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Sep 28, 2016 - Pressure Relief, and Potential Rattling Atoms in Y11Ni60C6. Yiming Guo and Daniel C. Fredrickson*. Department of Chemistry, University ...
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On the Functionality of Complex Intermetallics: Frustration, Chemical Pressure Relief, and Potential Rattling Atoms in Y11Ni60C6 Yiming Guo and Daniel C. Fredrickson* Department of Chemistry, University of WisconsinMadison, 1101 University Avenue, Madison, Wisconsin 53706, United States S Supporting Information *

ABSTRACT: Intermetallic carbides provide excellent model systems for exploring how frustration can shape the structures and properties of inorganic materials. Combinations of several metals with carbon can be designed in which the formation of tetrahedrally close-packed (TCP) intermetallics conflicts with the C atoms’ requirement of trigonal prismatic or octahedral coordination environments, as offered by the simple close-packings (SCP) of equally sized spheres. In this Article, we explore the driving forces that lead to the coexistence of these incompatible arrangements in the Yb11Ni60C6-type compound Y11Ni60C6 (cI154), as well as potential consequences of this intergrowth for the phase’s physical properties. Our focus begins on the structure’s SCP regions, which appear as C-stuffed versions of a AuCu3-type YNi3 phase that is not observed on its own in the Y−Ni system. DFT-Chemical Pressure (DFT-CP) calculations on this hypothetical YNi3 phase reveal large negative pressures within the Ni sublattice, as it is stretched to accommodate the size requirements of the Y atoms. In the Y11Ni60C6 structure, two structural mechanisms for addressing these CP issues appear: the incorporation of interstitial C atoms, and the presence of interfaces with CaCu5-type domains. The relative roles of these two mechanisms are investigated with the CP analysis on a hypothetical YNi3Cx series of C-stuffed AuCu3-type phases, the Y−Ni sublattice of Y11Ni60C6, and finally the full Y11Ni60C6 structure. Through these calculations, the C atoms appear to play the roles of relieving positive Y CPs and supporting relaxation at the AuCu3-type/CaCu5-type interfaces, where the cancellation occurs between opposite CPs experienced by the Y atoms in the two parent structures (following the epitaxial stabilization mechanism). The CP analysis of Y11Ni60C6 also highlights a sublattice of Y and Ni atoms with large negative CPs (and thus the potential for soft vibrational modes), illustrating how frustrated structures could lead to the full realization of the phonon glass−electron crystal concept.

1. INTRODUCTION As early as the 1920s, Linus Pauling noted that structures can arise in intermetallic compounds whose complexity is highly incongruent with the simple sphere packings of most elemental metals.1 Since Pauling’s original diffraction experiments on NaCd2 crystals, a series of phases with giant unit cells containing >1000 atoms has emerged, including not only NaCd2 (cF1,192),2 but also β-Al3Mg2 (cF1,168),3,4 Cu4Cd3 (cF1,124), 5 Sm 117 Co 56 Sn 116 (cF1,154), 6 Ta 39.5 Cu 3.9 Al 56.5 (cF5,908),7,8 and Ta39.1Cu5.4Al55.4 (cF23,134).7,8 Such compounds are rivaled only by quasicrystals9−11 in their complexity among inorganic materials, and have inspired numerous researchers to seek out regularities in their atomic arrangements.12−21 Diverse geometrical schemes have arisen from these efforts, including the extensive classification of structures based on the nesting of concentric polyhedra18,21 or recurring cluster units,16,17,19 and projections from higher-dimensional polytopes.15,20 Beyond the systematic classification of these structures, however, two major issues remain largely unresolved: what driving forces shape the specific features of these structures, and do their unique structures underlie similarly unique properties?22 © 2016 American Chemical Society

In this Article, we will see possible answers to these questions emerge through the theoretical Chemical Pressure (CP) analysis of the Yb11Ni60C6 type.23−25 Like a variety of other metal-rich ternary carbide structures, the Yb11Ni60C6 type represents a compromise between the preference of many intermetallic phases for tetrahedral close packed (TCP) arrangements, and the inability of C atoms to fit into the tetrahedral interstices of such packings.26 A solution to this conflict is the forced opening of tetrahedra to create octahedral holes in an intermetallic structure.27 The Yb11Ni60C6 type, however, demonstrates a more intriguing solution: the segregation of the structure into intermetallic and carbide domains (Figure 1), which closely recalls the theme of incompatible bonding or packing modes, such as polar vs nonpolar domains or radial vs periodic order, coexisting in giant intermetallic structures.8,28 This compound has several other features making it an excellent model system for complex intermetallics. When taking Y11Ni60C6 as a specific (and 4f-electron free) example, its 154Received: July 9, 2016 Published: September 28, 2016 10397

