Magnetic Full-Heusler Compounds for Thermoelectric Applications

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.


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, η max , is an increasing function of the dimensionless figure-of-merit, zT, expressed as: 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.
where T is the absolute temperature.The product S 2 σ 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 Â 10 À3 W/K 2 m), 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.

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 X 2 YZ, which generally crystallises in the L2 1 structure.The prototype of the L2 1 structure is Cu 2 MnAl (space group: Fm3m).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 L2 1 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 Cu 2 MnAl becomes the B2 structure.Its prototype is CsCl (space group: Pm3m).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: Im3m).There are other disordered phases, including the DO 3 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: BiF 3 , space group: Fm3m).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: Fd3m).
When the X' and Y atoms are of the same element, the chemical composition becomes XX' 2 Z, which crystallises in the X (XA or X a ) structure.This structure is called the inverse Heusler phase.The prototype is CuHg 2 Ti (or AgLi 2 Sb), and the space group is F43 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: F43 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, DO 3 or B32a structure.

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: ) where e, ε, ε F , f FD (ε, T), N k , τ(k, T),ℏ and σ ε, T ðÞ 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 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.
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]: where DOS is the electronic density of states.Adopting Eq. ( 8) for the partial DOS of the sp-electrons and d-electrons of Co 2 MnSi, 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,Co 2 TiAl is shown to exhibit the highest |S|of|À56| μV/K at 350 K among other compounds.It is observed that Co 2 TiSi, Co 2 TiGe and Co 2 TiSn 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 10 5 to 10 7 S/m.Among all the compounds, Co 2 MnSi exhibits the highest σ in the whole temperature range.The σ Reprinted from [37].Copyright 2010, with permission from Elsevier.(c) Reprinted from [53].Copyright 2017, with permission from Springer).value of Co 2 MnSi decreases from 4.6 Â 10 6 S/m at 300 K to 4.7 Â 10 5 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,Co 2 MnSi exhibits the highest PF (2.9 Â 10 À3 W/K 2 m at 500 K) among other compounds, which is comparable to that of a Bi 2 Te 3 -based material [86].Since Co 2 MnSi 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 Co 2 MnSi is needed.For this purpose, Li et al. [60,78] prepared a half-metallic Mn 2 VAl compound and measured its TE properties.Although Mn 2 VAl is a p-type material showing positive S, its highest PF (2.84 Â 10 À4 W/K 2 m at 767 K [78]) is lower than that of Co 2 MnSi.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 Co 2 TiSi, Co 2 TiGe and Co 2 TiSn (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 Co 2 TiSi, Co 2 TiGe and Co 2 TiSn [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]: 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 Co 2 TiSi, Co 2 TiGe and Co 2 TiSn (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 Co 2 CrAl, Co 2 MnAl, Co 2 MnSi, Co 2 FeAl and Co 2 FeSi [53].
It is suggested that the constant S value in the NM state for Co 2 TiSi, Co 2 TiGe and Co 2 TiSn (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 ε, T ðÞ 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 L2 1 , 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 Co 2 TiZ (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 Co 2 CrAl, Co 2 MnAl, Co 2 MnSi, Co 2 FeAl and Co 2 FeSi 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, Co 2 CrAl, Co 2 MnAl, Co 2 MnSi, Co 2 FeAl and Co 2 FeSi are not in the fully ordered L2 1 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 L2 1 structure.Recently, Li et al. [78] investigated the effect of structural disorder on the value of S for half-metallic Mn 2 VAl compounds by varying the B2 order degree.Figure 9(a) shows the measured S values for Mn 2 VAl 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 L2 1 order degree.To understand the reason for the difference in S between the L2 1 and B2 structures, the DOS of Mn 2 VAl with the L2 1 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 L2 1 structure, which is considered as the main reason for the higher S of the B2 structure than that of the L2 1 structure.Further increase in the B2 order degree is expected to yield a higher S for Mn 2 VAl.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 Co 2 CrAl, Co 2 MnAl, Co 2 MnSi, Co 2 FeAl, Co 2 FeSi and Mn 2 VAl 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 Co 2 MnSi, 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.  [52].Copyright 2017, with permission from American Physical Society).10(a)).Due to the high κ tot , the maximum zT value, zT max ,ofCo 2 MnSi 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 Co 2 TiSn (0.033 at 370-400 K) [38] and those of semi-metallic full-Heusler compounds (0.0052 at 300 K for Ru 2 NbAl [28] and 0.0027 at 300 K for Ru 2 VAl 0.25 Ga 0.75 [29]).

