3. Results and discussion
The steady-state temperature profiles along the length of the monolayer graphene nanoribbon are presented in Figure 3 with substantially smooth edges. Graphene nanoribbons, as defined herein, refer to, for example, single or multiple layers of graphene that have an aspect ratio of greater than about 5, based on their length and their width. Graphene nanoribbons may be prepared in either oxidized or reduced forms. When not otherwise specified herein, the term graphene nanoribbons should be interpreted to encompass both oxidized graphene nanoribbons and reduced graphene nanoribbons. Longitudinally opening, as defined herein, refers to, for example, opening of carbon nanotubes along their longitudinal axis to form graphene nanoribbons. Such longitudinal opening may be thought of as an unzipping reaction of the carbon nanotubes. Narrow graphene nanoribbons, as defined herein, refers to, for example, graphene nanoribbons having widths less than about 10 nanometers. Wide graphene nanoribbons, as defined herein, refers to, for example, graphene nanoribbons having widths greater than about 10 nanometers. In some cases, wide graphene nanoribbons have widths greater than about 100 nanometers. The graphene nanoribbon has transverse edges that have a so-called armchair configuration. The longitudinal edges of the graphene nanoribbon have a perfect zigzag configuration. The nanoribbon length is assumed to be 40 nanometers. The temperature has a nonlinear dependence on the distance from the given reference point in the direction of heat flow. More specifically, in the vicinity of the hot and cold regions, there exists a nonlinear dependence of the temperature with respect to the distance. This nonlinear dependence is caused by the finite-size effect arising from the graphene nanoribbon, given the fact that the characteristic length scale of the monolayer graphene nanoribbon is much smaller than the mean free path of phonons in graphene. The mean free path of phonons in the graphene nanoribbon can vary depending on the exact structure and dimensions of the monolayer graphene nanoribbon. For example, the mean free path of phonons in graphene can be up to 700 nanometers at room temperature. As a result, the resulting temperature gradient is very steep in the vicinity of the hot and cold regions so that Fourier’s law is no longer applicable to the monolayer graphene nanoribbon. This indicates that phonon transport within the monolayer graphene nanoribbon is not fully diffusive, and the thermal transport within the monolayer graphene nanoribbon is dominated by the mechanism of diffusive-ballistic heat conduction. In the regions between the hot slab and the cold slabs, the temperature has a more or less linear dependence on the distance, and thus the thermal conductivity can be determined by the temperature gradient.