Pressure-Induced Graphene-like Si Layer in Ductile RuSi2 with Ambient-Pressure Superconductivity and High Hardness

2. Computational Method

The structure searches of RuSi₂ materials under high pressure are based on the particle swarm optimization (PSO) algorithm as implemented in the crystal structure prediction CALYPSO code [54,55], which has successfully predicted the ambient-pressure and high-pressure structures of various systems [56,57,58,59]. In the pressure range of 0−300 GPa, we implemented the structure searches with unit cells including up to four formula units (f.u.). The structural geometrical optimization, electronic structure, Bader charge, electron localization functions (ELF) [60,61] and elastic constants calculations were performed through density functional theory in the Vienna ab initio simulation package (VASP) [62] within the Perdew−Burke−Ernzerhof (PBE) exchange correlation function of generalized gradient approximation (GGA) [63,64]. The Projector-Augmented Wave (PAW) [65] method was adopted to treat 4d⁷5s¹ and 3s²3p² as valence electrons for Ru and Si atoms, respectively. In addition, the cutoff energy of 800 eV for the plane-wave basis expansion of the electronic wave functions was implemented and appropriate Monkhorst−Pack k meshes for sampling in the Brillouin zone were set to ensure that all the enthalpy calculations were well converged to better than 1 meV/atom [66]. The crystal orbital Hamilton population (COHP) method was applied in LOBSTER code [67,68]. The phonon and electron-phonon coupling (EPC) matrix elements were calculated within density functional perturbation theory (DFPT) as implemented in the QUANTUM ESPRESSO package [69] with a kinetic cutoff energy of 80 Ry. In the first Brillouin zone, we have selected k-point meshes for Cmcm-RuSi₂ phase of 12 × 12 × 12 at 1 atm and 10 × 10 × 10 at 25, 50 and 75 GPa, respectively. The q-point meshes for Cmcm-RuSi₂ phase were set to 3 × 3 × 3 at 1 atm and 5 × 5 × 5 at 25, 50 and 75 GPa, respectively.

3. Results and Discussion

3.1. Structural Transition and Dynamical Stability

In the first instance, in order to determine the structural transition sequence and potential stable structures for RuSi₂ under high pressure, we carried out the CALYPSO code for thoroughly crystal structure searches. As a result, nine candidate low-energy structures with space groups of Cmca, Cmcm, P2/m, Immm, Cm, P3₂12, Imm2, C2 and P2₁/m were predicted at the pressure range of 0−300 GPa. The structural diagrams, together with detailed structural parameters of nine candidate phases under ambient pressure are summarized in Figure S1 and Table S1 of the Supplementary Information (SI). The previously known Cmca [49,70] and Immm [71] structures of transition metal disilicide RuSi₂ were successfully reproduced by our structure searches, demonstrating the reliability of the CALYPSO method. Subsequently, we constructed the enthalpy-pressure (H−P) curves of the low-energy phases and presented two possible decomposition routes (Ru + 2Si and RuSi + Si) with respect to Cmca phase, as shown in Figure 1. The phase stability of RuSi₂ compound is established at various pressures by judging the enthalpy difference relative to the solid constituent elements and stable compounds (i.e. Ru, Si solid phase, and RuSi compound). The corresponding known stable elements P6₃/mmc for Ru [72], Fd-3m, P6₃/mmc, and Fm-3m for Si [73], and compounds P2₁3 and Pm-3m for RuSi [44,45,74] under corresponding pressures were achieved from the previous literature. According to Figure 1, RuSi₂ adopts the orthorhombic β-FeSi₂ type configuration (space group: Cmca) as ground-state structure under ambient pressure, which is consistent with the previous experimental observations and theoretical researches [49,51,75]. As the pressure increases, the RuSi₂ is found to crystallize in another orthorhombic Cmcm phase, which is energetically favored over the Cmca-RuSi₂ above 50.97 GPa and maintains its stability up to 79.15 GPa. Noticeably, this Cmcm-RuSi₂ configuration is absolutely distinctive from previously reported metal disilicides MSi₂, such as hexagonal P6/mmm-type phase in MgSi₂ [35,36] and CaSi₂ [37,38], trigonal P-3m1 phase in RbSi₂ [34] and alkaline-earth metal disicides, trigonal I4₁/amd phase in LaSi₂ [40], rhombohedral R-3m phase in MgSi₂ [35,36], and orthorhombic Imma in MgSi₂ [35,36] and YSi₂ [41] Hence, this orthorhombic Cmcm phase is predicted to be a high-pressure ground-state structure in metal disilicides in the present work for the first time. Above 79.15 GPa, the RuSi₂ compound decomposes into RuSi + Si. Additionally, the considered pressure dependence of RuSi₂ decomposition reaction proves that RuSi₂ will not decompose into Ru + 2Si within the pressure range of 0−78.48 GPa (see Figure 1). Thus, compressing RuSi and Si is a potential synthetic route for RuSi₂ crystal. The above analysis demonstrates the thermodynamical stability of Cmca-RuSi₂ and Cmcm-RuSi₂ phases and provides their potential synthetic routes, which will stimulate the experimental synthesis. Subsequently, we focus on the structural, chemical bonding, electronic, superconducting and mechanical characters for the Cmca-RuSi₂ and Cmcm-RuSi₂ phases.

