Physical properties of bulk, defective, 2D and 0D metal halide perovskite semiconductors from a symmetry perspective

Metal halide perovskite-based nanostructures, nanosheets and nanoparticles at the forefront, show attractive optoelectronic properties, suitable for photovoltaics and light emission applications. Achieving a sounded understanding of these basic electronic and optical properties represents therefore a crucial step for the full technological exploitation of this class of semiconductors. The rapidly expanding chemical engineering and their unusual structural diversity is fascinating but also challenging for a rational description on par with those well-known for conventional semiconductors. In this sense, group theory-based symmetry analyses offer a general and rigorous approach to understand the properties of various bulk perovskites and perovskite-based nanostructures. In this work, we review the electronic and optical response of metal halide perovskite semiconductors using symmetry analysis from group theory, recalling the main results for the prototypical cubic Pm-3m lattice of AMX3 bulk perovskites (where A is cation, M metal and X halide), then extending the analysis to three cases of technological interest: AMX3 nanoparticles, A4MX6 isolated octahedra, A2MX4 layered systems, and recently introduced deficient halide perovskites (d-HP). On the basis of symmetry arguments, we will stress analogies and differences in the electronic and optical properties of these materials, as induced by the spatial confinement and dimensionality. Meanwhile, we will take advantage of this analysis to discuss recent results and debates from the literature, as the energetics of dark/bright states in the band-edge exciton fine structure of perovskite nanoparticles and nanosheets. From the present work, we also anticipate that the band-edge exciton fine structure of d-HP does not present optically dark states, in striking contrast to AMX3 nanoparticles and layered perovskites, a fact that can have important consequences on the photophysics of these novel perovskitoids.


Introduction
Metal halide perovskites have swiftly surged as promising semiconducting materials, during the last decade, thanks to their huge potential for optoelectronic applications. Since the pioneering demonstration in 2009 of their ability to convert light into current in dye sensitized solar cell architectures [1], and the decisive demonstration in 2012 of their inherent transport properties [2], halide perovskites have encountered an unprecedented success, most notably in photovoltaics and light emission. In photovoltaics, these materials have marked an impressive series of light-to-current efficiency records [3][4][5][6][7][8], with best performing perovskite-based single-junction and tandem cells showing certified 25.2% and 29.1% efficiency, respectively [9], and currently 16.1% for small modules [10]. Application of metal halide perovskites to light emission has also demonstrated comparable success, with reports of external quantum efficiencies overcoming 20% [11][12][13][14], challenging long-time established thin-film technologies. Halide perovskites present AMX 3 stoichiometry, with M corresponding to a divalent metal (Pb, Sn, Ge) and X to a halide, and their building Figure 1. Graphical schematization of different perovskite-derivate structures discussed in this work. At the center, the building block of the perovskite structure is depicted, corresponding to the MX6 octahedron and the A cation (as exemplified by methylammonium). The other investigated structures consist in: native 3D perovskite bulk structure and related dimensionally confined nanostructures, MX6 isolated octahedra, layered 2D halide perovskites in the Dion-Jacobson (DJ) and Ruddlesden-Popper (RP) phases, hollow [21][22][23] and deficient halide perovskites (d-HP) [24,25]. block is constituted by continuous network of MX 6 octahedra connected in corner-shared fashion (see figure 1). The A cation instead consists in a small organic molecule (methylammonium, or formamidinium) or an inorganic atom (Cs), which fills the cavities between the octahedral and guarantees the charge neutrality of the chemical unit. The astonishing performances of these materials are mainly related to their unique electronic structure that combines effective optical absorption [15], defect tolerance [16,17], and reduced charge recombination [18], which in turn determine their long photo-carrier diffusion [19,20] and effective light emission.
A peculiarity of these materials is their intrinsic chemical and structural tailorability, that is the possibility to widely tune their chemical composition. This allows to easily obtain complex nanostructures, ranging from bulk three-dimensional (3D) single crystals, layered two-dimensional (2D) nanosheets, down to 0D-like quantum dots, with size on the order of few tens of nanometers [26,27], or few nanometers, ideally down to isolated MX 6 octahedra [28,29]. Confinement of the charge carriers at the nanoscale imparts unique optoelectronic features to halide perovskites, with interesting implications for technologic applications. 0D halide perovskites nanoparticles revealed their high potential for various optoelectronic applications, including nowadays highest record efficiencies for quantum dot (16.6%) solar cells [30,31]. In 2D layered halide perovskites, the semiconducting MX 6 corner-shared octahedral structure is sandwiched between two layers of (traditionally) insulating organic spacers [32,33], with the consequent spatial and dielectric confinement imparting unique electronic and optical response to these systems [34]. The resulting sharp and effective excitonic emission is ideal for light emission application [35,36]. In addition, the improved stability of 2D perovskites, compared to their 3D analogues, is currently exploited as possible strategy for long-time stable photovoltaic devices [7,37]. The quest for new halide perovskite-derivatives and the exploration of low-dimension tailored nanostructures, also led recently to the design of the so called 'hollow' [21][22][23], and 'deficient' halide perovskite (d-HP) materials (see figure 1) [24,25]. In these systems, the introduction of large organic cations during the synthesis procedure results in the intentional formation of MX vacancies in the perovskite corner-shared pattern, but while for the case of 'hollow'-perovskites these defects are randomly distributed, for 'deficient'-perovskites they self-organize, as highlighted by the appearance at low angles of signals in the corresponding x-ray diffraction pattern. The local breaking of the MX 6 octahedral connectivity hardly affects the electronic properties and photovoltaic response of these materials, at least for low content of defects (~10%), but significantly improves the material stability compared to parental 3D systems, meanwhile reducing the content of toxic elements (Pb or Sn) lying at the M site [24,25,38].
