First-principles study on the superconductivity of doped zirconium diborides

Recent experiments [Barbero et al. Phys. Rev. B 95 , 094505 (2017)] have established that bulk superconductivity ( T c ∼ 8 . 3–8 . 7K) can be induced in AlB 2 -type ZrB 2 and HfB 2 , highly covalent refractory ceramics, by vanadium (V) doping. These AlB 2 -structured phases provide an alternative to earlier diamondlike or diamond-based superconducting and superhard materials. However, the underlying mechanism for doping-induced superconductivity in these materials is yet to be addressed. In this paper, we have used ﬁrst-principles calculations to probe electronic structure, lattice dynamics, and electron-phonon coupling (EPC) in V-doped ZrB 2 and consequently examine the origin of the superconductivity. We ﬁnd that, while doping-induced stress weakens the EPC, the concurrently induced charges strengthen it. The calculated critical transition temperature ( T c ) in electron (and V)-doped ZrB 2 is at least one order of magnitude lower than experiments, despite considering the weakest possible Coulomb repulsion between electrons in the Cooper pair, hinting a complex origin of superconductivity in it.

Similarly, Ni-doped NbB 2 becomes superconducting <6 K [28].Along with their high hardness mechanical properties, doping-induced superconductivity in these TMB 2 make them a commercially cheap alternative to nanocomposites that contain diamond and/or c-BN.Superconducting, superhard nanocomposites are important for industrial application in high-pressure cryogenic devices [30,31], wear-resistant parts of superconducting devices, superconducting micro-electro mechanical systems [32], superconducting radiofrequency accelerator cavities [33][34][35], quantum interference devices [36], photodetectors [37], infrared sensors [38], and electrical contacts to nanodevices [39].Despite the numerous studies to explain the origin of superconductivity in MgB 2 with AlB 2 -type structure [40][41][42][43], little is known about the doped TMB 2 family of materials [23].Consequently, from both perspectives of applications and fundamental material science, there is a need to understand the superconductivity in these doped TMB 2 phases.In view of that, we chose V-doped ZrB 2 as a representative member since it has the highest T c reported for this family of materials.
In its pristine form, ZrB 2 is not superconducting, but doping with V or Nb as low as 1% can induce superconductivity [27,29,44,45].Renosto et al. [29] reported that the highest T c recorded for Zr 1−x V x B 2 is 8.7 K for x = 0.04.Along with the multigap nature of the superconducting gap (2 ), they demonstrated that T c increases with increasing V content in Zr 1−x V x B 2 up to x = 0.04 and that further increase in V-doping levels causes T c to decrease, which implies the presence of a superconducting dome.From x-ray diffraction analysis, a dopant concentration of 4% V in ZrB 2 results in a uniaxial compressive strain of ∼0.3% along the c axis.Using microscopic characterization techniques such as nuclear magnetic resonance, superconducting quantum interference device magnetometry, resistivity, and muon-spin rotation measurements, Barbero et al. [27] established the bulk superconductivity (T c = 8.33 K) in Zr 0.96 V 0.04 B 2 along with an s-wave and the multigap nature of 2 , which was further validated by Jung et al. [46].
In this paper, we investigate the electronic structure, lattice dynamics, and electron-phonon coupling (EPC) in pristine and V-doped ZrB 2 using first-principles simulation to explore the origin of the superconductivity with <4% doping level.Our analysis reveals that, upon V doping, the Fermi level (E F ) shifts toward the conduction band, making V an effectively n-type dopant to ZrB 2 .This shifting of E F away from the well-known pseudogap region in the electronic structure of ceramics enhances the electronic densities of states (DOSs) at E F , [N (E F )], and improves the EPC.However, at experimentally relevant doping levels, our calculated superconducting transition temperature (T c ) is one order of magnitude lower than the reported experimental values, an indication for a complex origin of superconductivity in V-doped ZrB 2 .

