Nontrivial d-electrons driven superconductivity of transition metal diborides

Leveraging the progress of first-principles modelings in understanding the mechanisms of superconductivity of materials, in this work we investigate the phonon-mediated superconducting properties of transition metal diborides. We report that TaB2 and NbB2 show superconducting transition temperatures as high as 27.0 and 26.0 K at ambient conditions, respectively, comparable with those obtained for CaB2 or MgB2. By mode-by-mode analysis of the electron-phonon-coupling, we reveal that the high superconducting temperature of transition metal diborides is due mainly to the strong coupling between d electrons of the transition metals and the acoustic phonon modes along out-of-plane vibrations. This fact is distinct from that of CaB2 or MgB2, where the superconductivity stems mainly from the boron px and py orbitals, which couple strongly to the optical phonon modes dominated by in-plane B atomic vibrations. Further, we find that transition metal diborides present only a superconducting gap at low temperatures, whereas CaB2 or MgB2 are double superconducting gap superconductors. In addition, we investigate the strain effect on the superconducting transition temperatures of diborides, predicting that Tc can be further enhanced by optimizing the phonon and electronic interactions. This study sheds some light on the exploring high Tc boron-based superconductor materials.


Introduction
Superconductivity is a quantum phenomenon characterized by zero-resistivity and diamagnetism [1][2][3].Superconductors showing a high transition temperature, T c , are of great interest for scientific research and technological applications [4,5].In 1911, Onnes discovered that Hg becomes a superconductor for temperatures below 4.2 K [6].Afterwards, many materials showed a superconducting transition at different temperatures [7][8][9].According to the chemical components, we classify the materials as copper-based or iron-based superconductors and so on [10].Particularly, copper-oxides via alloying with other species (e.g.HgBaCaCuO) were found to have the superconducting transition temperature as high as 135 K at normal pressure [11][12][13][14].At the same time, iron-based layered materials, such as Sm[O 1−x F x ]FeAs, have T c of about 55 K [15][16][17].Cu and Fe-based compounds are unconventional superconductors that do not follow the BCS theory, and due to their complex electronic structures and strong anharmonic effect and relatively heavy atomic species, it is pretty challenging to disentangle the inner electrons pairing mechanism.On the other hand, hydrides are composed of light chemical elements and have a larger Debye temperature.Indeed, chalcogen and rare-earth hydrides (i.e.H 3 S, LaH 10 ) were recently confirmed to show the record high transition temperature of about 203 and 260 K, respectively [18][19][20][21][22].However, hydride superconductors require high pressures of up to several hundred GPa to stabilize the crystalline structure, reducing the range of potential applications.
In contrast to the superconductors at extreme conditions, the quest of high-T c superconductors at ambient condition is active.At the moment promising progress in this direction has been made in materials containing boron, such as boron-nitrides, boron-carbides and clathrates [23][24][25][26][27][28][29].In particular, diboride compounds (i.e.MB 2 ) possess simple crystal structures and can be easily synthesized [30][31][32].Indeed, diborides are layered materials where boron forms a honeycomb hexagonal lattice, and the metal atoms are located above the center of the hexagonal plane [33][34][35].In 2001, Nagamatsu et al reported that MgB 2 shows a compelling high superconducting transition temperature of 39 K at ambient conditions [36].Using first-principles simulations, Choi et al identified the electronic structures and phonon vibrational modes of MgB 2 and further revealed that the origin of the superconducting transition lies in boron σ bonding states that couple strongly with the in-plane B atomic vibrations [37].This fact leads to extensive electronic pairings and enhancement of the superconducting transition temperature [38][39][40].Recently, Yu et al utilized high-throughput calculations and materials screening methods to find a class of transition metal diborides that may show similar superconducting properties [41].In particular, it was measured that TaB 2 and NbB 2 show a T c in the range of about 1-10 K (also shown later), with almost the same order of magnitude of MgB 2 [42,43].However, so far, the origin of the superconductivity of transition metal diborides remains elusive.
