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.

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Chemical vapour deposition of two-dimensional materials typically involves the conversion of vapour precursors to solid products in a vapour-solid-solid mode. Here, we report the vapour-liquid-solid growth of monolayer MoS, yielding highly crystalline ribbons with a width of few tens to thousands of nanometres. This vapour-liquid-solid growth is triggered by the reaction between MoO and NaCl, which results in the formation of molten Na-Mo-O droplets. These droplets mediate the growth of MoS ribbons in the 'crawling mode' when saturated with sulfur. The locally well-defined orientations of the ribbons reveal the regular horizontal motion of the droplets during growth. Using atomic-resolution scanning transmission electron microscopy and second harmonic generation microscopy, we show that the ribbons are grown homoepitaxially on monolayer MoS with predominantly 2H- or 3R-type stacking. Our findings highlight the prospects for the controlled growth of atomically thin nanostructure arrays for nanoelectronic devices and the development of unique mixed-dimensional structures.