Two-gap topological superconductor LaB₂ with high T_c = 30 K

Abstract

Since two gap superconductivity was discovered in MgB₂, research on multigap superconductors has attracted increasing attention because of its intriguing fundamental physics. In MgB₂, the Mg atom donates two electrons to the borophene layer, resulting in a stronger gap from the σ band and a weaker gap from the π bond. First-principles calculations demonstrate that the two gap anisotropic superconductivity strongly enhances the transition temperature of MgB₂ in comparison with that given by the isotropic model. In this work, we report a three-band (B-σ, B-π, and La-d) two-gap superconductor LaB₂ with very high T_c = 30 K by solving the fully anisotropic Migdal–Eliashberg equation. Because of the σ and π–d hybridization on the Fermi surface, the electron–phonon coupling constant λ = 1.5 is significantly larger than the λ = 0.7 of MgB₂. Our work paves a new route to enhance the electron–phonon coupling strength of multigap superconductors with d orbitals. On the other hand, our analysis reveals that LaB₂ belongs to the weak topological semimetal category, leading to a possible topological superconductor with the highest T_c to date. Moreover, upon applying pressure and/or doping, the topology is tunable between weak and strong with T_c varying from 15 to 30 K, opening up a flexible platform for manipulating topological superconductors.

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New concepts

To date, available topological superconductors such as PbTaSe₂, PdBi, RhPb₂, as well as WS₂ and SnPbTe/Pb heterostructures all belong to S-wave single-gap BCS superconductors. In our work, we present a two-gap topological superconductor LaB₂, which is a totally new concept to this field. Therefore, the impact to the topological superconductor, Majorana fermion, and quantum computing research fields will be strong. So far, available candidates such as PbTaSe₂, PdBi, RhPb₂, as well as WS₂ and SnPbTe/Pb heterostructures published in high-impact journals in recent years all have T_c below 10 K. Such a low T_c causes tremendous difficulties for accurately measuring the possible Majorana fermions experimentally. Thus, it is of great importance to discover high-T_c topological superconductors. In this work, we report a record-breaking high T_c of 30 K in the topological superconductor LaB₂, comparing highly favorably over all the available candidates.

Introduction

Superconductivity is one of the most important and mysterious research fields not only because of the intriguing fundamental physics behind it but also because of its high potential for future industry. Yet the critical temperature T_c of superconductors is generally too low for practical applications. Thus enhancing the T_c of superconductors is an essential issue in condensed matter physics. One successful approach to raise T_c is by applying pressure. In recent years, nearly room temperature T_c have been found in hydrogen-rich hydrides under high pressure such as H₃S,¹ LaH₁₀,² and NaH₆.^3,4 The other approach to enhance the T_c of superconductors is through the anisotropic pairing or multigap mechanism such as in conventional superconductor magnesium diboride MgB₂.^5,6 In particular, the anisotropic pairing of unconventional superconductors, with the Cooper pairing different from the conventional s-wave pairing, has attracted extensive attention because of the many-body interactions among d-electrons in cuprate^7–10 and iron-based superconductors.^11,12 On the other hand, topological superconductors with p-wave pairing have also been predicted in Sr₂RuO₄.¹³

MgB₂, known as a classic example of an anisotropic pairing superconductor, has shown great promise for enhancing T_c through two-gap phonon-mediated superconductivity. The Fermi surfaces (FSs) of MgB₂ comprise σ bonds (s + p_x,y orbitals of B atoms) and π bonds (p_z orbitals of B atoms).^5,14–19 The two separated FSs in reciprocal space lead to two superconducting (SC) gaps Δ_σ ≈ 7 meV and Δ_π ≈ 2–3 meV at 0 K^20–23 with the gaps closed at the transition temperature T_c = 39 K.^20–23 Theoretical investigations demonstrate that the anisotropic effect increases the superconducting temperature of MgB₂ from 19 K to 55 K in the harmonic approximation. Additionally, the anharmonicity increases the phonon energy of the E_2g mode at the Gamma point from 62.7 meV to 75.9 meV and decreases the total EPC strength from λ = 0.73 to λ = 0.61. Consequently, T_c decreases from 55 K to 39 K,⁶ being in good consistence with the experimental T_c. The two-gap behavior exhibits exotic properties such as giant-paramagnetic response,²⁴ skyrmionic states,^25,26 hidden criticality,^27,28 and time-reversal symmetry breaking.^29,30

