### 1. INTRODUCTION

_{2}emissions is necessary. In response, research efforts in exploring environmentally sustainable energy alternatives, including biomass, solar, and wind power have intensified [1-5]. Concurrently, capturing waste heat from industrial processes has emerged as a viable strategy to address energy demands [6]. In this context, thermoelectric (TE) technology holds immense promise. TE devices offer a compelling solution for waste heat recovery by directly converting thermal energy into electricity [7,8]. TE materials possess the unique ability to directly convert temperature gradients into electrical power and vice versa. This unlocks exciting possibilities for capturing waste heat and seamlessly transforming it into usable electrical energy. Notably, TE technology operates without requiring moving parts or fluids, making it not only efficient but also environmentally friendly. Strategic placement of TE materials near heat sources, such as industrial furnaces or vehicle exhausts, allows for effective electricity generation from otherwise wasted thermal energy. However, the efficacy of TE devices is intrinsically linked to the properties of the constituent materials, highlighting the critical need for continuous advancements in TE material performance [9]. The dimensionless figure of merit,

*zT*, is a crucial parameter to quantify the TE performance of a material and is defined as given in Equation (1) [10-12].

*σ*,

*S*,

*κ*, and

_{e}*κ*are electrical conductivity, Seebeck coefficient, electronic thermal conductivity, and lattice thermal conductivity, respectively. Optimizing

_{l}*zT*necessitates maximizing the power factor (

*S*) while minimizing the thermal conductivity (

^{2}σ*κ*) [13]. However, this pursuit is hindered by the inherent interdependence between

*S*,

*σ*, and

*κ*[13]. Efforts to decouple these parameters and achieve a high

*zT*have involved strategies such as band engineering and nanostructuring [14-16].

*p*-type Bi

_{0.5}Sb

_{1.5}Te

_{3}being particularly notable due to its favorable valence band convergence, which significantly enhances its power factor [17,18]. This distinctive characteristic has fueled extensive research, particularly in doping studies, to further explore and improve the capabilities of this material.

_{0.5}Sb

_{1.5}Te

_{3}alloys. Yoo

*et al*. demonstrated that Ag doping effectively tunes the Fermi level, shifting it deeper into the valence band with increasing Ag content [19]. This shift resulted in enhanced bipolar conduction at high temperatures and a significant increase in peak

*zT*(around 0.9) at a shifted peak temperature of 523 K compared to 323 K for the undoped material.

*et al*. investigated the effects of 6 % ZnTe doping in single-crystal Bi

_{0.5}Sb

_{1.5}Te

_{3}fabricated using the Bridgman method with a hexagonal structure [20]. Their findings revealed degenerate semiconductor behavior, characterized by a decrease in electrical conductivity alongside an increase in the Seebeck coefficient. This interplay of properties led to an optimal

*zT*of 1.05 at 450 K.

*et al*. reported that Mn doping in Bi

_{0.5}Sb

_{1.5}Te

_{3}elevates the carrier concentration and power factor [21]. Furthermore, Mn dopants suppress the onset of the bipolar effect, leading to reductions in both lattice and bipolar thermal conductivities.

*et al*. focused on Pb doping in

*p*-type Bi

_{0.5}Sb

_{1.5}Te

_{3}[22]. Due to its low thermal activation energy for holes, Pb doping was found to be beneficial for enhancing carrier concentration. This process not only increases hole concentration but also expands the bandgap and effectively mitigates bipolar effects. Consequently, Pb doping significantly boosted the

*zT*value of Bi

_{0.5}Sb

_{1.5}Te

_{3}alloys, with a peak

*zT*of 1.17 achieved at a Pb concentration of 0.97 at% at 423 K.

*et al*., the present study employs the Single Parabolic Band (SPB) model to investigate the influence of Pb doping on the electron transport characteristics of Pb-doped Bi

_{0.5}Sb

_{1.5}Te

_{3}alloys. We conducted a detailed examination of band parameter alterations corresponding to various Pb doping levels. This analysis involved meticulous calculations of several key parameters, including

*m*

_{d}^{*},

*μ*

_{0},

*μ*, and the

_{W}*B*-factor. Our results revealed a significant increase in

*m*

_{d}^{*}with increasing Pb doping, culminating in a peak at 0.97 at% Pb. This trend was mirrored in the behaviors of

*μ*, the power factor, and

_{W}*B*-factor, all reaching their maxima at 0.97 at% Pb. Through precise manipulation of

*n*, we were able to achieve a remarkable enhancement in the

_{H}*zT*value of the 0.97 at% Pb sample, from 0.882 to 1.74.

