# Noise Optimization for MKIDs with Different Design Geometries and Material
Selections

Z. Pan, K. R. Dibert, J. Zhang, P. S. Barry, A. J. Anderson, A. N. Bender, B.
A. Benson, T. Cecil, C. L. Chang, R. Gualtieri, J. Li, M. Lisovenko, V.
Novosad, M. Rouble, G. Wang, and V. Yefremenko  (Corresponding author: Zhaodi
Pan)Z. Pan is with the Argonne National Laboratory, 9700 South Cass Avenue,
Lemont, IL, 60439, USA (email: panz@anl.gov)K. Dibert is with the University
of Chicago, 5640 South Ellis Avenue, IL, 60637, USA (email:
<EMAIL_ADDRESS>J. Zhang, A. N. Bender, T. Cecil, R. Gualtieri, J. Li,
M. Lisovenko, V. Novosad, G. Wang, and V. Yefremenko are with the Argonne
National Laboratory, 9700 South Cass Avenue, Lemont, IL, 60439, USA (emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>yefremenko@anl.gov)P. Barry is with Cardiff University, Cardiff CF10 3AT, UK
(email: barryp2@cardiff.ac.uk)A. J. Anderson and B. Benson are with Fermi
National Accelerator Laboratory, PO BOX 500, Batavia, IL 60510, the University
of Chicago, Chicago, IL, 60637, USA, and Kavli Institute for Cosmological
Physics, U. Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>bbenson@astro.uchicago.edu)C. L.
Chang is with the Argonne National Laboratory, Argonne, IL 60439 USA, the
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA, and
Kavli Institute for Cosmological Physics, U. Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA (email<EMAIL_ADDRESS>M. Rouble is with
McGill University, 845 Rue Sherbrooke O, Montreal, QC H3A 0G4, Canada (email:
<EMAIL_ADDRESS>
###### Abstract

The separation and optimization of noise components is critical to microwave-
kinetic inductance detector (MKID) development. We analyze the effect of
several changes to the lumped-element inductor and interdigitated capacitor
geometry on the noise performance of a series of MKIDs intended for
millimeter-wavelength experiments. We extract the contributions from two-level
system noise in the dielectric layer, the generation-recombination noise
intrinsic to the superconducting thin-film, and system white noise from each
detector noise power spectrum and characterize how these noise components
depend on detector geometry, material, and measurement conditions such as
driving power and temperature. We observe a reduction in the amplitude of two-
level system noise with both an elevated sample temperature and an increased
gap between the fingers within the interdigitated capacitors for both aluminum
and niobium detectors. We also verify the expected reduction of the
generation-recombination noise and associated quasiparticle lifetime with
reduced inductor volume. This study also iterates over different materials,
including aluminum, niobium, and aluminum manganese, and compares the results
with an underlying physical model.

###### Index Terms:

Two-level system (TLS), noise, microwave kinetic inductance detectors (MKIDs),
generation-recombination noise, optimization.

##  I Introduction

Future mm-wave experiments require ever-increasing detector counts to achieve
the necessary sensitivity to pursue ambitious science cases. Microwave kinetic
inductance detectors (MKIDs) are well suited to large-format detector arrays,
including photometer arrays [1, 2, 3] and on-chip filter-bank spectrometers
[4, 5, 6, 7]. To achieve photon-limited sensitivity for each detector, it is
necessary to understand each component of detector noise, including two-level
system (TLS) noise, shot noise from the generation and recombination (GR) of
quasiparticles, and amplifier noise.

TLS noise is caused by the coupling of a resonator to a thin amorphous solid
dielectric layer, where two-level tunneling states are thought to exist based
on a phenomenological model [8]. When atoms tunnel between two states, the
resulting dipole can couple to the electric field of the resonator, causing
excessive noise with a characteristic spectral shape proportional to
$f^{-1/2}$, where $f$ is the frequency of the noise. Though a full microscopic
understanding of TLS noise is yet to be established, previous studies have
shown that the TLS amplitude in MKIDs can be altered by changing the capacitor
geometry [9], and through substrate surface treatments prior to the deposition
of the metal films [10]. The generation-recombination noise of an MKID is
caused by the continual breaking and reforming of Cooper pairs within the
inductor. From the perspective of the resonator, the GR noise is also
dependent on inductor geometry and should decrease with reduced inductor
volume[11]. The impact of amplifier noise can be minimized by using amplifiers
with lower noise-equivalent temperatures or increasing the operating power
and, therefore, the resulting signal-to-noise ratio. We typically operate at
the power just below the bifurcation [12] threshold, where the power-dependent
nonlinear kinetic inductance starts to generate discontinuities in the
resonance.

In this paper, we fit detector noise power spectra to extract the TLS noise,
the GR noise, and the amplifier white noise for a series of detector designs
with different inductor volumes and interdigitated capacitor (IDC) geometries.
We repeat this analysis with different detector materials, including niobium,
aluminum, and manganese-doped aluminum (AlMn) with a lower superconducting
transition temperature $T_{c}$ than aluminum. The goal of this exercise is to
identify a set of geometric and material parameters that enable the tuning of
detector noise to achieve the requirements of a given an experiment. This has
immediate application both to our current efforts toward the development of
SPT-3G+ [13] and to future missions with more stringent noise requirements
such as an NEP at the level of $10^{-20}\mathrm{W\,/\sqrt{Hz}}$.

##  II Device Design and Fabrication

The MKID design used for this study is based on the prototype lumped element
KID design for SPT-3G+ [13]. We then modify component geometries and detector
materials to explore their impact on detector noise. Figure 1 shows a photo of
1$\times$0.5 inch device used for this study. The microstrip feedline is
capacitively coupled to five pixels, each composed of two MKIDs aligned to
orthogonal polarization optical modes. Operating as detectors, these
resonators are intended to couple to radiation via a feedhorn, which drives
the design of the resonator inductor to double as an efficient photon
absorber. Each inductor is coupled to an IDC whose geometry sets the unique
resonant frequency of each resonator on the device. Three types of devices
were fabricated, each designed to test one change to device geometry. These
include: A) a device with varying inductor volumes for all five pixels to
explore the impact on the GR noise, B) a device with a varying gap width
between IDC tines or equivalently varying IDC filling factor, and C) a device
with varying IDC tine width as well as the gap between tines. See Fig. 2 for
the design geometries of A), B), and C). The devices are fabricated via a
single-layer metal deposition on a high-resistivity silicon substrate. The
resonator geometry is defined via optical photolithography and a wet etch. The
silicon wafers are prepared with a buffered-HF etch to remove the native oxide
layer prior to metal deposition. Each six-inch wafer contains all of the
geometric design variations, so that the film material remains as consistent
as possible across all device types.

Figure 1: Microscope images of the type (A) chip described in the text with
different inductor volumes. Each pixel contains two detectors with two
inductors oriented in orthogonal directions and two associated capacitors to
form the resonators. The inductors are meandered metal lines, and the
capacitors utilize the interdigitated design. The capacitors have curved tines
that allow higher packing density in the final detector array. Bottom right
diagram is from [14].  Figure 2: Detector geometry variations. Here subfigures
A), B), and C) correspond to designs A), B), and C) discussed in the text.

##  III Noise Measurement and Modeling

Detector noise timestreams were measured with a homodyne single-tone setup
consisting of a microwave signal generator, quadrature demodulator, signal
splitter, attenuators, and a 200 kHz commercial ADC.

We measured our chips in a dilution refrigerator over a temperature range
spanning $\sim 8-300$ mK. The input line has attenuators located at 4 K with a
total attenuation of about 70 dB. The output signal is amplified by a
cryogenic low-noise amplifier with $\sim$ 30 dB gain and 6 K noise
temperature, as well as a room-temperature amplifier with $\sim$30 dB gain.

We report the noise power spectral densities (PSDs) calculated using the
local-gradient procedure in [15] and we have verified the consistency of this
method with the phase-shift method described in the same reference. Figure 3
shows example noise measurements for one detector taken at a microwave power
on the feedline of -110 dBm. At 8 mK, we observe in the noise PSD a clear
$1/\sqrt{f}$ slope associated with TLS noise [16] that decays with increasing
device temperature. Around 1000 Hz, we observe a rolloff in the PSD that is
related to the quasiparticle lifetime. The frequency of this rolloff shifts to
a higher frequency as the operating temperature increases since the
quasiparticle lifetime reduces at higher temperatures (Eq. 6) [17]. The noise
beyond 20 KHz is dominated by the white noise level of the cryogenic
amplifier. At higher temperatures, the detector’s $dI/df$ and $dQ/df$
responsivities reduce, which causes the white noise level in fractional
frequency shift ($df/f$) to increase when we convert the demodulated time-
ordered data to the $df/f$. We allow parameter $C$ to vary when fitting data
at different temperatures.

