Upward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe 2
J. R. Wang, Xiaofeng Xu, N. Zhou, L. Li, X. Z. Cao, J. H. Yang, Y. K. Li, C. Cao, Jianhui Dai, J. L. Zhang, Z. X. Shi, B. Chen, Zhihua Yang
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Upward curvature of the upper critical field and theV-shaped pressure dependence of T c in thenoncentrosymmetric superconductor PbTaSe J. R. Wang , Xiaofeng Xu † , N. Zhou , L. Li , X. Z. Cao , J.H. Yang , Y. K. Li , C. Cao , Jianhui Dai , J. L. Zhang , Z. X.Shi , B. Chen , Zhihua Yang Department of Physics and Hangzhou Key Laboratory of Quantum Matters,Hangzhou Normal University, Hangzhou 310036, China High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China Department of Physics, Southeast University, Nanjing 211189, China Department of Physics, University of Shanghai for Science & Tehcnology , Shanghai,China Xinjiang Technical Institute of Physics and Chemistry of Chinese Academy ofSciences, 40-1 South Beijing Road, Urumqi 830011, China
Abstract.
The temperature evolution of the upper critical field H c ( T ) inthe noncentrosymmetric superconductor PbTaSe was determined via resistivitymeasurements down to 0.5 K. A pronounced positive curvature in the H c - T phasediagram was observed in the whole temperature range below T c . The Seebeckcoefficient S ( T ) in the temperature range 5K ≤ T ≤ T c under hydrostatic pressure shows a markednon-monotonic variation, decreasing initially with the applied pressure up to P c ∼ † Electronic address: [email protected] pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe
1. Introduction
Superconductors lacking inversion symmetry have attracted tremendous researchinterest since the discovery of superconductivity in the noncentrosymmetric (NCS)CePt Si [1]. In the presence of inversion symmetry, Cooper pairs are either spin-singlet or spin-triplet, i.e, having well-defined parity. However, in NCS superconductors,the link between spatial symmetry and the Cooper pair spins may be broken and theadmixture of singlet-triplet pairing is possible [2, 3, 4]. This is because the antisymmetricspin-orbit coupling (SOC) in the NCS superconductors may split the Fermi surface intotwo sheets with distinct spin helicities [2, 5]. The splitting of the Fermi surface thereforefavors both the intra- and inter-band pairing, resulting in the admixture of spin-singletand spin-triplet components, with the ratio of these two being tunable by the strengthof SOC [2, 6]. Recently, NCS superconductors with strong
SOC were proposed aspotential platforms to realize so-called topological superconductivity [7, 8, 9]. Thecombination of strong SOC and broken inversion symmetry can give rise to topologicalsuperconductivity of class DIII [9].The NCS compound PbTaSe was recently reported to be superconducting below T c ∼ consists of stacked hexagonal TaSe layers alternating with Pb monolayers. In the TaSe layer, the Ta atom is located offsetfrom the inversion center. Electronic structure calculations reveal a single Dirac cone inthe Brillouin zone which is gapped by ∼ . The upper critical fieldof PbTaSe was determined by resistivity measurements down to 0.5 K [11, 12], andrevealed a prominent upward curvature, regardless of the criteria used to determine H c . Detailed modelling of the H c data further suggests two-band superconductivityin PbTaSe . The thermoelectric power (TEP) shows the nonlinearity in T and thedominance of negative charge carriers. Remarkably, our pressure study demonstrates aclear V-shaped pressure dependence of T c , similar to that recently observed in AFe As (A=K, Rb, Cs) [13, 14, 15]. The origin of this V-shaped diagram in PbTaSe is possiblydue to a Lifshitz transition under pressure, in contrast to the change of pairing symmetryin AFe As (A=K, Rb, Cs). Our findings may impose important constraints on anytheory of exotic superconductivity in this system and provide some useful indicationsfor the presence or absence of topological states. pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe
2. Experiment
Polycrystalline samples of PbTaSe were prepared by a solid state reaction in evacuatedquartz tubes, following similar procedures as given in Ref. [10]. Good crystallinity of thesamples was identified by a x-ray powder diffractometer. Traces of an impurity phaseof nonsuperconducting tantalum oxide, were still detectable. Resistivity was measuredby a standard four-probe lock-in technique in a Quantum Design PPMS equipped witha 9 Tesla magnet. For the TEP measurements, a modified steady-state method wasused in which a temperature gradient, measured using a constantan-chromel differentialthermocouple, was set up across the sample via a chip heater attached to one end of thesample [16, 17, 18]. The thermopower voltage was read out by a nanovoltmeter K2182Afrom Keithley Instruments. For the hydrostatic pressure measurements, a commercialpiston-type cell from Quantum Design was used, and Daphne 7373 oil was applied asthe pressure transmission media. For the resistivity curves under pressure, the samecontacts were used throughout the measurements such that the geometric errors in thecontact size were identical for different pressures.
