of 13
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
Characterization of the low electric field and zero-temperature two-level-system
loss in hydrogenated amorphous silicon
Fabien Defrance
*
and Andrew D. Beyer
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
Shibo Shu, Jack Sayers, and Sunil R. Golwala
California Institute of Technology, Pasadena, California 91125, USA
(Received 2 December 2023; accepted 14 February 2024; published 5 March 2024)
Two-level systems (TLS) are an important, if not dominant, source of loss and noise for superconducting
resonators such as those used in kinetic inductance detectors and some quantum information science plat-
forms. They are similarly important for loss in photolithographically fabricated superconducting mm-wave/THz
transmission lines. For both lumped-element and transmission-line structures, native amorphous surface oxide
films are typically the sites of such TLS in nonmicrostripline geometries, while loss in the (usually amorphous)
dielectric film itself usually dominates in microstriplines. We report here on the demonstration of low TLS loss
at GHz frequencies in hydrogenated amorphous silicon (a-Si:H) films deposited by plasma-enhanced chemical
vapor deposition in superconducting lumped-element resonators using parallel-plate capacitors (PPCs). The
values we obtain from two recipes in different deposition machines, 7
×
10
6
and 12
×
10
6
, improve on the
best achieved in the literature by a factor of 2–4 for a-Si:H and are comparable to recent measurements of
amorphous germanium. Moreover, we have taken care to extract the true zero-temperature, low-field loss tangent
of these films, accounting for temperature and field saturation effects that can yield misleading results. Such
robustly fabricated and characterized films render the use of PPCs with deposited amorphous films a viable
architecture for superconducting resonators and they also promise extremely low loss and high quality factor for
photolithographically fabricated superconducting mm-wave/THz transmission lines used in planar antennas and
resonant filters.
DOI:
10.1103/PhysRevMaterials.8.035602
I. INTRODUCTION
The standard tunneling model (STM) is the reference
model used to describe the properties of amorphous dielectrics
at low temperatures (below a few kelvins) [
1
,
2
]. According to
this model, amorphous dielectrics have defect states with two
physical configurations (e.g., locations of an atom or groups of
atoms) with different energies, forming a quantum two-state
(“two-level”) system with interstate tunneling [
1
,
3
]. When the
two-level systems (TLSs) present in amorphous dielectrics
have an electric dipole moment, they can couple to an oscillat-
ing electric field present in the dielectric (in superconductive
resonators used in qubits or kinetic inductance detectors, for
example) and convert some of the energy stored in the electric
field into phonon emission, resulting in dielectric loss and
noise. In superconducting qubits, dielectric loss is a major
source of decoherence, as first inferred by Martinis
et al.
[
4
].
In kinetic inductance detectors, dielectric loss adds noise and
can degrade responsivity and multiplexability [
5
,
6
], especially
in parallel-plate capacitor (PPC) geometries that are desirable
in some applications. In the superconducting transmission
line used in photolithographically fabricated planar antennas
and resonant bandpass filters at mm-wave/THz frequencies,
*
fabien.m.defrance@jpl.nasa.gov
dielectric loss determines attenuation length [
7
] and limits
spectral resolution [
8
]. Therefore, low-loss dielectrics are
highly desirable for diverse low-temperature applications.
Crystalline dielectrics have very low dielectric loss be-
cause of their low density of defects [
9
], but incorporation of
crystalline dielectrics into elements like PPCs or supercon-
ducting microstripline transmission lines requires complex
fabrication techniques [
10
,
11
]. When simpler techniques are
used to fabricate the single-layer equivalents [transmission
line resonators using coplanar waveguide (CPW), lumped el-
ement resonators using interdigitated capacitors (IDCs), and
superconducting CPW transmission line], they exhibit TLS
loss due to the formation of oxide at the exposed surfaces of
the superconducting material and substrate [
12
14
]. Choos-
ing superconducting materials having a weak reactivity with
oxygen, and thus producing a very thin oxide layer, yields
interesting results with materials such as rhenium or nitrides
(TiN, NbN, etc.) [
13
15
]. However, this solution limits de-
signs to use of a few superconductive materials.
By contrast, multilayer structures incorporating an eas-
ily deposited, low-loss, amorphous dielectric would have the
benefit of confining the electric field inside the dielectric,
drastically limiting the participation of surface oxides that
enhance the loss of CPW and IDC structures on crystalline
dielectric. Such multilayer structures would enable significant
progress in a wide range of low-temperature applications. KID
2475-9953/2024/8(3)/035602(13)
035602-1
©2024 American Physical Society
FABIEN DEFRANCE
et al.
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
development would benefit from the option of replacing large
IDCs limited by poorly controllable surface-oxide loss with
order-of-magnitude smaller PPCs limited by well-controlled
bulk dielectric film loss. Microstripline KID designs would
become less challenging to implement (cf. [
16
]). Low-
loss microstripline would also enable more sophisticated
planar antennas, higher resolution filters in superconduc-
tive spectrometers, and microstripline-based traveling-wave
parametric amplifiers using kinetic inductance. (The films
discussed in this paper have already been successful for the
last application [
17
].)
The main amorphous dielectrics whose GHz–THz
elec-
tromagnetic
behavior—believed to be due to electric-dipole-
coupled TLS—has been explored to date in the literature are
AlO
x
,SiO
2
,SiO
x
,SiN
x
, and hydrogenated amorphous silicon
(a-Si:H) (see [
18
] for an exhaustive review). Very recently,
hydrogenated amorphous silicon carbide (a-SiC:H [
19
]) and
amorphous germanium (a-Ge [
20
]) have been explored.
1
Prior
studies [
19
,
23
27
] show that a-Si:H dielectric loss can, in
general, be at least an order of magnitude lower than that
of AlO
x
,SiO
2
,SiO
x
, and SiN
x
(though low loss has been
achieved for SiN
x
[
28
], albeit with high stress). Loss in a-Ge
films are the lowest seen in the literature to date [
20
]. In
this paper, we demonstrate a-Si:H films with low-power loss
tangent 7 and 12
×
10
6
. These films improve on the best,
previous, robust measurements for a-Si:H [
23
] by a factor of
2–4,
2
and are comparable to measurements for a-Ge given
their uncertainties [
20
].
There is significant, potentially relevant literature on a-Si
and a-Si:H in three different contexts: as a photovoltaic, as an
example of subelectronic-band-gap optical absorption, with
impact on its use as a mirror coating in laser-interferometric
gravitational wave detectors, and as an exemplar of universal
STM behavior. We review this literature in Appendix
A
, leav-
ing it out of the body of the paper because it is not likely to be
relevant for reasons explained there, with the exception of the
results of Molina-Ruiz
et al.
[
29
]. We do offer a caveat to this
conclusion in Sec.
VII
.
Demonstrations of low loss have not usually addressed
robustness: stability of the deposition recipe over time, trans-
ferability between deposition machines, and suitability of the
recipe for combination with other fabrication steps. For exam-
ple, films made using the [
23
,
24
] recipe
in the same machine
a decade later yielded much higher loss tangent and low-loss
SiN
x
suffers high stress [
28
], rendering it challenging to in-
corporate in complex circuits. We demonstrate here
10
5
low-power loss tangent using two different combinations of
deposition technique and machine and our recipes are, by con-
struction, consistent with inclusion in PPCs and microstripline
[
17
].
1
We neglect epitaxially grown films (e.g., Al
2
O
3
[
21
,
22
], Si [
20
])
because epitaxy requirements on the substrate or the base film
present significant constraints on use in PPCs or microstripline.
2
The films presented in [
19
,
26
] could, after corrections and with
measurements at lower power, yield comparable low-power loss tan-
gent, but precise numbers at low power are not available. The most
robust numbers available for the same deposition process are in [
27
]:
3
.
6
×
10
5
(see Sec.
VI
).
Another challenge with measurements of such low loss is
systematic uncertainties. As described in Sec.
III
, the loss tan-
gent at 0 K and weak electric field, tan
δ
0
TLS
, characterizes the
intrinsic TLS loss of the film, with none of the two-level sys-
tems saturated by temperature or power. Most previous studies
of a-Si:H loss tangent characterized the resonator quality fac-
tor
Q
i
and inferred from it the loss tangent via tan
δ
=
Q
1
i
.
The quantity tan
δ
can be smaller than tan
δ
0
TLS
because of the
above saturation effects. It can also be larger because it may
include additional non-TLS losses (tan
δ
other
). Moreover, for
a-Si:H loss tangent studies using CPW resonators, separating
the TLS loss of a-Si:H from that of surface oxide layers can
add systematic uncertainty to the measurements of tan
δ
0
TLS
(e.g., Refs. [
18
,
30
]). We present in this article measurements
of tan
δ
0
TLS
that avoid these inaccuracies by design and by
measurement technique
3
: our LC resonator device design uses
PPCs to ensure surface oxide contributions are negligible and
our measurement technique fits for the dependence of res-
onant frequency
f
res
on temperature and of inverse quality
factor
Q
1
i
on readout power—two independent techniques
that yield broadly consistent results, with the former being
more robust.
II. TEST DEVICE DESIGN AND FABRICATION
A. Design
Each test device hosts six niobium (Nb) LC resonators
inductively coupled to a 50

