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Analysis and calibration techniques for superconducting resonators
Giuseppe Cataldo,1 , a) Edward J. Wollack,1Emily M. Barrentine,1Ari D. Brown,1S. Harvey Moseley,1and
Kongpop U-Yen1
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
A method is proposed and experimentally explored for in-situ calibration of complex transmission data for
superconducting microwave resonators. This cryogenic calibration method accounts for the instrumental
transmission response between the vector network analyzer reference plane and the device calibration plane.
Once calibrated, the observed resonator response is analyzed in detail by two approaches. The first, a
phenomenological model based on physically realizable rational functions, enables the extraction of multiple
resonance frequencies and widths for coupled resonators without explicit specification of the circuit network.
In the second, an ABCD-matrix representation for the distributed transmission line circuit is used to model
the observed response from the characteristic impedance and propagation constant. When used in conjunction
with electromagnetic simulations, the kinetic inductance fraction can be determined with this method with
an accuracy of 2%. Datasets for superconducting microstrip and coplanar-waveguide resonator devices were
investigated and a recovery within 1% of the observed complex transmission amplitude was achieved with
both analysis approaches. The experimental configuration used in microwave characterization of the devices
and self-consistent constraints for the electromagnetic constitutive relations for parameter extraction are also
presented.
I. INTRODUCTION
Recent advances in astrophysical instrumentation have
introduced superconducting microwave resonators for
readout and the detection of infrared light.1–4 An es-
sential element of the successful development of cameras
employing these resonant structures is to accurately and
efficiently characterize their properties. More broadly,
precision measurements of quality factors Q, of super-
conducting resonators are desired in the context of mi-
crowave kinetic inductance detectors (MKIDs) device
physics,5quantum computing6and the determination
of low-temperature material properties.7,8 At sub-kelvin
temperatures the quality factors are generally determined
by measuring and fitting the transmission data over the
relatively narrow bandwidth defined by the response of
a single resonator.9In this setting the fidelity of the ob-
served response can be limited by the intervening instru-
mentation between the device under test and the mea-
surement reference plane.
In physically modeling the electrical response of these
circuits, a lumped-element approximation can be applied
when the electrical size of the elements and their inter-
connections is small compared to a wavelength. This
consideration ensures that the structures are not self-
resonant. More generally, a transmission line approxi-
mation enables one to treat distributed elements with
one dimension greater than a fraction of a wavelength
in scale. However, when the electromagnetic fields and
phases within the circuit elements are uniform and un-
coupled, the lumped and distributed approaches are iden-
tical.10 It is not uncommon for “lumped elements” to
have a significant internal phase delay in practical mi-
crowave devices. For “capacitive” elements synthesized
a)Electronic mail: Giuseppe.Cataldo@NASA.gov
from electrically small transmission line lengths this ef-
fect can be small, but “inductive” elements inherently
possess a finite internal phase delay by design. The util-
ity of this iconic lumped picture primarily resides in its
conceptually simplicity as opposed to the end accuracy
and fidelity of the physical representation.
This being said, analytical methods based on lumped-
element approximations are commonly used to analyze
scattering parameters of resonators and compute their
Qfactors.11–16 These expressions are derived from LCR
circuit representations and are based on the assertion
of weak coupling between resonators to simplify the
analysis. These formulations employ simplified func-
tional forms for the transmission, S21 , and are suited
for approximating the response near resonance for high-
quality-factor lumped circuits.
A detailed review of fitting methods is offered in Ref. 9,
where several methods are presented for fitting the S21
magnitude to different models, neglecting the measured
phase. These methods do not deal with the alterations
that the data incur in the real measurement scenario,
such as crosstalk between cables and/or coupling struc-
tures, the resonator coupling ports not being coincident
with the reference plane of the measurements, and the
presence of and coupling to nearby resonators. Many of
the methods described that attempt to correct for these
effects on S21 remain sensitive to the details of fitting in
the complex plane.9,17,18 In this work we strive to address
these issues by using a transmission line representation to
simultaneously analyze multiple coupled resonators and
present a calibration methodology to remove non-ideal
instrument artifacts.
The resonator’s transmission amplitude and phase can
be characterized with a vector network analyzer (VNA).
Ideally, the VNA calibration reference plane or “test
port” would be directly connected to the device under
test (DUT); however, a cryogenic setting typically ne-
cessitates the use of additional transmission line struc-
2
tures to realize a thermal break between 300 K and the
cold stage (e.g., 0.3 K in this work, see Fig. 1). Other
ancillary microwave components – directional couplers,
circulators, isolators, amplifiers, attenuators, and asso-
ciated interconnecting cables – may be required to ap-
propriately prepare, excite, read out, and bias the de-
vice of interest. The phase velocities and dimensions of
such components change in cooling from room temper-
ature and therefore affect the observed response at the
device calibration plane. These intervening components
between the instrument and the device calibration plane
influence the complex gain and need to be appropriately
accounted for in the interpretation of the measured re-
sponse.
