Advances in half-Heusler alloys for thermoelectric power generation

Half-Heusler alloys are leading candidate materials for large scale thermoelectric power generation. In the past decade, substantial improvements have been made to the established n-types based on X IV NiSn, and exciting new p-types based on X V FeSb and ZrCoBi have been discovered. This has been achieved through careful experimental work and by increased fundamental insight. In particular, advances in materials synthesis and compositional optimisation guided by empirical modelling have been key to extracting more performance from half-Heuslers alloys. This review summarises recent progress for the main classes of half-Heusler alloys, including their fundamental electronic and thermal properties, processing, and thermoelectric performance. It also incorporates some of the expanding body of work focused on translating materials performance ( zT 4 1 at 773 K is now routinely possible) into generator devices.


Introduction
The world is currently undergoing a transition from an energy economy based on non-renewable fossil fuels to a system making use of a range of renewable generators including solar panels, wind turbines and hydroelectric plants, supplemented by nuclear energy. 1 While major changes are taking place in terms of primary energy generation, work on improving the overall efficiency of the world's energy infrastructure is important, which can be achieved by improving electricity transmission, improving device efficiency, and recovering waste energy lost as heat.3][4][5][6][7] The widespread application of TEG's would lead to a reduction in the energy production required by primary generators.][4][5][6][7] TEG's have temperature ranges in which they operate most effectively, due to the underlying material properties. 8The only commercial TEGs based on Bi 2 Te 3 for example operate well from 300-500 K, but their effectiveness begins to decrease at higher temperatures.For application involving higher temperatures a different material is required, such as PbTe and GeTe used in the Mars Perseverance rover.However, for large scale terrestrial applications, other requirements such as elemental abundance Institute of Chemical Sciences and Centre for Energy Storage and Recovery, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK.E-mail: j.w.g.bos@hw.ac.uk and cost are important. 9,10Other factors are scalability of materials processing, mechanical properties, temperature stability, matching n-and p-type compositions and ease of electrical and thermal contacting within the TEG devices. 42][13][14][15][16] In terms of outright performance, they are characterised by good electrical properties, making them suitable for power generation application, and are only held back by relatively high lattice thermal conductivities, which limit the overall heat to electricity conversion efficiency.Due to their good compromise between engineering properties and TE performance, HH alloys have received significant attention as potential thermoelectric materials.8][19][20][21][22][23][24][25][26][27][28][29] However, the difficulty in getting reproducible results and pushing zT above one meant that other materials systems, in particular skutterudites and tellurides gained more prominence. 6,7or designing and quantifying the potential of a thermoelectric material, the figure of merit (zT) is used: where S is the Seebeck coefficient, s is the electrical conductivity and k is the total thermal conductivity.zT is maximised when electrical transport is high (large power factor S 2 s) and when thermal transport is low (small k).k is made up of a lattice component (k l ) due to vibrations, an electronic component (k el ) due to heat carried by the charge carriers and in some cases a bipolar term (k bi ) for semiconducting systems.Within a given material, optimisation of zT involves adjusting the carrier concentration to find the optimal balance between a maximal S 2 s and minimal k el , whilst atomic substitutions (alloying) or nanostructuring can be used to reduce k l . 8he performance of a material is fundamentally determined by its electronic and phonon bandstructures.Another metric which describes this balance is the quality factor (B), originally established in works by Goldsmid, Chasmar and Stratton: [30][31][32] where m w is the weighted carrier mobility.In its simplest case, this is given by m w ¼ m 0 m Ã3=2 DoS , where m 0 is the carrier mobility in the limit of zero carrier concentration and m Ã DoS is the density of states effective mass.4][35] A singular focus on zT ignores the importance of S 2 s for power generation, and on this metric the HH alloys are leading.As discussed below the HH alloys have intrinsic fairly large k l values, and in terms of the quality factor, B, their good performance derives from the large m w .This contrasts with many chalcogenides where high B values are driven by glass-like k l values. 6p to 2014, the main research focus was on developing n-type materials based on X IV NiSn and p-type materials based on X IV CoSb.This involved developing strategies for alloying and carrier doping and resulted in zT B 1 and zT B 0.7 at 773 K for the n-types and p-types respectively. 14Grain size reduction through high energy ball milling was also established as a route to lower k l but was found to require small average grain sizes 0.2-0.3mm to be effective. 14,36,37The other route for optimisation that gained interest in the early 2010s was the use of phase segregation to embed favourable micro-and nanostructures for reduction of k l .][43][44][45] Recent advances include the rise of p-type compositions X V FeSb (X V = V, Nb and Ta) and ZrCoBi, each with zT 4 1 at 773 K. [46][47][48] It also includes the widespread use of single parabolic band (SPB) and Callaway modelling, 15 coupled to first principles calculations, which have placed understanding of the HH alloys on a much firmer footing.This manuscript will review these main recent developments and will conclude with consider the electronic structure to be related to III-V semiconductors, with an 18 electron [YZ] nÀ zinc blende sublattice charge balanced by the electropositive metal X n+ . 49This electron counting does not fully guarantee a semiconducting state, with HH phases with higher average atomic number sometimes found to be semimetallic (e.g.HfCoSb), due to the closing of the band gap. 19t should also be noted that d-orbitals from the X and Y metals form the dominant contributions to the valence and conduction band, 49 so the simple Zintl-model, with separate ionic X n+ and covalent [YZ] nÀ substructures, does not fully capture the chemical bonding. 52First principles calculations confirm that the crystal structure is stabilised by strong covalent bonding in the YZ sublattice but also by stabilisation linked to the ''ionic'' XZ rocksalt sublattice, 24,27 where (intuitively) site inversion would carry a large electrostatic penalty.4][55][56] Full occupancy of the interstitial site leads to the XY 2 Z full-Heusler structure as shown in Fig. 1.As indicated by defect energy calculations, inversion of X and Y is uncommon (e.g.o1% in X IV NiSn), 57 but there is evidence that this occurs for V 0.8 CoSb and VFeSb, 58,59 where V/Fe and V/Co are (almost) adjacent in the periodic table and hence likely to have similar site preferences.

