  Recently, hexagonal boron nitride (h-BN) has been shown to act as an ideal
substrate to graphene by greatly improving the material transport properties
thanks to its atomically flat surface, low interlayer electronic coupling and
almost perfect reticular matching. Chemical vapour deposition (CVD) is
presently considered the most scalable approach to grow graphene directly on
h-BN. However, for the catalyst-free approach, poor control over the shape and
crystallinity of the graphene grains and low growth rates are typically
reported. In this work we investigate the crystallinity of differently shaped
grains and identify a path towards a real van der Waals epitaxy of graphene on
h-BN by adopting a catalyst-free CVD process. We demonstrate the
polycrystalline nature of circular-shaped pads and attribute the stemming of
different oriented grains to airborne contamination of the h-BN flakes. We show
that single-crystal grains with six-fold symmetry can be obtained by adopting
high hydrogen partial pressures during growth. Notably, growth rates as high as
100 nm/min are obtained by optimizing growth temperature and pressure. The
possibility of synthesizing single-crystal graphene on h-BN with appreciable
growth rates by adopting a simple CVD approach is a step towards an increased
accessibility of this promising van der Waals heterostructure.
