Four-phonon scattering and multi-valley characteristics induce high thermoelectric performance in TlAgSe: a first-principles investigation

Abstract

Materials with intrinsically lower thermal conductivity and exceptional electrical properties are required for high-performance thermoelectric applications. The recently synthesized Zintl phase TlAgSe demonstrates lower thermal conductivity; however, it exhibits suboptimal thermoelectric performance. To gain a deeper understanding of the mechanism underlying the ultralow lattice thermal conductivity and evaluate the potential of thermoelectric performance of TlAgSe, the characteristics of chemical bonding, electronic band structures and phonon transport properties with high-order anharmonicity were systematically evaluated through first-principles calculations and the self-consistent phonon (SCPH) theory. Our findings reveal that hierarchical chemical bonding, high-order anharmonicity, and frequency renormalization are crucial factors contributing to the ultralow thermal conductivity of TlAgSe. Additionally, the multi-valley characteristics at the band edge, combined with polar optical phonon-dominated carrier scattering, lead to exceptional electrical properties. Consequently, peak ZT of ∼3.06 at a carrier concentration of 4.82 × 10¹⁹ cm⁻³ for p-type doping and ∼2.80 at a carrier concentration of 1.32 × 10¹⁹ cm⁻³ for n-type doping were achieved in TlAgSe at 600 K, highlighting its significant potential for high-performance thermoelectric applications.

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1. Introduction

Thermoelectric materials, which are capable of direct heat-to-electricity conversion via the Seebeck effect, have emerged as pivotal candidates for sustainable energy harvesting and solid-state refrigeration.^1,2 The energy conversion efficiency is quantified by the dimensionless figure of merit: ZT = S²σT/κ_tot, where S, σ, T, and κ_tot refer to the Seebeck coefficient, electrical conductivity, absolute temperature, and total thermal conductivity, respectively. The total thermal conductivity is an indicator of the heat transfer ability of a solid, which is mainly determined by contributions from phonons (κ_lat) and carriers (κ_ele), especially in semiconductor-based thermoelectric materials.^3,4 Historically, the interwoven constraints among these parameters—particularly the inherent conflict in simultaneously achieving a large Seebeck coefficient, high electrical conductivity and minimum total thermal conductivity—have imposed a fundamental limitation on the thermoelectric performance of materials in practical applications.^5,6

Contemporary advancements primarily leverage two material engineering paradigms: manipulation of electronic band structures^7–10 and optimization of phonon scattering.^11–13 On the electronic front, strategies such as band convergence,^14,15 resonant level in electronic density of states,^16,17 and lattice plainification^18,19 were successful in enhancing the Seebeck coefficient without reducing the electrical conductivity. Concurrently, phonon engineering—including hierarchical architecture design,^20,21 nanostructuring,^22,23 high-entropy alloying,^24,25 and enhanced anharmonicity^26–29—have successfully decoupled phonon and electron transport. Despite these advances, the state-of-the-art thermoelectrics suffer from thermodynamic stability and performance saturation. Crucially, achieving ultralow thermal conductivity alongside high band degeneracy in chemically simple systems remains a significant challenge.

Zintl phase compounds exhibit rich chemical bonding hierarchies and structural diversity, making them promising candidates for simultaneously achieving low thermal conductivity and high band degeneracy. The n-type Zintl phase CaMgGe exhibits strong lattice anharmonicity due to the coexistence of covalent and ionic bonds, resulting in low κ_lat and high ZT of 3.09 at 500 K.³⁰ In NaCdSb, the cage-like structure with Cd–Sb bonding exhibits pronounced anharmonicity, suppressing κ_lat and yielding an intrinsically high ZT of ∼1.3 at 673 K.³¹ The intrinsic p-type semiconductor TlCuSe exhibits a multiband electronic structure that enhances the power factor, while its weak chemical bonds and strong anharmonicity contribute to an ultralow thermal conductivity (∼0.25 W m⁻¹ K⁻¹), resulting in a peak ZT value of ∼1.9 at 643 K.³² Recently, Pathak et al. reported an ultralow thermal conductivity of ∼0.29 W m⁻¹ K⁻¹ for the Zintl phase TlAgSe at 300 K and ascribed it to the highly anharmonic lattice and low-energy optical phonons.³³ However, the underlying microscopy mechanism of phonon scattering responsible for such low thermal conductivity remains unclear. Furthermore, the thermoelectric performance of TlAgSe reported to date is relatively modest, and a systematic evaluation is still lacking, which is crucial for developing potential applications.

