Introduction

Recently, numerous attempts to design new intrinsically ultraincompressible (bulk modulus B > 250 GPa) and superhard (hardness H ≥ 40 GPa) materials evoked much interest in the synthesis of borides of transition metals1,2,3,4,5,6 because of their potential applications, such as cutting, drilling, machining and wear-resistant tools with enhanced properties over transition metal carbides7,8,9. In the transition metal borides, a high density of valence electrons of transition metal (Tm) should provide such compounds with high incompressibility, and strong covalent bonds between transition metals and boron should enhance the resistance against plastic deformation. Accordingly, diborides of osmium and rhenium have been synthesized10,11. However, although they possess high elastic moduli, the correctly measured load-invariant hardness of ReB2 and OsB2 is below 30 GPa. The weak Tm-B and Tm-Tm bonds are responsible for lattice instabilities observed in these diborides12.

Therefore, much attention turned to the synthesis of triborides or tetraborides of transition metals in which more boron atoms are expected to form a three-dimensional (3D) covalent bond network1,2,3,4,5,6. Using the “hardness models” based on the presumption that large elastic moduli guarantee high hardness, Wang et al.13 suggested that the 5d transition metal tetraborides, such as WB4 and MoB4, should be intrinsically superhard. However, WB4 synthesized by Gu et al. has load-invariant hardness less than 30 GPa3, as recently confirmed by Mohammadi et al.1. The instability issue raised by Liang et al.14 and Zhang et al.15 ruled out the 3D boron network in tetraborides of 5d transition metals. Instead, triborides were proposed to be experimentally accessible because of their thermodynamic, mechanical and dynamic stability, and because of the agreement of the simulated X-ray diffraction (XRD) pattern with the experimental one6. In recent experiments by Chen et al.16 and Tao et al.17, the theoretically proposed structure of WB3 was confirmed experimentally by XRD, high-resolution transmission electron microscopy (HRTEM) and Rietveld refinement. The load-invariant hardness was about 25.5 GPa17, thus WB3 is not intrinsically superhard. The physical origin of weakening of WB3 lies between the metallic W-B bonds which are responsible for the lattice instability under shear deformation6, being similar to what has been shown in ReB218. In view of these facts, the majority of recent attempts in the search for new superhard materials focused on transition metal borides with a 3D covalent bond network. After an exploration of several borides, researchers focused their attention on the orthorhombic TmB4 (Tm = Cr, Mn and Fe) in which each Tm atom is surrounded by a 3D covalent boron cage.

Several years ago, the high values of ideal shear strength of >50 GPa and hardness of ~48 GPa theoretically predicted for CrB4 by Niu et al.19,20, initiated many investigations into this material. However, the synthesized CrB4 has hardness only between 23.3 GPa21 and 28.6 GPa22 in agreement with the theoretical calculations by Li et al.23 Similar discrepancies between theoretical predictions and experimental results were found also for orthorhombic FeB4. Recently, Gou et al.24 reported a surprisingly high value of hardness for FeB4 of about 67 GPa for a “nanoindentation” depth of 20 to 40 nm. However, it is not clear if this value corresponds to the “asymptotic hardness” as recommended in ref. 25 for correctly measured load-invariant hardness that is obtained under conditions of fully developed plasticity26. This high value has been questioned by density functional theory (DFT) calculations of the anisotropic compressibility by Gou et al.27 and of the ideal strength by Li et al.28. More recently, Wang et al.29 reported a much lower hardness of synthesized FeB4 of 15.4 GPa in agreement with the theoretical calculations.

The question thus arises as to why the 3D covalent network in these tetraborides does not provide the presumed high hardness enhancement. For convenience, we use the nomenclature “Pearson symbol [space group number]” to denote the crystal structures, such as oI10[71] and oP10[58]. Although the oI10[71] structure was proposed for the experimentally synthesized CrB4 and FeB4, it has been latter found to be unstable, and should be substituted by oP10[58] structure19,27,28,30,31,32,33. The question why the low symmetry oP10[58] structure is energetically favorable and dynamically stable, remained so far unsolved. In order to clarify these questions, we shall study in the present work by means of DFT calculations the following problems: 1) What is the electronic origin of the stabilization from dynamically (phonon) unstable oI10[71] to dynamically stable oP10[58]? 2) What is the upper limit of mechanical strength of oP10[58]-TmB4? 3) Does the bond deformation path and electronic instability mode of TmB4 under shear resemble those of other borides such as ReB2 and WB3? 4) Could the three-dimensional covalent network in TmB4 support a higher plastic resistance?

