Endohedral gallide cluster superconductors and superconductivity in ReGa₅

Significance

The prediction of new superconductors remains an elusive goal. It is often chemists who find new superconductors, although it is difficult to translate the physics of superconductivity into chemical requirements for discovering new superconducting compounds. There are many strategies for finding new superconductors, one being to postulate that superconductivity runs in structural families. Here we show that a previously unappreciated structural family, the endohedral gallium cluster phases, is favored for superconductivity, and then use the understanding we develop to find a superconductor. More broadly, our work shows that molecule-based electron counting and stability rules can provide a useful chemistry-based design paradigm for finding new superconductors. Using these ideas to search for new superconductors will be of significant future interest.

Abstract

We present transition metal-embedded (T@Ga_n) endohedral Ga-clusters as a favorable structural motif for superconductivity and develop empirical, molecule-based, electron counting rules that govern the hierarchical architectures that the clusters assume in binary phases. Among the binary T@Ga_n endohedral cluster systems, Mo₈Ga₄₁, Mo₆Ga₃₁, Rh₂Ga₉, and Ir₂Ga₉ are all previously known superconductors. The well-known exotic superconductor PuCoGa₅ and related phases are also members of this endohedral gallide cluster family. We show that electron-deficient compounds like Mo₈Ga₄₁ prefer architectures with vertex-sharing gallium clusters, whereas electron-rich compounds, like PdGa₅, prefer edge-sharing cluster architectures. The superconducting transition temperatures are highest for the electron-poor, corner-sharing architectures. Based on this analysis, the previously unknown endohedral cluster compound ReGa₅ is postulated to exist at an intermediate electron count and a mix of corner sharing and edge sharing cluster architectures. The empirical prediction is shown to be correct and leads to the discovery of superconductivity in ReGa₅. The Fermi levels for endohedral gallide cluster compounds are located in deep pseudogaps in the electronic densities of states, an important factor in determining their chemical stability, while at the same time limiting their superconducting transition temperatures.

The prediction of new superconductors remains an elusive goal. Although one can analyze the superconductivity, once discovered, through materials physics-based “k-space” pictures based on Fermi surfaces, energy band dispersions, and effective interactions, often it is chemists, whose viewpoint is instead from “real space” rather than k-space, who find such superconductors in the first place (1, 2). Given the difficulty in making extrapolations between the physics of superconductivity and the chemical stability of compounds that will be superconducting, there are as many strategies for finding new superconductors as there are researchers looking for them (3–5). Most such search strategies fail, because the interactions that give rise to superconductivity can also lead to competing electronic states or can be strong enough to tear potential compounds apart (6, 7).

Table 1.

Selected Binary Phases with Endohedral Ga-clusters

Binary compounds	Structure type	Pearson symbol	Tc (K)	Reference
V8Ga41	V8Ga41	hR147	—	Girgis et al. (36)
Mo8Ga41	V8Ga41	hR147	9.8	Yvon (23)
Mo6Ga31	Mo6Ga31	mS148	8	Yvon (23)
ReGa5	ReGa5	oS48	2.3	This work
Rh2Ga9	Co2Al9	mP22	2.0	Shibayama et al. (22)
Ir2Ga9	Co2Al9	mP22	2.3	Shibayama et al. (22)
PdGa5	PdGa5	tI24	—	Grin et al. (29)

One chemical perspective for increasing the odds of finding superconductivity is to postulate that it runs in structural families. The perovskites are a well-known example of this in metal oxides, and in intermetallic compounds, the “122” ThCr₂Si₂ structure type is a good example (8–10). It is the discovery of these new structural families of superconductors that often leads, sometimes slowly or sometimes quickly, to advances in new superconducting materials. Here we show that a previously unappreciated chemical family, the endohedral gallium cluster phases, is a favored chemical family for superconductivity. Further, we analyze the occurrence and hierarchical structures of such phases from a molecular perspective and then use that perspective to predict the existence and structure of a previously unreported compound, ReGa₅. We find that compound and discover it to be superconducting.

