Superconducting boron allotropes

The search for elemental allotropes is an active research ﬁeld to get unusual structures with unique properties. The removal of metal atoms from pressure-induced stable binary compounds has become a useful method for obtaining elemental allotropes with interesting properties that otherwise would not be accessible at ambient pressure. Although three-dimensional boron allotropes have been studied extensively, none of those found so far are superconducting at ambient pressure. Here we propose that NaB 4 and Na 2 B 17 can be used as precursors to achieve superconducting boron allotropes at ambient pressure. First-principle swarm-intelligence structure search calculations identify several novel sodium borides (e.g., Na 3 B 2 , Na 2 B 3 , NaB 4 , and Na 2 B 17 ) under high pressure. Interestingly, the B atoms in I 4 / mmm NaB 4 and Pm Na 2 B 17 form three-dimensional frameworks with open channels, where Na atoms are located. After the removal of Na atoms, two hitherto unknown boron allotropes, named as I 4 / mmm B 4 and Pm B 17 , are stable at ambient pressure. They are metallic with superconducting critical temperatures of 19.8 and 15.4 K, respectively, becoming the highest ones among bulk boron allotropes. In addition, considering their predicted Vickers hardness of 27.3 and 26.8 GPa, they are also potential hard materials.

(Received 27 February 2020; revised manuscript received 11 April 2020; accepted 20 April 2020; published 11 May 2020) The search for elemental allotropes is an active research field to get unusual structures with unique properties. The removal of metal atoms from pressure-induced stable binary compounds has become a useful method for obtaining elemental allotropes with interesting properties that otherwise would not be accessible at ambient pressure. Although three-dimensional boron allotropes have been studied extensively, none of those found so far are superconducting at ambient pressure. Here we propose that NaB 4 and Na 2 B 17 can be used as precursors to achieve superconducting boron allotropes at ambient pressure. First-principle swarm-intelligence structure search calculations identify several novel sodium borides (e.g., Na 3 B 2 , Na 2 B 3 , NaB 4 , and Na 2 B 17 ) under high pressure. Interestingly, the B atoms in I4/mmm NaB 4 and Pm Na 2 B 17 form three-dimensional frameworks with open channels, where Na atoms are located. After the removal of Na atoms, two hitherto unknown boron allotropes, named as I4/mmm B 4 and Pm B 17 , are stable at ambient pressure. They are metallic with superconducting critical temperatures of 19.8 and 15.4 K, respectively, becoming the highest ones among bulk boron allotropes. In addition, considering their predicted Vickers hardness of 27.3 and 26.8 GPa, they are also potential hard materials. DOI: 10.1103/PhysRevB.101.174507

