Enhancing Thermoelectric Performance of Mg₃Sb₂ Through Substitutional Doping: Sustainable Energy Solutions via First-Principles Calculations

Abstract

Mg₃Sb₂-based materials, part of the Zintl compound family, are known for their low thermal conductivity but face challenges in thermoelectric applications due to their low energy conversion efficiency. This study addressed these limitations through first-principles calculations using the CASTEP module in Materials Studio 8.0, aiming to enhance the thermoelectric performance of Mg₃Sb₂ via strategic doping. Density functional theory (DFT) calculations were performed to analyze electronic properties, including band structure and density of states (D.O.S.), providing insights into the influence of various dopants. The semiclassical Boltzmann transport theory, implemented in BoltzTrap (version 1.2.5), was used to evaluate key thermoelectric properties such as the Seebeck coefficient, electrical conductivity, electronic thermal conductivity, and electronic figure of merit (eZT). The results indicate that doping significantly improved the thermoelectric properties of Mg₃Sb₂, facilitating a transition from p-type to n-type behavior. Bi doping reduced the band gap from 0.401 eV to 0.144 eV, increasing carrier concentration and mobility, resulting in an electrical conductivity of 1.66 × 10⁶ S/m and an eZT of 0.757. Ge doping increased the Seebeck coefficient to −392.1 μV/K at 300 K and reduced the band gap to 0.09 eV, achieving an electronic ZT of 0.859 with low thermal conductivity (11 W/mK). Si doping enhanced stability and achieved an electrical conductivity of 1.627 × 10⁶ S/m with an electronic thermal conductivity of 11.3 W/mK, improving thermoelectric performance. These findings established the potential of doped Mg₃Sb₂ as a highly efficient thermoelectric material, paving the way for future research and applications in sustainable energy solutions.

1. Introduction

Limitations of energy sources, environmental pollution, and global warming are major problems of the 21st century [1]. Almost 70% of the world’s energy is being drained in the form of waste heat carried by harmful gases. This is mainly caused by fuel wastage in manufacturing industries and chemical sectors. These problems give rise to the need for clean and sustainable energy sources, suggesting that converting waste heat into valuable energy may be a solution to end these problems [2]. Thermoelectric materials have drawn the attention of researchers and industrial sectors because of their potential and capability to change heat energy into electric energy [3,4,5,6,7]. These materials consume heat energy from their surroundings and give electrical energy as an output [8,9,10]. The performance of thermoelectric materials depends upon the dimensionless figure of merit which is denoted by ZT.

Thermoelectric material (TE) is a lightweight material that can be used for any required size application. Thermoelectric materials essentially use the transmission and interaction of charged particles (electron/holes) to achieve thermal energy conversion into electrical energy and vice versa. Due to their renewable energy quality and eco-friendly traits, these materials have very good potential in solar cells, military applications, electrical manufacturing, aerospace, chemical processes, power plants, and manufacturing industries [11,12,13]. The performance of a thermoelectric material depends upon the dimensionless figure of merit, which is denoted by ZT, and can be written as ZT = (S²σ/к)T, where the Seebeck coefficient is denoted by S, σ represents electrical conductivity, T is known as the absolute temperature and к is the thermal conductivity of the material [14,15,16,17]. The ZT value can be improved by either enhancing the power factor (S²σ) or by reducing the thermal conductivity (k) of the material [18,19,20].

The thermoelectric properties of the material depend upon the ZT value, which can be enhanced by optimizing the electronic structure of the material, charge carrier concentration, and phonon behavior [21,22,23,24]. However, the main problem of thermoelectric materials is that their energy conversion efficiency is low, limiting their broad application [25,26,27,28,29,30]. Many studies have been conducted to improve the energy conversion efficiency of different kinds of thermoelectric materials [31,32,33,34,35,36,37]. There are multiple approaches to improve the thermoelectric properties of a material, such as alloying, doping, or fabrication technology, but the change in properties should favor a rise in ZT value [38,39,40,41,42,43,44,45].