Thermal and practical hardware constraints can pose
significant but achievable calibration challenges at cryo-
genic temperatures. This can impact the data reduction
approach and examples in the literature exist where com-
plex measurements have been reduced to scalar quanti-
ties in the extraction of cryogenic resonator parameters.9
This is neither necessary nor desirable given that the
scattering parameter phase contains important informa-
tion which can be used to constrain the physical model
of the system.
Ideally, a scheme is desired that places the reference
plane of the measurement at the DUT which is identi-
cal to that used during calibration. To approximate this
goal, measurements of the calibration standards on mul-
tiple cool-downs could be used; however, this places con-
straints on the VNA stability and cryogenic instrument
repeatability, which at best are challenging, if not im-
practical, to achieve. To alleviate these concerns one can
place the necessary standards in the cooled test environ-
ment and switch between them. Variations on this theme
can be found in microwave probe stations, where the sam-
ple and calibration standards are cooled to 4 K by a con-
tinuous liquid helium flow or a closed-cycle refrigeration
system.19 This approach has been demonstrated at mil-
likelvin temperatures but necessitates the availability of a
low-power, high-performance microwave switch in the de-
sired test band.20 This three-standard Thru-Reflect-Line
calibration approach was expanded to include error cor-
rection for in-situ superconducting microwave resonator
characterization.21
For our metrology application, the measurement of the
circuit’s complex transmission, S21, is needed. A flexible
and in-situ calibration procedure is desired for multiple
resonators over a frequency range wide compared to the
resonator line width and inter-resonator spacing. To ex-
plain the approach presented here, the reader is guided
through a detailed description of the experimental set-up
(Section II), the motivation behind the transmission mea-
surement metrology (Section III), the calibration process
(Section IV), and the derivation of the resonator parame-
ters for two differing device datasets (Section V). Finally,
an ABCD-matrix fitting approach is used in extracting
transmission line materials properties and is described in
Section VI.
II. EXPERIMENTAL SET-UP
We present datasets for the detailed characterization
of two devices, the first with 2 resonators made on mi-
crostrip transmission lines and the second with 14 res-
onators in a coplanar waveguide (CPW) configuration.
The experimental apparatus used for the measurements
is depicted in Fig. 1. We describe in detail the first de-
vice to demonstrate the technique and then report on
the second device which contains multiple coupled res-
onators of varying quality factors as a demonstration of
the algorithm’s inherent robustness.
Figure 2a shows the packaged 2-resonator test device.
Here, a superconducting niobium (Nb) CPW feedline is
coupled to two microstrip stepped-impedance supercon-
ducting molybdenum-nitride (Mo2N) resonators.22 This
device was mounted inside a gold-plated copper (Cu)
package. Aluminum wire bonds were used to connect
the superconducting Nb CPW feedline to a Cu CPW
line on a printed circuit transition board. These CPW
lines were connected to Sub-Miniature version A (SMA)
coaxial connectors whose center-pins were soldered to the
printed circuit board. The packaged device was attached
to the cryostat cold stage and cooled to 330 mK in a
Helium-3 cryostat.
A series of coaxial cables and components routed the
device input and output lines from 330 mK to an Agilent
N5242A PNA-X network analyzer at room temperature.
A Weinschel Model-4M SMA attenuator (α3'7 dB)
with a measured return loss ≥20 dB from 2 to 4 GHz at
2 K was placed directly at the input of the device pack-
age to provide a matched termination for the test package
input. Improved signal-to-noise can be achieved through
the use of impedance-matched thermal blocking filters23
for realization of the terminations for the DUT. This con-
figuration would also be desirable or necessary for the
characterization of devices with low saturation power.
The output signal was routed via copper and supercon-
ducting niobium-titanium (NbTi) coaxial cables through
a CTD1229K PamTech circulator, which was mounted
on the 2-K stage of the cryostat. This circulator oper-
ates with a return loss ≥20 dB between 2.5-4 GHz. From
the isolator the output signal was routed to an AF S3-
02000400-08-CR-4 MITEQ amplifier mounted on the 2-K
stage.
III. MEASUREMENT METROLOGY MOTIVATION
We define the response at the “VNA reference plane”
as observed through a microwave path to the “device
calibration plane” as the instrument baseline (Fig. 1). In
a perfectly impedance-matched and lossless instrument,
the magnitude of the complex transmission amplitude,
S21, for a thru connection would be unity. In practice,
the instrument baseline has frequency structure and am-
plitude set by the details and quality of the intervening
transmission line components. These instrumental arti-
3
FIG. 1: Schematic of the experimental apparatus and
devices under test. The resonators are coupled to a
CPW feedline through coupling capacitors and their
response is observed through a VNA connected to the
DUT. Summary of the attenuators employed in each
measurement can be found in the table. The circulator
indicated by a * was only present for the 2-resonator
test configuration.
facts need to be addressed in calibration to provide an
unbiased measurement of the device response.
A Short-Open-Line-Thru (SOLT) calibration was used
to calibrate the PNA-X in 3.5-mm coax at the VNA ref-
erence plane defined in Fig. 1. After recording the com-
plex transmission measurements, an in-situ calibration
process was used in order to move the reference plane
to the device calibration plane. In the calibration pro-
cess adopted here we assume that the test device sees
a matched termination. For the 2-resonator data this
condition is realized by a 7-dB attenuator and isolator.
