# A Robust Light-Curve Diagnostic for Electron-Capture Supernovae and Low-Mass
Fe-Core-Collapse Supernovae

Masato Sato e-mail<EMAIL_ADDRESS>Astronomical Science Program, Graduate Institute for Advanced Studies, SOKENDAI 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  National Astronomical Observatory of Japan, National Institutes of Natural Sciences 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Nozomu Tominaga National Astronomical Observatory of Japan, National Institutes of Natural Sciences 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Astronomical Science Program, Graduate Institute for Advanced Studies, SOKENDAI 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Department of Physics, Faculty of Science and Engineering, Konan University, 8-9-1 Okamoto, Kobe, Hyogo 658-8501, Japan  Sergei I. Blinnikov NRC Kurchatov Institute 123182 Moscow, Russia  Dukhov Automatics Research Institute (VNIIA) 127055 Moscow, Russia  Marat Sh. Potashov NRC Kurchatov Institute 123182 Moscow, Russia  Keldysh Institute of Applied Mathematics RAS, 4 Miusskaya Square, 125047 Moscow, Russia  Takashi J. Moriya National Astronomical Observatory of Japan, National Institutes of Natural Sciences 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Astronomical Science Program, Graduate Institute for Advanced Studies, SOKENDAI 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia  Daichi Hiramatsu Center for Astrophysics | Harvard & Smithsonian 60 Garden Street, Cambridge, MA 02138-1516, USA  The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, USA

###### Abstract

Core-collapse supernovae (CCSNe) are the terminal explosions of massive stars.
While most massive stars explode as iron-core-collapse supernovae (FeCCSNe),
slightly less massive stars explode as electron-capture supernovae (ECSNe),
shaping the low-mass end of CCSNe. ECSNe was proposed $\sim 40$ years ago and
first-principles simulations also predict their successful explosions.
Observational identification and investigation of ECSNe are important for the
completion of stellar evolution theory. To date, only one promising candidate
has been proposed, SN 2018zd, other than the historical progenitor of the Crab
Nebula, SN 1054. We present representative synthetic light curves of low-mass
FeCCSNe and ECSNe exploding with energies in circumstellar media (CSM)
estimated with theoretically or observationally plausible methods. The
plateaus of the ECSNe are shorter, brighter, and bluer than those of the
FeCCSNe. To investigate the robustness of their intrinsic differences, we
adopted various explosion energies and CSM. Although they may have similar
bolometric light-curve plateaus, ECSNe are bluer than FeCCSNe in the absence
of strong CSM interaction, illustrating that multicolor observations are
essential to identify ECSNe. This provides a robust indicator of ECSNe because
the bluer plateaus stem from the low-density envelopes of their super-
asymptotic-giant-branch progenitors. Furthermore, we propose a distance-
independent method to identify ECSNe: $(g-r)_{t_{\rm PT}/2}<0.008\times t_{\rm
PT}-0.4$, i.e., blue $g-r$ at the middle of the plateau $(g-r)_{t_{\rm
PT}/2}$, where $t_{\rm PT}$ is the transition epoch from plateau to tail.
Using this method, we identified SN 2018zd as an ECSN, which we believe to be
the first ECSN identified with modern observing techniques.

Type II supernovae (1731) — Stellar evolution(1599) — Photometry(1234)

##  1 Introduction

Core-collapse supernovae (CCSNe) are the terminal explosions of massive stars,
triggered by the collapse of their cores. Collapsing cores are divided into
two mass-dependent types: Iron (Fe) cores, for which progenitors are massive
enough to ignite static Si burning, collapse because of photo-disintegration
and the stars explode as Fe-core-collapse supernovae (FeCCSNe, Colgate & White
1966; Bethe 1985; Woosley et al. 2002). Oxygen-neon-magnesium (ONeMg) cores,
for which progenitors are not massive enough to ignite static O burning, are
supported by electron degeneracy pressure. Degenerate ONeMg cores collapse
owing to electron capture by Mg and Ne, and the stars explode as electron-
capture supernovae (ECSNe, Miyaji et al. 1980; Nomoto et al. 1982; Nomoto
1984, 1987). The progenitors of ECSNe are likely to be less massive than those
of FeCCSNe, shaping the low-mass end of the CCSN-progenitor distribution.
However, the stellar mass range that forms ECSNe is not clear and may be
narrow or even non-existent for solar metallicity (Poelarends et al., 2008;
Langer, 2011; Doherty et al., 2015; Limongi et al., 2023).

If these progenitors retain a massive hydrogen-rich (H-rich) envelope, red
supergiants (RSGs) result in FeCCSNe, whereas super-asymptotic-giant-branch
(SAGB) stars result in ECSNe. Both types of explosion are observed as H-rich
type II supernovae (SNe II). After an initial shock-breakout flash, a plateau
phase is powered by the shock-heated H-rich envelope. The duration and
brightness of a plateau infer the explosion energy, pre-supernova radius, and
envelope mass of the star (Litvinova & Nadezhin, 1985; Popov, 1993; Eastman et
al., 1994; Kasen & Woosley, 2009). A tail phase following the plateau phase is
powered by the radioactive decay of 56Co, the daughter isotope of 56Ni
synthesized by explosive nucleosynthesis. Thus, the brightness of the tail is
proportional to the mass of the ejected 56Ni [$M{\rm(^{56}Ni)}$].

Observations of the light curves and spectra of SNe II reveal diversity in
properties such as the explosion energy ($\sim 10^{50}-10^{51}$ erg) and
$M{\rm(^{56}Ni)}$ ($\sim 10^{-3}-10^{-1}~{}M_{\odot}$) (Martínez et al.,
2022a, b, c; Anderson, 2019). Furthermore, recent observations of early light
curves and spectra indicate the presence of dense circumstellar media (CSM)
around progenitors, with estimated mass-loss rates ranging from
$\dot{M}=10^{-4}$ to $10^{-2}$ $M_{\odot}~{}{\rm yr}^{-1}$ (Yaron et al.,
2017; Morozova et al., 2017; Förster et al., 2018; Bruch et al., 2021, 2023;
Subrayan et al., 2023).

The explosion mechanisms of FeCCSNe are largely investigated with numerical
simulations. It has been shown that the FeCCSN of 15$M_{\odot}$ does not
explode when assuming spherical symmetry (Sumiyoshi et al., 2005). Recent
three-dimensional (3D) simulations demonstrated that multi-dimensional effects
like convection are key to successful explosions (Takiwaki et al., 2012),
although the explosion energy and amount of 56Ni are slightly lower than those
of observed SNe II (Burrows et al., 2019; Sieverding et al., 2020).

3D simulations of low-mass FeCCSNe of stars with zero-age main-sequence masses
$M_{\rm ZAMS}$ of $9-11M_{\odot}$ [s9.0–s11.0 models in Sukhbold et al.
(2016)] were performed by Burrows et al. (2019); Burrows & Vartanyan (2021),
who found successful explosions with energies of $\sim 10^{50}$ erg. Based on
the 3D simulations Wang & Burrows (2023) showed that such supernovae have 56Ni
yields of $M{\rm(^{56}Ni)}\sim 10^{-3}-10^{-2}M_{\odot}$. The progenitor radii
range $2.9\times 10^{13}-4.4\times 10^{13}$ cm. Their typical bolometric light
curves have also been calculated by Sukhbold et al. (2016), showing plateaus
of $\sim-15$ to $-18$ mag, tails of $\sim-12.5$ to $-15.5$ mag at the onset,
and a drop of $\sim 2.5$ mag from plateau to tail.

On the other hand, ECSN explosions have been successfully simulated with the
assumption of spherical symmetry owing to the low-density H/He envelopes of
their progenitors (Kitaura et al., 2006). First-principles simulations
revealed low explosion energies ($1.5\times 10^{50}$ erg) and small amounts of
56Ni ($0.002-0.004$ $M_{\odot}$) (Kitaura et al., 2006; Janka et al., 2008;
Wanajo et al., 2009). The elemental abundance ratios of ECSNe are consistent
with the Crab Nebula, a remnant of SN 1054 (Davidson et al., 1982; Nomoto et
al., 1982; Wanajo et al., 2009). Moreover, Tominaga et al. (2013); Moriya et
al. (2014) calculated ECSN light curves and found that they exhibit a plateau
with a bolometric brightness of $-15$ to $-16$ mag (as bright as that of SNe
II of RSG progenitors), a faint tail of $\sim-11$ mag at the onset, and a
large drop of $\sim 4$ mag from plateau to tail. These calculations were
performed using SAGB progenitor models with radii ranging $6.5\times
10^{13}-7.3\times 10^{13}$ cm.

Despite successful explosions in first-principles simulations, ECSN
observations remain elusive. Recently, Hiramatsu et al. (2021) proposed that
SN 2018zd is an ECSN based on observational features and indications such as
the light curve, explosion energy, dense CSM, nucleosynthesis, and pre-
supernova photometry. However, Callis et al. (2021) controversially proposed
that it is a low-mass FeCCSN because of a large $M{\rm(^{56}Ni)}$ using a
larger distance estimation than Hiramatsu et al. (2021). Therefore, a
distance-independent diagnostic method is required to firmly identify ECSNe.

The theoretical light curves of low-mass FeCCSNe and ECSNe have been
calculated by Kozyreva et al. (2021). They showed that ECSNe have a bluer
color until the middle of the plateau in the absence of factors affecting
plateau color, including explosion-energy variation and CSM. Meanwhile,
observations of SNe II (Yaron et al., 2017; Morozova et al., 2017; Förster et
al., 2018; Bruch et al., 2021, 2023) indicate the existence of dense CSM
corresponding mass-loss rates of $\dot{M}=10^{-4}-10^{-2}$ $M_{\odot}~{}{\rm
yr}^{-1}$ in the vicinity of the progenitors, and observations of RSG stars in
the Large Magellanic Cloud (Goldman et al., 2016) and Small Magellanic Cloud
(Yang et al., 2023) indicate mass-loss rates of $\dot{M}=10^{-6}$ to $10^{-5}$
$M_{\odot}~{}{\rm yr}^{-1}$ distributed up to $10^{-3}$ $M_{\odot}~{}{\rm
yr}^{-1}$. Also, the mass-loss rate of SAGB stars is expected to be $10^{-4}$
$M_{\odot}~{}{\rm yr}^{-1}$ from theoretical calculation (Poelarends et al.,
2008). Additionally, low-mass SNe II ($M_{\rm ZAMS}\leq 12.0~{}M_{\odot}$)
have an observed explosion-energy diversity of $E_{\rm exp}\sim
10^{50}-10^{51}$ erg (Martínez et al., 2022b), while the low explosion energy
theoretically estimated for low-mass FeCCSNe and ECSNe ($\sim 10^{50}$ erg,
Burrows et al. 2019; Burrows & Vartanyan 2021; Kitaura et al. 2006; Janka et
al. 2008).

