Structure variations within RSi2 and R 2 TSi3 silicides. Part I. Structure overview

Most articles dealing with RSi2 and R 2 TSi3 compounds are only interested in one specific compound or in a series of compounds with varying T elements while keeping R fixed (or vice versa). Here, the focus lies on the complete space of 2:1:3 and 1:2 silicides. In addition further variations of superstructures are revealed and focus on crystallographic properties.

The relationship between the large variety of the derivatives from AlB 2 and ThSi 2 aristotypes can be nicely explained within the group-subgroup scheme, also known as Bä rnighausen formalism (Bä rnighausen, 1980). The AlB 2 structure is one of the simplest inorganic structure types. It has hexagonal space group P6/mmm (No. 191) and its unit cell incorporates only the two Wyckoff sites 1a and 2d (Hofmann & Jä niche, 1935) occupied by one R atom on the Al site and two Si atoms on the B site, forming a two-dimensional Si network, similar to graphite. The unit cell of the ThSi 2 structure also has only two occupied Wyckoff positions (4a and 8e), but the Si sublattice forms a more complex 3D network (Brauer & Mittius, 1942).
Nowadays, 46 structure types derived from AlB 2 (Hoffmann & Pö ttgen, 2001) and four from ThSi 2 are known. They include binary and ternary intermetallic compounds with compositions RX 2 , RT 2 , RTX or R 2 TX 3 , where X is an element of the third or fourth group.
In this work, we systematize the occurrence of RSi 2 and R 2 TSi 3 compounds, where R = alkaline earth metal, lanthanide, actinide or member of the Sc group and T is a transition metal. We present 12 different structure types of these compounds derived from the AlB 2 type. Six of these structure types have not been considered by Hoffmann & Pö ttgen (2001). Additionally, we present three further structure types based on the tetragonal ThSi 2 type. One of these types is purely hypothetical and considers the possibility of ordered Si/ T positions in ThSi 2 -like structures. Furthermore, we order all structure reports for RSi 2 and R 2 TSi 3 compounds according to their R and T elements within an R-T grid. After analyzing all element combinations, we choose nine promising compounds not found in the literature and perform DFT calculations to evaluate the probability of a successful synthesis. We discuss peculiarities of the distribution of structure types among the RSi 2 and R 2 TSi 3 compounds, based on a mapping of symmetries on the R-T grid with corresponding symbols.

Methods
To gain a comprehensive overview of RSi 2 and R 2 TSi 3 compounds, we performed an extensive literature search by scanning the ICSD, SciFinder and Reaxys databases for all possible element combinations for T within the Cr to Zn groups and R within the Sc group, the alkaline earth metal, the lanthanides and the actinides. Only experiments at ambient conditions were considered. Additionally, we did not consider data sets if they were too incomplete, i.e. missing lattice parameters or an insufficient description of the symmetry. Additionally, we did not take incommensurately modulated structures into account, because these modulations mainly arise for nonstoichiometric disilicides within this family of compounds and because the descriptions do not conform with those of conventional symmetry. Please refer to Leisegang (2010), Kubata et al. (2005) and Dshemuchadse (2008) for further information. However, commensurable modulations are interpreted as superstructures. Table 1 contains the tabulated data of the composition of the compounds as well as their structure parameters, i.e. lattice parameters a and c, ratios c/a, formula units per unit cell, and structure type. These data were used without further refinement. The compounds, discussed within this article, are more than solid solutions as most of them exhibit ordered structures and, therefore, have distinct structure types compared to similar stoichiometries. Within this article, only the formula units and the deviation of the compounds within the range of R and T elements is of interest. Part II (Nentwich et al., 2020) will discuss and compare other parameters.
We used calculations based on density functional theory (DFT) to predict the stability of not yet reported RSi 2 and R 2 TSi 3 compounds. The formation energy ÁE tot is the difference of the total energy E tot of the compound and E tot of its elements, normalized to six atoms (R 2 Si 4 or R 2 TSi 3 ). Appendix B presents the space groups of the unary R crystals. The more negative the formation energy, the more thermodynamically favorable is the formation of that compound. We considered a formation energy of up to À25 meV per atom as potentially stable at room temperature. However, this assumption does not take into account potential energy barriers which might kinetically hinder the formation of the ground state. The projector-augmented wave (PAW) method (Kresse & Joubert, 1999) in spin-polarized Perdew-Burke-Ernzerhof parametrization (Perdew et al., 1996) was employed as implemented in the VASP code (Kresse & Furthmü ller, 1996). Total energies have been converged better than 10 À7 eV with a maximum kinetic energy of 320 eV for the planewave basis set and À-centered k-point meshes with spacings less than 0.02 Â 2 Å À1 . All structures have been fully relaxed, with respect to atomic positions as well as cell geometry within the space group, to forces less than 10 À3 V Å À1 . A Hubbard U correlation correction was not used because the Si framework with sand p-orbitals governs the stability of the structure and because it would complicate the comparability of the formation energies within the R 2 TSi 3 series.

Results and discussion
In this article, we treat the R 2 TSi 3 compounds as a distinct phase with a fixed composition and not as a solid solution. As ternary phase diagrams are scarce for these compounds, we checked all available data, in particular the thermodynamic assessment of Bodak & Gladyshevskii (1985), for compositional degrees of freedom in the corresponding phase diagram region and possibly prevailing solid solutions. Nevertheless, the vast majority of compounds were reported to form superstructures which, in general, allow only slight variations in stoichiometry. We discuss those structures as distinct phases due to the changes in symmetry at these particular compositions in the phase diagrams. Many ternary phase diagrams are often determined at elevated temperatures, which is beyond research papers Table 1 Alphabetically sorted list of RSi 2 and R 2 TSi 3 compounds and their crystal data.
R is an element of the alkaline earth metals, the scandium group, or the lanthanide or actinide series. T is a transition metal, Al or Si; thus a disilicide. The supercell can be identified by the formula units per unit cell. Lines written in blue indicate data sets not used for Fig. 9.        the scope of this work. The phase diagrams given by Bodak & Gladyshevskii (1985) are not at room temperature.

