# Measurement of the Low-temperature Loss Tangent of High-resistivity Silicon
with a High Q-factor Superconducting Resonator

M. Checchin<EMAIL_ADDRESS>Superconducting Quantum Materials and Systems
Center, Fermi National Accelerator Laboratory, Batavia IL 60510, USA  D.
Frolov  Superconducting Quantum Materials and Systems Center, Fermi National
Accelerator Laboratory, Batavia IL 60510, USA  A. Lunin  Superconducting
Quantum Materials and Systems Center, Fermi National Accelerator Laboratory,
Batavia IL 60510, USA  A. Grassellino  Superconducting Quantum Materials and
Systems Center, Fermi National Accelerator Laboratory, Batavia IL 60510, USA
A. Romanenko  Superconducting Quantum Materials and Systems Center, Fermi
National Accelerator Laboratory, Batavia IL 60510, USA

###### Abstract

In this letter, we present the direct loss tangent measurement of a high-
resistivity intrinsic (100) silicon wafer in the temperature range from $\sim
70$ mK to 1 K, approaching the quantum regime. The measurement was performed
using a technique that takes advantage of a high quality factor
superconducting niobium resonator and allows to directly measure the loss
tangent of insulating materials with high level of accuracy and precision. We
report silicon loss tangent values at the lowest temperature and for electric
field amplitudes comparable to those found in planar transmon devices one
order of magnitude larger than what was previously estimated. In addition, we
discover a non-monotonic trend of the loss tangent as a function of
temperature that we describe by means of a phenomenological model based on
variable range hopping conduction between localized states around the Fermi
energy. We also observe that the dissipation increases as a function of the
electric field and that this behavior can be qualitatively described by the
variable range hopping conduction mechanism as well. This study lays the
foundations for a novel approach to investigate the loss mechanisms and
accurately estimate the loss tangent in insulating materials in the quantum
regime, leading to a better understanding of coherence in quantum devices.

Superconducting quantum circuits based on cavity-quantum electrodynamics
architecture Wallraf  _et al._ (2004); Schoelkopf and Girvin (2008); Devoret
and Schoelkopf (2013) represent a leading technology for constructing quantum
processors and achieving quantum supremacy Arute and et al. (2019). This
technology utilizes the nano-fabrication processes developed by the
semiconductor industry to manufacture integrated microwave circuits. Silicon
(Si) is a vital material in integrated circuit technology and provides good
trade-off between losses in the milli-Kelvin regime and process
industrialization; therefore, it is the substrate of choice for
superconducting quantum bits fabrication.

Because of the high dielectric constant
($\varepsilon_{\text{r}}^{\prime}$=11.5 at low temperatures), a large fraction
of the electromagnetic energy is stored in the silicon substrate Wang  _et
al._ (2015); hence, its contribution to the overall device energy loss can be
substantial. Accurate knowledge of the Si loss tangent is thus pivotal for
correctly estimating dissipation in quantum devices.

High-resistivity silicon ($\rho\geq 5$ k$\Omega$-cm) at milli-Kelvin
temperatures and microwave frequencies is generally assumed to introduce
negligible dielectric losses compared to the native oxide and substrate-
metalization intermixing layers found in typical superconducting quantum
devices. However, no direct measurements of Si wafer loss tangent at milli-
Kelvin temperatures have been presented thus far. Therefore, a reasonable
large degree of uncertainty on the actual Si loss tangent value in the milli-
Kelvin range still exists. Loss tangent data of Si at milli-Kelvin
temperatures exist for silicon billets Bourhill  _et al._ (2019), which are
not a representative sample of the wafers used to fabricate quantum bits, and
for high-resistivity silicon at higher temperatures Krupka  _et al._ (2006).

In this letter, we present the direct measurement of the loss tangent of high-
resistivity floating zone (FZ) silicon wafers in the temperature range from
$\sim 70$ mK to $1$ K. We discover a non-monotonic trend of the loss tangent
of silicon with temperature and an increasing trend of the dissipation with
increasing electric field, that are inconsistent with known dissipation
mechanisms encountered at these temperatures, such as two-level systems (TLS)
absorption. Additionally, we demonstrate that the milli-Kelvin Si loss tangent
is one order of magnitude larger than previously extimated Woods  _et al._
(2019); Melville  _et al._ (2020).

Figure 1: Experimental set-up. (a) Three dimensional (3D) model of the
experimental setup. (b) Schematic of the experiment in the dilution
refrigerator.

Measurements were performed using a high quality factor (Q-factor) elliptical
superconducting niobium resonator hosting the Si sample in the high electric
field region. The fundamental mode $\text{TM}_{010}$ resonating at 2.6 GHz was
used to perform the sample characterization over the entire temperature range.

Elliptical superconducting resonators are typically adopted in particle
accelerators at liquid He temperatures to accelerate relativistic charged
particles because of their high efficiency (intrinsic Q-factor $Q_{0}\sim
10^{11}$) in producing accelerating gradients on the order of tens of MV/m
Romanenko  _et al._ (2014). Nonetheless, they also allow for $Q_{0}\gtrsim
10^{9}$ at $10$ mK Romanenko  _et al._ (2020), making them an ideal tool for
performing loss tangent measurements of insulators with a high degree of
accuracy at milli-Kelvin temperatures.

Fig. 1 (a) shows a 3D model of the experimental setup. The resonator was
secured to the mixing chamber plate of a dilution refrigerator (DR).
Temperature was recorded by two Ruthenium Oxide (RuOx) thermometers, one
attached to the mixing chamber plate and one attached to the sample-holder
transition flange, shown in green in Fig. 1 (a)—the latter is referred to as
sample thermometer. The input line had several stages of attenuation for a
total of $\sim 60$ dB, including the Eccosorb filters. The output line had
isolators and Eccosorb filters connected immediately after the device output
port and a 35 dB low-noise HEMT amplifier, which was thermalized to the
quasi-4 K plate (typically stable at approximately 2.4 K). A simplified scheme
of microwave connections in the DR is shown in Fig. 1 (b). The total gain of
the transmitted power line was measured to be approximately 62 dB, including a
warm 40 dB amplifier (not shown in the schematics of Fig. 1 (b)).

Intrinsic FZ Si(100) single-side polished wafers, with a thickness of 675
$\mu$m, were procured with room temperature resistivity of $10$ k$\Omega$-cm
and diced to 10 cm long and 2 mm wide strips. After dicing, one sample was
cleaned in an ultrasonic bath of isopropyl alcohol for 15 min, dried in ultra-
pure nitrogen, assembled to the resonator, and subsequently pumped to a vacuum
level of $p<10^{-5}$ Torr.

The loaded Q-factor ($Q_{\text{L}}$) was measured with a power ring-down
technique. A steady-state electromagnetic field was established in the
resonator and the transmitted power ($P_{\text{t}}$) free decay was recorded
as a function of time after shutting off the power fed to the device. The peak
electric field on the Si sample in the steady-state is measured to be
approximately $E_{\text{Si}}=10$ V/m, corresponding to an average number of
photons stored in the resonator in the order of $\langle n\rangle\sim 5\cdot
10^{9}$. This value was calculated by means of the proportionality factor
$\kappa=E_{\text{Si}}/\sqrt{P_{t}Q_{2}}=E_{\text{Si}}/(\omega\sqrt{\hbar\langle
n\rangle})=822$ V/(m-W1/2) obtained by finite-element simulations by the high-
frequency structure simulator (HFSS) program, where $Q_{2}$ is the quality
factor of the transmitted line antenna (Antenna 2 in Fig. 1 (a)). To be
noticed that independently on the number of photons in the resonator, the
maximum electric field experienced by the Si sample in this experiment is
comparable to that induced by plasma oscillations in the Si substrate of
typical transmon devices, which is found to be up to 20 V/m in correspondence
of the metalization edges Lisenfeld  _et al._ (2019).

We define $Q_{\text{L}}=-10\omega/\left(\text{ln}10\cdot
dP_{\text{t}}/dt\right)$, where $\omega$ denotes the angular frequency, and
$dP_{\text{t}}/dt$ is the angular coefficient of the transmitted power free
decay—with power measured in dBm. We then perform a linear regression to fit
the decay data to extract $dP_{\text{t}}/dt$ selecting a window of points
centered around 5 V/m. This strategy to measure $Q_{\text{L}}$ was implemented
to increase the measurement accuracy and circumvent the typical S21
measurement approach that is limited by distortions of the resonant peak and
appearance of side bands generated by microphonics Padamsee  _et al._ (2008)
as a consequence of the high $Q_{\text{L}}$. Additional details regarding this
measurement strategy are reported in Ref. [Romanenko and Schuster, 2017;
Romanenko  _et al._ , 2020].

In Fig. 2, we illustrate the loaded Q-factor measured as a function of
temperature. The blue dots represent the loaded Q-factor ($Q_{\text{L}}$) data
acquired in this study and plotted against the sample temperature. The light
blue diamonds show the intrinsic Q-factor of the resonator alone, without the
Si sample, as measured in Ref. [Romanenko  _et al._ , 2020]. At thermal
equilibrium, the sample could not be cooled below $74$ mK due to a not ideal
thermal path connecting the sample to the mixing chamber.

