Metallic alloys at the edge of complexity: structural aspects, chemical bonding and physical properties

Complex metallic alloys belong to the vast family of intermetallic compounds and are hallmarked by extremely large unit cells and, in many cases, extensive crystallographic disorder. Early studies of complex intermetallics were focusing on the elucidation of their crystal structures and classification of the underlying building principles. More recently, ab initio computational analysis and detailed examination of the physical properties have become feasible and opened new perspectives for these materials. The present review paper provides a summary of the literature data on the reported compositions with exceptional structural complexity and their properties, and highlights the factors leading to the emergence of their crystal structures and the methods of characterization and systematization of these compounds.


Introduction
Intermetallic compounds represent an important class of solid materials with numerous applications. Unlike simple alloys, whose crystal structures can be derived from the structures of elemental metals by statistical mixing in the atomic sites, intermetallic compounds crystallize with atomic arrangements different from those of the constituting elements. The formation of such structures is governed by a combination of geometric factors, such as the ratio of the atomic sizes, and electronic requirements, i.e. optimization of the chemical bonding. Since metallic systems typically demonstrate significant electron delocalization around atomic centers, chemical bonding in simple alloys lack directionality similarly to ionic bonding in salt-like compounds. Consequently, such alloys usually adopt crystal structures derived from 3D close packings. Different combinations of metallic and covalent bonds can though be present in intermetallic compounds, leading to increasing structural complexity which is reflected in the materials' properties. The latter range from metallic, originating from delocalized bonding, to narrow-gap semiconductors as the result of prevailing strongly localized chemical interactions.
The great diversity of the bonding patterns as well as spatial arrangement, frequently including atomic disorder, result in a rich and complex structural chemistry for intermetallic compounds. A particular acknowledgement of this complexity is the widely used classification of certain intermetallic structures as complex metallic alloys (CMAs) [1]. This class encompasses materials composed of metallic elements and crystallizing with large unit cells. The spatial scales of translational symmetry in such compounds greatly exceed the typical interatomic distances, leading sometimes to glass-like properties, e.g. abnormally low thermal conductivity [2]. Historically, the first metallic structures recognized as exceptionally complex were the binary intermetallic phases in the systems Na-Cd, Mg-Al, and Cu-Cd [3][4][5]. Although the compounds in these systems displayed relatively simple approximate compositions-NaCd 2 , Mg 2 Al 3 , Cu 4 Cd 3 -the crystal structures were hallmarked by huge unit cells with hundreds of atoms and extensive disorder. Since then, many more structures boasting extraordinary complexity have been discovered and characterized in different multicomponent systems [6] including the extreme case Al 55.4 Cu 5.4 Ta 39.1 with over 20 000 atoms per unit cell [7]. The application of the term 'intermetallic compound' has expanded to a wide range of substances showing peculiar metal-metal interactions and is not limited anymore to systems containing exclusively metallic elements, but is extended to representatives with some lighter main group elements. In the light of these developments, representatives of different families of compounds have been described as CMAs. Examples include metal-rich borides [8,9], carbides [10,11], and nitrides [12][13][14]. A special place in this family is reserved for Si and Ge. These two rather semi-metals have been frequently observed in formation of compounds isostructural to those of pure intermetallics [15,16].
Initial studies of CMAs were limited to elucidation of their crystal structures. The extreme structural complexity precluded theoretical investigations, owing to the lack of computational resources and suitable theoretical methods, whereas the frequently encountered disorder and hence varying compositions made directed synthesis and further physical property measurements exceedingly challenging. With the advance of theoretical calculations nowadays, first-principle investigations of CMAs have become possible [17][18][19]. The analysis of electronic structure and chemical bonding aided in rationalization of the sophisticated atomic arrangements, extending thereby the simple historical valence electron counting rules developed for various classes of intermetallics [20]. Developments in the field of mathematical topology and its applications to crystal structures enabled new ways of classifying complex intermetallic structures and put them in a broader context of 3D nets [21]. With optimization of the synthetic conditions, phase-pure samples of CMAs have become feasible. Examination of their physical and chemical properties have revealed that many of these compounds have potential applications as thermoelectric [22], catalytic [23], or construction materials [24]. The importance of complex intermetallic systems and the need for their systematic studies led to the creation of the European Network of Excellence complex metallic alloys (NoE CMA) in 2005 (now the European Integrated Center for the Development of New Metallic Alloys and Compounds [25]).
In this review paper, we will focus on the reported data on CMA with particular focus on their crystal structures and physical properties. As has been shown in the recent work [6] the number of reported compound decreases exponentially with the size of the unit cell. There is just a single representative with the unit cell volume over 100 000 Å 3 , around 100 over 10 000 Å 3 and more than 4500 over 1000 Å 3 . Having no doubts about the complexity of the extreme representatives, the lower limit for CMA or the border between 'simple' and 'complex' is still uncertain. Several approaches exist in the literature regarding how the structural complexity should be defined, in particular with respect to the size of the unit cell. One of such approaches suggests that an intermetallic compound can be regarded a CMA if there are more than 100 atoms in its primitive unit cell [26]. Although this definition suffers from arbitrariness, it allows easy analysis and screening of crystal structure databases. Herein, we will adhere to this interpretation of a CMA. However, compounds, which do not strictly satisfy the proposed definition but display interesting physical phenomena stemming from complex atomic arrangements in their crystal structures will also be highlighted.

Experimental and theoretical characterization of CMA
Intermetallic phases, including CMA, are usually obtained by high-temperature treatment of the constituting elements in inert reaction vessels, although some other approaches are widely used as well, such as spark plasma sintering [27], flux growth [28], and soft chemistry methods [29]. Production of new compounds in a multicomponent system usually requires exhaustive or selective exploration of the phase space. Although structural complexity per se does not necessarily impede preparation of single-phase samples, extensive dis order and possible homogeneity ranges in many complex intermetallic compounds complicate synthesis of pure materials with targeted compositions.
Preliminary examination of crystalline samples prepared using any of the methods mentioned above typically involves powder x-ray diffraction analysis (PXRD), whereas thorough crystallographic characterization is usually accomplished by means of single-crystal x-ray diffraction (SCXRD). For crystals of CMA, the reciprocal space is populated by a large number of Bragg reflections, which often hinders structure determination with laboratory x-ray diffraction instruments. High-resolution SCXRD measurements employing synchrotron facilities are frequently indispensable in these cases [7].
When suitable single crystals are not available, ab initio structure determination from PXRD data [30] or electron diffraction on microcrystals [31] can be utilized to generate and refine a model of the crystal structure.
Since x-ray diffraction provides distribution of electron density averaged over all unit cells in a measured sample, additional techniques may be necessary for unambiguous assignment of atomic chemical types or for resolving local structural features, especially in crystallographically disordered mat erials. Such techniques include, among others, neutron diffraction [32], nuclear magnetic resonance [33], and transmission electron microscopy [34].
After the crystal structure of an intermetallic compound has been established, further insights can be obtained through first-principle based methods, e.g. the factors responsible for the stabilization of a particular atomic arrangement or particular physical properties. Since ab initio treatment of highly complex and/or disordered materials is usually computationally demanding, idealized models or crystal structure fragments are commonly used for calculations [35][36][37]. Electronic instabilities inferred from such calculations often point toward perturbations of the translational order in the approximate model and enable explanation of the experimentally observed complexity. Depending on the employed computational approach, such instabilities can be described in terms of, e.g. excessive 'chemical pressure' [38,39] or under-optimization of chemical bonding [40][41][42].
Whereas electronic structure calculations provide information on energy landscapes in the ground state, entropy may be an important factor causing complex atomic arrangements in metallic materials at finite temperatures. Development of crystallographic disorder and hence distortion of translational symmetry normally takes place upon heating [43]. In the extreme case, crystallinity can be completely destroyed by rapid quenching of metallic melts with the formation of metallic glasses [44]. Glass-like properties are also observed for many crystalline CMAs with large unit cells. Besides affecting electronic and heat transport [2,45], perturbation of translational symmetry has a strong impact on cooperative electronic phenomena, such as magnetism. Spin-glass magnetic behavior has been reported in complex intermetallics with extensive disorder [46,47]. Nevertheless, disorder-free CMA phases can exhibit long-range magnetic order or superconductivity in the ground state, even when the unit cell sizes greatly exceed the length scales of the interatomic distances [48][49][50]. However, thorough characterization of physical properties for such materials remain challenging due to extreme complexity of their crystal structures and large number of bands in the electronic spectra.

