# Uncovering the chemical structure of the pulsating low-mass white dwarf SDSS
J115219.99+024814.4

A. D. Romero1, G. R. Lauffer1, A. G. Istrate2 and S. G. Parsons3
1Physics Institute, Universidade Federal do Rio Grande do Sul, Av. Bento
Goncalves 9500, Porto Alegre 91501-970, RS, Brazil
2Department of Astrophysics, Radboud University Nijmegen, P.O. Box 9010,
NL-6500 GL Nijmegen, the Netherlands
3 Department of Physics and Astronomy, University of Sheffield, Sheffield, S3
7RH, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Pulsating low-mass white dwarf stars are white dwarfs with stellar masses
between 0.30 M⊙ and 0.45 M⊙ that show photometric variability due to gravity-
mode pulsations. Within this mass range, they can harbour both a helium- and
hybrid-core, depending if the progenitor experienced helium-core burning
during the pre-white dwarf evolution. SDSS J115219.99$+$024814.4 is an
eclipsing binary system where both components are low-mass white dwarfs, with
stellar masses of 0.362$\pm$0.014 M⊙ and 0.325$\pm$0.013 M⊙. In particular,
the less massive component is a pulsating star, showing at least three
pulsation periods of $\sim$1314 s, $\sim$1069 s and $\sim$582.9 s. This opens
the way to use asteroseismology as a tool to uncover its inner chemical
structure, in combination with the information obtained using the light-curve
modelling of the eclipses. To this end, using binary evolutionary models
leading to helium- and hybrid-core white dwarfs, we compute adiabatic
pulsations for $\ell=1$ and $\ell=2$ gravity modes with Gyre. We found that
the pulsating component of the SDSS J115219.99$+$024814.4 system must have a
hydrogen envelope thinner that the value obtained from binary evolution
computations, independently of the inner composition. Finally, from our
asteroseismological study, we find a best fit model characterised by T${}_{\rm
eff}=10\,917$ K, M=0.338 M⊙, M${}_{\rm H}=10^{-6}$ M⊙ with the inner
composition of a hybrid WD.

###### keywords:

asteroseismology – white dwarfs – binaries

††pubyear: 2021††pagerange: Uncovering the chemical structure of the pulsating
low-mass white dwarf SDSS J115219.99+024814.4 –A

##  1 Introduction

White dwarfs (WDs) are the most common endpoint of stellar evolution. All
stars with initial masses below 7$-$12 M⊙ (e.g. Garcia-Berro et al., 1997;
Woosley & Heger, 2015; Lauffer et al., 2018), representing more than 95% of
the stars in the Milky Way, will end their lives as WDs. The WD population can
be divided into hydrogen-rich atmosphere objects (DA), that correspond to more
than 85% of all WDs, and hydrogen-deficient objects (non-DA), which show no
hydrogen in their atmospheres (see Fontaine & Brassard, 2008; Althaus et al.,
2010a).

For hydrogen-atmosphere WDs the mass distribution peaks at $\sim$ 0.6 M⊙,
which represents $\sim 84\%$ of the total sample (Kepler et al., 2007; Kepler
et al., 2015), exhibiting also a high-mass and a low-mass components. The low-
and high-mass WDs are most likely the result of close binary evolution, where
mass transfer and mergers commonly occur.

The low-mass tail in the DA mass distribution peaks at $\sim$ 0.39 M⊙ and
extends to stellar masses lower than 0.45 M⊙. White dwarfs with masses below
$\sim$0.30 M⊙ can only be formed through mass transfer in close binary
systems, since single star evolution is not able to form such remnants in the
Hubble time (Kilic et al., 2007; Istrate et al., 2016; Pelisoli & Vos, 2019).
These objects are known as extremely low-mass white dwarfs (ELMs). Low-mass
WDs are stars with stellar masses in the range of 0.30 $\leq$
M/M${}_{\odot}\leq$ 0.45. In addition to the binary formation channel, these
objects could also form as a result of strong mass-loss episodes during giant
stages for high metallicity progenitors. Noteworthy, low-mass WDs can harbour
either a pure-helium core (e.g. Panei et al., 2007; Althaus et al., 2013;
Istrate et al., 2014; Istrate et al., 2016) or a hybrid core, composed of
helium, carbon and oxygen (e.g. Iben & Tutukov, 1985; Han et al., 2000; Prada
Moroni & Straniero, 2009; Zenati et al., 2019).

Probably the only way to probe the inner chemical composition in detail is
through the pulsation period spectrum observed in variable stars. Each
pulsation mode propagates in a specific region providing information on that
particular zone, where its amplitude is maximum (Tassoul et al., 1990). Thus,
we can perform an asteroseismic study, where we compare the observed periods
with the theoretical period spectrum, computed using representative models, to
uncover the inner stellar structure (see e. g. Romero et al., 2012, 2019).

There are currently several families of pulsating WDs, that can be found in
specific ranges of effective temperature and surface gravity. They show
$g$-mode non-radial pulsations with periods range from minutes to a few hours
and variation amplitudes of millimag. The excitation mechanism acting on
pulsating WDs is a combination of the $\kappa$-mechanism, driven by an opacity
bump due to partial ionization of the main element in the outer layers (Dolez
& Vauclair, 1981; Winget et al., 1982), and the $\gamma$-mechanism, related to
the effect of a small value of the adiabatic exponent in the ionization zone
(Brickhill, 1991; Goldreich & Wu, 1999).

The location of the different classes in the Kiel diagram is depicted in
Figure 1. At high effective temperatures we find the GW Vir stars, with C/O
atmospheres, followed by He-rich atmosphere V777 Her and the C-rich atmosphere
DQV. Finally, the H-envelope pulsating WDs, known as ZZ Ceti stars, are
located at lower effective temperatures. For lower $\log(g)$, we find the
pulsating low-mass WDs, and the ELMs along with their progenitors, the pre-
ELMs. Even though the group of pulsating ELMs is considered a class on its
own, their instability strip is an extension of the ZZ Ceti instability strip
to lower surface gravities, as can be seen from Fig. 1.

Figure 1: The classes of pulsating WDs. The data was extracted from Fontaine &
Brassard (2008); Bognar & Sodor (2016); Pietrukowicz et al. (2017); Córsico et
al. (2019); Romero et al. (2019); Kupfer et al. (2019). For reference, we
include theoretical WD sequences with C/O core and masses of 0.878, 0.638 and
0.524 M⊙ (Romero et al., 2015) and He-core with stellar mass of 0.324 and
0.182 M⊙ (Istrate et al., 2016). The different symbols indicate the element
related to the excitation mechanism: hydrogen (circles), helium (triangle-up),
carbon and/or oxygen (triangle-left) and iron peak elements (squares). The
known variable low-mass WDs are indicated with a surrounding green circle. The
position of J115219.99$+$024814.4 is depicted with a triangle-right, with the
atmospheric parameters taken from Parsons et al. (2020) (see Table 1 for
details).

There are 11 pulsating ELMs known to date (green dots in Fig. 1, Hermes et
al., 2013b, a; Kilic et al., 2015; Bell et al., 2015, 2017; Pelisoli et al.,
2018). The pulsating ELMs are characterised by periods in the range of
$100-6\,300$ s, effective temperatures of $7\,800-10\,000$ K and a hydrogen
dominated surface composition (Córsico et al., 2019). In addition, there are
10 objects in the literature with stellar masses within the range 0.30 $\leq$
M/M${}_{\odot}\leq$ 0.45 (black-green dots in Fig. 1) that show photometric
variability with periods between 200 and 1300 s (Bognar & Sodor, 2016; Su et
al., 2017; Fuchs, 2017; Rowan et al., 2019). For 4 of them, the uncertainties
in the atmospheric parameters are quite large, leading to an uncertainty in
stellar mass of 0.1–0.4 M⊙ (Su et al., 2017; Rowan et al., 2019). The low
number of pulsating low-mass WDs as compared to canonical mass ZZ Ceti stars
could be due to some kind of fine tuning during the evolution of the
progenitor, but it is most likely due to the lack of studies focused on the
search for pulsations for objects in this stellar mass range.

Recently, Parsons et al. (2020) reported the discovery of the first pulsating
low-mass WDs in a compact eclipsing binary system (orbital period of 2.4 h),
which happens to have another low-mass WD as a companion. The binary nature of
the SDSS J115219.99$+$024814.4 system (hereafter J1152$+$0248) was first
reported by Hallakoun et al. (2016), based on K2 data from the Kepler mission.
Parsons et al. (2020) performed high-speed photometry observations with
HiPERCAM on the 10.4 m Gran Telescopio Canárias in five different bands, with
a total of 108 min of data, covering both primary and secondary eclipses. From
the high time-resolution light curves they found pulsation-related variations
from the cooler component with at least three significant periods. To
determine the mass and radius of each component in J1152$+$0248, they combine
radial-velocity determinations from X-shooter spectroscopy with the
information extracted from the primary and secondary eclipses in the light
curves (see their Table 1). The effective temperature was determined using two
techniques, i.e. by fitting the spectral energy distribution ($T_{\rm
eff}{\rm(SED)}$) and by modelling the light curves including the effects of
the eclipses ($T_{\rm eff}{\rm(Eclipse)}$). The stellar parameters obtained by
Parsons et al. (2020) for the pulsating component in J1152$+$0248 (hereafter
J1152$+$0248$-$V) are listed in Table 1.

Table 1: Stellar parameters presented in Parsons et al. (2020) for the pulsating component of the J1152$+$0248 eclipsing binary. Parameter | Value
---|---
M∗/M⊙ | $0.325\pm 0.013$
$R_{*}$/R⊙ | $0.0191\pm 0.0004$
$\log$ (g/[g cm-2]) | $7.386\pm 0.012$
Teff/K (SED) | $11\,100\pm^{950}_{770}$
Teff/K (Eclipse) | $10\,400\pm^{400}_{340}$

Based on the determination of the radius, Parsons et al. (2020) proposed that
J1152$+$0248$-$V is either a hybrid-core or a helium-core low-mass WDs with an
extremely thin surface hydrogen layer (MH/M${}_{\odot}<10^{-8}$). Different
inner chemical structures will influence the characteristic period spectrum of
a pulsating star. Therefore an asteroseismological study of this object can
shed some light on both the chemical composition and the mass of the hydrogen
envelope.

