Measuring the Stellar and Planetary Parameters of the 51 Eridani System A.
Elliott et al.

110 2023 10.1017/pasa.xxxx.xx

A. Elliott, Email<EMAIL_ADDRESS>
(Received xx xx xxxx; revised xx xx xxxx; accepted xx xx xxxx)

# Measuring the Stellar and Planetary Parameters of the 51 Eridani System

Ashley Elliott1 Tabetha Boyajian1 Tyler Ellis1 Kaspar von Braun2 Andrew W.
Mann3 and Gail Schaefer4 1Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA 70802, USA
2 Lowell Observatory, 1400 West Mars Hill Rd. Flagstaff, AZ, 86001, USA
3 Department of Physics and Astronomy, The University of North Carolina at
Chapel Hill, Chapel Hill, NC 27599, USA
4 The CHARA Array of Georgia State University, Mount Wilson Observatory, Mount
Wilson, CA 91023, USA

###### Abstract

In order to study exoplanets, a comprehensive characterization of the
fundamental properties of the host stars, such as angular diameter,
temperature, luminosity, and age, is essential, as the formation and evolution
of exoplanets are directly influenced by the host stars at various points in
time. In this paper, we present interferometric observations taken of directly
imaged planet host 51 Eridani at the CHARA Array. We measure the limb-darkened
angular diameter of 51 Eridani to be $\theta_{\rm LD}=0.450\pm 0.004$ mas and
combining with the Gaia zero-point corrected parallax, we get a stellar radius
of $1.45\pm 0.01$ R⊙. We use the PARSEC isochrones to estimate an age of
$23.2^{+1.7}_{-1.6}$ Myr and a mass of $1.550\pm 0.005$ M⊙. The age and mass
agree well with values in the literature, determined through a variety of
methods ranging from dynamical age trace-backs to lithium depletion boundary
methods. We derive a mass of $4.1^{+0.5}_{-0.4}$ MJup for 51 Eri b using the
Sonora Bobcat models, which further supports the possibility of 51 Eri b
forming under either the hot-start formation model or the warm-start formation
model.

###### keywords:

exoplanets, interferometry, fundamental properties, young stars

##  1 Introduction

Young, nearby stars are of major interest to scientists as they can provide
insight into stellar evolution, as well as planetary formation. The exoplanet
host, 51 Eridani has been an object of interest in the astronomical community
for a number of years. 51 Eridani is a F0 spectral type, $V=5.22$ mag star and
has a binary pair companion, GJ 3305A,B (both M0 dwarfs). 51 Eridani and GJ
3305A,B have a separation of 66′′ or roughly 2000 au, and the system
components are co-moving together in a hierarchical relationship (Feigelson et
al.,, 2006). The exoplanet, 51 Eri b, has a semi-major axis of $12^{+4}_{-2}$
au, an inclination of $133^{+14}_{-7}$ deg, an eccentricity of
$0.45^{+0.10}_{-0.15}$, and an orbital period of $32^{+17}_{-9}$ years, all
determined by fits of SPHERE and GPI data by Maire et al., (2019).

51 Eridani is a member of the $\beta$ Pictoris moving group ($\beta PMG$),
which is one of the youngest and closest moving groups to Earth. Zuckerman et
al., (2001) determined that the $\beta$ Pictoris Moving Group consisted of 17
star systems. Since then, there are now a few hundred candidate systems
belonging to the $\beta PMG$ (Miret-Roig et al.,, 2020). The age estimates for
the $\beta PMG$ include $23\pm 3$ Myr (Mamajek and Bell,, 2014) using the
lithium depletion boundary method and isochronal ages for FGMK stars, which is
consistent with the age estimate of 51 Eridani, $20\pm 6$ Myr (Macintosh et
al.,, 2015) using the group’s strong lithium depletion boundary age. A more
recent dynamical age estimate developed by Miret-Roig et al., (2020) presents
an age of $18.5^{+2.0}_{-2.4}$ Myr. This estimate is a dynamical trace-back
age that reconciles other trace-back estimates with other methods such as
lithium depletion, isochronal ages, and other dynamical estimates.

Macintosh et al., (2015) presented the discovery of the directly imaged planet
51 Eri b by the Gemini Planet Imager (GPI). Follow-up work by GPI and NIRC2 at
the W. M. Keck 2 telescope (Wizinowich et al.,, 2000) have confirmed and
refined the orbit and planetary properties of the system. The most current
study of this system by Dupuy et al., (2022) determined an upper limit for the
planet mass of $\geq 10.9$ MJup within a $2\sigma$ interval and an orbital
separation of $10.4_{-1.1}^{+0.8}$ au derived using joint fitting of the
Hipparcos-Gaia Catalog of Accelerations (HGCA-EDR3) proper motions and
relative astrometry. Samland et al., (2017) determined a planetary effective
temperature of $760\pm 20$ K, and a radius of $1.1^{+0.2}_{-0.1}$ RJup from
fitting spectro-photometry from VLT/SPHERE. Brown-Sevilla et al., (2023)
recently published updated parameters on 51 Eri b using observations from
VLT/SPHERE. A radiative transfer model fit using petitRADTRANS resulted in a
planetary effective temperature of $807\pm 45$ K along with a radius of
$0.93\pm 0.04$ RJup, and a mass of $3.9\pm 0.4$ MJup. For the mass of the
planet, Brown-Sevilla et al., (2023) reported three possible masses: $3.9\pm
0.4$ MJup as the nominal result, $2.4$ MJup as a mass using evolutionary
models by Baraffe et al., (2002) with an age of 10 Myr, and $2.6$ MJup with an
age of 20 Myr. These ages were taken from age estimates published by Lee et
al., (2022) and Macintosh et al., (2015), respectively.

Given the young age of the system (both star and planet) determined by several
literature sources, 51 Eri b is a perfect candidate to study a planet that is
both young and still being influenced by its initial conditions of formation
(Macintosh et al.,, 2015). There are two main planet formation scenarios that
most planets discovered fit into: the cold-start model and the hot-start
model. The cold-start model is described by core accretion and usually results
in lower entropy and a smaller radius of the planet. The hot-start model is
described by disk instability which results in a planet with a higher entropy,
a higher effective temperature, and a larger radius (Spiegel and Burrows,
2012a, ). A third formation scenario, cleverly named the warm-start model,
involves a combination of the hot and cold start formation criteria (Spiegel
and Burrows, 2012b, ). The warm-start model is a spectrum of initial
conditions motivated by the observed preference of core accretion with gas
giants forming closer to their stars and the recent observations of young
exoplanets that are hotter than the cold-start model predictions but colder
than the hot-start model predictions (Spiegel and Burrows, 2012b, ). A
planet’s luminosity can provide insights into its formation because it is a
function of age, mass, and initial conditions (Marley et al.,, 2007; Spiegel
and Burrows, 2012a, ). Macintosh et al., (2015)’s initial observations and
study of 51 Eri b noted that the core-accretion theory explains the formation
of this planet due to the derived low luminosity range (log($L/\rm
L_{\odot})=-5.4~{}\rm to-5.8$). Samland et al., (2017) explored all three
scenarios, ruling out the cold-start models due to the new luminosity ranges
(log($L/\rm L_{\odot})=-5.4~{}\rm to-5.5$) derived and found that the planet
mass favored the hot- or warm- start models. Dupuy et al., (2022) also ruled
out the cold-start formation with their derivation of a lower limit on the
initial specific entropy. Further study of this planet and its star can
continue to explain its formation.

