Coalescence of Andreev bound states on the surface of a chiral topological semimetal
V.D. Esin, Yu.S. Barash, A.V. Timonina, N.N. Kolesnikov, E.V. Deviatov
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Coalescence of Andreev bound states on the surface of a chiral topological semimetal
V.D. Esin, Yu.S. Barash, A.V. Timonina, N.N. Kolesnikov, and E.V. Deviatov
Institute of Solid State Physics of the Russian Academy of Sciences,Chernogolovka, Moscow District, 2 Academician Ossipyan str., 142432 Russia (Dated: February 16, 2021)We experimentally investigate the magnetic field dependence of Andreev transport through aregion of proximity-induced superconductivity in CoSi topological chiral semimetal. With increasingparallel to the CoSi surface magnetic field, the sharp subgap peaks, associated with Andreev boundstates, move together to nearly-zero bias position, while there is only monotonous peaks suppressionfor normal to the surface fields. The zero-bias dV /dI resistance value is perfectly stable with changingthe in-plane magnetic field. As the effects are qualitatively similar for In and Nb superconductingleads, they reflect the properties of proximized CoSi surface. The Andreev states coalescence andstability of the zero-bias dV /dI value with increasing in-plane magnetic field are interpreted as thejoined effect of the strong SOC and the Zeeman interaction, known for proximized semiconductornanowires. We associate the observed magnetic field anisotropy with the recently predicted in-planepolarized spin texture of the Fermi arcs surface states.
I. INTRODUCTION
Recently, chiral topological semimetals have been pre-dicted [1, 2] as natural generalization of Weyl semimet-als [3]. They are characterized by simultaneously bro-ken mirror and inversion symmetries and non-zero Chernnumbers. Chiral topological semimetals host new typesof massless fermions with a large topological charge,which lead to numerous exotic physical phenomena likeunusual magnetotransport [4], lattice dynamics [5], anda quantized response to circularly polarized light [6].In topological semimetals, the nontrivial topology re-sults in extensive Fermi arcs connecting projections ofbulk excitations on the side surface [3]. In a chiral topo-logical semimetal there is only one pair of chiral nodes ofopposite Chern numbers with large separation in momen-tum space. This leads to extremely long surface Fermiarcs [7], in sharp contrast to Weyl semimetals, which havemultiple pairs of Weyl nodes with small separation [3].Chiral topological semimetals can be realized, in par-ticular, in a family of transition metal silicides with achiral crystal structure, including CoSi, RhSi, RhGe, andCoGe single crystals, where CoSi is the mostly investi-gated material. Bulk band structure and extremely longFermi arcs have been confirmed by angle-resolved pho-toemission spectroscopy (ARPES) [7–10].In proximity to a superconductor, topological materi-als exhibit non-trivial physics that can in various casesresult in topological superconductivity and existence ofMajorana modes [11–13]. This concerns not only thetopological insulators [14–17], but also Weyl semimet-als, where the proximity was predicted to produce spec-ular Andreev reflection [18] (similar to the graphenecase [19, 20]), superconducting correlations and Majo-rana modes in the Fermi arcs [21, 22] as well as varioussuperconducting pairings decaying in the depth of thesample [23]. Topological surface states are responsible forJosephson current for long superconductor-semimetal-superconductor junctions [24–26], and Tomasch oscilla-tions within the region of proximity-induced supercon- ductivity [27].A proximity-induced superconductivity in chiral topo-logical semimetals with multifold fermions, such as CoSi,has been studied until now neither experimentally northeoretically. Although, a superconducting state in adoped chiral semimetal interfaced with the undopedone can generally take place and allow the existenceof topological superconductivity with surface Majoranafermions [28].Here, we investigate the magnetic field dependence ofAndreev transport through a region of proximity-inducedsuperconductivity in CoSi chiral topological semimetal.We observe sharp subgap peaks, which are usually as-cribed to Andreev bound state (ABS) positions. Evo-lution of these peaks depends on the magnetic field ori-entation: they are moving together to nearly-zero biasposition for parallel to the CoSi flake surface magneticfields, while there is only monotonous peaks suppressionin normal magnetic fields. Also, zero-bias dV /dI resis-tance value is perfectly stable in parallel magnetic field.These effects are qualitatively similar for In or Nb super-conducting leads, so they reflect properties of proximizedCoSi surface.The behavior of the peaks with increasing in-planemagnetic field can be interpreted as ABSs coalescencedue to the joined effect of spin-orbit coupling (SOC) andZeeman interaction. The effect is known for proximizedsemiconductor nanowires [29]. The observed magneticfield anisotropy can be associated with the Zeeman inter-action of the Fermi arcs states on (001) surface in CoSi,which have recently been predicted to be in-plane spinpolarized [30].
