  We investigate the electronic structure of graphene on a series of 2D
hexagonal nitride insulators hXN, X = B, Al, and Ga, with DFT calculations. A
symmetry-based model Hamiltonian is employed to extract orbital parameters and
spin-orbit coupling (SOC) from the low-energy Dirac bands of proximitized
graphene. While commensurate hBN induces a staggered potential of about 10 meV
into the Dirac bands, less lattice-matched hAlN and hGaN disrupt the Dirac
point much less, giving a staggered gap below 100 $\mu$eV. Proximitized
intrinsic SOC surprisingly does not increase much above the pristine graphene
value of 12 $\mu$eV; it stays in the window of (1-16) $\mu$eV, depending
strongly on stacking. However, Rashba SOC increases sharply when increasing the
atomic number of the boron group, with calculated maximal values of 8, 15, and
65 $\mu$eV for B, Al, and Ga-based nitrides, respectively. The individual
Rashba couplings also depend strongly on stacking, vanishing in
symmetrically-sandwiched structures, and can be tuned by a transverse electric
field. The extracted spin-orbit parameters were used as input for spin
transport simulations based on Chebyshev expansion of the time-evolution of the
spin expectation values, yielding interesting predictions for the electron spin
relaxation. Spin lifetime magnitudes and anisotropies depend strongly on the
specific (hXN)/graphene/hXN system, and they can be efficiently tuned by an
applied external electric field as well as the carrier density in the graphene
layer. A particularly interesting case for experiments is graphene/hGaN, in
which the giant Rashba coupling is predicted to induce spin lifetimes of 1-10
ns, short enough to dominate over other mechanisms, and lead to the same spin
relaxation anisotropy as observed in conventional semiconductor
heterostructures: 50\%, meaning that out-of-plane spins relax twice as fast as
in-plane spins.
