We present the x-ray diffraction (XRD) θ-2θ scan patterns of TaB₂, Ta₃ZrB₈, and ZrB₂ films in Fig. 1A. Three diffraction peaks are observed in the XRD pattern of TaB₂ film [power diffraction file (PDF): 65-0878], where the TaB₂ (001), Al₂O₃ substrate [21], and TaB₂ (002) peaks are seen at 2θ = 27.7°, 41.9°, and 56.5°, respectively. The sharp TaB₂ (001) and (002) peaks indicate that the synthesized TaB₂ film exhibits a strong preferred (001) growth orientation. Similar phenomena are also found in the XRD pattern of the ZrB₂ film (PDF: 65-3389), which displays 3 diffraction peaks at 2θ = 25.2°, 41.9°, and 51.8°, corresponding to ZrB₂ (001) peak, Al₂O₃ substrate peak, and ZrB₂ (002) peak, respectively, indicating the strong (001) preferred orientation of the ZrB₂ film. With Zr atoms (atomic radius, 145 pm) substituting for Ta atoms (atomic radius, 134 pm) in the TaB₂ crystal, a lattice expansion occurs, resulting in a lower-angle shift of the diffraction peak in the Ta₃ZrB₈ film compared to that of the TaB₂ film. Moreover, only 2 peaks are observed in the XRD pattern of the Ta₃ZrB₈ film: Ta₃ZrB₈ (001) and (002) peaks at 2θ = 26.5° and 54.1°, respectively, suggesting that the Ta₃ZrB₈ solid solution film has also strong (001) preferred orientation. The TaB₂, Ta₃ZrB₈, and ZrB₂ films all adopt the hexagonal crystal structure with the P6/mmm space group symmetry. The full widths at half maximum of the (001) peaks for the TaB₂, Ta₃ZrB₈, and ZrB₂ films are 0.44°, 0.24° and 0.29°, respectively, giving the estimated grain sizes of ~20, ~33, and ~28 nm using the Scherrer formula [22]. The high-resolution transmission electron microscopy (HRTEM) images of the TaB₂, Ta₃ZrB₈, and ZrB₂ films are shown in Fig. 1B and C, which verify their crystal structures. The interplanar spacing d values in Fig. 1B and D are measured to be 0.169 and 0.214 nm, which are close to the d values of TaB₂ (002) lattice plane (d₀₀₂ = 0.164 nm) and ZrB₂ (101) lattice plane (d₁₀₁ = 0.216 nm) given in PDF cards. Figure 1C exhibits a d value of 0.208 nm for the Ta₃ZrB₈ (101) plane, and this value is smaller than d₁₀₁ = 0.216 nm of ZrB₂ (101) plane and slightly larger than d₁₀₁ = 0.206 nm of TaB₂ (101) plane, indicating that these d values follow Vegard’s law, which is often obeyed by solid solution crystal structures.

The mechanical properties of TaB₂, Ta₃ZrB₈, and ZrB₂ films are characterized by means of nanoindentation device equipped with Berkovich diamond indenter. Figure 2A displays the measured load–displacement curves for these synthesized films. The forces loaded on the Berkovich diamond indenter at an indentation depth of 600 nm are 181.9, 217.4, and 164.9 mN for the TaB₂, Ta₃ZrB₈, and ZrB₂ films, respectively, in which the load force on the Ta₃ZrB₈ film is the highest, indicating that the deformation resistance of the Ta₃ZrB₈ film is stronger than those of the TaB₂ and ZrB₂ films. The hardness values of the TaB₂, Ta₃ZrB₈, and ZrB₂ films were estimated using the Oliver–Pharr method [23,24], and the results were evaluated from the indentation depth range of 50 to 120 nm (Fig. S1), producing the hardness values of 45.9 ± 1.0, 49.5 ± 2.2, and 33.0 ± 1.7 GPa for the TaB₂, Ta₃ZrB₈, and ZrB₂ films (Fig. 2B), respectively. These results show that the formation of Ta₃ZrB₈ solid solution enhances the hardness of superhard TaB₂. The elastic modulus of each sample can be obtained from the slope at the initial position of the unloading section [23] of the load–displacement curves in Fig. 2A, and the values are 512.9 ± 9.5, 579.2 ± 23.6, and 404.9 ± 14.1 GPa for the TaB₂, Ta₃ZrB₈, and ZrB₂ films, respectively, as also shown in Fig. 2B. Scanning electron microscope (SEM) morphology images in Fig. 2C reveal that indented TaB₂ and ZrB₂ films show radial cracks, while no cracks are seen around the indentation site of the Ta₃ZrB₈ film, showing enhanced toughness of the Ta₃ZrB₈ film compared to the TaB₂ and ZrB₂ films and demonstrating concurrent strengthening and toughening of the Ta₃ZrB₈ solid solution. We have checked the data for all 9 indentation points of each film and got the same toughness variation tendency. It is noted that nanoindentation is known to produce higher hardness than microindentation. However, the nanoindentation method is still an appropriate way to characterize the true hardness of thin film materials, because we must eliminate the substrate effect when the film materials are only a few micrometers. Besides, thin-film materials prepared by physical vapor sputtering usually occur good crystalline quality and high densification. Hence, the thin film materials should exhibit a higher hardness than the bulk materials, which usually achieved a slightly lower densification due to the introduction of pores during the sintering. We would like to emphasize that the solid solution strengthening and toughening effects were found in the transverse comparison concerning the mechanical properties of the 3 films using the unified test method.

