Abstract
The protection against backscattering provided by topology is a striking property. In twodimensional insulators, a consequence of this topological protection is the ballistic nature of the onedimensional helical edge states. One demonstration of ballisticity is the quantized Hall conductance. Here we provide another demonstration of ballistic transport, in the way the edge states carry a supercurrent. The system we have investigated is a micrometrelong monocrystalline bismuth nanowire with topological surfaces, that we connect to two superconducting electrodes. We have measured the relation between the Josephson current flowing through the nanowire and the superconducting phase difference at its ends, the current–phase relation. The sharp sawtoothshaped phasemodulated current–phase relation we find demonstrates that transport occurs selectively along two ballistic edges of the nanowire. In addition, we show that a magnetic field induces 0–π transitions and ϕ_{0}junction behaviour, providing a way to manipulate the phase of the supercurrentcarrying edge states and generate spin supercurrents.
Introduction
Reducing the size of a conductor usually decreases its conductivity because of the enhanced effect of disorder in low dimensions, leading to diffusive transport and to weak, or even strong localization. Notable exceptions occur when topology provides protection against disorder, such as in the quantum Hall effect or the recently discovered quantum spin Hall effect in twodimensional (2D) topological insulators^{1,2}. In the latter, crystalline symmetry combined with high spin–orbit coupling generate band inversion and onedimensional (1D) chiral edge states with perfect spinmomentum locking, that theoretically precludes backscattering along the edges. However, since the first evidences of edge state currents, both in the normal and superconductingproximitized states, demonstrating the robustness of ballistic conduction and spin polarization in the 1D edge states has remained a challenge^{3,4,5,6,7}. Here we provide a direct signature of ballistic 1D transport along the topological surfaces of a monocrystalline bismuth nanowire connected to superconducting electrodes. To this end, we measure the relation between the Josephson current I_{J} flowing through a nanostructure and the superconducting phase difference ϕ at its ends, the current–phase relation (CPR)^{8}. The CPR is an exquisite tool to discriminate between different transport regimes (ballistic, diffusive, tunnel).
The sharp sawtoothshaped CPR we find demonstrates that transport of Cooper pairs occurs ballistically^{9,10,11,12,13} along two edges of the nanowire, whose positions can be deduced from experiments in different magnetic field orientations. We also show that a Zeeman field induces 0–π transitions^{14} and ϕ_{0}junction behaviour^{15}, in agreement with recent theoretical predictions^{16}.
Results
Investigated samples
Monocrystalline bismuth nanowires were grown by sputtering (Methods section), and their crystalline orientation was determined by electron backscatter diffraction. Individual nanowires with (111) top facets were then selected to exploit the predicted topological edge states of those surfaces^{17}, brought about by the Bi lattice symmetries and high (eVrange) atomic spin–orbit coupling. The wires are narrow enough (thickness and width between 30 and 200 nm) that most normal state conduction is due to surfaces and edges, with only a few per cent contribution of the bulk states^{18,19}, as discussed in Supplementary Note 2. Furthermore, tightbinding simulations of a Bi nanowire with a rhombic section and top and bottom (111) facets, similar to the main nanowire reported here (Fig. 1 and Methods section), predict one edge state along each (111) facet, extending the result of Murakami^{17}. These edge states coexist with 2D metallic states on the nontopological (100) surfaces but nonetheless clearly dominate the local density of states (Fig. 1c).
