Multiband superconductivity and charge density waves in the Ni-doped topological semimetal ZrTe_2
Lucas E. Correa, Pedro P. Ferreira, Leandro R. de Faria, Thiago T. Dorini, Mário S. da Luz, Zachary Fisk, Milton S. Torikachvili, Luiz T. F. Eleno, Antonio J. S. Machado
MMultiband superconductivity and charge density waves in the topological semimetalNi-doped ZrTe L. E. Correa, ∗ P. P. Ferreira, † L. R. de Faria, T. T. Dorini, Z.Fisk, M. S. Torikachvili, L. T. F. Eleno, and A. J. S. Machado ‡ Universidade de S˜ao Paulo, Escola de Engenharia de Lorena,Departamento de Engenharia de Materiais, Lorena, S˜ao Paulo, Brazil Universit´e de Lorraine, CNRS, IJL, Nancy, France University of California, Irvine, California 92697, USA Department of Physics, San Diego State University, San Diego, California 92182-1233, USA (Dated: February 10, 2021)We carried out a comprehensive study of the electronic, magnetic, and thermodynamic propertiesof Ni-doped ZrTe . High quality Ni . ZrTe2 single crystals show the coexistence of charge densitywaves (CDW, T
CDW ≈
287 K) with superconductivity (T c ≈ . c ) and upper (H c ) critical magnetic fieldsboth deviate significantly from the behaviors expected in conventional single-gap s-wave supercon-ductors. However, the behaviors of the normalized superfluid density ρ s ( T ) and H c ( T ) can bedescribed well using a two-gap model for the Fermi surface, in a manner consistent with multi-band superconductivity. Electrical resistivity and specific heat measurements show clear anomaliescentered near 287 K, suggestive of CDW phase transition. Additionally, electronic structure calcu-lations support the coexistence of electron-phonon multiband superconductivity and CDW due tothe compensated disconnected nature of the electron- and hole-pockets at the Fermi surface, withvery distinct orbital characters, Fermi velocities, and nesting vectors, as well as the softening of theacoustic phonon modes. Our calculations also suggest that ZrTe is a non-trivial topological type-IIDirac semimetal. These findings highlight that Ni-doped ZrTe2 is uniquely important for probingthe coexistence of superconducting and CDW ground states in an electronic system with non-trivialtopology. I. INTRODUCTION
Transition metal dichalcogenides (TMDs) are layeredquasi-2 dimensional compounds of general compositionMX , where M is a transition metal (e.g. Zr, Hf, Ti andTa) and X is a chalcogen (S, Se and Te) [1–3]. These ma-terials have drawn a heightened level of attention recentlydue to a plethora of uncommon physical properties, in-cluding the coexistence of superconductivity and chargedensity wave (CDW) and emergent quantum states [4–10]. The latter are particularly relevant for applicationsincluding quantum information, spintronic devices andbattery systems [11–17]. Individual layers are composedof three atomic planes with X-M-X stacking. Bulk sam-ples and heterostructures are formed by stacked layersbound by van der Waals interactions. Therefore, theycan show different behaviors according to the stackingorder and metal coordination. In general, TMDs crystal-lize with hexagonal or rhombohedral symmetries, space-groups (SG) P /mmc , P m R m , and the M sitehas octahedral or trigonal prismatic coordination withthe strongly bond X layers. Such configuration is afriendly environment for the intercalation of other atomicspecies and complexes in the region between adjacentchalcogen planes (van der Waals gap), permitting tweak- ∗ Corresponding author: [email protected] † Corresponding author: [email protected] ‡ Corresponding author: ajeff[email protected] ing and fine-tuning of the host’s electronic ground state[18–24].The TMD ZrTe crystallizes in a trigonal structure(CdI -type, SG P m
1, number 164). The electronic be-havior is typical of semimetals, though detailed studiesare lacking [25–27]. Recent studies using high-resolutionangle-resolved photoemission spectroscopy (ARPES) anddensity functional theory (DFT) revealed details of theFermi surface that suggest the possibility of a non-trivialtopological order with origin in the strong spin-orbitcoupling of Te manifold [27, 28]. Interestingly, a topo-logical nodal line state in ZrTe was proposed furtherthrought nuclear magnetic resonance (NMR) combinedwith DFT calculations [29], establishing ZrTe as a topo-logical semimetal.An example of the striking effect of intercalation inthe van der Waals gap is the addition of small amountsof copper in ZrTe , yielding superconductivity with T c ≈ . [31] exhibit experimen-tal and band structure topology features reminiscent ofthe Cu x ZrTe compounds, resulting in a superconduct-ing state with T c ≈ . intercalated with Ni is a r X i v : . [ c ond - m a t . s up r- c on ] F e b reported in this work for the first time. High-quality sin-gle crystals were synthesized by isothermal chemical va-por transport (CVT). Electrical resistivity and specificheat data suggest the onset of a CDW ground state near287 K, followed by the onset of superconductivity withT c ≈ . and Ni-dopedZrTe are provided, showing electronic and phononic dis-persions that are consistent with the experimental find-ings and a non-trivial band-topology, putting ZrTe as atype-II Dirac semimetal candidate. II. METHODSA. Experimental Details
