Abstract
We employ firstprinciples calculations combined with selfconsistent phonon theory and Boltzmann transport equations to investigate the thermal transport and thermoelectric properties of fullHeusler compound Na_{2}TlSb. Our findings exhibit that the strong quartic anharmonicity and temperature dependence of the Tl atom with rattling behavior plays an important role in the lattice stability of Na_{2}TlSb. We find that soft TlSb bonding and resonant bonding in the pseudocage composed of the Na and Sb atoms interaction is responsible for ultralow κ_{L}. Meanwhile, the multivalley band structure increases the band degeneracy, results in a high power factor in ptype Na_{2}TlSb. The coexistence of ultralow κ_{L} and high power factor presents that Na_{2}TlSb is a potential candidate for thermoelectric applications. Moreover, these findings help to understand the origin of ultralow κ_{L} of fullHeusler compounds with strong quartic anharmonicity, leading to the rational design of fullHeusler compounds with high thermoelectric performance.
Similar content being viewed by others
Introduction
Thermoelectric (TE) materials are capable of converting heat energy into electricity without additional pollution, promising to alleviate the energy shortage and environmental pollution caused by energy use^{1,2,3}. Conventionally, the conversion efficiency of TE materials is measured by the figure of merit \(zT=\frac{{S}^{2}\sigma T}{{\kappa }_{{{{\rm{e}}}}}+{\kappa }_{{{{\rm{L}}}}}}\), in which S, σ, κ_{e}, κ_{L}, and T are the thermopower, electrical conductivity, electrical thermal conductivity, lattice thermal conductivity, and absolute temperature, respectively. In principle, highefficiency TE materials need to possess both a high TE power factor (PF = S^{2}σ) and low thermal conductivities (κ = κ_{e} + κ_{L}). The former involves band tuning^{4,5}, applying strain^{6}, and doping^{7,8}. Meanwhile, to capture an inherently high PF, the anisotropic electronic bands and multivalley band structure have been proposed^{9,10,11,12,13}, in which the dispersive part leads to high σ and carrier mobility μ, while the flat (multivalley) part induces large S. However, due to the mutual coupling of S and σ, it is difficult to improve the zT value by tuning a single parameter. To address this problem, common strategies are to suppress the κ_{L} of existing TE materials through introducing defects^{14,15}, alloying^{16,17}, nanostructures^{18}, and substructures^{19,20}, or to search for TE materials with inherently low κ_{L}.
Recently, high zT values of a series of fullHeusler compounds have been reported, such as Ba_{2}BiAu^{21}, Sr_{2}SbAu^{22}, Ca_{2}HgPb^{23}, and Ba_{2}AgSb^{24}, which is mainly attributed to the ultralow κ_{L}. Particularly, a very high zT value of 5 has been reported in fullHeusler compounds Ba_{2}BiAu^{21}. Hence, understanding the lattice dynamic behavior and phonon thermal transport in fullHeusler compounds is critical to gain insights into which can further suppress κ_{L} and enhance zT. However, existing theoretical studies only focus on fullHeusler compounds with lattice stability (no imaginary phonon frequencies) at 0 K. However, many materials with ferroelectriclike lattice instability also have good TE properties, such as Rb_{3}AuO^{25}, SnSe^{26}, Cu_{2}Se^{27}, and GeTe^{16,28}. However, the comprehensive understanding of fullHeusler compounds is hampered by the existence of imaginary phonon frequencies and the failure to consider higherorder anharmonic effects. Especially for some TE materials with imaginary phonon frequencies, the fourphonon (4ph) scattering processes even exceed threephonon (3ph) scattering processes^{16,29,30}.
In view of the above, we perform fistprinciples calculations, selfconsistent phonon (SCP) theory, and Boltzmann transport equations to investigate the thermal transport and TE properties in fullHeusler compound Na_{2}TlSb. We consider the contribution of quartic anharmonic renormalization of phonon frequencies to phonon group velocity υ_{ph} as well as the influence of 3ph and 4ph scattering to the phonon lifetimes τ_{ph}. The phonon spectrum, thermal conductivity spectrum κ_{L} (ω_{ph}), 3ph and 4ph scattering processes, etc. are investigated to uncover the microscopic mechanism of thermal transport. Our key finding is that a rational κ_{L} and temperature dependence can be obtained by including full quartic anharmonicity. Meantime, the ultralow κ_{L} is captured in Na_{2}TlSb due to the soft TlSb bonding and resonant bonding in the pseudocage composed of the Na and Sb atoms interaction. Additionally, the multivalley band structure increases the band degeneracy, resulting in a high PF in Na_{2}TlSb. High PF combined with ultralow κ_{L} means good TE performance, with the highest zT values of 2.88 (0.94) at 300 K in ptype (ntype) fullHeusler compound Na_{2}TlSb.
