of 7
Journal of Low Temperature Physics manuscript No.
Identifying drivers of energy resolution variation in a
multi-KID phonon-mediated detector
K. Ramanathan
·
T. Aralis
·
R. Basu Thakur
·
B.
Bumble
·
Y.-Y. Chang
·
O. Wen
·
S. R. Golwala
Received: 11/01/2021
Abstract
Phonon-mediated particle detectors employing Kinetic Inductance Detectors (KIDs)
on Silicon substrates have demonstrated both O(10) eV energy resolution and mm position
resolution, making them strong candidates for instrumenting next generation rare-event ex-
periments such as in looking for dark matter and for neutrino measurements. Previous work
has demonstrated the performance of an 80-KID array on a Si wafer, however current energy
resolution measurements show a
25
×
difference between otherwise identical KIDs – be-
tween 5 to 125 eV on energy absorbed by the KID. Here, we use a first principles approach
and attempt to identify the drivers behind the resolution variation. In particular, we analyze a
subset of 8 KIDs using the unique approach of pulsing neighboring KIDs to generate signals
in the target. We tentatively identify differences in quality factor terms as the likely culprit
for the observed variation.
Keywords
Kinetic Inductance Detector (KID), Athermal phonon, Energy resolution, Low
energy detector
1 KID Based Phonon Mediated Detectors
Kinetic Inductance Detectors, as first proposed by Day et al. [1], are excellent devices to
instrument rare-event search experiments due to their low energy threshold, inherent mul-
tiplexability, and straightforward cryogenic RF readout. Previous work by Moore et al. [2]
has demonstrated how patterning 20 KID resonators on a 4 cm
2
Silicon substrate enabled
both
O
(100) eV energy and sub-mm position reconstruction of external radiation incident on
the detector. Interacting particles within the bulk produce an athermal phonon population,
which propagate to the surface KID film and effect a change in the quasiparticle density
of the superconducting material by breaking Cooper pairs. The subsequent modified non-
linear kinetic inductance then modulates the RF transmission properties of the resonator
K. Ramanathan
·
T. Aralis
·
R. Basu Thakur
·
Y.-Y. Chang
·
S. R. Golwala
Division of Physics, Mathematics, & Astronomy, California Institute of Technology, Pasadena, CA 91125,
USA
E-mail: karthikr@caltech.edu
B. Bumble
Jet Propulsion Laboratory, Pasadena, CA 91107, USA
arXiv:2111.02587v1 [physics.ins-det] 4 Nov 2021
2
K. Ramanathan et al.
and by measuring said transmission (and in conjunction with Mattis-Bardeen (M-B) theory)
allows one to work back through the chain to figure out details of the original energy de-
posit. Subsequent work by Chang et al. [3] led to the fabrication of an 80-KID device on
a 75 mm diameter
×
1 mm thick Si substrate, as seen in Fig. 1, operated at 60 mK in an
Oxford Kelvinox 25 dilution refrigerator. This prototype detector couples all KIDs to a sin-
gle 300 nm wide coplanar waveguide (CPW) Niobium (
=
1.5 meV superconducting gap)
feedline. The capacitive and inductive elements are made of Aluminium (
=
200
μ
eV). The
resonators are intentionally over-coupled, with the capacitive quality factor Q
c
smaller than
the intrinsic quality factor
Q
i
so as to allow recovery of the phonon rising edge information.
All KIDs are nominally identical other than small changes in the inductor length to separate
their resonant frequencies by O(MHz) in the 3.05–3.45 GHz band. KID output is fed to a
cryogenic HEMT amplifier with a 3 K noise temperature and data is finally acquired using
an Ettus Research SDR. As designed the expected deposited energy resolution
σ
E
of the
detector is
<
20 eV across all KIDs.
Fig. 1: 80 KID device on a 3 in, 1 mm thick Si. substrate, mounted in a Gold plated Copper
box. Cartoon zoom of a single resonator element, showing 300 nm Niobium feedline cou-
pled to Aluminium KID composed of an interdigitated capacitor and meandering symmetric
coplanar inductor.