For the baseline removal approach described here to be
successful, the response of the attenuators, transmission
line media and amplifiers used in the system should be
smooth functions over the spectral band of interest rela-
tive to the test-device frequency response.
It is notable that in this implementation the feed trans-
mission line structure in the device under test plays a
dual role: it allows excitation and read-out of the sharply
peaked circuit response of interest, and off resonance
serves as a controlled impedance reference (thru line)
standard to specify the system’s gain as a function of
frequency. The underlying approach explored in this
work for baseline removal has its roots in a variety of
analogous instruments such as frequency-swept reflec-
tometry25 and spectral baseline correction in radio as-
tronomy.26,27 In particular, the differential reflectometer
configuration used for cavity characterization28 provides
insight in this context. In this method, the baseline is
explicitly removed on resonance by comparing the re-
sponse of a waveguide cavity with that of a short-circuit
via the difference port of a magic-T power divider. Ad-
justment of the short position in this bridge reflectometer
circuit reveals the interaction between the resonator and
the baseline shape. More importantly, this points to the
role of the resonator phase and baseline shape on the
observed scattering parameters at resonance.
IV. VNA TRANSMISSION DATA CALIBRATION
For a single resonator or multiple well-separated res-
onators, the response can be well reproduced by a
Lorentzian line shape. In the presence of multiple res-
onances, their mutual interaction as well as the interac-
tion with the continuum can result in a Fano spectral
response.29–33 This effect can also be experimentally ob-
served as an interaction between the resonators with the
relatively broad Fabry-Perot resonances resulting from
standing waves in the system. Such reflections produce
the dominant spectral variation in the observed instru-
ment baseline and can be mitigated by minimizing the
transmission line lengths and suitably terminating the re-
flections with matched attenuators and circulators at the
instrument termination plane. In calibrating the spectra,
an unbiased removal of these artifacts is desired.
As an example of the calibration process, we use
the data from the CPW feedline coupled to the two
molybdenum-nitride (Mo2N) resonators. Figure 3a
shows the real and imaginary components of the trans-
mission, S21, at the VNA reference plane as a function of
frequency. It can be seen that the characteristic scale of
the baseline variations is much larger than the resonator
response in frequency. To calibrate the VNA data in-situ,
the following steps are performed:
1. fit of the complex baseline (Fig. 3c);
2. normalization of the transmission’s real and imag-
inary parts (Fig. 3c-d);
3. correction for variations in gain and relocation of
the reference plane at the DUT by ratioing out the
complex baseline fit (Fig. 3e-f).
In general, the baseline can have an unpredictable
shape as determined by the details of the reflections oc-
curring throughout the instrument and its components.
The complex baseline, S21,bas, was modeled analytically
4
(a) Resonator chip, printed circuit
CPW transition board, and SMA
connector interface mounted in the
test package. The package cover has
been removed for clarity.
(b) Left: Layout for the 2-resonator chip. The stepped-impedance resonators are
coupled to a CPW feedline and are realized from low- and high-impedance
microstrip transmission lines. The “ground bridge” prevents excitation of the
asymmetric slotline mode on the CPW feedline structure. The 4-µm Nb
microstrip line uses SiO2as a dielectric insulator (0.1-µm thick) from the Mo2N
layer. This transmission line structure is not explicitly used at microwave
frequencies but enables millimeter wave coupling to a dual-slot antenna.22 Right:
A simplified cross-sectional view of the coplanar waveguide (top) and microstrip
transmission line (bottom) geometries used in modeling the electromagnetic
response. The permittivity for the dielectric layers and the permeability for the
lines are indicated. The arrows indicate the electric field and its dominant modal
symmetry. The microwave dielectric substrate (0.450±0.025-µm thick, 100-mm
diameter) is monocrystalline silicon.24
FIG. 2: Details of the 2-resonator device and packaging
and a Fourier series was found to be a convenient and
physically motivated representation of its response:
S21,bas =
n
X
j=1
Aj·exp (idjω).(1)
In Eq. (1) Ajis a complex coefficient, dja time delay and
ωthe angular frequency. In the case presented, n= 4
terms were found to be sufficient to adequately sample
the baseline and appropriately constrain its properties.
A fit to the 30,000 measured data points resulted in a
reduced χ2= 0.9973. Differences in experimental con-
figuration or changes in the desired calibration spectral
range may lead to alternative forms for Eq. (1).
The second step in data calibration consists of uniquely
specifying the complex gain amplitude between the VNA
reference plane and the device calibration plane so that
the S21 real and imaginary components in Fig. 3a lie be-
tween ±1. This normalization factor was found to be
approximately 6.32 by forcing the transmission ampli-
tude to be equal to unity far from the resonator response.
The normalized measurements and modeled data are pre-
sented in the Smith chart in Fig. 3d.
