Open access peer-reviewed chapter - ONLINE FIRST

# Magnetic Full-Heusler Compounds for Thermoelectric Applications

By Kei Hayashi, Hezhang Li, Mao Eguchi, Yoshimi Nagashima and Yuzuru Miyazaki

Submitted: February 4th 2020Reviewed: May 18th 2020Published: June 30th 2020

DOI: 10.5772/intechopen.92867

## 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.

### Keywords

• full-Heusler compounds
• half-metal
• spin-gapless semiconductor
• thermoelectric properties
• order degree

## 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.

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

ηmax=THTCTH1+zT11+zT+TC/TH,E1

where TH and TC 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.

zT=S2σκT,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.

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: Fm3¯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: Pm3¯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: Im3¯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: Fm3¯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: Fd3¯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 Xa) structure. This structure is called the inverse Heusler phase. The prototype is CuHg2Ti (or AgLi2Sb), and the space group is F4¯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: F4¯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 εi(k), the tensors of the Seebeck coefficient, S(T), electrical conductivity, σ(T), and carrier thermal conductivity, κe(T), can be expressed as:

ST=1eT+σεTεεFfFDεTεdεσT,E3
σT=+σεTfFDεTεdε,E4
κeT=1e2T+σεTεεF2fFDεTεdε1e2T+σεTεεFfFDεTεdε2σT,E5
σαβεT1Nki,ke2τkT2εikkαεikkβδεεik,αβ=xyz,E6

where e, ε, εF, fFD(ε, T), Nk, τ(k, T),and σεTare 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 κe(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, Stot(T), is calculated by

StotT=STσT/τ+STσT/τσT/τ+σT/τ=STσT+STσTσT+σT,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 Stot 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 Stot are presented, indicating that both n-type and p-type materials can be obtained from half-metallic full-Heusler compounds. The Stot is observed to increase with increasing temperature for almost all the compounds, which is the typical behaviour of metal. Furthermore, the Stot 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.

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 Stot values for the compounds were obtained. Hereafter, we use S to represent Stot. 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]:

S1DOSεFdDOSεdεε=εF,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.

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, SFM(T) and SNM(T), and that of the normalised magnetisation calculated by using the molecular field theory, M(T), a modified S value, SFM + NM, can be calculated according to the formula [38]:

SFM+NMT=SFMTσFMTMT+SNMTσNMT1MTσFM+NMT,E9

where σFM + 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 SFM + NM values for Co2TiSi, Co2TiGe and Co2TiSn (Figure 6) do not coincide with the measured values. The inconsistency between the SFM + NM values and the measured ones is also observed in the case of Co2CrAl, Co2MnAl, Co2MnSi, Co2FeAl and Co2FeSi [53].

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 σεTdescribed 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.

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 SFM + NM with the measured S. It is observed that the measured values of S are individually higher than the calculated value (SFM + 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.

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, κtot (Figure 10(a)). Similar to the case of common metals, a high κtot 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 κtot value (Figure 10(a)). Due to the high κtot, the maximum zT value, zTmax, of Co2MnSi is 0.039, which is obtained at temperatures above 900 K. Although this zTmax 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 Ru2VAl0.25Ga0.75 [29]).

It should be noted that the κtot of Co2MnSi is not equal to the carrier thermal conductivity, κe. The κe value can be calculated by using the Wiedemann-Frantz law, κe = 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 κe value of Co2MnSi was calculated and plotted in Figure 10(a). It can be observed from the figure that κe is only half as high as κtot. The rest is attributed to the lattice thermal conductivity, κl (=κtotκe), as shown in Figure 10(a), which amounts to a half of the κtot. This is contrary to the case of common metals where the κtot is mainly dominated by κe [88]. The non-negligible κl suggests that, for the theoretical evaluation of zT of the half-metallic full-Heusler compounds, the contribution of κl should not be ignored. Experimentally, the high contribution of κl to κtot indicates that the κtot of half-metallic full-Heusler compounds could be reduced by decreasing the κl.

## 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 εF 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 εF is at the top of the valence states and at the bottom of the conduction states.

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 εF. The position of εF 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-εF, where μ and εF 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-εF value means an increase in the hole/electron carrier concentration. Although the value of S at μ = εF, corresponding to the case of Mn2CoAl, is small, it is large at μ = εF − 0.184 and at μ = εF + 0.053 (pointed by orange and green arrows, respectively). The temperature dependences of S at μ = εF, μ = εF − 0.184 and μ = εF + 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.

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 εF, 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 κl. Figure 14 shows a flower-like microstructure of Co2Dy0.5Mn0.5Sn observed by Schwall et al. [43]. Although the chemical composition of Co2Dy0.5Mn0.5Sn coincides with that of the full-Heusler phase, Co2Dy0.5Mn0.5Sn 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 κl value of Co2Dy0.5Mn0.5Sn is lower than those of Co2MnSn and Co8Dy3Sn4.

He et al. [9] theoretically discovered a new class of stable nonmagnetic full-Heusler semiconductors with high PF and ultralow κl, 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 κl 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 Fe2V0.8W0.2Al 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.

## 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

 ηmax [−] 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 εi [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 κe [W/Km] carrier thermal conductivity tensor e = 1.60217662 × 10−19 C elementary charge ε [eV] electron energy εF [eV] Fermi level fFD [−] Fermi-Dirac distribution function Nk [−] total number of k-points τ [1/s] relaxation time ℏ = 6.582119569 × 10−16 eVs Dirac constant σ∼ [S/m] conductance spectrum tensor DOS [states/eV] density of states M [−] normalised magnetisation calculated by using molecular field theory μ [eV] chemical potential κe [W/Km] carrier thermal conductivity L [WΩ/K2] Lorentz number κl [W/Km] lattice thermal conductivity n [1/m3] carrier concentration

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Kei Hayashi, Hezhang Li, Mao Eguchi, Yoshimi Nagashima and Yuzuru Miyazaki (June 30th 2020). Magnetic Full-Heusler Compounds for Thermoelectric Applications [Online First], IntechOpen, DOI: 10.5772/intechopen.92867. Available from: