Heat flow in nanomaterials is an important area of study, with both fundamental and technological implications. However, little is known about heat flow in two-dimensional devices or interconnects with dimensions comparable to the phonon mean free path. Here we find that short, quarter-micron graphene samples reach B35% of the ballistic thermal conductance limit up to room temperature, enabled by the relatively large phonon mean free path (B100 nm) in substrate-supported graphene. In contrast, patterning similar samples into nanoribbons leads to a diffusive heat-flow regime that is controlled by ribbon width and edge disorder. In the edge-controlled regime, the graphene nanoribbon thermal conductivity scales with width approximately as BW 1.8±0.3 , being about 100 W m À 1 K À 1 in 65-nm-wide graphene nanoribbons, at room temperature. These results show how manipulation of two-dimensional device dimensions and edges can be used to achieve full control of their heat-carrying properties, approaching fundamentally limited upper or lower bounds.

The widely used 'silicon-on-insulator' (SOI) system consists of a layer of single-crystalline silicon supported on a silicon dioxide substrate. When this silicon layer (the template layer) is very thin, the assumption that an effectively infinite number of atoms contributes to its physical properties no longer applies, and new electronic, mechanical and thermodynamic phenomena arise, distinct from those of bulk silicon. The development of unusual electronic properties with decreasing layer thickness is particularly important for silicon microelectronic devices, in which (001)-oriented SOI is often used. Here we show--using scanning tunnelling microscopy, electronic transport measurements, and theory--that electronic conduction in thin SOI(001) is determined not by bulk dopants but by the interaction of surface or interface electronic energy levels with the 'bulk' band structure of the thin silicon template layer. This interaction enables high-mobility carrier conduction in nanometre-scale SOI; conduction in even the thinnest membranes or layers of Si(001) is therefore possible, independent of any considerations of bulk doping, provided that the proper surface or interface states are available to enable the thermal excitation of 'bulk' carriers in the silicon layer.

We investigate the effects of electron and acoustic-phonon confinement on the low-field electron mobility of thin square silicon nanowires (SiNWs) that are surrounded by SiO2 and gated. We employ a self-consistent Poisson-Schrödinger-Monte Carlo solver that accounts for scattering due to acoustic phonons (confined and bulk), intervalley phonons, and the Si/SiO2 surface roughness. The wires considered have cross sections between 3 × 3 nm 2 and 8 × 8 nm 2 . For larger wires, as expected, the dependence of the mobility on the transverse field from the gate is pronounced. At low transverse fields, where phonon scattering dominates, scattering from confined acoustic phonons results in about a 10% decrease of the mobility with respect to the bulk phonon approximation. As the wire cross-section decreases, the electron mobility drops because the detrimental increase in both electron-acoustic phonon and electron-surface roughness scattering rates overshadows the beneficial volume inversion and subband modulation. For wires thinner than 5 × 5 nm 2 , surface roughness scattering dominates regardless of the transverse field applied and leads to a monotonic decrease of the electron mobility with decreasing SiNWs cross section.

We present a calculation of the thermal conductivity of graphene nanoribbons (GNRs), based on solving the Boltzmann transport equation with the full phonon dispersions, a momentum-dependent model for edge roughness scattering, as well as three-phonon and isotope scattering. The interplay between edge roughness scattering and the anisotropy of the phonon dispersions results in thermal conduction that depends on the chiral angle of the nanoribbon. Lowest thermal conductivity occurs in the armchair direction and highest in zig-zag nanoribbons. Both the thermal conductivity and the degree of armchair/zig-zag anisotropy depend strongly on the width of the nanoribbon and the rms height of the edge roughness, with the smallest and most anisotropic thermal conductivities occurring in narrow GNRs with rough edges.

We present a calculation of the full thermal conductivity tensor for ͑001͒, ͑111͒, and ͑011͒ surface orientations of the silicon-on-insulator ͑SOI͒ nanomembrane, based on solving the Boltzmann transport equation in the relaxation-time approximation with the full phonon dispersions, a momentum-dependent model for boundary scattering, as well as three-phonon and isotope scattering. The interplay between strong boundary scattering and the anisotropy of the phonon dispersions results in thermal conduction that strongly depends on the surface orientation and exhibits marked in-plane vs out-of-plane anisotropy, as well as slight in-plane anisotropy for the low-symmetry ͑011͒ SOI. In-plane thermal conductivity is highest along ͓100͔ on Si͑011͒ and lowest in Si͑001͒ due to the strong scattering of the highly anisotropic TA modes with ͑001͒ surfaces. The room-temperature in-plane conductivities in ͑011͒ and ͑001͒ nanomembranes with thicknesses around 10 nm differ by a factor of 2, and the ratio can be much higher at lower temperatures or in rougher samples. We discuss surface facet orientation as a means of tailoring thermal conduction in low-dimensional nanostructrures and address the role of out-of-plane thermal conductivities in predicting vertical phonon transport in superlattices.