Results

Stabilization from oI10[71]- to oP10[58]-TmB4

We first searched the most stable structures of stoichiometric TmB4 by geometry optimizations of 26 commonly observed Tm-B, Tm-C, Tm-N, Tm-Al, Tm-Si ICSD structure types34. The geometries were optimized using DFT as implemented in the VASP code35,36. The details of the adopted initial structures of TmB4 are given in the supplementary materials. The formation energy/enthalpy was calculated from the chemical reaction Tm + 4B = TmB4. The energies of α−B and Tm in magnetic state were adopted as the reference ground states. Figure 1 shows the calculated formation enthalpy of CrB4 and FeB4. The most negative formation enthalpies of oP10[58] indicate that it is the most energetically favored structure. As listed in Table S1 of the supplementary materials, eight typical structures, such as oP10[58], oI10[71], oP10[59], hP20[194], hP10[194], mS10[12], mS20[14] and mS30[12], have been reported for transition metal tetraborides. In the present study, the oP10[58] is found to be the most stable one for both CrB4 and FeB4. The formation enthalpies obtained in the present study for the oI10[71] structure of CrB4 (ΔH = −0.301 eV/atom) and of FeB4 (ΔH = −0.143 eV/atom) are slightly higher than those calculated in previous publications of ΔH = −0.3098 eV/atom for CrB437, and of ΔH = −0.1698 eV/atom for FeB437. Note that the initial structure with the mS20[14] symmetry transforms into oP10[58] for both CrB4 and FeB4. Our evolutionary structure search scheme38,39 confirms further that oP10[58] is the most stable one. A systematic thermodynamic investigation of possible ground state structures with consideration of vibrational entropy has been performed on FeB4 by Kolmogorov et al.33 Their results indicate that the oP10[58]-FeB4 are thermodynamically stable in the considered temperature range, but it lies 3 meV/atom above the α−B <−> oP12-FeB2 tieline at T = 0 K while 10 meV/atom below the tieline at T = 900 K33. The decomposition energy from FeB4 to oP12-FeB2 and α-B are calculated to be positive (~0.01 eV/atom), while the energy from FeB4 to oP8-FeB and α−B are negative (~ −0.02 eV/atom), in agreement with previous calculations33.

Figure 1: Formation energies of TmB4 after geometrical optimizations.
figure 1

Formation energy of (a) CrB4 and (b) FeB4 calculated by DFT to determine the possible ground state phases of 26 commonly observed Tm-B, Tm-Al, Tm-P, Tm-O, Tm-S ICSD structure types and the newly reported tetraborides. The most stable structure is confirmed by our evolutionary search method.

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A similar trend of thermodynamic stability is also observed for MnB4 in oI10[71] and oP10[58] structures. The formation enthalpy of oI10[71]-MnB4 of −0.2704 eV/atom decreases slightly by 0.01 eV/atom (0.0097 eV/atom37) due to the transformation to the oP10[58] structure whose formation enthalpy is −0.2804 eV/atom. All these calculations suggest that oI10[71]-TmB4 is thermodynamically unstable with respect to oP10[58]-TmB4. Note that a Peierls distortion mechanism has been proposed as the origin of stabilization of mS20[14]-MnB4. However, such distortion will be hindered by temperature and the phonon assisted crossover observed from nonmagnetic Peierls insulator (mS20) to a magnetic Stoner metal (oP10)30. Taking CrB4 as a prototype, Fig. 2a,b show the typical bonding feature of oI10[71] and oP10[58]-TmB4. Although the boron network undergoes a significant distortion during the oI10[71] → oP10[58] transformation, the molar volume changes a bit and the simulated XRD figures show a similarity between oI10[71] and oP10[58], but some minor XRD peaks show up for oP10[58] structure which may be identified by experiments40.

Figure 2: Structure and properties at equilibrium.
figure 2

The equilibrium bond geometry of (a) oI10[71]- and (b) oP10[58]-CrB4 viewed along the crystallographic [001] direction, i.e. z axis in Cartesian coordinate system such that the x and y axes are along the crystallographic [100] direction and [010] direction respectively. The isosurfaces maps of the valence charge density difference (VCDD) correspond to +/−0.01 electrons/Bohr3, the large blue and small red spheres represent Cr and B atoms, respectively. Calculated phonon dispersion curves for (c) oI10[71]- and (d) oP10[58]-CrB4. The calculated COOP curve for (e) oI10[71]- and (f) oP10[58]-CrB4. The numbers close to the Cr atoms in Fig. 1a,b are the corresponding Bader charges. The bold numbers close to the line arrow in (a,b) are the neighbor distances between two neighboring Cr-Cr atoms in unit of Angstrom.

