Artificial two-level systems such as Josephson junction qubits coupled to superconducting microwave cavities1,2 have allowed for unprecedented break-throughs in quantum information processing. Inspired by these results, other solid state systems have been coupled to resonant circuits, such as double quantum dots in semiconductors3,4,5,6,7 as potential qubits, and molecules or impurity spin ensembles8,9,10,11 as quantum memories. In particular, carbon nanotubes (CNT) have recently demonstrated their potential as low-disorder one-dimensional electron systems12,13,14, which have been used to probe the physics of spin-orbit15 and electron-phonon coupling16, as well as to perform initialization and manipulation of spin qubits17. CNTs suspended over local gates not only offer a decoupling from the surface but also mechanical resonances with high quality factors18, creation of arbitrary local potentials and tight confinement with charging energies in excess of 50 meV (ref. 14).

Nevertheless, coupling CNTs to half-wave resonators7,19,20 and other types of LC circuits21,22,23,24 faces a significant challenge: achieving a low microwave loss device while preserving the ideal transport characteristics of pristine CNTs in a geometry, which allows full control over charge confinement. While CNT devices with clean transport spectra can be obtained using a growth-last approach25, the low yield and high temperatures (900 °C) involved prohibit the use of this method for fabricating superconductor-CNT hybrid devices. Recently, the transfer of CNTs from a growth substrate to a separate microwave device, followed by standard lithography has been demonstrated26, solving issues with material compatibility but not addressing limited yield and unclean transport spectra. As such, the combination of low-loss microwave circuits with clean CNTs has not been achieved so far, impeding the way toward experiments involving the interplay of electrons in CNTs with microwave photons.

Here we couple a locally tunable suspended CNT quantum device to an impedance-matching circuit based on superconducting transmission lines. Different to previous works7,19,20 where half-wave resonators are employed for dispersive and a minimal invasive measurement, our circuit is aimed at providing an efficient channel to transfer (collect) microwave radiation into (from) a quantum device. In addition, the circuit offers bandwidths (BW) in the MHz range even for device impedances on the order of 1 MΩ. These features, on one hand, allow us to perform high-BW measurements for deducing both conductance and susceptance changes in the quantum device at GHz frequencies, and on another hand, provide near unity collection of emitted radiation power for fast shot noise measurements. Through a mechanical transfer process27, we place the CNT on the finished microwave device in the last step. Such a selective assembly technique13 allows us to address the mentioned fabrication and yield issues while obtaining clean transport spectra in combination with low microwave loss circuits. We employ local gates to demonstrate a high degree of control over the formation of double dots in the ambipolar regime and perform radio frequency (RF) measurements of the device susceptibility. We are able to tune the inter-dot coupling strength and extract values on the order of gigahertz using the phase response of the microwave circuit. By performing simultaneous measurements of resistance and complex impedance, we observe good quantitative agreement between direct current (DC) conductance and RF measurements.


Mechanical transfer of CNT

A key advance in this hybrid-CNT device is the implementation of a mechanical transfer at ambient conditions. The same marks the last step of sample fabrication. After the complete RF and DC circuitries are fabricated, the chip is bonded on a printed circuit board and cleaned with a weak argon plasma to remove oxides and fabrication residue from the contacts. We grow CNTs on the fork-like structures of a separate chip, which allow a resist- and ebeam radiation-free mechanical transfer of the CNTs to the source/drain electrodes on a pillar structure (inset Fig. 1a). Forks are aligned with the pillar using an optical microscope of a micro-manipulator setup and successful transfers are monitored through voltage biased (200 mV) resistance measurements. We can then determine bandgap characteristics of transferred CNTs by measuring the conductance response to an applied gate voltage. Furthermore, we can remove tubes with unwanted characteristics, such as a metallic response, by applying a large bias voltage, and subsequently transfer and test another CNT on the same device. These capabilities make our method of transfer a deterministic one with unity device yield. Further details of the CNT growth and transfer method can be found in the Supplementary Note 2.