In addition to bulk MgB₂, several systems have been proposed to exhibit multigap superconductivity. For example, two to three SC gaps with the transition-temperatures T_c = 20 K, 70 K, and 34.5 K were reported in monolayer of MgB₂, CaB₂, and ScB₂.^31,32 The trilayers of CaB₄, GaB₄, BeB₄ and LiB₄ films have multigaps with T_c of 64 K, 51 K, 39 K and 32 K, respectively.^32,33 Four SC gaps with a T_c = 52 K were predicted in trilayer system MgB₄ above bulk MgB₂.³³ Other than the magnesium boride family, multiband superconductivity has also been predicted in the hole doped LiBC with T_c = 100 K,³⁴ bulk AlB₂ with T_c = 9 K,^16,35 and thin film AlB₂ with T_c = 17–26.5 K.^16,35 These predictions show that the honeycomb layer structure with sp₂ hybridized orbitals is essential for finding multigap superconductivity. To date, bulk MgB₂ has been confirmed as a two-gap superconductor,³⁶ while the other predicted multigap phonon-mediated systems are yet to be confirmed in future experiments.

In this work, we investigate the superconductivity in the LaB₂ compound in the same crystal structure as MgB₂ through first-principles calculations. The superconducting critical temperature T^McMillan_c = 22 K is obtained for LaB₂via the Allen–Dynes modified McMillan formula, while two-gap superconductivity with a notably higher T^aniso_c = 30 K is obtained through solving the fully anisotropic Migdal–Eliashberg equations. In addition, our analysis reveals LaB₂ to be a weak topological superconductor with the highest T_c in comparison with available candidates. The topology is tunable between weak and strong with superconducting T_c varying from 15 to 30 K upon applying pressure and/or doping, serving as a flexible candidate for topological superconductors.

Results and discussion

It is well known that magnesium diboride MgB₂ exhibits two-gap superconductivity with the highest ambient-pressure T_c to date. By substituting the magnesium atom with transition-metal elements, we systematically investigated the superconductivity of the transition-metal diboride XB₂. Among them, we found that LaB₂ can be stabilized in the same crystal structure as MgB₂ under ambient pressure, while it also shows two-gap superconductivity with a fairly high T_c of 30 K. LaB₂ is composed of alternate honeycomb boron and metallic lanthanum layers. The crystal structure with space group P6/mmm is shown in Fig. 1(a) and (b). The optimized lattice parameters a = 3.32 Å and c/a = 1.30 are obtained from geometry relaxation calculations based on density functional theory (DFT).

Fig. 1 LaB₂ hexagonal crystal structure and electronic properties. (a) Side view and (b) top view of the LaB₂ lattice structure. (c)–(e) Electronic band structure of LaB₂ projected onto the B-σ and B-π, La-d, and La-f orbital, respectively. The size of the colored spheres indicates the contributions from the atomic orbitals. (f)–(g) The Fermi surface near the H and A points is dominated by the boron π bond and boron σ bond, respectively. (h) The Fermi surface is dominated by the lanthanum d bond.

The atom-orbital-decomposed band structures depicted in Fig. 1(c)–(e) show the detailed characters of each electronic band. Near the Fermi level (E_F), the doubly degenerate boron-σ bands cross E_F from Γ to A, forming a small hole pocket around A (Fig. 1(c)). Similar to MgB₂, this slowly rising flat B-σ (sp²) band along Γ–A reflects weak interlayer interactions in LaB₂ with a slightly higher E_F due to the relatively higher ionicity of La. One boron-π (p_z) band also crosses E_F and forms a hole pocket around H (Fig. 1(c)). On the other hand, the contributions of lanthanum (La) to the electronic bands plotted in Fig. 1(d and e) demonstrate that three La-d orbitals: d_z², d_x²–y², and d_xy contribute notably to the bands near the Γ- and H-points, and along the H–A paths around E_F. Particularly, the La d_xy orbital takes part in the B-π hole pocket around H, while most of the La-f bands move from 2 to 5 eV above the Fermi level (Fig. 1(e)) with nontrivial hybridizations with the B-σ hole pocket around A near E_F.