II. COMPUTATIONAL DETAILS
Our analysis involves uncovering the effects of (i) lattice strain, (ii) carrier doping, and (iii) alloying within virtual crystal approximations (VCAs) of the electronic, vibrational, and their coupling in ZrB 2 using first-principles density functional theory simulations as implemented in the QUANTUM ESPRESSO code [47].Nonrelativistic norm-conserving pseudopotentials with the Perdew-Burke-Ernzerhof exchange-correlation functional are used in our simulations [48,49].The valence electronic configuration of Zr, V, and B are chosen as 4s 2 4p 6 5s 2 4d 2 , 3s 2 3p 6 4s 2 3d 3 , and 2s 2 2p 1 , respectively.For ground state calculations, the Brillouin zone is sampled by a grid of 14 × 14 × 13 k-points [50], and a kinetic energy cutoff for the plane-wave basis of ≈ 1090 eV is used.Dynamical matrices are computed over a 6 × 6 × 6 q-points grid using density functional perturbation theory.EPC was computed with the Wannier interpolated scheme [51]; 13 maximally localized Wannier functions [52,53] are constructed using the WANNIER90 library [54] within the EPW code [51].Further details are given in Sec.I in the Supplemental Material [55].The electron-phonon matrix elements are calculated on dense k-and q-grids of 25 × 25 × 25.We have used 300 and 0.5 meV as electron and phonon smearing widths, respectively.Within the Migdal-Eliashberg theory [56,57], the isotropic version of Eliashberg spectral function α 2 F (ω) is defined as (1) where N (E F ) is the electronic DOS at the Fermi level and g mnγ (k, q) are the electron-phonon matrix elements with electronic band indices m, n and phonon band index γ .The 1) is referred to nesting function (ζ q ) and is a property of the Fermi surface and hence has a pure electronic origin [51].After com-puting α 2 F (ω), electron-phonon interaction λ is evaluated by The superconducting transition temperature T c is calculated by the McMillan-Allen-Dynes formula [58,59]: where ω log is a logarithmically averaged phonon frequency which has replaced the Debye temperature D in the McMillan formula and is computed from α 2 F (ω) as In Eq. ( 3), μ * is a dimensionless parameter representing the Coulomb repulsion between electrons which form the Cooper pair.This repulsive parameter is hard to compute from the ab initio method, and for most superconductors, its value lies in the range 0.1-0.2[60][61][62].However, for d-band metals, Bennemann and Garland [63] Our estimates of μ * for doped ZrB 2 are in the range of 0.05-0.09with a doping level of 0.0 to 0.2 e − /cell.Nevertheless, we estimate T c for a wide range (0.0-0.1) of μ * in this paper.To model the V doping in ZrB 2 , VCA is used where the Zr atomic site in ZrB 2 is replaced with a virtual atom.The pseudopotential of the virtual atom is, in turn, represented by a composition average of the Zr and V pseudopotentials, i.e., V x virtual atom = xV V + (1−x)V zr , and the atomic mass of the virtual atom is taken as the average atomic mass of Zr and V, i.e., m virtual atom = xm V. + (1−x)m Zr .

