Free-electron effects on the optical absorption of the hybrid perovskite ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbI}}_3$ from first principles

Hybrid organic-inorganic perovskites, such as methylammonium lead tri-iodide (${\mathrm{MAPbI}}_3$), are interesting candidates for efficient absorber materials in next-generation solar cells, partly due to an unusual combination of low exciton-binding energy and strong optical absorption. Excitonic effects in this material have been subject to debate both for experiment and theory, indicating a need for better understanding of the screening mechanisms that act upon the electron-hole interaction. Here, we use cutting-edge first-principles theoretical spectroscopy, based on density-functional and many-body perturbation theory, to study atomic geometries, electronic structure, and optical properties of three ${\mathrm{MAPbI}}_3$ polymorphs and find good agreement with earlier results and experiment. We then study the influence of free electrons on the electron-hole interaction and show that this explains consistently smaller exciton-binding energies, compared to those in the material without free electrons. Interestingly, we also find that the absorption line shape strongly resembles that of the spectrum without free electrons up to high free-electron concentrations. We explain this unexpected behavior by formation of Mahan excitons that dominate the absorption edge, making ${\mathrm{MAPbI}}_3$ robust against free-electron-induced changes observed in other semiconductors.

Figure 1

Relaxed atomic geometries of orthorhombic, tetragonal, and cubic phases of ${\mathrm{MAPbI}}_3$. Ions are represented as gray (Pb), purple (I), brown (C), pink (H), and blue (N) spheres. Lattice constants $a,b$, and $c$ align with the [001], [010], and [001] directions, respectively.

Figure 2

Kohn-Sham band structure and density of states from $\mathrm{PBE}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$ (solid lines) and ${\mathrm{GW}}_0+\mathrm{SOC}$ (black circles) calculations for (a) orthorhombic, (b) tetragonal, and (c) cubic phases of ${\mathrm{MAPbI}}_3$. All conduction states are rigidly shifted to the ${\mathrm{GW}}_0+\mathrm{SOC}$ gap. The valence-band maximum is used as energy zero.

Figure 3

Polarization-averaged imaginary part of the frequency-dependent dielectric functions of ${\mathrm{MAPbI}}_3$, calculated within independent-quasiparticle approximation using $\mathrm{PBE}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$. Blue, green, and maroon curves correspond to orthorhombic, tetragonal, and cubic phases, respectively.

Figure 4

Polarization-averaged imaginary part of the frequency-dependent dielectric function of cubic ${\mathrm{MAPbI}}_3$. Results from independent-quasiparticle approximation, here $\mathrm{PBE}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$ (solid maroon line, identical to that in Fig. 3), are compared to the ${\mathrm{BSE}}_{\mathrm{el}}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$ approach (solid black line) that accounts for electron-hole interaction and to experimental [95, 96, 97] results (gray diamonds).

Figure 5

Polarization-averaged imaginary part of the frequency-dependent dielectric function of $C{\mathrm{MAPbI}}_3$ without free electrons (black) and with free-electron concentrations of $2.3\times {10}^{18}{\mathrm{cm}}^{-3}$ (red), $5.0\times {10}^{18}{\mathrm{cm}}^{-3}$ (orange), and $1.1\times {10}^{19}{\mathrm{cm}}^{-3}$ (blue). A dense hybrid 5:2:32.5 $\mathbf{k}$-point grid was used. The top panel shows the influence of BMS and BGR on the independent-quasiparticle spectrum ($\mathrm{PBE}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$). The bottom panel demonstrates the influence of free electrons, $n_c=1.1\times {10}^{19}{\mathrm{cm}}^{-3}$, on excitonic effects. The ${\mathrm{BSE}}_{\mathrm{el}}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$ spectrum without free electrons (black dashed line) is compared to data that include free-electron screening without (blue dot-dashed line) and with (blue dashed line) Pauli blocking. The violet curve approximately describes lattice screening via the low-frequency dielectric constant ${\varepsilon }_0=22.1$ in the model dielectric function [67, 98].

Figure 6

Burstein-Moss shift (red circles), band-gap renormalization (black squares), and sum of both (blue diamonds) as a function of free-electron concentration in the conduction band of $C{\mathrm{MAPbI}}_3$. A dense hybrid 5:2:32.5 $\mathbf{k}$-point grid was used. The red line is a curve fit to the BMS data of the form $E_{\mathrm{BMS}}=A{n}_c^{3/2}$.

Figure 7

Polarization-averaged imaginary part of the frequency-dependent dielectric function, computed using the ${\mathrm{BSE}}_{\mathrm{el}+\mathrm{fc}}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$ approach to account for excitonic effects. Results are shown for three different experimentally relevant free-electron concentrations and compared to data without free electrons and experiment [95, 96, 97]. As BMS and BGR are negligible for small values of $n_c$, they are only included for $n_c=3\times {10}^{18}{\mathrm{cm}}^{-3}$.

Figure 8

Exciton-binding energy in $C{\mathrm{MAPbI}}_3$ as a function of free-electron concentration, calculated using the ${\mathrm{BSE}}_{\mathrm{el}+\mathrm{fc}}+{\mathrm{\Delta }}_{{\mathrm{GW}}_0}+\mathrm{SOC}$ framework. Electronic and free-electron screenings of the electron-hole interaction are included. Data with (red) and without (blue) Pauli blocking are compared.