High pressure structural stability of the Na-Te system Open Access

Chalcogenide metal compounds as a kind of super ionic conductor is of great interest of its promising applications in solid state batteries fuel cells or solid state gas detectors and photo emissive ultraviolet light materials and power sources and so on.^1–3 Sodium telluride compounds have been researched in terms of experiment^4–8 and theory.^9–13 

In terms of experiment, previously, Kraus and Glass⁴ have investigated the phase diagram of Na-Te system by the method of thermal analysis. Three sodium telluride compounds are formed in that phase diagram, namely, Na₂Te, NaTe and NaTe₃. Soon afterwards, Petric and Pelton^7,8 have also got Na₂Te, NaTe and NaTe₃ by studying the thermodynamic properties of Na-Te system and pointed out that the existence of NaTe₃ and Na₂Te. The compounds Na₂Te₃ and NaTe₂ may exist at lower temperature.⁸ Bottcher and Keller⁵ have synthesized NaTe in a two-step reaction. The space group of NaTe is Pbcn. Schultz and Kochler⁶ have studied the solutions of sodium polytellurides in liquid ammonia by using UV-visible spectroscopy. Na₂Te, Na₂Te₂, Na₂Te₃ and Na₂Te₄ have been researched. At the same time, Schultz and Kochler⁶ also propose that Pellini and Guercigh have reported the formation of Na₂Te, Na₂Te₃ and Na₂Te₇ by thermal analysis methods in 1910. Chen et al.¹³ have explored the phase diagram and thermodynamic description for the Na-Te system within the CALPHAD framework. Three intermetallic compounds, Na₂Te, NaTe and NaTe₃ have been identified same as the results proposed by Kraus and Glass,⁴ Petric and Pelton.^7,8

In theory, Alay-e-Abbas et al.^9,10 have investigated the structural and electronic properties of Na₂Te using density function theory implemented in the WIEN2K code. Na₂Te crystallizes into a stable face centered cubic antifluorite (anti-CaF₂) structure (space group Fm-3m). The electronic properties indicate that Na₂Te is a direct band gap semiconductor. Alay-e-Abbas and Shaukat¹⁰ also have studied the optical properties of Na₂Te. In recent years, Kalarasse and Bennecer have researched the elastic, vibrational, dielectric and lattice dynamics properties of Na₂Te by the first principles study performed in ABINIT code. Zhang et al. have also discussed the elastic and lattice dynamic properties of Na₂Te as well as the structural, phonon and thermodynamics properties using the ABINIT code.

Although many different stoichiometry ratio sodium tellurium compounds have been proposed in experiment, only Na₂Te is investigated in theory. Considering the complexity of Na-Te system, we want to know what the exotic properties of Na-Te system under high pressure.

In this paper, in order to find all potential stable ground state structures of Na-Te system under extreme pressure, the evolutionary algorithm^14–19 method is used. The significate character of this methodology is the capability of predicting the stable structure with only the knowledge of the chemical composition. The evolutionary algorithm^14–21 method is performed with USPEX code which has been successfully applied to a variety of systems. Brillouin zone (BZ) sampling using a grid of spacing of 2π×0.05Å^-1 and a plane-wave basis set cutoff of 400 eV are sufficient for the initial search over structures. All structural optimizations and electronic calculations are performed using the Vienna ab initio simulation package (VASP),²² a plane wave code employing the projected-augmented wave (PAW) potentials²³ based on the density functional theory (DFT) with the Perdew-Burke-Ernzerhof (PBE)²⁴ parameterization of generalized gradient approximation (GGA). The hybrid function of Heyd, Scuseria and Ernzerhof (HSE06)^25,26 is also used to verify the results of band structures with GGA function. The plane-wave basis-set cutoff energy is set to 800 eV, and appropriate Monkhorst-Pack k-meshes²⁷ are employed with a resolution of 2π× 0.03 Å^-1 for Brillouin zone (BZ) sampling. Such conditions are adopted to ensure that all the enthalpy calculations are well converged to less than 1 meV per atom. In geometrical optimization, all forces on the atoms converge to less than 0.05 eV Å^-1. Phonon calculations are performed based on the supercell approach²⁸ using the PHONOPY code.²⁹ The electron localization function (ELF) and Bader charge analysis properties are performed using the (VASP).

