Spectroscopic evidence of mixed angular momentum symmetry in non-centrosymmetric Ru_7B_3
Soumya Datta, Aastha Vasdev, Ranjani Ramachandran, Soumyadip Halder, Kapil Motla, Anshu Kataria, Arushi, Rajeswari Roy Chowdhury, Ravi Prakash Singh, Goutam Sheet
SSpectroscopic evidence of mixed angular momentum symmetry innon-centrosymmetric Ru B Soumya Datta , Aastha Vasdev , Ranjani Ramachandran , SoumyadipHalder , Kapil Motla , Anshu Kataria , Arushi , RajeswariRoy Chowdhury , Ravi Prakash Singh , and Goutam Sheet ∗ Department of Physical Sciences, Indian Institute ofScience Education and Research Mohali, Sector 81,S. A. S. Nagar, Manauli, PO 140306, India and Department of Physics, Indian Institute of ScienceEducation and Research Bhopal, Bhopal 462066, India a r X i v : . [ c ond - m a t . s up r- c on ] F e b bstract Superconducting crystals with lack of inversion symmetry can potentially host unconventionalpairing. However, till date, no direct conclusive experimental evidence of such unconventional or-der parameters in non-centrosymmetric superconductors has been reported. In this paper, throughdirect measurement of the superconducting energy gap by scanning tunnelling spectroscopy, wereport the existence of both s -wave (singlet) and p -wave (triplet) pairing symmetries in non-centrosymmetric Ru B . Our temperature and magnetic field dependent studies also indicatethat the relative amplitudes of the singlet and triplet components of the order parameter changedifferently with temperature. In the BCS theory, it is assumed that the attractive interaction that leads to supercon-ductivity is isotropic in momentum space[1]. Consequently, the superconducting energygaps of BCS superconductors show s -wave (orbital angular momentum, l =0) symmetry. Incertain superconducting systems, the energy gap may become anisotropic in the momen-tum space, and show higher angular momentum symmetries like p -wave ( l =1)[2–4], d -wave( l =2)[5–7] etc. In certain other systems, existence of mixed angular momentum symmetry,where symmetries represented by different l are mixed, have also been possible[8–11]. Thephysics of such non- s -wave superconductors are not understood within the BCS formalism.If the crystal structure of a superconductor lacks a centre of inversion symmetry, parity isnot a good quantum number. In such a system an antisymetric spin-orbit coupling (ASOC)can exist. ASOC can in-principle remove the spin degeneracy of the Bloch states withsame k (crystal momentum), but opposite spins. In presence of ASOC, the orbital angularmomentum and spin angular momentum do not remain good quantum numbers any longer.Here, Pauli’s exclusion principle cannot restrict the symmetry of the Cooper pairs to beeither purely even-parity singlet or odd-parity triplet. Therefore, a complex mixed angu-lar momentum state becomes a possibility in a non-centrosymmetric superconductor[12].The unconventionality associated with such complex angular momentum symmetry of thesuperconducting order parameters might make the non-centrosymmetric superconductors(NCS) exhibit unusual behaviour in their electro-magnetic properties compared to thepurely s -wave superconductors. For example, they may display unusually high Pauli limit-ing fields[13], helical vortex states[15] and even topologically protected states[16]. Owing to ∗ [email protected] Si[13, 14],the study of such superconductors gained significant attention of the condensed matterphysics community[17–19].Despite a number of theoretical predictions of the possibility of the exotic superconduct-ing phases in non-centrosymmetric superconductors as discussed above, there has been noclear spectroscopic evidence of unconventionality in such superconductors studied till date.In this paper, we report our ultra-low temperature scanning tunneling microscopy and spec-troscopy results on a non-centrosymmetric superconductor Ru B . Ru B belongs to thespace group P mc and the cyclic crystallographic class C v [20]. Matthias et al. had firstreported superconductivity in Ru B in 1961[21]. However, owing to its low critical tem-perature, the system did not find much interest among the superconductivity community.Almost three decades later, the absence of the inversion symmetry in its crystal structurewas highlighted by Morniroli et al. [22]. In various transport and thermodynamic mea-surements in the past[23–25], it was seen that ∆ C e / ( γ n T c ) and 2∆ / ( k B T c ) in Ru B wereapproximately 1.4 and 3.3 respectively indicating a weak-coupling superconducting state.These measurements also indirectly indicated that a predominant fully gapped s -wave orderparameter could describe the superconducting state of Ru B well. However, as we note,certain special features of the data