3.2.1 Crystal Structure of MAPbI₃

MAPbI₃ crystal phases are categorized into (i) α phase (cubic), (ii) β phase (tetragonal), and γ phase (orthorhombic) [1–5]. Figure 3.2 explains the formation of the β and γ phases from the α phase MAPbI₃. The cubic structure in Figure 3.2 is identical to that of Figure 3.1, except that A is MA⁺. The tetragonal and orthorhombic phases are the most common non‐cubic perovskite structures, which are also characterized by a three‐dimensional network of corner‐sharing octahedra. In the tetragonal phase, only one tilting angle is different from zero, whereas all the tilting angles are nonzero in the orthorhombic phase. In the β phase, the Pb–I–Pb angle of the c‐axis is 180°, as in the cubic phase; however, the PbI₆ octahedra are off‐centered along the c‐axis and are rotated around the c‐axis. Moreover, the corner‐sharing octahedral BX₆ units in the β phase have a two‐story structure where neighboring octahedra are tilted inward and outward at each level (see also Figure 5.4c).

In the case of MAPbI₃, the room‐temperature phase is the β phase and the α phase exists only at high temperature (T > 327 K). At low temperatures, the orthorhombic phase appears. These phases exhibit a rather sharp transition at the corresponding phase transition temperatures (see Section 3.4).

Figure 3.3 shows the top view of the α, β, and γ phase crystals of MAPbI₃. The dotted lines in Figure 3.3 represent the unit cells of each structure, and the crystallographic orientations of each phase are indicated in the inset. The configurations of MA⁺ depicted in Figure 3.3 correspond to those obtained from density functional theory (DFT) calculations (see Section 5.3). In the α and β phase crystals, MA⁺ orientation is not fixed and its C–N axis is reorienting rapidly. However, these MA⁺ cations cannot be rotated freely due to the hydrogen bonding interaction between NH and I (i.e. NH⋯I), whereas, in the γ phase, the dynamic motion of MA⁺ is prohibited and MA⁺ is fully ordered (Section 3.5). In the Figures 3.3, the Pb–I–Pb angle in the α phase is 180°. In the DFT calculation results, the Pb–I–Pb bond angle varies with the size of the A‐site cation and the bond angle approaches 180° as the cation size becomes larger (Figure 5.14).

Figure 3.4 shows the X‐ray diffraction (XRD) spectra calculated for the different phases of MAPbI₃. For the XRD calculations, the crystal structures determined experimentally are used (see Section 3.2.2). The XRD spectrum of the α phase is characterized by a strong (100) peak at 2θ = 14.0°, which originates from the perovskite cage structure. In the tetragonal phase, the intense (100) peak in the α phase is split into (002) and (110). As indicated in Figure 3.3, the unit cell size of the β phase is larger than that of the α phase and the [100] direction of the α phase corresponds to the [110] direction in the β phase. For the c‐axis direction, the α and β phases have the same structure; however, because the β phase has a two‐story structure, (001) in the α phase corresponds to (002) in the β phase. The weak (211) diffraction peak at 2θ = 23.5° is a characteristic signature of the β phase formation; this peak is absent in the α phase.

In Table 3.1, the XRD 2θ peak positions for the different MAPbI₃ phases are summarized. In Table 3.1, the diffraction peak positions of δ‐phase formamidinium lead iodide (HC(NH₂)₂PbI₃, FAPbI₃) and CsPbI₃, as well as important secondary phase materials described in Section 3.2.3, are included.

3.2.2 Lattice Parameters of Hybrid Perovskites

Table 3.2 summarizes the structures and lattice parameters of various hybrid perovskite materials reported in Refs. [1, 3, 4, 6–8]. As mentioned above, the crystal structure of MAPbI₃ changes with temperature; however, the reported transition temperatures differ slightly. For the β → α transition, temperatures in the range 327–333 K have been reported [1–5], whereas the γ → β transition has been reported to occur in the range 161–165 K [1, 2, 5]. In Table 3.2, the α transition temperature of 327 K and β transition temperature of 165 K reported by Weller et al. [5] are indicated.

