  Tuning the bandgap in ternary III-V semiconductors via modification of the
composition or the strain in the material is a major approach for the design of
optoelectronic materials. Experimental approaches screening a large range of
possible target structures are hampered by the tremendous effort to optimize
the material synthesis for every target structure. We present an approach based
on density functional theory efficiently capable of providing the bandgap as a
function of composition and strain. Using a specific density functional
designed for accurate bandgap computation (TB09) together with a band unfolding
procedure and special quasirandom structures, we develop a computational
protocol efficiently able to predict bandgaps. The approach's accuracy is
validated by comparison to selected experimental data. We thus map the phase
space of composition and strain (we call this the ``bandgap phase diagram'')
for several important III-V compound semiconductors: GaAsP, GaAsN, GaPSb,
GaAsSb, GaPBi, and GaAsBi. We show the application of these diagrams for
identifying the most promising materials for device design. Furthermore, our
computational protocol can easily be generalized to explore the vast chemical
space of III-V materials with all other possible combinations of III- and
V-elements.
