Bonding and Stability of Ternary Structures in the CeT 2 Al 20 (T = Ta, W, Re) and YRe 2 Al 20 Alloys

: A-T-Al aluminides, where A = actinide, lanthanide or rare earth elements and T = transition metals, have attracted considerable attention as potential materials where heavy fermions may be formed. This led to the discovery of superconducting properties in cubic AT 2 Al 20 compounds with CeCr 2 Al 20 -type crystal structure. Other Al-rich aluminides, belonging to these A-T-Al systems, exhibited di ﬀ erent physical properties as a function of their crystal structure. Thus, predicting the stable structure of the Al-richest phase that will form in the A-T-Al systems is highly valuable. Stability of the crystal structures, forming in the CeT 2 Al 20 and YRe 2 Al 20 systems, was studied in current research using density functional theory (DFT) calculations. It is demonstrated that the total spin magnetic moment of the transition metal can be used as a descriptor for phase stability assessment in the AT 2 Al 20 systems, where T is a 5 d transition metal. Basing on crystallographic considerations, degree of distortion of the coordination polyhedrons, formed around T atoms, can be directly connected to the speciﬁc type of structure crystallizing in these systems.

Cubic AT 2 Al 20 structure is of particular importance since phases of this type exhibited unique physical properties [18,19]. The unit cell of the AT 2 Al 20 structure (CeCr 2 Al 20 -type) may be presented as a packing of coordination polyhedrons formed around heavy atoms. Around A-type atoms a polyhedron with 16 vertices occupied by Al atoms exists. Such polyhedron is of Frank-Kasper type [20]. Since the shortest A-A atomic distance is about 6Å, the A atoms are considered to be isolated from each other, and the structure can be classified as cage-type [18]. In addition to the AT 2 Al 20 , there are other structures classified as a cage-type, for example: the filled skutterudites [21] and the Laves phases [22]. In these systems, low-energy, large amplitude, localized anharmonic vibrations of cage-filling atoms are observed. This "rattling" of atoms affects thermal and transport properties of materials. It is believed that superconducting critical temperature, T c is enhanced by this effect in different cage-type structures, among them are CeCr 2 Al 20 -type compounds. For example, PrT 2 Al 20 (T = Ti, V) are exotic theory developed in [27,28]. This hypothesis can be demonstrated in Figure 2, presenting the reported structures that are formed in the Al-rich A-T-Al systems (where T = 5d transition metals). It is clear from this table that when T = W and Ta, cubic phases are formed in the AT 2 Al 20 alloys with essentially all A-type elements. When T = Re, the symmetry breaks, and the phases do not crystalize in the cubic symmetry. The novelty of current work is the demonstration of the ability to predict the stable structure of complex ternary Al-rich compounds in the A-T-Al systems where T = 5d transition metals. For the first time crystallographic analysis is presented, connecting the degree of distortion of coordination polyhedron formed around the transition metal and the specific crystal structure formed.
Metals 2020, 10, x FOR PEER REVIEW 3 of 14 The novelty of current work is the demonstration of the ability to predict the stable structure of complex ternary Al-rich compounds in the A-T-Al systems where T = 5d transition metals. For the first time crystallographic analysis is presented, connecting the degree of distortion of coordination polyhedron formed around the transition metal and the specific crystal structure formed.

Materials and Methods
DFT calculations were performed in the framework of the Full Potential + Linearized Augmented Plane Wave (FP-LAPW) method [35]. These calculations allow analysis of the relative stability of structures of considered phases. Total energy spin-polarized calculations of competing phases were performed. In this work spin orbit coupling (SOC) calculations were carried out as well since here, unlike previous work [27,28] the transition metal, Re, is much heavier than 3d transition metals. The results are presented with and without SOC calculations. WIEN2k code (Version 10.1) [36] was applied for calculations. In this code, the core states are treated as fully relativistic [36] and the valence states are considered using a scalar relativistic approach [37].