Figure 1: Device layout.
figure 1

(a) Design of an impedance-matching circuit based on two transmission lines with slightly different lengths l and d, both close to λ/4. The line d terminates in an open end while the central conductor of l connects to the source (S) of the CNT device. A false-colour SEM image is shown in b with left (LG), middle (MG) and right (RG) bottom gates. CNTs are transferred mechanically from fork structures to the bonded device chip in the final fabrication step, schematically shown in the inset of a. The sketch displays relative position of CNTs with fabricated gates and palladium contacts. The drain (D) is connected to the ground plane of the Niobium film, not visible here. (c) Equivalent matching circuit based on lumped elements. L and C are determined by the lengths l and d, while the phenomenological conductance GLoss captures the loss of the transmission lines. GCNT is the device conductance of the CNT.

Superconducting impedance-matching circuits

A transmission line based impedance-matching circuit, termed stub tuner28, can be realized by two transmission lines connected in parallel, with the device placed on one end, the other ending in an open circuit (Fig. 1a). Though simple lumped LC circuits can also be employed, they demand a lengthy iterative design procedure when used at frequencies of gigahertz due to stray capacitances of approximately femtofarads. By utilizing transmission lines, we are not only able to minimize parasitic effects, such as stray capacitances, but are also able to compensate for them, as they only change the effective length of the lines.

The microwave response of the stub tuner is mainly determined by the lengths of the two lines l and d, which are chosen such that a specific device impedance, called the matched load ZMatch, will be transformed to the characteristic impedance of the line Z0 at one specific resonance frequency. At this load, the reflection coefficient is minimized, implying maximum power transfer. To get the reflectance Γ, we calculate the impedance Zin of the hybrid device, consisting of a parallel combination of the two transmission lines28,29 as

γ=α+ is the propagation constant with α and being the loss and phase constants, respectively, the effective dielectric constant, c the speed of light, f the frequency, Z0 the transmission line impedance, and the complex admittance 1/ZCNT=Y=GCNT+iBCNT with GCNT conductance, and BCNT susceptance of the CNT device.

The circuit is sensitive to changes in both the conductance and susceptance, as shown in Fig. 2a, where we plot a set of reflection spectra for different load admittances Y, with lengths chosen for a resonance at 3 GHz and a matched load of 4.8 μS, assuming no losses. Varying GCNT from 3 to 4 μS increases both the depth and width of the resonance, but not the resonance frequency. In contrast, a change in the susceptance B mainly shifts the resonance, with a minor change in its depth. To illustrate the effects that losses have on this device, we can replace the circuit by an equivalent lossless one while introducing a phenomenological conductance GLoss parallel to the CNT load (Fig. 1c) and thus defining a total effective load Geff=GCNT+GLoss seen by the microwaves. In Fig. 2b, we plot Γ at the resonance as a function of Geff. The reflected signal is the smallest close to the matched load offering maximal power transfer. The latter is more important for measuring small noise-power emitted from the device. Due to the additional loss conductance GLoss, the CNT load which matches the circuit is smaller than GMatch and is given by . In particular, if GLoss>GMatch full matching is precluded and the reflection coefficient as a function of load becomes monotonic (see inset of Fig. 2b).

Figure 2: Calculated response of the matching circuit for different parameters.
figure 2

(a) Reflection spectra of a loss-less stub tuner with 4.8 μS matched load at 3 GHz for different loads. Changing the real part of the load changes the linewidth and depth of the resonance, while changing the imaginary part mainly shifts the resonance frequency. (b) The reflectance at the resonance frequency as a function of a total effective load (Geff=GCNT+GLoss) shows a non-monotonic response and the smallest reflected signal at the matched load. Its characteristic changes to a monotonic response for GLoss>GMatch as shown in the inset (same as the dashed box) for GLoss=6 μS.

