Structural chemistry of layered lead halide perovskites containing single octahedral layers

A comprehensive review of hybrid lead halide perovskites based on [PbX 4]∞ layers is presented. We use a crystallographic approach based on symmetry mode analysis to systematize key types of distortions, particularly octahedral tilting and layer shifts, in this extensive and diverse family of materials.


Introduction
Lead halide perovskites (LHPs) have recently revolutionised the field of solar cells, in addition to showing novel and promising properties in several other areas, such as luminescence, ferroelectricity etc. [1][2][3] The diversity of chemical composition and structural architecture in this enormous and rapidly expanding family of materials not only creates great opportunities for the synthetic and structural solid-state chemist, but also makes the field somewhat overwhelming for the newcomer. For those of us with long memories, the excitement and opportunities available for the solid-state chemist are somewhat reminiscent of the explosion in work on layered cuprate perovskites in the late 80's and early 90's, at the peak of the High-Tc superconductor revolution. Indeed, with the advent of 'hybrid' inorganic-organic systems in LHPs, the diversity of the field is clearly much greater than in more traditional inorganiconly systems. There have been many excellent reviews of the field of LHPs over the past few years, [4][5][6] which have focussed on various aspects from the underlying chemistry and crystal structure to optimisation of electronic and optical properties and further towards material processing and device manufacture. The purpose of the present review is to take a more crystallographically-oriented view of the state-of-the-art in the area of layered hybrid perovskites, LHPs, specifically those containing a single 'perovskite-like' octahedral layer of stoichiometry [PbX4]∞ (X = Cl, Br, I), in which the layers are separated by cationic organic moieties (A, Aʹ) to give overall compositions APbX4, A2PbX4 or AAʹPbX4. Even within this sub-field there are well over 250 crystal structures reported in the Cambridge Crystallographic Database (census date 11/11/10). Hence, we shall not refer to the copious body of work on three-dimensional (3D) perovskite structures, such (CH3NH3)PbI3 or the variety of '0-D' or '1-D' perovskite-related materials, based on chain-like structural fragments or isolated PbX6 octahedra. Moreover, in order to keep the review of a manageable size and digestible to the less-expert reader, we also limit our analysis to so-called (001)-cut layered perovskites: hence neither the related (110) or (111)-cut families nor the (001)-cut families with double, triple or higher-order perovskite-like layer thicknesses will be covered here.
We first briefly introduce the various families of perovskites, before proceeding to describe the types of structural distortion that exist in layered perovskites, then using these as a means of classification of the currently known examples. We shall primarily focus on the detailed structural nature of the inorganic layers [PbX4]∞ themselves, and then consider how this is influenced by the variety of A-site molecular cations which might determine the detailed architecture of these layers and their interactions: these features are ultimately the main driver influencing the physical properties of the resulting materials.

What is a perovskite?
The name 'perovskite' originated in the discovery of the mineral Perovskite, CaTiO3, in 1839. 7 This mineralogical curiosity later blossomed into arguably the most diverse and important class of compounds in solid state chemistry. The generic composition of perovskite may be regarded as ABX3, where A and B are 'large' and 'small' cations, respectively, and X is an anion. The aristotype crystal structure has cubic symmetry, space group Pm3 ̅ m, and consists of a cubicclose-packed array of A and X, with B occupying ¼ of the octahedral interstices, in an ordered manner (Fig. 1a). Note that the word 'cubic' here is used in two different senses. A 'cubic perovskite' does not necessarily adopt a cubic crystal system, and symmetry-lowering is the norm, due to the well-known tolerance factor and octahedral tilting effects; indeed Perovskite itself is orthorhombic! Moreover, there are many further variants on this basic compositional and crystal chemistry, and there is currently confusion and conflict in the literature regarding 'what exactly is a perovskite'. [8][9][10] This is unfortunate, but perhaps inevitable, in such a diverse field, and may require an international committee to propose some clear guidelines and definitions of nomenclature in this area. The use of the phrase 'layered perovskite' in the present work corresponds to the personal opinions and preferences of the authors, and it is not intended to impose on other authors.

