Yb 14 MSb 11 (M = Mn and Mg) are among the most efficient hightemperature p-type thermoelectric materials exhibiting a unitless thermoelectric figure of merit, zT, of 1.33 at 1275 K and 1.28 at 1175 K, respectively (1)(2)(3). The high zT depends on electronic and thermal transport, and a rationale for their high performance has recently emerged (4)(5)(6). The low thermal conductivity is directly linked to the complex and large crystal structure. The excellent electronic properties with high Seebeck coefficient and conductivity are linked to a multivalley Fermi surface. Both Yb 14 MnSb 11 and Yb 14 MgSb 11 share these attributes and show high zTs as a result. On the other hand, the Zn analog of the Yb 14 MSb 11 (generalized by 14-1-11) compounds, Yb 14 ZnSb 11 , has shown poor thermoelectric properties with a relatively high thermal conductivity and a low Seebeck coefficient (7). This is unexpected considering how similar the Zn compound is to the Mn and Mg analogs: All metals are 2+, all contain Yb and Sb, and the compounds are the same structure type. This prompts questions about how common high thermoelectric performance is within the vast space of isostructural materials forming in the Ca 14 AlSb 11 structure type (8). Here, we show that contrary to a previous report (7), Yb 14 ZnSb 11 has a high figure of merit (1.2 to 1.9 at 1175 to 1275 K) supported by first-principles computations, indicating similar band structures across the Zn, Mn, and Mg series. This marked improvement in figure of merit for Yb 14 ZnSb 11 is due to a new synthetic route leading to the reduction of competing side phase impurities. Beyond Yb 14 ZnSb 11 , we show that the multivalley band structure, which is favorable to thermoelectric properties, is universal to the compounds of the Ca 14 AlSb 11 structure type. Our work motivates the investigation of the many compositions that crystallize in the Ca 14 AlSb 11 structure type and showcases the importance of phase purity and the development of synthetic routes to obtain high-performance thermoelectric materials.
Band structure of Yb 14 ZnSb 11 The champion high-temperature p-type thermoelectric materials Yb 14 MnSb 11 and Yb 14 MgSb 11 (zT = 1.33 and 1.28, respectively) form in the Ca 14 AlSb 11 structure type, which can accommodate a number of different elements (8). Compounds of this structure type fall under the description of Zintl phases where simple electron counting rules can be used to rationalize the bonding and are considered semiconductors (9)(10)(11)(12). Figure 1 shows the crystal structure of Ca 14 AlSb 11 that contains eight formula units, each of which can be described as consisting of 14 Ca 2+ cations, a [AlSb 4 ] 9-tetrahedron, a Sb 3 7-linear unit, and four Sb -3 anions. The electronic properties of these materials will depend on their bonding and resultant electronic band structures.
Figure 2 shows the band structures and density of states of Yb 14 MgSb 11 , Yb 14 ZnSb 11 , and Yb 14 MnSb 11 obtained by density functional theory (DFT). The band structures of the three compounds can be described by the multiband model recently developed for Yb 14 Mg 1-x Al x Sb 11 (5). In this model, the high power factor comes from the combination of a low-effective mass band present at the point and a highly degenerate (N v = 8) heavy pocket of bands between N and P. As the temperature increases, the Fermi-Dirac distribution broadens, and this pocket of degenerate, heavy bands become involved in transport because of their proximity to the Fermi level. This leads to increases in effective mass with increasing temperature and thereby large Seebeck coefficients (5). The three compounds show very similar band structures and therefore should all have a high power factor. A closer look at the projected density of states and an analysis of the bonding in Yb 14 ZnSb 11 [Supplementary Materials, figs. S1 to S4] shows that the bands that drive the p-type transport are of Yb and Sb character with minimal contribution from Zn 2+ . Crystal orbital Hamilton population (COHP) analysis of the Yb 14 MSb 11 (M = Al, Mn, Mg, and Zn) series (fig. S2) shows that in all cases, the top of the valence band is dominated by Yb-Sb and Sb-Sb interactions with little to no contribution from M-Sb states. With this realization, it is of little surprise that the band structures share so many similarities at the valence band edge, and one would expect similar transport properties in these systems independent of the 2+ metal (Mn, Mg, and Zn).
