Kinetic inductance compact resonator with NbTiN micronwires

Superconducting microwave resonators are fundamental and versatile devices. With the advancement of superconducting micro-nano-processes in the last decades, high performance superconducting microwave resonators have been gradually realized.¹ Due to the ultra-low noise provided by the superconductivity and low temperature environment, a range of phenomena in superconducting microwave resonators can be harnessed. This enables advancements in fields such as quantum computing,² sensing,³ and parametric amplifying,⁴ with the quantum noise being less than or equal to the limit.

In a superconducting microwave resonator, the introduction of a nonlinear element such as a Josephson junction, or the kinetic inductance (KI) from a superconductor, gives rise to distinguishable energy levels, with the lowest energy gap described by the so-called Kerr nonlinearity. Josephson junctions can typically provide Kerr nonlinearities in excess of 1 MHz,⁵ which is important for qubit state differentiation. However, the dynamic range of Josephson junction based parametric amplifiers is limited by the undesired higher order nonlinear effects.⁶ In addition, Josephson junctions are sensitive to magnetic fields, and the Josephson inductance may be unexpectedly tuned by the complex electromagnetic environment, which limit the application scenarios for Josephson junction-based devices. Devices using the kinetic inductance of superconducting films demonstrate magnetic resistance, and the Kerr nonlinearity can be varied from^7,8 10⁻⁷ Hz to 100 kHz by the choice of materials and dimension design. Moreover, the nonlinearity effects caused by the material properties are less sensitive to the background magnetic noise.⁹ Thus, devices based on kinetic inductance have attracted increasing research interest, ranging from qubits¹⁰ to amplifiers⁴ and detectors.¹¹ High kinetic inductance superconducting films of NbN,¹² NbTiN,¹³ granular Al,¹⁰ and TiN⁸ were studied recently. As a ternary compound, NbTiN presents rich tunability in normal resistivity, critical temperature, crystallographic directions, and so on.¹⁴ These features make it achieve a high quality factor in resonators^15, $(>106)$⁠, high saturation input power in parametric amplifiers¹⁶  $(>−40dBm)$⁠, and high detection efficiency in single-photon detectors¹⁷ (nearly 95%).

For T ≪ T_c, the kinetic inductance of the superconducting films can be obtained⁴ as $Lk(I)=Lk(0)(1+I2I∗2+⋯)$⁠, where I_∗ sets the scale of the quadratic nonlinearity and is expected to be of the same order as the critical current. It shares the same form as the kinetic inductance of Josephson junctions. For the Josephson junction parallel circuit, its RCSJ¹⁸ (resistively and capacitively shunted junction) equation has the form of that of a driving cubic Duffing oscillator, in which the energy of nonlinear inductance corresponds to the potential energy. A driving cubic Duffing oscillator with mean-field approximation¹⁹ has been well-studied mathematically, and the bifurcation phenomenon is quite generic in nonlinear oscillators provided certain conditions are satisfied.²⁰ Moreover, the basic model resulted in a lot of interesting research,^21–25 such as the JPA (Josephson parametric amplifier), JBA (Josephson bifurcation amplifier), and JCA (Josephson chirped amplifier). Therefore, a pivotal inquiry pertaining to the strategic design and effective utilization of the nonlinear properties inherent in thin-film superconductors is carried out.

In this study, we characterized NbTiN films with large sheet kinetic inductance and experimentally demonstrated a high quality nonlinear resonator with a Kerr coefficient of around 10 Hz. Our experimental results suggest that by modulating the sputtering parameters, the NbTiN films exhibit high normal state resistivity and their kinetic inductors can be converted to super-inductors for microwave resonators.²⁶ Furthermore, the kinetic inductance of the NbTiN films can introduce sufficient Kerr-type nonlinearity to our resonators, which displayed the bifurcation and jump phenomenon precisely as seen in the cubic Duffing oscillator.

We deposited the NbTiN films by DC magnetron sputtering onto intrinsic, high-resistivity (⁠$≥20kΩ⋅$cm), ⟨100⟩-oriented 100 mm Si wafers in a high vacuum chamber with the base pressure down to 1.0 × 10⁻⁷ Pa. By changing the DC current flowing through the Nb target and the Ti target separately, we can control the deposition rate and the components of the thin films. The Ar/N₂ mixture ratio can significantly influence the characterization of the final films as well. To obtain a prominent kinetic inductance, the films are deposited at a thickness of ∼15 nm.

The transport properties of the thin films were measured by the standard four-probe method in a Quantum Design Physical Property Measurement System, down to T = 20 mK with a DC excitation of $≤1μ$A, and the thickness of the thin films were measured by x-ray reflection in a Bruker D8 Discover. Table I shows the deposition parameters and the characterization of the NbTiN films. In the resistivity and temperature curve, we define the normal resistance at 20 K and the superconducting critical temperature T_c at the half-height points of superconducting and normal transition, as shown in Fig. 1. The results indicate that increasing the flow of N₂ with a fixed flow of Ar can decrease the critical temperature of the NbTiN thin films and conversely increase the resistivity. For high nitrogen contents of NbTiN films, the same trends have been observed in many studies.^27,28 Similarly, the reduction of current through the targets can increase the resistivity and decrease the critical temperature, especially for a comparison group with a high nitrogen component, and result in a larger kinetic inductance. The injection of nitrogen atoms alters the lattice and electronic structure of the material, increasing lattice distortion and grain boundary scattering, which affects its superconductivity.¹⁴ 

