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Learn more: PMC Disclaimer | PMC Copyright Notice Fundam Res . 2025 Jan 11;6(2):1078–1085. doi: 10.1016/j.fmre.2024.12.025 Search in PMC Search in PubMed View in NLM Catalog Add to search Significant enhancement of hardness and stability in the Sc x Ta 1- x B 2 transition-metal diborides system Hang Zhai Hang Zhai a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China c International Center of Future Science, Jilin University, Changchun 130012, China Find articles by Hang Zhai a, b, c, 1 , Weiguang Gong Weiguang Gong a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China Find articles by Weiguang Gong a, 1 , Xi Shen Xi Shen b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Find articles by Xi Shen b, 1 , Xubin Ye Xubin Ye b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Find articles by Xubin Ye b , Xianqi Song Xianqi Song a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China c International Center of Future Science, Jilin University, Changchun 130012, China Find articles by Xianqi Song a, c , Peng Liu Peng Liu a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China Find articles by Peng Liu a , Dayu Yan Dayu Yan b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Find articles by Dayu Yan b , Youguo Shi Youguo Shi b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Find articles by Youguo Shi b , Richeng Yu Richeng Yu b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China d Songshan Lake Materials Laboratory, Dongguan 523808, China Find articles by Richeng Yu b, d , Xiaohui Yu Xiaohui Yu b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China d Songshan Lake Materials Laboratory, Dongguan 523808, China Find articles by Xiaohui Yu b, d, ⁎ , Quan Li Quan Li a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China c International Center of Future Science, Jilin University, Changchun 130012, China Find articles by Quan Li a, c, ⁎ , Yanming Ma Yanming Ma a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China c International Center of Future Science, Jilin University, Changchun 130012, China Find articles by Yanming Ma a, c, ⁎ Author information Article notes Copyright and License information a State Key Lab of Superhard Materials and Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, China b Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China c International Center of Future Science, Jilin University, Changchun 130012, China d Songshan Lake Materials Laboratory, Dongguan 523808, China ⁎ Corresponding authors. [email protected] [email protected] [email protected] 1 These authors contributed equally to this work. Received 2024 Oct 24; Revised 2024 Dec 13; Accepted 2024 Dec 24; Collection date 2026 Mar. © 2025 The Authors. Publishing Services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). PMC Copyright notice PMCID: PMC13069649 PMID: 41971785 Abstract Transition-metal borides have emerged as promising candidates for potential superhard materials due to their remarkable mechanical properties, favorable synthesis conditions and diverse physical properties. However, achieving the requisite hardness for classification as superhard materials in transition-metal borides remains a significant challenge. In this study, we employed a multicomponent strategy to optimize electron band filling, investigating the thermodynamic stability and mechanical properties of Sc x Ta 1− x B 2 (0 ≤ x ≤ 1) transition-metal diborides. Calculations predicted Sc 0.5 Ta 0.5 B 2 as the most promising composition, with its Fermi level located near the pseudogap, resulting in the lowest thermodynamic energy and superior mechanical properties. Experimentally, Sc 0.5 Ta 0.5 B 2 was successfully synthesized under ambient pressure, achieving a hardness of 31.5 GPa under a 4.9 N load, the highest hardness reported for single-phase transition-metal borides. Additionally, Sc 0.5 Ta 0.5 B 2 shows an electrical resistivity of 104.9 µΩ·cm at room temperature, indicating its excellent electrical conductivity. These findings provide critical insights for the design of novel superhard metal materials, achieved through electron filling modifications induced by multicomponent strategies. This approach expands the potential for developing cost-effective, superhard materials, with implications for advancing transition-metal diborides. Keywords: Metallic superhard materials, Transition-metal borides, Electron band filling, Mechanical properties, Vickers hardness, Pseudogap Graphical abstract Open in a new tab 1. Introduction The development of superhard materials is driven by the demand for abrasives and cutting tools in industrial