# Geometry dependence of TLS noise and loss in a-SiC:H parallel plate
capacitors for superconducting microwave resonators.

K. Kouwenhoven1,2<EMAIL_ADDRESS>terahertz.tudelft.nl G.P.J. van Doorn1
B.T. Buijtendorp2 S.A.H. de Rooij1,2 D. Lamers1 D.J. Thoen1 V. Murugesan1
J.J.A. Baselmans1,2 P.J. de Visser1 1Netherlands Institute for Space Research
(SRON), Niels Bohrweg 4, Leiden 2333 CA, Netherlands  2Department of
Microelectronics, Delft University of Technology, Mekelweg 4 2628 CD, Delft,
Netherlands

###### Abstract

Parallel plate capacitors (PPC) significantly reduce the size of
superconducting microwave resonators, reducing the pixel pitch for arrays of
single photon energy-resolving kinetic inductance detectors (KIDs). The
frequency noise of KIDs is typically limited by tunneling Two-Level Systems
(TLS), which originate from lattice defects in the dielectric materials
required for PPCs. How the frequency noise level depends on the PPC’s
dimensions has not been experimentally addressed. We measure the frequency
noise of 56 resonators with a-SiC:H PPCs, which cover a factor 44 in PPC area
and a factor 4 in dielectric thickness. To support the noise analysis, we
measure the TLS-induced, power-dependent, intrinsic loss and temperature-
dependent resonance frequency shift of the resonators. From the TLS models, we
expect a geometry-independent microwave loss and resonance frequency shift,
set by the TLS properties of the dielectric. However, we observe a thickness-
dependent microwave loss and resonance frequency shift, explained by surface
layers that limit the performance of PPC-based resonators. For a uniform
dielectric, the frequency noise level should scale directly inversely with the
PPC area and thickness. We observe that an increase in PPC size reduces the
frequency noise, but the exact scaling is, in some cases, weaker than
expected. Finally, we derive an engineering guideline for the design of KIDs
based on PPC-based resonators.

††preprint: APS/123-QED

##  I Introduction

Superconducting microwave resonators are one of the key elements of Kinetic
Inductance Detectors (KIDs) [1, 2, 3] and superconducting-Qubits [4, 5, 6].
The current resonators are typically based on planar structures such as
coplanar waveguides (CPW) and interdigitated capacitors (IDC). These planar
structures provide little capacitance per unit area since the fields are
spread between the substrate and air, which limits the packing density or
pixel pitch of these resonators [7, 8]. An alternative is the parallel plate
capacitor (PPC) [9, 10, 11, 12], for which the entire field is in a dense
($\epsilon_{r}\sim 10$) and/or thin dielectric layer. Such a PPC can
drastically shrink the size of the resonator, but the deposited dielectric
required to fabricate a PPC will likely increase the resonator’s microwave
loss and frequency noise level due to tunneling states in the dielectric.

The microscopic nature of these tunneling states is still unknown, but they
are typically assumed to arise from disorder in the crystalline lattice. Due
to the disordered lattice, one or multiple atoms, can tunnel between two
energetically similar states which is modeled by the standard tunneling model
(STM). In the context of disordered dielectrics this model often referred to
as the two-level system (TLS) model [13, 14, 15]. The TLSs can couple to the
electric field of the resonator through their electric dipole moment and
modify the dielectric constant of the material based on their individual
states. Averaging over all TLSs that are resonant with the resonator gives the
TLS contribution to the dielectric constant $\epsilon_{\rm{TLS}}$ [14, 15].

The real part of $\epsilon_{\rm{TLS}}$ introduces a resonance frequency shift
while the imaginary part introduces microwave loss. In addition, TLSs can
randomly switch states, which gives rise to a time-fluctuating dielectric
constant. This, in turn, causes the resonator’s resonance frequency to
fluctuate in time, introducing excess frequency noise.

Recent studies have focussed on developing and characterizing low-loss and
low-noise dielectrics [16, 17]. However, how the microwave loss and frequency
noise level scale with a PPC’s dimensions has not been experimentally
addressed.

In this work, we experimentally study the TLS-noise of PPC resonators based on
hydrogenated amorphous silicon carbide (a-SiC:H). Recent work showed that
a-SiC:H, depending on the exact deposition details, can be a low-loss, low
stress, deposited dielectric [17]. The exact deposition details for the
a-SiC:H used in this work are found in [17]. We vary two geometrical
parameters of the PPC: the plate area and the dielectric thickness. We present
an extensive dataset, which contains 56 devices, with a factor of 44 in area
variation and and a factor 4 in dielectric thickness.

In Section II we describe the resonator design, fabrication and the
experimental setup. In Section III.1 we focus on the TLS-induced microwave
loss and frequency shift and show that the dielectric between the PPC plates
contains lossy surface layers. In Section III.2 we focus on the resonator’s
frequency noise and how it scales with the PPC’s dimensions. We first discuss
the frequency noise spectrum and the power and temperature dependencies. Then,
we show that the noise level decreases when the area or thickness of the PPC
increases. In Section IV we draw conclusions on the practical applicability of
PPC-based resonators and discuss PPC-based KIDs.

