Potential high-Tc superconductivity in ZrB2 polymorph under pressure

Potential high-Tc superconductivity in ZrB2 polymorph under pressure

Computational Materials Science 176 (2020) 109517 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.el...

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Computational Materials Science 176 (2020) 109517

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Potential high-Tc superconductivity in ZrB2 polymorph under pressure a

a

a,b

a

Feifei Ling , Lingjuan Hao , Kun Luo , Zhikang Yuan , Yufei Gao ⁎ ⁎ Zhisheng Zhaoa, Yang Zhanga,b, , Dongli Yua, a b

a,b

a

T a,b

, Qi Gao , Yingmei Li

,

Center for High Pressure Science (CHiPS), State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, PR China Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao 066004, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Zirconium diboride Novel phases Electronic structure Superconductivity Electron-phonon coupling

The clear proof of superconductivity in transition-metal diborides was rarely reported. In the current work, the recently developed particle swarm optimization structural search method was utilized to propose three ZrB2 structures, namely, I41/amd-ZrB2, P42/mmc-ZrB2, and P4/nmm-ZrB2. The structural stability of the three novel ZrB2 phases was confirmed on the basis of elastic constant and phonon dispersion calculations. At ambient pressure, the mechanical properties of the I41/amd-ZrB2 and P42/mmc-ZrB2 phases are comparable to those of AlB2–ZrB2. Electron–phonon coupling (EPC) calculations revealed that the P4/nmm-ZrB2 phase is predicted to be a potential high-Tc superconductor with a calculated Tc of 12.7 K at 20 GPa. Moreover, significant pressureinduced EPC enhancement can also be found in the P4/nmm-ZrB2 phase. The maximum EPC constant λ and Tc under 600 GPa are 1.05 and 34.4 K, respectively.

1. Introduction Since the discovery of the superconductivity of MgB2 (Tc = 39 K) [1] in the AlB2 structure (P6/mmm), considerable efforts [2–12] have been devoted to searching for new superconductors in related compounds with a structure and chemistry similar to that of MgB2. Particularly some d-type transition-metal diborides (TMB2) [4], which are claimed to combine average coupling constants with comparable phonon frequencies to those of MgB2. The first superconductors reported in TMB2 compounds were NbB2 and MoB2, whose experimentally determined superconductivity critical temperatures were 3.87 and 7.45 K [13], respectively. Gasparov et al. [14] reported an experimentally observed superconducting transition of polycrystalline ZrB2 compounds with Tc = 5.5 K. However, these observations could not be confirmed in later studies [15–17]. Experimental studies [15,18,19] predicted that the occurrence of superconductivity is derived from the nonstoichiometry in TMB2 compounds. Similar situations apply for TaB2 and BeB2 compounds, whose superconductivity critical temperatures were experimentally determined to be 9.5 [20] and 0.7 K [21], respectively. However, the follow-up studies had reached contradictory conclusions [6,22,23]. In summary, to date, there is no clear proof of superconductivity that could be established for the majority of TMB2 materials. Numerous studies concentrating on electronic structure [5,24,25],

bond iconicity [26], Fermi surface [27], phonon spectra [28], and electron–phonon interaction [4,6,16] were performed to offer theoretical explanations for the lack of superconductivity in these TMB2 materials. Several important aspects affecting the superconductivity of TMB2 materials are summarized: (1) strong covalent interactions between boron and metal layers are enabled by the presence of d-electrons on metal; and (2) electron–phonon interaction plays an essential role in the occurrence of superconductivity for TMB2 materials, whose electron–phonon interaction is fairly weak. In general, pressure can effectively shorten the interatomic distance of materials, and consequently, significantly alter their electronic bonding states to modify the physical properties and/or induce the formation of new physical states. Experimental and theoretical research [29–33] confirmed that pressure can effectively regulate the phonon frequency and tune the electron–phonon coupling (EPC) in various materials. Therefore, pressure may be a promising method to acquire new superconductors or alter the superconducting properties of materials. In the case of ZrB2, experimental compression up to 50 GPa revealed no obvious phase transition [34]. The theoretical research performed by Ma et al. [35] found no high-pressure phase transitions up to 300 GPa. Two new ZrB2 structures were proposed by Pan et al. [36], but no superconductivity was reported. As an exploratory study, extensive particle swarm optimization (PSO) structural searches up to 500 GPa were performed to uncover the

⁎ Corresponding authors at: Center for High Pressure Science (CHiPS), State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, PR China (Y. Zhang). E-mail addresses: [email protected] (Y. Zhang), [email protected] (D. Yu).

https://doi.org/10.1016/j.commatsci.2020.109517 Received 8 November 2019; Received in revised form 21 December 2019; Accepted 5 January 2020 0927-0256/ © 2020 Published by Elsevier B.V.