Characterizing exoplanet host stars is key to understanding the exoplanet
itself. The directly measured fundamental properties of a star, i.e. radius,
effective temperature, and luminosity, lead to more precise characterization
of the exoplanet. The habitable zone around a planet is heavily influenced by
its host star, so in order to constrain the habitable zone boundaries and
planets’ effective temperature, one must know the host star’s fundamental
properties (von Braun and Boyajian, 2017a, ). The age of a star can provide
information about the formation of the exoplanet. There are several indirect
methods to determine stellar fundamental properties, such as atmospheric
modeling and stellar evolutionary models. These models have been shown to have
difficulties reproducing observations (Boyajian et al.,, 2012, 2013). Long
baseline optical/infrared interferometry provides high angular resolution
measurements of stars, allowing astronomers to calculate fundamental
properties without relying on models.

Interferometric observations allow us to measure the stellar angular diameter,
which when combined with the parallax, gives a direct measurement of the
stellar radii, one of the fundamental parameters of an astronomical object.
Optical interferometry achieves a higher resolution (on the order of milli-
arcseconds) than most large telescopes by combining light from several pairs
of telescopes that are separated across a variety of baselines, or the
separation between telescopes. The resolution capability of an interferometer
increases the precision of an angular diameter measurement which then lowers
the uncertainty of stellar parameters that can be derived from the angular
diameter, i.e. stellar effective temperature and linear radius.

Similar to 51 Eridani and its planet, several other systems of stars with
directly imaged exoplanets have been characterized with the CHARA Array. One
such system is the $\kappa$ Andromedae and its exoplanet, $\kappa$ And b by
Jones et al., (2016). Jones et al., (2016) took into account the oblate nature
and gravity darkening caused by $\kappa$ And’s rapid rotation through the use
of modeling interferometric observations. The model results were used to
determine fundamental stellar parameters, such as temperatures at the poles
and equator, surface gravities, luminosity, stellar age and mass, and
planetary age and mass.

Baines et al., (2012) used high resolution interferometric observations to
study HR 8799, which hosts 4 directly imaged companions (Marois et al.,,
2008). The classification of the 4 companions, which is highly dependent on
the age of the planet (inferred from the age of the star), is the subject of
debate amongst astronomers (see Table 1 in Baines et al., 2012). Baines et
al., (2012) combined the angular diameter, parallax, and photometry to
determine stellar parameters, such as linear radius, stellar effective
temperature, and luminosity. The effective temperature calculated revealed two
possible age scenarios of HR 8799: either the star is contracting onto the
zero-age main sequence (ZAMS) or expanding away from it. Baines et al., (2012)
used the Yonsei-Yale evolutionary models to investigate both possibilities.
The resulting young ages (less than 0.1 Gyr) from either scenario highly
favored the classification of planet for the 4 companions, not brown dwarf.
Additional recent works that use interferometric radii to determine planet
properties are as follows but not limited to: Caballero et al., (2022), Ellis
et al., (2021), and Ligi et al., (2019) with more references tabulated in the
von Braun and Boyajian, 2017b  compilation. Other works demonstrating the wide
utility of interferometric data are Korolik et al., (2023) and Roettenbacher
et al., (2022), which demonstrate analyzing stellar activity and co-alignment
of systems and Ibrahim et al., (2023) which studies the inner au disk of
Herbig Be star HD 190073.

In this paper, we present new stellar and planetary properties of the 51
Eridani system. This paper is organized as follows: the interferometric
observations in Section 2, the directly determined stellar properties in
Section 3, the derived stellar and planetary properties in Section 4, and a
discussion of the results presented in this paper in comparison with the
results published in the literature in Section 5.

##  2 Interferometric Observations

The Center for High Angular Resolution Astronomy (CHARA) is a long baseline,
optical/IR interferometer located at Mt. Wilson, CA. The CHARA Array (ten
Brummelaar et al.,, 2005) comprises of six 1-meter telescopes arranged in a
Y-shape configuration, with three arms: East (E), West (W), and South (S),
each having two telescopes labeled 1 (outermost) and 2 (innermost). Baselines
for the interferometer are formed by pairs of telescopes, such as E1/W1
(outermost East and West telescopes). We observed 51 Eridani in 2015 and 2016
using the Precision Astronomical Visible Observations (PAVO) beam combiner
(Ireland et al.,, 2008), which measures interference fringes over a 630-950 nm
dispersed bandwidth with 2 telescopes (as indicated in Table 1). We observed
51 Eridani in 2021 using the CLASSIC (Ten Brummelaar et al.,, 2013) beam
combiner, which operates in the near-infrared $H$-band. A log of observations
is listed in Table 1.

An observation sequence consists of the target star, or science star, and a
selection of calibrator stars. Observations of calibrator stars allow for the
removal of atmospheric and instrumentation noise from the observations of the
science star. Data were taken in the following pattern: calibrator star -
science star - calibrator star. This pattern is referred to as a bracket. The
observations taken adhere to a minimum 2 night, 2 calibrator, 2 baseline
requirement in order to reduce and/or eliminate unknown systematic errors in
the visibility data.

We use the Jean-Marie Mariotti Center Stellar Diameter Catalog (JSDC) (Chelli
et al., (2016), Bourgés et al., (2014))111Available at
https://www.jmmc.fr/english/tools/data-bases/jsdc-72/ to find suitable
calibrator stars for our science target. The calibrator stars are chosen such
that they are unresolved, nearby to the science star on the sky
($<10^{\circ}$), and have no known companions or rapid rotation. A summary of
observations can be found in Table 1 and a list of calibrator stars used can
be found in Table 2.

We reduce and calibrate data for each night of PAVO observations using the
PAVO software available through CHARA’s Remote Data Reduction Machine (Ireland
et al.,, 2008). In the calibration process, we found that that the calibrator
star, HD 29335, showed spurious results indicating that it was a bad
calibrator. The amount of usable observations diminished after the removal of
this calibrator from the observing sequence (seen in the parentheses in the
column “Brackets” in Table 1).

We reduce and calibrate data for each night of the CLASSIC observations using
the CLASSIC/CLIMB reduction software redfluor, accessed through CHARA’s Remote
Data Reduction Machine (ten Brummelaar, 2014a, , ten Brummelaar, 2014b, ). For
the first two nights of observations, only one observation of the science star
and one observation of a calibrator star was made. We found that the
calibrator was more resolved than the science star. Due to this trend, we
suspect that this calibrator (HD 26912) is a bad calibrator and did not use in
our analysis.

