Introduction

Superhard materials have attracted considerable attention in both fundamental and technological applications due to their superior mechanical properties such as high melting temperature, high hardness, high electrical and thermal conductivity1. Previously, it was generally accepted that the superhard materials are those strongly covalent bonded compounds formed by light elements (B, C, N and O), such as diamond2,3, c-BN4, BC55 and BC2N6 etc. These superhard materials are easily to form strong three-dimensional covalent bonding networks7,8. Although diamond is the known hardest material with a measured hardness at 60–120 GPa, but it reacts easily with iron-based materials. The hardness of cubic boron nitride (c-BN) is second only to that of diamond. However, it can be synthesised only under high pressure and high temperature conditions which needs great cost9. Therefore, great efforts have been devoted to exploring novel hard and ultra-incompressible materials10,11,12,13,14. Recently, it was reported that partially covalent heavy transition metal (TM) boride, carbide, nitride and oxide are found to be good candidates for superhard materials, such as ReB2, OsB2, CrB4 and WB415,16,17,18,19,20. These reports revealed that they all possess high bulk and shear moduli. Because the compounds formed by transition metal and light elements usually possess high valence electron density and directional covalent bonds and those covalent bonds are strong enough to improve the mechanical properties. Moreover, d valence electrons are considered to contribute to the hardness of transition-metal compounds. Further, these materials can be synthesized under lower pressure which leads to the low-cost synthesis condition and this is beneficial to their applications. Therefore, these pioneering studies open up a novel route for the search of novel superhard multifunctional materials.

The borides of rhodium are well known for their high melting temperature and hardness. The measured Vickers hardness of bulk RhB1.1 was 7-22.6 GPa, when the loads ranging from 0.49 to 9.81N21. Later, the 1.0 μm thick RhB1.1 film was studied by X-ray diffraction and it possesses a hardness of 44 GPa22. Past studies have identified two stoichiometric compositions: RhB (hexagonal structure, No. 194, P63/mmc). Wang et al. reported that when the pressure exceeds 22 GPa, RhB transforms from hexagonal RhB (anti-NiAs type) to the orthorhombic Pnma space group (FeB type)23, Rh2B (No.62, Pnma) has been determined that it possesses an orthorhombic structure24. In 1953, Richard et al. studied the crystal structure of Rh2B from X-ray rotation and Weissenberg photographs. Over the past 60 years, experimental equipment and technology have been improved dramatically, but there is no report about Rh2B in these years. This led us to the idea that if phase transition may occur in the Rh2B, which may bring about novel properties. Detailed structural, mechanical and electronic properties theoretical investigations of Rh2B are also seldom. Are there compounds with higher boron contents?High boron compounds did not reveal any new phases so far. The results are worth making the effort.

In this article, we report two new phases for Rh2B and RhB2 by the first-principles calculations. Our results show that the predicted new phase of Rh2B belongs to the monoclinic P21/m phase, which is energetically much more stable than the previously proposed Pnma structure in experiments. At the pressure of about 39 GPa, the P21/m phase transforms to C2/m phases. While the structure type of the new phase of RhB2 also belong to the monoclinic P21/m phase. Both of the two phases are dynamically and mechanically stable at ambient pressure. Further calculations are performed to study the properties of those high-pressure phases.

Results and discussion

At ambient pressure, the variable cell simulation revealed a monoclinic Rh2B-I (P21/m) structure as the most stable phase for Rh2B, as shown in Fig. 1 (a). The P21/m structure contains two Rh2B f.u. in a unit cell (a = 5.615 Å, b = 2.873 Å, c = 4.715 Å and β = 73.17 °), in which three inequivalent atoms Rh1, Rh2 and B occupy the Wyckoff 2e (0.928, 0.250, 0.769), 2e (0.561, 0.750, 0.230) and 2e (0.775, 0.250, 0.396) sites, respectively. Fig. 1 (b) along the y-axis and Fig. 1 (c) along the z-axis reveals a fundamental building block in the Rh2B-I structure. For RhB2, the predicted RhB2-II (P21/m) is the most thermodynamically stable phase among the considered structures. Fig. 1 (d) shows the structure of RhB2-II. RhB2-II belongs to a monoclinic (P21/m) space group containing two RhB2 f.u. in a unit cell (a = 6.044 Å, b = 3.057 Å, c = 6.116 Å and β = 100.72 °), in which Rh1, Rh2, B1, B2, B3 and B4 atoms occupy the Wyckoff 2e (0.306, 0.750, 0.043), 2e (0.292, 0.250, 0.448), 2e (0.015, 0.750, 0.397), 2e (0.963, 0.750, 0.086), 2e (0.582, 0.250, 0.279) and 2e (0.258, 0.250, 0.783) sites, respectively. From Fig. 1 (e) along the y-axis, it is seen that the boron atom sheets consist of triangle rings, within the triangle rings boron atom layer, the shortest B–B bond is 1.67 Å, which is smaller than the OsB2 (1.87 Å). At the same time, the boron atoms form a three dimensional space grid structure in Fig. 1 (f) and avoid the happening of the interaction between metal atoms.

