Electronic Structure and Thermoelectric Properties of Pseudoquaternary Mg2Si1-x-ySnxGey\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \

A theoretical study is presented on complex pseudoternary Bi-doped Mg2Si1-x-ySnxGey\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox{Mg}_{2}\hbox{Si}_{1-x-y}\hbox{Sn}_{x}\hbox{Ge}_{y}$$\end{document} materials, which have recently been revealed to reach high thermoelectric figures of merit (ZT) of ∼1.4. Morphological characterization by scanning electron microscopy and energy-dispersive x-ray spectroscopy indicated that the investigated samples were multiphase and that the alloy with nominal composition Mg2Si0.55Sn0.4Ge0.05\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox{Mg}_{2}\hbox{Si}_{0.55}\hbox{Sn}_{0.4}\hbox{Ge}_{0.05}$$\end{document} contained three phases: Mg2Si0.35Sn0.6Ge0.05\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox{Mg}_{2}\hbox{Si}_{0.35}\hbox{Sn}_{0.6}\hbox{Ge}_{0.05}$$\end{document} (Sn-rich phase), Mg2Si0.65Sn0.3Ge0.05\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox{Mg}_{2}\hbox{Si}_{0.65}\hbox{Sn}_{0.3}\hbox{Ge}_{0.05}$$\end{document} (Si-rich phase), and Mg2Si0.15Sn0.5Ge0.35\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox{Mg}_{2}\hbox{Si}_{0.15}\hbox{Sn}_{0.5}\hbox{Ge}_{0.35}$$\end{document} (Ge-rich phase). The electronic structure of all these phases was calculated in the framework of the fully charge self-consistent Korringa–Kohn–Rostoker method with the coherent potential approximation (KKR-CPA) to treat chemical disorder. Electron transport coefficients such as the electrical conductivity, thermopower, and the electronic part of the thermal conductivity were studied by combining the KKR-CPA technique with Boltzmann transport theory. The two-dimensional (2D) plots (as a function of electron carrier concentration and temperature), computed for the thermopower and power factor, well support the large thermoelectric efficiency detected experimentally. Finally, employing the experimental value of the lattice thermal conductivity as an adjustable parameter, it is shown that ZT ≈ 1.4 can be reached for an optimized Bi content near T ≈ 900 K in case of the nominal composition as well as the Sn-rich phase. The question of the effect of disorder on the convergence of the conduction bands and thus the electron transport properties is addressed through detailed examination of the Fermi surfaces.


INTRODUCTION
Mg 2 Si 1Àx Ge x ; Mg 2 Si 1Àx Sn x ; Mg 2 Ge 1Àx Sn x alloys have attracted much attention as they are composed of cheap (except germanium), abundant, and environmentally friendly (nontoxic) raw materials. They also have a high figure of merit and the lowest density amongst all efficient thermoelectrics. Mg 2 Si 1Àx Sn x has been found to be the most favorable in terms of thermoelectric energy conversion, as it has the highest thermal resistivity due to the maximum mass difference between its components. [1][2][3] However, quasiternary solid solutions Mg 2 Si 1ÀxÀy Sn y Ge z have attracted much less attention. Very recently, a high figure of merit ZT % 1.4 was measured in the Bi-doped Mg 2 Si 1ÀxÀy Sn y Ge z system. 4 On the other hand, morphological characterization carried out by scanning electron microscopy and energy-dispersive x-ray spectroscopy (EDX) indicated that the main feature of typical micrographs obtained in backscatter mode was a mosaic-like texture. 4 EDX maps for the different elements, i.e., Mg, Si, Sn, and Ge, showed that the distributions of Mg, Si, Sn, and Ge are not homogeneous, since there are areas richer in Si or Sn, suggesting the appearance of phase separation. Ge-rich inclusions were also found. It appears that, for the nominal alloy compound Mg 2 Si 0:55 Sn 0:4 Ge 0:05 : Bi x with x % 0:02 that exhibited the largest ZT value, three phases were detected: an Sn-rich phase, i.e., Mg 2 Si 0:35 Sn 0:6 Ge 0:05 (a = 6.598 Å ), an Si-rich phase, i.e., Mg 2 Si 0:65 Sn 0:3 Ge 0:05 (a = 6.475 Å ), and a Ge-rich phase, i.e., Mg 2 Si 0:15 Sn 0:5 Ge 0:35 (a = 6.569 Å ), with different relative contributions. The aforementioned lattice parameters were determined from Vegard's law using the following values for the end-compounds: 6.351 Å for Mg 2 Si, 6.763 Å for Mg 2 Sn, and 6.386 Å for Mg 2 Ge.
To investigate such a complex material from a theoretical point of view, we performed electronic structure calculations combined with Boltzmann transport modeling for four cases separately, i.e., for three alloy compositions close to the aforementioned phases and also for the nominal composition. The Korringa-Kohn-Rostoker (KKR) 5-8 method based on Green-function multiple scattering theory was used to compute the electronic band structure and kinetic parameters of electrons in the vicinity of the Fermi energy (E F ). The coherent potential approximation (CPA) was employed to account for chemical disorder effects (note that Si, Ge, Sn, and Bi occupy the same crystallographic site) on the electronic structure self-consistently. The linearized Boltzmann equation is implemented to determine electron transport coefficients such as the electrical conductivity r [used to extract the electron lifetime from experimental r exp ðTÞ], the Seebeck coefficient S, and the electronic part of the thermal conductivity j e : Quite recently, this procedure was successfully applied to n-type Mg 2 Si 1Àx Sn x thermoelectric material, 9 which has been intensively investigated 10-12 both experimentally and theoretically due to the so-called conduction-band convergence. 1,[13][14][15] In this work, we extend such theoretical investigations to the very promising pseudoquaternary Mg 2 Si 1ÀxÀy Sn x Ge y : Bi: This paper is organized as follows. The next section presents theoretical details of the electron transport coefficient calculations in terms of transport functions. A constant relaxation time approximation is employed for the electron scattering mechanism. ''Results and Discussion'' section reports on the computational results for the temperature-dependent transport properties, i.e., rðTÞ; SðTÞ; and j e ðTÞ, in Bi-doped Mg 2 Si 1ÀxÀy Sn x Ge y . This section also presents maps of the power factor PFðn; TÞ ¼ S 2 r and the figure of merit ZTðn; TÞ ¼ rS 2 =j (with j ¼ j e þ j l ) versus electron concentration n and temperature T, assuming experimental values of the lattice thermal conductivity j l . The article ends with ''Conclusion'' section.

