PHYSICAL REVIEW B 86, 094520 (2012)

Andreev bound state and multiple energy gaps in the noncentrosymmetric superconductor BiPd

Mintu Mondal,1 Bhanu Joshi,1 Sanjeev Kumar,1 Anand Kamlapure,1 Somesh Chandra Ganguli,1 Arumugam Thamizhavel,1

Sudhansu S. Mandal,2 Srinivasan Ramakrishnan,1,* and Pratap Raychaudhuri1,†1Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

2Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700032, India(Received 27 February 2012; revised manuscript received 28 May 2012; published 18 September 2012)

We report directional point contact Andreev reflection (PCAR) measurements on high-quality single crystalsof the noncentrosymmetric superconductor BiPd. The PCAR spectra measured on different crystallographicfaces of the single crystal clearly show the presence of multiple superconducting energy gaps. For point contactswith low resistance, in addition to the superconducting gap feature, a pronounced zero-bias conductance peak isobserved. These observations provide strong evidence for the presence of an unconventional order parameter inthis material.

DOI: 10.1103/PhysRevB.86.094520 PACS number(s): 74.20.Rp, 74.45.+c, 74.70.Ad, 74.20.Mn

The discovery of superconductivity in the noncen-trosymmetric superconductor1 (NCS) CePt3Si has generatedwidespread interest in this class of systems. In superconductorswhere inversion symmetry is present, the superconductingorder parameter (OP) is characterized by a distinct paritycorresponding to either a spin-singlet or a spin-triplet pairing.However in NCSs, the lack of inversion symmetry com-bined with antisymmetric (Rashba-type) spin-orbit coupling(ASOC)2 can cause an admixture of the spin-singlet andspin-triplet pairing.3 In the simplest situation of a single bandcontributing to superconductivity, this mixing is expected togive rise to a two-component OP. In a real system, the orderparameter would, therefore, have two or more components,depending on the complexity of the Fermi surface, giving riseto unusual temperatures and field dependences of supercon-ducting parameters.4–11

Despite numerous theoretical predictions, experimentalevidence of an unconventional superconducting state in NCSshas been very few, possibly due to the small spin-orbitcoupling. Thus, the vast majority of NCSs, (e.g., Re3W,Mg10Ir19B16, Mo3Al2C, and Re24Nb5) display predominantlyconventional s-wave behavior and occasionally multibandsuperconductivity.12–15 In some systems, such as CePt3Si(Ref. 16) and UIr (Ref. 17), the study of parity-broken super-conductivity is complicated by strong electronic correlationsand by the coexistence of magnetism. One notable exception isLi2Pt3B in which penetration depth18,19 and nuclear-magnetic-resonance20 measurements provide evidence for the existenceof nodes in the gap function. However, a direct spectroscopicevidence for the presence of an unconventional order parame-ter has not been reported for any of these materials.

In this paper, we report directional point contact Andreevreflection (PCAR) measurements on a BiPd single crystal(Ref. 21), which is a recent addition to the family of NCSs.Since the spin-orbit coupling depends on the square of theatomic number (Z) of the elements involved, the presence ofBi (Z = 83) is expected to result in a large spin-orbit couplingin this material. BiPd has a monoclinic crystal structure withlattice constants a = 5.63, b = 10.66, c = 5.68 A, α = γ

= 90◦, and β = 101◦. Recent thermodynamic and transportmeasurements21 on high-quality BiPd single crystals (residualresistivity, ρ ∼ 0.3 μ� cm and residual resistivity ratio ∼160)

revealed that the specific heat jump at Tc is smaller thanexpected for a BCS superconductor, suggesting the possibilityof multiple superconducting order parameters in this material.

Directional PCAR spectroscopy,22 i.e., where the conduc-tance spectra (dI/dV versus V ) are recorded by injecting acurrent from a normal metal through a ballistic point contactalong different crystallographic directions in the supercon-ductor, is a powerful tool to investigate the gap anisotropyin superconductors.23,24 In this paper, PCAR spectra wererecorded on a BiPd single crystal by injecting a current (I )either along b (I ‖ b) or perpendicular to b (I⊥b). The centralobservation from these studies is the presence of a pronouncedzero-bias conductance peak (ZBCP) in both crystallographicdirections, which coexists with more conventional gaplikefeatures. Our results strongly suggest that a spin-triplet OPcoexists with a spin-singlet OP in this material.

