A fresh class of superconducting and hard pentaborides

Hui Xie; Hong Wang; Fang Qin; Wei Han; Suxin Wang; Youchun Wang; Fubo Tian; Defang Duan

doi:10.1063/5.0157250

Abstract

On the basis of the current theoretical understanding of boron-based hard superconductors under ambient conditions, numerous studies have been conducted with the aim of developing superconducting materials with favorable mechanical properties using boron-rich compounds. In this paper, first-principles calculations reveal the existence of an unprecedented family of tetragonal pentaborides MB₅ (M = Na, K, Rb, Ca, Sr, Ba, Sc, and Y), comprising B₂₀ cages and centered metal atoms acting as stabilizers and electron donors to the boron sublattice. These compounds exhibit both superconductivity and high hardness, with the maximum superconducting transition temperature T_c of 18.6 K being achieved in RbB₅ and the peak Vickers hardness H_v of 35.1 GPa being achieved in KB₅ at 1 atm. The combination of these properties is particularly evident in KB₅, RbB₅, and BaB₅, with T_c values of ∼14.7, 18.6, and 16.3 K and H_v values of ∼35.1, 32.4, and 33.8 GPa, respectively. The results presented here reveal that pentaborides can provide a framework for exploring and designing novel superconducting materials with favorable hardness at achievable pressures and even under ambient conditions.

I. INTRODUCTION

Superconducting and superhard substances have been at the forefront of fundamental research and industrial applications. High-pressure syntheses and theoretical predictions1–12 have been performed to investigate the superconductivity of various materials such as copper/iron-based superconductors,13,14 as well as conventional Bardeen–Cooper–Schrieffer (BCS) superconductors. In view of the tremendous advances that have been achieved in enhancing the transition temperatures of conventional phonon-mediated superconductors, especially of highly compressed hydrides, it is natural to investigate superconductivity in other lightweight materials. This class of compounds containing light elements are excellent candidates for superhard materials.15

Borides have attracted considerable attention in research into superconductivity and hardness because of the diversity of their crystal chemistry.16 In early work, Japanese scientists reported superconductivity of MgB₂ at 39 K under ambient pressure,1 fueling enthusiasm for a search for high-temperature superconductivity in the compressed borides.17–24 Polycrystalline MgB₂ synthesized by high-pressure sintering is a superhard material25 with Vickers hardness H_v up to 4109.5 kg/cm². Thus, the idea of developing novel superhard materials using borides and with desired characteristics has gained traction.18,24,26–28 B₆C is a superconducting and superhard material, with a transition temperature T_c of ∼12.5 K and an H_v of ∼48 GPa at ambient pressure.29 Theoretical studies30 have revealed that α-BeB₆ is superconducting and superhard with T_c of 9 K and H_v of 46 GPa. At high pressure, the predicted T_c and H_v of β-BeB₆ are up to 24 K and 31 GPa, respectively. Du et al.23 performed electron–phonon coupling (EPC) and mechanical calculations demonstrating that ScB₆ phases have estimated transition temperatures and hardnesses of ∼2.2–5.8 K and 16.2–27.3 GPa under extreme conditions. Lately, Liu’s group have designed a novel class of clathrate-like superconductors, MB₇, with superior hardness, among which the T_c and H_v of KB₇ were estimated to be 26.2 K and 22.5 GPa at 1 atm. On the basis of previous results,16 borides tend to be hard or superhard materials, with these properties being derived from the covalent boron sublattice. However, although there have been extensive attempts to find high-temperature boron-based superconductors, theoretical and experimental success is yet to be realized. The lower T_c values of two- or three-dimensional borides compared with those of compressed hydrides has motivated us to perform a systematic investigation in this exciting field with a view to finding a new type of superconducting structure with high hardness.

