Recent progress in phosphide materials for thermoelectric conversion

Abstract

Thermoelectric materials allow for the interconversion of heat and electricity. Rapid progress has been made in materials performance but there is a continuing need for the exploration of new materials and disruptive thermoelectric effects. Metal phosphides are currently gaining increasing attention as potential thermoelectric materials. Structural complexity often leads to low thermal conductivity despite the low atomic mass of P, which underpins promising thermoelectric figures of merit, zT. Despite the ability of phosphorus to form strong covalent bonds, including polyanion bonding, a key challenge remains to embed better electrical properties and large power factors. A highest zT = 0.9 has been observed in n-type Cd₃P₂, whilst promising performance under applied fields has been observed in topological materials, e.g. in the Dirac metal TaP. In this review, we summarise recent materials developments and analyse some of the underlying thermoelectric parameters of metal phosphide materials.

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Robert J. Quinn

Dr Robert Quinn received his MChem degree in Chemistry from the University of Oxford (Exeter College) in 2017 and received his PhD from Heriot-Watt University in 2021 following study on Half-Heusler thermoelectric materials. He is currently a research associate on a Leverhulme Trust funded project, exploring metal phosphide phases as potential thermoelectric materials.

Jan-Willem G. Bos

Dr Jan-Willem Bos is an Associate Professor in Energy Materials at Heriot-Watt University. His research interests are in the interplay between composition, structure, and functional properties of extended solids. A large part of his current research is focused on the design of thermoelectric materials, including half-Heusler alloys and the emerging metal phosphides that are the topic of this review.

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1. Introduction

Thermoelectric technology can be used in power generation, mW-scale ambient heat scavenging and in thermal management, where the benefits of durability and absence of moving parts outweigh high costs and modest efficiencies.^1,2 A good example is the radioisotope thermoelectric generators (TEGs) used by NASA to power satellites and rovers. There continues to be a need for improved thermoelectric materials, based on sustainable elements and processing, lowering costs, and increasing power outputs and cooling performance of devices. For a single material, the thermoelectric performance is quantified by its figure of merit (zT):³Here S is the Seebeck coefficient, σ the electrical conductivity and κ the thermal conductivity, consisting of lattice κ_L and electronic κ_E components. Large zT is desirable, translating directly to higher device efficiency.⁴

Sustainability considerations and energy security have led to an enormous research effort to develop better thermoelectric materials.^5–9 This has been driven by new design concepts such as the phonon-glass electron-crystal, nanostructuring, electronic bandstructure engineering and by improved experimental characterisation of electrical and heat transport.^5–9 This has resulted in gradual increases in zT for a range of materials with highest values approaching zT = 3 for SnSe,^10–13 greatly exceeding the traditional limit of zT = 1. The wide range of heat sources (temperatures) means there is a need for a range of thermoelectric materials. These also need to satisfy engineering requirements such as good electrical and thermal contacting, long term stability and mechanical strength, as well as matched thermal expansion for the n- and p-type legs that make up the basic thermoelectric couples in devices.^5–9 As yet, there is no leading thermoelectric n/p-type couple that would enable widespread cost-effective power generation or replace Bi₂Te₃ for ambient harvesting or cooling applications. For example, most of the very high zT materials considered for power generation, including Bi₂Te₃ itself,^14,15 PbTe^16–20 and GeTe^21,22 use tellurium, which is rare and hence expensive. SnSe has exceptional zT based on its ultralow κ and good electronic properties when doped,^23–25 but has limited high temperature stability.²⁶ Half-Heuslers boast large power factors (S²σ) but their overall zT continues to be limited by large intrinsic thermal conductivity.^27,28

There is considerable interest in Zintl thermoelectrics based on Sb, with antimonide skutterudites reported with large S²σ and zT^29–33 and more recently Mg₃Sb₂ phases with high zT near room temperature in n-type samples.^34–38 There are several excellent reviews and book chapters covering Sb-based thermoelectrics,^39–44 which support very good zT values, often substantially exceeding unity.

Metal phosphides have attracted limited attention as thermoelectrics but is a group of materials with high potential.⁴⁵ As will be shown, structural complexity often leads to intrinsically low lattice thermal conductivities, hence providing a platform for potentially large zT values. In fact, for many of the explored materials the challenge is to improve the electrical properties and increase S²σ. From a chemical perspective, the wide range of accessible phosphorous anionic oxidation states allows for a range of structure types and chemical bonding. This includes P⁰/P⁻ with a high degree of covalent bonding and polyanion formation, and the isolated P³⁻ anion which affords more familiar ionic structure types. In addition to its chemical versatility, phosphorus is highly abundant (11th most abundant element in the earth's crust)⁴⁶ and relatively low-cost. It also has a lower atomic mass (20 amu) than most other common anions in thermoelectric compounds, offering potential advantages in terms of gravimetric power output. Metal phosphide structures include analogues of well-known thermoelectric candidate materials (in particular of skutterudites, clathrates and tetrahedrites), but also comprise a range of unique structures, with semiconducting or semimetallic bandstructures which are of interest for thermoelectric exploration.

One reason for the limited historical interest in metal phosphide thermoelectrics is through association with white phosphorus.⁴⁶ This material consists of weakly linked P₄ molecules and is highly pyrophoric, reacting violently with oxygen to form P₂O₅ and further with water to form H₃PO₄. The formation of highly toxic PH₃ gas is also a concern in synthesis involving phosphorus.⁴⁶ However, red phosphorus, effectively consisting of (P₄)_n oligomers, has much better stability and is commonly used for synthesis.⁴⁶ Once formed, metal phosphides can exhibit excellent stability,^47–50 in particular if more electropositive alkali and alkaline earth elements are avoided.

This review will start with an overview of the reported thermoelectric properties from the recent literature, and it will discuss some of the underlying electronic and thermal parameters. In subsequent sections a more detailed overview of the structures and properties of highlighted promising materials are given, including novel thermoelectric effects based on topological states. The review concludes with an outlook for the field. All structures shown in this review were drawn using the VESTA programme and are based on published crystallographic data.⁵¹

2. Evaluation of performance and underlying thermoelectric parameters

The electronic bandstructures of many known metal phosphide structures can be accessed via databases such as the Materials Project,⁵² and the electronic properties have been the subject of a recent review.⁴⁵ From a chemical bonding (Zintl) perspective, it is anticipated that the valence band has a large anion (P) component with conduction band more heavily weighted towards the metal orbitals. Most metal-phosphides are experimentally found to be p-type, consistent with small cation deficiencies leaving a shortage of electrons to fill the anion derived conduction band, hence leaving holes, and causing p-type conduction. Notable exceptions are the high zT phase Cd₃P₂ and the topological material TaP.^53–56

An overview of the reported thermoelectric parameters for a range of metal phosphides is given in Fig. 1. These include S²σ(T), κ_L(T) and zT(T), and the weighted mobility (μ_w), which was calculated from reported S(T) and σ(T) using an empirical model.⁶² The power output of a thermoelectric generator is related to S²σ, and as such it is an important engineering parameter. The materials reported so far have modest S²σ, typically achieving values between 0.8 and 1.2 mW m⁻¹ K⁻² above 300 K (Fig. 1a). A notable exception is CaCuP with S²σ = 1.75 mW m⁻¹ K⁻², possibly linked to its simple crystal structure.⁵⁰ Low values are found for Zn_0.75Cu₂P₈,^59,63 rich in P–P bonds and the clathrate Ba₈Cu₁₄Ge₆P₂₆,⁵⁷ which are both limited to S²σ < 0.4–0.6 mW m⁻¹ K⁻². In all cases, the reported S²σ are much smaller than state-of-the-art materials, that typically achieve power factors >4 mW m⁻¹ K⁻² and up to 10 mW m⁻¹ K⁻² for some low-defect half-Heusler phases.²⁷

