Poorer is better: towards robust, high performance Mg₂(Si,Sn) thermoelectric material by avoiding excess Mg

INTRODUCTION

As the global energy crisis escalates, the need for sustainable and efficient energy sources is becoming increasingly important^[1,2]. Current methods of power generation, such as coal and gas combustion, contribute significantly to environmental degradation and resource depletion. By converting (waste) heat into usable electrical energy, thermoelectric technology offers a promising solution to ease energy demand while reducing our carbon footprint^[3]. Thermoelectric (TE) generators are competitive solutions due to their high reliability, absence of mechanical components, extended operation lifespan and low maintenance demands. They have been used mainly in the aerospace field for space probes such as Voyager 1 and 2 (after decades of operation) or in Mars missions with Perseverance and Curiosity rovers^[4-6].

The efficiency of TE devices depends on the properties of the employed TE material, evaluated by the thermoelectric figure of merit $$ z T=\frac{s^{2} \sigma}{\kappa} T $$ at an absolute temperature T. Here, S represents the Seebeck coefficient, σ the electrical conductivity and κ the thermal conductivity^[7]. Maximizing the figure of merit is essential for the development of thermoelectric technologies, as it indicates the material’s ability to efficiently convert heat into electricity^[8,9] at a given temperature gradient. However, the value of zT is limited by the interdependence of the three essential transport properties: S, σ and κ, where κ should be low and the TE power factor PF = S²σ should be high. The Seebeck coefficient and electrical conductivity are largely governed by the charge carrier concentration (n) and the charge carrier mobility (µ). The charge carrier concentration depends on the defect types and concentrations of the semiconducting base material and can be adjusted by doping the material^[7]. Generally, TE materials achieve maximum zT for relatively high carrier concentrations (10¹⁹-10²⁰ cm^-3) and the addition of a well-chosen foreign atom will modify the Fermi level, resulting in higher carrier concentration^[7].

So far, significant advancements have been made in developing TE materials suitable for intermediate temperatures ranging from 500 K to 900 K, resulting in high figures of merit up to zT > 2. Noteworthy examples include PbTe^[10], Skutterudites^[11], half-Heusler compounds^[12] and Mg-based materials such as MgAgSb^[13,14], Mg₃Sb₂^[15,16] or Mg₂Si_1-xSn_x solid solutions^[17,18]. The latter meet several criteria for large scale applicability, including the utilization of lightweight and abundant raw elements (Mg, Si and Sn), low cost, environmental compatibility and excellent thermoelectric performance^[19,20]. N-type Mg₂Si_1-xSn_x solid solutions doped with antimony have been optimized showing excellent TE properties (zT_max ~ 1.4) arising from a degeneracy of the conduction bands (CB) for the composition range x = 0.6 to 0.7 and reduced lattice thermal conductivity due to alloying^[18,21,22]. The application potential has been demonstrated by the successful fabrication of prototypes by Kaibe et al.^[23] using a two-stage BiTe-silicide module, followed by the development of the first full Mg₂(Si,Sn) (used as both p- and n-type) device by Camut et al.^[24] which achieved conversion efficiencies of 12% with T_h = 550 °C and T_c = 30 °C and 3.6% with T_h = 400 °C and T_c = 25 °C, respectively. Later on, Wieder et al.^[25] developed a module combining p-type MgAgSb and n-type Mg₂(Si,Sn) leading to an improved efficiency of 6.4%.

Although Mg₂(Si,Sn)-based materials feature attractive TE properties and technological advances have been made, its stability remains a major drawback for thermoelectric devices which need to operate at high temperatures. Skomedal et al.^[26] studied the material stability at high temperature in air, showing the high sensitivity of the material to oxidation. A further well-known challenge is that the Mg stoichiometry in the Mg₂(Si,Sn) solid solutions is difficult to control^[27]. Mg content can decrease compared to the target stoichiometry as Mg has a lower melting point and a higher vapor pressure than the other elements involved and thus evaporates easily. This impacts the intrinsic Mg point defect densities and thereby the charge carrier concentration and the functional properties of the material^[28,29]. The use of excess Mg in the synthesis process (nominal starting composition Mg_2+δ(Si,Sn)) is usually chosen as a strategy to counteract the change in functional properties^[30-33]. However, Sankhla et al.^[34] studied the effect of heat treatment on the stability of material synthesized with excess Mg and demonstrated that changes in transport properties (decrease in carrier concentration) are linked to Mg loss. Recently, Duparchy et al.^[27] investigated room temperature stability in air of n- and p-type Mg₂(Si,Sn) solid solutions, demonstrating that the p-type material which had been synthesized without Mg excess is stable over time while the n-type degrades. This was traced back to the diffusion of loosely bound Mg in Sn-rich phases via Mg vacancies, leading to subsequent Mg oxidation at the surface and causing gradual changes in the integral material properties. Ghosh et al.^[35] showed that the dominant Mg diffusion path is through grain boundaries and Sankhla et al.^[36] developed a microscopic understanding for Mg loss and determined degradation kinetics.