Since the studied systems contain lanthanides and rare earth elements, the exchange correlation effects should be carefully examined. The exchange-correlation potential was calculated within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) [38]. The application of hybrid density functional theory method (HYB-DFT) [39,40] to similar systems with 3d-metals was discussed in [28], where HYB-DFT calculations were performed by the "exact exchange for correlated electrons" method. The hybrid functional replaces a fraction of DFT exchange with exact Hartree-Fock (HF) exchange and applied only to a selected set of lanthanides` or actinides` electrons, namely the 4f-or 5f-electrons inside the atomic sphere. The hybrid exchange-correlation energy functional + has the form [41]: where is the HF exchange, is the PBE exchange functional and is the PBE correlation functional. The mixing parameter α denotes the fraction of HF exchange replacing the PBE exchange. It has been shown that by increasing the amount of HF exchange, one increases the degree of electronic localization. It was also demonstrated that PBE nicely reproduces the stability of crystal structures for all considered systems and gives the results similar to HYB-DFT for all studied systems when α = 0.05. Larger magnitudes of α do not describe this phase competition in a correct way. It should be mentioned that nowadays also DFT plus Hubbard U (DFT + U) method is widely used for strongly correlated systems. In this approach, an orbital dependent field is introduced

Materials and Methods
DFT calculations were performed in the framework of the Full Potential + Linearized Augmented Plane Wave (FP-LAPW) method [35]. These calculations allow analysis of the relative stability of structures of considered phases. Total energy spin-polarized calculations of competing phases were performed. In this work spin orbit coupling (SOC) calculations were carried out as well since here, unlike previous work [27,28] the transition metal, Re, is much heavier than 3d transition metals. The results are presented with and without SOC calculations. WIEN2k code (Version 10.1) [36] was applied for calculations. In this code, the core states are treated as fully relativistic [36] and the valence states are considered using a scalar relativistic approach [37].
Since the studied systems contain lanthanides and rare earth elements, the exchange correlation effects should be carefully examined. The exchange-correlation potential was calculated within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) [38]. The application of hybrid density functional theory method (HYB-DFT) [39,40] to similar systems with 3d-metals was discussed in [28], where HYB-DFT calculations were performed by the "exact exchange for correlated electrons" method. The hybrid functional replaces a fraction of DFT exchange with exact Hartree-Fock (HF) exchange and applied only to a selected set of lanthanides" or actinides" electrons, namely the 4f -or 5f -electrons inside the atomic sphere. The hybrid exchange-correlation energy functional E PBE+αHF xc has the form [41]: where E HF x is the HF exchange, E PBE x is the PBE exchange functional and E PBE c is the PBE correlation functional. The mixing parameter α denotes the fraction of HF exchange replacing the PBE exchange. It has been shown that by increasing the amount of HF exchange, one increases the degree of electronic localization. It was also demonstrated that PBE nicely reproduces the stability of crystal structures for all considered systems and gives the results similar to HYB-DFT for all studied systems when α = 0.05. Larger magnitudes of α do not describe this phase competition in a correct way. It should be mentioned that nowadays also DFT plus Hubbard U (DFT + U) method is widely used for strongly correlated systems. In this approach, an orbital dependent field is introduced providing a correction for the self-interaction [42]. DFT + U significantly improve the results for band gap, optic, X-ray absorption or magnetic properties for insulating actinide oxides (such as UO 2 ), for oxides where the charge ordering is observed [43,44], and for some bromides (see, for example [45]). The detailed examination of the LDA + U method in the framework of a plane-wave pseudopotential approach may be found in [46].
We decided not to use this approach here for several reasons: (a) the Coulomb parameter, U, and exchange parameter, J, determined by first principle calculations do not always give reliable results for some Fe-based compounds (as an example) [47]; (b) our own experience [27,28,48,49] shows that the DFT in the GGA approximation with PBE functional very well reproduces the basic structural and electron spectrum properties of U-and Th-based binary, ternary and even quaternary compounds; (c) recent results [50][51][52] have demonstrated that DFT + U approach is not necessary for a correct description of U metal and mixing energies in the U-based compounds; (d) we consider the total energy differences to estimate the relative stability of compounds. In this case, possible GGA inaccuracies are canceled as it was demonstrated in [53].