Synthesis
HH materials are intermetallic compounds and are usually prepared from elemental precursors, which span a range of melting points, e.g.1855 1C for Zr and 232 1C for Sn.Given the presence of refractory elements, melting has historically been the favoured route for initial reaction.Most frequently this is done by arc-melting of elemental pieces in an argon atmosphere, although levitation melting has become more common in recent years. 60,61The main advantage of levitation melting is that the sample does not touch the sample environment (usually a Cu plate in arc melting) and the presence of eddy currents that improve the homogeneity of the melt. 48,60Arc-melting has been noted to contain different microstructures in different sections of the final ingot. 62A linked practical advantage of melt-based routes is that larger elemental pieces can be used avoiding issues with surface oxidation that can be significant in powder metallurgy.Nevertheless, powder-based routes can give good performing samples, but care has to be taken with precursors, and the lower temperatures also place limitations on grain growth. 59,63,64][67][68][69] After initial reaction it is common to anneal the sample, typically between 900-1000 1C, to equilibrate the sample and increase homogeneity.In the past couple of years however, what may be called 'out-of-equilibrium' processing has gained much prominence.Here, melting is followed by ball milling and rapid consolidation at high temperature and under applied pressure, often using spark plasma sintering (SPS), but no long duration heating is used. 47,48,70,71This can be a favourable route that maintains embedded nanostructures that would be annealed out at high temperatures. 72In some instances, most notably for ZrCoBi, the melting step is removed altogether, leading to an efficient synthetic protocol. 48Compared to the initial literature of the early 2000s, ball milling after HH phase formation has gained prominence and is now nearly always used.Reasons include promoting sample homogeneity and achieving a good particle size distribution that facilitates consolidation into dense ingots, whilst very small grain sizes are also beneficial for k l reduction. 36,37Using direct mechanical alloying of elemental precursors has been found to narrow the grain size distribution in a range of HH materials, 73 with HH and FH alloys forming during the milling stage. 48,73,74Ball milling needs to be carried out under an inert atmosphere, including the handling of the resulting fine powders, to protect the high surface area samples from oxidation.
Whether or not there is an annealing stage, all samples need to be in a highly dense form before property measurement.While samples can be cut from a melted piece with high density, 75,76 it is more common for the powdered sample to be consolidated using either hot-pressing or SPS, with much of the improvement in thermoelectric performance in recent years arising due to more careful consolidation of samples.In particular, SPS has become more prominent and enables very rapid densification. 77,78In conventional hot pressing, uniaxial pressure is applied to the sample in a graphite die which is Alternatively, these structures can be described as containing multiple face centred cubic sublattices that are offset with respect to each other, e.g. in the half-Heusler structure, X (red), Y (blue), empty and Z (purple) sublattices are offset by 1/4 along the body diagonal of the unit cell.This high symmetry arrangement, and possible partial occupancy of the second tetrahedral site, complicates reliable refinement of the atomic site occupancies in the half-Heusler structure.heated indirectly, for example using an induction coil. 79SPS applies a current directly through the graphite die, allowing for much more rapid heating of the sample.In some methods, such as flash-SPS, very short consolidation times on the order of minutes are possible. 80,81[84] 2.3.Electronic transport properties HH thermoelectric materials are small band gap semiconductors, with calculated band gaps typically ranging between 0.5-1 eV. 85he conduction band minimum typically lies at the X-point, while the valence band will either exist at the G-point or the L-point, and in some cases, both can be effectively degenerate, as illustrated for the ZrCoBi system (Fig. 2).In addition, the W-point can sometimes be close to the other valence band maxima.From this perspective, p-type materials have a larger potential for band-engineering to improve the thermoelectric properties, as degenerate band extrema allow for an increase in the density of states effective mass m Ã DoS À Á : where N V is the total carrier pocket degeneracy (where N b is the number of bands contributing and N o is the number of symmetry related pockets in the first Brillouin zone) in the electronic bandstructure and m Ã b is the band effective mass.Paradoxically, for good electronic performance a large m Ã DoS is desirable for large S, but a low m Ã b is desirable for high s.On balance a smaller m Ã b is better overall, yielding large m w . 86s shown in Table 1, m Ã DoS for HH materials can be quite large, in some cases exceeding 10 m e , with m Ã b typically between 1.5-3.0m e , which requires large amounts of doping to reach optimal n H up to orders of magnitude of 10 20 -10 21 cm À3 .HH materials typically have m H on the order of 10 cm 2 V À1 s À1 , with examples of single crystal ZrNiSn as large as 70 cm 2 V À1 s À1 and Ta 0.81 CoSb with m H as low as 1.4 cm 2 V À1 s À1 . 59,87Here, n H and m H are the carrier concentration and mobility from Hall measurements.
As collection of Hall data has become more frequent, so have detailed investigations into the dominant carrier scattering mechanisms.For semiconductors at low doping concentrations (n H o 10 17 cm À3 ) m H is expected to be limited by ionised impurity (II) scattering, which has a characteristic m H B T 1.5 behaviour. 88At high doping concentration (n H 4 10 18 cm À3 ), II scattering is minimal due to screening from the high number of charge carriers, and it is common to find scattering by phonons (acoustic phonon scattering; APS) to be dominant.This has a characteristic m H B T À1.5 temperature dependence for semiconductors and m H B T À1 in the metallic limit. 88A final intrinsic effect is alloy scattering (AS) due to size difference between alloying elements, with similar n H dependence to APS but a different characteristic m H B T À0.5 dependence. 89,90AS has been identified as the dominant carrier scattering mechanism in a number of HH phases, 91,92 with recent work indicating that this can be minimised by using alloying elements with similar sizes (minimising lattice strain). 93In other systems, the data suggests a dominant APS dependence, with the large S 2 s in HH materials attributed to a fundamentally weak coupling of electrons with acoustic phonons, leading to high m in spite of large m Ã DoS . 94lectronic transport can also be affected by ''extrinsic'' scattering contributions arising from the materials microstructure. 957][98] The most successful electronic transport models treat these materials as two-phase composites with bulk and GB phase contributions.The GB phase is usually modelled as a tunnel barrier (due to charge depletion in the GB region), accurately reproducing s(T) Fig. 2 First principles electronic bandstructures of XCoSb half-Heusler materials (including spin-orbit coupling contributions).As shown in (a) and (b), p-type doped ZrCoBi is expected to have carriers from the doubly degenerate L-point and G-point, leading to a substantial total degeneracy N V = 10.This is larger than many high-performance p-type half-Heusler phases, which have only L-point holes (N V = 8), such as ZrCoSb as shown in (c) and (d).By contrast, TiCoSb only has carriers from the triply degenerate G-point (N V = 3) as shown in (e) and (f) and is experimentally found to have the lowest power factor, S 2 s.Reproduced from Zhu et al.Nature Communications, 2018. 48n a range of thermoelectrics, including Mg 3 Sb 2 and SrTiO 3 .96,97 GB scattering is particularly relevant for p-type NbFeSb with a very sensitive processing dependence of S 2 s, caused by changes in s(T) 99 (also Section 3.1).It has also been observed in p-type ZrCoSb, where GB scattering is found to be substantial for low n H and small grain sizes (Section 3.4).100 The recent availability of ZrNiSn single crystals has enabled in-depth investigation of the electronic properties. This cludes collection of ARPES data on ZrNiSn that revealed a large intrinsic band gap B0.66 eV, comparable to DFT values (B0.5 eV) and unlike the B0.15 eV values usually found from transport data due to in-gap states resulting from interstitial Ni. 101 A comparison of single crystal and polycrystalline ZrNiSn 1Ày Sb y samples reveals a complex carrier scattering phase diagram.With increasing n H a crossover from II and GB scattering to APS and AS occurs near 300 K, whilst at high temperature the transition is from polar optical (PO) phonon scattering to APS and AS.These transitions occur near n H = 3-5 Â 10 20 cm À3 and are a consequence of the effective screening of II, GB and PO scattering at higher carrier concentrations.87