The presented CNT device has three recessed local bottom gates with additional source and drain contacts elevated by 150 nm (Fig. 1b). Bottom gates are separated by ≈200 nm from each other and from the source drain contacts producing a suspended length of the CNT of ≈800 nm. The transmission lines are patterned on a Nb film and yield a resonance frequency ωr/2π=2.9 GHz. All measurements are performed at the base temperature of the cryostat 20 mK (Supplementary Note 3). The source is connected to the central conductor of the transmission line (l branch) while the drain is connected to the ground plane. For the characterization of the devices, the CNT is tuned into the bandgap to present an infinite resistance and hence an open end. By measuring the reflection spectrum, it is possible to extract the relevant parameters (l, d, α and ). The extracted loss α=[0.0074,0.0082] m−1 for probe power in the range [−110,−140] dBm corresponds to internal quality factors of 10,000 to 9,000 for an equivalent half-wave resonator, showing that we are able to achieve low-loss microwave circuits in combination with CNT devices. Due to parasitic inductances from the fabricated contacts, the matching circuit has a lower effective GMatch≈1.6 μS compared with GLoss≈3.2 μS and operates therefore in the internal-loss-dominated regime. Measurements of the matching circuit response can be found in the Supplementary Note 4, as well as the measurement of an additional sample where full matching is demonstrated (Supplementary Note 5).

Measurements of CNT double quantum dots

Room temperature DC characterization during the device assembly allows us to perform several CNT mechanical transfer trials on the same device and choose nanotubes based on their gate dependence. In particular for semiconducting nanotubes, we can use the bottom gates to locally shift the Fermi level above or below the valence and conduction bands of the nanotube. The latter allows for creating gate-defined confinement potentials along the nanotube and hence for tuning the location, size and number of quantum dots13. With no RF power applied to the stub tuner and the middle gate VMG=0 V, we first measure the charge stability diagram of the CNT device. A gate sweep using left (VLG) and right gates (VRG) at VSD=10 mV DC bias is shown in the Fig. 3a. The current response clearly displays the ambipolar behaviour of quantum dots around a semiconducting gap of 30 meV with left and right gates tuning the CNT into n–n, n–p, p–n and p–p double dots. Here we have used a gate lever arm of ≈0.2 meV mV−1 extracted from the Coulomb diamond measurements. In addition, we observe the exact charge occupation of electrons and holes at corresponding gate voltages12. The high conductance in the n–n regime is possibly due to n-doping near the source/drain contacts. We also found p-doping for many samples for the same Pd contacts and do not exactly understand the nature of the observed contact doping.

Figure 3: Charge stability and RF reflectometry of CNT double quantum dots.
figure 3

(a) DC measurements of the CNT device at VSD=10 mV and VMG=0 show a semiconducting gap and the formation of bipolar double dots around it when VLG and VRG are swept. We observe the addition of electrons and holes in each regime starting from complete depletion in the bandgap. n–n double dots show relatively large conductance due to n-doping of the contacts. (b) Simultaneous measurements of current, reflected amplitude and phase at the resonance frequency in the p–p double-dot region at −10 mV bias indicated by a dashed box in a. Co-tunneling lines are clearly visible in all the plots. Inter-dot coupling lines not present in DC plot are visible in the amplitude and phase response due to frequency shifts at gate voltages marked by dashed circles. The arrow points at spurious gate-tunable lines most likely resulting from charge traps, which do not necessarily contribute to the current but do change the susceptance. A probe power of −110 dBm is applied.

We now perform simultaneous measurements of DC and RF reflectometry to obtain a collective response of the double quantum dots. Γ is measured near the resonance frequency with a probe power chosen to be so low that no out-of-equilibrium charges are induced when applying microwave signal to the stub tuner. Figure 3b shows a qualitatively similar honeycomb charge stability diagram in current, amplitude and phase responses taken at VSD=−10 mV bias. We clearly observe co-tunnelling lines3, long edges of the honeycomb, in all plots suggesting good sensitivity to impedance changes even in case of a circuit whose internal losses dominate the quality factor of the resonances. Here we note an important distinction of the stub tuner. For half-wave resonators, the RF signal for co-tunnelling lines strongly depend on the strength of the capacitive coupling to respective lead-dot transitions and their rates with respect to the resonance frequency30. DC-coupled stub tuner, in contrast, still responds through conductance changes that provide an external coupling by shunting the microwaves via the drain contact into the ground plane. This is further seen in Fig. 3b where larger current results in larger |ΔΓ| and Δφ (see Supplementary Fig. 3a,b). In addition, we also observe hybridized double dots at degeneracy, the boundary of two honeycombs at the two smaller edges, marked by dashed circles in Fig. 3b, in the phase and amplitude plots. The signal results from the susceptance changes caused by dipole coupling of the hybridized charge states to the microwave resonance. The responses at different charge degeneracies are different due to the distinct dot coupling energies tc, which are affected by all gate voltages in our sample.