Results
Structural stability
Na_{2}TlSb crystallizes in the facecentered cubic structure (space group Fm\(\overline{3}\)m [225]), where Na, Tl, and Sb take up the 8c, 4a, and 4b sites, respectively, as shown in Fig. 1d. The calculated lattice constants of the crystal cell is 7.485 Å, and the bond length of NaSb/Tl (TlSb) is 3.241 (3.743) Å. The dielectric tensors ϵ and Born effective charges Z* are listed in Supplementary Table 1. To investigate the dynamics stability, we calculated the HA and anharmonic phonon dispersion curves of Na_{2}TlSb, as shown in Fig. 1a. In HA approximations, the imaginary frequencies indicate that Na_{2}TlSb is dynamically unstable at 0 K. However, after considering the renormalization of phonon frequencies by the quartic anharmonicity, the imaginary frequencies disappear, indicating that Na_{2}TlSb is dynamics stable between 100 and 700 K. Furthermore, as shown in Supplementary Fig. 6, Na_{2}TlSb is on the convex hull in the ternary phase diagram, which indicates it is the minimum free energy structure under this component. Hence, the fullHeusler compound Na_{2}TlSb is most likely to be experimentally synthesized to form a stable structure. Additionally, a 20,000step AIMD is simulated to estimate the stability of Na_{2}TlSb at high temperatures, as shown in Supplementary Fig. 1. There is no significant change in free energy, indicating that Na_{2}TlSb is stable at 700 K. Furthermore, the mechanical stability is also estimated, and the elastic constants are list in Supplementary Table 1. Na_{2}TlSb meets the mechanical stability criteria of the cubic lattice structure, as written in
indicating Na_{2}TlSb is mechanically stable^{31}.
Phonon transport
For strongly anharmonic materials, quartic anharmonicity needs to be considered to obtain reasonable κ_{L} and temperature dependence^{32}. First, we investigate the impact of quartic anharmonicity on phonon frequencies ω_{ph}. We find that the phonon modes below 50 cm^{−1} shift up significantly with increasing temperature, as shown in Fig. 1a. Figure 1b, c reveal that the vibrations of Tl atoms dominate these phonon modes with imaginary frequencies. While the optical phonon branches, mainly contributed by the vibrations of Sb and Na atoms, is shifted upward only slightly. The vibrational behavior associated with the Tl atom is similar to that of the rattling modes in clathrates and skutterudites, whose phonon frequencies exhibit small values and strong temperature dependence^{33}. Actually, avoiding the overlap of acoustic and optical phonon modes is a clear sign of the rattling behavior of Tl atoms^{34}. The large mean square displacements (MSDs) and weak bonding of Tl atoms also confirm the above conclusion, as shown in Figs. 1e and 5b. Additionally, the phonon frequency shifts caused by cubic anharmonicity is not included in our calculations. Generally, the frequency shifts due to cubic anharmonicity is negative (Δω_{q} < 0) and much smaller than the frequency hardening induced by quartic anharmonicity^{33}. If the frequency shifts caused by cubic anharmonicity is considered, the hardening of lowlying phonon mode will be slightly suppressed, resulting in a slight decrease in the κ_{L}.
The imaginary HA phonon frequency will induce the phase transition of Na_{2}TlSb at low temperatures. To predict the phase transition temperature T_{c} of Na_{2}TlSb, we also calculated the temperature dependence of the squared frequency of the lowest acoustic phonon mode at X point, which is the softest mode in the region away from the Brillouin zone center, as shown in Supplementary Fig. 4. Above approximately room temperature, the temperature dependence of squared frequency can be rationally fitted by equation \({\Omega }_{{{{\bf{q}}}}}(T)={a}{({T}{{T}}_{{{{\rm{c}}}}})}^{2}\)^{35}, where T_{c} is the phase transition temperature. Applying the above equation, we obtain the phase transition temperature T_{c} as 83 K.
We calculate the temperaturedependence lattice thermal conductivities \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}}\) and \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) from 100 to 700 K for Na_{2}TlSb, as plotted in Fig. 2a. Due to the existence of imaginary frequencies, the solution of κ_{L} under HA approximations is numerically invalid. Hence, only the κ_{L} results of SCP approximations are provided. Compared with \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}}\), \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) is significantly reduced due to the effect of 4ph scatterings. The calculated \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}}\) and \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) are 0.91 and 0.44 Wm^{−1} K^{−1} at 300 K for Na_{2}TlSb, respectively. The values of \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) is ultralow, half that of quartz glass (κ_{L} ~0.9 WK^{−1} m^{−1}). To further explore the microscopic mechanism of heat conduction, we use the κ_{L} ~ T^{−α} to analysis the temperature dependence of κ_{L}. We evaluated the temperature dependence of κ_{L} from 200 to 700 K, since κ_{L} at extremely low temperature is mainly determined by the latticespecific heat C_{V} following the Debye T^{3} law^{29}. The \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}}\) exhibits anomalously temperature dependence, \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}} \sim {T}\)^{−0.52}, which deviates far from the universal law κ_{L} ~ T^{−1}. By including 4ph scattering, the temperature dependence of the \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) is enhanced and the value of α becomes 0.84. The increased temperature dependence of \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) further confirms the importance of the quartic anharmonicity for the materials with imaginary frequencies. Additionally, thermal expansion and additional phonongrain boundary scattering also play a crucial role in determining κ_{L} and corresponding temperature dependence^{32,36}. If the above factors are taken into account when estimating the phonon thermal transport properties, κ_{L} will decrease and temperature dependence of κ_{L} will be closer to the experimental results. Furthermore, we verify the feasibility of excluding the thermal expansion and cubic anharmonicity, we calculated the \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) considering the thermal expansion and discussed the effect of the cubic anharmonicity on the phonon frequencies. As shown in Supplementary Figs. 7, 8, \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) considering thermal expansion at 700 K is 4.1% lower than that without considering thermal expansion. Moreover, we observe that the phonon frequency shift caused by cubic anharmonicity at X point is very small, only 4.21 cm^{−1} at 700 K, and this value will not significantly change the phonon spectrum. Hence, it is rational to exclude the above factors to calculate κ_{L}.
