Thermoelectric materials have been widely applied in the energy conversion and solid-state refrigeration fields [1]. The figure of merit zT, which is defined as zT=S2σT/κe+κph, is used to evaluate the thermoelectric efficiency, where S, σ, κ_e, κ_ph, and T are the Seebeck coefficient, electrical conductivity, electronic thermal conductivity, lattice thermal conductivity, and absolute temperature, respectively [2,3]. A high zT value requires a high electrical conductivity and low thermal conductivity simultaneously, which is hard to produce owing to strong coupling between S, σ, and κ_e [4]. Hence, achieving proper electrical properties and low lattice thermal conductivity has become the principal route to obtaining high zT.

SrTiO₃ is a type of perovskite thermoelectric oxide with a structural formula ABO₃, which has attracted attention from many researchers owing to its high temperature stability and low cost. Pure strontium titanate, which has a band gap of 3.2 eV, is a typical insulating material [[5], [6], [7]] that can be transformed into an n-type thermoelectric material by donor doping. Element doping and reducing grain size are two effective ways to improve the figure of merit zT for SrTiO₃. Rare earth elements at the A-site [8] and transition elements at the B-site [9] can be used as doping elements for single-doping or co-doping. Hence, La doping can enhance the electric conductivity of SrTiO₃ [5]. Dy doping can effectively reduce the thermal conductivity of SrTiO₃ [10]. Nb doping can modify the energy band structure and enhance the electric conductivity of SrTiO₃ [11]. Single-element doping has achieved a zT of 0.37 at 1100 K for La doping [5], 0.25 at 973 K for Dy doping [12], and 0.37 at 1000 K for Nb doping [13]. The synergistic effects of dual-element doping can take advantages of multiple doping elements. Both La-Dy co-doped and La-Nb co-doped SrTiO₃ exhibit larger zT values than those of single-element doped samples. La-Dy co-doped SrTiO₃ can maintain a high zT value of 0.36 at 1073 K [14], and La-Nb co-doped SrTiO₃ materials have achieved an unprecedented zT of 0.6 at 1000 K [15]. Lattice thermal conductivity values can be limited by enhancing phonon scattering through the introduction of different types of scattering mechanisms, such as grain boundaries [16], interstitial atom point defects [17], oxygen vacancies [18,19], dislocations [20] and the introduction of pores [21]. High-energy ball milling is an effective way of producing nanoscale particles and synthesizing materials with high element doping concentrations regardless of the theoretical solid solution limit [[22], [23], [24]].

In this study, we combined multielement doping with burial sintering to improve the thermoelectric properties of SrTiO₃. Nanoscale Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ (x = 0, 0.05, 0.10, 0.15, and 0.20) powders were synthesized by high-energy ball milling. After sintering in a crucible filled with graphite powder, the bulk samples possessed complex structures, including shell-vesicular architectures, a high density of dislocations, and secondary phase particles. These complex structures greatly promoted carrier transport and limited the phonon transport, which simultaneously generated high electrical conductivity and low lattice thermal conductivity. The maximum zT value of 0.28 was achieved when x = 0.05, which gave the lowest total thermal conductivity of 2.95 W m^-1 K^-1 at 1100 K. A comparable zT value of 0.27 was also achieved when x = 0.2 at 1100 K, which indicates potential for a higher zT value from SrTiO₃ thermoelectric materials over a wide range of doping concentrations.