Yttrium borides form various stoichiometries with excellent thermoelastic, electronic, and mechanical properties, which can be systematically applied in various industries.16,27,31 YB₂, YB₄, YB₆, YB₁₂, and YB₆₆ are stable under ambient conditions, with H_v around 25.3–40.9 GPa.16,32,33 Furthermore, the T_c values of YB₆ and YB₁₂ are 1.8–12.8 and 4.7 K.17,34,35 We have performed extensive random structure searches for stable or metastable YB_2–12 compounds under high pressures and have discovered a novel superconductor, yttrium pentaboride, with a T_c of 12.3 K. In YB₅, B atoms are strongly covalently bonded to each other within B₂₀ cages, with Y atoms occupying the centers of these cages and acting as electron donors. A new group of pentaborides, MB₅ (M = Na, K, Rb, Ca, Sr, Ba, and Sc), isomorphic to the predicted YB₅, have also been estimated to exhibit high superconducting transition temperatures (T_cmax = 18.6 K) and hardness (H_vmax = 35.1 GPa). This study lays the foundation for exploring new classes of hard boron-based superconductors and provides valuable guidance for further experimental syntheses.

II. COMPUTATIONAL DETAILS

High-pressure structural predictions of YB_2–12 were performed as implemented in the AIRSS code36 at pressures of 1 atm, 50 GPa, and 100 GPa,36 generating ∼10 000 structures randomly. The CASTEP code,37 on-the-fly pseudopotentials, the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA)38 exchange-correlation functional, and 4s²4p⁶4d¹5s² (Y) and 2s²2p¹ (B) valence electrons were used during the random structure searches and geometrical optimizations. The kinetic energy cutoff and k-spacing were fixed appropriately at 800 eV and 2π × 0.03 Å⁻¹ in structure optimizations. Electronic characters were calculated using the Vienna Ab initio Simulation Package (VASP)39 and the all-electron projector-augmented wave method (PAW) pseudopotentials. The ab initio molecular dynamics (AIMD) simulations were carried out in the framework of the Nosé–Hoover method.40 The phonon dispersions were calculated with the supercell approach41 as used in the phonopy code.42 Electron–phonon coupling was computed by the Quantum ESPRESSO package,43 using a norm-conserving scheme with an energy cutoff of 80 Ry, a k-mesh of 24 × 24 × 24 and a q-mesh of 4 × 4 × 4. The superconducting transition temperatures of MB₅ were approximated with the Allen–Dynes modified McMillan equation44 $T_{c} = \frac{ω_{\log}}{1.2} \exp [- \frac{1.04 (1 + λ)}{λ - μ^{*} (1 + 0.62 λ)}],$ (1)where μ* is the Coulomb pseudopotential, and ω_log is the logarithmic average frequency and is defined as follows: $ω_{\log} = \exp [\frac{2}{λ} \int \frac{d ω}{ω} α^{2} F (ω) \ln ω] .$ (2)The EPC constant λ can be calculated as follows: $λ = 2 \int \frac{α^{2} F (ω)}{ω} d ω .$ (3)The elastic constants were computed using the VASP code, applying the strain–stress method. The mechanical moduli and H_v were derived from the elastic constants using the Voigt–Reuss–Hill approximation45 and the model proposed by Chen et al.46

III. RESULTS AND DISCUSSION

We investigated structures under ambient pressure and reproduced the previously known phases of YB₂, YB₄, YB₆, and YB₁₂, confirming the reliability of our predictions. The lattice parameters of the optimized structures are consistent with known experimental findings,16 as presented in Table S1 (supplementary material). The formation enthalpies ΔH of the predicted YB_2–12 stoichiometries were calculated with respect to the pure elements at 1 atm, 50 GPa, and 100 GPa [see Fig. 1(a)], from which the convex hull was constructed. The known $R \bar{3} m$ , Pnnm, and Cmca structures for boron47 and the hcp, dhcp, and P6₂22 structures for yttrium48 were adopted in their corresponding stable pressure. The results revealed that YB₂, YB₄, and YB₁₂, as expected, are stable at atmospheric pressure. Cubic YB₆ is metastable at ∼31 meV/atom above the convex hull, which does not show that our calculations are unreliable or that the material cannot be synthesized experimentally. The fact that the compound is in a metastable state means that the predicted effect can be observed in actual high-temperature experiments.49 At 50 GPa, the stable configurations remain the same, and I4/mmm–YB₆ emerges on the convex hull. Hexagonal YB₂ emerges as the most stable compound in the studied pressure ranges all along. With the pressure increased to 100 GPa, the stable structure of YB₄ changes to C2/m. Meanwhile, YB₁₂ deviates from the tieline and decomposes into YB₆ and B, whereas another unreported boron-rich composition P4/mmm–YB₅ is close to the convex hull (∼35 meV). The phonon bands of these structures manifest their dynamic stability, since the no imaginary frequency appears in the Brillouin zone (see Fig. S1, supplementary material). In the following discussion, the novel YB₅ structure is the main focus because it has relative high symmetry and is recoverable to ambient pressure.

