Impact of vacancies on structure, stability and properties of hexagonal transition metal diborides, M B 2 ( M = Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Mn, and Fe)

In this study, we have used density functional theory (DFT) calculations to characterize if and how defects influence the stability and electronic/mechanical properties of M B 2 (AlB 2 -type) for different transition metal M . From a point defect analysis including vacancies, interstitials, and anti-sites, we identify vacancies to be most favored, or least unfavored. To provide insight into possible vacancy ordering, we focus on vacancies on M - and B-sublattices for nine metals ( M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W), modelled both as disordered and ordered. We demonstrate and explain why vacancies have a significant impact for M from Group 4 (Ti, Zr, Hf), Group 5 (Nb, Ta) and 6 (Mo, W) with improved thermodynamical and dynamical stability as well as mechanical properties. This by diverging from the ideal composition through controlled off-stoichiometry in terms of vacancies in M - or B-deficient structures. Line compounds TiB 2 , ZrB 2 and HfB 2 account for B-poor or M - rich conditions by forming planar defects comprised of vacant B. This in contrast to the ordered M - and B vacancies identified for MoB 2 and WB 2 , with an optimal result at 33.33% M-and 25% B-vacancies, respectively, which significantly improves the stability and concurrent properties through elimination of antibonding states and minimization of non-bonding states. Similar behavior with enhanced stability and properties is demonstrated for NbB 2 and TaB 2 with an optimum around 10% M - and 17% B-vacancies, respectively.


Introduction
Metal boride compounds are structurally and compositionally diverse materials, as boron interacts with many of the elements in the periodic table, and typically possess a wide range of properties, such as high melting point and hardness, wear-and corrosion-resistance, and superconductivity, thus making them promising for many different applications [1][2][3].In the quest for designing materials with improved properties of relevance for industrial applications, novel elemental combinations are theoretically proposed through substitutional studies [4,5] or crystal structure prediction [6][7][8], and ultimately synthesized [4,[9][10][11].Another approach is to improve already known materials by tuning their composition [12][13][14][15].Both routes have been demonstrated appropriate for the hexagonally layered metal diboride MB 2 in the AlB 2 -type crystal structure.In recent studies, it has been demonstrated that TiB 2 and ZrB 2 , which traditionally have been considered as line compounds, can accommodate B-vacancies at B-deficient conditions through anti-phase boundary (APB) formation [16,17].This is in contrast to the defect vacancy formation energy indicating that vacancies are unfavorable [5].Here it should be noted that the defect formation energy only takes into account an isolated vacancy while the APB is composed of a vacant B-plane.Furthermore, when comparing calculated and measured lattice parameters of MB 2 , they are consistent for M from Group 3 (Sc, Y) and 4 (Ti, Zr, Hf), while being more deviating for M from Group 5 (V, Nb, Ta) and 6 (Cr, Mo, W).Reason for this have been suggested to be attributed to non-stoichiometric MB 2 [18][19][20][21].However, many questions remain unanswered, for example concerning possible vacancy formation in MB 2 , details whether it occur on the M-or B-sublattice and how they are distributed, and their potential impact on the stability and properties.These questions are related to challenges in determining the M/B ratios in the MB 2 structure and whether the reported composition is nominal, i.e., prior to synthesis, or ascribed to be measured for the synthesized MB 2 structure.Moreover, the distribution and arrangement of vacancies or defects in the structure is challenging to determine experimentally.
Motivated by these challenges and the lack of a systematic and thorough study beyond single point defects in MB 2 , we have used density functional theory (DFT) calculations to study the impact of defects in MB 2 (AlB 2 -type) for M from group 3 to 8. Starting from point defects we identify both M-and B-vacancies to be most likely.Furthermore, we go beyond single vacancies by considering both a disordered solid solution vacancy distribution as well as a large number of ordered vacancy structures for each MB 2 system and find that structures with ordered vacancies not only improve the thermodynamical stability but also the mechanical properties for selected MB 2 .