Octahedral tilting and Symmetry mode analysis
One ubiquitous type of distortion in cubic perovskites is 'tilting' of the octahedral BX6 units, which occurs due to a size mismatch between the A and B cations, governed by the Goldschmidt tolerance factor, t: Glazer 11 originally classified all the 'simple' tilts in cubic perovskites, and this was later updated by Howard and Stokes, using group-theoretical analysis 12 to give 15 possible simple tilt systems. The Glazer notation uses three lower case letters to specify the relative magnitudes of the tilts along the principal axes of the aristotype ('parent') unit cell. Superscripts, + or -, are used to specify whether these tilts occur 'in-phase' or 'out-of-phase' relative to each other, considering only a 2 × 2 × 2 array of corner-linked rigid octahedra. Thus, for example, Glazer tilt systems a 0 a 0 c + and a 0 a 0 care shown in Figure 1(a,b). These structural deviations from a high symmetry parent structure can be regarded as 'modes' of distortion (like normal vibrational modes), which are amenable to the application of grouptheoretical methods and representational analysis. [12][13][14] Indeed it is the application of these methods and, in particular, the advent of user-friendly graphical software, such as ISODISTORT 13 and AMPLIMODES 14 , amenable to non-expert users, that has allowed solidstate chemists to take a fresh look at structural phenomena of this type, in a much more rigorous and systematic way than was previously available. The a 0 a 0 c + and a 0 a 0 ctilts in Glazer notation can be described with irreducible representations (irreps) with labels M3 + and R4 + , respectively, using the notation of Miller and Love. 15 The distortions associated with these irreps correspond to the 'freezing-out' of phonon modes at specific points of the 1 st Brillouin zone of the parent cell. This has two important consequences for solid-state chemists: (i) if a suitable 'parent' model for a particular structure type can be derived, experimentally or otherwise, then structural distortions in 'real' examples of this structure type can be easily and systematically understood in terms of these constituent irreps. (ii) Since the capital letter (e.g. M or R) in the irrep label corresponds to a particular point in reciprocal space, these distortions can fairly easily be identified from a diffraction experiment, as they will give rise to particular types of supercell relative to the parent, higher-symmetry unit cell. Such 'symmetry mode analysis' is therefore an invaluable tool for the solid-state chemist in identifying common structural features across an otherwise apparently diverse range of (related) crystal structures. [12][13][14]16,17 We shall see that, by using the standard 'parent' phases for either RP (I4/mmm) or DJ (P4/mmm) structures, we can easily identify tilt modes, and other key types of distortion, unambiguously.
Throughout this work we use the on-line tool ISODISTORT to perform this analysis. 13 The symmetry labels for each type of distortion are dependent on the parent phase (and unit cell origin choice) used, but they will be self-consistent for a given sub-family of LHPs. These will be introduced, as required.

Layered perovskites
Here, we shall use the term 'layered perovskite' to mean a compound with a crystal structure that can be easily derived from the cubic perovskite structure by 'slicing' through octahedral apices along a particular crystallographic direction, and inserting additional species between the resultant layers. There are several common types of layered perovskite, of which two are relevant in this work. Ruddlesden-Popper (RP) phases and Dion-Jacobson (DJ) phases were originally observed in mixed metal oxides, 18,19 and these were identified as having generic compositions A2Aʹn-1BnX3n+1 and AAʹn-1BnX3n+1, respectively. For the n = 1 case consider here, the aristotype compounds can be taken as the tetragonal systems K2NiF4 (space group I4/mmm) 20 and TlAlF4 (space group P4/mmm), 21 respectively. It should be noted that, in addition to the compositional differences stated above, a key structural distinction between the original two families is in the nature of the relative positioning of adjacent [BX4]∞ layers. Thus, we can see ( Figure 2) that the adjacent layers in the DJ family are 'eclipsed' relative to each other (a coordinate displacement of (0,0)), whereas those in the RP family are staggered by (½, ½). A further important variant on these two structure types is the intermediate case, with a staggering of (0, ½) or (½, 0): here the parent phase is orthorhombic, with space group Ammm (taking c as the 'layer stacking' direction)(parent phase, NaWO2Cl2 22 ).
Octahedral tilting is also a recognised and common feature in layered perovskites, 23,24 and we shall describe this in terms of both Glazer-like notation and using irrep labels for the relevant tilt modes. In the case of single-layer layered perovskites, the tolerance factor is clearly of no direct relevance, although the nature of the interaction of the interlayer species with the [BX4]∞ framework will certainly influence the nature of tilting. When we use the word 'rotation', rather than tilt this specifically applies to a mode acting around the axis perpendicular to the layer direction.