II. SAMPLES AND TECHNIQUE
The initial CoSi material was synthesized from cobaltand silicon powders by 10 ◦ C/h heating in evacuated sil-ica ampoules up to 950 ◦ C. The ampoules were held atthis temperature for two weeks and then cooled down to
FIG. 1. (Color online) Top-view image of the sample withschematic diagram of electrical connections. A small (about100 µ m wide and 1 µ m thick) single-crystal CoSi flake isplaced on the pre-defined Nb leads pattern. We investigatetransport through 5 µ m long CoSi region between two 10 µ mwide superconducting leads: one lead is grounded, two otherleads are employed to apply current I and measure voltagedrop V , respectively. To obtain dV /dI ( I ) characteristics, thedc current is additionally modulated by a low ac component.All the wire resistances are excluded, which is necessary forlow-impedance samples. room temperature at 6 ◦ C/h rate. The obtaibed materialwas identified as CoSi with some traces of SiO by X-rayanalysis. Afterward, CoSi single crystals are grown fromthis initial load by iodine transport in evacuated silicaampoules at 1000 ◦ . X-ray diffractometry demonstratescubic structure of the crystals, also, X-ray spectral anal-ysis confirms equiatomic ratio of Co and Si in the com-position, without any SiO traces.To investigate transport through a region of proximity-induced superconductivity in CoSi topological semimetal,we use some modification of standard thin-flake samplepreparation technique [24–27, 31]. Topological semimet-als are essentially three-dimensional crystals [3], so onehas to use thick CoSi flakes. Small flakes can be easily ob-tained from the initial CoSi single crystal by a mechanicalcleaving method [26, 31]. We determine the flake surfaceas (001) one from standard magnetoresistance measure-ments [32]. Van der Waals forces are too weak to holda 1 µ m thick flake on the superconducting (niobium orindium) contacts, thus, it is pressed slightly with anotheroxidized silicon substrate, see Fig. 1. The contacts pat-tern consists from 10 µ m wide superconducting leads,which are defined by lift-off technique on the insulatingSiO substrate after magnetron sputtering of 150 nm Nbor thermal evaporation of 100 nm In.This procedure provides transparent junctions, sta-ble in different cooling cycles [24–27, 31]. In presentexperiment, both Nb-CoSi-Nb and In-CoSi-In junctionsare characterized by low (0.6 Ohm and 1.5 Ohm, re-spectively) junction resistances, which indicate broad( ≈ × µ m area) planar junctions with high junc-tion’s transparency. For our sample preparation tech- -0.5 0.0 0.50.600.650.70 d V / d I ( Ω ) V ( mV ) -1.0 -0.5 0.0 0.5 1.00.60.70.8 d V / d I ( Ω ) d V / d I ( Ω ) V ( mV ) In FIG. 2. (Color online) Typical examples of low-temperature dV /dI ( V ) characteristics for Nb-CoSi-Nb (blue curve) andIn-CoSi-In (red curve) junctions. The curves are qualitativelysimilar, superconducting gap positions are depicted by dashedlines. In addition, different subgap dV /dI ( V ) features areknown for finite-size NS junctions: while shallow oscillationsoriginate from Tomasch and MacMillan-Rowell geometricalresonances, sharp subgap peaks are usually associated withAndreev bound states [36, 45]. Inset demonstrates dV /dI ( V )temperature dependence for Nb-CoSi-Nb junction, the curvesare obrained at 1.4 K, 2.4 K, 3.6 K, 4.2 K, 5.0 K, 6.9 K, respec-tively. The flat dV /dI ( V ) curve above 7 K well correspondsto the gap estimation ∆ Nb ∼ nique, Andreev spectroscopy has been reliably demon-strated in Refs. [26, 27, 31]. Moreover, direct experi-mental comparison of Andreev and thermal regimes oftransport can be found in Ref. [31].We investigate transport through 5 µ m long CoSi re-gion between two superconducting leads (Nb-CoSi-Nb orIn-CoSi-In junctions), the connection scheme is depictedin Fig. 1: one lead is grounded, two other leads are em-ployed to apply current I and measure voltage V , respec-tively. To obtain dV /dI ( I ) characteristics, the dc current(within ± µ A, f = 7 . dV /dI ( I ))voltage components with a dc voltmeter and a lock-in,respectively. Due to the superconducting leads, all thewire resistances are excluded in Fig. 1, which is neces-sary for low-impedance samples. The measurements areperformed in the temperature interval 1.2 K – 4.2 K fortwo different magnetic field orientations in several coolingcycles. III. EXPERIMENTAL RESULTS