To probe into the dynamical stability of two thermodynamical stable structures under specific pressures, we have computed their phonon dispersion curves (see Figure S2 and Figure 7). It is evidently found that both the Cmca-RuSi₂ phase at atmospheric pressure and the Cmcm-RuSi₂ phase at the pressure range of 0−75 GPa are dynamically stable, as there is no imaginary frequency in the whole Brillouin zone. Although the Cmcm-RuSi₂ structure can be acquired under high pressure, it can be recovered to ambient pressure condition, thereby render it applicable in practical settings. Subsequently, we mainly focus on the the structural, chemical bonding, electronic, superconducting and mechanical features for the Cmca-RuSi₂ and Cmcm-RuSi₂ phases under atmospheric pressure.

3.2. Structural Character and Bonding Essence

The crystal structure diagrams for the Cmca-RuSi₂ phase at 1 atm and the Cmcm-RuSi₂ phase at 75 GPa were distinctly depicted in Figure 2 and homologous structural parameters were tabulated in Table S2. From Table S2, the lattice parameters (a = 10.170 Å, b = 8.108 Å, and c = 8.209 Å) of Cmca-RuSi₂ excellently align with previous results (a = 10.223 Å, b = 8.133 Å, and c = 8.250 Å) [49,51,75]. In this structure, there are two inequivalent Si atoms located at the Wyckoff sites 16g (0.372, -0.778, 0.057) and 16g (0.127, -0.551, 0.225), respectively, with Si-Si separations varying from 2.54 to 2.62 Å. All the Si atoms form a ladder-like Si-network framework with the planar quadrangle along the b axis and the angle between the nearest three Si atoms is nearly 90°. The shortest Si-Si distance (2.54 Å) is marginally longer than the those of Si (2.22 Å) atoms, d-Si (2.35 Å) [76], and allotrope Cmcm-Si₂₄ (2.33-2.41 Å) [76], but comparable to those of C49-WSi₂ [77] (2.56 Å) and B31-RhSi (2.53-2.69 Å) [78], expounding the contribution of Si-Si covalent bond to the structure construction. This result can be evidenced by the electron localization function (ELF) for the Cmca-RuSi₂ under ambient pressure, as displayed in Figure 3(a) and 3(b), which precisely describes the electron localization between Si atoms. Besides, each Ru atom is coordinated by eight Si atoms, forming a “RuSi₈” building block with Ru-Si separation range of 2.45-2.52 Å. The shortest Ru-Si distance (2.45 Å) is slightly longer than the sum of the covalent bond radii of Ru (1.25 Å) and Si (1.11 Å) atoms, indicating the covalent Ru-Si bond, which can be demonstrated by ELF in Figure 3a,b as well. For the Cmcm-RuSi₂ phase at 75 GPa, the lattice parameters are a = 6.693 Å, b = 4.052 Å and c = 3.866 Å, where Ru atoms occupying the Wyckoff sites 4c (-0.500, 0.307, 0.750) are neighbored by equivalent Si atoms locating at 8g (-0.832, 0.316, 0.750) sites. More charmingly, the Cmcm-RuSi₂ phase contains two interpenetrating honeycomb-like silicon-layer along the c-axis, which forms the silicon hexagonal ring with one metal atom Ru located at the center of hexagonal ring in the a-b plane, disclosing that metallic atom and nonmetallic silicon layer are in the same flat. This layered feature is distinct from previously