Pressure-Induced Structural Stabilities and Superconductivity in Rhodium Borides

Abstract

Transition metal borides have garnered significant research interest due to their versatile properties, including superconductivity and exceptional hardness. This study examines the stable crystal structures of Rhodium-Boron (Rh-B) compounds under high pressure using first-principles structural searching. Beyond the previously known Rh₂B, RhB₂, and RhB₄ phases, three new boron-rich phases—C2/m-RhB₆, Amm2-RhB₆, and Cmca-RhB₈—are identified, each characterized by three-dimensional covalent bonding networks. Their mechanical and thermodynamic stability is validated through elastic property assessments and phonon dispersion calculations. Surprisingly, these phases exhibit low bulk and shear moduli, ruling them out as candidates for hard materials. The metallic character of these borides is evident from their electronic density of states, which exhibits a sharp peak at the E_F-a signature often associated with superconducting systems. Indeed, our calculations predict T_c values of 8.93 K and 9.36 K for Amm2-RhB₆ and Cmca-RhB₈, respectively, at 100 GPa.

1. Introduction

Transition metal borides have attracted a lot of attentions for their facile synthesis at ambient pressure, superior mechanical properties, and good electrical conductivity. Transition metals have rich valence electrons and easily combine with Boron atoms to promote realization of new materials with remarkable properties under extreme conditions. The existing B-B directionally strong covalent bonding patterns and high valence electron density (VED) strengthen the hardness and thermal stability of transition metal borides. Especially, transition metal borides with rich boron content, such as ReB₂ [1], OsB₂ [2], RuB₂ [3], YB₄ [4], MnB₄ [5,6], WB₄ [7], CrB₄ [8,9], and others, can be experimentally synthesized at low pressure. This reduces synthesis costs and facilitates their application in various fields. In these borides, MgB₂ [10,11] has been discovered to have the highest superconducting critical temperature (T_c) (~39 K) at ambient pressure. Low compressibility is found in OsB₂, MnB₄, and WB₄ [12,13,14] due to their exceptionally high shear modulus. ReB₂ and WB₄ have high hardnesses up to 48 GPa and 46.2 GPa, respectively [15,16]. For these reasons, many scientists have dedicated themselves to the research of transition metal borides, aiming to discover new borides with outstanding properties. Consequently, exploring novel superhard transition metal boride materials remains essential.

The Rhodium-Boron (Rh-B) system is an intriguing area of study within the field of high-pressure material science. Rhodium (Rh) is a rare, highly corrosion-resistant transition metal, while boron (B) is a metalloid with exceptional hardness and a high melting point. The combination of Rh and B can potentially lead to materials with unique physical and chemical properties suitable for advanced technological applications. Recent developments in vacuum-based electron beam techniques have yielded phase-pure hexagonal RhB1.1, providing new opportunities for studying its fundamental properties. It can be characterized by X-ray diffraction and micro-indentation techniques. Hardness values exhibit a pronounced load dependence, increasing from 7 GPa (0.49 N) to 22.6 GPa (9.81 N) in Vickers tests [17]. Additionally, a 1.0 μm thick RhB1.1 film with a high hardness of 44 GPa [18] has been found, which means it can be considered as a potential superhard material. RhB initially crystallizes in the anti-NiAs structure at atmospheric pressure, transforming into the FeB-type phase above 22 GPa [19]. The stable phases of RhB₂ and Rh₂B, as low compressible materials, are also established by theoretical calculation under pressure [20]. Boron-rich RhB₄ has been investigated with the orthorhombic Pnnm phase. However, there is no phase transition observed in RhB₄ up to 100 GPa [21]. In addition, the mechanical and electronic properties of Rh₅B₄ under pressure have also been studied [22].

