# Ultra-sensitive Super-THz Microwave Kinetic Inductance Detectors for future
space telescopes

J.J.A. Baselmans<EMAIL_ADDRESS>THz Sensing group, Microelectronics
Department, Delft University of Technology, Mekelweg 1, 2628 CD - Delft, The
Netherlands  SRON - Netherlands Institute for Space Research, Niels Bohrweg 4
2333 CA - Leiden, The Netherlands  F. Facchin  THz Sensing group,
Microelectronics Department, Delft University of Technology, Mekelweg 1, 2628
CD - Delft, The Netherlands  A. Pascual Laguna  SRON - Netherlands Institute
for Space Research, Niels Bohrweg 4 2333 CA - Leiden, The Netherlands  J.
Bueno  SRON - Netherlands Institute for Space Research, Niels Bohrweg 4 2333
CA - Leiden, The Netherlands  D.J. Thoen  THz Sensing group, Microelectronics
Department, Delft University of Technology, Mekelweg 1, 2628 CD - Delft, The
Netherlands  V. Murugesan  SRON - Netherlands Institute for Space Research,
Niels Bohrweg 4 2333 CA - Leiden, The Netherlands  N. Llombart  THz Sensing
group, Microelectronics Department, Delft University of Technology, Mekelweg
1, 2628 CD - Delft, The Netherlands  P. de Visser  SRON - Netherlands
Institute for Space Research, Niels Bohrweg 4 2333 CA - Leiden, The
Netherlands  THz Sensing group, Microelectronics Department, Delft University
of Technology, Mekelweg 1, 2628 CD - Delft, The Netherlands

###### Abstract

Future actively cooled space-borne observatories for the far-infrared, loosely
defined as a 1–10 THz band, can potentially reach a sensitivity limited only
by background radiation from the Universe. This will result in an increase in
observing speed of many orders of magnitude. A spectroscopic instrument on
such an observatory requires large arrays of detectors with a sensitivity
expressed as a noise equivalent power NEP$\>=\>3\>\times
10^{-20}\>\mathrm{\;W/\sqrt{\mathrm{Hz}}}$. We present the design,
fabrication, and characterisation of microwave kinetic inductance detectors
(MKIDs) for this frequency range reaching the required sensitivity. The
devices are based on thin-film NbTiN resonators which use lens-antenna
coupling to a submicron-width aluminium transmission line at the shorted end
of the resonator where the radiation is absorbed. We optimised the MKID
geometry for a low NEP by using a small aluminium volume of $\approx$
1$\mathrm{\;\mu m}^{3}$ and fabricating the aluminium section on a very thin
(100 nm) SiN membrane. Both methods of optimisation also reduce the effect of
excess noise by increasing the responsivity of the device, which is further
increased by reducing the parasitic geometrical inductance of the resonator.
We measure the sensitivity of eight MKIDs with respect to the power absorbed
in the detector using a thermal calibration source filtered in a narrow band
around 1.55 THz. We obtain a NEP${}_{exp}(P_{abs})\>=\>3.1\pm 0.9\times
10^{-20}\>\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ at a modulation frequency of 200 Hz
averaged over all measured MKIDs. The NEP is limited by quasiparticle
trapping. The measured sensitivity is sufficient for spectroscopic
observations from future, actively cooled space-based observatories. Moreover,
the presented device design and assembly can be adapted for frequencies up to
$\approx$ 10 THz and can be readily implemented in kilopixel arrays.

##  I Introduction

Radiation in the far-infrared (FIR) part of the electromagnetic spectrum,
loosely defined as the 0.03-1 mm wavelength range, represents about half the
energy generated in the Universe since the Big Bang and includes information
from processes mostly invisible at other wavelengths (Dole  _et al._ , 2006).
Unfortunately, observations are notoriously difficult: The Earth’s atmosphere
is at best partially transparent. Even at the altitudes of stratospheric
balloons, line emission from residual water vapour can contaminate
astrophysical emission lines. Additionally, self-emission from the telescope
creates significant radiation loading, limiting instrument sensitivity. Only
an actively cooled space telescope with a temperature of about 4 K in
combination with background-limited detectors allows an instrument that is
limited only by the Universe background. A spectroscopic instrument on such an
observatory requires large-format detector arrays with a pixel count of
$\approx 10^{5}$ with a sensitivity expressed in a noise equivalent power
(NEP) of $\sim\>$3$\>\times 10^{-20}\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ (Farrah
_et al._ , 2019; Hailey-Dunsheath  _et al._ , 2021). The combination of
sensitivity, high radiation frequency, and pixel count presents a major
challenge for future detector systems. Several detector systems exist that are
progressing towards this goal.

Figure 1:  Detector geometry. (a) Micrograph of a single detector, consisting
of a NbTiN CPW line loaded with an IDC coupled to the readout line via a
coupling structure. Its shorted end consists of an aluminium section
fabricated on a thin SiN membrane, which is highlighted by the backlighting in
the micrograph. (b) Zoom onto the aluminium section on the SiN membrane with
the leaky-slot antenna coupled to a co-planar waveguide line (referred to as
THz line), where radiation absorption takes place. (c) Scanning electron
microscope image of the antenna centre showing the THz line in detail and the
NbTiN quasiparticle plug. The THz line dimensions are: ${w_{g}}$ = 1.2 $\mu$m
and ${w_{c}}$ = 0.4 $\mu$m. (d) Cross-sectional diagram of the detector
assembly (not to scale). The detector chip, as depicted in panels (a-c), is
coupled to a Si lens array using a spacer wafer with a Ta absorbing mesh, with
an opening aligned to the antenna to enable radiation coupling. The vacuum gap
$t$ is created using spin-on PermiNex® glue pillars as indicated.

Transition edge sensors (TESs) (Irwin and Hilton, 2005) are used in many
ground-based and balloon-borne observatories. Recent work (Audley  _et al._ ,
2016; Khosropanah  _et al._ , 2016; Williams  _et al._ , 2020) demonstrates
the progress in horn-coupled devices operating in a 60–110 $\mu$m wavelength
band reaching a NEP$\>\sim\>$3$\times 10^{-19}\mathrm{\;W/\sqrt{\mathrm{Hz}}}$
using a thermal calibration source (Williams  _et al._ , 2020). Nagler,
Sadleir, and Wollack (2020) suggest a possible route towards even better
sensitivities, reaching an electrical
NEP$\><10^{-20}\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ by means of a proximity-effect
TES.

