Band convergence and defect engineering synergistically revamping the carrier-phonon dynamics in Mg_3-xZn_xSb₂ solid solutions: an experimental and theoretical insights

Materials preparation of p-type Mg_1.8Zn_1.2Sb₂ and Ag_xMg_1.8-xZn_1.2Sb₂ pellets

The p-type Mg_1.8Zn_1.2Sb₂ and Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03, and 0.05) samples were prepared by a spark plasma sintering technique. Magnesium (Mg, 99.5%, metal turnings), zinc (Zn, 99.5%, metal powder), silver (Ag, 98.8%, metal powder), and antimony (Sb, 99.5%, metal powder) were weighed according to the stoichiometric ratio of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03 and 0.05). All elements of Mg, Sb, Ag, and Zn metal powders were grounded using mortar and pestle. The grounded powders were loaded into a cylindrical graphite die (inner diameter 13 mm) and subjected to sintering via spark plasma sintering technique at 873 K under the pressure of 40 MPa for 5 min holding time and cooling rate (50 ºC/min) to obtain dense disk-shaped pellets [Supplementary Scheme 1A and B].

Structural characterizations and TE transport property measurements

X-ray diffraction (XRD) was used for the phase composition of each sample characterized by the help of a PANalytical multipurpose diffractometer under CuKα radiation (λ = 1.5406 Å). High-resolution transmission electron microscopy (HR-TEM; JEOL JEM-2100 Plus with an operating voltage of 200 kV). The microstructure and compositional analysis of the samples was performed using high-resolution scanning electron microscopy (HR-SEM; Thermoscentific ApreoS) equipped with energy dispersive spectroscopy (EDS). Thermal conductivity (κ) was calculated by κ = ρDC_p, where ρ is the sample density (estimated based on the Archimedes method), D is the thermal diffusivity (measured by a laser flash apparatus (LFA 467 HT, NETZSCH)), and C_p is the specific heat (determined by differential scanning calorimetry thermal analyzer). The σ and S were simultaneously measured by a commercial ADVANCE RIKO ZEM-3 system under a helium atmosphere. The room temperature Hall measurement was measured by using ECOPIA HMS 5300. The sample’s carrier concentration (n) and mobility (μ) were calculated using the four-probe Van der Pauw method under a magnetic field.

Computational methods

All the periodic density functional calculations were computed by using the Vienna Ab initio Simulation Package (VASP), with the Projected Augmented Wave (PAW) method applied to account for the electron-ion interaction terms. The VASP package utilizes the Perdew-Burke Ernzerhof (PBE) exchange-correlation function in combination with PAW potentials. The convergence criteria of 0.03 eV/Å in force and 10^-5 eV in energy were adopted to optimize the supercells. The Monkhorst-Pack K-point mesh calculated the relaxation and self-consistent computations of the crystal structures^[41]. In general, the exchange-correlation energy comprises the exchange energy (from the Pauli exclusion principle) and the correlation energy (from Coulomb interaction effects), which are treated in terms of different approximations, expressed as E_total[ρ] = T_s[ρ] + E_ext[ρ] + E_H[ρ] + E_xc[ρ], where T_s[ρ] is kinetic energy (KE) of non-interacting electrons, E_ext[ρ] is external potential energy (PE), E_H[ρ] is classical Hartree energy, and E_xc[ρ] is exchange-correlation energy. In our work, the Monkhorst-Pack grid generates a uniform K-point grid of 3 × 3 × 2 for geometry optimization and 6 × 6 × 4 for electronic structure simulations. In addition, the basis set is a critical factor that directly determines the accuracy and efficiency of electronic structure prediction and is mainly based on the (I) plane wave basis set and (II) pseudopotentials. The formation energy (∆E_form) of the Zn and Ag at Mg₃Sb₂ is calculated by ∆E_form (ZnAg-Mg₂Sb₂) = E(_ZnAg-Mg2Sb2) - (E_Zn-Mg3Sb2 + E_Ag), where Zn-Ag-Mg₃Sb₂, E(_ZnAg-Mg3Sb2), and E_Ag denote the total energy of the Zn-Ag doped Mg₃Sb₂ system, the total energy of Zn@Mg₃Sb₂ without Ag dopant, and the total energies of Ag in the bulk phase, respectively. A plane-wave basis with a cutoff of 900 eV was used to calculate electron densities and wave functions. Electronic structure calculations, i.e., total density of states (TDOS), projected density of states (PDOS), band structure, and electron density analysis, were then simulated using the most stable geometries. All optimized structures presented in this study were taken from the VESTA packages.