Despite the similarities to Yb 14 MgSb 11 and Yb 14 MnSb 11 , Yb 14 ZnSb 11 has been reported thus far to exhibit poor thermoelectric performance with low electrical resistivity, low Seebeck coefficients, and high thermal conductivity (7). This is in clear contradiction with our first-principles band structure analysis. More recently, high efficiencies have been reported for the Yb 14-x La x ZnSb 11 and Yb 14-x Y x ZnSb 11 phases (13). These conflicting results suggest that there is something amiss in past work on Yb 14 ZnSb 11 (7). Systematic studies of Yb 14 MnSb 11 , the first compound of this structure type to show exceptional thermoelectric properties (14), have shown that the properties can be affected by impurities (2). In the original report on Yb 14 ZnSb 11 as part of the solid solution Yb 14 Mn 1-x ZnSb 11 , the phase purity of the Yb 14 ZnSb 11 pressed pellet sample was not characterized (7). In this report, the authors state that there is increasing amounts of Yb 9 Zn 4+x Sb 4 with increasing x in the solid solution Yb 14 Mn 1-x Zn x Sb 11 . The transport and thermal stability properties reported for Yb 14 ZnSb 11 match quite well with that reported for polycrystalline Yb 9 Zn 4+x Sb 9 (15). This suggests that the low Seebeck coefficient and relatively high thermal conductivities previously reported for Yb 14 ZnSb 11 are due to a sizeable Yb 9 Zn 4+x Sb 9 impurity in the sample. The effective removal of impurities in other structure types has been shown to lead to higher figures of merit for thermoelectric materials (1-3, 16, 17), and we show here that this is also true in the case of Yb 14 ZnSb 11 .
In early work on Yb 14 ZnSb 11 , Sn flux was used to grow single crystals from a combination of the elements (7,18,19). In these reports, reactions were performed with a large excess of Zn, in the ratios 14:6:11:86 (Yb:Zn:Sb:Sn). While this approach was very successful in growing single crystals (18,19), the excess of Zn moves the composition of the reaction closer to an adjacent ternary-phase Yb 9 Zn 4+x Sb 9 (15,20) (see Fig. 3, black arrow from Yb 14 MSb 11 ). As a result, Yb 9 Zn 4+x Sb 9 becomes a competing phase and a possible impurity when considering a large-scale flux growth of crystals (7). When considering the synthesis of polycrystalline Yb 14 ZnSb 11 , Zn dispersity becomes an even greater issue. As opposed to a flux growth in which atoms are highly mobile in a liquidous melt, here, the high melting points of the compounds that occupy this region of phase space prevent the formation of a melt, and atomic movement is reliant on solid-state diffusion. The diffusion process is slow, which makes the dispersity of elements within the reaction mixture even more important. In the Mg-and Mn-containing analogs of Yb 14 MSb 11 , polycrystalline samples were prepared from the elements with an excess of M that ranged from 5% for Mn to 20% excess in the Mg case (1,2). The excess of M resulted in the removal of Yb 11 Sb 10 impurities and an improvement in thermoelectric properties (1,2,21). Initial work on polycrystalline Yb 14-x RE x ZnSb 11 (RE = La, Y) took a similar approach, employing 100% excess of Zn (13). It is unexpected and notable that no Yb 9 Mn 4+x Sb 9 and Yb 9 Mg 4+x Sb 9 have ever been observed to the contrary of Yb 9 Zn 4+x Sb 9 . We turn to DFT to try to rationalize this observation. Figure 3 compares the reaction energy of Yb 9 M 4.5 Sb 9 in YbM 2 Sb 2 , Yb 14 MSb 11 and YbH 2 , focusing on the part of the phase diagram most relevant to M excess in Yb 14 MSb 11 . All computations compare compounds with Yb in a +2 state as we want to avoid the challenging treatment by DFT of mixed oxidation states of the rare-earth Yb. A negative reaction energy indicates a stable Yb 9 M 4.5 Sb 9 (i.e., which does not decompose in the three end members of the triangle). Figure 3 clearly indicates that Yb 9 Zn 4.5 Sb 9 is much more favored energetically than Yb 9 Mn 4+x Sb 9 and Yb 9 Mg 4+x Sb 9 . This explains why the Mn and Mg analogs could be formed with metal excess without major impurity formation and that the Zn 14-1-11 compound is inherently more difficult to form because of the competing Yb 9 M 4+x Sb 9 phase.