Ballistic CPR
To measure the CPR of the superconductor/bismuth nanowire/superconductor (S/Bi/S) junction, we inserted a singleBi nanowire with (111) surfaces into an asymmetric SQUID setup. The setup consists of a highcriticalcurrent Josephson junction, made of a W superconducting nanoconstriction, in parallel with a 1μmlong S/Bi/S junction characterized beforehand (Figs 1 and 3). In this configuration the modulation of the SQUID’s critical current by the magnetic flux yields the CPR of the junction with the smallest critical current^{20}. It is well known that the CPR of the superconductor/insulator/superconductor (SIS) Josephson junction is sinusoidal I_{J}(ϕ)=I_{c} sin ϕ. The CPR of a ballistic superconductor/normal metal/superconductor junction, on the other hand, is a characteristic sawtooth in a long junction (for which L>>ħv_{F}/Δ)^{9,10,11,12,13}, or segments of a sine in a short junction (Fig. 2c)^{20} (L is the length of the normal metal, v_{F} the Fermi velocity and Δ the superconducting gap). Any disorder smooths the CPRs, as illustrated in Fig. 2c for a junction in the diffusive regime, (that is, with a mean free path shorter than the length L). While the CPR of atomic point contacts^{20}, nanowirebased quantum dots^{21,22,23} and graphene^{24} have previously been measured in this way, the CPR of micrometrelong, quasiballistic channels has, to our knowledge, not been accessed.
The switching current of the asymmetric BiSQUID (Fig. 2a) clearly displays sawtoothshaped oscillations of amplitude 400 nA, superimposed on the 80 μA critical current of the nanoconstriction. The oscillation period of 9.5 G corresponds to a flux quantum Φ_{0}=h/2e through the SQUID loop area of 2 μm^{2}. Those oscillations demonstrate the sawtooth CPR characteristic of a long, perfectly connected ballistic channel. This is a key result of our paper. Eleven harmonics are visible in the Fourier transform of the signal at 100 mK. Figure 2b displays how the 1/n amplitude of the nth harmonic, I_{n}, expected for a perfect sawtooth, is exponentially damped by a small factor: I_{n}=I_{0}/n exp(−an), with a=0.19 at 100 mK and 0.25 at 1.2 K. Since the nth harmonics can be interpreted as the contribution to the supercurrent of the nth order Andreev reflection, the damping can be understood as due to an imperfect transmission encountered n times, as well as due to temperature T (refs 11, 12, 13), yielding the phenomenological expression I_{n}=I_{0}/nt^{2n} exp(−nT/E_{Th}). Here t is the modulus of the total transmission in the normal state. Other effects, such as high frequency noise, may also contribute to this decay. The damping coefficient at 100 mK yields a quasi perfect transmission t≥0.9, and we deduce the Thouless energy E_{Th}=4 K from the damping coefficient at 1.2 K.
Signatures of a second supercurrentcarrying path
Predicted by our tightbinding modelling of the wire (Fig. 1c), a second path can be identified upon closer inspection of the CPR: a wiggle with a smaller period is superimposed to the main sawtooth signal. The full critical current modulation is reproduced by adding to the main sawtooth the contribution of a second sawtooth with a four times smaller amplitude and a 10% smaller period than the first. This indicates that this second, ballistic, path is situated along the outer wire edge. The sum of these two sawtooths generates a beat period of ∼85 G (Supplementary Fig. 7 and Supplementary Note 4) which corresponds to a flux quantum through the wire. This second path is also apparent as a satellite peak in the Fourier transform next to the first harmonic (Fig. 2d), whose exponential decay exp(−0.45n)/n yields a transmission coefficient t∼0.75. The smaller supercurrent carried by the second channel could be due to a disordered region between the contact and this edge (inset Fig. 1a).
These findings of two distinct current paths, located at two opposite edges of the wire, are consistent with the twopath interference patterns of the critical current measured on the same wire before integration in the asymmetric SQUID configuration, for different orientations of the magnetic field (Fig. 3). The oscillation periods depend on the field direction, and correspond to one flux quantum through the surface enclosed by the two paths projected onto a plane normal to the field. It is thus consistent that the 100 G interference period in a perpendicular field B_{z} (corresponding to Φ_{0}/WL) is the same (given the uncertainty of the sample position in the magnet) as the beat period measured in the CPR (Supplementary Fig. 7). The critical current oscillations in an inplane magnetic field B_{//} was also measured for several angles. The period is determined only by the projection of the field on the normal to the wire axis, that is, the y component. The period Δ B_{y}=500 G=Φ_{0}/tL yields a nanowire thickness of t=40 nm, consistent with scanning electron microscope observations. We note that oscillating critical current patterns were also detected previously in Bi wires with unknown crystalline orientations^{25}, and were also interpreted as interference between two paths through the samples^{26}.