Quantum capacitance detectors (QCDs) (Echternach  _et al._ , 2013) are a less
mature technology, but are the most sensitive: they have shown single-photon
counting at 1.5 THz and background limited performance at power levels as low
as 10-20 W, corresponding to a
NEP$\;<\>$10${}^{-20}\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ using a 451 pixel array
of antenna-coupled detectors operating at 1.55 THz (Echternach, Beyer, and
Bradford, 2021).

Microwave kinetic inductance detectors (MKIDs) are microwave resonators whose
resonance frequency depends on the amount of radiation absorbed (Day  _et al._
, 2003). Antenna-coupled MKIDs have reached sensitivities of NEP = 3$\times
10^{-19}\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ for a 961 pixel array operating
around a radiation frequency of 850 GHz, read out by a single readout system
(Baselmans  _et al._ , 2017). Operation at radiation frequencies above 1 THz
with a similar sensitivity and array size was demonstrated by Bueno  _et al._
(2017, 2018). Lumped-element KIDs have almost reached similar sensitivities
for devices coupled to an on-chip filterbank (McGeehan  _et al._ , 2018).
MKIDs are especially attractive because of their ease of multiplexing,
allowing kilopixel arrays with only a single readout line. They are also the
only detectors whose sensitivity to ionising radiation has been tested
extensively for large arrays with relevant sensitivities (Karatsu  _et al._ ,
2019), with an extrapolated dead time in L2 of less than 4% depending on array
geometry. Additionally, Karatsu  _et al._ (2016) found negligible changes in
device performance after irradiating the devices with a proton beam simulating
the worst-case scenario for radiation absorption of a five-year observation in
L2.

In this paper, we present a MKID design that is scalable for a 0.1–10 THz
frequency band and for which we measure a NEP${}_{exp}(P_{abs})\>=\>3.1\pm
0.9\times 10^{-20}\>\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ at a modulation frequency
of 200 Hz averaged over all devices, using a thermal calibration source band-
pass filtered around 1.5 THz. Here, $P_{abs}$ is the THz power absorbed in the
detector. The design is optimised for low NEP and high responsivity, and will
be easily scalable to kilopixel arrays because the readout requirements and
fabrication are similar to earlier large system demonstrators (Baselmans  _et
al._ , 2017).

##  II Device design

We explain the device design referring to Fig. 1: The detector is an antenna-
coupled hybrid MKID made from a wide niobium-titanium-nitride (NbTiN) coplanar
waveguide (CPW) section loaded with an interdigitated capacitor (IDC)
fabricated on a silicon wafer. At the shorted end of the CPW, the MKID
consists of an aluminium section on a sillicon-nitride (SiN) membrane of 100
nm in thickness. The latter has a leaky-slot antenna and aluminium CPW line
(hereafter referred to as the THz line) to couple to and absorb the THz
radiation. The entire structure is a microwave resonator with a resonance
frequency ${F_{0}}\>\approx\>$3 GHz. Radiation absorption in the aluminium THz
line modifies the Cooper pair- and quasiparticle density which results in a
frequency shift of the resonator read-out using homodyne detection at
${F_{0}}$ (Day  _et al._ , 2003). Aluminium is chosen as the active material
for its superior performance in terms of NEP with respect to other materials.

The first step in optimising the MKID design is to minimise NEPGR, which is
the fundamental limit in MKID sensitivity, and is given by (Day  _et al._ ,
2003; de Visser  _et al._ , 2014):

$NEP_{GR}=\frac{2\Delta}{\eta_{pb}}\sqrt{\frac{n_{qp}V}{\tau^{*}_{R}}},$ (1)

where $\Delta$ is the energy gap of the superconductor, $n_{qp}$ the density
of thermally excited quasiparticles, $\tau^{*}_{R}$ the experimentally
observed quasiparticle recombination time (Flanigan  _et al._ , 2016), and $V$
the volume of the central strip of the THz line. To minimise NEPGR, we aim to
reduce the volume (Hailey-Dunsheath  _et al._ , 2021), which implies that the
THz line must be short while still absorbing all the power coupled in from the
antenna. This requires a thin and narrow central line. We use a central
conductor width ${w_{c}\;=\;0.4\mathrm{\;\mu m}}$ (Fig. 1c) fabricated from a
sputter-deposited 16 nm thick aluminium film with a sheet resistance of
$R_{s}\;=\;4.1\;\Omega/\square$, corresponding to a resistivity $\rho$ =
6.5$\cdot 10^{-8}\>\Omega$m, and critical temperature $T_{c}\>$= 1.54 K. To
match the line to the input impedance of the antenna,
$\mathrm{Z_{antenna}\;=\;141+36i\>\Omega}$, obtained from simulations in CST®,
we use a gap width ${w_{g}}$ = 1.2 $\mu$m, resulting in a characteristic
impedance of the THz line of $\mathrm{Z_{THz\;line}\;=\;194-103i\>\Omega}$.
The high imaginary component is caused by the high aluminium sheet resistance.
Despite this, we find a good match between the antenna and the THz line, with
the reflection coëfficient ${\mathbf{|}S_{\mathrm{11}}}\mathbf{|}$ = $-14.5$
dB. Subsequently we use simulations in SONNET® based upon the method described
by Endo  _et al._ (2020) to calculate the fraction of the power absorbed in
the THz central line with respect to the total power absorbed. We find a high
value of 91.5% caused by the high line impedance. Most of the remaining power,
8.2%, is absorbed in the ground plane, and the remaining losses are due to SiN
absorption and re-radiation. We note that quasiparticles created in the ground
plane do not contribute to the detector response because they diffuse away
from the THz line. A THz line length of 50 $\mu$m is sufficient to absorb 90%
of the radiation, the minimum THz line length in our devices is 113
$\mathrm{\;\mu m}$.