##  II Design, Fabrication and experimental setup

We use a lumped-element resonator design consisting of an a-SiC:H parallel
plate capacitor (PPC) and a meandered inductor, as shown in Fig. 1. The
resonators couple to the coplanar-waveguide (CPW) readout line with a small,
separate, capacitor with a bottom plate that is galvanically connected to the
central line of the CPW. The two ground planes of the CPW are electrically
balanced at regular intervals by bridges that galvanically connect the ground-
planes (not shown). The resonators, coupling lines, readout line, and bridges
are made from NbTiN with a critical temperature ($T_{\rm{c}}$) of $14.3$ K.

Figure 1: Optical micrograph of lumped element resonators with a-SiC:H
parallel plate capacitors. All metallic structures (pink) are NbTiN with a
$T_{\rm{c}}$ of 14.3 K. The perceived colour of a-SiC:H is different if the
a-SiC:H is on top of the NbTiN layer (i.e. on top of the bottom plate) and
varies due to thickness variations over the wafer. top: smallest (60$\times$16
$\mu$m) PPC resonator bottom: Biggest (200$\times$200 $\mu$m) PPC resonator.

We vary the capacitor area by a factor of 44 on each chip with 20 resonators.
To keep the resonant frequency within the range of our setup we reduce the
length of the inductor for the larger capacitors, as shown in Fig. 1.

We vary the PPC thickness by fabricating the exact same design with three
different dielectric thicknesses; 100 nm, 200 nm and 400 nm of a-SiC:H. The
designed resonance frequencies on the 200 nm wafer lie between 4.0 and 8.45
GHz and will shift down or up for the 100 nm and 400 nm wafers respectively.
For the 400 nm design, 4 resonators shifted beyond 9 GHz and can not be
measured with our current setup.

We fabricate the resonators on a 350 $\mu$m c-plane sapphire substrate using
optical contact lithography and reactive ion etching to pattern the
structures. The bottom PPC plate, inductor, readout lines and the bottom line
of the coupling structure are fabricated from a 220 nm thick reactive
magnetron sputtered layer of NbTiN [18, 19]. The next step is PECVD deposition
of a layer of a-SiC:H using a Novellus Concept One. Details regarding the
a-SiC:H deposition are discussed in [17]. The top PPC plate, readout line
bridges, and the top line of the coupling structure are fabricated from a
second layer of 150 nm NbTiN. To create a good galvanic contact between the
two NbTiN layers the wafer is cleaned with HF prior to the deposition of the
second NbTiN layer. The use of a sapphire substrate prevents surface erosion
due to the (short) over-etch times needed for patterning the NbTiN and SiC
layers.

In addition to the PPC resonators, we fabricate a set of planar reference
devices in the bottom NbTiN layer. These reference devices are used to
separate the noise contributions from the capacitor and inductor [20] which we
come back to in the discussion.

The samples are cooled down in a pulse-tube pre-cooled dilution refrigerator.
We use a box-in-box sample stage design [21, 22], to shield the sample from
stray light coming from the 3 K stage of the cooler. The sample stage is
surrounded by two magnetic shields, a superconducting Niobium shield, and a
Cryophy shield. A graphical representation of the set-up is presented in [8],
but in this case the sample holder is closed with a lid. Microwave
measurements are performed with a standard homodyne detection scheme. Before
reaching the sample, the input microwave signal is first attenuated at every
temperature stage. After the sample, the signal is amplified with a set of
low-noise amplifiers (LNAs), of which one is located at the 3 K stage. All
measurements are taken at a base-temperature of 100 mK unless otherwise
specified.

##  III Measurements

###  III.1 S21 measurements

Figure 2: a) Power dependent microwave loss $\tan\delta_{d}$ plotted against
the average number of photons in the resonator
($\left<n\right>\propto{|\vec{E}|^{2}}$), with $\vec{E}$ the electric field in
the dielectric. Measured datapoints, together with the fit to Eq. 2, are
highlighted for one KID of each thickness. For all other KIDs only the fitted
lines are presented. The insert shows that the TLS contribution to the
microwave loss $\tan\delta^{0}_{\rm{TLS}}=(\pi Pd_{0}^{2})/(3\epsilon)$ is
independent of the parallel plate area. The solid lines are the average
$\tan\delta^{0}_{\rm{TLS}}$. b) the temperature induced relative frequency
shift. Measured datapoints for one KID, together with the fit to Eq. 3, are
highlighted for one KID with the same PPC area but for different dielectric
thicknesses. For all other KIDs only the fitted lines are presented. The
fitted range is $\hbar\omega/k_{\rm{B}}-1\rm{K}$ which, for a KID resonating
at 4 GHz starts at $\sim$200 mK. c) Fitted TLS contribution $(\pi
Pd_{0}^{2})/(3\epsilon)$. ★ Datapoint from a 295 nm a-SiC:H microstrip
resonator with identical a-SiC:H deposition conditions [17], $(\pi
Pd_{0}^{2})/(3\epsilon)$ obtained from the loss tangent fit. The TLS
contribution is thickness dependent, which cannot be explained with a uniform
dielectric which assumes a constant TLS density ($P$) and dipole moment
($d_{0}$).

To characterize the resonators we measure the complex transmission parameter
$S_{21}$ with a vector network analyzer (VNA). We perform a frequency sweep to
obtain the resonance circle ($S_{21}$) for each resonator at different readout
powers and sample stage temperatures. The maximum readout power is set by the
bifurcation point of the resonator, above which the resonance circle starts to
show hysteric switching [23, 24].