A high-quality BiPd single crystal was grown by themodified Bridgman technique (for details about crystal growth,see Ref. 21). The directional point contact measurements wereperformed on a piece of single crystal cut into a rectangularparallelepiped shape of size 1 mm × 1.5 mm × 2 mm, whichhad large well-oriented faces on the (010) and (001) planes.The superconducting transition temperature Tc ∼ 3.62 K ofthe crystal was determined by measuring ac susceptibilityat 60 kHz using a two-coil mutual inductance technique.25

From the resistivity and specific heat measurements on asimilar crystal, we estimate the electronic mean-free path21 l ∼2.4 μm at low temperatures. The quality of the crystal was alsoconfirmed by observing de Haas–van Alphen oscillation.26

Before performing the point contact measurement, the crystalsurface was polished to a mirror finish. To make ballisticpoint contact, a mechanically cut fine tip made from 0.25-mmdiameter Ag wire was brought in contact with the crystalusing a differential screw arrangement in a conventionalsample-in-liquid 3He cryostat. Measurements were performedby making the contact on two different crystal faces: (i)(010) corresponding approximately to I ‖ b and (ii) (001)corresponding approximately to I⊥b. I -V characteristics ofthe junction formed between the tip and the sample weremeasured at different temperatures down to T = 0.4 K usinga conventional four-probe technique. The dI/dV versus V

spectra were obtained by numerically differentiating the I -V

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curves. For all spectra reported here, the contact resistance(Rc) in the normal state varied in the range of Rc ∼ 1–30 �.The corresponding contact diameter estimated usingthe Sharvin formula,27 d = ( 4ρl

3πRc)1/2 ∼ 100−500 A was

much smaller than l. Therefore, all our point contact spectraare taken in a ballistic limit. To further understand the natureof superconductivity, we have measured the upper criticalfield (Hc2) and its anisotropy along two crystallographic axes(H ‖ b and H⊥b) by measuring ac susceptibility as a functionof the magnetic field at different temperatures.

We first concentrate on the PCAR spectra at the low-est temperature. From a considerable amount of statistics,we observe two kinds of PCAR spectra, corresponding toI ‖ b and I⊥b, respectively.28 Figures 1(a) and 1(b) showrepresentative evolution with Rc of the (dI/dV ) versus V

spectra for I ‖ b and I⊥b, respectively. In both directions,the striking feature is the observation of a pronounced ZBCP,which coexists with more conventional gaplike features in thelow Rc contacts. In addition, for I ‖ b, clear coherence peaksassociated with superconducting gaps are observed around0.1 and 0.4 meV, respectively. For I⊥b, the corresponding

structures are observed at 0.4 and 0.8 meV, respectively. Asthe contact resistance is increased by gradually withdrawingthe tip in both directions, the ZBCP slowly vanishes, and werecover spectra with only gaplike features. To quantitativelyobtain the values of the superconducting energy gaps, we fit thespectra using a two-band Blonder-Thinkham-Klapwijk (BTK)model29,30 generalized to take into account broadening effects.In this model, the normalized conductance [G(V )/GN , whereGN = G(V � �)] is a weighted sum of the conductance oftwo transport channels [G1(V ) and G2(V )] arising from thetwo order parameters: G(V )/GN = (1 − w)G1(V )/G1N +wG2(V )/G2N . G1(V )/G1N and G2(V )/G2N are calculatedusing the generalized BTK formalism using the relative weightfactors of the two gaps (w), superconducting energy gaps(�1 and �2), the barrier potentials (Z1 and Z2), and thebroadening parameters (1 and 2) as fitting parameters. Allspectra can be fitted very well with this two-band model ifwe neglect the large ZBCP that arises for contact with lowRc. Analyzing more than 50 spectra along I ‖ b and I⊥b

[Figs. 1(c) and 1(d)], we observed that the dominant featureis a gap �1∼ 0.4 ± 0.1 meV, present along both directions.For I ‖ b, in about 50% of the spectra, we can clearly resolvea smaller gap �2 ∼ 0.1 ± 0.05 meV with w ∼ 0.2–0.6.On the other hand, in 50% of the spectra along I⊥b, wecan clearly resolve a larger gap �3 ∼ 0.8 ± 0.15 meVwith w ∼ 0.1–0.35. We did not obtain any spectra showingthe three gaps simultaneously in the same spectra. The largevariation in w and the dispersion in gap values arise fromsurface roughness, which limits our inability to precisely injecta current along a desired direction. Figures 1(e) and 1(f) showthe temperature dependence of the PCAR spectra for two rep-resentative low-Rc contacts, corresponding to I ‖ b and I⊥b,respectively. The ZBCP decreases with increasing temperatureand disappears at about 0.7Tc. At Tc (3.6 K), the spectraare featureless, and the conductance is independent of biasvoltage.