From these experimental results and first-principles calculations on defect densities^[37,38], it is clear that Mg₂(Si,Sn) exhibits a finite solubility range with respect to Mg, with compositions between Mg_2(1+δ1)(Si,Sn) and Mg_2(1+δ2)(Si,Sn) (δ₁ < δ₂) being in principle feasible. Here, Mg_2(1+δ1)(Si,Sn) and Mg_2(1+δ2)(Si,Sn) represent Mg-poor and Mg-rich thermodynamic states, which represent the lower and upper solubility limit, respectively, and differ by the concentration of intrinsic point defects. This finite width of presumable line phase is typical for TE materials and is, e.g., observed for BiTe and Mg₃Sb₂^[39-42], with phase widths δ₂-δ₁ < 0.001. The intrinsic defects with the lowest defect formation energies in Mg₂(Si,Sn) are Mg-related: Mg interstitials ($$ I_{M g}^{2+} $$) and Mg vacancies ($$ V_{M g}^{2-} $$)^[38]. Mg interstitials (Mg on the interstitial position) will be the dominant defect under Mg-rich conditions, contributing electrons^[27,38]. In Mg-poor samples, Mg vacancies will become more abundant, potentially with defect densities larger than Mg interstitials, acting as acceptor defects, compensating electrons. Hence, it is important to differentiate between: (i) excess Mg which refers to an non-soluble content of elemental Mg in the material, (ii) loosely bound Mg which is part of the crystal lattice but can be removed from it by changing the concentration of point defects and (iii) tightly bound Mg which refers to Mg that is bound in the crystal lattice of Mg-depleted Mg₂(Si,Sn) material and cannot be removed from it without initiating decomposition of the compound into its elemental constituents. Tightly bound Mg is very stable, while Sankhla et al.^[36] proved that loosely bound Mg is lost before the material starts to decompose and thus the fastest and most important material degradation mechanism. Hence, long-term stability requires the suppression of this loosely bound Mg from the material.

This might in principle be done by impeding Mg diffusion^[43], avoiding Mg loss by the use of coatings^[44], or potentially most easily by eliminating the loose Mg source. Duparchy et al.^[27] also demonstrated that p-type materials, which are synthesized without excess Mg and presumably Mg-poor after synthesis^[45,46], remain stable over years.

This study aims to overcome the fundamental origin of the instability of n-type Mg₂X by the synthesis of Mg-poor samples. We report for the first time, thermoelectric properties and microstructural analysis of optimized n-type Mg-poor Mg_2-δSi_0.3Sn_0.7 solid solutions. We find that - compared to the usually employed Mg-rich material - larger amounts of Sb dopant are required to increase the carrier concentration in Mg_2-δSi_0.3-ySn_0.7Sb_y-based solid solutions. This decreased dopant efficiency is expected due to the lower density of Mg interstitials and the formation of compensating defects^[47,48] originating from reduced amount of Mg in the nominal composition, but we demonstrate here that the carrier concentration required for optimal thermoelectric performance can be obtained. We furthermore show that the synthesized Mg-poor material undergoes a “self-adjusting” single-phase synthesis, insensitive to the nominal starting composition, leading to highly reproducible TE properties. Such phenomenon is explained by a loss of Mg-Sn melt during the pressure-assisted sintering, driving the material to the well-defined Mg-poor state by a slight adjustment of the Mg and Sn content of the samples.

Microstructural analysis of the samples shows a phase constitution comparable to that of Mg-rich samples and high temperature property measurements confirm that it is possible to synthesize a highly doped Mg-poor solid-solution, with good and reproducible TE properties, comparable to those of Mg-rich solid-solutions. Furthermore, extracting the microscopic material parameters, such as the effective mass and scattering constants, using a single-parabolic band analysis reveals similar results for Mg-poor and Mg-rich samples. However, comparing the synthesized Mg-poor material with synthesized Mg-rich samples that are fully Mg-depleted due to annealing shows a drastic performance reduction of the synthesized Mg-rich samples and detrimental grain boundary scattering, while the synthesized Mg-poor samples show only very limited grain boundary scattering. Besides being a possible solution towards improved material stability, Mg-poor material leads to better reproducibility as there is less chance for Mg loss during synthesis.

EXPERIMENTAL

RESULTS

N-type Mg_2-δSi_0.3-x-ySn_xSb_y (δ = 0.1, 0.05; x = 0.7; y = 0, 0.035, 0.05, 0.067) samples were synthesized and analyzed in this study. The main characteristic of the synthesized samples is the Mg content being deficient compared to what has been reported on Mg₂(Si,Sn) solid solutions so far and deficient also with respect to the nominal Mg:X (X= Si,Sn) of 2:1.

The phase purity of the samples was investigated using XRD. The X-ray diffractograms of synthesized Mg-poor undoped and doped samples are represented in Figure 1 in comparison to synthesized Mg-rich material. All diffractograms show the formation of phase-pure material which crystallizes in the fcc structure with Fm$$ \overline{3} $$m space group (sharp peaks, no peak overlapping). The square root of the normalized intensities is plotted to enhance the visibility of potential impurity phases. Moreover, the XRD patterns show no obvious peak splitting that would indicate any phase separation of the sample. For some samples a small asymmetry in the XRD peaks is observed, potentially due to variations in the Si:Sn ratio in Mg₂(Si,Sn). The diffractograms of Mg-poor doped samples are comparable to those of Mg-rich doped sample from Sankhla et al.^[57] that were synthesized using the same route and the same devices.

Figure 1. X-ray diffraction diffractogram of samples with varying Mg and Sb composition.

