SKIFFS: Superconducting Kinetic Inductance Field-Frequency Sensors for sensitive magnetometry in moderate background magnetic fields

Disordered superconductors such as NbTiN, TiN, and NbN have become ubiquitous in several fields of study due to their large kinetic inductance and resilience to large background magnetic fields.^1,2 Microwave kinetic inductance detectors^3–5 and superconducting nanowire single-photon detectors^6,7 fabricated from kinetic inductors are now routinely used in astronomy and imaging. Kinetic inductors can also be used in applications such as current-sensing,⁸ magnetometry,⁹ parametric amplification,^10,11 generation of frequency combs,¹² and superconducting qubits.^13,14

In this work, we take advantage of the kinetic inductance of a thin film of NbTiN to fabricate lumped-element resonators whose resonance frequencies are strongly dependent on the perpendicular magnetic field, changing by as much as 27 MHz for a field change of 1.8 μT. We demonstrate a method for the real-time measurement of AC magnetic fields based on phase-sensitive readout of microwave transmission through the resonators, finding a detection sensitivity of 1°/nT. Our Superconducting Kinetic Inductance Field-Frequency Sensors (SKIFFS) are able to operate in perpendicular background magnetic fields at least as large as 0.2 T and may find applications in quantum computation, where superconducting quantum interference devices (SQUIDs) based on Josephson junctions may not be applicable due to the large magnetic fields.

Our SKIFFSs are fabricated from a 7 nm NbTiN thin film (T_C ∼ 9 K, R_sheet = 252 Ω/□) DC-sputtered reactively on a c-axis sapphire substrate using a NbTi alloy target and an Ar/N environment.¹⁵ The device features are patterned using electron-beam lithography followed by reactive-ion etching with an SF₆/Ar plasma. A scanning electron micrograph of a SKIFFS is shown in Fig. 1(a). The sensor is a lumped-element microwave resonator fabricated from a 100 μm × 100 μm rectangular superconducting loop defined by a 5 μm wide line. Two 100 nm wide nanowires of length ℓ_nw are defined on the left and right arms of the loop as seen in the inset of Fig. 1(a). Care is taken to prevent current crowding at the ends of the nanowires^16,17 by linearly tapering from the micron-wide loop dimensions to the nanowires over a length of 10 μm. The resonator is completed by placing interdigitated capacitors at the center of the loop with fingers and gaps of width 1 μm. Each resonator is coupled to a microwave feedline using a coupling capacitor, which is also an interdigitated structure with four pairs of 10 μm wide fingers that are separated by 10 μm gaps.

A lumped-element circuit model of the SKIFFS is shown in Fig. 1(b). From a Sonnet¹⁸ simulation, we obtain an estimate of the resonator capacitance, C_R ≈ 0.2 pF. From the thin-film parameters, we estimate that the nanowires contribute an inductance of 360 pH/μm, while the micron-sized sections of the loop contribute an inductance of 7.2 pH/μm. Using these parameters, we estimate a resonance frequency of 5 GHz for a sensor with ℓ_nw = 10 μm.

The device reported in this work has six sensors with varying values of ℓ_nw from 7 to 12 μm that are coupled to a common microwave feedline. Following fabrication, the device is wirebonded to a copper printed circuit board equipped with microwave connectors and placed in the bore of an external magnet with the surface of the superconductor perpendicular to the magnetic field, B₀. Additionally, a home-made Helmholtz pair coil is attached to the sample holder such that the generated field, B_coil, is parallel to the external field. We use the external magnet to generate the moderate background magnetic fields and the coil to generate small additional magnetic fields. The magnetic field generated by the home-made coil was calibrated from the shift in the electron spin resonance line of phosphorus donor electron spins in silicon in a separate experiment.