Endohedral Gallium Clusters and Superconductivity

Elemental gallium, in group 13, is located at the Zintl border in the periodic table and is known in solid state chemistry for its tendency, due to its moderate electronegativity, to form compounds based on gallium clusters (11). (The Zintl border separates groups 13 and 14. In combination with electropositive metals, the elements in group 14 and to the right usually form compounds whose electronic structures are consistent with filled bonding, filled nonbonding, and empty antibonding levels, and therefore are electron precise, which is not generally the case for group 13 and to the left.) Previous investigations of binary alkali metal-Ga (A-Ga) solid state systems have resulted in the discovery of many new Zintl compounds, in which Ga_n clusters or molecules use the electrons donated from the alkali metals to satisfy their valence requirements (12). The large electronegativity differences between alkali metals and Ga always makes these A_mGa_n Zintl compounds valence-precise semiconductors, i.e., they display a relatively large band gap between occupied and unoccupied states, motivating the investigation of Zintl compounds as good thermoelectric materials above ambient temperature (13). Structurally, the Ga atoms in A_mGa_n systems form icosahedral (Ga₁₂) or octahedral (Ga₆) clusters, analogous to those found in borane chemistry (14). The gallium clusters in the Zintl phases are analogs to borane clusters and follow the same rules for the number of skeletal electrons required for stability. When replacing alkali metals with lanthanides or actinides (R) to form Ga-rich R_mGa_n compounds, the electronegativity differences between R and Ga are smaller than those between the alkalis and Ga, and the semiconducting band gap diminishes—sometimes to zero to yield metallic conductivity. The formation of exo-bonds to other clusters in vertex-sharing, edge-sharing, or face-sharing cluster hierarchies and the distortion of the clusters away from ideal deltahedral symmetries can also stabilize R_mGa_n compounds (15). Examples of the Ga clusters in these compounds can be seen in Fig. 1A.

Fig. 1.

Schematic structural relationships among different kinds of Ga-cluster compounds. (A) Ga metal reacts with alkali and alkali earth elements to yield Zintl phases such as K₃Ga₁₃, which has isolated Ga₁₂ icosahedral clusters, and Ba₅Ga₆, which has Ga₆ octahedral clusters. In Na₁₀NiGa₁₀, a transition metal (Ni)-centered endohedral Ni@Ga₁₀ cluster is found (19). (B) The combination of Ga plus R (R = lanthanide and actinide elements) leads to the formation of polar intermetallics, for example, PuGa₆, which contains Pu@Ga₁₂ clusters (15). (C) Centering Ga-clusters with transition metals stabilizes Ga-cluster compounds such as CoGa₃, which contains Co@Ga₈ square antiprism clusters (38). (D) Combining T-centered and R-centered clusters forms the unconventional superconductor PuCoGa₅, in which Pu@Ga₁₂ cuboctahedra share faces with neighboring Pu@Ga₁₂ and Co@Ga₈ (cube) clusters (16). (E) Adding more Ga atoms to T-Ga systems forms other Ga-rich compounds, such as the superconductor Mo₈Ga₄₁. In this compound, Mo@Ga₁₀ clusters are found (23).

The introduction of transition metals (T) to the centers of the gallium clusters to create T@Ga_n endohedral clusters reduces the cluster charge and is an important path to gallide chemical stability. For example, the Ni-centered Ni@Ga₁₀ cluster (Fig. 1A) yields the chemical stability of Na₁₀NiGa₁₀ (11). Of great interest for their electronic properties are the large number of thus-derived ternary A/R-T-Ga (A = alkali or alkali-earth; R = lanthanide or actinide; and T = late transition metal) compounds. An important class of superconductors has been discovered in this group. The actinide-based compound PuCoGa₅, for example, is assembled from metal-centered endohedral clusters: Pu-centered Ga cuboctahedra (Pu@Ga₁₂) and Co-centered Ga cubes (Co@Ga₈) (Fig. 1 B and C) and displays a very high critical temperature T_c= 18.5 K that increases to 22 K under pressure (16). The T_c= 2.8 K superconductor Sm₄Co₃Ga₁₆ similarly contains Sm@Ga₁₂ and Co@Ga₈ endohedral clusters that are isostructural with the Pu@Ga₁₂ and Co@Ga₈ clusters in PuCoGa₅; because the clusters are not present in a 1:1 ratio, the hierarchical architecture is more complex in this compound (17). Also important as heavy fermion superconductors are the In analogs of these phases, the CeMIn₅ (M = Co, Rh, Ir) family of compounds, which are iso-structural with PuCoGa₅; their study has considerably illuminated the understanding of the interplay between superconductivity and magnetism (18). Fig. 1 summarizes the structural relationships described here.

The electron transfer between cations and anions in the A-Ga or R-Ga systems is clearly primarily ionic due to the large electronegativity differences (19). It is much less obvious, however, to tell a priori how the electrons are transferred in Ga-rich T-Ga (T = transition metal) binary phases such as Mo₈Ga₄₁ and PdGa₅, because the electronegativities for late transition metals and gallium are similar (20, 21). Nonetheless, we can define here a set of electron counting rules and the relationships between electron counting and the hierarchical architectures of the endohedral clusters required for chemical stability through observation of the known phases. Further, we can establish an empirical relationship between electron counting and superconducting transition temperature in these compounds; we find that as the number of electrons per formula unit decreases, the critical temperature for superconductivity first increases and then decreases. Among the previously reported binary endohedral Ga cluster phases, Mo₈Ga₄₁ is a superconductor with T_c = 9.8 K (1); Mo₆Ga₃₁ is a superconductor with T_c = 8.0 K (1); Rh₂Ga₉ is a superconductor with T_c = 2.0 K; and Ir₂Ga₉ is a superconductor with T_c = 2.3 K (22).

