Introduction

Beryllium (Be) and boron (B) are elements that occupy adjacent positions in the periodic table, but they have quite different properties, as the former is a metal while the latter is a semiconductor under standard conditions. At ambient conditions, Be crystallizes in a hexagonal-closed-packed (hcp) structure and transforms into a body centered cubic (bcc) crystal structure at 1523 K or approximately 400 GPa1,2. It has gained considerable attention because of its simple atomic configuration and unusual behaviors. As it is highly transmissive to X-rays, Be is commonly used as a window material in X-ray equipment3. It is also one of the most effective moderators and reflectors for neutrons used in nuclear reactors4. In addition, its light weight, high melting point and high-strength performance result in it being widely used in aircraft, rockets, satellites and other aerospace applications, as well as in future fusion reactors5,6. Although Be has several negative characteristics, such as potential toxicity that poses a manufacturing challenge, controlled handling procedures, such as the use of a glove box, can enable the success industrial synthesis of beryllium-containing compounds7,8.

Boron is a semiconductor that becomes a metal and even a superconductor, when placed under high pressure9. Elemental boron has many allotropes that use the B12 icosahedron as the basic unit, of which α-Boron is the most common10. Boron-containing compounds have been investigated extensively due to their exotic physical properties, such as superconductivity, hardness and thermoelectric performance. For example, several transition metal diborides, such as OsB2, IrB2, ReB2 and BCx, possess high bulk and shear moduli and have been explored for use as potential super-hard materials11,12,13,14,15,16.

When beryllium and boron combine, various Be-B phases ranging from beryllium-rich to boron-rich will form. Some Be-rich phases, such as the Be2B phase, show a metallic character and are potential superconductors17. Another beryllium-rich phase, Be4B, is the only stable Be-rich phase at room temperature. Within the boron-rich compounds, BeB3, found only in the MgIn3 type (space group , NO.221, one formula unit per cell), is found to be not stable. BeB4, however, is highly stable at high pressures2.

Although beryllium diboride has been known and studied for more than half century18,19,20,21,22,23, there has still been no solid evidence that it exists with the stoichiometry BeB2. The first announced synthesis of BeB2.01 ± 0.03 with P6/mmm symmetry18 was later recognized to have a correct stoichiometry of BeB2.75 with a surprisingly complex crystal structure19. Very recently, Hermann, A. etal. systematically studied the binary Be-B phases at ambient and high-pressure conditions2,20. In their work, they found that the most stable structure for BeB2 at atmospheric pressure is cubic (space group , NO.216), with a diamondoid boron network and beryllium atoms occupying interstitial tetrahedral sites. This structure is semiconducting and can be understood in terms of the Zintl-Klemm concept, making it unique among Be-B phases. Other proposed beryllium di-boron phases can be classified into three crystal systems: (1) cubic, including the CaAl2 structure (space group , NO.227, eight units per cell, prototype MgCu224 ) and “AlLiSi” structures (space group , NO.21620); (2) hexagonal, including the MgB2 crystal structure (space group P6/mmm, NO.19120, one unit per cell, prototype AlB2) and the CaIn2 and MgGa2 structures (space group P63/mmc, NO.194, two units per cell25,26); and (3) orthorhombic, including the SrAl2 structure (space group Imma, NO.74, four units per cell, prototype CeCu227), another MgGa2 structure (space group Pbam, NO.55, eight units per cell28) and the “CaIn2” structure (space group Pnma, NO.6220). Many phases have already been found. At low pressure (under 160 GPa), the cubic phase (space group , NO.216) is the most stable, while the P63/mmc phase is dynamically unstable and can turn into low-symmetry Pnma. At pressures greater than 160 GPa, the P63/mmc phase is more stable2. There is still an open question regarding whether there exist other stable phases with low energy in the low pressure zone. Therefore, further theoretical studies, such as of the crystalline structure, electronic properties and mechanical properties, are important for this potential material.