Fig. 1 Summary of the thermoelectric properties of promising metal phosphides. Panel (a) shows the power factor (S²σ), (b) shows the weighted mobilities (μ_w); the dashed lines show a T^−1.5 acoustic phonon scattering dependence, (c) shows the lattice thermal conductivity (κ_L); the dashed lines show a T⁻¹ Umklapp scattering dependence, and (d) shows the figure of merit (zT). Please note that panel (b) and (c) are plotted on a log–log scale (Cd₃P₂,⁵⁵ Ba₈Cu₁₄Ge₆P₂₆,⁵⁷ Ag₆Ge₁₀P₁₂,⁵⁸ Zn_0.75Cu₂P₈,⁵⁹ CaCuP,⁵⁰ MgCuP,⁵⁰ YbCuZnP₂,⁴⁷ CaAg_0.9P,⁶⁰ EuIn₂PAs,⁶¹ ErCuZnP₂.⁴⁹).

The weighted mobility μ_w gives the fundamental electronic performance of a material, independent of carrier doping, and is defined as follows:⁶²

Here μ₀ is the carrier mobility in the limit of Hall carrier concentration, n_H → 0, is the density of states effective mass, which determines the magnitude of , and is the effective band mass of the charge carriers. N_v is the total valley degeneracy of the valence or conduction band extrema, which is the product of the degeneracy arising due to symmetry (N_o) and the degeneracy arising due to different bands (N_b). In the acoustic phonon scattering model μ₀ scales as , meaning μ_w scales as.^64,65 As such thermoelectric materials should ideally have light charge carriers , and a highly converged bandstructure near the Fermi energy (high N_v).⁷

Electronic bandstructure information available for metal phosphide phases is summarised in Table 1. Most compositions have band extrema at the Γ-point, which has the lowest possible N_o = 1. For example, n-type Cd₃P₂ has a single band at the Γ-point, with an exceptionally low leading to the highest Hall mobilities observed with μ_H ∼1600 cm² V⁻¹ s⁻¹ at room temperature.⁵⁵ The semimetallic p-type skutterudite CoP₃ is the only material in Table 1 that has similar low and high mobilities (Table 1).⁶⁶ The low for Cd₃P₂ results in a relatively low S, yielding modest μ_w, similar to the A(CuZn)P₂ 122-phases (Fig. 1b). The large zT observed for Cd₃P₂ is therefore mostly due to its low κ_L (Fig. 1c).

Table 1 Overview of electronic parameters for selected metal phosphide thermoelectrics. Data includes conduction type (n/p), location and degeneracy of the valence/conduction band, density of states effective mass derived using the single parabolic band model with acoustic phonon dominated scattering, reported Hall carrier concentrations (n_H) and mobilities (μ_H)

Composition	Type	Band		n H (cm^−3)	μ H (cm^2 V^−1 s^−1)
a see Section 3.2 on tetrahedrites.
Ag6Ge10−xGaxP12^67	p	Multiple^a	7.7–16.5	1.1–11 × 10^20	1.11–1.84
CaCuP^50,68	p	Γ (Nv = 2)	1.0	1.6 × 10^20	101
MgCuP^50,68	p	Γ (Nv = 1)	1.7	2.4 × 10^20	29
Cd3P2−δ^55	n	Γ (Nv = 1)	0.09	3.6–10.5 × 10^17	950–1690
Cd3P1−xAsx^53,54	n	Γ (Nv = 1)	0.05–0.11	6.1–28 × 10^17	850–9400
ErCuZnP2^49	p	Γ (Nv = 2 to 3)	0.88	2.7 × 10^20	53
YbCuZnP2^47	p	—	0.60	2.4 × 10^20	11.8
CoP3^66	p	Γ (Nv = 1)	0.11	3.26 × 10^19	748
CeFe4P12^66	p	—	0.16	1.42 × 10^19	24.9
CeRu4P12^66	p	—	2.26	1.8 × 10^20	6.0

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The best electronic performance so far is found for CaCuP, in part due to its second valence band (N_b = 2) at the Γ-point,⁶⁸ which affords a better trade-off between and mobility . Not all phosphide bandstructures are trivial. For example the tetrahedrite Ag₆Ge₁₀P₁₂ has a complex electronic structure, with multiple hole pockets in low symmetry positions, associated with lone pairs on Ge²⁺ cations.^67,69 Low dispersion (heavy) bands along the G–F and maxima located just outside the high symmetry Γ–H direction yield for Ag₆Ge_9.875Ga_0.125P₁₂.⁶⁷ The electronic quality of the Ag₆Ge₁₀P₁₂-based tetrahedrite phases is similar to Cd₃P₂ and the 122 phases (Fig. 1b).

An overview of the measured lattice thermal conductivities (κ_L) is given in Fig. 1c. Within the kinematic theory of heat transport, κ_L is given by:

Here, C_v is the heat capacity, v_s is the velocity of sound; related to the bond force constant (k) and average atomic mass (m); and Λ is the phonon mean free path. Materials with heavy atoms and weak bonds leading to low v_s, and strong anharmonicity leading to low Λ due to strong Umklapp scattering, are desired to achieve “intrinsically” low κ_L.^5,6,70 Phosphorus has a relatively low atomic mass and on that basis is not directly associated with low κ_L. Nevertheless, most compositions in Fig. 1c achieve κ_L < 2 W m⁻¹ K⁻¹ at elevated temperatures.

Using elastic properties from the Materials Project (Bulk and Shear moduli B and G),^52,75 initial approximations of κ_L for phosphide materials can be made. Longitudinal, transverse and average sound velocities (v_l, v_t and v_s), Debye temperatures (θ_D) and Grüneisen parameters (γ) were calculated²⁷ and are listed in Table 2.

Table 2 Overview of gravimetric densities (ρ), bulk (B) and shear (G) moduli from the Materials Project, calculated longitudinal, transverse, and average sound velocities (v_l, v_t and v_s), Debye temperatures (θ_D), Grüneisen parameters (γ) and reported (experimental) lattice thermal conductivities (κ_L)

Composition	Materials project reference	ρ (g cm^−3)	B (GPa)	G (GPa)	v l (m s^−1)	v t (m s^−1)	v s (m s^−1)	θ D (K)	γ	κ L (W m^−1 K^−1)/ref.
CaCuP	mp-8432	4.01	74	51	5951	3566	3946	443	1.36	4.6 ^50
CaAgP	mp-12277	4.89	61	43	4919	2965	3279	358	1.34	2.1 ^60
MgCuP	mp-7352	4.41	94	48	5986	3299	3677	445	1.67	2.6 ^50
SrCuP	mp-16321	4.81	66	46	5145	3092	3420	369	1.35	2.7 ^71
Zn3P2	mp-2071	4.54	65	35	4959	2777	3090	345	1.61	1.3 ^72
CaZn2P2	mp-9569	3.98	67	46	5678	3400	3762	417	1.37	3.3 ^73
Cd3P2	Ref.  ^55	5.60	64	18	3950	1760	1980	203	2.35	0.91 ^55
Ag6Ge10P12	mp-17862	5.09	65	36	4712	2659	2958	323	1.58	1.5 ^58
CoP3	mp-1944	4.38	149	115	8308	5124	5653	692	1.26	15.8 ^66
CeFe4P12	mp-16272	5.17	161	132	8074	5053	5565	689	1.21	14.0 ^66
CeRu4P12	mp-10069	5.74	167	110	7392	4378	4849	578	1.41	8.6 ^66
AgP2	mp-8200	4.86	62	26	4460	2313	2589	287	1.88	1.2 ^74