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Inorganic Chemistry

stronger resemblance to tetrahedrally close-packed (TCP) arrangements. In exploring the possible role of epitaxial stabilization in this compound, we will see a scheme emerge in which a rigid framework of TCP/SCP domains is created through interfaces tuned by carefully placed interstitial atoms. An unexpected outcome of the creation of this highly favorable framework, however, is a vast array of gaps in which potential rattling atoms appear, reminiscent of the cations of intermetallic clathrates38,39 and skutterudites.40,41 This theme of interpenetrating rigid and soft domains has parallels in other complex intermetallics, and suggests that these phases may offer unique avenues to phonon glass−electron crystal materials via the substructure approach.42,43 Figure 1. The Y11Ni60C6 structure (Yb11Ni60C6-type). The majority of the structure can be described as an intergrowth of C-stuffed AuCu3type YNi3 (yellow) and CaCu5-type YNi5 (purple) domains.

2. TECHNICAL PROCEDURES The DFT-Chemical Pressure analyses were carried out on the LDADFT electronic structures of various phases in the Y−Ni−C system, both observed and hypothetical. These analyses began with the geometrical optimization of the crystal structures with the Vienna Ab initio Simulation Package (VASP)44,45 in the high precision mode, using the ultrasoft LDA pseudopotentials provided with the package.46 The optimizations followed a two-step procedure: in the first step, the cell parameters are fixed and all atoms are allowed to relax within the unit cell; in the second step, both the cell parameters and the atomic coordinates are released simultaneously. Following the geometrical optimization of the structures, single point calculations were carried out with the ABINIT program47,48 at three different volumes (the equilibrium volume, plus slightly expanded and slightly contracted volumes; total volume range = 0.6%) to obtain the kinetic energy and electron densities, and the various components of the local potential needed for the construction of CP maps. The ABINIT calculations used the LDA exchangecorrelation functional of Goedecker, Teter, and Hutter, and Hartswigen−Goedecker−Hutter norm-conserving pseudopotentials.49,50 CP analysis was then performed with our group’s DFT-CP Package.51,52 The core undistorting procedure and Hirshfeld-inspired integration schemes were employed,32 using electron density profiles for free ions modeled after the structure’s Hirshfeld charges.53 The Hirshfeld charges were obtained by running a calculation with neutral

atom cubic unit cell is small enough to be amenable to DFT calculations whose results can be processed and interpreted with the CP method.29−32 Another advantage to considering Y11Ni60C6 as a model for complex intermetallics is that a hypothesis for its stability can be found in recent results for the Mn−Si−C system.33 In this earlier work, the structures of the newly synthesized Mn16SiC4 and Mn17Si2C4 were understood in terms of the complementary CP schemes of Mn-rich silicides and carbides. The Mn sublattices of these two types of compounds are respectively overly contracted and stretched as they accommodate their Si and C atoms. The formation of interfaces between silicide and carbide domains allowed these opposing stresses to cancel out, an effect we termed epitaxial stabilization by contrast with the stress occurring at most epitaxial interfaces.34,35 At first glance, the Y11Ni60C6 structure can be described in a similar manner, with blocks of a C-stuffed AuCu3-type YNi3 phase appearing as intergrown with fragments of YNi5, a CaCu5-type compound.36,37 The former is a classic simple close packed (SCP) phase, while the geometry of the latter bears a

Figure 2. The structure of Y11Ni60C6 depicted as an intergrowth of simple close packed (SCP, yellow) and more tetrahedrally close packed (TCP, purple) domains. (a) A hypothetical AuCu3-type YNi3 phase. (b) The SCP domain in Y11Ni60C6, as a C-stuffed truncated octahedral fragment of the hypothetical AuCu3-type YNi3 phase. The octahedral coordination environment of one interstitial C atom is highlighted by including its Y neighbor outside the cluster. (c) The CaCu5-type structure of YNi5. (d) The more TCP domain in Y11Ni60C6 viewed as a fragment from YNi5. (e) The full crystal structure of Y11Ni60C6 formed from the linkage of the SCP and TCP fragments through shared hexagonal faces, and the placement of scaffolding atoms at the Y1 and Ni1 sites (shown as isolated atoms). The SCP fragments pack in a bcc fashion, while the TCP fragments form connections between them running along the body diagonals. (f) The coordination polyhedron of theY2 site at the TCP/SCP interface. 10398

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Inorganic Chemistry atomic profiles obtained from the ABINIT homepage,54 and the charges were given as part of the CPpackage output. These Hirshfeld charges were then input into the Atomic Pseudopotentials Engine (APE) program55 to generate the ionic profiles. Further computational details, such as the energy cutoffs and k-point grids for the LDA-DFT calculations, as well as the validation of the ionicity used in the CP analyses, are given in the Supporting Information.