The above mentioned electronic bands form three Fermi surfaces as illustrated in Fig. 1(f)–(h). They are respectively the small hole pockets around A constructed mainly by the B-σ orbital with partially the La-f orbital, the paraboloid hole pockets around H mostly comprised by the B-π with some La d_xy, and the star-shaped cylinder Fermi surface surrounding Γ composed of the La d orbitals. With the above band character analysis, we have adopted the sp², p_z orbital of B atoms and d_z², d_x²–y², d_xy, f orbital of La atoms as the Wannier basis to build the tight-binding Hamiltonian and generate the Wannier band structure using Wannier90 code. This Wannier basis set can well reproduce the DFT band structures from −10 eV to 2 eV with high quality, and is used for the topological analysis as will be discussed later.

The superconductivity of LaB₂ is explored through the phonon dispersion and electron–phonon coupling (EPC) calculations based on density function perturbation theory (DFPT). Fig. 2(a) shows the phonon band structures with the atomic contributions resolved. The acoustic phonon modes in the low frequency region less than 16 meV are mainly derived from the vibrations of heavy La atoms coupled with light B atoms, while the light B atoms oscillate quickly and dominate the high frequency region above 32 meV. In between, there is a notable phonon band gap, indicating a low scattering rate between acoustic and optical phonons. The phonon density of states (DOSs) depicted in Fig. 2(d) shows the overall narrow bandwidth with high intensity for acoustic phonon bands, and wide bandwidth with low intensity for optical phonon bands. Fig. 2(b) shows the phonon-dependent EPC strength λ_qν of LaB₂. For the acoustic branches, there exist three Kohn anomalies around the M, L, and H points with the phonon energy significantly lowered to ∼10 meV. These three Kohn anomalies are associated with strong EPC strengths up to λ_M = 1.68, λ_L = 2.01, and λ_H = 0.6, respectively. For the optical branches, the phonon modes (point group E_2g) with significant EPC locate at ∼60 meV from Γ- to A-point. The maximum EPC strength λ_q = 0.46 here is much smaller than that of the Kohn anomalies in the acoustic branches. The Eliashberg spectral function presented in Fig. 2(c) demonstrates a strong peak from 10 to 16 meV with the Kohn anomalies dominating the lower part and with the relatively flat phonon bands dominating the higher part of the acoustic phonon bands. This strong acoustic phonon Eliashberg spectral function is closely related to the heavy atom La as can be seen from the atom-resolved phonon band structure in Fig. 2(a). As for the optical phonon modes, the Eliashberg spectral function is generally smaller than the acoustic counterparts. The integrated Eliashberg function over the low-frequency acoustic modes λ_A(ω) = 1.15 (Fig. 2(c)) is relatively strong because the Eliashberg spectral function is divided by the phonon frequency in the integral of eqn (5). Owing to the same reason, the integrated Eliashberg function over the high-frequency optical modes λ_O(ω) = 0.32 is lower than the acoustic counterpart, despite the wider bandwidth. Furthermore, the total EPC strength λ = 1.47 is calculated by integrating the Eliashberg spectral function over all phonon modes, in which the acoustic phonons contribute nearly 80% of the total λ.

Fig. 2 LaB₂ phonon related properties. (a) Phonon dispersion with atom resolved contributions. The size of the colored spheres illustrate the atomic contributions. (b) Phonon dispersion with EPC strength λ_qv indicated by the size of red spheres. (c) Eliashberg function α²F(ω) and integrated Eliashberg function λ(ω). (d) Phonon density of states (PHDOS).

To explore possible anisotropy and multigap superconductivity in LaB₂, we further solve the anisotropic Migdal–Eliashberg equations and analyze the contributions and anisotropic properties of EPC strength λ_nk for the electronic states forming the Fermi surfaces as depicted in Fig. 3. Because it has the same crystal structure as MgB₂, the Fermi surface (FS) of LaB₂ is similar to that of MgB₂ as can be seen in Fig. 3(a and b). The pipe-like FS forms a honeycomb-shape 2-dimensional network over the k_x–k_y plane with small Fermi pockets located within the honeycomb holes. Fig. 3(c and d) show that the strongest EPC (red) is distributed around the hexagon corners of the honeycomb network at the H-point. While the second strongest EPC (yellow and orange) distributes over the small Fermi pockets around the Γ- and A-points. The other regions over FS (green) show weak EPC strength only. These EPC maps demonstrate two distinct EPC features: the strongest red and the 2nd strongest yellow/orange regions depending on the k-points and the comprised atomic orbitals, evidencing the multigap superconductivity in LaB₂.