To further assess structural changes under indentation, we prepared a cross-sectional sample of the indented Ta₃ZrB₈ solid solution film by the focused ion beam (FIB) technique as illustrated in Fig. 3A1 and A2. The sample of the Ta₃ZrB₈ solid solution film was mounted on SEM stubs, and the selected indented area was coated with a layer of Pt metal to increase conductivity to enhance imaging quality and reduce damage to film surface by the Ga⁺ ion beam used for FIB cutting. To reduce the damage to the microstructure morphology of the cross-sectional sample, the Ga⁺ ion beam energy was selected to be 2 keV [25]. Ion beams etched grooves through the film thickness on both sides of the transverse axis of the indentation. Subsequently, along the direction of the white arrow in Fig. 3A2, the indentation cross-section was milled to a thickness of less than 30 nm to improve the transmittance during TEM characterization (Fig. 3A3). The TEM image of the cross-sectional sample is shown in Fig. 3B1, and a circular region with a diameter of about 126 nm beneath the indentation surface was selected for selected-area electron diffraction characterization (Fig. 3B1), which shows diffraction spots for the hexagonal crystal structure [26], in agreement with the structural analysis by the XRD measurements.

To further examine the impact of indentation loading on the crystal structure of the Ta₃ZrB₈ solid solution film, we took a close examination of an area directly underneath the indented surface. The HRTEM image of the cross-sectional sample (Fig. 3B2) reveals high-quality crystallinity without visible structural defects. Moreover, it is noted that the indented Ta₃ZrB₈ solid solution has a (001) interplanar spacing d of 0.326 nm, which is notably smaller than the corresponding (001) interplanar spacing d value of 0.343 nm taken at a site away from the indented surface (Fig. S2). This result shows that upon the elastic recovery after the removal of the indenter, the Ta₃ZrB₈ solid solution structure remains partially compressed but retains a high degree of crystallinity, which indicates that the strengthening and toughening observed in the experiments stem from intrinsic bonding characters of the chemically tuned solid solution structure.