I. INTRODUCTION
The preparation of high-temperature superconductors is of great importance for fundamental research and practical applications [1][2][3][4][5][6]. Recently discovered pressure-induced stable H-rich compounds (e.g., H 3 S [7] and LaH 10 [8,9]) broke the superconducting transition temperature record of 164 K in cuprates [10]. Notably, theoretical predictions play a leading role in accelerating these innovative discoveries [11][12][13]. Very recently, room-temperature or even higher-temperature superconductivity has been predicted through doping lithium into MgH 16 , forming a novel ternary compound, Li 2 MgH 16 [14]. However, a common characteristic of these hydrides is that their superconductivity emerges at more than one million times the atmospheric pressure. Therefore, the search for stable or metastable superconducting materials at ambient pressure is highly needed for practical application [15].
Some of pressure-induced stable compounds are recoverable to ambient pressure with interesting properties, especially for light mass elements, such as boron, carbon, and nitrogen.
They can be synthesized, since pressure allows them to overcome the required reaction barriers to be formed. For example, well-known superhard materials, such as BC 3 [16] and diamond [17], are synthesized at high pressures and become metastable at ambient pressure. On the other hand, pressureinduced stable binary compounds are also used as precursors to achieve elemental allotropes. For instance, two silicon allotropes, Si 24 [18] and P6/m-Si 6 [19], are obtained through a two-step synthesis methodology, removing the Na atoms from the pressure-induced stable Na 4 Si 24 and P6/m-NaSi 6 via a thermal "degassing" process. More interestingly, Si 24 has a proper band gap of 1.3 eV, showing broad applications as photovoltaic cells [18]. P6/m-Si 6 becomes a superconductor with a critical transition temperature of 12 K at ambient pressure [19]. In both structures the Si atoms form a threedimensional frame with open channels, where the Na atoms are located, which makes it easy for them to escape.
The study of boron allotropes has attracted great attention because of their electron deficiency, structural complexity, and unusual bonding pattern [20,21]. Up to now, at least 14 boron allotropes have been reported [22]. Most of them consist of B 12 icosahedra [23]. However, the arrangement of the icosahedra strongly modifies their electronic properties. For instance, I2 1 2 1 2 1 -B 60 , where B 12 icosahedra are linked by helical boron chains, is a metal, whereas Pnma-B 60 , in which B 12 icosahedra are interconnected by two-atom wide B ribbons, is a semimetal [24]. After compression, boron allotropes undergo complex structural transitions, caused by the appearance of novel boron units, which also modify their electronic properties [23,[25][26][27][28][29]. So far, no bulk superconducting boron allotrope has been found at ambient pressure.
Having in mind that pressure can stabilize unusual stoichiometric compounds [30][31][32] and that known sodium borides consist of covalent B frames [33][34][35], we consider the binary Na-B system to explore stable B-rich Na-B compounds with open channel frames, then to obtain metastable boron allotropes after releasing Na atoms. In this work, first-principle swarm-intelligence structural search method was employed to identify stable Na-B compounds at high pressures. Besides reproducing the known Na-B compounds (e.g., Na 3 B 20 [33] and NaB 15 [35]) at ambient pressure, four new Na-B compositions (e.g., Na 3 B 2 , Na 2 B 3 , NaB 4 , and Na 2 B 17 ) become stable under high pressure. As expected, two metastable I4/mmm B 4 and Pm B 17 allotropes are obtained from I4/mmm NaB 4 and Pm Na 2 B 17 precursors. Interestingly, these new compounds are predicted to be highly stable and even superconductors, becoming the first superconducting bulk boron allotropes at ambient pressure.

II. COMPUTATIONAL METHODS
To explore the stable Na-B phases under high pressure, we employ a swarm-intelligence based CALYPSO structure search method [36,37], which can find stable structures just depending on the chemical compositions. The geometry optimization and calculations of the electronic properties are performed with the Vienna ab initio simulation package (VASP) code [38] within the framework of density functional theory (DFT) [39,40]. The all-electron projector augmentedwave (PAW) [41] pseudopotentials with 2s 2 2p 6 3s 1 and 2s 2 2p 1 valence electrons for Na and B atoms, respectively, are used to describe the interactions between electrons and ions. The full-potential all-electron calculations of the equation of states for NaB 4 compound were performed with the full-potential linearized augmented plane-wave method as implemented in the WIEN2K code [42] to examine the validity of the selected PAW pseudopotentials under high pressure (Fig. S0). The Perdew-Burke-Ernzerhof (PBE) [43] functional within the generalized gradient approximation (GGA) [44] is used to account for the exchange-correlation energy. A cutoff energy of 800 eV and a Monkhorst-Pack [45] k-point grid with a reciprocal space resolution of 2π × 0.03 Å −1 in the Brillouin zone yield an excellent convergence for the Gibbs free energy. Electron-phonon coupling calculations within the density functional perturbation theory and the plane-wave pseudopotential method with ultrasoft pseudopotentials are implemented in the QUANTUM ESPRESSO package [46]. More detailed illustrations of structure search and computational details can be found in the Supplemental Material [47].