Schematic illustration of the crystal structures and XRD of Ag_xMg_1.8-xZn_1.2Sb₂

The Mg_1.8Zn_1.2Sb₂ samples expose the hexagonal crystal structures with a space group of p$$ \overline{3} $$m1. Zn is the suitable acceptor for boosting the density of holes, as well as enhancing the electronic transport properties, resulting in a narrow bandgap and manipulating the band structure in the Mg₃Sb₂ system. There are two distinct crystallographic sites of Mg atoms available in Mg₃Sb₂: Mg²⁺/(Mg1-octahedral) and [Mg₂Sb₂]^2-/(Mg2-tetrahedral). The Zn atoms are potential substitution for tetrahedral sites and proximity to neighbouring Sb atoms assists in an overall decrease in coulombic repulsion. Specifically, zinc atoms have lower electronegativity (χ = 1.65), shorter atomic radius (142 p.m.), and a smaller positive charge than magnesium atoms. The electronegativity difference (Δχ) between the substitution (Mg) and foreign atom (Zn) was a more important parameter to enhance the transport properties and produced internal defects. In general, the Δχ was calculated by |Δχ| = |χHost - χdopant|. The calculated Δχ between Zn at the Mg site is 0.29 and at the Sb site is 0.12 [Supplementary Table 1]. In addition, the smaller positive charges of Zn atoms are substituted at Mg sites to enhance the transport properties of Mg₃Sb₂. The differences in electronegativity and atomic mass with host and dopant sites are shown in Supplementary Figure 1A and B. Additionally, this contains multiple states compared to Mg₃Sb₂ and is situated below the Fermi level (E_F) which manipulates its band structure through a band convergence strategy. For further enhancing the TE transport properties, the heavy and aliovalent element Ag was introduced at Mg sites of Mg_1.8Zn_1.2Sb₂.

Figure 1A and B illustrates the hexagonal crystal structure of undoped and Ag-substituted Mg_1.8Zn_1.2Sb₂. The XRD image of Ag-substituted Mg_1.8Zn_1.2Sb₂ (x = 0, 0.01, 0.03, and 0.05) samples were shown in Figure 1C. The introduction of Ag in Mg_1.8Zn_1.2Sb₂ shows a lower angle shift in the XRD result, where a secondary Sb phase was observed at 42.4^o (JCPDS NO: 96-901-3011). As can be seen, the occurrence of Mg vacancies (V_Mg^2-) is due to the vaporization of Mg during heat treatment. The presence of secondary Sb can confirm the negative formation energy of low energy acceptor defect as V_Mg^2- at the lattice which act as acceptor defects (Mg_Mg = V_Mg^2- + 2h). These defects promote a greater number of holes or absorb excess electrons, and lead to the p-type pinning behavior through the introduction of energy states that are close to the VB^[42]. Here, the p-type materials pin the E_F, enabling the system in an equilibrium state via intrinsic acceptor defects and the material retains its p-type nature and reduces E_F (If n-type materials pin the E_F due to domination of intrinsic donor defects, the material remains n-type).

Figure 1. (A and B) crystal structures; (C and D) XRD and magnified XRD pattern of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03 and 0.05) samples. XRD: X-ray diffraction.

The ability to function as p-type TE semiconductors without any impurity substitution is caused by the formation of an adequate number of robust cation site vacancies^[43,44]. Figure 1D represents the expanded portion of XRD analysis, which confirms the lower angle shift leads to the expansion and tensile strain. In the Mg₃Sb₂ system, Zn has a large solubility limit (70%-90%) because of its (134 pm) minimal ionic radius compared with Mg (145 pm)^[45,46]. The increment of lattice parameters confirms the replacement of Mg²⁺ ions with Zn²⁺ and Ag⁺ ions at the Mg sites, which decreases the crystallite size and indicates the contribution of strain in the lattice.

However, the observed trend with XRD peak shifts upon increasing Ag content in Mg_1.8Zn_1.2Sb₂ can be attributed to competing effects of lattice expansion and contraction. Initially, Ag-substituted samples show a lower angle shift for x = 0.01 and 0.03 samples, due to a higher atomic radius of Ag (165 pm) than that of Mg (145 pm)^[45]. It confirms the lattice expansion leading to tensile strain, resulting in increasing lattice parameters (a = 4.484 Å to 4.508 Å and c = 7.232 Å to 7.259 Å) and unit cell volume (125.97 ų to 127.08 ų)[Supplementary Figure 1C and D]. After the Ag concentration exceeds x > 0.03, the peak shifted to a higher angle, due to the solubility limit, local structure distortions, and phase segregation, respectively^[47]. Beyond the solubility limit of Ag in Mg_1.8Zn_1.2Sb₂ system, the excess Ag occupies interstitial sites that expand the strain in the lattice. Also, this obtained trend of results were considered with the previously reported literatures^[48,49]. Furthermore, the microstructural properties and elemental distribution were analyzed by HR-SEM and EDS mapping [Supplementary Figure 2A-H]. Further, the EDS spectrum analysis and elemental compositions confirm the presence of Ag in the as-prepared compounds which is shown in Supplementary Figure 3A-D and Supplementary Table 2.

Carrier transport properties of Ag_xMg_1.8-xZn_1.2Sb₂

Figure 2A represents the schematic representation of the band diagram for undoped Mg₃Sb₂, Mg_1.8Zn_1.2Sb₂, and Ag-doped Mg_1.8Zn_1.2Sb₂ systems, which confirms the role of the heavy element Ag in the lighter Mg site led to E_F towards the VB and enhanced hole concentration, which reduces the band gap. Figure 2B indicates the VB convergence before and after heavy element substitution in the Mg_1.8Zn_1.2Sb₂ matrix. This scheme represents that, Ag substitution leads to narrowed band gap and enhance the valley degeneracy, respectively. Figure 2C shows the room temperature Hall carrier density of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0 - 0.05. The n and μ could be explained by the relation,

Figure 2. (A and B) schematic representation of band diagram (C) Hall carrier density, and (D) mobility of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03, and 0.05) samples.

