  Hexagonal boron nitride (hBN) has been grown on sapphire substrates by
ultra-high temperature molecular beam epitaxy (MBE). A wide range of substrate
temperatures and boron fluxes have been explored, revealing that high
crystalline quality hBN layers are grown at high substrate temperatures,
$>$1600$^\circ$C, and low boron fluxes, $\sim1\times10^{-8}$ Torr beam
equivalent pressure. \emph{In-situ} reflection high energy electron diffraction
(RHEED) revealed the growth of hBN layers with $60^\circ$ rotational symmetry
and the $[11\bar20]$ axis of hBN parallel to the $[1\bar100]$ axis of the
sapphire substrate. Unlike the rough, polycrystalline films previously
reported, atomic force microscopy (AFM) and transmission electron microscopy
(TEM) characterization of these films demonstrate smooth, layered,
few-nanometer hBN films on a nitridated sapphire substrate. This demonstration
of high-quality hBN growth by MBE is a step towards its integration into
existing epitaxial growth platforms, applications, and technologies.