A. Phase stability
Here we mainly focused on B-rich Na-B compounds, which have a tendency to form three-dimensional B frames. We conduct an extensive structure search on Na-B compounds with a variety of Na m B n (m = 1, n = 1-20; m = 2, n = 1, 3,5,7,9,11,13,15,17,19; m = 3, n = 2, 20) compositions at 0 K and the following pressures, 1 atm and 10, 25, 50, and 100 GPa. The relative thermodynamic stabilities of the considered Na m B n compounds are shown in the convex hull diagram [48] (Fig. S1). The thermodynamic stable phases sitting at the solid line are denoted by solid spheres, whereas the metastable phases, represented by open spheres, decompose into elemental Na [49,50] and B [23] solids or other Na m B n phases. At ambient pressure, the already known Cmmm Na 3 B 20 and I2 1 2 1 2 1 NaB 15 are reproduced in our structure search [35]. As can be seen in Fig. 1, I2 1 2 1 2 1 NaB 15 shows a much larger stable pressure range than Cmmm Na 3 B 20 . On the other hand, the predicted phases and phase transition pressures for NaB 3 are in excellent agreement with the results of Zhou et al. (Fig. S2) [34]. All these results demonstrate that our adopted structure search method and pseudopotentials are suitable for the Na-B system.
At higher pressures, several new B-rich stoichiometries (i.e., Na 2 B 3 , NaB 4 , and Na 2 B 17 ) emerge on the convex hull (Fig. S1). Their stable pressure ranges are shown in Fig. 1. In more detail, I4/mmm Na 2 B 3 is predicted to be stable above 15.4 GPa. For NaB 4 , there are two stable phases in the pressure range considered: Immm NaB 4 becomes stable at 38.4 GPa and then transforms into I4/mmm NaB 4 above 78.1 GPa. Na 2 B 17 , with a higher B content, stabilizes above 75.6 GPa. Na-rich C2/m Na 3 B 2 begins to be stable above 67.5 GPa. For compounds containing light elements, zero-point energy (ZPE) potentially has a large contribution to the total Gibbs free energy, which might influence their relative stability [51][52][53]. However, the inclusion of ZPE for the predicted sodium borides at the selected pressure of 100 GPa did not change their relative stabilities (Fig. S3). Phonon spectra calculations [54,55] demonstrate that all Na m B n compounds are dynamically stable, with no imaginary frequency modes at any high-symmetry direction in the whole Brillouin zone (Fig. S4).

B. Crystal structures
We mainly focus on the phases of NaB 4  phase is predicted to have an orthorhombic structure [space group Immm, 4 f.u. per cell, Fig. 2(a)]. There appears B 7 pentahedrons with B-B bond lengths of 1.65-1.80 Å at 50 GPa, and these pentahedrons make a B network via edge and face sharing. Each Na atom is 12-fold coordinated, with B forming a face-sharing Na-B polyhedra with Na-B bond lengths of 2.30-2.49 Å [ Fig. 2(b)]. The high-pressure NaB 4 phase has a tetragonal structure [space group I4/mmm, 4 f.u. per cell, Fig. 2(c)], which is isostructural to LiB 4 [56]. Strikingly, it shows B 18 icosahedrons [ Fig. 2 More interestingly, Na atoms form linear chains located in the skeletal channels. Notably, Na atom in Na 2 B 17 exhibits the highest coordination numbers (up to 19) among sodium borides. Electron localization function (ELF) [57] analysis shows that B-B bonds in three-dimensional frameworks of NaB 4 and Na 2 B 17 are covalent (Fig. S7); while the Na-B bonding is weakly ionic, which is also supported by the Bader charge analysis (Table S3). To analyze the relative bond strength between Na-B and B-B bonds, the integrated crystal orbital Hamilton populations (ICOHP) are employed as implemented in the LOBSTER package [58,59]. The resulting ICOHPs of Na-B and B-B pairs in I4/mmm NaB 4 are −0.033 and −3.410 eV/pair, respectively, indicating that the Na-B interaction is much weaker than that of B-B. A silicon allotrope, Si 24 , has been obtained from Na 4 Si 24 through removing Na 174507-3 atoms [18]. For comparison, the calculated ICOHPs of Na-Si and Si-Si pairs in Na 4 Si 24 are −0.066 and −2.544 eV/pair, respectively. This indicates that the Na-B interaction in NaB 4 is weaker than Na-Si in Na 4 Si 24 , whereas the B-B bond strength is stronger than Si-Si in Na 4 Si 24 . Similar results can be also seen in Pm Na 2 B 17 (Table S4).
Motivated by the unique structural characteristics of NaB 4 and Na 2 B 17 (covalent open B channels and Na located in the center of the open channel), we explore their energy and structural stability after removing Na atoms. The resulting structures, referred to as I4/mmm B 4 and Pm B 17 , are 0.24 and 0.25 eV/atom higher in energy than α-B 12 [23] at 1 atm (Table S5), respectively. These are lower formation enthalpies than P6/m-Si 6 (0.35 eV/atom) [19] relative to Si in the diamond structure at 1 atm, indicating that they are metastable and could be synthesized at certain conditions. Their structures are comparable to the B sublattice in I4/mmm NaB 4 and Pm Na 2 B 17 (their B-B covalent bonds are still maintained,  (Table S6) [60]. In addition, we have also performed molecular dynamical simulations [61] at 1000 and 1500 K with a time step of 1.0 fs to verify their thermal stability. The snapshots of the resulting structures clearly indicate that I4/mmm B 4 and Pm B 17 also remain stable at these temperatures (Fig. S8), which supports their practical applicability.
Already known B allotropes at ambient and high pressures present a myriad of different structural characteristics with various interesting properties. For example, α-B 12 , consisting of edging-sharing 20 boron trigons, is nonmetallic at ambient pressure [23]. High-pressure insulating ionic γ -B 28 phase consists of cationic B 2 pairs and anionic icosahedral B 12 containing 20 boron trigons, shows a reduction of the band gap with pressure similar to that of α-B 12 [23]. Metallic α-Ga-type B, transformed from α-B 12 above 74 GPa, which is regarded as the modification of B 12 icosahedra [27], is a superhard material. Under higher pressure (>375 GPa) B 10 phase presents a linear atomic chain combined with an isosceles triangle, exhibiting a high-T c superconductivity of 44 K at 400 GPa [29].  [62]. More recently, it has been reported that t-B 106 at ambient conditions is just slightly less stable than β-B [63], consisting of interstitial B atoms and interpenetrating icosahedra and α-B 12 [64]. However, compared with boron units of the known B allotropes at ambient and high pressures that we have briefly described above, the clathrate boron structural units in I4/mmm B 4 and Pm B 17 we are predicting in this work are further dismembered into smaller B subunits, which usually induce the presence of interesting properties. In the I4/mmm B 4 , B 18 icosahedron can be seen as the edge-sharing stacking of sixteen trigons and four hexagons [ Fig. 2(g)], while in Pm B 17 , B 11 -I has six trigons and six irregular squares. B 14 includes 16 trigons and 4 irregular squares. B 11 -II is composed of eight trigons, two irregular squares, and two pentagons [ Fig. 2(h)]. Considering the strong B-B covalent bonding, the threedimensional B framework in I4/mmm B 4 and Pm B 17 and the excellent hardness in B-based compounds [65][66][67][68], we have also explored the hardness of the predicted new boron allotropes with the strain-stress method [69] and the Voigt-Reuss-Hill approximation [70]. Based on the empirical Vickers hardness model H v = 2.0(k 2 G) 0.585 − 3.0, k = G/B, where G and B are the shear and bulk modulus [71], the calculated hardness values of I4/mmm B 4 and Pm B 17 are 27.3 and 26.8 GPa, respectively, indicating that they are good candidates as hard materials.