In the α and β phases of MAPbI₃, the internal angles of the unit cells are α = β = γ = 90°. In the high temperature α phase, however, the experimental lattice parameters of a and b are slightly shorter than that of c and the crystal structure becomes pseudo‐cubic with the average value of 6.313 Å [3]. The unit cell size of the β phase is larger by a factor of than that of the α phase (see Figure 3.3) and, therefore, a_β‐phase ∼  in Table 3.2. From the structure of Figure 3.2, we can also notice a relation of c_β‐phase ∼ 2c_α‐phase. Moreover, in the β phase, c/2 > a/. Accordingly, the lattice is expanded slightly along the c axis. This is the reason why the (100) XRD peak of the α phase splits into two peaks [i.e. (110) and (002)] in the β phase (see Figure 3.4). For the β phase, different lattice parameters of a = b = 8.874 Å and c = 12.670 Å have also been reported [4]. For the γ phase, the b axis is traditionally chosen as the z axis (see Figure 3.2), and thus c_β‐phase ∼ b_γ‐phase in Table 3.2.

Interestingly, MAPbBr₃ and MAPbCl₃ adopt the cubic α phase at room temperature. FAPbI₃ also shows a cubic structure at room temperature; however, this is a metastable phase, and its crystal structure gradually changes into the hexagonal δ phase [6] (see Section 3.2.3). The mixed‐cation hybrid perovskites of CsFAPb(I,Br)₃, used widely for solar cell fabrication, exhibit the cubic α phase [10, 11]. The α‐phase formation is also dominant in FAMAPb(I,Br)₃ alloys. The Sn‐containing perovskite alloys (MASn_xPb_1−xI₃)) show the pseudo‐cubic structures when x ≥ 0.5, while the β phase is observed at x < 0.5 [7].

3.2.3 Secondary Phase Materials

Hybrid perovskite thin films often contain secondary phases [10–13], such as PbI₂. The suppression of an unfavored impurity phase is critical for the formation of high efficiency devices [10, 12]. Figure 3.5 shows the crystal structures of the secondary phase materials including (a) PbI₂, (b) δ‐CsPbI₃, and (c) δ‐FAPbI₃. The dotted lines in Figure 3.5 represent the unit cell structure. The PbI₂ crystal has a two‐dimensional plate‐like structure, consisting of a PbI₆ edge‐sharing octahedron as indicated in Figure 3.5d [14, 15]. In particular, PbI₂ has the continuous stacking of a I–Pb–I unit and the separation of the Pb atom plane along the c axis is 7.0 Å. PbI₂ is the dominant phase formed after the exposure of the hybrid perovskite to humid air, and the hexagonal plate structure is generally confirmed after prolonged H₂O exposure [13, 16, 17].

The CsPbI₃ crystal adopts a δ phase shown in Figure 3.5b at room temperature [3]. This crystal has an orthorhombic symmetry consisting of a one‐dimensional crystal structure. The cross section of this one‐dimensional crystal is formed by double columns of edge‐sharing PbI₆ octahedra. The δ‐FAPbI₃ also has a one‐dimensional crystal structure propagating along the [100] direction; however, in this case, the linear crystal structure is made of face‐sharing octahedra with a single chain structure shown in Figure 3.5d [3].

Figure 3.6 shows the XRD spectra of PbI₂, δ‐CsPbI₃, and δ‐FAPbI₃, calculated from the crystal structures of Figure 3.5. In Figure 3.6, the XRD spectra of the secondary phases are compared with that of MAPbI₃ (β phase). The major XRD peak positions are also summarized in Table 3.1. All secondary phases indicate characteristic peaks at low diffraction angles of 2θ ≤ 13.1°, which are helpful for confirming the presence of impurity phases (Section 3.3.2).

