Thermoelectric technology is an energy conversion technique capable of converting heat directly into electricity and vice versa^[1], which has various applications including power generation^[2-3], refrigeration^[4-5], sensor^[6] and so on. The functional devices of thermoelectrics are typically composed of a series of n- and p-type semiconductors connected alternately in an electrical series and thermal parallel configuration^[7-8].

High-efficient thermoelectric devices require that both n- and p-type materials exhibit superior thermoelectric performance, which is measured by the dimensionless figure of merit (zT). zT is expressed as zT=S²σT/κ, where S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature and κ is the total thermal conductivity including the contributions from electronic thermal conductivity (κ_E) and lattice thermal conductivity (κ_L)^[9]. Obtaining high zT, however, is challenging because the electrical properties (S, σ, and κ_E) of charge carriers show a strong coupling effect. By modifying the electrons/holes transport properties for n-/p-type materials, band engineering^[10⇓-12] is demonstrated to be one of the successful strategies for decoupling the above electrical transport properties and enhancing zT.

In addition to assessing thermoelectric performance (zT), the long-term survival thermoelectric devices typically necessitate that both n- and p-type materials possess similar physical and chemical properties. In practice, the devices often use the same matrix material for governing the durability and fatigue life^[13-14]. For instance, Bi₂Te₃-based alloys^[4-5] are commonly employed as matrix materials in commercial thermoelectric refrigerators, owing to their excellent thermoelectric performance at room temperature^[15⇓-17]. Si-Ge alloys^[18-19] are applied to matrix materials in radioisotope thermoelectric generators (RTG) for the outstanding performance at high temperatures.

The discussion above suggests that the matrix materials with potential thermoelectric application need to possess both excellent electron and hole transport properties simultaneously. However, due to the opposite polarity of electrons and holes, the coexistence of electrons and holes (known as bipolar effect^[20]) in such materials would compensate for their respective contributions to thermoelectric performance, leading to a rapid decay of total thermoelectric performance above the thermal excitation temperature. This is commonly observed in narrow band gap thermoelectric materials, where zT values of the materials increase with temperature increasing until bipolar effect occurs.

Due to the similar band degeneracies, effective masses and deformation potential coefficients (Table 1), the narrow-gap Bi₂Te₃-based^[21-22] and Bi₂(Te,Se)₃-based^[23-24] thermoelectric materials suffer bipolar effect above room temperature, accompanied with a detrimental effect on thermoelectric performance. Since the bipolar effect becomes severe when the concentration of electrons is equal to that of holes, an effective doping either donor or acceptor could efficiently suppress the bipolar effect, improving thermoelectric performance at such temperature.

Recently, the band engineering^[25⇓-27] and the microstructure engineering^[17,23,28] strategies are employed to enhance the thermoelectric performance for Bi₂Te₃-based materials. The reported high peak zT values of 1.4 and 1.9 are achieved respectively in Bi_1.8Sb_0.2Te_2.7Se_0.3 + 15% (in mass) Te for n-type^[29] and melt-spun Bi_0.5Sb_1.5Te₃ (with excess Te) for p-type^[30], nearby the bipolar temperature. Therefore, the effects of modified band structure and modulated scattering charged carriers on the electron and hole transport properties are particularly important.

In this work, a two-band model, taking into account of the contribution of both majority and minority carriers, is utilized to quantify the crossover from single-band transport to two-band transport for Bi₂Te₃-based and Bi₂(Te,Se)₃-based thermoelectric materials with or without extrinsic doping. The band information of majority carriers can be identified in the single-band region, while the effect of minority is illustrated in the two-band region. Specifically, this work focuses on the influence of minority charged carriers on the Seebeck coefficient and Hall mobility in pristine Bi₂Te₃ single crystals along the in-plane direction. With further consideration of external scattering for charged carriers, the variations in the ratio of electron mobility to hole mobility in Bi₂Te_2.7Se_0.3 illustrate the manipulations on the bipolar effect.