In this work, we explore the pulsational properties of both hybrid- and
helium-core low-mass WD models representative for the case of
J1152$+$0248$-$V. Furthermore, we consider sequences with hydrogen envelopes
thinner than the value predicted by stable mass-transfer binary evolutionary
models. For the evolutionary computations we use the stellar evolution code
MESA (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018, 2019), while the
adiabatic pulsations are computed using GYRE stellar oscillation code
(Townsend & Teitler, 2013; Townsend et al., 2018). Using the theoretical
period spectra, we perform an asteroseismological study of J1152$+$0248$-$V to
uncover its inner structure.

This paper is organised as follows. In Section 2, we provide a description of
the evolutionary computations and the input physics adopted in our
calculations. In section 3 we present the pulsation computations. Section 4 is
devoted to study the pulsational properties of our low-mass WD models,
including a comparison between the hybrid and helium-core configurations. The
results of the asteroseismological study of J1152$+$0248$-$V are presented in
this section as well. Our final remarks are presented in Section 5.

##  2 Evolutionary Sequences

The evolutionary models presented in this work are computed using the open-
source binary stellar evolution code MESA (Paxton et al., 2011; Paxton et al.,
2013, 2015, 2018, 2019), version 12115 and are part of a grid of models
covering the mass interval where helium- and hybrid-core WDs overlap, i.e.
$\sim$0.32–0.45 M⊙ (Istrate et al. 2021a, in preparation). Since the aim of
this work is analyse the core-composition of J1152$+$0248$-$V using
asteroseismology, we only consider sequences compatible with its observed
effective temperature and radius. We present evolutionary sequences for a WD
mass of 0.325 M⊙, corresponding to the value obtained by Parsons et al.
(2020), and 0.338 M⊙ i.e. the maximum WD mass compatible within 1-$\sigma$.

###  2.1 Input physics

We compute binary evolutionary sequences using similar assumptions as in
Istrate et al. (2016). All models include rotation, with the initial
rotational velocity initialised such that the donor is synchronised with the
orbital period. We include magnetic braking for the loss of angular momentum
just for the donors leading to the formation of the helium-core WDs. For more
massive donors ($>$2.0 M⊙), which are considered for the progenitors of the
hybrid-core WDs, we assume that the magnetic braking stops operating. This
assumption follows from the conventional thinking that braking via a
magnetized stellar wind is inoperative in stars with radiative envelopes
(Kawaler, 1988). For all the models, we assume a mass transfer efficiency of
50 per cent, i.e. 50 per cent of the transferred mass is accreted by the
companion, while the rest leaves the system with the specific angular momentum
of the accretor.

Below, we briefly describe the main input physics considered in the
evolutionary computations and refer to Istrate et al. (2021a in preparation)
for more details. We consider an initial metallicity of $Z=0.01$, with a
helium abundance given by $Y=0.24+2.0\cdot Z$ and the metal abundunces scaled
according to Grevesse & Sauval (1998). Convection is modelled using the
standard mixing-length theory (Henyey et al., 1965) with a mixing-length
parameter $\alpha$ = 2.0, adopting the Ledoux criterion. A step function
overshooting extends the mixing region for 0.25 pressure scale heights beyond
the convective boundary during core hydrogen burning. In order to smooth the
boundaries we also include exponential overshooting with $f=0.0005$.
Semiconvection follows the work of Langer et al. (1983), with an efficiency
parameter $\alpha_{sc}$ = 0.001. Thermohaline mixing is included with an
efficiency parameter of 1.0. Radiative opacities are taken from Ferguson et
al. (2005) for 2.7 $\leq$ log T $\leq$ 3.8 and OPAL (Iglesias & Rogers, 1993,
1996) for 3.75 $\leq$ log T $\leq$ 8.7, and conductive opacities are adopted
from Cassisi et al. (2007).

The nuclear network used is cno$\\_$extras.net which accounts for additional
nuclear reactions for the CNO burning compared to the default basic.net
network. We consider the effects of element diffusion (e.g Iben & Tutukov,
1985; Thoul et al., 1994) on all isotopes and during all the stages of
evolution.

We adopt a grey atmosphere using the Eddington approximation on the evolution
prior to the cooling track, and the WD atmosphere tables from Rohrmann et al.
(2012) during the WD cooling stage.

Rotational mixing and angular momentum transport are treated as diffusive
processes as described in Heger et al. (2000), with an efficiency parameter fc
= 1/30 (Chaboyer & Zahn, 1992) and a sensitivity to composition gradients
parametrized by fμ = 0.05. We also include transport of angular momentum due
to electron viscosity (Itoh et al., 1987).

###  2.2 White dwarf formation history

Figure 2 shows the evolution of the donor star in the Kiel diagram leading to
a remnant mass of 0.338 M⊙ with a helium- (solid orange line) and hybrid-core
(blue dashed line). The evolution is computed from the zero-age main sequence
(ZAMS) until the remnant WD cools down to an effective temperature of 5 000 K.
In both cases, the companion star is treated as a point mass. We also mark on
the evolutionary sequences several important points. The moment when the donor
star overflow its Roche-lobe (RLOF), marking the beginning of the mass-
transfer phase, is depicted with a circle symbol, the end of the mass transfer
phase is marked with a star, the beginning and the end of the core-helium
burning phase are showed with the square and thin diamond symbol,
respectively. Finally, the beginning of the cooling track is represented by
the diamond symbol.

The helium-core WDs are formed by stripping mass when the donor star is on its
red-giant branch. The binary system consists of a low-mass donor star with a
initial mass of 1.3 M⊙, a main-sequence companion of 1.2 M⊙ and an initial
orbital period of $\sim$11.75 and $\sim$16.98 days, for the progenitor of the
0.325 M⊙ and 0.338 M⊙, respectively. After the end of the mass-transfer phase,
the remnant evolves through the so-called proto-WD phase, in which unstable
hydrogen burning leads to the occurrence of at least one hydrogen flash.

The initial binary configuration leading to the formation of the hybrid-core
WDs is a 2.3 M⊙ intermediate-mass donor star with a companion of 2.0 M⊙ and an
orbital period of 1.43 and $\sim$1.99 days, or the progenitor of the 0.325 M⊙
and 0.338 M⊙, respectively. The mass-transfer phase initiates during the
Hertzsprung gap. Unlike the case of the helium-core WD sequence, here the
mass-transfer ceases due to the core-helium ignition. Shortly after the mass-
transfer ended, the remnant starts the core-helium burning phase, which lasts
around 670 Myrs. Once the CO-core is formed, the proto-WD undergoes four
hydrogen shell flashes before finally settling on the cooling track.

Figure 2: The Kiel diagram showing the formation and the cooling evolution of
a 0.338 M⊙ He- (solid orange line) and hybrid-core (dashed blue line) WD. The
initial binary parameters are Mdonor = 1.3 M⊙, Maccretor =1.2 M⊙, Pinitial =
$\sim$16.982 days for the helium-core model, and Mdonor = 2.3 M⊙, Maccretor
=2.0 M⊙, Pinitial = $\sim$1.990 days for the hybrid-core model, at an initial
metallicity of Z=0.01. In both cases, the accretor is treated as a point mass.
The evolutionary tracks are computed from ZAMS until the effective temperature
of the WD reaches 5 000 K. The symbols mark various stages of evolution. The
beginning and the end of the mass-transfer phase are represented by the circle
and star symbol respectively, the beginning and the end of the core-helium
burning phase (in the case of the hybrid-core sequence) are depicted by the
square and thin diamond symbol respectively, and finally the diamond symbol
represents the beginning of the cooling track, i.e. the point when the
evolutionary track reaches its maximum effective temperature.

###  2.3 White dwarf cooling track

The WD sequences discussed in the previous section are formed through a
stable-mass transfer channel. While it is possible that the first mass-
transfer phase which leads to the formation of J1152$+$0248$-$V is stable, we
cannot rule out completely a common-envelope evolution. Additionally, the
short orbital period of J1152$+$0248 (2.4 h) suggests that the second mass-
transfer phase leading to the formation of the most massive component is
unstable and proceeds through a common-envelope evolution. This evolutionary
phase could possibly also affect the lower-mass component. Either way, there
is an uncertainty in the mass of the hydrogen envelope available at the
beginning of the cooling track resulting from the evolutionary history prior
to the observed stage of the system.

The mass of the hydrogen envelope is one of the main factors that influences
the cooling evolution of a WD. In order to take this uncertainty into account,
we also computed WD cooling sequences with hydrogen envelopes thinner than the
ones obtained from the binary evolutionary models described above. These
sequences were computed using relaxation methods available in MESA. The
initial conditions are taken at the point when the remnant reaches the
beginning of the cooling track. Using the stellar profile at this point, we
remove the desired amount of hydrogen, keeping the total mass unchanged (see
Istrate et al. 2021b, for details).

Figure 3: Stellar radius as a function of the effective temperature for both
hybrid- (dashed lines) and helium-core (solid lines) WD sequences with stellar
mass 0.338 M⊙. The symbols correspond to the determinations of the effective
temperature from observations (see table 1). The dotted vertical lines
indicate the blue and red edges of the observed instability strip for low-mass
WDs (see Fig. 1).  Figure 4: Same as Figure 3 but for the WD sequences with
stellar mass 0.325 M⊙.

Figures 3 and 4 show the WD radius as a function of the effective temperature
for sequences with helium- (top panels) and hybrid-core (bottom panels) and
stellar mass of 0.338 M⊙ and 0.325 M⊙, respectively.

For each core composition we consider different values of the hydrogen
envelope mass, starting from the value resulting from the binary evolution
down to 10-8 M⊙. In particular, for the stellar mass of 0.325 M⊙ and helium-
core, we also include a sequence with hydrogen-envelope mass of 10-10 M⊙.
Overplotted are the measured values of J1152$+$0248$-$V, both from the eclipse
and the SED fitting.

