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DOI: 10.1039/D0CP00496K
(Paper)
Phys. Chem. Chem. Phys., 2020, Advance Article

Zeeshan Muhammad†
^{ab},
Peitao Liu†^{c},
Rashid Ahmad^{ad},
Saeid Jalali Asadabadi^{e},
Cesare Franchini*^{c} and
Iftikhar Ahmad^{af}
^{a}Center for Computational Material Science, University of Malakand, Chakdara, Pakistan
^{b}Department of Physics, University of Malakand, Chakdara, Pakistan
^{c}University of Vienna, Faculty of Physics and Center for Computational Materials Science, Sensengasse 8, A-1090 Vienna, Austria. E-mail: cesare.franchini@univie.ac.at
^{d}Department of Chemistry, University of Malakand, Chakdara, Pakistan
^{e}Department of Physics, Faculty of Sciences, University of Isfahan, HezarGerib Avenue, Isfahan 81744-73441, Iran
^{f}Vice Chancellor, Gomal University, Dera Ismail Khan, Pakistan

Received
29th January 2020
, Accepted 27th April 2020

First published on 1st May 2020

We study the structural, electronic, and excitonic properties of mixed FAPb(I_{1−x}Br_{x})_{3} 0 ≤ x ≤ 1 alloys by first-principles density functional theory as well as quasiparticle GW and Bethe Salpeter equation (BSE) approaches with the inclusion of relativistic effects through spin orbit coupling. Our results show that the system volume decreases with increasing Br content. The quasiparticle band gaps vary from 1.47 eV for pure α-FAPbI_{3} to 2.20 eV for Br-rich α-FAPbBr_{3} and show stronger correlation with the structural changes. The optical property analysis reveals that the overall excitonic peaks are blue shifted with the Br fraction. Our results further reveal strong Br concentration dependence of the variation in the exciton binding energy (from 74 to 112 meV) and the carrier effective masses as well as the high frequency dielectric constants. These findings provide a way to tune the carrier transport properties of the material by doping, which could be utilized to improve the short circuit currents and power conversion efficiencies in multijunction solar cell devices.

According to recent studies, progress in the efficiency and chemical stability of perovskite based solar cells can be accomplished by the compositional chemical management of the A site having larger cations like cesium (Cs^{+}), methylammonium MA (CH_{3}NH_{3}^{+}) and formamidinium FA (CH(NH_{2})_{2})^{+},^{25–27,30,54} of the B site having smaller divalent inorganic cations e.g., Pb^{2+} and Sn^{2+},^{32,33,55} and of the X site containing monovalent metal halides including I^{−}, Br^{−} and Cl^{−}.^{18,19,22,28,29,36,37,56,57} For example, gradual substitution of the FA cation in MAPbI_{3} will reduce the band gap from 1.57 eV to 1.48 eV,^{30,58} thus yielding improved transport properties,^{59} long photoluminescence (PL) lifetimes,^{60} and lower recombination and device hysteresis with high PCEs.^{19,61,62} Similarly, slowly replacing the Pb^{2+} cation with less toxic Sn^{2+} at the B site will vary the band gap between 1.51 eV and 1.28 eV.^{55} The rapid oxidation of Sn^{2+} to Sn^{4+} especially in SnI_{2} and precursor solutions leads to high defect densities and short carrier diffusion lengths, which restricts its widespread use because of low PCEs.^{63,64} In order to have a wide range of variations in the electronic as well as in the optical properties, the mixing of metal halides at the X site provides an effective way to give such a fine-tuning property suitable for designing and improving the performance of multijunction solar cell devices. In addition to MAPb(I_{1−x}Cl_{x})_{3}^{18,56} and MAPb(I_{1−x}Br_{x})_{3},^{28,36,37,57} FAPb(I_{1−x}Br_{x})_{3}^{19,22,29} has also been successfully synthesized and demonstrated in solar cells. To date mixed hybrid perovskites containing the FAPb(I_{1−x}Br_{x})_{3} system have achieved the highest efficiency so far in perovskite based solar cells.^{17}