We further reduce the aluminium volume by using a single THz line coupled to
the antenna, in contrast to the dual THz line design from Bueno  _et al._
(2017). To prevent quasiparticle diffusion from the central line to the ground
plane, we use a small NbTiN plug creating an Andreev mirror close to the
antenna feed point (Andreev, 1964), as shown in Fig. 1c. The distance
$d\>=\>2.5\>\mathrm{\;\mu m}$ in Fig. 1c is motivated by two factors. First,
the THz current density, and with that the ohmic losses in the NbTiN plug,
strongly depend on the distance $d$ from the start of the THz line, which we
simulate in CST®. For $d\>>\>2.5\>\mathrm{\;\mu m},$ we find that the losses
become independent of $d$ and are limited only by losses in the aluminium
film. Second, quasiparticle creation is most intense at the start of the
narrow section of the THz line where the current density peaks. Initial
quasiparticle excitations will relax to the gap edge of aluminium in about
$t_{II}$ =1.3 ns (Kozorezov  _et al._ , 2000), which corresponds to a
diffusion distance $d=\sqrt{\frac{t_{II}}{2\rho N_{0}e^{2}}}\approx 1.9\;\mu
m$, using $N_{0}$ the single particle density of states in aluminium
$N_{0}=1.08\>\cdot 10^{47}\>\mathrm{m^{-3}J^{-1}}$ and $e$ the electron
charge. Hence, $d$ must be larger than this value to prevent quasiparticles
created at the start of the narrow section, which have not yet relaxed to the
gap energy of the aluminium to escape over the NbTiN Andreev barrier.

A low ${NEP_{GR}}$ alone is not sufficient for the detector to actually reach
this sensitivity experimentally. It is also crucial that the measured noise,
and thereby the NEP, is dominated by photon fluctuations and not by excess
noise sources. We therefore maximise the photon noise power spectral density
as measured with an MKID, which is given by (de Visser  _et al._ , 2014)

$S_{\theta}^{P}=2P_{abs}hF\frac{(d\theta/dP_{abs})^{2}}{1+(2\pi
f\tau^{*}_{R})^{2}}$ (2)

when using the MKID phase $\theta$ as observable, as is the case in this work.
Here, we omit the photon bunching term which is negligible for the power and
frequency used in this work. The first term represents the photon number
fluctuations, with $F$ the photon frequency, $h$ Planck’s constant, $f$ the
modulation frequency, and $d\theta/dP_{abs}$ the MKID phase responsivity to
the absorbed power. It is useful to reformulate this equation in terms of
quasiparticle excitations, as an MKID measures photon fluctuations indirectly
by measuring the quasiparticles created. Using
$dN_{qp}/dP{{}_{abs}}=\tau^{*}_{R}\eta_{pb}/\Delta$,
$P{{}_{abs}}=N_{qp}\Delta/(2\tau^{*}_{R}\eta_{pb})$ (Flanigan  _et al._ ,
2016) and the Mattis-Bardeen result for ${d\theta/dn_{qp}}$ (Mattis and
Bardeen, 1958), we get

$S_{\theta}^{P}=\frac{n_{qp}\tau^{*}_{R}\eta_{pb}}{1+(2\pi
f\tau^{*}_{R})^{2}}\frac{hFV}{\Delta}\cdot\left(-\frac{\alpha\beta
Q}{|{\sigma}|V}\frac{d\sigma_{2}}{dn_{qp}}\right)^{2},$ (3)

where the first two terms express the fluctuations in quasiparticle number due
to photon fluctuations and the last term represents the MKID responsivity to
quasiparticle number changes, $d\theta/dN_{qp}$ where $N_{qp}=n_{qp}V$ is the
total number of quasiparticles. Furthermore,
$\beta{\;=1+\frac{2t/\lambda}{\sinh{2t/\lambda}}}\sim\;$2 for thin films, with
$\lambda$ the bulk magnetic penetration depth and $t$ the film thickness,
$|\sigma|$ the absolute value of the aluminium surface conductance
$\sigma=\sigma_{1}-i\sigma_{2}$, which depends on the energy gap,
quasiparticle density, and microwave frequency (Mattis and Bardeen, 1958). $Q$
is the resonator quality factor, $\alpha=L_{k,c}/L_{tot}$, which is the ratio
of the kinetic inductance of the THz central line with respect to the total
inductance. We maximise $S_{\theta}$ in three ways: (i) We increase both
$\tau^{*}_{R}$ and $\eta_{pb}$ by placing the THz line on a very thin membrane
(Fyhrie  _et al._ , 2016), in contrast with the 1$\>\mu m$ thick membrane used
by Bueno  _et al._ (2017). This enhances the re-trapping of 2$\Delta$ phonons
emitted by quasiparticle recombination. The result is an increase in the
phonon escape time $\tau_{esc}$ which increases the experimental quasiparticle
recombination time
$\tau_{R}^{*}=0.5\cdot\tau_{R}\cdot(1+\tau_{esc}/\tau_{B})$. Here $\tau_{R}$
is the single particle recombination time as defined by Kaplan  _et al._
(1976) and ${\tau_{B}}$ = 0.26 ns is the pair-breaking time in aluminium
(Guruswamy, Goldie, and Withington, 2014). The membrane thickness is
approximately 100 nm, which is limited by its mechanical strength. We find
$\tau_{esc}/\tau_{B}\>$=0.65 using a simple phonon-trapping model (de Visser
_et al._ , 2021), which is a factor four increase with respect to a solid
substrate. A larger ratio of $\tau_{esc}/\tau_{B}$ also increases $\eta_{pb}$;
we estimate $\eta_{pb}$ = 0.37 based on the work of Guruswamy, Goldie, and
Withington (2014), which will be smaller for the same THz line on a solid
substrate. (ii) We minimise the THz central line volume $V$ as discussed
before, and (iii) we maximise the kinetic inductance fraction $\alpha$. For
the kinetic inductance of the THz line itself, this is automatically achieved
by using a narrow and thin THz line. However, $\alpha$ in Eq. 2 is defined
with respect to the entire resonator and a $\lambda/4$ distributed resonator
has a significant geometric inductance. To reduce the geometric inductance, we
include a large IDC to ground in the MKID as shown in Fig. 1a, following
Noroozian  _et al._ (2009). The IDC-section has a much larger capacitance-to-
inductance ratio than a CPW line, and therefore the same resonance frequency
is obtained with less parasitic inductance in the NbTiN section. The effect is
greatest for the shortest THz lines: $\alpha\approx 0.6$ for a 113
$\mathrm{\;\mu m}$-long THz line, while a device without the IDC would have
$\alpha\approx 0.42$. This is a relatively small difference because the THz
line itself has a very high (kinetic) inductance due to its small dimensions
and high sheet resistance.

