Strategies for the High Average <i>zT</i> in the Electron-Doped SnTe

Park, Hyunjin; Lee, Kyu Hyoung; Heo, Minsu; Kim, Sang-il; Kim, Hyun-Sik

doi:https://doi.org/10.1155/2023/5512034

International Journal of Energy Research

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5512034 | https://doi.org/10.1155/2023/5512034

Strategies for the High Average zT in the Electron-Doped SnTe

Hyunjin Park,¹Kyu Hyoung Lee,²Minsu Heo,¹Sang-il Kim,¹and Hyun-Sik Kim¹

Academic Editor: Raja Sen

Received09 Aug 2023

Revised10 Oct 2023

Accepted24 Nov 2023

Published15 Dec 2023

Abstract

We propose a new strategy to achieve a high average zT in electron-doped SnTe by applying the two-band (TB) and single parabolic band (SPB) models to the electronic transport properties of Sn_0.97M_0.03Te (, In, Bi, and Sb) reported in the literature. To achieve a high average zT at temperatures from 300 to 823 K, both zT at 300 and 823 K should be high with a steadily increasing zT over the temperatures. The p-type SnTe is known to have a light valence band and a heavy valence band that are approximately 0.40 eV apart. The Bi-doped SnTe exhibits one of the highest zT among all the other doped samples at 300 K (0.09) and the highest zT at 823 K (0.9), with a steadily increasing zT in between. The TB model confirms the presence of the resonant state at 300 K which is responsible for the high zT at 300 K. The B-factor, which is related to the theoretical maximum zT, calculated by the SPB model indicates a steady increase in zT with increasing temperature. The temperature-dependent B-factor of the Bi-doped SnTe suggests that the initial position of its Fermi level at 300 K calculated by the TB model may be responsible for the temperature coefficient of the B-factor, which determines the zT at 823 K. According to the SPB model, experimental zT of 0.9 of the Bi-doped SnTe can be further improved to 1.03 (14% improvement) at 823 K upon carrier concentration optimization. To achieve a high average zT in SnTe, electron doping with a dopant that forms a resonant state and placing the Fermi level at the light valence band in the vicinity of the heavy valence band maximum are both essential.

1. Introduction

Efforts to prevent accelerating global warming are continuing through research on various sustainable energy harvesting technologies. They have made significant contributions to reducing greenhouse gas emissions, which are the primary cause of global warming [1]. Thermoelectric technology, one of the environmentally friendly energy harvesting technologies, is attracting much attention as it can generate electricity directly from waste heat without combusting fossil fuels [2]. The current efficiency of thermoelectric power generation devices is not yet high enough to compete with conventional coal-fired power plants. The efficiency of thermoelectric devices is closely linked to the thermoelectric figure-of-merit (zT) of the materials used in the devices. As a result, many studies have focused on increasing the zT of thermoelectric materials. The zT (=S²σT/κ) depends on the Seebeck coefficient (), electrical conductivity (), temperature ( in Kelvin), and thermal conductivity (). The is, in turn, a sum of electronic contribution to thermal conductivity () and lattice thermal conductivity () [2, 3]. The S²σ in zT is called the power factor (PF), and it determines the electron transport properties of the material. As shown in the zT equation, the zT is maximized when the PF is high and the is low.

However, it is difficult to increase the PF because the and of the material have a trade-off relationship with a changing Hall carrier concentration () [2, 4–6]. For example, as the increases, the of a material increases while its decreases. For this reason, the PF of a thermoelectric material peaks at a specific , and optimizing the for the highest PF is an important goal in thermoelectric research. In addition, many band engineering strategies have been shown to be effective in improving the PF by bypassing the trade-off between the and . For example, the band convergence strategy increases the number of the Fermi pockets (valley degeneracy) of electronic bands that contribute mainly to electronic transport by aligning two bands similar in energy. When two bands approach each other in energy, the density-of-state effective mass () increases, leading to an enhancement of the without a decrease . This increase in is due to the increase in the number of the Fermi pockets, which does not lead to a decrease in [7–9]. The is the only parameter in the zT equation that can be lowered without significantly affecting other parameters in zT. The reduction can be achieved through complex crystal structuring and nano-/meso- or all-scale architecturing [10, 11].

