Graphene exhibits extraordinary electronic and mechanical properties, and extremely high thermal conductivity. Being a very stable atomically thick membrane that can be suspended between two leads, graphene provides a perfect test platform for studying thermal conductivity in two-dimensional systems, which is of primary importance for phonon transport in low-dimensional materials. Here we report experimental measurements and nonequilibrium molecular dynamics simulations of thermal conduction in suspended single-layer graphene as a function of both temperature and sample length. Interestingly and in contrast to bulk materials, at 300 K, thermal conductivity keeps increasing and remains logarithmically divergent with sample length even for sample lengths much larger than the average phonon mean free path. This result is a consequence of the two-dimensional nature of phonons in graphene, and provides fundamental understanding of thermal transport in two-dimensional materials.

We derive expressions of interatomic force and heat current for many-body potentials such as the Tersoff, the Brenner, and the Stillinger-Weber potential used extensively in molecular dynamics simulations of covalently bonded materials. Although these potentials have a many-body nature, a pairwise force expression that follows Newton's third law can be found without referring to any partition of the potential. Based on this force formula, a stress applicable for periodic systems can be unambiguously defined. The force formula can then be used to derive the heat current formulas using a natural potential partitioning. Our heat current formulation is found to be equivalent to most of the seemingly different heat current formulas used in the literature, but to deviate from the stress-based formula derived from two-body potential. We validate our formulation numerically on various systems described by the Tersoff potential, namely three-dimensional silicon and diamond, two-dimensional graphene, and quasi-one-dimensional carbon nanotube. The effects of cell size and production time used in the simulation are examined.

A detailed understanding of the connections of fabrication and processing to structural and thermal properties of low-dimensional nanostructures is essential to design materials and devices for phononics, nanoscale thermal management, and thermoelectric applications. Silicon provides an ideal platform to study the relations between structure and heat transport since its thermal conductivity can be tuned over 2 orders of magnitude by nanostructuring. Combining realistic atomistic modeling and experiments, we unravel the origin of the thermal conductivity reduction in ultrathin suspended silicon membranes, down to a thickness of 4 nm. Heat transport is mostly controlled by surface scattering: rough layers of native oxide at surfaces limit the mean free path of thermal phonons below 100 nm. Removing the oxide layers by chemical processing allows us to tune the thermal conductivity over 1 order of magnitude. Our results guide materials design for future phononic applications, setting the length scale at which nanostructuring affects thermal phonons most effectively.

The majority-vote model with noise on Erdös-Rényi's random graphs has been studied. Monte Carlo simulations were performed to characterize the order-disorder phase transition appearing in the system. We found that the value of the critical noise parameter qc is an increasing function of the mean connectivity z of the random graph. The critical exponents β/ν, γ/ν and 1/ν were calculated for several values of z, and our analysis yielded critical exponents satisfying the hyperscaling relation with effective dimensionality equal to unity.