Nb coplanar waveguide (CPW)
feedline that is used for readout. Each resonator is com-
posed of two parallel plate capacitors (PPCs) in series with
an inductor, as described in Fig.
1
. To facilitate their iden-
tification, the six resonators are grouped in frequency into
two triplets, centered on 0.85 GHz and 1.55 GHz. Within
each triplet, the resonators are designed to have resonant
frequencies separated by 5%. The physical dimensions of the
resonator components and their predicted/simulated electrical
values are listed in Table
I
. Because we initially had a conser-
vative expectation of obtaining TLS loss tangent tan
δ
10
4
and thus internal quality factor
Q
i
10
4
, we designed the
coupling quality factor,
Q
c
, to also be close to 10
4
to maximize
the coupling efficiency [
5
].
Q
c
and
f
res
were designed and
3
The utility of PPCs has been known for some time [
4
]andwas
emphasized in [
23
] and, more recently, in [
30
]. The applicability
of the frequency-shift technique was first pointed out in [
14
,
31
,
32
],
with [
31
,
32
] both (1) explaining that readout power saturation of
TLS does not impact frequency shift vs temperature because it is
sensitive to the full ensemble of TLS, while quality factor is only
sensitive to TLS within a linewidth of
f
res
(temperature saturation
is of course the means by which the frequency shift data yield the
low temperature loss tangent), and (2) recognizing the convenience
of the frequency shift technique relative to measuring quality factor
at low power. Reference [
18
] reiterated these points. This technique
is, however, only usable when the superconducting film has high
enough
T
c
that the dependence of kinetic inductance on temperature
does not dominate over the TLS effect, a caveat that may explain the
prevalence of
Q
i
measurements in the literature. In particular, half
of the literature results cited in Sec.
VI
use the Al superconductor,
which can suffer this effect.
035602-2
CHARACTERIZATION OF THE LOW ELECTRIC FIELD ...
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
FIG. 1. (a) Exploded view, not to scale, showing the geometry of the LC resonators used to measure the loss tangent and internal quality
factor of a-Si:H. The CPW feedline is composed of a center conductor on the top Nb layer and ground electrodes on the bottom Nb (“ground
plane”) layer separated by the a-Si:H dielectric film to be characterized. The resonator consists of an inductor that runs parallel to the feedline
center conductor on the top Nb layer and a series pair of PPCs formed by plates on the Nb top layer and the ground plane. (The a-Si:H layer is
only 800 nm thick, while the inductor-top plate gap is 22 μm and the CPW center conductor and gap are 20 μm and 12 μm wide, respectively,
so the structure is approximately planar in spite of the vertical separation of the center conductor and ground plane.) A 44 μm gap separates the
ground plane plate from the surrounding ground plane, though this gap is not strictly necessary because this electrode acts as a virtual ground.
The inductor couples the CPW feedline to the LC circuit and a gap in the ground plane below the inductor mitigates magnetic screening that
would reduce the inductance value and feedline coupling. The gap also limits parasitic capacitance with the ground plane. (b) Lumped element
circuit equivalent. The CPW feedline has an impedance
Z
0
=
50

, the two capacitances
C
0
correspond to the two PPCs,
L
is the inductance,
and
k
represents the mutual inductance with the CPW.
simulated using Sonnet software and the very useful method
developed by Wisbey
etal.
[
33
]. The unexpectedly high values
of measured
Q
i
led to
Q
i

Q
c
, causing very deep, overcou-
pled resonances.
B. Fabrication
We fabricated the devices at the NASA Jet Propulsion Lab-
oratory’s MicroDevices Laboratory (MDL) and at the Caltech
Kavli Nanoscience Institute (KNI) clean room facility. For
each recipe, we fabricated four devices simultaneously on the
same 4-in. high resistivity silicon wafer. In the rest of this
article, we will identify each of the four devices (for each
recipe) with an index going from 1 to 4. Figure
2
describes
the main fabrication steps.
Previous studies demonstrated that TLS impacting res-
onator behavior reside primarily in oxide layers localized
at interfaces (metal-vacuum, metal-dielectric, and dielectric-
vacuum) [
12
14
,
34
,
35
]. In our PPC geometry, the electric
field created by the resonator is confined between the ca-
pacitor plates and only the metal-dielectric interfaces are
relevant. Therefore, we took particular care to eliminate any
oxide layers at these interfaces: we used buffered oxide etch
(BOE) on all silicon surfaces prior to metal-film deposition,
we preceded depositions of a-Si:H in the JPL ICP-PECVD
machine by Ar
+
ion milling (not possible in the KNI PECVD
machine), and we did the same with depositions of Nb on
a-Si:H.
We tested many different recipes for a-Si:H, mainly vary-
ing the gas composition, gas flow, rf power, temperature, and
deposition technique. The two recipes (A and B; Table
II
)
we present here are among those giving the lowest TLS loss
values.
TABLE I. Designed dimensions and electrical parameters of the six resonators present on each device. For the PPCs,
w
and

correspond
respectively to the width and length of the top plates, while, for the inductor,
w
and

are the width and length of the inductor line.
C
0
is the
capacitance of each of the two PPCs, so the total capacitance is
C
=
C
0
/
2(
C
0
is calculated using the capacitor dimensions and the permittivity
of silicon). The distance between the inductor and the CPW feedline is set to 22 μm to obtain
Q
c
10
4
.
PPCs
Inductor
w