The details of our calculations are following: the used muffin-tin radii (R MT ) were equal to 2.5 Bohr for all atoms (Ta, W, Re, Y and Ce) except Al, for which R MT was chosen equal to 2.2 Bohr. This choice is dictated by the necessary condition to avoid overlapping between the atomic spheres in the volume optimization procedure. For the basis-set size at each point, the cut-off parameters R MT K max = 7 was used, where K max is the magnitude of the largest k vector in the wave function expansion. A maximum l, l max , equal to 10 was taken in the expansion of the radial wave functions inside the atomic sphere to represent the valence states. The cut-off energy, separating core and valence states was chosen to be equal to −7.0 Ryd for all atoms. This condition is necessary for minimization of leakage of the electron core states into the interstitial region. All integrations over the Brillouin zone (BZ) were done using TEMPS scheme implemented in WIEN2k with the broadening parameter in Fermi energy calculations equal to 0.002 that corresponds to~40 • C. It was found that the number of k-points in the first BZ equal to 300 is enough to obtain the accuracy of the self-consistent calculations for the energy of the system not less than~10 −3 Ryd. It should be noted that for the Ce-Re-Al system, energies were calculated with higher accuracy (~10 −4 Ryd), due to the fact that energy differences between competing structures are smaller than 10 −3 Ryd. Keeping in mind that the stability of the structure is determined by comparison of the energies of structures in equilibrium conditions, volume optimization was carried out for each structure aiming at minimal total energy determination. Using a set of energies and corresponding volumes of the unit cells, the equilibrium volume of the unit cell of all considered phases was obtained by fitting the data to Murnaghan equation of state [54]. The total energies of the competing structures of the phases were compared for the optimized structures. Such procedure not only allows confirmation of the stability of experimentally observed structures but also gives possibility to compare experimental and theoretical lattice parameters of the observed structures, providing an opportunity to assess the quality of calculations.

Results and Discussion
In the current research, the hypothesis of structure stability, suggested in [27,28], was checked for the row of 5d transition metals. Ce with electron configuration of [Xe]4f 1 5d 1 6s 2 was chosen as representative of the A atom type in the studied AT 2 Al 20 composition and the 5d transition metals changed from Ta to Re. A = Y was chosen due to its different than Ce electron configuration, i.e., [Kr] 4d 1 5s 2 , without f -electrons in its outer shell.
Comparing the minimal energies of the competing structures is not a trivial task since the number of atoms in each structure should be taken into account. DFT calculations are performed for the unit cells rather than for the conventional lattice cells, therefore, transferring from conventional lattice cell to Metals 2020, 10, 422 6 of 13 a unit cell reduces the number of atoms for the orthorhombic structure (1c) and its structural derivatives (2c and 3c) by two and for the cubic structure by four. The tetragonal unit cell will remain unchanged.
It can be understood from Table 2 that the total number of atoms in the tetragonal unit cell is twice the number of atoms containing the orthorhombic 1c structure. Thus, in order to compare the minimal energies for these structures, the total energy of the tetragonal structure should be divided by 2. Similarly, the total energies of the orthorhombic 2c and 3c should be divided by 2 or 3, respectively. Table 2. Total number of each type of atoms in the conventional lattice cell (denoted by the space group notation) and in the corresponding unit cell used for the density functional theory (DFT) calculations. Orthorhombic structure is referred by "Ortho". Since the cubic structure crystalizes in the AT 2 Al 20 composition while all other structures possess AT 2 Al 10 composition, the excess of pure Al atoms that obtained experimentally (in [26][27][28], for example), should be accounted when the relative stability of these phases is considered. The cubic structure has 20 Al atoms more than the other structures. Therefore, the formation energy of pure Al-metal structure (FCC), multiplied by 20, should be added to both tetragonal and orthorhombic structures to keep the composition constant. The calculated total energy for pure Al at equilibrium volume per atom was found to be equal to −485.646 Ry. In order to obtain the energy preference of one structure over the other the following equation should be used: The energy differences for all competing structures for the Ce-T-Al and Y-Re-Al systems at AT 2 Al 20 composition are presented in Table 3, with and without SOC calculations. For the Re based systems, calculations of the structural derivatives are also presented. ∆E shows the preference of the structure with respect to the cubic one, i.e., more negative ∆E value points on more stable structure. It can be seen that the cubic structure is preferred in the CeTa 2 Al 20 and CeW 2 Al 20 alloys. In the CeRe 2 Al 20 alloy, breakage of symmetry occurred and the tetragonal structure was found to be the stable one. When the A atom type is Y, the orthorhombic 3c structure is stable. Results of calculations were compared with the experimental data [1,[55][56][57]. In the CeT 2 Al 20 (T = Ta and W) systems, the stable structure was determined as cubic (i.e., CeCr 2 Al 20 type), while for the CeRe 2 Al 20 and YRe 2 Al 20 alloys the stable structures of the Al-richest ternary phases were found to be tetragonal (CaCr 2 Al 10 type) and orthorhombic 3c (LuRe 2 Al 10 -type), respectively. As mentioned, our results presume additional pure Al phase in these two alloys to preserve studied here AT 2 Al 20 composition . Table 4 presents the comparison of the experimental unit cell parameters, as reported in the literature and the values retrieved from the calculations. The results are in excellent agreement.