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A plot of experimental κ_L against the calculated v_s is given in Fig. 2. This shows a very clear correlation between calculated v_s and measured κ_L, as expected from eqn (5). The high-zT material Cd₃P₂ has v_s ∼2000 m s⁻¹ and has κ_L < 1 W m⁻¹ K⁻¹ at 300 K. The next lowest κ_L < 2 W m⁻¹ K⁻¹ materials are AgP₂ (v_s ∼ 2600 m s⁻¹), Zn₃P₂ (v_s ∼ 3100 m s⁻¹) and tetrahedrite Ag₆Ge₁₀P₁₂ (v_s ∼ 3000 m s⁻¹), whilst CaCuP with its large μ_w has v_s ∼ 4000 m s⁻¹ and κ_L ∼ 4 W m⁻¹ K⁻¹ at 300 K.

Fig. 2 Measured lattice thermal conductivity (κ_L) plotted against calculated velocity of sound (v_s) for selected metal phosphide thermoelectrics (see Table 2 for source data).

In general, a large v_s is caused by strong responses to compression and shear strain (large B and G, and hence strong bonds) and when material density is low. Inspection of Table 2 suggests that the trend in v_s (κ_L) is driven by the elastic moduli with changes in density having a smaller role. We note that these calculated values provide a general guide to the elastic properties but are far from exact and need confirmation from measurements.

Another important parameter is the bonding anharmonicity, quantified by the Grüneisen parameter (γ), which is a measure of the strength of phonon Umklapp scattering. For Cd₃P₂ the calculated γ from fitting the phonon bandstructure is 2.3–2.4, highlighting substantial anharmonic bonding, which combined with heavier mass of Cd results in the low observed κ_L.⁵⁵ Compared to CaCuP, with γ ∼1.35 and its much higher v_s, the relatively large κ_L can be attributed to stronger bonds with reduced anharmonicity. A final point is that the phosphide skutterudites have by far the largest bulk moduli, coupled to low γ, leading to the highest v_s and κ_L values in Table 2.

Fig. 1d shows the figures of merit for leading phosphide materials. The highest reported zT = 0.9 at 673 K was reported for n-type Cd₃P₂ in 2022, linked to decent μ_w but largely driven by its low κ_L.⁵⁵ The best performance amongst the larger group of p-type materials is zT ∼ 0.7 at 750 K for the tetrahedrite Ag₆Ge₁₀P₁₂, benefitting from several optimisation studies^58,69 since the original report in 2017.⁴⁸ Clathrates based on Ba₈Cu₁₆P₃₀ have been reported with zT ∼ 0.6 at 800 K.^57,76 The A(CuZn)P₂ 122-phases reach zT = 0.6 at 800–1000 K,^47,49 whilst 111 phases, including CaCuP reach zT = 0.5 at 800 K.⁵⁰

3. Phosphide thermoelectric materials

In this section, we will examine the leading phosphide materials in more detail, focusing on their crystal structure and bonding characteristics as well as thermoelectric performance. This will start with ZnCu₂P₈, a structure dominated by P–P bonding in Section 3.1. Section 3.2 will cover phosphide analogues of established phonon-glass electron-crystal systems (clathrate, skutterudite and tetrahedrite). Section 3.3 and 3.4 will focus on materials with 122 and 111 compositions, which do not exhibit P–P bonding. Section 3.5 will examine the n-type phases Cd₃P₂ and TaP, which display exceptional thermoelectric properties at high and low temperatures respectively.

3.1 ZnCu₂P₈ – a low-cost material with P–P bonding

ZnCu₂P_8, first reported in 2020,¹⁵ is a good illustration of the range of bonding that phosphorus can exhibit. From an electron counting perspective, assuming Zn²⁺ and Cu⁺ (both closed shell d¹⁰), the P atoms must average to [P₈]⁴⁻, requiring a large amount of P–P bonding.

The P–P network is quite complex (Fig. 3), although at its simplest, half the P atoms have three single P–P bonds and a single lone pair (i.e. P⁰), and the other half have two P–P bonds with two lone pairs (i.e. P⁻). Zn exists in a relatively uniform tetrahedral P coordination, while Cu has a more heavily deformed tetrahedral P environment. This neat separation of Zn and Cu sites may not be present in real samples however, with Cu⁺/Zn²⁺ cation disorder being attributed to electronic and thermal transport features.⁶³

Fig. 3 Crystal structure of ZnCu₂P₈. Panel (a) shows the tetrahedral coordination of Zn (grey), Cu (blue) and the P–P network (red). Panel (b) shows the lone pair behavior of P (green), where P atoms with one lone pair coordinate to Cu (blue), and P atoms with 2 lone pairs bond to Zn (red). Panel (b) reproduced with permission from Nuss et al., Zeitschrift für Anorganische und Allgemeine Chemie, 2020.⁶³

Ab initio calculations show a strong dependence of the electronic structure and bandgap on Cu/Zn inversion.⁶³ This is reflected in the measured properties in the two studies on this material. ZnCu₂P₈ prepared using ball milling has an order of magnitude larger ρ_{300 K},⁵⁹ compared to high-temperature synthesis, which is found to have metal-like ρ(T) and S(T) above 100 K (but still with relatively high ρ_{300 K} ∼50 mΩ cm, S_{300 K} ∼200 μV K⁻¹),⁶³ suggesting a higher level of p-type doping, potentially linked to different Cu/Zn inversion or subtle deviations from nominal stoichiometry.⁶³ Overall, the performance of ZnCu₂P₈ is limited by a large ρ(T) with highest observed S²σ = 0.11 mW m⁻¹ K⁻² at 673 K.⁵⁹ In order to optimise the thermoelectric performance, the Cu⁺/Zn²⁺ ratio was varied in the nominal Zn_1−xCu_2+xP₈ (−0.25 ≤ x ≤ 0.25) series.⁵⁹ For x < 0 (increasing Zn – adding electrons), an increase in ρ(T) was observed, yielding a highly resistive material. For x > 0 improvements to S²σ were found to be possible. To minimise CuP₂ impurities observed in the Cu-rich Zn_1−xCu_2+xP₈ samples, Zn_0.75Cu₂P₈ was synthesised, yielding S²σ = 0.38 mW m⁻¹ K⁻² at 673 K, and a highest reported zT_{673 K} = 0.25. The most promising feature of ZnCu₂P₈ is its very low κ = 1.05–1.60 W m⁻¹ K⁻¹ at 300 K, decreasing to as low as ∼0.8 W m⁻¹ K⁻¹ at 673 K. Despite the modest overall performance, ZnCu₂P₈ is a fascinating thermoelectric material with complex chemical bonding offering further possibilities for optimisation. It is based on abundant low-cost elements and has good stability under ambient conditions.^59,63

3.2 Phosphide analogues of high-zT phonon-glass electron-crystal thermoelectrics: tetrahedrite, clathrate and skutterudite materials

The phonon-glass electron-crystal concept (PGEC) is one of the key guiding ideas of modern thermoelectrics research.⁷⁷ Its aim is to unite the low thermal transport of a structural glass with the good electrical properties of a crystalline material.

