$GW$ quasiparticle band gap of the hybrid organic-inorganic perovskite ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbI}}_3$: Effect of spin-orbit interaction, semicore electrons, and self-consistency

We study the quasiparticle band gap of the hybrid organic-inorganic lead halide perovskite ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbI}}_3$, using many-body perturbation theory based on the $GW$ approximation. We perform a systematic analysis of the band gap sensitivity to relativistic spin-orbit effects, to the description of semicore Pb-$5d$ and I-$4d$ electrons, and to the starting Kohn-Sham eigenvalues. We find that the inclusion of semicore states increases the calculated band gap by 0.2 eV, and self-consistency on the quasiparticle eigenvalues using a scissor correction increases the band gap by 0.5 eV with respect to the $G_0{W}_0$ result. These findings allow us to resolve an inconsistency between previously reported $GW$ calculations for ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbI}}_3$. Our most accurate band gap is 1.72 eV, and is in good agreement with the measured optical gap after considering a small excitonic shift as determined in experiments.

Figure 1

Polyhedral representation of the crystal structure of ${\mathrm{MAPbI}}_3$. The Pb atoms are represented by the large blue spheres at the center of each octahedron. The small red spheres located at the shared corners of the octahedra are the I atoms. The MA cations are shown using ball-and-stick models at the center of the cuboctahedral cavities (C is light blue, N is brown, and H is pink). The structure is viewed along the $b$ lattice vector.

Figure 2

Convergence of the quasiparticle band gap of ${\mathrm{MAPbI}}_3$ with respect to the energy cutoff for unoccupied states [(a) and (c)], and with respect to the plane-wave kinetic energy cutoff for the polarizability [(b) and (d)]. Panels (a) and (b) refer to SR calculations, while (c) and (d) are for FR band gaps. In each panel, we show the results for all four combinations of pseudopotentials considered here: Pb and I without semicore states (“w/o s,” black empty circles), Pb with semicore (“w s”) and I without semicore states (black filled circles), Pb without semicore and I with semicore states (blue empty circles), and Pb and I with semicore states (blue filled circles). These data correspond to a $2\times 2\times 2\Gamma$-centered Brillouin zone mesh, and a 136 eV plane-wave cutoff for the exchange self-energy. The plane-wave cutoff for the polarizability in (a) and (c) is set to 82 eV, and the number of total bands in (b) and (d) is set to 1000.

Figure 3

(a) Fully relativistic quasiparticle corrections of the DFT/LDA band gap, (b) the exchange self-energy, and (c) the correlation self-energy of states near the band edges and throughout the Brillouin zone, as a function of DFT/LDA eigenvalues. The key to the symbols and the calculation parameters are the same as in Fig. 2. All calculations are performed for the fully optimized crystal structure of ${\mathrm{MAPbI}}_3$.

Figure 4

(a)–(c) Convergence of the self-consistent scissor correction to the band gap calculated for the (a) experimental and (b) fully optimized crystal structures of ${\mathrm{MAPbI}}_3$, and of the (c) fully optimized crystal structure of ${\mathrm{CsPbI}}_3$ with the number of iterations. The first iteration corresponds to a $G_0{W}_0$ calculation using DFT/LDA eigenstates and eigenvalues (i.e., no scissor correction). (d)–(f) Evolution of the quasiparticle band gap calculated for the (d) experimental and (e) fully optimized crystal structures of ${\mathrm{MAPbI}}_3$, and (f) the fully optimized crystal structure of ${\mathrm{CsPbI}}_3$ throughout the SS-$GW$ iterative procedure. The blue dotted lines show the progression of the calculation at each iteration: the vertical blue lines represent the $G_0{W}_0$ calculation performed at each iteration, while the horizontal blue lines are the scissor shifts applied at each iteration. The iterative procedure stops when the $G_0{W}_0$ correction vanishes, that is when the small and large circle do coincide on the black dotted line. All $GW$ calculations are performed using the calculation parameters listed at the end of Sec. 3.