presented in [23, 24] were ignored while making a claimfor absence of unconventional superconductivity in Ru B . The most intriguing among suchspecial features was a kink in the field dependent ρ - T data[24], beyond which the supercon-ducting transition curves split into two parts. The two parts exhibited significantly differentsensitivity to the applied magnetic field and led to two dramatically different field scales forthe upper critical fields ( H c ∼ et al. discussed the possibility of a mixed angular momentum symmetry of the superconductingorder parameter in Ru B . In addition, though it was ignored by the authors, a possiblesignature of unconventional pairing was also present in the specific heat data as presentedin [23]. More recently, Cameron et al. performed small-angle neutron scattering [26] onRu B and reported that the orientation of the vortex lattice in Ru B strongly dependson the history of the applied magnetic field thereby indicating the possibility of a brokentime-reversal symmetry in the order parameter. In order to probe the true order parametersymmetry of Ru B , we carried out detailed and direct temperature and magnetic field3ependent scanning tunneling spectroscopy measurements on Ru B . The analysis of suchdata reveals spectroscopic signature of an order parameter with mixed angular momentumsymmetry.The single crystals used for our measurements showed a bulk superconducting transi-tion at 2.6 K (Figure 1(b)). The scanning tunnelling microscopy (STM) and spectroscopy(STS) experiments were performed in a Unisoku system with RHK R9 controller, insidean ultra-high vacuum (UHV) cryostat at ∼ − mbar pressure. The lowest temperaturedown to which the measurements were performed was 300 mK. The STM is also equippedwith a superconducting solenoid capable of producing a magnetic field up to 11 T. Since thesingle crystals could not be cleaved using the standard cleaving technique (optimized forlayered materials only), we cleaned the surface by reversed sputtering for 30 minutes withArgon (Ar) ion in-situ inside an integrated UHV preparation chamber. Following that, weimmediately transferred the sample to the scanning stage at low temperature. The Tungsten(W) tip which was prepared outside by electrochemical etching was also cleaned in UHV bybombarding it with a high-energy electron-beam. This process helped us probe the pristinesurface of Ru B . In the inset of Figure 1(b) we show an STM topographic image showingthe distinctly visible crystallites with average grain size ∼ S - S ) captured at ran-domly chosen points on the surface of Ru B at ∼
310 mK. A visual inspection reveals thatbased on their overall shapes, the spectra can be distinctively divided into two categories.The first type ( S , S and S ) shows coherence peaks around ± S , S and S ) of second type showcoherence peaks at ± dI/dV ∼ V =0) below that. We have also analysed these spectra within a single band ‘ s -wave’model[1] using Dyne’s formula[27]: N s ( E ) ∝ Re (cid:18) ( E − i Γ) √ ( E − i Γ) − ∆ (cid:19) . The tunnelling current isgiven by I ( V ) ∝ (cid:82) + ∞−∞ N s ( E ) N n ( E − eV )[ f ( E ) − f ( E − eV )] dE . Here, N s ( E ) and N n ( E )are the density of states (DOS) of the superconducting sample and the normal metallic tiprespectively, while f ( E ) is the Fermi-Dirac distribution function. Γ is the Dyne’s parameterthat takes care of broadening of DOS. The theoretical plots thus generated are shown asblack lines on the experimental data points in the Figure. It is seen that the spectra S , S S give ∆ of the order of 0.31 meV, 0.29 meV and 0.24 meV respectively while spectra S , S and S provide the values 0.48 meV, 0.42 meV and 0.49 meV respectively for thesame analysis. It is also noted that, while the first group of spectra ( S , S and S ) showsreasonably good fitting with the single gap ‘ s -wave’ model (albeit with large Γ), the secondgroup ( S , S and S ) exhibits a significant departure from that.The above-mentioned discrepancy between the experimental spectra and the spectragenerated theoretically within a single-band ‘ s -wave’ model, prompted us to consider otherpossible symmetries of the order parameter. To perform such an analysis, we modifiedDyne’s equation by introducing a more general expression[28] of ∆( θ ) than an isotropic∆. The modified Dyne’s equation reads as N s ( E, θ ) ∝ Re (cid:18) ( E − i Γ) √ ( E − i Γ) − (∆ (cid:48) Cos ( nθ )) (cid:19) . Here, θ is the polar angle (w.r.t. (001)) and the integer n can be 0, 1 or 2 for s , p and d wave symmetries respectively. The expression for tunnelling current is also modified to I ( V ) ∝ (cid:82) + ∞−∞ (cid:82) π N s ( E, θ ) N n ( E − eV )[ f ( E ) − f ( E − eV )] dθdE . In Figure 1(c) we show theexperimental