In the present work, potential crystal structures of BeB2 were extensively explored using ab initio evolutionary simulations. A new phase with Cmcm symmetry was uncovered. The crystal structures, phonon dispersions, mechanical properties, phase transitions and electronic structures of this phase as well as other available phases were investigated. The band structures and density of states reveal that the new phase is metallic. Further study on the mechanical properties using three different methods was used to predict the Vickers hardness of the newly discovered Cmcm phase. Charge density difference maps and Mulliken population analysis, as well as COHP diagrams, were also carried out to analyze the character of the bonding between the atoms in Cmcm BeB2. The superconductivity of the new Cmcm phase was also investigated.

Results

Crystal Structure

In a previous work, several phases were found both experimentally and theoretically. However, only four structures are stable according to the calculated phonon spectra in the present work. These are cubic (space group , NO.216), orthorhombic (space group Pnma, NO.62), orthorhombic (space group Pbam, NO.55) and hexagonal (space group P63/mmc, NO.194). In the present work, a new stable phase with space group NO.63 was uncovered by performing systematic structure searches for BeB2. Its optimized equilibrium lattice parameters, axial ratio c/a and unit cell volume are shown in Table 1. The crystal structure information of all other known dynamical stable phases are presented in Table 22,20. Table 1 reveals that the conventional unit cell of the new Cmcm phase contains twelve atoms in total with two inequivalent Wyckoff positions 4c and 8f for Be and B atoms, respectively.

Table 1 Optimized structual data for the new phase of BeB2
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Table 2 Optimized structural data for other known phases of BeB2
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The schematic crystal structure of the Cmcm phase is shown in Figures 1, 2 and 3, where yellow and blue spheres represent Be and B atoms, respectively. Figure 1 shows the unit cell, while Figure 2 shows the super-cell of the same crystal structure. Figure 2(b) is perpendicular to the x-axis. There are two parallel columns formed by boron-centered and beryllium-centered coordination polyhedra that are marked by green and blue colors, respectively. Each boron atom is surrounded by four beryllium atoms that form a distorted tetrahedral configuration. Eight boron atoms embrace a beryllium atom, forming a complex polyhedron composed of 3 rectangles and 6 triangles. Along the z-axis, the structure can be viewed as several layers, as shown in Figure 2 (b) and (c). Six boron atoms connect to each other in a hexagonal ring, where 4 atoms are coplanar (denoted as B and B*) and the other 2 atoms are out of this plane (denoted as A and A*). Be atoms form another layer (denoted as C) between the B and B* layers along the z-axis. The upper image of Figure 2 (a) is taken from the A, B, C, B* and A* layers, whereas the lower image is composed of the B*, A*, C*, A and B layers. The distance between two A layers is 5.120 Å. The configurations of boron-centered and beryllium-centered coordination polyhedra are displayed in Figure 3. The distances of the three types of Be-B bonds are 1.923 Å, 2.021 Å and 2.105 Å, respectively, as shown in Figure 3 (a). The Boron-Boron bond lengths are 1.763 Å (the blue bond) and 1.657 Å (the green bond), as shown in Figure 3 (b).

Figure 1
figure 1

Unit cell of the crystal structure of the new Cmcm phase.

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Figure 2
figure 2

The crystal structure of the new Cmcm phase projected along different directions.

(a) view along the direction of the z-axis. (b) view along the direction of the x-axis. (c) view along the direction of the y-axis.

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Figure 3
figure 3

The coordination polyhedrons of the new Cmcm phase.

(a) The coordination polyhedrons for the B atoms. (b) The coordination polyhedrons for the Be atoms.

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Phonon Dispersion

The phonon dispersion curves of the new phase as well as of other previously proposed crystalline structures were calculated to determine their dynamical stability, as shown in Figure 4. The new Cmcm phase is dynamically stable, as there are no imaginary phonon frequencies detected in the whole Brillouin zone. For the previously proposed phases2,20, the Pbam, Pnma, and P63/mmc phases are dynamically stable according to the same rule. However, the appearance of imaginary frequencies indicates that the P6/mmm, and Imma phases mentioned in previously published articles2,20 are dynamically unstable.

Figure 4
figure 4

Phonon dispersion curves of available BeB2 phases.