The obtained lattice parameter of Mg_1.95Si_0.3Sn_0.7 (a = 6.643 Å) is in close agreement with literature values of undoped Mg₂Si_0.3Sn_0.7 (a = 6.64 Å^[46], 6.62 Å^[60], 6.66 Å^[61]). The obtained lattice parameter of the Mg_1.95Si_0.233Sn_0.7Sb_0.067 sample (a = 6.669 Å) is higher than the one of undoped Mg-poor solid solution. A systematic increase of a with Sb content suggests the dopant incorporation in the matrix. Sb is substituting on the X position and no secondary phases such as Sb or Mg₃Sb₂ are observed in the XRD pattern. All densities and lattice parameters are given in Table 1. The weighted residuals of the Rietveld refinement are given in Supplementary Table 1. Examining the sample densities reveals no observable trend in relation to Mg content.

The microstructure of the samples was investigated using SEM combined with EDS to observe possible phase separation due to unmixing of the solid solution^[62], to identify secondary phases and to analyze the possible differences originating from the reduced Mg content. For all the synthesized samples, microstructure and phase constitution are very similar, as illustrated in Figure 2. Additional images are provided in the Supplementary Materials [Supplementary Figure 3]. Backscattered electron images done by SEM, we do not observe precipitated elemental Si or Sn for any of the samples as might be expected due to the Mg-deficient nominal stoichiometry. Besides, we observe a relatively homogeneous matrix with globular, sub-structured Si-rich Mg₂X inclusions distributed all over the sample [Figure 2D], which are not detected in XRD. This is explained by their low area fraction and variable composition which leads to small and smeared signals. Furthermore, such Si-rich Mg₂X regions are not due to unmixing but rather a consequence of an incomplete reaction, as is typical for this material system when prepared by ball milling^[27,34,57], which originate probably from the relatively slow dissolution of Si into Mg₂X during milling and sintering^[31,63]. As shown in Supplementary Figure 4, when comparing as-sintered and annealed SEM/EDS images of Mg_1.95Si_0.3Sn_0.7 sample, one can clearly see that the Si-rich Mg₂X regions are dissolving in the matrix. As we know that having such Si-rich regions will dissolve with time and do not impact the thermoelectric material properties, we remain with the current, optimized synthesis route. Finally, no secondary phases containing Sb are observed for the doped sample, implying good incorporation of the dopant into the host structure. As shown by Nolas et al.^[47], Mg₂Si can dissolve up to 37 mol.% of Sb. According to our results, we believe that Sb solubility limit in Mg₂Si_1-xSn_x with x~0.3 is above 6.7 mol.% as no Sb impurities were detected in the samples.

Figure 2. Low and high magnification BSE-SEM images of (A and C) Mg_1.95Si_0.3Sn_0.7, (B and D) Mg_1.95Si_0.233Sn_0.7Sb_0.067. The BSE-SEM images show the microstructure of undoped and doped Mg-poor material. One can see that globular, sub-structured inclusions are visible in both samples. Those inclusions are Si-rich Mg₂X precipitated. Some contrast is also visible in both matrixes. Similar contrast is also visible in Mg-rich material as shown in Figure 2. Such contrast is due to slight Si:Sn variations. EDS mapping of the Si-rich regions are given in Supplementary Figure 2. EDS: Energy dispersive X-ray spectroscopy; BSE-SEM: backscattered electron-scanning electron spectroscopy.

Inductively coupled plasma atomic emission spectroscopy (ICP-AES) has been performed on a Mg_1.95Si_0.3Sn_0.7 for a more accurate quantification of the respective elements. Such measurements have a high reliably, leading to low standard deviations (less than 1 at.%) as demonstrated by Ai et al.^[64] and He et al.^[65]. We obtain Mg = 66.68 at.%, Si = 10.23 at.% and Sn = 23.09 at.%, while according to the nominal composition, Mg = 66.1 at.%, Si = 10.17 at.% and Sn = 23.73 at.% are expected. The measured values correspond to a nominal stoichiometry of Mg_2.001Si_0.307Sn_0.693, where the content of Si + Sn was normalized to unity. Essentially, the ICP measurement reveals a Mg:X (X: Si,Sn) ratio quite close to 2:1, even though the material was synthesized with a 2.5% deficiency of the Mg content. Furthermore, we also see a small, but measurable increase of Si:Sn with respect to the starting composition. Indeed, if we quantify how much Sn is missing compared to the nominal Si:Sn ratio, it amounts to 2.2% of the overall X component which almost exactly balances the nominal Mg under-stoichiometry.

The main carrier type and functional homogeneity of Mg_1.95Si_0.3Sn_0.7 and Mg_1.95Si_0.233Sn_0.7Sb_0.067 was determined by a room temperature surface Seebeck coefficient mapping using a PSM as shown in Figure 3A and B. The Seebeck coefficient spatial map is showing n-type conduction for Mg-poor undoped and doped material, respectively. The Seebeck coefficient obtained by PSM measurement matches with the bulk Seebeck coefficient measured at room temperature using the temperature-dependent in-house developed device [Figure 3A] when assuming an usual underestimation between 10% and 20% of the measured values in the PSM due to the cold finger effect^[66]. The functional homogeneity is quantified by a frequency distribution of the Seebeck coefficient distribution profile [Supplementary Figure 5]. The full width half maximum values are 15% and 4% for the undoped and doped sample, respectively. This indicates that the doped sample exhibits better functional homogeneity than the undoped one, plausibly because the Seebeck coefficient is more sensitive to small variations in charge carrier concentration due to local compositional fluctuations at low carrier concentration levels. We also observe a gradient across the sample, potentially due to a slightly inhomogeneous temperature profile during sintering^[46] but the difference is around 5% across the whole sample and hence doesn’t occlude interpretation of the bulk measurements.