Frank-Kasper phases
Frank-Kasper (FK) phases, polytetrahedral or topologically close packed (TCP) structures, represent one of the most populated groups [6] of intermetallics. In contrast to close packed structures of pure metals Mg (hcp) or Cu (ccp), FK phases are formed by two or more metals with distinct atomic sizes leading to different coordination numbers and certain variety of structural motifs. Ideal FK polyhedra have coordination numbers (CN) 12, 14, 15 and 16 (figure 1) possessing local symmetries I h , D 6d , D 3h and T d , respectively. However, the real solid-state compounds frequently exhibit a significant degree of distortion. Classic FK phases include in line with quite simple and wide spread, e.g. Laves or A15 phases, a good number of really complex formations. A broad overview of existing and potentially existing, but not yet experimentally confirmed FK patterns can be found in the work of Sikirić [51]. Here we focus on the most complex representatives and their occurrence in the periodic table. This family includes also a large number of quasicrystals and their approximants, which will be considered in a separate section.
Among the classic FK only so-called M, P and R phasesrepresented by Nb 24 Ni 21 Al 7 (52 atoms/cell) [52], Cr 10 Mo 25 Ni 21 (56 atoms/cell) [53], and Co 5 Cr 2 Mo 3 (159 atoms/cell) [54] respectively, can be assigned to moderately complex. All these alloys can be presented as direct structural derivatives of the σ phase (such as CrFe, 30 atoms/cell) [55] and are closely related to each other with nearly identical numbers of atoms per corresponding primitive unit cells. They exhibit a full set of FK type polyhedra with the smallest atoms having icosahedral coordination (CN = 12). The larger ones have CN = 14, 15 and 16, with 14 being the most frequent. The R phase is more common, though 90% of the known representatives contain Si/Ge or, alternatively, small atoms like ferrous metals, and, on the other hand, forms almost exclusively with W, Mo, Mn, or V. Participation of different trans ition metals has been observed solely in the form of impurities (up to 5 at.%) [56]. The single exception in this d metal rich family is an s-p system (s, p and d refer to the valence shell electrons of the constituting atoms) representative ε-Mg 2 Al 3 [57].
Although s-p systems are well-represented in the simple FK structures, e.g. Laves phases [58], they are rarely observed in the complex formations. Besides the ε-Mg 2 Al 3 , only the µ-type related phases K 7 Cs 6 [59] and Mg 4 Zn 7 [60] can be assigned to that group. It shall also be noted that Mg plays a special role in these systems being able to form compounds with both p and d elements extending the FK family. K 7 Cs 6 is a unique representative being formed solely by s elements. Its structure is closely related to the µ phase with identical coordination polyhedra and even their ratio. Due to participation of the large alkali metals its unit cell is about five times larger (~2300 Å 3 ) than that one of the µ phase (~500 Å 3 ). This type also offers another unique combination for the FK phases an  13 Au 19 Ga 8 , MgFe 6 Ge 6 , Cs 6 K 7 and MgZn 2 , respectively. spd system Li 8 (Al,Cu) 18 [61]. Mg 4 Zn 7 is significantly more complex compared to the classic Laves phase MgZn 2 [62]. This compound keeps nearly the same 2:1 ratio of the icosahedral/Friauf units, however, with the minor presence of the two remaining FK polyhedra.
The remaining complex FK alloys include Mn 77 Fe 4 Si 1 [63], Mn 45 Co 40 Si 15 [64,65], V 41 Ni 36 Si 23 [66] and V 2 (Co,Si) 3 [67] types and show major Si contents. Three formers are exclusive representatives of their own structure types while the last one contains six alloys. Mn 77 Fe 4 Si 19 (K phase), Mn 45 Co 40 Si 15 (X phase), and the above-mentioned R type alloys are closely related, all representing plane-layered TCP structures. Both V 2 (Co,Si) 3 and V 41 Ni 36 Si 23 represent monoclinic derivatives with the latter being the most complex within the related compounds. Both structures can also be described in terms of rumpled layers stacked together resembling those of the K or R phase. The V 2 (Co,Si) 3 type is the most represented and interestingly does not contain any other silicides but solely Al and Mg compounds [67,68]. As in two previous groups, Mg forms a binary solid solution Mg 2+x Ir 3−x [68] introducing a new transition metal to the family of complex FK phases.

Beyond Frank-Kasper-an extended nestle of complexity
In contrast to classic FK phases a good number of complex intermetallics in general follows FK building principles but contain a minor amount of structural peculiarities, e.g. defects, disorders or moderate changes in atomic coordination, e.g. octahedral fragments, moving them slightly away from close packed structures. Though their classification in this group is somewhat arbitrary, the majority of polyhedra in the crystal structure (but not all) are of FK type. Each of the subgroups here contain some characteristic features, though any clear separation is hardly possible, and a reasonable overlap can be observed.

Giant unit cells
The classic examples include a few so-called Samson phases, until recently the most complex representatives of intermetallics. Three of them, ~β-Mg 2 Al 3 [4,37], ~NaCd 2 [3], and ~Cu 4 Cd 3 [5] (all compositions are non-stoichiometric), contain over 1000 atoms per crystallographic unit cell, the majority of coordination numbers are in the range 12-16 and represended by icosahedra or Friauf polyhedra as the main coordination units. Although, the structures contain polyhedra with unusual coordination numbers 13 and 11 and exhibit both positional and occupational disorders leading to a less dense packing. Similarly, ~Cu 4 Cd 3 is complemented with pentagonal prisms leading to octahedral voids. This compound shows key features of a Mackay type approximant crystal (see below). A few other examples-Mg 21 Zn 25 [69], Al 69 Ta 39 [70], and Mg 3 Cr 2 Al 18 type [71,72] are the cases with all coordination numbers being 12, 14, 15 or 16 but not all coordination polyhedra satisfying FK criteria, though they include features of bcc type packing, e.g. capped pentagonal or hexagonal prisms introducing octahedral voids. A group of closely related compounds with the (approximate) composition AB 6 also belongs to this family. The prominent examples are ~Na 44 Tl 7 [73], ~Mg 6 Pd [74] and the fully ordered Mg 44 Ir 7 [75] type phases. This series partially belongs to a larger family of γ-brass derivatives and will be discussed from a different viewpoint below. ~Na 44 Tl 7 and ~Mg 6 Pd remind of ~Cu 4 Cd 3 due to the presence of a large number of pentagonal prisms, but contain at least one position with lower coordination numbers, nine and ten, respectively. The ordered variant of the ~Mg 6 Pd type has later been observed in the Mg-Ir system with the stoichiometry closer to 7:1-Mg 29 Ir 4 [76].

