Band engineering in Mg₃Sb₂ by alloying with Mg₃Bi₂ for enhanced thermoelectric performance

Abstract

Mg₃Sb₂–Mg₃Bi₂ alloys show excellent thermoelectric properties. The benefit of alloying has been attributed to the reduction in lattice thermal conductivity. However, Mg₃Bi₂-alloying may also be expected to significantly change the electronic structure. By comparatively modeling the transport properties of n- and p-type Mg₃Sb₂–Mg₃Bi₂ and also Mg₃Bi₂-alloyed and non-alloyed samples, we elucidate the origin of the highest zT composition where electronic properties account for about 50% of the improvement. We find that Mg₃Bi₂ alloying increases the weighted mobility while reducing the band gap. The reduced band gap is found not to compromise the thermoelectric performance for a small amount of Mg₃Bi₂ because the peak zT in unalloyed Mg₃Sb₂ is at a temperature higher than the stable range for the material. By quantifying the electronic influence of Mg₃Bi₂ alloying, we model the optimum Mg₃Bi₂ content for thermoelectrics to be in the range of 20–30%, consistent with the most commonly reported composition Mg₃Sb_1.5Bi_0.5.

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Conceptual insights

In thermoelectrics, band engineering through alloying has been very successful by focusing on the straightforward benefits of band convergence. On the other hand, the engineering of band mass is more complicated as the band mass is typically correlated with changes in the band gap. While charge carriers with small band mass typically have higher weighted mobilities needed for efficient thermoelectrics, the smaller band mass is often associated with a smaller band gap that leads to detrimental bipolar conduction at high temperatures and a lower peak zT. Although a larger band gap may allow a higher peak zT at a higher temperature, at lower temperatures the zT is likely to be lower because of the higher effective mass. The conceptual insight here is that for a material with limited thermal stability the band gap can be reduced to improve zT. This concept highlights the importance to consider the thermal stability and both the n- and p-type weighted mobilities as a prerequisite for band engineering.

Introduction

Thermoelectric materials have the ability to convert heat directly into electricity without any mechanical motion.^1,2 Since the main obstacle for wide-use of thermoelectrics is the low conversion efficiency limited by the materials, material research has been focused on developing high efficiency materials.

The maximum power and efficiency achievable by a particular thermoelectric material is determined by their figure of merit zT = (S²σ)/κT where S is the Seebeck coefficient, σ is the electrical conductivity, κ is the thermal conductivity and T is the absolute temperature. Strategies to enhance zT can be largely categorized into two different methods. One is by reducing the lattice thermal conductivity, for example, by increasing phonon scattering with alloying.^3–8 The other is by changing the electronic structure, i.e. band engineering.^9–13

A recently developed new material where both strategies were successfully used to reach a high zT ≈ 1.5 is n-type Mg_3+xSb₂Bi_0.49Te_0.01, first reported by Tamaki et al.¹⁴ and followed by other groups.^15–17 Excess-Mg nominal compositions (x as high as 0.2) were used to synthesize n-type compounds and take advantage of the multi-valley conduction band as predicted by density functional theory (DFT) calculations (Fig. 1).¹⁴ The thermodynamic explanation behind this excess-Mg strategy to overcome the persistent p-type behavior^18–21 was later clarified by Ohno et al.¹⁶ Mg₃Bi₂-alloying was introduced to reduce the lattice thermal conductivity, because Mg₃Sb₂ and Mg₃Bi₂ are expected to form solid solutions. Te doping was used to optimize the carrier concentration.