The ab initio evolutionary algorithm USPEX, which can synchronously find stable stoichiometries and corresponding structures in multi-component systems are used for finding stable Na-Te compounds under pressure lower than 140 GPa. In our extensive research, we reproduce successfully the earlier structures of Fm-3m, Pnma and P6₃/mmc Na₂Te. Simultaneously, a novel structure with different stoichiometries is uncovered at extreme pressure. Compound such as Na₈Te₂ is theoretically stable. We also perform the structural prediction methods on the known molecular NaTe, NaTe₃, NaTe₂ and Na₂Te₃ under pressure lower than 140 GPa. The thermodynamic convex hull curves of Na-Te system, which are a complete set of phase stable against transformation into any other phase and decomposition into any set of other phases are plotted in Fig. 1. As clearly see from the figure, Fm-3m Na₂Te and Pbcn NaTe are stable at ambient pressure. When the pressure is increasing up to 20 GPa, Imma Li₈Te₂ emerges on the convex hull, and Na₂Te is still the global minimum compound. However, with increasing the pressure, NaTe becomes the unstable structure at 80 GPa. Therefore, we deduce that NaTe is stable up to at least 60 GPa at. It should be mentioned that P4/mmm NaTe₃ emerges on the convex hull at 80 GPa and disappears while the pressure rising to 120 GPa. As a result, we elucidate that NaTe₃ adopts the P4/mmm phase between 80 and 100 GPa. Simultaneously, we find that Na₂Te₃ and NaTe₂ are not existent at 0 K. This result is conflicting to previous studies.^4,7,8 The discrepancy may be attributed to the negligence of temperature effects in our calculations.

The thermodynamic properties of Na₂Te, NaTe, Na₈Te₂ and NaTe₃ are investigated furtherly to ensure the stable pressure range. The enthalpy difference curves and the formation enthalpy as s function of pressure are shown in Fig. 2. For Na₂Te, Beister et al.³⁰ predict phase transformation sequence: Fm-3m→Pnma→P6₃/mmc with transition occurred at 2.3 and 5.3 GPa, respectively. From our result (Fig. 2(a)), we can see that Fm-3m Na₂Te changes first to the Pnma structure at 1.5 GPa, then the Pnma phase and P6₃/mmc structure are not separated clearly. With increasing pressure, it transforms into the Imma structure at 85.5 GPa. The motif of crystal structure for Imma Na₂Te is depicted in Fig. 3(a), noticed that this phase is a layer structure along bc-plane. Noticeably, we find that along the bc-plane, the Na atoms and the Te atoms form the layer stacking of ABAB…. The crystal structure of NaTe has been investigated. However, its phase transition under high pressure is still unknown. We study the enthalpy difference curve of NaTe, which is shown in the supplementary material (Fig. S1). For NaTe, high-pressure Pmmm phase is uncovered, which is presented in Fig. 3(b). From Fig. 3(b) we can see that Pmmm NaTe is also a layer structure. The convex hull curves show us that Na₈Te₂ is stable above at least 20 GPa, in order to obtain the accurate pressure range, the formation enthalpy of Na₈Te₂ is drawn in Fig. 2(c). It is clearly revealing that Imma Na₈Te₂ is stable in the pressure range of 27 to 140 GPa. For Imma Na₈Te₂ (Fig. 3(c)), this structure is isostructural with the Imma Li₈Te₂.³¹ Moreover, we also calculate the formation enthalpy of P4/mmm NaTe₃ to obtain the specific pressure range (Fig. 2(d)). From Fig. 2(d), it is asserted that P4/mmm NaTe₃ (Fig. 3(d)) is stable above 84 GPa. Eventually, we determine that P4/mmm NaTe₃ is stable between 84 and 100 GPa. For P4/mmm NaTe₃, we find that it also a layer structure along bc-plane, and we can see that the Na atoms locate in the vertex position, Te atoms forming an octahedron between two Na ion layer.

Dynamic property of the phase is one of the essential criteria for considering the structure stability. The density functional perturbation theory is performed to probe the dynamic stability of the predicted Na-Te compounds at 0 K and different pressures. Fig. 4 displays the calculated phonon spectrums of Imma Na₂Te, Pmmm NaTe, Imma Na₈Te₂ and P4/mmm NaTe₃, respectively_. It can be seen that there is no sign of imaginary frequencies in the entire Brillouin zone (BZ), which indicates that the structures of above mentioned are dynamical stable. Meanwhile, we also calculate the phonon spectrum of Pbcn NaTe (Fig. S2 of the supplementary material), indicating that the structure is unstable at 0 K. Therefore, we describe the formation enthalpy of NaTe, which is shown in Fig. 2(b). It is clearly revealing that Pmmm NaTe is stable above at least 17.7 GPa. Then, we deduce the high-pressure phase of NaTe is Pmmm phase (stable above 17.7 GPa).

Mechanical stability is a basic factor for the existence of a structure. In this paper, the elastic constants of several new structures are determined by applying the stress-strain method. The results are displayed in the Table I. Our results for elastic constants matrix Cij satisfy the Born-Huang stability criteria.³² Therefore, these structures are mechanically stable in the whole pressure range studied.