Change in calculated S for Co 2 TiSi 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
It should be noted that the κ tot of Co 2 MnSi 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 Co 2 MnSi 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 À 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 .

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 Mn 2 CoAl, 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.
Owing to the nearly zero S values, Mn 2 CoAl cannot be used as a TE material; however, there is a possibility of achieving high |S|inMn 2 CoAl 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 Mn 2 CoAl.We calculated the S value for the partially substituted Mn 2 CoAl, as shown in Figure 13(a).The calculation was based on a rigid band model; thus, the electronic structure of the partially substituted Mn 2 CoAl is assumed to be the same as that of Mn 2 CoAl.In the figure, the horizontal axis is μ-ε F , where μ and ε F are the chemical potential (i.e., the Fermi    To achieve high |S| values for Mn 2 CoAl, it is important to retain its SGS characteristic.Galanakis et al. [90] theoretically investigated the effects of structural disorder on the electronic structure of Mn 2 CoAl.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 Mn 2 CoAl.Also, Xu et al. [91] reported that an as-prepared Mn 2 CoAl compound is non-stoichiometric and contains the Mn-Co antisite defect.In a case where Mn 2 CoAl 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  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 Ba 2 AuBi and Ba 2 HgPb 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 Ba 2 BiAu.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 Fe 2 VAl-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 Fe 2 V 0.8 W 0.2 Al 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 highefficiency TE power generation devices.

Figure 1 .
Figure 1.Schematic figure of a thermoelectric (TE) power generation device.

Figure 2 .
Figure 2. Crystal structures of full-Heusler compounds.The Strukturbericht symbol and a prototype structure are written above and below each crystal structure, respectively.

Figure 3 .
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).

Figure 6 .
Figure 6.Temperature dependence of the calculated S of Co 2 TiZ (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).

Figure 7 .
Figure 7. Change in calculated S for Co 2 TiSi 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).

Figure 8 .
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 10 (
Figure 10(b) shows the temperature dependence of zT for Co 2 MnSi calculated using the PF value (Figure 5(b)) and the κ tot value (Figure10(a)).Due to the high κ tot , the maximum zT value, zT max ,ofCo 2 MnSi 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 Co 2 TiSn (0.033 at 370-400 K)[38] and those of semi-metallic full-Heusler compounds (0.0052 at 300 K for Ru 2 NbAl [28] and 0.0027 at 300 K for Ru 2 VAl 0.25 Ga 0.75[29]).It should be noted that the κ tot of Co 2 MnSi is not equal to the carrier thermal conductivity, κ e .The κ e value can be calculated by using the Wiedemann-Frantz

Figure 9 .
Figure 9. (a) Temperature dependence of the measured S of Mn 2 VAl with the B2 order degree of 27 and 66%.(b) Measured S values of Mn 2 VAl at 767 K plotted against the B2 order degree.(Reprinted from [78].Copyright 2020, with permission from IOP Publishing).
level) of the partially substituted Mn 2 CoAl and the Fermi level of Mn 2 CoAl, 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 Mn 2 CoAl, is small, it is large at μ = ε F À 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 À 0.184 and μ = ε F + 0.053 are shown in Figure13(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 .
Figure 13.(a) Calculated S value at 300 K for the partially substituted Mn 2 CoAl plotted as a function of μ-ε F , where μ and ε F are the Fermi levels of partially substituted Mn 2 CoAl and that of Mn 2 CoAl, respectively.The highest |S| values are obtained at μ = ε F À 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 À 0.184 and μ = ε F + 0.053.
shows a flower-like microstructure of Co 2 Dy 0.5 Mn 0.5 Sn observed by Schwall et al. [43].Although the chemical composition of Co 2 Dy 0.5 Mn 0.5 Sn coincides with that of the full-Heusler phase, Co 2 Dy 0.5 Mn 0.5 Sn is not in a single phase.It consists of two major phases: half-metallic Co 2 MnSn and ferromagnetic Co 8 Dy 3 Sn 4 phases.This phase separation is induced by rapid cooling from the liquid phase.Consequently, the κ l value of Co 2 Dy 0.5 Mn 0.5 Sn is lower than those of Co 2 MnSn and Co 8 Dy 3 Sn 4 .