Topological insulators are a newly discovered phase of matter characterized by gapped bulk states surrounded by conducting boundary states1,2,3. Since their theoretical discovery, these materials have encouraged intense efforts to study their properties and capabilities. Among the most striking results of this activity are proposals to engineer a new variety of superconductor at the surfaces of topological insulators4,5. These topological superconductors would be capable of supporting localized Majorana fermions, particles whose braiding properties have been proposed as the basis of a fault-tolerant quantum computer6. Despite the clear theoretical motivation, a conclusive realization of topological superconductivity remains an outstanding experimental goal. Here we present measurements of superconductivity induced in two-dimensional HgTe/HgCdTe quantum wells, a material that becomes a quantum spin Hall insulator when the well width exceeds dC = 6.3 nm (ref. 7). In wells that are 7.5 nm wide, we find that supercurrents are confined to the one-dimensional sample edges as the bulk density is depleted. However, when the well width is decreased to 4.5 nm the edge supercurrents cannot be distinguished from those in the bulk. Our results provide evidence for supercurrents induced in the helical edges of the quantum spin Hall effect, establishing this system as a promising avenue towards topological superconductivity. In addition to directly confirming the existence of the topological edge channels, our results also provide a measurement of their widths, which range from 180 nm to 408 nm.
Topological superconductors, like topological insulators, possess a bulk energy gap and gapless surface states. In a topological superconductor, the surface states are predicted to manifest as zero-energy Majorana fermions, fractionalized modes that pair to form conventional fermions. Owing to their non-Abelian braiding statistics, achieving control of these Majorana modes is desirable both fundamentally and for applications to quantum information processing. Proposals towards realizing Majorana fermions have focused on their emergence within fractional quantum Hall states8 and spinless p + ip superconductors9, and on their direct engineering using s-wave superconductors combined with topological insulators or semiconductors10,11. Particularly appealing are implementations in one-dimensional (1D) systems, where restriction to a single spin degree of freedom combined with proximity to an s-wave superconductor would provide the basis for topological superconductivity12. Effort in this direction has been advanced by studies of nanowire systems13,14,15,16,17,18 and by excess current measurements on InAs/GaSb devices19.
An attractive route towards a 1D topological superconductor originates from the 2D quantum spin Hall (QSH) insulator. This topological phase of matter was recently predicted20,21 and observed22,23 in transport measurements of HgTe/HgCdTe quantum wells thicker than a critical thickness dC = 6.3 nm. Due to strong spin–orbit coupling the bulk bands of the system invert, crossing only at the edges of the system to form 1D counterpropagating helical modes. Time-reversal symmetry ensures protection of these modes against elastic backscattering over distances shorter than the coherence length24. The helical nature of the edge modes makes them a particularly appealing path towards topological superconductivity, owing to the intrinsic elimination of their spin degree of freedom. Here we report measurements of supercurrents induced in HgTe/HgCdTe quantum well heterostructures. As the system enters the QSH regime we find that these supercurrents become confined to the topological edge modes, verifying their existence and providing a microscopic picture of the QSH state.
Our approach consists of a two-terminal Josephson junction, with a rectangular section of quantum well located between two superconducting leads (Fig. 1). At a given bulk carrier density, the presence or absence of helical edge channels influences the supercurrent density profile across the width of the junction. In the simplest case the supercurrent density is uniform throughout the device, and edge channels are indistinguishable from bulk channels (Fig. 1a). This behaviour would be expected for a non-topological junction (quantum well width smaller than dC), or in a topological junction (quantum well width larger than dC) far from the bulk insulating regime.
In a topological junction, decreasing the bulk carrier density brings the device closer to the QSH insulator regime (Fig. 1b). Scanning superconducting quantum interference device (SQUID) measurements suggest that over a range of bulk densities the QSH edge channels coexist with bulk states, and can carry considerably more edge current than would be expected for a non-topological conductor25. In the two-terminal configuration, this helical edge contribution appears as peaks in the supercurrent density at each edge. When the bulk density becomes sufficiently low, these edge peaks are the only features in the supercurrent density (Fig. 1c). Then the supercurrent is carried solely along the helical edges, and the system is in the regime of the quantum spin Hall superconductor.
Placing such a Josephson junction in a perpendicular magnetic field B provides a way to measure the supercurrent density in the quantum well. In general, the maximum supercurrent that can flow through a Josephson junction is periodically modulated by a magnetic field. Typically, the period of the modulation corresponds to the magnetic flux quantum Φ0 = h/2e. In our junctions this period matches the area of the HgTe region plus half the area occupied by each contact, a result of the Meissner effect. The particular shape of the critical current interference pattern depends on the phase-sensitive summation of the supercurrents traversing the junction26. In the case of a symmetric supercurrent distribution, this integral takes the simple form: Here LJ is the length of the junction along the direction of current, accounting for the magnetic flux focusing from the contacts.