γ-brass superstructures
γ-brass is a class of intermetallics that formally belongs to the Hume-Rothery phases (alloys) whose stability is defined by valence electron concentration and geometric requirements. Historically the name comes from the Cu-Zn alloys having ~33-39 w.% Cu contrary to α, β and other mixed types long before the era of the x-ray structural analysis. Due to its mechanical properties and gold-like appearance, brasses are used in multiple daily applications. The crystal structure of Cu 5 Zn 8 was first solved in 1931 [77] and followed by several isostructural findings whose compositional analysis revealed an empirical 21 valence electrons/13 atoms rule as the electronic stability criterion. Further intensive explorations in the field of intermetallics [16,78,79] revealed that γ-brass motifs are not so rare and the stability ranges are not so strict or welldefined. The classical γ-brasses are moderately complex, but the identical structural motifs are base or major building units in a number of CMAs. The classical γ-brass cluster itself can be described with two approaches based on the central tetrahedral core. According to the first one the cluster consist of a central tetrahedron surrounded by a bigger tetrahedron followed by an octahedron and finally a cuboctahedron (figure 2 left).
The second one appears more native for homoatomic specimens and consists of four interpenetrating icosahedra around one common tetrahedron (figure 2 right). A detailed overview of the γ-brass cluster distribution in intermetallics has been published recently [80], so we will mainly focus on the main tendencies and peculiarities within the most complex cases.
The majority of these cases contain γ-brass clusters combined with other cluster types, so strict electronic requirements are not applicable in general. The most closely related is γ′brass-a 2 × 2 × 2 F superstructure with the NaTl type of cluster packing, i.e. polyhedral Zintl phase (see chapter 7). Some of them, in fact, fulfill electronically an extended Zintl phase formalism, e.g. Li 21 Si 5 can be represented as [Li 22 Si 4 ] 4+ [Li 2 0 Si 5 ] 4− [16] . γ′-brass structures may include slightly modified building units including additional, distorted or missing shells including the centered 27 atom bcc type cluster, 29 atom α-Mn type cluster, 22 (30) atom clusters with missing(extra) inner tetrahedron etc. The most numerous representatives of this family belong to the Dy 4 CoCd [81] and the Sm 11 Cd 45 type [82] being on the other hand also the most electron rich. The former almost exclusively consist of γ-brass clusters with single transition metal atoms filling the voids. A truly pure F type γ-brass is observed only in Cu 40.5 Sn 11 type [83] with four compositionally different 26 atoms clusters filling the space. The 26 atom cluster in combination with different clusters (not limited to γ-brass ones) has been observed in a plenty of other F type representatives including the above mentioned Samson phases-~Mg 6 Pd [74], ~Na 44 Tl 7 [73], or Al 69 Ta 39 [70]. The most complex in this group is perhaps Ba 16 Na 204 Sn 308+x with up to 542 atoms per cell [84]. Competing derivatives with classical 26 atom cluster include a quite represented Th 6 Mn 23 [85] and Sc 11 Ir 4 [86] types. It is worth noting that the γ-brass cluster has never been observed for pure metals and is also quite rare in compounds and even more rarely observed as a formally isolated structural fragment. In this light, homoatomic Li 26 clusters in Li 13 Na 29 Ba 19 [87] represent a unique case where all atoms of the same type are concentrated in γ-brass units (figure 2 right). A few other elements (Al, Cd, Cu, Li, Mn and Zn) still form identical/distorted homoatomic clusters as a part of larger structural formations [83,[88][89][90][91][92], though only two of them belong to complex-Cu 40.5 Sn 11 type [83] and (FeNi)Zn 6.5 [92].

Polyicosahedral Li formations
A small but diverse group in this family is represented by socalled s-bonded intermetallics-compounds and alloys formed strictly by s elements. In spite of negligible differences in electronegativity, elements of groups I and II form a decent number of alloys including intragroup compounds. Their majority is represented by simple structure types with, as a rule, up to 20 atoms per unit cell, while complex and extremely complex cases are also not rare here. Not at least due to the difference in the atomic sizes, heavier alkaline earth metals form a number of binary phases with Li exhibiting clear separation of the atomic roles. Starting from the binary BaLi 4 [93], Li tends to form polytetrahedral clusters growing into (poly)icosahedral clusters. This tendency is also kept in the ternary regions, e.g. Li-Na-Ba or Li-Ca-Ba systems, leading to extremely large unit cells. The 'simplest' compounds in this group, the binary Sr 19 Li 44 [94] and Ba 19 Li 44 [95], contain ~250 atoms per unit cell (~9000 Å 3 ) and double fused Li icosahedral clusters. The same units are observed in the ternaries Li 33.3 Ba 13.1 Ca 3 and Li 18.9 Na 8.3 Ba 15.3 (up to 900 atoms per unit cell and over 30 000 Å 3 ) [96]. Surprisingly, the role of Ca or Na in these phases can be played by the more electronegative In resulting in the isostructural compound BaLi 1.06 In 1.16 [97]. This example shows how diffuse sometimes the border between formally polar and unpolar formations is. Li feels equally comfortable with both the active metals and the main group elements, but in both cases tends to form homoatomic motifs separated from the rest of the structure where it may also play some roles. The last case in this subgroup, Li 13 Na 29 Ba 19 (~20 000 Å 3 ) [14] shows complete segregation of Li polyicosahedral units forming the γ-brass clusters. All these compounds exhibit coordination numbers from 12 to (rarely) 17 and display the majority of FK type polyhedra. However, due to significant difference in the atomic sizes, larger empty Ba 6 octahedra can also be observed.

Quasicrystals, approximants and approximantlike structures
After their discovery four decades ago [98], quasicrystals (QCs) received broad recognition and started penetrating various areas of fundamental research and daily applications [99]. Though aperiodic ordering has been observed in a variety of unexpected areas and scales (e.g. in the so called soft QC systems [100][101][102]), all classical solid state QCs belong almost exclusively to intermetallics, while a few metalloids (Si, Ge and recently Te) can also be involved [103]. Due to practically unlimited unit cells (due to the absence of the translational symmetry) all QCs are by default considered complex. They can be systematized based on the rotational symmetry and thereafter on local structural motifs. Two main classes can be outlined-formations with 3-and 2-dimensional quasi periodic ordering, each of them containing a number of subgroups. QCs are in turn followed by slightly less complex structurally related but fully crystalline approximant crystals (ACs). The latter serve as useful hints for understanding of the atomic ordering in aperiodic formations and are related to the corresponding quasicrystals according to a q/p = 2a R (p + qτ)/ (2 + τ) ½ [104], where q/p (two neighboring Fibonacci numbers) denotes the order of the approximant, and τ is the golden mean (τ = (√5 + 1)/2. So higher are the q and p , so closer is the atomic ordering to the approaching QC phase, while those of the order 2/1 (and higher) already contain all necessary building units present in QCs.