III. RESULTS AND DISCUSSION
Our optimized lattice parameters a = b = 3.17 Å and c = 3.54 Å for a unit cell (u.c.) of ZrB 2 as well as in-plane B-B bond length (d B−B = 1.83 Å) and out-of-plane Zr-B (d Zr−B = 2.55 Å) are in good agreement with previous theory and experiments [64][65][66].The electronic band structure of pristine ZrB 2 shows a Dirac-like linear band dispersion along the M-K, K-, and -A high-symmetry k-path [see Fig. 1(a)].The linear dispersed band along the M-K symmetry line is ∼2 eV, which is rarely observed in Dirac materials.The features in the band structure match with previously reported ones [67,68].The orbital projected electronic DOS shows a dominant role of the Zr-4d orbital toward the conduction bands, while a B-2p orbital is more dominant at valence bands along with a contribution from a Zr-4d orbital [see Fig. 1(b)], again in good agreement with a previous report [69].The Fermi level (E F ) is located near a pseudogap region of the electronic DOS [Fig. 1 The unit cell of ZrB 2 consists of three atoms which results in a total of nine phonon branches, three acoustic and six  I].The phonon DOSs [F (ω)] and the isotropic Eliashberg spectral functions [α 2 F (ω)] are displayed in Fig. 1(d), which shows that spectral patterns of α 2 F (ω) replicate F(ω).Using Eq. ( 2), we estimate the average value of the EPC constant (λ) to 0.126, in agreement with previous theoretical and experimental reports [71][72][73].It is noted here that, unlike in MgB 2 , the E 2g phonon in ZrB 2 is at higher energy than the B 1g one at the point.Thus, the contribution of E 2g phonon modes to the α 2 F (ω) of ZrB 2 is weaker than that of MgB 2 [74].Together with low N (E F ), energetics of E 2g phonon modes in vibrational  [55].The facts that the experimental n-type carrier density in ZrB 2 of the order of 10 21 electrons cm −3 [67] is consistent with the carrier densities considered in our simulations and that the E F D condition is satisfied suggest that the Eliashberg formalism is adequate to describe the electron-phonon interaction and consequently the superconductivity in ZrB 2 [75].The weak EPC in pristine ZrB 2 (λ = 0.126) results in a superconducting temperature in the mK range.With the Coulomb repulsion constant μ * = [0, 0.1], the upper limit of the transition temperature T c using the McMillan-Allen-Dynes equation is <0.04K (Fig. S8 in the Supplemental Material [55]), which explains the absence of the superconductivity >1 K in pristine ZrB 2 [76].
The electronic structure of V-doped ZrB 2 is investigated using both supercell and VCA approaches.In the supercell approach, we considered two different supercells (i) 2 × 2 × 2 (24 atoms) and (ii) 3 × 3 × 3 (81 atoms) and substituted one V atom for Zr, resulting in a V-doping concentration of 12.5% (Zr 0.875 V 0.125 B 2 ) and 3.7% (Zr 0.963 V 0.037 B 2 ), respectively.Upon V doping in ZrB 2 , the E F moves away from the pseudogap region into the conduction band [Fig. 2 z .We also note the splitting of degenerate Zr-d xz and d yz orbitals upon V doping in ZrB 2 .Band structure analysis by Zhai et al. [23] further suggests that E F of Zr 0.963 V 0.037 B 2 is very close to the d xy + d yz band at the A high-symmetry k-point with a low dispersion width.
Similar numbers are obtained when using VCA, with optimized lattice parameters a (c) of Zr 0.963 V 0.037 B 2 and Zr 0.875 V 0.125 B 2 being 3.16 Å (3.52 Å) and 3.15 Å(3.48 Å), respectively.This corresponds to in-plane (out-of-plane) compressive strain of −0.31% (−0.56%) and −0.63 (−1.7%), respectively, as compared with pristine ZrB 2 .The larger compressive strain out-of-plane as compared with in-plane agrees qualitatively well with the experimental observations where strain along the c axis is well resolved, while almost no in-plane strain is noted [29].The electronic band structure and DOS analysis revealed that the E F moves away from the pseudogap region into the conduction band with a corresponding increase in N (E F ) for V-doped ZrB 2 (see Sec. VI in the Supplemental Material [55]).Using VCA, N (E F ) of Zr 0.963 V 0.037 B 2 and Zr 0.875 V 0.125 B 2 are 0.32 and 0.46 states eV −1 cell −1 , respectively.Comparison of the electronic structure calculated at 4% of V doping using the supercell approach and VCA reveals similar results, while at 12.5% of doping levels, N (E F ) from VCA calculations [N (E F ) VCA = 0.46 states eV −1 cell −1 ] is significantly lower than achieved with the supercell approach [N (E F ) SC = 0.52 states eV −1 cell −1 ].
