# Supercurrent diode effect in thin film Nb tracks

N Satchell  School of Physics and Astronomy, University of Leeds, Leeds, LS2
9JT, United Kingdom  PM Shepley  School of Physics and Astronomy, University
of Leeds, Leeds, LS2 9JT, United Kingdom  MC Rosamond  School of Electronic
and Electrical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom
G Burnell<EMAIL_ADDRESS>School of Physics and Astronomy, University
of Leeds, Leeds, LS2 9JT, United Kingdom

###### Abstract

We demonstrate nonreciprocal critical current in 65 nm thick polycrystalline
and epitaxial Nb thin films patterned into tracks. The nonreciprocal behavior
gives a supercurrent diode effect, where the current passed in one direction
is a supercurrent and the other direction is a normal state (resistive)
current. We attribute fabrication artefacts to creating the supercurrent diode
effect in our tracks. We study the variation of the diode effect with
temperature and magnetic field, and find a dependence with the width of the Nb
tracks from 2-10 $\mu$m. For both polycrystalline and epitaxial samples, we
find that tracks of width 4 $\mu$m provides the largest supercurrent diode
efficiency of up to $\approx 30\%$, with the effect reducing or disappearing
in the widest tracks of 10 $\mu$m. We propose a model based on the limiting
contributions to the critical current density to explain the track width
dependence of the induced supercurrent diode effect. It is anticipated that
the supercurrent diode will become a ubiquitous component of the
superconducting computer.

##  I Introduction

The supercurrent diode is an analogue to rectification in semiconductor pn
junctions, where current is allowed to flow only in one direction. In the
supercurrent diode, the critical current of the device ($I_{c}$) is
nonreciprocal ($I_{c}^{+}\neq I_{c}^{-}$), leading to the situation where a
dissipationless supercurrent can pass in one direction, but upon reversing the
current direction, the device becomes resistive. It has been suggested that a
supercurrent diode may become a useful component for digital processing in a
superconducting computer.

Nonreciprocal $I_{c}$ behavior has been predicted and observed prior to the
common use of the term supercurrent diode effect. Some notable early
experimental works include superconducting foils under external magnetic
fieldsSwartz and Hart (1967) and reports in superconductor-ferromagnet hybrid
systemsTouitou  _et al._ (2004); Vodolazov  _et al._ (2005); Morelle and
Moshchalkov (2006); Papon, Senapati, and Barber (2008).

Theoretically, a nonreciprocal $I_{c}$ can manifest in Josephson devices with
anomalous $\phi_{0}$ phase differencesGrein  _et al._ (2009); Halterman  _et
al._ (2022) or when the barrier materials have symmetry breakingZhang  _et
al._ (2022); Davydova, Prembabu, and Fu (2022). A common requirement of such
predictions are that the barriers contain exotic materials such as
ferromagnetic multilayersEschrig (2015); Braude and Nazarov (2007); Heim  _et
al._ (2013); Goldobin, Koelle, and Kleiner (2015); Shukrinov, Rahmonov, and
Sengupta (2019); Mazanik  _et al._ (2020); Meng  _et al._ (2022), topological
insulatorsBobkova  _et al._ (2016); Alidoust and Hamzehpour (2017); Li  _et
al._ (2022), Weyl semimetalsAlidoust (2018); Alidoust and Halterman (2020);
Kulikov  _et al._ (2020); Sinha (2020), materials with large spin-orbit
couplingAlidoust (2020); Alidoust, Shen, and Žutić (2021); Monroe, Alidoust,
and Žutić (2022), or that the electrodes are unconventional
superconductorsAmin, Omelyanchouk, and Zagoskin (2001); Liu, Zhou, and Tao
(2021); Xie, Efetov, and Law (2022). Several recent experiments have reported
Josephson diode effect in devices with exotic barrier materialsPal  _et al._
(2022); Baumgartner  _et al._ (2022); Wu  _et al._ (2022); Golod and Krasnov
(2022); Jeon  _et al._ (2022); Turini  _et al._ (2022).

In addition to the Josephson diode effect are reports and predictions for an
intrinsic supercurrent diode effect. Theoretically, an intrinsic effect can
occur in superconducting material systems with symmetry breaking, such as
broken inversion symmetryDaido, Ikeda, and Yanase (2022); Yuan and Fu (2022);
Karabassov  _et al._ (2022); Zhai  _et al._ (2022); Daido and Yanase (2022) or
broken time-reversal symmetryYuan and Fu (2022); Scammell, Li, and Scheurer
(2022); Wang, Wang, and Wu (2022). Therefore, much attention has been paid to
observations of supercurrent diode effect in exotic materials systems such as
multilayers with noncentrosymmetric structure Ando  _et al._ (2020); Wakatsuki
_et al._ (2017); Zhang  _et al._ (2020); Narita  _et al._ (2022), and systems
with time-reversal symmetry breakingLin  _et al._ (2022). In order to fully
understand the interesting and novel new physics of such systems, it is
important to be able to distinguish between supercurrent diode effect which is
intrinsic to the material, and which is induced as a result of device
fabrication.

Induced supercurrent diode effect can occur in tracks of $s$-wave
superconductors without intrinsic inversion or time-reversal symmetry
breaking, such as NbHou  _et al._ (2022); Suri  _et al._ (2022); Ustavschikov
_et al._ (2022); Gutfreund  _et al._ (2023). The induced supercurrent diode
effect is attributed to non-perfect device fabrication leading to
imperfections in the edges of the tracks. Therefore, the edges of the tracks
are unlikely to be identical, which can lead to the unequal generation,
penetration and expulsion of vortices on the opposite edges of the tracks,
resulting in rectificationVodolazov and Peeters (2005); Hou  _et al._ (2022).
The ability to induce a supercurrent diode effect without requirements for
exotic materials is useful from a practical point of view, since established
industrial processes tend to be based on $s$-wave superconductors such as Nb.

In this work, we report measurements on Nb thin films patterned into tracks of
varying widths. From our field and temperature dependent measurements of
induced supercurrent diode effect, we report three key experimental
observations. Firstly, our Nb tracks are 65$\,\mathrm{nm}$ thick, refining the
thickness in which supercurrent diode effect manifests. Secondly, by varying
the width of the tracks, we observe a dependence of the diode effect with
track width. Thirdly, we establish that single crystal epitaxy is not a
condition for diode effect by measuring both polycrystalline and epitaxial
samples. We report our observed diode effect as an efficiency parameter
$\eta=\frac{I_{c}^{+}-I_{c}^{-}}{I_{c}^{+}+I_{c}^{-}}$. Finally, we propose a
model to explain our observed track width dependence in terms of the critical
current densities in our devices and conclude that induced supercurrent diode
effect is a particularly important consideration in track widths where vortex
penetration contributes significantly to the overall critical current density.

##  II Methods

Nb films are deposited by dc magnetron sputtering in the Royce Deposition
System Roy . The magnetrons are mounted below, and confocal to, the substrate
with source-substrate distances of 134 mm. The base pressure of the vacuum
chamber is 3$\times 10^{-9}\,\mathrm{mBar}$ with the substrate at room
temperature and 1$\times$10${}^{-8}\,\mathrm{mBar}$ with the substrate at
1000∘C. Nb is grown at a rate of 0.06 nm/s at an Ar (6N purity) gas pressure
of 3.6$\times$10${}^{-3}\,\mathrm{mBar}$ to a nominal thickness of 65 nm. The
growth rate and film thicknesses are checked by x-ray reflectivity. The first
Nb sample was deposited at room temperature on Si/SiOx substrate and the
second Nb sample was deposited at elevated temperature of 1000∘C onto a single
crystal $a$-plane Al2O3 substrate to promote epitaxial growth.

Samples are fabricated into tracks using a direct laser writer to define
resist masks in S1813 photoresist, and reactive ion etching at 130 W in a 1:2
Ar:SF6 plasma to etch the Nb films. After fabrication, devices are measured in
a continuous flow ${}^{4}\mathrm{He}$ cryostat with 3$\,\mathrm{T}$ horizontal
superconducting Helmholtz coils. Traditional 4-point-probe transport geometry
is used to measure the current-voltage characteristic of the tracks with
combined Keithley 6221-2182A current source and nano-voltmeter in pulse mode
with 1 ms pulses and a 1$\%$ duty cycle to avoid a reduced retracking $I_{c}$
due to heating. Varying the duty cycle from 1-50$\%$ resulted in no changes to
the reported $I_{c}$. The schematic of the fabricated devices and measurement
geometry is shown in Figure 1 (a).