Therefore, we calculated and investigated the multicolor light curves of low-
mass FeCCSNe and ECSNe as follows. First, we adopted the physical quantities
estimated in previous studies with theoretically or observationally plausible
methods such as the explosion energy from explosion simulation and mass-loss
rates from progenitor observation or calculation to derive representative
light curves. Then, we adopt wide ranges of physical quantities to robustly
discriminate between ECSNe and low-mass FeCCSNe.

This paper is structured as follows. We introduce the models and methods in
Section 2. We show the resulting light curves in Section 3: the representative
light curves in Section 3.1 and light curves with a wide range of physical
quantities in Section 3.2. We discuss their robust characteristics and propose
an ECSN diagnostic method in Section 4. Finally, we present our conclusions in
Section 5.

##  2 Model & Method

We adopted SAGB and RSG progenitor models to calculate light curves for ECSNe
and low-mass FeCCSNe, respectively. On top of the both progenitor models, we
attached a CSM structure following Moriya et al. (2018), with density

$\rho_{CSM}(r)=\frac{\dot{M}}{(4\pi v_{\rm wind})r^{-2}},$ (1)

up to $r=r_{\rm out}$, where $r$ is the distance from the center of the star,
$v_{\rm wind}$ is the wind velocity, and $r_{\rm out}$ is the CSM radius. For
the wind velocity, acceleration was considered using a simple $\beta$ velocity
law:

$v_{\rm wind}(r)=v_{\rm 0}+(v_{\rm\infty}-v_{\rm 0})(1-\frac{R}{r})^{\beta},$
(2)

where $v_{\rm 0}$ is the initial velocity of the wind, $v_{\rm\infty}$ is the
terminal velocity of the wind, and $R$ is the progenitor radius.

We adopted the SAGB progenitor models from Tominaga et al. (2013). Since SAGB
stars experience uncertain mass loss and third dredge-up from thermal pulses,
we adopted various envelope masses $M_{\rm env}$ and H abundances in the
envelope $X({\rm H})_{\rm env}$. Nucleosynthesis was adopted from Wanajo et
al. (2009). We adopted an inner mass cut at $1.364$ or $1.370~{}M_{\odot}$ to
reduce the calculation cost, although it is estimated as $1.362~{}M_{\odot}$
in the first principles simulation (Kitaura et al., 2006). This small
difference does not substantially affect the light curve properties. An
explosion energy of $E_{\rm exp}=1.5\times 10^{50}$ erg from the first-
principles simulation (Kitaura et al., 2006) and mass-loss rate of
$\dot{M}=10^{-4}$ $M_{\odot}~{}{\rm yr}^{-1}$ from the calculation of
Poelarends et al. (2008) are adopted for the representative light curves, as
listed in Table 1. We also calculated models with wide parameter ranges, as
listed in Table 2.

Table 1: Parameter sets for the representative ECSN light curves $M_{\rm env}$ | $X({\rm H})_{\rm env}$ | $R$ | Mass cut | $E_{\rm exp}$ | $M{\rm(^{56}Ni)}$ | $\dot{M}$ | $r_{\rm CSM}$ | $v_{\rm 0}$ | $v_{\rm\infty}$ | $\beta$
---|---|---|---|---|---|---|---|---|---|---
[$M_{\odot}$] |  | [$10^{13}$ cm] | [$M_{\odot}$] | [$10^{50}$ ergs] | [$M_{\odot}$] | [$M_{\odot}~{}{\rm yr}^{-1}$] | [cm] | [km/s] | [km/s] |
3.0 | 0.2 | 7.0 | 1.370 | 1.5 | 0.002 | $10^{-4}$ | $10^{15}$ | 1 | 10 | 3
3.0 | 0.7 | 7.1 | 1.370 | 1.5 | 0.002 | $10^{-4}$ | $10^{15}$ | 1 | 10 | 3
4.7 | 0.7 | 7.2 | 1.370 | 1.5 | 0.002 | $10^{-4}$ | $10^{15}$ | 1 | 10 | 3
Table 2: Parameter sets for the ECSN light curves with wide physical quantity ranges $M_{\rm env}$ | $X({\rm H})_{\rm env}$ | $R$ | Mass cut | $E_{\rm exp}$ | $M{\rm(^{56}Ni)}$ | $\dot{M}$ | $r_{\rm CSM}$ | $v_{\rm 0}$ | $v_{\rm\infty}$ | $\beta$
---|---|---|---|---|---|---|---|---|---|---
[$M_{\odot}$] |  | [$10^{13}$ cm] | [$M_{\odot}$] | [$10^{50}$ ergs] | [$M_{\odot}$] | [$M_{\odot}~{}{\rm yr}^{-1}$] | [cm] | [km/s] | [km/s] |
2.0 | 0.7 | 6.5 | 1.364/1.370 | $0.6-14.2$ | $0.002$ | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
3.0 | 0.2 | 7.0 | 1.364/1.370 | $0.6-12.6$ | $0.002-0.003$ | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
3.0 | 0.5 | 7.0 | 1.364/1.370 | $0.6-10.4$ | $0.002-0.003$ | $10^{-4}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
3.0 | 0.7 | 7.1 | 1.364/1.370 | $0.6-10.5$ | $0.002-0.003$ | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
4.7 | 0.5 | 7.3 | 1.364/1.370 | $0.6-13.7$ | $0.002$ | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
4.7 | 0.7 | 7.2 | 1.364/1.370 | $0.6-13.7$ | $0.002-0.003$ | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$

For the representative light curves, we adopted the RSG progenitor models of
s9.0, s10.0, and s11.0 ($M_{\rm ZAMS}=9.0,~{}10.0,$ and $11.0~{}M_{\odot}$
respectively) in Sukhbold et al. (2016), explosion energies of $E_{\rm
exp}=0.9$, $1.5$, and $1.5\times 10^{50}$ erg, for each progenitor
respectively, from Burrows & Vartanyan (2021), and a mass-loss rate of
$\dot{M}=10^{-6}$ $M_{\odot}~{}{\rm yr}^{-1}$ (Goldman et al., 2016; Yang et
al., 2023). The adopted parameters are listed in Table 3. We also calculated
models with wide parameter ranges, as listed in Table 4. In both calculations,
the hydrogen abundance in the progenitor envelope is approximately $0.6$,
comparable to the $X({\rm H})_{\rm env}=0.7$ models for SAGB stars. We adopted
mass cuts for $M_{\rm ZAMS}=9.0$, $10.0$, and $11.0~{}M_{\odot}$ progenitors
and $M{\rm(^{56}Ni)}$ for $M_{\rm ZAMS}=9.0$, $9.25$, $9.5$, and
$11.0~{}M_{\odot}$ progenitors from the results of 3D simulation (Burrows &
Vartanyan, 2021; Wang & Burrows, 2023), otherwise from the results of
spherical simulation (Sukhbold et al., 2016). We placed 56Ni at the bottom of
the ejecta.

We focus on the plateau in this study, the duration of which is extended by
heating from radioactive decay (Kasen & Woosley, 2009) and is dependent on the
56Ni yield, that is, the mass cut and explosion energy. However, this
extension is negligible for ECSNe and low-mass FeCCSNe ($\lesssim 4\%$
according to equation (10) in Kasen & Woosley 2009) as 56Ni yields are small
($\sim 10^{-3}-10^{-2}~{}M_{\odot}$, Kitaura et al. 2006; Janka et al. 2008;
Wanajo et al. 2009; Wang & Burrows 2023). We did not consider 56Ni mixing
because it has little impact on plateau duration and luminosity (Nakar et al.,
2016; Moriya et al., 2015; Kozyreva et al., 2019).

Examples of progenitor density profiles as a function of radius are shown in
Figure 1. The progenitor radii are listed in Tables 1 to 4. SAGB stars have an
extended and lower-density envelope compared to those of RSGs.

Table 3: Parameter sets for the representative light curves of low-mass FeCCSNe Progenitor model | $R$ | Mass cut | $E_{\rm exp}$ | $M{\rm(^{56}Ni)}$ | $\dot{M}$ | $r_{\rm CSM}$ | $v_{\rm 0}$ | $v_{\rm\infty}$ | $\beta$
---|---|---|---|---|---|---|---|---|---
| [$10^{13}$ cm] | [$M_{\odot}$] | [1050 erg] | [$M_{\odot}$] | [$M_{\odot}~{}{\rm yr}^{-1}$] | [cm] | [km/s] | [km/s] |
s9.0 | 2.9 | 1.350 | 0.9 | 0.007 | $10^{-6}$ | $10^{15}$ | 1 | 10 | 3
s10.0 | 3.6 | 1.490 | 1.5 | 0.03 | $10^{-6}$ | $10^{15}$ | 1 | 10 | 3
s11.0 | 4.0 | 1.510 | 1.5 | 0.03 | $10^{-6}$ | $10^{15}$ | 1 | 10 | 3
Table 4: Parameter sets for the light curves of low-mass FeCCSNe with wide physical quantity ranges Progenitor model | $R$ | Mass cut | $E_{\rm exp}$ | $M{\rm(^{56}Ni)}$ | $\dot{M}$ | $r_{\rm CSM}$ | $v_{\rm 0}$ | $v_{\rm\infty}$ | $\beta$
---|---|---|---|---|---|---|---|---|---
| [$10^{13}$ cm] | [$M_{\odot}$] | [1050 erg] | [$M_{\odot}$] | [$M_{\odot}~{}{\rm yr}^{-1}$] | [cm] | [km/s] | [km/s] |
s9.0 | 2.9 | 1.350 | $0.5-13.7$ | 0.007 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s9.25 | 2.8 | 1.390 | $0.6-10.0$ | 0.01 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s9.5 | 2.9 | 1.410 | $0.9-10.0$ | 0.02 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s9.75 | 3.1 | 1.450 | $0.7-9.8$ | 0.01 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s10.0 | 3.6 | 1.490 | $1.2-13.1$ | 0.03 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s10.25 | 3.8 | 1.470 | $1.9-13.1$ | 0.03 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s10.5 | 3.8 | 1.480 | $2.0-13.3$ | 0.02 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s10.75 | 3.9 | 1.470 | $1.7-12.8$ | 0.03 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s11.0 | 4.0 | 1.510 | $1.4-13.3$ | 0.03 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s11.25 | 4.0 | 1.530 | $2.0-13.3$ | 0.02 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s11.5 | 4.0 | 1.500 | $1.7-12.9$ | 0.02 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s11.75 | 4.3 | 1.590 | $2.0-13.2$ | 0.02 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
s12.0 | 4.4 | 1.530 | $1.5-12.6$ | 0.03 | $10^{-6}-10^{-2}$ | $10^{14}-10^{16}$ | $1-5$ | 10 | $1-5$
Figure 1: Density profiles of select SAGB and RSG models.