According to the Bardeen–Cooper–Schrieffer (BCS) theory,²⁹ the kinetic inductance per square L_k□ of the film is L_k□(T = 0) = ℏR_□/πΔ(0), where R_□ is the normal state resistance per square when the film is in the normal state and Δ(0) ≃ 1.764k_BT_c is the superconducting energy gap at T = 0 K. We estimated the kinetic inductance per square as shown in Table I, and it is reasonable to let Δ(T) ≈ Δ(0) for³⁰  $Δ(T)=Δ(0)tanh[1.74(1−T2Tc2)0.5]$⁠, for which the temperature in the dilution refrigerators can reach 10 mK. The changes in the quasiparticle density of states would deviate from the BCS prediction,³¹ which are induced by the short elastic scattering length of the high-resistive electromagnetic response of NbTiN. However, this significant deviation occurs at higher temperatures (almost 0.2T_c), and it still agrees well with the BCS prediction in the low-temperature range.

The fabrication process for our device mainly consists of three steps. First, we deposited an almost 12-nm-thick NbTiN film onto an intrinsic, high-resistivity silicon wafer. Second, the NbTiN micronwires and the marks were defined using a direct-write laser lithography system (Heidelberg DWL66+) using a positive AZ703 photoresist, and the pattern was then transferred by dry etching using reactive ions of CF₄ plasma. Finally, the capacitors and CPW transmission line were delineated by a second round of photolithography. Following this, a 120-nm-thick aluminum film was deposited using ultra high vacuum e-beam evaporation systems for the lift-off process. The upper and lower layers, shown in Fig. 2(b), are connected to each other by the 5 × 5 μm² NbTiN pads on either side of the micronwire. Figure 2(c) shows the microscopic photo of the device, in which the inductor is 13.5 μm in length and 1 μm in width. The 530 × 530 μm² capacitors are separated from the ground by a 10 μm wide gap (see the supplementary material for the SEM and AFM photos). After the fabrication, the wafer was diced into 5 × 5 mm² chips and then bonded into a copper sample box. As shown in Fig. 2(d), the sample was mounted on a cold finger installed at the mixing chamber stage of a Bluefors LD400 dilution refrigerator at a base temperature of 10 mK.

Figure 3(a) shows that the measurement data for the nonlinear resonators at low on-chip power are in very good agreement with the Lorentz curve. As shown in Table II, L_k,theory is the kinetic inductance estimated by $Lk=(lw)R□ℏπΔ$ and L_k,VNA is the kinetic inductance extrapolated from VNA measurement data by substituting the capacitance simulation value. The internal quality factor of the two resonators is greater than 10⁴ despite the wafer not undergoing any treatment. For the external quality factor, the design of our resonators exhibits weak coupling. The average photon number³⁶ in the resonators can be calculated by $n=4Q2PinQcℏωr2$⁠, where P_in is the on-chip power and ω_r is the resonator angular frequency.

In order to extract the Kerr nonlinearity coefficients, we performed power scans of the devices to be able to estimate the average microwave photon number in the resonator $n$ and resonant frequency shift Δf, and the linear fitting result of the Kerr nonlinearity coefficients is shown in Fig. 3(c). Furthermore, we took the same measurement after three months of aging in the open air, keeping the cabinet temperature at 25 °C. We observed a systematic frequency shift dip of about 10 MHz (see Table II) toward lower frequency for resonator R₁. The frequency shift could result from the oxidation on the surface of the NbTiN films, which would lead to a reduction in the thickness of the superconducting layer and hence an increase in the kinetic inductance. After exposure to air for three months, the oxide layer on the surface of the aluminum film underwent full oxidation and had reduced lattice defects that account for the increase in Q_i.

The normalized frequency ratio is shown in Fig. 4(a), defined as Ω = ω_d/ω_r, where ω_d represents the driving frequency. This graph depicts the bifurcation response, which is theoretically derived from the cubic Duffing oscillator model.²⁰ The direction of the sweep, either forward or backward, influences the distinct dynamical trajectories observed. The dotted line corresponds to unstable solutions, and the red (blue) solid line occurs during jumping when sweeping forward (backward). We performed forward and backward continuous wave sweep in the device. The transmission response curve in Fig. 4(b) shows the hysteresis region at the −64dBm on-chip power.

In conclusion, we first study the superconducting characterization of the NbTiN films. Reducing the target current and increasing the N₂ flow rate can result in a higher resistivity, which corresponds to higher kinetic inductance. Then, we designed and fabricated nonlinear superconducting lumped resonators with strong coupling. The nonlinear resonators demonstrated a high internal quality factor (Q_i ≥ 10⁴) and Kerr coefficients (K/2π ∼10 Hz). By reducing the inductor linewidth to tens of nanometers, the Kerr nonlinearity is expected to reach several hundred kHz. The response curve displays the bifurcation hysteresis region at high power. This can help us understand more about the nonlinear mechanism of the KI-resonator. Furthermore, the KI-resonator is a promising candidate for high saturation power parametric amplifiers due to the high intrinsic quality factor and wide Kerr nonlinearity range.