Electronic structure and thermoelectric properties of p-type Ag-doped Mg₂Sn and Mg₂Sn_1-xSi_x (x = 0.05, 0.1)

Thermoelectric generators (TEGs) are solid state energy converters with no moving parts that have a high power density and excellent long-term reliability and require low or no maintenance. The efficiency of thermoelectric materials is defined by the figure of merit (zT = α²σT/κ), where α is the Seebeck coefficient, σ is the electrical conductivity (=1/ρ, where $ρ$ is the electrical resistivity), and κ is the thermal conductivity. In order to design highly efficient thermoelectric materials, one maximizes the product α²σ, known as the power factor, and minimizes κ. However, since zT is composed of inter-related materials properties, improving one property can degrade the others,^1,2 and a maximum zT can only be obtained by optimizing all three thermoelectric properties simultaneously.

One of the promising thermoelectric materials for power generation is Mg₂X (X = Sn, Si), because of its low cost, abundance, and the fact that good zT values have been reported on n-type Mg₂Sn_1−xSi_x.^3,4 The band gap, electrical conductivity, and Hall resistivity of non-intentionally doped Mg₂Sn were reported in 1948 by Robertson⁵ and in 1955 by Blunt.⁶ The band gap of Mg₂Sn is about 0.33 eV at 0 K,⁶ though values between 0.16 eV and 0.33 eV are given in the literature,^7,8 and depend on the measurement techniques used. The thermal conductivity of pure Mg₂Sn was also reported in 1968 by Martin and Danielson.⁹ In contrast to the n-type material, only a limited number of studies have focused on p-type Mg₂Sn_1−xSi_x. The sample preparation techniques require the presence of an additional amount of Mg to compensate for the very high vapor pressure of that element, so that intentional defect doping by varying the Mg/Sn stoichiometry is difficult to control in this system. Ag-doped p-type Mg₂Sn material has been studied by Savvides and co-workers.¹⁰ Ag is also an acceptor in Mg₂Si.¹¹ Isoda et al.¹² studied double-doping Mg₂Sn_0.75Si_0.25 with Ag and Li and prepared their samples by liquid solid reaction followed by hot pressing. Tada et al. recently reported thermoelectric properties of sodium acetate and metallic sodium doped Mg₂Sn_0.75Si_0.25.¹³ 

Here we also use Ag as an acceptor and address the issue of which site Ag substitutes for in Mg₂Sn and Mg₂Sn_1-xSi_x(x = 0.05, 0.1). In Mg₂Si, Ag was predicted to either substitute for Mg¹⁴ or for both Mg and Si,¹⁵ whereas recent synchrotron studies¹⁵ showed that Ag can migrate from the Si site when temperature increases. We report Bloch spectral function and density of states calculations for Mg₂Sn and Mg₂Sn_0.9Si_0.1 showing that if Ag substitutes for Sn (Ag_Sn), it is likely to form a resonant level like Sn in Bi₂Te₃ (Ref. 16) or Tl in PbTe.¹⁷ On the other hand, if Ag substitutes for Mg (Ag_Mg), an almost rigid-band-like behavior is calculated, and the thermopower of Ag-doped Mg₂Sn at a given carrier concentration should not differ from that calculated for Mg₂Sn using rigid-band models.¹⁸ In addition, we show that a small addition of Si (10% in our case) only slightly affects the electronic structure of the system, thus similar thermoelectric properties should be expected. In the experimental part, we show galvanomagnetic, thermoelectric, and thermomagnetic transport data of Ag-doped Mg₂Sn_xSi_1−x (x = 0, 0.05, 0.1) samples and conclude that no signature of a resonant level is observed, which suggests that Ag substitutes for Mg. Still, a power factor in excess of 20 μW cm⁻¹K⁻² is observed between 300 and 400 K for the sample with x = 0 and an optimum doping level of 6 × 10¹⁹cm⁻³.