We adopted the one-dimensional multi-group radiation hydrodynamics code STELLA
to calculate light curves because it solves radiation hydrodynamics with non-
equilibrium prescription between gas and radiation (Blinnikov & Bartunov,
1993; Blinnikov et al., 1998, 2000), which is required in the low-density
regions, i.e., the SAGB envelope and CSM. STELLA solves time-dependent
equations about the angular moments of averaged intensity over up to 100
fixed-frequency bins, providing an spectral energy distribution (SED) at each
timestep. We adopted the standard 100 frequency bins ranging over $1-50,000$
Å. Bolometric light curves are produced, integrating the SED at each time
step. Light curves with any filter could be calculated by convolving the SED
with the corresponding response function. In this paper, we provide light
curves for the $u$-, $g$-, and $r$-bands of Sloan Digital Sky Survey (Fukugita
et al., 1996), and $B$\- and $V$-bands of Johnson-Cousins (Bessell, 2005).

##  3 Results

First, we present the representative light curves of ECSNe and low-mass
FeCCSNe adopting the typical CSM profiles and explosion energies, in Section
3.1. We then present the light curves resulting from wide ranges of physical
parameters, varying CSM profile, and explosion energy in Section 3.2.

###  3.1 Representative Light Curves

Figure 2 shows the light curves of ECSNe and low-mass FeCCSNe from
representative progenitors with typical explosion energies and CSM structure.
The left-hand panels show the bolometric, $u$-band, $g$-band, and $r$-band
light curves and the $u-g$ and $g-r$ evolution of low-mass FeCCSNe, and the
right-hand panels show those of ECSNe.

Figure 2: Bolometric, $u$-band, $g$-band, and $r$-band light curves and $u-g$
and $g-r$ color evolution of low-mass FeCCSNe and ECSNe of representative
progenitors, explosion energy, and CSM.

The representative FeCCSNe of $M_{\rm ZAMS}=10.0$ and $11.0~{}M_{\odot}$ show
long ($\sim 150-160$ days) and faint ($\sim-15$ mag in bolometric) plateaus,
and the lower-mass FeCCSN of $M_{\rm ZAMS}=9.0~{}M_{\odot}$ shows a shorter
but still long ($\sim 130$ days) and fainter ($\sim-14$ to $-15$ mag in
bolometric) plateau.

The ECSNe have shorter and brighter plateaus than the low-mass FeCCSNe. The
duration and brightness of plateaus range $70-110$ days and $-16$ to $-17$ mag
in bolometric, respectively. The plateaus of ECSNe are longer and slightly
fainter if the $M_{\rm env}$ is larger or $X({\rm H})_{\rm env}$ is higher.
The short and bright plateaus of ECSNe stem from the low-density and extended
envelopes of SAGB stars according to the scaling derived in Popov (1993);
Eastman et al. (1994).

Furthermore, the ECSNe show bluer plateaus than the FeCCSNe. FeCCSNe show a
rapid color evolution to red ($\gtrsim 0.06$ mag/day in $u-g$) from just after
the shock breakout until approximately $30$ days after explosion. In contrast,
ECSNe show a slow color evolution to red ($\lesssim 0.04$ mag/day in $u-g$)
for the $M_{\rm env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ model and
the $M_{\rm env}=4.7~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ model, and
almost a constant color for the $M_{\rm env}=3.0~{}M_{\odot}$ and $X({\rm
H})_{\rm env}=0.2$ model from just after the shockbreakout until approximately
$30$ days after explosion. This is consistent with the results of Kozyreva et
al. (2021).

###  3.2 Light Curves with Wide Physical Quantity Ranges

In this section, we provide and compare the light curves of ECSNe and low-mass
FeCCSNe with wide ranges of physical quantities for three cases, depending on
the presence and duration of the CSM interaction: models without strong CSM
interaction in Section 3.2.1, models with CSM interaction shorter than the
plateau duration in Section 3.2.2, and models with CSM interaction longer than
the plateau duration in Section 3.2.3.

####  3.2.1 Models Without Strong CSM Interaction

Figure 3 shows bolometric, $u$-band, $g$-band, and $r$-band light curves, and
the evolution of $u-g$ and $g-r$ of an ECSN and an FeCCSN showing no strong
CSM interaction. Here, we chose an ECSN from an SAGB star of $M_{\rm
env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm
exp}=1.0\times 10^{50}$ erg, and an FeCCSN from an RSG of $M_{\rm
ZAMS}=9.0~{}M_{\odot}$ exploding with $E_{\rm exp}=3.6\times 10^{50}$ erg. In
both models, no CSM exists around the progenitors.

They exhibit similar bolometric light curves. This implies that it is
difficult to distinguish ECSNe from low-mass FeCCSNe solely with bolometric
light curves. This could explain why ECSNe have not been identified in
previous transient surveys. The relations of the explosion energy $E_{\rm
exp}$ and envelope mass $M_{\rm env}$ between ECSN and FeCCSN giving similar
bolometric light curves are discussed in Appendix A.

Figure 3:  Bolometric, $u$-band, $g$-band, and $r$-band light curves and the
$u-g$ and $g-r$ evolution of an ECSN and FeCCSN showing no strong CSM
interaction. The ECSN is from an SAGB star of $M_{\rm env}=3.0~{}M_{\odot}$
and $X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm exp}=1.0\times 10^{50}$
erg, and the FeCCSN is from an RSG of $M_{\rm ZAMS}=9.0~{}M_{\odot}$ exploding
with $E_{\rm exp}=3.6\times 10^{50}$ erg. In both models, no CSM exists around
the progenitors.

On the other hand, the ECSN and FeCCSN show different multicolor light curves.
The ECSN is brighter than the FeCCSN in the blue bands ($u$\- and $g$-band)
with similar light curves in the $r$-band. The difference is more prominent in
the $u$-band than in the $g$-band. As a result, the ECSN is bluer in the $u-g$
and $g-r$ than the FeCCSN, similar to the representative light curves (Section
3.1).

####  3.2.2 Models with Short and Strong CSM Interaction

Figure 4 shows bolometric, $u$-band, $g$-band, and $r$-band light curves, and
the $u-g$ and $g-r$ evolution of an ECSN and FeCCSN with short and strong CSM
interaction. We adopted the same progenitors and explosion energies as the
models in Section 3.2.1 but attached dense and marginally extended CSM. We
adopted the ECSN from an SAGB star of $M_{\rm env}=3.0~{}M_{\odot}$ and
$X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm exp}=1.0\times 10^{50}$ erg
with a CSM of $\dot{M}=10^{-2}$ $M_{\odot}~{}{\rm yr}^{-1}$, $r_{\rm
CSM}=10^{15}$ cm, $v_{\rm 0}=5$ km/s, $v_{\rm\infty}=10$ km/s, and $\beta=3$,
and the FeCCSN from an RSG of $M_{\rm ZAMS}=9.0~{}M_{\odot}$ exploding with
$E_{\rm exp}=3.6\times 10^{50}$ erg with a CSM of the same parameters as the
ECSN.

The ECSN shows a fainter bolometric peak than the FeCCSN, where the
interaction between the SN ejecta and CSM enhances the luminosity. Their light
curves are similar in the later epochs after the CSM interaction ends
(approximately 60 days for the ECSN and 40 days for the FeCCSN after the
explosion). The similar late plateau results from the lower explosion energy
of the ECSN compared to the FeCCSN, as in Section 3.2.1 (Appendix A). The
lower explosion energy results in a lower SN ejecta velocity, weaker CSM
interaction, and thus a fainter peak. The ECSN exhibits a bluer color than the
FeCCSN after the CSM interaction until the plateau ends.

Figure 4:  Bolometric, $u$-, $g$-, and $r$-band light curves and the $u-g$ and
$g-r$ evolution of an ECSN and FeCCSN showing strong CSM interaction that ends
before the plateau does. The ECSN is from an SAGB star of $M_{\rm
env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm
exp}=1.0\times 10^{50}$ erg with a CSM of $\dot{M}=10^{-2}~{}M_{\odot}~{}{\rm
yr}^{-1}$, $r_{\rm CSM}=10^{15}$ cm, $v_{\rm 0}=5$ km/s, $v_{\rm\infty}=10$
km/s, and $\beta=3$. The FeCCSN is from an RSG of $M_{\rm
ZAMS}=9.0~{}M_{\odot}$ exploding with $E_{\rm exp}=3.6\times 10^{50}$ erg with
a CSM of the same parameters as the ECSN.

####  3.2.3 Models with Long and Strong CSM Interaction

Figure 5 shows bolometric, $u$-band, $g$-band, and $r$-band light curves, and
the $u-g$ and $g-r$ evolution of an ECSN and an FeCCSN showing strong and
long-lasting CSM interaction. We chose an ECSN from an SAGB star of $M_{\rm
env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm
exp}=1.0\times 10^{50}$ erg and an FeCCSN from an RSG of $M_{\rm
ZAMS}=9.0~{}M_{\odot}$ exploding with $E_{\rm exp}=1.4\times 10^{50}$ erg. We
adopted a CSM with the same parameters in both models: $\dot{M}=10^{-2}$
$M_{\odot}~{}{\rm yr}^{-1}$, $r_{\rm CSM}=3\times 10^{15}$ cm, $v_{\rm 0}=5$
km/s, $v_{\rm\infty}=10$ km/s, and $\beta=3$.

The ECSN and FeCCSN have similar bolometric and multicolor light curves. The
reason for this is discussed in Appendix B. In contrast to the models in
Section 3.2.1 and 3.2.2, no color difference between the ECSN and FeCCSN are
found in these models.