Fig. 3 EPC distribution on Fermi surface and contributions to superconducting gaps of LaB₂. (a) Top and (b) side view of the Fermi surface (FS) in the primitive BZ. Dash lines in (b) show the high symmetry lines in BZ. (c) Top and (d) side view of the momentum-resolved EPC at FS. The red color region around k_z = 0.4 KH indicates the maxima EPC region. (e) Top and (f) side view of the momentum-resolved superconductivity gap at T = 5 K. The red color region around k_z = 0.4 KH indicates the maximum SC gap region.

The corresponding EPC-induced superconducting gaps in the Fermi surfaces over the BZ are presented in Fig. 3(e and f). Good correspondence with the EPC maps in Fig. 3(c and d) can be clearly seen, showing that the electronic states on the Fermi surface with higher EPC contribute strongly to the superconducting gaps. Overall, one can see two main features in the superconducting gap distributions: (I) the large gap (Δ_nk ≃ 5 meV) regions around the zone center and H-point at k_z ≃ 0.4 indicated by red color; (II) the middle gap (Δ_nk ≃ 4 meV) areas denoted by yellow and orange colors around Γ and A points. Combining these two main features in the superconducting-gap maps (Fig. 3(e and f)) with the strongest and second strongest EPC distributions (Fig. 3(c and d)), we demonstrate clearly the two-gap superconductivity in LaB₂ similar to MgB₂.

To quantify the EPC distribution over the BZ presented in Fig. 3(c and d), the EPC distribution ρ_nk as a function of the EPC strength λ_nk is integrated through

and is depicted in Fig. 4(a). It can be seen clearly that there exist two distinct peaks at λ₁ ∼ 0.75 and λ₂ ∼ 1.5. The former is a low-intensity peak with a narrow width while the latter is a high-intensity peak with a wide width. According to the orbital character analysis based on Fig. 1(c–e) and 3(c, d), the weak λ₁ peak is mainly composed of the B-σ band (A-point) as well as the B-π and La-d_z² hybridized band (Γ-point) along the Γ–A–Γ k-path crossing the Fermi level. These two bands (B-σ and B-π La-d_z² hybridized band) contribute equally to the EPC λ₁ with the same gap (Fig. 3(e, f) and 4(b)). This is the reason why we observe only two EPC peaks in Fig. 4(a) even though we see three bands (FSs) in Fig. 1(c–h). As for the strong λ₂ peak, the main contribution is comprised of the B-π band hybridizing with two La-d orbitals (d_x²–y² and d_xy) (Fig. 1(c–e)) around the H-point along the L–H–A k-path crossing the Fermi level, which shows a very strong EPC nature. Different from MgB₂, the EPC strengths of these B-π La-d hybridized bands is much stronger than that of the B-σ bond in LaB₂. The strongly enhanced EPC of the B-π bond by hybridization with La-d orbitals demonstrates the potential of d-orbitals in superconductivity.

Fig. 4 EPC strength distribution, superconducting gap and DOS of LaB₂ (a) EPC distribution ρ_nk as a function of the EPC strength λ_nk. (b) Superconductivity gap distribution at the Fermi surface from T = 3 K to T = 30 K. Blue and red dash curves are obtained from fitting the superconductivity gap equation using the maximum gap distribution. (c) Quasiparticle density of states at T = 5 K, 15 K, 25 K, 30 K (blue, red, yellow, and purple curves, respectively). (d) Zoom in plot of (c). Gray arrows indicate the 2 gaps.