In this work, the Bi₂Te₃ and Bi₂Te_2.7Se_0.3 specimens were synthesized with the stoichiometric ratios by accurately weighing high-purity (99.99%) bismuth (Bi), tellurium (Te), and selenium (Se) elements. The elements were then mixed and loaded into the tapered end (with a 60° conical tip) of a glass tube with an inner diameter of 12 mm. The tube was evacuated and sealed, and the mixture was suspended in a furnace with vertically distributed temperature gradient. Based on the pseudo-binary phase diagram of Bi₂Te₃ and Bi₂Se₃, the furnace temperature was initially set at 1173 K (above the melting point) for 6 h. Subsequently, the temperature was cooled down to 893 K over a period of 2 h. Following this, a slow cooling process at a rate of 2 K/h was implemented until the temperature at each region of the ingot gradually descended below the melting point.

The crystal structure of Bi₂Te₃ belongs to the hexagonal crystal system, and its space group is R$\bar{3}$m (Fig. 1). In the crystal structure of Bi₂Te₃, every five atomic layers form a charge-neutral layer. Along the c-axis, these five atomic layers are arranged in the order of Te(1)-Bi-Te(2)- Bi-Te(1), with the same atomic species within each layer. The numbers in parenthesis represent different Te positions, and Te atoms and Bi atoms form octahedral coordination. Upon alloying with 10% Bi₂Se₃, Se atoms replace some of Te positions, and the formation energy of Se vacancies is lower, rendering their electron donors unable to absorb two electrons. Intra-layer Bi-Te forms polar covalent bonds, while inter-layer Te(1)-Te(1) exhibits weak van der Waals bonding. Consequently, Bi₂Te₃ displays pronounced anisotropy. The in-plane electrical conductivity of n-type Bi₂Te₃ is six times higher than the out-of-plane direction; in contrast, it is only three times higher in the out-of-plane direction for p-type Bi₂Te₃^[31-32]. Therefore, the crystal orientation significantly influences the thermoelectric performance. The Bi₂Te₃ and Bi₂Te_2.7Se_0.3 single crystals prepared in this study are primarily investigated in the in-plane direction. It can be observed from the powder X-ray diffraction (XRD) patterns that there is no obvious impurity phase in the material (Fig. 2(a)), while it can also be concluded from the single crystal XRD patterns that the materials are well oriented along the (00l) planes (Fig. 2(b)).

The electrons and holes in solids follow Fermi-Dirac distribution^[33]. The fundamental parameters related to the charge carrier transport can be obtained by solving the Boltzmann transport equation in equilibrium^[34-35]. The following equations used for the simulation calculations in this work can all be found in the relevant literatures^{[33-34,36 -37]}. F_i in the following equations represents the Fermi integral of power exponent i.

where η is the reduced Fermi level (η=E_F/k_BT), E_F is the Fermi level, k_B is the Boltzmann constant (1.380649× 10^-23 J/K), n is the carrier concentration (cm^-3), m_d* is the density of states effective mass (g), h is the Plank constant (6.6260693×10^-34 J·s) and e is the elementary charge (1.602176634×10^-19 C). The Seebeck coefficient (S) is a function of η, so the change of m_d* can be judged by the Seebeck coefficient versus the carrier concentration^[34]. It should be noted that the Hall carrier concentration (n_H) calculated by the Hall coefficient is not equal to the actual carrier concentration, and the two can be converted by the formula n_H=n/A, where A is the Hall factor (Eq. 5)^[34]. A similar Hall mobility (μ_H) can also be obtained from Eq. 6, where μ₀ is the mobility in the case of non-degeneracy limit (Eq. 7)^[37]. Here, ħ is the reduced Plank constant (ħ=h/2π), C_l is the longitudinal elastic constant, m_I* is the inertial effective mass, m_b* is the band effective mass, Ξ is the deformation potential coefficient (eV) characterizing the effectiveness of acoustic phonons to scatter charge carriers^[33].

The thermal conductivity of the material is composed of the electronic thermal conductivity and the lattice thermal conductivity. The electronic thermal conductivity can be obtained by the Wiedemann-Franz law (Eq. 8). Similar with S, the Lorenz constant (L) is related to the position of the Fermi level which can be obtained by Eq. 9^[34].