Figure 5:  Bolometric, $u$-band, $g$-band, and $r$-band light curves and the
$u-g$ and $g-r$ evolution of an ECSN and FeCCSN showing strong CSM interaction
lasting after the plateau ends. The ECSN is from an SAGB star of $M_{\rm
env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm
exp}=1.0\times 10^{50}$ erg with a CSM of $\dot{M}=10^{-2}~{}M_{\odot}~{}{\rm
yr}^{-1}$, $r_{\rm CSM}=3\times 10^{15}$ cm, $v_{\rm 0}=5$ km/s,
$v_{\rm\infty}=10$ km/s, and $\beta=3$, and the FeCCSN is from an RSG of
$M_{\rm ZAMS}=9.0~{}M_{\odot}$ exploding with $E_{\rm exp}=1.4\times 10^{50}$
erg with a CSM of the same parameters as the ECSN.

##  4 Discussion

###  4.1 The Blue Plateau of ECSNe

In this section, we discuss the origin of the bluer color of ECSNe than
FeCCSNe found in Section 3.2.1 and 3.2.2. Here we chose the ECSN from an SAGB
star of $M_{\rm env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ exploding
with $E_{\rm exp}=1.0\times 10^{50}$ erg, and the FeCCSN from an RSG of
$M_{\rm ZAMS}=9.0~{}M_{\odot}$ exploding with $E_{\rm exp}=3.6\times 10^{50}$
erg shown in Figure 3. In both models, no CSM exists around the progenitors so
that we could investigate the intrinsic color.

Figure 6 shows the evolution of the photospheric temperature $T_{\rm ph}$ and
color temperature $T_{\rm col}$ of the ECSN and FeCCSN. While $T_{\rm ph}$ and
$T_{\rm col}$ decline with time in both the ECSN and FeCCSN, those of the ECSN
are higher than those of the FeCCSN. This indicates that the higher $T_{\rm
ph}$ of the ECSN causes the bluer color.

Figure 6:  Evolution of the photospheric temperature $T_{\rm ph}$ and color
temperature $T_{\rm col}$ of the ECSN and FeCCSN, for which light curves are
shown in Figure 3. The ECSN is from an SAGB star of $M_{\rm
env}=3.0~{}M_{\odot}$ and $X({\rm H})_{\rm env}=0.7$ exploding with $E_{\rm
exp}=1.0\times 10^{50}$ erg, and the FeCCSN is from an RSG of $M_{\rm
ZAMS}=9.0~{}M_{\odot}$ exploding with $E_{\rm exp}=3.6\times 10^{50}$ erg. In
both models, no CSM exists around the progenitors.

Figure 7 shows the time evolution of the photospheric radii $r_{\rm ph}$ and
recombination front radii $r_{\rm rec}$ of the ECSN and FeCCSN. The
recombination front $r_{\rm rec}$ is defined as the outermost radius where the
ionization fraction $x$ is higher than $0.1$. During the plateau, the
photosphere of the ECSN is mostly located deeper than the recombination front
($r_{\rm ph}<r_{\rm rec}$, until approximately 55 days after the explosion),
whereas the radii of the photosphere and the recombination front of the FeCCSN
are almost same ($r_{\rm ph}\simeq r_{\rm rec}$, after approximately 20 days
after the explosion).

Figure 7: Time evolution of the photospheric radius $r_{\rm ph}$ and the
recombination front $r_{\rm rec}$ of the ECSN and FeCCSN in Figure 3.

The left panel of Figure 8 shows the temperature structures of the ejecta of
the ECSN and FeCCSN at 40 days after the explosion. The ECSN has a higher
$T_{\rm ph}$ than recombination temperature as a result of the relation
$r_{\rm ph}<r_{\rm rec}$, while the FeCCSN has almost the same $T_{\rm ph}$
and recombination temperature because of the relation $r_{\rm ph}\simeq r_{\rm
rec}$.

Figure 8:  Temperature (left panel) and density (right panel) structures of
the ejecta of the ECSN and FeCCSN in Figure 3 at 40 days after the explosion.
The radii of the photosphere $r_{\rm ph}$ and the recombination front $r_{\rm
rec}$ are indicated by vertical dotted and dashed lines, respectively.

The right panel of Figure 8 shows the density structures of the ejecta of the
ECSN and FeCCSN at 40 days after the explosion. The ECSN has a largely low
density around the recombination front $r_{\rm rec}$ resulting in $r_{\rm
ph}<r_{\rm rec}$. Conversely, the FeCCSN has high density around $r_{\rm
rec}$, resulting in $r_{\rm ph}\simeq r_{\rm rec}$. Therefore, the blue
plateau of the ECSN is caused by the low-density ejecta of the ECSN, which
originates from the low-density envelope of the SAGB star.

Kozyreva et al. (2021) also show the blue plateau of ECSNe and claim that the
reason is because the recombination of ECSN hardly settles because of the low-
density envelope of a SAGB star, making the photosphere recede slowly. Indeed,
we can find that the $r_{\rm rec}$ of the ECSN recedes more slowly than that
of the FeCCSN (Figure 7), agreeing that recombination of the ECSN hardly
settles. However, we find that the $r_{\rm ph}$ and $r_{\rm rec}$ of the ECSN
do not match in fact. The $r_{\rm ph}$ of the ECSN is located deeper than the
$r_{\rm rec}$, resulting in the bluer color of the ECSN.

###  4.2 ECSN Diagnostic Method

In this section, we present a new diagnostic method for ECSNe. First, we
define a pair of observational quantities:

  1. 1.

$t_{0}$: a time at which the $g$-band absolute magnitude reaches $-13$ mag, as
presented in Moriya et al. (2018). $t_{0}$ is set as the origin of time in
this section.

  2. 2.

$t_{\rm PT}$: the time of the mid-point between the end of the plateau and the
start of the tail. $t_{\rm PT}$ is derived by fitting equation (1) of Valenti
et al. (2016) to the calculated $g$-band light curve.

Figure 9 shows the relation between $t_{\rm PT}$ and the $g-r$ color at
$t=t_{\rm PT}/2$ [$(g-r)_{t_{\rm PT}/2}$]. The points indicate ECSN models
(blue crosses) and FeCCSN models (orange circles). While the left panel shows
all the models (736 ECSNe and 1266 FeCCSNe) calculated in this work, the right
panel shows the models not showing strong CSM interaction at $t_{\rm PT}/2$
(442 ECSNe and 861 FeCCSNe), that is, the models without strong CSM
interaction (Section 3.2.1) and part of the models with short and strong CSM
interaction (Section 3.2.2). In the right panel, the ECSNe are clearly
separated from the FeCCSNe and located at

$(g-r)_{t_{\rm PT}/2}<0.008\times t_{\rm PT}-0.4,$ (3)

reflecting their bluer plateau. This is a novel diagnostic method because it
is independent of the distance.

Figure 9: Diagnostic method to select ECSNe. The horizontal axis shows $t_{\rm
PT}$ and the vertical axis shows the $g-r$ color at $t_{\rm PT}$/2 for all
models (left panel) and for models showing no strong CSM interaction at
$t_{\rm PT}/2$ (right panel). SN 2018zd is overplotted in the right panel with
a dark-red diamond.

We apply our diagnostic method to SN 2018zd, which is proposed as an ECSN
(Hiramatsu et al., 2021) and a low-mass FeCCSN (Callis et al., 2021). The
disagreement is caused by uncertainty in the distance to SN 2018zd. The
diagnostic method can be applied because the spectral evolution of SN 2018zd
(Extended Data Figure 4 of Hiramatsu et al. 2021) indicates that CSM
interaction ends earlier than $t_{\rm PT}$/2. We estimate $t_{\rm PT}=125$
days and $(g-r)_{t_{\rm PT}/2}=0.45$. SN 2018zd is shown with a dark-red
diamond in the right panel of Figure 9. SN 2018zd clearly satisfies the
criterion of equation (3). This indicates that SN 2018zd is an ECSN, according
to the distance-independent diagnostic method.

We present alternative indicators, in which $B-V$ is adopted instead of $g-r$
and several epochs (e.g., $t=40$ days) are adopted instead of $t_{\rm PT}/2$,
in Appendix C. We propose a procedure of observations to find ECSNe in
Appendix D.

##  5 Conclusion

We have presented multicolor light curves of ECSNe and low-mass FeCCSNe. We
first adopted the physical quantities estimated in previous studies with
theoretically or observationally plausible methods to reveal representative
light curves, and then adopted wide physical quantity ranges to investigate
the robustness of their intrinsic differences. We adopted SAGB progenitor
models with the envelope masses of $M_{\rm env}=2.0-4.7~{}M_{\odot}$ from
Tominaga et al. (2013) for ECSNe and RSG progenitor models with the zero-age
main-sequence masses of $M_{\rm ZAMS}=9-12~{}M_{\odot}$ (s9.0–s12.0) from
Sukhbold et al. (2016) for FeCCSNe. We also considered various CSM structures.
We used the multi-group radiation hydrodynamics code, STELLA (Blinnikov &
Bartunov, 1993; Blinnikov et al., 1998, 2000) to calculate light curves.

The representative light curves of ECSNe show brighter and shorter plateaus
than those of low-mass FeCCSNe. This stems from the low-density and extended
envelopes of SAGB stars according to the scaling derived in Popov (1993);
Eastman et al. (1994).

The calculations with wide physical quantity ranges revealed that an ECSN and
a low-mass FeCCSN can display similar bolometric light curves. This may
explain why we could not identify ECSNe in previous transient surveys. ECSNe
show bluer plateaus than FeCCSNe if no strong CSM interaction is taking place
at the epoch.

The bluer plateaus of ECSNe originate from the low-density envelopes of SAGB
stars. The low-density envelope results that the photosphere of an ECSN is
mostly located deeper than the recombination front during the plateau, whereas
the radii of the photosphere and the recombination front are almost same in an
FeCCSN. Therefore, the photospheric temperature of an ECSN is higher than the
recombination temperature, whereas the photospheric temperature of an FeCCSN
is almost same as the recombination temperature. This causes the bluer
plateaus of ECSNe than those of FeCCSNe.

We proposed a new diagnostic method to select ECSNe. In the method, the
transition time from plateau to tail phase ($t_{\rm PT}$) and the $g-r$ at
$t_{\rm PT}/2$ [$(g-r)_{t_{\rm PT}/2}$] are adopted. ECSNe show bluer color
than low-mass FeCCSNe at $t_{\rm PT}$/2 in the absence of a strong CSM
interaction. The novel aspect of this method is that we can distinguish ECSNe
from low-mass FeCCSNe independently of distance.

We apply the method to SN 2018zd, which has been proposed as a promising ECSN
candidate (Hiramatsu et al., 2021) and as a low-mass FeCCSN (Callis et al.,
2021). The disagreement between the propositions is primarily caused by the
different estimation of its distance, which results in different ejected 56Ni
mass estimations. Our new method indicates that SN 2018zd is an ECSN.