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structure constructed from 2D kinked B2–B1–B2 atom layers separated by Zr–Zr atom layers along the c axis. Table 1 presents the optimized lattice constants and atomic positions of the corresponding structures of ZrB2 under various pressure conditions. To evaluate the stability of the proposed structures, the elastic constants of the crystals are calculated by the finite strain technique implemented in CASTEP. The three predicted structures of ZrB2 are all tetragonal structures, so there are six independent elastic constants C11, C12, C13, C33, C44, and C66, which are given in Table 2. The mechanical stability criteria for the tetragonal structures are as follows: Cii > 0, (i = 1, 3, 4, 6), (C11 − C12) > 0, (C11 + C33 − 2C13) > 0, and [2(C11 + C12) + C33 + 4C13] > 0 [50,51]. As shown above, all of the three new crystal structures satisfy the generalized elastic stability criteria at their corresponding pressures. The absence of imaginary frequencies in the entire Brillouin zone (Fig. 2) also demonstrates that the three novel structures of ZrB2 are dynamically stable at their corresponding pressures. The existence of the imaginary frequency for the P4/nmm-ZrB2 at low-pressure conditions (< 20 GPa) suggests that it is dynamically unstable below 20 GPa. The relative enthalpy difference curves of the three proposed ZrB2 phases with respect to the AlB2–ZrB2 phase in the pressure range of 0–1000 GPa are presented in Fig. 3. The ReB2-type and RuB2-type ZrB2 unraveled by Pan et al. [36] are also considered for comparison. Fig. 3 confirms that the enthalpies of the predicted I41/amd-ZrB2 and P42/ mmc-ZrB2 are obviously lower than those of ReB2-type and RuB2-type ZrB2. At ambient pressure, the enthalpies of I41/amd-ZrB2 and P42/ mmc-ZrB2 are slightly higher than that of the AlB2–ZrB2 phase by approximately 57 and 64 meV/atom, respectively. In combination with the structural stability analysis, we conclude that I41/amd-ZrB2 and P42/mmc-ZrB2 are metastable phases at ambient pressure. It is worth mentioning that the enthalpy of the P4/nmm-ZrB2 phase decreases with increasing pressure, and becomes more stable than the AlB2–ZrB2 phase above 800 GPa. In terms of the current state of technological prowess, the highest static pressure currently available in the lab exceeds 1 TPa, and the dynamic pressure of higher than 100 TPa can be achieved by laser-based compression [52]. This outcome suggests that P4/nmm-ZrB2 is very likely to be synthesized under high pressure and can be significantly favorable for applications under extreme conditions.

