[1]Lukas Nulens These authors contributed equally to this work. These authors
contributed equally to this work.

[1]Quantum Solid-State Physics, Department of Physics and Astronomy, KU
Leuven, Celestijnenlaan 200D, Leuven, B-3001, Belgium 2]Experimental Physics
of Nanostructured Materials, Q-MAT, CESAM, Université de Liège, Allée du 6
Août 19, Sart Tilman, B-4000, Belgium 3]Quantum Device Physics Labaratory,
Department of Microtechnology and Nanoscience, Chalmers University of
Technology, Goteborg, SE-412 96, Sweden

# Catastrophic magnetic flux avalanches in NbTiN superconducting resonators

<EMAIL_ADDRESS><EMAIL_ADDRESS>Nicolas Lejeune  Joost
Caeyers  Stefan Marinković  Ivo Cools  Heleen Dausy  Sergey Basov  Bart Raes
Margriet J. Van Bael  Attila Geresdi  Alejandro V. Silhanek  Joris Van de
Vondel  *  [  [

###### Abstract

The impact of the magnetic field penetration on the resonance frequency of
NbTiN superconducting resonators is investigated by a combination of magneto-
optical imaging and high-frequency measurements. At temperatures below
approximately half of the superconducting critical temperature, the
development of magnetic flux avalanches manifests itself as jumps in the
resonance frequency of the coplanar resonators. A clear change in the rate of
decreasing resonance frequency with magnetic field is observed when a magnetic
perforation event regime sets the transition between a Meissner-like phase in
the ground plane and flux injection into the central feedline. These regimes
are directly visualized by magneto-optical imaging and the impact of
avalanches in the ground plane and the resonator is discerned. At high working
temperatures, a smooth flux penetration is shown to prevail. We propose some
hints and strategies to mitigate the influence of avalanches on the response
of the resonators and to improve the magnetic resilience of coplanar
resonators. Our findings demonstrate that superconducting resonators represent
a valuable tool to investigate the magnetic flux dynamics in superconducting
materials. Moreover, the current blooming of niobium-based superconducting
radio-frequency devices makes this report timely by unveiling the severe
implications of magnetic flux dynamics.

###### keywords:

Coplanar waveguide resonators, Superconductivity, Magneto-optical imaging,
Avalanches

##  1 Introduction

Superconducting coplanar waveguide (CPW) resonators have become an essential
component of quantum circuits due to their ability to readout different qubit
systems. These CPW resonators combine a conventional fabrication method with
superior quality factors needed to perform circuit quantum electrodynamics.
During the last decade, careful design and material selection has resulted in
quality factors reaching values up to $10^{6}-10^{7}$ paper_megrant2012planar
; paper_vissers2010low . In order to obtain this high performance, the CPW
resonators must be screened from external damping sources among which magnetic
flux quanta play a particularly detrimental role. Although efficient magnetic
screening can be achieved, this is not always a viable option since some qubit
implementation schemes such as spin ensembles in solid-state systems
paper_kubo2010strong ; paper_schuster2010high ; paper_amsuss2011cavity ,
phase-slip qubits paper_astafiev2012coherent ; paper_peltonen2018hybrid ;
paper_mooij2005phase ; paper_mooij2006superconducting , and trapped electrons
paper_schuster2010proposal ; paper_bushev2011trapped , require inevitable
exposure to a magnetic field. For these systems, it is crucial that the
resonator characteristics remain unaffected under an external magnetic field.
Unfortunately, previous studies have shown that increasing the magnetic field
leads to a decrease in the quality factor as well as the resonance frequency
of the resonator Song_resonator_field2009 ; paper_bothner2011improving ;
paper_bothner2012reducing ; paper_bothner2012magnetic ;
paper_chiaro2016dielectric ; paper_kroll2019magnetic . Consequently, the
identification of new strategies and designs for increasing the magnetic field
resilience of superconducting CPW resonators remains a subject of major
technological significance.
The deleterious effect of magnetic field on the resonator characteristics can
be attributed to the penetration of magnetic flux quanta or vortices
Raes2012_singlevortexmotion . Their interaction with the induced radio-
frequency (RF) currents gives rise to the absorption of energy and
consequently dissipation which translates into a decrease in the resonator
performance. This indicates that not only the amount of vortices penetrating
the CPW resonator is relevant but also their exact location. In particular,
the presence of vortices should be avoided in regions of maximal RF currents
and/or magnetic field Nsanzineza_impact_singlevortex2014 . Two approaches can
be adopted with the aim to lessen vortex-induced losses: (i) hinder their
movement, and (ii) decrease their number in the critical regions. Several
works have focused on the first approach by creating artificial pinning sites
paper_bothner2011improving ; paper_bothner2012reducing ;
paper_bothner2012magnetic ; paper_song2009reducing ; paper_kroll2019magnetic .
Alternatively, it has been shown that reducing the ground plane
paper_bothner2017improving , or dividing the ground plane into fractal
structures paper_graaf2012magnetic ; paper_de2014galvanically  permits to
decrease the demagnetization effects and decrease the field penetration.