3. Results And Discussion
Figure 1(a) presents the zero-field resistivity of PbTaSe sample from 350 K down to2 K. Zero resistivity of the superconducting transition is clearly seen, with the low- T transition expanded in Fig. 1(b). We also note that our transition is slightly broaderand the residual resisitivity ratio (RRR) is somewhat smaller than that reported inRef. [10], mostly because of the higher disorder level or slight off-stoichiometry of Sein our samples, considering the high vapor pressure of Se in the reaction. The high- T resistivity rises linearly with temperature above ∼
150 K. This is consistent withits Debye temperature Θ D =143 K estimated from the heat capacity measurement,indicating the dominance of electron-phonon scattering at this T -range [19]. Thesuperconducting transition at various magnetic fields was studied by the fixed-fieldtemperature sweeps down to He temperatures, presented in Fig. 1(c). In this study,four different criteria have been used to extract the critical temperature at each magneticfield, as exemplified in Fig. 1(b). These criteria include: The onset T onsetc , 90% of thenormal state ρ n T c , middle point T c and the zero resistance T zeroc .The resultant H c - T diagram from these four criteria is cumulatively presented inFig. 2(a). This diagram features, unlike most of superconductors whose H c has anegative/downward curvature, the evident upward curvatures for all four lines. Notethis positive curvature is present for all T below T c and it becomes very steep forthe T onsetc line at low T . This overall upward curvature can not be explained by theGinzburg-Landau theory, nor by the one-band Werthamer-Helfand-Hohenberg (WHH)model [20]. In Ref. [10], the formula H c ( T )= H c (0)(1- t / ) / was applied to fitthe data between 2.4 K to T c . However, we show in Fig. 2(b) that this formulaactually deviates from the experimental data below 1.8 K. Instead, it is possible to pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe H c has also beenobserved in some two-gap superconductors, e.g., LaFeAsO . F . [21], MgB [22] andCa (Pt As )((Fe − x Pt x ) As ) [23]. To analyze our experimental data on a quantitativefooting, we now fit the H c curve with the two-band theory outlined in Ref. [24]. Theequation of H c ( T ) for a two-band superconductor is given by: a [ lnt + U ( h )][ lnt + U ( ηh )] + a [ lnt + U ( h )]+ a [ lnt + U ( ηh )] = 0 , (1)where a =2( λ λ - λ λ )/ λ , a =1+( λ - λ )/ λ , a =1-( λ - λ )/ λ , λ = p ( λ − λ ) + 4 λ λ , h = H c D φ T , η = D D and U ( x ) = ψ ( x + ) − ψ ( ). ψ ( x ) here is the digamma function. λ , λ denote the intraband coupling constants, and λ , λ are the interband couplingconstants. D and D are the diffusivity of each band. The small η may imply a muchstronger scattering on one of the bands. Hence, there are totally six free parametersin the fitting process, λ , λ , λ , λ , D and η . Within this theoretical framework,we fit our experimental data from the 90% of ρ n criterion, as shown in Fig. 2(b). Thistwo-band theory can overall fit the experimental data within the error bar and the resul-tant parameters are given in the figure. As can be seen, the intraband coupling slightlydominates the interband coupling while η ≪
1, indicating that electrons on one band aremore scattered than on the other. However, it can not be concluded that the two-bandsuperconductivity is wholly responsible for the upward curvature of H c in this material,in view of the number of free parameters in the fit. In fact, we also fit the data from50% of ρ n criterion and obtain λ = λ =0.5, λ = λ =0.4, D =0.000575 and η =0.071,also suggestive of the much stronger scattering on one band. Although these H c datacan not exclusively rule out other possible scenarios, which will be discussed later, theydo indicate that two-band superconductivity appears to be a plausible mechanism inPbTaSe [25].Let us discuss other possible scenarios for the upward curvature in H c . In theorganic superconductor (TMTSF) PF , H c also has a pronounced upward curvatureand no sign of saturation was observed down to 0.1 K [26]. However, its H c at T → , its H c at0.5 K is still much lower than its Pauli limit, suggesting that the orbital effect dominatesin limiting its H c . One theory accounting for the strong upward curvature observed inthe cuprates [27, 28] was associated with the quantum critical point in these systems[29]. In our system, however, there is no clear sign to show it is in proximity to thequantum critical regime.The Seebeck coefficient or TEP reveals important