Like MAPbX₃, MASnX₃ also exhibits a very rich phase diagram as a function of temperature, i.e., the crystal structure goes from cubic, tetragonal, orthorhombic, and monoclinic to triclinic class by cooling.^{[17,23,27–29]} Hao et al. reported a tetragonal CH₃NH₃SnI₃ phase at room temperature.^[2] It was found that CH₃NH₃SnI₃ compound crystallizes into monoclinic phase at low temperature experimentally. However, the unit cell of monoclinic phase deviates from the orthorhombic crystal class very slightly.^[28,29] The complicated phase transition behavior of CH₃NH₃SnI₃ is attributed to the disordering of (CH₃NH₃)⁺ cations and the distortions of (SnX₃)⁻ octahedra at finite temperature. The geometrical structure of the hybrid compound is illustrated in Fig. 1. In perovskite CH₃NH₃SnI₃, each Sn atom is surrounded by six I atoms, with four I atoms in the equatorial direction and two I atoms in the apical direction. The dipolar organic methylammonium ions are embedded in the octahedron SnI₆ cages. The previous work^[30] revealed that arrangements along different directions play an important role in the stability and electronic structure. Thus the wake interaction between inorganic and organic molecule must be taken into account as performed in VASP with the nonlocal vdW density functional. We first estimate the equilibrum lattice constants of the hybrid perovskite compound by using DFT + D2 calculations in the frame of the PBE–GGA density functional, and the results are listed in Table 1, which are consistent with previous results obtained by Feng and Xiao.^[12] Meanwhile, the calculated averaged bond length and the bond angles of Sn–I–Sn along the equatorial and apical direction are shown in Table 2, which are in agreement with previous results.^[11] From Table 1, the calculated bond lengths of Sn–I for orthorhombic phase in CH₃NH₃SnI₃ perovskite are about 3.09 Å and 3.12 Å–3.13 Å which are consistent with the experimental data.^[12] The corresponding Sn–I–Sn angles are 89.77° and 94.86° in the equatorial and the apical direction respectively. These results reveal that CH₃NH₃SnI₃ has more distorted structure than CH₃NH₃PbI₃, which may be the origin of ferroelectricity.

Table 1.

Table 1.

Table 1. Lattice constants of orthorhombic phase in perovskite CH3NH3SnI3 compared with Feng and Xiao’s work.[12] . Compound This work/Å Feng and Xiao’s work a b c a b c orthorhombic 8.56 12.41 8.43 8.556 12.428 8.326

Table 1. Lattice constants of orthorhombic phase in perovskite CH3NH3SnI3 compared with Feng and Xiao’s work.[12] .

For orthorhombic phase in perovskite CH₃NH₃SnI₃, the energy band structure is shown in Fig. 2, and the calculated energy band gap is 1.27 eV by using HSE06 functional method as shown in Table 2. Our calculations reveal that the CH₃NH₃SnI₃ is a direct gap semiconductor, and the calculated results are consistent with those reported in previous research.^[31] Figures 2(a) and 2(b) show the band structures of the orthorhombic phase in CH₃NH₃SnI₃ calculated by standard DFT and HSE06 functional methods respectively. For this phase, they all exhibit a direct band gap at the Γ point of the Brillouin zone. The difference could be seen easily from Figs. 2(a) and 2(b), the calculated band gaps are 0.74 eV by standard density functions theory and 1.27 eV by HSE06 method, respectively, which are listed in Table 2.

Fig. 2.

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	Fig. 2. Band structures for orthorhombic phase in CH3NH3SnI3 perovskites by standard DFT and HSE06 calculations, respectively. The Brillouin zone is also shown.

Table 2.

Table 2.

Table 2. Calculated band gaps of DFT-D2 and HSE06, the bond lengths of Sn–I and the bond angles of Sn–I–Sn along the equatorial and apical direction. . Band gap/eV Sn–I–Sn bond angel/(°) Bond lengths of Sn–I/Å DFT-D2 HSE06 equatorial apical Cal. cell Exp. cell 0.74 1.27 89.78 94.86 3.09-3.14 3.09-3.19

Table 2. Calculated band gaps of DFT-D2 and HSE06, the bond lengths of Sn–I and the bond angles of Sn–I–Sn along the equatorial and apical direction. .