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Phase Transition

Figure 5 gives the calculated relative enthalpy as a function of pressure for all stable phases of BeB2, relative to the phase. The inserted figures show the stable Cmcm phase and the phase of BeB2, before and after the phase transition at 13 GPa, respectively. As mentioned before, the latter has been considered in previous works to be the most stable phase at atmospheric pressure, having a structure with a diamondoid boron network and beryllium atoms occupying the interstitial tetrahedral sites. The polyhedron representation of the phase has also been marked, as shown in the bottom-right corner of Figure 5.

Figure 5
figure 5

Calculated enthalpy as a function of pressure for all mechanical and dynamical stable phases of BeB2.

(a) The Cmcm phase transforms into the phase at a pressure of 13 GPa. (b) The coordination polyhedrons of the known phase. The blue spheres represent B atoms and the yellow spheres represent Be atoms.

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As shown in Figure 5, the enthalpy of the newly uncovered phase increases with pressure within the studied pressure range up to 50 GPa, while the enthalpies of the other three stable phases (Pnma phase, Pbam phase, P63/mmc phase) have an inverse trend. It can be concluded that the Cmcm phase is more competitive than the previously known most stable phase at pressures up to 13 GPa. When the pressure exceeds 40 GPa, the Pnma phase and P63/mmc phase have almost similar enthalpies and both of them become more favorable than the new Cmcm phase. The cubic structure ( phase) continues to have the greatest stability in the pressure range from 13 GPa to 50 GPa. These results are consistent with the results previously found by Andreas Hermann etal.2

First-principles methods were used to calculate the formation enthalpy of the new phase Cmcm (No.63) at different pressures. The structural parameters of each phase are listed in Table 3. The enthalpy of formation is calculated as follows:

As shown in Table 4, when the pressure is 1 atm, the formation enthalpy of the Cmcm phase is −56 meV/atom, which is within the range of formation enthalpies (−40 meV to −125 meV) of some proposed models for BeB2.79 phases20. This may explain why the BeB2.79 phase with a very complex structure has been found experimentally, rather than the pure BeB2 phase with the simple Cmcm structure. It also indicates that the proposed models of BeB2.79 phases20 with enthalpies less than −56 meV/atom are more competitive. Upon comparing the formation enthalpy of the new phase (− 0.056 eV/atom) with that of the most stable known phase (−0.016 eV/atom), it may be observed that the former is more competitive than the latter. In short, the Cmcm phase is more thermodynamically stable than any other known BeB2 phase.

Table 3 The parameters for structure used in the formation enthalpy calculation
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Table 4 Formation enthalpy for BeB2 phases at P = 1 atm
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Using the same approach, the formation enthalpy of the Cmcm phase at 10 GPa and 15 GPa was calculated to be −39.7 meV/atom and −32.2 meV/atom, respectively. By comparing to the value at 1 atm (−56.267 meV/atom), it may be seen that the formation enthalpy value increases continuously with the pressure, indicating that increasing pressure is not beneficial for the synthesis of the Cmcm phase. All of these conclusions agree well with those obtained from the pressure phase diagram2,20 and the results will provide guidance for experimental work on BeB2 synthesis.

In addition, to provide more “physics” of the phase transition, we have calculated the solid-solid structural phase transitions between the Cmcm and phases at 13 GPa by using the VC-NEB method29. Figure 6 illustrate the snapshots from a dynamical trajectory collected from transition path sampling connecting the Cmcm phase and the phase. The energy barrier of the transition was calculated to be 0.25 eV/atom.

Figure 6
figure 6

Snapshots from a dynamical trajectory collected from transition path sampling connecting (a) the Cmcm phase to (f) the phase.

Black dashed line means a bond disappears while purple solid line means a bond created compared to the previous snapshot. Boron (Beryllium) atoms are shown in green (gray) colors.

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Discussion

Mechanical Properties

Mechanical properties, including the elastic constants, bulk modulus, shear modulus, Young's modulus, Poisson's ratio, are the main bases for choosing and designing materials. Therefore, all these properties were calculated using the first-principles approach in this work. The elastic constants of all structures under the ambient pressure Cij were calculated with the strain-stress method combined with Hooke's Law, as implemented in the CASTEP code30. The results are listed in Table 5. In order to determine the mechanical stability of the new predictions, Cij has to satisfy the elastic stability criteria31.

