Tolerance factors of hybrid organic–inorganic perovskites: recent improvements and current state of research

S. Burger , M. G. Ehrenreich and G. Kieslich *
Department of Chemistry, Technical University of Munich, Lichtenbergstraße 4, 85748 Garching, Germany. E-mail:

Received 17th June 2018 , Accepted 7th September 2018

First published on 7th September 2018

In 2014 we applied Goldschmidt's concept of ionic tolerance factors to the large family of hybrid organic–inorganic perovskites. Initially seen as a guiding concept for the discovery of new hybrid organic–inorganic perovskites, the tolerance factor concept has also proven to be a valuable tool for understanding and manipulating the phase stability and properties of existing phases. Since our initial report, there have been many research examples in which tolerance factors were used to understand the existence and stability of certain hybrid organic–inorganic perovskites, while the concept itself has been continuously improved. Here we give an update on the current state of the concept, reviewing the different improvements that have been made over the past few years and drawing on topical examples in which tolerance factors have played a major role.


Among the recent research directions in materials science, the research on organic–inorganic networks that adopt a perovskite motif is one of the fastest-moving areas. Materials such as [CH3NH3]PbI3, [(NH2)2CH]PbI3 and [CH3NH3]PbBr3 have led to breath-taking discoveries in optoelectronics, resulting in a new paradigm in solar cell research.1–6 Likewise, related organic–inorganic perovskites such as [(C3H7)4N]Mn(C2N3)3, [NH3NH2]Zn(HCOO)3, [C6N2H14](NH4)I3 and [(CH3)3NH]Cd(N3)3 show fascinating ferroelectric-to-paraelectric phase transitions and tuneable mechanical properties, and provide great opportunities as working media in mechanocalorics.7–11 Having this collection of properties on hand, it is no surprise that much effort has recently been devoted to the discovery and synthesis of new hybrid ABX3 perovskite-type materials.

Like all-inorganic solid-state perovskites, ABX3 hybrid organic–inorganic perovskites adopt a three-dimensional ReO3-type framework formed by the B-cation and X-anion with the A-site cation sitting in the open void of the network. Focusing on the connectivity of the ReO3-network of different hybrid perovskites, it is possible to distinguish between I3O0-type and I0O3-type perovskites, following the nomenclature suggested by C. N. R. Rao and A. K. Cheetham for organic–inorganic (coordination) networks.12 This classification highlights both the 3D connectivity of the ReO3-type framework and the type of chemical bond involved with “Ix” in the nomenclature reflecting the dimensionality of the inorganic network and “Oy” reflecting the dimensionality of the coordination network. While I3O0-type perovskites can be described as materials with metal-cation–anion interactions in the 3D ReO3-type network (Fig. 1a),13 I0O3-type perovskites exhibit a framework composed of metal-cations and molecular anions (Fig. 1b–e),8,9,14,15 or molecular cations and anion interactions (Fig. 1f).11

image file: c8ta05794j-f1.tif
Fig. 1 Schematic representation of some typical hybrid perovskites: (a) [CH3NH3]PbBr3, a representative of I3O0 hybrid perovskites and (b) [NH3NH2]Zn(HCOO)3, (c) [C3N2H5]Mn(H2POO)3, (d) [PPN]Mn[Au(CN)2]3, (e) [NH(CH3)3]Cd(N3)3 and (f) [H2DABCO](NH4)Br3, as representatives for the chemical flexibility of I0O3 perovskites (DABCO = 1,4-di-azabicyclo[2.2.2]octan and PPN = bis(triphenylphosphine)iminium).

