  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 SiO$_2$ and gated. We employ a self-consistent
Poisson-Schr\"{o}dinger-Monte Carlo solver that accounts for scattering due to
acoustic 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 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 $\times$ 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.