To investigate the electronic properties of Na-Te system, the band structures along the high-symmetry directions in the BZ at different pressures for P6₃/mmc Na₂Te, Imma Na₂Te, Pmmm NaTe, Imma Na₈Te₂, and P4/mmm NaTe₃ are calculated as presented in Fig. 5. At ambient pressure, Fm-3m Na₂Te is a direct band gap semiconductor. However, the P6₃/mmc structure has an indirect band gap of 0.976 eV at 60 GPa (Fig. 5(a)). With increasing the pressure, the band-gap is decreasing in Pnma and P6₃/mmc phase. Interestingly, Imma Na₂Te is a direct band gap of 1.869 eV (at 120 GPa). Simultaneously, the band gap increases with pressure while in the high-pressure structure (Imma phase). The band gap as a function of pressure for Imma Na₂Te is displayed in Fig. 6. Given the well-known shortcomings (underestimate band gap) of standard DFT approaches such as the LDA and GGA function,^33–35 the band structure of Imma Na₂Te is calculated using the hybrid function of HSE06. Clearly, the conclusion is consistent with the result obtained using DFT. In order to understand precisely the abnormal phenomenon, the partial density of states (PDOS) of Imma Na₂Te is presented in Fig. 5(f). Clearly, we can see that the Te p states dominate the top of the valence band, and the conductor band is dominated by Na s, Na p and Te d states, respectively. Compression leads to band broadening, then, increases the splitting between Te p states and Na p and Te d states. The understanding may similar to CsCdF₃,³⁶ which is an indirect-gap insulator under ambient condition with the gap increasing under pressure. Simultaneously, we find from the Fig. 5(f) that there is a strong hybridization between 4 and 8 eV for Na p state and Te d state. That may be the reason for the band gap increasing.

For NaTe (Fig. 5(c)), it is clearly shown that the presence of several bands crossing at the fermi level exhibiting a metallic character of Pmmm NaTe under high pressure. It can be seen from the Fig. 5(d) that Imma Na₈Te₂ is metallize in the entire pressure range studied. The high-pressure P4/mmm NaTe₃ structure is also metallic as presented in Fig. 5(e).

To gain detailed insight into the bonding character of Na-Te system, we have calculated the electronic localization function (ELF), which is shown in Fig. 7. From Fig. 7, we can see that the ELF value is smaller than 0.5 between Na and Te atoms, which illustrates that the possible ionic bond formation between Na and Te atoms. In order to further clearly understand the interaction between Na and Te atoms, we have calculated the Bader charge analysis (the atoms in molecules method)³⁷ to evaluate the charge transfer. The number of electrons of the Na-Te system is listed in Table II. From Table II, we can clearly see that Na atoms are the electronic donors. However, for Na₈Te₂, we find that two Na atoms neither loss electron nor get electron given the calculation error. That may be due to the irregular collection of Na atoms. This behavior is similar to the Imma Li₈Te₂.³¹ 

In summary, we have extensively explored the stable structures and possible stoichiometries in the Na-Te system by first-principles evolutionary crystal structure prediction at pressures up to 120 GPa. Several novel structures, namely, Imma Na₂Te, Pmmm NaTe, Imma Na₈Te₂ and P4/mmm NaTe₃, are found under high pressure. We determine that the Fm-3m Na₂Te transforms to Pnma structure at 1.5 GPa, however, we do not accurately distinguish the Pnma and P6₃/mmc structures. Then, above 85.5 GPa, Na₂Te adopts the Imma phase, and is stable up to at least 140 GPa. We also find that the previous structure Pbcn NaTe is unstable at 0 K, and predict that the high-pressure Pmmm NaTe is stable between 17.7 and 60 GPa. Na₈Te₂ is stable above 27 GPa. The dynamic characteristics and the mechanical properties of the structures of the above-mentioned are investigated to prove the stability of these structures. Interestingly, we find that the band gap of Imma Na₂Te increases with an increase of pressure. This result may be caused by the increasing of splitting between Te p states and Na s, Na p and Te d states. The other cause may be the strong hybridization between Na p state and Te d state. The ELF and Bader charge analysis show that the formation of ionic bonds between Na and Te atoms for Na-Te system, and the Na atoms are the electron donors.

See supplementary material for the enthalpy difference curve of NaTe and the delineative phonon dispersive curves for Pbcn NaTe at 0 K and 0 GPa.

This work was supported by the 111 Project (No. B12011), the National Natural Science Foundation of China (Nos. 51632002, 51572108, 11574109, 11634004, 11674122, 11404134 and 11574112), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_15R23), National Found for Fostering Talents of basic Science (No. J1103202), Parts of calculations were performed in the High Performance Computing Center (HPCC) of Jilin University.

Dedicated to Prof. Guangtian Zou on the occasion of his 80th birthday.