Fitting the resonance dip $(|S_{21}|)$ to an analytical model [25] provides
the resonator’s resonance frequency $f_{r}$ and the internal ($Q_{i}$) and
coupling ($Q_{c}$) quality factors as a function of readout power and
temperature. The resonator’s intrinsic loss is given by $Q_{i}$, which is
related to the loss tangent of the dielectric as

$\tan\delta_{d}=\frac{1}{pQ_{i}},$ (1)

assuming that the resonator’s microwave loss is dominated by the dielectric
material. Here $p$ is the participation ratio of the dielectric, given by
$p=w^{e}_{d}/w^{e}$ where $w_{e}$ is the total electric energy stored in the
resonator and $w^{e}_{d}$ is the energy stored in the dielectric. For a PPC
where the thickness of the dielectric is small compared to the dimensions of
the plates the fringing fields are negligible, so we can assume that the
entire electric field is in the dielectric and $p=1$.

We measure $Q_{i}$ over a wide power range, from the bifurcation point of the
resonator down to the single-photon level. The measured datapoints of three
resonators, one for each PPC thickness, are presented in Fig. 2(a). The STM
predicts a power dependent microwave loss,

${\tan\delta_{d}}=\frac{\pi
Pd_{0}^{2}}{3\epsilon}\tanh{\frac{\hbar\omega_{r}}{2k_{\rm{B}}T}}\left(1+\frac{{|\vec{E}|}}{{E_{c}}}\right)^{-\beta}+{\tan\delta_{\rm{HP}}},$
(2)

which should be geometry independent. Here $P$ is the TLS density, $d_{0}$ the
TLS dipole moment, $\epsilon$ the dielectric constant of the TLS hosting
material, ${|\vec{E}|}$ is the electric field in the dielectric, $\omega_{r}$
is the resonance frequency, $T$ is the temperature, and $E_{\rm{c}}$ is the
critical field above which the TLSs start to saturate with $\beta=0.5$ in the
STM. The TLS contribution $(\pi Pd_{0}^{2})/(3\epsilon)$ is often presented as
the TLS contribution to the loss tangent at zero temperature:
$\tan\delta^{0}_{\rm{TLS}}$. At high powers the loss no longer follows the STM
and saturates to $\tan\delta_{\rm{HP}}$. We fit Eq. 2 to the measured internal
quality factor $Q_{i}$. The fits for all resonators are given in Fig. 2(a).
The fitted TLS contribution $\tan\delta^{0}_{\rm{TLS}}$ of each resonator is
plotted against the PPC area in the inset of Fig. 2(a), which shows that
$\tan\delta^{0}_{\rm{TLS}}$ is independent from the PPC area.

For the wafer with 200 nm of a-SiC:H, we observe that some resonators have a
much lower power-handling ($<$ -10dB with respect to the mean power handling)
compared to devices on the same chip. For these devices there are not enough
datapoints to accurately fit Eq. 2 and they are therefore omitted from the
analysis. To extend the dataset we measured a second chip of the same design
and fabrication run. The full dataset, with all measured devices, is available
in the reproduction package [26].

Next, we analyse the KID resonance frequency as a function of temperature by
varying the temperature of the sample stage with a PID-controlled heater, from
$25.0$ mK to $1.0$ K. The STM predicts a temperature dependent shift of the
KID resonance frequency ($\Delta
f_{0}/f_{0}=-\frac{1}{2}\Delta\epsilon^{\prime}/\epsilon$ ),

$\frac{\Delta
f_{0}}{f_{0}}=p\frac{Pd_{0}^{2}}{3\epsilon}\left[\mathrm{Re}\left\\{\Psi\left(\frac{1}{2}+\frac{\hbar\omega}{2\pi
jk_{\rm{B}}T}\right)\right\\}-\log{\frac{\hbar\omega}{2\pi
k_{\rm{B}}T}}\right],$ (3)

which should be geometry independent. Here $\Delta f_{0}=f(T)-f_{0}$ where
$f_{0}$ is the resonance frequency at $T=0$, $\Psi$ the complex digamma
function, and $p$ the participation ratio as in Eq. 1. We fit Eq. 3 to the
measured resonance frequencies, highlighted for one resonator of each
dielectric thickness in Fig. 2(b). For all other resonators only the fitted
line is presented.

Equation 3 contains two regimes, for $k_{\rm{B}}T<\hbar\omega$ the digamma
term dominates while for $k_{\rm{B}}T>\hbar\omega$ the logarithmic term
dominates. In the first regime ($k_{\rm{B}}T<\hbar\omega$) we have a low
signal-to-noise ratio, so we limit the fit to the range
$k_{\rm{B}}T>\hbar\omega$. Since the measured temperature range is far below
the $T_{\rm{c}}$ of NbTiN (14.3 K) there is no quasiparticle contribution to
the measured frequency shift [27, 28].

The only TLS parameters that impact the TLS induced microwave loss and
resonance frequency shift are the TLS density ($P$) and the TLS dipole moment
($d_{0}$). The fitted TLS contribution $(\pi Pd_{0}^{2})/(3\epsilon)$ from
both Eq. 2 and Eq. 3, for all devices, are plotted in Fig. 2(c). There is a
clear, close to 1:1 agreement between the obtained TLS contribution $(\pi
Pd_{0}^{2})/(3\epsilon)$ from both measurements.

