Dimensionality of superconductivity in the transition metal pnictide WP
Angela Nigro, Giuseppe Cuono, Pasquale Marra, Antonio Leo, Gaia Grimaldi, Ziyi Liu, Zhenyu Mi, Wei Wu, Guangtong Liu, Carmine Autieri, Jianlin Luo, Canio Noce
DDimensionality of superconductivity in the transition metal pnictide WP
Angela Nigro, Giuseppe Cuono, ∗ Pasquale Marra,
3, 4, † Antonio Leo,
1, 5, 6
Gaia Grimaldi,
6, 1
Ziyi Liu, Zhenyu Mi, Wei Wu, Guangtong Liu,
7, 8
Carmine Autieri,
2, 6
Jianlin Luo,
7, 8, 9 and Canio Noce
1, 6 Dipartimento di Fisica "E.R. Caianiello", Università degli Studi di Salerno, I-84084 Fisciano (Salerno), Italy International Research Centre Magtop, Institute of Physics,Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan Department of Physics, and Research and Education Center for Natural Sciences,Keio University, Hiyoshi, Kanagawa, 223-8521, Japan NANO_MATES Research Center, Università degli Studi di Salerno, I-84084 Fisciano (Salerno), Italy Consiglio Nazionale delle Ricerche CNR-SPIN, UOS Salerno, I-84084 Fisciano (Salerno), Italy Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
We report theoretical and experimental results on transition metal pnictide WP. The theoretical outcomesbased on tight-binding calculations and density functional theory indicate that WP exhibits the nonsymmorphicsymmetries and is an anisotropic three-dimensional superconductor. This conclusion is supported by magnetore-sistance experimental data as well as by the investigation of the superconducting fluctuations of the conductivityin the presence of a magnetic external field, both underlining a three dimensional behavior.
I. INTRODUCTION
The discovery of superconductivity under external pres-sure in the chromium arsenide CrAs stimulated consider-able efforts in the quest of superconductivity in other bi-nary pnictides at ambient pressure.
The CrAs belongs tothe family of transition metal pnictides with chemical for-mula MX (with M=transition metal and X=P, As, Sb), andit has an orthorhombic MnP-type crystal structure at ambi-ent conditions. Soon after this discovery, a new memberof the same family, the transition metal phosphide MnP, hasbeen grown up. Both MnP and CrAs become superconduct-ing under external pressure, and exhibit a similar temperature-pressure phase diagram with a superconducting dome and the presence of magnetic phases which can coexist withsuperconductivity.
More recently, a new superconductor of the same serieshas been produced, namely the tungsten phosphide WP, witha bulk superconductivity appearing at 0.84 K at ambientpressure. So far, WP is the only known example of a 5 d tran-sition metal phosphide with a nonmagnetic groundstate. Inthis compound, the spatial extension of the W 5 d orbitals in-duces a large overlap and a strong coupling with the neigh-boring p -orbitals, which results in a distortion of the crystalstructure which is more pronounced in comparison with thatof CrAs and MnP. Moreover, the spin-orbit coupling of W5 d electrons is stronger than that of 3 d electrons of the CrAsand MnP. The primitive cell of WP contains four W and four P atoms,with each W atom surrounded by six nearest-neighbour Patoms, and located at the centre of the face-sharing WP octahedra, shown in Fig. 1. Four of the six bonds are in-equivalent due to the space group anisotropy. In this class of materials, the nonmagnetic phase presentshole-like branches of the Fermi surface with two-dimensional(2D) dispersion, together with electron-like branches with afull three-dimensional (3D) character.
