# Phonon-Mediated Quasiparticle Lifetime Renormalizations in Few-Layer
Hexagonal Boron Nitride

Håkon I. Røst<EMAIL_ADDRESS>Department of Physics and Technology,
University of Bergen, Allégaten 55, 5007 Bergen, Norway.  Simon P. Cooil
Department of Physics and Centre for Materials Science and Nanotechnology,
University of Oslo (UiO), Oslo 0318, Norway.  Anna Cecilie Åsland  Department
of Physics, Norwegian University of Science and Technology (NTNU), NO-7491
Trondheim, Norway.  Jinbang Hu  Department of Physics, Norwegian University of
Science and Technology (NTNU), NO-7491 Trondheim, Norway.  Ayaz Ali
Department of Smart Sensor Systems, SINTEF DIGITAL, Oslo, 0373, Norway.
Takashi Taniguchi  International Center for Materials Nanoarchitectonics,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.
Kenji Watanabe  Research Center for Functional Materials, National Institute
for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.  Branson D. Belle
Department of Smart Sensor Systems, SINTEF DIGITAL, Oslo, 0373, Norway.  Bodil
Holst  Department of Physics and Technology, University of Bergen, Allégaten
55, 5007 Bergen, Norway.  Jerzy T. Sadowski  Center for Functional
Nanomaterials, Brookhaven National Laboratory, Upton, New York, 11973, USA.
Federico Mazzola  Department of Molecular Sciences and Nanosystems, Ca’
Foscari University of Venice, 30172 Venice, Italy.  Justin W. Wells
<EMAIL_ADDRESS>Department of Physics, Norwegian University of Science
and Technology (NTNU), NO-7491 Trondheim, Norway.

###### Abstract

Understanding the collective behavior of the quasiparticles in solid-state
systems underpins the field of non-volatile electronics, including the
opportunity to control many-body effects for well-desired physical phenomena
and their applications. Hexagonal boron nitride (hBN) is a wide energy bandgap
semiconductor, showing immense potential as a platform for low-dimensional
device heterostructures. It is an inert dielectric used for gated devices,
having a negligible orbital hybridization when placed in contact with other
systems. Despite its inertness, we discover a large electron mass enhancement
in few-layer hBN affecting the lifetime of the $\pi$-band states. We show that
the renormalization is phonon-mediated and consistent with both single- and
multiple-phonon scattering events. Our findings thus unveil a so-far unknown
many-body state in a wide-bandgap insulator, having important implications for
devices using hBN as one of their building blocks.

Keywords: Hexagonal Boron Nitride, Graphene Heterostructures, Many-Body
Interactions, Electron-Phonon Coupling, ARPES.

Department of Physics, Norwegian University of Science and Technology (NTNU),
NO-7491 Trondheim, Norway. Department of Electronic Engineering, Faculty of
Engineering & Technology, University of Sindh, Jamshoro, 76080, Pakistan.
Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche, Trieste
I-34149, Italy. Department of Physics and Centre for Materials Science and
Nanotechnology, University of Oslo (UiO), Oslo 0318, Norway. ARPES, Many-Body
effects, electron-phonon coupling

Hexagonal Boron Nitride (hBN) is an inert layered compound that has gained
significant attention for its compatibility with the vast majority of low-
dimensional van der Waals (vdW) materials 1, 2, 3, 4, 5, 6, 7, 8. It is
strikingly similar to graphene both in lateral size, crystalline structure,
and Debye frequency, but due to its dissimilar sub-lattices, it hosts a wide
energy band gap separating the valence and conduction bands 9, 10. For the
engineering of vdW heterostructures embedded in the form of devices, hBN has
proven to be a key building block due to its large capacitive coupling and
current tunneling barrier 11, 12, 13, 7, 14. Furthermore, its chemical
inertness, large energy bandgap, and high phonon energies have made it one of
the most common dielectrics for use in state-of-the-art, low-dimensional
devices that require atomic-scale flatness, and negligible interface doping
and scattering 1, 3, 15, 16, 17.

Recently, hBN has been predicted to host a strong electron-phonon coupling
which can compromise the performance of hBN-derived electronic devices 18, 19.
However, experimental proof of such couplings has so far been lacking. Herein,
we investigate the many-body effects of few-layer hBN supported on graphene.
We discover and quantify the predicted electron-phonon coupling, and also an
additional scattering effect that, together, significantly renormalizes the
hBN $\pi$-band states. The latter effect is found to be consistent with a
scattering process involving multiple phonons. Thus, our findings are crucial
for understanding the many-body states in hBN-based devices, and for achieving
increased control over their performances.

Figure 1: Exfoliated hBN on graphene/SiC. a: The preparation of few-layer hBN
by exfoliating from a bulk crystal and transferring onto graphene/SiC using a
polymer (PMMA) film. b: PEEM micrograph of hBN on graphene/SiC. The region
encompassed by the dashed line (green) is the relevant hBN region used for
further analysis. c: $\mu$-LEED from a $1.5~{}\mu$m diameter area on the hBN
region marked in b, collected with electron energy 35 eV. d: LEEM I-V spectra
collected from the circular areas marked in b. The dips in the I-V curves
reveal that the hBN (orange) consists of 8 layers, while the substrate (grey)
is mainly monolayer graphene. The curves have been normalized to the maximum
intensity at the mirror electron microscopy energy threshold ($\approx 0.4$
eV). The hBN curve has been scaled ($\times 6$) and offset in intensity for
improved readability.

Few (5$-$10) layer hBN flakes were exfoliated from bulk material and
transferred via a polymer stack onto a substrate of epitaxial graphene on
6H-SiC(0001) (Fig. 1a). The material was then heated in ultrahigh vacuum
(pressure $\leq 1\times 10^{-9}$ mbar) to remove any residues of polymer from
the transfer process (see the Supporting Information for details 20). Using
photoemission electron microscopy (PEEM), we selected a high-quality hBN flake
of approximately $35\times 20~{}\mu\text{m}^{2}$ lateral size (Fig. 1b). The
crystalline quality across the flake was ascertained from small-area low-
energy electron diffraction ($\mu$-LEED, 1.5 $\mu\text{m}$ diameter spot),
showing no appreciable variation across the full flake area (Fig. 1c). The
diffraction pattern taken at an electron energy of 35 eV revealed six first-
order spots as expected for a stack of rotationally aligned hBN layers.