Figure 3. Surface Seebeck map of (A) Mg_1.95Si_0.3Sn_0.7 and (B) Mg_1.95Si_0.233Sn_0.7Sb_0.067, respectively, measured by the Seebeck microprobe. The scaling of the spot structure where S is outside the matrix range is the same as the Si-rich inclusions visible in the SEM micrographs. SEM: Scanning electron microscopy.

The temperature-dependent transport properties of the samples Mg_1.9Si_0.3Sn_0.7, Mg_1.95Si_0.3Sn_0.7, Mg_1.95Si_0.233Sn_0.7Sb_0.067-I, Mg_1.95Si_0.233Sn_0.7Sb_0.067-II, Mg_1.95Si_0.25Sn_0.7Sb_0.05 and Mg_1.95Si_0.265Sn_0.7Sb_0.035 in comparison to a synthesized Mg-rich doped solid solution Mg_2.06Si_0.385Sn_0.6Sb_0.015 by Sankhla et al.^[57] are shown in Figure 4. Transport properties of the solid solution samples measured in the heating and cooling mode may differ for the first measured cycle^[46], therefore cooling data are plotted here. Both the temperature-dependent Seebeck coefficient and electrical conductivity of the undoped sample exhibit a typical semiconductor behavior while doped samples behave similar to heavily doped semiconductors. The temperature-dependent Seebeck coefficients of all samples plotted in Figure 4A are negative, indicating n-type-dominated conduction in agreement with the surface Seebeck scans [Figure 3]. The two undoped Mg-poor samples (Mg_1.95Si_0.3Sn_0.7 and Mg_1.9Si_0.3Sn_0.7) have very similar Seebeck coefficients, indicating nearly identical charge carrier concentration, in particular as the Seebeck coefficient is very sensitive to the carrier concentration for low n. This in turn indicates that the intrinsic defect density is very similar for both samples, which makes it plausible that the samples are indeed Mg-poor and not in some intermediate state with respect to Mg content. Undoped Mg-poor solid solutions are expected to exhibit weaker n-type behavior (lower carrier concentration and larger n-type Seebeck) compared to undoped Mg-rich Mg₂(Si,Sn), due to a larger density of Mg vacancies, which act as acceptor defects, and lower densities of donating Mg interstitials, both changes reducing the number of free electrons. Supplementary Figure 6 shows the carrier concentration mapping (obtained using 2PB band modeling using Seebeck coefficient values) for undoped Mg-rich and Mg-poor Mg₂(Si,Sn) with n ≈ 1.3 × 10²⁵ m^-3 for both samples, indicating a larger variation in carrier concentration for Mg-rich material (∆n_Mg-poor = 0.08 × 10²⁵ m^-3 < ∆n_Mg-rich = 0.3 × 10²⁵ m^-3, where Δn represents the respective standard deviation). The exact reason for this remains to be confirmed, but the result might be interpreted as stronger variation in carrier concentration due to locally varying, beginning Mg-loss of the Mg-rich sample.

Figure 4. Experimental (filled symbols) and simulated (solid lines) temperature dependent (A) Seebeck coefficient, (B) electrical conductivity, (C) thermal conductivity, (D) lattice and bipolar thermal conductivity, (E) power factor and (F) figure of merit for different doping and Mg concentrations for Mg_2-δSi_0.3-ySn_0.7Sb_y. Theoretical results were obtained from a single parabolic band (SPB) model. Only samples where the Hall effect was measured, were modelled. For some samples, the height of the error bars is lower than the size of the symbol, hence not visible.

Thermal excitation of the minority charge carriers is clearly visible for the undoped sample (Mg_1.95Si_0.3Sn_0.7) at high temperatures as the (absolute) Seebeck coefficient starts decreasing. Such behavior is also visible in Mg_1.95Si_0.265Sn_0.7Sb_0.035 when excitation of the minority carriers is visible at temperatures over 650 K. In contrast, samples highly doped with Sb (y = 0.05, 0.067) exhibit a linear trend, suggesting that the Seebeck coefficient is dominated by the majority carriers. The trend of the electrical conductivity is opposite to the behavior of the Seebeck coefficient, with a higher variance in amplitude. In contrast to the observations by Kato et al.^[67] and Sankhla et al.^[57], grain boundary scattering is barely visible for the Mg-poor samples. A maximum power factor of 43.3 µWcm^-1K^-2 at 640 K was achieved for Mg_1.95Si_0.233Sn_0.7Sb_0.067.

Besides this, for undoped samples, the temperature dependent thermal conductivity [Figure 4C] first decreases with increasing temperature as expected and then increases due to the bipolar effect for temperatures above 473 K. The thermal conductivity increases by adding Sb due to the larger electronic contribution. The combined lattice and bipolar contribution [Figure 4D] of the undoped sample shows a sharp increase above 500 K which is due to the bipolar contribution. However, for the doped sample, the lattice thermal conductivity is greatly reduced compared to undoped samples. The high power factor and low thermal conductivity of Mg_1.95Si_0.233Sn_0.7Sb_0.067 lead to a relatively high experimental figure of merit of 1.2 ± 0.16 at 719 K [Figure 4F].