As known well, MAPbI₃ shows strong degradation in humid air and the perovskite crystal is converted into a completely different phase by H₂O incorporation (Figure 1.12) [13, 18, 19]. Figure 3.7 shows the crystal structures of two hydrate phases of CH₃NH₃PbI₃·H₂O [20] and (CH₃NH₃)₄PbI₆·2H₂O [21], determined experimentally. The black lines indicate the unit cell structures. The hydrate phases have monoclinic structures, and the CH₃NH₃PbI₃·H₂O phase has a one‐dimensional PbI₆ structure, which is the same as δ‐CsPbI₃ in Figure 3.5b (i.e. double chain of edge‐sharing PbI₆ octahedron). In (CH₃NH₃)₄PbI₆·2H₂O, the PbI₆ octahedron is completely separated, indicating a zero‐dimensional structure [22]. The configurations of the MA⁺ and H₂O molecules in these hydrate phases can be interpreted by the strong hydrogen bonding interactions generated by H₂O: namely, (i) I atoms (OH⋯I) and (ii) MA⁺ (NH⋯O) [22, 23]. Consequently, the OH bonds are oriented in the direction of the I atoms, whereas a NH⋯O(H₂)⋯HN local structure is generated between MA⁺ and H₂O [22].

Figure 3.8 shows the XRD spectra of the hydrate phases calculated from the structures of Figure 3.7. The XRD peak positions of Figure 3.8 are also summarized in Table 3.1. The formation of the hydrate structures can be characterized by low‐angle diffraction peaks of (100) at 8.7° in CH₃NH₃PbI₃·H₂O and at 11.5° in (CH₃NH₃)₄PbI₆·2H₂O [16, 18, 19].

3.3.1 Tolerance Factor of Hybrid Perovskites

For the calculation of t using Eq. 3.1, the actual r_A, r_B, and r_X values are required. Table 3.3 summarizes the ionic radii that can be applied for the t calculation of hybrid perovskites. In Table 3.3, for the r_A of organic cations, the values calculated by Kieslich et al. [25] are indicated, whereas the r_B and r_X represent those reported by Shannon [31], except for Sn²⁺ [29, 30].

Although representative ionic radii are described in Table 3.3, the calculation of ionic radii is in general difficult as an ion is not a rigid sphere and its size depends on the peripheral conditions. For example, r_B changes with the choice of r_X due to the effect of the electronegativity [28], and the coordination number of r_A (i.e. 6‐fold or 12‐fold coordination) also influences the value [31]. (In Table 3.3, the r_A values of K⁺, Rb⁺, and Cs⁺ represent those of the sixfold coordination reported in Ref. [31].) Moreover, in the case of organic cations, the cation has a nonspherical geometry and hydrogen bonding further varies the bond length, making the determination of r_A difficult. For the r_A of organic cations in Table 3.3, the results have been obtained using a rather simple rigid‐sphere model assuming free rotation of the organic cations [25].

Due to the uncertainty of the exact cation and anion sizes, the calculation of an accurate t value is a rather challenging problem. In fact, a variety of ionic radii have been applied for the t estimation of hybrid perovskite [25–29, 32]. Nevertheless, t is a vital parameter, allowing the interpretation of perovskite stability and device performance. In the following, we discuss the t values of hybrid perovskites, calculated according to the representative values indicated in Table 3.3.

For an understanding of structural stability of perovskites, another structural parameter, octahedral factor μ, is often considered. The μ is a simple parameter calculated by the ratio of r_B and r_X [30]:

(3.2)

Figure 3.10 shows the relation of t and μ, calculated for different hybrid perovskites. In Figure 3.10, in addition to (t, μ) of monocation perovskites (MAPbX₃, FAPbI₃, and CsPbI₃), those of mixed‐cation perovskites are shown. The t of the mixed‐cation perovskites can be calculated by considering the effective radius of r_A (i.e. r_A,eff) according to

(3.3)

where x indicates the molar ratio with r_A1 and r_A2 being the radii of two different A‐site cations. The effective radius of the X‐site anions (r_X,eff) for APb(I,Br)₃ alloys can also be estimated in a similar manner.