3D quasiperiodic formations
3D aperiodic formations or icosahedral quasicrystals can be divided into three main groups based on Bergman (B) [105], Tsai (T) [106] and Mackay (M) [107] type clusters as their core building units. All of them can be represented in terms of multiple endohedral clusters starting from an icosahedron (B and M types, figure 3) or positionally/orientationally disordered tetrahedron (T type, figure 3). Initial studies showed that these three types are quite well separated electronically based on valence electron count, showing just a minor overlap for the B and T types (1.9-2.1 e/a). However, recent discoveries proved the almost complete overlap of their areas of existence [103,108]. On the other hand, the valence electron count for the transition metals has always been in part controversial leading to somewhat arbitrary numbers. According to this classification QCs fall in the area between the Hume-Rothery alloys and the electron precise Zintl phases sharing the space with a few other groups that will be discussed below. They possess features of all these classes and therefore cannot be strictly assigned to any of them. Chemically, these three types are separated based on the representative elements and their combination and not less important the 'forbidden' elements. For example, B and T types are strictly separated by the active metals involved, with Na belonging to the B type and Ca to the T type. On the other hand, the M type always contains either Al or Zn but no electropositive (group I-III) metals. Rare earth metals are, as a rule, representatives of the T type, though can be found in the B type but solely in the simultaneous presence of both Mg and Zn [109,110]. Li and Mg can (sometimes simultaneously) participate in both the cationic and the anionic substructures and therefore may also be observed in the both B and T types but in different roles [111,112].
The B type is frequently ascribed to the FK type since its approximants belong to tetrahedrally close packed structures and exhibit identical polyhedra with the coordination numbers 12, 14, 15 and 16, as discussed above, and interestingly all of them in the same structure simultaneously. The vast majority of stable QC phases of this type contain Mg, and a few unique cases were observed with Li [113], Na [103] and in the Zr-Ti-Ni system [114]. The T type has many common features with the B type e.g. identical second and third shells and similar fifth shell but atoms types participating in each of them are different (figure 3). The central position is different quantitatively (12 versus 4 atoms) but due to significant orientational disorder the inner tetrahedron in the T type emulates a cuboctahedron moving at least little closer to the B type. The fourth shells are completely different, while the fifth one in the T type can be considered as a complex derivative of the B type. Cluster types in each of these types are identical at the body center and at the origin. The M type in this view is definitely more complex as a clear difference of the origin and the body cluster shells become evident above the third level. Icosidodecahedra at the origin are surrounded by rhombicosidodecahedra sharing triangles with the central icosidodecahedron and preventing its further expansion.
All types' approximants of the order 1/0, 1/1 and 2/1 are known [115][116][117][118], with all of them of the order 1/1 and higher being complex enough to be considered in this work. A unique 3/2-2/1-2/1 approximant has been detected and most important refined solely for the Bergman type [119]. A rhombohedral distortion variant of the 2/1 approximant of the T type [120] and a tetragonal superstructural variant of the i-Li-Cu-Al [121] are also known in Ca 3 Cd 17 Al and τ-Al 56 (Cu, Zn) 11 Li 33 , respectively. It shall be noted that some of the discovered isostructural phases have never been formally considered as approximants due to missing (not detected) QCs, e.g. Na-Au-Tt phases [122]. On the other hand some systems contain multiple candidates for the approximant position [103]. Limited cases are known where the cluster packing does not allow a clear identification of the approximant type being a combination or a derivative of the known types. For example, Al 67 Pd 11 Mni 4 Si 7 [123] and Al 67−0.125x Pd 11+0.375 x Mn 14−1.25x Re x Si 8 solid solution [124] are considered as 1/1 approximants of the Mackay type QCs; however, a detailed analysis of their cluster shells revealed one cluster sequence being with little exclusions close to the Bergman type while the other one is of a new type starting from an icosahedron followed by cubocta hedral motifs pointing towards a new M subtype. Similar deviations from the classic M type have also been observed in Mo 7 Sn 12 Zn 40 /V 7 Sn 12 Zn 40 [125,126] An even more complex case has been observed in Al 28.3 Co 7.7 Ge 0.5 Pd 4.6 [127] containing three different pseudo-Mackay clusters only partially following the shell sequence in the classical Mackay type approximants.

2D quasiperiodic structures
2D quasiperiodic structures exhibit in-layer quasiperiodic ordering combined with a periodic stacking of those layers. Multiple symmetry-based subgroups of 2D QCs are known with heptagonal (h, 7-), octagonal (o, 8-), decagonal (d, 10-) and dodecagonal (dd, 12-fold) being the most frequently observed. Heptagonal quasicrystals have not been yet experimentally confirmed and the potential approximants include relatively simple borides and carbides [115]. A few octagonal QCs have been detected as metastable phases during rapid solidification [128][129][130][131]. One possibly stable dodecagonal QC has been observed for a tantalum telluride [132] and some metastable dd phases were observed in transition metals bulk alloys or thin films [133][134][135]. A few dd-QC approximants have been observed in the same Ta-Te system exhibiting pretty large unit cells with some analogy to the 3D approximants of the different orders [136,137]. The structural motifs in dd-QCs and approximants are represented by squares and equilateral triangles (less frequent empty hexagons due to diso rders) with local five-and six-fold centers ( figure 4).
The vast majority of the 2D formations belong to the decagonal systems including the historical discovery [98]. Most of them are preferably Al based while Zn-and rarely Ga-based formations could also be observed [115]. Since d-QCs are mostly represented by transition and post transition elements, their area of existence partially overlaps with the above-discussed 3D QCs of the M type. d-QCs with the broad stability ranges have been observed in two particular systems with the late 3d metals-Al-Ni-Co and Al-Cu-Co, while most others are restricted to nearly point compositions [138]. A completely chemically isolated case is a so called FK d-QC phase in the RE-Mg-Zn systems [139,140] with Mg 4 Zn 7 [60] serving as the closest crystalline approximant. It is worth noting that RE-Mg-Zn systems are known for the B type i-QC phases and particularly in Y-Mg-Zn both d and i phases have been found to coexist [141]. TEM invest igation revealed that this d-QC form a standard 2-layer model and is best described with the help of additional interstitial layers leading to the observed ~25% increase in the layer thickness [140].
d-QC approximant structures (figure 5), in analogy with the corresponding quasicrystals, are also characterized by the layer periodicity (roughly 4Å per each pair along the tenfold axis) and, in analogy with the icosahedral approximants, by the order. With a unique exception, all the structures are orthorhombic or monoclinic with the β angle approaching pentagonal ≈108°. Most of the structures with 4 and higher layer periodicity satisfy all requirements for complex alloys including various approximants of the orders 1/1 1/1, 2/1 1/1 and 3/2 2/1 [142][143][144][145], while certain representatives contain over 500 [146] or even 1000 (Al 474 Cr 108 , Al 477. 44 Cr 63.42 Fe 51.42 [31,145,147]) atoms per unit cell moving towards the level of complexity of a quasicrystal. A number of Mg 4 Zn 7 -related structures, potential d-QC approximants has been detected in the systems where actual d-QCs has never been found. If Li 12.5 Mg 13.6 Zn 39. 5 Al 15.1 [148] is still in likely expected area of habitancy of d-QCs, Na 8 Au 10 Ga 7 [149] (figure 5) exists right next to an i-QC, no quasiperiodic phases have ever been detected in the Na-Au-In system to explain the structural motifs in Na 8 Au 11 In 6 [150]. All the above already contain multiple pentagonal motifs, formally satisfying the geometric requirements for an approximant structure, and therefore may serve as hints for further FK d-QC explorations.