reported planar structures, such as MgB₂ [21], MoB₂ [22], NaC₂ [23], YC₂ [24], CeP₂ [28] and LaP₂ [29], where the hexagonal-ring nonmetallic layer and metallic atom generate two alternate layers and are distributed in different flats. In silicon hexagonal-ring, the there are two kinds of Si-Si bond lengths, corresponding to 2.298 and 2.461 Å, while the Si-Si bond length between the silicon layers is 2.456 Å. These three Si-Si bond lengths are elongated into 2.417, 2.668 and 2.655 Å, respectively, when the pressure decreases to the ambient pressure. Additionally, ten silicon atoms are coordinated around one Ru atom, forming a distorted RuSi₁₀ dodecahedron with four different Ru-Si bond lengths of 2.347, 2.431, 2.435 and 2.440 Å at 75 GPa, compared with the elongated Ru-Si bond separations of 2.523, 2.615, 2.554 and 2.683 Å under atmospheric pressure. Both of Si-Si and Ru-Si bond distances provide an explanation of the Si-Si and Ru-Si covalent bonds in the Cmcm-RuSi₂ crystal at ambient pressure and high pressure, which can also be substantiated by the electron localization in the intermediate regions of the Si-Si and Ru-Si bonds, as depicted in Figure 3c–f. In short, based on the overall analyses of bond length and ELF, the Cmca-RuSi₂ and Cmcm-RuSi₂ phases under ambient pressures as well as the Cmcm-RuSi₂ phase under high pressure are characterized by a combination of the covalent Ru-Si and Si-Si bonds.

The formation of strong covalent Ru−Si and Si−Si interactions in these two stable configurations of RuSi₂ can be quantitatively disclosed through the calculated crystal orbital Hamilton population (COHP) curves and integral crystal orbital Hamilton population (ICOHP) at the Fermi level, as displayed in Figure 4. The calculated ICOHP values for Ru−Si and Si−Si pairs in Cmca-RuSi₂ under 1 atm are −1.84 and −2.41 eV for per pair, respectively. Moreover, the predicted ICOHP values for the Ru−Si bonds in Cmcm-RuSi₂ at 1 atm and 75 GPa is −1.66 and −1.68 eV for per pair, respectively, with the corresponding ICOHP values of the Si−Si pairs −2.25 and −2.23 eV for per pair. Afterward, the predicted ICOHP values for the Si−Si pairs (−2.41, −2.25 and −2.23 eV for per pair) in both RuSi₂ structures are comparable to those of covalent Si−Si pairs in Na₄Si₂₄ (−2.544 eV/pair) [76], P−P pairs in LaP₂ (−2.32 eV/pair) [29], P1−P2 pairs in Ce₂P₃ (−2.41 eV/pair) [28], B2−Β2 pairs in LiB₄ (−2.55 eV/pair) [79] and B2−Β2 pairs in NaB₄ (−2.55 eV/pair) [80], reconfirming strong covalent Si−Si bond in both configurations. From the consequences above, it can be conspicuously found summarized into two aspects. On the one hand, the strength of the covalent Si−Si bond is stronger than that of the Ru−Si covalence in both RuSi₂ configurations as the estimated ICOHP values for the Si−Si pairs are slightly lower than those of the Ru−Si pairs. On the other hand, compared with the layered Cmcm-RuSi₂ phase, Cmca-RuSi₂ possesses the stronger Ru−Si and Si−Si covalent bonds, which should be the main driving force that the Cmca-RuSi₂ phase have more magnificent incompressibility than the Cmcm-RuSi₂ structure.

