Introduction
The superconducting quantum interference device (SQUID) has been used very successfully for magnetometry and voltage or current measurements in the fields of medicine, metrology and other fields of research1, 2. It combines two quantum properties of superconductivity: the tunnelling of Cooper pairs through a nonsuperconducting medium (the Josephson effect3) and flux quantization in a superconducting loop. The direct-current version of this device (DC-SQUID) is composed of a superconducting loop having two Josephson junctions. Its most striking property is that the maximum superconducting current flowing through the device can be periodically modulated by the magnetic flux entering the loop4, with a period equal to the flux quantum. Recently, miniaturized versions of these devices have been used to implement phase qubits5 or to measure quantum magnetization reversal of nanoparticles6 and single-molecule magnets7.
There has also been enormous progress in the understanding of the electronic and magnetic properties of isolated molecular systems, revealing intriguing new physics. In the early experiments, the measuring devices were composed of two nanoelectrodes and a bridging molecule, allowing the measurement of the electronic transport through single molecules8. In such devices, measurements are performed at the molecular level, the observables are referred to molecular orbitals, and not to Bloch waves, as in massive materials. New rules were then found for these systems and it became possible to directly probe the quantum properties of the molecule. For example, the tunnelling of electrons through molecular junctions can show the presence of Coulomb-blockade8, Kondo coupling9, 10 and shell filling11, depending on the intensity of coupling between the molecule and the electrodes. Recent progress in improving the contact of CNTs to superconducting nanoelectrodes has shown that such molecules can accommodate a superconducting current12 and behave as gate-controlled Josephson junctions13, 14. These CNT junctions are in the quantum-confined regime and can thus act as gate-controlled QDs, that is, devices reminiscent of the well-studied superconducting single-electron transistors15, but with strong quantum confinement. It was also predicted that a reverse superconducting current (Josephson current) would take place in a junction involving tunnelling through a QD populated with an odd number of electrons16, 17, 18, 19, 20, 21. Such a junction is also called a 
-junction because the spin ordering of the Cooper pair tunnelling through the junction is reversed; that is, the Cooper pair wave function is multiplied by a phase factor ei
.
We combine here the research area of SQUIDs with that of electronic transport through molecules in order to design a new detector: a SQUID with molecular Josephson junctions made out of CNTs. This system allows us to address the problem of resonant tunnelling through a QD having discrete energy levels and coupled to superconducting electrodes. The SQUID has the advantage of linking the phase across both junctions with the magnetic flux entering the loop. It then yields insight into the current–phase relation across a QD coupled to superconductors. In particular, a gate-controlled transition from the normal to the -junction can be observed. Owing to the geometrical aspects of CNTs, such SQUIDs are also very promising in the study of spin states of an individual magnetic molecule placed on one of the two CNT Josephson junctions.
Results and discussion
SQUID design and CNT junction
We designed and built the first CNT-SQUID as presented in Fig. 1a and described in the Method section. The superconducting SQUID loop is made from Al, and has a critical temperature Tc 
1.2 K. If not stated differently, all measurements were performed at a cryostat temperature of about 35 mK. The CNT-SQUID contains two CNT-based superconducting transistors, which have been described previously13, 14 and can be modelled by a QD between two superconducting leads (Fig. 1b). The position of the quantum levels can be tuned with gate voltages VG1 and VG2. When a quantum level is aligned with respect to the Fermi energy of the superconducting leads ('on' state in Fig. 1b), a supercurrent of a few nanoamperes can flow by resonant tunnelling through the CNT. When the quantum levels are far from the Fermi energy ('off' state in Fig. 1b), the supercurrent is strongly reduced (typically by a factor of 10 to 103). However, we show in the following that the supercurrent is not completely suppressed. This effect, not seen before13, might be due to a Josephson effect via a non-zero electron density between the levels of the CNT and higher-order tunnel processes. Further studies are needed to understand this effect.