The Ni . ZrTe single crystals were prepared byisothermal CVT. A precursor Ni . Zr alloy was preparedinitially by arc melting the stoichiometric amounts ofhigh purity Ni and Zr in a Ti-gettered Ar atmosphere.The resulting material was ground together with Te fora Ni . ZrTe nominal composition, pelletized, sealed ina quartz tube under 150 mmHg of UHP Ar, and reactedat 950 ◦ C for 48 hs. This material was quenched in wa-ter, an important step to avoid the formation ZrTe , re-ground, pelletized again, and treated again at 950 ◦ C for48 hs. The Ni . ZrTe pellet was placed in a quartz tubecontaining ≈ . of iodine, which served as thetransport agent, and sealed under vacuum. This tube wasplaced horizontally in a box furnace, the temperature wasraised to 1000 ◦ C and left there for 10 days. Given theoff-stoichiometry composition of the charge, a gradientof the thermochemical potential forms around the pellet,creating the necessary condition for chemical transportvapor. The tube was quenched again to avoid the forma-tion of Te-rich phases and thin single crystals could beeasily identified growing out of the pellet. The typicaldimensions of the larger crystals were ≈ × × . .The Ni . ZrTe composition for the crystals of thisstudy was determined by energy dispersive spectroscopy(EDS). While the 1:2 ratio for Zr:Te is the same for allcrystals, there is some variance in the concentration ofNi for crystals from the same batch, though the variancewithin each crystal is minimal. θ − θ X-ray diffractionscans of the flat facets showed only the (00 l ) reflectionsof the ZrTe structure, corresponding to the basal plane,which confirmed the phase purity and the single crys-tallinity. The Ni intercalation does not change the X-raypattern any significant way. A rocking curve centered onthe (002) reflection revealed a full-width spread at half-maximum within 0.05 ◦ (data not shown), suggestive ofhigh crystallinity. The lattice parameter c for Ni . ZrTe was determined using the programs Powdercell [33] andHighScore Plus from Panalytical, yielding c = 6 .
610 ˚A, avalue consistent with previous measurements [27].
FIG. 1. Powder x-ray diffraction of Ni . ZrTe . The insetshows the unit cell of NiZrTe (Ni: green; Zr: red; Te: blue). B. DFT calculations
First-principles electronic-structure calculations wereconducted within the Kohn-Sham scheme of the Den-sity Functional Theory (DFT) [34, 35], using ultra-soft pseudopotentials [36], as implemented in Quantum espresso [37, 38], and some auxiliary post-processingtools [39, 40]. The calculations were performed using thegeneralized gradient approximation (GGA) by Perdew-Burke-Ernzerhof (PBE) [41] for the exchange and cor-relation (XC) effects. All numerical and structural pa-rameters were converged and optimized to guarantee aground state convergence of 10 − Ry in total energy and10 − Ry/a (a ≈ .
529 ˚A) in total force acting on nuclei.Phonon dispersions were obtained by Density FunctionPertubation Theory [37, 38]. The dynamical matrix wascalculated using a 3 × × q -point grid and a smearingvalue of 158 K according to the Fermi-Dirac distribution.In order to simulate different compositions of Ni x ZrTe ,we used fully-relaxed 2 × × III. RESULTS AND DISCUSSIONA. Characterization of the normal andsuperconducting states
The temperature dependence of the electrical resistiv-ity ρ ( T ) for Ni . ZrTe in the ab -plane is shown in Fig.2a. Superimposed to the metallic behavior, two anoma-lies can be identified. First, a clear superconducting tran-sition with onset at T c ≈ . c was taken from the temperature where an ex-trapolation of ρ ( T ) from the normal phase and a fit to the ρ ( T ) data below T c start to diverge. The ρ ( T ) = 0 valueis reached near 2.7 K. Secondly, a hump centered near287 K, taken from the discontinuity in dρ/dT , is sugges- (a) (b) FIG. 2. (a) Resistivity dependence with temperature in zero magnetic field. The insets show the ρ ( T ) curve in (i) low-temperature regime and (ii) high-temperature regime. The inset (iii) presents the dρ/dT curve, evidencing the CDW transitionat T CDW ≈
287 K. (b) Temperature dependence of specific heat for Ni . ZrTe . The inset depicts the CDW anomaly at hightemperatures. tive of the formation of a CDW [43–47]. In fact, the be-havior of ρ ( T ) for Ni . ZrTe reflecting the coexistenceof the superconducting and CDW ground states is rem-iniscent, for example, of the behavior found in the low-dimensional compounds 2H-Pd x TaSe and HfTe [9, 48].The temperature dependence of the specific heat forNi . ZrTe is shown in Fig. 2b. A fit of the C p /T vs T data at low temperatures to γ + βT yields values for theSommerfeld coefficient γ and the phonon term β equal to4.79 mJ/molK and 0.99 mJ/mol, respectively, the lattercorresponding to a Debye temperature Θ D ≈
380 K. Incontrast to the ρ ( T ) data, there is no clear feature sug-gestive of a superconducting transition in the C p data,which raises the question of whether superconductivityin Ni . ZrTe is a bulk phenomenon. However, the C p vs T data of Fig. 2b show a clear feature with onset near287 K, consistent with the ρ ( T ) curve and the onset of aCDW ground state.In order to determine the lower and upper criticalfields, H c and H c , respectively, we carried out isother-mal magnetization curve measurements from 1.9 to 5.0 K,and measurements of ρ ( T ) near T c in fields up to 1400 Oe,as shown in Fig. 3. In spite of the specific heat lackingany clear anomalies due to superconductivity, the mag-netization curves below T c (Fig. 3a), and the drive ofthe resistivity transitions to lower temperatures with field(Fig. 3b) are both solid evidence for superconductivityin Ni . ZrTe .The temperature dependence of H c for Ni . ZrTe isshown in Figure 3c. The H c values were taken from onsetof the transitions in ρ ( T ). The behavior of H c ( T ) in thedirty limit, within the framework of the single-band BCStheory, was modeled by Werthamer-Helfand-Hohenberg(WHH) model [49]. Taking the derivative dH c /dT | T = T c and using the expression µ H c (0) = − . T c dH c dT | T = T c (1)the fit yields the blue line in Fig. 3, which obviouslyunderestimates the values of H c . The upturn in theH c ( T ) data, which starts to become more prominentnear T/T c ≈ .