Although at macroscopic scale the strain is relatively low (−0.23 to −0.35%), our bond-length distribution analysis from supercell approach (Zr 0.963 V 0.037 B 2 ) calculations revealed that the maximum change in bond lengths of Zr-B in the supercell is as high as −1.9%.Thus, we examine the material properties of ZrB 2 over a wide range of uniaxial compressive strain (ε ||c ), e.g., 0.3, 0.5, 1, 3, and 5%.The electronic band structure revealed that the bands at the A symmetry k-point are becoming closer to E F upon compressive stress, and at ε ||c = 5%, a new Fermi pocket appears at the A point of the Brillouin zone (see Sec. III in the Supplementary Material [55] for band structure, DOS, and Fermi surface plot).We do not find any substantial changes in N (E F ) upon uniaxial compressive strain along the c axis.Phonon band and F(ω) analysis reveals that, upon compressive strain, phonon frequencies shift toward higher energy, known as phonon hardening, as seen in Figs.3(a) and S7 in the Supplemental Material [55]), and hence, no imaginary modes of vibrations are observed even if the strain (ε ||c ) is relatively high (= 5%).Recent theoretical results suggest that a thin film of layered borides like MgB 4 and InB 4 will go for structural phase transition with a relatively smaller biaxial compressive strain of −0.8 and 3%, respectively [77,78].Similarly, layered AlB 2 can maintain structural stability from only −1.5% (compressive) to 7.5% (tensile) strain [79,80], suggesting that bulk ZrB 2 is structurally more stable to compressive stress than the mentioned layered structure of borides.
Next, we examine the effect of electron doping of ZrB 2 .It is noteworthy that the highest value of T c is observed for 4% doping of V/Nb [29], where each V or Nb atom donates one electron per atom for the conduction.This corresponds to a doping of 0.04 electrons per primitive unit cell.However, to reveal trends and clarify, the electronic and vibrational properties were computed self-consistently for a range of charge doping levels (n), i.e., 0.037, 0.1, 0.125, 0.15, 0.2, and 0.4 electrons/cell, leading to an electron doping concentration of 1.2 × 10 21 to 1.2 × 10 22 cm −3 in ZrB 2 .A compensating jellium background is used to neutralize the charge in the cell.In this paper, we simulate electron doping for two different cases: (i) full relaxation of both structural parameters and ions and (ii) only relaxation of ions while keeping the unit cell static.In the former case, we find an expansion of the unit cell lattice upon carrier doping, contrary to experimental observations; hence, we present the results from case (ii).Details of case (i) are discussed in Sec.VII in the Supplemental Material [55].
Electron doping of ZrB 2 results in E F moving away from the pseudogap region and into the conduction band (see Fig. S12(A) in the Supplemental Material [55]), causing an increase in N (E F ), as shown in Fig. 4(e).At a carrier doping of 0.1 e − /cell, E F is close to the d xz /d yz bands at the A point in the similar condition to the supercell calculation of Zr 0.963 V 0.037 B 2 [44].This is shown in Fig. S11 in the Supplemental Material [55].Analysis of phonon bands and F(ω) in Figs.4(a) and S13 in the Supplemental Material [55] shows both acoustic and optical phonon modes shift toward lower energy, henceforth referred to as phonon softening, upon electron doping in ZrB 2 .The -point phonon frequencies of optical phonon modes with different doping concentrations are tabulated in Table I and quantify the phonon softening at the -point for electron-doped ZrB 2 .
With increased doping levels comes phonon softening which, in turn, leads to increased overall spectral weightage of α 2 F (ω).The spectral weight of lower frequency transverse acoustic (TA-1) phonons increases up to a doping level of 0.1 e − /cell and remains constant afterwards, while contribution from high-frequency transverse acoustic (TA-2) and longitudinal acoustic (LA) modes increase continuously with increase in doping levels [see inset of Fig. 4(b)].It is important to note here that a small but noticeable difference in α 2 F (ω) is observed between two different lattice relaxation methods (see Figs. 4(b) and S14 in the Supplemental Material [55]).While contribution from acoustic phonons to α 2 F (ω) is fixed after a doping level of 0.1 e − /cell, the contribution of optical phonons to α 2 F (ω) steadily increases with the doping level increment [see Fig. 4(b)].