Therefore, it is legit-
FIG. 1. The orthorhombic crystal structure of tungsten phosphideWP with space group Pnma. Orange and blue spheres indicate Wand P ions respectively, with nonequivalent lattice positions of the Wions labeled as W A , W B , W C , W D . Face-sharing WP octahedraare shaded in gray. imate to ask what is the dimensionality of transport in thesesystems in the superconducting phase. It is well-known thatthis depends on the structure of the order parameter in the k -space associated with the points of the Fermi surface. In par-ticular, when the hole-like surfaces exhibits a vanishing orderparameter, the system can be considered a 3D isotropic super-conductor with the critical field H c approximately equal inall directions. On the other hand, if the electron-like surfaceexhibits a vanishing order parameter, then the system can beconsidered a 2D superconductor, with the critical field paral-lel to a specific plane H c much larger than the one in the per-pendicular direction H ⊥ c . Finally, when the superconductivitycomes from both branches of the Fermi surface, the systemwill be an anisotropic 3D superconductor with H c /H ⊥ c (cid:54) = 1 . a r X i v : . [ c ond - m a t . s up r- c on ] S e p Although many interesting studies have been so far reportedon these and other similar compounds, both theoretical and experimental, only a limited amount of informationon WP single crystal system is until now available. The aimof this paper is to fill this gap, providing an electronic structurecalculations and a consistent description of resistivity mea-surements, looking at experiments carried out on WP singlecrystal samples. In particular, we will focus on isothermalmagnetoresistance R ( H ) measurements performed at differ-ent magnetic field directions with respect to the a -axis, and onthe resistivity measurements R ( T ) at different applied mag-netic fields, which reveal the superconducting fluctuationsaround the critical temperature T c .We point out that magnetoresistance measurements per-formed as a function of the applied field direction will giveremarkable information about the properties of WP. In-deed, the comparison between theoretical models for 2D andanisotropic 3D superconductors, will provide indicationabout the anisotropy of the upper critical field and the effec-tive mass, revealing the dimensional behavior of WP.Furthermore, the analysis of the temperature dependence ofthe resistivity fluctuations under the application of a magneticfield, through the scaling procedure obtained theoretically byUllah and Dorsey model, will help to discriminate again be-tween 2D and 3D nature of WP.Here, we will prove that these experimental outcomes to-gether with the theoretical background will give important in-sights on the properties of WP, unveiling relevant microscopicaspects of this material, suggesting that this compound can beconsidered a 3D anisotropic superconductor. The paper is or-ganized as follows: In the next section we will describe theexperimental methods adopted to synthesize WP; In sectionIII, we present the theoretical calculations and the experimen-tal data about magnetoresistance and paraconductivity mea-surements; Finally, the last section will be devoted to the dis-cussion of the results and to the conclusions. II. SAMPLE PREPARATION METHOD ANDEXPERIMENTAL DETAILS
We grew high quality WP needle-like single crystals bychemical vapor transport method, which has been also usedto grow other pnictides.
The starting WP polycrystallinepowders and iodine were placed in a quartz tube sealed un-der high vacuum in a two-zone furnace with a temperaturegradient from ◦ C to ◦ C for one week. Than thesample temperature was raised up to ◦ C in a one-monthperiod. Using this method, we grew single WP single crys-tals with a typical dimension of . × . × . . Aswe determined by the analysis of XRD patterns, the b -axis di-rection is parallel to the longest direction of the sample. Thestoichiometry of the sample was determined by scanning elec-tron microscope and indicates a ratio between W and Patoms, up to experimental errors. Further details on the fab-rication procedure and on the structural, compositional, andtransport characterizations are reported elsewhere. The elec-trical resistance measurements below were performed by
Parameters Values t A W A − . t A W A . t A W A − . t A W A − . t A W A . t A W A − . t A W A . t A W A − . t A W A . t A W B . t A W C . t A W C . TABLE I. Values of the hopping parameters of the tight-bindingminimal model (energy units in eV). the standard four-probe technique in a top-loading Helium-3refrigerator with a superconducting magnet with fields up to
15 T . III. THEORETICAL AND EXPERIMENTAL RESULTS
In this section we will report and discuss the theoreticalsimulation by employing density functional theory (DFT) andtight-binding approach, as well as the experimental results onthe resistivity of WP single crystals.
