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3D integration and measurement of a semiconductor double quantum dot with a high-impedance TiN resonator

Abstract

One major challenge to scaling quantum dot qubits is the dense wiring requirements, making it difficult to envision fabricating large 2D arrays of nearest-neighbor-coupled qubits necessary for error correction. We describe a method to ameliorate this issue by spacing out the qubits using superconducting resonators facilitated by 3D integration. To prove the viability of this approach, we use integration to couple an off-chip high-impedance TiN resonator to a double quantum dot in a Si/SiGe heterostructure. Using the resonator as a dispersive gate sensor, we tune the device down to the single electron regime with an SNR = 5.36. Characterizing the individual systems shows 3D integration can be done while maintaining low-charge noise for the quantum dots and high-quality factors for the superconducting resonator (single photon QL = 2.14 × 104 with Qi ≈ 3 × 105), necessary for readout and high-fidelity two-qubit gates.

Introduction

One major challenge for noisy intermediate-scale quantum (NISQ)-era superconductor and semiconductor qubit systems lies in the wiring interconnect problem1,2. Unlike classical processors, where of order 103 signal lines operate 109 transistors, all solid state qubit platforms require at least one independent control line per qubit. In cryogenic solid state platforms, Josephson effect qubits, such as the transmon, capacitively shunted flux, or fluxonium qubit, have a modest overhead of 1–3 lines per qubit for control and readout3,4,5,6. Semiconductor spin qubits in accumulation-mode gate-defined quantum dots require between 9 and 13 bias lines per qubit to form the necessary electrostatic environment, with anywhere from 1 to 5 of those lines requiring 1 GHz bandwidth to perform high-fidelity qubit control7,8,9,10,11,12. In order to implement the surface code for quantum error correction, a 2D grid of N × N nearest-neighbor coupled qubits is minimally necessary13. For semiconductor qubits, this requirement is of great practical concern, as the need for dense networks of sub-100 nm sized electrodes to form a large array of coupled spins is major interconnect engineering challenge with current proposals requiring several technical innovations before becoming feasible14.

Fortunately, circuit quantum electrodynamics (cQED) provides a framework where the extreme wiring density requirements for quantum dot spin qubits may be alleviated by using superconducting resonators to mediate long range 2-qubit interactions, providing room for the necessary wiring to form the individual qubit structures. This scheme requires strong qubit-cavity coupling (gκ, γ) to be viable past the structural engineering to fabricate the devices; here, g is the electron–photon coupling strength, γ is the qubit dephasing rate, and κ is the cavity photon loss rate. Recent work has shown hybrid cQED architectures can reach the strong coupling to the charge, valley-orbit, and spin degrees of freedom as well as facilitate interactions between spin or superconducting qubits, with most demonstrations relying on high-impedance resonators to increase the coupling strength (\(g\propto \sqrt{{Z}_{r}}\))15,16,17,18,19,20,21. Using cavity mediated two-qubit gates reduces the array problem to fabricating N × N copies of single qubit structures (linear dot arrays), which is routine in the lab.

Vertical integration further ameliorates the interconnect problem by allowing some large components to be placed vertically off-chip in a 3D architecture. This approach uses thermomechanical bonding to create electrical contacts between two or more dies using indium22,23. Currently, 3D integrated circuits are the workhorse interconnect solution for high-qubit count superconducting quantum processors, facilitating demonstrations of quantum supremacy and large quantum volumes in solid state quantum processors24,25. In this paper, we demonstrate the viability of 3D integration for spin qubits using a Si/SiGe double quantum dot (DQD) and a vertically integrated high-Q, high-impedance TiN resonator used for dispersive gate charge sensing without degradation in performance of either component.