Based on this understanding, we designed and synthesized the previously unreported binary endohedral cluster compound ReGa₅ and found it to be superconducting at a critical temperature T_c = 2.3 K. Our electronic structure calculations show that the Fermi level of ReGa₅ is located within a pseudogap in density of electronic states (DOS), which, as is seen in the other binary endohedral gallium cluster superconductors, is required for the chemical stability of the compound. Given that superconducting transition temperatures should be higher for materials with a higher density of electronic states and that the location of the Fermi energy within a deep pseudogap is a requirement for chemical stability in the endohedral gallide phases, superconductivity and structural stability can be seen to compete in this family. Nonetheless, a compromise is clearly met between the two competing factors in the real materials, resulting in a large family of superconducting endohedral gallium cluster compounds.

Electron Counting Rules for Ga-Rich Compounds and a Molecular Perspective on Ga-Clusters in Solids

Mo₈Ga₄₁ and Mo₆Ga₃₁.

In the crystal structure of Mo₈Ga₄₁, the most striking features are Mo atoms inside 10-atom Ga clusters, i.e., Mo@Ga₁₀ endohedral clusters (23, 24). These endohedral clusters are arranged such that an almost regular cube of Mo atoms is found. The Mo@Ga₁₀ clusters share all their vertex Ga, an architecture that creates a Ga cuboctahedron in the interstitial space between clusters that is itself centered by a Ga atom in Mo₈Ga₄₁; this compound can thus be written as Ga(MoGa₅)₈. The whole architecture is strongly reminiscent of an A-site deficient perovskite oxide, i.e., Ga_1/8Mo@Ga_10/2 ∼ A_xTO₆/₂, although with 10-vertex-connected dodecahedral Mo@Ga₁₀ clusters rather than 6-vertex-connected M@O₆ octahedral clusters. The perovskite structure is known to host many important superconductors, ranging from the high T_c copper oxides to low T_c Na_0.23WO₃, and in analogy Mo₈Ga₄₁ is also superconducting (25). The overall symmetry of the thus-arranged endohedral clusters in Mo₈Ga₄₁ is rhombohedral, which is one of the variants of the many possible distortions of the simple cubic lattice found in oxide perovskites (8).

In the crystal structure of the related superconducting cluster phase Mo₆Ga₃₁, two of the Mo@Ga₁₀ endohedral clusters are fused together, such that 4 of the 10 Ga are shared between two Mo, in an edge sharing motif (23). The four peripheral Ga’s shared between endohedral clusters are on the vertices of a square, creating an overall face sharing motif of double clusters. These double clusters share vertices with other double clusters to create a mixed corner sharing plus edge sharing architecture. This kind of double cluster architecture in Mo₆Ga₃₁ is again reminiscent of the motif found in other superconducting phases: in this case, the family of Chevrel structure-derived phases made from corner and face-sharing Mo₆S₈ clusters (26).

To investigate the electronic factors behind the vertex-sharing Mo@Ga₁₀ cluster architecture in Mo₈Ga₄₁, we begin with the electronic structure of the hypothetical model compound Mo₈Ga₄₀, which is based on removing the Ga that is in the interstitial region between the Mo@Ga₁₀ clusters in Mo₈Ga₄₁. Hypothetical Mo₈Ga₄₀ (i.e., MoGa₅) made only of the endohedral clusters sharing corner Ga, was then subject to complete structural optimization using Vienne Ab Initio Simulation Package (VASP) (27). The electronic structure of this compound is shown in Fig. 2A. We find that the Fermi level of MoGa₅, which has 21e- (6 from Mo and 3 × 5 from Ga) is located in a pseudogap in the electronic DOS. An important question to next consider is how the endohedral Mo atom affects the stability of the Ga₁₀-cluster. To get further insight, then, extended Hückel theory was used to analyze isolated molecular “Ga₁₀” and “MoGa₁₀” clusters (28). Fig. 2 (Top Left) illustrates the crystal orbital energy diagrams for the two cases evaluated (the primitive unit cell used contains one 10-atom Ga-cluster or one 11-atom Mo-centered Ga-cluster at the corners of the cell) at the Γ-point in the Brillouin zone, with the orbital energies given relative to the corresponding Fermi levels. We find that after inserting the Mo atom into the Ga₁₀ cluster to create an endohedral cluster, the degenerate orbitals at E_F in the Ga₁₀ cluster are split significantly in energy, resulting in E_F (the Fermi energy) for Mo@Ga₁₀ being in an energy gap rather than in a partially occupied state with a significant DOS, and stabilizing the cluster. These crystal orbital energy diagrams provide a rationale for how Ga₁₀-clusters are stabilized through the presence of endohedral transition metal elements.