Fig. 4(b) shows the calculated superconductivity gap energy distribution versus temperature from 3 to 30 K. Overall, the two-gap features can be clearly seen. Close to the zero temperature limit, the two gap values are ∼3.9 and ∼5.1 meV for Δ₁ and Δ₂, respectively. While above T = 20 K, both the gap values decrease to zero. By using the BCS-like gap equation non-linear fit for the calculated superconducting gap functions, we obtain two curves for Δ₁ and Δ₂ as functions of temperature. The two gap curves converge to 0 at T = 30 K. Fig. 4(c and d) shows the superconductivity density of states at T = 5, 15, 25, 30 K. At T = 5 K, one can clearly see the two peaks at ±3.9 meV and ±5.1 meV. At T = 30 K, both the high and low energy SC gaps are nearly vanishing owing to the approaching superconducting transition temperature. Based on our fitting gap functions in Fig. 4(b) and superconductivity DOS in Fig. 4(c) and (d), we conclude the two-gap superconducting T_c = 30 K.

Another important research area in borophene–metal systems is topological superconduction. The Dirac nodal line and topological surface state have been discovered in MgB₂ owing to the spatial inversion symmetry and time-reversal symmetry (PT symmetry).³⁷ MgB₂ is topologically nontrivial not only electronically, but also phononically. The phononic Weyl nodal line and phononic surface arc states have been reported in MgB₂.³⁸ In addition, with Mg being replaced by other metal atoms, TiB₂ and ZrB₂ have also been theoretically reported to show PT symmetry protected electronic Dirac nodal lines.³⁹ The topological surface state and Dirac nodal line in ZrB₂ have further been detected by angle-resolved photoemission spectroscopy (ARPES).⁴⁰ On the one hand, these interesting electronic topological results are cornerstones for hosting Majorana fermions. On the other hand, the phononic topological results can support phonon-related phenomena such as electrical resistance, heat conduction, and phonon waveguides.³⁸

To explore the topological properties in LaB₂, we performed topological calculations including the Wilson loop, topological index, inverted band structure, Greens function, and surface states. Here we demonstrate the main conclusion in the topological aspect given from the Wilson loop calculations for the topological index . The other calculated results are presented in the ESI.† As can be seen in Fig. 5(a)–(d), the Berry phase of the Wannier center shows even winding numbers along k_x and k_y in BZ. On the contrary, the corresponding winding numbers are odd in both the k_z = 0 and k_z = 0.5 planes (see red lines and numbers in Fig. 5(e) and (f)). Consequently, LaB₂ belongs to the weak topological insulator category^41,42 with the index . This index also implies that except for the (0001) surface, the topological surface states exist in the other surfaces of LaB₂. Detailed topological surface states of LaB₂ can be found in the ESI.† Moreover, this topological index could be tuned from a weak to strong topological insulator by applying strain or doping, as will be discussed later.

Fig. 5 Wannier charge center (WCC) evolution (Wilson loop) in three-axis planes of BZ for time-reversal invariant LaB₂. (a) for k_x = 0.0. (b) for k_x = π/a. (c) for k_y = 0.0. (f) for k_y = π/b. (e) for k_z = 0.0 (b) for k_z = π/c. The solid red line is the reference line for determining the topological invariance. Combining all together leads to the weak topological insulator state for LaB₂.

To date, the superconducting T_c of the available topological superconductors such as PbTaSe₂,^43,44 PdBi,^45,46 RhPb₂,⁴⁷ and WS₂⁴⁸ are all below 10 K. Such low superconducting temperatures cause tremendous difficulties for accurately measuring the possible Majorana fermions experimentally.^43,49 Thus our predicted high-T_c of 30 K in topological superconductor LaB₂ is of great importance. By applying external pressures and doping levels, we carry out the three-phase diagram calculations and present the results in Fig. 6. The superconducting temperature calculated using the Allen–Dynes modified McMillan formula varies from ∼15 K to ∼30 K, indicating T_c can be further enhanced under certain conditions. Besides, the topological behavior analyzed by the topological numbers is also tunable from weak to strong topology. Both the tunable T_c and topology demonstrate that LaB₂ is a rarely found convincing candidate in the topological superconductor research field.

Fig. 6 Three-phase diagram of topological superconductor LaB₂ under various pressures and doping levels. Red dots and green triangles denote strong and weak topological insulators, respectively. Blue squares indicate the superconducting temperatures of LaB₂ calculated using the Allen–Dynes modified McMillan formula with the doping and pressure conditions indicated by the black dotted lines. Tunable T_c ranging from ∼15 K to ∼30 K and topological phases from weak to strong topological insulator are shown.