It is evident that different supercurrent densities JS(x) in the junction can give rise to different interference patterns ICmax(B). The flat supercurrent density of a trivial conductor corresponds to a single-slit Fraunhofer pattern |(sin(πLJBW/Φ0))/(πLJBW/Φ0)|, characterized by a central lobe width of 2Φ0 and side lobes decaying with 1/B dependence (Fig. 1a). As helical edges emerge, this single-slit interference evolves towards the more sinusoidal oscillation characteristic of a SQUID (Fig. 1b). The central lobe width shrinks to Φ0 when only edge supercurrents remain, with the side lobe decay determined by the widths of the edge channels (Fig. 1c). Measuring the dependence of ICmax on B therefore provides a convenient way to measure the distribution of supercurrent in a junction. To quantitatively extract JS(x) from the measured quantity ICmax(B) we follow an approach where non-zero ICmax(B) minima are ascribed to an asymmetric supercurrent distribution27. Although other effects may lead to non-zero minima in ICmax(B), we consider here only the possibility of an odd component in JS(x). Full details of the extraction procedure can be found in the Supplementary Information.
To study how supercurrents flow in the QSH regime, we measure a Josephson junction consisting of a 7.5-nm-wide quantum well contacted by titanium/aluminium leads28. Our contact lengths are each 1 μm, and the contact separation is 800 nm. The junction width of 4 μm is defined by etched mesa edges. A voltage VG applied to a global top gate allows us to tune the carrier density in the junction. At each value of VG and B, the critical current ICmax is determined by increasing the current through the junction while monitoring the voltage across the leads. The behaviour observed in this device is reproducible in several other similar junctions, as reported in the Supplementary Information.
As a function of the top-gate voltage, the overall behaviour of the junction evolves between two extremes. At more positive gate voltage and higher bulk density, the critical current envelope strongly resembles a single-slit pattern (Fig. 2a). This type of interference suggests a nearly uniform supercurrent density throughout the sample, confirmed by transformation to the JS(x) picture (Fig. 2b). This nearly flat distribution indicates that the quantum well is in the high carrier density regime of an essentially trivial conductor.
At more negative gate voltage and lower bulk density, the critical current envelope becomes close to a sinusoidal oscillation (Fig. 2c). The shift towards a SQUID-like interference pattern corresponds to the development of sharp peaks in supercurrent density at the mesa edges (Fig. 2d).
We can track this evolution in a single device by measuring the critical current envelope at a series of gate voltages. As the top gate is varied from VG = 1.05 V to VG = −0.45 V, the maximum critical current decreases from 505 nA to 5.7 nA. At the same time, the overall critical current behaviour shows a narrowing of the central interference lobe, from 2Φ0 at positive gate voltages to Φ0 at negative gate voltages (Fig. 3a, b). The side lobes additionally become continuously more pronounced, indicating the confinement of supercurrent to channels at the edges of the junction (Fig. 3c, d). The normal resistance, measured at large bias to overcome superconductivity, increases from 160 Ω to 3,000 Ω over the range of this transition (Fig. 3e). Although it is possible to gate further towards depletion, the critical currents become too small to reliably measure and no meaningful supercurrent density can be extracted.
At the most negative gate voltage, VG = −0.45 V, we can estimate the widths of the supercurrent-carrying edge channels using a Gaussian line shape (Fig. 3f). Using this method, we find widths of 408 nm and 319 nm for the two edges. Our measurements of edge widths in another device with similar dimensions, as well as one with a 2 μm mesa width, show edges as narrow as 180 nm (Supplementary Information). These width variations, as well as the normal-state resistance that is low compared with the resistance h/2e2 for two ballistic 1D channels, suggest the presence of additional edge modes or of bulk modes coupled too weakly across the junction to carry supercurrent.
To provide further evidence that the observed edge supercurrents are topological in nature, we next turn to a heterostructure with a quantum well width of 4.5 nm. In this device, the well width is smaller than the critical width dC, so that the sample is not expected to enter the QSH regime. Near zero top-gate voltage and a normal resistance of 270 Ω, the critical current interference pattern has a maximum of 243 nA and resembles a single-slit envelope (Fig. 4a, b). On energizing the top gate and decreasing the bulk density, the single-slit pattern persists. In contrast to the wide well sample, this behaviour corresponds to a supercurrent density that remains distributed throughout the junction even as the normal resistance rises to several kΩ (Fig. 4c–f). As the edge supercurrents are present only when the well width is larger than dC, we conclude that our observations provide evidence for induced superconductivity in the helical QSH edge states.
By studying Fraunhofer interference, our measurements provide detailed information about the supercurrent distribution in HgTe quantum wells. In the quantum spin Hall regime, this interferometry confirms the existence of topological edge channels associated with the quantum spin Hall insulating state. Our observed supercurrent distributions additionally provide the first direct measurements of the helical edges’ spatial extent. In general, our application of this Fourier imaging technique to HgTe quantum wells could be widely adopted as a method to elucidate the microscopic structure of topological materials. In our devices, the observation of Josephson supercurrents through the helical edge channels establishes the HgTe/HgCdTe system as a platform in which to pursue topological superconductivity and Majorana bound states, whether through following existing theoretical proposals or those yet to be formulated29,30,31.
We acknowledge A. Akhmerov and J. D. Sau for theoretical discussions. We acknowledge financial support from Microsoft Corporation Project Q, the NSF DMR-1206016, the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, the DOE SCGF Program, the German Research Foundation (DFG-JST joint research programme ‘Topological Electronics’), and EU ERC-AG programme (project 3-TOP).
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Journal of Low Temperature Physics (2018)