In this work, we investigate the origin of phonon-mediated superconducting properties of transition metal diborides using first-principles calculations and mode-by-mode analysis of the electron-phonon coupling.Based on the McMillan-Allen-Dynes (MMAD) formalism, we report that TaB 2 and NbB 2 have a superconducting transition temperature as high as 27.0 and 26.0 K at ambient conditions (see figure 1), respectively, which are comparable with those of CaB 2 or MgB 2 .Combining the systematic analysis of projected energy-band structures and atomic vibrational-dependent phonon spectra, we reveal that the superconductivity occurring in transition metal diborides is distinct from the main-group element diborides (e.g.CaB 2 or MgB 2 ).The high T c of TaB 2 and NbB 2 is due mainly to the strong coupling between d electrons of the transition metals and the acoustic phonon modes along the out-of-plane vibrations, whereas the superconductivity of main-group element diborides is due mainly to the boron p x and p y orbitals coupled to the in-plane B-dominated polarized optical phonon modes.At the same time, for ScB 2 , YB 2 and VB 2 , we find a relatively low T c (about 1 K) due to the weak electron-phonon coupling strength.
Furthermore, we investigate the temperature-dependent superconducting gaps using the anisotropic Migdal-Eliashberg (ME) theory.For transition metal diborides, only a superconducting gap is observed at low temperatures, in striking contrast with the main-group element diborides since they show double superconducting gaps.In addition, we explore the strain effect on the superconducting transition temperature of diborides.We find that T c of TaB 2 and NbB 2 increases and then decreases with strain strength, whereas the T c of CaB 2 decreases monotonously.This suggests that we can modulate the phonons and the electronic interactions to enhance the superconducting transition temperature in the boron-based materials.

Theory of phonon-mediated superconductivity
We are employing first-principles density functional theory calculations to investigate the phonon-mediated superconducting properties of boron-based compounds as implemented in the Quantum-Espresso and EPW packages [54][55][56][57].The electron-phonon coupling matrix elements in a solid are defined by [58] where h is the reduced Planck constant, M is the atomic mass, ω qν is the phonon frequency associated with the wave-vector q and branch ν, ψ nk and ψ mk+q are the Kohn-Sham wave functions with band index n or m and wave-vector k or k + q, respectively.Furthermore, dV SCF /dµ qν is the derivative of the self-consistent potential with respect to the atomic vibrational displacements, and ϵ qν is the eigenvector corresponding to phonon modes (qν).Accordingly, one can obtain the lattice vibrational mode-dependent electron-phonon coupling strength by [59,60]  For comparison, we also present the available experimental Tc (orange panes with different color strength) [44][45][46][47][48][49][50][51][52][53].For clarity, we point to the references where the experimental data are taken from in the top-right corner through labels in the square brackets.We note that for TaB2 and NbB2 crystalline materials, we extracted the experimental data from several different measurements, and all show a Tc in the range of about 1 to 10 K.
where N(ε F ) is the density of states at the Fermi level ε F and Ω BZ is the volume of 1 st Brillouin zone (BZ).Therefore, the isotropic Eliashberg spectral function α 2 F(ω) can be calculated according to [61,62] and where λ = qν λ qν and ω log is defined as the logarithmic average of the phonon energy.In general, the compounds composed by light chemical species have large ω log .Finally, the superconducting transition temperature within the MMAD formalism can be expressed as [63][64][65] where µ * is an adjustable parameter associated with the Coulomb pseudopotentials of a system [63,64].On the other hand, using the anisotropic ME theory, the T c can also be obtained in terms of the superconducting gap equations, which can be written as [56,57,66] where T is the temperature and iω j is the fermion Matsubara frequency.Hence, through solving these equations self-consistently, we can obtain the superconducting transition temperature using the properties of

Details of first-principle calculations
We leverage the above formalism to compute the electron-phonon-related properties and the superconducting transition temperatures of the investigated boron-based materials.We start by optimizing the ionic coordinates and the lattice constants using first-principles density functional theory calculations [55,[67][68][69].The exchange-correlation potential is evaluated with the generalized gradient approximation under the norm-conserving pseudopotentials in terms of the Perdew-Burke-Ernzerhof 's functional [70,71].
