**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*<sup>F</sup> 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*<sup>F</sup> 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 *ε*F. The position of *ε*<sup>F</sup> 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*<sup>F</sup> are the chemical potential (i.e., the Fermi

**Figure 11.** *Schematic illustration of DOS for spin-gapless semiconductors (SGSs) and half-metals (HMs).*

**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

*Temperature dependence of (a) measured* κ*tot,* κ*<sup>e</sup> and* κ*<sup>l</sup> and (b) evaluated* zT *of Co2MnSi.*

*(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].*

It should be noted that the *κ*tot of Co2MnSi is not equal to the carrier thermal conductivity, *κ*e. The *κ*<sup>e</sup> value can be calculated by using the Wiedemann-Frantz

0.0027 at 300 K for Ru2VAl0.25Ga0.75 [29]).

*Copyright 2020, with permission from IOP Publishing).*

*Magnetic Materials and Magnetic Levitation*

**Figure 9.**

**Figure 10.**

**74**

To achieve high |*S*| values for Mn2CoAl, it is important to retain its SGS charac-

Other examples considered here are the full-Heusler compounds having low values of *κ*l. **Figure 14** shows a flower-like microstructure of Co2Dy0.5Mn0.5Sn

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 *κ*<sup>l</sup> value of Co2Dy0.5Mn0.5Sn is lower than

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*<sup>l</sup> 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 consider-

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 semiconduc-

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

*Flower-like microstructure of Co2Dy0.5Mn0.5Sn. (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,*

tors [93]. Despite the long historical investigation, Hinterleitner et al. [94] discovered quite recently that a metastable Fe2V0.8W0.2Al thin film exhibits

novel findings on the full-Heusler compounds can be achieved.

observed by Schwall et al. [43]. Although the chemical composition of

teristic. 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

*Magnetic Full-Heusler Compounds for Thermoelectric Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.92867*

via partial substitutions.

**Figure 14.**

**77**

*with permission from WILEY-VCH).*

those of Co2MnSn and Co8Dy3Sn4.

ably high *zT* of 5 can be achieved at 800 K.

**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).*

#### **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*<sup>F</sup> are the Fermi levels of partially substituted Mn2CoAl and that of Mn2CoAl, respectively. The highest |*S*| values are obtained at* μ *=* ε*<sup>F</sup> 0.184 and at* μ *=* ε*<sup>F</sup> + 0.053, as denoted by orange and green arrows, respectively. (b) Temperature dependence of* S *at* μ *=* ε*F,* μ *=* ε*<sup>F</sup> 0.184 and* μ *=* ε*<sup>F</sup> + 0.053.*

level) of the partially substituted Mn2CoAl and the Fermi level of Mn2CoAl, respectively. A negative/positive *μ*-*ε*<sup>F</sup> 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*<sup>F</sup> 0.184 and at *μ* = *ε*<sup>F</sup> + 0.053 (pointed by orange and green arrows, respectively). The temperature dependences of *S* at *μ* = *ε*F, *μ* = *ε*<sup>F</sup> 0.184 and *μ* = *ε*<sup>F</sup> + 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.

#### *Magnetic Full-Heusler Compounds for Thermoelectric Applications DOI: http://dx.doi.org/10.5772/intechopen.92867*

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 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 *κ*<sup>l</sup> 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 *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*<sup>l</sup> 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.

#### **Figure 14.**

*Flower-like microstructure of Co2Dy0.5Mn0.5Sn. (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).*

level) of the partially substituted Mn2CoAl and the Fermi level of Mn2CoAl, respectively. A negative/positive *μ*-*ε*<sup>F</sup> 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*<sup>F</sup> 0.184 and at *μ* = *ε*<sup>F</sup> + 0.053 (pointed by orange and green arrows, respectively). The temperature dependences of *S* at *μ* = *ε*F, *μ* = *ε*<sup>F</sup> 0.184 and *μ* = *ε*<sup>F</sup> + 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

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

*Temperature dependence of the measured (a)* σ*, (b)* S *and (c)* n *of Mn2CoAl. (Reprinted from [19]. Copyright*

applications.

**76**

**Figure 12.**

**Figure 13.**

*2020, with permission from American Physical Society).*

*Magnetic Materials and Magnetic Levitation*

**Nomenclatures**

PF [W/K<sup>2</sup>

*k* [m<sup>1</sup>

*L* [WΩ/K<sup>2</sup>

**Author details**

University, Sendai, Japan

provided the original work is properly cited.

*n* [1/m<sup>3</sup>

**79**

*η*max [] maximum TE efficiency *zT* [] dimensionless figure-of-merit

*Magnetic Full-Heusler Compounds for Thermoelectric Applications*

*DOI: http://dx.doi.org/10.5772/intechopen.92867*

*T* [K] absolute temperature *S* [V/K] Seebeck coefficient *σ* [S/m] electrical conductivity *κ* [W/Km] thermal conductivity

m] power factor

] wavenumber *S* [V/K] Seebeck coefficient tensor *σ* [S/m] electrical conductivity tensor *κ*<sup>e</sup> [W/Km] carrier thermal conductivity tensor

*<sup>e</sup>* = 1.60217662 <sup>10</sup><sup>19</sup> C elementary charge *ε* [eV] electron energy *ε*<sup>F</sup> [eV] Fermi level

*τ* [1/s] relaxation time = 6.582119569 <sup>10</sup><sup>16</sup> eVs Dirac constant

DOS [states/eV] density of states

*μ* [eV] chemical potential

*f*FD [] Fermi-Dirac distribution function

*M* [] normalised magnetisation calculated by using

molecular field theory

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

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Department of Applied Physics, Graduate School of Engineering, Tohoku

\*Address all correspondence to: hayashik@crystal.apph.tohoku.ac.jp

*Nk* [] total number of *k*-points

*σ*~ [S/m] conductance spectrum tensor

*κ*<sup>e</sup> [W/Km] carrier thermal conductivity

] carrier concentration

] Lorentz number *κ*<sup>l</sup> [W/Km] lattice thermal conductivity

*ε<sup>i</sup>* [eV] electronic energy–wavenumber dispersion curve of the *i*-th band

#### **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).*

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 highefficiency TE power generation devices.

### **Acknowledgements**

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.

*Magnetic Full-Heusler Compounds for Thermoelectric Applications DOI: http://dx.doi.org/10.5772/intechopen.92867*
