Magnetic Full-Heusler Compounds for Thermoelectric Applications

Abstract

Full-Heusler compounds exhibit a variety of magnetic properties such as non-magnetism, ferromagnetism, ferrimagnetism and anti-ferromagnetism. In recent years, they have attracted significant attention as potential thermoelectric (TE) materials that convert thermal energy directly into electricity. This chapter reviews the theoretical and experimental studies on the TE properties of magnetic full-Heusler compounds. In Section 1, a brief outline of TE power generation is described. Section 2 introduces the crystal structures and magnetic properties of full-Heusler compounds. The TE properties of full-Heusler compounds are presented in Sections 3 and 4. The relationship between magnetism, TE properties and order degree of full-Heusler compounds is elaborated.

1. Introduction

Thermoelectric (TE) power generation using TE devices is one of the key technologies to solve global energy problem, owing to its availability of direct conversion of thermal energy into electricity [1, 2, 3]. A schematic figure of a TE device is shown in Figure 1. It consists of n- and p-type TE materials connected in series electrically with metal electrodes and arranged thermally in parallel. The TE materials are wedged between ceramic plates. When one side of the device is heated and the other side is cooled, electrons and holes in the n- and p-type TE materials, respectively, diffuse from the hot side to the cold side, thus generating a flow of electric current.

Figure 1.

Schematic figure of a thermoelectric (TE) power generation device.

To commercialise TE devices, there is a need to improve their TE efficiency. The maximum TE efficiency, η_max, is an increasing function of the dimensionless figure-of-merit, zT, expressed as:

$${\eta }_{\max }=\frac{T_{\mathrm{H}}-T_{\mathrm{C}}}{T_{\mathrm{H}}}\frac{\sqrt{1+\mathit{zT}}-1}{\sqrt{1+\mathit{zT}}+T_{\mathrm{C}}/T_{\mathrm{H}}},$$E1

where T_H and T_C are the heating and cooling temperature, respectively. The dimensionless figure-of-merit, zT, is determined by TE properties (S: Seebeck coefficient, σ: electrical conductivity, κ: thermal conductivity) of the individual TE materials in the device.

$$\mathit{zT}=\frac{S^2\sigma }{\kappa }T,$$E2

where T is the absolute temperature. The product S²σ is called the power factor (PF), which is a measure of electric power generated using the TE material. To achieve high TE efficiency (standard levels for practical use are zT > 1 and PF > 2 × 10⁻³ W/K²m), high S, high σ and low κ are required. To meet these requirements, a variety of TE materials have been explored, such as chalcogenides, skutterudites, clathrates, silicides, Zintl compounds, half-Heusler compounds and oxides [1, 2, 3]. Most of these materials are semiconductors because in general they have high S than metals. However, recent theoretical and experimental studies have revealed that metals, in particular, half-metallic full-Heusler compounds have relatively high S as well as high σ. In addition, their junction with a metal electrode is robust compared to that of semiconductors, which is also an advantage.

In Section 2, the crystal structures and magnetic properties of full-Heusler compounds are introduced. Sections 3 and 4 demonstrate that magnetic full-Heusler compounds are promising for the TE power generation device.

2. Crystal structures and magnetic properties of full-Heusler compounds

The physical properties of full-Heusler compounds depend on their crystal structures. As shown in Figure 2, there are several types of crystal structures with different order degrees [4, 5, 6]. The full-Heusler compounds have four interpenetrating fcc sublattices, and each sublattice consists of the X, X’, Y or Z atom. The X, X’ and Y atoms are transition metals, whereas Z is a main group element. In some cases, the Y atom is a rare earth element or an alkaline earth metal.

Figure 2.

Crystal structures of full-Heusler compounds. The Strukturbericht symbol and a prototype structure are written above and below each crystal structure, respectively.

When the X and X’ atoms are of the same element, the chemical composition of the compounds is written as X₂YZ, which generally crystallises in the L2₁ structure. The prototype of the L2₁ structure is Cu₂MnAl (space group: Fm$\overline{3}$m). The Cu atoms occupy the 8c (1/4 1/4 1/4) site, whereas the Mn and Al atoms occupy the 4b (1/2 1/2 1/2) and 4a (0 0 0) sites, respectively. The L2₁ structure is a highly ordered structure of the full-Heusler compounds. Disorder among the Cu, Mn and/or Al atoms, that is, antisite defects, gives rise to different crystal structures. In a case where the Mn and Al atoms are evenly located at the 4b and 4a sites, the Cu₂MnAl becomes the B2 structure. Its prototype is CsCl (space group: Pm$\overline{3}$m). In a fully disordered phase, all the atoms are randomly distributed in the 8c, 4b and 4a sites, thus resulting in the A2 structure. In such a structure, all the sites are equivalent, which are expressed as a bcc lattice (prototype: W, space group: Im$\overline{3}$m). There are other disordered phases, including the DO₃ and B32a structures. The former is caused by the random distribution of the X, X’ and Y atoms at the 8c and 4b sites (prototype: BiF₃, space group: Fm$\overline{3}$m). In the B32a structure, the 8a (0 0 0) and 8b (1/2 1/2 1/2) sites are occupied by the X/Y and X’/Z atoms, respectively. The prototype is NaTl (space group: Fd$\overline{3}$m).

When the X’ and Y atoms are of the same element, the chemical composition becomes XX’₂Z, which crystallises in the X (XA or X_a) structure. This structure is called the inverse Heusler phase. The prototype is CuHg₂Ti (or AgLi₂Sb), and the space group is F$\overline{4}$3 m. In the structure, the X and Z atoms occupy the 4d (3/4 3/4 3/4) and 4a (0 0 0) sites, respectively, and the X’ atoms occupy the 4b (1/2 1/2 1/2) and 4c (1/4 1/4 1/4) sites.

In addition to the above ternary full-Heusler compounds, there are quaternary full-Heusler compounds, XX’YZ, which crystallise in the Y structure (prototype: LiMgPdSn or LiMgPdSb, space group: F$\overline{4}$3 m). The X, X’, Y and Z atoms are situated at the 4d, 4b, 4c and 4a sites, respectively, occupying one of the fcc sublattices. It should be noted that the inverse Heusler and the quaternary full-Heusler phases are ordered phases, and any disorder among the constituent atoms causes a structural change; the structure changes to the B2, A2, DO₃ or B32a structure.

Earlier theoretical studies demonstrated a half-metallic nature in full-Heusler compounds [7, 8]. Since then, many studies have been dedicated to investigate the electronic and magnetic properties of ternary and quaternary full-Heusler compounds. It has been revealed that full-Heusler compounds exhibit a variety of electronic properties; they exhibit the properties of semiconductors [9, 10, 11, 12, 13, 14, 15, 16, 17, 18], spin-gapless semiconductors (SGSs) [19, 20, 21, 22, 23, 24, 25, 26], semimetals [27, 28, 29], metals [30, 31, 32, 33, 34] and half-metals (HMs) [32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78]. Considering the magnetic properties, they have been reported to exhibit nonmagnetism [9, 10, 11, 14, 15, 16, 17, 18], ferromagnetism [12, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78], ferrimagnetism [13, 30, 35, 47, 59, 60, 67] and antiferromagnetism [25, 26, 34]. The full-Heusler, inverse Heusler and quaternary Heusler compounds obey the Slater-Pauling rule [79, 80, 81]: the total spin magnetic moment per unit cell scales with the total number of valence electrons in the unit cell.

3. Thermoelectric properties of half-metallic full-Heusler compounds

In this section, we present some of the theoretical and experimental studies on the TE properties of half-metallic full-Heusler compounds. The TE properties can be calculated on the basis of the Boltzmann transport equations [82, 83, 84]. Using the electronic energy-wavenumber dispersion curve of the i-th band ε_i(k), the tensors of the Seebeck coefficient, S(T), electrical conductivity, σ(T), and carrier thermal conductivity, κ_e(T), can be expressed as:

$$\mathit{S}\left(T\right)=-\frac{1}{\left|e\right|T}\frac{{\int }_{-\infty }^{+\infty }\stackrel{\sim }{\mathit{\sigma }}\left(\epsilon ,T\right)\left(\epsilon -{\epsilon }_{\mathrm{F}}\right)\left(-\frac{\partial f_{\mathrm{FD}}\left(\epsilon ,T\right)}{\partial \epsilon }\right)\mathrm{d}\epsilon }{\mathit{\sigma }\left(T\right)},$$E3

$$\mathit{\sigma }\left(T\right)={\int }_{-\infty }^{+\infty }\stackrel{\sim }{\mathit{\sigma }}\left(\epsilon ,T\right)\left(-\frac{\partial f_{\mathrm{FD}}\left(\epsilon ,T\right)}{\partial \epsilon }\right)\mathrm{d}\epsilon ,$$E4