Figure 2: The loaded quality factor as a function of the sample temperature is
shown in blue. The resonator intrinsic Q-factor versus temperature at 2.6 GHz
was reported in Ref. [Romanenko  _et al._ , 2020] and it is shown in light
blue.

The loss tangent of the sample is calculated as follows:

$\dfrac{1}{Q_{\text{S}}}=\dfrac{p_{\text{{Si}}}}{Q_{\text{{Si}}}}+\dfrac{p_{\text{SiO}_{2}}}{Q_{\text{SiO}_{2}}}=\dfrac{1}{Q_{\text{L}}}-\dfrac{1}{Q_{0}}-\dfrac{1}{Q_{1}}-\dfrac{1}{Q_{2}}\text{,}$
(1)

where $Q_{1}=5.8\cdot 10^{9}$ and $Q_{2}=6.5\cdot 10^{11}$ represent the
external Q-factors of the antennas (Antenna 1 and Antenna 2 in Fig. 1 (a))
measured for the same setup in a liquid helium bath at $1.5$ K, as described
in Ref. [Melnychuk  _et al._ , 2014], whereas $p_{\text{Si}}=9\cdot 10^{-4}$
and $p_{\text{SiO}_{2}}=3\cdot 10^{-9}$ denote the participation ratios of
silicon and the native silicon oxide layer, defined as follows:
$p_{\text{diel}}=\int_{V_{\text{diel}}}\varepsilon_{\text{diel}}|\mathbf{E}|^{2}dV_{\text{diel}}/\int_{V}\varepsilon_{0}|\mathbf{E}|^{2}dV$.
Both values were calculated using the HFSS program.

It is important to highlight that the measured loaded Q-factor is dominated by
the sample loss, whereas the other contributions are negligible. In fact,
$Q_{0}$ is approximately one order of magnitude higher than $Q_{\text{L}}$ at
the lowest temperature, while for temperature levels approaching 1 K, it is
two orders of magnitude higher (see Fig. 2), and both $Q_{1}$ and $Q_{2}$ are
at least one order of magnitude higher than $Q_{\text{L}}$. This implies that
the measurement strategy implemented has a high level of accuracy and it is
not affected by dissipation mechanisms extrinsic to the sample under study.

The loss tangent of SiO2 at milli-Kelvin temperatures is experimentally known
to be approximately $1/Q_{\text{SiO}_{2}}\simeq 5\cdot 10^{-3}$ Martinis  _et
al._ (2005). For the geometry under study, the participation ratio of the SiO2
layer is calculated to be $p_{\text{SiO}_{2}}=3\cdot 10^{-9}$. Therefore, SiO2
contribution to $Q_{\text{L}}$ is $Q_{\text{SiO}_{2}}/p_{\text{SiO}_{2}}\sim
10^{11}$, hence negligible compared to that of silicon. We can then define the
silicon loss tangent as:

$\dfrac{1}{Q_{\text{Si}}}=\dfrac{1}{p_{\text{Si}}Q_{\text{S}}}\text{.}$ (2)

Fig. 3 shows the loss tangent of silicon as a function of the sample
temperature. At the lowest temperature, 74 mK, the loss tangent is equal to
$2.7\cdot 10^{-6}$, which is in agreement with measurements performed on
silicon billets in whispering gallery mode configuration Bourhill  _et al._
(2019). In contrast, our experiment recorded higher values compared to
indirect estimations based on measurements and simulations of planar devices
Woods  _et al._ (2019); Melville  _et al._ (2020), which are in the order of
$10^{-7}$, one order of magnitude lower than our experimental observations.

Figure 3: Silicon loss tangent as a function of temperature. The blue solid
line is a fit of the VRH model to the experimental data. The shaded area
represent the experimental error of the measurement. The inset is a zoom of
the data with temperature on a linear scale.

We observe a non-monotonic dependence of the silicon loss tangent on
temperature, which does not resemble the two-level system (TLS) temperature
dependence. The value of $1/Q_{\text{Si}}$ decreases with decreasing
temperature, reaching a minimum at approximately 80 mK. Below 80 mK, the trend
is opposite, and the loss tangent increases as the temperature decreases with
a steeper slope. This trend is better appreciated in the inset of Fig. 3,
where the temperature is shown on a linear scale. The shaded areas show the
experimental error associated with the measurement. This was calculated
through error propagation, where the error in $Q_{\text{L}}$ was propagated
from the square root of the variance of the angular coefficient that was
calculated by the linear regression routine; the errors in $Q_{0}$, $Q_{1}$,
and $Q_{2}$ were assumed to be $10$% while that in $p_{\text{Si}}$ was
estimated to be $25$%, and it was calculated through HFSS simulations assuming
$\pm 2$ mm sample misalignment and
$11.5\leq\varepsilon_{\text{r}}^{\prime}\leq 11.9$.

To further shed light into the dissipation mechanism in place as a function of
temperature, we calculate the electric field dependence of the loss tangent at
constant temperature. We select three temperatures above, equal, and below the
minimum of the loss tangent as a function of T, nominally 74 mK, 82 mK, and
200 mK. And we calculate the silicon loss tangent by means of Eq. 1, where the
loaded Q-factor is inversely proportional to the local slope of the
transmitted power ring-down data
($Q_{\text{L}}=-10\omega/\left(\text{ln}10\cdot dP_{\text{t}}/dt\right)$). In
Fig. 4, we report the parametric plot of Si loss tangent versus peak electric
field $E_{\text{Si}}$. Interestingly, the electric field dependence of the
loss tangent is also not in agreement with TLS-driven dissipation.
$1/Q_{\text{Si}}$ decreases with decreasing field, and saturates to a constant
value for $E_{\text{Si}}\lesssim 10$ V/m. The observed trend is independent on
temperature—for temperatures above, equal, and below the loss tangent minimum
the field dependence is unchanged. This finding points against the possibility
that the upward trend of $1/Q_{\text{Si}}$ for temperatures below 80 mK is due
to TLS-type of losses.

Figure 4: Loss tangent as a function of the peak electric field in the sample
measured at fixed temperature (74 mK, 82 mK, and 200 mK). The shaded areas
represent the measurement error estimated as discussed in the text above. The
second $x$ axis reports the number of photons in the resonator.

We exclude the possibility that the observed trend as a function of the
electric field is due to an increase in temperature of the sample because of
thermal feedback. To prove this, we performed a series of free decays
experiments at fixed temperature (82 mK) with increasing input power
($P_{\text{i}}$) values, the experimental data is shown in Fig. 5. The
thermalization time of the sample thermometer was observed to be roughly
$\tau_{\text{T}}\sim 2$ min, while the loaded decay time of the resonator plus
sample was measured to be in the order of $\tau_{\text{L}}\sim 0.1$ s. Since
$\tau_{\text{L}}\ll\tau_{\text{T}}$, if any thermal feedback effect was at
work then differences in the field dependence as a function of the steady-
state $P_{\text{i}}$ would be expected. On the contrary, the electric field
dependence of the loss tangent is independent on the steady-state condition,
meaning that no feedback effect is playing a role in the measurements
performed.

Figure 5: Loss tangent as a function of the peak electric field on the sample
measured at 82 mK for different steady-state $P_{\text{i}}$ values. The shaded
areas represent the measurement error estimated as discussed in the text
above. In the inset we report the power ring down data as a function of time.
The color palette of the data reported matches the colors of the main plot.
The second $x$ axis reports the number of photons in the resonator.

As discussed above, the experimental findings reported in this paper give
insights into an unexplored dissipation mechanism of intrinsic Si at milli-
Kelvin temperatures. Based on our experimental findings, the source of
dissipation should be sought in mechanisms that are not TLS-related. Rather,
we expect that conduction losses could be the cause of the observed
temperature and field dependencies. Including conduction losses in the
picture, the dielectric constant takes the form:

$\varepsilon=\varepsilon_{\text{r}}^{\prime}-i\left(\varepsilon_{\text{r}}^{\prime\prime}+\dfrac{\sigma}{\omega\varepsilon_{0}}\right)\text{,}$
(3)

where $\varepsilon_{\text{r}}^{\prime}$ and
$\varepsilon_{\text{r}}^{\prime\prime}$ represent the real and imaginary parts
of the complex dielectric permittivity, whereas $\sigma$ denotes the low
temperature conductivity. The loss tangent is thereafter expressed as follows:

$\dfrac{1}{Q_{\text{Si}}}=\dfrac{1}{Q_{\text{TLS}}}+\dfrac{\sigma}{\omega\varepsilon_{0}\varepsilon_{\text{r}}^{\prime}}\text{,}$
(4)

where the first term
$1/Q_{\text{TLS}}=\varepsilon_{\text{r}}^{\prime\prime}/\varepsilon_{\text{r}}^{\prime}$
denotes the dielectric loss tangent dominated by dielectric dissipation (TLS),
whereas the second term describes the conduction losses. Due to the rationale
discussed above, we can deem the TLS contribution of the Si oxide negligible.