Thermal transport properties
Compared to other state-of-the art thermoelectric materials, which benefit from ultralow (o1 W m À1 K À1 ) k, 102,103 the HH family of materials typically have large S 2 s and relatively high k values.This is partly due to large k l as outlined above, but also due to large electronic (k el ) values, consistent with the large reported S 2 s values of these materials.Within the HH thermoelectric literature, experimental k l (T) data is typically treated using the Debye-Callaway model to describe the temperature dependence, with k l (T) defined as: 91,104,105 where v s is the velocity of sound, y D is the Debye temperature and t ph À1 is the total phonon scattering rate.Some publications also make use of the simplified Slack model. 104,106While work Table 1 Electronic bandstructure information of selected HH materials and experimental density of states effective mass (m Ã DoS , in electron mass units) obtained from Hall data as outlined in Appendix A. Here, N b is the number of bands at the valence or conduction band extrema, N 0 is the number of equivalent points in the Brillouin zone and N v is the total valley degeneracy  Table 2 Overview of 300 K velocity of sound data for well-studied HH compositions.Here, a HH is the lattice parameter.v l , v t and v avg are the longitudinal, transverse, and average velocities of sound.Y D is the Debye temperature and g is the Gru ¨neisen parameter.The latter two were obtained from v l and v t data as outlined in Appendix A. k Umklapp l is the 300 K lattice thermal conductivity for defect-free compositions, calculated using the Debye-Callaway model and assuming only Umklapp scattering.In many cases, experimental acoustic velocity of sound data is available (AC), other sources include calculations (DFT; materials project, MP) and for HfNiSn from mechanical measurements on calculating k l from the phonon bandstructure exists, [107][108][109][110][111][112][113] it is more typical to use approximations based on calculated elastic properties or experimental acoustic properties to estimate v s , y D and t ph .The most fundamental description of t ph considers Umklapp scattering, which can be written as: 105 1 where g is the Gru ¨neisen anharmonicity parameter, G is the Shear modulus and V 0 is the volume per atom.Assuming only Umklapp scattering and grain boundary scattering equivalent to grain size of 100 mm, k l (300 K) for HH materials can be calculated to be around 20-40 W m À1 K À1 as shown in Table 2 and Fig. 3. Experimental velocity of sound values obtained directly from acoustic measurements have been used where possible and used to calculate g and G (see Appendix).The volume per atom has been calculated using the reported lattice parameter (a HH ).
As the scattering rate, t U À1 increases, k l is expected to decrease, which is illustrated well in Fig. 3a, with the outliers being ZrCoBi and HfCoSb due to their low y D .TaFeSb also has low y D and is another outlier, but is not included in Fig. 3, since only the average velocity of sound was reported. 47In the plot of k l vs. g in Fig. 3b, there is a larger spread due to lower v t for TiNiSn and V 0.8+d CoSb (yielding a larger g) versus their second row analogues.This is ultimately corrected for by a larger G, giving a relatively smooth dependence in Fig. 3a.Naturally these values only represent an upper limit for k l , and the measured experimental values don't get close to 40 W m À1 K À1 .Within the Debye-Callaway model, phonon scattering can be expected to arise due to grain boundaries, the existence of point defects and electron-phonon interactions.Even small effects can have a large effect on k l .In ZrNiSn for example, introducing a point defect scattering term G = 0.02 is enough to reduce k l from 37 W m À1 K À1 to 17 W m À1 K À1 , which is the value expected from a 2% Hf impurity (which is typical in commercial elemental Zr).Small amounts of interstitials are also sufficient to embed this amount point defect disorder and have been linked to strongly suppressed k l values in X IV NiSn and X IV CoSb systems. 63,114n addition to k l , HH thermoelectrics usually have significant k el , due to the large carrier concentrations and bipolar transport at high temperatures.The total k el is given by: 115 Here, the first term is the Wiedemann-Franz (k WF ) electronic thermal conduction due to the flow of electron (s n ) and hole (s p ) currents, which simplifies to k WF = LsT if the Lorenz numbers (L n/p ) for electrons and holes are identical. 115The second term is the bipolar thermal conductivity (k bi ), which is linked to carrier recombination due to the presence of opposing electron and hole currents. 115Here S n/p are the partial Seebeck coefficients of the n-and p-type carriers.The measured S is given by: 115 k bi can be significant at high temperatures for materials with small band gaps, where a large number of minority carriers can be easily excited.From eqn (7), this also leads to a simultaneous degradation of S, leading to a rapid drop in the thermoelectric figure of merit, zT.In the context of the HH thermoelectrics, this situation applies to the n-type X IV NiSn materials, which have small band gaps, E g B 0.5 eV from DFT calculations and B0.15 eV from transport and optical measurements. 116Most p-types have much larger band gaps and are not strongly affected.

Overview of materials efficiency: zT and zT avg
The most commonly used metric to quantify the thermoelectric performance of a single material is the materials figure of merit (zT).A larger zT is expected to translate into higher efficiency in a thermoelectric device.However, a more appropriate metric of performance is the temperature average of zT over the gradient the TEG operates under.For this review we have used the following equation: 117,118 zT avg ¼ Fig. 4 shows zT and zT avg for some of the most promising HH materials found in the literature.The best n-type materials based on X IV NiSn achieve large zT at 600-900 K (Fig. 4a), reaching a peak due to a gradual decline in S(T) and increase in k bi (T) at high temperatures.The n-type X IV CoSb and X V0. 8 CoSb examples shown have a lower zT, but they maintain a growth in zT up to 1100 K, potentially making them more suitable at higher temperatures than the X IV NiSn systems.
The best p-type samples all show a gradual increase in zT at least up to 1200 K, at their best achieving zT B 1.5.The overall lower zT for the p-types means that where zT avg would approach 1 at 800 K for the best n-type materials, it takes until 1200 K to reach similar zT avg for p-type HH phases.From Fig. 4c and d, it is clear that the p-type X IV CoSb materials have been overtaken rapidly by the new wave of ZrCoBi and X V FeSb systems.

Recent developments in half-Heusler materials
In the following an overview of the main recent materials development will be given, starting with the new p-type materials, followed by good n-types and then other compositions of interest.
3.1.X V FeSb (X V = V, Nb and Ta) (p-type) In 2014, promising p-type thermoelectric properties were reported for (V 0.6 Nb 0.4 ) 1Àx Ti x FeSb, with moderately large S 2 s around 3 mW m À1 K À2 and quite small k l below 3 W m À1 K À1 , yielding peak zT 900K = 0.8 for x = 0.2. 119The good performance in this series was attributed to large m Ã DoS ¼ 10 m e and consequently large S(T), combined with significant X-site alloy point defect scattering suppressing k l .This 'heavy hole' behaviour is the main characteristic of the X V FeSb family (and for the other p-type HH materials) and arises due to a doubly degenerate L-point valence band maxima with N v = 8 (Table 1). 119,120Consequently, high doping levels between 10 21 -10 22 cm À3 are required to optimise the thermoelectric performance at high temperatures, which was not achieved in this initial report.It only took one year for the same group to report zT 1200K = 1.5 for the composition Fig. 4 Literature zT values of the best n-type and p-type half-Heusler thermoelectric materials.Panels (a) and (b) show zT and the average zT avg of n-type samples as a function of temperature. 142,148,172,174,179,183,191,242Panels (c) and (d) show zT and the average zT avg of p-type samples. 47,48,75,120,121NiSn systems have large low temperature zT, but their increase slows above 800 K due to onset of bipolar transport linked to the relatively small band gap.

Materials Advances Review
Open Access Article.Published on Nb 0.88 Hf 0.12 FeSb. 121This material removes all V from the system, resulting in a decrease in m Ã DoS to 6.9 m e reducing the doping required to reach the optimal S 2 s, with the increased m H resulting in a near doubling of S 2 s to between 5-6 mW m À1 K À2 .In addition to p-type doping the material, Hf substitution introduces X-site alloy point defect scattering, maintaining low k l .Further success was found in TaFeSb, for which the composition Ta 0.74 V 0.1 Ti 0.16 FeSb uses the same approach of combined carrier doping and phonon scattering using Ti, with a small fraction of V used to further suppress k l and push zT 973K from 1.39 to 1.52. 47he largest peak zT reported for X V FeSb materials comes from the (Nb 1Àx Ta x ) 0.8 Ti 0.2 FeSb series, with both x = 0.36 and 0.4 reaching zT 1200K = 1.6, 48As can be seen in Fig. 4c however, the magnitude of zT is lower for the (Nb 1Àx Ta x ) 0.8 Ti 0.2 FeSb series closer to room temperature (leading to a lower zT avg in Fig. 4d), perhaps due to the weaker mass disorder provided by Nb.
21][122][123] As first reported in 2016, careful processing allows for S 2 s 300K 4 10 mW m À1 K À2 to be achieved in Nb 0.95 Ti 0.05 FeSb, using a hot pressing temperature of 1100 1C to eliminate the parasitic electrical resistance from grain boundary scattering observed with lower pressing temperatures, as illustrated in Fig. 5. 99 This was later also demonstrated for the analogous systems with Zr and Hf dopants. 123In addition to the very large S 2 s, the k l (T) achieved in Ta 0.74 V 0.1 Ti 0.16 FeSb is extremely low for a HH material, starting at k l,300K = 2.3 W m À1 K À1 decreasing to k l,300K = 1.6 W m À1 K À1 , with mass disorder from Ti/V substitution proving to be effective at suppressing k l . 47 range of synthetic routes have been used to prepare X V FeSb HH materials.The initial reports on (V 0.6 Nb 0.4 ) 1Àx Ti x FeSb and Nb 1Àx Hf x FeSb used levitation melting, mechanical milling, SPS and a short annealing step at a low temperature of 750 1C. 119,121he TaFeSb based samples were prepared directly from the elements using ball milling (in a shaker mill), followed by direct current hot-pressing (similar to SPS) for only two minutes at 850 1C, with no subsequent annealing step. 47This report does not establish whether the HH phase forms during the milling step, or whether reaction takes place during hot pressing with the assistance of finely dispersed powder precursors.In spite of the well documented benefit on s RT of a 1100 1C densification temperature for Nb 1Àx Ti x FeSb (Fig. 5), highly doped Ta 1Àx Ti x FeSb samples seem to suffer no detriment from the relatively low 850 1C pressing temperature (although x o 0.06 samples seem to suffer from grain boundary effects). 47This suggests that these systems have a different grain boundary chemistry.Note that due to the higher melting point of Ta compared to Nb, a priori smaller grain sizes and a more pronounced impact on s RT might have been expected.However, this does not appear to be the case.To conclude, the X V FeSb materials are blessed with naturally excellent electronic properties and thermal transport, which can be easily suppressed using well established mass disorder techniques.

