High-kinetic inductance NbN films for high-quality compact superconducting resonators

TL;DR

This paper demonstrates high-quality, compact NbN superconducting resonators with high kinetic inductance, stable performance over time, and potential for scalable quantum circuits.

Contribution

It introduces NbN thin-film resonators with high internal quality factors and high impedance, suitable for scalable quantum technologies.

Findings

Internal quality factor above 10^5

High impedance (>2kΩ) in compact footprint

Stable performance over nine months

Abstract

Niobium nitride (NbN) is a particularly promising material for quantum technology applications, as entails the degree of reproducibility necessary for large-scale of superconducting circuits. We demonstrate that resonators based on NbN thin films present a one-photon internal quality factor above 10$^5$ maintaining a high impedance (larger than 2k$\Omega$), with a footprint of approximately 50x100 $\mu$m$^2$ and a self-Kerr nonlinearity of few tenths of Hz. These quality factors, mostly limited by losses induced by the coupling to two-level systems, have been maintained for kinetic inductances ranging from tenths to hundreds of pH/square. We also demonstrate minimal variations in the performance of the resonators during multiple cooldowns over more than nine months. Our work proves the versatility of niobium nitride high-kinetic inductance resonators, opening perspectives towards the fabrication of compact, high-impedance and high-quality multimode circuits, with sizable interactions.

Tables

Table 1. Table 1: Measured and estimated main parameters of the deposited superconducting thin films. The kinetic inductivity of the films is estimated according to the procedure presented in Appendix  A .

Film Properties
Ar/N${}_{\text{2}}$	$T_C$	$R_{\square }$	$L_{\text{k,0}}$	$j_c$
(sccm)	(K)	($\Omega$/$\square$)	(pH/$\square$)	(A/cm${}^{\text{2}}$)
80/0.5	5.0	106	34.5	–
80/1	5.2	122	38.2	$1.23\times {10}^6$
80/2	7.3	151	34.9	$1.36\times {10}^6$
80/3	7.5	196	44.4	$1.66\times {10}^6$
80/4	6.5	225	52.5	$1.51\times {10}^6$
80/5	6.0	267	57.2	$1.21\times {10}^6$
80/6	5.8	310	76.8	$1.05\times {10}^6$
80/7	5.6	362	91.3	–
80/8	4.2	518	173.3	–

Table 2. Table 2: Superconducting devices main parameters for the films presented in Tab.  1 . For each film composition and resonator inductor width, the table reports only the values of the devices closest to critically coupled conditions ( κ ≈ γ 𝜅 𝛾 \kappa\approx\gamma ).

Device Design		Device Properties
	Width	${\omega }_0/2\pi$	$\kappa /2\pi$	$\gamma /2\pi$	$K/2\pi$	$F{\delta }_{\text{TLS}}^0$	$n_c$	$\Delta f_{\text{age}}$
ID	(nm)	(GHz)	(kHz)	(kHz)	(Hz)	$(\times {10}^{-5})$		(MHz)
801-500	500	5.58	86.95	57.52	- 2.40	1.02 $\pm$ 0.16	34.14 $\pm$ 12.97	3.42
801-250	250	4.95	112.87	68.00	- 19.58	1.40 $\pm$ 0.05	9.69 $\pm$ 4.35	–
802-500	500	5.87	74.51	61.77	- 3.05	1.00 $\pm$ 0.03	47.07 $\pm$ 11.08	3.78
802-250	250	5.17	40.59	66.21	- 5.56	1.22 $\pm$ 0.27	42.84 $\pm$ 48.82	–
803-500	500	5.95	91.88	95.96	- 4.17	1.83 $\pm$ 0.12	1.74 $\pm$ 1.58	4.83
803-250	250	5.15	50.42	66.66	- 11.50	1.31 $\pm$ 0.07	44.38 $\pm$ 16.56	–
804-500	500	5.49	123.21	95.31	- 7.13	1.69 $\pm$ 0.20	77.03 $\pm$ 34.87	4.00
804-250	250	4.74	45.52	90.26	- 9.59	1.88 $\pm$ 0.094	64.42 $\pm$ 22.21	–
805-500	500	5.72	105.78	91.09	- 5.77	1.53 $\pm$ 0.15	47.82 $\pm$ 29.05	3.76
805-250	250	5.06	59.69	73.91	- 13.94	1.41 $\pm$ 0.07	51.64 $\pm$ 14.33	–
806-500	500	4.93	76.049	111.30	- 5.17	2.16 $\pm$ 0.17	48.56 $\pm$ 18.93	3.28
806-250	250	4.37	64.83	132.39	- 14.88	2.84 $\pm$ 0.19	22.71 $\pm$ 12.56	–
807-500	500	6.47	64.81	42.17	- 18.46	0.67 $\pm$ 0.05	94.46 $\pm$ 50.58	–
807-250	250	6.59	49.06	62.92	- 82.39	1.43 $\pm$ 0.63	2.23 $\pm$ 4.27	–
808-500	500	3.34	45.63	233.55	- 6.23	6.72 $\pm$ 0.29	30.84 $\pm$ 12.42	–

