In order to produce the fraction x = 0.25 and 0.75, the minimum SQS supercell should contain 12 atoms. In this study, we consider mixing on the anion sublattice of the rocksalt crystal structure. For each fraction, the atom positions are randomly exchanged between the atoms, terminated until the atomic arrangement of a supercell converges to the closest CFs of the random solution and the error is estimated by ². The CFs of pairs and triplets are listed in Table 1.

Table 1.

Table 1.

Table 1. Pair and multi-site correlation functions of SQS structures for Mg2GexSn1−x solid solutions. . x = 0.25 x = 0.5 x = 0.75 Figure Random SQS-12 SQS-24 Random SQS-12 SQS-24 Random SQS-12 SQS-24 Π2,1[06] 0.25 –0.0833 0.1667 0 0 0 0.25 –0.0833 0.1667 Π2,2[03] 0.25 –0.25 0.1667 0 –0.3333 0 0.25 –0.25 0.1667 Π2,3[12] 0.25 0 0.25 0 0 –0.1667 0.25 0 0.25 Π2,4[06] 0.25 0.25 0.1667 0 –0.3333 0.3333 0.25 0.25 0.1667 Π2,5[12] 0.25 0.0833 0.3333 0 0 0 0.25 0.0833 0.3333 Π2,6[04] 0.25 –0.25 0.375 0 1 0 0.25 –0.25 0.375 Π2,7[24] 0.25 0 0.2917 0 0 –0.1667 0.25 0 0.2917 Π3,1[08] 0.125 –0.125 0 0 0 0 0.125 –0.125 0 Π3,2[12] 0.125 –0.2917 0 0 0 0 0.125 –0.2917 0 Error 0 1.0 0.0764 0 1.2222 0.1667 0 1.0 0.0764

Table 1. Pair and multi-site correlation functions of SQS structures for Mg2GexSn1−x solid solutions. .

For the SQS-12 supercell, the errors of the CF are 1.0 for x = 0.25 (0.75) and 1.22 for x = 0.5 accounting for the 7th nearest neighbors. Though SQS-12 possesses many pair correlations that match the random alloy statistics, the discrepancies are relatively large. In order to minimize error, we doubled the SQS supercell, and the CFs of SQS-24 are also listed in Table 1. Compared to SQS-12, the usage of SQS-24 supercell reduces the error largely with more pair correlations matching the random alloy statistics for all these three proportions. For x = 0.25 (0.75), the error of the 7th nearest neighbors is only 0.0764. Although enlarging the SQS supercell may further reduce the error, this size represents a good compromise between accuracy and the computational requirements associated with the TE transport property calculations. Thus, in the following, we choose the SQS-24 for the electronic and TE property studies.

The special quasi-random structures with 24 atoms for these three solid solutions are shown in the following Fig. 1. The optimized lattice constants are a = 8.23 Å, b = 8.24 Å, c = 9.51 Å for Mg₂Ge_0.25Sn_0.75, a = 5.47 Å, b = 9.61 Å, c = 11.49 Å for Mg₂Ge_0.5Sn_0.5, and a = 8.23 Å, b = 8.24 Å, c = 9.51 Å for Mg₂Ge_0.75Sn_0.25, respectively. Correspondingly, the volume of Mg₂Ge_0.25Sn_0.75 is 608.79 ų, which is slightly larger than that of 580.89 ų for Mg₂Ge_0.5Sn_0.5 and 556.80 ų for Mg₂Ge_0.75Sn_0.25.

Fig. 1.

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	Fig. 1. (color online) The SQSs of Mg2GexSn1−x solid solutions with (a) x=0.25, (b) x=0.5, and (c) x=0.75, respectively. The number of atoms in the SQS unit cell is 24.

Using these SQSs, we calculated the phonon dispersion curves, as shown in Fig. 2, in comparison with the counterparts of pure Mg₂Ge and Mg₂Sn. Clearly, there are no imaginary frequencies existing in all these phonon dispersion relations, indicating the dynamical stability of these SQSs. Compared to the pure Mg₂Sn and Mg₂Ge, the slopes of acoustic modes of solid solutions are reduced largely, reflecting a lower phonon group velocity. The low-frequency optical modes are mixed with the acoustic branches in Mg₂Ge_0.5Sn_0.5 and Mg₂Ge_0.75Sn_0.25, showing the strong phonon scattering between them. Both the low group velocity and strong phonon scattering are beneficial to reducing the lattice thermal conductivity.

Fig. 2.

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	Fig. 2. (color online) The calculated phonon dispersion curves of Mg2GexSn1−x at (a) x=0.25, (b) x=0.5, and (c) x=0.75, respectively. The corresponding results of Mg2Sn (red solid lines in panel (a)) and Mg2Ge (blue solid lines in panel (c)) are also shown for comparison.

