Figure 1. Schematic diagram of a specific heat flux imposed through one individual graphene nanoribbon in a shape of rectangle. It is assumed and indicated that the property being transported in the graphene nanoribbon flows from the red region to the blue regions. Linear momentum of the atoms is transferred to maintain a constant heat flux between the red region and the blue regions.
The length of the monolayer graphene nanoribbon is typically significantly greater than its width, but of course, there are exceptions. A specific heat flux imposed through the graphene nanoribbon modeled in this study is depicted schematically in Figure 2 in a shape of square. Unless otherwise stated, the graphene nanoribbon is 5.4 nanometers in width, 40 nanometers in length, and 0.335 nanometers in thickness. The structure of the graphene nanoribbon is divided into specific regions by defining hot and cold slabs. Heat is continually being transferred from the hot slab to the cold slabs so as to equalize the temperature within the graphene nanoribbon. The temperature gradient can be determined in the direction of heat flow, since the total energy is conserved in the heat conduction process. Accordingly, the thermal conductivity can be determined from the temperature gradient obtained and the heat flux imposed. The basic rate equation of the heat conduction process within the graphene nanoribbon is known as Fourier's law of heat conduction. The ratio of heat flux to the slope of the temperature profile is proportional to the thermal conductivity of the graphene nanoribbon. Fourier's Law therefore provides the definition of the thermal conductivity of the graphene nanoribbon, and forms the basis of the general method that can be used to determine its value. The thermal conductivity of the monolayer graphene nanoribbon is proportional to the effective mean free path for phonon scattering. Therefore, the thermal conductivity variation over a range of the effective mean free path for phonon scattering can be represented as a straight line. Additionally, the intrinsic thermal conductivity of the two-dimensional carbon-based material can be derived from the intercept of the straight line by extrapolation from the case with infinite length.