The large 3ph and 4ph SRs are one of the main reasons for the ultralow κ_{L} of Na_{2}TlSb, as shown in Fig. 2b. Similar to 3ph scattering, 4ph scattering is also important in phonon thermal transport. It is evident from the calculations that 4ph SRs even exceeds 3ph SRs below 50 cm^{−1}. The strong 4ph SRs can be attributed to the strict constraints on the 3ph scattering phase space by the selection rule^{37}. Similar to BAs, the 3ph scattering phase spaces are restricted by the selection rule due to the large phonon band gap, resulting in a weakened 3ph SRs in Na_{2}TlSb. Meantime, the selection rule has little effect on the 4ph scattering phase spaces, resulting in large 4ph SRs. The SRs curve equal phonon frequencies is also plotted by the purple line in Fig. 2b, which means that the phonon lifetime τ_{ph} is equal to the vibrational period of the phonon quasiparticle. The phonon quasiparticle image is invalid if the τ_{ph} is less than one vibrational period, i.e., the phonon annihilates before completing one vibration^{29}. As shown in Fig. 2b, all 3ph and most 4ph scattering distributions are below the curve, indicating that our BTE scheme is valid. The κ_{L} spectrum and frequency cumulative κ_{L} indicate that the major contributions to the κ_{L} are the phonon branches below 50 cm^{−1}. Meantime, the phonon branches below 50 cm^{−1} are the major contributions to the suppressed κ_{L} calculated with SCP+3,4ph relative to SCP+3ph, which is consistent with strong 4ph SRs below 50 cm^{−1}. Figure 2d show the cumulative κ_{L} as the function of maximum MFP. The result shows that heatcarrying phonons have considerable MFPs at 300 K. The maximum MFP of \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}}\) is about 210 nm, while the maximum MFP of \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) is only about 80 nm. Through nanostructuring, the \({\kappa }_{3,4{{{\rm{ph}}}}}^{{{{\rm{SCP}}}}}\) can be lower, e.g., as low as 0.27 WK^{−1} m^{−1} when the maximum MFP is limited to 10 nm.
The phonon group velocity υ_{ph} at 100 and 300 K is shown in Fig. 3a. The small υ_{ph} further confirms the ultralow κ_{L} of Na_{2}TlSb. Relative to 100 K, the υ_{ph} of 300 K is slightly larger below 50 cm^{−1}, which is consistent with the change of the phonon dispersion curves form 100 to 300 K. The observed small specific heats C_{V} further indicates the existence of ultralow κ_{L}, as shown in Fig. 3b. Figure 3c show the decomposed 3ph scattering rate. The combination (splitting) processes dominate the lowfrequency (highfrequency) 3ph SRs below 100 cm^{−1}. Due to the large phonon band gap from 100 to 130 cm^{−1}, it is difficult for lowfrequency phonon modes to be excited to highfrequency phonon modes through the combination processes. To obey the energy conservation constraint, highfrequency phonons can only be generated by the threephonon scattering process of highfrequency phonons. Hence, the 3ph SRs of highfrequency phonons exhibit the same regularity as those of lowfrequency phonons, e.g., the 3ph SRs of lowfrequency (highfrequency) modes above 130 cm^{−1} are mainly determined by the combination (splitting) processes. Figure 3d show the decomposed 4ph scattering rates. The redistribution processes dominate the 4ph scattering processes because it is easier to satisfy the selection rule^{37}. With respect to the 4ph scattering processes, due to the strong redistribution processes, the splitting processes at low frequencies is suppressed and weaker than the combination processes. It is only at very high phonon frequencies that the splitting processes are larger than the combination processes.
To further understand the microscope mechanism of the ultralow κ_{L}, we calculated the Gr\(\ddot{u}\)neisen parameter γ and 3ph scattering phase space W_{3ph}, as shown in Fig. 4a, b. Compared with the optical phonon modes, the acoustic phonon modes have larger γ, indicating that the acoustic phonon modes have stronger cubic anharmonicity. This result is consistent with our previous analysis that the rattling behavior of Tl atoms have stronger anharmonicity. Figure 4b, c show the large 3ph and 4ph scattering phase spaces, indicating that Na_{2}TlSb have large anharmonic scattering rates. Meanwhile, The decomposed anharmonic (3ph and 4ph) scattering phase spaces exhibit a similar law to the decomposed anharmonic scattering rates. Figure 4d show the decomposed 4ph SRs into normal and Umklapp processes. The Umklapp processes dominate the 4ph SRs of the entire phonon modes, indicating that the 4ph scattering processes mainly suppress heat conduction, which is consistent with the above analysis of κ_{L}. The Umklapp processes are much larger than the normal processes, which also confirms the above analysis, as shown in the inset of Fig. 4d.