Equations

$$L_{\text{k}}(T)=\frac{R_{\square }\hslash }{\pi \Delta }\frac{1}{\tanh (\frac{\Delta }{2k_{B}T})}.$$

$$H=\hslash {\omega }_0{a}^{†}a+\hslash \frac{K}{2}(a^{†}a{)}^2,$$

$$S_{21}=1-\frac{\kappa }{\kappa +\gamma }\frac{e^{i\varphi }}{\cos \varphi }\frac{1}{1+2i(\delta -\xi n)},$$

$$K\propto \frac{L_{\text{k,0}}{\omega }_0^2}{(j_{c}wt{)}^{n_{\text{fr}}}},$$

$$\frac{1}{Q_i}=$$

$$+\frac{\alpha }{\pi }\sqrt{\frac{2\Delta }{h{f}_r}}\frac{n_{\text{qp}}(T)}{n_s(0)\Delta }.$$

$$\frac{\Delta f}{f_r}=$$

$$\displaystyle-\ln{\frac{hf_{r}}{2\pi k_{B}T}}\bigg{)}-\alpha\frac{\Delta L_{\text{k}}}{L_{\text{k}}},$$

$$\frac{\Delta L_{\text{k}}}{L_{\text{k}}}=\frac{\lambda^{2}(T)-\lambda^{2}(0)}{\lambda^{2}(0)}\propto\bigg{(}\frac{k_{B}T}{\Delta}\bigg{)}^{4}.$$

$$H=\hslash {\omega }_0{a}^{†}a+\hslash \frac{K}{2}(a^{†}a{)}^2,$$

$$S_{21}=1-\frac{\kappa }{\kappa +\gamma }\frac{e^{i\varphi }}{\cos \varphi }\frac{1}{1+2i(\delta -\xi n)},$$

$$\delta \equiv \frac{{\omega }_{\text{d}}-{\omega }_0}{\kappa +\gamma },\xi \equiv \frac{|{\tilde{\alpha }}_{in}{|}^{2}K}{\kappa +\gamma },n\equiv \frac{|\alpha {|}^2}{|{\tilde{\alpha }}_{in}{|}^2},$$

$$\frac{1}{2}=\bigg{(}\delta^{2}+\frac{1}{4}\bigg{)}n-2\delta\xi n^{2}+\xi^{2}n^{3}.$$

$$|{\tilde{\alpha }}_{in}{|}^2=\frac{\kappa }{(\kappa +\gamma {)}^2}\frac{P_{\text{in}}}{\hslash {\omega }_0},$$

$$K=\xi \frac{(\kappa +\gamma )}{|{\tilde{\alpha }}_{in}{|}^2}.$$

$$L_{\text{k}}$$

$$\displaystyle~{}\approx L_{\text{k,0}}~{}\bigg{(}1+\frac{1}{n_{\text{fr}}}\bigg{(}\frac{I}{I_{c}}\bigg{)}^{n_{\text{fr}}}\bigg{)}.$$

$$K\propto \Delta L{\omega }_0^2\propto \frac{L_{\text{k,0}}{\omega }_0^2}{I_c^{n_{\text{fr}}}},$$