Where e is the charge of an electron, and R_H is Hall coefficient. Here, positive n values indicate the domination of holes in the prepared system, which confirms that all the prepared samples exhibit a p-type semiconductor behavior at room temperature. Mg_1.8Zn_1.2Sb₂ shows a low hole n of 6.40 × 10¹⁸ cm^-3; the introduction of Ag leads to an increase in the number of holes. This enhanced n confirms that Ag⁺ substitution replaces the [Mg₂Sb₂]^2- layers, which is an octahedral site of Mg₃Sb₂ with polar covalent bonding. Particularly, the Ag_0.05Mg_1.75Zn_1.2Sb₂ sample obtains the highest n of 8.19 × 10¹⁹ cm^-3. In comparison with previous reports, Ag substitution at Mg sites increases the n, due to the convergence of bands producing excess holes near the VB and attaining a narrow band gap, which is confirmed by band structure calculation^[50]. Therefore, these results confirm that the foreign element (Ag) substitution enhances the domination of the ionized impurity scattering mechanism at 303 K^[51]. The introduction of Ag leads to improving the density of states (DOS) with band convergence, which increases the electrical transport properties via manipulation of the hole’s concentration and enhances the material’s TE performance. Figure 2D shows the Hall μ. In general, the μ can be expressed as Equation 3, where m* is an effective mass of the electrons, and $$ \langle\tau\rangle $$ is the average relaxation time. Here, relaxation time is a combination of m*, temperature, and energy carriers, which is expressed as τ $$ \propto $$E^rT^s (m^*)^t. The highest μ of 145 cm²/Vs is obtained for the Ag_0.05Mg_1.75Zn_1.2Sb₂ at room temperature, which is 56% higher than the undoped sample. Specifically, the high μ weakens the polar covalent bonding and the carriers move faster than regular, due to the reduction of carrier scattering^[52]. It may arise via secondary impurities (Sb) and intrinsic Mg vacancies, which place interstitial sites between Mg and Sb. This weak polar covalent bonding in the [Mg₂Sb₂]^-2 layer helps to improve the S²σ. Thus, the increasing trend of μ at 303 K confirms the Zn, and substitution of Ag confirms the domination of carrier scattering mechanism from a mixed scattering of ionized impurity and acoustic phonon scattering. Also, Ag at Mg sites softens the chemical bonds of the Mg_1.8Zn_1.2Sb₂ system^[34].

The σ was measured for Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0 - 0.05) samples [Figure 3A]. In general, the σ of the material was estimated by σ = neμ, where e indicates electronic charge. For Mg_1.8Zn_1.2Sb₂, the σ increases from 77 S/cm to 157 S/cm at 303 K - 753 K, and the increasing trend of σ indicates the typical non-degenerate semiconductor behavior^[30]. Further, the σ of Ag-substituted Mg_1.8Zn_1.2Sb₂ increases, which confirming the aliovalent substitution enhances the hole concentration. Simultaneously, isovalent Zn increases μ via ionized impurity scattering^[40]. The σ values were improved after substituting Ag at Mg sites from 38 to 225 S/cm at room temperature, which was 193% greater than undoped Mg_1.8Zn_1.2Sb₂. To be specific, the Ag_0.05Mg_1.75Zn_1.2Sb₂ sample shows a drastic improvement in σ of 225 S/cm at 303 K due to a significant improvement of n and μ. However, when the temperature increased, the downward trend of σ indicates the typical heavily doped degenerate semiconductor behavior of Ag-substituted samples. Thus, the Ag atom could migrate to the Mg_1.8Zn_1.2Sb₂ lattice and take the sites of Mg atoms and interstitial sites, which induces additional domination of minority carriers and secondary Sb at higher temperatures. This typical phenomenon resulting a reduction of σ via low hole concentration (i.e., hole trapping)^[53,54].

Figure 3. Temperature dependence of (A) electrical conductivity; (B) Seebeck coefficient; (C) Fermi energy; and (D) S vs. σ for Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03 and 0.05) samples.

The obtained n and the measured S values are positive, indicating that holes are the majority carrier, and electrons are minority carriers in undoped and Ag-substituted Mg_1.8Zn_1.2Sb₂ samples, which are displayed in Figure 3B. In general, the S can be evaluated by the semiclassical Mott-Jones formula^[55]