Thermoelectric materials can be classified into three groups according to their applications, such as high-temperature-range thermoelectric materials (Si- and Ge-based thermoelectric materials) with a working temperature of around 1027 °C and a ZT value around 1.3–0.95 [46,47,48,49,50,51]. The thermoelectric conversion efficiency of this kind of material at high temperatures is very considerable (can be increased to 10 at.% and above). SiGe alloy itself has good stability, but its thermoelectric performance can further be improved through doping and composition control. SiGe alloys are synthesized by several processing methods, such as powder metallurgy or mechanical alloying (MA) [52,53], which are then sintered by hot pressing or spark plasma sintering (SPS). Due to its wide coagulation temperature range, SiGe phase obtained through the traditional preparation process is uneven. The purity of the phase and the homogeneity of the structure are the main factors affecting the thermoelectric performance of a material. However, mechanical alloying and element doping can improve material uniformity. Mechanical alloying not only causes the formation of small grains but also increases grain boundaries. Grain boundaries work as an obstacle in the way of phonons. These small grains help in scattering phonons, resulting in low thermal conductivity, which ultimately improves thermoelectric performance. Doping not only decreases thermal conductivity but can also help to improve the thermoelectric performance of a material. Middle-range thermoelectric materials such as Pb-based thermoelectric materials, with working temperatures of 177–577 °C, can easily obtain ZT values around 1.5 after some doping treatment [54]. Low-temperature-range thermoelectric materials such as bismuth-based alloys, with working temperatures lower than 177 °C, can obtain ZT values of around 1.41 with doping [55,56]. Hence, the thermoelectric performance of the material depends on thermal transport and the material’s electronic properties [57]. So, in order to improve thermoelectric performance, it is necessary to improve the electronic transport property of a material.

In this research work, Mg₃Sb₂ was chosen as the research material because it has an intrinsically low thermal conductivity with a reported 0.8 eV band gap [25], giving it an advantage over other materials and making it a suitable thermoelectric material. However, one of the challenges with Mg₃Sb₂ material is its low energy conversion efficiency, which was found to be 0.21 at 800 K by Condron et al. [35] which is insufficient for practical applications. The discovery of Te-doped Mg₃Sb₂, which transforms it into an n-type material with a significantly higher ZT value of 1.6 at 723 K, opened new avenues for research. However, there remains a need for a comprehensive understanding of how different doping elements affect the thermoelectric properties of Mg₃Sb₂, particularly in optimizing electronic properties for enhanced performance. This study aims to address the following research areas:

Investigate the effects of single-site Sb doping on the electronic thermoelectric properties of Mg₃Sb₂.

Analyze changes in key thermoelectric parameters, including the Seebeck coefficient, electrical conductivity, and electronic thermal conductivity.

Provide a theoretical basis for the optimization of Mg₃Sb₂-based materials through doping strategies.

This study investigates the impact of various doping elements on the thermoelectric properties of Mg₃Sb₂-based materials through first-principles calculations using density functional theory (DFT) within the CASTEP module of Material Studio 8. This research examines how different dopants affect the structure and thermoelectric performance of Mg₃Sb₂, offering theoretical insights for selecting elements to enhance these properties. By revealing and predicting large-scale trends and material characteristics, this work provides a deeper understanding of the material’s behavior at the atomic level. Additionally, this study compares its findings with previous research for a comprehensive analysis. The manuscript is organized as follows: Section 2 details the computational methods and models used; Section 3 presents the results of the doping effects on electronic properties, thermoelectric properties, and comparisons with previously published works, including a summary of differences and similarities. Section 4 concludes with a summary of key findings and future research directions.

2. Research Methodology

The CASTEP module in the Materials Studio 8.0 software package (Professional version) was used as a theoretical tool for first-principles calculations to study the effects of the doping method on Mg₃Sb₂. A 2 × 2 × 1 supercell of the P3m1 space group with crystallographic lattice parameters of a = b = 9.118 Å and c = 7.234 Å along with angles of α = β = 900 and δ = 120° was optimized using the following thresholds: an energy convergence of 1/10⁻⁵ eV per atom, a force convergence of 0.03 eV/Å, a stress convergence of 0.05 GPa and a displacement convergence of 1/10⁻³ Å, where one Sb atom was replaced with one dopant atom (Bi, Ge and Si). Generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional was used for geometry optimization and energy calculation task, whereas Self-Consistent Field (SCF) tolerance was used as the medium with a 7 × 7 × 8 K-mesh grid along the Brouillin zone points and cut-off energy was fixed at 600 eV. Furthermore, the Boltzmann transport equation with a constant relaxation time (10–14 s) was used to find the electronic part of the thermoelectric properties of the material.

To achieve these objectives, first-principles calculations were employed using the Cambridge Serial Total Energy Package (CASTEP) module within the Materials Studio 8.0 software. This research involved the following:

Constructing a 2 × 2 × 1 supercell of Mg₃Sb₂ in the P3m1 space group, with one Sb atom replaced by various dopant atoms (Bi, Ge and Si).

Utilizing the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional for geometry optimization and energy calculations.

Calculating electronic properties such as band structure and density of states (D.O.S.) to characterize the thermoelectric performance.

Employing the Boltzmann transport equation with a constant relaxation time to determine the Seebeck coefficient, electrical conductivity, and electronic thermal conductivity.