To support the scenario of intrinsic strengthening and toughening in Ta₃ZrB₈ solid solution, we further examined factors that may influence the hardness and toughness of the film samples, including residual compressive stress in the film, grain size, tissue phase around grain boundaries, indentation-induced structural transformation, and dislocation generation and movement. Residual compressive stress in film samples tends to enhance measured hardness [27]. We calculated the intrinsic stress of as-deposited films using the Stoney equation [28] and obtained −2.99 ± 0.23, −1.02 ± 0.16, and −1.39 ± 0.11 GPa for the TaB₂, Ta₃ZrB₈, and ZrB₂ films, respectively. It is seen that the Ta₃ZrB₈ solid solution film hosts the lowest intrinsic compressive stress yet the highest measured hardness, suggesting that compression stress is not a determining factor for the strengthening of Ta₃ZrB₈ solid solution over TaB₂ or ZrB₂. Highly crystallized and oriented film specimens comprising grains with a size of 20 nm or larger are known to exhibit minimal grain-boundary effect and possess hardness close to that of high-quality single crystals [29]. The grain sizes for the synthesized TaB₂, Ta₃ZrB₈, and ZrB₂ films in this work are ~20, ~33, and ~28 nm, respectively. Therefore, grain size is not a main influencing factor in enhancing the mechanical properties of the Ta₃ZrB₈ solid solution film. It has been reported that a tissue phase may improve hardness; for example, a boron-rich tissue phase at the grain boundaries of TiB_2.4 films has shown to contribute to notable improvement of hardness because the high cohesive strength of the tissue phase prevents grain-boundary sliding [30]. Moreover, such tissue phases may also help improve toughness via inhibition of crack propagation and promotion of grain-boundary sliding [31]. However, no tissue phase was found in the crystal structure of Ta₃ZrB₈ solid solution. Another possible source of structural toughening is phase transformation during indentation, which dissipates a large amount of deformation energy and deflects or even stops the crack propagation [32]. However, the results in Fig. 3B clearly show that there is no structural transformation in the Ta₃ZrB₈ solid solution film during the nanoindentation test, thereby removing this possibility. In addition, a recent work reported positive effect of stacking fault and twinning on the plastic deformation of interstitial intermetallic compounds [33]. In fact, for TMB₂ with high crystalline quality, the material deformation mode at room temperature usually transitions from complete elasticity to suddenly fracture [7]; thus, there is probably no stacking fault and twinning-induced plasticity under the conditions reported in our present work. We finally evaluate the role of dislocation in affecting the mechanical properties. Dislocation pinning is a major mechanism for hardness improvement in metallic solid solutions, and dislocation slip dominates plasticity of metallic systems. However, for ceramics similar to TM compounds studied in this work, dislocation generation and motion are not major deformation modes due to the very high energy cost associated with the breaking and rearrangement of the strong (pure or mixed) covalent bonds. Instead, incipient plastic formation is usually dominated by brittle bond breaking, followed by crack propagation. We have analyzed the area underneath the indented surface using the selected-area inverse fast Fourier transform method, and the results in Fig. 3B3 show no dislocation in this area. Some literatures reported that TMB₂ films with the (001) preferred orientation make it difficult to form dislocation [26,30], which hints that dislocation may not be a major relevant factor for solid solution strengthening and toughening in Ta₃ZrB₈ solid solution. Moreover, we likewise obtain the scanning TEM high-angle annular dark-field (STEM-HAADF) picture (Fig. 3C1) beneath the indentation surface, which illustrates the high crystalline quality of Ta₃ZrB₈. Subsequently, the corresponding geometric phase analysis is executed to observe the strain distribution of film after indentation test. The [0001] and [1010] are defined as x and y axes, respectively. The stain maps of horizontal normal strain (e_xx) and vertical normal strain (e_yy) are computed and displayed in Fig. 3C2 and C3, which exhibit the compression stain (blue color) along the direction perpendicular to (0001) plane and the tensile strain (red color) along the direction parallel to (0001) plane, indicating a complex strain state of Ta₃ZrB₈ under berkovich indentation deformation mode. Below, we carry out a systematic and in-depth theoretical evaluation of the structural deformation modes and stress responses of Ta₃ZrB₈ solid solution under indentation loading conditions, and the results are compared to those of TaB₂ and ZrB₂ to reveal the atomistic mechanisms responsible for the strengthening and toughening caused by the chemical tuning of bonding charge on the main load-bearing bonds in the solid solution crystal.

We checked the stability of the Ta₃ZrB₈ solid solution by evaluating its formation enthalpy and phonon spectrum. The obtained negative mixing enthalpy of −95.173 meV/atom indicates the energetic stability, and a lack of imaginary phonon modes in the entire Brillouin zone of Ta₃ZrB₈ (Fig. S3) shows the dynamic stability. These results help to explain the successful experimental synthesis of this compound and establish its robust viability for practical applications.