Fig. 2 demonstrates typical examples of low-temperature dV /dI ( V ) characteristics for Nb-CoSi-Nb(blue curve) and In-CoSi-In (red curve) junctions. Thecurves are qualitatively similar, superconducting gap po- B || V ( mV ) d V / d I ( Ω ) V ( mV ) FIG. 3. (Color online) Evolution of dV /dI ( I ) subgap regionfor two different orientations of magnetic field. (a) Nb-CoSi-Nb junction, the field is oriented normally to the flake’s plane.All subgap features are gradually suppressed, dV /dI level isincreasing in magnetic field. (b) Nb-CoSi-Nb junction in par-allel magnetic field. The zero-bias value dV /dI ( I = 0) is sta-ble, ABS peaks (arrows) monotonously come to nearly-zerobias. (c) Qualitatively similar behavior for In-CoSi-In junc-tion: The zero-bias value dV /dI ( I = 0) is very sensitive tonormal magnetic field 48 mT, while the dependence is weakfor the in-plane magnetic field of the same value. sition is well defined, there are pronounced subgap fea-tures for junctions of both types.For the In-CoSi-In junction, the gap value well corre-sponds to the known [33] one ∆ In ∼ Nb ∼ dV /dI ( V ) curve isflat above 7 K for the Nb-CoSi-Nb junction, which isbelow the bulk 9 K value. Observation of well definedsuperconducting gap is a direct confirmation of Andreevregime [34, 35] of transport for both type junctions.In the Andreev regime, different subgap dV /dI ( V ) fea-tures, which can be seen in Fig. 2, are known for finite-size conductors [36]. The developed wide central struc- ture in dV /dI reflects the proximity-induced gap [36, 37],e.g. in the topological surface state [38, 39]. Shal-low oscillations originate [40] from Tomasch [41, 42] andMacMillan-Rowell [43, 44] geometrical resonances. Incontrast, sharp subgap peaks are usually associated withAndreev bound states [36, 45].Our main experimental result is the difference in theABS evolution for two different orientations of magneticfield, as it is demonstrated in Figs. 3, and 4.For Nb-CoSi-Nb junction, central region of dV /dI ( I )curves is shown in Fig. 3 (a) and (b) for different magneticfields. If the field is oriented normally to the flake’s plane,all subgap features are gradually suppressed in Figs. 3(a). dV /dI level is monotonously increasing, no specialtraces can be observed for ABS resonances, as it is shownby colormap in Fig. 4 (a) and by the dV /dI ( I = 0) mag-netic field scan in Fig. 4 (b). This behavior is usual forthe superconductivity suppression in magnetic field [35].In contrast, the zero-bias value dV /dI ( I = 0) is stablein parallel magnetic field, while the width of the centralregion is gradually decreasing, see Figs. 3 (b). SubgapABS peaks monotonously come to nearly-zero position,they are coalescing together at approximately 2 T, seeFig. 4 (c). The stability of zero-bias level dV /dI ( I = 0)below 2 T is also demonstrated by the dV /dI ( I = 0)magnetic field scan in Fig. 4 (d) for parallel magneticfield.Thus, evolution of dV /dI ( I ) curves is drastically dif-ferent for two field orientations. Qualitatively similar be-havior can be observed for In-CoSi-In junctions, as it isdepicted in Fig. 3 (c). The zero-bias value dV /dI ( I = 0)is very sensitive to normal magnetic field 48 mT, whilethe dependence is weak for the in-plane magnetic fieldof the same value. dV /dI ( I = 0) magnetic field scanssupport this similarity in the insets to Fig. 4 (b) and(d), despite much smaller (about 40 mT) onset field forIn-CoSi-In junctions. IV. DISCUSSION
As a result, we observe coalescence of dV /dI subgapABS peaks at nearly-zero bias and perfect stability ofthe zero-bias dV /dI ( I = 0) value only for the parallel tothe CoSi flake surface magnetic fields. These effects arequalitatively similar for In or Nb superconducting leads,so they reflect behavior of proximized CoSi surface.ABS behavior in Fig. 4 (c) strongly resembles the onetaking place for proximized semiconductor nanowires,whose conductance spectra have been extensively stud-ied both experimentally [45–56] and theoretically (see,e.g. Ref 29 and references therein). In the absence of anintrinsic Zeeman splitting as well as an external magneticfield, non-topological ABSs usually appear as two sym-metric subgap peaks [36, 45]. However, when the Zee-man field increases in the presence of a pronounced SOC, In(B ||) B ( mT ) FIG. 4. (Color online) (a) Detailed evolution of dV /dI ( V )level in normal magnetic field for Nb-CoSi-Nb junction. Nospecial traces can be observed for ABS. (b) dV /dI ( I = 0) levelis monotonously increasing in normal magnetic field scan forthe Nb-CoSi-Nb junction (main field) and for the In-CoSi-Inone (inset). (c) Subgap ABS peaks monotonously come tonearly-zero position in parallel magnetic field, they are coa-lescing together at approximately 2 T, as depicted by yellowdashed lines. (d) Zero-bias level dV /dI ( I = 0) is stable inparallel magnetic field below 2 T for the Nb-CoSi-Nb junc-tion (main field) and below 40 mT for the In-CoSi-In one(inset). the finite-energy Andreev states can coalesce togetherand form near-zero-energy midgap states [45]. The the-ory predicts that the topological phase transition, wherealmost-zero-energy Andreev bound states transform intothe Majorana modes, takes place at the Zeeman field wellexceeding the coalescence point. It is difficult to exper-imentally distinguish between the topological and non-topological scenarios under such conditions, since theyboth result in the zero-bias peak [29]. The main qual-itative statement of this theoretical model can be alsoapplied to two-dimensional systems [28, 57–60].Similar physical mechanism can be responsible for ABScoalescence together at the surface of chiral semimetalCoSi in proximity to two Nb superconducting leads:(i) Although surface states are generally two-dimensional, a preferable direction is defined by Fermiarcs on a particular crystal surface [27]. This shouldbe especially significant for chiral semimetals with longFermi arcs [7–9].