⇦Chinese Physics Letters, 2018, Vol. 35, No. 1, Article code 016401 Ground State Structures of Boron-Rich Rhodium Boride: An Ab Initio Study * Bin-Hua Chu(初斌华)1**, Yuan Zhao(赵元)1, Jin-Liang Yan(闫金良)1, Da Li(李达)2 Affiliations 1School of Physics and Opto-Electronic Engineering, Ludong University, Yantai 264025 2State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012 Received 18 September 2017 *Supported by the Natural Science Foundation of Shandong Province under Grant Nos ZR2016AP02, ZR2016FM38 and ZR2016EMP01, the Innovation Project of Ludong University under Grant No LB2016013, the Open Project of State Key Laboratory of Superhard Materials of Jilin University under Grant No 201605, and the National Natural Science Foundation of China under Grant Nos 11704170 and 61705097.
**Corresponding author. Email: chubinhua0125@126.com Citation Text: Chu B H, Zhao Y, Yan J L and Li D 2018 Chin. Phys. Lett. 35 016401 Abstract A new phase of RhB$_{4}$ is predicted based on first-principles calculations. The new phase belongs to the orthorhombic $Pnnm$ space group, named as $o$-RhB$_{4}$, and analysis of the calculated enthalpy shows that $o$-RhB$_{4}$ belongs to the orthorhombic $Pnnm$ space group. The calculated phonon band structure shows that the orthorhombic $Pnnm$ $o$-RhB$_{4}$ structure is stable at ambient pressure. We expect that the phase transition can be further confirmed by experiments. DOI:10.1088/0256-307X/35/1/016401 PACS:64.60.Ej, 64.60.Bd, 64.60.-i © 2018 Chinese Physics Society Article Text Superhard materials ($H_{\rm v}\ge40$ GPa) are widely used as abrasives, coatings' cutting and polishing tools, coatings and abrasives due to their superior mechanical properties.[1] Previously, it was generally accepted that the superhard materials are those strongly covalent bonded compounds formed by light elements. Unfortunately, almost all these materials (diamond,[2] cubic BN,[3] and BC$_{2}$N,[4] etc.) are expensive because they either occur naturally in limited supplies or have to be made at high pressure synthetically.[5] Therefore, great efforts have been devoted to exploring new types of hard and ultra-incompressible materials. Transition metals were previously known to form borides with high boron content. 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,[6-10] such as ReB$_{2}$,[11] WB$_{4}$,[12] RuO$_{2}$.[13] Therefore, these pioneering studies open up a novel route for the search of new superhard materials. The borides of rhodium are well known for their relatively high melting temperature, hardness, and brittleness. Studies have identified two stoichiometric compositions: RhB (hexagonal structure of the anti-NiAs type, No. 140, $I4/mcm$),[14] Rh$_{2}$B (orthorhombic structure, No. 62, $Pnma$).[15] Recently, bulk RhB$_{1.1}$ was synthesized and the measured Vickers hardness was 7–22.6 GPa, depending on the loads ranging from 0.49 to 9.81 N.[16] Later, RhB$_{1.1}$ films (film thickness of 1.0 μm) were deposited on SiO$_{2}$ substrates, and the films were considered to be superhard because of their 44 GPa Vickers hardness.[17] It is of fundamental interest to explore other compositions beyond these two in the structure-rich family of the Rh-B system. There is a strong need to clarify compounds with higher boron contents. In this Letter, we extensively investigate the ground-state structure of RhB$_{4}$ by the ab initio particle swarm optimization (PSO) algorithm on crystal structural prediction. This method has been successfully applied to many other systems. First principles calculations are performed to investigate the total energy, lattice parameters, bulk modulus, hardness, and density of states for the novel phases. In addition, other borides with various stoichiometries are also studied for comparison. The evolutionary variable-cell simulations for RhB$_{4}$ were performed at different pressures with systems containing one to four formula units (f.u.) in the simulation cell using the USPEX code.