To obtain the temperature variation in the superconductingenergy gaps, we analyze the temperature dependence of twopoint contacts along the two directions with large Rc (Fig. 2)where the ZBCP is suppressed. Consistent with the notationused for the superconducting energy gaps, we denote thebarrier parameters and the broadening parameters as Z1 and 1

(associated with �1), Z2 and 2 (associated with �2) and, Z3,3 (associated with �3). A comparison between the single-and the two-gap fits [Figs. 2(a) and 2(d)] of the spectra atthe lowest temperatures shows that a single gap is clearlyinadequate to fit the spectra. Figures 2(b) and 2(e) show thetwo-gap fits of the spectra at various temperatures for I ‖ b

and I⊥b, respectively. As expected, Z1, Z2, and Z3 [Figs. 2(g)and 2(h)] are temperature independent31 for both I ‖ b andI⊥b. The temperature variation in 1, 2, and 3, on theother hand, is more complicated. Formally, the broadeningparameters are introduced as an inverse lifetime32 of theexcited quasiparticles. From this perspective, one expects thisparameter to be small at low temperatures and to increaseclose to Tc due to recombination of electron and holelikeBogoliubons. This is consistent with the temperature variationin 1 in both Figs. 2(g) and 2(h). However, 2 and 3

decrease with temperature. To understand this discrepancy,we note that, phenomenologically, the broadening parameters

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and (d) I⊥b. Temperature variation in the PCAR spectra for (b) I ‖ b and (e) I⊥b. (c) and (f) show the temperature dependence of thesuperconducting energy gaps extracted from (b) and (e), respectively; the solid black lines are the expected BCS variation in �. (g) and (h)show the temperature variation in the broadening parameters and the barrier parameters obtained from the best fits of the spectra in (b) and (e),respectively. (i) Temperature variation in the relative weight factor (w) for the two gaps used in the best fit of the spectra in (b) and (e). Theblue solid line in panel (c) shows the real part of the ac susceptibility (χ ) as a function of temperature showing the superconducting transitionof BiPd.

take into account all nonthermal sources of broadening, suchas a distribution of gap function resulting from an anisotropicgap function23 and instrumental broadening. For a stronglyanisotropic gap function, with an increase in temperature,intraband scattering can partially smear out the gap anisotropythereby causing the broadening parameter to decrease. In thepresent case, we can also not rule out the possibility that thediscrepancy is an artifact arising from the fact that our fitsassume k-independent �1 and �2 where the anisotropy ofthe gap functions is ignored. Figures 2(c) and 2(f) show thetemperature dependence of �1, �2, and �3, and Fig. 2(i)shows the relative weight factors corresponding to the two-gapfits. It is instructive to note that �1 has a similar temperaturevariation for both I ‖ b and I⊥b and closes at Tc, confirmingthat this gap is associated with the same gap function. ForI ‖ b, w remains constant with temperature, whereas, �2

decreases rapidly at low temperatures and forms a tail towardsTc as expected for a multiband superconductor. For I⊥b, w

decreases with increasing temperature, and above 1.6 K, allthe spectra can be effectively fitted with a single gap �1.

We now focus on the origin of the ZBCP in the low-resistance spectra. Since ZBCP can arise from several origins,it is important to analyze the observed ZBCP in BiPd critically.First, we look for extrinsic origins of the ZBCP that are notassociated with genuine spectroscopic features. It has beenshown that, in the case where the point contact is not purely inthe ballistic limit, ZBCP can arise from the current reaching