Full text

I Introduction

The possibility of tuning the magnitude of the inductance in superconducting circuits is paramount to achieve the high-degree of control and flexibility required for quantum technology Kjaergaard et al. (2020). Two platforms have been mostly used to achieve high values of inductance: arrays of Josephson junctions (JJs) Scigliuzzo et al. (2022); Manucharyan et al. (2009); Kuzmin et al. (2019); Puertas Martínez et al. (2019) or high–kinetic inductance disordered thin films Annunziata et al. (2010); Maleeva et al. (2018). JJ devices are characterized by very low dissipation rates, and for this reason they are routinely used in circuit quantum electrodynamics (cQED) Kono et al. (2020); Wang et al. (2022). Usually, JJ-based circuits still exhibit large nonlinearities. To dilute such nonlinearity suited for applications such as quantum–limited parametric amplifiers, arrays of JJs sacrifice compactness and add fabrication overhead White et al. (2015); Macklin et al. (2015).

Kinetic inductance is a promising novel resource for quantum technology Winkel et al. (2020); Day et al. (2003); Macklin et al. (2015); Ho Eom et al. (2012). For instance, superconducting high–kinetic inductance (high–$L_{\text{k}}$) thin films have been used for several cryogenic applications, ranging from detectors Day et al. (2003); Gol´tsman et al. (2001); Cabrera et al. (1998) to amplifiers Ho Eom et al. (2012); Parker et al. (2022), and filters Liu and Houck (2017); Sigillito et al. (2017). In dirty superconductor thin films, the inductance depends on the Cooper pairs carrier density Annunziata et al. (2010), and it can be tuned by controlling the chemical composition of the films during the deposition. From a cQED perspective, the advantages of high–$L_{\text{k}}$ are manifold. Thin films give the designers the freedom to operate with low– or high–impedance, and thus a higher range of achievable capacitive-coupling between different circuital elements Devoret, Girvin, and Schoelkopf (2007). In turn, this allows exploring several regimes of light-matter interaction, from the strong to more exotic ultrastrong couplings Kockum et al. (2019); Forn-Díaz et al. (2019). Given the compact nature of high–$L_{\text{k}}$ film resonators, thin films can reduce the dimensions of both readout and control apparatus, and facilitate the realization of many-modes quantum devices Niepce, Burnett, and Bylander (2019). Key to the success of this technology within a quantum framework is attaining regimes of low internal losses, where long coherence times can be maintained.

Many disordered superconductors have shown high–kinetic inductance, e.g., NbN Anferov et al. (2020); Niepce, Burnett, and Bylander (2019), NbTiN Parker et al. (2022); Samkharadze et al. (2016); Ho Eom et al. (2012), and TiN Leduc et al. (2010), or materials composed by microscopic effective Josephson arrays such as the emergent granular Aluminum (grAl) Grünhaupt et al. (2019). Several groups manufactured weakly-disordered thin films that present considerable quality factors Vissers et al. (2010); Leduc et al. (2010). Consequently, thin films with moderate $L_{\text{k}}$ proved useful for quantum computing tasks Krantz et al. (2019), in particular for optical communication Shaw et al. (2017), quantum key distribution Takesue et al. (2007); Boaron et al. (2018), and quantum teleportation Bussiéres et al. (2014).

Large-scale quantum applications will, however, also require a wide range of $L_{\text{k}}$ tunability and high control and reproducibility of the superconducting building blocks. In this work, we demonstrate that we fully control the kinetic inductance of NbN films (from $30$ to $170\,\rm{pH}/\square$), maintaining a very-high resonator quality in excess of $10^{5}$ at low photon number, with a high and reliable degree of reproducibility. We identify the saturable two-level systems (TLS) as the main limiting factor to single-photon lifetime Scigliuzzo et al. (2020). Furthermore, our devices show no significant film degradation (ageing) over the course of different cooldowns nine months apart.