The sample grown at room temperature has a superconducting $T_{c}$ of 8.75 K
and a residual-resistivity ratio ($RRR$) of 2.8, giving an estimate for the
mean free path ($\ell$) of 6 nm – indicating a polycrystalline
microsctructure. The second sample has a higher $T_{c}$ of 9.05 K and a $RRR$
of 30, giving an estimate for $\ell$ of 96 nm, consistent with an epitaxial
microstructure. The increased $\ell$ is expected for the epitaxial Nb due to
the decrease in crystallographic defects such as grain boundaries. The
properties of our polycrystalline Nb thin films are reported
elsewhereQuarterman  _et al._ (2020).

##  III Results

Figure 1 shows the supercurrent diode effect in the polycrystalline sample
with 4 $\mu$m wide track. Figure 1 (b) shows the current-voltage ($I-V$)
characteristic of our track at applied fields where the diode effect is found
to be maximum. $I_{c}^{+}$ and $I_{c}^{-}$ are extracted from the $I-V$ when
the voltage reaches a small threshold value. Large nonreciprocal $I_{c}^{+}$
and $I_{c}^{-}$ can be seen in the $I-V$ characteristic. When the field
polarity is reversed, the nonreciprocal $I_{c}^{+}$ and $I_{c}^{-}$ are also
reversed. In the normal state of the device, the $I-V$ curve shows a slight
non-linear dependence, which we attribute to Joule heating as a result of the
large current densities. Our measurements at lower current densities (e.g. in
wider tracks or at warmer temperatures) show $I-V$ curves following the
expected linear metallic behavior.

Figure 1 (c) shows the out-of-plane applied field dependence of $I_{c}^{+}$
and $I_{c}^{-}$. Due to the diode effect, unexpectedly, the maximum values of
$I_{c}$ do not occur at zero applied field. We attribute this to the vortex
penetration field, which is discussed later. The presented field sweep is
acquired by sweeping from negative to positive field. A similar curve with the
same features is obtained by sweeping from positive to negative field. The
sign of the $I_{c}^{+}$/$I_{c}^{-}$ maximum with positive/negative field does
not change with temperature or field sweep direction and appears to favor
having the $I_{c}^{+}$ ($I_{c}^{-}$) maximum in positive (negative) field.
Across the 11 samples in our study showing the diode effect, when keeping the
device mounting and wiring geometry the same, $I_{c}^{+}$ ($I_{c}^{-}$)
maximum appears in positive (negative) field with a ratio of 8:3. The
$I_{c}^{+}$/$I_{c}^{-}$ maximum with positive/negative field can be reversed
by swapping the current wiring direction (Figure 1 (a)). Figure 1 (d) shows
the extracted diode efficiency $\eta$, where peak efficiency is achieved
around the fields corresponding to $I_{c}^{+}$/$I_{c}^{-}$ maximum. The
maximum in $\eta$ corresponding to the maximum in $I_{c}^{+}$ is a common
feature of our data, however we believe that this is a consequence of how
$\eta$ is defined rather than being a fundamental feature of the induced
supercurrent diode. We note a sight asymmetry and digitization in the high
field $\eta$ values in this device, Figure 1 (d). The asymmetry in $\eta$ at
high fields is present in some of our devices and may indicate that the diode
effect persists to high fields in some devices. The digitization of $\eta$ at
high fields is a result of determining $I_{c}$ from the finite current step
size in the I–V measurements.

Our interpretation of the origin of the supercurrent diode effect relies upon
our samples being in the limit where the critical current is determined by the
vorticies in the system, as opposed to being the depairing current. From the
$I_{c}(B)$ dependence, it is possible to extract the maximum super-heating
field of the Meissner stateHou  _et al._ (2022), $B_{s}$. The inset to Figure
1 (c) shows the low field $I_{c}$ with linear fits (dashed lines). $B_{s}$
corresponds to the interpolated intercept of the fits to the low field data,
when the $B$ offset is taken into account. From the four linear fits shown in
Figure 1 (c) inset, we obtain $B_{s}=10\pm 1$ mT. The expressionHou  _et al._
(2022), $B_{s}=\phi_{0}/(\sqrt{3}\pi\xi w$), provides an order of magnitude
estimate for $B_{s}$, where $\phi_{0}$ is the flux quantum and $\xi$ is the
Ginzburg–Landau coherence length ($\xi=11.6$ nm for polycrystalline thin film
NbQuarterman  _et al._ (2020)). For $w=4~{}\mu$m, $B_{s}=8.2$ mT, in
approximate agreement with our experimental findings. This indicates strongly
that in the region where the diode effect is observed the critical current is
determined by the vorticies.

Figure 2 (a) shows a scanning electron microscope image of the epitaxial 65 nm
thick Nb sample patterned into 4 $\mu$m wide track. To avoid charging during
imaging, a thin Au layer is deposited over the whole sample. Imaging of
multiple tracks reveals the following: firstly, some hardened resist residue
remains on the devices at the sidewalls. We do not believe that the resist
residue contributes to the observed diode effect; secondly, all our devices
have a low line-edge roughness in the range of $10-40$ nm with no significant
differences between opposite edges. This implies that the fabrication
artefacts believed to be responsible for the induced diode effect are subtle.

Figure 2 shows the supercurrent diode effect in the epitaxial 65 nm thick Nb
sample patterned into 4 $\mu$m wide track at 5 K. From the comparison with the
polycrystalline device presented in Figure 1 of the same track width and
measurement temperature, the epitaxial device shares many of the same features
with the following differences. From the I-V characteristic, Figure 2 (b), the
residual resistance of the epitaxial sample is lower than the polycrystalline
sample. This sample also displays slightly non-linear dependence, consistent
with Joule heating. The tracks patterned from the epitaxial sample has an
$I_{c}$ about twice that of the polycrystalline track, shown in Figure 2 (b)
and (c). The increased $I_{c}$ in the epitaxial sample is most likely due to
the combination of increased gap (higher $T_{c}$), decreased grain boundaries,
and increased mean free path compared to the polycrystalline sample. As shown
in Figure 2 (d), the measured maximum $\eta$ in this condition is smaller than
the polycrystalline sample.

We next study the track width dependence of the supercurrent diode effect in
the polycrystalline and epitaxial Nb samples. For polycrystalline Nb, the
London penetration depth $\lambda_{L}=96$ nmQuarterman  _et al._ (2020),
providing an estimate of the Pearl penetration depth
$\lambda_{P}=2\lambda_{L}^{2}/t\approx 300$ nm. For comparable epitaxial Nb,
$\lambda_{L}$ tends towards bulk Suter  _et al._ (2005) providing an estimate
$\lambda_{P}\approx 50$ nm. Other works have considered tracks in the limit
$w/\lambda_{P}=1/25$Suri  _et al._ (2022) and $w/\lambda_{P}=2$Hou  _et al._
(2022), however here we explore a new limit where the width of the tracks are
far greater than the Pearl penetration depth, $w>>\lambda_{P}$, covering
between $10<w/\lambda_{P}<200$. In this new limit, we observe a significant
superconducting diode effect and report a width dependence in our samples.

Figure 3 shows the full track width and temperature dependence of the
supercurrent diode effect in the epitaxial and polycrystalline Nb samples.
Considering first the epitaxial sample, Figure 3 (a), two samples of at $w=7$
and $10~{}\mu$m did not show finite $\eta$. In these two tracks, $I_{c}$ at
low temperatures exceeded the maximum output of our current source (100 mA),
limiting the range of temperatures we could measure. For the narrower samples
which showed $\eta$, the temperature dependence show similar trends for all
samples, with the largest $\eta$ at 1.8 K, and $\eta$ decreasing with warming.

Figure 3 (b) and (c) presents the track width dependence of $\eta$ at fixed
temperature. At 1.8 K, a clear peak in $\eta$ is found for the $4~{}\mu$m
track, with $\eta$ decreasing linearly for narrower or wider tracks. At 5 K,
the peak in $\eta$ is broader, with $w=4$ and $5~{}\mu$m showing similar
$\eta$. Again, for narrower or wider tracks $\eta$ decreases linearly with
width. Linear fits to the decay of $\eta$ for tracks of $4,5,6$ and $7~{}\mu$m
are presented as dashed lines.