In order to seize and quantify the impact of the aforementioned mitigation
strategies, the complete sample surface should be taken into account. To that
end, a macroscopic technique capable of visualizing the magnetic flux
penetration and structural weak points is essential. Magneto-optical imaging
(MOI) has proven to be an excellent tool for examining macroscopic structures
such as superconducting CPW resonators with $\mu$m spatial resolution
paper_lange2017high ; paper_ghigo2007evidence . In order to make morphometric
criteria-based decisions for field-resilient CPW resonators, the macroscopic
image of a full device must be complemented and compared to high-frequency
transmission measurements.

In this work, we investigate the effect of the magnetic field penetration on a
superconducting CPW resonator. To that end, we designed a sample containing
two $\lambda/4$ resonators able to be characterized by magneto-optical imaging
and high-frequency measurements at several temperatures. Our results show that
both observed resonances follow the expected temperature behavior determined
by the kinetic inductance of the material. We analyzed the field dependence of
the high-frequency measurements and observed several different regimes in the
behavior of the resonance frequency. These regimes were directly correlated to
the MOI observations, with a particular focus on the noisy behavior in the
resonance frequency that can be attributed to the development of
thermomagnetic instabilities precursors of abrupt flux avalanches. Based on
our findings, we suggest several ways to optimize the field resilience of
future CPW resonators.

##  2 Results and Discussion

###  2.1 Sample design

The superconducting resonators investigated in this work consist of a 100 nm
thick NbTiN layer sputtered on a sapphire substrate, as schematically
represented in Figure 1(a). A critical temperature of 16.5 K was obtained
using SQUID magnetometry on a reference layer of 100 nm. Several identical
samples were fabricated in order to allow comparison between the MOI and RF
measurements. In order to fit the complete resonator structure in the field of
view of the MOI setup a substrate size of $(5\times 4)\text{ mm}^{2}$ with a
superconducting region of interest of $(2.5\times 2.5)\text{ mm}^{2}$ was
chosen. In this superconducting region two overcoupled, hanger-type
$\lambda/4$ resonators with length 4089 $\mu m$ and 3953 $\mu m$ were
capacitively coupled to a central feedline. The central conductor has a width
$w$ = 20 $\mu$m with a separation from the groundplane of $s$ = 10 $\mu$m
resulting in a 50 $\Omega$ impedance.

Figure 1: (a) A schematic representation of the investigated sample. In the
superconducting NbTiN region indicated in dark grey, two $\lambda/4$
resonators are capacitively coupled to a central feedline. (b) The $S_{21}$
transmission parameter of the longest resonator plotted for different fixed
temperatures between 8.8 K and 16.3 K. The color change from blue to red
indicates the increase in temperature. (c) For both resonators indicated in
panel a the resonance frequency as a function of the temperature is indicated
from blue to red with increasing temperatures. The dashed black line
corresponds to a fit of both resonance frequencies as described in the text.
The data points corresponding to the transmission measurement shown in panel b
are indicated with the corresponding colored arrow.