The transport performance of narrow-band gap semiconductors can be predicted by appropriately weighted summation of the transport of each independent band^[33]. The corresponding calculation relationship between conduction band and valence band is shown in Eq. 11, and the difference of Fermi levels between conduction band and valence band is the band gap (E_g) of the material. The total Seebeck coefficient (S_total) and conductivity (σ_total) can be obtained by the following equations:

where the subscripts n and p denote the n- and p-type charged carriers. The total Hall coefficient (R_H,total) is weighted by the square of the electrical conductivity of each band from which the total Hall carrier concentration (n_H,total) and Hall mobility (μ_H,total) can be defined^[33]:

The total thermal conductivity (κ_total) in the multi- band case has an additional bipolar thermal conductivity κ_bi besides the lattice and electronic thermal conductivity^[33].

The electrical properties were characterized using an electrical performance measurement system^[38-39] for Seebeck coefficient, electrical resistivity, and Hall coefficient. The Seebeck coefficient was determined by the slope of the thermopower with temperature gradient (with a temperature difference of approximately 5 K across the sample). The electrical resistivity and Hall coefficient were both measured using the van der Pauw method under a reversible magnetic field. The thermal conductivity was determined by measuring the thermal diffusivity (D) using the LFA457 and LFA467 instruments (NETASCH). The thermal conductivity (κ) was then calculated using the formula κ=ρC_pD, where C_p is the specific heat capacity estimated using the Dulong-Petit law, and ρ is the material density.

The thermoelectric transport properties of Bi₂Te₃ and Bi₂Te_2.7Se_0.3 can be estimated by two-band model. Witting et al.^[24] have used the two-band model to simulate and calculate the Pisarenko lines of pure bismuth telluride material. In this work, the part of the two-band region is supplemented for comparison. The parameters used in the estimation are shown in Table 1.

The estimations are realized by shifting the Fermi level from the conduction band to the valence band. In this way, the Hall coefficient dependent Seebeck coefficient is just head to tail to form a closed curve that looks like the Arabic numeral “8” as shown in Fig. 3(a). Similarly, the Hall mobility vs. Hall coefficient form a closed curve as well (Fig. 3(b)). For the same R_H, the curve gives two possible S or μ_H, because of the compensation between majority and minority carriers. When the minority carriers are almost negligible, the Hall coefficient can be approximated as the inverse of the majority carrier concentration (Eq. 15), which is consistent with the approximation of the single parabolic band model. In this single-band region, the Seebeck coefficient and mobility increase with the increase of Hall coefficient, without obvious cancellation effect of minority carriers. When minority carriers gradually increase, the Hall coefficient decreases and gradually tends to 0 due to the reverse Hall effect by minority carriers. In such two-band region, the Seebeck coefficient and mobility also decrease significantly due to the opposite sign between majority and minority carriers.

Fig. 3(c, d) give the conventional Pisarenko lines for better comparing the difference between the single-band region and the two-band region. When the material is heavily doped, the two-band curves (solid lines) of the Seebeck coefficient and Hall mobility as a function of Hall carrier concentration are consistent with the curves estimated by the single-band model (dashed lines). Accordingly, the density of states effective mass m_d* can be estimated from the change of Seebeck coefficient as a function of Hall carrier concentration. At 300 K, the density of states effective mass of both n- and p-type Bi₂Te₃ is 1.06m_e (Fig. 3(c)), which is consistent with similar analysis results in other studies^{[24,40 -41]}. With the decrease of Hall carrier concentration, the curve gradually deviates from the estimation by single parabolic band. The deviation for Bi₂Te₃ occurs when the carrier concentration is 2×10¹⁹ cm^-3.

Similar estimations were carried out for Bi₂Te_2.7Se_0.3, as shown in Fig. 4. It is found that the density of states effective mass of both n- and p-type Bi₂Te_2.7Se_0.3 increase to 1.4m_e compared to Bi₂Te₃, which indicates the solid solution of Bi₂Se₃ can improve the band effective mass of Bi₂Te₃. At the same time, the deviation from the dash lines for Bi₂Te_2.7Se_0.3 occurs when the carrier concentration is 7×10¹⁸ cm^-3, which demonstrates that the solid solution of Bi₂Se₃ expands the band gap and inhibits the bipolar effect.