To gain a comprehensive understanding of the mechanisms underlying the ultralow thermal conductivity and thermoelectric potential in TlAgSe, first-principles calculations combined with self-consistent phonon theory and the Boltzmann transport equation were employed. The results reveal exceptional thermoelectric performance arising from its unique hierarchical bonding architecture and multi-valley electronic band structure. Quartic anharmonicity, driven by bond hierarchy and phonon frequency renormalization, significantly suppresses lattice thermal conductivity to approximately ∼0.31 W m⁻¹ K⁻¹ at 300 K when four-phonon scattering processes are included, showing excellent agreement with experimental measurements. This finding underscores the critical role of higher-order anharmonic effects and phonon frequency renormalization in achieving ultralow thermal conductivity in solid-state materials. The calculated electronic structure identifies TlAgSe as a direct-bandgap semiconductor with multi-valley characteristics at both valence and conduction band edges, leading to high band degeneracy and sharply peaked density of states, and contributing to superior electrical transport properties. A peak ZT of ∼3.06 is achieved at 600 K under p-type doping (∼4.82 × 10¹⁹ cm⁻³), surpassing the n-type counterpart (ZT ∼2.80), primarily due to the preservation of high Seebeck coefficients despite a moderate reduction in electrical conductivity. This study not only establishes TlAgSe as a promising candidate for high-performance thermoelectric applications in both p- and n-type configurations but also provides actionable insights for performance optimization through band and phonon engineering.

2. Computational methods

The involved first-principles calculations were conducted within the projector augmented wave (PAW) method as implemented in the Vienna Ab initio Simulation Package (VASP).³⁴ The generalized gradient approximation (GGA) in the scheme of revised Perdew–Burke–Ernzerhof (PBEsol)³⁵ was utilized to optimize the crystal structure and atomic coordinates. The Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional^36,37 was adopted to obtain the electronic band structure. The LOBSTER code was utilized to evaluate the chemical bonding strength through the crystal orbital Hamiltonian population (COHP).³⁸ The ab initio molecular dynamics (AIMD) simulations³⁹ were performed within the canonical ensemble using a Nose–Hoover thermostat (NVT) at 300 and 600 K with a time step of 1 fs and a total time 3 ps. The ALAMODE code^40–42 was used to obtain the second-, third- and fourth-order interatomic force constants (IFCs). The finite displacement method was utilized to extract the second-order IFCs from configurations. The accurate third- and fourth-order IFCs were obtained using machine-learning techniques within the compressed sensing lattice dynamics (CSLD) method.⁴³ The cutoff interactions up to the 10th and 4th nearest neighbors were adopted to calculate the anharmonic third- and fourth-order IFCs, respectively. The temperature-dependent phonon spectrum and phonon renormalization were calculated based on self-consistent phonon (SCPH) theory.^44,45 After obtaining the second-, third-, and fourth-order IFCs, the thermal transport properties were calculated using the ShengBTE package.^46,47 The electrical transport properties with a variety of scattering mechanisms were calculated through Onsager coefficients as implemented in the ab initio scattering and transport (AMSET) code,⁴⁸ which is widely used for both two-dimensional and bulk thermoelectric materials.^49–52 Details of the methods can be found in SI.