Chalcogenides that include Bi, Sb, Sn, or Ge elements are currently the most promising materials for thermoelectric applications. Among these materials, SnTe has recently emerged as a popular choice in the development of clean energy sources due to its Pb-free composition [12–14]. Although SnTe has a crystal structure and electronic structure similar to that of PbTe, its zT is lower than that of PbTe due to its high intrinsic defect concentrations, narrow bandgap, and high energy difference between the highest and the 2^nd highest valence bands [14]. Banik et al. demonstrated that Mg alloying in SnTe significantly improved the because of the reduction in the energy difference between the highest valence band (light valence band) and the 2^nd highest valence band (heavy valence band) [15]. The energy difference between the light valence band and the heavy valence band is approximately 0.40 eV at room temperature [14]. Tan et al. reported that Hg alloying in SnTe increased the bandgap and decreased the energy difference between the light and heavy valence bands, simultaneously. In other words, Hg alloying promotes band convergence while suppressing bipolar conduction [16]. Tan et al. showed that Mn alloying in SnTe tuned the valence bands and exhibited an higher than expected from the band convergence between the light and heavy valence bands [17]. Recently, Zhao et al. reported a high zT of 0.9 at 823 K in Bi-doped SnTe (Sn_0.97Bi_0.03Te) [14]. This was achieved by electron doping with Bi, which effectively maximized the electronic transport properties of SnTe. The experimental and of the Bi-doped SnTe coincided with the highest (~53 μV K^-1) at the optimum () calculated using the two-band (TB) model at 300 K. Other dopants (Ga, In, and Sb) also decreased the of SnTe, but it was the Bi dopant that optimized the for the highest S in Sn_0.97M_0.03Te (, In, Bi, and Sb). However, the advantages of Bi doping in SnTe relative to other dopants (Ga, In, and Sb) were not fully explored. The potential impact of resonant state formation via Bi doping was also not considered.

In this study, we first evaluated the effect of electron doping (Ga, In, Bi, and Sb) on SnTe using the TB model, which considers one light valence band and one heavy valence band. By comparing the Seebeck–Pisarenko ( vs. ) calculated via the TB model and the experimental data of Sn_0.97M_0.03Te (, In, Bi, and Sb), we found that the Bi-doped SnTe generated a resonant state on top of electron doping. Once we confirmed the resonant state formation via Bi doping, we then performed the single parabolic band (SPB) model calculations for Sn_0.97M_0.03Te (, In, Bi, and Sb) to evaluate how different electron doping affects the electronic band parameters when a single parabolic band was assumed. The reason that we first employed the TB model instead of the SPB model was that the SPB model could only provide the information about the density-of-states effective mass () change, but not the reason behind it. We calculated the temperature-dependent , nondegenerate mobility (), weighted mobility (), and B-factor of Sn_0.97M_0.03Te (, In, Bi, and Sb) using the SPB model [18]. From the temperature-dependent band parameters, we analyzed the reason behind the high zT of Bi-doped SnTe at high temperatures (~823 K). We found that the resonant state formation and temperature coefficient of B-factor are responsible for the high zT (~0.9) of Sn_0.97Bi_0.03Te at 823 K. The temperature coefficient of the B-factor that is favorable to all temperature ranges can be achieved if the Fermi level () is placed at the light valence band in the vicinity of the heavy valence band maximum. We also found that the zT of the Bi-doped sample can be improved to 1.03 (14% improvement) if the is tuned to . Although the average zT of the Bi-doped sample is the highest among other samples after the optimization, the SPB model shows that the theoretical maximum zT of the Sb-doped sample (1.13) is even higher than that of the Bi-doped sample at 823 K.

2. Method

2.1. SPB Model

The temperature-dependent and of the Sn_0.97M_0.03Te (, In, Bi, and Sb) samples were obtained from Zhao et al. [14]. With the SPB model, the is defined as follows [19]:

The Fermi integral of order () in Equation (1) is defined as in Equation (2) [20, 21].

The is defined by and as shown in Equation (3) [20, 21].

At a consistent temperature, the lines that vary with can be matched to the empirical findings by tuning the in Equation (3). The SPB model allows for the anticipation of under varied conditions not explored in the experiments.

The was calculated from the experimental -dependent at different temperatures (from Zhao et al.) using the SPB model [5, 14]. According to the SPB model under acoustic phonon scattering assumption, the is defined in terms of the and the Fermi level () (Equation (4)) [22].

The was fitted to the experimental -dependent using Equations (3)–(5).

The is defined as in Equation (5) [23].

The B-factor is determined from the , , and as shown in Equation (6), where , , and are the electric charge, Planck’s constant, and Boltzmann’s constant, respectively [23].

2.2. TB Model

With the two-band model, the and are defined as in Equations (7) and (8), where the lower subscript and mean the light valence band and the heavy valence band [24].

, the Seebeck coefficient of the light-mass valence band, is defined as in Equation (9) [24].

, the chemical carrier concentration of the light-mass valence band, is defined as in Equation (11) [24].

, the hall factor of light-mass valence band, is defined as in Equation (12) [24].

, the mobility of light-mass valence band, is defined as in Equation (13) [24].