85, is in sharp contrast with the quadraticbehavior of the WHH theory. Actually, the positive cur-vature of µ H c below T c is an ubiquitous feature inmultiband superconductivity [50–57].Ignoring the interband impurity scattering channel andtaking the orbital pair-breaking effect, we can fit the H c within a two-band model derived from quasi-classic Us-adel equations [58], given by0 = a [ln t + U ( h )] [ln t + U ( ηh )]+ a [ln t + U ( h )] + a [ln t + U ( ηh )] , (2)where a , a and a are defined by the intraband ( λ and λ ) and interband ( λ and λ ) coupling strength be-tween bands 1 and 2, h = H c D / φ T , η = D /D , and U ( x ) = ψ ( x + 1 / − ψ (1 / ψ ( x ) is the digammafunction, D and D are the intraband electronic diffu-sivities of bands 1 and 2 in the normal-state, and φ isthe magnetic flux quantum. The diffusivity ratio η de-termines the curvature of H c ( T ).As seen in Figure 3c, the H c ( T ) data can be fit verywell with this two-band model. The upper critical fieldat T = 0 yielded by Equation 2 is H c (0) = 5 .
05 kOe.The resultant diffusivity and coupling parameters are η = 19 . λ = 0 . λ = 0 . λ = 0 .
02 and λ = 0 .
01. The value found for the diffusivity ratio re-flects a large difference in the electron mobility of theFermi surface sheets involved in the superconductivity,originating the positive curvature that deviates from the (a) (b)(c) (d)
FIG. 3. (a) Magnetization dependence with the applied magnetic field for Ni x ZrTe . (b) Magnetoresistance with appliedmagnetic field. (c) Upper critical field as a function of reduced temperature. (d) Normalized superfluid density as a functionof reduced temperature at zero magnetic field. The red dashed line shows the expected behaviour for an isotropic single-gappairing mechanism. traditional WHH-like single-band behavior, and reflectsthe suppression of the small superconducting gap withmagnetic field. Moreover, values of λ ij from the fit sug-gest that the intraband coupling is one order of magni-tude higher than the interband scattering, also favoringthe emergence of the two-gap superconductivity [59].The values of H c were extracted from the M ( H )isotherms of Fig. 3a, by using a ∆ M = 10 − emu cri-terion, i.e. the values of H c at each temperature weretaken when the difference between the data and the ex-trapolated linear regions reached 10 − emu. The tem-perature dependence of the superfluid density ρ s ( T ) forNi . ZrTe , normalized to its zero temperature value,is shown in Fig. 3d. In the framework of the Londonapproximation, ρ s ( T ) = H c ( T ) /H c (0). The lower crit-ical field behavior of Fig. 3d diverges quite significantly from the quadratic behavior expected from the single-band model (dashed line). A robust upturn in H c be-comes quite noticeable near T /T c ≈ .
75. Anomalousupturns in H c of this type have consistently been foundin two-band superconductors [60, 61].For an isotropic two-gap superconductor in the Meiss-ner state, the normalized superfluid density at low tem-peratures can be expressed as [62] ρ s ( T ) = 1 − c (cid:18) π ∆ S (0) k B T (cid:19) / e − ∆ S (0) k B T − (1 − c ) (cid:18) π ∆ L (0) k B T (cid:19) / e − ∆ L (0) k B T , (3)where ∆ S and ∆ L are the small and large superconduct-ing gap, respectively, and c is the superconducting weightparameter of ∆ S . The solid line in Figure 3(d) shows thefit to the superfluid density using Equation 3. The goodfit is consistent with the existence of two superconductinggaps at the Fermi surface. The estimated parameters forNi . ZrTe are ∆ S (0) = 0 .
15 meV, ∆ L (0) = 1 .
