††thanks: These authors contributed equally to the work††thanks: These authors
contributed equally to the work††thanks: These authors contributed equally to
the work

# The effect of niobium thin film structure on losses in superconducting
circuits

Maxwell Drimmer  Department of Physics, ETH Zürich, 8093 Zürich, Switzerland
Quantum Center, ETH Zürich, 8093 Zürich, Switzerland  Sjoerd Telkamp
Department of Physics, ETH Zürich, 8093 Zürich, Switzerland  Quantum Center,
ETH Zürich, 8093 Zürich, Switzerland  Felix L. Fischer  Department of Physics,
ETH Zürich, 8093 Zürich, Switzerland  Quantum Center, ETH Zürich, 8093 Zürich,
Switzerland  Ines C. Rodrigues  Department of Physics, ETH Zürich, 8093
Zürich, Switzerland  Quantum Center, ETH Zürich, 8093 Zürich, Switzerland
Clemens Todt  Department of Physics, ETH Zürich, 8093 Zürich, Switzerland
Quantum Center, ETH Zürich, 8093 Zürich, Switzerland  Filip Krizek  Institute
of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00 Praha 6, Czech
Republic  Dominik Kriegner  Institute of Physics, Czech Academy of Sciences,
Cukrovarnická 10, 162 00 Praha 6, Czech Republic  Christoph Müller  Institute
of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00 Praha 6, Czech
Republic  Werner Wegscheider  Department of Physics, ETH Zürich, 8093 Zürich,
Switzerland  Quantum Center, ETH Zürich, 8093 Zürich, Switzerland  Yiwen Chu
Department of Physics, ETH Zürich, 8093 Zürich, Switzerland  Quantum Center,
ETH Zürich, 8093 Zürich, Switzerland

(August 28, 2024)

###### Abstract

The performance of superconducting microwave circuits is strongly influenced
by the material properties of the superconducting film and substrate. While
progress has been made in understanding the importance of surface preparation
and the effect of surface oxides, the complex effect of superconductor film
structure on microwave losses is not yet fully understood. In this study, we
investigate the microwave properties of niobium resonators with different
crystalline properties and related surface topographies. We analyze a series
of magnetron sputtered films in which the Nb crystal orientation and surface
topography are changed by varying the substrate temperatures between room
temperature and 975 K. The lowest-loss resonators that we measure have quality
factors of over $10^{6}$ at single-photon powers, among the best ever recorded
using the Nb on sapphire platform. We observe the highest quality factors in
films grown at an intermediate temperature regime of the growth series (550 K)
where the films display both preferential ordering of the crystal domains and
low surface roughness. Furthermore, we analyze the temperature-dependent
behavior of our resonators to learn about how the quasiparticle density in the
Nb film is affected by the niobium crystal structure and the presence of grain
boundaries. Our results stress the connection between the crystal structure of
superconducting films and the loss mechanisms suffered by the resonators and
demonstrate that even a moderate change in temperature during thin film
deposition can significantly affect the resulting quality factors.

##  I Introduction

The effort to build large-scale quantum processors has emphasized how studying
materials can improve the lifetime and coherence of quantum systems [1, 2, 3].
Superconducting microwave resonators are often used to investigate loss in
superconducting qubits because they are generally simpler to fabricate and
measure while all sources of relaxation and decoherence affecting the
performance of a resonator will also impact the performance of a qubit made
from the same material [4]. Additionally, there are other applications that
benefit from superconducting resonators with low levels of microwave
dissipation, such as parametric amplifiers [5], quantum sensors [6], and
microwave kinetic inductance detectors (MKIDs) [7] used for astronomy and
particle physics [8].

Measurements of superconducting resonators can distinguish between different
sources of loss by determining the losses dependence on temperature and
applied power. Dissipation at low microwave power and low temperature is
especially relevant for quantum information processing [9]. In this regime, a
common limiting loss mechanism is dielectric loss due to two-level systems
(TLS), and extensive work has been conducted in different material systems to
locate and quantify this loss [10]. Oxides and impurities in close vicinity to
the superconductor are a commonly observed cause of degradation of
superconducting properties as they can host TLS [11, 12]. Many investigations
attempting to reduce TLS loss have focused on the removal of unwanted oxides
and impurities from different interfaces through etching or cleaning steps.
Pre-deposition cleaning steps of the substrate to avoid metal-substrate
interface losses [13, 14, 15, 16, 17, 18], as well as post-deposition
processing steps of both substrate and deposited metal to reduce losses at the
exposed surfaces [19, 20, 21, 12, 22, 11, 23, 24] have been investigated
intensively. However, the connection between the superconductor film
structure, which is determined by the deposition process, and the microwave
loss suffered by circuits is not as well understood. Differences in the
deposition process have been previously studied by some authors [15, 25, 26]
who also report changes in microwave properties, but an accurate description
of the effect of film crystal properties on TLS loss as well as other loss
mechanisms has not yet been established.

Our study focuses on niobium, commonly used in the superconducting circuits
community [1] because it has the highest critical temperature and critical
magnetic field of any elemental superconductor [27]. Furthermore, mono-
crystalline Nb growth is possible at sufficiently high temperature and
optimised growth conditions [25]. Sapphire substrates offer good thermal as
well as chemical stability and have a small lattice mismatch with the Nb
lattice, which allows for epitaxial growth [28]. These key properties make Nb
on sapphire suitable material platform to study the effect of crystallinity
changes on superconducting resonator performance.

We present a study in which we systematically investigate the effect of the Nb
crystal properties and interface epitaxy on the behavior and performance of
superconducting microwave resonators. In the first part of this study, we
investigate the influence of the Nb growth temperature on the film
crystallinity as well as surface morphology and interface epitaxy towards the
sapphire substrate. The differences that we observe are then linked to the DC
superconducting transport characteristics. In the second part, we characterize
coplanar waveguide (CPW) resonators and explore their low-temperature
microwave properties. Specifically, we investigate the relationship between
thin film crystallinity as well as general morphology and the single-photon
quality factors that we measure. Furthermore, we study power-independent
losses and relate them to the grain structure of the superconducting films.

##  II Experimental design

For this study, five thin film depositions were performed at various
temperatures while all of the other deposition conditions were kept the same.
The Nb thin films were sputtered in an Ultra High Vacuum (UHV) deposition
chamber using DC magnetron sputtering [29]. The base pressure of the
sputtering chamber was below $2\times 10^{-10}$ mBar, indicating a low
impurity background. The pressure during the deposition was 8.8$\times
10^{-3}$ mBar and the distance between the 2 inch Nb target and the substrate
was 110 mm. The power applied was 125 W corresponding to a stable growth rate
of 0.27 nm/s at room temperature determined from in-situ Quartz Crystal
Microbalance measurements. The effective growth rate decreased linearly with
increasing temperature due to partial re-evaporation of adatoms from the
surface. After growth, the thickness was recalibrated using profilometry and
Transmission Electron Microscopy (TEM) measurements and the growth time was
adjusted to achieve a thickness of around 185 nm for all films. The substrate
temperature shown in Tab. 1 was set based on pyrometer readings of the
temperature of the middle of the wafer.