Nowotny chimney ladder phases
Nowotny phases or Nowotny chimney ladder structures (NCL) [151] is a common name for a family of, as a rule, binary compounds formed between transition metals of groups 4-9 and p block elements of groups 13-15. The two components form separate sub-lattices. The transition metal provides a 4-fold chimney-like helix, which hosts a spiral ladder of the p block element. In spite of the intriguing structural feature the majority of the NCL phases are pretty simple containing a few tens of atoms per unit cell (prototype compound RuGa 2 -24 [152], Ru 2 Sn 3 -20 [153], Ir 3 Ga 5 -24 [154], Ir 4 Ge 5 -36 [155]). Although, it does not prevent unique representatives to accommodate 200-300 atoms reaching unit cell volumes of 3000-4000 Å 3 [156,157]. The most complex representatives-Rh 10 [156,161], and Mn 27 Si 47 [157] still exhibit the same feature-a smaller but 'higher-fold' helix of a p element inside of a larger four-fold helix of a transition metal. In other words, the structures consist of two incommensurate helices and can be systematized by a number of simple primitive tetragonal blocks along the c axis (10, 11, 13, 15, 17 and even 27) naturally leading to extremely elongated unit cells with c/a = 8 and up to 21. Most of the compounds are well ordered, not at least due to electronic restrictions-14 valence electrons per transition metal [162]. Whereas some solid solubility in both d and p parts can be tolerated it is still strictly controlled by the electronic criteria [158,163]. Interestingly, these substitutions have been found to follow Fibonacci sequence [163] that is rare besides quasicrystals however was observed in a series with similar building principles-preferentially homoatomic honeycomb Au networks [164]. A detailed invest igation of a selected solid solution RuGa x Sn y (8 + 3x + 4y = 14) revealed by means of electron microscopy that at certain Ga/Sn ratios the unit cell c parameter becomes incredibly large (up to 270 Å) [163] pointing towards either very fine long-range ordering of the third component or to incommensurate modulation.

Polyanionic structures in polar intermetallic compounds
Intermetallic compounds with sizeable transfer of the electron density between the constituting elements are usually referred to as polar. Typical examples include binary and multinary phases containing highly electropositive elements, such as alkali, alkaline-earth, or rare-earth metals (the so called 'active metals') and metallic or metalloid elements with high electronegativity.
Crystal structures of polar intermetallic compounds can often be formally broken down into cationic and anionic fragments. Whereas the cationic substructure typically comprises isolated cations of the corresponding active metals, the anionic part may display extensive covalent bonding between the chemical species, resulting in polyanions of different dimensionalities. The presence of such structural units frequently yields complex atomic arrangements due to the inherent combination of different bonding patterns within a given crystal structure.
Chemical compositions and structural peculiarities of many polar intermetallic compounds can be rationalized within the Zintl concept, widely used in solid state chemistry [165]. In the frame of this approach, a complete electron transfer is assumed from the atoms of the electropositive metals to the polyanionic part of the structure. In an ideal Zintl phase, such redistribution of electron density leads to a state where all atoms adopt a stable closed shell (octet) electronic configuration. The formation of covalent bonds, i.e. 'extra' electron pairs, within the polyanionic substructure serves as a way of saturating the electron count of the bonded atoms. With all the atoms adopting an octet state, the chemical bonding in Zintl phases gets optimized, i.e. there appears an energy gap between the occupied bonding and empty antibonding electronic states. From the physics perspective, this should yield a semiconducting ground state, which is indeed observed for many Zintl phases. The Zintl method offers a simple electron counting scheme, which can be especially efficiently applied to compounds of main group elements and, to some extent, to transition metal-bearing phases [166]. Although polarity of chemical bonding is especially pronounced when highly electropositive and electronegative elements are combined in one compound, polyanions can be formally identified in crystal structures of various intermetallic phases, including the classical CMAs described in the previous chapters. Polyanionic building blocks can adopt atomic arrangements similar to those encountered in common intermetallic types, e.g. gamma brasses [82,[167][168][169] and FK phases or quasi crystalline phases and approximants [170,171], even in crystal structures that do not formally belong to these types. The large diversity of possible anionic units gives rise to numerous compositions with unusual crystal structures or even unique structure types. The most complex crystal structures demonstrating polyanions are found with the elements of group 12-15.
Giving an overview of all reported polyanionic structures in CMAs is hardly possible and would go beyond the scope of this contrib ution. Therefore, in the following chapters, we will focus on the main bonding principles leading to structural complexity in intermetallic phases and highlight some of the physical property studies performed for the discussed representatives.