Figure 3: Sample fractional frequency shift noise PSD for a resonator. The
$1/\sqrt{f}$ feature at lower stage temperature corresponds to TLS noise and
decreases as we increase the operating temperature. The flat white noise level
in the mid-frequency range corresponds to the generation and recombination of
quasiparticles. The white noise floor above the GR rolloff at a few kHz is
mostly system noise dominated by amplifier noise.

Each noise PSD is fit with the following model [18]:

$S_{xx}(f)=\left(\frac{A+Bf^{-n}}{1+(2\pi f\tau)^{2}}+C\right),$ (1)

where $S_{xx}$ is the frequency noise PSD, $A$ is the GR noise component,
$Bf^{-n}$ is the $1/\sqrt{f}$ noise (mostly dominated by TLS in our system),
$\tau$ is the quasiparticle lifetime, and $C$ is the system white noise level.

To break the degeneracy of parameters, we note that the GR noise level $A$
does not depend on temperature because the temperature dependence of $N_{qp}$
and $\tau$ cancels [11]. Thus, we can fit for $A$ using high-temperature data
where the TLS component is suppressed, and then fit for $B$ in low-temperature
data using a model with fixed $A$. We found that our low-temperature data is
consistent with $n\sim 0.5$ and therefore fixed $n=0.5$ to better expose the
geometry dependence of parameter $B$. The quasiparticle lifetime, $\tau$, and
the system white noise level, $C$, are not degenerate with other parameters
since they correspond to distinct features: the rolloff and the flat region at
high frequency ($\sim 10^{5}$ Hz).

Some of the AlMn detectors were found to have a non-negligible resonator ring-
down time $\tau_{\text{res}}={Q}/{\pi f_{0}}$, where $Q$ is the resonator
quality factor, and $f_{0}$ is the resonant frequency. Since the ring-down
time is assumed to be negligible in Eq. (1), we re-introduced the rolloff term
$1+(2\pi f\tau_{\text{res}})^{2}$ to the denominator of Eq. (1) for AlMn
detectors with $\tau_{res}$ from independent measure of $Q$ and $f_{0}$ [11].
The AlMn resonators also have low bifurcation power [19], which in some cases
was only slightly above the system white noise floor. As a result, we
collected limited data for a fraction of AlMn resonators within a small range
of stage temperature and bias powers relative to the datasets taken for other
materials. For a robust extraction of the fit parameters, we fit multiple
noise curves taken for the same resonator at different temperatures and powers
simultaneously.

##  IV Generation-recombination Noise Optimization

The generation-recombination noise contribution to the quasiparticle
fluctuation PSD is expected to follow:

$S_{N}(f)=\frac{4N_{qp}\tau}{1+(2\pi f\tau)^{2}},$ (2)

where $N_{qp}$ is the number of quasiparticles. The numerator of this
expression should scale with detector volume $V_{L}$ since
$N_{qp}=n_{qp}V_{L}$, where $n_{qp}$ is the quasiparticle number density. We
seek to demonstrate the dependence of the numerator on inductor volume.

(a)  X-polarization detectors, aluminum, type A  (b)  Y-polarization
detectors, aluminum, type A

Figure 4: Quasiparticle number fluctuation PSD vs. inductor volume for X and Y
polarizations. We separate X and Y due to different inductor geometries.

We fit the noise PSDs of aluminum detectors with varying inductor volume (Type
A from Section II), and extracted the GR noise level ($A$ in Eq. (1)). This
parameter theoretically has dependence
$A=4N_{qp}\tau\left(dx/dN_{qp}\right)^{2}$, where $x=df/f$. To isolate the
linear dependence of $4N_{qp}\tau$ on inductor volume, we divide out the
$dx/dN_{qp}$ term by recognising that

$\frac{dx}{dN_{qp}}=\frac{dx}{dT}\frac{dT}{dN_{qp}}=\frac{1}{V_{L}}\frac{dx}{dT}\frac{dT}{dn_{qp}},$
(3)

and making use of expression for quasiparticle number density:

$n_{qp}=2N_{0}\sqrt{2\pi k_{B}T\Delta}\exp\left(-\Delta/k_{B}T\right),$ (4)

where $N_{0}$ is the single spin density of states at the Fermi level
($1.72\times 10^{10}\mathrm{\mu m}^{-3}eV^{-1}$ for aluminum), $k_{B}$ is the
Boltzmann constant, and $\Delta=1.76k_{B}T_{c}$ is the energy gap for
aluminum. Fig. 4 shows $4N_{qp}\tau$ as a function of inductor volume for the
aluminum devices. The data was taken at 0.2 K where the TLS noise is reduced.
We grouped the detectors by polarization alignment, as the inductor geometry
differs slightly between polarization. This figure shows that the GR noise has
a positive correlation with inductor volume. We note that X and Y detectors
are different (Fig. 2A). The observed differences between X vs. Y
polarizations are consistent with different excess quasiparticles arising from
different coupling to the RF bias between the two classes of detectors. The
coupling quality factors $Q_{c}$ for X detectors are $(1.0\pm 0.1)\mathrm{e}5$
and $(1.1\pm 0.4)\mathrm{e}4$ for Y detectors, which are different by a factor
of $\sim$10\. The difference in $Q_{c}$ leads to a difference in bifurcation
power of $\sim$20 dB. With a similar amount of power at the feedline, X
detector noise may have more contribution from quasiparticles agitated by the
bias power that makes the noise for X detectors deviate from that for Y and a
linear dependence on volume. As to the power dependence for Y, we noted that Y
detectors have more TLS noise contribution compared to X detectors, probably
due to different internal power for Y or the different geometry. And the
sloped TLS noise can have some degeneracy with the flat GR noise term. As we
increased the power, we saturate more of the TLS component and reduce the
systematics from the TLS component, leading to a reduction of fit A values.

##  V Two-level System Noise Optimization

We fit noise PSDs for aluminum detectors of varying IDC tine gap width (Type B
from II) and IDC tine width (Type C from II) to extract the TLS amplitude ($B$
from Equation 1).

Fig. 5a shows a clear decrease in TLS amplitude as the IDC tines are spaced
further apart. With a larger gap between the IDC tines, the electric field is
smaller, resulting in weaker coupling to the dipole states in the amorphous
solid.

Coupling of the field to the TLS is expected to cause a shift in both
quadratures of the dielectric response function [20]. Assuming a log-uniform
distribution of two-level tunneling states, the resonance frequency shift can
be calculated from the modification to the dielectric [21]. We can measure the
quantity $F\delta$ by fitting resonant frequency vs. temperature to a TLS
model [16]. Here $F$ is the filling factor, which defines the fraction of the
power coupled through the dielectric, and $\delta$ is the loss tangent for the
dielectrics. We measured the TLS loss for the resonators and plotted the fit
$B$ values vs. $F\delta$ in Fig. 5b, where a clear correlation is observed. We
repeated these measurements for niobium resonators with detector type C and
found a similar positive correlation between $B$ and $F\delta$ in Fig. 5c.

(a)  Fit $B$ vs. capacitor finger gap, aluminum, type B  (b)  Fit $B$ vs.
$F\delta$, aluminum, type B  (c)  Fit $B$ vs. $F\delta$, niobium, type B

Figure 5: TLS noise’s dependence on capacitor finger gap width and $F\delta$.
(a) and (b) are for aluminum, and (c) is for niobium.

To minimize TLS noise, we should reduce the filling factor in dielectrics or
surface states. While intuitive ways such as changing capacitor gap size
exist, a more systematic way for this optimization is through simulations
using electromagnetic simulation software, such as HFSS
111https://www.ansys.com/products/electronics/ansys-hfss. We can simulate the
filling factors for potential TLS locations such as dielectrics, surface oxide
layers, or interface layers and study their correlation with the observed TLS
noise levels.

There are additional external parameters for tuning the TLS noise, such as the
driving power and the operating temperature. The phenomenological TLS model
predicts the dependence on power and temperature as [22]

$S_{xx}\propto P^{\alpha}f^{-1/2}T^{\beta}\tanh\left(hf_{0}/2k_{B}T\right),$
(5)

where $P$ is the power, $f$ is the frequency, $\beta$ is an empirically
derived exponent, and $f_{0}$ is the resonant frequency. We fit our measured
TLS noise levels parametrized by $B$ qualitatively to the model in Eq. (5) and
plot the data and dashed fit lines in Fig. 6. The fit values are
$\alpha=-0.3\pm 0.1$ and $\beta=0.8\pm 0.2$. Similar to [22], our data cannot
distinguish temperature dependence between this model and a power law of
$T^{-0.6}$. Ref. [22] reported $\alpha=-0.46\pm 0.05$ and $\beta=-0.14\pm
0.02$ for niobium and the temperature range of 0.12 to 1.2 K. Our measurements
for aluminum from 0.01 to 0.32 K indicate the exponents can change with
material and temperature. We note that the physical mechanism of the TLS noise
remains unclear and further study is needed beyond the phenomenological model.
To reduce TLS noise, we can operate at a high bias power just below the onset
of bifurcation [12], and raise the stage temperature ensuring that the white
noise level and resonator quality factor are not degraded.

