Revisiting the stable structure of the Cu$_{4}$ complex in silicon

The photoluminescence (PL) spectrum of Cu-containing silicon has a sharp zero-phonon (ZP) band at 1.014 eV. The luminescence center corresponding to this band is called Cu$_{PL}$ and is known to have the local $C_{3v}$ symmetry. A recent measurement by ultrahigh-resolution PL spectroscopy revealed that the Cu$_{PL}$ center is a Cu$_{4}$ complex. Later, it was shown, by first-principles calculations, that the structure was Cu$_{(s)}$Cu$_{3(i)}$, that is, a complex composed of three interstitial Cu$_{(i)}$ atoms around a substitutional Cu$_{(s)}$ atom. This complex (called $C$-type) has the desired symmetry. However, in this study, we show that the lowest-energy structure is different. The tetrahedral structure Cu$_{4}$, called $T$-type, has the lowest energy, with the value being 0.26 eV lower than that of $C$-type. Between these two types, there is an energy barrier of 0.14 eV, which allows $C$-type to exist in a metastable state. Details of the electronic properties of the $T$-type complex are given, by comparing with $C$-type and other isovalent complexes such as Li$_{4}$. Whereas the Cu$_{4}$ tetrahedron is incorporated in silicon in a manner compatible with the tetrahedral network, it also has its own molecular orbitals that exhibit metallic characteristics, in contrast to other complexes. The ZP of the PL spectrum is very likely ascribed to the backflow mode of the Cu$_{4}$ tetrahedron.