The device and sample holder assembly are cooled to a temperature of 1.9 K, and microwave transmission through the central feedline is monitored using a network analyzer with a microwave power of −72 dBm at the device. All six resonances are found in the range of 3.945–5.230 GHz. Microwave powers exceeding −64 dBm were observed to distort the resonance lineshapes of some of the resonators. This behavior has been observed previously and is a signature of the large kinetic inductance of the superconductor.^19,20 For the remainder of this work, we focus on one of the SKIFFSs with ℓ_nw = 10 μm and a resonance frequency near 4.2 GHz.

The loaded quality factor of the resonator depends strongly on the perpendicular magnetic field, B₀. In order to study this dependence, we monitor the resonance near 4.2 GHz as B₀ is swept from 10 to 200 mT. The results are shown in Fig. 2(a), where we have extracted the loaded quality factor as a function of the background magnetic field. As B₀ is increased, the quality factor drops from 1000 at B₀ = 10 mT to 200 at B₀ = 200 mT. The inset shows microwave transmission through the device for B₀ = 60 mT, producing a quality factor of 600. We remark here that the resonator is strongly overcoupled due to the choice of coupling capacitor parameters. A higher loaded quality factor at zero field can readily be obtained by adjusting the dimensions of the interdigitated capacitor coupling of the resonator to the microwave feedline toward critical coupling. Additionally, higher quality factors should be possible by operating the resonator at lower temperatures, where T/T_C ≪ 0.1. Quality factors between 10 000 and 100 000 have previously been reported for comparable thin films of NbTiN in the presence of moderate background fields at 300 mK (Ref. 2).

Next, we set B₀ = 60 mT and apply small magnetic fields using our home-made coil. The results are shown in Fig. 2(b). As the magnetic field of the coil, B_coil, is increased, the resonance frequency, f_R, shifts continuously from its value at 60 mT as

where $fR0=4.182 GHz$ is the resonance frequency with B_coil = 0 μT and B₀ = 60 mT. The functional form of the shift in the resonance frequency has been observed in other devices.^1,9,21–23 The resonance frequency shows a maximum shift of 27 MHz as B_coil is increased to 1.8 μT. When B_coil exceeds 1.8 μT (∼10 flux quanta), the resonance frequency abruptly jumps back to 4.182 GHz and continues to change as before. This behavior can be understood by noting that a screening current, i_loop, is generated in the superconducting loop in order to keep the magnetic flux threading the loop constant as B_coil is changed. This current modulates the kinetic inductance of the superconducting nanowires, L_R, thereby resulting in a change in the resonance frequency of the SKIFFS resonator. Eventually, as B_coil is increased, i_loop exceeds the critical current of the nanowires. From the data, we estimate the maximum i_loop to be ∼2 μA. This results in the formation of a normal metal which breaks the superconducting loop, allowing additional magnetic flux to thread the loop as the normal section of the loop returns to the superconducting state. Once the loop returns to the superconducting state, there is no longer a screening current since there is no difference between the magnetic flux threading the loop and the applied magnetic flux due to B_coil.

Three observations support this interpretation of the abrupt jumps in f_R as B_coil is swept. The first is the functional form in Eq. (1), which is similar to the functional form of a kinetic inductor modulated by a DC current. The screening current, i_loop, generated in the superconducting loop is related to the applied magnetic flux, B_coil, by the relation

where $LTloop$ is the total loop inductance, A_loop is the loop area, Φ₀ is the magnetic flux quantum, and n is the integer number of flux quanta threading the superconducting loop.⁹ The linear relationship between i_loop and B_coil provides an explanation for the similarity between the functional form of the change in resonance frequency, δf_R(B_coil), and the functional form of a kinetic inductor that is modulated by a DC current.^1,9,21–23 Second, we have observed that the largest value of B_coil that can be screened before the abrupt jump occurs is strongly dependent on temperature and is significantly reduced at higher temperatures. For instance, while ∼1.8 μT of magnetic field can be screened at a temperature of 1.9 K with a microwave power of −72 dBm, that value drops to only ∼1.2 μT at a temperature of 4.2 K. This observation supports our interpretation since the critical current of superconductors depends strongly on temperature for temperatures above ∼0.1T_C.²⁴ Third, we have also observed that higher microwave powers reduce the maximum B_coil that can be supported. This observation also supports our interpretation since larger microwave powers lead to larger microwave currents, which in turn limit the largest i_loop that can be supported in the nanowire. We note that similar abrupt jumps have been observed in the frequency of resonators tuned by nanoSQUIDs fabricated from Nb constrictions.²⁵ 