The continuous exploration of the chemical space associated to metal halide perovskites calls for fundamental understanding of their physics and photophysics, inherently complicated by the interplay of many features like dynamic disorder [39][40][41], spin-orbit coupling (SOC) [42], many-body effects [43,44], etc. Theoretical analysis of halide perovskites at the atomistic scale hence requires involved and computationally expensive methods, in order to properly describe the properties of these compounds, including many-body approaches, such as those based on the GW approximation [43,44] and the solution of the Bethe-Salpeter Equations [45], molecular dynamic simulation techniques [39-41, 46, 47], methods including electron-phonon interactions, etc [48][49][50]. On the opposite side of the barricade, the use of simplified methods from the theory of conventional semiconductors, e.g. effective mass-based approaches [51,52] or tight-binding models [53], have been successfully used in the past to describe the electronic and optical properties of halide perovskite structures. In particular, group theory-based symmetry analysis revealed a powerful tool to achieve a robust insight into the basic properties of these semiconductors [54][55][56]. With special focus on the optoelectronic properties of metal halide perovskites, in this work we show how symmetry analyses can anticipate important properties and support the interpretation of complex experiments, with references to the recent literature. Recalling the main results from group theory for the cubic α-phase of the CH 3 NH 3 PbI 3 halide perovskite [54,55], taken as reference for 3D systems, we extend the symmetry analysis to the three cases of technological interest depicted in figure 1, namely perovskite nanoparticles, layered 2D halide perovskites and the d-HP introduced by Mercier and collaborators [24,25].
The manuscript is organized as follows. We first discuss the electronic structure of these systems, in terms of hybridization of the orbitals of the composing atoms (linear combination of atomic orbitals (LCAO)) and in absence of SOC, in order to highlight the differences in the electronic structure induced by the dimensionality and spatial confinement. Then, we consider the effect of SOC, particularly important when heavy atoms (Sn, Pb) occupy the M site, and discuss the corresponding electronic structures and optical properties, from a symmetry viewpoint, with reference to recent results from the literature.

Dimensionality and confinement effects on the electronic structure of halide perovskites
The α-phase of metal halide perovskite systems belongs to the Pm-3m space group and is the reference structure for the 3D bulk perovskites. When considering the rotational disorder of the organic cation, the 'pseudo' version of this cubic phase is stable at room temperature for the prototypical hybrid CH 3 NH 3 PbCl 3 and CH 3 NH 3 PbBr 3 perovskites, while being accessible above 54 • C for CH 3 NH 3 PbI 3 [57], and at even higher temperatures for the all inorganic CsPbI 3 compound [58]. Thanks to its simplicity, this model structure represents the ideal case for a detailed symmetry analysis, meanwhile conveying important information on the properties of halide perovskites under realistic operative conditions. Single particle electronic wavefunctions of molecules and crystals can be conveniently expressed on the basis of the wavefunctions of the constituting atoms, in the formal frame of the LCAO [59]. Additionally, for crystalline solids, translational symmetry can be further exploited, which imposes the single particle states to have Bloch form [59,60]: where u nk (r) is a function with the same periodicity as the crystalline lattice, while e ikr accounts for the difference in the phase of the wavefunction in different cells. Because of the presence of the k-vector in equation (1), one must consider the symmetry of the system both in the real and reciprocal space. The strength of group theory analysis is that it determines which are the allowed combination of atomic orbitals in equation (1), without need to explicitly solve the Schrodinger equation. Furthermore, it provides a robust classification of the wavefunctions ψ n,k (r) on the basis of irreducible representations (IR) of the space group of interest. This is practically done by evaluating the inner product χ site ⊗ χ phys , involving the site-reducible representation (χ site ), associated to the symmetry and multiplicity of the specific atomic positions in the cell, and the IR associated to the physical observable of interest (χ phys ), in this case the symmetry of the atomic orbitals (s-, p-, etc.). Notably, χ site contains also the information about the phase relation e ikr of equation (1). In the general case of both halide and oxide perovskites, the A cation is not involved in the formation of the frontier orbitals. For halide perovskites, we can hence conveniently express the LCAO expansion of the single particle electronic states on the minimal basis of the frontier s-and p-orbitals of the metal M and halide X, here-on named s(M), p(M), s(X), p(X). In our representation of figure 2(a) [54,55], M lies in the Wyckoff position 'a' (0,0,0) and its site symmetry χ site belongs to the total-symmetric IR, Γ 1 + and R 1 + , at the points Γ and R of the corresponding Brillouin zone in figure 2(b). The character table of the Pm-3m space group is reported in the supporting information (available