As we know, the content of boron in transition metal borides is very important to their crystal structures and physical properties under pressure. The pressure could change the bonding characterization of atoms and stimulate the formation of unusual stoichiometric compounds. Still, there is scant knowledge on Rhodium borides with richer B content, although some borides (such as ReB₂ [1], WB₄ [2], and so on) have been well studied. The rich B atoms of stable phases easily form a complex three-dimensional network, which benefits their unique physical properties. So far, the high-pressure phase stabilities of B-rich Rh-B compounds have not been discussed. Therefore, it is very important to investigate the stability of B-rich Rh-B system under high pressure.

To find stable phases of Rh-B compounds under pressure, the phase stabilities and electronic and mechanical properties of Rh_xB_y system (x = 2, y = 1; x = 1, y = 1–8, 10, 12) under pressures of 0–200 GPa were studied. The exploration of crystal structures was based on the particle swarm optimization algorithm combined with first-principles calculations. In Rh-B compounds, the structures of RhB (P6₃/mmc phase) and Rh₂B (Pnna phase) synthesized in experiments [23,24] were well reproduced by our calculations. However, in our calculations, new stable stoichiometries of RhB₆ and RhB₈ were found under pressure. The detailed structure features, electronic properties, and mechanical properties of stable phases in Rh-B systems were investigated under pressure.

2. Computational Methods

In this work, structure searches for Rh-B system were employed at 0 GPa, 100 GPa, and 200 GPa by the CALYPSO code based on the particle swarm optimization methodology [25,26]. The accuracy and efficiency of CALYPSO (version 7.0) have been validated through successful structure predictions for a wide range of compounds [27,28,29]. The computational investigations were carried out employing the VASP software package (version 5.4.4.18Apr17) [30], where structural optimizations and energy calculations were performed within the GGA-PBE exchange-correlation function [31]. A plane-wave cutoff energy of 650 eV was employed to ensure convergence of the electronic structure. The 4d⁸5s¹ and 2s²2p¹ were treated as the valence electrons for Rh and B, respectively. A convergence threshold of energy of 10⁻⁵ eV/atom and force of 10⁻³ eV/Å were set in the optimization process. Monkhorst-Pack k-point meshes with grid spacings of 0.04 Å⁻¹ and 0.02 Å⁻¹ were employed for structural optimization and self-consistent energy calculations, respectively. The dynamical stability was evaluated through phonon calculations employing the supercell-based finite displacement method as implemented in the Phonopy package (version 2.41.0) [32]. The elastic constants were determined using the strain-energy method, which involves applying small deformations to the equilibrium lattice and fitting the resulting energy-strain relationship. Based on the Voigt-Reuss-Hill approximation, we derived the key mechanical parameters: bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio [33]. The Vickers hardness (H_v) was calculated by employing Chen et al.’s empirical model [34] based on the computed elastic moduli.

The phonon-mediated superconducting properties were computed within the framework of density-functional perturbation theory (DFPT) using a plane-wave pseudopotential approach, as implemented in the Quantum ESPRESSO package (version 7.0) [35]. In the calculations, a kinetic cutoff energy of 90 Ry was used by a series of tests, which established the convergency of energy. A Q-mesh of 2 × 4 × 2 and k-mesh of 8 × 16 × 8 for P2₁/m-Rh₂B, q-mesh of 4 × 4 × 4 and k-mesh of 16 × 16 × 16 for C2/m-Rh₂B and P6₃/mmc-RhB, q-mesh of 4 × 4 × 2 and k-mesh of 16 × 16 × 8 for C2/m and Amm2-RhB₆, and k-mesh of 16 × 16 × 16 for C2/m-Rh₂B and P6₃/mmc-RhB were used in the first Brillouin zone (BZ) for the electron–phonon matrix calculations. The superconducting temperature T_c of stable phases of Rh-B compounds was calculated via the Allen–Dynes-modified McMillan equation [36].