Results and discussion

Design of a 2D array of semiconductor qubits using 3D integration

In Fig. 1a we illustrate a 3-tier stack integration scheme that is conceivable using state of the art 3D integration22,26,27. The base die, a superconducting multichip module (SMCM), serves as a routing layer for high density control wiring, possibly incorporating control and readout circuitry such as cryogenic CMOS, superconducting amplifiers or photodetectors28,29,30. Superconducting through-silicon vias (TSVs) bring qubit control lines to the the qubit die to the SMCM using indium bump bonds to create electrical contact27. In our design, low-Q readout resonators are integrated onto the qubit chip as they are more tolerant to higher internal losses resulting from multilayer processing necessary to form the spin qubit structures. Readout is done via a mutual inductance to a nearby shorted signal line with the resonator probed in reflection. The top die consists of a network of high impedance, low loss resonators used for cavity mediated two-qubit interactions between distant quantum dot qubits.

Fig. 1: Design of a 3D integrated cQED quantum dot processor.
figure 1

a Diagram of a cQED spin qubit processor utilizing a 3-tier stack. b Layout of an exchange only qubit cQED processor unit cell with a 0.16 mm2 footprint. Top die consists of four λ/2 high-impedance resonators with λ/4 voltage taps. Base die consists of nine control lines (I), two readout resonators (II, one centered, 1/2 in each lower corner of the unit cell), and two qubits (III, one centered and 1/4 in each corner). Control and readout lines are routed to the backplane of the die using through-silicon vias (TSVs) with a 10 × 20 μm2 footprint27. c A simplified circuit schematic with the elements in red on the qubit die and the elements in gray on the coupler die. d A 64 qubit processor tiled out from the base tile design fitting in a 4.6 × 1.6 mm2 area.

Figure 1b shows the CAD design of a lattice unit cell consisting of two qubits, four coupling bus resonators with voltage bias taps (IV, gray die), and two readout resonators (II, red die) fitting in a 0.16 mm2 top down area across two die. Thirteen TSVs (hatched purple, in Fig. 1b) route bias lines for each triple quantum dot vertically into each unit cell. Figure 1c provides a simplified circuit schematic of the unit cell with the elements on the qubit die colored in red and the elements on the coupler die colored in gray. Using this design Fig. 1d shows how a 64 qubit processor can be tiled out in a 4.6 × 1.6 mm2 footprint. This design builds upon demonstrated technology and assumes nominal materials parameters that are possible for high-kinetic inductance films (e.g., quantum dot gate design, TSV or indium bump lithographic dimensions, sheet kinetic inductance, etc.)22,27,31,32,33. We emphasize here this design requires fabrication process development to ensure thermal budget compatibility between the TSV process and the quantum dot fabrication.

Device design

For our experiment, we limit the architecture to the 2-tier stack consisting of a quantum dot qubit die and high-impedance cavity coupler die as illustrated in Fig. 2a. Optical and scanning electron micrograph images of the two dies prior to bonding can be seen in Fig. 2b. The base die consists of a Si/SiGe heterostructure in which accumulation-mode, gate-defined quantum dots are formed. The cQED coupler die consists of a high-kinetic inductance, high-impedance TiN λ/2 coplanar waveguide resonator. To minimize parasitic photon loss from coupling of the resonator to the 24 dot bias leads, we fabricate low impedance control lines using buried coplanar waveguides (BCPWs) with a characteristic impedance of Zg ≈ 1Ω. The low characteristic impedance ameliorates unwanted loss out the dot leads 34.

Fig. 2: Experimental realization of a 3D integrated dot device.
figure 2

a Upper panel shows a cross section of the 2-tier stack consisting of the high-impedance resonators and Si/SiGe qubit die. Lower shows a simplified circuit diagram for the device. b Left panel shows a dark field optical microscope image of a completed qubit die chip. Gold squares are underbump metalization consisting of Ti/Pd/Ti/Pt/Au used to facilitate galvanic contact between the qubit and coupling resonator die. Periodic sections of niobium are removed that lay under the resonator to minimize the added capacitance. Niobium shorting straps are used to hold the dot gates at equipotential during fabrication and packaging. They are opened at star points located at the red × marks using a diamond scribe and micromanipulator. Inset: Si/SiGe mesa and resonator landing pad (gold square above mesa) prior to e-beam lithography of the quantum dot gates (scale = 100 μm). Right panel shows optical and scanning electron micrographs of the TiN resonator die (TiN center pin is false colored in blue). Blue circles on the qubit and cavity chips indicate where the resonator center pin contacts are made.