We then performed stress–strain calculations for TaB₂, ZrB₂, and Ta₃ZrB₈ under indentation loading condition comprising a shear strain constrained by a coexisting loading-induced normal compression. Here, we focus on the case of indentation on the (001) oriented crystal, which is the preferred growth orientation of the synthesized film samples. The (001) crystal plane has a 6-fold bonding symmetry as illustrated in Fig. 4A for the [1-10] directions. We assessed that indentation strength by calculating normal compression constrained shear stress response to strains along the high-symmetry (001)[1-10] slip directions, where the evenly distributed dense bonding network serves to sustain the indentation deformation applied to specimens in the (001) orientation. The calculated peak stress of the corresponding stress–strain curve is defined as the indentation strength, which is directly related to measured hardness [34,35]. Our calculated results (Fig. 4B) show that the indentation strengths of the (001) oriented TaB₂, ZrB₂, and Ta₃ZrB₈ are 42.7, 38.8, and 45.7 GPa, respectively; moreover, the Ta₃ZrB₈ crystal exhibits extended strain range that is notably longer than those of TaB₂ and ZrB₂, signaling considerably improved ductility and toughness as measured by the area under the stress–strain curves [36]. These results show that intrinsic stress–strain relation of the Ta₃ZrB₈ solid solution can produce the long-sought simultaneous strengthening and toughening, which deviates considerably from Vegard’s law that describes properties of a solid solution via a linear interpolation relation between the respective properties of the constituent pure materials [37]. In addition, the indentation stress–strain relation in the (001)[110] slip direction was also evaluated (Fig. S4), where the peak stress for TaB₂, ZrB₂, and Ta₃ZrB₈ are 42.9, 35.6, and 40.3 GPa, respectively, which support the main conclusion of this work. Moreover, it is noted that the stronger resistance in the [1-10] slip direction determines the indentation response of the specimen in the (001) orientation with a 6-fold crystal symmetry spanning a network of strong directions capable of resisting indentation deformation applied in the (001) plane, which is similar to the situation reported for OsB₂ [38]. It needs to be emphasized that the results obtained from indentation stress–strain curve are usually in good agreement with the experimental hardness measured by well-controlled nanoindentation test. This calculation method obviously differs from the results derived from uniaxial shear strain–stress curve due to the difference of crystal deformation mode. The latter makes the crystal slip along a direction in the fixed crystal plane without normal compressive stress, which is usually unsuitable for direct comparison with indentation hardness measurements. For example, the uniaxial shear strength of ZrB₂ along the (0001)[10-10] slip direction, which is equivalent to the (001)[1-10] slip direction, is 46.3 GPa [39], while the biaxial indentation strength of ZrB₂ along the (001)[1-10] slip direction in our work is 38.8 GPa, and this value is much closer to the indentation hardness of ~33.0 GPa.

We further examined the evolution of bond lengths and bonding charge under the indentation deformation to unveil the mechanism for the strengthening and toughening of Ta₃ZrB₈ solid solution. Analysis of the bond length variations in deformed Ta₃ZrB₈ (Fig. 5) shows that all the B–B bonds along with a small set of TM–B bonds are the main load bearers under the indentation shear strains, because these bonds exhibit the largest elongation in response to the strains and are therefore responsible for setting the limit on the load-bearing capacity of the material. Consequently, we focus on the charge evolution in the boron layer. Results (Fig. 6 and Fig. S5) show that at equilibrium, there is no obvious difference in the electron distribution along the B–B bonds between TaB₂ and Ta₃ZrB₈; however, under rising strains, a deformation-induced charge redistribution develops along the B–B bonds. In solid solution Ta₃ZrB₈, the difference of the charge distributions along all B–B bonds under large strain becomes more pronounced. In every direction of the boron 6-membered ring, there are some enhanced B–B bonds in the solid solution, compared to the base compound TaB₂. Selected B–B bonds in the [210] and [120] directions retain more electron under large strains in Ta₃ZrB₈, and some B–B bonds in the [1-10] direction are much less charge depleted under large strains after Zr doping, leading to a more than 20% strain range increase from 0.17 in TaB₂ to 0.21 in Ta₃ZrB₈, generating the enhanced peak stress and related strain range. We also examined the charge distribution on the TM–B bonds due to their crucial role in sustaining the 3-dimensional TMB₂ crystal structure. As shown in Fig. S6, the Ta–B load-bearing bonds along the slip direction in Ta₃ZrB₈ are enhanced compared to those in TaB₂, contributing to the strengthening and toughening of the solid solution.