high-pressure structures of ZrB2. Three novel ZrB2 structures were predicted in combination with the first-principle calculations. The structural characteristics, stability, electronic structure, electron–phonon interaction, and superconductivity under different pressure conditions were systematically investigated. 2. Computational methods The well-developed CALYPSO code [37] was carried out to search for the new ZrB2 structures. This code is based on PSO, with simulation cell sizes of 1–4 formula units (f.u.) at 0, 200, 400, and 600 GPa, respectively. CALYPSO is designed to predict metastable or stable structures only knowing the chemical composition of a given compound at certain external conditions [38–41]. The structural relaxation and total–energy calculation were carried out using the density functional theory implemented in the CASTEP code [42,43]. The interaction between the ion and the electron was treated by ultrasoft pseudopotentials. The 4d25s2 and 2s22p1 were treated as the valence electrons of Zr and B atoms, respectively. A plane wave energy cutoff of 450 eV was used in the calculation to achieve a total energy convergence within 1 meV per atom. A k-point spacing (2π × 0.015 Å−1) was used to generate Monkhorst-Pack k-point grids for Brillouin zone sampling. Electronic band structures were performed by using density functional theory with Perdew–Burke–Ernzerhof generalized gradient approximation [44] in the Vienna ab initio Simulation Package code [45]. Phonon spectra were performed through a supercell approach by using PHONOPY code [46]. The EPC calculations were employed through the Quantum ESPRESSO package [47] with ultrasoft pseudopotentials. The dynamical matrices and EPC were implemented using density functional perturbation theory in the linear response method [48]. Kinetic energy cutoff values were adopted to be 40 Ry for the three new structures of ZrB2. The k-grid and q-grid were integrated over the Brillouin zone as 20 × 20 × 12 and 5 × 5 × 3, respectively, according to the scheme [49] proposed by Monkhorst and Pack. 3. Results and discussion 3.1. Structural features and stability Our simulations indicate that the stable structure for ZrB2 is definitely the AlB2-type crystalline structure (P6/mmm space group, No.191), which is consistent with experimental results [49]. Hereafter, the crystalline structure is denoted as AlB2–ZrB2. Besides the lowest energy structures, two new structures with energies approximately 60 meV/atom higher than those of the AlB2–ZrB2 structure are obtained at normal pressure. These two predicted ZrB2 structures are tetragonal and isostructural with the metastable α-TiB2 and β-TiB2 [41], respectively. We denote these two metastable structures as I41/amd-ZrB2 and P42/mmc-ZrB2, respectively. Under the pressure of 400 GPa, another new ZrB2 polymorph with tetragonal structure was uncovered, which is denoted as P4/nmm-ZrB2. The three new predicted structures of ZrB2 are shown in Fig. 1. In I41/amd-ZrB2 (Fig. 1a), the B and Zr atoms have only one equivalent position. Each B atom is composed of sp2 hybridization with the three nearest B atoms. The B atomic layers are composed of several zigzag B chains arranged in parallel. Two adjacent B layers are perpendicularly oriented with each other. The Zr atomic layer is inset between these two adjacent B layers. The P42/mmc-ZrB2 structure (Fig. 1b) is somewhat similar to that of I41/amd-ZrB2. In the P42/mmcZrB2 structure, two styles of B atoms form a tri-coordinate with the nearest-neighbor and next-nearest-neighbor B atoms. The B layers are also composed of zigzag B chains arranged in parallel. In contrast with I41/amd-ZrB2, the P42/mmc-ZrB2 structure can be viewed as an alternative stacking of two perpendicularly oriented double B layers. The Zr atomic layers are inset between two adjacent B layers. As shown in Fig. 1c, P4/nmm-ZrB2 exhibits a sandwich-like layer

3.2. Mechanical properties The mechanical properties of the three predicted structures were investigated. The experimental stable AlB2-ZrB2 structure was also considered for comparison. The elastic constants (Cij), bulk modulus (B), and shear modulus (G) were evaluated. The brittle-or-ductile behavior was estimated using Pugh's ratio (B/G). The theoretical Vickers hardness was systematically explored using two semiempirical methods proposed by Šimůnek [54] and Chen [54]. Table 2 illustrates these results. It can be seen that Šimůnek’s semiempirical model could accurately predict the hardness of ZrB2 compounds. The calculated hardness for the AlB2-ZrB2 structure at 0 GPa is 24 GPa, which is in satisfactory agreement with the experimentally reported values of 23 GPa [49]. The calculated hardness for I41/amd-ZrB2 and P42/mmc-ZrB2 structure are both 23.8 GPa that be comparable with that of AlB2-ZrB2. The B/G values of such phases are 1.08 and 1.07, smaller than the critical value of 1.75, indicating that they are intrinsically brittle. Interestingly, P4/nmm-ZrB2 exhibits distinct mechanical properties from I41/amd-ZrB2 and P42/mmc-ZrB2. As we have discussed above, the P4/ nmm-ZrB2 structure could exist stably above 20 GPa, therefore its mechanical properties at 20 GPa were discussed. Its calculated hardness at 20 GPa is 17.4 GPa, which is much lower than that of AlB2-ZrB2 (26.4 GPa at 20 GPa). Combined with its B/G value of 1.93, indicating that the P4/nmm-ZrB2 phase has excellent deformability. In general, B represents the resistance to volume change by loading pressure, and G represents the resistance to deformations upon shear stress. Traditionally, it is accepted that the shear modulus is closely 2

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(a)

c a

c

c b

a

b

b

a

(b)

c

c b

a

c a

b

a

b

(c)

b

c a

c

b

a

Fig. 1. Three new predicted structures of ZrB2: (a) the crystal structure of I41/amd-ZrB2 (No.141) at ambient pressure, (b) the crystal structure of P42/mmc-ZrB2 (No.131) at ambient pressure, and (c) the crystal structure of P4/nmm-ZrB2 (No.129) at 20 GPa. The B1, B2, Zr1, and Zr2 atoms are represented by orange, purple, green, and cyan balls, respectively. Table 1 Calculated lattice parameters of the a, b, and c, and atom positions for the predicted ZrB2 structures. Structure

