Anisotropic vortices on superconducting Nb(110)
Artem Odobesko, Felix Friedrich, Song-Bo Zhang, Soumyajyoti Haldar, Stefan Heinze, Bjorn Trauzettel, Matthias Bode
AAnisotropic vortices on superconducting Nb(110)
Artem Odobesko, ∗ Felix Friedrich, Song-Bo Zhang, SoumyajyotiHaldar, Stefan Heinze, Bj¨orn Trauzettel,
2, 4 and Matthias Bode
1, 4, 5 Physikalisches Institut, Experimentelle Physik II,Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany Theoretische Physik IV, Institut f¨ur Theoretische Physik und Astrophysik,Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany Institut f¨ur Theoretische Physik und Astrophysik,Christian-Albrechts-Universit¨at zu Kiel, Leibnizstr. 15, 24098 Kiel, Germany W¨urzburg-Dresden Cluster of Excellence ct.qmat, Germany Wilhelm Conrad R¨ontgen-Center for Complex Material Systems (RCCM),Universit¨at W¨urzburg, Am Hubland, 97074 W¨urzburg, Germany
We investigate the electronic properties of type-II superconducting Nb(110) in an external mag-netic field. Scanning tunneling spectroscopy reveals a complex vortex shape which develops fromcircular via coffee bean-shaped to elliptical when decreasing the energy from the edge of the su-perconducting gap to the Fermi level. This anisotropy is traced back to the local density of statesof Caroli–de Gennes–Matricon states which exhibits a direction-dependent splitting. Oxidizing theNb(110) surface triggers the transition from the clean to the dirty limit, quenches the vortex boundstates, and leads to an isotropic appearance of the vortices. Density functional theory shows thatthe Nb(110) Fermi surface is stadium-shaped near the Γ point. Calculations within the Bogoliubov-de-Gennes theory using these Fermi contours consistently reproduce the experimental results.
PACS numbers: 74.25.Ha
Introduction—
In the recent past, we have witnessed agrowing interest in the physical properties of vortices inunconventional and topological superconductors, as theyallow for the emergence of Majorana zero modes (MZMs)[1–3] which may potentially be used for applications inquantum computation [4]. Although some success hasbeen achieved on heavy electron iron-based superconduc-tors [5, 6], the unambiguous spectroscopic identificationof MZMs remains demanding, as their spectroscopic dis-tinction from trivial quasiparticle excitations at experi-mentally accessible temperatures still constitutes signifi-cant challenges [7–9].In vortices, such trivial excitations occur when the elec-tron mean free path (cid:96) is much larger than the vortexdiameter, i.e., the superconducting coherence length ξ .As predicted by Caroli, de Gennes, and Matricon in the1960s [10], Andreev reflection inside the vortex gives riseto a discrete set of bound states. Their separation in en-ergy is given by ∆ /E F , where 2∆ is the width of thesuperconducting gap and E F is the Fermi energy, oftenresulting in values in the µ eV range only [11]. At suchlow splittings, the CdGM states above and below theFermi level can hardly be resolved even if measurementsare performed at milli-Kelvin temperatures, as they ther-mally merge into one broad peak which appears to beenergetically positioned at zero bias [12–14].Furthermore, the spatial distribution of specific CdGMstates inside the vortex scales with their angular momen-tum µ , resulting in a wave function which peaks at a dis-tance r µ ≈ | µ | /k F away from the vortex core [11, 15].However, the specific spatial and energy distribution ofCdGM bound states in a given material depends on vari- ous parameters, such as the pairing anisotropy, spin-orbitcoupling, Fermi surface anisotropy, or vortex–vortex in-teractions [16–18]. In fact, the resulting patterns of vor-tex bound states can be quite complex [19, 20]. Fur-thermore, in the dirty limit ( (cid:96) (cid:46) ξ ), scattering processeslead to energetic broadening and eventually