The Nb films were grown on epi-ready 420 um thick, (0001) sapphire wafers with
a diameter of 2 inches. The substrates were pre-baked at 500 K for 8 hours in
the load lock and subsequently outgassed for 20 minutes at growth temperature
in the UHV chamber before growth.

An overview of the growth parameters for all wafers is shown in Tab. 1.

Wafer | A | B | C | D | E
---|---|---|---|---|---
Temperature [K] | 300 | 550 | 745 | 900 | 975
Pressure [mBar$\times 10^{-3}$] | 8.8 | 8.8 | 8.8 | 8.8 | 8.8
Power [W] | 125 | 125 | 125 | 125 | 125
Table 1: Overview of the sputtering conditions of the five wafers investigated
in this study.

From this series of Nb thin films, we fabricated chips containing CPW
resonators [30]. One chip features four open-circuit $\lambda/2$-resonators of
different lengths which are coupled to a central waveguide in the "hanger" or
"notch" geometry [4, 31] (see Appendix Fig. 7). We pattern this design using a
photo-lithography process followed by a $\text{SF}_{6}$-based reactive-ion
etch. After etching, the excess resist is stripped and the wafer is diced into
chips which are packaged and measured in a dilution refrigerator. For more
details about fabrication and measurement, see Appendices C and D.

##  III Results

###  III.1 Nb thin film crystal properties

The crystal properties of a sputtered superconducting film can be affected by
varying the substrate temperature during the deposition. If the substrate
temperature is relatively low, deposited material forms a poly-crystalline
film consisting of small crystal grains without a preferred orientation. At
higher temperature, adatoms on the surface are supplied with increased energy
that results in enhanced surface mobility. This leads to self-assembly into
larger grains that are oriented in the most energetically favorable way, which
in turn has an effect on the film morphology as described by a Thornton Zone
Diagram [32, 33].

Fig. 1 (a) shows X-Ray diffraction (XRD) traces of the Nb films listed in Tab.
1. Various Nb diffraction lines are indicated in the figure. Film A that was
deposited at room temperature shows signatures of all the commonly observed
diffraction lines of poly-crystalline Nb, namely 110, 200, 211, and 220 [34].

Figure 1: (a) XRD radial scans of the wafers listed in Table 1. The traces are
offset with respect to each other for clarity. The sharp peaks that are
associated with the mono-crystalline sapphire substrate are labeled with a
star. (b) XRD polefigures of the 110 Bragg peak showing the orientation
distribution of the [110] direction for films deposited at various
temperatures.

We observe a clear crossover in crystallinity from poly-crystalline at room
temperature (Wafer A) to mono-crystalline at 975 K (Wafer E). The
crystallographic changes become apparent at 550 K (Wafer B), where the [211]
crystal orientation disappears and exclusively the [110] out-of-plane
orientation is observed. At 745 K (Wafer C) the dominant out-of-plane crystal
orientation changes to the [111] direction, which is not visible in Wafers A
or B. Finally, at 975 K the 222 peak is the only visible peak associated with
Nb in the XRD spectrum which indicates that we have grown a mono-crystalline
film with [111] out-of-plane orientation. We do not directly observe the 111
peak because it is a forbidden reflection for Nb.

This crystallographic crossover from poly-crystalline to mono-crystalline is
confirmed by the XRD polefigures shown in Fig. 1 (b). In this figure, a
stereographic projection of the diffraction signal from the 110 lattice planes
is shown, from which we can also asses the in-plane crystal orientations. The
signal in the center of the figure corresponds to an out-of-plane [110]
orientation. For Film A, the figure shows an almost homogeneous distribution
of diffraction intensity. This indicates that there is no directional
preference of the crystal grains and is consistent with a poly-crystalline
film. Starting at 550 K, dominant crystal grain orientations are formed as
indicated by discrete features in the polefigure. At 745 K, highly intense
diffraction spots appear which are consistent with the [111] out of plane
orientation. The three observed peaks correspond to the three-fold rotation
axis around the [111] direction. However, a weak contribution in the center of
the polefigure remains at 745 K as well as some minor contributions at 900 K
that are not aligned with the dominant orientation. At 975 K, a completely
mono-crystalline structure with [111] out-of-plane orientation is formed and
no other orientations can be detected within the probed area of several mm2.

The deposition temperature also influences the surface roughness of the
sputtered Nb film. The surface topography was measured using atomic force
microscopy (AFM) for each individual wafer directly after growth and is shown
in Fig. 2 (a). The surface of the wafer grown at room temperature clearly
consists of small grains with no apparent preferential orientation. This grain
structure of Nb has been frequently observed on a large variety of substrates
such as Si [35] and GaAs [29]. A distinct triangular pattern appears at 550 K
that becomes more and more evident with increasing temperature. The schematic
in Fig. 2 (c) illustrates how a top view of the Nb (111) lattice planes could
result in the observed triangular geometry. Our data suggests that the crystal
domains grow in size and consequently change the surface roughness and
morphology in a complex process of self-assembly during growth. The effect on
the root-mean-square (RMS) surface roughness calculated from the 2$\times$2
$\mu$m AFM areas is plotted in Fig. 2 (b). It is evident from this plot that
Wafer B has the lowest surface roughness.

Figure 2: An overview of the effect of deposition temperature on Nb surface
morphology studied by AFM topography images. In (a), Wafer A shows the
commonly observed randomly oriented grain structure of Nb while the wafers
grown at higher temperature start to display a triangular pattern that can be
associated with the [111] direction viewed from the top (as indicated in (c)).
Figure (b) shows the $R_{q}$ surface roughness parameter associated with each
wafer.

Another key feature that is known to have an effect on superconducting
microwave resonator performance is the quality of the interface between the
superconductor and the substrate [36]. We investigated the Al2O3-Nb interface
of Wafers A, B and D using Scanning Transmission Electron Microscopy (STEM).
In Fig. 3 (a), it can be observed that for films grown at room temperature, a
high-quality non-epitaxial interface is obtained that does not show any signs
of oxides or chemical intermixing. The poly-crystalline character of the Nb
layer is further substantiated by the ring-shaped pattern observed in the fast
Fourier transform (FFT) image in the top right corner. At intermediate growth
temperatures, it can be seen in Fig. 3 (b) that an interface with a high
degree of epitaxy between the sapphire (0001) and the Nb (110) crystals forms.
Some variation in grain alignment was observed on the length scale of the
lamella window ((1.5 $\mu$m) for Wafer B. In Fig. 3 (c), the Nb to sapphire
interface of Wafer D is shown. For this wafer, no grain boundaries were
observed on the length scale of the lamella window and the Nb (111) to
sapphire interface seems fully epitaxial. Therefore, we conclude that the
(200) peak observed in the XRD measurements shown in Fig. 1 is only a very
minor portion of the total crystal structure. A crystallographic model of the
Nb lattice made using crystal visualization software (VESTA [37]) is overlaid
with a zoomed-in part of the figure and shows agreement with the image. The
intense highly symmetrical features of the FFT in the in-set of Fig. 3 (c)
clearly confirm the single crystalline nature and high degree of epitaxy in
the measured portion of the film. The strain relaxation by the formation of a
periodic array of misfit dislocation can be deduced from the Bragg filtered
image of Wafer D shown in Fig. 3 (c).