The BCPWs consist of a coplanar waveguides made from niobium, a SiO2 dielectric layer deposited over the center conductor, and a capping niobium layer connecting the two ground electrodes. These BCPWs can be seen in the lower half of the Si/SiGe die in Fig. 2b. The addition of the capping ground plane over the control wiring suppresses cross capacitances between control lines as well as the parasitic leakage capacitance to the resonator. The cavity coupler die consists of a λ/2 TiN CPW with a λ/4 segment shunted by a large parallel plate capacitor on the quantum dot die used for low frequency voltage biasing34. TiN was chosen for its high-kinetic inductance (Lk), high critical magnetic field, and high internal Q allowing for Zr > 1kΩ while maintaining Qi > 105 33,35,36.

In order to maximize the probability of reaching strong coupling, we push the resonator into the high-impedance regime using a 50 nm-thick TiN film with a 500 nm wide center pin, and 10.25 μm gap, as shown in Fig. 2b. Using COMOSL Multiphysics to simulate the 3D structure’s influence on the CPW line capacitance and the nominal kinetic inductance of the TiN film (Lk ≈ 9pH/□) we estimate a characteristic impedance of Zr = 575Ω with target fundamental frequency of 4 GHz. The high impedance is expected to increase the charge-photon coupling rate by a factor of 3.4 over 50Ω resonators. Higher impedance can be engineered by reducing the TiN center pin thickness and width (Lk 1/wt) but was not pursued for this study32. We find the estimates of the inductance and capacitance of the TiN resonator are in good agreement with the experimentally measured range of resonance frequencies 3.985–4.115 GHz from several different samples with nominally identical resonator dimensions.

Galvanic contact between both dies is facilitated by underbump metal pads made from Ti/Pt/Au on the resonator die and Ti/Pd/Ti/Pt/Au on the quantum dot die, which do not form native oxides and can be seen in the optical images in Fig. 1b. To avoid unwanted damping caused by power dissipation of the microwave currents in the resonator to the normal metal pads, they are placed at voltage antinodes and at the end of the quarter wave DC tap where nominally no microwave currents exist for the λ/2 mode, thus preserving the high internal quality factor 22,37.

For these data a single layer of aluminum gates are used to form the double quantum dot18,38. Underneath the gate stack is an ultra thin (~1.7 nm thick) SiO2 layer grown by nitric acid oxidation of silicon (NAOS), which has been shown to be higher quality than similar thickness thermal oxides39. A scanning electron micrograph with an overlaid Thomas-Fermi simulation of the induced electron gas can be seen in Fig. 3a. The two dimensional electron gas (2DEG) is induced in an 8.6 nm-thick silicon quantum well made from 800 ppm 28Si ~42 nm below the surface. As designed, the gates labeled (P1,P2) are intended for accumulating a double quantum dot and the gates labeled (B1:B2:B3) serve to tune the various tunnel barriers. The other gates labeled (S1:S2:S3:S4) help corral the charges to upper portion of the plunger gates and mitigate unwanted transport currents. The superconducting resonator has a galvanic connection to the gate S1 as a single-layer variant of a split-gate coupler design, which serves to decouple the cavity pin voltage from the neighboring quantum dot easing tuning constraints for two-qubit samples40. Tuning the device as intended, a double quantum dot (DQD) can be formed and sensed under P1 and P2 using the TiN resonator as shown in Fig. 3c. In the many electron regime the DQD has tunnel rates comparable to the resonator frequency resulting in a visible interdot transition shown in the inset of Fig. 3c.