Table 5 Calculated values of elastic constants Cij (in GPa), Bulk modulus B (in GPa), Shear modulus G (in GPa), Young's modulus E (in GPa), Poisson's Ratio σ, B/G ratio and Vick's Hardness (in GPa) calculated by Xing Qiu Chen's34 formula for stable phases of BeB2 at 0 GPa and 0 K
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The mechanical stability condition and elastic constants of the orthogonal structure are positive. However, the following inequalities must also be satisfied to indicate stability:

As shown in Table 5, these values all meet the mechanical stability criteria, initially confirming that the Cmcm phase is mechanically stable.

Based on the Voigt-Reuss-Hill approximation method32,33, we can find the appropriate bulk modulus B and shear modulus G using the elastic constants. In addition, the values of Young's modulus E and Poisson's ratio σ can be calculated using the following formula:

Usually, the bulk modulus B is used to characterize a material's resistance to volume deformation against external pressure, while the shear modulus G measures a material's ability to resist shear strain. Young's modulus E measures the resistance against longitudinal tensions. Among the 4 mechanical and dynamical stable BeB2 phases, the Cmcm phase has the lowest value of bulk modulus and the second lowest shear modulus, revealing that it has low resistance to compression and shear strain. In addition, the elastic constant C22 (282 GPa) is significantly smaller than C11 (492 GPa) and C33 (690 GPa), revealing that resistance along the b axis is much smaller than along the a and c axes.

In order to evaluate the ductility of the material, the B/G values were calculated and are listed in Table 5. Higher B/G values greater than 1.75 correspond to a ductile material, while values less than 1.75 correspond to a brittle material34. As shown in Table 5, the B/G value of the Cmcm phase is approximately 1.01, suggesting that it is very brittle.

Poisson's ratio reflects the strength of the covalent bond to some extent. A small Poisson's ratio (σ = 0.13) indicates that the Cmcm phase is more intensely covalently linked than the Pbam phase (σ = 0.29) and P63/mmc phase (σ = 0.18). These results show that the Cmcm phase is likely to be a type of potential ultra-incompressible material, despite its bulk modulus and shear modulus not being very high.

Finally, the hardness of the Cmcm phase was calculated in three different ways. Using a recently proposed simple empirical hardness formula Hv = 2(G3/B2)0.585–335, the Vickers hardness of these phases was calculated. As the result shows, the Cmcm phase has a hardness value of 36.8 GPa, approaching the critical value of a super-hard material, 40 GPa. It also reveals that the predicted hardnesses of the Pnma phase and the phase are approximately 41.4 GPa and 42.3 GPa, respectively, implying that both of them are potential super-hard materials. The hardness value for the Cmcm phase is also calculated to be 25.2 GPa using th he formula of hardness given by Artem R. Oganov36 (details can be found in the computational methods section).

The hardness of the new phase was also calculated by the microscopic hardness model proposed by F.M. Gao et al.37,38 (details can be found in the computational methods section). The calculated parameters and hardness of the Cmcm phase calculated by this model are listed in Table 6. It is found that the total hardness of the Cmcm phase is only 13.8 GPa with this method.

Table 6 Parameters related to hardness and the value of Vickers hardness for Cmcm phase and the total hardness is 13.8 GPa
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Chen's model gives the highest values (36.8 GPa) from these three models followed by Oganov's model (25.2 GPa) while Gao's model provides the lowest values (13.8 GPa), which may be caused by their applicability to the boron compounds and with metals. The scattered values of hardness predicted from these three different models reveal that although it has a simple crystal structure, the electronic structure and bonding characters of the newly uncovered Cmcm phase of BeB2 are quite complex, as will be analyzed below.

Electronic Properties

The electronic properties of the newly discovered phase, including energy band structures, total and partial density of states (DOS) and charge density maps, were calculated and are shown in Figure 7,8,9.

Figure 7
figure 7

The electronic band structure and density of states for the Cmcm phase.

The dashed transverse line denotes the Fermi level.

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Figure 8
figure 8

Density of states for BeB2 in various stable phases.

From left to right: Cmcm, , P63/mmc, Pnma and Pbam. The dashed lines represent the position of the Fermi level.

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Figure 9
figure 9

Electron density difference plot (e/Å3) for the Cmcm phase at a select slice of (100).