THEORETICAL AND COMPUTATIONAL DETAILS
Boltzmann transport theory was employed to calculate the electron transport properties. In this approach, transport quantities can be expressed in compact form as 16 ; where The transport function rðEÞ is the key quantity in Boltzmann transport modeling, since together with the chemical potential l, it contains all the materialrelated information. Generally, it is a tensor and has the form where is the Kronecker product. The transport function contains two functions designating the electronic properties of the system, namely the band structure E n ðkÞ and the electron lifetime of bands s n ðkÞ;, which describe possible scattering processes.
In standard density functional theory (DFT) calculations, it is impossible to take into account all these electron diffusion effects. [17][18][19][20] Fortunately, it has been shown 9,21,22 that assuming the function s n ðkÞ to be a constant s 0 (the constant lifetime approximation) often gives results in close agreement with experiment.
In practice, the k-space volume integration in Eq. 3 is replaced by surface integration: where the isoenergetic surfaces SðEÞ are calculated using a marching cube algorithm 23 on a threedimensional (3D) mesh. In this work, the Brillouin zone was inscribed into a box divided into 512,000 voxels (80 Â 80 Â 80). Using Eq. 4 and the regular structure relation jv vj ¼ 1=3v 2 , the transport function can be represented as In general, the chemical potential l is calculated from the density of states (DOS) function. Next, employing a rigid-band model, 24 the effect of doping can be simulated by adding extra carriers n into the semiconducting system where n 0 corresponds to the number of electrons filling the valence bands. In the case of the Bi-doped system, our analysis was based on the KKR-CPA electronic band structure calculated for Mg 2 Si 1ÀxÀy Sn x Ge y with 2% of Bi diluted on the Si/ Ge/Sn site, and the Fermi level was found above the conduction-band edge. To investigate different magnitudes of doping, E F was varied along these electronic bands to mimic carrier concentrations in the range 10 19 cm À3 < n < 10 21 cm À3 .
The lifetime s 0 used in this work was calculated by matching the experimental and calculated conductivity This matching was done using the experimental conductivity for the sample with 2% Bi and calculated for four different compositions, separately (for all temperatures independently). This yielded four different s 0 ðTÞ relations with an almost linear dependence on temperature; therefore only two values, corresponding to T = 300 K and T = 800 K, are presented in Table I.