Such a separation into two sub-classes of three-dimensional hybrid perovskites with the general formula ABX3 is useful when focusing on the intrinsic properties that come from the nature of the [BX3] framework. For instance, the semiconducting behaviour of [CH3NH3]PbI3 originates from covalent lead iodide interactions,16 whereas the insulating behaviour of [CH3NH3]Mn(HCOO)3 is based on coordinative manganese oxygen bonds and in turn small energy band dispersion.17 On the other hand, there are properties for which such a separation into two subclasses is less advantageous. For instance, it was shown that molecular building units on the A-site and/or X-site both lead to low energy lattice modes that are thermally accessible.18–20 These lattice modes gain importance when phase transitions as a function of temperature and pressure are focused on. Particularly the propensity of the A-site cation to form strong hydrogen bonding interactions with the negatively charged framework has been revealed to be of outmost importance in this context.21,22 For example, the temperature-driven order–disorder transitions in [NH3NH2]Zn(HCOO)3 are directly related to hydrogen bonding interactions between the [NH3NH2]+ cation and the negatively charged [Zn(HCOO)3] framework.8 Similarly, hydrogen bonding interactions have been identified to play a role in the low-temperature crystal chemistry of [CH3NH3]PbI3.23 It must be emphasized here that these hydrogen bonding interactions originate from the molecular nature of the A-site cation and are inherently absent in all-inorganic perovskites, drawing a clear line between hybrid perovskites and their all-inorganic analogues.24,25 For a general overview that discusses the properties which come from the diversity of hybrid perovskites, we like to refer to some illuminating reviews that have appeared recently on this topic.26–29

When looking at more fundamental design principles towards new organic–inorganic frameworks that adopt a perovskite architecture, separation into subclasses becomes even less useful. In fact, the perovskite motif as such allows for applying the same geometrical considerations to I3O0- and I0O3-type perovskites. These geometrical considerations are at the heart of the guiding principle towards new (hybrid) perovskites: Goldschmidt's concept of ionic tolerance factors (TFs). Already in 2001, Mitzi considered applying Goldschmidt's concept to the family of lead halides, qualitatively estimating the available space for molecular A-cations.30 It was then in 2014 when we applied Goldschmidt's concept to hybrid perovskites,31 describing the size of molecular A-cations such as [CH3NH3]+ and [(CH3)2NH2]+ based on a semi-empirical approach. Since our initial report, this concept has been continuously improved and widely used in the search for new hybrid perovskites. The purpose of this research update is to review the most important improvements in the tolerance factor concept for hybrid perovskites over the past few years, drawing on some topical research examples where TFs have been explicitly applied. We would further like to note that we have used the terminology hybrid perovskites throughout the manuscript, overarching both sub-classes, I3O0 and I0O3, of organic–inorganic perovskite-type materials.

The development of tolerance factors

In 1926, Goldschmidt formulated the tolerance factor concept for inorganic solid state perovskites; the relationship between the ionic radii of the A-site cation, B-site cation and X-site anion obtained by applying simple trigonometry is given in eqn (1) and Fig. 2:32
image file: c8ta05794j-t1.tif(1)
with rA being the ionic radii of the A-site cation, rB the ionic radii of the B-site cation and rX the ionic radii of the X-site anion, respectively (Fig. 2). By treating all ions as hard spheres, Goldschmidt found that for TFs between 0.8 and 1, the perovskite structure is expected to form, with distorted structures usually observed when TFs decrease. This concept has played a central role in the development of oxide perovskites and is a Madelung-type equivalent for the perovskite structure – maximising enthalpic interactions through a high packing density. For all inorganic solid-state perovskites, this concept has been reviewed several times leading to an optimised TF window of approximately 0.87 and 1.33–36 The success of the TF approach and its wide use comes from the manifold properties that can be found in different perovskite compounds and in turn the large interest of the materials science community in the synthesis and characterization of new and optimized perovskite-type materials. These properties evidently originate from the availability of manifold inorganic perovskite compounds with different compositions and properties that can be synthesised such as PbZrxTi1−xO3,37 BaTiO3,38,39 BiFeO3,40 CsPbX3,41 LaxCa1−xMnO3[thin space (1/6-em)]42 or various metal-oxo perovskites for catalysis43 to name just a few. Consequently, inorganic solid-state perovskites are one of the most important materials families in condensed matter physics and materials science.44,45

image file: c8ta05794j-f2.tif
Fig. 2 Schematic of the perovskite structure (middle) and the trigonometric relationship of the ionic radii of the A-site, B-site and X-site that is used to derive the tolerance factor equation.

