Ab Initio Electronic Gaps of Ge Nanodots: The Role of Self-Energy Effects Margherita Marsili,* ,† Silvana Botti, ‡,¶,# Maurizia Palummo, §,# Elena Degoli, ∥,# Olivia Pulci, §,# Hans-Christian Weissker, ⊥,# Miguel A. L. Marques, ¶,# Stefano Ossicini, ∥,# and Rodolfo Del Sole §,# † Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universita ́ di Padova, via Marzolo 8, I-35131 Padova, Italy ‡ Laboratoire des Solides Irradie ́ s, E ́ cole Polytechnique, CNRS, CEA-DSM, 91128 Palaiseau, France ¶ Institut Lumiè re Matie ̀ re, UMR5306 Universite ̀ de Lyon 1-CNRS, Universite ̀ de Lyon, F-69622 Villeurbanne, Cedex, France § NAST and Dipartimento di Fisica, Universita ̀ di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, I-00133 Rome, Italy ∥ Istituto Nanoscienze CNR and Dipartimento di Scienze e Metodi dell’Ingegneria, Universita ́ di Modena e Reggio Emilia, I-42122 Reggio Emilia, Italy ⊥ Aix-Marseille University, CNRS, CINaM UMR 7325, 13288 Marseille, France # European Theoretical Spectroscopy Facility (ETSF) ABSTRACT: Nanostructuring of a material leads to enor- mous effects on its excited state properties. This study, through the application of different state-of-the-art ab initio theoretical tools, investigates the effect of size on the electronic gap of germanium nanocrystals highlighting similarities and differ- ences with respect to equivalent silicon nanostructures. We performed both GW and ΔSCF calculations for the determination of their electronic structure. While it is known that ΔSCF corrections to the Kohn−Sham gap vanish for extended systems, the two approaches were expected to be equivalent in the limit of small clusters. However, it has been recently found that for hydrogenated Si clusters the ΔSCF gaps are systematically smaller than the GW ones, while the opposite is true for Ag clusters. In this work we find that the GW gaps are larger than the ΔSCF ones for all the Ge dots, with the exception of the smallest one. Such crossing between the ΔSCF and the GW gap values was not expected and has never been observed before. Moreover, also for hydrogenated Si nanocrystals we found a similar behavior. The origin of this crossing might be found in the Rydberg character of the LUMO of the smallest clusters and can also explain the qualitative differences in the comparison between GW and ΔSCF found in the previous studies. ■ INTRODUCTION The possibility of tailoring the electronic and optical properties of nanostructures simply by changing their size opens the way for the application of these systems in a variety of different fields, ranging from opto-electronics to photovoltaic devices. Many different materials, such as Si, CdTe, and III−V materials have seen their electronic gap engineered through the control of the size of the structures they constituted. Empirical or ab initio calculations, at the level of density functional theory (DFT), are able to capture qualitatively this effect. Nonetheless, to describe it in a more accurate way, it has often been necessary to go beyond standard ground state DFT calculations, using the so-called ΔSCF method within the DFT framework, 1 or the more sophisticated, but computation- ally demanding, GW approximation within the many-body perturbation theory (MBPT) framework. 2,3 For a long time the two theoretical approaches have been considered almost equivalent in the limit of small cluster diameters. However, recent studies on Si 4 and Ag clusters 5 have shown that the two methods are not equivalent. Moreover the discrepancy between the two methods is qualitatively different for the two systems: the ΔSCF gaps are larger than the GW ones in the Ag clusters case, while, on the contrary, in Si clusters the ΔSCF gaps are smaller than the GW ones. Recent scanning-tunneling spectroscopy (STS) experi- ments 6,7 have measured the strong dependence of the electronic gap of Ge nanodots on their size. As expected, quantum-confinement effects lead to a strong increase of the gap when the dot diameter was reduced. Despite the simplicity of these systems and the analogy with their, more extensively studied, Si counterparts, only ΔSCF calculations of their electronic gap have been reported. 8 On Ge, no systematic investigation of their electronic gap within the MBPT framework has been carried out yet. In this work, we present calculations of quasiparticle gaps for hydrogenated Ge nanocrystals of increasing size, with diameters ranging from 0.6 to 1.6 nm, using both ΔSCF-LDA and the perturbative GW method. Received: December 10, 2012 Revised: June 10, 2013 Published: June 13, 2013 Article pubs.acs.org/JPCC © 2013 American Chemical Society 14229 dx.doi.org/10.1021/jp3121269 | J. Phys. Chem. C 2013, 117, 14229−14234