Figure 2:  Detector chip and experimental setup. (a) A microscope image of the
detector chip prior to lens array alignment, showing the 27 different
detectors and the four different geometries used, which vary in THz line
length. We note that the centre pixel is one of the designs with the shortest
THz line length. (b) Photograph of the detector assembly after lens array and
spacer wafer mounting. (c) Cross-sectional CAD render of the experimental
setup. The upgrades with respect to the setup of de Visser  _et al._ (2014)
are highlighted in the insert.

The last step in our effort to maximise the photon noise (and GR noise)
visibility is to reduce the TLS noise, which we achieve by widening the
capacitive section of the resonator, including the IDC (Gao  _et al._ , 2008).
We use a central line width $w$ = 40$\mathrm{\;\mu m}$ and a gap width of
$g\>=\>$8$\mathrm{\;\mu m}$. Wider structures become impractically large
because of the onset of self-resonances within the IDC itself. Using a
narrower gap increases the IDC capacitance, but will increase the TLS noise
(Wenner  _et al._ , 2011). We therefore limit the ratio $w/(w+2g)$ to about
0.70. We note that an increase in $Q$ does not provide a useful optimisation,
because both the photon noise level and the TLS noise level scale with
$Q^{2}$.

For the radiation coupling, we use a leaky wave antenna (Neto, 2010) because
this antenna allows for relatively large feature sizes, enabling high
radiation frequencies (Yurduseven, Llombart Juan, and Neto, 2016), and because
it must be fabricated on a membrane, compatible with our design optimisation
discussed above. The antenna feed is placed in the second focus of a
synthesised elliptical Si lens with a diameter of 1.550 mm. The antenna uses a
small vacuum gap $t\ll\lambda$ to launch a leaky wave into the Si (Neto,
2010). The antenna design is optimised for 1.5 THz resulting in a slot opening
angle of $\theta$ = 45 ∘, a slot length of $H$ = 200 $\mathrm{\;\mu m},$ and a
vacuum gap of $t$ = 4 $\mu$m, as shown in Fig. 1b,d. The entire design of the
KID can be optimised for frequencies up to $\sim$ 10 THz by changing the leaky
slot dimensions and vacuum gap height.

##  III Fabrication and assembly

The device fabrication is largely identical to that of Bueno  _et al._ (2017)
and we use UV lithography and dry etching to define most of the structures. We
start with a $\mathrm{\rho\>=\>10\>k\Omega cm}$ Si wafer on which we deposit
150 nm SiN on both sides using low-pressure chemical vapour deposition
(LPCVD). The SiN is patterned to define the membrane areas on the front side
of the wafer and etch masks on the back side, which are used in the last
fabrication step to remove the Si, creating a free-standing membrane.
Subsequently, we sputter deposit 207 nm NbTiN
($\mathrm{T_{c}\>=\>15\>K\>,\rho\>=\>228\cdot 10^{-8}\;\Omega m}$) to define
the resonators. The NbTiN etch requires a short over-etch which removes
approximately 50 nm from the SiN. To define the THz line and antenna we
sputter deposit a 16 nm aluminium film and pattern it in two steps: First we
perform UV contact lithography and wet etching to define the readout line
bridges and a large patch of aluminium covering the membrane area. We then use
electron beam lithography and wet etching to define the final patterns,
similarly to Mirzaei  _et al._ (2021), resulting in a high-resolution
definition of the structures as shown in Fig. 1c. The small black structures
on the aluminium are small voids created during the 15-minute 175 ∘C bake of
the PMMA e-beam resist, likely caused by thin-film dewetting (Thompson, 2012).
In the last step, we open the membranes by potassium-hydroxide (KOH) etching
(8 hours in 75∘C KOH:H2O 1:4), followed directly with a RCA2 clean (10 minutes
in 70∘C HCl:H2O2:H2O 1:1:5) to remove potassium ions on the back side of the
membrane.

Figure 3:  Noise, response and NEP for a single detector. (a) Measured phase
noise power spectral density for several source powers, increasing from bottom
to top. There is a clear transition from detector-limited noise (lower curves)
to photon-noise-limited noise, characterised by a white spectrum rolled-off
with $\tau^{*}_{R}$ (highest curves). The insert shows the $\tau^{*}_{R}$ for
the same source powers. (b) KID phase response around the lowest source power
used to obtain the NEP, $P_{s}$ = 0.145 aW ($P_{abs}$ = 5.63 zW), which
corresponds to a radiator temperature of 4 K. The data represent a measurement
obtained by sweeping the radiator temperature from 3.95 K to 6.21 K and back
to 3.95 K. The data is plotted on a semi-logarithmic scale to show the absence
of any response at low powers. The fits and fitted range are indicated as
well. (c) MKID response and fits for increasing and decreasing radiator
temperature around a higher source power, $P_{s}$ = 12.2 fW. We indicate the
absorbed power as well as the absorbed power in panels (b) and (c) for
convenience. (d) Measured NEP for the same set of powers, now expressed in
power absorbed in the detector. The legend in absorbed power is also valid for
panel (a).