Fig. 2.

Electronic structures of Ga-cluster–based binary phases from a molecular perspective. (Left) The isolated clusters, showing for each: above, the Ga_n clusters and then below, the TGa_n endohedral clusters. (Center) The molecular energy level diagrams for the isolated Ga-clusters and the T-centered endohedral Ga-clusters, obtained using the extended Hückel theory. (Two different minimal basis sets involving Slater-type single-zeta functions for s and p orbitals and double-zeta functions for d orbitals were used.) (Right) The electronic DOS generated by VASP based on the optimized crystal structures of Mo₈Ga₄₁, Rh₂Ga₉, and PdGa₅.

Rh₂Ga₉ (Ir₂Ga₉) and PdGa₅.

Superconducting Rh₂Ga₉ and Ir₂Ga₉ both crystalize in the Co₂Al₉-type structure (22). The Ga clusters in these compounds are single-capped square antiprismatic Ga₉ (or Al₉) clusters with endohedral transition metal atoms, creating T@Tr₉ (T = Co, Rh, or Ir and Tr = Al or Ga) endohedral clusters. These endohedral clusters are assembled in edge-sharing zig-zag strands along one crystallographic axis (the c axis in the monoclinic unit cell) and share corners between strands. Thus, in this Co₂Al₉ structure type, the T@Tr₉ clusters share both corners and edges. As was the case for Mo₈Ga₄₁, we calculated the electronic structure of Rh₂Ga₉ and find that the Fermi level is again located in a pseudogap in the density of states at an electron count of 22.5 e- per RhGa_4.5 (Rh₂Ga_9/2) unit (Fig. 2, Middle Right). A sharp, deep pseudogap is seen about 0.25 eV below the Fermi level, associated with 22 e- per RhGa_4.5. Similarly to what we observed for Mo@Ga₁₀ clusters from molecular orbitals calculations, we find that the Rh-centered Ga₉ cluster is more stable than the empty Ga₉ cluster due to the splitting in energy of degenerate orbitals at the Fermi level (Fig. 2, Middle Left).

PdGa₅ crystalizes in a tetragonal crystal structure (29). In binary PdGa_5, each palladium atom is coordinated by 10 gallium atoms in the form of a bicapped tetragonal antiprism, forming Pd@Ga₁₀ clusters. The Pd@Ga₁₀ clusters share edges (8 Ga) and vertices (2 Ga). The Fermi level in the electronic structure of PdGa₅ is located in a pseudogap with 25 e-/PdGa₅, again in analogy to what is seen in the other gallate cluster compounds. A sharp narrow gap about 1.5 eV above the Fermi level is associated with 26 e- per Pd. Thus, just as we find in the other endohedral cluster compounds, the Pd atoms play an important role in splitting the orbitals of Ga₁₀ to place the Fermi level in the gap and thus yield chemical stability.

From these and similar analyses, considering Ga-rich binary phases, we find that electron-deficient compounds such as Mo₈Ga₄₁ prefer vertex-sharing of the Ga clusters, whereas electron-rich (26e-) compounds like PdGa₅ favor edge-sharing of the clusters. A single formula for the formation of stable T-Ga compounds can therefore be found. The formula is TGa_(n-1/2*m+l) (n = number vertices of the T-centered cluster; m = shared vertices; l = isolated Ga atoms in interstitial positions). Moreover, the superconducting transition temperature changes for the endohedral cluster compounds as one progresses in the transition metal series from Mo, to Rh/Ir, to Pd. Noting the missing members of the series in both electron count and structure, we thus postulated the existence of several possible new compounds and set out to synthesize them and test their properties.

Structural and Physical Properties of the Previously Unreported Superconductor ReGa₅

The Synthesis of ReGa₅ and Phase Information.