C. Electronic and superconducting properties
Much effort has been made to explore the superconductivity of B allotropes [72][73][74][75]. To be noted, two-dimensional (2D) boron allotropes become superconducting due their unique structure and inherent metallity [72,73] [74], and 17.9 K for 2D boron layer [75]). The calculated electron-phonon coupling parameters λ of I4/mmm B 4 and Pm B 17 at 1 atm are 0.65 and 0.67, respectively, comparable to 0.61 in MgB 2 [83]. As clearly revealed in Fig. 3(g), the electron-phonon coupling (EPC) constant of I4/mmm B 4 mainly arises from the contribution of low-frequency phonon modes below 23.9 THz, giving the dominant contribution (97.8%) to the integral EPC constant λ. The same is observed in Pm B 17 , in which low-frequency vibrations below 27.1 THz contribute 96.7% of the total λ [ Fig. 3(h)]. Therefore, the superconducting mechanism is quite simi-lar to that of the superconducting Li 6 P electride [84], but different from the high-frequency H-derived vibrations of high-T c H 3 S [7,12], H 3 Se [85], and CaYH 12 [86] superconductors and intermediate-frequency H-derived vibrations of H 4 Te [87].

IV. CONCLUSIONS
In summary, four hitherto unknown Na-B compounds (i.e., Na 3 B 2 , Na 2 B 3 , NaB 4 , and Na 2 B 17 ) are identified with first-principle swarm-intelligence structural search calculations. The arrangements of B atoms in these compounds evolve sequentially with increasing the B content from onedimensional B linear chains to B 6 octahedra, B 7 pentahedra, and three-dimensional B frameworks. Removing Na atoms from I4/mmm NaB 4 and Pm NaB 17 , two metallic I4/mmm B 4 and Pm B 17 allotropes become very stable at ambient conditions. Besides being potential hard compounds, among the already known three-dimensional boron allotropes, they show the highest superconducting transition temperatures of 19.8 and 15.4 K. Our work provides an effective way to get superconducting boron allotropes.