Although the FAPb(I_{1−x}Br_{x})_{3} alloy has been experimentally studied, a thorough in-depth theoretical description and understanding of the structural parameters, electronic density of states (DOS), band gaps, carrier excitations and exciton binding energies has not been fully addressed. In the present work, the FAPb(I_{1−x}Br_{x})_{3} system is studied with density functional theory (DFT), which proves to be a powerful tool for modeling and simulations of materials.^{65,66} The commonly used generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE)^{67,68} provides structural lattice parameters close to the experimental values, but it is well known that DFT fails to describe the excited state properties such as quasiparticle (QP) band gaps and excitonic properties.^{69,70} An additional complication in relativistic HOIPs is the need to include spin–orbit coupling (SOC) effects, which causes further reduction of the bandgaps.^{46,71–77} Moreover, the use of hybrid functionals like HSE06 including SOC does not satisfactorily improve the issue of band gap underestimation.^{76} The state-of-the-art approach to improve the band gap is the GW method,^{78,79} which provides a good approximation in evaluating the self-energy operator as contraction of the one-body Green's function (G) with the screened Coulomb interaction (W). The GW method has been widely used for many systems ranging from elemental semiconductors^{69,70,80} to HOIPs,^{46,71,72,74–77,81} yielding band gaps in better agreement with experiments.

Since during the optical excitation the interactions of the electron–hole pairs create excitons which strongly couple to the incoming photons, this leads to substantial modification of the optical spectrum both below and above the quasiparticle band gaps. To explore the excitonic peaks in the optical spectra, it is necessary to account for electron–hole pair interactions, which is normally done by solving the Bethe–Salpeter equation (BSE) on top of GW calculations.^{46,82–85}

By means of density functional theory and many-body methods with the inclusion of spin–orbit coupling, we have performed a systematic first-principles study of the structural, electronic, and optical properties of the FAPb(I_{1−x}Br_{x})_{3} alloy with 0 ≤ x ≤ 1. Our results show that as the Br concentration increases, the volume of the FAPb(I_{1−x}Br_{x})_{3} alloy decreases, the band gap increases from 1.47 eV for pure α-FAPbI_{3} to 2.20 eV for Br-rich α-FAPbBr_{3}, and the excitonic peaks are blue shifted. Our results further reveal strong Br concentration dependence of the exciton binding energy (varies from 74 to 112 meV) and the carrier effective masses as well as the high frequency dielectric constants. Our findings open the way to fine-tune the carrier transport properties of the material by chemical substitution and provide important insights to improve short circuit currents and power conversion efficiencies in solar cell devices.

For the accurate electronic structure, in particular the band gap, we carried out delicate one-shot G_{0}W_{0} calculations using Γ-centered 4 × 4 × 2 k-point sampling with 2200 empty bands, a dielectric cutoff energy of 150 eV and 128 points on the frequency grid. The electron–hole interactions were accounted for by solving the BSE for the polarizability. 16 occupied and unoccupied QP energies were employed in constructing the BSE Hamiltonian and the screened interactions W were from the preceding G_{0}W_{0} calculations. The broadening parameter used in evaluation of the polarizability was chosen to be 0.1 eV. For the detailed GW + BSE methodology that the VASP code follows, we refer the readers to ref. 96.

It is known that to obtain converged optical spectra and exciton binding energies, many k points are normally needed.^{46,82} However, for the relatively large systems studied here, this is already prohibitive for the G_{0}W_{0} calculations. To address this issue, we followed the strategy proposed in ref. 46, where the optical spectra and exciton binding energies were calculated by a simplified approximation of BSE [termed as model BSE (mBSE)]. In mBSE calculations, the QP energies from G_{0}W_{0} calculations are approximated by the PBE one-electron energies with a scissor operator on the unoccupied orbitals such that the PBE one-electron gap matches the G_{0}W_{0} calculated band gap. This turns out to be a good approximation for the systems studied here (see good agreement between G_{0}W_{0} + SOC and PBE + SCISSOR + SOC band structures in Fig. 4). In addition, the expensive calculations of the dielectric function matrix from G_{0}W_{0} are approximated by a simple analytical model dielectric function within the diagonal element approximation:^{46,82,96,97}