As expected, the radius for the hybrid-core sequences is smaller than for the
helium-core sequences, for the same hydrogen envelope mass. Intriguingly, the
observations are not compatible with a thick hydrogen envelope, i.e. that
obtained from binary evolution computations. For sequences characterised with
a stellar mass of 0.338 M⊙ (see Fig. 3) the hydrogen-envelope mass consistent
with the radius and effective temperature from Parsons et al. (2020) is
between $10^{-4}$ M⊙ and $10^{-6}$ M⊙ if we consider a hybrid-core. For the
sequences with a helium-core, the hydrogen envelope mass is below $10^{-5}$
M⊙. For sequences with stellar mass of 0.325 M⊙ (see Fig. 4), the hydrogen-
envelope mass needs to be even smaller to fit the observations, between
$10^{-5}$ M⊙ and $10^{-8}$ M⊙ for sequences with a hybrid-core and $10^{-5}$
M⊙ and $10^{-10}$ M⊙ for sequences with a helium-core. Thus, J1152$+$0248$-$V
has a hydrogen envelope thinner than that obtained from evolutionary
sequences, independently of the central chemical composition.

##  3 Adiabatic Pulsations

The pulsational properties in stars are dominated by the characteristic
frequencies, i.e. the Brunt-Väisälä ($N^{2}$) and the Lamb ($L_{\ell}^{2}$)
frequencies (Cowling, 1941; Unno et al., 1989). The Brunt-Väisälä frequency
represents the oscillation frequency of a convective bubble around the stable
equilibrium position. It is closely related with gravity modes since the
restoring force is gravity. In the case of low-mass WDs, the gravity modes
correspond to the long period (low frequency) modes, with periods $\gtrsim
200$ s for $\ell=1$. The $N^{2}$ frequency is given by (Brassard et al.,
1991):

$N^{2}=\frac{g^{2}\rho}{P}\frac{\chi_{T}}{\chi_{\rho}}\left(\nabla_{\rm
ad}-\nabla+B\right)$ (1)

where $g$, $\rho$ and $P$ are the gravitational acceleration, the density and
the pressure, respectively. $\nabla_{\rm ad}$ and $\nabla$ are the adiabatic
and the temperature gradient respectively, and $\chi_{T}$ and $\chi_{\rho}$
are defined as,

$\chi_{T}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}$ (2)

$\chi_{\rho}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}$ (3)

The term $B$ is called the Ledoux term and it is defined as,

$B=-\frac{1}{\chi_{T}}\left(\frac{\partial\ln P}{\partial
lnX_{i}}\right)_{\rho,T}\frac{d\ln X_{i}}{d\ln P}$ (4)

where $X_{i}$ is the chemical abundance of the element $i$. The Ledoux term
$B$ gives the explicit contribution of the change of chemical composition,
thus it contributes in the region where chemical transitions are found
(Brassard et al., 1992a).

The Lamb frequency is given by:

$L_{\ell}^{2}=\frac{(\ell+1)\ell}{r^{2}}c_{S}^{2}$ (5)

where $c_{S}$ is the adiabatic sound speed. The Lamb frequency is inversely
proportional to the time that a sound wave takes to travel a distance $2\pi
r/\ell$. It is closely related to pressure modes since the restoring force is
the pressure gradient. Pressure modes correspond to short period (high
frequency) modes with periods of $\lesssim 10$ s for low-mass WDs.

For a chemically homogeneous and radiative star, the period spectrum is
characterised by a constant period separation, known as the asymptotic period
spacing (Tassoul et al., 1990),

$\Delta\Pi_{a}=\frac{\Pi_{0}}{[\ell(\ell+1)]^{1/2}}$ (6)

where

$\Pi_{0}=2\pi^{2}\left(\int_{r_{1}}^{r_{2}}\frac{N(r)}{r}dr\right)^{-1}$ (7)

In the case of WDs, the inner chemical structure is far from homogeneous,
showing composition gradients (Althaus et al., 2010b). In this case, the
asymptotic period spacing corresponds to the period spacing at the limit of
very large values of $k$ (Tassoul et al., 1990).

###  3.1 Numerical computations

For our pulsational computations we employed GYRE (Townsend & Teitler, 2013;
Townsend et al., 2018), an open–source stellar oscillation code in its
adiabatic form, version 5.0. Although GYRE is integrated within MESA, it was
used as an stand-alone package. We performed a scan using a linear grid over a
period range of $80-2000$ s with alpha_osc$=100$ and alpha_exp and alpha_ctr
set to 50. The boundary conditions and variables were set to UNNO and DZIEM,
respectively111The inlist and all its configuration settings can be found in
here [link to be updated].

We compute pulsations for helium- and hybrid-core models in the effective
temperature range of $13\,000$ and $8\,000$ K, which covers the empirical
instability strip. We compute the period spectrum for $\ell=1$ and $\ell=2$
gravity modes with periods in a range of $80\ {\rm s}\leq\Pi\leq 2\,000$ s.

###  3.2 Internal composition and the characteristic frequencies

To compare the pulsation properties of helium- and hybrid-core WD models, we
chose one template model for each central composition, with stellar mass of
0.338 M⊙, effective temperature of $10\,000$ K and hydrogen envelope mass of
$10^{-4}$ M⊙. In the top panel of Figures 5 and 6 we depict the chemical
profiles for a helium-core and a hybrid-core template models, respectively.
Both template models show a pure hydrogen envelope, since gravitational
settling had enough time to separate the elements in the outer layers. For the
hybrid-core model we have a second chemical transition (He/C/O transition)
around $r/R\sim 0.4$, where the helium abundance decreases towards the centre
while the carbon and oxygen abundance increase.

The presence of chemical transitions, where the abundances of nuclear species
vary considerably in radius, modifies the conditions of the resonant cavity in
which modes should propagate as standing waves. Specifically, the chemical
transitions act as reflecting walls, trapping certain modes in a particular
region of the star, where they show larger oscillation amplitudes. Trapped
modes are those for which the wavelength of their radial eigenfunction matches
the spatial separation between two chemical transitions or between a
transition and the stellar centre or surface (Brassard et al., 1992b; Bradley
et al., 1993; Córsico et al., 2002).

Figure 5: Internal profiles and characteristic frequencies for a helium-core
cooling sequence with M=0.338 M⊙, T${}_{\rm eff}=10\,000$ K and M${}_{\rm
H}=10^{-4}$ M⊙. Top panel: the mass fraction of hydrogen and helium as a
function of the relative radius. Bottom panel: the logarithm of the Brunt-
Väisälä (full blue line) and the Lamb (dashed orange line) characteristic
frequencies as a function of the relative radius.  Figure 6: Internal chemical
composition and characteristic frequencies for a hybrid-core sequence with
M=0.338 M⊙, T${}_{\rm eff}=10\,000$ K and M${}_{\rm H}=10^{-4}$ M⊙. Top panel:
mass fraction of hydrogen, helium, carbon and oxygen as a function of the
relative radius. Bottom panel: the logarithm of the Brunt-Väisälä (full blue
line) and Lamb (dashed orange line) characteristic frequencies as a function
of the relative radius.

The bottom panels of Figures 5 and 6 show the run of the logarithm of the
Brunt-Väisälä and Lamb frequencies as a function of radius for the template
models. The overall behaviour of $\log N^{2}$ is a general decrease with
increasing stellar depth, eventually reaching small values in the deep core,
as a consequence of degeneracy. As expected, the bumps in $\log N^{2}$
correspond to the chemical transition, where the chemical gradients have non-
negligible values (Tassoul et al., 1990). For the helium-core template model,
the Brunt-Väisälä frequency shows only one bump corresponding to the H/He
transition at $r/R\sim 0.9$. For the hybrid-core template model, in addition
to the bump at the base of the hydrogen envelope, the Brunt-Väisälä frequency
shows a structure corresponding to the He/C/O transition around $r/R\sim 0.4$.
This transition is wide and shows three peaks due to the structure of the
chemical gradients. We expect that the different profiles for the Brunt-
Väisälä frequency for helium-core and hybrid-core models will impact the
pulsation properties, for example the period spectrum and the period spacing.

The Lamb frequency is only sensitive to the H/He transition at the bottom of
the hydrogen envelope, as it is the case for ZZ Ceti stars (Romero et al.,
2012). Thus, we do not believe the pressure modes could give information on
the inner regions, if ever detected in low-mass WD stars.

##  4 Results

###  4.1 Pulsational properties

Figure 7:  The forward period spacing ($\Delta\Pi_{k}=\Pi_{k+1}-\Pi_{k}$) as a
function of the period for modes with $\ell=1$ for helium-core (top panels)
and hybrid-core (bottom panels) models, with stellar mass 0.338 M⊙. Each
column corresponds to a different hydrogen envelope mass. In each plot we show
the period spacing for four effective temperatures along the cooling sequence,
$12\,000$, $11\,000$, $10\,000$ and 9000 K.

As previously mentioned, for a chemically homogeneous and radiative star, the
forward period spacing ($\Delta\Pi_{k}=\Pi_{k+1}-\Pi_{k}$) would be constant
and given by the asymptotic period spacing, defined in equation 6. However, in
the stratified inner structures, as the ones found in low-mass WDs, the
forward period spacing deviates from a constant value, particularly for low-
radial order modes. In this section we will focus on models with stellar mass
0.338 M⊙. Similar results are found for models with stellar mass of 0.325 M⊙.

Figure 7 shows the forward period spacing as a function of the period for
$\ell=1$ modes. Top panels correspond to models with helium-core while bottom
panels show the results for hybrid-core models. Finally, each column
corresponds to sequences with different values for the hydrogen-envelope mass
and each curve shows the forward period spacing at a different effective
temperature along the cooling curve, from $12\,000$ K to 9000 K.

For the helium-core models the distribution for the forward period spacing as
a function of the period, shows a simple trapping cycle characteristic of one-
transition models (Brassard et al., 1992b; Córsico & Althaus, 2014), with
defined local minima, as shown in the top panels of Figure 7. For the hybrid-
core models (bottom panels in Fig. 7) the pattern in the forward period
spacing is more complex largely due to the influence of the He/C/O transition
in the Brunt-Vaisälä frequency. Even though we expect the H/He transition to
be dominant in the mode selection process, the presence of the second,
broader, transition is not negligible. The differences in the pattern of the
forward period spacing can be used to determine the inner composition of low-
mass WDs, if enough consecutive periods are detected.

