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.