  Ti1-xMgxN(001) layers with 0.00 {\leq} x {\leq} 0.49 are deposited on
MgO(001) by reactive magnetron co-sputtering from titanium and magnesium
targets in 5 mTorr pure N2 at 600 {\deg}C. X-ray diffraction {\omega}-2{\theta}
scans, {\omega}-rocking curves, {\phi}-scans, and high resolution reciprocal
space maps show that the Ti1-xMgxN layers are rock-salt structure single
crystals with a cube-on-cube epitaxial relationship with the substrates:
(001)TiMgN{\parallel} (001)MgO and [100]TiMgN{\parallel} [100]MgO. Layers with
thickness d = 35-58 nm are fully strained, with an in-plane lattice parameter
of 4.212{\pm}0.001 {\AA} matching that of the MgO substrate, while the
out-of-plane lattice parameter increases with x from 4.251 {\AA} for TiN(001)
to 4.289 {\AA} for Ti0.51Mg0.49N(001). This yields a relaxed lattice parameter
for Ti1-xMgxN(001) of ao = (1-x)aTiN + xaMgN - bx(1-x), where aTiN = 4.239
{\AA}, aMgN = 4.345 {\AA}, and the bowing parameter b = 0.113 {\AA}. In
contrast, thicker Ti1-xMgxN(001) layers with d = 110-275 nm are partially (pure
TiN) or fully (x = 0.37 and 0.39) relaxed, indicating a critical thickness for
relaxation of 50-100 nm. The in-plane x-ray coherence length is large (100-400
nm) for fully strained layers with 0.00 {\leq} x {\leq} 0.45 but drops by an
order of magnitude for x = 0.49 as the composition approaches the phase
stability limit. It is also an order of magnitude smaller for thicker (d {\leq}
110 nm) layers, which is attributed to strain relaxation through the nucleation
and growth of misfit dislocations facilitated by glide of threading
dislocations.
