  The dilute-nitride ternary compound GaP$_{1-x}$N$_{x}$ is highly attractive
to monolithically integrate pseudomorphic red light-emitting devices and
photovoltaic cells with the standard Si technology because it is lattice
matched to Si with a direct band gap of $\approx1.96$ eV for $x=0.021$. Here,
we report on the chemical beam epitaxy of GaP$_{x}$N$_{1-x}$ alloys on
nominally $(001)$-oriented GaP-on-Si substrates. The incorporation of N into
GaP$_{1-x}$N$_{x}$ was systematically investigated as a function of the growth
temperature and the fluxes of the N and P precursors, 1,1-dimethylhydrazine
(DMHy) and tertiarybutylphosphine (TBP), respectively. We found that the N mole
fraction exhibits an Arrhenius behavior characterized by an apparent activation
energy of $(0.79\pm 0.05)$ eV. With respect to the fluxes, we determined that
the N mole fraction is linearly proportional to the flux of DMHy, and inversely
proportional to the one of TBP. All results are summarized in an universal
equation that describes the dependence of \textit{x} on the growth temperature
and the fluxes of the group-V precursors. The results are further illustrated
in a growth diagram that visualizes the variation of the chemical composition
as the growth temperature and the flux of DMHy are varied. This diagram also
shows how to obtain single-phase and flat GaP$_{1-x}$N$_{x}$ layers, as certain
growth conditions result in chemically phase-separated GaP$_{1-x}$N$_{x}$
layers with rough surface morphologies. Last, our results demonstrate the
feasibility of chemical beam epitaxy for the synthesis of single-phase and flat
GaP$_{x}$N$_{1-x}$ layers with N mole fractions up to about $x=0.04$, a value
well above the one required for the lattice-matched integration of
GaP$_{1-x}$N$_{x}$-based devices on Si.