Figure 1: Device and operation scheme.
a, Typical device geometry of the CNT-SQUID with two lateral gates G1 and G2 (coloured in gold). The atomic force microscope (AFM) image shows the SQUID loop (grey), which is interrupted by the two CNT Josephson junctions with a length of about 200 nm. The single-walled CNT with a diameter of about 1 nm was located using an AFM, and Pd/Al (3/50 nm) aligned electrodes were deposited over the tube using electron-beam lithography. b, Energy level schematics of a QD between two superconducting leads. The energy level spacing 
E and the superconducting gap 2g are indicated. The position of the quantum levels can be tuned with a gate voltage VG. Only when a quantum level is adjusted to the Fermi energies of the superconducting leads (green curve) can a strong supercurrent flow between the leads. c, Schematics of the CNT-SQUID with two nanotube junctions, which can be tuned with the gate voltages VG1 and VG2. In case I, both junctions have a quantum level adjusted to the Fermi energy of the leads (on-resonance) and maximal supercurrent can flow through the device. In cases II and III, one and two junctions are tuned off-resonance, respectively.
Electronic transport properties and Kondo effect
Before turning our attention to the CNT-SQUIDs, we have to characterize the electronic transport properties of our CNT junctions. Particularly important is the interplay between Kondo correlations and superconductivity, which has recently motivated theoretical16, 17, 18, 19, 20, 21, 22 and experimental studies23, 24, 25. The Kondo correlations are due to a magnetic exchange interaction between a localized magnetic moment and free conduction electrons. In order to minimize the exchange energy, the conduction electrons tend to screen the magnetic moment and the ensemble forms a spin singlet. It has been predicted that a Kondo resonance in a spin 1/2 QD with an odd electron number can coexist with a superconducting state. Indeed, for strongly hybridized QD states, it has been theoretically predicted16 that the Josephson coupling could be enhanced by the Kondo resonance. This effect partially offsets the reduction in Josephson coupling due to the Coulomb repulsion energy Uc. The Kondo resonances of each QD can be studied independently by measuring the differential conductance dI/dV as a function of source–drain voltage Vsd and lateral gate voltage VG while keeping the chemical potential of the other QD at a constant value. Whereas Vsd shifts the Fermi energy of the left lead with respect to the right one, the lateral gate voltage VG tunes the position of the quantum levels with respect to both Fermi energies of the superconducting leads. The measured conductance map of dI/dV versus Vsd and VG exhibits the typical features of Coulomb diamonds (Fig. 2b), which are connected by Kondo 'ridges'23 of enhanced conductance. The enhanced conductance at zero bias results from the Kondo resonance when there are an odd number of electrons on the QD. Note that the Kondo ridges appear at the same gate voltages as the occurrence of a superconducting state (Fig. 2c). Moreover the maximum supercurrent coincides with the most prominent Kondo ridge (labelled 
in Fig. 2b), a fact that supports the enhancement of superconductivity by the Kondo resonance16. The tunnel rate 
1 meV/h (where h is Planck's constant), the Coulomb energy Uc 6 meV, and the energy-level spacing E 9 meV are obtained from the size and shape of the Coulomb diamonds. Using E = hvF/2kL, where vF = 8.1 
105 m s-1 is the Fermi velocity in the CNT, k = 1 for orbital and spin degeneracies, k = 2 for only spin degeneracy, and L is the length of the CNT. We obtained L = 186 nm for k = 1, which is in agreement with the length of 200 nm estimated using atomic force microscopy (AFM). This suggests that our CNTs have orbital and spin degeneracies. However, the gate dependence does not exhibit a clear fourfold symmetry as expected for such degenerate QDs11 but rather an even–odd behaviour26, which may have two origins. First, channel mixing might occur at the transparent contacts. More probable, however, is the influence of defects induced either by the interaction of the CNT with the substrate, or by structural imperfections inside the nanotube. These defects lower the symmetry of the system and thereby lift the orbital degeneracy27.