And for the heavy-mass valence band, the Seebeck coefficient (S_H), chemical carrier concentration (), Hall factor (), and mobility () are defined as follows [24, 25]:

where is the hall coefficient [25],

Table 1 shows the band parameters that we used in this study [24, 25].

3. Results and Discussion

Figure 1(a) shows the experimental -dependent of the pristine and electron-doped (Ga, In, Bi, and Sb) SnTe. The measurements from Zhao et al. are presented with filled symbols [14], while those from other literature are presented with empty symbols [16, 26–28]. The solid line in Figure 1(a) is the -dependent calculated for the pristine SnTe using the TB model (1 light valence band + 1 heavy valence band). The light valence band (LVB) is 0.18 eV apart from the lowest conduction band edge, and it is approximately 0.38 eV higher than the heavy valence band (HVB). For the of LVB and HVB, 0.13 m₀ and 1.92 m₀ are used, respectively. Considering the narrow band gap () between the LVB and the conduction band minimum, LVB is set to be nonparabolic (with the band nonparabolicity parameter of at 300 K). For simplicity, the HVB with of 0.56 eV () is assumed parabolic. These details of SnTe band structure incorporated in the TB model closely follow those adopted by Zhang et al. [24] (Table 1).

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The -dependent calculated by the TB model (dark grey solid line in Figure 1(a)) decreases until (shaded in blue) and suddenly peaks at around . The abrupt increase in the for in the range from to (shaded in light blue) is closely related to the presence of the HVB 0.38 eV below the LVB. As the increases the Fermi level at the LVB enters a regime where contribution from HVB becomes significant. However, the Fermi level that corresponds to the local maximum (33 μV K^-1) at is not exactly at the HVB maximum. The blue-shaded and light blue-shaded regions in Figure 1(a) are translated in terms of Fermi levels in Figure 1(c).

Electron doping is effective in decreasing the of SnTe. According to the pristine SnTe and Sn_0.97M_0.03Te (, In, Bi, and Sb) samples (filled symbols) in Figure 1(a), the experimental of Sn_0.97M_0.03Te (, In, Bi, and Sb) samples are all lower than that of SnTe (). The of the Sn_0.97Ga_0.03Te sample is in the region where the HVB is dominant (no shading). The of the Sn_0.97In_0.03Te and Sn_0.97Bi_0.03Te samples are in the region where both HVB and LVB contribute to electronic transport (shaded in light blue). The of the Sn_0.97Sb_0.03Te sample is in the region where only LVB is dominant (shaded in blue). While the -dependent measurements for the pristine SnTe, Sn_0.97Ga_0.03Te, and Sn_0.97Sb_0.03Te can be described with the TB model calculation results (solid line), those for Sn_0.97In_0.03Te and Sn_0.97Bi_0.03Te are deviated from the TB model significantly. It is well known that In doping in SnTe generates resonant state within the valence band which causes an increase in higher than expected from the TB model [24]. However, the formation of resonant states via Bi doping is not well understood, as experimental results have been contradictory. For example, the Bi-doped SnTe measurements reported by Vedeneev et al. and Galushchak et al. closely follow the TB model results, which suggests that Bi doping only alters the Fermi level without changing any band parameters of LVB and HVB [27, 28]. However, the experimental of the Bi-doped SnTe samples from Zhao et al., Tan et al., and Zhou et al. are higher than the TB model prediction at their experimental . This indicates that the Bi doping alters the band structure of SnTe. Possible resonant state formation in SnTe by Bi doping has been proposed recently by Shenoy and Bhat from density functional theory (DFT) calculations [29, 30]. Additionally, we have estimated how the Seebeck–Pisarenko (-dependent ) would change as the energy offset between the LVB and HVB decreases. The Seebeck–Pisarenko lines with energy offset being smaller than 0.38 eV are shown in light grey lines. As the energy offset decreases, the hump near (when the energy offset is 0.38 eV) shifts to lower and disappears when the energy offset is zero. Even when the LVB and HVB are converged (light grey dotted line), the experimental data for Bi-doped SnTe (with high Bi doping content) do not agree with the TB model results. Based on the reasons outlined above, it is suggested that resonant state formation is likely to occur in Bi-doped SnTe with high Bi-doping concentrations. Vedeneev et al. doped Bi at a concentration of 1 at.%, while other samples have a Bi doping content of 2 at.% or higher. The amount of Bi doping content, combined with the doping site (Bi can replace either Sn or Te), which may change with the doping content, must have resulted in the contradicting results for the Bi-doped samples as shown in Figure 1(a).