We identify the dependence of the loss tangent with temperature with the
second term of Eq. 4, and particularly with the occurrence of an electron
hopping mechanism, likely variable range hopping (VRH) Mott (1968); Apsley and
Hughes (1974, 1975). VRH conduction may take place at low temperatures where
localized states close to the Fermi level within $\sim\kappa_{\text{B}}T$ can
contribute to the overall conduction. Electrons subjected to an electric field
$\mathbf{E}$ hop between states separated by the shorter four-dimensional
distance—three spatial coordinates and one energy coordinate, the so-called
hopping range ($\mathcal{R}$), defined as:

$\begin{split}&\mathcal{R}=2\alpha
R+\dfrac{W+e\mathbf{R}\cdot\mathbf{E}}{\kappa_{\text{B}}T}\\\
&W=\dfrac{3^{4}}{4^{4}\pi g(\epsilon_{\text{F}})R^{3}}\text{,}\end{split}$ (5)

where $\alpha$, $W$, and $R$ are the wave-function localization parameter,
average energy difference between the states Mott (1968); Mott and Davis
(1979), with $g(\epsilon_{\text{F}})$ the density of localized states within
$\sim\kappa_{\text{B}}T$ around the Fermi energy $\epsilon_{\text{F}}$, and
the spatial distance between the states involved, respectively. The VRH
conductivity can then be defined as proportional to the difference between the
hopping probability with and against the field $\mathbf{E}$, $\langle
P^{+}\rangle$ and $\langle P^{-}\rangle$:

$\sigma_{\text{h}}\sim\left|\langle P\rangle^{+}-\langle
P\rangle^{-}\right|=\left|e^{-\text{min}\,\mathcal{R}^{+}}-e^{-\text{min}\,\mathcal{R}^{-}}\right|\text{.}$
(6)

In the zero field approximation, the distance $R$ that minimizes $\mathcal{R}$
in Eq. 5 always increases as the temperature lowers and conduction becomes
dominated by hops between levels that are closer in energy even if far apart,
since the tunneling contribution (first right-hand side term in the first
formula of Eq. 5) becomes exponentially suppressed compared to the energy
activation term (second right-hand side term in the first formula of Eq. 5).
Under this approximation, the conductivity is expected to decrease with
$\sim\text{exp}\left(-T^{-1/4}\right)$, and for $T\rightarrow 0$ it
exponentially approaches zero Mott (1968); Mott and Davis (1979).

On the contrary, when $E>0$ then $R$ increases with different rates whether
hops are happening with or against the electric field, with $R^{-}$ growth
rate being the largest. In return, due to the $1/(R^{-})^{3}$ dependence of
$W^{-}$ and the increasing weight of $-eR^{-}E$ that lowers the hopping
activation energy, the VRH process is governed by $\mathcal{R}^{-}$ since
growing slower than $\mathcal{R}^{+}$. As in the $E=0$ case,
$\sigma_{\text{h}}$ decreases with decreasing $T$. However, by further
lowering the temperature, the condition $W^{-}<eR^{-}E$ is eventually met and
electron hopping against $\mathbf{E}$ is not anymore limited by the activation
energy. Consequently, the $\mathcal{R}^{-}$ range reverts its trend and
decreases with decreasing temperature, resulting in a sudden increase of
$\sigma_{\text{h}}$ (increase of $\langle P^{-}\rangle$, decrease of $\langle
P^{+}\rangle$) and forming a minimum that eventually translates in a minimum
of $1/Q_{\text{Si}}$ versus $T$.

We fit the experimental data as a function of temperature by means of a
phenomenological model based on the VRH mechanism described above. We rewrite
Eq. 4 assuming two conduction loss channels, specifically VRH (defined by
$\sigma_{\text{h}}$) and a residual term (defined by $\sigma_{0}$) that allows
for a better fit outcome. Eq. 4 is rewritten as:

$\dfrac{1}{Q_{\text{Si}}}=\dfrac{\sigma_{\text{h}}+\sigma_{0}}{\omega\varepsilon_{0}\varepsilon_{\text{r}}^{\prime}}\text{,}$
(7)

where the hopping conductivity is calculated as:

$\sigma_{\text{h}}=\dfrac{\left|J^{+}-J^{-}\right|}{E}\text{,}$ (8)

with $J=2e\gamma g(\epsilon_{\text{F}})\kappa_{\text{B}}TR\langle P\rangle$
and $\gamma$ the hopping attempt frequency Mott (1968); Mott and Davis (1979).

The least square regression routine is ran by numerically calculating the
values $R^{+}$ and $R^{-}$ that minimize $\mathcal{R}^{+}$ and
$\mathcal{R}^{-}$ respectively, and by keeping $\alpha$, $\gamma$,
$g(\epsilon_{\text{F}})$, and $\sigma_{0}$ as free parameters. The fit is
reported in Fig. 3. As shown, the model describes the data exhaustively within
the experimental error, returning the following values:
$\alpha^{-1}=1.05~{}\mu\text{m}$, $\gamma=11.4$ THz,
$g(\epsilon_{\text{F}})=1.33\cdot 10^{13}~{}\text{eV}^{-1}\text{cm}^{-3}$, and
$\sigma_{0}=0.52~{}\mu$S/m. We interpret the $\sigma_{0}$ contribution as the
manifestation of excitation of free-carriers in conduction band likely
generated by cosmic rays absorption—due to the large dimension of the sample,
a non-negligible fraction of excitations in conduction band are expected. Due
to the lack of literature on VRH in high-resistivity intrinsic silicon, we
compare the values returned by the fitting routine with those of amorphous
silicon. The $\gamma$ value obtained falls within the range
$\gamma\simeq(3-15)$ THz reported in Ref. [Pfeilsticker  _et al._ , 1978],
while $g(\epsilon_{\text{F}})$ and $\alpha$ are smaller than what reported for
amorphous silicon where
$g(\epsilon_{\text{F}})\simeq(10^{16}-10^{23})~{}\text{eV}^{-1}\text{cm}^{-3}$
and $\alpha^{-1}\simeq(0.1-10)~{}\text{nm}$ Pfeilsticker  _et al._ (1978);
Pichon and Rogel (2011). These findings imply that i) as expected the number
of available states in high-resistivity intrinsic silicon are less and
separated by a larger distance compared to amorphous silicon, and ii) that the
states involved in the hopping conduction mechanism are not well localized.

The VRH mechanism can qualitatively describe the experimental data as a
function of the electric field as well, and for low $E$ values the
conductivity is expected to increase with field as
$\sigma_{\text{h}}\sim\text{sinh}(eRE/\kappa_{\text{B}}T)$ Mott and Davis
(1979). However, the fitting routine does not interpolate the data
exhaustively, suggesting that a generalized first principles theoretical model
is needed to describe accurately the field dependence observed. Nevertheless,
the model here presented provides a first glimpse into the suspected loss
mechanism at play at milli-Kelvin temperatures.

In conclusion, we described an accurate method to directly measure the loss
tangent of insulating materials in wafer form using a high Q-factor
superconducting resonator. We reported the direct measurement of the loss
tangent of a high-resistivity silicon wafer in the temperature range from
$\sim 70$ mK to 1 K. Furthermore, we showed that the loss tangent of high-
resistivity Si in the milli-Kelvin range is one order of magnitude higher than
that previously indirectly estimated from measurements and simulations of
planar devices Woods  _et al._ (2019); Melville  _et al._ (2020), although it
is in agreement with values obtained from whispering gallery measurements of
Si billets Bourhill  _et al._ (2019). In addition, we discovered a non-
monotonic behavior of the loss tangent dependence on temperature with a
minimum at approximately 80 mK, that we interpret as the occurrence of the
sign change in the against-field hopping range ($\mathcal{R}^{-}$) slope as a
function of temperature. We also observe a decreasing trend of the loss
tangent with decreasing electric field with saturation below $\sim 10$ V/m,
that can be only qualitatively described by the VRH mechanism. More
experimental studies are on the way to fully characterize the dissipation of
silicon in the milli-Kelvin regime and uncover the detailed nature of the
underlying mechanisms at play.

The methodology developed in this study will serve as a tool for detailed
investigation of losses in dielectrics at milli-Kelvin temperatures, and guide
the selection of materials for the fabrication of high-coherence quantum
devices.

This material is based on work supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359.