Key thermoelectric materials exhibiting PGEC behaviour are the skutterudites (e.g. LaFe₃CoSb₁₂)⁷⁸ and clathrates (e.g. Ba₈Ga₁₆Ge₃₀)⁷⁹ consisting of a conducting framework and rattling ions (La³⁺/Ba²⁺ in the compositions above), causing glass-like κ_L, whilst the materials are crystalline. Tetrahedrites are chalcogenides (e.g. Cu₁₂Sb₄S₁₃)⁸⁰ with complex structures, characterised by extremely low κ_L linked to out-of-plane rattling vibrations of three-fold coordinated Cu ions.⁸¹ There are phosphide analogues of all three of these important thermoelectric structure types, as illustrated in Fig. 4. These are discussed below in the following sections.

Fig. 4 Crystal structures of phosphide analogues of established phonon-glass electron-crystal thermoelectric materials. Panel (a) shows the Ag₆Ge₁₀P₁₂ tetrahedrite structure. Panel (b) shows the Ba₈Cu₁₆P₃₀ clathrate structure. Panel (c) shows the (unfilled) CoP₃ skutterudite structure with [P₄]⁴⁻ squares (Ba = green, Co = blue, Cu = orange, Ag = grey, Ge = dark green, P = red).

3.2.1. Tetrahedrite phosphides. The ‘tetrahedrite’ group spans a large number of minerals with the (simplified) general formula A₆(B₄C₂)D₄Y₁₂Z,⁸² where A is Cu/Ag, B/C are late transition metals, D is a pnictide (cation) and Y/Z are S or Se. Tetrahedrite chalcogenides based on Cu₁₂Sb₄S₁₃ are promising thermoelectric materials, with reported zT values as high as 1.1, driven by very low κ_L < 0.3 W m⁻¹ K⁻¹.^80,83–86

Ag₆Ge₁₀P₁₂ can be written as Ag₆(Ge₄Ge₂)Ge₄P₁₂Vac to match the tetrahedrite formula, where Vac is a vacancy in the structure. The presence of this vacancy is significant because it leads to the formation of an isolated [Ag₆]⁴⁺ cluster.⁸⁷ Valence balancing to reach a semiconducting state requires a mixture of oxidation states for Ge, leading to a formal assignment of [Ag₆]⁴⁺(Ge₄²⁺Ge₂⁴⁺)Ge₄⁴⁺P₁₂³⁻.⁴⁸

Ag₆Ge₁₀P₁₂ is a p-type conductor, with semiconducting ρ(T) and large S_{300 K} ∼ 250 μV K⁻¹ (over 350 μV K⁻¹ in single crystals⁴⁸).^48,58,88 Hole concentrations are sensitive to non-stoichiometry, controlled by nominal excess of P during synthesis. In one study 3% excess yielded the best electronic properties with largest S²σ ∼ 1.1 mW m⁻¹ K⁻² above 500 K.⁵⁸ Higher P excess leads to increased impurity phases, increased κ_L and degraded thermoelectric performance. Isovalent alloying with Cu and Sn in Ag_6-xCu_xGe_10−ySn_yP₁₂ has been found to lead to a reduction in κ_L, but the level of Cu substitution is limited due to deterioration of the electronic performance.⁸⁸ Hole doping using Ga in Ag₆Ge_10−xGa_xP₁₂ is effective and can control the hole concentration, with some additional suppression of κ_L.⁶⁷

The is rather large for Ag₆Ge_10−xGa_xP₁₂, between 8 and 16m_e depending on the Fermi level, due to the combination of a low dispersion band in the G–F direction and the contribution of a lower energy valence band in the Γ–H direction.⁶⁷ The complex band structure and measured values are illustrated in Fig. 5.

Fig. 5 Illustration of band convergence in Ag₆Ge_10−xGa_xP₁₂. Panel (a) highlights the dependence of on the hole concentration (Fermi energy). Panel (b) compares against optimal doping for a range of thermoelectric systems. Panels (c–f) show the multiple bands that contribute to thermoelectric performance at different doping levels. Reproduced under a Creative Commons CC-BY license from Namiki et al., Materials Today Sustainability, 2022.⁶⁷ References in Panel (a) correspond to the original manuscript.

As is the case for most phosphide materials, high performance in the tetrahedrites is derived from low thermal transport. In single crystal Ag₆Ge₁₀P₁₂, κ_L ∼ 1.75 W m⁻¹ K⁻¹ is observed at 300 K, which is reduced to 1.3 W m⁻¹ K⁻¹ for polycrystalline samples, with lowest observed values ∼0.8 W m⁻¹ K⁻¹ in Ag_5.7Cu_0.3Ge₁₀P₁₂.^48,67,88 One explanation for the low κ_L is the weak interaction between the covalent Ge–P framework and the enclosed [Ag₆Ge₄] clusters, in which large thermal displacement parameters give Ag⁺ a liquid-like behaviour.⁴⁸ Alternatively, the [Ag₆]⁴⁺ cluster has been considered to act as a low energy “rattler”, leading to suppression of acoustic phonon modes and low sound velocities.^58,69

At present, the highest reported figures of merit are for Ag₆Ge₁₀P₁₂ + 3% P and the Ag₆Ge_10−xGa_xP₁₂ series, which have zT_{750 K} ∼ 0.7.^58,69

3.2.2. Clathrate phosphides. A small number of P-based clathrates have recently been reported in the literature,⁸⁹ and this is an area of growing interest. One of the leading thermoelectric compositions is the p-type type-I clathrate Ba₈Cu₁₄Ge₆P₂₆, which benefits from low κ_{300 K} ∼0.85 W m⁻¹ K⁻¹ decreasing to 0.7 W m⁻¹ K⁻¹ at 1000 K.⁵⁷ In this material Ge is required to give the material a semiconducting electron count relative to the parent Ba₈Cu₁₆P₃₀ phase and also serves to suppress κ_L. Ba₈Cu₁₆P₃₀ has been reported to be metallic but with a large S_{300 K} ∼50 μV K⁻¹,⁹⁰ whilst off-stoichiometric Ba₈Cu_15.5P_30.5 is reported to be semiconducting.⁹¹ This suggests that adjusting the Cu/P content is a possible route to control the thermoelectric properties. For Ba₈Cu₁₄Ge₆P₂₆, a high intrinsic ρ(T) leads to a low S²σ, which never exceeds 0.56 mW m⁻¹ K⁻².⁵⁷ Fortunately, the low thermal conductivity leads to a good peak zT_{812 K} ∼ 0.63, which could be increased with further electronic optimisation. Similar zT_{800 K} = 0.62 has been reported for Ba₈Cu₁₄Zn₂P₃₀,⁷⁶ where the Cu/Zn ratio can be used to optimise doping between metallic Ba₈Cu₁₆P₃₀ and a highly resistive Ba₈Cu_11.4Zn_5.6P₃₀ phase. Further work on clathrate phosphides would be of considerable interest due to their inherently low κ_L. Although probably not directly useful for thermoelectric applications, a gold phosphide clathrate, Ba₈Au₁₆P₃₀ has been reported to have extremely low κ_L, as low as 0.15 W m⁻¹ K⁻¹ at 400 K, attributed not only to the high atomic mass of Au, but also to twinning of crystallites on the nanoscale, leading to increased boundary scattering.⁹² As prepared, this material is metallic and engineering the electron count of Ba₈Au₁₆P₃₀ to find a semiconducting state would be of interest. BaNi₂P₄, BaCu₂P₄ and SrNi₂P₄ form crystal structures which deviate from the familiar type-I clathrate.⁹³ Further work on BaNi₂P₄ found a stronger than predicted interaction between Ba and the covalent framework.⁹⁴ While these phases are quite metallic, they point to a large underexplored field still in its infancy when compared to Si and Ge based clathrate thermoelectrics with many more new compositions to be discovered.