The number of vdW-bonded hBN layers in the flake was ascertained from low-
energy electron reflectivity (LEER) measurements (Fig. 1d). The incident,
coherent electron beam was tuned in the range 0-10 eV, and the corresponding
electron reflectivity at each energy was recorded from micrographs of the
surface. The averaged I–V characteristics from areas on and adjacent to the
hBN flake (circles in Fig. 1b) reveal dips in the reflected intensity at
energies beyond the mirror electron microscopy energy threshold at $\approx
0.4$ eV. These dips represent the transmission of electrons into discrete and
unoccupied states above the vacuum energy level. In the case of graphene or
hBN, the minima are given by the unoccupied $\pi^{*}$-states of each atomic
layer. The number of minima $n$ observed can hence be linked directly to the
$n+1$ stacked layers present 21, 22, 23, 24. From the substrate region (grey),
one dominant dip ($\approx 2.1$ eV) can be observed between the higher
intensity regions (0.5 and 6.0 eV), suggesting the substrate is mainly
monolayer graphene on top of $(6\sqrt{3}\times
6\sqrt{3})\text{R}30^{\circ}$-reconstructed SiC(0001) 21. In comparison, the
hBN flake (orange) shows 7 dips originating from 8 stacked layers 23. For the
electronic structure measurements, we will refer to this 8-layer region from
now on.

The electronic structure of 8-layer hBN on graphene is mapped out at room
temperature in Fig. 2 for binding energies near the top of the $\pi$-band
($E_{\text{VBM}}$). The occupied bandstructure as shown was measured as a
function of constant energy and simultaneously across all momentum vectors of
the first Brillouin zone (BZ), using an aberration-corrected momentum
microscope with its energy and momentum resolutions set to 50 meV and
$0.02~{}\text{Å}^{-1}$, respectively 25, 26, 20. The measured $\pi$-states
reveal several signatures of electron scattering. Notably, a broadening
appears along the $\bar{\Gamma}-\bar{\text{K}}$ direction at approximately
$E_{\text{VBM}}+1$ eV (blue arrows) where also two faint and linearly
dispersing features appear (grey arrow/circles), intersecting with the hBN
$\pi$-band. Additional strong signs of broadening (green arrows) can be seen
along the $\bar{\text{K}}-\bar{\text{M}}$ direction for energies closer to the
valence band maximum (VBM).

Sudden broadenings of the electronic structure like the ones mentioned are
typical hallmarks of electron-boson interactions 27, 28, 29, 30. They signify
a direct change in the quasiparticle lifetime $\tau$ – or said differently, a
reduction of the time an electron uses to fill a photohole 28. This phenomenon
has already been thoroughly studied in hBN’s sister compound graphene 31, 32.
Therein, spectroscopic signatures of lifetime renormalizations at large
binding energies away from the Fermi level ($E_{\text{F}}$) have been reported
32. Their origin has been debated33, but the evidence is pointing towards a
strong electron-phonon coupling (EPC) in the $\sigma$-bands, mediated by the
sudden onset of electron density of states (eDOS) at their band maxima 34, 35.
The EPC renormalizes the $\sigma$-bands with a large mass-enhancement (i.e.
with a mass-enhancement parameter $\lambda\approx 1$), manifesting itself as
energy broadenings and ‘kinks’ 34. Additional broadenings near the $\sigma$\-
and $\pi$-band extremum points from phonon-mediated intra-band
($\pi\rightarrow\pi$) and inter-band ($\sigma\rightarrow\pi$) scattering have
also been reported 35.

To ascertain the origins of the renormalizations observed from the hBN
$\pi$-band, their magnitudes and binding energies were studied in more detail.
In Fig. 2c, the measured $\bar{\Gamma}-\bar{\text{K}}-\bar{\text{M}}$ wedge of
the BZ is plotted along with the energy band half-linewidths
($\propto\tau^{-1}$) along these high-symmetry directions. The maximum
broadening along $\bar{\Gamma}-\bar{\text{K}}$ appears at $\approx 175$ meV
lower binding energy than the $\pi$-band vH singularity at $\bar{\text{M}}$.
Interestingly, this energy separation matches roughly with the expected
scattering energy $\hbar\omega_{\text{D}}^{\text{BN}}$ of the longitudinal
optical phonon modes of hBN 19, 36. This is illustrated in Fig. 2c by the
semi-transparent, blue interaction region overlaid on the measured
$\pi$-bandstructure in this energy range. Given its demonstrated occurrence in
graphene 35, we postulate that the described renormalization comes from intra-
band $\pi_{\text{BN}}$-$\pi_{\text{BN}}$ electron-phonon scatterings
originating at the vH singularity ($\bar{\text{M}}$) of hBN.

Figure 2: The electronic structure of hBN on graphene/SiC. a: A volumetric 3D
rendering of the hBN electronic structure within the 1st BZ. A binding energy
range of 0-1.5 eV is shown relative to the maximum of the valence ($\pi$)
band. Prominent scattering interactions have been marked by arrows. b: A
sequence of constant energy ($k_{\text{x}}$  _vs._ $k_{\text{y}}$) surfaces at
binding energies relative to the valence band maximum (VBM). Each surface is
shown as a $1/3$ slice of a full pie chart. The scattering interactions from a
have been marked (arrows and circles). c: Valence band ($\pi$) dispersion
extracted along the $\bar{\Gamma}-\bar{\text{K}}-\bar{\text{M}}$ high-symmetry
directions, together with the extracted half-linewidth (in eV). The ARPES data
has been overlaid with a simple nearest-neighbor tight binding calculation
(dashed blue). Also marked are the measured energies for the VBM and the
$\pi$-band van Hove (vH) singularity of hBN. The semi-transparent (blue)
rectangle marks an interaction region of $\Delta
E=2\hbar\omega_{\text{D}}^{\text{BN}}$ around the vH energy.

A more rigorous analysis of the observed energy renormalization is partially
prevented by the faint, linear artifacts that intersect with the hBN
$\pi$-band at a similar binding energy (blue arrows). Still, their effect is
very local in the energy and momentum phase diagram. Naively, these artifacts
can be mistaken for the $\pi$-band of the underlying graphene layer. However,
the graphene $\pi$-band is only nearly linear close to the $\bar{\text{K}}$
point, which is found at a radius of approximately $1.7~{}\text{Å}^{-1}$
relative to $\bar{\Gamma}$ in both graphene and hBN 37, 38. Selective area
photoemission measurements of the hBN flake and the adjacent graphene region
showed that the two materials were more or less aligned rotationally on top of
each other (within $\approx 5^{\circ}$). Given the small misalignment, the
linear features were situated at the wrong distance from $\bar{\Gamma}$, and
thus could not be a part of the graphene $\pi$-band itself 39, 40.

Additionally, the true graphene $\pi$-band should cross the Fermi level 31.
However, a curvature analysis 41 of the
$\bar{\Gamma}-\bar{\text{K}}-\bar{\text{M}}$ wedge from Fig. 2c revealed that
the anomalous features were visible exclusively at binding energies larger
than $E_{\text{VBM}}$, i.e., where the hBN has a finite eDOS (see the
Supporting Information 20). This signifies that the observed anomalies involve
hBN-dependent transitions that are inelastic in energy and/or momentum. Based
on similar observations in few-layer graphene, we infer that they are
signatures of secondary electrons ejected from the limited number of
unoccupied final states available 42, 43, 44. Alternatively, these could be
from Umklapp-scattered electrons ejected from the underlying graphene,
interacting with the hBN as they pass through the flaked material on their way
out into vacuum 45.