the completequenching of the CdGM states [21]. Thus, detailed in-vestigations of the electronic structure inside supercon-ducting vortices can help to better understand and tellapart trivial from topological states.In this paper, we present results of a scanning tun-neling spectroscopy (STS) study of vortex bound stateson clean and oxygen-reconstructed Nb(110) in an ex-ternal out-of-plane magnetic field. Whereas no boundstates are found on the oxygen-reconstructed surface, awell pronounced zero-bias peak appears in the vortexcore of clean Nb(110). Differential conductance d I/ d U maps measured at constant tip–sample distance revealthat the spatial distribution of the local density of states(LDOS) crucially depends on the bias voltage. Whileround-shaped vortices are visualized when the bias is setclose to the edge of the superconducting gap, a two-foldanisotropy is observed at zero bias with an intermediatecoffee bean-shaped form (the shape of a roasted coffearobusta bean, to be precise). The elliptical shape of theLDOS maps is qualitatively explained by the anisotropyof the Fermi surface which exhibits a stadium-like shapearound the Γ point of the surface Brillouin zone (SBZ).Using this anisotropic Fermi surface as an input, the ex-perimental data are modeled by solving self-consistentlythe Bogoliubov-de Gennes (BdG) equation. Experimental procedures—
STM and STS measure- a r X i v : . [ c ond - m a t . s up r- c on ] J u l -5 0 5[001][110] ba ≈ 0.55
40 nm [001][110](b)(a) [001][110] U = -2 mV150 nm U = -0.8 mV U = -1 mV U = 0 mV U = 0 mV(f)(d) (g)(e) -5 -3 -1 1 3 501 d I / d U ( no r m . ) d ef g U (mV)(c) (h)(j) U (mV) - - d i s t an c e L f r o m v o r t e x c o r e ( n m ) U (mV)-5 0 5(m) a l ong [ ] a l ong [ ] lowhigh d I/ d U s i gna l (k) FIG. 1. (a) Topography of clean Nb(110). (b) Atomic resolution scan with indicated unit cell. (c) Tunneling spectrum ofthe superconducting gap. (d)-(g) Constant-separation d I/ d U maps taken at the voltages marked by colored circles in (c) at µ H = 100 mT. Whereas the vortices appear round-shaped at a bias voltage close to the edge of the gap, they deform intoellipses when approaching zero bias. (h) Stacked representation showing the bias-dependent LDOS inside the vortex in theenergy window from 0 to 2 mV. (j) Fermi level ( U = 0 V) d I/ d U map revealing an elliptical shape elongated along the [001]direction with an eccentricity b/a = 0 . ± .
1. (k),(m) Waterfall plot of tunneling spectra taken along the lines labeled b and a in (j), respectively. With increasing distance from the vortex core along the [110] direction the zero-bias peak splits into tworidges, resulting in the appearance of an X-shaped feature as indicated by two arrows and dashed lines in (k). In contrast, alongthe [001] direction only a vanishing of the peak without any splitting is observed (m). Scan parameters: (a)-(m) I set = 200 pA; T = 1 . U set = − U set = −
10 mV; (d)-(g) U set = −
100 mV; (h)-(j) U set = −
10 mV; (k)-(m) U set = − ments are performed in a home-built setup (base tem-perature T = 1 . µ H = 12 T along the surface normal. Spectro-scopic measurements are conducted by a lock-in tech-nique with U mod = 0 . ÷ . f = 890 Hz. Nb(110) was cleaned by a seriesof high temperature flashes up to T fl ≈ ◦ C. As de-scribed in detail in Ref. 22, lower flash temperatures re-sult in two intermediate and less-ordered oxygen recon-structions, i.e., NbO x phase-I at T fl ≤ ◦ C and NbO x phase-II for 2000 ◦ C ≤ T fl ≤ ◦ C [23].
Experimental results—
Figure 1(a) shows the constant-current STM topography image of a clean Nb(110) sur-face with occasional oxygen-reconstructed patches whichappear as dark spots ( <
10% of the surface area). Theatomic resolution scan reported in Fig. 1(b) confirms theexpected lattice constant a [001] = 330 pm. Nb is a type-II superconductor with T Nbc = 9 . .
