Advances in modelling and simulation of halide perovskites for solar cell applications

In the ABX₃ halide perovskite structure, the large monovalent cation A is bonded with the nearest twelve X halide anions, forming an AX₁₂ cuboctahedron, while the smaller divalent metal cation B forms corner-sharing BX₆ octahedra in a 3D framework. It should be noted that the A cation can be organic, forming hybrid organic–inorganic halide perovskites, and inorganic, leading to all-inorganic halide perovskites. At a glance, it seems that there are a great number of combinatorial cases simply with monovalent A and divalent B cations based on the periodic table. However, there are some geometric factors restricting the formation of stable 3D halide perovskite structures. The first is the Goldschmidt tolerance factor [45], t, defined as,

$$\begin{eqnarray}&&t=\displaystyle \frac{{r}_{A}+{r}_{X}}{\sqrt{2}({r}_{B}+{r}_{X})}\end{eqnarray} \tag{ 1 }$$

where r_A, r_B and r_X are the ionic radii of the A, B, and X ions, respectively. It is empirically known that with t = 1 for a perfect cubic lattice, $0.8\leqslant t\leqslant 1.0$ for most stable perovskite structures with tetragonal, orthorhombic and rhombohedral (or trigonal) lattices. Otherwise, non-perovskite structures are formed: a hexagonal structure is formed when t > 1, and different structures are found when t < 0.8 [46, 47].

Regarding the halide ions (X = F⁻, Cl⁻, Br⁻, I⁻), typical examples of B-site metal cations are B = Pb²⁺, Sn²⁺, Ba²⁺, Ge²⁺, and monovalent A-site cations include organic CH₃NH₃⁺ (MA⁺), HC(NH₂)₂⁺ (FA⁺) and CH₃CH₂NH₃⁺ (EA⁺) cations and inorganic Cs⁺ and Rb⁺ cations. By using the calculated tolerance factors, Kieslich et al [48] demonstrated that 2352 amine–metal–anion compounds are possible, of which 180 halide compounds (and 562 organic anion-based compounds) can be stable perovskites with 0.8 < t < 1.0 (figure 1). These are too many cases to consider in computational work. In such a situation, another constraint factor is necessary.

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The second constraint is known as the octahedral factor, defined as the ratio of the ionic radii of the B cation and the halide anion, μ = r_B/r_X [49]. This reflects the stability of the BX₆ octahedron, and should be in the range of $0.44\leqslant \mu \leqslant 0.9$ for a stable perovskite. Together with the tolerance factor, the octahedral factor can provide a constraint to select stable halide perovskites among many possible cases. For instance, we can draw a 2D map of the ionic radii of A cations (Rb⁺, Cs⁺, MA⁺, FA⁺, EA⁺) and X anions (F⁻, Cl⁻, Br⁻, I⁻), which are marked along the abscissa and ordinate in figure 2, for typical B-site metal cations of Pb²⁺, Sn²⁺ and Ba²⁺. Their ionic radii with proper coordination numbers (CN = 12, 6, 6 for A, B, X ions, respectively) are provided in [50, 51]. In figure 2, we can see the intersections located within the polygonal (shaded) region formed by tolerance limits of these constraint factors, which indicate a potentiality of stable perovskite formation with corresponding A, B, X elements, and those outside the region, which indicate a failure of perovskite formation. It should be noted that there is an ambiguity in defining ionic radii due to the nonsphericity of the organic molecular A cation and the reduced electronegativity of heavy halide anions, and thus some modifications have been proposed, such as effective molecular A cation radii [48] and modified B-site metal cation radii from empirical bond lengths [47].

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Once a certain compound with ABX₃ formula is proven to form a stable halide perovskite due to its appropriate tolerance and octahedral factors, we should also consider the possibility of modification in the shape of BX₆ octahedra. In fact, BX₆ octahedra can be tilted or distorted because of the slight rotation or displacement of constituent atoms or molecules at finite temperature (and possibly pressure). It is worth noting that octahedral tilting is known to be associated with antiferroelectricity, while octahedral distortion caused by cation off-centring yields ferroelectricity (i.e. the creation of spontaneous electric polarization) that can enhance charge carrier separation and allow the photovoltage to exceed the band gap thousands of times over [52, 53]. Such octahedral tilting or distortion may lose the symmetry of a crystalline unit cell, being closely related with phase transition driven by temperature [54]. In general, the archetype cubic perovskite structure is observed at high temperature, and when temperature decreases, lower symmetry phases are found in the order of tetragonal $\to$ orthorhombic $\to$ monoclinic and/or rhombohedral.

For the typical case of hybrid organic–inorganic halide perovskite MAPI, which has been playing a major role in advancing PSC technology, its cubic α-phase with space group Pmm is only found at high temperature, and phase transition to the tetragonal β-phase (I4/mcm) occurs at 327.4 K, and to the orthorhombic γ-phase (Pna2₁) at 162.2 K [55]. In the case of MAPI, the orientation of the MA cation is of great importance in determining the octahedral tilting and the crystalline phase [53, 56]. The organic MA⁺ cation has two modes of reorientation: (i) methyl and/or ammonium rotation around the C−N axis and (ii) whole molecular rotation of the C−N axis itself [53]. In the cubic phase, three different orientations are identified as $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ for the C−N axis, which lower lattice symmetry from the cubic Pmm to the pseudocubic Pm and $R3m$ phases, and the $\langle 110\rangle$ model was found to be the most energetically stable configuration by ab initio MD [56] and structural optimization [57] in agreement with experiment [53]. By performing ab initio MD simulations, the activation barriers were calculated to be 117 meV for reorientations of the axis [58] and 13.5 meV for ion rotation [59, 60]. For the tetragonal phase, the C−N bonds in MA distribute in a vertical way, and in a parallel way for the orthorhombic phase, as shown in figure 3. In these non-cubic phases, their unit cells ($\sqrt{2}a\times \sqrt{2}a\times 2a$ supercell) contain four formula units (48 atoms), being larger than the cubic phase with one formula unit (12 atoms) [61].

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Temperature-driven phase transitions were also observed for all-inorganic halide perovskite ABX₃ (A = Cs, Rb; B = Pb, Sn; X = F, Cl, Br, I) (see [62] and references therein). For CsSnI₃ typically, phase transitions from the perovskite cubic α-phase (${Pm}\bar{3}22m$) upon decreasing temperature occur to the tetragonal β-phase (P4/mbm) at 446 K and to the orthorhombic γ- and non-perovskite Y-phases (Pnma) at 373 K [63], while for CsPbI₃, phase transition occurs only to the non-perovskite orthorhombic Y-phase at 583 K [64]. It is worth noting that for CsSnI₃ two orthorhombic structures coexist at room temperature, of which the yellow Y-phase characterized by an edge-connected 1D double-chain is not useful for solar energy applications [65]. Unlike the hybrid perovskite, the all-inorganic halide perovskites do not have rotational molecular disorder in the A-site, and instead the flexibility associated with the inorganic octahedra network solely causes the phase transition. By performing the phonon calculations for CsBX₃ (B = Pb, Sn) in cubic phase, Yang et al [62] revealed that these inorganic halide perovskites exhibit lattice instabilities in the cubic phase, confirmed by the phonon soft mode (imaginary frequencies) at the Brillouin zone boundary, and thus the octahedral tilting and distortion that is by definition antiferroelectric in nature can occur spontaneously. A quantitative picture for phase diversity can be obtained by calculating the free energy through phonon calculations, which will be discussed later.