Finally, the complex baseline, S21,bas , was removed
from the data through Eq. (1) to eliminate the influ-
ence of reflections and move the reference plane to the
DUT. By looking at the topology of the instrument,
the error matrices introduced by the connections be-
tween the reference plane and the device under test in
the cryostat are in series. From the properties of sig-
nal flow graphs,25,34 the uncalibrated transmission coef-
ficient, S21, at the VNA reference plane is therefore the
product of the following scattering matrix elements:
S21 =S21,bas ·S21,res,(2)
where S21,res refers to the desired calibrated response of
the resonator device under test.
V. PHENOMENOLOGICAL RESONATOR MODEL
With the baseline correction applied, the calibrated
data (Figure 3e) were modeled in order to extract the
characteristic resonant frequencies and line widths. The
feedline and resonator model for the packaged device can
be represented as a realizable causal filter as follows:
S21,res = 1 +
M
X
j=1
a0,j +a1,j xj+a2,j x2
j+. . .
1 + ixj+b2,j x2
j+. . . ,(3)
where, for each j= 1, ..., M (Mbeing the total num-
ber of resonators), xj=Qtot,j ·(ω/ωo,j −ωo,j /ω) and
5
FIG. 3: Measurement calibration overview. a) Complex S21 data for the DUT as observed at the VNA reference
plane. b) From the perspective of the DUT, the magnitude of the response is uncalibrated due to the amplifiers,
attenuators, and other microwave elements in the system. c) The transmission spectrum is normalized by forcing its
amplitude to be equal to unity far from the resonator responses. d) The normalized transmission data are shown on
a Smith chart. The baseline variation with frequency can be seen as a deviation in the S21 from unity far from the
resonator response. e) After removing the complex baseline and specifying the device-calibration-plane phase, the
calibrated data are used to fit an analytical model, and the residuals are shown. f ) The Smith chart shows the two
resonance loops going through (1,0) after calibration.
6
Qtot,j is the total loaded quality factor defined as Qtot,j =
ωo,j /Γj. For each resonator, therefore, the fitting pa-
rameters are: the ak,j and bk ,j complex coefficients (k=
0,1,2, ...,j= 1, ..., M ), the resonance frequency ωo,j, and
the full width at half power Γj. Here we use k= 2 in
specifying the order of the polynomial.
The functional form of Eq. (3) exhibits several features.
First, its second-order terms in both the numerators and
denominators allow for reproduction of the resonator re-
sponses, thereby enabling the representation of a physi-
cally realizable, distributed circuit network.35 Second, its
causality is assured by the degree of the numerator being
not greater than that of the denominator, which allows
the functional form to satisfy the Kramers-Kronig rela-
tions.36 Far removed from the resonators, S21,res →1+i0
and the transmission represents an ideal thru line. It
was verified that our second-order model approached this
limit without additional terms to avoid increasing the
number of parameters and the computational effort. Fi-
nally, this functional form enables one to simultaneously
fit any number of resonators while formally taking into
account their physical interactions.
Rewriting xjin the equivalent form, xj=Qtot ·
(ωo/ωj)(ωj/ωo+ 1) ·(ωj/ωo−1), reveals a commonly
employed approximation, xj≈2Qtot ·(ωj/ωo−1), near
resonance. This form can be viewed as an asymptotic ex-
pansion in powers of 1/Qtot which modifies the position
of the resonant frequency in the complex plane.37 When
Eq. (3) is evaluated in this limit the expressions reduce to
the first-order lumped-circuit treatments described in the
literature for a single resonator in transmission.11,12,38
However, we find that to faithfully reproduce the interac-
tions between resonators over a wide spectral bandwidth
the full functional form presented here is required.
A. Analysis of the 2-resonator dataset
Equation (3) was used to fit the calibrated data by
means of a least-squares curve fitting routine based on a
trust-region reflective Newton method.39,40 Constraints
were given in the form of global lower and upper bounds
for each Γ >0 and ωo. The starting guess for each Γ
was chosen close to the actual value, which can be read-
ily determined by estimating each resonator’s spectral
width. The initial guess for the resonance frequency,
on the other hand, corresponds to the frequency val-
ues where the observed S21 has minima. The results are
shown in Fig. 3e (black line) for both components. The
residuals on the real and imaginary parts are comparable
to the normally-distributed noise level in the measured
data with a reduced χ2= 0.9987 and a standard devia-
tion of σ= 0.0155. A Smith chart is provided in Fig. 3f,
which shows that the two circles touch each other in (1,0)
with a relative rotation of about 7◦caused by the phase
delay in the feedline length between the two resonators.
Because of the noise in the measured data and the resid-
ual systematic errors in the baseline fit, the data around
(1,0) converge to this point within a radius of 0.04. The
estimated values for the fitting parameters of each res-
onator are summarized in Table I.
The internal and coupling Qfactors were calculated
using the following equations:41,42
Qi,j =Qtot,j
1−Dj
,(4)
Qc,j =Qtot,j
Dj
,(5)
where Djrepresents the diameter of the circle associated
with each resonator (Fig. 3f). The Q-factor values are
shown in Table II.
