  We study room temperature spin transport in graphene devices encapsulated
between a layer-by-layer-stacked two-layer-thick chemical vapour deposition
(CVD) grown hexagonal boron nitride (hBN) tunnel barrier, and a few-layer-thick
exfoliated-hBN substrate. We find mobilities and spin-relaxation times
comparable to that of SiO$_2$ substrate based graphene devices, and obtain a
similar order of magnitude of spin relaxation rates for both the Elliott-Yafet
and D'Yakonov-Perel' mechanisms. The behaviour of
ferromagnet/two-layer-CVD-hBN/graphene/hBN contacts ranges from transparent to
tunneling due to inhomogeneities in the CVD-hBN barriers. Surprisingly, we find
both positive and negative spin polarizations for high-resistance
two-layer-CVD-hBN barrier contacts with respect to the low-resistance contacts.
Furthermore, we find that the differential spin injection polarization of the
high-resistance contacts can be modulated by DC bias from -0.3 V to +0.3 V with
no change in its sign, while its magnitude increases at higher negative bias.
These features mark a distinctive spin injection nature of the
two-layer-CVD-hBN compared to the bilayer-exfoliated-hBN tunnel barriers.