The enhancement in λ is almost linear with the doping level, and it increased from 0.126 to 0.240 with an increase in the doping level from 0.0 to 0.2 e − /cell [see Fig. 4(c)].Even though the phonon softening causes enhancement in λ, it has a detrimental effect on ω log , which is lowered with increased doping levels [see Fig. 4(d)].The quantitative difference in estimated λ and ω log with two different lattice relaxed configurations is clearly visible in Figs.4(c) and 4(d).This can be attributed to a larger amplitude in phonon softening when the system is allowed for complete relaxation of both lattice parameters and ions (see Fig. S13 in the Supplemental Material [55]).
The phonon softening in electron-doped ZrB 2 is obvious.However, the results from VCA are slightly different.We find phonon hardening in Zr 0.963 V 0.037 B 2 and Zr 0.875 V 0.125 B 2 as compared with ZrB 2 .The only exception is a small phonon softening of B 1g phonons within VCA (see Table I and Sec.VI in the Supplemental Material [55]).Despite overall phonon hardening, we find an increase in spectral weight of α 2 F (ω) in Zr 0.963 V 0.037 B 2 and Zr 0.875 V 0.125 B 2 .This indicates that the electron states at the Fermi level N (E F ) dictate EPC in doped ZrB 2 .To explain the enhancement in electron-phonon interaction and phonon softening for electron-doped ZrB 2 , we have calculated the Fermi nesting function ζ q [see Fig. 5].The nesting function ζ q is the imaginary part of electronic susceptibility, responsible for the Fermi surface topology and one of the major terms in the Migdal-Eliashberg picture of electron-phonon interaction.By Eqs. ( 1) and ( 2), the increment in the nesting function can enhance the electron-phonon FIG. 5. (a)-(e) q x -q y projected nesting function (ζ q ) computed over the irreducible wedge of the Brillouin zone of doped ZrB 2 with doping levels of 0.037, 0.1, 0.125, 0.15, and 0.2 e − /cell, respectively.Clearly, an enhancement in ζ q with increase in doping levels is observed.interaction, and this argument is supported by reported calculations [81,82] on electron-phonon interaction.
The nesting function is one of the reasons behind the softening of the phonon modes and can be responsible for the charge density wave state [83][84][85].Further, the Fermi nesting function is associated with the imaginary part of electronic susceptibility [86,87], closely related to the Fermi surface topology [88], and strongly influences the electronphonon interactions [89].With an increase in doping levels, we observed the appearance of electron-type Fermi pockets around the A points with the overall Fermi surface becoming broader (see Fig. S12(B) in the Supplemental Material [55]).It indicates that the Fermi surface topology has changed with electron doping, which also reflects in the broader peak in the Fermi nesting function [see Fig. 5].In the electrondoped ZrB 2 case, the enhanced nesting function ζ q further explains the phonon softening at higher doping.The estimated λ values for Zr 0.963 V 0.037 B 2 and Zr 0.875 V 0.125 B 2 are 0.147 and 0.199, which are close to 0.150 (0.037 e − /cell) and 0.197 (0.125 e − /cell) for electron-doped ZrB 2 [see Fig. 4(c)].Similarly, computed ω log for the same cases are 511.9 and 486.3K within VCA, in close agreement with values of electrondoped ZrB 2 [see Fig. 4(d)].We also note that, in the case where atomic configurations (i.e., cell parameters + ions) of bare electron-doped ZrB 2 could relax completely, a noticeable difference in λ and ω log is found, particularly at higher doping levels [see Figs.4(c) and 4(d)].Now we calculate the superconducting transition temperature (T c ) using Eq. ( 3) and also estimate T c by solving the isotropic Eliashberg gap [ (T)] equations [60,90].In the latter case, (T) is interpolated by the cubic-spline technique, and T c is the temperature at which (T ) = 0. We find that both methods provide similar values of T c (see Figs. 6 and S17 and S18 in the Supplemental Material [55]).We start by analyzing T c with lowest extreme of μ*.At the experimental doping level (4% of V ↔ 0.037 e − /cell), the computed T c (μ * = 0.0) is 0.14 K [see Fig. 6(a)].This is much lower than the reported experimental values of 8.33 to 8.7 K. Despite an increase in the doping level to n = 0.2 e − /cell, T c is estimated to be 1.8 K.It is important to note that the abovementioned T c is the maximum extreme value, with a decrease of T c with increasing μ* [see Figs.6(b) and 6(c)].For μ * = 0.05, the computed T c is in the range of 0.001 to 0.328 K with a doping level of 0.037-0.2e − /cell.Based on these results, the calculated T c , for equal or even higher than experimental doping levels, is at least one order of magnitude lower than experimental values despite using an extremely minimal value of μ*.The experimental value of T c is obtained for a doping level of 0.4 e − /cell [≈ 10 22 cm −3 ] and μ * = 0.0 [see Fig. 6(a)].However, it does not appear reasonable to equate μ* to zero at this high doping level.Using computed N (E F ) for the doping level of 0.4 e − /cell in Eq. ( 5), μ* is estimated to be 0.11.These high values of μ* result in a T c ≈ 0.8 K, one order of magnitude smaller than the experimental value of 8.3-8.7 K.