$${\mathit{\kappa }}_{\mathrm{e}}\left(T\right)=-\frac{1}{e^{2}T}{\int }_{-\infty }^{+\infty }\stackrel{\sim }{\mathit{\sigma }}\left(\epsilon ,T\right){\left(\epsilon -{\epsilon }_{\mathrm{F}}\right)}^2\left(-\frac{\partial f_{\mathrm{FD}}\left(\epsilon ,T\right)}{\partial \epsilon }\right)\mathrm{d}\epsilon -\frac{1}{e^{2}T}\frac{{\left\{{\int }_{-\infty }^{+\infty }\stackrel{\sim }{\mathit{\sigma }}\left(\epsilon ,T\right)\left(\epsilon -{\epsilon }_{\mathrm{F}}\right)\left(-\frac{\partial f_{\mathrm{FD}}\left(\epsilon ,T\right)}{\partial \epsilon }\right)\mathrm{d}\epsilon \right\}}^2}{\mathit{\sigma }\left(T\right)},$$E5

$${\stackrel{\sim }{\sigma }}_{\mathit{\alpha \beta }}\left(\epsilon ,T\right)\equiv \frac{1}{N_k}\sum _{i,k}\frac{e^2\tau \left(k,T\right)}{{\mathrm{\hslash }}^2}\frac{\partial {\epsilon }_i\left(k\right)}{\partial k_{\alpha }}\frac{\partial {\epsilon }_i\left(k\right)}{\partial k_{\beta }}\delta \left(\epsilon -{\epsilon }_i\left(k\right)\right),\left(\alpha ,\beta =x,y,z\right),$$E6

where e, ε, ε_F, f_FD(ε, T), N_k, τ(k, T),$\hslash$ and $\stackrel{\sim }{\mathit{\sigma }}\left(\epsilon ,T\right)$ are the elementary charge, electron energy, Fermi level, Fermi-Dirac distribution function, total number of the k-points, relaxation time, Dirac constant and conductance spectrum tensor, respectively. It is difficult to calculate the relaxation time; hence, the calculation of TE properties generally gives S(T), σ(T)/τ and κ_e(T)/τ [84]. In context to magnetic materials, the electronic states of the majority and minority spin electrons are considered. Assuming that τ for the majority and minority spin electrons is the same, the total S for the magnetic materials, S_tot(T), is calculated by

$$S_{\mathrm{tot}}\left(T\right)=\frac{S_{\uparrow }\left(T\right){\sigma }_{\uparrow }\left(T\right)/\tau +S_{\downarrow }\left(T\right){\sigma }_{\downarrow }\left(T\right)/\tau }{{\sigma }_{\uparrow }\left(T\right)/\tau +{\sigma }_{\downarrow }\left(T\right)/\tau }=\frac{S_{\uparrow }\left(T\right){\sigma }_{\uparrow }\left(T\right)+S_{\downarrow }\left(T\right){\sigma }_{\downarrow }\left(T\right)}{{\sigma }_{\uparrow }\left(T\right)+{\sigma }_{\downarrow }\left(T\right)},$$E7

where S and σ with the up- and down-arrow subscripts those evaluated from the electronic states of the majority and minority spin electrons, respectively.

Figure 3(a) and (b) shows the temperature dependence of the calculated S_tot for ternary and quaternary half-metallic full-Heusler compounds, respectively. To calculate the electronic band, the full-potential linearised augmented plane wave (FLAPW) method was employed, adopting the local spin density approximation (LSDA) or the generalised gradient approximation in the Perdew-Burke-Ernzerhof parametrisation (PBE-GGA) as the local exchange-correlation potential. As seen in the figure, the negative and positive S_tot are presented, indicating that both n-type and p-type materials can be obtained from half-metallic full-Heusler compounds. The S_tot is observed to increase with increasing temperature for almost all the compounds, which is the typical behaviour of metal. Furthermore, the S_tot is observed to attain values as high as several tens of μV/K. These values are lower than those of TE semiconductors but higher than those of common metals, demonstrating the potential of half-metallic full-Heusler compounds as high-temperature TE materials.

Figure 3.