online at stacks.iop.org/JPMATER/03/042001/ mmedia). The halide atoms lie in the Wyckoff position 'd' (1/2,0,0) and their reducible site symmetry χ site belong to Γ 1 + + Γ 3 + and R 4 − IR, at the Γ and R points, respectively. The combination of these site symmetries with the IR of the s-(A 1g ) and p-(T 1u ) orbitals of the composing M and X atoms results in the LCAO hybridization in figures 2(c) and (d), highlighting strikingly different atomic combination at the Γ and R point of the Brillouin zone. In Γ, figure 2(c), the symmetry allowed atomic orbitals combinations are the s(M)-s(X) and p(M)-p(X) ones, corresponding to Γ 1 + and Γ 4 − IR, while mixed s-p combinations are symmetry forbidden, as they belong to different IR. In R, figure 2(d), the s(M)-p(X) and p(M)-s(X) combinations are instead allowed and correspond to R 1 + and R 4 -IR, respectively. Electronic structure calculations based on density functional theory (DFT) and many-body GW approximation performed on the α-phase of CsPbI 3 perovskites clearly showed direct band gap at the R point of the primitive cubic Brillouin zone [42,44,54,61], with the valence and conduction band edges belonging to R 1 + and R 4 -IR, respectively [54]. The smaller transition energy at R compared to Γ is easily explained in terms of the specific orbital hybridization highlighted in figures 2(c) and (d) [42,61,62]. In fact, the overlap between s(M)-p(X)/p(M)-s(X) atomic orbitals in figure 2(d) is expected to be larger than the overlap between s(M)-s(X) and the p(M)-p(X) in figure 2(c), because of the different spatial extension of the involved atomic orbitals. As result, the destabilization (stabilization) associated to the valence (conduction) band is expected to be larger in R than in Γ. Furthermore, symmetry anticipates that, in absence of SOC, the conduction band edge of metal halide systems is triply degenerate, being associated to a three-dimensional IR. This prediction should be reasonably accurate for AMX 3 perovskites containing light elements, while being definitely inaccurate in presence of tin and lead [42]. For the sake of example, for Germanium-based CH 3 NH 3 GeCl 3 perovskite, spin-orbit effects on the order of 0.15 eV were predicted from DFT [62].
Layered halide perovskites can be obtained by cutting the 3D bulk structure along specific crystallographic directions, with <100> family in figure 3(a) representing the most widely studied class of layered halide perovskites. Symmetry analysis of a free-standing monolayer 2D A 2 MX 4 model system in figure 3(a) can be conveniently carried out using layer-group symmetry as a first step, that is, considering only one isolated periodic plane and neglecting plane stacking. <100> layered perovskites belong to tetragonal p4/mmm layer-group, with an electronic band gap at M in the corresponding first Brillouin zone, reported in figure 3(b). Before to evaluate the product χ site ⊗ χ phys , for the present case, it is worth mentioning that a symmetry reduction from cubic to tetragonal, is often observed for 3D perovskites as a result of a low temperature structural phase transition [57]. This reduction splits the triplet of halide atoms in the AMX 3 motif into an atom occupying a Wyckoff position along the tetragonal axis and a doublet of in-plane atoms. In the A 2 MX 4 2D reference layer the quadruplet of halogen atoms yields a doublet of apical atoms and a doublet of equatorial atoms, figure 3(a), resulting in different site symmetries, χ site . As shown in figures 3(c) and (d), p xy and p z orbitals are split, as they belong to E u and A 2u IR, respectively, with consequently different definition of χ phys . The character tables for single and double groups of the p4/mmm layer-group at the Γ and M points are related to the ones of the P4/mmm space group reported in the supporting information. The metal cation site symmetries for p4/mmm layer-group correspond to Γ 1 + and M 1 + , at the Γ and M point of the 2D first Brillouin zone, reported in figure 3(b). The equatorial halides (X eq ) lie at Wyckoff positions 'c' and their site symmetries correspond to Γ 1 + + Γ 2 + and M 5 -. For the apical halide (X ap ), lying at the 'd' Wyckoff position, the site symmetry IR correspond to Γ 1 + + Γ 3 and M 1 + + M 3 -. The LCAO scheme of an isolated perovskite layer, as computed from the χ site ⊗ χ phys inner product at the Γ, figures 3(c), and (M) points of the Brillouin zone, figure 3(d), nicely parallels the hybridization found for the 3D case. At the Γ point, symmetry allowed combinations involve mixing between s(M)-s(X) and p(M)-p(X) orbitals, having Γ 1 + and Γ 5 -IR, respectively. At M, instead, these combinations are not allowed anymore, while s(M)-p(X) and p(M)-s(X) combinations are permitted, corresponding to M 1 + and M 5 -IR. Most notably, the electronic structure of layered perovskites shows an important difference, compared to the 3D case, namely that the original triply degenerate p(M)-s(X) combination found in the 3D system, figure 2(d), splits in a doubly degenerate state belonging to Γ 5 and M 5 -IR and a non-degenerate state belonging to Γ 3 and M 3 -IR, at the Γ and M point of the first Brillouin zone, respectively. This means that in layered systems, the p xy (M)-s(X) hybridization inside the layer and p z (M)-s(X) along orthogonal direction are decoupled, as they belong to two different IR, a fact that becomes important when the anisotropy of the optical properties is considered [63].