The La, Dy, and Nb, co-doped SrTiO₃ polycrystalline powder samples with nominal chemical compositions of Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ (x = 0, 0.05, 0.10, 0.15, and 0.20) were prepared by a mechanical alloying process. Appropriate quantities of the reagent powders, SrO (Aladdin, AR), La₂O₃ (Energy, 99.99 % purity), Dy₂O₃ (Aladdin, 99.99 % purity), TiO₂ (Anatase, Aladdin, 99 % purity), and Nb₂O₅ (Aladdin, 99 % purity), were first hand-mixed with an agate mortar and pestle. Then, the mixture was ground for total 120 h with a planetary high-energy ball mill (Pulverisette 4, FRITSCH, Germany) at 200 r min^-1 with tungsten carbide jars and balls to synthesize the doped SrTiO₃ nanopowders. The milling period was composed of 50 min running and 10 min pause. The mass ratio between balls and the mixture was 25:1, with the total mass of balls 500 g. The size of the tungsten carbide balls were 20 mm, 10 mm, 5 mm, and 1 mm in diameter for fully reaction. The synthesized powders were first uniaxially pressed under a pressure of 10 MPa to obtain Φ 20 mm preforms, which were further densified by cold isostatic pressing under a compressive stress of 250 MPa in oil. Finally, burial sintering processes were performed on the pristine samples, i.e., each of the preforms was placed into an alumina crucible filled with condensed graphite powder and heated at 1473 K for 3 h and then at 1773 K for 5 h in a muffle furnace. The bulk disk samples were cut into cuboid bars with dimensions of 2.5 mm × 2.5 mm × 12 mm for thermoelectric property measurements and small disks with dimensions of Φ 12.7 mm × 1 mm for thermal conductivity measurements. All the operation steps were executed in air, including reagent powder weighing, uniaxial pressing, ball milling, and bulk sintering.

The phase compositions of both the powder and bulk samples were characterized by X-ray diffraction (XRD, Empyrean, PANalytical, Netherlands) with CuK_α radiation at a scanning rate of 2°/min from 20° to 80°. Rietveld refinements of all the XRD patterns were performed with the use of the FullProf software. The cross-sectional macrostructures were examined using an optical microscope (Olympus GX51, Olympus Corp., Japan). The morphologies of both the powders and freshly fractured bulk samples were characterized with a field emission scanning electron microscope capable of performing energy-dispersive X-ray spectroscopy (FESEM/EDS, SUPARR 55, Carl Zeiss, Germany), which was operated in secondary electron mode at an accelerating voltage of 15 kV. Backscatter observations and mappings were conducted by an electron probe microanalysis (EPMA, JXA-8530 F Plus, JEOL, Japan) device capable of performing wavelength dispersive X-ray spectroscopy (WDS). Detailed microstructures of the sample were characterized with a transmission electron microscope (TEM, Talos F200x, FEI, USA) operating at an accelerating voltage of 200 kV.

The Seebeck coefficient and electrical conductivity measurements were performed on a commercial thermoelectric transport property measuring system (LSR-3, Linseis, Germany) from 300 to 1100 K in a helium atmosphere. The Hall coefficients (R_H) were measured under a reversible magnetic field from 300 to 770 K by the van der Pauw technique with a Hall measurement system (8400 Series HMS, Lake Shore, USA). The Hall carrier densities (n_H) and Hall carrier mobilities (μ_H) were calculated via n_H = 1/(eR_H) and μ_H = R_H/ρ, respectively.

The thermal conductivities were calculated using the formula κ = D × ρ × Cp, where D is the thermal diffusion coefficient measured by laser flash analysis (LFA 427, Netzsch, Germany), ρ is the geometric density measured by the Archimedes method, and C_p is the specific heat capacity measured by a differential scanning calorimetry (DSC, STA 449F3, Netzsch, Germany) device.

The electrical and thermal transport properties were measured along the same direction. The error bars in the measurements of the Seebeck coefficients, electrical conductivities and thermal conductivities were estimated to be 5 %, 10 % and 7 %, respectively.

The ideal SrTiO₃ has a cubic perovskite structure with a space group of Pm 3¯m. The ground state atomic and electronic structure of La, Dy, and Nb co-doped SrTiO₃ were calculated by first-principles density functional theory using the Vienna ab initio simulation package (VASP) [25,26]. These calculations were performed by spin-polarized density functional theory (DFT) with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional within the generalized gradient approximation (GGA) [25]. The projected augmented wave (PAW) potential was used to describe the ion-electron interaction [27,28]. The valence atomic configurations are 4s²4p⁶5s² for Sr, 2s²2p⁴ for O, 3s²3p⁶3d²4s² for Ti, 4s²4p⁶4d⁴5s¹ for Nb, 5s²5p⁶5d¹6s² for La, and 4s²4p⁶4d¹⁰4f¹⁰5s² for Dy. The energy cutoff of the plane-wave basis was 450 eV. During geometry optimization, numerical convergence was achieved with a tolerance of 1.0×10-5eV in energy and 0.2 eVnm-1 in force. The Brillouin zone was sampled with a 3×3×1 Monkhorst-Pack k-point grid for geometry optimization and electronic structure calculations.