Figure 2: Correlation between Kondo effect and superconductivity.
a, Colour-scale representation of a differential conductivity dI/dV map measured at zero bias at 34 mK as a function of the lateral gate voltages VG1 and VG2 and at a backgate voltage VBG = 0 V. A magnetic field of Hz = 50 mT was applied perpendicular to the SQUID plane in order to suppress the superconductivity of the leads. The effect of cross-capacitance was subtracted in situ. The dotted line indicates the gate voltage range of VG1 that is further studied in b and c, whereas the dotted square encloses the region that is further studied in Fig. 5b and c. b, dI/dV map as a function of VG1 and source–drain voltage Vsd. VG2 = –6 V and Hz = 50 mT. Note that a and b have the same colour code for dI/dV. The dotted lines in b indicate the Coulomb diamonds, which are connected by Kondo ridges labelled 
, 
, 
and . The conductances in the middle of the Kondo ridges are 1.02, 0.72, 0.46 and 1.45 e2/h for , , and , respectively. (See Supplementary Information for a discussion of the determination of the Kondo temperature.) c, Differential resistivity dV/dI map of the same VG1 region and for VG2 = -6 V, but for Hz = 0. A supercurrent is observed in the black regions, which correspond to the Kondo ridges in b.
CNT-SQUID characteristics
The operation of the CNT-SQUID is based on the quantum phase interference of the supercurrent flowing through two CNT-based superconducting transistors13 in a superconducting ring (Fig. 1c). The position of the quantum levels in each junction can be individually tuned using the two lateral gate voltages VG1 and VG2, and the transparency of the CNT contact barriers can be globally adjusted using the backgate voltage VBG.
In order to find the gate voltages required to adjust the levels of each CNT junction with respect to the Fermi energy of their contacts, we measured the differential conductance dI/dV at zero bias as a function of VG1, VG2 and VBG in a field of Hz = 50 mT to drive the superconducting leads to the normal state (Figs. 2a and 3a). The maps follow the stability diagram of two QDs in parallel, that is, a chequerboard pattern with high-conductance states having values of the order 4e2/h (where e2/h is the conductance quantum) centred on points where both CNTs are on-resonance, and low-conductance states (typically between 0.1 and 0.5 e2/h), where both CNTs are off-resonance.
Figure 3: Correlation between normal-state conductance and superconducting switching current.
a, Colour-scale representation of a typical differential conductivity dI/dV map at 34 mK and a backgate voltage VBG = -6 V as a function of the lateral gate voltages VG1 and VG2. A magnetic field of Hz = 50 mT was applied perpendicular to the SQUID plane in order to suppress the superconductivity of the leads. The effect of cross-capacitance was subtracted in situ. b,c, Colour-scale representations of the switching current Isw at 30 mK as a function of VG1 and VG2 at the same backgate voltage. Magnetic fields Hz = 0 and 1.3 mT were applied in b and c, respectively. The latter corresponds to half a flux quantum (h/4e). Isw is maximal or minimal when both junctions are tuned on- or off-resonance, respectively (the three cases I, II and III of Fig. 1c are indicated). Each pixel corresponds to a single measurement of Isw.
Full size image (27 KB)At small applied fields, the CNT junctions transmit a supercurrent. We define the switching current Isw as the maximum dissipationless current that the Josephson junctions can pass through the device. Isw is measured by ramping the current at a constant sweep rate from zero to Isw, at which point a voltage drop is measured across the junction. Fig. 3b,c presents Isw as a function of VG1 and VG2 in zero field and in a perpendicular magnetic field Hz = 1.3 mT corresponding to a magnetic flux, penetrating the SQUID loop, of 
e = 0/2 = h/4e where 0 is the magnetic flux quantum. Comparing the dI/dV map (Fig. 3a) with the Isw map (Fig. 3b,c) shows that Isw is maximal (about 6 nA) at maximal conductance in the normal state, that is, when both CNTs are on-resonance. Isw is about two times smaller when only one CNT is on-resonance. Typical voltage versus current characteristics and field modulations of Isw are presented in Fig. 4. The overall flux modulation of the CNT-SQUID can be understood by using a standard SQUID model2, which predicts the field dependence of the maximal supercurrent, called the critical current Ic:
Figure 4: CNT-SQUID characteristics.
a,c, Voltage versus current curves at three applied fields corresponding to zero (red), a quarter (green) and a half (blue) flux quantum for the situations when both junctions are tuned on-resonance (a) or off-resonance (c) a and c correspond to maximal and minimal switching currents in Figs. 3b and 2a, respectively. The current was swept at sweep rates of 0.5 nA s-1 and 5 pA s-1 for a and c, respectively. Note that each curve is a single sweep (no data averaging was performed). b,d, Field modulation of the switching current Isw for the situations in a and c, respectively. Each point in b and d corresponds to a single measurement taken at frequencies of 1 and 0.1 Hz, respectively. The field periodicity times the area of the SQUID loop is in good agreement with a flux quantum (h/2e).