In Figure 3.10, the red circles indicate (t, μ) pairs that generate single perovskite phases, whereas the open circles indicate the conditions that lead to the formation of non‐perovskite phases (i.e. δ‐CsPbI₃ and δ‐FAPbI₃). Moreover, the blue circles represent the formation of the α‐phase/δ‐phase mixtures, as reported for Cs_xFA_1−xPbI₃ [10] and Cs_xFA_1−xPb(I_0.6Br_0.4)₃ [11]. In general, materials with t = 0.8–1.0 are considered to adopt a perovskite structure [24], while the orthorhombic and hexagonal δ‐phases are formed at t < 0.8 and t &gt; 1.0, respectively. The result of Figure 3.10 essentially agrees with this picture, even though the absolute t values of CsPbI₃ (0.807) and FAPbI₃ (0.987) are slightly inconsistent. More importantly, the analysis of Figure 3.10 indicates a rather narrow region (t = 0.91–0.98) for the formation of the single perovskite phase. Thus, the center position of t = 0.945 appears to be the most favorable value for the stabilization of perovskite architecture. In other words, for Cs_xFA_1−xPbI₃ alloys, the Cs content of x ∼ 0.2 provides an optimum [10, 33] and a large deviation from the optimum enhances the phase separation into the α and δ phases. The best device performance has also been obtained for hybrid perovskite absorbers in the consistent range t = 0.94–0.98 [10]. Conversely, a wide range of μ is possible for the perovskite formation, provided that t is in the range t = 0.91–0.98.

3.3.2 Tolerance Factor of Mixed‐Cation Perovskites

Mixed‐cation hybrid perovskites are now the main materials applied to solar cell fabrication [10, 12, 24, 27, 33–36], and the t value provides a guiding principle for designing proper multi‐cation perovskites. Figure 3.11a shows t of APbX₃ calculated for different r_A,eff and r_X,eff and, by following the result of Figure 3.10, the t range 0.91–0.98 is highlighted with the optimum of t = 0.945 in red. In Figure 3.11a, the guidelines are indicated for the APbX₃ alloys with A = MA⁺, FA⁺, and Cs⁺, and X = I⁻ and Br⁻. From similar calculations, the corresponding t contour lines are obtained for FA_1−yMA_yPb(I_1−xBr_x)₃ (Figure 3.11b) and Cs_yFA_1−yPb(I_1−xBr_x)₃ (Figure 3.11c). Since the r_X values of I⁻ and Br⁻ are similar, r_X,eff does not vary significantly with the Br content x. In contrast, the r_A of Cs⁺ is quite small, compared with FA⁺. Accordingly, the t value of the alloys is governed predominantly by r_A,eff, and the selection of an ideal t can be realized through the mixing of A‐site cations. As shown in Figure 3.11a, in MAPb(I_1−xBr_x)₃, the perovskite structure is maintained in the entire x range, whereas CsPb(I,Br)₃ and FAPb(I,Br)₃ show unsuitable t, independent of x. For the Cs‐ and FA‐based materials, therefore, the control of r_A,eff is critical.

In FA_1−yMA_yPbI₃, the incorporation of smaller MA⁺ moves t toward the favorable value, generating a stable cubic perovskite phase. A small amount of MA⁺ (y &gt; 0.1) is sufficient to cross the α‐phase/δ‐phase boundary (see Figure 3.11b). With the inclusion of Br⁻, the minimum amount of MA⁺ for the α‐phase formation increases slightly. Since FAPbI₃ exhibits the smallest band gap (E_g) among APbI₃ perovskites (see Figure 4.9), FA_1−yMA_yPb(I_1−xBr_x)₃ alloys are generally formed using small x and y to enhance the longer wavelength optical response. Thus, FAMAPb(I,Br)₃ absorbers are often made near the border of t = 0.98 [36].