(a)  Fit $B$ vs. driving power, aluminum, type B  (b)  Fit $B$ vs. operating
temperature, aluminum, type B

Figure 6: TLS noise’s dependence on driving power and temperature. We have
included ten detectors in each plot for five different gap sizes and two
polarizations. The design is type (B) in Fig. 2. We added dashed fit lines to
Eq. 5 and got $\alpha=-0.3\pm 0.1$ and $\beta=0.8\pm 0.2$.

##  VI Quasiparticle lifetime

The quasiparticle lifetime is another parameter we can fit from the $S_{xx}$
rolloff frequency. It is related to the recombination time $\tau_{R}$, which
for low temperatures is approximately given by [23]. The recombination time is
summarized in [23]:

$\tau_{R}=\frac{\tau_{0}}{\sqrt{\pi}}\left(\frac{k_{B}T_{c}}{2\Delta}\right)^{5/2}\sqrt{\frac{T_{c}}{T}}e^{\Delta/k_{B}T}=\frac{\tau_{0}}{n_{qp}}\frac{N_{0}\left(k_{B}T_{c}\right)^{3}}{2\Delta^{2}},$
(6)

where $\tau_{0}$ is a material-dependent characteristic electron-phonon
interaction time and can be modified by impurity scattering. Note that one
recombination is associated with the disappearance of two quasiparticles so
that $\Gamma_{R}^{*}=2\Gamma_{R}F_{\omega}^{-1}$[24], where $\Gamma_{R}$ is
the recombination rate, $\Gamma_{R}^{*}$ is the time constant for a small
quasiparticle perturbation to decay (quantity measured experimentally), and
$F_{\omega}$ is the phonon trapping factor that accounts for pair breaking by
emitted phonon from the recombination. If we neglect phonon trapping,
$\Gamma_{R}^{*}=2\Gamma_{R}$, and the resulting measured quasiparticle
lifetime $\tau=\tau_{R}/2$, where $\tau_{R}$ is defined in Eq. (6). We plot
the dependence of the quasiparticle lifetime as a function of temperature in
Fig. 7 and then fit it to the model in Eq. (6) with the factor of two
correction using temperatures above 0.24 K. The fit value for $\tau_{0}$ is
0.3$\pm$0.1 $\mu$s, which is of the same order of magnitude as for previous
measurements [23, 25] but somewhat smaller. Ref. [26] found impurities can
change quasiparticle recombination time, which could mean there are impurities
in our aluminum. Another uncertainty comes from $T_{c}$ and $\Delta$. We found
the $\tau_{0}$ in the model fit sensitive to the preset $T_{c}$ and $\Delta$
values, and that using $T_{c}=1.2$ K and $\Delta=1.76k_{B}T_{c}$ gives
$\tau_{0}=0.6\,\mu$s for the fit. The current $T_{c}=1.33$ K we used for this
analysis is from a four-wire temperature sweep, which is close to our
measurement of 1.37 K by fitting $f$ vs. $T$ to a Mattis-Bardeen model [21].
In the model, $\Delta=1.76\,k_{B}T_{c}$ may not be accurate for a thin film,
and we neglected phonon trapping. We note these uncertainties for our
$\tau_{0}$ result. The quasiparticle lifetime below $\sim$0.2 K is limited by
other relaxation mechanisms, such as nonequilibrium quasiparticle excitations
[27, 28].

Figure 7: Quasiparticle lifetime vs. operating temperature. We have included
ten detectors in each plot for five different gap sizes and two polarizations.
Plotted is for aluminum type (B) in Fig. 2. We fit data above 0.24 K to the
model in Eq. 6 with the correction.

##  VII AlMn Noise Results

The AlMn samples we used are the same as the ones reported in [19]. We can
reach a lower $T_{c}$ than pure aluminum by doping aluminum with manganese,
and baking provides yet another knob to tune their $T_{c}$. Low-Tc MKIDs are
interesting for spectrometers because they open up below 100 GHz for a few
redshifted CO lines and are also useful for MKIDs-based dark matter searches
that require higher sensitivity. The aluminum and AlMn films were sputtered
using high-purity aluminum and Mn doped aluminum with 1050 ppm (parts per
million) and 1150 ppm doping levels, respectively. The superconducting
transition temperatures for the aluminum sample, 1050 ppm AlMn, and 1150 ppm
AlMn are 1.37 K, 0.73 K, and 0.61 K, respectively. The 1050 ppm sample
subsequently went through a baking process at 180 ${}^{\circ}\mathrm{C}$ for
10 minutes, which shifted its $T_{c}$ from 0.73 K pre-baking to 0.78 K after
baking. Our goals with the AlMn samples are to understand the dependence of GR
noise level and quasiparticle lifetime on the doping and baking conditions.

The measured noise PSDs for the AlMn resonators are similar to those for the
aluminum resonators shown in Fig. 3, but the PSD curves are noiser especially
in the low frequency region below about 1 kHz. Since the GR noise level $A$,
in the temperature and bias power range where we took data, is independent of
temperature or bias power, we fit the set of PSD curves for an AlMn resonator
simultaneously with a single parameter $A$, while the $\tau_{res}$ corrections
were applied independently to each noise curve. The quasiparticle lifetime for
each noise curve, $\tau$, was also obtained from this global fit. The GR noise
level can in principle be expressed in quasiparticle number fluctuations,
$4N_{qp}\tau$, by factoring out $dx/dN_{qp}$ from $A$. However, we did not do
this for the AlMn resonators because $N_{0}$ for AlMn was not known to us.

Fig. 8 shows the GR noise level $A$ in fractional frequency shift for
different doping and baking configurations. We observe an increase in GR noise
with increasing Mn doping levels, and we find that baking makes the GR noise
slightly worse for the 1050 ppm AlMn resonators. Clearly, these observations
are still limited by statistics because of the limited dataset.

We plot the fitted quasiparticle lifetime $\tau$ as a function of the reduced
temperature $T/T_{c}$ in Fig. 9. All samples show saturated $\tau$ when
$T/T_{c}<0.1$, but their rolloff start at different reduced temperatures.
Doped aluminum with impurities can be analyzed with more complex theories
developed by Zittartz, Bringer, and Müller-Hartmann [29], or Kaiser [30].
Barends et al. [26] obtained some qualitative conclusions applying these
models but did not model their data quantitatively. $\tau$ for AlMn in Fig. 9
show wider scatter compared with the Al resonators shown in Fig. 7. This is
because the AlMn resonators have lower bifurcation power than pure aluminum,
which required a reduction in the bias power for the measurement and
consequently decrease the signal-to-noise level.

We can draw qualitative conclusions from the $\tau$ vs. $T/T_{c}$ dependence
shown in Fig. 9. In the same figure, the modeled quasiparticle lifetime
$\tau=\tau_{R}/2$ using Eq. (6) are also shown. $\tau_{0}$’s are chosen that
the theoretical curves match the corresponding roll-off edges visually.
Assuming all samples are BCS superconductors, the samples doped with manganese
have $\tau_{0}$ about an order of magnitude smaller than pure aluminum. This
reduction in $\tau_{0}$ is not unexpected through its dependence on impurities
[31]. We realize that the BCS modeling may not fully apply here: O’Neil et al.
[32] demonstrated that the density of states of AlMn remains BCS-like, but
additional sub-gap states and gap edge smearing are also present.

Figure 8: Histograms of the GR noise level $A$ in $S_{xx}$ for pure aluminum
and AlMn with different Mn concentrations.  Figure 9: Quasiparticle lifetime
vs. reduced temperature $T/T_{c}$ for pure aluminum and AlMn with different Mn
concentrations. We have added model curves using Eq. (6) with the samples’
measured $T_{c}$. The $\tau_{0}$ values for the model are chosen such that the
curves follow the falling edges at higher temperatures. The data is too noisy
for a good fit, but we can still tell $\tau_{0}$ for AlMn is an order of
magnitude smaller than pure aluminum if we assume a BCS description.

Finally, we use the GR noise level $A$ to constrain $\tau_{0}$ and $N_{0}$.
With Eq. (4) and Eq. (6), we can decompose the dependence of $A$ into:

$A=4N_{qp}\tau\left(dx/dN_{qp}\right)^{2}=\frac{\tau_{0}}{N_{0}}\frac{2}{V_{L}}\left(\frac{dx}{d\rho}\right)^{2}\frac{(k_{B}T_{c})^{3}}{2\Delta^{2}},$
(7)

where $\rho=n_{qp}/N_{0}=2\sqrt{2\pi k_{B}\Delta}\exp{(-\Delta/k_{B}T)}$. We
can put constraints on $\tau_{0}/N_{0}$ using Eq. (7) since all other terms
can be measured or are known from the design. The resulting histograms for
$\tau_{0}/N_{0}$ are summarized in Fig. 10. The $\tau_{0}/N_{0}$ values for
pure aluminum are about two orders of magnitude larger than those for AlMn,
indicating that the density of states $N_{0}$ for AlMn is higher than that for
aluminum since the $\tau_{0}$’s only differ by one order of magnitude. Ref
[33] reported a calculation of local density of states for Al with impurities.
One caveat is that we have assumed all the samples are BCS-like.