[18-20] The ab initio calculations were performed using the density functional theory within the generalized gradient approximation (GGA),[21] as implemented in the Vienna ab initio simulation package (VASP).[22-24] The energy cutoff 450 eV and appropriate Monkhorst–Pack $k$ meshes were chosen to ensure that enthalpy calculations are well converged to better than 1 meV/f.u. For hexagonal structures, the ${\it \Gamma}$-centered grids were used. Formation enthalpy was calculated from $\Delta H=E({\rm RhB_{4}})-E({\rm Rh})-4E({\rm B})$, where $E$ represents one formula unit (f.u.) total energy of each solid phase. The solid phase of boron is obtained from its $\alpha$ phase. Elastic constants were calculated by the strain–stress method. Elastic constants were obtained from evaluations of the stress tensor generated by small strains using the density-functional plane wave technique as implemented in the CASTEP code[25] and from the calculated elastic constants $C_{ij}$, and the bulk modulus $B$ and shear modulus $G$ were further estimated using the Voigt–Reuss–Hill (VRH) approximation.[26] The phonon calculations were carried out using a supercell approach as implemented in the PHONON code.[27] In the present work, for RhB$_{4}$, the predicted $o$-RhB$_{4}$ is the most thermodynamically stable phase among the considered structures. Here $o$-RhB$_{4}$ belongs to the orthorhombic $Pnnm$ space group containing four RhB$_{4}$ f.u. in a unit cell ($a=10.618$ Å, $b=5.426$ Å, and $c=2.876$ Å), in which five inequivalent atoms B$_{1}$, B$_{2}$, B$_{3}$, B$_{4}$ and Rh occupy the Wyckoff 4g (0.362, 0.406, 0.000), 4g (0.486, 0.668, 0.000), 4g (0.064, 0.395, 0.000), 4g(0.082, 0.725, 0.000) sites, and 4g (0.290, 0.779, 0.000), respectively.
cpl-35-1-016401-fig1.png
Fig. 1. Crystal structure of $o$-RhB$_{4}$.
cpl-35-1-016401-fig2.png
Fig. 2. Formation enthalpies versus pressure for different rhodium boride structures.
To explore the thermodynamic stability of $o$-RhB$_{4}$ for future experimental synthesis, the formation enthalpy of the predicted $o$-RhB$_{4}$ structure with respect to the separate phases is examined by following the reaction route $\Delta H=E({\rm RhB_{4}})-E({\rm Rh})-4E({\rm B})$. The Rh (space group: $Fm\bar{3}m$) and $\alpha$-boron (space group: $R\bar{3}m$)[28] were chosen as the referenced phases. In addition, the WB$_{4}$-type,[29] CrB$_{4}$-type[30] and OsB$_{4}$-type[31] structures were also considered. Figure 2 presents the enthalpy curves of RhB$_{4}$ structure relative to the (Rh+4B) within the given pressure range. From Fig. 2, it can be clearly seen that WB$_{4}$-type, CrB$_{4}$-type and OsB$_{4}$-type structures have positive formation enthalpies, indicating their thermodynamical instabilities at 0 GPa. The $o$-RhB$_{4}$ has the lowest negative formation enthalpy, and the formation enthalpy of predicted $o$-RhB$_{4}$ is much lower than that of the reacting substance (Rh+4B) by 0.36 eV per formula at ambient pressure, indicating that it can be directly synthesized by elementals Rh and B. With the increase of pressure, the enthalpy of $o$-RhB$_{4}$ is always the lowest among all the candidate structures. At zero temperature, a stable crystalline structure requires all phonon frequencies to be positive. Therefore, we performed the full phonon dispersion calculation for $o$-RhB$_{4}$ at 0 GPa. As shown in Fig. 3, no imaginary phonon frequency was detected in the whole Brillouin zone, indicating the dynamical stability of the $o$-RhB$_{4}$ structures.
cpl-35-1-016401-fig3.png
Fig. 3. Phonon dispersion curves at 0 GPa for $o$-RhB$_{4}$ structure.