ZrCoBi (p-type)
In 2018, ZrCoBi 0.65 Sb 0.15 Sn 0.2 was reported to have the then largest zT 973K = 1.42 for any p-type HH phase. 48This composition benefits from a large N v = 10, which arises due to degenerate G and L (Table 1) valence band maxima, which is an inherent property of the stoichiometric ZrCoBi phase (Fig. 2).This convergence translates to large m Ã DoS , with the reported Hall data suggesting that m Ã DoS increases from around 9 m e to 13 m e with Sn substitution up to x = 0.2 in the ZrCoBi 1Àx Sn x solid solution (Table 1). 48,86A combination of Sn and Sb alloying is successful in suppressing k l and optimising the carrier concentration, with the parent composition already having a markedly lower k l than many HH phases due to the high mass of Bi and a low velocity of sound, v s (Table 2).As was the case for TaFeSb, 47 the ZrCoBi samples were prepared via ball milling the constituent elements and then direct current hot pressing for 5 minutes at 900 1C. 48In this report the HH phase was found to form directly after the milling step.No further experimental work has emerged on p-type X IV CoBi materials since, although two separate computational investigations of X IV CoBi materials also find near degenerate G and L point valence band maxima in TiCoBi Fig. 5 The effect of hot-pressing temperature on the electronic properties of Nb 0.95 Ti 0.05 FeSb.As seen in (a) hot-pressing at 1100 1C leads to over double the power factor, S 2 s at 300 K, relative to pressing at 950 1C.As and HfCoBi. 124,125It was discovered, however, that n-type ZrCoBi is also a promising thermoelectric material, reaching zT 973K = 1 for ZrCo 0.9 Ni 0.1 Bi 0.85 Sb 0.15 , again utilising a ball-milling to pressing synthetic protocol. 86As before Bi/Sb alloying is used to suppress k l , but interestingly Co/Ni alloying also reduces k l significantly, despite minimal expected mass and strain differences.The reported m Ã DoS ¼ 7 m e is quite large considering the low band degeneracy of the X-point, suggesting a large m b , consistent with the reported low mobility.In the previous year n-type ZrCo 1Àx Pd x Bi was reported with substantially lower zT, due to a combination of significantly larger k l and a much lower m Ã DoS ¼ 4:4 m e . 126The samples in this report were prepared using arc-melting, ball-milling and rapid uniaxial hot-pressing.

X IV CoSb (X IV = Ti, Zr and Hf) (p-type)
Before the rapid emergence of the high zT p-type X V FeSb and ZrCoBi materials, X IV CoSb materials were by far the most promising candidate p-type HH phases.zT 1073K = 1.0 had been achieved for Ti 0.2 Hf 0.8 CoSb 0.8 Sn 0.2 in 2012, with the same achieved in a 2013 follow up report on Ti 0.12 Zr 0.44 Hf 0.44 CoSb 0.8 Sn 0.2 , with both utilising X-site alloying and milling to suppress k l . 127,128Once again, these p-type compositions benefit from high m Ã DoS due to the doubly degenerate L-point valence band maximum (Table 1), although S 2 s did not exceed 3.0 mW m À1 K À2 in these materials, with a fundamentally low m H in this system.The modest S 2 s is a serious drawback for these compositions, as there are few clear-cut ways to improve the electronic properties and zT, whereas a range of strategies are available to suppress k l .In 2015 zT was pushed up to 1.15 at 973 K for the composition Ti 0.25 Hf 0.75 CoSb 0.85 Sn 0.15 , by optimising n H and X-site alloying, with phase segregation into Ti-rich and Hf-rich regions credited with further suppression of k l . 75Another important detail in this paper is while m Ã DoS $ 10:7 m e remains largely constant in the Ti 1Àx Hf x CoSb solid solution, m H is around 3 times lower in TiCoSb than HfCoSb, which makes the expensive element Hf key to high performance. 75tudies into the stability of the phase segregation revealed both zT and microstructure to remain unchanged upon repeated temperature cycling. 129,130][133][134][135][136][137] Since 2015, there hasn't been any increase in zT in the p-type X V CoSb materials, although work on these materials continues. 138,139n extensive investigation into the Zr 1Àx Hf x CoSb 1Ày Sn y system determined that the temperature dependence of hole mobility is consistent with alloy scattering (m H B T À0.5 ), in stark contrast to the X V FeSb materials, which have temperature dependences consistent with acoustic phonon scattering. 92he above reports generally make use of arc-melting to react elemental pieces, although a 2021 report eliminated the melting step, simply ball-milling elemental precursors and hot-pressing at 1000 1C. 140This study equalled the record with zT 975K = 1.1 for this family with the composition Ti 0.4 Hf 0.6 -CoSb 0.83 Sn 0.17 , with suppressed k l and S 2 s. 140 While this is a positive result, it also illustrates that k l can't be suppressed further using Ti without also suppressing the electronic performance.
3.4.X IV CoSb (X IV = Ti, Zr and Hf) (n-type) The X IV CoSb materials are unusual in that good zT values are possible in both n-and p-type doped compositions.The n-type materials have the X-point conduction band minimum shared amongst all 18 electron HH phases (Fig. 2).Despite the fact that X IV CoSb materials have long been known to be intrinsically ntype, 141 reports on competitive zT have only emerged in the last 4-5 years. 93,142Initially, Ni substitution was used to dope Zr 0.5 Hf 0.5 CoSb, with zT 1073K = 1 achieved in Zr 0.5 Hf 0.5 Co 0.9 Ni 0.1 Sb, with moderate S 2 s vales exceeding 3.25 mW m À1 K À2 and once again suppressed k l due to X-site alloying. 142A m Ã DoS $ 6 m e was confirmed in subsequent reports on other n-type compositions, with (Zr 1Àx Hf x ) 1Ày Nb y CoSb having m Ã DoS $ 6:5 m e . 93,100Although the Nb doped materials don't exceed zT = 1, the initial report used Callaway analysis to show that the X-site alloying in these materials suppresses k l without causing any reduction in m H .This is consistent with the earlier observation that minimising the size mismatch between dopant/alloying elements helps maintain good m H values, 93 whilst mass disorder effectively reduces k l .These studies have also shown that n-type TiCoSb is substantially more promising than its p-type counterpart, 139 with the composition Ti 0.6 Hf 0.3 Ta 0.1 CoSb reaching zT 973K = 0.9. 143 subsequent study on (Hf 0.3 Zr 0.7 ) 0.94 Nb 0.06 CoSb established that s 300K can be heavily suppressed by grain boundary scattering, introduced by intense ball-milling used to reduce grain size. 100his leads to semiconducting behaviour near 300 K, while at high temperatures degenerate semiconducting behaviour is observed.Hence, as was seen for NbFeSb, care must be taken when processing HH phases to prevent grain boundary scattering of charge carriers.At present all n-type X IV CoSb materials reported use a melting, milling and pressing procedure, but this may simply reflect the novel nature of these compositions and a lack of detailed processing studies.