structures, since its structure has mostly tetrahedral interstices. Therefore, following the structural description above, a majority of the Y11Ni60C6 structure can be understood in terms of the intergrowth of two different packing types. The Y atom sitting at the center of the hexagonal faces in Figures 2b and 2d plays then a dual role in the structure, as its environment combines SCP (yellow) and more TCP (purple) regions, resulting in a hybrid coordination polyhedron (Figure 2f). As is evident in Figure 2e, this framework of SCP and TCP units is not space-filling, but instead leaves openings for two additional sites: the Y1 and Ni1 positions. The coordination polyhedra of these atoms are defined chiefly by the geometrical requirements of the SCP/TCP framework, suggesting that these atoms are present to buttress the intergrowth framework against collapse rather than to enjoy tailor-made environments (a theme we will return to with an interest in properties below). In summary, the Y11Ni60C6 structure can be viewed as a SCP/TCP intergrowth of AuCu3-type YNi3 and CaCu5-type YNi5 fragments, with additional scaffolding sites and interstitial C atoms. Curiously, a search in the Y−Ni−C ternary phase diagram indicates that while YNi5 in CaCu5 type is an experimentally observed phase, the AuCu3-type YNi3 parent structure of the SCP units in Y11Ni60C6 does not form. While a phase with the stoichiometry YNi3 in the system is observed, it adopts a quite different structure type (Be3Nb type, based on the intergrowth of the CaCu5-type and Laves phase-type slabs).56 Since the AuCu3-type YNi3 structure is unstable on its own, it can be inferred that the YNi3 fragments in Y11Ni60C6 must be somehow stabilized by other structural features of the ternary phase. The origin of this stabilization will provide an entryway to understanding the existence of Y11Ni60C6.

3. THE Y11Ni60C6 STRUCTURE AS AN INTERGROWTH OF THE CaCu5 AND AuCu3 TYPES Over the course of this Article, we will see that Y11Ni60C6 (Im3̅m, cI154) has much to tell us about how interfaces and interstitial atoms can stabilize otherwise unobserved geometrical motifs in intermetallic systems. To begin, let us examine the structure of this phase in more detail. Despite the relatively large number of atoms in its unit cell, it can be easily reduced to building blocks based on two simple, common structure types: the AuCu3 and CaCu5 types. Starting with these component structures greatly simplifies the description of Y11Ni60C6. The AuCu3 structure type is an fcc variant (Figure 2a) in which different elements occupy the corner and face-center positions. One does not have to search far in the structure of Y11Ni60C6 to find this pattern (Figure 2b): the origin of the Y11Ni60C6 unit cell lies in the center of a Ni6 octahedron, which is in turn enclosed in a cube of Y atoms, in effect creating a unit cell of a hypothetical AuCu3 type YNi3 phase. An additional layer of Ni atoms extends this pattern further, creating a fragment of the AuCu3 type that has the shape of a truncated octahedron. The C atoms of Y11Ni60C6 lie in the Ni square pyramids under the square faces of this unit (with additional Y atoms completing octahedral coordination environments around the C atoms). These C-stuffed AuCu3 type clusters are then arranged in a body-centered cubic fashion in the full crystal structure of Y11Ni60C6, with a monolayer of atoms occurring between them. The second key structure for understanding Y11Ni60C6 is the CaCu5-type (Figure 2c), in which the Cu atoms are stacked in alternating kagome (blue) and honeycomb (green) layers. Its Ca atoms, meanwhile, are located at the centers of the hexagons in the honeycomb layers in large 18-vertex polyhedra. A look at any of the spaces between the AuCu3-type clusters along the cell diagonals reveals small fragments of this second structure type (Figure 2d): in each case, two hexagons of Ni derived from the honeycomb nets of the CaCu5 type are separated by a smaller hexagon to make short hexagonal antiprisms, while the Y atoms occupy the centers of the honeycomb hexagons, analogous to the Ca positions in CaCu5. In Y11Ni60C6, both the AuCu3-type and CaCu5-type fragments have hexagonal faces based on a ring of Ni atoms centered by Y atoms. The majority of the crystal structure is built from these two fragment types intergrowing with each other through the sharing of these hexagonal faces (Figure 2e). Adding a CaCu5-type fragment to each of the hexagonal faces of the AuCu3-type clusters, and vice versa, leads to a bcc arrangement of AuCu3-type domains joined through CaCu5type linkers. The coexistence of these two structure types in this phase is unexpected from the point of view of atomic packing modes. The AuCu3 structure type is a classic member of the simple close packed (SCP) phases, which contains a large number of octahedral interstices. The CaCu5 type, on the other hand, is more reminiscent of tetrahedrally close packed (TCP)