ε^{−1}(|G|) = 1 − (1 − ε^{−1}_{∞})e^{−|G|2/4μ2},
| (1) |

By performing PBE + SCISSOR + mBSE calculations, we are able to check the k-point convergence by adopting a dense k-point mesh up to 18 × 18 × 9. Our test calculations indicate that the spectra and exciton binding energies are well converged on a 16 × 16 × 8 k-point mesh with an accuracy of about 3 meV in the exciton binding energies [see Fig. 9(b) and (c)]. Therefore, the 16 × 16 × 8 k-point mesh was used for all mBSE calculations. The exciton binding energies were calculated by the energy difference between the G_{0}W_{0} band gap and the first bright mBSE eigenvalue. Furthermore, for comparison purposes, the exciton binding energies were also estimated by using the Wannier Mott model:^{98} E_{b} ≈ 13.6m_{r}*/ε_{∞}^{2}, where m_{r}* is the reduced mass .

r_{eff} = xr_{Br} + (1 − x)r_{I},
| (2) |

(3) |

(4) |

Generally, for ABX_{3} perovskite-like structures, the tolerance factor is in the range 0.80 ≤ t ≤ 1.06 and the octahedral factor lies between 0.442 and 0.895.^{99,102} The calculated effective tolerance factor for the FAPb(I_{1−x}Br_{x})_{3} system ranges from 0.987 to 1.008, thus confirming the perovskite structure as summarized in Table 1. The octahedral factor values also affirm the stable perovskite structure as they lie between 0.541 and 0.607 for the FAPb(I_{1−x}Br_{x})_{3} composition.

Br fraction x | 0 | 0.17 | 0.33 | 0.50 | 0.67 | 0.83 | 1 |
---|---|---|---|---|---|---|---|

Goldsmith tolerance factor t | 0.987 | 0.990 | 0.993 | 0.997 | 1.000 | 1.004 | 1.008 |

Octahedral factor λ | 0.540 | 0.550 | 0.561 | 0.572 | 0.583 | 0.594 | 0.607 |

Table 2 summarizes the calculated lattice parameters of the FAPb(I_{1−x}Br_{x})_{3} system for x = 0, 0.17, 0.33, 0.50, 0.67, 0.83, and 1. The calculated averaged ground state lattice parameters for FAPbI_{3} (a = 6.38 Å) and FAPbBr_{3} (a = 6.01 Å) using the PBE functional are in good agreement with experimental data, i.e., 6.36 Å and 5.99 Å for FAPbI_{3} and FAPbBr_{3}, respectively,^{29,39,103,104} and the relaxed structure of α-FAPbI_{3} is shown in Fig. 1. In the full relaxation of the structures, the deviation from tetragonal symmetry is caused by the steric effect of the FA^{+} cation.

Composition x | a (Å) | b (Å) | c (Å) | α (°) | β (°) | γ (°) |
---|---|---|---|---|---|---|