C
0
w

Lf
res
(μm)
(μm)
(pF)
(μm)
(μm)
(nH)
(MHz)
Q
c
Lower
265
1236
42
20
2920
1.84
810
8
×
10
3
frequency
245
1223
39
20
2920
1.84
844
8
×
10
3
triplet
225
1210
35
20
2920
1.84
882
8
×
10
3
Higher
265
1236
42
20
760
0.54
1489
2
×
10
4
frequency
245
1223
39
20
760
0.54
1554
2
×
10
4
triplet
225
1210
35
20
760
0.54
1627
2
×
10
4
035602-3
FABIEN DEFRANCE
et al.
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
FIG. 2. Schematic showing main device fabrication steps.
(1) Deposition and patterning (etchback) of a 190 nm thick Nb film
on a 375 μm thick, 100 mm diameter, high resistivity silicon wafer,
forming the ground plane (also including the return conductors of the
CPW feedline) and the bottom plate of the PPCs. (2) Deposition of a
800 nm thick layer of a-Si:H using plasma-enhanced chemical vapor
deposition (PECVD; Caltech KNI) for recipe A and inductively
coupled plasma PECVD (ICP-PECVD; JPL MDL) for recipe B.
(3) Deposition and patterning (etchback) of a 400 nm thick Nb
film on top of the a-Si:H layer, forming the CPW feedline center
conductor, the inductor, and the PPC top plates. We deposit all the
metal films using rf magnetron sputtering at JPL MDL with a 6-in.
target.
III. THEORETICAL MODEL FOR THE EFFECT
OF TWO-LEVEL SYSTEMS ON
RESONATOR PARAMETERS
TLS present in dielectrics can couple to a time-varying
electric field via their electric dipole moments. They can also
emit phonons. The transfer of energy from the electric field
to phonon emission is a form of loss for the dielectric. The
circuit’s total loss can be expressed as [
36
]
tan
δ
=
tan
δ
other
+
j
F
j
tan
δ
j
,
(1)
with tan
δ
the total loss tangent, tan
δ
j
the loss tangent of each
lossy dielectric,
F
j
the filling factor of each lossy dielectric
(indicating the portion of the device’s total energy stored in
each dielectric material), and tan
δ
other
representing additional
non-TLS loss mechanisms. The circuit loss tangent, tan
δ
,is
equivalent to
Q
1
i
, the inverse of the internal quality factor,
which is a quantity easily measurable when the electric circuit
is a resonator. In the rest of this article, all loss tangents are
small compared to unity, so we use the approximation tan
δ
δ
.
The dielectric loss tangent
δ
j
is not a single number, how-
ever, because saturation effects cause it to depend on electric
field strength (i.e., stored energy in the resonator, which is de-
termined by readout power) and temperature. Moreover, TLS
affect not just the imaginary part of the dielectric constant—
i.e., cause loss—but they also affect the real part—i.e., cause
a frequency shift. At microwave frequencies and low tem-
peratures, the standard tunneling model (STM) yields the
following expressions for these two effects (e.g., Ref. [
31
]):
Q
1
i
=
δ
other
+
j
F
j
δ
0
j
,
TLS
tanh
(
hf
res
2
k
B
T
)
[
1
+
(
|

E
j
|
E
c
,
j
)
2
]
β
j
,
(2)
f
res
(
T
)
f
res
(0)
f
res
(0)
=
j
F
j
δ
0
j
,
TLS
π
{
Re
[

(
1
2
hf
res
(0)
2
j
π
k
B
T
)]
ln
(
hf
res
(0)
2
π
k
B
T
)}
,
(3)
with
δ
0
TLS
the “intrinsic” or “asymptotic” (zero temperature,
low electric field) loss tangent,
h
and
k
B
the Planck and
Boltzmann constants, respectively,
T
the temperature of the
dielectric,

E
j
the electric field inside the
j
th dielectric film
under consideration,
E
c
,
j
the critical electric field for TLS
saturation for the
j
th dielectric film,

the complex digamma
function, and
β
j
an exponent determined by the TLS density
of states in dielectric film
j
. This exponent is 0.5 for a loga-
rithmically uniform density of states [
1
], but the experimental
literature yields values between 0.15 and 0.35 [
29
,
37
].
Variants on Eq. (
3
) are seen in the literature. The sign of
hf
res
(0)
/
(2
j
π
k
B
T
) can be either
or
+
because

(
z
)
=

(
z
)
and therefore Re[

(
z
)]
=
Re[

(
z
)]. The denominator of the
last fraction is sometimes written as
k
B
T
only. The absence of
2
π
arises from the underlying TLS theory [
1
], but it results
in a frequency offset. A careful demonstration of this formula
by Gao ([
31
] Appendix G) shows that the denominator of the
last fraction must be 2
π
k
B
T
in order for the right part of the
equation to be equal to zero when
T
=
0K.
Since TLS couple via electric fields, only capacitive ele-
ments contribute to TLS loss [
30
,
38
]. Because we use PPCs,
the electric field is confined between the capacitor plates. We
assume no oxides, and therefore no TLS, remain at the metal-
substrate interfaces given the aforementioned oxide-removal
steps. Though [
30
] argued for the importance, in PPC mea-
surements, of accounting for TLS residing in the parasitic
capacitance of the inductor, we argue our geometry differs
substantially from the one they present, leading to a much
smaller participation of surface oxides. Specifically, consid-
ering the half of the resonator that is at positive voltage during
a particular half period (1) the vast majority of field lines
emanating from the inductor (present because it is not a virtual
ground along its entire length) terminate on the ground plane
or the PPC bottom plate, passing through the same a-Si:H
being tested with the PPC, because the adjacent PPC top plate
TABLE II. Deposition recipes for a-Si:H. The gas ratio (SiH
4
/
Ar) corresponds to the relative flow rate of the two gases. The two recipes
use two different machines and deposition processes. Recipe A uses an Oxford Plasmalab System 100 at Caltech KNI for PECVD, while
recipe B uses an Oxford Plasmalab System 100 ICP 380 at JPL MDL for ICP-PECVD. Safety restrictions prevented the use of a pure SiH
4
atmosphere at Caltech KNI.
Gas ratio Temperature Gas pressure Gas flow rf power ICP power
Recipe (SiH
4
/Ar)
(°C)
(mTorr)
(sccm)
(W)
(W)
Deposition time
Process
Facility
A5%
/
95%
350
800
250
10
N/A
27