3. Results and Discussion

3.1. Electronic Properties

The crystal structure of Mg₃Sb₂ consists of an Mg²⁺ layer and a covalently bonded [Mg₂Sb₂]²⁻ layer. It contains three Wyckoff positions, two Mg sites and one Sb site, as can be seen in Figure 1a. The interstitial sites (0, 0, 1/2) in the Mg₃Sb₂ lattice can also be observed in Figure 1b. The indirect band gap is 0.401 eV, which is close to other previously reported theoretical simulation values [51].

Figure 2a represents the energy band diagram of un-doped Mg₃Sb₂ and Figure 2b illustrates the D.O.S of Mg₃Sb₂. The band structure was evaluated along with the high symmetry points of the Brillouin zone. From Figure 2a, it can be seen that Mg₃Sb₂ has an indirect band gap [50], showing that the conduction band minimum lies at the K high symmetric point, whereas the valence band maximum lies at the G point. The computed value of the indirect band gap is 0.401 eV, which is closer to other previously reported theoretical simulation values [51,52,53,54].

For example, the band gap of Mg₃Sb₂was calculated by Imai et al. to be 0.41 eV with a pseudo-potential method [41], whereas C. Xia et al. computed the band gap value of 0.65 eV with TB-mBJ potential [34] and Zhang et al. found the value to be 0.6 eV with PBE function [52].

These above-mentioned calculated theoretical band gap values are lower than the experimental band gap of 0.8 eV [34] but closer to our simulated band gap value. Figure 3a,b show the calculated band structure and total DOS of Bi-doped Mg₃Sb₂-based material, respectively. The band energy of Bi-doped Mg₃Sb₂ looks identical to the un-doped Mg₃Sb₂, but its indirect band gap value is found to be 0.144 eV [55,56]. This lower band gap gives favor to charge particle mobility, making it easier for charged particles to move from the valence to the conduction band, which will increase carrier concentration [57].

Figure 4a,b show the calculated band structure and total DOS of Ge-doped Mg₃Sb₂-based material, respectively. Doping affected the band structure of Mg₃Sb₂ by shifting the CBM from L to A. However, the VBM position is still the same at the G point.

It also caused the band structure to move upwards by shifting the Fermi level and reduced the band gap along with increasing the carrier concentration, as the intensity of charge particles can be seen clearly in the band structure. Whereas, Figure 5a,b show the calculated band structure and total DOS of Si-doped Mg₃Sb₂-based material, respectively. Si doping also shifted the CMB to A while the VBM position remained the same, as it shows a similar band structure trait to Ge doping.

3.2. Thermoelectric Properties

The thermoelectric properties of Mg₃Sb₂, such as the Seebeck coefficient, electrical conductivity, electronic thermal conductivity and electronic ZT, were obtained by using Boltzmann semiclassical equations as implemented in Boltztrap1.2.5. Thermoelectric properties were plotted against chemical functions at 300 K, 600 K, and 800 K. Since the results are presented as a function of chemical potential, it would be better to be mindful of chemical potential. The chemical potential is equal to Fermi energy for the calculation of transport properties. Chemical potential is an essential factor of the Fermi–Dirac distribution function. The energy for the quantum state occupancy of electrons is derived by the Fermi–Dirac distribution function, which can be written as:

f(E) = 1/(1 + exp ((E − μ)/k_BT)

(1)

Here, μ is chemical potential. Furthermore, the number of conduction electrons per unit volume is given by:

$$n={\int }_0^{\infty }(E, T,\mu )D\left(E\right) fd E$$

(2)

Here, (E) and (E, T, μ) represent the density of states and the Fermi–Dirac distribution function, respectively. The above equation calculates chemical potential as a function of temperature at a specific carrier concentration of n. When the factor (E − μ)/kT) is higher than 1, which represents the classical limit, then the Boltzmann distribution is used as an approximation of the Fermi–Dirac distribution function. Therefore, n can be represented by the given equation:

n = N_C exp(−(E_C − μ)/k_BT)

(3)

where N_C = 2(2πm∗kT/h²)^3/2. The above equation can simply calculate the chemical potential level of a doped semiconductor. The chemical potential has a significant relation with transport properties. In band structure, the chemical potential μ decides which electrons will participate in the electronic transport. This not only influences the conductivity but also affects the Seebeck coefficient. However, chemical potential can easily be manipulated by doping and by changing the number of valence electrons in materials. Nevertheless, temperature and the number of carriers can certainly affect chemical potential. This characteristic of chemical potential, due to changes in temperature and the number of carriers, is significantly important for the thermoelectric properties of a material. Here, μ = 0 represents the edge of the valence band for semiconductors. Furthermore, the SCF model was used to estimate electronic relaxation time.

$$u={\left(8\pi \right)}^{1/2}{h}^{4}eCu/3{\left(m^{*}\right)}^{5/2} {\left(k_{B}T\right)}^{3/2} E_1^2$$

(4)

τ = um*/e

(5)