Electronic structure calculations for the Ag-doped Mg₂Sn and Mg₂Sn_0.9Si_0.1 were performed using the Korringa-Kohn-Rostoker (KKR) method with the coherent potential approximation (CPA) applied to account for the chemical disorder,¹⁹ as implemented in the Munich SPRKKR package.²⁰ The experimental anti-fluorite crystal structure and lattice parameter (6.765 Å) for Mg₂Sn were used,³ and the lattice parameter for Mg₂Sn_0.9Si_0.1 was adjusted according to the Vegard's law. A regular k-mesh was used in calculations, using about 1500 k-points for the self-consistent cycle and 60 000 for the density of states (DOS) and Bloch spectral functions (BSF) calculations (in the irreducible part of the Brillouin zone). The crystal potential was constructed in the framework of the local density approximation (LDA), using the Vosko, Wilk and Nussair formula²¹ for the exchange-correlation part. As already discussed in literature,^8,18,22 the standard local density exchange-correlation energy functionals yield negative band gap values for the case of Mg₂Sn, a problem that we observed in the present work as well. The conduction band minimum at X point is located too low on the energy scale, which yields a semi-metallic ground state due to the overlap of conduction and valence bands, similar to the previous reports.^8,18,22 However, since the effect is limited mostly to a rigid shift of the whole band structure,⁸ it is still possible to discuss the character of the impurity states, in which we are interested in the present work, bearing in mind that the position of the Fermi level (E_F) will be too deep in the valence band.

Fig. 1 presents the total and the Ag DOS of Ag-doped Mg₂Sn, assuming Ag_Sn substitution. The above-mentioned band gap problem leads to the formation of a pseudo-gap instead of the real band gap in the background Mg₂Sn DOS. The partial DOS of Ag impurity for the Ag_Sn substitution case shows a peak near E_F, which suggests the formation of a resonant state. Such a resonant state could result in enhancement of the thermopower of the doped system^1,16,17 compared to the thermopower of a solid with the pure Mg₂Sn bands. The orbital character of the peak is p-like, which is different from the well-known case of resonant Tl impurity in PbTe,¹⁷ where the resonant level originates from 6 s electrons of Tl.²³ Also, the peak is considerably broader, which means a higher level of hybridization with the host electronic states. It is worth noting the existence of accompanying, low-energy 'hyper-deep' resonant state, located around −5 eV. This time, similar to the PbTe:Tl case, the DOS peak coming with this state is s-like. The semi-core 4 d filled shell is located also around −5 eV and will not contribute to the transport properties of the system.

Ag_Mg substitution in Mg₂Sn gives a completely different picture. The partial DOS of Ag, shown in Fig. 2, is very flat and does not show any signs of resonances, both in deep and close to E_F energies. Thus, this time Ag is expected to behave as a regular acceptor: its presence in the crystal should only shift E_F.

Due to the band gap problem, it is not evident which type of conductivity (n or p) should be expected in the aforementioned cases. To have a better insight into the effect Ag-doping has on the electronic structure, Bloch spectral density functions (BSF) A(k,E) were calculated. BSF are a generalization of the dispersion relation for disordered systems^19,23 and can be interpreted as a wavevector-resolved density of states. For a perfect crystal without impurities, A(k,E) at selected k-points is a Dirac delta function of energy, being non-zero only at (k,E) points where the electronic state has an energy eigenvalue. Thus, the collection of (k,E) peak points of BSF can describe electron dispersion relation E(k). In disordered systems, due to the alloy scattering of electrons, electronic bands are smeared, and for majority of cases, BSF take the form of the Lorentz function where the peak of BSF corresponds to the center of the band and the full width at half maximum (FWHM) value Γ corresponds to the life time of the given electronic state²⁴ (which is finite due to alloy scattering):

More intense scattering leads to broader BSF and shorter electronic lifetimes.

To visualize the band structure using BSF, we plot in Figs. 3 and 4, the 2-dimensional projections of the BSF, with the color marking the BSF values, as a function of both selected k-values (abscissa) and energy (ordinate). The black points, corresponding to BSF values greater than 300 Ry⁻¹, generally follow the binary Mg₂Sn dispersions E(k), whereas the presence of Ag is responsible for the smearing of the band structure. In Fig. 3, we can identify three main valence bands of Mg₂Sn (for better resolution, the fourth lower lying s-like band is not shown on the figure), two of which are touching the valence band maximum at Γ point, with the third band split off by the spin-orbit interaction. The value of this relativistic p^3/2-p^1/2 band splitting, around 0.5 eV, is in good agreement with the previous calculations (0.52 eV (Ref. 18) and 0.46 eV (Ref. 8)) as well as measurements (0.48 eV (Ref. 7) and 0.60 eV (Ref. 25)). As was recently found,¹⁸ spin-orbit interactions play a role in the transport properties of p-type Mg₂Sn. Besides splitting of the bands at Γ point, the spin-orbit coupling modifies the band curvature, strongly reducing the DOS effective mass of holes and decreasing thermopower of Mg₂Sn.¹⁸ 