As is well known, the standard DFT usually underestimates the band gap of semiconductors due to the inherent lack of derivative discontinuity and the self-correlation error of electrons.^[32] Therefore, results (1.27 eV) obtained by the hybrid functional within the framework of density functional theory (DFT-D2) method are closer to the experimental data (1.3 eV).^[9] We ascribe the difference between the calculated data and experimental data to the following reasons: (i) the experimental materials are flawed, while the calculated structure is considered as an ideal model; (ii) though HSE06 calculations improve the exchange action, the improved exchange effect is still from a semi-empirical theory. According to the calculated band structure, the carrier effective mass values are shown in Table 3. The effective mass values of orthorhombic phase in perovskite CH₃NH₃SnI₃ at Γ along three principal directions, i.e., [1 0 0], [0 1 0], and [0 0 1] direction are evaluated respectively by using Eq. (1) as the effective mass values of hole and electron are associated with the slope of the band edge. The calculated results compared with those of Feng’s work are listed in Table 3. From the Table 3, it is clearly seen that carrier effective mass values along three principal directions are anisotropic, which are consistent with Feng and Xiao’s^[12] results. Owing to our calculated band gap more closer to the experimental data, we believe that the effective mass values of electrons and holes are considered as more reliable results. Also, the results reveal that the orthorhombic phase in CH₃NH₃SnI₃ shows that the hole effective mass is smaller than the electron’s.

Table 3.

Table 3.

Table 3. Calculated effective mass values of hole and electron of CH3NH3SnI3 for orthorhombic phase compared with the results of CH3NH3SnI3 (orthorhombic phase) calculated by Feng and Xiao’s work.[12] . Direction [11] [11] Orthorhombic x 1.77 3.74 0.35 0.59 y 0.15 0.81 0.11 1.14 z 1.75 5.77 0.26 0.53

Table 3. Calculated effective mass values of hole and electron of CH3NH3SnI3 for orthorhombic phase compared with the results of CH3NH3SnI3 (orthorhombic phase) calculated by Feng and Xiao’s work.[12] .

The density of states (DOS), and the projected density of states (PDOS) of CH₃NH₃SnI₃ are shown in Fig. 3. The calculated results show that the valence band maximum (VBM) for orthorhombic phase in CH₃NH₃SnI₃ is mainly due to the 5p orbitals of the I and partly due to the 5s orbital of Sn, while in the conduction band minimum (CBM), the main contribution is from the 5p orbitals of Sn and little contribution is from the 5p orbitals of I. Also, it is noted that the organic cations (CH₃NH₃)⁺ are far from the Fermi level. Therefore, the emergency of the photocurrent is mainly generated by the electron transition from 5p orbitals of I and 5s orbital of Sn to the 5p orbitals of Sn. A similar structure such as CH₃NH₃PbI₃ also shows that the strong hybridization of s–p orbital in valence band maximum (VBM) plays an important role in the structure stability and the small effective mass.^[24] More detailed analyses of the chemical bonding mechanisms of those hybrid organic–inorganic pervoksite materials can be found in previous publications.^[33–36]

Fig. 3.

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	Fig. 3. The calculated PDOS from top to bottom in CH3NH3SnI3 perovskite.

In order to further illustrate our results, the calculated projected DOS in valence band maximum and conduction band minimum are shown in Fig. 4 to explain the electron and hole carrier effective mass values along three principal directions. Here, we present the theoretical discovery of a new relationship between the carrier effective mass anisotropy and the electronic DOS anisotropy along three principal directions. As mentioned above, the valence band maximum (VBM) is mainly due to the I-p states and the conduction band minimum (CBM) is mainly due to the Sn-p states. Thus we just discuss the electron occupation probabilities of I-p states in valence band maximum and Sn-p states in conduction band minimum, respectively, as shown in Fig. 4. The above Table 2 reveals that the hole effective mass values of orthorhombic phase in CH₃NH₃SnI₃ along the x, y, and z directions are 0.35, 0.11, and 0.27 respectively. From Fig. 4(a), increasing sequence of hole effective mass values along different directions is which is consistent with the decreasing sequence of electron occupation probabilities in the p orbital of I along the x, y, and z directions p_x < p_z < p_y in valence band maximum. Similarly the calculated values of electron effective mass , and are 1.77, 0.15, and 1.75 respectively. Increasing sequence of electron effective mass values along different directions is which is also consistent with the decreasing sequence of electron occupation probabilities in the p orbital of Sn along the x, y, and z directions as follows: p_y > p_z > p_x in the bottom conduction-band depicted in Fig. 4(b). There might be such a relationship that the higher the occupation probabilities in the p orbital along the x, y, and z directions, the smaller the effective mass is, hence the higher the carrier mobility.