A. Theoretical calculations
The WP is a system which exhibits nonsymmorphic sym-metries. It is well-known that the nonsymmorphic sym-metries in the Pnma structure are responsible for exotictopological behaviors like the topological nonsymmorphiccrystalline superconductivity,
2D Fermi surface topology, Dirac topological surface states and topologically-drivenlinear magnetoresistance. A detailed analysis of the effectsof the nonsymmorphic symmetries on the fermiology of thiscompound and a tight-binding minimal model fitted to theDFT band structure has been reported elsewhere. Here, weprovide further investigation, and we report a low energytight-binding model in order to calculate the hopping parame-ters at the Fermi level and give an indication about the dimen-sionality of the energy spectra. We restrict to the representa-tive subspace of one d -orbital for every W atom of the unit celland consider only non-vanishing projected W-W hopping am-plitudes. In Table I we report the values of the parameters ofthe tight-binding minimal model, in which we have includedthe nearest-neighbor hopping terms along x , y and z direction.The parameters t lmnα i ,α j corresponds to the hopping amplitudes FIG. 2. Fit of the DFT bands (red lines) using the tight-bindingmodel (blue lines) along the high-symmetry path of the orthorhombicBrillouin zone. The Fermi level is at zero energy. between sites α i and α j (where i, j = W A , W B , W C , W D asin Fig. 1) along the direction l x + m y + n z . From the examination of Table I, we note that the dominantparameters along the x , y , and z directions are t A W A = t x = − .
284 eV , t A W A = t y = − .
195 eV , and t A W C = t z = 0 .
319 eV . Moreover, the values of these parametershave the same order of magnitude, which indicates that theWP is fully 3D with moderate anysotropy. For complete-ness, in Fig. 2 we show the fit of the DFT bands using thetight-binding minimal model along the high-symmetry pathof the orthorhombic Brillouin zone. From this fit we infer thatthe model well captures all the symmetries along the high-symmetry lines of the Brillouin zone.Other evidence of the 3D behavior of the WP is provided bythe analysis of the Fermi surface obtained through DFT calcu-lations. The DFT calculations have been performed by usingthe VASP package, treating the core and the valence elec-trons within the Projector Augmented Wave method with a
400 eV cutoff for the plane wave basis. The obtained Fermisurface is formed by a 3D branch around the center of theBrillouin zone, and two hole-like 2D sheets centered aroundthe SR high-symmetry line at ( k x , k y )= ( π , π ) in the ab plane.The presence of two hole-like 2D sheets is favored by non-symmorphic symmetries. Using the Fermisurfer code, weshow in Fig. 3 the Fermi surface of the WP in the normalphase. The various colors indicate the different Fermi veloci-ties, as shown in the color bar legend, with the highest Fermivelocities coming from the central 3D surface.We see that the Fermi surface is formed by the four bandsthat cut the Fermi level, as shown in the panels (a)-(d), andthis result suggests that WP is an highly anisotropic 3D metal.However, we point out that in the superconducting phase the FIG. 3. Fermi surface of WP in the first Brillouin zone with spin-orbit coupling. In panels (a), (b), (c) and (d) we show the contribu-tions of the four different bands that cut the Fermi level. The colorcode denotes the Fermi velocity. degree of anisotropy could change because the anisotropydoes not depend only on the bare electron band structure,but also on the superconducting coupling. Interestingly, asimilar configuration in a borocarbide compound gives riseto a larger superconducting coupling within the 3D brancheswith respect to the 2D sheets of the Fermi surface. We canspeculate that WP is highly anisotropic in the normal statebut it could be less anisotropic in the superconducting state,with a strongest superconducting order parameter for the 3Delectron-like branch, with the hole-like 2D sheets weakly con-tributing to the superconducting order parameter. B. Experimental results
A possible way to determine the dimensionality of a super-conductor is represented by the study of the angular depen-dence of the upper critical field H c . It is well known indeedthat this measurement may be indicative of a truly 2D or ananisotropic 3D system. Within the Tinkham model, the be-havior of the H c ( θ ) , for a 2D sample, exhibits the followingangular dependence (cid:12)(cid:12)(cid:12)(cid:12) H c ( θ ) sin( θ ) H c (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) H c ( θ ) cos( θ ) H ⊥ c (cid:19) = 1 . (1)On the other hand, in the 3D case, and within the Ginzburg-Landau theory, the angular dependence of the critical field iswritten as (cid:18) H c ( θ ) sin( θ ) H c (cid:19) + (cid:18) H c ( θ ) cos( θ ) H ⊥ c (cid:19) = 1 , (2)where H c , H ⊥ c , being the critical fields measured respec-tively in the parallel and perpendicular direction with respectto the sample surface, and H c ( θ ) the critical field measuredat an angle θ with respect to the normal to the sample surface.