Temperature dependence of the calculated S_tot for half-metallic full-Heusler compounds. Their crystal structures are also shown. The calculation of the electronic structure was performed using the full-potential linearised augmented plane wave (FLAPW) method with local spin density approximation (LSDA) or generalised gradient approximation in the Perdew-Burke-Ernzerhof parametrisation (PBE-GGA).

The temperature dependence of S for several half-metallic Co-based full-Heusler compounds was determined by Balke et al. [37] and Hayashi et al. [53]. For the measurements, the S_tot values for the compounds were obtained. Hereafter, we use S to represent S_tot. As shown in Figure 4(a)–(c), the Co-based full-Heusler compounds exhibit negative S in the order of several tens of μV/K. For metals, the sign of S is well explained by Mott’s formula [85]:

Figure 4.

Temperature dependence of the measured S of several Co-based full-Heusler compounds. ((a) and (b) Reprinted from [37]. Copyright 2010, with permission from Elsevier. (c) Reprinted from [53]. Copyright 2017, with permission from Springer).

$$S\propto -\frac{1}{\mathrm{DOS}\left({\epsilon }_{\mathrm{F}}\right)}{\left.\frac{\mathrm{d}\mathrm{DOS}\left(\epsilon \right)}{\mathrm{d}\epsilon }\right|}_{\epsilon ={\epsilon }_{\mathrm{F}}},$$E8

where DOS is the electronic density of states. Adopting Eq. (8) for the partial DOS of the sp-electrons and d-electrons of Co₂MnSi, it was obtained that in half-metallic full-Heusler compounds, the itinerant sp-electrons contribute more to S than the localised d-electrons [53]. In Figure 4, Co₂TiAl is shown to exhibit the highest |S| of |−56| μV/K at 350 K among other compounds. It is observed that Co₂TiSi, Co₂TiGe and Co₂TiSn exhibit a characteristic temperature dependence of |S|; the value of |S| increases with increasing temperature and becomes constant at temperatures above 350 K. This characteristic behaviour is further discussed later in this section.

The half-metallic full-Heusler compounds are predicted to have high electrical conductivity σ owing to their metallic properties; hence, they are considered to be superior to the semiconductors. Figure 5(a) shows the temperature dependence of the measured σ for several Co-based full-Heusler compounds [53]. The σ values of the compounds are observed to be high, ranging from 10⁵ to 10⁷ S/m. Among all the compounds, Co₂MnSi exhibits the highest σ in the whole temperature range. The σ value of Co₂MnSi decreases from 4.6 × 10⁶ S/m at 300 K to 4.7 × 10⁵ S/m at 1000 K. This is a typical electrical conductivity-temperature relation in metals. From the S and σ values (shown in Figures 4(c) and 5(a), respectively), the PF was calculated and plotted in Figure 5(b) [53]. Owing to the high S and high σ, Co₂MnSi exhibits the highest PF (2.9 × 10⁻³ W/K²m at 500 K) among other compounds, which is comparable to that of a Bi₂Te₃-based material [86]. Since Co₂MnSi exhibits a negative S, it could be a potential n-type TE material. Thus, to develop a TE device using full-Heusler compounds, a p-type counterpart to Co₂MnSi is needed. For this purpose, Li et al. [60, 78] prepared a half-metallic Mn₂VAl compound and measured its TE properties. Although Mn₂VAl is a p-type material showing positive S, its highest PF (2.84 × 10⁻⁴ W/K²m at 767 K [78]) is lower than that of Co₂MnSi. Thus, there is a need to explore more p-type half-metallic full-Heusler compounds with high PF.

Figure 5.

(a) Measured σ and (b) PF of several Co-based full-Heusler compounds as a function of temperature. (Reprinted from [53]. Copyright 2017, with permission from Springer).

Here, the temperature dependence of S for the various full-Heusler compounds is discussed. Comparing the calculated S values for Co₂TiSi, Co₂TiGe and Co₂TiSn (Figure 3(a)) with the measured values (Figure 5(b)), it is obtained that not only the temperature dependence but also the sign of the S values are different. As mentioned earlier, the measured S value is almost constant at temperatures above 350 K; however, the calculated values do not display such relation. To explain this difference, Barth et al. [38] considered the difference in the electronic structure of the ferromagnetic (FM) state and nonmagnetic (NM) states. They obtained that the FM-NM phase transition occurs around 350 K for Co₂TiSi, Co₂TiGe and Co₂TiSn [38]. Using the temperature dependence of S for the FM and NM states, S_FM(T) and S_NM(T), and that of the normalised magnetisation calculated by using the molecular field theory, M(T), a modified S value, S_FM + NM, can be calculated according to the formula [38]:

