Band structures of delafossite transparent conductive oxides from a self-consistent GW approach Fabio Trani, 1,2,3 Julien Vidal, 4,5,2 Silvana Botti, 5,1,2 and Miguel A. L. Marques 1,2 1 Laboratoire de Physique de la Matière Condensée et Nanostructures (LPMCN), Université Lyon 1, CNRS, Domaine Scientifique de la Doua, 69622 Villeurbanne, France 2 European Theoretical Spectroscopy Facility (ETSF) 3 Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy 4 Institute for Research and Development of Photovoltaic Energy (IRDEP), UMR 7174, CNRS/EDF/ENSCP, 6 quai Watier, 78401 Chatou, France 5 Laboratoire des Solides Irradiés (LSI), École Polytechnique, CNRS, CEA-DSM, 91128 Palaiseau, France Received 27 May 2010; revised manuscript received 7 July 2010; published 19 August 2010 We present a comparative study of the electronic band structures of the compounds CuMO 2 M =B,Al,In,Ga which belong to the family of delafossite transparent conductive oxides. The theoretical approaches we use are the standard local-density approximation LDA to density-functional theory, LDA + U, hybrid functionals, and perturbative GW on top of LDA or self-consistent Coulomb hole plus screened exchange calculations. The latter approach, state-of-the-art theoretical approach for quasiparticle band struc- tures, predicts direct band gaps that are compatible with experimental optical gaps only after including the strong polaronic and excitonic effects present in these materials. For what concerns the so-called band-gap anomaly of delafossite compounds, we find that GW approaches yield the same qualitative trends with increas- ing anion atomic number as the LDA: accounting for the oscillator strength at the absorption edge is the key to explain the experimental trend. None of the methods that we applied beyond the simple LDA is in agreement with the small indirect gaps found by many early experiments. This supports the recent view that the absorp- tion bands identified as a sign of the indirect experimental gaps are likely due to defect states in the gap and are not a property of the pristine material. DOI: 10.1103/PhysRevB.82.085115 PACS numbers: 71.20.-b, 71.45.Gm, 78.20.-e, 71.15.Qe I. INTRODUCTION Transparent conductive oxides TCOs are wide band-gap semiconductors characterized by large free carrier densities. These carriers are created by either intrinsic or extrinsic dop- ing, giving to TCOs both low resistivity and transparency in the visible energy window. The technological applications of these materials are wide, ranging from their use as transpar- ent contacts in flat panel displays, 1 to photovoltaic devices. 2 The charge carriers are usually electrons. Indeed, the most common examples of TCOs are electron n-doped SnO 2 , In 2 O 3 , and ZnO. Hole p-type conductivity in TCOs was much harder to achieve but it was ultimately found in CuAlO 2 thin films. 3 A few years later, bipolar either n- or p-type conductivity was discovered in one element of the same family, namely, CuInO 2 . 4,5 These spectacular achieve- ments opened the way for the fabrication of TCO p-n junctions, 6 and to the development of a new technology en- tirely based on “invisible circuits,” the so-called transparent electronics, 7–9 with many innovative applications stemming from it, such as stacked solar cells, transparent screens, or functional windows that generate solar electricity. The materials responsible for such amazing properties be- long to a particular class of Cu ternary oxides appearing in nature in the delafossite crystal phase, CuMO 2 , where M is a group-III element. 10 Their crystal structure is characterized by parallel planes composed of M and O atoms linked by dumbbell Cu atoms, yielding a strong anisotropy in the elec- tronic properties. Since the discovery of their relatively high conductivity, delafossite copper oxides have been studied ex- tensively both from the theoretical and the experimental point of view, and are the object of a raising interest from the scientific community, especially for their applications in thin- film solar-cell technology. From the experimental point of view, delafossites have been subject to conductivity and optical measurements, argu- ably the most important properties for their use as TCOs. The oldest optical experiments for CuAlO 2 pointed to a large difference between the direct and indirect band gap. 3,11–18 Analogous results followed for CuInO 2 Refs. 4, 19, and 20 and CuGaO 2 . 21 However, the most recent experimental work 22–24 suggests that all early results should be reanalyzed in view of the fact that most of the samples used in experi- ments were thin films. Indeed, the discrepancy between the gap measured for thin films and single-crystal samples hints at the fact that strain might play an important role. Moreover, one should consider that even if the extraction of the direct band gap from inspection of the absorption onset is fairly straightforward, the identification of the indirect band gap is considerably hindered by the inevitable presence of defect bands in the samples. Recent accurate single-crystal measurements 22–24 lead to a reduction in the difference be- tween direct and indirect band gap and an overall opening of the band gap. From the theoretical side, Laskowski et al. 25 and Chris- tensen et al. 26 showed that absorption at the direct edge of copper delafossites is dominated by huge excitonic effects about 0.5 eV. Similar excitonic effects were also found in experiments for CuScO 2 . 23,27 This fact should be always taken into account when comparing calculated quasiparticle energies and optical measurements, as the optical and quasi- particle gaps differ by the exciton binding energy. Finally, various experiments 28–31 point to the importance of small PHYSICAL REVIEW B 82, 085115 2010 1098-0121/2010/828/08511511 ©2010 The American Physical Society 085115-1