**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 [82–84]. 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, *κ*e(*T*), can be expressed as:

$$\mathbf{S}(T) = -\frac{\mathbf{1}}{|\varepsilon|T} \frac{\int\_{-\infty}^{+\infty} \tilde{\sigma}(\varepsilon, T)(\varepsilon - \varepsilon\_{\mathrm{F}}) \left(-\frac{\partial f\_{\mathrm{FD}}(\varepsilon, T)}{\partial \varepsilon}\right) d\varepsilon}{\sigma(T)},\tag{3}$$

$$\sigma(T) = \int\_{-\infty}^{+\infty} \tilde{\sigma}(\varepsilon, T) \left( -\frac{\partial f\_{\rm FD}(\varepsilon, T)}{\partial \varepsilon} \right) \mathrm{d}\varepsilon,\tag{4}$$

than those of TE semiconductors but higher than those of common metals, demonstrating the potential of half-metallic full-Heusler compounds as high-temperature

*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).*

*Magnetic Full-Heusler Compounds for Thermoelectric Applications*

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

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

*<sup>S</sup>*<sup>∝</sup> � <sup>1</sup>

DOSð Þ *ε*<sup>F</sup>

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 halfmetallic 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

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 temperature dependence of *S* for several half-metallic Co-based full-Heusler

d DOSð Þ*ε* d*ε*

 *ε*¼*ε*<sup>F</sup>

, (8)

TE materials.

**Figure 3.**

this section.

**69**

of *S* is well explained by Mott's formula [85]:

$$\begin{split} \kappa\_{\varepsilon}(T) &= -\frac{1}{e^{2}T} \Big|\_{-\infty}^{+\infty} \tilde{\sigma}(\varepsilon, T)(\varepsilon - \varepsilon\_{\mathrm{F}})^{2} \Big( -\frac{\partial f\_{\mathrm{FD}}(\varepsilon, T)}{\partial \varepsilon} \Big) \mathrm{d}\varepsilon \\ &- \frac{1}{e^{2}T} \frac{\Big\{\int\_{-\infty}^{+\infty} \tilde{\sigma}(\varepsilon, T)(\varepsilon - \varepsilon\_{\mathrm{F}}) \left( -\frac{\partial f\_{\mathrm{FD}}(\varepsilon, T)}{\partial \varepsilon} \right) \mathrm{d}\varepsilon \right\}^{2} , \end{split} \tag{5}$$

$$\tilde{\sigma}\_{a\beta}(\varepsilon, T) \equiv \frac{1}{N\_k} \sum\_{i,k} \frac{e^2 \tau(k, T)}{\hbar^2} \frac{\partial e\_i(k)}{\partial k\_a} \frac{\partial e\_i(k)}{\partial k\_\beta} \delta(\varepsilon - e\_i(k)), (a, \beta = \infty, y, z), \tag{6}$$

where *e*, *ε*, *ε*F, *f*FD(*ε*,*T*), *Nk*, *τ*(*k*,*T*), and *σ*~ð Þ *ε*, *T* 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

$$\mathcal{S}\_{\text{tot}}(T) = \frac{\mathcal{S}\_{\uparrow}(T)\sigma\_{\uparrow}(T)/\tau + \mathcal{S}\_{\downarrow}(T)\sigma\_{\downarrow}(T)/\tau}{\sigma\_{\uparrow}(T)/\tau + \sigma\_{\downarrow}(T)/\tau} = \frac{\mathcal{S}\_{\uparrow}(T)\sigma\_{\uparrow}(T) + \mathcal{S}\_{\downarrow}(T)\sigma\_{\downarrow}(T)}{\sigma\_{\uparrow}(T) + \sigma\_{\downarrow}(T)},\tag{7}$$

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

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

**Figure 3.**

magnetic moment per unit cell scales with the total number of valence electrons in

**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 [82–84]. 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

�<sup>∞</sup> *<sup>σ</sup>*~ð Þ *<sup>ε</sup>*, *<sup>T</sup>* ð Þ� *<sup>ε</sup>* � *<sup>ε</sup>*<sup>F</sup>

*σ*~ð Þ� *ε*, *T*

*σ*~ð Þ *ε*, *T* ð Þ *ε* � *ε*<sup>F</sup>

�<sup>∞</sup> *<sup>σ</sup>*~ð Þ *<sup>ε</sup>*, *<sup>T</sup>* ð Þ� *<sup>ε</sup>* � *<sup>ε</sup>*<sup>F</sup>

*<sup>∂</sup>εi*ð Þ*<sup>k</sup> ∂k<sup>β</sup>*

where *e*, *ε*, *ε*F, *f*FD(*ε*,*T*), *Nk*, *τ*(*k*,*T*), and *σ*~ð Þ *ε*, *T* 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