According to [28], the value of the magnetic moment of the transition metal may be used to predict the stable structure of the ternary Al-rich phase that will form in the AT 2 Al 20 alloys (where T = 3d transition metals and A = lanthanide with 4f electron or no f electrons in theirs shells beyond the totally filled inert gas shell). Minimal value of the magnetic moment corresponds to a large splitting between the e g and t 2g states due to the fact that majority of 3d-electrons occupy the lowest energy levels. According to the Theory of Coordination Compounds (TCC) [58], the large splitting between the e g and t 2g energies corresponds to a strong crystal field of the transition metal neighborhood and, thus, to a more stable structure. It should be noted that this "rule" was applied to AT 2 Al 20 alloys where A was either with 4f -electrons that are well localized or without f -electrons in the outer shell, therefore, spectroscopic and magnetic properties were independent of the environment [28].  [56] Calculated magnetic moments of the 5d transition metals, in studied here systems for each competing structure, are presented in Table 5. The behavior of the magnetic moments of the 5d transition metals complies with suggested for the 3d transition metals theory. It can be seen that the lowest value of the magnetic moment corresponds to the stable structure formed in each system. For the CeT 2 Al 20 (T = Ta, W), the lowest value is obtained for the cubic structure, while when T = Re the tetragonal structure possesses the lowest value. For the YRe 2 Al 20 alloy, even if we sum up the magnetic moments of both Re (1) and Re (2) type atoms, the orthorhombic 3c structure has the lowest value of magnetic moment of Re, see Table 5. It should be noted that in all other discussed structures, besides orthorhombic 3c, transition metal occupies only one Wyckoff site, marked as Re (1) in Table 5.
Confirmation of the fact that f -electrons do not influence the stability trends in considered systems may be found through the analysis of Density of States (DOS) for electrons. As an example, the calculated local partial DOS for the 4f -states of Ce and 5d-states of Re are presented in Figure 3 for three main competing structures (i.e., cubic, tetragonal and orthorhombic 1c). The 4f electrons peak of Ce is localized above the Fermi energy level. These electrons do not contribute to the bonding with 5d electrons of Re. For the studied here Y-based compounds similar analysis of partial DOS (not presented here) has shown that only small amount of promoted Y f -states in the background of the rest of atoms exists, and they also do not influence the competition between the structures. Thus, the 4f electrons of Ce and promoted f -states of Y do not influence the structure stability. This conclusion underlines that Metals 2020, 10, 422 8 of 13 the magnetic moment of the transition metal can be used as a descriptor to predict the stable Al-rich ternary structure that will form in the AT 2 Al 20 alloys when T is 3d or 5d transition metal. Since magnetic moment of T atom type is the one which dictates the stability, coordination polyhedron formed around this atom in each structure should dictate the geometry of the structure (i.e., cubic, tetragonal or orthorhombic). The difference between the discussed here structures can be expressed in terms of distortion of the coordination polyhedron formed around T, and might also effect the stability according to the Jahn-Teller effect and TCC theory [58,59]. In all structures discussed in this manuscript (including their structural derivatives), icosahedron (with coordination 12) is formed around T atoms. In cubic structure (of the CeCr 2 Al 20 type), this icosahedron is the closest to perfect one because only Al atoms occupy its vertices. Although these Al atoms belong to two different Wyckoff positions, the distances from the center of the icosahedron (occupied by the T atom) to its vertices are almost equal, see Table 6. In other structures discussed in this manuscript, icosahedrons formed around T atoms contain two A-type atoms and 10 Al atoms, as shown in Table 6, thus, they are distorted. Degree of polyhedral distortion was calculated following [60] and is presented in Table 6. According to [60], the volume of icosahedron is equal to 2.5362r 3 , where r denotes the center-to-vertex distance for a regular/ideal polyhedron. The polyhedral distortion was calculated as follows: where V i is the volume of the ideal icosahedron (which belong to the cubic structure in our case), and V r is the volume of the regular icosahedron. It should be noted that out of all reported in the database [34] representatives of the discussed structure types we have chosen for comparison T and A atom types with similar atomic radius, since identical atoms do not form all these structures, as discussed here and in our previous publications [17,[26][27][28]. It can be clearly seen that in each structure the degree of polyhedral distortion is different with tetragonal structure having the highest degree of distortion relatively to the orthorhombic ones. Combining this understanding with the above made conclusion it can be proposed that the magnetic moment of the T atom imposes the required degree of distortion of the polyhedron which as a result will impose the geometry of the unit cell (i.e., highest distortion-tetragonal, smaller distortion-orthorhombic and zero distortion-cubic). For differentiation between three possible orthorhombic structures (1c, 2c, 3c), crystallographic analysis should be extended. In Figure 4, chains of interconnected icosahedrons, as formed in the unit cells of three orthorhombic structures, are presented. In the 1c orthorhombic structure, T and A atom types occupy one Wyckoff position each. Thus, icosahedrons, connected through the vertices occupied by the A type atoms, form parallel to c direction rows, see Figure 4a. In the 2c orthorhombic structure, T atom occupies one Wyckoff position and A atom occupies two different Wyckoff positions-thus there are two independent options to interconnect these icosahedrons. Consequently, icosahedrons form zigzag-like chains along the same c direction, as shown in Figure 4b. In the 3c orthorhombic structure, both T and A atoms occupy two independent Wyckoff positions. Around one position, icosahedron is formed exactly as in the 1c structure, i.e., it includes two A atoms occupying the same Wyckoff site, see Re (2) in Table 6. Thus, such icosahedrons form straight row with their neighboring icosahedrons. The second position of T atom (Re (1) in Table 6) is surrounded by the same, as in the 2c structure, icosahedra (with A-type atoms at two different Wyckoff positions) which imposes it to interconnect in a zigzag shape. Thus, in this structure both rows and zigzags exist, imposing larger unit cell, see Figure 4c. Variety in the sequence of the coordination icosahedra provide an opportunity to have several derivatives of the orthorhombic structure as a function of degree of distortion imposed by the T and A atom types.