Figure 10: $\tau_{0}/N_{0}$ histograms for pure aluminum and AlMn with
different Mn concentrations.

##  VIII Conclusions

In this paper, we explored and validated a few directions to control the noise
components of MKIDs. For aluminum detectors, we were able to tune the GR noise
by changing the inductor volume of our detectors and observed a positive
correlation between the quasiparticle fluctuation noise and inductor volume.
We can reduce the TLS noise using three methods: 1) lowering the capacitor
filling factor, 2) raising the operating temperature, and 3) increasing the
driving power. We compared the temperature and power dependence of TLS with
models and correlated the TLS noise with $F\delta$ measured using a TLS
dielectric loss model. We did the TLS study for both niobium and aluminum
resonators. We also explored the noise components’ dependence on manganese
doping with AlMn detectors. We found manganese doping increases the GR noise
level. Assuming a BCS description, we observed that AlMn samples prefer a
lower $\tau_{0}$ and a higher $N_{0}$ than aluminum samples. The noise for our
samples can be controlled below the photon noise limit for ground-based
photometer applications (e.g., SPT3G+). However, we need more thorough
optimization for space applications or spectrometers with narrower bands that
require lower noise. This study provides a few directions for future noise
optimization.

##  IX Acknowledgements

Work at Argonne, including the use of the Center for Nanoscale Materials, an
Office of Science user facility, was supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, and Office of High
Energy Physics, under Contract No. DE-AC02-06CH11357. Zhaodi Pan is supported
by ANL under the award LDRD-2021-0186. Maclean Rouble acknowledges funding
from the Natural Sciences and Engineering Research Council of Canada and
Canadian Institute for Advanced Research.

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of out-of-budget epochs across all reports.

For a report $\rho$ with attribution function $A$ over $k$ epochs, we also
define:

(52)
$\displaystyle\Delta^{\max}(\rho_{r}):=\max_{\mathbf{F},\mathbf{F}^{\prime}\in\mathcal{P}(\mathcal{I}\cup\mathcal{C})^{k}:\forall
i\in[k],\mathbf{F}_{i}^{\prime}=\mathbf{F}_{i}\text{ or
}\mathbf{F}_{i}^{\prime}=\emptyset}\|A(\mathbf{F})-A(\mathbf{F}^{\prime})\|_{1}$

That is, $\Delta^{\max}(\rho_{r})$ is the maximum L1 change that happens when
we remove any number of device-epochs from the database. By comparison, the
global sensitivity $\Delta(\rho_{r})$ is the maximum change that happens when
we remove a single device-epoch from the database. For certain attribution
functions, such as last touch attribution,
$\Delta^{\max}(\rho_{r})=\Delta(\rho_{r})$, as detailed in Lemma4.

Then, we have:

$\displaystyle\left\|\mathbb{E}[\mathcal{M}(D)-Q(D)]\right\|_{1}\leq\frac{\mathcal{M}_{0}(D)+\sigma\ln(1/\beta)/\sqrt{2}}{\kappa}\max_{r\in
R}\Delta^{\max}(\rho_{r})$

###### Proof.

There are two differences compared to the proof of Thm. 1.

First, $\mathcal{M}_{0}(D)$ is an unbiased estimate of $\tilde{Q}_{0}(D)$,
which is an upper bound on the number of reports containing at least one out-
of-budget epoch.

Second, we can use $\tilde{Q}_{0}(D)$ to bound the bias as follows:

(53)
$\displaystyle\left\|\mathbb{E}[\mathcal{M}(D)-Q(D)]\right\|_{1}=\|\tilde{Q}(D)-Q(D)\|_{1}$
(54) $\displaystyle=\|\sum_{r\in
R}A(F_{r,e_{1}},\dots,F_{r,e_{k}})-A(D_{d_{r}}^{e_{1}},\dots,D_{d_{r}}^{e_{k}}\|_{1}$
(55) $\displaystyle\leq\sum_{r\in
R}\|A(F_{r,e_{1}},\dots,F_{r,e_{k}})-A(D_{d_{r}}^{e_{1}},\dots,D_{d_{r}}^{e_{k}}\|$
(56) $\displaystyle\leq\sum_{r\in R:\forall
i\in[k],F_{r,e_{i}}=D_{d_{r}}^{e_{i}}}\|A(F_{r,e_{1}},\dots,F_{r,e_{k}})-A(D_{d_{r}}^{e_{1}},\dots,D_{d_{r}}^{e_{k}}\|_{1}$
(57) $\displaystyle+\sum_{r\in R:\exists i\in[k]:F_{r,e_{i}}\neq
D_{d_{r}}^{e_{i}}}\|A(F_{r,e_{1}},\dots,F_{r,e_{k}})-A(D_{d_{r}}^{e_{1}},\dots,D_{d_{r}}^{e_{k}}\|_{1}$
(58) $\displaystyle\leq\sum_{r\in R:\exists
i\in[k]:F_{r,e_{i}}=\emptyset}\|A(F_{r,e_{1}},\dots,F_{r,e_{k}})-A(D_{d_{r}}^{e_{1}},\dots,D_{d_{r}}^{e_{k}}\|_{1}$
(59) $\displaystyle\leq\sum_{r\in R:\exists
i\in[k]:F_{r,e_{i}}=\emptyset}\Delta^{\max}(\rho_{r})$ (60)
$\displaystyle\leq(\tilde{Q}_{0}(D)/\kappa)\max_{r\in
R}\Delta^{\max}(\rho_{r})$

We conclude with a Laplace tail bound as in Thm. 1.

∎

###### Theorem 3 (Sensitivity of counting queries).

Consider a query $(d_{r},E_{r},A_{r},\rho_{r})_{r\in R}$ augmented by a
counting query as in Thm. 1 or Thm. 2. Take a report
$\hat{\rho}_{r}:D\mapsto(\rho_{r}^{0}(D),\rho_{r}(D))\in\mathbb{R}^{m}$ where
$\rho_{r}^{0}(D)=\kappa\cdot\sum_{e\in
E_{r}}\mathds{1}[D_{d_{r}}^{e}=\emptyset]$ (Thm. 1 ) or
$\rho_{r}^{0}(D)=\kappa\cdot\mathds{1}[\exists e\in
E:D^{e}_{d_{r}}=\emptyset]$ (Thm. 2).

Take $x=(d,e,F)\in\mathcal{X}$. We have:

(61)
$\displaystyle\Delta_{x}(\hat{\rho}_{r})\leq\kappa\cdot\mathds{1}[d=d_{r},e\in
E_{r}\text{ and }F\neq\emptyset]+\Delta_{x}(\rho_{r})$

This means that every requested device-epoch that has budget left and contains
data should pay additional budget for the DP count.

###### Proof.

First, we have:

(62)
$\displaystyle\Delta_{x}(\hat{\rho}_{r})\leq\Delta_{x}(\rho_{r}^{0})+\Delta_{x}(\rho_{r})$

because for all $D,D^{\prime}$ such that $D^{\prime}=D+x$ we have
$\|\hat{\rho}_{r}(D^{\prime})-\hat{\rho}_{r}(D)\|_{1}\leq\|\rho_{r}^{0}(D^{\prime})-\rho_{r}^{0}(D)\|_{1}+\|\rho_{r}(D^{\prime})-\rho_{r}(D)\|_{1}\leq\Delta_{x}(\rho_{r}^{0})+\Delta_{x}(\rho_{r})$.

Second, we have:

(63) $\displaystyle\Delta_{x}(\rho_{r}^{0})=\begin{cases}\kappa&\text{if
}d=d_{r},e\in E_{r}\text{ and }F\neq\emptyset\\\
0&\text{otherwise}\end{cases}$

Indeed, consider $D,D^{\prime}$ such that $D^{\prime}=D+x$.

  * •

If $F=\emptyset$, $d\neq d_{r}$, or $e\not\in E_{r}$ we have
$\rho_{r}^{0}(D)=\rho_{r}^{0}(D^{\prime})$ for all such $D,D^{\prime}$ so
$\Delta_{x}(\rho_{r}^{0})=0$.