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# Synchronization of small-scale magnetic features, blinkers, and coronal
bright points

Zahra Shokri  Department of Physics, Faculty of Science, University of Zanjan,
University Blvd., 45371-38791, Zanjan, Iran  Nasibe Alipour  Department of
Physics, Faculty of Science, University of Zanjan, University Blvd.,
45371-38791, Zanjan, Iran  Hossein Safari  Department of Physics, Faculty of
Science, University of Zanjan, University Blvd., 45371-38791, Zanjan, Iran
Pradeep Kayshap  Vellore Institute of Technology (VIT), Bhopal University,
Kothri Kalan, Sehore, M.P., India  Olena Podladchikova  Physikalisch-
Meteorologisches Observatorium Davos, World Radiation Center, 7260, Davos
Dorf, Switzerland  Giuseppina Nigro  Department of Physics, University of
Calabria, P. Bucci 31C, 87036 Rende (CS), Italy  Durgesh Tripathi  Inter-
University Centre for Astronomy and Astrophysics, Post Bag - 4, Ganeshkhind,
Pune 411007, India

###### Abstract

We investigate the relationship between different transients such as blinkers
detected in images taken at 304 Å, extreme ultraviolet coronal bright points
(ECBPs) at 193 Å, X-ray coronal bright points (XCBPs) at 94 Å on AIA, and
magnetic features observed by HMI during ten years of solar cycle 24. An
automatic identification method is applied to detect transients, and the YAFTA
algorithm is used to extract the magnetic features. Using ten years of data,
we detect in total 7,483,827 blinkers, 2,082,162 ECBPs, and 1,188,839 XCBPs,
respectively, with their birthrate of about $1.1\times 10^{-18}$ ${\rm
m}^{-2}{\rm s}^{-1}$, $3.8\times 10^{-19}$ ${\rm m}^{-2}{\rm s}^{-1}$, and
$1.5\times 10^{-19}$ ${\rm m}^{-2}{\rm s}^{-1}$. We find that about 80% of
blinkers are observed at the boundaries of supergranules, and 57% (34%) are
associated with ECBPs (XCBPs). We further find that about 61–80% of transients
are associated with the isolated magnetic poles in the quiet Sun and that the
normalized maximum intensities of the transients are correlated with
photospheric magnetic flux of poles via a power law. These results
conspicuously show that these transients have a magnetic origin and their
synchronized behavior provides further clues towards the understanding of the
coupling among the different layers of the solar atmosphere. Our study further
reveals that the appearance of these transients is strongly anti-correlated
with the sunspots cycle. This finding can be relevant for a better
understanding of solar dynamo and magnetic structures at different scales
during the solar cycle.

Solar photosphere (1518); Solar transition region (1532); Solar corona (1483);
Solar dynamo (2001); Solar cycle (1487); Quiet sun (1322)

. .

##  1 INTRODUCTION

The inhomogeneous complex nature of the quiet Sun (QS) is predominantly due to
the presence of a large variety of small-scale magnetic features. It is
believed that these small-scale magnetic features are essentially the sources
of magnetic energy, which can be dissipated to heat the upper atmosphere
(Amari et al., 2015). Transients observed at different spatio-temporal scales,
e.g., blinkers (Harrison, 1997; Subramanian et al., 2012), explosive events
(e.g., Gupta & Tripathi, 2015), coronal bright points (CBPs, Vaiana et al.,
1973; Golub et al., 1974; Habbal & Withbroe, 1981), active region transient
brightenings (ARTBs, Berghmans et al., 2001), Hi-C brightenings (e.g.,
Subramanian et al., 2018; Rajhans et al., 2021), transient loops in the core
of active regions (Tripathi, 2021), active region jets (e.g., Chifor et al.,
2008; Mulay et al., 2016), etc. in the different atmospheric layers are
associated with photospheric magnetic features. Due to such associations and
various other properties of the transients, the process of magnetic
reconnection is considered to be one of the key mechanisms for their
generation. These transients observed ubiquitously throughout the solar
atmospheric are further considered to play an essential role in transferring
the mass and energy within the solar atmosphere (Parker, 1988; Schrijver et
al., 1997; Moore et al., 1999; Winebarger et al., 2002; Yamauchi et al., 2005;
Alipour et al., 2012; Priest, 2014; Upendran & Tripathi, 2021) and formation
of the solar wind including switchbacks (Hassler et al., 1999; Tu et al.,
2005; Zank et al., 2020; Tripathi et al., 2021; Liang et al., 2021; Upendran &
Tripathi, 2021).

Blinkers were first identified by Harrison (1997) and are defined as small-
scale brightenings in the transition region. They are observed both in active
regions (ARs) and QS, with AR blinkers being more abundant than those in QS
(Parnell et al., 2002). They were primarily identified in spectral lines such
as O V 629 Å, O IV 554 Å, O III 599 Å, He II 304 Å, and He I 584 Å, covering a
temperature range of 5.0$\times 10^{4}$ K, to 2.5 $\times 10^{5}$ K (Harrison,
1997; Berghmans et al., 1998; Harrison et al., 1999). Blinkers observed at 629
Å, 554 Å, and 599 Å show a typical size of 20-50 Mm2 with a lifetime 1-40 min
(Harrison, 1997; Bewsher et al., 2002). Berghmans et al. (1998) showed that
blinkers with sizes within the range of 40–400 Mm2 are also recognizable at
304 Å images with lifetimes about 2–60min.

X-rays and extreme ultraviolet (EUV) CBPs mostly contain small-scale loops
(Vaiana et al., 1973; Golub et al., 1974; Habbal & Withbroe, 1981). Most of
CBPs show similar morphological structures to large ARs with the small
magnetic opposite polarity (Madjarska, 2019). Using the Skylab X-ray
observations, Golub et al. (1974) determined that 100 CBPs occur on the
visible disk of the Sun at every time. Also, they found the lifetime of CBPs
ranging from 2 to 48 hr. Alipour & Safari (2015) investigated the statistical
properties of the CBPs in EUV emissions for 4.4 years. They showed that CBPs
have a typical size of 130 Mm2 and a lifetime ranging from a few minutes to
more than three days. Also, they estimated the average number of 572 (ranging
from 427 to 790) CBPs covering 2.6$\%$ solar visible disk at any time. Most
(about 90%) of CBPs occur at the supergranular cell boundaries and junctions
(Yousefzadeh et al., 2016). Recently, Hosseini Rad et al. (2021) calculated a
total energy-loss flux for the system of CBPs about $(4.84\pm 1.60)\times
10^{3}$ erg cm-2 s-1. To estimate the contribution of QS small-scale events in
the heating of the corona, they assumed a scale-free behavior for the energy
of CBPs. Then, they obtained a total energy-loss flux by extrapolating the
power-law distribution to the nanoflare region (events with energies greater
than $10^{24}$ erg). The energy-loss flux for the system of CBPs estimated
0.5–8.1% of the quiet coronal plasma’s heat flux.

It is accepted that magnetic features play an important role in forming most
events in the solar atmosphere. The magnetic fluxes observe in the form of
bipole and unipole region at the solar surface (Bellot Rubio & Orozco Suárez,
2019). A bi-pole refers to the two or a group of different opposite polarities
that occur in a small region. Magnetic flux also occurs in the unipolar
regions that are not distinctly coupled with any other pole(s) of opposite
polarity in the vicinity. Bewsher et al. (2002) showed that most blinkers
occur above the regions of unipole or bipole with a strength of more than 10
G. Also, most of CBPs are associated with bipoles’ magnetic fields (Madjarska,
2019). Sattarov et al. (2002) identified photospheric poles with strength
above 20 G and sizes between 5.5″ and 55.2″. They showed that the bipole
numbers remained approximately constant (from 1992 to 2001). They also
confirmed an anti-correlation of CBPs with sunspots.

There have been numerous studies on blinkers as well as CBPs on their physical
and plasma properties. However, no work systematically studies the
relationship between blinkers and CBPs observed in EUV and X-rays to the best
of our knowledge. The relationship between transients and small-scale magnetic
features is the main aim of this paper. Therefore, to identify, track and
classify them, we use an automatic feature identification method based on the
invariant and unique properties of the Zernike moments (ZMs) and the support
vector machine (SVM) classifier. We have used this method essentially since
the magnitude of ZMs is invariant under rotation and image normalization
(Alipour et al., 2012; Javaherian et al., 2014; Alipour & Safari, 2015;
Honarbakhsh et al., 2016; Yousefzadeh et al., 2016; Raboonik et al., 2017;
Alipour et al., 2019).

For the above-described purpose, we have extended the automatic identification
method of Alipour & Safari (2015) for the blinkers, ECBPs, and XCBPs by
analyzing Atmospheric Imaging Assembly (AIA)/Solar Dynamics Observatory (SDO)
images in three channels (namely 304, 193, and 94 Å). We aim to derive the
statistical properties of blinkers, ECBPs, and XCBPs, as well as their
relations. We also study the statistics of the photospheric magnetic features
and their relationship with transients in the solar cycle 24.

This paper is organized as follows: a primary overview of data analysis and
the automatic identification algorithms are explained in Section 2. The
statistical analysis and results are presented in Section 3. Finally, we
summarize and conclude in Section 4.

##  2 Observations and methodology

###  2.1 Data

The AIA on board SDO provides the solar atmospheric images at seven EUV and
two ultraviolet (UV) wavelengths (Lemen et al., 2012; Boerner et al., 2012).
AIA at 171, 193, and 211 Å filters cover a temperature range 0.6 to 2 MK
(plasma at QS temperature), 335, 94, and 131 Å filters include temperature up
to 10 MK (hot plasma), and 304 Å filter is sensitive to plasma at 50000 K
(most originate from the transition region). We used the full-disk synoptic
images taken at He II 304 Å, Fe XII 193 Å, and Fe XVIII 94 Å filters. Coronal
features such as ECBPs and XCBPs are detectable at 193 Å EUV and 94 Å X-ray
emissions, respectively. However, we note that emissions at 94 Å are
contaminated signals from hot Fe XVIII line and cooler components of Fe X and
Fe XIV (O’Dwyer et al., 2010; Del Zanna et al., 2011; Aparna & Tripathi,
2016). Also, blinkers as the transition region features are observable at 304
Å emissions. The data set includes the sequence of co-spatiotemporal images
with time intervals of 24 h. We analyzed the statistical properties related to
transients for ten years from 2010 June until 2019 December. We obtained the
co-spatiotemporal HMI continuum and line-of-sight (LOS) magnetogram images and
applied a method to track solar photospheric flows on the intensity continuum
images to determine the supergranular cell boundaries.