Methods
All first-principles calculations were performed by means of density functional theory (DFT) and the projector augmented wave method [22,23], as implemented within the Vienna ab-initio simulation package (VASP) version 5.4.1 [24][25][26].We used the generalized gradient approximation (GGA) as parameterized by Perdew-Burke-Ernzerhof (PBE) [27] for treating the electron exchange and correlation effects.For Cr-, Mn-, Fe-based phases, the spin-polarized GGA version with multiple magnetic spin configurations was considered.A schematic illustration of considered colinear magnetic spin configurations for MB 2 in the AlB 2 -type structure is shown in Fig. S1.A plane-wave energy cut-off of 400 eV was used and the Brillouin zone was integrated by Monkhorst-Pack special k-point sampling [28].The total energy is minimized through relaxation of unit-cell shape, volume, and internal atomic positions until satisfying an energy convergence of 10 − 6 eV/atom and force convergence of 10 − 2 eV/Å.Structures with vacancies of configurational order were generated using the alloy theoretic automated tool kit (ATAT) [29] while structures containing vacancies with configurational disorder were modelled using the special quasi-random structure (SQS) method [30].
The formation energy for a general M 1− z B x phase is given by where, E(M 1− x B x ), E(M), and E(B) represent the calculated total energy of M 1− x B x , M, and B, respectively.For MB 2 with vacancies on either the M-or-B-lattice, the use of ΔE f is not very informative since explicit information about possible decomposition of M 1− z B 2− y into other binary phases is missing.An approach most appropriate to reveal whether formation of M 1− z B 2− y is favorable with respect to MB 2 and other binary phases and elemental structures is the formation enthalpy ΔH cp .It is calculated using where the equilibrium simplex is comprised by the set of most competing phases being identified using a linear optimization procedure [31,32].Competing phases considered in this work are listed in We approximate contribution from other temperature-dependent effects, e.g., lattice vibrations and electronic entropy, to the formation enthalpy to be negligible as such contribution from a phase, significant or not, tend to be cancelled out in the Gibbs free energy of formation term [33].Both ordered and disordered vacancies on M-and B-sites have been considered.For a disordered distribution, elevated temperature leads to a decreased Gibbs free energy due to contribution from configurational entropy, as approximated by where the entropic contribution, ΔS, when assuming an ideal solution of vacancies on the M-or B-site, is given by where k B is the Boltzmann constant and x is the concentration of vacancies.Density of states (DOS) and crystal orbital Hamilton population (COHP) were retrieved using the LOBSTER code [34][35][36][37] where the calculated band-structure energy is reconstructed into orbital interactions.Positive COHP values indicate an antibonding interaction, and negative COHP values indicate a bonding interaction.Mechanical properties were retrieved through stress-strain calculations with preand postprocessing using the VASPKIT code [38].
Visualization of atomic structures was done with the VESTA code [39].

Stability and structure of MB 2 (AlB 2 -type)
The M-B binary prototype structures considered within this work are listed in Table S2, in total amounting to 44 different structures acquired from the Inorganic Crystal Structure Database (ICSD) [40], the Materials Project database [48], from M-B phase diagrams [41], and from recently published studies of M-B systems, e.g., including crystal structure predictions using evolutionary algorithms [6][7][8][42][43][44][45][46][47].The corresponding calculated formation energy ΔE f (Eq.( 1)) for each structure combined with all considered M is shown in Fig. 1(a) for M = Sc, Y, Ti, Hf, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Mn, Fe, where the composition is expressed as M 1-x B x .The stable crystal structures in each M-B system are connected to form a so-called convex hull, represented by the solid blue circles and blue lines.Phases above the convex hull are considered not stable, or at best metastable, and are represented by grey crosses.We find a general trend where the minimum value of ΔE f in each M 1-x B x system is shifted towards the M-rich region (to the left) when the valance of M increases.A suggested explanation thereof will be discussed in relation to the electronic structure and bonding analysis presented further below.
In this work, focus is on MB 2 in the AlB 2 -structure, illustrated in Fig. 1(b), with hexagonal metal layers interleaved by boron layers arranged in a honeycomb lattice.In Fig. 1(a), MB 2 (AlB 2 ) is indicated at x = 2/3 by either a green diamond if stable or a red square if above the convex hull.Out of the 13 considered M-elements, 7 are on the hull and considered stable (M = Sc, Y, Ti, Zr, Hf, V, Nb), one is close to stable (M = Ta, 15 meV/atom above convex hull), and 5 are further away from the hull (M = Cr, Mo, W, Mn, Fe).In those six M-B systems where MB 2 (AlB 2 ) is above the convex hull (red squares), compositionally equivalent but structurally different phases are found at lower energies, and in four systems (M = Ta, Cr, Mo, and W) these are also found on the convex hull.This contrasts with the experimental literature where an AlB 2 -type structure have been reported for all systems investigated herein, except for M = Fe.Detailed information and a comparison of the crystal structure and stability of these five crystallographically different MB 2 structures are shown in Fig. S2 for all 13 considered M. Reasons for this theoretical and experimental inconsistency, and potential explanations thereof will be the focus of this work.