production, as well as their critical roles in defense and aerospace applications. Diamond and cubic boron nitride, recognized as the toughest known materials, have historically met numerous industrial requirements. However, their widespread use is constrained by diamond's low thermal stability and the high synthesis cost of cubic boron nitride [ [1] , [2] , [3] , [4] , [5] , [6] ]. This limitation has driven a vigorous search for alternative materials that offer exceptional stability and strength, while also being cost-effective to synthesize. Transition-metal (TM) borides have emerged as strong contenders for superhard materials, with transition metals introducing a dense array of valence electrons into the compounds, enhancing their resistance to compression, while boron atoms form strong covalent bonds that provide resistance to both elastic and plastic deformation [ [7] , [8] , [9] , [10] , [11] ]. Following this principle, various transition metal borides, including OsB 2 and WB 4 , known for their high electron density and boron content, have been successfully synthesized under ambient pressure [ 12 , 13 ]. Furthermore, in contrast to diamond and cubic boron nitride, the inclusion of transition metals in these borides imparts a variety of multifunctional physical properties, such as electrical conductivity, superconductivity, and magnetic effects. However, the metallic bonding in transition metal borides compromises their mechanical strength [ 14 , 15 ], resulting in a hardness of < 30 GPa [ [16] , [17] , [18] , [19] , [20] ]. This highlights an urgent demand to enhance the hardness of transition metal borides and to investigate the mechanisms underlying the augmentation of their mechanical properties [ 5 , 21 , 22 ]. Atomic-scale substitution can influence the degree of electronic state localization by altering the electronic structure of the constituent elements, potentially leading to an enhancement in the hardness of materials [ [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] ]. Transition metal diborides (TMB 2 ) offer a significant advantage due to their ability to host a sufficient number of electrons to form strong boron covalent bonds and thus possess excellent mechanical properties [ 34 , 35 ]. These diborides avoid the flexible and versatile three-center boron bonding states that are often found in boron-rich compounds [ 36 , 37 ]. Diborides of early transition metals from the IIIB–VIB groups crystallize in an isostructural arrangement with hexagonal symmetry, comprising alternating planar sheets of graphite-like boron layers and closely packed metal layers, typically adopting the AlB 2 -type phase [ 38 ]. For diborides of late transition metals from the VIIB and VIIIB groups, they crystallize in hexagonal and orthorhombic symmetries, featuring puckered layers of metal and boron atoms, adopting the ReB 2 -type and OsB 2 -type phases, respectively [ 15 ]. Typically, two constituent transition-metal diborides with identical structural types and similar electron configurations are expected to form a uniform and continuous new ternary phase, avoiding the precipitation of a second phase, which could disrupt the electronic structure of the constituent phase. Among these three structural types, elements applicable to AlB 2 -type structures are cost-effective and offer more versatile options, making them a promising material platform for exploring multiple reinforcement strategies [ [39] , [40] , [41] , [42] ]. Herein, TaB 2 , with a high Vickers hardness (∼26 GPa), is selected as the matrix material [ 43 ], while scandium (Sc) is chosen as a substituent element owing to notable disparities in atomic radii and valence electrons between Sc (1.41 Å, 3 p 6 4 s 2 3 d 3 ) and Ta (1.34 Å, 5 p 6 6 s 2 5 d 3 ). This variance has the potential to significantly enrich the electronic state localization of the constituent phase, possibly producing ternary compounds with high hardness [ 28 , 29 ]. A set of ternary transition–metal borides, denoted as Sc x Ta 1−x B 2 ( x = 0.0–1.0), is designed via a multicomponent strategy and substitutional method. The obtained results reveal that differences in the occupation of bonding energy states of these isostructural compounds lead to significantly affect mechanical properties. Among them, Sc 0.5 Ta 0.5 B 2 , with its Fermi level (E F ) lying within the pseudogap, exhibits the most negative formation energy and superior mechanical characteristics due to the highest degree of electron state localization. Notably, we successfully synthesized Sc 0.5 Ta 0.5 B 2 under ambient pressure, achieving a hardness of 31.5 GPa and an electrical resistivity of 104.9 µΩ·cm at room temperature. This ternary compound demonstrates a remarkable enhancement in both hardness and stability compared to its binary boride counterparts. 