### 2. EXPERIMENTAL

### 2.1 Materials and Methods

*S*) and Hall carrier concentration (

*n*) from the research conducted by Wang

_{H}*et al*., with

*S*recorded at 323 K and

*n*at room temperature.

_{H}*k*,

_{B}*e*,

*η*and

*F*(

_{n}*η*) correspond to the Boltzmann constant, electron charge, reduced chemical potential, and the Fermi integral of order n, respectively, with

*F*(

_{n}*η*) defined in Equation (3).

*ε*,

*T*and

*h*denote the reduced energy of carrier, temperature in Kelven, and Planck's constant, respectively. By manipulating

*m*

_{d}^{*}in Equation (4) [24], we generated a Seebeck Pisarenko plot. Furthermore, we employed Equation (5) [25] to compute

*μ*, establishing a relationship between

_{H}*μ*and η as well as the non-degenerate mobility (

_{H}*μ*

_{0}). This approach allowed us to align the

*n*-dependent

_{H}*μ*lines with the experimental

_{H}*μ*through precise adjustment of

_{H}*μ*

_{0}.

*Ξ*is characterized in Equation (6) [24]. Here,

*m*

_{d}^{*}and

*μ*

_{0}were estimated via Equations (4-5).

*N*represents the number of valley degeneracy and was calculated using a value of six in the valence band of Bi

_{v}_{0.5}Sb

_{1.5}Te

_{3}reported by Lee

*et al*. [24].

*C*represents a longitudinal elastic constant and was calculated using a value of 54.7 GPa in Bi

_{l}_{0.5}Sb

_{1.5}Te

_{3}reported by Kim

*et al*. [26].

*L*denotes the Lorenz number and

*σ*represents the electrical conductivity.

*B*-factor can be derived from Equation (9), where

*κ*denotes the lattice thermal conductivity and

_{l}*μ*refers to the value determined in Equation (8) [27].

_{W}*Γ*) is defined as the ratio of

_{CvB}*κ*to that of the pure phase (

_{l}*κ*

_{l}^{p}). This relationship can be mathematically expressed using the disorder scaling parameter (

*u*), as shown in Equation (10). Consequently,

*Γ*can be estimated from

_{CvB}*u*, as presented in Equation (11) [28,29,30].

*θ*denotes the Debye temperature,

_{D}*V*represents the average atomic volume of the alloy,

*h*signifies Planck's constant, and

*v*stands for the average phonon velocity.

### 3. RESULTS AND DISCUSSION

### 3.1. Calculation of Density-of-States Effective Mass, *m*_{d}^{*}

_{d}

*S*) on the Hall carrier concentration (

*n*) is investigated in Bi

_{H}_{0.5}Sb

_{1.5}Te

_{3}+

*x*at% Pb (

*x*= 0.00, 0.49, 0.65, 0.81, 0.97, and 1.3) at 323 K (Fig 1(a)). Experimental

*S*values are represented by symbols, while the continuous line shows the calculated

*n*dependence of

_{H}*S*using the Single Parabolic Band (SPB) model. The close agreement between the symbols and the line indicates a good fit of the model.

_{0.5}Sb

_{1.5}Te

_{3}(

*x*= 0.00),

*n*is 2.6 × 10

_{H}^{19}cm

^{-3}, corresponding to an

*S*of 193.41 μV K

^{-1}. With 0.49 at% Pb doping,

*n*increases to 3.6 × 10

_{H}^{19}cm

^{-3}, while

*S*decreases to 180.30 μV K

^{-1}. This trend demonstrates that increasing Pb doping leads to a rise in

*n*and a concurrent decrease in

_{H}*S*. Further Pb addition systematically reduces

*S*. For example, Bi

_{0.5}Sb

_{1.5}Te

_{3}with 1.3 at% Pb has an

*S*of 122.92 μV K

^{-1}, a 36.45% decrease from the undoped sample. Conversely, experimental

*n*values show a pronounced increase with Pb content, reaching 9.2 × 10

_{H}^{19}cm

^{-3}for the 1.3 at% Pb composition, a 253.85% increase from pure Bi

_{0.5}Sb

_{1.5}Te

_{3}.