The lattice distortion energies of different doping ratios were calculated using the following equation:

where E₀ is the total energy of Sr_0.9La_0.05Dy_0.05TiO₃ and E_x is the total energy of Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ (x = 0, 0.05, 0.1, 0.15, and 0.2).

In this study, the concentrations of La and Dy for all the formulations were fixed at 5 mol.%; hereafter, the La and Dy contents will not be specified. The synthesized powders Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ (x = 0, 0.05, 0.10, 0.15, and 0.20) are denoted as LaDy-Nb0, LaDy-Nb05, LaDy-Nb10, LaDy-Nb15, and LaDy-Nb20 and the corresponding XRD patterns are shown in Fig. 1(a). After high-energy ball milling for 120 h, the major diffraction peaks were indexed as cubic SrTiO₃ (PDF #35-0734) structures for all Nb doping levels. Secondary phase peaks are indexed as TiO₂ (PDF #21-1272), which is a residue from the precursor materials. The average particle size was calculated from the XRD patterns by the Scherrer equation to be approximately 15 nm. To investigate the influence of the Nb doping on the crystal structure, the XRD patterns were fitted by Rietveld refinement. Successive expansion of the lattice parameters were observed in response to increases of the Nb content (Fig. 1(b)), which indicated successful replacement of the smaller Ti⁴⁺ ions (60.5 pm) at the B-sites of the perovskite structure with larger Nb⁵⁺ ions (64 pm) [13]. An SEM image of the Sr_0.9La_0.05Dy_0.05Ti_0.8Nb_0.2O₃ powder sample (Fig. 2) showed nanoparticle clusters with an average grain size of ∼10 nm, which agreed well with the particle size determined from the XRD results. After high-energy ball milling for 120 h, the La, Dy, Nb co-doped SrTiO₃ nanopowders were synthesized, which indicated that high-energy ball milling can be used to prepare multielement-doped SrTiO₃ powders with high donor contents.

Fig. 3 shows XRD patterns of the as-sintered bulk samples. Compared with XRD patterns of the powder sample, the sharp diffraction peaks indicate completion of the crystallization process. An expansion of the (110) peaks from 31.75° to 33.0° is shown in Fig. 3(b). As the Nb content increased, the ion substitution caused the diffraction peaks to shift to lower angles, which was consistent with the XRD results of the powder sample. Considering the XRD patterns of both the powder and bulk Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ samples, all three elements were successfully doped into the lattice and did not change the basic perovskite structure. Most of the second phase peaks in Fig. 1(a) were absent in Fig. 3; however, two diffraction peaks at 2θ = 44.2° and 64.6° appeared, which might belong to rutile TiO₂ (PDF #21-1276)

To identify this second phase in the bulk samples, element composition analysis is performed via EPMA and listed in Table 1. Fig. 4 shows a backscattering electron image of the polished transverse surface of the LaDy-Nb0 sample. Some dark gray particles (e.g., Point A) with grain sizes less than 5 μm located in the pale matrix (e.g., Area B). The EDS results indicate that the dark gray particles were TiO₂, which corresponds to the XRD patterns. The element maps (marked by the red rectangle) indicate an evident aggregation of Ti and uniform distributions of Sr, La, and Dy in the matrix, which further confirmed that TiO₂ is a secondary phase. The actual compositions of all the samples were consistent with the nominal chemical compositions. When the results of the EMPA were compared with the XRD analysis, the presence of the TiO₂ second phase was also verified in other samples. The rutile TiO₂ probably results from the anatase phase transformation, which has a reaction temperature greater than 873 K [29].