Figure 1(b) shows the experimental -dependent of the pristine and electron-doped (Ga, In, Bi, and Sb) SnTe (in symbols). Some of the Bi-doped samples (presented in Figure 1(a)), which did not report their measurements, were omitted in Figure 1(b). The TB model result is shown in the solid line. No abrupt change in the is observed in the TB model result. The appearance of HVB at high decreases the rate of decrease with increasing (>10²⁰ cm^-3). The absence of a hump in the calculated -dependent (unlike the calculated -dependent in Figure 1(a)) can be explained by the different electrical conductivity-weighted averaging of the contributions to and from the LVB and HVB. It is to be noted that the -dependent measurements of all the samples have reasonable agreements with the TB model. This includes the In-doped and Bi-doped SnTe samples. If their experimental μ_H are lower than those predicted by the TB model, we cannot claim the formation of resonant state via In or Bi doping. In that case, the enhancement would be explained with the conventional - trade-off relation. In other words, the experimental -dependent of the Bi-doped SnTe samples in Figure 1(b) would have lower than the TB model result at the given if the of the HVB have become heavier with Bi doping. Because the enhancement of In or Bi-doped SnTe greater than calculated by the TB model (Figure 1(a)) is not accompanied by corresponding reduction in (hence ), and the enhancement of Bi-doped SnTe cannot be explained by the band convergence between the LVB and HVB (light grey lines in Figure 1(a)), it is very likely that the high experimental (71 μV K^-1) of Sn_0.97Bi_0.03Te is due to the resonant state formation on top of the electron doping.

Figure 1(c) shows the schematic band structure of the pristine SnTe and the Fermi levels (in unit of k_BT) of the pristine and Sn_0.97M_0.03Te (, In, Bi, and Sb) calculated by the TB model (in dashed lines). The Fermi level () of 0 is set to be at the LVB maximum (in yellow). The energy difference () between the LVB maximum and HVB maximum (in green) is 14.6 (0.38 eV at 300 K). The blue- and light blue-shaded regions in Figure 1(c) correspond to those shown in Figures 1(a) and 1(b). The only difference is that the shaded regions in Figure 1(c) are expressed in terms of the and not the . The is a better indicator of the degeneracy than the , as the can be misleading due to its dependence on . The decreases with increasing until the is 11 above the HVB maximum (shaded in blue). However, once the between the and the HVB maximum is smaller than 11, the contribution from the HVB kicks in, and the improves with an increasing . This continues until the is 17 below the HVB maximum (shaded in light blue). When the between the and the HVB maximum is greater than 17, the contribution from the LVB becomes negligible, and the decreases again as the increases (no shading). The of both In-doped and Bi-doped samples are in the region where both LVB and HVB contribute to transport. Resonant state formation and optimally placed Fermi level are the physical reasons behind the high zT of In-doped (zT~0.1) and Bi-doped samples (~0.09) at 300 K. The zT of Bi-doped sample (zT~0.9) becomes higher than the In-doped sample (~0.6) at higher temperatures.

The band structure of SnTe is well-known at 300 K (Figure 1(c)). However, the band structure at high temperatures is hardly studied, making the TB model calculation of SnTe at high temperatures difficult. Once we confirmed the existence of the resonant state in the In-doped and Bi-doped samples at 300 K using the TB model, we switched the evaluation tool to the single parabolic band (SPB) model, for simplicity. The reason that we switched from the TB model to the SPB model is that the SPB model is simpler and more intuitive than the TB model while capturing important characteristics of the electronic band structures. When electronic transport properties are fitted by the TB model, we need to fit 4 different band parameters (2 effective masses and 2 deformation potentials for the LVB and HVB). In contrast, only two band parameters (1 effective mass and 1 deformation potential) need to be fitted when the SPB model is utilized. Therefore, the effect of resonant state formation and electron doping can be more intuitively understood with the SPB model than with the TB model.

Figure 2(a) shows the experimental and calculated as a function of at 300 K [14]. By fitting Equations (1) and (3) to the experimental data (in filled symbols), the calculated -dependent (in lines) are achieved. The pristine SnTe has the highest (). The decreases as Ga, In, Bi, and Sb are doped to SnTe. The of Sn_0.97Ga_0.03Te () is only 40% smaller than that of SnTe, while the of Sn_0.97Sb_0.03Te () is approximately 95% smaller than that of SnTe. The of pristine SnTe (~29 μV K^-1) decreases to 25 and 17 μV K^-1, respectively, as Ga and Sb are doped. However, the significantly improves when In (66 μV K^-1) and Bi (71 μV K^-1) are doped into SnTe. From the TB model results (Figure 1), we know that the improvements in In-doped and Bi-doped samples are due to the resonant state formation and the appropriately positioned where the HVB starts to contribute to the electronic transport (shaded in light blue in Figure 1(c)).