The bonding charge redistribution caused by introducing the solute atom (Zr) with an EN in proper contrast with that of the solvent atom (Ta) enhances the main load-bearing B–B bonding network while still retaining sufficiently strong TM–B bonding network, thereby extending the strain range and enhancing the peak stress, leading to the concurrent strengthening and toughening of Ta₃ZrB₈ solid solution. It is noted that there exist many other parameters that affect mechanical properties, and, among them, valence electron concentration (VEC) is the most influential factor to regulate the mechanical properties of materials [40]. Recently, VEC is also shown to be a convincing indicator to regulate the mechanical properties of TMB₂ by checking the elastic constants (C_ij), bulk modulus (B), and shear modulus (G). Specifically, theoretical hardness (H_Tian) and brittleness are enhanced as VEC decreases from a broad perspective. However, within a narrow range of VEC, the elastic constants for different systems are often scattering widely with following a regular distribution pattern, which suggests the need to introduce a secondary yet significant factor, such as EN. In our present work, we examined the values of key parameters underlying the mechanical properties for the 3 samples to evaluate the effect of VEC variation, as shown in the Table. It is seen that Zr atoms introduced into TaB₂ make the VEC go down by 0.25 compared with TaB₂. Among these quantities, C₁₁ and C₃₃ measure a material’s ability to resist linear compressions along different crystallographic axes [41], and B represents the material’s ability to resist volume change [42]; these parameters of Ta₃ZrB₈ are between the values of ZrB₂ and TaB₂, indicating an interpolation relation of volume deformation resistance. Moreover, C₄₄ and G are related to the material’s ability of shear resistance [19], and Ta₃ZrB₈ hosts weak shear resistance owing to its small C₄₄ and G. The ductility for Ta₃ZrB₈ has a slight improvement inferred from its small G/B despite the intrinsic brittle feature of the samples. Hence, VEC has a visible impact on these key material parameters. Nevertheless, the theoretical hardness values obtained using Tian’s model based on elastic moduli exhibit the variation trend of ZrB₂ > Ta₃ZrB₈ > TaB₂, which is evidently different from the experimental hardness variation trend and is better described by considering the effect of EN and indentation strain–stress relations. This situation highlights 2 important points: (a) EN plays a more significant role in regulating the mechanical properties of Ta₃ZrB₈ solid solution compared to the variation of VEC within a narrow range caused by the Ta/Zr substitution; (b) compared with commonly used C_ij, indentation stress–strain relations provide a more appropriate foundation for evaluating the mechanical responses probed by nanoindentation. These findings expand the traditional solid solution approach to a new realm of chemically tuning strong covalent bonding materials. More importantly, this strengthening and toughening solid solution protocol constructed by introducing metal atoms with relatively low EN is also effective in other material systems. For instance, the EN of Zr (1.33) is lower than that of Ti (1.54), and as Zr atoms are doped into TiN lattice, the Ti_1−xZr_xN (x = 0.41) film exhibits a slightly higher hardness and H³/E² value than the corresponding results for TiN [43]. Similar phenomena have been reported in MoVN [44] and CrVN [45], which show a strengthening and toughening effect, as V atoms with relatively lower EN are introduced into parent TMN (Mo₂N or CrN) phases. The crystal structure and chemical bonding of TMN (or TMC) are different from those of TMB₂ in that TMN (or TMC) usually adopts the rock-salt structure with TM–TM metallic bonds and TM–N (or TM-C) ionic bonds, while, in contrast, TMB₂ crystallizes in hexagonal structure with alternately stacked metal layers and B layers, consisting of not only the TM–TM metallic bonds and TM–N (or TM–C) ionic bonds but also B–B covalent bonds. Considering that the B–B covalent bonds are the strongest among all chemical bonds in TMB₂, when the EN of TM is changed, the reinforced B–B covalent bonds produce more marked positive effects on the strength and toughness of TMB₂, although the optimized TM–B ionic bonds also contribute. For TMN, the change of EN only regulates the TM–N ionic bonds, which are similar to the TM–B bonds in this work, and the solute metal atom with lower EN in TMN may also improve the TM–N ionic bonds between solvent metal and N atoms, leading to the strengthening and toughening phenomena in TMN systems. Furthermore, we also notice that the EN of B (2.04) is much lower than that of N (3.04), and as a result, the EN deviation between metal and B atoms is smaller than that between metal and N atoms. Hence, B atoms are more sensitive to the variation of the EN of metal atoms than N atoms; as a result, the optimization of TM–N bonds may be inferior compared to TM–B bonds. Even so, the EN of solute metal atoms may still play an effective role in regulating broader classes of materials.