The TLS data in Fig. 2 shows a clear thickness dependence where a thicker
dielectric has both a lower TLS induced microwave loss (Fig. 2a) and a smaller
TLS induced resonance frequency shift (Fig. 2b), as summarized by Fig. 2c. The
insert of Fig. 2a highlights that the TLS induced microwave loss does not
depend on the area of the PPC and is thus only a function of the thickness of
the dielectric. The thickness dependence cannot be explained by a uniform
dielectric, which assumes a constant TLS density ($P$) and dipole moment
($d_{0}$) in Eq. 2 and Eq. 3. Instead the medium between the PPC plates likely
contains one or more surface layers which have a higher TLS density or a
stronger TLS dipole moment (or both) than the bulk dielectric. The TLS
contribution we obtained in Fig. 2 is then the thickness weighted average of
the bulk dielectric material and surface layers. Changing the thickness of the
dielectric changes the bulk to surface layer ratio, yielding a different
effective TLS contribution $(\pi Pd_{0}^{2})/(3\epsilon)$.

###  III.2 Noise measurements

Figure 3: a) Frequency noise spectra for resonators with different plate sizes
and dielectric thickness, at the same internal electric field of
$|\vec{E}|=1.13\times 10^{3}$ V/m ($\epsilon_{r}=10$). The TLS noise level
shows a clear area and thickness scaling. The spectra typically show a 1/$f$
spectrum up to 0.1-1 kHz. The roll-of in the noise level at the resonator ring
time is visible at higher frequencies ($>10^{4}$ Hz). b) Frequency noise level
of one PPC resonator ($A=200\times 200$ $\mu$m, $t=400$ nm, $f_{r}=5.15$ GHz)
as a function of $|\vec{E}|$ which follows the expected $|\vec{E}|^{-1}$
scaling from the tunneling models [15, 29]. c) TLS noise level at 10 Hz vs.
parallel plate area ($A_{\rm{PPC}}$) for all three a-SiC:H thicknesses at the
same internal electric field. The solid lines are fits $c\cdot
A_{\rm{PPC}}^{-\alpha}$. From thinnest to thickest dielectric layer, the
slopes ($\alpha$) are $1.04\pm 0.12$, $0.73\pm 0.10$, and $0.72\pm 0.09$. The
uncertainty range is given by the standard deviation error on the fit-
parameters $\alpha$ and $c$. The dotted line indicates the slope of the
expected $S_{f}\propto A_{\rm{PPC}}^{-1}$ scaling. d) Observed thickness
scaling of the TLS noise level. The points are the obtained from the fitted
$S_{f}(A_{\rm{PPC}})/f^{2}$ relations in panel c), at the smallest PPC area
(▲), the median area of $6.6\times 10^{-9}$ $\rm{m}^{2}$ (◆) and the largest
PPC area (▼). The dashed line indicates a $t^{-1}$ scaling with respect to the
400 nm median area point.

We perform noise measurements at the resonance frequency of the resonator with
a homodyne detection scheme [21]. The noise measurement consists of two time
streams: one of 40 seconds sampled at 50 ksample/s and one of 0.5 seconds
sampled at 1 Msample/s. The fluctuations in the IQ-coordinates are translated
to phase ($\theta$) and amplitude ($A$) with respect to the resonance circle
[30]. We estimate the PSD from both time streams, and stitch them at 20 kHz,
The resonance circle relates the measured phase noise to frequency noise as

$S_{f}/f_{r}^{2}=\frac{S_{\theta}}{(4Q_{l})^{2}}$ (4)

where $S_{\theta}$ is the PSD in the phase coordinate. The resonance frequency
$f_{r}$ and loaded quality factor $Q_{l}=(1/Q_{c}+1/Q_{i})^{-1}$ are obtained
from the resonance circle, see sec. III.1. We focus on the frequency noise
from the phase coordinate, which has a higher TLS-noise level compared to the
amplitude coordinate at powers where $|\vec{E}|>E_{\rm{c}}$ [31].

The frequency noise spectra for a microwave resonator for which the total
electric field volume $V$ contains a TLS host volume $V_{h}$ is given by [14,
15]

$\frac{S_{f}(f)}{f^{2}}\sim\frac{\int_{V_{h}}S_{\epsilon}(f,\vec{E},T)|\vec{E}|^{4}dV}{4\left(\int_{V}\epsilon|\vec{E}|^{2}dV\right)^{2}}$
(5)

where $S_{\epsilon}$ describes the fluctuations in the real part of the
dielectric constant ($\epsilon_{\rm{TLS}}$) due to random switching of TLSs.
If we assume a uniform distribution of TLS in the dielectric volume of the PPC
$V_{\rm{PPC}}$, then $V_{h}=V=V_{\rm{PPC}}$ and the frequency noise level
should scale as $V_{\rm{PPC}}^{-1}$. The derivation for the interacting
tunneling model [29] shows that $S_{\epsilon}$ has a $1/f$ spectrum and in
strong electric fields ($|\vec{E}|\gg E_{\rm{c}}$) scales with
$|\vec{E}|^{-1}$, matching the empirical relations of the STM [14, 15]. The
interacting TLSs do however result in a temperature dependent noise level that
is not explained by the STM: at low temperatures ($k_{\rm{B}}T\ll\hbar\omega$)
and strong-field ($|\vec{E}|\gg E_{\rm{c}}$) $S_{f}\propto T^{(1-\mu)/2}$
while at high temperatures ($k_{\rm{B}}T\gg\hbar\omega$) $S_{f}\propto
T^{\mu-1}$ where $\mu\approx 0.3$.