3. Results and Discussion

To find stable structures of Rh-B compounds under pressure, the stoichiometries of the Rh_xB_y system (x = 2, y = 1; x = 1, y = 1–8, 10, 12) with cells of 1–4 formula units (f.u.) in size are fully explored up to 200 GPa by CALYPSO code [25,26]. In order to study the thermodynamical stability of the Rh-B system, the formation enthalpies were calculated as

$$∆H=[H\left({Rh}_x{B}_y\right)-xH\left(Rh\right)-yH\left(B\right)]$$

Here, H (Rh_xB_y), H (Rh), and H (B) are the enthalpies of Rh_xB_y, Rh, and B, respectively. The cubic phase of Rh [37] and the hexagonal α-B [38] are established as the referential phases. Based on the definition of formation enthalpies of Rh-B compounds, the phase stabilities of various stoichiometries in the Rh-B system are displayed in Figure 1a, in which the convex hull is constructed relative to solid Rh and B. The dots on the solid line indicate thermodynamic stability, while the hollow symbols above the convex hull represent metastable structures in the Rh-B system. The most stable structure corresponds to the lowest formation enthalpy against decomposing into Rh and B at the given pressure. As showed Figure 1a, the formation enthalpies for Rh-B compounds are negative except RhB₆ and RhB₈ at 0 GPa. Rh₂B and RhB were located on the solid line of the convex hull, which indicated their thermodynamic stability. With increasing pressure up to 100 GPa, RhB unexpectedly became unstable and disappeared from the solid line. Surprisingly, thermodynamic stability was found for two new compounds, RhB₆ and RhB₈, which indicated that RhB₆ and RhB₈ could be experimentally synthesized by metal Rh and B. As pressure continued to increase up to 200 GPa, the compounds remained stable. In order to establish the phase transition, Figure 1b exhibits the pressure ranges of the stable compounds.

For Rh₂B, pressure-dependent phase transitions from the P2₁/m to C2/m phase were observed at 29 GPa (Figure 2a), which is consistent with Ref. [19]. RhB crystalizes in the hexagonal P6₃/mmc phase (Figure 3a) and keeps stable below 19 GPa (Figure 2b). The predicted new monoclinic C2/m-RhB₆ could be stable above 36 GPa (Figure 2c), which could be experimentally synthesized by Rh₂B and element B. Under high pressure, the monoclinic phase C2/m transforms into the orthorhombic Amm2 phase at 95 GPa. The C2/m phase structures feature four molecules per unit cell, including the RhB₁₀ structural unit. From Figure 3b, the B atoms in the RhB₁₀ structural unit of the C2/m phase are not evenly distributed around the Rh atom. Under high pressure, the number of boron atoms surrounding the Rh atom increases from 10 to 12. The orthorhombic Amm2 phase consists of RhB₁₂ polyhedra, where each Rh atom is connected to 12 neighboring B atoms (Figure 3c). Notably, it exhibits an interesting RhB₁₂-B-RhB₁₂ sandwich stacking order. Along the a-axis, the crystal structure exhibits alternating B hexagonal planes interconnected through RhB₁₂ hexagonal columns, forming a three-dimensional framework. So far, the metal hexa-borides have exhibited a wide range of unusual physical properties, which mainly contributed to their crystal phases [39,40,41]. The new predicted stable crystal phases of RhB₆ are the same at those of ZrB₆ under pressure, for which their B atom arrangement is also similar. For RhB₈, it stabilizes as the orthorhombic Cmca phase above 93 GPa, which has no phase transition under higher pressure (Figure 2d). Figure 3e shows the Cmca phase RhB₁₂ polyhedral and B parallel hexagonal planes similar to Amm2-RhB₆. However, there are two-layer zigzag B₄ chains between two RhB₁₂ hexagonal columns. The structural information of the new predicted phases is described in

Table 1. The calculated optimized equilibrium lattice parameters a, b, and c and the atom position of stable phases of the Rh-B system at different pressures.