###  2.2 Resonator characteristics

The resonance frequencies, $f_{r}$, at 7.06 GHz and 7.44 GHz corresponding to
each resonator manifest themselves as an absorption dip in the $S_{21}$
transmission spectrum at 1.8 K. The resonance dips showed a pronounced
skewness likely due to an impedance mismatch caused by the lengthy wire bonds
($\sim$ mm) required to fit the high-frequency setup paper_doyle2008lumped ;
paper_grabovskij2008situ ; paper_khalil2012analysis . In Figure 1(b) the
magnitude of the transmission is shown for different temperature values. As
the temperature increases from lower (blue) to higher (red) values, the
resonance frequency shifts to lower values and the full width at half maximum
increases, indicating an increase in losses. Figure 1(c) displays the
temperature dependence of both resonance features. The resonance frequencies
7.06 GHz and 7.44 GHz with corresponding internal quality factors of $\sim
2000$ and $\sim 800$ were obtained following the fitting routine described in
Ref. paper_probst2015efficient .

In general, the resonance frequency of a $\lambda/4$ resonator is determined
by the expression $f_{r}=1/(4l\sqrt{L^{\prime}C^{\prime}})$, where $l$
corresponds to the resonator length, and $L^{\prime}$ and $C^{\prime}$ are the
inductance and capacitance per unit length, respectively
Book_pozar2011microwave . The resonance frequencies were originally designed
based solely on a geometric inductance contribution of
$L^{\prime}_{g}=4.19\cdot 10^{-7}$ H/m. However, a kinetic inductance
contribution should also be taken into consideration. Specifically, this
kinetic inductance can be expressed as a function of the temperature-dependent
superconducting energy gap, $\Delta(T)$ Book_tinkham2004introduction ;
paper_annunziata2010tunable :

$L^{\prime}_{k}(T)=\frac{l}{w}\frac{R_{\square}h}{2\pi^{2}\Delta(T)}\frac{1}{tanh\left(\frac{\Delta(T)}{2k_{B}T}\right)},$
(1)

with $l$ and $w$ corresponding to the length and width of the resonator,
$R_{\square}$ the normal state sheet resistance, $h$ and $k_{B}$ are Planck’s
and Boltzmann’s constants, and $T$ is the temperature
Book_tinkham2004introduction ; paper_annunziata2010tunable . By inserting an
interpolation formula for the temperature-dependent superconducting energy gap
$\Delta(T)\approx\Delta(0)\tanh(1.74\sqrt{T_{c}/T-1})$ following the BCS-
theory in Equation 1, the temperature dependent resonance frequency can be
fitted with fitting parameters $L^{\prime}_{k}(T=0),T_{c}$ and
$L^{\prime}_{g}$. The fit for both resonators is indicated in dashed black
lines in Figure 1(c), with parameters $L^{\prime}_{g}=3.82\cdot 10^{-7}$ H/m,
$L^{\prime}_{k}=3.73\cdot 10^{-8}$ H/m, $T_{c}=17.27$ K and
$L^{\prime}_{g}=4.16\cdot 10^{-7}$ H/m, $L^{\prime}_{k}=5.46\cdot 10^{-8}$
H/m, $T_{c}=17.21$ K for the shortest and longest resonator respectively. The
excellent fit demonstrates that the resonance frequency serves as a very
sensitive probe for any variation of the superconducting energy gap.

###  2.3 Magnetic field response of the resonators

Let us investigate the response of the resonators as a function of the applied
out-of-plane magnetic field ($\mu_{0}H_{a}$). Figure 2(a) shows the resonance
frequency of the longest resonator after a zero-field cooling process and
subsequent sweeping of the magnetic field at the fixed temperature of 5 K. For
the sake of clarity, we have unfolded the magnetic field cycle in such a way
that the abscissa can be associated to the measurement chronology. Note that
as magnetic field increases, the resonant frequency shifts down with an
average rate smaller than the subsequent increase rate of $f_{r}$ when
decreasing field. In other words, $f_{r}(\mu_{0}H_{a})$ exhibits an
irreversible response when cycling the applied magnetic field. This hysteric
behavior of $f_{r}$ is a universal feature which has been shown for Nb
paper_bothner2012magnetic ; paper_bonura2006magnetic , NbN
paper_yu2021magnetic , YBCO paper_lahl2003nonlinear , MgB2
paper_ghigo2010mechanisms , and Al paper_borisov2020superconducting  CPW
resonators and can be linked to the inhomogeneous distribution of current,
$I_{rf}$, shaking an inhomogeneous magnetic flux distribution. As shown in the
supplementary figure 1 the internal quality factor has a very similar magnetic
field dependence. This demonstrates that the same interplay defines
dissipation in these resonators.