The trapping period, i.e. the period difference between two consecutive minima
in $\Delta\Pi_{k}$, is longer for lower effective temperatures and thinner
hydrogen envelopes. This effect is much more evident for the helium-core
models than for the hybrid-core models.

In general, we expect the forward period spacing to increase with decreasing
effective temperature and thinner envelopes (Brassard et al., 1992b). This can
be better explained in terms of the asymptotic period spacing, given by
equation 6. The higher values for $\Delta\Pi_{a}$, and $\overline{\Delta\Pi}$,
for lower effective temperatures results from the dependence of the Brunt-
Väisälä frequency as $N\propto\sqrt{\chi_{T}}$, with $\chi_{T}\rightarrow 0$
for increasing degeneracy ($T\rightarrow 0$) (see for instance Romero et al.,
2012). In addition, the asymptotic period spacing, and the mean period
spacing, are longer for thinner hydrogen envelopes (Tassoul et al., 1990).

If the stellar mass is fixed, the asymptotic period spacing for the helium-
core model is longer than that for the hybrid-core model, for the same
effective temperature and hydrogen envelope mass. This is related to the
dependence of the Brunt-Väisälä frequency on the surface gravity $N\propto g$,
where $g\propto$ M/R2. In this case, the radius is smaller for the hybrid-core
models which leads to a larger value of the surface gravity (see Figs. 3 and
4).

Figure 8: Period difference ($\Pi_{\rm he}-\Pi_{\rm hy}$) as a function of the
radial order for $\ell=1$ modes. The stellar mass is fixed to 0.338 M⊙.

Each plot corresponds to a different hydrogen envelope mass. We consider four
effective temperatures.

Finally, given the different chemical structure of helium- and hybrid-core WD
models, we expect different values for the period spectra as well. Figure 8
show the period difference between helium- and the hybrid-core models,
$\Pi_{\rm he}-\Pi_{\rm hy}$ as a function of the radial order $k$, for
harmonic degree $\ell=1$. Each column corresponds to a different hydrogen
envelope mass, while the curves in each panel correspond to four different
effective temperatures along the 0.338 M⊙ WD cooling track. In all cases, the
difference in the periods between the helium- and hybrid-core models are
between $\sim 50$ s, for $k\lesssim 10$ and short periods, and $\sim 200$ s
for $k\gtrsim 30$ and longer periods. Note that the period difference is
always positive, meaning that, for a given radial order, the period for the
helium-core model is larger than the one corresponding to the hybrid-core
model. This is expected, since the periods, and the period spacing, vary as
the inverse of the Brunt-Väisälä frequency, and thus the surface gravity (Eq.
7). As the period also increases with the cooling age, the period difference
is larger for lower effective temperatures, particularly for modes with radial
order $k>20$. For future reference, we list the periods for $\ell=1$ and
$\ell=2$, corresponding to the models considered in Figure 8 in tables 4, 5, 6
and 7.

###  4.2 Asteroseismology of J1152$+$0248$-$V

For J1152$+$0248$-$V Parsons et al. (2020) detected three pulsation periods,
with the period of $\sim 1314$ s being the one showing the highest amplitude.
The list of observed periods and their corresponding amplitudes are listed in
Table 2.

Table 2: Observed pulsation periods and the corresponding amplitudes for J1152$+$0248$-$V (Parsons et al., 2020). Period [s] | amplitude [ppt]
---|---
$1314\pm 5.9$ | $33.0\pm 1.3$
$1069\pm 13$ | $9.8\pm 1.3$
$582.9\pm 4.3$ | $8.9\pm 1.3$

In order to find the theoretical model that better matches the observed
periods, we use a standard $\chi^{2}$ approach where we search for a minima of
the quality function, defined as:

$\chi^{2}=\sum_{i=1}^{3}\left(\frac{\Pi^{(i)}_{\mathrm{obs}}-\Pi^{(i)}_{\mathrm{th}}}{\sigma^{(i)}_{\mathrm{obs}}}\right)^{2}$
(8)

where $\Pi^{(i)}_{\mathrm{obs}}$ and $\sigma^{(i)}_{\mathrm{obs}}$ are the
observed periods and their associated uncertainties in $g_{s}$-band, and
$\Pi^{(i)}_{\mathrm{th}}$ are the theoretical periods. In our fit, we consider
the mode with the highest detected amplitude, as an $\ell=1$ mode, since the
amplitude is expected to decrease with increasing harmonic degree due to
geometric cancellation (Robinson et al., 1982). For the remaining two observed
periods we allowed them to be fitted by $\ell=1$ and $\ell=2$ modes. Finally,
we restrict the possible seismological solutions to those models that fit the
observed periods within the uncertainties reported by Parsons et al. (2020)
(see Table 2). The results of our seismological fit are depicted in Figure 9.
The value of $\chi^{2}$ (colour scale) is shown as a function of the effective
temperature and the mass of the hydrogen envelope. Top (bottom) panels
correspond to models with stellar mass 0.338 M⊙ (0.325 M⊙). Models with
helium-core are depicted on the left panels, while the ones with hybrid-core
are shown in the right panels. The dashed-line rectangle indicates the region
where the models are compatible with the radius and effective temperature
determinations from Parsons et al. (2020). As can be seen from this figure,
there are several families of solutions. For a stellar mass 0.325 M⊙, the best
fit models are mainly characterised by thick hydrogen envelopes, $\sim
10^{-4}$ M⊙ and low effective temperatures, $>9\,000$ K, for both core
compositions. Thin hydrogen-envelope solutions, with hydrogen mass below $\sim
10^{-5}$ M⊙, are found for models with stellar mass 0.338 M⊙ and hybrid-core,
for all the effective temperature range.

Figure 9: The value of $\chi^{2}$ (colour scale) as a function of the
effective temperature and the mass of the hydrogen-envelope. Top panels
correspond to models with stellar mass 0.338 M⊙, while bottom panels show the
results for 0.325 M⊙ models. Both helium-core (left panels) and hybrid-core
(right panels) structure are considered. The dashed-line rectangle indicates
the region of the Teff\- Hydrogen envelope plane restricted by the
determinations of radius and effective temperature from Parsons et al. (2020).

If we combine our seismological results with the restrictions from the radius
and the effective temperature presented in Parsons et al. (2020) (dashed-line
rectangle in Fig. 9), the number of possible seismological solutions is
largely reduced. In particular, the models characterised by a hybrid-core
structure are the ones satisfying both criteria.

Table 3: Asteroseismological models for the low-mass WD J1152$+$0248$-$V. The effective temperature, hydrogen-envelope mass and central composition are listed in columns 2, 3 and 4, respectively. We list the theoretical periods that better fit the observed periods in column 4, along with the values of $\ell$ and k, in columns 6 and 7. The value of the quality function $\chi^{2}$ is listed in column 8.  # |  $T_{\rm eff}$ [K] | R$/$R⊙ | M∗/M⊙ | MH/M⊙ | XC |  $\Pi_{\rm th}$ [s] | $\ell$ | k |  $\chi^{2}$ [s]
---|---|---|---|---|---|---|---|---|---
1 | $10\,917$ | 0.0187 | 0.338 | $10^{-6}$ | C/O | 1315.4 | 1 | 21 | 0.296
|  |  |  |  |  | 1063.3 | 2 | 30 |
|  |  |  |  |  | 582.01 | 2 | 15 |
2 | $10\,904$ | 0.0187 | 0.338 | $10^{-6}$ | C/O | 1316.1 | 1 | 21 | 0.305
|  |  |  |  |  | 1063.8 | 2 | 30 |
|  |  |  |  |  | 582.3 | 2 | 15 |
3 | $10\,929$ | 0.0187 | 0.338 | $10^{-6}$ | C/O | 1314.9 | 1 | 21 | 0.318
|  |  |  |  |  | 1062.8 | 2 | 30 |
|  |  |  |  |  | 581.8 | 2 | 15 |
4 | $11\,717$ | 0.0195 | 0.325 | $10^{-6}$ | C/O | 1316.1 | 1 | 21 | 0.345
|  |  |  |  |  | 1066.4 | 2 | 30 |
|  |  |  |  |  | 581.3 | 2 | 15 |

The best fit models for J1152$+$0248$-$V are listed in Table 3. These models
fit the observed periods within the uncertainties and are in agreement with
the determinations of radius and effective temperature presented by Parsons et
al. (2020). Note that all the models are characterised by a hydrogen-envelope
mass of $10^{-6}$ M⊙ and a hybrid-core. Thus, we can conclude that
J1152$+$0248$-$V is a low-mass white dwarf star with a hybrid-core and a thin
hydrogen-envelope.

##  5 Conclusions

In this work we explore the inner chemical structure of the pulsating low-mass
white dwarf in the J1152$+$0248 system using asteroseismology as a tool. This
is the first pulsating low-mass WD in an eclipsing binary system with another
low-mass WD which allows to constrain the mass and radius of each component
independently of evolutionary models.

We use low-mass WDs models with a stellar mass of 0.325 and 0.338 M⊙ and a
helium- and hybrid core respectively, resulting from fully binary evolutionary
computations. In addition, to account for the uncertainty in the hydrogen
envelope mass due to the evolution through a common envelope channel, we use
WD sequences with the same stellar mass but with a hydrogen envelope thinner
than the canonical value obtained from stable mass-transfer, i.e. from 10-4 to
10-10 M⊙. For all these sequences, we calculate adiabatic pulsations for
effective temperatures in the range $13\,000$ K $\leq T_{\rm eff}\leq$ 8000 K.

We perform a study on the pulsating low-mass WD in the binary system
J1152$+$0248 to uncover its inner chemical composition. By comparing the
determinations of the radius and effective temperature presented by Parsons et
al. (2020) with theoretical models (Istrate et al. 2021b, in preparation), we
find that the variable component of J1152$+$0248 must have a hydrogen envelope
thinner than that predicted from stable-mass transfer binary evolution
computations.

From the asteroseismological study we find a best fit model characterised by
M∗=0.338 M⊙, $T_{\rm eff}=10\,917$ K, MH/M${}_{\odot}=10^{-6}$ and a hybrid-
core composition. In particular, all local minima of $\chi^{2}$ correspond to
models with hybrid-core and a thin hydrogen-envelope.