X IV NiSn (X IV = Ti, Zr and Hf) (n-type)
The X IV NiSn compositions are very much the original HH thermoelectric, 17,18,20,21,[24][25][26][27][144][145][146][147] with the 2005 report of zT 700K = 1.5 driving a significant amount of interest in HH thermoelectrics. 148 As sown in Fig. 4, zT values from this report have never been fully matched, contrary to the expected evolution of material optimisation, which sees a gradual increase in zT over the years.
The X IV NiSn phases have some unique properties relative to the other HH alloys.The X IV NiSn materials have the lowest reported m Ã DoS $ 3 m e amongst the high zT materials, which arises due to a combination of low N v = 3 and low m Ã b $ 1:7 m e (Table 1).This means that lower doping levels are required to optimise S 2 s, and in several instances the best performance comes for B1-2% atomic substitution, [148][149][150] contrasting with 15-20% required for the p-types.
2,[151][152][153][154][155][156][157][158][159][160][161][162][163][164][165][166] The interstitial site is the vacant 8-coordinate (tetrahedral coordination to X and Z) site in the material (sometimes referred to as the Wyckoff 4d or tetrahedral site), which is fully occupied in the FH structure (Fig. 1). 49Even stoichiometric X IV NiSn materials commonly form with a few percent occupation of the interstitial site, with an X IV /Sn binary phase formed to compensate. 41,43,45,57,72,160reparing compositions with intentional metal excess, X IV Ni 1+x Sn is possible, although there is a limit to x, past which a FH phase begins to form. 151,154,155Both interstitial Ni acting as point defects or FH nanoinclusions have been claimed to be useful for scattering phonons. 63,72The pursuit of FH nanoinclusions was a major focus for X IV NiSn investigations in the early 2010s. 14The presence of interstitial Ni is also cited as reducing the observed electronic band gap. 71,116-Site phase segregation.Initial success in reproducing the large reported zT values came from work focusing on X-site phase segregation.This is based on the limited mutual solidity of Ti and Zr/Hf, whereas Zr and Hf with a more similar size are fully miscible. 167Reports by Schwall and Balke, and Populoh et al. yielded zT 830K = 1.2 with the low k l underpinning the good performance linked to phase segregation. 38,4069][170][171] Phase segregation is not simply a case of the formation of two distinct phases, with Ti alloyed compositions containing several HH phases discernible by diffraction.This can be ascribed in large part to low expected diffusion rates of the X IV site atoms, leading to regions with kinetically trapped compositions resulting from the melting stage. 39,168Interestingly, SEM and SXRD data for Ti 0.5 Zr 0.25 Hf 0.25 -NiCu y Sn prepared by a powder route (never exceeding 850 1C) shows an increasing homogenisation on the X IV site with increasing Cu content, 64 which was attributed to the presence of a molten Cu-Sn phase that improves the reaction kinetics during synthesis.
The approach which was able to reduce k l the most used a convoluted synthetic protocol consisting of arc-melting, induction-melting, annealing, an undisclosed ''densification aid'', ball-milling and finally hot-pressing. 172This work also aimed to exploit X-site phase segregation by initial heating above the spinodal line and subsequent heat treatment just below to force nanoscale segregation.However, the paper itself does not provide direct microscopy confirmation.Nevertheless, the composition Ti 0.5 Zr 0.25 Hf 0.25 NiSn (+DA) was reported to have a very low k l B 1.8 W m À1 K À1 at 300 K, while maintaining a large S 2 s, enabling zT 823K = 1.5. 172Outside of this report, the group is able to consistently achieve zT 800K 4 1 with the assistance of a ''densification aid'' that is considered to improve electrical contacting of the grains. 173,174In the composition Ti 0.5 Zr 0.5 NiSn for example, the addition of the ''densification aid'' increases S 2 s at 700 K from B2.5 W m À1 K À2 to B4 W m À1 K À2 , and coupled to a low k l this enables zT 4 1. 174 Interstitial metals and FH inclusions.Investigations into the nature of excess Ni in these materials has progressed since our 2014 review. 146][177] In well-annealed samples (T 4 800 1C), the solubility of interstitials is highest in TiNiSn (x max = 0.06-0.08 at 900 1C), 160 whilst x max r 0.03 for ZrNiSn and HfNiSn. 178The solubility of Ni interstitials increases with temperature, consistent with the entropy stabilised nature of this defect. 53Full-Heusler (nano)inclusions can be stabilised by (rapid) cooling from the melt, but exposure to T 4 800 1C leads to the gradual dissolution of FH inclusion, 72 and conversion to interstitials and distinct (macroscopically segregated) FH phases that are discernible in diffraction.Due to the link to processing (i.e.cooling from the melt), the experimental 4d site occupancies can far exceed equilibrium values, with electronic properties recorded for ZrNi 1+x Sn with x 4 0.1. 114,161Unpublished neutron powder diffraction data from our group confirms x r 0.18 in arc-melted ZrNi 1+x Sn ingots, although this decreases to x = 0.03 when annealed at 900 1C (in agreement with the phase diagram), with the excess Ni forming FH regions visible by diffraction.The trapping of large amounts of excess Ni (in out of equilibrium samples) likely explains most of the variations in reported TE properties for nominally similar samples, but with different processing (thermal) histories.The impact of Ni interstitials and inclusions is profound with strong reductions of k l and electron mobility evident. 72,160roperty measurements have confirmed the significance of interstitials on the thermoelectric properties, with excess Ni suppressing k l by up to 70% in the TiNi 1+x Sn and ZrNi 1+x Sn system. 72,160,161,175Callaway analysis demonstrates that Ni/Cu interstitials have a similar impact to vacancies with additional disorder contribution due to bond formation. 63,160For the Ti 1Àx Ta x Ni 0.92 Sn system, the intentional Ni deficiency (with the aim of obtaining stoichiometric TiNiSn) leads to an increase in both k l and m H , with the improved electronic properties leading to a net increase in zT (Fig. 6). 71At low temperatures, the Ni deficiency leads to a substantial increase in s, while at higher temperatures S is enhanced, leading to a minimum of 35% increase in S 2 s(T) across the entire temperature range.
Using appropriate processing, excess Ni can also be stabilised as FH (nano)inclusions in the HH matrix, acting an extrinsic source of phonon and electron scattering.Chai et al. reported success in preparing composite ZrNi 1.1 Sn with significantly improved thermoelectric properties relative to the stoichiometric composition, attributed largely to a dispersion of nanoscale FH inclusions. 44,72,153,167DTA analysis suggested the FH phases 'dissolve' into the HH matrix above 800 1C, and reform with a smaller domain size when cooled. 72These nanoscale FH inclusions suppress k l by up to 40%, leading to a B40% increase in zT max from 0.54 to 0.75, compared to the first heating cycle.In addition to these melt-based routes, the Poudeu group has reported extensively on nanocomposite HH samples prepared using a high-temperature solid-state route. 42,156,157Carrier filtering was reported to improve S 2 s and reductions in k l were observed, with a highest zT = 0.8 at 800 K.
Recently it was found that W nanoinclusions are highly effective in reducing k l by 20% while simultaneously enhancing S 2 s by up to 33%.Hf 0.6 Zr 0.4 NiSn 0.99 Sb 0.01 + 5 wt% W has peak zT 773K = 1.4. 179This would seem to be a successful implementation of the nanoinclusion effect (i.e. carrier filtering and reductions in k l ) that had been previously considered to be possible with nanoscale FH inclusions.The introduction of W in these samples was achieved by ball milling W nanoparticles into premade HH samples, followed by SPS for densification.The cycling stability was confirmed for 10 cycles up to 1073 K, suggesting the composition is sufficiently stable for application.
While m Ã DoS in ZrNiSn and HfNiSn have been well established as being close to 3.0 m e , the value for TiNiSn based materials is less clear. 114Data on Sb-doped TiNi 1+x Sn samples from 2018 show a similar m Ã DoS $ 2:8 m e , which seems to be robust up to 700 K. 175 Data from high performance Ti 1Àx Ta x Ni 0.92 Sn samples however are more consistent with m Ã DoS $ 4:5 m e . 71Given that both reports place m H between 20-30 cm 2 V À1 s À1 at room temperature, there does not appear to be a significant increase in m Ã b (which would lead to a significant decrease in m H ). Considering eqn (3), the increase in m Ã DoS should be attributed to an increase in N v .Our reported data on TiNiCu 0.075 Sn, which uses interstitial Cu to n-type dope the material, gives a relatively large values of B5.5 m e . 63Similarly Hall data for the Ti 0.5 Zr 0.25 Hf 0.25 NiCu y Sn series conform to m Ã DoS ¼ 4:1 m e , including the undoped sample. 64Some data in the Ti 1Àx Hf x NiSn 1Ày Sb y series also suggests m Ã DoS ¼ 3:9 m e . 180While the large m Ã DoS in the Ti 1Àx Ta x Ni 0.92 Sn system was not addressed, first principles calculations on the interstitial Cu-doped materials suggest the appearance of a second conduction band minimum in the electronic bandstructure, supporting the idea that the increase in m Ã DoS is caused by to an increase in N v . 64iven the high volume of publications on the X IV NiSn system, a number of protocols outside of melting, ball-milling and hot-pressing/SPS exist, with TiNiSn being the target of most experimentation.Microwave synthesis has reported for TiNiSn, achieving a peak zT 740K = 0.45. 65In 2019 Zilber et al. successfully grew TiNiSn single crystals from a Sn flux, with lower n H and larger S than typically reported in polycrystalline samples. 181Similarly ZrNiSn 1Ày Sb y crystals can be grown and extracted by centrifuging, with improved m H compared to high quality polycrystalline samples. 87elf-propagating synthesis for exothermic processes allows for the heat produced from initial reaction to drive the reaction to completion, as has been demonstrated for ZrNiSn. 66,69In a custom pressing setup, the heat released from the reaction (initiated by Tungsten electrode) was enough to densify the pellet, with a T max = 1668 K achieved by the self-propagating reaction. 66he large literature pool also allows for comparisons to be made in how the HH phases form, which is important when planning microstructure and defects.ZrNiSn and HfNiSn are known form congruently from the molten state, while TiNiSn prefers to form the TiNi 2 Sn FH phase. 155,176,177Similarly TiNiSn decomposes when heated above 1450 K.The rapid solidification achieved by melt-spinning is claimed to be able to suppress formation of TiNi 2 Sn from the molten state. 182It remains the case that melting, milling and hot-pressing/SPS give the best thermoelectric performance.
Overall, for the X IV NiSn compositions, it remains challenging to optimise the thermoelectric properties.This is related to the tendency for significant deviations from nominal stoichiometry, phase segregation and the small electronic bandgap, which leads to significant bipolar transport at high temperatures.Elimination of interstitial Ni is desirable as this would increase the electronic bandgap, hence reducing the minority carrier conduction and improving the high temperature zT.