Hybrid layered perovskites
With the advent of hybrid perovskites, inorganic solid-state chemistry had already paved the way for a useful description of the structural architectures of these compounds in terms of octahedral tilting and other distortions such as intra-octahedral distortion indices. 12,[25][26][27] However, the inclusion of non-spherical, and often highly anisotropic, molecular species at the inter-layer A sites opens up a new level of complexity in hybrid layered perovskites. One particular feature of note is the much greater tendency for 'slippage', 'shift' or 'staggering' of adjacent perovskite blocks relative to each other, such that the conventional criteria used in recognising RP versus DJ phases can no longer be applied simply. We'll use 'layer shift' from now on, to describe this. In fact, it has already been recognised that a range of degrees of shift of adjacent inorganic layers are observed in LHPs which span the RP-DJ regime. 28 We therefore choose a definition of 'RP' and 'DJ' solely in terms of the degree of layer shift, rather than the original additional differences in A/B stoichiometry. In accord with Tremblay et al. 28 we shall refer to any layered LHP, regardless of stoichiometry, having an inter-layer offset close to (½, ½) as RP-like ('near-RP' or nRP), any having an offset near (0, 0) as DJ-like (nDJ) and any having an offset near (0, ½) as DJ2-like (nDJ2).
Criteria similar to those of Tremblay will be used to define how close the structures are to one of the ideal types, by use of a layer shift parameter, . We shall also see that layer shift can also be easily understood in terms of symmetry mode analysis, with specific modes occurring commonly, regardless of other simultaneous types of distortion.
The next sections describe a comprehensive survey and classification of all structurally well-characterised (001)-cut LHPs of stoichiometry APbBr4 or A2PbBr4. These structures are taken from the CCDC 29 up to 11/11/20. In our initial survey, we began by classifying the unit cell metrics of all the known structures in relation to either of the generic structure types RP, DJ or DJ2. It was immediately apparent that, although there is a wide diversity of variants spanning these ideal 'end-members', several common themes in types of distortion and types of supercell emerge.

A classification of (001)-cut LHPs of stoichiometry APbBr4 or A2PbBr4
We find it convenient to classify this vast array of structures in terms of the observed unit cell metrics, and their relationship to the parent RP or DJ parents. In particular we shall use the number of octahedral layers per unit cell repeat as a key discriminator. In other words, regardless of whether the resultant layer shift looks 'RP-like', 'DJ-like' or 'DJ2-like', we'll aim to derive all the structure types from either a RP parent (I4/mmm), for those with two or more layers per unit cell, or a DJ parent (P4/mmm) for those with one layer per unit cell. In many cases, it is equally feasible to use the alternate parent phase, leading to an equivalent result. For the case of LHPs, the parent tetragonal unit cell metrics are aRP ~ aDJ ~ 5.5 -6.5 Å, for chlorides to iodides, with cRP, cDJ obviously being variables, dependent on the nature of the organic moieties.
In all the Tables, we refer to the individual structures using CCDC deposition numbers.
We start with the two-layer cases as these more usefully illustrate some of the structural principles observed. The parent phase is the K2NiF4 type, i.e. with two adjacent, fully staggered layers forming the unit cell repeat perpendicular to the layer direction.
Throughout the following survey it is worth noting that the 'layered perovskite' version of this, also of nDJ2 type. A third case (1186561) was a very early example of a LHP, and was refined with disordered octahedra: the authors stated possible 'unresolved superlattice structure', and we agree that this structure probably does contain octahedral tilting modes, so does not formally belong in this section; in fact, a subsequent re-determination is included later (200737 in section 3.2).
Note there is one further possible symmetry for the M5⁻ mode: M5⁻(a,0), which we shall see in the next section. The relative directions of the shifts should be clear from Figure   3.  It can be seen that the (a,0) mode results in layers shifted towards DJ type, (a,a) towards DJ2 type, and (a,b) provides both degrees of freedom. Note that each mode also allows a distortion of the octahedra, which is illustrated in (a) by the differing lengths of the trans-octahedral edges.

Metrics ~ 2 aRP × 2 aRP in the layer plane
These structures are detailed in Table 2. A very common distortion of a cubic perovskite unit cell (unit cell parameter ap) is an approximately 2 ap × 2 ap supercell caused by octahedral tilting. Such effects are seen, for example, in both of the Glazer systems in Figure 2. It comes as no surprise that such features are also commonplace in layered perovskites. We note that unit cell volume is effectively doubled in all these derivatives but the c-axis remains equivalent to that of the parent phase (i.e. not doubled relative to the RP parent, but still encompassing two adjacent octahedral layers per c-axis repeat).
In addition, the body-centring is lost. It can be seen that there is a diversity of resultant space groups. As discussed above, we are now anticipating that structures may contain two particular types of distortion of the [PbX4]∞ layers (i.e. octahedral tilting and layer shift). Our classification therefore considers those structures with layer shifts and octahedral tilting, either independently or cooperatively, starting from the simplest to the more complex.
Insert Table 2 here Table 2. Summary of experimentally known structures with two octahedral layers per unit cell (derived from RP parent) and √2aRP × √2aRP cell metrics in the layer plane.