Table 1: Summary of Observations: The date of observation in UT time is listed in the first column on the left. The wavelength of observation is shown in the second column. The baseline of observation is in the third column. The number of brackets is listed in the fourth column. The number in parentheses is the actual number of brackets used for data analysis due to the bad calibrators HD 29335 and HD 26912 (marked with a ∗ in the table). Finally, the calibrators used for each night are listed in the right-most column. Date [UT] | Wavelength | Baseline | Brackets | Calibrators
---|---|---|---|---
2015-10-11 | 0.630-0.950 $\mu$m | E1/W1 | 7 (5) | HD28375, HD27563, HD29335∗
2015-10-12 | 0.630-0.950 $\mu$m | S2/E2 | 9 (7) | HD28375, HD27563, HD29335∗
2015-11-06 | 0.630-0.950 $\mu$m | E2/W1 | 3 (1) | HD27563, HD29335∗
2016-11-09 | 0.630-0.950 $\mu$m | W2/E2 | 3 (1) | HD27563, HD29335∗
2021-08-25 | 1.6731 $\mu$m | S1/E1 | 1 (0) | HD26912∗
2021-08-26 | 1.6731 $\mu$m | S1/E1 | 1 (0) | HD26912∗
2021-08-27 | 1.6731 $\mu$m | S1/E1 | 4 (3) | HD29248, HD28736
Table 2: Summary of calibrator stars: The $R$ magnitudes and $\theta_{\rm LD}$ are taken from the JMMC Stellar Diameters Catalog v2 (Chelli et al.,, 2016). HD 29335 and HD 26912 are bad calibrators (marked with a ∗). Calibrator |  $R$ mag |  $\theta_{\rm LD}$ [mas]
---|---|---
HD 28375 | 5.531 | $0.179\pm 0.006$
HD 27563 | 5.891 | $0.170\pm 0.006$
HD 29335∗ | 5.345 | $0.225\pm 0.006$
HD 26912∗ | 4.250 | $0.287\pm 0.026$
HD 29248 | 8.687 | $0.111\pm 0.003$
HD 28736 | 5.979 | $0.348\pm 0.008$

##  3 Directly Determined Stellar Properties

An interferometer produces interference fringes that allow us to measure two
basic pieces of information: the amplitude of the fringes and the phase shift
of the peaks. The amplitudes allow us to measure visibilities, which describe
the fringe contrast of the interference pattern. The visibility of a star can
tell us about the star’s size, shape, and any surface features. Visibilities
are measured between 0 and 1, with a visibility of 1 indicating an unresolved
star and a visibility of 0 indicating a completely resolved star.

When observing a star, it is important to remember that a star’s brightness is
not uniform across the star. Optical depth and effective temperature gradients
across the disk of a star combine to create an effect called limb darkening.
To an observer, a star is brighter in the center and then as you move out from
the center, the brightness goes down until it is 0 at the apparent edge of the
star. A star’s uniform disk diameter does not take into account the limb
darkening effect while the limb darkened diameter does.

In order to determine the angular diameter of a star, the visibility squared
as a function of baseline ($B$), angular diameter (either uniform,
$\theta_{\rm UD}$, or limb darkened, $\theta_{\rm LD}$), and wavelength
($\lambda$) is used to fit the interferometric visibilities using Equation 1.

$V^{2}=\Biggl{[}\left(\frac{1-\mu}{2}+\frac{\mu}{3}\right)^{-1}\cdot\left((1-\mu)\cdot\frac{J_{1}(x)}{x}\right)\\\
+\left(\mu\sqrt{\frac{\pi}{2}}\cdot\frac{J_{3/2}}{x^{3/2}}\right)\Biggr{]}^{2}$
(1)

where $x=\frac{\pi B\theta}{\lambda}$ and $\mu$ is the limb-darkening
coefficient (LDC). The first iteration of this fit is applied using a $\mu$
value of 0, which corresponds to a uniform disk diameter, $\theta_{\rm UD}$.

To estimate the bolometric flux ($F_{\rm bol}$) of 51 Eridani , we use the
spectral-energy distribution using available photometry from Gaia (Gaia
Collaboration,, 2020; Lindegren et al.,, 2021), the Two-Micron All-Sky Survey
(2MASS; Skrutskie et al.,, 2006), Tycho-2 (Høg et al.,, 2000), the Wide-field
Infrared Survey Explorer (WISE; Cutri and et al.,, 2014), and the General
Catalog of Photometric Data (GCPD; Mermilliod et al.,, 1997). We combine the
spectrum of 51 Eridani from the STIS Next Generation Spectral Library (NGSL;
Heap and Lindler,, 2007) (covering the near UV and optical) with one from Cool
Stars library (Rayner et al.,, 2009) (covering the near-infrared), filling any
additional gaps with a stellar atmosphere (Allard et al.,, 2011). We scaled
the combined spectrum to match the photometry, yielding an absolutely
calibrated spectrum. The final fit is shown in Figure 1 and our bolometric
flux measurement is in Table 3. More details on this fitting procedure,
including our treatment of systematic uncertainties in the photometry and
spectra are given in Mann et al., (2015) and Mann et al., (2016).

Figure 1: Spectral energy distribution and spectrum of 51 Eridani. The black
line is the NGSL and SpeX spectrum, with grey regions indicating a model.
Photometry is shown in red, with horizontal error bars indicating the width of
the filter, while vertical show the photometric uncertainty. Synthetic
photometry from the spectrum is shown in green. The bottom panel shows the
difference between the spectrum and photometry in units of standard
deviations.

From the angular diameter and $F_{\rm bol}$, the stellar effective
temperature, $T_{\rm eff}$, is calculated using the following form of the
Stefan-Boltzmann equation:

$T_{\rm eff}=2341\cdot\Biggr{(}\frac{F_{\rm bol}}{\theta^{2}}\Biggl{)}^{1/4}K$
(2)

In Equation 2, the bolometric flux is in units of
$10^{-8}\rm~{}erg~{}s^{-1}cm^{-2}$ and $\theta$ is in units of mas. In the
first initial calculation of $T_{\rm eff}$, $\theta$ refers to the uniform
disk angular diameter. In order to account for limb darkening, we use the Limb
Darkening Coefficients table (Claret and Bloemen,, 2011) to solve for a linear
$R$-band LDC with initial guesses of the star’s $T_{\rm eff}$, $\log g$, and
[Fe/H]. This value is used in a subsequent $\theta_{\rm LD}$ fit (Equation 1)
and a new temperature is derived using Equation 2 where $\theta$ is now
referring to the limb-darkened angular diameter, $\theta_{\rm LD}$. We iterate
on this process twice when no changes are seen. We determine the error for
$\theta_{\rm LD}$ sampling the angular diameter in a Monte Carlo simulation,
varying on a normal distribution of the uncertainties for wavelength ($\pm 5$
nm), calibrator diameters ($5\%$ of the diameter), and limb-darkening
coefficients ($\pm 0.02$).

A stellar luminosity, $L_{\star}$ is calculated using Equation 3

$L_{\star}=4\pi D^{2}F_{\rm bol}$ (3)

where $D$ is the distance (seen in Table 3) and $F_{\rm bol}$ is the
bolometric flux (seen also in Table 3). The distance, $D$, is calculated using
the zero-point corrected parallax, $33.44\pm 0.08$ mas (Gaia Collaboration,
(2020), Lindegren et al., (2021)). This yields a distance of $D=29.93\pm 0.07$
pc. The linear radius is then calculated using this new distance and Equation
4 below:

$R_{\star}=\frac{\theta_{\rm LD}~{}D}{2}$ (4)

where $\theta_{\rm LD}$ is the limb-darkened angular diameter and $D$ is the
zero-point corrected distance. The final parameters are presented in Table 3.
We calculate the errors for $T_{\rm eff}$, $L_{\star}$, $D$, and $R_{\star}$
using standard error propagation.