where k_B is Boltzmann constant, h is Plank constant, $$ m_{d}^{*} $$ is DOS carrier m*, and e is an elemental charge. The S is directly proportional to m* and indirectly proportional to the n. Here, compared with the undoped sample, the Ag_0.05Mg_1.75Zn_1.2Sb₂ sample attains a higher n of 8.19 × 10¹⁹ cm^-3. $$ m_{b}^{*}=N_{V}^{2 / 3} m_{b}^{*} $$, where N_V is the valley degeneracy and $$ m_{d}^{*} $$ is the average m* of a single valley (N_V = 1), which can be used to calculate $$ m_{d}^{*} $$. The S of Mg_1.8Zn_1.2Sb₂ is 234.7 µV/K at 303 K, and 103 µV/K at 753 K. After Ag substitution at Mg_1.8Zn_1.2Sb₂, the S trend was directly opposite to the undoped sample, which increases with temperature, indicating non-degenerate semiconductor behavior. Specifically, a convex-like trend of S indicates the influence of minority charge carriers, secondary Sb, and interstitial Ag atoms, which play a major role at high temperatures^[56]. Therefore, orbital overlap and the strengthening of chemical bond covalency and smaller electronegativity of Ag atom results in an enhanced DOS m* via VB convergence, which helps to enhance the S at high temperatures. This finding emphasizes that, the bipolar diffusion effect is possible when VB electrons are thermally excited to the conduction band (CB) even at relatively lower temperatures^[57].

Consequently, the peak of S shifts to lower temperatures while the σ continuously increases with Ag concentration. Also, the Ag_0.01Mg_1.79Zn_1.2Sb₂ shows a drastic improvement in the S of 283 µV/K at 553 K and 257 µV/K at 753 K; it might be due to the enhancement of VB convergence, which exhibits a narrow band gap and lowers the energy separation. This increases the S at high temperatures by enhancing the electronic DOS and inhibiting bipolar conduction^[58]. Therefore, the introduction of VB convergence is a significant advantage in improving the S and TE performance over a broad temperature range. This indicates that, the interstitial silver (Ag) atoms, minority charge carriers, and secondary Sb have significant impacts at high temperatures. Ag-substituted samples have a S approximately 150% higher than undoped Mg_1.8Zn_1.2Sb₂, due to hole excitation into heavier bands, enhancing VB convergence^[59]. Thus, this finding suggests that VB convergence adds a substantial benefit to the S and TE performance across a wide temperature range.

The contribution of impurity or secondary phase was confirmed by the reduced E_F ($$ \eta=E_{F} / k_{B} T $$) using the S (σ × exp$$ {\left[\frac{|S|}{k_{B} / e}-2\right]} $$) and σ ($$ \ln \left[exp \left(\frac{\sigma}{\sigma E_{0}}\right)-1\right] $$) values, as shown in Figure 3C. At room temperature, positive values indicated an increased hole concentration. At higher temperatures, negative values suggested the influence of minority charge carriers, highlighting the significant role of secondary Sb in contributing to bipolar conductivity. Figure 3D represents the σ and S vs. Ag content at 303 K, which confirms the typical semiconductor behavior of the as-prepared samples^[60].

Density functional theory representation of Ag_xMg_1.8-xZn_1.2Sb₂

The periodic density functional calculations were performed to gain insight on the TE properties of Mg₂ZnSb₂ compounds. The optimized geometries and formation energy of undoped, Zn, and Zn-Ag substituted Mg₃Sb₂ systems are shown in Supplementary Figure 4A-E. Here, the computed band diagrams, PDOS plots, and electron density difference of Mg₂ZnSb₂ and Ag-Mg₂ZnSb₂ systems are provided in Figure 4. The VB at the high symmetry point Γ and the CB at A indicate the indirect nature of the band explained in Figure 4A^[61]. The VB and CB of Mg₂ZnSb₂ and Ag- Mg₂ZnSb₂ systems are composed of the p-orbital of Sb and s-orbital of Mg, respectively. The obtained temperature-dependent S trend has been significantly changed after Ag doping due to the following phenomenon: VB convergence, which is highly consistent with the previous reports^[62,63]. It is a potential strategy, where the multiple VB maxima (VBM) (i.e., light and heavy VBs) of the material exist at nearly the same energy level which enhances the DOS near the E_F, resulting in enhanced S. On the other hand, the convergence of multi-band improves the mobility of holes and increases the σ. Therefore, the significant enhancement of S and σ enhances the overall power factor (PF) and TE performance. This VB convergence can be introduced via doping impurities, alloying, or strain engineering, respectively^[64]. The Mg₃Sb₂ intrinsically exhibits a wide band gap of ~0.6-0.8 eV and lower valley degeneracy (N_v = 1), which renders the TE performance^[65]. In this present investigation, the S of undoped Mg_1.8Zn_1.2Sb₂ sample decreased with temperature indicating the semiconductor behavior, whereas Ag-doped samples show an increasing trend with temperature due to the following reasons. Here, the substitution of isovalent Zn atom replaces ~60% Mg atoms and only settles in the tetrahedral [Mg₂Sb₂]^2- sites, resulting in reducing the band gap from 0.28 eV (Mg₃Sb₂) to narrow the band gap of 0.16 eV (Mg_1.8Zn_1.2Sb₂) with increased valley degeneracy [Supplementary Figure 5A]^[63]. The DOS and PDOS for the Mg₃Sb₂ system explain the presence of Mg-s, Mg-p, Sb-s, and Sb-p orbitals, as shown in Supplementary Figure 5B-D. Substituting the heavy element Ag⁺ at anionic Mg2 site of Mg_1.8Zn_1.2Sb₂ system introduces additional acceptor levels and exhibits the convergence of heavy VB Γ and lighter VB A near E_F of the band structure. The VB modification enhances the DOS and improves the TE performance of the Ag-substituted Mg_1.8Zn_1.2Sb₂ system^[45]. In addition, this phenomenon helps increase the DOS m* of the system which significantly improves the S of 103 µV/K to 257 µV/K with a temperature of Ag-substituted Mg₂ZnSb₂ system.

