## 4 Theory of the Thermal Conductivity of Graphene and GNR
Naturally occurring carbon materials are made up of two stable isotopes of 12C (\(\sim\)99%) and 13C (\(\sim\)1%). The change in the isotope composition significantly influences the crystal lattice properties. Increasing the \"isotope doping\" leads to a suppression of the thermal conductivity in graphene and GNRs of up to two orders of magnitude at RT due to the enhanced phonon-point defect (mass-difference) scattering [90; 91; 92; 93; 94; 95]. Partial contribution of _LA_, _TA_ and _ZA_ phonons to the thermal conductivity in graphene is a subject of numerous debates. Some researchers argued that _LA_ and _TA_ phonons carry most of the heat in suspended graphene [41; 42; 43; 49; 72; 84; 85], while others suggested that _ZA_ modes are dominate heat carriers [37; 54; 55; 60; 96]. Supporting the graphene by substrates or ripples in suspended graphene flakes increases the scattering of _ZA_ modes and decreases their partial contribution to the thermal conductivity [37; 97]. Graphene and GNRs also demonstrated an intriguing dependence of the thermal conductivity on their geometrical parameters: lateral sizes and shapes [41; 42; 43; 49; 68; 69; 70; 71; 85]. Using the BTE approach, [PERSON] _et al._[43] have demonstrated that RT thermal conductivity of a rectangular graphene flake with 5 \(\upmu\)m width increases with length \(L\) up to \(L\sim\) 40-200 \(\upmu\)m and converges for \(L>\) 50-1000 \(\upmu\)m in dependence on the phonon boundary scattering parameter \(p\) (see Figure 2a). The dependence of the thermal conductivity on \(L\) is non-monotonic, which is explained by the interplay between contribution  
Figure 2: **(a)** Dependence of the thermal conductivity of the rectangular graphene ribbon on the ribbon length \(L\) shown for different specular parameters \(p\). The ribbon width is fixed at \(d=\) 5 \(\upmu\)m; **(b)** Dependence of the thermal conductivity of the rectangular graphene ribbon on the ribbon length \(L\) shown for different ribbon width \(d\). The specular parameter is fixed at \(p=0.9\). Note in both panels an unusual non-monotonic length dependence of the thermal conductivity, which results from the exceptionally long phonon mean free path of the low-energy phonons and their angle-dependent scattering from the ribbon edge. Reprinted with permission from [43]. Copyright 2012 American Chemical Society. to the thermal conductivity from two groups of phonons: participating and non-participating in the edge scattering [43]. The exceptionally large mean free path (MFP) of the acoustic phonons in graphene is essential for this effect. The increase in the flake width or phonon edge scattering (see Figure 2a,b) attenuates the non-monotonic behavior. It disappears in circular flakes or flakes with very rough edges (with specular parameter \(p<0.5\)). Strong dependence of thermal conductivity on the flake length was recently confirmed experimentally. [PERSON] _et al._[98] reported on the logarithmically divergent room temperature thermal conductivity in suspended graphene flakes with \(L\sim 0.01\)-\(10\)\(\upmu\)m [98]. A number of studies [68, 69, 70] employed the MD simulations to investigate the length dependence of the thermal conductivity in graphene and GNRs. The converged thermal conductivity in graphene was found for \(L>16\)\(\upmu\)m in Ref. [68]. In Refs. [69, 70], the thermal conductivity increases monotonically with an increase of the length up to \(2.8\)\(\upmu\)m in graphene [70] and \(800\) nm in GNRs [69]. The obvious length dependence in graphene and GNRs can be attributed to the extremely large phonon mean free path \(\Lambda\sim 775\) nm [5], which provides noticeable length dependence even for flakes with micrometer lengths. The available values of phonon thermal conductivity in SLG, few-layer graphene and GNRs are presented in Table 1 at RT (if not indicated otherwise). Readers interested in a more detailed description of theoretical models for the heat conduction in graphene materials are referred to other reviews [2, 13].