3. Results and discussion

3.1 Crystal structures and electronic structures

The crystal structure of TlAgSe belongs to the Pnma space group, as shown in Fig. 1(a). In the unit cell, the Tl⁺ ions are located in the cavities formed by edge-shared (AgSe₄)⁻ tetrahedra. The optimized lattice constants of a = 8.48 Å, b = 4.69 Å, and c = 7.66 Å are consistent with the experimental values (a = 8.88 Å, b = 4.88 Å, and c = 7.77 Å).³³ Fig. 1(b) shows the coordination environments of Ag and Tl with neighboring Se atoms. The variation in Ag–Se bond lengths within (AgSe₄)⁻ tetrahedra (ranging from 2.693 Å to 3.060 Å) indicates the anisotropic bonding strength and non-uniform interatomic force constants. In contrast, the Coulomb repulsion from the 6s² lone-pair electrons on Tl⁺ leads to the formation of coplanar and nearly identical Tl–Se bonds. Fig. 1(c) displays the charge density distribution in TlAgSe. Overlapping pink isosurfaces indicate covalent bonding, whereas the isolated spheres refer to the ionic interactions. Clearly, both covalent and ionic chemical bonds can be found in (AgSe₄)⁻ tetrahedra, illustrating the coexistence of covalent and ionic bonds. The isolated spheres surrounding the Tl and Se atoms suggest the formation of ionic bonds, which are weaker than the covalent Ag–Se bonds.

Fig. 1 (a) Crystal structures of TlAgSe. (b) Coordination environment of Ag and Tl atoms with Se atoms. The values indicate the bond lengths. (c) Charge density of TlAgSe. (d) Calculated electronic band structure and atomic projected density of states (PDOS). (e) Crystal orbital Hamilton population (COHP) and integrated crystal orbital Hamilton population (ICOHP) of TlAgSe.

Utilizing the HSE06 hybrid functional, the electronic band structure and projected density (PDOS) of states of TlAgSe were calculated, as displayed in Fig. 2(d). The valence band maximum (VBM) and conduction band minimum (CBM) are located along the Γ–X direction, suggesting that TlAgSe is a direct-bandgap semiconductor with a band gap of 1.24 eV. The VBM is mainly contributed to by the Tl-6s, Ag-4d, and Se-2p orbitals, while the CBM predominantly originates from the Tl-6p, Ag-4s, and Se-2p orbitals. Notably, multi-valley characteristics are observed within a narrow energy range in both the VBM and CBM. The primary VBM lies along the Γ–X direction, with a second valley located along the Γ–Y direction, approximately 13 meV lower in energy than the VBM. A third valley located at the Γ point is about 102 meV below the VBM. A similar feature is found in the conduction band, where the second valley located at the T point lies ∼15 meV above the CBM. Compared to SnSe, a record high ZT thermoelectric material, with 162 meV energy separation between the VBM and its second valence valley, TlAgSe exhibits a higher degree of band convergence at both band edges. These multi-valley band structures result in a sharp peak in the PDOS, and thus increase the effective mass of the carriers, which is beneficial for obtaining larger Seebeck coefficients in both p- and n-type doped TlAgSe. Furthermore, both the valence and conduction band extrama (VBM/CBM) lie along high-symmentry lines in the Brillouin zone, inducing larger band degeneracy that enables enhanced Seebeck coefficients while maintaining carrier mobility. These characteristics work together to greatly enhance the power factor in both p- and n-type doped TlAgSe, resulting in higher ZT values.

Fig. 2 (a) Finite-temperature phonon dispersion, (b) phonon density of states at 300 K and 600 K, and mode-dependent Grüneisen parameters of TlAgSe. (c) Comparison of anisotropic atomic displacement parameters and experimental values.³³ (d) Anisotropic relative regular residuals (RRR) of Tl, Ag, and Se atoms in TlAgSe as a function of atomic displacement.

The bonding strength under different coordination environments was evaluated through the crystal orbital Hamiltonian population (COHP),³⁸ as shown in Fig. 1(e). The projected COHP (pCOHP) is an indicator of bonding and anti-bonding states between atom pairs: the positive and negative –pCOHP refer to the bonding and anti-bonding states, respectively. The Tl–Se bond is a stable ionic bond, while the chemical bond between Ag and coordinated Se atoms presents varying strengths, as displayed by the integrated COHP (–ICOHP). Obviously, significant anti-bonding states are observed near the VBM in the Ag–Se bonds, which are composed of Ag-4d and Se-2p orbitals, as displayed in Fig. 1(d). Those anti-bonding states lead to a decrease in the bonding strength of the (AgSe₄)⁻ tetrahedron and result in strong lattice anharmonicity. The variation of Ag–Se bond lengths in the (AgSe₄)⁻ tetrahedron refers to the hierarchical bonding strength between Ag and Se atoms. Specifically, the longer Ag–Se1 bond presents a weaker ionic bonding strength, while the compressed Ag–Se3 bond is stronger due to the covalent bonding character, as displayed in Fig. 1(c). Clearly, the coexistence of ionic and covalent bonds suggests a complex and hierarchical chemical interaction network in TlAgSe. The differing bond lengths among Ag and its neighboring Se atom introduce structural complexity, which is crucial for enhancing the anharmonicity of TlAgSe.