03 meV, c = 9 %, and H c (0) = 35 Oe. Alternatively, for thecase of a single anisotropic superconducting gap, the c parameter would be 0 or 1. However, given that both∆ S and ∆ L terms have non-zero values, the possibilitythat the anomalous upturn in H c ( T ) stems from a singleanisotropic order parameter can be ruled out.The dashed line in Fig. 3d shows the expected behaviorfor ρ s ( T ) if the Ni . ZrTe superconductivity stemmedfrom a single-gap, considering an isotropic Fermi veloc-ity and s-wave pairing symmetry [63–65]. The strongdiscrepancy between the two models excludes the singleisotropic s-wave gap scenario. In light of the impossibil-ity of matching the H c ( T ) and H c ( T ) behaviors to asingle-gap model, and the success of the two-gap modelto fit the data, the latter gains credence, without theneed to deal with unconventional pairing symmetries orstrong anisotropic effects for the Fermi surface. B. CDW and superconductivity interplay:first-principles calculations
Given the strong possibility of multiband supercon-ductivity and CDW coexistence/competition in Ni-dopedZrTe2, as yielded by the resistivity, magnetization andheat capacity data, we carried out first-principles calcu-lations of the electronic and phononic structures in orderto gain insights on the microscopic mechanisms in thissystem. The projected density of states for Ni x ZrTe ( x = 0 . , . , . , . , .
00) is presented in Figure 4.The total density of states at the Fermi level ( N E F )including spin-orbit coupling effects of pure ZrTe is0.985 states/eV. Of these carriers, 62% are derived fromZr-3d manifold and 34% corresponds to Te-5p orbitals.As the Ni intercalating content increases, the Ni-3dstates create a localized band around − . − x = 0 .
25, for instance, thetotal DOS increases to 1.62 states/eV, favoring the emer-gence of spontaneous symmetry breaking mechanismssuch as superconductivity and other electronic instabili-ties. At the same time the contribution of the Te-p man-ifold at the Fermi surface reaches 64 % of the total DOS.Therefore, the main role of the Ni ions as they populatethe van der Waals gap is the stabilization of the trans-ferred charge to the Te sites, gradually enhancing theelectronic correlation at the Fermi surface, consistentlywith the doping level. This mechanism is similar to the
FIG. 4. Projected density of states for different Ni-dopedNi x ZrTe compounds. formation of strong Cu-Se bonds in Cu x TiSe [19].From the estimated total density of states at the Fermilevel for Ni . ZrTe , we can evaluate the superconduct-ing critical temperature using McMillan’s equation [66], T c = Θ D .
45 exp (cid:20) − . λ ) λ − µ ∗ (1 + 0 . λ ) (cid:21) , (4)where Θ D is the Debye temperature, λ is the electron-phonon coupling constant, and µ ∗ is the Coulomb pseu-dopotential, which measures the strength of the electron-electron Coulomb repulsion [67]. The electron-phononcoupling constant and Coulomb pseudopotential, respec-tively, are estimated to be 0.70 and 0.17 according to λ ≈ ( γ exp /γ calc ) − µ ∗ ≈ { . N ( E F ) / [1 + N ( E F )] } [68, 69], and the estimated critical temperature from Eq.4 is T c ≈ .
28 K, close to the 4.1 K value from the ρ ( T )data. The proximity of the measured and estimated val-ues is consistent with the electron-phonon interactionsbeing the driving mechanism for superconductivity.In particular, it is reasonable to assume that the elec-tronic ground state of Ni . ZrTe corresponds to theenergy-momentum dispersion of ZrTe , with a small dis-placement of chemical potential due to Ni- d hybridiza-tion with Te- p orbitals. For comparison, the electronicband-structure without and with spin-orbit effects andthe Fermi surface topography of ZrTe is presented inFigure 5. The calculated electronic ground state is inexcellent agreement with recent spectral density mapsobtained by high-resolution ARPES measurements [27]. (a) (b)(c) (d) (e)(f) (g) (h) FIG. 5. Electronic band structure (a) without and (b) with spin-orbit coupling effects along high-symmetry points at the firstBrillouin zone projected onto Te- p (blue) and Zr- d (red) orbitals. (c)-(e) the three independent sheets of the ZrTe Fermisurface. The color map shows the contribution of Te-5p (blue) and Ni-3d (red) manifold to the electronic wave function. (f)-(h)Fermi surface projected onto the Fermi velocity.
The typical semimetal behavior of ρ ( T ) data for pureZrTe2 [70] is consistent with the 3 distinct bands crossingthe Fermi level . Two hole-like bands ( α and α ) crossthe Fermi level around Γ and A high-symmetry points,giving rise to quasi-cylindrical pockets along the out-of-plane direction in the Fermi surface, and an additionalelectron-like band ( β ) can be found around the L di-rection in the first Brillouin zone, composing multipledisconnected pockets with hexagonal symmetry. We canobserve a clear predominance of Zr-d orbitals in the con- duction, electron band ( β pocket) and a majority contri-bution of Te-p orbitals in valence, hole bands ( α and α pockets). The disconnected multiband nature with verydistinct orbital projections, Fermi velocities, and nestingvectors support two-band superconductivity conjecturein this system [56, 68, 71–75], consistently with the ex-perimental findings. In fact, the different band Fermi ve-locities of ZrTe favors the emergence of such unconven-tional superconducting magnetic properties due to thepresence of multiple condensates with dissimilar compet-ing characteristic lenghts [76].It is interesting to note that the electronic structureand electron-phonon parameters for ZrTe are nearlyidentical to TiSe [77–80]. TiSe is composed by Se-4phole-pockets at the center of its first Brillouin zone (Γ)and Ti-3 d electron-pockets located at L high symmetry-point with electron-phonon coupling constant ≈ .
