Structural, mechanical, and electronic properties of Rh2B and RhB2: first-principles calculations

The crystal structures of Rh2B and RhB2 at ambient pressure were explored by using the evolutionary methodology. A monoclinic P21/m structure of Rh2B was predicted and donated as Rh2B-I, which is energetically much superior to the previously experimentally proposed Pnma structure. At the pressure of about 39 GPa, the P21/m phase of Rh2B transforms to the C2/m phases. For RhB2, a new monoclinic P21/m phase was predicted, named as RhB2-II, it has the same structure type with Rh2B. Rh2B-I and RhB2-II are both mechanically and dynamically stable. They are potential low compressible materials. The analysis of electronic density of states and chemical bonding indicates that the formation of strong and directional covalent B-B and Rh-B bonds in these compounds contribute greatly to their stabilities and high incompressibility.

. 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 RhB 1.1 was 7-22.6 GPa, when the loads ranging from 0.49 to 9.81N 21 . Later, the 1.0 μm thick RhB 1.1 film was studied by X-ray diffraction and it possesses a hardness of 44 GPa 22 . Past studies have identified two stoichiometric compositions: RhB (hexagonal structure, No. 194, P6 3 /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 , Rh 2 B (No.62, Pnma) has been determined that it possesses an orthorhombic structure 24 . In 1953, Richard et al. studied the crystal structure of Rh 2 B 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 Rh 2 B in these years. This led us to the idea that if phase transition may occur in the Rh 2 B, which may bring about novel properties. Detailed structural, mechanical, and electronic properties theoretical investigations of Rh 2 B 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 Rh 2 B and RhB 2 by the first-principles calculations. Our results show that the predicted new phase of Rh 2 B belongs to the monoclinic P2 1 /m phase, which is energetically much more stable than the previously proposed Pnma structure in experiments. At the pressure of about 39 GPa, the P2 1 /m phase transforms to C2/m phases. While the structure type of the new phase of RhB 2 also belong to the monoclinic P2 1 /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.
We calculated the formation enthalpy of the considered structural candidates of Rh 2 B and RhB 2 in the pressure range of 0-100 GPa. The formation enthalpy of Rh x B y with respect to the pure phases is investigated by the following equations Δ H = H RhxBy -xH Rh -yH B . 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 Rh 2 B and RhB 2 structure relative to the (Rh + α B) within the given pressure range. From Fig. 2, it can be clearly seen that the Rh 2 B-I has the lowest negative formation enthalpies at ambient pressure, and the enthalpy of Rh 2 B-I is much lower than that Rh 2 B in experiment by ∼ 0.16 eV per formula. This indicates that Rh 2 B-I is thermodynamically stable and can be synthesized in experiments. When the pressure is higher than 39 GPa, the phase transition from P2 1 /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 Rh 2 B. The experimentally observed phase is a metastable phase with impurities. This can explain no presence of our predicted phases of Rh 2 B in experiments. For RhB 2 , the predicted RhB 2 -II is the most thermodynamically stable phase in our calculations, no further phase transition was observed in the high pressure range.
To check the dynamical stabilities of the currently predicted Rh 2 B-I and RhB 2 -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 Rh 2 B-I and RhB 2 -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.
Elastic constants are essential for understanding the mechanical properties of a crystal. We calculated the zero-pressure elastic constants C ij of the two phases and the elastic constants C ij are listed in Table 1. For a stable structure, C ij has to satisfy Born-Huangcriteria 27 : For a monoclinic crystal, the independent elastic stiffness tensor consists of thirteen components of C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 , C 23 , C 15 , C 25 , C 35 , and C 46 . The mechanical stability criteria is given by:   11 ] } 0 15  As summarized in Table 1, the Rh 2 B-I and RhB 2 -II phases satisfy all mechanical stability criteria, indicating that they are mechanical stable at ambient pressure. The calculated C 33 values are bigger than that of C 11 and C 22 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 Rh 2 B-I and RhB 2 -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 stiff 28 . The ratio value of B/G is commonly used to describe the ductility or brittleness of materials, with 1.75 as 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 Rh 2 B-I. Indicating there is a strong covalent interaction between the B and Rh atoms in RhB 2 -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.
To gain more detailed information about the bonding character, we calculated the electronic localization function (ELF) of Rh 2 B-I , RhB 2 -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 bonds 31 . 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 Rh 2 B-I. This is consistent with the analysis of their DOS. The system of covalent bonds in the RhB 2 -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 RhB 2 -II. In Fig. 5(c) there are strong bonds between Rh and B in C2/m phase.

Conclusions
In summary, we have predicted a new phase Rh 2 B-I at ambient pressure for Rh 2 B and a new phase RhB 2 -II for RhB 2 through the ab initio calculation. The two new phases all belong to monoclinic P2 1 /m structure. Besides, both of the two phases are dynamically and mechanically stable at ambient pressure. Rh 2 B-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 P2 1 /m and C2/m phases for Rh 2 B.   For RhB 2 , RhB 2 -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 Rh 2 B and RhB 2 were performed at ambient pressure as implemented in the USPEX code [32][33][34] . The structure relaxation was performed by using the density functional theory implemented in the Vienna ab initio simulation package (VASP) code [35][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 code 39 and the bulk modulus, and shear modulus were thus estimated by using the Voigt-Reuss-Hill approximation 40 . The phonon calculations were carried out by using a supercell approach as implemented in the PHONON code 41 . The details of convergence tests have been described elsewhere [42][43][44][45] .