Figure 3: STEM images of Nb to sapphire interfaces of wafers grown at various
temperatures. (a) A bright field STEM image in which a high-quality non-
epitaxial interface is observed between the sapphire substrate and the poly-
crystalline Nb film grown at room temperature (Wafer A). (b) A similar image
of the interface between the Nb grown at 550 K and the sapphire substrate
(Wafer B). In (c) the interface for Wafer D including a crystallographic model
showing the Nb crystal lattice is shown. The zone axis differs between the
images. In (d), a Bragg filtered image of a portion of the interface in (b) is
shown in which the formation of sporadic misfit dislocations is indicated.

###  III.2 DC transport characterization

Before fabricating the resonators, the critical temperature and magnetic field
of the superconducting Nb films were measured to benchmark the quality of the
growth. Low-temperature DC transport measurements were performed in a Van der
Pauw geometry using standard lock-in techniques to obtain the Tc and Bc of our
wafers. All of the films have critical temperatures around $9.25\pm 0.1$K,
which is close to the maximum bulk value of Niobium of 9.3 K [38]. We found
that the critical magnetic field was consistently higher for the films with a
lower degree of crystallinity. This can be attributed to superconducting
vortices that are formed in the mixed state ($B$>$B_{c1}$) and are pinned on
grain boundaries in the material [39]. More details on these measurements can
be found in Appendix G.

The mean free path $l_{\mathrm{MFP}}$ and the Ginzburg-Landau (GL) coherence
length $\xi_{GL}(0)$ are determined from these measurements since they can be
used to identify possible sources of decoherence. Using the phase diagram
shown in supplementary Fig. 16, we extrapolate the GL Bc2 value to find the
critical magnetic field at 0 K. The GL coherence length was calculated using
Eq. 15 and the mean free path is determined by measuring the wafer’s
resistivity at 10 K and using Eq. 16. The resulting values are shown in Tab.
2.

| Wafer A | Wafer B | Wafer C | Wafer D | Wafer E
---|---|---|---|---|---
$\rho_{\mathrm{10K}}$ [$\mu\Omega$cm] | 4 | 2.3 | 1.1 | 0.77 | 0.74
$l_{\mathrm{MFP}}$ [nm] | 9 | 16 | 34 | 48 | 50
$\xi_{\mathrm{GL}}(0)$ [nm] | 14 | 20 | 23 | 23 | 23
Table 2: The residual resistivity at 10 K $\rho_{\mathrm{10K}}$, the mean free
path length $l_{\mathrm{MFP}}$ derived from $\rho_{\mathrm{10K}}$, and the
Ginzburg-Landau coherence length $\xi_{\mathrm{GL}}(0)$ determined from the
critical magnetic field values shown in Fig. 16.

###  III.3 Microwave characterization

The losses of microwave resonators were measured by analyzing the S21
scattering response and extracting the linewidth and resonance frequency of
each resonance. The different physical mechanisms which contribute to the
observed loss rate can be distinguished by varying the applied microwave power
and temperature [4]. According to similar studies in literature, we expect
dielectric losses due to TLS to be the dominant source of loss in the low
temperature and single-photon power regime [12, 11]. In addition to losses
from TLS, which scale strongly with applied microwave power, we consider loss
mechanisms that do not depend on power. These include temperature-dependent
losses due to thermal quasiparticles, but also temperature-independent losses,
e.g. due to external pair-breaking radiation or parasitic modes within the
resonators [17, 40, 12, 41]. Therefore, we include $\delta_{\mathrm{PI}}(T)$,
a power-independent (PI) loss term, in our model:

$\frac{1}{Q_{\mathrm{int.fit}}(T,P)}=\frac{1}{Q_{\mathrm{int}}^{\mathrm{TLS.full}}(T,P)}+\delta_{\mathrm{PI}}(T)$
(1)

Other authors have employed this practice in different variants, some
explicitly modeling the temperature dependence of PI losses [24, 19], some
splitting the total loss into a high-power and a low-power regime [18, 42, 11,
43]. The full expression for our TLS loss model
$\frac{1}{Q_{\mathrm{int}}^{\mathrm{TLS.full}}(T,P)}$ is presented in Appendix
6.

Figure 4: (a) Temperature and power dependence measured of the quality factor
of the starred resonator from Wafer B marked in (b). Data is shown on the left
while the model that was fit to that data is on the right. (b) The top panel
is the low-temperature, single-photon power internal quality factors extracted
from all resonators that we measured, excluding those with fabrication
defects. The bottom panel shows the TLS losses extracted from the power and
temperature dependence of the internal quality factor for each of the
resonators. The TLS loss determined from fitting one of the resonators from
Wafer B was subject to large error, because at ten photons in the resonator
the internal quality factor exited the critical coupling regime (see Appendix
F.1).

Fig. 4 (a) compares a measurement of the internal quality factor of one
resonator from Wafer B to Eq. 1. The excellent agreement indicates that our
model captures the temperature and power dependence of the measured quality
factors (see Appendix F for the same comparison for all resonators shown in
Fig. 4 (b) ). As predicted by TLS theory, the internal quality factor is
highest at high power and has a non-straightforward temperature dependence
originating from the competition of TLS saturation and TLS interactions [19,
10, 24]. Additionally, fitting to our model allows us to distinguish TLS loss
from PI loss. In Fig. 4 (b) we show the low-temperature single-photon quality
factors for each resonator that we measured together with the extracted TLS
losses. We excluded any resonators with visible fabrication defects. The yield
of working resonators varied for each chip but was always $\geq 50\%$.
Resonators from Wafer B, grown at 550 K, show substantially higher quality
factors and lower TLS losses than resonators grown at the other temperatures.
The highest single-photon $Q_{\mathrm{int}}$ we measured was $1.1\pm 0.1\times
10^{6}$, which to our knowledge is among the highest ever measured single-
photon $Q_{\mathrm{int}}$ for a niobium on sapphire resonator [44, 45].

We also fabricated resonators from Wafer E, but observed more than one order
of magnitude lower internal quality factors than for the other wafers. We
attribute this to the high number of defects that exist in wafers grown at 975
K (more detail can be found in Appendix A). The formation of these defects is
likely a result of surface adhesion problems during film deposition at these
high temperatures. We observe only small variations in internal quality factor
between Wafers A, C, and D that are on the order of variations between
resonators from the same wafer.

The fitted PI losses are shown as a function of temperature in Fig. 5 (a). We
observe a clear temperature dependence of the PI loss for Wafers C and D while
the PI loss appears constant for Wafers A and B. We further observe
differences in the temperature-independent contribution to
$\delta_{\mathrm{PI}}$ that we attribute to slight differences in the
measurement environment. These differences are rather small compared to the
temperature-dependent variations and compared to the loss from TLS. In Fig. 5
(b), the temperature dependence of the fractional frequency shift $\Delta
f_{r}/f_{0}=(f_{r}(T)-f_{r}(T\approx 0))/f_{r}(T\approx 0)$ shows a similar
trend, where $f_{r}$ is the resonator frequency extracted from fitting the raw
data. This behavior will be further explored in the discussion section. The PI
losses are smaller than the TLS losses for resonators from the Wafers A and B
at all temperatures. For Wafers C and D, we observe that the PI loss dominates
over the TLS loss at the highest temperatures that we measured.