Table 3: Stellar and planetary parameters for 51 Eridani and 51 Eri b. Each property is listed in the left column, the corresponding value in the center column, and the source for each property in the right column. We use $\rm R_{\odot}=6.957\times 10^{8}$ m and $\rm L_{\odot}=3.846\times 10^{33}\rm~{}erg~{}s^{-1}$. Property | Value | Source
---|---|---
$\theta_{\rm LD}~{}[\rm mas]$ | $0.450\pm 0.004$ | Section 3
$F_{\rm bol}~{}[\rm 10^{-8}~{}erg~{}s^{-1}~{}cm^{-2}]$ | $20.5\pm 0.3$ | Section 3
$\rm Parallax~{}[\rm mas]$ | $33.44\pm 0.08$ |  Gaia Collaboration, (2020),
|  | Lindegren et al., (2021)
$\rm Distance~{}[pc]$ | $29.93\pm 0.07$ |  Gaia Collaboration, (2020),
|  | Lindegren et al., (2021)
$\rm[Fe/H]~{}[dex]$ | $0.13_{-0.02}^{+0.03}$ | Swastik et al., (2021)
log $g$ $[\rm cm~{}s^{-2}]$ | $(4.07-4.10)\pm 0.21$ | Arentsen et al., (2019)
$T_{\rm eff}~{}[\rm K]$ | $7424\pm 45$ | Section 3
$L_{\star}~{}[\rm L_{\odot}]$ | $5.72^{+0.09}_{-0.10}$ | Section 3
$R_{\star}~{}[\rm R_{\odot}]$ | $1.45\pm 0.01$ | Section 3
$\rm Age~{}[Myr]$ | $23.2^{+1.7}_{-1.6}$ | Section 4.1
$\rm Mass~{}[\rm M_{\odot}]$ | $1.550\pm 0.005$ | Section 4.1
$\rm Planet~{}Mass~{}[M_{\rm Jup}]$ | $4.1^{+0.5}_{-0.4}$ | Section 4.1
Figure 2: The top plot shows the calibrated squared visibilities and
uncertainties from PAVO (in grey circles) and CLASSIC (in green squares). The
PAVO data are binned in equal spacing and the weighted average is shown (in
black circles). The dashed blue line is the visibility model (Eq. 1) with the
limb darkened coefficient of $\mu=0.4516\pm 0.0200$. The bottom plot displays
the residuals from PAVO (unbinned in grey, binned in black) and CLASSIC (in
green).The limb-darkened angular diameter is $0.450\pm 0.004$ mas.

##  4 Modeled Stellar and Planetary Properties

Using our measured effective temperature, luminosity, and radius, we estimate
the age and mass of the star using stellar evolutionary models. Improved age
estimates for the star means the age of the planet, 51 Eri b, is also improved
given that the system is coeval. We then estimate the planet’s mass through
evolutionary modeling and provide further insight on how the planet formed.

51 Eridani falls in a unique spot in the HR diagram. Previous works
(summarized in the Introduction of Mamajek and Bell, 2014) indicated that the
$\beta PMG$ consisted of pre-main sequence (PMS) stars. Mamajek and Bell,
(2014) analyzed the kinematics of the $\beta PMG$ and concluded that the
majority of the A0-F0 stars are either near or on the zero-age main sequence
(ZAMS), which includes 51 Eridani.

###  4.1 Age and Mass Using Isochronal Modeling

To estimate the age and mass of 51 Eridani, we use the stellar evolution
models: the PAdova and TRieste Stellar Evolution Code (PARSEC) (Bressan et
al.,, 2012) and the Garching Stellar Evolution Code (GARSTEC) (Weiss and
Schlattl,, 2008).

We use the PARSEC version 1.2S model222Available at
http://stev.oapd.inaf.it/cgi-bin/cmd developed by Bressan et al., (2012) to
create stellar isochrones given stellar priors. We generated isochrones for a
range of ages between 0 and 50 Myr in steps of 0.5 Myrs in log space using a
metal fraction of $Z=0.0165$ (Table 3). We interpolated the outputted
isochrones to obtain a finer grid of points along each isochrone track. We
then performed two separate two-dimensional interpolations over $\log(L/$L⊙)
and $T_{\rm eff}$; one to extract an age and the other to extract a mass. We
use a Monte Carlo simulation to obtain the errors for the age and mass
estimates by sampling the $T_{\rm eff}$ and luminosity in a normal
distribution. The posterior distributions are shown in Figure 3. The final
solution results in an age of $23.2^{+1.7}_{-1.6}$ Myr and a mass of $1.550\pm
0.005$ M⊙.

Figure 3: Corner plot for the PARSEC model displaying the results of a Monte
Carlo simulation of 5000 iterations to estimate the age and mass of 51
Eridani. The dashed vertical lines indicate the quantiles for each histogram:
$16\%$, $50\%$, and $84\%$ from left to right. The left-most line corresponds
to the lower bound uncertainty. The middle line corresponds to the accepted
value. The right-most line corresponds to the upper-bound uncertainty. The
solutions are as follows: an age of $23.2^{+1.7}_{-1.6}$ $\rm Myr$ and a mass
of $1.550\pm 0.005$ $\rm M_{\odot}$.

We then used the GARSTEC models (Weiss and Schlattl,, 2008) through the
implementation of bagemass. bagemass (Maxted et al.,, 2015) is a program
written in Fortran that uses stellar evolution models to estimate the mass and
age of a star and produces posterior probability distributions calculated
using Bayesian methods. The GARSTEC model chosen has an $\alpha$ mixing length
of 1.78 and a He-enhancement value of 0. The priors used are $T_{\rm eff}$,
$\log(L/$L⊙), and [Fe/H] (Table 3). We fixed the metallicity and the age is
given a range between 0 and 50 Myr. The posterior distributions are shown in
Figure 4. The final solution determined gives an age of $40_{-4}^{+5}$ Myr and
a mass of $1.60_{-0.02}^{+0.01}$ M⊙.

Figure 4: The above plot displays the corner plot for the GARSTEC model run
through bagemass. The dashed vertical lines indicate the quantiles for each
histogram: $16\%$, $50\%$, and $84\%$ from left to right. The left-most line
corresponds to the lower bound uncertainty. The middle line corresponds to the
accepted value. The right-most line corresponds to the upper-bound
uncertainty. The solutions are as follows: an age of $40_{-4}^{+5}$ Myr and a
mass of $1.60_{-0.02}^{+0.01}$ M⊙.

