3.1. Structural properties of FABI3 (M = Sn, Ge or Pb)
Since these perovskites reported to crystallize in cubic structure [27, 28].The crystal structures of FABI3 in these computations were supposed to be the cubic and have the space group: Pm3m, which is composed with 12 atoms with the geometry illustrated in Fig. 1. In the unit cell, The Formamidinium, CH (NH2)2 cation is positioned at the center of the unit cell and the B atom is presented at the cubic corners, while the I anions reside in the face-centered positions as illustrated in Fig. 1.
The optimization results of lattice parameter for FABI3 (B = Sn, Ge or Pb) are presented in Table 1, we plot in Fig. 2 the energy of FABI3 (M = Sn, Ge or Pb) as a function of the lattice parameter,In Fig. 2, we illustrate the energy of FABI3 (M = Ge) as a function of the lattice parameter. Indeed, Fig. 2 shows that the optimized value of lattice parameter set at ‘a’ = 6.3Å, for FAGeI3. While, It is seen that the minimum value of the total energy (-322,56152763Ry) set at ‘a’ =6.35Å, for FASnI3, which is the optimum lattice parameter for this material. While for the perovskite FAPbI3, the optimum lattice parameter is located at a = 6.5Å.
Our investigated optimized lattice parameter of the unit cells for the considered perovskite FASnI3 by the PBE-GGA approximation, have been found acceptable when compared with the experimental and theoretical results, Table 1, summarize our results of the calculated lattice parameter and the available experimental results existing in literature of the perovskite FASnI3.It is worth to note that GGA-PBE functional overestimates the experimental lattice parameter.
Our calculated lattices parameters are also demonstrated to reduce with the change of Pb by Sn, and Sn by Ge, respectively, this alteration in structural characteristics of this perovskite can be beneficial to tune its electronics properties. In addition, by comparing the energies of the tow perovskites materials (-322,56152Ry) for FASnI3 and (-171,28411Ry) for FAGeI3, it is clear that Sn-based material is the most stable Lead-free photovoltaic material, which is in accordance with previous results.
The impact of Lattice parameter on the band gap FABI3 (B = Sn, Ge or Pb)
To investigate the impact of varying lattice parameter ‘a’ on the energy of band gap of FABI3 (B = Sn, Ge or Pb). We investigate and discuss the impact of lattice parameter on the band structure of the perovskite materials FABI3 (B = Sn, Ge or Pb). We illustrate the different band structures, the DOS and PDOS of each perovskite for various lattice parameters. We provide our findings for the material FAGeI3, when choosing the optimized lattice parameter: ‘a’ = 6.2 Å, 6.3 Å and 6.5 Å. For FASnI3, we fixed the lattice parameter at: ‘a’ = 6.2Å, 6.35Å and 6.5Å to study the different band structures and DOS (density of states) of these perovskites. On the other hand, for the FAPbI3, we choose the values of the lattice parameter a = 6.4Å, a = 6.5Å and 6.6Å.
3.2. Investigation of FASnI3
To inspect the impact of the lattice parameter on the band gap of the FASnI3, we illustrate in Fig. 3 (a), the band structure of FASnI3 for the lattice parameter ‘a’ = 6.2Å. This figure illustrate that FASnI3 has a semi-conductor behavior. In addition, this figure demonstrates that this perovskite has a direct band gap, at R-point, being similar to the conventional perovskites [36, 37]. Moreover, the band gap for the lattice Parameter ‘a’ = 6.2Å is around 1.2eV. While, the DOS and PDOS of the FASnI3 is presented in Fig. 3 (b) for ‘a’ = 6.2Å. The same value of band gap energy found in Fig. 3 (a) is validated using the DOS for FASnI3.
When choosing the optimized value of the lattice parameter, ‘a’ = 6.35 Å, we provide the band structure in Fig. 3(a), which demonstrate that the band gap is about 1.36eV. this value is in reasonable agreement with the available experimental results of FASnI3 1.41eV [38]. The theoretical studies of other researchers found the band gap of FASnI3: 0.65eV in Ref. [8], when using GGA-PBE approach, and 0.702eV when applying the HSE06 approach in the same reference. 0.96eV in Ref. [39]. The calculated theoretical the band gaps energy under estimate the experimental values. Also, this figure shows a direct band gap located at R-point. As it is expected, In Fig. 3 (b) the DOS validates the band gap found in Fig. 3 (a). Moreover, the bottom of conduction band BCB is the combination of Sn orbitals and I orbitals, while the Top of valance band TVB is mainly generated by I orbitals. This obtained band gap of this perovskite makes this perovskite a potential option for the solar cells applications.
When the lattice parameter was fixed at ‘a’ = 6.5 Å, we provide in Fig. 3 (a) our findings, Indeed Fig. 5 (a), showing the band structure of FASnI3, it is clear that this material is a semi-conductor having a direct the band gap of 1.52eV. Moreover, the DOS and PDOS of this material is illustrated in Fig. 3 (b) corresponding the lattice parameter ‘a’ = 6.5 Å. Also, Fig. 3 (b) validates the findings of Fig. 3 (a) of a direct band gap at the R-point.