11

PECVD
KNI
B
100%
/
0%
350
10
30
50
300
26

06

ICP-PECVD
JPL
035602-4
CHARACTERIZATION OF THE LOW ELECTRIC FIELD ...
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
sits at a larger voltage, (2) field lines emanating from the PPC
top plate strongly prefer to terminate on the PPC bottom plate
rather than on the inductor because the former is much closer
(800 nm vs 22 μm), and (3) any field lines that do travel from
the PPC top plate to the inductor primarily pass through the
higher

a-Si:H rather than through surface oxides. The same
arguments can be made for the half of the resonator that is at
negative voltage. Together, these points imply that the bulk of
the inductor’s parasitic capacitance is subject to the same TLS
as the PPC and the fraction of field lines terminating on the
inductor and passing through surface oxides is a small fraction
of the already fractionally small parasitic capacitance of the
inductor.
We thus assume the entirety of the resonator stored energy
resides inside the a-Si:H film, which gives a filling factor
F
=
1 and allows us to rewrite Eqs. (
2
) and (
3
)as
Q
1
i
=
δ
other
+
δ
0
TLS
tanh
(
hf
res
2
k
B
T
)
[
1
+
(
|

E
|
E
c
)
2
]
β
,
(4)
f
res
(
T
)
f
res
(0)
f
res
(0)
=
δ
0
TLS
π
{
Re
[

(
1
2
hf
res
(0)
2
j
π
k
B
T
)]
ln
(
hf
res
(0)
2
π
k
B
T
)}
.
(5)
Equation (
4
) implies that, due to their low energy, TLS
saturate at high temperature (
hf
res

2
k
B
T
) and under high
electric field (
|

E
|

E
c
), reducing the dielectric loss [
32
]. A
measurement of
Q
1
i
as a function of temperature and/or elec-
tric field can, in principle, be fit to Eq. (
4
) to determine
δ
other
,
E
c
,
δ
0
TLS
, and
β
. Equation (
5
) indicates that the frequency
shift suffers no field dependence and has a characteristic tem-
perature dependence—behaviors of great practical utility for
robustly determining
δ
0
TLS
, as we will show.
While fitting to Eq. (
4
) may seem the most straightforward
approach to finding
δ
0
TLS
, it can be challenging. Doing so
requires determining
Q
i
, which can be difficult if the com-
plex
S
21
(
f
f
res
) trajectory deviates significantly from ideal
behavior and/or if
Q
i

Q
r
(the total quality factor
Q
r
of a
resonator is related to its internal quality factor
Q
i
and its
coupling quality factor
Q
c
by the relation
Q
1
r
=
Q
1
i
+
Q
1
c
).
More importantly, such a fit requires extensive and sensitive
data:
|

E
|
must be swept from
|

E
|
E
c
to
|

E
|
E
c
to fit
the
|

E
|
dependence, these data must show clear plateaus at
high and low
|

E
|
to determine
δ
other
and the normalization
of the
|

E
|
dependence, and either the data must be taken at
T

hf
res
/
k
B
so the TLS are unsaturated or one must apply a
correction for the tanh[
hf
res
/
(2
k
B
T
)] dependence (see Fig.
3
).
Data over a wide range in
T
would maximize the robustness of
the temperature correction, but taking such data is experimen-
tally time consuming and not usually done in the literature. It
may also be difficult to obtain sufficient signal to noise at the
low readout power required for
|

E
|
E
c
and it may not be
possible to reach the
|

E
|
E
c
regime before the supercon-
ductor becomes nonlinear (when the current approaches the
critical current) or pair-breaking takes place due to dissipation
in the superconductor arising from
δ
other
.
In this article, we take the approach—which has not been
seen in the literature—of fitting both Eqs. (
4
) and (
5
) and
FIG. 3. Log-log plot illustrating Eq. (
4
).
Q
1
i
converges towards
δ
other
+
δ
0
TLS
tanh[
hf
res
/
(2
k
B
T
)] at low
|

E
|
/
E
c
and towards
δ
other
at
high
|

E
|
/
E
c
. The exponent
β
only changes the steepness of the
dependence on
|

E
|
;
β
=
0
.
5 was used here, corresponding to a
log-uniform TLS density of states. The different curves correspond
to different values of
hf
res
/
(2
k
B
T
). The plot illustrates that taking
data in the fully unsaturated (in
T
and
|

E
|
) limit yields the best
lever arm on determining the true value of
δ
0
TLS
from
Q
1
i
data
and that taking
T
-saturated data necessitates applying a correction
factor tanh[
hf
res
/
(2
k
B
T
)]. For reference,
hf
res
/
(2
k
B
T
)
0
.
96 [and
tanh(0
.
96)
=
0
.
74] for
f
res
=
1 GHz at
T
=
25 mK. Our data probe
the range
hf
res
/
(2
k
B
T
)
[0
.
08
,
0
.
16], requiring a factor of roughly
6–13 correction for
T
saturation.
comparing the results in order to better understand the sys-
tematics associated with the two methods. We conclude that
the frequency-shift approach [Eq. (
5
)] is far more robust.
IV. EXPERIMENTAL SETUP
We cooled the devices using a Chase Cryogenics closed-
cycle
3
He
/
3
He
/
4
He sorption cooler mounted to the second
stage of a Cryomech PT415 cryocooler. Each device resided in
its own box sealed to prevent optical radiation from breaking
quasiparticles or heating the substrate. A magnetic shield,
residing at 4 K and consisting of two layers of Amuneal
A4K material, enclosed the devices to limit the impact of
Earth’s magnetic field. A combination of stainless steel and
NbTi semirigid coaxial cables carried the readout signal to the
devices, with 30 dB and 10 dB in-line attenuators at 4 K and
0.35 K, respectively, to block 300 K thermal noise. Similar
NbTi coax carried the signal exiting each device to a cryogenic
low-noise amplifier (LNA) at 4 K, with a noise temperature
of approximately 5 K, followed by stainless steel coax back
to 300 K. Additional LNAs at 300 K ensured the cryogenic
LNA dominated the system noise. We monitored the device
temperature using a Stanford Research System (SRS) SIM921
reading a Lakeshore Germanium Resistance Thermometer
(GRT) located next to the devices. The temperature was varied
between 240 mK and 450 mK using a SRS SIM960 analog
PID controller supplying a current to a 10 k

heater on the
mechanical stage holding the devices. A Copper Mountain
Technologies SC5065 Vector Network Analyzer (VNA) per-
formed measurements of
S
21
(
f
) using the above signal chain.
We used the Python module SCRAPS [
39
]tofitthe
S
21
(
f
)
data to standard forms (e.g., Ref. [
5
]) to extract the resonance
035602-5
FABIEN DEFRANCE
et al.
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
FIG. 4. (a), (b) Data and best-fit TLS model [Eq. (
5
)] for [
f
res
f
res
(0)]
/
f
res
(0) as a function of temperature. (c), (d) Data and best-fit
TLS model [Eq. (
4
)] for
Q
1
i
data as a function of electric field. (a), (c) Devices A(1) and A(2); (b), (d) devices B(1) and B(2). Much of
the literature uses photon number instead of
|