With respect to the environmental impact, there is concern about the toxicity of lead in the major halide perovskites such as MAPI and CsPbI₃, promoting an extensive search for alternatives. Although there has been extensive research to try simple substitution of Pb²⁺ in the ABX₃ structure with alternative divalent cations, it has turned out to be challenging. There is also research interest in another approach to search for multivalent elements. This resulted in low-dimensional perovskite structures such as ${{\rm{A}}}_{3}^{{\rm{I}}}$ B ${}_{2}^{{\rm{III}}}$ X₉ ('3–2–9') layered (2D), ${{\rm{A}}}_{3}^{{\rm{I}}}$ B ${}^{{\rm{II}}}$ X₅ ('3–1–5') single-chained (1D) and ${{\rm{A}}}_{4}^{{\rm{I}}}$ B ${}^{{\rm{II}}}$ X₆ ('4–1–6') isolated octahedra (0D), and double perovskite structures ${{\rm{A}}}_{2}^{{\rm{I}}}$ B ${}^{{\rm{I}}}$ B ${}^{{\rm{III}}}$ X₆ ('2–1–1–6') [66, 67]. Xiao et al [68] have reported their recent work on Pb-free halide perovskites with such different structural dimensionalities, which have a wide range of band gaps and PCEs. In particular, 3D double perovskites such as Cs₂AgBiX₆ (X = Br, Cl), called elpasolites, have received much attention as promising absorber materials [69–73]. Volonakis et al [74] have reported a series of double perovskites with ${{\rm{B}}}^{{\rm{I}}}$ = Cu, Ag, Au and ${{\rm{B}}}^{{\rm{III}}}$ = Bi, Sb, predicting their band gaps in the range from 0.5 to 2.7 eV for the Bi system and from 0.0 to 2.6 eV for the Sb system. So far, there has been no report for double perovskite-based solar cells, which might be due to difficulties in synthesizing uniform thin films of the correct phase and chemical composition.

There is an ever-increasing need to improve the performance of solar cells, represented by higher efficiency and greater stability. Although the PCE of MAPI-based solar cells can be as high as 22%, being comparable with sc-Si solar cells, their instability in air is relatively low, which could be a major barrier to commercialization. Making solid solutions or mixing at the A-site and/or X-site with two or more similar elements is an efficient way to satisfy the need for enhancing performance by increasing the stability, improving the charge carrier transport, and tuning the band gap [31, 75]. Experimentally, it is much easier to realize such mixing than to prepare double perovskites. There have been several first-principles investigations into perovskite solid solutions such as hybrid perovskites, MAPb(I_1−xBr_x)₃ [57, 76–78], MAPb(I_1−xCl_x)₃ [79] and (Cs/MA/FA)PbI₃ [80], and inorganic perovskites, CsPb(X_1−xY_x)₃ (X, Y = I, Br, Cl) [81] and (Rb_xCs_1−x)SnI₃ [82]. When mixing larger I⁻ anions with smaller Br⁻ or Cl⁻ anions, the crystalline lattice can be reduced and a phase transition from the tetragonal to the cubic can occur at room temperature. Meanwhile, mixing MA⁺ cations with larger FA⁺ cations can tune the octahedral tilting with the proper Goldschmidt tolerance factor.

To perform modelling of solid solutions, we can use the supercell (SC) method, with the sacrifice of high computational cost. When employing the pseudopotential plane wave method, the virtual crystal approximation (VCA) approach can be used as an alternative, to reduce the computational cost, but the accuracy or reliability should be checked carefully before going into the main calculation. In the VCA approach, the virtual atom is introduced and its pseudopotential is constructed by averaging the relevant potentials and wave functions of constituent atoms [83, 84]. There are some technical problems with these methods: random distribution of alloying atoms for the SC method, and local effects from the interactions between the constituent atoms for the VCA method. It has been proven that the VCA approach can reliably reproduce the lattice constants in cubic phase as a linear function of mixing content, i.e. Vegard's law, for hybrid halide perovskites [78, 79] (figure 4).

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In general, standard DFT exchange-correlation (XC) functionals within local or semi-local approximations, such as local density approximation and generalized gradient approximation (GGA), severely underestimate the band gap of inorganic semiconductors, due to the inherent nature of DFT as a ground state theory and the artificial self-interaction between electrons, although they can provide reliable structures and stabilities. Surprisingly for MAPI, the GGA functionals (Perdew-Burke-Ernzerhof (PBE) [89] or PBE for solid: PBEsol [90]) can yield a band gap in excellent agreement with experimental values within ±0.1 eV [77–79, 91]. This is due to a fortuitous error cancellation between the GGA underestimation and the overestimation by the lack of SOC [92–96], which inversely indicates the importance of relativistic effects in halide perovskites. From analysis of the partial density of states (PDOS) for halide perovskites ABX₃, the valence band maximum (VBM) consists of an antibonding coupling of  X p-states and B s-states, while the conduction band minimum (CBM) is dominated by B p-states (see figure 5). It should be noted that for MAPI the majority of occupied molecular orbitals of MA are found deep (∼5 eV) below the VBM and a minority (but not negligible amount) are found ∼0.5 eV below the VBM, indicating hydrogen bonding interactions between the organic MA moiety and the inorganic PbI₆ octahedra [97]. In APbX₃ perovskites the heavy Pb atom has a significant SOC effect of lowering the CBM, whereas the lighter Sn atom has a relatively weak SOC effect, resulting again in the large underestimation of the band gap by GGA functionals for ASnX₃ perovskites [98, 99]. Moreover, the GGA functionals cannot accurately describe dispersion of the valence band even in Pb-based perovskites.

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The relativistic effect can be approximated by first-order scalar relativistic (SR) and higher order SOC contributions. Umari et al [97] demonstrated that SR-DFT barely changes the band obtained by DFT, while SOC-DFT largely lowers the conduction bands for hybrid iodide perovskites MAPbI₃ and MASnI₃. When the SOC effect is considered, splittings of the conduction bands, known as the Rashba effect, occur as a result of interaction between the magnetic moment (spin) of the electron and the local electric field [100]. In fact, this electromagnetic force displaces electrons in the momentum space, acting on up and down spins in opposite directions. For the case of cubic halide perovskites, neglecting SOC yields direct band gaps at the band edge points of R and M in the Brillouin zone, as shown in figure 6. When SOC is turned on, both the valence and conduction bands around R split into symmetrical valleys for the case of MAPI (see figure 6(a)). More importantly, since the splitting is much more evident in the Pb 6p conduction band compared with the I 5p valence band, the CBM slightly shifts along the ${\rm{R}}\to {\rm{\Gamma }}$ line, resulting in a vertical energy difference of 25 meV between the CBM and VBM at the R point [94]. Thus the band gap changes into the indirect mode, which affects light absorbing function as we will discuss later. Motta et al [94] revealed that the orientation of the MA molecular cation has the same effect on the band gap transition. For the all-inorganic halide perovskites like CsPbI₃, however, such relativistic spin-splitting does not occur due to a centre of inversion symmetry. Although the SOC effect causes the separation of Pb 6p into p_1/2 and p_3/2, no splitting of the band extrema is observed at the high symmetry points, as shown in figure 6(b).

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Since both the standard DFT and SOC treatment based on local or semi-local XC functionals are insufficient to describe the electronic structures of halide perovskites, it is necessary to adopt more delicate approaches. One way is to use a screened hybrid functional such as HSE06 [101, 102], which has a tendency to overestimate the band gap of hybrid and all-inorganic perovskites, and thus by combining with SOC treatment can give a reasonable estimation. More profoundly, as in other inorganic semiconductors, an accurate description of electronic structure can be provided by considering many-body interactions within the GW approximation [97, 103, 104] or within the random phase approximation (RPA) [105]. The quasiparticle self-consistent GW (QSGW) approach could yield the large overestimation of band gaps as 2.68 ∼ 2.73 eV (1.55 eV) for MAPI (MASnI₃), while SOC + GW, by combining with SOC can give the closest band gaps of 1.67 eV (1.10 eV) to the experimental value of 1.60 eV (1.20 eV) [97, 103]. In addition, inclusion of SOC in the GW calculation increases band dispersion so significantly that the standard parabolic approximation can no longer be applied to the band extrema, which differs from experimental results indicating parabolic band extrema in MAPI.