The age and mass estimates determined by the PARSEC model and the GARSTEC
model are vastly different, with the GARSTEC model estimating an age almost
twice that of the PARSEC model. The GARSTEC model could not return the prior
distributions for $T_{\rm eff}$ and log($L/$L⊙) within the uncertainties or
stay within the range of [Fe/H] priors given. Looking at the posterior
distributions, we see that the GARSTEC model returns a bimodal solution and
skews the priors, reflecting the model not returning the priors sufficiently
well enough. This bimodal distribution could be due to the star’s location on
the HR diagram. Although both models incorporate the PMS phase and the ZAMS
phase of stellar evolution, the position of this star being right at the
transition point could cause each model to either favor the PMS side or the
early ZAMS side of this transition point. However, the PARSEC model is able to
return the priors well and produce a single mode solution. We adopt for the
final results of this work, the PARSEC age and mass of $23.2^{+1.7}_{-1.6}$
Myr and $1.550\pm 0.005$ M⊙.

###  4.2 Analysis of the Planetary Companion 51 Eri b

The next step in the analysis of 51 Eri b is to estimate a mass using the
assumption that the planet is the same age as the star (see Section 4.1). We
use the Sonora Bobcat models (Marley et al.,, 2021) which were developed to
study L-, Y-, and T- type brown dwarfs and self-luminous exoplanets. 51 Eri b
is considered to be a self-luminous planet but it has been debated on whether
51 Eri b is an L- or T- type brown dwarf given its unique strong methane
absorption features seen in T- type brown dwarfs and similar near-IR colors to
L-type brown dwarfs (Macintosh et al.,, 2015).

The Sonora Bobcat models produce evolutionary tables that hold either age,
mass, or bolometric luminosity fixed. For this paper’s purpose, we use the
table holding mass fixed. In our analysis, we use the planet’s effective
temperature of $T_{\rm eff}=807\pm 45$ K (Brown-Sevilla et al.,, 2023). Figure
5 shows the Sonora Bobcat models as age vs $T_{\rm eff}$ along with the
position of 51 Eri b. To determine a mass, we first perform a one-dimensional
interpolation to create a finer grid of points for each of the iso-mass lines.
We then perform an additional two-dimensional interpolation over the effective
temperature and age and find the corresponding mass. To obtain uncertainties
for the mass, we ran a Monte Carlo simulation, sampling the age and $T_{\rm
eff}$ on a normal distribution. For the age, an asymmetrical normal
distribution was created to account for the uneven uncertainties. Figure 6
shows the posterior distribution for this simulation. The final mass of 51 Eri
b is $4.1^{+0.5}_{-0.4}$ MJup.

Figure 5: The planet iso-mass models from Marley et al., (2021). Iso-mass
models were created using the The Sonora-Bobcat model evolutionary tables that
hold mass fixed. A one-dimensional interpolation was done to create a finer
grid of points for each line. Then, a two-dimensional interpolation was done
over $T_{\rm eff}$ and age to determine a planetary mass. For results, see
Figure 6 and Table 3. Figure 6: Corner plot for the Sonora-Bobcat planet iso-
mass model displaying the restuls of a Monte Carlo simulation of 5000
iterations to estimate the mass of 51 Eri b. The dashed vertical lines
indicate the quantiles for each histogram: $16\%$, $50\%$, and $84\%$ from
left to right. The left-most line corresponds to the lower bound uncertainty.
The middle line corresponds to the accepted value. The right-most line
corresponds to the upper-bound uncertainty. The final results of this
simulation give a mass of $4.1^{+0.5}_{-0.4}$ MJup. See Section 4.2 for
details.

##  5 Discussion

###  5.1 Stellar Parameters

We measure a limb-darkened angular diameter for 51 Eridani of $\theta_{\rm
LD}=0.450\pm 0.004$ mas (Section 4). We also measure an angular diameter of
$\theta_{LD}=0.425\pm 0.026$ mas using a limb-darkening coefficient of
$\mu=0.2582$ from our CLASSIC data alone, which is consistent with the PAVO
diameter within $1\sigma$. The PAVO measurement has a precision of less than
$1\%$, compared to the CLASSIC measurement which has a precision of just over
$6\%$. This is due to the greater spatial frequency coverage obtained with
PAVO’s multiple wavelengths channels, as well as sampling further down the
visibility curve to higher spatial frequencies. More coverage, especially at
higher spatial frequencies, ensures that the visibility squared fit is more
accurate. Simon and Schaefer, (2011) measure an angular diameter of 51 Eridani
also using CHARA’s CLASSIC beam combiner. Their measurements were taken in
both $H$\- and $K^{{}^{\prime}}$-band on the E1/W1 baseline (314 m). Simon and
Schaefer, (2011)’s angular diameter of $\theta_{\rm LD}=0.518\pm 0.009$ mas is
over $7\sigma$ off from our PAVO angular diameter. We suspect that this
disagreement is due to the observation strategy used by Simon and Schaefer,
(2011) who observed on a single baseline (E1/W1) and only used one calibrator
in their analysis. This observation strategy is dangerously insensitive to
verifying whether or not the single calibrator they used is good. In contrast,
our measurements were taken using multiple calibrators over multiple nights
(see Table 1). The angular diameter of 51 Eridani is near the resolution limit
($\frac{\lambda}{2B}$) of the CHARA Array on the longest baselines in the
near-infrared. Analysis in the literature shows that interferometric
measurements of under-resolved stars tend to systematically over-estimate
their angular diameters (Casagrande et al., (2014), White et al., (2018),
Tayar et al., (2022)). However, our observations using CLASSIC data alone do
not support this trend as they are in agreement within $1\sigma$ of the PAVO
observations.

Simon and Schaefer, (2011) also performed an analysis to estimate the age and
mass of 51 Eridani. This analysis used the pre-main sequence (PMS) evolution
models developed by Siess et al., (2000) and the Yonsei-Yale (Y2) models (Yi
et al.,, 2003). Simon and Schaefer, (2011) compared their absolute angular
diameter (angular diameter scaled to a distance at 10 pc) with those predicted
by both models and found an average age of $13\pm 2$ Myr and a mass of
$1.75\pm 0.05$ M⊙. Our age is almost twice that of Simon and Schaefer, (2011)
and our mass is $4\sigma$ off of Simon and Schaefer, (2011)’s. If we scale our
angular measurement to a distance of 10 pc, it would fall closer to the older
isochrones in Figure 3 in Simon and Schaefer, (2011) for the same $V-K$,
suggesting that the difference in angular diameter seems to be the main source
of discrepancy. Additional errors are subject to arise in transforming the
theoretical properties calculated by the models to observational quantities,
such as magnitudes and color indexes from color tables. Comparing our directly
measured $T_{\rm eff}$ and luminosity to the model predictions ensures that
any ambiguous errors that result from the color tables will not affect our
results.

Simon and Schaefer, (2011) also used a $V-K$ vs. $M_{K}$ diagram to estimate
age and mass of 51 Eridani. The mass results were around $0.2$ M⊙smaller than
their initial analysis of comparing the model-predicted angular diameters and
yielded older ages (around 5 Myr), creating a noted discrepancy between the
two methods for age and mass. However, the $V-K$ vs. $M_{K}$ result estimates
align with our results.

The age we determine with the PARSEC model ($23.2^{+1.7}_{-1.6}$ Myr) agrees
with the age estimates found in the literature of both the star itself and
that of the $\beta PMG$. These age estimates have been determined through a
variety of methods from isochronal ages to lithium depletion methods. Mamajek
and Bell, (2014) conclude with an isochronal age of $22\pm 3$ Myr, where their
value compares favorably to several lithium depletion boundary ages from Binks
and Jeffries, (2014) ($21\pm 4$ Myr) and Malo et al., (2014) ($26\pm 3$ Myr).