RESULTS AND DISCUSSION
It is worth noting that the electrical conductivity and thermopower measured for the multiphase samples with different Bi contents show that the r(T) and S(T) curves no longer change when the Bi concentration exceeds 2% (likely indicating the solubility limit for this dopant). Figure 1 shows that a difference in the Seebeck coefficient is visible only for the sample with 1% nominal Bi content, whereas the samples with higher Bi concentrations, namely 1.75%, 2%, and 2.5%, exhibit very similar S(T) dependences. Consequently, the experimental thermopower for these three Bi contents fits the theoretical S(T) curve for n % 2:5 Â 10 20 cm À3 , which corresponds to 1.8% nominal Bi content. Figure 2 indicates that the measured electrical conductivity increases with Bi content up to 2%. There is no difference between the samples with 2% and 2.75% Bi concentration. The above-mentioned differences in the experimental behaviors of the thermopower and electrical conductivity with Bi content suggest different solubility limits of Bi in different phases.
KKR-CPA calculations resulted in an energy gap between valence and conduction bands in all considered phases. However, as expected from the local density approximation (LDA) employed in this work, such computations tend to underestimate bandgap values. To enable reliable analysis of temperature-dependent electron transport parameters, the calculated gap was then broadened to the experimental value. 25 Electronic structure calculations of Mg 2 (Si-Sn-Ge) show that the important differences among the results obtained for the three considered phases Table I   Electronic Structure and Thermoelectric Properties of Pseudoquaternary Mg 2 Si 1ÀxÀy Sn x Ge y -Based Materials (also compared with the nominal composition) essentially concern the two lowest-lying conduction bands. Inspecting the dispersion curves for highsymmetry directions in more detail, it appears that the relative position of the two conduction bands near the X point is mostly affected by the alloy composition. One can tentatively connect the mutual shift of these bands with the variable lattice parameter. These electronic structure features are best illustrated in Fig. 3. It can be noticed that the Sn-rich and nominal phases exhibit the smallest energy separation between the two conduction bands (near the X point). Therefore, as already shown 9 for the Mg 2 (Si-Sn) system, this electronic structure behavior presumably enhances the thermoelectric properties of these alloys. When these two bands tend to converge near the X point (the band separation almost vanishes for phases ph 1 and ph N, as shown in Fig. 3), the thermopower indeed reaches its highest values (Fig. 4a) among the considered phases. The influence of the conduction band separation on the Seebeck coefficient is best seen at lower temperature and lower carrier concentration (in particular below 3 Â 10 20 cm À3 ). As a consequence, these electronic band features have an effect on the power factor (Fig. 5). For 2% Bi content and at temperature of 900 K, the power factor reaches a value about 15% higher for phase ph N (or ph 1) with respect to the worst one (ph 2), which exhibits the largest conduction band separation near the X point.
To estimate the dimensionless figure of merit ZT in these four phases, the lattice part of the thermal conductivity (j l ) was estimated from experimental data. 4 Accordingly, j l was chosen to decrease linearly with temperature from a value of j l ¼ 2:0 W m À1 K À1 at 300 K to j l ¼ 1:25 W m À1 K À1 at 800 K. Besides, the thermal conductivity was taken to be the same for all investigated compounds, allowing coherent comparison among the results obtained for the four phases, to focus on the contribution of the electronic quantities to ZT. The resulting ZT maps depending on the carrier concentration and temperature are presented in Fig. 6. It can be noticed that the highest ZT values were found for n % 1:5 Â 10 20 cm À3 (corresponding to ca. 1% Bi) and T % 900 K for the Sn-rich or nominal   Table I for exact alloy compositions) (Color figure online).
phases, in line with the results obtained for the power factor (Fig. 5). As mentioned above, the Sirich phase, which exhibits large splitting between the two conduction bands (Fig. 3, top), also has the worst ZT among the four presented alloy compositions (Fig. 4b).

CONCLUSIONS
We have shown that the KKR-CPA method combined with the Boltzmann transport approach can be successfully adopted for theoretical study of thermoelectric properties in the complex pseudoternary n-type Mg 2 Si 1ÀxÀy Sn x Ge y system, which contains a high degree of chemical disorder (alloying and doping). The implemented procedure (within the constant relaxation time approximation) satisfactorily reproduces the dependence of electron transport properties, such as the thermopower and power factor, on temperature and carrier concentration. It has been found that the high efficiency of thermoelectric conversion measured in Bi-doped nominal Mg 2 Si 0:53 Sn 0:4 Ge 0:05 ; which actually consists of three phases: Mg 2 Si 0:35 Sn 0:6 Ge 0:05 (Sn-rich phase), Mg 2 Si 0:65 Sn 0:3 Ge 0:05 (Si-rich phase), and Mg 2 Si 0:15 Sn 0:5 Ge 0:35 (Ge-rich phase), is likely due to the fact that two of them (ph 1 and ph N) exhibit similar thermoelectric parameters in the high temperature range. The ZT values versus temperature and carrier concentration were estimated using the lattice part of the thermal conductivity as an adjustable parameter. ZT = 1.4 was achieved near 900 K at the experimental value of j l : It has also been shown that the Bi dopant presumably reaches its solubility limit close to 2%, preventing the achievement of the best working concentration.