The mechanical stability of TlAgSe were evaluated based on elastic constants calculated through the stress–strain method, as listed in Table S1. Due to its orthorhombic crystal structure, TlAgSe possesses six independent elastic moduli. The calculated values satisfy the Born criteria for mechanical stability:⁵³ C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁ > C₁₂, C₁₁ + C₃₃ > 2C₁₃, and 2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃ > 0, suggesting the good mechanical stability of TlAgSe. AIMD simulations of the total energy and the initial and final atomic configurations as a function of time-step at 300 K and 600 K (Fig. S1) show no obvious bond breaking or structural reconstruction during the simulations, suggesting a high thermodynamic stability of TlAgSe up to a medium temperature.

3.2 Phonon dispersion and lattice thermal conductivity

Heat transfer in a semiconductor is mainly contributed to by lattice vibrations.^10,54–56 Strong four-phonon anharmonic effects result from the existence of hierarchy chemical bonding, leading to significant renormalization of phonon frequencies.^57,58 Fig. 2(a) displays the calculated phonon dispersion at finite temperature along the high-symmetry directions. The absence of negative frequencies indicates the dynamic stability of TlAgSe. The phonon spectrum consists of 36 phonon branches, including 3 acoustic branches and 33 optical branches, due to the 12 atoms in the unit cell. The frequencies harden with increasing temperature when fourth-order anharmonic renormalization is considered, resulting in an increase in phonon group velocity. Notably, the coupling between acoustic and optical branches can be observed at around ∼30 cm⁻¹, which favours enhanced phonon scattering and lowers the thermal conductivity. Fig. 2(b) shows the projected phonon density-of-states (PhDOS) and mode-dependent Grüneisen parameters. The left panel illustrates the different contributions of the constituent atoms to the phonon frequency in TlAgSe. Heavy Tl and Ag atoms mainly contribute to the acoustic phonon branches, while Se atoms predominantly contribute to the optical phonon branches. The phonon frequency at 600 K is hardened compared to that at 300 K, suggesting strong anharmonicity at elevated temperatures. The right panel of Fig. 2(b) shows the mode-dependent Grüneisen parameters. The larger Grüneisen parameters for both acoustic and optical phonon modes illustrate the significant anharmonic behavior in TlAgSe, which originates from the hierarchical bonding character.

Fig. 2(c) displays the temperature-dependent mean-squared displacement (MSD) of each element in TlAgSe. Clearly, direction-dependent MSD is observed, suggesting anisotropic atomic vibrations and the presence of hierarchical chemical bonding in TlAgSe. The Ag atom presents significant atomic displacement, especially along the x-direction, as illustrated by the inserted displacement ellipsoid at 600 K. This enhanced displacement is attributed to the weak interaction between Ag and Se atoms along the x-direction within the (AgSe₄)⁻ tetrahedron, exhibiting a rattler-like behavior. Moreover, Tl atoms present relatively larger MSDs along all three directions, resulting from their weak bonding environment and the lone pair of electrons of Tl⁺. These anisotropic displacements align well with the refined atomic displacement parameter reported by Pathak et al.³³ The hierarchical bonding strength and rattler-like behavior contribute to increased phonon scattering, thereby leading to a reduction of the lattice thermal conductivity of TlAgSe.