II Device Design and Fabrication

We fabricate planar lumped LC resonators etching 13 nm-thick NbN film, with typical impedance ${Z=\sqrt{L_{r}/C_{r}}}$ of 2 k$\Omega$, where $L_{r}$ and $C_{r}$ are inductance and capacitance of the resonator respectively. The fabrication begins with a 2 minutes dip in 40% HF bath to remove the native oxide and possible contamination from the surface of an intrinsic, high–resistivity ($\geq 10~{}k\Omega$cm), $\braket{100}$–oriented 100 mm Si wafers. There follows an NbN films bias sputtering Dane et al. (2017) at room temperature in a Kenosistec RF sputtering system. After subsequent deposition of Ti/Pt alignment markers by optical lift-off process and dehydration step at 150∘C for 5 minutes, 80 nm-thick CSAR positive e-beam resist is spin coated at 4000 rpm on the wafer, and baked at 150∘C for 5 minutes. With an electron beam lithography (Raith EBPG5000+ at 100 keV) step, the devices are patterned on the resist through development in amyl acetate for 1 minute, followed by rinsing in a solution 9:1 MiBK:IPA. The pattern is then transferred to the NbN using CF${}_{\text{4}}$/Ar mixture and reactive ion etching with a power of 15 W for 5 minutes. The resist is stripped by means of Microposit remover 1165 heated at 70∘C. Finally, the wafer is coated with 1.5 $\upmu$m AZ ECI 3007 positive photolithography resist for devices protection and diced.

The advantages of bias sputtering Dane et al. (2017), i.e., application of a RF bias voltage on the substrate, resides in ion bombardment during the film deposition, which causes a reduction of superconducting critical temperature and grain size, permitting the deposition of a polycrystalline material. The polycrystallinity improves the superconductor homogeneity and it eases the device realization. The film behaves as an amorphous material with respect to etching procedures, and simultaneously maintains the advantageous properties of crystalline superconductors. For example, the low electron–phonon interaction time is barely changed, with important repercussions, such as enhanced maximum count rate in superconducting single–photon detectors Dane et al. (2017). To optimize the films properties, we fine-tune the nitrogen to argon partial pressures in the chamber across several fabrication runs, with pressure and substrate bias chosen to give the lest roughness in the films. These steps also improve the reproducibility and yield of the samples.

III Experimental Setup

After the fabrication, we glue the chips on a copper support with PMMA and wire-bond it to a customized printed circuit board. The copper support is then mounted on the cold finger installed at the mixing chamber stage of a BlueFors LD250 dilution refrigerator at a base temperature of 10 mK, see Fig. 1(a).

In order to test multiple devices in a single cooldown and to increase the experimental throughput, the devices are connected in transmission configuration through cryogenically operating coaxial switches (Radiall, R573 series), sharing both input and output lines [see Fig. 1(a)].

Each sample is composed by 7 resonators, capacively coupled to a common $50\,\Omega$ coplanar transmission line, as Fig. 1 (b). The output line is connected to two cryogenic insulators (Low Noise Factory, LNF-ISISC4-8A series) to attenuate thermal noise injection from the amplifier, and a cryogenic low–noise amplifier (Low Noise Factory, LNF-LNC4-8C series) operating at the 4 K stage of the cryostat. Input and output lines are then connected at room temperature to a Vector Network Analyzer (Rohde & Schwarz, ZNA26 series) to acquire the scattering parameters.

The notch (or hanger McRae et al. (2020)) configuration of resonators has been chosen for a precise estimation of the internal quality factor of the devices – they self-calibrate with respect to the transmission baseline – and it allows frequency multiplexing McRae et al. (2020). Moreover, the microwave feedline can be probed in DC to estimate the resistance–temperature characteristic curve and the critical temperature $T_{C}$. The latter has been measured with a closed loop cryostat (PhotonSpot inc.) with 800 mK base temperature.

IV Results

IV.1 NbN films composition, critical temperature and kinetic inductance

The kinetic inductance per square $L_{\text{k}}$ of the film can be expressed as Annunziata et al. (2010):

[TABLE]

Here, $R_{\square}$ is the sheet resistance when the film is in the normal state, $T$ is the superconducting film temperature, and $\Delta$ is the superconducting band gap that, according to BCS theory, can be approximated to $\Delta\simeq\text{1.764}~{}k_{B}T_{C}$ for $T\ll T_{C}$ with $T_{C}$ the critical temperature.

Fig. 2 reports $T_{C}$ and $L_{\text{k}}(T)$, both measured and estimated according to Eq. (1), as a function of the N2 flow during the spattering deposition. The deposition conditions, critical temperatures, sheet resistance, estimated kinetic inductance, and critical current density $j_{c}$ of the thin films are reported in Tab. 1. Increasing the N2 concentration in the films causes an increment of impurities embedded in the films during the sputtering process. At low N2, we notice a raise in the critical temperature. While also the resistivity increases, the ratio between these two in Eq. (1) does not compensate, causing a little drop in $L_{\rm k}$ in the proximity of the stochiometric condition, which for this deposition setting are found approximately at 2.5 sccm N2 flux. Larger N2 concentration increases the film disorder, causing a raise of film resistivity and a drop of critical temperature, both boosting the kinetic inductance.