Figure 2: (a) Resonance frequency as a function of the applied magnetic field
$\mu_{0}H_{a}$ obtained at $T=5$ K. The magnetic field has been swept to
complete a full loop $0\rightarrow 5\rightarrow-5\rightarrow 0$ mT. The inset
shows a zoom-in of the framed region. (b) The derivative
$df_{r}/d\mu_{0}H_{a}$ extracted from panel a makes the noisy and smooth
regime apparent. (c) The local magnetization as a function of the applied
magnetic field inside the region schematically represented in the inset. The
blue, red, and grey data points correspond with the magnetization averaged
over the central conductor of the meander, the gap separating the central
conductor and the ground plane, and the ground plane (pitch) between the
meander structure respectively. (d) The upper row of the panel shows magneto-
optical images corresponding to the magnetic field indicated in panel a. The
Meissner currents are indicated with grey arrows in the image corresponding to
point I. The lower row of the panel shows differential images revealing the
change of magnetic flux between two consecutive field steps. The nucleation
point where the magnetic perforation takes place is indicated with a dotted
circle.

A close inspection of Figure 2(a) allows us to identify several different
features which should be associated with the peculiarities of the magnetic
field penetration in the complete structure. As we will demonstrate below, in
order to identify the non-local magnetic origin of these features it is
essential to map the magnetic field distribution in the entire device (shown
in Figure 2(d)).

Figure 2(a) shows that the resonance frequency maximizes at $\mu_{0}H_{a}=0$
mT, where little magnetic flux (mainly due to frozen earth magnetic field)
remains trapped in the resonators. In the field range $0\text{
mT}<\mu_{0}H_{a}<0.4$ mT, screening Meissner currents circulate in the ground
plane (point (I) in Figure 2(a,d)). Note that the ground plane and the
resonators are capacitively coupled to the central excitation line. In other
words, the ground plane and resonators are physically disconnected from the
excitation line and surround it. Therefore, the structure can be regarded as
topologically equivalent to a ring surrounding a disk. Due to demagnetization
effects, the Meissner currents circulating clockwise in the ground plane (see
grey arrows in image (I) of Figure 2(d)), produce a magnetic field of opposite
polarity than the applied field all along the inner border of the ground plane
and a small positive field inside the central hole (where the feed line sits).
As long as $0\text{ mT}<\mu_{0}H_{a}<0.4$ mT the resonators are protected by
the screening currents in the ground plane and $f_{r}$ gently decreases as
magnetic field increases.
At $\mu_{0}H_{a}=0.5$ mT a peculiar event, known as magnetic perforation takes
place. This event consists of a sudden breakdown of the screening currents in
the ground plane with a simultaneous injection of magnetic flux inside the
central hole paper_olsen2007avalanches ; paper_shvartzberg2019quasiperiodic ;
paper_jiang2020selective . This abrupt change of the magnetic field at the rim
of the resonators marks the start of a steeper magnetic field dependence of
$f_{r}$. The injection of the magnetic field becomes more apparent in the
differential image (i.e. subtraction of two images taken a two consecutive
magnetic fields) shown in Figure 2(d) corresponding to point (I) in which the
nucleation point of the magnetic perforation has been indicated with a dotted
circle.
As the magnetic field is progressively increased beyond the magnetic
perforation point ($0.5\text{ mT}<\mu_{0}H_{a}<5$ mT), $f_{r}(\mu_{0}H_{a})$
exhibits a noisy behavior with a global linear decrease as the magnetic field
increases. This noisy regime is better seen in the derivative
$df_{r}/d\mu_{0}H_{a}$ shown in Figure 2(b). The origin of this jumpy behavior
can be traced back to the development of magnetic flux avalanches caused by
thermomagnetic instabilities RevModPhys_criticalstate . A snapshot of the
magnetic landscape after triggering some finger-shaped flux avalanches at the
point labeled (II) is shown in Figure 2(d). These avalanches are neither fully
reproducible paper_johansen2007reproducible ; paper_qureishy2017dendritic  nor
totally random and could equally originate at border spots with high degree of
perfection or defects paper_brisbois2016magnetic ; paper_jiang2020selective .