Structural relationships
The many structure types within compounds RSi 2 and R 2 TSi 3 compounds are related to each other according to their space groups and occupied Wyckoff positions. Starting from the highest symmetric structure, different perturbations induce symmetry reductions. Bä rnighausen diagrams are the perfect tool to visualize these group-subgroup relationships in a simple and descriptive way. Fig. 1 presents the full Bä rnighausen diagram for the RSi 2 and R 2 TSi 3 compounds analyzed in this work. This diagram is partially based on a diagram by Hoffmann & Pö ttgen (2001), but is greatly extended.
The presented Bä rnighausen diagram would allow for further group-subgroup transitions; thus the authors cannot exclude the existence of further structure types within the RSi 2 and R 2 TSi 3 compounds and thus also additional branches in the diagram. However, the space groups we present here already have a high number of free parameters. The extension of the diagram by further symmetry reduction accompanied with further degrees of freedom without losing the rough lattice and symmetry is challenging.
Our diagram provides information about the type of transition (klassengleiche with perpetuation of lattice symmetry, translationengleiche with perpetuation of translational symmetry and isomorphous with perpetuation of both), the change of the lattice (direction and distance), the characteristics of the structure (space group, structure type and Wyckoff positions) as well as the absolute occurrence of the structure types in the literature. Additionally, Fig. 2 visualizes the atom arrangements of the different structures and presents their relationships in a hierarchical structure similar to the Bä rnighausen diagram. In contrast, it focuses on the structural models and only shows these branches that include new structure types compared to Hoffmann & Pö ttgen (2001). Appendix A includes tables with Wyckoff positions of all structure types taken into account within this article (Tables 2,  3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17).
3.1.1. Compounds deduced from the AlB 2 structure type.
First, we will present the relationships of RSi 2 and R 2 TSi 3 compounds derived from the AlB 2 structure. The lattice parameters are in the range of a h % 3.8-4.2 Å and c h % 3.9-4.5 Å , which is much higher than for the parent structure AlB 2 itself (a AlB 2 = 3.00 Å , c AlB 2 = 3.24 Å ).
Hoffmann & Pö ttgen (2001) gave an overview of the hexagonal and orthorhombic transitions of AlB 2 -related compounds. Only three of Hoffmann's Bä rnighausen branches are applicable for the stoichiometries addressed here (RSi 2 and R 2 TSi 3 ). We identify further structure types not discussed by Hoffmann & Pö ttgen (2001), analyze the relationships of all structure types in the following paragraphs and show the new structure types in the Bä rnighausen diagram (Fig. 2). Our Bä rnighausen diagram (Fig. 2) thus exhibits four main branches which result from interactions with a T element or an Si vacancy &.
The first branch of the Bä rnighausen diagram describes the symmetrical relationships between the hexagonal derivatives of the AlB 2 type. Fig. 2 shows that Ce 2 CoSi 3 (Gordon et al., 1997) has the same structural motif as the aristotype. The difference is the ordering of the T atoms resulting in isolated [Si 6 ] rings, see top right of Fig. 2. Only a certain part of this pattern is visible in the unit cell of Ce 2 CoSi 3 and in other structure types of the RSi 2 and R 2 TSi 3 compounds, indicated by red bonds. Besides [Si 6 ] rings, [T 2 Si 4 ] hexagons also occur, with the T atoms opposing each other in the ring. This ordering change indicates the doubling of the unit-cell parameter a in the Ce 2 CoSi 3 type and an isomorphous symmetry reduction. If the Si atoms are shifted along the c direction, the layers are no longer perfectly planar, but puckered. This arrangement can be described with the same space group as Ce 2 CoSi 3 , but with half-occupied Wyckoff site 12o, instead of fully occupied 6m, known as the structure type U 2 RuSi 3 . Fig. 2 shows both structure types within one subfigure with the different Si positions indicated by a series of atoms.
Compared to their ideal crystallographic positions, the Er 2 RhSi 3 (P6 3 /mmc) type (Gladyshevskii et al., 1992)   and i is isomorphic. Fig.   2 comprises the respective structure plots. The fourth branch of AlB 2 -like compounds comprises the superstructures caused by interplays with vacancies (R further klassengleiche reduction of the symmetry of the Ce 2 CoSi 3 or U 2 RuSi 3 type. The reported noncentrosymmetric structure for Er 2 RhSi 3 (P62c) (Chevalier et al., 1984) assumes additional distortions of the [Si 6 ] rings and their centering R atoms by decoupled x and y coordinates resulting in a translationengleiche symmetry reduction of centrosymmetric Er 2 RhSi 3 (P6 3 /mmc).
The second branch only includes the Ho 2 PdSi 3 structure type (Tang et al., 2011) with monoclinic space group I112/b (Nentwich et al., 2016). This structure contains eight Si/T layers with stacking sequence ABCDBADC. Each layer exhibits the same Si/T occupation pattern as the Ce 2 CoSi 3 type. The [T 2 Si 4 ] rings of adjacent layers are shifted and rotated by multiples of 60 around the c axis with respect to each other. The 12-fold coordinated R elements are located on two different Wyckoff positions, either coordinated by two Models of the different observed structure types within RSi 2 and R 2 TSi 3 compounds (unit cell outlined in black). The AlB 2 -like structures are depicted such that the view onto the two-dimensional R network is almost identical. The common structure pattern of the ordered AlB 2 -like structures (gray frame at right top) is highlighted with a light-gray frame and red Si/T bonds. The structure types Ce 2 CoSi 3 and U 2 RuSi 3 are almost identical. In contrast to the U 2 RuSi 3 type, the Si atoms of the Ce 2 CoSi 3 type are on the highly symmetric z = 1 2 position. This is highlighted by the blurred Si location along the c direction. The tetragonal structures (gray frame at center top) compose a 3D Si/T subnetwork with incomplete hexagons at the faces (highlighted in orange). The structures are connected according to their symmetry relations (dashed lines, if the transition is not minimal; labels comprise the lattice transformation). largest structures within the AlB 2 Bä rnighausen diagram. The atoms are assumed to be on the ideal crystallographic position, without any distortions, although the space group would allow this. The transition from AlB 2 type to Ho 2 PdSi 3 involves several symmetry reduction steps, detailed in Fig. 1.