Motivated by the large interest of the materials science community in the search for new hybrid perovskite materials, we extended Goldschmidt's concept to hybrid perovskites in 2014.31 The challenge was to find a simple and similarly applicable concept of describing the size of molecular cations such as [CH3NH3]+ or [(NH2)2CH]+: what is the ionic radius of a molecular cation and which description is the most appropriate one? Furthermore, the anisotropy that comes with the use of molecular anions such as [COO], [N3] or [Au(CN)2] was important to include. In this context a semi-empirical approach seemed to be the most elegant way, combining experimental inputs with reasonable simplifications: under the assumption of free rotation of the A-site molecule around the centre of mass together with experimental X-ray bond lengths, we estimated effective ionic radii reff,A for a series of molecular A-site cations. For instance, the effective radius reff,A of [C(NH2)3]+ (guanidinium) is obtained by identifying the centre of mass of the molecule (the C atom), and reported X-ray crystallographic data of a [C(NH2)3]+-based perovskite was used to measure the distance from C to N – the non-hydrogen atom farthest away from C. To this distance, the Shannon reported radius of N is added to obtain the reff,A of [C(NH2)3]+, presenting the largest possible spherical radius for a hypothetical fast rotating guanidinium cation. The simplification is clear: despite [C(NH2)3]+ being a planar cation, a spherical shape is approximated, and errors from X-ray crystallographic data must be considered. The anisotropy of molecular X-anions is taken into account by treating these as cylinders, assuming rotation around the main axis of the cylinder and the description of these with an effective height.14,31 Based on the obtained set of effective ionic radii for molecular A-site cations (see Table 2, left column shows a selection of reff,A from our original work) and effective heights of molecular B-site anions, TFs of every A–B–X permutation can be calculated.46 The applicability of TFs has been shown in our initial report,31 where TFs for different hybrid Pb iodides and Mn formates have been calculated and compared to those of experimentally observed crystal structures. For TFs between 0.8 and 1.0, mainly hybrid perovskites where observed, whereas TFs > 1 or TFs < 0.8 predominantly led to low-dimensional structures (Fig. 3).

Table 1 Improved ionic radii of a selection of B-site metal cations typically found in hybrid perovskites as given by Palgrave47 compared to the radii reported by Shannon.48 All values in Å. HS = high spin
Cation Shannon Palgrave et al.
Ionic radius Iodides Bromides Chlorides
Pb2+ 1.19 1.03 0.98 0.99
Mg2+ 0.72 0.75 0.72 0.67
Ni2+ 0.69 0.57 0.58 0.60
Mn(HS)2+ 0.83 0.72 0.72 0.73
Yb2+ 1.02 0.93 0.88 0.86

Table 2 Ionic radii of typical A-site cations found in hybrid perovskites compared with radii reported by Kieslich et al.31 and computationally derived radii as suggested by Becker et al.52 All values in pm
A-site cation Kieslich et al. Becker et al.
r eff,A r eff,A
[NH4]+ 146 170
[NH3OH]+ 216 226
[CH3NH3]+ 217 238
[NH3NH2]+ 217 220
[(CH2)3NH2]+ 250 284
[(NH2)2CH]+ 253 277
[C3N2H5]+ 258 303
[(CH3)2NH2]+ 272 296
[C(NH2)3]+ 278 280
[(CH3)4N]+ 292 301

image file: c8ta05794j-f3.tif
Fig. 3 Showcasing the applicability of the TF concept for the series of hybrid lead iodides and manganese formates as reported in 2014.31 For both families, a perovskite structure is experimentally observed for TFs between 0.8 and 1, using the effective radii as given in the initial publication. Open symbolds indicate reported compounds with a nonperovskite structure.

When looking for the origin of the largest inaccuracies of the TF concept under the assumption that the formula itself is a reasonable approach, the attention is immediately drawn towards the question how accurate are the tabulated ionic radii? Consequently, the most important improvements that have been reported in the past few years concern the definition and description of effective ionic radii for the A-site and B-site cations. Following this thought, Palgrave and co-workers have made an important step forward in improving the concept for hybrid perovskites with X being a halide anion.47 By recognising the different bonding situations of halides in comparison to ionic oxides and fluorides,48 they have revised ionic radii of divalent metals when bound to different halides. For instance, they found that the rB of Pb2+ in a hypothetical [A]PbI3 perovskite is better approximated with rPb2+ = 103 pm, compared to the radius reported by Shannon (rPb2+ = 119 pm), which was derived based on lead oxides. A similar conclusion can be drawn for bromides and chlorides, and the differences along the halide series to the Shannon radii are shown in Table 1. Additionally, Palgrave and co-workers suggested including the well-known octahedral factor μ as an additional parameter when screening new hybrid perovskites.47,49μ accounts for a possible size-mismatch between the B-cation and X-site, putting further constraints on the combination of B-site cations and X-site anions. In the context of ionic radii of halide-based systems, P. M. Woodward and co-workers recently reinvestigated Pb–X bond lengths in the solid solutions Cs1−xRbxPbBr3 and Cs1−xRbxPbCl3, revising bond valence parameters50 for the bond valence method and thus obtaining reasonable Pb–X bond lengths (with X being a halide) to determine revised tolerance factors.51 The obtained trend of band-gap as a function of the tolerance factor for both series is remarkable, suggesting that the bond valence method is a suitable approach for all-inorganic halide perovskites. One key feature of this approach is the linear correlation of the calculated TFs with specific characteristics of the perovskite material, which in turn might uncover local structural deviations to explain interesting material behaviours. However, despite considering different classes of materials, hybrid and inorganic perovskites, respectively, the approaches by Palgrave and Woodward result in relatively similar TFs, e.g. 0.92 and 0.953 for CsPbBr3 and 0.93 and 0.948 for CsPbCl3, which are well within the TF window of ABO3 perovskites as discussed by Woodward in 2001.33