Electronic structure
Figure 5a show the electronic band structure and density of states (DOS) of Na_{2}TlSb. Na_{2}TlSb is an indirect band gap semiconductor with the conduction band minimum (CBM) at the L point and the valence band maximum at the K − Γ high symmetry line. The calculated band gap is 0.08 eV using the PBEsol functional with SOC, which is in the range commonly found in TE materials. In Na_{2}TlSb, the CBM is mainly contributed by Sb and Tl atoms, while the VBM is almost entirely contributed by Sb atoms. The CBM has high electronic band dispersion, which will lead to high electrical conductivity σ. Meantime, the observed remarkable band asymmetry at CBM further manifests a good TE performance in ntype Na_{2}TlSb^{38}. A multivalley band structure at VBM results in high degeneracy. Additionally, the second (Γ − X line) and third (W point) highest valleys of the valence band are very close to the VBM, by doping, we can achieve a higher degeneracy. A large degree of degeneracy means that ptype Na_{2}TlSb has a high Seebeck coefficient S. Meanwhile, multiple valleys create additional conduction channels, resulting in high σ. Figure 5b show the projected crystal orbital Hamilton population (pCOHP) analysis. At the VBM, Na−Sb and Tl−Sb show bonding states, while Na−Tl presents antibonding states. Obviously, the Tl atoms are weakly bound to other atoms, which is consistent with the MSD results.
Since the band gap of materials is usually underestimated using PBE functional, we recalculated the band gap accurately with HSE06 functional with SOC, as shown in Supplementary Fig. 2. The band gap of HSE06 functional with SOC is 0.30 eV. Since the electronic structures of HSE06 and PBEsol functional are not obviously different except for the band gap for Na_{2}TlSb, the band gap of the HSE06 functional is used to obtain rational electronic transport properties. The carrier relaxation time τ of Na_{2}TlSb is calculated by considering the ADP, IMP, and POP scattering mechanisms, as shown in Supplementary Fig. 3.
Electron transport
Figures 6,7 show the electronic transport parameters for ptype and ntype Na_{2}TlSb, respectively. We observe that σ is proportional to carrier concentration n and inversely proportional to temperature T. The former is due to an increase in the number of carriers participating in the conduction process due to an increase in the concentration n. The latter is attributed to the increased scattering rates dominated by the electronphonon interaction caused by the temperature increase, which is consistent with the results in Supplementary Fig. 3. The ntype Na_{2}TlSb exhibits higher σ relative to the ptype Na_{2}TlSb, which is consistent with our previous analysis of the larger electronic band dispersion at the CBM. Unlike σ, the S decreases with increasing n at the same T, while the S increases with increasing T at the same n. Apparently, ptype Na_{2}TlSb exhibits a large S due to the large degeneracy at the VBM. In general, κ_{e} and σ vary in the same law with n due to the increase in heat carrier, which is consistent with the trend in Fig. 6c and d. Due to the coexistence of a large S and σ, we obtain a high TE power factor. Particularly, S have single peak at 500 and 700 K due to the bipolar effect^{39}. The above phenomenon is also observed in PbTe and PbSe with small band gaps^{40,41}. The strong bipolar effect severely suppresses the S, leading to a decrease in TE performance. As a result, the highest PF is obtained at 500 K because the bipolar effect increases with temperature. At the optimal doping concentration n, the power factor is 2.26 (2.92) m Wm^{−1} K^{−2} for ntype Na_{2}TlSb at 300 (500) K. Due to the multivalley structure at the VBM, ptype Na_{2}TlSb has a large power factor, e.g., 9.84 and 12.76 mW m^{−1} K^{−2} at 300 and 500 K, respectively. Moreover, the strong quartic anharmonicity of Na_{2}TlSb affects not only phonon thermal transport properties but also electronic transport properties. Concretely, strong quartic anharmonicity leads to a hardening of phonon frequency, thereby reducing the electronphonon coupling strength. Furthermore, the reduced electronphonon coupling strength results in higher carrier mobility and larger electrical conductivity^{42}. Hence, if the influence of quartic anharmonicity on electronic transport is included, the TE performance of Na_{2}TlSb will be further improved.
The combination of ultralow κ_{L} and high TE power factor in the Na_{2}TlSb captures an anomalously high zT ~4.81 for hole doping at n_{h} ~ 7 × 10^{19} cm^{−3} and 500 K, as shown in Fig. 8a. Meanwhile, we predict the zT ~ 2.88 at n_{h} ~ 4 × 10^{19} cm^{−3} and 300 K, which is also ultrahigh value for almost TE materials at room temperature. Due to the existing of large electronic band dispersion and remarkable band asymmetry at CBM, the zT is also very high for ntype Na_{2}TlSb, e.g., 0.94 (1.48) at n_{e} ~ 3 × 10^{18} (7 × 10^{18}) cm^{−3} and 300 (500) K, as shown in Fig. 8b. The optimum n_{e} for ntype Na_{2}TlSb is far small related to n_{h} in the ptype case, which means that ntype case is easier to achieve good TE performance. Hence, we also recommend ntype Na_{2}TlSb as a potential TE material. Additionally, the zT value calculated based on the result of \({\kappa }_{{{{\rm{3ph}}}}}^{{{{\rm{SCP}}}}}\) is also given as the lower limit. Nonetheless, Na_{2}TlSb also exhibits good TE properties, e.g., 1.05 (3.72) for ntype (ptype) case at 500 K. Additionally, thermal radiation^{43}, airinduced thermal convection^{44}, and the effect of grain boundary scattering on carrier mobility^{45} can lead to degradation of TE performance in experiments. Hence, if the above factors are taken into account, the calculated zT value will be slightly lower. Moreover, we only use the BTE to compute the electronic transport properties of Na_{2}TlSb at specific doping concentrations without considering specific dopability, which also means that further experimental and theoretical explorations are required. Additionally, to determine the thermodynamic