In the polycrystalline Nb samples, Figure 3 (d), tracks in the width regime
$3\leq w\leq 5~{}\mu$m follow a similar temperature trend where $\eta$ is
largest for temperatures of 4 or 5 K, and decreases from the maximum value as
the temperature is cooled or warmed. Tracks with $w\geq 7~{}\mu$m show the
largest $\eta$ at the lowest temperature, with $\eta$ decreasing as the
temperature is warmed. Considering the track width dependence of $\eta$ at
fixed temperature, Figure 3 (e) and (f), $\eta$ shows the largest value for
tracks of $w=3$ or $4~{}\mu$m. Upon increasing $w$, $\eta$ decreases linearly
between $7\leq w\leq 10~{}\mu$m at 1.8 K and between $5\leq w\leq 10~{}\mu$m
at 1.8 K. The $5~{}\mu$m track at 1.8 K is a outlier to this trend. Linear
fits to the decay of $\eta$ for tracks of $7,8$ and $10~{}\mu$m are presented
as dashed lines.

In an attempt to further understand the origin of the diode effect and the
role of vorticies, we perform initialization experiments on one of our tracks.
Figure 4 shows the epitaxial 65 nm thick Nb sample with 5 $\mu$m wide track at
6 K. (a-f) shows the initial state of the device after zero applied field cool
and the conditions necessary to initialize the supercurrent diode effect in
the device. At the zero field cooled condition in Figure 4 (a), the device
already shows a small diode effect of $\eta=6\%$, presumably due to the small
trapped flux in the superconducting magnet. Figure 4 (a-f) present the results
of performing sequentially larger field sweeps. Starting from zero applied
field, we observe a crossover in $I_{c}^{+}$/$I_{c}^{-}$ at approximately 1
mT. The crossover at 1 mT, where $\eta=0$, likely corresponds to the true zero
field condition. Trapped magnetic flux is a common feature of superconducting
coils. Here, we choose not to perform background subtraction of this effect
and we present the dataset ‘as measured’. Due to the larger field step size
used to acquire the data presented in Figures 1 and 2, the small trapped flux
is not visible in those datasets. A further consequence of the trapped flux is
that our devices will have been cooled through the superconducting transition
in a small magnetic field, which will contribute flux vorticies in the as
cooled state. The dependence of the diode effect on field cooling was not
investigated in this study. The maximum in $I_{c}^{+}$ is at about 4 mT. As in
Figures 1 and 2, the maximum diode effect, $\eta$, corresponds to the maximum
in $I_{c}^{+}$. At larger fields the effect reduces, and $I_{c}^{+}$ =
$I_{c}^{-}$ at about 20 mT.

In Figure 4, comparing the out and return field sweep directions, there is a
hysteretic behavior in $I_{c}$, and hence $\eta$. Considering Figure 4 (e),
the hysteresis is particularly observable in $I_{c}^{+}$ between 10 and 20 mT,
but is present over the whole field range where $\eta\neq 0$. The hysteretic
behavior with field history suggests that the diode effect is sensitive to the
field history of the track. We expect that the preceding field history
establishes different vortex states in the track, which therefore influences
the magnitude of $\eta$. The largest hysteresis between 10 and 20 mT is
reproduced in subsequent larger field sweeps (Figure 4 (f)).

##  IV Discussion

From the presented experimental data, we can report three key experimental
observations. Firstly, at 65 nm thick, our films are in a distinct mid-range
thickness with respect to other works in the field Ando  _et al._ (2020); Hou
_et al._ (2022). In this mid-range thickness we have shown that a diode effect
is possible. Secondly, by varying the width of the tracks, we observe an
unexpected dependence of the diode effect with track width. Thirdly, we extend
previous work Hou  _et al._ (2022), which considered only epitaxial Nb and
demonstrate that the supercurrent diode effect is present in both
polycrystalline and epitaxial Nb.

In our polycrystalline Nb, the coherence length is 11.6 nm Quarterman  _et
al._ (2020), which is much less than the thickness of the 65 nm film studied
here. Our results therefore suggest that the supercurrent diode effect
reported here does not rely upon subtle interfacial effects, which may play an
increasing role when film thickness is comparable to the coherence length. An
added benefit of using thicker films is that at 65 nm the Nb has a near bulk
$T_{c}$, which makes our devices useful for integrating with Nb based
computing schemes operating at 4.2 K.

Our data shows that the induced supercurrent diode effect depends upon the
width of the track. From our data, we can estimate the upper limit of track
width for the supercurrent diode effect. For the polycrystalline film,
extrapolating the linear fit shown in Figures 3 (e) and (f) $\eta=0$ for $w$
between about 12 and $13~{}\mu$m. In the epitaxial film, $\eta=0$ is observed
at 5 K for $w=7\mu$m. Extrapolating the data at 1.8 K suggests that $\eta=0$
for $w\approx 8~{}\mu$m. In the epitaxial sample, $\eta$ also reduces in
narrowest tracks, but this trend is not reproduced in the polycrystalline
sample.

We propose that the induced diode effect does result from asymmetric
properties in the two edges of our tracks. This asymmetry causes unequal
generation, penetration and expulsion of vortices on the two edges of the
tracks, resulting in the observed rectification. Each edge has a different
critical current density required for penetration of magnetic vortices,
$j_{\textit{pen}}$. This could be induced by the asymmetric edge roughness
mechanism of Hou et al. Hou  _et al._ (2022), or more generally by any local
changes in the superconducting gap such as, most probably, edge profile. The
diode effect in our tracks therefore relies on being in a regime where the
critical current density ($j_{c}$) of the tracks is limited by flux vortice
entry. We know this to be the case from our analysis of the $I_{c}(B)$
dependence of Figure 1 (c), inset. The requirement for track to go normal is
therefore flux penetrating the track from the edge barriers continuously and
flux vortices moving freely across the track without being pinned.

There are three possible mechanisms to limit the critical current desnity
$j_{c}$ of our superconducting tracks. The ultimate limit is the depairing
current, $j_{\textit{pair}}$. Where the critical current of the track is
limited by flux vortices, the critical current density is limited by the
greater of the current density required for penetration of magnetic vortices,
$j_{\textit{pen}}$, and the current density required to depin the vortices,
$j_{\textit{pin}}$.

For our Nb tracks, we can reasonably assume that $j_{\textit{pair}}$ is an
intrinsic material property and $j_{\textit{pin}}$ is a property of the film,
both of which do not depend on track width. On the other hand, it has been
shown that $j_{\textit{pen}}$ depends on the track width ($w$) according
toIl’in  _et al._ (2010)

$j_{\textit{pen}}=\frac{2\pi B_{\textit{pen}}}{\mu_{0}\sqrt{dw}},$ (1)

where $d$ is the film thickness, $B_{\textit{pen}}$ is the minimum field for
vortex penetration and $\mu_{0}$ is the magnetic permeability of free space.
Figure 5 shows the experimentally determined maximum nominal $j_{c}$ at 5 K
for both the epitaxial and polycrystalline Nb tracks. $j_{\textit{pen}}$ is
fitted to the polycrystalline dataset with $B_{\textit{pen}}$ a free fitting
parameter, returning a best fit value of $11\pm 1$ mT. Note that
$B_{\textit{pen}}$ is likely a combination of the applied and self magnetic
fields in our system. Also plotted are the experimentally determined
$j_{\textit{pair}}$ from Il’in et al. for narrow polycrystalline Nb
tracksIl’in  _et al._ (2010) and $j_{\textit{pin}}$ from Zhu et al. for a 110
nm polycrystalline Nb filmZhu, Lockhart, and Turneare (1994).

Considering the fit to the polycrystalline Nb tracks, it is possible to divide
Figure 5 into three width regions. For narrow tracks occupying region I
($w<w_{1}$), the critical current density in the track is limited by
$j_{\textit{pair}}$. Since vortices do not contribute to $j_{c}$, our
conjecture is that it is not possible for fabrication to induce a supercurrent
diode effect in this region. For intermediate region II ($w_{1}<w<w_{2}$), the
critical current is limited by $j_{\textit{pen}}$. In this region, the track
goes normal when the transport current equals $j_{\textit{pen}}$, and a diode
effect can be induced by device fabrication if $j_{\textit{pen}}$ is
asymmetric on each edges. For wide tracks in region III ($w>w_{2}$), the
critical current is no longer limited by $j_{\textit{pen}}$. Here, upon
applying a transport current, first vortices enter the track at the edges at
transport current exceeding $j_{\textit{pen}}$. However, these vortices become
pinned, preventing the track from going normal. It is only when the transport
current exceeds $j_{\textit{pin}}$ that vortices can move and the track goes
normal. In region III, the induced diode effect should be suppressed as
$j_{\textit{pin}}$ is an intrinsic material property and therefore likely to
be uniform across the width of the track.