In agreement with the DOS result and the resonant state behavior interpretation, the Ag_Sn impurity strongly affects the Mg₂Sn bands, leading to a considerable band broadening (Fig. 4, left). The Ag electronic states couple mainly to the valence states around Γ point, and E_F moves deeper into the valence band block, which allows the classification of Ag_Sn as an acceptor impurity. The spectral function of the system at k = (0.115, 0.115, 0.115) 2π/a, i.e., about one fourth into the Brillouin zone along the Γ-L direction, where one of the valence bands is crossing E_F, is shown at the bottom panel in Fig. 4. As is characteristic for the resonant behavior, the spectral functions are considerably broadened: the FWHM of the upper BSF is around 0.2 eV, resulting in relatively short life time τ_imp = 3 × 10⁻¹⁵ s at that k-point. Although the overall resonant effect is not as strong as in the case of the Tl resonant impurity in PbTe,²³ still the Ag_Sn substitution is not expected to follow the rigid band behavior, and the transport properties of the Mg₂Sn_1-xAg_x system are expected to differ from Na or Li doped Mg₂Sn.

The BSF of Mg_2-yAg_ySn (Ag_Mg substitution) show rigid-band-like behavior, in agreement with the DOS analysis. A direct comparison of the spectral functions between Mg₂Sn_1-yAg_y (first column in Fig. 4) and Mg_2-yAg_ySn (second column in Fig. 4) demonstrates that in the second case sharp and well-defined “virtual” energy bands are formed, meaning that Ag_Mg does not lead to excessive electron scattering. The bottom panel of Fig. 4 compares spectral functions, plotted for the same k–point. The contrast between resonant-like (Mg₂Sn_1-yAg_y) and rigid-band-like (Mg_2-yAg_ySn) behaviors is evident. For the Ag_Mg substitution, the corresponding lifetime is of the order of τ_imp ≈ 10⁻¹³ s, longer than even the electron-phonon scattering relaxation time (τ_ph ≈ 10⁻¹⁴ s (Ref. 26) for n-type Mg₂Si_1-xSn_x). Thus, assuming Matthiessen's formula τ ⁻¹_total = τ ⁻¹_ph + τ ⁻¹_imp, the contribution from τ_imp in the total relaxation time is small, showing that in the Ag_Mg substitution case, scattering on Ag impurities is less important. Also, the BSF width does not depend much on energy around E_F here, supporting the constant relaxation time approximation, used frequently for first principles thermopower calculations of Mg₂Si-Sn based systems.^18,22,26

In the third column of Fig 4, we present BSF for the Si-alloyed system, Mg_2-yAg_ySn_0.9Si_0.1 for the Ag_Mg substitution case. An addition of Si increases disorder in the system leading to a small additional blurring of the band structure. The BSF at selected k-points (lowest panel in Fig. 4) also show a delta-like behavior with little larger FWHM values, proving the existence of sharp and well-defined electronic bands. The slightly broader BSF mean that slightly stronger alloy scattering should be observed in Mg_2-yAg_ySn_0.9Si_0.1 compared to Mg_2-yAg_ySn. However, since the curvature of the bands does not markedly change from the Mg_2-yAg_ySn case, the thermopower of both Si-free and Si-alloyed samples at similar carrier concentrations should be close. Si addition is more important at higher Si concentrations and for n-type materials, where it leads to conduction band convergence effect (see Ref. 26 and references therein).

The comparison between this analysis and the experimental data on Ag-doped Mg₂Sn presented in next paragraph will clarify the acceptor nature of Ag in Mg₂Sn.