The third branch comprises the orthorhombic derivatives of the AlB 2 type. The starting point for further reductions is an orthohexagonal setting with space group Cmmm and Wyckoff sequence 2a, 4k. This setting is still a missing link (Hoffmann & Pö ttgen, 2001), meaning that no report about a compound with this structure has been found. This space group has independent lattice parameters a and b -in contrast to all previous structure types -causing a translationengleiche symmetry reduction and making it an important starting point for five further structure types.
One of them is Ba 4 Li 2 Si 6 (von Schnering et al., 1996), which has perfectly ordered Si/T layers with the same occupational pattern as the Ce 2 CoSi 3 type. As in the Ho 2 PdSi 3 structure type, the Si/T atoms are perfectly ordered and form an ABCD stacking sequence, which is consistent with the two differently coordinated R sites as mentioned before. Accompanied with the anisotropic available space of the R site surrounded by one [T 2 Si 4 ] and one [Si 6 ] ring, its z component is not on the ideal crystallographic position resulting in a puckering of the R and Si/T layers. Identical R elements are connected along the former hexagonal a direction. These structural changes are accompanied with three consecutive klassengleiche symmetry reductions doubling the a and b parameters and quadrupling the c parameter.
A second structure type is U 2 RhSi 3 (Pö ttgen & Kaczorowski, 1993) with space group Pmmm (No. 47). Its Si/T atoms are partially ordered and only shifted along the b direction. These shifts induce a break in translational symmetry and a klassengleiche reduction. The Ho 2 PdSi 3 , Ba 4 Li 2 Si 6 and Ca 2 AgSi 3 structure types (Gordon et al., 1997) have perfectly ordered Si/T layers and the same local arrangements around the R atoms. The R elements of the same Wyckoff site are connected along the orthorhombic a direction. These structural changes indicate the doubling of lattice parameters and a klassengleiche transition from structure type U 2 RhSi 3 . Hoffmann & Pö ttgen (2001) have already reported a second structure type with the same space group as U 2 RhSi 3 , but with a different Wyckoff sequence, namely Er 3 &Si 5 . This type represents the disordered nonstoichiometric disilicides. In addition to the disordered ones, we also found reports about ordered versions. The otherwise very detailed review by Hoffmann & Pö ttgen (2001) did not discuss these variants, which form due to vacancy ordering. According to the real stoichiometry of RSi 1.67 , one Si atom is regularly missing in the Si hexagons (Roge et al., 1995). This arrangement can be realized by a hexagonal and a orthohexagonal setting . The hexagonal setting will be discussed in the fourth branch. The orthohexagonal arrangement requires a triplication of the a parameter. We will refer to this setting as Ho 3 &Si 5 type. We prepared a list of its atomic parameters in space group P1 (No. 1) and inserted it to the software FINDSYM (Stokes & Hatch, 2005), which determined the highest possible space group as Pmm2 (No. 25). We changed the setting to P2mm (No. 25) for a better comparability to its supergroup Pmmm (No. 47). Thus, the triplication causes a translationengleiche and a klassengleiche symmetry reduction, which is accompanied with potential shifts of all atoms within the a,b plane.
The fourth branch comprises the ordered R 3 &Si 5 structures, which are not related to the disordered Er 3 &Si 5 type within the Bä rnighausen diagram.
d 'Avitaya et al. (1989) described a ffiffi ffi 3 p Â ffiffi ffi 3 p low-energy electron diffraction (LEED) pattern of Er 3 &Si 5 thin films. Iandelli et al. (1979) determined the space group of this arrangement for Yb 3 &Si 5 as P62m (No. 189), only allowing the x parameter of R and Si to deviate from its ideal crystallographic position. To consider the underlying symmetries of this arrangement, the cell needs to be enlarged and rotated with respect to the AlB 2 unit cell using an isomorphous symmetry reduction. The location of the vacancy on an independent Wyckoff site is accompanied by a further translationengleiche symmetry reduction and an origin shift from space group P6/mmm to P62m.
Another model proposed by Stauffer et al. (1992) is based on the aforementioned arrangement, but every second Si/T layer is rotated by 120 around c. We determined the space group of this vacancy ordering as P62c, assuming that only the occupational pattern of the Si lattice would adapt, without changing the atomic positions. This results in a doubling of the c parameter, accompanied by a klassengleiche transition. The first reports concerning this arrangements used the compound Er 3 &Si 5 . However, this type name is already used for the disordered nonstoichiometric disilicides. Thus, we will refer to this structure type as Tb 3 &Si 5 in accordance with the report by Luo et al. (1997).
We did not consider cells based on the Ho 3 &Si 5 type with doubled c parameter, as it is only reported for the ffiffi ffi 3 p Â ffiffi ffi 3 p type cells. Further remarks. Gordon et al. (1997) reported a further superstructure for Ce 2 PdSi 3 with doubled lattice parameter a and quadrupled c, but did not focus on the specific space group. Therefore, we could not implement this report for the construction of the Bä rnighausen diagram. During the literature research we additionally found structures of the EuGe 2type with space group P31m (No. 164). This structure type is very similar to the AlB 2 type, but with a puckered Si sublattice, inducing a translationengleiche transition. Reports about this structure type refer to binary alkaline earth disilicides at nonambient conditions (Evers et al., 1977b;Bordet et al., 2000;Brutti et al., 2006) or with mixed R sites (Eisenmann et al., 1970;Evers et al., 1979) as well as theoretical considerations about the puckering only (Gemming & Seifert, 2003;Gemming et al., 2006;Enyashin & Gemming, 2007;Flores-Livas et al., 2011). As these reports do not meet the requirements of experiments at ambient conditions, we did not consider this group of compounds within this work.