Based on the understanding above, we realized that that there were no compounds known with 22 electrons per T atom and we thus attempted their synthesis. Loading compositions of Re_1.5Ga_98.5 and Re₃Ga₉₇ (Re: powder, 99.999%, Alfa Aesar; Ga: sponge, 99.995%, Alfa Aesar) about 1 g total mass, were sealed into evacuated SiO₂ jackets (<10⁻⁵ Torr) to protect them from air during heating. The samples were heated to 900 °C for 24 h at a speed of 3 °C/min, followed by cooling to 500 °C at a rate of 4 °C/h, and annealed at this temperature for 2 d, after which the containers were spun in the centrifuge at 1,200 × g for 10 s. Both loading compositions give the same previously unreported compound, ReGa₅ according our subsequent analysis, with Ga-flux as a minor impurity. Looking for the analogous cluster compound based on Ru was not successful; Ru_1.5Ga_98.5 and Ru₃Ga₉₇ reacted using the same procedure yielded only the known compound RuGa₃. This 17 electron per cluster compound is semiconducting with extensive edge-sharing of Ru@Ga₉ endohedral clusters; the Fermi energy falls in a deep, wide gap in the density of states (Fig. S2).

Fig. S2.

Calculated DOS of RuGa₃, showing the Fermi level located at the edge of a band gap.

Fig. S1.

Powder X-ray diffraction pattern for polycrystalline ReGa₅ sample.

The Crystal Structure of ReGa₅ and Comparison with PdGa₅ and MoGa₅.

To obtain insights into the detailed structure of the new Re-based Ga-rich compound found, single crystals were investigated. The results of single crystal X-ray diffraction characterization of a specimen extracted from a single crystal sample of nominal composition ReGa₅ are summarized in Tables S1 and S2 and the crystal structure of ReGa₅ is shown in Fig. 5. ReGa₅ crystallizes in an orthorhombic crystal structure in space group Cmce (space group 64) and displays a previously unobserved structure type. It can be described in terms of single-capped square antiprismatic coordination polyhedra of Ga around the transition metal atoms, forming Re@Ga₉ endohedral clusters. These clusters are structurally analogous to those found for Ru@Ga₉ in RuGa₃. The architecture of the new compound is different, however, from those previously observed. Four neighboring Ga atoms on one of the square faces of a Re@Ga₉ cluster are shared with a neighboring Re@Ga₉ cluster, creating an overall double cluster architecture. Each Re@Ga₉ double cluster then also shares 4 vertex Ga with neighboring clusters in a corner sharing geometry, and, finally, a “capping” Ga is left coordinated to only a single Re.

Table S1.

Single crystal crystallographic data for ReGa₅ at 293(2) K

Refined formula	ReGa5
F.W. (g/mol);	534.8
Space group; Z	Cmce (No.64); 8
a (Å)	9.2127(5)
b (Å)	10.1043(5)
c (Å)	9.2321(5)
V (Å3)	859.40(8)
Absorption correction	Multiscan
Extinction coefficient	0.0005(1)
µ (mm−1)	58.739
θ range (°)	2.992–29.576
hkl ranges	−12≤ h ≤ 12
	−14≤ k ≤ 14
	−12≤ l ≤ 12
No. reflections; Rint	6,243; 0.0309
No. independent reflections	639
No. parameters	36
R1; wR2 (all I)	0.0540; 0.1154
Goodness of fit	1.142
Diffraction peak and hole (e−/Å3)	5.161; –5.945

Table S2.

Atomic coordinates and equivalent isotropic displacement parameters of ReGa₅

Atom	Wyckoff	Occupancy	x	y	z	Ueq
Re1	8f	1	0	0.3435(1)	0.0119(1)	0.007(1)
Ga2	8c	1	1/4	1/4	0	0.028(1)
Ga3	16g	1	0.3412(2)	0.4877(2)	0.6595(2)	0.047(2)
Ga4	8f	1	0	0.2842(1)	0.2787(1)	0.014(1)
Ga5	8f	1	0	0.0922(2)	0.0749(2)	0.061(2)

U_eq, one-third of the trace of the orthogonalized U_ij tensor (Ų).

Fig. 5.

The calculated electronic structure for ReGa₅: the physics-based picture. (Left) The total DOS as a function of energy near the Fermi energy (E = 0) obtained from LDA calculations in WEIN-2k with spin-orbit coupling (SOC) included. (Right) The corresponding energy dispersion of the bands in selected directions in the orthorhombic Brillouin zone.