Before discussing the thermoelectric properties, it is necessary to investigate the microstructures of the sintered bulk samples, and the LaDy-Nb20 bulk sample is selected as a typical example. Fig. 5(a) shows an optical micrograph of the LaDy-Nb20 bulk sample along the axial direction. A vesicular architecture embedded in a dense shell (approximately 400-500 μm thick) was observed. Further observations of the shell and vesicular architectures by SEM are shown in Fig. 5(b) and (c). The average grain sizes for the shell and vesicular architectures were approximately 4 and 10 μm, respectively. A schematic diagram of the sintering process is shown in Fig. 5(d). As the temperature increased, the densified powders first formed a grain shell in the surface layer as the free surface is more likely to crystallize. Such a shell structure can act as a frame for the sample and simultaneously limit both the growth of the grains on the shell and shrinkage of the whole sample, which results in an inner porous structure and larger inner grains during the high-temperature sintering procedure. The densities of the sintered samples with Nb doping levels of 0, 5, 10, 15, and 20 mol.% were 5.0, 4.91, 4.95, 4.97, and 5.12 g cm^-3, respectively. The density of LaDy-Nb0 was 91 % of its theoretical density and the other samples were approximately 88 % of their theoretical densities.

Fig. 5. Typical cross-sectional microstructures of the LaDy-Nb20 sintered samples: (a) optical microscope image after polishing; (b, c) SEM images of fresh fractures for different area; (d) a model of the sintering process.

Fig. 6(a) and (b) shows bright-field TEM images of the LaDy-Nb20 sample, from which a high dislocation density can be clearly observed. Such high dislocation densities are mainly attributed to the existence of oxygen vacancies, wherein vacancy diffusion facilitated the climb of dislocations [30,31]. The oxygen vacancies resulted from the burial sintering process, in which graphite powder acted as a reducing agent at high temperature. The corresponding defect reaction equations are expressed as follows [6,19,32]:

(3)$\text{O}_{\text{O}}^{\times }\to \text{V}_{\text{O}}^{\bullet \bullet }+\frac{1}{2}{{\text{O}}_{2}}\text{+2e }\!\!'\!\!\text{ }$

According to a previous study [33], the valance change of the Ti ions from +4 to +3 and Nb ions from +5 to +4 can also capture some electrons. These oxygen vacancies facilitated dislocation climbing, gliding, and tangling into complex dislocation network [30]. Furthermore, nanoscale particle was found on the matrix (Fig. 6(b)), selected area electron diffraction (SAED) patterns (Fig. 6(c)) further confirmed the secondary phase to be rutile TiO₂, which is consistent with the XRD analysis. The effects of such a complex structure, which includes a shell-vesicular architecture, high density of dislocations, and a secondary phase, on both electrical and thermal conductivity will be discussed later.

The carrier concentration calculated from Hall measurements (Fig. 7) exceeded 10²¹ cm^-3 and the negative sign indicates that electrons are the main charge carriers for all the samples. Furthermore, the carrier concentration was almost independent of temperature for all tested samples, showing typical character of a degenerate semiconductor. The average carrier concentrations of samples were 2.39 × 10²¹, 3.66 × 10²¹, 4.85 × 10²¹, and 5.83 × 10²¹ cm^-3 for the LaDy-Nb05, LaDy-Nb10, LaDy-Nb15, and LaDy-Nb20 samples, respectively, and increased as the Nb content was increased. From the band structure calculation (Fig. S2 in Supporting Information), the Femi level was determined to lie in the conduction band only for La and Dy co-doping, and shifted deeper as the Nb content was increased, leading to a higher carrier concentration. The linear changes of the carrier concentrations are consistent with the changes of the density of states (Fig. S3).