Comparing the width dependence of $j_{c}$ in Figure 5 to our experimental data
on the width dependence of the supercurrent diode effect in Figures 3 we can
make the following observations and predictions. We expect that the induced
diode effect will be suppressed in region III, which for the polycrystalline
dataset is at $w\approx 13~{}\mu$m. This width corresponds well with our
extrapolation to $\eta=0$ in Figures 3 (e) and (f). We can also use the model
to predict that $w=600$ nm is the upper boundary of region I for
polycrystalline Nb. Therefore tracks narrower than about 600 nm should not
show induced supercurrent diode effect, however fabrication of tracks that
narrow would require a change in fabrication methodology. Finally, applying
similar analysis to the epitaxial Nb tracks, we conclude that both
$j_{\textit{pen}}$ and $j_{\textit{pin}}$ must be higher in the epitaxial Nb.
Interestingly, a higher $j_{\textit{pen}}$ would suggest that that region I
extends to wider epitaxial tracks tracks compared to polycrystalline tracks,
which would be consistent with our observation of reduced $\eta$ in the
narrowest epitaxial tracks in Figures 3 (b) and (c).

##  V Conclusions

In conclusion, we report on induced supercurrent diode effect in 65 nm thick
polycrystalline and epitaxial Nb films patterned into tracks. Consistent with
previous works, the superconductor itself does not require any intrinsic
inversion symmetry breaking and the layer can be significantly thicker than
the coherence length. Our experimental results and model suggests that in
order to distinguish between intrinsic and induced supercurrent diode effect
in a materials system, the critical current density in the fabricated devices
should be limited by $j_{\textit{pair}}$ and not by flux vortices. Therefore,
claims of intrinisic diode effect should only be made for devices limited by
$j_{\textit{pair}}$ (occupying region I of Figure 5). This is particularly
relevant for the study of exotic materials to isolate the proposed intrinsic
effect from the fabrication induced effect we report here.

For technological applications, it would be possible to manipulate the devices
edges to create local differences in $j_{\textit{pen}}$, for example by
engineering edge roughness or by changing the superconducting properties of
one edge by ion implantation, and therefore control induced diode effects.

Our largest reported diode efficency is $\eta\approx 30\%$ in a 4 $\mu$m wide
track at 5 K. We report that the supercurrent diode effect can be observed in
Nb tracks over a range of track widths, applied out-of-plane applied fields
and temperatures. Our results show a track width dependence, where for the
widest tracks in this study, we report that the diode effect reduces or
disappears altogether. We present a model of the critical currents in the
track to account for our observed width dependence.

###### Acknowledgements.

The work was supported financially through the following EPSRC grant:
EP/V028138/1. We acknowledge support from the Henry Royce Institute. We would
like to thank N.O. Birge for helpful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

N Satchell: Investigation (lead); methodology (lead); conceptualization
(equal); writing - original draft (lead); visualization (lead); formal
analysis (lead); writing - review and editing (equal); funding acquisition
(supporting). PM Shepley: Investigation (supporting); methodology
(supporting); writing - review and editing (equal). MC Rosamond: Investigation
(supporting); methodology (supporting); writing - review and editing (equal).
G Burnell: Project administration (lead); funding acquisition (lead);
conceptualization (equal); methodology (supporting); investigation
(supporting); writing - review and editing (equal).

## DATA AVAILABILITY

The datasets generated during the current study are available in the
University of Leeds repository, DOI: 10.5518/1293.

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Figure 1:  Supercurrent diode effect in polycrystalline 65 nm thick Nb
patterned into a 4 $\mu$m wide track, measured at 5 K. (a) Schematic cross
section of the Nb track device showing the applied field and measurement
current direction (not to scale). (b) Current-voltage characteristic of the
device measured at $\pm$3 mT applied out-of-plane field. (c) Extracted
critical currents ($I_{c}^{+}$ and $I_{c}^{-}$) with out-of-plane applied
field, insert shows low field region with fitting to the model described in
the text. The uncertainty in determining $I_{c}$ is the current step size and
is smaller than the data points. (d) Diode efficiency $\eta$ corresponding to
the dataset in (c).

Figure 2:  Supercurrent diode effect in epitaxial 65 nm thick Nb. (a) Scanning
electron microscope image of a 4 $\mu$m wide track. (b-d) Electrical transport
characteristic of a 4 $\mu$m wide track, measured at 5 K. (a) Current-voltage
characteristic of the device measured at $\pm$7 mT applied out-of-plane field.
(b) Extracted critical currents ($I_{c}^{+}$ and $I_{c}^{-}$) with out-of-
plane applied field. The uncertainty in determining $I_{c}$ is the current
step size and is smaller than the data points. (c) Diode efficiency $\eta$
corresponding to the dataset in (b).

Figure 3:  Supercurrent diode effect in epitaxial and polycrystalline 65 nm
thick Nb patterned into tracks (a,d) The maximum diode efficiency parameter,
$\eta$, with temperature for a series of tracks with varying width for the
epitaxial and polycrystalline samples respectively. (b,c,e,f) Corresponding
track width dependence of $\eta$ at (b,e) 1.8 K and (c,f) 5 K. Solid lines in
(a,d) represent guides for the eye, while dashed lines in (b,c,e,f) show
linear fits to the decay of $\eta$ for the widest tracks.

Figure 4:  Initialization of the supercurrent diode effect in epitaxial 65 nm
thick Nb patterned into 5 $\mu$m wide track at 6 K. (a-e)
$I_{c}^{+}$/$I_{c}^{-}$ with sequentially larger field sweeps on a semi-log
scale. Before each sweep the sample was briefly warmed above $T_{c}$ and
cooled again in zero applied field. The uncertainty in determining $I_{c}$ is
the current step size and is smaller than the data points.

Figure 5:  Width dependence of the critical current densities in Nb tracks.
Experimental data for the nominal critical current density at 5 K for all
devices in this study. The polycrystalline tracks are fit to Equation 1, with
a best fit value $B_{\textit{pen}}=11\pm 1$ mT. Also shown are literature
values for polycrystalline Nb depairing currentIl’in  _et al._ (2010)
($j_{\textit{pair}}$) and depinning currentZhu, Lockhart, and Turneare (1994)
($j_{\textit{pin}}$). We define three regions depending on which mechanism
determines the critical current density of our tracks, see text.


[type=editor, auid=000,bioid=1, orcid=0000-0003-0339-5098]

[1]

1]organization=Ioffe Institute, addressline=Polytekhnicheskaya st., 26,
city=St. Petersburg, postcode=194064, country=Russia

[cor1]Corresponding author

# Cometary outbursts in the Oort cloud

D. V. Belousov<EMAIL_ADDRESS>[  A. K. Pavlov
<EMAIL_ADDRESS>
###### Abstract

Comet nuclei in the outer Solar system are constantly irradiated by cosmic
rays at low temperatures. Accumulated high concentrations of radicals can
undergo fast recombination with significant heating of cometary surface
layers. We present the model of comet activity at large heliocentric distances
caused by the recombination of radicals. We found that the considered
mechanism can cause activity of comets in distant regions of the Solar system,
even at the Oort cloud distances. Outbursts in distant comet reservoirs can be
a new source of dust and ice particles contributing to the recently discovered
anomalous diffuse light in the cosmic extragalactic background optic light.
The orbits of small-radii comets in the Oort cloud are highly influenced by
cometary outbursts. This effect may account for the observed decrease in the
number of small-radius long-period comets.

###### keywords:

Comets Kuiper beltCosmic rays Ices

##  1 Introduction

The outer solar system includes two sources of cometary nuclei: the Kuiper
belt and the Oort cloud. The surface temperatures of comets range from 30-60 K
in the Kuiper belt to about 10 K in the Oort cloud. Due to low temperatures,
internal processes that comets tend to have close to the Sun (e.g., water ice
sublimation) are suppressed. When considering the chemical and physical
evolution of ice at low temperatures, cosmic ray irradiation plays a
significant role in changing the properties of ice. In the Kuiper belt, comet
nuclei are irradiated by solar cosmic rays (SCR) and galactic cosmic rays
(GCR) with the modulated LIS (Local Interstellar Spectrum). The SCR flux in
the Kuiper belt region is highly decreased compared to the inner part of the
Solar system. In the Oort cloud, galactic cosmic rays with the LIS constantly
bombard cometary surfaces. Irradiation by energetic particles generates
radicals and ions in ice at low temperatures. Since the mobility of radicals
decreases substantially with temperature, comet nuclei in the outer Solar
system can accumulate high concentrations of radicals. Indeed, experiments on
ice irradiation with protons, electrons, neutrons, and gamma-photons
demonstrate efficient accumulation of radicals in ice at temperatures 10-100 K
(Moore et al., 1983; Carpenter, 1987; Shabalin et al., 2003; Zhu et al., 2021;
Pavlov et al., 2022). Radicals are highly reactive at elevated temperatures,
so even a slight rise in the ambient temperature of irradiated ice can cause
instant exothermic recombination of radicals (Shabalin et al., 2003). Besides,
a high concentration of radicals is unstable to spontaneous recombination
(Shabalin et al., 2003). When the concentration of radicals reaches 1%, the
irradiated sample can suddenly ignite due to the recombination of radicals
(Moore and Hudson, 1992).