When decreasing the magnetic field from $\mu_{0}H_{a}=5$ mT, the local
magnetic field inverts its polarity at the border of the sample even before
the applied field changes polarity Brandt1993_edgeH ; Zeldov1994_edgeH ;
McDonald1996_edgeH . In this case, antivortices nucleate at the border of the
sample and annihilate with the trapped vortices. At the beginning, this
process takes place smoothly and leads to an effective removal of magnetic
flux from the resonators. Since no flux avalanches occur in this regime, the
$f_{r}(\mu_{0}H_{a})$ curve does not exhibit any jumpy behavior (see point
(III) in Figure 2(a,d)). The fact that the flux is efficiently removed from
the border of the resonators due to the pronounced decrease of the total
magnetic field at the sample’s borders gives rise to a lower density of
vortices when decreasing field than for increasing field for the same applied
field Brandt1993_edgeH ; Zeldov1994_edgeH ; McDonald1996_edgeH . This
irreversible behavior, in turn, leads to a steeper increase of
$f_{r}(\mu_{0}H_{a})$ as $\mu_{0}H_{a}$ decreases as compared to the up sweep
branch. Note that in this regime flux avalanches are no longer triggered
although those created previously leave a frozen imprint on the magnetic
landscape and may act as channels for the penetration of antivortices
paper_motta2011visualizing . The differential imaging shown in Figure
2(d)-(III) clearly demonstrates that no avalanches develop in this regime.
The smooth regime finishes at a certain field $\mu_{0}H^{*}\approx 2.5$ mT.
Further decreasing the applied field beyond this point, triggers avalanches of
flux with opposite polarity to the applied field. This is naturally
accompanied by a jumpy $f_{r}(\mu_{0}H_{a})$ response all the way down to
$\mu_{0}H_{a}=-5$ mT. Snapshots of the actual magnetic landscape and the
differential imaging for point (IV) within this regime are shown in Figure
2(d). A zoom of the transition from smooth to jumpy evolution of
$f_{r}(\mu_{0}H_{a})$ is shown in the inset of Figure 2(a), corresponding to
the framed area. Note here that abrupt jumps which tend to increase $f_{r}$
are followed by smooth linear regions where $f_{r}$ decreases. This behavior
can be explained as follows: after the event of a flux avalanche, the magnetic
field and current at the border of the resonator are relaxed, and therefore
$f_{r}$ increases. As $\mu_{0}H_{a}$ increases further, the system builds up
magnetic pressure (local field and current increases at the border of the
resonator), and therefore $f_{r}$ slowly decreases.
It is interesting to note the presence of a kink in the vicinity of the middle
zone of the noisy regime in panel (a), indicated by a red arrow. In order to
identify the origin of this feature, we have measured the local magnetic field
$B$ in different regions of the device and computed the local magnetization
$\mu_{0}M=B-\mu_{0}H_{a}$. The resulting magnetization $\mu_{0}M$ as a
function of the applied field is shown in Figure 2(c) for the resonator (blue
dots), the gap separating the resonator from the ground-plane (red downwards
triangles), and the ground plane in between the meander shape resonator (grey
upwards triangles). These regions are indicated with the same color code in
the inset of Figure 2(c). This figure shows that the kink in the $f_{r}(H)$
curve arises at the same moment when the local magnetization of the ground
plane starts to increase as a consequence of the development of magnetic flux
avalanches in the ground plane. These events relieve the magnetic pressure
everywhere in the resonator and tend to slow down the decrease of the
resonance frequency. A selection of MOI images of the resonator meander
showing the point avalanches is available in Supplementary Figure 2.

The magnetic and magnetization profiles over the magnetic sweep of the region
indicated in the inset of Figure 2(c) can be found in supplementary video 1.
Similarly, the evolution of the $B(\mu_{0}H_{a})$ curve as a function of the
measurement location along the same region can be found in supplementary video
2.