2. Computational and experimental methods 2.1. Computational method We used the AlB 2- type structure ( P 6/ mmm ) as the prototype to enumerate candidate Sc x Ta 1-x B 2 (0 ≤ x ≤ 1) structures by substituting Ta sites with Sc atoms. The permutation and combination method implemented in the CALYPSO code was employed to generate all possible substitutional configurations, considering 1-8 formula units with 21 distinct substitutional concentrations. After relaxation of all these structures via ab initio density functional theory (DFT) total-energy calculation, we chose the structure with the lowest energy for further analysis. All first-principles calculations were conducted using the Vienna ab initio simulation package code (VASP) [ 44 ], adopting the generalized gradient–approximation functional proposed by Perdew, Burke, and Ernzerhof (GGA-PBE) for the exchange-correlation energy, along with a plane-wave basis set [ 45 ]. The electron–ion interaction was described using the projector augmented–wave method [ 46 ], with valence-electron configurations of 2 s 2 2 p 1 , 3 p 6 4 s 2 3 d 3 , and 5 p 6 6 s 2 5 d 3 for B, Sc, and Ta atoms, respectively. The total energy of the structure was minimized by relaxing structural parameters using a conjugate gradient optimization method. A Monkhorst–Pack k-mesh spacing of 0.2 Å and an energy cutoff of 600 eV were used in the calculations [ 47 ]. Energy convergence was achieved at ∼1 meV per atom, with residual stresses and forces in fully relaxed structures below 0.1 GPa and 0.001 eV/Å, respectively. Elastic constants were calculated using the strain–stress method and the elastic moduli were derived from the Voigt–Reuss–Hill averaging scheme [ 48 ]. The Vickers hardness was calculated based on Tian's hard model ( Hv = 0.92 k 1.137 G 0.708 ) [ 49 ]. The atomic bonding information was obtained by calculating the COHP using the LOBSTER code [ 50 , 51 ]. Stress-strain relations under various loading conditions are calculated under a biaxial stress state, which includes a shear stress and a normal compressive stress component. The shape of the deformed cell and atomic relaxation are determined completely at each step through the constrained structural optimization. The starting position for each strain step is taken from the relaxed coordinates of the previous strain step and with a small increment of 0.01 in each step to ensure the quasistatic strain path. At each step, the applied shear strain is fixed to determine the shear stress σ x s , while the other five independent components of the strain tensors and all the atoms inside the cell are simultaneously relaxed until the normal stress component σ s s reaches a specified value, σ s s = σ x s tan ϕ where ϕ∼ 68° corresponding to the prominent Vicker shear stress state. σ s s ≈ 0 and σ x s ≈ 0 are associated with pure shear and tensile strains, respectively. The stress responses along various high-symmetry crystal orientations under tensile, shear, and Vickers shear strains were computed, and the lowest peak stress for each type of strain defines the weakest deformation direction [ [48] , [49] , [50] , [51] , [52] , [53] , [54] , [55] , [56] ]. 2.2. Experimental method Pure scandium, tantalum, and boron powders in a ratio of 1:1:2.5 were thoroughly ground together using an agate mortar and pestle to obtain a uniform mixture. The excess boron powder was included to ensure the thermodynamic stability of the Sc 0.5 Ta 0.5 B 2 structure. The resulting mixture was then pressed into a 500 mg pellet using a hydraulic press under a force of 10,000 pounds. These pellets were placed in an arc melting furnace, applying an AC of > 70 A under high-purity argon at ambient pressure. Powder X-ray diffraction (XRD) analysis was performed using a Huber diffraction meter with Cu Kα1 radiation in the 2 θ range of 10° to 100°, with a step size of 0.005° Selected area electron diffraction (SAED) patterns and a scanning transmission electron microscopy high-angle annular dark-field (STEM HAADF) images were obtained using an ARM200F (JEOL, Tokyo, Japan) transmission electron microscope equipped with double Cs correctors (CEOS GmbH, Heidelberg, Germany) for the condenser lens and objective lens. ABF and HAADF images were acquired at acceptance angles of 11.5–23.0 and 90–370 mrad, respectively. The available spatial resolution for each STEM image is better than 78pm at 200 kV. Vickers hardness was measured using a microhardness tester on well-sintered bulk samples under different applied loads of 0.25–4.9 N, with a dwell time of 15 s under each load. Vickers hardness was calculated using Eq. 1 : H v = 1854.4 F / a 2 , (1) where F is the loading force applied on sample in Newtons (N), a is the average length of the two diagonals of each indent in micrometers (µm), and H v is Vickers hardness in gigapascals (GPa). The temperature dependence of the electrical resistivity, ρ(T) , of the compounds was measured at the temperature range of 2−300 K using a standard four-probe method on a physical property measurement system (PPMS7, Quantum Design). 