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Effects of polarization on the band-structure of delafossite transparent conductive oxides

We use hybrid functionals and restricted self-consistent GW, state-of-the-art theoretical approaches for quasiparticle band structures, to study the electronic states of delafossite Cu(Al,In)O$_2$, the first p-type and bipolar transparent conductive oxides. We show that self-consistent GW gives remarkably wider band gaps than all the other approaches used so far. Accounting for polaronic effects in the GW scheme we recover a very nice agreement with experiments. Furthermore, the modifications with respect to the Kohn-Sham bands are strongly k-dependent, which makes questionable the common practice of using a scissor operator. Finally, our results support the view that the low energy structures found in optical experiments, and initially attributed to an indirect transition, are due to intrinsic defects in the samples.