4. CHEMICAL PRESSURE ISSUES IN AuCu3-TYPE YNi3 In the previous section, we saw that the Y11Ni60C6 structure contains well-developed fragments of a AuCu3-type YNi3 phase, a compound that does not form in the binary Y−Ni system on its own. Let us begin our analysis of the driving forces underlying Y11Ni60C6 with a look at what issues such an SCP YNi3 compound would face as an independent phase. One clue is found in the observation that AuCu3-type phases do in fact occur in the related Y−Pd and Y−Pt systems, such as YPd3 and YPt3.57 As Ni, Pd, and Pt all have the same valence electron count, this nonexistence of YNi3 would seem to have more to do with atomic size, the role of which can be analyzed using the DFT-Chemical Pressure (DFT-CP) method. We thus carried out a DFT structural optimization of a hypothetical AuCu3-type YNi3 phase, and performed a CP analysis on it. The resulting CP scheme is shown in Figure 3, where the Ni sublattice is traced out with blue cylinders, and the Y positions are highlighted with transparent red spheres. The local pressures experienced along various directions by each atom are represented with CP anisotropy surfaces that follow standard conventions: black lobes signify directions along which the contraction of the structure is energetically favorable (negative pressure), whereas white lobes indicate a desire for expansion (positive pressure). The size of each lobe is proportional to the sum of the pressures felt by the atom along that direction. The CP scheme for this YNi3 phase (Figure 3) shows a clear competition between Y−Ni and Ni−Ni interactions. The Y atoms’ CP surfaces appear as cuboctahedral arrangements of white protrusions, one pointing toward each of the 12 Ni neighbors surrounding each Y atom. The Ni CP surfaces also 10399

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Figure 4. Hypothetical series of C-stuffed derivatives of the AuCu3 type with composition YNi3Cx for CP analysis. (a) AuCu3-type YNi3, acting as the basic structure. (b) CaTiO3-type YNi3C, where all Ni6 octahedra are C-stuffed. (c) Y4PdGa12-type Y4Ni12C, with two out of eight Ni6 octahedra in a 2 × 2 × 2 supercell occupied (vertices and body center). (d) K2PtCl6-type Y2Ni6C, in which four out of the eight Ni6 octahedra in the supercell are occupied (cell corners and face centers).

Figure 3. Chemical Pressure (CP) scheme for a hypothetical AuCu3type YNi3 phase. Each atom is overlaid with a CP anisotropy surface, a radial plot that gives the sum of the pressures experienced by that atom along each direction. The distance of the surface from the atom along any given direction gives the magnitude of the sum, while the color gives the sign: black for negative (calling for the structure’s contraction), white for positive (pushing for the structure’s expansion).

exhibit white lobes pointing along Y−Ni contacts, confirming that these interactions experience positive pressure and would benefit from the expansion of the structure. However, the Ni atoms also display black lobes which are directed along Ni−Ni contacts, indicating that the distances at these contacts are overly long. The CP picture of AuCu3-type YNi3 can then be interpreted in terms of simple atomic size arguments: the unit cell contains both Y−Ni and Ni−Ni close contacts, whose distances are restricted to be equal by symmetry. However, since the optimal Y−Ni distance is longer than that of a Ni−Ni one (the atomic radii of Y and Ni are 1.80 and 1.24 Å, respectively), they are forced to compromise, leading to overly short Y−Ni contacts (white, positive CP lobes) and overly long Ni−Ni ones (black, negative CP lobes). Replacing Ni with Pd or Pt would serve to decrease the discrepancy in the ideal bond lengths, providing us with a rationale for why YPd3 and YPt3 adopt the AuCu3 type, but YNi3 does not. So far, with the help of CP analysis, we have established that YNi3 in the AuCu3 type is destabilized due to the difference in atomic radii between Y and Ni. Comparing this isolated structure with the AuCu3-type fragment in Y11Ni60C6, two major structural differences become apparent: First, the latter contains interstitial C atoms in some of the octahedral voids. Second, in the latter, the AuCu3-type fragments form an intergrowth with the CaCu5-type YNi5 fragments through shared Y@Ni6 hexagonal faces. In the following, we will discuss these two factors individually, and investigate their roles in stabilizing both the AuCu3-type fragments and the structure as a whole.

Figure 5. CP schemes calculated for the hypothetical YNi3Cx series. For Y4Ni12C and Y2Ni6C, only one-eighth of the conventional cubic unit cell is illustrated. As more interstitial C is stuffed into the YNi3 basic structure, the positive CP on the Y atoms and the negative CP on the Ni atoms are both relieved, at the expense of increasingly large positive CP on the C atoms.