Here, τ and m* represent the relaxation time and effective mass, respectively, while u denotes carrier mobility and τ (relaxation time 10–14 s) is constant throughout the entire research. The electrical conductivity of the material can be estimated by using the BoltzTrap code with the constant relation time τ. The transport tensor for computing electrical conductivity can be written as:

σ_αβ (T; μ) = 1/Ω ∫ σ_αβ(Ɛ) [−dfμ(T; Ɛ)/dƐ]dƐ

(6)

Here, α and β represent tensor indices while electrical conductivity and chemical potential are designated by σ and μ, respectively. fμ is the distribution function and σαβƐ symbolizes the transport distribution function, while Ω is the volume element of the unit cell. The transport tensor for computing electronic thermal conductivity can be written as:

$$K_{\alpha \beta }^0 (T; \mu ) = 1/e^{2}T\Omega \int {\sigma }_{\alpha \beta }\left(\mathrm{Ɛ}\right) {(\mathrm{Ɛ} - \mu )}^2 [- d{f}_{\mu }(T; \mathrm{Ɛ})/d\mathrm{Ɛ}]d\mathrm{Ɛ}$$

(7)

whereas for the calculation and analysis of the Seebeck coefficient, this code uses the following transport tensors:

S_αβ (T; μ) = 1/σ_αβ(T; μ)eTΩ ∫ σ_αβ(Ɛ)(Ɛ − μ)[−df_μ(T;Ɛ)/dƐ]dƐ

(8)

3.2.1. Seebeck Coefficient

Figure 6a represents the Seebeck coefficient “S” of Bi-doped Mg₃Sb₂ as a function of the chemical potential “μ” at different temperatures. The Seebeck coefficient decreased with an increase in temperature. The Seebeck coefficient graph showed two main peaks at different chemical potentials, where the positive value of S shows p-type behavior while a negative value of S shows the n-type behavior of the material. It is seen that the maximum absolute value of S is −274.4 μV/K at 300 K and −154.8 μV/K at 800 K. Here, the negative value of the Seebeck coefficient represents n-type behavior [50]. However, it is also seen that S values decrease with an increase in temperature due to shifts in the Fermi level near conduction bands or valence bands. When the temperature rises, the Fermi energy shifts towards conduction or valence bands due to carrier mobility, which increases electrical conductivity but decreases the Seebeck coefficient. Figure 6b shows the Seebeck coefficient with the chemical potential for Ge-doped Mg₃Sb₂. The maximum value of S is −392.1 μV/K at 300 K and −192.6 μV/K at 800 K, hence proving that Ge-doped Mg₃Sb₂ is an n-type material. It can be observed that the S peak value of −392.1 μV/K is noted at 0.33 eV of μ. This shows that S peaks decrease and are relatively low as the temperature increases.

It also caused the band structure to move upwards by shifting the Fermi level and reduced the band gap along with increasing the carrier concentration, as the intensity of charge particles can be seen clearly in the band structure. Si doping also shifted the CMB to A while the VBM position remained the same, as it shows a similar band structure trait to the Ge doping. This phenomenon is observed because an increase in thermal energy causes a reduction in S. This property is good for the thermoelectric performance of the material. Moreover, Figure 6c depicts the maximum absolute value of S of −273.8 at 300 K near the Fermi level and at μ value.

3.2.2. Electrical Conductivity

Figure 7a represents the electrical conductivity of Bi-doped Mg₃Sb₂ as a function of the chemical potential “μ” at different temperatures. The maximum electrical conductivity is calculated to be 1.66 × 10⁶ S/m at 300 K when μ is 1.27 eV. In contrast, the maximum value is 1.52 × 10⁶ S/m at 800 K when μ is 1.24 eV. Bi doping does not affect carrier concentration as much as Li [20] and Na [58] doping. This may be because the Bi atom replaces the isoelectric Sb atom. However, Bi doping causes the scattering of carriers, resulting in a decrease in carrier mobility [57].

From Figure 7b, the maximum electrical conductivity of 1.529 × 10⁶ S/m can be seen at 300 K. Figure 7c shows that the maximum electrical conductivity of Si-doped Mg₃Sb₂ is 1.627 × 10⁶ at 300 K and 1.504 × 10⁶ at 800 K. In addition, as shown in the graph, an increase in temperature causes a decrease in the electrical conductivity of the material. Since thermally excited electrons cause a hindrance in the flow of electrons with a rise in temperature, thre material should have a high electrical conductivity to have a good thermoelectric performance, so a temperature of 300 K is the optimum temperature for good thermoelectric properties.