The measured frequency spectra ($S_{f}(f)/f^{2}$) of 6 KIDs, with different
PPC areas and dielectric thicknesses, are presented in Fig. 3(a) at a
temperature of 100 mK.

The frequency noise spectra show a clear $1/f$ behaviour at lower frequencies,
which transitions to something more akin to $f^{-1/2}$ typically between 0.1-1
kHz. Both behaviors have been observed before in different measurements and
device architectures [32, 28, 14, 33], but typically not simultaneously. The
frequency at which the transition happens varies between resonators and
resonator designs, see Appendix. A. To compare the noise levels of different
resonators we fit the $1/f$ spectrum between 5-50 Hz, where the spectrum is
always fully $1/f$, and use the fitted 10 Hz point as the TLS-noise level
reference.

We measure the noise spectra over a range of read powers, with 2 dB steps. In
general, the electric field in the TLS-medium is proportional to the readout
power on the through line as $|\vec{E}|\propto P_{\rm{read}}^{-1/2}$. For a
PPC the internal electric field is directly calculated from the read power and
geometry as

$|\vec{E}|=2\sqrt{\frac{Q^{2}}{Q_{c}}\frac{P_{\rm{int}}}{\omega_{r}\epsilon}\frac{1}{V_{PPC}}}.$
(6)

where $\omega_{r}$ is the resonance frequency of the resonator, $\epsilon$ the
dielectric constant and $V_{\rm{PPC}}$ the volume of the dielectric in the
PPC. The internal power $P_{\rm{int}}$ is given by
$P_{\rm{int}}=\frac{2}{\pi}\frac{Q^{2}}{Q_{c}}P_{\rm{read}}=\left<n\right>(\hbar\omega_{r}^{2})$.
The frequency noise level at 10 Hz for one PPC resonator ($A=200\times 200$
$\mu$m, $t=400$ nm, $f_{r}=5.15$ GHz), for different E-field is given in Fig.
3(b) to illustrate that we indeed observe the expected $\propto|\vec{E}|^{-1}$
relation.

In addition, we have measured the temperature dependence of the frequency
noise. The frequency noise spectra and noise levels at 10 Hz for temperatures
between $25-800$ mK, and at different internal powers, are discussed in
Appendix C and plotted in Fig. 9. At all measured powers, where $|\vec{E}|\gg
E_{c}$ the frequency noise level follows the temperature dependence predicted
by the interacting tunneling model, see Appendix C for details.

Neither the STM or interacting tunneling model [29] predict the $f^{-1/2}$
region of the noise spectra we typically observe for superconducting
resonators [32, 28, 14]. The $f^{-1/2}$ roughly follows the expected
$|\vec{E}|$ and temperature scalings of the $1/f$ region [29], which suggests
that the $f^{-1/2}$ region has a similar field and temperature dependence as
the TLSs and is therefore interesting for follow up.

For a uniform dielectric where the TLSs are uniformly distributed over the
dielectric volume we expect a frequency noise level that scales as
$S_{f}\propto V_{\rm{PPC}}^{-1}$. Separated in the two geometrical parameters
of the PPC; PPC area and thickness, we expect that frequency noise level
scales the same with area $S_{f}\propto A_{PPC}^{-1}$ as with thickness
$S_{f}\propto t^{-1}$.

To see the effect of the PPC dimensions on the TLS-noise level we need to
compare the noise levels at equal electric field in the resonator. For each
resonator we translate the measured read power points to the E-field in the
dielectric (Eq. 6). We interpolate between these measured E-field points,
using the known $S_{f}\propto|\vec{E}|^{-1}$ relation, to find the noise level
at a desired E-field, see Fig. 3(b). Details of the interpolation are
discussed in Appendix B. The TLS-noise level, at 10 Hz, vs. the PPC plate area
at $|\vec{E}|=1.13\times 10^{3}$ V/m, assuming a $\epsilon_{r}=10$ for
a-SiC:H, is given in Fig. 3(c).

The frequency noise level in Fig. 3(c) scales as a power law with parallel
plate area as expected. We fit the relation $S_{f}=\beta\cdot
A_{\rm{PPC}}^{-\alpha}$, with $A_{\rm{PPC}}$ the PPC area, and find $\alpha$
is $1.04\pm 0.12$ for 100 nm, $0.73\pm 0.10$ for 200 nm, and $0.72\pm 0.09$
for 400 nm. The STM predicts $S_{f}\propto A_{\rm{PPC}}^{-1}$, as in Eq. 5.
Although the obtained area scalings agree, at least qualatitavely, with the
predicted scaling, the 100 nm film behaves differently than the 200 and 400 nm
films, which is inconsistent with a uniform dielectric.

The resonance frequency scaling $S_{f}\propto f_{0}^{\mu}$ from the
interacting tunneling model [29], with $\mu\approx 0.3$ would only introduce a
scatter on the datapoints in Fig. 3(c) since the resonance frequency is, by
design, not correlated with the PPC area.