Figure 1.(a) Enthalpies of formation of predicted YB_2–12 as functions of B concentration at 1 atm, 50 GPa, and 100 GPa. Open, half-filled, and filled symbols represent unstable, metastable, and stable stoichiometries, respectively, relative to the elementary substance. (b) Crystal structures of P4/mmm–YB₅. (c) The B₂₀ unit. Y and B atoms are presented as golden and green spheres. The cutoff distance between B–B bonds is 2.0 Å.

Yttrium pentaboride adopts a tetragonal P4/mmm space group [see Fig. 1(b)], where B atoms occupy 4n and 1d positions bonding to each other within the B₂₀ cage [see Fig. 1(c)], and Y atoms lies at the crystallographic 1b position occupying the centers of the B₂₀ cages. This structure is similar to that of cubic YB₆, in which B atoms constitute a boron octahedral network. The straight chains of boron octahedrons are arranged in a three-dimensional frame. This three-dimensional clathrate-like B₂₀ structure with metal atoms filling the B cages is also found in NaB₅, KB₅, RbB₅, CaB₅, SrB₅, BaB₅, and ScB₅ through isostructural metal atom replacement at 100 GPa, with each material showing thermal stability (see Fig. S2, supplementary material). Phonon calculations reveal their dynamic stability at ambient pressure (Fig. S3, supplementary material). The electron localization function (ELF)50 was calculated to map the electron pair probability and understand interatomic interactions. No charge localization appears between boron and metal atoms, providing evidence for M–B ionic bonding (see Fig S4, supplementary material). Bader charge analysis was also performed to calculate the transferred charge between atoms (see Table S2, supplementary material). The metal atoms serve as electron donors to the boron sublattice, which is also supported by the ELF results. The ELF values between B–B bonds in the three-dimensional frameworks of the metal pentaborides are larger than 0.7, implying the formation of covalent bonds. Among them, the ELF between two adjacent boron octahedrons is approximately equal to or even greater than 0.9, which is greater than that between B5 (the B atom shared between adjacent boron octahedrons) and its neighboring atoms, which has a range of 0.7–0.8. The integrated crystal orbital Hamilton population (ICOHP) of adjacent B–B pairs with bond lengths of 1.6–2.0 Å was calculated using the LOBSTER package51 to further confirm their covalent characteristics. The negative ICOHP values of B–B pairs in the above structures indicate B–B covalent interactions. As displayed in Fig. S5 (supplementary material), the ICOHP values of B1–B2 (B3–B4) pairs in MB₅ are in the range of −7.8 to −5.4 eV/pair, which indicates strong covalent bonding. The B1–B3 (B1–B4, B2–B3 or B2–B4) bonds are fairly strong, with ICOHP values of −5.4 to −2.7 eV/pair, stronger than or approximately equal to that of B5–B1 (B5–B2, B5–B3, or B5–B4, −3.6 to −2.9 eV/pair) bonds. The ICOHP values are consistent with the ELF results.