Figure 4. (A and B) The computed band structures; (C and D) PDOS; and (C1 and D1) extracted PDOS (-4 to 4); and (E and F) electron density difference of Mg₂ZnSb₂ and Ag- Mg₂ZnSb₂ systems. The yellow and red area denotes the loss and gain of electrons. The electron densities of the doped material were taken from -4.12 × 10^-4 e/Å. PDOS: Projected density of states.

In this case, the VBM was located at the Γ and A points displayed in Figure 4B, indicating a significant improvement in the electrical transport behavior of Mg₃Sb₂ after the substitution of Zn, and Ag into the Mg2 sites. It also enriches the number of bands in the DOS, which lowers the thermal excitation of the electrons and enables more opportunities for electron transitions, thereby enhancing the electrical transport characteristics in the low-temperature domain and agreeing with experimental results. In addition, Supplementary Figure 6A-D shows the optimized geometries of Zn at Mg and Sb sites, and Zn-Ag at Mg and Sb sites of Mg₃Sb₂.The bonding behavior between the foreign dopants (i.e., Zn and Ag) in the Mg₃Sb₂ has been revealed from the PDOS analysis. Figure 4C and D shows the VB and CB regions of the energy diagram, which were composed of different orbitals such as Mg-s, Mg-p, Sb-s, and Sb-p, Zn-p, Zn-d, Ag-p, and Ag-d, respectively. Figure 4C1 and D1 represents the extracted PDOS confirming the presence of s, p, d orbitals of Mg, Sb, Zn, and Ag. Additionally, the Mg-p states hybridized with the Sb-s from -0.5 eV to -2.2 eV in the VB region, which clarifies the bonding between Mg and Sb at Mg₃Sb₂. Supplementary Figure 7A-C shows the TDOS of Mg₂ZnSb₂ and Ag-Mg₂ZnSb₂. The simulation results agreed with the experimental results, which effectively increased the electron DOS near the E_F. The incorporation of foreign elements into Mg₃Sb₂ is expected to enhance the active sites within the system, thus leading to an improvement in TE performance. Moreover, the electron density difference analysis was used to probe the atomic bonding behavior of Ag with Mg₂ZnSb₂, as presented in Figure 4E-F1. Furthermore, the electron density difference in Figure 4E1 shows that enhanced electron density is redistributed between Ag/Zn and Mg/Sb atoms of Mg₃Sb₂, which illustrates that the strong chemical bonding (i.e., Zn-Sb/Mg and Zn-Ag-Sb/Mg) between the dopant elements and the matrix compound, which stabilizes the system. The electron density contour plot in [Figure 4E1 and F1] shows the spherical electron localization between (i) Ag and Sb, (ii) Ag and Mg, and (ii) Zn and Sb elements due to their p and d lone pair interference. These results highlight that the synergistic effect between the foreign and the host material, i.e., Zn-Sb, Zn-Mg, Ag-Sb, and Ag-Mg, could significantly tune the electronic structure and thereby improve the TE properties. Also, it is interesting to note that the substitution of Ag into the Mg₂ZnSb₂ enhances the electron density. The substitution of Ag into the Mg₂ZnSb₂ system increases the active sites of the matrix.

Figure 4E1 shows the Zn-d states, which strongly overlap with Sb-s and Mg-s from -0.2 eV to -1.3 eV of the VB region, implying that Zn-Sb and Zn-Mg bonding structures of Mg₂ZnSb₂. It is also observed that, the Ag-d orbitals hybridize with the Sb-p and Mg-p near the Fermi and VB regions from -0.21 eV to -0.95 eV, illustrating the Ag-Sb and Ag-Mg bonding states of Ag at Mg₂ZnSb₂. Furthermore, the Sb-s and Mg-p orbitals hybridize with the Zn-d orbitals from the VB region from ~ -0.51 eV to -1.25 eV indicating the formation of a covalent bond between Zn, Sb, and Mg. Figure 4F1 shows the bottom of the near Fermi region, which is contributed by the intermixing of p-d orbitals of Ag-Sb hybridized bands. This difference in the electronic DOS pattern is responsible for the enhanced m* due to the incorporation of Ag in the Mg₂ZnSb₂ structure.

