Towards understanding the electronic structure of the simpler members of two-dimensional halide-perovskites

In this paper, we analyze the band structure of two-dimensional (2D) halide perovskites by considering structures related to the simpler case of the series, (${\mathrm{BA})}₂{\mathrm{PbI}}₄$, in which ${\mathrm{PbI}}₄$ layers are intercalated with butylammonium $[\mathrm{BA}={\mathrm{CH}}₃$(${\mathrm{CH}}₂$)${}₃{\mathrm{NH}}₃]$ organic ligands. We use density-functional-theory (DFT) based calculations and tight-binding (TB) models aiming to discover a simple description of the bands within 1 eV below the valence-band maximum and 2 eV above the conduction-band minimum, which, including the energy gap, is about a $\mathrm{\ensuremath{\Delta}}E=5$ eV energy range. The bands in this $\mathrm{\ensuremath{\Delta}}E$ range are those expected to contribute to the transport phenomena, photoconductivity, and light emission in the visible spectrum, at room and low temperature. We find that the atomic orbitals of the butylammonium chains have negligible contribution to the Bloch states which form the conduction and valence bands in the above defined $\mathrm{\ensuremath{\Delta}}E$ range. Our calculations reveal a rather universal, i.e., independent of the intercalating BA, rigid-band picture inside the above $\mathrm{\ensuremath{\Delta}}E$ range characteristic of the layered perovskite ``matrix'' (i.e., ${\mathrm{PbI}}₄$ in our example). Besides demonstrating the above conclusion, the main goal of this paper is to find accurate TB models which capture the essential features of the DFT bands in this $\mathrm{\ensuremath{\Delta}}E$ range. First, we ignore electron hopping along the $c$ axis and the octahedral distortions and this increased symmetry (from ${\mathrm{C}}₂$ to ${\mathrm{C}}₄$) halves the Bravais lattice unit cell size and the Brillouin zone unfolds to a ${45}^{\ensuremath{\circ}}$ rotated square and this allows some analytical handling of the 2D TB Hamiltonian. The Pb $6s$ and I $5s$ orbitals are far away from the above $\mathrm{\ensuremath{\Delta}}E$ range and, thus, we integrate them out to obtain an effective model which only includes hybridized Pb $6p$ and I $5p$ states. Our TB-based treatment (a) provides a good quantitative description of the DFT band structure, (b) helps us conceptualize the complex electronic structure in the family of these materials in a simple way, and (c) yields the one-body part to be combined with appropriately screened electron interaction to describe many-body effects, such as excitonic bound states.