To evaluate the significance of the fourth-order IFCs in TlAgSe, the anisotropic potential energy wells and relative regular residual (RRR) analysis were conducted.⁵⁹ Fig. S2 shows the calculated anisotropic potential energy wells for each atom. Clearly, the Ag atom along the x-direction possesses the shallowest potential well, followed by the Tl atom, whose potential wells along all three directions are relatively shallow, suggesting a weak Ag–Se bonding strength along the x-axis, a weak Tl–Se bond, and the existence of rattler-like vibrations for both Tl and Ag atoms. The Se atoms exhibit deeper wells along three axis directions, suggesting stronger bonding. The variation in potential well depths illustrates the hierarchical nature of the chemical bonding, which is a key factor for intrinsically low lattice thermal conductivity in solids. The RRRs of TlAgSe shown in Fig. 2(d) reveal that Tl (z axis), Ag (x, z axis), and Se (x, z axis) atoms show a “∼”-shape curve, underscoring the dominance of three-phonon scattering. Conversely, the RRRs for Tl (x, y axes), Ag (y axis), and Se (y axis) atoms present “w”– or “m”-shape characteristics, illustrating significantly quartic anharmonicity.^58,59 Consequently, four-phonon scattering emerges as an essential mechanism in TlAgSe, illustrated by a strong high-order anharmonic effect.

The aforementioned discussion collectively suggests the intrinsically low lattice thermal conductivity (κ_lat) of TlAgSe, which is an essential parameter for thermoelectric performance evaluation.^60,61 The κ_lat of TlAgSe was evaluated by taking into account the effects of the quartic anharmonic renormalization of phonon frequencies by solving the Boltzmann transport equation as implemented in the ShengBTE code. The convergence test for lattice thermal conductivity was conducted, as displayed in Fig. S3(a). Fig. 3(a) shows the κ_lat of TlAgSe over the temperature range of 300–600 K. At room temperature, κ_lat considering only three-phonon (3ph) scattering is 0.72 W m⁻¹ K⁻¹, while upon incorporation of four-phonon (4ph) scattering, κ_lat is reduced to ∼0.31 W m⁻¹ K⁻¹. This value is remarkably lower than those of established intrinsically low κ_lat compounds, such as SnSe,^62,63 BiSbSe₃,⁶⁴ TlSe,⁶⁵ and TlInTe₂.⁶⁶ The inclusion of 4ph scattering correlates well with the reported experimental value (∼0.29 W m⁻¹ K⁻¹ @ 300 K), confirming the importance of high-order phonon scattering and finite-temperature phonon frequency renormalization in accurately predicting κ_lat, especially in systems with intrinsically ultralow thermal conductivity. Notably, a systematic deviation between calculated and experimental results becomes pronounced at elevated temperatures. This discrepancy likely originates from additional carrier scattering mechanisms in polycrystalline samples, particularly grain boundary scattering and nano-precipitate interactions.³³ Considering the inherent anisotropic crystal structure of the TlAgSe, the direction-dependent κ_lat is shown in Fig. S3(b). Clearly, the lowest κ_lat can be found along the x-direction, which results from the rattler-like vibrations of Tl and Ag atoms along the x-direction, as illustrated by the shallow potential wells (Fig. S2) and larger MSD (Fig. 2(c)).

Fig. 3 (a) Comparison of the calculated and experimental³³ lattice thermal conductivity as a function of temperature. (b) Frequency-dependent group velocity. (c) Cumulative and differential lattice thermal conductivity as a function of phonon frequency. The vertical dashed line refers to the cutoff frequency of the acoustic branch. (d) Normalized lattice thermal conductivity as a function of phonon mean free path.

Fig. 3(b) plots the mode-dependent group velocity of TlAgSe at 300 K. Small group velocities are usually observed in weak chemical bonding interactions present in intrinsically low thermal conductivity compounds, such as AgCuTe,⁶⁷ BiSeI,¹³ MgAgSb,⁶⁸ and TlAgTe.^69,70 For optical phonon branches, most group velocities are lower than 1000 m s⁻¹, attributed to the weak chemical bonding between Ag and coordinated atoms. In contrast, the group velocities of acoustic phonon branches are higher than those of optical phonon branches, attributed to the stronger chemical bonds between Tl and Se atoms.

The cumulative and differential lattice thermal conductivity plots of the TlAgSe as a function of phonon frequency at 300 K are displayed in Fig. 3(c). The contributions of the acoustic phonon mode are 76.0%, 69.5%, and 69.7% to the lattice thermal conductivity along x-, y-, and z-directions, respectively, confirming their dominant role in heat transport of TlAgSe. A sharp reduction in differential lattice thermal conductivity occurs uniformaly along all the three directions at the acoustic cutoff frequency. This trend illustrates the effective suppression of lattice thermal conductivity in these directions, attributable to strong coupling between acoustic and optical phonons. For the optical phonon modes, the lower frequency range (<60 cm⁻¹) mainly contributes to the lattice thermal conductivity along all three directions. Fig. 3(d) displays the normalized cumulative lattice thermal conductivity as a function of the phonon mean free (MPF) path along the three axes at 300 K. To achieve 50% suppression of lattice thermal conductivity, the required MPF values are 5.0 nm, 11.5 nm, and 10.9 nm along the x-, y-, and z-directions, respectively. These results provide a strategy through which the lattice conductivity of TlAgSe can be further reduced through nanostructuring.