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As the next step we calculated the phonon dispersion curves as shown in Fig. 2c,d for CrB4, (Figs S1 and S2 for FeB4 and MnB4). Our phonon calculations suggests that oI10[71] is dynamically unstable for all three tetraborides, while the oP10[58] structure is dynamically stable (see Fig. 2c,d for CrB4). Similar phonon instabilities for oI10[71]-FeB4 and oI10[71]-MnB4 are presented in the supplementary materials as Figs S1 and S2. The phonon dispersion relation of oI10[71]-TmB4 exhibits at the Γ-point imaginary phonon frequencies, thus showing its dynamic instability at T = 0 K. This result is in agreement with the previous studies of FeB4 by Kolmogorov et al.33 It should be noted that the oI10[71]-TmB4 structure comprised of tetragonal boron network has been previously examined within the extended Huckel method. It has been concluded that maximum binding in the 3d series is achieved for Cr, and that the electron-rich Mn, Fe, Co and Ni tetraborides may be unstable in this configuration41. Our present results clearly show that the dynamic instability applies for all the three oI10[71]-TmB4. In contrast, the oP10[58]-CrB4, oP10[58]-MnB4 and oP10[58]-FeB4 phases are stable as there are no imaginary modes.

With focus on the phonon-assisted transformation from oI10[71] to oP10[58], we attempted to find the electronic origin of the stabilization from higher symmetry structures to those with lower symmetry. After comparing the electronic structure of oI10[71]- and oP10[58]-CrB4, we found that the transformation from oI10[71] and oP10[58] induces a highly directional electronic partition at two inequivalent sites of the boron atoms in the oP10[58] structure: one with minor charge transfer of ~0.12 electrons, and the other one with a higher charge transfers of ~0.32 electrons. Eight equivalent boron sites are found in oI10[71] structure with an average Bader charge of ~0.22 electrons. The stronger anisotropy of charge transfer in oP10[58] structure indicates its higher electronic polarization and localization as compared to oI10[71] one. The Bader charge density analysis42 shown in Fig. 2 confirms the different charge transfer at crystallographic sites of Cr for oI10[71] structure (+0.90 electrons) and oP10[58] structure (+0.88 electrons), suggesting a slightly higher ionic contribution for the former. The anisotropy of Bader charges can be attributed to the changes from regular to distorted boron network where metal atoms are donating electrons to stabilize the oP10[58] structure. The analysis of crystal orbital overlap population (COOP) curves of oI10[71]- and oP10[58]-CrB4 shown in Fig. 2e,f, provides a further confirmation of slightly higher COOP(Cr-B) of 0.28 in oP10[58] structure as compared with oI10[71] structure (0.26). Note that an average COOP value of bonds is used to do the comparison for convenience. Similar higher values of COOP(Fe-B) (0.23) and COOP(Mn-B) (0.20) are also shown for oP10[58] structure as compared to oI10[71] structure (0.21 and 0.18 for Fe-B and Mn-B bonds respectively) (refer to Figs S3 and S4 in supplementary materials). This suggests a slightly stronger bond strength for oP10[58] structure. When observing the structures in Fig. 2a,b one notices much “dense” bond network in oP10[58] structure as compared with oI10[71], which is a qualitative but illustrative confirmation of the analysis of the electronic structure.

The calculated valence charge density difference (VCDD) of oI10[71]- and oP10[58]-CrB4 shown in Fig. 2 provide additional support of their stabilization by crystal field splitting of the d orbitals. By comparing the isosurface of VCDD of the oI10[71]-CrB4 (Fig. 2a) with that of oP10[58]-CrB4 (Fig. 2b) we see a big difference in the shape of VCDD isosurface at the metal sites: the VCDD isosurface show anisotropic shape in oP10[58]-CrB4, whereas they are relatively isotropic in oI10[71]-CrB4. This suggests that although oP10[58] has a lower lattice symmetry, it possess a stronger electronic directionality (see Fig. 2a,b), in agreement with the analysis of charge transfer and COOP analysis in the preceding paragraph. In order to get such electronic partition of d orbitals, the boron network in oI10[71] structure needs to reorganize its positions to transform into oP10[58] structure by rotation operations. Furthermore, the boron-boron bonds indicated by the black arrows in oP10[58] structure (Fig. 2b) are absent in the oI10[71] structure, as seen by the hollow sites in oI10[71] in Fig. 2a.

