# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 3 Time-Domain Differential and Frequency-Resolved Raman
### Frequency-Resolved Raman (FR-Raman)  
For the FR-Raman technique, as shown in Figure 6, an amplitude-modulated square-wave laser with different frequencies is used for both sample heating and Raman signals collecting [25]. Figure 6a shows that durations of the laser excitation time and the thermal relaxation time are the same. The temperatures at the beginning and end of the laser excitation time are frequency-dependent. Figure 5: (a) The relationship between the reconstructed silicon Raman spectrum per cycle and the Fourier number. (b) The decreasing trend of the normalized Raman intensity against the Fourier number. (c) The decreasing trend of the Raman shift against the Fourier number. Reprinted from [23], with permission from © The Optical Society. At very high frequencies, shown in Figure 6b, the temperature rise in the laser excitation time and the temperature fall in the thermal relaxation time are almost negligible, that is, the temperature of the sample can be assumed to be constant in the whole process. Furthermore, this state is named as \"quasi-steady state\". At very low frequencies, shown in Figure 6c, the laser excitation time is long enough for the temperature rising to a steady-state temperature. Furthermore, the rising period, which was much smaller than the heating period, can be neglected. As a result, the sample temperature can be seen as a constant during the excitation time. Furthermore, this state is termed \"steady state\". The temperature rise values at these two states are then taken as \(\Delta T_{\text{qs}}\) and \(\Delta T_{\text{s}}\), where \(\Delta T_{\text{qs}}=\Delta T_{\text{s}}\)/2. As the heating effect increases with the decrease of frequency, Raman intensity decreases and Raman peak redshifts. The Gaussian distribution function is used to fit the Raman peaks to obtain precise Raman properties: intensity, Raman shift, and linewidth. The variation of these properties against the modulation frequency can be fitted to determine the thermal diffusivity of a sample. The determined thermal diffusivities of Si based on Raman intensity and Raman shift are \(9.57\times 10^{-5}\) m\({}^{2}\)/s and \(11.00\times 10^{-5}\) m\({}^{2}\)/s, respectively, which agree well with literature value. The FR-Raman technique not only provides a novel way to probe transient thermal transport with very high temporal resolution, but also can be used to characterize the anisotropic thermal conductivities of materials without the need of optical absorption and temperature coefficient [26]. The example given here is for the measurement of black phosphorus (BP). As shown in Figure 7a, the armchair and zigzag directions of the suspended BP are aligned along an edge of a square dent. Figure 7b shows that the sample is irradiated by a modulated laser beam and the corresponding Raman spectra are collected. Then, as shown in Figure 7c,d, Raman spectra of the sample before and after cutting are collected upon continuous wave (CW) laser irradiation with different laser powers. The power differential of the Raman shift \(\Phi_{1}\) and \(\Phi_{2}\) are determined. For these two parameters, \(\Phi_{1}\) depends on the armchair thermal conductivity \(\kappa_{\text{AC}}\) and the zigzag thermal conductivity \(\kappa_{\text{ZZ}}\), while \(\Phi_{2}\) mainly depends on \(\kappa_{\text{AC}}\). These two values are linearly related to the average temperature rise in the heating region \(\Delta T_{1}\) and \(\Delta T_{2}\), which are obtained from the ANSYS simulation results shown in Figure 8a,b. Afterwards, as shown in Figure 8c, \(\eta\) (\(\kappa_{\text{ZZ}}\)/\(\kappa_{\text{AC}}\)) can be determined by interpolating \(\Phi_{2}\)/\(\Phi_{1}\) to the simulated curve of \(\Delta_{2}\)/\(\Delta T_{1}\) and \(\eta\). Figure 6: (**a**) Mechanism of frequency-resolved (FR)-Raman. (**b**) Temperature variation at quasi-steady state. (**c**) Temperature variation at very low frequency (close to steady-state). Reprinted from [25], with permission from © The Optical Society.