We used the LOS magnetograms to study the statistics of magnetic patches and
their relationship with blinkers, ECBPs, and XCBPs. To this end, we used the
HMI-LOS magnetograms with the spatial sampling of 2.4 arcsec/pixel at
1024$\times$1024 pixels. The HMI instrument on board SDO provides different
types of surface magnetic fields (Schou et al., 2012; Yeo et al., 2013, 2014;
Couvidat et al., 2016). The HMI noise levels depend on the HMI data product
and location on the solar surface. We used more than 12 G and 20 G (i.e., HMI
threshold values) to locate positive and negative polarities at the solar
surface. These thresholds are higher than the maximum noise level of about 10
G for LOS magnetograms.

###  2.2 Identification of AIA transients

Identification methodologies of the solar features have been often employed
based on the intensity characteristics at different observational wavelengths
and determining a threshold value by the trial-and-errors (Bewsher et al.,
2002; Subramanian et al., 2012). Due to irregular variation of the intensity
related to each feature (in different times, positions, and sizes),
determining a single threshold seems to be accompanied by missing/false
detection. Recently, automatic detection methods based on image moments and
SVM to identify solar features have been developed (Alipour et al., 2012;
Javaherian et al., 2014; Alipour & Safari, 2015). In this method, first, we
extract information from the image moments covering the morphology, intensity,
and geometry of features (blinkers, CBPs, etc.). Second, we fed the
information to an SVM classifier to detect the objects. The method developed
in this paper is a follow-up to the previous work presented by Alipour &
Safari (2015), which identified the CBPs (mainly at 193 Å) in EUV images taken
by AIA. Here, we adopted the automatic identification method of Alipour &
Safari (2015) to recognize the blinkers at 304 Å and XCBPs at 94 Å filters in
addition to identifying ECBPs at 193 Å. Below we provide a summary of the
automatic identification procedure applied here.

  * -

We performed a separate automatic identification classifier for each event,
including event class (positive class) and non-event class (negative class).

  * -

We collected (by visual inspection) a set of events by random surveying the
data during ten years of observations. We selected the events based on well-
known characteristics as (a) intensity criteria: localized intensity
enhancement both temporally and spatially (Alipour & Safari, 2015), (b) size
criteria: events with a length scale equal to or larger than 4.8″ and smaller
than 60″, and (c) morphology criteria: the small-scale loop-like or point-like
structures. We selected about 1000 events according to the above criteria for
each positive class.

  * -

We also collected some non-event in the negative class, including the regions
without small-scale transient phenomena, parts of large-scale coronal loops,
and regions outside of events. Figure 1 (left panel; a-b) shows the sample of
original images related to blinkers and non-events at 304 Å filter.

  * -

We computed the ZMs (with the order ($p$) and repetition number $(q)$) for
each sub-image. The repetition number satisfies the condition that
$|q|-p\leqslant 0$ is considered an even number. Alipour et al. (2019) showed
that for an AR image, the ZMs with $p_{\rm max}<10$ reconstruct the overall
shape of the AR. The morphological details of the AR were carefully
reconstructed by increasing the order number up to 31. They also showed that
for the $\rm p_{\rm max}(>31)$, the reconstructed images slightly deviated
from the original one (see their Figure 1). We used $\rm p_{\rm max}=5$, which
is enough to reconstruct the overall shape of a small-scale event. Figure 1
(right panel; a-b) shows the reconstructed images corresponding to blinkers
and non-blinkers with different shapes and morphology, respectively.

  * -

The algorithm automatically scans the full-disk images by a moving box in
which the maximum intensity is located on the center of the box. SVM
classifier picks up the events with their locations on the solar disk.

###  2.3 Performance of identification Method

To evaluate the performance of automatic identification for transients, we
computed various scores by applying the elements of a confusion matrix. The
number of positive (P) and negative (N) cases in the data set are important
terminologies for the confusions matrix Powers (2011). The elements of the
confusion matrix are true positive (TP: events correctly identified), false
positive (FP: non-events incorrectly identified), true negative (TN: non-
events correctly identified), and false negative (FN: events incorrectly
identified) (Fawcett, 2006). The important scores are precision (positive and
negative), recall (positive and negative), f1 score (positive and negative),
accuracy, Gilbert Skill Score (GS), Heidke skill score (HSS1, HSS2), and True
Skill Statistic (TSS). Table 1 gives the formulae for scores. For further
details, refer to Barnes & Leka (2008); Mason & Hoeksema (2010); Bloomfield et
al. (2012); Bobra & Couvidat (2015); Raboonik et al. (2017); Alipour et al.
(2019). The accuracy, precision, and f1 are scores for class-balanced
classifiers in which the number of events and non-events are approximately the
same. Also, the HSS and GS scores suffer the class-imbalanced problem. The TSS
is an essential metric to measure the performance of the class-imbalanced
classifiers.

Here, we investigated the performance of three independent classifiers to
identify blinkers, ECBPs, and XCBPs. To compute the skill scores as the
measurement of performance of a classifier, we collected a database (by visual
inspection) that includes more than 2000 events and 2000 non-events for each
classifier. The database for each classifier contains features with various
backgrounds, maximum intensity, shape, morphology, structures, and scales
collected within a solar cycle. The training data set includes about 70
percent (randomly selected) of the events (positive) and non-events (negative)
classes. For the test data set, we used the remaining 30 percent of both
positive and negative classes. We consider a sample event and non-event only
in either the training or testing data set. We performed this randomized
sampling for the training and test sets more than 100 times. We first
calculated the elements of the TP, FP, TN, and FN for the test set of each
trial. Then, using these elements, we computed the various skill scores. A
similar analysis was performed to determine the performance of automatic
detection for ECBPs and XCBPs. Table 2 represents the average and standard
deviation of the recall, precision, $f_{1}$ score (positive and negative),
accuracy, Gilbert Skill Score (GS), Heidke Skill Score (HSS1 and HSS2), and
True Skill Statistic (TSS) scores for automatic identification of blinkers,
ECBPs, and XCBPs. We trained classifiers with various backgrounds, maximum
intensities, shapes, morphologies, structures, and scales. Therefore, we
expect the present identification method for the transients is less affected
by the background variations. Some of the previous identification methods
(Sattarov et al., 2002; Hara & Nakakubo-Morimoto, 2003) for the transient
events applied the intensity thresholds that highly depend on the background
variation during a solar cycle. Applying this identification method to the
SoHO/EIT image, Alipour & Safari (2015) detected 670 CBPs compared to 450
events reported by Sattarov et al. (2002) at the same data set.

Table 1:  Definition of different skill scores.  Score | Formula
---|---
Recall (positive and negative) | $\rm{recall}^{+}=\dfrac{{\rm{TP}}}{\rm{TP}+\rm{FN}}$
| ${\rm{recall}}^{-}=\dfrac{\rm{TN}}{\rm{TN}+\rm{FP}}$
Precision (positive and negative) | ${\rm{precision}^{+}}=\dfrac{\rm{TP}}{\rm{TP}+\rm{FP}}$
| ${\rm{precision}^{-}}=\dfrac{\rm{TN}}{\rm{TN}+\rm{FN}}$
$f_{1}$ score (positive and negative) | $f_{1}^{+}=\dfrac{2\times\rm{precision}^{+}\times\rm{recall}^{+}}{\rm{precision}^{+}+\rm{recall}^{+}}$
|
$f_{1}^{-}=\dfrac{2\times\rm{precision}^{-}\times\rm{recall}^{-}}{\rm{precision}^{-}+\rm{recall}^{-}}$
Accuracy | $\rm{accuracy}=\dfrac{\rm{TP}+\rm{TN}}{\rm{TP}+\rm{FN}+\rm{TN}+\rm{FP}}$
Heidke Skill Score (HSS) | $\rm{HSS}_{1}=\dfrac{\rm{TP}-\rm{FP}}{\rm{TP}+\rm{FN}}$
|
$\rm{HSS}_{2}=\dfrac{2\times[(\rm{TP}\times\rm{TN})-(\rm{FN}\times\rm{FP})]}{(\rm{TP}+\rm{FN})\times(\rm{FN}+\rm{TN})+(\rm{TN}+\rm{FP})\times(\rm{TP}+\rm{FP})}$
Gilbert Skill Score (GS) | $\rm{GS}=\dfrac{\rm{TP}-\rm{CH}}{\rm{TP}+\rm{FP}+\rm{FN}-\rm{CH}},$
|
$\rm{CH}=\dfrac{(\rm{TP}+\rm{FP})\times(\rm{TP}+\rm{FN})}{\rm{TP}+\rm{FN}+\rm{TN}+\rm{FP}}$
True Skill Statistic (TSS) | $\rm{TSS}=\dfrac{{\rm{TP}}}{\rm{TP}+\rm{FN}}-\dfrac{{\rm{FP}}}{\rm{FP}+\rm{TN}}$
Table 2:  Average and standard deviation of the precision (Precision+ and Precision-), recall (Recall+ and Recall-), $f_{1}$ score (positive and negative), accuracy, Gilbert Skill Score (GS), Heidke Skill Score (HSS1 and HSS2), and True Skill Statistic (TSS) scores for automatic identification of blinkers, ECBPs, and XCBPs. Scores |  |  |  |  |
---|---|---|---|---|---
Features |  |  Blinkers |  ECBPs |  XCBPs |
|  $\rm{Recall}^{+}$ |  $0.91\pm 0.01$ |  $0.91\pm 0.01$ |  $0.96\pm 0.01$ |
|  $\rm{Recall}^{-}$ |  $0.97\pm 0.01$ |  $0.97\pm 0.01$ |  $0.89\pm 0.01$ |
|  $\rm{Precision}^{+}$ |  $0.96\pm 0.01$ |  $0.97\pm 0.01$ |  $0.90\pm 0.01$ |
|  $\rm{Precision}^{-}$ |  $0.91\pm 0.01$ |  $0.92\pm 0.01$ |  $0.96\pm 0.01$ |
|  $f_{1}$ score positive |  $0.93\pm 0.01$ |  $0.94\pm 0.01$ |  $0.93\pm 0.01$ |
|  $f_{1}$ score negative |  $0.94\pm 0.01$ |  $0.94\pm 0.01$ |  $0.93\pm 0.01$ |
|  $\rm{Accuracy}$ |  $0.93\pm 0.01$ |  $0.94\pm 0.01$ |  $0.93\pm 0.01$ |
|  Heidke Skill Score $(\rm{HSS}_{1})$ |  $0.87\pm 0.01$ |  $0.88\pm 0.01$ |  $0.86\pm 0.01$ |
|  Heidke Skill Score ($\rm{HSS}_{2}$) |  $0.87\pm 0.01$ |  $0.88\pm 0.01$ |  $0.86\pm 0.01$ |
|  Gilbert Skill Score (GS) |  $0.77\pm 0.01$ |  $0.79\pm 0.02$ |  $0.76\pm 0.02$ |
|  True Skill Statistic (TSS) |  $0.87\pm 0.01$ |  $0.88\pm 0.01$ |  $0.86\pm 0.01$ |