Structures with no octahedral tilting, but with layer shifts
We include in Table 2, and subsequent Tables, the parameter  which describes the extent of layer shift between neighbouring layers (highlighted in Figure 3). There are two distinct types of layer shift which may be present in layered perovskites.
These can be described and classified conveniently in the language of symmetry mode analysis. The first type is represented by a 'gamma mode' (i.e. acts at the Brillouin zone centre), usually designated 5 + . It acts simply to slide adjacent layers in the same sense relative to each other along one crystallographic axis, and results in a lowering of symmetry to monoclinic ( Figure 4). The resultant  values for this type of distortion can be simply calculated from the unit cell metrics (see, for example, the equations in section 3.2.2), although direct graphical measurement (for example using Crystalmaker 102 ) also gives a good approximation. It will be seen that this type of monoclinic distortion is a common feature in LHPs, leading to bridging of the RP to DJ regimes.  Figure 3) and (b) the analogous Γ5 + mode. Note that the former leads to orthorhombic symmetry, whereas the latter corresponds to a monoclinic distortion.
The second type of layer shift mode is the antiferrodistortive M5⁻ type introduced in section 3.1.1. i.e. a shift of adjacent octahedral layers in opposite directions relative to an axis perpendicular to the layer direction (Figures 3, 4). The mode, acting alone, leads to a lowering of symmetry from tetragonal to orthorhombic, and so the corresponding  values can be calculated straightforwardly, from the associated difference in x, y, or z coordinates. The first, and unique, example here is 1852626, which occurs in space group Cmcm. This example is a high temperature (413 K) polymorph of a phase that appears at ambient temperature in space group Pbcn (1845548). It has no octahedral tilting, just the M5⁻(a,0) displacive mode (Figure 3), which is distinct from those seen in section 3. (as occurs in the first example) and, in addition, a purely displacive 5 + mode, which leads to an additional displacement along the a-axis, and a monoclinic distortion.

Structures with a single type of octahedral tilt and no layer shift
The structure 1863837, in space group P42/ncm, is a unique example of one of the simplest types of distortion in this family (Balachandran 30 reported five examples of oxides with this structure type). The structure exhibits an out-of-phase tilting of octahedra around the ab-plane, but the direction of this tilt alternates in adjacent layers ( Figure 5(a)) hence retaining the tetragonal symmetry. The corresponding tilt mode is designated X3 + (a,a) (see ESI for further details of some of these tilt mode descriptions).
It is necessary to use an extended Glazer-like notation to describe the tilts in these systems that contain two adjacent octahedral layers which, while they may be symmetry-related, may also contain opposite directions, or signs, of the corresponding tilts. In the adapted Glazer-like notation, e.g. as used by Hayward, 103 the tilt system here is a⁻b 0 c 0 /b 0 a⁻c 0 . Aleksandrov 25 undertook a comprehensive group-theoretical analysis of tilting in RP phases, using a different, but equivalent, notation and designated this tilt system 00/0⁻0; we shall use the Glazer-like notation. A further two examples (1934896 and 1992692) are also based on a single X3 + tilt mode, but there are two key distinctions from the previous structure: first, the X3 + mode has a different OPD, and is designated X3 + (0,a): the octahedral tilt system is a⁻a⁻c 0 /-(a⁻a⁻)c 0 . This structure type is the most common tilted type reported amongst the inorganic oxide analogues by Balachandran. 30 Second, there is an additional, purely displacive ( − ), mode which leads to a polar space group, Aba2 or Cmc21 indeed, the former compound has been demonstrated to exhibit ferroelectricity. 31 There are nine examples in Table 2 (commencing 2016195) of phases exhibiting a single X2 + (0,a) rotation mode and no other significant mode. This results in unit cell metrics cRP × 2 aRP × 2 aRP and space group Cmca (alternatively 2 aRP × 2 aRP × cRP and non-standard space group Acam). Note that this, by coincidence only, is the same space group as for the examples discussed, with active mode X3 + (0,a). The tilt system here can be designated a 0 a 0 c/a 0 a 0 c, with no tilting relative to the layer-plane but rotations around the axis perpendicular to the layers, with each layer having the same degree of rotation ( Figure 5(b)). Note that we do not use the superscript notation for the c-axis in the case of layered perovskites with single octahedral layers (as Glazer's original concept explicitly relies on octahedra being linked in the third direction). We use 'c' to mean 'rotated perpendicular to the layer direction' and c 0 to mean 'no rotation'. 104 The symbol (-c) is used if the 2 nd layer is rotated contrary to the first (which is only really relevant if there is a partial layer shift from the ideal RP parent).
The next subset of structures (commencing 2003637) involves the same single rotation mode (X2 + (0,a)) but the symmetry is lowered to Cmc21. These ten examples are simple derivatives of the corresponding Cmca subset above, but they have an additional displacive mode (5⁻) acting along the c-axis, which leads to the polar space group.
Nevertheless, they have  = (0.5,0.5) and can be regarded as RP. There is often a considerable distortion of the PbX6 octahedra present in these structures which leads to the observation that the polar axis (c) is significantly shorter than the other in-plane axis (b), in each case.