The total density of states (DOS) and projected density of states of Mg₂Ge_xSn_1−x solid solutions are shown in Fig. 3, compared with those of pure Mg₂Sn and Mg₂Ge. In general, all these densities of states are featured by four well-resolved peaks, distributed on both sides of Fermi level. From Fig. 3(a), the slope of the total DOS at VBM is larger than that of CBM for any compositions, indicating that the effective mass of carriers of p-type is larger than that of n-type. Since the effective mass is proportional to the Seebeck coefficient (S), the value of S for p-type Mg₂Ge_xSn_1−x is also larger than that of n-type. From the partial density of states (PDOS) as shown in Figs. 3(b)–3(f), the peak at CBM is largely observed by the Mg-s orbital, while the peak at VBM nearby is mainly contributed by the Sn-p orbital for Mg₂Ge₂₅Sn_0.75 and Ge-p orbital for Mg₂Ge_0.75Sn_0.25 respectively, indicating the occurrence of charge transfers from Mg to Sn or Ge. Through the Bader analysis,^[46] we can obtain the charge transfer quantitatively, where about two electrons are transferred from Mg atom to Sn atom in Mg₂Ge_0.25Sn_0.75 or to Ge atom in Mg₂Ge_0.75Sn_0.25. In Mg₂Ge_0.5Sn_0.5, the Sn and Ge atoms are almost equivalent in sharing the electrons. Further analysis shows that these electrons are mainly from the Mg-3s orbital. As x increases from 0.25 to 0.75, the band gap between VBM and CBM is enlarged from 0.35 eV to 0.44 eV. These values lie in the experimental results for the pure Mg₂Sn of 0.3 eV and Mg₂Ge of 0.6 eV.^[47] It is noted that the band gap of Mg₂Ge_0.5Sn_0.5 calculated by the ordered supercell is 0.23 eV,^[19] which is slightly lower than that of the pure Mg₂Sn and the present result of 0.4 eV.

Fig. 3.

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	Fig. 3. (color online) (a) shows the total DOS for Mg2GexSn1−x at x = 0, 0.25, 0.5, 0.75, 1. Panels (b)–(f) show the projected DOS of Mg2GexSn1−x solid solutions. All the Fermi levels have been set at zero.

The Debye temperature (Θ_D) describes atomic vibrations to be elastic waves and it is related to the sound velocity.^[48] The Debye temperature can be expressed as follows:

where h and k_B are Planck and Boltzmann constants, respectively, n is the number of atoms in one unit cell, Ω is the volume per unit cell, and v_m is the averaged sound velocity. Approximately, the averaged sound velocity can be calculated by the following expression

where is the transverse elastic wave velocity, (ρ is the density of a material) and is the longitudinal elastic wave velocity. B and G are the bulk and shear modulus, which can be determined by elastic constants following B = (C₁₁ + 2C₁₂)/3 and G = (C₁₁ − C₁₂ + 3C₄₄)/5, respectively. The elastic constants C₁₁,C₁₂, and C₄₄ are obtained by enforcing small strains on the equilibrium unit cell based on Hooke’s law.^[49] Our calculated elastic moduli, sound velocities and Debye temperature of Mg₂Ge_xSn_1−x are listed in Table 2. As x increases, the values of v_t and v_l show decrease trends. Mg₂Ge_0.25Sn_0.75 has the lowest Debye temperature due to the fact that Θ_D is proportional to v_m according to Eq. (1).

Table 2.

Table 2.

Table 2. The calculated elastic constants (Cij, in unit GPa), bulk modulus (B, in unit GPa), shear modulus (G, in unit GPa), density (ρ, in units g/cm3), volume (V, in units Å3), transverse sound velocity (vt, in units m/s), longitudinal sound velocity (vl, in units m/s), mean sound velocity (vm, in units m/s), Debye temperature (ΘD, in unit K), and minimum lattice thermal conductivity (κmin) of Mg2GexSn1−x (x = 0.25, 0.5, 0.75) solid solutions. . Phase C11/GPa C12/GPa C44/GPa ρ/(g/cm3) V/Å3 B/GPa G/GPa vt/(m/s) vl/(m/s) ΘD/K κmin Mg2Ge0.25Sn0.75 83.87 22.04 28.34 3.41 25.38 42.65 29.37 2935 4898 329 0.69 Mg2Ge0.5Sn0.5 90.02 22.10 31.80 3.30 24.27 44.74 32.66 3146 5173 358 0.76 Mg2Ge0.75Sn0.25 90.42 22.43 32.27 3.19 23.14 45.09 32.96 3214 5283 372 0.80

Table 2. The calculated elastic constants (Cij, in unit GPa), bulk modulus (B, in unit GPa), shear modulus (G, in unit GPa), density (ρ, in units g/cm3), volume (V, in units Å3), transverse sound velocity (vt, in units m/s), longitudinal sound velocity (vl, in units m/s), mean sound velocity (vm, in units m/s), Debye temperature (ΘD, in unit K), and minimum lattice thermal conductivity (κmin) of Mg2GexSn1−x (x = 0.25, 0.5, 0.75) solid solutions. .

