# Energy and Charge Transport in 2D Atomic Layer Materials: Raman-Based Characterization
## 5 Probing of Conjugated Hot Carrier Transport  
In most of the work on Raman study of energy transport in 2D materials, hot carrier diffusion is not considered, although this effect could be critically important, especially for tightly focused laser spot (\(<\)0.5 \(\upmu\)m diameter). Figure 16 shows the physics of hot carrier diffusion. The sample is irradiated by a laser, the energy of which is higher than the bandgap of MoS\({}_{2}\). Thus, electrons are excited to the conduction band while leaving holes in the valence band. Then a fast thermalization process (about 10\({}^{-12}\) s) happens, and hot carriers dissipate part of the energy to other electrons and lattice. This process is neglected due to the very short time. The second process is hot carrier diffusion, in which the remaining photon energy carried by electrons is diffused out of the laser spot area before recombining with holes. As this process is typically in nanoseconds, it should be taken into consideration. Afterwards, electrons and holes recombine because of Coulomb attraction, the energy is released by exciting phonons at the same time. The phonons then dissipate the energy with the sample and through layers down to the substrate.  
Figure 15: (**a**,**b**) \(\Theta\) for different \(\kappa\) and \(D\) values obtained from simulation and experiments of the 13 nm-thick sample under (**a**) 20\(\times\) and (**b**) 100\(\times\) objective lenses. (**c**) Measured \(\kappa\) and \(D\). (**d**) The sensitivity variation of temperature rise to \(\kappa\) and \(D\) under different energy transport states. Reprinted from [34], with permission from Elsevier, 2019.  
For steady state, the generation and diffusion of heat and hot carriers are governed by two partial differential equations. The first one is the carrier diffusion equation to determine the hot carrier concentration \(\Delta N(r,t)\) (cm\({}^{-3}\)):  
\[D\
abla^{2}\Delta N-\frac{\Delta N}{\tau}+\frac{\partial n_{0}}{\partial T_{\rm CW }}\frac{\Delta T_{\rm CW}}{\tau}+\Phi\alpha=0, \tag{9}\]  
where \(D\) (cm\({}^{2}\)-s\({}^{-1}\)) is carrier diffusion coefficient, \(\tau\) (s) is electron-hole recombination time of the sample, \(\Phi\) (photons per cm\({}^{3}\) per s) is incident photon flux of the laser source, and \(n_{0}\) (cm\({}^{-3}\)) is the equilibrium free-carrier density at temperature \(T\). The second equation is the thermal diffusion equation which involves the free carrier density:  
\[\kappa\
abla^{2}\Delta T_{\rm CW}+(h\
u-E_{g})\Phi\alpha+\frac{E_{g}\Delta N}{ \tau}=0, \tag{10}\]  
where \(\Delta T(r,t)\) (K) and \(E_{g}\) (eV) are temperature rise and bandgap energy of the sample, and \(h\
u\)is photon energy of the laser source. Due to the hot carrier diffusion effect, the real heating area will be larger than the laser irradiating area, and is highly related to the hot carrier diffusion length (\(L_{\rm D}=\sqrt{\tau D}\)). As a result, when the laser spot size is large enough, the hot carrier diffusion will have negligible effect on the heating area.  
For transient state, Yuan et al. used a picosecond laser to characterize the thermal transport for supported samples Yuan et al. (2018). The laser pulse (13 ps) is so short that the heat conduction becomes very weak. Then, five transport states in both time and space domains are constructed. \(\kappa\) and \(R\) values are determined by taking \(D\) into consideration. Zobeiri et al. used a nanosecond laser to study the thermal transport for suspended samples Zobeiri et al. (2018). Three transport states in both time and space domains are constructed. \(\kappa\) of the suspended sample is determined by taking \(D\) into consideration. As shown in Figure 15a, due to the large laser spot size under 20\(\times\) objective lens, the effect of \(D\) on \(\kappa\) of the sample is very tiny. While under 100\(\times\) objective lens, shown in Figure 15b, due to the relatively small laser spot size, the effect of \(D\) cannot be neglected.