Similar to the case of the graphene $\sigma$-band, the strong eDOS increase
set naturally by the hBN $\pi$-band maximum at the $\bar{\text{K}}$ point is
expected to enable phonon-mediated energy renormalizations 32, 34, 19. This is
also corroborated by our complex self-energy $\Sigma$ analysis: along the
$\bar{\text{K}}-\bar{\text{M}}$ direction an abrupt increase in half-linewidth
near the VBM can be observed, being a typical hallmark of EPC 27, 29, 28. At
these energies the gradient of the $\pi$-band dispersion is small, and its
measured energy half-linewidth translates to the imaginary part of the self-
energy ($\imaginary\Sigma$) directly 30, 28, 46. The real part of the self-
energy ($\real\Sigma$) can be found from the discrepancy between the measured
band position and the theoretically expected, non-interacting bandstructure:
$\real\Sigma\equiv E(\mathbf{k})-\varepsilon(\mathbf{k})$ 27.

The non-interacting electronic band $\varepsilon(\mathbf{k})$ can be obtained
using various approximations. For example, the $\pi$-band of mono- or
multilayer hBN can be described by a tight-binding (TB) approximation, which
is subsequently fitted to experimental results or first-principles
calculations to obtain reasonable hopping parameters 37, 39, 47. However, the
resultant $\varepsilon(\mathbf{k})$ can only be used as an approximation at
best, as the fitting will ignore all finite $\real\Sigma$ contributions from
any many-body interactions. Alternatively, the full bandstructure of the hBN-
on-graphene system could be calculated from first principles, but this
requires detailed knowledge of the hBN-substrate interaction and the stacking
sequence and rotation of the hBN layers, which goes well beyond the scope of
this work 48, 49.

Figure 3: Energy renormalization near the hBN $\pi$-band maximum. a: The
measured electronic structure of the hBN $\pi$-band, overlaid with the
renormalized band position (green dots) as determined by curve fitting, and
the unrenormalized band (white line) from a Kramers-Kronig analysis. b: The
self-consistent $\real\Sigma$ and $\imaginary\Sigma$ of the $\pi$-band
corresponding to the spectrum shown in a. Both experimental $\Sigma$
contributions (gray) have been overlaid by a Kramers-Kronig transformed
equivalent (blue), estimated from the opposite experimental component. An
abrupt energy renormalization (arrows) on a background of electron-impurity
and electron-electron interactions (dashed lines) can be distinguished from
both $\Sigma$ components.

In general, discrepancies between the measured and calculated electronic
structure will occur, and their magnitude will depend on the type and level of
approximation used. A solution that allows us to circumvent this problem and
properly establish the non-interacting ‘bare’ energy band
$\varepsilon(\mathbf{k})$, is to make no rigorous assumptions about its energy
dispersion. Instead, the fact that $\real\Sigma$ and $\imaginary\Sigma$ are
causally related can be exploited, so that one is determined from the other
via a Kramers-Kronig (K-K) transformation. This methodology has been
previously adopted for graphene 32, 34, and is discussed in detail in Refs. 50
and 51. A first guess for $\varepsilon(\mathbf{k})$ was found from a nearest-
neighbor TB calculation using the parameters $t_{1}=2.92$ eV and
$\Delta_{\text{BN}}=4.3$ eV 9, 52. Then, keeping the energy and momentum of
the VBM fixed, the shape of $\varepsilon(\mathbf{k})$ was adjusted to satisfy
causality between the $\Sigma$ components via the K-K transformation (see the
Supporting Information for details 20). The resulting, non-interacting
$\pi$-band and $\Sigma$ components are shown in Fig. 3.

At a glance, a prominent energy renormalization can be readily distinguished
from both the $\real\Sigma$ and $\imaginary\Sigma$ near the VBM (arrows in
Fig. 3b). Its characteristic functional shape and weakly increasing background
towards larger binding energies indicate the presence of both electron-phonon
and electron-electron interactions 27, 29. Indeed, had this feature originated
from a crossing between hBN and the anomalous band features near the vH
energy, it would have existed exclusively as a local broadening across the
intersectional region 45, 35. Instead, the observed energy renormalization
persists at larger binding energies as expected 28.

A closer inspection of the measured $\Sigma$ components revealed a fine
structure of multiple, distinct spectral signatures at well-defined binding
energies. Hence, the $\Sigma_{\text{ph}}$ due to EPC was estimated from the
data by subtracting the electron-impurity and electron-electron scattering
contributions as shown in Fig. 3b (details in the Supporting Information 20).
The resulting $\real\Sigma_{\text{ph}}$, with its leading features labeled, is
shown in Fig. 4a (blue dots). We note that the renormalized and un-
renormalized bands could not be unambiguously distinguished from one another
at energies smaller than the instrument resolution (50 meV). Hence the first
few data points of $\real\Sigma_{\text{ph}}$ have been omitted 53.

Figure 4: Signatures of electron-phonon interactions near the hBN $\pi$-band
maximum. a: The experimentally determined, real self-energy contribution from
electron-phonon interactions $\real\Sigma_{\text{ph}}$ (in blue), overlaid
with a best-fit model (purple line) and its corresponding Eliashberg function
$\alpha^{2}F(\omega)$ (in green) as estimated by a MEM analysis. An
experimental error $\sigma=5$ meV for each data point was estimated from the
measured noise level and is shown by the size of the data point spheres.
Additionally, the hBN Debye energy $\hbar\omega_{\text{D}}^{\text{BN}}\approx
175$ meV has been indicated relative to the VBM (dashed vertical line). b: The
integrated mass-enhancement factor $\lambda(\omega)$ of the electron-phonon
coupling, estimated from the $\alpha^{2}F(\omega)$ shown in a.

From the resultant $\real\Sigma_{\text{ph}}$, several distinct and peaked
features can be readily distinguished. When referenced to the VBM, the first
two appear at energies similar to the in-plane acoustic phonons of few-layer
hBN 36. A minor feature can be resolved at approximately the hBN Debye energy
$\hbar\omega_{\text{D}}^{\text{BN}}$ (dashed line) 19. Surprisingly, three
more features can also be seen at energies larger than
$\hbar\omega_{\text{D}}^{\text{BN}}$, the first one at approximately
$E_{\text{VBM}}+250$ meV.

