Abstract
Carbon nanotubes (CNTs) are not intrinsically superconducting but they can carry a supercurrent when connected to superconducting electrodes^{1,2,3,4}. This supercurrent is mainly transmitted by discrete entangled electron–hole states confined to the nanotube, called Andreev bound states (ABS). These states are a key concept in mesoscopic superconductivity as they provide a universal description of Josephsonlike effects in quantumcoherent nanostructures (for example molecules, nanowires, magnetic or normal metallic layers) connected to superconducting leads^{5}. We report here the first tunnelling spectroscopy of individually resolved ABS, in a nanotube–superconductor device. Analysing the evolution of the ABS spectrum with a gate voltage, we show that the ABS arise from the discrete electronic levels of the molecule and that they reveal detailed information about the energies of these levels, their relative spin orientation and the coupling to the leads. Such measurements hence constitute a powerful new spectroscopic technique capable of elucidating the electronic structure of CNTbased devices, including those with wellcoupled leads. This is relevant for conventional applications (for example, superconducting or normal transistors, superconducting quantum interference devices^{3} (SQUIDs)) and quantum information processing (for example, entangled electron pair generation^{6,7}, ABSbased qubits^{8}). Finally, our device is a new type of d.c.measurable SQUID.
Main
First conceived of four decades ago^{9}, ABS are electronic analogues of the resonant states in a Fabry–Pérot resonator. The cavity is here a nanostructure and its interfaces with superconducting leads play the role of the mirrors. Furthermore, these ‘mirrors’ behave similarly to optical phaseconjugate mirrors: because of the superconducting pairing, electrons in the nanostructure with energies below the superconducting gap are reflected as their timereversed particle—a process known as Andreev reflection. As a result, the resonant standing waves—the ABS—are entangled pairs of timereversed electronic states, which have opposite spins (Fig. 1a); they form a set of discrete levels within the superconducting gap (Fig. 1b) and have fermionic character. Changing the superconducting phase difference ϕ between the leads is analogous to moving the mirrors and changes the energies E_{n}(ϕ) of the ABS. In response, a populated ABS carries a supercurrent (2e/ℏ)(∂ E_{n}(ϕ)/∂ ϕ) through the device, whereas states in the continuous spectrum (outside the superconducting gap) have negligible or minor contributions in most common cases^{5}. Therefore, the finite set of ABS generically determines Josephsonlike effects in such systems. As such, ABS play a central role in mesoscopic superconductivity and can be seen as the superconducting counterpart of the Landauer channels for the normal state: in both cases, only a handful of them suffices to account for all of the transport properties of complex manyelectron systems such as atomic contacts or CNTs. In effect, the ABS concept quantitatively explains the Josephson effect in atomic contacts^{10}; it also explains tunnelling spectroscopy of vortex cores and surface states in some superconductors^{11}. However, there has been so far no detailed direct spectroscopic observation of individual ABS. Interest in such spectroscopy has increased with recent proposals for using ABS as quantum bits^{8}, and Andreev reflection as a source of entangled spin states^{6}.
Nanotubes are particularly good candidates for the observation of ABS. First, CNT–superconductor hybrid systems are expected to show a small number of ABS, and the typical millielectronvolt energy scales involved in nanotube devices are comparable to conventional superconducting gaps. These are favourable conditions for a wellresolved spectroscopy experiment. Second, given the length of CNTs, it is possible to introduce a tunnel probe that enables straightforward tunnelling spectroscopy^{12}. Furthermore, CNTs are of fundamental interest as nearly ideal, tunable onedimensional systems in which a wealth of phenomena (for example Luttingerliquid behaviour^{13}, Kondo effects^{3,14} and spin–orbit coupling^{15}) has been observed and the rich interplay of these effects with superconducting coupling has attracted a lot of interest^{16,17,18,19,20,21,22}.