As the key factor in light absorbing properties, the band gap of the halide perovskites can be tuned by different means: (i) static volume change, (ii) temperature change, and (iii) chemical substitution. When decreasing the volume by the action of a small perturbation or by increasing pressure, the band gap was found to monotonically decrease in DFT calculations with the additional benefit of transition from indirect to direct band gap for MAPI [106, 107]. Also, it was found that the out-of-phase band-edge states are stabilized as the lattice expands [60]. Regarding the temperature effect, there has been no theoretical work carried out yet, although the band gap was observed in temperature-dependent photoluminescence to decrease with decreasing temperature from 1.61 eV at 300 K to 1.55 eV at 150 K for MAPI [108]. We focus on the third method of chemical substitution at each site of ABX₃, which is more appropriate for materials design than the physical volume effect [34, 109, 110].

The A-site cation of MAPI can affect only indirectly the band gap by changing the crystal structure due to its molecular orbitals far away from the VBM. When replacing the MA cation with a smaller molecular cation such as NH₄⁺ or H⁺, the band gap was found to lower by 0.3 eV with contraction of cubic lattice from 6.29 to 6.21 Å in the QSGW + SOC calculation for NH₄PbI₃ [103], and to be less than 0.3 eV with a lattice constant of 6.05 Å for HPbI₃ [60]. It is important to recognize the relationship between the ionic radii and the local structure. In fact, both NH₄PbI₃ and HPbI₃ adopt non-perovskite chain or layer structures due to their tolerance factors of lower than 0.81, which give rise to instability of the octahedral networks with respect to tilting. Replacing MA with the larger FA can also result in structural distortions [111]. Meanwhile, substituting inorganic cation Cs⁺ with a smaller ionic radius leads to a widening of the band gap to 1.73 eV in cubic CsPbI₃ [112–114], being suitable for the top cell material in a tandem cell. However, it is difficult to form its cubic phase at room temperature. In this situation, it is desirable to adopt the mixing technique, which can promise the synergistic effect of enhancing stability against moisture and maintaining efficiency. Mixing can be done in various ways, such as MA/FA, Cs/MA, Rb/Cs [82], Cs/MA/FA [80], and Rb/Cs/MA/FA [115].

Substitution or mixing on the B-site can directly alter the conduction band dominated by B p-states without the severe change in crystal structure. For example, replacing Pb with isovalent Sn in MAPI reduces the band gap by ∼0.3 eV due to a slight downward shift of the CBM since the Sn 5p-states are weaker than the Pb 6p-states [116]. It also induces a phase transition from the tetragonal I4 cm phase for MAPI to the pseudocubic P4 mm phase for MASnI₃, being beneficial to the light absorber. However, Sn²⁺ is less chemically stable in the octahedral network due to its ease of oxidation to Sn⁴⁺, resulting in the degradation of PSCs. Another isovalent Ge²⁺ is further unstable owing to the lower binding energy of its 4p electrons. When mixing Pb with Sn to form solid solutions MA(Sn_1−xPb_x)I₃, the band gap was found to follow the quadratic function of mixing content x (see figure 7), such that a band inversion occurred associated with a systematic change in the atomic orbital composition of the conduction and valence bands [116–118].

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Since the X-site anion dominates the valence bands, its substitution can be expected to change the VBM [119]. In going from I to Br to Cl, the valence band composition changes from $5p$ to $4p$ to $3p$, resulting in a monotonic increase in electron binding energy (higher ionization potential). Accordingly, the band gap increases from 1.5 eV for MAPI to 2.10 eV on Br [78] to 2.70 eV on Cl substitution [79]. For the case of FAPbX₃, the substitution of Br for I increases the band gap from 1.48 to 2.23 eV as found in experiment [120]. For the former case of MA(Sn_1−xPb_x)I₃, mixing I with Br or Cl can give band gap changes according to the quadratic function of mixing content (figure 7),

$$\begin{eqnarray}&&{E}_{g}(x)={E}_{g}(0)+[{E}_{g}(1)-{E}_{g}(0)-b]x+{{bx}}^{2}\end{eqnarray} \tag{ 2 }$$

where b is the bowing parameter reflecting the fluctuation degree in the crystal field and the nonlinear effect from the anisotropic binding [78, 79, 121]. If the bowing parameter is zero, the band gap could follow Vegard's law, as with the lattice constant. Comparing MAPb(I_1−xBr_x)₃ and MAPb(I_1−xCl_x)₃, they are 0.185 eV (0.33 eV in experiment [121]) and 0.873 eV respectively, indicating the larger compositional disorder and low miscibility for the latter case (figures 7(b) and (c)).

In the halide perovskites, the major charge carriers are conduction electrons and holes created by several external factors. They can couple with each other by an electrostatic interaction to form a kind of quasiparticle, namely an exciton. Therefore, the effective masses of these charge carriers and the exciton binding energy are of importance in checking the suitability for the charge transport layer in solar cells.

Effective masses of conduction electrons and holes can be calculated from the lower conduction band and upper valence band near the R point for the cubic phase, which can be approximated as a parabolic function of momentum k as follows

$$\begin{eqnarray}&&E=\displaystyle \frac{{{\hslash }}^{2}}{2{m}^{* }}{k}^{2}\Longrightarrow {m}^{* }={{\hslash }}^{2}{\left[\displaystyle \frac{{\partial }^{2}E}{\partial {k}^{2}}\right]}^{-1}.\end{eqnarray} \tag{ 3 }$$

This parabolic approximation can be slightly broken when considering the SOC effect in the hybrid halide perovskites as discussed above. Therefore, the calculated effective masses depend somewhat on the theoretical method and the complexity of band dispersion. The GW + SOC calculation yielded average values of ${m}_{e}^{* }/{m}_{e}=0.19$ (0.28) and ${m}_{h}^{* }/{m}_{e}=0.25$ (0.13) for cubic MAPI (MASnI₃) [97], whereas PBE with the inclusion of a van der Waals (vdW) correction [122] can give similar values of ${m}_{e}^{* }/{m}_{e}=0.20$ (0.34) and ${m}_{h}^{* }/{m}_{e}=0.23$ (0.43) for MAPI (MAPbCl₃) [61, 79]. The electron effective masses are larger than those of GaAs (0.07) and CdTe (0.11), whereas the holes have distinctly lower masses (0.5 and 0.35, respectively), predicting a high hole mobility in the hybrid halide perovskites. For the cubic CsSnX₃, the hole effective masses were predicted to decrease when going from Cl to Br to I by QSGW calculations [104]. Since the mobility is inversely proportional to the effective mass, the halide perovskites are expected to exhibit ambipolar and large diffusion lengths for electrons and holes [95].

An exciton can be viewed as a H atom with differences attributed to (i) a replacement of the electron and nucleus masses by the effective masses and (ii) an introduction of a dielectric constant, ε, of the material. Then, the exciton binding energy can be obtained using the effective masses and dielectric constant in the following equation

$$\begin{eqnarray}&&{E}_{b}=\displaystyle \frac{{m}_{e}{e}^{4}}{2{(4\pi {\varepsilon }_{0})}^{2}{{\hslash }}^{2}}\displaystyle \frac{{m}_{r}^{* }}{{m}_{e}}\displaystyle \frac{1}{{\varepsilon }^{2}}\approx 13.56\displaystyle \frac{{m}_{r}^{* }}{{m}_{e}}\displaystyle \frac{1}{{\varepsilon }^{2}}\,({\rm{eV}})\end{eqnarray} \tag{ 4 }$$

where ${m}_{r}^{* }$ ($1/{m}_{r}^{* }=1/{m}_{e}^{* }+{m}_{h}^{* }$) is the reduced effective mass. The static dielectric constant can be calculated by the DFPT method without or with electron–hole coupling, as will be discussed below. If this modelling is valid as the extending radius of the lowest bound state (${a}^{* }=\varepsilon \tfrac{{m}_{e}}{{m}_{r}^{* }}{a}_{B}$, where a_B is the Bohr radius) is larger than the lattice constant, the exciton is the weak exciton, known as the Mott–Wannier exciton. Otherwise, it is a Frenkel exciton, being extremely localized. For MAPI (MAPbBr₃), the exciton binding energy E_b and effective radius ${a}^{* }$ were calculated to be 45 (99) meV and 3.0 (1.9) nm [78], indicating that the excitons are likely to be of the Mott–Wannier type.