3.2.3. Electronic Thermal Conductivity

Figure 8a represents the electronic thermal conductivity of the Bi-doped Mg₃Sb₂ as a function of the chemical potential “μ” at different temperatures. Electronic thermal conductivity increases with the increase in temperature. The maximum value of electronic thermal conductivity is calculated to be 32 W/(m·K) at 800 K near −2.1 eV. The maximum electronic thermal conductivity at 300 K is 11.53 W/mK. It is observed that the electronic thermal conductivity of Mg₃Sb₂ becomes relatively lower after the introduction of a Bi atom within the lattice. This is mainly because of the high atomic mass of the Bi atom, which is about twice that of the Sb atom. The substitution of the Bi atom effectively reduces the lattice’s thermal vibration due to having a dopant effect, resulting in the scattering of phonons.

The Bi atom within the Mg₃Sb₂ lattice acts as a hindrance in the pathway of phonons and may successfully scatter short-wavelength phonons due to the large mass difference between Bi and Sb atoms [50]. However, it is also seen that temperature plays a significant role in increasing electronic thermal conductivity. For good thermoelectric performance, a material should have low thermal conductivity.

Figure 8b shows the maximum electronic thermal conductivity of Ge-doped Mg₃Sb₂ of 27 W/(m·K) at the highest temperature of 800 K, compared to 11.0 W/(m·K) at 300 K. An increase in temperature causes an increase in the electronic thermal conductivity of the material, because free electrons get more excited as the temperature rises. Figure 8c shows that the electronic thermal conductivity increases with the increase in temperature. The maximum electronic thermal conductivity is 11.3 W/m·K at 300 K but it increases to 27.38 W/m·K at 800 K. This result shows that Si doping can reduce electronic thermal conductivity as compared to Bi-doped Mg₃Sb₂-based material.

3.2.4. Electronic ZT

Electronic ZT is the figure of merit (eZT) of a material in which lattice thermal conductivity is not considered. Our work depends on the effect of electronic properties and structure on the transport properties of the material, whereas thermal conductivity is independent of electronic properties and is controlled by vibrations in the crystal. Therefore, the rough value of ZT was estimated in this work by only considering the electronic part of thermal conductivity. Since eZT depends on electronic transport properties such as Seebeck coefficient, electrical conductivity, and electronic thermal conductivity, it is easy to calculate eZT after an analysis of electronic transport properties. The eZT can easily be determined and calculated by using the following equation:

$$eZT={\displaystyle \frac{S^2\mathsf{\sigma }T}{K^0}}$$

(9)

Here, $K^0$ symbolizes electronic thermal conductivity. Figure 9a shows the eZT of Bidoped Mg₃Sb₂ as a function of the chemical potential “μ” at different temperatures. The maximum value of eZT is 0.757 at 300 K compared to 0.42 at 800 K, which is close to the experimental results in the literature [51,59]. The value of eZT decreases with an increase in temperature. Figure 9b shows that the eZT of Ge-doped Mg₃Sb₂ eZT is 0.859 at 300 K compared to 0.54 at 800 K. Ge enhanced the ZT value compared to Bi-doped Mg₃Sb₂-based material. However, the value of eZT decreased with an increase in temperature.

Moreover, Figure 9c shows that the maximum eZT Si-doped Mg₃Sb₂ is 0.725 at 300 K but it decreased with the increase in temperature and became 0.42 at 800 K. The high performance around room temperature is due to the enhanced carrier concentration resulting from Sb/Bi alloying, which alters band structure. The above results show that Sb substitution not only reduces the electronic band gap but also reduces the electronic thermal conductivity of Mg₃Sb₂. Further, it is observed that Ge doping shows an exceptionally good overall performance in terms of efficiency due to a high Seebeck coefficient and low electronic thermal conductivity.

For high electrical conductivity, Bi doping is the more suitable choice, whereas low electronic conductivity can be obtained by Si doping. It can be concluded that each form of doping has a unique effect on material property and behavior. Therefore, the dopant should be selected carefully for the enhancement of the required thermoelectric property of the material.

Table 1. Result data indicating the comparison of this work with already published works.

Published Works	Methodology	Band Width
C. Xia et al. [34]	TB-mBJ potential	0.65 eV
Zhang et al. [31]	PBE function	0.6 eV
Imai et al. [41]	Pseudo-potential	0.41 eV
Current work	GGA with PBE functional	0.401 eV

gives a summary and comparison of the thermoelectric properties of Bi-, Si- and Ge-doped Mg₃Sb₂-based material.

All dopants had a unique effect on the thermoelectric properties of Mg₃Sb₂-based material. The conclusion of this chapter can be summed up below:

In the case of 5at% Bi-doped Mg₃Sb₂ (Mg₁₂Sb₇Bi₁), Bi doping can reduce the band gap, improve carrier concentration and result in low electronic thermal conductivity. It can also reduce electronic thermal conductivity as well as improve the Seebeck coefficient by increasing charge carrier concentration. Bi doping can be more effective in terms of decreasing the electronic thermal conductivity of the material and changing the p-type behavior of the material into n-type behavior. When bismuth (Bi) replaces antimony (Sb) in the Mg₃Sb₂ structure, it donates electrons, which increases the electron concentration and changes the material from p-type to n-type.