E
|
; Fig.
6
in Appendix
B
shows plots (c) and (d) as a function of photon number to facilitate
comparison with published results. Table
III
lists the best-fit parameters. Disagreements between the model and the data are discussed in the
text; in particular, how we extract
δ
0
TLS
and
δ
other
in the presence of such disagreements. The
f
res
(
T
) data were taken at a readout power (at the
device) of approximately
100 dBm (
|

E
|≈
500 V
/
m). The
Q
1
i
(
|

E
|
) data were taken at 246 mK.
frequency
f
res
and quality factors
Q
r
,
Q
i
, and
Q
c
for each value
of
T
and readout power (electric field).
We varied the readout power applied by the VNA to the
feedline,
P
read
, over the approximate range
150 dBm to
70 dBm. Because the SNR decreases with readout power,
we had to compensate by reducing the IF bandwidth of the
VNA, which results in a proportional increase in measurement
time. Each factor of 10 decrease in IF bandwidth yielded a
10 dB reduction of noise floor and a factor of 10 increase in
measurement time. We used an IF bandwidth of 1 kHz above
100 dBm, reducing it to 10 Hz at the lowest readout powers.
V. RESULTS
Figure
4
shows fits of the measured
f
res
(
T
) and
Q
1
i
(
|

E
|
)
to Eqs. (
5
) and (
4
), respectively. Table
III
shows the inferred
fit parameters
δ
0
TLS
,
δ
other
,
E
c
, and
β
. Figure
5
summarizes and
compares the
δ
0
TLS
values from all the fits for all the devices.
To calculate the electric field
|

E
|
from the readout
power, we use the following equation, which is derived in
Appendix
B
:
|

E
|=
1

A
0
d
P
read
2
π
f
res
Q
2
r
Q
c
,
(6)
where
A
0
is the top plate area,

is the permittivity of a-
Si:H,
d
=
800 nm is the distance separating the plates of
the capacitors, and
P
read
is the readout power at the device.
We calculated
A
0
=
w
×

from the dimensions in Table
I
.
We assumed
/
0
=
11
.
68, the experimental value found for
crystalline silicon, for the a-Si:H relative permittivity be-
cause the systematic uncertainty due to any deviation from
this value is negligible given the large logarithmic range in
|

E
|
. Across multiple chips (those presented in this article
and others), we measured
f
res
variations smaller than 4%.
Since
f
res
=
1
/
(2
π
LC
) and
C
=
C
0
/
2
=

A
0
/
2
d
, the corre-
sponding variation in
/
d
would be less than 8%. Inductance
variations could also affect
f
res
, but simulations indicate that
such variations could only occur if the position of the inductor
and/or its width varied by more than 2 μm, ruled out by the
photolithography accuracy.
035602-6
CHARACTERIZATION OF THE LOW ELECTRIC FIELD ...
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
TABLE III. Best-fit parameters for fits to data shown in Fig.
4
. Two resonances are missing in devices A(2) and B(2) due to fabrication
yield. The values with an asterisk (
) are considered unreliable. The text explains the fits in more detail and assesses the reliability of the fitted
parameters. Of particular interest is the consistency of
δ
0
TLS
obtained from the frequency-shift and quality-factor data.
Data set
f
res
(
T
)
Q
1
i
(
|