Charge carrier transport can be characterized by the diffusion length L_d, defined as the average length a carrier travels before recombination. This can be calculated using the diffusivity D and charge carrier lifetime τ by ${L}_{d}=\sqrt{D\tau }$. The diffusivity can be directly related to mobility (μ) by the Einstein relation μ = Dq/k_BT due to the fact that transport is limited by carrier scattering and carrier effective mass. L_d must be long enough for the charge carriers to reach the contacts in solar cells, and these are reported to be considerably longer in MAPI than in other semiconductors. This can be partly attributed to the defect-tolerance of hybrid halide perovskites [123]. Providing effective masses of <0.2m_e for MAPI, the carrier mobility was calculated to be modest (<100 cm² (V·s)^–1) compared to the inorganic semiconductors Si or GaAs (>1000 cm² (V·s)^–1) [124]. Such limitation of carrier mobility can be caused by strong scattering. Zhao et al [125] have found that by applying the PBE + SOC approach the electron–acoustic phonon couplings in MAPI are weak, and charge carriers are scattered predominantly by charged defects or impurities.

The frequency-dependent complex dielectric functions can be calculated within DFPT without the electron–hole coupling effects. Although such excitonic effects can be calculated by solving the Bethe–Salpeter equation for the two-body Green function without or with local field effect [126], ignoring electron–hole coupling still yields quite a reasonable result in good agreement with experiment for the small band gap semiconductors [88]. Given a self-consistent QSGW potential, the polarizability $P({\boldsymbol{q}},\omega )$ is obtained using the RPA, and then the inverse dielectric function is obtained from ${\varepsilon }^{-1}={\left[1-\nu ({\boldsymbol{q}})P({\boldsymbol{q}},\omega )\right]}^{-1}$ where $\nu ({\boldsymbol{q}})$ is the bare Coulomb interaction [127]. The macroscopic dielectric function ${\varepsilon }_{{M}}$ is given as follows

$$\begin{eqnarray}&&{\varepsilon }_{{M}}=\mathop{\mathrm{lim}}\limits_{q\to 0}\displaystyle \frac{1}{{\varepsilon }_{{\boldsymbol{G}}=0,{\boldsymbol{G}}^{\prime} =0}^{-1}}\end{eqnarray} \tag{ 5 }$$

Then, the frequency-dependent photoabsorption coefficient is given as

$$\begin{eqnarray}&&\alpha (\omega )=\displaystyle \frac{2\omega }{c}\sqrt{\displaystyle \frac{{\left[{{\rm{Re}}}^{2}{\varepsilon }_{{M}}(\omega )+{{\rm{Im}}}^{2}{\varepsilon }_{{M}}(\omega )\right]}^{1/2}-{\rm{Re}}{\varepsilon }_{{M}}(\omega )}{2}}.\end{eqnarray} \tag{ 6 }$$

Figure 8 shows the macroscopic dielectric functions and the photoabsorption coefficients as functions of photon energy in hybrid halide mixed perovskites MAPb(I_1−xCl_x)₃, calculated using PBE + vdW-D2 functional within VCA as the mixing content x increases [79]. The real part is related to the polarizability of the medium in response to an oscillating electric field. For the hybrid solid solutions MAPb(I_1−xBr_x)₃ and MAPb(I_1−xCl_x)₃, the static dielectric constants decrease as the mixing content x increases [78, 79]. The rotation of molecular cation MA⁺ is of some importance in the explanation of these variation tendencies. In fact, the static dielectric constant, ${\varepsilon }_{s}={\mathrm{lim}}_{\omega \to 0}{\rm{Re}}{\varepsilon }_{{M}}(\omega )$, is dominated by the rotational motion of the MA molecule that has an intramolecular dipole, while the high-frequency or optical dielectric constant, ${\varepsilon }_{\infty }\approx {\varepsilon }_{{M}}(\omega \sim {10}^{15}\,{\rm{Hz}})$ is related to the vibrational polar phonons of the lattice (see figure 8(c)) [92]. The MA molecular dipoles screen the Coulombic interaction between photo-excited electrons and holes in the Pb–I sublattice [128–130], leading to a reduction of exciton binding. The MA cation is situated in a cuboctahedral cage, and thus, when going from I to Br to Cl, the size of the cage decreases, inducing a restriction on molecular motion and a decrease in static dielectric constant with a reduction of exciton screening. In accordance with these tendencies, the onset of photoabsorption and the first peak shift to a higher photon energy, i.e. shorter wavelength light, with increasing mixing content. There exist other first-principles calculations of optical properties for CsSnI₃ [131], CsPbX₃ [132], MASnX₃ [133] and MABaI₃ [134].

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It is crucial to get a precise insight into phonon dispersions in the halide perovskites in order to understand material processes such as ionic transport, and the recombination and scattering of charge carriers, as well as material stabilities. Phonon dispersion is computationally accessible via lattice dynamics calculations, in which the dynamic matrix is constructed via the Hessian matrix as the second derivatives of the total energy with respect to the atomic displacements that can be obtained directly from DFT calculations. The Hessian matrix (or interatomic force constants) can be calculated either in real space by performing force calculations on a series of symmetry-inequivalent displaced structures with the use of a supercell [139, 140] or in reciprocal space using perturbation theory (e.g. DFPT [85]). Diagonalizing the dynamic matrix yields a set of eigenvectors (phonon modes) and eigenvalues (phonon frequencies) [36]. Lattice dynamics in DFT is one of the most expensive calculations and, thus, classical or semi-classical MD based on interatomic potential functions [141] can be used to overcome the limitations of DFT.

For the hybrid perovskite MAPI in the cubic, tetragonal and orthorhombic phases, the phonon bands were calculated in the harmonic approximation [142, 143]. Confirming that the calculated phonon spectra were in quite good agreement with the experimental data, these calculations indicate that the lowest bands below 100 cm⁻¹ are assigned to the bending and stretching of the Pb−I bonds, i.e. diagnostic modes of the inorganic cage. The medium bands in the range from 100 to 150 cm⁻¹ are the modes coupled to the molecular motion, and the highest frequency branches in the range from 300 to 3300 cm⁻¹ are associated with the vibration and bond stretching of the MA cation. For other hybrid halide perovskites MAPbX₃ (X = Br, Cl), similar arguments were found to be valid, putting special emphasis on the reorientation of the MA cations, which plays a key role in shaping the vibrational spectra of the different compounds [53, 144–146]. It was found that MAPI exhibits double-well instabilities at the Brillouin zone boundary and a remarkably short phonon quasiparticle lifetime, being associated with low thermal conductivity. Also, the optical phonon scattering is stronger than the acoustic scattering at room temperature in these materials [52, 145, 147]. Vibrational entropy was proved to play a crucial role in determining the stable phase, e.g. between the pseudocubic and tetragonal phase for MAPI, such that the materials favour structures that maximize the number of soft intermolecular interactions as temperature rises [148].

It is important to understand the anharmonic effects in the phase transition of halide perovskites. When the potential energy of a crystalline solid is expanded as a Taylor series of ionic displacements, only the second term (${d}^{2}U/{{dr}}^{2}$) is considered in the harmonic approximation with the temperature-independent frequencies and infinite lifetime, while the effects of temperature and first-order anharmonicity can be considered in the QHA [150]. It is associated with the imaginary (or negative) frequency, i.e. soft mode. While all the phonon modes have positive frequencies in the tetragonal and orthorhombic phases (figure 9(a)), there are two imaginary frequency acoustic modes in the cubic phase, of which the centres are around the ${\rm{R}}[\displaystyle \frac{2\pi }{a}(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})]$ and ${\rm{M}}[\tfrac{2\pi }{a}(\tfrac{1}{2},\tfrac{1}{2},0)]$ points, corresponding to the $\langle 111\rangle$ and $\langle 110\rangle$ directions (figure 9(b)). As a common feature of the perovskites, these soft modes indicate a double-well characteristic in the potential energy surface, and thus a dynamic instability of the cubic structure as observed in inelastic x-ray scattering measurements [52]. Such instabilities represent antiferroelectric distortions associated with collective rotation and tilting of the corner-sharing octahedral framework that can be observed directly in MD [59, 151]. In the double-well potential, the cubic phase is a saddle point between two equivalent broken-symmetry phases (figure 9(d)), and the barriers are 37 and 19 meV for the R and M modes, being comparable to k_BT at room temperature [147].The phase transitions from cubic structure can be understood as a condensation of these soft modes, as proved by solving the time-dependent Kohn–Sham equation describing the nuclear motion in the double-well potential [147, 152, 153].