X V0.8 CoSb (X V = V, Nb and Ta) (n-type)
4][185][186][187][188] As a stoichiometric phase, these materials are expected to be metallic with the possibility of ferromagnetism. 189In 2016, the material VCoSb was synthesised with S(T) significantly larger than what is typical of a fully metallic sample, although n H was quite large at B1.6 Â 10 22 cm À3 . 185The properties of this phase and the subsequently investigated NbCoSb and TaCoSb phases were discovered to be due to X-site vacancies, 187,188 which reduce the overall electron count without depleting electronic states in the valence band, which is easily understood through the valence electron counting X n+ [YZ] nÀ description introduced earlier.To reach a theoretically semiconducting state one would need to reach the composition X V0.8 CoSb (Fig. 7).Each of the X V CoSb phases forms with a different vacancy fraction, with the V phase forming preferentially as V 0.87 CoSb, the Nb phase as Nb 0.85 CoSb, while the Ta phase forms close to being 'fully' vacant at Ta 0.81 CoSb. 59n H is dependent on the vacancy fraction as has been demonstrated for the Nb 0.8+d CoSb system, 183 so V 0.87 CoSb is highly degenerate while Ta 0.81 CoSb is close to being a semiconductor. 59There is some level of control that can be achieved, as demonstrated in Nb 0.8+d CoSb, which can be doped between 0.5-6.5 Â 10 21 cm À3 by varying the nominal vacancy fraction. 183One complication to doping control is the dependence of d on the annealing temperature. 190elative to the X IV NiSn materials, m Ã DoS for these materials is quite large, ranging from 10 m e for V 0.87 CoSb to 5.9 m e for Ta 0.81 CoSb (Table 1). 59,183,191The lower magnitude of m H (B4 cm 2 V À1 s À1 for Nb 0.8+d CoSb) may point to an increase in m Ã b , but calculated bandstructures for both NbCoSb and Nb 0.8 CoSb suggests more bands can contribute than the singly degenerate X-point minimum that is commonly observed (Fig. 7). 188Assuming a doubly degenerate conduction band as shown in Table 1, m Ã b for Nb 0.8+d CoSb is calculated to be B2.33 m e , which is a large value compared to n-type X IV NiSn.
One unusual behaviour in this family is the very low k l in V 0.87 CoSb below 2 W m À1 K À1 , which would expected to be larger than the Nb and Ta phases due its lower formula mass. 59,191The relative acoustic behaviour of the V 0.8+d CoSb and Nb 0.8+d CoSb materials have been confirmed to be similar, with no significant effect caused by vacancies observed through experiment, which would point to a reduced k l , with measured v s in the Nb 0.8+d CoSb varying by no more than AE2% between 0 o d o 0.2. 183,191As of yet there is no clear explanation for this extremely low value.One can note that the experimental acoustic properties point to a large g for these materials (Table 2), which correlates with strong Umklapp scattering and low k l as shown in Fig. 3.3][194][195] Over large distances the vacancies are disordered, but at the nanoscale the vacancies are short-range ordered (SRO), offering a route to reduce k L . 192Furthermore, Monte Carlo simulated vacancy ordering coupled to DFT supercell calculations points to widening of the electronic CBM (increase in m Ã b ) relative to the ordered Nb 0.8 CoSb and the presence of Co-defect states in the band gap, with electron diffraction simulations confirming existence of SRO in real samples. 195A more detailed discussion on the vacancy ordering in these materials can be found in the 2021 review by Xia et al. 12 The majority of reports on the X V0.8 CoSb materials have utilised a similar synthetic protocol, with melted elemental pieces being optionally ball-milled and then consolidated by hot-pressing/ SPS. 183,184However, a powder route consisting of solid-state reaction and hot pressing achieves similar performance. 59Finally, a recent report on V 0.855 Ti 0.1 CoSb used a direct ball milling and consolidation approach and achieved zT values larger than other reported V 0.8+d CoSb samples. 191This suggests that for the group V based materials, these relatively straightforward powder based routes may be more feasible than for X IV NiSn, where the best performing samples are invariably prepared using melting routes.