3.2.3. Skutterudite phosphides. The A_xBX₃ skutterudite structure (named after CoAs₃) can be described in several ways. One viewpoint is a network of very strongly tilted corner-sharing BX₆ octahedra with up to 25% of void sites filled with rattler A ions. Another is a large primitive array of B (e.g. Co³⁺) ions, inside of which 75% of sites are filled by [X₄]⁴⁻ square rings with X–X bonding. The remaining 25% empty sites are available for filling with rattler ions to reduce κ_L. Antimonide skutterudites have attracted enormous interest and are prototypical PGEC materials, with high reported zT values approaching 2 after a sustained push to increase performance.^30,95 Phosphide skutterudites by contrast appear less promising, and not only because of their lower average atomic mass, large Bulk moduli and higher κ_L (Fig. 2 and Table 2).

Unfilled CoP₃ is an unusual material with a semiconducting ρ(T) but with a magnitude typical of conductive metallic samples (∼3 μΩ m between 5–1000 K), which is consistent with the low reported S(T) of no more than 40 μV K⁻¹.^66,96 Early band structure calculations into CoX₃ skutterudite materials observed highly linear valence bands crossing a pseudo-band gap,^97,98 as illustrated for CoP₃ in Fig. 6. This unusual semimetallic bandstructure may be the cause of poor thermoelectric performance in CoP₃.^30,31 The material is further held back by a large κ, although κ_L can be suppressed through the addition of La as “rattlers” in the vacant site,⁹⁶ with a reported reduction in κ_L from ∼15.8 W m⁻¹ K⁻¹ in CoP₃ to ∼3 W m⁻¹ m⁻¹ for La_0.24CoP₃.

Fig. 6 Electronic band structure of skutterudite CoP₃. The material is largely gapped between −2 and 0 eV, but has one highly disperse quasi linear valence band at the Γ-point. Reproduced with permission from Llunell et al., Physical Review B, 1996.⁹⁸

More promising is p-type CeFe₄P₁₂, where Co has been replaced by Fe to balance the charge donation from the Ce⁴⁺ rattlers, maintaining a semiconducting electron count.^66,99 Interestingly, below 200 K S(T) is slightly negative, with thermal excitations leading to p-type conduction at higher temperatures. The peak S²σ_{900 K} = 0.7 mW m⁻¹ K⁻² is substantially larger than for CoP₃, although κ remains large at all temperatures, limiting zT to ∼0.08 at 800 K.⁹⁹ CeRu₄P₁₂ similarly shows strong p-type semiconducting properties, with the decrease of S(T) on heating pointing to a more significant bipolar character than the Fe analogue.⁶⁶ Analysis of room temperature Hall data using the SPB model assuming acoustic phonon scattering, reveals that the switch from Co(Fe) to Ru leads to a substantial increase in from 0.11–0.16m_e to 2.26m_e as shown in Table 1.

3.3 The 1 : 2 : 2 phases – EZn₂P₂ and derivatives

A substantial number of EM₂P₂ compositions have been reported, with much interest generated on the back of the discovery of high-temperature superconductivity in the Fe pnictides. Generally compositions with M = Zn and Mn are found to be semiconducting. The two most common structure types are hexagonal CaAl₂Si₂ (illustrated in Fig. 7a) and tetragonal ThCr₂Si₂/CeGa₂Al₂ (shown in Fig. 7c).

Fig. 7 Common crystal structures adopted by 1 : 2 : 2 phosphide thermoelectric materials. Panel (a) shows the CaZn₂P₂ structure (CaAl₂Si₂-type; Ca = cyan, Zn = grey, P = red), Panel (b) shows the α-BaZn₂P₂ structure (Ba = green), (c) Panel shows the β-BaZn₂P₂ (ThCr₂Si₂-type); structure and Panel (d) shows the α-EuIn₂P₂ structure (Eu = yellow, In = Blue).

In terms of thermoelectric performance, most work has focused on hexagonal p-type CaZn₂P₂ and YbCuZnP₂ derived compositions. In these materials the valence states can be simplified as Ca²⁺(Zn²⁺)₂(P³⁻)₂ and Yb³⁺Cu⁺Zn²⁺(P³⁻)₂.⁴⁷ The tetragonal ThCr₂Si₂ structure-type has more potential for intralayer P–P bonding, which would lead to a decrease in the oxidation state from P³⁻. Observation of this structure with P–P bonding is limited to larger E cations, such as Ba.

As-synthesized CaZn₂P₂ has a high ρ ∼ 2 Ω.m and S ∼ 300 μV K⁻¹ at 300 K, consistent with a large bandgap and little intrinsic doping,⁴⁷ with the Goldsmid-Sharp approximation suggesting E_g ∼ 0.6 eV, although recent DFT calculations expect E_g = 1.4 eV.¹⁰⁰ Attempts to p-type dope via Cu substitution lead to a decrease in ρ(T) at all temperatures, but S(T) is also heavily suppressed at low temperatures, keeping S²σ < 0.02 mW m⁻¹ K⁻².⁷³ However, a sharp (and so far unexplained) increase in S above 600 K leads to a moderate S²σ_{970 K} = 0.5 mW m⁻¹ K⁻² in CaZn_1.9Cu_0.1P₂.⁷³ A similar temperature dependence has also been observed in Cu-doped Zn₃P₂, with sharp increases in S occurring above 600 K, coinciding with a discontinuity in σ(T).⁷² Mn²⁺ can be used instead of Zn²⁺ due to localisation of the 3d⁵ electrons, also yielding semiconducting materials with similar behaviour but a lower max zT,⁷³ but brings the prospect of coupling to magnetism.^101,102

The performance of YbCuZnP₂-based compositions with 1 : 1 mixtures of Cu⁺ and Zn²⁺ (valence balanced through incorporation of Yb³⁺) is much more promising. YbCuZnP₂ was found to have near constant ρ(T) ∼ 22 μΩ m (5 orders of magnitude smaller than CaZn₂P₂) above 300 K and a high temperature S ∼ 160 μV K⁻¹ at 970 K, leading to S²σ_{970 K} ∼ 1.1 mW m⁻¹ K⁻². More recently the thermoelectric properties of other RECuZnP₂ materials (RE = Pr, Nd, Er) have been explored, in a study combining theory and experiment.⁴⁹ While not directly reported by the authors, ErCuZnP₂ displayed the largest S²σ of any reported phosphide material at that time, with S²σ_{780 K} ∼ 1.5 mW m⁻¹ K⁻².⁴⁹

Also promising is the low κ_L for this structure type, consistently below 3 W m⁻¹ K⁻¹ at 300 K, approaching 1 W m⁻¹ K⁻¹ at 1000 K in the case of CaZn₂P₂ (Fig. 1). At present, YbCuZnP₂ has the largest reported zT_{970 K} ∼ 0.63 in this family of materials,⁴⁷ although at 800 K ErCuZnP₂ is reported with larger zT = 0.5.⁴⁹ Controlled doping (particularly in CaZn₂P₂) and employing isovalent alloying strategies may lead to further improvements.