Figure 5: Power-independent loss (a) and fractional frequency shift (b) of
each resonator measured in Fig. 4 (b) as a function of temperature.

##  IV Discussion

The differences that we observed between microwave measurements of our wafers
can be interpreted within the context of material properties of our films
presented in subsection III.1. From the XRD characterization shown in Fig. 1
it can be seen that the biggest change in crystallinity occurs between Wafers
A and B and that comparatively little change occurs between Wafers C and D.
This interpretation of the XRD data is in line with the values of
$l_{\mathrm{MFP}}$ and $\xi_{GL}(0)$ shown in Tab. 2 which indicate that grain
boundaries become less frequent in Wafers grown at larger temperatures. The
AFM data displayed in Fig. 2 demonstrates a less straightforward trend, but we
can clearly see that Wafer B has a far lower surface roughness than the other
films. This could be attributed to the observation that only the niobium film
on Wafer B is primarily oriented in [110] direction while Wafers C and D are
dominantly oriented in the [111] direction.

The power-dependant response of our resonators indicates that the dominant
mechanism limiting our single-photon-power low-temperature internal quality
factors is loss due to TLS. We achieve the lowest TLS loss for Wafer B, which
features the lowest surface roughness, suggesting that the roughness of the
film has an effect on TLS loss. The exact mechanism that describes how surface
morphology affects the TLS loss in our study is unclear. In films with larger
surface roughness, the TLS loss could be higher due to an increased volume of
dielectric material or the presence of sharp surface features that concentrate
the electrical field distribution at the surface. Furthermore, changes in
surface morphology observed in the AFM or different dominant crystal
orientations could favor the growth of one niobium oxide species over another.
This would lead to a change in the TLS loss because certain oxide species are
known to host higher densities of TLS [46].

It is however premature to correlate dielectric loss exclusively with the
roughness of a film. For instance, changes in surface roughness between Wafers
C and D are not clearly reflected in their dielectric losses. Furthermore, the
average TLS loss of Wafer A is 1.5 times that of Wafer D, even though Wafer D
shows a slightly higher surface roughness. To explain these differences, we
suggest a separate contribution to the overall TLS loss that depends on grain
structure. Oxides are known to penetrate metallic films along grain boundaries
[47], potentially hosting additional TLS. Furthermore, disorder in the crystal
due to the grain boundaries themselves could also lead to higher TLS loss
[10]. In contrast, contributions from other common sources of TLS such as the
substrate-air and metal-substrate interfaces are likely negligible in our
study from qualitative arguments. The substrate-air interface is completely
unchanged between different chips and the results of our STEM study convince
us that the metal-substrate interface has no oxides and no visible structural
damage for both high and low-temperature depositions.

Other authors who investigated the effect of different growth conditions on
microwave dissipation report higher internal quality factors/T1 times for
growths at 770 K [26] or at 1070 K [25] compared to room-temperature growths.
Furthermore, in other material systems, highly ordered films displayed lower
losses than granular films [15]. Both types of studies only compared the
extremes of a continuous spectrum of growth conditions and grain structures.
Our results do not contradict these findings, instead, we report a temperature
"sweet spot" in our growth series where the combination of all the factors
that affect TLS loss produces the highest low-energy quality factors.

We can also interpret our measurements to learn about sources of loss other
than TLS. The temperature-dependant behavior shown in Fig. 5 is likely a
result of thermally populated quasiparticles, since only this loss mechanism
both increases the dissipation and decreases the resonance frequency of a
resonator as a function of temperature [4]. In our low-temperature regime, the
density of quasiparticles depends on the film’s quasiparticle trapping rate
which is related to the grain boundary density [48]. We argue that as the
growth temperature exceeds 550 K, our wafers enter a regime where the
quasiparticle trapping is reduced to an extent that we see an effect on
resonator frequencies and losses. However, beyond Wafer C there does not seem
to be a big difference in the trapping of quasiparticles. We speculate that
the reason behind this behavior lies in changes in the trapping mechanism of
thermal quasiparticles. The global amount of grain boundaries/scattering sites
decreases from Wafer A to D, as indicated by the MFP increasing. The vortex
trapping behavior resulting from these grain boundaries likely affects the
films less as films become more mono-crystalline, as evidenced by the
coherence length increasing from Wafer A to Wafer C and then staying constant
through Wafer E. We theorize that Wafers A and B are relatively unaffected by
thermal quasiparticles up to $\sim$500 mK because of quasiparticle trapping at
grain boundaries. However, in Wafers C and D this trapping mechanism is no
longer able to suppress thermal quasiparticles and as a result we observe
power independent losses that increase with temperature.

##  V Conclusion

In the first section of our study, we show that the crystallographic structure
of Nb on sapphire films is drastically affected by changing the deposition
temperature. This resulted in films ranging from a poly-crystalline structure
(Sample A) at room temperature to a completely mono-crystalline character at
the highest temperature of 975 K (Sample E). With increasing temperature, the
films have fewer grain boundaries and the crystal domains undergo an
increasing degree of ordering in the preferred [111] orientation. These
differences in crystallinity also lead to variation in surface roughness, most
notably in Wafer B, which has the lowest surface roughness.

These differences in material properties lead to significant variations in the
microwave behavior of our superconducting resonators. Specifically, we found
the best single-photon quality factors of over one million in the resonators
made from Wafer B, which was grown at an intermediate temperature of 550 K.
These quality factors are state-of-the-art for Nb on sapphire resonators [44,
45], and are within an order of magnitude of the state-of-the-art for niobium
resonators on any substrate [12]. We believe that the TLS loss rate could be
further reduced by optimizing the fabrication process by including steps like
a HF etch to remove the processing oxide [12, 11, 24], etching trenches [42,
49, 50], encapsulating the resonator [23, 51], or optimizing the CPW geometry
for low electric field participation at material interfaces [42, 50, 52]. Our
results could also be strengthened by measuring more resonators on more
samples, as the TLS losses we report vary from resonator to resonator.

This result indicates that TLS losses in Nb qubits on sapphire are reduced by
increasing the substrate temperature by only 250 K above room temperature. A
similar optimum might also exist for other material systems and are a subject
for further investigation. Growing at moderate temperatures will significantly
decrease the number of grain boundaries in the film while also leading to
relatively low surface roughness. Our work indicates that these factors could
reduce the TLS losses by up to an order of magnitude. A relatively small
temperature increase during growth could be applied in many different
sputtering systems, since it does not put as stringent requirements on the
amount of background impurities in the system as the high-temperature growth
of, for example, Wafer D does.

We also observe a temperature dependence in our resonator measurements that we
attribute to the degree of crystallinity of the films. Our data displays two
distinct behavioral regimes which we suspect arise from the relation between
the quasiparticle dynamics in the film and the grain boundary density. This
could be an important consideration for studying quasiparticle trapping
dynamics and could be further investigated by future experiments, for example
those with a magnetic field degree of freedom. Finally, the experiment would
additionally benefit from x-ray photoelectron spectroscopy (XPS) and TEM
electron-energy-loss spectroscopy (EELS) analysis to determine the composition
of the surface oxide for each wafer.

