Due to the increasingly environmental crisis, development of sustainable clean energy becomes a key challenge in global scientific research^[1]. Based on the Seebeck effect, thermoelectrics could directly convert heat into electricity without involving any moving parts and emissions, which has been considered as a solution for the crisis^[2,3,4,5,6]. The conversion efficiency of thermoelectric materials mainly depends on the thermoelectric figure of merit, ZT=S²T/ρ(k_E+k_L), where S,ρ, T,k_E and k_Lare the Seebeck coefficient, the electrical resistivity, the absolute temperature, the electronic and lattice thermal conductivities, respectively. Thus, a high S, a low ρand a low (k_E+k_L) are expected for a superior ZT.

However, the strong coupling among S,ρandk_Evia the carrier concentration, band structure and charge scattering, results in a simultaneous optimization of these parameters to be difficult^[7,8]. The newly developed strategy of band engineering^[9,10] successfully decouples the correlation, leading to a significant enhancement in power factor (S²/ρ) and then an increased ZT in various materials, such as PbTe^[11], SnTe^[12,13], GeTe^[14,15,16], SnSe^[17], Mg₂Si^[18]and half-Heusler^[19]. Alternatively, minimizing k_L^[20], the only independent parameter determining ZT, through nanostructuring^{[21,22,23,24,25,26,27,28,29]}, point defects^{[30,31,32,33,34]}, dislocations^[11,35-36], liquid phonons^[37,38], low sound velocity^[39,40,41], lattice anharmonicity^[42,43,44], and low cutoff frequency of acoustic phonons^[45], has effectively resulted in a ZT-enhancement as well.

Ternary chalcogenides I-V-VI₂ compounds (where I=Cu, Ag or alkali metal; V=Sb, Bi; and VI=S, Se, Te) have aroused widespread interests for thermoelectric applications due to their intrinsically low k_L stemming from the strong anharmonicity induced by lone s² pair electrons of the group V cations^[46]. So far, the study of their thermoelectric properties mainly focuses on the Ag-based compounds, particularly Ag(Sb/Bi)(Te/Se)₂^[23,47-55]. The peak ZTs of 2.0 and 1.2 for AgSbTe₂ and AgSbSe₂ have been realized through band engineering and carrier concentration optimization^[48,56], respectively, which are considered as potentially lead-free alternatives for p-type PbTe.

As one of a few n-type semiconductors among this family of compounds, the crystal structure of AgBiSe₂ transforms from hexagonal to rhombohedral and then to cubic with increasing temperature^{[57,58,59,60]}, as shown in Fig. 1(a). The cubic AgBiSe₂ shows highly disorder at cation sites, which are beneficial for the low k_L through the strong phonon scattering by point defects^[61]. Recently, numerous efforts have been put on its thermoelectric performance. In order to enhance the power factor (S²/ρ), Nb-^[52], Ge-^[62], In-doping^[63] at cation site or Te-^[64], Cl-doping^[65] at anion site are carried out, and an enhanced ZT up to 0.7 is obtained in single phase AgBiSe₂ at 773 K. It is well known that a realization in the highest possible ZT, even using the successful strategies mentioned above, strongly depends on an optimization in carrier concentration^[66,67,68]. Therefore, an investigation on the transport properties in a broad carrier concentration range, is essential to fully assess the potential for thermoelectric applications.

This work focuses on the systematic investigation of the thermoelectric properties of (Ag₂Se)_1-x(Bi₂Se₃)_x alloys. A broad carrier concentration ranging from 1.0×10¹⁹ cm^-3 to 5.7×10¹⁹ cm^-3 is obtained through the control of x, which enables a comprehensive evaluation of the electronic transport properties based on a single parabolic band transport with acoustic scattering. Moreover, the underlying physics including the scattering mechanism, deformation potential coefficient, density-of-states effective mass and optimal carrier concentration are discussed in details. The work offers a fundamental understanding of the transport properties of (Ag₂Se)_1-x(Bi₂Se₃)_x.

Polycrystalline (Ag₂Se)_1-x(Bi₂Se₃)_x(0.4≤x≤0.62) was synthesized using Ag₂Se and Bi₂Se₃ as the starting materials. Both Ag₂Se and Bi₂Se₃ were synthesized by sealing stoichiometric quantity of Ag (99.999%) and Se (99.999%), Bi (99.999%) and Se (99.999%) in vacuum quartz ampoules, melting at 1273 K and 1123 K for 4 h, respectively, quenching in cold water, and then annealing at 843 K for 3 d. The obtained Ag₂Se and Bi₂Se₃ ingots were weighed according to the stoichiometric ratio, melted at 1173 K for 10 h and annealed at 843 K for 2 d. Pellet samples (>96% of the theoretical density) with ~12.0 mm in diameter were obtained by an induction heating hot press system at 800 K for 30 min under a uniaxial pressure of ~ 80 MPa.