  * •

If $F\neq\emptyset$, $d=d_{r}$ and $e\in E_{r}$ we have:

    * –

For Thm. 1,
$\|\rho_{r}^{0}(D^{\prime})-\rho_{r}^{0}(D)\|_{1}=\|\kappa\cdot\sum_{\hat{e}\in
E_{r}}\mathds{1}[(D^{\prime})_{d_{r}}^{\hat{e}}=\emptyset]-\mathds{1}[D_{d_{r}}^{\hat{e}}=\emptyset]\|=\kappa\cdot\mathds{1}[F=\emptyset]-\mathds{[}\emptyset=\emptyset]=\kappa$.
This is true for all such $D,D^{\prime}$, so
$\Delta_{x}(\rho_{r}^{0})=\kappa$.

    * –

For Thm. 2,
$\|\rho_{r}^{0}(D^{\prime})-\rho_{r}^{0}(D)\|_{1}=\|\kappa\cdot\mathds{1}[\exists\hat{e}\in
E_{r}:(D^{\prime})^{\hat{e}}_{d_{r}}=\emptyset]-\mathds{1}[\exists\hat{e}\in
E_{r}:D^{\hat{e}}_{d_{r}}=\emptyset]\|\leq\kappa$. Moreover, this max is
attained for $D=\\{(d_{r},\hat{e},F),\hat{e}\in
E_{r}\setminus\\{e\\}\\},D^{\prime}=\\{(d_{r},\hat{e},F),\hat{e}\in E_{r}\\}$.
Hence $\Delta_{x}(\rho_{r}^{0})=\kappa$.

∎

###### Theorem 4 (Sensitivity for certain histogram attribution functions).

Consider an attribution function $A$ of the following form. First, $A$
attributes a positive value $a_{\mathbf{F}}(f)$ to each relevant event $f\in
F_{1}\cap F_{A}\cup\dots\cup F_{k}\cap F_{A}$. Next, each event is mapped to a
one-hot vector $H(f)\in\mathbb{R}^{m}$ (i.e., $H(f)\in\\{0,1\\}^{m}$ and
$\|H(f)\|_{1}=1$). Finally, the attribution is the weighted sum:

(64) $\displaystyle A(F_{1},\dots,F_{k})=\sum_{i=1}^{k}\sum_{f\in F_{i}\cap
F_{A}}a_{\mathbf{F}}(f)\cdot H(f)$

We define:

(65) $\displaystyle
A^{\max}:=\max_{\mathbf{F}\in\mathcal{P}(\mathcal{I}\cup\mathcal{C})^{k}}\sum_{i=1}^{k}\sum_{f\in
F_{i}\cap F_{A}}a_{\mathbf{F}}(f)$

Consider any attribution report $\rho_{r}$ with attribution function $A$ with
output in $\mathbb{R}^{m}$.

  * •

If $m=1$ or $k=1$, we have

(66) $\displaystyle\Delta(\rho_{r})\leq\Delta^{\max}(\rho_{r})\leq A^{\max}$

Moreover, if there exists
$\mathbf{F}^{\max}=(\emptyset,\dots,\emptyset,\\{f_{0}\\},\emptyset,\dots,\emptyset)$
containing a single relevant event $f_{0}\in F_{A}$ such that $A^{\max}$ is
attained, i.e., $a_{\mathbf{F}^{\max}}(f_{0})=A^{\max}$, then

(67) $\displaystyle\Delta(\rho_{r})=\Delta^{\max}(\rho_{r})=A^{\max}$

  * •

If $m\geq 2$ and $k\geq 2$, we have:

(68) $\displaystyle\Delta(\rho_{r})\leq\Delta^{\max}(\rho_{r})\leq 2A^{\max}$

Moreover, if there exists
$\mathbf{F}^{\max}=(\emptyset,\dots,\emptyset,\\{f_{0}\\},\emptyset,\dots,\\{f_{1}\\},\emptyset)$
and
$\mathbf{F}^{\prime\max}=(\emptyset,\dots,\emptyset,\\{f_{0}\\},\emptyset,\dots,\emptyset)$
such that $a_{\mathbf{F}^{\max}}(f_{0})=A^{\max}$,
$a_{\mathbf{F^{\prime}}^{\max}}(f_{1})=A^{\max}$ and $H(f_{0})\neq H(f_{1})$,
then:

(69) $\displaystyle\Delta(\rho_{r})=\Delta^{\max}(\rho_{r})=2A^{\max}$

###### Proof.

Consider a report $\rho_{r}$ with such an attribution function $A$. First, we
observe that $A(\emptyset)=0\in\mathbb{R}^{m}$, because of Eq. 64.

We start by upper bounding $\Delta^{\max}(\rho_{r})$. Take
$\mathbf{F},\mathbf{F}^{\prime}\in\mathcal{P}(\mathcal{I}\cup\mathcal{C})^{k}:\forall
i\in[k],\mathbf{F}_{i}^{\prime}=\mathbf{F}_{i}\text{ or
}\mathbf{F}_{i}^{\prime}=\emptyset$.

  * •

If $m=1$, for any event $f$ we have $H(f)=1$. Since $a_{\mathbf{F}}(f)\geq 0$,
we have: $\sum_{i=1}^{k}\sum_{f\in F_{i}\cap F_{A}}a_{\mathbf{F}}(f)\cdot
H(f)-\sum_{f\in F_{i}^{\prime}\cap F_{A}}a_{\mathbf{F^{\prime}}}(f)\cdot
H(f)\leq\sum_{i=1}^{k}\sum_{f\in F_{i}\cap F_{A}}a_{\mathbf{F}}(f)\cdot 1\leq
A^{\max}$ and $\sum_{i=1}^{k}\sum_{f\in F_{i}\cap F_{A}}a_{\mathbf{F}}(f)\cdot
H(f)-\sum_{f\in F_{i}^{\prime}\cap F_{A}}a_{\mathbf{F^{\prime}}}(f)\cdot
H(f)\geq-\sum_{f\in F_{i}^{\prime}\cap F_{A}}a_{\mathbf{F^{\prime}}}(f)\cdot
1\geq-A^{\max}$. Hence, $\|A(\mathbf{F})-A(\mathbf{F}^{\prime})\|_{1}\leq
A^{\max}$, and thus $\Delta^{\max}\leq A^{\max}$.

  * •

If $k=1$, we have $\mathbf{F}^{\prime}=F_{1}\text{ or }\emptyset$. In the
first case, $\|A(\mathbf{F})-A(\mathbf{F}^{\prime})\|_{1}=0\leq A^{\max}$. In
the second case,

(70) $\displaystyle\|A(\mathbf{F})-A(\mathbf{F}^{\prime})\|_{1}$
$\displaystyle=\|A(\mathbf{F})\|_{1}$ (71) $\displaystyle\leq\sum_{f\in
F_{1}\cap F_{A}}a_{\mathbf{F}}(f)\|H(f)\|_{1}$ (72) $\displaystyle\leq
A^{\max}$

Hence $\Delta^{\max}\leq A^{\max}$.

  * •

If $m\geq 2$, we have:

(73) $\displaystyle\|A(\mathbf{F})-A(\mathbf{F}^{\prime})\|_{1}$
$\displaystyle=\|\sum_{i=1}^{k}\sum_{f\in F_{i}\cap
F_{A}}a_{\mathbf{F}}(f)\cdot H(f)$ (74) $\displaystyle\hskip
8.53581pt-\sum_{f\in F_{i}^{\prime}\cap F_{A}}a_{\mathbf{F^{\prime}}}(f)\cdot
H(f)\|_{1}$ (75) $\displaystyle\leq\sum_{i=1}^{k}\sum_{f\in F_{i}\cap
F_{A}}a_{\mathbf{F}}(f)\|H(f)\|_{1}$ (76)
$\displaystyle+\sum_{i=1}^{k}\sum_{f\in F_{i}^{\prime}\cap
F_{A}}a_{\mathbf{F^{\prime}}}(f)\|H(f)\|_{1}$ (77) $\displaystyle\leq
2A^{\max}$

This is true for any such $\mathbf{F},\mathbf{F}^{\prime}$, so
$\Delta^{\max}\leq 2A^{\max}$.

Next, we lower bound $\Delta^{\max}$.

  * •

If $m=1$ or $k=1$, and if there exists
$\mathbf{F}^{\max}=(\emptyset,\dots,\emptyset,\\{f_{0}\\},\emptyset,\dots,\emptyset)$
such that $a_{\mathbf{F}^{\max}}(f_{0})=A^{\max}$, we have

(78) $\displaystyle\Delta^{\max}(\rho_{r})$
$\displaystyle=\max_{\mathbf{F},\mathbf{F}^{\prime}\in\mathcal{P}(\mathcal{I}\cup\mathcal{C})^{k}:\forall
i\in[k],\mathbf{F}_{i}^{\prime}=\mathbf{F}_{i}{or}\mathbf{F}_{i}^{\prime}=\emptyset}\|A(\mathbf{F})-A(\mathbf{F}^{\prime})\|_{1}$
(79) $\displaystyle\geq\|A(\mathbf{F}^{\max})-A(\emptyset)\|_{1}$ (80)
$\displaystyle=\|A^{\max}\cdot H(f_{0})-0\|_{1}$ (81) $\displaystyle=A^{\max}$

(in fact this is true even when $m\neq 1$ and $k\neq 1$).