Figure 3 shows the calculated orbital decomposed electronic density of states (DOS) for CrB4 and FeB4 in oI10[71] and oP10[58] structures in their spin polarized states. Figures S5–S7 in the supplemental materials provide additional DOS curves for TmB4. Both structures show metallic bonding because of finite value of DOS at the Fermi level (EF), which originates mostly from the d-orbitals of Tm and the p-orbitals of B. In the oI10[71]-CrB4, the DOS around EF (0.88 states/eV cell) is slightly higher than that in oP10[58]-CrB4 (0.80 states/eV cell), indicating a stronger “splitting” of bonding and antibonding states for oP10[58]-CrB4, in agreement with the preceding COOP analysis. A similar feature is observed for MnB4, but does not apply for oI10[71] and oP10[58]-FeB4 in which the Fermi level is located at the left shoulder of a peak in the DOS. Such differences will be shown to be responsible for their different shear moduli in the following section. Interestingly, we find in Fig. 3 a distinct change of the majority of and dxy orbitals (which mainly contribute to the in-plane rotation) below and above EF when the structure changes from oI10[71] to oP10[58] for both CrB4 and FeB4, suggesting that crystal field splitting of d orbitals is responsible for the phonon-assisted stabilization from oI10[71] to oP10[58] as discussed above. In case of CrB4, the first peak of dx2−y2 below the EF move upward by 0.1 eV during the transformation from oP10[58] to oI10[71], while the first peak of dxy below EF (at −1.167 eV) nearly disappears. For FeB4 however, the unexpected peaks show up at the Fermi level for oP10[58] structure, but not for oI10[71] structure.

Figure 3: Orbital-decomposed electronic density of states of 5d orbitals of transition metals.
figure 3

(a) oP10[58]-CrB4, (b) oI10[71]-CrB4, (c) oP10[58]-FeB4 and (d) oI10[71]-FeB4. The transformation from oI10[71] to oP10[58] corresponds to a change of DOS shape and relative position of the decomposed d orbitals, which is more pronounced for FeB4. The reference Cartesian coordinates was chosen such that the x, y and z axes are along the crystallographic [100] direction, [010] direction and [001] direction respectively. The vertical dashed lines indicate the Fermi levels.

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Mechanical properties of oP10[58]-TmB4

We next explore the mechanical properties of the oP10[58]-TmB4 with Tm changing from Cr to Mn and to Fe. The calculated elastic constants of oP10[58]-TmB4 are listed in Table S2 of the supplemental materials and compared with the results from other publications. It is shown that our present calculations are in good agreement with the others, and all the three orthorhombic tetraborides meet the requirements of elastic stability43 The Voigt average bulk and shear moduli are derived from elastic constants and compared with other hard materials in Table 119,23,27,28,31,32,44,45,46,47,48. It is seen that all three oP10[58] tetraborides are ultra-incompressible materials owing to their high bulk moduli of more than 250 GPa. A relatively high value of shear moduli is observed for oP10[58]-CrB4 (267 GPa) and oP10[58]-MnB4 (247 GPa), suggesting they are stiff. However, a much lower value found for oP10[58]-FeB4 (192 GPa), indicates that it may be less stiff than the other two tetraborides, in agreement with the aforementioned DOS analysis that suggests a stronger metallic bonding for FeB4.

Table 1 Voigt bulk modulus BV, shear modulus GV, and ideal strength (minimum tensile strength σmin and shear strength τmin) of transition metal tetraborides (TmB4, Tm = Cr, Mn, Fe) calculated by first principles methods.
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Well established criterion for whether a solid is ductile or brittle is if dislocation embryos can be nucleated from an atomically sharp crack prior to the crack propagation49,50. If this is the case at a given temperature, the solid is considered to be intrinsically ductile49,50. This distinction of the mechanical behavior is characterized by the ratio of the shear to bulk modulus G/B, based on the consideration that B is a measure of the resistance to fracture and G the resistance to plastic deformation49,51,52. The critical G/B ratio which separates ductile and brittle materials is around 0.57, i.e. if G/B < 0.57 the material is ductile, and it is brittle when G/B > 0.5751,52. In view of the relatively high ratio of G/B ≈ 0.957 for oP10[58]-CrB4, 0.879 for oP10[58]-MnB4, 0.674 for oP10[58]-FeB4, we conclude that all these tetraborides are brittle49,51. Even in brittle solids, a brittle to ductile transition is possible at an elevated temperature where the nucleation of dislocation embryos becomes possible prior to cleavage propagation50.