All aforementioned structure types will be termed AlB 2like in the following sections. By studying the atomic coordinates of the addressed space groups, we observed that the R research papers elements form a rigid frame for the structure, as they are mostly the heaviest and largest elements in the structure and, thus, the most immobile. This also means that the Si/T atoms are more mobile and thus puckering of these layers is rather common.

Compounds deduced from ThSi 2 structure type.
Compounds of the ThSi 2 type (Brauer & Mittius, 1942) crystallized in space group I4 1 /amd (No. 141), see gray box of Fig. 2 (with tetragonal lattice parameters a t % a h , c t % 13.4-14.4 Å ). The Si/T atoms form a complex 3D network, in contrast to the 2D honeycombs in AlB 2 . So far, the only reported variation of the ThSi 2 type is the GdSi 2 structure (Perri et al., 1959b;Binder, 1960) with independent lattice parameters a and b. This degree of freedom causes a translationengleiche symmetry reduction to space group Imma (No. 74).
If the ThSi 2 or GdSi 2 type structures exhibit Si vacancies, these do not order regularly and only cause partially occupied Wyckoff positions. The proportion of vacancies is generally 10% (RSi 1.8 ), thus almost one Si ion per tetragonal or orthorhombic unit cell is vacant. The resulting structures remain in the original space group and are called ThSi 2 -defect and Nd& x Si 2Àx , respectively.
In contrast to the distortive modulation of ThSi 2 , we did not find evidence for a tetragonal superstructure induced by ordering. This absence may be partially due to the small number of reports concerning tetragonal R 2 TSi 3 compounds [18 structure reports in ten articles (Gordon et al., 1997;Albering et al., 1994;Kaczorowski & Noë l, 1993;Lejay et al., 1983;Chevalier et al., 1986;Li et al., 2008;Mayer & Felner, 1973b;Pö ttgen & Kaczorowski, 1993;Raman & Steinfink, 1967;]. In order to shed light on a potential ordering, we constructed a tetragonal superstructure based on geometrical, chemical and electronic considerations. First, every Si atom has exactly one T element in its coordination. Second, every T element is coordinated by exactly three Si atoms. Third, every zigzag chain fulfills the 1:3 ratio of T:Si (zigzag chains explained in Section 3.2). And fourth, shortrange periodicity is mandatory; thus, no doubling of the unit cell along the c direction is expected. By choosing an arbitrary atom within the tetragonal Si/T network as the first T element, only two positions unfold positioning the next T element. Two atomic arrangements resulted following the aforementioned conditions. We transferred these patterns onto the simple space group P1 (No. 1) and imported them into the tool FINDSYM (Stokes & Hatch, 2005) to determine the space group. Both variants proved to be identical and to exhibit the space group C222 1 (No. 20). We will refer to this new structure type with eight instead of four formula units as POTS (proposed ordered, tetragonal structure). The gray box in Fig. 2 visualizes the Si/T-ordering. As this structure has not been reported so far for R 2 TSi 3 compounds, we decided to perform DFT calculations to estimate its stability, see Section 3.3.
These three structure types introduced in this section (x3.1.2) will be addressed as ThSi 2 -like in the following.

Structure description
The hexagonal and the tetragonal subgroups of RSi 2 and R 2 TSi 3 compounds do not seem to be symmetrically related at first glance. The AlB 2 -like compounds exhibit graphite-like 2D networks of planar Si/T hexagons, whereas the Si/T atoms of ThSi 2 -like compounds form 3D networks. Still, the structures show similarities due to the trigonal coordination of the Si atoms. Fig. 3 illustrates the Si/T atoms in trigonal prisms, the 12-fold coordinated R atoms (connectors in black) and the Si/ T zigzag chains (bonds in red/orange) in both structures.
Not only are the hexagonal honeycombs similar to graphite but also the tetragonal 3D network. The typical net exists simultaneously in planes perpendicular to the tetragonal a t and b t directions which are interconnected by bonds along the c t direction. More precisely, two consecutive Si/T zigzag chains are rotated by 90 along the c t direction, thereby spanning the (100) t and (010) t faces of the unit cell and causing incomplete hexagons (see the orange bonds in the ThSi 2 structure type in Fig. 2). This additional symmetry degree of freedom causes a slight deformation of the trigonal Si/T arrangement in the tetragonal network. The Si-T bonds along the c t direction (in orange, interchain) elongate in comparison to the intrachain bonds (in red), see Fig. 3. Further, the angle within the zigzag chains increases, whereas the other two angles decrease Differences in the arrangement of Si/T (blue) zigzag chains in hexagonal (left) and tetragonal (right) RSi 2 and R 2 TSi 3 compounds. The consecutively added zigzag chains (red bonds) in hexagonal compounds always lie within the same plane, whereas in tetragonal compounds these layers are rotated by 90 along the bonds shown in orange. The 12-fold coordination of the R elements is highlighted for one atom ,as an example, with bonds shown in black.

Figure 4
Overview of the literature reports of RSi 2 and R 2 TSi 3 crystals. The number of reports is visualized with numbers and colors (few to very frequent: red -yellow -green -blue -purple). Additionally, to predict the stability for selected unreported structures, this study performed DFT calculations for the highlighted compounds (black circles).
(between bonds shown in red and orange). Therefore, the chains with stronger bonds are slightly flattened compared to the ideal structure with perfect trigonal coordination. These structural differences between hexagonal and tetragonal structure types cause different crystal symmetries that permit a common origin in the Bä rnighausen diagram for the RSi 2 and R 2 TSi 3 compounds.