Turning the attention to the A-site cation and the approaches to derive an effective radius for molecular A-cations, Becker et al. approached the challenge of obtaining more precise ionic radii of molecular cations computationally.52 By looking at the total electron density of the gas phase energy-minimised states of several ions and taking the isocharge radius of [NH4]+ as a reference, they calculated the effective ionic radii reff,A of 18 molecular cations. Their results are intrinsically consistent due to the applied method, showing a trend of larger effective ionic radii in comparison to our initially suggested reff,A (see Table 2 for comparison between reff,A from our initial approach and the computationally derived ionic radii by Becker and co-workers). In combination with the revised radii by Palgrave for the B-cation and the inclusion of the octahedral factor μ, they obtained a shifted TF window, i.e. 0.9 < TF < 1.12,47 which shows a general good agreement with our initial estimations. According to the authors, 93% of the already known stable 3D perovskites fit within their stated TF stability range which clearly indicates the powerful descriptive nature of this revised and similarly consistent concept, e.g. inaccuracies that might occur from experimental errors based on X-ray crystallographic data are absent in their computational approach. They suggested at this point 106 unknown compounds for which TFs fulfill the stability criteria for a 3D perovskite (see Fig. 4). It can therefore be concluded here that when calculating TFs, a consistent set of reff,A and the B-metal site should be used, of which the approach by Becker for A-site cations with the improvements made by Palgrave for B-site cations seems the most consistent approach at the current state.

image file: c8ta05794j-f4.tif
Fig. 4 2D mapping of octahedral factors and calculated TFs using the effective ionic radii as given by Becker et al. for which the empirical stability range is found to be 0.9 < TF < 1.12. This plot suggests the existence of 106 hitherto unknown compounds (green crosses: reported 3D hybrid perovskites, red crosses: low dimensional phases, and unreported compounds: diamonds). Figure adapted from ref. 52.

Finally and only recently, the effective radii of A-site cations were further refined by explicitly taking into account the anisotropy of the molecular A-cation.53 Saliba et al. introduced the globularity factor g = S/Seq in which the actual surface S of the molecular cation is related to the volume of the cation Seq when treated as an ideal sphere. Notably, the “real” surface and respective volume have been derived computationally with [(NH2)2CH]PbI3 as a reference with TF = 1. This approach seems useful when the molecular complexity and anisotropy of the A-cation are high, e.g. in the case of guanidinium, [C(NH2)3]+ and formamidinium [(NH2)2CH]+. Importantly, however, this approach has only been validated for [C(NH2)3]+ so far and requires further validation.

Research examples

We now want to discuss a few selected research examples of the recent literature in which the TF concept has either been directly applied, leading to an understanding of the observed results or indicates the presence of a more complex underlying chemistry related to the thermodynamic landscape of the investigated material.