limit of achievable dopant concentrations, we estimated defect solubility for the formation of ptype and ntype semiconductors. We considered several possible neutral defects, including Na vacancy, Tl vacancy, Au in place of Tl (Au_{Tl}), Hg in place of Tl (Hg_{Tl}), and Pb in place of Tl (Pb_{Tl}). The calculated defect formation energy (\(\Delta {{E}}_{{{{\rm{F}}}}}^{{{{\rm{def}}}}}\)) of Na (Tl) vacancy in Na_{2}TlSb is 0.10 (0.27) eV/defect in Na (Tl) poor condition. The above values are comparable to Nadoped PbTe (0.27 eV/defect)^{46}, where the hole doping concentration can achieve 10^{20} cm^{−3} at room temperature^{41,47}. The calculations indicated that the optimum hole concentration can be obtained in Na_{2}TlSb. Furthermore, the \(\Delta {{E}}_{{{{\rm{F}}}}}^{{{{\rm{def}}}}}\) of Au_{Tl} (Hg_{Tl}) is 0.75 (0.19) eV/defect. These values suggested that Hg_{Tl} is more likely to reach the optimum value of hole concentration, while Au_{Tl} is difficult. Actually, since Au (Hg) are as heavy as Tl, their introduction is expected to maintain the Tldominated soft phonons and strong anharmonicity, thereby preserving an ultralow κ_{L} in ptype Na_{2}TlSb. For these reasons, the use of Au_{Tl} and Hg_{Tl} would be a strategic doping mechanism to best trigger ptype performance despite the toxicity of Hg and the low doping solubility of Au. Similarly, the introduction of Pb_{Tl} is also expected to maintain ultralow κ_{L} in ntype Na_{2}TlSb. Hence, we also calculated the \(\Delta {{E}}_{{{{\rm{F}}}}}^{{{{\rm{def}}}}}\) of Pb_{Tl} to be −0.18 eV/defect, which indicates that Pb_{Tl} is easier to achieve the optimal doping concentration. The above results strongly indicate that the doping concentration required for optimal zT can be achieved through defects.
Discussion
In summary, we investigate the thermal and TE transport properties in Na_{2}TlSb using the firstprinciples calculations combined with SCP theory and Boltzmann transport equations, which explicitly include the phonon frequency shift and 4ph scattering caused by quartic anharmonicity. Our results indicate that the ultralow κ_{L} of Na_{2}TlSb can be explained by the small υ_{ph} and strong 3ph and 4ph scattering. The Tl atom with rattling behavior has strong temperature dependence and cubic and quartic anharmonicity, which plays an important role in the phonon without imaginary frequencies. Additionally, the lowfrequency fourphonon scattering rates in Na_{2}TlSb can match or even exceed the threephonon scattering rates. Meanwhile, the multivalley band structure at VBM increases the band degeneracy, resulting in a high TE power factor in ptype Na_{2}TlSb. Additionally, due to large electronic band dispersion and remarkable band asymmetry at CBM, the ntype Na_{2}TlSb exhibit a high σ. By considering ADP, POP, and IMP scattering, we capture a rational electronic relaxation time and transport properties. As a consequence, the ntype and ptype Na_{2}TlSb show a high TE figure of merit, whose values are 1.48 and 4.81 at the optimal carrier concentration and 500 K. Our study reveals the important role of quartic anharmonicity in phonon thermal transport, which contributes to our comprehensive understandings of ultralow κ_{L} microscopic mechanism in fullHeusler compounds. At the same time, we also provide ideas for the rational design of highperformance TE materials.
Methods
Firstprinciples calculation and CSLD
We perform firstprinciple calculations employing the VASP code^{48,49}, using the planewave basis set and projector augmentedwave method to simulate the potentials of ions cores and valence electrons^{50}. The exchangecorrelational interactions is dealt by the PerdewBurkeErnzerhof revised for solids (PBEsol) functional^{51} of the generalized gradient approximation (GGA)^{52}. We use 520 eV as the kinetic energy cutoff, and Γcentered 12 × 12 × 12 kpoint meshes to sample the whole Brillouin zone. The structure of Na_{2}TlSb is fully relaxed until the energy and HellmannFeynman force convergence criterion are less than 10^{−8} eV and 1 × 10^{−4} eV Å^{−1}, respectively. Throughout the thermal transport calculations, we consider the nonanalytic part of the dynamics matrix, which is derived using the dielectric tensor ϵ and Born effective charges Z* calculated by the density functional perturbation theory (DFPT)^{53}. All required harmonic (HA) and anharmonic interatomic force constants (IFCs) are calculated in the 2 × 2 × 2 supercell with 6 × 6 × 6 kpoint meshes, implemented in the ALAMODE package^{35,54}. Additionally, we also calculated the HA phonon dispersion curves within the 2 × 2 × 2 and 3 × 3 × 3 supercells to check the convergence of IFCs, as shown in Supplementary Fig. 4. It can be observed that the HA phonon dispersion curves agree well with each other, indicating the harmonic IFCs exacted form the 2 × 2 × 2 supercell are enough to capture convergence results. Furthermore, it can be deduced that anharmonic IFCs exacted from 2 × 2 × 2 supercell are also convergent, since anharmonic IFCs generally converge faster than harmonic IFCs. Specifically, the HA IFCs were extracted in three configurations produced by the finitedisplacement method^{55}. To obtain the displacement and force datasets required for anharmonic (cubic and quartic) IFCs, we use 4000step ab initio molecular dynamics (AIMD) simulation with 2 fs time step and 300 K temperature to capture 80 snapshots first. On this basis, we obtain 80 quasirandom configurations by applying the random displacement of 0.1 Å to each atom in the 80 snapshots. Finally, we extract displacement and force information obtained from 80 quasirandom configurations to derive anharmonic IFCs via the compressive sensing lattice dynamics (CSLD) method^{56}. In the CSLD calculations, we consider all (third) nearest neighbor interactions for third (fourth to sixth) order IFCs. Our calculations use 80 quasirandom configurations, more than used in previous work on Tl_{3}VSe_{4}^{57} and cubic SrTiO_{3}^{35}, are sufficient to capture accurate aharmonic IFCs that produce convergent results.