We propose a procedure of observations to find ECSNe: 1) explore SNe II
showing blue-colored multicolor light curves, 2) check if the strong CSM
interaction is taking place using spectroscopic observations, and 3) diagnose
if it is an ECSN or not by applying the new method.

We thank Masaomi Tanaka, Keiichi Maeda, and Tomoya Takiwaki for helpful
discussions. Numerical computations were in part carried out on PC cluster at
Center for Computational Astrophysics, National Astronomical Observatory of
Japan. SIB and MSP thank the Russian Foundation for Basic Research (RFBR) and
the Deutsche Forschungsgemeinschaft (DFG), Joint RFBR-DFG project number
21-52-12032.

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##  Appendix A Why ECSNe and low-mass FeCCSNe without strong CSM interaction
or with short and strong CSM interaction exhibit similar bolometric light
curves

ECSNe and low-mass FeCCSNe can have similar bolometric light curves if no
strong CSM interaction takes place (Section 3.2.1) or if the strong CSM
interaction ends during the plateau (Section 3.2.2). According to equations
(9) and (10) of Eastman et al. (1994), we derive relations among the explosion
energy $E_{\rm exp}$, envelope mass $M_{\rm env}$, and pre-supernova radius of
such an ECSN and low-mass FeCCSN. Let us assume the plateau duration
$\tau_{\rm pl}$ and luminosity $L_{\rm pl}$ of an ECSN and a low-mass FeCCSN
are the same, i.e.,

$\tau_{\rm{pl,Fe}}=\tau_{\rm{pl,EC}}$ (A1)

and

$L_{\rm{pl,Fe}}=L_{\rm{pl,EC}},$ (A2)

where EC and Fe denote ECSN and FeCCSN respectively. Using these equations,
the relations among the explosion energy, envelope mass, and pre-supernova
radius are derived as

$\frac{E_{\rm{exp,EC}}}{E_{\rm{exp,Fe}}}\sim\left(\frac{R_{\rm{RSG}}}{R_{\rm{SAGB}}}\right)^{\frac{5}{4}}$
(A3)

and

$\frac{M_{\rm{env,EC}}}{M_{\rm{env,Fe}}}\sim\left(\frac{R_{\rm{RSG}}}{R_{\rm{SAGB}}}\right)^{\frac{3}{4}}.$
(A4)

The progenitor radii of the ECSN and FeCCSN of the models discussed in Section
3.2.1 are $7.1\times 10^{13}$ and $2.9\times 10^{13}$ cm, respectively. Hence,
the ratio of $E_{\rm exp}$ is $\sim 0.3$ and the ratio of $M_{\rm env}$ is
$\sim 0.5$, which is consistent with the ratios of their parameters ($E_{\rm
exp}$ ratio $\sim 0.3$ and $M_{\rm env}$ ratio $\sim 0.4$).

##  Appendix B Why ECSNe and low-mass FeCCSNe with long and strong CSM
interactions exhibit similar bolometric and multicolor light curves

The ECSN and FeCCSN discussed in Section 3.2.3 exhibit similar bolometric and
multicolor light curves. The observational quantities which characterize the
light curves are peak luminosity $L_{\rm pk}$, peak color $T_{\rm pk}$, and
the duration of CSM interaction $\tau_{\rm CSM}$ if they show long-lasting
strong CSM interaction that ends after the plateau does. $L_{\rm pk}$ is
expressed as (Chevalier & Irwin, 2011)

$L_{\rm pk}=L_{\rm pk}(E_{\rm exp},M_{\rm ej},\dot{M}).$ (B1)

$T_{\rm pk}$ is roughly expressed as

$T_{\rm pk}=T_{\rm pk}(L_{\rm pk}(E_{\rm exp},M_{\rm ej},\dot{M}),r_{\rm
CSM}),$ (B2)

if we assume that the radiation is approximately blackbody and that the
photosphere is located near the outermost region of the CSM. $\tau_{\rm CSM}$
is roughly expressed as

$\tau_{\rm CSM}=\tau_{\rm CSM}(E_{\rm exp},M_{\rm ej},r_{\rm
CSM})\propto\frac{r_{\rm CSM}}{\sqrt{E_{\rm exp}/M_{ej}}}$ (B3)

because CSM interaction lasts during the shock propagation in the CSM. There
are three free quantities independent of the pre-supernova structure among the
four physical quantities $M_{\rm ej}$, $E_{\rm exp}$, $\dot{M}$, and $r_{\rm
CSM}$. Only $M_{\rm ej}$ depends on the pre-supernova structure because ECSNe
have lower $M_{\rm ej}$ than low-mass FeCCSNe. Therefore, the three
observational quantities $L_{\rm pk}$, $T_{\rm pk}$, and $\tau_{\rm CSM}$ can
be same in ECSNe and FeCCSNe if a certain relation of $E_{\rm exp}$,
$\dot{M}$, and $r_{\rm CSM}$ is satisfied.

##  Appendix C Other indicators of ECSNe

We have presented a new diagnostic method to select ECSNe in Section 4.2, in
which the transition time from plateau to tail $t_{\rm PT}$ and the color
$g-r$ at $t_{\rm PT}/2$ are adopted. Here, we also present an alternative
indicator in which the color index $B-V$ is adopted instead of $g-r$. Figure
10 shows the indicator. The horizontal axis shows the $t_{\rm PT}$ and the
vertical axis shows $B-V$ at $t_{\rm PT}/2$ [$(B-V)_{t_{\rm PT}/2}$]. ECSNe
can be selected with a criterion,

$(B-V)_{t_{\rm PT}/2}<0.0089\times t_{\rm PT}-0.36.$ (C1)

Figure 10:  An indicator of ECSNe in which $B-V$ is adopted instead of $g-r$.

As to be discussed in Appendix D, in the procedure of observations, we need to
know if the CSM interaction ends before $t_{\rm PT}/2$ from spectroscopic
observations although we can only know $t_{\rm PT}$ after the plateau ends.
This is crucial, especially when we diagnose far or ubiquitous objects,
because frequent spectroscopic observations are difficult.

To overcome this, we adopt $g-r$ or $B-V$ at specific epochs
($t=40,~{}45,~{}50,~{}55,~{}60,$ and $65$ days; time origin as $t_{0}$,
defined in Section 4.2) instead of $(g-r)_{t_{\rm PT}/2}$. Figure 11 and 12
show the relation between $t_{\rm PT}$ and $g-r$, and $t_{\rm PT}$ and $B-V$,
respectively, for each epoch. The models with earlier plateau ends or later
CSM-interaction ends than each epoch are not shown. The colors of ECSNe and
FeCCSNe are generally different, but they are similar if the adopted epoch is
close to $t_{\rm PT}$ (left side of each panel). This infers that earlier
spectroscopy is preferred but also often suffers from CSM interaction.
Therefore, we suggest that spectroscopy around $t=50$ days is preferential if
the opportunity of spectroscopic follow-ups is limited.

|  |
---|---|---
|  |
Figure 11:  ECSN indicators based on $t_{\rm PT}$ and $g-r$ at 40, 45, 50, 55, 60, and 65 days. The horizontal axis shows $t_{\rm PT}$ and the vertical axis shows $g-r$ at each epoch. In each panel, models with earlier $t_{\rm PT}$ or later CSM-interaction ends than each epoch are not shown.  |  |
---|---|---
|  |
Figure 12:  Same as Figure 11 but for $B-V$.

##  Appendix D The procedure of observations

We propose a procedure of observation to identify ECSNe as follows:

  1. 1.

Explore SNe II showing blue plateaus, which are the ECSN candidates using the
selection method presented below (Figure 13 and 14), in which the evolution of
$g-r$ or $B-V$ is adopted.

  2. 2.

From spectroscopic observations, check whether the CSM interaction ends
earlier than $t_{\rm PT}/2$.

  3. 3.

If the CSM interaction ends before $t_{\rm PT}/2$, identify ECSNe with
equation (3) or (C1). If the CSM interaction lasts beyond $t_{\rm PT}/2$ but
ends at later epochs in the plateau, identify ECSNe using Figure 11 or 12.

We present methods to select SNe II showing blue color, which are ECSN
candidates, in which the evolution of $g-r$ or $B-V$ is adopted. Figures 13
and 14 show the criteria of ECSN candidates as well as the $g-r$ (Figure 13)
and $B-V$ (Figure 14) evolution of ECSNe (blue lines in left panels) and
FeCCSNe (orange lines in right panels) though the light curves at epochs with
ongoing CSM interaction or after the plateau are not shown. If a SN II shows
bluer light curves than the criteria, we propose spectroscopic observations to
check if the CSM interaction is ongoing.

Figure 13:  A selection criterion of ECSN candidates, in which the $g-r$
evolution is adopted. The $g-r$ evolution of ECSNe (blue lines in the left
panel) and FeCCSNe (orange lines in the right panel) are also shown though the
light curves when the CSM interaction is taking place or after $t_{\rm PT}$
are not shown. If a SN II shows bluer color than the criterion, we propose
spectroscopic follow-ups to check if CSM interaction is taking place.

Figure 14:  Same as Figure 13 but $B-V$ is adopted.


# Two-dimensional few-atom noble gas clusters in a graphene sandwich

Manuel Längle1,2∗, Kenichiro Mizohata3, Clemens Mangler1,
Alberto Trentino1,2, Kimmo Mustonen1, E. Harriet Åhlgren1,
Jani Kotakoski1†
1University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna,
Austria
2University of Vienna, Vienna Doctoral School in Physics, Boltzmanngasse 5,
1090 Vienna, Austria
3University of Helsinki, Department of Physics, P.O. Box 43, FI-00014 Finland
∗E-mail<EMAIL_ADDRESS>†E-mail<EMAIL_ADDRESS>
> Van der Waals atomic solids of noble gases on metals at cryogenic
> temperatures were the first experimental examples of two-dimensional
> systems. Recently such structures have also been created on surfaces under
> encapsulation by graphene, allowing studies at elevated temperatures through
> scanning tunneling microscopy. However, for this technique, the
> encapsulation layer often obscures the actual arrangement of the noble gas
> atoms. Here, we create Kr and Xe clusters in between two suspended graphene
> layers, and uncover their atomic structure through direct imaging with
> transmission electron microscopy. We show that small crystals ($N<9$)
> arrange as expected based on the simple non-directional van der Waals
> interaction. Crystals larger than this show some deviations for the
> outermost atoms, possibly enabled by deformations in the encapsulating
> graphene lattice. We further discuss the dynamics of the clusters within the
> graphene sandwich, and show that while all Xe clusters with up to $N\sim
> 100$ remain solid, Kr clusters with already $N\sim 16$ turn occasionally
> fluid under our experimental conditions with an estimated pressure of ca.
> 0.3 GPa. This study opens a way for the so-far unexplored frontier of
> encapsulated two-dimensional van der Waals solids with exciting
> possibilities for fundamental condensed matter physics research and possible
> applications in quantum information technology.