The Electronic Structure of ReGa₅

According to the electron counting rules described above, the schematic electronic structure diagram for ReGa₅ is shown in Fig. 3. We consider two formula units of ReGa₅, which are the contents of the primitive unit cell. The two Re atoms per unit cell bring 10 d orbitals to the electronic system, whereas the two sets of Ga₄ on the corners of the shared square faces in the double cluster bring 32 Ga sp orbitals. Strong interactions occur within the Ga₄ portion, a reflection of the multiple Ga-Ga contacts in the structure. In the scheme here, there are 8 + x low-lying Ga levels for the two Ga₄ and 4 − y low-lying Ga levels for 2 isolated Ga atoms, 12 being the minimum number of Ga levels needed to make the 22 occupied orbitals per Re₂Ga₁₀. For Re₂Ga₁₀, then, the total number of electrons is: 2 × 7e-/Re + 10 × 3e-/Ga = 44 e-/Re₂Ga₁₀. This means that 22 bonding orbitals in the cluster would be fully occupied by electrons, and the Fermi level of ReGa₅ should be located in a gap or pseudogap in the DOS. To test whether these molecule-like the electron counting rules work for ReGa₅ when considered in the context of the electronic structure expected from a physics-based electronic picture, the electronic structure of ReGa₅ was calculated by use of the program WIEN2k with spin-orbit coupling included. Dramatically, the Fermi level is exactly located in the deep pseudogap in the DOS, just as expected from our molecular picture. Thus, unlike the case for regular superconductors, where having the Fermi level on a peak in DOS is preferred for superconductivity, a different kind of electronic structure is found for ReGa₅. The WEIN2k calculations allow for the electronic structure to be described in more detail than is available in the molecular picture: the valence and conduction bands barely cross the Fermi energy in ReGa₅, at different places in the Brillouin zone (Fig. 5), in an electronic structure that is reminiscent of that of the semimetal WTe₂ (30). Specifically, one conduction band comes down in energy, crossing the Fermi level and then increasing in energy again, between the Z-G-X points, whereas one valence band increases in energy, crosses the Fermi level, and then comes down again, again between the Z-G-X points.

Fig. 3.

Motivation for searching for superconductors in the Re-Ga systems based on electron counting and cluster architectures. Mo₈Ga₄₁ contains vertex-sharing 10-coordinate Ga-clusters; Rh₂Ga₉ and PdGa₅ contain both vertex-sharing and edge-sharing clusters. Before the current work there were no known compounds in this family with 22 electrons per transition metal.

Fig. 4.

The crystal structure of ReGa₅. ReGa₅ crystalizes in an orthorhombic structure with space group Cmce (S.G. 64). (green, Ga; pink, Re.) (A) This view emphasizes the shapes of the clusters. (B) This view emphases the vertex-sharing of the clusters and the square faces with four-corner Ga atoms that are shared to create the double cluster architecture.

Superconductivity in ReGa₅.

The resistivity of ReGa₅ undergoes a sudden drop to zero at 2.3 K, characteristic of superconductivity. In correspondence with ρ(T), the magnetic susceptibility [χ_mol(T)] measured in a field of 10 Oersteds after zero field cooling decreases from its normal state value at 2.3 K and shows large negative values, characteristic of an essentially fully superconducting sample. To prove that the observed superconductivity is intrinsic to the ReGa₅ compound the superconducting transition was characterized further, through specific heat (C_p) measurements. The Inset of Fig. 6B shows C_p/T vs. T² in the temperature range 1.85–4 K under a magnetic field of 5T (open squares) and 0T (blue circles). The zero field data were fit by using the formula C_p/T = γ + βT², yielding the electronic specific heat (Sommerfeld) coefficient γ = 4.68(7) mJ⋅mol⁻¹⋅K⁻² and the phonon specific heat coefficient β = 0.38(1) mJ⋅mol⁻¹⋅K⁻⁴. The latter quantity is related to the Debye temperature (Θ_D) through Θ_D = (12π⁴nR/5β)^1/3, and the estimated Debye temperature for ReGa₅ is thus 314(2) K. This temperature is only 6 K lower than the Debye temperature for Ga metal (Θ_D = 320 K) (31). Fig. 6B presents the temperature dependence of the zero-field electronic specific heat C_el/T. The good quality of the sample and the bulk nature of the superconductivity are strongly supported by the presence of a large anomaly in the specific heat at T_c = 2.1 K, in excellent agreement with the Tc determined by ρ(T) and χ_mol(T). An equal entropy construction gives the specific heat jump ΔC_el./γT_c = 1.6, which is slightly larger than expected for a weak-coupling BCS superconductor, where it is 1.43 (32, 33). Using the McMillan equation (34), we calculate the electron-phonon coupling constant λ_ep = 0.51. [For this calculation the Coulomb repulsion constant was taken as μ* = 0.13 (μ* = 0.13 falls in the range 0.1–0.15 used in the literature) (35).] Having both the electron-phonon coupling constant λ_ep and the Sommerfeld coefficient γ_, the density of states at the Fermi energy can be calculated from N(E_F) = 3γ/[π²k²_B(1 + λ_ep)]. The N(E_F) obtained for ReGa₅ is low, N(E_F) = 1.3 states eV⁻¹ per formula unit and agrees with band structure calculations.