Regarding the samples with lower dopant concentrations (y = 0.035 and y = 0.05), zT values are higher than the highly doped sample at temperatures from room temperature to 600 K. These values are comparable to zT values of the Mg-rich reference sample. The decrease in zT at higher temperatures is attributed to the influence of minority charges. Overall, the power factor is highest for y = 0.067, followed by the Mg-rich sample, with y = 0.05 showing better properties than y = 0.035. Therefore, the choice between y = 0.05 or y = 0.067 dopant content will depend on the specific application temperature requirements.

DISCUSSION

To prevent material degradation, coatings were initially explored. Focused research has been conducted on coatings, yet no suitable coating has been found for this material system^[68-71]. First of all, the typical mismatch in the coefficient of thermal expansion presents significant challenges when oxides are employed, particularly under thermal cycling. Furthermore, elemental Mg, which drives material degradation is highly reactive. Consequently, many oxides such as Al₂O₃ that have a lower formation enthalpy than MgO might not work as a coating in the long run, as demonstrated by Deshpande et al.^[72]. Therefore, in our work, we propose the synthesis of Mg-poor materials to prevent direct Mg loss, addressing the degradation mechanism described by Sankhla et al.^[34], Duparchy et al.^[27] and Ghosh et al.^[35].

The synthesis of Mg-poor Mg_1.95Si_0.3Sn_0.7 has been successful, leading to the formation of a typical n-type semiconductor. The microstructural analysis proves that no elemental Si or Sn remains; only Si-rich secondary phases within the solid solution system Mg₂(Si,Sn) were formed [Figure 2]. Furthermore, the synthesized Mg-poor sample do not show any Mg₂Si and Mg₂Sn unmixing. These findings contradict the speculation that Mg deficiency initiates unmixing of the solid solution, raised by Yasseri et al.^[63]. ICP analysis was conducted to determine the elemental composition of the undoped Mg_1.95Si_0.3Sn_0.7 sample, resulting in a composition of Mg_2.001Si_0.307Sn_0.693, close to Mg₂(Si,Sn), with a higher Mg and lower Sn content than the nominal one. We propose that these differences between nominal and measured composition are the consequence of a self-adjusting stoichiometric balance during the synthesis process, in particular, during uniaxial hot pressing. Indeed, according to the Mg-Si-Sn phase diagram^[73], a sample that contains more Sn than the Sn-rich limit according to the miscibility gap^[62,73] and is Mg deficient, is located in a three-phase region, with elemental Si, Sn-rich Mg₂X and liquid Mg-Sn (for temperatures above ~210 °C) as coexisting phases. According to the phase diagrams provided in Figure 9 from the work by Orenstein et al.^[73], Mg_1.95Si_0.3Sn_0.7 is not located inside the Sn-rich Mg₂X phase, but we note that our samples have been sintered at 700 °C, i.e., larger than the temperatures at which the ternary isothermal phase diagrams are known and point out that the miscibility gap closes with increasing temperature, and, secondly that Mg₂X samples have been synthesized repeatedly as single-phase materials inside the thermodynamically predicted miscibility gap, presumably due to stabilization by coherent interfaces^[63,74]. It is therefore highly plausible that the sintering step leads to the formation of Mg₂(Si,Sn) and Mg-Sn, with the latter being expelled during the pressure-assisted sintering as it is liquid. We do not observe elemental Si (see Figure 2). As Mg₂Si is tighter bound than Mg₂Sn, Si might replace Sn in the structure as long as there is X excess. According to the suggested procedure, one would expect a sample with reduced Sn content (as observed) and at the lower solubility limit of Mg in Mg_2(1+δ1)X. The very similar Seebeck data for the two undoped sample supports this. The ICP results with a Mg content slightly larger than 2 is not a contradiction to the sample composition corresponding to the lower solubility limit of Mg but a result of the limited accuracy of the measurement and an indication that the phase width with respect to Mg is smaller than the accuracy of the ICP measurement. For undoped materials, we are not aware of any report on the phase width, but for doped Mg₂X a value of Δδ_Mg = 0.008 was reported, smaller than typical uncertainties of less than 1 at.% for ICP^[64].

All three Mg-poor doped samples were examined by XRD and SEM/EDS [Figures 1 and 2], demonstrating successful dopant incorporation, as no visible Sb-rich phase precipitated were detected. Furthermore, Seebeck coefficient measurements [Figure 3] revealed differences in the Seebeck coefficient between undoped Mg_1.95Si_0.3Sn_0.7 and doped Mg_1.95Si_0.233Sn_0.7Sb_0.067 (-350 µV/K vs. -80 µV/K). These observations suggest that Sb was effectively incorporated on the Si/Sn site, with the doped sample exhibiting relatively low local functional fluctuations. The thermoelectric properties of the Mg-poor material improve significantly with doping, resulting in a high power factor and an overall figure of merit comparable to that of Mg-rich material compounds^[57] [Figure 4F]. Also, samples Mg_1.95Si_0.233Sn_0.7Sb_0.067 - I and II are very similar microstructure- and property-wise, indicating a high reproducibility of the synthesis route. Indeed, while the charge carrier concentration for Mg-rich samples might depend on the details of the sintering step (temperature, duration)^[31,57] due to Mg loss, using Mg-poor materials might lead to a very high reproducibility as no excess or loosely bound Mg is lost during the sintering step.