Two-dimensional (2D) van der Waals atomic crystals are one of the simplest
conceivable solids, described to a large extent with the simple pairwise
interaction approximated by the Lennard-Jones potential. Such structures
formed by noble gas atoms have been experimentally studied since the 1960’s at
cryogenic temperatures (?). Since the van der Waals interaction between noble
gas atoms is isotropic, and only depends on the inter-atomic distance, the 2D
crystals assume a close-packed hexagonal structure. This was confirmed through
ultralow-energy electron diffraction experiments already in 1975 (?). First
real space images of 2D noble gas crystals were recorded via scanning
tunneling microscopy in 1998 on a graphite surface (?).

Studies on van der Waals structures were for a long time limited to cryogenic
temperatures due to the weakness of the van der Waals interaction. Only after
the discovery of the impermeability of graphene to the smallest gases in 2008
(?), it became possible to create such structures at elevated temperatures by
trapping noble gases within the van der Waals gap between graphene and the
supporting surface. Such Xe clusters have already been imaged using scanning
probe microscopy (?, ?). However, the graphene encapsulation complicates
imaging the atomic structure of the crystallites which has even lead to
misidentification of such structures in the past (?, ?). It has been also
shown that Kr can be implanted under the first graphite layer, essentially
into the interface between graphene and bulk graphite (?). Implantation deeper
into graphite should also be possible, but the rigidity of the thicker
graphite layers on both sides of the implanted species (as compared to a
single layer) might make it impossible to form solid structures without
considerable damage to the surrounding lattice.

Here, we create 2D few-atom noble gas clusters by implanting Kr and Xe between
suspended graphene sheets using ultralow-energy ion irradiation, and image
them at room temperature using atomic-resolution scanning transmission
electron microscopy (STEM), complemented by electron energy loss spectroscopy
for confirming the chemical identity of the atoms. Our samples consist of both
mechanically exfoliated bi- and few-layer graphene as well as commercial
graphene grown via chemical vapor deposition (CVD). For the CVD-grown samples,
two graphene monolayers were stacked on top of each other to create an
artificial double layer with a random misorientation between the graphene
crystals. This allows us to compare samples where the van der Waals gap has
never been exposed to air or contamination to samples where the two layers are
joined under ambient conditions, and different moiré periodicities. In all
cases, similar noble gas clusters were found with no discernible difference.
Samples were irradiated with singly charged ultralow-energy Kr or Xe ions to
introduce the noble gases. For Kr-irradiated samples, successful implantation
between two graphene layers was achieved at 60 eV, and for Xe-irradiated
samples at 55 eV and 65 eV, close to the simulated lower bound for Xe
implantation of 70 eV (?). One additional sample was implanted using 30-eV Xe
ions at the University of Vienna using a plasma source within the vacuum
system (?) connected to the Nion UltraSTEM 100 microscope used here for
imaging. These energies are close to the 25 eV that has been shown to allow
implanting He, Ne and Ar into the van der Waals gaps of graphene-covered metal
surfaces (?).

After the implantation, the samples were imaged via STEM at 60 kV using either
the high or the medium angle annular dark field (HAADF and MAADF) detectors.
Before being inserted into the vacuum system connected to the microscope, the
samples were baked in vacuum at ca. 150∘C for at least 8 h. Some samples were
additionally cleaned before imaging with a laser at the microscope column (?).
Larger scale images of implanted samples with many clearly visible clusters
are shown in Fig. S1 and Fig. S2 along with the recorded spectroscopic
signatures confirming the elemental identification as Kr and Xe, respectively.

Due to their significantly higher atomic number as compared to the carbon
atoms of graphene, both Kr and Xe are easily visible in clean areas of the
sample in the $Z$-contrast (?) STEM images. Clusters with
$N_{\mathrm{Kr}}\in[2,12]$ and $N_{\mathrm{Xe}}\in[2,13]$ atoms are shown in
Fig. 1a-d along with the simulated energy-optimized structures. All clusters
appear flat, and assume a hexagonal close-packed structure. The same cluster
shapes are found both for Xe as well as Kr, and in samples using both
exfoliated graphene and graphene grown via CVD. Therefore, there is no
plausible way that the clusters could contain any other element that remains
invisible in our microscopy images.

The hexagonal close-packed structure minimizes the number of lower-coordinated
atoms while at the same time maximizing the number of nearest neighbors within
the cluster (see also Fig. S3 in the supplementary material), demonstrating
that the interaction between the atoms within each cluster is strong enough to
dominate its shape under these conditions. Up to $N<9$, all clusters in the
experiment are found also in the optimal structure as predicted by the
simulations, although in some cases also other structures with same $N$ are
observed (see Fig. S4). Most notably, the Kr9 cluster also appears in a shape
where the atoms are arranged along two lines. In this unexpected Kr9 structure
the number of van der Waals bonds with approximately the nearest neighbor
distance ($d_{nn}$) is only 14 (in one case even 13), whereas in the ideal
structure it is 17. In addition to this specific case of Kr9, the atoms in the
outer rim of clusters with $N\geq 9$ appear to have less defined positions
(also seen in Fig. 1b & c). These deviations from the ideal structure may
arise from factors other than the interaction between the noble gas atoms (for
example from deformations in the encapsulating graphene structure, as
discussed below). We also point out that one Kr10 configuration, as well as
Kr11,12 all show the same cluster with occasional detachment and attachment of
one or two atoms (the image series is shown in Fig. S5). In many cases, we
also observe shape changes both in the simulations (Fig. S6), as well as in
the experimental images (Fig. S4). Although we did not find Kr6, Xe11 or Kr13
within our images of small crystals, there is no reason to assume that they
would be less stable than the clusters shown in Fig. 1.

Figure 1:  Structure of small clusters. (a) Filtered STEM-ADF images (see
Methods) of Kr (top row) and Xe (bottom row) clusters with $N\in[2,12]$,
including all cases where the experimentally observed structure corresponds to
that with the lowest energy according to simulations. The corresponding
simulated structures after energy optimization are shown in the middle row for
Xe (Kr has the same structure, see Fig. S7 for all simulated structures).
Carbon atoms are shown with small gray dots, and Xe atoms with large dots
connected with lines. (b) Example STEM-ADF images of nine-atom Kr and Xe
clusters along with the simulated lowest-energy configuration for Kr. (c)
Example STEM-ADF image of a eleven-atom Kr cluster along with the simulated
lowest-energy configuration for Kr. (d) Example STEM-ADF image of a thirteen-
atom Xe cluster and the corresponding lowest-energy structure for Xe. All
images have a size of $2\times 2$ nm2.

Nearest neighbor distances for the clusters are shown in Fig. 2a. Both
experimental and simulated results show clearly the same trend, where the
smallest clusters have the shortest interatomic distances, with the strongest
effect seen in the experimental data for the Xe clusters. In all cases, the
values are fairly constant with a slowly increasing trend for cluster sizes of
$N\geq 10$. Comparing $d_{nn}$ to the corresponding pressures estimated via
simulations (Fig. 2b) shows that the smallest clusters are under a pressure of
up to 1 GPa, whereas for the larger clusters this drops to around 0.3 GPa.
These values are in the same range, but somewhat smaller than 1 GPa that has
been estimated for 10 nm-sized clusters experimentally (?), and significantly
lower than 30 GPa estimated for smaller He and Ar clusters computationally
(?). However, in neither of those cases were the clusters encapsulated between
two graphene sheets.

The increasing pressure when the cluster size decreases is likely due to the
higher necessary local curvature for graphene required for accommodating the
smaller clusters. Additionally, the pressure for the smallest experimental
clusters appears to increase even faster than what is predicted by the
simulations. This can be related to the size of the point defect trapping the
clusters, as discussed below.

Figure 2: Interatomic distances and 3D to 2D transformation. (a) Measured
interatomic distances $d_{nn}$ (full circles) in the experimentally observed
Kr (green) and Xe (blue) clusters as a function of the cluster size. Each data
point is averaged over all images where the structure was clearly visible,
taking always the mean value per image. The errorbars show the standard
deviation of the data. Open symbols correspond to the simulation results. (b)
Relationship between the pressure (plotted on logarithmic scale) exerted on
the simulated 2D noble gas structure and the interatomic separation. The lines
are guides for the eye. (c) Sequence of images showing selected steps from
structural optimization starting with the twelve-atomic 3D Kr cluster in a
graphene sandwich, which turns into a 2D structure. The change in energy from
left to right between the configurations is $-5.564$, $-0.565$ and $-0.033$
eV, respectively.

To understand the influence of the graphene sandwich on the shape of the
crystals, we carried out further structure optimization for cluster sizes up
to 18 atoms (Fig. S8). These show, that in all cases, the optimum shape for
the cluster (even at zero temperature) is three-dimensional for clusters
larger than three atoms. However, when the cluster is placed inside the
graphene sandwich, it relaxes into a two-dimensional structure. All clusters
retain the 2D shape between the graphene sheets, even up to 61 atoms, which
was the maximum simulated cluster size. The change between the 3D and 2D
structures is associated with an energy cost of ca. $60$ meV/atom for the Xe18
cluster. This energy is compensated by the interface energy between the
cluster and graphene, and the lower deformation energy for the graphene
sandwich in the flatter configuration. An upper estimate for the interface
energy (per atom) can be obtained by calculating the energy associated with an
individual noble gas atom placed within two graphene sheets (total energy
minus the energy of the similarly deformed graphene sheets without the
additional atom). This is ca. 149 meV for Kr and ca. 66 meV for Xe, which
shows that the interface energy indeed has a similar order of magnitude as the
energy difference of the 3D and 2D clusters. Indeed, when graphene is removed
from around the simulated 2D clusters, they transform to the 3D shape during
further relaxation.