There are a number of other interesting 1 : 2 : 2 compositions that do not have the CaAl₂Si₂ structure type. LaMnCuP₂ for example crystallises in the BaCu₂S₂-type structure (Fig. 7b), with the large La³⁺ cation stabilised by the an increased coordination number (CN) of 7 by phosphide ions.¹⁰³ This is supported by the existence of α-BaZn₂P₂ which also forms with the BaCu₂S₂-type structure.¹⁰⁴ Through higher temperature reaction conditions, β-BaZn₂P₂ was produced with the tetragonal ThCr₂Si₂-type structure which places Ba in a higher CN = 8 environment.¹⁰⁴ Initial thermoelectric property measurements of the α-phase confirm semiconducting ρ(T) and S_{300 K} > 120 μV K⁻¹. With these observations in mind, there should be a large compositional and structural range for further investigation.

One final composition of note is EuIn₂As_2−xP_x, which differs from the above structures because it has direct In–In bonding (Fig. 7d).⁶¹ Focussing on the P-rich compositions, the samples are degenerate semiconductors, with a largely temperature independent S²σ ∼ 0.5 mW m⁻¹ K⁻² between 300–800 K. Alloying leads to a substantial reduction in κ_L, consistent with mass and strain disorder, with a reduction from κ_{L 300 K} = 6.7 W m⁻¹ K⁻¹ in EuIn₂P₂ to ∼3 W m⁻¹ K⁻¹ in alloyed EuIn₂PAs. With this beneficial alloying contribution, zT_{773 K} ∼ 0.25 is achieved. Other isoelectronic EuIn₂P₂-type phosphides have been shown to exist experimentally,¹⁰⁵ which may also have promising thermoelectric properties.

3.4 The 1 : 1 : 1 phases – CaCuP, CaAgP and derivatives

A number of semiconducting phosphides with 1 : 1 : 1 composition have been reported. Assuming ionic P³⁻, the electron donating metals are required to have formal +1 and +2 oxidation states. One larger group of materials satisfying this requirement is the Nowotny-Juza I-II-V phases, which are cubic semiconductors with a valence electron count of 8.¹⁰⁶ Their structures are closely related to the heavily investigated half-Heusler thermoelectrics,²⁷ differing only in preferred site occupancies.¹⁰⁶ An example of a phosphide Nowotny-Juza phase is LiZnP,¹⁰⁷ whose structure is shown in Fig. 8a. Here electronegative P occupies the pseudo-cubic site, whereas late transition metals such as Ni occupy this position in the 18 electron half-Heusler structure.¹⁰⁶ Unfortunately, exploration of these phases is difficult, with the alkali metal leading to instability in ambient conditions. A phosphide analogue of MgAgSb, with zT ∼ 1.4 at temperatures below 550 K,^108,109 and good initial device properties,¹¹⁰ would be of interest but has not been reported.

Fig. 8 Crystal structures of selected 1 : 1 : 1 metal phosphides. Panel (a) shows the cubic Nowotny-Juza phase LiZnP; Panel (b) shows CaAgP, consisting of CaP₅ and AgP₄ polyhedra; Panel (c) shows CaCuP with BN-like [CuP]²⁻ layers spaced by Ca²⁺ cations; Panel (d) shows MgCuP with edge-linked CuP₄ tetrahedra in the b direction (P = red, Li = yellow, Zn = dark grey, Ca = cyan, Ag = light grey, Cu = blue and Mg = lime green).

The remainder of this section will focus on the thermoelectric properties of the 111-compositions CaAg_0.9P, MgCuP, CaCuP and SrCuP.

The silver phosphide CaAgP has recently attracted interest as a possible nodal-line Direct semimetal,^111,112 although the presence of a predicted band-gap depended on the exchange correlation functional, with PBE-GGA giving a semimetallic bandstructure,^111,112 whereas a hybrid-functional gives a low band-gap semiconductor.¹¹³ The CaAgP structure, originally discovered in 1979,¹¹⁴ consists of edge-linked CaP₅ square based pyramids (Fig. 8b).

In our report, we observe large S(T) values supporting a semiconducting bandstructure, with the metallic ρ(T) likely arising due to Ag vacancies introducing holes into the material, with nominal CaAgP containing 3% Ag impurity and an Ag-site occupancy of ∼0.9 from PXRD data.⁶⁰S²σ(T) values in the targeted off-stoichiometric CaAg_0.9P composition reach a peak of 1.25 mW m⁻¹ K⁻² at 550 K, decreasing at higher temperatures due to a bipolar decrease in S(T). The thermal properties of CaAg_0.9P are promising, with a low κ_{L 340 K} ∼ 1.8 W m⁻¹ K⁻¹ decreasing to a minimum of κ_{L 600 K} ∼ 1.2 W m⁻¹ K⁻¹, although it appears that there is a significant bipolar contribution to κ at higher temperatures. Moderate electronic performance and low κ_L leads to a peak zT_{600 K} ∼ 0.43, which is promising for a largely unoptimized system. The thermoelectric properties of CaAgP have recently (the paper appeared during the proof stages of this review) been the subject of a computational study, confirming experimental findings.¹¹⁵

Following on from CaAgP, we reported the thermoelectric properties of MgCuP and CaCuP, based on sustainable elements and with low gravimetric densities.⁵⁰ These compositions each have different crystal structures (shown in Fig. 8c and d) and are different from CaAgP, again highlighting the structural diversity of the phosphides. CaCuP forms with hexagonal [CuP]²⁻ layers isostructural with graphene or BN layers, separated by Ca²⁺ cations.¹¹⁶ Where CaCuP can be viewed as pseudo-2D, the MgCuP structure consists of chains of edge-linked CuP₄ tetrahedra in the b-direction giving a pseudo-1D character to the structure (Fig. 8d).¹¹⁷ Electronic bandstructure calculations for both phases exist in the literature, with calculated E_g > 1 eV for both phases.⁶⁸ An experimental follow-up, in the context of transparent semiconductors, found degenerate p-type behaviour arising due to Cu deficiency in CaCuP,¹¹⁸ consistent with our observations.

Both CaCuP and MgCuP are p-type and have a very similar zT between 300–800 K, reaching ∼0.5 at 790 K, although CaCuP has much better electronic properties while MgCuP has a lower κ_L.⁵⁰ At present CaCuP boasts the largest high temperature S²σ of any phosphide material reported in the literature, with a maximum S²σ_{790 K} = 1.75 mW m⁻¹ K⁻² (Fig. 1a) and an average value of 1.62 mW m⁻¹ K⁻² between 340–790 K, which is helpful for power generation. Unsurprisingly, CaCuP has one of the larger κ_L covered in this review, with κ_{L 340 K} ∼ 3.9 W m⁻¹ K⁻¹, compared to 2.4 W m⁻¹ K⁻¹ for MgCuP, decreasing to 1.8 W m⁻¹ K⁻¹ and 1.5 W m⁻¹ K⁻¹ at 790 K respectively. From a thermoelectric perspective, CaCuP is the more promising of the two phases due to its better electronic performance and the absence of detrimental bipolar carrier effects at high temperatures.⁵⁰ A further complication with the MgCuP phase was found with increased amounts of Cu₃P and CuP₂ impurity phases observed upon heating, pointing to partial decomposition of the majority phases.⁵⁰ However, this effect appears to be reversible on cooling with observation of reversible hysteretic effects in the measured S(T) and ρ(T) on temperature cycling.