Although very weak inter-planar electronic coupling is expected, as due to the large spatial separation between planes, usually exceeding the radius of the s,p(M) s,p(X) atomic orbitals, vertical coupling may be further analyzed using 3D periodic space groups. In this frame, different patterns in the inter-plane packing can be relevant in terms of symmetry analysis. In fact, according to the nomenclature of oxide perovskites, the 2D layered systems can be classified in Dion-Jacobson (DJ), Ruddlesden-Popper (RP), in relation to the presence of one or two interlayer cations per formula unit, as shown in figure 1 [64]. Additionally, Aurivilius phase can also be found in presence of covalent network between the two-dimensional oxide perovskite slabs, but this case has not been observed in halide perovskites up to now. For halide perovskites, DJ and RP terms are conventionally used in the literature to define different inter-planar stacking, with DJ indicating a displacement of two consecutive perovskite planes close to an in-plane (0,0) translation and RP indicating a displacement close to a (1/2,1/2) translation. Within a tetragonal frame, we may define reference A 2 MX 4 DJ and RP phases belonging to P4/mmm and I4/mmm space groups, respectively. Interestingly, we notice that these bulk layered crystallographic structures are formed by periodically repeating the <100> free-standing slab along the stacking axis. Therefore, the symmetry properties of these DJ and RP reference phases can be understood by mapping their 3D Brillouin zone onto the 2D Brillouin zone of the related p4/mmm layer space group. The case of the Cs 2 PbI 4 reference RP phase is treated in the following from that perspective (vide infra).
Finally, we discuss the case of 'defective' halide perovskites, in terms both of its possible exploitation for photovoltaics, as related to its improved stability compared to 3D lead-halide perovskites, and as reference model to discuss the perturbation of the electronic structure of halide perovskites, as due to point defects (e.g. the MX vacancy). Dealing with this second aspect, it is worth to mention that the d-HP structure by Mercier et al is most likely not the most representative test-case for defects but it has the advantage to be realistic and is simple enough to allow a symmetry-based analysis. In particular, we will see how the introduction of a point defect results in the relaxation of the spectroscopic selection rules. As anticipated, Leblanc et al synthetized and characterized samples of deficient CH 3 NH 3 PbI 3 and HC(NH 2 ) 2 PbI 3 perovskites, reporting XRD patterns consistent with a self-organization of the defects. From their diffraction measurements, these authors proposed the crystalline model reported in figure 4(a), belonging to the tetragonal P4/m space group and having cell 5 times larger than for parental pristine Pm-3m space group α-CH 3 NH 3 PbI 3 and α-HC(NH 2 ) 2 PbI 3 [24,25]. MX vacancies self-organize along channels parallel to the c-axis, with ideally empty channels in correspondence of one depleted metal cation each five chemical units. For this system, symmetry analysis should be carried out at the A point of the first Brillouin zone of the primitive tetragonal lattice [25], as this corresponds to the R point of the parental Pm-3m space group (see supporting information). The complete evaluation of the product χ site ⊗ χ phys for this system reveals quite complicated, because of the presence of the same chemical species in different Wyckoff positions, as classified in figure 4(b, c), and because the symmetry reduction relaxes the constraints, allowing many more possible atomic orbitals hybridizations, compared to the Pm-3m lattice. The resulting symmetry adapted LCAO at the A point of the Brillouin zone is summarized in figure 4(c), with the full scheme reported in the supporting information. Most notably, in figure 4(c) we can find again similar interactions as in the cubic Pm-3m lattice. For the valence band, the hybridization of p(X eq/ap )-s(M) is still allowed for A 1 -IR, with the X eq and X ap halides responsible for the hybridization within the tetragonal plane and along the c-axis, respectively. For the conduction band, the symmetry reduction brings to a splitting of the original triply degenerate s(X)-p(M) combination into a two-fold p xy (X)-s(M) combination with A 3 + + A 4 + symmetry and non-degenerate p z (X)-s(M) combination of A 1 + symmetry. On the basis of this LCAO scheme, the effect of MX vacancy channels along c-axis can be easily anticipated, as this interrupts the MX bonding alternation in the ab-plane after four MX 6 octahedra, as highlighted in figure 4(a), while preserving continuous MX channels along the c-direction. As a result, the valence band is expected to be less sensitive to the introduction of MX vacancies, as it is characterized by electronic dispersion both within the tetragonal plane and along the orthogonal c-axis. Similarly, the conduction state with A 1 + symmetry, involving s(X ap )-p z (M) hybridization along c, preserves in large part its electronic communication, while the A 3 + + A 4 + conduction state, involving in-plane s(X eq )-p xy (M) hybridization will be significantly destabilized by the defect channels, as discussed in [25]. Summarizing, introduction of MX vacancy in d-HP results in a splitting of the triply degenerate p(M)-s(X) conduction states into doubly degenerate in-plane and non-degenerate out-of-plane components, as for the reference 2D case, in figure 3(d). However, while in the latter, the electrons are fully delocalized in the tetragonal ab-plane, in the case of d-HP, they preserve the full electronic conjugation along the c-axis, hence likely presenting more one dimensional-like electronic and optical properties.