In previous studies, the role of Bi in enhancing zT has mostly been attributed to Sb–Bi disorder that reduces thermal conductivity by disorder scattering of phonons.^14,15,17 However, alloying Mg₃Sb₂ with Mg₃Bi₂ also has a significant impact on the electronic structure.¹⁴ This influence is readily anticipated by the fact that Mg₃Bi₂ is a semi-metal while Mg₃Sb₂ is a semiconductor.^{18–20,22,23}Fig. 1 illustrates the features of the electronic structure that is expected to change with the alloying of Mg₃Bi₂, based on DFT calculations. In addition to the reduction of the band gap, the band effective mass of individual bands is also expected to decrease with Mg₃Bi₂ alloying. Such correlation between and E_g is a commonly observed tendency explained with the interaction between the valence and conduction bands.²⁴ Furthermore, the shift of bands due to alloying, in general, could cause band convergence that is beneficial for thermoelectric properties.¹² These changes in the electronic structure suggest that there would be an optimum Mg₃Bi₂ content. More Mg₃Bi₂ would result in increased mobility due to the lighter ; however, a reduced band gap would compromise the peak zT value because of the onset of bipolar conduction being shifted to lower temperatures.^25,26 In addition to the competing changes in electronic properties, the lattice thermal conductivity of the compounds will have a minimum when the fraction of Bi and Sb are nearly equal, due to the alloy-scattering of phonons.

Fig. 1 Schematic of changes in the Mg₃Sb₂ band structure due to Mg₃Bi₂ alloying. Alloying leads to a smaller band gap, and also a reduced band effective mass which leads to a higher mobility. The valence band in Mg₃Sb₂ is a single-valley, whereas the conduction band (CB1) is a multi-valley with a degeneracy of six. The minimum of the CB1 band is in the LAΓM plane, but not at a high symmetry point; when CB1 is plotted along the M–L direction, it apparently appears to have a higher minimum energy than the K band.^15,23 The true minimum of the CB1 is significantly lower in energy than the K band.¹⁴ The dashed lines illustrate how the Vegard's law would predict the bands to shift (with respect to the valence band). The solid-solution behavior of the material is evidenced from diffraction (see ESI†).

Closely related to understanding the bipolar-limited peak zT is the contrasted transport properties of the valence and conduction bands. For example, if the conduction band has a significantly higher weighted mobility than the valence band, the n-type material would have less contribution from minority carriers than for the p-type material (for a given E_F). Therefore, quantifying the weighted mobility of the n- and p-type bands is an essential prerequisite for engineering the band structure with an optimum content of Mg₃Bi₂. The contrasted transport properties of n- and p-type have been understood so far only in a rather qualitative matter.

In this study, by modeling both n- and p-type, and also both Mg₃Bi₂-alloyed and non-alloyed Mg₃Sb₂ samples, we illustrate how the DFT-predicted band structure is consistent with the trend of changes in Seebeck effective masses and mobilities seen in transport properties. We then explain how the optimum Mg₃Bi₂ content balances the benefits from increased weighted mobility and reduced lattice thermal conductivity with the disadvantage of a decreased band gap, which enables us to show the origin of the highest peak zT composition in Mg_3+x(Sb,Bi)_2−yTe_y.

Band structure analyzed with and μ₀

A useful metric that relates the band structure to experimental measurements is the Seebeck effective mass which is extracted by combining thermopower and Hall measurements (Pisarenko plots).²⁶ By comparing and the mobility parameter μ₀ between n- and p-type, or Mg₃Bi₂-alloyed and non-alloyed Mg₃Sb₂ samples, it is possible to infer differences in the band structure. For example, if is found larger in the n- than p-type, it would indicate a higher density-of-states in the conduction band. Largely, two distinct cases could lead to such a higher density-of-states: presence of either a heavier band (i.e. a small curvature at the band edge, heavy ) or multiple bands (e.g. band-degeneracy). In the case of a heavier band, μ₀ would typically be found smaller (assuming that other properties such as the deformation potentials²⁷ are similar). On the other hand, when multi-valley bands are contributing, μ₀ is identical to that of just a single valley (in the limit of negligible intervalley scattering). If μ₀ is found similar in both n- and p-type despite a large difference in , then the difference in density-of-states could be attributed to the presence of multiple bands. Therefore, with μ₀ and pairs, we can see if the transport properties are consistent with the DFT-predicted band structure.