It is rather surprising, that the small clusters remain stationary at room
temperature for long enough to allow their imaging in the microscope. In fact,
one could even expect that all noble gas atoms would escape through edges from
between the two graphene layers before they are able to form solid structures.
While loosing all atoms is unlikely due to the large size of the samples (up
to tens of $\mu$m for exfoliated samples and up to even mm for the CVD-grown
ones) compared to atomic length scales, the motion of noble gases in the
graphene sandwich warrants a discussion. To study this, we turn to room
temperature molecular dynamics simulations. Our simulations of individual Kr
and Xe atoms show that they are both highly mobile in a graphene sandwich with
estimated speeds of ca. $316$ m/s and $233$ m/s, respectively. For three-atom
clusters this changes to ca. $334$ m/s and $231$ m/s and for the Kr7 cluster
to ca. $369$ m/s at $300$ K. (The corresponding speeds in gas under normal
conditions, according to ideal gas law, would be $180$ m/s and $130$ m/s,
respectively, for two dimensions). Therefore, for all simulated clusters the
speed within the graphene sandwich would be too high for imaging with the
experimental time resolution of ca. 1 s per image. The simulations further
reveal that despite the weakness of the interaction between the noble gas
atoms, already at $N=3$, all atoms within the clusters move together rather
than separating (also seen for Kr7 in Fig. S6). This is in good agreement with
our experimental results, where we only rarely observe detachment of
individual atoms from the clusters.

Obviously, regardless of the fast expected migration, the clusters remain
static for long enough for us to be able to image them. This suggests that the
experimentally observed clusters are pinned to their positions either by
corrugations associated with the surface contamination (many clusters appear
at the edges of contamination) or by defects (?, ?), some of which are
expected to lead to out-of-plane distortion of graphene (?). To test this
hypothesis, we created two different defects in the graphene sandwich, and
simulated the migration of a Xe3 cluster in both cases. We point out that
these serve simply as examples of defects that would lead to either negative
or positive Gaussian curvature in the graphene structure, and other defects
may in reality be more likely to result from the low energy ion irradiation.
In the first case, we introduce an inter-layer covalent bond, which locally
brings the two graphene sheets together (Fig. 3a,c). This leads to the cluster
avoiding the defect position and instead migrating everywhere else. In the
second case, we introduced an inverse Stone-(Thrower)-Wales defect (?) that
leads to negative curvature in graphene. Here, the Xe3 cluster is clearly
attracted to the defect, and once trapped in its vicinity, it remains there
(Fig. 3b,d). Although it can not be directly seen from the images, in both
cases all atoms moved together throughout the whole simulation. We assume that
similar to these simulations, also in our experiments the observed clusters
are attracted to defect sites. Indeed, the ultralow-energy irradiation which
is used to introduce the noble gas atoms into the graphene sandwich also by
necessity creates defects into it. Some of them in turn are associated with a
negative curvature of the graphene sheet, which provides a favorable pocket
for a small noble gas cluster. Point defects are indirectly visible both in
single and double layer parts of the sample in Fig. S9a (by covalently bound
heavier impurity atoms (?)), which additionally shows that noble gas clusters
are only trapped in the double layer, highlighting that they can only be
captured within the van der Waals gap between two graphene layers. The size of
the defect-related deformation may be a cause for the apparent unexpectedly
high pressure for the smallest Xe clusters; the smallest clusters may fit into
the deformation without further bending graphene by being squeezed tighter
together leading to less bending in graphene but tighter pressed Xe atoms.

Interestingly, we also see occasionally that the clusters in the experiments
jump between different sites in the graphene sandwich, even in areas void of
visible contamination. As an example, Fig. 3e shows the trajectory of a Xe8
cluster that was observed jumping between five different locations. Another
example of a jumping cluster is shown in Fig. S10, where the cluster
eventually merges with a stationary one. We assume that the movement of the
clusters is indirectly triggered by the imaging electron beam through
scattering-related changes in the atomic configurations in the defects (?)
that serve as the pinning sites. These pinning sites cannot arise from the
moiré superstructure of the two graphene layers, because we observe similar
behavior for samples regardless of the relative orientation of the graphene
sheets (and even for mechanically exfoliated graphene with AB stacking). Our
interpretation is also in line with Ref. (?), where individual W atoms were
observed to similarly jump between discrete positions in graphene, assumed to
be associated with point defects. The main difference is that the W atoms
migrate on top of graphene, whereas the noble gas clusters are confined within
the van der Waals gap, which presumably has a significant influence on the
associated energy barriers.

Figure 3: Cluster migration. (a) Trajectory of a Xe3 cluster in a graphene
sandwich with an interlayer bond-defect, simulated for ca. $1$ ns via room
temperature molecular dynamics. (b) Trajectory of a Xe3 cluster in a graphene
sandwich where one of the layers contains an inverse Stone-(Thrower)-Wales
defect, simulated for ca. $1$ ns. For both simulations, the positions for the
cluster atoms are shown for every $0.1$ ps, and the defect location is marked
with an open circle. (c) Side view of graphene with the interlayer bond, where
covalent bonds between the carbon atoms are shown with thin lines. (d) Side
view of graphene with the inverse Stone-(Thrower)-Wales defect, and the Xe3
cluster (shown with small balls). (e) Experimentally measured trajectory of an
eight-atomic Xe cluster overlaid on a STEM-ADF image recorded at the end of
the image sequence. The position of the cluster at the beginning of the
sequence is marked with a white arrow, and the final position at the top of
the image is visible due to the higher contrast of the Xe atoms. Positions for
each of the images in the sequence are shown with semitransparent white
circles, and the jumps between the images are marked with semitransparent
white lines. The five distinct locations are highlighted with the rectangles.

One interesting question regarding the pressure-induced stability of 2D van
der Waals atomic structures is under which conditions they remain solid. The
solid-fluid phase transitions carry information of the size-dependent
interaction strength within the noble gas clusters, being of high relevance
for future studies. This is also interesting, since from a purely energetic
point of view, one would expect smaller clusters to have a lower melting point
and therefore to display liquid-like characteristics before the larger ones.
However, in the case of our clusters sandwiched between two graphene sheets,
this is counterbalanced by the increasing pressure for the decreasing cluster
sizes, as discussed above. Nevertheless, it is natural to expect that there is
a pressure- and temperature-dependant size limit, beyond which the energy cost
of the undercoordinated atoms at the rim (corresponding to the surface energy
of a 3D structure) becomes too large, and the structure undergoes a
transformation into a three-dimensional shape. Phase diagrams suggest a solid-
to-liquid transformation at room temperature for pressures slightly above 1
GPa for Kr and around 0.5 GPa for Xe (?). Therefore, one would expect larger
Kr clusters to loose their solid shape, which we indeed observe (Fig. 4).
Although according to simulations, both Xe61 and Kr61 clusters should retain
the 2D shape, experimental images (Fig. 4) show a brighter area within the Kr
structure, which may indicate that the structure is not strictly two-
dimensional. Indeed, Kr structures larger than $N\sim 16$ already start to
show less defined atomic positions (at the rim already earlier, see Fig. 1a),
and shapes that resemble liquid more than small crystallites. However, Xe
clusters remain two-dimensional and exhibit characteristics of solid
crystallite structures (clearly discernible atomic positions with long-range
order) up to much larger sizes, although the estimated pressure is only 0.3
GPa. In fact, as shown in Fig. S9b, the Xe clusters start to loose their solid
characteristics only at the size of $N\sim 100$ atoms.

As is evident in all images of clearly liquid-like structures (Fig. 4, Fig.
S9b), the solid-to-fluid transition is also associated with loosing the strict
two-dimensionality. Although the energy per atom decreases faster for 3D
clusters than for 2D ones (Fig. S8), this energy gain is countered with a cost
for larger deformation in graphene, which is larger for Xe than for Kr due to
its larger size. This may provide a partial explanation for why Xe remains
solid for much larger structures than Kr. Finally, we point out that as can be
seen in Fig. 4, the larger clusters rotate during imaging, ruling out epitaxy
with the graphene lattice.

Figure 4: Structure of larger clusters. (Top) Filtered STEM-MAADF images of a
Xe51 cluster within AB-stacked bilayer graphene that remains crystalline
throughout the entire experiment (over $8$ min). The numbers on the images
indicate the frame number within the image sequence and the corresponding
time. The small images next to them show a fast Fourier transform of each
image with overlaid hexagons highlighting the spots corresponding to graphene
and to the noble gas cluster and the misorientation angle between them. These
angles for each image frame are shown in the plot in the lower right corner.
(Bottom) Filtered STEM-HAADF images of a Kr cluster of a similar size. The
provided example of a fast Fourier transform shows the orientation of the two
graphene sheets in the double layer sandwich (misorientation of 25∘). Due to
the lack of crystallinity, the Kr cluster does not show characteristic spots.

Creation of small solid state noble gas clusters in a graphene sandwich, as
was demonstrated here, opens the so-far unexplored frontier of encapsulated 2D
van der Waals atomic solids. Simulations show that the small clusters would
assume a 3D shape without the graphene encapsulation, even at zero
temperature, and that noble gas clusters are extremely mobile in the graphene
sandwich, unless pinned to a position due to a deformation in at least one of
the encapsulating graphene layers. Although small clusters remain solid for
both Kr and Xe, larger Kr clusters ($N>16$) start loosing their solid
structure whereas Xe clusters remain solid up to $N\sim 100$. Larger solid
structures may be achieved by providing more structural rigidity for example
through increasing the number of the encapsulating layers, which can be
expected to increase the pressure. Combined with in situ techniques for
example allowing experimentation from cryogenic to elevated temperatures,
encapsulated 2D noble gas structures provide exciting possibilities for
studies in fundamental condensed matter physics ranging from research on the
growth and atomic-scale dynamics in solids to phase transitions and
topological defects. They may also lead to well-defined systems for example
for quantum information research.

## Online content

Supplementary material includes description of methods, overview images and
electron energy loss spectra, a schematic presentation of the observed atomic
structures, images of additional observed shapes of small Kr and Xe clusters,
time series of a Kr cluster changing size over time, simulations of shape
changes of a cluster, all different simulated 2D and 3D shapes, simulation
results for the energetic differences between 2D and 3D clusters and an
additional example of an image series of a Xe cluster that jumps within the
graphene sandwich.

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### Methods

Samples. Few-layer graphene samples were fabricated using both mechanical
exfoliation as well as stacking single-layer graphene grown via CVD
(EasyTransfer graphene by Graphenea, Inc.) on top of each other. Mechanically
exfoliated samples were transferred using the typical transfer procedure (?)
to Quantifoil Holey Carbon TEM grids with a hole size for the suspended sample
of either 1.2 or 2.0 $\mu$m, depending on the sample. CVD-samples were
transferred to the same grids following the manufacturer instructions (two
consecutive transfers onto the same grid were carried out to create the
necessary double layer).