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The soft phonon modes are also found in the all-inorganic halide perovskites. For the case of CsSnI₃ perovskite, the temperature-dependent lattice dynamics were carried out for the four different phases within the QHA, revealing the strong anharmonic effects such as soft modes [65, 154]. However, the calculated phase transition temperatures fail in quantitative agreement with experimental values, requiring further studies to eliminate these discrepancies. Yang et al [62] systematically investigated the phonon dispersions of CsSnX₃ and CsPbX₃ (X = F, Cl, Br, I) perovskites, verifying the phase instabilities of cubic structures related to the spontaneous octahedral tilting. In all eight cases, double-well potentials as a function of distortion amplitude Q, given by

$$\begin{eqnarray}&&E(Q)={{aQ}}^{2}+{{bQ}}^{4}+O({Q}^{6})\end{eqnarray} \tag{ 10 }$$

with fitting parameters a and b, were found for soft phonon modes with barrier heights ranging from 108 to 512 meV, assessing the chemical and thermodynamic driving forces for these instabilities. When compared with the hybrid halide perovskites, it is surprising that the soft mode is found even at the zone centre point Γ (figure 9(c)), which is related to the ferroelectric distortion. Marronnier et al [155, 156] thoroughly investigated this anomaly at Γ for the cubic and tetragonal phases of CsPbI₃. As a polar mode, the soft phonon mode found at Γ for cubic CsPbI₃ is linked to the displacements of the Cs⁺ cation in one direction and of the I⁻ anion in the opposite direction. In spite of such displacements, ferroelectricity has not been observed at a macroscopic scale due to the oscillations along this polar soft mode. Volume relaxation with tight convergence thresholds (10⁻⁴ Ry Bohr⁻¹ for the force and 10⁻¹⁴ for the phonon self-consistency) and frozen phonon calculations remove the soft modes at Γ. Based on those results, it was concluded that the strongly anharmonic mode at Γ will not condense at lower temperatures, whereas the remaining phonon instabilities at the edge points of M and R are responsible for soft modes that condense at lower temperatures to induce the phase transition [155].

Forming solid solutions or alloys by mixing two or three equivalent elements at the same site can provide a direct and easy route to improving the performance of materials. As such, perovskite solid solutions were found to have a controlled band gap and improved stability; in lead-based hybrid halide perovskites APbX₃, the best efficiency can be obtained by mixing organic cations (MA/FA) on the A-site and halides on the X-site. In such mixing cases, it is important to understand whether the solid solution is stable against phase separation in the entire range of composition, where configurational entropy can play an important role in describing the disorder of site occupancy [36, 77]. Thermodynamic miscibility for mixing two components, e.g. (1 − x)ABX₃ + xA^'BX${}_{3}^{{\prime} }$, can be calculated by the Helmholtz free energy difference given as

$$\begin{eqnarray}&&{\rm{\Delta }}F(x,T)={\rm{\Delta }}U(x,T)-T{\rm{\Delta }}S(x).\end{eqnarray} \tag{ 11 }$$

The first term, internal energy difference, is written as follows [157]

$$\begin{eqnarray}&&{\rm{\Delta }}U(x,T)=\displaystyle \frac{{\displaystyle \sum }_{k}{\rm{\Delta }}{U}_{k}(x){g}_{k}{e}^{-{E}_{k}(x)/{k}_{B}T}}{{\displaystyle \sum }_{k}{g}_{k}{e}^{-{E}_{k}(x)/{k}_{B}T}}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{U}_{k}(x)={E}_{k}(x)-[(1-x){E}_{{{ABX}}_{3}}+{{xE}}_{{{A}}^{{\prime} }{{BX}}_{3}^{{\prime} }}]\end{eqnarray} \tag{ 13 }$$

where E_k(x) is the DFT total energy of the alloy with mixing content x and configuration number k, and g_k is the degeneracy representing the number of configurations with the same energy E_k. At room temperature, the internal energy of mixing can be well fitted into the subregular solution expression [76, 157]

$$\begin{eqnarray}&&{\rm{\Delta }}U(x)={\rm{\Omega }}x(1-x),\,{\rm{\Omega }}=\alpha +\beta x\end{eqnarray} \tag{ 14 }$$

with the fitting parameters, α and β. Then, the entropy of the alloy is given in the ideal solution limit as follows

$$\begin{eqnarray}&&{\rm{\Delta }}S(x)=-{k}_{B}[x\mathrm{ln}x+(1-x)\mathrm{ln}(1-x)]\end{eqnarray} \tag{ 15 }$$

which is expected for a random alloy at high temperatures.

Brivio et al [76] have provided the thermodynamic insight for photoinstability in the hybrid halide solid solutions MAPb(I_1−xBr_x)₃. By generating different configurations (using the SOD code [158]) and applying the generalized quasichemical approximation method [159], they calculated the internal energy, configurational entropy, Helmholtz free energy of solid solutions and phase diagram as functions of the alloy composition (figure 10). At low temperatures, the free energy curve is asymmetric and positive, indicating the existence of a miscibility gap. When the temperature increases, the curve becomes symmetric and negative since the probability of sampling all possible configurations increases. From the calculated phase diagram, it is clear that there exists a stable region where the solid solution can be formed in a stable state against phase separation; e.g. at 300 K, the alloy cannot be formed in the region of 0.19 < x < 0.68, which is the miscibility gap. Also, the spinodal and binodal points are found, and thus the alloy can present metastable phases in the intervals of 0.19 < x < 0.28 and 0.58 < x < 0.68 at 300 K. Although it is difficult to directly compare these findings with experiment, some indirect experimental evidence such as blue-shift in optical absorption from I-rich to Br-rich compositions were found. For the all-inorganic perovskites such as CsPb(I_1−xBr_x)₃ [81] and Cs_xRb_1−xSnI₃ [82], similar arguments are valid.

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It is well known that the hybrid halide perovskites are weak on external actions such as humidity, ultraviolet light and heat. Also, whether or not they are stable compounds with respect to chemical decomposition is an important issue. If they are intrinsically stable, the degradation of PSCs can be suppressed by thoroughly protecting the device from the external actions. To check the chemical stability, the following reaction is generally suggested for chemical decomposition of ABX₃ perovskites

$$\begin{eqnarray}&&{{\rm{ABX}}}_{3}\rightleftharpoons {{\rm{BX}}}_{2}+{\rm{AX}}.\end{eqnarray} \tag{ 16 }$$

Then, the Gibbs free energy difference between the products and the reactants, ${\rm{\Delta }}G={G}_{{{ABX}}_{3}}-({G}_{{{BX}}_{2}}+{G}_{{AX}})$, is an estimation of the chemical stability.

Under conditions of zero temperature and zero pressure, simply the DFT total energy difference, i.e. formation energy, can be calculated with this aim [78, 79, 160]. It was found that for MAPI all the possible phases are unstable for chemical decomposition with PBE functional but including the vdW correction gives the orthorhombic phase as stable. As shown in figure 11(a), substituting Cs for MA, Sn for Pb, and Br and Cl for I changes the formation energy from negative to positive, indicating the enhancement of intrinsic chemical stability [160]. Systematic study on mixing I with Br or Cl in MAPI can provide the turning point for the mixing content where the chemical decomposition changes from exothermic to endothermic; 0.2 for MAPb(I_1−xBr_x)₃ [78] and 0.07 for MAPb(I_1−xCl_x)₃ [79] (figure 11(b)), which agreed well with the experimental findings.