Figure 1
figure 1

Crystal structures.

The green spheres represent the B atoms and the gray ones represent the Rh atoms. (a) (b) (c) Rh2B-I and (d) (e) (f) RhB2-II.

Full size image

We calculated the formation enthalpy of the considered structural candidates of Rh2B and RhB2 in the pressure range of 0-100 GPa. The formation enthalpy of RhxBy with respect to the pure phases is investigated by the following equations ΔH = HRhxBy – xHRh – yHB. The Rh (space group: Fm-3m)25 and α-Boron (space group: R-3m)26 were chosen as the referenced phases. Fig. 2 presents the enthalpy curves of Rh2B and RhB2 structure relative to the (Rh + αB) within the given pressure range. From Fig. 2, it can be clearly seen that the Rh2B-I has the lowest negative formation enthalpies at ambient pressure and the enthalpy of Rh2B-I is much lower than that Rh2B in experiment by ∼0.16 eV per formula. This indicates that Rh2B-I is thermodynamically stable and can be synthesized in experiments. When the pressure is higher than 39 GPa, the phase transition from P21/m phase to C2/m phase, because C2/m phase is more energetically stable in high pressure. In experiment, they used high purity rhodium metal and relatively pure boron (98.8% with slight amounts of iron and carbon)24. They directly mix and sinter the samples for Rh2B. The experimentally observed phase is a metastable phase with impurities. This can explain no presence of our predicted phases of Rh2B in experiments. For RhB2, the predicted RhB2-II is the most thermodynamically stable phase in our calculations, no further phase transition was observed in the high pressure range.

Figure 2
figure 2

Formation enthalpy-pressure diagrams.

Calculated enthalpies per unit of various structures relative to the (Rh + αB) as a function of pressure range from 0–100 GPa. (a) Rh2B-I and (b) RhB2-II

Full size image

To check the dynamical stabilities of the currently predicted Rh2B-I and RhB2-II, we have calculated their phonon dispersion curves. A stable crystalline structure requires all phonon frequencies to be positive, as seen in Fig. 3 (a,b), the absence of any imaginary phonon frequency in the whole Brillouin zone for Rh2B-I and RhB2-II indicate the dynamical stabilities of the two structures at ambient pressure. In Fig. 3 (c), the calculated phonon band structure shows no soft phonon, further confirming the stability of C2/m phase at 50 GPa.

Figure 3
figure 3

The phonon-dispersion curves.

(a) Rh2B-I at 0 GPa, (b) RhB2-II at 0 GPa and (c) C2/m phase at 50 GPa respectively.

Full size image

Elastic constants are essential for understanding the mechanical properties of a crystal. We calculated the zero-pressure elastic constants Cij of the two phases and the elastic constants Cij are listed in Table 1. For a stable structure, Cij has to satisfy Born–Huangcriteria27: For a monoclinic crystal, the independent elastic stiffness tensor consists of thirteen components of C11, C22, C33, C44, C55, C66, C12, C13, C23, C15, C25, C35 and C46. The mechanical stability criteria is given by:

Table 1 Elastic Constants, Bulk Modulus (GPa), Shear Modulus (GPa), B/G ratio, Young’s modulus Υ (GPa) and Poisson’s ratio υ of Rh2B-I and RhB2-II at zero pressure.
Full size table

As summarized in Table 1, the Rh2B-I and RhB2-II phases satisfy all mechanical stability criteria, indicating that they are mechanical stable at ambient pressure. The calculated C33 values are bigger than that of C11 and C22 in two structures, implying that the resistance to deformation along the c-direction is stronger than that along the a-direction and b-direction. The calculated bulk modulus of Rh2B-I and RhB2-II is 238 and 255 GPa, respectively, both phases can be grouped into incompressible materials. Besides the bulk modulus and shear modulus, Young’s modulus could also provide a good measure of the stiffness of materials. The Young’s modulus Y is obtained by the equation: Y = (9GB)/(3B + G). Young’s modulus is defined as the ratio of stress and strain and is used to provide a measure of the stiffness of materials in the range of elastic deformation. When the value of Y is large, the material is stiff28. The ratio value of B/G is commonly used to describe the ductility or brittleness of materials, with 1.75 as the critical value29. Higher (or lower) B/G value than the criteria indicates that the materials is ductile (or brittle). The B/G value of Rh2B-I is 2.74, exceeding the critical value and implying its ductile nature. RhB2-II also behaves in a ductile manner. The value of the Poisson’s ratio υ is indicative of the degree of directionality of the covalent bonds. The Poisson’s ratio υ is obtained by the equation: υ = (3B-2G)/ 2(3B + G). The typical υ value is 0.1 for covalent materials and 0.33 for metallic materials, respectively30. The Poission’s ratio of RhB2-II (0.277) is smaller than that of Rh2B-I (0.377), indicating that the directionality degree of covalent bonding of RhB2-II is stronger than that of Rh2B-I. The directionality of covalent bonding plays an important role in the hardness of materials.