Fig. 3: Evaluation of charge noise and readout SNR in a 3D integrated DQD.
figure 3

a An SEM of the single-layer double quantum dot (DQD) aluminum gate structure with the resonator connected to S1. Overlaid is a Thomas-Fermi simulation of a DQD in the few electron regime. b DC transport, gate referred noise measurements in the device. The black trace is acquired using a peak location monitoring method over a 21 h period (red is smoothed using a moving mean). The teal and green traces are extracted by measuring current fluctuations at the maximum first derivative point of a Coulomb blockade peak for 510 s. c Dispersive gate charge sensing of a double quantum dot formed under P1 and P2 down to single electron occupancy using the high-impedance TiN resonator as a dispersive gate readout. Inset: A tunnel coupled many electron double quantum dot with nL,R ≈ 10. d Generalized quadrature histogram showing the visibility of tunneling resonance (orange) vs blockade (teal). Here, a the drive power is ~−95 dBm on chip and yields an SNR ≈ 5.36. Solid lines are Gaussian fits to the data.

A third dot can be formed under B2 by repurposing P1 and P2 as barrier gates. For each dot, we use bias spectroscopy to extract the lever arm of the nominal chemical potential gate and find typical lever arms ranging between α = 0.20–0.25 eV/V (an example can be seen in supplemental Fig. 3a). In general, we observe good electrical confinement for each dot with charging energies ranging from EC = 3–5 meV (3 meV for the B2 dot and 5 meV for the P1/P2 dots) with orbital slitting EOrb ≈ 1 meV in the few to single electron regime. Additional parasitic quantum dots were observed under certain bias conditions, likely due to uncontrolled accumulation of 2DEG under the gate electrodes41 and are a source of low frequency instabilities in the device. These instabilities prevented tunnel coupling at frequencies comparable to the resonator frequency in the single electron regime.

Quantum dot charge noise characterization

We next measure the charge noise spectrum the dots experience in order to estimate the charge dephasing rate and evaluate the prospect for strong coupling between the charge and photon degrees of freedom with this design. To do this we use two methods: Coulomb blockade peak tracking and voltage-to-current transduction42. Using both methods allows for noise to be characterized over 7 orders of magnitude from 10−5–102 Hz. The first method is performed by repeatedly sweeping the plunger gate voltage (see supplemental material for details) of the dot over the course of a day to monitor its location with a repetition rate of ~0.1 Hz. The data are then fit to a thermally broadened conductance peak defined by43

$$I({V}_{{{\mbox{g}}}})={I}_{{{\mbox{offset}}}}+\frac{{A}_{0}}{4{k}_{{{\mbox{b}}}}{T}_{{{\mbox{e}}}}}\cosh {\left(\frac{\alpha ({V}_{{{\mbox{g}}}}-{V}_{{{\mbox{offset}}}})}{2{k}_{{{\mbox{b}}}}{T}_{{{\mbox{e}}}}}\right)}^{-2}.$$
(1)

From the fits the we extract an electron temperature of Te ≈ 200 mK (upper bound), peak current (\(\frac{{A}_{0}}{4{k}_{{{\mbox{b}}}}{T}_{{{\mbox{e}}}}}\)), where A0 is a fitting factor that depends on the biasing of the dot (\({A}_{0}={V}_{{{\mbox{SD}}}}4{k}_{{{\mbox{b}}}}{T}_{{{\mbox{e}}}}{G}_{\max }={e}^{2}{V}_{{{\mbox{SD}}}}({{{\Gamma }}}_{{{\mbox{r}}}}| | {{{\Gamma }}}_{{{\mbox{l}}}})\)), and the peak offset voltage (Voffset). Using the extracted offset voltage for each scan, we generate a corresponding time series from which the power spectral density of the offset voltage can be computed as shown by the black (raw) and red (smoothed) traces in Fig. 3d. We observe a 1/f2 power law dependence in the offset voltage at frequencies below 1 mHz similar to that observed in stadium style devices44, characteristic of Brownian motion in the fluctuating variable45. One potential mechanism for the observed spectrum is a very small hysteresis in the peak’s location due to the raster-style sweep causing it to systematically increase (or decrease) in gate voltage. The noise floor of the measurement is due to the perfect zero lag auto-correlation of the time series used to compute the power spectrum. This feature generates a large white noise spectrum (the FFT of a delta-like peak at zero lag) and dominates the high frequency part of the spectrum 46.