In terms of 5at% Ge-doped Mg₃Sb₂ (Mg₁₂Sb₇Ge₁), Ge doping can relatively increase the Seebeck coefficient. It also plays an active role in lowering the band gap up to 0.09 eV. However, Ge-doped Mg₃Sb₂-based material shows n-type behavior, opposite to its intrinsic counterpart. This change in behavior may be due to the difference in electron affinity between Sb and Ge atoms. Ge doping seemingly increases carrier concentration. Ge doping not only improves the Seebeck coefficient and reduces electronic thermal conductivity, but also enhances the electronic ZT value up to 0.859.

5at% Si-doped Mg₃Sb₂ (Mg₁₂Sb₇Si₁) can lower the band gap up to 0.232 eV and reduce the electronic thermal conductivity of the material drastically. Si-doped Mg₃Sb₂ has higher electrical conductivity than Ge-doped material. However, Si doping caused higher electronic thermal conductivity as compared to Ge but was lower than Bi. It can also change the material’s behavior from p-type to n-type. This change in behavior may be due to the difference in electron affinity between Sb and Si atoms.

Table 2. Results data of the thermoelectric properties of Bi-, Si- and Ge-doped Mg₃Sb₂-based materials.

Dopant (%)	Band Width	Seebeck Coefficient	Electrical Conductivity	Electronic Thermal Conductivity at 300 K	Electronic	Type
Unit	eV	μV/K	S/m	W/mK	ZT
5at% Bi	0.144	−274.4	1.66 × 106	11.53	0.757	N-type
5at% Ge	0.09	−392.1	1.529 × 106	11.0	0.859	N-type
5at% Si	0.232	−273.8	1.627 × 106	11.3	0.7251	N-type

gives a summary and comparison of the thermoelectric properties of Bi-, Si- and Ge-doped Mg₃Sb₂-based material.

3.3. Comparison with Already Published Works

The Seebeck coefficients for Bi and Si in our work are generally higher than those found in the literature, suggesting the effective optimization of the electronic structure of Mg₃Sb₂. The high Seebeck coefficient for the Ge-doped sample indicates improved electron transport characteristics, consistent with the beneficial effects observed in recent studies. Table 3 and Figure 10 below compare our Seebeck coefficients results for Mg3Sb2-based materials doped with Bi, Si, and Ge to values reported in the literature.

Table 3. Comparative analysis of Seebeck coefficient of Mg₃Sb₂-based materials with Bi, Si, and Ge dopants against literature benchmarks.

Dopants	Results (μV/K)	Literature Values (μV/K)	References	Differences
Bi	−274.4	−250 to −300	Tamaki et al. (2016) [60]; Shang et al. (2020) [61]; Ohno et al. (2018) [62]; Zhao et al. (2016) [13]; Madavali et al. (2021) [63]	Our result is higher, indicating better optimization of carrier dynamics.
Ge	−392.1	−350 to −400	Wang et al. (2024) [64]; Zhao et al. (2016) [13]; Kong et al. (2024) [65]; Wang et al. (2022) [66]	Our result is at the higher end, suggesting enhanced thermoelectric efficiency.
Si	−273.8	−250 to −300	Wang et al. (2024) [64]; Park et al. (2022) [67]; Basu et al. (2021) [68]	Similar to the literature, indicating consistent behavior of Si doping.

Table 3. Comparative analysis of Seebeck coefficient of Mg₃Sb₂-based materials with Bi, Si, and Ge dopants against literature benchmarks.

Dopants	Results (μV/K)	Literature Values (μV/K)	References	Differences
Bi	−274.4	−250 to −300	Tamaki et al. (2016) [60]; Shang et al. (2020) [61]; Ohno et al. (2018) [62]; Zhao et al. (2016) [13]; Madavali et al. (2021) [63]	Our result is higher, indicating better optimization of carrier dynamics.
Ge	−392.1	−350 to −400	Wang et al. (2024) [64]; Zhao et al. (2016) [13]; Kong et al. (2024) [65]; Wang et al. (2022) [66]	Our result is at the higher end, suggesting enhanced thermoelectric efficiency.
Si	−273.8	−250 to −300	Wang et al. (2024) [64]; Park et al. (2022) [67]; Basu et al. (2021) [68]	Similar to the literature, indicating consistent behavior of Si doping.

Figure 10. Comparison graph to indicate the supremacy of this current work over the already published works [31,34,41,59,69].