To better understand the observed peaks and quantify their individual
contributions to the overall EPC, their Eliashberg function
$\alpha^{2}F(\omega)$ and corresponding electron mass-enhancement factors
$\lambda_{n}$ were estimated. The former approximates the DOS of the
interacting phonon modes present, and the latter quantifies the strengths of
their coupling to the electrons 28. Using a maximum entropy method (MEM)
procedure 54, 55, 53, $\alpha^{2}F(\omega)$ was extracted from the data20, and
the cumulative mass-enhancement from the different phonon modes estimated as
$\lambda(\omega)=\sum_{n}\lambda_{n}=2\int_{0}^{\omega}\frac{\differential\omega^{\prime}}{\omega^{\prime}}\alpha^{2}F(\omega^{\prime})$.
The results, along with the corresponding best-fit to
$\real\Sigma_{\text{ph}}$, have been shown in Fig. 4. The energies and
$\lambda_{n}$ of the electron-phonon interactions have been summarized in
Table 1.

Immediately, the two lowest energy features in $\alpha^{2}F(\omega)$ can be
correlated to the peaks 1 and 2 as observed from $\real\Sigma_{\text{ph}}$.
Both signify a strong EPC in the acoustic energy regime, having mass-
enhancement factors $\lambda_{1}=0.39$ and $\lambda_{2}=0.21$, respectively.
Such strong coupling to the acoustic modes has already been predicted
theoretically in hBN systems thicker than 1 ML 18. Also, the observed,
dominant feature (3) above the Debye energy is faithfully re-asserted with a
$\lambda_{3}=0.19$. The smaller features ($i$-$iii$), although visible from
the measured $\real\Sigma_{\text{ph}}$, were not unambiguously resolved by the
MEM analysis when accounting for the thermal broadening of the measurements.

Table 1: A summary of the peak assignments from Fig. 4, along with their energies, contributions to $\lambda$, and suggested scattering phonons. The entries in parenthesis are distinguishable from $\real\Sigma$ but were not resolved by the MEM analysis. The phonon mode suggestions have been based on their measured and calculated energy values from Refs. 36 and 56. Peak |  $\hbar\omega$ [meV] | $\lambda_{n}$ | Phonon mode(s)
---|---|---|---
1 | $60\pm 12$ | 0.39 | TA${}_{\text{BN}}$/ZO${}_{\text{BN}}$
2 | $115\pm 9$ | 0.21 | TA${}_{\text{BN}}$/ LA${}_{\text{BN}}$/ZO${}_{\text{BN}}$
($i$) | $175\pm 16$ | minor | TO${}_{\text{BN}}$/ LO${}_{\text{BN}}$
3 | $245\pm 5$ | 0.19 | $2\times\text{LA}_{\text{BN}}$
($ii$) | $315\pm 16$ | minor | $2\times\text{TO}_{\text{BN}}$
($iii$) | $345\pm 13$ | minor | $2\times\text{LO}_{\text{BN}}$

Based on their energies, the observed peaks above the Debye energy cannot be
explained by single-phonon scattering alone. Their presence, however, hints at
the possibility of multi-phonon scattering events. Consecutive electron
scattering by multiple phonons has been predicted from theory, suggesting that
substantial scattering rates may occur in insulating and semiconducting polar
materials 57, 58. Additionally, experimental signatures of multi-phonon
scattering in few-layer hBN have been suggested from inelastic electron
tunneling spectroscopy (IETS) measurements 36. However, a mode-specific
quantification of any multiple-phonon interactions with electrons has, until
now, not been presented. From the known phonon energies of hBN 36, 56 we are
able to suggest different two-phonon combinations to explain the observed
higher-order peaks above the Debye energy. These, along with suitable phonon
modes for the EPCs at lower energies, have also been summarized in Table 1.

While electron-phonon scattering between the hBN and the underlying graphene
cannot be ruled out completely, we can render it unlikely in the present case.
Previous ARPES studies of graphene-on-hBN have indicated that coupling of the
two materials should lead to large (i.e., Fröhlich) polaron formation at the
interface59. In contrast, no such signatures of large polaron formation can be
observed here 20, 60. Examining other exfoliated hBN flakes with different
thicknesses and rotational alignment with the substrate led to the same
conclusion. Alternatively, one could suggest that the EPC peaks presented in
Table 1 resulted from electron-phonon scattering between the hBN and graphene
bands 35. However, inter-material EPC for the current system is expected to be
weak when compared to the theoretical EPC of few-layer hBN 18, and the mass-
enhancement demonstrated from our analysis.

We conclude that the apparent correspondence between the measured EPCs and
scattering with the graphene is coincidental. We reiterate, however, that our
MEM analysis was unable to properly distinguish all of the engaging hBN phonon
modes. This was in part caused by the achievable instrumental resolution, but
primarily by the thermal broadening at room temperature. For instance, further
investigating the potential coupling to the lowest-energy modes using a more
sensitive method, e.g. helium atom scattering61, 62, could provide additional
insight. However, this would require larger-area single crystalline samples to
be achieved.

In summary, we have reported the existence of several EPC processes that
renormalize the $\pi$-band electronic structure of hBN. Together, they cause
an increase in the electron scattering rate and a decrease in energy state
lifetime. This may have severe implications for the electron transport in any
hBN-adjacent conducting layers, e.g. in vdW heterostructures. Thus, our work
not only sheds light on the complex many-body ground state of few-layer hBN;
it also provides valuable insight into the possible scattering mechanisms that
may hamper the performances of hBN-based electronic devices.

##  1 Supporting Information

Further details on the hBN sample preparation, the measurement parameters, the
curvature analysis of the hBN $\pi$-states, and the interpretation of the
measured self-energies.

##  2 Acknowledgements

The authors acknowledge financial support from the Research Council of Norway
(RCN) through project no. 324183, 315330, 262633, 280788, and 245963. K.W. and
T.T. also acknowledge support from JSPS KAKENHI (Grant Numbers 19H05790,
20H00354, and 21H05233). Part of the research was performed at the ESM 21-ID-2
beamline of the National Synchrotron Light Source II, a U.S. Department of
Energy (DOE) Office of Science User Facility operated for the DOE Office of
Science by Brookhaven National Laboratory under Contract number DE-SC0012704.
This work also used the resources of the Center for Functional Nanomaterials,
Brookhaven National Laboratory, which is supported by the U.S. Department of
Energy, Office of Basic Energy Sciences, under Contract number DE-SC0012704.
We would also like to thank T. Balasubramanian, T.-Y. Chien, A. Grubišić-Čabo,
A. Ettema, T. Frederiksen, B. Hellsing, J. Manson, D. Pohlenz, A. Qaiumzadeh,
A. Sudbø, and E. Thingstad for fruitful discussions.