The electronic structure is crucial to the understanding of physical properties of materials. The electronic density of states (DOS) and the atom resolved partial density of states (PDOS) of the two phases at 0 GPa and C2/m phase at 50 GPa are shown in Fig. 4. In Fig. 4 (a), there is a deep valley at about −7 eV. It is a pseudogap of DOS, which is the borderline between the bonding and antibonding states. There are no clear overlap of rhodium’s d electron and boron’s p electron in range of −7 and 1 ev. In Fig. 4 (b) rhodium and boron atoms form strong covalent bonds as displayed by the much overlap of rhodium’s d electron and boron’s s electron, boron’s p electron curves in comparison with that of Rh2B-I. Indicating there is a strong covalent interaction between the B and Rh atoms in RhB2-II. In Fig. 4 (c), there is a deep valley at about −9 eV. There are no clear overlap of rhodium’s d electron and boron’s p electron in range of −7 and 5 ev. The finite electronic DOS at the Fermi level indicates a metallic feature for the three phases.

Figure 4
figure 4

The total and partial densities of states.

(a) Rh2B-I phase at 0 GPa, (b) RhB2-II phase at 0 GPa and (c) C2/m phase at 50 GPa respectively.

Full size image

To gain more detailed information about the bonding character, we calculated the electronic localization function (ELF) of Rh2B-I , RhB2-II at 0 GPa and C2/m phase at 50 GPa. The ELF was employed to understand the electron pairing and localization of the crystal structure. It should be noted that ELF is useful in distinguishing metallic, covalent and ionic bonds31. The ELF is a contour plot in real space where different contours have values ranging from 0 to 1. ELF = 1 is that there is no chance of finding two electrons with the same spin. From Fig. 5, we can see clearly that the strong bonds between Rh and B in Rh2B-I. This is consistent with the analysis of their DOS. The system of covalent bonds in the RhB2-II is significantly anisotropic, where the neighbor boron atoms form very powerful covalent bonds within the planar triangle unit, whereas Rh–B bonds are appreciably weaker. Therefore, these strong covalent bonding will increase the structural stabilities and high bulk moduli of RhB2-II. In Fig. 5(c) there are strong bonds between Rh and B in C2/m phase.

Figure 5
figure 5

Contours of the electronic localization function (ELF).

Electron localization function isosurface maps for (a) Rh2B-I at 0 GPa, (b) RhB2-II at 0 GPa and (c) C2/m phase at 50 GPa respectively.

Full size image

Conclusions

In summary, we have predicted a new phase Rh2B-I at ambient pressure for Rh2B and a new phase RhB2-II for RhB2 through the ab initio calculation. The two new phases all belong to monoclinic P21/m structure. Besides, both of the two phases are dynamically and mechanically stable at ambient pressure. Rh2B-I is energetically much superior to the previously proposed Pnma structure in the experiment. At the pressure of about 39 GPa, a phase transition occurs between the P21/m and C2/m phases for Rh2B. For RhB2, RhB2-II is the most thermodynamically stable phase in our calculations, no further phase transition was observed in the high pressure range.

Computation details

The evolutionary variable-cell simulations for Rh2B and RhB2 were performed at ambient pressure as implemented in the USPEX code32,33,34. The structure relaxation was performed by using the density functional theory implemented in the Vienna ab initio simulation package (VASP) code35,36,37. The exchange correlation functional was treated by the generalized gradient approximation (GGA)38 with the projector-augmented wave (PAW) potential. The tested energy cutoff 450 eV was used, the k-points amplings in the Brillouin zone were performed using Monkhorst-Pack method (for the hexagonal structures, a Gamma-centered k-points was used) to ensure that enthalpy calculations are well converged with energy differences of less than 1meV/per atom. For each candidate structure, the atomic positions, bond lengths and cell parameters were fully optimized. Elastic constants were carried out using the CASTEP code39 and the bulk modulus and shear modulus were thus estimated by using the Voigt-Reuss-Hill approximation40. The phonon calculations were carried out by using a supercell approach as implemented in the PHONON code41. The details of convergence tests have been described elsewhere42,43,44,45.

Additional Information

How to cite this article: Chu, B. et al. Structural, mechanical and electronic properties of Rh2B and RhB2: first-principles calculations. Sci. Rep. 5, 10500; doi: 10.1038/srep10500 (2015).