Halide-based hybrid perovskites (I3O0)

Firstly, we would like to stay within the area of three-dimensional halide-based hybrid perovskites which have gained a large amount of research attention due to the outstanding properties of [CH3NH3]PbI3 as a light absorber in thin film solar cells. The TF of the most studied compound [CH3NH3]PbI3 lies well within the TF window of a stable perovskite phase in all the abovementioned approaches. Direct calorimetric measurements show, however, that [CH3NH3]PbI3 is thermodynamically unstable with respect to its components, and consequently much research has been devoted to stabilizing [CH3NH3]PbI3 chemically.54 The use of [(NH2)2CH]+ to form [(NH2)2CH]PbI3 leads to a TF close to or slightly above 1 (depending on the concept used), and polymorphism is observed experimentally under ambient conditions. The [(NH2)2CH]PbI3 perovskite phase is metastable leading to its transformation to a low-dimensional phase, visible by a change of colour from black (perovskite) to yellow (low-dimensional).16 The transformation itself seems to be driven by a strain relaxation process which is caused by the size mismatch of the [(NH2)2CH]+ cation and the void within the [PbI3] lattice,55 effects that are captured in the TF concept. It is therefore no surprise, however, that the perovskite phase can be stabilised by partial substitution with [CH3NH3]+ on the A-site, or in other words, by moving the TF of the compounds into the stable regime. This approach was exploited by Zhu et al.56 for the preparation of [(NH2)2CH]0.85Cs0.15PbI3, (see Fig. 5). Notably, both parent phases CsPbI3 and [(NH2)2CH]PbI3 exhibit TFs slightly too small (Cs+) and slightly too large ([(NH2)2CH]+) to form a stable perovskite phase; however, averaging the radii leads to a TF within the stability criteria for [(NH2)2CH]0.85Cs0.15PbI3. This finding is fascinating by itself, suggesting a relatively flexible inorganic [PbI3] framework in which lattice strain can be balanced by choosing the right A-site cations. Grätzel and co-workers57 then continued the idea of tuning the stability by investigating triple cation perovskites consisting of [CH3NH3]+, [(NH2)2CH]+ and Cs+, leading to a novel compositional strategy to develop perovskite solar cells bearing state-of-the-art performance characteristics. A possible explanation for the high turn-over efficiency of such phases could be the rigidification of the lattice, in turn decreasing carrier combination pathways. Shortly after, other researchers followed the idea of replacing the A-site cation with other molecular cations with different sizes to match the TF concept. For instance, Bakr et al. investigated the incorporation of ethylammonium [NH3C2H5]+ into [CH3NH3]PbI3.58 They seemingly obtained materials with stoichiometries such as [NH3C2H5]0.17[CH3NH3]0.83PbI3, finding optimised optoelectronic properties based on increased carrier lifetimes. Similarly, the incorporation of large guanidinium [N(CH2)3]+ cations was investigated, leading to improved materials stability paired with high turn-over efficiencies and averaged TFs within the stability window.59 In the light of some recent discoveries, however, questions regarding the defect chemistry in such materials arise. For instance, Kanatzidis reported on the incorporation of ethylenediammonium [NH3C2H4NH3]2+ enhancing the photovoltaic performance of the working materials.60 Shortly after, they thoroughly studied the defect chemistry of such A-site solid solutions, revealing a complex defect chemistry behaviour related to materials series such as [CH3NH3]1−x[NH3C2H4NH3]x(Sn)1−0.7x(I)3−0.4x and [CH3NH3]1−x[NH3C2H4NH3]x(Pb)1−0.7x(I)3−0.4x,61 with blue shifted band gaps (hollow perovskites). When further increasing the size of the A′-site cation in solid-solutions with the general stoichiometry image file: c8ta05794j-t2.tif, the formation of low dimensional structures is favoured. These low-dimensional structures depict the hybrid analogue of Ruddlesden–Popper and Dion–Jacobson-type phases.62,63 It should be highlighted here that during the synthesis of such complex materials, particular care must be taken for the analysis of the crystal structure and composition. It was only recently realized that the decomposition reaction of dimethylformamide (DMF) into dimethylammonium (DMA) and the formate anion (DMF → DMA + HCOO) plays a crucial role.64,65 It seems that DMA can itself act as a molecular A-site cation for balancing the TF of the compound to fulfill the stability criteria. For instance, the perovskite phase of CsPbI3 can potentially be stabilized by the incorporation of DMA, very similar to the approach schematically shown in Figure 4.66,67
image file: c8ta05794j-f5.tif
Fig. 5 Schematic showing how the stability criteria of TF can be used to stabilize the metastable phases of FAPbI3 and CsPbI3 by the formation of A-site solid solutions. Reprinted with permission from ref. 56. Copyright (2016) American Chemical Society.