In Cs_yFA_1−yPb(I_1−xBr_x)₃, the incorporation of small Cs⁺ has a key role in forming the structurally stable perovskite. By Cs⁺ incorporation, the perovskite unit cell shrinks and the (100) XRD peak shifts toward a greater 2θ angle [10, 11]. Since t changes significantly with the Cs content y, t can be tuned toward more structurally stable regions by controlling y. With the optimum Cs⁺ content of y = 0.2, an ideal t of ∼0.95 can be realized (Figure 3.11c). In fact, a number of studies confirm that y = 0.1–0.3 in Cs_yFA_1−yPb(I,Br)₃‐based alloys is essential for generating stable perovskite structures with some great advantages [10–12, 33–35]; i.e. (i) phase impurities can be avoided, (ii) perovskite formation becomes less sensitive to process conditions, and (iii) decomposition into PbI₂ is further suppressed. An x value of ∼0.2 is also beneficial for maintaining E_g sufficiently low (see Figure 4.10a). Since MAPbI₃ shows lesser thermal stability due to the desorption of small MA⁺ at a low temperature of 100 °C [37, 38], research into more thermally stable CsFAPb(I,Br)₃ has become popular.

The discussion of t can further be applied for triple‐cation‐mixed α‐phase perovskites, CsFAMAPb(I,Br)₃. Figure 3.12 shows two XRD spectra for the Cs content x of 0 and 0.05 in Cs_x(FA_0.83MA_0.17)_1−xPb(I_0.83Br_0.17)₃ [12]. When x = 0, the alloy is FA_0.83MA_0.17Pb(I_0.83Br_0.17)₃ with t = 0.98. As confirmed from Figure 3.11b, the composition of this alloy is in the boundary and the XRD shows the formation of δ‐FAPbI₃ and PbI₂ (see a dotted circle in Figure 3.12). However, the Cs⁺ incorporation with x = 0.05 pushes t away from the boundary (t = 0.97) and eliminates the phase impurities. As a result, even a small amount of Cs⁺ leads to a drastic improvement in forming a suitable perovskite crystal.

3.5.1 Orientation of Center Cations

Based on powder neutron diffraction experiments, Weller et al. determined the precise location of MA⁺ and FA⁺ within the framework of PbI₃⁻ at different temperatures [5, 6]. Figure 3.15 shows the refined crystal structures of (a) γ phase (100 K), (b) β phase (180 K), (c) α phase (352 K) of MAPbI₃, and (d) α phase of FAPbI₃ (298 K) [5, 6]. In the MAPbI₃ γ phase, the MA⁺ cations are fully ordered. Specifically, the CN bonds lie normal to the b‐axis (see Figure 3.3), and the C → N vector has an alternating order in neighboring unit cells along the b axis direction. The gray area in Figure 3.15a indicates 90% probability ellipsoid (time‐averaged atom positions) for the H atoms of MA⁺. Even though the H atoms show the relatively large freedom at both ends of MA⁺ at 100 K, the orientation of MA⁺ is essentially fixed because of the hydrogen bonding between NH₃ and I (i.e. NH⋯I). The strong hydrogen bonding in the γ phase can be evidenced from the short NH⋯I distances of 2.6–2.8 Å. In this orthorhombic phase, the tilting of the PbI₆ octahedra results in Pb–I–Pb angles of 161.94° (along the b axis) and 150.75° (along the a and c axes).