A systematic study of the pulsational properties and the observed period
spectrum of low-mas WDs can give valuable information on the inner structure
of these objects. This will help in disentangling the two WD populations
coexisting in the mass interval $\sim$0.32–0.45 M⊙ which in turn leads to
clues of the underlying progenitor population.

## Data Availability

As an effort to promote open science, the results and codes to reproduce our
results are available at https://zenodo.org/communities/mesa/.

## Acknowledgements

ADR and GRL acknowledges the support of the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001, and by Conselho
Nacional de Desenvolvimento Científico e Tecnológico - Brazil (CNPq). AGI
acknowledges support from the Netherlands Organisation for Scientific Research
(NWO). SGP acknowledges the support of a Science and Technology Facilities
Council (STFC) Ernest Rutherford Fellowship. Special thanks to Pablo Marchant,
Josiah Schwab and Jocelyn Goldstein for their help with numerical debugging
and to the developers of MESA and GYRE for their continuous efforts to improve
and extend these codes. This research has made use of NASA’s Astrophysics Data
System and open source software as the Python 3 language (Python Core Team,
2015), the Ipython kernel (Pérez & Granger, 2007), the Jupyter Project
(Kluyver et al., 2016) and the packages Matplotlib (Hunter, 2007), Pandas
(McKinney, 2010) and the Scipy ecosystem (Virtanen et al., 2019).

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##  Appendix A Theoretical periods

$M_{H}/M_{\odot}$ | $10^{-4}$ | $10^{-5}$ | $10^{-6}$
---|---|---|---
$\ell$ | k | He | C/O | He | C/O | He | C/O
1 | 1 | 204.948 | 146.886 | 218.643 | 147.610 | 220.319 | 148.418
1 | 2 | 221.260 | 192.002 | 273.896 | 239.660 | 291.871 | 252.223
1 | 3 | 292.114 | 252.597 | 296.524 | 255.774 | 356.006 | 305.175
1 | 4 | 364.331 | 322.131 | 369.816 | 326.066 | 376.420 | 328.547
1 | 5 | 420.666 | 362.298 | 442.175 | 391.085 | 442.595 | 392.939
1 | 6 | 465.821 | 405.630 | 507.842 | 447.835 | 512.597 | 468.370
1 | 7 | 518.050 | 469.162 | 560.626 | 494.718 | 583.536 | 541.930
1 | 8 | 578.586 | 529.785 | 606.713 | 557.217 | 650.957 | 592.739
1 | 9 | 636.102 | 573.359 | 665.303 | 607.547 | 704.944 | 632.512
1 | 10 | 682.723 | 620.481 | 727.082 | 659.543 | 752.211 | 684.640
1 | 11 | 739.842 | 677.741 | 786.677 | 710.230 | 810.293 | 739.861
1 | 12 | 798.462 | 723.525 | 838.796 | 749.740 | 873.722 | 778.522
1 | 13 | 849.341 | 756.759 | 892.586 | 794.649 | 937.811 | 838.306
1 | 14 | 902.294 | 808.744 | 954.758 | 856.940 | 997.247 | 898.565
1 | 15 | 960.397 | 864.535 | 1014.63 | 912.796 | 1051.18 | 952.676
1 | 16 | 1014.40 | 919.221 | 1067.36 | 959.175 | 1106.24 | 991.396
1 | 17 | 1066.15 | 959.235 | 1122.26 | 1002.96 | 1168.11 | 1052.52
1 | 18 | 1122.16 | 1004.73 | 1182.63 | 1062.39 | 1231.03 | 1116.87
1 | 19 | 1177.61 | 1061.12 | 1240.57 | 1121.12 | 1289.88 | 1166.22
1 | 20 | 1229.32 | 1121.06 | 1294.31 | 1165.97 | 1343.50 | 1203.20
2 | 1 | 120.014 | 100.316 | 141.344 | 100.940 | 142.736 | 101.493
2 | 2 | 142.587 | 111.277 | 160.508 | 139.347 | 185.489 | 150.999
2 | 3 | 185.975 | 151.547 | 187.659 | 152.494 | 209.625 | 176.847
2 | 4 | 225.657 | 191.494 | 231.584 | 194.754 | 232.453 | 195.717
2 | 5 | 254.034 | 211.840 | 272.358 | 229.803 | 273.019 | 231.131
2 | 6 | 281.506 | 237.353 | 307.956 | 260.380 | 314.097 | 273.528
2 | 7 | 314.200 | 273.648 | 334.955 | 288.159 | 354.935 | 316.423
2 | 8 | 349.925 | 308.696 | 364.899 | 326.241 | 319.257 | 347.570
2 | 9 | 379.496 | 336.314 | 400.421 | 358.342 | 418.988 | 372.001
2 | 10 | 408.479 | 365.660 | 435.919 | 386.702 | 449.058 | 401.706
2 | 11 | 443.206 | 396.338 | 468.693 | 416.106 | 484.623 | 439.694
2 | 12 | 475.033 | 425.943 | 497.913 | 452.092 | 521.580 | 476.215
2 | 13 | 504.023 | 461.149 | 531.857 | 483.738 | 557.625 | 498.294
2 | 14 | 536.607 | 486.195 | 567.956 | 503.483 | 590.826 | 524.268
2 | 15 | 569.643 | 506.188 | 600.466 | 533.089 | 621.325 | 560.439
2 | 16 | 599.506 | 538.593 | 630.662 | 570.281 | 655.268 | 596.169
2 | 17 | 630.599 | 572.313 | 664.302 | 597.936 | 691.754 | 617.974
2 | 18 | 663.345 | 594.209 | 699.097 | 619.426 | 727.368 | 649.520
2 | 19 | 694.175 | 617.784 | 731.092 | 652.467 | 759.583 | 684.916
2 | 20 | 724.441 | 652.464 | 762.392 | 689.291 | 791.026 | 716.668
2 | 21 | 756.787 | 684.847 | 795.637 | 714.925 | 825.950 | 738.317
2 | 22 | 787.930 | 709.047 | 829.417 | 737.077 | 861.764 | 774.935
2 | 23 | 818.226 | 731.327 | 861.601 | 772.178 | 895.634 | 813.010
2 | 24 | 849.940 | 764.288 | 893.184 | 810.806 | 927.753 | 848.125
2 | 25 | 881.227 | 800.000 | 926.496 | 843.600 | 960.895 | 870.022
2 | 26 | 911.804 | 833.115 | 959.922 | 863.817 | 995.764 | 902.046
2 | 27 | 943.122 | 853.700 | 991.721 | 894.125 | 1030.54 | 939.114
2 | 28 | 974.435 | 879.080 | 1023.75 | 930.100 | 1063.74 | 966.625
2 | 29 | 1005.23 | 913.682 | 1057.21 | 956.318 | 1096.41 | 989.506
2 | 30 | 1036.10 | 941.506 | 1089.98 | 978.538 | 1130.42 | 1026.26

Table 4: Period values for $\ell=1$ and $\ell=2$ modes corresponding to models
with M${}_{*}=$ 0.338 M⊙, effective temperature of $12\,000$ K and hydrogen
envelope mass of 10-4, 10-5, and 10-6 M⊙.

$M_{H}/M_{\odot}$ | $10^{-4}$ | $10^{-5}$ | $10^{-6}$
---|---|---|---
$\ell$ | k | He | C/O | He | C/O | He | C/O
1 | 1 | 207.777 | 151.919 | 233.957 | 152.645 | 235.524 | 153.322
1 | 2 | 234.935 | 193.714 | 276.833 | 243.947 | 307.762 | 266.071
1 | 3 | 308.929 | 266.516 | 311.893 | 268.497 | 361.733 | 309.919
1 | 4 | 380.793 | 335.386 | 389.048 | 342.310 | 391.382 | 343.863
1 | 5 | 433.845 | 372.487 | 463.274 | 408.527 | 463.749 | 411.409
1 | 6 | 481.618 | 420.763 | 528.221 | 461.074 | 537.098 | 489.063
1 | 7 | 538.336 | 486.755 | 577.219 | 513.621 | 610.573 | 564.377
1 | 8 | 602.632 | 547.503 | 629.465 | 582.118 | 676.213 | 616.106
1 | 9 | 655.766 | 598.093 | 692.730 | 635.549 | 725.608 | 655.982
1 | 10 | 707.296 | 647.314 | 756.426 | 681.750 | 779.270 | 707.562
1 | 11 | 769.074 | 697.705 | 814.497 | 731.479 | 843.416 | 768.330
1 | 12 | 825.765 | 746.133 | 866.850 | 782.429 | 909.524 | 813.564
1 | 13 | 877.491 | 789.346 | 928.721 | 827.532 | 973.853 | 868.318
1 | 14 | 936.544 | 836.128 | 992.887 | 884.697 | 1032.09 | 927.724
1 | 15 | 994.745 | 890.138 | 1049.75 | 944.978 | 1087.01 | 992.624
1 | 16 | 1047.83 | 952.948 | 1104.48 | 1003.92 | 1149.66 | 1032.11
1 | 17 | 1104.56 | 1002.48 | 1166.22 | 1038.52 | 1215.34 | 1085.88
1 | 18 | 1162.76 | 1035.48 | 1227.62 | 1092.55 | 1277.54 | 1149.55
1 | 19 | 1216.77 | 1091.03 | 1283.50 | 1156.59 | 1333.40 | 1211.86
1 | 20 | 1272.22 | 1153.26 | 1340.63 | 1216.10 | 1392.04 | 1256.28
2 | 1 | 120.711 | 103.726 | 149.787 | 104.375 | 151.814 | 104.829
2 | 2 | 151.834 | 112.262 | 162.269 | 141.279 | 194.448 | 158.655
2 | 3 | 195.703 | 159.298 | 197.606 | 160.112 | 211.759 | 179.466
2 | 4 | 234.191 | 198.316 | 243.168 | 203.875 | 243.337 | 204.548
2 | 5 | 261.843 | 218.039 | 284.536 | 239.630 | 285.963 | 241.754
2 | 6 | 291.547 | 246.112 | 318.537 | 267.939 | 328.974 | 285.360
2 | 7 | 327.190 | 283.652 | 345.423 | 299.045 | 370.474 | 328.794
2 | 8 | 362.990 | 318.254 | 379.569 | 339.961 | 404.184 | 360.358
2 | 9 | 390.988 | 350.257 | 416.755 | 375.420 | 431.958 | 388.493
2 | 10 | 424.473 | 383.250 | 452.823 | 402.049 | 466.541 | 417.648
2 | 11 | 459.804 | 409.554 | 483.925 | 429.733 | 504.598 | 454.562
2 | 12 | 490.304 | 438.938 | 516.320 | 467.240 | 542.571 | 490.865
2 | 13 | 521.968 | 475.075 | 553.706 | 498.686 | 578.574 | 513.862
2 | 14 | 556.885 | 500.928 | 589.250 | 519.210 | 610.434 | 540.633
2 | 15 | 588.750 | 521.371 | 620.508 | 550.428 | 644.063 | 580.230
2 | 16 | 620.001 | 555.159 | 654.070 | 590.481 | 681.677 | 621.644
2 | 17 | 653.702 | 590.195 | 690.425 | 625.170 | 719.126 | 649.981
2 | 18 | 686.347 | 622.643 | 724.303 | 648.467 | 753.175 | 671.000
2 | 19 | 717.267 | 644.476 | 756.359 | 674.238 | 785.390 | 707.347
2 | 20 | 750.538 | 671.115 | 790.585 | 711.447 | 821.198 | 745.512
2 | 21 | 783.051 | 704.597 | 825.706 | 743.906 | 858.469 | 768.119
2 | 22 | 814.218 | 737.499 | 859.058 | 764.570 | 893.692 | 799.575
2 | 23 | 846.959 | 757.240 | 891.796 | 797.047 | 926.893 | 838.324
2 | 24 | 879.365 | 785.944 | 926.331 | 835.814 | 961.390 | 878.146
2 | 25 | 911.008 | 822.649 | 960.885 | 872.099 | 997.639 | 907.292
2 | 26 | 943.351 | 858.165 | 993.755 | 900.244 | 1033.59 | 933.155
2 | 27 | 975.716 | 887.886 | 1027.15 | 924.922 | 1067.91 | 970.154
2 | 28 | 1007.49 | 908.910 | 1061.95 | 960.303 | 1101.84 | 1008.85
2 | 29 | 1072.09 | 940.675 | 1095.65 | 997.176 | 1137.52 | 1034.44
2 | 30 | 1103.79 | 975.775 | 1128.77 | 1024.26 | 1173.68 | 1060.13