Crystal structure
Fig. 2 shows calculated lattice parameters a and c of MB 2 (AlB 2 type structure) and these are in close agreement with values from previous theoretical work [5,48].Deviations are only found when M is a magnetic element (M = Cr, Mn, Fe) and a closer examination reveals that values retrieved from the Materials Project database [48] are either from a FM or a non-magnetic (NM) structure, while values from Moraes et al. [5] are assumed to be NM since no other information is given about magnetism.In this work, we have modelled magnetism in MB 2 by considering not only the ferromagnetic (FM) state but also six anti-ferromagnetic (AFM) configurations by expanding the unit cell both in-as well as out-of-plane.Fig. S1 provides detailed information about considered magnetic spin configurations and Table S1 shows their corresponding calculated data, including a comparison of calculated and experimentally reported lattice parameters.CrB 2 , MnB 2 , and FeB 2 are all found with an AFM spin configuration of lowest energy (Table S1), and the lattice parameter for the magnetic state of lowest energy is included in Fig. 2. For CrB 2 , a unit cell expanded in-plane (AFM5) is found to be of lowest energy and its calculated lattice parameters are within 1% when compared to experimentally reported values.We also find a unit cell expanded in-plane (AFM6) to be of lowest energy for MnB 2 where the in-plane lattice parameter a match those reported for synthesized MnB 2 while the calculated out-of-plane lattice parameter c is about 2.5% smaller than those measured.
In Fig. 2, we also show reported experimental lattice parameters for MB 2 (AlB 2 ).Measured values have been acquired from multiple references (values and references listed in Table S3) and their spread is represented by the red area which encompass min and max values for each M. Note that experimental values are retrieved for MB 2 in both bulk and thin film form, where the latter can include strained samples depending on the choice of substrate.
A high agreement between theoretical and experimental data is found for M from Group 3 (Sc, Y) and 4 (Ti, Zr, Hf) for both a and c.Noticeable is that the measured lattice parameters also show a rather small spread for both a and c (Table S3).This is indicative of a welldefined material and MB 2 from Group 3 and 4 are also known as line compounds.When M is from Group 5 (V, Nb, Ta), the agreement between theory and experiment is still good for Period 3 (V) though an increased deviation is observed when moving to Period 4 (Nb) and 5 (Ta).Both NbB 2 and TaB 2 have been reported to have a significant homogeneity region, however, detailed information about compositional quantification within the MB 2 structure is quite uncertain [19,49,50].When M is from Group 6 (Cr, Mo, W) or 7 (Mn) we find a significant difference between theory and experiment for c when M = Mo, W, and Mn.For M = Cr and Mn, one could argue that this deviation may stem from how magnetism is treated in calculations, however, this argument is not valid for Mo and W. In a recent work by Fuger et al. [21], the lattice parameters of a sub-stoichiometric WB 2-y (WB 1.47 ) thin film was demonstrated to correlate both with calculated values for WB 2-y with disordered B-vacancies (WB 1.5 ) and with those reported experimentally by Woods et al. [51].

Point defects in MB 2 (AlB 2 -type)
From the trends in calculated and measured lattice parameters of MB 2 (AlB 2 -type) shown in Fig. 2, we find a strong indication for line compounds for M from Group 3 and 4, a wider range of lattice parameters for M from Group 5, and clear deviations for M from Group 6 and 7. Possible explanations for these deviations have been suggested to be related to boron vacancies in MB 2 , i.e., MB 2-y [5,14,15,18,20,21,[52][53][54][55][56], but little is said about other possible defects.We therefore choose to also consider other plausible types of defects in the ideal MB 2 (AlB 2 ) structure.We initially choose a small 2 × 2 × 2 supercell and consider different vacancy, interstitial and anti-site defects.Examples of anti-site defects are a B atom populating an M-site, an M atom populating a B-site, and site exchange of an M and B atom.Considered defects are illustrated in Fig. S3.The defect formation energy is calculated using where E supercell (defect) and E supercell (MB 2 ) are the total energies of the supercell with and without the defect, respectively, and the last term corresponds to the considered defect adding or removing N i atoms of a chemical potential μ i .Here μ i is represented by the calculated energy of M or B in their ground-state crystal structure.
The calculated defect formation energies are shown in Fig. 3.In general, interstitial and anti-site defects of both M and B as well as the site exchange are found to be energetically unfavorable for all M in MB 2 , although they become less unfavorable with increasing valence of M. The most unstable defect is found to be anti-site with M on the B-site.For all systems, the vacancy defects are of lowest energy, where a B-vacancy generally is of lower energy, with W being the only exception.Focusing on vacancy defects, ΔE defect is positive when M is from Group 3 (Sc, Y) and 4 (Ti, Zr, Hf), i.e., vacancy formation is unfavored.This observation is consistent with the data shown in Fig. 2, with well-defined lattice When M is from Group 5 (V, NB, Ta), 6 (Cr, Mo, W), 7 (Mn), and 8 (Fe), we find a significant decrease of ΔE defect , and for four systems (Ta Mo, W, Fe) it is negative, i.e., it is energetically favorable to create a vacancy.These systems are also those for which there is a deviation between experimental and calculated lattice parameters, and where there is a spread in reported lattice parameters, as shown in Fig. 2. It should be noted that the defective structures considered in Fig. 3 only serve an illustrative purpose, and may not represent the optimal configurations.Hence, other defect configurations cannot be ruled out, but the results suggest that both M-and B-vacancies could have an impact on the composition of MB 2 , and consequently its properties.