Elemental combinations and stability analysis of missing links with DFT calculations
During the literature search, we collected numerous structure reports of various RSi 2 and R 2 TSi 3 compounds. Fig. 4 gives an overview of the reported compounds according to their appearance within the R-T grid. In this R-T diagram, we marked the number of reports with different colors, see Fig. 4. This diagram does not include the elements of the Zn group as those compounds were only analyzed at elevated temperatures (Demchenko et al., 2002;Malik et al., 2013;Nasir et al., 2010;Romaka et al., 2012;Salamakha et al., 1998), which are out of the scope of this article. Additionally, we did not find any reports which include R 2 CrSi 3 compounds. We assume that certain electron configurations are necessary for the formation of R 2 TSi 3 compounds. Furthermore, some elements rarely appear within the R 2 Si and R 2 TSi 3 compounds, such as Sm and Yb, which are highly volatile (Cao, 2014, private communication), Tc, which has a very low radio-active halflife and is very scarce (Holleman & Wiberg, 2007), or Pm, which is radioactive (Cao, 2014, private communication;Frontzek, 2014, private communication). The interest in using La and Lu was lower as most of the research aimed for the magnetic properties that do not exist for these two elements (Frontzek, 2014, private communication). The cost of the elements seems to play a subordinate role, e.g. the more expensive Rh (89 000 USD per kg) compounds were analyzed more frequently than the ones containing Ir (36 000 USD per kg) (Haynes, 2012).

Figure 7
Overview of the R 2 TSi 3 compounds that were analyzed systematically by the same first author according to their R element (see color code). Some of the results were published in more than one article: Mayer:2 (Mayer & Tassa, 1969;Mayer & Felner, 1972, 1973a, Szlawska (Szlawska et al., 2007(Szlawska et al., , 2009(Szlawska et al., , 2016, 2012, Li (Li et al., 1997,b, 1999, 2002a, 2008, Frontzek (Frontzek et al., 2004(Frontzek et al., , 2006Frontzek, 2009), Pottgen (Pö ttgen & Kaczorowski, 1993Pö ttgen et al., 1994), Majumdar (Majumdar et al., 1998(Majumdar et al., , 1999a(Majumdar et al., ,b, 2000. overview of RSi 2 series with the corresponding authors and R elements. This summary shows the high interest in the lanthanide compounds compared to R elements of the alkaline earth metals and the actinides. Fig. 6 shows a similar illustration of T series within the R 2 TSi 3 compounds. Sorted by T element and author, the corresponding R elements are highlighted. Within the 3d elements the largest variety was analyzed, mostly in combination with La and Ce. In contrast, the heavy lanthanides were more favored when 4d elements were used, which have been intensively studied. Finally, Fig. 7 shows the R series, sorted by R element and author, with highlighted T elements. Again, the focus on the 3d elements as well as La and Ce is clear. The most complete investigations were carried out for U and Th, which emphasizes their importance for reactor technology.
By studying the R-T diagram of Fig. 4 one main question arises: What are the stability relationships of those R 2 TSi 3 compounds that are missing? To clarify this question, we sorted the compounds according their R element and discuss the Co, Rh and Pt series in the following sections.
We assumed ordered structures as DFT cannot evaluate mixed positions, except in the framework of virtual crystal approximations (VCA) using potential mixing. We adapted the structure type of the adjacent compounds within the R-T grid or used the highly symmetric Ce 2 CoSi 3 structure type with space group P6/mmm (No. 191) as the basis for the unknown compounds. Table 17 summarizes the formation energies and lattice parameters all considered compounds. We will compare the formation energy of an unreported compound with those of similar reported compounds to evaluate its relative stability.
The DFT results of all models indicate metallic structures, although the DFT band gap problem may suppress the appearance of small band gaps. Thus, all structures have an intrinsic buffer of electronic states at the Fermi level to account for stability considerations of the T coordination within the ionic Si/T subnetwork according to molecular orbital theory, see Nentwich et al. (2020).
The first compound of interest is Nd 2 CoSi 3 . The series of Nd compounds is fairly complete, compare Fig. 4, for example, with reported Nd 2 RhSi 3 (Chevalier et al., , 1984Szytuła et al., 1993;Mitsufuji et al., 1996;Gribanov et al., 2010;Zajdel et al., 2015), which is the 4d analog compound to Nd 2 CoSi 3 . Additionally, we found comments on this compound in two publications, but without any information concerning property, structure and phase purity (Chevalier et al., 1984;Szytuła et al., 1993). The formation energies and existing structure types of La 2 CoSi 3 and Ce 2 CoSi 3 serve as references. Furthermore, the likewise hypothetical compound Pr 2 CoSi 3 was also calculated. The blue markers in Fig. 8 show the respective formation energies ranging from À4.61 eV to À4.37 eV. The lowest energy results for R = Ce and the highest for R = Nd. As the formation energy of Pr 2 CoSi 3 lies in between the reported compounds, we expect it to be stable. The energy difference between Nd 2 CoSi 3 and La 2 CoSi 3 (the reported compound with highest energy) is 25 meV per atom. This corresponds to the tolerance limit; thus, we conclude that Nd 2 CoSi 3 could also be stable. This conclusion is supported by the reports of Mayer & Tassa (1969) and Felner & Schieber (1973) on Pr 2 Co 0.8 Si 3.2 and Nd 2 Co 0.8 Si 3.2 . They also synthesized samples with higher T content, which lead to 'the disappearance of the AlB 2 type phase, and the X-ray patterns obtained could not be interpreted' (Mayer & Tassa, 1969). Nevertheless, we think that the synthesis of Pr 2 CoSi 3 and Nd 2 CoSi 3 and the interpretation of the corresponding X-ray patterns would be successful nowadays due to improved hardware and measurement techniques. Additionally, an enhanced thermal treatment would certainly improve the crystal quality regarding the Si/T ordering. Thus, we advise reinvestigating the R 2 TSi 3 compounds discussed by Mayer & Tassa (1969), with R = La, Ce, Pr, Nd, Sm, Gd and T = Fe, Co, Ni.