A high electronic density of states (DOS) is a guideline for successful searches for new superconducting materials, and therefore the projected DOS of MB₅ at atmospheric pressure was calculated. Note that all the studied compounds are metallic, as presented in Fig. 2. In the alkali metal (AMB₅) and the light alkaline earth metal (AEMB₅) pentaborides, the greatest contribution to the DOS close to the Fermi energy E_f is provided by the B-p orbital [see Figs. 2(a)–2(e)], although the M-d orbital at E_f does play a non-negligible role in their metallicity. (It should also be mentioned that the electronic DOS of alkali metal pentaborides has a van Hove singularity near E_f, implying fairly strong electron–phonon interactions associated with boron phonon modes.) By contrast, in BaB₅ and the rare-earth borides ScB₅ and YB₅, the DOS values are dominated by both the M-d and B-p orbitals [see Figs. 2(f)–2(h)]. In addition, ScB₅ exhibits a particularly high DOS at the Fermi level (see Fig. S6, supplementary material).

Figure 2.Calculated electronic projected DOS for P4/mmm-MB₅ (M = Li, Na, K, Rb, Ca, Sr, Ba, Sc, and Y) at 1 atm. The black line where the energy is equal to zero is the Fermi level.

The conclusions regarding the electronic DOS are further confirmed by the results for electron–phonon coupling (EPC). The calculated Eliashberg spectral function α²F(ω) and the EPC parameter λ(ω) at ambient pressure are shown in Fig. 3. Clearly, EPC in the AMB₅, BaB₅, and ScB₅ is much more prominent than in the light AEMB₅ and YB₅. The calculated EPC values, logarithmic average phonon frequency ω_log, electronic DOS near the Fermi energy N(E_f), and superconducting transition temperatures T_c are presented in Table I. The estimated λ values of NaB₅ and KB₅ are 0.73 and 0.64, respectively, and the corresponding values of T_c are 17.5 and 14.7 K when the Coulomb potential parameter μ* equals 0.1. According to the phonon DOS (PHDOS) illustrated in Figs. 3(a) and 3(b), vibrations can be separated into low-frequency and high-frequency regions. The low-frequency modes (<8 THz) contribute 52% and 35%, respectively, comprising a mixture of vibrations from M (M = Na and K) and B atoms. EPC calculation for RbB₅ gives a relatively large λ of 0.78, which is primarily associated with the high-lying frequencies (>6 THz) of boron (∼63%) [see Fig. 3(c)], yielding an estimated T_c of 18.6 K. The estimated T_c values for CaB₅ and SrB₅ are 6.6 and 6.8 K, respectively. For the calculated λ of 0.49 and 0.50, as presented in Figs. 3(d) and 3(e), the high-frequency B vibrational modes contribute about 88% and 90%, respectively. The high ω_log and N(E_f) values cannot considerably improve T_c, whereas the weak electron–phonon interaction limits the superconductivity of these compounds. In BaB₅, ω_log is slightly lower than in the CaB₅ and SrB₅ phases. However, T_c reaches 16.3 K, flowing from the higher λ of 0.73. The contribution of 88% to the total EPC is closely related to the high-frequency vibrations. As can be seen, although ScB₅ has a stronger EPC interaction and higher N(E_f) than the other pentaborides, its ω_log is quite low, which is caused by λ and is not favorable to T_c. The estimated superconducting transition temperature is ∼14.2 K. YB₅ is also predicted to be superconducting, with a T_c of 12.3 K, and corresponding ω_log and λ of 403.8 K and 0.66, respectively. We note that the T_c values of borides are not strongly dependent on the contributions of the DOS of boron atoms to the total DOS at the Fermi energy or the electron–phonon interactions of boron to the total λ.

Figure 3.Eliashberg spectral function α²F(ω), EPC parameter λ(ω), and PHDOS of MB₅ at 1 atm.

Boride	λ	ω_log	N(E_f) (states Ry⁻¹ f.u.⁻¹)	T_c (K)
NaB₅	0.73	454.3	4.94	17.5
KB₅	0.64	535.6	5.05	14.7
RbB₅	0.78	420.0	5.07	18.6
CaB₅	0.49	604.7	5.15	6.6
SrB₅	0.50	574.9	5.21	6.8
BaB₅	0.73	420.4	6.07	16.3
ScB₅	0.90	242.1	7.51	14.2
YB₅	0.66	403.8	5.67	12.3

Table 1. Calculated EPC parameter λ, logarithmic average phonon frequency ω_log, electronic DOS at the Fermi level N(E_f), and superconducting transition temperature T_c of P4/mmm-MB₅, with a Coulomb pseudopotential μ* = 0.1 at 1 atm.