At the bottom step of the ladder for spatially confined perovskite-derivate structures, we obviously find the isolated MX 6 octahedron, corresponding to the case where the octahedral connectivity is interrupted along all directions. The symmetry of this system can be analyzed as well [55], taking advantage of the lacking of translational symmetry and adopting point group symmetry, corresponding to Oh, for an ideal geometry with equal MX bond lengths and all angles of 90 and 180 degrees. The determination of the product χ site ⊗ χ phys is routinely performed, with the only difference with respect to the previous 3D, 2D and d-HP cases that no phase factors as in equation (1) should be considered. The reducible site symmetry of the halides is clearly six-dimensional, corresponding to A 1g + E g + T 1u . The corresponding hybridization scheme for the isolated MX 6 octahedron is reported in figure 4(d), globally showing more symmetry allowed LCAO combinations, compared to the simple 3D and 2D case. For instance, while the mixing of s(M) and p(M) states is still forbidden, as in the case of both the 3D and 2D structures, mixing between both s(X) and p(X) is allowed for the isolated octahedron, under three different IR (A 1g , E g , T 1u ). Similarly, while the combinations between s(M)-s(X) and s(M)-p(X) states are forbidden in the 3D Pm-3m case for the same point of the Brillouin zone (see figures 2(c) and (d)), these are allowed for the isolated octahedron. Altogether, the lack of translational symmetry in the case of one MX 6 octahedron and presence of more halides per chemical units compared to the 3D and 2D cases results in more complex electronic hybridization scheme. Most noteworthy, the electronic structure of the MX 6 octahedron still allows for the s(M)-p(X) and p(M)-s(X) with A 1g and T 1u symmetry respectively, which in the 3D case correspond to the frontier orbitals [54]. One should however take care in assigning the frontier highest-occupied molecular orbital and lowest unoccupied molecular orbitals levels of the isolated MX 6 octahedron to A 1g and T 1u symmetry, as this is not general. Maughan et al in fact proposed different assignments for isolated SnI 6 octahedra [65], as well as in solid state, for the case of vacancy order cesium tin-iodide perovskite [65,66]. As the present paper mainly focuses on lead-based systems, we will discuss later the specific case of Cs 4 PbBr 6 , a system with crystalline structure consisting in isolated PbBr 6 octahedra, spaced by Cs cations, showing striking efficient photoluminescence quantum efficiencies (PLQY~45%) [67,68].
Before to conclude, we briefly comment on the symmetry analysis of AMX 3 nanoparticles, representing a special case, where the octahedral connectivity is still present, although limited to~tens of chemical units in length, and where the size and shape of the nanoparticle itself can influence the optical and electronic properties of the system. In this sense, the electronic density of states of perovskite quantum nanostructures also exhibit delta-like features but the optoelectronic properties of these 0D nano-objects are strongly dependent on the symmetry properties of the parental perovskite lattice. Due the large number of atoms (typically about 50 000 atoms for a cubic CsPbBr 3 quantum dot with a lateral size of 12 nm), the description of the electronic structure shall rely on empirical methods. One of the advantages of a fully atomistic supercell tight-binding model is that it does not require any approximation on the QD shape, leading to simulations using very realistic geometries and scaling up to a few millions atoms [69]. However, the available symmetry-based perovskite tight-binding model [53] has not yet being extended from the 3D lattice to a 0D perovskite nanostructure. The most common description for semiconductor nanostructure relies on a k.p multiband effective mass approximation for the description of the bulk electronic structure and envelope functions allowing to account for the size and shape of the nanostructure: where ψ i (r) is a monoelectronic wavefunction of the nanostructure, u n,k0 (r) are the bulk Bloch functions closed to a critical point k 0 of the Brillouin zone and f i,n (r) are slowly varying functions at the nanoscale, incorporating the information on the size and shape of the nanoparticle. It appears clearly that a proper account of the symmetry for ψ i (r) is not straightforward, provided that u n,k0 (r) are related to double space group IRs of the 3D lattice, while f i,n (r) correspond to simple point group IRs of the nanostructure shape.
The proper symmetry-based treatment of this new type of tensorial product between a mesoscopic and atomistic quantities has been proposed only recently by M.A. Dupertuis and coworkers with the introduction of a general formalism of maximal symmetrization and reduction of fields [70,71]. However for early computations of III-V nanostructure electronic wavefunctions [72], the shapes of the nanostructures were considered in many cases to be roughly compatible with the four-fold axes of the zinc-blende, avoiding elaborate symmetry-based procedures. In other cases involving nanostructure cylindrical shapes with the infinite axis parallel to one of the zinc-blende four-fold axes [73], it was shown that a few approximations on the bulk k.p Hamiltonian followed by a unitary transformation of the Bloch functions, allows a complete approach of the symmetry of ψ i (r), including strain, piezoelectric and non-linear piezoelectric effects [74]. A similar methodology can be used for inhomogeneous and strained wurtzite quantum nanostructures [75]. From that perspective, the first symmetry-based approach to ψ i (r) for a perovskite nanostructure was including a complex dependence of the bulk Bloch functions u n,k0 (r) on the cubic, tetragonal or orthorhombic perovskite lattice structural instabilities, but assuming cubic nanostructure shapes for a simplified treatment [76]. This approach was quickly extended by various authors [77,78]. Conversely, Nestoklon and coworkers were considering a reduction of symmetry due to the anisotropy cubic, tetragonal or orthorhombic nanostructure shapes, but relying on a basis of cubic bulk Bloch functions [79]. More detailed approaches to the computation of symmetry-based ψ i (r) have not yet been implemented for more realistic perovskite nanostructures, such as cuboids. To illustrate such a situation, let us consider a nanostructure with a tetragonal shape associated to a tetragonal distortion of the perovskite lattice. The simplest situation corresponds to the nanostructure four-fold axis being parallel to the tetragonal axis of the perovskite lattice, and a more complex case to the two axes perpendicular to each other. One may assume in this last case, that the incompatibility between the two axes directions simply results in some approximate descent of symmetry toward an orthorhombic symmetry. But, the proper methodology is more complex and requires to design an optimal basis related to combinations of Bloch functions as well as ultimately reduced envelope functions in this new basis [70].