$$S_{\mathrm{FM}+\mathrm{NM}}\left(T\right)=\frac{S_{\mathrm{FM}}\left(T\right){\sigma }_{\mathrm{FM}}\left(T\right)M\left(T\right)+S_{\mathrm{NM}}\left(T\right){\sigma }_{\mathrm{NM}}\left(T\right)\left\{1-M\left(T\right)\right\}}{{\sigma }_{\mathrm{FM}+\mathrm{NM}}\left(T\right)},$$E9

where σ_FM + NM is the modified electrical conductivity of a mixture of FM and NM states weighted by using M(T). Although the above consideration is plausible, the calculated S_FM + NM values for Co₂TiSi, Co₂TiGe and Co₂TiSn (Figure 6) do not coincide with the measured values. The inconsistency between the S_FM + NM values and the measured ones is also observed in the case of Co₂CrAl, Co₂MnAl, Co₂MnSi, Co₂FeAl and Co₂FeSi [53].

Figure 6.

Temperature dependence of the calculated S of Co₂TiZ (Z = Si, Ge, Sn) considering the FM-NM phase transition. In the calculation, the chemical potential at T = 0, μ(0), was set to (a) ε_F and (b) 150 meV below ε_F. (Reprinted from [38]. Copyright 2010, with permission from American Physical Society).

It is suggested that the constant S value in the NM state for Co₂TiSi, Co₂TiGe and Co₂TiSn (Figure 4(b)) is governed by the relaxation time rather than by the electronic structure [38]. The S value is calculated by using Eq. (1), where both the numerator and denominator of the fraction are functions of relaxation time τ(k, T); τ is included in both numerator and denominator of the fraction through $\stackrel{\sim }{\mathit{\sigma }}\left(\epsilon ,T\right)$ described in Eq. (6). However, in the calculation, the τ in the numerator and denominator cancels each other. In addition, the total S is calculated assuming that τ for the majority and minority spin electrons is the same (Eq. (7)). The neglected τ in Eqs. (1) and (7) could be a reason for the difference in the temperature dependence of the calculated and measured S. Another possible reason for this discrepancy is the method employed in calculating the electronic structure. The calculation results shown in Figures 3 and 6 are based on the LSDA or PBE-GGA. The use of the onsite Hubbard interaction in combination with PBE-GGA, namely, PBE + U or GGA + U [51, 55, 70, 73], and the Tran-Blaha modified Becke-Johnson (TB-MBJ) [64, 73] gives electronic structures different from that obtained using the LSDA or PBE-GGA, which may lead to a temperature dependence of S well-fitted to the measured one.

Also, defect and/or disorder in full-Heusler compounds affect the temperature dependence, as well as the sign of S, which could be another reason for the discrepancy in the temperature dependence of S. The structure model used for the calculation in Figures 3 and 6 is the L2₁, X or Y structure, which is highly ordered phases, devoid of any defect, for the ternary and quaternary full-Heusler compounds. Popescu et al. [52] investigated the effect of several defects on the temperature dependence of S for Co₂TiZ (Z = Si, Ge, Sn) in the FM state. As shown in Figure 7, off-stoichiometric defects, such as Co vacancy and the substitution of excess atoms at a particular site, change the sign of S.

Figure 7.

Change in calculated S for Co₂TiSi with several off-stoichiometric defects such as (a) Co vacancy (V_Co), (b) excess Co atoms at the Si site (Co_Si), (c) excess Co atoms at the Ti site (Co_Ti) and (d) excess Si atoms at the Ti site (Si_Ti). (Reprinted from [52]. Copyright 2017, with permission from American Physical Society).