Structures with a single type of octahedral tilt and layer shift
There are several further examples of structures (237190-1975109) containing a single tilt mode, X3 + (0,a); however, this is now supplemented by a further key mode, designated M5⁻ (0,-b), which describes a shift of adjacent octahedral layers in opposite directions along the b-axis ( Figure 3). This type of superposition of two modes may be written as X3 +  M5⁻. The resulting space group is orthorhombic, Pbcn. Again, such an 'antiferrodistortive' displacement means that the octahedral layers are no longer in perfectly staggered configuration relative to each other. In other words, such structures to some extent fall between the extremes of 'ideal RP' and 'ideal DJ' types. In these cases, where the unit cell contains two adjacent layers per unit cell repeat, we choose to consider them to be derived from the RP, rather than DJ, parent structure but, of course, the degree of layer shift will dictate whether these examples might be regarded as nDJ or nRP in the classification introduced by Tremblay et al.. 28 The parameter  therefore comes into play here (defined in these orthorhombic cases as simply the difference in y-parameters between two Pb atoms in neighbouring layers, shown in Figure 3). As can be seen, the majority of the examples here are nDJ. Note that it is convenient in simple supercells with "2 × 2" in-plane metrics to determine the  parameters relative to the supercell axes, but  =  in these cases.
There are also several further, lower symmetry structures that are derived from a single rotation described by the X2 + mode, but with additional modes leading to lower symmetry space groups. This set consists of eight examples (1938882 onwards), which is a key mode, which could be described For the nDJ structures these should perhaps be described as a 0 a 0 c/a 0 a 0 (-c). It should also be noted that the distortive effect of the M5⁻ mode is often more dominant than the octahedral rotation mode (perhaps a manifestation of the stereochemically active Pb 2+ lone pair); an example is 1938883. It can be seen, in even in these relatively 'simple' examples of LHPs, that the unambiguous assignment of Glazer-like tilt systems is not as straightforward as it is in the traditional inorganic layered perovskite families. There is further one example (1914148) of a combination of X2 + (0,a) rotation with a different shift mode, M5⁻ (b,0), which naturally leads to space group Pbcm; in this case the rotational mode is again near zero.

Structures with two types of octahedral tilt and no layer shift
We now consider structures within this family of unit cell metrics which accommodate two distinct types of tilt mode. This subset is very common, with 24 examples (commencing 1938881). It has contributions from the two modes we have seen individually: X2 + (0,a) and X3 + (b,0), and results in unit cell metrics 2 aRP × 2 aRP × cRP and space group Pbca. The tilt system can be regarded as a⁻a⁻c/-(a⁻a⁻)c. It is perhaps not surprising that this tilt system is common, as it resembles the most common tilt system in 3D oxide perovskites, a⁻a⁻c + (or GdFeO3 type).

Structures with two types of octahedral tilt and layer shift
Finally, we describe several classes of structure having, simultaneously, two tilts and one or two layer shift modes. The first subset has metrics cRP × 2 aRP × 2 aRP and space group either C2/c (centrosymmetric) or its polar derivative Cc. There are nine examples (1826587-956552). These structures have monoclinic, rather than orthorhombic, unit cells which means they have the additional degree of freedom, described by the 5 + strain mode, whereby the adjacent layers are permitted to slide 'inphase' relative to each other leading to shifts intermediate between RP and DJ. The two tilt modes in this case are designated X2 + (0,a) and X4 + (0,a). This leads to the tilt system a⁻a⁻c/a⁻a⁻c for unshifted layers; however, the same issue of how to describe this taking into account nRP versus nDJ arises. Due to the additional complexity here, we'll use the ideal 'unshifted' tilt system. At first sight, this may resemble the a⁻a⁻c/-(a⁻a⁻)c system above. However, looking closely at the relationship between directions of tilts in neighbouring layers ( Figure 6) the distinction between the X4 + and X3 + tilt mode is clear. Figure 6. Comparison of the (a) X3 + (0,a) and (b) X4 + (0,a) modes, derived from the RP parent. Two layers are plotted, separated by half a unit cell. Notice that the top (yellow) layer tilt pattern is identical for each, but in the bottom (blue) layer tilts change in relative sense. The corresponding Glazer-like notation is a ⁻ a ⁻ c 0 /-(a ⁻ a ⁻ )c 0 and a ⁻ a ⁻ c 0 /a ⁻ a ⁻ c 0 , respectively.
The remaining structures in Table 2

Metrics ~ 2 aRP × aRP or 2 aRP × 2 aRP in the layer plane
There are several structures with two octahedral layers per unit cell which also have one doubled cell axis in the layer plane, and some which have both axes doubled. These are presented in Table 3. We shall save the larger supercell structures for section 3.3.
Insert Table 3 here Table 3. Summary of experimentally known structures with two octahedral layers per unit cell (derived from RP parent) and at least one axis in the layer plane doubled.