Table 5: Period values for $\ell=1$ and $\ell=2$ modes corresponding to models
with M${}_{*}=$ 0.338 M⊙, effective temperature of $11\,000$ K and hydrogen
envelope mass of 10-4, 10-5, and 10-6 M⊙.

$M_{H}/M_{\odot}$ | $10^{-4}$ | $10^{-5}$ | $10^{-6}$
---|---|---|---
$\ell$ | k | He | C/O | He | C/O | He | C/O
1 | 1 | 209.850 | 157.321 | 252.328 | 157.951 | 254.801 | 158.549
1 | 2 | 253.518 | 195.924 | 280.175 | 248.189 | 326.917 | 283.059
1 | 3 | 329.045 | 283.798 | 331.577 | 285.471 | 366.984 | 315.927
1 | 4 | 398.872 | 349.747 | 411.843 | 361.596 | 412.719 | 362.727
1 | 5 | 449.743 | 386.155 | 487.861 | 248.295 | 490.081 | 433.524
1 | 6 | 501.311 | 439.089 | 550.052 | 476.945 | 567.284 | 513.238
1 | 7 | 564.292 | 507.595 | 598.176 | 536.112 | 642.441 | 588.261
1 | 8 | 629.238 | 566.539 | 658.379 | 608.287 | 703.193 | 639.718
1 | 9 | 679.188 | 625.059 | 725.480 | 662.554 | 752.760 | 680.114
1 | 10 | 739.471 | 673.875 | 790.184 | 704.021 | 815.755 | 733.170
1 | 11 | 802.594 | 718.873 | 845.394 | 756.443 | 884.549 | 800.999
1 | 12 | 856.823 | 772.083 | 904.980 | 819.879 | 953.196 | 856.930
1 | 13 | 914.781 | 828.101 | 972.970 | 868.674 | 1017.90 | 905.083
1 | 14 | 977.584 | 871.154 | 1036.10 | 918.761 | 1075.56 | 964.206
1 | 15 | 1034.07 | 923.283 | 1092.89 | 982.771 | 1138.42 | 1034.76
1 | 16 | 1091.60 | 988.449 | 1155.57 | 1045.98 | 1208.10 | 1074.30
1 | 17 | 1153.16 | 1042.38 | 1222.19 | 1076.96 | 1277.21 | 1124.88
1 | 18 | 1211.49 | 1072.43 | 1284.28 | 1129.61 | 1341.23 | 1190.77
1 | 19 | 1268.95 | 1127.11 | 1344.28 | 1199.66 | 1402.32 | 1264.19
1 | 20 | 1330.48 | 1189.77 | 1408.16 | 1267.85 | 1468.56 | 1325.45
2 | 1 | 121.586 | 107.287 | 157.518 | 107.919 | 163.156 | 108.312
2 | 2 | 163.416 | 113.564 | 167.306 | 143.524 | 203.910 | 168.026
2 | 3 | 206.981 | 168.965 | 209.618 | 169.896 | 216.181 | 183.062
2 | 4 | 243.544 | 205.790 | 256.828 | 214.873 | 257.004 | 215.468
2 | 5 | 271.659 | 226.439 | 298.421 | 250.869 | 302.091 | 254.650
2 | 6 | 304.270 | 256.870 | 329.841 | 277.285 | 347.023 | 299.360
2 | 7 | 343.101 | 295.672 | 359.639 | 312.151 | 388.134 | 342.354
2 | 8 | 376.609 | 329.099 | 397.374 | 355.056 | 418.378 | 374.155
2 | 9 | 406.703 | 365.975 | 436.177 | 392.490 | 450.250 | 406.368
2 | 10 | 443.943 | 400.944 | 471.367 | 418.069 | 489.031 | 434.251
2 | 11 | 478.181 | 424.514 | 502.886 | 445.030 | 529.062 | 470.434
2 | 12 | 509.339 | 453.685 | 540.512 | 483.040 | 568.103 | 506.993
2 | 13 | 545.056 | 489.279 | 579.302 | 514.825 | 603.410 | 532.999
2 | 14 | 580.024 | 516.482 | 613.501 | 538.859 | 637.183 | 562.265
2 | 15 | 612.022 | 540.797 | 647.314 | 572.300 | 675.967 | 604.029
2 | 16 | 646.904 | 574.901 | 685.322 | 612.391 | 716.491 | 646.414
2 | 17 | 981.976 | 609.741 | 722.939 | 649.241 | 755.304 | 682.219
2 | 18 | 714.771 | 647.571 | 757.683 | 684.100 | 790.917 | 702.311
2 | 19 | 749.324 | 679.309 | 793.296 | 703.049 | 827.350 | 735.388
2 | 20 | 784.642 | 696.728 | 831.114 | 737.108 | 866.985 | 777.381
2 | 21 | 818.136 | 728.585 | 868.816 | 774.163 | 907.360 | 807.014
2 | 22 | 852.723 | 764.778 | 904.819 | 802.704 | 946.459 | 832.156
2 | 23 | 888.137 | 792.796 | 941.137 | 828.950 | 984.102 | 871.453
2 | 24 | 922.537 | 815.698 | 979.211 | 866.280 | 1021.99 | 914.490
2 | 25 | 957.299 | 850.262 | 1017.29 | 905.561 | 1061.17 | 951.177
2 | 26 | 992.689 | 887.230 | 1053.93 | 942.024 | 1101.43 | 976.345
2 | 27 | 1027.87 | 923.354 | 1090.87 | 966.846 | 1141.57 | 1011.69
2 | 28 | 1062.61 | 950.145 | 1129.15 | 998.759 | 1180.46 | 1053.86
2 | 29 | 1098.14 | 975.476 | 1167.12 | 1038.43 | 1219.08 | 1088.51
2 | 30 | 1133.72 | 1011.07 | 1204.56 | 1074.88 | 1258.10 | 1110.10

Table 6: Period values for $\ell=1$ and $\ell=2$ modes corresponding to models
with M${}_{*}=$ 0.338 M⊙, effective temperature of $10\,000$ K and hydrogen
envelope mass of 10-4, 10-5, and 10-6 M⊙.