A more recent study on the $\beta PMG$ from Miret-Roig et al., (2020)
concludes with an age estimate of $18.5_{-2.4}^{+2.0}$ Myr through dynamical
age trace-back methods using precise Gaia DR2 astrometry and ground based
radial velocities. Couture et al., (2023) developed a method to correct the
trace-back age method that reduces the systemic errors from a combination of
uncorrected gravitational red-shift and convective blue-shift absolute radial
velocity measurements and the random errors from the parallax, proper motion,
and additional radial velocity measurements. Their final resulting age is
$20.4\pm 2.5$ Myr for the $\beta PMG$. Our age estimate compares favorably to
Couture et al., (2023)’s dynamical age estimates of the $\beta PMG$.

###  5.2 A quick look at available TESS data

The Transiting Exoplanet Survey Satellite (TESS: Ricker et al., (2015))
observed 51 Eridani in two sectors: Sector 5 and Sector 32. Sepulveda et al.,
(2022) investigated the variability seen in the TESS light curve data and
concluded that 51 Eridani is a $\gamma$-Dor pulsator. We downloaded the
available 2 minute cadence data using the Lightkurve Collaboration et al.,
(2018). The data were flattened and binned. From these data, we constructed a
periodogram of the processed TESS data to search for significant exoplanet
signals. We extracted significant frequencies and tested to see if a possible
planet signal could be found. There were no significant possible planet
signals. The light curves did show significant variability which does
corroborate the findings of Sepulveda et al., (2022) in 51 Eridani being a
$\gamma$-Dor pulsator.

###  5.3 Planetary Parameters

We derive a mass for 51 Eri b to be $4.1^{+0.5}_{-0.4}$ MJup through the
Sonora Bobcat evolutionary models (Section 4.2). Other studies have estimated
the mass of 51 Eri b through a variety of methods, such as atmospheric
modeling and evolutionary modeling. Several works present exploration of both
methods, i.e. using atmospheric modeling to obtain planetary parameters such
as effective temperature, surface gravity, luminosity, and radius, and using
these parameters along with usually an adopted age in evolutionary models to
estimate a mass.

The initial discovery announcement of 51 Eri b (Macintosh et al.,, 2015)
performed atmospheric modeling to estimate the planet parameters, such as
effective temperature and luminosity. Macintosh et al., (2015) used two
separate models, a cloud-free and a partial cloudy model to obtain these
parameters. The cloud-free model resulted in a $T_{\rm eff}=750$ K and a
log($L/\rm L_{\odot})=-5.8$. The partial cloudy model gave a $T_{\rm eff}=700$
K and a log($L/\rm L_{\odot})=-5.6$. Macintosh et al., (2015) determined that
the luminosity derived from the models does not change much with either model
and used the luminosity along with age to estimate a mass based off of
formation models. Given hot-start formation, the mass of 51 Eri b is $\sim 2$
MJup. With a cold-start formation, the mass falls in a range of 2-12 MJup. Our
mass of $4.1$ MJup is in the middle of the two estimates.

Samland et al., (2017) presented the first spectro-photometric measurements in
the $Y$\- and $K$\- bands of 51 Eri b using VLT/SPHERE. This study used
atmospheric modeling using petitCODE - Cloudy to obtain planetary parameters.
The resulting temperature, $T_{\rm eff}=760\pm 20$ K, radius of
$1.11^{+0.16}_{-0.14}$ RJup, and surface gravity of $4.26\pm 0.25$ (cgs-units)
are used in a variety of methods to obtain mass estimates. Samland et al.,
(2017) used surface gravity and radius relations to get a mass of
$9.1^{+4.9}_{-3.3}$ MJup which is outside the uncertainty range of our mass.
Samland et al., (2017) also used radius and effective temperature relations to
derive a luminosity and a planet formation model which relates luminosity and
mass. The range in luminosity rules out the cold-start formation model and
resulted in masses between 2.4 and 5 MJup for the hot-start model and a larger
spread of masses between 2 and 12 MJup for the warm-start model. Our mass also
agrees with both mass ranges for the hot-start and warm-start models.

Dupuy et al., (2022) performed a cross calibration of Hipparcos-GAIA
astrometry and the orbit fitting code orvara. Their results provided orbital
parameters as well as an upper limit on the mass of 51 Eri b of $<11$ MJup.
Through private communication, another run of the orvara fit was made using
the new age and mass of 51 Eridani presented in this paper. The new upper
limit $2\sigma$ constraint on the mass of 51 Eri b is 9.5 MJup. Our mass of
$4.1$ MJup clearly falls under this upper limit. Dupuy et al., (2022)’s result
also indicated that the cold-start formation is ruled out given the derived
initial entropy. They suggested that 51 Eri b formed similarly to other
directly imaged planets which indicate either the warm or hot-start scenarios.

The most recent study of 51 Eri b, done by Brown-Sevilla et al., (2023),
revisited the work by Samland et al., (2017) and obtained new observations of
51 Eri b with VLT/SPHERE at a higher S/N than Samland et al., (2017)’s
original data. Brown-Sevilla et al., (2023) used a new atmospheric retrieval
code, petitRADTRANS, to update the $T_{\rm eff}$ and the resulting posterior
distributions for the log$g$ and planet radius to obtain a mass of $3.9\pm
0.4$ MJup. Brown-Sevilla et al., (2023) labeled this result as the nominal
model. Brown-Sevilla et al., (2023) also performed an analysis using
evolutionary models from Baraffe et al., (2002) with two separate ages: 10 Myr
(Lee et al.,, 2022) and 20 Myr (Macintosh et al.,, 2015), where they found
masses of $2.4$ MJup and $2.6$ MJup respectively. The mass presented in this
work ($4.1_{-0.4}^{+0.5}$ MJup) agrees well with the nominal result obtained
by Brown-Sevilla et al., (2023).

Our mass falls within the ranges of both the hot-start and the warm-start
models as described by both Samland et al., (2017) and Brown-Sevilla et al.,
(2023). With our analysis, we can rule out a purely core accretion model as
the main formation mechanism. Either scenario is likely but future
observations and analyses are needed to further test each model and develop a
more conclusive argument.

##  6 Conclusion

In this work, we present interferometric observations of the directly imaged
exoplanet host star 51 Eridani taken with both the PAVO and CLASSIC beam
combiners at the CHARA Array. These observations resulted in a highly precise
angular diameter measurement of 51 Eridani, $\theta_{\rm LD}=0.450\pm 0.004$
mas. From this angular diameter, we calculate the effective temperature,
$T_{\rm eff}=7424\pm 45$ K, luminosity, $L_{\star}=5.7\pm 0.1$ L⊙, and linear
radius, $R_{\star}=1.45\pm 0.01$ R⊙. We use the PARSEC isochrones to derive an
age of $23.2^{+1.7}_{-1.6}$ Myr and mass of $1.550\pm 0.005$ M⊙ of 51 Eridani.
For 51 Eri b, we estimate a mass of is $4.1^{+0.5}_{-0.4}$ MJup using the
Sonora/Bobcat evolutionary models.