High purity elemental Mg(99.98%), Sn(99.999%), Si(99.9999%), and Ag(99.9999%) were used to synthesize Mg₂Sn_1-xSi_x (x = 0, 0.05, and 0.1), either unintentionally doped by defects or doped with Ag (0.5 at. %, 1 at. %, 2 at. %, and 2.5 at. %). Excess Mg was added to compensate for the loss of Mg by evaporation. Adding too much excess Mg leads to the formation of a Mg+Mg₂Sn eutectic phase, while too little excess Mg leads to the formation of a Sn + Mg₂Sn eutectic phase.¹⁰ This precludes us from using adjustments in the Mg/Sn ratio to try to force the Ag atoms onto one lattice site or the other. To be consistent, we label our samples (Mg_1-yAg_y)₂Sn or (Mg_1-yAg_y)₂Sn_1-xSi_x as if the Ag was substituting for Mg, a fact we will justify a posteriori. To minimize the loss of Mg and protect the samples from any reactions with quartz ampoules, sealed graphite or boron nitride crucibles were used. Stoichiometric amounts of raw elements with appropriate excess Mg were loaded in the crucibles inside of quartz ampoules, which were sealed under a pressure less than 10⁻⁶Torr. The samples were slowly heated up to 1123 K, cooled down to 923 K at 1 K/min and then annealed for 2 days. To keep the experimental consistency, five parameters, the amount of Mg (1.4 g), the melting and annealing temperatures and their durations, were maintained the same for all material synthesis. Fig. 5 shows the picture of an Mg₂Sn sample. Phase purity was checked by X-Ray Diffraction (XRD) and Differential Scanning Calorimetry (DSC). Fig. 6(a) shows XRD patterns of Mg₂Sn_1−xSi_x (x = 0, 0.05, 0.1). Only clear Bragg peaks of anti-fluorite structure (space group, Fm3m) are observed with no secondary phases. With the addition of Si, which has a smaller ionic radius than Sn, the peaks are shifted to the right according to the amount of Si, indicating that Si successfully substituted on the Sn-site. To confirm the absence of second phases in Mg₂Sn, DSC was used to trace the latent heat, because loss of Mg may form Sn segregation or Sn-Mg₂Sn eutectic phases. The absence of latent heat peaks at the melting temperatures of 475 K (Sn-Mg₂Sn eutectic) and 503 K (Sn segregation), respectively, is shown in Fig. 6(b). Ag-doped Mg₂Sn_1-xSi_x samples also show no secondary peaks or a very small Sn-Mg₂Sn eutectic peak around 470 K. It should be noted that Mg in Mg₂Sn_1-xSi_x starts to oxidize above 700 K, so that the absence of secondary phases with a melting point above 700 K could not be ascertained by DSC, which is otherwise a much more sensitive method to identify secondary phases than XRD.

The resulting samples were cut into parallelepipeds for thermoelectric properties measurements, which were performed using a customized cryostat. The four-probe resistivity ρ was measured with an AC resistance bridge. The major errors in the absolute value of ρ stem from measurements of the sample geometry and are estimated to be about 10%. The temperature-dependence and other relative errors stem from current and voltage measurements and are much smaller (10⁻⁴) than geometrical errors. The thermopower α values were measured using the conventional static heater and sink method. Very thin wires (diameter = 25 μm) of copper and constantan were used as thermocouples to minimize heat losses. Thus, the estimated error in measuring α is less than 3%, mostly due to noise in the voltage measurements and uncertainties in thermocouple calibration. Hall resistivity ρ_H and adiabatic Nernst–Ettingshausen voltage were also measured as described previously²⁷ in transverse magnetic fields from −14 kOe to +14 kOe. The adiabatic Nernst coefficient is converted into the isothermal Nernst coefficient (N) also as described in Ref. 27. The origin of errors in thermomagnetic measurements is also mainly in measuring sample's geometry. Estimated errors for Hall and Nernst measurements are 5%.

Binary Mg₂Sn shows a large temperature-independent positive α from 120 K to 180 K (Fig. 7(a)). The ρ_H is linear in the applied field H and corresponds to a hole concentration about 8.0 × 10¹⁶ cm⁻³ (Fig. 7(c)) in the same temperature range. Above 180 K, α decreases and switches to negative at 280 K, due to the thermal excitation of minority electrons and bipolar transport. The α values obtained here are in good agreement with that reported by Chen and Savvides.¹⁰ On the other hand, Mg₂Sn_0.95Si_0.05 and Mg₂Sn_0.9Si_0.1 show fairly small positive α at low temperatures compared with that of Mg₂Sn, and the Mg₂Sn_0.9Si_0.1 sample becomes n-type at 100 K, suggesting that the density of n-type defects increases with increasing Si content; it has been suggested in the Mg₂Si and Mg₂Ge systems that such defects might be interstitial Mg atoms.²⁸ The ρ of the unintentionally doped samples is reported in Fig. 7(b). All samples show typical semiconductor-like behaviors: ρ decreases as temperature increases. The samples become more conductive at low temperature with the increase in concentration (x) of substitutional Si. Fig. 7(d) shows (R_He)⁻¹ (where R_H is the Hall coefficient) for the three samples around room temperature obtained from the ρ_H measurement. The ρ_H for all three samples showed a linear behavior with negative slopes, suggesting that electrons are dominant at these temperatures. Therefore, (R_He)⁻¹ is indicative of the excess electron concentration. This result is consistent with the negative values of the thermopower in the same temperature regime. Again, the Hall coefficient confirms the conclusion that the increase in x leads to an increase in the number of electrons in the samples.