To resolve offset voltage noise below the noise floor of the peak tracking method, we use a second method where we measure fluctuations in the transport current at the point of maximum voltage-to-current transduction and compute the associated current noise power spectrum42,47. Assuming a linear transfer function between voltage and current we can convert the current noise power spectrum to a gate referred voltage power spectrum by computing

$${S}_{V}(f)={\left|\frac{dI}{d{V}_{{{\mbox{g}}}}}\right|}^{-2}{S}_{I}(f),$$
(2)

where SV(I) are the power spectral densities of the voltage (current) and \(\frac{dI}{d{V}_{{{\mbox{g}}}}}\) is the derivative at the point along the blockade peak where the gate voltage was set for the data acquisition. Using this method we are able to extract the offset voltage noise spectrum between 2 × 10−3–100 Hz as shown by the teal trace in Fig. 3b. A moving average filter is used to reduce statistical fluctuations in the data without distorting the overall shape of the signal46.

The same data can be subdivided into 10 s traces where each trace’s power spectrum can be averaged producing the dark green spectrum from 0.1–100 Hz in Fig. 3b, in agreement with the smoothed data from the single time series. In most spectra from this device we observe a prominent Lorentzian spectrum at high frequencies with a characteristic switching time between 1 and 10 Hz. The spectra measured have an amplitude at f = 1 Hz of SV(f) = 10 − 36 μV2/Hz comparable with other work48. Alternatively, one can cast in terms of chemical potential fluctuations: Sμ(f) = α2SV(f) = 0.6 − 1.5 μeV2/Hz, or in terms of offset charge49: \({S}_{{{\mbox{C}}}}(f)={(e/{E}_{{{\mbox{C}}}})}^{2}{S}_{\mu }(f)=2.4-6\times 1{0}^{-2}\,\,{{\mbox{m}}}{e}^{2}/{{\mbox{Hz}}}\,\). These values corroborate recent work that suggests reducing the volume of deposited dielectrics above the quantum dots reduces charge noise47. The variation in measured values is somewhat dependent on the tuning of the device, with the lowest numbers corresponding to small source-drain bias VSD ≤ 50 μV (while maintaining \(| \frac{dI}{d{V}_{{{\mbox{g}}}}}| > 100\,\,{{\mbox{pA}}}/{{\mbox{mV}}}\,\)) and the amplitude of the low frequency switcher. The two methods measured noise amplitudes and exponents that are similar where they cross over around 1–10 mHz, demonstrating they are complementary techniques to measuring noise in the quantum dot. Elimination of unwanted nearby 2DEGs through use of screening gates could improve the low frequency instability and minimize parasitic switchers coupled to the intended dots50. Using measured spectrum, we estimate the charge dephasing rate for an ideal charge qubit at zero detuning with tunnel coupling 2t = hfr as 49:

$${\left.\frac{\gamma }{2\pi }\approx \frac{{d}^{2}{E}_{01}}{d{\epsilon }^{2}}\right|}_{\epsilon = 0}{S}_{\mu }(1\,{{\mbox{Hz}}})\approx \frac{{S}_{\mu }(1{{\mbox{Hz}}}\,)}{2t}\approx 9-22\,\,{{\mbox{MHz}}}\,.$$
(3)

This suggests for a device where the dots are placed optimally relative to the cavity electrode, strong coupling to the charge degree of freedom should be possible.

Characterization of the dispersive gate readout

To evaluate the resonator as a dispersive gate readout, we measure the the signal to noise ratio (SNR) for observing tunneling resonances where the electron has high susceptibility to the cavity photon’s electric field. We first tune a dot-lead transition to maximize the phase shift in the microwave tone at a fixed power of −95 dBm on chip. Next, we take 104 measurements of the demodulated IQ voltages when the electron is biased in Coulomb blockade or tunneling resonance with a 50 ms integration time sampled at 100 kSa/s. We then define the measurement axis along the centroids of the raw demodulated IQ blobs and define zero to be the midpoint between the blobs analogous to how one might threshold for singlet-triplet blockade readout38. As shown in Fig. 3d, by taking a histogram of the data along the Q quadrature we find the blockade peak voltage is well described by a Gaussian process (solid lines are fits), while the tunneling peak undergoes an additional non-Gaussian process evidenced by the asymmetry of the peak and substantial deviation from the fit. This is possibly due to the low frequency switcher observed in the transport noise data causing the location of the peak to telegraph in voltage space. Curiously, the non-Gaussian shoulder is not present at lower drive powers, suggesting the process is stimulated from the microwave energy in the resonator (see supplemental Fig. 4 for more details).