the critical current33 (Ic) of the point contact. However, inour case, such a possibility can be trivially ruled out for tworeasons. First, as we have shown before, our contact is well inthe ballistic limit even after considering the error associatedwith our determination of contact diameter from Rc. Moreimportantly, the conductance spectra at currents larger thanIc cannot contain any spectroscopic information. In our case,however, we observe clear signatures of the superconductingenergy gap at bias voltages much higher than the voltagerange where the ZBCP appears. Other origins of ZBCPinclude (i) magnetic scattering,34,35 (ii) proximity-induced pairtunneling36 (PIPT), and (iii) the Andreev bound state7,37 (ABS)when the superconductor has an unconventional symmetry.The ZBCP resulting from magnetic scattering is expectedto split under the application of magnetic field, and PIPTsshould get suppressed at small fields on the order of 0.1 T. InFigs. 3(a) and 3(b), we show the evolution of the ZBCP withthe magnetic field (H ) applied perpendicular to the junction(i.e., H ‖ I ) for two contacts with I ‖ b and I⊥b, respectively.Unlike the case when H is parallel to the junction, for thisorientation of H , the ZBCP arising from the ABS is notexpected to split but gradually reduces and disappears at highfields. We observe that the ZBCP for both I ‖ b and I⊥b

persists at moderately high fields and does not show anysplitting with the magnetic field. This effectively rules outmagnetic scattering and PIPT as origins of the ZBCP but isconsistent with the expectation for ABSs. We would also like

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to point out that, although the narrow ZBCP observed in ourexperiments has a superficial similarity with that originatingfrom the Josephson effect in a superconductor-superconductortunnel junction, such a possibility is extremely remote in ourexperimental configuration where the contact is establishedbetween a normal metal tip and a superconducting singlecrystal. Although it is possible for some contacts to formaccidental grain boundary Josephson junctions through a tinybroken piece of the crystal, which comes in a series duringthe contact formation process, it is statistically impossiblefor this to happen for all large area contacts that we havemeasured in the course of this paper. We, therefore, concludethe ZBCPs observed here are manifestations of the ABSoriginating from an unconventional component of the orderparameter in this material. Further confirmation of the ABSorigin of the ZBCP comes from its evolution with contact size.Since the mean size of the ABS is on the order of the dirtylimit coherence length (ξ 0), the ZBCP originating from theABS gradually disappears as the contact diameter becomessmaller than ξ 0. From the upper critical field (Hc2) measuredwith H ‖ b and H⊥b [Fig. 4(b)], assuming the simplestsituation of a triangular Abrikosov vortex lattice existing inBiPd, we estimate ξ 0 to be on the order of 20 and 17 nm,respectively. In Fig. 4(a), we plot the height of the ZBCP(defined as the difference between the experimental zero-biasconductance and the zero-bias conductance obtained from thegeneralized two-band BTK fit) as a function of d calculated

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FIG. 5. (Color online) (a) Schematic of the spin-split Fermisurface due to spin-orbit coupling, forming the + (outer surface)and − (inner surface) helicity bands. The spin eigenstates (shown byarrows) point parallel and antiparallel to g(k), respectively. (b) and(c) One possible realization of the gap functions corresponding to�− and �+ corresponding to a particular choice of the vector g(k).Darker (red) and lighter (blue) regions correspond to positive andnegative values of the gap function, respectively. Note that the smallmonoclinic distortion of BiPd has been neglected in the symmetry ofthe gap functions.

using the Sharvin formula.27 The ZBCP disappears for d � 20and d � 32 nm for I ‖ b and I⊥b, respectively.38 We believethat the slightly larger critical diameter for I⊥b compared toξ 0 results from several approximations used in this analysis.First, the determination of d from Rc is necessarily anapproximation, which does not take into account the irregularshape of a real contact or the effect of the barrier potential thatcould exist between the tip and the superconductor. Second,the determination of ξ 0 assumes a triangular vortex lattice,which might not necessarily be the case for a superconductorwith unconventional pairing symmetry. Considering the errorsinvolved with these approximations and the fact that thecriterion for the disappearance of the ZBCP with contact size isonly valid within a factor on the order of unity, the qualitativetrend of the ZBCP with d is in excellent agreement with thetheoretical expectation for the ABS. We, therefore, concludethat the ZBCP in BiPd originates from the ABS resulting froman unconventional OP for which the phase varies on the Fermisurface.