	P (GPa)	Lattice Constants (Å)	x	y	z	Site
P21/m-Rh2B	0	a = 5.587	0.926	0.250	0.231	Rh (2e)
		b = 2.813	0.563	0.750	0.770	Rh (2e)
		c = 4.684	0.776	0.250	0.604	B (2e)
C2/m-Rh2B	50	a = 8.429	0.421	0	0.095	Rh (4i)
		b = 2.722	0.287	0	0.360	Rh (4i)
		c = 7.073	0.123	0	0.652	B (4i)
P63/mmc-RhB	0	a = b = 3.378	0.667	0.333	0.75	Rh (2a)
		c = 4.140	0	0	0	B (2c)
C2/m-RhB6	0	a = 14.01	0.191	0.5	0.152	Rh (4i)
		b = 2.777	0.288	0.5	0.514	B (4i)
		c = 4.821	0.919	0.5	0.177	B (4i)
			0.962	0.5	0.836	B (4i)
			0.359	0.5	0.262	B (4i)
			0.098	0	0.348	B (4i)
			0.982	0	0.326	B (4i)
Amm2-RhB6	150	a = 5.688	0	0.5	0.408	Rh (2a)
		b = 2.567	0.261	0.5	0.071	B (4c)
		c = 5.245	0.252	0	0.258	B (4c)
			0.5	0	0.103	B (2b)
			0.5	0	0.408	B (2b)
Cmca-RhB8	100	a = 5.288	0.5	0	0.5	Rh (4a)
		b = 13.271	0.5	0	0.007	B (16g)
		c = 2.908	0.75	0.711	0.25	B (8e)
			0.5	0.218	0.948	B (8f)

.

All positive phonon frequencies indicate the stability of the crystals. Phonon dispersion calculations performed via the finite displacement method confirm the dynamical stabilities of predicted RhB₆ and RhB₈. No imaginary modes were found in the phonon frequencies of RhB₆ and RhB₈ within their stability pressure ranges (Figure 4). Notably, orthorhombic RhB₈ exhibited no imaginary phonon modes in the BZ at ambient pressure, suggesting its potential recoverability under ambient conditions.

Elastic parameters provide fundamental insights into key mechanical properties, including phonon dispersion relations, bonding characteristics, and mechanical stability criteria. Meanwhile, they provide valuable information for estimating material hardness. The elastic parameters have been listed in

Table 2. The elastic constants C_ij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), and hardness H_V (GPa); Poisson’s ratio ν of stable Rh-B compounds at zero pressure; and other theoretical data.

Structures		C11	C22	C33	C44	C55	C66	C12	C13	C23	B	G	E	ν	HV
P21/m-Rh2B	this work	483	348	363	81	75	57	158	154	189	243	84	227	0.344	4.7
P21/m-Rh2B	Ref. [17]	339	350	527	56	57	73	187	143	132	238	87	232	0.337
C2/m-Rh2B	this work	506	328	369	64	53	50	148	138	198	239	74	202	0.360	3.3
P63/mmc-RhB	this work	439		303	155			204	247		283	106	282	0.334	7.6
	Ref. [19]	438		342	172			223	256		296	102
C2/m-RhB6	this work	467	448	360	123	34	145	128	181	94	288	143	354	0.242	13.6
Amm2-RhB6	this work	487	330	501	151	65	130
Cmca-RhB8	this work	392	472	515	15	82	116	190	133	159	258	56	156	0.399	2.3
P63/mmc-ReB2	Ref. [43]	643		1035	263			159	129		344	364	642	0.21
Amm2-ZrB6	Ref. [44]	683	698	623	215	295	125	65	94	101	280	230	543	0.178	40.1
Cmcm-ZrB6	Ref. [44]	660	669	601	220	283	139	66	86	94	269	231	540	0.165	24.6

. Moreover, as seen from Table 2, all four phases of RhB compounds are stable, except for Amm2-RhB₆ at ambient pressure, according to mechanical stability criteria [42].