2 Energy resolution estimation
The energy resolution can be estimated using a novel in-array technique without use of a
known energy external radiation source. By pulsing a source KID with large readout power,
one creates a non-equilibrium quasiparticle population within the device. Recombination
then generates phonons that propagate out into the substrate. These are then absorbed by
other target KIDs, like a regular particle interaction. Fig. 2 details this process and one can
see the response of neighboring KIDs to a square 20
μ
s square pulse. We can then apply an
optimal filter (OF) estimate for the resolution [4] given by:
σ
2
=
d fJ
(
f
)
|
̃
s
(
f
)
J
(
f
)
|
2
[
d f
|
̃
s
(
f
)
|
2
J
(
f
)
]
2
(1)
Identifying drivers of energy resolution variation in a multi-KID phonon-mediated detector
3
where ̃
s
is Fourier transform of a pulse signal time stream and J is the power spectral den-
sity of a corresponding noise stream. For the analysis presented here we selected a subset
of 8 resonators on the prototype device, chosen to be neighbors in frequency space from
3130–3170 MHz, and took 2 data runs — a series of pulses applying
>
10dBm to neighbor-
ing resonators, and corresponding noise time streams, taken over readout powers ranging
from -25 dBm to -7 dBm. General analysis and processing details can be found in e.g. Ref.
[5]. Applying the OF framework shows large variation in measured energy resolution — the
best performing KIDs hit the design goal of
σ
E
<
20 eV while the worst performing devices
have
σ
E
>
100 eV. This variation is not explicable under an amplifier-limited noise model.
Fig. 2:
Left:
Driving a KID with a square pulse of
>
10 dBm readout power over a 20
μ
s
window (pink shaded region), in this case the 3142 MHz resonator (green line), results in
quasiparticle production and subsequent absorption in neighboring resonators, much like a
regular particle interaction.
Right:
S21 view of different datasets taken using the 3142 MHz
resonator. The indicated pulse in this case was one received by pulsing another neighboring
resonator.
3 Impedance mismatches and measuring quality factors
The actual transmission spectra of real devices, e.g. in Fig. 3
Top
, shows deviations in
the resonance circle from the expected transmission
S
21
=
1
(
Q
r
/
Q
c
)
/
2
jQ
r
x
(with
x
(
f
f
r
)
/
f
r
) at resonance for total quality factor
Q
r
, where
Q
1
r
=
Q
1
c
+
Q
1
i
. These lead to
an asymmetric line shape even at low-power. Khalil et al. [6] attribute this to an impedance
mismatch between the input and output lines of the resonator. They quantify this by intro-
ducing an imaginary component to the coupling quality factor, parametrized by a rotation
angle
φ
and leading to a modified description of the transmission:
S
21
(
f
) =
ae
2
π
j f
τ
[
1
(
Q
r
/
Q
c
cos
φ
)
e
j
φ
1
+
2
jQ
r
x
]
(2)
where we have also included a complex feedline attenuation term
a
, a feedline delay term
τ
,
and f
r
is the resonant frequency. Measuring individual terms in Eq. 2 is achieved by fitting
to the raw transmission spectrum, as seen in Fig. 3
Bottom
, and is potentially degenerate in
solution. A related, but purely empirical way of estimating quality factors is by a) directly
4
K. Ramanathan et al.
Fig. 3:
Top Left:
S21 Transmission across all studied resonators, showing the characteris-
tics dips. Note the uneven overall level and upward spiking features.
Top Right, Bottom:
Resonance circles and transmission spectra of the 3138 MHz resonator before and after the
impedance match corrections outlined by Khalil et al., showing the effect of the
φ
rotation
and scaling by cos(
φ
) (referenced in Eq. 2) to recovering the expected Lorentzian transmis-
sion feature of a resonator.
measuring the diameter of the resonance circle to evaluate
Q
r
/
Q
c
cos
φ
, and b) identifying
the frequency direction of the resonance circle, knowing the readout frequency of each data
point, and estimating
δ
f
r
/
f
r
δ
x
, extracting a quality factor ratio through:
δ
S
21
Q
2
r
Q
c
cos
φ
(
δ
1
Q
i
2
j
δ
f
r
f
r
)
(3)
=
Q
2
r
Q
c
cos
φ
1
2
δ
S
21
δ
x
4 Device Performance and Discussion
We can then use the M-B relations
δ
(
1
/
Q
i
)
ακ
1
n
qp
and
δ
f
r
/
f
r
≈−
ακ
2
n
qp
/
2 [7], where
κ
1
,
2
are temperature sweep derived parameters that help us move to a quasiparticle number
n
qp
basis. So the final prescription is to compute the expected resolution using the raw noise
and pulse data in the electronics basis from Eq. 1 and then use the empirically derived
quality factors and measured transmission line parameters to convert
σ
E
to a quasiparticle
basis. Fig. 4 shows the example of the
κ
1
direction energy resolution for various resonators
at a -16 dBm readout power. As the
Left
plot shows, the resolution appears to be driven by
differences in the quality factor ratio. Accounting for RF transmission and specifics of the
pulse shape and noise spectrum, one can recover the expected linear relationship between
the measured
σ
2
nqp
and its constituent components. Looking at the
Q
c
,
Q
i
, and
φ
terms in
Identifying drivers of energy resolution variation in a multi-KID phonon-mediated detector
5
Fig. 4: Expected energy resolution from measurement in the
κ
1
direction at -16 dBm readout
power, expressed in quasiparticle units, for the various studied resonators.