*<sup>σ</sup>*↑ð Þ *<sup>T</sup> <sup>=</sup><sup>τ</sup>* <sup>þ</sup> *<sup>σ</sup>*↓ð Þ *<sup>T</sup> <sup>=</sup><sup>τ</sup>* <sup>¼</sup> *<sup>S</sup>*↑ð Þ *<sup>T</sup> <sup>σ</sup>*↑ð Þþ *<sup>T</sup> <sup>S</sup>*↓ð Þ *<sup>T</sup> <sup>σ</sup>*↓ð Þ *<sup>T</sup>*

where *S* and *σ* with the up- and down-arrow subscripts those evaluated from the

**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

*<sup>∂</sup> <sup>f</sup>* FDð Þ *<sup>ε</sup>*, *<sup>T</sup> ∂ε* � �

<sup>2</sup> � *<sup>∂</sup> <sup>f</sup>* FDð Þ *<sup>ε</sup>*, *<sup>T</sup> ∂ε* � �

*<sup>∂</sup> <sup>f</sup>* FDð Þ *<sup>ε</sup>*, *<sup>T</sup> ∂ε* � �

*<sup>∂</sup> <sup>f</sup>* FDð Þ *<sup>ε</sup>*, *<sup>T</sup> ∂ε* � �

n o<sup>2</sup>

d*ε <sup>σ</sup>*ð Þ *<sup>T</sup>* , (3)

d*ε*, (4)

d*ε*

d*ε*

*<sup>σ</sup>*ð Þ *<sup>T</sup>* , (5)

*δ ε*ð Þ � *εi*ð Þ*k* ,ð Þ *α*, *β* ¼ *x*, *y*, *z* , (6)

*<sup>σ</sup>*↑ð Þþ *<sup>T</sup> <sup>σ</sup>*↓ð Þ *<sup>T</sup>* , (7)

the unit cell.

conductivity, *κ*e(*T*), can be expressed as:

*Magnetic Materials and Magnetic Levitation*

*<sup>S</sup>*ð Þ¼� *<sup>T</sup>* <sup>1</sup>

*<sup>κ</sup>*eð Þ¼� *<sup>T</sup>* <sup>1</sup>

*<sup>σ</sup>*~*αβ*ð Þ� *<sup>ε</sup>*, *<sup>T</sup>* <sup>1</sup>

**68**

*Nk* X *i*, *k*

*<sup>S</sup>*totð Þ¼ *<sup>T</sup> <sup>S</sup>*↑ð Þ *<sup>T</sup> <sup>σ</sup>*↑ð Þ *<sup>T</sup> <sup>=</sup><sup>τ</sup>* <sup>þ</sup> *<sup>S</sup>*↓ð Þ *<sup>T</sup> <sup>σ</sup>*↓ð Þ *<sup>T</sup> <sup>=</sup><sup>τ</sup>*

j j*e T*

*σ*ð Þ¼ *T*

*e*<sup>2</sup>*T*

*<sup>e</sup>*<sup>2</sup>*τ*ð Þ *<sup>k</sup>*, *<sup>T</sup>* ℏ2

� 1 *e*<sup>2</sup>*T* Ð <sup>þ</sup><sup>∞</sup>

ðþ<sup>∞</sup> �∞

ðþ<sup>∞</sup> �∞

Ð <sup>þ</sup><sup>∞</sup>

*<sup>∂</sup>εi*ð Þ*<sup>k</sup> ∂k<sup>α</sup>*

same, the total *S* for the magnetic materials, *S*tot(*T*), is calculated by

electronic states of the majority and minority spin electrons, respectively.

*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).*

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 *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]:

$$\mathcal{S}\infty - \frac{1}{\mathcal{D}\mathcal{OS}(\varepsilon\_{\mathcal{F}})} \frac{\mathrm{d}\,\mathrm{DOS}(\varepsilon)}{\mathrm{d}\varepsilon}\Big|\_{\varepsilon=\varepsilon\_{\mathcal{F}}},\tag{8}$$

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 halfmetallic 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

value of Co2MnSi decreases from 4.6 � <sup>10</sup><sup>6</sup> S/m at 300 K to 4.7 � <sup>10</sup><sup>5</sup> 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

*Magnetic Full-Heusler Compounds for Thermoelectric Applications*

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

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, *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

*<sup>S</sup>*FMþNMð Þ¼ *<sup>T</sup> <sup>S</sup>*FMð Þ *<sup>T</sup> <sup>σ</sup>*FMð Þ *<sup>T</sup> M T*ð Þþ *<sup>S</sup>*NMð Þ *<sup>T</sup> <sup>σ</sup>*NMð Þ *<sup>T</sup>* f g <sup>1</sup> � *M T*ð Þ

dependence, as well as the sign of *S*, which could be another reason for the

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,

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 *σ*~ð Þ *ε*, *T* 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

m at 500 K) among other compounds, which is

*<sup>σ</sup>*FMþNMð Þ *<sup>T</sup>* , (9)

m at 767 K [78]) is lower than that of Co2MnSi. Thus, there is

the highest PF (2.9 � <sup>10</sup>�<sup>3</sup> W/K<sup>2</sup>

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

PF (2.84 � <sup>10</sup>�<sup>4</sup> W/K<sup>2</sup>

formula [38]:

**71**

Co2FeAl and Co2FeSi [53].