From Table 2, the largest values of C₁₁ in P2₁/m-, C2/m-Rh₂B, P6₃/mmc-RhB, C2/m-, Amm2-RhB₆, and Cmca-RhB₈ reveal their higher linear incompressibility along the a-axis than that along b- and c-axes. Amm2-RhB₆ and Cmca-RhB₈ have the largest C₃₃ values, indicating their low compressibility along the c-axis. Meanwhile, the estimated bulk modulus B, shear modulus G, and Young’s modulus E are also consistent with other calculated data [17,19,42,43]. In comparison to ReB₂ and ZrB₆, the stable Rh-B compounds have lower values of C₁₁, C₂₂, C₃₃, and shear modulus, which maybe play important roles in their compressibility. Hardness is another important parameter representing the mechanical properties of materials. The hardness of stable Rh-B compounds under ambient conditions has been successfully calculated and is listed in Table 2 using an empirical model recently proposed by Chen et al. [34]. Notably, the stable Rh-B phases display unexpectedly low hardness values compared to isostructural transition metal borides. The calculated low hardness of these stable phases aligns with their relatively small Young’s modulus and large Poisson’s ratio. A low Young’s modulus indicates a weak ability to resist tension and pressure within the range of elastic deformation, while the large Poisson’s ratio suggests weak directional covalent bonding, which reduces the material’s hardness.

In order to investigate the electronic properties of stable Rh-B compounds, their partial densities of states (PDOSs) are investigated under pressure. As shown in Figure 5, all thermodynamically stable Rh-B phases exhibit finite densities of states at the Fermi level (N(E_F)), confirming their metallic character. The density of states of Rh-d and B-2p occupied most of total density of states at the Fermi level. Moreover, the similar shapes of the density of states of these stable compounds were observed in the whole energy range, exhibiting hybridizations between Rh-3d and B-2p orbitals. Notably, stable Rh-B compounds lacked the characteristic deep pseudo gap valleys near the Fermi level, which were evidenced by their finite density of states across the entire energy range. Therefore, the relatively weak covalent bonding formed by Rh-3d and B-2p to other borides decreases their hardness. The decrease of the total DOS at the Fermi level for Amm2-RhB₆ and Cmca-RhB₈ is accompanied by the appearance of the pseudogap around the Fermi level, which is helpful to increase the stability of orthorhombic RhB₈. To further investigate the chemical bonding in RhB₆ and RhB₈, electron localization functions (ELFs) were calculated to help visualize B-B bonding. A high ELF (0.5–1) indicates that the electrons are highly localized and strong covalent bonds exist, while a low ELF (0–0.5) demonstrates that the electrons are highly delocalized and ionic bonds exist. Figure 6 displayed ELFs for the (100) plane of Amm2-RhB₆ and (001) plane of Cmca-RhB₈ at 150 GPa. Strong covalent bonding of B-B in Amm2-RhB₆ and appreciably weaker B-B bonding in Cmca-RhB₈ was observed. The strong covalent bonding helps to increase the stability of RhB₆ and RhB₈.

Based on their metallicity and electronic structural characteristics of stable Rh-B compounds under pressure, electron-coupling calculations were applied to investigate the superconductivity of Rh₂B, RhB, RhB₆, and RhB₈ at different pressures. It was applied to different compounds [45,46] to indicate the reasonability of the calculations.

Table 3. Predicted electron–phonon coupling and superconducting properties of stable Rh-B compounds at pressure. T_c is estimated by Allen–Dynes-modified McMillan equation.

	Pressure (GPa)	Nf (states/ev/f.u.)	ωlog (K)	λ	Tc (K)
P21/m-Rh2B	0	1.182	221.8	0.411	1.08
	50	0.971	291.2	0.335	0.37
C2/m-Rh2B	50	1.005	246.8	0.406	1.22
P63/mmc-RhB	0	0.189	281.8	0.251	0.03
C2/m-RhB6	50	0.619	506.7	0.408	2.39
Amm2-RhB6	100	0.662	468.7	0.566	8.93
	150	0.608	494.2	0.505	6.32
Cmca-RhB8	100	0.685	616.2	0.528	9.36
	150	0.639	698.5	0.473	6.92
	200	0.608	705.8	0.450	5.45