To illustrate the control over the confinement potential of the double quantum dots in this clean CNT device, VMG is used to tune the tunnelling barrier between the two quantum dots. In the DC measurements, the strength of the tunnelling coupling is visible as the separation between the charge triple points with the larger value corresponding to a stronger coupling or weaker barrier (Supplementary Fig. 5). For a quantitative analysis, the phase response of the stub tuner can be measured using a weak probe power (−130 dBm) near the hybridization of two charge states (m, n+1) and (m+1, n) as shown in Fig. 4a. We operate in the zero-bias regime, allowing the dots to stay in equilibrium and rule out any conductance changes which could affect the resonance response. Such a phase shift is shown in Fig. 4c close to (2,2) to (1,3) hole transition. We infer the frequency shifts Δf from the phase variations which are almost linearly correlated near resonance (see Fig. 4e, error <10% in our case). Following a semiclassical model describing the coupling of a qubit with frequency to a resonator with frequency ωr (Supplementary Note 7), the dispersion Δf of the resonator is given by

Figure 4: Reflectometry response of inter-dot coupling.
figure 4

(a) Energy levels of a double quantum dot with an inter-dot hybridization energy of tc, forming a two-level system with energy ωd. (b) The stub tuner resonance experiences a negative dispersive shift Δf for ωr<2tc due to its coupling to the two-level system. (c) The phase response measured at the resonance frequency as a function of VLG and VMG yields a pronounced inter-dot transition due to hybridization of the charge states (2,2) and (1,3). (d) A cut indicated by the dashed arrow in c with a fit to equation (2) to extract the inter-dot coupling energy tc=20 GHz. (e) The stub tuner phase response at VMG=130 mV around the dot degeneracy (open circles) and uncoupled dots (solid circles) marked by points 2 and 1 in c, respectively. (f) Extracted tc and Γtot as a function of VMG. Error bars represent uncertainties in the least square fitting. (g) The maximum phase response ΔΦ indicated in d is suppressed at larger VMG due to increasing Γtot marked by the grey area. (h) The reflected power response at VMG=260 mV displays near-resonant absorption despite negligible dispersion. Solid lines are the fits to equation (2). Symbols have the same meaning as in e.

where Γtot=γ/2+Γφ with γ and Γφ are the effective relaxation and dephasing rates of the hybridized double dot, respectively, Δ=ωdωr and being the detuning, g0 the zero temperature coupling strength with the resonator and the polarization of the double-dot transition at electronic temperature T. We use the equation to first extract g0 in a regime where tc is large (VMG is small) so that ωd2tcωr, Γtot. This yields a dependence of the frequency shift Δf proportional to , now independent of Γtot. A fit with this equation to the data at VMG=130 mV is shown in Fig. 4d, yielding g0/2π=37 MHz. We find the same g0 for VMG=110 mV supporting the assumption that Γtot is relatively small in this regime. Fixing this g0 for phase responses at other VMG, we plot the extracted tc and Γtot in Fig. 4f and observe a reduction of tc on increasing VMG reflecting a reduction in the tunnel coupling strength between the dots. The phase response starts to be suppressed for VMG >200 mV due to increasingly fast double-dot relaxation, yielding Γtot>tc. The inverse dependence of Γφ on tc has been seen in similar systems7,31 and could be due to the 1/f charge noise environment32 of the nanotube. The sign of the frequency shift always remains negative, further signifying that the resonator energy is always <2tc. Increasing VMG to >250 mV, we do not notice any dispersion because of the large Γtot. However, the hybridized dots still show a response similar to Fig. 2a, now only in the reflectance amplitude. For VMG=260 mV, the stub tuner response for coupled and uncoupled dots regimes is presented in Fig. 4h. The fit to the resonance at the double-dot degeneracy (open circles) shows a smaller depth and α=0.0086, m−1 compared with the one in the uncoupled regime (solid circles) with α=0.0082, m−1. This behaviour is a result of an added loss channel that is, absorption from the two-level hybridized dots when 2tc becomes comparable to ωr, which is also consistent with the change in the resonance depth due to the conductance increase for our device parameters (Supplementary Figs 3a and 6b).