Realistic electronic structure of metal halide frames in presence of SOC and optical response
After discussing the effect of dimensionality and point defects on the atomic orbital interaction in halide perovskite frames, we consider here a more realistic description of the electronic structure of these systems, by introducing the effect of SOC. As the most performant halide perovskites are based on heavy tin and lead metal, SOC turns out to be very important and therefore should not be regarded as simple perturbation of the electronic structure but should be explicitly accounted [42,62]. Inclusion of SOC requires the introduction of double groups, which are deduced from the product of the D 1/2 = E 1/2g IR associated to the electron spin with the IR of the simple space group. Once this is done for the case of the cubic Pm-3m lattice metal halide perovskite, one obtains that the valence band edge is finally associated to R 6 + IR, while the triply degenerated conduction band minimum, splits into two-fold R 6 − (E 1/2u ) and four-fold R 8 − (E 3/2u ) bands. These predictions are nicely confirmed by periodic DFT calculations in figure 5(a), performed on a cubic Pm-3m lattice of the CsPbI 3 perovskite [42,58]. Indeed, explicit calculations clearly show a splitting of 1.62 eV for the conduction band edge at the R point of the first Brillouin zone.
Symmetry allows also to establish whether an optical transition is permitted or not. The rate for the probability of an optical transition from the initial state i to a final state f is given by the Fermi golden rule [59]: where ψ f and ψ i are the corresponding wavefunctions, r is the perturbing dipole operator and the Dirac delta expresses the condition for energy conservation following the absorption of a photon with energy ℏϖ.
Optical transitions are allowed if the inner product of the IR of the initial and final states χ i ⊗ χ f contains the IR of the dipole operator (corresponding to T 1u in the Pm-3m space group) [81]. In table 1, we report the symmetry analysis of the optical transition for the case of the Pm-3m AMX 3 perovskite lattice, starting from simple band-to-band picture of the optical absorption processes and progressively introducing the effect of excitonic interactions and SOC. Band-to-band transitions are related to the excitation of one electron from the valence to the conduction band-edge single particle states. Without SOC, the direct product of the IR of the valence and conduction band edges contains the IR of the perturbing dipole operator (T 1u ) and the transition is therefore allowed, triply degenerate and isotropic. In the Wannier picture, the excitonic transition from ground state to the excited state is instead related to an additional Coulomb interaction term between electron and hole in the excited state. The IR of the lowest energy excited state hence corresponds to the product of the IR of the valence and conduction band edges with a S-like envelope function (A 1g ). Ultimately, for the lowest excited state of the Wannier progression the selection rules correspond to those of the single particle transitions but notice that this is not necessarily the case for higher components of the Wannier series, since the symmetry of the envelope function for the electron-hole correlated motion has to be included [54]. Inclusion of SOC brings to an important difference, as the lowest energy excited state corresponds in this case to an allowed triplet (triply degenerate) T 1u transition, plus a forbidden singlet A u transition. This has not to be confused with the typical terminology used for organic semiconductors, where the triplet is indeed dark, while the singlet is bright [82]. The latter picture in fact relies on the separation of the radial and spin degrees of freedom, which is reasonable in Carbon-based semiconductors, while the results in table 1 are obtained by directly considering spinors and the total electron momentum. Hence, in the present case the terms 'singlet/triplet' are no more referred specifically to the pure-spin part of the wavefunction, but to a mixed space-spin description of the wavefunction. In light of the small exciton binding energy reported for 3D lead-halide perovskite thin films and single crystals, on the order of 10 meV [83][84][85], excitonic effects are generally regarded as irrelevant in the photophysics of these semiconductors at room temperature. However, there is growing interest in understanding excitonic properties (fine structure, coupling with phonons, etc) for perovskite nanoparticles. These systems can be derived from the prototypical 3D perovskite structure, by reducing the corner-shared MX 6 octahedral connectivity down to few nanometers, corresponding to tens of AMX 3 consecutive chemical units [29]. Correspondingly, the spatial confinement of the electronic structure on the scale of the De-Broglie wavelength imparts unique features, like the enhancement electron-hole interaction, promising for light application [13,14]. In this context, the observation of one, two or three lines in the optical emission spectrum of CsPbBr 3 cubic-like nanocrystals, was initially attributed to possible underlying cubic, tetragonal and orthorhombic crystallographic structures (see figures 5(b) and (c)) [76]. In fact, the short-range exchange interaction between E 1/2u (CB) and E 1/2g (VB) monoelectronic edge states is expected to influence the Bloch function symmetries and lead to the exciton bright triplet splitting typical of respectively Oh, D4h and D2h exciton point symmetries. It is worth to note that in [76,80], the theoretical analysis of the optical response of lead-bromine related nanoparticles is performed on the basis of the 3D cubic model ( figure 2). This fact is in part justified by the fact that the continuum corner-shared octahedral connectivity, although limited to the nanometer scale, is expected to result in single particle states delocalized throughout all the nanoparticle, resembling Block waves in the parental 3D case. Another interpretation for the appearance of several lines in the optical spectrum of CsPbBr 3 nanoparticles is related to the effect of the perovskite nanocrystal shape anisotropy on the long-range part of the exchange interaction, influencing the Wannier exciton envelope function [79]. Further symmetry reductions were considered in another contribution [78], in which the high brightness of the emission was attributed to the unusual localization of the exciton dark state above the bright triplet due to a strong Rashba effect. However, the only direct observation of a perovskite nanocrystal dark state performed so far for HC(NH 2 ) 2 PbBr 3 by using an external magnetic field, is yielding a more classical ordering, as shown in figure 5(c) [80]. In this work, the authors demonstrate that a dipole-forbidden exciton component is present in the low-energy part of the spectrum and gains optical intensity under the presence of an external magnetic field (see figure 5(d)). This funding nicely agrees with the A 1u + T 1u fine-exciton structure predicted from the 3D case (table 1), with the external magnetic field mixing the optically forbidden A 1u component with the triply degenerate allowed T 1u components. The high brightness of the bright triplet emission was rather attributed to a second order phonon relaxation process. The dark-bright exciton splitting computed from the long-range exchange interaction was also in fair agreement with the experimental observations [80].