The details of the electronic transports (Seebeck coefficient, electronic resistivity and Hall coefficient) measurements are given in the literature^[69]. The thermal conductivity was estimated viaκ = dC_pD, where d is the geometric density, C_p is the heat capacity, and D is the thermal diffusivity. Parameter D was measured using a laser flash technique with the Netzsch LFA457 system, and C_p was determined by the Dulong-Petit limit and assumed to be temperature independent. All the transport property measurements were carried out under helium atmosphere from 300 K to 700 K. The measurement uncertainty for each transport property (S, ρand κ) is about 5%.

Phase composition and microstructures were identified by X-Ray Diffraction (XRD) and Scanning Electron Microscope (SEM, Phenom Pro) equipped with an Energy Dispersive spectrometer. The longitudinal (v_L) and transverse (v_T) sound velocities were measured on the pellet samples at room temperature using an ultrasonic pulse receiver (Olympus-NDT) equipped with an oscilloscope (Keysight). Infrared Fourier transform spectroscopy (Bruker Tensor 2) was used to measure the optical reflectance coefficient for estimating the optical band gap^[70].

According to the phase diagram of Ag₂Se-Bi₂Se₃ system^[57] (Fig. 1(b)), (Ag₂Se)_1-x(Bi₂Se₃)_x shows a single phase region at x ranging from 0.4 to 0.62. This tunable composition in a broad range suggests an availability of carrier concentration tuning for thermoelectric (Ag₂Se)_1-x(Bi₂Se₃)_x. Powder XRD patterns for (Ag₂Se)_1-x(Bi₂Se₃)_x(0.4≤x≤0.62) are shown in Fig. 2. It is found that the single phase region confines x to be from 0.48 to 0.56, since Ag₂Se and Bi₂Se₃ impurity phases are observed for the samples with x≤0.46 andx≥0.58, respectively. As shown in Fig. 3(a), the locally enlarged XRDpatterns for the samples with single phase (0.48≤x≤0.56) show that the diffraction peaks shift to higher angles with increasing x. The dependence of lattice parameters on x in Fig. 3(b) shows that both a and c decrease with increasing x, which can be understood by the introduction of cation vacancy as x>0.5.

To further confirm the phase purity, the microstructure for the samples with 0.48≤x≤0.58 is characterized by SEM (Fig. 4(a, c-g)), and the mapping on the composition by EDS is carried out on the samples with x=0.48 (Fig. 4(b)) and 0.58 (Fig. 4(h)). No impurity phase can be observed for the samples with 0.5≤x≤0.56. Two types of precipitate are observed in the sample withx=0.48, as shown in Fig. 4(a). The EDS mappings (Fig. 4(b)) show black and white precipitates which are mainly composed of Ag-Se and Bi, respectively. The ratios of Ag:Bi:Se are about 60.1 : 9.3 : 30.6 and 8.9 : 84 : 7.1, respectively, which suggest Ag₂Se and Bi impurities. Moreover, Bi₂Se₃-rich precipitates identified by EDS as well, are observed in the sample 0.58, respectively. Both XRD and SEM results suggest the single phase (Ag₂Se)_1-x(Bi₂Se₃)_x region of x being 0.5-0.56. Therefore, single phased (Ag₂Se)_1-x(Bi₂Se₃)_xmaterials with 0.5≤x≤0.56 are focused on.

As shown in Fig. 5(a), the Hall carrier concentration (n_H) for (Ag₂Se)_1-x(Bi₂Se₃)_x is found to significantly increase with increasing x. It can be understood by the increase of x would lead to the formation of cation vacancies, which further drives the evaporation of Se due to its high vapor pressure. The deficiency of Se anions would lead to an increase in electron concentration. As a result, an increase in carrier concentration from 1.0×10¹⁹ cm^-3 to 5.7×10¹⁹cm^-3 is obtained with increasing x from 0.5 to 0.56. Such a broad carrier concentration range enables a detailed assessment on the electronic transport properties, and a comprehensive understanding of fundamental material parameters. The optical measurements for all the samples are shown in Fig. 5(b). The estimated optical band gap (E_g) is about 0.45 eV, which is in good agreement with literatures (0.5-0.6 eV)^[65,71].

Temperature dependent Hall coefficient (R_H) and Hall mobility (µ_H) for (Ag₂Se)_1-x(Bi₂Se₃)_x are shown in Fig. 6(a). A sharp decrease in µ_H and R_H around 550 K are found, which can reasonably be attributed to the phase transition between rhombohedral and cubic structures. At temperatures apart from phase transition temperature, R_H in three phases keeps nearly temperature independent, suggesting a single band transport behavior in a degenerated state. The decrease in μ_H with increasing temperature via μ_H~T^-1.5, indicates the charge carrier scattering by acoustic phonons.