3.3. Electronic Structure and Superconductive Property

To further delve into the electronic structures of the Cmca-RuSi₂ phase at 1 atm as well as the Cmcm-RuSi₂ phase at 1 atm and 75 GPa, we calculated their electronic band structure (BS) and density of states (DOS), as constructed in Figure 5. Note that the Cmca-RuSi₂ structure under ambient pressure displays a semiconductor behavior characterized by a direct band gap of 0.485 eV, in conformity with the preceding theoretical calculations reported by Zhang et al [49]. On the contrary, as the pressure increase, the conduction band and the valence band of Cmcm-RuSi₂ phase pass through the Fermi level (E_F) and overlap, implying its metallic character. In reality, a semiconductor-metal transformation or metallization occurs under high pressure, along with the structural transition from Cmca-RuSi₂ to Cmcm-RuSi₂. This phenomenon metallization induced by high pressure can also be appreciated in other systems, such as MgH₂ [81], CaLi₂ [82] and SiH₄ [83]. Moreover, the Cmcm-RuSi₂ phase still maintains the metallic characteristic when the pressure is released into the ambient pressure, as shown in Figure 5(b). On the other hand, the semi-conductive and metallic traits in Cmca-RuSi₂ and Cmcm-RuSi₂ can be elucidated in the right side of Figure 5a–c. On the basis of the total and partial density of states (TDOS and PDOS) of both RuSi₂ structures, it is obviously achieved that the Ru-4d state make the dominant contribution to TDOS. Apart form the Ru-4d state contribution, the Si-3p state also contributes to TDOS at a certain extent. The hybridization effect between the Ru-4d and Si-3p states near both sides of the E_F disclose the Ru-Si covalent bond in both RuSi₂ phases, which in excellent accord with the previous analyses of bond length, ELF, and COHP. Eventually, the chemical bonding in the Cmca-RuSi₂ and Cmcm-RuSi₂ configurations comprises a complex combination of the covalent Ru-Si and Si-Si bonds with the Si-Si bond stronger than that of the Ru-Si bonding. More noticeably, as presented in the left sides of Figure 3b,c, the electronic BS exhibits multiple characters in proximity to E_F, comprising of the hole pockets around T/S/U points, electron pockets around Z/S points and flat band along the Y-S-X direction. In view of the atypical BS, we further dig into the topological structure of Fermi surfaces (see Figure 6), which serve to unveil the electronic performance at the E_F. The emerge of three bands in Cmcm-RuSi₂ indicates excellent electrical conductivity and the third band should be attributed to the flat band along the Y-S-X, further secured by the decreased energy approaching the EF at the ambient pressure. These electronic traits proposed from the BS and Fermi surfaces have been comprehensively investigated and deemed as imperative signs for superconductivity.