E
|
)
f
res
(
T
)
Q
1
i
(
|

E
|
)
f
res
δ
0
TLS
δ
0
TLS
δ
other
E
c
f
res
δ
0
TLS
δ
0
TLS
δ
other
E
c
(MHz)
(
×
10
6
)(
×
10
6
)(
×
10
6
)(V
/
m)
β
(MHz) (
×
10
6
)(
×
10
6
)(
×
10
6
)(V
/
m)
β
Device A(1)
Device A(2)
786
12.0
8.7
7.9
2.5
0.30
785
11.0
21.5
*4.8
*2.8
*0.10
818
12.0
6.4
7.8
6.4
0.49
817
12.0
21.0
*4.7
*2.6
*0.10
859
12.0
16.2
*8.2
*5.7
*0.10
857
12.0
21.0
*5.1
*6.1
*0.10
1506
12.0
15.3
*6.2
*4.8
*0.10 1505
11.0
15.1
*3.1
*2.5
*0.10
1565
13.0
16.3
*4.8
*2.4
*0.10
1564
11.0
15.7
*3.3
*2.6
*0.10
1652
13.0
16.1
*4.5
*4.4
*0.11
Device B(1)
Device B(2)
824
8.3
*7.1
28.0
*4.2
*0.46
809
7.9
8.9
0.9
3.4
0.20
855
8.3
8.7
0.2
5.8
0.27
847
8.6
8.2
0.3
5.4
0.33
895
3.5
*4.8
13.7
*5.3
*0.35
893
8.4
6.9
0.6
7.3
0.40
1545
5.0
8.1
4.1
7.2
0.27
1603
8.1
7.9
0.8
8.3
0.32
1583
4.4
5.7
2.4
9.4
0.60
1685
8.3
7.3
4.1
8.2
0.35
1680
8.3
6.0
0.6
15.4
0.60
A.
δ
0
TLS
The best-fit values of
δ
0
TLS
range from 6
.
4
×
10
6
to
21
.
5
×
10
6
for devices A(1) and A(2) and from 3
.
5
×
10
6
to 8
.
9
×
10
6
for devices B(1) and B(2). The values obtained
from frequency shift and quality factor data are within a factor
of two of one another.
Among the resonators fabricated using a given recipe (both
across resonator frequency and across chips), the frequency-
shift values of
δ
0
TLS
are generally more consistent than the
quality-factor values. This difference arises because there are
a number of systematic deviations from expected behavior
present only in the quality-factor data and fits while the
frequency-shift data seem to follow the model well. Figure
4
shows the
δ
0
TLS
(low power) plateau is generally visible but the
FIG. 5. Values of
δ
0
TLS
obtained from quality-factor data (orange)
and frequency-shift data (blue) for (a) device A(1), (b) device A(2),
(c) device B(1), and (d) device B(2). The relative difference between
the two fitted values of
δ
0
TLS
is indicated in the gray box above.
δ
other
(high power) plateau is not always and, in some cases,
the data are independent of power. While we can eliminate
some of these systematic deviations by excluding the last 5–15
data points, some remain. We consider three different types of
deviations in turn.
In the cases of the device B(1) resonances at 824 MHz and
895 MHz, we see quality factor independent of readout power
over the regime where the model fits the data. This behavior
makes the data insensitive to
δ
0
TLS
, so we consider the
δ
0
TLS
values for these resonators unreliable and we mark them with
an asterisk in Table
III
.
The other resonators on devices B(1) and B(2) show
Q
1
i
rising at high power (field), presumably due to readout power
generation of quasiparticles [
24
]. In the cases of the B(1)
855 MHz and B(2) 847 MHz resonances, the inverse quality
factor shows a significant decrease at high power before the
rise, while the other device B(1) and B(2) resonances do not
[excluding B(1) 824 MHz and B(1) 895 MHz, already dis-
cussed above]. We find the
δ
0
TLS
values for the B(1) 855 MHz
and B(2) 847 MHz resonance quality factor fits are more
consistent with their frequency-shift fits, suggesting this dif-
ference in behavior causes these resonators’ quality factor fits
to be more reliable (smaller systematic uncertainty).
The device A(1) and A(2) resonators display no high-
power plateau and in fact show a downward deviation from
the expected saturation behavior. This behavior could be an
effect of readout power too, now due to the modification of
the quasiparticle distribution function
f
(
E
) by readout power
[
40
]. Independent of the explanation, this behavior makes it
impossible to fit for
δ
other
.
Overall, it is also not known for certain why devices A(1)
and A(2) show primarily a drop in
Q
1
i
with field, while
devices B(1) and B(2) show a rise, though we may speculate
that variations in Nb film properties are the cause.
Our conclusion from the above results and discussion is
that the frequency-shift data and fits determine
δ
0
TLS
far more
reliably than do quality factor data and fits. After averaging
035602-7
FABIEN DEFRANCE
et al.
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
TABLE IV. Degradation of
δ
0
TLS
with time as evidenced by
δ
0
TLS
measurements from frequency-shift data taken at different dates. For
device A(1), the old and new measurements are separated by 53
months, while, for device B(1), they are separated by 18 months.
Device A(1)
Device B(1)
f
res
New
δ
0
TLS
Old
δ
0
TLS
f
res
New
δ
0
TLS
Old
δ
0
TLS
(MHz) (
×
10
6
)(
×
10
6
) (MHz) (
×
10
6
)(
×
10
6
)
786
12.0
10.0
824
8.3
7.5
818
12.0
9.9
855
8.3
7.7
859
12.0
9.5
895
3.5
4.0
1506
12.0
11.0
1545
5.0
4.5
1565
13.0
N
/
A
1603
8.1
7.5
1652
13.0
11.0
1685
8.3
7.6
these values for each recipe, we obtain
δ
0
TLS
12
×
10
6
for
recipe A and
δ
0
TLS
7
×
10
6
for recipe B.
B.
δ
other
The fitted values of
δ
other
are only valid when a high
readout power plateau is visible. For devices A(1) and A(2),
only the A(1) 786 MHz and A(1) 818 MHz resonances show
this plateau, yielding, respectively, values of 7
.
8
×
10
6
and
7
.
9
×
10
6
. For devices B(1) and B(2), all resonances show the
plateau and yield values of 0
.
2
×
10
6
to 28
×
10
6
. We see that
non-TLS losses vary enormously, even within a single device,
and are often non-negligible compared to TLS loss, even at
low readout powers. We do not know for certain the cause
of this variation, but we again may propose film property
variations. Certainly, at the low values of
δ
0
TLS
obtained here,
it is clear that non-TLS sources of loss may become important
or dominant and thus they deserve future investigation.
C. Evolution of
δ
0
TLS
and
δ
other
with time
We fabricated devices A(1) and A(2) in October 2017
and devices B(1) and B(2) in March 2020. We measured
devices A(1) and B(1) twice: first in June 2018 and September
2020, respectively, and second in March 2022. [We did not
remeasure devices A(2) and B(2).] Between measurements,
we stored the devices in the ambient laboratory atmosphere:
air rather than N
2
atmosphere, standard air conditioning with
generally stable humidity and temperature but no specific
additional humidity or temperature control, and no use of
dessicant.
Table
IV
shows the
δ
0
TLS
measurement evolution. We see
a 19% increase for device A(1) and a 6% increase for device
B(1) (both averaged over the resonators on the device). We
can translate these changes to rates of increase per unit time
of 0
.
36
×
10
6
/
month for device A(1) and 0
.
33
×
10
6
/
month
for device B(1). We thus observe that
δ
0
TLS
increases with time
and that the rate of increase is remarkably consistent between
the two devices, with an average value of 0
.
35
×
10
6
/
month.
We may speculate that exposure to laboratory air results in
uncontrolled uptake of oxygen, hydrogen, or water by the
a-Si:H films that gives rise to an increase in TLS density,
but it is also possible that an increase in TLS density arises
from evolution of the physical structure of the amorphous
films simply due to thermal activation. We would need to
undertake comparisons to a set of devices stored in more
controlled environments (vacuum, dry N
2
atmosphere, dry air
atmosphere, finer temperature control) to narrow down the
cause. These changes are fairly modest, but it is sensible to
take simple precautions (e.g., storage with dessicant) to try to
prevent such degradation in the future.
Concerning
δ
other
, the evolution with time seems more ran-
dom. In September 2020, we measured the high power
Q
1
i
for device B(1). Compared to the
δ
other
fits obtained from
March 2022 data and presented in Table
III
, four resonators
show differences smaller than a factor of 2, while two res-
onators (at 824 MHz and 895 MHz) show increases by factors
of 16 and 3, respectively. We observe that the resonators that
show the largest degradations are the ones for which the values
of
δ
other
are largest, absolutely, and large compared to
δ
0
TLS
.
The dramatic differences between both the values and
the degradation of
δ
other
for resonators on the same device
suggest that the cause is not general atmospheric or thermal
conditions. Local changes in film properties, such as stress,
perhaps aggravated by thermal cycling during test, seem a
more likely culprit. Diagnostic measurements sensitive to lo-
cal film properties—profilometry or AFM—may be useful in
identifying a specific cause. It remains to be seen whether
these seemingly occasional large absolute values and degra-
dations of
δ
other
will prove to be a significant practical issue
for large arrays of such resonators (i.e., resonator yield).
Jointly, the variations of
δ
0
TLS
and
δ
other
with time imply that
one must take care to fully recharacterize all resonators during
each cooldown: values from previous cooldowns may not be
sufficiently accurate.
D. Measurements of
E
c
and
β
Recall that
β
is related to the TLS density of states;
β
=
0
.
5 corresponds to a log-uniform density of states. The values
of
β
seen in the literature, varying from 0.15 to 0.35 [
29
,
37
],
motivate us to also fit for
β
rather than assume the naive value
of 0.5. We require
β
[0
.
10
,
0
.
60] to cover both the naive
value and the prior literature values.
The position and slope of the
|

E
|
-dependent region of the
Q
1
i
(
|

E
|
) data between the two plateaus (see Figs.
3
and
4
)
determines the values of
E
c
and
β
, respectively, via Eq. (
4
).
As noted previously, the plateaus are not cleanly visible in
all the data, resulting in unreliable values of
E
c
and
β
.We
note these cases with an asterisk in Table
III
. Considering
only the reliable fits, we find 0
.
2
<β<
0
.
6, with the
β
=
0
.
6
values potentially limited by the allowed range for
β
.The
median values of
E
c
and
β
for devices B(1) and B(2) for
the reliable fits are 7.3 V
/
m and 0.33, respectively. The
β
values are thus consistent with the literature. (
E
c
values are not
available from the literature in general, presumably because,
for non-PPC geometries, determining it requires sophisticated
modeling of the variation of
|

E
|
over the resonator at a given
P
read
.) However, we find
E
c
and
β
to be fairly degenerate,
potentially giving rise to the large variation in
β
we see. Recall
that our data only probe values of
hf
res
/
(2
k
B
T
)