B. Analysis of the 14-resonator dataset
A 14-resonator CPW dataset was also studied (see
Fig. 4a-b). In this example, the resonators span a spec-
tral range of about 45 MHz with two pairs of resonators
strongly interacting with each other, namely resonators
5-6 and 9-10. The analysis of this dataset was challeng-
ing because, for high values of Qtot, the denominator of
Eq. (3) approaches an indeterminate form of the type
[0 · ∞], which can result in numerical instability. In addi-
tion, the parameters of interest span 6 orders of magni-
tude leading to an ill-conditioned Hessian matrix for the
system.
Equation (3) was used to fit the calibrated data by
means of a least-squares curve fitting routine based on
a trust-region reflective Newton method with a diago-
nal preconditioning of the conjugate gradient. To help
the algorithm with convergence, all Γjand ωo,j (j=
1,...,14) were provided with lower and upper bounds.
Namely, 0 <Γj≤Γmax and ωmin ≤ωo,j ≤ωo,max
for j= 1,...,14. Here, Γmax = 10 MHz, well above
the largest Γ1(≈1.6 MHz). The minimum angular fre-
quency is ωmin = 2.5040 GHz and the maximum reso-
nance frequency, ωo,max, was set not to exceed a distance
of 10 ×Γ14 from the last resonance frequency. For the
highest-Qresonators (i.e., the highest frequencies in the
recorded transmission spectra) the model is more sensi-
tive to changes in ωo,j. This choice for the bounds helped
the algorithm in identifying solutions within the desired
range of parameters and converged after <300 iterations.
The model represented by Eq. (3) recovers the cal-
ibrated data within an accuracy of <1% (Fig. 4c-d)
with a reduced χ2= 0.9985 and a standard deviation
σ= 0.0027, thus proving to be a robust method to an-
alyze numerous, strongly coupled resonators. In this
dataset, the range of quality factors spans more than
three orders of magnitude, e.g., 2,300 < Qc<630,000.
A particular challenge noted in this existing dataset was
that the observed spectral range was suboptimal. From
an experimental design perspective, the phase variation
of the DUT and the baseline need to be appropriately
sampled and to possess differing spectral signatures to
enable independent reconstruction from the measured
7
TABLE I: Parameter summary for the 2-resonator analytical modela
j a0,j a1,j a2,j b2,j ωo,j /(2π) Γj/(2π)
[−] [−] [−] [−] [−] [GHz] [kHz]
1−0.7345 + i0.1029 −0.0440 + i0.0879 −0.0008 −i0.0009 −0.0120 −i0.0048 2.9121 . . . ±7×10−9139.02 ±0.01
2−0.5496 −i0.0463 −0.4737 + i2.1868 −2.1104 −i2.4638 30.0366 −i13.2458 2.9670 . . . ±60 ×10−950 ±2
aIn the analysis double precision (16 digits) was used; however, for clarity here we only show the leading digits.
TABLE II: Qfactors for the 2-resonator analytical
model
j Qtot,j Qi,j Qc,j
[−] [−] [−] [−]
1 20,948 ±1 107,583 ±2 26,013 ±1
2 59,400 ±180 109,500 ±330 129,800 ±390
dataset. By using the modeled response as noiseless in-
put to the fitting algorithm, it was found that the spec-
tral range of this dataset should have been extended by
30% to minimize the calibration error with this algo-
rithm. This increase in spectral range would enable the
responses of the resonator wings and the baseline to be
sufficiently decoupled to achieve an unbiased amplitude
calibration.
VI. ABCD-MATRIX MODEL
The phenomenological resonator model based on
Eq. (3) and discussed in the previous section is useful
and sufficient if one needs to know the center frequen-
cies and the resonators’ Qfactors. Clearly, there are
many quasi-TEM structures such as microstrip, CPW, or
other transmission line types which can lead to the ob-
served response. Since the phenomenological resonator
model is blind to the details of the realization, to probe
the internal structure of the circuit a distributed trans-
mission line representation needs to be employed. Here,
the impedance, propagation constant, and lengths are
used to characterize the circuit response in an ABCD or
chain-matrix formulation. In this process, specifically,
the impedance and propagation constant are used as pa-
rameters to fit the transmission data.
Although coupling to higher-order modes (e.g., losses
to radiation, surface waves, and parasitic coupling be-
tween elements) can influence the underlying network
topology, in the formulation used here we assume single-
mode propagation and rely upon the specification of
the circuit network. This process is related to “de-
embedding” or the process of inferring the response of
a device under test when electrical properties of the in-
tervening structure (in this case, the feedline coupling
structure) are known.43 Elegant implementations of such
concepts in the context of scalar millimeter-wave circuit
metrology are described in Refs. 44 and 45.
To model the 2-resonator device response below the
instrument termination plane (Fig. 1), 29 transmission
line elements were used. For each line section the
physical lengths, l, were known to a micron accuracy
from the photolithographic mask, while the character-
istic impedances, Zo, and the propagation constant, γ,
were simulated with HFSS (High Frequency Structure
Simulator). For simplicity, the two coupling capacitors
were modeled as lumped elements and their capacitances
also simulated with HFSS (Cc1≈53 fF and Cc2≈23 fF).