Figure 5A represents the PF of undoped and Ag-doped Mg_1.8Zn_1.2Sb₂ samples, which is calculated by PF = S²σ. Regardless of the temperature, compared with Mg_1.8Zn_1.2Sb₂, Ag-substituted samples show the increasing trend of PF due to the combination of large S and σ, respectively. When increasing the concentration of Ag (x = 0, 0.01, 0.03, and 0.05), the PF value of 167 µW/mK², 299 µW/mK², 456 µW/mK², and 467 µW/mK² was maximized at 753 K. Specifically, the highest PF observed for Ag_0.05Mg_1.75Zn_1.2Sb₂ of 467 µW/mK² at 753 K, which is ~180% larger than the Mg_1.8Zn_1.2Sb₂ (167 µW/mK²). This result confirms that Ag-substitution at Mg sites of Mg_1.8Zn_1.2Sb₂ yields the best control over σ and S to optimize the improved PF at higher temperatures.

Figure 5. (A) Temperature-dependent power factor (B) weighted mobility of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03, and 0.05).

Figure 5B represents the μ_W of the as-prepared samples. In addition, the bipolar conduction plays a pivoted role in TE performance at high temperatures, further it can be realized by the μ_W. The μ_W was calculated using measured electrical resistivity and S values by the Drude-Sommerfield free electron model and analyzed phonon scattering mechanisms, which are calculated by^[66],

where k_B/e = 86.3 µV/K, T is an absolute temperature, and ρ is electrical resistivity. This is also a widely used parameter to confirm the scattering mechanisms such as grain boundary scattering at room temperature, ionized impurity scattering (T^3/2) at room - 450 K, acoustic phonon scattering (T^-3/2) occurring above 450 K, and so on. Here, compared with undoped Mg_1.8Zn_1.2Sb₂ the μ_W of all the Ag-substituted samples decreases with increasing temperature, which confirms the domination of acoustic phonon scattering (T^-0.65 to T^-1.13). Thus, this result helps to correlate with the presence of defects at a concerning temperature. To be specific, the Mg_1.8Zn_1.2Sb₂ and Ag_0.03Mg_1.77Zn_1.2Sb₂ system has strong anharmonicity which is a significant contributor to its low thermal conductivity below ~1 W/mK. The κ_Lof the samples decreases from ~0.73 W/mK to ~0.56 W/mK for Mg_1.8Zn_1.2Sb₂ and Ag_0.03Mg_1.77Zn_1.2Sb₂ at 753 K, which is ~23% lower than undoped sample. Typically, the independent κ_L served as the key parameter for attaining elevated TE performance through the incorporation of defects in the lattice, offering a novel degree of freedom for modifying the physical parameters to improve the thermal transport properties.

This result confirms that the Ag-substituted Mg_1.8Zn_1.2Sb₂ samples have a lower μ_W value than the undoped Mg_1.8Zn_1.2Sb₂ at room temperature [i.e., 51 to 22.5 cm²/Vs]. This is due to the domination of ambient temperature defects such as grain boundaries and ionized impurities. While increasing the measuring temperature, all Ag-substituted samples show higher μ_W values because of microstructural defects such as secondary phases, dislocations, and point defects, respectively. This trend also confirms the domination of acoustic phonon scattering after 453 K. In this, the carrier scattering behavior of the as-prepared samples was confirmed, and the decreasing trend of μ_W due to the domination of phonon scattering. At 753 K, the μ_W was decreased for all the Ag-substituted samples.

Phonon transport properties and defect analysis

The total thermal conductivity (κ_T) was calculated by κ_T = DρC_p, where D is thermal diffusivity, C_p is specific heat capacity, and ρ is the density of as-prepared pellets. Figure 6A displays the temperature-dependent κ_T of undoped and Ag-substituted Mg_1.8Zn_1.2Sb₂ solid solutions, which obtained the κ_T is less than 1 W/mK. Further, the point defect was introduced via substituting Ag at Mg sites, which helps to reduce the κ_T. However, κ_T increases with Ag content at 303 K, which indicates the domination of electronic thermal conductivity (κ_e) (κ_T = κ_L + κ_e). In contrast, the κ_T of Ag-substituted Mg_1.8Zn_1.2Sb₂ samples was reduced at higher temperatures. The lowest κ_T of 0.55 W/mK at 303 K and 0.96 W/mK at 753 K for undoped Mg_1.8Zn_1.2Sb₂. The Ag_0.03Mg_1.77Zn_1.2Sb₂ sample exhibits the κ_T of 0.693 W/mK at 753 K due to increased carrier heat conduction in the lattice, which is 28% lower than the undoped sample. Figure 6B is the κ_e calculated via the Wiedemann-Franz relationship, κ_e = LσT, where L is the Lorenz number, calculated by a single parabolic band (SPB) model with acoustic phonon scattering assumption. The κ_e increases significantly with the Ag content in the entire temperature range, being consistent with the change of σ. At room temperature, the κ_e of 0.03 W/mK for undoped and 0.12 W/mK for Ag_0.05Mg_1.75Zn_1.2Sb₂ sample, respectively. The undoped Mg_1.8Zn_1.2Sb₂ obtained the κ_e of 0.22 W/mK at 753 K, which is higher than that of Ag-substituted samples.

Figure 6. (A-C) Temperature dependence of thermal transport characterization for Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03 and 0.05), (D) comparison of κ_L with reported work^[68,69].