The assembled detector array consists of the device chip, a 250 $\mathrm{\;\mu
m}$ Si spacer wafer, and a laser-machined Si lens array commercially obtained
from Veldlaser. A cross-sectional sketch of the detector assembly is given in
Fig. 1d, and a photograph is shown in Fig. 2b. The spacer wafer top has a 40
nm $\beta$-Ta absorbing mesh $(R_{s}\>=\>\mathrm{51\>\Omega/\square}$) with a
design similar to that used by Baselmans  _et al._ (2017) that acts as a low-
pass filter at the MKID readout frequencies and as THz radiation absorber to
eliminate stray radiation. Above each antenna, a 600 $\mu$m diameter hole
allows radiation coupling to the antenna. On the other side, the spacer wafer
has 4 $\mathrm{\;\mu m}$ thick lithographically defined pillars made from
PermiNex®, and a spin-on photo-sensitive glue used to define the vacuum gap
$t$ as indicated in Fig. 1d and to provide a permanent bond between the chip
and the spacer wafer. To assemble the final detector, we align the spacer
wafer with the lens array and glue them together with two spots of 2216 epoxy
(see Fig. 2b), which is cured for 30 minutes at 90∘C. At this temperature the
PermiNex® is unaffected. In the next step, we align the lens-spacer assembly
to the chip and press the assembly together using a spring-loaded tool. The
PermiNex® is cured for 15 minutes at 180 ∘C to create a permanent bond between
spacer and chip, which is found to be resilient against more than 20 thermal
cycles down to 120 mK. A photograph of the entire detector chip without the
lens array is given in Fig. 2a. The chip has 27 detectors, with four different
device designs that differ in THz line length, which can be 913, 663, 238, or
113 $\mathrm{\;\mu m}$. This corresponds to $V$ = 6.0, 4.4, 1.6, and 0.85
$\mathrm{\;\mu m}^{3}$ respectively, including the small volume near the NbTiN
plug. All devices are designed with a coupling Q factor
$Q_{c}$=120$\cdot\mathrm{10^{3}}$.

##  IV Experiment

In the experiment, we measure the sensitivity, NEP${}_{exp}(P_{s},f)$, at
several powers from a thermal radiator source $P_{s}$ using the setup depicted
in Fig. 2. We denote with $f$ the post-detection modulation frequency of the
spectral shape of the NEP. We mount the detector assembly in a holder which
itself is mounted inside a light-tight box mechanically anchored to the cold
stage of an adiabatic demagnetisation refrigerator (ADR). All interfaces and
venting holes of the holder and light-tight box are equipped with labyrinths
coated with a radiation-absorbing layer from carbon-loaded epoxy and SiC
grains. A thermal radiation source weakly coupled to the 3 K stage of the
cooler enclosed in a 3 K box is placed directly above, and can be heated to 30
K. In the first experiment, we used a 24 mm diameter radiator with a radiation
absorbing layer from carbon-loaded epoxy and 0.5 mm SiC grains, as shown in
Fig. 2c. Radiation coupling between the source and the detector is achieved
through a 1.85 mm aperture at 42 mm from the lens surface. The use of the
aperture reduces the throughput to the detectors, and therefore higher
temperatures are needed to reach the same amount of source power at the
detectors. This reduces the sensitivity of the radiation power to errors in
the measured temperature. Furthermore, it allows a check as to whether the
coupled power matches the beam pattern of the detector. We use eight filters
in total —mounted on the radiator box, the light-tight box, and the sample
holder— to define a pass-band of around 1.55 THz. These are the exact same
filters as discussed by de Visser  _et al._ (2014). The second experiment was
identical, except that we used a 40 mm diameter radiator with 1 mm SiC grains
and no aperture, to cross-check our results: In this case, the antenna pattern
does not play a role because the opening angle to the radiator is much larger
than the beam opening angle. Also, the total power coupled to the detector
chip largely exceeds the power coupled to the single-mode antenna beams. Stray
coupling to this excess radiation will result in overestimation of the
coupling efficiency. In all cases, a polariser is used on the holder to allow
only radiation in co-polarisation with respect to the antenna. We note that,
because of the increased sensitivity of the detectors, we needed to upgrade
the light-tight box with respect to the setup in de Visser  _et al._ (2014)
—as indicated in the insert of Fig. 2c— with better stray-light control around
the filter mount. We use a CRYOPHY® outer shield and niobium inner shield at
3K surrounding the light-tight box and radiator to limit time-dependent
magnetic fields. The theoretical coupling between the absorbing aluminium
volume in the MKID aligned with the aperture is calculated using a CST
simulation of the antenna beam and the tool provided by Zhang  _et al._ (2019)
to be $\eta_{opt,calc.}\>=\>$4.3%. The reason for the low coupling is the
(intended) large spill-over loss between the small pin-hole aperture and the
lens–antenna beam patterns. This includes the losses in the ground plane of
the THz line and the antenna to THz line mismatch, which are discussed above.
For the 40 mm diameter radiator without aperture, we find
$\eta_{opt,calc.}\>=\>$76%.

We began experiments with a frequency sweep with the chip at 120 mK and the
radiator at 3 K, which represents fully dark conditions. We find 23 out of 27
MKIDs with an average $Q_{i}=0.23\cdot 10^{6}\pm 0.13\cdot 10^{6}$ and an
average $Q_{c}=6\cdot 10^{4}\pm 2\cdot 10^{4}$ at a readout power of $-118$
dBm. We selected a subset of the detectors that couple well to the aperture.
For each detector we obtained NEP${}_{exp}(P_{s},f)$ using

$NEP_{exp}(P,f)=\sqrt{S_{\theta}(f)}\left(\frac{d\theta}{dP}\right)^{-1}\sqrt{1+(2\pi
f\tau_{R}^{*})^{2}},$ (4)