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Figure 7 shows the calculated band structure along high symmetry directions, as well as the total and partial DOS of the optimized Cmcm structure from first-principles calculations within the GGA scheme. The overlapping of the valence bands and conduction bands around the Fermi level suggests that the new phase has a clear metallic character, which is confirmed by the finite value of total DOS at the Fermi level.

Figure 7 also plots the partial DOS of the novel Cmcm phase. It reveals that the total DOS of the upper part of the valence bands (from −9.5 eV to the Fermi level) is mainly contributed by the B-p state, while that of the conduction bands come from both the B-p and Be-p states. The Be-s state also contributes to the DOS of the conduction band (above 4 eV), while making a very slight contribution to that of the valence band. The B-s state contributes to the lower part of the valence band (less than −7.5 eV) and to the upper part of the conduction band, especially in the 4 eV to 8 eV region of the total DOS. As discussed above, there is significant hybridization of the s and p states from both Be and B in the region of 4 eV to 8 eV, implying the tendency to form covalent bonds between Be and B atoms.

Figure 8 plots the total DOS for all stable phases at ambient pressure, corresponding to the Cmcm, , P63/mmc, Pmma and Pbam phases, from left to right. The dashed lines represent the position of the Fermi level. The first panel is the newly uncovered Cmcm phase and the others are other known dynamically stable phases. Filling electrons at the Fermi level (approximately 1.18 states/eV/el.) shows that the new phase has an obvious metallic character, while the cubic phase with a band gap of 0.95 eV indicates its semiconductor property at low and intermediate pressures2. Except for the phase, all other known phases are metallic, as shown in the last three panels of Figure 8.

Figure 9 shows electron density difference maps for the Cmcm phase on the selected slice (100) plane. It can be seen that electrons are gathered at the positions of the B atoms and especially between Boron-Boron bonds, as indicated by the region in red in the top left panel and its enlarged bottom left panel. After removing the atoms, it is clearly seen that electrons are transferred from the Be atoms to the B atoms, as shown in the right two panels. From the electron gathering region between Boron-Boron, one can speculate that there is a strong covalent interaction.

In order to give some insight into these bonding characters, an atomic and bond Mulliken population analysis, which can provide a good way to quantitatively evaluate the charge transfer chemical bond strength of the studied system, was performed and analyzed by the CASTEP code, with the results listed in Tables 7 and 8. From Table 7, it can be clearly seen that electrons transfer from the Be atoms to the B atoms in the Cmcm phase, as the charges for Be and B are 0.69 and −0.34, respectively.

Table 7 Atomic Mulliken populations for Cmcm phase
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Table 8 Bond populations and bond lengths (Å) for Cmcm phase
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Now let us turn to the bond Mulliken population analysis results. A high nonzero value of overlap population indicates that there is a strong covalent character of the bond, while a small value close to zero shows that there is weak or no interaction between two related atoms and a negative value indicates that the atoms cannot form a bond39. From Table 8, it may be seen that the bond population ranges from 0.20 to 1.63 for the Cmcm phase. The maximum number, 1.63, exists between Boron-Boron bonds, indicating their strong covalent character. However, the Be-B bond has very weak covalence, as its Mulliken population is only 0.20. The population of Be-Be is −0.24, suggesting that there are no bonds forming. All of these Mulliken population results are consistent with the conclusions drawn from the electron density difference maps. As discussed in the mechanical properties section of this work, the hardness of the Cmcm phase may be a result of the high population value of the Boron-Boron bonds.

Some techniques such as the crystal orbital overlap population (COOP)40,41 and its analogous crystal orbital Hamilton population (COHP)42,43 can provide a straightforward view of orbital-pair interactions; based on these techniques, it is possible to analyze and interpret the bonding situation in solid-state materials. To elucidate the bonding situations in this new BeB2 phase, we performed crystal orbital Hamilton population (COHP) analysis, which partitions the band structure energy (in term of the orbital pair contributions) into bonding, nonbonding and anti-bonding energy regions within a specified energy range. Figure 10 shows the resulting –pCOHP as a function of energy for the new phase. Positive values of –pCOHP describe bonding energy regions, whereas negative values describe anti-bonding energy regions. As seen in the COHP diagrams in Figure 10, there appear to be obvious Be-B bonding states at the Fermi level, while that is not the case for the B-B combination, indicating that the interactions between Be-B bonds in the unit cell are stronger than those of B-B bonds, despite the fact that a single Be-B bond is weaker than a single B-B bond, as shown by the Mulliken population. Above or below the Fermi level (in the range of −6 to 6 eV), the COHP plots of the Be-B and B-B combinations are clearly dominated by bonding states, which shows that the new phase has a favorable stability performance.