easier. The effect of C-stuffing is immediately seen even in the least C-rich phase of the series, Y4Ni12C. Comparing its CP scheme with that of the unstuffed phase (Figures 5a,b) reveals that when an interstitial C is introduced into a Ni6 void, all surrounding Y and Ni atoms respond in the shapes of their CP lobes. For the six Ni atoms surrounding the C atom, the negative CP lobes on the C-side become significantly reduced, while the negative CP on the opposite side (not pointing to an interstitial C atom) remains largely unchanged (cf. Figure 5a). In addition, small positive CPs between the Ni and C appear. The inclusion of the C atom has thus, at a local level, relieved the overly long contacts within the Ni sublattice. The eight surrounding Y atoms experience similar CP relief, as the positive CPs of the Ni cores become more involved in Ni−C interactions (see images of the contact volumes in the

5. ROLE OF C INTERSTITIALS To investigate how introducing interstitial C atoms affects the CP issues of a AuCu3-type YNi3 phase, we carried out analyses on a series of hypothetical YNi3Cx phases, with C atoms placed with different frequencies in the structures’ Ni6 octahedra (Figure 4): AuCu3-type YNi3 (x = 0), Y4PdGa12-type Y4Ni12C (x = 0.25), K2PtCl6-type Y2Ni6C (x = 0.5), and CaTiO3-type YNi3C (x = 1). This series provides a spectrum of phases at different levels of C-stuffing, so that the effect of C insertion can be clearly identified from the trends in their CP schemes. The results of the CP analysis of this series are summarized in Figure 5, with only C-stuffed octants of the 2 × 2 × 2 supercells of Y4Ni12C and Y2Ni6C shown to make comparison 10400

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Inorganic Chemistry Supporting Information). On each Y atom, the positive CP lobes pointing toward the Ni atoms of a C@Ni6 octahedron have almost disappeared. Since each Y in Y4Ni12C has two C@ Ni6 neighbors on opposite sides (see Figure 4c), this results in CP relief along the axis connecting the two C neighbors, while positive CP on the equator remains unresolved, producing a planar, snowflake-shaped distribution of positive CP on the Y atom (Figure 5b). When more C atoms are introduced, these trends continue: each C atom stuffed into a Ni6 octahedral void reduces the magnitude of the CP lobes in its local environment (both its direct Ni neighbors and the eight Y atoms that the Ni octahedra interact with). By the time we reach the most C-rich structure (x = 1), these local features merge, resulting in a structure whose major tension is between well-distributed negative CPs within the YNi3 framework and positive CPs concentrated on the C atoms. The large positive pressures in the C atoms in YNi3C, however, reveal that this is not a story of simple relief, but of a transfer of the CP tension: achieving nearly optimized Ni−Ni and Y−Ni contacts requires the incorporation of a substantial quantity of C atoms (in YNi3C the Y:C ratio is 1:1), each of which experiences significant positive CP.58 At lower levels of C incorporation, the negative CPs in the empty Ni6 octahedra hint at the prevalence of more TCP geometries in the Ni-rich side of the Y−Ni system, with their tighter interstitial spaces.

Figure 6. Evaluation of the extent of epitaxial stabilization between the AuCu3- and CaCu5-type fragments in the Y−Ni sublattice of Y11Ni60C6. (a) CP scheme for the Y coordination environment in a hypothetical AuCu3-type YNi3 phase. (b) The corresponding scheme for a Y site in CaCu5-type YNi5. (c) The CP scheme of the Y−Ni sublattice of Y11Ni60C6 structure (C atoms excluded from calculation) zoomed-in on the Y2 environment at the TCP/SCP interface. Note that significant pressures remain in the intergrowth.

allowing the Y@Ni6 hexagons in the AuCu3-type YNi3 {111} and YNi5 {001} planes to fuse (Figure 7).

6. ROLE OF EPITAXIAL INTERFACES The stuffing of C atoms into octahedral holes is not the only way in which AuCu3-type YNi3 is modified in the structure of Y11Ni60C6. The fragments of YNi3 are also intimately intergrown with domains of CaCu5-type YNi5. How might this intergrowth serve to reinforce the AuCu3-type clusters? One pathway is epitaxial stabilization: CP relief through the formation of interfaces between structures with complementary CP schemes. The AuCu3 and CaCu5 types appear well-poised for such an effect. The Y atoms in the AuCu3-type YNi3 experience positive CPs as they stretch the Ni sublattice, while previous analyses of the CaCu5 type suggest that the Y atoms in YNi5 will show predominately negative CPs. In this section, we explore how interfaces between AuCu3- and CaCu5-type domains could allow the cancellation of these opposing CPs. We begin by taking a closer look at the CP schemes of the Y environments in the two simple structures. For the hypothetical AuCu3-type phase, the familiar tension between positive Y−Ni and negative Ni−Ni CPs is again seen (Figure 6a, this time viewed from a different angle in preparation for latticematching with the CaCu5-type). In the CP scheme of the CaCu5-type YNi5 (Figure 6b), the most prominent features are the large negative CP lobes on the central Y atom pointing directly up and down, indicating a strong preference for the structure to contract. While the Y atom also bears smaller positive lobes pointing toward its Ni neighbors in the perpendicular directions, the net CP on the Y is negative: overall, the Y is too small for its coordination environment in this structurethe reverse of the situation faced by the Y in the AuCu3-type phase. The opposite CPs the Y atoms would experience in AuCu3type YNi3 and CaCu5-type YNi5 provide a motive for their intergrowth. However, for the AuCu3- and CaCu5-types, it is difficult to find common atomic planes for epitaxial matching. In the structure of Y11Ni60C6, this problem is solved by breaking up the AuCu3- and CaCu5-types into fragments,