Structures with a single type of octahedral tilt and no layer shift
Two quite high symmetry derivatives (1588974 and 1552603, space group Imma) have cell metrics aRP × 2 aRP × cRP. These are of particular interest as they are rare examples of AAʹPbX4 stoichiometries (i.e. having two distinct, ordered interlayer cations). These structures cannot be derived directly from the RP parent phase, so we use the (0, ½)shifted parent in space group Ammm. 22 Indeed, they are perfect examples of DJ2 type, but they also have a single octahedral tilt mode. From the Ammm parent, the tilt mode is designated T3 + , and the corresponding tilt system is a + b 0 c 0 /-(a + )b 0 c 0 , although we note that 1552603 was modelled with disorder of the Br ligands. It should also be noted that there is no possibility of an a + tilt mode in an RP-derived structure (i.e. there is no suitable irrep of the space group I4/mmm).

Structures with more complex tilts and layer shifts
Five further structures (1915486-659016), space group P21/n, metrics aRP × 2 aRP × cRP, display a more complex set of distortions (starting from the RP parent phase these are designated 3, which is essentially a tilt, and 4, a octahedral distortion) and there is additional symmetry lowering due two different types of layer shift; M5which acts along the b-axis, and 5 + (monoclinic distortion) which acts along a. The resulting tilts and displacements are shown in Fig. 8; although the symmetry is too low to define a rigorous tilt system, it is reminiscent of the a + b 0 c 0 /-(a + )b 0 c 0 type above. A further structure (1841680) in P21/c (aRP × cRP × 2 aRP) has a similar resultant structure.  The final three structures (1552604-1841683) in this section have metrics 2 aRP × cRP × 2 aRP. These larger, low symmetry unit cells have a diversity of allowed distortion modes, but often it is reasonable to pick out the most significant ones. The highest symmetry example (1552604) has space group Pnnm. This has the M5⁻ displacive mode leading to a nDJ2 structure, and tilt modes leading to a system a + b 0 c/ -(a + b 0 c). The other two structures have four and two unique Pb sites, respectively. The former is close to RP type and has only a rotation mode around the b-axis, but there are significant, and differing, distortions of each of the Pb sites. The latter is borderline, RP-DJ with a + -like tilts, but again, other distortion modes are significant.

Structures with one octahedral layer per unit cell (derived from DJ parent)
We chose to discuss structures with two octahedral layers per unit cell rather than a single layer first, not because they are 'simpler' but because they offer a much greater diversity of constituent distortions modes, from single tilt or displacement types to types with much greater degrees of freedom. In fact, the single layer sub-family, discussed in this section, has far fewer degrees of freedom, but nevertheless has surprisingly few examples of 'high symmetry' structures (only one centrosymmetric and orthorhombic, for example). In contrast, despite the structural diversity described in section 3.1, it can be noted that there is only a single example of a triclinic structure there. In this section we shall see that the vast majority of examples have monoclinic symmetry, and several derivatives of these have triclinic symmetry. In fact, 87 out of 108 structures in this section correspond to the same basic structure type! We'll derive these single layer structures (Table 4) from the DJ-type parent (space group P4/mmm). As we shall see, the common features highlighted in section 3.1, viz. octahedral tilting and layer shift modes, also occur here, but the mode labels used to describe them are necessarily different (i.e. different parent Brillouin zone). It is therefore helpful to point out the different labels used to describe the corresponding tilt modes between the two sub-  Table 4 here   Table 4. Summary of experimentally known structures with one octahedral layer per unit cell (derived from DJ parent). A thorough study of the possible combination of tilt modes in DJ phases has been given by Aleksandrov, 25 and a briefer version, in the context of hybrid systems, by Li et al. 104 Layer shift modes in these systems are described by symmetry-lowering to monoclinic or triclinic (strain modes, 5 + for example). Note that there is no option of the antiferrodistortive layer shift mode (corresponding to the M5⁻ mode prevalent in section 3.1, using the I4/mmm parent) in this section, although many of the structures exhibiting those modes could equally well be derived from the P4/mmm parent by a Z5mode, which leads to doubling of the number of layers per unit cell.

Structures with no tilts but layer shifts, or tilts but no shifts
There are two examples (993479 and 1871404) with layer shift, but no tilting; both can be regarded as nRP due to the layer shift (5 + mode). The second of these has an unusually large octahedral distortion. There is only one example of a structure type with a single octahedral layer per unit cell, with no layer shift but with octahedral tilting: this is perhaps surprising and contrasts with the common occurrence of such tilted/unshifted structure types in inorganic DJ phases. The example (120686) has the M3 + rotation mode and resulting tilt system a 0 a 0 c.