3. Results and discussion 3.1. Thermodynamic stability of Sc x T a 1−x B 2 (0 ≤ x ≤ 1) compounds at ambient pressures ScB 2 and TaB 2 adopt the hexagonal P 6 /mmm lattice, comprising alternating planar sheets of graphite-like boron layers and closely packed metal layers, as illustrated in Fig. 1a . The crystal structure analysis by particle swarm optimization (CALYPSO) method [ [56] , [57] , [58] , [59] , [60] , [61] , [62] , [63] ] was employed to generate cation-substituted P 6 /mmm phase structures with various compositions. The structures of Sc x Ta 1− x B 2 (0 ≤ x ≤ 1) are isostructural and exhibit a similar electronic structure, characterized by the presence of pseudogaps. However, they differ in electron band fillings, with the E F of ScB₂ lying below the pseudogap, while that of TaB₂ lying above it. Among these compounds, the E F of Sc 0.5 Ta 0.5 B 2 nearly lies within the pseudogap, indicating a more stable electronic structure. The positioning of E F relative to the pseudogap influences the electronic structure. When E F lies above the pseudogap, it results in the partial occupation of antibonding states, while E F below the pseudogap leads to the occupation of bonding states, both contributing negatively to the stability. Their schematic view is shown in Fig. 1b . To further assess the thermal stability of Sc x Ta 1− x B 2 structures, we calculated their relative formation energies compared to ScB 2 and TaB 2 . The relative formation energy per atom of Sc x Ta 1− x B 2 was determined using the following formula: Δ E (Sc x Ta 1− x B 2 ) = E (Sc x Ta 1− x B 2 ) − xE (ScB 2 ) − (1 − x ) E (TaB 2 ), where E represents the calculated energy per atom at zero temperature and pressure. Fig. 1c presents the normalized enthalpies per atom of the most energetically favorable structures at each stoichiometry (the convex hull). The majority of Sc x Ta 1− x B 2 ( x = 0.0–1.0) compounds are on the convex line corresponding to thermodynamically stable phases. Among these compounds, Sc 0.5 Ta 0.5 B 2 exhibits the lowest relative enthalpy per atom, achievable through synthesis from ScB 2 and TaB 2 . Typically, achieving 50% substituting with a large size mismatch between the atoms of solute and solvent is not feasible in ternary transition metal boride compounds due to the solubility limit [ 13 , [64] , [65] , [66] ]. It is worth noting that this is achieved in the Sc x Ta 1− x B 2 ( x = 0.0–1.0) system due to its unique electronic structure characteristics. Fig. 1. Open in a new tab (a) Polyhedral representation and coordination environments of transition-metal diborides, with blue spheres indicating transition-metal atoms and red spheres representing boron atoms. (b) Schematic illustration of the Fermi level positions in ScB 2 , TaB 2 and Sc 0.5 Ta 0.5 B 2 . (c) Calculated formation enthalpies of Sc x Ta 1- x B 2 ( x = 0.0–1.0) compounds relative to ScB 2 and TaB 2 at ambient pressure. 3.2. Mechanical properties of Sc x Ta 1−x B 2 (0 ≤ x ≤ 1) compounds To explore the elastic mechanical properties, we evaluated the elastic modulus of Sc x Ta 1− x B 2 ( x = 0.0–1.0) compounds by introducing small strains near the equilibrium state. Initially, we investigated the effect of varying components on Young's modulus ( E ), shear modulus ( G ), bulk modulus ( B ), and compound hardness, as illustrated in Fig. 2a – c . Young's modulus ( E ) measures the tensile or compressive stiffness of the material, while the shear modulus ( G ) quantifies the ability of a material to resist deformation caused by shear stresses involving elongation and bending of bonds. Both moduli exhibit a distinct upward trend when varying x from 0 to 0.5 and a subsequent decline as x reaches 1, suggesting that Sc 0.5 Ta 0.5 B 2 possesses superior mechanical properties among the Sc x Ta 1− x B 2 ( x = 0.0–1.0) compounds, which is consistent with its thermodynamic stability. The bulk modulus ( B ) assesses the resistance of the material to volume change and is directly linked to the valence-electron concentration. Our results show that as the Ta concentration increases, a corresponding