As shown in Fig. 4(c, d), the uncertainty in Hall mobility for Bi₂Te_2.7Se_0.3 is mainly attributed to the differences in mobility μ₀ caused by various factors, such as the change in scattering mechanisms and the possible anisotropy. The single crystal quality of n-type Bi₂Te_2.7Se_0.3 has the additional significant effect on the mobility. To reproduce and compare the experimental data collected from different literatures^{[42⇓⇓⇓⇓⇓⇓-49]} and obtained in this work, the different mobility ratios (μ_0,CB/μ_0,VB) of conduction band and valence band are used in the model (μ_0,VB is approximated as a constant for avoiding too many variables). Changing the mobility ratio has little effect on the single-band region of the Seebeck and Hall coefficient. However, when the proportion of minority carriers gradually increases, the proportion of minority carriers conductivity (depending on μ₀) increases accordingly, so that the compensate effect of minority carriers on Seebeck coefficient in two-band region is related to μ₀.

Fig. 5 shows the electrical conductivity (σ) dependence of thermoelectric performance estimated from the single- and two-band models. When the conductivity is lower than 200 S·cm^-1, the Seebeck coefficient decreases sharply against the curve of the single-band model, which corresponds to the bipolar effect where carriers in both conduction and valence bands contribute significantly to transport (Fig. 5(a)). Due to the bipolar thermal conductivity, the total thermal conductivity (κ) in the low conductivity region increases dramatically (Fig. 5(b)), where the lattice thermal conductivity κ_L of Bi₂Te_2.7Se_0.3 single crystal along the in-plane orientation is estimated to be 0.9 W·m^-1·K^-1. It should be noted that the lower thermal conductivity for poly-crystalline samples (include parts of single-crystalline samples) could be understood by the existence of additional scattering sources for phonon transport, such as point defects and grain boundaries.

The maximum power factor (PF) along in-plane direction of Bi₂Te_2.7Se_0.3 is 40-50 μW·cm^-1·K^-2 at room temperature^[48] (Fig. 5(c)). The room temperature zT values of n- and p-type Bi₂Te_2.7Se_0.3 in the literature are quite dispersed due to the wide-difference in μ₀ and κ (Fig. 5(d)), because microstructure engineering is often employed to reduce κ_L and improve zT. The room temperature zT of the I-doped n-type Bi₂Te_2.7Se_0.3 in this work is expected to achieve a peak of about 0.7 with σ ranging from 600 to 700 S·cm^-1. For the better-quality single crystal, the maximum zT is almost above 0.9 at room temperature.

The thermoelectric transport properties of polycrystalline Bi₂Te_2.7Se_0.3 in different literature are also collected, as indicated by the star symbols in Fig. 4 and Fig. 5. The existence of a large number of grain boundaries increases the scattering strength of charge carriers. In single-band region, Seebeck coefficient is insensitive to the grain boundary scattering, which can significantly reduce the conductivity and mobility of polycrystalline Bi₂Te_2.7Se_0.3 (Fig. 4(d)). The lower textures of the polycrystal material would also reduce the conductivity of Bi₂Te_2.7Se_0.3 and thus reduce the power factor (Fig. 5(a, c)). At the same time, the existence of grain boundary also reduces the lattice thermal conductivity through additional phonon scattering (Fig. 5(b)). The literature works^[54⇓-56] often introduce point defects into polycrystalline materials to further reduce the lattice thermal conductivity for improving the thermoelectric performance of Bi₂Te_2.7Se_0.3. Therefore, it can be concluded from Fig. 5(d) that although zT values of Bi₂Te_2.7Se_0.3 materials are very dispersed due to the existence of strong anisotropy and different scattering mechanisms, zT values of single crystal and polycrystalline materials are nearly comparable.

This work employs the single- and two-band models to comprehensively illustrate the variations in thermoelectric transport properties of Bi₂Te₃ and Bi₂Te_2.7Se_0.3 in a wide range of Fermi level. Utilizing these models not only facilitate the effective analysis of fundamental material parameters such as m_d*, κ_L, and μ₀, but also provide a more intuitive representation of the changes in material properties under the influence of minority carriers, which is non-negligible in single-band model. This work presents accurate values for m_d* and κ_L, as well as the effect of μ₀ ratio on thermoelectric performance parameters. This offers a convenient tool for transport performance analyses when bipolar effect become severe in narrow gap thermoelectrics.