Calculation of transport properties
We calculate the temperaturedependence anharmonic phonon energy eigenvalues, including the offdiagonal elements of phonon selfenergies. On top of SCP calculations, the thermal transport parameters is solved based on the phonon BTE, as employed in the FourPhonon package^{58,59,60}. Generally, the computational cost of 4ph scattering is expensive in most materials. Hence, to ensure sufficient convergence, we use the available 4ph phase spaces as the criterion for calculating the 4ph SRs. We use 12 × 12 × 12 qmesh to capture the κ_{L}, and the numbers of available 3ph and 4ph scattering processes have reached approximately 1.8 × 10^{6} and 1.6 × 10^{10}, respectively. The numbers of available 4ph scattering processes are much larger than LiCoO_{2}^{60}, indicating that present calculations are sufficient to obtain an accurate κ_{L}. Finally, we obtained the κ_{L}, which is defined as
where k_{B} is the Boltzmann constant, ℏ is the reduced Planck’s constant, V is the volume of the unit cell, N_{q} is the number of wave vectors, q and ν are signs of the phonon modes, ω_{qν} is frequency, υ_{qν} is the group velocity. The F_{qν} is defined as
where τ_{qν} and Δ_{qν} are the phonon lifetime of singlemode relaxation time approximation and quantity displaying the population deviation of the iterative solution. To examine the stability of Na_{2}TlSb at 700 K, we performed 20,000step AIMD simulations with a time step of 2 fs.
The electronic band structure, crystal orbital Hamilton population (COHP), highfrequency dielectric constants ϵ_{∞}, and deformation potentials are calculated using PBEsol functional with 12 × 12 × 12 kpoint meshes. The elastic constants C_{ij}, static dielectric constants ϵ_{s}, and effective polar phonon frequency ω_{po} are calculated using DFPT. The above materials’ parameters are listed in Supplementary Table 1. The accurate band gap is obtained using HSE06 functional and 12 × 12 × 12 kpoint meshes. The electronic band structure is recalculated in uniform 135 × 135 × 135 kpoint grids to obtain the electron relaxation time τ_{e} and electronic transport parameters, as performed in AMSET code^{61}. Since Na_{2}TlSb contains a heavy Tl element, the spinorbit coupling (SOC) is also included in the calculations of electronic transport properties. The τ_{e} is calculated by including the fully anisotropic acoustic deformation potential (ADP) scattering, polar optical phonon (POP) scattering, and ionized impurity (IMP) scattering. The electronacoustic phonon and electronoptical phonon interaction is treated by ADP and POP scattering. For details of scattering rates, please refer to the Supplementary Methods.
Data availability
All data were available from the corresponding authors upon reasonable request.
Code availability
The related codes are available from the corresponding authors upon reasonable request.
References
Sootsman, J. R., Chung, D. Y. & Kanatzidis, M. G. New and old concepts in thermoelectric materials. Angew. Chem. Int. Ed. 48, 8616–8639 (2009).
Yue, T., Xu, B., Zhao, Y., Meng, S. & Dai, Z. Ultralow lattice thermal conductivity and anisotropic thermoelectric transport properties in Zintl compound βK_{2}Te_{2}. Phys. Chem. Chem. Phys. 24, 4666–4673 (2022).
Zhao, Y. et al. High thermopower and potential thermoelectric properties of crystalline LiH and NaH. Phys. Rev. B 95, 014307 (2017).
Liu, W. et al. Convergence of conduction bands as a means of enhancing thermoelectric performance of nType \({{{{\rm{Mg}}}}}_{2}{{{{\rm{Si}}}}}_{1x}{{{{\rm{Sn}}}}}_{x}\) solid solutions. Phys. Rev. Lett. 108, 166601 (2012).
Wang, H., Gibbs, Z. M., Takagiwa, Y. & Snyder, G. J. Tuning bands of PbSe for better thermoelectric efficiency. Energy Environ. Sci. 7, 804–811 (2014).
Lv, H. Y., Lu, W. J., Shao, D. F. & Sun, Y. P. Enhanced thermoelectric performance of phosphorene by straininduced band convergence. Phys. Rev. B 90, 085433 (2014).
Li, J. et al. The roles of Na doping in BiCuSeO oxyselenides as a thermoelectric material. J. Mater. Chem. A 2, 4903–4906 (2014).
Kim, G.H., Shao, L., Zhang, K. & Pipe, K. P. Engineered doping of organic semiconductors for enhanced thermoelectric efficiency. Nat. Mater. 12, 719–723 (2013).