Figure 7. The common structural motif in the AuCu3-type YNi3 (hypothetical) and CaCu5-type YNi5 used as interface for their intergrowth in Y11Ni60C6. Note that the match in atomic arrangement only occurs locally and does not extend to the entire plane, preventing a simple layered intergrowth from forming.

To focus on the role these shared hexagons play in Y11Ni60C6, we performed a DFT-CP calculation on its Y−Ni sublattice (with the C atoms excluded). In Figure 6c, we zoomin on the CP features surrounding the Y atom at the interface between the AuCu3-type and CaCu5-type domains, where the cancellation of pressures should be strongest. As might be expected from the hybrid geometry of this Y atom’s coordination environment, the CP distribution around this site bears similarities to those of both of the two simpler structures. The Y atom exhibits large positive CP features pointing into the AuCu3-type domain (yellow), similar to those calculated for the corresponding region of AuCu3-type YNi3. On the YNi5 side (purple), the Y atom bears negative CP lobes analogous to those in the CaCu5-type phase, though with a smaller magnitude. With positive and negative CPs dominating on opposite sides of the Y atom, its net CP magnitude of 326 GPa is lower than it would have been in either the YNi3 or YNi5 phases (1067 and −663 GPa, respectively). In terms of individual Y−Ni interactions, however, the Y atom at the interface does not have a convincingly improved CP scheme. While the negative CPs on the YNi5 side have decreased somewhat, the positive CPs pointing in the AuCu3-type domain have increased relative to YNi3 on its own (Figure 6a vs 6c). 10401

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Inorganic Chemistry This result is far from the simple cancellation of positive and negative Y CPs that we expected at the interface, but can be understood in terms of how atomic motions affect the balance of interactions here. Raw, unintegrated CP maps consist of a bath of negative pressure in the interstitial regions of a structure surrounding intense and overall positive pressures on the ion cores. The interatomic pressures in the CP schemes are obtained through the integrating volumes of the CP map for the various contacts, while positive or negative CP lobes arise depending on whether the ion cores or negative background play a larger role for each contact. At the interface, the Y atom moves out of the AuCu3-type domain, in line with the CP gradient it experiences, while nearby Ni−Ni contacts are largely unchanged. This motion of the Y leads to negative pressure in the map originally assigned to the lengthened Y−Ni contacts in the region being transferred to neighboring Ni−Ni contacts which are nearly as short as before (see Figures S4 and S5 in the Supporting Information). With this diminished electronic support for the Y−Ni contacts, positive CPs grow along these interactions despite the longer distances. In other words, a simple elongation of these Y−Ni contacts is not enough to relieve their positive CPs; the Ni−Ni contacts of the Y coordination environment must also expand to provide a looser grip. In the next section, we will see that insertion of C atoms facilitates this loosening of the Y environment very effectively.

Figure 8. The role of C insertion in facilitating the intergrowth of AuCu3- and CaCu5-type fragments in Y11Ni60C6. (a) CP scheme for the Y−Ni sublattice (C excluded from calculation), focused on the Y2 site’s coordination environment at the interface region. (b) The corresponding CP scheme following the insertion of C atoms. The CP surfaces of the C atoms are not shown, as their magnitudes would obscure the other features in the plot (see text). (c) The inserted C atoms drawn in the context of the full AuCu3-type fragment, illustrating how they contribute to the expansion of the Y2 coordination environment.