Metals 2020, 10, x FOR PEER REVIEW 9 of 14 of distortion of the polyhedron which as a result will impose the geometry of the unit cell (i.e., highest distortion-tetragonal, smaller distortion-orthorhombic and zero distortion-cubic). For differentiation between three possible orthorhombic structures (1c, 2c, 3c), crystallographic analysis should be extended. In Figure 4, chains of interconnected icosahedrons, as formed in the unit cells of three orthorhombic structures, are presented. In the 1c orthorhombic structure, T and A atom types occupy one Wyckoff position each. Thus, icosahedrons, connected through the vertices occupied by the A type atoms, form parallel to c direction rows, see Figure 4a. In the 2c orthorhombic structure, T atom occupies one Wyckoff position and A atom occupies two different Wyckoff positions-thus there are two independent options to interconnect these icosahedrons. Consequently, icosahedrons form zigzag-like chains along the same c direction, as shown in Figure 4b. In the 3c orthorhombic structure, both T and A atoms occupy two independent Wyckoff positions. Around one position, icosahedron is formed exactly as in the 1c structure, i.e., it includes two A atoms occupying the same Wyckoff site, Summarizing, it can be proposed that in the A-T-Al systems (where A-type atoms have 4f electrons or no f-electrons in the outer shell) the stability of the Al-richest ternary phase can be predicted by the magnetic moment of the T atom. In fact, this parameter prognosticates the degree of distortion of the coordination polyhedron formed around T atom which stabilizes the structure. When the needed distortion is achieved, the geometry of the unit cell is set following the interconnection of the icosahedrons formed around the T atom types. Table 6. Calculation of the degree of distortion of the icosahedron formed around T atom in the discussed structures. Following [60], r parameter is the average distance from the center to the vertices of the coordination polyhedron. Distances are given in Å. For calculations, atomic models of EuTa 2 Al 20 , NdRe 2 Al 10 , LnOs 2 Al 10 , GdRe 2 Al 10 and YRe 2 Al 10 (taken from [34]) were used as prototypes of cubic, tetragonal and 1c, 2c, 3c orthorhombic structures, respectively, containing A and 5d T atoms. electrons or no f-electrons in the outer shell) the stability of the Al-richest ternary phase can be predicted by the magnetic moment of the T atom. In fact, this parameter prognosticates the degree of distortion of the coordination polyhedron formed around T atom which stabilizes the structure. When the needed distortion is achieved, the geometry of the unit cell is set following the interconnection of the icosahedrons formed around the T atom types. Table 6. Calculation of the degree of distortion of the icosahedron formed around T atom in the discussed structures. Following [60], r parameter is the average distance from the center to the vertices of the coordination polyhedron. Distances are given in Å . For calculations, atomic models of EuTa2Al20, NdRe2Al10, LnOs2Al10, GdRe2Al10 and YRe2Al10 (taken from [34]) were used as prototypes of cubic, tetragonal and 1c, 2c, 3c orthorhombic structures, respectively, containing A and 5d T atoms.

Conclusions
In [27,28], it was proposed that the magnetic moment of transition metal can be used for assessment of the stability of the Al-rich ternary structures forming in the A-T-Al systems, where T = 3d transition metal and A has either no f electrons or 4f electrons in its outer shell. In current research, a successful attempt to generalize this theory was made. CeT2Al20 (T = Ta, W, Re) and YRe2Al20 compositions were studied by DFT calculations in order to validate the proposed theory for

Conclusions
In [27,28], it was proposed that the magnetic moment of transition metal can be used for assessment of the stability of the Al-rich ternary structures forming in the A-T-Al systems, where T = 3d transition metal and A has either no f electrons or 4f electrons in its outer shell. In current research, a successful attempt to generalize this theory was made. CeT 2 Al 20 (T = Ta, W, Re) and YRe 2 Al 20 compositions were studied by DFT calculations in order to validate the proposed theory for compounds containing 5d transition metals. T = Re is a special case, since in the row of 5d transition metals, similar to Mn in the row of 3d transition metals, with this element the symmetry breaks. Generally speaking, we prove that the cubic symmetry is stable for Al-rich ternary structures forming in the A-T-Al systems (with AT 2 Al 20 composition), if is T situated in the columns 5B and 6B in periodic table while for T situated in 7B column, the symmetry breaks and less symmetric structures with AT 2 Al 10 composition crystallize. These results were compared with structural data reported in the database and found to be in excellent agreement [35]. Calculation of magnetic moments of T elements for all competing structures in each studied system allowed concluding that indeed the lowest values of the magnetic moments correspond to the stable structures obtained by total energies calculations. Analysis of DOS has shown that 4f -electrons peaks of A are localized above the Fermi energy level and do not contribute to the bonding with 5d-electrons of T. In fact, the value of magnetic moment of T atom type imposes specific degree of distortion on coordination polyhedron formed around transition metal. Consequently, this distortion imposes specific geometry on the crystal structure formed, due to the necessity to interconnect these polyhedrons. Crystallographic analysis of distortion of coordination polyhedra, around T atoms in these systems, reasons the formation of the structural derivatives of the orthorhombic phase (referred as 1c, 2c and 3c) formed with T = Re. It can be seen that as a function of this distortion, sequence of the polyhedral changes-imposing larger unit cell.