transient and steady-state measurements are conducted for this technique. The transient measurement consisted of an excitation period (\(t_{\text{e}}\)) and a thermal relaxation period (\(t_{\text{r}}\)). Furthermore, \(t_{\text{r}}\) is required to be long enough for the sample to completely cool down upon coming of next laser cycle. Multiple cycles are used to collect sufficient Raman signal. As shown in Figure 4a, during the excitation period, there is a temperature rise and Raman signal is also collected. With the increase of sample temperature, the Raman intensity decreases, the peak is  
Figure 4: **(a)** The temperature evolution of the sample, and corresponding variations of Raman peak intensity (\(I\)), peak shift (\(\omega\)) and linewidth (\(\Gamma\)). **(b)** Temporally accumulative Raman spectra of one laser pulse cycle in Case 1 and Case 2. Reprinted from [23], with permission from \(\copyright\) The Optical Society. redshifted and the linewidth broadens. During \(t_{\rm r}\), as the laser is turned off, no Raman signal is collected in this period. Comparing the two cases, the excitation period in Case 2 is a bit longer than Case 1, while the thermal relaxation periods are the same. Due to the longer excitation time, the temperature increases further, and the corresponding Raman spectrum also varies. Figure 4b shows the temporally accumulative Raman spectra in the two cases. It can be seen that the Raman intensity in Case 2 is higher than that in Case 1, and a Raman peak position softening is observed. By combining the above Raman signal variation and further physical data analysis, the temperature evolution of the sample can be obtained to determine the thermal diffusivity of the sample [23]. In the work of TD-Raman development, the thermal diffusivity of a Si cantilever is measured. In the experimental setup, one end is connected to bulk base and the other end is heated by a modulated laser. As the length of the sample is much larger than the diameter or thickness of the sample, a one-dimensional (1D) model is used for the heat conduction with the governing equation as:  
\[\rho c_{p}\frac{\partial\Delta T}{\partial t}=\kappa\frac{\partial^{2}\Delta T }{\partial x^{2}}+q_{0}, \tag{1}\]  
where \(\rho\) is the density, \(c_{p}\) is the specific heat of the sample, \(\Delta T\) is the temperature rise, \(\kappa\) is the thermal conductivity, \(x\) is the distance away from the heat sink, and \(q_{0}\) is the heat generation per unit volume induced by the laser heating. The spatially averaged temperature rise can be obtained based on the following equation:  
\[\overline{\Delta T}(t)=\frac{2q_{0}L^{3}}{\kappa(x_{2}-x_{1})}\sum\limits_{m= 1}^{\infty}\frac{1}{m^{4}\pi^{4}}\bigg{[}1-\exp(-\frac{m^{2}\pi^{2}\alpha t}{L ^{2}})\bigg{]}\big{(}\cos\frac{m\pi}{L}x_{1}-\cos\frac{m\pi}{L}x_{2}\big{)}^{ 2}, \tag{2}\]  
where \(\alpha\) is the thermal diffusivity, and \(L\) is the sample length. The normalized temperature rise is then obtained as:  
\[\overline{\Delta T}^{*}=\frac{\overline{\Delta T}(t)}{\overline{\Delta T}(t \rightarrow\infty)}=\frac{\sum\limits_{m=1}^{\infty}\frac{1}{m^{4}\pi^{4}} \bigg{[}1-\exp(-\frac{m^{2}\pi^{2}\alpha t}{L^{2}})\bigg{]}\big{(}\cos\frac{m \pi}{L}x_{1}-\cos\frac{m\pi}{L}x_{2}\big{)}^{2}}{\sum\limits_{m=1}^{\infty} \frac{1}{m^{4}\pi^{4}}\big{(}\cos\frac{m\pi}{L}x_{1}-\cos\frac{m\pi}{L}x_{2} \big{)}^{2}}. \tag{3}\]  
As the variations of Raman signal are linearly related to temperature rise, the Raman spectra obtained in experiment are employed to determine the average temperature rise of the sample in the heated area.