3.3. Investigation of FAGeI3
To investigate the impact of varying the value of the lattice parameter on the band gap value of the FAGeI3, we presented the obtained findings in Fig. 6, In fact, Fig. 5 (a) present band structure of FAGeI3 for the lattice parameter ‘a’ = 6.2 Å. This figure shows that FAGeI3 is a semi-conductor. In addition, this figure indicates that this solar perovskite has direct band gap energy at the R-point for this lattice parameter ‘a’ = 6.2Å, reaches the value 1.73eV. The DOS and PDOS of FAGeI3 is presented in Fig. 5 (b) for the lattice parameter ‘a’ = 6.2Å. The DOS approves the results of band gap showed in Fig. 5 (a).
When choosing the optimized lattice parameter, a = 6.3Å, we presented the obtained band structure in Fig. 5 (a) showing that a band gap of 1.72eV was obtained. This band gap match well the experimental band gap 2.29eV [40].the band gap energy value 1.04eV is obtained in Ref. [8], when using GGA-PBE, and the value 1.414eV in the same reference, when applying the HSE06 approach. While, Fig. 5 (b) provides the density of states confirms the findings found in Fig. 5 (a), with a direct band gap located in the R-point. In addition, the bottom of conduction band is generated by both Ge and I orbitals, while, the Top of valance band TVB is mainly created with I orbitals.
For the lattice parameter, a = 6.5Å, we present the obtained the band structure in Fig. 5 (a) for FAGeI3. This figure demonstrates that FAGeI3 is a semi-conductor with the direct band gap of 1.79eV. Indeed, the DOS and PDOS of this material is illustrated in Fig. 5 (b) for the fixed lattice parameter a = 6.5 Å. the DOS approves similar band gap energy of band structure found in Fig. 5 (a).
3.4. Investigation of FAPbI3
In order to shows the influence of the lattice parameter ‘a’ on the energy band gap of FAPbI3, we provide our obtained findings in Fig. 6 (a), In fact, Fig. 6 (a) shows band structure of FAPbI3 for the lattice parameter ‘a’ = 6.4Å. Figure 6 (a) demonstrates that FAPbI3 is a semi-conductor. In addition, the band structure indicates that this perovskite has direct band gap at the R-point for this lattice a = 6.4Å reaches the value 1.43eV. Furthermore, the total and partial density of states (DOS and PDOS) of FAPbI3 is presented in Fig. 6 (b) for a = 6.4Å. The same value of band gap energy found in Fig. 6 (a) is validated with this figure for FAPbI3. In addition, the Pb orbitals are dominant in the bottom of conduction band BCB. While, the Top of valance band TVB is mainly generated with the combination of Pb orbitals and I orbitals.
When fixing the optimized lattice parameter, a = 6.5Å, we provides the band structure in Fig. 6 (a) demonstrate that the band gap energy is about to 1.61eV, which is in best agreement with the experimental result 1.53eV in Ref. [41]. This obtained band gap value is suitable for the photovoltaic applications. The theoretical studies of other researchers obtained the band gap energy of FAPbI3: 1.57eV in Ref. [8], when using GGA-PBE, and 2.533eV when applying the HSE06 approach in the same reference. In addition, in Fig. 6 (b) we illustrate the DOS and PDOS of FAPbI3 for the lattice parameter a = 6.5Å, the DOS validates the band gap of band structure in Fig. 6 (a). Moreover, it is clear that the Bottom of conduction band BCB is mainly generated with Pb orbitals, while the Top of valance band TVB is mainly created by the combination of Pb orbitals and I orbitals.
For the value of lattice parameter, a = 6.6Å, we presents the obtained band structure of the material FAPbI3 in Fig. 6 (a) demonstrating that the band gap energy is about 1.79eV. In addition, Fig. 6 (a) illustrates that FAPbI3 has a band gap, located at R-point, being similar to the conventional perovskites. Indeed, in Fig. 6 (b), we presented the density of states confirms the findings of band structure in Fig. 6 (a), with a band gap at the R-point. In addition, it is clear that in general the Bottom of the conduction band BCB is created by the combination of Pb orbitals and I orbitals, while the top of valance band TVB is mainly generated by I orbitals.
To inspect the dependency of the band gap as a function of lattice parameter ‘a’, in fact, we obtained the total energy of FABI3 perovskite, when varying the lattice parameter a (Å), see Fig. 4. From Fig. 4, it is clear that the band gap value increases, when increasing the lattice parameter for the three perovskites: FASnI3, FAGeI3, and FAPbI3. In addition, it is also found that for fixed lattice parameter values lower than 6.4Å, the band gap increases, when we switch from one perovskite to another: FAPbI3 FASnI3, FAGeI3 respectively, as a result of change the Pb by Sn and Sn by Ge.
3.5. Effect of the spin orbit coupling (SOC) on band gap energy
In order to complete this study, we investigated the impact of the spin-orbit coupling on the electronic properties, precisely on the band gap of FABl3 (B = Sn, Ge or Pb), we provide the obtained results of the band gap energy value when including SOC. It is worth to note that the values of band gap energies are calculated for the optimized lattice parameters of each solar perovskite. Table 2 summarizes the obtained results of band gap of FABl3 (B = Sn, Ge or Pb) with and with no SOC. it is observed that the impact of adding the SOC is to reduce the band gap of perovskites: FASnI3, FAGeI3, and FAPbI3. such results of this reduction of band gap energy are in accordance with the previous results of literature [42]