View all Tables

Compared with high-temperature hydrogen-based superconductors, it is obvious that the EPC parameter λ and ω_log of the borides are much lower owing to the higher mass of boron. However, only a limited difference is observed in N(E_f), a manifestation of the fact that it is the low values of λ and ω_log that lie at the root of the poorer superconductivity of the borides. However, borides can be stabilized at attainable pressures or even ambient pressure, indicating their higher application potential.

To investigate the mechanical characteristics of the MB₅ compounds, we calculated their elastic constants C_ij under ambient conditions, from which we derived the bulk modulus B, shear modulus G, Young’s modulus Y, Poisson’s ratio ν, and Vickers hardness, as presented in Table II. We estimated the hardness of the novel pentaborides to accurately evaluate their incompressibility using Chen’s theoretical model46 as follows: $H_{v} = 2 {(k^{2} G)}^{0.585} - 3,$ (4)where k is the ratio of the shear modulus to the bulk modules (Pugh’s modulus ratio). It is found that the calculated H_v of the MB₅ compounds is in the range of 7.9–35.1 GPa, with KB₅ showing the highest hardness, which can be attributed to its high values of k and G. Other pentaborides RbB₅, CaB₅, SrB₅, and BaB₅ exhibit superior mechanical properties, with H_v above 30 GPa. Furthermore, charge transfer affects the hardness. From the δ(e) values of the metal atoms in MB₅ with high hardness (H_v > 30 GPa) presented in Table S2 (supplementary material), one finds that the metal atoms in AMB₅ and AEMB₅ transfer approximately (0.6–0.7)e and (1.2–1.4)e, respectively. It seems that this is a reliable new way to identify hard superconducting materials among metastable structures with high symmetry.

Borides	C₁₁	C₃₃	C₄₄	C₆₆	C₁₂	C₁₃	B	G	Y	ν	H_v
NaB₅	421.2	398.6	74.9	113.6	112.9	8.1	165	117	284	0.21	18.7
KB₅	424.4	331.7	125.4	181	53	30.3	155	155	384	0.13	35.1
RbB₅	340.5	323.7	119.1	177.2	85.5	24.3	141	139	314	0.13	32.4
CaB₅	541.1	415.7	107.8	158.1	51.5	15.8	183	159	370	0.16	30.0
SrB₅	504.7	347.7	116.5	193.7	46.6	33.9	173	162	370	0.14	33.1
BaB₅	399.5	330.1	133.4	207.4	92.2	40.2	162	156	355	0.14	33.8
ScB₅	530.8	433	53.8	100.6	107.3	52.7	211	108	276	0.28	11.1
YB₅	564.2	375.2	59.6	162.5	78.7	49.7	202	123	307	0.25	15.8

Table 2. Calculated elastic constants C_ij (GPa) and derived bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus Y (GPa), Poisson’s ratio ν, and hardness H_v (GPa) at ambient pressure. The elastic constants of all the MB₅ compounds satisfy the mechanical stability criterion.⁵²

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IV. CONCLUSIONS

Our extensive structure searches combined with first-principles calculations have revealed the appearance of a metastable clathrate-like B₂₀ structure in tetragonal MB₅ (M = Na, K, Rb, Ca, Sr, Ba, Sc, and Y), adopting P4/mmm symmetry. In this structure, metal atoms act as donors to transfer charges to B, and B atoms form a three-dimensional boron octahedral covalent network. Density of states calculations show that the pentaborides are metallic and that the electronic states occupying the Fermi level come mainly from boron atoms. Calculated electron–phonon couplings and elastic constants indicate that the cage-like B₂₀ structure can be viewed as a structural model for hard superconductors, with an estimated highest T_c of 18.6 K and a maximum H_v of 35.1 GPa at 1 atm. It is worth mentioning that KB₅, RbB₅ and BaB₅ are superconducting and hard materials, in which the corresponding superconducting critical temperatures are estimated to be around 14.7, 18.6, and 16.3 K and the Vickers hardness to be 35.1, 32.4, and 33.8 GPa, respectively. Our work provides a reference for future experiments and demonstrates that metastable borides hold considerable promise as superconductors with superior hardness at achievable pressures.

