Pressure-induced evolution of structures and phase transition of technetium diboride Open Access

Superhard materials have become an indispensable component of engineering technology due to their exceptional properties, making them highly valuable in various applications, such as abrasives, cutting tools, and wear-resistant coatings.^1–4 Consequently, extensive research efforts have been dedicated to the pursuit of novel hard materials. In general, the incorporation of light elements (B, C, N, and O) into electron-rich transition metals represents a reliable approach for synthesizing high hardness materials.⁵ The synthesis of ReB₂, OsB₂, RhB_1.1, Re₂C, TcB₂, and OsN₂^6–14 under extreme conditions has been widely reported. Among the materials mentioned, TcB₂ stands out for its exceptional mechanical properties and chemical inertness, making it an attractive candidate for superhard materials.^13,15–25

According to previous reports, technetium diboride has a rhenium diboride (hP6-TcB₂) type structure under ambient conditions. Earlier in the experiment, Trzebiatowski and Rudzinski¹³ synthesized TcB₂ by pressing boron and technetium powders into pellets and sintering in a vacuum. TcB₂ was indexed as of hexagonal symmetry, and each conventional cell contains two units and belongs to the P6₃/mmc space group. Later, Wang¹⁸ evaluated the structure stability and electronic characteristics of TcB₂ with P6₃/mmc (hP6-TcB₂) and Pmmn (oP6-TcB₂) by first-principles calculations under environmental conditions. The results show that hP6-TcB₂ is the ground state structure of the two elastically stable structures. In addition, the calculated C₃₃ of hP6-TcB₂ is as high as 1022 GPa and the G/B value is 0.87, indicating that it is a potential superhard material. Immediately, Aydin and Simsek¹⁹ investigated the electronic structure and mechanical properties of hP6-TcB₂ and hP3-TcB₂ at zero pressure. It is found that hP6-TcB₂ is more energetically favorable than hP3-TcB₂, and both of them are non-spin polarized materials. In addition, both hP6-TcB₂ and hP3-TcB₂ satisfy the elastic stability criterion and are considered to be elastically stable. On the basis of the previously proposed elastically stable structures, Deligoz et al.²⁰ calculated the phonon dispersion curves of hP6-TcB₂, hP3-TcB₂, and oP6-TcB₂ by direct methods and confirmed that they are all dynamically stable.

Furthermore, the formation enthalpies (with reference to Tc + B) of hP6 and oP6 at 0–100 GPa were investigated by Zhong et al.²¹ The results show that the formation enthalpies of the considered structures under zero pressure are all negative, implying that they are thermodynamically stable. It is also found that hP6-TcB₂ is structurally stable in the range of 0–100 GPa. Subsequently, Kuang et al.²² selected seven possible transition metal boride structures for probing and discussed the thermodynamic stability of these structures at zero pressure. The obtained formation enthalpies indicate that hP6-TcB₂, oP6-TcB₂, hR9-TcB₂, and hP3-TcB₂ are all thermodynamically stable and can be synthesized at ambient pressure. Moreover, they further explored the relative formation enthalpies of hP6-TcB₂ and hP3-TcB₂ at 0–200 GPa. The results indicate that hP6-TcB₂ remains stable over a wide range of 0–147.5 GPa and transforms into hP3-TcB₂ phase at around 147.5 GPa.

In summary, previous studies have speculated the structure of TcB₂ based on known abundant transition metal diborides.^18–26 However, the resulting phase transition sequence may not offer a complete overview and still lacks a complete understanding of the mechanical behavior of TcB₂ under high pressures. In order to obtain a more precise phase transition sequence of technetium diboride under different pressures, we conducted a comprehensive exploration of the crystal structure of TcB₂ across an extensive pressure range (0–400 GPa) by using the particle swarm optimization (PSO) algorithm²⁷ in the CALYPSO code.²⁸ This approach is effectively capable of finding stable or meta-stable structures only depending on the given chemical composition. By employing the PSO method over an extensive pressure range, we find that hP6-TcB₂, hP3-TcB₂, oP6-TcB₂, hR9-TcB₂, and the orthorhombic phase Cmcm (oC12-TcB₂) all exhibit thermodynamic and dynamic stability under ambient conditions, in which the hP6-TcB₂ structure remains stable from 0 to 174.9 GPa, and it will transform to the hP3-TcB₂ phase with the further increase of pressure.