65 [80],very similar to ZrTe . This compound is widely investi-gated, and it can be regarded as a prototype for the studyof the connection between CDW and superconductivityin low-dimensional materials [6, 20, 24, 81–84]. The on-set of a CDW phase transition in TiSe is marked by ananomaly in the electrical resistivity at T = 202 K that isquite similar to the 287 K anomaly in Ni . ZrTe [85].It is noteworthy to point out that the physical nature ofCDWs in general is controversial. Theoretical and ex-perimental studies suggest that rather than CDWs, thephenomena being observed in the bulk properties canbe the result of exciton condensation (electron-hole cou-pling) leading to insulating behavior [86, 87], Jahn-Tellereffect (electron-phonon coupling) [88] or hybrid exciton-phonon modes [89, 90].Given the similarity of the electronic ground states andelectron-phonon coupling strength for ZrTe and TiSe ,we argue that ZrTe is able to harbor a hidden electronicCDW phase transition through an exciton-phonon-drivenmechanism. Tuning the density of states at the Fermilevel, the Couloumb electron-hole interaction can giverise to electron-hole bound states, i.e. exciton quasipar-ticles. If the binding energy of the electron-hole pair isgreater than the energy difference between the maximumof the valence band and the minimum of the conductionband, the system becomes energetically unstable, under-going an excitonic condensate with phase coherence anda periodicity defined by the wave-vector that connectsthe valence and conduction sheets [87]. The exciton for-mation will be favored in systems with low density ofstates and compensated electron- and hole-type carriersat the Fermi surface, as shown in the case of Ni . ZrTe .However, the manifestation of electron-phonon super-conductivity suggests that the parent CDW state couldbe mediated by a combination of phonons and excitonicfluctuations. The phonon dispersion of ZrTe , withoutSOC effects, obtained using Density Functional Pertur-bation Theory (DFPT) [91, 92] is shown in Figure 6.A narrow softening of the acoustic phonon branches oc-curs along Γ–A. Such Kohn anomaly [93] could inducea Peierls distortion, where a structural phase transitiontakes place due to the softening down to zero of a partic-ular phonon mode [94]. Within this hybridized exciton-phonon scenario, increasing the exciton binding energywould significantly enhance the softening of the phononspectrum, which promotes the structural instability [89].Therefore, it is plausible that ZrTe could harbor hid-den CDW instabilities which can be brought forwardby small Ni-doping levels, as suggested by the observedanomalies in ρ ( T ) and C p ( T ) near 287 K. The band-structure calculation suggests that the intercalation of Ni FIG. 6. ZrTe phonon dispersion and phonon DOS. Thereis a narrow softening of the acoustic phonon branches alongΓ–A. Ni intercalation could enhance such Kohn anomaly dueto increased screening effects on the Fermi surface, leading toa structural phase transition and to a lower symmetry phase. in the ZrTe structure fine-tunes the Fermi surface topog-raphy by enhancing the Te- p contribution at the Fermilevel, creating a delicate balance between the electron-phonon and electron-hole interactions necessary for thesubtle competition between superconductivity and CDWtransition. C. Band structure topology
Additionally, due to the strong spin-orbit coupling ofthe Te-p wavefunction, the electronic dispersion is heav-ily affected by SOC effects. It is possible to observeelectronic broken degeneracies over the entire extent offirst Brillouin zone in the vicinity of the Fermi level withSOC (see Figures 5a-b). However, the double group ir-reductible representations of the band-crossing degenera-cies along Γ–A direction (out-of-plane), two-fold degen-erate due to inversion- and time-reversal-symmetries, areprotected by C rotational-symmetry of the D d point-group along the k z axis [31], including spin degree-of-freedom. Thus, the linear band crossings above E F alongΓ–A are allowed, resulting in Dirac cones that can be ob-served in the electronic band-structure.The group analysis of the electronic dispersion alongthe Γ–A high-symmetry line, without (a) and with (b)SOC effects, is shown in Fig. 7. In the absence of spin-orbit coupling, the bands with E u and E g representa-tions are fourfold degenerate and the E g states have alow dispersion along the out-of-plane direction. Equallyimportant, there is a band-crossing between the E g andA g