Polyanions of group 12 elements
We start our discussion with polyanions of group 12 elements-Zn, Cd, and Hg. These metals show relatively high electronegativities and often act as electron acceptors when combined with active metals. In addition, they demonstrate a notable tendency to form homoatomic clusters, typically linked together yielding extended 3D polyanions. The complex atomic arrangement in the Samson phase NaCd 2 , described in chapter 4.1, is an example of this structural chemistry. Another CMA from the same binary system is Na 26 Cd 141 [172]. Although, this compound does not demonstrate structural disorder and is apparently a line phase, its primitive hexagonal unit cell accommodates 167 atoms. The Cd species in the structure are strongly interlinked with 6, 8, 9, or 10 nearest neighbors.
Two moderately big families of binary CMAs with group 12 polyanions belong to the Gd 13 Cd 58 [173][174][175][176][177][178][179][180][181][182] and Sm 11 Cd 45 structure types [82,176,183,184]. The former can be described as a hexagonal quasicrystalline approximant [182]. This type is adopted by a number of zincides, cadmides, and mercurides. The Sm 11 Cd 45 structure bears similarities with γbrasses and is observed for some cadmides and mercurides, but has not been reported for Zn compounds. In contrast to the structure of γ-brass, which is made up of 26-atom clusters respectively [82]. Similarly to most other polar intermetallic compounds with Zn and its congeners, the anionic substructure in the Sm 11 Cd 45 type can be viewed as a 3D framework.
Among the binary mercurides (amalgams), the two compounds Na 11 Hg 52 [188] and Ba 20 Hg 103 [189] demonstrate especially complex crystal structures. The Na structure can be described as consisting of infinite chains of face-sharing polyhedral clusters encapsulating Na atoms and arranged in a pseudo-hexagonal rod packing [188]. The structure of Ba 20 Hg 103 adopts cubic symmetry and boasts four kinds of fused clusters resembling sodalite cages (truncated octahedra) [189]. Interestingly, both structures display no significant crystallographic disorder and owe their complexity solely to peculiarities of the chemical bonding. Introduction of Cd in the structure of Ba 20 Hg 103 results in an isotypic compound, Ba 20 Cd 4 Hg 99 , whereas inclusion of Zn slightly changes the filling and local arrangement in one of the clusters and yields a different composition, Ba 20 Zn 5 Hg 99 [189].
Mixing of different group 12 elements can sometimes lead to complex structures not observed for compositions with a single element of this group. Examples of CMAs resulting from partial ordering of the atoms constituting their polyanions are CaZn 1.31 Hg 3.69 [190], crystallizing in the derivative of the Ba 20 Hg 103 type, and BaZn 0.6 Hg 3.4 [191], adopting a new structure type with a mixed occupied Zn/Hg polyanionic framework related to the Hg substructure in the moderately complex amalgam Rb 3 Hg 20 [191,192].
In a similar way, addition of an element that does not tend to mix with M (M = Zn, Cd, Hg) or changes the electronic count and thereby affects the chemical bonding can be employed to create multinary CMAs. In this case, partial or complete ordering of M and the extra element in the polyanionic substructure may lead to new topologies.
Thus, several binary compounds with relatively simple crystal structures have been reported in the Ca-Zn system. However, a complex ternary intermetallic phase with the composition Ca 21 Ni 2 Zn 36 can be grown by adding a small amount of Ni to the Ca-Zn melt [28]. In this compound, Zn forms icosahedra around the Ni atoms, decorated by additional Zn atoms and interconnected by Zn-Zn bridges ( figure 7).
By introducing Li to the Ca-Zn binary system, two complex structures with the compositions Ca 12 Li x Zn 59−x and Ca 15 Li x Zn 75−x can be produced [193]. In the cluster representation, both structures denote different stacking variants of dimers of face-sharing hypho-icosahedra (i.e. icosahedra with three missing vertices each) and trimers consisting of two hypho-icosahedra sandwiching a regular icosahedron. Even for the binary combinations that are known to form complex structures, transition to ternary and multinary systems upon addition of extra elements proved to be a viable way to produce new CMAs. Introduction of Cu or Al to the Ca-Cd system results in two structures with large unit cells: Ca 10 Cu 2 Cd 27 [194] and the already mentioned in chapter 5.1 Ca 3 AlCd 17 [120]. Whereas the former composition, crystallizing in a monoclinic superstructure of the 1/1 Bergman approximant, displays atomic sites with mixed and partial occupation, the latter one is well-ordered and represents a rhombohedrally distorted variant of the 2/1 Tsai-type approximant. Other examples of complex ternary structures with group 12 polyanions are occupationally-disordered Na 49 Sn 37.5 Cd 58.5 [195], displaying empty icosahedral clusters in the structure along with other polyhedral units, and perfectly ordered Ce 20 Mg 19 Zn 81 , related to the structure of Ba 20 Hg 103 [196].
Geometrical restraints of the polyanionic frameworks may also be used to vary the structural complexity. An illustration of this approach is the structure of K 29 NaHg 48 , which crystallizes differently from the binary Na or K mercurides [50]. Apparently, the significantly mismatching sizes of the K + and Na + ions constituting the cationic substructure yield a new atomic arrangement which can efficiently accommodate both types of cations. The structure contains two sorts of Hg 12 clusters surrounded by isolated K species: icosahedral clusters centered by Na and hexagonal antiprismatic clusters centered by K ( figure 8). Magnetization measurements indicate a superconducting transition in this compound at about 2.5 K.
We conclude this chapter with a short mention that the chemistry of Zn and its congeners in the realm of intermetallics is sometimes replicated by Mg. Examples include the isostructural GdMg 5+x [197] and (Ce, Y)Mg 5−x [198,199], crystallizing in a defect Sm 11 Cd 45 variant.