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

Acknowledgment. We thank Dr. Yunxian Liu and Dr. Zihao Huo for AIMD calculations and many stimulating discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12104127 and 22131006), the Doctoral Starting Up Foundation of Hebei Normal University for Nationalities (Grant No. DR2020001), the Clean Energy (Carbon Peaking and Carbon Neutrality) Industry Research Institute of Chengde (Grant No. 202205B090), and the Natural Science Foundation of Shandong Province (Grant No. ZR2020QA060). Parts of the calculations were performed at the High Performance Computing Center of Shandong Bailing Cloud Computing Co., Ltd.

References

[1] J.Nagamatsu, N.Nakagawa, Y.Zenitani, J.Akimitsu, T.Muranaka. Superconductivity at 39 K in magnesium diboride. Nature, 410, 63-64(2001).

[2] X.Huang, T.Cui, Y.Ma, Y.Liu, F.Tian, D.Li, H.Yu, D.Duan, B.Liu. Pressure-induced decomposition of solid hydrogen sulfide. Phys. Rev. B, 91, 180502(2015).

[3] D.Duan, T.Cui, B.Liu, Z.Shao, Y.Liu, Y.Ma. Structure and superconductivity of hydrides at high pressures. Natl. Sci. Rev., 4, 121-135(2016).

[4] S. I.Shylin, M. I.Eremets, I. A.Troyan, V.Ksenofontov, A. P.Drozdov. Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system. Nature, 525, 73(2015).

[5] Y.Wang, D.Duan, Z.Shao, X.Xiao, H.Xie, D.Li, B.Liu, H.Song, T.Cui, F.Tian. High-temperature superconductivity in ternary clathrate YCaH₁₂ under high pressures. J. Phys.: Condens. Matter, 31, 245404(2019).

[6] S.Jiang, X.Feng, C. J.Pickard, Z.Zhang, Y.Yao, T.Cui, S. A. T.Redfern, H.Xie, V. Z.Kresin, D.Duan, H.Song. Hydrogen pentagraphenelike structure stabilized by hafnium: A high-temperature conventional superconductor. Phys. Rev. Lett., 125, 217001(2020).

[7] C. J.Pickard, M.Du, D.Duan, M. J.Hutcheon, Z.Zhang, A. M.Shipley, Y.Yao, V. Z.Kresin, T.Cui, H.Song. Design principles for high-temperature superconductors with a hydrogen-based alloy backbone at moderate pressure. Phys. Rev. Lett., 128, 047001(2022).

[8] Y.Li, R. J.Hemley, S.Meng, J. S.Tse, B.Yang, S.Lu, I. I.Naumov, H.Liu. Superconductivity in dense carbon-based materials. Phys. Rev. B, 93, 104509(2016).

[9] F.Peng, R. J.Needs, C. J.Pickard, Q.Wu, Y.Ma, Y.Sun. Hydrogen clathrate structures in rare earth hydrides at high pressures: Possible route to room-temperature superconductivity. Phys. Rev. Lett., 119, 107001(2017).

[10] A. K.Mishra, M.Somayazulu, R. J.Hemley, M.Baldini, H.Liu, Y.Meng, Z. M.Geballe, M.Ahart. Synthesis and stability of lanthanum superhydrides. Angew. Chem., Int. Ed., 57, 688-692(2018).

[11] A. P.Drozdov, S.Mozaffari, D. A.Knyazev, V. S.Minkov, M.Tkacz, D. E.Graf, L.Balicas, P. P.Kong, F. F.Balakirev, M. I.Eremets, S. P.Besedin, V. B.Prakapenka, E.Greenberg, M. A.Kuzovnikov. Superconductivity at 250 K in lanthanum hydride under high pressures. Nature, 569, 528-531(2019).