To obtain , the reduced Fermi-level η = E_F/k_BT is first found from the experimental thermopower by solving the following equation (assumes relaxation time τ ∝ E^−1/2):²⁸

where F_i(η) is the Fermi–Dirac integral:The η determined this way with experimentally measured thermopower could be referred to as an effective value η_eff in the sense that we do not necessarily assume the band to be a single parabolic band (eqn (1) is often derived with such an assumption²⁸). Then the Seebeck effective mass is determined from:Here, n_H is the measured Hall carrier concentration. It is seen from eqn (3) that, for a given η_eff, increases with carrier concentration (∝ n_H^2/3) indicating that is related to the density-of-states. We use this procedure on a multiple of samples with different doping levels (i.e. different Te content). For samples with the identical type of carriers (n- or p-type) and Mg₃Bi₂ content, appears to be nearly constant (Fig. 2a), indicating that is a stable way to characterize the conduction and valence bands.

Fig. 2 (a and b) Seebeck effective mass of the Mg₃Sb₂ (open circles) and Mg₃Sb₂–Mg₃Bi₂ alloy (filled circles) samples. (c) Weighted mobility and (d) Mobility parameter as a function of temperature. The lines show a fit to the μ_w ∝ T^−3/2 dependency (phonon scattering). All data points are from temperatures where bipolar conduction is not significant, representing the property of majority carriers. For weighted mobility, the low-temperature points where the mobility is grain-boundary limited^32,33 was excluded. The p-type Mg₃SbBi data were taken from ref. 18. All other data are from the current study.

The mobility parameter μ₀ can be extracted from weighted mobility μ_w, which is an essential parameter for thermoelectrics that determines the magnitude of conductivity for a given reduced Fermi-level:

Here m_e is the mass of an electron. The weighted mobility μ_w, is determined by using η = η_eff in eqn (4) (Fig. 2c). Then, μ₀ follows (Fig. 2d) from:using determined previously (Fig. 2a and b). μ₀ is the mobility of a carrier at k_BT higher than the band edge.

It is seen that n-type Mg₃Sb₂ has a larger than p-type samples (Fig. 2b), while μ₀ is similar (Fig. 2d). This is a signature consistent with the multi-valley conduction band and single valley valence band structure (Fig. 1). It is this multi-valley feature of the conduction band that is mostly responsible for the superior μ_w, and thus the better thermoelectric performance of the n-type. The ratio of the between n- and p-type is about 1.9, somewhat smaller than (N_V = 6)^2/3 = 3.3 which indicates that the of the individual conduction pocket could be lighter than the valence band pocket rather than being similar.

The Mg₃Bi₂ alloying effect is seen by the monotonic decrease in with increasing Mg₃Bi₂ content (Fig. 2b), accompanied with an increase in μ₀ (Fig. 2d). This trend is seen in both n- and p-type samples, and suggests the band masses are becoming lighter with alloying, as opposed to band convergence which should give a peak in .¹² Lighter band masses typically give better μ_w for carriers scattered by phonons, as is indeed observed in Fig. 2c. An increased μ_w is always beneficial for thermoelectrics (other than that it usually accompanies a reduced band gap), as we can see from the definition of the dimensionless material quality factor B:^26,29–31

where κ_L is lattice thermal conductivity. B determines the maximum zT obtainable at a given temperature when the carrier concentration is optimized. For example, B of n-type Mg₃Sb₂ at 600 K is increases from 0.2–0.25 to 0.5–0.6 after alloying with 25% of Mg₃Bi₂, predicting an increase in the maximum zT from 0.6 to 1.3 at 600 K. Half of the increase in B is attributed to the increase in μ_w. Therefore, Mg₃Bi₂-alloying is seen to lead to better performance not just because of the lattice thermal conductivity reduction discussed in the literature, but also because of the significantly improved electronic properties.

Band gap and bipolar effects analyzed with the peak thermopower

Alloying with Mg₃Bi₂ is expected to reduce the band gap, as can be seen in Fig. 1. The alloying effect on the band gap can be analyzed by the Goldsmid-Sharp gap, together with a correction method. The Goldsmid-Sharp gap E_GS is found from the peak thermopower (S_max) with respect to temperature and the temperature at which it occurs (T_max).