Figure 10
figure 10

The crystal orbital Hamilton population (COHP) analysis of the bonding interactions in BeB2 based on plane-wave calculations using the newly introduced pCOHP method.

(a) for the Be-B bond. (b) for the B-B bond. All energies are shown relative to the Fermi level Ef.

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Superconductivity Properties

Encouraged by the relatively simple binary MgB2 having a superconductivity transition temperature of 39 K44 and the controversy regarding the reported superconductivity properties of BeB2 and BeB2.7519,21, we also calculated the superconductivity properties of the Cmcm phase of BeB2. The Tc can be estimated from the Allen-Dynes modified McMillan equation45,

which has been found to be highly accurate for materials with an EPC constant λ<1.546, where ωlog is the logarithmic average of the phonon frequency and μ* is the effective Coulomb repulsion and was assumed to be constant at 0.1. The calculated spectral function α2F(ω) and integrated λ(ω) of the Cmcm phase are plotted in Figure 11. Our results reveal that the Cmcm phase exhibits fairly low superconductivity properties, with a Tc of only 0.1 K. These results shed light on the controversy regarding the reported superconductivity properties of BeB2 or BeB2.75. The synthesized sample may contain both BeB2 and BeB2.75 phases. When the BeB2 phase dominates, an absence of superconductivity would be observed, as shown in Ref. [21], while when the BeB2.75 phase dominates, superconductivity appears.

Figure 11
figure 11

The calculated spectral function α2F(ω) and integrated λ(ω) of the Cmcm phase.

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Methods

Ab initio evolutionary simulations were run using the USPEX (Universal Structure Predictor: Evolutionary Xtallography) code47,48,49.

The USPEX code depends on VASP (Vienna ab initio simulation package)50 to achieve global optimization to calculate the enthalpy of crystal structures and explore the lowest enthalpy phase of a given elemental composition. Here, we used the USPEX code to search for stable compounds and structures with a fixed chemical composition of BenB2n (n = 1 to 5); the Cmcm phase comes from the 25th structure with the stoichiometry of Be2B4. During the structure search, USPEX selects a whole range of 50 generations to calculate, with each generation containing 50 individuals. The settings used for the variation operators are as follows: 60% of each generation was used to produce the next generation by heredity, 20% comes from soft mutations, 10% is produced randomly from space groups and the rest is produced through lattice mutations. The minimum length of any lattice vector was defined as 2.0 Å. The cutoff for USPEX relaxation and the k-points for resolution were 318 eV and 2π × 0.02 Å−1, respectively.

First-principles51 calculations were carried out using the density functional theory (DFT) approach by applying a generalized gradient approximation (GGA) for the exchange correlation functional52,53. We applied the Ultrasoft pseudo-potential introduced by Vanderbilt54 and the k-point samplings in the Brillouin zone were performed using the Monkhorst-Pack Scheme. The convergence tests used a kinetic energy cutoff of 600 eV and a k-point of 13 × 7 × 8 for the predicted Cmcm phase in the geometry optimization calculations. The self-consistent convergence of the total energy was 5 × 10−7 eV/atom, the maximum force on each atom was below 0.01 eV/Å and the maximum atomic displacement was below 5 × 10−4 Å. The phonon dispersion curves were plotted using the super-cell calculation method55 applied in the Phonopy program56. The calculation of the elastic constant and Mulliken overlap populations was carried out using the CASTEP code57. From the calculated elastic constants Cij, the polycrystalline corresponding bulk modulus B and shear modulus G were calculated using the Voigt-Reuss-Hill approximation32,33. In addition, Young's modulus E and Poisson's ratio σ were obtained by the equations E = (9G·B)/(3B + G) and σ = (3B−2G)/(6B + 2G), respectively.