IV.2 Internal quality factor and Kerr nonlinearity

We model each cavity as a Kerr nonlinear resonator, described by the Hamiltonian:

[TABLE]

where $\hat{a}$ ($\hat{a}^{\dagger}$) is the bosonic creation (annihilation) operator, $\omega_{0}$ is the resonant frequency of the cavity, and $K$ the Kerr nonlinearity. These devices are also characterized by photon loss events, whose rate is given by $(\kappa+\gamma)$, with $\kappa$ the external coupling and $\gamma$ the internal losses. The resonator is driven at a frequency $\omega_{\text{d}}$ of intensity $P_{\rm in}$.

For our devices, we measure the transmitted power $S_{21}$ of the resonator hanged to the feedline. To extract the parameters of a Kerr resonator in hanger configuration, we use input–output theory Eichler and Wallraff (2014), and obtain:

[TABLE]

where the interdependent variables $\phi$, $\delta$, and $\xi$ depend on $K$, $\kappa$, $\gamma$, $\omega_{0}$, on the drive frequency $\omega_{\text{d}}$, on the photon number $n$, and on other internal parameters of the resonator that can be independently measured, as detailed in Appendix B. As it will be detailed later, the internal dissipation rate $\gamma$ has a nontrivial power dependence. Hence, we perform power scans of the devices to be able to estimate the total loss rate $(\kappa+\gamma)$ with respect to the average number of photons in the resonator $\braket{n_{\text{ph}}}$. We first estimate the resonator frequency $\omega_{0}$, as it corresponds to the dip in frequency of $S_{21}$ in the low-power regime. Then, to retrieve $\kappa$, $\gamma$ and $K$, we use a global fit routine at all powers, to increase the accuracy of the extrapolated values. In the linear regime at low-power, we also benchmark our results for $\kappa$ and $\gamma$ using the python package by Probst et al. Probst et al. (2015), finding no significant discrepancies with respect to our routine.

In order to precisely estimate the internal quality factor and reduce fit uncertainty, it is more convenient to approach a critical coupling condition $\kappa\approx\gamma$, where the internal quality factor approximately matches the external coupling McRae et al. (2020). As the internal quality factor is not known a priori, we take full advantage of notch configuration to engineer multiple couplings $\kappa$ for devices on the same chip. Fourteen hanger resonators are fabricated per each film, seven with inductor width of 250 nm and seven with inductor width of 500 nm. In Fig. 3 we present the measured and fitted transmitted power $S_{21}$ using Eq. (3) through the feedline for one of the critically coupled resonators as a function of the probe frequency and for several input powers.

Figure 4(a) reports the extracted quality factors of the 105 tested resonators in the linear regime (for Kerr-induced frequency shift much smaller than the resonator line width). The datapoints represent the averaged internal quality factor estimated for the seven resonators with different couplings $\kappa$ to the feedline [see Fig. 7(a) for the plot of the extracted $Q_{i}$ for all the measured resonators]. The error bars represent the standard deviation between the internal quality factor of the different devices. We also include in the error bar the uncertainty deriving from the global fit of each resonator, although we notice that they are negligible with respect to the parameter spreading due to fabrication.

To quantify the self-Kerr nonlinearity $K$ we reach regimes of large photon numbers, where the effect of the nonlinearity is comparable to the resonator linewidth. In Fig. 4(b), we reported the measured $K$ of the critically coupled tested resonators ($\kappa\approx\gamma$) for each film and width. The data for the resonators with 500 nm (250 nm) wide inductor are shown with triangles (circles). The self-Kerr nonlinearity increases with $L_{\text{k}}$, with a clear offset between the two inductor widths, as also reported in Anferov et al. (2020). This is in agreement with the relation

[TABLE]

obtained by replacing $I_{c}=j_{c}wt$, in the Kerr equation [see Appendix B, Eq. (15)], where $j_{c}$ is the critical current density of the thin film, and $w$ and $t$ are respectively the width and thickness of the inductor wire of the resonators. From this equation, we notice that the difference in width of the nanowire inductors clearly affects the value of $K$.