Full size image (54 KB)Here, Ici are the critical currents of each Josephson junction (i = 1 and 2), which can be tuned independently with the gate voltages VG1 and VG2. Ic is an upper bound of the experimentally observed switching current Isw, because the former neglects environmental effects like temperature, electronic noise and quantum effects. The field modulation depth is maximal when Ic1 and Ic2 are comparable (Figs. 3 and 4), but it is minimal when the difference between both current magnitudes is maximal. For short junctions, the maximal supercurrent at zero temperature for a single-channel junction is given by Ici = eg/
, where 2g is the superconducting gap and is the reduced Planck constant. Because of proximity effects between the Al and the Pd contact layers, the effective gap of the contacting bilayer is reduced to 2g 0.12 meV (data extracted from the temperature dependence of dI/dV(T); see (see Supplementary Information). We yield Ici = 15 nA, which is further reduced when the asymmetry of the barrier is taken into account28. Experimentally, we do not measure Ici, but a maximal switching current Isw 3 nA per junction for the device presented here. The highest value achieved so far with our single-walled CNT junction is Isw 5 nA. The discrepancy between Ici and Isw is due to the fact that the latter is a stochastic variable influenced by temperature, electronic noise and quantum effects. The effect of the electromagnetic environment can drastically reduce Isw, especially for small, underdamped, current-biased Josephson junctions29. Another manifestation of the electromagnetic environment is that the superconducting branch exhibits a small bending towards a non-zero voltage (Fig. 4a,c) due to retrapping effects30.
CNT 
-junction
A closer investigation of the field modulation shows strong deviations from equation (1). In particular, when the contact barriers are increased with the backgate voltage, a reduced Kondo effect is found (see Supplementary Information) and, under certain gate voltages, the contrast of the interference fringes drops to zero and finally exhibits a phase shift, leading to a minimum switching current at zero field (Fig. 5). Such an effect, in which the minimum energy state of one of the Josephson junctions (the so-called -junction) is obtained for a phase difference of instead of zero, has been the object of intense studies in the past decade. It has been reported experimentally as a consequence of d-wave superconductivity31, when tunnelling occurs through a ferromagnetic layer32, 33. Local control of the sign of a Josephson current was also proved possible34 by tuning the local density of state of an SNS junction35. Reminiscent of a ferromagnetic impurity in a Josephson junction, it was predicted that a reverse Josephson current would take place in a junction involving tunnelling through a QD populated with an odd number of electrons16, 17, 18, 19, 20, 21. For a strong Kondo effect (Fig. 3), the Josephson coupling is expected to be positive (0-junction) because the localized spin is screened due to the Kondo effect. On the other hand, for a weak Kondo effect (Fig. 5), the large on-site interaction only allows the electrons in a Cooper pair to tunnel one by one via virtual processes in which the spin ordering of the Cooper pair is reversed, leading to a negative Josephson coupling (-junction).
Figure 5: Gate-controlled 
-junction CNT-SQUID characteristics.
![Figure 5 : Gate-controlled |[pi]|-junction CNT-SQUID characteristics.](/nnano/journal/v1/n1/thumbs/nnano.2006.54-f5.jpg)
a, Field modulation of the switching current Isw for a grounded backgate (VBG = 0 V), where a weak Kondo coupling was observed. Using the lateral gate voltage VG1 while keeping the gate voltage VG2 at 12.5 V, the flux modulation can be driven from an even (0-junction SQUID, black) to an odd (-junction SQUID, blue) curve. In the vicinity of the phase reversal, the field modulation exhibits a clear distortion from a sine dependence, suggesting a peculiar current phase relation in that case. Each point corresponds to a single Isw measurement (no averaging). b–e, Colour-scale representations of the maximum switching current Isw (b and d) and normalized field modulation maps (c and e) as a function of the lateral gate voltages VG1 and VG2 for two different backgate voltages VBG = 0 (a and b) and 1 V (d and e). Each pixel corresponds to a single measurement of Isw. The VG1–VG2 zone in b and c is indicated by the dotted square in Fig. 2a. Red and blue regions in c and e correspond to even (0) and odd () flux modulation, respectively. b,c and d,e show the situations when only one or two junctions are tuned via a -transition, respectively. Note that in the centre of e, both junctions have a -shift, leading to an even flux modulation. The sizes and positions of the corresponding Kondo ridges in the normal state (see Supplementary Information) are schematically indicated by dotted lines in c and e.