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Figure 2(b) shows the calculated 300 K of pristine SnTe and Sn_0.97M_0.03Te (, In, Bi, and Sb) obtained from Figure 2(a). The of pristine SnTe is 1.67 m₀ (SPB model). According to the TB model, the of the LVB and HVB in SnTe were 0.13 and 1.92 m₀ (Figure 1). The of SnTe is approximately 5.2 k_BT (~0.13 eV at 300 K) below the HVB maximum (Figure 1(c)). This means that the majority of electronic contributions to SnTe come from the HVB. Consequently, its estimated by the SPB model (1.67 m₀) is comparable to the of the HVB in the TB model (1.92 m₀). Although the in Figure 2(b) is estimated while assuming there is only one parabolic band contributing to transport, it directly corresponds to the actual band structure and the position of its Fermi level we obtained from the TB model (Figure 1(c)). The Sb-doped SnTe which has the lowest has its Fermi level positioned at the LVB farthest from the top of the HVB (Figure 1(c)). The of Sn_0.97Ga_0.03Te is 1.03 m₀ (SPB model). The Ga-doped sample has a smaller (~17.7) than the pristine SnTe (~19.8) in Figure 1(c), meaning that the is closer to the HVB maximum. This indicates that the LVB is more populated in the Ga-doped sample. When the contributions from the two bands are approximated to that from one band, the effective mass is expected to be lighter than that of the pristine SnTe. The of Sn_0.97In_0.03Te and Sn_0.97Bi_0.03Te are 2.04 and 1.30 m₀ (SPB mode), respectively. The of both In-doped and Bi-doped samples are in the region where both LVB and HVB actively participate in the electronic transport. One difference between the In-doped and Bi-doped samples is the location of their relative to the HVB maximum. The of the In-doped sample is slightly below the HVB maximum, indicating that the HVB is the major band. The of the Bi-doped sample is above the HVB maximum, indicating that the LVB is the major band in the sample. The difference in the major band affects the resulting estimated by the SPB model. For example, the of the In-doped sample is heavier than that of the pristine SnTe due to the additional resonant state formation effect, but the of the Bi-doped sample is lighter than the pristine SnTe even with the resonant state. The of the pristine SnTe and Sn_0.97M_0.03Te (, In, Bi, and Sb) obtained by using the SPB model are closely related to the electronic band structure incorporated in the TB model as shown in Figure 1(c). Therefore, the SPB model is an effective tool for understanding the effect of electron doping and resonant state formation.

Figure 2(c) shows the temperature-dependent calculated for the pristine SnTe and Sn_0.97M_0.03Te (, In, Bi, and Sb) using the SPB model. The band structure provided in Figure 1(c) for the pristine SnTe can be changed with doping or temperature [31]. For example, the increase with an increasing temperature observed in the pristine SnTe is related to the possible change in the between the LVB and HVB at high temperatures. This increase in with temperature is also observed in Ga-doped and Sb-doped samples. However, the of the In-doped sample at 823 K (1.96 m₀) is lighter than that calculated at 300 K (2.04 m₀). This shows that the effect of resonant state formation only observed at 300 K is stronger than the LVB and HVB converging in energy at high temperatures [32]. On the contrary, the of the Bi-doped SnTe at 300 K (1.30 m₀) is lighter than that obtained at 823 K (1.63 m₀). The convergence of the LVB and HVB at high temperatures due to Bi doping has a larger impact on the band structure than the resonant state formed by Bi doping near 300 K.

Figure 3(a) shows the experimental and calculated as a function of at 300 K [14]. By fitting Equations (3) and (4) to the measurements (in filled symbols), the calculated -dependent (in lines) are calculated. For the in Equation (5), the obtained in Figure 2 are used. Experimental of the pristine SnTe (34 cm² s^-1 V^-1) is the lowest among other doped SnTe samples at 300 K. As the decreases with electron doping, the corresponding significantly increases. For example, the of In-doped sample is approximately 44 cm² s^-1 V^-1, which almost doubles to 87 cm² s^-1 V^-1 in the Bi-doped sample. Moreover, the of Sb-doped sample amounts to 539 cm² s^-1 V^-1. However, the obtained from the measurements do not necessarily follow the trend of the experimental . This is because is the mobility of the sample when the sample is defectless and its is close to its nondegenerate regime.