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Figure 5: TOC graphic


# The Pre-He White Dwarfs in Eclipsing Binaries. IV. WASP 1814+48 with
Multiperiodic Pulsations

Jae Woo Lee1, Kyeongsoo Hong2, Hye-Young Kim3 and Jang-Ho Park1
1Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
2Institute for Astrophysics, Chungbuk National University, Cheongju 28644,
Republic of Korea
3Department of Astronomy and Space Science, Chungbuk National University,
Cheongju 28644, Republic of Korea E-mail<EMAIL_ADDRESS>
(Accepted 2022 ———. Received 2022 ———; in original form 2022 )

###### Abstract

For the EL CVn candidate 1SWASPJ181417.43+481117.0 (WASP 1814+48), we secured
the first spectroscopic observations between 2015 April and 2021 March. Using
the echelle spectra, the radial velocities (RVs) of the primary star were
measured with its atmospheric parameters of $T_{\rm eff,1}=7770\pm 130$ K and
$v_{1}$$\sin$$i=47\pm 6$ km s-1. We fitted our single-lined RVs and the TESS
light curve simultaneously. From the binary modeling, we determined the
following fundamental parameters for each component: $M_{1}=1.659\pm 0.048$
$M_{\odot}$, $R_{1}=1.945\pm 0.027$ $R_{\odot}$, and $L_{1}=12.35\pm 0.90$
$L_{\odot}$ for WASP 1814+48 A, and $M_{2}=0.172\pm 0.005$ $M_{\odot}$,
$R_{2}=0.194\pm 0.005$ $R_{\odot}$, and $L_{2}=0.69\pm 0.07$ $L_{\odot}$ for
WASP 1814+48 B. The surface gravity of $\log g_{2}=5.098\pm 0.026$ obtained
from $M_{2}$ and $R_{2}$ is concurrent with 5.097$\pm$0.025 computed directly
from the observable quantities. WASP 1814+48 B is well-matched with the 0.176
M⊙ white dwarf (WD) evolutionary model for $Z=0.01$. The metallicity and our
Galactic kinematics indicate that the program target is a thin-disk star. The
whole light residuals after removal of the binary trend were analyzed and
found to oscillate at a total of 52 frequencies. Among these, most of the low
frequencies below 24 day-1 are aliases and orbital harmonics. The five
significant frequencies between 32 and 36 day-1 are the pulsation modes of
WASP 1814+48 A located in the $\delta$ Sct domain on ZAMS, and the high
frequencies of 128$-$288 day-1 arise from WASP 1814+48 B in the pre-He WD
instability strip. Our results reveal that WASP 1814+48 is the fifth EL CVn
star that is composed of a $\delta$ Sct-type primary and a pre-ELMV (extremely
low-mass pre-He WD variable).

###### keywords:

binaries: eclipsing - binaries: spectroscopic - stars: fundamental parameters
- stars: individual (1SWASPJ181417.43+481117.0) - stars: oscillations
(including pulsations).

††pagerange: The Pre-He White Dwarfs in Eclipsing Binaries. IV. WASP 1814+48
with Multiperiodic Pulsations–6††pubyear: 2021

##  1 INTRODUCTION

Extremely low-mass ($\la 0.3$ M⊙) white dwarfs (ELM WDs) are the remnants of
stars that cannot burn helium (He) in their cores. Because the Universe does
not have enough time to produce them by single-star evolution, the He-core ELM
WDs are thought to be formed by considerable mass loss at the red giant branch
phase of stellar binaries before He burning (Marsh, Dhillon & Duck 1995;
Kilic, Stanek & Pinsonneault 2007). In this case, they can provide information
about the past evolution of the precursor binaries. We have been performing
time-series spectroscopy on eclipsing binaries (EBs) containing an ELM WD
precursor (pre-ELM WD) as possible (Lee et al. 2020). The main purpose of the
observations is to measure the fundamental parameters of the interesting rare
objects, combining existing or new photometric data, and to present their
evolution scenario. The promising targets for this subject are EL CVn-type
binaries, which are post-mass transfer stars comprising an A/F main sequence
and a pre-ELM WD companion (Maxted et al. 2014; van Roestel et al. 2018).

Five out of $\sim 70$ EL CVn stars are pulsating EBs that exhibit possible
multiperiodic oscillations arising from both a $\delta$ Sct (or $\gamma$
Dor)-type primary and a pre-ELMV companion (Hong et al. 2021; Kim et al.
2021). Because the fundamental parameters such as masses and radii can be
measured accurately and in detail, spectroscopic and eclipsing binaries
provide us with an opportunity to test and refine the evolutionary models of
stars (Hilditch 2001; Torres, Andersen & Giménez 2010). At the same time,
pulsating stars help to probe and constrain their interior physics from core
to surface through asteroseismology (Antoci et al. 2019; Aerts 2021). Thus,
the stellar pulsations in EBs are of great value, because the synergy between
the two kinds of variables can clearly enhance our understanding of stellar
physics (Murphy 2018; Kurtz 2022).

We focus on the EL CVn-type star 1SWASPJ181417.43+481117.0 (WASP 1814+48; TIC
420947520; Gaia EDR3 2122136709525411072), which was recognized by Maxted et
al. (2014) to be an eclipsing variable with a period of 1.7994305 $\pm$
0.0000005 days. From archival WASP photometry, they found that the binary
target is a totally-eclipsing detached system with a mass ratio of $q$ = 0.134
$\pm$ 0.015, relative radii of $r_{1}$ = 0.2565 $\pm$ 0.0025 and $r_{2}$ =
0.0247 $\pm$ 0.0004, and surface brightness and luminosity ratios of $J$ =
3.000 $\pm$ 0.064 and $L_{2}/L_{1}$ = 0.0277 $\pm$ 0.0002, respectively.
Further, the effective temperatures of the brighter, more massive primary star
and its companion were obtained to be $8000\pm 300$ K and $12,500\pm 1800$ K,
respectively, by comparing the observed and synthetic flux distributions. This
work is the fourth in a paper-series on the pre-ELM WDs in EBs (Lee et al.
2020; Lee, Hong & Park 2022; Hong et al. 2021). We analyze in detail our high-
resolution spectra and the TESS photometric data of WASP 1814+48, and report
the discovery of multiple types of pulsations originating from the EB system.

##  2 TESS PHOTOMETRY AND ECLIPSE TIMINGS

Highly precise photometry of WASP 1814+48 has been performed by the TESS
mission (Ricker et al. 2015) from 2019 July 18. We downloaded the 2-min
cadence data taken during Sector 14, 25-26, and 40-41 from
MAST111https://archive.stsci.edu/ and used the simple aperture photometry
(SAP-FLUX) data in this study. The flux measurements were detrended and
normalized by fitting a second-order polynomial to the out-of-eclipse portion
of each sector’s light curve, and they were converted to magnitude units. The
resultant observations are displayed in the top panel of Figure 1. A total of
91,240 individual points were obtained for the five sectors. The crowdedness
factor CROWDSAP reported in the TESS data is the ratio of the target flux to
the total flux in a photometric aperture. This can be used to see if nearby
stars are observed in the same pixel as the target, where a value of 1 means
that there is no contamination. The CROWDSAP value for WASP 1814+48 is
0.9900$\pm$0.0024 on average for these sectors.