When approaching the Sun, a comet increases its brightness by reflecting
sunlight on dust and ice particles ejected in a halo. This process can be
gradual or abrupt. The latter is called a cometary outburst (Gronkowski,
2007). Cometary outbursts have been detected far beyond the water-ice
sublimation boundary of about 3 AU (Meech et al., 2009). Amorphous-to-
crystalline ice transition is supposed to be the primary source of cometary
outbursts at distances from the Sun of up to 14 AU. However, recent detections
of comets at much greater heliocentric distances, namely C/2017 K2
(PANSTARRS), C/2010 U3 (Boattini) and C/2014 UN271 (Bernardineli-Bernstein),
demonstrate the need for other mechanisms of comet activity (Bouziani and
Jewitt, 2022). In Pavlov et al. (2022), we showed that the recombination of
radicals can potentially cause comet activity at large heliocentric distances.
Since recombination reactions occur at very low temperatures, comets can
demonstrate activity even in the Oort cloud.

Herein, we consider cometary outbursts in distant regions of the Solar system.
To evaluate the feasibility of cometary activity in large heliocentric
distances, we developed the model of cometary outbursts induced by the
recombination of free radicals. The model is described in Section 2. The
results of numerical simulations are presented in Section 3 and discussed in
Section 4. The conclusions are summarized in Section 5.

##  2 Model description

###  2.1 Basic equations

We use models for describing processes inside a comet from Prialnik and Bar-
Nun (1987), Orosei et al. (1999) and Marboeuf et al. (2012). All listed models
were developed for comets approaching the Sun for heliocentric distances less
than 14 AU. Since free radicals constantly accumulate in the upper surface
layer of a comet during irradiation, we can treat radicals as a renewable
source of energy inside a comet. This very feature of free radicals makes the
recombination of radicals the perfect sustainable driving force for the
process inside a comet at distances where solar irradiation is strongly
decreased (Pavlov et al., 2022).

In our simulations, we use the following model of comet activity. The
recombination of radicals increases the temperature of dust and ice in a
layer. The transition from amorphous to crystalline ice starts slowly at about
120 K and proceeds quickly at 130 K (Schmitt et al., 1989). It is well-known
that amorphous ice traps molecules of volatile gases (e.g., CO) at low
temperatures (Bar-Nun et al., 1985). The release of trapped gases is the
accompanying process of the amorphous-to-crystalline ice transition. Owing to
the high rate of recombination of radicals and large gas capacities of
amorphous ice, the release of volatiles can produce high-pressure zones
beneath the comet’s surface. When the gas pressure exceeds the tensile
strength of the comet material, the rupture of the comet layers above the
high-pressure zones occurs with the ejection of dust and ice particles
(Gronkowski and Wesołowski, 2015).

According to current models, comets formed in cold regions with efficient
condensation of volatile gases. The experimental and theoretical works show
that amorphous ice dominates the comet material. From observations, CO and CO2
gases are the most abundant components in a comet coma besides water.
Therefore, our model includes four constituents of cometary nuclei: dust,
water ice, CO, and CO2. The initial ice composition is determined by the dust-
to-ice mass ratio $\mu$, the mass fraction of trapped gases in amorphous ice
$f_{n}$ ($n$ = CO, CO2) and porosity $\Psi$. The mass density of a cometary
layer $\rho$ is the sum of all compounds:

$\rho=\rho_{\mathrm{d}}+\rho_{\mathrm{a}}+\rho_{\mathrm{c}}+\sum_{n}(\rho_{n}^{\mathrm{v}}+\rho_{n}^{\mathrm{c}})$
(1)

where $\rho_{\mathrm{d}}$, $\rho_{\mathrm{a}}$, $\rho_{\mathrm{c}}$,
$\rho_{n}$ are the mass density of dust, amorphous ice, crystalline ice, and
volatiles $n$ ($\mathrm{c}$ stands for the ice state and $\mathrm{v}$ for the
gas state), respectively. The effect of water vapour on the energy and local
gas pressure at the temperatures considered is insignificant.

The thermodynamic evolution of the cometary nucleus obeys the global energy
conservation equation:

$\rho c\frac{\partial T}{\partial t}=\nabla(k\nabla
T)-\sum_{n}c_{n}\boldsymbol{J}_{n}\nabla
T-\sum_{n}H_{n}^{\mathrm{s}}Q_{n}^{\mathrm{s}}+Y_{\mathrm{ac}}+Y_{\mathrm{rec}}$
(2)

where $c$ is the specific heat capacity, and $k$ is the thermal conductivity
of a cometary material. The values of $c$ and $k$ are calculated according to
Marboeuf et al. (2012). For refractories, we use the specific heat capacity of
enstatite (Bouziani and Jewitt, 2022). The heat conductivity of the cometary
material is strongly affected by the Hertz factor $h$ (Marboeuf et al., 2012).
The typical value for $h$ lies in the 10-1 to 10-3 range.

The (2) term of Eq. 1 describes the heat diffusion through a comet layer. The
(3) term denotes the convection, where $c_{n}$ and $\boldsymbol{J}_{n}$ are a
specific heat and a gas flux of volatiles $n$, respectively. The (4) term
accounts for the energy consumption during sublimation and energy release
during condensation, where $Q_{n}^{\mathrm{s}}$ is a gas source and
$H_{n}^{\mathrm{s}}$ is the latent heat of $n$ gas sublimation. The (5) term
describes the energy of the amorphous-to-crystalline ice transition, taking
into account the energy loss due to the sublimation of trapped gases:

$Y_{\mathrm{ac}}=\rho_{\mathrm{a}}\lambda\left(H^{\mathrm{cr}}-\sum_{n}f_{n}H_{n}^{\mathrm{s}}\right)$
(3)

where $H^{\mathrm{cr}}$ is the latent heat of crystallization and $\lambda$ is
the rate of crystallization (Schmitt et al., 1989). The (6) term accounts for
the energy released during the recombination of radicals:

$Y_{\mathrm{rec}}=\sum_{x}H_{x}^{\mathrm{rec}}K_{x}{N_{x}}^{2}$ (4)

where $H_{x}^{\mathrm{rec}}$ is the energy of the recombination reaction per
one radical of type $x$, $N_{x}$ is the concentration of radicals and $K_{x}$
is the rate of recombination.

The recombination of radicals obeys the bimolecular kinetic equation:

$\frac{\partial N_{x}}{\partial t}=-K_{x}(T){N_{x}}^{2}$ (5)

$K_{x}$ depends on the temperature $T$ according to the Arrhenius law:

$K_{x}=K_{0,x}e^{-U_{\mathrm{a},x}/k_{\mathrm{b}}T}$ (6)

with the activation energy $U_{\mathrm{a},x}$ and the Boltzmann constant
$k_{\mathrm{b}}$.

The model considers radicals only in ice component of comet matter with the
following reactions of radicals:

$\mathrm{H}+\mathrm{H}=\mathrm{H}_{2}$ (7)

$\mathrm{OH}+\mathrm{OH}=\mathrm{H_{2}O_{2}}$ (8)

The surface boundary condition for Eq. 2 is:

$(1-A)F_{\mathrm{b}}=\varepsilon\sigma T^{4}+k\nabla T$ (9)

where $A$ is the bolometric Bond albedo, $\varepsilon$ is the infrared surface
emissivity and $\sigma$ is the Stefan-Boltzmann constant. The value
$F_{\mathrm{b}}$ accounts for all possible sources of external heating of
comet nuclei. The solar radiation dominates at distances $r_{\mathrm{h}}$ up
to 103 AU. For regions beyond 103 AU (the Oort cloud), the contribution from
the cosmic microwave background ($F_{\mathrm{CMB}}$), the galactic disk
radiation ($F_{\mathrm{disk}}$) and the radiation of supernovae or passing
luminous O, B stars ($F_{\mathrm{OB,SN}}$) should be taking into account
(Stern and Shull, 1988), so:

$F_{\mathrm{b}}=\frac{L_{\mathrm{s}}}{4\pi
r{{}_{h}}^{2}}+F_{\mathrm{CMB}}+F_{\mathrm{disk}}+F_{\mathrm{OB,SN}}$ (10)

where $L_{\mathrm{s}}$ is the solar luminosity.