In Fig. 3(d) we compare the fitted noise level for the different dielectric
thicknesses. Since the slopes in Fig. 3(c) differ, we compare the noise level
at different PPC surface areas; the smallest area (▲), the median area (◆),
and the largest area (▼). If we compare the data from the 200 and 400 nm
films, which show a similar area scaling ($\sim A_{\rm{PPC}}^{-0.7}$), we see
that the frequency noise level scales roughly with the expected
$t_{\rm{PPC}}^{-1}$ scaling for a uniform dielectric.

In Sec. III.1, we concluded that the dielectric contains surface layers based
on the thickness-dependent microwave loss and resonance frequency shift. If
these surface layers have a similar effect on the frequency noise, we expect a
frequency noise scaling with thickness stronger than that of a uniform
dielectric ($S_{f}\propto t^{-1}$). The data in Fig. 3d does not show a
thickness dependence that is unambiguously stronger than $S_{f}\propto
t^{-1}$, which implies that the surface layers affect the noise differently
than the microwave loss and frequency shift discussed in Sec. III.1. The
origin of this difference is unknown. Barends et al. [28] made a similar
observation: the resonance frequency shift and frequency noise respond
differently to thicker and thicker dielectric ($\rm{SiO_{x}}$) layers
deposited on top of a CPW resonator.

##  IV Discussion and Conclusions

In addition to the PPC designs, each wafer contains single-layer reference
devices: co-planar waveguide (CPW) and Interdigitated Capacitor (IDC)
resonators. The measured microwave loss and frequency noise of these devices
is presented in Appendix. A. The frequency noise against internal power is
plotted in Fig. 4. Calculating the E-field inside the TLS-medium for a CPW or
IDC is non-trivial due to the degeneracy in $p\tan{\delta_{d}}$ (Eq. 1). This
means we cannot quantitatively compare the TLS properties of the substrate to
the a-SiC:H but we can show that the PPC resonator is dominated by a different
TLS-contribution than the CPW and IDC resonators.

The inductor’s loss and frequency noise contribution can be analyzed by
comparing the IDC and CPW resonators which have an equal line and gap width
and therefore the same field distribution in the substrate TLS layer [20]. The
difference between the CPW and IDC resonators is then the inductor in the IDC
resonator. Figure 4 shows that the CPW and IDC have similar frequency noise
levels, which means the contribution of the inductor is negligible. The IDC
and PPC resonators have the same inductor, but a different lumped element
capacitor. Compared to the IDC designs, the PPC resonators show a 15-20 dB
higher TLS-noise level in the $1/f$ region of the spectrum. This means the
noise of the PPC resonators is dominated by the a-SiC:H capacitor.

In the measured microwave loss and resonance frequencies in Fig. 2, we
observed a thickness dependency that can be explained by a volume between the
PPC plates that is not one uniform dielectric bulk material but contains one
or more surface layers. Since these surface layers have a different
$\tan\delta_{\rm{TLS}}$ than the bulk, they could have an effect on the
frequency noise of the PPC as well, where the noise level scales stronger with
the thickness of the dielectric than with the PPC area. The measured frequency
noise in Fig. 3 does, however, not show a clear indication that we see this
effect; see Sec. III.2. This could mean that the surface layers affect the
noise differently than the microwave loss and frequency shift.

The location or nature of these surface layer(s) is unknown. They can be i.e.
surface oxides on the metal capacitor plates or growth effects of the a-SiC:H.
To find the $\tan\delta_{\rm{TLS}}$ of the bulk dielectric we would need to
extend the dielectric thickness range far enough to reach two extremes: thin
enough to be dominated by the surface layers and thick enough to be limited by
the properties of the bulk dielectric. A way to mitigate these surface layers
might be found by exploring various cleaning steps on the bottom NbTiN plate
of the capacitor.

One of the applications of superconducting resonators is the Kinetic
Inductance Detector (KID). Since the noise level scales with $P_{\rm{int}}$,
KIDs are always operated at their maximum internal power
$P_{\rm{int}}^{\rm{max}}$, at the edge of bifurcation [23, 24]. The
bifurcation point, and thus $P_{\rm{int,max}}$, typically depends on the
critical current in the narrow inductor lines and does not depend on the
capacitor geometry. For a set $P_{\rm{int,max}}$, the internal E-field inside
the capacitor will depend on the capacitor geometry, see Eq. 6. This gives us
two PPC dimension scalings, one for the internal E-field, and one for the
dielectric volume ($V_{\rm{PPC}}$) in the resonator as discussed before:

$S_{f}\propto\frac{1}{\vec{|E|}}\cdot\frac{1}{V_{\rm{PPC}}}$ (7)

where $|\vec{E}|\propto 1/\sqrt{\omega_{r}V_{\rm{PPC}}}$. The two dielectric
volumes factors $V_{\rm{PPC}}$ partly cancel and a

$S_{f}\propto\sqrt{\omega_{r}/V_{\rm{PPC}}}$ (8)

scaling remains for a dielectric with uniform TLS density. However, if we look
at the measured data in Fig. 3, we see that, for the 200 and 400 nm thick
films, we have a geometric scaling

$S_{f}\propto A_{\rm{PPC}}^{-\alpha}\cdot t^{-\beta}<V_{\rm{PPC}}^{-1},$ (9)

since $\alpha\approx 0.72-0.73$ (see Fig. 3(c)) and $\beta\approx 1$ (see Fig.
3(d)). The resulting geometric volume scaling, which is weaker than
$V_{\rm{PPC}}^{-1}$, reduces the combined geometry and E-field scaling in Eq.
8.