53 meV [Fig. 1(c)]. We care-fully investigate the spatial variation of the electronicstructure around vortex cores by mapping the differentialconductance d I/ d U at various energies E = eU withinthe superconducting gap. To avoid set-point–related ar-tifacts, all measurements are performed by stabilizingthe STM tip at a set-point bias voltage correspondingto an energy far within the normal metallic regime, i.e.,at U set = −
100 mV. After recording the z -trace along onescan line at this bias voltage the feedback is switched off,the bias voltage is set to measurement parameters, andthe tip is approached towards the surface by a fixed value∆ z = 180 pm. Afterwards the d I/ d U signal is measuredalong the (shifted) z -trace. This procedure increases thetunneling current and thereby improves the signal-to-noise ratio. At the same time, it guarantees that thed I/ d U maps presented in Figs. 1(d)-(g) are measured ata constant tip–sample separation.Figures 1(d)-(g) show the Abrikosov lattice formed onclean Nb(110) in a magnetic field µ H = 100 mT at fourselected tunneling voltages indicated by colored circlesin Fig. 1(c). Some qualitative differences regarding theappearance of the vortices are immediately evident. At U = − U = 2 mV). As the voltage is reduced the LDOS in-tensity becomes (i) larger inside the vortex than in thesurrounding superconducting region, it (ii) splits along[110] direction (1.2 mV ≥ U ≥ . U ≤ . I/ d U map of a vortex measuredat the Fermi level ( U = 0 mV). Its shape clearly deviatesfrom a circle (hatched line) and rather corresponds to anellipse (thick white line) with the semi-major a and semi-minor b axes oriented along the [001] and [110] direction,respectively. The ratio b/a amounts to ≈ .
55. We wouldlike to emphasize that the orientation of the ellipse doesnot depend on the presence of defects, such as step edges,and stays the same at higher magnetic fields.Figures 1(k) and (m) show the spatial evolution of tun-neling spectra measured along the axes b and a . Thespectrum measured at the vortex center is characterizedby a strong zero-bias peak, representing CdGM stateswhich are thermally blurred into a single peak [19, 20].However, along the two high-symmetry axes we observequalitatively different transitions from the peaked spec-trum measured at the vortex center to the fully gappedspectrum far away. Whereas the zero-bias peak splits inenergy into two ridges if measured along the [110] direc-tion [marked by two black arrows and hatched lines inFig. 1(k)], a behavior expected for CdGM state [17, 24],no such splitting is detected along a , i.e., the major axesalong the [001] direction [Fig. 1(m)].A detailed investigation of the electronic structure in-side vortices observed on ordered clean Nb(110), theless-ordered NbO x phase-II, or the disordered NbO x phase-I [22] reveals that the existence and shape of 5 0 5 U (mV) L ( n m ) d I/ d U s i gna l low (e) high4 2 0 2 U (mV) . . d I / d U s i gna l ( no r m . ) on vortex core: Nb(110) NbO x phase II NbO x phase I off vortex Nb(110)0 mV0.8 mV1 mV (b) ( с ) (d) (a)(a) FIG. 2. (a) Tunneling spectra measured on different Nb sur-faces in an external magnetic field: (black) the fully gappedtunneling spectrum of clean Nb(110) far away from any vor-tex; (blue) intense zero-bias peak in the center of the vor-tex on clean Nb(110); (green) less intensive peak in a vortexcore on NbO x phase-II; (red) Ohmic behavior in vortex coreon NbO x phase-I. Stabilization parameters: U set = − I set = 100 pA. Panels (b)-(d) show differential conductanced I/ d U maps of a single vortex on NbO x phase-I imaged at(b) U = 1 mV, (c) U = 0 . U = 0 mV. (e) Tun-neling spectra measured along the line in (d). No zero-biasfeature is observed. Stabilization parameters: U set = − I set = 100 pA, µ H = 100 mT, T = 1 . ZBP features critically depends on the surface quality.Fig. 2(a) presents tunneling spectra measured in vortexcores on all three surfaces. For comparison, the blackcurve in Fig. 2(a) shows the spectrum of superconduct-ing clean Nb(110) far away from a vortex, clearly pre-senting a superconducting gap. Evidently, the spectralfeatures observed around zero bias exhibit marked, sur-face structure-dependent differences. Whereas the ZBPintensity measured on clean Nb(110) (blue) clearly ex-ceeds the normal conducting DOS, indicating the pres-ence of CdGM states in the vortex core, the ZBP is lesspronounced on the NbO x phase-II (green) and essentiallyabsent for the