Kazuki
Imasato
^{a},
Stephen Dongmin
Kang
^{ab},
Saneyuki
Ohno‡
^{ab} and
G. Jeffrey
Snyder
*^{a}
^{a}Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA. E-mail: jeff.snyder@northwestern.edu
^{b}Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, CA 91125, USA

Received
20th October 2017
, Accepted 20th November 2017

First published on 20th November 2017

Mg_{3}Sb_{2}–Mg_{3}Bi_{2} alloys show excellent thermoelectric properties. The benefit of alloying has been attributed to the reduction in lattice thermal conductivity. However, Mg_{3}Bi_{2}-alloying may also be expected to significantly change the electronic structure. By comparatively modeling the transport properties of n- and p-type Mg_{3}Sb_{2}–Mg_{3}Bi_{2} and also Mg_{3}Bi_{2}-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 Mg_{3}Bi_{2} 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 Mg_{3}Bi_{2} because the peak zT in unalloyed Mg_{3}Sb_{2} is at a temperature higher than the stable range for the material. By quantifying the electronic influence of Mg_{3}Bi_{2} alloying, we model the optimum Mg_{3}Bi_{2} content for thermoelectrics to be in the range of 20–30%, consistent with the most commonly reported composition Mg_{3}Sb_{1.5}Bi_{0.5}.

## Conceptual insightsIn 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. |

The maximum power and efficiency achievable by a particular thermoelectric material is determined by their figure of merit zT = (S^{2}σ)/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+x}Sb_{2}Bi_{0.49}Te_{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} Mg_{3}Bi_{2}-alloying was introduced to reduce the lattice thermal conductivity, because Mg_{3}Sb_{2} and Mg_{3}Bi_{2} 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 Mg_{3}Sb_{2} with Mg_{3}Bi_{2} also has a significant impact on the electronic structure.^{14} This influence is readily anticipated by the fact that Mg_{3}Bi_{2} is a semi-metal while Mg_{3}Sb_{2} 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 Mg_{3}Bi_{2}, 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 Mg_{3}Bi_{2} 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 Mg_{3}Bi_{2} content. More Mg_{3}Bi_{2} 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 Mg_{3}Sb_{2} band structure due to Mg_{3}Bi_{2} 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 Mg_{3}Sb_{2} 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 Mg_{3}Bi_{2}. 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 Mg_{3}Bi_{2}-alloyed and non-alloyed Mg_{3}Sb_{2} 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 Mg_{3}Bi_{2} 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−y}Te_{y}.

To obtain , the reduced Fermi-level η = E_{F}/k_{B}T is first found from the experimental thermopower by solving the following equation (assumes relaxation time τ ∝ E^{−1/2}):^{28}

(1) |

(2) |

(3) |

Fig. 2 (a and b) Seebeck effective mass of the Mg_{3}Sb_{2} (open circles) and Mg_{3}Sb_{2}–Mg_{3}Bi_{2} 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 Mg_{3}SbBi data were taken from ref. 18. All other data are from the current study. |

The mobility parameter μ_{0} 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:

(4) |

(5) |

It is seen that n-type Mg_{3}Sb_{2} has a larger than p-type samples (Fig. 2b), while μ_{0} 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 Mg_{3}Bi_{2} alloying effect is seen by the monotonic decrease in with increasing Mg_{3}Bi_{2} content (Fig. 2b), accompanied with an increase in μ_{0} (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}

(6) |

E_{GS} = 2e|S_{max}|T_{max} | (7) |

Using the μ_{w} and E_{GS} extracted for both n- and p-type samples with different Mg_{3}Bi_{2} content, we find the true gap 0.54 eV (±0.05 eV due to uncertainty in T_{max}) for Mg_{3}Sb_{2} and its linear decrease with increasing Mg_{3}Bi_{2} content in the Mg_{3+x}Sb_{2−y}Bi_{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 Mg_{3}Bi_{2} content extrapolates to a negative gap at pure Mg_{3}Bi_{2}, 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 Mg_{3}Bi_{2}. 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 Mg_{3}SbBi (similar Mg_{3}Bi_{2} 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.

Experimentally it has been noticed that Mg_{3}Sb_{2} 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 Mg_{3}Bi_{2} content, we assume the maximum operation temperature for Mg_{3}Bi_{2}-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 Mg_{3}Sb_{2}. As shown in Fig. 4, the zT values of Mg_{3}Sb_{2} 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 Mg_{3}Sb_{2}, 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 Mg_{3}Bi_{2} up to 25% is purely beneficial without compromises in any transport properties. Alloying Mg_{3}Bi_{2} much more than this amount results in a lower zT peak value at a lower temperature as shown by the case of Mg_{3+x}Sb_{1.00}Bi_{0.99}Te_{0.01}.

The optimum Mg_{3}Bi_{2} content is found by collectively considering the impact of Mg_{3}Bi_{2} alloying on the band-gap, μ_{w} and also lattice thermal conductivity. We linearly interpolate μ_{w} and E_{g} with respect to Mg_{3}Bi_{2} 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 Callaway^{35} and Klemens.^{36}

Here θ

Fig. 5 Peak zT as a function of the Mg_{3}Bi_{2} alloyed with Mg_{3}Sb_{2}. 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 Mg_{3}Bi_{2} content. The optimum is found in the range of 20–30% Mg_{3}Bi_{2} 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} |

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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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