Stability of multiple M-and B-vacancies in MB 2 (AlB 2 -structure)
Considering that the defect energies for both M-and B-vacancies are the most favorable among all defects considered in Fig. 3, we focus solely on vacancy defects on either the M-or B-site and go beyond a single vacancy.For synthesized off-stochiometric MB 2 , there is generally no information provided concerning if vacancies are randomly distributed or if they form ordered vacancy structures, nor is it always shown what the actual M and B content is in the MB 2 phase.The one exception is the recent discovery of line defects in TiB 2 , ZrB 2 , and CrB 2 which are comprised of entire B (10-10) planes being absent [16,17].
To identify possible vacancy structures in MB 2 , we start from the ideal AlB 2 -structure and consider both ordered (generated using ATAT) and disordered/solid-solution (SQS modelling) vacancies on the M-and B-sublattices, targeting the energetically most optimal configuration of a vacancy defect at a given composition, or vacancy concentration in M 1-z B 2 and MB 2-y .The stability of M-deficient M 1-z B 2 or B-deficient MB 2-y structures is here investigated by calculating the formation enthalpy ΔH cp (Eq.( 2)) for ordered vacancies and SQS-modelled disordered vacancies.For the latter we have also estimated the contribution from configurational entropy at 1000 K (Eq.( 4)) to the Gibbs free energy of formation ΔG cp (Eq.( 3)).Here it should be noted that the competing phases considered are those listed in Table S2, excluding here identified stable defect (vacancy) structures and others previously identified in the literature.The ideal AlB 2 -type structure is included as a competing phase, though the two MoB 3 -protpype structures (hR24 and hP16) are not, since they build upon the AlB 2 -type structure with ordered M-vacancies.Moreover, many theoretical studies considering vacancy defects in MB 2 express stability in terms of formation energy ΔE f (Eq.( 1)), i.e., a comparison with the elements M and B [9,21,54,57,58].This is unfortunate since ΔE f only gives an indication whether MB 2 , with or without defects, is energetically stable with respect to M and B, and thus fails to include information of any possible decomposition of M 1− z B 2− y into other binaries.An example of this is given in Fig. S4 where, e.g., ΔE f becomes less negative upon increasing Nb-vacancy  S3 for experimental data along with References).concentration even though ΔH cp becomes negative, i.e., indicating stability of Nb 1-z B 2 .Fig. 4 displays ΔH cp and ΔG cp for M-and B-vacancies in MB 2 with M = Ti, Zr, Hf, V, Ta, Nb, Cr, Mo, and W. For each M, we have considered 460 structures with ordered M-vacancies (M 1-z B 2 ) and 950 structures with ordered B-vacancies (MB 2-y ).Every grey cross corresponds to a structure with a unique ordered vacancy configuration.Open circles correspond to disordered vacancies.We find three distinctly different results depending on which Group M belongs to.
For M from group 4 (M = Ti, Zr, Hf in Fig. 4(a-c)), ordered as well as disordered vacancies on both the M-and B-sublattices are energetically unfavored as seen from the increase of ΔH cp and ΔG cp when deviating from the ideal MB 2 structure upon increased vacancy concentration.This agrees with TiB 2 , ZrB 2 , and HfB 2 all being considered as typical line compounds and it also aligns with the well-defined lattice parameters shown in Fig. 2. Hence, no homogenously distributed vacancies on any of the M-or B-sublattices are to be expected for M from Group 4. Despite no structure of deviating composition being stable, ΔH cp < 0, we find two interesting structures of low energy at y = 2/3 (x = 0.571), i.e., M 3 B 4 (or MB 1.333 ) and M 12 B 16 (or MB 1.333 ).These are illustrated in Fig. 5 with (a) every third B-layer vacant and (b) a B-deficient antiphase boundary (APB), where the atomic layers are shifted half of a unit cell along c when crossing the defect region.The latter is illustrated by the red area in Fig. 5(b).Detailed structural information for M 3 B 4 and M 12 B 16 is found in Tables S4-S6.This finding with vacant B-planes does resemble the APBs reported in Refs.[16,17] for TiB 2 and ZrB 2 with defects being comprised of vacant B (10-10) planes.An illustrative picture of such APBs is shown in Fig. S5.Compared to the AlB 2 structure, the APB structures have both vacant B-planes and a translation of half of the unit cell.The reason for not finding the previously reported specific APBs in this study, among considered vacancy structures, can be related to limited unit cells composed of too few atoms when using ATAT, and that the crystal structure for generating ordered vacancy structures is based on the ideal AlB 2 and hence not likely accounting for crystal translations of half of the unit cell in any of the generated structures.
