Supplementary Information

Supplementary Information Quantum supercurrent transistors in carbon nanotubes Pablo Jarillo-Herrero, Jorden A. van Dam, Leo P. Kouwenhoven Device Fabrication The nanotubes were grown by chemical vapour deposition (CVD) 1 on degenerately p- doped silicon wafers (heavy p-type, or n-type, doping is necessary to make the substrate conductive at low temperature so that it acts as a backgate) with 25nm thermally grown oxide. For the catalyst, 4mg of Fe(NO 3 ) 3 9H 2 O, 2mg of MoO 2 (acac) 2 (SigmaAldrich), and 3mg of Alumina nanoparticles (Degussa Aluminum Oxide C) were mixed in 3ml of methanol and sonicated for ~1hr. The resulting liquid catalyst is deposited onto the substrate with.5μm 2 openings in the PMMA resist and blown dry. After lift-off in acetone, the substrate with patterned catalyst is placed in a 1-inch quartz tube furnace and the CVD is carried out at 9 C with 7sccm H 2, 52sccm CH 4 for 1 min. Argon is flown during heating up and cooling down. The methane and hydrogen flows have been optimised to obtain long and clean nanotubes (~1μm) without amorphous carbon deposition. The nanotubes are located by atomic force microscope (AFM) inspection and e-beam lithography is then carried out to pattern electrodes over the nanotubes. Our electrodes are customized for each device, and we typically choose straight and uniform sections of nanotubes in areas free of residues. The metal electrodes are deposited via e- beam evaporation and a typical thickness of 1nm Ti and 6nm Al is used. The Ti is used to have good contact to the CNT and the Al to have a value of the critical temperature well above the base temperature of our dilution refrigerator. After evaporation and liftoff, no further imaging of the devices is carried out: we have observed that AFM imaging after metal deposition can damage/perturb the devices, increasing their resistance. An image from our computer design program, showing the designed electrodes on top of the AFM picture of one of our CNT devices is shown in Fig. S1. (Note that the AFM markers are deformed due to the high temperatures during CVD growth). The electrodes contact the nanotubes with 1% yield and the devices are then probed to determine their metallic/semiconducting character and room temperature resistance. Only metallic nanotubes with room-temperature R<5kΩ are further studied in these experiments. Filtering system In this section we describe our filtering system. A filtering system is necessary to prevent electronic noise from reaching the sample (as much as possible) since this suppresses strongly the critical current. As mentioned in the main text we use three filters in series for each of the four measurement wires attached to a nanotube: a copper-powder filter (CuF), a π-filter and a two-stage RC filter (RCF). Three filters are used in order to cover the entire spectrum, from low frequency up to the microwave regime. CuFs are widely