The challenge with Yb 14 ZnSb 11 is therefore to design a synthetic route that homogenizes the reactants while staying close to stoichiometry to avoid forming Yb 9 M 4+x Sb 9 . Inspired by previous work on the synthesis of Yb 14 MnSb 11 and Yb 14 MgSb 11 (3), we used the binary phases, YbH 2 , Yb 4 Sb 3 , and ZnSb intimately mixed on stoichiometry to form 14-1-11, followed by reaction and consolidation in a twostep Spark Plasma Sintering (SPS) process. This route maintains stoichiometry and improves reaction interfaces required for direct formation of the targeted ternary phase instead of binary intermediates. The YbH 2 not only provides a high-dispersity source of Yb to provide a stoichiometric reaction but also provides a partially reducing atmosphere as the hydride decomposes to elemental Yb and H 2 gas at elevated temperatures (1). A slice of the densified pellet used in all the thermoelectric measurements described below was analyzed by room temperature powder x-ray diffraction (PXRD) and is shown in Fig. 4. All reflections were indexed as the tetragonal I4 1 /acd phase Yb 14 ZnSb 11 without any unassigned intensities that could be attributed to Yb 2 O 3 , Yb 11 Sb 10 , or Yb 9 Zn 4+x Sb 9 .
The unit cell parameters determined from Rietveld refinement of phase-pure Yb 14 ZnSb 11 were a = 16.6105(1) Å and c = 21.9554(2) Å, with a unit cell volume of 6057.65(6) Å 3 . This is larger than what was previously reported for single-crystal samples [a = 16.562(3) Å, c = 21.859(5) Å, V = 5995.9(16) Å 3 collected at 90 K] (18,19) and likely be due to data collection temperature. Elemental analysis by energy-dispersive spectroscopy (EDS) confirmed that the sample was phase-pure Yb 14 ZnSb 11 (fig. S10). The lattice parameters of multiple samples of polycrystalline Yb 14 ZnSb 11 prepared by this route described above and from the elements are in good agreement. Using the same method using different batches of Yb 4 Sb 3 showed phase-pure Yb 14 ZnSb 11 and confirms that the reproducibility of this route is high (see figs. S7 to S9 and table S1).
Previous literature on Yb 14 ZnSb 11 (7) suggested a possible decomposition of the phase above 800 K with a 2% mass loss and endotherm observed above that temperature. Thermometric gravimetry and S2).
The thermal conductivity of Yb 14 ZnSb 11 is shown in Fig. 5A, and a simplified model of the electronic band structure is shown in Fig. 5B. The thermal conductivity of Yb 14 ZnSb 11 has the unique "S" shaped temperature dependence that is seen with other analogs of Yb 14 MSb 11 (M = Mn and Mg). The low temperature maximum is attributed to the distribution of carriers from the heavy to the light valence band in the two-band model previous described (5). At a temperature of ~1150 K, there is an increase due to the onset of bipolar conductance (1,5,21). In comparison to Yb 14 MnSb 11 and Yb 14 MgSb 11 , the thermal conductivity of the Zn analog is higher over the entire temperature range. Using the Wiedemann-Franz law ( total = l + LT), the lattice thermal conductivity ( l ) can be separated from the electronic contribution (LT). Here, sigma () is the electrical conductivity, T is absolute temperature, and L is the Lorenz number, which can be estimated using the Seebeck coefficient (22). Removing the electronic component from the thermal conductivity of both analogs, the higher thermal conductivity of the Zn analog can be attributed to the lower electrical resistivity (shown below) as the calculated lattice thermal conductivities of these analogs are very similar. As the method used to estimate L relies on the Seebeck coefficient and assumes a constant effective mass, there are slight deviations in l at temperature ranges where the effective mass is changing because of a change in bands contributing to transport (5,22).
Figure 6A provides the electrical resistivity from both Van der Pauw and off-axis four-probe measurements from room temperature to 1275 K. The measured electrical resistivity of Yb 14 ZnSb 11 peaks to 5.30 milliohms•cm at 1250 K and then bends over because of bipolar conductance. The electrical resistivity measured by the off-axis four-probe instrumental arrangement agrees very well with the values obtained by the Van der Pauw method. The electrical resistivity here is higher than that previously reported for both single-crystal and Y-or La-substituted polycrystalline samples (7,13). The differences are attributed to impurities present in polycrystalline samples previously reported (7,13).