Figure 3.15b shows one orientation of MA⁺ in the MAPbI₃ β phase at 180 K with 50% probability ellipsoids. The β‐phase exhibits a smaller octahedral tilting with a Pb–I–Pb angle of 157.92° (along the a and b axes) and 180° (along c axis). In the β phase, the C–N axis of MA⁺ is preferentially directed toward the face center position of the distorted cubic structure because of the hydrogen bonding interaction (NH⋯I distance: 3.2 Å); however, the cation is disordered around the c‐axis (see also Figure 3.3). The molecular dynamics (MD) simulations also confirm the preferential orientation of MA⁺ even at room temperature [39–43], and the dynamical motion of MA⁺ is strongly correlated with the I atoms due to the presence of NH⋯I interaction. Indeed, neutron scattering [41] and ultrafast infrared‐vibrational spectroscopy [42] experimentally confirmed the strong coupling of MA⁺ with PbI₃⁻. Moreover, a large‐scale MD simulation of MAPbI₃ shows that roughly half of MA⁺ is participating in the hydrogen bonding [40].

At a high temperature of 352 K, the perovskite crystal becomes cubic with all Pb–I–Pb angles being 180°. The 50% probability ellipsoid of the α phase in Figure 3.15c shows that the Pb atom position is fixed, whereas the I atoms move with greater freedom. Hydrogen bonding interaction becomes weaker at 352 K (NH⋯I distance: 3.1–3.5 Å), and MA⁺ is orientationally more disordered, rotating rapidly near the unit cell center.

In Figure 3.15d, the 30%‐probability ellipsoid representation of α‐FAPbI₃ (cubic) at room temperature is shown. In this crystal, the atomic configuration of PbI₃⁻ is almost the same as that of the cubic MAPbI₃ (see Figure 3.15c). In α‐FAPbI₃, however, the N–(C)–N axis of FA⁺ is directed toward the center of the cube face and the free motion of FA⁺ is hindered effectively by the strong NH⋯I bonding that occurs at both ends of FA⁺ (see dotted lines in Figure 3.15d). As a result, the NH⋯I distance in α‐FAPbI₃ (2.8–3.0 Å) is considerably shorter, if compared with MAPbI₃ at the same temperature. Thus, the hydrogen bonding interaction is significant in α‐FAPbI₃, and the preferential rotation of FA⁺ around the N‐(C)‐N axis is the dominant motion in FAPbI₃ [44, 45].

In the DFT calculations, the hydrogen bonding in APbI₃ discussed above is well reproduced [37, 46, 47] (see Figure 5.3).

3.5.2 Relaxation of Center Cations

The organic center cations reorient rapidly within the inorganic scaffold, even though organic cations interact strongly with the PbI₃⁻ framework through hydrogen bonding. It is now well established that MA⁺ and FA⁺ in PbI₃⁻ show similar reorientation with a time scale of 0.5–14 ps at room temperature [1, 39–42, 44, 45, 48–50]. Figure 3.16 shows the variation of MA⁺ reorientation time (τ_reo) in MAPbI₃ with temperature, as summarized by Fabini et al. [45]. In Figure 3.16, τ_reo determined by nuclear magnetic resonance (NMR) [49], GHz spectroscopy [1], and quasi‐elastic neutron scattering (QENS) [41, 50] is shown. As confirmed from Figure 3.16, all the measurements show consistent results and the relaxation time becomes longer with decreasing temperature as the thermal motion becomes less significant. It can be observed that the γ → β transition varies τ_reo significantly, whereas the effect of the β → α transition on τ_reo is negligible. This stems from the fact that the crystal structure of the γ phase is quite different from those of the β and α phases.

Because the free motion of the organic cations is prohibited by the intense hydrogen bonding, τ_reo in Figure 3.16 needs to be interpreted as the time interval from one configuration with the hydrogen bonding to another with a similar configuration, yet with a different orientation. In other words, MA⁺ jumps between preferential orientations within the PbI₃⁻ cage [41, 42]. Moreover, MA⁺ is wobbling around the crystal axis before the reorientation [42]. Accordingly, the atomic arrangements, represented by Figure 3.15b,d, are important for the interpretation of hybrid perovskite crystals.