When M is from Group 5 (V, Nb, Ta in Fig. 4(d-f)) the formation of vacancies is being favored (ΔH cp ≤ 0), or close to favored (0 ≤ ΔH cp < 50), with increasing stability in the V-Nb-Ta series.For V 1-z B 2 in Fig. 4(d) the stable region (ΔH cp ≤ 0) for ordered M-vacancies is 0 ≤ z < 0.15 or up to 15% vacant V-sites.In Fig. 4(e), the stable region for ordered vacancies for Nb 1-z B 2-y is wider than for V 1-z B 2-y , with 0 ≤ z < 0.22, i.e., up to 22% vacant Nb, and also includes 0 ≤ y < 0.2 or up to 10% vacant B-sites.Notably, the stable range for vacant Nb would not have been identified if one were to look at the formation energy only, as demonstrated in Fig. S4(a).Disordered vacancies are also found stable but within a more narrow compositional range.
For Ta 1-z B 2-y in Fig. 4(f), stable defect structures of ordered character are found in an even wider compositional range, 0 ≤ z < 0.28 and 0 ≤ y < 0.34, which corresponds to 28% vacant Ta-and 17% vacant B-sites, respectively.Again, disordered vacancies are also found stable, with ΔG cp < 0, for z and y less than 0.2 (20% vacant Ta) and 0.32 (16% vacant B).Note that ordered vacancies, in particular on the Msublattice, are still energetically favored when compared to a disordered substitution where configurational entropy is accounted for at 1000 K. Schematic illustrations of selected low-energy structures found for Nb 1-z B 2-y and Ta 1-z B 2-y are shown in Fig. 6(a) (B-vacancies) and Fig. 7(a) (M-vacancies).Detailed structural information is found in Tables S7, S8, S10 and S11.The low-energy B-and M-deficient structures are all characterized by B-and M-vacancies distributed in such a way to simultaneously maximize both the distance between the vacant sites and the number of B-B and M-M bonds, respectively.
When M is from Group 6 (M = Cr, Mo, W in Fig. 4(g-i)), MB 2 in the ideal AlB 2 -structure is far from stable.This might seem counterintuitive since these phases have all been reported experimentally.Here it is important to note that all three phases, and especially MoB 2 and WB 2 , show significant deviation when comparing calculated and measured lattice parameters, as demonstrated in Fig. 2. Furthermore, neither MoB 2 or WB 2 is dynamically stable as seen in Figs.S14 and S15.Altogether, this is indicating that synthesized phases may not be composed of the ideal stoichiometric AlB 2 composition.Still, measured peaks from X-ray diffraction (XRD) do support the AlB 2 -structure, although the peak positions are slightly shifted as compared to the calculated structure.A potential explanation for this can be found in Fig. 4(g-i) where B-and Mdeficient structures in all three systems are found with a decrease in ΔH cp with increasing vacancy concentration, i.e., vacancies are stabilizing the structure.For both MoB 2 and WB 2 we find either stable or close to stable structures with ordered vacancies of B (around x = 0.6) or M (at x = 0.75).
Calculations for M = Cr is shown in Fig. 4(g), with results shown for non-magnetic configurations of CrB 2 .Since an AFM spin configuration (AFM5) is found to be of lowest energy for the stoichiometric phase, as demonstrated in Table S1, we have considered different magnetic spin configurations (Fig. S1) for a few selected ordered vacancy Cr 1-z B 2-y structures of low ΔH cp .These reveal that the trends are qualitatively similar to the non-magnetic results shown in Fig. 4(g), but with slightly lower ΔH cp .Overall, both M-and B-vacancies in CrB 2 lead to a decrease in ΔH cp , i.e., the formation enthalpy is less positive.We further note that observed trends in ΔH cp for CrB 2 , together with results in Fig. 3, indicate that vacancies have similar impact on the stability trend for CrB 2 as for NbB 2 and TabB 2 .Next, we focus on M = Mo and W, which are the elements found with ΔE defect < 0 for a single M-or B-vacancy in Fig. 3 and with a calculated lattice parameter c deviating significantly compared to measured values (Fig. 2).Both MoB 2 and WB 2 is found to with a significant decrease in ΔH cp with increasing vacancy concentration, as seen in Fig. 4(h and i In addition to the results shown in Fig. 4 for calculated stability of MB 2 upon M-and B-vacancy formation, demonstrating that ΔH cp decreases, most clearly for MoB 2 and WB 2 , with increasing vacancy concentration, we have investigated the dynamical stability at ideal and offstoichiometric compositions.Figs.S6-S15 shows phonon dispersion plots for the ideal AlB 2 -type structure of MB 2 .We find that all structures are dynamically stable with the exception for the dynamically unstable MoB 2 and WB 2 in Figs.S14 and S15.Furthermore, in Figs.S22-S27 we show phonon dispersion for selected low-energy vacancy structures of MoB 2 and WB 2 .These are all found to be dynamically stable.The phonon dispersion for selected low-energy vacancy structures of TaB 2 are also found to be dynamically stable, as seen in Figs.S16-S21.This altogether indicates that incorporation of vacancies on either M-or Bsites in MB 2 have a positive impact on both the thermodynamical and dynamical stability.