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In Figure 3(b), the used to calculate the theoretical -dependent at 300 K (Figure 3(b) in lines) are presented for the pristine SnTe and doped SnTe samples. The of pristine SnTe is 107 cm² s^-1 V^-1, and when SnTe is doped with Ga, In, Bi, and Sb, the is calculated to be 136, 92, 182, and 2320 cm² s^-1 V^-1, respectively. Because the is a band parameter, its value estimated by the SPB model is also affected by the position of the in the band structure (Figure 1(c)). The of the pristine SnTe, Ga-doped, and In-doped SnTe are positioned at the HVB. On the contrary, the of the Bi-doped and Sb-doped samples are placed at the LVB. Because the is inversely proportional to the , the of the Bi-doped and Sb-doped samples, whose major band in the LVB, are higher than those of other samples. The reason that the of the Sb-doped sample (2320 cm² s^-1 V^-1) is much higher than that of the Bi-doped sample (182 cm² s^-1 V^-1) is that the contribution from the HVB is almost negligible in the Sb-doped sample. When Figure 3(b) is compared to Figure 2(b), we can see that the trend of the change with different dopants (Figure 2(b)) is opposite to the trend of the change (Figure 3(b)). For example, In doping decreases the of SnTe while increasing the when compared to those of the pristine SnTe. If the decrease in is less than expected from the increase, we can suspect that band engineering is applied to overcome the - trade-off relation. In order to closely evaluate the effect of resonant state formation or change in between the LVB and HVB, the needs to be characterized, which is an -weighted (Equation (5)).

Figure 3(c) shows the calculated as a function of temperature. All samples exhibit a decreasing trend in as the temperature increases. The decrease in is mostly due to the increased carrier-phonon interaction at high temperatures. The carrier-phonon interaction is expressed as a deformation potential (), and it is also inversely proportional to . In addition, Zhou et al. reported that the of the LVB and HVB decreased at high temperatures [25]. The decreases in and are the primary reasons for the decrease with temperature. Different samples have different rates of decrease with an increasing temperature. The temperature coefficient of for the pristine SnTe and doped SnTe are shown in Table 2.

The temperature coefficient of () is calculated by Equation (16) below.

The and are the at 823 K and that at 300 K, respectively. Because the Bi-doped does not have the at 823 K, is used instead with of 423 K. The of the pristine SnTe, Ga-doped, and In-doped SnTe samples, whose are initially at the HVB at 300 K, are lower than those calculated in the Bi-doped and Sn-doped samples. As the increases with Ga and In dopings, corresponding decrease. However, for the Bi-doped and Sb-doped samples, whose are initially at the LVB at 300 K, the decrease in (from Bi-doped to Sb-doped sample) significantly increases the (by more than 13 times in magnitude). Physical mechanism behind the observed is not straightforward as it is correlated to the temperature coefficients of and for both LVB and HVB and to how the between the LVB and HVB changes with an increasing temperature. Nonetheless, it is important to note that the of the Sb-doped sample is the highest, because this result hints that the should be placed in the LVB for the higher thermoelectric performance.

Figure 4(a) shows at 300 K calculated by substituting (Figure 2(b)) and (Figure 3(b)) into Equation (5) [18]. The is a better indicator of band engineering than the alone. This means that the can be used to identify a wider range of band engineering effects, including band convergence and resonant state formation. The of In-doped and Bi-doped samples (≥267 cm² s^-1 V^-1) are higher than those of other samples (≤231 cm² s^-1 V^-1) at 300 K. The high of In-doped and Bi-doped samples can be attributed to the resonant state formation as confirmed in Figure 1(a). The of the Bi-doped sample (1.30 m₀) was lighter than that of the pristine sample (1.67 m₀) in Figure 2(a). However, when the is considered, we see that the of the Bi-doped sample (270 cm² s^-1 V^-1) is higher than that of the pristine SnTe (231 cm² s^-1 V^-1) by approximately 17%. The filled bars in Figure 4(a) show the experimental PF reported at 300 K [14]. The experimental PF is positively correlated with the of a sample. This means that as the of a sample increases, the PF of the sample also tends to increase. The experimental PF of the Bi-doped sample is the highest among all the samples, with a value of 158 μW cm^-1 K^-2. This is consistent with the fact that the of the Bi-doped sample is also the highest (270 cm² s^-1 V^-1). The theoretical maximum PF is directly proportional to the , while the experimental PF is not. This means that the can only be used to predict the theoretical maximum PF, and not the actual experimental PF. The theoretical maximum PF predicted by the SPB model (using the in Figure 4(a)) is presented in empty bars. According to the SPB model, the PF of the Bi-doped sample can be further improved by more than 51.3% (15.8⟶23.9 μW cm^-1 K^-2) after appropriate tuning.

