# The Impacts of Random Distributed Vacancy Defects in Steady-State Thermal Conduction of Graphene
## 2 Materials and Methods
### Thermal Conductivity  
The thermal conductivity in matter is one of the natural properties. For graphene, optothermal Raman, and electrical measurements are performed to test the thermal conductivity of suspended graphene layers from exfoliation or chemical vapor deposition (CVD). Besides, MD numerical simulation and analytical methods based on the Boltzmann transport equation in valence force field are explored to determine the precise value of thermal conductivity. In Table 1, the ranges of thermal conductivity for graphene are listed from different experimental tests and simulation methods. Large deviation is demonstrated for thermal conductivity from 600 W/mK to 10000 W/Mk. However, the results of [PERSON] [21; 36], [PERSON] [25] and [PERSON] [31] are more approximated consistent to the measurements results of [PERSON] [8], [PERSON] [10], [PERSON] [14], and [PERSON] [12] than the result of [PERSON] [29], as shown in Figure 1. Therefore, the values of thermal conductivity for graphene are settled in the reported range of Balandin [8] and Nika [21]. The heat in graphene is mostly carried by acoustic phonons rather than by electrons. The heat flux along a graphene flake can be computed as [21]:  
\[ \begin{array}{ll}\overset{\rightarrow}{W}&=\sum\limits_{s,\overset{ \rightarrow}{q}}\overset{\rightarrow}{v}(s,\overset{\rightarrow}{q})\hbar \omega_{s}(\overset{\rightarrow}{q})N(\overset{\rightarrow}{q},\omega_{s}( \overset{\rightarrow}{q}))\\ &=\sum\limits_{s,\overset{\rightarrow}{q}}\overset{\rightarrow}{v}(s, \overset{\rightarrow}{q})\hbar\omega_{s}(\overset{\rightarrow}{q})n( \overset{\rightarrow}{q},\omega_{s})\end{array}  \tag{8}\]The above equation can be used to calculate the thermal conductivity with the actual dependence on the phonon frequency \(\omega_{s}(\overrightarrow{q})\), the phonon velocity \(\overrightarrow{v}(s,\overrightarrow{q})\), and the number of the phonon with the certain frequencies \(n(\overrightarrow{q},\omega_{s})\). \(\overrightarrow{v}\hbar\omega\) is the energy carried by one phonon and \(N(\omega,\overrightarrow{q})\) is the number of phonons in the heat flux. From the macroscopic definition:  
\[W_{\alpha}=-\kappa_{\alpha\beta}(\
abla T)_{\beta}\hbar L_{x}L_{y} \tag{9}\]  
where \(L_{x}=d\) is the width of graphene flake, \(L_{y}\) is the sample length, and \(h\) is supposed to be 0.35 nm as the thickness of graphene. The thermal conductivity tensor \(\kappa_{\alpha\beta}\) in Equation (9) is expressed as:  
\[\kappa_{\alpha\beta}=\frac{1}{\hbar L_{x}L_{y}}\sum_{s,\overrightarrow{q}} \tau_{tot}(s,\overrightarrow{q})v_{\alpha}(s,\overrightarrow{q})v_{\beta}(s,\overrightarrow{q})\frac{\partial N_{0}(\omega_{s})}{\partial T}\hbar\omega_{ s}(\overrightarrow{q}) \tag{10}\]  
where \(\tau_{tot}\) is the combined phonon relaxation rate in graphene. Moreover, \(\alpha,\beta\) are the subscripts that represent the elements in the matrix of thermal conductivity tensor. Similarly, the diagonal element of the thermal conductivity tensor is computed as:  
\[\kappa_{\alpha\alpha}=\frac{1}{\hbar L_{x}L_{y}}\sum_{s,\overrightarrow{q}} \tau_{tot}(s,\overrightarrow{q})v^{2}(s,\overrightarrow{q})\cos^{2}\varphi \frac{\partial N_{0}(\omega_{s})}{\partial T}\hbar\omega_{s}(\overrightarrow {q}) \tag{11}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **K (W/mK)** & **Method** \\ \hline [PERSON] [8] & 2000–5000 & RO, exfoliated \\ [PERSON] [10] & 1300–2800 & RO, exfoliated \\ [PERSON] [14] & 1500–5000 & RO, CVD \\ [PERSON] [12] & 2500 & RO, CVD \\ [PERSON] [13] & 600 & RO, exfoliated \\ [PERSON] [44] & 1100 & Electrical self-heating, exfoliated \\ [PERSON] [45] & 600 & Electrical, exfoliated \\ [PERSON] [25] & 4000 & Ballistic, width dependence \\ [PERSON] [21] & 2000–5000 & VFF, BTE, width dependence \\ [PERSON] [36] & 1000–5000 & RTA, size dependence \\ [PERSON] [29] & 8000–10000 & MD, square graphene sheet \\ [PERSON] [31] & 1400–2400 & BTE, length dependence \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal conductivity of graphene in reference.