## Acknowledgements

We thank M. Bahrami Panah, I. Kladarić, H. Doeleman, T. Schatteburg, L.
Michaud, E. Planz, S. Andresen, and Y. Yang for valuable discussions, ScopeM
and specifically P. Zeng for lamella preparation by FIB and TEM support and M.
Sousa (IBM) via the Binning and Rohrer Nanotechnology Center (BRNC) for TEM
support. Funding: This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 948047). This project was additionally
supported by the Swiss National Science Foundation (SNSF), Czech Science
Foundation (Grant No. 22-22000M), MEYS CR grant nr.
CZ.02.01.01$/$00$/$22$\\_$008$/$0004594, and Czech Academy of Sciences
(project no. LQ100102201). Author contribution: M.D., S.T., and F.F. conceived
the experiment. S.T. grew the wafers with technical support from C.T.. S.T.
F.K., D.K., and C.M. performed and analyzed the XRD measurements. S.T. and
F.K. preformed and analyzed the STEM measurements. S.T. and F.F. characterized
the wafers using DC transport and AFM measurements. I.R. and F.F. designed the
resonators. M.D. fabricated and packaged the devices. M.D. and I.R.
constructed the microwave measurement setup. M.D. and F.F. performed the
microwave measurements. F.F. performed the fitting and analyzed the data with
support from M.D. and I.R.. M.D., S.T., F.F. interpreted the results and wrote
the manuscript, which was revised by all authors. The work was supervised by
Y.C. and W.W..

##  Appendix A Growth Defects at High Temperatures

At growth temperatures above 900 K, we observe high densities of point-like
growth defects. We propose that reduced adhesion of Nb atoms at high growth
temperatures is the cause of these defects. Any impurity or defect on the
sapphire substrate surface can lead to localized spots where the Nb does not
nucleate homogeneously and ultimately results in a hole in the film. The
density of defects that we observed is always high at high growth temperatures
but can vary from wafer to wafer. We observed this variation even though the
substrate preparation was the same for all growths. While for wafers of type D
grown at 900 K we can achieve low defect densities, this was not possible for
wafers grown at 975 K (Wafer E). For this wafer, even after multiple attempts,
the defect density remained high. Resonators that are fabricated on wafers
with visibly high defect densities show orders of magnitude lower single-
photon power low-temperature internal quality factors (1.0$\times 10^{4}$ and
1.2$\times 10^{4}$ for two resonators on Wafer E) compared to our other
devices. We exclude these devices from our main study, as the mechanisms
limiting them are exclusive for the highest temperatures and no clear trend is
visible.

A qualitative comparison of the density of these defects between a high-
temperature and a low-temperature wafer is shown in Fig. 6.

Figure 6: Optical microscope images of adhesion defects. (a) shows a wafer
with low defect density grown at room temperature. (b) shows a wafer with a
very high defect density grown at 900 K.

##  Appendix B Resonator Design

Superconducting microwave resonators serve as a proxy for the characterization
of materials for application in superconducting qubits. A common
implementation of a microwave resonator is a coplanar waveguide. A single
resonator consists of a center conductor of width $w_{\mathrm{cc}}$ separated
from a lateral ground plane by trenches of width $w_{\mathrm{gap}}$. Our chip
design features four open-circuit $\lambda/2$-resonators of lengths $l_{i}$
for $i=1-4$ and one common feedline, contained in a
$$10\text{\,}\mathrm{m}\mathrm{m}$\times$10\text{\,}\mathrm{m}\mathrm{m}$$
square chip. Each resonator is of a different length to separate their
response in frequency space and to allow the measurement of all four
resonators with one read-out signal. A length $l_{\mathrm{coupling.i}}$ of
each resonator runs in parallel to the feedline with a spacing of
$d_{coupling}$. In this layout, each resonator is coupled capacitively to the
common feedline, with the coupling strength being controlled by varying
$l_{\mathrm{coupling.i}}$ and $d_{coupling}$. A schematic of the design is
shown in Fig. 7. For all chips, $w_{cc}=$10\text{\,}\mu\mathrm{m}$$,
$w_{gap}=$6\text{\,}\mu\mathrm{m}$$,
$d_{\mathrm{coupling}}=$12\text{\,}\mu\mathrm{m}$$, and
$l_{i}=\left(8.782,8.160,7.609,7.120\right)$mm are kept constant between
different chips. For this choice of $l_{i}$, we determined resonance
frequencies of $\left(7,7.5,8,8.5\right)$GHz by simulating our design using
AnsysEM. We vary the coupling length $l_{\mathrm{coupling.i}}$ in a range of
$$25\text{\,}\mu\mathrm{m}$-$350\text{\,}\mu\mathrm{m}$$ between different
chips and also within each chip. As a result, we measured coupling quality
factors of $Q_{\mathrm{c}}\approx 4\times 10^{4}-4\times 10^{6}$. Internal
quality factors can only be determined with small error bars if they are
within the critical coupling regime [53] $Q_{\mathrm{int}}\sim Q_{\mathrm{c}}$
(see Appendix F.1). In our chip design, each resonator is coupled to the
feedline with a different $l_{\mathrm{coupling.i}}$ such that the chip covers
a broad range of possible internal quality factors. As a consequence, some
resonators on our chips, for instance one resonator on Wafer B and Wafer D,
were not in the critical coupling regime, which prevented us from extracting
an accurate estimate of the internal quality factors.

Figure 7: (a) A diagram with the important geometrical parameters in our chip
design. (b) An optical microscope image of a chip from Wafer B after
fabrication.

##  Appendix C Resonator Fabrication

The wafers are cleaned by being sonicated in acetone and then in isopropanol
for two minutes each. Then the wafers are baked at 180∘ C for two minutes
before we spin a $\sim$1.5 $\mu$m thick layer of AZ 5214E-JP image-reversal
photoresist and bake at 105∘ C for one minute. We use an i-line soft contact
masked photolithography process to expose the wafers and develop the resist in
AZ 726 MIF for three minutes.

After rinsing in water and drying, we load the wafers into an Oxford
Instruments Plasmalab 80 Plus reactive-ion etcher. We flow five sccm of
$\text{SF}_{6}$ and use an RF power of 100 W while illuminating the wafer with
a helium-neon laser and monitoring the intensity of the reflected spot. When
the intensity of the spot drops (typically after five minutes of etching), we
continue etching the wafers for between 20-75 seconds in order to account for
any anisotropy of the plasma. Subsequent STEM measurements of cross-sections
(shown in Fig. 8) of these resonators demonstrate the sidewall profile of this
procedure.

After etching, the excess resist is stripped in 70∘ C N-Methyl-2-pyrrolidone
(NMP) for at least 30 minutes before the wafers were sonicated in NMP, then
acetone, and finally isopropanol for one minute each and blown dry. A Disco
DAD 3221 Dicing Saw is used to cut the wafer into 10x10 mm chips. We use small
pieces of a 100 $\mu$m indium sheet to press our chip onto the corner
pedestals of a QCage.24 sample holder from Quantum Machines and use aluminum
wirebonds to connect the waveguide launchers to the transmission line of the
package.