Left:
Resolution
as related to the empirically measured
Q
2
/
|
Q
c
|
. Differences in this quantity appear to drive
the variation in resolution.
Right:
Accounting for RF transmission, the defined x-axis quan-
tity should be linearly equivalent to the variance of the measured energy and this relationship
is clearly demonstrated in the data.
Fig. 5: Internal quality factor (
Left
, coupling quality factor (
Center
), and impedance mis-
match
φ
(
Right
, for all 8 studied resonators.
turn, as in Fig. 5, we note that the devices that show the greatest deviation from the pack, e.g.
the 3138 MHz resonator, are the same ones that show the largest difference in resolution in
Fig. 4. In addition we notice certain unexpected negative quantities for the coupling quality
factor, though this is offset by a corresponding negative scaling term. Converting from a
quasiparticle basis to a substrate deposited energy resolution, where we include the KID
volume
V
and account for phonon to quasiparticle conversion efficiency via
η
ph
0
.
3 we
6
K. Ramanathan et al.
have that the best and worst performing KIDs have a resolution of:
σ
eV
=
σ
nqp
×
×
V
/
η
ph
(4)
=
σ
best
= (
1
.
3
)
2
×
200
μ
eV
×
30000
μ
m
/
0
.
3
16 eV
σ
worst
= (
2000
)
2
×
200
μ
eV
×
30000
μ
m
/
0
.
3
400 eV
The currently unexplained differences in the quality factors, whether due to impedance
Fig. 6:
Left:
S21 transmission of a 300 nm wide Niobium CPW feedline for different physi-
cal configurations of the Cu box, as measured in a 4K fridge.
Right:
Holder lid with applied
Eccosorb layer held on by cirlex clamps.
matching issues or inherent differences between resonators appear to drive the variation in
measured energy resolution. One hypothesis for the observed behavior is the presence of
box modes, i.e. electromagnetic coupling between the device and its metallic holder box,
sourcing the changing performance. Some support for this hypothesis was established by
measuring the transmission for a feedline only device, as seen in Fig. 6, in different physical
configurations. In the closed box configuration (red line), one observes numerous spectral
line features, indicative of these box modes. Other spectral features are apparent with the
device lid off (green line), but these modes are completely removed after applying a thin
mm layer of Eccosorb dielectric foam absorber (blue line). However there is risk that
this can degrade resonator quality factors, because the Eccosorb can remain at an elevated
temperature and act like a blackbody load on the device.
5 Conclusion and Future Work
In this letter we have highlighted an ongoing concern in deploying large scale KID arrays,
in that the energy resolution across devices is inconsistent even with identical designs. We
used an empirical method to extract the quality factors for the resonators and were able to
reconstruct the measured energy resolution from its constituent components. We briefly dis-
cussed a possible source of this variation as arising from box modes but accurately pinning
it down will require further experimental testing and simulation. Eliminating the resolution
variation between KIDs will be a necessary step towards deploying detectors with
O
(100)
KIDs and realizing the promise of
O
(10) eV energy resolution necessary for future precise
rare-event searches.
Identifying drivers of energy resolution variation in a multi-KID phonon-mediated detector
7
Acknowledgements
We acknowledge the support of the following institutions and grants: NASA, NSTGRO
80NSSC20K1223; Department of Energy, DE-SC0011925F; Fermilab, LDRD Subcontract 672112
References
1. P. K. Day et. al., Nature 425, 817 (2003) DOI:10.1038/nature02037
2. D. C. Moore et. al., Appl. Phys. Lett. 100, 232601 (2012) DOI:10.1063/1.4726279
3. Y.Y. Chang et. al., Journal of Low Temp. Phys. Vol. 193, pp 1199–1205 (2018)
DOI:10.1007/s10909-018-1900-9
4. S. R. Golwala, Dissertation (Ph.D.), University of California at Berkeley (2000)
5. B. Cornell, Dissertation (Ph.D.), California Institute of Technology (2018)
6. M. S. Khalil et. al., Journal of Appl. Phys. 111, 054510 (2012) DOI:10.1063/1.3692073
7. J. Gao, Dissertation (Ph.D.), California Institute of Technology (2008)