In order to demonstrate the application of SKIFFS for magnetometry, we use the experimental setup shown in Fig. 3(a). A microwave probe tone is applied at the center of the SKIFFS resonance, f_R, and transmission through the device is monitored using phase-sensitive detection. An IQ mixer downconverts the microwave probe tone to DC such that the phase of microwave transmission through the device can be computed from its in-phase and quadrature components. For small changes in magnetic field, δB_coil, the phase of the microwave transmission, δϕ, follows linearly with a slope that depends on the quality factor of the resonator. We bias the SKIFFS with B_coil ≈ 1.3 μT such that the device sensitivity is ∼31 MHz/μT. Figures 3(b) and 3(c) show the results of applying sinusoidal magnetic fields with amplitude of δB_coil = 25 nT peak-to-peak at frequencies of 1 kHz and 100 Hz, respectively. The phase of the transmitted microwaves shows oscillations which follow δB_coil. From the peak-to-peak change in the phase, we find a phase sensitivity of δϕ/δB_coil ≈ 1°/nT, in good agreement with an estimate of 0.9°/nT from the independently measured quality factor of 600 and the bias point of 31 MHz/μT.

We estimate the noise of SKIFFS magnetometry by measuring their phase response to a small 500 Hz sinusoidal magnetic field. Next, we subtract the sinusoidal component from the measured phase response and compute the power spectral density of the remaining noise. The results of this procedure are shown in Fig. 4 with and without a background magnetic field, B₀, in addition to the sinusoidal excitation. For comparison, we have also included independent measurements of the magnetic field fluctuations in our experimental setup using dynamical decoupling noise spectroscopy,^26–29 where the power spectral density is inferred from the phase accumulated due to magnetic field fluctuations present during an electron spin resonance experiment with phosphorus donor spins in silicon at 340 mT.

The square-root power spectral density is a factor of 2 larger in the presence of a background magnetic field, likely due in part to fluctuations in the external magnetic field and vibration-induced noise in our experimental setup. Measurements at both fields show f^−0.5 dependence, indicating the presence of 1/f noise power in the detection similar to low-temperature SQUIDs.³⁰ The agreement between the magnetic field noise measured using SKIFFS and electron spins suggests that our measurement of the SKIFFS noise may be limited by fluctuations in the external magnetic field.^31,32 Further work is necessary to separate the baseline noise of SKIFFS from the fluctuations in the external magnet.

The value of the power spectral density in Fig. 4 is $∼10 pT/Hz$ (⁠$50 μΦ0/Hz$⁠) at 1 kHz, indicating that our devices are at worst one to two orders of magnitude more noisy than typical low-temperature SQUIDs at comparable frequencies in zero background field.³³ Previous work^34,35 has shown that two-level fluctuators can lead to a 1/f noise power which has a typical value of $SB1/2(1 Hz)=1 μΦ0/Hz$⁠, indicating the potential for improved sensitivity of our devices by two orders of magnitude. Kinetic inductance magnetometry with a noise floor of $30 fT/Hz$ has previously been reported in a device fabricated from NbN and operated in a shielded environment with zero background field.⁹ The inset in Fig. 4 shows a close-up of the noise measurement near 1 kHz, where spurious peaks can readily be identified. The peaks at 1 and 1.5 kHz are due to the spurious second and third harmonics of the 500 Hz excitation from our waveform generator. The peak at 1.175 kHz arises from the magnet stabilization system attached to our external magnet.