Figure 6C illustrates the κ_L part of as prepared undoped and Ag-substituted Mg_1.8Zn_1.2Sb₂ samples. The κ_L was calculated by subtraction of κ_e from the measured κ_T (κ_L = κ_T - κ_e). In this present investigation, defect engineering was introduced with the substitution of heavy element Ag at lighter Mg sites of Mg_1.8Zn_1.2Sb₂. Based on the formation energy, Ag atoms can replace the tetrahedral Mg2/[Mg₂Sb₂]^2- sites and some excess Ag atoms occupy the interstitial sites of Mg_1.8Zn_1.2Sb₂ system^[45]. Here, the larger atomic radius of Ag is ~165 pm which is lower than Mg (~145 pm), creates mass contraction and enhances the phonon scattering via point defects. This also causes some lattice distortion via the introduction of various wavelengths (short to long) of lattice defects. The charge imbalance between Mg²⁺ and Ag⁺ creates the formation of Mg vacancies (produces excess holes), and occupancy of interstitial sites. Therefore, heavier Ag⁺ substitution at Mg²⁺ sites creates V_Mg and introduces different types of defects such as point defects, dislocation, stacking faults, grain boundaries, and strain in the lattice via atomic dislocations. These defects, identified using HR-TEM, Inverse Fast Fourier Transform (IFFT), and subsequent strain analysis, strengthen the phonon transport properties^[45,67].

The above-mentioned defects promotes the scattering of different wavelength phonons, thus lowering the thermal conductivity of Ag-substituted samples at elevated temperatures. Figure 6D shows the comparison of κ_L with previous reports and present work, which confirms that Ag substitution in the Mg_1.8Zn_1.2Sb₂ solid solution significantly strengthens the various frequency phonon scattering and reduces the κ_L of 0.56 W/mK at 753 K for the Ag_0.03Mg_1.77Zn_1.2Sb₂ sample^[68,69]. Here, it should be noted that the Ag substitution at the Mg site emphasizes the significance of 0D point defects in the lattice via atomic dislocations, which strengthen the phonon transport properties. The high-frequency phonons were scattered via point defects and the relaxation time is $$ \tau_{P D}^{-1} $$~ω⁴. In this instance, it can be observed that Mg vacancy and interstitial Ag atoms exhibit a significant capacity to diminish κ_L at similar point defect ratios.

However, the role of the bipolar effect is dominant via ionized impurity scattering which enhances the κ_e at 303 K. Due to the contribution rate of phonon-electron scattering than phonon-phonon scattering, the enhancement of κ_L was obtained at 303 K. After increasing the temperature, the Ag_xMg_1.8-xZn_1.2Sb₂ samples exhibit lower κ_L, which is also the contribution of κ_e.

Further, the microstructure details of Ag_0.03Mg_1.77Zn_1.2Sb₂ sample were investigated by high-resolution TEM characteristics, shown in Figure 7A-F. The intrinsic defects such as grain boundaries (long), dislocations (mid), and stacking faults (short) have been identified using microstructural high-resolution TEM images with Fast Fourier Transform (FFT) and IFFT patterns, respectively. In addition, the lattice strain has been explored via geometric phase analysis (GPA) with HR-TEM results. The mass contraction between Mg and Ag leads to lattice distortion and produces point defects. The existence of stacking faults, which change the atomic arrangement via the scattering of short wavelength phonons and serve as phonon scattering centers via mass and strain fluctuations, was discovered by utilizing the IFFT pattern from the HR-TEM image [Supplementary Table 3]. The heavy element Ag substitution at lighter Mg sites of Mg_1.8Zn_1.2Sb₂ solid solution creates point defects in the form of dislocations and scatter mid-wavelength phonons, respectively^[70]. Further, the different orientations of grains and grain boundaries scatter long wavelength phonons near room temperature. The d-spacing values of 0.26 nm (110), 0.27 nm (110), and 0.35 nm (011) indicate that each grain is crystallized with random orientation shown in Figure 7A. These grain boundaries help scatter the long-wavelength phonons, which are determined by Klemens and Matthiessen’s expression $$ \frac{1}{\tau_{G B}}=\frac{v}{L_{G}} $$, where ν is the Poisson ratio and L_G is grain size. Figure 7B represents the IFFT images of dislocation clusters and perfect crystal lattice, which are extracted from Figure 7C. This indicates that Ag substitution introduces different wavelengths of defects and plays a major role in Mg_1.8 Zn_1.2Sb₂ solid solution. In Figure 7D and E, the obtained dislocation indicates the heavy Ag substitution in the undoped system leads to changes in the arrangement of atoms, which complicates the atom’s interaction and strengthens the phonon scattering [Scheme 1A and B]^[10]. Furthermore, the presence of stacking fault represents Ag substitution interrupting the order of stacked planes in a matrix [Figure 7F]. The obtained defects, such as stacking faults and dislocations, scatter the mid-wavelength phonons and reduce κ_L of the as-prepared samples^[71,72]. Supplementary Figure 8 shows the calculated microstrain vs. κ_L, which confirms the heavy element Ag at anionic Mg2 sites enhances the microstrain significantly scatters the phonons, and reduces the κ_L of as-prepared samples.