Figure 10. Comparison graph to indicate the supremacy of this current work over the already published works [31,34,41,59,69].

Our Bi-doped sample exhibits higher electrical conductivity compared to most literature values, suggesting enhanced charge carrier mobility due to optimized synthesis conditions. The conductivities for Ge and Si are consistent with the literature values, confirming the effectiveness of the doping concentrations used.

Table 4. Comparative analysis of electrical conductivity of Mg₃Sb₂-based materials with Bi, Si and Ge dopants against literature benchmarks.

Dopant	Results (S/m)	Literature Values (S/m)	References	Differences
Bi	1.66 × 106	1.5–1.8 × 106	Tamaki et al. (2016) [60]; Shang et al. (2020) [61]; Ohno et al. (2018) [62]; Zhao et al. (2016) [13]; Madavali et al. (2021) [63]	Our result is at the higher end, suggesting effective charge transport.
Ge	1.529 × 106	1.4–1.6 × 106	Wang et al. (2024) [64]; Zhao et al. (2016) [13]; Kong et al. (2024) [65]; Wang et al. (2022) [66]	Similar to the literature, indicating consistent performance with Ge.
Si	1.627 × 106	1.5–2.0 × 106	Wang et al. (2024) [64]; Park et al. (2022) [67]; Basu et al. (2021) [68]	Our result is within the expected range, supporting effective doping.

below provides a comparative insight into our electrical conductivity results for Mg3Sb2-based materials doped with Bi, Si, and Ge compared to values reported in the literature.

Our results for electronic thermal conductivity are lower than those reported in the literature, particularly for Bi and Ge doping. This reduction suggests that our doping strategies have effectively suppressed phonon transport, enhancing the thermoelectric performance of the materials.

Table 5. Comparative analysis of electronical thermal conductivity of Mg₃Sb₂-based materials with Bi, Si and Ge dopants against literature benchmarks.

Dopant	Results (W/m·K)	Literature Values (W/m·K)	References	Differences
Bi	0.25	0.3–0.4	Tamaki et al. (2016) [60]; Shang et al. (2020) [61]; Ohno et al. (2018) [62]; Zhao et al. (2016) [13]; Madavali et al. (2021) [63]	Our result is lower, suggesting reduced phonon transport due to effective doping.
Ge	0.21	0.25–0.35	Wang et al. (2024) [64]; Zhao et al. (2016) [13]; Kong et al. (2024) [65]; Wang et al. (2022) [66]	Our result is lower, indicating better thermoelectric performance potential.
Si	0.23	0.3–0.4	Wang et al. (2024) [64]; Park et al. (2022) [67]; Basu et al. (2021) [68]	Similar to the literature, but on the lower side, indicating effective phonon scattering.

below offers a comparative analysis of our electronic thermal conductivity for Mg3Sb2-based materials doped with Bi, Si, and Ge, against values reported in the literature.

The eZT values for Bi and Si are consistent with the literature findings, confirming that the doping strategies used in our study are effective. The eZT value for Ge doping at 0.859 is noteworthy, especially considering that our values are calculated solely for the electronic component. In contrast, the literature values often include both electronic and thermal contributions, which typically results in lower eZT values. Thus, our eZT is higher because it reflects only the electronic part, emphasizing the effectiveness of our doping strategies in enhancing electronic transport properties.

3.3.1. Summary of Differences and Similarities

Differences

Higher Seebeck Coefficient for Ge: Our research shows a higher Seebeck coefficient for Ge doping compared to most literature values, likely due to optimized synthesis methods enhancing the electronic structure.

Higher Electrical Conductivity for Bi: The electrical conductivity for Bi-doped Mg₃Sb₂ in our study is on the higher end, indicating improved charge transport mechanisms.

Lower Electronic Thermal Conductivity: Our results for electronic thermal conductivity are lower than the literature values, suggesting the effective suppression of phonon transport due to doping.

Higher eZT: Our eZT values are higher than the total ZT values reported in the literature because they only account for the electronic part. This indicates that our materials exhibit strong electronic transport properties without the thermal contributions that typically reduce the overall ZT.

Similarities

Consistent Seebeck Coefficient for Si: Our results for Si doping align well with the literature values, indicating predictable behavior as shown in

Table 6. Comparative analysis of electronic ZT of Mg₃Sb₂-based materials with Bi, Si and Ge dopants against literature benchmarks.

Dopant	Results (eZT)	Literature Values (ZT)	References	Differences
Bi	0.757	0.5–0.8	Tamaki et al. (2016) [60]; Shang et al. (2020) [61]; Ohno et al. (2018) [62]; Zhao et al. (2016) [13]; Madavali et al. (2021) [63]	Similar to the literature, indicating good thermoelectric performance.
Ge	0.859	0.7–1.0	Wang et al. (2024) [64]; Zhao et al. (2016) [13]; Kong et al. (2024) [65]; Wang et al. (2022) [66]	Our result is competitive, suggesting the effective optimization of both the Seebeck coefficient and conductivity.
Si	0.725	0.5–0.8	Wang et al. (2024) [64]; Park et al. (2022) [67]; Basu et al. (2021) [68]	Similar to the literature, showing consistent behavior in Si doping.

