**2. Properties of Mg2X compounds**

(*S* is thermopower or Seebeck coefficient; *σ* is electrical conductivity; κ is thermal

conductivity, and *T* is absolute temperature) up to 1.3 were obtained by several research groups [3–7]. Many researchers believe that there is possibility for further improvement. Now

Already in the 1960s, it was shown that Mg2X compounds (X = Si, Ge, Sn) and their solid solutions are promising compounds for thermoelectric energy conversion [8, 9]. Very high values of *ZT* are reported in Refs. [10, 11]. However, later the interest to these compounds has been almost vanished until the last decade. A new wave of research activity on Mg2X compounds was initiated by information about high figure of merit achieved in Mg2Si-Mg2Sn solid solutions and growing interest to environment-friendly materials for thermoelectric energy

The maximum conversion efficiency of thermoelectric generator *η* is determined by dimen-


*T T ZT*

*H C*

max

is the reason of slow progress in the development of efficient thermoelectrics.

minimize lattice thermal conductivity, yielding high values of parameter *Z*.

to simultaneously maximize electronic parameters, characterized by power factor *S*<sup>2</sup>

h

264 Thermoelectrics for Power Generation - A Look at Trends in the Technology

1 1 ,

*H*

(1)

(2)

*σ*, and to

*T*

1

*H C*

where *TH* and *TC* are temperatures at hot and at cold junctions of thermoelectric generator thermopile. is the dimensionless figure of merit, averaged over working temperature range Δ*T* = *TH* – *TC*. The semiconductor physics theory gives the following estimate for parameter *Z*

> 3 \* 2

> > lat

where *m*\* is the effective mass of electron state density (DOS), *µ* is the free charge carriers' mobility, and *κ*lat is the lattice thermal conductivity. One can see that a good thermoelectric material will have heavy effective mass, high charge carriers' mobility, and low lattice thermal conductivity. However, in fact coefficients determining Z are strongly interdependent. Thermoelectric materials with high DOS typically have low mobility. Introducing a disorder to suppress the thermal conductivity usually leads to decrease of charge carriers' mobility. This

The unique characteristics of an electronic band structure of Mg2X compounds make possible to explore the combination of two approaches to optimize the thermoelectric performance of such materials: the band structure engineering and the alloying [2, 5]. The combination allows

( ) <sup>=</sup> , *m µ <sup>Z</sup>* k

*<sup>T</sup> <sup>T</sup> ZT*

+ +

considerable efforts are directed to the development of a matching p-type material.

conversion.

[13]:

sionless figure of merit *ZT* [12]:

## **2.1. Physical properties and crystal structure of Mg2X**

The basic properties of Mg2X compounds are shown in **Table 1**. Melting temperature and energy gap, *E*g, are typical for so-called middle temperature range thermoelectrics (600 < *T* < 1200 K). The materials, especially Mg2Si, have very low density, *d*. Therefore, the ratio for Mg2Si is the highest among commercial thermoelectrics. This is advantage for applications, where weight is a significant factor. High *ZT* in Mg2Si can be related to high electron (*µn*) and low hole (*µp*) mobility. Their values at room temperature are shown in **Table 1**. Mg2Ge has the highest electron mobility, but the electron to hole mobility ratio is lower in comparison to that for Mg2Si. Mg2Sn has the highest effective mass of DOS. It should be noted that among Mg2X compounds, Mg2Sn has the highest hole mobility with small difference between election and hole mobility. This suggests that p-type thermoelectrics based on Mg2X alloys should contain a large fraction of Mg2Sn.


**Table 1.** Some parameters of Mg2X compounds.

Phase diagrams for the systems of magnesium and carbon groups of elements are well known [14]. Each phase diagram contains only one chemical compound of Mg2X-type and two eutectic points. Mg2X compounds crystallize with cubic, CaF2-type, structure (space group Fm3m) [16, 20]. In Mg2X structure, the fluorine atom is replaced by the magnesium atom and the calcium atom is replaced by X atom (**Figure 1**). Each atom of the X group is surrounded by eight magnesium atoms in a regular cube. The bond in all these compounds is covalent [18]. Lattice parameters of compounds are presented in **Table 1**.

**Figure 1.** Mg2X crystal structure.

#### **2.2. Energy spectra of current carriers in Mg2X**

Fundamental parameters of the electronic structure of the Mg2X compound can be obtained from optical and electronic transport property measurements on high quality single crystals. Comprehensive review of transport properties and electronic energy structure for Mg2X compounds is given in Ref. [21].