Kindled by the metallic and significant band structure characters in Cmcm-RuSi₂ under ambient and high pressures coupled with the outstanding superconductivity in the layered framework in MgB₂ [21], MoB₂ [22], NaC₂ [23], KP₂ [27], CeP₂ [28], and LaP₂ [29] as well as alkali, alkaline-earth and rare-earth metal disilicides, analogous to the Cmcm-RuSi₂ phase, we assume that the layered Cmcm-RuSi₂ structure should be a promising superconductor. Subsequently, we calculated the phonon dispersion curves, the projected phonon density of states (PHDOS) and Eliashberg spectral function α²F(ω) together with the electron−phonon integral constant λ of Cmcm-RuSi₂ at 1 atm, 25 GPa, 50 GPa, and 75 GPa with the intention of illuminating their potential superconductivity, presented in Figure 7. As analysed above, its dynamical stability could be verified within the pressure range of 0−75 GPa as a consequence of the absence of imaginary phonon frequency throughout the whole Brillouin zone at 1 atm, 25 GPa, 50 GPa, and 75 GPa. In the PHDOS, the phonon vibrational modes of Cmcm-RuSi₂ can mainly be divided into three districts on the basis of the phonon region. The vibrations of Ru atoms mainly contribute to the low-frequency phonon regions (< 9 THz), while the high-frequency phonon modes (10−15 THz) are associated with silicon layers because of Ru atom heavier than Si atom. It is worthy noting that the intermediate-frequency branches stem from the coupled vibrations of Ru atom and Si layer. For the EPC parameter λ, the low-frequency vibrations by Ru atoms around below 9 THz contribute 82.4%, 77%, 74% and 67.5% at 1 atm, 25 GPa, 50 GPa and 75 GPa of the total λ, respectively, while the high-frequency phonon bands above 10 THz associated with light Si atom vibrations only account for 13.2%, 19.7%, 20.5% and 22.5% in Cmcm-RuSi₂, respectively. In addition, the corresponding coupled vibrations from Ru atoms and Si layer contribute 4.4%, 3.3%, 6.5% and 10% at 1 atm, 25 GPa, 50 GPa and 75 GPa of the total λ, respectively. This directly denotes that the vibrations from Ru atoms combined with the coupled vibrations between Ru atoms and Si layers play significant roles in the superconductivity of Cmcm-RuSi₂.

The electron-phonon coupling parameters λ, logarithmic average phonon frequency ω_log, electron density of states at the Fermi level N(E_f) and superconducting transition temperature T_c of Cmcm-RuSi₂ at 1 atm, 25 GPa, 50 GPa and 75 GPa were also calculated in Table 1. The superconducting critical temperature T_c values are estimated by using the Allen–Dynes modified McMillan equation:

$$T_c={\displaystyle \frac{{\omega }_{\mathrm{bg}}}{1.2}}\exp \left[{\displaystyle \frac{1.04\left(1+\lambda \right)}{\lambda -{\mu }^{*}\left(1+0.62\lambda \right)}}\right]$$

where, the Coulomb pseudopotential parameters μ* are set to typical 0.10 and 0.13, respectively. As summarized in Table 1, the predicted EPC parameter λ for the Cmcm-RuSi₂ crystal is 0.4 at 75 GPa and displays a monotonous decrease with the increment of the pressure. Accordingly, the maximal EPC parameter λ value of 0.91 emerges in Cmcm-RuSi₂ under ambient pressure, which is comparable to 0.93 for RaSi₂ at 0 GPa, 1.1 for CaSi₂ at 0 GPa, 0.92 for Al₂Si₃ at 10 GPa, 1.01 for P6₃/mmc-Y₃Si, but larger than 0.81 for P6/mmm-MgSi₃ at 0 GPa, 0.808 for Cmmm-Si₄ at ambient pressure, 0.897 for P6/m-NaSi₆ at 0 GPa, 0.799 P6/m-Si₆ at 0 GPa. To compare with the former binary conventional superconductors, we draw a overall diagram which displays a combination of the T_c and stabilized pressure for established superconductors, as shown in Figure 8. Additionally, the detailed T_c values

For the conventional superconductors at different pressures are summarized in Table S3. It is undoubtedly found that the Cmcm-RuSi₂ structure are estimated to be 10.07 K at atmospheric pressure, which is comparable to 13.7 K in MgSi₃ , 13 K in P6/m-NaSi₆ and 12 P6/m-Si₆ at 0 GPa, 14 K for CaSi₂ at 15 GPa, 10.5 K for P6/mmm-CeP₂ at 16.5 GPa, and 13.6 K for CrB₂ at 60.7 GPa, but higher than other transition metal disilicides, such as RbSi₂ (9 K) [34], MgSi₂ (~7 K) [35,36], BaSi₂ (6.8 K) [39], LaSi₂ (2.5 K) [40] and YSi₂ (0.5 K, 50 GPa) [41]. In short, the novel Cmcm-RuSi₂ configuration at ambient pressure possesses the secondarily highest T_c among the binary metal disilicides, merely lower than that of CaSi₂ (14 K) under 15 GPa. In addition, it is worthy mentioning that the superconducting critical temperature T_c in Cmcm-RuSi₂ is significantly higher than those of the corresponding metallic Cmcm-Si₄ (2.2 K) [87] and Ru (0.51 K) [91,92] at ambient pressure, which tremendously advance the superconductivity for the related elemental substances.