The Hall carrier concentration and mobility of the Yb 14 ZnSb 11 pellet are shown in Fig. 6 (B andC). The carrier concentration starts at 1.03 × 10 21 holes cm -3 at 295 K and decreases with increasing temperature to 6.47 × 10 20 at 600 K, finally reaching a minimum of 4.88 × 10 20 at 850 K. After that temperature, the carrier concentration begins to increase, which is attributed to the onset of bipolar conductance (1,5). The temperature dependence of the carrier concentration is consistent with the electrical resistivity measurements presented above. Yb 14 ZnSb 11 shows comparable room temperature The PXRD pattern for Yb 14 ZnSb 11 (blue) synthesized from the stoichiometric reaction of the binaries, YbH 2 , Yb 4 Sb 3 , and ZnSb, and the calculated diffraction pattern from the single-crystal structure (black) (18). Full Rietveld refinement is provided in figs. S7 to S9 and lattice parameters in table S1. The Hall mobility starts around 5 cm 2 V -1 s -1 and remains relatively level until 700 K where the mobility begins to decrease rapidly down to 1 cm 2 V -1 s -1 . This temperature corresponds to the same point at which the thermal conductivity begins to decrease. This correlation suggests that the decrease in mobility is the transition of holes from the light band at to the lower-mobility, heavy band between N and P (5). The room temperature carrier mobility of this analog is double that of Yb 14 MnSb 11 (2.2 cm 2 V -1 s -1 ), which is atttibuted to the smaller unit cell volume, leading to better orbital overlap (2,14,21). Despite the lower carrier concentrations, Yb 14 MgSb 11 has slightly lower carrier mobilities (4.0 to 4.70 cm 2 V -1 s -1 ) than Yb 14 ZnSb 11 at room temperature likely due to the larger unit cell volume of the Mg analog (6150 Å 3 ) (1, 3).
Figure 7A provides the temperature-dependent Seebeck coefficients of Yb 14 ZnSb 11 from both two-probe and off-axis four-probe measurements. The two-probe Seebeck coefficient of Yb 14 ZnSb 11 was measured on a custom instrument at JPL, allowing for direct comparison to previously reported values for Yb 14 MnSb 11 and Yb 14 MgSb 11 across the entire operating temperature range (1)(2)(3)23). The Seebeck coefficient starts at 32.27 V K -1 at 311 K and increases linearly until it levels off to a peak of 211.25 V K -1 at 1181 K. After that point, the Seebeck coefficients begin to decrease attributed to the onset of bipolar conductance (22). The temperature dependence is consistent with both the electrical resistivity and carrier concentration described above. Yb 14 ZnSb 11 has a lower-peak Seebeck coefficient than the Mn and Mg analogs (234 and 225 V K -1 , respectively) (1,2,21). This can be attributed to the lower effective mass of the bands that make up the valence band of Yb 14 ZnSb 11 below the Fermi level. The inflection point in the Seebeck coefficient at high temperature corresponds with an estimated bandgap of 0.50 eV by the Goldsmid-Sharp method (24). This agrees with the calculated bandgap of 0.43 eV and is evidence of Yb 14 ZnSb 11 having a smaller bandgap than that of the Mg (0.53 eV) or Mn (0.59 eV) analogs and a similar gap to that of Yb 14 AlSb 11 (0.46 eV) (5,14,25,26).
The off-axis four-probe Seebeck coefficient was measured with a commercial instrument, allowing for comparison to other materials measured by this more common method. As outlined in previous work, the measured Seebeck coefficients can be highly dependent on the method used (3,22,27,28). The off-axis four-probe method is susceptible to the systematic exaggeration of the Seebeck coefficients as temperature increases. This is commonly referred to as the "cold-finger" effect and derives from the movement of heat away from the surface of the material into the thermocouple (22,27,29). During the measurement, the thermocouple assembly on the hot side is at an overall lower temperature than the surface of the hot side of the material. Because of this, heat can be drawn away from the surface of the material and into the thermocouple. In addition to phonons, electronic carriers from the surface of the material are also drawn into the thermocouple probes. Both these carry thermal energy and can lower the surface temperature of the material. Despite the lowering of the surface temperature, the bulk of the sample on the hot side is at a higher temperature and a greater thermal gradient. The thermocouple probe measures the surface temperature and not the temperature of the bulk sample. This can lead to the thermal gradient being underestimated, leaving an overestimation of the Seebeck coefficient (V/T). This effect has been outlined as having a temperature dependence that increases with temperature (22).