The seven (Mg_1−yAg_y)₂Sn_1−xSi_x (y ≠ 0) samples are all p-type at all temperatures, with hole concentrations reported in Fig. 8(a). Except for the two low-doped cases, all samples have between 6 × 10¹⁹ and 8 × 10¹⁹cm⁻³ of holes. The hole concentration p of Ag (0.5 at. %)-doped Mg₂Sn sample (6.0 × 10¹⁹ cm⁻³ at 300 K) is almost identical with what was reported by Chen (6.1 × 10¹⁹ cm⁻³ at 300 K).¹⁰ An interesting finding is that roughly 4 times the amount of Ag is required in order to obtain a similar order of hole concentration in (Si 5 at. %:Ag 2 at. %) and (Si 10 at. %:Ag 2 at. %) when compared to Mg₂Sn:Ag (0.5 at. %). Consistently with the behavior of the y = 0 samples, it appears that adding Si adds n-type defects, and it requires more Ag to compensate them in this case. The hole concentrations of heavily-doped (Mg_1-yAg_y)₂Sn_1−xSi_x (x = 0, 0.05, 0.1) are almost temperature independent up to 300 K. For the lightly doped (Mg_1−yAg_y)₂Sn_0.95Si_0.05 (y = 0.005) and (Mg_1−yAg_y)₂Sn_0.95Si_0.05 (y = 0.01), the hole concentration appears to gradually increase as the temperature increases, but this is an artifact. Only the values below 100 K represent the actual hole concentration; the values above 100 K reflect the fact that the Hall coefficient decreases with increasing temperature, not because of an increase in hole concentration but because of the thermal generation of minority electrons. For these samples, the onset of bipolar conduction intensifies above 200 K (Fig. 8(a)). For simplicity, we adopt henceforth the hole concentration at 80 K as a label for each sample.

Fig. 8(b) shows the measured thermopower α as a function of temperature. All the samples with p > 10¹⁹cm⁻³ show α to vary linearly with temperature. Two samples with p ∼ 10¹⁷cm⁻³ have α bending over around 200 K, consistent with the appearance of bipolar conduction, as seen in Fig. 8(a), due to the thermally excited minority carriers. Consistently, all of the Ag-doped samples with p > 10¹⁹cm⁻³ show increasing ρ with increase of temperature (Fig. 8(c)), the typical behavior of degenerately-doped semiconductors, and in contrast to the ρ of the undoped Mg₂Sn_1−xSi_x samples (Fig. 7(b)). The Hall mobility μ decreases with increasing temperature (Fig. 8(d)), suggesting phonon scattering as the dominant mechanism. Mg₂Sn_0.95Si_0.05:Ag (p = 6.5 × 10¹⁹ cm⁻³) displays nearly the same mobility as that of Mg₂Sn:Ag (p = 6.0 × 10¹⁹ cm⁻³) with the almost identical carrier concentration. The μ of Mg₂Sn_0.95Si_0.05:Ag (p = 7.1 × 10¹⁹ cm⁻³) is slightly lower than that of Mg₂Sn:Ag (p = 7.7 × 10¹⁹ cm⁻³). Mg₂Sn_0.9Si_0.1:Ag (p = 6.7 × 10¹⁹ cm⁻³) shows the lowest mobility and the highest ρ among all the samples with p > 10¹⁹cm⁻³. All this suggests the appearance of alloy scattering, which scales^29,30 with the quite large difference of electronegativity between Sn and Si (the Pauling electronegativity of Sn is 1.96 and that of Si is 1.90 (Ref. 31)).