  * •

If $m\geq 2$ and $k\geq 2$, and there exists $f_{0},f_{1}$ such that removing
$f_{1}$ shifts the attribution to $f_{0}$, and $H(f_{0})\neq H(f_{1})$, then:

(82) $\displaystyle\Delta^{\max}(\rho_{r})$
$\displaystyle\geq\|A(\mathbf{F}^{\max})-A(\mathbf{F}^{\prime\max})\|_{1}$
(83) $\displaystyle=\|A^{\max}\cdot H(f_{0})-A^{\max}\cdot H(f_{1})\|_{1}$
(84) $\displaystyle=2A^{\max}$

We now focus on $\Delta(\rho_{r})$. First, we have
$\Delta(\rho_{r})\leq\Delta^{\max}(\rho_{r})$, because if we note
$N:=\\{\mathbf{F},\mathbf{F}^{\prime}\in\mathcal{P}(\mathcal{I}\cup\mathcal{C})^{k}:\exists
i\in[k]:\mathbf{F}^{\prime}_{i}=\emptyset\wedge\forall j\neq
i,\mathbf{F}^{\prime}_{i}=\mathbf{F}_{i}\\}$ and
$N^{\max}:=\\{\mathbf{F},\mathbf{F}^{\prime}\in\mathcal{P}(\mathcal{I}\cup\mathcal{C})^{k}:\forall
i\in[k],\mathbf{F}^{\prime}_{i}=\mathbf{F}_{i}\text{ or
}\mathbf{F}^{\prime}_{i}=\emptyset\\}$ we have $N\subset N^{\max}$.

Second, the pairs of databases $\mathbf{F}^{\max},\mathbf{F}^{\prime\max}$
exhibited in Eq. 78 and Eq. 82 happen to belong to both $N$ and $N^{\max}$, so
the upper bounds hold. ∎

#### Instantiation.

In particular, the upper bounds from the previous lemma apply when the
attribution function $A$ distributes a predetermined conversion value across
impressions (e.g., last-touch, first-touch, uniform, etc.), maps each
impression to a bin (e.g., $H(f)$ is a one-hot encoding of one of $m$ campaign
identifiers), and then sums up the value in each bin. The resulting report
$\rho_{r}(D)\in\mathbb{R}^{m}$ contains a histogram of the total attributed
conversion value per bin.

The first tightness result (Eq. 67) applies if there exists an impression that
can be fully attributed. The second tightness result (Eq. 69) applies if there
exists two impressions $f_{0},f_{1}$ with different one-hot encodings, such
that removing $f_{1}$ shifts the maximum attribution value $A^{\max}$ to
$f_{0}$ (e.g., in last-touch attribution).

Note that we allow $A^{\max}$ to have any value, and we don’t require every
database to be fully attributed. This is a slight generalization of (DGK+23,
10), which defines an attribution rule that requires $\sum_{i=1}^{k}\sum_{f\in
F_{i}\cap F_{A}}=1$.


11institutetext: Department of Physics, National and Kapodistrian University
of Athens, 15783 Athens, Greece
11email<EMAIL_ADDRESS>22institutetext: Department of Physics,
University of Ioannina, 45110 Ioannina, Greece 33institutetext: University of
Thessaly, 35100 Lamia, Greece

# High-resolution observations with ARTEMIS/JLS and the NRH††thanks: This work
is dedicated to the memory of Costas Caroubalos (1928-2021), founder of the
ARTEMIS radiospectrograh.

IV. Imaging spectroscopy of spike-like structures near the front of type-II
bursts

S. Armatas 11 C. Bouratzis 11 A. Hillaris 11 C.E. Alissandrakis 22 P. Preka-
Papadema A. Kontogeorgos 1133 P. Tsitsipis 33 X. Moussas 11

(Received …..; accepted ……)

###### Abstract

Context. Narrowband bursts (spikes) are very small duration and bandwidth
bursts which appear on dynamic spectra from microwave to decametric
frequencies. They are believed to be manifestations of small-scale energy
release through magnetic reconnection.

Aims. We study the position of the spike-like structures relative to the front
of type-II bursts and their role in the burst emission.

Methods. We used high-sensitivity, low-noise dynamic spectra obtained with the
acousto-optic analyzer (SAO) of the ARTEMIS-JLS solar radiospectrograph, in
conjunction with high-time-resolution images from the Nançay Radioheliograph
(NRH) in order to study spike-like bursts near the front of a type-II radio
burst recorded at the west limb during the November 3, 2003 extreme solar
event. The spike-like emission in the dynamic spectrum was enhanced by means
of high-pass-time filtering.

Results. We identified a number of spikes in the NRH images. Due to the lower
temporal resolution of the NRH, multiple spikes detected in the dynamic
spectrum appeared as single structures in the images. These spikes had an
average size of $\approx$ 200″ and their observed brightness temperature was
1.4 to $5.6\times 10^{9}$ K, providing a significant contribution to the
emission of the type-II burst front. At variance with a previous study on the
type-IV associated spikes, we found no systematic displacement between the
spike emission and the emission between spikes. At 327.0 MHz, the type II
emission was located about 0.3 R⊙ above the pre-existing continuum emission,
which, in turn, was located 0.1 R⊙ above the western limb.

Conclusions. This study, combined with our previous results, indicates that
the spike-like chains aligned along the type II burst MHD shock front are not
a perturbation of the type II emission, as in the case of type IV spikes, but
a manifestation of the type II emission itself. The preponderance of these
chains, together with the lack of isolated structures or irregular clusters,
points towards some form of small-scale magnetic reconnection, organized along
the type-II propagating front.

###### Key Words.:

Sun: corona – Sun: radio radiation – Sun: activity – Sun: Radiation
mechanisms: non-thermal

††offprints: S. Armatas

##  1 Introduction

Type II radio bursts are the electro-magnetic signature of magnetohydrodynamic
(MHD) shock fronts propagating in the solar corona; these are driven either by
CMEs, front or flank, or by flare blasts (Vršnak & Cliver, 2008; Pick &
Vilmer, 2008). These MHD shocks accelerate high-energy electrons from which
the type-II burst emission originates. This emission appears on the dynamic
spectra, in the metric and higher wavelengths, as slowly drifting lanes (the
backbone) from high to low frequencies (Roberts, 1959; Krueger, 1979), thus
tracing the MHD outward propagation. Furthermore, the shock-associated
suprathermal electrons generate the fine structures observed near the front of
the type-II bands. These include the so-called herringbones (Roberts, 1959)
and slow-drift fiber-like structures (Chernov, 1997; Chernov et al., 2007b).

Figure 1: Measurement of the duration and size of an individual spike in the
NRH 1D recording. Panel (b) shows the NRH total normalized flux in the east-
west direction and panel (c) the temporal profile at 327 MHz. In panel (d) we
present the Artemis-JLS/SAO time profile at 327 MHz for comparison. The red
curves in panels (b) and (c) trace the gaussian fit used in the duration and
width measurement of the spikes. The bright bands above and below the main one
are due to sidelobes. Figure 2: Event SOL2003-11-03T09:43:20 recorded by the
ARTEMIS-JLS/ASG and ARTEMIS-JLS/SAO receivers. The Dynamic spectrum of the
entire event from ARTEMIS-JLS/ASG is presented in the upper panel. In the
middle panel, we show the SAO dynamic spectrum and in the lower panel, its
time derivative inside the 5 min box of the upper panel. The dashed white
horizontal lines on the spectrum mark three of the frequencies (327, 410.5,
and 432 MHz) of the Nançay radioheliograph (NRH) channels. The boxes indicate
selected segments of spike chains, (A, B, C & D) which are studied in this
work (shown enlarged in figures 3, 4, and 5).

A new type of type-II associated fine structure was reported in a recent work
by Armatas et al. (2019, hereafter Paper I). Using the high-sensitivity (10 ms
resolution) receiver of the ARTEMIS/JLS111Appareil de Routine pour le
Traitement et l’Enregistrement Magnetique de l’ Information Spectral/Jean-
Louis Steinberg (former ARTEMIS-IV) radio spectrograph (Caroubalos et al.,
2001; Kontogeorgos et al., 2006a), they detected narrowband spike-like
structures, which were shorter than 100 ms and embedded in type II burst
harmonic emission. An interesting result of Paper I was that most of the
spikes detected were aligned in chains tracing the type II lanes with very few
isolated spike-like structures observed. The average parameters of the type-II
associated spike-like structures were found to be very close to the
corresponding parameters of type IV-associated spikes. A similar fine
structure was also recorded by the LOFAR for the 2014 August 25 type-II radio
burst (Magdalenić et al., 2020) and by the IPRT/AMATERAS for the 2014 April 25
event (Tan et al., 2019).