We have calculated the effective doping efficiency (η_dop) [Table 2] of Sb from the (measured) carrier concentration n and the nominal dopant concentration under the assumption that each Sb atom replaces one Si/Sn atom and provides one electron, using $$ \eta_{d o p}=\frac{n}{c_{S b}} $$ with $$ c_{\mathrm{Sb}}=\frac{4 * y}{a^{3}} $$ the dopant concentration. The effective doping efficiency is much lower for the synthesized Mg-poor samples than for the synthesized Mg-rich materials, which could be caused by Sb becoming electrically inactive (i.e., does not act as a donor anymore). However, if this was the case, comparing the different Mg-poor samples we should see an increase of the doping efficiency with decreasing Sb content, which is not observed. Also, density functional theory (DFT) calculations from Ayachi et al.^[75] indeed show a change from +1 to neutral for Sb_Si and Sb_Sn for high Fermi levels (i.e., large carrier concentrations), but these charge transition levels are similar for Mg-rich and Mg-poor materials, hence, Sb becoming inactive is probably not the reason for the decrease in doping efficiency. A more plausible reason could be the formation of compensating defects with increasing Sb content. This has been observed by Kato et al.^[29] and Dasgupta et al.^[48] who showed experimentally that Mg vacancies increase with increasing Sb doping, for nominal Mg-rich samples. Also for Mg-poor samples an increase of p-type Mg vacancies is expected with increasing carrier concentration caused by increasing Sb content, leading to a partial compensation of the free electrons from Sb addition.

The transport data of Mg_1.95Si_0.233Sn_0.7Sb_0.067-II (sample on which Hall measurement was performed) were investigated for differences in the microscopic material parameters using a single parabolic Band (SPB) model with respect to previously reported data for Mg-rich samples and those extracted from the samples after annealing, i.e., presumably Mg-poor^[57]. This composition was chosen as it leads to the best thermoelectric performance in the material. As no thermal conductivity measurements were performed on that sample, the data of Mg_1.95Si_0.233Sn_0.7Sb_0.067-I (with almost identical electrical properties) was utilized instead.

The SPB model can be used for highly doped samples of Mg₂Si_1-xSn_x with x~0.7 due to this composition being closely located to convergence of CBs, as described in detail in many studies^{[21,57,76,77]}. In our case we used x = 0.7 + y with y being the Sb content (Sb is comparable in size to Sn rather than Si). The basic parameters of this model are the reduced chemical potential (η), the mobility parameter (μ₀) and the density of states effective mass ($$ m_{D}^{*} $$), governed by

Here k_B represents Boltzmann’s constant, h is Planck’s constant, and $$ F_{i}=\int_{0}^{\infty} \frac{\varepsilon^{i} d \varepsilon}{1+\exp [\varepsilon-\eta]} $$ are the Fermi integrals of the order of i, and the reduced chemical potential η is given by $$ \eta=\frac{E_{\mathrm{F}}}{k_{\mathrm{B}} T} $$ where E_F is the Fermi level. For the calculation we have assumed a scattering parameter of λ = 0 corresponding to the energy dependence of scattering with acoustic phonons (AP) and alloy scattering (AS)^[78,79].

Using the room temperature Hall coefficient, we calculate the density of states effective mass as function of temperature assuming the carrier concentration to be constant and corresponding to the room temperature value. Figure 5A shows that the density of states effective mass is constant over temperature, verifying the validity of the SPB model here without considering any contribution from the valence band (VB) or a second CB. The room temperature charge carrier concentration (n_H), the Hall mobility (μ_H) and the density of states effective mass ($$ m_{D}^{*} $$) of Mg_1.95Si_0.233Sn_0.7Sb_0.067-II from this work and various other samples from the literature^[57], namely Mg_2.06Si_0.385Sn_0.6Sb_0.015 before Mg loss, Mg_2.06Si_0.385Sn_0.6Sb_0.015 after intermediate Mg loss and Mg_2.06Si_0.385Sn_0.6Sb_0.015 after Mg loss - fully Mg depleted, are summarized in Table 3.

Figure 5. (A)Density of states effective mass as a function of temperature for the Mg_1.95Si_0.233Sn_0.7Sb_0.067-II sample. The effective mass was obtained from room temperature Hall measurement, assuming a constant carrier concentration; (B) Weighted mobility (μ_w) and Hall mobility (μ_H) of Mg_2.06Si_0.385Sn_0.6Sb_0.015 and Mg_1.95Si_0.233Sn_0.7Sb_0.067-II.

The Hall mobilities for Sample 1 [Mg-poor] and Sample 2 [Mg-rich] are roughly comparable as shown in Figure 5B. with the Sample 1 [Mg-poor] showing slightly higher mobilities. This is potentially due to larger Sn content in that sample, leading to slightly reduced Alloy scattering. Notably, we do not observe a reduced mobility of the Sample 1 [Mg-poor], in contrast to results from Kato et al.^[67] who observed that Mg-poor doped Mg₂Si have a lower Hall mobility than Mg-rich Mg₂Si. The weighted mobilities exhibit a similar behavior as the Hall mobilities, in line with similar density of states effective mass and hence electronic band structure between the samples. Finally, a higher weighted mobility indicates higher achievable power factor for Sample 1 [Mg-poor] compared to Sample 2 [Mg-rich].