spectrum S along with theoretical plots considering isotropic ‘ s -wave’ ∆ (redline) and anisotropic ‘ p -wave’ ∆ (blue line). It is clear that the spectrum, especially the‘V’-shaped part of that between the coherence peaks, is better described by the ‘ p -wave’symmetry. It is also interesting to note that, the extracted value of ∆ (0.47 meV) for such fitdoes not differ much with the same from the best ‘ s -wave’ fit (0.48 meV). Such ‘V’-shapedspectra are often seen for superconductors with possible unconventional symmetries andare well described by the Tanaka-Kashiwaya model[28] we used here. For example, fromtunnelling spectroscopic study on SmFeAsO . , Millo et al. [29] associated such a shapewith an unconventional order parameter. To note, it was also reported there that some ofthe STM spectra could also be fitted well within pure ‘ s -wave’ model but with significantlysmaller ∆ and relatively higher Γ – a situation similar to our group-I spectra.Now we focus on the spectrum S , which belongs to group I and fits reasonably wellwith single ‘ s -wave’ gap (and with relatively large Γ). A closer inspection, however, revealsthat there is a small discrepancy between the experimental data and the ‘ s -wave’ modelspectrum. We investigated the evolution of this discrepancy with temperature. Temper-ature dependence of ∆ approximately followed the BCS prediction (Figure 2(c)) with ∆ = 0.3 meV. The broadening parameter Γ did not change much within this range. The5eparture of the experimental spectrum from the s -wave model rapidly decreased with in-creasing temperature and at around 750 mK the discrepancy almost disappeared (Figure2(a)). To illustrate this effect clearly, we show two spectra, one at 338 mK and another at961 mK along with their theoretical ( s -wave) fits in Figure 2(b). This observation indicatesthe possibility of a mixed angular momentum symmetry in the order parameter where theamplitudes corresponding to different l vary differently with temperature. This also ex-plains why Fang et al. did not find any signature of unconventionality in their lower criticalfield ( H c ) studies[24] down to 1.2 K, which is well above the temperature window where wecould notice deviation from s -wave behaviour in our data. We have also performed magneticfield dependence of another spectrum ( S ) from the same group I (Figure 3(a)). The keyobservation is that while the spectra deviate appreciably from ‘ s -wave’ theoretical curves atlow fields, above 10 kG the spectra resembled far more closely to the ‘ s -wave’ predictions.For demonstrating this effect more clearly, we show the spectra and respective ‘ s -wave’ fitsat zero field and at 16 kG in Figure 3(b). The evolution of the extracted gap (∆) andbroadening parameter (Γ) with the applied magnetic field are shown in Figure 3(c).In Figure 4(a) we demonstrated the detailed temperature dependent study of spectra S which belong to group II. At ∼
740 mK a small zero-bias conductance peak (ZBCP)appeared. It became more and more pronounced up to ∼ p -wave’ component in the orderparameter where the interface normal and the lobe-direction of the ‘ p -wave’ maintain anacute angle between them[30, 31]. Since the surface of our Ru B has crystallites withrandom orientations and the tip is engaged randomly at different points, this condition cannaturally be satisfied some times.Based on the discussion above, if Ru B has a mixed angular momentum symmetry inits order parameter, then we would expect the unconventional component to be present ingroup-I spectra too. To understand this aspect in details, we used an ‘ s + p -wave’ model toanalyse the spectra S . In this model, the effective gap is given by ∆ s + p = ∆ s + ∆ p Cosθ .In Figure 4(b) we present the spectrum S (black circles) at 338 mK with numericallygenerated spectra considering ‘ s + p -wave’ symmetry (blue line). Pure ‘ s -wave’ fit (red6ine) is also shown for comparison. It is clear that the mixed angular momentum symmetryprovides a better description of the data. Temperature evolutions for the amplitudes of thetwo components ∆ s and ∆ p extracted from the above ‘ s + p -wave’ fittings for the wholespectra S are presented in Figure 4(c). Not as a surprise, while the convensional ∆ s followsa smooth BCS like dependence upto 2 K, the smaller ∆ p sharply drops and goes beyondour measurement resolution at 0.9 K.It should be noted here that the superconducting gap is the manifestation of a phase-coherent macroscopic condensate. Therefore, the measured ∆ should ideally be uniqueirrespective of the measurement techniques. Since the system is metallic in nature, anyrole of special surface states is ruled out. The T c and the H c that we measure for all ourspectra match well irrespective