The Seebeck coefficient measured by the off-axis four-probe method starts at approximately the same value as the two-probe measurements [39.7 V K -1 at 335 K (four-probe) versus 34.0 V K -1 at 328 K (two-probe)], and that trend holds for the values below 600 K. Above 600 K, the Seebeck coefficients begin to deviate from the two-probe measurement, and by 1000 K, the four-probe measurement has reached a value of 210.8 V K -1 . As temperature increases, the deviation between the two measurement methods increases. The four-probe measurement reaches a maximum of 261.3 V K -1 at 1273 K, a 27.7% larger Seebeck coefficient than what was measured by the two-probe method (204.71 V K -1 at 1261 K). A graph of the deviation in Seebeck coefficients for two-probe and off-axis four-probe as a function of temperature is provided in fig. S15. Although the Seebeck coefficients measured from the four-probe method is systematically larger at high temperatures, the calculated bandgap by the Goldsmid-Sharp method from the maximum values is only 0.66 eV (24). Considering that DFT has a tendency to underestimate bandgaps, this is likely within error of the calculated gap of 0.43 eV (30).
Pisarenko plots using the two-probe measurements at 400 and 900 K (Fig. 7, B andC) provide an effective mass of 1.91 m 0 that increases with temperature to 2.35 m 0 because of the contribution of the second valence band at higher temperatures, confirming the multiband nature of transport within this analog of Yb 14 MSb 11 (5). The effective mass at 400 K is between what can be calculated from previously reported two-probe transport data for the Mn and Mg analogs (2.89 and 1.88 m 0 ), but as temperature increases and the Fermi-Dirac distribution broadens, the effective mass of the Zn analog becomes smaller than that of the other two analogs (3.45 and 2.78 m 0 ) (1,2,21). The lower effective mass of Yb 14 ZnSb 11 is consistent with the higher carrier mobilities seen in this analog. Using the fourprobe Seebeck coefficient measurement, the same effective mass can be calculated at 400 K; however, at 900 K, the larger measured Seebeck coefficient gives a higher calculated effective mass of 2.70 m 0 . Although the higher effective mass from the four-probe is possibly inflated because of the cold-finger effect, the value is not unreasonable considering the similarities in electronic structure between the Zn, Mn, and Mg analogs.
Because of the high Seebeck coefficient and low electrical resistivity, Yb 14 ZnSb 11 shows a high power factor of 8.4 W cm -1 K -2 from 1075 to 1150 K (fig. S16), rivaling the best reported results of Yb 14 MnSb 11 (9.3 W cm -1 K -2 at 1250 K) and surpassing Yb 14 MgSb 11 (7.5 W cm -1 K -2 from 1150 to 1200 K). Because of its complex structure, the thermal conductivity remains low to give maximum calculated zTs of 1.2 at 1175 K using the values from a two-probe Seebeck coefficient and Van der Pauw resistivity. This is comparable to the efficiency of Yb 14 MgSb 11 and slightly below that of Yb 14 MnSb 11 measured by the same methods (1, 2). The maximum zT calculated from values measured by an off-axis four-probe reaches 1.9 at 1275 K (Fig. 8). This is the highest reported efficiency for any material of the Ca 14 AlSb 11 structure type and rivals the highest reported zTs of any p-type high-temperature material. Because of the possibility of systematic errors in the measurement of high-temperature Seebeck coefficients depending on experimental setup, the zT calculated from the off-axis four-probe measurements is likely exaggerated but is still effective in identifying high-efficiency materials. The availability of commercial four-probe instruments makes them a powerful tool for the measurement of high-temperature Seebeck coefficients. However, the systematically large values measured by this method should be taken into consideration when making comparisons to Seebeck coefficients measured by other methods. Both experimentally determined zTs from this work show how the effective removal of impurities through improved synthetic methods can result in large improvements in thermoelectric performance. To confirm the reproducibility, additional samples were synthesized and measured using the two-probe Seebeck coefficient and Van der Pauw resistivity and can be found in Supplementary Materials fig. S17.
Theory and experimental data have demonstrated that the Yb 14 MSb 11 with M = Zn, Mn, and Mg show very similar electronic structure and transport properties that result in similarly high thermoelectric performance (Fig. 9). The low zT observed previously (7) originated from impurities forming easily for the Zn analog. This raises the question of how general the excellent electronic properties for other compositions within the Ca 14 AlSb 11 structure type are. Figure 10 shows the band structure computed within DFT for Yb 14 MPn 11 where M = Mg, Zn, and Cd and Pn = As, Sb, and Bi. We are not reporting on the Mn version here, which should have a more challenging electronic structure due to magnetic ordering. The similarity between the band structures of these compounds despite their widely different elemental constituents is notable. They all show the valence band features of one curvy band at with a series of lower-energy flatter bands between N and P with high degeneracy that are responsible for the high thermoelectric performances. Even when Yb is replaced by Ca, the band structure retains these features (see fig. S18). This type of universality in favorable band structures is seen in other highsymmetry families of thermoelectric materials (31) such as lead chalcogenides (32,33), tin chalcogenides (34,35), cubic GeTe (36), and half Heusler phases (37,38) and in the general class of gapped metals (39). Further tuning and optimization of the thermoelectric performance is possible by varying the pnictogen from As to Bi, as the bandgaps and offsets between the two valence bands change in a systematic fashion. Our work indicates that many opportunities remain unexplored in the 14-1-11 field when the challenges associated with synthesis of these previously unexplored phases are overcome and that many members of the Ca 14 AlSb 11 should exhibit exceptional thermoelectric perform ance if adequately doped (5).