The TESS eclipse times of WASP 1814+48 and their errors were determined using
the Kwee & van Woerden (1956) method. These are presented in Table 1, where
Min I and II present the primary and secondary eclipses, respectively, at
orbital phases 0.0 and 0.50. In order to obtain the updated linear ephemeris
of the binary star and to phase its time-series data, we used the primary
minimum epochs in the following least-squares solution:

$\mbox{Min I}=\mbox{BJD}~{}2,459,010.506694(27)+1.79943078(17)E.$ (1)

The 1$\sigma$-error values for each coefficient are given in the parentheses.
The timing residuals from the linear ephemeris appear as $O-C$ in Table 1. The
TESS light curve phased with Equation (1) is depicted in the middle panel of
Figure 1.

##  3 GROUND-BASED SPECTROSCOPY AND DATA ANALYSIS

The spectroscopic observations of WASP 1814+48 were performed with the
Bohyunsan Observatory Echelle Spectrograph (BOES; Kim et al. 2007), attached
to the 1.8-m telescope in the Bohyunsan Optical Astronomy Observatory (BOAO),
Korea. A total of 31 spectra were secured for seven nights between 2015 April
and 2021 March. Their wavelength coverage ranges from 3600 to 10,200 $\rm\AA$
with a resolving power of $R=30,000$. Each exposure time of our target star
was 40 min, resulting in a signal-to-noise (S/N) ratio of approximately 20
around 4500 $\rm\AA$. This corresponds to 0.015 of the eclipsing period, so
orbital smearing was not considered. All observed spectra were reduced by
following the IRAF standard data-reduction procedures (Hong et al. 2015),
which include flat fielding, de-biasing, extraction, and wavelength and flux
calibration.

Figure 2 shows the trailed spectra of WASP 1814+48 in the Mg II $\lambda$4481
region. In this figure, the S-wave feature of the more massive primary
component (WASP 1814+48 A) is clearly shown, while there is no sign of the
hotter secondary star (WASP 1814+48 B). To measure the radial velocities (RVs)
from the observed spectra, we used the cross-correlation function (CCF) method
(Simkin 1974; Tonry & Davis1979) implemented in the RaveSpan software
(Pilecki, Konorski & Gorski 2012; Pilecki et al. 2017). In the run, most of
metallic and Balmer lines are difficult to measure, so we selected the
spectral region of Mg II $\lambda$4481 that is useful for determining the RVs
of the EL CVn-type stars (Lee et al. 2020 for WASP 0131+28; Hong et al. 2021
for WASP 0843-11; Lee, Hong & Park 2022 for WASP 1625-04). Template spectra
were taken from the BOSZ stellar atmosphere
models222https://archive.stsci.edu/prepds/bosz (Bohlin et al. 2017) matching
the effective temperature ($T_{\rm eff,1}$) and surface gravity ($\log g_{1}$)
of the primary component discussed below. We measured the RVs of WASP 1814+48
A and they are presented in Figure 3 and Table 2. The RV measurements were
fitted to a sine wave (Lee et al. 2018), in order to obtain the spectroscopic
orbit of WASP 1814+48 A. The results from this calculation are summarized in
Table 3, where $\gamma$ is the systemic velocity, $K_{1}$ and $a_{1}$$\sin$$i$
are the velocity semi-amplitude and semimajor axis of the primary component,
and $f$(M) is the mass function. The orbital period $P$ = 1.79943078 days was
taken from Eq. (1).

To determine the rotational velocities ($v_{1}$$\sin$$i$) and surface
temperature ($T_{\rm eff,1}$) of WASP 1814+48 A, we selected five absorption
lines, Ca II K $\lambda$3933, Hγ $\lambda$4340, Fe II $\lambda$4383, Hβ
$\lambda$4861, and Hα $\lambda$6563, which are useful spectral indicator for
A0$-$F0 type stars (Gray & Corbally 2009). From the BOES spectra, each
absorption region was combined using the FDBinary code (Ilijić et al. 2004) to
obtain a better S/N spectrum. Maxted et al. (2014) estimated the surface
temperature of WASP 1814+48 A to be $8000\pm 300$ K, based on the surface
brightness ratio and the observed flux distribution. Therefore, the
reconstructed spectrum was compared to the synthetic spectra with a
temperature range of 6500 K$-$9000 K (in steps of 10 K) and the projected
rotational velocity range of 10$-$200 km s-1 (in steps of 1 km s-1). The
synthetic spectra were interpolated using the stellar models from the BOSZ
spectral library (Bohlin et al. 2017) by adopting the solar metallicity and
the surface gravity of $\log g_{1}$ = 4.1 (cf. Section 4). We obtained both
atmospheric parameters of WASP 1814+48 A by performing a $\chi^{2}$ grid
search that minimizes the difference between the synthetic and reconstructed
spectra and averaging the values found in each region (Hong et al. 2017; Lee,
Hong & Park 2022). As a consequence, $T_{\rm eff,1}=7770\pm 130$ K and
$v_{1}$sin$i=47\pm 6$ km s-1. In Figure 4, the reconstructed spectrum from the
FDBinary code is presented with the best-fit model.

##  4 BINARY MODELING

As with the archival WASP observations, the TESS light curve of WASP 1814+48
shows a box-shaped primary eclipse and an ellipsoidal variation. The shape
resembles that of EL CVn, its class prototype, and there are no noticeable
differences between each sector. The observed depths for the primary and
secondary eclipses are 0.031 mag and 0.016 mag, respectively, in the TESS
photometry, and 0.037 mag and 0.017 mag in the WASP one. In both the WASP and
TESS light curves, the phase difference between Min I and Min II implies that
WASP 1814+48 is in a circular orbit. To obtain improved binary parameters, we
solved the TESS photometric data with our RV curve using the detached mode 2
of the Wilson-Devinney program (Wilson & Devinney 1971; van Hamme & Wilson
2007; hereafter W-D) and applied a mass ratio ($q$)-search method (e.g., Lee,
Hong & Kristiansen 2019). The binary modeling was done in a manner similar to
the single-lined eclipsing system HW Vir (Lee et al. 2009) and the EL CVn-type
star WASP 0131+28 (Lee et al. 2020).

In the synthesis of the RV and light curves, the surface temperature of WASP
1814+48 A was given as $T_{1}$ = 7700 $\pm$ 130 K from our spectroscopic
analysis, as discussed in the previous section. The albedos ($A$) and gravity-
darkening exponents ($g$) for both components were all set to 1.0 from their
temperatures. The logarithmic limb-darkening (LD) law was adopted and its
bolometric ($X$, $Y$) and monochromatic ($x$, $y$) coefficients were
interpolated from the values of van Hamme (1993) incorporated into the W-D
program. Because the TESS passband is not included in the binary code, we used
the LD parameters for Cousins $I_{\rm c}$-band. Furthermore, a circular orbit
($e$ = 0) and a synchronous rotation ($F_{1}=F_{2}$ = 1.0) were applied. In
this article, WASP 1814+48 A and B are denoted by subscripts 1 and 2,
respectively.