An isolated MX 6 octahedron represents another example of 0D nanostructure, where the spatial confinement results from the total interruption of the corner-shared connectivity of the MX 6 octahedra. An interesting technological case for lead-based systems corresponding to this situation is Cs 4 PbBr 6 , a system crystallizing in the space group R-3c and consisting in chemically unconnected octahedral separated by Cs cations [86]. This system has attracted considerable attention, due to reported photoluminescence quantum yield ranging between 40% and 45% [67,68,87], resulting from the enhancement of the electron-hole interaction, as due to the confinement of the electronic structure on the isolated Cs 4 PbBr 6 octahedron. In this sense, the interruption of the octahedral connectivity seems to play a crucial role in determining the positive emission properties of this system, although alternative explanations have been proposed in the literature, as related to defects or emission from other phases [88]. As we aim here just to provide a symmetry perspective for a real case of lead-based system with isolated octahedral, we disregard about more complex questions and we focus on the electronic structure of R-3 c phase of Cs 4 PbBr 6 , as computed from DFT (see supporting information). Our calculations nicely agree with previous results from the literature [68,87], showing that in absence of SOC the material is an indirect semiconductor but with direct transition in Γ which lies just few meV higher in energy. Focusing on the transition at Γ, our calculations predict valence and conduction band edges belonging to A 2g and A 1u IR, respectively, hence resulting in a dipole allowed transition of A 2u symmetry, both in the band-to-band and in the Wannier exciton frame (see table 1), consistently with the effective emission from these materials. Further inclusion of SOC, via the introduction of double group, results in exciton transition at Γ corresponding to E 1/2g ⊗ E 1/2u ⊗ A 1g = A 1u + A 2u + E u . Group theory analysis hence predicts that the exciton fine structure of Cs 4 PbBr 6 contains one dark state (A 1u ) and three bright states, one polarized along the rhombohedral axis of C 3 -symmetry (A 2u ) and two degenerate components with in-plane polarization (E u ), similar to that of CsPbBr 3 nanoparticles undergoing towards tetragonal distortions. However, because of the enhanced electron-hole interaction for isolated PbBr 6 octahedra, it is reasonable to expect much larger splitting of the exciton fine structure for this case than for 3D nanoparticles. In this frame, spectroscopic measurements of the excitonic fine structure of Cs 4 PbBr 6 could clarify the mechanism of light emission in this system.
We now focus on the role of SOC on the electronic structure and optical properties of bulk layered 2D perovskites. We showed before that the symmetry reduction from the 3D cubic space group to the layer-group already brings to the splitting of the triply degenerate conduction band edge into a two-fold state (M 5 -) and a non-degenerate state (M 3 -), as illustrated in figure 6(a). Further inclusion of SOC results in additional splitting of the former band, in two states with M 6 and M 7 symmetry. When considering interlayer stacking, in the specific case of the RP phase, one finds that the M point of the 2D first Brillouin zone of the p4/mmm layer-group maps into the X and P points of the 3D first Brillouin zone of the I4/mmm space group, with X and P differing in the electronic interaction between planes. DFT calculation for RP of Cs 2 PbI 4 reported in figure 6(b) predict in fact a~1.2 eV SOC-induced splitting of the conduction band at X point of the Brillouin zone, in X 2 and X 3 − + X 4 − IR. We may notice almost flat dispersions of the electronic bands along the X → P direction, both in absence and when including SOC, that demonstrates the weakness of the electronic coupling between inorganic planes, because of the large inter-planar distance. The interlayer coupling is only revealed by a small splitting of the CB (X 3− + X 4− ) in the absence of SOC and related to the centering of the unit cell for the I4/mmm conventional cell. It is now well-known that spatial confinement of the frontier orbital in an atomically-thin layer and the dielectric contrast between the inorganic and the organic components enhance electron-hole interaction [34], resulting in exciton binding energy in the order of 450 meV [52]. As a result, excitons dominate the optoelectronic response of 2D layered perovskite. Thus, it is worth analyzing the symmetry of the lowest energy excitonic transition in order to determine its fine-structure. In this case, the IR of the exciton ground state, determined as in table 1 from the direct product of the IR of the valence (E 1/2g ) and conduction (E 1/2u ) band edges with a S-like envelope function (A 1g ), results in A 1u + A 2u + E u IR for the exciton fine structure. This corresponds to the Γ 1 -, Γ 2 and Γ 5 assignment, often reported in the early literature, as illustrated in figure 6(c) [89][90][91][92]. Hence, as in the 3D case, the ground state exciton fine structure consists in a dark (A 1u ) state and three bright states, two with in-plane polarization (E u ) and one with out-of-plane polarization (A 2u ), in agreement with earlier theoretical predictions [63]. Further considering a symmetry reduction due to a monoclinic lattice distortion, as for the case of the actual crystallographic structure of the (PEA) 2 PbI 4 (PEA = phenylethynyl-ammonium) RP phase, leads to a modified symmetry decomposition for the band-edge exciton fine structure in the C 2h IR: 2A u + 2B u . This is essentially predicting that the expected purely dark state is mixed with the out-of-plane polarization singlet (A 1u → A u and A 2u → A u ), leading to two weakly observable emission lines close together in good agreement with recent experimental observation, which underlines the important role of the dark state on the photophysics of (PEA) 2 PbI 4 [93]. For these systems an unusual bi-exciton dynamics was measured using time-dependent photoluminescence and it was hence proposed that a dark (or weakly allowed) excitonic state is involved in the bi-exciton emission, as schematized in figure 6(d).