0 | 6.33 | 6.51 | 12.66 | 90.00 | 92.80 | 90.00 |

0.17 | 6.31 | 6.50 | 12.32 | 90.00 | 93.05 | 90.00 |

0.33 | 6.29 | 6.52 | 11.99 | 90.00 | 93.28 | 90.00 |

0.50 | 6.30 | 6.37 | 12.00 | 90.00 | 93.34 | 90.00 |

0.67 | 6.30 | 6.17 | 12.09 | 90.00 | 93.49 | 90.00 |

0.83 | 6.01 | 6.17 | 12.27 | 90.00 | 94.51 | 90.00 |

1 | 5.96 | 6.16 | 11.92 | 90.00 | 94.90 | 90.00 |

Fig. 2 shows the overall decrease in volume of the 1 × 1 × 2 supercell from 520.9 Å^{3} to 454.2 Å^{3} as the Br content increases, which is consistent with the decreasing experimental lattice constants.^{19,29} Also, the Pb–X (where X = I and Br) bond length reduces from 3.26 Å to 2.96 Å, which can be attributed to the smaller ionic radius of the Br atom, which influences the interplanar geometry. Moreover, the electronic charge distribution is much stronger around the Br atom due to its higher electronegativity value (2.96) as compared to I (2.66). Thus, the heavy Pb atom will interact strongly with the Br atom due to the high electronegativity difference, which leads to the reduction in the bond length.^{105}

Br fraction x | 0 | 0.17 | 0.33 | 0.50 | 0.67 | 0.83 | 1 | |
---|---|---|---|---|---|---|---|---|

Bandgap E_{g} (eV) |
PBE | 1.43 | 1.50 | 1.52 | 1.62 | 1.72 | 1.79 | 1.80 |

PBE + SOC | 0.41 | 0.52 | 0.57 | 0.61 | 0.71 | 0.82 | 0.87 | |

GW + SOC | 1.47 | 1.63 | 1.72 | 1.79 | 1.94 | 2.11 | 2.20 | |

Expt. | 1.47,^{29,106} 1.48^{19,30,58} |
— | — | — | — | — | 2.22,^{29} 2.23,^{19} 2.26^{104} |

The G_{0}W_{0} + SOC calculated band gaps for various Br fractions are plotted in Fig. 3. Such a linear trend in the band gaps with the x composition in the alloy can be expressed by the quadratic equation known as Vegard's law:^{107,108}

E_{g}[FAPb(Br_{x}I_{1−x})_{3}] = E_{g}[FAPbI_{3}] + (E_{g}[FAPbBr_{3}] − E_{g}[FAPbI_{3}] − b)x + bx^{2},
| (5) |

E_{g}(x) = 1.47 + 0.71x + 0.02x^{2},
| (6) |

Fig. 3 Linear variation in the G_{0}W_{0} + SOC calculated band gaps with respect to the Br concentration. |

The G_{0}W_{0} + SOC derived QP band structures along the high-symmetry points Γ(0, 0, 0), M(0.5, 0.5, 0) and X(0.5, 0, 0) for each FAPb(1_{1−x}Br_{x})_{3} configuration are shown in Fig. 4. The band structures are plotted by using the Wannier interpolation scheme,^{109,110} in which the Wannier orbitals are constructed on a Γ-centered 4 × 4 × 2 k-point mesh and all the relevant orbitals are included for the initial representation of the Kohn Sham states. With the 1 × 1 × 2 supercell, both the valence band maximum (VBM) and conduction band minimum (CBM) fold to the M-point and the band gaps are found to be direct in nature. Fractional deviations from the M-point are observed for the VBM and CBM due to the presence of Rashba–Dresselhaus splitting.^{111–113} The dispersion of both the conduction band and valence band slightly increases with the Br content, leading to different effective masses for electrons and holes near the CBM and VBM. Such dispersions can be understood by the enhanced orbital interaction with decreasing structural volume.

Fig. 4 G_{0}W_{0} + SOC calculated QP band structures obtained by Wannier interpolation^{109,110} (black lines) superposed by the PBE + SOC band structures with a scissor operator to match the G_{0}W_{0} band gap (red lines). The Fermi level is set to zero. |

To study the carrier transport properties for the FAPb(1_{1−x}Br_{x})_{3} alloy, we calculated the effective masses m* for electrons and holes around the CBM and VBM, respectively, by fitting the dispersion relation on the basis of the parabolic approximation

(7) |

Br fraction x | Effective mass (M–Γ) | Effective mass (M–X) | Average | m_{r}* |
|||
---|---|---|---|---|---|---|---|

m_{e}*/m_{0} |
m_{h}*/m_{0} |
m_{e}*/m_{0} |
m_{h}*/m_{0} |
m_{e}*/m_{0} |
m_{h}*/m_{0} |
||