Our sample is described in Fig. 1. A CNT is well connected to two superconducting metallic contacts 0.7 μm apart, leaving enough space to place a weakly coupled tunnel electrode in between. The electrodes are made of aluminium with a few nanometres of titanium as a sticking layer (see Supplementary Information for details); they become superconducting below ∼1 K. The two outer contacts are reconnected, forming a loop. A magnetic flux threaded through the loop produces a superconducting phase difference ϕ=(2e/ℏ) across the tube. By measuring the differential conductance of the tunnel contact at low temperature (T∼40 mK) we observe (see Figs 2a and 3a) welldefined resonances inside the superconducting gap. The energies of these resonances strongly depend on the voltage applied on the backgate of the device, and vary periodically with the phase difference across the CNT, a signature of ABS. From the raw measurement of the differential conductance between the tunnel probe and the loop we can extract the density of states (DOS) in the tube (see for example Fig. 2b) through a straightforward deconvolution procedure (see Supplementary Information). Figure 2 shows the dependence of the ABS spectrum on the flux in the loop at a fixed gate voltage. By d.c.biasing this device at a point that maximizes ∂ I/∂ (see Fig. 2a), it can be used as a SQUID magnetometer that combines the advantages of refs 23 and 3. Being nanotubebased, our SQUID should be able to detect the reversal of magnetic moments of only a few Bohr magnetons^{3}. At the same time, the present device can be read out with a d.c. current measurement (similar to ref. 23) and requires a single gate voltage, making it easier to operate than ref. 3. The gatevoltage dependence of the DOS shows a pattern of resonance lines (Fig. 3b) that is more or less intricate depending on the strength of the coupling to the leads (see Supplementary Information).
We now show that the ABS observed in this device arise from the discrete molecular levels in the CNT. For this we describe our nanotube phenomenologically as a quantum dot coupled to superconducting leads (see Supplementary Information for a detailed discussion of the model). The essential physics of ABS in this system is already captured when one considers a single orbital of the quantum dot filled with either one or two electrons. Owing to the Pauli exclusion principle, these two electrons have opposite spins and can thus be coupled by Andreev reflection. Furthermore, the doubly occupied state is higher in energy by an effective charging energy that can be determined from the experimental data. Hence, the minimal effective model consists of a spinsplit pair of levels (SSPL), the parameters of which are the splitting , the mean position of the SSPL relative to the Fermi level (which is controlled by the gate voltage) and the coupling to the leads (see Supplementary Fig. S1a). Previous theoretical work^{24,25} has shown that there can be up to four ABS, symmetric (in position, but not in intensity) about the Fermi Level. For sufficiently large (respectively, ), however, the two outer (respectively, all) ABS merge with the continuum and are no longer visible in the spectrum^{24,25,26}.
We now discuss the dependence of the ABS energies on the gate voltage V_{g}. The ABS appear as facing pairs of bellshaped resonances centred at and with their bases resting against opposite edges of the superconducting gap (see the green dashed curves in Fig. 3b.) For large enough the inner resonances cross at the Fermi energy, forming a loop (Fig. 3b). Such loops are a distinct signature of SSPL in this model (spindegenerate levels () cannot give loops). Most of the features observed in Fig. 3b can be identified as such pairs of bellshaped resonances corresponding thus to different SSPL in the nanotube.
Closer inspection reveals however that adjacent resonances are sometimes coupled, forming avoided crossings (as indicated by diamond symbols in Figs 3b and 4), so that we need to consider the case where two SSPL contribute simultaneously to the spectral properties within the superconducting gap. For this, we extend the model to two serially connected quantum dots each containing a SSPL, with a significant hopping term in between. This model is fairly natural, given that the centre tunnel probe electrode is likely to act as an efficient scatterer. The full description of the model, the derivation of the retarded Green function from which we obtain the spectral properties, and the parameters used to produce the theoretical panels in Figs 3 and 4 are detailed in the Supplementary Information. Assuming for simplicity that all states in the two dots are identically capacitively coupled to the gate and that the couplings to the leads are independent of V_{g}, we can locally reproduce most features of the gatevoltage dependence of the DOS, and simultaneously the flux dependence at fixed V_{g} (see Fig. 4). By summing contributions of independent SSPL and pairs of coupled SSPL (that is isolated orbitals and coupled pairs of quantumdot orbitals), we can also reproduce the observed dependence on an extended V_{g} range (see Fig. 3b,c, and discussion in the Supplementary Information).