The wave function cutoff for the plane-wave basis set is set at 100 Ry [72].For electronic self-consistent processes, the sampling of the Brillouin zone is a Monkhorst-Pack grid of 14×14×14 points [73].The convergence threshold of the total energy is 10 −12 Ry/Bohr, the convergence threshold of interatomic forces is 10 −10 Ry/Bohr, and the convergence threshold of electronic self-consistency is 10 −16 Ry/Bohr.As to the phonon spectral calculations [74], we sample the 1 st -BZ with a 4×4×4 grid and the convergence threshold of self-consistent potentials is set as 10 −14 Ry.In this case, the phonon frequencies of investigated diborides at the Γ point start from zero, confirming the structural stability.Finally, we used the EPW code to investigate the electron-phonon coupling coefficients and the superconducting properties [56,57].We performed a Wannier function interpolation to evaluate the electron-phonon coupling matrix elements [66,75,76].Therefore, the initial coarse grids are set at 8×8×8 for the k-mesh and 4×4×4 for the q-mesh, respectively.Whereas, to obtain the converged transport properties, we interpolated the k-mesh and the q-mesh to the dense 48×48×48 and 24×24×24 grids, respectively.Finally, we have explored the possibility of emerging magnetism in the investigated materials.Our calculations indicate that the ground state is non-magnetic as both the total (sum of the magnetic moments) and the absolute magnetisations (sum of the absolute values of magnetic moments) vanish.
We note that for TaB 2 and NbB 2 , we extracted the experimental data from several different measurements which we report with color strength.For clarity, the corresponding references for the experimental data are indicated in the top-right corner in the square brackets.All measured T c for investigated diborides fall in the range of about 1 to 10 K. Indeed, such large variance of the experimental values is most-likely ascribed to the sample preparation and handling, where the materials purity plays a significant role in the superconducting transition temperature.The experiments on TaB 2 reported different T c and the discrepancies are justified based on sample preparation [44,46,[48][49][50].The same behavior also occurs for NbB 2 .In general the earlier experiments are reporting the lower transition temperatures.However, we are unable to find more recent evidence on these materials that might shed light on the observed experimental behavior.At the same time, our modeling structures are ideal crystal materials with no defects and impurities.Therefore, the calculated T c might be higher than the experimental values (see figure 1).We expect this work could stir novel interest in these boron compounds.
In ab-initio simulations, in general, the superconducting transition temperature depends on the choice of the parameters of such as Coulomb pseudopotential (µ * ) according to the McMillan formula or the superconducting gap equations.In our cases, to be consistent with previous studies [63,77], we fix µ * = 0.1 for all the investigated materials.At the beginning we have examined the parameters setting by comparing our results with reference [37] for MgB 2 , and then use a similar set of parameters to investigate the superconducting properties of transition metal diborides.At the same time, we have also compared the electronic band structures obtained from the Wannierisation process and the original DFT calculations and found that they are nearly identical (see the Supplementary data).Interestingly, we find that within first-principles simulations, all investigated transition metal diborides become superconducting at low temperatures, and the calculated transition temperatures T c are consistent with the experimental evidence since both are in the same order of magnitude.In particular, we find that TaB 2 and NbB 2 show the highest T c among them, as large as 27.0 and 26.0 K, respectively, obtained from the anisotropic ME method.This is higher than the results obtained using MMAD theory calculations, which omit the structural difference between the in-plane and out-of-plane direction of the diborides.Whereas, the T c of ScB 2 , YB 2 and VB 2 for MMAD and ME calculations are relatively low, around 0.4 (1.2), 0.6 (2.4) and 0.2 (3.0) K, respectively, where the values in the brackets corresponding to anisotropic calculations.In figure 1, we have also compared T c of transition metal diborides with the main-group element diborides.Taking CaB 2 as an example, which has a similar crystal structure and electronic properties as MgB 2 , we obtained the T c of about 25.7 (70.0)K, in agreement with the previous predictions by Yu and Choi et al [39,41].