Defining SNR in terms of the separation of the demodulated peaks (Δ) and the blockade peak standard deviation (σb), we find SNR = Δ/σb ≈ 5.36, which compares favorably to PCB integrated reflectometry methods while using somewhat lower powers51. We note using σb instead of σt inflates the SNR by roughly 20%, as we empirically find the tunneling peak standard deviation (σt) is larger. Compared to other resonator measurement techniques, which define SNR as the power SNR (\({{{\mbox{SNR}}}}_{P}={({{\Delta }}/{\sigma }_{b})}^{2}\)) we are notably much lower with SNRP = 28.7 compared to SNRP > 1000 for similar integration times38,52. The likely reason for the large difference is due to the substantially weaker coupling strength between the resonator and the quantum dots in our system. Based on S1 vs. P1(2) measurements, the lever arm of the resonator gate to the quantum dots is quite small (≈0.05αP1(2)), causing the corresponding difference in the IQ signal during tunneling events to be substantially smaller than plunger coupled devices.

Attempts to tune the coupling strength by tuning the resonator gate voltage40 are hampered by reductions in the loaded Q, as discussed later in the text. We emphasize this issue is not intrinsic to the 3D architecture but rather a bug of the single-layer gate layout resulting in poor placement of the quantum dots relative to the cavity electrode. This unfortunately prevents direct measurement of the charge-photon coupling rate as we need 104 photons in the resonator to resolve the electron tunneling resonances. Typical direct coupling measurements require single photon probe powers to minimize driving effects on the qubit15. Use of an overlapping gate architecture, which has more precise placement of the quantum dots will substantially improve this aspect of the device performance. Additional optimizations such as using heterodyne detection with fast sampling DACs or quantum limited superconducting amplifiers can also improve the SNR through noise mitigation 29,38.

Noise characterization of the TiN resonator

To characterize the effectiveness of our improved leakage suppression technique, we performed a systematic study of the quality factor with the device tuned in the (0,0) charge configuration while keeping the 2DEG reservoirs accumulated. Figure 4a shows the normalized S212 response function of the TiN resonator as a function of the voltage bias on the resonator pin. At high bias (VS1 > 450 mV), a substantial degradation in QL is observed, due to induced 2DEG under the resonator gate substantially damping the cavity mode. Curiously, line cut comparisons of zero voltage bias (teal line) and sub-accumulation bias (pink line) show the linewidth of the resonator is substantially increased at finite voltage bias (see inset to Fig. 4a).

Fig. 4: TiN resonator characterization.
figure 4

a Measurement of the resonator fundamental transmission spectra as a function of gate voltage. Inset: extracted Lorentzian fits to the teal and pink traces at VS1 = 0 mV and VS1 = 300 mV, respectively. b Extracted loaded quality factor as a function of gate voltage showing an anomalous decrease in the loaded quality factor upon voltage biasing the center pin. A narrow plateau in QL occurs between VS1 = ± 2 mV where the additional loss is less than the coupling quality factor. c Power dependence of the loaded quality factor at 0 mV and 300 mV center pin bias. We extract a single photon internal quality factor of Qi ≈ 3 × 105 consistent with radiative losses from the large CPW geometry56. We estimate degradation of the internal quality factor to Qi ≈ 3 × 104 at VS1 = 300 mV. Error bars are the 95% confidence intervals of the fit extracted QL. d Phase noise power spectral density of the resonator with VS1 = 0 mV by probing the cavity on resonance with the dots empty. We extract a 1/fβ with β = 1.98 noise spectrum in the phase noise for frequencies below 0.1 Hz. e Upper: Resonator frequency shift (δfr = fr(VS1 = 0) − fr(VS1)) as a function of voltage bias on the cavity pin. We observe a nonmonotonic modulation in the center frequency. Lower: Zoom-in of a bias region in which multiple TLS-cavity interactions are observed. f Spectral motion of two TLS interacting with the TiN resonator over several hours.