Other HH phases
The best performing new and existing materials have been described above, but exploration of new phases has turned up some interesting results.7][198] The bandstructure of NbCoSn appears contain a doubly degenerate X-point conduction band minimum, giving N v = 6, which appears to be partially responsible for m Ã DoS ¼ 6:1 m e in the NbCoSn 1Àx Sb x series (Table 1). 197,198At present the largest zT is found for X-site alloying between Nb and Ta, yielding zT 1000K = 0.75 for Nb 0.4 Ta 0.6 CoSn 0.94 Sb 0.06 . 196Better low temperature performance has been observed when using Pt as both n-type dopant and phonon scatterer in NbCo 0.95 Pt 0.05 Sn, due to the large mass difference between Co and Pt. 197While it would be useful to use V for X-site alloying, the VCoSn phase does not appear to readily form. 131While there have been reports of the phase being observed by powder diffraction, the evidence presented is less than convincing. 199,200It has been suggested that this composition is difficult to reach through normal processing routes, but might be made through non-equilibrium processes. 201Finally, p-type doping has also been demonstrated using scandium doping in Nb 1Àx Sc x CoSn, however these samples have relatively poor performance, with the largest zT 879K = 0.13 reported for Nb 0.95 Sc 0.05 CoSn. 2024][205][206] As synthesised, these materials are p-type and are characterised by spontaneous Ni deficiency (as much as 15%). 207They also have small band gaps (E g B 0.3 eV), leading to substantial bipolar transport at low temperatures, ultimately limiting the maximum achievable zT values.A large study comparing R = Sc, Dy, Ho, Er, Tm and Lu, confirmed the intrinsic p-type behaviour with semiconductor like S(T) and r(T). 205Typical peak S 2 /r = 1-1.5 mW m À1 K À2 for the RNiSb materials with highest reported zT B 0.3 at 650 K. 205,208,209 Careful analysis of the bipolar transport, suggested much higher electron than hole mobilities. 204Subsequent attempts at n-type doping exploiting Te in ScNiSb 1Àx Te x yielded large S 2 /r = 4 mW m À1 K À2 , demonstrating the potential of n-type RNiSb systems.Substitution of Te is also correlated to an increasing m Ã DoS , providing another example of the significant impact of a doping element on the underlying bandstructure. 206o zT values were reported, but the chemical flexibility for substitution on the X site and potential to manipulate the Ni deficiency (concentration and potentially SRO through heat treatment), suggest that these materials are promising for further exploration.
Another interesting development is the exploration of ''double HH'' phases, which combine an equal mixture of 17 electron and 19 electron HH phases. 112For example the model material proposed in 2019 was [TiFeSb] 0.5 [TiNiSb] 0.5 , each of which forms a HH-like phase independently.Experimental realisation of [TiFeSb] 1Àx [TiNiSb] x was successful, showing that VEC o 18 were p-type and VEC 4 18 were n-type as would be expected, and that isoelectronic alloying using Hf could be used to reduce k l . 210ome unusual points emerged, such as the large reduction in k l just above and below x = 0.5 and massive apparent m Ã DoS values up to 23.8 m e , which was later confirmed by a separate group. 211he emergence of defective HH phases and double-HH phases has provided an extended range of potential compositions to explore, which has allowed for the application of computational screening, beyond the common style of reporting the first principles properties of stoichiometric XYZ HHs. 112,212The emergence of these phases coincide with the development of effective band structure techniques, that allow supercell bandstructures to be compared to stoichiometric parent phases. 213,214

Recent developments in half-Heusler thermoelectric generators
1][222][223][224] Most of this published work is from research institutions but there has been industrial work producing HH materials and generators, most notably from GMZ energy in the USA, and the Isabellen Hutte Heusler company and Vacuum Schmelzen in Germany.A vital aspect in translating materials to TEG performance is the electrical and thermal contacting.In particular, the requirements for electrical contacting are strict in order to maintain as much of the intrinsic performance of the HH materials.The effective zT for a ''contacted'' material is given by: 4 This leads to strict criteria on permissible contact resistances, e.g. for a L = 2 mm height leg, r c o 5 mO cm 2 to keep losses o5-10%.In addition to a low contact resistance, the contacts also need to be stable under operating conditions, i.e. no chemical reaction with the TE materials, often necessitating the use of barrier layers. 4In terms of reviewing the literature, comparison of published results on TEG modules is difficult as they depend on dimensions of the legs, measurement conditions, heat flow through the legs and minimising heat losses.An overview of reported power densities and efficiencies for a range of HH based TEG modules is given in Table 3.One of the first papers to address upscaling and device manufacture originated from the Fraunhofer Institute in 2014. 215This work used vacuum melting and casting to prepare kilogram scale ingots of optimal n-type Zr 0.4 Hf 0.6 NiSn 0.98 Sb 0.02 and p-type Zr 0.5 Hf 0.5 NiSb 0.8 Sn 0.2 .The HH materials had zT B 0.8 (n-type) and B0.5 (p-type) at 773 K, respectively.TEG modules containing 7 n/p-couples were fabricated using Cu contacts and a brazing process.These were exposed to repeated thermal cycling up to 873 K and thermal shock without degradation of performance.The power density was 3.2 W cm À2 (area of thermoelectric) from a 527 K gradient, with the cold side near room temperature.The efficiency was estimated to be B5% with an average TEG module ZT B 0.4.
Zhang et al. demonstrated a 1 kW TEG for use in automotive applications. 216The HH materials used were based on milled nanostructured X IV NiSn and X IV CoSb with zT = 1 (773 K) and zT = 0.9 (973 K), respectively.The production process and different stages in the TEG module development are illustrated in Fig. 8.A Ag-Cu brazing alloy was used to bond the legs directly to the Cu leads through heating in vacuum at 825 1C.Unicouple generators produced 5.3 W cm À2 (500 K gradient), while a module consisting of 28 couples generated 4 W cm À2 (500 K gradient).The reduced power density of the module is caused by parasitic losses, attributed to thermal losses between the inert atmosphere steel encapsulation and TEG device (Fig. 8).The final TEG that was embedded in the exhaust stream of a heavy-duty vehicle used an impressive number of 400 TEG modules and could generate 1 kW electrical output (power density B2.8 W cm À2 ) with a 2.1% heat-to-electricity efficiency (340 K gradient, with an average 823 K exhaust temperature).
Reflecting the development of the new p-type compositions, a shift has occurred towards X V FeSb as the counterpart to X IV NiSn.The initial materials development study by Fu et al. reported a power density B2 W cm À2 from an 8 couple TEG module (650 K gradient). 121A subsequent study geared towards improving the device performance yielded a measured 2.1 W cm À2 power density and 8.3% efficiency from a 655 K gradient. 218imulations as part of this work suggest a maximum efficiency of 10.5% is possible, but this would only be possible at with a reduction of power density.The work reporting ultrahigh 300 K power factors in NbFeSb translates to a very large TEG power density B20 W cm À2 (500 K gradient), albeit measured on a single contacted bar of material (and not in a TEG geometry). 99TaFeSb shows similar high power densities based on measurement of a single leg. 47Further work on NbFeSb in a unicouple device using a high-performance X IV NiSn composition as the n-type, enables a device power density of 6.2 W cm À2 (500 K gradient), outperforming a comparable unicouple device with a X IV CoSb p-type leg (5.5 W cm À2 ). 122Furthermore, the NbFeSb material used in that study did not contain Hf and is hence only about 1/6 of the cost of the alternative X IV CoSb p-types (typical 30-70% Hf on the X-site), offering a significant cost saving.Subsequent improvements in device contacting and geometry enabled a power density 8.6 W cm À2 at 6% efficiency under a 570 K gradient. 224his is a clear demonstration that the outstanding performance of NbFeSb can be translated to working TEG devices.
Self-propagating synthesis is a promising route to produce large quantities of materials that has recently been applied to HH materials. 67,68Using this processing, 200 gram batches of Zr 0.5 Hf 0.5 NiSn 0.985 Sb 0.015 and Zr 0.5 Hf 0.5 CoSb 0.8 Sn 0.2 were prepared and consolidated using conventional hot pressing. 217hese were then fabricated into TEG modules containing 8 unicouple pairs.This study covered a range of leg geometries, where the ratio of leg height over area (L/A) can be used to favour either efficiency (increase L/A) or power density (decrease L/A).Limiting measured values in a 680 K gradient are B2.4 W cm À2 (B8% efficiency) and B1.7 W cm À2 (9.5% efficiency).Simulations suggest an upper limit of B3.1 W cm À2 and 10.5% for the power density and efficiency, respectively. 217However, it is worth noting that the performance of these modules is limited by relatively large contact resistances, B30 mO cm À2 for both n-and p-type legs.A final noteworthy study focused on TiNiSn (hence avoiding the use of Hf) as the n-type material and NbFeSb as the p-type.Despite a low zT 873K B 0.4 for both n-and p-types, a fabricated TEG with 16 pairs showed a respectable power density of B3.1 W cm À2 from a 575 K gradient, reflecting the large S 2 s of the HHs. 219