Another interesting compound is Eu 2 RhSi 3 . The Rh series is well represented in the R-T diagram and its 3d analog Eu 2 CoSi exists. However, the R element Eu supposedly only forms a compound with Co, but not with Rh (Mayer & Tassa, 1969;Mayer & Felner, 1973a). We also modeled R 2 RhSi 3 compounds with R elements Gd, Tb, Dy and Ho again and used the formation energies of existing structures as references. For the Rh series, the formation energies range from À6.68 eV to À4.34 eV, with the not yet reported Eu 2 RhSi 3 having the highest formation energy. Both tested symmetriesthe higher symmetric Ce 2 CoSi 3 and the lower symmetric Er 2 RhSi 3 -gave almost the same results, for formation energies (À4.34 eV) and interatomic distances [d a (R,R) % 4.13 Å , d c (R,R) % 4.27 Å ]. The formation energy of Eu 2 RhSi 3 differs from the second highest formation energy of Ho 2 RhSi 3 by 160 meV per atom which exceeds the limit of 25 meV per atom, see green markers in Fig. 8. Therefore, the Eu 2 RhSi 3 compound in Ce 2 CoSi 3 or Er 2 RhSi 3 structure type is significantly less stable.
The third compound of interest is Eu 2 PtSi 3 . In the R 2 PtSi 3 series only a few element combinations have not yet been experimentally confirmed. Nevertheless, we identified missing compounds for R between Nd and Gd. Due to the radioactivity and low abundance of Pm and the volatility of Sm, we chose the Eu compound for further investigation. In analogy to the Rh series, we additionally chose R = Gd, Tb, Dy as Formation energies of some R 2 TSi 3 compounds in different structure types. references for formation energy and structure. In addition we modeled the not-yet-reported compound Ho 2 PtSi 3 . We decided to calculate the compounds in the reported Er 2 RhSi 3 (P62c) symmetry and additionally in the higher symmetric type Ce 2 CoSi 3 as well as in the lowest possible symmetry P1 (No. 1) to evaluate the influence of the degrees of freedom onto the formation energies. The energies for the R 2 PtSi 3 compounds range from À6.18 eV to À5.11 eV, see orange markers in Fig. 8. Except for Eu, the energies of different compounds and also different structure types are very similar. As expected, the energies of the lower symmetric Er 2 RhSi 3 structure types are always lower than those of the highly symmetric type Ce 2 CoSi 3 , due to the additional degrees of freedom in atomic positions. The spread is between 0 meV for Gd and 28 meV for Ho per atom and about additional 1 meV going down to P1 (No. 1). The energies of the low-symmetric versions of the R 2 PtSi 3 compounds are even lower than that of existing Gd 2 PtSi 3 . The formation energy of the (still) hypothetical Ho 2 PtSi 3 in Ce 2 CoSi 3 type structure is 33 meV per atom higher than that of Gd 2 PtSi 3 , thus this high-symmetry type is certainly not stable. However, the lower symmetry types will very probably be stable. The formation energy of Eu 2 PtSi 3 is 14 meV per atom higher than for Gd 2 PtSi 3 ; therefore, the compound is in the two considered symmetries most probably accessible as the thermodynamically stable phase. On the one hand, these data show that in some cases (Eu 2 RhSi 3 , Eu 2 PtSi 3 and Gd 2 PtSi 3 ) the formation energy hardly changes for different structure types. On the other hand, the formation energy of different structure types may change so strongly that our relative limit of 25 meV per atom is by far exceeded and only the lower symmetric variations may be stable. This is the case for Tb 2 PtSi 3 , Dy 2 PtSi 3 and Ho 2 PtSi 3 .
After analyzing those three R series, we discovered further characteristics in the R-T diagram worth studying for different reasons. Compound La 2 PdSi 3 attracted our attention because Chaika et al. (2001) and Behr et al. (2008) have already successfully synthesized this compound, but did not determine the lattice parameters or structural information during their investigations. We performed DFT calculations for La 2 PdSi 3 using the Ce 2 CoSi 3 structure type as well. The formation energy is lower than for the chemically similar compound La 2 CoSi 3 which was reported in the ordered structure type Ce 2 CoSi 3 . Thus, we conclude that the Ce 2 CoSi 3 type may be a stable configuration for La 2 PdSi 3 , next to the disordered AlB 2 type. The relaxed parameters a = 8.34 Å and c = 4.38 Å are very close to the lengths expected from the adjacent compounds La 2 RhSi 3 and Ce 2 PdSi 3 (a % 8.25 Å , c % 4.3 Å ). We recommend checking La 2 PdSi 3 for indicators of an ordered Si/T site, e.g. satellite reflections.
Furthermore, we wondered which structure would arise for stoichiometric BaSi 2 . Most reported space groups of BaSi 2 are orthorhombic (Imai & Watanabe, 2010;Evers, 1980;Janzon et al., 1970;Kitano et al., 2001;Migas et al., 2007;Schä fer et al., 1963;Evers et al., 1977bEvers et al., , 1978a and do not fit into our Bä rnighausen diagram and are, therefore, not listed in Table 1 nor depicted in Figs. 4 and 9. The only exception is a hexagonal phase determined by Gladyshevskii (1959). In fact, the original sample had Li impurities and exhibits the structure type Ba 4 Li 2 Si 6 , discovered by von Schnering et al. (1996). This finding explains the discrepancy with the tetragonal phases of the related alkaline earth compounds CaSi 2 and SrSi 2 , e.g. Evers et al. (1977a,b). We tested both an hexagonal and a tetragonal variant for BaSi 2 to evaluate which symmetry is more stable. Additionally, we modeled SrSi 2 in both the hypothetical AlB 2 and the already reported ThSi 2 structure type to compare the formation energies. As expected, the formation energy of tetragonal SrSi 2 is lower than the one of hexagonal SrSi 2 . The energies for both BaSi 2 models are almost identical (À2.06 eV) and, thus, expected to be equally stable. Nevertheless, these data alone are not sufficient to convey the stability of BaSi 2 to SrSi 2 as the elements Ba and Sr are too different. Furthermore, given the degrees of freedom, the tetragonal model of BaSi 2 relaxed into an orthorhombic lattice with differences in lattice parameters a and b in the order of 0.4%. It should be noted that the a parameters of hexagonal and tetragonal symmetry differ for both BaSi 2 and SrSi 2 compounds (see Table 18), although they are alike for dimorphic compounds of the family, e.g. GdSi 2 .