Fig. 6.

Characterization of the superconducting transition of ReGa₅. (A) χ_v (T) measured in a 10 Oe applied magnetic field from 1.8 to 6 K with zero-field cooling. (Inset) Resistivity vs. temperature over the range of 2–50 K measured in different applied magnetic fields. (B) Temperature dependence of the electronic specific heat C_el of ReGa₅. The sample was measured with (µ₀H = 5T) and without magnetic field, presented in the form of C_p/T (T), and the electronic part was obtained from heat capacity at µ₀H = 5T. (Inset) Temperature dependence of specific heat C_p of ReGa₅ sample measured with (5T) and without magnetic field, presented in the form of C_p/T (T²).

Superconductivity in the Endohedral Gallide Cluster Family and Comparison with Matthias’ Rules.

The concepts and results described here are presented in Fig. 7, which summarizes the structural and electronic character of the endohedral gallide superconductor family. The correlation of the superconducting transition temperatures with the chemical and structural characteristics is shown. The plot includes the superconductor discovered here, ReGa₅, the previously known superconductors, and the nonsuperconductors V₈Ga₄₁ (36) and PdGa₅ (29). The best superconductors are found for the endohedral molybdenum gallides, whose hierarchal cluster architectures and electron counts are intermediate in the context of the full family. To determine whether it is solely the electron count that governs the superconducting T_c, the T_cs can be compared with the calculated electronic DOS for the materials in the family. This comparison is shown in Fig. 8, which is essentially an extension of Matthias’ rules for intermetallic transition metal-based superconductors, where T_c simply tracks the d-state derived electronic density of states at E_F (37). The current case is not as straightforward because the chemistry and structures are more complex and all compounds are required to be near a minimum in the density of states to be chemically stable. Nonetheless a fruitful comparison can be made. Thus, Fig. 8 shows the observed superconducting T_c vs. electron count and the calculated DOS at E_F vs. electron count for the endohedral gallide superconductors. The plot is scaled so that the peaks for T_c and the calculated DOS near Mo₈Ga₄₁ (∼21.4 e- per transition metal) coincide. If Matthias’ rules for intermetallic superconductors generally held in this family, i.e., if T_c were solely determined by the electronic density of states, then the two curves would coincide over the whole range. They do not, coinciding nicely for small electron counts but not for high electron counts. The implication is that both the electron count and the architecture of the endohedral cluster packing play an important role in determining T_c, the former at low electron counts and the latter at the higher electron counts; i.e., if the DOS was the dominant factor in determining T_c for the gallide cluster superconductors, then the T_c for PdGa₅ should be comparable to that for Mo₈Ga₄₁. It is not.

Fig. 7.

The structural and electronic characteristics of the binary endohedral gallide cluster superconductor family. The horizontal axis is the number of electrons (e-) per transition metal and the vertical axis is the superconducting transition temperature (T_c). The formulas of the compounds are shown. The endohedral clusters shown in the insets illustrate the crossover from corner sharing to edge sharing cluster architectures as a function of electron count.

Fig. 8.

Comparison of the superconducting T_c vs. electron count and the calculated electronic DOS at E_F vs. electron count for the endohedral gallide cluster superconductors. The plot has been scaled so that the peaks for T_c and the calculated DOS near Mo₈Ga₄₁ (∼21.4 e- per transition metal) coincide. If Matthias’ rule for intermetallic superconductors generally held in this family, then the two curves would coincide over the whole range; they do for small electron counts but not for high electron counts. (Inset) Calculated DOS near E_F (displaced along the vertical axis for clarity) for the endohedral gallide superconductors.

Conclusion

The endohedral gallide cluster compounds are presented as a chemical family that is favorable for superconductivity. Based on the electronic structures of isolated transition metal-centered endohedral clusters, a molecule-based electronic understanding of the materials is developed that relates their chemical stability, formulas, and the hierarchical architecture of the clusters when found in solid compounds. The empirical electron counting rules developed for the stability of T-Ga binary compounds in this family, TGa_(n-1/2*m+l) (T = transition metal, n = vertices of the T-centered cluster; m = shared vertices; l = isolated Ga atom), revealed the lack of superconducting examples at 22 electrons per transition metal. This led to the investigation the Re-Ga chemical system. Validating the molecule-based viewpoint of this family of materials, the previously unreported compound ReGa₅ was found, structurally characterized, and analyzed by electronic structure calculations. Further, resistivity, heat capacity and magnetic susceptibility measurements revealed ReGa₅ to be a superconductor with a T_c ∼ 2.3 K. This work shows that selection of potential superconducting materials based on molecular electron counting rules, here demonstrated in a family of endohedral cluster compounds, is a useful chemistry-based design paradigm for finding new superconductors. Extension of this concept to additional materials families will be of significant future interest.