Effect of M-and B-vacancies on selected properties
The trends observed in Fig. 4 demonstrates a decrease in ΔH cp upon vacancy formation for M from Group 5 (VB 2 , NbB 2 , TaB 2 ) and 6 (CrB 2 , MoB 2 , WB 2 ), while for M from group 4 (TiB 2 , ZrB 2 , HfB 2 ) ΔH cp increases.To understand why this is the case, and the possible impact vacancies may have on the crystal structure and on electronic and mechanical properties, we here choose to analyze both the ideal structure/composition and a few selected low-energy ordered vacancy structures in more detail.

Electronic structure and chemical bonding
The electronic structure is analyzed in terms of density of states (DOS), and corresponding chemical bonding is analyzed on the basis of a crystal orbital Hamilton population (COHP).Fig. 8 shows DOS and COHP for the ideal AlB 2 -type structure of MB 2 .For M from Period 4 (Sc to Fe) in Fig. 8(a), we find that the Fermi level, E f , in DOS is shifted upwards with increasing valence of M, with a local minimum for TiB 2 and maximum for MnB 2 .Along the same series in Fig. 8(b), we find COHP curves with none or little contribution around E f for TiB 2 and VB 2 .This is in contrast to both ScB 2 , which have unfilled bonding states above the Fermi level, and MB 2 of higher valence M, e.g., CrB 2 and MnB 2 , with filled anti-bonding states at the Fermi level.These are examples of structures with a non-optimized bonding.Similar observations are found for M from Period 5 (Y to Mo) and 6 (Hf to W).Furthermore, the minima in DOS along with an optimized COHP for TiB 2 , ZrB 2 and HfB 2 coincides with the minimum values of ΔE f in Fig. 2.This is in contrast to MB 2 for M = Cr, Mn, Fe, Mo, and W, which are found with filled anti-bonding states at E f and found above the convex hull in Fig. 2. In addition, the minimum values of ΔE f for M-B in Fig. 1 is shifted to more M-rich regions.Altogether this indicates that the presence of too many electrons result in unoptimized bonding conditions which, in turn, impact the calculated energy and ultimately the stability.
Both M-and B-vacancies have a positive impact on the stability for MB 2 with M = Nb, Ta, Mo, and W in Fig. 4. In Fig. 9, we show the impact from M-and B-vacancies on DOS and COHP with focus on two systems, NbB 2 and MoB 2 .For the B-deficient structures, Nb 16 B 28 (12.5% B-vacancies) and Mo 4 B 6 (25% B-vacancies), we find E f located at or close to a local minimum in Fig. 9(a and c) and also observe a decrease in the number of states at E f as compared to the ideal structure.Corresponding COHP curves for NbB 2 in Fig. 8(b) show anti-bonding states at E f for the ideal NbB 2 , originating mainly from Nb-B interactions.These are eliminated for Nb 16 B 28 due to a shift in E f .For MoB 2 , not all bonding states are filled at E f for the ideal structure.With the introduction of B-vacancies, all bonding states are filled, without populating any antibonding states nor having too much nonbonding states at E f .
For M-deficient structures, Nb 15 B 36 (16.67%Nb-vacancies) and Mo 6 B 18 (33.33%Mo-vacancies), the fewer electrons available results in a shift of E f , as compared to the ideal structure, where DOS in Fig. 9(a  and c) shows a decreased number of states at E f .This is most clear for Mo 6 B 18 .Analysis of the COHP curves show fewer non-bonding states up to E f for both Nb 15 B 36 and Mo 6 B 18 as compared to the ideal MB 2 structures (Fig. 9(b and d)).Similar results are also found for bonding analysis of M-and B-deficient TaB 2 and WB 2 in Fig. S28 where the ideal WB 2 have anti-bonding states at E f which are eliminated for the vacancy structures.Altogether, DOS and COHP in Figs. 9 and S28 indicates that both M-and B-vacancies have a positive impact on the bonding characteristics, which explain the observed decrease in ΔH cp in Fig. 4(e, f, h  and i).

Lattice parameters
In Fig. 2, we show that calculated and experimentally reported lattice parameters deviates slightly for TaB 2 and NbB 2 and significantly for MoB 2 and WB 2 .In particular, this concerns the lattice parameter c for the latter two phases, as demonstrated in Fig. 2(b).These four systems are also those which show a decrease in ΔH cp for M-and B-vacancies as compared to their ideal MB 2 structure.To find out the impact of M-or B-  vacancies on the lattice parameters, we choose to investigate the five ordered vacancy structures of lowest energy for each unique composition in Fig. 4. Detailed structural information for these structures is found in Tables S7-S14.We choose to express the extracted lattice parameter a and c by folding it back to the corresponding ideal AlB 2stuctrure.
Fig. 10 shows the lattice parameters for NbB 2 and MoB 2 with M-and B vacancies, with black crosses representing ordered structures, open blue circles disordered vacancies, and data acquired from various experimental reports as red triangles.Here we note that information in the literature for reported composition for synthesized structures is sometimes a bit vague which can be related to challenges in measuring the B-content in the actual MB 2 structure and not in the entire sample.Despite this, interesting trends are revealed.First to note is that ordered and disordered vacancies do follow the same trend for both M-and Bdeficient compositions.
For NbB 2 , we find a decrease in a while c is close to constant with increasing Nb-vacancy concentration, whereas an increased B-vacancy concentration leads to a decrease in c but with almost no change in a.These trends are, for both Nb-and B-deficient regimes, in line with experimentally reported data.The homogeneity regime suggested by Nunes et al. [19], from NbB 1.86 (7.14% B-vacancies) to Nb 0.857 B 2 (14.29%Nb-vacancies) is consistent with results obtained here for ordered and disordered vacancies, both with respect to the stability results shown in Fig. 4(e) and lattice parameters given in Fig. 10(a).To reveal whether it would be possible or not to distinguish a structure containing disordered vacancies from one with disordered vacancies, we have simulated X-ray diffractograms for a few selected low-energy structures with ordered and disordered M-or B vacancies, also including the ideal MB 2 structure for reference, see Figs.S30 and S31.A comparison of these indicates that it may be hard to discriminate whether B-vacancies are ordered or disordered, which is expected.This since ordered B-vacancies do not result in any additional diffraction peaks nor impact the intensity.When it comes to ordered Nb-vacancies, on the other hand, additional peaks appear (see Fig. S31(c)), especially below 2θ = 25 • .Hence, when analyzing a X-ray diffractogram these additional peaks should not be mistaken for a secondary phase but could indeed be an indication for NbB 2 with ordered Nb-vacancies.Analysis of lattice parameters for TaB 2 in Fig. S29(a) along with simulated diffractograms in Figs.S34 and S35 reveals similar results as for NbB 2 , i,e., experimental lattice parameters that matches a phase with B-vacancies, though with difficulties in telling whether the vacancies are ordered or disordered.
For MoB 2 , we notice that reported lattice parameters seems to be close to constant despite differences in measured composition.Again, measuring the B-content could be challenging, and we therefore presume that the real composition of MoB 2 do not differ as much as indicated in the literature.Interestingly, the calculated lattice parameters for both ordered and disordered B-vacancies follow the same trend with increasing vacancy concentration, and also coincide with experimentally reported values at x = 0.6 (y = 0.5 in MoB 2-y or 25% B-vacancies).
Corresponding simulated X-ray diffractograms in Fig. S32 again indicate that it is difficult to discriminate ordered B-vacancies from disordered.A look at the lattice parameters for MB 2 with ordered Mo-vacancies, a configuration that is found to be close to stable at x = 0.75 (33.33%Mo-vacancies) in Fig. 4(h), shows that a is smaller while c is larger than values reported.Based on the excellent agreement between calculated and experimental lattice parameters reported herein (Figs. 2 and 9(b)), we assume that ordered Mo-vacancies are unlikely.If they are to be found, corresponding simulated X-ray diffractograms in Fig. S33 demonstrates strong additional diffraction peaks between 2θ = 20 and 30 • .We have not found any indication for peaks matching these in the literature.Similar analysis of WB 2 , shown in Figs.S29(b), S36, and S37, and making a comparison with experimental data, indicates that B-vacancies are most likely to be found in synthesized WB 2 .Note that the ordered W-vacancy structures found stable in Fig. 4(i) at x = 0.75 do no match reported lattice parameters nor measured XRD data.If this can be related to synthesis far from this composition or not remains to be answered.Based on this we suggest experimental efforts to verify the existence of the predicted stable W 0.67 B 2 (or W 6 B 18 or WB 3 ).If it would be possible to synthesize, the simulated XRD pattern in Fig. S37 gives an indication for possible diffraction peaks.