Figure 7. Microstructural and GPA strain analysis of Ag_0.03Mg_1.77Zn_1.2Sb₂ sample: (A) HR-TEM image with grain boundaries; (B) IFFT pattern with dislocations of 7(C); (C) HR-TEM image with dislocations; (D)HR-TEM with defects (E) IFFT pattern of selected portion with dislocations and stacking faults (F) HR-TEM image with stacking faults (G) HR-TEM image with strain (G1) strain distribution (ε_xy); (H) HR-TEM image with high magnitude strain distribution (H1) high magnitude strain distribution field of 7(H). GPA: Geometric phase analysis; HR-TEM: high-resolution transmission electron microscopy; IFFT: inverse fast fourier transform.

Scheme 1. (A) Phonon scattering in perfect crystal (B) defect structure.

In addition, the GPA has been performed to find the interaction of strain and displayed in Figure 7G-H1. Figure 7G and H are the selected strain fields by GPA strain analysis from HRTEM analysis. Figure 7G1-H1 represents the presence of strain clusters and grain boundaries in the axes of ɛ_xx, and ɛ_xy with a scale bar of -1 to 1. The positive value indicates the high magnitude strain in the lattice^[45,73]. This result confirms the influence of Ag in the lattice enhancing high-magnitude lattice strains and strengthening the phonon scattering. Thus, the obtained various defects and lattice strain significantly enhanced the phonon scattering through short to long-wavelength phonons, resulting in lower κ_L of 0.56 W/mK at 753 K for the Ag_0.03Mg_1.77Zn_1.2Sb₂ sample. Scheme 2A and B represents the phonon scattering at a lower and higher temperature range. In general, the long wavelength/low-frequency phonon dominates the scattering at near room temperature range, when temperature increases the wavelength of phonons will decrease and scattering of phonons also increases, which significantly reduces the κ_L of prepared samples.

Scheme 2. (A and B) phonon scattering at low and high temperatures.

Figure 8A represents the κ_L comparison graph of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03, and 0.05) at three different temperature ranges: 303 K, 503 K, and 753 K. From this result, the κ_L value of undoped and Ag-substituted samples gradually decreased with enhancing the doping content and temperature. Figure 8B shows the comparison of PF and zT with Ag content at 753 K. Here, the Ag-substitutedMg_1.8Zn_1.2Sb₂ samples show an enhancing trend with temperature due to the drastic enhancement of the S at high temperatures (at 753 K). Figure 8C shows the temperature-dependent zT of Ag-substituted samples. The convergence of multi-band improves the mobility of holes and increases the σ. Therefore, the significant enhancement of S and σ enhances the overall PF and TE performance.

Figure 8. Temperature dependence of (A) κ_L at 303K, 503 K, and 753 K (B) power factor vs. zT (C) figure of merit of Ag_xMg_1.8-xZn_1.2Sb₂ (x = 0, 0.01, 0.03 and 0.05) and (D) zT comparison^{[28,36,37,74-77]}.

In this investigation, the Ag-substituted Mg₂ZnSb₂ compound exhibits the band convergence of heavy VBs Γ and lighter VB A near E_F of the band structure. Benefiting from the improvement of PF and the significant reduction in κ_L, the Ag-substituted Mg_1.8Zn_1.2Sb₂ samples show considerably increased zT at the entire temperature range. The sample Ag_0.03Mg_1.77Zn_1.2Sb₂ shows a peak zT of 0.5 at 753 K, which is ~285% compared to undoped Mg_1.8Zn_1.2Sb₂ (zT = 0.13). This suggests that heavy element substitution is an effective strategy for enhancing the TE transport properties of p-type Mg_1.8Zn_1.2Sb₂-based solid solution. Figure 8D illustrates the comparison graph of a zT values with reported works. This work suggests that the substitution of Ag via multiple scattering strategies leads to synergistic enhancement in the TE performance of Mg₃Sb₃-based compounds. Compared with reported single and co-doped Mg₃Sb₃-based compounds, this work achieved the highest zT of ~0.5 at 753 K for Ag_0.03Mg_1.77Zn_1.2Sb₂. Furthermore, the present work has been compared with previous findings in terms of PF, κ_L, and zT as shown in Supplementary Table 4^{[37,45,73-78]}. This indicates that Ag substitution at Mg site of Mg_1.8Zn_1.2Sb₂-based compounds exhibits enhanced carrier and decreased phonon transport properties.

Figure 9 represents the TE efficiency (η) vs. zT of Ag_0.03Mg_1.77Zn_1.2Sb₂ sample with temperature difference (303-753 K). The maximum η has been expressed as $$ \eta=\frac{\Delta T}{T_{h}} \frac{\sqrt{1+Z T}-1}{\sqrt{1+Z T}+T_{c} / T_{h}} $$. The theoretical η is determined by the calculated zT of the prepared sample and the temperature difference between cold (T_c) and hot sides (T_h). Here, the maximum TE efficiency of 4.9% has been achieved for Ag_0.03Mg_1.77Zn_1.2Sb₂ sample with comparable zT (0.5 K at 753 K).

Figure 9. Thermoelectric efficiency vs. zT of Ag_0.03Mg_1.77Zn_1.2Sb₂ sample.