###  2.4 Identification of magnetic patches from HMI magnetograms

Most transients are associated with unipolar or bipolar magnetic fields. So,
we analyzed the statistics of the magnetic sources related to the features. To
extract the small-scale magnetic poles, we used the Yet Another Feature
Tracking Algorithm (YAFTA) developed based on the downhill method (DeForest et
al., 2007). We selected the negative and positive poles above 12 and 20 G with
a length scale of about 4.8″ to 60″. We extracted two nearest opposite
polarities with a separation less than 16.8″ as a bipolar magnetic feature. To
this end, we analyzed about 100 randomly selected ECBPs and their related
magnetic bipoles. For most cases, we observe that the two nearest opposite
polarity poles with a separation distance (from the boundary to boundary along
with both the Sun-x and Sun-y axis) are less than 16.8″ (for both $\Delta x$
and $\Delta y$) form a bipolar magnetic feature (Figure 2).

##  3 Results and Discussions

###  3.1 Statistics of transients

In this work, first, we applied an automatic identification method to
recognize the blinkers (at 304 Å), ECBPs (at 193 Å), XCBPs (at 94 Å) from
SDO/AIA images. Blinkers form on the transition region; however, ECBPs and
XCBPs are the transient coronal features. Using a region-growing algorithm, we
extracted pixels related to blinkers, ECBPs, and XCBPs. We considered an event
(an ECBP and/or an XCBP) as the coronal counterpart of one or multiple
blinkers if that blinker(s) appeared at a distance (boundary to boundary of
features) less than 7.2″ (empirically via trial and error) from the event. So,
we studied the connection of blinkers, ECBPs, and XCBPs together. Figure 3
shows lightcurves (with time interval 12 s) of transients at three AIA filters
at 304, 193, and 94 Å. We observed a blinker with the coronal counterpart
features (ECBP and XCBP) is along with the lightcurves that were extracted
with the simultaneous intensity enhancements at three filters versus the time
(Figure 3(a)). Lightcurves indicate the simultaneous occurrence of a blinker
and ECBP at 304 and 193 Å, respectively (Figure 3(b)). Figure 3(c) represents
a blinker without any coronal counterpart features. We also observed that
groups (two or more) of blinkers are associated with a single coronal feature
(Figure 4). Figure 5 represents a composite triple-filter image (consisting of
a 304, 193, and 94 Å) with positions of blinkers observed on 2018 August 26 at
00:00 UT in 0.95$R_{\odot}$ (the area inside the red circle) at the solar
disk. We found that out of 2785 blinkers (green points), about 1840 (yellow
points) are associated with ECBPs, and about 978 blinkers (blue points) have
occurred with both ECBPs and XCBPs. Furthermore, we observed 46 ECBPs (purple
points) without detected blinkers. However, our investigation (by visual
inspection) showed that the method did not detect 25 blinkers associated with
ECBPs.

Applying the automatic identification method, we observed 2198$\pm$478 (mean
and standard deviation for daily number) QS blinkers for each $0.95R_{\odot}$
on the full-disk image. For the whole of the Sun’s surface (about
2.2$\times$area0.95disk), we determined the total number of blinkers (sum of
the number of blinkers at QS and ARs) at every moment. So, we estimated
5643$\pm$674 (average and one standard deviation) blinkers at every moment,
which is twice the previous study (Harrison, 1997) at the O V SoHO/CDS
observations. Brković et al. (2001) followed a similar approach with an
iterative scheme. They showed that the thresholds affect the birthrate. So,
the number of blinkers increases to 20000. In this case, they showed that most
blinkers were covered with a single-pixel (1.67″) or somewhat overlapping
features. In the present method, we recognized events with individual pixels
and diameters larger than 4″ (to avoid the noise) without overlapping regions.
Also, two or more features with a distance less than 4.8″ are considered as a
single event. We detect an average number of 612$\pm$152 ECBPs and 349$\pm$64
XCBPs for $0.95R_{\odot}$ images. So, we estimated 1346$\pm$334 and
767$\pm$141 of ECBPs and XCBPs for the whole of the Sun’s surface,
respectively. To compare the area (surface) number density of QS and ARs
blinkers, we first computed the QS area projected onto the Lambert cylindrical
equal-area map. Then, we obtained the area number density of blinkers by
dividing the QS and ARs blinkers number by the QS and ARs area, respectively.
We obtained the surface density of blinkers about 0.001$\pm$0.0002 $\rm
Mm^{-2}$ and 0.002$\pm$0.0005 $\rm Mm^{-2}$ for QS and ARs, respectively.
These indicate that ARs’ blinkers are more abundant than the QS (Parnell et
al., 2002).

Figure 6 (left panel) shows the month-averaged time series of sunspot numbers
(SILSO data, Royal Observatory of Belgium, Brussels), QS blinkers, ECBPs, and
XCBPs during ten years within the solar cycle 24. We observed that transients
and sunspots have very strong anti-correlation (Pearson correlations -0.93$<$
r $<$-0.88). In general, one may ask if the sunspot numbers and sizes increase
during a cycle, the QS area decreases, which leads to the anti-correlation of
QS transients with sunspots. We investigated the correlation between the area
(surface) number density of transients and sunspots to address this question.
Figure 6 (right panel) represents the month-averaged time series of sunspot
numbers, the area number density of QS blinkers, ECBPs, and XCBPs during the
cycle. We observed that the area number density of QS transients has a very
strong anti-correlation (Pearson correlations -0.92$<$ r $<$-0.88) with
sunspots.

Using the ten years of AIA data with a cadence of 24 h, we identified in total
7,483,827 blinkers, 2,082,162 ECBPs, and 1,188,839 XCBPs. Our analysis shows
that about 57% and 34% of blinkers are associated with ECBPs and XCBPs,
respectively. Figure 7 shows the percent of blinkers are associated with ECBPs
(blue line) and XCBPs (red line) during the solar cycle 24. We observed that
these percentages remain approximately constant during the cycle. The
remaining blinkers with no coronal counterpart may be associated with
chromospheric features, which are not investigated here. Subramanian et al.
(2012) showed that two-third of blinkers are associated with the coronal
counterpart features. These multiple connections between blinkers and
counterpart features provide the link between the transition region and
corona, and, hence, they play an important role in the mass and energy
transfer in the solar atmosphere. Indeed, it seems the chromospheric upflows
and coronal down-flows associated with blinkers may provide the emergence and
preservation of the temperature gradient in the transition region.

In order to obtain the events’ birthrate, first, we tracked events within a
rectangular box (with the size of 600$\arcsec\times$600$\arcsec$ with a
cadence of 12 sec during 6 h (2019 September 00:00 UT- 06:00 UT). Then, we
obtained the birthrate for events by dividing the number of events by the size
of the box and the time interval (6 hours). We obtained the birthrate of
$1.1\times 10^{-18}$ ${\rm m}^{-2}{\rm s}^{-1}$, $3.8\times 10^{-19}$ ${\rm
m}^{-2}{\rm s}^{-1}$, and $1.5\times 10^{-19}$ ${\rm m}^{-2}{\rm s}^{-1}$ for
blinkers, ECBPs, and XCBPs, respectively.