Hicks, L. D. & Dresselhaus, M. S. Effect of quantumwell structures on the thermoelectric figure of merit. Phys. Rev. B 47, 12727–12731 (1993).
Parker, D., Chen, X. & Singh, D. J. High threedimensional thermoelectric performance from lowdimensional bands. Phys. Rev. Lett. 110, 146601 (2013).
Yue, T., Sun, Y., Zhao, Y., Meng, S. & Dai, Z. Thermoelectric performance in the binary semiconductor compound A_{2}Se_{2} (A = K, Rb) with hostguest structure. Phys. Rev. B 105, 054305 (2022).
Feng, Z., Fu, Y., Yan, Y., Zhang, Y. & Singh, D. J. Zintl chemistry leading to ultralow thermal conductivity, semiconducting behavior, and high thermoelectric performance of hexagonal KBaBi. Phys. Rev. B 103, 224101 (2021).
He, J., Xia, Y., Naghavi, S. S., Ozoliņš, V. & Wolverton, C. Designing chemical analogs to pbte with intrinsic high band degeneracy and low lattice thermal conductivity. Nat. Commun. 10, 719 (2019).
Li, Z., Xiao, C., Zhu, H. & Xie, Y. Defect chemistry for thermoelectric materials. J. Am. Chem. Soc. 138, 14810–14819 (2016).
Hu, L., Zhu, T., Liu, X. & Zhao, X. Point defect engineering of highperformance bismuthtelluridebased thermoelectric materials. Adv. Funct. Mater. 24, 5211–5218 (2014).
Xia, Y. & Chan, M. K. Y. Anharmonic stabilization and lattice heat transport in rocksalt βGeTe. Appl. Phys. Lett. 113, 193902 (2018).
Chen, Z. et al. Low lattice thermal conductivity by alloying SnTe with AgSbTe_{2} and CaTe/MnTe. Appl. Phys. Lett. 115, 073903 (2019).
He, J., Girard, S. N., Kanatzidis, M. G. & Dravid, V. P. Microstructurelattice thermal conductivity correlation in nanostructured PbTe0.7S0.3 thermoelectric materials. Adv. Funct. Mater. 20, 764–772 (2010).
Tadano, T., Gohda, Y. & Tsuneyuki, S. Impact of rattlers on thermal conductivity of a thermoelectric clathrate: a firstprinciples study. Phys. Rev. Lett. 114, 095501 (2015).
Nolas, G. S., Poon, J. & Kanatzidis, M. Recent developments in bulk thermoelectric materials. MRS Bull. 31, 199–205 (2006).
Park, J., Xia, Y. & Ozoliņš, V. High thermoelectric power factor and efficiency from a highly dispersive band in ba_{2}BiAu. Phys. Rev. Appl. 11, 014058 (2019).
Wang, W. et al. Low lattice thermal conductivity and high figure of merit in ntype doped fullHeusler compounds X2YAu (X = Sr, Ba; Y = As, Sb). Int. J. Energy Res. 45, 20949–20958 (2021).
Hu, Y., Jin, Y., Zhang, G. & Yan, Y. Electronic structure and thermoelectric properties of full Heusler compounds Ca2YZ (Y = Au, Hg; Z = As, Sb, Bi, Sn and Pb). RSC Adv. 10, 28501–28508 (2020).
Wang, S.F., Zhang, Z.G., Wang, B.T., Zhang, J.R. & Wang, F.W. Intrinsic ultralow lattice thermal conductivity in the fullHeusler compound Ba_{2}AgSb. Phys. Rev. Appl. 17, 034023 (2022).
Zhao, Y. et al. Quartic anharmonicity and anomalous thermal conductivity in cubic antiperovskites A_{3}BO(A = K, Rb; B = Br, Au). Phys. Rev. B 101, 184303 (2020).
Li, C. W. et al. Orbitally driven giant phonon anharmonicity in SnSe. Nat. Phys. 11, 1063–1069 (2015).
Liu, H. et al. Copper ion liquidlike thermoelectrics. Nat. Mater. 11, 422–425 (2012).
Wu, D. et al. Origin of the high performance in getebased thermoelectric materials upon bi2te3 doping. J. Am. Chem. Soc. 136, 11412–11419 (2014).
Zhao, Y. et al. Lattice thermal conductivity including phonon frequency shifts and scattering rates induced by quartic anharmonicity in cubic oxide and fluoride perovskites. Phys. Rev. B 104, 224304 (2021).
Xia, Y. et al. Highthroughput study of lattice thermal conductivity in binary rocksalt and zinc blende compounds including higherorder anharmonicity. Phys. Rev. X 10, 041029 (2020).
Wu, Z.j et al. Crystal structures and elastic properties of superhard IrN_{2} and IrN_{3} from first principles. Phys. Rev. B 76, 054115 (2007).
Xia, Y., Pal, K., He, J., Ozoliņš, V. & Wolverton, C. Particlelike phonon propagation dominates ultralow lattice thermal conductivity in crystalline Tl_{3}VSe_{4}. Phys. Rev. Lett. 124, 065901 (2020).
Tadano, T. & Tsuneyuki, S. Quartic anharmonicity of rattlers and its effect on lattice thermal conductivity of clathrates from first principles. Phys. Rev. Lett. 120, 105901 (2018).
He, J. et al. Ultralow thermal conductivity in full Heusler semiconductors. Phys. Rev. Lett. 117, 046602 (2016).