To find a possible explanation for this severe underestimation of T c by the isotropic Eliashberg theory, we have calculated momentum-resolved electron-phonon interactions for various doping levels (see Fig. S19 in the Supplemental Material [55]).In all doping levels, the distribution of electron-phonon interaction has only a single peak for which spreading increases with doping.The maximally doped ZrB 2 has a shorter spread, reflecting its lesser anisotropic nature than MgB 2 [60].Unlike MgB 2 , doped ZrB 2 does not have isolated multiple sheets of the Fermi surface and is quite small, resulting in being more isotropic with lesser electron-phonon strength.These features of doped ZrB 2 provides a McMillan-Allen-Dynes T c one order lower than that of MgB 2 using the isotropic Eliashberg theory [60].With more isotropic electron-phonon interaction, the nearly continuous Fermi surface and lower isotropic transition temperature indicate that the solution of the anisotropic Eliashberg equations will not result in the increase in theoretical estimation of superconducting transition temperature for doped ZrB 2 as they do in pristine MgB 2 [60,91,92].It is important to note that V doping in ZrB 2 creates localized d-band defect states near the Fermi level (see Fig. 2), which could not be completely replicated in the electronic structure of electron-doped ZrB 2 nor within the VCA.Often, localized defect states enhance the average EPC strength (λ) in materials [93,94].The formation of a VB 2 -like local region in Zr 1−x V x B 2 may further enhance the average EPC strength as VB 2 has a relatively larger EPC strength (λ VB2 = 0.28) and almost the same logarithmic average of phonon frequencies (ω log ) as ZrB 2 [95,96].Also, the effect of local inhomogeneity of V in Zr 1−x V x B 2 cannot be completely ignored, as it can play a nontrivial role in enhancing the superconducting transition temperature as observed in Cu-doped Bi 2 Se 3 [97,98].All these phenomena together or individually may influence the average EPC strength (λ) and consequently the determination of the superconducting transition temperature (T c ).Thus, we suggest a mechanism beyond those investigated earlier is responsible for the observed superconductivity.
Experiments suggest that superconductivity in V-doped ZrB 2 have bulk nature.To eliminate the role of compositional fluctuations in V-doped ZrB 2 , Renosto et al. [29] carried out temperature (T)-dependence measurements of the specific heat (C P ) at zero applied magnetic field.A clear jump in C p /T vs T 2 is observed, which was consistent with their magnetization (M) vs T and resistivity (ρ) vs T measurements.Similar observations were found for Nb-doped ZrB 2 [44].Both Renosto et al. [29] and Marques et al. [44] found that, at low temperature, the electronic contribution to specific heat at log scale [ln(C e /γ T c )] as a function of T c /T shows a clear deviation from typical Bardeen-Cooper-Schrieffer (BCS)-type behavior.The change in C/γ T c from nonsuperconducting to superconducting phase was determined to 0.42 [29] and 0.15 [44], values that are considerably smaller than the weakcoupling BCS theory value (∼ 1.43).The superconducting gap [ (0)]-to-temperature (T) ratio in Zr 0.97 V 0.03 B 2 , being 2 (0)/k B T c ≈ 8.95, is much higher than the BCS theory of ≈ 3.5 [45].Along with the experiments discussed above, our theoretical results hint that superconductivity in V-doped ZrB 2 is of complex origin.