*m*

_{d}^{*}) for Bi

_{0.5}Sb

_{1.5}Te

_{3}+

*x*at% Pb (

*x*= 0.00 to 1.3) at 323 K (Fig 1(b)) increases compared to the pristine compound (

*x*= 0.00) with Pb introduction.

*m*

_{d}^{*}exhibits a monotonic rise from Bi

_{0.5}Sb

_{1.5}Te

_{3}to Bi

_{0.5}Sb

_{1.5}Te + 0.97 at% Pb, reaching a maximum value of 1.37

*m*(electron rest mass), a 22.25% increase. Interestingly, Bi

_{e}_{0.5}Sb

_{1.5}Te

_{3}+ 0.49 at% Pb and Bi

_{0.5}Sb

_{1.5}Te

_{3}+ 0.65 at% Pb share the same

*m*

_{d}^{*}(1.23

*m*), deviating from the expected trend. Furthermore,

_{e}*m*

_{d}^{*}unexpectedly decreases from 1.37

*m*to 1.31

_{e}*m*for Bi

_{e}_{0.5}Sb

_{1.5}Te

_{3}+ 1.3 at% Pb compared to Bi

_{0.5}Sb

_{1.5}Te

_{3}+ 0.97 at% Pb. These observations suggest that

*m*

_{d}^{*}does not strictly correlate with increasing Pb content. Instead, it may reach a plateau or even decrease beyond certain doping levels. This behavior highlights the influence of Pb doping on material properties, particularly the significant effect on effective mass at varying dopant concentrations.

### 3.2. Calculation of Non-Degenerate Mobility, *μ*_{0}

*μ*) with

_{H}*n*for Bi

_{H}_{0.5}Sb

_{1.5}Te

_{3}+ x at% Pb (

*x*= 0.00 to 1.3) at 323 K. Symbols represent experimental

*μ*while lines depict values calculated from the SPB model, presenting good agreement. The highest

_{H}*μ*(169.85 cm

_{H}^{2}V

^{-1}s

^{-1}) is observed for Bi

_{0.5}Sb

_{1.5}Te

_{3}+ 0.65 at% Pb (cyan symbol).

*μ*exhibits a decrease from the pristine state to Bi

_{H}_{0.5}Sb

_{1.5}Te

_{3}+ 0.49 at% Pb, followed by an increase to Bi

_{0.5}Sb

_{1.5}Te

_{3}+ 0.65 at% Pb. A sharp decline in

*μ*is observed with increasing Pb content from 0.65 to 0.81 at% (169.85 cm

_{H}^{2}V

^{-1}s

^{-1}to 146.23 cm

^{2}V

^{-1}s

^{-1}). Further increase in Pb content (0.81 to 1.3 at%) leads to a more gradual decrease (146.23 cm

^{2}V

^{-1}s

^{-1}to 132.61 cm

^{2}V

^{-1}s

^{-1}).

*μ*

_{0}) for various Bi

_{0.5}Sb

_{1.5}Te

_{3}+ x at% Pb (

*x*= 0.00, 0.49, 0.65, 0.81, 0.97, and 1.3) at 323 K. The pristine Bi

_{0.5}Sb

_{1.5}Te

_{3}(

*x*= 0.00) exhibits a

*μ*

_{0}of 230.5 cm

^{2}V

^{-1}s

^{-1}. Interestingly,

*μ*

_{0}remains unchanged with 0.49 at% Pb doping. However,

*μ*

_{0}increases to 234.5 cm

^{2}V

^{-1}s

^{-1}at 0.65 at% Pb, indicating a complex relationship between doping and mobility. This trend becomes non-linear with further doping. At 0.81 at% Pb,

*μ*

_{0}decreases to 208.0 cm

^{2}V

^{-1}s

^{-1}. This seesaw behavior reflects the intricate interplay between Pb concentration and carrier transport. Interestingly,