0 | 0.212 | 0.267 | 0.225 | 0.279 | 0.218 | 0.273 | 0.121 |

0.17 | 0.227 | 0.288 | 0.234 | 0.298 | 0.231 | 0.293 | 0.129 |

0.33 | 0.232 | 0.279 | 0.228 | 0.287 | 0.230 | 0.283 | 0.127 |

0.50 | 0.253 | 0.296 | 0.255 | 0.304 | 0.254 | 0.300 | 0.138 |

0.67 | 0.275 | 0.336 | 0.277 | 0.372 | 0.276 | 0.354 | 0.155 |

0.83 | 0.369 | 0.467 | 0.364 | 0.431 | 0.367 | 0.449 | 0.202 |

1 | 0.351 | 0.412 | 0.353 | 0.404 | 0.352 | 0.408 | 0.189 |

Fig. 5 Calculated m_{e}*/m_{0} and m_{h}*/m_{0} ratios as a function of the Br content, where m_{e}*(m_{h}*) is the effective electron (hole) mass and m_{0} is the mass of a free electron. |

Fig. 6 depicts the G_{0}W_{0} + SOC calculated partial density of states (PDOS) to reveal the possible origin of the band gap variations. One can see that the electronic states near the band gap are mostly governed by the orbital overlap of the BX_{6} octahedron. The organic part does not contribute to the Fermi energy but acts as a charge compensating center, because the organic molecule weakly interacts with the inorganic part via possible hydrogen bonding through the organic cation groups. Thus, for the sake of clarity, around the band gap only the contributions from the orbitals (s and p) of Pb, I and Br atoms are presented and analyzed. From the PDOS, the valence band is mainly composed of X-p orbitals, mixed in minor fractions with B-s orbitals, while the conduction band is mainly contributed by B-p orbitals, partly hybridized with X-s states. The gradual replacement of I with Br causes the Br-4p orbital to dominate at the VBM, which tends to strongly hybridize with Pb-s states as compared to I-p states. This pushes the CBM towards the high energy region and thus increases the band gap. Such orbital behavior has also been observed in a DFT study of the MAPb(1_{1−x}Br_{x})_{3} system.^{57}

Fig. 6 G_{0}W_{0} + SOC calculated partial density of states for the FAPb(I_{1−x}Br_{x})_{3} system. The Fermi level is set to zero. |

Previously, it has been revealed that the polaronic effects in HOIPs contribute to excitonic features due to the interacting longitudinal optical phonons with electronic states of the carriers, thereby forming polarons, which influences the dielectric behavior and exciton binding energies.^{38,46,48,114} However, polaronic effects are not considered here, since they are beyond the scope of this work. Fig. 7 shows the imaginary part of the dielectric function Im[ε(ω)] with increasing Br composition along with the corresponding optical transition oscillator strengths. One can see that Im[ε(ω)] for all compositions shows a typical three peak feature in the considered energy range: two peaks are above the G_{0}W_{0} band gap and one peak is below the G_{0}W_{0} band gap, which is an exciton peak resulting from the electron–hole interactions upon photoexcitation. Our PBE + SCISSOR + mBSE calculated spectra for x = 0, x = 0.67, and x = 1 are in good agreement with available experimentally measured data,^{115–118} in particular the first two peaks. The spectrum of x = 0 from M. Kato et al.^{116} is largely blue-shifted for the first two peaks as compared to that of T. J. Whitcher et al.^{115} and P. F. Ndion et al.^{117} as well as our simulated one. As the Br fraction increases from x = 0 to x = 1, the optical spectra are systematically blue-shifted along with the gradual reduction in the overall amplitude of oscillator strengths. This trend is consistent with the increasing band gaps and can be attributed to variation in the electronic charge distribution of X-p states with changing halide component as evident from the PDOS of Fig. 6.

