Magnetic field tunable spectral response of kinetic inductance detectors
F. Levy-Bertrand M. Calvo U. Chowdhury A. Gomez J. Goupy A. Monfardini








florence.levy-bertrand@neel.cnrs.fr
Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Groupement d'Intérêt Scientifique KID, Grenoble and Saint Martin d'Hères, France


Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Groupement d'Intérêt Scientifique KID, Grenoble and Saint Martin d'Hères, France


Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Groupement d'Intérêt Scientifique KID, Grenoble and Saint Martin d'Hères, France


Centro de Astrobiología (CSIC-INTA), Ctra. Torrejon-Ajalvir km.4, 28850 Torrejon de Ardoz, Spain


CEA/DRF/IRIG – Grenoble  - France



Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Groupement d'Intérêt Scientifique KID, Grenoble and Saint Martin d'Hères, France

§ ABSTRACT 
We tune the onset of optical response in aluminium kinetic inductance detectors from a natural cutoff frequency of 90 GHz to 60 GHz by applying an external magnetic field. The change in spectral response is due to the decrease of the superconducting gap, from 90 GHz at zero magnetic field to 60 GHz at a magnetic field of around 3 mT. We characterize the  variation of the superconducting gap, the detector frequency shift and the internal quality factor as a function of the applied field. In principle, the magnetic field tunable response could be used to make spectroscopic measurements. In practice, the internal quality factor behaves hysteretically with the magnetic field due to the presence of vortices in the thin superconducting film. We conclude by discussing possible solutions to achieve spectroscopy measurements using kinetic inductance detectors and magnetic field.


Magnetic field tunable spectral response of kinetic inductance detectors
    A. Monfardini
    September 9, 2024
========================================================================






Kinetic Inductance Detectors (KID), based on planar superconducting resonators <cit.>, are popular detectors for astrophysical observations <cit.> and interesting devices for physics studies <cit.>.
One of the current challenges in millimetre astrophysics observations is to achieve a given degree of spectral resolution without sacrificing the large field-of-view of the current cameras. 
Here we explore a solution to achieve that goal by tuning the spectral response of KID with an external magnetic field. We present the first demonstration of the optical response of KID under a variable magnetic field. We also evaluate the effects of the applied magnetic field on the resonators quality factors and conclude by discussing future improvements of our initial KID design.






KID are a particular implementation of superconducting resonators. They are planar LC-resonant circuits made of superconductor thin films deposited on an insulating substrate, optimized for photon detection. The photon detection principle consists in monitoring the resonance frequency shift that is proportional to the incident power. Incident radiation breaks down Cooper pairs, generating quasi-particles and modifying the kinetic inductance resulting in an shift of the resonance frequency f=1/(2π√(LC)) where  C is the capacitance and L=L_K+L_G is the total inductance, i.e. the sum of the kinetic and geometric inductances. The internal quality factor of the resonator decreases  with the number  of generated quasi-particles.

Figure <ref> shows a schematic view of the experimental set-up. The array of KID is made of a 200 nm aluminum film deposited on a high-resistivity silicon wafer. Four wire resistivity measurements performed on the same 200 nm aluminum film give a critical temperature T_c∼1.2K and a perpendicular critical field H_c∼4.5mT.  Each KID is coupled via its inductor to the readout line. The inductor L is a Hilbert shape, sensitive to all in-plane polarization <cit.>. The lines are 4 μm wide. A magnetic field is applied perpendicular to the KID using a custom Helmholtz coil. The KID and the coil are cooled to approximately 100 mK in a dilution refrigerator with optical access. Illumination is controlled by a Martin-Puplett spectrometer <cit.> at room temperature and reaches the KID through a suitable series of optical filters and lenses <cit.>. 



The bottom panel of figure <ref> displays the measured spectral response of  KID as a function of the incident optical frequency at different magnetic fields. The response extends from the 2Δ superconducting gap up to the low-pass filter frequency (180 GHz). At zero magnetic field the 2Δ superconducting gap equals 90 GHz, in agreement with the BCS-value  3.52k_BT_c/h∼88GHz. When increasing the magnetic field the 2Δ-gap decreases, increasing the band response from 90-180 GHz to 60-180 GHz at about 3 mT. The level of noise in the spectra, observable outside the response band, below 2Δ and above 180 GHz, increases with the magnetic field. Within the limits of noise and measurement accuracy, about ∼1 GHz resolution for the optical frequency, the common-band response appears identical. So, in principle, by subtracting the response measured at 3 mT from that measured at zero magnetic field, we could obtain the response of the 60-90 GHz band. Extrapolating a little further, subtracting the responses measured at very close magnetic fields would give access to the response of a highly resolved spectral band: at 2 mT, a 0.001 mT step, would give a spectral resolution R=ν/Δν of about 8000 at 80  GHz (e.g. a spectral band of 0.01 GHz). 

In practice, it is not so straightforward to access the spectroscopic signal. In a magnetic field, the frequency shift is due both to changes of the optical load, the signal of interest, and of the magnetic field. The latter are due to variations in the kinetic inductance L_k, resulting from the change of the 2Δ superconducting gap as L_K=ħ R_□/(πΔ) where R_□ is the sheet resistance in the normal state <cit.>.