The detailed description of the ABCD-matrix model can
be found in Appendix A.
Substituting the numerical values of each transmis-
sion line element into Eq. (A4) and cascading through
Eq. (A3) yields the ABCD parameters of the entire trans-
mission line circuit. These parameters were used in
Eq. (A2) to evaluate the modeled transmission response,
S21. A challenge of this method lies in the difficulty of
fully specifying the elements of the system at cryogenic
temperatures. To simplify the extent of the system to
be characterized, the instrument calibration plane was
chosen to approximate a matched termination as close as
feasible to the packaged device under test.
In addition, as a cross-check of the baseline correction
obtained by fitting, the elements leading to this instru-
mental response were computed with the ABCD-matrix
model and agreed to within a 5% accuracy with the mea-
sured transmission line coaxial-cable lengths given nomi-
nal literature values for the teflon dielectric. Experimen-
tal investigation revealed that the superconducting NbTi
thermal break cable was the limiting element in the un-
corrected baseline response. In fitting, it was found that
the characteristic impedance of this line was ≈60 Ω. This
section of cable was subsequently inspected and a gap in
the teflon dielectric was noted to be consistent with these
observations.
A. Transmission line parameter extraction
For the transmission line structures the characteristic
impedance, Zo, and propagation constant, γ, are func-
tions of the effective permittivity, εr,eff, and permeability,
µr,eff, in the medium. In particular, Zois a function of the
transmission line geometry proportional to the relative
wave impedance in the medium, Zn≡(µr,eff/εr,eff)1/2,
8
FIG. 4: Results for the 14-resonator CPW dataset. a) Normalized transmission data. b) Smith chart representation
of the normalized transmission data. c) Calibrated transmission data and analytical model. The achieved residuals
are smaller than 1%. d) Smith chart showing the calibrated 14-resonator data wrapping around (1,0). Note: The
14-resonator CPW dataset was acquired in two discrete sections and a small gap in the measured transmission
spectra is present at ≈2.535 GHz.
and γis defined such that:
γ2≡ − ω
c2
·(εr,eff ·µr,eff) = −ω
c2
·n2
eff,(6)
where ωis the angular frequency, cthe speed of light in
vacuum, and neff is the effective index in the medium.
For microstrip, the effective permittivity and permeabil-
ity are explicitly defined as:
εr,eff ≡C(ε1, ε2)
C(εo, εo), µr,eff ≡L(µ1, µ2)
L(µo, µo),(7)
where the capacitance, C, and the inductance, L, per unit
length are measures of the electromagnetic energy stored
in the transmission line.46,47 To find the effective con-
stitutive relations, these functions are evaluated in the
presence and absence of the dielectric and superconduct-
ing media, respectively. Here, ε2refers to the monocrys-
talline silicon substrate, ε1to the superstrate dielectric,
µ1to the centerline, µ2to the ground plane metallization,
and εoand µoare the permittivity and permeability of
free space (see Fig. 2b). For other transmission line struc-
tures, analogous formulations of Eq. (7) can be defined
for the constitutive parameters.
This parameterization results in εr,eff and µr,eff becom-
ing implicit functions of the line’s materials properties
9
and cross-sectional geometry. The imaginary component
of µr,eff was taken as zero. An effective dielectric loss tan-
gent ≈1.8×10−5was observed in fitting the observed
microstrip resonator spectra. This is consistent with the
bound on the dielectric loss tangent, <5×10−5, at
2.2 K for a similar (1.45-µm-thick) silicon sample used
as a Nb microstrip ring resonator at 4.7 GHz. As dis-
cussed in Ref. 48, upon cooling to cryogenic tempera-
tures the bulk resistivity silicon (ρ≥1 kΩ −cm for these
samples) freezes out and the intrinsic loss mechanisms in
silicon dominate at microwave frequencies. The achiev-
able thicknesses and high uniformity (±0.013 µm) of the
monocrystalline silicon wafers24 employed in this inves-
tigation enable its use as an ultra-low-loss controlled-
impedance microwave substrate. The observed loss in
the circuit can have contributions from the bulk proper-
ties of the silicon layers, the bisbenzocyclobutene (BCB)
wafer bonding agent, two-level systems in the silicon ox-
ide layer, and conversion to surface and radiation modes.
In addition, residual coupling to the electromagnetic en-
vironment could contribute to the observed loss, in par-
ticular for higher Qfactors. Although a detailed break-
down of the contributions to the observed loss is possible
by investigating the performance as a function of geome-
try and temperature, this was not attempted here. Thus,
in ascribing the entire observed loss to the dielectric, the
above-mentioned loss tangent represents a conservative
upper bound to that of monocrystalline silicon.