To highlight the dependencies of the self-Kerr $K$ from both kinetic inductance and inductor width, we plot in Fig. 7(b) $|K|/\omega_{0}^{2}$.

IV.3 Origin of the internal dissipation

All devices, except the ones with the highest $L_{\text{k}}$, show $Q_{i}$ around $10^{5}$ in the single-photon regime. For the devices with the largest $L_{\text{k}}=170$ pH/$\square$, we argue that the nitrogen concentration of the film is approaching the limiting value of the superconductor–normal metal transition (SNT) Burdastyh et al. (2020). This argument is corroborated by the observation that the resonators with 250 nm wide inductors did not exhibit any superconducting transition while reaching base temperature, while the $w$ = 500 nm devices show a lower internal quality factor in the range of $10^{4}$ at single photon regime. These 170 pH/$\square$ resonators, however, present $Q_{i}$ in excess of $10^{5}$ at a thousand photon number, in line with the other films, suggesting that the dissipation can be mainly attributed to two-level system (TLS) hosted in highly disordered films McRae et al. (2020); Müller, Cole, and Lisenfeld (2019). While TLS nature is still unclear, it is believed to be related to atoms tunneling between two sites of a disordered solid Phillips (1972); w. Anderson, Halperin, and c. M. Varma (1972); Gao (2008).

In order to describe the various mechanisms contributing to the internal quality factor as a function of input power, we write Pappas et al. (2011); Niepce, Burnett, and Bylander (2019); Scigliuzzo et al. (2020):

[TABLE]

In this equation, $\delta_{0}$ is the residual loss rate of the resonator, i.e. the sum of the other loss contributions which are not described by TLS nor quasi-particle loss models. The second contribution is due to TLS, which according to the TLS model Gao (2008) generates a power and temperature dependent resonator loss. Here, $F$ is defined as the filling factor (the ratio between the electric field threading the TLS and the total electric field), $n_{c}$ is the characteristic photon number of TLS saturation, and $\delta^{0}_{\text{TLS}}$ is the intrinsic TLS loss. The third contribution is due to quasi-particles, where $\alpha$ is the ratio between kinetic and total inductance, $n_{\text{qp}}(T)=n_{s}(0)\sqrt{2\pi k_{B}T\Delta}e^{-\Delta/k_{B}T}$ is the temperature dependent population of quasi-particles Barends et al. (2011), and $n_{s}(0)$ is the Cooper pairs zero energy density of states. We assumed $\alpha$ to be 1, as the kinetic inductance contribution to the resonant frequency largely dominates the geometric inductance.

Fig. 5 reports the measured quality factor for representative resonators of different films, close to critical coupling conditions, $\kappa\approx\gamma$, as a function of the photon number. We then fit the model in Eq. (5), and we find that the TLS contribution dominates the internal quality factor. We report the value of $F\delta^{0}_{\text{TLS}}$ and $n_{c}$ obtained from the fits in Tab. 2. Comparing them to what previously found in Niepce, Burnett, and Bylander (2019); Yu et al. (2021), we find similar fitting values for the exponent $\beta$, close to 0.2 for all the tested devices. However, as reported in Tab. 2, the characteristic photon number of TLS saturation, $n_{c}$, is estimated to range between 30 and 100, a much larger value than the one reported for similar films in NbN Niepce, Burnett, and Bylander (2019), which we attribute to a discrepancy in the photon number expressions [see Eq. (12), in accordance with Yu et al. (2021)].

To further show the dominance of TLS, we investigated the resonator’s quality factor evolution at different operating temperatures. The cryostat was slowly warmed up from a base temperature of 15 mK to a temperature of almost 1 K in a controlled way, and the resonator spectrum was acquired at $\braket{n_{\text{ph}}}\approx 50$. The evolution of the extracted internal quality factor $Q_{i}$ and resonator resonant frequency $\omega_{0}$ are reported in Fig. 6(a).