The electronic band structures were calculated within DFT, using the Vienna Ab initio Simulation Package (VASP) (40,41) with the Perdew-Burke-Ernzerhof (PBE)-generalized gradient approximation (GGA) functional and adopting the projector augmented-wave (PAW) (42,43) approach. The primitive structures were relaxed until the forces are less than 0.01 eV/Å. During the relaxations and static runs, the wave functions were expanded on a plane-wave basis set up to an energy cutoff of 520 eV, and the Brillouin zone was sampled using a 2 × 2 × 2 Monkhorst-Pack k-point grid. A non-self-consistent field calculation on an 8 × 8 × 8 k-point grid was performed to calculate the density of states (DOS). Then, by using the BoltzTraP (44,45) software, the eigenvalues were interpolated on a 10-times denser grid to obtain the DOS. The Yb pseudopotential has the 4f electrons frozen in the core (i.e., not as valence electrons). This was motivated by a previous x-ray photoelectron spectroscopy study, which indicated that Yb-4f states in Yb 14 MnSb 11 and Yb 14 ZnSb 11 are below the Fermi level by more than 0.5 and 1 eV, respectively (19). We also performed some tests with the PAW Yb pseudopotential providing f electrons as valence electrons and applying different Hubbard U values on the f states. The typical U values used in previous work on other Yb antimonides (23) lead to f states, much lower than the Fermi level, confirming that f states could be neglected. We also applied different Hubbard U values on the d states of Mn. Given that U values in range (2,5) eV give a similar band structure (see fig. S4), we will consider in the following only the one obtained with U = 3 eV, which is a typical U value for Mn in many compounds.
Yb 14 MnSb 11 is a ferromagnetic material with the Curie temperature (Tc) of 53 K (46). Transport measurements and thermoelectric operations are performed at much higher temperature leading to a paramagnetic state (14). In this work, all the results are performed on an antiferromagnetic spin ordering on the manganese atoms. Although the statistical effects of spin ordering are not included in this approach model, the antiferromagnetic ordering captures two important features of paramagnetism, the spin polarization and zero total magnetic moment.
We performed bonding analysis using the COHP and Lobster (47). The default basis functions of the "pbeVasp-Fit2015" basis set were used. The charge spilling considering all occupied states arrived at around 1.4% for all compounds. The total spilling considering both occupied and unoccupied states was around 13% for Mn and Mg compounds and 13% for Al and Zn compounds. Because the following analysis will mainly be performed for the occupied states and the charge spilling shows a very low value, this rather high total spilling should not influence our conclusions.
The Yb-M-Sb phase diagrams have been computed with similar parameters and with the GGA-PBE functional and with a +U value of 3 eV on Mn. All Mn-containing compounds were computed in both the states in which the initial magnetic moments of the Mn atoms were aligned ferromagnetic and antiferromagnetic. In all cases tested, the antiferromagnetic state had the lower final energy. Only Yb 2+containing compounds are considered, as modeling the different oxidation state of Yb (+2 and +3) would be challenging for DFT. This allows us to use the PAW pseudopotential with frozen f electrons as well for the phase stability computations. We did not use Yb metal as an end-member of the phase diagram but instead used YbH 2 . The Yb 9 M 4+x Sb 9 was calculated as Yb 9 M 4.5 Sb 9 , which represents one M interstitial atom in the Yb 9 M 4 Sb 9 unit cell (M = Zn, Mn, and Mg). Pymatgen (48) was used for generating input files for VASP and post-processing the outputs such as plotting the band structure on the standard high-symmetry k-point path (49), plotting the DOS or the COHP, and building the phase diagram.