Fig. 7 PBE + SCISSOR + mBSE calculated imaginary part Im[ε(ω)] of the dielectric function of the FAPb(I_{1−x}Br_{x})_{3} alloy. The corresponding G_{0}W_{0} derived fundamental gap E_{g}(G_{0}W_{0}) and optical transition oscillator strengths are represented by dashed lines and brown histograms, respectively. The available experimental data on FAPbI_{3},^{115–117} FAPbIBr_{2}^{117} and FAPbBr_{3}^{118} are shown for comparison. Note that the Im[ε(ω)] data shown here from ref. 117 are indirectly calculated from the experimentally measured refractive index and extinction coefficient. |

Since changes in the halide components affect the bound state of an electron–hole pair, we evaluated the exciton binding energies E_{b} within Wannier–Mott theory by using the effective mass approximation as well as via mBSE. The resulting exciton binding energies are summarized in Table 5 and depicted in Fig. 8. One can observe that the mBSE calculated exciton binding energies almost show linear behavior as the Br concentration increases and this trend is in general captured by Wannier–Mott theory, though the values deviate. The mBSE calculated E_{b} for α-FAPbI_{3} (74 meV) is significantly large as compared to the experimental value of 2.4 meV,^{119} whereas for the pure α-FAPbBr_{3} the calculated E_{b} (112 meV) is in reasonable agreement with the experimental value of 170 meV.^{103} For the latter, Wannier–Mott theory yields a better value of E_{b} towards the experimental value. From Wannier–Mott theory, we can infer that the overall increase in E_{b} with increasing Br concentration in FAPb(I_{1−x}Br_{x})_{3} is both affected by increasing m_{r}* and decreasing ε_{∞}. The smaller value of E_{b} in the I-rich configuration requires smaller energy to dissociate the electron–hole pair, which in turn leads to high optical absorption and high PCE as compared to the Br-rich material.

Br fraction x | 0 | 0.17 | 0.33 | 0.50 | 0.67 | 0.83 | 1 |
---|---|---|---|---|---|---|---|

High frequency dielectric constant ε_{∞} |
5.320 | 4.929 | 4.746 | 4.613 | 4.447 | 4.266 | 4.173 |

E_{b} (Wannier Mott model) |
58 | 72 | 77 | 88 | 107 | 151 | 149 |

E_{b} (mBSE) |
74 | 85 | 90 | 95 | 99 | 108 | 112 |

Br fraction x | 0 | 0.17 | 0.33 | 0.50 | 0.67 | 0.83 | 1 |
---|---|---|---|---|---|---|---|

ε_{∞} |
5.320 | 4.929 | 4.746 | 4.613 | 4.447 | 4.266 | 4.173 |

μ (Å^{−1}) |
1.055 | 1.060 | 1.070 | 1.073 | 1.081 | 1.095 | 1.109 |

SCISSOR (eV) | 1.062 | 1.095 | 1.133 | 1.171 | 1.233 | 1.283 | 1.319 |

Fig. 9(a) shows a comparison of the imaginary part of the dielectric function calculated from G_{0}W_{0} + BSE and PBE + SCISSOR + mBSE on a 4 × 4 × 2 k-point mesh. One can see that PBE + SCISSOR + mBSE in general reproduces very well the spectra obtained from G_{0}W_{0} + BSE, in particular the first bright peak, suggesting that PBE + SCISSOR + mBSE is a suitable method to describe the optical spectra for FAPbI_{3} with an accuracy that is comparable with the more demanding G_{0}W_{0} + BSE.

Fig. 9(b) and (c) display the convergence of the optical spectra and exciton biding energies calculated from PBE + SCISSOR + mBSE with respect to the number of k points. Clearly and expectedly, the convergence is very slow. Our calculations indicate that the spectra and exciton binding energies are converged with a 16 × 16 × 8 k-point mesh, with an accuracy of about 3 meV in the exciton binding energies.

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## Footnote |

† These authors contributed equally to this work. |

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