Number of channels and Josephson current amplitude
The amplitude of the measured supercurrent as well as its resilience up to high magnetic fields (B_{max}=0.5 T) (Fig. 3, Supplementary Fig. 6 and ref. 25) are consistent with a small number of channels, each of them confined to an extremely narrow region in space (within <4 nm=Φ_{0}/B_{max}L). The maximum supercurrent through one ballistic channel is I_{1}=π Δ /Φ_{0}=e Δ /2ħ∼250 nA for a short junction (that is, much shorter than ξ_{s}=ħv_{F}/Δ≈600 nm). It is smaller for a long junction^{13}, of the order of ev_{F}/L≈100±30 nA for L=1 μm and v_{F}=6±2 × 10^{5}m s^{−1}. This value of v_{F}, deduced from the Fourier spectrum of the CPR at 1 K, is in qualitative agreement with the values deduced from the dispersion relation of 1D edge states of (111) Bi triangular surfaces, measured by photoemission in ref. 27. The critical current of the nanowire, given by the modulated current amplitude of 400 nA (Fig. 2a), thus implies that at most six perfectly transmitted channels carry the supercurrent. A reasonable assumption is that one path contains three to four quasi perfect channels, each with the same sawtoothshaped CPR, and all situated at the inner edge of the wire, on the bottom (111) facet. They could be associated to the orbitals p_{x},p_{y},p_{z} of Bi, as suggested by Murakami^{17}, or could also run along the edges of few parallel terraces at the facet edge. The smaller contribution of the second path is attributed to one or two other channels of smaller transmission, at the outer edge of the top (111) facet (see sketch in Fig. 2b).
Phase shifts of the CPR induced by an inplane magnetic field
The purely 1D nature of the edge Andreev bound states that carry the supercurrent across the nanowire implies that they are insensitive to orbital dephasing and offers the possibility to explore the effect of a Zeeman field on the phase of the Josephson current. A Zeeman field can induce a crossing of the Andreev levels, turning an energy maximum into a minimum^{28}: this causes a sign change of the CPR, or equivalently a π shift of its phase. 0–π transitions are expected when dephasing by the magnetic field equals dephasing by the propagation time through the wire, that is, when the Zeeman energy equals the Thouless energy, g_{eff} μ_{B}B=ħ<v_{F}>/L. Such 0–π transitions are visible in Fig. 4a,b,e, as phase jumps in the CPR plotted as a function of a magnetic field in the (111) plane, either perpendicular or parallel to the wire axis. The characteristic field B_{x,y}∼600 G, B_{z}∼400 G between two successive 0–π transitions yields an effective g factor g_{eff}≈30–100, consistent with the high g factors of some bands in Bi (ref. 29) as well as recently found for the surface states with ARPES experiments^{30} in high magnetic field. We note that penetration of vortices in the superconducting electrodes would also lead to phase jumps. This is, however, unlikely as no sign of hysteresis was found in our data. This realization of a 0–π transition induced by the Zeeman field is possible because the junction is long, contains few channels and the g_{eff} are high enough that the transition occurs at a magnetic field below the superconducting electrodes’ (relatively high) critical field.