In the context of our conclusions regarding the effect of crystal structure on
the microwave properties of our resonators, it is crucial to verify that our
fabrication process was reproducible and did not lead to any unwanted
variation in the measured resonators. Although effort was made to perform the
fabrication process exactly the same for each chip, small variations in for
example thickness of the films did require slightly different dry etching
times. We show the cross-section of the etched trench for Wafers A and B in
Fig. 8. From this figure, it can be seen that a very similar etch profile is
obtained in both chips. This result further substantiates that the fabrication
process was indeed reproducible and the observed differences in microwave
characterization are indeed related to the crystal structure rather than chip-
to-chip variation in fabrication.

Figure 8: BF STEM images showing the cross-sectional profile of resonators
after fabrication from Wafer B (a) and Wafer A (b).

##  Appendix D Microwave Measurement Details

We conduct microwave measurements in a dilution refrigerator. Our sample is
thermalized to the base plate of the cryostat. The temperature of our sample
is estimated from a thermometer on the base plate of the cryostat. The
microwave response of the resonators is analyzed with a vector network
analyzer (VNA) (Keysight, P5004A). During the microwave characterization of
our samples, we apply microwave signals with powers ranging from
$-80\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$ to
$10\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$ at frequencies between
$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and
$12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and record the magnitude and phase
of the transmitted signal. We repeat this procedure at temperatures between
($8\text{\,}\mathrm{m}\mathrm{K}$, $400\text{\,}\mathrm{m}\mathrm{K}$). Some
measurements were repeated to check that the samples were well-thermalized and
that the results were reproducible.

The input signal enters the fridge and is subject to 60 dB of attenuation
installed on different stages of the cryostat as well as a low-pass filter
(K$\&$L Microwave, 6L250-12000/T26000-OP/O) and an eccosorb filter (Laird,
Eccosorb CR-110) before reaching the sample. The sample and eccosorb filter
are inside of a mu-metal magnetic shield (Amuneal, A4K Can). After the sample,
the signal passes through another eccosorb and low-pass filter as well as two
isolators (LNF, ISISC4-12A) in series mounted at the mixing chamber plate. The
output signal is further amplified at the 4 K stage by a high electron
mobility transistor (HEMT) in addition to a room-temperature amplifier. A
diagram of the setup is shown in Fig. 9.

Figure 9: A schematic of the measurement chain inside of the dilution
refrigerator. iso x2: Two isolators in series. K$\&$L: Low-pass filter. ecco:
Eccosorb filter. H: HEMT.

In addition to the attenuation from the attenuators, the applied microwave
power is further reduced by loss in the cables and attenuation due to
microwave reflection at the connectors. To find the total attenuation of the
input line we perform a power calibration measurement [54]. While measuring
over a broad bandwidth of several gigahertz, we assume that the the noise of
the detected signal is dominated by the HEMT thermal noise. We calculate
$P_{\text{HEMT.0 dBm}}$, the signal power at the HEMT while applying
$0\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$, by multiplying the noise power of
the HEMT (expressed as a noise RMS voltage) with the signal-to-noise ratio
(SNR) of the measured signal at $0\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$ of
applied power. Assuming a 50 $\Omega$ impedance,

$\displaystyle P_{\text{HEMT.0 dBm}}$
$\displaystyle=10\log\left[\frac{\left(V_{\text{Noise
RMS}}^{\text{HEMT}}*\text{SNR}_{\text{0 dBm}}\right)^{2}}{50/1000}\right]$ (2)
$\displaystyle V_{\text{noise RMS}}^{\text{HEMT}}$
$\displaystyle=\sqrt{P_{\text{Noise.HEMT}}*50}$ $\displaystyle
P_{\text{Noise.HEMT}}$ $\displaystyle=k_{\mathrm{B}}T_{\mathrm{HEMT}}\Delta f$

We take noise of the HEMT $P_{\text{Noise.HEMT}}$ from the noise temperature
($T_{\mathrm{HEMT}}=2$ K) stated in the datasheet of the amplifier. The
intermediate frequency bandwidth $\Delta f$ is set when measuring with the VNA
and in our case was $5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. The SNR is
calculated by measuring the magnitude of the S21 response of a selected
frequency range 50 times and dividing the average signal by the standard
deviation. By assuming another $2\text{\,}\mathrm{d}\mathrm{B}$ of attenuation
between the HEMT and the sample caused by installed circulators, we find the
signal power at the sample to be:

$P_{\text{on\ chip}}[\text{dBm}]=2\text{dB}+P_{\text{HEMT.0
dBm}}+P_{\text{applied}}$ (3)

The attenuation is given by:

$\text{Att}[\text{dB}]=P_{\text{on\ chip}}-P_{\text{applied}}$ (4)

The resulting attenuation for one measurement setup is plotted in Fig. 10.

Figure 10: A representative attenuation spectrum over a range that covers the
designed frequencies of our resonators. A narrow dip corresponding to a
resonance is visible at $\approx 8.3$ GHz

##  Appendix E Microwave Loss Model

From the standard tunneling model [55, 56], the temperature and power
dependence of the internal quality factor due to TLS loss can be described by
[57]

$\displaystyle\frac{1}{Q_{\mathrm{int}}^{\mathrm{TLS}}}=\delta_{\mathrm{Loss.TLS}}\frac{\tanh\left(\frac{\hbar\omega_{r}}{2k_{\mathrm{B}}T}\right)}{\sqrt{1+\frac{n_{\mathrm{ph}}^{\alpha}(P)}{n_{\mathrm{s}}(T)}}}$
(5)

where $\delta_{\mathrm{Loss.TLS}}=\sum^{N}_{i}p_{i}\delta_{\mathrm{TLS.i}}$ is
the sum over different lossy regions $i$, each with loss tangent
$\delta_{\mathrm{TLS.i}}$ and participation ratio $p_{i}$. The participation
ratio of a volume $i$ is the fraction of the total electric field strength
concentrated in that volume
$p_{i}=\int_{V_{i}}\frac{\epsilon_{i}}{2}\left|\mathbf{E}_{i}(\mathbf{r})\right|^{2}/E_{\mathrm{tot}}\
\mathrm{d}\mathbf{r}$ [36]. $n_{\mathrm{ph}}(P)$ is the power-dependent
intracavity photon number, $\alpha$ is a fitting parameter that is introduced
to account for the effect of non-uniform field distributions over the length
of the transmission line that makes up the resonator [58, 24],
$\omega_{r}=2\pi f_{r}$ is the angular resonance frequency of the resonator
and $n_{\mathrm{s}}(T)$ is the photon number at which the TLS saturate.

We cannot quantitatively distinguish between the contributions of different
dielectrics to our total loss in this study. To isolate loss contributions
from different interfaces (e.g. the substrate-air interface), we would need to
vary the cross-sectional geometry of our resonators [4], while to
differentiate between loss contributions from different dielectric materials
we would need to identify the dielectric species present on each wafer by
means of XPS or EELS [11]. Therefore, we only fit the overall sum of
dielectric losses $\delta_{\mathrm{Loss.TLS}}$ in our model. This description
is still sufficient in our case, as we have qualitative arguments to exclude
contributions of different interfaces (see the fourth paragraph of Sec. IV).