$M_{H}/M_{\odot}$ | $10^{-4}$ | $10^{-5}$ | $10^{-6}$
---|---|---|---
$\ell$ | k | He | C/O | He | C/O | He | C/O
1 | 1 | 211.971 | 163.184 | 272.688 | 163.728 | 278.506 | 164.253
1 | 2 | 276.632 | 198.670 | 286.577 | 253.161 | 349.212 | 303.724
1 | 3 | 352.387 | 305.461 | 355.981 | 307.717 | 375.098 | 324.564
1 | 4 | 418.508 | 365.398 | 439.196 | 385.267 | 440.661 | 386.324
1 | 5 | 469.126 | 404.819 | 516.118 | 451.060 | 523.307 | 460.588
1 | 6 | 526.313 | 461.883 | 573.907 | 497.554 | 604.686 | 542.008
1 | 7 | 595.997 | 532.423 | 627.193 | 563.328 | 680.545 | 615.165
1 | 8 | 657.552 | 589.357 | 695.003 | 637.210 | 738.264 | 665.314
1 | 9 | 711.747 | 653.526 | 766.831 | 686.424 | 797.295 | 704.490
1 | 10 | 780.216 | 697.500 | 833.735 | 731.373 | 868.130 | 768.783
1 | 11 | 843.534 | 745.433 | 893.015 | 793.537 | 942.846 | 847.662
1 | 12 | 901.897 | 808.247 | 961.966 | 869.583 | 1019.19 | 918.324
1 | 13 | 968.255 | 875.514 | 1036.37 | 928.897 | 1094.93 | 970.636
1 | 14 | 1035.46 | 925.732 | 1106.85 | 979.084 | 1164.70 | 1031.76
1 | 15 | 1097.32 | 977.308 | 1172.29 | 1045.33 | 1233.01 | 1100.78
1 | 16 | 1162.30 | 1043.90 | 1241.03 | 1107.19 | 1306.57 | 1139.70
1 | 17 | 1230.74 | 1099.27 | 1314.57 | 1145.56 | 1383.92 | 1213.45
1 | 18 | 1296.20 | 1132.24 | 1387.83 | 1214.51 | 1462.30 | 1297.58
1 | 19 | 1360.81 | 1197.61 | 1458.53 | 1291.93 | 1539.71 | 1374.64
1 | 20 | 1429.04 | 1270.06 | 1527.91 | 1369.63 | 1614.85 | 1424.85
2 | 1 | 122.673 | 111.022 | 161.731 | 111.709 | 177.473 | 112.048
2 | 2 | 177.534 | 115.239 | 179.106 | 146.305 | 211.221 | 178.889
2 | 3 | 219.601 | 181.074 | 224.211 | 182.664 | 226.761 | 188.896
2 | 4 | 254.109 | 214.244 | 273.063 | 228.523 | 274.394 | 229.265
2 | 5 | 283.806 | 237.817 | 313.728 | 263.906 | 322.327 | 270.651
2 | 6 | 320.563 | 270.466 | 343.764 | 289.721 | 369.126 | 316.276
2 | 7 | 361.292 | 310.114 | 378.605 | 328.255 | 408.989 | 358.240
2 | 8 | 392.415 | 342.644 | 419.762 | 372.913 | 440.332 | 391.682
2 | 9 | 428.280 | 384.076 | 460.811 | 410.346 | 478.305 | 423.698
2 | 10 | 467.738 | 417.823 | 496.879 | 433.479 | 507.736 | 451.141
2 | 11 | 502.082 | 439.454 | 532.901 | 463.571 | 564.517 | 493.970
2 | 12 | 537.746 | 471.766 | 575.233 | 507.100 | 608.865 | 537.519
2 | 13 | 577.290 | 510.640 | 617.606 | 544.610 | 651.321 | 570.193
2 | 14 | 614.507 | 543.309 | 656.751 | 574.361 | 690.583 | 602.753
2 | 15 | 650.556 | 573.042 | 694.888 | 610.217 | 731.419 | 647.504
2 | 16 | 689.501 | 608.268 | 736.295 | 655.098 | 775.176 | 697.196
2 | 17 | 728.692 | 646.436 | 779.002 | 698.230 | 820.347 | 733.340
2 | 18 | 765.618 | 688.715 | 820.721 | 730.570 | 865.547 | 756.089
2 | 19 | 804.125 | 724.033 | 861.022 | 752.829 | 909.721 | 799.923
2 | 20 | 843.683 | 741.825 | 901.474 | 797.069 | 952.601 | 844.521
2 | 21 | 882.106 | 778.876 | 943.447 | 839.523 | 995.140 | 871.128
2 | 22 | 920.408 | 820.648 | 986.213 | 865.190 | 1038.45 | 909.888
2 | 23 | 959.520 | 851.564 | 1028.20 | 898.880 | 1082.74 | 959.228
2 | 24 | 998.867 | 876.304 | 1069.13 | 645.024 | 1127.70 | 1003.20
2 | 25 | 1037.62 | 916.100 | 1110.60 | 988.842 | 1172.86 | 1032.01
2 | 26 | 1076.42 | 958.321 | 1153.10 | 1019.67 | 1217.46 | 1070.84
2 | 27 | 1116.01 | 995.860 | 1195.53 | 1052.01 | 1261.71 | 1117.93
2 | 28 | 1155.22 | 1022.68 | 1237.56 | 1095.90 | 1305.52 | 1157.13
2 | 29 | 1194.12 | 1057.21 | 1279.30 | 1137.63 | 1349.21 | 1187.67
2 | 30 | 1233.58 | 1097.80 | 1321.30 | 1166.17 | 1393.53 | 1231.34

Table 7: Period values for $\ell=1$ and $\ell=2$ modes corresponding to models
with M${}_{*}=$ 0.338 M⊙, effective temperature of  $9\,000$ K and hydrogen
envelope mass of 10-4, 10-5, and 10-6 M⊙.


# Substrate enhanced superconductivity in Li-decorated graphene

T. P. Kaloni1, A. V. Balatsky2,3,4, and U. Schwingenschlögl1
<EMAIL_ADDRESS>1PSE Division, KAUST, Thuwal 23955-6900,
Kingdom of Saudi Arabia
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA
3Center for Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA
4Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm Sweden

###### Abstract

We investigate the role of the substrate for the strength of the electon
phonon coupling in Li-decorated graphene. We find that the interaction with a
$h$-BN substrate leads to a significant enhancement from $\lambda_{0}=0.62$ to
$\lambda_{1}=0.67$, which corresponds to a 25% increase of the transition
temperature from $T_{c0}=10.33$ K to $T_{c1}=12.98$ K. The superconducting
gaps amount to 1.56 meV (suspended) and 1.98 meV (supported). These findings
open up a new route to enhanced superconducting transition temperatures in
graphene-based materials by substrate engineering.

###### pacs:

74.78.Db, 63.20.Dj, 81.05.Uw, 31.15.Ar

##  I Introduction

Recent observations in alkali-doped graphene have opened exciting venues to
superconductivity accomplished by doping mouri . Most theoretical estimates of
the electron-phonon coupling so far have assumed suspended graphene as a base,
since this geometry makes calculations more direct and less computationally
costly. However, most of the engineered superconducting graphene samples use a
substrate. Hence, it is important to characterize the role of the substrate on
the superconductivity in atomically thin graphene. We have performed a first-
principles study of the role of the substrate on the phonon spectrum and the
electron-phonon coupling and find not only that the interaction with the
substrate is relevant but that in the case of a $h$-BN substrate the electron-
phonon coupling can be enhanced by as much as 9% so that $T_{c}$ can be
expected to reach 12.98 K, a 25% increase. This observation points to a new
direction in the search for novel superconducting materials: substrate-
engineered superconductivity, where the nascent superconducting states are
significantly enhanced by the coupling to a properly chosen substrate.

Graphite intercalated compounds are characterized by a nearly free electron
band, which upon increased doping crosses the Fermi energy ($E_{F}$).
Empirically, there are intercalated compounds that exhibit superconductivity
with a transition temperature of a few to about 10 K. An empirical correlation
between the crossing of the chemical potential and the onset of
superconductivity was first put forward in Ref. Littlewood  and subsequently
was called the “Cambridge criterion” balatsky . Superconductivity in Ca-
intercalated bilayer graphene has been predicted with a sizable $T_{c}=11.5$ K
by analyzing this criterion in Refs. balatsky ; jishi . Recently, the
prediction has been verified experimentally for Ca-intercalated graphene on
either the Si or the C face of a SiC substrate, finding $T_{c}=7$ K Li .
Experimentally, it also has been observed that KC8 graphite valla  and
K-intercalated few layer graphene on SiC are superconducting jacs , where
theoretical arguments for superconductivity in the latter material have been
presented in Ref. kaloni-epl . To complete the list of superconducting C
allotropes we also mention that undoped single and multiwall nanotubes exhibit
superconductivity with a sizable $T_{c}\sim 10$ to 12 K Tang ; Takesue .
Today, the highest value of $T_{c}=38$ K is observed experimentally in Cs3C60
Alexey .

It was already pointed out that not all intercalants lead to an enhanced
$T_{c}$ in graphite Littlewood . A high $T_{c}$ can be obtained when the
distance between the intercalated atom and the graphene plane is small so that
the deformation potential is large Boeri . Our observations are consistent
with this mechanism: The distance between the intercalant and the graphene
plane is 2.62 Å, 2.47 Å, and 2.26 Å for non-superconducting BaC6 Boeri1 ,
superconducting SrC6 with $T_{c}=1.65$ K Boeri1 , and superconducting CaC6
with $T_{c}=11.5$ K Emery ; Ellerby , respectively, see Table I.

compound | distance | $T_{c}$
---|---|---
BaC6 | 2.62 Å | 0 K
SrC6 | 2.47 Å | 1.65 K
CaC6 | 2.26 Å | 11.5 K
Table 1: Compound, perpendicular distance of the intercalated atom from the
center of the C hexagon, and superconducting transition temperature.

In this paper we report a first-principles study of the electron-phonon
coupling to estimated the values of $\lambda$ and $T_{c}$ for Li-decorated
suspended graphene and Li-decorated graphene on a $h$-BN substrate. We show
that the presence of the substrate enhances the electron-phonon coupling and
superconducting transition temperature, which reflects a significant impact of
the interaction of the electronic states with the substrate on the phonon
mediated superconductivity in doped graphene.

##  II Results and discussion

The unit cell of Li-decorated monolayer graphene comprises 6 C atoms and 1 Li
atom in a $\sqrt{3}\times\sqrt{3}R30{{}^{\circ}}$ geometry, where the Li atom
lies above the center of the C hexagon in a distance of 1.76 Å, slightly
smaller than the value reported in Ref. mouri . The possible reason for the
latter is inclusion of the van der Waals interaction in our calculations,
which is expected to provide a correct interlayer spacing. The structural
arrangements of Li-decorated graphene suspended and supported by a $h$-BN
substrate are presented in Figs. 1(a) and 1(b). The electronic band structures
obtained for $\sqrt{3}\times\sqrt{3}R30{{}^{\circ}}$ suspended graphene
without and with Li-decoration are shown in Figs. 2(a) and 2(b). It is well
known that the C $\pi$ and $\pi^{*}$ orbitals form a Dirac cone at the Fermi
energy. Due to Brillouin zone backfolding, the Dirac cone appears at the
$\Gamma$-point and not at the K-point as in the case of the primitive unit
cell of graphene.

Figure 1: Crystal structure of Li-decorated graphene (a) suspended and (b)
supported by a $h$-BN substrate.