The investigation of the variation trends with the increasing pressure serve as an appropriate pathway for a deeper understanding of the superconductivity driven in Cmcm-RuSi₂, as intuitively exhibited Figure 9. Regarding the broad stable pressure range obtained in Cmcm-RuSi₂, we have implemented calculations to estimate its pressure-dependent superconductivity. As the pressure decreases, a successive increase presents in the T_c values, eventually reaching 10.07 K at atmospheric pressure. Manifestly, two substantial parameters (λ and ω_log) play a vital role in determining T_c, displaying the contrasting trends with pressure. The increment in T_c with decreasing pressure results from the synchronous enhancement in λ, which dominates the evolution of T_c with pressure. The growth in λ should be related to the increasing DOS at the Fermi energy upon the reducing pressure, which further originate from the decrease of charges of Ru-4d orbital transferred from silicon layer, as tabulated in Table S4. In the Cmcm-RuSi₂ phase, one Ru atom gain 1.38 e at 75 GPa and reduces to 1.06 e under ambient pressure. The charge transferred from nonmetallic Si to metallic Ru is feasible for the reason that the electronegativity value of Si (1.90) element is lower than that of Ru element according to the periodic table of the elements. The anion character in metallic element Ru is rarely seen in binary metal silicides. In contrast, the reducing process of ω_log with the decreasing pressure should be attributed to the elongation of Si−Si bonds within the configuration. Upon the reducing compression, the variation trend of λ (increment) and ω_log (diminution) is similar to those in Ce₂P₃, CeP₂ [28]and Sc₂P [93].

As the representative superconductor in the component of 1:2 in binary metallic compounds, MgB₂ encompasses a similar honeycomb B layer, differing from Cmcm-RuSi₂. It is necessary to unlock the discrepancy in the structural characters, electronic properties, the superconducting mechanism and T_c values between both phases at ambient pressure. In regard to the structural trait, the honeycomb B layer and Mg atoms are located in the alternate layers along the c axis in MgB₂ whereas the graphene-like Si layer and the Ru atoms are in the same plane with Ru atom sited at the center of the Si hexatomic ring. In the case of the metallic character, MgB₂ mainly stems from the B-2p orbital while the current Cmcm-RuSi₂ should be ascribed to the Ru-4d state. The high frequency for MgB₂ derives from the graphene-like B layer with the superconductivity of MgB₂ dominated by the electron-phonon coupling of B atoms [94], while the transition Ru-4d orbital electrons also take part in the coupling. This comparison not only points toward a possibility to uncover different superconducting sources in the traditional prototype phases, such as H₃S, CaH₆ and LaH₁₀, but also widens the cognition of the elemental contributions in superconductivity. As for the T_c value, MgB₂ is higher than Cmcm-RuSi₂, which can be summarized as the following reasons: (a) The MgB₂ has an open B layer without the B−B interlayer covalence while Cmcm-RuSi₂ contains an closed Si layer with the interlayer covalent Si−Si bond; (b) the bond strength of Si-Si bond in Cmcm-RuSi₂ is weaker than that of MgB₂, as demonstrated by the calculated ICOHP (−2.25 per pair Si−Si in comparison with −6.92 per pair B−B); (c) the ω_log value for Cmcm-RuSi₂ is lower than that of MgB₂ (167.97 Κ with respect to 450 Κ) [94], which results from the lower phonon frequencies of Ru and Si atoms lower than those of Mg and B atoms, as related to the larger atomic mass of Ru and Si, respectively.