The TaB₂ (ZrB₂) film was grown on a double-side polished Al₂O₃ (0001) substrate using a TaB₄ (ZrB₄) target (60 mm in diameter, 99.95% pure) in a direct current (DC) magnetron sputtering system with a base pressure of 5 × 10⁻⁴ Pa. Growth took place in Ar (99.999% pure) gas environment at a constant DC power of 200 W for 180 min; meanwhile, the target-to-substrate separation, work pressure, substrate temperature, substrate bias voltage, and substrate rotation speed were set at 8 cm, 0.8 Pa, 600 °C, −80 V and 10 rpm, respectively. The Ta_1−xZr_xB₂ solid solution films were deposited by magnetron cosputtering individual TaB₄ and ZrB₄ targets. The DC power for the ZrB₄ target remained 200 W, while the DC power for the TaB₄ target was set at 150 W, and the other sputtering parameters were kept the same as those described above. The film thicknesses are ~1.1, ~1.6, and ~2.6 μm, respectively. Chemical contents for as-deposited films were characterized by x-ray photoelectron spectroscopy (ESCALAB-250) using Al Kα as the x-ray source. The x-ray photoelectron spectroscopy results show that the ratio of Ta and Zr content in the synthesized Ta_1−xZr_xB₂ film is 3:1. The θ-2θ XRD scans used a Bragg–Brentano (Bruker D8) diffractometer with a monochromatic Cu Kα source (λ = 0.15418 nm) to characterize the crystal structure of the as-deposited films. The measurement was performed in locked coupled mode with 40-mA current and 40-kV voltage. Readings were recorded at a scan speed of 0.5 s/step in a 2θ range from 20° to 60°. For the obtained solid solution film, the indentation cross-section was fabricated using an FIB system (FEI Strata 400S) with a coarse milling condition of 30 keV and a final polishing condition of 2 keV. The HRTEM (JEOL 2010F and JEM 2100F), operated at 200-kV accelerating voltage, was utilized to confirm the crystal structure of the samples. The aberration-corrected STEM (Titan Cubed Themis G2300) was used to obtain the HAADF picture of atomic phase of films. Nanoindentation tests were performed by an MTS Nano Indenter G200 system equipped with continuous stiffness measurement mode [46]. During nanoindentation test, the loading function was displacement controlled with a constant rate of 10 nm/s. A maximum indent depth of 600 nm for each film was reached by a Berkovich pyramidal diamond probe (apex angle = 142.3°) with a nominal tip radius of 150 nm, and the fused quartz standard of known hardness and elastic modulus was used as calibration [23]. Nine load–displacement curves were acquired in each film and analyzed following the Oliver and Pharr method to obtain the hardness values. The distance of each indentation was set at 30 μm to avoid the effect of plastic deformation around the indentation [47]. The morphology of indentation of film was observed by an SEM (Hitachi SU8010) at 2-kV high voltage.

Structural optimization of the TMB₂ compounds in P6/mmm symmetry (no. 191) was performed using Vienna Ab Initio Simulation Package code [48], using the periodic boundary conditions, projector-augmented wave potentials [49], and the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation [50] for exchange correlation energy with a plane-wave basis set. For TaB₂, ZrB₂, and Ta₃ZrB₈ of 2 × 2 × 2 supercell, the conjugate gradient optimization method was employed to relax atomic positions and structural parameters, with the total energy of the structure converged to 10⁻⁶ eV/atom and residual forces on each atom less than 0.005 eV/Å. A 20 × 20 × 20 Monkhorst–Pack k-point grid and a 500 eV cutoff energy were used in the calculations to determine the structural optimization, charge distribution, and stress responses. Indentation strength was obtained from the calculated Berkovich stress–strain relationships. The stress–strain curves were obtained by fixing the applied strain along specific loading paths and relaxing the lattice vectors and atomic positions step by step, and the peak stress before structural failure gives the indentation strength [51,52]. During the stress–strain calculations, the shear (σ_zx) and normal compressive (σ_zz) stress component obey the relation σ_zz = σ_zxtanϕ, where ϕ is the centerline-to-face angle of the indenter [53]. The stress–strain calculations have proved to be suitable for estimating indentation strengths of diverse TM light-element compounds, providing accurate description of strength and atomistic deformation modes [14,54–56]. We have adopted this approach in this work to assess the chemically tuned solid solution strengthening and toughening phenomena and mechanisms for dual-TMB₂. The lattice dynamical property is evaluated by employing density functional perturbation theory with 96 atoms per cell. Elastic constants C_ij were extracted by introducing small finite distortions on the lattice to determine the elastic tensor to extract elastic constants. The polycrystalline elastic modulus and polycrystalline shear modulus were calculated using the Voigt–Reuss–Hill approximation.