where $P$ is the radiation power, given now by $P_{s}$, which is the power
from the source that can couple to a single mode. ${P_{s}}$ is calculated from
the radiator temperature and the measured filter transmission using the
formalism explained in detail in Appendix B of Ferrari  _et al._ (2018).
Figure 3a shows the phase noise power spectral density $S_{\theta}(f)$ for one
representative device, which has a resonance frequency of 2.74 GHz and an
aluminium length of 913 $\mathrm{\;\mu m}$. The spectrum is obtained from 64
seconds of time domain data sampled at 50 kilosample/sec with the readout
tuned to the MKID resonance frequency while maintaining a constant black body
temperature. At low powers, shown by the lower (blue) lines, we observe a
noise spectrum with a 1/f slope. At increasing power, the noise level at
frequencies f $<$ 1 kHz increases sharply until the noise becomes white for
$P_{s}\>>$ 1 fW. The insert in the figure shows $\tau_{R}^{*}$ for the same
powers obtained from a fit to the cross power spectral density of the
resonator amplitude and phase noise (de Visser  _et al._ , 2012), which is
given in Appendix VIII.1. We observe a saturation of the recombination time at
around 0.3 ms and a reduction for increasing powers as expected. Subsequently,
we obtain the responsivity $d\theta/dP_{s}$ by a linear fit to the MKID phase
response upon a small change of the black-body temperature. Care was taken to
sweep the radiator temperature slowly to prevent hysteresis caused by a
temperature difference between the thermometer and the radiating surface. In
Fig. 3b, we show the response measurement and fit around the lowest $P_{s}$ of
0.145 aW (corresponding to a 4 K radiator temperature). We fit the response
curve over a range of 1 aW $<\>P_{s}\><$ 0.1 fW (which corresponds to a range
in absorbed power 0.039 aW $<\>P_{abs}\><$ 39 aW, as we show later): the high
power limit is used so as to reach a statistically significant response which
is still linear; the low end was chosen so as to reduce the sensitivity to
drifts. It is clear from the figure that there is negligible hysteresis and
that the fits are excellent over the whole power range spanning a factor 1000,
with no response below $P_{s}$ = 1 aW. This proves that the setup is light
tight; radiation leaks would allow unfiltered radiation to couple to the
detector. The total integrated power per mode for a 4 K radiator is $10^{6}$
times larger than the power in the narrow band around 1.5 THz, and it is
strongly temperature dependent. Even a weak coupling to broad-band stray
radiation would therefore result in a significant response of the detector at
4 K ($P_{s}$ = 0.145 aW) in Fig. 3b. For higher powers, we fit a symmetric
power range around the power at which we measure the noise, as shown in Fig.
3c.

A possible source of systematic error is the radiator temperature readout: The
detector response starts at $P_{s}$ = 1 aW, corresponding to a radiator
temperature of 6.3 K, where a 50 mK temperature change would result in a 10%
change in $P_{s}$. For the full aperture setup, the response starts at a
temperature of 4.5 K, which is due to the higher coupling. At this
temperature, a 50 mK temperature change will result in a 20% change in
$P_{s}$. We therefore performed a cross-calibration between the Cernox®
radiator thermometer and another similar thermometer. The temperature
difference between the two is less than 15 mK in the temperature range of the
experiment, contributing less than 3% uncertainty in radiator power.

We can now obtain NEPexp using Eq. 4, and use it to determine the detector
coupling efficiency $\eta_{opt}$ using the photon noise from the radiator as
an absolute calibration source (Ferrari  _et al._ , 2018). This procedure is
only valid if the MKID is photon-noise limited, which is the case for source
powers exceeding 5 fW; here the spectrum is white with a single Lorentzian
roll-off (Janssen  _et al._ , 2013). The optical efficiency is given by
(Ferrari  _et al._ , 2018)

$\eta_{opt}=\frac{\int{2P_{s,\nu}h\nu d\nu}+\int{4\Delta
P_{s,\nu}/\eta_{pb}}d\nu}{NEP_{exp}^{2}-\int{2P_{s,\nu}h\nu
F_{\nu}O_{\nu}d\nu}},$ (5)

where $F_{\nu}$ is the filter transmission, $O_{\nu}$ is the photon occupation
number, and $P_{s,\nu}$ is $P_{s}$ per unit frequency $\nu$. The result is
given in the insert of Fig. 4a. We get
$\eta_{opt}\>\mathrm{=0.039}\>\pm\>$0.004 with very little scatter between the
individual datapoints. We can now obtain the experimental NEP as a function of
the power absorbed, NEP${}_{exp}(P_{abs},f),$ using Eq. 4 with
$P=P_{abs}=\eta_{opt}P_{s}$. The result is shown in Fig. 3d. At the lowest
powers, the NEP at $f$ = 200 Hz reaches a value of
NEP${}_{exp}(P_{abs}f\>\mathrm{=\>200\>Hz})\;=\;3.3\pm 0.3\times
10^{-20}\;\mathrm{\;W/\sqrt{\mathrm{Hz}}}$. At lower frequencies, the NEP is
higher because of the 1/f noise. For higher powers, the NEP spectral shape
becomes flatter, and for $P_{abs}\>>\>$53 aW the NEP is frequency independent
down to 1 Hz. In Fig. 4a, we give
NEP${}_{exp}(P_{abs},f\>\mathrm{=\>200\>Hz})$ together with the theoretical
value of the photon noise background from the radiation source and obtain a
result very similar to those of de Visser  _et al._ (2014),Baselmans  _et al._
(2017) and Ferrari  _et al._ (2018), but with ten times greater sensitivity.
For a comparison of the NEP as a function of the absorbed power as shown in
Fig. 4a for all measured MKIDs, we refer to Appendix VIII.3. For all MKIDs,
the microwave readout power absorbed by the quasiparticle system,
$P_{abs,qp}$, is significant, especially at low absorbed powers: For example,
at the lowest value of $P_{abs}$ = 5.63 zW, we estimate that
$P_{abs,qp}\;\simeq$ 20 aW. At values of $P_{abs}\;\geq$ 1 fW, $P_{abs,qp}$
and $P_{abs}$ become similar.

Figure 4:  NEP and coupling efficiency. (a) The NEP at 200 Hz, as indicated by
the dots in panel (d), as function of absorbed power using the pinhole setup
for the same devices as in Fig.3. The black line shows the theoretical curve.
The absorbed power is evaluated from the source power using the measured
efficiency as indicated in the insert, and the mean value of the points,
$\eta_{opt}$ = 0.0389 $\pm$ 0.004, is given by the line. The statistical error
bars are smaller than the dots. (b) The measured
NEP${}_{exp}(P_{abs},\>f\;\mathrm{\>=\>200Hz})$ for the pinhole experiment, as
displayed in Fig. 2c, together with the dark NEP obtained at the same
modulation frequency using a change in bath temperature to obtain the
response. (c) The averaged coupling efficiencies for the two experiments. The
red dots represent the results for the large radiator, where all detectors
couple almost identically to the source as expected; the expected coupling is
given by the line. The black dots give the coupling for the pinhole setup,
where only the best aligned detector is expected to reach the theoretical
coupling value given by the black line. We note that, in both cases, the
measured coupling exceeds the theoretical value by the same factor 1.6. (d)
The measured NEP${}_{exp}(P_{abs},\>f)$ for the pinhole experiment at the
lowest radiator temperature for all measured MKIDs. We observe no dependence
of the NEP on aluminium volume. We note that panel (a) and Fig.3 are given for
the detector with F0 = 2.74 GHz.

