==== Front Sci Rep Sci Rep Scientific Reports 2045-2322 Nature Publishing Group UK London 37380870 37642 10.1038/s41598-023-37642-8 Article First-principles demonstration of band filling-induced significant improvement in thermodynamic stability and mechanical properties of Sc1-xTaxB2 solid solutions Mopoung Kunpot 1 Ektarawong Annop Annop.E@chula.ac.th 12 Bovornratanaraks Thiti 1 Alling Björn 3 1 grid.7922.e 0000 0001 0244 7875 Extreme Condition Physics Research Laboratory and Center of Excellence in Physics of Energy Materials, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, 10330 Thailand 2 grid.7922.e 0000 0001 0244 7875 Chula Intelligent and Complex Systems, Faculty of Science, Chulalongkorn University, Bangkok, 10330 Thailand 3 grid.5640.7 0000 0001 2162 9922 Theoretical Physics Division, Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden 28 6 2023 28 6 2023 2023 13 105043 5 2023 25 6 2023 © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Mixtures of different metal diborides in the form of solid solutions are promising materials for hard-coating applications. Herein, we study the mixing thermodynamics and the mechanical properties of AlB2-structured Sc1-xTaxB2 solid solutions using the first-principles method, based on the density functional theory, and the cluster-expansion formalism. Our thermodynamic investigation reveals that the two diborides readily mix with one another to form a continuous series of stable solid solutions in the pseudo-binary TaB2 -ScB2 system even at absolute zero. Interestingly, the elastic moduli as well as the hardness of the solid solutions show significant positive deviations from the linear Vegard’s rule evaluated between those of ScB2 and TaB2. In case of Sc1-xTaxB2, the degrees of deviation from such linear trends can be as large as 25, 20, and 40% for the shear modulus, the Young’s modulus, and the hardness, respectively. The improvement in the stability as well as the mechanical properties of Sc1-xTaxB2 solid solutions relative to their constituent compounds is found to be related to the effect of electronic band filling, induced upon mixing TaB2 with ScB2. These findings not only demonstrate the prominent role of band filling in enhancing the stability and the mechanical properties of Sc1-xTaxB2, but also it can potentially open up a possibility for designing stable/metastable metal diboride-based solid solutions with superior and widely tunable mechanical properties for hard-coating applications. Subject terms Mechanical properties Metals and alloys Atomistic models Electronic structure Second Century Fund (C2F), Chulalongkorn UniversityThailand Science Research and Innovation Fund, Chulalongkorn UniversityIND66230003 Ektarawong Annop Grants for Development of New Faculty Staff, Ratchadaphiseksomphot Fund, Chulalongkorn UniversityDNS 66_002_23_001_3 Ektarawong Annop http://dx.doi.org/10.13039/100007684 Asahi Glass Foundation http://dx.doi.org/10.13039/501100004704 National Research Council of Thailand NRCT5-RSA63001-04 Bovornratanaraks Thiti Swedish Government Strategic Research Area in Materials Science on Functional Materials, Linköping UniversitySFOMatLiU No. 2009 00971 Alling Björn Swedish Foundation for Strategic Research through the Future Research Leaders 6 programFFL 15-0290 Alling Björn http://dx.doi.org/10.13039/501100004359 Vetenskapsrådet 2019-05403 Alling Björn Knut and Alice Wallenberg Foundation, SwedenKAW-2018.0194 Alling Björn issue-copyright-statement© Springer Nature Limited 2023 ==== Body pmcIntroduction Double-metal diborides with chemical formula of M1-x′Mx″B2, where M′ and M″ are two distinct metallic elements (Mg, Al, Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Ru, Os, and Re, for example) and 0 ⩽ x ⩽ 1, have increasingly gained interest in the recent years as promising hard-coating materials for cutting tools, thanks to their high thermal and chemical stabilities as well as good mechanical properties1–13. It has lately been shown from the theoretical and experimental aspects that, by adding the second metallic element