3.4. Mechanical Property and Hardness

With the high development in the superconductor, it necessitates the eminent mechanical property and hardness integrated with spectacular ductility. The elastic constants for the Cmca-RuSi₂ and Cmcm-RuSi₂ phases at 1 atm in conjunction with the Cmcm-RuSi₂ configuration under ambient pressure and 75 GPa were computed through the identical strain−stress means [95], as corroborated in Table 2. Both phases are dynamically stable under focused pressures as they satisfy the mechanical stability criteria for the orthorhombic crystal system [96]. Thereafter, the homologous shear modulus G, bulk modulus B, B/G, Young’s modulus E and Poisson’s ratio ν of both phases can be attainable from the elastic constants based on the Voigt–Reuss–Hill (VRH) method [97] (see Table 2). In the Cmca-RuSi₂ configuration under ambient pressure, the current estimated elastic constants C_ij, bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio v are in conformity with the prior theoretical consequens [49], clarifying that the methodology utilized in present work is convincing. Generally speaking, the C₁₁, C₂₂ and C₃₃ values in material denote the resistance to compression along the corresponding a-, b- and c-axis. From Table 2, Cmcm-RuSi₂ under 75 GPa possesses ultra-high C₁₁, C₂₂ and C₃₃ values, which are much higher than those of P6/mmm-ScB₂ [98], explaining that this phase behaves excellent incompressibility along a-, b- and c-axis. The calculated B, G and E values of Cmcm-RuSi₂ under 75 GPa are 410.13, 184.99 and 482.44 GPa, respectively, which are significantly higher than those of the Cmca-RuSi₂ phase at ambient pressure (B, G and E values are 173.53 , 107.99 and 268.30 GPa, respectively), corroborating that this phase transition induced by compression enhances the mechanical properties of RuSi₂ crystal to a great extent. Moreover, the value of B/G can estimate the brittle or ductile manner of a solid: a high ratio of B/G (> 1.75) indicates the ductility, while a low ratio (< 1.75) indicates the brittleness according to the Pugh criterion [99,100]. For transition metal silicides, the improvement of ductility can greatly improve their industrial work efficiency, because their actual productions as high-temperature materials are limited by their own brittleness [101,102,103]. As listed in Table 2, the the B/G value of Cmca-RuSi₂ at 1 atm is 1.61, uncovering the brittle nature whereas the B/G values of Cmcm-RuSi₂ at 1 atm and 75 GPa are 2.03 and 2.22, corresponding to ductile character, respectively. This phenomenon confirms the brittle-to-ductile transformation in RuSi₂, accompanied by the structural transformation under high pressure, which is analogous to NbSi₂ [104] and MoN₂ [105].

Hardness should be characterized as a substantial mechanical property of a material and can be used to measure the ability to resist local deformation, especially plastic deformation, indentation or scratches. In addition, the orientation bond in a material is an important index to measure its hardness, and it can be expressed by Pugh’s ratio G/B [99,100]. A larger G/B value corresponds to a higher hardness. As summarized in Table 2, the G/B values of Cmca-RuSi₂ and Cmcm-RuSi₂ phases under atmospheric pressure are 0.64 and 0.49, respectively. However, Gao et al. indicated that the vital problem in digging into the hardness of materials is how to describe the strength of internal bonds of the materials. Thus, we further probe into the hardness values of both structures employing the hardness model proposed by Gao et al [106,107]. The Vickers hardness could be calculated with the following formula:

$$H_{\nu }={\left\{\prod ^{\mu }{\left[740P^{\mu }{\left(V_b^{\mu }\right)}^{-{\displaystyle \frac{5}{3}}}\right]}^{n^{\mu }}\right\}}^{{\displaystyle \frac{1}{\sum n^{\mu }}}}$$

where

$$V_b^{\mu }={\left(d^{\mu }\right)}^3/\sum \left[{\left(d^v\right)}^3{N}_b^{\mu }\right]$$