It is known that high elastic moduli do not guarantee high resistance against plastic deformation because upon finite shear electronic instabilities and transformations to softer phases with lower shear resistance often occur9,26. The anisotropic ideal shear strength of oP10[58]-TmB4 may be obtained from the calculated stress-strain relationships along different crystallographic planes and directions. The slip systems of (110)[1–10] and (010)[101] were reported as the weakest ones for oP10[58]-CrB4 by Li et al.23 and by Niu et al.19, respectively. A similar disagreement exists also for the weakest slip system in FeB4, where (110)[001] has been reported by Li et al.28 but (111)[11–2] by Zhang et al.48. We studied all the four slip systems in more detail. Our results suggest that the (110)[1–10] slip systems is the weakest one for CrB4, whereas (110)[001] is the weakest for FeB4 and MnB4. Figure 4a–c show the calculated dependence of the stress, magnetic moment and total energy on strain for oP10[58]- CrB4, oP10[58]- MnB4 and oP10[58]- FeB4 along their weakest slip systems. The minimum strength for the three TmB4 is shown in Table 1 and compared with previous results for these tetraborides19,23,27,28,30,31,32, and with those of WB3 and MoB36, ReB212, OsB244, B6O45, c-BN46 and diamond45,47. It is seen that the minimum shear strengths of oP10[58]-CrB4 of 36.2 GPa in the (110)[1–10] slip system are comparable to those of WB3 (37.7 GPa) and ReB2 (34.4 GPa), but they are much lower than those of c-BN (58.3 GPa46). This suggests that CrB4 cannot be intrinsically stronger than WB3 in view of the fact that its strength is between that of WB3 and ReB2. These results are in agreement with the recent experiments in which the load-invariant (“asymptotic”) Vickers hardness of CrB4 (23.3 GPa21 and 28.6 GPa22) was found to be comparable to that of ReB2. The high hardness of 48 GPa for CrB4 “predicted” by Niu et al.19 on the basis of the “hardness model” is not justified. Similar conclusions apply also for MnB4 and FeB4 because a much lower ideal shear strength is found for oP10[58]-MnB4 (35.1 GPa) and oP10[58]-FeB4 (25.7 GPa), although Niu et al.19 predicted a high value of hardness of 41.5 GPa for MnB4. The reason is simply the fact that the “theoretical models of hardness” compares in fact elastic stiffness, and do not account for electronic instabilities upon a finite shear18,26.

Figure 4: Analysis of shear deformations.
figure 4

(a) Stress-strain, (b) magnetic moment-strain and (c) energy-strain relationships for oP10[58]- CrB4, oP10[58]- MnB4 and oP10[58]- FeB4 along the weakest shearing path. The deformation paths are indicated in the figures. The snapshots of deformed bond structure and VCDD viewed along crystallographic [001] direction, i.e. z axis in Cartesian coordinate system at shear strain of (d) γ = 0.0000, (e) γ = 0.1717 (at peak) and (f) γ = 0.4002 (after instability) for CrB4 along the weakest (110)[1–10] slip system, and (g) γ = 0.0000, (h) γ = 0.2434 (at peak), and (i) γ = 0.4002 (after instability) for FeB4 along the weakest (110)[001] slip system. The black arrows in (d–i) indicate the position of bond instability.

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Bond deformation mechanism of oP10[58]-TmB4

In view of the mechanical weakness of oP10[58]-TmB4, we investigated the bond deformation paths and electronic instability mode of oP10[58]-TmB4 along the weakest shear path. Figure 4a,c show that both stress-strain and total energy-strain curves are smooth, i.e. no distinct electronic instabilities occur as in the cases of ReB218 and WB36, but not for the change of magnetic moment for FeB4. Figure 4d–f show the variation of the VCDD from the equilibrium (γ = 0.0000), to a shear strain γ = 0.1717 corresponding to the stress maximum and at shear strain γ = 0.4002 after the shear instability for CrB4 in the weakest (110)[1–10] slip system. One can see the change of in-plane lengths of the B1–B2 and B3–B4 bonds with respect to (001) planes from ~1.85 Å at equilibrium to ~2.11 Å at maximum stress. With the elongation of B1–B2 and B3–B4 bond lengths, the magnitude of charge depletion between them increases gradually before the stress reaches the peak value of about 25.7 GPa (see Fig. 4e). After the peak stress, the B2 atom and B4 atom come closer together and form B2–B4 bonds with length of ~1.97 Å at a strain of 0.4002, while the B1 and B3 atoms further separate. The shear deformation after the peak stress is accompanied by a stress decrease to about 5 GPa (Fig. 4a).