2D Tinkham model3D Ginzburg-Landau model H c ( m T ) θ (degree) Γ =1.4 FIG. 4. Angular dependence of the superconducting upper criti-cal field H c at . . Black circles and green squares represent themeasured transition fields (defined by the 90% criterion) with pos-itive and negative field polarity, respectively. The full lines are thetheoretical fits to the experimental data for the angular dependenceof the critical field. Red and blue lines represent the theoretical de-pendencies according respectively to the Ginzburg-Landau model fora 3D superconductor with anisotropic effective mass, and to the Tin-kham model for a 2D superconductor. In our experiments, we find the anisotropy in the upper crit-ical field H c ( θ ) when the field angle is rotated away from the a -axis. Extracting the values of H c ( θ ) from the resistive tran-sition at several angles, and T c chosen at the 90% of normalstate resistance, we infer the full H c ( θ ) plot as given in Fig. 4.As it can be observed, the H c ( θ ) experimental data are muchbetter described by the anisotropic Ginzburg-Landau theory(red line), suggesting an anisotropic 3D environment for thesuperconductivity in WP. Interestingly, from these data weare able to also infer the degree of the anisotropy Γ looking atthe following ratio Γ = H c H ⊥ c , (3)We find that Γ = 1 . from which we may also estimate theeffective mass ratio as m ∗⊥ /m ∗ = Γ ; This ratio is m ∗⊥ /m ∗ (cid:39) , suggesting a moderate mass anisotropy.Since H c /H ⊥ c > , we infer that WP may be consid-ered a 3D anisotropic superconductor, and thus we expectthat both electron and hole-like branches of the Fermi sur-faces contribute to superconductivity. For completeness, wepoint out that 3D anisotropic superconductivity has been in-tensively investigated in the last years because of its devia-tions from BCS theory even in superconductors with electron-phonon coupling. The study of thermal fluctuation effects turns out to be an-other experimental tool to identify 3D rather than 2D thermalfluctuations. Moreover, it may offer several hints to under-stand relevant properties of the compounds under investiga-tion, such as the occurrence of pronounced dissipation in themixed state, detrimental for applications, as well as to pro-vide fundamental information about the nature of the super-conducting state. µ H(mT)1.52.53.54.55.56.57.58.59.510.5 σ / σ n T (K) Δ FIG. 5. The normalized excess conductivity plotted as a func-tion of the temperature in a magnetic field µ H ranging from . to . . It is well known that the understanding of superconductingfluctuations of conductivity around the transition temperature,in the presence of an applied field, requires a rather complexanalysis. However, in a sufficiently strong magnetic field, thepaired quasiparticles are confined within the lowest Landaulevel (LLL) and, consequently, transport is restricted to thefield direction. In this case, the effective dimensionality of thesystem is reduced and the effect of fluctuations becomes moreimportant. Specifically, the width of the temperature rangearound T c , for a measurable excess conductivity, increaseswith the applied magnetic field as the in-field Ginzburg num-ber G i ( H ) given by G i ( H ) = G i (0) / (cid:20) HH c (0) (cid:21) / , (4) G i ( H ) = G i (0) / (cid:20) HH c (0) (cid:21) / , (5)for 3D and 2D systems, respectively. Here, H c (0) = − T c dH c /dT | T = T c is the zero-temperature Ginzburg-Landauupper critical field, whereas G , i (0) are the zero-fieldGinzburg numbers given by G i (0) = 12 (cid:18) k B T c E c (cid:19) , G i (0) = k B T c E F . (6)In these formulas E F is the Fermi energy, E c is the con-densation energy within a coherence volume given by E c =( B c (0) / µ )( ξ (0) / Γ(0)) , B c (0) is the zero temperaturethermodynamic critical field, ξ (0) is the zero temperaturein-plane coherence length, and µ is the vacuum magneticpermeability. Ullah and Dorsey calculated the fluctuation conductivityusing the LLL approximation and the self-consistent Hartreeapproximation, including contributions up to the quartic termin the free energy. The resulting scaling law for the con-ductivity in the magnetic field, in terms of unspecified scalingfunctions f and f , valid for 3D and 2D superconductors, T c ( K ) µ H (mT) ∆ T C ( m K ) µ H (mT)