To gain further understanding of the mechanism of low thermal conductivity of TlAgSe, the thermal transport behavior was systematically investigated. Fig. 4 shows the weighted phase spaces of TlAgSe under the three-phonon (WP3) and four-phonon (WP4) scattering mechanisms. The weighted phase space depicts the available phonon scattering channels among all modes; namely, a larger weighted phase space implies more scattering channels and a higher probability of phonon scattering.^71,72 As depicted in Fig. 4(a), the weighted phase space values under 4ph scattering are obviously higher than those in the 3ph scattering scenario, signifying heightened prevalence of four-phonon scattering processes under energy-conservation constraints.

Fig. 4 (a) Total weighted scattering phase space of the three-phonon (WP3) and four-phonon (WP4) scattering processes in TlAgSe as a function of phonon frequency. (b) Weighted phase space associated with the splitting and recombination processes within the three-phonon scattering process. (c) Weighted phase space encompassing the splitting, recombination, and redistribution processes in the four-phonon scattering process. (d) Scattering rates for the isotope, boundary, three-phonon, and four-phonon scattering processes. (e) Scattering rates pertaining to the splitting and recombination processes in the three-phonon scattering scenario. (f) Scattering rates related to the splitting, recombination, and redistribution processes in the four-phonon scattering process.

Fig. 4(b) and (c) show the phonon frequency-dependent WP3 and WP4 contributions of TlAgSe under different scattering channels. The WP3 primarily originates from the combination process in the low-frequency region, with the main contribution from the acoustic phonon mode. This phenomenon is ruled by the conservation of energy and momentum in phonon scattering, preferentially enabling 3ph combination processes that convert low-frequency phonons into their high-frequency counterparts, thereby suppressing the available phase space for phonon splitting processes.^71,73 Consequently, the scattering channel for the splitting process dominates in the mid to high frequency range, as displayed in Fig. 4(b). For 4ph scattering, the splitting processes diminish in the low-frequency region (<30 cm⁻¹) and then intensify with increasing frequency, which results from the energy-conservation constraints.⁷⁴ Therefore, the redistribution processes play the predominant role in the low-frequency region. The splitting processes occur primarily across the higher frequency region through relaxed momentum conservation criteria, emerging as the dominant mechanism for the suppression of lattice thermal conductivity. Thus, low-frequency phonons are primarily governed by 3ph combination and 4ph redistribution processes, whereas high-frequency phonon generation is dominated by 4ph splitting processes.

Fig. 4(d) displays the phonon frequency-dependent scattering rates of TlAgSe across multiple mechanisms at 300 K, including isotope, boundary, three-phonon (3ph), and four-phonon (4ph) scattering. The 4ph scattering dominates the low- and medium-frequency regions, which results from the coupling of the acoustic and optical phonons induced by the lone-pair electrons of the Tl atom. At high frequencies, the contribution of 3ph scattering becomes comparable to that of 4ph, as illustrated in Fig. 4(e) and (f), emphasizing the critical role of optical modes in enhancing anharmonic scattering and reducing lattice thermal conductivity. Specifically, the low-frequency scattering is predominantly driven by 3ph combination processes alongside 4ph redistribution processes, where strong 4ph redistribution suppresses splitting by limiting quantum states for emitted phonons and enforcing energy-momentum conservation.⁷⁴ Conversely, in the medium-frequency range, scattering rates originate from both splitting and redistribution processes in 4ph scattering, as illustrated in Fig. 4(f), resulting from the redistribution process more readily meeting scattering selection rules.^75,76