used in dilution refrigerator measuring setups. They are typically used to suppress the high frequency noise (f 1GHz), lowering the effective electron temperature. Our CuF filters consist of ~1.5m long manganine wires, and give an attenuation 5 db at 1GHz. The π-filters cover the intermediate frequency range (~1 MHz - 2 GHz). The two-stage RC filters are useful in the range few khz 1 MHz and are widely used to measure small critical currents. The scheme of the RC filter can be seen in Fig. S2a, and a picture of the actual filter in Fig. S2b. The advantage of a two-stage versus a single stage RC filter is that it provides an attenuation of 4dB per decade (instead of 2dB/decade) above a certain cut-off frequency. An example of a used configuration is: R 1 = 82Ω, R 2 = 1.2kΩ; C 1 = 2nF, C 2 = 4.7nF, which gives a cut-off frequency in the ~1 khz range. Additional data & Fano resonances Here we show additional data from a different device. Figure S3 shows a dv/di versus (I,V G ) plot, similar to Fig. 2a of the main paper. Both multiple Andreev reflection and a modulation of the critical current as a function of gate voltage are clearly visible. In the center of the figure a sharp resonance can be seen, similar also to the sharp resonances in Fig. 2a. Such resonances have been observed and discussed in the context of carbon nanotubes strongly coupled to the leads 2, 3 and attributed to Fano resonances 4, although their origin have not been fully established. Two interfering channels are needed for Fano resonances to occur, a strongly coupled one and a weakly coupled one. The weakly coupled one can be an impurity, an inner shell in a mutiwall tube or a weakly coupled tube in a thin rope. It has also been suggested that intrinsic resonances may arise for individual single wall tubes 3, due to an asymmetric coupling of the two orbital channels in carbon nanotubes. We have examined the diameters of our samples and they are in the 2 to 7nm range. While single wall nanotubes of 2-3 nm are usually obtained with our CVD growth method 1, a diameter like 7nm is more rare. We note however that single wall NTs grown by CVD up to 13nm in diameter have been reported 5, 6. Therefore it cannot entirely be excluded that the Fano resonances are due to the fact that those tubes measured are not individual single wall tubes. Nevertheless, the conductance of our devices in the normal state gets very close to, but doesn t exceed, 4e 2 /h, similar to Ref. [3]. Future studies are necessary to clarify the precise origin of the Fano resonances. Multiple Andreev Reflection In order to visualize the regions of supercurrent flow, figure 2 (main text) shows differential resistance plots as a function of current bias. Because of this, the multiple Andreev reflection (MAR) lines move up and down along the plots (e.g., Fig. 2a,c). Figure S4 shows the differential conductance, di/dv (in log scale), versus measured source-drain voltage and gate voltage (black/dark red is low di/dv and yellow is high di/dv, the white features at low V are due to the conversion from current biased to voltage biased near supercurrent). The features in Fig. S4 are in good agreement with previous MAR results in carbon nanotubes 7. For example, the Andreev reflection peaks at 2Δ g and 2Δ g /2 are clearly visible OFF resonance, while they become smeared ON resonance.

Also, as predicted theoretically 7-9, the subgap structure becomes very complex in the vicinity of the resonances. It is worth noting that, depending on the resonance studied, we observe 5-7 MAR peaks, whereas in ref. 7 only 2-4 MAR peaks are observed. This may be due to insufficient noise filtering (no supercurrent was observed in ref. 7) or to a shorter inelastic scattering length in the multiwall nanotube studied in ref. 7, preventing the observation of higher order MAR processes. Magnetic field dependence The application of a magnetic field, B, suppresses superconductivity in the electrodes and, thus, suppresses the proximity effect associated-features in the transport through the nanotube. As an example we show in Figs. S5a,b the suppression of MAR and I C with B (shown for the device in Fig. S2; other devices exhibit the same behaviour). Figure S5a corresponds to the ON resonance case, while fig. S5b corresponds to the OFF resonance case. As mentioned in the main text, the differential resistance in the ON-resonance case is lower when the leads are superconducting. On the other hand, in the OFF-resonance case, there is a large peak in dv/di at low energies. [Note that the vertical scale is very different for the two figures.] Similar peaks in dv/di (although smaller in magnitude) have been observed previously 1 and attributed to electron-electron interactions. While we cannot rule out such effects (for example a small Coulomb interaction effect), it has been shown 7 that a non-interacting model which takes into account only a resonant level in between two superconducting leads, can yield also such an enhancement of the differential resistance at low energies in the OFF-resonance case. A more detailed study, both theoretical and experimental, should shed light on the relative importance of each of the possible effects accounting for these peaks. Quality factor and how to increase the switching current As we mention in the main text, the behaviour of our CNT devices is reminiscent of that of small, underdamped, current-biased Josephson unctions. Whether a Josephson unction is underdamped or overdamped can be estimated by calculating the quality factor, Q, for the unction, taking into account its electromagnetic environment. Fig. S6 shows the electrical circuit representing our device. The circuit is basically an extension of the familiar RCSJ model which includes also the environment, represented by a parallel capacitor C and a series resistor R. The unction itself is represented by an ideal Josephson unction with a capacitor C and a resistor R in parallel. The unction is biased by a current I and we measure the voltage V. V is the voltage across the unction. The circuit is described by the following equations: V I = CV + C V + I sinδ + (1) R V = V + R C V h V = δ 2e V + + I sinδ (2) R (3)