The origin of the intriguing superconducting properties in diborides
Next, we elucidate the origin of the superconducting properties of transition metal diborides.Looking at the MMAD formula, equation ( 5), T c is proportional to the logarithmic average of phonon energy (ω log ) and depends on the electron-phonon coupling constant (λ).Table 1 shows the ab initio calculated λ and ω log of the diboride superconductors investigated here.We notice that TaB 2 , NbB 2 and CaB 2 have significant electron-phonon coupling coefficients, whereas ScB 2 , YB 2 and VB 2 show a relatively small value of the  electron-phonon coupling coefficients.We find that the ω log of TaB 2 , NbB 2 and CaB 2 is lower than that of ScB 2 , YB 2 and VB 2 .Therefore, we conclude that in the investigated TaB 2 , NbB 2 and CaB 2 materials, T c is determined by the large electron-phonon coupling strength.
To present a detailed mode-by-mode analysis of the coupling between electrons and phonons in the high T c materials, TaB 2 , NbB 2 and CaB 2 , in figures 2(a), (b) we begin by displaying the phonon spectra and the vibrational density of states (VDOS) of TaB 2 , respectively.At the same time, in the inset of figure 2(b), we specify the high symmetry points in the first BZ.To identify the lattice vibrational effects, we have MMADe a projection of the in-plane (x − y) and the out-of-plane (z) vibrational modes on the phonon dispersions for different ions, where we use different colors to indicate the corresponding atomic vibrational contributions.We find that the high-frequency optical phonon modes (ω >400 cm −1 ) come from the boron vibrations due to the light elemental mass, whereas the low-frequency acoustic modes (50-200 cm −1 ) stems from the Ta vibrations due to the relative heavy atoms.Clearly, there is a forbidden gap between acoustic and optical modes, which can be further noticed in the VDOS, as shown in figure 2(b), where, in red, we report the total VDOS (red) and, in gray, the B-dominated phonon modes contributions.However, at Γ, as the phonon frequency ω approaches 0, both B and Ta contribute to the long wave-length acoustic mode since it represents the collective vibrations of the unit cell.In figure 2(c), we present the mode-dependent electron-phonon coupling strength (λ qν ) of TaB 2 , where red circles correspond to a significant value of the λ qν , i.e. a strong electron-phonon coupling.We find that around the H point, for the out-of-plane acoustic modes (ZA), TaB 2 shows a electron-phonon coupling coefficient with λ qν reaching up to 3.3.The phonons in these modes come mainly from the Ta atomic vibrations along the out-of-plane direction.For the optical modes, the electron-phonon coupling strength is weak.In the inset, we plot the atomic vibrational polarization vector at H for ZA mode.Indeed, we find that Ta vibrates mainly along the z direction, and the vibrational magnitude of Ta is much stronger than that of B, as indicated by the length of the arrows.
In figures 2(d) and (f), focusing now on NbB 2 , we show the projected phonon dispersions, the VDOSs, and the mode-dependent electron-phonon coupling strength, respectively.We find that the high-frequency optical branches dominated by B in the first BZ are analogous to that of TaB 2 .In contrast, the maximum phonon energy of low-frequency Nb-dominated acoustic branches for NbB 2 is higher than that of TaB 2 due to the relative light niobium compared with tantalum, resulting in the forbidden gap of the former being smaller than the latter.Further, we find that near H, for the ZA vibrational modes, a larger electron-phonon coupling coefficient of about 1.3, although smaller than the maximum value of TaB 2 , is shown.In the inset of figure 2(f), we plotted the atomic vibrational vector corresponding to the ZA mode at H and found that the magnitude of Nb vibrating along the z direction remains much larger than the in-plane vibrations of B.