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Figure 4(b) shows the calculated as a function of temperature. All samples show a decrease in as temperature increases, with the decrease rate being the smallest for the Sb-doped sample. The temperature coefficient of is much smaller than the temperature coefficient of , making it insignificant. Therefore, the decrease in with an increasing temperature mostly stems from the temperature coefficient of (Figure 3(c)). The of the Ga-doped (52 cm² s^-1 V^-1), Bi-doped (51 cm² s^-1 V^-1), and Sb-doped samples (51 cm² s^-1 V^-1) are the highest at 823 K, with values that are higher than the other samples. The of the Sb-doped sample is the lowest at 300 K, but it increases sharply at higher temperatures. This is noteworthy because it suggests that placing the at the LVB at 300 K may be a promising way to improve the thermoelectric properties of materials at high temperatures. The of the Bi-doped sample, which is the highest at 300 K, remains high even at 823 K, which further supports this hypothesis.

Figure 5(a) shows the calculated B-factor of for the pristine SnTe and Sn_0.97M_0.03Te (, In, Bi, and Sb) at 300 K. The B-factor is calculated through Equation (6) using the calculated (Figure 4(a)) and the measured [14]. The measured of the pristine and doped SnTe are given in Figure 5(b). According to Equation (6), the B-factor is directly proportional to the ratio of the to . The trend in the B-factor is similar to that in the for the doped SnTe samples (Figure 4(a)). This is because the of all the doped samples is similar (~1.9 W m^-1 K^-1). Because all the doped samples are doped with the same doping content (3.0 at.%), the of the doped samples being comparable is reasonable. Despite its much higher of 231 cm² s^-1 V^-1 than the Ga-doped sample (142 cm² s^-1 V^-1), the B-factor of pristine SnTe (0.047) becomes comparable to that of the Ga-doped sample (0.049). This is because the pristine SnTe has a much higher thermal conductivity (3.1 W m^-1 K^-1) than the Ga-doped sample (1.9 W m^-1 K^-1). The B-factor is a measure of intrinsic band and phonon transport properties, and it is related to the theoretical maximum zT. However, the B-factor is not directly proportional to the theoretical maximum zT. The Bi-doped sample has the highest B-factor (0.095) among the pristine and doped SnTe samples at 300 K. This suggests that the zT of the Bi-doped sample can be the highest at 300 K when the is optimized.

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(c)

(d)

Figure 5(c) shows the experimental zT of the pristine and doped SnTe at 300 K [14]. Here, the zT of the In-doped sample (zT~0.10) is slightly higher than the Bi-doped sample (0.09). However, based on the B-factor calculated in Figure 5(a), the zT of the Bi-doped sample can be higher than that of the In-doped if its is tuned optimally. Figure 5(d) shows the -dependent zT of the pristine and doped SnTe samples at 300 K. The experimental results are plotted in filled symbols, and the calculated results (SPB model) are presented in solid lines. Most of the pristine and doped samples have their theoretical maximum zTs at higher than 10¹⁹ cm^-3. We have approximated the contributions from the LVB, HVB, and the resonant state of electron-doped SnTe samples with the SPB model and predicted their maximum zT and optimum . While the SPB model may not produce a perfectly accurate prediction, the error is likely to be small. Thus, the -depedent zT predicted by the SPB model can serve as a useful guide for experimentalists in tuning . Additionally, the prediction error in the SPB model decreases as the increases (when the Fermi level is deep within the HVB, where the discrepancy between the SPB model and the TB model is minimized). In other words, except for Sb-doped SnTe, which has the lowest , other electron-doped SnTe samples with high would have their -dependent zT accurately predicted by the SPB model. However, the zT of the Sb-doped sample is estimated to be at its maximum near 10¹⁸ cm^-3. This has to do with the of the Sb-doped sample being at the LVB with a negligible effect from the HVB (Figure 1(c)). On the contrary, the theoretical zT of the Bi-doped sample peaks at , even though its is at the LVB. In this case, the HVB maximum adjacent to the of the Bi-doped sample is responsible for the difference. Based on the SPB model, the theoretical maximum zT of the Bi-doped sample (zT~0.311) is lower than that of the In-doped sample (0.312). This is as predicted from the calculated B-factor in Figure 5(a). The significantly higher theoretical maximum zTs of the In-doped and Bi-doped samples than those of other samples can be attributed to the formation of the resonant states. For the Bi-doped sample, its zT can be improved by ~244% (0.09⟶0.31) when the is tuned to .