Note that a single superconducting terminal is sufficient to induce ABS in a quantum dot (in which case, of course, there can be no supercurrent, see refs 27, 28). Given this, and in light of our analysis, we believe that some features observed in refs 29, 30 that were tentatively explained as outofequilibrium secondorder Andreev reflection can now be reinterpreted as equilibrium ABS spectroscopy on a quantum dot well connected to one superconducting lead, as in refs 27, 28, with the second lead acting as a superconducting tunnel probe.
The agreement between experiment and theory in Figs 3 and 4 shows that ABS spectra constitute an entirely new spectroscopic tool for quantum dots and CNTs. This spectroscopy provides extremely detailed information, in particular about the relative spin state of the nanotube levels, without requiring high magnetic fields. Note that, in contrast to the usual Coulomb blockade spectroscopy of quantum dots, the energy resolution is here essentially independent of the temperature (as long as k_{B}T≪Δ) and of the strength of the coupling to the leads. It should therefore allow the exploration of the transition between the Fabry–Pérot (where the Luttingerliquid physics is expected to play a role^{18,19}) and the Coulomb blockade regimes in CNT. We also expect this new technique to be able to provide key insights in cases where simple charge transport measurements are not sufficient to fully probe the physics at work. In particular, it should allow detailed investigation of the competition between superconductivity and the Kondo effect^{16} that arises for stronger couplings to the leads. Furthermore, used in combination with an inplane magnetic field, it could also probe spin–orbit interactions^{20,21,22}. Finally, it should be emphasized that although our phenomenological model successfully describes the observed experimental data, further theoretical work is needed to establish a truly microscopic theory that should predict the level splittings from the bare manybody Hamiltonian.
The information extracted from such spectroscopy may also help to optimize fieldeffect transistors, SQUIDs or even nanoelectromechanical devices based on nanotubes, by better understanding how current is carried through the device. It could also be used for evaluating recently proposed devices for quantum information processing such as entangled electron pair generation by crossed Andreev reflection^{6} or ABSbased quantum bits^{8}. Regarding the latter, our observation of tunable ABS is heartening even though the measured spectroscopic linewidth (30–40 μeV fullwidth at halfmaximum) seems to question the feasibility of such qubits (if it were intrinsic to the sample, it would correspond to subnanosecond coherence time). The present linewidth is however likely to be caused simply by spurious noise in the experimental setup. More investigations are needed to assess the potential of nanotube ABS as qubits.
Change history
17 November 2010
In the version of this Letter originally published online, there were several errors in Figure 1 and 'ϕ=(2e/h)Φ' should have read 'ϕ=(2e/ℏ)Φ' in the fourth paragraph of the text. These errors have now been corrected in all versions of the Letter.
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Acknowledgements
This work was partially supported by ANR project ANR07BLAN0240 SEMAFAET, C’Nano project SPLONA and Spanish MICINN under contracts NAN200729366 and FIS200804209. The authors gratefully acknowledge discussions with the Quantronics group, O. Auslaender, J. C. Cuevas, R. Egger, M. Grifoni, T. Kontos, H. le Sueur, A. MartínRodero, P. Simon and C. Strunk.
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JD.P. and C.Q.H.L. fabricated the sample. JD.P., C.Q.H.L. and P.J. designed and carried out the experiment. JD.P., C.Q.H.L., P.J., C.B. and A.L.Y. analysed the data and wrote the paper. P.M. provided the equipment for the nanotube synthesis.
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Pillet, JD., Quay, C., Morfin, P. et al. Andreev bound states in supercurrentcarrying carbon nanotubes revealed. Nature Phys 6, 965–969 (2010). https://doi.org/10.1038/nphys1811
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DOI: https://doi.org/10.1038/nphys1811
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