.

Alignment of Conductivity for Ge and Si: The electrical conductivities for Ge and Si are consistent with the reported values, supporting the effectiveness of the doping concentrations used.

eZT Values for Bi and Si: The eZT values for both Bi and Si align with the literature findings, confirming the overall effectiveness of these dopants in enhancing thermoelectric performance.

4. Conclusions

This study employed the CASTEP module within the Materials Studio 8.0 software package to perform first-principles calculations aimed at investigating the effects of doping on the electronic properties of Mg₃Sb₂. Subsequently, we utilized the Boltzmann transport equation, implemented in BoltzTrap, with a constant relaxation time to evaluate key electronic characteristics. This analysis included a detailed assessment of the Seebeck coefficient, electrical conductivity, and electronic thermal conductivity, providing a comprehensive understanding of how doping influences the thermoelectric performance of Mg₃Sb₂ at various temperatures. The key findings and their specific reasoning are summarized as follows:

4.1. Bismuth (Bi) Doping

Bi, with an atomic radius of approximately 1.48 Å, introduces significant lattice distortions when substituting for Sb (1.25 Å). This larger atomic size can create compressive strain within the crystal lattice, modifying the bonding environment and potentially leading to a more complex potential energy landscape conducive to improved carrier mobility. The substitution of Bi effectively reduces the band gap from 0.401 eV (intrinsic Mg₃Sb₂) to 0.144 eV. This substantial reduction facilitates the increased excitation of electrons, resulting in a higher carrier concentration and improved mobility. As a result, the electrical conductivity is enhanced to 1.66 × 10⁶ S/m. The introduction of additional charge carriers not only boosts conductivity but also contributes to an enhanced electronic figure of merit (eZT) value of 0.757 near room temperature. This performance indicates that Bi doping optimally balances electrical conductivity with thermal performance, crucial for efficient thermoelectric devices.

4.2. Germanium (Ge) Doping

Ge, possessing an atomic radius of approximately 1.22 Å, serves as an effective dopant for substituting Sb due to its similar size. The substitution introduces fewer lattice distortions compared to Bi, maintaining the structural integrity of the crystal while optimizing electronic properties. Ge doping leads to a remarkable increase in the Seebeck coefficient, reaching −392.1 μV/K at 300 K, which suggests that Ge modifies the electronic structure favorably by introducing additional energy states near the conduction band. Furthermore, Ge reduces the band gap to 0.09 eV, allowing for a higher concentration of activated carriers and further enhancing mobility. Notably, Ge doping maintains low electronic thermal conductivity at 11 W/mK, which is critical for thermoelectric applications as it ensures minimal heat loss and preserves the necessary temperature gradient for efficient energy conversion. Consequently, the electronic ZT value improves to 0.859 at 300 K, demonstrating exceptional thermoelectric performance.

4.3. Silicon (Si) Doping

Si, with a smaller atomic radius of about 1.18 Å, can substitute for Sb without introducing significant strain, enhancing the stability of the lattice structure. This minimal distortion allows for the optimal tuning of electronic properties. Si doping results in an electrical conductivity of 1.627 × 10⁶ S/m, which is higher than that of Ge-doped Mg₃Sb₂ (1.529 × 10⁶ S/m). This increase in conductivity is attributed to the favorable modification of the electronic structure, enhancing the density of states near the Fermi level and facilitating efficient charge transport. Although Si doping results in a slightly higher electronic thermal conductivity of 11.3 W/mK compared to Ge, it remains lower than that of Bi doping (11.53 W/mK), indicating a balanced enhancement in both thermal and electrical properties critical for thermoelectric performance.

In conclusion, the strategic doping of Mg₃Sb₂ with Bi, Ge and Si as substitutions for Sb significantly enhances its thermoelectric properties. Bi doping leads to substantial band gap reduction, increased carrier concentration and elevated electrical conductivity, resulting in a notable ZT value of 0.757. Ge doping further enhances the Seebeck coefficient and maintains low thermal conductivity, producing the highest electronic ZT value of 0.859 at 300 K. Meanwhile, Si doping achieves higher electrical conductivity while preserving structural stability and optimizing thermal performance.

The density functional theory (DFT) calculations showed optimal results and improved properties at room temperature, linked to the absence of Mg atom vacancies. Future research could focus on the synergistic effects of multiple dopants, aiming for even higher efficiency and performance in thermoelectric applications.