rise in the valence-electron concentration of Sc x Ta 1− x B 2 ( x = 0–1) system exists, resulting in a linear increase in the bulk modulus. Subsequently, we evaluated the hardness of Sc x Ta 1− x B 2 ( x = 0–1) compounds, as shown in Fig. 2d . The results reveal that the hardness of Sc x Ta 1− x B 2 compounds near x = 0.5 exceeds those of ScB 2 and TaB 2 . Fig. 2. Open in a new tab Calculated (a) Young's modulus, (b) shear modulus, (c) bulk modulus, and (d) hardness as a function of composition x in Sc x Ta 1- x B 2 . Detailed data are provided in Table S1. To investigate resistance to plastic deformation and ultimate mechanical strengths, we determined stress responses along various deformation paths and under different types of strains. The lowest peak stress in each case defines the corresponding ideal strength, which represents the minimum stress required to plastically deform a perfect crystal under that type of strain type. Fig. S1 presents the calculated peak stress values for Sc 0.5 Ta 0.5 B 2 along high-symmetry crystal orientations under tensile, pure shear strain, and Vickers indentation deformation conditions, compared with those of ScB 2 and TaB 2 . The results offer a quantitative description of the peak value, range, and trend of the stress response for ScB 2 , Sc 0.5 Ta 0.5 B 2 , and TaB 2 . Notably, the stress disparity among these compounds exceeds 10 GPa, indicating their anisotropic stress responses. The lowest stress peaks significantly increase from ScB 2 to Sc 0.5 Ta 0.5 B 2 before decreasing to TaB 2 under all types of strain, positioning Sc 0.5 Ta 0.5 B 2 as the strongest compound among them with the most robust overall strength performance. We examined the stress–strain relations of ScB 2 , TaB 2 , and Sc 0.5 Ta 0.5 B 2 under tensile, shear, and Vickers shear strains along selected deformation direction where the lowest stress peak (i.e., the ideal strength) is obtained. Additionally, we calculated the ideal strength of ZrB 2 with an electron equivalent to that of Sc 0.5 Ta 0.5 B 2 for comparison, as Zr positions midway between Sc and Ta on the periodic table. The lowest peak stresses under tensile strain, obtained along the ‹1 1 ¯ 0› direction for ScB 2 , TaB 2 , and ZrB 2 are 34.3, 39.1, and 33.4 GPa, respectively. These materials exhibit a considerably softer deformation mode with a large range of deformation, in which structural variations exhibit a degree of flexibility across this large range of strain due to the presence of metallic bonds. Notably, the tensile strength of Sc 0.5 Ta 0.5 B 2 is higher than that of both ScB 2 and TaB 2 , with a peak stress of 40.7 GPa along the ‹110› direction of Sc 0.5 Ta 0.5 B 2 , as shown in Fig. 3a . This demonstrates that Sc 0.5 Ta 0.5 B 2 undergoes a significantly stiffer deformation mode with a broader deformation range. Fig. 3. Open in a new tab Calculated stress–strain responses for (a) tensile, (b) pure shear, and (c) Vickers shear deformations of ScB₂, TaB₂, Sc ₀.₅Ta₀.₅B₂, and ZrB₂ along their respective weakest deformation directions. Loading conditions for the TMB₂ structures under (d) tensile, (e) pure shear, and (f) Vickers shear strains. We next evaluated the structural response of ScB 2 , TaB 2 , ZrB 2 , and Sc 0.5 Ta 0.5 B 2 under pure shear deformation, as shown in Fig. 3b . The ideal shear strengths of ScB 2 and TaB 2 occur in the (001) [1 1 ¯ 0] and (001) [110] directions, with peak stresses of 31.6 GPa and 28.9 GPa, respectively. Notably, the ideal shear strength of Sc 0.5 Ta 0.5 B 2 is significantly higher than that of ScB 2 and TaB 2 , with the lowest stress peak of 40.3 GPa obtained along the (001) [1 1 ¯ 0] shear direction. ScB 2 and TaB 2 exhibit a creep-like deformation mode over large strains, whereas Sc 0.5 Ta 0.5 B 2 maintains a steady, high level of stress followed by a sudden stress collapse just beyond its peak value. This behavior is characteristic of materials with robust covalent bonding, typical of those formed by light elements. We then investigated the stress–strain relations of ScB 2 , TaB 2 , ZrB 2 , and Sc 0.5 Ta 0.5 B 2 under Vickers indentation deformation, corresponding to loading conditions commonly encountered in indentation measurements [ 67 , 68 ]. The calculated results for these compounds are shown in Fig. 3c . The ideal Vickers shear strengths of ScB 2 , TaB 2 , and ZrB 2 are observed in the (1 1 ¯ 0) [001] direction, with peak stresses of 25.2, 31.3, and 32.4 GPa, respectively. The ideal Vickers shear strength of Sc 0.5 Ta 0.5 B 2 reaches 41.9 GPa along the (1 1 ¯ 0) [110], significantly surpassing those of ScB 2 and TaB 2 . This indicates considerably enhanced indentation strength and hardness of Sc 0.5 Ta 0.5 B 2 . We present in Fig. 3d - f the load conditions of TMB 2 under the tensile, pure shear, and Vickers shear strains. The results show that the mixed bonds tilted in the deformation direction undergo elongation, with their bond strength serving as a key factor affecting mechanical strength. Moreover, the ideal tensile, shear, and Vickers shear strength of Sc 0.5 Ta 0.5 B 2 is significantly higher than that of ScB 2 and TaB 2 , indicating a more robust stress response reflecting the intrinsic crystal stiffness of the bonding network in Sc 0.5 Ta 0.5 B 2 because of its more thermodynamically stable structure. 