We conclude this symmetry analysis focusing on the expected optical properties of d-HP. This class of perovskite systems has been only recently discovered and therefore its optical response has not been subjected to detailed experimental characterization, as in the case of 3D and 2D halide perovskites, yet [24,25]. In this sense, symmetry arguments can already anticipate important differences with respect to the pristine 3D and layered 2D cases. Predictions from the previous Section are schematized in figure 7(a), nicely confirmed by the band structure from periodic DFT calculations in figure 7(b). In absence of SOC, DFT nicely confirms that the triply degenerate conduction band edge splits into doubly degenerate in-plane A 3 + + A 4 + state and non-degenerate out-of-plane A 1 + state, with the latter being more stable. In addition, symmetry anticipates that SOC induces further splitting for the higher A 3 + + A 4 + component of the conduction band, as confirmed by DFT, (0.47 eV computed splitting). From these results, we can anticipate that the band-edge exciton fine structure of deficient perovskites only consists in optically bright states, in striking contrast to 3D and 2D structures, where one dark state was found. Indeed, the IR of the lowest energy excitonic state, as estimated from the inner product of the IR of the valence (E 1/2g ) and conduction (E 1/2u ) band edges with a S-like envelope function (A g ) results in 2A u + E u IR, where the first two components have out-of-plane polarization, while the latter doubly degenerate component has in-plane polarization. In other words, the decrease of symmetry, as due to the presence of the defect in the crystal lattice, results in a relaxation of the selection rules, with the dark A 1u state of the parental 3D compound acquiring optical intensity from the T 1u states of the pristine systems. Further experimental investigation of these systems under magnetic field, as done in [80], and performed on samples with controlled content of defects are recommended, in order to confirm the current prediction. In this sense, it would be interesting to investigate how the absence of optically dark excited state can affect the photophysics of deficient perovskites, considering the role of such states in the case of 3D perovskites [80], and 2D perovskites [93].

Conclusions
Detailed understanding of the photophysics of metal halide perovskites is a crucial step both for the full exploitation of these interesting semiconductors to fields where their potential has been widely assessed, namely photovoltaics and light emission, and also for their future usage for innovative applications, as lasing, sensing, spintronic, etc. In this frame, more and more involved experimental characterization studies of the optoelectronic response of halide perovskite materials have been reported in the recent literature, unveiling their excited state dynamics in the ultrafast regime, and/or under the effect of external fields [76,78,80,84,85,93]. All these spectroscopic characterizations in turn greatly benefit from theoretical support and interpretation from symmetry analysis, traditionally used in the past to understand the properties of conventional inorganic semiconductors. Here, we provided a detailed symmetry analysis of halide perovskite structures of technological interest, namely the α-phase AMX 3 perovskites, as reference for nanoparticles, layered 2D perovskites and the recently introduced 'deficient' perovskites, as interesting study case to understand the effect of point defects in the crystalline structure of halide perovskitoids, down to isolated MX 6 octahedra, for the case of Cs 4 PbBr 6 emitter. We highlighted how the change in symmetry, reflecting the change in dimensionality or the introduction of defects, modifies the electronic structure and optical response of metal halide perovskite materials. Meanwhile, we reviewed some of the recent results from the literature, as the current debate on the energetic order of the band-edge excitonic fine structure of perovskite nanoparticles [76,78,80]. Furthermore, we predict the exciton fine structure of pure R-3c Cs 4 PbBr 6 crystals and d-HP structures, anticipating that these may show significant differences with respect to the 2D and 3D case, as due to confinement of the electronic state in one PbBr 6 octahedron (Cs 4 PbBr 6 ) or due to the perturbation of the ideal lattice associated to point defects (d-HP). For d-HP in particular, we predict the presence of four bright excited states in the band-edge exciton fine structure, as result of the relaxation of symmetry rules due to the inclusion of point defect. We hope this work will become a useful reference to frame symmetry properties of metal halide perovskite in a broad context and will inspire similar analyses and additional spectroscopic measurement for these and other cases of technological interest.