where δ is the phase difference across the unction, I its critical current, and we have made use of the Josephson relations. Differentiating and substituting eqs. (2) and (3) in (1) we arrive to the following differential equation: h R C h 1 h 1 + + C δ + I RC + + RCC + I = I e R cosδ δ δ sinδ 2 2e R 2e which represents the dynamics of the phase and can be intuitively viewed as the motion of a particle of mass [C(1 + R/R ) + C )] in a tilted washboard potential 2e/ħ(I sin(x)-i) in the presence of friction. The plasma frequency for small oscillations is given by and the quality factor given by w p = 2eI R h C 1 + R + C Q = w p 1 h 1 RC + 2e I R In early experiments (11,12) on superconducting SETs as well as in our CNT QD experiments, the electrodes contacting the island are superconducting, therefore R is very small (ideally zero), resulting in high quality factors and a strong supression of the measured critical current. For our devices, we estimate: C ~ few ff (effective capacitance of the QD), R 2 kω (diff. resistance at the switching current value), I ~ 6 na. C in our case is mainly given by the capacitance between the bonding pads (with area = (3μm) 2 ), which is difficult to estimate. An upper bound is given by a parallel plate capacitor formula C ~ 1 pf (for 25 nm SiO 2 thickness), but the actual number is probably much lower, in the 1 ff to 1 pf range. R is also hard to estimate, since it should be in principle zero (our four-probe wires consist of two pairs of bonding wires bonded on top of two superconducting bonding pads-leads, which contact the CNT). In any case, for values of R from zero to ~ 2 Ω, the Q factors we obtain are Q 5, consistent with having an underdamped unction and, consequently, a strongly supressed critical current. Later experiments (13) performed with superconducting SETs intentionally increased R to several hundred Ω (by inserting non-superconducting elements in the leads) and also inserted on-chip capacitors to increase C. This resulted in critical currents much higher, aproaching the theoretical limit. We expect that the introduction of a controlled electromagnetic environment in a similar manner in superconducting circuits incorporating CNT QDs will also result in increased measured critical currents.

Charging effects: Electronic transport through a quantum dot can be classified into three categories depending on the ratio of the dot coupling to the leads, represented by Γ, with respect to the charging energy, U: i) hγ << U (Closed QD regime) Charging effects dominate transport (Coulomb blockade) ii) hγ U (Intermediate transparency regime) Charging effects important, but higher-order tunneling processes significant too (cotunneling and Kondo effect). iii) hγ >> U (Open QD regime) Quantum interference of non-interacting electrons. Our nanotube devices operate in the open QD regime. Of course, in order to be able to speak of QDs, the condition ΔE > hγ must be fulfilled, where ΔE is the energy spacing between different levels (CNT shells in our case). The conductance, G, vs gate voltage V G, looks very different in each of these three regimes. In regime (i) G vs V G displays sharp Coulomb peaks with G max << 2e 2 /h, spaced by the addition energy, E add = U + ΔE (multiplied by the gate voltage to bias conversion factor). The charging energy is typically large and the Coulomb diamonds are well-resolved. In regime (ii), G vs V G also exhibits Coulomb peaks, but these are typically broadened, with G max 2e 2 /h, and asymmetric due to Kondo effect in the odd N side of the peak, where N is the number of electrons in the QD. The charging energies are much smaller than in (i) and the Coulomb diamonds look fuzzy. For nanotubes one typically finds groups of four closely spaced peaks separated by a large gap from the next group of four peaks (see, for example, refs. 15, 24). In regime (iii), G vs V G exhibits a series of broad resonances, with G max 4e 2 /h (reflecting the two modes of the CNT band structure), symmetric and with a spacing proportional to the energy level spacing (because the charging energy is negligible). In our case, these last conditions are met: the energy level spacing calculated from Fig. 1b agrees with the nanotube length in between electrodes; and the maximum conductance is very close to 4e 2 /h. In regime (i), N changes by 1 when varying the gate voltage across a Coulomb peak. In regime (iii), however, the average number of electrons in the CNT QD is changed by 4 electrons across each resonance (average because, since the QD is open, charge fluctuations are important). This can be viewed as shrinking the 4 closely spaced peaks (e.g., in ref. 24) until they form a single resonance, while keeping the gate voltage spacing in between groups of four peaks constant. This is our experimental situation. It becomes clear then that, if charging effects were relevant for our experiment, each conductance resonance (and the corresponding max. supercurrent resonance) should be split in four closely spaced resonances. Moreover, the conductance in that case would not be higher than 2e 2 /h. Note that the gate voltage spacing between critical current resonances is the same as the spacing between conductance resonances in the normal state, which means that they stem from the same origin, and are not due to charging effects. Although we cannot give an estimate of the smallness of U, from the