This prediction can be demonstrated with our CNT-SQUID because it links the phase across both junctions to the magnetic flux entering the loop. When one of the two junctions has a negative Josephson coupling, the flux modulation is odd, that is, Isw has a minimum at zero field. However, when both junctions have a positive or a negative Josephson coupling, the flux modulation is even, that is, Isw has a maximum at zero field. Because the number of electrons in the QD is odd in the middle of a Kondo ridge and even at the outside, a gate voltage allows us to vary the QD junction from a positive to a negative Josephson coupling. Fig. 5 details the situations when only one (Fig. 5b,c) or two (Fig. 5d,e) junctions are tuned via a -transition. Such CNT devices are therefore very promising in circumstances where gate-controlled phase-relations are needed. For example, theoretical works have shown that a -Josephson junction has a protection against decoherence36. Consequently, a -junction SQUID can be well suited to implementing a qubit with a long coherence time37.
Prospects for detecting single magnetic moments
In designing a CNT-SQUID, our motivation is to use it as a detector for magnetization switching of the magnetic moment of a single molecule. The aspect ratio of CNTs makes them ideal for coupling to single nanometre-sized objects. For example, it has been shown that a semiconducting nanotube properly functionalized and operated at the conduction threshold has the ability to sense the binding of a single molecule by electrostatic coupling38. In this paper, we present a device that we want to use to detect the switching of few magnetic moments. Indeed, SQUIDs are the most sensitive magnetic flux detectors2, 39. The magnetic flux variation is related to the magnetization change M associated with the reversal of magnetic moments: 
, where 
is the flux coupling factor determined by the geometry of the device and the sample. The flux sensitivity of a SQUID is limited by the quantum limit, which has been achieved experimentally by several groups2, 40. However, the high sensitivity of such SQUIDs cannot be used to detect the magnetization reversal of single nanoparticles or molecules because of the very small flux coupling factor 
. An improvement was achieved with planar microbridge DC-SQUIDs (Fig. 6a). A 3-nm particle was placed on top of a microbridge with a cross-section of 50 20 nm2 (Fig. 6b). The coupling was strong enough to study 103 magnetic moments41, 42. However, the flux coupling factor was rather poor (Fig. 6b). In the case of a CNT junction (Fig. 6c), a nanometre-sized molecule could be placed directly on the CNT, which has a cross-section of about 1 nm2. We therefore expect a nearly optimized coupling factor , because the molecule size and the junction cross-section are comparable (Fig. 6d). The precise calculation of is difficult in such a near-field situation for which the dipolar approximation is not valid. However, a rough estimate of the magnetic signal of a Mn12 molecule, sitting on the CNT, yields a flux variation of 10-4 flux quantum, which should be within the sensitivity of our measurements (see Supporting Information for the first estimation of the flux sensitivity of CNT-SQUIDs).
Figure 6: Schematics of single-molecule studies using CNT-SQUIDs.
a, Schematic of a planar microbridge DC-SQUID on which a ferromagnetic particle is placed. The SQUID detects the flux through its loop, produced by the sample magnetization. b, Cross-section (50 20 nm2) of a microbridge junction on which a 3-nm particle is placed. The magnetic field lines are shown in green. The flux coupling is poor because of the large mismatch between the particle size and the junction cross-section. c, Schematic of one of the two junctions of a CNT-SQUID separating the superconducting (SC) leads. A nanometre-sized molecule sits on top of the CNT. d, Cross-section (1 nm2) of a CNT junction on which a 0.6-nm molecule is placed. The flux coupling is optimized because the molecule size and the junction cross-section are comparable.