In this work, we extend the study presented in Paper I by using combined
ARTEMIS-JLS and Nançay Radioheliograph (NRH) to study spike-like bursts at the
front of a type-II radio burst. Out of the four events studied in Paper I,
there is only one, observed on November 3 2003 (SOL2003-11-03T09:43:20), that
was recorded by the Nançay Radioheliograph. The present article is organized
as follows: In Sect 2, we present the observations, data selection, and
analysis. In Sect. 3, we give our results and we conclude with a discussion in
Sect. 4.

##  2 Observations and data analysis

The basic data used in this study are high and medium time resolution dynamic
spectra recorded by the ARTEMIS-JLS solar radio-spectrograph at Thermopylae
(Caroubalos et al., 2001, 2006; Kontogeorgos et al., 2006b). ARTEMIS-JLS
consists of a 7 m moving parabolic dish covering the metric range and a dipole
aerial adapted to the decametric range which was added in October 2002. Two
receivers operate in parallel: a sweep frequency analyzer (ASG) covering the
650-20 MHz range in 630 channels with a cadence of 10 samples/sec and a high
sensitivity multi-channel acousto-optical analyzer (SAO), which covers the
270-450 MHz range in 128 channels with a high time resolution of 100
samples/sec. The narrow-band, high-time-resolution SAO recordings are used in
the analysis of the fine temporal and spectral structures. As in previous
works, a high pass Gaussian filter of 1 s width in time was applied to enhance
the fine structures. The broadband, medium-time-resolution data of the ASG, on
the other hand, are used for the detection and analysis of radio emission from
the base of the corona to $\sim 2$R⊙. We note that the ASG data were used in
the overview of event SOL2003-11-03T09:43:20 in Paper I.

Images of the radio emission were obtained from the Nançay Radio Heliograph
(NRH) (Kerdraon & Delouis, 1997; Klein & Kerdraon, 2011), which records two-
dimensional (2D) images of the Sun at five frequencies (164, 236.6, 327,
410.5, and 432 MHz) with a time resolution of 150 ms. All five frequencies are
within the spectral range of the ASG, while the last three are also within the
range of the SAO; the latter were used in conjunction with the high resolution
dynamic spectra in our combined type II associated spike-like bursts analysis.

We obtained 2D NRH images from the Nançay site222https://rsdb.obs-nancay.fr/.
From the 2D images, we computed 1D images by integrating in the (east-west) EW
and (north-south) NS directions; these are more convenient for comparison with
dynamic spectra (see Figs. 3, 4, and 5) and facilitates the measurement of the
duration.

On the 150 ms NRH 1D images, we measured the duration and the size of spike-
like bursts. The identification of individual bursts was done by inspection,
and the width was measured after fitting the positional and spectral profiles
with a Gaussian at full width at half maximum (FWHM, see Fig. 1); the method
is similar to the one used by Bouratzis et al. (2016, their Fig. 3) in their
study of type-IV associated spikes.

##  3 Results

In this section, we present results from the analysis of simultaneous SAO/NRH
observations. We start with an overview of the 2003 November 03 event and we
proceed with the study of spikes.

###  3.1 Overview of the 2003 November 03 event

The SOL2003-11-03T09:43:20 complex event on 2003 November 03 took place in a
period of extremely high activity on the Sun between 26 October and 18
November 2003 (Veselovsky et al., 2004; Chertok et al., 2005; Gopalswamy,
2006; Ishkov, 2006); it was associated with a GOES 2F/X3.9 class limb flare in
AR 10488 at heliographic cooordinates 08N∘77W∘ (Chertok & Grechnev, 2005;
Veronig et al., 2005, 2006).

An overview of the radio activity of the event based on the ARTEMIS-IV/JLS ASG
receiver and Wind/WAVES was presented in Alissandrakis et al. (2005). Paper I
provided more details of the type II event, which drifted at 1.6 MHz
$\rm{s}^{-1}$ in the 500–110 MHz range. Metric wavelength observations of the
event were also analyzed by Dauphin et al. (2005, 2006); Vršnak et al. (2006);
Chernov et al. (2007a); Auraß et al. (2013); Auraß (2014), and Chernov &
Fomichev (2021).

In the upper panel of Fig. 2, we present the dynamic spectrum of the type II
burst with some of its accompanying activity in the range 30-530 MHz, from
recordings of ARTEMIS-JLS/ASG with 150 ms time resolution. In the middle
panel, we present details of the type II front in the harmonic within the 5
min white box of the upper panel, this time from SAO recordings with 10 ms
resolution. We note that the type II emission was strong enough to saturate
the receiver in parts of the SAO spectrum. The lower panel of Fig. 2 shows the
time derivative of the dynamic spectrum, which enhances the fine details, both
obtained by the ARTEMIS-JLS/SAO.

The type II harmonic emission started at 09:51:35 UT above 500 MHz, well
within the Artemis-JLS/SAO frequency range and the 327, 410.5, 432 MHz Nançay
radioheliograph channels. It exhibited rich fine structure. Our study is
focused on the spike-like burst chains of the harmonic, as the fundamental was
outside of the Artemis-JLS/SAO frequency limits. In the same figure, we marked
four regions (A, B, C and D) with spikes that are presented in detail in Sect.
3.2. We note that in addition to spikes, we detected pulsation-like structures
just above the saturation region of the dynamic spectrum, covering a good part
of the harmonic emission (see Fig. 2).

Figure 3: Spike chain A recordings from SAO and NRH. Upper panel: SAO dynamic
spectrum for spike chain A (box in Fig. 2). The corresponding frequency of the
NRH is marked on the right. The next panel shows the NS 1D NRH images at 327
MHz. In the last panel, we show NRH (blue) and SAO (red) normalized total flux
time profiles at 327 MHz and at 150 ms resolution are presented; a second SAO
high-resolution (10 ms) profile (red dotted) is included for comparison.
Twelve spike-like structures, numbered (1-12), are marked by vertical dashed
lines. Figure 4: Spike chain B recordings from SAO and NRH. Same as in Fig. 3
for the Spike Chain B at 410.5 MHz. Three spike-like structures, numbered
(13-15), are marked by vertical dashed lines. Figure 5: Spike chain D
recordings from SAO and NRH. Same as in Fig. 3 for the Spike Chain D at 327
MHz. Figure 6: NRH & SAO images at the beginning of the type II emission at
327.0 MHz. Top: Sequence of NRH 2D images near the start of the Type II at 327
Mz. The color table of each image is normalized to the corresponding minimum
and maximum intensity. The black arch marks the photosheric limb. Bottom: Flux
as a function of time for NRH (blue) and for SAO (red). The time range of the
images is marked by the double arrow.  Figure 7: Source positions from
09:52:00 to 09:52:16 UT at 327.0 MHz and from 09:52:27 to 09:53:00 UT at 236.6
MHz, before (red) and during (black) the type II. The black arch marks the
photospheric limb. Figure 8:  Brightness temperature, size, and position of
sources as a function of time at 327.0 MHz (left). Size is the geometric
average of the axes of the fitted ellipse. Straight lines are the result of
linear regression. Positions of the sources on the plane of the sky after
09:52:08 UT (right). Spike sources are marked as red asterisks in both plots.
Figure 9: Running difference images of the 12 identified spikes at 327 MHz.
The red asterisks mark the position of the maximum of the radio source, the
blue asterisks that of the running difference. The white arch corresponds to
the solar limb. Figure 10: Running difference images of the 3 identified
spikes at 410.5 MHz. Same annotation as in Fig. 9

###  3.2 Temporal fine structure in dynamic spectra and images

Due to the frequency drift of the type II, a relatively small fraction of the
spikes detected in the SAO spectra are visible in the fixed frequency NRH
images. The detection of the spikes was done by inspection of the peaks in the
NRH flux and 1D images and comparison with the SAO time profiles. The process
is illustrated in the Figs. 3 and 4, where vertical dashed lines indicate the
detected spike-like structures in the flux profiles, the 1D images, and the
dynamic spectra.

Twelve spike-like bursts which are part of two chains that trace the type II
lane front at 327 MHz were identified in spectral region A of Fig. 2. They are
numbered 1-12 in Fig. 3 which, in addition to the dynamic spectrum, shows the
NS 1D NRH intensity as a function of position and time. The bottom panel of
the figure shows plots of normalized flux as a function of time for both
instruments.

Near the type II front at 410.5 MHz (region B), three spike-like bursts were
detected and marked 13, 14, and 15 in Fig. 4. The structures beyond spike
number 15 on Fig. 4 were excluded as they were part of the pulsation-like
structures just above the SAO saturation area (see Fig. 2). The dynamic
spectrum at 432 MHz (region C) reached saturation just after the appearance of
the type II front, thus we could see no spikes there. The burst in spectral
region D of Fig. 2, which preceded the harmonic band of the type II at 327
MHz, resemble extremely narrowband type IIIs rather than spikes; nonetheless,
we included it in our study for the purposes of comparison with the spike-like
structures (Fig. 5).