Figures 6(a) and 6(b) show the calculated B-factor and measured as a function of temperature. The B-factor in Figure 6(a) is calculated by using the temperature-dependent (Figure 4(b)) and (Figure 6(b)). The B-factor of all the samples increases with an increasing temperature. This is because the decreases as the temperature increases and the B-factor scales with (Equation (6)). From the fact that the B-factor increases with temperature, we can predict that the theoretical maximum zT of the pristine and doped SnTe will increase with temperature. The temperature-dependent B-factor in Figure 6(a) has two interesting observations. Firstly, the samples with their at the HVB (nonshaded region in Figure 1(c)) or LVB (shaded in blue in Figure 1(c)) with negligible contribution from other bands have a low B-factor at low temperatures (pristine; Ga-doped and Sb-doped samples). However, the rate of B-factor increase with temperature is high for these samples. So the B-factors of the Sb-doped (0.46) and Ga-doped samples (0.41) become the highest and the second highest among all the samples at 823 K. Secondly, the In-doped and Bi-doped samples whose is at the region where both HVB and LVB contribute (shaded in light blue in Figure 1(c)) have high B-factor until ~700 K. Their high B-factor near 300 K is due to the resonant state formation. The rate of B-factor increase with temperature for the Bi-doped sample suddenly increases near 550 K while that of the In-doped sample decreases. As the temperature increases, contribution from another band adjacent in energy increases. For the Bi-doped sample, the contribution from the HVB increases with temperature, while for the In-doped sample, the contribution from the LVB increases with temperature. Therefore, it is advantageous to place the at LVB at 300 K for the improved thermoelectric performance at higher temperatures. The highest experimental zT of ~0.9 at 823 K for the Bi-doped sample can be attributed to the following factors: the resonant state formation at 300 K, the position of the in the region where both the LVB and HVB contribute to transport, and the fact that is actually on the LVB and not the HVB. From the highest B-factor of the Sb-doped sample at 823 K, we expect the highest theoretical maximum zT in the Sb-doped sample. However, even when the is optimally tuned, the average zT of the Bi-doped sample will be higher than that of the Sb-doped sample as the B-factor of the Sb-doped sample is only higher than that of the Bi-doped sample at temperatures higher than 823 K.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figures 6(c)–6(g) show the experimental (in symbols) and calculated zT (in lines) as a function of and temperature. A comparison of the experimental and calculated optimum for the maximum zT at a specific temperature can be used as a guide for further zT improvement. The theoretical maximum zT of the Sb-doped sample (zT~1.13) at 823 K is the highest followed by the Ga-doped (1.06) and Bi-doped samples (1.03). However, even when the is optimized, the average zT, which is more important for the efficiency of the thermoelectric device, will be the highest in the Bi-doped sample because of the steady increase in the theoretical maximum zT over the whole temperature range. Another 14% increase in the zT of the Bi-doped sample (0.9⟶1.03) is possible at 823 K if the is tuned. Additionally, a more than 61% increase in the zT is also possible in the Sb-doped sample (0.7⟶1.13). For the highest zT at high temperature, the should be placed at one of the valence bands where the contribution from the other band is not strong at 300 K. To achieve the highest zT at high temperature, the should be placed in a valence band where the contribution from the other band is not strong at 300 K. However, to achieve the highest average zT, placing at the LVB near the HVB maximum is advantageous.

4. Conclusion

In summary, we evaluated the effect of electron doping in p-type SnTe using Ga, In, Bi, and Sb as dopants (Sn_0.97M_0.03Te (, In, Bi, Sb)) using the two-band (TB) and single parabolic band (SPB) models. We found that In-doped and Bi-doped samples formed the resonant states near 300 K because their experimental data significantly deviated from the Hall carrier concentration ()-dependent Seebeck coefficient () calculated using the TB model (one light valence band (LVB) and a heavy valence band (HVB) participating in electronic transport). We also identified where the Fermi levels () of the doped samples are positioned. The of the Sb-doped sample was at the LVB while that of the Ga-doped sample was at the HVB further apart from the maximum of the other band. However, the of both In-doped and Bi-doped samples were in the region where both LVB and HVB are active. We then estimated the band parameters such as weighted mobility () and B-factor of the doped samples using the SPB model as a function of temperature. According to the temperature-dependent B-factor, which is related to the theoretical maximum zT, the In-doped and Bi-doped samples with the resonant state formation near 300 K and their η positioned where both the LVB and HVB participate in transport exhibited high average zT than other samples. Especially, the Bi-doped sample whose was at the LVB had the higher theoretical maximum zT (zT~1.03) than the In-doped sample (0.90) at 823 K. Choosing a dopant that forms a resonant state at 300 K and engineering the position of so that it is at the LVB but still close to the HVB maximum are important strategies for achieving a high average zT in SnTe.

Data Availability

Data are available upon request.

Conflicts of Interest

There is no conflict of interest.

Authors’ Contributions

Hyunjin Park, Kyu Hyoung Lee, and Minsu Heo contributed equally to this work.

Acknowledgments

This research was supported by the Nano·Material Technology Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2022M3H4A1A04076667). This work was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00212959).

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Copyright © 2023 Hyunjin Park et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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