In contrast with the transition metal diborides, taking CaB 2 as a case study of the main-group element boron-based compounds, we show in figures 2(g), (i), the projected phonon dispersions, the VDOSs, and the mode-dependent electron-phonon coupling strength of CaB 2 , respectively.We find that the highest frequency of CaB 2 of about 650 cm −1 is lower than that of TaB 2 and NbB 2 .However, the B atomic vibrations dominate the optical branches in all the investigated boron-based materials.Meanwhile, we noticed that the maximum energy of acoustic branches dominated by calcium is higher than the minimum of the optical modes.Notably, we find that around Γ point, a maximum electron-phonon coupling λ qν = 6.4 is observed for the in-plane B-dominated optical modes.Therefore, the higher T c of CaB 2 is mainly due to the strong electron-phonon coupling occurring in the in-plane optical vibrational modes.This fact differs from the case of TaB 2 and NbB 2 , where the ZA modes at H dominate the electron-phonon coupling.In the inset of figure 2(i), we plotted the atomic vibrational vector corresponding to the optical mode at Γ for CaB 2 and find that the phonon eigenvectors due to the B vibrations are mainly distributed in the x − y plane.
On the other hand, to elucidate the atomic orbital effect on the superconducting properties of investigated boron-based compounds, we calculate the projected band structures (figures 3(a), (d)) and the partial density of states (PDOS) (figures 3(b), (e)) for TaB 2 and NbB 2 , respectively.In the figure, the atomic s, p and d orbitals contributions to the bands are labeled by different colors.We are also showing in figures 3(c), (f) the strength of electron and phonon interactions (λ nk ), and accordingly, the band and wave-vector dependent electron-phonon coupling strength with respect to a specific Bloch state can be calculated by [56,57,66] where λ nk,mk+q (ω) is the anisotropic electron-phonon coupling strength, which can be written as [58] λ nk,mk+q where ω j is the phonon energy.By exchanging phonons, a pair of electrons at states k and −k would be scattering to (k + q) and −(k + q) states.For convenience, in figures 3(a) and (d), we labeled the bands with a series of numbers.We find that for TaB  Reverting to the main-group element boron-based compounds, retaking CaB 2 as an example, in figures 3(g), (h) and (i), we show the projected band structures, the partial density of states, and the band and wave-vector dependent electron-phonon coupling strength (λ nk ), respectively.At the Fermi surface, the energy bands are determined by boron atoms.For instance, the bands of (1, 1 near L come mainly from the B p z orbital.Furthermore, according to figure 3(h), the B p orbitals populate mostly the density of states at the Fermi level.Looking at figure 3(i), the strong coupling between electrons and phonons occurs mainly from the B p x and p y orbitals interacting with the in-plane B-dominated optical modes.Therefore, the superconducting behavior of transition metal and main-group element diborides originates from the pairing of different electronic orbitals mediated by different phonon modes.We now investigate the superconducting gap of diborides using the anisotropic ME theory [56,57,66].Figures 4(a), (c) show the ∆ nk as a function of temperature for TaB 2 , NbB 2 and CaB 2 , respectively.There, the red shaded area at each temperature represents the statistic distribution ρ(∆) for superconducting gaps at various electronic states.At the same time, the solid line corresponds to the evolution of ρ(∆) with temperature.At T = 0, the superconducting gap of TaB 2 or NbB 2 is smaller than that of CaB 2 .As the temperature increases, the superconducting gap decreases.Moreover, for TaB 2 or NbB 2 , there is only a single superconducting gap, whereas for CaB 2 , two superconducting gaps are observed.This is because, for the main-group element boron-based compounds, the B p orbitals form either σ or π bonds and both couple to the corresponding phonon modes (although the former is much stronger than the latter).That said, the superconducting gap for the transition metal diborides is due to the d orbitals, which are coupled to the cation-dominated out-of-plane acoustic modes.Our calculations of T c , in particular for transition metal diborides of TaB 2 and NbB 2 based on ME theory, are consistent with the results obtained using the MMAD formula (equation ( 5)).