In order to detect rapidly the switching of magnetization, we often use the SQUID as a threshold detector41, 42. This method, called the cold mode, consists in biasing the SQUID close to the switching current. The magnetization reversal triggers a transition of the SQUID from the superconducting to the normal state because of the fast magnetization change and/or the associated dissipation. A dV/dt pulse can then be easily detected on the current lead biasing the SQUID because of the hysteretic behaviour of the SQUID. The sensitivity of the cold mode scales roughly with the inverse of the switching current; we therefore expect a strong improvement with the CNT-SQUIDs, which have a tunable switching current in the nano and picoampere regions instead of the microampere region of previous SQUIDs41, 42.
Another important feature of the CNT-SQUIDs concerns the ability to tune the coupling between the detector and sample. Indeed, the supercurrent through the junction can be switched on and off using the lateral gate. In the off state, the magnetic molecule is decoupled from the measuring device and it can evolve without decoherence coming from the device. In order to measure the magnetization state of the molecule, the SQUID is then switched on. This should have important consequences as it allows us to limit the back-action of the CNT-SQUID on the quantum state of a single molecule magnet.
Further improvement of the flux sensitivity could be achieved with a new readout scheme, which probes the SQUID's nonlinear inductance rather than its resistance43. The SQUID is driven with a microwave frequency a.c. current near a bifurcation point where two oscillation states exist44. These superconducting dynamical states differ in amplitude and phase, and are associated with zero d.c. voltage. They can be distinguished by measuring a change in amplitude or phase of the a.c. voltage across the SQUID43. In this non-dissipative, dispersive SQUID magnetometer, the switching from one dynamical state to the other would signal a change in spin state of the molecule. As the working temperature of our devices was limited to a few hundred millikelvin, further improvements could be achieved by using other superconducting materials with higher Tc, such as Nb, Sn and NbTi.
In conclusion, the CNT-SQUIDs provide a new generation of ultrasensitive magnetometers of nanometre-sized samples. Such devices also offer the opportunity to test interesting physical phenomena ranging from Kondo physics to -junctions, and pave the way for non-locality experiments by generating pairs of entangled electrons in a nanotube45, 46, 47.
Methods
In order to build a CNT-SQUID as presented in Fig. 1a, we started from a degenerately n-doped silicon substrate with a 350-nm-thick thermally grown SiO2 layer on top, which was used as a backgate. Single-walled CNTs were prepared by the laser vaporization method48 at Rice University. They were dispersed in water by sonication using sodium dodecyl sulphate as surfactant. The CNTs were deposited using a combing technique, which allows good control of the CNT density and orientation on the substrate49. The silica surface was first functionalized using a standard silanization technique, leading to a self-assembled monolayer of aminopropyltriethoxysilane (Aldrich). The substrate was then dipped for 5 min in the dispersion of CNTs and withdrawn at a constant velocity of 200 
m s-1. The sample was thoroughly washed in distilled water in order to remove the surfactant from the nanotubes. The nanotube location was imaged by AFM, and aligned e-beam lithography was carried out to pattern the SQUID loops and the contacts. The fork geometry for the loop allowed us to fabricate both junctions from the same nanotube. Metal electrodes were deposited using electron-gun evaporation and a thickness of 3 nm Pd followed by 50 nm Al was used. Pd provides high-transparency contacts to the CNTs50. Al is a superconductor widely used in nanoscale devices, having a critical temperature of about 1.2 K. Only devices with resistance below 30 k
and no significant gate effect at room temperature were used for our studies. In addition to the backgate, two lateral gates G1 and G2 were aligned to each device, allowing us to tune independently the electronic properties of each CNT junction (Fig. 1a).
We fabricated about 100 CNT-SQUIDs and 300 CNT superconducting transistors using CNTs, ropes of CNTs and multiwalled CNTs. Only devices having individual CNTs are presented here. About 30% of all devices worked similarly to the presented one. Fabrication failure was caused mainly because of misalignment of the contact pads and lateral gates, low-conductance nanotube–metal contacts, and semiconducting CNTs. The properties of the superconducting contacts, and our filtering system are discussed in the Supplementary Information.
Author contributions
J.-P.C. fabricated the devices, and W.W. conceived and performed the experiments with help from J.-P.C. and V.B. All authors discussed the results and commented on the manuscript.