We note at this point that the time resolution of the Artemis-JLS/SAO is more
than a factor of ten higher than that of the NRH; therefore, certain features
that were detected and measured as single bursts in the NRH 1D images
correspond, in fact, to two or more bursts in the dynamic spectrum. An example
is shown in the lower panels of Figs. 3, 4, and 5 where the time profiles, at
327 and 410.5 MHz, of the NRH and the SAO at 150 ms resolution are plotted,
with an SAO high-resolution (10 ms) profile for comparison. In addition,
several NRH peaks correspond to multiple high resolution SAO peaks.

Due to the difference in time resolution, the measured duration from the 1D
NRH images is expected to be greater than the average duration of spikes
measured by the SAO. Thus, the duration of the spike-like structures was found
to be in the range of 150-600 ms, while in Paper I, the average duration was
reported to be less than 100 ms. This is due to the fact that individual
spikes recorded by the ARTEMIS/JLS SAO receiver cannot be clearly identified
in the 1D plot of the Nançay Radioheliograph, as discussed above.

###  3.3 Burst positions and structure

NRH 2D images give important information on the position of the emitting
sources. Figure 6 shows images at the very beginning of the type II emission
at 327.0 MHz. We note that as the type II emission increases, the relative
intensity of the pre-existing continuum source diminishes; moreover, the type
II source is higher than the continuum source, displaced by about 5.3′ or
0.3R⊙, whereas their position angles differ by 13°. The projected location of
the pre-type II emission was 0.1R⊙ above the west limb.

For a more detailed measurement of source positions and sizes, we performed a
least-squares fit with a Gaussian model. The positions derived in this way
from the 327.0 MHz images are plotted in Fig. 7, where, again, the shift
between the type II and the pre-existing emission is clearly visible.

Table 1: Parameters of spikes measured from 2D NRH images. Frequency (MHz) | 327 | 410.5 | 327
---|---|---|---
Region | A | B | D
Source position, (″) |  |  |
EW | 1240$\pm$12 | 1240$\pm$ 3 | 1225
NS | 350$\pm$ 9 | 330$\pm$ 1 | 380
Source size, (″) |  |  |
Bmaj | 235$\pm$9 | 205$\pm$2 | 285
Bmin | 175$\pm$8 | 150$\pm$4 | 200
Position Angle (∘) | -58$\pm$4 | -55$\pm$3 | -58
$T_{b}$, (109 K) |  |  |
Range | 1.4 - 4.5 | 3.3 - 5.6 | 1.3
Table 2: NRH beam size. Frequency (MHz) | 164 | 236 | 327 | 410.5 | 432
---|---|---|---|---|---
Region |  |  | A | B | C
Major Axis | 480″ | 329″ | 236″ | 193″ | 171″
Minor Axis | 205″ | 145″ | 99″ | 85″ | 78″
Position Angle | -55∘ | -55∘ | -55∘ | -55∘ | -55∘

The source positions at 236.6 MHz are also plotted in Fig. 7. Although this
frequency is outside the SAO range, the ASG spectrum (Fig. 2, top) shows
clearly the type II front; moreover, the NRH images show a lot of short
timescale structures, some of which might be spikes. As in the case of 327.0
MHz, the type II source was located higher than the pre-existing continuum,
this time displaced by about 6.4′, or 0.4 R⊙ and their position angles are
almost identical. Overall, both the continuum and the Type II sources at 236.6
MHz were above the corresponding 327.0 MHz structures, as expected; the type-
II positions are consistent with Dauphin et al. (2006, their Fig. 7).

An important question is whether the spike emission is at the same location as
the emission between spikes or, as in the case with Type IV associated spikes,
it is located at the periphery of the background emission (Bouratzis et al.,
2016). To investigate this issue, we plotted in Fig. 8 the source parameters,
deduced from Gaussian fit, as a function of time (left) and their positions on
the sky plane (right), marking the spikes by red asterisks. We note that after
the initial intensity rise (09:52:08 UT), most sources fall on a straight
line, which reflects a systematic displacement to the west ($X$ plot in Fig.
8, left), consistent with the upward motion of the type II front. Spikes 1-9
also fall along this line, whereas spikes 10-12 do not. A careful inspection
of the upper panel of Fig. 3 reveals that the latter are part of a different
spike chain, apparently associated to a different part of the shock front than
the one which produced spikes 1-9.

Images of the 15 identified SAO/NRH spikes are given in Fig. 9 for 327.0 MHz
and Fig. 10 for 410.5 MHz. For better visibility, and also to show position
differences between the spike source and the preceding emission, we used
running difference images and marked the peak position of the source in the
original and in the subtracted image. A preliminary remark is that some spikes
are double (# 1, 10, 14), with the two sources close to each other – these are
probably due to separate spikes at slightly different positions that are not
resolved temporally by the NRH. In conformity with our previous conclusion,
there appears to be no large shift between the spike and the pre-spike
emission, with the exception of spikes 7 and 10.

The parameters derived from the Gaussian fit are given in Table 1. The source
size is of the order of 150″ to 240″; thus the sources are not fully resolved,
as their axes are not much larger than the NRH beam size (given in Table 2 for
all frequencies). This can be seen in their elongated shape in Figs. 9 and 10.
Their brightness temperature is on the order of 109 K, which is a lower limit
both due to the beam size and refraction effects.

##  4 Discussion and conclusions

We examined the characteristics of type II spike-like bursts detected from
high temporal resolution recordings of ARTEMIS-JLS/SAO combined with imaging
from the Nançay Radioheliograph. Complementing our Paper I results, we
estimated the position, size, duration, and shift of the structures from NRH.

The small number of fifteen spike-like structures in the type II front that
were identified in both instruments is due to the frequency drift of the type
II bursts, which shifts the spike-like sources outside of the NRH channel;
therefore, relatively short segments of the spike chains detected by the SAO
were recorded by the NRH.

We found that the durations measured from NRH data with a sampling interval of
150 ms exceeded the corresponding values from the ARTEMIS-JLS/SAO data. This
indicates that most single spike burst measurements of the NRH recordings
correspond to groups of spikes detected in the SAO dynamic spectrum. The size
of spikes was in the 150″-240″ range, not fully resolved in the NRH images,
whereas their observed brightness temperature was in the 1.4 to $5.6\times
10^{9}$ K range.

Our detailed investigation of source positions at 327.0 MHz showed that the
type II emission was located $\sim 0.3$ R⊙ above the pre-existing continuum
source; similar results were obtained for 236.6 MHz. Most type II emission
peaks were located along a straight line, indicating upward motion of the
source. The same trend was observed for all the spike sources, with the
exception of three, which were part of a different spike chain.

We found no substantial position difference between spike and inter-spike
emission, thus type II spikes appear to be associated with the entire emitting
region. This is in contrast to the type IV spikes that are located at the
periphery of the background source (Bouratzis et al., 2016) and, hence, are a
perturbation on top of the continuum emission. This indicates that spikes
constitute the principal radio signature of the MHD shock type-II radio
emission (see Paper I), with their chains delineating the shock front
emission; this front appears continuous on low-resolution dynamic spectra.
Still, the type-II and type-IV associated spikes are quite similar (see Table
1 in Paper I), possibly hinting towards a similar emission mechanism.

We found evidence that different spike chains make up parallel drifting lanes.
These combined, probably correspond to the “band-split of the band-split”
image of the type-II burst front, reported by Magdalenić et al. (2020), since
the continuity of the front line depends on the dynamic spectrum resolution,
as mentioned in the previous paragraph. A possible interpretation of the
multiple lanes responsible for the band split is provided in Chernov &
Fomichev (2021), asserting that it may be due to several sources formed at
different positions of the large-scale type-II shock front.

One of the prime candidates for the interpretation of spikes is the emission
associated to plasma instabilities due to electron beams originating in a
sequence of small-scale reconnections associated to different regions of the
shock front. The preponderance of spike-like burst chains that are almost
parallel to one another and the scarcity of isolated bursts and irregular
clusters of spikes at the fronts of the type-II bursts suggest some type of
organized structure, as opposed to random turbulence driven reconnection.
Thus, triggering along the front of the MHD shock disturbance might be at the
origin of such processes.

###### Acknowledgements.

The authors wish to thank the radio monitoring service at LESIA (Observatoire
de Paris) for providing data used for this study, in particular our colleagues
from the Observatoire de Paris-Meudon and the NRH staff. Data from GOES, NOAA
and SOHO-LASCO were obtained from the respective data bases; we are grateful
to all those who contributed to the operation of these instruments and made
the data available to the community. We also wish to thank the Onassis
Foundation for financial support (Grant 15153) for the continued operation of
the ARTEMIS-JLS radio spectrograph and the University of Athens Research
Committee for Grant 15018.

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