We investigate the effects of electron and acoustic-phonon confinement on thelow-field electron mobility of thin square silicon nanowires (SiNWs) that aresurrounded by SiO$_2$ and gated. We employ a self-consistentPoisson-Schr\"{o}dinger-Monte Carlo solver that accounts for scattering due toacoustic phonons (confined and bulk), intervalley phonons, and the Si/SiO$_2$surface roughness. The wires considered have cross sections between 3 $\times$3 nm$^2$ and 8 $\times$ 8 nm$^2$. For larger wires, as expected, the dependenceof the mobility on the transverse field from the gate is pronounced. At lowtransverse fields, where phonon scattering dominates, scattering from confinedacoustic phonons results in about a 10% decrease of the mobility with respectto the bulk phonon approximation. As the wire cross-section decreases, theelectron mobility drops because the detrimental increase in bothelectron--acoustic phonon and electron--surface roughness scattering ratesovershadows the beneficial volume inversion and subband modulation. For wiresthinner than 5 $\times$ 5 nm$^2$, surface roughness scattering dominatesregardless of the transverse field applied and leads to a monotonic decrease ofthe electron mobility with decreasing SiNWs cross section.
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