Based on Debye theory, the lattice thermal conductivity is proportional to 1/T at low temperature and reaches the minimum value at high temperature. When the temperature is higher than the Debye temperature, the lattice thermal conductivity can be well estimated using the minimum lattice thermal conductivity (κ_min), where the phonon scattering is dominated by the Umklapp process. The κ_min can be calculated according to the following expression^[50]

with V the average volume per atom. The results are also listed in Table 2. Among these three solid solutions, Mg₂Ge_0.25Sn_0.75 has the lowest κ_min of 0.69 W · m⁻¹ · K⁻¹, originating from its larger V and lower acoustic velocities. This value is somewhat lower than the experimental data of ∼ 1.2 W · m⁻¹ · K⁻¹ for Mg₂Ge_0.25Sn_0.75 at 723 K.^[22] Since the minimum thermal conductivity should be regarded as the lower limit of the lattice thermal conductivity for solid solutions, it provides predictive theoretical guidance for methods to further reduce the lattice thermal conductivity, such as the fabrication and synthesis of nanostructures.^[51]

The TE transport properties of Mg₂Ge_xSn_1−x solid solutions were calculated by semi-classical Boltzmann theory with CSTA. Using the W–F formula, κ_e = LσT, the zT can be further expressed as zT = rS²/L, where L = 2.45 × 10⁻⁸ W · Ω/K² is the standard Lorenz number and r = κ_e/(κ_e + κ_l) ≤ 1. Thus, in order to obtain the zT value, one still needs to calculate the σ and S parameters. It is well known that the carrier concentration of a TE material with a good performance is in the range of 10¹⁹ to 10²⁰ carriers per cm⁻³.

Thus, in Fig. 4, we show the calculated S as a function of carrier concentration at this range. According to the definition of Seebeck coefficient,

Fig. 4.

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Fig. 4. (color online) Transport coefficients map of Mg2GexSn1−x as a function of carrier concentration n and temperature T for x = 0.25 (top), 0.50 (mid), 0.75 (bottom), respectively. Panels (a)–(c) show the Seebeck coefficient; panels (d)–(f) show the electronic conductivity; panels (g)–(i) show the power factor. The negative scale of x axis stands for the n-type, while the positive scale represents the p-type.

the S is proportional to the effective mass of carrier, m^∗. From Fig. 4, generally, the S of p-type Mg₂Ge_xSn_1−x solid solutions is larger than that of n-type covering the whole range of concentrations considered here, consistent with the larger valence-band effective mass. We note that there is a flexion of the S curve in these solid solutions. As temperature increases, the absolute value of S decreases at low concentration but increases at high concentration. Actually, at high concentration, S increases with increasing temperature due to the proportionality between S and T . However, at low concentration, the bipolar effect^[52] plays an important role in the contribution of S, which originates from the two types of carriers introduced by a small band gap. The dipolar effect limits the TE performance, which should be reduced as much as possible in applications. This character also appears in the pure Mg₂Sn material.^[53]

From the electronic structures above, σ/τ can be easily calculated as a function of carrier concentration and temperature. However, in order to isolate the electrical conductivity σ, one should use the relation, σ = (σ/τ) × τ, based on the CSTA, where the scattering time τ should be known. In typical electron–phonon regime, τ is proportional to T⁻¹n^−1/3. Thus, in the following, we use the experimental data to determine τ and obtain the electrical conductivity σ. For p-type Mg₂Ge_xSn_1−x (x = 0.25, 0.5), the experimental data from Jiang et al.^[23] is used. In this literature, the Ag-doped Mg_1.98Ag_0.02Ge_0.4Sn_0.6 solid solution has been successfully synthesized. We select the same typical temperature of 400 K as that in Ref. [19]. At 400 K, the Seebeck coefficient S is about 203 μV/K. Compared with the calculated S from Fig. 4(a), the corresponding carrier concentration is 9.4 × 10¹⁹ cm⁻³. Using the experimental electrical conductivity of 2.00 × 10⁴ S/m and the calculated σ/τ, we obtain a relaxation time of τ = 0.56 × 10⁻⁵T⁻¹n^−1/3. For n-type, we choose the data from Liu et al.^[22] at 300 K. Similarly, we obtain the τ = 3.9 × 10⁻⁵T⁻¹n^−1/3. Due to the lack of experimental data for Mg₂Ge_0.75Sn_0.25, we adopt the same scattering time as that for Mg₂Ge_0.25Sn_0.75. The calculated electrical conductivities for these three types of solid solution are shown in Figs. 4(d)–4(f). Evidently, the σ of the n-type is several times higher than that of the p-type for all three solid solutions.