In this work, the crystal structure prediction was performed in the range of 0–400 GPa containing up to 4 formula units (f.u.) per simulation cell by using the PSO algorithm²⁷ as implemented in the CALYPSO code.²⁸ This method can predict stable or meta-stable crystal structures based on given chemical compositions under specific external conditions. More details of this structural search method can be found in Refs. 27 and 28. So far, it has been successfully applied not only to element solids, but also to binary and ternary compounds.^29–32 

The ab initio optimizations and theoretical simulations are performed using ultrasoft pseudopotential, as implemented in the Cambridge Serial Total Energy Package (CASTEP).^33,34 The generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE)³⁵ is used to describe the exchange-correlation function form, and the valence electron interactions with 2s²2p¹ and 4d⁵5s² are selected for the analysis of the electron configurations for B and Tc atoms, respectively. To ensure that the convergence condition is achieved, geometry optimization is performed until the energy and force differences are less than 5 × 10⁻⁶ eV/atom and 0.01 eV/Å, respectively. For the integration over the Brillouin zone, the Monkhorst–Pack k-point meshes³⁶ are 21 × 21 × 7 for hP6-TcB₂, 12 × 12 × 9 for hP3-TcB₂, 8 × 13 × 9 for oP6-TcB₂, 22 × 22 × 5 for hR9-TcB₂, and 4 × 18 × 18 for oC12-TcB₂. In addition, the phonon calculations were performed by using a supercell approach with the finite displacement method³⁷ in the CASTEP code, and 8 × 8 × 4 and 10 × 10 × 8 k-meshes were used for the force calculation of hP6-TcB₂ and hP3-TcB₂ supercells, respectively.

After full geometric optimization, we present the structural parameters of different TcB₂ phases in Table I, along with other theoretical data.^{13,19–22,25,26} As shown, the calculated a₀ and c₀ of hP6-TcB₂ are very close to the experimental values, with a difference of only 0.03% and 0.07%, respectively. To further evaluate their thermal stability, the formation enthalpy of the identified structures at zero pressure was calculated. The formula for the ΔH of TcB₂ is as follows: ΔH = H (TcB₂) − H (Tc) − 2H (B), where boron in α phase and technetium in a P6₃/mmc state were used as the reference structures. The obtained results are listed in Table II. It is evident that hP6-TcB₂, hP3-TcB₂, oP6-TcB₂, hR9-TcB₂, and oC12-TcB₂ all have negative formation enthalpies at zero pressure. Among them, hP6 has the relatively lowest formation enthalpies (−1.34 eV/f.u.) at zero pressure and is the most stable phase under ambient conditions. Meanwhile, the formation enthalpies of hP3-TcB₂, oP6-TcB₂, hR9-TcB₂, and oC12-TcB₂ are −0.18, −1.18, −1.16, and −0.86 eV/f.u., respectively, which are consistent with other theoretical data.^{15,18,19,21,22,25,26}

To determine the phase transition sequence of TcB₂ at 0 K, the enthalpy differences of hP3-TcB₂, oP6-TcB₂, hR9-TcB₂, oC12-TcB₂ with respect to the hP6-TcB₂ phase were obtained within the pressure range of 0–400 GPa. The calculated results are drawn in Fig. 1. As shown, hP6-TcB₂ possesses the lowest formation enthalpy at zero pressure and is considered to be the ground state. Moreover, the structural stability of hP6-TcB₂ endures until 174.7 GPa. As the pressure continues to rise, the hP6 phase transforms into the hP3 phase, and the latter remains structurally stable up to 400 GPa. The crystal structures of them are illustrated in Fig. 2.

Meanwhile, the dynamic stability of hP6-TcB₂ and hP3-TcB₂ under pressure is further discussed, and the calculated phonon dispersion curves under different pressures are presented in Fig. 3. The results show that hP6-TcB₂ and hP3-TcB₂ exhibit no imaginary phonon frequencies throughout the entire Brillouin zone at 0–180 and 160–400 GPa, respectively, suggesting that these phases are dynamically stable in a large range of pressures. Furthermore, Fig. 4 illustrates the pressure–volume curve of TcB₂ under pressure up to 400 GPa by fitting the data energy-volume to the Vinet equation of state.³⁸ As shown, the volume from hP6-TcB₂ to hP3-TcB₂ is reduced by 4.66%, which means that the structural transformation between them should be a first-order phase transition.