bands, occurring at approximately 0.7 eV above theFermi level. Including the SOC effects, the degeneracy isbroken, abruptly changing the electronic dispersion. TheE u will give rise to the double group R ± representation,while the E g derived states will result in low dispersive (a) (b) FIG. 7. Irreducible representations and parity analysis alongΓ–A direction of ZrTe (a) without and (b) with SOC effects. electronic-states with R ± , representation. As a result, wewill have the formation of two linear band-crossings thatresemble tilted Dirac cones between 0.6 eV and 0.8 eVderived from the crossings between R , and R repre-sentations. At the same time, SOC-broken degeneracyat the E g states results in parity inversion, promoting atopologically non-trivial gap.Therefore, Ni x ZrTe is a possible candidate to be clas-sified as a non-trivial topological type-II Dirac semimetal,in contrast to recent studies claiming a topological nodalline state [29]. However, a comprehensive experimentalcharacterization of the topological properties is still inorder, and the role of Ni doping needs to be exploredfurther. IV. CONCLUSIONS
In summary, we have investigated the normal and su-perconducting phases of Ni . ZrTe single crystals byelectrical resistivity, specific heat, and magnetizationmeasurements. The electrical resistivity and magnetiza-tion data show undoubtedly the onset of superconductiv-ity with T c ≈ . . ZrTe adds tothe small list of materials where superconductivity andCDW compete for the Fermi surface. The temperaturedependence of the upper critical field and the normal-ized superfluid density can be consistently fitted with atwo-gap model, suggesting that there are two supercon-ducting order parameters. The DFT calculations sup-port the experimental findings remarkably well, and theyshow that ZrTe can be classified as a non-trivial topo-logical type-II Dirac semimetal. The electronic structurecalculations indicate that the main result for the inclu-sion of Ni in the van der Waals gap is the stabilizationof the transferred charge to the Te sites, gradually en-hancing the electronic correlation at the Fermi surface,according to the doping level, opening the way for elec-tronic instabilities and spontaneous symmetry-breakingmechanisms. Additionally, the electron-phonon couplingstrength combined with the Fermi surface topology andthe softening of the acoustic phonon modes in ZrTe areconsistent with an electron-phonon multiband supercon-ductivity and an exciton-phonon driven mechanism forthe CDW condensation. Given the uniqueness of the elec-tronic structure in Ni-doped ZrTe , and the coexistenceof superconductivity and CDW ground states, such com-pounds hold the promise to be excellent probing groundsin the investigation of coherent and non-trivial quantumstates. ACKNOWLEDGMENTS
We gratefully acknowledge the financial supportof the Funda¸c˜ao de Amparo `a Pesquisa do Estadode S˜ao Paulo (FAPESP) under Grants 2018/10835-6,2020/08258-0, 2016/10167-8, 2018/08819-2, 2019/14359-7 and 2019/05005-7. This study was also financed inpart by the Coordena¸c˜ao de Aperfei¸coamento de Pes-soal de N´ıvel Superior (CAPES) - Brazil - Finance Code001. The research was conducted using high-performancecomputing resources made available by the Superin-tendˆencia de Tecnologia da Informa¸c˜ao (STI), Univer-sidade de S˜ao Paulo. The authors also thank Michel L.M. dos Santos for fruitful discussions. [1] S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev,and A. Kis, Nature Reviews Materials , 17033 (2017).[2] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,and M. S. Strano, Nature Nanotechnology , 699 (2012).[3] M. Chhowalla, H. S. Shin, G. Eda, L.-J. Li, K. P. Loh,and H. Zhang, Nature Chemistry , 263 (2013).[4] A. H. C. Neto, Physical Review Letters , 4382 (2001).[5] T. Valla, A. V. Fedorov, P. D. Johnson, P. A. Glans,C. McGuinness, K. E. Smith, E. Y. Andrei, and H. Berger, Physical Review Letters , 086401 (2004).[6] E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G.Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong,and R. J. Cava, Nature Physics , 544 (2006).[7] B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger,L. Forr´o, and E. Tutiˇs, Nature Materials , 960 (2008).[8] B. T. Zhou, N. F. Yuan, H.-L. Jiang, and K. T. Law,Physical Review B , 180501 (2016). [9] D. Bhoi, S. Khim, W. Nam, B. Lee, C. Kim, B.-G. Jeon,B. H. Min, S. Park, and K. H. Kim, Scientific Reports , 1 (2016).[10] G. H. Han, D. L. Duong, D. H. Keum, S. J. Yun, andY. H. Lee, Chemical Reviews , 6297 (2018).