3.3 Thermoelectric transport properties of TlAgSe

The carrier transport properties of TlAgSe were systematically investigated with multiple carrier scattering mechanisms considered, including acoustic deformation potential (ADP) scattering, polar optical phonon (POP) scattering, and ionized impurity (IMP) scattering. The overestimation of the mobility can be addressed through the hybrid HSE06 functional, which improves the agreement with experimental mobility and the fully first-principles calculations performed using the EPW code. The related parameters to perform the AMSET calculation are listed in Table S1. The carrier-concentration-dependent scattering rates are shown in Fig. S4(a). The POP scattering rates decrease and the IMP scattering rates increase with increasing carrier concentration, while the ADP scattering rates exhibit trends in both the p- and n-type doping that are nearly carrier concentration independent. Notably, the POP scattering is significantly higher than ADP and IMP scattering in the low-carrier-concentration regime, which results from the polar chemical bonds inducing strong coupling interaction between carriers and optical phonons. The IMP scattering intensifies with increasing carrier concentration and becomes predominant in very high concentrations. Overall, the total scattering rate undergoes a decline first and then tends to increase over the whole carrier concentration range for both the p- and n-type doping. The lower scattering rates and smaller effective mass of electron carriers result in higher carrier mobility in n-type doping, as presented in Fig. S4(b) and Table S2, suggesting the superior electrical conductivity of n-type doped TlAgSe.

The thermoelectric transport properties in TlAgSe as a function of carrier concentration at different temperatures for p- and n-type doping are presented in Fig. 5. The electrical conductivity (σ) is proportional to the carrier mobility (µ) and carrier concentration (n). As illustrated in Fig. S4(b), the carrier mobility increases with increasing carrier concentration, enabling more carriers to participate in transport and thus significantly enhancing electrical conductivity. Notably, the σ for electrons is considerably larger than that for holes, as depicted in Fig. 5(a), which originates from the smaller scattering rates (Fig. S4(a)), larger carrier mobility (Fig. S4(b)) and smaller effective mass of the conduction band (Fig. 1(c) and Table S2).

Fig. 5 Evaluated thermoelectric properties of p-type and n-type TlAgSe at 300, 450, and 600 K. (a) Electrical conductivity (σ), (b) Seebeck coefficient (|S|), (c) power factor (PF), (d) electronic thermal conductivity (κ_ele), (e) total thermal conductivity (κ_tot), and (f) figure of merit (ZT) values.

Different from the carrier-dependence of σ, the Seebeck coefficient (S) is proportional to the density of states effective mass and inversely proportional to the carrier concentrations. Clearly, |S| declines with increasing carrier concentration at a fixed temperature, as shown in Fig. 5(b), while |S| increases with increasing temperature at constant carrier concentration. The larger |S| is usually anticipated from the larger density of states effective mass of carrier. The calculated effective mass for VBM is larger than that for CBM, as listed in Table S2. Thus, the Seebeck coefficient of p-type doping is significantly larger than that in n-type doping at the same temperature and carrier concentration, resulting from the flatter valence band and smaller energy separation between the valleys than in the conduction band, as seen in Fig. 1(d).

Due to the opposing dependence of σ and |S| on carrier concentration, the optimized power factor (PF = σS²) can be anticipated over the evaluated carrier concentration range. The carrier-concentration-dependent PF values at 300, 450, and 600 K for p- and n-type TlAgSe are presented in Fig. 5(c). Evidently, the PF first increases and then decreases with increasing carrier concentration at each temperature. The PF of n-type doping TlAgSe is higher than that of p-type doping across the same carrier concentration and temperature range, illustrating the predominant contribution of electrical conductivity to PF. This outcome stems from the synergistic effect of the effective mass at the CBM, which reduces the scattering rate and enhances both the mobility and PF under n-type doping conditions. Notably, the peak PF of n-type doping reaches ∼85.2 µW cm⁻² K⁻¹ at 600 K, significantly exceeding the ∼64.6 µW cm⁻² K⁻¹ observed in p-type doping, suggesting higher output power potential in the TlAgSe-based thermoelectric modules.