The elastic constants of solids and their associated properties can provide essential insights into understanding the mechanical performance of materials. Thus, we initially calculated the elastic constants of hP6-TcB₂, hP3-TcB₂, oP6-TcB₂, hR9-TcB₂, and oC12-TcB₂ at 0 GPa by a stress–strain method.^39,40 The obtained results, together with the calculated elastic constants of hP3-TcB₂ at 180 GPa and other relevant theoretical data,^{15,18,19,21,22,25,26} are presented in Table II. As shown, our obtained results at 0 GPa are consistent well with the available theoretical data. Moreover, it is well known that the mechanical stability of different crystal structures can be assessed using the Born–Huang stability criteria.⁴¹ After careful comparison, hP6-TcB₂, oP6-TcB₂, hR9-TcB₂, and oC12-TcB₂ all meet their mechanical stability criteria, indicating they should be mechanically stable at 0 GPa. Furthermore, the high-pressure phase hP3-TcB₂ is mechanically stable at 180 GPa because its elastic constants also satisfy the mechanical stability criteria at this pressure.

The obtained Young's modulus E and Poisson's ratio σ are also listed in Table III. Generally, the higher Young's modulus E, the stronger the resistance to compression and tension within the elastic deformation. Therefore, the order of stiffness for TcB₂ at 0 GPa should be hP6 > hR9 > oP6 > oC12 > hP3. In addition, Poisson's ratio can be used to evaluate the degree of covalent bond directionality, with larger Poisson's ratio, the weaker directionality of the covalent bond. Thus, the degree of TcB₂ covalent bond directionality at 0 GPa is hR9 > hP6 > oP6 > oC12 > hP3. In order to more intuitively describe the elastic properties of different TcB₂ phases, the correlations between the bulk modulus B, the shear modulus G, Young's modulus E, and hardness H_V under different structures are presented in Fig. 5. Moreover, at the pressure of 180 GPa, Young's modulus of hP3-TcB₂ is 950 GPa and Poisson's ratio is 0.33.

To enhance our understanding of the electronic structure of TcB₂, we depicted the density of states distribution for the hP6 phase at 160 GPa and the hP3 phase at 180 GPa in Fig. 6. As shown, the two high-pressure phases of TcB₂ show electron accumulation near the Fermi level, highlighting their inherent metallic characteristics. The notable electron density of states (DOS) values at the Fermi energy level in both phases can be attributed to the contribution of Tc-3d and B-2d orbitals. When focusing on the energy interval under consideration, the total DOS of the two phases exhibits a similar basic composition, almost both being Tc-3d with a B-2p orbital superposition shape. This indicates that both phases possess strong Tc-3d and B-2p hybridization, forming strong covalent bonds.

The crystal orbital Hamilton population (COHP) curve enables the assessment of the relative contribution of a specific bond to the overall bonding energy of the system, thereby facilitating the analysis of the interaction strength between two distinct types of bonds, namely, B–B bonds and Tc–B bonds. In addition, integrated crystal orbital Hamilton population (ICOHP) provides energy values after integration up to the Fermi level, which can be used to visually quantify the strength of covalent bonds. Using the LOBSTER post-processing program,^44,45 we calculated of the COHP and ICOHP of the B–B and B–Tc bonds in the hP6 and hP3 phases. The obtained results are also illustrated in Fig. 6. As shown, two distinct categories of B–B bonds and two distinctive types of B–Tc bonds can be distinguished in the hP6 phase. Among them, the B–B bond with a length of 1.60 Å (ICOHP = 8.79 eV) constitutes most of the total B–B bonds. Another type of B–B bond with a length of 2.81 Å, which involves B atoms closer between two folded boron layers, forms a small amount of covalent bonding (ICOHP = 0.38 eV). A similar trend is observed for the B–Tc bonds, where there are two types of Tc–B bonds with lengths of 2.05 and 2.06 Å. Although there is only a small difference of 0.01 Å between them, their covalent bond strengths are almost twice as different (ICOHP = 4.37 eV and ICOHP = 2.46 eV, respectively).