In Fig. 4b, we give the coupling efficiency for all MKIDs for the aperture-
coupled radiator by the black dots, and the theoretical coupling for a well-
aligned detector by the grey square. We observe that the MKID with the highest
resonance frequency has the maximum coupling, as expected as this is the
detector in the chip centre as can be seen in Fig. 2a. The measured
$\eta_{opt}\>=\>$7%, exceeding the expected value of 4.3% by a factor 1.63. We
observe six MKIDs with similar coupling but some scatter, which can be
explained by a small misalignment of the aperture, causing unequal coupling to
the six detectors at equal distance from the centre device. The red dots
represent the results with the large radiator; we now see that all detectors
couple identically with a mean value of $\eta_{opt}\>=\>1.27\;\pm\>0.04$,
exceeding the expected value by a factor 1.68. The fact that both experiments
yield an identical coupling with respect to the theoretical value implies that
both setups are equivalent and systematic errors such as an error in the beam
pattern calculation, in-band stray radiation coupling with the large radiator,
or a difference in temperature between the radiator surface and thermometer
can be excluded. The higher measured transmission is therefore most likely
caused by a higher transmission of our eight quasi-optical filters compared to
the data provided by their supplier QMCI®, which gives a peak transmission of
13.5%. A potential reason is that QMCI® performs the filter measurements at
room temperature.

In Fig. 4c we give NEP${}_{exp}(P_{abs},\>f\>\mathrm{\>=\>200Hz})$ for eight
KIDs in the pinhole experiment and find a mean value of
NEP${}_{exp}(P_{abs},\>f\mathrm{\>=\>200Hz})\>=\>3.1\pm 0.9\times
10^{-20}\>\mathrm{\;W/\sqrt{\mathrm{Hz}}}$. We also give the dark NEP
(Baselmans  _et al._ , 2008; Janssen  _et al._ , 2014). To obtain the dark
NEP, we measure the response of the MKIDs to a change in chip temperature
while keeping the radiator in dark conditions, that is, $\mathrm{T_{BB}}$ = 3
K. Under these conditions, the number of quasiparticles in the aluminium
$\mathrm{N_{qp}}$ can be calculated from the chip temperature, the volume of
the aluminium section of the resonator, and the energy gap (Janssen  _et al._
, 2014). Rewriting $\mathrm{N_{qp}}$ in terms of FIR power allows the dark NEP
to be calculated using Eq. 4 by replacing the responsivity term
$d\theta/\delta P$ by the dark responsivity given by Baselmans  _et al._
(2008):

$\frac{d\theta}{dP_{dark}}=\frac{\eta_{pb}\tau^{*}_{R}}{\Delta}\frac{d\theta}{dN_{qp}(T)},$
(6)

where we obtain $\tau^{*}_{R}$ using a fit to the cross power spectral density
of the phase- and amplitude noise as discussed above. For the operating
temperature of 120 mK, a good fit cannot always be obtained. We find a mean
value of NEP${}_{{dark}},\>f\mathrm{\>=\>200Hz})\>=\>3.8\pm 1.3\times
10^{-20}\>\mathrm{\;W/\sqrt{\mathrm{Hz}}}$, in good agreement with
NEP${}_{exp}(P_{abs}\>f\mathrm{\>=\>200Hz})$. A similar agreement between dark
and optical NEP was found by Janssen  _et al._ (2014). Finally, panel (d)
shows the NEP${}_{exp}(P_{abs},f)$ at the lowest radiator power for the same
set of MKIDs. All have a similar sensitivity.

##  V Discussion

The observed reduction in photon noise level for $P_{abs}\><\>$228 aW (see
Fig. 3a) is inconsistent with a NEP limited by a thermal reservoir of residual
quasiparticles, which was the conclusion in previous work (de Visser  _et al._
, 2014). However, it is consistent with quasiparticle trapping as discussed by
de Rooij  _et al._ (2021). The optimisation of the MKID responsivity, as
discussed in the context of Eq. 3, is now especially relevant. A high photon
noise level allows for the device to remain background limited even when the
photon-noise level drops. The use of a thin membrane, increasing both
$\tau^{*}_{R}$ and $\eta_{pb}$, is therefore an important optimisation. The
downside is a strong reduction in instantaneous dynamic range when compared to
the results of Baselmans  _et al._ (2017). We find an instantaneous dynamic
range of a few hundred aW, with readout re-tuning the devices operate up to 30
fW absorbed power.

Interestingly, we observe in Fig. 4 that the
NEP${}_{exp}(P_{abs},\>f\mathrm{=\>200\>Hz})$ is independent of the aluminium
volume. More precisely, we observe that both the noise $S_{\theta}$ and
responsivity $d\theta/dP$ are independent of volume. For the noise, this is
not surprising, because we are limited in this regime by amplifier noise and
device noise, but the responsivity is expected to scale with the inverse of
the volume, as can be seen from Eqs. 2 and 3. We find from the dark NEP
measurement that $d\theta/dN_{qp}$ does scale inversely with volume as
expected. We also find that the low temperature quasiparticle lifetime
$\tau^{*}_{R}$ scales inversely with volume. A smaller effect is that $\alpha$
is slightly smaller for the smallest volume devices due to parasitic
inductance in the IDC, and the net result is the observed volume independence
of the responsivity to power. Interestingly, the observed volume (or line
length length) dependence in $\tau^{*}_{R}$ is not seen in the Appendix of de
Rooij  _et al._ (2021).