Mechanical properties
Despite the difficulties of directly comparing calculated and measured mechanical properties, our intent is to show that vacancies that result in a deviation from the ideal composition do have an impact on the properties.We choose to focus on NbB 2 , TaB 2 , MoB 2 , and WB 2 , which are the systems where vacancies are energetically favored as compared to ideal AlB 2 structure (Fig. 4).In Fig. 11 we present calculated data for the bulk, Young's and shear modulus along with theoretical hardness, comparing the ideal MB 2 with selected low-energy structures containing ordered M-or B-vacancies.For NbB 2 and TaB 2 we find a decreased bulk modulus with increasing vacancy concentration.For both MoB 2 and WB 2 with M-vacancies we see a clear decrease in the bulk modulus whereas B-vacant structures is found with a bulk modulus similar to the ideal structure.Of more significance is that for the Mo and W system we find an increased hardness and Young's and shear modulus with increasing M-or B-vacancy concentration, while a similar trend is not as clear for the Nb and Ta system.Nonetheless, there is a maximum at lower vacancy concentrations for Nb and Ta (y ≈ 0.25 and z ≈ 0.2) and at a higher vacancy concentration for Mo and W (y ≈ 0.5 and z = 0.33).Noteworthy is that the maximum value for here investigated vacancy range, coincide with the minimum energy in Fig. 4.This further strengthen the suggestion that improved mechanical properties are attainable with synthesis of substoichiometric MB 2 with vacancies on the B-or M-site.

Conclusion
We have used first-principles calculations to investigate the phases stability, crystal structure, and properties upon formation of various defects in MB 2 of an AlB 2 -type structure, focusing on ordered and disordered M-and B-vacancies.Our calculations combined with a close comparison to experimental information reveal that: (i) Calculated lattice parameters for ideal MB 2 matches those reported experimentally for M = Sc, Y, Ti, Zr, Hf, V, Cr but deviates to various extent for M = Nb, Ta, Mo, W, Mn.The observed differences thus indicates that the ideal composition may not be the case for the latter metals.(ii) M-and B-vacancies for M from Group 4 (M = Ti, Zr, Hf) is unlikely if distributed homogenously in the structure.A B-deficient sample may result in line defects, or antiphase boundaries, as indicated herein and reported elsewhere.However, such defect will not influence measured lattice parameters.(iii) Both ordered and disordered M-and B-vacancies for M from Group 5 (M = Nb, Ta) and 6 (M = Cr, Mo, W) leads to an increased stability, where ordered vacancies are lowest in energy even when configurational entropy is taken into account for the disordered configurations.(iv) For NbB 2 and TaB 2 , calculated lattice parameters at both M-and B-deficient conditions matches those reported experimentally.(v) For MoB 2 and WB 2 , calculated lattice parameters at B-deficient conditions matches those reported experimentally when the Bvacancy concentration is 25%.Ordered B-vacancies also result in close to stable structures.(vi) Mechanical properties are improved for NbB 2 , TaB 2 , MoB 2 , and WB 2 upon formation of both M-and B-vacancies.

Fig. 1 .
Fig. 1.(a) Convex hull for the M− B systems.Filled circles are stable phases on the convex hull and crosses are phases above the convex hull.Filled diamonds represent stable MB 2 phases, with the AlB 2 type structure, on the convex hull.Filled squares represent MB 2 phases, with the AlB 2 type structure, above the convex hull.(b) Hexagonal unit cell of MB 2 in the AlB 2 type structure.Each unit cell consists of one metal M (blue) and two boron atoms (green).