Based on the analysis of optical and electronic transport data, supplemented by results of band structure calculations, the band structure of Mg2X compounds was proposed [16, 22–27]. **Figure 2** shows schematically the most important characteristic of this band structure near to Fermi energy. The valence band of the compounds is similar to the valence band of Si and Ge. It consists of two degenerate bands (*V*1, *V*2) with different effective masses (1 \* and 2 \* )and

a third band (*V*3) split below the two other bands by gap *E*2 due to spin-orbital interaction. The maximums of valence bands are located at Γ-point of a Brillouin zone. The conduction band consists of two subbands *CL* and *CH* of light ( \* ) and heavy ( \* ) electrons with their

minimums located at X-point of a Brillouin zone. These subbands are separated by energy gap 1 = <sup>Χ</sup> − (Χ). There is a third conduction band *C* with minimum at Γ-point, separated by gap *E*0 from the top of valence bands. However, *E*0 is considerably larger than indirect band gap *Eg*; therefore, *C* band has no direct effect on thermoelectric properties of Mg2X compounds. Theoretical calculations confirmed this structure except for the fact that these calculations did not take into account spin-orbital interaction [16, 25–27].

Location of conduction band minimum at the X-point is favorable for thermoelectric performance of a material. In this case, the effective mass of DOS is six times heavier than inertial mass. Because of that n-type Mg2Si has high electrical conductivity and high thermopower. On the other hand, the valence band structure does not have such favorable thermoelectric features. The maximum of the valance band is at Γ-point; thus, the inertial mass and effective mass of DOS are not different. The valence band has three subbands, one of which split due to spinorbital interaction [28]. This splitting extends with the increasing atom mass.

**Figure 2.** Schematic band structure of Mg2X. For Mg2Si and Mg2Ge light electron band (CL) lies below heavy electron band (CH), as shown in the picture. In the case of Mg2Sn the heavy electron band CH is below the light electron band CL.

Parameters of band structure for Mg2Sn, Mg2Ge, and Mg2Si are presented in **Table 2**. The values of indirect band gap *Eg* determined from electrical conductivity temperature dependence (*Eg T*) and from optical data (Eg 0 ) are in good agreement. *E*1 and *E*2 are gaps between the conduction and valence subbands, respectively. *E*0 is a direct band gap value. According to definition of *E*1, it is positive for Mg2Ge and Mg2Si, where the low-lying conduction band has smaller effective mass. The opposite situation is in Mg2Sn, where *E*1 is negative. The effective mass of conduction band ( \* ) is shown for a low-lying subband, i.e. \* for Mg2Ge and Mg2Si, while

 \* —in the case of Mg2Sn. The temperature coefficient of a band gap is shown in the last column.


**Table 2.** Parameters of Mg2X band structure (presented in **Figure 3**).

**Figure 1.** Mg2X crystal structure.

compounds is given in Ref. [21].

<sup>Χ</sup> −

1 =

**2.2. Energy spectra of current carriers in Mg2X**

266 Thermoelectrics for Power Generation - A Look at Trends in the Technology

consists of two subbands *CL* and *CH* of light (

not take into account spin-orbital interaction [16, 25–27].

Fundamental parameters of the electronic structure of the Mg2X compound can be obtained from optical and electronic transport property measurements on high quality single crystals. Comprehensive review of transport properties and electronic energy structure for Mg2X

Based on the analysis of optical and electronic transport data, supplemented by results of band structure calculations, the band structure of Mg2X compounds was proposed [16, 22–27]. **Figure 2** shows schematically the most important characteristic of this band structure near to Fermi energy. The valence band of the compounds is similar to the valence band of Si and Ge.

a third band (*V*3) split below the two other bands by gap *E*2 due to spin-orbital interaction. The maximums of valence bands are located at Γ-point of a Brillouin zone. The conduction band

minimums located at X-point of a Brillouin zone. These subbands are separated by energy gap

by gap *E*0 from the top of valence bands. However, *E*0 is considerably larger than indirect band gap *Eg*; therefore, *C* band has no direct effect on thermoelectric properties of Mg2X compounds. Theoretical calculations confirmed this structure except for the fact that these calculations did

Location of conduction band minimum at the X-point is favorable for thermoelectric performance of a material. In this case, the effective mass of DOS is six times heavier than inertial mass.

\* ) and heavy (

(Χ). There is a third conduction band *C* with minimum at Γ-point, separated

\* and 2

\* ) electrons with their

\* )and

It consists of two degenerate bands (*V*1, *V*2) with different effective masses (1

## **2.3. Thermal conductivity of Mg2X compounds**

**Figure 3** shows temperature dependencies of reciprocal thermal conductivity of pure Mg2X compounds. One can see that reciprocal thermal conductivity can be described satisfactory by a linear law and residual reciprocal thermal conductivity is zero within experimental uncertainty. The most probable reason for observed difference in data of different authors is dependence of reciprocal thermal conductivity on deviation from stoichiometry.