###  2.4 Temperature response of the resonators

As shown in the previous section, flux avalanches have a critical impact on
the overall performance of the CPW resonators. In Figure 3 we identify the
transition from the flux avalanche regime to smooth magnetic penetration using
both the magnetic field dependent resonance frequency and the MOI images.
Panel (a) shows the magnetic flux penetration after zero-field cooling for
three different temperatures (5, 9, and 12 K) at the same applied field
$\mu_{0}H_{a}=1.1$ mT. Finger-like avalanches clearly visible at 5 K are no
longer present at 9 and 12 K. Indeed, it is well known that the occurrence of
thermomagnetic instabilities is restricted to low temperatures (roughly for
$T<T_{c}/2$). At $T=9$ K, it is observed that border defects act as magnetic
faucets from where magnetic flux is injected into the ground plane, a
phenomenon which has been described in detail in
Ref.paper_brisbois2016magnetic . At $T=12$ K, the sample has been largely
penetrated by the magnetic field and as the magnetic field increases, non-
penetrated white regions shrink and converge to form discontinuity lines where
the supercurrents change direction paper_gurevich2000nonlinear .

Figure 3: Temperature dependence of the magnetic flux avalanche activity. (a)
Magnetic landscape after zero-field cooling at three different temperatures
(5, 9 and 12 K) for the same applied field $\mu_{0}H_{a}=1.1$ mT. (b)
Hysteresis loop of the resonance frequency $f_{r}(\mu_{0}H_{a})$ obtained at 4
K. The region where (anti)flux avalanches are triggered when decreasing
(increasing) from a maximum (minimum) applied field, are highlighted with a
grey background. (c) Threshold field $\mu_{0}H^{*}$ as a function of
temperature as determined by the onset of the noisy regime in
$f_{r}(\mu_{0}H_{a})$ measurements (red triangles) and by magneto-optical
imaging (blue triangles). The dotted line indicates the field of 0.2 mT at
which the images in panel (d) have been selected. The experiment has been
conducted twice at 5 K and 9 K resulting in additional blue
($\bigtriangledown$,$\bigtriangleup$) datapoints. (d) Magnetic landscape after
zero-field cooling at three different temperatures (5, 9 and 12 K) for the
same applied field $\mu_{0}H_{a}=0.2$ mT.

As we have pointed out above, the magnetic flux avalanche activity becomes
more apparent when decreasing the applied field below certain threshold value
$\mu_{0}H^{*}$ after reaching full penetration. This effect is a direct
consequence of the high energy release produced by the vortex-antivortex
annihilation which facilitates the development of avalanches with opposite
polarity than the applied field. The threshold field $\mu_{0}H^{*}$ at which
anti-avalanches are triggered can be identified either as the onset of jumpy
$f_{r}(\mu_{0}H_{a})$ response (shadowed region in panel (b)) or can be
directly obtained from the magneto-optical images. Figure 3(c) shows the
threshold field $\mu_{0}H^{*}$ as a function of temperature obtained by these
two independent methods for the decreasing ($\bigtriangledown$) and increasing
($\bigtriangleup$) branches of the hysteresis loops. To ensure consistency, we
repeated the experiment during a second cooldown at 5 K and 9 K. As a result,
Figure 3(c) displays multiple data points for these temperatures. As $T$
increases $\mu_{0}H^{*}$ progressively diminishes showing no magnetic flux
avalanche activity above 10 K. Panel (d) shows MOI images acquired in the
decreasing branch of the hysteresis loop at $\mu_{0}H_{a}=0.2$ mT which
corresponds to the dotted line shown in panel (c) and illustrating the
presence of antiflux avalanches at 5 and 9 K and their absence at 12 K.

A complete animation of the evolution of the magnetic landscape during the
magnetic loop can be found in the supplementary video 3.

##  3 Conclusion

We explored NbTiN CPW resonator structures using both magneto-optical imaging
and RF transmission experiments. A clear and unambiguous correspondence is
demonstrated between both independent methods. This allows us to directly link
the complex magnetic field penetration to the changes in the resonance
frequency in an applied magnetic field. Major attention is drawn to magnetic
flux avalanches, which drastically impact the resonance frequency of the
device when triggered. Based on the measurements and analysis presented above,
we can devise a roadmap for design optimization of the superconducting
resonators with the goal of improving their resilience and performance under
perpendicular magnetic fields:

  * •

Ground plane. In order to increase the threshold field needed for magnetic
perforation, it is essential to decrease the local field at the outer rim of
the ground plane. In the Meissner state this local field can be approximated
paper_zeldov1994magnetization  as $\approx\sqrt{W/t}\mu_{0}H_{a}$, where $W$
is the width of the ground plane and $t$ is the thickness of the
superconducting film. Therefore, the desired effect can be achieved by
reducing the lateral dimension $W$ of the ground plane.

