The in-plane thermal conductance of the graphene nanoribbons at different temperatures is shown in Fig. 2. The graphene nanoribbons that are supported on the silicon carbide substrate are about 380 nm in length. The width varies from 40 to 280 nm. The theoretical results obtained for the thermal conductance of graphene nanoribbons within which phonon transport is ballistic are also presented in Fig. 2. The theoretical phonon transport model used for determining the maximum thermal conductance is available in the literature [33, 34]. This parameter defines the upper limit for ballistic transport in the graphene nanoribbons. The results indicate that acoustic phonons are transmitted within the graphene nanoribbons in a substantially quasi-ballistic regime, even at room temperature. The characteristics of phonon transport in much longer graphene nanoribbons have been investigated [33, 34]. To better understand the physics of phonon transport within the structure, the experimental results obtained for much longer graphene nanoribbons are also presented in Fig. 2 for comparison. A much larger length scale of the graphene nanoribbons causes a significant decrease in thermal conductance, making phonon transport in the diffusive regime.
The thermal conductivity of graphene nanoribbons depends strongly upon their length in the ballistic regime and decreases with decreasing the length due to phonon-boundary scattering [35, 36]. In this study, the values of thermal conductivity are determined on the basis of an equation which defines its dependence on the measured electrical resistance of the opposite thermometer probes. The in-plane thermal conductivity of the graphene nanoribbons at different temperatures is shown in Fig. 3. The experimental results obtained for much longer graphene nanoribbons are also presented in Fig. 3 for comparison. As the width of the graphene nanoribbons increases, there is an increase in the thermal conductivity. The maximum thermal conductivity is lower than that obtained in the experiments, especially at lower temperatures.
As the dimensions of graphene nanoribbons decrease, there is a transition from the diffusive transport mode to the ballistic transport mode, which can be predicted with various models [32]. This transition causes a decrease in the thermal conductivity of graphene nanoribbons, which is similar to a decrease in carrier mobility during charge transport in the quasi-ballistic regime, for example, related to a short channel transistor [37, 38]. The thermal conductivity of a graphene nanoribbon with infinite width can be expressed as a summation of ballistic thermal conductivity and diffusive thermal conductivity by referring to the following formula:
$$k{\mathbf{=}}{\sum\limits_{p} {\left( {{{\left( {l{C_{p,ballistic}}} \right)}^{{\mathbf{-}}1}}{\mathbf{+}}{{\left( {{k_{p,diffusive}}} \right)}^{{\mathbf{-}}1}}} \right)} ^{{\mathbf{-}}1}} \approx {C_{ballistic}}{\left( {{l^{{\mathbf{-}}1}}{\mathbf{+}}2{{\left( {\pi \lambda } \right)}^{{\mathbf{-}}1}}} \right)^{{\mathbf{-}}1}}$$
1
in which p is the phonon mode, l is the length of the graphene nanoribbon, Cp,ballistic is the ballistic thermal conductance corresponding to a phonon mode, kp,diffusive is the diffusive thermal conductivity corresponding to a phonon mode, w is the width of the graphene nanoribbon, δ is the thickness of the graphene nanoribbon, and λ is the phonon mean free path.
The characteristics of phonon transport in the graphene nanoribbons that are supported on the silicon carbide substrate are investigated, when the width of the graphene nanoribbons is larger than the phonon mean free path. The results obtained for the thermal conductivity of the graphene nanoribbons are presented in Fig. 4 for different lengths of the graphene nanoribbons. Both the results obtained from the experiments and predicted by the above model are presented in Fig. 4, when the graphene nanoribbons are about 2.0 and 2.4 µm in width, respectively. The upper limits of thermal conductivity for ballistic transport in the graphene nanoribbons are also included in Fig. 4. A wide temperature range is considered, varying from 100 to 300 K. There is substantial agreement between the experimental results and the theoretical results obtained by the model. The thermal conductivity measured in the experiments and predicted by the model is much lower than the upper limit determined by the theory of ballistic heat conduction, especially for the longer graphene nanoribbons. In contrast, ballistic transport effects are considerable for the shorter graphene nanoribbons.
The width of the graphene nanoribbons is an essential factor involved in the heat conduction process within the carbon-based material due to phonon-boundary scattering [39, 40]. The characteristics of phonon transport in the graphene nanoribbons that are supported on the silicon carbide substrate are investigated, when the width of the graphene nanoribbons is smaller than the phonon mean free path within the carbon-based material. The results obtained for the thermal conductivity of the graphene nanoribbons are presented in Fig. 5 for different widths of the graphene nanoribbons. The graphene nanoribbons are about 380 nm in length. The width varies from 40 to 280 nm. The thermal conductivity increases in dependence upon the width increase of the graphene nanoribbons. Various empirical models [41, 42] have been utilized to determine the effect of the width of a graphene nanoribbon on the thermal conductivity due to the phonon scattering arising from the crystallographic disorder of the edges of the graphene nanoribbon. The effective thermal conductivity of a graphene nanoribbon can be expressed as:
$${k_{effective}} \approx {C_{ballistic}}{\left( {{\beta ^{{\mathbf{-}}1}}{{\left( {\frac{{Rq}}{w}} \right)}^n}{\mathbf{+}}{k^{{\mathbf{-}}1}}} \right)^{{\mathbf{-}}1}}$$
2
in which β is a thermal conductivity parameter, Rq is the root mean square roughness of the edges, and n is a constant exponent. Both the results obtained from the experiments and predicted by the above model are presented in Fig. 5. A very good agreement of the results obtained from the experiments in comparison with the results predicted by the model is achieved. More specifically, the accuracy of the model is demonstrated by a good agreement between the experimental results obtained from thermal conductivity measurements and the theoretical results obtained by the model. However, the model may fail to make accurate predictions of the thermal conductivity of graphene nanoribbons with a degree of crystallographic disorder of the edges [24]. Overall, a very good agreement of the experimental results in comparison with the theoretical results is achieved.
A uniform roughness of the edges is used in the above empirical model, making it difficult to make accurate predictions about the thermal conductivity of graphene nanoribbons under extreme conditions of their structure [24]. The classical theory of transport processes is based on the Boltzmann transport equation. This equation is still being used to model transport phenomena in many fields, and many important results can therefore be derived [43, 44]. Numerical simulations are performed by solving the Boltzmann transport equation based on the complete phonon dispersion relations [45, 46] to gain insight into the characteristics of heat conduction within the graphene nanoribbons. Both phonon-boundary scattering and quasi-ballistic phonon propagation are accounted for, and there is no need to use a uniform roughness for the edges of the graphene nanoribbons. Both the results obtained from the experiments and predicted by the Boltzmann transport equation are presented in Fig. 6, in which the in-plane thermal conductivity of the graphene nanoribbons with different widths is expressed as a function of temperature. Good agreement between the theoretical and experimental results is achieved. However, the Boltzmann transport equation is ineffective in gaining insight into the physics of phonon scattering arising from the crystallographic disorder of the edges, for example, when phonon transport is localized within the carbon-based material [47, 48]. Based on the kinetics and atomic structure consideration, a material with high crystalline and strong interactions, composed of light atoms, such as diamond and graphene, is expected to have large phonon conductivity. Solids with more than one atom in the smallest unit cell representing the lattice have two types of phonons, i.e., acoustic and optical. Acoustic phonons are in-phase movements of atoms about their equilibrium positions, while optical phonons are out-of-phase movement of adjacent atoms in the lattice. Optical phonons have higher energies or frequencies, but make smaller contribution to conduction heat transfer, because of their smaller group velocity and occupancy.