interstitial C atoms at the interface. As has been discussed in Section 3, these C atoms fill in octahedral holes in the SCP fragment. Their placement has further significance, however: they cap the Ni−Ni contacts of the Y2 coordination polyhedra that link the Y@Ni6 hexagon to the upper Ni triangle in this environment. As the Ni octahedra expand to accommodate the C atoms in their centers, much of the positive CP in their cores is transferred from Y−Ni to Ni−C and Ni−Ni interactions, soothing the positive Y−Ni pressures present in the C-free structure (see Figures S4 and S5 in the Supporting Information). The inclusion of the C atoms also has a more far-reaching influence on the CPs of their surrounding Ni atoms. In a manner similar to that observed before for transition metalstuffed AuCu3-type gallide lattices,60 the presence of C atoms alleviates the large negative CPs along the Ni−Ni contacts tracing out their octahedral environments. As these C-filled octahedra expand, the central octahedron of the AuCu3-type unit is compressed isotropically through the collective effect of the eight TCP/SCP interfaces surrounding each SCP region. This compression leads to the healing of the Ni−Ni negative CPs in this central octahedron, and naturally precludes its occupation by additional C atoms. All of these factors help explain why the CP scheme on the AuCu3 side of the interfaces is relieved with the addition of C. A similar relief appears to occur on the CaCu5-type side as well, with the large negative CP feature of Figure 8a being smaller in Figure 8b. This effect can be traced to the sinking of the Y atom into the CaCu5-type domain (now free to move in this way without surrendering space in the CP map to the Ni−Ni interactions on the other side), as is evidenced in the decrease in the Y−Y distance within the CaCu5-type fragments from 3.57 Å in the Y−Ni sublattice alone to 3.46 Å in the full Y11Ni60C6 structure. When we discussed C-insertion into the AuCu3-type phase in Section 5, we saw that CP relief in the Y−Ni sublattice was coupled to a buildup of positive CP on the C atoms. A similar effect is also seen in the CP scheme of Y11Ni60C6, with the net CPs on the C atoms being extremely positive: 1977 GPa, even higher than that calculated for the fully C-stuffed YNi3C phase (1324 GPa). Compared to the situation in YNi3C, however, these C atoms make up a much smaller fraction of the crystal structure, with the CP relief they support in their surroundings evidently being worth the sacrifice. Based on these intense

7. Y11Ni60C6: SYNERGY OF STABILIZING FORCES Up until now we have examined how two separate effects in Y11Ni60C6 might stabilize the appearance of AuCu3-type YNi3 clusters in its structure: C-stuffing and epitaxial intergrowth. On their own, the favorability of these features was found to be limited. Interstitial C atoms help expand the Ni sublattice and reduce the strain arising from the size difference between the Y and Ni in the AuCu3 type, but a large amount of C is required to achieve satisfactory CP relief in the Y−Ni sublattice, and this arrangement places large positive CPs on all of these C atoms. Epitaxial intergrowth, on the other hand, leads to neutralization of the net atomic CPs on Y atoms, but the positive and negative features remain spatially segregated, meaning that individual intermetallic contacts are still strained. We now turn to whether enhanced stabilization arises when the C incorporation and SCP/TCP intergrowth act in concert, rather than individually. To explore this possibility, we begin with the CP schemes for the environment surrounding the interfacial Y2 site in the Y11Ni60 sublattice (Figure 8a), and watch how the scheme changes as the C atoms are added to form the full Y11Ni60C6 structure (Figure 8b). The C-free CP scheme appears just as before, with the Y atom bearing intense white lobes pointing in the YNi3 region and smaller black ones pointing into the YNi5 domain. Upon adding the C atoms, this Y site’s CP distribution changes dramatically, particularly on the AuCu3-type side. The positive features pointing upward from the Y atom and within the shared Y@Ni6 hexagon have been replaced with a small negative protrusion, whose insignificant magnitude is suggestive of nearly optimized Y−Ni interactions.59 The negative CPs on the lower side of the Y have also become reduced relative to those of the C-free structure. Expanding our view to encompass a larger fraction of the structure helps explain why the incorporation of C makes such a big difference in the CP scheme. In Figure 8c, we show the broader context of the Y2 site by drawing the outline for the entire truncated cuboctahedral SCP fragment, along with the 10402

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modes.51 The strongly anisotropic black CP surfaces on the Y1 and Ni1 sites trace out a NbO-like61 framework (Figure 9c), along the edges of which atomic motions should be particularly facile. This sublattice of negative CP could play the part of a phonon-scattering substructure, a route earlier proposed for the creation of phonon glass−electron crystal materials for thermoelectric applications.43

pressures, it could be interesting to attempt substituting the C atoms with smaller N ones in an attempt to synthesize a nitride analogue of this structure.

8. POTENTIAL FUNCTIONALITY FROM FRUSTRATION The majority of the Y11Ni60C6 structure can be understood in terms of SCP-TCP intergrowth driven by synergy between Cinterstitials and epitaxial stabilization (136 of 154 atoms/unit cell). The remaining atoms, at the Y1 and Ni1 positions, are geometrically very different, seeming to occupy voids left by the TCP/SCP intergrowth framework, rather than belonging to a familiar geometrical motif. Y1 lies in the holes between neighboring SCP clusters along the a, b, and c axes, inside a 20coordinate polyhedon. The Ni1 sites, on the other hand, form square rings around the Y1 atoms (and can be seen as extending the kagome layers of the CaCu5-type fragments). They are coordinated by 12 Ni and 2 Y atoms that form a bicapped hexagonal antiprism (a 14-vertex Frank−Kasper polyhedron). What role do these atoms play in the structure? The CP schemes for these two sites confirm the viewpoint of their serving as space-filling scaffolding atoms (Figure 9). Both