To check the elastic stability of $o$-RhB$_{4}$, we calculated the elastic constants by the strain–stress method at zero pressure. Table 1 lists the calculated elastic constants of $o$-RhB$_{4}$. The theoretical values and available experimental data of other rhodium borides, OsB$_{4}$, MnB$_{4}$, MoB$_{4}$ and CrB$_{4}$ are all listed in this Table. For a given crystal structure, it should also satisfy the elastic stability criteria if it is stable. The elastic stability criteria for orthorhombic crystal can be displayed as follows: $$\begin{align} &C_{11}> 0,~C_{22}>0,~C_{33}>0,\\ &C_{44}>0,~C_{55}>0,~C_{66}>0,\\ &[C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23})]>0,\\ &(C_{11}+C_{22}-2C_{12})>0,\\ &(C_{11 }+C_{33}-2C_{13})>0,\\ &(C_{22}+C_{33}-2C_{23})>0. \end{align} $$ As listed in Table 1, it is clear that $o$-RhB$_{4}$ satisfies these conditions, confirming that it is mechanically stable under ambient conditions. Thus $o$-RhB$_{4}$ is both mechanically and dynamically stable at ambient pressure. The bulk modulus and the shear moduli in this table are calculated by the Voigt–Reuss–Hill approximation method.[32] In addition, Young's modulus $Y$ and Poisson's ratio $\upsilon$ are obtained by $E=(9BG)/(3B+G)$ and $\upsilon=(3B-2G)/(6B+2G)$. As seen in Table 1, the calculated $C_{22}$ for $o$-RhB$_{4}$ is larger than $C_{11}$ and $C_{33}$, implying that the resistance to deformation along the $b$-direction is stronger than $a$-direction and $c$-direction. To further compare the incompressibility of rhodium borides, Fig. 4 shows the relationship between the volume and pressure. The experimental data of WB$_{4}$ are obtained by Romans and Krug.[33] We can obtain that the incompressibility of $o$-RhB$_{4}$ is similar to that of RhB and WB$_{4}$. However, the shear modulus is a better indicator of potential hardness than bulk modulus. The calculated shear moduli of $o$-RhB$_{4}$ and the theoretical values of RhB, OsB$_{4}$, MoB$_{4}$, MnB$_{4}$ and CrB$_{4}$ are 146, 87, 218, 210, 241 and 260 GPa, respectively.[14,30,31,34] This suggests that $o$-RhB$_{4}$ may be a potential superhard material. In addition, The value of Poisson's ratio ($\upsilon$) is indicative of the degree of directionality of the covalent bonding. The typical $\upsilon$ values are 0.1 for covalent materials and 0.33 for metallic materials, respectively.[35] The directionality of covalent bonding plays an important role in the hardness of materials. From Table 1, it can be seen that all $\upsilon$ is below 0.33 and $o$-RhB$_{4}$ has a similar Poisson's ratio to that of OsB$_{4}$. Small Poisson's ratio indicates a high degree of covalent bonding, which contributes to the materials' hardness. All of these excellent mechanical properties strongly suggest that $o$-RhB$_{4}$ is a potential candidate for superhard materials.
cpl-35-1-016401-fig4.png
Fig. 4. The pressure dependence of unit cell volume for $o$-RhB$_{4}$.
Table 1. Calculated elastic constants (GPa), bulk modulus $B$ (GPa), shear modulus $G$ (GPa), Young's modulus ${\it \Upsilon}$ (GPa), Poisson's ratio $\upsilon$, and $B/G$ ratios of $o$-RhB$_{4}$, RhB, OsB$_{4}$, MoB$_{4}$, MnB$_{4}$ and CrB$_{4}$.