The mass ratio is considered the most important parameter in binary modeling.
We can directly calculate the velocity semi-amplitudes ($K_{1}$ and $K_{2}$)
and the mass ratio ($q=K_{1}/K_{2}$) from double-lined RV curves. However,
WASP 1814+48 B was too faint to measure its RVs. Hence, we computed a series
of binary models for assumed mass ratios below 0.3, the so-called $q$-search
procedure. The sum of squares of residuals ($\sum W(O-C)^{2}$) showed a global
minimum around $q$ = 0.104. The $q$ value was adjusted to obtain the best-fit
parameters of the eclipsing system. The final result is presented in Table 4,
and the synthetic light and RV curves are displayed as solid curves in the
middle panel of Figure 1 and the upper panel of Figure 3, respectively. The
residuals between the observations and the binary model are plotted in the
bottom panels of each figure. We can see our modeling result that describes
the light and RV curves very well. The parameters’ errors in Table 4 were
obtained following the method introduced by Southworth et al. (2020) and
applied by Lee, Hong & Kim (2021) for the highly-precise TESS data.

The fundamental stellar parameters of WASP 1814+48 A and B were first computed
from our simultaneous light and velocity solution. The results are summarized
in Table 5. We used the solar temperature and bolometric magnitude of $T_{\rm
eff}$⊙ = 5780 K and $M_{\rm bol}$⊙ = +4.73, respectively. The bolometric
corrections (BCs) were made using the empirical calibration between $\log
T_{\rm eff}$ and BC presented in Flower (1996) and later corrected in Torres
(2010). Using an apparent visual magnitude of $V$ = 10.673 $\pm$ 0.008 and the
color excess of $E$($B-V$) = 0.028 $\pm$ 0.007 (Stassun et al. 2019), we
determined the distance of the target star to be 536 $\pm$ 20 pc. Within
3$\sigma$-error values, our distance coincides with 586 $\pm$ 5 pc inverted
from the GAIA EDR3 parallax of 1.706 $\pm$ 0.013 mas (Gaia Collaboration et
al. 2021) and 584 $\pm$ 4 pc estimated from the GAIA EDR3 measurements by
Bailer-Jones et al. (2021).

The synchronous rotations for the component stars were calculated to be
$v_{\rm 1,sync}=54.70\pm 0.77$ km s-1 and $v_{\rm 2,sync}=5.46\pm 0.15$ km s-1
from the binary period $P$ and their radii $R_{1,2}$. Since both values of
$v_{\rm 1,sync}$ and $v_{1}$$\sin$$i$ coincide close to the margins of errors,
we believe the primary star is currently in a state of synchronized rotation
or nearly so.

##  5 PULSATIONAL CHARACTERISTICS

The physical parameters of WASP 1814+48 in Table 5 indicate that the primary
and secondary components are $\delta$ Sct and pre-ELMV candidates,
respectively (Maxted et al. 2013; Wang, Zhang & Dai 2020; Hong et al. 2021).
To find the pulsation signatures present in the TESS observations, we applied
a multifrequency analysis to the entire light curve residuals from our binary
model. The PERIOD04 software of Lenz & Breger (2005) was conducted up to the
Nyquist limit of about 360 day-1. The periodogram for WASP 1814+48 is
presented in Figure 5, where two dominant signals are clearly visible around
33 day-1. In addition, as shown in the inset box, there are the orbital
frequency ($f_{\rm orb}$ = 0.55573 day-1) and its multiples ($Nf_{\rm orb}$,
$N$ is an integer) in the low-frequency domain. We performed an iterative pre-
whitening process (Lee et al. 2014), and extracted 52 frequencies with an S/N
amplitude ratio larger than about 4.0 (Breger et al. 1993). The results of
this process are summarized in Table 6, and the uncertainties of each
parameter were computed following the method proposed by Kallinger, Reegen &
Weiss (2008). The amplitude spectra after pre-whitening the first two
frequencies and then 38 frequencies are illustrated in the middle and bottom
panels of Figure 5, respectively.

Of the extracted signals, possible orbital harmonics and combination
frequencies were carefully examined using the frequency resolution of $\Delta
f\simeq$ 0.002 day-1 (Loumos & Deeming 1978). They are remarked in the last
column of Table 6. Most frequencies lower than 24 day-1 may be either alias
effects or orbital harmonics up to $42f_{\rm orb}$. The integer multiples of
the orbital frequency can arise from tidally excited pulsations by the
companions in eccentric binary systems such as the heartbeat star KOI-54
(Welsh et al. 2011). However, our binary model favors the conclusion that WASP
1814+48 is in a circular orbit. Thus, it is difficult to regard the observed
multiples of $Nf_{\rm orb}$ as stellar oscillations excited by tidal
interaction. We think the aliases and the orbital harmonics result from
insufficient removal of the binary effects and the systematic trends in the
TESS data.

To cross-check the significant frequencies found from the entire TESS
observations and to examine their variations with time, we independently
analyzed the light curve residuals of each sector in the same way as before.
The resultant frequencies were compared with those in Table 6. Only $f_{1}$,
$f_{2}$, $f_{6}$, and $f_{9}$ were detected in all sectors, and $f_{11}$ in
all but Sector 14. The five frequencies were stable within the standard
deviation of $\sim$0.002 day-1. The remaining frequencies except for some
orbital harmonics ($f_{\rm orb}$, $2f_{\rm orb}$, $3f_{\rm orb}$) were not
found in most sectors. On the other hand, if the high frequencies between 128
and 288 day-1 result from WASP 1814+48 B with about 2 % light contribution to
the EB system, they are highly diluted by the light of WASP 1814+48 A. Then,
these signals would have been too weak to detect in each sector with the
frequency resolution of $\sim$0.055 day-1. As the pre-He WD transits the
larger and more massive primary star during the secondary eclipses, we
analyzed the entire secondary-eclipse residuals (orbital phases
0.454$-$0.546). The primary-eclipse data were also analyzed for comparison.
Figure 6 presents the amplitude spectra of both eclipse phases in the
frequency region of 100$-$300 day-1. Two high-frequency signals of 127.2627
and 135.5987 day-1 were detected only in the secondary eclipse phase. This
implies that the main source of the high frequencies is the secondary
companion WASP 1814+48 B.