Performing a similar scan over a range VS1 = ± 400 mV, we observe a dramatic change in the loaded quality factor with the magnitude of the applied electric field ( VS1), as shown in Fig. 4b. The origin of the anomalous loss mechanism upon voltage biasing the resonator is unclear but likely cannot be attributed to the induced 2DEGs in under the accumulation gates LA or RA, as degradation occurs regardless of the sign of the bias. To extract the internal Qi, we perform power sweeps at two voltage biases, shown in Fig. 4c. Assuming the high power limit at zero voltage bias is defined by the explicit coupling capacitances for probe and readout, we extract Qc = 2.31 × 104 with Qi(VS1 = 0mV) ≈ 3 × 105 and Qi(VS1 = 300 mV) ≈ 3 × 104. To our knowledge, the extracted Qi at zero voltage bias is the highest measured Qi in a superconducting-semiconductor hybrid system.

It has recently been proposed that high-impedance resonators may exhibit lowered quality factors due to enhanced phase noise rather than true energy loss, due to fluctuations in the kinetic inductance from charge noise resulting in a corresponding frequency modulation53. To see if this was present, we measured the phase noise at low frequencies by probing the transmission phase on resonance (at time t = 0) over the course of several hours. We compute the corresponding phase noise power spectral density, as shown in Fig. 4d. We observe a 1/f2 dependence of the phase noise PSD below 0.1 Hz, atypical of high Q superconducting resonators54,55. Additionally, when performing the voltage bias studies, we observed the cavity resonance frequency is a nonmonotonic function of the applied voltage bias to the center pin with maximum frequency modulation of 14 kHz, as shown in Fig. 4e. Several reproducible TLS-cavity crossings, inferred by shifts of the resonator frequency, are observed over the ±400 mV tuning range explored. These data suggest the effect is present, but due to their small magnitude (≈3–5 kHz) are insufficient to explain the factor of ~10 reduction in Qi with application of voltage bias. We observe these TLS are not fixed in location in voltage space and undergo time dependent spectral diffusion over hours-long timescales, illustrated by the data in Fig. 4f. At this time it is unclear if these defects originate from the quantum dot die or the TiN resonator die or if any of the noise between the two systems is correlated.

In summary, we have described a 3D integration approach to hybrid superconductor-semiconductor quantum processor. We demonstrated such an integration scheme is viable as our system had nearly all necessary ingredients for long range coupling: high single photon quality factor cavities with high-impedance resonators and low-charge noise quantum dots. The remaining ingredient, strong charge-photon coupling may be achieved by using a gate stack that more precisely places the quantum dots relative to the cavity electrode, such as the linear overlapping gate array. Using impedance engineering, single photon loaded quality factors as high as 2.14 × 104 were measured with an estimated Qi ≈ 3 × 105 placing a photon loss rate bound of κ/2π = fr/Qi ≈ 15 kHz at 4 GHz, which can be easily utilized by lowering the coupling capacitance to probe the resonator (or eliminating it entirely). While the noise data acquired suggests dephasing rates below 30 MHz are possible, potentially sufficient for strong coupling, further improvements to the charge dephasing rates will be desirable for high-fidelity operation in a quantum processor. This may be achieved either by limiting the impact of charge noise on the qubit operation through Hamiltonian engineering or improvements in the dielectrics used for fabricating the quantum dots. One possible route, plausible given the resonator characterization data, is to utilize voltage biased high-impedance resonators and phase noise measurements to determine dielectrics with low-charge noise.

Methods

Packaging, sample loading procedure, and failure analysis

Devices are packaged via aluminum wire bonding in a hybrid PCB and metal enclosure designed to limit parasitic chip modes out to 20 GHz for non-2-tier stack samples (see supplement for details). The shorting strap structures on the quantum dot chip are scratched away at star points with a diamond scribe that is silver epoxied to a wedge bonder tip allowing use of the wire bonder as a micromanipulator. The silver epoxy ensures the tip is well grounded and subsequent SEM inspection of many non-MCM devices indicate there is no risk of electrostatic discharge (ESD) from this process.