0
.
075, well
away from saturation of the temperature dependence. Data at
lower temperature may thus provide a larger sloped region,
improving the determination of
β
and
E
c
. We thus conclude
035602-8
CHARACTERIZATION OF THE LOW ELECTRIC FIELD ...
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
that our results are consistent with prior data but that we would
need lower-
T
data to constrain
β
more tightly than prior work.
VI. COMPARISON TO LITERATURE RESULTS
AND DISCUSSION
To provide context for our results, we review the literature
on comparably low-loss, depositable dielectrics.
Comparing these
δ
0
TLS
results with previously published
measurements is not always straightforward, in large part
because prior measurements rely almost exclusively on
Q
1
i
rather than
f
res
data. Considering resonator design, a non-PPC
geometry (e.g., CPW, microstripline) results in a design-
specific value of
F
, the TLS fill factor, which is generally
calculated and reported to obtain a value of
δ
0
TLS
.Lessfre-
quently, if ever, reported is the
E
c
value because, as noted
above, determining it requires somewhat sophisticated mod-
eling of the variation of
|

E
|
over the device and how the
variation in field saturation with position determines the over-
all
Q
1
i
. Considering measurement technique, determining the
true value of
δ
0
TLS
requires either data taken over a sufficiently
broad range of
|

E
|
(usually done) as well as data taken at suffi-
ciently low
T
such that the temperature dependence saturates
(generally but not always possible) or data taken at a wide
enough range of
T
that the temperature dependence can be
fitted for (possible but rarely done).
We consider in turn other studies of a-Si:H and of other
depositable low-loss dielectrics (a-SiC:H and a-Ge).
A. a-Si:H
Among the existing studies of dielectric loss in a-Si:H,
the one from O’Connell
et al.
[
23
] is most straightforward
to compare because they used PPCs also. They found
Q
1
i
[22
25]
×
10
6
for resonance frequencies close to 6 GHz
and at 100 mK. A plot shown in the article also indicates
that
δ
other

δ
0
TLS
and that
|

E
|
E
c
. We apply a modest cor-
rection for incomplete temperature saturation (from Eq. (
4
),
Q
1
i
(
|

E
|
E
c
)
δ
0
TLS
tanh[
hf
res
/
(2
k
B
T
)]) to their values of
Q
1
i
to obtain
δ
0
TLS
[25
28]
×
10
6
.
Mazin
et al.
[
24
] used Al microstrip resonators. Be-
cause they operated below 100 mK and at about 9 GHz,
tanh[
hf
res
/
(2
k
B
T
)]
1. Their data show a clear low-power
plateau, indicating
δ
other

δ
0
TLS
. We thus extract from their
data the value
δ
0
TLS
60
×
10
6
.
Bruno
et al.
[
25
] used Nb lumped-element LC resonators.
While measurement details are lacking, it is reasonable to in-
fer that
δ
TLS
=
25
×
10
6
was obtained at
T
=
4
.
2Kfor
f
res
=
10
.
5 GHz. The correction for tanh[
hf
res
/
(2
k
B
T
)]
0
.
06 is
large, yielding
δ
0
TLS
420
×
10
6
.
Molina-Ruiz
et al.
[
29
] measured Al CPW resonators on
a-Si at 10 mK and 4–7 GHz. While this material is not hydro-
genated (H is undetectable, with an upper limit of
<
0
.
1%), we
include it in discussion of a-Si:H because it can be considered
to be on a continuum with a-Si:H. They also used
Q
1
i
mea-
surements across a wide range of readout powers to extract
δ
0
TLS
. They tried to correct for the contribution to
δ
0
TLS
of TLS
loss of oxide layers at exposed surfaces or at film interfaces by
measuring resonators of the same geometry fabricated directly
on a bulk crystalline Si substrate, introducing a systematic
uncertainty. The value obtained for their lowest-loss deposi-
tion recipe,
δ
0
TLS
=
3
.
3
±
3
.
5
×
10
4
, is interestingly low, but
it has a large fractional uncertainty for this reason. We note
that this film has low acoustic loss tangent, 21
×
10
6
, though
Molina-Ruiz
etal.
[
29
] also show that acoustic and rf loss have
different microscopic sources, with a thinner film showing
lower acoustic loss and higher rf loss, so there is no guarantee
that the true rf loss is comparable to the low acoustic loss.
Buijtendorp
et al.
[
19
,
26
] and Hähnle
et al.
[
27
] provide
the most recent results on a-Si:H loss. While Buijtendorp
et al.
[
19
,
26
] provide the more extensive study of a-Si:H, their
range of readout power is not large enough to identify the
low- and high-power plateaus, so
δ
0
TLS
cannot be accurately
determined. They also make significant corrections for surface
oxides that introduce scatter and make it difficult to identify
a single number or robust upper limit. Hähnle
et al.
[
27
],
by contrast, provide
Q
i
data covering the full range needed,
yielding
δ
0
TLS
=
3
.
6
±
0
.
5
×
10
5
(after subtracting
δ
other
;
tanh[
hf
res
/
(2
k
B
T
)]
1for
f
res
=
6 GHz and
T
=
60 mK).
Hähnle
et al.
[
27
] references [
26
] for deposition parameters,
so we assume the former used the lowest loss recipe from the
latter.
It is clear from the above summary that many authors have
demonstrated a-Si:H films with
δ
0
TLS
<
10
4
. It is of obvious
interest to identify under what conditions these results can
be obtained and what causes the remaining variability. We
summarize in Table
V
deposition recipes and measured
δ
0
TLS
for our devices and the above literature results. It is clear
that
δ
0
TLS
<
10
4
seems achievable with a variety of PECVD
techniques and recipes and that these various recipes exhibit
less than an order of magnitude variation in
δ
0
TLS
. However,
no patterns connecting
δ
0
TLS
to PECVD deposition technique
or parameters readily emerge.
Due to the large uncertainty on
δ
0
TLS
for e-beam evapo-
rated a-Si films, it is not yet clear whether such films offer
comparably low
δ
0
TLS
. Molina-Ruiz
et al.
[
29
] show that a
lower substrate temperature during deposition dramatically
increases both rf and acoustic loss, while we do not observe
a similar dependence for PECVD films, so it does seem that
PECVD is a more robust process for achieving
δ
0
TLS
<
10
4
for a-Si:H. (We note that PECVD films of SiO
2
and SiN
x
exhibit significantly higher loss tangents, with the previously
noted exception of the high-stress film in [
28
], so PECVD
alone is not sufficient.)
This comparison with the previously published measure-
ments shows that the average value of a-Si:H
δ
0
TLS
for our
recipe B is about four times lower than that measured by
O’Connell
et al.
[
23
], eight times lower than that measured
by Mazin
et al.
[
24
], of the order of a decade lower than that
measured by Molina-Ruiz
et al.
[
29
], and five times lower
than that measured by Hahnle
et al.
[
27
]. We thus believe we
have obtained the lowest-loss a-Si:H films in the literature.
Moreover, the modestly poorer loss obtained from recipe A
implies our
δ
0
TLS
is fairly robust against changes in deposition
machine and recipe.
B. Other depositable low-loss dielectrics: a-SiC:H and a-Ge
While our focus in this paper has been on a-Si:H because,
out of the widely studied depositable dielectrics AlO
x
,SiO
2
,
035602-9
FABIEN DEFRANCE
et al.
PHYSICAL REVIEW MATERIALS
8
, 035602 (2024)
TABLE V. Summary of a-Si:H deposition recipes and measured
δ
0
TLS
for our devices and previously published results. a-Si:H films
measured by Mazin
et al.
and O’Connell
et al.
were both fabricated at University of California Santa Barbara (UCSB) using a recipe identical or
very close to the one detailed by Lucero in his Ph.D. thesis [
41
], Appendix B.4.6 (private communication from Mazin). HD PECVD stands for
high-density plasma-enhanced chemical vapor deposition. While most of the above publications use PECVD to deposit a-Si:H, Molina-Ruiz
et al.
use e-beam evaporation instead to deposit a-Si and thus gas ratio and gas flow are not applicable. O’Connell
et al.
do not provide precise
film thicknesses but indicate their films are hundreds of nm thick, comparable to the other films presented in this table. As noted in the text,
Buijtendorp
et al.
[
19
,
26
] and Hähnle
et al.
[
27
] use the same deposition parameters, but the former do not provide precise
δ
0
TLS
values, while
the latter does not provide deposition details. We assume Hähnle
et al.
[
27
] used the lowest loss recipe from Buijtendorp
et al.
[
26
].
Gas ratio Temperature Gas pressure Gas flow
Thickness
δ
0
TLS
Ref
/
Facility
Process
(SiH
4
/Ar)
(°C)
(mTorr)
(sccm) Deposition time
(nm)
(
×
10
6
)
Caltech KNI
PECVD
5%
/
95%
350
800
250
27