Space group

a (Å)

b (Å)

c (Å)

Wyckoff position

x

y

z

AlB2-ZrB2 (Exp. [49])

P6/mmm

3.17

3.17

3.53

AlB2-ZrB2 (0 GPa) (Our work)

P6/mmm

3.166

3.166

3.540

I41/amd-ZrB2 (0 GPa)

I41/amd

3.333

3.333

11.062

P42/mmc-ZrB2 (0 GPa)

P42/mmc

3.332

3.332

11.069

P4/nmm-ZrB2 (20 GPa)

P4/nmm

3.2153

3.2153

5.6691

Zr (1a) B (2d) Zr (1a) B (2d) Zr (4a) B (8e) Zr1 (2c) Zr2 (2e) B1 (4 h) B2 (4i) Zr (2c) B1 (2a) B2 (2c)

0 1/3 0 1/3 0.5 0.5 0 0 0.5 0 0 0.5 0

0 2/3 0 2/3 1.0 1.0 0.5 0 0.5 0.5 0.5 0.5 0.5

0 1/2 0 1/2 −0.25 −0.66809 0 0.75 0.66798 0.58209 0.32807 0 0.83156

3

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Table 2 Calculated elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), B/G and hardness (GPa) of AlB2-type and new predicted ZrB2 phases under different pressure values. Crystal

C11

C12

C13

C33

C44

AlB2-ZrB2 (0 GPa) AlB2-ZrB2 (20 GPa) I41/amd-ZrB2 (0 GPa) P42/mmc-ZrB2 (0 GPa) P4/nmm-ZrB2 (20 GPa)

568 695 470 474 414

49 84 135 132 224

120 179 90 90 195

437 550 592 591 386

255 316 227 231 181

C66

B

G

B/G

HvExp

H va

Hvb

236.4 282.1 222.1 223.9 140.5

1.015 1.11 1.08 1.07 1.93

23 [49]

262 258 167

240.0 313.5 240.1 240.3 271.0

24.0 26.4 23.8 23.8 17.4

45.1 45.0 40.1 40.6 13.7

HvExp The experimental hardness value. Hva The calculated hardness with Šimůnek’s semiempirical model [53]. Hvb The calculated hardness with Chen’s semiempirical model [54].

Fig. 2. Calculated phonon dispersions of the three new structures for ZrB2: (a) I41/amd-ZrB2 at ambient pressure, (b) P42/mmc-ZrB2 at ambient pressure, and (c) P4/ nmm-ZrB2 at 20 GPa.

amd-ZrB2 and P42/mmc-ZrB2 are somewhat similar to those of AlB2–ZrB2 and reveal metallic character as there are some bands cross over the Fermi level (Ef). It can be clearly seen that the PDOS of Zr 4d and B 2p electrons have similar shapes from approximately −2 to −3 eV. This finding indicates the presence of significant hybridization between Zr 4d and B 2p orbitals for the I41/amd-ZrB2 and P42/ mmc-ZrB2 structures. This fact also shows a strong covalent interaction between the Zr and B atoms. As shown in Fig. 4b and c, the typical feature of these two borides is that there is a deep valley, namely, the pseudogap located at the Fermi level. The Ef is lying on the pseudogap in I41/amd-ZrB2 and P42/mmc-ZrB2 suggesting that the p-d bonding states started to be saturated. The presence of strong covalent bonds and the nearly full occupation of bonding states results in the high modulus and high hardness in these two ZrB2 compounds. In addition, the σ-bands (p-x,y) along the G–Z paths are completely occupied and far below the Fermi level for the I41/amd-ZrB2 and P42/mmc-ZrB2 structures (Fig. 4b and c). Hence, I41/amd-ZrB2 and P42/mmc-ZrB2 lose the ability to split the degeneracy of this band through EPC [55], thereby indicating that they could be nonsuperconducting phases. The electronic band structure and PDOS of P4/nmm-ZrB2 (Fig. 4d) at 20 GPa are somewhat different from those of AlB2–ZrB2. Two bands cross the Fermi level for P4/nmm-ZrB2, which reveals its metallic character. The PDOS for B-px,y, and B-pz orbitals are relatively flat over the energy window of −4.5 to 4.5 eV, indicating the weak hybridization between Zr 4d and B 2p orbitals, and revealing the weak covalent interaction between the Zr and B atoms. Traditionally, the formation of covalent bonding between transition metals and light elements (B, C, N) is the key factor for the high hardness of the transition-metal borides, carbides, and nitrides. The relative weak covalent interaction between the Zr and B atoms in P4/nmm-ZrB2 is responsible for its low hardness. In addition, it was concluded that the loss of covalent character within the metal-boron bonding is the key factor detrimental to superconductivity in transition–metal diborides [55]. Moreover, the B 2p electrons have a large contribution to the PDOS at the Fermi level (the right panel of Fig. 4d), which quite closely resembles that of MgB2. A highly degenerate flat band that originates from the B-px,y orbitals along the G–Z path appears near the Fermi level, which might help to enhance the electron–phonon interaction [56,57]. These indications of potential