Structural investigations must be performed with care, oftentimes a challenging task when only thin films are available. For instance, Xu et al. initially reported on increasing the moisture stability of [CH3NH3]PbI3 by the incorporation of two pseudohalide SCN anions, thereby challenging the TF concept: the TF of a potential [CH3NH3]PbI(SCN)2 phase is significantly smaller than 0.8 due to the large size of the SCN anions. However, structural characterisation remained challenging as suggested by the authors and shortly after, Hillebrecht and co-workers found that [CH3NH3]PbI(SCN)2 crystallises in the K2NiF4-type structure, again a hybrid representative of a Ruddlesden–Popper phase,68,69 confirming the applicability of the TF concept. This on the one hand shows the increasing complexity of structural investigations when introducing structural complexity in the X-site by the use of molecular anions, but at the same time, underpins the importance of the TF concept in understanding crystal formation of different phases.70,71

Based on the current literature of hybrid perovskites of the type I3O0, TFs seem to be extremely useful in explaining stability criteria for existing and potentially new hybrid perovskite phases. Interestingly, the number of studies that explicitly use TFs for the study of Sn-based compounds which can be seen as less-toxic alternatives to Pb-based perovskites is limited.72 This is not a big surprise, having in mind that TFs have mainly been applied to Pb-based perovskites for improving their stability by manipulating TFs thereby fulfilling size criteria for stability. The metastability of Sn2+-based materials, however, mainly originates from the low potential for the oxidation of Sn2+ to Sn4+ and not from size criteria. Similarly, TFs of double perovskites with the general formula A2BB′X6 which present another way forward in the preparation of lead-free alternatives have not been in the broad focus of attention, and it is questionable if general applicable size criteria can be formulated in these cases. However, despite efforts in the synthesis of hybrid double perovskites,73,74 it has been pointed out recently that orbital-symmetry requirements for the formation of direct band-gaps are difficult to fulfill in double perovskites.75,76 When looking at more complex systems with semiconductive properties, it is important to realize that TFs also capture hybrid organic–inorganic Ruddlesden–Popper77 and Dion–Jacobson78,79 phases in which (2D) perovskite slabs are separated by interlayer motifs typically consisting of A′-cations that are too large to fit into the ReO3-cavity: while the 2D perovskite layers fulfill the TF stability criteria, the separation layers typically involve A′-cations with a relatively large organic backbone, e.g. [CH3(CH2)3NH3] (n-butylammonium), that would lead to TFs significantly larger than 1.30,80,81 For instance, Kanatzidis and co-workers reported on the successful preparation of [CH3(CH2)3NH3]2[CH3NH3]n−1PbnI3n+1 (n = 1, 2, 3, and 4) perovskite thin films, in which n-butylammonium as the very large A-cation separates perovskite layers which fulfill the TF criteria.63

Despite the high agreement of TFs with experimental observations, it is important to realize that in situations where [CH3NH3]+ (or similarly [N(CH2)3]+) is replaced by intermediate-sized cations with different charges, [NH3C2H4NH3]2+ (ethylenediammonium), the TF concept reaches its limitations. In these situations, e.g. the above-mentioned hollow perovskites, the calculation and interpretation of TFs should be performed with care. These findings highlight the importance of in-depth structural studies on defective systems based on high resolution powder X-ray diffraction and X-ray total scattering experiments.53,82 In fact, the complexity of the underlying thermodynamic landscape and in turn the impact of generally weak interactions are nicely seen when looking at the ternary system [NH3NH2]+–Pb2+–I, recently investigated by Miller and co-workers. The authors observe a range of different materials with different compositions such as [NH3NH2]15Pb3I21 and [NH3NH2]PbI3.83 Focusing on [NH3NH2]PbI3, a low-dimensional structure is observed, despite a TF within the window at which a perovskite should be formed. In [NH3NH2]PbI3, a low-dimensional structure is favoured due to a large number of hydrogen bonding interactions between neighbouring [NH3NH2]+ cations, which were observed in related materials.8 Therefore it can be concluded that TFs seem to describe dominant interactions originating from the 3D ReO3-like framework, i.e. ionic interactions, but the propensity of weak interactions originating from the molecular nature of the A-cations challenges this concept. From a chemical viewpoint, however, a seeming breakdown of the TF concept can be the start of interesting chemical interactions and sharpens the chemist's perception of the role of weak molecular interactions.