9. CONCLUSIONS In this Article, we have used Y11Ni60C6 as a model system for the giant cubic intermetallic structures that represent one pinnacle of complexity in solid state chemistry. Following the theme of incompatible bonding or packing modes in these intermetallic structures, Y11Ni60C6 has regions with two very distinct arrangements of atoms: C-stuffed domains of a simple close packed (SCP) structure and domains of the CaCu5-type that are more tetrahedrally close packed (TCP). One clue to the driving forces stabilizing this arrangement was the correspondence of the SCP domain with a hypothetical AuCu3-type YNi3 phase, which does not form on its own. This observation led us to focus on how the context of these YNi3 fragments in the Y11Ni60C6 structure addresses bonding issues that a bulk AuCu3-type phase would encounter. After exploring the roles of C interstitials and SCP-TCP interfaces separately, a synergy between these two structural features emerged. Here, C-insertion loosens the grip of the AuCu3-type fragments on the Y atoms at the SCP-TCP interfaces, allowing them to relax into the CaCu5-type domains. The CP relief experienced by Y11Ni60C6 relative to its simpler components can be seen as a variant of the epitaxial stabilization mechanism proposed previously for the formation of Bergman clusters in Ca10Cu2Cd27,62 as well as intermetallic/ carbide intergrowths in Mn16SiC4 and Mn17Si2C4.33 As in these other cases, two incompatible structural motifs are driven into close proximity by the opportunity for the cancellation of opposite CPs at the interface. The necessity of the support of C interstitials for this CP relief to arise here, however, carries a cautionary message for future syntheses attempting to use epitaxial stabilization to create new compounds: the favorability of these interfaces may depend on the coordinated motions of atoms rather than individual atoms moving along their CP gradients. Another lesson that can be derived from Y11Ni60C6 is the potential utility of frustrated crystal structures. When two incompatible structures intergrow to form more complex ones, it is nearly inevitable that gaps will arise that must be filled with scaffolding atoms. The CP schemes of the Y1 and Ni1 sites in Y11Ni60C6 suggest that these sites can have quite suboptimal coordination environments. The large negative CPs that emerge may underlie soft vibrational modes that could result in atomic rattling, somewhat like the cations of intermetallic clathrates.23,24 Low lattice thermal conductivity in this and other complex structures based on geometrically incompatible domains may arise not just from the sheer size of their unit cells, but from advantageous features at the interfaces between their domains.

Figure 9. CP analysis of the scaffolding sites in Y11Ni60C6. Both (a) Y1 and (b) Ni1 experience overall large negative CP, showing that they are too small for their respective coordination environments. The largest positive lobes on the coordinating Ni atoms in (b) correspond to near collisions between Ni atoms from neighboring CaCu5-type fragments growing from the same AuCu3-type cluster. (c) The framework of negative pressures defined by the Y1 and Ni1 sites.

sites exhibit uniformly negative CP surfaces, with the magnitudes being particularly pronounced along certain directions. The Y1 site’s CP plot is flatted and square-shaped, with the largest features directed toward the Ni1 atoms emanating from the nearby CaCu5-type fragments (Figure 9a). The Ni1 CP distributions are severely elongated toward the Y atoms capping their Ni hexagonal antiprisms. The absence of positive CP features suggests that these atoms are too small for these coordination environments. From these CP results, the Y1 and Ni1 sites appear to serve to fill a vacuum in the structure’s gaps, rather than to achieve CP relief themselves. While the Y1 and Ni1 sites do not further our theme of CPdriven intergrowth, they are significant for another reason: they offer the possibility of interesting physical properties. In our earlier examination of the connection between CP features and phonon frequencies, we saw that atoms held in place between opposing negative CPs are associated with soft vibrational

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b01645. 10403

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Additional computational details; LDA-DFT optimized coordinates and total energies for all structures analyzed with the DFT-CP method; phonon band structure calculations on selected compounds for the verification of the treatment of ionicity; table of net CPs and Hirshfeld charges calculated for the symmetry-distinct sites for all structures analyzed; plots of the sum of the contact volumes around key atoms in selected structures (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Joshua Engelkemier for conversations on CP and phonon analysis, and Dr. Brandon Kilduff for insights into how changes in the shapes of contact volumes can underlie structural distortions in intermetallics. We are also grateful to Dr. Timothy Stacey for preliminary work in rationalizing the Yb11Ni60C6 structure type, and Katerina Hilleke for her exploration of the applicability of the μ3-acidity model to this compound. We gratefully acknowledge the financial support of the US DOE Office of Science Early Career Program (DESC0003947) through the Office of Basic Energy Sciences. This research involved calculations using computer resources supported by National Science Foundation Grant CHE0840494.

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