In fitting, for each transmission line type (high or low)
a global characteristic impedance and phase velocity were
employed. The low-impedance lines have a width of
60 µm, whereas the high-impedance lines have a width
of 18 µm. Electromagnetic simulations were performed
with HFSS for the CPW and microstrip configurations
shown in Fig. 2b. A study of the boundary conditions
for the 18-µm line width is summarized in Table III,
whose last column represents the configuration used in
the device under test. To incorporate the effects of the
kinetic inductance in the simulation, a surface inductance
per square, L
k, was introduced.49 Waveports matched
to the transmission line impedance and an S-parameter
amplitude convergence ∆S < 0.01 were employed in the
finite-element simulations. By comparing to analytical
expressions for microstrip geometries, a conservative sys-
tematic error of <2% was estimated for the simulated
parameters derived from HFSS. Similar estimates arise
from consideration of the magnitude of the constitutive
relations derived from the simulations in the limit where
the dielectric materials are replaced by free space and
metallization layers are perfect-electric field boundaries.
To study the resonator circuit’s response in detail, a
fit of the S21,res data depicted in Fig. 3e to Eq. (A2) was
performed. The fitting parameters were Zo,neff, and Cc
for each of the two resonators’ transmission line sections.
From Zo, the medium’s relative wave impedance, Zn, can
be determined as follows:
Zn≡µr,eff
εr,eff 1/2
=Zo
Zo,HFSS
(8)
for each of the transmission lines used in the resonator
realization. Here, Zo,HFSS represents the transmission
line characteristic impedance simulated using HFSS with
perfect E-field boundaries for conductors and the permit-
tivity of free space for dielectrics.
From Eq. (6) and Eq. (8) it follows that the effective
permittivity and permeability are related by
εr,eff =neff/Zn, µr,eff =neff ·Zn.(9)
These scaling relationships between εr,eff and µr,eff were
enforced by the algorithm during fitting. The resulting
transmission response computed with this methodology
[see Eq. (A2)] agrees with the measured data within 1%,
similarly to the accuracy previously found through the
evaluation of Eq. (3). This analysis approach is analo-
gous to that implemented in Ref. 50 to explicitly link the
transmission line parameters which depend upon the de-
tailed sample geometry to the interaction of the fields in a
homogeneous bulk material sample in an electromagneti-
cally self-consistent manner. For example, a non-physical
dielectric function would be encountered if µr,eff ≡1 was
tacitly adopted for the superconducting material.51,52
A parameter of interest for superconducting resonator-
based detectors is the kinetic inductance fraction,53 α,
which is defined as the ratio of the transmission line’s
kinetic inductance, L(µ1, µ2)−L(µo, µo), relative to the
total inductance, L(µ1, µ2). With these definitions and
through the use of Eq. (7), one obtains:
α= 1 −1
µr,eff
.(10)
The results found for αare in quantitative agreement
with the values derived from the HFSS simulations (Ta-
ble III) and can be found in Table IV.
VII. CONCLUSIONS
This work presented a methodology to calibrate the
transmission response of superconducting microwave res-
onators at cryogenic temperatures and described two
methods to analyze the resulting dataset. The phe-
nomenological and ABCD-matrix methods recovered the
measured transmission data with the same 1% level of
accuracy, thus providing a numerical validation of the
general approach. The derived Qfactors in either ap-
proach were statistically indistinguishable.
It is interesting to compare the number of parameters
required by each method. The first analysis method em-
ploys a rational polynomial function of degree kwith 2k
complex and 2 real parameters for each resonator. A
second-order function with 6 parameters per resonator
was shown to recover the measured data to within a 1%
10
TABLE III: HFSS simulations – Superconducting microstrip transmission line (18-µm line width) a
Transmission line conductor, model definition Perfect-E Perfect-E Nb Mo2N
Line surface inductance, L
k(line) – – 0.12 4.3 [pH/]
Line metallization thickness – – 0.25 0.07 [µm]
Ground plane conductor, model definition Perfect-E Perfect-E Nb Nb
Ground plane surface inductance, L
k(ground) – – 0.12 0.13 [pH/]
Ground plane metallization thickness – – 0.50 0.50 [µm]
Superstrate relative permittivity, ε1/εo1.000 1.000 1.000 1.000 [–]
Substrate relative permittivityb,ε2/εo1.000 11.55 11.55 11.55 [–]
Transmission line impedance, Zo,HFSS 8.5(0) 2.6(1) 3.1(1) 7.8(4) [Ω]
Relative wave impedance, Zn= (µr,eff/εr,eff)1/21.00(0)c0.307 0.366 0.922 [–]
Effective index, neff = (εr,eff ·µr,eff)1/21.00(2) 3.293 3.999 10.67 [–]
Effective relative permittivity, εr,eff =neff/Zn1.00(2) 10.73 10.93 11.57 [–]
Effective relative permeability, µr,eff =neff ·Zn1.00(2) 1.0(1) 1.463 9.844 [–]
Kinetic inductance fraction, α= 1-1/µr,eff 0.00(2) 0.0(1) 0.317 0.898 [–]
aThe upper portion of the table contains the simulation input parameters and the lower portion the computed and derived results.
bFor the simulations presented, the input dielectric function for silicon was specified as lossless for simplicity.
cThe last significant digit computed in HFSS simulations is indicated with parentheses for selected entries. Deviation from zero is
suggestive of the magnitude of the systematic error encountered in simulation of the transmission line structures.