Structures with metrics
We recall (section 3.1.2) that the corresponding tilt system (a⁻a⁻c/-(a⁻a⁻)c) is also the most common in the RP-derived structures, but in that case it is common for these compounds to retain orthorhombic symmetry and have perfect RP-like staggering of adjacent layers. This is symmetry-disallowed in the single layer structures, where combination of these two tilt modes naturally leads to monoclinic symmetry.
Nevertheless, it is permissible for these structures to be close to DJ-type, with  values close to (0,0). In fact, the full range of  values spanning nDJ to nRP is observed (Table   4 and Figure 10). The difference between the P21/a and P21/c types is simply a choice of crystallographic setting, and has no consequence. The remaining structures in this sub-section fall into three sub-groups of the above (P21, P1 ̅ and P1). They have the same tilt system, aac, but additional distortions; the additional flexibility in unit cells angles describes the tendency towards structures with layer shifts away from the DJ-RP line.  Table  4. Note that Δ consists of two components (Δ1, Δ2) but by symmetry Δ1= Δ2 therefore only one of these is shown.

Structures with 2 aDJ or higher order supercells
There are few more complex supercells derived from the single layer DJ parent. The first (1883687) has metrics aDJ × 2 aDJ × cDJ . The key distortion mode is a X3 + (a,0) tilt mode ( Figure 11)

More complex derivatives
In addition to the final example above, a few more complex structures have unit cells where at least one axis has a metric larger than 2aRP or 2cRP. Ultimately, we find the most complex superstructure yet reported, with a unit cell volume 16  Insert Table 5 here informative. In 995699, we immediately recognise the X2 + (0,a) mode (i.e. octahedral rotation around the c-axis) and the M5⁻(b,0) shift mode, which acts along b, leading to an nDJ structure ( Figure 13) and tilt system a 0 a 0 c/a 0 a 0 c⁻. In addition, the type of 'rippling' distortion referred to above is also observed here. Looking down the b-axis a 'sinusoidal' rippling of the [PbX4]∞ layers can be seen, with a repeat length of four octahedra (Figure 14a). Acting alone, the X2 + and M5⁻ modes would produce a unit cell of metrics type 2 aRP × 2 aRP × cRP and space group Pbcm: we note from section 3.1.2 that this type of distortion has not been seen, in isolation. The full mode label for this X2 + /M5⁻ combination is (0,a|b,0). The additional supercell expansion is caused by the new layer rippling feature, which is described by modes with labels 3 and Y4. This type of distortion was first noted in our recent example (TzH)2PbCl4 170 which has a 'triple ripple' rather than a 'double ripple' (Figure 14b). The following six Pbca examples adopt the same structure type.
Naturally, the amplitudes of octahedral rotation, layer shift and 'rippling' are variable within this family with 1838616, for example showing almost zero octahedral rotation (X2 + ) and having a smaller M5⁻ shift, leading to nRP status. The next structure in Table   5 (724584) is a derivative of this structure type but with additional degrees of freedom.
Although an X3 + rotation mode is permitted in this symmetry it has effectively zero amplitude; the resultant structure is nDJ, with tilt system close to a 0 a 0 c/a 0 a 0 c⁻.
1995236 has a structure related to those above, but with an additional doubling of the b-axis. Taken together, the three structures in Figure 14 show a trend where the layer rippling feature produces axes of repeat lengths of four, six and eight octahedra, i.e.
22, 32 and 42 times the parent). 1831521 has 2 aRP × 22 aRP × cRP metrics and space group P21/c. The structure is close to DJ, and has X3 + tilt, M5⁻ shift and more complex distortions.   The two largest supercell derivatives have unit cell volumes of 12 and 16 times the RP parent, and both have four octahedral layers per unit cell repeat. The first is 1521060, having metrics 32 aRP × 2 aRP × 2 cRP. This complex structure has seven independent Pb sites but does appear to have a relatively high pseudo-symmetry when viewed down the c-axis (Fig 17). The nDJ character is clear from this view, although there are slight shifts, consecutively from one layer to the next. Each layer also has the conventional M3 + rotation and M5 + tilt modes, when derived from the DJ parent, but there is an additional complex mode, which undulates the layers slightly.

All of the analysis in sections 3.1-3.3 is based on the architecture of the [PbX4]∞ layers
only, and makes no reference to the nature of the interlayer organic moieties; indeed, in most cases, it was carried out without knowledge of these! In the crystallography of purely inorganic perovskites and layered perovskites, [23][24][25]30 Table 4), which exhibits an order-disorder transition of the 4-aminopiperidine, but this hardly changes the distortions within the [PbI4]∞ layer itself. In this section, we will consider structural trends across this family of compounds, and we highlight a few interesting cases of the nature of the interlayer cations' influence on the overall structure. This is not intended to be an exhaustive survey or rationalisation; rather, we hope it encourages workers in the field to consider some of the structural principles we have introduced in this paper, in further understanding existing compounds, and possibly in designingin particular features in future work.