M″ to the diboride compounds M′B2, the stabilities and properties of the resulting double-metal diborides M1-x′Mx″B2 can be significantly improved and thus become superior to those of the constituent diboride compounds, either M′B2 or M″B23–8,10–13. For instance, alloying TaB2 with ZrB2 to form thin films of Ta1-xZrxB2 results in improvement in hardness and toughness of the films, relative to those of TaB2 and ZrB2 films7,10,13. Another example is that introducing Cr (Al) atoms into ZrB2 (TiB2) films increases wear, oxidation, and corrosion resistances of the films, and the mechanical properties of the resulting off-stoichiometric films of Zr1-xCrxBy (Ti1-xAlxBy) can be improved or tuned via controlling the films’ composition6,8,11. The enhancement of stabilities and mechanical properties of M1-x′Mx″B2 with respect to M′B2 and M″B2 has been suggested to be directly related to the changes in the number of valence electrons filling bonding and antibonding electronic states of the material14–16, controlled by variation of M′B2 and M″B2 contents, as recently demonstrated in Sc1-xVxB2 exhibiting superior thermodynamic stability and mechanical properties, especially hardness, as compared to ScB2 and VB212. Apart from the double-metal diborides, such a band-filling scenario has theoretically been found to play a crucial role in determining the thermodynamic stability and mechanical properties of some single-metal diborides − for example, AlB217 and TaB218,19. In the case of TaB2, the effect of band filling can be triggered by partial substitution of vacancies for B atoms in the diboride. Despite the increase in the number of broken bonds around the vacancies tending to destabilize the diboride, the replacement of some B atoms in TaB2 by vacancies reduces the number of electrons occupying the diboride’s antibonding states enhancing its stability. The two effects arising from the formation of vacancies on the boron sublattice of TaB2 counterbalance and eventually result in stabilization of B-deficient TaB2-x over a small range of x (0.167 ≲ x ≲ 0.25) in thermodynamic equilibrium. Besides, the shear strength, stiffness, hardness of thermodynamically stable TaB2-x are superior to those of TaB218,19. Hypothetically, replacing some Ta atoms of TaB2 by any group-III or group-IV metallic element (Sc, Y, Ti, Zr, or Hf) to form solid solutions of Ta-containing double-metal diboride should as well yield a decrease in the number of electron occupying the antibonding states of the material without a need for forming the vacancies, and thus give rise to the aforementioned band filling-induced enhancement of thermodynamic stability and mechanical properties of the solid solutions with respect to their constituent compounds, as theoretically predicted for Sc1-xVxB2 solid solutions. Most often, Ta-containing double-metal diborides, represented by mixtures of TaB2 and group-IV metal diboride (TiB2, ZrB2, HfB2) are of interest and considered in the literature7,10,13,16,20–26, whereas solid solutions of TaB2 and group-III metal diboride (ScB2 or YB2) are much less studied2,27,28. We are therefore inspired to examine using first-principles approaches the mixing thermodynamics of ScB2 and TaB2, both of which crystallize in the hexagonal space group of P6/mmm (AlB2-type structure), as well as the mechanical properties of Sc1-xTaxB2 solid solutions. Our thermodynamic considerations reveal a mixing tendency of Sc and Ta atoms, residing on the metal sublattice of Sc1-xTaxB2. This thus enables formation of single-phase solid solutions of Sc1-xTaxB2 across the entire composition range (0 ⩽ x ⩽ 1) at low temperatures. Interestingly, the values of the shear and Young’s moduli as well as the hardness of Sc1-xTaxB2 are observed to largely deviate in the positive direction from the linear mixing trends drawn between those ScB2 and TaB2 (Vegard’s law), which can be directly interpreted in terms of electronic band filling. Additionally, our prediction reveals that Sc1-xTaxB2 exhibits more superior mechanical properties than ScB2 and TaB2, and