[11] N. Zibouche, A. Kuc, J. Musfeldt, and T. Heine, Annalender Physik , 395 (2014).[12] F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasub-ramaniam, Nature Photonics , 899 (2014).[13] E. Yang, H. Ji, and Y. Jung, The Journal of PhysicalChemistry C , 26374 (2015).[14] S. Wu, Y. Du, and S. Sun, Chemical Engineering Journal , 189 (2017).[15] A. Krasnok, S. Lepeshov, and A. Al´u, Optics Express , 15972 (2018).[16] A. David, P. Rakyta, A. Korm´anyos, and G. Burkard,Physical Review B , 085412 (2019).[17] B. Lucatto, D. S. Koda, F. Bechstedt, M. Marques, andL. K. Teles, Physical Review B , 121406 (2019).[18] K. E. Wagner, E. Morosan, Y. S. Hor, J. Tao, Y. Zhu,T. Sanders, T. M. McQueen, H. W. Zandbergen, A. J.Williams, D. V. West, et al. , Physical Review B ,104520 (2008).[19] R. A. Jishi and H. M. Alyahyaei, Physical Review B ,144516 (2008).[20] E. Morosan, K. E. Wagner, L. L. Zhao, Y. Hor, A. J.Williams, J. Tao, Y. Zhu, and R. J. Cava, Physical Re-view B , 094524 (2010).[21] A. Kiswandhi, J. S. Brooks, H. B. Cao, J. Q. Yan,D. Mandrus, Z. Jiang, and H. D. Zhou, Physical Re-view B , 121107 (2013).[22] T.-R. Chang, P.-J. Chen, G. Bian, S.-M. Huang,H. Zheng, T. Neupert, R. Sankar, S.-Y. Xu, I. Belopolski,G. Chang, et al. , Physical Review B , 245130 (2016).[23] D. M. Guzman, N. Onofrio, and A. Strachan, Journal ofApplied Physics , 055703 (2017).[24] S. Kitou, A. Nakano, S. Kobayashi, K. Sugawara,N. Katayama, N. Maejima, A. Machida, T. Watanuki,K. Ichimura, S. Tanda, et al. , Physical Review B ,104109 (2019).[25] R. De Boer and E. H. P. Cordfunke, Journal of Alloysand Compounds , 115 (1997).[26] A. H. Reshak and S. Auluck, Physica B: Condensed Mat-ter , 230 (2004).[27] I. Kar, J. Chatterjee, L. Harnagea, Y. Kushnirenko,A. Fedorov, D. Shrivastava, B. B¨uchner, P. Mahadevan,and S. Thirupathaiah, Physical Review B , 165122(2020).[28] P. Tsipas, D. Tsoutsou, S. Fragkos, R. Sant, C. Alvarez,H. Okuno, G. Renaud, R. Alcotte, T. Baron, and A. Di-moulas, ACS nano , 1696 (2018).[29] Y. Tian, N. Ghassemi, and J. H. Ross Jr, Physical Re-view B , 165149 (2020).[30] A. J. S. Machado, N. P. Baptista, B. S. De Lima,N. Chaia, T. W. Grant, L. E. Corrˆea, S. T. Renosto,A. C. Scaramussa, R. F. Jardim, M. S. Torikachvili, et al. ,Physical Review B , 144505 (2017).[31] P. P. Ferreira, A. L. Manesco, T. T. Dorini, L. E. Correa,G. Weber, A. J. S. Machado, and L. T. F. Eleno, arXivpreprint arXiv:2006.14071 (2020).[32] B. S. de Lima, R. R. de Cassia, F. B. Santos, L. E. Correa,T. W. Grant, A. L. R. Manesco, G. W. Martins, L. T. F.Eleno, M. S. Torikachvili, and A. J. S. Machado, SolidState Communications , 27 (2018). [33] W. Kraus and G. Nolze, Journal of Applied Crystallog-raphy , 301 (1996).[34] P. Hohenberg and W. Kohn, Physical Review , B864(1964).[35] W. Kohn and L. J. Sham, Physical Review , A1133(1965).[36] A. Dal Corso, Computational Materials Science , 337(2014).[37] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-cioni, I. Dabo, et al. , Journal of physics: Condensed mat-ter , 395502 (2009).[38] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau,M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni,D. Ceresoli, M. Cococcioni, et al. , Journal of Physics:Condensed Matter , 465901 (2017).[39] A. Kokalj, Journal of Molecular Graphics and Modelling , 176 (1999).[40] M. Kawamura, Computer Physics Communications ,197 (2019).[41] J. P. Perdew, K. Burke, and M. Ernzerhof, PhysicalReview Letters , 3865 (1996).[42] K. Okhotnikov, T. Charpentier, and S. Cadars, Journalof cheminformatics , 1 (2016).[43] C. Berthier, P. Molini´e, and D. J´erome, Solid State Com-munications , 1393 (1976).[44] R. A. Craven and S. F. Meyer, Physical Review B ,4583 (1977).[45] J. Yang, W. Wang, Y. Liu, H. Du, W. Ning, G. Zheng,C. Jin, Y. Han, N. Wang, Z. Yang, et al. , Applied PhysicsLetters , 063109 (2014).[46] X. Zhu, Y. Cao, J. Zhang, E. Plummer, and J. Guo,Proceedings of the National Academy of Sciences ,2367 (2015).[47] K. K. Kolincio, M. Roman, M. J. Winiarski,J. Strychalska-Nowak, and T. Klimczuk, Physical Re-view B , 235156 (2017).[48] S. J. Denholme, A. Yukawa, K. Tsumura, M. Nagao,R. Tamura, S. Watauchi, I. Tanaka, H. Takayanagi, andN. Miyakawa, Scientific Reports , 45217 (2017).[49] N. R. Werthamer, E. Helfand, and P. C. Hohenberg,Phys. Rev. , 295 (1966).[50] L. Lyard, P. Samuely, P. Szabo, T. Klein, C. Marcenat,L. Paulius, K. H. P. Kim, C. U. Jung, H.-S. Lee, B. Kang, et al. , Physical Review B , 180502 (2002).