## 3 Intrinsic Thermal Conductivity of Graphene  
We start by summarizing the state of the art in the fundamental understanding of the thermal properties of graphene, focusing on recent theoretical and experimental reports. In 2008, it was discovered at the University of California--Riverside that graphene has extremely high intrinsic thermal conductivity \(K\), which can exceed that of carbon nanotubes (CNTs) [1, 2, 3, 4, 5]. FLG retains excellent thermal properties [1,2,31]. Graphite, which is the 3D bulk limit for FLG with the number of layers n \(\rightarrow\infty\), is still an outstanding heat conductor with the intrinsic \(K\approx\) 2000 W/mK at RT. For comparison, \(K\approx\) 430 W/mK for silver and is much lower for silver nanoparticles used in TIMs. The first experimental studies of thermal conductivity of graphene were carried out using an original non-contact optothermal technique based on Raman spectroscopy (see Figure 1a,b). Taking into account the temperature shift of Raman G-peak, the temperature profiles for large-area suspended graphene flakes were determined. The thermal conductivity values were extracted from numerical simulations, taking into account temperature profiles and actual size and shape of the flakes. It was established that graphene demonstrates very high thermal conductivity \(K\), exceeding 3000 W/mK near RT for large graphene flakes [1, 2, 3, 4, 5]. The measurements were performed with large-area suspended graphene layers exfoliated from bulk graphite. The development of the optothermal technique was instrumental for carrying out the thermal measurements with graphene. In the optothermal technique, the heating power, \(\Delta P\), is provided with the laser light focused on a suspended graphene layer connected to heat sinks at both ends [1]. The temperature rise, \(\Delta T\), in response to the dissipated power, \(\Delta P\), is determined with a Raman spectrometer. The Raman G peak in graphene's spectrum exhibits strong temperature, T, dependence. The calibration of the spectral position of G peak with temperature has to be performed by changing the sample temperature while using low laser power to avoid local heating. During the thermal conductivity measurements the suspended graphene layer is heated by the increasing laser power. The local temperature rise in graphene is measured as \(\Delta T=\Delta\omega_{G}/\varphi_{G}\), where \(\varphi_{G}\) is the temperature coefficient of the Raman G peak in the relevant temperature range [1]. The heat dissipated in graphene can be determined by measuring the integrated Raman intensity of G peak [4, 5] or by a detector placed under the graphene [32, 33]. Due to the fact that the optical absorption in graphene depends on the wavelength of the light [1, 2, 34, 35, 36], and because it is affected by strain, defects, and multiple reflections for graphene suspended over the trenches, one should measure the optical absorption in the specific experimental conditions. The dependence of the light absorption in graphene on the wavelength results from the many-body effects [34, 35, 36]. A correlation between \(\Delta T\) and \(\Delta P\) for graphene samples with a given geometry gives the thermal conductivity value via solution of the heat diffusion equation. The suspended portion of graphene is needed for determining \(\Delta P\), forming a two-dimensional (2-D) heat front propagating toward the heat sinks, and reducing thermal coupling to the substrate. The thermal conductivity data in the Raman optothermal technique can be extracted for the graphene and FLG flakes of arbitrary shape via solution of the heat-diffusion equation using the finite element method [31]. Independent studies conducted by other research groups [32, 33] also utilized the non-contact optothermal technique but modified it via addition of a power meter under the suspended portion of the graphene samples. It was found that thermal conductivity of the suspended high-quality CVD graphene exceeded 2500 W/mK at 350 K, and it was as high as \(K\approx\) 1400 W/mK at 500 K [32].