NbO x phase-I (red).This reduction of the ZBP is accompanied by a moreisotropic appearance of vortices, as evidenced by thed I/ d U map of a vortex on the NbO x phase-I presentedin Fig. 2(b-d), which remains circular throughout the en-tire energy range within the superconducting gap. Fur-thermore, the tunneling spectra presented in Fig. 2(e)which were taken along the line in (d) across the vor-tex reveal the absence of any feature potentially relatedto bound states. These findings suggest that the ori-gin of the anisotropy observed for vortices on the cleanNb(110) surface results from the presence of the CdGMstates. Their absence on the less ordered NbO x phase-Iis consistent with a cross-over from the clean to the dirtylimit [21]. If we assume for the clean Nb(110) surfacea mean free path close to the bulk value, (cid:96) Nb ≥
100 nm[25], it clearly exceeds the coherence length ξ Nb ≈
38 nm.Since the oxygen-reconstructed surfaces of Nb are lessordered, additional scattering of quasiparticles may sub-stantially decrease the mean free path and quench theCdGM states.
Discussion—
Anisotropic vortices were first observedby Hess et al . on superconducting NbSe [19, 20]. Threescenarios were discussed as potential reasons: (i) vortex–vortex interactions, (ii) anisotropic pairing, and (iii) ananisotropy of the Fermi surface [16, 17, 24]. The effectof vortex–vortex interactions becomes relevant for spac-ings comparable with or smaller than the London pen-etration depth λ . Under this condition the stray field-mediated interaction between a given vortex and adja-cent flux lines may cause a significant distortion of itsshape [17]. Furthermore, as the spacing approaches thecharacteristic coherence length ξ , the hybridization ofthe nearest neighbor quasiparticle wave functions mayalso affect the LDOS [18]. However, both λ Nb ≈
39 nmand ξ Nb ≈
38 nm are at least four times smaller thanthe Abrikosov lattice constant ( ≈
160 nm) observed at µ H = 100 mT. Therefore, we rule out that scenario (i)is responsible for the observed elliptical shape in Nb.The two-fold rotational symmetry of (110) surfacesof body-centered cubic crystals, such as Nb, certainlyimplies inequivalent electronic properties along the twohigh-symmetry directions [001] and [110]. Since both sce-narios, (ii) and (iii), are intimately related to the surfaceelectronic structure of the respective superconductor, ei-ther of the two effects may—in principle—be responsiblefor the anisotropic dispersion of bound states inside thevortex. However, experiments performed on crystallineNb planar tunnel junctions suggest that the anisotropyof the superconducting gap does not exceed 10% of theaverage value (∆ = 1 .
53 meV) [26, 27], insufficient toexplain the strong anisotropy observed in Fig. 1(j).To explain the experimental observations, we calcu-late the Fermi surface of clean Nb(110) using the plane-wave-based VASP [28, 29] code within the projectoraugmented-wave method [30, 31]. The generalized gra-dient approximation of Perdew-Burke-Ernzerhof [32, 33]is used for the exchange correlation. Details of theorymethods and examples of the good agreement betweentheoretical and measured LDOS can be found in Ref. 22.The calculated Fermi surface of the Nb(110) surface isplotted in Fig. 3(a). Multiple Fermi surface pockets cen-tered around the various high-symmetry points Γ, N, S,and H can be recognized [22]. Since STM is most sensi-tive to electronic states in vicinity to the Γ point of theSBZ, with other states decaying exponentially with in-creasing k (cid:107) [34, 35], we expect that the stadium-shapedcontours (red) in Fig. 3(a) contribute most to the mea-sured signal. Therefore, we employ similar contours toapproximate the Fermi surface within an effective modeldescribed in the following.In this model, we consider an electron gas on a 2D h i gh l o w 2 0 22 0 2 (a) a l ong [ ] a l ong [ ] (e) 10 0 10 U ( Δ ) x ( a ) y ( a ) x ( n m ) E = Δ E = Δ 10 1010 10654315105051015 H 1 0 1101 k a / π y k a / π x (b)(d)
60 00.4120240 3000180 ( с ) E = Δ E = Δ E = Δ E = Δ E = FIG. 3. (a) Calculated Fermi surface of Nb(110). Red linesdenote pockets around the Γ point. (b) Hypothetical Fermisurface and (c) Fermi velocity of the dominant Fermi contour.(d) Calculated LDOS along two principal directions crossingthe vortex core. (e) Excitation energy-dependent LDOS pro-files after thermal broadening with k B T (cid:39) . . square lattice, with different effective masses, m x /m y ≈ .