[51] F. Hunte, J. Jaroszynski, A. Gurevich, D. Larbalestier,R. Jin, A. Sefat, M. A. McGuire, B. C. Sales, D. K.Christen, and D. Mandrus, Nature , 903 (2008).[52] H. Lei, D. Graf, R. Hu, H. Ryu, E. S. Choi, S. W. Tozer,C. Petrovic, et al. , Physical Review B , 094515 (2012).[53] Y. Li, Q. Gu, C. Chen, J. Zhang, Q. Liu, X. Hu, J. Liu,Y. Liu, L. Ling, M. Tian, et al. , Proceedings of the Na-tional Academy of Sciences , 9503 (2018).[54] F. B. Santos, L. E. Correa, B. S. de Lima, O. V. Cigarroa,M. S. da Luz, T. Grant, Z. Fisk, and A. J. S. Machado,Physics Letters A , 1065 (2018).[55] T. Shang, A. Amon, D. Kasinathan, W. Xie, M. Bobnar,Y. Chen, A. Wang, M. Shi, M. Medarde, H. Yuan, et al. ,New Journal of Physics , 073034 (2019).[56] C. Xu, B. Li, J. Feng, W. Jiao, Y. Li, S. Liu, Y. Zhou,R. Sankar, N. D. Zhigadlo, H. Wang, et al. , PhysicalReview B , 134503 (2019).[57] A. Majumdar, D. VanGennep, J. Brisbois, D. Chareev,A. V. Sadakov, A. Usoltsev, M. Mito, A. V. Silhanek, T. Sarkar, A. Hassan, et al. , Physical Review Materials , 084005 (2020).[58] A. Gurevich, Physical Review B , 184515 (2003).[59] P. J. F. Cavalcanti, T. T. Saraiva, J. A. Aguiar, A. Vagov,M. Croitoru, and A. A. Shanenko, Journal of Physics:Condensed Matter , 455702 (2020).[60] M. Angst, R. Puzniak, A. Wisniewski, J. Jun, S. M.Kazakov, J. Karpinski, J. Roos, and H. Keller, Phys-ical Review Letters , 167004 (2002).[61] C. Ren, Z.-S. Wang, H.-Q. Luo, H. Yang, L. Shan, andH.-H. Wen, Physical Review Letters , 257006 (2008).[62] M.-S. Kim, J. A. Skinta, T. R. Lemberger, W. N. Kang,H.-J. Kim, E.-M. Choi, and S.-I. Lee, Physical ReviewB , 064511 (2002).[63] B. S. Chandrasekhar and D. Einzel, Annalen der Physik , 535 (1993).[64] A. Carrington and F. Manzano, Physica C: Superconduc-tivity , 205 (2003).[65] R. Prozorov and R. W. Giannetta, Superconductor Sci-ence and Technology , R41 (2006).[66] W. L. McMillan, Physical Review , 331 (1968).[67] W. L. McMillan and J. M. Rowell, Physical Review Let-ters , 108 (1965).[68] P. P. Ferreira, F. B. Santos, A. J. S. Machado, H. M.Petrilli, and L. T. F. Eleno, Physical Review B ,045126 (2018).[69] K. H. Bennemann and J. W. Garland, in AIP ConferenceProceedings , Vol. 4 (American Institute of Physics, 1972)pp. 103–137.[70] Y. Aoki, T. Sambongi, F. Levy, and H. Berger, Journalof the Physical Society of Japan , 2590 (1996).[71] A. Floris, A. Sanna, S. Massidda, and E. K. U. Gross,Physical Review B , 054508 (2007).[72] D. J. Singh, Physical Review B , 174508 (2013).[73] D. J. Singh, PloS One , e0123667 (2015).[74] A. Bhattacharyya, P. P. Ferreira, F. B. Santos, D. T.Adroja, J. S. Lord, L. E. Correa, A. J. S. Machado,A. L. R. Manesco, and L. T. F. Eleno, Physical ReviewResearch , 022001 (2020).[75] Y. Zhao, C. Lian, S. Zeng, Z. Dai, S. Meng, and J. Ni,Physical Review B , 104507 (2020).[76] Y. Chen, H. Zhu, and A. A. Shanenko, Physical ReviewB , 214510 (2020). [77] A. Zunger and A. J. Freeman, Physical Review B ,1839 (1978).[78] C. M. Fang, R. A. De Groot, and C. Haas, PhysicalReview B , 4455 (1997).[79] R. Bianco, M. Calandra, and F. Mauri, Physical ReviewB , 094107 (2015).[80] T. Das and K. Dolui, Physical Review B , 094510(2015).[81] A. F. Kusmartseva, B. Sipos, H. Berger, L. Forro, andE. Tutiˇs, Physical Review Letters , 236401 (2009).[82] T. Rohwer, S. Hellmann, M. Wiesenmayer, C. Sohrt,A. Stange, B. Slomski, A. Carr, Y. Liu, L. M. Avila,M. Kall¨ane, et al. , Nature , 490 (2011).[83] E. M¨ohr-Vorobeva, S. L. Johnson, P. Beaud, U. Staub,R. De Souza, C. Milne, G. Ingold, J. Demsar, H. Sch¨afer,and A. Titov, Physical Review Letters , 036403(2011).[84] L. Li, E. O’Farrell, K. Loh, G. Eda, B. ¨Ozyilmaz, andA. C. Neto, Nature , 185 (2016).[85] F. J. Di Salvo, D. E. Moncton, and J. V. Waszczak,Physical Review B , 4321 (1976).[86] T. Pillo, J. Hayoz, H. Berger, F. L´evy, L. Schlapbach,and P. Aebi, Physical Review B , 16213 (2000).[87] H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. De-spont, M. Garnier, H. Beck, P. Aebi, L. Patthey,H. Berger, et al. , Physical Review Letters , 146403(2007).[88] K. Rossnagel, L. Kipp, and M. Skibowski, Physical Re-view B , 235101 (2002).[89] J. van Wezel, P. Nahai-Williamson, and S. S. Saxena,Physical Review B , 165109 (2010).[90] H. Watanabe, K. Seki, and S. Yunoki, Physical ReviewB , 205135 (2015).[91] S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. , 1861 (1987).[92] X. Gonze, Phys. Rev. A , 1086 (1995).[93] W. Kohn, Physical Review Letters , 393 (1959).[94] D. L. Duong, M. Burghard, and J. C. Sch¨on, PhysicalReview B92