The saturation photon number $n_{\mathrm{s}}(T)$ is proportional to
$\left(\tau_{1}(T)\tau_{2}(T)\right)^{-1}$, the inverse of the averaged
relaxation and decoherence times of the TLS ensemble [59, 19, 24]. According
to the spin-boson model, we expect the relaxation time to follow a thermal
distribution
$\tau_{1}\propto\tanh\left(\frac{\hbar\omega_{r}}{2k_{\mathrm{B}}T}\right)$,
while we model decoherence from TLS-TLS and TLS-phonon interactions as a
simple inverse polynomial relation $\tau_{2}\propto T^{-\beta}$ [57, 60, 19,
24]. This yields the expression $n_{\mathrm{s}}(T)\approx
DT^{\beta}\coth\left(\frac{\hbar\omega_{r}}{2k_{\mathrm{B}}T}\right)$ which
describes the temperature dependence of the saturation photon number. Our TLS
model (Eq. 6) includes four fit parameters: the exponents $\alpha$ and
$\beta$, a proportionality constant of the saturation photon number $D$, and
the TLS loss $\delta_{\mathrm{Loss.TLS}}$ [24]:

$\displaystyle\frac{1}{Q_{\mathrm{int}}^{\mathrm{TLS.full}}}=\delta_{\mathrm{Loss.TLS}}\frac{\tanh\left(\frac{\hbar\omega_{r}}{2k_{\mathrm{B}}T}\right)}{\sqrt{1+\frac{n_{\mathrm{ph}}^{\alpha}(P)}{DT^{\beta}}\tanh\left(\frac{\hbar\omega_{r}}{2k_{\mathrm{B}}T}\right)}}.$
(6)

To model power-independent but potentially temperature-dependent losses
besides TLS loss, we introduce a fifth, temperature-dependent fitting
parameter $\delta_{\mathrm{PI}}$, resulting in Eq. 1 as our final loss model.

To apply Eq. 1 to our data, we need to determine the photon number in the
resonator $n_{\mathrm{ph}}(P)$ as a function of applied microwave power. We
can convert microwave power to photon number using the expression [58, 61]

$n_{\mathrm{ph}}=\frac{P_{\mathrm{in}}}{hf_{\mathrm{r}}^{2}}.$ (7)

where $h$ is the Planck constant. The power in the resonator
$P_{\mathrm{in}}$, also referred to as the ‘circulating power’, can be derived
from input-output theory [62] to be given as [58, 61]

$P_{\mathrm{in}}=P_{\mathrm{on\
chip}}\frac{Q_{\mathrm{int}}^{2}Q_{\mathrm{c}}}{\pi\left(Q_{\mathrm{int}}+Q_{\mathrm{c}}\right)^{2}}.$
(8)

where $P_{\mathrm{on\ chip}}$ is the applied power after attenuation. In cases
where the response of the resonator shows duffing behavior a different
formalism for the calculation of the photon number in the resonator is used
(see Appendix 12)

We did not model the effect of dielectric loss on the resonance frequency,
because the accuracy of our frequency measurements and fitting procedure was
only sufficient to resolve relatively large frequency shifts (on the order of
$\Delta f_{r}/f_{0}\approx 10^{-5}$, likely caused by excess quasiparticles)
but was not sufficient to resolve the comparably smaller fractional frequency
shifts caused by TLS (which are on the order of $\Delta f_{r}/f_{0}\approx
10^{-6}$ [17, 24]).

##  Appendix F Fitting procedures

###  F.1 Fitting S21 microwave data

We analyze the complex S21 scattering matrix element of the microwave signal
that is sent through the signal chain. From transmission line theory one can
derive the S21 response of the equivalent lumped LC-circuit for a notch-type
configuration [30, 31]

$S^{ideal}_{21}=1-\frac{\left(Q_{l}/\left|Q_{c}\right|\right)e^{i\phi}}{1+2iQ_{l}\left(f/f_{r}-1\right)},$
(9)

with the loaded quality factor $Q_{\mathrm{l}}$, the coupling quality factor
$Q_{\mathrm{c}}$, the frequency $f$, and the resonance frequency $f_{r}$. We
include asymmetries in the response due to an impedance mismatch in the phase
$e^{i\phi}$ of $Q_{\mathrm{c}}$ and fit the phase $\phi$ [31, 63]. We extend
the ideal model by adding a complex prefactor to account for an imperfect
background with a cable damping $a$, a frequency-dependent cable delay $\tau$,
and an initial phase offset $\alpha$ [31]

$S^{real}_{21}=ae^{i\alpha}e^{-2\pi
if\tau}\left[1-\frac{\left(Q_{l}/\left|Q_{c}\right|\right)e^{i\phi}}{1+2iQ_{l}\left(f/f_{r}-1\right)}\right].$
(10)

While $Q_{\mathrm{c}}$ describes energy dissipation into the waveguide through
the coupling channel, we are mainly interested in the internal quality factor
$Q_{\mathrm{int}}$, which describes all other sources of loss, including
material losses. We can determine $Q_{\mathrm{int}}$ from the relation
$Q_{\mathrm{l}}^{-1}=Q_{\mathrm{int}}^{-1}+\Re(Q_{\mathrm{c}}^{-1})=Q_{\mathrm{int}}^{-1}+|Q_{\mathrm{c}}|^{-1}\cos{\phi}$,
following the diameter correction method formalism [63]. By fitting the
measured scattering responses with Eq. 10, we extract the external and
internal quality factors and the resonance frequency of each resonator as a
function of temperature and applied microwave power.

We perform a two-step least-squares fit of our data with Eq. 10. As a first
step, we remove the data around the resonance peak and fit the surrounding
background with the expression for the imperfect background in Eq. 10. As a
second step, we use the parameters of the fitted background as initial guesses
and fit the complete model, including a refit of the background parameters. We
used a modified version of the stlab [64] fitting routine which employs the
Python library lmfit. This method automatically estimates the standard
deviation for all of our fits. The individual steps are shown in Fig. 11.

Figure 11: (a) shows the initial background fit of the resonance circle. (b)
shows the resonance circle with the background removed. (c) shows the final
fit of the resonance circle including the final background refit. (d) shows
the final fit of the signal magnitude. The data is taken from one resonator of
Wafer B at -50 dBm applied power.

In Fig. 12 we show a comparison between the fitted quality factors obtained
from the commonly used circle fit method [31] and the direct fitting of Eq. 10
for one representative power trace of one resonator from Wafer B. Both methods
fit the same physical model but with different numerical procedures. The
obtained values only differ significantly at the lowest applied powers, where
the circle fit method yields a strongly divergent coupling quality factor.
Since divergent behavior of $Q_{\mathrm{c}}$ is unphysical, we trust the
values obtained by the direct fit more and use the direct fit as our standard
fitting procedure.

Figure 12: Comparison of the direct fitting routine adapted from [64] to the
fitting routine employed by Probst et al. [31]. (a) Fitted internal quality
factor versus power for one resonator from Wafer B. (b) Fitted coupling
quality factor for the same resonator Figure 13: (a)-(d) show the same fitting
routine as displayed in Fig. 11 but using the nonlinear model from Eq. 12. The
data is taken from one resonator of Wafer C at -26 dBm applied power.