In summary, we have operated a RF superconducting impedance-matching circuit to measure CNT quantum dots in a hybrid device fabricated using a mechanical transfer of CNTs. The transfer employed here not only allows for a deterministic assembly and unity yield of complex RF devices, but also for the selection of CNTs with specific properties (metallic or semiconducting), reuse of the same circuit with different tubes and incorporation of desirable contact materials. We demonstrate the ability to locally control the confinement potential along the length of the suspended nanotube and to form dots with a precise number of charges. The high symmetry in the ambipolar charge stability around the bandgap indicates low disorder in the CNT system.

In addition, the matching circuit enables a comparatively simple extraction of the RF device impedance. We have shown that one can quantitatively deduce admittance changes in microsiemens resolution by measuring the complex reflectance Γ. This sensitivity can be used to deduce basic parameters of a clean CNT double-dot operating as a charge qubit, such as the inter-dot tunnel coupling strength and the relaxation rate. More importantly, conductances can be deduced using a simple analytic formula with measurement BW reaching (4/π)frZ0GMatch estimated using equation (1) near matching. We show an extraction of G from the RF amplitude response for a similar device, now at full matching, in the Supplementary Note 5. We find Coulomb diamond plots quantitatively similar to its DC counterpart and demonstrate a BW up to 2 MHz and a reflectance down to −40 dB for device GCNT=3.9 μS. The reliable high-BW extraction of G at GHz frequencies holds promises for probing quantum charge-relaxation resistance, which can deviate from its usual DC counterpart described by the Landauer formula33,34.

We demonstrate near matching with a substantial in and out microwave coupling in the measured device. For example, we see from Supplementary Fig. 3a that we achieve |Γ|≈−8 dB at GCNT≈0.4 μS. This relates to a reflectance probability of 16 %, hence 84 % is transmitted into the matching circuit and CNT device. Taking into account the internal loss described by GLoss=3.2 μS yields a substantial power transmission of 10 % from a 2.5 MΩ device to a 50 Ω transmission line. This is beneficial for high throughput detection of emission noise from the quantum device defined in the CNT wire and shot noise measurements35. The presented ability to combine an intricate RF circuit with a pristine suspended CNT will invite novel studies on devices with engineered mechanical, electrical and photonic degrees of freedom.


Sample fabrication

The device is patterned on a 150-nm thick Nb film, sputtered on an undoped Si substrate with 170 nm of SiO2, using photolithography and subsequent dry etching. The centre conductor of the transmission line is 12-μm wide, while gaps are 6-μm wide, yielding a calculated Z0=49 Ω. The geometric lengths l and d of the stub tuner are 10.66 mm and 10.36 mm, respectively. Spurious modes due to the T-junction are suppressed using on-chip wire bonds. The local bottom gates (Ti/Au of 5/35-nm thickness and 60-nm width) are defined by electron beam lithography and recessed in the SiO2 using dry anisotropic and isotropic etching with CF4 and Ar/CHF3, respectively (total recess depth≈100 nm). A recess of depth of 4 μm is etched via dry etching with SF6/O2 around the Pd (100 nm) contacts to facilitate mechanical transfer27. A modified micro-manipulator is used for CNT transfer at ambient conditions using optical microscopy. Contact resistances of CNT are generally found in excess of 100 kΩ (ref. 13), which may be attributable to remaining oxide on the contacts and contact geometry. We obtained 20% yield of transferring a single tube. More than one tube transferred at the same time due to multiple tubes on the same fork showed high conductance and could be removed by applying a large source drain bias.

Data analysis

We extract the relevant parameters such as l,d and α of the device by fitting the resonance at zero conductance using the following equation

where the phase factor p accounts for impedance mismatches in the setup and Γ0 for an offset. We could not exactly determine the electronic temperature of the device and assume T=0 K for all fits in Fig. 4d. For VMG<160 mV, extraction of relatively smaller Γtot is unreliable due to large detuning Δ from the stub tuner resonance.

Additional information

How to cite this article: Ranjan, V. et al. Clean carbon nanotubes coupled to superconducting impedance-matching circuits. Nat. Commun. 6:7165 doi: 10.1038/ncomms8165 (2015).