While the Hall mobility (μ_H) is affected by the carrier concentration, the mobility parameter (μ₀) is used for scattering analysis to understand the differences in carrier mobility as independent of carrier density. We modelled μ₀ making use of the low and high temperature conductivity measurement by considering scattering processes of charge carriers with acoustic phonons, lattice disorder due to alloy scattering and grain boundaries. Indeed, acoustic phonon scattering is the most relevant scattering mechanism for highly doped samples at high temperatures^[80]. Alloy scattering is included as it is a relevant mechanism in solid solutions^[78]. Grain boundary scattering has been shown to be relevant in several Mg-based TE materials^[81-83] especially for samples that have experienced Mg-loss^[34,84]. We assume that the scattering mechanisms are independent of each other meaning that the mobilities follow Matthiessen’s rule^[76]:

The acoustic phonon scattering term ($$ \mu_{0}^{A P} $$) is given by

where ħ is the reduced Plank constant, ρ is the theoretical mass density, v_l is the longitudinal velocity of sound (7,680-2,880x m²s^-1 for Mg₂Si_1-xSn_x), E_Def is the deformation potential, which characterizes the interaction between charge carriers and phonons. The single valley effective mass m_s is obtained from $$ m_{s} =\frac{m_{D}^{*}}{N_{V}^{2 / 3}} $$ with N_V = 6.

The alloy scattering mobility ($$ \mu_{0}^{A S} $$) is given by:

with N₀ being the number of atoms per unit volume, x being the Sn + Sb fraction at the X site and E_AS the alloy scattering potential.

Last but not least, the mobility constant of grain boundary scattering ($$ \mu_{0}^{G B} $$) is defined by:

where B is the grain size, which is kept constant (B = 5 µm)^[57] and E_B is the potential barrier at the grain boundary (called barrier height).

Using E_AS = 0.5 eV from literature, E_Def and E_B as remaining temperature-independent unknowns can be extracted by fitting the modeled mobility parameter to the “experimental” $$ \mu_{0}=\mu_{H}\left(\frac{F_{-\frac{1}{2}}(\eta)}{2 F_{0}(\eta)}\right)^{-1} $$. Doing so we obtain E_Def =11 eV [Figure 6], a deformation potential constant similar to those extracted from modelling the mobility of Sample 2 [Mg-rich] (E_Def = 9.8 eV ) and Sample 2 [after Mg loss] (11 eV) extracted from Sankhla et al. study^[57]. Liu et al.^[76] demonstrated high mobility of Mg-rich Mg₂(Si,Sn) due to the low deformation potential varying between 8.77 and 9.43 eV, while Agrawal et al.^[85] found a deformation potential of 9.32 eV, both being comparable to what was obtained for Mg-poor Mg₂(Si,Sn) in this study. This difference of ~10% in the deformation potentials could be easily attributed to experimental uncertainty^[59], confirming that there is no significant difference between Mg-rich and our Mg-poor material.

Figure 6. Comparison between experimental and calculated mobility parameter assuming acoustic phonon scattering, alloy scattering and grain boundary scattering for the sample Mg_1.95Si_0.233Sn_0.7Sb_0.067-I. The deformation potential was set at 11 eV, the alloy scattering potential at 0.5 eV and the results for four different barrier heights are shown: 0 meV (no GB scatter), 100 meV and 131 meV (reference values, used by Sankhla et al.^[57] in their study) and 60 eV, resulting in a good fit to the experimental data. The inset represents merged low-temperature data and high temperature data of electrical conductivity of Mg_1.95Si_0.233Sn_0.7Sb_0.067-I. The low-temperature data were fitted to the high-temperature ones with a constant factor assuming device uncertainty of 10% to fit the high temperature data for further modeling of the scattering parameters.

The low temperature electrical resistivity data presented in Figure 6 were used for the mobility parameter analysis. Indeed, one would expect a convex increase at the left-hand curvature due to grain boundary scattering which is not really the case. Fitting the mobility data down to 150 K, we estimate a barrier height for grain boundary scattering of E_B = 60 meV, significantly below values extracted from Sankhla et al.^[57] who obtained E_B = 100 meV for Sample 2 [Mg-rich], which increased to E_B = 131 meV after experiencing some Mg loss {Sample 2 [after Mg loss]}. For Sample 2 [Mg-depleted] an even higher barrier height was indicated by the positive slope of σ(T)^[34]. Modelling results with these barrier heights are shown in Figure 6, showing that barrier heights extracted from Sample 2 [Mg-rich] predict a stronger mobility reduction towards low temperatures than is observed for Sample 1 [Mg-poor]. In combination with the absence of indications for grain boundary scattering for all Mg-poor samples of this study [Figure 4], we can rule out a large impact of grain boundary scattering for the here synthesized Mg-poor samples, in contrast to observations on Sample 2 [Mg-rich], and particularly to Sample 2 [after Mg loss]. Note that the employed model is inadequate to model mobilities at very low temperatures as Equation (9) forces the total mobility to 0 for any finite barrier height for temperature towards 0.

Overall, with respect to the SPB parameters, Sample 2 [after Mg loss] and Sample 2 [Mg-depleted] show a clear trend of decreasing mobility, increasing GB scattering and increasing deformation potential with Mg-loss while Sample 1 [Mg-poor] is parameter-wise relatively close to Sample 2 [Mg-rich]. An important finding is that synthesized Sample 1 [Mg-poor] has not the same properties as the by annealing Sample 2 [Mg-depleted].