The effect of structural disorder on S for Co₂CrAl, Co₂MnAl, Co₂MnSi, Co₂FeAl and Co₂FeSi has been obtained, as shown in Figure 8 [53]. The figure compares the calculated S_FM + NM with the measured S. It is observed that the measured values of S are individually higher than the calculated value (S_FM + NM). Considering the crystal structure, Co₂CrAl, Co₂MnAl, Co₂MnSi, Co₂FeAl and Co₂FeSi are not in the fully ordered L2₁ structure; most of them crystallise in the disordered B2 and/or A2 structures. This result implies that the B2 and/or A2 structures exhibit higher S than the L2₁ structure. Recently, Li et al. [78] investigated the effect of structural disorder on the value of S for half-metallic Mn₂VAl compounds by varying the B2 order degree. Figure 9(a) shows the measured S values for Mn₂VAl with the B2 order degree of 27 and 66%. The S values for the structure having 66% B2 order degree are observed to be higher than those for 27% B2 order degree in the entire measurement temperature range. In addition, it is observed that the S value increases with increasing the B2 order degree (Figure 9(b)). The increase in the B2 order degree means an increase in the disorder between the V and Al atoms, that is, a decrease in the L2₁ order degree. To understand the reason for the difference in S between the L2₁ and B2 structures, the DOS of Mn₂VAl with the L2₁ and B2 structures was calculated by using the Korringa-Kohn-Rostoker method. It was obtained that the B2 structure exhibits a steeper DOS of the majority-spin sp-electrons than the L2₁ structure, which is considered as the main reason for the higher S of the B2 structure than that of the L2₁ structure. Further increase in the B2 order degree is expected to yield a higher S for Mn₂VAl. The modulation of the order degree can be a key strategy to enhance the S value of the half-metallic full-Heusler compounds; the disorder in Co₂CrAl, Co₂MnAl, Co₂MnSi, Co₂FeAl, Co₂FeSi and Mn₂VAl gives rise to the higher S. To establish this strategy, the effects of the order degree, not only on S but also on σ, should be investigated for several half-metallic full-Heusler compounds.

Figure 8.

Comparison between the calculated S_FM + NM and the measured S at 300 K for several Co-based full-Heusler compounds. (Reprinted from [53]. Copyright 2017, with permission from Springer).

Figure 9.

(a) Temperature dependence of the measured S of Mn₂VAl with the B2 order degree of 27 and 66%. (b) Measured S values of Mn₂VAl at 767 K plotted against the B2 order degree. (Reprinted from [78]. Copyright 2020, with permission from IOP Publishing).

Considering the TE performance of the half-metallic full-Heusler compounds, not only PF but also zT are important. To evaluate the zT of Co₂MnSi, we obtained the temperature dependence of the total thermal conductivity, κ_tot (Figure 10(a)). Similar to the case of common metals, a high κ_tot was obtained. It decreases with increasing temperature from 79 W/Km at 300 K to 21 W/Km at 1000 K. Figure 10(b) shows the temperature dependence of zT for Co₂MnSi calculated using the PF value (Figure 5(b)) and the κ_tot value (Figure 10(a)). Due to the high κ_tot, the maximum zT value, zT_max, of Co₂MnSi is 0.039, which is obtained at temperatures above 900 K. Although this zT_max value is far below the standard level of zT = 1, it is higher than that of Co₂TiSn (0.033 at 370–400 K) [38] and those of semi-metallic full-Heusler compounds (0.0052 at 300 K for Ru₂NbAl [28] and 0.0027 at 300 K for Ru₂VAl_0.25Ga_0.75 [29]).

Figure 10.

Temperature dependence of (a) measured κ_tot, κ_e and κ_l and (b) evaluated zT of Co₂MnSi.

It should be noted that the κ_tot of Co₂MnSi is not equal to the carrier thermal conductivity, κ_e. The κ_e value can be calculated by using the Wiedemann-Frantz law, κ_e = LσT, where L is the Lorentz number. Evaluating the L value on the basis of the single parabolic band model [87] and using the measured σ value (Figure 5(a)), the κ_e value of Co₂MnSi was calculated and plotted in Figure 10(a). It can be observed from the figure that κ_e is only half as high as κ_tot. The rest is attributed to the lattice thermal conductivity, κ_l (=κ_tot − κ_e), as shown in Figure 10(a), which amounts to a half of the κ_tot. This is contrary to the case of common metals where the κ_tot is mainly dominated by κ_e [88]. The non-negligible κ_l suggests that, for the theoretical evaluation of zT of the half-metallic full-Heusler compounds, the contribution of κ_l should not be ignored. Experimentally, the high contribution of κ_l to κ_tot indicates that the κ_tot of half-metallic full-Heusler compounds could be reduced by decreasing the κ_l.