IV. CONCLUSIONS
In summary, we have examined the origin of superconductivity in electron-doped AlB 2 -type ZrB 2 within Migdal-Eliashberg formalism using various approaches.We found that doping-induced uniaxial compressive strain along the c direction causes a weakening of the EPC owing to acoustic and low-frequency phonon hardening which suppress superconductivity.Contrary to compressive strain, electron-dopinginduced carriers strengthen the EPC due to enhancement of densities of electronic states at the Fermi level as well as softening of both acoustic and optical phonons.Even though the calculated EPC constant in ZrB 2 is enhanced upon carrier doping, it is not sufficiently large to explain the superconducting transition temperature of 8.3-8.7 K within standard phonon-mediated superconductivity theory.The highest value of calculated superconducting temperature is one order of magnitude lower with respect to reported experimental transition temperatures, which indicates a complex origin of the superconductivity in ZrB 2 .The results indicate that further theoretical and experimental studies are required to understand the origin of this superconductivity.Electron doping in V-doped ZrB 2 can be related to a relatively localized V-4d orbital, whose role in the superconductivity warrants further investigation.The nontrivial topologicallike bands appearing near the Fermi level of ZrB 2 , which are robust upon doping, may be important for this superconductivity.
(a)], resulting in increased N (E F ).The calculated N (E F ) for Zr 0.963 V 0.037 B 2 and Zr 0.875 V 0.125 B 2 are 0.34 and 0.52 states eV −1 cell −1 , respectively.The atom projected DOS for Zr 0.963 V 0.037 B 2 shows that V-3d states are more localized than Zr-4d states near the Fermi level [Figs.2(b)-2(c)], with a major contribution from d 2

FIG. 2 .
FIG. 2. (a) Electronic density of states (DOS) per unit cell of V-doped ZrB 2 obtained with two different supercell dimensions: 3 × 3 × 3 for Zr 0.963 V 0.037 B 2 (blue solid line) and 2 × 2 × 2 for Zr 0.875 V 0.125 B 2 (magenta dashed line).(b) Atom and (c) orbital projected DOS of V and Zr atoms in Zr 0.963 V 0.037 B 2 .The inset in (c) shows orbital-projected Zr states around the Fermi level (E F ).

FIG. 4 .
FIG. 4. (a) Phonon density of states (DOS) [F(ω)] of electrondoped ZrB 2 with different electron-doping concentrations (n in units of e − /cell).(b) The isotropic Eliashberg spectral function [α 2 F (ω)] for various electron-doped levels considered in this paper.Inset of figure (b) is enlarged version of α 2 F (ω) as a function of high acoustic phonon energies.Note that both (a) and (b) are obtained for the configuration at which only ionic degrees of freedom were allowed (see text for details).(c)-(e) Electron-phonon coupling constant (λ), logarithmic average of phonon frequencies (ω log ), and electronic DOS at the Fermi level [N (E F )] as a function of doping concentrations, respectively.For virtual crystal approximation (VCA), n represents the V-doping concentration (x) in Zr 1−x V x B 2 .

TABLE I .
Computed -point phonon frequencies of optical phonons at various doping levels.Frequencies corresponding to Zr 1−x V x B 2 are obtained within VCA. are responsible for a weaker EPC strength in ZrB 2 than MgB 2 .This weaker EPC is the primary cause of absence of superconductivity in ZrB 2 .The Debye temperature ( D ) is estimated to be 560 K, in agreement with the experimental value of 555.6 K by Renosto et al. [29].For details, see Sec.II in the Supplemental Material spectra