3.3. Intrinsic hardening of Sc 0.5 Ta 0.5 B 2 compounds In TMB 2 compounds, the bonding interactions involve TM–TM, B–B, and TM–B interactions [ 69 ]. To better understand the origin of bonding behaviors in AlB 2 -type TMB 2 compounds, we analyzed the electron localization function (ELF) for ScB 2 , TaB 2 , and Sc 0.5 Ta 0.5 B 2 in different planes where these interactions occur. Fig. S2a illustrates that the charges in the (001) plane of ScB 2 , TaB 2 and Sc 0.5 Ta 0.5 B 2 predominantly exhibit delocalization. In the (110) plane of ScB 2 , TaB 2 , and Sc 0.5 Ta 0.5 B 2 , as shown in Fig. S2b, charge localization occurs between B atoms, indicating the presence of strong covalent B–B bonds. While it is weakened between TM and B atoms and each TM atom donates part of its electrons to the B atoms, resulting in a combination of ionic and covalent TM–B bonds. In general, the bonding behavior in TMB 2 entails a combination of ionic, covalent, and metallic natures, with the strength of covalent bonds fundamentally affects the mechanical properties of TMB 2 compounds. The total density of states of ScB 2 , TaB 2 and Sc 0.5 Ta 0.5 B 2 are shown in Fig.4a , revealing that the E F of Sc 0.5 Ta 0.5 B 2 nearly resides within the pseudogap, indicating its lowest electronic state at E F among the Sc x Ta 1− x B 2 ( x = 0, 0.5, 1) compounds. A lower E F implies that more electrons participate in bonding and become localized. Furthermore, the E F of Sc 0.5 Ta 0.5 B 2 is located just below the top of bonding state as confirmed by the crystal orbital Hamilton population (COHP) analysis of Sc 0.5 Ta 0.5 B 2 , depicted in Fig. S3. If additional electrons were present, they would occupy the antibonding states. Sc 0.5 Ta 0.5 B₂ thus exhibits the highest number of bonding electrons and, consequently, resulting in the strongest bond strength compared to ScB₂ and TaB₂. Fig. 4. Open in a new tab (a) Total density of states (DOS) for ScB₂, TaB₂, and Sc ₀.₅Ta₀.₅B₂, with the red arrow indicating the pseudogap position. (b) Negative integrated crystal orbital Hamilton population (–ICOHP) values for B–B, TM–B, and TM–TM bonds in ScB₂, TaB₂, and Sc₀.₅Ta₀.₅B₂, illustrating the relative bond strength and covalent character of these interactions. To qualitatively assess the covalent bond strength of TM–TM, B–B, and TM–B interactions in these compounds, we analyzed the integral COHP values, where more negative values indicate stronger covalent interactions. As shown in Fig. 4b , the resulting -ICOHPs of TM–TM bonding in Sc 0.5 Ta 0.5 B 2 are higher than those of TaB 2 and ScB 2 .This is further evidenced by increased charge localization within the transition-metal atomic layer in the ELF of Sc 0.5 Ta 0.5 B 2 (Fig. S2a). Similarly, the -ICOHPs of B–B bonds in Sc 0.5 Ta 0.5 B 2 are higher than that of the ScB 2 and TaB 2 . Regarding the TM–B bond, the Sc –B bond value in Sc 0.5 Ta 0.5 B 2 is higher than that of ScB 2 , while the Ta–B bond is comparable to that in TaB 2 . Collectively, these findings indicate that Sc 0.5 Ta 0.5 B 2 exhibits stronger covalent bonding compared to ScB 2 and TaB 2 , which enhances its resistance to both elastic and plastic deformation. The present findings in these compounds showcase the importance of the superior bonding configuration created by the electron band filling; in particular, the E F lying within the pseudogap plays a prominent role in stabilizing and strengthening the crystal structure, leading to the more robust and stiffer mechanical characters of Sc 0.5 Ta 0.5 B 2 compared to ScB 2 and TaB 2 . Recent studies have demonstrated that lattice distortion plays a crucial role in solid solution hardening [ 70 , 71 ]. If this effect can be introduced and optimized within the current research framework, the mechanical properties of transition-metal diborides are expected to be further improved. 