###  3.2 Supergranular cell boundaries

It seems that blinkers may mostly occur at the boundaries of the supergranular
cells. We applied a ball-tracking method based on the tracking solar
photospheric flows on HMI continuum images to recognize the supergranular cell
boundaries (Potts et al., 2004; Attie & Innes, 2015; Attie et al., 2016). We
determined the supergranular cell boundaries for a field of view
840$\arcsec\times$480$\arcsec$ on 2019 December 4 (00:00 to 00:30 UT). Figure
8 shows the position of the peak intensity of 432 blinkers (red circles) and
supergranular boundaries. By comparing the location of blinkers and
supergranular cell boundaries, we found that about 80% of blinkers appeared on
boundaries and the remaining 20% located within the cells, which is in
agreement with Yousefzadeh et al. (2016) in which about 90% of the CBPs
appeared on the supergranular cell boundaries.

As blinkers have a short life and are pretty localized and considering that
these features are analogous to those of flares and microflares (even if the
amount of the involved energy is different), it might be possible to speculate
that blinkers are generated through a mechanism similar to that which produces
flares, i.e., magnetic reconnection (Giovanelli, 1946; Dungey, 1953), but at a
different scale. With the help of high-resolution imaging and spectroscopic
observations, it is widely reported that the blinkers are the results of
magnetic reconnection (Priest, 1999; Marik & Erdélyi, 2002; Doyle et al.,
2004; Subramanian et al., 2008; Young et al., 2018). The blinker location at
the supergranular cell boundaries might lead us to consider blinkers as events
likely related to a reconnection mechanism occurring between adjacent magnetic
structures with footpoints rooted at the supergranular boundaries. It
generally makes sense to expect the more significant magnetic field variation
among adjacent magnetic structures, making their border a favored candidate
location for magnetic reconnection. Indeed, the horizontal flow dominates and
sweeps the magnetic field towards the boundaries of supergranules. Hence, this
process (horizontal flow) produces mixed polarity magnetic fields at the
boundaries – the favorable topology of the magnetic field for magnetic
reconnection. Hence, we can say that the boundaries of supergranules are the
most favorable region for magnetic reconnection. Also, we showed that about
34% of blinkers were related to the X-ray features (XCBPs). These may support
categorizing some blinkers as small-scale flaring events. This speculation
certainly needs to be supported by appropriate future observations. In
addition, it is crucial to contemplate those reconnection mechanisms, such as
tearing type instabilities (Furth et al., 1963) that have already been
considered in the context of the solar corona (see, e.g., Matthaeus & Lamkin,
1986; Strauss, 1988; Velli & Hood, 1989; Shibata & Tanuma, 2001; Dahlburg et
al., 2009; Pucci & Velli, 2014; Del Sarto et al., 2016; Li et al., 2016; Singh
et al., 2019; Betar et al., 2020), and three-dimensional reconnection models
have been proposed (see Priest, 1984; Janvier, 2017).

###  3.3 Statistics of magnetic features

By applying the YAFTA code on the LOS HMI magnetograms, we extracted both QS
and ARs small-scale magnetic poles with sizes in the range of 4.8″ to 60″ for
the two thresholds ($>$ 12 G and $>$ 20 G). Figure 9 shows ARs’ boundaries
(green contours) and all positive (yellow contours) and negative (blue
contours) poles and bipoles (red boxes) in the LOS magnetogram (above 12 G).
As we observed in the figure, out of 3890 poles, about 1842 of them are formed
as bipoles.

Figure 10(a-b) depicts the month-averaged time series of sunspots and magnetic
features (bipole and poles). We found a strong anti-correlation (Pearson
correlation r=-0.76 to -0.71) between the QS bipoles (for both thresholds) and
sunspots. We also observed a strong anti-correlation (r=-0.78 to -0.72)
between the QS poles and sunspots. We also obtained a strong anti-correlation
between the area number density of QS small-scale magnetic features with
sunspots, which confirms the anti-phased behavior. The number density was
obtained by dividing the number of QS small-scale magnetic features to the QS
area projected to the Lambert cylindrical equal-area map. The densities are
still anti-correlated with sunspots, showing that this anti-phase is very
likely not simply due to a decrease in the QS area during the solar maximum.
This outcome might provide some insight into the solar dynamo mechanism.

We obtained very strong positive correlations (r$\geq$0.80) between the
transients and QS bipoles. Previous studies (Preś & Phillips, 1999; Madjarska
et al., 2003; Ugarte-Urra et al., 2004) showed that most CBPs associated with
QS bipoles. The negative correlation between the QS bipoles and sunspots
conforms to the anti-cycle variation of transients (Golub et al., 1979; Davis,
1983; Harvey, 1985). The anti-phased behavior of the QS small-scale magnetic
features and transients with sunspots and the locations of the transients at
the supergranular boundaries are two important findings that verified the
previous studies. Based on the identification of magnetic features of the
whole solar cycle 23 from MDI/SoHO, Jin et al. (2011) and Jin & Wang (2012)
obtained the anti-phase property for the QS small-scale magnetic features with
sunspots. Furthermore, Jin et al. (2011) stated that the small-scale magnetic
features are generated by some local turbulent dynamo represented as an anti-
phase with the global dynamo (see also Singh et al., 2018).

The idea of a local and a global dynamo as two conceptually distinct dynamos
has been proposed in the past years because of their diverse range of activity
and because of the different properties in time, i.e., temporal coherence in
the global dynamo, whereas time incoherence is in the local dynamo (Proctor et
al., 1995; Childress & Gilbert, 1996; Cattaneo & Hughes, 2001; Ossendrijver,
2003). In recent years the idea that only one dynamo mechanism is operating in
the Sun is finding its place in the community, and the point addressed in more
recent works (Tobias & Cattaneo, 2013; Pongkitiwanichakul et al., 2016; Nigro
et al., 2017) is how does one get a chaotic turbulent mess to have
organization at large scale. Assuming this more recent scenario and the
observed anti-correlation of the QS small-scale magnetic features with
sunspots, the reason for this anti-correlation can be a self-organization
process of the emerging magnetic flux into large-scale structures (i.e., ARs
and hence sunspots) that should occur during the solar maxima, when magnetic
energy increases. This large-scale self-organization might be due to the
magnetic helicity inverse cascade by subtracting magnetic energy stored in the
small-scale magnetic structures.

Furthermore, we studied the correlation between the small-scale magnetic
features of the full-disk and sunspots by adding the statistics of ARs’ small-
scale magnetic features to the QS for different thresholds. Figure 10(c) shows
a very weak anti-correlation (r=-0.19) between bipoles (for the threshold
$>$12 G) and sunspots. We observed a strong positive correlation (r=0.61)
between bipoles and sunspots for the threshold $>$20 G (Figure 10(d)). We also
found a strong positive correlation between poles (Figure 10(e-f)) and
sunspots. Indeed, they both should provide their contribution to the large-
scale components of the global magnetic field of the Sun (e.g., dipole and
quadrupole component, assuming a decomposition of the global solar magnetic
field on a spherical harmonic basis), thus poles and sunspots turn out to be
in phase.

To study the synchronization of transients and their associated magnetic
features, we considered 4.8″ and 7.2″ as the maximum distances (boundary to
boundary) from magnetic poles to blinkers and coronal features, respectively.
Out of 12,195,147 (5,694,525) detected poles, about 45% (28%) of them are the
components of bipoles for a threshold above 12 G (20 G). We observed that 33%
(13%), 55% (26%), and 59% (32%) of QS blinkers, ECBPs, and XCBPs are
associated with QS bipoles for a threshold above 12 G (20 G), respectively.
Also, we found that about 31% (25%), 26% (28%), and 25% (30%) of blinkers,
ECBPs, and XCBPs, appear above regions where a unipole dominates at the
threshold above 12 G (20 G), respectively. Overall, 61%, 76%, and 80% of
blinkers, ECBPs, and XCBPs appear above regions where one or both polarities
at the threshold above 12 G and these percentages decrease to 37%, 51%, and
58% with increasing the threshold to 20 G. Bewsher et al. (2002) showed that
about 88% of blinkers (observed at O V images) appeared above one pole or both
polarities regions ($>$ 10 G and size $>$ 10 MDI pixels).

Figure 11 represents the scatter plot of the daily number of transients and
the daily number of QS poles (with a threshold above 12 G). We observed that
the daily number of blinkers, ECBPs, and XCBPs correlated with poles via the
power-law relation ($y\propto x^{\alpha}$, where $y$ and $x$ indicate the
transient numbers and pole numbers, respectively) with exponents
1.57$\pm$0.03, 1.84$\pm$0.03, and 1.36$\pm$0.01, respectively. This relation
is useful to estimate the number of transients as a function of the
photospheric QS magnetic field.