Fig. 2 .
Fig. 2. Calculated lattice parameters (a) a and (b) c as function of M for MB 2 in the AlB 2 -type structure, where data shown is from this work (filled green circles), from Moraes et al. [5] (filled squares) and from the Materials Project database [48] (filled triangles).Experimental values are given by the red area, the range being represented by reported min and max values (see TableS3for experimental data along with References).

Fig. 3 .
Fig. 3. Calculated defect formation energies for potential defects in MB 2 (AlB 2 ) created in an ideal 2 × 2 × 2 supercell with focus on (a) M and (b) B. Vacancy defects are represented by circles, different interstitials by squares, and antisite defects by triangles.The blue crosses correspond to site exchange of M and B.

Fig. 4 .
Fig. 4. Calculated stability for M and B-vacancies for MB 2 in the AlB 2 -type structure.Formation enthalpy (ΔH cp ) and Gibbs free energy of formation (ΔG cp ) for ordered (crosses) and solid solution vacancies (circles).The vertical dashed line represents the ideal MB 2 composition.Filled green diamonds represent stable MB 2 phases and filled red squares represent MB 2 phases not stable or at best metastable.Horizontal dashed line represents the set of most competing phases.Selected lowenergy structures with ordered vacancies are indicted by arrows.

Fig. 5 .
Fig. 5. Schematic illustration of selected low-energy structures found for TiB 2-y , ZrB 2-y , and HfB 2-y when y = 2/3 (x = 0.571 in M 1-x B x ).Blue, green, and red atoms represent metal M, boron, and boron vacancies, respectively.Primitive unit cell (left) and side view of the crystal (right) for B-vacant structures with (a) every third B-layer vacant and (b) an antiphase boundary represented by the red area.

Fig. 6 .
Fig. 6.Schematic illustration of selected low-energy B-vacant structures found for (a) NbB 2-y and TaB 2-y , and (b) MoB 2-y and WB 2-y .Blue, green, and red atoms represent metal M, boron, and boron vacancies.Primitive unit cell (left), top view of defect B-layer (middle), side view of the crystal (right) for each Bvacant structure.
).At B-deficient conditions, a minimum in ΔH cp is reached around y = 0.25 for MB 2-y (x = 0.6 in M 1-x B x ) which corresponds to 25% vacant B. Three representative low-energy structures found for y = 0.25 and 0.3 are illustrated in Fig. 6(b), displaying chains of B accommodating Bvacancies.For M-deficient conditions, ΔH cp reaches a minimum at z = 1/3 (x = 0.75 in M 1-x B1 x ) for Mo and W, which corresponds to 33.33% M-vacancies.All the structures of lowest-energy at z = 1/3 are composed of M-layers, being Mo or W, having the metal arranged in a honeycomb lattice with the M-vacancy centered in the hexagon (Fig. 7(b)).These structures are close to degenerate in energy which can be explained by having equivalent M-deficient layers being differently stacked relative each other.At x = 0.75 and for M = Mo there are three structures found close to stable with ΔH cp ≈ +15 meV/atom.For M = W and at x = 0.75, we also identify multiple stable structures, i.e., with negative ΔH cp .An example of such a structure with ordered W-vacancies is W 6 B 18 shown in Fig. 7(b) with ΔH cp = -12 meV/atom.Detailed structural information for selected low-energy structures is found in Tables S9, S10, S13, and S14.

Fig. 7 .
Fig. 7. Schematic illustration of selected low-energy M-vacant structures found for (a) Nb 1-z B 2 and Ta 1-z B 2 , and (b) Mo 1-z B 2 and W 1-z B 2 .Blue, orange, and green atoms represent metal M, metal vacancies, and boron.Primitive unit cell (left), top view of defect M-layer (middle), side view of the crystal (right) for each Mvacant structure.

Fig. 8 .
Fig. 8. (a) Total and partial density of states for MB 2 in the AlB 2 -type structure.(b) COHP bonding analysis of M-B, M-M and B-B interactions, and total for MB 2 in the AlB 2 -type structure.The filled regions of the plot represent occupied states up to the Fermi level which is set to 0 eV.

Fig. 9 .
Fig. 9. Density of states and COHP bonding analysis for ideal, B-deficient, and M-deficient (a,b) NbB 2 and (c,d) MoB 2 .(a,c) Total and partial density of states and (b, d) COHP bonding analysis of M-B, M-M and B-B interactions, and total bond contribution.The filled regions of the plot represent occupied states up to the Fermi level which is set to 0 eV.

Fig. 10 .
Fig. 10.Calculated lattice parameters a and c for M-and B-vacancies in (a) NbB 2 and (b) MoB 2 as function of B-content x in M 1-x B x .Data is displayed for ordered (grey crosses) and disordered (blue open circles) vacancy structures, and experimental values are given for reported compositions (red open triangles).

Fig. 11 .
Fig. 11.Calculated mechanical properties as function of composition for selected low-energy structure with ordered B-and M-vacancies in MB 2 (M = Nb, Ta, Mo, W).(a) Bulk modules, (b) Young's modulus, (c) shear modulus, and (d) Vickers hardness.