SI Text

Calculation Methods.

VASP.

Structure optimization was completed using VASP, which uses projector augmented-wave (PAW) pseudopotentials that were adopted with the Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA), in which scalar relativistic effects are included. The conjugate gradient algorithm was also applied. The energy cutoff was 400 eV. Reciprocal space integrations were completed over a 7 × 7 × 7 Monkhorst-Pack k-points mesh with the linear tetrahedron method. With these settings, the calculated total energies converged to less than 0.1 meV/atom.

Extended Huckel theory.

Two different minimal basis sets involving Slater-type single-zeta functions for s and p orbitals and double-zeta functions for d were applied. The parameters for Ga are 4s: ζ = 1.77, Hii = –14.58 eV; and 4p: ζ = 1.55, Hii = –5.60 eV. The parameters for Mo are 5s: ζ = 1.96, Hii = –8.77 eV; 5p: ζ = 1.90, Hii = –5.60 eV, and 4d: ζ1 = 4.54 (c1 = 0.5899), ζ2 = 1.90 (c2 = 0.5899), Hii = –11.06 eV. The parameters for Rh are 5s: ζ = 2.14, Hii = –9.17 eV; 5p: ζ = 2.10, Hii = –3.97 eV, and 4d: ζ1 = 5.54 (c1 = 0.5823), ζ2 = 2.40 (c2 = 0.6405), Hii = –12.71 eV. The parameters for Pd are 5s: ζ = 2.19, Hii = –8.64 eV; 5p: ζ = 2.15, Hii = –2.68 eV, and 4d: ζ1 = 5.98 (c1 = 0.5535), ζ2 = 2.61 (c2 = 0.6701), Hii = –12.65 eV.

WIEN2k.

The electronic structure (DOS and band structure) of ReGa₅ was calculated using the WIEN2k code with spin orbit coupling, which has the full-potential linearized augmented plane wave method (FP-LAPW) with local orbitals implemented. For treatment of the electron correlation within the generalized gradient approximation, the electron exchange-correlation potential was used with the parametrization by Perdew et al. (i.e., the PBE-GGA). For valence states, relativistic effects were included through a scalar relativistic treatment, and core states were treated fully relativistically. The structure used to calculate the band structure was based on the single crystal data. The conjugate gradient algorithm was applied, and the energy cutoff was 500 eV. Reciprocal space integrations were completed over a 9 × 9 × 9 Monkhorst-Pack k-points mesh with the linear tetrahedron method. With these settings, the calculated total energy converged to less than 0.1 meV/atom. The electronic structures for the V, Mo, Rh, and Pd gallides were calculated based on the reported experimental structures, using similar methods.

Experimental Details.

Phase analysis.

All samples were examined by powder X-ray diffraction for identification and phase purity on a Bruker D8 Advance powder diffractometer using Cu radiation (λ_Kα = 1.5406 Å) and a Lynxeye detector. The scattered intensities were recorded as a function of Bragg angle (2θ) with an exposure time of 1 s per step in a rotation mode, ranging from 5° to 110°.

Structure determination.

Single crystal data were measured using a Bruker Apex II diffractometer with Mo K_α radiation (λ = 0.71073 Å). Data were collected over a full sphere of reciprocal space with 0.5° scans in ω with an exposure time of 10 s per frame. The 2θ range extended from 4° to 70°. The SMART software was used for data acquisition. Intensities were extracted and corrected for Lorentz and polarization effects with the SAINT program. Empirical absorption corrections were accomplished with XPREP, which is based on modeling a transmission surface by spherical harmonics using equivalent reflections with I > 3σ(I).With the SHELXTL package, the crystal structure was solved using direct methods and refined by full-matrix least squares on F². All crystal structure drawings were produced using the program VESTA.

Physical property measurements.

The magnetization measurements were performed in a 10 Oe applied field using a Quantum Design Superconducting Quantum Interference Device (SQUID) magnetometer, over a temperature range of 1.8–6 K. The magnetic susceptibility is defined as χ = M/H where M is the measured magnetization in emu and H is the applied field in Oe. The resistivity and specific heat measurements were measured using a Quantum Design Physical Property Measurement System (PPMS) from 1.85 to 300 K with and without an applied field. Resistivity measurements were made in the standard four-probe configuration, and the specific heat measurements were performed on a polycrystalline sample of approximate weight 10 mg.