Tuning the superconducting transition temperature by strain
Finally, we investigate the strain effect on the superconducting properties of boron-based compounds.To introduce a finite strain, we modulate the lattice constants of the material in the out-of-plane (i.e.z) direction and then fully relax the in-plane lattice parameters and atomic coordinates to minimize the total energy of the system and reduce any residual force.The amount of strain introduced is given in terms of (c − c 0 )/c 0 with c and c 0 being the distorted and un-distorted lattice constants in the out-of-plane direction, respectively.In our case, all phonon frequencies calculated are positive, indicating that the investigated material under strain remains stable.In figures 5(a) and (b), we show the superconducting transition temperatures (T c ) of TaB 2 , NbB 2 and CaB 2 as a function of strain using either the MMAD or the ME formulas calculations, respectively.We find that both methods present the same trend of T c , i.e. the T c of TaB 2 and NbB 2 increases and then decreases, but for CaB 2 , it decreases with the strain.This fact can be due to the enhanced electron-phonon coupling constant (λ) of transition-metal diborides at a medium strain range, and finally, T c declines due to the over-expanded atomic distances, as shown in figure 5(c).In contrast, for CaB 2 , λ decreases monotonously as the strain increases.
To illustrate the distinct strain-dependent behaviors of T c of transition metal and main-group element diborides, in figure 5(d), we show the in-plane (solid dots) and the out-of-plane (hollow dots) lattice parameters as a function of strain for TaB 2 (red), NbB 2 (blue) and CaB 2 (green).As expected by Poisson's effect, we find that the in-plane and the out-of-plane lattice constants vary oppositely with the strain.As the out-of-plane lattice constant increases, the in-plane lattice constant decreases so that the p x and p y orbital wavefunction overlapping for the in-plane borons is enlarging.According to equation (2) in section 2, the electron-phonon coupling constant is proportional to the g-factor but inversely proportional to the corresponding mode frequencies.In the case of CaB 2 , since the electron-phonon transition matrix element in equation ( 1) is mainly determined by the in-plane p x and p y orbitals of borons which couple strongly to the in-plane B vibrations, g is increasing.Simultaneously, we find the phonon frequencies for the in-plane vibrational optical modes around Γ also increase with strain (labeled by upper-arrow), as shown in figure 5(e).Therefore, the decreased electron-phonon coupling constant for CaB 2 is a combined effect of the two quantities.In the case of TaB 2 (or NbB 2 ), we note that the electron-phonon coupling matrix is dominated by coupling out-of-plane acoustic modes around H with the d-orbitals of transition metals.Indeed, looking at figure 5(f), the lattice vibrational frequencies of TaB 2 around H decrease with the strain (down arrow), thus resulting in the increasing of electron-phonon coupling constants.

Conclusions
In summary, we clarified the origin of the phonon-mediated superconducting properties of transition metal diborides.Based on the McMillan-Allen-Dynes and anisotropic ME theory calculations, we found that TaB 2 and NbB 2 have T c as high as 27.0 and 26.0 K at ambient conditions, respectively, comparable with CaB 2 or MgB 2 , whereas ScB 2 , YB 2 and VB 2 have a relatively low T c of about 1 K due to the weak electron-phonon interactions.Using first-principles simulations, we presented a mode-by-mode analysis of the electron-phonon coupling coefficients and revealed that the high T c of TaB 2 and NbB 2 comes mainly from the strong coupling between d electrons of the transition metals and the acoustic phonon modes of out-of-plane atomic vibrations.This fact is distinct from the main-group element diborides, where the superconductivity originates from the boron p x and p y orbitals by interacting with the in-plane B-dominated optical phonon modes.In addition, we investigated the evolution of superconducting gaps with temperature using the anisotropic ME theory.We found that the transition metal diborides show a single superconducting gap, strikingly contrasting with the main-group element diborides since they are double superconducting gaps.Further, we found the T c of TaB 2 and NbB 2 increases and then decreases with the strain strength, while the T c of CaB 2 decreases monotonously.This suggests a strategy to enhance the superconducting transition temperature by engineering the phonon structures and electronic interactions.Our calculated results are consistent with the experimental values.This study will shed some light on exploring high T c boron-based superconductor materials.