According to the above discussion on R_H, a single parabolic band model with acoustic phonon scattering is used to understand the transport properties and underlying physical parameters. Assuming a single band valley (band degeneracy of 2), temperature dependent effective deformation potential coefficient (E_def) and density of state effective mass (m^*) are shown in Fig. 6(b). It is found that the changes of both carrier concentration and temperature have negligible effects on both E_def and m^* for the samples in three phases, indicating a rigid band behavior for (Ag₂Se)_1-x(Bi₂Se₃)_x. An m^* of ~0.51 m_e and a E_def of ~33 eV are obtained for the samples in the hexagonal structure. The obtained m^* is in good agreement with literature[65]. m^* and E_def in the rhombohedral phase are about 0.54 m_e and 29 eV, respectively, while about 4 m_e and10 eV in the cubic phase.

The Hall carrier concentration dependent Hall mobility, and Seebeck coefficient are shown in Fig. 6(c, d) respectively. Based on estimated average m^* and E_def at each temperature (in Fig. 6(b)), the SPB model with the acoustic phonon scattering enables a reasonable prediction on both Seebeck coefficient and Hall mobility at three different temperatures. Moreover, this model prediction agrees well with available literature results^[52,64-65].

Temperature dependent Seebeck coefficient and electricalresistivity for (Ag₂Se)_1-x(Bi₂Se₃)_x are shown in Fig. 7(a, b), respectively. Both Seebeck coefficient and electrical resistivity decrease with increasing x due to the increase of Hall carrier concentration. Moreover, both of them increase with increasing temperature, showing a degenerate semiconducting behavior. Figure 7(a, b) also include the Seebeck coefficient and electrical resistivity for pristine^[64] and In-doped^[63] AgBiSe₂ from literatures for comparison, respectively. It is seen that the lowest Seebeck coefficient and electrical resistivity obtained in this work is comparable to those of In-doped sample. The Seebeck coefficient for all the samples is negative, indicating a n-type conduction, which is consistent with the Hall coefficient measurements (Fig. 6(a)).

Figure 7(c, d) show the temperature dependent total thermal conductivity (κ) and lattice thermal conductivity (κ_L) for (Ag₂Se)_1-x(Bi₂Se₃)_x, respectively. κ_L is obtained by subtracting the electronic thermal conductivity (κ_E) from the total thermal conductivity via the Wiedemann-Franz Law, κ_E=LT/ρ, where L is the Lorenz factor determined by the SPB model with the acoustic phonon scattering. Both κ and κ_L decrease with increasing temperature. The κ slightly increase with increasing x due to the increased carrier concentration leading to higher κ_E. It is seen that the κ_L is lower than 0.7 W/(m·K) in the entire temperature range, mainly resulting from the strong anharmonicity^[42,46,72] and strong phonon scattering by disorded cations^[42,61,72]. The κ_L at high temperatures in the cubic phase is found to be as low as~0.4 W/(m·K), showing great potential for thermoelectric applications.

The sound velocities (v_s) of longitudinal (v_L) and transverse (v_T) branches, as the important parameters determining k_L, are measured and listed in Table 1. Moreover, the physical parameters including Debye temperature (q_D), bulk modulus (B) and Grüeneisen parameter (g) are estimated based on the measured v_L and v_T^[73,74], as listed in Table 1. It is shown that the change in sound velocities for all the samples does not exceed 5%, which is within the measurement uncertainty range. This excludes the influence of lattice softening on k_L. This material shows a very low sound velocity of ~1500 m/s, one of the lowest sound velocities among thermoelectrics^[39,75]. Moreover, the Grüneisen parameter (γ) as large as 2 indicates the strong anharmonicity^[46]. Both the low sound velocity and the strong lattice anharmonicity could be the origin for the low κ_L observed.

Temperature dependent figure of merit (ZT) for (Ag₂Se)_1-x(Bi₂Se₃)_x is shown in Fig. 8(a). ZTs for all the samples increase with increasing temperature and x, due to the carrier concentration optimization. Thus, a peak ZT of 0.5 at 700 K is achieved for the sample with the highest carrier concentration. The SPB model further enables a prediction in carrier concentration dependent ZT at different temperatures using the experimental k_L (Fig. 8(b)). The predicted ZT agrees well with the experimental data as shown in Fig. 8(b). It is found the highest carrier concentration obtained in this work is very close to the optimal one. According to the calculated band structure^[71], engineering the band should enable a possibility for further enhancing the thermoelectric performance.

In this work, a composition control for (Ag₂Se)_1-x(Bi₂Se₃)_xwith x in the range of 0.5-0.56 enables the carrier concentration ranging in (1.0-5.7)×10¹⁹cm^-3, which allows a systematical understanding on the fundamental physical parameters determining its transport properties. It is shown that the electronic transport properties can be well understood by a single parabolic band model. The carrier concentration is well optimized in this work, leading to a peak ZT of 0.5 with the help of intrinsically low lattice thermal conductivity. This work offers a fundamental understanding on the material physics affecting the thermoelectric performance, which could guide the further improvements for this material.