Nano-assembled open quantum dot nanotube devices

A pristine suspended carbon nanotube is a near ideal environment to host long-lived quantum states. For this reason, they have been used as the core element of qubits and to explore numerous condensed matter physics phenomena. One of the most advanced technique to realize complex carbon nanotube based quantum circuits relies on a mechanical integration of the nanotube into the circuit. Despite the high-quality and complexity of the fabricated circuits, the range of possible experiments was limited to the closed quantum dot regime. Here, by engineering a transparent metal-nanotube interface, we developed a technique that overcomes this limitation. We reliably reach the open quantum dot regime as demonstrated by measurements of Fabry-Perot interferences and Kondo physics in multiple devices. A circuit-nanotube alignment precision of ± 200 nm is demonstrated. Our technique allows to envision experiments requiring the combination of complex circuits and strongly coupled carbon nanotubes such as the realization of carbon nanotube superconducting qubits or ﬂ ux-mediated optomechanics experiments.


growth chips
The CNTs are grown on custom cantilever comb chips (cf.Supplementary Figure S1) which allows us to transfer CNTs suspended between cantilevers onto the circuit chips.An overview of such a cantilever chip is shown in Supplementary Figure S1.

Supplementary Note 2. E-beam induced hydrocarbon deposition minimization
The SEM imaging of the carbon nanotubes during the localization step is expected to lead to e-beam induced hydrocarbons (HC) deposition on the nanotube [1].The presence of HC at the Nevertheless in our situation we expect the HC deposition on the suspended nanotubes to be very low, and this for several reasons.It was shown that HC deposition is mainly due to the low energy (< 50 eV) secondary electrons (SE) [2] which result from the interaction of the primary electrons with atoms.Having a low number of SEs in the vicinity of the nanotube is thus a key point to minimize the HC deposition at its surface.In our situation, the section of the nanotube which is transferred into the circuit corresponds to the center of a 30µm long suspended nanotube.This means that the amount of atoms in the vicinity of the transferred section of the nanotube is extremely low.Consequently the production of SEs close to the transferred section of the nanotube is expected to be much lower than the one in the neighborhood of a nanotube lying on a substrate.
In addition, we use a relatively high acceleration voltage (8 kV), which results in a lower number of SE out of the interaction with the nanotube in comparison to a low acceleration voltage beam (1 − 2 kV) [1].In Supplementary Figure S2, we show that the acceleration voltage of the e-beam has a strong influence on the amount of HC deposited at the surface of a silicon substrate.One visible consequence is that the contrast between the suspended nanotube and the background is low, while the electron count on the cantilevers is completely saturated (white on SEM picture in Supplementary Figure S1).This indicates the low amount of SEs produced in the vicinity of the nanotube.
A second important factor is the amount of precursors species for HC formation, mainly coming from the residual vacuum and the surfaces in the chamber [1].We thus work at low pressure (P = 2.5 × 10 −6 mbar) and because the nanotube is suspended and is consequently far away from surfaces, one can expect a lower HC deposition on the nanotube.
The parameters used during the exposure are: • Acceleration voltage: 8 kV 3 Supplementary Note 3. Radiative thermal annealing (RTA)process and temperature estimation  SEM (degasing at 16 W, 5 min).For this purpose, the chip is moved far away from the e-beam column and the halogen lamp (cf.Supplementary Figure S3).The circuit is then placed about 5 mm below the lamp and the chip is radiatively heated at a lamp power of 5.6 W for a time varying from 25 to 100 min.A second heating step is performed at 11 W. The heating time for this step has been varied between 5 min and 45 min.The CNT junctions are characterized after each step once the chip has cooled down again.
Measuring the current through the CNT during RTA allows us to monitor the physical processes at the CNT-metal interface and to find the optimum duration which varies for each sample.In general, there is only little contact improvement below 5 min.For RTA beyond 30-45 min the measured current remains nearly constant for all devices independent of both powers.
We therefore consider 30-45 min to be the best RTA duration.A further heating at 16 W did not show any contact improvement and led to higher junction resistance in some cases.
For a better understanding of the RTA process it is essential to know the temperature at the surface of the circuit chip.Performing a standard temperature measurement using a thermocouple would be inaccurate because the thermal dissipation through the device substrate and the sample holder is radically different from the dissipation via the thermocouple wires.Therefore, in order to estimate the temperature at the surface of the circuit chip during RTA we relied on the melting temperature of aluminium bond wires.We micro-bonded aluminium bond wires across the chip and performed heating tests at different lamp powers inside the SEM at a pressure in the low 10 −5 mbar range and for 30 min each (cf.Supplementary Figure S4).At a power of 26.2 W the bonds still remained intact and only the surface is partially molten while they are completely molten at 29.5 W. Putting these values in reference with the melting temperature of bulk aluminium of 660 • C and assuming a linear relationship between lamp power and chip temperature we can roughly estimate the sample temperature to 200-300 • C at 11 W and 100-150 • C at 5.6 W. This temperature range is much smaller than the typical growth temperature of 800-900 • C for CNTs grown on top of metal electrodes.Because aluminium wire bonds are much larger than the typical dimensions of the circuit (30 µm width compared to about 1 µm for the circuit electrodes), and aluminium has one of the largest thermal conductance among the used metals, we expect the thermal dissipation to be stronger in the aluminium wire bonds than at the surface of the circuit electrodes.For this reason, our method to estimate the temperature only provides lower bounds of the real temperature at the surface of the circuit.Moreover, the exact surface temperature strongly depends on the used metal.
4 Supplementary Note 4. Further examples for gate dependence measurements after current-induced annealing and RTA We measured the gate dependence after current-induced annealing and after different radiative thermal annealing steps for more devices as well.All devices with a narrow gap in the gate dependence showed a behavior similar to the device shown in figure 4 in the main part.
Figure S5: Gate dependence of two other devices at a source-drain voltage V sd = 0.1 V before the RTA and after the two RTA steps.a. Device #9.b.Device #12.This device has only been measured at higher voltages than V sd = 0.1 V before the second RTA step and the current has been scaled to V sd = 0.1 V, assuming ohmic behavior.