The electrical conductivity of La, Dy, and Nb co-doped SrTiO₃ materials with different Nb doping levels are plotted as a function of temperature from 300 to 1100 K (Fig. 8(a)), and show that the materials had obvious metallic behaviors. Ultrahigh values of the electrical conductivity were achieved in the LaDy-Nb20 sample, ranging from 5300 S cm^-1 at 300 K to 672 S cm^-1 at 1100 K. The electrical conductivity of LaDy-Nb20 at 1100 K was twice as high as that of the La-doped samples [5] or Nb-doped samples [13], and four times as high as that of the La and Dy co-doped samples [34]. Such a high electrical conductivity can be attributed to the high doping level that supplies a high carrier concentration of 10²¹ cm^-3 and a complex microstructure that leads to a high carrier mobility [2]. Generally, electrical conductivity can be calculated by the equation σ = neμ, where n is the electron carrier concentration, e is the electron charge, and μ is the carrier mobility. According to the results of the carrier concentration (Fig. 7), the carrier concentration mainly depends on the doping level and is temperature independent, and the temperature dependence of σ is mainly related to the carrier mobility μ(T). Phonon scattering mechanisms, especially acoustic phonons (μ_s-T^―1.5), and impurity scattering mechanisms (μ_i-T^1.5) are the two main aspects that influence carrier mobility [15]. After solving the first derivative i of “lnσ-lnT” (as shown in the inset image of Fig. 8(a)), the relationships between σ and T ⁱ indicate similar scattering mechanisms for all the samples, and thus similar carrier mobilities. Hence, the influence of Nb content on the electrical conductivity depends on the variation of carrier concentration. The acoustic phonon scattering is arguably the main scattering mechanism, and the perturbations from the impurities promote carrier transport at low temperature. As discussed before, the complex microstructures of the sintered samples, contain vesicular architectures with large grains and shell structures with small grains. Large grains have better electron transport properties than small grains because of fewer grain boundaries, reduced grain scattering, and more effective transport routes [35]. The samples with vesicular architectures have lower densities than previously reported [36], the low density does not have an obvious negative effect on the electrical conductivity. The calculated value of the carrier mobilities for the LaDy-Nb05, LaDy-Nb10, LaDy-Nb15, and LaDy-Nb20 samples were 5.72, 5.25, 5.35, and 5.68 cm² V^-1 s^-1 at 300 K, respectively. All these values were 2.5 times as high as those of the La and Nb co-doped samples [15] and those of the Nd single doped samples [37] at the same temperature.

The temperature dependence of the Seebeck coefficient of Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ is shown in Fig. 8(b). The negative Seebeck coefficient indicates an n-type electrical transport behavior, and the Seebeck coefficient decreased with increasing temperature. As the Nb content increased, the absolute values of |S| monotonically decreased from 206 μV K^-1 for LaDy-Nb0 to 120 μV K^-1 for LaDy-Nb20 at 1100 K, and the change rate at |S| decreased as the Nb content was increased. The value of |ΔS| between LaDy-Nb0 and LaDy-Nb05 is approximately 40 μV K^-1 at 1100 K, whereas the value of |ΔS| decreased to be 8 μV K^-1 between LaDy-Nb15 and LaDy-Nb20 at the same temperature. To understand this phenomenon, the expression of the Seebeck coefficient for degenerate semiconductors is as follows [2]:

where k_B, h, n, and m* are the Boltzmann constant, Plank constant, carrier concentration and the density-of-states (DOS) effective carrier mass, respectively. From this formula, m* and n are the two parameters that influence S under different element doping levels. According to Wunderlich et al. [38], the DOS effective mass will not obviously change under heavy doping conditions. Similar results can be found in the single-element Nb-doped SrTiO₃ samples [13]. The effective masses calculated from the Eq. (4) are 4.53m₀, 4.58m₀, 4.85m₀, and 5.03 m₀, for LaDy-Nb05, LaDy-Nb10, LaDy-Nb15, and LaDy-Nb20 bulk samples, respectively, where m₀ = 9.0 × 10^-9 kg is the mass of an electron. Despite the fact that the effective masses of samples increased as the Nb-doping content was increased, the main factor that influences S is the carrier concentration, which has a negative relationship with the Nb content.