The practical applicability of PPC resonators for KIDs depends on how their
noise level compares to the standard planar structures. Figure 4 shows the
measured $S_{f}/f^{2}$ of a 200 nm PPC KID and of the planar reference designs
fabricated on the same chip. In addition, Fig. 4 contains datapoints from two
different MKID designs operated at their maximal internal power. The first is
a compact LEKID design for an optical to near-IR energy-resolving pixel based
on an IDC [8]. The second point is for ultra-sensitive antenna-coupled THz
KIDs based on planar structures [3].

Figure 4: TLS noise level at 10 Hz vs. internal power for different resonator
types. The PPC, IDC and CPW resonators are fabricated on the same chip, where
the PPC and IDC share the same inductor design. The PPC has a plate area of
6000 $\mu\rm{m}^{2}$ (60$\times$100 $\mu$m) There is a strong increase in
noise level between the planar devices deposited on the crystalline substrate
(CPW, IDC) and the devices based on the a-SiC:H parallel plate capacitor, but
no difference between the planar devices themselves. The markers are the
frequency noise of an optical to near-IR energy-resolving KID (pixel pitch 150
$\mu$m [8] and an ultra-sensitive far-infrared KID (pixel pitch $\sim$1.5 mm)
[3], both operated at their highest internal power.

Compared to the IDC of the LEKIDs in [8], which have 2 $\mu$m fingers and
gaps, the 200 nm PPC of Fig. 4 with 60$\times$100 $\mu$m sides has a 42 times
higher capacitance. This means that we can replace the IDC of [8] with a
12$\times$12 $\mu$m PPC to get the same KID resonance frequencies. Assuming
the noise scaling from Eq. 8, this would add roughly 8 dB of frequency noise
to the 200 nm PPC line in Fig. 4. Taking the dimensions of the inductor into
account, we can use the a-SiC:H PPC to reduce the area required for a KID by a
factor 10 at the cost of roughly 20 dB of extra frequency noise.

The second use-case are ultra-sensitive KIDs for the far-infrared (FIR), where
one of the limiting factors is the frequency noise at low frequencies
($0.1-10$ Hz) [3]. The FIR KIDs [3] have a TLS noise level far below both the
PPC and the reference resonators, see Fig. 4. The noise level of these KIDs is
so low because they are based on wide planar structures (central line of 40
$\mu$m with 8 $\mu$m gaps) that leverage the $W_{tot}^{-1.6}$ scaling of the
frequency noise [34, 35]. Compared to the expected $V_{\rm{PPC}}^{-1/2}$
scaling for a PPC resonator with a uniform dielectric, the CPW has a much
stronger geometrical scaling. This means that for FIR KID arrays, which are
not limited by the pixel pitch ($\sim$ 1.5 mm), PPC based resonators are at a
disadvantage.

To reduce the frequency noise level of a PPC-based resonator, a promising
route would be to investigate the dielectric material properties and eliminate
any possible TLS surface layers

## Data availability

full dataset and reproduction package available on Zenodo:
https://doi.org/10.5281/zenodo.10159731.

## Acknowledgments

We acknowledge Nick de Keijzer and Robert Huiting for their work on the 100 mK
sample stage. This work is financially supported by the Netherlands
Organisation for Scientific Research NWO (Projectruimte 680-91-127)

Colorblind and grayscale safe color scheme from Paul Tol’s Notes:
https://personal.sron.nl/~pault/

##  Appendix A Reference devices (CPW and IDC)

Figure 5: Microscope image of one of the Interdigitated capacitor (IDC)
reference resonators. The IDC has 2 $\mu$m defined fingers and gaps. Single-
layer planar resonator fabricated in the bottom NbTiN layer which has a
$T_{c}$ of 14.3 K.

In addition to the parallel plate capacitor designs each wafer contains
reference designs based on the work [20]. In addition to three PPC resonators,
this chip contains three Interdigitated capacitor (IDC) resonators, and three
Coplanar-Waveguide (CPW) resonators. A microscope image from one of the IDC
resonators is presented in Fig. 5. The IDC has fingers and gaps of 2 $\mu$m
and the CPW has a center line and gap of 2 $\mu$m (2-2-2) such that they have
the same electric field distribution. Both resonators should then probe the
TLS defects in the substrate in the same way, see Eq. 1. The difference in
loss and frequency noise between the CPW and IDC can then be attributed to the
effects of the inductor of the IDC resonator, if one takes the stray
capacitance in the inductor into account [20].

The microwave loss ($(pQ_{i})^{-1}$) for one of each resonator types is
plotted in Fig. 6. Since the layer thickness of the substrate TLS-layer that
gives the TLS induced loss of the CPW an IDC [15, 36] is unknown, the
participation ratio $p$ is unknown and we cannot extract $\tan\delta_{d}$ from
the CPW and IDC measurements. The CPW and IDC show a similar noise level, so
the loss introduced by the inductor is negligible. The PPC on the other hand,
has roughly an order of magnitude higher $\tan\delta_{\rm{TLS}}$ compared to
the CPW resonator which shows that the PPC resonator is dominated by the
$\tan\delta_{\rm{TLS}}$ of the PPC and the contribution of the inductor is
negligible.