In addition, we characterize the contribution of the TLS and quasiparticle using the frequency shift of the resonators at different temperatures [see Fig. 6(b)]. Indeed, we have that Scigliuzzo et al. (2020):

[TABLE]

where $\Psi$ is the digamma function Pappas et al. (2011) and $\frac{\Delta L_{\text{k}}}{L_{\text{k}}}$ is the kinetic inductance change. Since $L_{\text{k}}(T)=\mu_{0}\lambda^{2}(T)(l/wd)$, where $\lambda(T)$ is the temperature dependent London penetration depth, and as reported in Lee and Lemberger (1993) for $T\leq 0.4~{}T_{C}$, $[\lambda(T)/\lambda(0)-1]\propto(T/T_{C})^{2}$, we obtain

[TABLE]

From the two fits of Eqs. (5)–(6) reported in Fig. 6, we extracted a thin film critical temperature $T_{C}$ of 7.4 K, and TLS contribution $F\delta^{0}_{\text{TLS}}$ of 1.8 $10^{-5}$, both in accordance with the previously obtained results reported in Fig. 2 and Fig. 5. We notice that for higher temperature the internal quality factor increases due to the saturation of TLS fluctuators, and then it starts dropping due to the losses caused by quasi-particles population. Due to the large $T_{C}$ of the characterized film ($T_{C}\approx 7.5~{}$K), this effect becomes dominant only at a temperature higher than 700 mK, i.e. at roughly 10% of $T_{C}$. The role played by TLS and quasi–particles is also confirmed by the temperature evolution of the resonator frequency shift $\Delta f$ [see Fig. 6(c)]. Below 400 mK $\Delta f$ is caused solely by TLS fluctuators, while at larger temperatures, the quasi-particles induced shift dominates.

IV.4 Ageing characterization

Finally, to address the problem of thin films ageing, in particular the effect of the niobium oxide native layer on the internal quality factor Santavicca and Prober (2015); Medeiros et al. (2019); Verjauw et al. (2021), the devices were tested after nine months from the initial measurements, and in the same exact configuration. The devices were kept wire-bonded to the PCBs and were stored in a controlled N${}_{\text{2}}$ atmosphere. We observed a systematic dip frequency shift of about $\Delta f_{\text{age}}$ ranging between 3 and 5 MHz [see Tab. 2] towards lower frequency for all the measured resonators. In Fig. 5 are collected the power scans of the internal quality factors of the tested resonators, with respect to average number of photons, measured right after fabrication [in blue] and after nine months [in red]. While for the low-$L_{\text{k}}$ devices the quality factor dropped slightly due to ageing, for larger $L_{\text{k}}$ films this effect is less pronounced, with the internal quality factor remaining almost unchanged after nine months. All the resonators remain, however, dominated by coupling to TLS, with internal quality factors at large number of photons in the excess of $10^{5}$.

V Discussion and Conclusion

We have demonstrated high ($Q_{i}>10^{5}$) internal quality factor, high–kinetic inductance superconducting resonators operating at single-photon regime based on NbN thin films superconductor. The NbN was bias-sputtered to increase kinetic inductance and device yield due to its poly-crystalline nature, at the expenses of a reduced critical temperature.

Both stochiometric and non-stochiometric NbN films were deposited by varying atmospheric deposition conditions. After characterizing the deposition rates, films of equal thickness of $\sim$13 nm were sputtered. We fabricated and characterized 105 compact LC resonators multiplexed in hanger configuration and with kinetic inductance ranging from 30 pH/$\square$ to 170 pH/$\square$, as shown in Fig. 7(a). While for low photon number the internal quality factor presents a slight dependence from kinetic inductance, at high photon numbers the resonators quality factors approach $10^{6}$, suggesting a clear TLS induced loss mechanism.

By fitting the high–power scattering parameter of the resonators we also estimate the self-Kerr nonlinearity $K$ for the different tested films. Qualitatively, as depicted in Fig. 7(b), we observe a linear dependence of $K$ with respect to $L_{\text{k}}$. The sel-Kerr $K$, being below 100 Hz, is about four orders of magnitude lower than that of resonators made with Josephson junctions arrays Krupko et al. (2018).

In conclusion, the results show a clear dominance of TLS–induced losses across the board, with lower kinetic–inductance films having larger internal quality factors, lower self-Kerr nonlinearity and less robustness with respect to ageing effects. The overall performance make the technology particularly appealing for those applications relying on low–nonlinearity, high–quality factor and large average photon number, such as parametric amplifiers, readout resonators, and photon detectors.