During review of this manuscript, Moll et al. reported on the thermoelectric properties of SrCuP,⁷¹ which is isostructural to CaCuP. SrCuP was prepared using a mechanical alloying route and is more susceptible to oxidation than CaCuP with substantial Sr₁₀(PO₄)₆O impurities and visible degradation of the SPS pellet in ambient conditions. In terms of its thermoelectric performance, SrCuP is a p-type degenerate semiconductor with peak S²σ_{700 K} = 0.7 mW m⁻¹ K⁻² and a minimum κ_{L 600 K} ∼ 1.3 W m⁻¹ K⁻¹. The highest zT reported was zT_{600 K} = 0.2, but improvements should be possible if better quality samples can be produced.

3.5 n-type compositions – Cd₃P₂ and topological magneto–thermoelectric TaP

A recent report on n-type Cd₃P₂ found the best thermoelectric performance for a phosphide material reported thus far, with zT_{673 K} ∼ 0.91.⁵⁵ Promising performance with zT_{300 K} = 0.2 was originally reported as early as 1968 by Masumoto and Isomura,^53,54 where alloying with As was found to lead to zT_{300 K} ∼ 0.33, in large part due to suppression of κ_L. The crystal structure of Cd₃P₂ is a distortion of the cubic Ag₂O₃-type structure, with symmetry lowered from cubic to tetragonal. The structure consists of edge and vertex linked CdP₄ tetrahedra (Fig. 9a), while there are three distinct 6-coordinate environments for the P³⁻ anions.

Fig. 9 Crystal structures of (a) Cd₃P₂ and (b) TaP (P = red, Cd = pink and Ta = brown). The Cd₃P₂ structure is tetragonal distortion of the cubic Ag₂O₃ structure. TaP consists of edge linked TaP₆ trigonal prisms, separated by corner-sharing TaP₄ tetrahedra.

At present, temperature dependent thermoelectric property data is only available from the 2022 study.⁵⁵μ_H has been confirmed to be quite high from multiple sources,^53–55,119 with values ∼1500 cm² V⁻¹ s⁻¹ at room temperature due to a low , although μ_H can be lower when grain boundaries are present.⁵⁵ This low electron mass arises from a single highly dispersive conduction band minima at the Γ-point. Typical n_H values are on the order of 10¹⁷–10¹⁸ cm⁻³ at 300 K,^53–55 which is on the lower end for thermoelectric materials. As expected from low extrinsic carrier doping, these materials have substantial intrinsic carrier effects, with a highly temperature dependent n_H(T) and a strong bipolar decrease in S(T) with increasing temperature.⁵⁵ The calculated band gap for the material is 0.46 eV. In the most highly doped sample, S²σ reaches large values for a system with such low , with a maximum of 1.3 mW m⁻¹ K⁻² at 673 K.

As already discussed, the real driver behind the high zT is that κ_L < 1 W m⁻¹ K⁻¹ at all temperatures in the most recent report,⁵⁵ (or ∼2 W m⁻¹ K⁻¹ in the original reports^53,54). First principles calculations and acoustic measurements find low v_s are largely responsible for low κ_L.⁵⁵ In the phonon density of states, the acoustic phonon behaviour is dominated by the vibration of Cd atoms, while P atoms contribute more heavily to optical phonons.

Alloying in the Cd₃P_1−xAs_x solid solution was carried out in the 1970,^53,54 confirming the formation of a complete solid solution. Across the solid solution, all samples are n-type and degenerately doped. The alloying dependence of κ_L shows a decrease from either end member consistent with expected alloying behaviour, however an increase near x = 0.5 suggests partial ordering on the P/As site. The thermoelectric properties of single crystal Cd₃As₂ are interesting, showing strong magneto–thermoelectric effects attributed to linear dispersion in the conduction band.¹²⁰ The room temperature thermoelectric values for Cd₃P₂ with zero applied field are better however, so an examination of Cd₃P₂ in applied fields could prove to be interesting.

On the subject of magneto–thermoelectric effects, n-type TaP displays particularly interesting properties,⁵⁶ perhaps not in the context of power generation, but rather with the observation of the ‘quantised thermoelectric Hall effect’ (QTHE). In short, Weyl Semi-Metals (WSM) benefit from an energy independent density of states g(E) which increases with applied magnetic fields, protected by the topological behaviour of the Weyl nodes. Since S is proportional to the logarithmic derivative of g(E), the application of a magnetic field allows an applied field to induce large S(B). In single crystal TaP, the QTHE has successfully been demonstrated, with S_xx(9T) > 1000 μV K⁻¹ observed at ∼45 K (Fig. 10b). As a semimetal, the applied field induced a substantial magnetoresistance in the sample, but at the maximum reported field of 9T, the power factor S_xx²σ exceeded 50 mW m⁻¹ K⁻² at ∼45 K (Fig. 10f).⁵⁶ Such values are not ordinarily obtained for thermoelectric materials in thermodynamic equilibrium. The practicality of harvesting heat or applying the Peltier effect at 45 K under a 9T magnetic field is challenging and may prevent exploitation, however the results are exceptional.

Fig. 10 Magneto–thermoelectric effect in TaP. Panel (a) illustrates the experimental setup used to measure the Seebeck coefficient in the x-direction (S_xx). Panel (b) shows the temperature dependence of S_xx at applied fields B_z between 2.5–300 K, and panel (c) highlights the 2.5–6 K region. Panel (d) shows the dependence of S_xx on the applied field at fixed temperatures, with panel (e) showing the same data on logarithmic axes. Finally, panel (f) shows the power factor in the x-direction as a function of temperature at fixed magnetic fields, showing the very large S_xײ/ρ_xx > 50 mWm⁻¹ K⁻² near 45 K in 9T. Reproduced under a Creative Commons CC BY license from Han et al., Nature Communications, 2020.⁵⁶

4. Discussion

Metal phosphides are gaining increasing attention with a larger selection of materials now being investigated for their thermoelectric performance. This has enabled more detailed insight into the underlying thermoelectric properties of metal phosphides. The current highest figures of merit are zT_{673 K} = 0.9 for n-type Cd₃P₂ and zT_{750 K} = 0.7 for p-type tetrahedrite Ag₆Ge₁₀GaP₁₂. In both cases, the performance is underpinned by low κ_L and relatively modest electronic properties (compared to the best performing thermoelectrics, such as Bi₂Te₃, SnSe, PbTe, GeTe, skutterudites and half-Heuslers).^5,7,9 Besides these two compositions, there are several other p-type compositions that reach zT = 0.5–0.6 at 700 K, including 111-type CaCuP and 122-type YbCuZnP₂. As observed for antimonide Zintl compositions, n-type conduction is relatively rare in metal phosphides. This is likely linked to the relative ease for metal deficiency in these compositions. One major conclusion is that despite the low atomic mass of P, low κ_L is commonly observed. This is linked to relatively “weak” chemical bonding (low k in eqn. (5)) in these materials, with a clear correlation between calculated v_s and measured κ_L (Fig. 2). Searching for crystal structures with low bulk moduli (weak and anharmonic bonding) is therefore a promising route towards finding higher zT materials. Where calculations are not available, searching for crystal structures with mixtures of shorter and longer bonds as part of a single coordination environment might be a fruitful route.²⁴