3.4. Experimental synthesis and characterization of Sc 0.5 Ta 0.5 B 2 compounds To validate our theoretical predictions, Sc 0.5 Ta 0.5 B 2 was synthesized by sintering uniformly mixed Sc, Ta, and B powders using an argon melting at ambient pressure. XRD analysis ( Fig. 5 a) revealed that the Bragg peaks of Sc 0.5 Ta 0.5 B 2 are positioned between those of ScB 2 and TaB 2 . This indicates that ScB₂ and TaB₂ form a single-phase miscible solution at high concentrations, with no evidence of secondary phase formation. The phase of Sc 0.5 Ta 0.5 B 2 shares the hexagonal symmetry of its parent phases, TaB 2 and ScB 2 . A discernible shift in lattice parameters toward larger values is observed with increased Sc content, consistent with the larger atomic radius of Sc (1.61 Å) compared to Ta (1.43 Å). The lattice parameters of Sc 0.5 Ta 0.5 B 2 were determined to be a = b = 3.095 Å, c = 3.393 Å, falling between those of ScB 2 ( a = b = 3.147 Å, c = 3.511 Å) and TaB 2 ( a = b = 3.059 Å, c = 3.289 Å), in agreement with Vegard's law and confirming a 1:1 ratio of Sc to Ta in the compound. Fig. 5. Open in a new tab Experimental synthesis and structural characterizations of TaB 2 , ScB 2 , and Sc 0.5 Ta 0.5 B 2 : (a) XRD patterns; (b) SAED pattern along the [ 1 ¯ 10] zone axis; (c) STEM HAADF image showing the atomic-scale dispersion of Sc and Ta atoms; (d) Intensity linescan profiles of L1 and L2 from (c), highlighting the distribution of Sc and Ta within the lattice. The microstructure of Sc 0.5 Ta 0.5 B 2 was further investigated using transmission electron microscopy. SAED pattern along the [1 1 ¯ 0] zone axis ( Fig. 5 b) confirmed the hexagonal structure of Sc₀.₅Ta₀.₅B₂, consistent with the XRD results. The absence of a secondary phase indicates that extrinsic hardening mechanisms, such as precipitation hardening or dispersion hardening, do not contribute to the observed mechanical properties of Sc 0.5 Ta 0.5 B 2 . STEM HAADF imaging ( Fig. 5 c) demonstrated that Sc and Ta atoms remain confined within their respective layers, with no diffusion into the boron atomic layers. In Fig. 5 d, the intensity linescan profiles along the L1 and L2 directions confirm that the dispersion of Sc and Ta atoms at the atomic scale. Besides, the atomic columns with brighter contrast represent the heavy element Ta and the darker ones represent the light element Sc. High substitution ratios have been previously reported in ternary compounds such as W₀.₅Ta₀.₅B, where the atomic radii of W (1.30 Å) and Ta (1.34 Å) show minor differences. In contrast, the significant atomic radius mismatch between Sc and Ta in Sc₀.₅Ta₀.₅B₂ likely contributes to its enhanced mechanical properties, as larger solute-solvent atomic mismatches in ternary compounds are generally correlated with improved performance. Vickers hardness measurements were conducted under applied loads of 0.49, 0.98, 1.96, 2.9, and 4.9 N. As shown in Fig. 6a , Sc₀.₅Ta₀.₅B₂ exhibits a Vickers hardness of 40.5 GPa under a 0.49 N load, significantly exceeding that of its parent phases, ScB₂ (16.4 GPa) and TaB₂ (34.6 GPa). As the applied load increases, the hardness reduces gradually to an asymptotic value, a trend similarly observed for other applied loads. At a high load of 4.9 N, the measured hardness of TaB 2 is 25.2 GPa, which aligns well with the previous experimental result (25.6 GPa) [ 4 ]. Notably, the near-asymptotic Vickers hardness of Sc 0.5 Ta 0.5 B 2 stabilizes at 31.5 GPa, representing a significant enhancement over both ScB 2 (9.9 GPa) and TaB 2 (25.2 GPa). This value also surpasses the hardness of many well-known single-phase transition metal borides with high mechanical strength ( Fig. 6b ). Although the formation of a second phase (extrinsic harden) is another approach to improving mechanical properties, single-phase materials are typically preferred due to their uniformity, durability, and long-term performance, which can be compromised by the introduction of a second phase. Furthermore, intrinsic multi-component materials such as Sc₀.₅Ta₀.₅B₂ enable precise tuning of hardness by modifying the interactions between constituent components, offering a more consistent and reliable enhancement of mechanical performance [ 9 , [64] , [65] , [66] , 72 , 73 ]. Fig. 6. Open in a new tab (a) Vickers microindentation hardness of Sc 0.5 Ta 0.5 B 2 , ScB 2 and TaB 2 measured under applied loads ranging from 0.49 to 4.9 N. (b) Correlation between boron content and Vickers hardness under a load of 4.9 N for the Sc 0.5 Ta 0.5 B 2 sample, comparison to with other single-phase transition-metal borides (refer to Table S2 for details). Fig. 7 displays the measured temperature dependence of the electronic resistivity of the Sc 0.5 Ta 0.5 B 2 sample in comparison to ScB 2 and TaB 2 . It can be clearly observed that three curves rise continuously