TABLE IV: Transmission line parameter extraction –
2-resonator ABCD-matrix model
Line Zo,HFSS Zoneff α
[Ω] [Ω] [−] [−]
1, low 2.7(0)a2.47 ±0.01 10.697 ±0.571 0.898 ±0.008
1, high 8.5(0) 7.70 ±0.10 11.143 ±0.336 0.901 ±0.007
2, low 2.7(0) 2.47 ±0.01 10.701 ±0.162 0.898 ±0.004
2, high 8.5(0) 7.70 ±0.03 11.147 ±0.084 0.901 ±0.003
aThe last significant digit computed in HFSS simulations is
indicated with parentheses.
accuracy. The second method, based upon an analysis of
the circuit’s distributed network, was implemented via
cascaded ABCD matrices. Here, 2 complex and 3 real
parameters were necessary to compute the ABCD ma-
trices of each transmission line section used to represent
the resonator structure and the feedline. In addition, the
line lengths were taken as known and the constraint be-
tween the wave impedance and the effective index was
enforced in fitting. Broadly speaking the two approaches
have comparable underlying complexity.
While the phenomenological model is useful in pro-
viding the values of the resonators’ central frequencies
and widths, the ABCD-matrix method provides insight
into the circuit’s internal structure when used in con-
junction with electromagnetic simulation tools, to make
an explicit linkage between the model and the underlying
geometric details of the transmission media in use. This
approach can be of particular value in extracting the line
parameters required for precision circuit design. More
importantly, the ABCD-matrix model naturally enables
the treatment of distributed transmission line structures
without the approximations typically employed in con-
ventional lumped-circuit analysis. The accuracy of these
analysis methods exceeds that of a simple lumped-circuit
approximation over the frequency span and parameter
range experimentally explored.
ACKNOWLEDGMENTS
We acknowledge financial support from the NASA
ROSES/APRA program and the Massachusetts Insti-
tute of Technology “Arthur Gelb” fellowship. We would
like to thank Amil Patel for sample fabrication as well
as David Chuss, Negar Ehsan, Omid Noroozian, Jack
Sadleir, and Thomas Stevenson for helpful conversations
and contributions to this work.
Appendix A
The ABCD matrix for a two-port network is defined in
terms of the total voltages and currents as follows:34,54
"Vin
Iin #="A B
C D #"Vout
Iout #.(A1)
The scattering parameter S21, also known as the trans-
mission amplitude, t, can be directly related to the
ABCD parameters as:54
S21 =2Zl
AZl+B+CZlZs+DZs
,(A2)
11
where Zland Zsrepresent the load and source
impedances, respectively. The ABCD matrix of the cas-
cade connection of multiple networks is equal to the prod-
uct of the ABCD matrices representing the individual
two-ports, that is
"A B
C D #=Y
i"AiBi
CiDi#i
.(A3)
Here, the subscript iis used to indicate the different ma-
trices as well as the ABCD parameters of each matrix.
In particular, the ABCD parameters of a two-port cir-
cuit represented by a transmission line are:34,54
"A B
C D #T L
="cosh(γl)Zo·sinh(γ l)
1/Zo·sinh(γl) cosh(γl)#.(A4)
In Eq. (A4), Zois the characteristic impedance, γthe
propagation constant, and lthe line length. The deter-
minant of this matrix is unity.
Each resonator is made of a low-, high-, and low-
impedance transmission line connected to a coupling ca-
pacitor (Fig. 2b). The resonator’s low-impedance trans-
mission line section termination was modeled as an open-
circuited line, i.e., Zin,1=Z1coth γ1l1, while the high-
and second low-impedance transmission lines were spec-
ified by34
Zin,j =Zj
Zin,j−1cosh(γjlj) + Zjsinh(γjlj)
Zin,j−1sinh(γjlj) + Zjcosh(γjlj), j = 2,3.
(A5)
Finally, the total impedance as seen through the coupling
capacitors is Ztot = 1/(iωCc) + Zin,3. Here, the index j
specifies the internal lines in the stepped-impedance res-
onator. The ABCD matrix associated with the complete
resonator structure is therefore:
"A B
C D #res
="1 0
1/Ztot 1#.(A6)
The individual resonators are connected to the feedline
via a T-junction at their respective electrical delays. The
following expression,
"A B
C D #T−junc
="A B
C D #T L "A B
C D #res "A B
C D #T L
,
(A7)
allows the resonators to be cascaded with the feedline
sections, whose ABCD matrices are expressed through
Eq. (A4). The resulting two-port circuit is used to
compute the frequency response of the chip. The wire
bonds, transition board, connectors, and coaxial cables
between the device reference and the instrument termi-
nation planes were added in a similar fashion to inves-
tigate the baseline response. For simplicity we do not
explicitly treat the parasitic reactance at the transmis-
sion line junctions; however, this detail can readily be in-
corporated in the formulation and can be of importance
at higher frequencies. The complex phase velocity and
kinetic inductance fraction of the transmission line con-
figuration employed can be extracted from the response
of this composite ABCD-matrix for the system.
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