The most common tilt systems
It can readily be seen from Tables 2 and 4 that some unit cell metrics and space groups occur very frequently, and our analysis in section 3.3 reveals clearly that these symmetries can be rationalised by the underlying octahedral tilt modes and resultant tilt systems. The two most common are based on 2 × 2 supercells within the layer plane, and space groups P21/a (equivalently P21/c) in Table 4 (commencing no. 641641) and Pbca and C2/c (plus Cc) in Table 2 (commencing 1938881) to these structures, we see a marked tendency in Table 2 Table 4, a much wider variety of amines is accommodated, most commonly aliphatic amines, but also some aliphatic and other diamines. It seems that this type of tilt system (which is closely related to the most common GdFeO3-type structure in conventional cubic perovskites) is intrinsically quite stable, and robust to many different types of interlayer species, especially given the additional degree of flexibility available via layer-shift modes. (reported only as either 'eclipsed' or 'staggered').

Cl vs Br vs I
In the case of 1,6-diaminohexane, the chloride (1914631, Table 2) adopts a nDJ2 structure type. Although this symmetry permits both octahedral rotations and tilts (tilt system formally a⁻a⁻c/-(a⁻a⁻c⁻) the tilt mode amplitude is near zero, and the only significant modes are the X2 + rotation and the M5⁻ and 5 + shifts. In contrast, the bromide and iodide are isostructural, exhibiting nRP structures, with the common a⁻a⁻c tilt system (150501/2, Table 4), differing only slightly in the degree of layer shift.
In each case the tilt mode is significant. A key difference between the chloride and the bromide/iodide is that in the former the cation adopts a fully stretched (all trans) conformation, whereas in the latter pair the terminal C-C bond has a gauche kink. This leads to an interesting 'inverse' variation in the interlayer distances: Cl (12.32 Å) > Br all these interesting features are discussed in more detail by Louvain et al. 121 For n-butylamine a more diverse range of phases has been reported, which are discussed in section 3.5.

Homologues and isomers
Differences in amine structure can have a significant impact on bonding motifs, spatial arrangements and subsequently structural distortions. Due to the large diversity in amines that have been utilised in these materials only some general observations regarding structurally-related amines will be discussed. The simplest of these are

APbX4 and AA'PbX4 compositions
It can be noted that the

Polymorphs and phase transitions
We have already highlighted interesting polymorphic behaviour in the cystamine-  1944739, a 0 a 0 c/a 0 a 0 c, respectively) they retain centrosymmetricity. In each case the low temperature phase has ordered organic moieties, but it is suggested that the lower steric hindrance in BA and 2-FBA permits the differences in crystal packing.
Ferroelectricity has been confirmed in 2-FBA, and Xiong's group have also observed related behaviour in several other fluorinated amine systems. 32,45,50 3-Fluoro-Nmethylbenzylamine also displays an order-disorder transitions, but retains centrosymmetricity in both phases, this time from an unusual HT phase with a layer shift but no tilts (1852626,  Table 4). Structure 1962913 (section 3.1.2) is an unusual case, which shows a transition from a high symmetry phase with no octahedral tilting or layer shifts, but displaying antiferrodistortive displacements of the Pb atoms and disordered piperidinium cations at high temperature (352 K) to a layered phase with fivecoordinate Pb at low temperature. 105 Billing has described several phase transitions in the series of alkylammoniumtemplated materials (RNH3)2PbI4, with n-butyl, n-pentyl or n-hexyl chains. 71 These do not involve disorder of the alkyl chain, but instead exhibit changes in packing of these chains, which induce shift/tilt transitions of the [PbI4]∞ layers. For example, 665693 (Table 2) transforms to 665691 (Table 4)   (MHy)2PbI4 (MHy = methylhydrazinium) is of interest as MHy is the smallest organic cation to be incorporated into any structure in this review. It undergoes three phase transitions versus temperature, mediated by order-disorder of the MHy, which also leads to some interesting physical properties. 34 The phases are 1937296 (Table 2) which displays only the M5⁻ shift mode (note there is reported to be an isostructural phase transition between two phases with this symmetry), 1937299 and 1937297 (Table 5) both of which display the unique 'triple tilt' C-mode (see section 3.3 and Fig. 15).
In addition to the differences in conformation of linear chain amines which can affect Cl vs Br vs I analogues (section 3.4.2), these changes may also occur within the same compound, as a function of temperature. An example is (DAB)2PbCl4 (DAB = 1,4diaminobutane). Courseille et al. 94 report a complex structure at ambient temperature (1305732), and suggest a simpler structure (DJ-like, a⁻a⁻c) above RT; a full structural analysis of this phase is required, however.
Finally, there are intriguing cases where an in situ chemical reaction takes place, for example, reaction of alykynyl or alkenyl amines with Br2 or I2 126,144 leads to addition of Br2 across the unsaturated C-C bond. In both cases (955778 to 1048947 and 955776 to 955777) the tilt system (the apparently very robust a⁻a⁻c, Table 4

Summary and conclusions
The main aim of this comprehensive review is to provide a more systematic approach