Reactions were done in 10-g batches using a stoichiometric 4:3, Yb:Sb ratio. In an Ar-filled glove box [<0.5-parts per million (ppm) O 2 ], Yb metal (Edgetech 99.99%) was filed using a metal rasp. These filings were added to a 55-cm 3 tungsten carbide grinding vial (SPEX Sample Prep.) with two 12.7-mm-diameter tungsten carbide balls. Ground Sb shot (5N plus, 99.999%) was then added to the vial. The vial was closed and further sealed in mylar under Ar before being removed from the glove box. The mixture was milled for three rounds of 30 min with a thorough scrape under inert atmosphere (<0.5-ppm O 2 ) in between each round using a chisel to help minimize cold welding and elemental loss as well as improve mixing. The resultant powder was then sealed under Ar in a Nb tube and then further jacketed in fused silica under vacuum (<0.5 mtorr). The reaction was annealed for 12 hours at 850°C in a tube furnace. The resultant black powder was analyzed by PXRD. In most cases, a mixture of Yb 4 Sb 3 and Yb 11 Sb 10 (<10% from Rietveld refinement) was observed. This mixture of phases represents a single point on the Yb-Sb phase diagram and was treated as a homogeneous distribution of those two phases. PXRD of the Yb 4 Sb 3 used can be found in the Supplementary Materials (fig. S5).
Reactions were performed in 5-g batches using a 1:1 ratio of Zn:Sb. In an Ar-filled glove box (<0.5-ppm O 2 ), Zn pieces (Columbus Chemical Industries 99.98%) and ground Sb shot were combined in a 65-cm 3 stainless steel grinding vial with two 12.7-mm balls (SPEX Sample Prep.). The closed mill was further sealed under Ar in mylar and then milled for four rounds of an hour with 15 min off in between each round. Because Zn is much less prone to cold welding in comparison to Yb, no scrapes were used. After this milling, the majority ZnSb phase can be identified by PXRD. To remove any small amounts of unreacted Sb and Zn portions, the sample was annealed in a Nb tube under inert atmosphere at 450°C for 12 hours. The resultant gray powder was identified as phase-pure ZnSb by PXRD, which can be seen in fig. S6.
Reactions were loaded with a total of 5 g with ZnSb, Yb 4 Sb 3 (prepared as described above as a mixture of Yb Yb 11 Sb 10 is introduced because of elemental loss in the Yb 4 Sb 3 reaction but is easily compensated for by using the following threevariable, three-equation, system of equations to solve for coefficients a, b, and c (Eqs. 1 to 3). Here, the phase fractions of Yb 4 Sb 3 and Yb 11 Sb 10 from PXRD of that reaction mixture will be converted to mole percent () using the molecular weights of the two compounds. Once the above reaction was balanced, the powders were weighed in an Ar-filled glove box (<0.5-ppm O 2 ) and added to a 65-cm 3 stainless steel grinding vial with two 12.7-mm-diameter stainless steel balls. The reaction was further sealed under Ar in mylar before being removed from the glove box and milled for three rounds of 30 min. The reaction was scraped with a chisel under inert atmosphere in between rounds 2 and 3. The resultant black powder was loaded into a 12.7-mm-diameter graphite die (Cal Nano). The reaction was performed directly in an SPS instrument (Dr. Sinter Lab Jr., Fuji Corp.) by a two-stage process. The reaction was completed at an initial temperature of 600°C for 30 min under 5 kN of force and active vacuum. The progress of the reaction can be monitored with pressure as the YbH 2 reacts, forming H 2 gas as a by-product. After 30 min, the pressure increased to 6.5 kN, and the temperature was ramped to 850°C where it was held for another 20 min to consolidate the sample. The resultant black pellet was over 98% of the theoretical crystallographic density by the Archimedes method. This two-step process was chosen to avoid the vaporization of Zn at high temperatures before it had the opportunity to fully react.
Phase identification and purity of polycrystalline samples was analyzed PXRD using a Bruker D8 Eco Advanced with Cu K radiation ( = 1.54 Å) and a Ni filter to remove Cu K. Diffraction experiments were performed at room temperature using a zero-background offaxis quartz plate. Diffraction patterns were analyzed by Rietveld refinement using the JANA 2006 software package (50).
The sample composition was analyzed by Z contrast using scanning electron microscopy (SEM, Thermo Fisher Quattro ESEM). Elemental distribution and total content were analyzed by EDS (Bruker Quantax) using a Yb 14 MgSb 11 single crystal as the Yb and Sb standard. A Zn standard was prepared from a piece of Zn shot (99.9999%, Johnson Matthey).