with measured temperature from 2 to 300 K and the electrical resistivity of Sc 0.5 Ta 0.5 B 2 , TaB 2 , and ScB 2 at room temperature is 104.9 µΩ·cm, 68.6 µΩ·cm and 39.4 µΩ·cm, respectively. The electrical resistivity of TaB 2 at room temperature is consistent with previous experimental observations (∼ 68 µΩ·cm) [ 74 ]. The conducting characteristic of Sc 0.5 Ta 0.5 B 2 is comparable to other classic metallic materials such as WB 4 [ 75 ], indicating its excellent electrical conductivity. Additionally, the electrical resistivity of Sc 0.5 Ta 0.5 B 2 is higher than that of ScB 2 and TaB 2 , that is consistent with our theoretical results where Sc 0.5 Ta 0.5 B 2 has the fewest delocalized electrons. Fig. 7. Open in a new tab Measured electrical resistivity of the Sc 0.5 Ta 0.5 B 2 sample, comparison to the ScB 2 and TaB 2 phases. 4. Conclusion In summary, a series of ternary transition–metal borides, Sc x Ta 1− x B 2 ( x = 0–1), was designed using a multicomponent strategy. Comprehensive first-principles calculations were conducted to evaluate the relative formation energy and mechanical strength of Sc x Ta 1− x B 2 structures with varying electron band occupations. Our findings reveal that Sc 0.5 Ta 0.5 B 2 , characterized by near-complete band occupation, exhibits the superior mechanical performance among the Sc x Ta 1− x B 2 compounds, primarily due to its robust covalent bonding states. To validate the theoretical predictions, the ternary Sc 0.5 Ta 0.5 B 2 compound was successfully synthesized using the argon arc melting technique at ambient pressure. Vickers hardness measurements confirmed that Sc 0.5 Ta 0.5 B 2 achieved a hardness of 31.5 GPa, the highest reported for single-phase transition-metal borides. Moreover, it demonstrated excellent electrical conductivity, with an electrical resistivity of 104.9 µΩ·cm at room temperature. This study offers fundamental insights into enhancing material properties through multicomponent design strategies. The diversity of transition-metal diborides with the AlB₂-type configuration highlights the vast potential for exploring ternary diborides. The findings offer guidance for the design and synthesis of ternary transition-metal diborides with exceptional mechanical strength and superhard properties. Declaration of competing interest The authors declare that they have no conflicts of interest in this work. Acknowledgments This work is supported by the National Key Research and Development Program of China ( 2021YFA1400503, 2023YFA1608901 and 2021YFA400300), Natural Science Foundation of China (T2325013, 52288102, 12347162, 12375304 and 12304268), Program for Jilin University Science and Technology Innovative Research Team. Reported calculations utilized computing facilities at the High-Performance Computing Center of Jilin University. Biographies Hang Zhai received her Ph.D degree in 2023 at Jilin University. She is currently a postdoctor at the State Key Laboratory of Superhard Materials & International Center of Computational Method and Software, College of Physics, Jinlin University. Her main research interests are crystal structure prediction and structural chemistry. Xiaohui Yu is currently a professor at Institute of Physics (IOP), Chinese Academy of Sciences (CAS). He received his Ph.D. degree from University of Science and Technology of China in 2011. His research focuses on the regulation of structure and physical properties under high-pressure extreme conditions. Quan Li is a professor at State Key Laboratory of Superhard Materials & International Center of Computational Method and Software, College of Physics, Jilin University. He received his Ph.D. degree from Jilin University in 2011. His research interest mainly focuses on structural design and mechanical properties of superhard materials. Yanming Ma is a professor at Jilin University. He received his Ph.D. degree from Jilin University in 2001, and then spent several years in National Research Council of Canada and Eidgenössische Technische Hochschule Zürich carrying out postdoctoral researches. His research interest mainly focuses on development of computational methods for crystal structure prediction and electronic structure calculations, and the use of these methods to explore material structures and properties under high pressure. Footnotes Peer review under the responsibility of Editorial Board of Fundamental Research. Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.fmre.2024.12.025 . Contributor Information Xiaohui Yu, Email: [email protected]. Quan Li, Email: [email protected]. Yanming Ma, Email: [email protected]. Appendix. 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