Structure $C_{11}$ $C_{22}$ $C_{33}$ $C_{44}$ $C_{55}$ $C_{66}$ $C_{12}$ $C_{13}$ $C_{23}$ $B$ $G$ $Y$ $\upsilon $ $B$/$G$
$o$-RhB$_{4}$ This work 497 571 530 242 238 210 139 130 97 244 146 365 0.25 1.67
RhB Theory[14] 438 342 172 223 256 310 87 238 0.37 3.56
OsB$_{4}$ Theory[31] 612 576 630 152 349 178 128 245 294 218 524 0.204 1.35
MoB$_{4}$ Theory[34] 505 936 189 141 103 285 210 506 0.2 1.36
MnB$_{4}$ Theory[30] 277 241 560 0.161 1.15
CrB$_{4}$ Theory[30] 275 260 593 0.141 1.06
cpl-35-1-016401-fig5.png
Fig. 5. Total and partial densities of states for $o$-RhB$_{4}$.
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 $o$-RhB$_{4}$ phases at 0 GPa is shown in Fig. 5. In Fig. 5, there is a deep valley at about $-$13 eV. It is a pseudogap of DOS, which is the borderline between the bonding and antibonding states. Rhodium and boron atoms form strong covalent bonds as displayed by the noticeable overlap of rhodium's $d$ electron and boron's $s$ electron, boron's $p$ electron curves in $o$-RhB$_{4}$. It is indicated that there is a strong covalent interaction between the B and Rh atoms in $o$-RhB$_{4}$. The dotted line is the Fermi level. It can be seen that $o$-RhB$_{4}$ is metallic because of the finite value of the DOS at the Fermi level, which originates mostly from $p$ electrons of boron and $d$ electrons of rhodium. In summary, structure searches based on first principles calculation allow us to identify a novel boron-rich $o$-RhB$_{4}$ with orthorhombic $Pnnm$ space group. Phonon dispersions and elastic constant analysis suggest that $o$-RhB$_{4}$ is stable at ambient condition. The calculated PDOS results demonstrate a strong hybridization between Rh-$d$ and B-$p$ electrons. References Synthesis and Design of Superhard MaterialsProperties of diamond under hydrostatic pressures up to 140 GPaSuperhard hexagonal transition metal and its carbide and nitride: Os, OsC, and OsNMechanical properties of cubic BC2N, a new superhard phaseAdvancements in the Search for Superhard Ultra-Incompressible Metal BoridesMechanical and electronic properties of diborides of transition 3d–5d metals from first principles: Toward search of novel ultra-incompressible and superhard materialsOsmium Diboride, An Ultra-Incompressible, Hard MaterialLow-compressibility and hard materials Re B 2 and W B 2 : Prediction from first-principles studyCorrelation between hardness and elastic moduli of the ultraincompressible transition metal diborides RuB2, OsB2, and ReB2Semiconducting Superhard Ruthenium MonocarbideSynthesis of Ultra-Incompressible Superhard Rhenium Diboride at Ambient PressureStability and Strength of Transition-Metal Tetraborides and TriboridesElastic properties of potential superhard phases of RuO 2 Novel High-Pressure Phase of RhB: First-Principles CalculationsThe crystal structure of Rh 2 BNew Hard and Superhard Materials: RhB 1.1 and IrB 1.35Superhard Properties of Rhodium and Iridium Boride FilmsCrystal structure prediction using ab initio evolutionary techniques: Principles and applicationsHow Evolutionary Crystal Structure Prediction Works—and WhyNew developments in evolutionary structure prediction algorithm USPEXAccurate and simple analytic representation of the electron-gas correlation energyAb initio molecular dynamics for liquid metalsNorm-conserving and ultrasoft pseudopotentials for first-row and transition elementsEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setFirst-principles simulation: ideas, illustrations and the CASTEP codeThe Elastic Behaviour of a Crystalline AggregateIonic high-pressure form of elemental boronOrigin of hardness in WB4 and its implications for ReB4, TaB4, MoB4, TcB4, and OsB4Unusual rigidity and ideal strength of CrB 4 and MnB 4Ultra-incompressible Orthorhombic Phase of Osmium Tetraboride (OsB 4 ) Predicted from First PrinciplesCrystal structures and elastic properties of superhard Ir N 2 and Ir N 3 from first principlesComposition and crystallographic data for the highest boride of tungstenStructural Modifications and Mechanical Properties of Molybdenum Borides from First PrinciplesSynthesis and Design of Superhard Materials
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