Supplementary Note 5. Additional low temperature measurements
In this section we present further cryogenic measurements of different devices.Unless specified otherwise the measurements were carried out at the base temperature of the cryostat (30 mK).
These cryogenic measurements complement the ones from figure 5 in the main text and emphasize that we can reach highly transparent contacts between the CNT quantum dot and the leads in multiple devices (cf.Fabry-Pérot resonances (Supplementary Figure S7) and Kondo ridges (Supplementary Figure S8)), while the observation of fourfold shell filling in Supplementary Figure S6 demonstrates the cleanliness of our devices.In these measurements the differential conductance value is measured for the whole chip and includes the resistance of the circuit electrodes.The exact line resistances on chip vary with the electrode material but are usually in the range of few kΩ.6 Supplementary Note 6. Device details In Supplementary Table S1 we provide details about the devices presented in the main text and supplementary notes.Device #12 has been excluded in figure 4e because it has not been measured at V sd = 0.1 V before heating.7 Supplementary Note 7. Flux mediated single photon coupling estimation We consider a circuit where a SQUID (Superconducting Quantum Interference Device) built out of a carbon nanotube [3,4] is integrated in a microwave cavity similarly to what has been realized with aluminium suspended beams [5,6].In such systems, the photon energy depends on the mechanical oscillations of the carbon nanotube via the modulation of the magnetic flux threading the SQUID loop.The flux-dependent resonance frequency of the cavity reads: Where L and C are the inductance and capacitance of the bare cavity respectively, L S is the inductance of the SQUID, ϕ S = Φ S /2π is the reduced magnetic flux threading the SQUID loop, ϕ 0 = Φ 0 /2π = ℏ/2e is the reduced magnetic flux quantum, and I C,JJ is the critical current of a single Josephson junction.Here we neglected the linear inductance of the arms of the SQUID.
The single-photon, single-phonon optomechanical coupling rate resulting from this interaction is: where γ is a geometrical factor taking into account the shape of the mechanical mode, which is of the order of 1. B ∥ is the in-plane magnetic field, considering a CNT oscillating out-of-plane.
l is the length of the suspended CNT, and x ZP F is the zero-point fluctuations of the CNT mechanical resonator.
Importantly, the critical current of a Josephson junction made out of a CNT is typically of 1-30 nA, which means that the current i ZP F in the cavity central conductor associated to a single photon in a standard geometry can switch the SQUID to its resistive state.To prevent the switching of the SQUID junctions, one can either reduce i ZP F by increasing the impedance of the bare cavity, or position the SQUID away from the current anti-node of the cavity to reduce the microwave current perceived by the SQUID junctions.Combining these two strategies, and using the model developed in [7], one obtains a single-photon optomechanical coupling g 0 = 1 MHz, while maintaining a critical photon number n C ≃ 10.For this estimation, we used the following

Figure S1 :
Figure S1: Overview of a cantilever chip with the panorama scanning electron microscope (SEM) image taken to locate the suspended CNTs (marked by arrows).The left CNT has been integrated into device #4.The chip measures 3 mm in width and the scale bar measures 10 µm.

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Figure S2: The lower amount of HC deposition the higher is the acceleration voltage is due to the fact that secondary electrons are produced deeper under the surface of the substrate at high acceleration voltage.

Figure S3 :
Figure S3: Image of the radiative thermal annealing setup inside the SEM with a. e-beam column, b.Ta shield and c. halogen lamp with filament.

Figure S4 :
Figure S4: SEM image of a circuit chip with aluminium bond wires.a.Before RTA, b.After RTA at 26.2 W, c.After RTA at 29.5 W. Scale bar: 50 µm.

Figure S6 :
Figure S6: Double quantum dot charge stability diagram at source-drain voltage V sd ≈ 0 V for device #11 showing clear four fold shell filling.The second and fourth gate out of five gates below the suspended CNT are varied while V g1 = V g3 = −2 V and V g5 is floating.The device is similar to the one presented in figure 2a in the main text.The measurement is performed at 30 mK.

Figure S7 :
Figure S7: Source-drain bias V sd vs. gate voltage V g conductance map of device #3 with Fabry-Pérot resonances for positive gate voltage, measured at 150 mK.

Figure S8 :
FigureS8: Source-drain bias V sd vs. gate voltage V g conductance map with a zoom into the negative gate voltage range of the map shown in figure5ain the main text (device #4).Kondo ridges are visible between every second diamond and towards -3 V we can observe a slight transition from SU(2) to SU(4) Kondo effect.The measurement is performed at 30 mK.

Table S1 :
Overview of the devices presented in figure4in the main text and in the supplementary notes.t Ar denotes the time of argon milling, t air the time of air exposure between argon milling and loading of the sample in the SEM, t heat1 the radiative thermal annealing time at 5.6 W and t heat2 the time at 11 W.