Fig. 4.

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	Fig. 4. Detailed PDOS from top to bottom for orthorhombic phase in CH3NH3SnI3 perovskite. The left panel shows the VBM, and the right panel shows the CBM.

In order to evaluate the transfer efficiency of the carrier in perovskite CH₃NH₃SnI₃, we systematically investigate the carrier mobility of CH₃NH₃SnI₃. From experimental measurement, the system of CH₃NH₃SnI₃ was reported to have high electron mobility (μ_e ∼ 2320 cm²· V⁻¹·s⁻¹),^[8] which is higher than CH₃NH₃PbI₃ (μ_e ∼ 66 cm²· V⁻¹·s⁻¹).^[8] The charge mobility is calculated according to Eq. (2). It should be noticed that reliable band structure is highly essential for the calculations of carrier mobilities. All the calculated results are listed in Table 4. Figure S1 (see Appendix A) shows the energy changes of VBM and CBM band edge of CH₃NH₃SnI₃ with the applied strain along the x, y, and z directions, respectively. By simulating the linear relation between band edge energy (E_edge) and strain (δ) exerted in the x, y, and z directions, we can obtain the deformation potential (DP) constant E₁. The DP constant is a characterization of the coupling strength of the electron or hole to the acoustic phonon. It is found that the DP constant E₁ values of electrons and holes are comparable in CH₃NH₃SnI₃ perovskite (see Table 4). The elastic matrix constant along the y direction is 637 kbar (1 bar = 10⁵ Pa), which is about twice that along the z direction, and the effective mass values for the hole and electron are the minimal (along z direction) among three directions. Therefore, it is clearly seen that the carrier mobility along the y direction is highest because of high DP and small effective masses. From Eq. (2), we can see the main difference between the electron and hole mobilities in CH₃NH₃SnI₃ derives from the small effective masses, rather than the DP constant E₁. It can be clearly seen that CH₃NH₃SnI₃ perovskite shows high carrier mobilities of 14470 cm²· V⁻¹·S⁻¹ and 5445 cm²· V⁻¹·S⁻¹ for holes and electrons, respectively, which are rather higher than the experimental data (2230 cm²· V⁻¹·S⁻¹).^[8] The difference can be attributed to the following reasons: (i) there are inevitably impurities and defects in the experimental material; (ii) the phonons have little influence on the charge transport properties at low temperatures; (iii) the Sn^{2 +} ion can be oxidized into Sn^{4 +} state easily, which may underestimate the carrier mobilities in the experimental measurement. The carrier mobility (17 and 545, 11 and 1663 cm²· V⁻¹·S⁻¹) for holes and electrons in the x and z direction are two or three orders lower than that along the y direction, respectively, and so is the anisotropic carrier mobility. Our calculated carrier mobility may lay the solid foundation and the theoretical support for the application of the device.

Table 4.

Table 4.

Table 4. Calculated values of deformation potentials E1c and E1v for electron and hole, respectively, elastic constant (cii) , and carrier mobilities μe and μh for electron and hole, respectively at the Γ point of the Brillouin zone. Three directions are considered for the orthorhombic phase. . Direction E1c/eV E1v/eV Cii/kbar μe/103 cm2· V−1·S−1 μe/103 cm2· V−1·S−1 Orthorhombic x −6.25 −8.23 432 0.017 0.545 y −9.13 −8.26 637 5.445 14.47 z −7.39 −6.13 382 0.011 1.663

Table 4. Calculated values of deformation potentials E1c and E1v for electron and hole, respectively, elastic constant (cii) , and carrier mobilities μe and μh for electron and hole, respectively at the Γ point of the Brillouin zone. Three directions are considered for the orthorhombic phase. .