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At finite temperature, the vibrational contributions were considered by performing lattice dynamics calculations (mostly via DFPT [85]), revealing that interestingly for MAPI and MASnI₃ there is no significant change in Gibbs free energy between the reactant and the products, while for CsSnI₃ its free energy decreases faster than that of CsI and SnI₂. In other work involving MAPI using experimental data for chemical potentials, the free energy difference was calculated to be 0.16 eV fu^–1 at 300 K, indicating that the finite temperature effects tend to stabilize the perovskite structure against spontaneous chemical decomposition under standard thermodynamical conditions [161]. For mixing Cs with Rb in CsPbI₃, the free energy differences tend to increase as the Rb content x increases, with a turning point. This turning point decreases as the temperature increases, as shown in figure 11(c). The phase diagram for chemical decomposition of CsPbI₃ into PbI₂ and CsI can be obtained as shown in figure 11(d), indicating that CsPbI₃ is stable in the temperature range from 0 to 600 K and in the pressure range from 0 to 4 GPa [149], fitting well with experiment.

The most important defects in the halide perovskites ABX₃ are the twelve intrinsic point defects: the vacancies (${{\rm{V}}}_{{\rm{A}}}$, ${{\rm{V}}}_{{\rm{B}}}$, ${{\rm{V}}}_{{\rm{X}}}$), the interstitials (A_i, B_i, X_i) and the antisites (${{\rm{A}}}_{{\rm{B}}}$, ${{\rm{A}}}_{{\rm{X}}}$, ${{\rm{B}}}_{{\rm{A}}}$, ${{\rm{B}}}_{{\rm{X}}}$, ${{\rm{X}}}_{{\rm{A}}}$, ${{\rm{X}}}_{{\rm{B}}}$). For each point defect, various charge states are considered. By using equation (17), the formation energies of each point defect with various charge states are calculated as functions of Fermi energy (E_F) at different thermodynamic conditions reflecting the growth conditions. In general, equilibrium conditions for the product with its pure constituents (e.g. MAPI is in equilibrium with MA, Pb and I₂) give the constraints for the relation between their chemical potentials, resulting in two different scenarios: halide-rich and halide-poor growth conditions. Therefore, two diagrams for defect formation energy can be obtained under these conditions. From these defect formation energy diagrams, it is possible to identify the dominant defect that has the lowest formation energy and thermodynamic charge transition levels.

In the case of tetragonal MAPI, the dominant point defects were found to be the acceptor-type lead vacancy (${{\rm{V}}}_{{\rm{Pb}}}^{-2}$) under I-rich conditions [169, 172, 173], and donor-type ${\mathrm{MA}}_{i}^{+1}$ under I-poor conditions [172]. Buin et al [174] had slightly different findings, such that the major acceptor defects are ${{\rm{V}}}_{{\rm{Pb}}}^{-2}$, ${{\rm{V}}}_{{\rm{MA}}}^{-1}$ and ${{\rm{I}}}_{{\rm{i}}}^{-1}$, whereas the donor defects are ${{\rm{V}}}_{{\rm{I}}}^{+1}$ and ${\mathrm{Pb}}_{{\rm{i}}}^{+2}$. Among these, defects ${{\rm{V}}}_{{\rm{I}}}^{+1}$, ${\mathrm{MA}}_{{\rm{i}}}^{+1}$ and ${{\rm{V}}}_{{\rm{MA}}}^{-1}$ were found to possess the lowest formation energies over the entire band gap. The low formation energy of ${{\rm{V}}}_{{\rm{Pb}}}$ is related to the energetically unfavourable s–p antibonding coupling [172]. In the case of MAPbBr₃ and MAPbCl₃ with cubic phases, ${\mathrm{Pb}}_{{\rm{Br(Cl)}}}^{-3}$ has a somewhat comparable formation energy to ${{\rm{V}}}_{{\rm{Pb}}}$ [173, 175]. Interestingly, the dominant defects that have the low formation energies exhibit shallow transition levels in the band gap, while the defects that have higher formation energy (and thus are difficult to form) would have deep-trap levels. In addition, all vacancies yield shallow traps and resonances within the band, implying that charge carriers can still move easily to the VBM and CBM [174]. It has been reported that only interstitial iodine defect I_i is a deep trap and non-radiative recombination centre for holes [176, 177], pointing out the strong dependence of transition levels on the selection of XC functional (HSE + SOC should be used) [178]. Therefore, it can be suggested that point defects in the hybrid halide perovskites should not contribute to the density of deep traps (recombination centres) that directly controls the diffusion length of charge carriers. However, this can be changed according to the growth conditions. For example, the formation energy of deep-level antisite ${\mathrm{Pb}}_{{\rm{I}}}$ may be low enough to significantly contribute to the density of recombination centres under I-rich conditions [173, 174]. In this sense, the optimal growth conditions should be halide-poor for MAPI, and halide-rich for MAPbBr₃ and MAPbCl₃ [173].

The experimentally observed unintentional doping for both n-type and p-type MAPI without any added dopant can be explained by the creation of neutral vacancy pair defects, i.e. Schottky defects and Frenkel defects. It was found that Schottky defects such as ${{\rm{V}}}_{{{\rm{PbI}}}_{2}}$ and ${{\rm{V}}}_{{\rm{MAI}}}$, which may dominate defect formation in the halide perovskites under stoichiometric growth conditions, do not make a trap state [179]. The elemental defects derived from Frenkel defects play the role of unintentional doping sources, implying that n-type and p-type doping can be controlled by carefully choosing the proper atomic composition in the growth procedure [180]. Such vacancy pair defects were also considered in the water-intercalated and mono-hydrated MAPI in order to reveal the mechanism of PSC degradation upon exposure to moisture [169]. From the calculated binding energy of the vacancy pair defects, it was concluded that the formation of ${{\rm{V}}}_{{{\rm{PbI}}}_{2}}$ from the vacancy point defects ${{\rm{V}}}_{{\rm{I}}}^{-1}$ and ${{\rm{V}}}_{{\rm{Pb}}}^{+2}$ is spontaneous in these three compounds, while the formation of ${{\rm{V}}}_{{\rm{MAI}}}$ is less favourable than the formation of individual vacancies ${{\rm{V}}}_{{\rm{I}}}^{-1}$ and ${{\rm{V}}}_{{\rm{MA}}}^{+1}$ in the hydrous compounds. Moreover, all these Schottky defects exhibit deep-trap levels in the hydrous compounds (figure 12), giving evidence of degradation of PSCs under humid conditions [169].

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Defect calculations have been performed for other halide perovskites. By using the GW + SOC method, Li et al [181] investigated the defect physics of cubic CsPbI₃. They found that under Pb-rich conditions the vacancy point defects ${{\rm{V}}}_{{\rm{Pb}}}$ and ${{\rm{V}}}_{{\rm{I}}}$ are the dominant acceptor and donor defects and can pin the Fermi energy in the middle of the band gap due to their comparable formation energies. Under Pb-poor conditions, the acceptor ${{\rm{V}}}_{{\rm{Pb}}}$ has a high density and thus can generate a high density of hole carriers. While the antisite ${\mathrm{Pb}}_{{\rm{Cs}}}$ acts as the recombination centre under the Pb-rich condition, there are no such centres with a low-energy formation energy under Pb-poor conditions [181]. There exist recent theoretical works on defect physics and chemistry for the hybrid layered perovskite (CH₃NH₃)₂Pb(SCN)₂I₂ [182], the bismuth-based lead-free double perovskites [183], and defect passivation in the hybrid perovskites using quaternary ammonium halide anions and cations [184].

Once the intrinsic ionic defects have formed inside solids, they can migrate according to the minimum energy pathways, as the hybrid perovskites exhibit ionic charge transport as well as electronic conduction in experiments. Such ionic diffusion in the halide perovskites is closely related to the long-term stability and PCE of PSCs. In particular, the anomalous hysteresis of the J–V curves, which is a severe disadvantage of PSCs, and giant switchable PV effects, are importantly invoked by ion migrations [185]. The chemical decomposition of MAPI upon moisture exposure can also be explained partly by ion diffusion. In simulations, the main tasks are to identify the migrating species and their migration pathways, and to quantitatively determine the underlying energetics, on the condition that the formation of such ionic defects has been already clarified.