aspects of charge conductionin a material [30, 31]. Figure 3 shows the Seebeck coefficient S ( T ) of PbTaSe from350 K down to 5 K. Evidently, its TEP, of modest magnitude, is negative over theentire T -range studied, indicating the dominance of electron carriers. In a single-bandmetal, according to the Sommerfeld theory, TEP is given by S =- π k B e TE F , where E F pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe T -linear Seebeck coefficient. In reality,the situation may become complicated due to the presence of other excitations. In ametal with both electrons and holes, the electron and hole partial Seebeck coefficientsadd together according to their weights in conduction [30, 31], i.e., S =( σ h σ ) S h +( σ e σ ) S e . S h and S e are the Sommerfeld values (both linear in T ) given above. Provided thatelectron and hole partial weights in conduction also change with T , a nonlinear-in- T Seebeck coefficient can naturally emerge. Indeed, from ab initio calculations [10], theFermi surface of PbTaSe consists of both electron and hole pockets. On the otherhand, since the electron pocket dominates, we may apply the single-band Sommerfeldequation quoted above to make a crude estimate of the Fermi energy of this material.From Fig. 3, the linear fit at low T gives E F ∼
480 meV.Shown in Fig. 4(a) is the pressure dependence of the resistivity of PbTaSe up to ∼
25 kbar. With increasing pressure, the resistivity of the sample decreasessystematically, while the pressure seems to have little bearing on its overall T dependence. The resistivity at room temperature changes by about 15% at 25 kbar.Interestingly, although the metallicity of the sample is enhanced by the pressure, its T c shows a nonmonotonic variation, from an initial decrease with pressure to an increaseonce across a critical pressure, as demonstrated in Fig. 4(b) and 4(c). Note that T onsetc of this sample is about 0.2 K lower than that presented in Fig. 1(b). This discrepancymay arise from the sample off-stoichiometry even if they were harvested from the samebatch. Utilizing the same criteria as in Fig. 1(b), we plot the pressure dependence of T c in Fig. 5(a). Remarkably, regardless of the criteria used, T c is firstly reduced by thepressure up to P c ∼ T c , similar to theeffects of hydrostatic pressure on iron-based superconductors AFe As (A=K, Rb, Cs)[13, 14, 15].The origin of this T c reversal under pressure has been extensively discussed by F.Tafti et al . in their works [13, 14, 15]. This includes a change of the pairing symmetryor a Lifshitz transition across P c , i.e., an abrupt change of Fermi surface topology. InAFe As (A=K, Rb, Cs), they argued that the V-shaped pressure dependence of T c wasthe result of a change in pairing symmetry from d -wave state below P c to an s ± above P c . A Lifshitz transition was ruled out in their data, as there were no visible changes ineither the Hall or resistivity data across P c . Following the same procedure given in Ref.[15], we plot the inelastic scattering, defined as ρ (T=20 K)- ρ where ρ is the residualresistivity at each pressure, as a function of pressure in Fig. 5(b). We should mentionthat the choice of T =20 K is arbitrary but any other cuts above T c do not affect theresults. The plot of ρ (20K) in the inset also shows a similar dependence. In contrast toAFe As (A=K, Rb, Cs), the inelastic scattering in PbTaSe displays a sharp drop at acritical pressure somewhere between 5 kbar to 10 kbar (the shaded area in the figure),where T c also shows a sharp inversion. This makes a compelling case that the T c reversalin this system is most likely due to a Lifshitz transition around P c (5-10 kbar). pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe
4. Conclusion
To summarize, we report a strong positive curvature in the H c - T diagram of the NCSsuperconductor PbTaSe . Its Seebeck coefficient was found to be negative in sign andto vary non-linearly with T . All these experimental data appear to be consistent witha picture of two-band superconductivity in PbTaSe , while other possible mechanismshave also been discussed. Interestingly, the recent theoretical proposal of the largeSeebeck coefficient arising from topological Dirac fermions has not been seen in oursamples [33]. The pressure dependence of the superconducting transition T c shows aclear ’V’ shape, which most likely results from a Lifshitz transition under pressure.Further studies are highly desired to explore the possible parity-mixed state by usingultralow- T thermal conductivity and/or penetration depth measurements.
