On-Chip Microwave Filters for High-Impedance Resonators with Gate-Defined Quantum Dots

Circuit quantum electrodynamics (QED) employs superconducting microwave resonators as quantum buses. In circuit QED with semiconductor quantum-dot-based qubits, increasing the resonator impedance is desirable as it enhances the coupling to the typically small charge dipole moment of these qubits. However, the gate electrodes necessary to form quantum dots in the vicinity of a resonator inadvertently lead to a parasitic port through which microwave photons can leak, thereby reducing the quality factor of the resonator. This is particularly the case for high-impedance resonators, as the ratio of their total capacitance over the parasitic port capacitance is smaller, leading to larger microwave leakage than for 50-$\mathrm{\Omega }$ resonators. Here, we introduce an implementation of on-chip filters to suppress the microwave leakage. The filters comprise a high-kinetic-inductance nanowire inductor and a thin-film capacitor. The filter has a small footprint and can be placed close to the resonator, confining microwaves to a small area of the chip. The inductance and capacitance of the filter elements can be varied over a wider range of values than their typical spiral inductor and interdigitated capacitor counterparts. We demonstrate that the total linewidth of a 6.4 GHz and approximately 3-$\mathrm{k}\mathrm{\Omega }$ resonator can be improved down to 540 kHz using these filters.

Figure 1

(a) Optical image with false-color shading of the central area of the device, showing the superconducting high-impedance nanowire resonator (in red) and the QD gates of a full device (in yellow). (b) Simplified electrical circuit of the resonator and its surrounding components. The resonance modes of this device can be understood using three sections of coplanar waveguides with appropriate capacitance and inductance per unit length, ${\tilde{C}}_r$ and ${\tilde{L}}_r$, respectively. A half-wave mode $\lambda /2$ couples the DQDs at each end of the resonator in antiphase, whereas a quarter-wave mode $\zeta /4$ also exists where both DQDs are coupled in phase (only one side of the mode is shown for simplicity). The resonator is probed in transmission through the “in” and “out” ports with coupling capacitances $C_{\mathrm{in}}$ and $C_{\mathrm{out}}$. Because of the physical footprint of the DQD gates at each end, an extra capacitance $C_g\gg \{C_{\mathrm{in}},C_{\mathrm{out}}\}$ causes the microwave energy to escape from the resonator primarily through the gate fanout lines. To prevent irreversible loss, modeled here by 50-$\mathrm{\Omega }$ resistors, low-loss microwave filters are fabricated on chip to reflect the microwaves back into the resonator. Filters act as ac grounds through a bias-tee effect. The path of energy escaping the $\lambda /2$ mode is represented by red arrows, with double-ended arrows representing a reflection back into the resonator.

Figure 2

(a) Optical image of a planar filter with one line. Each filter can be thought of as an $LC$ microwave bias tee. The inductor $L_f$ consists of a superconducting high-kinetic-inductance nanowire made from the same film as the resonator. The capacitor $C_f$ has an interdigitated geometry. Both components are low loss, thanks to superconducting metals and the absence of amorphous dielectrics. The nanowire inductor is much smaller than an equivalent spiral inductor and does not require looping around a bondpad, allowing the filter to be placed closer to the active area. The interdigitated capacitor is still relatively large. (b) Optical image of a thin-film filter with 15 gate lines [with the same scale as Fig. 2]. The thin-film capacitor can be made a lot smaller than its interdigitated equivalent and straddles multiple gate lines at once, thereby dramatically reducing the footprint and simplifying the microwave hygiene. In this implementation, the top capacitor plate is electrically floating to further simplify the integration. The large $8.5C_f$ series capacitance of the capacitor plate to the ground plane acts as a short for the relevant frequencies. (c) Stitched optical image of a full device chip (4 mm by 2.8 mm) with thin-film filters. To single out the effects from the capacitive loading of the resonator by the gates while minimizing any other loss mechanisms, we use prototyping chips built from the same processed wafers as the full devices, but the quantum-dot areas do not include any of the implanted regions, the gate oxide, or the implant contact pads. The insets show electron microscope images of the nanowire resonator dc tap intersection, the prototyping fine gates mockup and full device fine gates. The mockup and full devices have identical capacitive load, $1.8\mathrm{fF}$ per DQD. (d) Material stack of a thin-film capacitor prototyping chip. See Appendix pp1 for details.