Tadano, T. & Tsuneyuki, S. Selfconsistent phonon calculations of lattice dynamical properties in cubic SrTiO_{3} with firstprinciples anharmonic force constants. Phys. Rev. B 92, 054301 (2015).
Ravichandran, N. K. & Broido, D. Unified firstprinciples theory of thermal properties of insulators. Phys. Rev. B 98, 085205 (2018).
Ravichandran, N. K. & Broido, D. Phononphonon interactions in strongly bonded solids: selection rules and higherorder processes. Phys. Rev. X 10, 021063 (2020).
Markov, M., Rezaei, S. E., Sadeghi, S. N., Esfarjani, K. & Zebarjadi, M. Thermoelectric properties of semimetals. Phys. Rev. Mater. 3, 095401 (2019).
Gong, J. J. et al. Investigation of the bipolar effect in the thermoelectric material CaMg2Bi2 using a firstprinciples study. Phys. Chem. Chem. Phys. 18, 16566–16574 (2016).
Pei, Y.L. & Liu, Y. Electrical and thermal transport properties of Pbbased chalcogenides: PbTe, PbSe, and PbS. J. Alloys Compd. 514, 40–44 (2012).
Wang, H., Pei, Y., LaLonde, A. D. & Snyder, G. J. Heavily doped ptype PbSe with high thermoelectric performance: an alternative for PbTe. Adv. Mater. 23, 1366–1370 (2011).
Fetherolf, J. H., Shih, P. & Berkelbach, T. C. Conductivity of an electron coupled to anharmonic phonons. arXiv:2205.09811, https://doi.org/10.48550/arXiv.2205.09811 (2022).
Kraemer, D. et al. High thermoelectric conversion efficiency of MgAgSbbased material with hotpressed contacts. Energy Environ. Sci. 8, 1299–1308 (2015).
Zhu, Q., Song, S., Zhu, H. & Ren, Z. Realizing high conversion efficiency of Mg3Sb2based thermoelectric materials. J. Power Sources 414, 393–400 (2019).
Hu, C., Xia, K., Fu, C., Zhao, X. & Zhu, T. Carrier grain boundary scattering in thermoelectric materials. Energy Environ. Sci. 15, 1406–1422 (2022).
He, J. et al. Role of sodium doping in lead chalcogenide thermoelectrics. J. Am. Chem. Soc. 135, 4624–4627 (2013).
Wang, X. et al. Sodium substitution in lead telluride. Chem. Mater. 30, 1362–1372 (2018).
Kresse, G. & Furthmüller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758–1775 (1999).
Csonka, G. I. et al. Assessing the performance of recent density functionals for bulk solids. Phys. Rev. B 79, 155107 (2009).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal properties from densityfunctional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001).
Tadano, T., Gohda, Y. & Tsuneyuki, S. Anharmonic force constants extracted from firstprinciples molecular dynamics: applications to heat transfer simulations. J. Phys. Condens. Matter 26, 225402 (2014).
Esfarjani, K. & Stokes, H. T. Method to extract anharmonic force constants from first principles calculations. Phys. Rev. B 77, 144112 (2008).
Zhou, F., Sadigh, B., Åberg, D., Xia, Y. & Ozoliņš, V. Compressive sensing lattice dynamics. II. Efficient phonon calculations and longrange interactions. Phys. Rev. B 100, 184309 (2019).
Zeng, Z. et al. Nonperturbative phonon scatterings and the twochannel thermal transport in Tl_{3}VSe_{4}. Phys. Rev. B 103, 224307 (2021).
Feng, T., Lindsay, L. & Ruan, X. Fourphonon scattering significantly reduces intrinsic thermal conductivity of solids. Phys. Rev. B 96, 161201 (2017).
Feng, T. & Ruan, X. Quantum mechanical prediction of fourphonon scattering rates and reduced thermal conductivity of solids. Phys. Rev. B 93, 045202 (2016).
Han, Z., Yang, X., Li, W., Feng, T. & Ruan, X. FourPhonon: an extension module to ShengBTE for computing fourphonon scattering rates and thermal conductivity. Comput. Phys. Commun. 270, 108179 (2022).
Ganose, A. M. et al. Efficient calculation of carrier scattering rates from first principles. Nat. Commun. 12, 1–9 (2021).
Momma, K. & Izumi, F. VESTA: a threedimensional visualization system for electronic and structural analysis. J. Appl. Crystallogr. 41, 653–658 (2008).
Acknowledgements
This research were supported by the National Natural Science Foundation of China under Grant No. 11974302, No. 12174327, No. 92270104, and the Graduate Innovation Foundation of Yantai University, GIFYTU under Grant No. KGIFYTU2213.
Author information
Authors and Affiliations
Contributions
The research was conceived and designed by YZ and ZD. Calculations on stabilities and thermoelectric properties were conducted by TY. Analysis of the data was performed by TY, JN, and SM. Methodology and supervision, project administration is ZD. All authors discussed the results and contributed to writing the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Yue, T., Zhao, Y., Ni, J. et al. Strong quartic anharmonicity, ultralow thermal conductivity, high band degeneracy and good thermoelectric performance in Na_{2}TlSb. npj Comput Mater 9, 17 (2023). https://doi.org/10.1038/s41524023009704
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41524023009704
This article is cited by

A QuantumChemical Bonding Database for SolidState Materials
Scientific Data (2023)