E_GS = 2e|S_max|T_max (7)

This relation is associated with the fact that thermopower stops increasing with temperature due to the onset of minority carrier conduction in heavily doped semiconductors. E_GS is, however, not the true gap E_g but an effective gap because minority carrier conduction also depends on their weighted mobility. For example, if the minority carriers have low μ_w relative to the majority carriers, the bipolar effect is suppressed, leading to higher S_max and E_GS for a given true band gap (therefore, E_GS being higher than E_g is a desirable characteristic of a thermoelectric material). It has been demonstrated previously how to determine the true gap by correcting E_GS using the weighted mobility ratio between the majority and minority carriers.³⁴

Using the μ_w and E_GS extracted for both n- and p-type samples with different Mg₃Bi₂ content, we find the true gap 0.54 eV (±0.05 eV due to uncertainty in T_max) for Mg₃Sb₂ and its linear decrease with increasing Mg₃Bi₂ content in the Mg_3+xSb_2−yBi_y solid solution (Fig. 3a). As expected from the contrast in weighted mobilities, the uncorrected band gap E_GS (open circle in Fig. 3a) of n- is much higher than that of the p-type. By using the correction method,³⁴ consistent estimates of the true band gap E_g were obtained. The decreasing trend in the band gap with respect to Mg₃Bi₂ content extrapolates to a negative gap at pure Mg₃Bi₂, which is indeed reported to show semi-metallic behavior.^{18–20,22,23}

Fig. 3 (a) The reduction in band gap due to alloying with Mg₃Bi₂. Open circles represent the Goldsmid gap estimated by eqn (7). The solid circles are estimations of the true gap by using a correction method that uses the μ_w ratio between n- and p-type carriers.³⁴ The true gaps individually estimated from n- and p-type samples coincide. Linear interpolation of the trend for the true gap is shown by the dashed line. (b) Thermopower peaks with respect to temperature shown for n- and p-type Mg₃SbBi (similar Mg₃Bi₂ content). The peak thermopower shows a higher value at a higher temperature in n-type, which indicates a higher μ_w for the n-type carriers.

The advantage of the high contrast in n- and p-type μ_w's is readily noticeable in the comparison of n- and p-type thermopower curves with respect to temperature (Fig. 3b). S_max and T_max are higher in n-type, which is beneficial in reaching a higher peak zT, at the expense of lower values in the p-type.

The optimum Mg₃Bi₂ content for thermoelectrics

Whether a larger band gap, which leads to a higher peak zT at a higher temperature, could be useful depends on the stable temperature range of the material. The maximum operation temperature must be decided to gauge the benefits of widening the band gap.

Experimentally it has been noticed that Mg₃Sb₂ is not stable over long periods above 750 K due to the high vapor pressure of Mg, which leads to a decrease in the solubility of the n-type dopants (i.e. Te) and thus a loss of n-type carriers.¹⁶ Therefore, for deciding the optimum Mg₃Bi₂ content, we assume the maximum operation temperature for Mg₃Bi₂-alloyed samples to be around 700–750 K, and limit the optimization space to where the peak zT is found below this temperature.

We note that the expected peak zT temperature is above the maximum operation temperature in Mg₃Sb₂. As shown in Fig. 4, the zT values of Mg₃Sb₂ samples are still increasing after the maximum operation temperature. A higher zT value would be achievable if the sample were to reach a higher temperature (which is not possible due to stability). Because the peak zT is stability-limited rather than band-gap limited in Mg₃Sb₂, it is beneficial to reduce the band gap until the peak zT temperature is within the stability range and gain an increased μ_w. Because of this stability limit, we found that alloying Mg₃Bi₂ up to 25% is purely beneficial without compromises in any transport properties. Alloying Mg₃Bi₂ much more than this amount results in a lower zT peak value at a lower temperature as shown by the case of Mg_3+xSb_1.00Bi_0.99Te_0.01.

Fig. 4 zT curves of Mg₃Sb₂ and Mg₃Sb₂–Mg₃Bi₂ alloy n-type samples. Around the thermal stability limit (700–750 K), the zT of Mg₃Sb₂ samples show an increasing trend. With the addition of Mg₃Bi₂, the peak is shifted to lower temperatures near or below the stability limit.