Hybrid perovskites of the type I0O3

We now want to discuss I0O3-type perovskites, in which the ReO3-type framework is formed by molecular X-anions, e.g. [NH3NH2]Zn(HCOO)3, [C3N2H5]Mn(H2POO)3 and [PPN]-Mn[Au(CN)2]3 (Fig. 1b–f). Due to the underlying coordination chemistry of the divalent metal cation and the X anion, such materials fall well within the large family of coordination polymers. As mentioned above, the difference in the underlying chemistry has large implications on the materials properties, with free lone pairs located at the molecular anions oftentimes acting as anchors for hydrogen bonding interactions.17,21,22,27 Despite the chemical complexity coming from the molecular nature of the X-anion, it is fascinating to observe that the TF concept still holds good for the majority of these materials. For instance, materials families such as formates, azides, dicyanometallates and dicyanoamides form perovskite-type materials of which most of the materials exhibit TFs that fall within the window between 0.8 and 1 (see Fig. 3 for Mn-based formates).15,29,46,84 An exciting finding of the recent literature in this context is the discovery of a new family of I0O3-type hybrid perovskites based on the hypophosphite [H2POO]-anion, extending the parameter space by establishing a new molecular X-anion.14 Compounds such as [N(CH2)3]Mn(H2POO)3, [(NH2)2CH]Mn(H2POO)3 and [C3N2H5]Mn(H2POO)3 highlight the applicability of the TF concept and at the same time, show the robustness of TFs towards cage distortions. To assess the distortions of the materials in more detail, the authors suggested the introduction of a distortion factor, enabling the comparison of the distortion of the different ReO3-type 3D networks related to their observed volume. Such an analysis has been previously suggested qualitatively for formate-based perovskites, leading to a classification in which large cage distortions and different binding situations of the formate anions are captured.85,86 Importantly, such a classification also includes materials families in which monoatomic or very small A-cations are used as it is the case in perovskite-type materials such as RbMn(HCOO)3 and (NH4)Cd(HCOO)3. Very similar to the abovementioned halides, solid-solutions on the A-site can be used to tune dielectric properties, whilst staying within the TF window. Examples are the series [NH3NH2]1−x[CH3NH3]xMn(HCOO)3 and [NH3NH2]1−x[NH3OH]xZn(HCOO)3 in which the temperature of the ferroelectric-to-paraelectric phase transition temperature can be tuned by varying x.87,88 Interestingly, such solid-solutions offer the possibility to incorporate A-cations which do not form a perovskite-structure in their parent phase, e.g. [NH3OH]+ in [NH3NH2]Zn(HCOO)3. The key behind the tuneable properties is the incorporation of small structural distortions, which directly achieves the balance between hydrogen-bonding interactions, and configurational and vibrational entropy – parameters that go beyond the TF concept. A similar conclusion can of course be drawn for related materials such as azides and dicyanamides.89,90 Importantly and very similar to halide based perovskites, the energy landscape seems very shallow, and consequently, tuning the reaction temperature and solvent has a large impact on the obtained materials, again emphasizing the role of thermodynamic parameters. This was nicely shown by Mączka and co-workers, who investigated the effect of solvent, temperature and pressure within the family [NH3NH2]M(HCOO)3 with M = Mn, Zn, Co and Fe.91 For M = Zn and Mn, a perovskite structure is observed whilst M = Co and Fe lead to the formation of a low-dimensional structure, despite the TF concept suggesting the formation of perovskite-type phases for all divalent metals.8 It was shown that for this family, vibrational entropy is an important parameter, flattening the energy landscape and in turn enabling experimentalists to assess polymorphism by changing reaction conditions.92 Vibrational entropy in general has been identified as playing an important role in such materials, being the driving-force behind temperature driven phase transitions.19,93 In the rigid body approach, the TF is not designed to capture such effects, and polymorphism in [NH3NH2]Zn(HCOO)3 and potential phase transitions in general are tough to predict. Furthermore, comparisons can be drawn between the low dimensional structure of the [NH3NH2]Zn(HCOO)3 polymorph and the abovementioned [NH2NH3]PbI3 phase, in which hydrogen bonding interactions between [NH3NH2]+ cations are observed. In general however, TFs are a very good guideline that gives experimentalists the first idea about the A–B–X permutation for which a perovskite-type structure is expected. The increased parameter space in I0O3 hybrid perovskites, however, raises challenges in the targeted incorporation of distinct materials properties, while the balance of weak interactions such as dispersion interactions, hydrogen bonding interactions, and configurational and vibrational entropy together with the large library of molecular cations opens intriguing opportunities in caloric applications at the same time.10,94,95