within a certain range of x it becomes superhard with hardness exceeding 40 GPa.Figure 1 Energies of mixing (Δ Emix) at T = 0 K of ordered and disordered solid solutions of Sc1-xTaxB2 evaluated with respect to ScB2 and TaB2. Red crosses denote the cluster-expansion (CE) predicted Δ Emix of 20240 ordered Sc1-xTaxB2 solid solutions. Open black circles represent the density-functional-theory (DFT) calculated Δ Emix of 1241 ordered Sc1-xTaxB2 solid solution, included in the final CE. Blue squares stand for the DFT-calculated Δ Emix of 5 disordered Sc1-xTaxB2 solid solutions, modeled by the SQS technique29. Thick black lines, connecting large filled black circles, indicate the ground-state lines of ordered Sc1-xTaxB2 solid solutions, derived from the DFT calculations. Results and discussion Mixing thermodynamics of Sc1-xTaxB2 We as a first step assess the alloying behavior of Sc and Ta atoms in the solid solutions of Sc1-xTaxB2. To this end, the cluster expansion of the total energy of Sc1-xTaxB2 according to the mathematical foundation of Sanchez, Ducastelle, and Grastias30 is performed, together with the Connolly-Williams method31, to determine the effective interactions between Sc and Ta. In the present work, our cluster-expansion model utilizes a total of 33 effective interactions (1 zerolet, 1 singlet, 19 pair, and 12 triplet interactions) and it fits the DFT-derived total energies of 1241 out of 20420 structures of ordered Sc1-xTaxB2, as the input of the model, with the leave-one-out cross validation score of 10.235 meV/f.u. The obtained effective interactions are then used for prediction of the total energies of the remaining 19179 structures of ordered Sc1-xTaxB2, not included in building the model. Figure 1 displays the energies of mixing (Δ Emix) at T = 0 K of ordered and disordered solid solutions of Sc1-xTaxB2, calculated with respect to ScB2 as well as TaB2. The negative values of Δ Emix of Sc1-xTaxB2, where 0 ⩽ x ⩽ 1, indicate that ScB2 and TaB2 readily mix with each other, therefore resulting in the formation of Sc1-xTaxB2 solid solutions. As can be seen also from Fig. 1, Δ Emix of ScB2, TaB2, and ordered Sc1-xTaxB2 at different fixed values of x lie on the convex hull (a series of thick black lines connecting large filled blacked circles), suggesting that they are thermodynamically stable and promising candidates of ground-state structures for Sc1-xTaxB2. Also, we note here that the convex hull of the pseudo-binary ScB2 -TaB2 system, predicted by the cluster-expansion model, agrees with that derived from the DFT calculations. This, together with the relatively low cross-validation score of 10.235 meV/f.u., ensures the predictive ability of the model. Besides the ground-state structures of Sc1-xTaxB2, whose Δ Emix lie on the convex hull, we observe that Δ Emix of the lowest-energy structure of ordered Sc1-xTaxB2 of a given composition x, predicted to be unstable at T = 0 K and visualized in Fig. 1 as open black circles, lie only slightly above the convex hull by a few meV/f.u. Such tiny differences in Δ Emix between the lowest-energy structures of unstable-ordered Sc1-xTaxB2 and the convex hull are comparable to the numerical accuracy of our DFT total-energy calculations of Sc1-xTaxB2 and smaller than the cross-validation score of our cluster-expansion model. Our results and analyses on the mixing thermodynamics of ScB2 and TaB2 thus implies the thermodynamic stability of ordered Sc1-xTaxB2 over the entire composition range (0 ⩽ x ⩽ 1) even at T = 0 K. Our prediction of the mixing tendency of Sc and Ta atoms, residing on the metal sublattice of Sc1-xTaxB2, is qualitatively in line with the theoretical results, previously reported in the literature2,27,28. It is also worth mentioning that the formation of single-phase substitutional solid solutions of Sc1-xTaxB2 can be straightforwardly interpreted by the Hume-Rothery rules stating that the formation of solid solutions can be expected, if the atomic sizes and the electronegativity of the constituent elements differ by less than 15% and 0.4, respectively, and