The electronic band structure of Li-decorated graphene is found to be modified
significantly as compared to that of pristine graphene. The nearly free
electron Li $s$ band crosses the Fermi level, due to charge transfer from Li
to C. As a consequence, the “Cambridge criterion” is satisfied and the system
should be a superconductor. We will comment later on this phonomenon by
analyzing the strength of the electron-phonon coupling. A gap of 0.38 eV opens
1.56 eV below the Fermi level, as to be expected kaloni-cpl ; Farjam . In Fig.
2(b) the partially occupied parabolic bands indicated by arrows are due to Li
$s$ states, compare Figs. 2(a) and 2(b). It has been reported that the carrier
density in Li-decorated monolayer and Li-intercalated multilayer graphene with
and without substrate can differ by a factor of 100 from that of pristine
graphene kaloni-cpl .

Figure 2: Electronic band structure of (a) suspended C6, (b) suspended C6Li,
(c) C6 on $h$-BN, and (d) C6Li on $h$-BN. Li $s$ bands crossing the Fermi
level are indicated by arrows.

For Li-decorated graphene on $h$-BN, see Fig. 1(b), the separation between
graphene and the substrate is found to be 3.39 Å, which is close to the values
for superlattices of graphene and $h$-BN as well as graphene on a $h$-BN
substrate. The perpendicular distance of the Li atom to the graphene plane is
1.77 Å. The lost sublattice symmetry (only each third C hexagon is occupied by
a Li atom) is responsible for a band gap of 90 meV. This value agrees well
with previous reports Quhe ; Naveh ; kaloni-jmc , which also applies to the
fact that the B and N states appear far away from the Fermi level. The
electronic band structure in Fig. 2(d) clearly shows that a nearly free
electron Li $s$ band crosses the Fermi level, satisfying the “Cambridge
criterion” and thus pointing to superconductivity in the system. The nature
and magnitude of the gap at the $\Gamma$-point just below the Fermi level are
similar to Fig. 2(b).

At this point we assume that the basic mechanism of the superconductivity in
the suspended and supported cases is the same as in Ca-intercalated graphene,
i. e., electron-phonon driven pairing balatsky . It has been proposed that the
dopant-induced soft phonon modes contribute substantially to the electron-
phonon coupling Mazin ; Littlewood  and it is known that the motion of the
adatom is responsible for about half of the coupling, while the other half is
due to the C atoms Calandra ; Sanna ; Rosenmann . The presence of the Li $s$
states around the Fermi energy alone cannot be sufficient to give a large
electron-phonon coupling mouri , but the coupling to the out-of-plane C
vibrations plays an important role due to transitions between the C $\pi^{*}$
and Li $s$ states. The Li $s$ band enhances the coupling Boeri  and, hence,
the transition temperature. For this reason, we calculate the phonon
dispersion, see Fig. 3(a), and $\alpha^{2}F(\omega)$, see Fig. 3(b), and
estimate the strength of the electron-phonon coupling $\lambda$ using Eq. (2).

For the phonon dispersion of Li-decorated suspended graphene we find that most
modes between 300 cm-1 and 500 cm-1 are due to a mixture of Li and out-of-
plane C vibrations. The pure out-of-plane modes appear from 500 cm-1 to 900
cm-1 and higher energy C-C stretching modes from 900 cm-1 to 1515 cm-1. The
modes from 300 cm-1 to 500 cm-1 are responsible for the electron-phonon
coupling. This also can be seen from $\alpha^{2}F(\omega)$ as addressed in
Fig. 3(b). Experimentally, for pristine graphene the frequency of the G-mode
is 1580 cm-1 mouri1 , which softens to 1515 cm-1 under Li decoration. The
softening can be attributed to charge transfer from Li to graphene and the
induced stronger electron-phonon coupling, in agreement with findings for the
molecular/atomic charge transfer in graphene Rao ; carbon . We obtain for the
electron-phonon coupling $\lambda=0.62$ and estimate for the superconducting
transition temperature $T_{c}=10.33$ K. This value is slightly higher than
that of Ref. mouri , since we take into account the van der Waals interaction
to achieve an accurate distance to the Li atom.

Figure 3:  Electron-phonon dispersion of Li-decorated graphene (a) suspended
and (c) supported by a $h$-BN substrate. (b,c) Corresponding Eliashberg
functions.

Our central result is that an enhancement of the superconductivity in Li-
decorated graphene can be achieved by the application of a $h$-BN substrate.
The phonon modes between 100 cm-1 and 300 cm-1 at the $\Gamma$-point are
attributed to the Li vibrations, out-of-plane C vibrations, and $h$-BN
substrate, see Fig. 3(c). There are also substrate modes around 850 cm-1 as
well as higher energy modes between 900 cm-1 and 1430 cm-1, which are due to
both the substrate and C-C stretching. The softening of the modes as compared
to the suspended system is due to the interaction with the substrate, as
observed experimentally, for example, in graphite supported by Ni(111) new .
The modes in the range from 100 cm-1 to 300 cm-1 are responsible for a shift
in $\alpha^{2}F(\omega)$ see Fig. 3(d), and enhancement of $\lambda$ and
$T_{c}$. We obtain $\lambda=0.67$ (as compared to $\lambda=0.62$ in the
suspended system) and thus a higher $T_{c}=12.98$ K, which is a 25% increase
with respect to the suspended system. The clearly indicates that one can take
full advantage of the substrate to boost $T_{c}$. Similarly, it has been
observed experimentally in the FeSe0.5Te0.5 superconductor that $T_{c}$ is
enhanced by 15% if the material is supported Johnson , while the details of
the band structure are very different in the present material. The most likely
reason for the obtained enhancement of $T_{c}$ by the application of a $h$-BN
substrate are stronger spin fluctuations due to the lattice mismatch of 1.4%.
Finally, we estimate the superconducting gap $\Delta_{sc}$ by the relation
$1.75k_{B}T_{c}=\Delta_{sc}$ Parker , where $k_{B}$ is the Boltzmann constant.
We obtain for Li-decorated suspended and supported graphene, respectively,
values of 1.56 meV and 1.98 meV.

##  III conclusion

In conclusion, using density functional theory we have investigated the role
of the substrate for the electron-phonon coupling in Li-decorated suspended
and supported graphene. We find that the interaction with a $h$-BN substrate
significantly enhances the electron-phonon coupling to $\lambda=0.67$ as
compared to $\lambda=0.62$ in the suspended case. The transition temperature
thus is enhanced by 25% to 12.98 K. The superconducting gap for the suspended
and supported systems is found to be 1.56 meV and 1.98 meV, respectively. Our
results show that graphene-based nanomaterials can be tailored by properly
choosing the substrate to robustly increase the superconducting transition
temperature.

###### Acknowledgements.

We thank G. Profeta for fruitful discussions. This work is supported by US
DOE, ERC-DM-321031, and VR.

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##  IV Appendix

###  IV.1 Computational details

All the results are obtained from density functional theory in the local
density approximation. The van der Waals interaction is taken into account via
Grimme’s scheme grime . We use the Quantum-ESPRESSO code paolo  with norm-
conserving pseudopotentials and a plane wave cutoff energy of 70 Ryd. A
Monkhorst-Pack $32\times 32\times 1$ k-mesh is employed for the optimization
of the lattice parameters and the ionic relaxation and a $48\times 48\times 1$
k-mesh for refining the electronic structure. We achieve an energy convergence
of $10^{-7}$ eV and a force convergence of 0.002 eV/Å. Li-decorated monolayer
graphene is modeled by a $\sqrt{3}\times\sqrt{3}R30{{}^{\circ}}$ supercell
with $a=b=2.26$ to that a Li atom is added on each third hollow site. Phonon
frequencies are determined by density functional perturbation theory for
evaluating the effects of the adatoms on the phonon spectrum, using the scheme
described in Ref. Mod . The phonon dispersion is calculated with a $24\times
24\times 1$ k-mesh. We study the effect of the substrate on the strength of
the electron-phonon coupling and the transition temperature for a supercell
with Li-decorated monolayer graphene on top of $h$-BN with $a=b=4.32$ Å and
$c=15$ Å (to avoid artificial interaction due to the periodicity). Note that
graphene on $h$-BN can be synthesized due to the small lattice mismatch of
only 1.4% and interacts only weakly with the substrate Dean ; Xue ; Yang . By
construction of the supercell of the suspended system, with 6 C atoms and 1 Li
atom, there are 21 phonon modes, whereas we have 39 modes for the supported
system.

###  IV.2 Superconducting transition temperature

The Allen-Dynes formula dynes ; allen , which is a modification of McMillan’s
formula mcmillan , is used to calculate

$T_{c}=\frac{<\omega>_{\log}}{1.20}\exp\Big{(}-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\Big{)}.$
(1)

The terms $<\omega>_{\log}$, $\lambda$, and $\mu^{*}$ are the logarithmic
frequency average, electron-phonon coupling constant, and effective Coulomb
repulsion, respectively. Moreover, the dimensionless parameter

$\lambda=2\int_{0}^{\infty}\frac{d\omega\alpha^{2}F(\omega)}{\omega}$ (2)

measures the strength of the Eliashberg function

$\alpha^{2}F(\omega)=N_{\uparrow}(0)\frac{\sum_{kk^{\prime}}|M_{kk^{\prime}}|^{2}\delta(\omega-\omega_{q})\delta(E_{k})\delta(E_{k^{\prime}})}{\sum_{kk^{\prime}}\delta(E_{k})\delta(E_{k^{\prime}})},$
(3)

where $k$ and $q$ represent the electron band index and phonon wave number,
respectively. In addition, $N_{\uparrow}(0)$ is the single-spin density of
states at the Fermi surface and $M_{kk^{\prime}}$ is the matrix element for
electron-phonon coupling. The effective Coulomb repulsion (also called Coulomb
pseudopotential) is given by anderson

$\frac{1}{\mu^{*}}=\frac{1}{\mu}+\ln\left(\frac{\omega_{el}}{\omega_{ph}}\right),$
(4)

where $\omega_{el}$ is the plasma frequency and $\omega_{ph}$ the frequency
cutoff in $\alpha^{2}F(\omega)$. The Coulomb coupling $\mu$ is given by the
product of the density of states at the Fermi surface and the matrix element
of the screened Coulomb interaction averaged over the Fermi surface. We use
$\mu^{*}=0.115$ in agreement with the experimental observation of the critical
temperature for bulk CaC6 Emery