11

800
12
JPL MDL
ICP-PECVD 100%
/
0%
350
10
30
26

06

800
7
O’Connell
et al.
[
23
] HD PECVD 66%
/
33%
100
2
45
3

15

Unknown
26
Mazin
et al.
[
24
]
HD PECVD 66%
/
33%
100
2
45
3

15

200
60
Bruno
et al.
[
25
]
PECVD 100%
/
0%
250
200
Unknown
Unknown
Unknown
420
Molina-Ruiz
et al.
e-beam evap
N/A
50
/
225
/
450
<
5
×
10
6
N/A
19

106

59–317

330
±
350
(no H) [
29
]
Hähnle
et al.
[
27
]PECVD5%
/
95%
350
1000
500
7

0

250
±
15
36
±
5
SiO
x
,SiN
x
, and a-Si:H, it generally gives the lowest loss tan-
gents, there are other depositable dielectrics with comparably
low loss and thus of potentially comparable utility.
Buijtendorp
et al.
[
19
] studies hydrogenated amorphous
silicon carbide, a-SiC:H, deposited using PECVD. They
measure
δ
0
TLS
=
30
±
0
.
4
×
10
6
at
f
res
=
7
.
4 GHz and
T
=
60 mK, requiring no tanh[
hf
res
/
(2
k
B
T
)] correction. These
results are comparable to the same authors’ results for a-Si:H
in Hähnle
et al.
[
27
], noted above.
Kopas
et al.
[
20
] have demonstrated similar low loss
tangents for a-Ge. They use Nb CPW resonators deposited
on a 1 μm thick a-Ge film deposited by thermal evapo-
ration. The temperature (40 mK) and resonance frequency
(6.3 GHz and 7.3 GHz) of their measurements ensure that
tanh[
hf
res
/
(2
k
B
T
)]
1. Their range of readout power reaches
the low-power plateau, so their low-power loss tangent
measurement
Q
1
i
=
[11
13]
×
10
6
corresponds to
δ
0
TLS
+
δ
other
. Their data do not clearly reach the high-power plateau
that would yield
δ
other
, so the measurements are an upper limit
on
δ
0
TLS
. Because the measured loss tangent is so low, they
subtract contributions from TLS at various interfaces and in
the substrate based on calculated fill factors and literature
values [
34
]: 5
.
8
×
10
6
,1
.
1
×
10
6
,0
.
8
×
10
6
, and 2
.
6
×
10
7
for the metal-substrate, substrate-air, and metal-air interfaces
and silicon substrate loss, respectively. They obtain
δ
0
TLS
+
δ
other
=
[4
.
7
4
.
9]
×
10
6
for a-Ge alone, which we interpret
as an upper limit on
δ
0
TLS
. However, no uncertainties on these
large subtractions are provided and it is not clear that these
subtractions are valid, as [
34
] employed TiN resonators on
crystalline Si while these authors consider Nb resonators on
an a-Ge film on crystalline Si. Thus
δ
0
TLS
<
11
×
10
6
is the
more robust result.
VII. CONCLUSION
We have reported low-field, low-temperature TLS loss
(
δ
0
TLS
) values of 7
×
10
6
and 12
×
10
6
for two similar but
far from identical deposition recipes, obtained in two different
CVD deposition systems at two different sites. These values
are robust and subject to minimal modeling uncertainty thanks
to the use of PPCs, multiple resonators spanning a factor of 2
in frequency, and frequency-shift data. We have also deter-
mined
δ
0
TLS
from quality-factor data and found those values
to be broadly consistent with the frequency-shift results, but
we have explained why quality-factor data is more difficult
to model and thus the resulting values subject to greater sys-
tematic uncertainty. Our lowest frequency-shift values of TLS
loss are a factor of 4 lower than the best previously published,
robust results for a-Si:H and are comparable to results for a-Ge
[
20
]. They also depend only modestly (factor of
1
.
5) on
deposition machine and details of deposition technique. Com-
parison to the literature reveals no clear reason for the very
low observed
δ
0
TLS
values, though it does suggest that CVD
is a robust process for achieving
δ
0
TLS
<
10
4
with a-Si:H.
A variety of fields employing superconducting devices with
dielectric film capacitors may benefit from this development.
We have also characterized, with varying levels of robust-
ness, the other parameters
δ
other
,
E
c
, and
β
that impact the
quality factor of resonators using these films. We observed
degradation of both
δ
0
TLS
and
δ
other
with time when no special
storage measures are taken, motivating more precautions in
the future.
Lastly, we have demonstrated that the technique of using
frequency-shift data with PPC resonators is a highly robust
way to study TLS loss and that the use of quality-factor data
requires careful attention to the temperature and field values
at which the data are taken. In the latter case, to infer
δ
other
,
E
c
, and
β
—the latter two of which inform us about the micro-
scopic nature of the TLS—a large dynamic range in
|

E
|
must
be probed, covering both the low and higher power plateaus.
Other effects (quasiparticle loss and/or heating) may make
such analyses challenging.
Future work will study the dependence of the above TLS
and non-TLS parameters on deposition parameters in the ma-
chines used for this work, comparisons of these parameters
for resonators fabricated using other machines at other sites
but with efforts made to mimic deposition recipes, correlation
of rf (
1 GHz) and mm-wave/THz (
100–400 GHz) loss,
035602-10