In FeB4 and MnB4, the weakest link does not appear along the (110)[1–10] shear. Therefore we concentrate on the deformation within the (110)[001] slip system. In view of the similarity between FeB4 and MnB4, we shall take FeB4 as an illustrative example. Figure 4g–i show the variation of the VCDD isosurfaces in equilibrium (γ = 0.0000), at a shear strain of γ = 0.2434 corresponding to the maximum stress, and at γ = 0.4002 after the shear instability of FeB4, respectively. No significant bond weakening or breaking appear before the shear stress reaches the maximum value at strain of about 0.25. After the shear instability, a multiple out-of-plane bond weakening and breaking appears between the {001} planes (see the regions marked by black arrows in Fig. 4h,i). In the following orbital analysis we shall show that such bond weakening is attributed to the electronic instability between the d orbitals of Fe and p orbitals of B, which are responsible for the out-of-plane bonding between {001} planes.

Electronic origin of lattice instability under shearing

For that purpose, we analyze the change of orbital-decomposed DOS of d orbitals of Tm and p orbitals of boron in TmB4 before and after lattice instability. Figure 5a–f shows the orbital-decomposed DOS of CrB4 at equilibrium, at a strain of 0.1717, and at a strain of 0.4002. Upon shear but before the lattice instability, the occupied dxy states located below EF move towards EF (see the red curve between −2 eV and 0 eV in Fig. 5a,b), while the peak of dxy above EF move downwards and split into two parts between 0 and 1 eV, and between 1 and 2 eV. The splitting feature is also observed for the dz2 states above EF (see blue curve between 0 eV and 2 eV in Fig. 5a,b). During this process, the shape of the other three d-states around EF does not change significantly. After the lattice instability (see Fig. 5c), most of the dxy and dz2 states between −2 eV and 0 eV have vanished, and both contribute substantially to peaks in the DOS between 0 and 2 eV above EF. At the same time, the pseudogap in the pz states of boron became narrower and finally vanished, while px and py increase their contribution at EF after the lattice instability. At the peak stress, all three p orbitals contribute to the states at EF. Therefore, the change of the DOS arising from the dxy orbitals of Cr and from px and py orbitals of B are mainly responsible for the in-plane bond weakening of B1–B2 and B3–B4 bonds within the x–y plane, as shown above.

Figure 5: Orbital-decomposed electronic density of states of oP10[58]-CrB4.
figure 5

(a,d) at equilibrium, (b,e) at a strain of 0.1717, (c,f) at a strain of 0.4002. The reference Cartesian coordinates was chosen such that the x axis is normal to crystallographic (1–10) plane, the y axis is along the crystallographic [110] direction and the z axis along the [001] direction respectively. The black arrows in (a–c) indicate the change of dxy state, and those in (d–f) show the change of pseudogap at Fermi level.

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Figure 6 shows the orbital-decomposed EDOS of the (110)[001] slip system of FeB4 in equilibrium, at a strain of 0.2434 corresponding to the stress maximum, and at strain 0.4002 after the shear instability. In equilibrium, dz2 states show a pseudogap at EF whereas the other four d orbitals contribute mostly to the finite value at EF. Before the instability, the peaks of all five orbitals between −6 eV and 0 eV move towards EF, and at the maximum stress one profound peak of dz2 character is seen at the Fermi level giving large contribution to the states at EF, while nearly all peaks of other spin-up orbitals above the Fermi level vanish. After the lattice instability, the states of all spin-down orbitals move further upwards, whereas the dxz spin-down states appear above Fermi level. The appearance of boron pz states at the Fermi level and the decrease of peaks of px and py states can be clearly seen in Fig. 6d–f. Therefore we can conclude that the electronic instability of FeB4 during (110)[001] shear is mainly due to the changes of the energies of states of Fe dz2 and dxz orbitals, and of pz orbitals of B. MnB4 shows a similar instability as FeB4, therefore we do not discuss it here in any further detail (see Fig. S8 in supplemental materials).