The temperature dependence of the power factor calculated by the Seebeck coefficient and electrical conductivity (PF = S²σ) is shown in Fig. 8(c). Owing to the high electrical conductivity at low temperature and high Seebeck coefficient at high temperature, the power factor for all the samples shows a parabola-like curve, and the maximum values for the designed Sr_0.9La_0.05Dy_0.05Ti_1-xNb_xO₃ (x = 0, 0.05, 0.10, 0.15, and 0.20) are 1.25 mW m^-1 K^-2 at 550 K, 1.10 mW m^-1 K^-2 at 600 K, 0.87 mW m^-1 K^-2 at 600 K, 0.88 mW m^-1 K^-2 at 750 K, and 1.00 mW m^-1 K^-2 at 750 K, respectively. As the Nb content was increased, the peak values of the PF shifted to high temperatures owing to rapidly enhanced electrical conductivity and the gradual variations in the Seebeck coefficient.

The temperature dependence of total thermal conductivity (κ_tot) and the corresponding electron thermal conductivity (κ_e) for all the bulk samples are shown in Fig. 8(d). The lattice thermal conductivity (κ_ph), which excludes contributions of electrons on the thermal transport are shown in Fig. 8(e). For all the samples, the total thermal conductivity decreased as the temperature was increased. At 300 K, κ_tot decreased from 8.38 W m^-1 K^-1 for LaDy-Nb20 to 6.02 W m^-1 K^-1 for LaDy-Nb05. The total thermal conductivities of the LaDy-Nb0, LaDy-Nb05, and LaDy-Nb10 samples were similar. However, the thermal conductivities of all the samples tended to be approximately equal at high temperatures. The total thermal conductivity changed from 2.95 W m^-1 K^-1 for LaDy-Nb05 to 3.5 W m^-1 K^-1 for LaDy-Nb20 at 1100 K. The electronic thermal conductivity was estimated with the use of the Wiedemann-Franz law, κ_e = σ × L × T, where L = 2.44 × 10^-8 V² K², which is the Lorenz number for degenerated semiconductors [4]. In accord with the electrical conductivity, LaDyNb-20 has the highest κ_e values (from 4.39 W m^-1 K^-1 at 450 K to 1.86 W m^-1 K^-1 at 1100 K), and LaDy-Nb0 has the lowest κ_e value (from 1.09 W m^-1 K^-1 at 300 K to 0.41 W m^-1 K^-1 at 1100 K) among all the samples. As the Nb content increased, the average proportion of κ_ph/κ_tot decreased from 82 % for LaDy-Nb0 to 39 % for LaDy-Nb20. When the temperature was lower than 700 K, all the samples exhibited relatively high lattice thermal conductivities, and the lattice thermal conductivities decreased rapidly with increasing temperature. When the temperature was higher than 700 K, the lattice thermal conductivity of all the bulk samples tended to be steady. To discuss the factors influencing the thermal conductivity, the relaxation time (τ) has been discussed and can be expressed as follows [34]:

where the subscripts of U, GB, and PD mean the scattering mechanism of phonon-phonon Umklapp process, grain boundary, and point defect, respectively. The Umklapp process is generally obscured by scattering from the effects of impurities and is thus ignored in the following discussion. In the low temperature regime, the average phonon frequency is low. Long-wavelength phonons are the main mode of thermal transport and are scattered by grain-boundaries. The grain boundary scattering can be given by [34]:

where v_g is the phonon group velocity and L is average grain size. The average grain size of complex shell-vesicular architectures for all the samples are nearly same. All the samples contained a refined shell structure, with a grain size of 4 μm, which effectively scattered phonons, and vesicular architecture, with a grain size of 10 μm, which had less influence on phonon scattering. However, the vesicular architectures contained a large number of pores, having sizes of several micrometers (Fig. 5), which limited the heat transport and enhanced the scattering effect of vesicular architectures. Conversely, the second phase TiO₂ in the LaDy-Nb0 and LaDy-Nb05 samples was confirmed to effectively reduce the thermal conductivity owing to the grain sizes of approximately 1 μm [15]. When the temperature was higher than the Debye temperature (513 K) [39], the phonon wavelength decreased considerably, which indicates that the phonon mean-free-path was much shorter than the grain size, i.e., τGB-1→0. Under these conditions, point defects have a major effect on phonon scattering. To clarify the effects of point defects on the thermal conductivity, we assume that SrTiO₃ is a single crystal system to simplify the discussion, and τPD can be expressed as [34,40]:

where V is the unit atom volume, v_p is the phonon phase velocity, ΓM and ΓS are the scattering parameters related to mass fluctuation and strain field fluctuation, f_i is the fractional occupation of atoms with mass difference Δm_i and radius difference Δr_i, residing on a site with average mass and radius m and r, respectively. The point defects come from substitution of La, Dy, and Nb atoms and oxygen vacancies. Owing to the mass differences, all these substitutions enhance ΓM and reduce the relaxation time. The masses of La and Dy are approximately 1.5 and 2 times as heavy as Sr atoms, and Nb atoms are twice as heavy as Ti atoms. Oxygen vacancies provide the maximum mass and radius contraction, i.e., the maximum ΓM and ΓS, representing an ideal type of point defects for phonon scattering [18]. With the use of a function of Tj to fit the lattice thermal conductivity curve over the temperature range of 300-700 K, the index of j changes from -0.57 to -1.23 for the samples with higher Nb content. According to the XRD results, as the Nb content increased, the higher lattice parameters indicated intensification of the lattice distortion, i.e., enhancement of ΓS. Therefore, Nb can be thought of as an effective point defect scattering source [33] owing to enhancement of both ΓM and ΓS. Notably, the samples gradually expanded as the temperature increased, which alleviated lattice distortion at high temperature, reducing ΓS. This result is a reasonable interpretation of the stable lattice thermal conductivity at temperatures higher than 700 K. Furthermore, high density dislocations have considerable effects on the mid-frequency phonon scattering [20], and will be studied in the future work. As a result, the LaDy-Nb20 sample exhibits the lowest κ_ph values (∼1.6 W m^-1 K^-1 from 700 to 1100 K).

The temperature dependence of the dimensionless figure-of-merit zT is shown in Fig. 8(f). With increasing temperature, the zT values of all the samples increased. When the temperature was lower than 700 K, the zT values decreased with increasing Nb concentrations. From 700 to 1100 K, the power factors of the samples with high doping levels (LaDy-Nb15, LaDy-Nb20) increased. The total thermal conductivities of the samples with high doping levels (LaDy-Nb15, LaDy-Nb20) decreased faster than those of the samples with low doping levels (LaDy-Nb0, LaDy-Nb05, LaDy-Nb10). Thus, as the temperature increased, the zT values of the samples with high doping levels increased faster than those of the samples with low doping levels and the zT values of LaDy-Nb0 and LaDy-Nb10 finally stabilized. Because LaDy-Nb05 has the lowest thermal conductivity of the tested samples it achieved the highest zT value of 0.28 at 1100 K. LaDy-Nb20 also achieved a comparable zT value of 0.27 at the same temperature.

La, Dy, and Nb co-doped SrTiO₃ powders with different Nb contents have been successfully synthesized by a high-energy ball milling method. After a graphite powder burial sintering process, the sintered bulk samples had complex structures with exterior shell and internal vesicular and high dislocation densities. Such structures produced an ultrahigh electrical conductivity of ∼5300 S cm^-1 at 300 K, a low lattice thermal conductivity of ∼1.6 W m^-1 K^-1 from 700 to 1100 K for LaDy-Nb20, and the corresponding total thermal conductivity was 3.5 W m^-1 K^-1. In addition to the above complex structures, the existence of rutile TiO₂ phases in the LaDy-Nb05 bulk sample further reduced the total thermal conductivity to be 2.95 W m^-1 K^-1. By varying the Nb content, maximum zT values of 0.28 for LaDy-Nb05 and 0.27 for LaDy-Nb20 at 1100 K were achieved. These findings suggest potential for developing the thermoelectric properties of SrTiO₃ through multielement doping and complex structural designs.