4. Future prospects of magnetic full-Heusler compounds as potential thermoelectric materials

In this section, we introduce other full-Heusler compounds to demonstrate the potentials of magnetic full-Heusler compounds in TE applications. First, we consider the full-Heusler SGSs as an example. Schematic illustrations of the DOS of SGSs and HMs are shown in Figure 11. The DOS of SGSs has an open band gap in one spin electron and a closed gap in the other. Since the Fermi level ε_F is located just at the closed gap, the electron or hole concentration in SGSs is expected to be less than that in HMs. One of the investigated SGSs is the full-Heusler Mn₂CoAl, which crystallises in the X structure (the inverse Heusler phase). The variation of its σ, S and carrier concentration, n, with temperature is shown in Figure 12, as determined by Ouardi et al. [19]. It can be observed that the σ and n vary slightly with the temperature, which is attributed to the typical behaviour of gapless semiconductors [89]. In addition, the S value is nearly equal to 0 μV/K. The reduced Seebeck effect indicates the occurrence of electron and hole compensation, which is the evidence that ε_F is at the top of the valence states and at the bottom of the conduction states.

Figure 11.

Schematic illustration of DOS for spin-gapless semiconductors (SGSs) and half-metals (HMs).

Figure 12.

Temperature dependence of the measured (a) σ, (b) S and (c) n of Mn₂CoAl. (Reprinted from [19]. Copyright 2020, with permission from American Physical Society).

Owing to the nearly zero S values, Mn₂CoAl cannot be used as a TE material; however, there is a possibility of achieving high |S| in Mn₂CoAl by tuning the position of ε_F. The position of ε_F can be varied via partial substitution, which increases the hole or electron carrier concentration in Mn₂CoAl. We calculated the S value for the partially substituted Mn₂CoAl, as shown in Figure 13(a). The calculation was based on a rigid band model; thus, the electronic structure of the partially substituted Mn₂CoAl is assumed to be the same as that of Mn₂CoAl. In the figure, the horizontal axis is μ-ε_F, where μ and ε_F are the chemical potential (i.e., the Fermi level) of the partially substituted Mn₂CoAl and the Fermi level of Mn₂CoAl, respectively. A negative/positive μ-ε_F value means an increase in the hole/electron carrier concentration. Although the value of S at μ = ε_F, corresponding to the case of Mn₂CoAl, is small, it is large at μ = ε_F − 0.184 and at μ = ε_F + 0.053 (pointed by orange and green arrows, respectively). The temperature dependences of S at μ = ε_F, μ = ε_F − 0.184 and μ = ε_F + 0.053 are shown in Figure 13(b), which again demonstrates that high |S| values can be achieved for both p-type and n-type regions. These calculation results prove the full-Heusler SGSs as potential materials for TE applications.

Figure 13.

(a) Calculated S value at 300 K for the partially substituted Mn₂CoAl plotted as a function of μ-ε_F, where μ and ε_F are the Fermi levels of partially substituted Mn₂CoAl and that of Mn₂CoAl, respectively. The highest |S| values are obtained at μ = ε_F − 0.184 and at μ = ε_F + 0.053, as denoted by orange and green arrows, respectively. (b) Temperature dependence of S at μ = ε_F, μ = ε_F − 0.184 and μ = ε_F + 0.053.

To achieve high |S| values for Mn₂CoAl, it is important to retain its SGS characteristic. Galanakis et al. [90] theoretically investigated the effects of structural disorder on the electronic structure of Mn₂CoAl. It was obtained that the SGS characteristic is not conserved in the presence of Mn-Co, Mn-Al and Co-Al antisite defects. Instead of the closed gap, low DOS intensity emerges in the electronic structure of the majority spin electrons around ε_F, indicating that the disorder induces half-metallic characteristics in Mn₂CoAl. Also, Xu et al. [91] reported that an as-prepared Mn₂CoAl compound is non-stoichiometric and contains the Mn-Co antisite defect. In a case where Mn₂CoAl is not an SGS, the |S| cannot be increased via partial substitutions.

Other examples considered here are the full-Heusler compounds having low values of κ_l. Figure 14 shows a flower-like microstructure of Co₂Dy_0.5Mn_0.5Sn observed by Schwall et al. [43]. Although the chemical composition of Co₂Dy_0.5Mn_0.5Sn coincides with that of the full-Heusler phase, Co₂Dy_0.5Mn_0.5Sn is not in a single phase. It consists of two major phases: half-metallic Co₂MnSn and ferromagnetic Co₈Dy₃Sn₄ phases. This phase separation is induced by rapid cooling from the liquid phase. Consequently, the κ_l value of Co₂Dy_0.5Mn_0.5Sn is lower than those of Co₂MnSn and Co₈Dy₃Sn₄.