*μ*

_{0}plateaus at around 208.0 cm

^{2}V

^{-1}s

^{-1}for higher Pb doping levels (0.97 and 1.3 at%), suggesting a saturation effect. The highest

*μ*

_{0}(234.5 cm

^{2}V

^{-1}s

^{-1}) is achieved at 0.65 at% Pb, while the lowest (208.0 cm

^{2}V

^{-1}s

^{-1}) is found at 0.81 at% Pb. Despite these variations, the overall

*μ*

_{0}values remain relatively close across all compositions. This emphasizes the importance of precise Pb doping control to achieve desired mobility characteristics in Bi

_{0.5}Sb

_{1.5}Te

_{3}.

*Ξ*) calculated using the SPB model at 323 K for Bi

_{0.5}Sb

_{1.5}Te

_{3}thermoelectric alloys with varying Pb doping concentrations (as determined by Equation (6)).

*Ξ*signifies the interaction strength between charge carriers and phonons [31]. The pristine Bi

_{0.5}Sb

_{1.5}Te

_{3}(0.00 at% Pb) exhibits the highest

*Ξ*of 16.06 eV. Introducing 0.49 at% Pb doping reduces

*Ξ*to 14.30 eV, a decrease of 10.7%, marking the largest change observed. As Pb doping increases further,

*Ξ*shows a steady decline. Compositions with 0.65, 0.81, and 0.97 at% Pb exhibit similar

*Ξ*values in a range of 14.17 - 14.18 eV. A further increase in Pb content to 1.3 at% leads to a slight decrease in

*Ξ*to 13.91 eV. A higher

*Ξ*value indicates a stronger interaction between charge carriers and phonons, consequently hindering charge mobility. The observed decrease in

*Ξ*with increasing Pb doping (up to 0.97 at%) suggests that Pb doping suppresses this interaction in Bi

_{0.5}Sb

_{1.5}Te

_{3}, potentially leading to enhanced charge mobility.

### 3.3. Calculation of Weighted Mobility, *μ*_{W}

_{W}

*μ*) calculated using the SPB model (symbols) and the experimental power factor (bars) for Bi

_{W}_{0.5}Sb

_{1.5}Te

_{3}+ x at% Pb (

*x*= 0.00, 0.49, 0.65, 0.81, 0.97, and 1.3) at 323 K. The data reveal a positive correlation between

*μ*and the power factor, consistent with the predictions of Equation (8).

_{W}_{0.5}Sb

_{1.5}Te

_{3}(

*x*= 0.00),

*μ*is 273.47 cm

_{W}^{2}V

^{-1}s

^{-1}and the power factor is 27.13

*μ*cm

_{W}^{-1}K

^{-2}. Doping with 0.49 at% Pb leads to a modest increase in both

*μ*(283.06 cm

_{W}^{2}V

^{-1}s

^{-1}) and the power factor (28.87

*μ*cm

_{W}^{-1}K

^{-2}), suggesting improved thermoelectric performance. A further increase in Pb content to 0.65 at% results in a more significant rise in

*μ*(320.30 cm

_{W}^{2}V

^{-1}s

^{-1}) and the power factor (32.77

*μ*cm

_{W}^{-1}K

^{-2}). However, this trend is not monotonic: increasing the Pb content to 0.81 at% reduces

*μ*(304.75 cm

_{W}^{2}V

^{-1}s

^{-1}) and the power factor (31.09

*μ*cm

_{W}^{-1}K

^{-2}), highlighting a non-linear relationship between Pb doping and thermoelectric properties.

*μ*reaches a maximum of 333.54 cm

_{W}^{2}V

^{-1}s

^{-1}and the power factor attains a value of 33.28

*μ*cm

_{W}^{-1}K

^{-2}. This composition exhibits a 21.97% increase in

*μ*and a 22.66% increase in the power factor compared to the undoped sample, indicating it has the optimal doping level for thermoelectric efficiency at 323 K. Increasing the Pb concentration further to 1.3 at% leads to decreases in both

_{W}*μ*(311.87 cm

_{W}^{2}V

^{-1}s

^{-1}) and the power factor (29.49

*μ*cm

_{W}^{-1}K

^{-2}). This underscores the importance of precise doping control. Among the investigated compositions, 0.97 at% Pb doping maximizes both

*μ*and the power factor, establishing it as the optimal concentration for thermoelectric performance [32,33].