exhibited the relevant superconducting parameters the EPC parameters λ, logarithmic average frequency ω_log, the values of density of states at the Fermi level, and superconducting temperature T_c at different pressures. The results indicated that Rh₂B, RhB, RhB₆, and RhB₈ have quite low superconducting temperatures close to zero under high pressure due to the low logarithmic average frequency and electron–phonon coupling parameter. C2/m- and Amm2-RhB₆ at 100 GPa has 8.93 K and 9.36 K with EPC parameters of λ 0.505 and 0.566, respectively, in which λ plays an important role in the superconducting temperature in transition borides. Though RhB₆ has a similar layered structures to that of MgB₂, it has a low T_c in comparison to MgB₂ [47].

In order to the identify the effect of pressure on the superconducting temperature, the T_c of RhB₈ under pressure was calculated. The density of states of RhB₈ at the Fermi level decreased from 0.685 at 100 GPa to 0.608 at 200 GPa, and the logarithmic average frequency ω_log was 616.2 K at 100 GPa and 705.8 K at 200 GPa. With pressure increasing from 100 GPa to 20 GPa, the EPC parameter in RhB₈ decreased from 0.528 to 0.450, which led to the superconducting temperature changing from 9.36 K to 5.45 K. In order to investigate the superconducting mechanism of the Rh-B system, Figure 7 presents the Eliashberg EPC spectral function α²F(ω) and its integral of Amm2-RhB₆ at 100 and 150 GPa. It is seen that the phonon modes are separated into two parts: the acoustic phonon modes (low frequencies < 10 THz) dominated by vibrations of Rh atoms and the optical phonon modes (high frequencies) from the vibrations of B atoms. When the pressure increases from 100 GPa to 150 GPa, the electron–phonon coupling parameter decreases from 0.56 to 0.55, indicating the low influence of pressure on the superconducting transition temperature. The underlying mechanisms stimulate more comprehensive exploration for the superconducting properties of transition metal borides.

4. Conclusions

In conclusion, first-principles structural searching was extensively explored to obtain the stable crystal structures of Rh-B compounds under high pressure. The stable pressure ranges of the stable Rh-B phases were established. Based on the previously predicted Rh₂B, RhB₂, and RhB₄, three new phases of the boride-rich compounds RhB₆ and RhB₈ (C2/m-RhB₆, Amm2-RhB₆, and Cmca-RhB₆) were established under pressure for the first time. Three-dimensional covalent bonding networks were found in the boride-rich RhB₆ and RhB₈. Based on their elastic properties and the calculation of phonon dispersion, the C2/m-, Amm2-RhB₆, and Cmca-RhB₈ were found to have mechanical and thermodynamic stability. Surprisingly, the three stable phases of RhB₆ and RhB₈ had a low bulk and shear modulus, and the hardness was not regarded as indicating potential hard materials. Meanwhile, the electronic structures and densities of states of the stable phases of Rh-B under pressure were also investigated. From the shapes of the densities of states, no deep pseudo-gap valleys near the Fermi level were seen, similar to other high-hardness transition metal borides. Electronic structure calculations revealed substantial densities of states at E_F for the stable Rh-B phases, demonstrating their metallic character. Surprisingly, the phonon dispersion and electron–phonon coupling calculations suggest that Amm2-RhB₆ and Cmca-RhB₈ at 100 GPa have superconducting temperatures of 8.93 K and 9.36 K, respectively. For other stable Rh-B compounds, their superconducting temperatures are very low, close to zero. Moreover, the superconducting temperatures decrease with pressure for RhB₆ and RhB₈. The existence of superconducting transition metal borides will provide diversity to the family of superconductors. The establishment of phase regions of stable Rh-B compounds provides a synthesized route of transition for metal borides under certain extreme conditions. The findings will benefit from future theoretical and experimental research in the field.