The optimum Mg₃Bi₂ content is found by collectively considering the impact of Mg₃Bi₂ alloying on the band-gap, μ_w and also lattice thermal conductivity. We linearly interpolate μ_w and E_g with respect to Mg₃Bi₂ content as an empirical estimation. For lattice thermal conductivity, we describe the approximate trend by using the relation between κ_L,alloy and κ_L,pure proposed by Callaway³⁵ and Klemens.³⁶

Here θ_D is the Debye temperature, Ω is the volume per atom, ν is the average speed of sound, and Γ is the scattering parameter mainly caused by mass fluctuation. Γ can be calculated by the modified Klemens model.⁵ Then, by combining these data, we calculate the peak zT below 725 K as a function of Mg₃Bi₂ content (Fig. 5). We find the optimum Mg₃Bi₂ composition in the range of 20–30% which is indeed close to the experimentally successful composition reported in the literature.¹⁴

Fig. 5 Peak zT as a function of the Mg₃Bi₂ alloyed with Mg₃Sb₂. Model calculation for the peak zT below 725 K (stability temperature) is indicated with the solid line, while the peak zT within an unrestricted temperature limit is shown with the dashed line. It is seen that the peak zT is stability-temperature-limited up to 30–40% of Mg₃Bi₂ content. The optimum is found in the range of 20–30% Mg₃Bi₂ content. Experimental values are plotted together for comparison (circles). The open circle corresponds to a sample with lower than typical lattice thermal conductivity and thus a higher zT.³⁷

Conclusion

The electronic structure of Mg₃Sb₂ and its alloy with Mg₃Bi₂ was studied by analyzing the transport properties with an effective mass approach. Heavier Seebeck effective mass and larger weighted mobility in n- than p-type were observed, which is a signature of the higher band degeneracy of the conduction band. Mg₃Sb₂–Mg₃Bi₂ alloys showed a decrease in Seebeck effective mass and increase in weighted mobility, indicative of a larger curvature in each individual band (lighter band mass). This change accompanied a decrease in the band gap. Since the band gap of the Mg₃Sb₂ (0.5–0.6 eV) makes the zT peak at a temperature higher than the stability limit, we find the Mg₃Bi₂-alloying to be electronically beneficial for thermoelectrics despite the reduction in band gap. By empirically modeling the transport property changes due to alloying, we find the optimum Mg₃Bi₂ content in the range of 20–30%, which is in good agreement with the experimental observation.

Experimental method

Pure elemental materials of magnesium turnings (99.98%, Alfa Aesar), antimony shots (99.9999%, Alfa Aesar), bismuth granules (99.997%, Alfa Aesar) and tellurium lumps (99.999%, Alfa Aesar) were loaded according to the nominal composition and sealed into stainless-steel vials with two half inch stainless-steel balls inside the argon-filled glove box. Mechanical reaction was conducted by high energy-ball milling with a high-energy mill (SPEX 8000D) for two hours. The reacted powder was extracted from vials and transferred into a graphite die under argon atmosphere to prevent the oxidization of the materials. Subsequently, induction heating rapid hot press was conducted for 1 hour at 873 K and 45 MPa under argon gas flow to consolidate pellet samples.³⁸ Pressed samples were gradually cooled down to room temperature with no load. The Seebeck coefficient of each sample was measured with Chromel-Nb thermocouples in a two-probe configuration under high vacuum.³⁹ The Hall coefficient and electric resistivity was measured simultaneously using a 4-point probe van der Pauw technique with a 2 T magnetic field under high vacuum. Thermal diffusivity D was measured by using the flash method with a Netzsch LFA 457 under dynamic argon atmosphere. The thermal conductivity κ was calculated by κ = D × C_p × d, where d is the geometrical density and C_p is the heat capacity. Average values of the heat capacity reported in previous studies (Tamaki et al.¹⁴ for Mg₃Bi₂-alloyed samples, Bhardwaj et al.⁴⁰ for non-alloyed Mg₃Sb₂ samples) were used for the calculations.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors would like to acknowledge support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences through the Solid-State Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center (DE-SC0001299), and also from the NASA Science Mission Directorate's Radioisotope Power Systems Thermoelectric Technology Development program. KI acknowledges support from Funai Foundation for Information Technology.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7mh00865a

‡ Present address: Institute of Physical Chemistry, Justus-Liebig-University Giessen, Heinrich-Buff-Ring 17, D-35392 Giessen, Germany.

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