Figure 3

Summary for planar filter prototypes. The $\zeta /4$ mode frequencies lie between 3 and 4 GHz, while the $\lambda /2$ mode frequencies lie between 6 and 7.5 GHz. The experimental splits are designed to separate the problems caused by insufficient or defective filtering from those caused by poor microwave hygiene. For each prototype, up to three variants are tested. A “no gates” variant, not shown in the table, is meant as a control experiment to measure frequencies and linewidths in the absence of gate loading. Typical values yield ${\kappa }_{\zeta /4}/2\pi =0.3\mathrm{MHz}$ and ${\kappa }_{\lambda /2}/2\pi =0.9\mathrm{MHz}$ for the two modes. The “with gates” variant mimics the full devices. In certain cases, after initial measurements, extra wirebonds shorting the gates to the ground plane are added close to the resonator area and chips are then remeasured, resulting in the “with wirebond surgery” variant. This sanity check procedure is useful to verify that the failure of chips is due to insufficient filtering or poor microwave hygiene, and not due to other problems like a resonator defect. The color coding is a subjective assessment of whether or not the linewidth is optimal, with green being very close to ideal ($\lesssim 0.5\mathrm{MHz}$), yellow being not ideal but still $\lesssim 4\mathrm{MHz}$, and red being $>10\mathrm{MHz}$. For this set, $L_f\approx 114\mathrm{nH}$ and $C_f\approx 0.1\mathrm{pF}$. The “expected” column is a binary assessment of whether the linewidth should be narrow ($✓$) or broad ($\mathsf{X}$) based on the presence or absence of filtering. See the main text for discussion.

Figure 4

Summary for thin-film filter prototypes. (a) The experimental splits are designed to test the impact of the capacitor area, by changing the length of the capacitor (as shown in the inset), and include results from two very slightly different designs (v1 and v2) and ${\mathrm{Si}\mathrm{N}}_z$ deposition powers (100 W and 150 W). The capacitor $C_f=1\mathrm{pF}$ has a width of $6\mu \mathrm{m}$ (top plate width of $320\mu \mathrm{m}$) and length of $100\mu \mathrm{m}$, while the ${\mathrm{Si}\mathrm{N}}_z$ thickness is 30 nm. (b) The $\lambda /2$ mode frequencies lie between 6 and 7.5 GHz. Two groups are observed: good devices with ${\kappa }_{\lambda /2}/2\pi <1.25\mathrm{MHz}$, and poor devices with ${\kappa }_{\lambda /2}/2\pi >2\mathrm{MHz}$. The reason for their failure is unknown, but in two cases we observe that the resonance is recovered using wirebond surgery with linewidths 1.1 MHz and 0.7 MHz. The best performing device achieved ${\kappa }_{\lambda /2}/2\pi =0.540\mathrm{MHz}$, or a quality factor of 11 900. (c) Linewidths for the $\zeta /4$ mode. The dc tap capacitor for this mode is scaled up in size compared with the gate line capacitors to improve the ac grounding of the mode (see main text). While the half-wave mode is insensitive to losses through the dc tap because of its symmetry, the quarter-wave mode can lose energy through both the gate lines and the dc tap, making the contributions from the different filters convoluted for this mode. In all plots, some points are slightly offset horizontally for clarity.

Figure 5

Angled-view scanning electron microscope image of the thin-film capacitor structure overlapping a step and the ${\mathrm{Nb}}_y{\mathrm{Ti}}_{1-y}\mathrm{N}$ film.

Figure 6

Numerical model of the resonator and gate filters implemented using the software qucs. RLCG elements represent waveguides with arbitrary inductance per unit length $\mathtt{L}$ and capacitance per unit length $\mathtt{C}$ (in SI units). The corresponding device elements are delimited by dashed boxes. The model includes effects from the diamond-shaped pads at the end of the resonator narrow section, capacitive loading by the gates, and various waveguide impedances. In some cases, the waveguide dimensions $w$ and $s$ are indicated in the element name in units of microns for convenience. The narrow resonator sections res1, res2, and restap1 have $w=120\mathrm{nm}$ and $s=17\mu \mathrm{m}$. The diamond-shaped pad w6s23respad1 has $w=6\mu \mathrm{m}$ and $s=23\mu \mathrm{m}$. While the bare impedance of the narrow resonator section consisting of res1 and res2 is quite high at about $4.5\mathrm{k}\mathrm{\Omega }$, the resonator is so small that the capacitive loading by the surrounding elements brings the effective impedance down to approximately $3.2\mathrm{k}\mathrm{\Omega }$. This last value is obtained by replacing the w6s23respad1, res1, and Cg1 elements, and their symmetric counterparts, with a single RLCG element of the same total length, fixing $\mathtt{L}\mathtt{=}\mathtt{9.60}\times {\mathtt{10}}^{\mathtt{-}\mathtt{4}}$ and yielding $\mathtt{C}\mathtt{=}\mathtt{9.4}\times {\mathtt{10}}^{\mathtt{-}\mathtt{11}}$.

Figure 7

Measurement setup for the ${}^3\mathrm{He}$ system. The picture shows a wirebonded device with crossbonds mounted on the PCB.