_{W}### 3.4. Calculation of *B*-factor

*B*-factor, a crucial parameter in Bi

_{0.5}Sb

_{1.5}Te

_{3}, is 0.39 for the pristine material. Introducing 0.49 at% Pb doping increases the

*B*-factor slightly to 0.42. A more pronounced effect is observed at 0.65 at% Pb, where the

*B*-factor significantly rises to 0.56. This suggests a substantial alteration in material properties due to Pb inclusion. Interestingly, the

*B*-factor plateaus at 0.81 at% Pb, with minimal change from the previous value. This implies a threshold beyond which additional Pb has minimal impact. The

*B*-factor then exhibits a significant increase to 0.87 at 0.97 at% Pb doping, representing the largest change observed. This highlights a highly responsive behavior at this concentration. However, a slight decrease to 0.74 is observed at 1.3 at% Pb. Analysis reveals that the

*B*-factor peaks at 0.97 at% Pb, a 121.46% increase compared to the pristine state. This peak presents the most pronounced response to Pb doping within the studied range, providing insights into the optimal Pb concentration for tuning material properties.

*κ*) of Bi

_{l}_{0.5}Sb

_{1.5}Te

_{3}+ x at% Pb (

*x*= 0.00 to 1.3) at 323 K. Equation (7) was used to determine the electronic thermal conductivity (

*κ*), which was then subtracted from the total thermal conductivity to obtain

_{e}*κ*. The pristine Bi

_{l}_{0.5}Sb

_{1.5}Te

_{3}exhibits a

*κ*of 0.56 W m

_{l}^{-1}K

^{-1}. Introducing 0.49 at% Pb doping leads to a minor reduction in

*κ*to 0.54 W m

_{l}^{-1}K

^{-1}. As the doping concentration increases to 0.65 at%,

*κ*further decreases to 0.46 W m

_{l}^{-1}K

^{-1}, indicating a trend of decreasing

*κ*with increasing Pb doping. A small decrease to 0.43 W m

_{l}^{-1}K

^{-1}is observed at 0.81 at% Pb, suggesting a deceleration of the reduction rate. A significant decrease in

*κ*to 0.31 W m

_{l}^{-1}K

^{-1}occurs at 0.97 at% Pb, marking a critical point where Pb addition significantly suppresses lattice thermal transport. However, increasing Pb doping to 1.3 at% deviates from this trend, resulting in a slight rise in

*κ*to 0.34 W m

_{l}^{-1}K

^{-1}. Analysis reveals that the most pronounced reduction in

*κ*(44.93% compared to undoped Bi

_{l}_{0.5}Sb

_{1.5}Te

_{3}) occurs at 0.97 at% Pb doping, where

*κ*reaches its minimum value. This highlights the optimal Pb concentration for minimizing

_{l}*κ*, which is crucial for enhancing thermoelectric performance by reducing phonon heat conduction. This result is attributed to the formation of point defects, caused by the difference in ionic radius and mass between Pb and Bi/Sb [22]. This difference leads to lattice strain and the creation of energetically favorable defect sites.

_{l}*zT*) and the Hall carrier concentration (

*n*) for Bi

_{H}_{0.5}Sb

_{1.5}Te

_{3}+ x at% Pb compositions (

*x*= 0.00 to 1.3) at 323 K. The symbols represent experimentally determined

*zT*values, while the lines depict theoretical

*zT*values from the SPB model calculations, demonstrating good agreement. The pristine Bi

_{0.5}Sb

_{1.5}Te

_{3}exhibits a

*zT*of 1.01 with an

*n*of 1.35 × 10

_{H}^{19}cm

^{-3}. Introducing 0.49 at% Pb leads to a slight increase in

*zT*to 1.06 at an

*n*of 1.55 × 10

_{H}^{19}cm

^{-3}. A notable rise in

*zT*to 1.30 is observed at 0.65 at% Pb doping (

*n*= 1.36 × 10

_{H}^{19}cm

^{-3}). Increasing Pb content to 0.81 at% maintains

*zT*at 1.31 (

*n*= 1.46 × 10

_{H}^{19}cm

^{-3}). A significant jump in

*zT*to 1.74 (

*n*= 1.27 × 10

_{H}^{19}cm

^{-3}) occurs at 0.97 at% Pb doping, representing the peak performance. However, increasing the Pb content further to 1.3 at% reduces