Figure 6: Measured loss tangent $\tan\delta_{d}=(pQ_{i})^{-1}$ for a PPC, IDC,
and CPW based resonator. The scatter at high $Q_{i}$ is higher due to the
uncertainty in fitting deep $S_{21}$ dips ($\text{min}(S_{21})=Q/Q_{i}$). We
fit Eq. 2 to the measured datapoints with powers below the bifurcation point
of the resonator.

The measured frequency noise spectra ($S_{f}/f^{2}$), for one of each
resonator types, are plotted in Fig. 7. The IDC and CPW resonators have near
identical noise spectra, while the PPC resonator has a 15-20 dB higher noise
level. As before, we conclude that the contribution of the inductor is
negligible, and the frequency noise of the PPC is dominated by the a-SiC:H
PPC. Note that the point at which the spectra changes to $f^{-1/2}$ is at
lower frequencies for the CPW and IDC resonators, around 10 Hz. For the PPC
resonators this is transition lies between $10^{2}-10^{3}$ Hz, see Fig. 7.

Figure 7: Resonator frequency noise $S_{f}/f^{2}$ from the measured phase
noise $S_{\theta}$ power spectral density of each resonator at the same
internal power $P_{\rm{int}}=-50$ dBm. The PPC has a plate area of 6000
$\mu\rm{m}^{2}$ (60$\times$100 $\mu$m)

##  Appendix B Frequency noise spectra analysis

There are several analysis steps that have to be taken to arrive at Fig. 3(b)
from the measured frequency noise spectra in Fig. 3(a). First the $1/f$ region
of the spectra is fitted to a linear $a\cdot f^{-b}$ relation from 5 - 50 Hz.
The frequency noise at 10 Hz is then calculated with the fitted parameters
$(a,b)$ and has an uncertainty given by the standard deviation error on those
parameters. The result is the frequency noise at 10 Hz at the measured read
powers for each KID. The read power is translated to the internal E-field
inside the PPC through Eq. 6. This yields the points in Fig. 8, which follow
the expected $|\vec{E}|^{-1}$ relation [15, 29].

Figure 8: Fractional frequency noise at 10 Hz versus the internal electric
field of the PPC. Each point is retrieved by fitting the $1/f$ region of the
frequency noise spectra and calculating the 10 Hz point from the fitted
parameter. The error bar of each point is the calculated uncertainty based on
the fit parameter error standard deviation. The red line is a fit with
$a\cdot|\vec{E}|^{-b}$, with the uncertainty region based on the fit error
standard deviation. The red point is the retrieved frequency noise at 10 Hz
and desired E-field.

To get the frequency noise at a desired E-field ($|\vec{E_{des}}|$), we fit
the expected $a\cdot|\vec{E}|^{-b}$ relation to these datapoints, with
$b\approx 1$. Using the fitted function parameters we obtain the frequency
noise level at the desired field strength, with an uncertainty given by the
standard deviation error in the fitted parameters ($a$,$b$). The total error
of the point $S_{f}(f=10\text{ Hz},|\vec{E}|=|\vec{E_{des}}|)$ in Fig. 8 is
the propagated error of the uncertainties in the $S_{f}(f)$ and
$S_{f}(f=10\text{ Hz},|\vec{E}|)$ fits.

##  Appendix C Temperature dependence

The interacting tunneling model [29] predicts the following temperature
dependent TLS-noise level.

$S_{f}\propto\begin{cases}T^{-(1+\mu)},&T\ll\omega_{r}\text{ in weak field}\\\
T^{(1-\mu)/2},&T\ll\omega_{r}\text{ in strong field}\\\
T^{\mu-1},&T\gg\omega_{r}\end{cases}$ (10)

We measure the frequency noise at different temperatures between 25 mK and 1 K
for all three resonator designs (PPC, IDC, CPW). The frequency spectra at one
internal power, and the frequency noise at 10 Hz vs. internal power for a PPC
resonator are plotted in Fig. 9. For all powers the resonator is operated in
the strong field condition of Eq. 10, where one expects a smooth crossover
between the limits of Eq. 10 around $\hbar\omega=k_{\rm{B}}T$. In Fig. 9 the
frequency noise indeed shows a backbend below $\hbar\omega=k_{\rm{B}}T$. This
temperature dependence is not explained by the STM [15], but can be explained
by the interacting tunneling model presented in [29].

The spectrum at $f>$ 100 Hz follows a $f^{-1/2}$ relation at the lowest
temperatures. The slope of this region changes with temperature resulting in
an almost white spectrum, between 1 kHz and the resonator roll-off, at 800 mK.
The $f>$ 100 Hz region roughly follows the temperature dependence of Eq. 10.
Neither the $f^{-1/2}$ region of the spectrum nor its temperature dependence
is explained by the STM or interacting tunneling models.

Figure 9: Fractional frequency noise of a PPC resonator for different bath
temperatures and internal powers at 10 Hz. The internal powers fall within the
strong field condition of Eq. 10. The vertical line is based on the resonance
frequency at 100 mK: $T=\hbar\omega/k_{\rm{B}}$. The dotted lines connect
measured datapoints to highlight backbend below $\hbar\omega=k_{\rm{B}}T$.

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