Figure 6: Orbital-decomposed electronic density of states of oP10[58]-FeB4.
figure 6

(a,d) at equilibrium, (b,e) at a strain of 0.2434, (c,f) at a strain of 0.4002. The reference Cartesian coordinate was chosen such that the x axis is normal to crystallographic (1–10) plane, the y axis is along the crystallographic [110] direction and the z axis along [001] direction respectively. The black arrows in Fig. (a–c) indicate that the major contributions at Fermi level changes from dxy to dz2 state during shear, while arrows shown in Fig. (d–f) show the change of pz state at Fermi level.

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The electronic instability of TmB4 (Tm = Cr, Mn and Fe) upon shear due to d orbitals of Tm and p orbitals of B does not resemble the cases of ReB218 and WB36. The important difference is that in TmB4 the covalent bond networks are mainly responsible for the shear instability, while metal-boron or metal-metal bonds are the carrier of the shear instability in ReB2 and WB2. Although the three dimensional covalent network has been proposed to be responsible for high hardness in transition metal borides for a long time, our results show uniquely that the 3D boron network in the tetraborides may not be strong enough to provide these materials with high plastic resistance. Because the strength of a real crystalline material is limited by a variety of defects, such as dislocations, cracks, grain boundaries and others, it is usually orders of magnitude smaller than that of an ideal crystal. The Peierls–Nabarro (PN) stress of dislocations provides a realistic explanation on plastic resistance of a real crystal and can be correlated to shear moduli and ideal shear strength53,54,55 as , where α = 2π and K = G/(1−ν) for edge dislocation and K = G for screw dislocation, b is the Burgers vector, ν is the Poisson ratio, and ζ is the dislocation half-width which can be expressed as , where τmaxis the ideal shear strength. Thus, the plastic resistance of a crystal depends mainly on the shear modulus, ideal shear strengths, lattice topology, bonding nature and deformation mode. The present work emphasizes the necessity to conduct a combinational analysis of the lattice stability, of the values of shear moduli, of the anisotropic shear strength, of the complexity of lattice topology and its changes during shear, of the nature of chemical bonding, and others which are critical parameters in Peierls–Nabarro dislocation model of plastic lattice resistance53,54,55. Despite a century of research on dislocation mediated plasticity, a systematic connection to those quantities derived from first principle calculations has not been rationalized, inducing massive applications of a single parameter e.g. elastic moduli or ideal shear strength, to directly quantify the mechanical strength and hardness of a real material3,4,9,10,11,12,19,23,24,28.

Discussion

In summary, we carried out comprehensive density functional theory calculations to determine the thermodynamic, mechanical and phonon stabilities, the deformation paths and the electronic instability modes of orthorhombic TmB4. The electronic structure calculations reveal that the transformation from oI10[71]-TmB4 to oP10[58]-TmB4 can be explained by the variation of the in-plane electronic structure within (001) planes and the formation of new boron-boron bonds at hollow sites by distortion of boron networks. These processes significantly enhance the electronic hybridization of d orbitals of Tm with p orbitals of B by crystal field splitting. Depending on the different deformation paths (slip systems), different orbitals are responsible for the electronic instability upon finite shear. The relatively low shear moduli and ideal strengths of TmB4 suggest that these materials cannot be intrinsically superhard. The instability of weak 3D boron covalent networks is found to be responsible for the weakness of oP10[58]-TmB4. These results illustrate the importance and necessity of a combinational analysis of a variety of parameters related to plastic deformation of the crystalline materials, and their combination in the attempt to design new intrinsically hard and superhard materials.

Methods

First principles calculations

The present first-principles density functional theory (DFT) calculations of the formation energy of TmB4, Tm = Cr, Mn and Fe were done by means of the VASP code35 with the projector augmented wave method36 employed to describe the electron-ion interaction and the generalized-gradient approximation of Perdew-Burke-Ernzerhof (PBE)56 for the exchange correlation term. The integration in the Brillouin zone has been done on special k grids for the phases that were under consideration, which were determined according to the Monkhorst-Pack scheme, the energy cutoff of 600 eV, and the tetrahedron method with Blöchl corrections for the density of states and smearing methods for the stress calculations. The verification of the reliability of our calculations has been done by calculating the total energies, equilibrium lattice parameter, and bulk modulus of TmB4 and compared with previous theoretical and experimental values in the present work, and of a number of other materials in our earlier work. In the present studies, all DFT calculations are performed with spin-polarized scheme because of the appearance of magnetic momentum during the deformation. Nevertheless, in cases where no magnetic momentum has been found, additional non-spin-polarized calculation is used to check its possible influence.