Changes in the comet layer composition are obtained by solving the mass
conservation equation for each component of comet matter considering
amorphous-to-crystalline ice transition, gas diffusion,
sublimation/condensation:

$\frac{\partial\rho_{\mathrm{c}}}{\partial t}=\lambda\rho_{\mathrm{a}}$ (11)

$\frac{\partial\rho_{\mathrm{a}}}{\partial t}=-\lambda\rho_{\mathrm{a}}$ (12)

$\frac{\partial\rho_{n}^{\mathrm{v}}}{\partial
t}+\nabla\boldsymbol{J}_{n}=f_{n}\lambda\rho_{\mathrm{a}}+Q_{n}^{\mathrm{s}}$
(13)

$\frac{\partial\rho_{n}^{\mathrm{c}}}{\partial t}=-Q_{n}^{\mathrm{s}}$ (14)

The gas flux of volatiles ($n$ = CO, CO2) is given by:

$\boldsymbol{J}_{n}=G_{n}\nabla P_{n}$ (15)

where $P_{n}$ is the partial pressure of gas $n$ and $G_{n}$ is the gas
diffusion coefficient. The gas flow though porous comet matter can be: 1) free
molecular (Knudsen regime), 2) viscous or 3) mixture of them depending on the
Knudsen number. The model considers cylindrical pores with the tortuosity of
$\sqrt{2}$ following Marboeuf et al. (2012).

The surface boundary condition for Eq. 13 is $P_{n}=0$. The sublimation of
crystalline ice is not included since it is insignificant at the considered
temperatures. Diffusion of dust is also not taken into account.

The ejection of volatiles from the amorphous ice leads to unsaturation
conditions when the local pressure ($P_{n}$) is greater or less than the
saturated gas pressure $P_{n}^{\mathrm{s}}$. The re-establishment of
saturation conditions by means of gas condensation (when
$P_{n}>P_{n}^{\mathrm{s}}$) or ice sublimation (when
$P_{n}<P_{n}^{\mathrm{s}}$) occurs at much smaller timescales than the
amorphous-to-crystalline ice transition. The model assumes that if the time of
condensation/sublimation is less than the chosen time-step of the model, the
re-establishment of saturation conditions occurs instantly. The deviation of
the local pressure from the saturated pressure is added/subtracted to/from the
density of the condensed gas to guarantee mass conservation (Davidsson, 2021).

The release of trapped gases from the amorphous ice results in gas pressure
growth. The pressure continues increasing until the gas flux from amorphous
ice reaches the balance with the diffusion flux casting through the comet
layer. The values for the pressure in subsurface areas can be calculated using
Eq. 2-15. When subsurface pressure exceeds the tensile strength of the comet
matter, fractures can develop. This effect may lead to the intense ejection of
comet matter over cavities with high gas pressure (Gronkowski and Wesołowski,
2015; Prialnik and Sierks, 2017). Here, we assume that the outburst goes off
when subsurface pressure exceeds the tensile strength.

In the numerical calculations, we use a fully implicit scheme with a finite
volume method for Eq. 2 and 13 in 1D spherical geometry. The time step is
constantly adjusted during calculations in order to restrict changes in
pressure and temperature.

###  2.2 Initial conditions and model parameters

Here, we consider comet nuclei located in the Oort cloud. Although the
composition of comets in the Oort cloud is unknown, observed dynamically "new"
comets may represent comets in the Oort cloud. Since the model simulates the
evolution of upper layers of comets, where effective irradiation by cosmic
rays occurs, all the parameters listed hereafter describe the surface layers.
Initially, the volatile gases CO and CO2 are present only as trapped in the
amorphous ice. We use this constraint to distinguish this model from the model
of cometary activity at large heliocentric distances based on the sublimation
of volatile ice. In addition, the sublimation of condensed and trapped gases
has a similar effect on the total energy of cometary activity (see Eq. 2). The
molar abundances of the gases CO and CO2 relative to water are both 10 $\%$
(i.e., the mass ratios of CO and CO2 to water ice are 0.16 and 0.25,
respectively). We chose these values for $f_{\mathrm{CO}}$ and
$f_{\mathrm{CO_{2}}}$ for two reasons: 1) trapped volatile molecules in high
concentrations make the amorphous-to-crystalline ice transition energy
neutral. Thus, recombination reactions are the main mechanism of comet
activity in this model; 2) the molar abundance of interstellar CO2 ices is in
the range of $10-23\%$ with approximately the same value for CO (Gerakines et
al., 1999). In the model dust to ice mass ratio varies from the ice-rich
$\mu=0$ to the ice-poor $\mu=4$ cases. The pore radius is set to 1 $\mu$m
following Gronkowski and Wesołowski (2015). The intense GCR irradiation in the
upper comet layer can produce a dense crust enriched with organic molecules
and dust (Strazzulla and Johnson, 1991). Here, we assume the dense
irradiation-produced crust is made of refractories (organics and dust) with
low porosity $\Psi=0.1$ and a density close to 3000 kg m-3. Under the crust,
comet matter is supposed to be very porous ($\Psi$ = 0.65). All parameters
used in simulations are listed in Table 1. The observations of the
67P/Churyumov–Gerasimenko show that comets tend to be low-strength bodies with
tensile strength as low as a few Pa (Basilevsky et al., 2016). However, the
large-scale structure of long-period comets can significantly differ from that
of short-period comets. In addition, the tensile strength could depend on the
depth of a comet layer. In simulations, we use the maximum estimated value of
the large-scale tensile strength of comet matter 104 Pa (Reach et al., 2010)
following Gronkowski and Wesołowski (2015).

Free radicals can be activated: 1) when the ambient temperature increases
(induced recombination) or 2) when a critical concentration of radicals is
reached (spontaneous recombination). The model considers only induced
recombination. The activation energy of H radicals (Eq. 6) varies from 0.01 eV
to 0.11 eV (Flournov et al., 1962; Kirichek et al., 2017). At temperatures
above 50 K, H radicals are present only in trace amounts compared to hydroxyl
radicals. OH radicals are stable at low temperatures and react violently at
about 100 K with an activation energy of 0.25 eV (Johnson and Quickenden,
1997; Pavlov et al., 2022). The recombination of hydroxyl radicals starts at
the temperature of 100 K, so the rate of recombination at temperatures below
100 K is negligible. Thus, we do not consider the mutual recombination of OH
and H radicals in our model; the parameters of this reaction are poorly known
from the experiments. According to the model, the comet layer must be heated
by at least one kelvin to induce the recombination of H radicals. Parameters
for Eq. 6 are from (Pavlov et al., 2022) and (Flournov et al., 1962). Long-
term irradiation at low temperatures leads to the accumulation of high
radiation doses in surface layers of comets. GCRs (mainly protons) effectively
deposit energy in the upper 10 meters of the comet’s surface. Gronoff et al.
(2020) modelled the irradiation of a comet ($\rho$ = 538 kg m-3, $\mu$ = 4) by
GCR and obtained the absorbed dose profile. A particular layer of a comet is
able to accumulate radicals with a concentration not exceeding the critical
value. From experiments, spontaneous recombination in the proton-irradiated
water ice occurs at the concentration of H radicals equal to about 1$\%$ (to
the number of water ice molecules)(Moore and Hudson, 1992). Using the yield of
H radicals in the proton-irradiated amorphous ice 7$\times$10-3 eV-1 from
Moore and Hudson (1992) and the deposition dose rate in the Oort cloud from
(Gronoff et al., 2020), the time of the critical concentration accumulation
varies from 102 Myr at 1 meter to 103 Myr at 10 meters beneath the comet
surface. The first centimetres of a cometary nucleus reach the critical
concentration of radicals within the 5 Myr period. This effect may favour the
formation of a dense crust.

In the model, we use two profiles of radicals distribution. The first profile
corresponds to the moment when the high concentration of radicals is achieved
directly under the dust layer. The second profile corresponds to the
accumulation of equilibrium saturated concentration of radicals in surface
layers (Fig. 1). According to Shabalin (1998), the saturated concentration of
radicals can be independent of the irradiation dose rate at low temperatures.
As a result, the upper cometary layers can have a similar saturated
concentration of radicals. Here, we assume that only the upper 10 meters of
the comet nucleus accumulate a high concentration of radicals during the
evolution of the Oort cloud based on the dose deposition rates (Gronoff et
al., 2020). We propose that the initial temperature gradient in the upper
10-meter layer is insignificant. The concentrations of radicals are close to
or less than the critical concentration ($\approx 1\%$). The upper 10 cm layer
is a dust layer without radicals.