where μ labels all different types of covalent bonds in the system, $P^{\mu }$ is the Mulliken overlap population of the μ type bond, ${\nu }_b^{\mu }$ is its volume, $n^{\mu }$ is the number of μ type bonds, $d^{\mu }$ is its bond length, and $N_b^{\mu }$ is the bond number of type μ per unit volume. On account of the aforementioned empirical equation, the estimated hardness values of Cmca-RuSi₂ and Cmcm-RuSi₂ phases reach 31.74 and 26.76 GPa, respectively (see Table 3). The hardness value of Cmca-RuSi₂ under ambient pressure is consistent with the previous theoretical results [49]. The predicted H_v values of Cmca-RuSi₂ (31.74 GPa) and Cmcm-RuSi₂ (26.76 GPa) are comparable to the experimental measurements of β-SiC (34 GPa) [106], BP (33 GPa) [108], TiC (32 GPa) [109] β-Si₃N₄ (30 GPa) [109] and transition metal disilicide WSi₂ (28.5 GPa) [110], but higher than elemental solid Si (12 GPa) [106]. Hence, it can be concluded that both Cmca-RuSi₂ and Cmcm-RuSi₂ are potential high hardness materials. In comparison with Cmcm-RuSi₂, Cmca-RuSi₂ has a higher hardness, which can be ascribed to the stronger Ru-Si and Si-Si covalent bonds and better orientation than those of Cmcm-RuSi₂.

4. Conclusions

To sum up, our comprehensive first-principles structure searches in RuSi₂ within the pressure range of 0−300 GPa have disclosed the emerge of the peculiar Cmcm-RuSi₂ phase at 50.97 GPa. This Cmcm-RuSi₂ configuration contain an extraordinary graphene-like silicon layer, which is dynamically stable and simultaneously leads to exceptional superconductivity and high hardness under ambient pressure, despite it is recovered from the high pressure. On the one hand, its predicted T_c value could reach 10.07 K at ambient pressure, comparable to the previous proposed P6/m-NaSi₆ (13 K) and CaSi₂(14 K)。On the other hand, the Vickers hardness value is 26.76 GPa, which is comparable to WSi₂ (28.5 GPa). Furthermore, its excellent conductivity stem from the the strong interaction between Ru-4d electrons and the phonons B−B and coupled Ru−Si vibrations, while its high hardness can be ascribed to the 3D robust Ru−Si and Si−Si directional covalent framework. In addition, the Cmcm-RuSi₂ phase is a ductile material. The ambient-pressure ground Cmca phase is a narrow-gap semiconductor with a estimated hardness of 31.7 GPa.

The special graphene-like silicon layer in the transition-metal disilicides Cmcm-RuSi₂ phase broaden the existing silicon-layer space beyond alkali, alkaline-earth and rare-earth metal disilicides. The removal of metal atoms can be utilized to acquire silicon framework such as layered silicon, silicon clathrates and other silicon allotropes. This mode has uncovered a succession of novel elemental structures, such as I4/mmm-B₄, Pm-B₁₇ [80] and P6/m-Si₆ [33], which is destined to promote the quest for the analogous silicon laminar structures and distinct silicon motifs (eg. 3D silicon clathrate framework and 1D silicon chains) in elemental silicon and transition-metal silicides. On the other hand, this layered network in Cmcm-RuSi₂ provide a wonderful bedding for adding light element adatoms to layered superconductors, which could enhance the superconductivity for the parent Cmcm-RuSi₂ configuration. This method has been practically employed recently to improve the superconductivity in very diverse ultrathin materials, ranging from monolayer MgB₂ to truly 2D atomic sheets doped graphene and transition metal dichalcogenides.

The Cmcm-RuSi₂ structure has the secondarily highest superconducting transition temperature among the metal disilicides, MSi₂ (M = metal), merely lower than that of CaSi₂ (14 K), which facilitate the in-depth investigation on the superconductivity mechanism in the metal disilicides in both theory and experiment.

The anion transition-metal in Cmcm-RuSi₂ is rarely reported in preceding binary compounds that contains metal and nonmetal elements, which widens the physical and chemical cognition.