Future analysis is encouraged using the JWST data taken in 2022 with NIRCAM
and MIRI. Additional spectra can provide more constraints on planet
composition, effective temperature, and luminosity when used with the
atmospheric retrievals developed for JWST. Brown-Sevilla et al., (2023)
suggested the use of the Mid-Infrared ELT Imager and Spectrograph (Quanz et
al.,, 2015) which will increase the wavelength coverage up to 13$\mu m$.
Observations of other members of the $\beta PMG$ can help further the age
constraints on these objects. In Table A1 in Alonso-Floriano et al., (2015)’s
paper, they list 185 members and candidate members of the $\beta PMG$. At
least two F type stars are observable with PAVO at the CHARA Array and with
future instruments being commissioned, such as the SPICA instrument (Mourard
et al., (2017), Mourard et al., (2022)), the increased sensitivity will allow
for more targets in this group to be observed. In addition, there are at least
5 stars that are observable in the southern sky with GRAVITY at VLTI (GRAVITY
Collaboration et al.,, 2017).

## Acknowledgements

This work is based upon observations obtained with the Georgia State
University Center for High Angular Resolution Astronomy Array at Mount Wilson
Observatory. The CHARA Array is supported by the National Science Foundation
under Grant No. AST-1636624 and AST-2034336. Institutional support has been
provided from the GSU College of Arts and Sciences and the GSU Office of the
Vice President for Research and Economic Development.

CHARA telescope time was granted by NOIRLab through the Mid-Scale Innovations
Program (MSIP). MSIP is funded by NSF.

CHARA Array time was granted through the NOIRLab community-access program
(NOIRLab Prop. ID: 2021A-0141; PI: T. Boyajian).

CHARA Array time was granted through the NOIRLab community-access program
(NOIRLab Prop. ID: 2021A-0247; PI: T. Ellis).

We thank Dr. Gururaj A. Wagle for his assistance in understanding
interpolation and Henry Ngo for the extremely helpful suggestions in debugging
interpolation code. We also thank Nageeb Zaman and Dr. Jonas Klüter for their
assistance in developing an asymmetrical normal distribution for Monte Carlo
simulations used in this work.

This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI : 10.26093/cds/vizier). The original description of
the VizieR service was published in 2000, A&AS 143, 23 (Ochsenbein et al.,,
2000).

This research has made use of the Jean-Marie Mariotti Center JSDC catalogue
333Available at http://www.jmmc.fr/catalogue_jsdc.html.

This research has made use of the Jean-Marie Mariotti Center Aspro
444Available at http://www.jmmc.fr/aspro service.

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA’s Science Mission Directorate.

This research made use of Lightkurve, a Python package for Kepler and TESS
data analysis (Lightkurve Collaboration et al.,, 2018).

## Financial Support

A.E. and T.S.B. acknowledge support by the National Support Foundation under
Grant No. AST-2205914.

## Conflicts of Interests

None

## Data Availability

All interferometric data are available in the CHARA archive555Available at
https://www.chara.gsu.edu/observers/database. The TESS observations are
available in the Barbara A. Mikulski Archive for Space Telescopes (MAST)
archive.

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# Solid argon as a possible substrate for quasi-freestanding silicene

S. Sattar1 R. Hoffmann2 U. Schwingenschlögl1,
<EMAIL_ADDRESS>+966(0)544700080 1PSE Division, KAUST,
Thuwal 23955-6900, Kingdom of Saudi Arabia  2Department of Chemistry and
Chemical Biology, Cornell University, Baker Laboratory, Ithaca NY 14853-1301
(USA)

###### Abstract

We study the structural and electronic properties of silicene on solid Ar(111)
substrate using ab-initio calculations. We demonstrate that due to weak
interaction quasi-freestanding silicene is realized in this system. The small
binding energy of only $-32$ meV per Si atom also indicates the possibility to
separate silicene from the solid Ar(111) substrate. In addition, a band gap of
$11$ meV and a significant splitting of the energy levels due to spin-orbit
coupling are observed.

silicene, substrate, separation, argon

##  I Introduction

Silicene is based on a two-dimensional honeycomb Si lattice, similar to
graphene, attracting interest by the predicted linear dispersion near the
Dirac point and various potential applications in electronic devices. The
structure is not perfectly two-dimensional, but a finite buckling mixes some
$sp^{3}$ hybridization into the $sp^{2}$ states. The weaker Si-Si bonding, as
compared to the C-C bonding in graphene 2dbands , strongly complicates the
synthesis. Still, silicene has been grown on thin film ZrB2, resulting in
electronic properties that are different from the expectations for
freestanding samples zrb2_1 . Both the buckling and the electronic properties
can be modified by epitaxial strain, which hints at a strong interaction with
the substrate. Silicene on Ir(111) has been investigated experimentally and
theoretically in Ref. ir111_si .

It has been demonstrated that Si nanoribbons can be grown on Ag(110)
substratea si_ag1101 ; si_ag1102  and the electronic structure has been
investigated by angular resolved photoelectron spectroscopy si_ag1103 .
Epitaxial growth with a highly ordered honeycomb structure on Ag(111) has been
confirmed by scanning tunneling microscopy ag111_si1 . The Si nearest neighbor
distance of $1.9\pm 0.1$ Å, obtained by line analysis of the microscopy data,
also points to strong interaction with the substrate. A systematic study of Si
superstructures on Ag(111) has been performed in Ref. ag111_si3,  using low
energy electron diffraction, scanning tunneling microscopy, and ab-initio
calculations. Further results from scanning tunneling microscopy have been
reported in Refs. ag111_si4 ; ag111_si5 ; ag111_si6 , confirming that the
quasiparticles in silicene behave as massless Dirac fermions. However,
experiments indicate that the Dirac nature is perturbed by symmetry breaking
due to the substrate ag111_si7 . This argumentation is supported by ab-initio
results, which lack a Dirac dispersion for various stable and metastable
structures of silicene on Ag(111) ag111_si8 ; ag111_si9 .

Graphene has been separated from SiC(0001) substrate, on which the binding
energy experimentally amounts to 106 meV Ebg_SiC . As Si bonds are usually
weaker than C bonds, to separate silicene from a substrate probably a
significantly smaller energy will be necessary. Hexagonal boron nitride grap-
hbn  and SiC(0001) grap-sic1 ; grap-sic2  are known substrates for graphene
and therefore have been studied also for silicene by ab-initio calculations,
finding that the Dirac cone is preserved, though slightly doped in the case of
hydrogenated SiC(0001) si-hbn . For a superlattice of silicene and hexagonal
boron nitride a binding energy of 57 meV per Si atom has been predicted
theoretically si-hbn2 . The electronic properties of silicene on II-VI and
III-V semiconducting (111) substrates, including AlAs, AlP, GaAs, GaP, ZnS,
and ZnSe have been investigated in some detail si-semisub ; new1 ; new2 ,
finding n-doping on metal terminated and p-doping on non-metal terminated
surfaces.