47, in two principal directions and with an s -wave pair-ing interaction. In order to mimic a truly anisotropicband structure, we further consider an isotropic spin-orbit interaction, which can be either of Rashba or Dres-selhaus type [36, 37]. This spin-orbit interaction resultsin two split Fermi surfaces. The larger Fermi surfacetakes a stadium shape and carries a strong anisotropicFermi velocity, see Fig. 3(b), while the smaller one is cir-cular. We set the chemical potential at a value where thestadium-shaped Fermi surface dominates. Additionally,we apply a quantized magnetic flux and impose mag-netic periodic boundary conditions on the lattice. Then,we solve the resulting BdG equations self-consistently onthe lattice and calculate the LDOS around a vortex coreat a low temperature set to k B T (cid:39) . , where ∆ is the pairing potential in the absence of the magneticfield. We present the results in Fig. 3(d-e) and providemore details about the simulation in Ref. 23. At low ex-citation energies, the LDOS is enhanced at the vortexcore with an elliptical shape. This changes to a coffee-bean shape at intermediate excitation energies and toa round shape at high excitation energies, in excellentqualitatively agreement with experimental observations.Our analysis confirms that the anisotropy of the Fermisurface is carried over to the spatial structure of the ex-cited states within the vortices. Depending on detailsof the anisotropy of the Fermi surface and the resultingFermi velocities and the spin-orbit coupling, a distinctanisotropy of the CdGM states can be observed even ina system with an isotropic order parameter. Conclusion—
In conclusion, we have presented STSstudies of vortex bound states of the clean Nb(110) sur-face in an external magnetic field. Our results revealthat the vortex shape in d I/ d U maps depends on the en-ergy, transforming from circular to a coffee-bean shape,and finally to an ellipses when approaching E F . Weprovided evidence that the observed energy-dependentspatial distribution originates from Caroli-de Gennes-Matricon states which inherit their anisotropy from theNb(110) Fermi surface. By oxidizing the Nb(110) surfacewe could trigger the transition from the clean to the dirtylimit, accompanied by the complete disappearance of anyanisotropy in the LDOS due to the quenching of the vor-tex bound states. These findings will be useful in thefuture for distinguishing trivial from topological vortexbound states by, for example, their spatial distributionor by introducing disorder.This research was supported by the DFG (throughSPP1666 and SFB 1170 “ToCoTronics”, the W¨urzburg-Dresden Cluster of Excellence ct.qmat, EXC2147,project-id 390858490, and the Elitenetzwerk BayernGraduate School on “Topological Insulators”. S.J.H. andS.H. thank the Norddeutscher Verbund f¨ur Hoch- undH¨ochstleistungsrechnen (HLRN) for providing computa-tional resources. ∗ corresponding author:[email protected][1] Ø. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod,and C. Renner, “Scanning tunneling spectroscopy ofhigh-temperature superconductors,” Rev. Mod. Phys. , 353–419 (2007).[2] D. A. Ivanov, “Non-Abelian statistics of half-quantumvortices in p-wave superconductors,” Phys. Rev. Lett. ,268–271 (2001).[3] R. M. Lutchyn, J. D. Sau, and S. 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