Since the model considers the induced recombination of radicals, comet surface
layers must experience external heating to provoke recombination reactions. In
extremely distant regions (i.e., the Oort cloud), the impact between comets
and the variation in solar radiation during the orbital motion is
insignificant. However, close supernovae or a passing O/B star can heat
cometary nuclei in the Oort cloud (Stern and Shull, 1988). Stern and Shull
(1988) showed that the passage of very bright stars (classes O and B) in the
vicinity of the Oort cloud during the evolution of the Solar system $\approx$
4.5 Byr resulted in at least one heating event, which caused the temperature
of each cometary nucleus to rise to 16 K. Also, close supernovae cause
periodic heating to 30 K for all comets in the Oort cloud. The layer with
radicals must be heated by at least 1 K for $U_{\mathrm{a,H}}$ = 0.01 eV or
reach the temperature T = 30 K for $U_{\mathrm{a,H}}$ = 0.11 eV in order to
induce the recombination of H radicals.

Table 1: Parameters of the model. Parameter | Value
---|---
Comet radius R, m | $10^{3}$
$\Psi$ | $0.65$
$r_{p}$, m | $10^{-6}$
$h$ | $10^{-3}-0.1$
$\epsilon$ | $0.96$
$A$ | $0.04$
$f_{CO},f_{CO_{2}}$ | $0.16,0.25$
Initial temperature $T_{0}$, K | $10$
Initial concentrations of radicals $N_{0,x}$ | $0-1\%$
$U_{\mathrm{a,H}}$, eV | $0.1-0.11$
$U_{\mathrm{a,OH}}$, eV | $0.25$
Figure 1: Sample distributions of radicals in the comet located in the Oort
cloud. The black solid curve represents the case of the maximum concentration
beneath the surface dust layer (1). The grey solid curve shows the
distribution of radicals when radicals reach the saturated value (2). The dose
deposition rates are from Gronoff et al. (2020).

##  3 Results

###  3.1 Pressure building

Numerical calculations have been carried out using the wide range of possible
values for the Hertz factor $h$, the dust-to-ice mass ratio $\mu$, the
activation energy of H radicals $U_{\mathrm{a,H}}$ (Table 1) and implementing
two different spatial distributions of radicals (Fig. 1).

We first examine the comet nucleus with the first distribution of radicals.
The pressure distribution under the comet surface during the comet activity
caused by the recombination of radicals is shown in Fig. 2. Here, the maximum
concentration of radicals is 1$\%$ (relative to the number of water ice
molecules; the first distribution Fig. 1). Our simulations show that the
efficient pressure growth occurs near the source of gas – the amorphous-to-
crystalline ice transition front. In addition, effective sublimation and
condensation of CO gas take place at the H recombination front with the
concomitant local gas pressure building (Fig. 2). In the model, we assume that
the ice transition front position determines the depth where the initial
concentration of amorphous ice decreased by 10$\%$. Similarly, positions of H
and OH recombination fronts are determined by a local drop in the
concentration of radicals by 10$\%$. According to Fig. 2, the gas pressure
accumulated below the comet crust exceeds the tensile strength of comet
material ($\approx$ 104 Pa), so the comet crust and all material above
cavities with the high gas pressure can be ejected.

Figure 2: Pressure building inside a comet caused by the recombination of
radicals. Red dashed curves depict (from left to right) positions of the
amorphous-to-crystalline transition, the OH and H radicals’ recombination
fronts. Calculations were conducted using the first distribution of radicals
(see Fig. 1) with the maximum concentration of 1$\%$, parameters from Table 1
and $h$ = 0.1, $\mu$ = 1, $U_{\mathrm{a,H}}$ = 0.03 eV. Here and further in
all figures, the time of the outburst is counted with the onset of the
recombination of H radicals.

The propagation of recombination and amorphous-to-crystalline transition
fronts, as well as the location of zones with gas pressure exceeding the
tensile strength, are shown in Fig. 3. The initial pressure rise takes place
at the ice transition front within a few seconds after the onset of H
recombination. Further propagation of volatiles is governed by effective gas
diffusion owing to deeper layers heating by H and OH radicals. According to
Fig. 3, only 0.5 meters of a comet nucleus undergoes an ice transition, while
high-pressure zones can extend up to about 2.5 meters.

Figure 3: Physical evolution of a comet during the outburst. The positions of
amorphous-to-crystalline transition (AC) and OH, H radicals’ recombination
fronts are presented by blue curves. The positions of the deepest layers with
the gas pressure > 104 Pa are shown by the red curve. The purple square
depicts the initial rise in gas pressure coinciding with the ice transition
front. Initial conditions and parameters are as in Fig. 2, except
$U_{\mathrm{a,H}}$ = 0.01 eV.

The total pressure of CO and CO2 gases achieves its maximum value within 10 –
100 seconds (Fig. 4). We found that the maximum gas pressure depends slightly
on the activation energy of H radicals (see Fig. 4). The pressure values
accumulated in subsurface cavities depending on the dust-to-ice mass ratio
$\mu$ and various values of the Hertz factor $h$ are presented in Fig. 5.
Based on a comparison with the tensile strength of comet matter, we conclude
that the surface layers of comets with $\mu=0-3$ and a high concentration of
radicals can experience outbursts. This effect is slightly influenced by the
Hertz factor (Fig. 5).

Figure 4: Pressure building at the amorphous-to-crystalline ice transition
front as a function of time for different activation energies of H radicals.
The initial conditions and parameters of simulations are as in Fig. 2, but
various $U_{\mathrm{a,H}}$. Figure 5: Maximum gas pressure accumulated under
the surface of a comet, depending on the dust-to-ice ratio $\mu$ for three
values of the Hertz factor. The black dashed line represents the tensile
strength of the comet material (Reach et al., 2010). The initial conditions
and parameters of simulations are as in Fig. 2, but for $U_{\mathrm{a,H}}$ =
0.01 eV and various $h$ and $\mu$.

Concentrations of radicals sufficient to cause the significant pressure
building in the subsurface layers are presented in Fig. 6. To achieve gas
pressure exceeding the tensile strength, the upper comet layer (just below the
dust layer) must accumulate a concentration of radicals in the ice component
above 0.7 $\%$.

Figure 6: Maximum gas pressure accumulated under the surface of a comet as a
function of maximum concentration of radicals in ice (the first distribution,
see Fig. 1), for four values of the dust-to-ice mass ratio $\mu$. The black
dashed line represents the tensile strength of the comet material (Reach et
al., 2010). Calculations were carried out using parameters from Table 1 and
$h$ = 0.1, $U_{\mathrm{a,H}}$ = 0.01 eV.

We now consider the second distribution of radicals. We find that the
effective pressure building follows the recombination fronts (Fig. 7). Due to
the low activation energy of H radicals, the recombination front of H radicals
effectively casts through a comet, simultaneously heating cometary layers to
approximately the same temperature (since we use the uniform saturated
concentration of radicals). If the concentration of radicals is enough to
induce ice transition, all layers undergoing recombination of radicals can
accumulate high gas pressure. Calculations show that the total pressure rise
depends slightly on the activation energy of H radicals. According to Fig. 7,
diffusion of volatiles results in the accumulation of high gas pressure in
layers lying deeper than the recombination fronts.

Figure 7: As Fig. 2, but for the second distribution of radicals (see Fig. 1)
with the saturation concentration of 1$\%$, $h$ = 0.1, $\mu$ = 1 and
$U_{\mathrm{a,H}}$ = 0.01 eV. The positions of H and OH recombination fronts
coincide on this plot.

From Fig. 8, comets with the saturation concentration of radicals above 0.6
$\%$ (to the number of water ice molecules) are surface layers that can
experience burst activity. Compared to the first distribution of radicals,
recombination of high saturation concentration of radicals creates higher
pressure in subsurface cavities and causes instabilities in cometary layers
with $\mu=0-4$.

Figure 8: As Fig. 6, but for the second distribution of radicals (see Fig. 1).

###  3.2 Ejected mass

The main goal of the present study is to assess the efficiency of the
recombination reaction as a possible mechanism of cometary outbursts at large
heliocentric distances. So, the valuable outcome of the model is the mass loss
of a comet nucleus at the Oort cloud distances.