Ar exists as solid at low temperature, with short range, weak, and attractive
London dispersion forces responsible for the molecular bonding argonforce .
The temperature-pressure phase diagram has been studied in Refs. young75 ;
macrander77 , demonstrating that a face centered cubic structure is stable
below a temperature of 84 K. Despite various attempts, so far no suitable
substrate could be identified such that the characteristic electronic
structure of silicene would not be perturbed dramatically on it dimoulas13 .
This is probably the consequence of too high binding energies on the tested
substrates. For example, values of 89 and 76/84 meV per atom have been
reported for hexagonal boron nitride and Si/C-terminated SiC(0001),
respectively si-hbn . In this context, we study the possibility of utilizing
solid Ar(111) as a substrate and analyze the consequences on the electronic
properties of silicene. We will argue that silicene on solid Ar(111) is quasi-
freestanding. In addition, in Refs. new3 ; new4  a buffer layer of solid noble
gas has been used to deposit metal clusters by soft landing and subsequent
evaporation of the noble gas. A similar approach with solid Ar(111) on top of
the desired substrate can provide access to growth of silicene on essentially
any substrate.

Figure 1: Top view (left; along c-axis) and side view (right; along x-axis) of
silicene on Ar(111).

##  II Computational method

All calculations are performed using density functional theory in the
generalized gradient approximation (Perdew-Burke-Ernzerhof parametrization)
and ultrasoft pseudopotentials, as implemented in the Quantum-ESPRESSO package
QE . The plane wave cutoff energy for pure Ar is set to 476 eV and for all
other systems to 544 eV. In the self-consistent calculation of pure Ar a
Monkhorst-Pack $32\times 32\times 32$ k-mesh is employed, whereas for all
other systems a $32\times 32\times 1$ k-mesh is used. To achieve a high
resolution, dense $64\times 64\times 64$ and $64\times 64\times 1$ k-meshes
are used for calculating the density of states (DOS). An energy convergence of
$10^{-5}$ Ry and a force convergence of $10^{-4}$ Ry/Bohr are achieved.
Calculations are performed with and without spin-orbit coupling (SOC) and with
and without van der Waals (vdW) interaction london . In the zoomed band
structures shown in the following we use $\delta K=(0.002;0.002;0)$ with
$K=(1/3;1/3;0)$. We consider a slab geometry with silicene on one side of an
Ar(111) slab, which inherits hexagonal symmetry from the two subsystems.

##  III Results and Discussion

The optimized lattice parameter of solid Ar is $5.36$ Å, which leads to an Ar-
Ar distance of $5.36/\sqrt{2}$ Å$=3.79$ Å. On the other hand, for freestanding
silicene we obtain $3.86$ Å in agreement with Ref. sili . For the combined
argon-silicene (ArSi) system we set the lattice parameter to $3.79$ Å, i.e.,
silicene is subject to a lattice mismatch of 1.9%. Figure 1 shows for the ArSi
system a buckling of 0.53 Å (distance between the bottom and top atomic
layers), which is slightly higher than predicted for freestanding silicene
(0.46 Å) in Ref. sili1 . We note that an artificially planarized structure
with the same Si-Si bond length as the ground state buckled structure is only
32 meV per Si atom higher in energy. The Ar(111) substrate consists of six Ar
layers. The top layer is arranged such that two thirds of the atoms are
located below Si atoms and the last third is located below the center of a Si
hexagon, which turns out to minimize the energy. After structural optimization
without vdW interaction, the Si-Si bond length is 2.27 Å with a bond angle of
$115^{\circ}$, in agreement with Refs. sili1 ; sili2 ; sili3 . We obtain a
distance of $4.3$ Å between the top Ar layer and the silicene, whereas the
interlayer spacing in the substrate amounts to 3.6 Å. Taking into account the
vdW interaction, we obtain the same buckling but reduced interlayer spacings
of $3.4$ Å and $3.1$ Å, respectively.

Figure 2:  Band structure and DOS of solid Ar (mg: midgap). Figure 3:  Band
structure and DOS of silicene without vdW interaction (no band gap without SOC
and $E_{g}=2$ meV with SOC).

Figure 4: Band structure and DOS of the ArSi system. Top: Without vdW
interaction ($E_{g}=2$ meV without SOC and no band gap with SOC). Bottom:
Including vdW interaction ($E_{g}=11$ meV without SOC and $E_{g}=10$ meV with
SOC).

The band structure and DOS of solid Ar are shown in Fig. 2 and those of
freestanding silicene in Fig. 3. Without SOC we obtain for silicene the
characteristic linear dispersion of the $\pi$ and $\pi^{*}$ bands around the K
point 2dbands , reflecting massless Dirac fermions. The inclusion of SOC opens
a band gap of 2 meV, which is small, but much larger than in the case of
graphene (due to the stronger SOC) and agrees with Refs. sili3 ; si-hbn2 .
When we turn on the vdW interaction we obtain virtually identical results to
Fig. 3 both without and with SOC.

For the combined ArSi system the band structure and DOS are presented in Fig.
4. Without vdW interaction and without SOC we obtain again a band gap of 2
meV, indicating minor influence of the substrate despite the fact that the
interlayer spacing between Ar and Si is large. SOC splits the bands near the K
point, one forming a perfect Dirac cone and one showing a band gap of 2 meV.
We note that the band gap of silicene can be tuned by applying an external
electric field, because the insulating Ar does not screen electric fields. The
binding energy $(E_{\rm ArSi}-E_{\rm Ar}-E_{\rm Si})/2$ per Si atom between
silicene and the substrate amounts to $-3$ meV, both without and with SOC.

Switching on the vdW interaction enhances the band gap to 11 meV without and
10 meV with SOC. This is much larger than in graphene but for many potential
device applications still too small. For example, for metal-oxide-
semiconductor field-effect transistors a sizeable band gap is required for a
good on-off ratio and low power dissipation device . If SOC is included, both
split bands show energy gaps, where the splitting at the K point is of similar
magnitude as seen in the top part of Fig. 4. The larger band gap reflects the
relevance of the vdW interaction in the hybrid system, in contrast to
freestanding silicene, while SOC splits the bands near the Fermi level. The
binding energy per Si atom accordingly is enhanced to $-32$ meV, which,
however, is still very small. In particular, it is much less than reported for
silicene on inert hexagonal boron nitride si-hbn2 . Due to the weak
interaction, we conclude that solid Ar(111) will support quasi-freestanding
silicene.

##  IV Conclusion

In conclusion, we have discussed the structure and electronic properties of
silicene on solid Ar(111). It turns out to be critical to take into account
the vdW interaction to obtain realistic results. We have shown that the Dirac
cone of freestanding silicene remains intact on Ar(111), which points to a
weak interaction with the substrate. In fact, we obtain for the binding energy
a small value of $-32$ meV per Si atom, indicating a quasi-freestanding nature
of silicene on Ar(111). Any other substrate employed so far has resulted in
fundamental perturbations of the Dirac states, which is not the case on solid
Ar according to our simulations. It is likely that separation of silicene from
this substrate is possible.

###### Acknowledgements.

Fruitful discussions with T. P. Kaloni are gratefully acknowledged. We thank
the subeditor for bringing to our attention the work of Weaver and coworkers
on noble gas substrates.

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