Contributions

S.F. conceived the experiments with inputs from P.S., developed the recipes and optimized the deposition and characterization techniques. S.F., I.N.A. and F.O. fabricated the devices. S.F., S.Y.B.d.P. and V.J. measured the resonators. S.F. analyzed the data with inputs from M.S., F.M. and P.S.. S.F., F.M. and P.S. wrote the manuscript with inputs from all authors. E.C. supervised the work.

Acknowledgements

This research was funded by Swiss National Science Foundation through the Sinergia programme grant number 177165, 2018—2022. P.S. acknowledges the project grants NCCR SNF 51AU40-1180604 and SNF project 200021-200418.

Appendix A Estimation of kinetic inductance

The kinetic inductance of the films was estimated both using classical BCS theory and via Sonnet simulations, by simulating the resonant frequency of the tested chips and comparing it to the measured value at low photon number.

The chip designs were slightly different for the various films, to have the resonant frequencies of the devices in the same frequency range, and to ensure low reflections by matching as much as possible the microwave feedline to a 50 $\Omega$ impedance line. We used overall five designs.

For each design, we simulated three resonators with Sonnet to estimate their resonant frequency at three different sheet kinetic inductivity values $L_{\text{s}}$, see Fig. 8(a). Given that the resonator capacitance remains constant, the resonant frequency should follow a $\omega_{0}\propto 1/\sqrt{L_{\text{k}}}$ dependence. By plotting the hyperbola passing through these three points, as shown in Fig. 8(b), it is possible, by measuring at low photon number the resonant frequency of the resonator under test $\omega_{0}/2\pi$, to estimate the kinetic inductance of the film. To increase the robustness of such method, we average the obtained $L_{\text{k}}$ for the three resonators [see Fig. 8], allowing us to reduce even further the uncertainty associated to the kinetic inductivity of the films.

Appendix B Equations derivation

The transmission spectra from the main feedline was measured at different input power. The nonlinear effects of the kinetic inductance of the film, produces an effective Kerr behavior of the resonators (see Fig. 3).

The nonlinear Kerr Hamiltonian for superconducting resonators can be written as Anferov et al. (2020):

[TABLE]

where $K\propto\Delta L\omega_{0}^{2}$ is Kerr–nonlinearity, $\Delta L$ inductance change and $\omega_{0}$ fundamental resonator frequency. Following the same formalism of Eichler and Wallraff (2014); Anferov et al. (2020) using input–output theory leads to the scattering parameter for hanger resonators of:

[TABLE]

with

[TABLE]

$\tilde{\alpha}_{in}\equiv\sqrt{\kappa}\alpha_{in}/(\kappa+\gamma)$, $\kappa\equiv\omega_{0}/Q_{\text{c}}$ being the coupling to the feedline, $\gamma\equiv\omega_{0}/Q_{\text{i}}$ the internal resonator losses. $\phi$ is a parameter used to take into account impedance mismatch between the feedline and the resonator [also called the $\phi$ rotation method ($\phi$RM) Gao (2008)] and $n$ is calculated as solution of the equation:

[TABLE]

Cryogenic attenuated input lines were calibrated at base temperature and an extra contribution of -8 dB was estimated from the coaxial lines, for a total attenuation of -68 dB. Nonlinear effects described hereon have been analyzed from the assumption that the input power is known within an error of $\pm$2 dB. The average photon number $\braket{n_{\text{ph}}}=|\tilde{\alpha}_{in}|^{2}$ in the resonator

[TABLE]

together with Eq. (12), allows to estimate the Kerr nonlinearity $K$ from the fitting functions parameters $\xi$ and $n$:

[TABLE]

Using the generalized relation of current dependence of kinetic inductance derived in Clem and Kogan (2012), in the limit for small $I/I_{c}$, we get:

[TABLE]

Substituting Eq. (14) in the relation for Kerr:

[TABLE]

where $L_{\text{k,0}}$ is the kinetic inductance of the superconducting film at zero bias current, $n_{\text{fr}}=2.21$ is the exponential of the fast relaxation limit function defined in Clem and Kogan (2012), $I$ is the bias current, which is proportional to the pump power $P_{\text{in}}$. In order to compare the nonlinearity of the tested devices, we have measured the critical current of the coupling feedline so to have an estimate of the critical current $I_{c}$.