**Figure 3.** Temperature dependence of reciprocal thermal conductivity of pure Mg2X compounds: 1, 2—Mg2Si [19, 21]; 3 —Mg2Ge[19]; 4, 5—Mg2Sn[19, 32].

## **3. Solid solutions of Mg2X compounds**

#### **3.1. Mg2X-based solid solutions**

As one can see from **Table 1**, Mg2X compounds have relatively high thermal conductivity, which should be decreased to make these compounds practically useful thermoelectrics. However, decreasing in thermal conductivity should not lead to a considerable decrease of charge carriers' mobility. Thermal conductivity can be reduced by selective scattering of phonons and electrons by point defects through forming solid solutions (alloys) between these isostructural compounds.

There is a continuous series of solid solutions in the system Mg2Si-Mg2Ge [9]. Phase diagrams of Mg2Si-Mg2Sn and Mg2Ge-Mg2Sn have wide peritectic region in the middle composition range [15, 33]. Until recently, it was commonly accepted that solid solutions exist only at compositions *x* < 0.4 and *x* > 0.6 for the Mg2Si1-*x*Sn*x* system, and at *x* < 0.3, *x* > 0.5 for the Mg2Ge1-*x*Sn*x*system. However, it has been demonstrated that solid solutions of any composition can be produced avoiding liquid stage by mechanical alloying.

**Figure 4** shows dependences of lattice parameter (*a*) on alloys composition (*x*). In Ref. [33], it was shown that *a*(*x*) dependence follows to Vegard's law for the whole composition range of the Mg2Si-Mg2Sn system.

Efficient Thermoelectric Materials Based on Solid Solutions of Mg2X Compounds (X = Si, Ge, Sn) http://dx.doi.org/10.5772/65864 269

**Figure 4.** Lattice parameter *a* vs. solid solution composition *x* dependences: 1—Mg2Sn1-*x*Ge*x* [15]; 2—Mg2Si1-*x*Ge*x* [9]; 3— Mg2Si1-*x*Sn*x* [33]; 4—Vegard's law.

#### **3.2. Thermal conductivity of Mg2X solid solutions**

**2.3. Thermal conductivity of Mg2X compounds**

268 Thermoelectrics for Power Generation - A Look at Trends in the Technology

—Mg2Ge[19]; 4, 5—Mg2Sn[19, 32].

**3.1. Mg2X-based solid solutions**

isostructural compounds.

the Mg2Si-Mg2Sn system.

**3. Solid solutions of Mg2X compounds**

can be produced avoiding liquid stage by mechanical alloying.

**Figure 3** shows temperature dependencies of reciprocal thermal conductivity of pure Mg2X compounds. One can see that reciprocal thermal conductivity can be described satisfactory by a linear law and residual reciprocal thermal conductivity is zero within experimental uncertainty. The most probable reason for observed difference in data of different authors is

**Figure 3.** Temperature dependence of reciprocal thermal conductivity of pure Mg2X compounds: 1, 2—Mg2Si [19, 21]; 3

As one can see from **Table 1**, Mg2X compounds have relatively high thermal conductivity, which should be decreased to make these compounds practically useful thermoelectrics. However, decreasing in thermal conductivity should not lead to a considerable decrease of charge carriers' mobility. Thermal conductivity can be reduced by selective scattering of phonons and electrons by point defects through forming solid solutions (alloys) between these

There is a continuous series of solid solutions in the system Mg2Si-Mg2Ge [9]. Phase diagrams of Mg2Si-Mg2Sn and Mg2Ge-Mg2Sn have wide peritectic region in the middle composition range [15, 33]. Until recently, it was commonly accepted that solid solutions exist only at compositions *x* < 0.4 and *x* > 0.6 for the Mg2Si1-*x*Sn*x* system, and at *x* < 0.3, *x* > 0.5 for the Mg2Ge1-*x*Sn*x*system. However, it has been demonstrated that solid solutions of any composition

**Figure 4** shows dependences of lattice parameter (*a*) on alloys composition (*x*). In Ref. [33], it was shown that *a*(*x*) dependence follows to Vegard's law for the whole composition range of

dependence of reciprocal thermal conductivity on deviation from stoichiometry.