Trielides (Group 13)
Similarly to Zn and its congeners, elements of group 13 (triels, Tr) readily form homoatomic polyanionic moieties. Such anionic structures frequently possess subunits with icosahedral symmetry, which explains another name of triels-'icosagens' [200]. The largest number of complex polyanionic structures is found in the intermetallic compounds with triels (such compounds are referred to as trielides). This can be in part explained by the aptness of homoatomic bond formation, high chemical affinity of triels to most elements, including noble metals, ease of substitutional disorder, and low toxicity of all of triels, except  the heaviest member, thallium Tl, which encouraged their studies. In addition, small spatial requirements for the atoms of the lightest group 13 element, boron B, make it easy to realize different patterns of atomic packings in crystal structures. All this leads to great structural complexity, which is observed even for binary compounds. Furthermore, the crystal structure of the tetragonal and rhombohedral β modifications of elemental boron can be classified as complex [201][202][203]. Both modifications can accommodate some metal atoms, which is sometimes accompanied by formation of defects in the boron framework [204][205][206]. Some boron-rich binary systems do not adopt intercalated variants of boron modifications, but can be structurally related to them [207][208][209]. Although, in most of these systems, a high boron to metal ratio makes their description as polyanionic structures somewhat inappropriate, and their treatment as interstitial compounds should be applied, simple electron counting rules, e.g. within Wade's approach [210] indicate the importance of electron transfer from the interstitial atoms for the stability of the boron frameworks [207,208].
Heavier congeners of boron build homoatomic polyanions in binary compounds as well. The exceedingly complex Samson phase with the approximate composition Mg 2 Al 3 was described in chapter 4.1. Other complex binary structures include Na 22 Ga 39 [211], Na 7 Ga 13 [212,213], Na 7 In 11.8 [214], Na 15 In 27.4 [215], Na 4.95 In 9.22 [216], K 22−x In 39+x [217], K 39 In 80 [218], K 17 In 41 [219], and representatives of the K 8 In 11 type with the general composition A 8 Tr 11 (A = K, Rb, Cs; Tr = Ga, In, Tl) [220][221][222][223][224]. Most of them contain icosahedral Tr clusters along with other Tr units of different geometries. The only outliers in this group, not displaying regular icosahedral units, are Na 7 In 11.8 , with the polyanionic structure based on icosioctahedral In 16 and nido-icosahedral (i.e. with one missing vertex) In 11 clusters; and the K 8 In 11 -type phases, bearing isolated all capped trigonal prisms Tr 11 (figure 9). The formal charge of the latter structural unit implies that the A 8 Tr 11 compounds are not electron-balanced, according to the notation (A + ) 8 [Tr 11 ] 7− (e − ), and should show metallic properties, which is observed experimentally [224]. By removing excessive electrons, semiconducting properties can be achieved, e.g. in Cs 8 Ga 11 Cl, in which Cl adopts a cubic environment of the Cs atoms and is not bonded to any of the Ga species [224]. In contrast, in the ternary derivative Cs 8 Tl 11 Pd x (x ≈ 0.8), the Pd atoms are accommodated inside the Tl 11 cluster [225].
Mixing of two different triels in the anionic substructure typically results in substitution variants of the binary structure types, as in Na 17 Ga 29 In 12 , adopting the K 17 In 41 structure [219]. Introduction of an electronegative element from groups 11-12 or 14 either leads to a ternary derivative of a known binary structure (e.g. Na 35 Cd 24 [231,232], crystallizing isotypically in a rhombohedrally distorted variant of the Bergman phase (see chapter 5.1), which can be alternatively described as a stuffed derivative of the β modification of boron. Note that Mg behaves here as an analog of the group 12 elements.
It is worth noting that Au appears to serve as an efficient trigger of structural complexity in trielides. Although gold compounds are known to display different sorts of polyanionic units with varying degrees of complexity, most of such phases are not classified as CMAs following the definition used in the present work [164,[233][234][235][236][237][238][239][240]. Yet, many gold-bearing trielides demonstrate exceedingly large unit cells with a high number of atoms. Besides the above-mentioned examples, such compounds are represented by, e.g. Na 8 Au 11 In 6 , containing pentagonal bipyramids AuAu 5 In linked by extra Au and In species into a 3D polyanionic framework [150]; Na 128 Au 81 Ga 275 , possessing icosahedral and fused double-icosahedral clusters building a framework by different interconnection modes [241]; and structurally related CsAu 1.4 Ga 2.8 and CsAu 2 Ga 2.6 , hallmarked by the presence of Au/Ga tetrahedral stars in their crystal structures [242]. In contrast to the above-given examples with extended polyanions, another complex goldcontaining trielide, Na 12 K 38 Tl 48 Au 2 , demonstrates isolated [Tl 7 ] 7− and [Tl 9 ] 9− clusters, along with isolated monatomic Au − ions [243] (figure 10). Formal charge partitioning results in the electron-balanced formula (Na + ) 12 (K + ) 38 ([Tl 7 ] 7− ) 3 ([Tl 9 ] 9− ) 3 (Au − ) 2 . Transport measurements indicate that this composition is a metallic Zintl phase [243].
The latter compound provides an example of another common approach to highly complex structures, which involves combining elements of significantly different atomic sizes in the cationic sublattice. Other relevant examples are the structures of A 3 Na 26 In 48 (A = K, Rb, Cs) [244], adopting a defect K 29 NaHg 48 type, with interlinked empty In 12 icosahedra and Na-centered NaIn 12 hexagonal antiprisms, Li 3 Na 5 Ga 19.56 [245] and Rb 0.6 Na 6. 25 Ga 20.02 [246], exhibiting fused double Ga icosahedra, and Na 13 K 4 Ga 50−x , displaying trimeric units of condensed Ga icosahedra [247,248].  [252] all bear transition-metal-centered MTr 10 clusters. However, whereas these clusters are isolated in the former structure, in the latter two, they are encapsulated within large fullerene-like cages of different composition, based on Tr and A atoms.
A vast family of transition-metal-bearing aluminides RE 6 M 4 Al 43 , where RE can be a rare-earth metal, calcium, or uranium and M is an element of groups 4-6 [253][254][255][256][257][258][259] represents another example of structural complexity. The crystal structure of the prorotypic Ho 6 Mo 4 Al 43 can be described as consisting of fused Ho-centered HoAl 13 , and Mo-centered MoAl 10 and MoAl 13 polyhedra (figure 11). The polyanionic framework exhibits extensive Al-Al bonding. The rare-earth atoms are located in the vertices of stacked Kagomé nets. Magnetic measurements on some of the representatives indicate competing magnetic interactions, possibly due to the structure-imposed geometric frustration [257,258].
In the above-given examples of transition-metal-containing trielides, the high triel to transition metal ratio results in large spatial separation between the transition metal atoms. By increasing the transition metal content, metal-metal bonding mediated by overlap of the partially filled d-states can be realized. Complex transition-metal-rich phases with triel-based polyanions include, among others, SrNi 7.9 In 5.1 , crystallizing in an orthorhombic superstructure of the NaZn 13 type [260], and the related compounds Ce 7 Rh 18 Ga 11 and Ce 8 Rh 23 Ga 11 , described as complex intergrowths of Mg 2 Cu 3 Si-and CeCo 3 B 2 -type slabs [261].

Tetrelides (group 14)
The structural chemistry of polar intermetallic compounds with the group 14 elements (tetrels, Tt) resembles that observed in the systems within groups 12 and 13, e.g. formation of extended 3D homoatomic frameworks, as well as shares common structural features with pnictides (compounds with the group 15 elements), e.g. occurrence of polyanionic units of lower dimensionalities. The lightest group 14 element, carbon, is unique in many respects, and the structural chemistry of carbides will not be discussed in this chapter. Homoatomic Tt-Tt interactions in tetrelide polyanions typically occurs via two-center two-electron bonding, which makes it possible to apply the Zintl approach for crystal structure rationalization.
A number of rather complex layered structures are known in the quaternary systems RE-TM-Tr-Tt (RE = rare-earth metal, and TM = 3d element) [307] that, however, do not completely satisfy the complexity criteria of this review. Though, upon crystallization from liquid Al, a pseudoternary CMA Er 44 Mn 55 (Al/Si) 237 can be prepared [48]. In the crystal structure of this material, five symmetrically independent Mn atoms are surrounded by the statistically mixed Al/Si species, with coordination numbers of 6, 8 (×2), 9, and 11. The corresponding polyhedra