fact that we do not observe either in the normal or superconducting state any signatures of charging effects, we conclude that these are negligible. References: 1. Kong, J., Soh, H. T., Cassell, A. M., Quate, C. F. & Dai, H. J. Synthesis of individual single-walled carbon nanotubes on patterned silicon wafers. Nature 395, 878-881 (1998). 2. Zhang, Z., Dikin, D. A., Ruoff, R. S. & Chandrasekhar, V. Conduction in carbon nanotubes through metastable resonant states. Europhys. Lett. 68, 713-719 (24). 3. Babic, B. & Schonenberger, C. Observation of Fano resonances in single-wall carbon nanotubes. Phys. Rev. B 7, 19548 (24). 4. Fano, U. Effects of Configuration Interaction on Intensities and Phase Shifts. Phys. Rev. 124, 1866-1878 (1961). 5. Li, Y. M. et al. Growth of single-walled carbon nanotubes from discrete catalytic nanoparticles of various sizes. J. Phys. Chem. B 15, 11424-11431 (21). 6. Cheung, C. L., Kurtz, A., Park, H. & Lieber, C. M. Diameter-controlled synthesis of carbon nanotubes. J. Phys. Chem. B 16, 2429-2433 (22). 7. Buitelaar, M. R. et al. Multiple Andreev reflections in a carbon nanotube quantum dot. Phys. Rev. Lett. 91, 575 (23). 8. Levy Yeyati, A., Cuevas, J. C., López-Dávalos, A. & Martín-Rodero, A. Resonant tunneling through a small quantum dot coupled to superconducting leads. Phys. Rev. B 55, R6137-R614 (1997). 9. Johansson, G., Bratus, E. N., Shumeiko, V. S. & Wendin, G. Resonant multiple Andreev reflections in mesoscopic superconducting unctions. Phys. Rev. B 6, 1382-1393 (1999).

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AFM markers Figure S1. Jarillo-Herrero et al. Designed electrodes Nanotube 1 µm

a b 1cm R 1 R 2 C 1 C 2 R 1 C 1 R 2 C 2 Figure S2. Jarillo-Herrero et al.

5 25-25 -5-2. -1.5-1. -.5 V G (V) Figure S3. Jarillo-Herrero et al. I (na)

25 125-125 -25 ON OFF ON OFF ON OFF -3-2.75-2.5 V G (V) Figure S4. Jarillo-Herrero et al. 2 g 2 g /2 V ( µ V)

a b 3 5 ON OFF 25 15 I (na) I (na) -25-15 -5-2 -1 1 2 B (mt) -3-2 -1 1 2 B (mt) 3 6 9 12 dv/di (kω) 2 4 6 dv/di (kω) Figure S5. Jarillo-Herrero et al.

R I V C V C R Figure S6. Jarillo-Herrero et al.