Figure 1 .
Figure 1.The calculated superconducting transition temperatures of MB2-type diborides (with M = Ta, Nb, Sc, Y, V, and Ca) using first-principles simulations based on the McMillan-Allen-Dynes formula (blue panes) and the anisotropic Migdal-Eliashberg theory (green panes).For comparison, we also present the available experimental Tc (orange panes with different color strength)[44][45][46][47][48][49][50][51][52][53].For clarity, we point to the references where the experimental data are taken from in the top-right corner through labels in the square brackets.We note that for TaB2 and NbB2 crystalline materials, we extracted the experimental data from several different measurements, and all show a Tc in the range of about 1 to 10 K.

Figure 2 .
Figure 2. (a), (d), (g) The phonon spectra, where for in-plane (x − y plane) and out-of-plane direction (z direction) atomic vibrational contributions are denoted in color (see legend); (b), (e), (h) the vibrational density of states (VDOS), where B and metal atomic contributions are distinguished; (b-inset) sketch of the first Brillouin Zone and the high-symmetry points for these crystal structures; (c), (f), (i) the mode-dependent electron-phonon coupling strength (λqν ), where the insets correspond to the atomic vibrational vectors of out-of-plane acoustic phonon modes at H for TaB2 (top panel) and NbB2 (middle panel), and in-plane optical phonon mode at Γ for CaB2 (down panel), respectively.Note that the atomic vibrational magnitude is proportional to the length of the arrow.
2 and NbB 2 , many energy bands, dominated by different atomic orbitals, are crossing the Fermi surface.The bands of (1, 1 ′ , 1 ′ ′ and 1 ′ ′ ′ ) are determined mainly by tantalum or niobium d xz and d yz which are degenerated and occupied equally.The bands of (2, 2 ′ and 2 ′ ′ ) come mainly from the tantalum or niobium d x 2 −y 2 and d xy where both have equivalent occupations.The bands labeled by (3 and 3 ′ ) originate in the tantalum or niobium d z 2 , respectively.However, both the s and p orbitals of Ta and Nb play a negligible role in the band structures around the Fermi surface.Simultaneously, we find boron p x or p y orbitals contribute to the bands (4, 4 ′ , 4 ′ ′ and 4 ′ ′ ′ ), whereas boron s and p z states contributes much less than the former.The different atomic orbitals contributions can be seen more clearly in the PDOS as shown in figures 3(b), (e) for

Figure 4 .
Figure 4. Ab initio calculated superconducting gaps (∆ nk ) using the anisotropic Migdal-Eliashberg theory for (a) TaB2, (b) NbB2 and (c) CaB2, where dashed line indicates the evolution of superconducting gaps with temperature and the red shaded area corresponds to the statistic distribution ρ(∆) for different electronic states at a given temperature.

Figure 5 .
Figure 5.The strain engineering of superconducting transition temperatures (Tc) using (a) McMillan-Allen-Dynes and (b) Migdal-Eliashberg formula calculations, (c) and (d) electron-phonon coupling constant (λ) and in-plane (solid dots) and out-of-plane (hollow dots) lattice constants as a function of strain for investigated TaB2 (red), NbB2 (blue) and CaB2 (green), where we defined the strain by (c − c0)/c0 with c and c0 being the distorted and undistorted lattice constants in the out-of-plane direction, respectively.(e) and (f) the phonon dispersions of CaB2 and NbB2 under −2% and zero strain, respectively.

Table 1 .
Ab initio calculated electron-phonon coupling constant (λ) and the logarithmic average of phonon energy (ω log , in unit of meV) for investigated diborides superconductors.