In comparison, the high-pressure phase hP3-TcB₂ exhibits a structurally more concise arrangement. The flat hexagonal boron layers overlap with the technetium layers, with technetium atoms positioned directly below the hexagonal rings [see Fig. 6(b)], resulting in simpler bonding characteristics within the crystal. Notably, the neighboring B–B bonds showcase robust covalent bonding traits, featuring an ICOHP value of 8.65 eV. However, the covalent bonding strength of the Tc–B bonds that connect boron layers with the transition metal layers is comparably weaker, with an ICOHP value of 3.78 eV. For both hP6-TcB₂ and hP3-TcB₂, the strong covalent bond formed by B–B bonds serves as the primary factor for maintaining the structural stability. However, the close connection between B atoms constitutes the compact B layer, so covalent bonding strength in B–Tc bonds is comparatively weaker, yet not negligible when compared to B–B bonds. Moreover, it is noteworthy that the hardness of the hP6 phase is significantly greater than that of hP3, which may be attributed to the hP6 phase having a zigzag configuration of the B atomic framework. However, under pressure, the zigzag B framework in the hP6 phase is replaced by a flatter B layer, resulting in enhanced interlayer sliding between B and Tc layers. This characteristic is also reflected in the hP3 phase, which exhibits a smaller shear modulus of 114 GPa.

In order to further comprehensively investigate the bonding characteristics between the two high-pressure phases, the electron localizability can be quantified through the Pauli principle, so as to more intuitively understand the electron distribution in distinct regions.⁴⁶ The electron localization density of hP6-TcB₂ and hP3-TcB₂ is plotted in Fig. 7, which can visualize the bonding states between different atoms. In general, high Electron Localization Function (ELF) values are considered to form a stronger localization feature in the region, while low ELF values correspond to gaps in an atomic orbital. As can be seen in Fig. 7, the B–B bonds in both different phases of TcB₂ have high ELF values, indicating the existence of strong covalent bonds between the immediately neighboring B atoms. In addition, the ELF values between the adjacent Tc–B are around 0.5, indicating that there are partial covalent bonds in Tc–B, which is consistent with our analysis from the density of states and COHP.

In general, when the Gibbs free energy reaches its minimum value, the material exists in a stable phase. Thus, through computation and analysis of the Gibbs free energy, one can anticipate the phase transition behavior depicted in the phase diagram. The high-pressure and high-temperature phase diagram of TcB₂ are shown in Fig. 8. It is obvious that temperature has a significant effect on the structural stability of TcB₂. With the increase of temperature, the transition pressure of hP6-TcB₂ → hP3-TcB₂ gradually decreases. At a room temperature of 300 K, the phase transformation from hP6-TcB₂ to hP3-TcB₂ occurs at about 164 GPa.

By employing a combination of particle swarm optimization for structure prediction and density functional theory calculations, we successfully predicted the stable phases of TcB₂ under both ambient and high pressures and investigated their structural stability over a wide range of pressures. The phase transition sequence of hP6-TcB₂ → hP3-TcB₂ under pressure was established, with a transition pressure of approximately 174.9 GPa. The calculation of elastic constants and phonon dispersion curves indicates that both hP6-TcB₂ and hP3-TcB₂ exhibit mechanical and dynamic stability in a large range of pressures. Moreover, the hP6-TcB₂ demonstrates a higher bulk modulus and lower Poisson's ratio, suggesting its potential for low-pressure compression applications. For hP3-TcB₂, the transition from the folded B layer to a flatter B layer under pressure results in an increased ease of sliding between the B layer and the transition metal layer. This is also reflected in its low hardness and shear modulus. Electronic structure calculations show that both hP6-TcB₂ and hP3-TcB₂ exhibit characteristics of metallic behavior. Furthermore, the B–B bonds in these two phases contribute to the majority of the covalent bonds in the system, which is crucial for its structural stability. Finally, the high-pressure and high-temperature phase diagram of TcB₂ was constructed for the first time using the QHA method. The results show that the transition pressure of hP6-TcB₂ → hP3-TcB₂ gradually decreases with rising temperature.

This work was supported by the Natural Science Basic Research Program of Shaanxi Province under Grant No. 2024JC-YBQN-0044, the National Natural Science Foundation of China under Grant No. 11904282, and the Doctoral Scientific Research Foundation of the Xi’an University of Science and Technology under Grant No. 2018QDJ029.

The authors have no conflicts to disclose.

Yi X. Wang: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). H. Wu: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Wu N. Xie: Investigation (equal); Methodology (equal); Visualization (equal). Xiao F. Wang: Investigation (equal); Methodology (equal); Software (equal). Shao W. Sun: Investigation (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Jian B. Gu: Formal analysis (equal); Investigation (equal); Methodology (equal).

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