  Band alignment between two materials is of fundamental importance for
multitude of applications. However, density functional theory (DFT) either
underestimates the bandgap - as is the case with local density approximation
(LDA) or generalized gradient approximation (GGA) - or is highly
computationally demanding, as is the case with hybrid-functional methods. The
latter can become prohibitive in electronic-structure calculations of
supercells which describe quantum wells. We propose to apply the DFT$+U$
method, with $U$ for each atomic shell being treated as set of tuning
parameters, to automatically fit the bulk bandgap and the lattice constant, and
then use thus obtained $U$ parameters in large supercell calculations to
determine the band alignment. We apply this procedure to
InP/In$_{0.5}$Ga$_{0.5}$As, In$_{0.5}$Ga$_{0.5}$As/In$_{0.5}$Al$_{0.5}$As and
InP/In$_{0.5}$Al$_{0.5}$As quantum wells, and obtain good agreement with
experimental results. Although this procedure requires some experimental input,
it provides both meaningful valence and conduction band offsets while,
crucially, lattice relaxation is taken into account. The computational cost of
this procedure is comparable to that of LDA. We believe that this is a
practical procedure that can be useful for providing accurate estimate of band
alignments between more complicated alloys.