We remark here on practical considerations when using SKIFFS for magnetometry and other applications. First, the abrupt jumps shown in Fig. 2(b) limit the dynamic range of the device. The largest magnetic field that can be applied before an abrupt jump in the resonance frequency is a function of the geometry of the superconducting loop. The dynamic range can be extended by increasing the width of the nanowires. However, this choice comes at the expense of a loss in sensitivity, as wider nanowires reduce the kinetic inductance non-linearity of the loop.²¹ In order to compensate for the reduced sensitivity, the loop area can be increased. Second, the operation of SKIFFS can be extended to even higher background magnetic fields by appropriately designing the superconducting loop parameters. For our thin-film, we estimate a zero-field London penetration depth of 450 nm. By choosing parameters for the superconducting loop which are smaller than the penetration depth, the magnetic field resilience can be extended.² Superconductors with large (>1 μm) London penetration depths such as granular aluminum³⁶ may extend the use of SKIFFS to higher magnetic fields. Third, we have demonstrated readout of the microwave phase transmission through the SKIFFS devices. The detection sensitivity of the phase transmission is linearly dependent on the quality factor of the resonator. As a result, higher detection sensitivity can be achieved by ensuring that the loaded quality factor of the resonators remains high. For our devices, higher loaded quality factors can be achieved by adjusting the parameters of the coupling capacitor. Further improvement should be possible by operating at temperatures T/T_C ≪ 0.1, where losses due to thermally excited quasiparticles and thermally activated motion of trapped flux vortices are significantly reduced. Finally, we note that even though we have applied small magnetic fields using a home-made coil, similar results can be obtained by driving DC currents through the central microwave feedline such that magnetic fields due to the currents couple into the SKIFFS loop. This provides a method of biasing SKIFFS on-chip, similar to the bias lines used for traditional SQUIDs.

The ability to detect small magnetic fields in the presence of large background fields provides a useful tool for several areas of research. For example, in the field of quantum computation, previous work has shown that small magnetic field fluctuations can lead to loss of quantum control of electron spins of donors in silicon as well as trapped ions on timescales longer than few milliseconds due to undesirable fluctuations in the background magnetic fields that provide the Zeeman splittings which are necessary for the experiments.^31,32,37 In these settings, SKIFFS can be used to monitor the background magnetic fields and compensate for fluctuations. One approach is to build a magnetic-field-locked microwave source. In this application, the microwaves transmitted through a SKIFFS resonator can be used to drive spin rotations by appropriate frequency conversion through a phase-locked loop. More generally, SKIFFS can be used as resonators that can be tuned using magnetic fields instead of DC currents, and similar structures have been demonstrated for resonators which are capacitively coupled to a microwave feedline.²⁵ 

In summary, we have demonstrated the use of superconducting kinetic inductance field-frequency sensors for magnetometry in the presence of moderate background magnetic fields. We obtained resonance frequency shifts of 27 MHz in response to changes in the magnetic field of 1.8 μT. The real-time measurement of AC magnetic fields can be implemented by using phase-sensitive readout of microwaves transmitted through the sensors with a sensitivity of 1°/nT. We find that our sensors are at most one to two orders less sensitive in comparison with low-temperature superconducting quantum interference devices operating in zero background field. We anticipate potential applications of our sensors wherever small changes in a large background field must be accurately monitored.

See supplementary material for a detailed derivation of Eq. (2) and estimates of the loop inductance and sensitivity of our devices.

We acknowledge helpful discussions with Shyam Shankar. Devices were designed using the CNST nanolithography toolbox³⁸ and fabricated in the Princeton Institute for the Science and Technology of Materials Micro/Nano Fabrication Laboratory and the Princeton University Quantum Device Nanofabrication Laboratory. Our work was supported by the NSF, in part through Grant No. DMR-1506862, and in part through the Princeton MRSEC (Grant No. DMR-1420541).