From the overview presented here, the main challenge is to improve the thermoelectric power factor in metal phosphides. Table 1 shows that investigated phosphides have and μ_H values that are typical for semiconducting materials. The main issue appears to be that μ₀ (for a given ) lags behind that of thermoelectrics with large S²σ,⁶² hence compromising μ_w. Several materials have μ_w around 50–100 cm² V⁻¹ s⁻¹, with the clear outlier being CaCuP with 180 cm² V⁻¹ s⁻¹ which outperforms every other phosphide reported at all temperatures (Fig. 1b). However, even for CaCuP, the μ_w is only about 1/3 of high-power factor materials, such as the half-Heuslers^27,121–123 and antimonide skutterudites.^30–33 For CaCuP, the high μ_w is likely linked to its relatively simple crystal structure. The observation of large μ_w(T) for tetrahedrite Ag₆Ge₁₀P₁₂ is unexpected given its complex structure with unusual separation into weakly interacting [Ag₆Ge₄] clusters within a [Ge₆P₁₂] framework.⁴⁸ This suggests that the framework of vertex-sharing GeP₄ tetrahedra is effective in promoting hole transport. The low μ_w(T) for the two structures containing high degrees of P–P bonding (Ba₈Cu₁₄Ge₆P₂₆ and Zn_0.75Cu₂P₈), suggests that electrons in P–P bonds are localised and that this is not favourable for obtaining good electrical properties. There are several theoretical predictions of high thermoelectric performance, including in CuP₂,¹²⁴ and BaAu₂P₄.¹²⁵ These compositions or the features supporting good properties (bond hierarchy and low κ_L for BaAu₂P₄) are worthy of further investigation. Another interesting composition that was mentioned in the context of low κ_L is AgP₂ (Section 2). As prepared, this material is too insulating, but it could be of interest if it could be doped. Hall data reveal a low and μ_H = 50 cm² V⁻¹ s⁻¹.⁷⁴ Another potentially interesting silver phosphide is Ag₃SnP₇, which has a low κ_L = 1.5 W m⁻¹ K⁻¹ with no further thermoelectric characterisation.¹²⁶

Assuming that electrical transport is dominated by acoustic phonon scattering (APS), the magnitude of μ₀ within the SPB model is related to the longitudinal velocity of sound, (v_l), the band mass , the inertial carrier mass (m_I) and the deformation potential (E_def):²⁷

Here, E_def quantifies the coupling strength of the electrical transport to the crystal lattice (acoustic phonons). Collapsing the electrical transport of the materials in Table 1 onto one effective single band, assuming , and using the v_l from Table 2 enables an estimate of E_def. For CaCuP and MgCuP E_def = 13.4 eV and 7.9 eV respectively. For Cd₃P₂ and YbCuZnP₂ values come out closer to 40 eV, although in both cases the temperature dependence of σ and μ_H don't match a typical degenerate semiconductor. At this point it is unclear if acoustic phonon scattering is indeed the dominant scattering mechanism, with optical phonon and ionised impurity scattering also likely to contribute, but the obtained values of E_def are fairly large for thermoelectric materials.^28,127–129 Further work on understanding electronic transport (carrier scattering) in these materials is needed and can provide avenues for further materials discovery and optimisation.

There are considerations for thermoelectric materials beyond zT, which limit application in popular thermoelectric systems. This includes finding matching n- and p-types, good mechanical properties, electrical and thermal contacting, stability under temperature cycling and during exposure to high temperatures.^5,9 Most of the phosphides discussed in this review are at least stable on the bench in ambient conditions, but it seems likely that some sort of encapsulation is needed for high-temperature operation.

A number of studies report thermogravimetric (TGA) and differential scanning calorimetry (DSC) data, generally under inert conditions (typical gas flow for TGA/sealed cups for DSC). For example, SrCuP is stable up to 1273 K from DSC,⁷¹ but as discussed in Section 3.5, does readily oxidise in ambient conditions. By contrast MgCuP and CaCuP have good stability on the bench,⁵⁰ but a (reversible) decomposition occurs for MgCuP above 573 K. The Ba₈Cu₁₄Ge₆P₂₆ clathrate has good stability in inert conditions up to its melting point at 1100 K.⁵⁷ Investigation of Ag₆Ge₁₀P₁₂ in an open atmosphere showed decomposition above 750 K due to the evolution of P₄ gas.⁴⁸ By contrast, this composition is stable up to the melting point of 1000 K when heated inside a sealed quartz tube. Similarly Cd₃P₂ shows significant mass loss above 700 K in TGA. However, the sample can be consolidated using hot-pressing at 923 K.⁵⁵ ZnCu₂P₈ shows no thermal events during DSC measurements up to 960 K, above which mass loss is detected consistent with P₄ gas loss leaving behind the maximum oxidation state phases Cu₃P and Zn₃P₂.⁶³

Materials design can be used to remedy stability issues as demonstrated for Cu₂Se, where Cu ion conduction can be suppressed, improving stability under operating conditions.¹³⁰ The high zT in Cd₃P₂ is an outstanding result showing that good performance is possible, but it won't be possible to use this composition due to the toxicity of Cd. Some of the 122 and 111 phase contain more abundant and benign constituent elements. With further exploration of phosphide materials, electrical contacting, and measurement of mechanical and thermal properties will be essential to determining viability for device application.

5. Outlook

Metal phosphides have only very recently started to attract attention and are now approaching figures of merit, zT = 1 in both p- and n-type compositions. Whilst the overall performance lags behind those of state-of-the-art materials, this is a promising starting point for further exploration. Metal phosphides comprise a large class of materials with great structural and compositional (bonding) diversity afforded by the flexible oxidation state of P, from covalent P⁰/P⁻¹ to more ionic materials with P³⁻ anions. Hence, there is huge scope to discover better performing thermoelectrics.

The current best performing metal phosphides have low κ_L and are limited by relatively low μ_w. The low μ_w in the materials studied so far appears due to a relatively poor mobility (μ₀), when compared to thermoelectric materials with high S²σ. When analysed within the APS limit, large E_def are found, suggesting a strong coupling with the lattice. Even if APS is not the dominant scattering mechanism, this result demonstrates that the carrier scattering is relatively high, compared to materials with comparable . This leaves improving μ_w through , and in particular by increasing N_v as increasing degrades the mobility, as the most direct route to improve performance (eqn (2)–(4)). Investigation of high-zT tetrahedrite Ag₆Ge₁₀P₁₂ has found a highly converged band structure, whilst the large compositional range of 122- and 111-phases could afford opportunities for increasing N_v through solid-solution formation. It is also likely that materials with weaker APS (reduced electronic scattering) can be prepared. Targeted design of these materials would require a better understanding of the various carrier relaxation mechanisms. The observation of huge thermoelectric responses in the Weyl semimetal TaP, although driven by field induced increases in S, suggests that large power factors are possible in metal phosphides. Other now more routinely used optimisation strategies focused on defect-engineering,^6,8 including finding optimal dopants, control of intrinsic defects, nanostructuring, etc, have also not yet been widely applied to metal phosphides.

To conclude, metal phosphides remain a largely unexplored frontier area for thermoelectrics research, with many structures left to be discovered and explored. There is a clear need to improve understanding of the interplay between structure, bonding, and thermoelectric properties in metal phosphides, so that materials exploration can be accelerated. This should include using materials informatics and effective descriptors for high performance and first principles work focused on gaining microscopic understanding. There is no fundamental reason why higher zT values cannot be achieved. Given the established issues with high-temperature stability, this should focus on sub-ambient, room and moderate temperature applications.

Author contributions

Data analysis and visualisation (R. J. Q); manuscript writing – original draft (R. J. Q); conceptualisation, funding, project supervision, manuscript writing–review & editing (J. W. G. B.).

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

The Leverhulme Trust is acknowledged for funding the research on metal phosphide thermoelectrics (RPG-2020-177).

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