The usual conversion of phase difference into a charge (super) current can be supplemented by a conversion into a spin current in the presence of strong spin–orbit interaction. Consequently, a magnetic field can shift the CPR by a phase ϕ_{0} (refs 15, 26). This effect is related to the spinsplitting of Andreev states induced by spin–orbit interactions, that generates spindependent velocities, and is expected to be enhanced in quantum spin Hall ribbons with asymmetric edges^{16}. It is also predicted to be greatest in a field parallel to the spin–orbit field. Such ‘ϕ_{0}junctions’ are visible in Fig. 4a,b,d, in the form of an oscillating phase of the CPR with increasing field, outlined by the winding hatched curve, with a stronger effect in a field perpendicular to the wire axis (compare Fig. 4a,b).
Discussion
The CPR of a Binanowirebased Josephson junction demonstrates that conduction occurs along ballistic channels confined at two edges of the wire’s (111) facets. Whereas narrow edge states were predicted for a bilayer of (111) Bi (ref. 17), it was not obvious that such states should exist in thicker samples, until 1D edge states were detected by scanning tunnelling microscopy recently, in two layerdeep pits at the (111) surface of bulk Bi crystals^{31}. Our work shows that in the superconducting proximity regime, the contribution of the nanowire’s two ballistic edge states outweighs that of the more numerous diffusive channels on the wire’s nontopological surfaces. That is because the contribution to the supercurrent is proportional to the conductance times Thouless energy^{32}, leading to a critical current of I_{c,1ch,diff} for a diffusive channel (with l_{e} the mean free path), and I_{c,1ch,bal}∝1/L for a ballistic channel. Thus a diffusive channel carries a supercurrent smaller than a ballistic channel by a factor (L/l_{e})^{2}∼100, where l_{e} is the mean free path. Moreover, in addition to being ballistic rather than diffusive, topological edge states are better coupled to the superconducting electrodes due to their spinmoment locking that should produce perfect Andreev retroreflections^{33}. By contrast, the slower diffusive carriers also undergo regular (as opposed to Andreev) reflections on opaque barriers, which decrease the supercurrent they can carry. Our results show that proximityinduced superconductivity can be exploited to reveal and select ballistic protected edge states in systems where topological and nontopological conduction paths coexist. Moreover, even if the topological character of the edge states carrying the supercurrent is not yet directly proven, our results indicate an effective mean free path for the edge states much greater than 1 μm, whereas the mean free path of the surface and bulk states, whose contribution dominates the transport in the normal state, does not exceed 200 nm (Supplementary Note 1). This result in itself points towards a topological protection for these edge states. We are now working on microwave experiments on these same wires that, by exploring the nature of the level crossing at ϕ=π, should be more definite on this issue^{34,35,36}.
Methods
Sample fabrication and measurement
Bismuth nanowires were grown by RFsputtering a Bi target of 99.9999% purity onto Si substrates at 473 K and in an argon pressure of 10 mBars. A Bi film of 400 nm deposited at a rate of 0.7 nm s^{−1} in these conditions exhibits sparse arrays of nanowires, typically 100 μm long and distant by 10 μm from one another. High resolution transmission electron microscope observations indicated high quality single crystals, of rhombohedric or hexagonal sections as well as clear facets, with typical width of 50–300 nm. The nanowires were then dry deposited on an oxidized silicon substrate. After optical selection, their crystalline orientation was determined using electron backscatter diffraction. An example is shown in Supplementary Fig. 2. This characterization was performed at the extremity of the nanowires to avoid electron beaminduced damage of the Bi wire. We also checked on similar nanowires that their crystalline orientation was constant as we changed the position of the beam spot, as is expected for single crystals. At this stage, we selected nanowires whose top surface was determined to be oriented perpendicular to the trigonal [111] axis. Electrical connections were then made using gallium focused ion beam (FIB)deposited superconducting tungsten wires^{37}. For all the measured nanowires, the lengths between the W lines were chosen to be greater than 1 μm to avoid possible superconducting contamination. Previous studies using the same setup showed that this could be an issue for wires whose length is below 200 nm. For the same reasons, we minimized the total exposure time of the nanowires under the FIB to a single scan at high scanning rate and low magnification. We connected nine segments with different lengths, from a total of three such nanowires. Their resistances and lengths are summarized in the Supplementary Table 1, and show low contact resistance on average.