Pnictides (Group 15)
Elements of group 15 (pnictogens, Pn) frequently form polyanionic units in their compounds with metals (metal pnictides). Homoatomic clusters or cages are not as common for pnictides as for the elements of groups 12-14, and the occurrence of such building units becomes less typical upon going from the lightest (P) to the heaviest pnictogens (Bi) (note that nitrogen is deliberately excluded from the discussion due to the very different chemistry it exhibits). Examples of pnictide CMAs with isolated cage-like polyanionic blocks include the binary Zintl phases A 3 As 7 (A = Li, K, Rb) [308][309][310] and A 3 Sb 7 (A = Rb, Cs), as well as the solid solution K 3 As 7−x Sb x [311]. Although they adopt several different structure types, all of these structures accommodate [Pn 7 ] 3− clusters, typically arranged in a close-packed manner. These clusters are isoelectronic and isostructural to the P 4 S 3 molecule and can be viewed as consisting of fused pentagonal and trigonal rings. Similar [As 7 ] 3− clusters are observed in the crystal structure of a complex ternary Zintl phase Cs 4 Zn(As 7 ) 2 (figure 13) [312]. In this compound, the Zn atoms are sandwiched between the As 7 units forming isolated [Zn(As 7 ) 2 ] 4− anions with Zn in distorted tetrahedral environment of the As atoms.
Isolated polyanionic clusters are also observed in the notably complex structure of the isotypic Zintl compounds Sr 3 Sn 2 Pn 4 (Pn = P, As) [313,314]. In this case, though, no Pn-Pn bonding is taking place, but single bonds between the Sn atoms emerge instead. Ethane-like Sn 2 Pn 6 units in staggered conformation link by sharing the Pn vertices to form isolated six-membered rings [Sn 12 Pn 24 ] 36− , comprising the anionic substructure.
This example demonstrates that the aptness of some p elements to form homoatomic bonds can be successfully employed to provide an extra degree of complexity to polyanions in pnictides. Other examples of CMAs resulting from the application of this strategy include Eu 7 Ga 6 Sb 8 (Ga-Sb + Ga-Ga bonds) [315] and K 8 In 8 Ge 5 As 17 (Ge-As + In-As + Ge-Ge + As-As bonds) [316], both showing quasi-2D polyanionic slabs in their crystal structures. Cagelike 3D anionic frameworks hallmarked by extensive disorder were found in the crystal structure of Ba 23 (M 1−x Ge x ) 20 Sb 25−δ (M = Ga, In), which can be regarded as a derivative of the clathrate-I type [45]. Structural complexity and defect-driven disturbance of periodicity result in glass-like thermal conductance in these compounds, with the lattice thermal conductivity as low as 0.2-0.4 W m −1 K −1 in the temperature range between 323 and 823 K.
The clathrate-I type atomic arrangement can also be realized in the form of a complex orthorhombic superstructure adopted by coinage-metal-bearing pnictide clathrates Ba 8 M 16 P 30 (M = Cu, Au) [317,318]. Similarly to other clathrates and related cage compounds, these materials exhibit inherently low thermal conductivity. By proper chemical doping, the thermoelectric figure of merit ZT in these phases can be increased up to about 0.6 at T ≈ 800 K [319].
While homoatomic polyanionic cages are rare in pnictides, extended linear or 2D homoatomic moieties are frequently observed [320][321][322]. In some cases, packing effects or electronic instabilities, e.g. Peierls distortion, result in structural breakdown of the propagating chains or sheets with the decrease of dimensionality and the formation of isolated building blocks, such as linear or bent oligomers or polyatomic rings. First-principle calculations indicate that the tendency of the homoatomic extended pnictides structures to undergo spontaneous bond length alternation is correlated with the extent of the s-p orbital mixing and diminishes upon going down the group [321,323].
A good illustration of this trend is the large and actively studied family of pnictides with the general composition The complex crystal structure of the A 14 MPn 11 leads to a low thermal conductivity in these phases, similarly to other CMAs. By optimization of the charge carrier concentration, high values of electrical resistivity and Seebeck coefficient can be achieved, resulting in good thermoelectric performance at high temperatures [349,350]. Thus, Yb 14 MnSb 11 is considered one of best thermoelectric materials for T > 1000 K with a dimensionless figure of merit, ZT, of about 1.0 in this temperature region [22]. In addition, magnetic properties of the Mn-containing representatives of this family have been actively studied. Most of the antimonide and bismuthide members were found to order ferromagnetically at temperatures T ≈ 35-65 K and display colossal magnetoresistance, with the latter being related to nearly half-metallic band structures in these phases [329,339,[341][342][343][351][352][353][354]. Introduction of additional magnetic species, e.g. rare-earth ions, lead to emergence of new magnetic interactions and give rise to various low-temperature phenomena, such as glassy magnetic transitions and magnetic clustering [49,355,356].
Recognition of Zintl pnictides as potential materials for high-temperature power generation has resulted in discovery of many new complex structures with polyanionic units. The two isotypic phases Ba 21 Cd 4 Sb 18 [357] and Eu 21 Zn 4 Sb 18 [358] belong to this group of materials and display exceedingly sophisticated mode of the polyanion construction ( figure 15). Besides the isolated (i.e. not bonded to the transition metal atoms) Sb 3− and [Sb 2 ] 4− units, the structure contains oligomeric polyanions [M 4 Sb 12 ] 26− (M = Cd, Zn) consisting of four edge-and corner-sharing MSb 4 tetrahedra and terminated by a 'dangling' Sb-Sb dumbbell.
Many metal pnictides display various combinations of anionic subunits within one crystal structure, which naturally leads to high complexity of the atomic arrangements. In addition, compounds of pnictogens with transition metals (TM) can be stabilized by direct TM-TM interactions of two-center or multicenter character, realized via the d-electrons from the TM penultimate shells. In the realm of Zintl phases, this principle is exemplified by the ternary arsenides K 38 Nb 7 As 24 and Cs 9 Nb 2 As 6 [359]. The anionic substructure of the former compound accommodates isolated [NbAs 4 ] 7− tetrahedra and [Nb 2 As 6 ] 9− dimers of edge-sharing tetrahedra. Similar dimers serve as the unique polyanionic type in the structure of Cs 9 Nb 2 As 6 . The Nb atoms in these dimeric units have an average oxidation state of +4.5. One unpaired electron which provides unusual chemical bonding between the Nb atoms in the dimer reveals itself in the magnetization measurements, which show expected paramagnetic behavior.
A notable case of pnictide CMAs with extended polyanions are the structurally similar compounds Ba 2 Mn 1−x Bi 2 (x ≈ 0.15) and Ba 2 Zn 1−x Sb 2 (x ≈ 0.3) [363]. Both structures demonstrate transition-metal centered Pn coordination polyhedra linked in various modes to form star-like subunits propagating in one direction. In addition, homoatomic twocenter and hypervalent Pn-Pn bonding is observed. However, whereas Ba 2 Mn 1−x Bi 2 (x ≈ 0.15) represents a highly-reduced compound with an average Mn oxidation state of about +1.5 and is expected to be metallic, Ba 2 Zn 1−x Sb 2 (x ≈ 0.3) nearly conforms to the perfect Zintl electron count and exhibits a (pseudo)gap in its electronic spectrum.

Summary
The present contribution summarizes basic underlying mechanisms causing the emergence of CMAs. Structural complexity in intermetallic systems is a frequently observed phenomenon originating from compositional or chemical bonding inhomogeneities. The driving forces for the formation of CMAs is the optimization of atomic packing, e.g. by release of geometrical strains and fulfilling dense occupation of space, or enhancement of chemical bonding interactions by electronic optimization. The former usually involves alleviation of crystallographic defects by, e.g. complete or partial ordering of distinct chemical species or realization of certain stacking sequences of akin building blocks. In chemistry, the concept of 'chemical pressure' has been widely used to rationalize such phenomena [38,39]. Optimization of chemical bonding commonly results in separation of a crystal structure into regions with different patterns of interactions, such as two-center and multi-center bonds, or chemical bonding with different degrees of polarity [40][41][42]. The latter allows description of certain crystal structures in terms of polyanions. Elements of group 12-15 are the most common constituents of anionic building fragments in intermetallics, as they easily form chemical bonds with many elements and may show homoatomic bonding as well. Upon going from group 12 to group 15, the tendency to form extended 3D polyanionic units becomes less pronounced, leading to structural moieties of lower dimensionalities. In such cases, particular chemical bonding patterns may propagate along specific crystallographic directions, creating 'chemical bonding anisotropy' [364]. Recognition of repeating structural units or templates in CMAs allow reduction of their complex atomic arrangements to simpler known structures, which sometimes requires employment of methods inherited from mathematical topology, graph-theoretical approaches or number theory [21,188,200,365]. At the same time, developments in computational methods make first-principle calculations on complex intermetallics more and more feasible. Yet, satisfactory theoretical description of exceptionally complex representatives remain problematic, as such compounds challenge the approaches of classic crystallography implemented in computational algorithms. Even determination of their structures often calls for the use of large-scale facilities, such as synchrotron sources, as laboratory equipment is often not capable of providing complete data [7]. As the scales of periodicity start exceeding interatomic distances significantly, the properties of such compounds become similar to those of glass-like materials. This trend is reflected in low thermal conductivity and glassy magnetic behavior in some of the reported measurements [2,45,47]. CMAs offer therefore a platform for design of new thermoelectric and magnetically frustrated systems. Although at present, detailed physical properties are not available for many CMAs, with the advancement of experimental and theor etical methods, investigations in this hard-to-tackle yet prolific family of materials may bring new fascinating results.