FIG. 6. The critical temperature as a function of the applied mag-netic field µ H . The error bars correspond to the transition width ∆ T c evaluated from the 10-90% criterion. The inset refers to thetransition width ∆ T c versus µ H . The red solid line is the linearbest fit to the experimental data. respectively. The expression for the fluctuations of the con-ductance σ are ∆ σ ( H ) = (cid:20) T H (cid:21) / f (cid:20) B T − T c ( H )( T H ) / (cid:21) , (7) ∆ σ ( H ) = (cid:20) TH (cid:21) / f (cid:20) A T − T c ( H )( T H ) / (cid:21) , (8)for 3D and 2D systems, respectively. In these expressions,known as the Ullah-Dorsey scaling law equations, A and B are appropriate constants characterizing the material under in-vestigation.We would like to stress that these scaling laws have beenfound in high-temperature superconductors above a character-istic field µ H LLL of the order of few teslas, and in iron-based superconductors, with a measured field µ H LLL =6 – . Moreover, it has recently shown that paraconduc-tivity of K-doped SrFe As , in the presence of the magneticfield, obeys the 3D Ullah-Dorsey scaling law, in a suitablerange of applied magnetic field. In Fig. 5 we show the normalized excess conductivity ∆ σ H ( T ) /σ n curves for a WP single crystal in applied mag-netic field up to . and with direction parallel to the a -axis. In particular, the excess conductivity due to fluctua-tion effects near the superconducting transition is defined as ∆ σ H ( T ) = σ H ( T ) − σ n ( T ) , with σ H ( T ) the sample con-ductivity and σ n ( T ) the normal state conductivity. In Fig. 5the data has been obtained by the excess conductance calcu-lated as ∆Σ H ( T ) = Σ H ( T ) − Σ n ( T ) , with Σ H ( T ) the mea-sured conductance and Σ n ( T ) the normal state conductance.In the temperature range investigated the normal conductanceis temperature-independent Σ n = 41 Ω − .Fig. 6 shows the critical temperature T c as a function ofthe applied magnetic field. The errors bars correspond to thetransition width determined from the 10-90% resistance dropcriterion. The inset shows the transition width ∆ T c obtainedby the 10-90% criterion as a function of the applied magnetic -0.05 0.00 0.0510 -2 -1 µ H(mT)4.55.56.57,58.59.510.5 Δ σ / σ n ( H / T ) / ( m T / K - / ) (T-T c (H))/(TH) (K mT -2/3 ) FIG. 7. Scaling plots of the excess conductivity as function of ( T − T c ( H )) / ( T H ) / in magnetic fields µ H , in the range from 1.5 to . , for the 3D Ullah-Dorsey model of the paraconductivitydescribed by Eq. (7). field on a log-log scale. A power law behavior H − α is in-ferred, with an exponent α = 0 . very close to the value / predicted for field induced fluctuation effects in a 3D super-conductor. Therefore, this result suggests a 3D behavior ofthe conductivity fluctuations.In Fig. 7 the scaled excess conductivity ∆ σ D ( H ) is plot-ted for the case of 3D scaling. Notice that the 3D scalingbehavior has been calculated from the normalized excess con-ductivity ∆ σ H ( T ) /σ n curves shown in Fig. 5, using T c ( H ) values reported in Fig. 6. For fields ≥ , the data exhibit areasonable scaling behavior of the fluctuations around T c ( H ) .Hence, this result indicates that, at sufficiently high fields, thefluctuation conductivity is well described within the 3D LLLapproximation, and that the field µ H LLL is of the order of in the superconducting phase.As we noted before, the relevant parameter which quantifiesthe fluctuation strength in a superconductor is the Ginzburgnumber given by Eq. (6). In order to estimate the 3D Ginzburgnumber for the WP, the thermodynamic critical field µ H c (0) may be inferred by the jump of specific heat at T c given by µ ∆ C/T c = ( µ H c (0) /T c ) , whereas the in-plane coher-ence length ξ by the slope near T c of the out