The devices are then loaded into a Leiden CF-450 dilution refrigerator with a base temperature of 50 mK. Substantial care is taken to ensure the sample, fridge wiring, and experimentalist remain grounded during the load to minimize ESD risk to the sample. Use of a grounded dissipative floor tiling under the cryostat, grounding wrist straps, antistatic coats, tools, gloves, humidity control (35–55%RH), and ionizing fans do not appear to substantially mitigate the ESD risk to the sample in our experimental setup (load yield less than 10%). Compared to standard 50Ω resonator devices we observe increased catastrophic failure (discerned by gate to gate shorts) from ESD likely due to the high resistance nature of the TiN resonator pin at room temperature resulting in damage during packaging or loading. Delamination of a completed 2-tier stack device showed no apparent ESD from the thermomechanical bonding process itself to the quantum dot gate stack, corroborating ESD is a packaging or loading issue rather than a fabrication problem. Passive on-chip ESD protection measures such as subdegenerate phosphorous doping between bond pads or freeze out TiN resistors to chip ground33 would serve as a future route to improving sample yield.

Measurement details

A detailed measurement setup and fridge wiring schematic is provided in the supplement. Transport current measurements are done using a battery-powered DL1211 transimpedance amplifier with 109 gain and an effective bandwidth of ≈2 kHz. The output signal is sent to a SR560 voltage preamplifier with unity gain and a 10 kHz cutoff two pole low-pass filter and sampled by an NI-DAQ 6216 with sampling rates between 1 and 10 kSa/s. Microwave characterization measurements of the resonator are done using an Agilent N5230A vector network analyzer. Homodyne detection of the double quantum dot via the resonator are done using an Agilent E8257D PSG Analog Signal Generator with a power level corresponding to ~−95 dBm on chip (≈1.7 × 104 intracavity photons). The signal is amplified by cryogenic and room temperature amplifiers, filtered, and then demodulated by a Marki 0416 IQ mixer. The demodulated DC voltages are filtered and sent to a pair of SR560 voltage preamplifiers at unity gain with a 100 kHz two pole low-pass filter and are then sampled by an NI-DAQ 6216 at 100 kSa/s with 5 kSa per point for the data shown resulting in an SNR ≈ 5.36.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Code availability

All data analysis code and COMSOL simulation are available from the corresponding author on reasonable request.

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Acknowledgements

We thank L.F. Edge for providing the Si/SiGe heterostructure used in this work. We acknowledge discussions on modeling 3D architecture structures in COMSOL with E. Leonard, M. Beck, and C. Liu. We acknowledge helpful discussions with J. Kerckhoff. Research was sponsored in part by the Army Research Office (ARO) under Grant Numbers W911NF-17-1-0274. We acknowledge the use of facilities supported by NSF through the UW-Madison MRSEC (DMR-1720415) and the MRI program (DMR-1625348). Work done at MIT Lincoln Laboratory was funded in part by the Assistant Secretary of Defense for Research & Engineering under Air Force Contract No. FA8721-05-C-0002. The views, conclusions, and recommendations contained in this document are those of the authors and are neither necessarily endorsed nor should they be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO) or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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N.H., D.R., D.Y., J.L.Y., W.D.O., R.M. and M.A.E. designed and planned the experiment. N.H. fabricated the quantum dot chip and performed the measurements. D.R., D.Y., J.L.Y. and R.D. developed and fabricated the TiN resonators and the multichip bonding process. R.D. performed the indium bump bonding between the resonator and qubit chips. N.H., R.M. and M.A.E. wrote the manuscript with input from all the other authors.

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Correspondence to M. A. Eriksson.

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Holman, N., Rosenberg, D., Yost, D. et al. 3D integration and measurement of a semiconductor double quantum dot with a high-impedance TiN resonator. npj Quantum Inf 7, 137 (2021). https://doi.org/10.1038/s41534-021-00469-0

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