*zT*to 1.56 (

*n*= 1.30 × 10

_{H}^{19}cm

^{-3}). Therefore, the optimal

*zT*of 1.74 is achieved at 0.97 at% Pb doping, highlighting the critical role of precise Pb doping for maximizing thermoelectric efficiency in Bi

_{0.5}Sb

_{1.5}Te

_{3}alloys at this temperature. To attain the anticipated optimal

*zT*of 1.74, adjusting the carrier concentration by incorporating excess Te emerges as a promising approach. Kim

*et al*. effectively mitigated hole formation arising from SbTe anti-site defects within Bi

_{0.5}Sb

_{1.5}Te

_{3}, culminating in the realization of the theoretically predicted maximum

*zT*of 1.05 [34]. Notwithstanding variations in experimental methodologies, excess Te incorporation is as a viable strategy that minimally impacts band parameters.

### 3.5. Calculation of Scattering Parameter from Callaway & von Baeyer (CvB) Model

*κ*for Bi

_{l}_{0.5}Sb

_{1.5}Te

_{3}+

*x*at% Pb (

*x*= 0.00, 0.49, 0.65, 0.81, 0.97, and 1.3) to the

*κ*of the pristine Bi

_{l}_{0.5}Sb

_{1.5}Te

_{3}samples (

*κ*

_{l}^{P}). The sample with

*x*= 0.00 serves as the reference (

*κ*

_{l}^{P}) and has a

*κ*/

_{l}*κ*

_{l}^{P}ratio of unity, corresponding to undoped Bi

_{0.5}Sb

_{1.5}Te

_{3}. The

*κ*/

_{l}*κ*

_{l}^{P}ratio directly reflects the influence of point defects on phonon scattering, evident in its decrease with increasing x up to 0.97. At

*x*= 0.49, 0.65, and 0.81, the ratio exhibits a relatively gradual decrease to 0.96, 0.82, and 0.78, respectively. However, a sharp drop to 0.55 occurs at

*x*= 0.97, which is approximately 45 % lower than the pristine phase. This trend does not persist, as the ratio increases again at

*x*= 1.3.

*Γ*) for Bi

_{0.5}Sb

_{1.5}Te

_{3}+ x at% Pb (

*x*= 0.00, 0.49, 0.65, 0.81, 0.97, and 1.3). These parameters were calculated using Equations (10-11) and the

*κ*/

_{l}*κ*

_{l}^{P}ratio presented in Fig 5(a). In contrast to the trend of the

*κ*/

_{l}*κ*

_{l}^{P}ratio, Γ exhibits a sharp increase to 0.221 at

*x*= 0.97 and decreases to 0.163 again at

*x*= 1.3. In agreement with the observed decrease in

*κ*, the influence of point defects appears to be maximized at a specific composition,

_{l}*x*= 0.97.

### 4. CONCLUSIONS

*p*-type Bi

_{0.5}Sb

_{1.5}Te

_{3}alloys using the Single Parabolic Band model reveals the crucial impact of dopant concentration on thermoelectric performance. The study identifies 0.97 at% as the optimal Pb doping level for maximizing thermoelectric efficiency at 323 K. This composition yields the highest weighted mobility, power factor, and

*B*-factor while minimizing lattice thermal conductivity. The observed 44.93 % reduction in lattice thermal conductivity aligns well with predictions from the Callaway-von Baeyer model, highlighting the significant influence of point defect scattering. These factors culminate in a theoretical peak

*zT*of 1.74 for the 0.97 at% Pb-doped sample, marking a substantial improvement over the undoped material. Our findings underscore the importance of precise doping control in optimizing the thermoelectric properties of Bi

_{0.5}Sb

_{1.5}Te

_{3}alloys and provide valuable insights for the development of high-performance thermoelectric materials for room-temperature applications. Future work should focus on further exploring the interplay between doping concentration, microstructure, and thermoelectric performance to advance the development of these promising materials.