Fig. 3. Relative enthalpy difference curves of various ZrB2 structures with respect to the AlB2–ZrB2 phase as a function of pressure.

related to the hardness of the material. The direct evidence for this conclusion could be shown in current work. Specifically, the P4/nmmZrB2 structure possesses a very large bulk moduli of 271 GPa, while the corresponding hardness value is as small as 17.4 GPa. In contrast, from the current calculation, it is found there is a general hardness trend in the sequence of AlB2-ZrB2 (20 GPa) > AlB2-ZrB2 (0 GPa) > P42/ mmc-ZrB2 (0 GPa) = I41/amd-ZrB2 (0 GPa) > P4/nmm-ZrB2 (20 GPa). This trend is almost the same with that of their shear modulus. 3.3. Electronic properties The electronic properties of the three ZrB2 phases were investigated by analyzing their electronic band structures and the partial density of states (PDOS) (Fig. 4). The experimentally stable AlB2–ZrB2 phase was also considered for comparison. The agreement with earlier calculations [5,24] of the electronic band structures and PDOS for the AlB2–ZrB2 phase is excellent. The band structures and PDOS for I41/ 4

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Fig. 4. Band structures and PDOS for (a) AlB2–ZrB2 at ambient pressure, (b) I41/amd-ZrB2 at ambient pressure, (c) P42/mmc-ZrB2 at ambient pressure, and (d) P4/ nmm-ZrB2 at 20 GPa. The projections are made on the B-s, B-px,y, and B-pz orbitals within the band structures. The sizes of the green squares, red circles, and blue triangles represent the characteristic weights of the B-s, B-px,y, and B-pz orbitals, respectively.

intermediate-frequency branches (8–13 THz) resulted from the coupled vibrations of Zr and the two B atoms in the unit cell. The high-frequency modes (13–22 THz) primarily originated from the B atoms. The highfrequency modes (13–22 THz at 20 GPa and 35–46 THz at 600 GPa) revealed a large phonon linewidth along the G–Z path (Fig. 5a and d), with a high contribution to the Eliashberg spectral function α2F(ω) and EPC parameter λ. The large gap in the phonon DOS of TMB2 with AlB2 structure would lead to the decoupling of the transition metal and B vibrations [16]. In contrast to these TMB2 compounds [6,16], the broadening of the B peaks is somewhat similar to that of MgB2 which reveals the signature of strong EPC [4] for the P4/nmm-ZrB2 phase. Therefore, the calculated EPC parameter λ is 0.73, which is considerably larger than that of AlB2–ZrB2. The logarithmic average

superconductivity stimulated us to conduct EPC calculations for the P4/ nmm-ZrB2 phase under various pressure conditions. 3.4. Superconductivity The phonon dispersions, total and projected phonon DOS (PHDOS), and EPC calculations were carried out to explore the superconductivity of the three new phases. Only P4/nmm-ZrB2 was proven to be superconducting. As previously indicated, the P4/nmm-ZrB2 phase is dynamically stable above 20 GPa. The superconductivity of the P4/nmmZrB2 was investigated at 20 and 600 GPa (Fig. 5). The PHDOS can be separated into three regions (Fig. 5b). The low-frequency modes (0–8 THz) are mainly related to vibrations of heavy Zr atoms. The 5