Conclusions
Significant progress has been made in the development of HH thermoelectrics with the emergence of new high-performing ptype materials based on X V FeSb and ZrCoBi, and further development of existing n-types based on X IV NiSn.Application of these materials in TEG modules has led to the demonstration of promising power outputs and efficiencies.Looking ahead there are a number of areas that deserve attention.Further work on translating materials into TEGs is needed, including underpinning work on electrical and thermal contacting and stability studies.A related area is the development of scalable synthetic routes to mass produce HH alloys with good performance.These more applied activities should be underpinned by further materials development work as there continues to be significant potential for further improvements.One area of interest is the elimination of parasitic effects, for example grain boundary resistances, enabling the intrinsic materials performance to be extracted.Another area is new design concepts, in particular building on the much-improved understanding of the electronic structure, [225][226][227] and the emergence of new classes of more complex HH alloy compositions. 112The complex HH alloys behave opposite to the conventional XYZ compositions: they have low thermal conductivities but also relatively poor power factors.Uniting large power factors and low thermal conductivities remains the key challenge in HH materials development.The largely harmonic bonding limits the strength of phonon-phonon scattering, leaving structural disorder as the most plausible route to achieve low k l .A possible interesting route is to focus on grain boundary engineered composite samples, where the boundary region is exploited to suppress phonon transmission and reduce k l .This has been demonstrated for graphene decorated skutterudites and titanium oxides, 228,229 whilst dislocation networks have been exploited in Bi 2 Te 3 . 230In principle, these approaches can be applied to HH alloys, and the key challenge is to find appropriate additives and processing routes to manipulate the microstructure.In terms of homogenous materials, advances might come from the design of materials with favourable bandstructures, enabling large power factors at lower doping levels, hence limiting the electronic thermal conductivity. 227One possible route to enhance S without affecting s is the introduction of resonant states, leading to a ''spike'' in the electronic density of states.This concept has been explored for X IV NiSn with some evidence of a favourable impact on the power factor and thermoelectric performance. 231,232As already remarked, the similar conduction band structures for XYZ HH compositions limits opportunities for band engineering in the ntypes.In the p-types, there is more diversity and hence more opportunity for bandstructure engineering.ZrCoBi and NbFeSb are both systems that naturally benefit from large valley and orbital degeneracies.Alloying has been widely explored to reduce the lattice thermal conductivity but can also lead to substantially different bandstructures.Recent work using unfolded supercell calculations has shown that new bands can emerge in substituted HH compositions (e.g. in vacancy systems, 170 for interstitials 58 and complex compositions 233 ) and this is another possible route for performance improvements.This is an area where theory could lead experiments by identifying new compositions with promising electronic bandstructures.
To conclude, great progress has been made over the past decade, driven by improvements in materials preparation and through better understanding from first principles and semiempirical modelling.Many avenues of research remain open for exploration in the future and some of these will undoubtedly lead to better materials performance.The current generation of materials is already good enough for niche applications in thermoelectric waste heat recovery.Improved performance, or increased penalties for carbon emissions, may lead to increased industrial interest for large scale application in thermoelectric power generation.
where V 0 is the volume per atom.This equation can be found in a number of different forms, expressing V 0 in terms of the number of atoms, using Plancks constant and moving around the 2p factor, or even moving v s inside of the brackets.The final elastic property required to estimate Umklapp scattering is the Gruneisen anharmonicity parameter (g), which quantifies bond anharmonicity.Different g can be found in the literature, but given the information we have available the acoustic g is used, calculated using the equivalent expressions: 113,245 g ¼ where n is the Poisson ratio.A more involved procedure to determine g uses the phonon bandstructure to calculate g(q), which can be done through DFT. 247ithin the SPB model, HH materials are typically treated with acoustic phonon scattering as the dominant carrier scattering mechanism.For clarity, the SPB model uses the following equations with only acoustic phonon scattering: 91,248 F i ðZÞ ¼ (S12)

Fig. 1
Fig. 1 Unit cells of the (a) XYZ half-Heusler, (b) XY 2 Z full-Heusler, (c) YZ zinc blende and (d) XZ rock salt structure types.These are all based on a face centred cubic lattice of Z atoms (purple) but with different fillings of octahedral (red, X atoms) and tetrahedral (blue, Y atoms) sites.Alternatively, these structures can be described as containing multiple face centred cubic sublattices that are offset with respect to each other, e.g. in the half-Heusler structure, X (red), Y (blue), empty and Z (purple) sublattices are offset by 1/4 along the body diagonal of the unit cell.This high symmetry arrangement, and possible partial occupancy of the second tetrahedral site, complicates reliable refinement of the atomic site occupancies in the half-Heusler structure.

5 3.13 92 a
Based on 19 electron NbCoSb with potentially larger N b in the defective Nb 0.8 CoSb composition.b Cu interstitials may increase band degeneracy at the conduction band.c Data suggests a higher N v for this sample, see Section 3.5.d Data suggests that m Ã DoS increases with Sn substitution, as discussed in Section 3.2.

Fig. 6
Fig.6The effect of nominal Ni deficiency in the Ti 1Àx Ta x Ni 1Àd Sn system, in samples with identical carrier concentration, n H .As seen in (a), nominal Ni deficiency causes a substantial increase in m H at 300 K and enhances S(T) at high temperatures.This leads to a significant increase in power factor, S 2 s as seen in (b).Reproduced from Ren et al. 2019, Advanced Electronic Materials.71

Fig. 7
Fig. 7 Calculated electronic band structures of (a) 19 electron NbCoSb and (b) 18 electron Nb 0.8 CoSb.NbCoSb is metallic, although the gap in the lower energy states is typical of semiconducting half-Heusler materials, with an L-point valence band maximum and an X-point conduction band minimum.Nb 0.8 CoSb contains these features but has additional conduction band states at the U-point, which could be responsible for the increased m Ã DoS compared to NbCoSn.Reproduced from Zeier et al. 2017, Chemistry of Materials.188 where Z is the reduced Fermi energy, n H and m H are the Hall concentration and Hall mobility determined directly from Hall measurements.Z is typically solved numerically from S. m Ã DoS is typically determined from a series of samples at fixed temperature by plotting S vs. n H in a Pisarenko plot and fitting for a fixed m Ã DoS , although a single datapoint can be used.The weighted mobility (m w ) is a measure of the fundamental mobility of a material independent of n H and can be defined as:248,249 m w ¼ m 0 where m 0 is the n H = 0 limit within the SPB model.From this relationship, multiple highly dispersive bands are preferable to a single low dispersion band with the same m Ã DoS .A relationship compatible with S(T) and s(T) is expressed as:249

Table 3
Reported power densities from half-Heusler TEG module fabrication studies.Where reported, the efficiencies are on the order of 5-10%