Ion irradiation. Most samples were irradiated at the University Helsinki using
the $500$ kV ion implanter KIIA operated at $20$ kV with an electrostatic
deceleration lens slowing the ions down to $20-100$ eV. To reduce sample
contamination during the implantation process, some of the samples were baked
in air for $1$ h at $300^{\circ}$C and inserted warm into the irradiation
setup. For Kr-irradiated samples, successful implantation between two graphene
layers was observed at 60 eV irradiation, and for Xe-irradiated samples at
$55$ eV and $65$ eV.

Additional irradiations were carried out within the vacuum system CANVAS (?)
at the University of Vienna, which connects the Nion UltraSTEM 100 with other
experimental stations. Here, a plasma source was used to create a Xe beam. The
energy of the plasma was measured by biasing a Faraday cup and measuring the
current as a function of the bias voltage. For implantation, the beam was
decelerated using sample bias until the energy of the ions was ca. 30 eV.
Results from these irradiations are shown in Fig. S9, whereas all other
results are from samples irradiated at the University of Helsinki.

Electron microscopy and spectroscopy. All samples were imaged at the
University of Vienna with the Nion UltraSTEM 100 dedicated scanning
transmission electron microscopy (STEM) instrument at $60$ kV using either the
high or the medium angle annular dark field (HAADF and MAADF) detector with
annular ranges of $60-200$ mrad and $80-300$ mrad, respectively. The
convergence semi-angle was $30$ mrad, and a typical beam current ca. $30$ pA.
Electron energy loss spectroscopy was carried out with the same instrument
with a Gatan PEELS 666 spectrometer retrofitted with an Andor iXon 897
electron-multiplying charge-coupled device camera using a collection semi-
angle of ca. $35$ mrad. Before being inserted into the vacuum system connected
to the microscope, the samples were baked in vacuum at ca. 150∘C for at least
8 h. Some samples were additionally cleaned before imaging with a laser at the
microscope column ($50$ $\mu$s laser pulse, power $20$ mW, wavelength $473$ nm
and spot size ca. $700$ $\mu$m2). The images were calibrated using the auto
scaling algorithm (?) based on the lattice spacing of graphene, and the
positions of the noble gas atoms were detected with the automatic blob
detector (?). The samples remained at room temperature during imaging. Due to
the excellent conductive properties of graphene, no charging of graphene or
the sandwiched structures took place during imaging.

Image processing. Image filtering was done by applying a double Gaussian (?)
and a Fourier filter. For the double Gaussian filter, the inner radius was set
as 0.15 times the distance from the centre to the first set of graphene spots
in the fast Fourier transform (FFT) and the radius of the outer Gaussian was
set as 1.3 times the respective distance to the outer set of FFT spots. In the
Fourier filter, the spots corresponding to graphene were deleted and thus
lattice was removed after back-transformation.

Atomistic simulations. The molecular dynamics simulations were conducted using
the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The
periodic system consisted of two $10\times 10$ nm2 AB-stacked graphene layers
(7872 C atoms) with the interlayer distance of $3.39$ Å relaxed to a local
energy minimum with the Polak-Ribiere modified conjugate gradient algorithm
(?).

The defected graphene sheets with an interlayer bond and a defect with
negative curvature were created by introducing the defect to the relaxed AB-
stacked graphene and re-relaxing the system. After the initial relaxations,
the noble gas atoms were introduced between the graphene layers in the $2$D
form as observed in the experiments and the system was relaxed again.

Optimised $3$D clusters were obtained from these $2$D forms by annealing the
cluster without graphene at $80$ K over $40$ ps followed by cooling down to
$1$ K during $800$ ps and relaxation. The rapid annealing and slow cooling was
introduced to escape possible local energy minima.

For observing the cluster transformation into $2$D, two optimised $3$D
clusters (Kr6 and Kr12) were individually placed between the relaxed graphene
sheets and then relaxed again.

The room temperature dynamics of a single atom, a three-atom cluster and a
seven-atom cluster were simulated by first ramping the temperature from $0$ K
to $300$ K over $0.2$ ns, after which the simulation was run at $300$ K for
another $1$ ns. The output was printed every $500$ steps with a time step of
$0.2$ fs.

The pressure of the Kr and Xe clusters were calculated from a $4\times 5$ atom
semiperiodic cell. The bond lengths were scaled to the selected value and the
forces ($F_{y}$) between atoms taken from the initial structure along the
nonperiodic direction when conjugent gradient energy minimisation was started.
Pressure was calculated with $P_{y}=F_{y}/A$, where $A$ is the 2D area given
by the distance between two neighboring atoms and the height of the van der
Waals gap containing the cluster ($3.4$ Å for Kr, $3.3$ Å for Xe).

Kr and Xe interactions were described via the Lennard-Jones potential, with
the Kr-Kr and Xe-Xe parameters $\varepsilon_{\mathrm{Kr}}=30.7$ meV,
$\sigma_{\mathrm{Kr}}=3.6233$ Å and $\varepsilon_{\mathrm{Xe}}=44.1$ meV,
$\sigma_{\mathrm{Xe}}=3.9460$ Å) (?) and the Kr-C and Xe-C with parameters
($\varepsilon_{\mathrm{Kr-C}}=10.7$ meV, $\sigma_{\mathrm{Kr-C}}=3.5116$ Å and
$\varepsilon_{\mathrm{Xe-C}}=12.8$ eV, $\sigma_{\mathrm{Xe-C}}=3.673$ Å)
calculated using the Lorenz-Berthelot mixing rules (?, ?). Carbon-carbon
interaction was described by the AIREBO potential (?), including the van der
Waals interaction.

## Acknowledgments

Funding: Funding through the Austrian Science Fund (FWF) within projects
P31605, P34797 and M2596, and generous grants for computational resources from
the Vienna Scientific Cluster are gratefully acknowledged.

## Author contributions

M.L. and J.K. designed the experiments. M.L. prepared the samples and carried
out the ion irradiation with the help of K.M. and H.Å. H.Å. carried out the
molecular dynamics simulations and analysed that data together with J.K. M.L.
and A.T. carried out the microscopy with the help from C.M., K.M. and J.K.
M.L. and J.K. analyzed the experimental data, plotted the figures, and wrote
the first draft. All authors were involved in writing the manuscript. J.K.
supervised the project.

## Competing interests

The authors declare no competing interests.

## Data and materials availability

All original data used in producing the presented results is available in the
supplementary material and through the University of Vienna repository Phaidra
(?).

## Supplementary Material

Figure S1: STEM-MAADF overview image of an area containing several Kr clusters
(some of which are highlighted by numbered rectangles with magnified versions
shown at the bottom) and spectral map. The electron energy loss spectral map
was recorded over a cluster with approximately $50$-$70$ atoms, shown on the
bottom of Fig. 4. The contrast here corresponds to the integrated spectrum for
each pixel. The signal from the Kr $M_{4,5}$ core loss peak arises from the
area spatially located at the cluster, as can be seen from the spectrum
obtained by subtracting the graphene signal (from within the area marked by
the white rectangle) from the signal corresponding to the area marked by the
black rectangle (the spectra were normalised to the values before the peak
onset at 89 eV).  Figure S2: STEM-MAADF overview image of an area containing
several Xe clusters (some of which are highlighted by numbered rectangles with
magnified versions shown at the bottom) and spectral map. The electron energy
loss spectral map was recorded over a Xe6 cluster (same cluster as Xe6 in Fig.
1 of the main article). The contrast here corresponds to the integrated
spectrum for each pixel. The Xe $M_{4}$ and $M_{5}$ core loss peaks arise from
the area spatially located at the cluster, as can be seen from the spectrum
obtained by subtracting the graphene signal (from within the area marked with
a white rectangle) from the signal corresponding to the area marked by the
black rectangle (the spectra were normalized to the values before the peak
onset at 672 eV. Two very bright pixels, possibly caused by cosmic rays
hitting the detector, were replaced by their average surrounding.  Figure S3:
Schematic presentation of the observed atomic structures. The numbers indicate
the size of the cluster (positions marked with $i\leq N$ constitute the
cluster with $N$ atoms). For each added atom, the bonds with the same color as
the circle are new compared to the cluster with one less atom. The position
shown with gray outline and black number $6$ corresponds to the position of
the sixth atom in Xe6 in the observed cluster, whereas the colored circle with
white $6$ is the position of the sixth atom in all clusters with $N>6$. In
contrast, the colored circle with white $13$ corresponds to the observed
structure of Xe13, whereas the gray circle with black $13$ would lead to a
more symmetric structure but with the same number of nearest neighbor
interactions.  Figure S4: Filtered STEM-ADF images (see Methods) of small Kr
and Xe clusters displaying all additionally observed shapes in our
experiments. All images have an area of $2\times 2$ nm2. Left are Kr and right
are Xe clusters. The overlaid values for $n$ show the number of nearest-
neighbor pairs for each configuration.  Figure S5: Filtered STEM-HAADF images
(see Methods) of a Kr cluster changing size over time. The labels correspond
to the frame numbers within the image sequence and the corresponding time. The
second images series was recorded 8 min 32 s after the first frame of the
first sequence.  Figure S6: Snapshots of a room temperature molecular dynamics
simulation showing configuration changes for a Kr7 cluster.  Figure S7:
Lowest-energy configurations of simulated cluster shapes in 2D and 3D.  Figure
S8: Energy difference between 2D and 3D clusters. The lowest-energy 3D
structures were obtained by removing the graphene encapsulation and relaxing
the clusters again after annealing.  Figure S9: Filtered STEM-MAADF images
(see Methods) showing (a) a step edge between a double layer and a single
layer and (b) Xe clusters with $N\approx 100$ implanted in a double layer. In
panel (a) white rectangles mark some positions where Si heteroatoms have been
trapped at defects in both areas, whereas black rectangles show locations of
noble gas clusters, that are exclusively in the double layer area. In panel
(b), black rectangles show solid Xe clusters and the white ones mark clusters
that have a liquid-like structure. The larger solid cluster has $N=93$ atoms,
whereas the smallest liquid-like structure has only a few atoms more. Both
images have been recorded on the same sample prepared from CVD graphene,
which–in contrast to all other samples–was implanted with Xe clusters with a
plasma source at the University of Vienna, as described in Methods. Both scale
bars are 5 nm. Figure S10: Filtered STEM-MAADF images (see Methods) of a Xe6
cluster jumping between different positions, and finally merging with another
cluster. The labels correspond to the frame numbers within the image sequence
and the corresponding time. The black squares on frames $0$ and $11$ mark the
area shown in frames $4-9$ and $20-26$.