Subsequently, we use the chemical similarity of Ba and Sr to evaluate which orthorhombic structure type is more favorable for compound Sr 2 AgSi 3 , as it is the only alkaline earth compound that has not yet been synthesized. Both, the Ba 4 Li 2 Si 6 type of (Ba,Eu) 2 AgSi 3 and the Ca 2 AgSi 3 type are reasonable. We excluded other structure types as other chemically similar compounds only crystallize in those two structures. Here, chemically similar means a noble metal T and R preferring the +II oxidation state (e.g. alkaline earth metals, Eu and Yb). For T = Ag, Sr 2 AgSi 3 is the only alkaline earth compound that has not yet been synthesized.
As a reference, we used Ba 2 AgSi 3 , also in both structure types. For Ba 2 AgSi 3 , the respective formation energies exhibited a clear preference for the reported Ca 2 AgSi 3 type R-T diagram of the RSi 2 and R 2 TSi 3 compounds. The color of the markers symbolizes the range of ordering n, see Section 3.4. If the structure is disordered (AlB 2 , ThSi 2 , GdSi 2 ), then n = 0 and the symbol is gray. If the structure is ordered, the range of ordering accords to the number of stacks along c in the unit cell. Up to three markers on one grid position are possible, representing different publications. structure. However, the formation energies for both Sr 2 AgSi 3 models are almost identical with a value of À2.83 eV, therefore we conclude that both structure types are equally stable. The formation energy of Sr 2 AgSi 3 is slightly lower than that of Ba 2 AgSi 3 , which supports a stable structure.
Finally, we consider the potential tetragonal R 2 TSi 3 superstructure as determined in Section 3.1. We did not find reports on this ordered tetragonal structure and expect that it is energetically unfavored. Only a few articles on suitable compounds exist, mainly containing Th compounds (Albering et al., 1994;Lejay et al., 1983;Chevalier et al., 1986;Li et al., 2008;Kaczorowski & Noë l, 1993;Pö ttgen & Kaczorowski, 1993) as well as U 2 CuSi 3 (Albering et al., 1994;Lejay et al., 1983;Chevalier et al., 1986), La 2 AlSi 3 (Raman & Steinfink, 1967), Ce 2 AuSi 3 (Gordon et al., 1997), Er 2 CuSi 3 and Nd 2 AgSi 3 . We chose Nd 2 AgSi 3 for better comparability, as several compounds with either Nd or Ag have already been examined in the previous discussions. To compare our hypothetical tetragonal superstructure with an existing structure, we chose the hexagonal Ce 2 CoSi 3 type, since the most obvious tetragonal ThSi 2 type exhibits mixed positions. We further took the disilicide NdSi 2 into account in both ThSi 2 and AlB 2 type structures.
Please note that the lattice parameters of the POTS type (calculated) are related to those of the ThSi 2 type (experimental) by rotation and elongation by a factor of % ffiffi ffi 2 p . Thus, the interatomic distances of both tetragonal structure types of Nd 2 AgSi 3 are approximately the same a ThSi2 = 4.12 Å % 4.21 Å = a POTS / ffiffi ffi 2 p . For Nd 2 CuSi 3 , we compared three different symmetries, the high symmetry Ce 2 CoSi 3 , experimentally confirmed Er 2 RhSi 3 ðP62cÞ and low symmetry P1 (No. 1). The lattice parameters of all three models are a = 8.06 Å and c % 4.24 Å , which is in good agreement with the experimental values [Er 2 RhSi 3 ðP62cÞ-type].
The formation energies of Nd 2 AgSi 3 stoichiometry are À3.69 eV for the Ce 2 CoSi 3 type and À3.72 eV for the tetragonal superstructure. With an absolute formation energy which is lower by 0.30 eV per atom, the tetragonal type is clearly favored. In general, the superstructural order for tetragonal symmetries may be suppressed for further reasons. On the one hand, the 3D Si/T network itself may present kinetic barriers. On the other hand, the entropy of mixing may hinder structural ordering more severe for the degeneracies of the 3D Si/T network than for the planar stacking of hexagonal symmetries. Fig. 9 gives an overview of the scatter of structure types within the RSi 2 and R 2 TSi 3 compounds. This figure adapts the R-T grid of Fig. 4 with symbols announcing symmetry and range of order. To quantify the ordering within the different structure types, we defined the range of order as zero if the Si/ T atoms do not order and otherwise as the number of Si/T layers along c in the unit cell. The range of order is highlighted by the color of the marker. The symmetry is marked by shape: hexagon for hexagonal AlB 2 -like, open star for orthorhombic AlB 2 -like, diamond for tetragonal ThSi 2 , elongated diamond for orthorhombic GdSi 2 . For technical reasons, this diagram shows at most three reports of the same compound (left, right, bottom). Our algorithm chooses the datasets with the highest as well as the lowest a parameter and an additional dataset with a different structure type, to depict the most significant variations. Fig. 9 visualizes the range of order in dependence on the atomic number of the R and T cations; it depicts the following trends:

Structure distribution
First, most of the compounds in the grid exhibit an hexagonal AlB 2 -like lattice. The other lattice types are mainly determined by the included R and T element. For example, the orthorhombic GdSi 2 structure type arises exclusively for lanthanide disilicides. The tetragonal lattice is dominant for R = Th compounds as well as for the disilicides with light rare earth elements. Additional compounds with tetragonal lattice are Ce 2 AuSi 3 , Nd 2 AgSi 3 and Er 2 CuSi 3 , all possessing a noble metal T element. Thus, the Fermi level of the T element affects the structural stability, see Nentwich et al. (2020).
Furthermore, the completely ordered orthorhombic structure types Ca 2 AgSi 3 and Ba 4 Li 2 Si 6 are only reported for R 2 TSi 3 compounds with the monovalent ions T = Ag, Au and the divalent ions R = Ca, Ba, Eu, Yb (Cardoso Gil et al., 1999;Sarkar et al., 2013). The partially ordered structure type U 2 RhSi 3 additionally arises for U 2 PdSi 3 (Chevalier et al., 1996). Here, we do not consider the compound Ba 2 LiSi 3 itself, since Li does not accord with our limitations to the T elements. Thus, the ordered orthorhombic AlB 2 -like structure types are more probable if the T element is a monovalent atom and if the R element prefers the +II oxidation state -as for the alkaline earth metals.