the crystal structures of the constituent elements must be similar32. For this particular case, the constituent elements of Sc1-xTaxB2 are ScB2 and TaB2, both of which are isostructural. Also, we find that the atomic radius of Ta is larger than that of Sc by about 8.7%, and the difference between their electronegativity is 0.14 only. This analysis, based on the Hume-Rothery rules, further strengthens the reliability of our results on the mixing thermodynamics of Sc and Ta atoms in Sc1-xTaxB2.Figure 2 Effective pair and triplet interactions between transition-metal atoms residing on the metal sublattice of Sc1-xTaxB2 for several short-range coordination shells, derived from the final CE. By inspecting the effective pair and triplet interactions between the metal atoms for several short-range coordination shells of the metal sublattice in Sc1-xTaxB2, as extracted from the cluster-expansion model and shown in Fig. 2, we find that the magnitudes of the pair interactions for the first, second, and third coordination shells are, respectively, 38.593, 57.154, and 12.442 meV/f.u. The strength of these pair interactions are relatively much stronger than those of the remaining pair and triplet interactions, whose magnitudes are lower than 7 meV/f.u. Because of the strong pair interactions between the metal atoms for the first three coordination shells in Sc1-xTaxB2, any Sc and Ta atoms constituting the solid solutions tend to be surrounded by metal atoms of the opposite type, residing in their first, second, and third coordination shells, in order to lower their total energies. That is, Sc1-xTaxB2 displays short-range chemical ordering of Sc and Ta atoms, leading to thermodynamic stabilization of ordered Sc1-xTaxB2, at low temperatures. It is worth noting that, as Sc1-xTaxB2 is subjected to high-temperature conditions, the ordered patterns of Sc and Ta atoms can become disordered due mainly to the increasingly strong contribution of mixing entropy (Δ Smix), arising from random distribution of Sc and Ta atoms on the metal sublattice of Sc1-xTaxB2. Therefore, it is of interest to estimate the temperature, at which Sc and Ta atoms, residing on the metal sublattice of Sc1-xTaxB2, undergoes such an order-to-disorder transition across the whole composition range. In the present work, we do so by comparing the Gibbs free energies of mixing (Δ Gmix) of disordered solid solutions of Sc1-xTaxB2 to those of the ordered ones. Given that the effects of pressure, lattice vibrations, and electronic excitations are neglected, Δ Gmix as a function of temperature T and chemical composition x can be expressed as;1 ΔGmix(x,T)=ΔEmix(x)-TΔSmix(x). To estimate Δ Gmix for disordered Sc1-xTaxB2, the models of Sc1-xTaxB2 created by the SQS technique29 are employed, together with the DFT calculations, to obtain Δ Emix of the disordered solid solutions (blue shaded squares shown in Fig. 1), and their Δ Smix are analytically derived from the mean-field approach12. That is,2 ΔSmix(x)=-kB[xln(x)+(1-x)ln(1-x)]. Note that, since our Δ Emix of disordered Sc1-xTaxB2 are calculated at 5 discrete compositions (x = 0.167, 0.333, 0.5, 0.667, and 0.833), they together with Δ Emix of ScB2 (x = 0) and TaB2 (x = 1) are interpolated by the cubic spline function and then combined with the term -T Δ Smix, where Δ x = 0.01, to estimate Δ Gmix for disordered Sc1-xTaxB2. Ordered solid solutions of Sc1-xTaxB2 are, on the other hand, assumed to exhibit the zero-Kelvin properties. Thus, Δ Smix for ordered Sc1-xTaxB2 are approximated to be zero, and their Δ Gmix are provided solely by Δ Emix of the ground-state structures of Sc1-xTaxB2. That is, the DFT-derived ground-state lines of ordered Sc1-xTaxB2, shown in Fig. 1. Figure 3 illustrates the curves of Δ Gmix for ground-state ordered and disordered structures of Sc1-xTaxB2 at four selected temperatures. We observe that Δ Gmix curve of disordered Sc1-xTaxB2 lies below that of ground-state ordered Sc1-xTaxB2 across the entire composition range at T ≳ 1337 K. This also implies that, above such a critical