The electronic thermal conductivity (κ_ele) results from the thermal motion of carriers and presents the same dependence of electrical conductivity on carrier concentration. The relationship between electronic thermal conductivity and electrical conductivity follows the Wiedemann–Franz law: κ_ele = LσT, where L is the Lorentz number. Fig. 5(d) displays κ_ele as a function of carrier concentration at different temperatures for p- and n-type TlAgSe. Clearly, κ_ele increases with increasing carrier concentration, which is in accordance with the trend of electrical conductivity (Fig. 5(a)). The superior electrical conductivity of n-type doping led to higher κ_ele than that in p-type doping. In wide bandgap semiconductors, the total thermal conductivity (κ_tot) is mainly composed of two parts: κ_tot = κ_lat + κ_ele. As previously discussed, TlAgSe is intrinsically characterized by low lattice thermal conductivity. Therefore, the significant increase in κ_ele becomes the predominant contributor to κ_tot in the high-carrier-concentration region, as displayed in Fig. 5(e).

By integrating the assessment of electrical and total thermal transport properties, the temperature-dependent figure-of-merit (ZT) as a function of carrier concentration for both p- and n-type TlAgSe is shown in Fig. 5(f). The ZT value initially increases and subsequently decreases with increasing carrier concentration, resulting from the continuous rise in κ_tot. At a given temperature, p-type doping achieves higher ZT values compared to n-type doping due to the moderate PF values and the lower κ_tot. For p-type doping, the peak ZT increases from ∼1.66 at 300 K to ∼3.06 at 600 K, with optimal carrier concentrations of ∼2.87 × 10¹⁹ cm⁻³ and ∼4.82 × 10¹⁹ cm⁻³, respectively. While for n-type doping, a relatively lower ZT (∼1.42 @ 300 K) can be obtained in n-type doping since the larger κ_ele induced by the excellent electrical conductivity. These results highlight TlAgSe as a potential candidate for high-performance thermoelectric materials.

4. Conclusion

This comprehensive study establishes TlAgSe as a pioneering thermoelectric material through first-principles calculations and Boltzmann transport analyses. The orthorhombic Pnma structure of TlAgSe features synergistic bonding heterogeneity. Shallow potential wells and large anisotropic mean squared displacements of Ag and Tl atoms reveal the rattler-like vibration and strong anharmonicity. All these factors lead to ultralow lattice thermal conductivity. Dynamic stability is confirmed by non-imaginary phonon modes and AIMD simulations. Furthermore, quartic anharmonicity, evidenced through relative regular residual analysis, activates dominant four-phonon scattering. As a result, κ_lat is reduced to 0.31 W m⁻¹ K⁻¹ at 300 K (a 56% reduction compared to predictions based on only three-phonon scattering) via enhanced phonon splitting processes above 30 cm⁻¹ and redistribution at lower frequencies. The multi-valley band structure enables superior electronic transport, resulting in exceptional electrical transport properties. A peak ZT of ∼3.06 is achieved in p-type TlAgSe at 600 K (∼4.82 × 10¹⁹ cm⁻³), outperforming the n-type counterpart (∼2.80 @ 600 K), owing to a favorable balance between Seebeck coefficients and moderate electrical conductivity. This work not only addresses TlAgSe as a candidate with high potential for high-performance thermoelectric applications but also provides critical insights and guidelines for performance optimization.

Conflicts of interest

The authors declare no conflict of interest.

Data availability

The data that supports the findings of this study are available from the corresponding authors upon reasonable request.

Supplementary information (SI): supplementary methods; Table S1: the parameters used in the electronic transport calculation with AMSET code; Table S2: the effective mass of VBM and CBM; Fig. S1: the total energy of AIMD simulations; Fig. S2: the anistropic potential energy well for each element. Fig. S3: the convergency tests for lattice thermal conductivity calculation. Fig. S4: the hole and electron scattering rates and mobilities. See DOI: https://doi.org/10.1039/d5ta09360k.

Acknowledgements

This work was supported by the Natural Science Foundation of Henan Province (No. 242300420634), the Research Foundation for Advanced Talents of Henan University of Technology (No. 2022BS067 and 2022BS068), the National Natural Science Foundation of China (No. 12504034 and 12204156), the Key R&D and Promotion Special Project (Science and Technology Research) in Henan Province (No. 232102211068), the Innovative Funds Plan of Henan University of Technology (No. 2022ZKCJ15), the Natural Science Foundation of Hebei Province (No. E2024106001), and the Science Research Project of Hebei Education Department (No. QN2025363).

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