The low impact of grain boundary scattering in the synthesized Mg-poor material is a crucial finding, proving that synthesized Mg-poor materials are comparable to Sample 2 [Mg-rich] reported in the literature and clearly different than Sample 2 [Mg-depleted] which was initially Mg-rich. Our data for Sample 1 [Mg-poor] clearly show that substantial grain boundary scattering is not related to the Mg content as such. Instead, it might be explained by several reasons. First of all, it could be that it is not the composition that determines the effect of the grain boundaries but rather different grain boundary nature. In the case of Mg-depletion, many Mg vacancies form towards the grain boundaries, disturbing the transport while for synthesized Mg-poor sample, the grain boundaries could be Si/Sn rich from the self-adjusting synthesis, leading to less vacancies. Second, de Boor et al.^[86] attributed grain boundary scattering to MgO formation and observed a correlation between MgO content and barrier height, for Mg₂Si that was nominally Mg-rich. Such MgO will only form rapidly if excess Mg is available in the material. Finally, such low impact of grain boundary scattering could be explained by a different Si to Sn content at the grain boundaries between synthesized Mg-poor and Mg-depleted samples which could influence the barrier height.

The thermoelectric figure of merit for a material that can be determined by an SPB model can be written as $$ z T=\frac{S^{2} }{L+(\psi \beta)^{-1}} $$ with $$ \psi=\frac{8 \pi e}{3}(\frac{2m_{0}k_{B}}{h^{2}})^{1.5} F_{0} $$ and the material quality factor $$ \beta =\frac{(\frac{m_{D}^{*}}{m_{0}})^{1.5}\mu _{0}T^{0.25}}{\kappa _{lat}} $$. This material quality factor β, unlike the figure of merit, can be used to evaluate the potential thermoelectric performance of a TE compound independently of whether the optimum carrier concentration has been adjusted as β is independent of the carrier concentration^[57,58,87]. The Mg_1.95Si_0.233Sn_0.7Sb_0.067 sample exhibits more or less the same material quality factor β as the Mg-rich materials from the literature [Figure 7A] while a lower material quality factor was extracted for Mg-rich materials after experiencing Mg loss. From the zT(n) curve plotted in Figure 7B, for a Mg-poor doped sample in comparison to Mg-rich doped (with and without Mg loss), it is indicated that the charge carrier concentration is not yet optimized using an Sb dopant concentration of 6.7% for the Mg-poor materials, and an increase up to zT ≈ 1.4 at 700 K might be achievable with a lower doping content. On the other hand, it is also well-known that the SPB model overestimates zT significantly towards lower carrier concentrations as it ignores the impact of the minority carriers. Indeed, the samples with lower dopant concentrations (y = 0.035, 0.05) show a reduced figure of merit, which can be understood from the onset of minority carrier effects above 600 K in the Seebeck coefficient and the thermal conductivity. While a two or three band model would be required for a quantitative assessment, we deduce that only a small increase of zT by further adjusting the carrier concentration in the match between theory and model can be expected. Our findings combined with those by Sankhla et al.^[57] suggest that, although Mg-loss (due to Mg sublimation from an initially Mg-rich composition) can degrade the material properties strongly, combining an initial Mg-deficient compositions with optimized doping can achieve properties similar to those of optimized Mg-rich materials.

Figure 7. (A) Predicted temperature-dependent material parameter of Mg-poor doped and Mg-rich doped samples; and (B) SPB prediction (line) and experimental data (symbols) for the figure of merit versus carrier concentration of Mg-poor and Mg-rich doped samples at 700K. SPB: Single parabolic band.

CONCLUSIONS

We have demonstrated the successful and reproducible synthesis of single phase Mg-poor n-type Mg₂(Si,Sn) TE materials, and discovered that the material undergoes a self-adjusting synthesis. In fact, it appears that Mg-poor material synthesis is insensitive to the precise nominal composition, which makes the material synthesis suitable for upscaled synthesis. Up to now, Mg-rich compositions were typically employed, first to compensate for Mg loss during synthesis and second to achieve high carrier concentrations. We show here that synthesized Mg-poor samples have highly reproducible and spatially homogeneous thermoelectric properties, presumably due to avoiding loss of excess or loosely bound Mg and second that a sufficiently high carrier concentration can be achieved, despite a reduced dopant efficiency. Moreover, analysis of the transport properties reveals that optimal doping of synthesized Mg-poor solid solutions can achieve transport properties with microscopic parameters comparable to those of synthesized Mg-rich compositions. Furthermore, synthesized Mg-poor samples were compared to samples that were initially synthesized Mg-rich, but experienced Mg-loss, showing that the performances’ degradation of the latter was linked to increased grain boundary scattering while in this work we showed that, when the material is synthesized Mg-poor, grain boundary effects are negligible and the material performances are good. Hence, synthesized Mg-rich Mg-depleted and synthesized Mg-poor sample behave differently, leading to very different properties and microscopic parameters. Overall, we show that Mg-poor Mg₂(Si,Sn) materials can exhibit similar or even better TE properties and disprove key arguments for Mg-excess, paving the way for Mg-poor materials, which could exhibit better chemical stability.