temperature, Sc and Ta atoms residing on the metal sublattice of Sc1-xTaxB2 configurationally disorder, and Sc1-xTaxB2 is thermodynamically stable in the form of single-phase disordered solid solutions for the whole composition range.Figure 3 Gibbs free energies of mixing (Δ Gmix) of ground-state ordered (black filled circles) and disordered (red filled squares) structures of Sc1-xTaxB2 solid solutions, evaluated with respect to ScB2 and TaB2, at T = 400, 800, 1200, and 1600 K. Note also that, because of the use of the mean-field approach to estimate Δ Smix for disordered Sc1-xTaxB2, modeled by the SQS method, the configurational order-to-disorder transition temperature of Sc1-xTaxB2 can be overestimated by 20–30%, approximately33–35. This can be explained by the absence of the short-range ordering of Sc and Ta atoms in the SQS models of Sc1-xTaxB2 and the error in the mean-field estimated Δ Smix. The other sources of uncertainty in prediction of the order-to-disorder transition temperature may be attributed to the neglect of effects, arising for example from lattice vibrations and electronic excitations. It has been shown that, without explicitly taking into account the effect of lattice vibrations, the configurational order-to-disorder transition temperature of a given alloy system can be overestimated by up to 30%36,37. Based on this error analysis, one can reasonably expect that the order-to-disorder transition temperature of Sc1-xTaxB2, derived here from the mean-field approach, is overestimated by a few hundred Kelvin. Despite ordered Sc1-xTaxB2, predicted to be thermodynamically stable at T ≲ 1337 K, disordered solid solutions of Sc1-xTaxB2 may be experimentally achieved in the form of thin solid films at moderate-to-low temperatures by using, for example, magnetron sputtering techniques enabling kinetic limitation4,6–8,10,11,13, and thus the as-fabricated films of disordered Sc1-xTaxB2 could remain metastable under ambient conditions.Figure 4 Electronic density of states of ordered (black solid line) and disordered (red dashed line) Sc1-xTaxB2 with x = 0.5, 0.667, 0.833, and 1. The vertical dashed line at 0 eV in (a)−(d) indicates the highest occupied electronic state. Figure 5 Equilibrium lattice parameters (a and c) of ordered (red shaded circles) and disordered (blue shaded squares) solid solutions of Sc1-xTaxB2, where 0 ⩽ x ⩽ 1 and Δ x = 1/3. The black dashed lines are evaluated between ScB2 (x = 0) and TaB2 (x = 1) indicate the linear Vegard’s law. Electronic, structural and mechanical properties of Sc1-xTaxB2 As demonstrated and discussed in the theoretical works on metal diborides, previously published in the literature12,14–16,18,19,38, the negative values of Δ Gmix for both ordered and disordered Sc1-xTaxB2 solid solutions, indicating their thermodynamic stability relative to ScB2 and TaB2, can be directly explained by the changes in the number of electrons filling bonding and antibonding states of Sc1-xTaxB2. Due to the interactions between metal atoms arranging themselves in the simple hexagonal geometry2, as in the case of metal diborides, the electronic density of states of the diborides displays a valley-like feature, frequently separating the bonding states from the antibonding states, around the Fermi level14,15,18,38–40. For ScB2, the valley is located above the Fermi level, implying that there exist still unfilled bonding states of the material12,15,18,39. This can also be confirmed by considering the corresponding chemical bondings of the compound, visualized through its crystal orbital Hamilton population (−COHP)18,41. As can be seen from Fig. S1, the −COHP bonding analysis for ScB2 clearly show non-zero bonding states at the Fermi level. On the other hand, the electronic density of states of TaB2 apparently reveal that the valley lies below the Fermi level15,18,19,39,42. This indicates that, for TaB2, not only the bonding states of the material are fully occupied, but some of its antibonding states resulting particularly from the interactions between the 5d orbitals of Ta atoms and