Conclusion

In summary, we have shown that LaB₂ is a three-band two-gap topological superconductor with T^aniso_c = 30 K by using DFPT calculations and solving the fully anisotropic Migdal–Eliashberg equations. First-principles DFPT calculations show that LaB₂ is dynamically stable with T^McMillan_c = 22 K from the Allen–Dynes modified McMillan formula. By including the anisotropic effect in the Migdal–Eliashberg equations, the T^aniso_c of LaB₂ can be significantly raised up to 30 K. The calculated SC gap Δ_σ(T = 0 K) and Δ_π(T = 0 K) are 5.1 meV and 3.9 meV, respectively. In comparison with MgB₂, the enhanced λ in LaB₂ is most likely due to the hybridization between the La-d orbitals and borophene π bonds. The topological index indicates that LaB₂ belongs to the weak TI category. We suggest to adopt the d orbitals as an extra factor to enhance the electron–phonon coupling strength for multigap superconductors. Furthermore, we also show that under given pressure and doping conditions, the topology and T_c in LaB₂ are both tunable from weak to strong and from ∼15 K to ∼30 K, respectively. Our work thus paves a new platform for manipulating topological superconductors.

Methods

Electronic property calculations are performed using the Vienna ab initio simulation package (VASP)^50–53 adopting the projector-augmented plane wave (PAW)^54,55 pseudopotential with the Perdew–Burke–Ernzerhof (PBE)⁶⁴ functional based on density functional theory (DFT). A k-mesh of 18 × 18 × 18 and kinetic energy cutoff of 400 eV are adopted in the electronic calculations. Ab initio phonon calculations are preformed using the Quantum Espresso code⁵⁶ based on density functional theory (DFT). Pseudopotentials with the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional are generated by the Optimized Norm-Conserving Vanderbilt Pseudopotential (ONCVPSP) code.⁵⁷ The plane wave basis with the cutoff kinetic energy of 80 Ry are used. 24 × 24 × 24 k-mesh and 6 × 6 × 6 q-mesh with Methfessel–Paxton smearing of 0.04 Ry⁵⁸ are adopted in the self-consistent electron and phonon calculations based on density function perturbation theory (DFPT). Superconducting critical temperatures T_c are calculated not only by using the Allen–Dynes modified McMillan formula⁵⁹ in eqn (2)–(5) but also by solving the anisotropic Migdal–Eliashberg equations on the imaginary-axis²⁰ in eqn (6) and (7).The logarithmic average phonon frequency ω_ln and the Eliashberg function α²F areThe contribution of electron–phonon coupling from different phonon frequencies, the λ(ω) is calculated throughTo solve the anisotropic Migdal–Eliashberg equations, the EPW (electron–phonon Wannier, electron–phonon coupling using Wannier function)^20,60,61 and Wannier90⁶² codes are utilized. The sp², p_z orbitals of B atoms and d_z², d_x²–y², d_xy, f orbitals of La atom are included in the projection of the Wannier basis with a uniform 12 × 12 × 12 k-mesh. Furthermore, for the EPC interpolation, a k-grid of 60 × 60 × 60 and q-grid of 30 × 30 × 30 are used. Matsubara frequencies for fermions, iω_j = (2j + 1)πT, are set with a cutoff energy of 0.3 eV greater than four times of the maximum phonon energy. The effective Coulomb potential and convergence threshold less 10⁻⁴ eV are used in solving the Eliashberg equations. To study topological properties, iterative surface Green's function and Wilson loop simulations are carried out via the WannierTools package⁶³ with the same tight-binding Wannier Hamiltonian.

Data availability

The data that support the findings of this study are available from the authors upon reasonable request.

Author contributions

H. T. J. conceived the project. C. H. C. and Y. S. L. carried out the DFPT calculations and topological analysis, respectively. A. H. supervised the calculations. All authors contributed to discussions and the final manuscript.

Conflicts of interest

The authors declare no competing financial or non-financial interests.

Acknowledgements

This work was supported by the Ministry of Science and Technology, Taiwan. H.-T. J. also thanks support from NCHC, CINC-NTU, AS-iMATE-109-13, and CQT-NTHU-MOE, Taiwan.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3nh00249g

‡ These authors contributed equally to this work.

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