(ii) An inhomogeneous potential is formed at the sur-face of CoSi by two interfaces with the Nb leads, provid-ing a platform for confined ABSs. Superconducting cor-relations can be efficiently induced in topological surfacestates even for several-micrometer-long junctions [24–26, 61, 62].(iii) As it is well known, the condition of a pronouncedSOC is satisfied for the standard topological semimetalsurface states [3]. The relative strength of SOC in chiraltopological semimetal CoSi is of the order of millielec- tronvolts due to the weak SOC on the Co 3 d and Si 3 p orbitals [30]. Since such an energy scale of the SOC-induced band splitting is much less than the characteris-tic interband separation (see, for example, Figs. 3A andS4 in Ref. [63]), the effects of SOC is characterized assmall in forming the CoSi band structure and the topo-logical surface states.This does not concern, however, the influence ofSOC on the superconducting effects, where significantlysmaller energetic scale near the Fermi surface comes tothe fore. The SOC-induced splitting near the Brillouinzone center (the Γ point), estimated experimentally as ≈ . dV /dI peakson the magnetic field direction in Figs. 3, and 4. Thisanisotropy should be associated with the spin polariza-tion of the Fermi arc surface states in CoSi.A spin polarization of the Fermi arcs is known to takeplace in a number of topological Weyl semimetals due toa strong spin-momentum locking. Thus the spin polar-ization of the arcs, that has been discovered in TaAs, liescompletely in the plane of the (001) surface and reaches80% [65]. Recent theoretical studies of chiral topologi-cal semimetal CoSi have shown that, in disregarding theeffects of SOC, the Fermi arcs are spin degenerate [2].However, the spin-orbit interaction lifts the spin degen-eracy of the surface states leading to their in-plane spinpolarization on the (001) surface, with strongly corre-lated and predominantly antiparallel spin textures in theneighboring Fermi arcs [30].This fully supports our interpretation of the experi-mental results in Fig. 4 (a). The in-plane spin polariza-tion of the Fermi arc surface states in CoSi allows theZeeman interaction only with the in-plane componentsof the magnetic field and not with its normal to the sur-face component. The finite-energy ABS peaks coalescetogether due to the interaction of the spin textures ofindividual arcs with increasing in-plane Zeeman field, inthe presence of SOC.While the topological transition to the state with Ma-jorana fermions is generally expected to occur at a Zee-man field well exceeding the coalescence point [29] (about2 T in Fig. 4 (c)), the zero-bias dV /dI ( I = 0) level sta-bility is destroyed above the same 2 T field in Fig. 4 (d),probably due to the close value of the Nb film criticalfield. Therefore, the topological transition point seems tobe unreachable in our samples, although the surface Ma-jorana fermions are generally allowed in superconduct-ing topological chiral semimetal CoSi [28] and we cannotexperimentally distinguish between the topological andnon-topological scenarios in Fig. 4 (c).Although the direct and inverse magnetoelectric Edel-stein effects are known to be generally present in non-centrosymmetric materials and superconductors [66–77],we have detected no conclusive evidence of them in ourstudy. V. CONCLUSION
The Andreev transport through a region of proximity-induced superconductivity in topological chiralsemimetal CoSi has demonstrated the coalescenceof ABS peaks to nearly-zero bias position and thestability of the zero-bias dV /dI resistance value withincreasing parallel to the flake surface magnetic field.Normal to the surface field only monotonously sup- presses the peaks. We associate the striking magneticfield anisotropy with the Zeeman interaction of thein-plane polarized spin texture of the Fermi arcs in CoSi.The Andreev states behavior is interpreted as the joinedeffect of the strong SOC and the Zeeman interaction,known for proximized semiconductor nanowires. Thetopological transition point to the state with Majoranafermions seems to be unreachable in our samples.
ACKNOWLEDGMENTS
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