Surpassing the Resistance Quantum with a Geometric Superinductor

Superinductors have a characteristic impedance exceeding the resistance quantum $R_Q\approx 6.45\mathrm{k}\mathrm{\Omega }$, which leads to a suppression of ground-state charge fluctuations. Applications include the realization of hardware-protected qubits for fault-tolerant quantum computing, improved coupling to small-dipole-moment objects, and the definition of a new quantum-metrology standard for the ampere. In this work, we refute the widespread notion that superinductors can only be implemented based on kinetic inductance, i.e., using disordered superconductors or Josephson-junction arrays. We present the modeling, fabrication, and characterization of 104 planar aluminum-coil resonators with a characteristic impedance up to 30.9 $\mathrm{k}\mathrm{\Omega }$ at 5.6 GHz and a capacitance down to $\le 1$ fF, with low loss and a power handling reaching ${10}^8$ intracavity photons. Geometric superinductors are free of uncontrolled tunneling events and offer high reproducibility, linearity, and the ability to couple magnetically—properties that significantly broaden the scope of future quantum circuits.

Figure 1

(a) An analytical comparison of the characteristic impedance (blue) and first resonance frequency (yellow) for a straight wire ($d/w=10$, dashed) and a coil ($p=1\mu \mathrm{m}$, $\rho =1$, solid) in vacuum (${ϵ}_{\mathrm{eff}}=1$). While the mutual inductance of the coil allows for an impedance increase with length, it only reduces the resonance frequency by a constant factor. (b) The normalized current distribution for a coil with $p=1\mu \mathrm{m}$, $n=60$, and $d_{\mathrm{in}}=6\mu \mathrm{m}$. The simulated points (yellow, the line is a guide to the eye) follow approximately the analytical model for a $\lambda /2$ resonator (blue). The inset shows the current-amplitude distribution, where red corresponds to high current values and blue to low values. (c) The admittance-frequency response for the same coil in vacuum. The yellow dots are obtained from FEM simulations. The zero crossing (black dot) represents the first resonance frequency. The blue line is the admittance for a LC circuit, in agreement with the simulated data at low frequencies (dark yellow area). Above, $Y$ displays additional resonances, which can be taken into account with the equivalent circuits shown in the insets. The coil resonator can be modeled as an LC circuit around $f_0$, below which it behaves as a pure inductor.

Figure 2

(a) The main nanofabrication steps for each substrate (for details, see Appendix pp1). (b) A scanning-electron-microscopy image of the coil resonators inductively coupled to the shorted coplanar waveguide feed line. (c) An enlarged view of an aluminum-coil resonator with pitch 300 nm on SOI-BE. (d) A schematic of the measurement setup. The sample is bonded to a printed circuit board, mounted in a copper sample box and cooled down to 10 mK in a cryogenic dilution refrigerator. It is shielded by two magnetic shields and a radiation shield. The incoming signal from the vector-network analyzer (VNA) is attenuated by approximately 82 dB (considering attenuators and cable losses) and passes through one low-pass filter (LPF) and a circulator. Two ECCOSORB filters are attached to the input and output of the circulator. The outbound signal passes through another circulator for isolation and a band-pass filter (BPF) before being amplified by a low-temperature amplifier (HEMT) and further room-temperature amplifiers.

Figure 3

(a) The measured $Q_i$ of a representative overcoupled coil ($p=300$ nm, $n=155$, $Q_e=6\times {10}^4$ on SOI-BE) as a function of the mean intraresonator photon number $n_P$. The blue dots are values extracted from a Lorentzian fit and the band represents the 90% confidence interval. The yellow line is a fit to a TLS model. The inset shows the frequency shift $\delta f=0.24$ mHz per photon, obtained from a linear fit (yellow) of the measured data (blue dots). (b),(c) The blue dots are measured values for the same device as in (a). The yellow lines represent the fits to a BCS model, which yields $L_k=58.0$ nH.

Figure 4

The first row shows all measured resonance frequencies for coils with different pitches on the three different substrates. The data follow the trend of Eq. (5), with ${ϵ}_{\mathrm{eff}}$ as the only fit parameter. The second row represents the characteristic impedances calculated using the formula $Z_C=2\pi f_0(L_g+L_k)$, where $f_0$ is the measured resonance frequency of each coil and $L_g$ and $L_k$ are from Eqs. (2) and (4), respectively. The lines are deduced from the corresponding fit in the first row. While it is possible to surpass $R_Q$ on silicon, lower-permittivity substrates allow for significantly higher frequencies.

Figure 5

(a) The extracted capacitance per unit radius for three different substrates. The dots represent the average capacitance that would be added by increasing the radius of the coil by 1 $\mu \mathrm{m}$. The larger error bar for SOI results from the pitch dependence shown in the inset. (b) The inferred scaling of the characteristic impedance as a function of the pitch at a fixed frequency of $f_0\approx 10.7$ GHz. The dashed lines indicate the impedance scaling for each substrate (silicon in yellow, SOI in green, SOI-BE in blue) considering the average ${ϵ}_{\mathrm{eff}}$ value extracted from the fits in Fig. 4. The bands account for the permittivity uncertainty from the fit. For SOI, the strong pitch dependence of ${ϵ}_{\mathrm{eff}}$ is taken into account with a quadratic interpolation.