the 2p orbitals of B atoms are also filled by electrons39,43, see also Fig. S2 illustrating the −COHP bonding analysis for TaB2. Figure 4 shows the electronic density of states around the Fermi level of Sc1-xTaxB2 with x = 0.5, 0.667, 0.833, and 1. We find that partial substitution of Sc atoms for Ta atoms in TaB2, leading to the formation of Sc1-xTaxB2 solid solutions, can reduce the number of electrons occupying the diboride’s antibonding states. As the value of x in Sc1-xTaxB2 decreases from 1 to 0.5, the Fermi level shifts toward the valley’s bottom and the number of electronic states at the Fermi level decreases as compared to that of TaB2. Such a shift of the Fermi level toward to the valley due to the partial replacement of some Ta atoms in TaB2 by Sc atoms is an indication of a decrease in the number of electron filling the material’s antibonding states, and it is verified, for example, by the −COHP bonding analysis for the ordered solid solution of Sc0.5Ta0.5B2, as shown in Fig. S3. It is worth noting here that our results on the −COHP bonding analysis for ScB2 and TaB2 are in good agreement with those, recently reported by Dahlqvist et al.18. The depletion of electrons in the antibonding states is in turn expected to result in the strengthening of bonding between the atoms constituting Sc1-xTaxB2. The shift of the Fermi level toward the valley and the reduction of the number of electronic states at the Fermi level can also be observed for ScB2, when some Sc atoms residing on the metal sublattice of the compound are replaced by Ta atoms (not shown). Nevertheless, we note that the changes in the electronic properties in this case is related to an increase in the number of electrons filling the bonding states of the material. We note further that the rigid-band model can herein be assumed when describing any change in the electronic properties of Sc1-xTaxB2, induced by variation of ScB2 and TaB2 contents. Besides, we observe that, for a given value of x, the electronic density of states around the Fermi level of disordered Sc1-xTaxB2 imitates that of ordered Sc1-xTaxB2 (see Fig. 4), suggesting that the configuration of Sc and Ta atoms residing on the metal sublattice has minimal impact on the electronic properties of Sc1-xTaxB2. Based on these findings, we propose that such an effect of band filling, induced by alloying TaB2 with Sc and vice versa, is essentially responsible for the thermodynamic stability of Sc1-xTaxB2 solid solutions with respect to their constituent compounds. According to our prediction, the stability of Sc1-xTaxB2 is expected to be at maximum at x ≈ 0.5. This is because, at such a composition, its Fermi level is positioned very close to the valley’s bottom and the number of electronic states at the Fermi level is minimum. The band filling-induced improvement in the bond strength between the atoms constituting Sc1-xTaxB2 may be characterized by negative deviation of their lattice parameters a and c from the linear mixing trends, drawn between the parameters a and c of ScB2 and TaB2. As can be seen from Fig. 5, the parameters a and c of (either ordered or disordered) Sc1-xTaxB2 are negatively deviating from the linear Vegard’s rule by up to 0.7% and 0.9%, respectively. On the other hand, the parameters a and c of disordered Sc1-xTaxB2 differ from those of ordered Sc1-xTaxB2 by less than 0.05% and 0.4%, respectively. To the best of our knowledge, we are not aware of any experimental and theoretical works reporting the values of a and c for Sc1-xTaxB2, except for ScB2 and TaB2. We note that the reliability of our approach used to derive the properties of ScB2 and TaB2, including the lattice parameters, has already been demonstrated and discussed in our theoretical works on Sc1-xVxB212 and B-deficient TaB2-x19, in which our calculated values of lattice parameters (both a and c) of the two diborides were compared and found to be in excellent agreement with the existing experimental and theoretical data, available in the literature38,43–48.Figure 6 Formation energies (Δ Edefect-form) of