Based on the model of Ipatov (2012) and Gronkowski and Wesołowski (2015), when
local gas pressure exceeds the tensile strength of a cavity, instabilities can
develop, resulting in the formation of cracks or even the total removal of
subsurface layers into a comet coma. To eject layers, the gas pressure must
exceed the sum of the weight of layers above the cavity and the tensile
strength. According to our results, cavities with a gas pressure high enough
to reject surface layers can be formed up to $\approx$ 10 meters below the
surface caused by the induced recombination of radicals. The active surface
area of a comet experiencing an outburst can be obtained from observations.
Kelley et al. (2022) found that the ejected mass during the outburst of long-
period comet C/2014 UN271 (Bernardinelli-Bernstein) at a distance of 20 AU
originated from the cylindrical volume with a diameter of about 130 m. This
value is comparable to the diameters of observed pits on the surface of short-
period comets, such as 67P/Churyumov-Gerasimenko, 17P/Holmes (Vincent et al.,
2015; Reach et al., 2010). The thickness of cometary layer experiencing
activity is determined by the maximum depths of subsurface layers with the
high gas pressure: $\approx$ 2 and 10 m for the first and second distributions
of radicals, respectively. With a dust layer density of 3000 kg m-3 and an
average density of comet matter of 500 kg m-3 ($\mu$ = 1), the ejected mass is
equal $(0.7-2.7)\times 10^{9}$ kg. This value lies within the mass range of
observed cometary outbursts (Gronkowski and Wesołowski, 2015).

##  4 Discussion

###  4.1 A possible source of distant comet activity

We have shown that comet nuclei in distant regions of the Solar system can
become active due to the recombination of free radicals accumulated in surface
layers under cosmic ray irradiation. For a wide range of possible parameters
for comet matter composition, the gas pressure in subsurface cavities
overcomes the large-scale tensile strength. The main contribution to the gas
pressure is CO gas ejected from the amorphous ice. High gas pressure creates
instabilities and even causes the total rejection of surface layers. In
addition, gases ejected from the amorphous ice can flow both to the surface of
a comet and diffuse into deeper layers (Fig. 2, 7). Local heating due to the
recombination of radicals manages efficient gas diffusion and prevents the
instant condensation of gases. This effect leads to the enrichment of
underlying layers with volatiles. Recombination of radicals could be one of
possible sources of the observed activity of long-period comets (C/2017 K2
(PANSTARRS), C/2010 U3 (Boattini) and C/2014 UN271 (Bernardineli-Bernstein))
at large heliocentric distances. Our model can be used to simulate the surface
activity of long-period comets, Kuiper belt objects and Trans-Neptunian
objects. An important indicator that recombination reactions do occur on the
surface of small bodies in the outer Solar system should be the presence of
substances such as H2O2 and O2 (Johnson and Quickenden, 1997). The James Webb
Space Telescope can be utilized to detect radiation products on the surface of
small bodies in the outer Solar system.

###  4.2 Model constrains

In our calculations, we assume that radicals can be accumulated only in the
ice fraction of the comet matter. However, radicals can be also produced in
the refractories (dust and organics) during the irradiation. The accumulation
of radicals in organic molecules was discovered under proton, alpha and
neutron irradiation at low temperatures (Carpenter, 1987; Kaiser et al., 1997;
Shabalin et al., 2003). The critical concentration of H atoms in methane
molecules is $6\pm 3\%$ (Kaiser et al., 1997). On the other hand, there are no
experimental results on the accumulation of radicals in dust. Since the mass
fraction of dust in the comet matter can be comparable to or even overcome the
fraction of water ice, radicals in dust can contribute substantially to the
processes on cometary surfaces. It is well known that the decay of
radionuclides continuously irradiates the inner layers of comets, generating
radicals in much deeper layers than considered in our model. We expect changes
in chemical and physical composition due to the recombination of accumulated
radicals in the entire volume of a comet. It is worth mentioning that the
accumulation and recombination of radicals, as well as values of critical and
saturation concentrations depend on the ice matrix (Johnson and Quickenden,
1997). There are no experimental results on the accumulation of radicals in
cometary analogues. So, in our calculations, we used experimental results for
pure ice. Nevertheless, experiments are desirable to simulate the irradiation
of cometary matter.

From observations, comets are rather heterogeneous bodies with chemical
compositions and physical conditions changing from place to place. As a
result, comets can experience activity in different areas where ice with
radicals meets conditions to start chain recombination reactions. We saw in
simulations that even a slight rise in ambient temperature can start
propagation of the recombination front, so it is crucial to properly model the
heating of the surface, taking into account all surface irregularities.

In our model, we used absorbed doses of GCRs from (Gronoff et al., 2020). It
should be pointed out that Gronoff et al. (2020) used in calculations LIS
based on the Badhwar–O’Neill model. The flux of low-energy particles used in
Gronoff et al. (2020) are greatly overestimated compared to Boschini et al.
(2020), who used the GalProp–HelMod model and several instruments, including
Voyager 1, to constrain LIS. We estimated that the new LIS would decrease the
dose rate only in the upper 10 cm of a comet. In addition, GCR spectra could
change during the Solar system evolution, owing to supernovae and the early
Sun’s activity (Gronoff et al., 2020).

###  4.3 Astronomical implication

The studied mechanism of comet activity driven by the recombination of
radicals substantially increases the range of heliocentric distances where
comets can experience activity. We now point out two highly significant
applications of comet activity at large distances from the Sun.

Debris disks, detected around main-sequence stars, contain a lot of small dust
and ice particles. The influences of Poynting-Robertson and solar wind drugs
cause a constant loss of dust particles. On the other hand, the outgassing of
cometary nuclei and collision between planetesimals generate enough dust
particles to sustain detected debris disks (Chen et al., 2008). The Sun has a
relatively dust-poor disk compared to extrasolar debris disks. The sources of
interplanetary dust particles in the outer solar system region are long-period
comets and Kuiper belt objects (Poppe et al., 2019). Ice sputtering and
Poynting-Robertson drugs become inefficient with the increasing distance from
the Sun. As a result, distant cometary outbursts in the regions beyond the
Kuiper belt can create a large amount of stable ice and dust particles. It is
worth pointing out that the model of the debris disk around the Sun assumes a
sharp decline in the number of Kuiper belt objects at distances > 50 AU (Petit
et al., 2011). This boundary of the Kuiper Belt is probably due to observation
limitations (Fraser et al., 2023). In this case, dust particles ejected from
small bodies at heliocentric distances beyond 50 AU could be the indicator of
comet-like objects. Based on our results, cometary outbursts in distant
regions of the Solar system, where mutual collisions between small bodies are
insignificant, can produce a large number of dust and ice particles. In the
case of the Oort cloud, generated dust particles could periodically be
forwarded to the inner parts of the Solar system during the evolution of the
Oort cloud. Looking for new sources of dust particles is extremely important
in light of the detection of anomalous diffuse light of unknown origin in the
cosmic extragalactic background optic light (Lauer et al., 2022; Carleton et
al., 2022).

An important stage of cometary outbursts is a jet of gas and dust erupting
through newly formed cracks in the comet crust. When leaving the comet
surface, this jet creates the non-gravitational acceleration (Marsden et al.,
1973). Since cometary nuclei in the Oort cloud have relatively small orbital
velocities, the non-gravitational acceleration can perturb the orbits of
comets in addition to well-known gravitational mechanisms. In the article
(Belousov and Pavlov, 2023), we examined the influence of cometary outbursts
on the stability of comet orbits in the Oort cloud. We showed that cometary
nuclei with a radius of less than 1 km can be expelled from the Oort cloud due
to a single outburst. This effect could lead to the observed decrease in the
number of the small-radius long-period comets (Boe et al., 2019).

##  5 Conclusions

We have developed the model of comet activity at large heliocentric distances
caused by the recombination of radicals accumulated under cosmic ray
irradiation at low temperatures. The main conclusions are the following:

  * •

Our model demonstrates that the recombination of radicals can drive cometary
outbursts at large heliocentric distances and predicts comet activity even in
the Oort cloud with a total ejected mass similar to the observed outbursts.

  * •

Recombination reactions can lead to significant chemical changes in the comet
matter.

  * •

The ejection of large amounts of dust and ice particles beyond the Kuiper Belt
may contribute to the recently discovered anomalous diffuse light of unknown
origin.

  * •

Cometary outbursts in the Oort cloud may lead to the observed depletion in the
number of small-radius long-period comets.

## Acknowledgements

This work has been supported by the grant of the Russian Science Foundation,
RSF 24-22-00257. The authors want to thank Gennady Vasilyev (Ioffe Institute)
for helpful comments on the local interstellar spectrum and the absorbed dose
of cosmic rays.

## Declaration of Competing Interest

The authors declare no conflicts of interest associated with this manuscript.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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