Eames et al [186] investigated the diffusions of vacancy points (ions), not interstitials, in cubic MAPI, based on the fact that in ABX₃ halide perovskites vacancy-mediated ion diffusion is the most acceptable diffusion process while interstitial migration has not been observed in experiments. It should be noted that experiments probing ion migration, however, typically cannot distinguish between different migration pathways. They suggested three vacancy-mediated ion migration pathways including I⁻ migration along the octahedron edge, Pb²⁺ migration along the diagonal direction and MA⁺ hopping into a neighbouring vacant site (figure 13(a)). The activation energies for these ionic migrations were calculated to be 0.58, 2.31 and 0.84 eV, respectively [186]. When compared with the measured values for hysteresis, the calculated value for I⁻ is in good agreement with the measured value, 0.60−0.68 eV, whereas for Pb²⁺ and MA⁺ vacancies they are higher than the measured values, confirming that it is difficult for these ion migrations to occur [186]. It should be noted that there is a wide spread of activation energies for ion migrations in both theory and experiments [187]. Azpiroz et al [188] considered three kinds of vacancy (${{\rm{V}}}_{{\rm{I}}}$, ${{\rm{V}}}_{{\rm{Pb}}}$, ${{\rm{V}}}_{{\rm{MA}}}$) and the iodine interstitial (I_i) in tetragonal MAPI, emphasizing that the migration of MA is responsible for the observed J–V hysteresis. The activation energies of these defects migrations were calculated to be 0.08, 0.46 and 0.80 eV along similar pathways for the three vacancies ${{\rm{V}}}_{{\rm{I}}}$, ${{\rm{V}}}_{{\rm{MA}}}$, ${{\rm{V}}}_{{\rm{Pb}}}$ and 0.08 eV along the c axis for I_i. In the case of MAPbBr₃, they are 0.09 and 0.56 eV for ${{\rm{V}}}_{{\rm{Br}}}$ and ${{\rm{V}}}_{{\rm{MA}}}$, respectively. Haruyama et al [189] calculated the activation energies as 0.32/0.33 eV for ${{\rm{V}}}_{{\rm{I}}}$/${{\rm{V}}}_{{\rm{I}}}^{+}$ and 0.57/0.55 eV for ${{\rm{V}}}_{{\rm{MA}}}$/${{\rm{V}}}_{{\rm{MA}}}^{-}$ in tetragonal MAPI, 0.55/0.50 eV for ${{\rm{V}}}_{{\rm{I}}}$/${{\rm{V}}}_{{\rm{I}}}^{+}$ and 0.61/0.57 eV for ${{\rm{V}}}_{{\rm{FA}}}$/${{\rm{V}}}_{{\rm{FA}}}^{-}$ in trigonal FAPbI3, and 0.32/0.32 eV for Cl impurity ${{\rm{I}}}_{{\rm{Cl}}}$/${{\rm{I}}}_{{\rm{Cl}}}^{+}$ in cubic MAPbI_3−xCl_x. In these calculations, the common conclusion is that halide anions and MA or FA cations are the major ionic carriers due to their relatively low activation energies, which is consistent with the experimental findings and can explain the J–V hysteresis [190, 191]. Based on the dilute diffusion theory, it was suggested that replacement of MA with the larger cation FA can suppress the hysteresis and prevent ageing in PSC performance [189].

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Recently, we investigated vacancy-mediated ion migrations in cubic MAPbX₃ (X = I, Br, Cl), and further water molecular diffusion in their water-intercalated and mono-hydrated phases, aiming to uncover the role of water molecules in the chemical decomposition and thus in the degradation of PSC performance [192]. It was found that the insertion of water into MAPI reduces the activation energies of ion migrations (figure 13(c)), indicating the easy decomposition of the hybrid halide perovskite under humid conditions. Also, the activation barriers for ion and water migrations become higher from X = I to Br to Cl. These results indicate that the decomposition of halide perovskite occurs through a multi-step process such as from water intercalation to hydration and to decomposition, identifying the crucial role of the water molecule in this process. Egger et al [193] investigated hydrogen migration in tetragonal MAPI to explain the hysteresis and material stability. Using the supercells containing a hydrogen impurity with different charge states, they found that the crystal structure may be relaxed significantly for charged hydrogen impurities, collective iodide displacements can enhance proton diffusion, and the migration barriers for proton transfer are relatively low. Based on these findings it was suggested that, in the hybrid halide perovskites, the hydrogen-like defects, introduced either extrinsically or intrinsically, may be mobile and play an important role in explaining the hysteresis effects and stability issues [193].

In the standard simulation techniques using the 3D periodic boundary condition, the surfaces can be modelled by a slab with a supercell, which consists of atomic layers and a vacuum layer. The number of atomic layers and the thickness of the vacuum layer must be checked to ensure that the DFT total energy difference, e.g. the surface formation energy, is not influenced by these modelling parameters. By applying ab initio atomistic thermodynamics, the formation energies of surfaces with various indices and terminations are calculated as functions of chemical potentials of constituent elements at finite temperature and pressure, producing a surface phase diagram.

The structural and electronic properties of tetragonal MAPI surfaces with various indices and terminations have been investigated using DFT calculations with the rev-vdW-DF functional and the inclusion of the SOC effect [194, 195]. Among low-index surfaces considered in that work, tetragonal (110) and (001) surfaces are flat nonpolar surfaces consisting of alternately stacked neutral [MAI]⁰ and [PbI₂]⁰ planes, while (100) and (101) surfaces are composed of charged [MAPbI]²⁺ and [I₂]²⁻ planes, and [MAI₃]²⁻ and [Pb]²⁺ planes, implying large reconstruction or defect formation. Several types of PbI_x polyhedron terminations were considered but not MA terminations. Identifying the constraint relations between the chemical potentials, the surface phase diagrams were drawn for the four types of surface by calculating the Gibbs free energy differences as functions of the chemical potentials of I and Pb. Through these diagrams, stable terminations in each index surface can be determined for different growth conditions, i.e. Pb-rich (poor) and I-rich (poor) conditions. Also, the relaxed structures and electronic properties were analysed in detail. Based on those calculations, it was concluded that a vacant termination is more stable than the PbI₂-rich flat termination on all the surfaces under thermodynamic equilibrium growth conditions of bulk MAPI. The flat terminations on the (001) and (110) surfaces were found to have surface states above the bulk valence band (possibly attracting photogenerated holes), which can be efficient intermediates of hole transfer to the adjacent hole transport materials with smaller energy loss [194].

As discussed above, the trapping of charge carriers at defects on the surfaces and grain boundaries is one of the detrimental factors in PSC performance, because it causes nonradiative recombination loss, causes carrier lifetime to deteriorate, and initiates J-V hysteresis. Uratani and Yamashita [196] investigated the types of surface defects responsible for carrier trapping in tetragonal MAPI (001) surfaces by performing HSE + SOC calculations. They constructed (2 × 2) surface models with a vacuum region of 15 Å and three types of termination, i.e. MAI, flat and vacant (figure 14(a)). Several kinds of intrinsic point defect such as vacancies (${{\rm{V}}}_{{\rm{I}}}$, ${{\rm{V}}}_{{\rm{MA}}}$, ${{\rm{V}}}_{{\rm{Pb}}}$), interstitials (I_i, Pb_i) and antisites (${\mathrm{Pb}}_{{\rm{I}}}$, ${\mathrm{Pb}}_{{\rm{MA}}}$) were identified to be formed on each terminated surface, and their formation energies were calculated for I-rich, moderate and Pb-rich conditions. The energy levels of defect states were also drawn in comparison with the VBM and CBM of each system. Using the calculation results, they concluded that under I-rich conditions iodine interstitials I_i on flat and vacant surfaces are responsible for the carrier trapping, while under Pb-rich conditions ${{\rm{V}}}_{{\rm{I}}}$ on vacant surfaces and Pb_i defects are the carrier traps. Therefore, the Pb-rich conditions are better than the I-rich conditions in terms of PV performance, which is consistent with experiments.