The samples were then cooled to 100 mK and their critical current versus magnetic field was measured using a lockin detection technique.
After this first characterization step, the CPR was measured on one segment (s_{1}JU segment of Supplementary Table 1) using the asymmetric SQUID technique^{20} with a reference junction made of a W constriction. To this end, a FIBdeposited W wire was added between the two tungsten electrodes, in parallel to the Bi nanowire (Fig. 1), and was subsequently etched with the Ga+ beam while measuring the total resistance between the contacts, until the total resistance reached 190 Ω, corresponding to a constriction resistance of 300 Ω. The critical current of the SQUID was deduced from the average of 100 to 400 measurements of the switching current, using a counter synchronized to a current ramp at 180 Hz, and triggered by the jump in the sample resistance. In those measurements, the sample resistance was measured at 100 kHz with a lockin detector operating with a time constant of 1 ms.
Tightbinding simulation of bismuth nanowire
Tightbinding simulations were performed on a 1D infinite geometry corresponding to the stacking of five (111) Bi bilayers, seven atoms wide, arranged to provide a simplified model of the rhombic section of the nanowire studied in this paper. The z direction therefore corresponds to the [111] direction or trigonal axis in Bi’s rhombohedral lattice, which was determined experimentally by electron backscatter diffraction. The infinite direction of the nanowire and the direction of the lateral surface were chosen in order to match two criteria: they should be of high symmetry since nanowire growth mechanisms favour high symmetry surface orientation, and should reproduce the rhombic section that was inferred by scanning electron microscope observation. The retained geometry corresponds to the (100) lateral surface and a growth direction along the bisectrix axis. This corresponds to the case studied by Murakami in the 1 bilayer thickness limit and 20 atoms wide. We find a sharp density of states along two edges, that correspond to zigzag type A edges (defined in ref. 31) on the top surface, and the B type edges on the bottom surface: those are the edges whose atoms are less coupled to the bulk.
The Hamiltonian matrix was written according to the sp3 tightbinding model of Liu and Allen^{38} using the KWANT python package^{39}. The local density of states at the Fermi level was computed by summing the local spectral weights in a window of 0.1 eV around the Fermi level. A smaller window would yield even higher densities of states on the acute edges.
Data availability
All data presented in the main paper and supplement are available from the authors upon request.
Additional information
How to cite this article: Murani, A. et al. Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry. Nat. Commun. 8, 15941 doi: 10.1038/ncomms15941 (2017).
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Acknowledgements
We acknowledge fruitful discussions with M. Aprili, S. Bayliss, C. Beenakker, S. Bergeret, D. Carpentier, J. Jobo, M. Houzet, F. Konschelle, J. Meyer, J. Enrique Ortega, P. Simon, T. Wakamura, A. Yazdani and financial support from CNRS, ANR MASH (ANR12BS040016), ANR DYMESIS (ANR2011IS0400101), ANR DIRACFORMAG (ANR14CE320003) and ANR JETS (ANR16CE30002901). Yu.A.K. acknowledges financial support by RFBR and Moscow region grant 144803664.
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A.M., R.Del., R.Deb., H.B. and S.G. conceived the experiment and performed the measurements. A.K., Yu.A.K., V.T.V. and I.I.K. grew the Bi nanowires and characterized the nanowires with transmission electron microscope, A.M., A.K., S.S. and S.G. fabricated the circuits using FIB, F.B. and A.M. characterized the crystalline orientation of the nanowires. A.M., H.B. and S.G. wrote the manuscript with input from all authors. A.C., A.M. and H.B. conducted the numerical simulations.
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Murani, A., Kasumov, A., Sengupta, S. et al. Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry. Nat Commun 8, 15941 (2017). https://doi.org/10.1038/ncomms15941
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