Figure 12 shows the cumulative distribution function (CDF) for the normalized
maximum intensities (maximum intensity divided to the quiet background
intensity) of QS transients and related magnetic flux of poles, as well as
their scatter plots. To avoid the projection effects, we selected the events
and their related magnetic flux within a central region with the size of
960$\arcsec\times$960$\arcsec$. We observed that the tail of the distributions
follows the power-law behaviors (Figure 12 left and middle panels). To obtain
the power-law index ($\alpha$), we applied a maximum-likelihood fitting for
data without binning (Clauset et al., 2009). This power-law behavior in the
distribution tails can be due to a self-similar feature in the reconnection
process that generates blinkers, analogous to what has been observed for
flares and nanoflares as the system of self-organised criticality (Lin et al.,
1984; Crosby et al., 1993; Krucker & Benz, 1998; Parnell & Jupp, 2000;
Klimchuk et al., 2009; Farhang et al., 2018, 2019). We also observed that
(Figure 12 right panel) the normalized maximum intensity of transients are
correlated with the magnetic flux (the linear fits in log-log scale with
slopes $\beta$ about 1.31, 0.94, and 0.97 (red lines) in the scatter plots)
for blinkers, ECBPs, and XCBPs, respectively. To test the consistency for the
power laws and correlations (slope of scatter plots), suppose $x$ and $y$
satisfy the power-law-like distributions as $N(x)\propto x^{-\alpha_{x}}$ and
$N(y)\propto y^{-\alpha_{y}}$, respectively. Then, considering $y\propto
x^{\beta}$ and after some algebra manipulation (e.g., Aschwanden, 2011;
Javaherian et al., 2017), we give the slope as
$\beta=(\alpha_{x}-1)/(\alpha_{y}-1)$. we obtained the slope about 1.49, 0.84,
and 0.91 for the relation between the magnetic flux and the normalized maximum
intensities for blinkers, ECBPs, and XCBPs, respectively.

##  4 conclusion

Transients (such as blinkers, ECBPs, and XCBPs) are the fundamental
atmospheric phenomena that are mostly connected. These synchronized
atmospheric features may transfer mass and energy to the different layers.
Also, the chromospheric up-flows and coronal down-flows associated with
blinkers may provide the emergence and preservation of the temperature
gradient in the transition region.

Here, we adopted an automated identification procedure based on ZMs and SVM
classifier (Alipour & Safari, 2015) to detect blinkers, ECBPs, and XCBPs at
304, 193, and 94 Å AIA images, respectively. Classifiers contain features with
various backgrounds, maximum intensities, shapes, morphologies, structures,
and scales. The TSS was obtained greater than 0.86 as the performance of
classifiers. Also, we applied the YAFTA algorithm on HMI magnetograms to
extract the magnetic features such as poles and bipoles. We provide a summary
of the main results as follow:

  * -

We obtained an average number of 2198$\pm$478 blinkers for each image (in a
0.95$R_{\odot}$ of the solar surface) for ten years at solar cycle 24.
Therefore, we estimated about 5643$\pm$674 blinkers for the Sun at every
moment with a birthrate of $1.1\times 10^{-18}$ ${\rm m}^{-2}{\rm s}^{-1}$.

  * -

By analyzing ten years of AIA data, we identified in total 2,082,162
(1,188,839) ECBPs (XCBPs). We also determined an average number of
1364$\pm$334 (767$\pm$141) at every moment for the entire of the Sun with the
birthrate $3.8\times 10^{-19}$ ${\rm m}^{-2}{\rm s}^{-1}$ ($1.5\times
10^{-19}$ ${\rm m}^{-2}{\rm s}^{-1}$) for ECBPs (XCBPs).

  * -

Our investigation shows that roughly 57% and 34% of 7,483,827 blinkers at
transition regions are associated with ECBPs and XCBPs, respectively.
Therefore, these characteristics suggest that the transients are synchronized
together at different layers of the solar atmosphere. In other words, the
transients at different solar upper atmospheric layers tie up together
probably with a similar physical process.

  * -

We showed that the majority of the blinkers occur near the supergranular cell
boundaries, i.e., most of the blinkers are occurring in magnetic reconnection-
prone regions. Hence, this observational finding indicates (indirectly) that
blinkers are possibly forming due to magnetic reconnection.

  * -

We observed a very strong anti-correlation between the number of sunspots,
transients, and QS bipoles during solar cycle 24. Assuming this anti-
correlation is not a visible effect due to the background intensity
variations, it can be due to an organization of the magnetic flux into large-
scale structures as a consequence of a magnetic helicity inverse cascade that
takes place when the magnetic energy increases during solar maxima. Indeed,
when the magnetic energy increases via the omega effect, the magnetic energy
is more efficiently channeled towards large scales, thus forming coronal loops
and active regions. Consequently, the small-scale structures might have less
energy and/or be in an inferior number, thus producing this observed anti-
correlation.

  * -

This study showed that roughly 61%, 76%, and 80% of blinkers, ECBPs, and XCBPs
are associated with QS poles at the photosphere, respectively.

  * -

The anti-phased behavior of the QS magnetic bipoles and transients with
sunspots and the locations of the transients at supergranular boundaries
confirm the local dynamo as an important mechanism for the evolution of quiet
photosphere magnetic fields.

  * -

We observed the power-law relation ($y\propto x^{\alpha}$) with $\alpha$
equals $1.57\pm 0.03$, $1.84\pm 0.03$, and $1.36\pm 0.01$ for the daily number
of blinkers, ECBPs, and XCBPs with the daily number of QS poles, respectively
(Figure 11).

  * -

We showed that the normalized maximum intensities of QS transients correlated
with the magnetic fluxes via the power-law functions. The slopes of the power-
law functions are verified by the power-law behavior for the normalized
maximum intensities and magnetic flux of poles distributions (Figure 12). The
power-law behavior observed in the normalized maximum intensity distribution
of transients leads us to consider these events as energy releases due to a
self-similar process in analogy with what has been already observed for
flares.

In the next step, we would like to investigate the detection method for the
small brightenings such as campfires from AIA/SDO and EUI on Solar Orbiter
(Chitta et al., 2021; Berghmans et al., 2021) to determine the role of the
small-scale events in the heating of the coronal plasma.

Figure 1:  (a) Samples of blinkers observed by SDO/AIA at 304 Å and
reconstructed images with the maximum order number $p_{\rm max}$=5. (b)
Samples of the non-blinker regions and reconstructed images.

Figure 2:  Two samples of ECBPs and corresponding magnetograms. The red line
indicates the distance between the opposite polarities. The scale line about
16.8″ presents at the top of the magnetogram.  Figure 3: Lightcurves
corresponding to blinkers in three filters 304, 193, and 94 Å with time
interval 12s. (a) Intensity enhancement in 304, 193, and 94 Å simultaneously
occurred blinker, ECBP, and XCBP at the position (x=-210″ ,y=-123″). (b)
Simultaneous intensity enhancement in 304 and 193 Å occurred blinker and ECBP
in the position (x=112″ , y=179″). (c) Intensity enhancement in 304 Å which
occurred blinker in the position (x=-119″ , y=80″).

Figure 4:  Samples for the group of two (first row) and three (second row)
blinkers associated with ECBP and XCBP.

Figure 5: The SDO/AIA composite triple-filter image (304, 193, and 94 Å) on
2018 August 26 00:00 UT. Location of QS blinkers (color points) recognized by
presented automatic algorithm. The position of 46 ECBPs and/or XCBPs (purple
points) detected without the related blinkers. To more avoid the high noisy
features in the detection, we used the region inside the 0.95$R_{\odot}$ (red
circle).

Figure 6: The month-averaged time series of the sunspot numbers (dotted line),
QS blinkers, ECBPs, and XCBPs (left panel), area number density of QS
blinkers, ECBPs, and XCBPs (right panel) during ten years within solar cycle
24. The blue shaded area indicates the standard deviation of values (for a
month) each time.

Figure 7: The percent of blinkers are associated with ECBPs (blue line) and
XCBPs (red line) during the solar cycle 24.

Figure 8:  SDO/HMI magnetogram with a field of view
840$\arcsec\times$480$\arcsec$. The location of 432 blinkers is marked with a
red cycle. Superposed cell boundaries obtained by the ball-tracking method
were mapped to an HMI magnetogram with a yellow line. Most blinkers are found
close to supergranular boundaries.

Figure 9: The LOS magnetogram on 2018 August 26 00:00 UT including ARs
boundaries (green points), positive poles (yellow points), negative poles
(blue points), and bipoles (red boxes).

Figure 10: The month-averaged time series of the sunspot numbers (dotted
line), QS bipoles, bipoles (QS and ARs), and poles (QS and ARs) for threshold
above 12 G (left panel) and 20 G (right panel) are shown. The blue shaded area
indicates the standard deviation of values (for a month) each time.

Figure 11:  Scatter plot of the daily number of QS poles (above 12 G) and the
daily number of blinkers, ECBPs, XCBPs. We fitted a power-law curve $y\propto
x^{\alpha}$ (red line) for each data.

Figure 12: (Left panel) The cumulative distribution function (CDF) of the
normalized intensities of transients (maximum intensity divided to the quiet
background intensity), (Middle panel), CDF for the magnetic flux (in Mx unit)
of poles associated with transients for the threshold above 12 G, (Right
panels) Scatter plot for the magnetic flux of poles and normalized maximum
intensity of transients. We fitted the power-law relation (purple lines) to
the tails of distributions in which $\alpha$ indicates the power-law index. We
fitted a linear function (red line) for each scatter plot (in log-log scale)
at the regions corresponding to the tail of the distribution of the normalized
maximum intensity.

## Acknowledgement

We thank NASA/SDO, HMI & AIA science teams, for providing data used here. We
also gratefully thank the anonymous referee for very useful comments and
suggestions that improved the manuscript.

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