of plane uppercritical field, dH ⊥ c /dT | T = T c given by ξ = (cid:115) Φ (cid:46) (cid:18) πT c µ (cid:12)(cid:12)(cid:12)(cid:12) dH ⊥ c dT (cid:12)(cid:12)(cid:12) T = T c (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (9)In terms of the measured quantities µ ∆ C/T c , dH ⊥ c /dT | T = T c , Γ , and T c , the Ginzburg number can beexpressed as G i (0) = 2 (cid:18) k B Γ∆ C (cid:19) (cid:18) πT c µ Φ dH ⊥ c dT (cid:12)(cid:12)(cid:12) T = T c (cid:19) . (10)The specific heat jump µ ∆ C/T c = 120 J / m K ofWP single crystal has been already measured and reportedelsewhere, while the slope dH ⊥ c /dT | T = T c = −
22 mT / K has been obtained by the H ⊥ c ( T ) extracted from the datain the inset of Fig. 7 and Γ ≈ . . Thus, we found that G i (0) ≈ − , which is very small compared, for in-stance, with the value ≈ − observed in the iron-selenidesuperconductor, whereas it is comparable with the value ob-served in the low-temperature superconductor niobium. Another relevant superconducting parameter is theGinzburg-Landau parameter κ given by κ = µ (cid:112) µ ∆ C/T c dH ⊥ c dT (cid:12)(cid:12)(cid:12) T = T c , (11)that for our sample is κ ≈ . , which is again of the sameorder of magnitude of the value measured for niobium. IV. CONCLUDING REMARKS
In conclusion, we have synthesized superconducting singlecrystals of WP, and investigated electrical transport proper-ties. The synthesis of WP single crystals was accomplishedthrough the chemical vapor transport method which has beenproved to be successfully to grow transition metal pnictides.From our analysis, we find that the angular dependence of theupper critical field exhibits a smooth behavior. Looking atthe normal state, we extract a rather large anisotropy, whilein the superconducting state the upper critical field shows ananisotropy
Γ = 1 . , largely lower than that found, for in-stance, in iron sulfides and organic superconductors. Wenote that this value for Γ corresponds to an estimated effectivemass anisotropy equal to m ∗⊥ /m ∗ ≈ . Moreover, the mag-netoresistance measurements performed at different appliedmagnetic field angle reveal a 3D behavior differently from the2D character found in iron selenide. On the other hand, the fit of the superconducting fluctuations of the conductivity, bymeans of Ullah-Dorsey theory, suggests again a 3D scalinglaw rather than a 2D behavior. Therefore, these experimentaldata, supplemented by the theoretical theories used to fit theirtrend, indicate that the WP can be considered an anisotropic3D superconductor.It is worth stressing that these results are corrobo-rated by ab-initio electronic structure calculations that showanisotropic hopping parameters, whose values clearly indicatea 3D behavior. Interestingly, most of the density of states atthe Fermi energy is contributed by W 5 d electrons, also sug-gesting that the superconductivity is originated from the con-densation of electrons coming from the transition metal ion.Nevertheless, further theoretical and experimental studiesare needed to determine the pairing symmetry and the cor-responding superconducting mechanism, as well as the roleplayed by W 5 d electrons in stabilizing the superconductingphase. ACKNOWLEDGMENTS
P. M. is supported by the Japan Science and TechnologyAgency (JST) of the Ministry of Education, Culture, Sports,Science and Technology (MEXT), JST CREST Grant No. JP-MJCR19T, by the (MEXT)-Supported Program for the Strate-gic Research Foundation at Private Universities TopologicalScience (Grant No. S1511006), and by JSPS Grant-in-Aidfor Early-Career Scientists (Grant No. 20K14375). G. C. ac-knowledges financial support from “Fondazione Angelo DellaRiccia”. C. A. and G. C. are supported by the Foundation forPolish Science through the International Research Agendasprogram co-financed by the European Union within the SmartGrowth Operational Programme. ∗ [email protected] † [email protected] W. Wu, J. Cheng, K. Matsubayashi, P. Kong, F. Lin, C. Jin, N.Wang, Y. Uwatoko, and J. Luo,
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