Figure <ref> illustrates the frequency variations due to both the magnetic field and the change in optical load. The figure shows Vector Network Analyser (VNA) response of a KID under two optical loads and two magnetic fields. 
The red and blue curves correspond, respectively, to measurements under a high or low optical load, with the 300 K window of the cryostat either closed by a dark plastic cap  or closed by a mirror. 
The dark plastic cap acts as blackbody source at about 300 K (T_bb∼300K). 
The mirror, which reflects the emission from the coldest stages of the cryostat, acts as a very cold blackbody source at about 0 K  (T_bb∼0K). 
Varying the magnetic field from 0 to 2.7 mT, the resonance frequency shifts by about 2 MHz (from left to right panel). Due to the optical signal, the resonance frequency shifts by 6 kHz at 0 mT and by 50 kHz at 2.7 mT. 









To access the absolute spectroscopic signal preliminary calibrations in magnetic field are required. Figure <ref> shows the variation of the superconducting gap, the frequency and the quality factor of KID with respect to the magnetic field. Within the errors bars both the 2Δ superconducting gap and the δ f/f relative frequency shift remain identical while ramping up and down the magnetic field. The magnetic field dependence of the superconducting gap follows the following formula valid for thin film <cit.> (i.e. for d/λ<<1 where d is the thickness and λ is the magnetic penetration depth):


    Δ(H)=Δ_0√(1-(H/H_c)^2)

where Δ_0∼ 90 GHz is the gap at zero magnetic field and H_c∼ 4.5 mT is the critical field. The relative frequency shift under an almost constant small optical load is adjusted with the following formula valid for δ f<< f:

    δ f(H)/f=-α_0/2[1-√(1-(H/H_c)^2) ]

where α_0=L_K^0/(L_K^0+L_G)∼ 1.4% is the ratio of the kinetic inductance over the total inductance at zero magnetic field. The expression <cit.> is the usual, δ f/f=-α/2×δ L_k/L_K, with the magnetic field dependence of L_K inserted. The value of α is in agreement with the ones estimated from the actual resonance frequency f_L_K+L_G, the resonance frequency  simulated with the SONNET software <cit.> without any kinetic inductance f_L_G  and the following formula α=1-(f_L_K+L_G/ f_L_G)^2, resulting in α∼1-4%. The value of α is low because the kinetic inductance of a 200 nm thick film of aluminum is small <cit.>. 
The internal quality Q_i  strongly varies with the magnetic field from about 10^6-10^5 down to 10^3 and its value depends on the history of the magnetic field. This behavior is due to the presence of vortices in the resonator meander. The vortices are in the so-called plastic regime, in which they are alternatively pinned and mobile when sweeping the magnetic field <cit.>. Vortices develop in aluminum thin films because they are type-II superconductors contrary to bulk aluminum <cit.>.






To achieve spectroscopy using kinetic inductance detectors and magnetic field the first point to address is the vortex issue. The theory predicts that in thin superconducting films, vortices develop above a magnetic field perpendicular to the films H_0=πΦ_0/(4w^2) where Φ_0 is the quantum of magnetic flux and w is the width of the  superconducting line <cit.>. Experiments on 200 nm thick Nb superconducting lines validate the formula <cit.>, and, show that  the critical field for the vortex nucleation, H_0 drops drastically from about 200 mT in bulk Nb to below 1 mT for a 200 nm thick Nb film. For superconducting thin films, the width of the lines must be of the order of 500 nm to avoid the formation of vortices up to 5 mT. Thus, future developments of spectroscopy using kinetic inductance detectors and magnetic field requires a new KID design and e-beam lithography.


An other point to address is the magnetic field generation. The diameter of the magnetic coil must scale up with the diameter of the field of view which is proportional to the focal plane diameter. One solution is to implement a large standard coil at room temperature, around the cryostat.  The magnetic field at the center of the coil is H=μ_0 I N/d where μ_0 is the vacuum permittivity, I is the current, N is the number of turns and d is the diameter of the coil. For a diameter of 30 cm, a current of 1 A, and 3000 turns, the magnetic field equals 6 mT. This solution is the simplest from a cryogenic point of view, but the magnetic field on the KID may not be sufficiently homogeneous.



We tune the spectral response of kinetic inductance detectors with a magnetic field. The change in spectral response is due to the decrease of the superconducting gap when increasing the magnetic field. Our results suggest that it may be possible to achieve high-resolution spectroscopy over a wide field of view using KID and a magnetic field. The main pending limitation is the formation of vortices in the lines of the KID. This can be solved by reducing the lines width. The spectral resolution could reach several thousands depending also on the superconducting gap steepness. The spectral bands could be adjusted with the superconducting material and the magnetic field. Aluminum films with a critical temperature of T_c∼1.2 K and H_c∼5 mT give access to the relevant 50-100 GHz band. Tantalum with a critical temperature of T_c∼4.5 K and H_c∼80 mT could, in principle, give access to a 200-400 GHz band. 


We acknowledge the contribution of  G. Donnier-Valentin and of T. Gandit, respectively, for the design and for the realization of the Helmotz superconducting coil. We acknowledge the contribution of O. Bourrion for the electronic acquisition. We thank B. Sacépé for excellent discussions. We acknowledge the overall support of the Cryogenics and Electronics groups at Institut Néel and LPSC. This work has been partially supported by the French National Research Agency through the LabEx FOCUS Grant No. ANR-11-LABX-0013 and the EU13s Horizon 2020 research and innovation program under Grant Agreement No. 800923 (SUPERTED).  A. G. acknowledges financial support from PID2022-137779OB-C41 funded by the Spanish MCIN/AEI/10.13039/501100011033.


The data that support the findings of this study are available from the corresponding author upon reasonable request.



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