B vacancy in the dilute limit of ordered (red shaded circles) and disordered (blue shaded squares) solid solutions of Sc1-xTaxB2, where 0 ⩽ x ⩽ 1 and Δ x = 1/3. The effect of electronic band filling has lately been theoretically shown to play an important role also in determining the mechanical properties of metal diborides12,15,16,19. In case of double-metal diborides, like Sc1-xTaxB2, the effect can be triggered for example by controlling the contents of ScB2 and TaB2 and/or by the presence of structural defects. Recent theoretical works on metal diborides18,19 revealed that, for TaB2 crystallizing in the AlB2-type structure, the effect of band filling can be triggered by partial substitution of vacancies for B atoms. The presence of sufficiently small amount of such vacancies in TaB2 results in the depletion of electrons in the antibonding states of the compound, and thus the thermodynamic stabilization of B-deficient TaB2-x, where 0.167 ≲ x ≲ 0.25, even at absolute zero with superior mechanical properties as compared to those of stoichiometric TaB219. For this reason, we are motivated to examine the formation of B vacancy (in the dilute limit) in the solid solutions of Sc1-xTaxB2, as characterized by the defect formation energy (Δ Edefect-form). For a given configuration of Sc1-xTaxB2, Δ Edefect-form can be evaluate from;3 ΔEdefect-form=Edefect-(Edefect-free-μB), where Edefect-free and Edefect denote the total energies of defect-free and defective Sc1-xTaxB2, respectively, and μB is the chemical potential of B, which is derived from the total energy per atom of α-rhombohedral B. Herein, the structural model of any defect-free Sc1-xTaxB2 is modeled in within a 144-atom supercell, and the corresponding defective structure is created by removing a single B atom from the defect-free supercell of Sc1-xTaxB2. As can be seen from Fig. 6, showing Δ Edefect-form of B vacancy in the dilute limit of ordered and disordered Sc1-xTaxB2 with x ranging from 0 to 1, Δ Edefect-form of B vacancy is about −0.4 eV/defect for TaB2. The negative sign for Δ Edefect-form of TaB2 indicates that, for this particular compound, a small amount of B vacancies is thermodynamically favored, and some B atoms residing on the boron sublattice of the diboride are thus readily be substituted by vacancies. Nevertheless, as one-sixth of Ta atoms residing on the metal sublattice of the diboride are replaced by Sc atoms, giving rise to the formation of either ordered or disordered solid solution of Sc0.167Ta0.833B2, Δ Edefect-form of B vacancy becomes positive, and its value considerably increases up to about +3.8 eV/defect with x decreasing from 0.833 to 0.333, before it slightly decreases to a value of +3.3 eV/defect at x equal to 0. The change from the negative sign to the positive one of Δ Edefect-form of B vacancy in the dilute limit of the diboride, as Ta atoms are (partially or fully) substituted by Sc atoms, indicates that introducing Sc atoms into TaB2 can hinder the formation of B vacancies in the pseudo-binary alloy system of Sc1-xTaxB2. Accordingly, the concentration of B vacancies in Sc1-xTaxB2 with x<1 is expected to be very tiny, as compared to that in TaB2. By considering this, together with the positive and high values of Δ Edefect-form for other types of structural defects in ScB2 and TaB2 in the dilute limit as reported in the previous studies of Dahlqvist et al.18 and Ektarawong et al.19, it is reasonable for us to study the mechanical properties of Sc1-xTaxB2 solid solutions at different fixed values of x without taking into account any structural defects in the models of ordered and disordered Sc1-xTaxB2. Figures 7 and 8 illustrate the values of the elastic moduli and constants as well as the hardness of ordered and disordered Sc1-xTaxB2 with x = 0, 0.167, 0.333, 0.5, 0.667, 0.833, and 1. Just as in the case of lattice parameters a and c, we are not aware of any experimental and theoretical works on metal diborides reporting the values of the elastic moduli and constants as well as the hardness for Sc1-xTaxB2, where 0