SEM images of a portion of the consolidated sample after measurements show a single homogeneous phase. Backscattered electron images show no Z contrast within the sample. Any contrasted regions in the backscattered images corresponded to regions of pullout in the secondary electron images. The elemental distribution and composition of Yb 14 ZnSb 11 were analyzed by EDS. The elemental composition from EDS was found to be Yb 14.4(5) Zn 1.0(7) Sb 10.5(4) from an average of 15 points. The minor deviations in elemental composition in comparison to nominal are due to the overlap of high-intensity Yb and Sb peaks with low-intensity Zn peaks (51). X-ray mapping of the sample shows a homogeneous distribution of the elements throughout the sample (fig. S10).
The thermal stability of Yb 14 ZnSb 11 was analyzed by TG/DSC (STA 449 F3, Netzsch) using a SiC furnace and Al 2 O 3 pans. After establishing baselines, a small piece of polished pellet was placed in the pan and heated at 10 K/min from room temperature to 1100°C and back down under Ar flow. This heating and cooling cycle was repeated four times consecutively. To avoid oxidation of the sample, four pump-purge cycles were used before starting the measurement. The purging gas flow was limited to an Ar (20 ml/min; 99.999%, Praxair) protective flow over the measurement equipment. A polished Zr ribbon was placed in a loop on top of the highest heat shield to remove minor amounts of oxygen in the purge gas flow.
Thermal diffusivity was measured on densified pellets using Laser Flash Analysis (Netzsch LFA 475 Microflash) under Ar flow. The fully densified pellet was sliced into a thin disk (>1.5 mm) and polished until a uniform level surface was achieved on both sides. The density of this disk was measured using the Archimedes method with toluene as the liquid. The density of the sample by the Archimedes method using toluene was 8.29(3) g/cm 3 , which is over 98% dense when considering the larger unit cell volume of this sample of Yb 14 ZnSb 11 [V = 6057.65(6) Å 3 , calc = 8.38951 (8) (1,2,25). Here, MM is the molar mass of the respective compounds. The coefficient of thermal expansion for Yb 14 MnSb 11 was used to estimate the temperature dependence of density (2,25). Thermal diffusivity as a function of temperature is provided in the Supplementary Materials (fig. S19). These were all combined to give the total thermal conductivity according to the equation: = D × C p × (D = measured diffusivity; C p = heat capacity adjusted for molar mass; = the temperature-adjusted density from Yb 14 MnSb 11 ) (52). The uncertainty of the thermal conductivity is estimated to be ±8%, considering the uncertainties from D, C p , and .
Resistivity and Hall carrier concentrations were measured at the Jet Propulsion Laboratory (JPL) using the Van der Pauw method with a current of 100 mA and a 1.0-T magnet on a specialized high-temperature instrument (53). Pictures of the experimental setup can be seen in fig. S20. The heating and cooling curves for resistivity are provided in fig. S21. Uncertainty for Hall carrier concentrations and electrical resistivity are estimated to be ±5% (53).
The Seebeck coefficients were measured at JPL using a custom instrument that uses the light pipe method with tungsten-niobium thermocouples under high vacuum (23). Pictures of the experimental setup can be seen in fig. S20. Heating and cooling curves are provided in fig. S22. Uncertainty in two-probe Seebeck coefficient is estimated at ±2% (23).
The Seebeck coefficient and electrical resistivity were measured on a portion of the sample after measurement at JPL. The sample was shaped into a bar (approximately 10 mm by 1 mm by 3 mm) and polished, so all sides were parallel using sandpaper. Images of the bar sample for the off-axis four-probe configuration before and after measurement along with the resistivity and Seebeck heating and cooling data are provided in figs. S22 to S24. The Seebeck coefficient and electrical resistivity were measured with an off-axis four-probe arrangement with Pt thermocouples interleaved with carbon film on the bar using a Linseis LSR-3 instrument. The measurement was performed under static He atmosphere after three prior pump/purge cycles (P min < 20 mtorr) and heated in a high-temperature infrared furnace. A polished Zr ribbon was placed inside the susceptor to act as an oxygen sponge, protecting the sample integrity. The ribbon became blackened and brittle upon completion of the measurement. The uncertainty in electrical resistivity measurement is estimated to be ±7% (54). Treating the two-probe Seebeck coefficient measurement as the "correct" value, uncertainty in the off-axis four-probe measurement can be estimated as +2%/-30% (54). The uncertainty is asymmetric because of the cold-finger effect that leads to greater Seebeck coefficient values at elevated temperatures. The uncertainty in electrical resistivity can be estimated at ±7% considering contributions from probe spacing, tip radius, and sample dimensions (54).
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