Figure 14.

Flower-like microstructure of Co₂Dy_0.5Mn_0.5Sn. (a) Elemental mappings, (b) combined image of elemental mappings shown in (a). (c) Line scan along the line indicated in (b). (Reprinted from [43]. Copyright 2012, with permission from WILEY-VCH).

He et al. [9] theoretically discovered a new class of stable nonmagnetic full-Heusler semiconductors with high PF and ultralow κ_l, attributed to atomic rattling. The compounds contain alkaline earth elements (Ba, Sr or Ca) in the X sublattice and noble metals (Au or Hg) and main group elements (Sn, Pb, As or Sb) in the Y and Z sublattices, respectively. The κ_l value of Ba₂AuBi and Ba₂HgPb was obtained to be lower than 0.5 W/Km at 300 K. At higher temperatures, it was close to the theoretical minimum, that is, the amorphous limit of 0.27 W/Km [92]. Park et al. [16] further examined the TE properties of Ba₂BiAu. They predicted that considerably high zT of ∼5 can be achieved at 800 K.

Finally, there are many ternary and quaternary full-Heusler compounds yet to be explored. Among the full-Heusler compounds, nonmagnetic Fe₂VAl-based compounds have been intensively investigated as one of the potential TE semiconductors [93]. Despite the long historical investigation, Hinterleitner et al. [94] discovered quite recently that a metastable Fe₂V_0.8W_0.2Al thin film exhibits extremely high zT of ∼6 at 350 K as a result of its high S. The crystal structure of the thin film is reported to be the disordered A2 structure, which could be the reason for its high S, as in the cases of several half-metallic Co-based and Mn-based full-Heusler compounds [53, 78]. If the disorder in structure contributes to the high S, then the strategy of enhancing zT by controlling structural disorder would be applicable to the other full-Heusler compounds. Herewith, more conventional and novel findings on the full-Heusler compounds can be achieved.

To explore the potentials of the full-Heusler compounds, theoretical studies are vital to minimise the experimental tasks. Figure 15 exhibits a plot of S versus σ/τ at 1000 K for several Co-based and Mn-based full-Heusler compounds, as calculated by Li et al. [60]. Furthermore, recent advancements in machine learning dispel the difficulty in searching novel full-Heusler compounds [95, 96]. Combining such calculations with experiments, we can effectively discover magnetic full-Heusler compounds with much higher TE efficiency, which promises the realisation of high-efficiency TE power generation devices.

Figure 15.

Calculated S versus σ/τ at 1000 K for several Co-based and Mn-based full-Heusler compounds. The grey curves indicate PF/τ. (Reprinted from [60]. Copyright 2018, with permission from Elsevier).

Acknowledgments

We greatly acknowledge the financial supports from the Thermal and Electric Energy Technology Foundation and from the Tsinghua-Tohoku Collaborative Research Fund.

Conflict of interest

We declare that there is no conflict of interest.

Nomenclatures

η_max [−]

maximum TE efficiency

zT [−]

dimensionless figure-of-merit

T [K]

absolute temperature

S [V/K]

Seebeck coefficient

σ [S/m]

electrical conductivity

κ [W/Km]

thermal conductivity

PF [W/K²m]

power factor

ε_i [eV]

electronic energy–wavenumber dispersion curve of the i-th band

k [m⁻¹]

wavenumber

S [V/K]

Seebeck coefficient tensor

σ [S/m]

electrical conductivity tensor

κ_e [W/Km]

carrier thermal conductivity tensor

e = 1.60217662 × 10⁻¹⁹ C

elementary charge

ε [eV]

electron energy

ε_F [eV]

Fermi level

f_FD [−]

Fermi-Dirac distribution function

N_k [−]

total number of k-points

τ [1/s]

relaxation time

$\hslash$ = 6.582119569 × 10⁻¹⁶ eVs

Dirac constant

$\stackrel{\sim }{\mathit{\sigma }}$ [S/m]

conductance spectrum tensor

DOS [states/eV]

density of states

M [−]

normalised magnetisation calculated by using molecular field theory

μ [eV]

chemical potential

κ_e [W/Km]

carrier thermal conductivity

L [WΩ/K²]

Lorentz number

κ_l [W/Km]

lattice thermal conductivity

n [1/m³]

carrier concentration