The corresponding power factors (PF), S²σ, can then be obtained, which are shown in Figs. 4(g)–4(i). The general characteristics of these curves are similar with one peak at each temperature. With the increase of carrier concentration, the value of PF increases firstly to the maximum value and then decreases. This is because PF is mainly dominated by σ at low carrier concentrations, while S plays an important role in PF at high carrier concentrations. The maximum value of PF is 0.8mW/mK² for p-type Mg₂Ge_0.25Sn_0.75 at 500 K and 5.96 mW/mK² for n-type Mg₂Ge_0.5Sn_0.5 at 700 K, which is in good agreement with the experimental data, i.e., 0.9mW/mK² at 500 K for p-type Mg₂Ge_0.4Sn_0.6^[23] and 5.4 mW/mK² for n-type Mg₂Ge_0.25Sn_0.75.^[22]

We show in Fig. 5 the calculated electronic thermal conductivity, κ_e, as a function of carrier concentration from 10¹⁹ cm⁻³ to 10²⁰ cm⁻³ and temperature from 500 K to 800 K, according to the Wiedemann–Franz relation, κ_e = LσT . In general, the κ_e increases monotonically with increasing the carrier concentration for both p-type and n-type, similar to the results of σ. As the concentration increases from 10¹⁹ cm⁻³ to 10²⁰ cm⁻³, the κ_e of the n-type solid solutions is increased from ∼ 0.2 W · m⁻¹ · K⁻¹ to ∼ 2 W · m⁻¹ · K⁻¹, in agreement with the experimental values from ∼ 1.1 W · m⁻¹ · K⁻¹ to ∼ 1.3 W · m⁻¹ · K⁻¹ for Mg₂Ge_0.25Sn_0.75 with temperature increasing from 298 K to 723 K.^[22] The κ_e of the p-type is lower than that of the n-type by an order of magnitude, which is increased from ∼ 0.05 W · m⁻¹ · K⁻¹ to ∼ 0.2 W · m⁻¹ · K⁻¹ with the concentration increasing from 10¹⁹ cm⁻³ to 10²⁰ cm⁻³. The experimental κ_e for the p-type Mg₂Ge_0.5Sn_0.5 can be approximated by the difference between the κ and κ_l, which is increased from ∼ 0.05 W · m⁻¹ · K⁻¹ to ∼ 0.15 W·m⁻1 ·K⁻¹ with temperature increasing from 500 K to 700 K.^[23] Our theoretical results are also consistent with these observations.^[22,23] The total thermal conductivity can then be evaluated by the combination of κ_e and κ_l.

Fig. 5.

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	Fig. 5. (color online) Calculated κe of Mg2GexSn1−x with (a) x = 0.25, (b) x = 0.5, and (c) x = 0.75 as a function of the carrier concentration n and the temperature T, respectively. The negative scale of x axis stands for the n-type, while the positive scale represents the p-type.

All the TE parameters are unified by zT value to measure the TE performance. In Fig. 6, we show the calculated zT values of Mg₂Ge_xSn_1−x solid solution as a function of temperature from 500 K to 800 K and carrier concentration from 10¹⁹ cm⁻³ to 10²⁰ cm⁻³. It can be seen that the zT value of the n-type Mg₂Ge_xSn_1−x is much higher than that of the p-type, which can be mainly attributed to the much higher PF for the n-type. The maximum of zT value reaches 2.45 for the n-type Mg₂Ge_0.5Sn_0.5 at 800 K with a carrier concentration of 3.28 × 10¹⁹ cm⁻³, mainly due to combination of high PF and low lattice thermal conductivity. We note that this value is slightly higher than that of 2.25 for Mg₂Ge_0.5Sn_0.5 in Ref. [19], which is obtained by the ordered supercell method. Similarly, the Mg₂Ge_0.25Sn_0.75 and Mg₂Ge_0.75Sn_0.25 also show high TE performance with zT of 2.31 and 2.40, respectively. These results are higher than the experimental data of 1.4 for Mg₂Ge_0.25Sn_0.75 at a different carrier concentration of 3.0 × 10²⁰ cm⁻³. At the same concentration, we obtain a zT value of 0.9. The differences between the theoretical and experimental values may originate from the CSTA or temperature effects or a combination. The maximum zT value of 0.62 for the p-type Mg₂Ge_xSn_1−x solid solution appears at x = 0.25, which is still about four times lower than that of the n-type.

Fig. 6.

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	Fig. 6. (color online) Calculated zT map of Mg2GexSn1−x with (a) x = 0.25, (b) x = 0.5, and (c) x = 0.75 as a function of the carrier concentration n and the temperature T, respectively. The negative scale of the x axis stands for the n-type, while the positive scale represents the p-type.