Abstract
Coupling carbon nanotube devices to microwave circuits offers a significant increase in bandwidth (BW) and signaltonoise ratio. These facilitate fast noninvasive readouts important for quantum information processing, shot noise and correlation measurements. However, creation of a device that unites a lowdisorder nanotube with a lowloss microwave resonator has so far remained a challenge, due to fabrication incompatibility of one with the other. Employing a mechanical transfer method, we successfully couple a nanotube to a gigahertz superconducting matching circuit and thereby retain pristine transport characteristics such as the control over formation of, and coupling strengths between, the quantum dots. Resonance response to changes in conductance and susceptance further enables quantitative parameter extraction. The achieved near matching is a step forward promising highBW noise correlation measurements on high impedance devices such as quantum dot circuits.
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Introduction
Artificial twolevel systems such as Josephson junction qubits coupled to superconducting microwave cavities^{1,2} have allowed for unprecedented breakthroughs in quantum information processing. Inspired by these results, other solid state systems have been coupled to resonant circuits, such as double quantum dots in semiconductors^{3,4,5,6,7} as potential qubits, and molecules or impurity spin ensembles^{8,9,10,11} as quantum memories. In particular, carbon nanotubes (CNT) have recently demonstrated their potential as lowdisorder onedimensional electron systems^{12,13,14}, which have been used to probe the physics of spinorbit^{15} and electronphonon coupling^{16}, as well as to perform initialization and manipulation of spin qubits^{17}. CNTs suspended over local gates not only offer a decoupling from the surface but also mechanical resonances with high quality factors^{18}, creation of arbitrary local potentials and tight confinement with charging energies in excess of 50 meV (ref. 14).
Nevertheless, coupling CNTs to halfwave resonators^{7,19,20} and other types of LC circuits^{21,22,23,24} 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 growthlast approach^{25}, the low yield and high temperatures (∼900 °C) involved prohibit the use of this method for fabricating superconductorCNT hybrid devices. Recently, the transfer of CNTs from a growth substrate to a separate microwave device, followed by standard lithography has been demonstrated^{26}, solving issues with material compatibility but not addressing limited yield and unclean transport spectra. As such, the combination of lowloss 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 impedancematching circuit based on superconducting transmission lines. Different to previous works^{7,19,20} where halfwave 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 highBW 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 process^{27}, we place the CNT on the finished microwave device in the last step. Such a selective assembly technique^{13} 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 interdot 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.
Results
Mechanical transfer of CNT
A key advance in this hybridCNT 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 forklike structures of a separate chip, which allow a resist and ebeam radiationfree 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 micromanipulator 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.
Superconducting impedancematching circuits
A transmission line based impedancematching circuit, termed stub tuner^{28}, 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 Z_{Match}, will be transformed to the characteristic impedance of the line Z_{0} 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 Z_{in} of the hybrid device, consisting of a parallel combination of the two transmission lines^{28,29} as
γ=α+iβ 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, Z_{0} the transmission line impedance, and the complex admittance 1/Z_{CNT}=Y=G_{CNT}+iB_{CNT} with G_{CNT} conductance, and B_{CNT} 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 G_{CNT} 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 G_{Loss} parallel to the CNT load (Fig. 1c) and thus defining a total effective load G_{eff}=G_{CNT}+G_{Loss} seen by the microwaves. In Fig. 2b, we plot Γ at the resonance as a function of G_{eff}. The reflected signal is the smallest close to the matched load offering maximal power transfer. The latter is more important for measuring small noisepower emitted from the device. Due to the additional loss conductance G_{Loss}, the CNT load which matches the circuit is smaller than G_{Match} and is given by . In particular, if G_{Loss}>G_{Match} full matching is precluded and the reflection coefficient as a function of load becomes monotonic (see inset of Fig. 2b).
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 ω_{r}/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 halfwave resonator, showing that we are able to achieve lowloss microwave circuits in combination with CNT devices. Due to parasitic inductances from the fabricated contacts, the matching circuit has a lower effective G_{Match}≈1.6 μS compared with G_{Loss}≈3.2 μS and operates therefore in the internallossdominated 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 gatedefined confinement potentials along the nanotube and hence for tuning the location, size and number of quantum dots^{13}. With no RF power applied to the stub tuner and the middle gate V_{MG}=0 V, we first measure the charge stability diagram of the CNT device. A gate sweep using left (V_{LG}) and right gates (V_{RG}) at V_{SD}=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 voltages^{12}. The high conductance in the n–n regime is possibly due to ndoping near the source/drain contacts. We also found pdoping for many samples for the same Pd contacts and do not exactly understand the nature of the observed contact doping.