  * •

Resonators. If flux avalanche-induced noise needs to be suppressed, a possible
solution consists in reducing the pitch of the meanders. Indeed, it has been
shown in Ref. paper_denisov2006onset  that thermomagnetic instabilities are
progressively suppressed as the width of the superconductor decreases and are
fully absent below certain threshold value. This is similarly exemplified in
the field resilience of nanowire NbTiN resonators paper_samkharadze2016high .
Alternatively, by introducing artificial roughness along the borders of the
superconducting device it should be possible to shrink the $H-T$ region where
avalanches occur paper_cerbu2013vortex ; paper_brisbois2016magnetic .

  * •

Artificial defects. It has been demonstrated that the introduction of periodic
pinning such as through holes can be beneficial for reducing the losses caused
by the flux shaking. However, this may only hold at high enough temperatures
where flux avalanches do not take place. Indeed, it is known that this kind of
pinning may actually favor the triggering of magnetic flux avalanches and
deteriorate the performance of the device at low temperatures
paper_menghini2005dendritic ; paper_motta2014controllable . Another
interesting approach consists in adding pinning centers in the resonator there
where the RF local field maximizes, as proposed in Ref.
paper_ghigo2004microwave .

  * •

Ratchet potential. It may be worth exploring the possibility to keep the
magnetic flux at bay by introducing asymmetric pinning landscapes forming flux
lenses paper_lee1999reducing . Although the efficiency of this sort of ratchet
potentials have been demonstrated at relatively low frequencies, their
performance at GHz seem to be less promising paper_dobrovolskiy2020upper .

It is interesting to mention that some high-$T_{c}$ superconductors such as
YBa2Cu3O7 do not exhibit much flux avalanche activity, at least at field rates
normally used in the laboratory. However, the possibility exists that the
rapidly oscillating RF field and the associated huge rate of change stimulate
the otherwise absent flux avalanches in these materials. Therefore, it would
be interesting to extend the present work to perform MOI and frequency
measurements simultaneously. This endeavour risks to be highly non-trivial
since the proximity of the Al mirror and the magnetic indicator needed for MOI
measurements may substantially distort the performance of the resonators. If
nevertheless one succeeds in this task, it will be possible to discern if the
RF excitation itself promotes or stimulates magnetic flux avalanches
paper_awad2011flux .

##  4 Methods

###  4.1 Sample fabrication

The resonators were fabricated on a double side polished (430 $\pm$ 25) $\mu$m
thick C-plane Al2O3 wafer. First, the wafer was cleaned using a standard 5:1:1
RCA process of deionized water, ammonia water, and H2O2 respectively and
heated to 80 ∘C for 10 minutes Kirby2008 . Subsequently, one side of the wafer
was covered by 100 nm NbTiN by sputtering on a target of Nb81.9Ti18.1N
Burton2015  in a near-UHV sputter system, under an Ar:N2 partial gas pressure
ratio of 60:37. Prior to, and during deposition the wafer was annealed at 660
∘C for 30 minutes. Subsequently, the coplanar waveguide resonators are
patterned on the sample side using a PMMA A4 mask, pre-baked at 180 ∘C. An
e-spacer 300Z conduction layer is applied to mitigate charging issues that may
arise. After development in IPA:H2O 4:1, an Ar/Cl (4:50 sccm) ICP plasma
etches the metal Mahashabde2020 ; Niepce2020 . Finally, the remainder of the
resist was removed by a heated aceton.