Lastly, a fascinating class of hybrid perovskites moved into the focus of attention of the materials science community that fall within the sub-class of I0O3 hybrid perovskites: metal-free ferroelectric perovskites. In these materials, the three dimensional ReO3-type framework is built of [NH4]+ cations and X-anions such as Br and I and are charge balanced by divalent A-site cations such as [(CH2)6N2(CH3)H]2+, [(CH2)6N2(OH)H]2+, [(CH2)3(CH)N2H5]2+ and [(CH2)5(CH)2N2H4]2+.11 Some of these materials exhibit a complex structural behaviour as a function of temperature, with the rotation of the A-cation similar to the temperature activated disorder observed for [CH3NH3]+ in [CH3NH3]PbI3. Initially reported in 2003 by Bremner and co-workers,96 it was only recently found that large spontaneous polarization exists in [(CH2)6N2(CH3)H](NH4)I3, competing with the classical ferroelectric BaTiO3. In these materials, the bonding situation within the network challenges the TF concept: whilst in some cases the TF concept is fulfilled, e.g. size criteria seem to dominate the energetic landscape, in other cases low dimensional, so-called hexagonal perovskites are preferred due to the possibility to form intermolecular interactions between A-site cations. It seems that due to the unusual bonding situation within the ReO3-lattice, weaker interactions become even more important, and the TF concept should be carefully applied. However, for TFs > 0.9, mainly 3D perovskites are observed, pointing to the general applicability of the concept across various types of hybrid perovskites.

A bright future?

After reviewing the most remarkable research examples that have been reported in the past three years, it is only natural to ask where can TFs take us in the future? When discussing this question, it is important to realise that the TF concept itself only depicts a broad guideline towards new hybrid perovskites and is far away from being a thermodynamic rule. As outlined above, the beauty of this guideline lies in its simplicity, making it possible to calculate TFs from available literature data within minutes. The abovementioned study by Becker and co-workers is only one study out of few in which TFs suggest the existence of many undiscovered phases.46,47,52,97,98 At the same time, the concept sharpens our chemical intuition, not only for hybrid perovskites, but also for the complexity of organic–inorganic materials in general. For instance, when the TF of a hypothetical A–B–X permutation indicates the existence of a perovskite-type material but is experimentally not observed, or vice versa, it can be interesting to ask why? This can be the start of fascinating discoveries such as strong entropic effects or related phenomena.92–94 In this regard, hybrid perovskites can remind us about the importance of lattice entropy in organic–inorganic materials. Following this thought, the TF concept is a rigid-body approach, intrinsically neglecting dynamic effects. In other words, rigid-body guidelines such as the TF concept (or similarly the reticular chemistry approach for metal–organic frameworks) provide a toolbox to understand and build network materials based on enthalpic interactions, whilst dynamic effects hidden in vibrational and configurational entropy are intrinsically neglected. The long-term goal might therefore be to find design principles, or at least common structural anomalies, that allow for assessing entropic effects more generally. The first step towards this goal can be database assisted approaches that help screening for trends in symmetry-breaking phenomena. Similarly, in I0O3 perovskites, tilts and shifts are observed that are not present in all inorganic perovskites, and despite inspiring studies, a broad overview of such effects is still missing.99,100 Whether such studies will lead to further improvement of the TF concept is an open question for ongoing research, and it will be exciting to see how this development goes on. Therefore, when looking at the development of hybrid perovskites in the past few years, it can be summarised that the tolerance factor concept is an overarching guiding principle for chemists and physicists of different areas that can lead experimentalists in the right direction towards new hybrid perovskite materials, who should, however, always keep the limitations of such a concept in mind.

Conflicts of interest

There are no conflicts to declare.


GK and ME would like to thank the “Fonds der chemischen Industrie” for financial support through the Liebig Fellowship scheme. The authors are very grateful for insightful discussions with Ian Sharp and Anthony K. Cheetham.


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