In addition, we used three different methods to calculate hardness. These three methods were proposed by Xing Qiu Chen35, Artem R. Oganov36 and Faming Gao et al.37,38, respectively. The description of Chen's model can be found in the text. To use the formula of hardness given by Artem R. Oganov36, we need to use the structure file POSCAR (which must contain an element symbol line) and set the parameters for goodBonds, valence and valence electrons. For main group elements, only the outermost electrons are considered as valence electrons under normal circumstances. Hence, the numbers of valence electrons for Be and B atoms are 2 and 3.

Details of Gao's model are described below. The Vickers hardness of complex crystals can be calculated by a geometric average of all bonds as follows,

For the Cmcm phase, the total hardness is given by

Because no d-orbital valence electrons are involved in the chemical bonds, the hardness of each bond for Cmcm phase can be expressed by:

where du is the length of the bond, is the valence electron density (which can be calculated by

where ZBe and ZB are the valence electron numbers of the Be and B atoms constructing Be-B or B-B bonds, NBe and NB are the nearest coordination numbers of the Be and B atoms, Nj is the number of j bond in the unit cell and V is the volume of the unit cell) and is the Phillips ionicity of the bond. According to the generalized ionicity scale, the Phillips ionicity can be obtained from the following formula,

where fh is the population ionicity scale of the chemical bond, p is the overlap population of the bonds and pc is the overlap population of the bonds in a specified pure covalent crystal (here 0.57 is adopted).

For the complex crystal compounds, we considered three effects on hardness: the covalent component, the ionic component and the small metallic component. First, we defined a factor of metallicity fm as nm/ne for a simple-structured compound, where nm and ne are the numbers of electrons that can be excited at the ambient temperature and the total number of valence electrons in the unit cell, respectively. According to the electronic Fermi liquid theory, the thermally excited electron number nm can be described by the product of DF and the energy width kT, where k is the Boltzmann constant and T is the temperature. At the ambient temperature, kT is equal to 0.026 eV. Therefore, fm can be written as:

When the chemical bonds of a crystal are greater than or equal to two, we refer to it as a complex crystal. For the metallicity of complex crystals, the fm can be calculated by

where (or ) is the thermally excited electron number of Be or B atoms in the u-type bond, NMA (or NMB) is the number of chemical bonds with a metallic component around Be or B atoms and is the number of valence electrons per u-type bond.

To elucidate the bonding information in this new phase, we adopted a variant of the familiar COHP approach that stems from a PW calculation and was dubbed “projected COHP” (pCOHP)58,59. In this approach, all of the projection and analytic methods are implemented in a standalone computer program that processes PAW parameters and self-consistent results from VASP.

The calculation of the electron-phonon coupling (EPC) parameter λ was performed using the pseudo-potential plane-wave method within the density functional perturbation theory (DFPT)60 as implemented in the Quantum Espresso package61 by using Von Barth-Car type norm-conserving pseudo-potential with cutoff energies of 80 and 320 Ry for the wave functions and the charge density, respectively. A 7 × 4 × 4 q-point mesh in the first Brillouin zone was used in the EPC calculation.

Conclusions

In summary, a new stable phase of BeB2 with the space group of Cmcm was discovered by using ab initio evolutionary simulations. The Cmcm phase has a lower enthalpy than any previously proposed phase. The new structure is mechanically and dynamically stable, as determined by checking the calculated elastic constants and phonon dispersions, while several previously proposed phases (cubic: ; hexagonal: P6/mmm; orthorhombic: Imma) were found to be dynamically unstable. The Cmcm phase may transform to the cubic phase when the pressure exceeds 13 GPa. The calculated electronic band structure and density of state suggest that the uncovered new phase is metallic. Scattered hardness values calculated from three models suggest the complex electronic and bonding features of the Cmcm phase. The charge density difference maps and the Mulliken population analysis reveal that there are strong covalent interactions between the B atoms. The COHP diagrams show that the total interaction of Be-B bonds is stronger than that of B-B bonds. The Cmcm phase exhibits fairly low superconductivity properties, with a calculated Tc of approximately 0.1 K. The current theoretical predictions will most likely promote further experimental and theoretical investigation on the Be-B system.