Some resonators show Duffing-type nonlinear behavior at the highest
measurement powers that our standard fit was not able to capture (see Fig.
13). The Duffing-type behavior is likely caused by a Kerr nonlinearity and can
be modeled as a shift in the resonance frequency of the resonator with
microwave power [65]. This shift is introduced as a squared term in the
equation of motion for the mode amplitude [54]:

$\dot{\alpha}+\left(i\left(\omega_{r}+\beta|\alpha|^{2}\right)+\frac{\kappa}{2}\right)\alpha=0,$
(11)

where $\alpha$ is the mode amplitude, $\beta$ the Kerr anharmonicity,
$\kappa=\frac{f_{r}}{Q_{\mathrm{l}}}$ is the total decay rate. For a
derivation of this expression without the anharmonicity term, refer to Sec.
7.2 in [62]. We can express the scattering parameter in terms of the mode
amplitude $\alpha=|\alpha|e^{-i\psi}$ with a phase $\psi$ [66]

$\displaystyle S_{21}^{\mathrm{ideal.duffing}}$
$\displaystyle=1+\mathrm{i}\sqrt{\frac{\kappa_{\mathrm{c}}}{2}}\frac{|\alpha|e^{-i\psi}}{\sqrt{\Phi_{\mathrm{applied}}}}e^{i\phi}$
(12) $\displaystyle\psi$
$\displaystyle=\operatorname{atan}2\left(\frac{2\left(\Delta-\beta|\alpha|^{2}\right)}{\kappa}\right)$

with $e^{i\phi}$ being introduced to account for impedance mismatches,
$\Delta=\omega-\omega_{r}$ and
$\Phi_{\mathrm{applied}}=\frac{P_{\mathrm{applied}}}{hf_{\mathrm{r}}}$ the
photon number flux due to the power that is applied to the resonator. From the
coupling decay rate $\kappa_{\mathrm{c}}=\frac{f_{r}}{|Q_{\mathrm{c}}|}$ and
the total decay rate $\kappa$ we can determine the internal decay rate
$\kappa_{\mathrm{int}}=\frac{f_{r}}{Q_{\mathrm{int}}}$ by applying again the
diameter correction method
$\kappa=\kappa_{\mathrm{int}}+\kappa_{\mathrm{c}}\cos{\phi}$. The mode
magnitude is given by $|\alpha|=\sqrt{n_{\mathrm{ph}}}$ where
$n_{\mathrm{ph}}$ can be found as the smallest real positive root of the mode
equation rewritten as a third-order polynomial [54]

$\beta^{2}n_{\mathrm{ph}}^{3}-2\Delta\beta
n_{\mathrm{ph}}^{2}+\left(\Delta^{2}+\frac{\kappa^{2}}{4}\right)n_{\mathrm{ph}}-\kappa_{\mathrm{e}}\Phi_{\mathrm{applied}}=0.$
(13)

For all methods of fitting the S21 microwave data, we observe a strong
decrease in fit accuracy when the internal quality factor exceeds the external
quality factor. For $Q_{\mathrm{int}}\gg Q_{\mathrm{c}}$, the resonator enters
the overcoupled regime [53] where coupling losses dominate over internal
losses. In this regime the resolution of $Q_{\mathrm{int}}$ becomes
increasingly low and the uncertainty of the fit increases. For this reason, we
excluded fitted internal quality factors for which $Q_{\mathrm{int}}\geq 10\
Q_{\mathrm{ext}}$. This was not a problem for Wafers A and C, however, one
resonator of Wafer B and one resonator of Wafer D could not be fitted for this
reason. We note here that measuring in a strongly undercoupled regime
$Q_{\mathrm{int}}\ll Q_{\mathrm{c}}$ is not ideal either, as high microwave
powers are necessary to introduce photons in the resonator. Driving the whole
system at high power renders the system more susceptible to heating and other
sources of noise.

At the highest microwave powers that we measured, we observe behavior that is
not easily fitted by our standard loss model. The resonance circle starts to
appear deformed and/or features jumps due to nonlinear dissipative behavior
[67]. If the resonance circle only features a jump, marked as behavior (1) in
Fig. 14, the internal quality factor can still be extracted with Eq. 12. If
the resonance circle features other distortions, like caving in towards the
jump, marked as behavior (2) for -6.0 dB and 0.0 dB in Fig. 14 (a), the fit
will return incorrect results as the numerical method tries to fit a regular
circle (or in the case of duffing behavior, a regular circle with a jump to
the deformed data). Even though it is expected that the internal quality
factor decreases at high power [68], the high-power internal quality factors
shown in Fig. 14 (b) are likely an underestimation due to the aforementioned
fitting errors.

Figure 14: High-power behavior of one resonator from Wafer D. In the raw data
(a), the resonance circle starts to feature a jump (1) and caves in towards
the jump (2) around -6.0 dB. The abrupt jumps (1) are a sign of duffing
behavior and can be fitted, however, our fit can not account for the
deformation of type (2). In (b) we show the decrease in internal quality
factor at the highest powers calculated by our fitting routine. The star marks
the onset of behavior (2).

###  F.2 Fitting of Loss Models

As described in Appendix E, our loss model includes four fitting parameters
accounting for TLS loss and one for temperature-dependent loss. We bound the
values of the fitting parameters $\alpha$ and $\beta$ to a reasonable range
based on the results of Refs. [19, 24] for better convergence. A non-linear
least-squares fit is performed using the lmfit package in Python. The
$Q_{\mathrm{int}}$ values the model is fitted to are weighted by their
corresponding standard deviation. We do not include internal quality factors
data points from the highest powers beyond the point where the internal
quality factor starts to decrease (marked with a star in Fig. 14 (b)), as this
behavior is not modeled by Eq. 1 and, as we argue in the previous section, the
fit uncertainty is high.

In Fig. 15, we show the results of fitting the power and temperature-dependent
data of all resonators included in Fig. 4 (b). The model was able to reproduce
the data well in all cases.

Figure 15: Comparison between data and model for all resonators from included
in Fig. 4 (b)

##  Appendix G DC transport characterization

The full data set from which the values for $\xi_{GL}(0)$ shown in Tab. 2 are
calculated is plotted in Fig. 16. Measurements were done at liquid helium
temperatures in the Van der Pauw geometry using lock-in amplifiers to measure
the relatively low resistances of the films. The bias current was set to 20
$\mu$A in order to get good signal to noise ratios. The critical temperature
found for all five wafers was around 9.25$\pm 0.1$ K.

The critical magnetic field at zero K was determined directly from our data by
using [69]

$B_{c2}(0)=\left.0.69T_{c}\frac{dB_{c2}}{dT}\right|_{T=T_{c}}.$ (14)

The Ginzburg-Landau coherence length can then be calculated using

$B_{c2}(T)=\frac{\phi_{o}}{2\pi\xi_{GL}^{2}(T)}.$ (15)

We also determine the mean free path [70]

$\rho\ell=3.72\times 10^{-6}\mu\Omega\mathrm{cm}^{2}$ (16)

which is determined by measuring the wafer’s resistivity at 10 K. The
resulting values shown in Tab. 2.

Figure 16: DC low-temperature superconducting characteristics of the our films
as a function of temperature and magnetic field. The critical temperature
$T_{C}$ is indicated as the point at which the critical magnetic field is zero
and is the same for all films.

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