**Figure 5** shows the experimental values of lattice thermal conductivity of Mg2Si1-*x*Sn*x* [34], Mg2Ge1-*x*Sn*x*[34], Mg2Si1-*x*Ge*x* [35] alloys, and the results of calculations according to procedure, described in Refs. [34, 36]. In alloys, thermal conductivity sharply decreases with the addition of a small amount of second compound, while it has a weak dependence on the composition in the middle composition range 0.2 < *x* < 0.8. One can see that the lowest thermal conductivity can be achieved in the system Mg2Si1-*x*Sn*x* due to the maximum mass difference between the compounds. Consequently, this system is the most favorable from the point of view of thermoelectric energy conversion.

**Figure 5.** Lattice thermal conductivity of alloys at room temperature: 1–Mg2Si1−*x*Ge*x*, 2–Mg2Ge1−*x*Sn*x*, 3–Mg2Si1−*x*Sn*x*. Symbols: experiment 1–[35]; 2, 3–[34]; lines–calculation [35].

#### **3.3. Dependency of energy gap on solid solution composition**

Besides lower thermal conductivity, the solid solutions of Mg2X provide opportunity to further enhancement of thermoelectric properties by electronic band structure engineering. **Figure 6** shows dependences of energy gap of Mg2X alloys vs. composition [9, 15, 37–40]. From the study of the Mg2Si-Mg2Ge system [9]—one can conclude that energy gap is practically independent of alloy composition.

**Figure 6.** Energy gap *Eg* of alloys as a function of composition *x*. 1—Mg2Si1−*x*Ge*x* [9]; 2—Mg2Sn1−*x*Ge*x* [15]; 3—Mg2Si1−*x*Sn*<sup>x</sup>* [37].

The situation is very different in other two alloy systems. The Mg2Ge-Mg2Sn system was studied by Busch et al. [15]. Notwithstanding the very narrow band gaps of Mg2Sn and Mg2Ge, it can be concluded that band gap dependence on solid solution composition is nonlinear. Our results of the band gap study of Mg2Ge1-*x*Sn*x* solid solutions confirm this behavior. The situation is the same in the Mg2Si-Mg2Sn system [37–40]. Zaitsev et al. [37] proposed that there is band inversion in Mg2Si1 *x*Sn*x* solid solutions. It means that in Mg2Si and Mg2Sn conduction bands *CL* and *CH* change their positions. Band inversion hypothesis was confirmed theoretically. Fedorov et al. [38] showed that the lowest conduction bands of Mg2Si and Mg2Ge are formed by Si or Ge states, whereas that of Mg2Sn was formed by Mg states. **Figure 7** shows schematically dependence of relative positions of heavy electrons and light electrons conduction bands, as well as, the top of the valence band on the Mg2Si1-*x*Sn*x* alloy composition. The lower panel explains the occurrence of kink on dependence of the band gap on composition at the inversion point. According to calculation in Ref. [5], composition dependence on light electron subband position in Mg2Si1-*x*Sn*x* is nonlinear, while corresponding dependence of position of heavy electron subband is linear. Therefore, actual dependence of band gap on composition is more complex. Nevertheless, the scheme shown in **Figure 7** illustrates correctly the essential physics. At the composition value, corresponding to band inversion point, the minima of heavy and light electrons subbands have equal energy. From the point of view of thermoelectricity, such situation is favorable, because DOS increases without decreasing in electron mobility. Such degeneration of subbands occurs at certain composition and certain temperature, so this favorable situation is very limited.

**Figure 7.** Schematic dependence of relative positions of heavy and light electron conduction subbands as well as the top of the valence band on the composition of the Mg2Si1−xSnx alloy (the upper panel). Explanation of origin of kink on band gap dependence (the lower panel).

Calculations show that the most favorable situation realizes when heavy electrons subband lays higher [41]. Another advantage of this situation is the absence of interband scattering [40].

#### **3.4. Synthesis technology and doping**

**3.3. Dependency of energy gap on solid solution composition**

270 Thermoelectrics for Power Generation - A Look at Trends in the Technology

of alloy composition.

[37].