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resulted in the maximum critical temperature of 34.4 K under the pressure of 600 GPa. If μ* was to be considered from 0.13 to 0.1 (Fig. 6), then the value of Tc would change from 9.95 K to 12.7 K at 20 GPa, and from 29.8 K to 34.4 K at 600 GPa. 4. Conclusion In conclusion, various ZrB2 compounds were systematically investigated on the basis of the PSO algorithm combined with first-principle calculations. Three metastable phases, namely, I41/amd-ZrB2, P42/mmc-ZrB2, and P4/nmm-ZrB2 were proposed. The I41/amd-ZrB2 and P42/mmc-ZrB2 phases could be stable at ambient pressure with the mechanical properties comparable to that of AlB2-ZrB2. The P4/nmmZrB2 phase was predicted to be stable above 20 GPa and energetically favored relative to the experimentally stable AlB2–ZrB2 structure at an extreme pressure condition of 800 GPa. The electron–phonon coupling calculations revealed that the P4/nmm-ZrB2 phase is a superconductor with a Tc of 12.7 K at 20 GPa. The pressure-enhanced electron–phonon interaction was observed in the P4/nmm-ZrB2 phase. An unexpected high Tc of 34.4 K was predicted under the pressure of 600 GPa. It is expected that this comprehensive structure search can stimulate future theoretical and experimental investigations on high-Tc transition-metal diboride superconductors. CRediT authorship contribution statement Feifei Ling: Conceptualization, Data curation, Investigation, Validation, Visualization, Writing - original draft. Lingjuan Hao: Data curation, Software, Methodology. Kun Luo: Supervision, Methodology, Formal analysis. Zhikang Yuan: Supervision, Investigation, Formal analysis. Yufei Gao: Supervision, Formal analysis. Qi Gao: Software, Validation, Writing - original draft. Yingmei Li: Software, Validation, Writing - original draft. Zhisheng Zhao: Software, Methodology. Yang Zhang: Formal analysis, Writing - review & editing. Dongli Yu: Formal analysis, Funding acquisition, Writing - review & editing.

Fig. 5. Calculated phonon dispersions for P4/nmm-ZrB2 under pressure of (a) 20 GPa and (d) 600 GPa. The red solid circles show the phonon linewidth with a radius proportional to its strength. The middle panels (b) and (e) show the contributions of the Zr and B atoms to the projected phonon DOS (PHDOS) at 20 and 600 GPa, respectively. The Eliashberg phonon spectral function α2F(ω) and electron–phonon integral λ(ω) of the P4/nmm-ZrB2 under 20 and 600 GPa are shown in (c) and (f), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 51772263 and 51072174). References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Superconductivity at 39 K in magnesium diboride, Nature 410 (2001) 63–64. [2] K.P. Bohnen, R. Heid, B. Renker, Phonon dispersion and electron-phonon coupling in MgB2 and AlB2, Phys. Rev. Lett. 86 (2001) 5771–5774. [3] J. Kortus, I.I. Mazin, K.D. Belashchenko, V.P. Antropov, L.L. Boyer, Superconductivity of metallic boron in MgB2, Phys. Rev. Lett. 86 (2001) 4656–4659. [4] R. Heid, B. Renker, H. Schober, P. Adelmann, D. Ernst, K.P. Bohnen, Lattice dynamics and electron-phonon coupling in transition-metal diborides, Phys. Rev. B 67 (2003) 180510. [5] A.L. Ivanovskii, Band structure and properties of superconducting MgB2 and related compounds (a review), Phys. Solid State 45 (2003) 1829–1859. [6] P.P. Singh, Theoretical study of electron-phonon interaction in ZrB2 and TaB2, Phys. Rev. B 69 (2004) 094519. [7] A. Floris, G. Profeta, N.N. Lathiotakis, M. Lüders, M.a.L. Marques, C. Franchini, E.K.U. Gross, A. Continenza, S. Massidda, Superconducting properties of MgB2 from first principles, Phys. Rev. Lett. 94 (2005) 037004. [8] Y. Singh, A. Niazi, M.D. Vannette, R. Prozorov, D.C. Johnston, Superconducting and normal-state properties of the layered boride OsB2, Phys. Rev. B 76 (2007) 214510. [9] Y. Ma, Y. Wang, A.R. Oganov, Absence of superconductivity in the high-pressure polymorph of MgB2, Phys. Rev. B 79 (2009) 054101. [10] S.T. Renosto, H. Consoline, C.a.M. Dos Santos, J. Albino Aguiar, S.-G. Jung,

Fig. 6. Superconducting transition temperature Tc as a function of μ* for P4/ nmm-ZrB2 at 20 and 600 GPa.

frequency ωlog is 331.6 K. At the Coulomb pseudopotential μ* = 0.1, the resultant Tc value for P4/nmm-ZrB2 under 20 GPa is 12.7 K based on the Allen-Dynes modified McMillan equation. The phonon modes became harder with increasing pressure (Fig. 5d). The evidence of the pressure-enhanced EPC phenomenon was observed (Fig. 5f). The EPC parameter λ underwent an obvious increase from 0.76 to 1.05, which 6

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[11]

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