Second, tetragonal LaSi 2 does not follow the hexagonal symmetry of the disilicides with third group elements Sc and Y. This phenomenon illustrates the affiliation of Sc and Y to the heavy and of La to the light rare earth elements (RÖ MPP Online, 2011).
Third, with increasing atomic number of R within the lanthanide disilicides, three structure types succeed each other. The tetragonal ThSi 2 type is the dominant one for light rare earth elements (Ce-Eu), followed by the orthorhombic GdSi 2 type in the intermediate range and the hexagonal AlB 2 type for the heavy rare earth elements (according to the classification by Sitzmann; RÖ MPP Online, 2011). This development is present in all samples independent of their thermal treatment, see Nentwich et al. (2020). This meets an observation of Mayer et al. (1967): upon heating the samples to 1600 C, they discovered two phase transformations, one from AlB 2 type to GdSi 2 type and another one from GdSi 2 type to ThSi 2 type. These transformations are reversible. A decreasing atomic number within the lanthanide group is accompanied with a significantly increasing radius and therefore with a higher space requirement. Increased thermal lattice vibrations at higher temperatures also cause higher space requirements. Thus, annealing has the same effect as decreasing the atomic number of R.

research papers 4. Conclusions
We present an extensive literature study of the RSi 2 and R 2 TSi 3 compounds crystallizing in AlB 2 -and ThSi 2 -like structures complemented by DFT calculations. The local similarities between these structures, e.g. threefold planar coordination of the Si/T atoms, twelvefold coordination of the R elements, are highlighted and discussed. Additionally, we systematized the structure data and arranged them in a Bä rnighausen diagram showing the relationships between structure types. We were able to determine the space groups of the ordered nonstoichiometric disilicides as piezoelectric P62m (No. 189),P62c (No. 190) and P2mm (No. 25).
According to Bodak & Gladyshevskii (1985), compounds La 2 FeSi 3 , La 2 CoSi 3 , La 2 NiSi 3 , Ce 2 CuSi 3 and Ce 2 NiSi 3 form a solid solution of structure type AlB 2 (disordered Si/T sites). Nevertheless, as evident from the discussion, we conclude that superstructures are expected to be the thermodynamic equilibrium structures, although they may be hard to synthesize, as they require obtaining the exact chemical composition on the one hand and for a careful thermal treatment on the other hand.
Comparison of the symmetry distribution within the R-T grid showed a special characteristic of the structure types Ca 2 AgSi 3 and Ba 4 Li 2 Si 6 . These structure types only arise if R has the formal +II oxidation state and T is either Au or Ag. Additionally, these structures are reported to have ionic character, whereas all other compounds are reported to be metallic. The given R-T diagram also shows a transition from tetragonal ThSi 2 to orthorhombic GdSi 2 to hexagonal AlB 2 type within the lanthanide disilicides with increasing atomic number of R. The structure types behave similarly with increasing temperature when respective crystals are heated.
Figs. 5 to 7 emphasize the number of systematic investigations of the RSi 2 and R 2 TSi 3 compounds. On the one hand, these systematic investigations reduce systematic errors. On the other hand, the author's expectations may also have an impact on the evaluation (such as the structure type).
Concluding the DFT analysis, hypothetical compounds Ho 2 PtSi 3 , Pr 2 CoSi 3 , Eu 2 PtSi 3 and Nd 2 CoSi 3 are suggested to be stable, whereas Eu 2 RhSi 3 will be unstable. Due to the positive results for Pr 2 CoSi 3 and Nd 2 CoSi 3 , we recommend reinvestigating the R 2 TSi 3 compounds reported by Mayer & Tassa (1969), with R = La, Ce, Pr, Nd, Sm, Gd and T = Fe, Co, Ni (originally with R 2 T 0.8 Si 3.2 stoichiometry). To complete the crystal structure information of La 2 PdSi 3 , we predict the lattice parameters a = 8.34 Å and c = 4.38 Å in a Ce 2 CoSi 3 type structure. With respect to the question whether Sr 2 AgSi 3 prefers the Ca 2 AgSi 3 or the Ba 4 Li 2 Si 6 structure type, both models result in almost identical formation energies of À2.83 eV and are equally stable from a theoretical point of view. Likewise, BaSi 2 may exhibit hexagonal as well as tetragonal symmetry, as the formation energy of both models is À1.03 eV. In comparison, the potential tetragonal superstructure is less favorable than a highly symmetric hexagonal structure. The results of this work do not exclude the existence of structures that are equally or more stable than the ones presented here. The solid solutions with disorder at the Si/T position may always present potential candidates for the ground state of a specific R 2 TSi 3 compound.
At this point, the question of particular driving forces for a certain type of symmetry and the multiplicity of the superstructure symmetry types and structure types remains. This question will be addressed in the second part of this work (Nentwich et al., 2020) focusing on the electronic structure.

APPENDIX B Fundamentals of the DFT calculations
To calculate the formation energies with DFT, it is necessary to know the energy of the components that make up the compound. Table 19 contains a list of the underlying singleelement compounds used to calculate the formation energies in Table 18.  T 4b x T % 0 y T % 4 12 z T % 1 4 Si 4b x Si,1 % 0 y Si,1 % 2 12 z Si,1 % 1 4 Si 8c x Si,2 % 1 4 y Si,2 % 1 12 z Si,2 % 0 Table 18 Formation energies (eV) and lattice parameters (Å ) calculated with DFT.
Formation energies are given for R 2 Si 4 and R 2 TSi 3 compounds, respectively (same amount of atoms within calculated range). Compounds marked with * have already been reported in the literature.  Table 16 Wyckoff positions of the orthorhombic structure type GdSi 2 with space group Imma (No. 74) and lattice parameters a % a t , c % c t .

Element
Wyckoff symbol x y z    Table 19 Space groups of the unary R crystals used for standardization of the formation energies.