of whether they fall under Group I or II. Therefore, it can beconcluded that, all the spectra falling under two distinct groups differ with each other basedon which component of the order parameter contributes predominantly for a particularcrystallite that the measurement is performed on.In summary, we performed scanning tunnelling spectroscopy on single-crystal Ru B andrecorded several spectra which can be broadly categorised into two groups, group-I and II.Group-I consists of spectra which are shallow in shape. They show overall good agreementwith single gap ‘ s -wave’ symmetry except at very low temperatures where they deviate fromsuch pure s -wave model. Smaller superconducting gap and larger broadening parameter arecharacteristics of these spectra. Group-II spectra are broader in shape and show sharpercoherence peaks. They significantly deviate from the predictions of ‘ s -wave’ model but atheoretical model with ‘ p -wave’ symmetry shows better agreement. The superconductinggap is larger, and the broadening parameter is very small for such spectra compared to thosebelonging to the first group. The temperature dependences of the spectra belonging to boththe groups indicate the presence of a mixed ‘ s + p -wave’ symmetry in the order parameterwhere the two components have different temperature dependence.We thank Tanmoy Das for his useful comments. R.P.S. acknowledges the financial sup-port from the Science and Engineering Research Board (SERB)-Core Research Grant (grantNo. CRG/2019/001028). G.S. would like to acknowledge financial support from the Swar-7ajayanti fellowship awarded by the Department of Science and Technology (DST), Govt.of India (grant No. DST/SJF/PSA-01/2015-16). [1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. , 1175 (1957).[2] H. R. Ott, H. Rudigier, T. M. Rice, K. Ueda, Z. Fisk, and J. L. Smith,
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H = 10 GZFCW M ( e m u / g ) T (K)FCC (b)(a) (c)
FIG. 1. (a)
Six representative tunnelling spectra ( S - S ) plots (colour points) along with corre-sponding numerically generated spectra under single gap s -wave model (black lines). The extractedfitting parameters ∆ and Γ are shown also for each spectrum. (c) Spectrum S with both best sin-gle gap ‘ s -wave’ and ‘ p -wave’ fit alongwith corresponding extracted parameters. The temperature( T ) ∼
310 mK for all spectra. (b)
Bulk magnetization ( M ) data in both zero field cool warming(ZFCW) and field cool cooling (FCC) condition with 10 G magnetic field. inset : STM topographicimage of the sample. S ( d I/ d V ) N ( a . u . ) V (mV) 2 K338 mKexp fit (a) (b)(c) -1.0 -0.5 0.0 0.5 1.00.40.81.21.62.0 D = 0.275 meV , G = 0.105 meV S1:
T = 961 mK D = 0.31 meV , G = 0.1 meVS1: T = 338 mK ( d I/ d V ) N ( a . u . ) V (mV)
H = 0expfit D BCS D G D , G ( m e V ) T (K)
FIG. 2. (a)
Temperature (T) dependence of tunnelling conductance spectra S (colour lines) withtheoretical fits (black lines) in absence of any magnetic field. (b) Spectra S at 340 mK and alsoat 960 mK along with corresponding fitting parameters, where better fit at higher temeperatureis visible. (c) Evolution of ∆ and Γ with temperature, extracted from the plot (a) along with anideal BCS trend of ∆ for comparison. a) (b)(c) -1.0 -0.5 0.0 0.5 1.00.51.01.52.0 D = 0.1 meV , G = 0.176 meVS3: H = 16 kG D = 0.24 meV , G = 0.075 meVS3: H = 0 kG ( d I/ d V ) N ( a . u . ) V (mV)
T = 0.31 Kexpfit -1.0 -0.5 0.0 0.5 1.001234567 S fitexp ( d I/ d V ) N ( a . u . ) V (mV) 21 kG 0 kG D G D , G ( m e V ) H (kG)
FIG. 3. (a)
Magnetic field ( H ) dependence of tunnelling conductance spectra S (colour lines)with theoretical fits (black lines) all measured at T ∼
310 mK. (b)
Spectra S in the environmentof H = 0 and H = 16 kG field, along with corresponding fitting parameters. H (cid:107) c -axis of thecrystal and a better fit at higher field is visible. (c) Evolution of ∆ and Γ with magnetic field,extracted from the plot (a). a) (b)(c) -1.50 -0.75 0.00 0.75 1.5003691215 ( d I/ d V ) N ( a . u . ) V (mV) S -1.0 -0.5 0.0 0.5 1.00.250.500.751.001.25 s+p-wave fit D s = 0.28 meV , D p = 0.1 meV , G = 0.09 meV T = 338 mKH = 0S - exp s-wave fit D = 0.31 meV, G = 0.1 meV ( d I/ d V ) N ( a . u . ) V (mV) D S D P D ( m e V ) T (K) S1 FIG. 4. (a)
Temperature (T) dependence of tunnelling conductance spectra S incorporatinggradual appearance and disappearance of the peak like feature. (b) Spectrum S along with bestpure s -wave (red line) and mixed s + p -wave (blue line) fits with corresponding extracted parameters∆ and Γ. (c) Evolution of ∆ s and ∆ p with temperature, extracted from the s + p -wave fits ofspectra S . Ideal BCS trends of ∆ are also shown for comparison.. Ideal BCS trends of ∆ are also shown for comparison.