5. Acknowledgement
The authors would like to thank N. E. Hussey, C. M. J. Andrew, C. Lester, A. F.Bangura, Xin Lu, Zengwei Zhu, Xiaofeng Jin for valuable discussions. This workwas supported by the National Key Basic Research Program of China (Grant No.2014CB648400) and by NSFC (Grant No. 11474080, 11104051, 11104053). X.X. wouldalso like to acknowledge the auspices from the Distinguished Young Scientist Funds ofZhejiang Province (LR14A040001).
References [1] Bauer E, Hilscher G, Michor H, Paul C, Scheidt E W, Gribanov A, Seropegin Y, Noel H, SigristM and Rogl P 2004
Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. B et al Phys. Rev. B Phys. Rev. Lett.
Phys. Rev. Lett. Nat. Mater. Phys. Rev. B Rev. Mod. Phys. Phys. Rev. B Phys. Rev. B Phys. Rev. B et al Nat. Phys. et al Phys. Rev. B et al Phys. Rev. B Phys.Rev. B pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe [17] Wakeham N, Bangura A F, Xu X, Mercure J F, Greenblatt M and Hussey N E 2011 Nat. Commun. Sci. Rep. Phys. Rev. et al
Nature
Phys. Rev. Lett. Phys. Rev. B Phys. Rev. B R H = σ h R h + σ e R e ( σ h + σ e ) , where σ e ( h ) and R e ( h ) are the conductivityand Hall coefficient of e ( h ) carriers, respectively (Note that R e and R h are opposite in sign).The resultant Hall effect may become very small due to the substantial cancellation of hole andelectron components.[26] Lee I J, Naughton M J, Danner G M and Chaikin P M 1997 Phys. Rev. Lett. Phys. Rev. Lett. et al Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett.
Phys. Rev. Lett.
J. Phys.: Condens. Matter Phys. Rev. Lett. pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe T (K) (a) (b)
T (K) T zeroc T T T onsetc (c) T (K) 0 T2.4 T
Figure 1. (Color online) (a) Zero-field resistivity of PbTaSe measured from 350 Kdown to 2 K. The data were renormalized to the 350 K value. Resistivity shows aclear deviation from high- T linearity (red straight line) below ∼
150 K, marked by thearrow. (b) A close-up view of the low- T superconducting transition, illustrating thefour criteria used to determine T c at each field. (c) Superconducting transition undervarious magnetic fields. pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe H c ( T ) H c2
90% two-band fit fit from Ref. [10]
T (K) = =0.5; = =0.4D =0.00051; =0.049(b) H c ( T ) T (K)onset90%50%zero (a)
Figure 2. (Color online) (a) Temperature dependence of the upper critical field H c determined from Fig. 1(c) by using the criteria in Fig. 1(b). (b) The experimental H c data using the 90% of ρ n criterion, along with the two-gap fitting described in the text.The blue thin line shows the failure of the fit to the equation H c ( T )= H c (0)(1- t / ) / ,where t is the reduced temperature T /T c , used in Ref. [10]. pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe S / T ( V / K ) T (K) S ( V / K ) T (K)S( V/K)=-0.077*T
Figure 3. (Color online) The temperature dependence of the Seebeck coefficientmeasured from 350 K down to 5 K. The typical error bars, estimated from differentruns with varying heater currents, were given at certain temperatures. The straightred line is the linear fit to the low- T data. The inset shows the plot of S/T as afunction of temperature. pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe (c) ( m c m ) T (K) (b) ( m c m ) T (K) ( m c m ) T (K)(a)
Figure 4. (Color online) (a) shows the resistivity under several pressures, measuredby the same electrical contacts. Panels (b) and (c) expand the low- T superconductingtransitions under different pressure values. pward curvature of the upper critical field and the V-shaped pressure dependence of T c in the noncentrosymmetric superconductor PbTaSe ( T = K ) ( m c m ) P (kbar) ( T = K )- ( m c m ) P (kbar) T c ( K ) T onsetc T T T zeroc Figure 5. (Color online) (a) shows T c , estimated by four criteria used in Fig. 1(b),as a function of pressure. (b) The inelastic scattering, defined as ρ (T=20 K)- ρ where ρ is the residual resistivity at each pressure, as a function of pressure. The criticalpressure where the Lifshitz transition occurs is somewhere in the shaded area. Theinset shows ρρ