#### **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).*

#### **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).*

the measured *σ* for several Co-based full-Heusler compounds [53]. The *σ* values of the compounds are observed to be high, ranging from 10<sup>5</sup> to 10<sup>7</sup> S/m. Among all the compounds, Co2MnSi exhibits the highest *σ* in the whole temperature range. The *σ*

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

value of Co2MnSi decreases from 4.6 � <sup>10</sup><sup>6</sup> S/m at 300 K to 4.7 � <sup>10</sup><sup>5</sup> 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 � <sup>10</sup>�<sup>3</sup> W/K<sup>2</sup> m 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 � <sup>10</sup>�<sup>4</sup> W/K<sup>2</sup> m 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, *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]:

$$\mathcal{S}\_{\rm FM+N\mathcal{M}}(T) = \frac{\mathcal{S}\_{\rm FM}(T)\sigma\_{\rm FM}(T)\mathcal{M}(T) + \mathcal{S}\_{\rm N\mathcal{M}}(T)\sigma\_{\rm N\mathcal{M}}(T)\{1 - \mathcal{M}(T)\}}{\sigma\_{\rm FM+N\mathcal{M}}(T)},\tag{9}$$

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].

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 *σ*~ð Þ *ε*, *T* 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

the measured *σ* for several Co-based full-Heusler compounds [53]. The *σ* values of the compounds are observed to be high, ranging from 10<sup>5</sup> to 10<sup>7</sup> S/m. Among all the compounds, Co2MnSi exhibits the highest *σ* in the whole temperature range. The *σ*

*(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).*

*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*

**Figure 4.**

**Figure 5.**

**70**

*2017, with permission from Springer).*

*Magnetic Materials and Magnetic Levitation*

**Figure 6.**

*Temperature dependence of the calculated* S *of Co2Ti*Z *(*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*<sup>F</sup> and (b) 150 meV below* ε*F. (Reprinted from [38]. Copyright 2010, with permission from American Physical Society).*

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 Co2Ti*Z* (*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 *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

*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).*

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

*Magnetic Full-Heusler Compounds for Thermoelectric Applications*

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

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.

compounds.

**73**

**Figure 8.**

**Figure 7.**

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

#### **Figure 7.**

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 Co2Ti*Z* (*Z* = Si, Ge, Sn) in the FM state. As shown in **Figure 7**, off-stoichiometric defects, such as Co vacancy and the substitution of

*Temperature dependence of the calculated* S *of Co2Ti*Z *(*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*<sup>F</sup> and (b) 150 meV below*

ε*F. (Reprinted from [38]. Copyright 2010, 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

excess atoms at a particular site, change the sign of *S*.

*Magnetic Materials and Magnetic Levitation*

**Figure 6.**

**72**

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

#### **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).*

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, *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.

law, *κ*<sup>e</sup> = *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*<sup>e</sup> value of Co2MnSi was calculated and plotted in **Figure 10(a)**. It can be observed from the figure that *κ*<sup>e</sup> is only half as high as *κ*tot. The rest is attributed to the lattice thermal conductivity, *κ*<sup>l</sup> (=*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*<sup>e</sup> [88]. The non-negligible *κ*<sup>l</sup> suggests that, for the theoretical evaluation of *zT* of the half-metallic full-Heusler compounds, the contribution of *κ*<sup>l</sup> should not be ignored. Experimentally, the high contribution of *κ*<sup>l</sup> to *κ*tot indicates that the *κ*tot of half-metallic full-Heusler compounds could be reduced

*Magnetic Full-Heusler Compounds for Thermoelectric Applications*

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

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

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

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

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

by decreasing the *κ*l.

conduction states.

**Figure 11.**

**75**

**thermoelectric materials**

**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).*

**Figure 10.** *Temperature dependence of (a) measured* κ*tot,* κ*<sup>e</sup> and* κ*<sup>l</sup> and (b) evaluated* zT *of Co2MnSi.*

**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 Ru2VAl0.25Ga0.75 [29]).

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

law, *κ*<sup>e</sup> = *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*<sup>e</sup> value of Co2MnSi was calculated and plotted in **Figure 10(a)**. It can be observed from the figure that *κ*<sup>e</sup> is only half as high as *κ*tot. The rest is attributed to the lattice thermal conductivity, *κ*<sup>l</sup> (=*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*<sup>e</sup> [88]. The non-negligible *κ*<sup>l</sup> suggests that, for the theoretical evaluation of *zT* of the half-metallic full-Heusler compounds, the contribution of *κ*<sup>l</sup> should not be ignored. Experimentally, the high contribution of *κ*<sup>l</sup> to *κ*tot indicates that the *κ*tot of half-metallic full-Heusler compounds could be reduced by decreasing the *κ*l.