The phonon dispersion

The phonon calculations were performed within the harmonic approximation using the direct method57 based on the calculated non-vanishing Hellman-Feynman forces employing the Phonopy code58. To confirm our results, we also used a linear response method based on the perturbation theory as implemented in VASP code.

The elastic constants

For orthorhombic crystals (Group numbers 16–74), there are nine independent elastic constants usually referred to as c11, c22 , c33 , c44 , c55 , c66 , c12 , c13 , and c23. In the following, we give a set of simple strain configurations and the corresponding strain-energy density variations ΔE/V0. (r1) ε = (δ,0,0,0,0,0) with relation , (r2) ε = (0,δ,0,0,0,0) with relation , (r3) ε = (0,0,δ,0,0,0) with relation , (r4)ε = (0,0,0,δ,0,0) with relation , (r5)ε = (0,0,0,0,δ,0) with relation , (r6)ε = (0,0,0,0,0,δ) with relation , (r7)ε = (δ,δ,0,0,0,0) with relation , (r8)ε = (δ,0,δ,0,0,0) with relation , (r9)ε = (0,δ,δ,0,0,0) with relation .

The stress-strain relationships

First, the atomic basis vectors of a given crystal cell were projected onto the Cartesian coordinates R with one axis vector being parallel to the imposed strain direction for the tension. For the shear, one axis vector was perpendicular to the slip plane and another one was parallel to the slip direction in that plane. Afterwards, the crystal has been incrementally deformed by transforming the unstrained atomic basis vector matrices R to the strained ones R′ using the deformation matrices.

where e1 = exx, e2 = eyy, e3 = ezz, e4 = ezy + eyz, e5 = ezx + exz, and e6 = eyx + exy are the strain components in the Voigt notation. The diagonal strain components exx, eyy, and ezz represent the tension while the off-diagonal components represent the shear. In order to keep the crystal under a stress state of uniaxial tension or shear, the strained cell has been relaxed for both the atomic basis vectors and for the atom coordinates inside the unit cell by keeping the applied strain component fixed and relaxing the other five strain components until their conjugate stress components i.e., Hellmann–Feynman stresses reached negligible values. Such a relaxation scheme is accomplished by using a slightly modified VASP code with specific constraints of strain components. To ensure that the strain path is continuous, the starting position at each strain step has been taken from the relaxed coordinates of the previous strain step. In the instance of having a large strain, the crystal symmetry may be changed and the Brillouin zone significantly deformed. Therefore we adopt a high energy cutoff of 600 eV and verified the convergence of the calculations of the stress-strain curves by using different meshes of k points. Although the spin-polarized calculation does not have big impact on the shear strength on FeB4 and CrB4, we performed the spin-polarized calculations for a comparison because of the significant magnetic effect on the results of MnB4 and possible appearance of magnetic moment during severe deformation.

Evolutionary Structure Searches

Structure searches were performed using the open-source evolutionary algorithm (EA) XtalOpt38,39 along with the default parameter set from ref. 38. EA runs were carried out on the TmB4 with 2, 3 and 4 formula units in the primitive cell. In this algorithm a new offspring is procreated as soon as an individual is optimized and the parents are chosen from a population based pool. The population size were around 735, 225, 110 for the 2, 3 and 4 formula unit cells of CrB4, respectively, and 200, 225, 160 for the 2, 3 and 4 formula unit cells of MnB4, respectively. In each run the same low enthalpy structure was found at least 3 times (often more) before the run was terminated. A six-step structural-optimization scheme was used for all runs, and each step employed the geometry of the structure from the previous step for the initial geometry. Only the ions were allowed to relax in the first two steps, while the last four steps also allowed the lattice parameters to relax. The precision of the calculation was increased at each step. The calculations were performed using the PBE exchange-correlation functional, the PAW method and an energy cut-off of 450 eV in the final step. The lowest energy structures from each search were optimized using the aforementioned computational settings, to obtain a more accurate energy ranking.

Additional Information

How to cite this article: Zhang, R. F. et al. Crystal Field Splitting is Limiting the Stability and Strength of Ultra-incompressible Orthorhombic Transition Metal Tetraborides. Sci. Rep. 6, 23088; doi: 10.1038/srep23088 (2016).