###  4.2 RF measurements

High-frequency measurements were performed using a Keysight P5003B Streamline
Vector Network Analyzer in a Janis 3He cryostat. The input line was attenuated
with 60dB, which was distributed across different temperature stages within
the cryostat. The sample was pasted onto an oxygen-free copper holder and
connected via wire bonding to an RF holder with a sample space of (7 × 4) mm2.
A 4-8 GHz circulator/isolator and 4-8 GHz high-electron-mobility transistor
(LNF-LNC4_8C) were connected at the 4K stage after the output line of the
sample, amplifying the signal back to the second port of the VNA. The magnetic
field was applied perpendicular to the sample using a superconducting magnet,
as illustrated in Figure 1a. To compensate for the remanent field, the sample
was field cooled ($-0.2$ mT) before every measurement cycle. A measurement
cycle consists of sweeping the magnetic field as follows : $0\rightarrow
5\rightarrow-5\rightarrow 5\rightarrow 0$ mT completing a full loop. Initial
field cycles were executed with a driving power of $-40$ dBm and compared to a
higher driving power of $0$ dBm. As there was no significant difference all
measurements were executed at $0$ dBm.

###  4.3 MOI technique

Direct visualization of the magnetic flux landscape was obtained by magneto-
optical imaging. This technique is based on the Faraday rotation of linearly
polarized light in a $3~{}\mu$m-thick Bi-doped yttrium iron garnet (indicator)
with in-plane magnetic domains, placed on top of the investigated sample
paper_koblischka1995magneto . Since the rotation of polarization is
proportional to the local magnetic field $B_{z}$ at the indicator, the use of
an analyzer oriented perpendicularly to the initial direction of polarization
results in images where the intensity is proportional to $B_{z}$. The images
are acquired with a CCD camera and have a pixel size of $1.468\times
1.468~{}\mu$m2. Post-image processing was done to remove the inhomogeneous
illumination and field-independent background, using the ImageJ software. More
information about the MO imaging setup can be found in Ref.
paper_shaw2018quantitative . Low temperature MO measurements are performed in
a closed-cycle cryostat and the external magnetic field was applied through a
copper coil. MO imaging allows to record spatial maps of the magnetic flux and
thus obtain direct information on the location and size of each event.

###  4.4 SQUID magnetometry

Magnetic measurements were carried out in a superconducting quantum
interference device magnetometer (Quantum Design SQUID VSM MPMS3). The DC
magnetization versus temperature was measured in a parallel applied field upon
heating the sample from 1.8 to 300 K under a remanent magnetic field in the
order of 0.01 mT. The critical temperature $T_{c}$ was defined as the highest
temperature with a measurable diamagnetic magnetization contribution on top of
the linear background. A demagnetization procedure was conducted at 300 K
prior to the measurement by oscillating the field from -7 Tesla to near zero
to minimize the remanent magnetic field from the superconducting magnet
paper_blunt1991investigation .

##  5 Data availability

The data that support the plots of this paper and other findings within this
study are available from the corresponding author upon reasonable request.

##  6 Code availability

The codes used for this study are available from the corresponding author upon
reasonable request.

Acknowledgments The authors acknowledge support from the EU COST action
SUPERQUMAP CA21144, the Fonds de la Recherche Scientifique - FNRS under the
grants Weave-PDR T.0208.23 and CDR J.0176.22, the Research Foundation Flanders
(FWO, Belgium), Grants No. G0A0619N and 11K6523N, and the KU Leuven C1 program
C14/18/074. N. L. acknowledges support from FRS-FNRS (Research Fellowships
FRIA) and S. M. acknowledges support from FRS-FNRS (Research Fellowships ASP).
I. C. and A. G. acknowledge support by the European Union’s H2020 research and
innovation program, grant no. 804988 (SiMS) and 828948 (AndQC). The authors
thank Christian Haffner for his valuable support regarding the experimental
development of the high-frequency setup.

Author contributions L.N. and B.R. designed the devices, I.C. and A.G.
fabricated the devices, N.L. and S.M. executed MOI measurements, L.N., J.C.
and H.D. executed RF measurements, S.B. executed SQUID magnetometry. The
manuscript was written by L.N., N.L., A.V.S. and J.V.V. with the help from all
other authors. All authors discussed the results and reviewed the manuscript.
J.V.V., M.J.V.B., B.R. and A.S. initiated and supervised the research.

Competing interests The authors declare no competing interests.

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