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 outofequilibrium 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 V_{SD}=−10 mV bias. We clearly observe cotunnelling lines^{3}, 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 halfwave resonators, the RF signal for cotunnelling lines strongly depend on the strength of the capacitive coupling to respective leaddot transitions and their rates with respect to the resonance frequency^{30}. DCcoupled 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 t_{c}, 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, V_{MG} 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 zerobias 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 ω_{r} (Supplementary Note 7), the dispersion Δf of the resonator is given by
where Γ_{tot}=γ/2+Γ_{φ} with γ and Γ_{φ} are the effective relaxation and dephasing rates of the hybridized double dot, respectively, Δ=ω_{d}–ω_{r} and being the detuning, g_{0} the zero temperature coupling strength with the resonator and the polarization of the doubledot transition at electronic temperature T. We use the equation to first extract g_{0} in a regime where t_{c} is large (V_{MG} is small) so that ω_{d}⩾2t_{c}≫ω_{r}, Γ_{tot}. This yields a dependence of the frequency shift Δf proportional to , now independent of Γ_{tot}. A fit with this equation to the data at V_{MG}=130 mV is shown in Fig. 4d, yielding g_{0}/2π=37 MHz. We find the same g_{0} for V_{MG}=110 mV supporting the assumption that Γ_{tot} is relatively small in this regime. Fixing this g_{0} for phase responses at other V_{MG}, we plot the extracted t_{c} and Γ_{tot} in Fig. 4f and observe a reduction of t_{c} on increasing V_{MG} reflecting a reduction in the tunnel coupling strength between the dots. The phase response starts to be suppressed for V_{MG} >200 mV due to increasingly fast doubledot relaxation, yielding Γ_{tot}>t_{c}. The inverse dependence of Γ_{φ} on t_{c} has been seen in similar systems^{7,31} and could be due to the 1/f charge noise environment^{32} of the nanotube. The sign of the frequency shift always remains negative, further signifying that the resonator energy is always <2t_{c}. Increasing V_{MG} to >250 mV, we do not notice any dispersion because of the large Γ_{tot}. However, the hybridized dots still show a response similar to Fig. 2a, now only in the reflectance amplitude. For V_{MG}=260 mV, the stub tuner response for coupled and uncoupled dots regimes is presented in Fig. 4h. The fit to the resonance at the doubledot 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 twolevel hybridized dots when 2t_{c} becomes comparable to ω_{r}, 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).
Discussion
In summary, we have operated a RF superconducting impedancematching 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 doubledot operating as a charge qubit, such as the interdot tunnel coupling strength and the relaxation rate. More importantly, conductances can be deduced using a simple analytic formula with measurement BW reaching (4/π)f_{r}Z_{0}G_{Match} 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 G_{CNT}=3.9 μS. The reliable highBW extraction of G at GHz frequencies holds promises for probing quantum chargerelaxation resistance, which can deviate from its usual DC counterpart described by the Landauer formula^{33,34}.
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 G_{CNT}≈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 G_{Loss}=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 measurements^{35}. 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.
Methods
Sample fabrication
The device is patterned on a 150nm thick Nb film, sputtered on an undoped Si substrate with 170 nm of SiO_{2}, using photolithography and subsequent dry etching. The centre conductor of the transmission line is 12μm wide, while gaps are 6μm wide, yielding a calculated Z_{0}=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 Tjunction are suppressed using onchip wire bonds. The local bottom gates (Ti/Au of 5/35nm thickness and 60nm width) are defined by electron beam lithography and recessed in the SiO_{2} using dry anisotropic and isotropic etching with CF_{4} and Ar/CHF_{3}, respectively (total recess depth≈100 nm). A recess of depth of 4 μm is etched via dry etching with SF_{6}/O_{2} around the Pd (100 nm) contacts to facilitate mechanical transfer^{27}. A modified micromanipulator 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 Γ_{0} 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 V_{MG}<160 mV, extraction of relatively smaller Γ_{tot} 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 impedancematching circuits. Nat. Commun. 6:7165 doi: 10.1038/ncomms8165 (2015).
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Acknowledgements
We thank J. Gramich for experimental assistance and acknowledge technical support from the FIRST Center for Micro and Nanoscience. We acknowledge financial support by the ERC project QUEST, the EC project SE2ND, the NCCR QSIT and the Swiss National Science Foundation.
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V.R. and G.P.H. contributed equally to the work. V.R. and G.P.H. performed process development, sample fabrication, experiments and data analysis. M.J. and T.H. contributed to the fabrication process development and measurements. M.M. and C.H. developed the transfer process and integrated the carbon nanotubes. A.N. provided the theory support. G.P.H., A.W. and C.S. devised, initiated and supervised the work. V.R. and G.P.H. wrote the manuscript with input from all authors.
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Supplementary Figures 16, Supplementary Notes 17, and Supplementary References (PDF 8770 kb)
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Ranjan, V., PueblaHellmann, G., Jung, M. et al. Clean carbon nanotubes coupled to superconducting impedancematching circuits. Nat Commun 6, 7165 (2015). https://doi.org/10.1038/ncomms8165
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DOI: https://doi.org/10.1038/ncomms8165
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