A better understanding of the trapping mechanism and the transition between
trapping-dominated hybrid MKIDs and devices limited by quasiparticle creation
due to the readout power will be important for further optimisation, which is
especially relevant for the 1/f noise. If trapping is caused by surface
states, as argued by de Rooij  _et al._ (2021), device optimisation is
possible using thicker aluminium films as this will reduce trapping and
increase $\tau^{*}_{R}$. Interestingly, thicker films will also increase
$\tau^{*}_{R}$ as well as $\eta_{pb}$ due to enhanced $2\Delta$ phonon
recycling caused by the longer dwell time of the phonons in the film. This
might partially compensate the responsivity reduction caused by an increase in
the volume and reduction in $\alpha$.

It is interesting to consider the intrinsic energy resolution for the
possibility of single photon counting in the FIR. This resolution is given by
Moseley, Mather, and McCammon (1984): $dE\simeq
NEP_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})\cdot\sqrt{\tau^{*}_{R}}$. We find an
energy resolution averaged over all MKIDs of $dE/h$ = 0.5 $\pm$ 0.2 THz, with
a best value of $dE/h$ = 0.24 THz for one of the devices with the smallest
volume. Reliable single photon detection should therefore be possible for
frequencies above a few THz. For more details, we refer to Appendix VIII.2.

##  VI Conclusions

In conclusion, we present a new MKID design, where radiation coupling and
absorption take place in a small aluminium volume fabricated on a 100 nm SiN
membrane embedded in a NbTiN resonator. The device is optimised for high
quasiparticle responsivity and low NEP. We obtain a mean
NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})\>=\>3.1\pm 0.9\times
10^{-20}\>\mathrm{\;W/\sqrt{\mathrm{Hz}}}$ with a time constant varying from
0.33 msec to 0.03 msec. The NEP is limited by 1/f noise and quasiparticle
trapping, limiting the performance in dark conditions at low modulation
frequencies. At absorbed powers of 53 aW and higher the device reaches
background-limited operation with a flat noise level down to 1 Hz. The antenna
design, device geometry, and assembly using PermiNex® spin-on glue allows
scaling to frequencies up to $\approx$ 10 THz. Importantly, creating kilopixel
arrays from the presented device design is straightforward; the fabrication
and readout are largely identical to those in the work presented by Bueno  _et
al._ (2018). All presented data, measurement data, and analysis scripts will
be made available in Baselmans  _et al._ (2022)

###### Acknowledgements.

P.J.dV. is financially supported by the Netherlands Organisation for
Scientific Research NWO (Veni Grant 639.041.750 and Projectruimte 680-91-127).

##  VII References

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##  VIII Appendix

###  VIII.1 Cross power spectral density and lifetime fits

In Fig. 3a of the main text we show in the insert the apparent quasiparticle
recombination time $\tau^{*}_{R}$. These datapoints were obtained from the
Lorentzian fits to the cross power spectral density of the noise measured at
each radiator power. The data and fits that resulted in the values of the
recombination time are given in Fig.5.

Figure 5:  Cross power spectral density at each source power and the fits
resulting in the apparent recombination time as indicated. Only at the lowest
radiator powers does the photon-noise signal disappear and we cannot fit the
data with a single Lorentzian roll-off.

###  VIII.2 Energy resolution

Using NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})$ and the lifetimes obtained
from the fits to the photon noise cross power spectral density as discussed
above, we can obtain an estimate of the energy resolution,
$dE=NEP_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})\cdot\sqrt{\tau^{*}_{R}}$. The
result is given in Fig. 6. We find an energy resolution averaged over all
MKIDs of $dE/h$ = 0.5 $\pm$ 0.2 THz and a clear improvement towards the
smaller device volumes, which results from the fact that the smaller devices
have a shorter quasiparticle lifetime and a similar NEP.

Figure 6:  Estimated energy resolution for all MKIDs discussed in this work,
given as a function of the MKID resonance frequency.

###  VIII.3 NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})$ as function of
absorbed power

Figure 7:  NEP as function of absorbed power for all measured devices. Top
row: Data of all measured MKIDs using radiation coupling via a small pinhole
to the small radiator source. The left figure gives the data for all KIDs
identical to Fig3.e in the main text. The centre and right plots give the NEP
divided by the photon noise limit, showing more clearly that the measured NEP
can be below the photon noise limit for 10 aW $<\>P_{abs}\><$ 100 aW. Bottom
row: Similar data, but now using the large radiator coupled without pinhole to
the detector chip. We observe that the region where the experimental NEP is
below the photon noise limit scales with $P_{abs}$, and not with radiator
temperature.

In Fig. 4a of the main manuscript we give
NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})$ for the MKID with a resonant
frequency of 2.74 GHz. The other measured MKIDs show a similar behaviour, as
shown in the top left figure of Fig. 7. However, in the power interval between
10 and 100 aW, we see for most detectors that the measured
NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})$ is slightly lower than the
photon noise limit. We re-plot the same data normalised to the photon noise
limit in the other two panels in the top row to make this more clear. To
experimentally address this issue, we repeat the measurement using a large (40
mm diameter) radiator coupled to the detector chip without the small aperture
used in the rest of the experiments. We observe an identical behaviour, which
shows both the same limited NEP and an NEP lower than the photon noise limit
in the same 10-100 aW absorbed power range. This correspondence in power and
not in temperature very clearly excludes possible issues with the filter stack
or thermometer: The coupling efficiency with the large black body is 1.27,
whereas it is 7% with the aperture, that is, about a factor 16 difference This
implies that the same absorbed power levels are obtained at different radiator
temperatures. This is very clear from the rightmost figures: At 6K we observe
in the top panel that NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})$ is below
the photon noise limit. However, in the bottom panel
NEP${}_{exp}(P_{abs},f\>=\>\mathrm{200\>Hz})$ is identical to the background
limit. To further exclude issues with the thermometer, we recalibrated our
thermometer as discussed in the main text. The exact mechanism of the
discussed effect remains uncertain, but it could be related to the fact that
the photon arrival rate is 1 kHz for 1 aW, respectively. This is of the order
of the bandwidth given by the recombination time,
$\frac{1}{2\pi\tau^{*}_{R}}$, which is 500 Hz for $\tau^{*}_{R}$=0.3 msec.