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Surface modelling and simulations are also important for understanding the interaction of halide perovskites with water, which can be related with the chemical decomposition, especially of MAPI, under humid condition. Such water-assisted chemical decomposition of MAPI can be explained by hydrolysis [60] or hydration [192]. In these processes, the initial step is the adsorption of water on the MAPI surface and its diffusion (or penetration) into the bulk. Liu and co-workers explored the effects of water on the detailed structure and properties of MAPI surfaces based on ab initio MD using the Gaussian-type double-ζ polarized basis sets and the PBE + vdW XC functional [198]. Tetragonal MAPI (001) surfaces with a MA⁺ termination were considered as the base surfaces. It was found that the water adsorption energy on the MAPI (001) surface is about 0.30 eV and the diffusion barrier of a water molecule from the surface to the inside region is only about 0.04 eV, indicating that the water can easily penetrate into the bulk. Koocher et al [197] found that water adsorption is greatly affected by the orientation of the MA⁺ cations close to the surface. The activation barriers of water penetrations were calculated to be 0.023 eV in the case of P⁺ (the CH₃-end of the MA molecule towards the top surface) and 0.271 eV in the case of P⁻ (the NH₃-end) on the MAI-terminated (001) surface (figure 14(b)). Therefore, it could be suggested that controlling orientation via poling or interfacial engineering can enhance the moisture stability. Choosing the MAI-terminated (110) surface in MAPI and the MABr-terminated (110) surface in MAPbBr₃, Zhang et al [199, 200] calculated the adsorption energies of water molecules to be 1.74 and 1.51 eV with PBE, and 1.87 and 1.67 eV with PBE + vdW on MAPI and MAPbBr₃ surfaces, respectively. They suggested that the deprotonation of the MA cations followed by the desorption of the CH₃NH₂ molecules is a primary step in the degradation mechanism.

In PSCs, the halide perovskite photoabsorbers are sandwiched between the electron transport layer (ETL) and hole transport layer (HTL), so that the photoexcited electrons and holes can be efficiently transported to the collecting electrodes through these layers. In the case of MAPI, for example, it is in contact with a mesoporous TiO₂ scaffold as the ETL on one side and with organic Spiro-OMeTAD material as the HTL on other side, thus creating heterojunction interfaces at these contacts [201]. Such interfaces can be considered as large defects, where nonradiative recombination of electrons and holes may occur, resulting in a decrease in PV performance. In fact, charge carrier extraction from the absorber to the transporter can be obstructed at the interfaces by several factors such as interfacial energy barriers originating from imperfect band alignment and charge carrier recombination driven by interfacial defect traps [202]. On the other hand, the interfaces can be weak against the infiltration of external particles such as water molecules and hydrogen atoms or protons, which can trigger or facilitate the degradation of PSCs. Therefore, understanding interface phenomena and further engineering interface structure and composition are very important for improving PSC performance.

Mosconi and co-workers investigated the interfaces of TiO₂ to MAPI and also MAPbI_3−xCl_x with a small Cl content (∼4%) using SOC-DFT calculations [188, 203, 204]. The most stable tetragonal (110) and pseudocubic (001) surfaces of MAPI and MAPbI_3−xCl_x were selected and the model systems were made by $(3\times 5\times 3)$ supercells with slab stoichiometries MA₆₀Pb₄₅I₁₅₀ and MA₆₀Pb₄₅I₁₃₅Cl₁₅, which were fully optimized by performing atomic relaxations. These optimized supercells were deposited onto a (5 × 3 × 2) slab of anatase TiO₂ (101) surface made by 120 TiO₂ units. The lattice mismatches were found to be +0.36% and +1.92% for the (110) surfaces, and +0.75% and −1.85% for the (001) surfaces, along the TiO₂ a and b directions, respectively. The binding of perovskite to TiO₂ substrate was found to be through the chemical bond between the perovskite halide atoms and the under-coordinated Ti atoms of the TiO₂ surface. From the simulation results, they conclude that MAPI and MAPbI_3−xCl_x tend to grow (110) surfaces on TiO₂ due to their higher binding energies to TiO₂ than (001) surfaces, and interfacial Cl atoms increase the binding energy of the MAPbI_3−xCl_x (110) surface compared to MAPI. Through the electronic structure calculations, it was revealed that the interaction of the perovskite with TiO₂ changes the interface electronic structure to a stronger interfacial coupling between the Ti $3d$ and Pb $6p$ conduction band states and to a slight upshift of TiO₂ conduction band energy [203]. In order to investigate the effect of interface defects, vacancies ${{\rm{V}}}_{{\rm{I}}}$ and ${{\rm{V}}}_{{\rm{MA}}}$ were created at the perovskite surface exposed to the vacuum (HTL contact) and at the oxide-contacting surface, based on the already established fact that under working conditions ${{\rm{V}}}_{{\rm{I}}}$ should diffuse towards the HTL and ${{\rm{V}}}_{{\rm{MA}}}$ towards the ETL side [188]. When compared with the non-defective interface, the outermost valence band states of the perovskite intrude into the band gap of the TiO₂ substrate, while the conduction band states are found inside the manifold of the oxide's conduction states. The presence of defects at interfaces was found to modify the band alignment, cause the bending of the perovskite bands close to the TiO₂ surface, and create the trap states [188].

For the case of all-inorganic perovskites, the CsPbBr₃/TiO₂ heterostructure has been investigated using the PBEsol+HSE06 functional [205]. From the experimental observations for CsPbBr₃/TiO₂ interfaces, the interface-modified CsBr and PbBr₂ layers were confirmed to have been made during the synthesis, leading to the corresponding CsBr/TiO₂ and PbBr₂/TiO₂ interfaces. From the analysis of the charge population, it was found that large charge transfer occurs between Cs and O atoms, with much smaller charge transfer between Br and Ti atoms, being caused by the interface effects. Accordingly, the charge difference was plotted before and after the interface formation: charge accumulation occurs on Ti and O atoms, but depletion on Cs and Br atoms, implying electron transfer from the perovskite to the TiO₂. From the integrated local DOS projected along the direction perpendicular to the interface, the staggered gap offset junction was observed for both interfaces, revealing the driving force of the charge transfer through the interfaces. Moreover, the band gap in the regions far away from the interface is almost consistent with that of individual CsPbBr₃ or TiO₂, whereas in the vicinity of the interface it decreases. The interface electronic states were found to be located on the conduction band edge of CsPbBr₃, indicating the shallow gap states. It was suggested that these shallow gap states may increase the absorbed photons and promote optical absorption [205].

Regarding the degradation issue, the interaction between the MAPI surface and a liquid water environment was investigated by making a slab interface model consisting of MAPI surface and water molecules and performing ab initio MD simulations [206]. The perovskite surfaces were made by cutting (2 × 2) slabs from the bulk tetragonal MAPI crystal, leading to (001) surfaces with three kinds of termination: MAI, PbI₂ and PbI₂-defective terminations. To generate the defective surface, six PbI₂ units were removed out of a total of eight PbI₂ units from the PbI₂-terminated (001) surface [194]. The vacuum region was filled with water molecules, the numbers of which were determined from the experimental density of liquid water as 284, 226 and 235 for MAI-terminated, PbI₂-terminated and PbI₂-defective surfaces, respectively. By analysing the MD trajectories, the MAI-terminated slabs were found to undergo rapid solvation according to a nucleophilic substitution of I by water, desorption of MA molecules, and thus net dissolution of a MAI molecule. In contrast, the PbI₂-terminated surfaces were stable against such a solvation process, although the percolation of a water molecule can occur to form a hydrated bulk phase. It was also concluded that PbI₂ defects may facilitate the solvation of the surface and initiate the degradation of the entire perovskite.