Isothermal Nernst coefficient (N) coefficients were used to determine the relaxation time scattering parameter λ, following the “method of the four coefficients.”³² λ is defined as the exponent of the energy dependence of the relaxation time τ, which is assumed to follow the power law $τ∝Eλ$ (for acoustic phonon scattering λ is −0.5, for polar optical phonon scattering it is +0.5 and for ionized impurity scattering +1.5). λ can be calculated from α, N, and μ = R_H/ρ for degenerate semiconductors with a parabolic band dispersion:³² 

The values of λ for the Ag-doped Mg₂Sn_1−xSi_x (x = 0, 0.05, 0.1) samples show various values in the range −0.7 ≤ λ ≤ 0.7 (Fig. 9(a)), indicating that the combination of acoustic phonon, optical phonon and alloy scattering limits the mobility in the samples, in agreement with the temperature dependence of μ, but also that, within a few error bars of the measurements (estimated to be of the order of 0.5), the relaxation time is essentially constant vis-a-vis energy (i.e., λ ≈ 0).

The measured thermopower results at room temperature for samples studied in this and previous^10,12,13 work are plotted in Fig. 9(b) as a function of the hole concentration (the Pisarenko plot²) and compared to the theoretical calculations for Mg₂Sn, taken from Ref. 18. The measured data points agree very well with the calculated curve, which is based on the use of a rigid band model for the valence band of Mg₂Sn, including the relativistic band structure, and using λ ≈ 0, a hypothesis justified a posteriori in the light of the data in Fig. 9(a). This shows that all the impurities used as dopants in the present and in the other studies (i.e., Ag, Li, Na) behave as classical, rigid-band-like acceptors. In view of theoretical calculations results, presented in Sec. II, this suggests that Ag substitutes for Mg rather than Sn. Also, the Si-substituted samples follow the same trend, in agreement with our theoretical observations of similar electronic structures of Ag-doped Mg₂Sn and Mg₂Sn_0.9Si_0.1. This observation also justifies a posteriori the formula (Mg_1−yAg_y)₂Sn_1−xSi_x used in this text.

The power factor α²/ρ of the tested samples is shown in Fig. 10. The two lightly doped samples, Mg₂Sn_0.95Si_0.05:Ag (p = 0.5 × 10¹⁷ cm⁻³) and Mg₂Sn_0.95Si_0.05:Ag (p = 4.3 × 10¹⁷ cm⁻³) show very small values of α²/ρ, while samples with p > 10¹⁹cm⁻³ show much higher values of α²/ρ, ranging from 8 μWcm⁻¹K⁻² to 22 μWcm⁻¹K⁻² at 400 K. The maximum α²/ρ of 22 μWcm⁻¹K⁻² at 380 K has been obtained for the Mg₂Sn:Ag with an optimal doping level of p = 6.0 × 10¹⁹cm⁻³. Alloying with Si results in suboptimal power factors because alloy scattering decreases the mobility.

Summarizing the KKR-CPA band structure calculations, Ag shows p-type behavior in Mg₂Sn and Mg₂Sn_1−xSi_x. If Ag is located on the Sn site, it can form a resonant level, and consequently modifications of transport properties of the Ag-doped system can be expected compared to intrinsic Mg₂Sn. On the other hand, if Ag substitutes for Mg, it behaves as a classical p-type dopant, simply transferring charge to the host Mg₂Sn bands, which remain rigid. The addition of 10% of Si to the system was calculated to have little effects on the valence bands, beyond resulting in additional alloy scattering.

To summarize the experiments, p-type Mg₂Sn, Mg₂Sn_0.95Si_0.05, and Mg₂Sn_0.9Si_0.1 were prepared by a co-melting method with sealed crucibles. Si substitution for Sn leads to a higher density of n-type defects, and also significantly decreases the mobility, as expected, especially in the 10 at. % alloys. Based on the four measured material properties (N, S, p, and ρ), we conclude that the combination of acoustic phonon, optical phonon, and alloy scattering is found to be dominant in the extrinsic samples and that the total relaxation time is essentially energy-independent. According to Pisarenko relation, no resonant level behavior was observed, which is in line with electronic structure features calculated for Ag substituting on the Mg site. Even so, the maximum power factor is 22 μWcm⁻¹K⁻² at 380 K in Mg₂Sn: Ag doped at an optimal density of p = 6.0 × 10¹⁹cm⁻³.

This project is supported by NSF-CBET/DOE joint program on thermoelectricity, project Seebeck CBET 1048622. This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center. B.W. and J.T. were supported by the Polish National Science Center (NCN, Project No. DEC-2011/02/A/ST3/00124).