We can now put these observations in the proper perspec-tive. For a NCS, ASOC leads to a term of the form αg(k) · σ inthe Hamiltonian, where α is the spin-orbit coupling constant,σ is the Pauli matrices, and the vector g(k), representingthe orbital direction, obeys the antisymmetric property suchthat g(k) = −g(−k). In general, the explicit form ofg(k) is determined by details of the crystal structure. TheASOC breaks the spin degeneracy, which leads to two bandscharacterized by ± helicities for which the spin eigenstatesare either parallel or antiparallel to g(k). This is schematicallyillustrated in Fig. 5. These helicity eigenstates are, therefore,

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coherent superpositions of spin-up and spin-down eigenstates.The superconducting gap functions for the intraband pairs are�+(k) and �−(k) for the respective helicity bands [Figs. 5(b)and 5(c)]. When the ASOC is large, interband pairing issuppressed, and in such a case, the superconducting transitiontemperature is maximized6 when the quantization directionof the triplet symmetry becomes parallel to g(k). Since thepairing occurs between intraband electrons only and the bandsare of helicity eigenstates, the superconducting gap functionis an admixture of spin-singlet and spin-triplet symmetries:�(k) = [�s(k)I + �t (k)g(k) · σ ](iσy), where I is the 2 × 2identity matrix, g(k) is the unit vector pointing along g(k), and�s(k) and �t(k) are the singlet and triplet amplitudes of thegap function, respectively. A PCAR experiment will, thus, seetwo gap functions, �±(k) = �s(k) ± �t (k) where each gapis defined on one of the two bands formed by the degeneracylifting of the ASOC. In general, both the singlet and the tripletcomponents of the order parameter can be anisotropic and caneven change sign over the Fermi surface. An ABS is formed asa helical edge mode11,39–41 for each k when |�t (k)| > |�s(k)|.In such a situation, on one of the bands, say �−(k) can changesign [Fig. 5(b)], giving rise to nodes in the superconductinggap function for the band with negative helicity. Although atpresent, our understanding is clearly limited by the lack ofexperimental or theoretical information on the topology ofthe Fermi surface and the g(k) appropriate for the monoclinicstructure for BiPd, based on the experimental data, we proposethe following scenario. Since �1 ∼ 0.4 meV is observed forboth I ‖ b and I⊥b and has a similar temperature dependencein both directions, this is likely to originate from one of the gapfunctions associated with �+. On the other hand �2 and �3

are likely to both be associated with a strongly anisotropicgap function (�−) for which the observed gap values aredifferent for the two different directions of current injection.Whereas, in principle, �1, �2, and �3 could also arise from

a multiband scenario containing three different bands, thisis an unlikely possibility for the following reasons. First,a simple multiband scenario consisting of multiple s-wavegap functions on different Fermi sheets cannot explain theexistence of the pronounced ZBCP that we observe in ourdata. Second, we do not observe �2 and �3 simultaneously inany of our spectra despite the surface roughness that producesa significant scatter in their individual gap functions for bothdirections of injection current. It is, therefore, unlikely that�2 and �3 arise from two different gap functions on differentFermi sheets. In this context, we recall that ABS is observedonly when tunneling occurs in the basal plane42 of the crystalof Sr2RuO4 whose OP has pure spin-triplet symmetry. On thecontrary, the ABS observed here in the BiPd compound occursfor both I ‖ b and I⊥b. We believe the main reason behindthis difference is that, although pure triplet symmetry breaksthe crystal symmetry, the mixed singlet-triplet OP in NCSs canrestore the full crystalline symmetry,18 and hence, OP nodesin NCSs can appear along all crystalline directions.

To summarize, we report possible evidence for the mix-ing of spin-triplet and spin-singlet OPs in the NCS BiPd.Furthermore, the presence of the pronounced ABS observedfrom the ZBCP suggests that the pair potential associatedwith the triplet OP is large enough to produce a sign changein at least one of the gap functions. Despite the absence oftheoretical or experimental information on the Fermi surfacethat somehow limits our interpretation of experimental results,we believe that this observation is an important step towardsrealizing Majorana Fermionic modes, which are predictedto exist in the vortex core of NCSs.8 We believe that ourpaper will motivate further investigations on the precise natureof the order parameter symmetry in this interesting NCSsuperconductor.

We thank D. Agterberg for valuable feedback on this paper.

*ramky@tifr.res.in†pratap@tifr.res.in1E. Bauer, G. Hilscher, H. Michor, C. Paul, E. W. Scheidt,A. Gribanov, Y. Seropegin, H. Noel, M. Sigrist, and P. Rogl, Phys.Rev. Lett. 92, 027003 (2004).

2L. P. Gorkov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001).3P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev.Lett. 92, 097001 (2004).

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