Besides lower thermal conductivity, the solid solutions of Mg2X provide opportunity to further enhancement of thermoelectric properties by electronic band structure engineering. **Figure 6** shows dependences of energy gap of Mg2X alloys vs. composition [9, 15, 37–40]. From the study of the Mg2Si-Mg2Ge system [9]—one can conclude that energy gap is practically independent

**Figure 6.** Energy gap *Eg* of alloys as a function of composition *x*. 1—Mg2Si1−*x*Ge*x* [9]; 2—Mg2Sn1−*x*Ge*x* [15]; 3—Mg2Si1−*x*Sn*<sup>x</sup>*

The situation is very different in other two alloy systems. The Mg2Ge-Mg2Sn system was studied by Busch et al. [15]. Notwithstanding the very narrow band gaps of Mg2Sn and Mg2Ge, it can be concluded that band gap dependence on solid solution composition is nonlinear. Our results of the band gap study of Mg2Ge1-*x*Sn*x* solid solutions confirm this behavior. The situation is the same in the Mg2Si-Mg2Sn system [37–40]. Zaitsev et al. [37] proposed that there is band inversion in Mg2Si1 *x*Sn*x* solid solutions. It means that in Mg2Si and Mg2Sn conduction bands *CL* and *CH* change their positions. Band inversion hypothesis was confirmed theoretically. Fedorov et al. [38] showed that the lowest conduction bands of Mg2Si and Mg2Ge are formed by Si or Ge states, whereas that of Mg2Sn was formed by Mg states. **Figure 7** shows schematically dependence of relative positions of heavy electrons and light electrons conduction bands, as well as, the top of the valence band on the Mg2Si1-*x*Sn*x* alloy composition. The lower panel explains the occurrence of kink on dependence of the band gap on composition at the inversion point. According to calculation in Ref. [5], composition dependence on light electron subband position in Mg2Si1-*x*Sn*x* is nonlinear, while corresponding dependence of position of heavy electron subband is linear. Therefore, actual dependence of band gap on composition is more complex. Nevertheless, the scheme shown in **Figure 7** illustrates correctly the essential physics. At the composition value, corresponding to band inversion point, the minima of heavy and light electrons subbands have equal energy. From the point of view of thermoelectricity, such situation is favorable, because DOS increases without decreasing in electron mobility. Such

There are several methods to produce Mg2X compounds. One of them is direct co-melting [8, 10]. This method has some limitation due to the large difference in melting temperature of components and high magnesium vapor pressure. It is necessary to pay a special attention to magnesium losses due to evaporation and segregation of the components (especially for Mg2Sn).

Another way to produce these compounds is through a solid-state reaction. Mg2X compounds have negative heat of formation, i.e. the formation reaction is exothermic [42–45]. However, oxide films on Mg particles prevent the reaction. Therefore, it is necessary to pay attention to the purity of components and avoid oxidization during mixing. Alternative manufacturing route has been developed for magnesium silicide derivatives [46]. Elemental powders were mixed in stoichiometric proportions, cold pressed into cylindrical preforms and heated in an oxygen-free environment to initiate the exothermic reaction. Reaction products were additionally heat treated for homogenization. Dense sinters can be produced by hot uniaxial pressing of the obtained powders under moderate temperature and pressure conditions.

Several advantages were identified in the proposed technology: relatively short time of synthesis, possibility of *in-situ* or *ex-situ* doping and grain size control.

Single crystals of Mg2X compounds can be easily produced by any methods of directed crystallization.

It is hard to produce homogeneous solid solutions via a liquid phase through co-melting of the components. One of the problems is related to large difference in masses of magnesium, silicon, germanium, and tin atoms. Without stirring, segregation by specific weight occurs. The other problem relates to phase diagrams of solid solutions, which have large difference in liquidus and solidus curves in a wide range of compositions [33]. Therefore, compositional segregation occurs during crystallization as well. In order to homogenize alloys, a long-term annealing is necessary. The necessary homogenization annealing time is determined by diffusion processes, which depend on temperature and crystallite size. Temperature cannot be high due to magnesium evaporation. In order to shorten the annealing time, hot pressing can be utilized. Ingots of alloys are crushed into powder and then powder is pressed in a vacuum. The finer grain size the less time for homogenization is needed [47]. Annealing is not required for the samples produced from nanosize particles.

Recently, mechanical alloying in the ball mill followed by spark plasma sintering (SPS) has become the most popular preparation technique for this solid solution.

As mentioned above, the figure of merit Z is function of free charge carriers' concentration. Optimal concentration yielding maximum *ZT* value is equal to about 1019 to 1020 cm−3. Theoretical and experimental investigations of a doping impurity effect in Mg2Si for a wide range of impurity elements (B, Al, N, P, Sb, Bi, Cu, Ag, Au) were made by Tani and Kido [48, 49]. As, P, Sb, Bi, Al, and N were suggested as n-type dopants whereas Ga is suggested as p-type dopant. For In, Ag, Cu, and Au, the doping effect, i.e. a resulting conduction type, depends on the site in lattice where a doping atom will occupy. Actually, Ag-doped samples show p-type of conductivity. In Mg2Sn-rich solid solutions, impurities Na, Li, Ga, Ag and at low concentration Al and In act as p-type dopants [50–53].