##  6 DISCUSSION AND CONCLUSIONS

It is known from archival WASP photometry (Maxted et al. 2014) that WASP
1814+48 is a candidate EL CVn star. For the target star, we obtained the first
spectroscopic observations using the BOES echelle spectrograph attached to the
BOAO 1.8-m reflector. From the total 31 spectra, the RVs and atmospheric
parameters of the cooler, more massive primary star were measured, and its
surface temperature and rotation velocity were determined to be $T_{\rm
eff,1}=7770\pm 130$ K and $v_{1}$sin$i=47\pm 6$ km s-1, respectively. The
spectroscopic measurements were solved with the high-precision photometric
data from the TESS mission. The combined solution demonstrates that WASP
1814+48 is an EL CVn-type detached EB with masses of $M_{1}=1.659\pm 0.048$
$M_{\odot}$ and $M_{2}=0.172\pm 0.005$ $M_{\odot}$, radii of $R_{1}=1.945\pm
0.027$ $R_{\odot}$ and $R_{2}=0.194\pm 0.005$ $R_{\odot}$, and luminosities of
$L_{1}=12.35\pm 0.90$ $L_{\odot}$ and $L_{2}=0.69\pm 0.07$ $L_{\odot}$. The
component stars fill $f_{1}=49\%$ and $f_{2}=36\%$ of their inner Roche lobe,
respectively.

The surface gravity of the secondary companion can be calculated directly from
the light and RV parameters without knowledge of its mass and radius
(Southworth, Wheatley & Sams 2007) and is represented as:

$g_{2}={{GM_{2}}\over{R^{2}_{2}}}={{2\pi}\over
P}{{K_{1}(1-e^{2})^{1/2}}\over{r^{2}_{2}\sin i}}$ (2)

The observable quantities of $P$, $r_{2}$, $i$ and $K_{1}$ were taken from
Table 4. This calculation results in $\log g_{2}$ = 5.097$\pm$0.025, which is
well-matched with the 5.098$\pm$0.026 obtained from the secondary’s mass and
radius presented in this article.

Following the solar values333($X_{\odot}$, $Y_{\odot}$, $Z_{\odot}$) =
($-$8.0, 0.0, 0.0) kpc and ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$) = (9.58,
10.52, 7.01) km s-1 and the procedure applied by Lee et al. (2020), the
Galactic space motion of WASP 1814+48 was computed from our system velocity
($\gamma$) and the GAIA EDR3 measurements (position, parallax, and proper
motion). The obtained velocity components were $U=-18.4\pm 0.3$ km s-1,
$V=252.8\pm 0.6$ km s-1, and $W=9.1\pm 0.3$ km s-1, which correspond to a
total space velocity of 253.6 $\pm 0.8$ km s-1. Also, the Galactic orbit’s
eccentricity and angular momentum in the $z$ direction were obtained to be
$e_{\rm G}$ = 0.1813 $\pm$ 0.0003 and $J_{\rm z}$ = 1980 $\pm$ 17 kpc km s-1,
respectively. The position of WASP 1814+48 B in the $U-V$ and $J_{\rm
z}-e_{\rm G}$ planes falls within the thin-disk population described by Pauli
et al. (2006), indicating that our program target has the kinematics of thin-
disk stars.

Using the fundamental stellar parameters of WASP 1814+48, we studied the
evolutionary history of the EB system in terms of the H-R and $\log T_{\rm
eff}-\log g$ diagrams. The locations of the primary (A) and secondary (B)
stars are presented as star symbols in Figure 7, while the oblique dash-dotted
and dashed lines denote the instability strips of $\gamma$ Dor and $\delta$
Sct variables. WASP 1814+48 A resides inside the $\delta$ Sct region on the
ZAMS, which implies that it is a $\delta$ Sct candidate. In both diagrams, the
black dotted, dashed, and solid lines represent the evolutionary sequences of
He-core WD stars with metallicities of $Z$ = 0.001, 0.01, and 0.02,
respectively, for masses of $M$ = 0.182 $M_{\odot}$, 0.176 $M_{\odot}$, and
0.171 $M_{\odot}$ (Istrate et al. 2016). We can see that WASP 1814+48 B with
0.172 $\pm$ 0.005 $M_{\odot}$ is in good agreement with the 0.176 $M_{\odot}$
WD model for $Z$ = 0.01. The result matches well with the thin-disk population
classified by our Galactic kinematics. The lifetime ($t$) of the binary star
in a constant luminosity phase was estimated to be about 1.2 $\times$ 109 yr
using the $M_{\rm WD}-t$ relation (Chen et al. 2017).

The whole light residuals from our binary model were analyzed using the
software Period04 to find multiperiodic frequencies in our target star. We
found two dominant oscillations at $f_{1}$ = 33.70479 day-1 and $f_{2}$ =
32.80546 day-1, corresponding to about 42.7 min and 43.9 min, respectively. As
a consequence of the pre-whitening process, WASP 1814+48 oscillated in a total
of 52 frequencies that satisfied our criterion of S/N $\ga$ 4.0. Most signals
in the low frequency region of $<$ 24 day-1 may be sidelobes due to incomplete
binary modeling and instrumental artifacts in the TESS observations, but not
tidally excited modes and $\gamma$ Dor pulsations. The five frequencies
between 32 and 36 day-1 originated from WASP 1814+48 A located in the $\delta$
Sct instability domain, as shown in Figure 7. From the pulsation frequency-
density relation of $Q_{i}$ = $f_{i}$$\sqrt{\rho/\rho_{\odot}}$, we determined
their pulsation constants to be $Q_{1}$ = 0.014 days, $Q_{2}$ = 0.014 days,
$Q_{6}$ = 0.015 days, $Q_{9}$ = 0.013 days, and $Q_{11}$ = 0.014 days in
pressure ($p$) modes of $\delta$ Sct type with $Q<0.04$ days (Breger 2000;
Antoci et al. 2019). Moreover, the period ratios of $P_{\rm pul}/P_{\rm
orb}=0.0155\sim 0.0171$ are within the threshold of 0.09 $\pm$ 0.02 for
$\delta$ Sct EBs pulsating in $p$ modes (Zhang, Luo & Fu 2013). On the other
hand, the frequency signals between 128 and 288 day-1 may be pulsation modes
related to WASP 1814+48 B in the pre-He WD instability strip (Córsico et al.
2019). The periods and pulsation constants for the five high frequencies are
in ranges of $5.0<P_{\rm pul}<11.2$ min and $0.017\leq Q\leq 0.038$ days,
respectively. These results make WASP 1814+48 a very promising target for
asteroseismology, consisting of a $\delta$ Sct-type primary and a pulsating
pre-He WD companion.

## Acknowledgments

We would like to thank the BOAO staffs for assistance during our spectroscopy.
This work includes data collected by the TESS mission, which were obtained
from MAST. Funding for the TESS mission is provided by the NASA Explorer
Program. This research was supported by the KASI grant 2022-1-830-04. K.H. was
supported by the grants 2019R1A2C2085965 and 2020R1A4A2002885 from the
National Research Foundation (NRF) of Korea.

## DATA AVAILABILITY

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

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Figure 1: TESS observations of WASP 1814+48 distributed in BJD (top panel) and
orbital phase (middle panel). The circles are individual measures and the
solid line represents the synthetic curve obtained with our binary model. The
corresponding residuals are plotted in the bottom panel.  Figure 2: Phase-
folded trailed spectra of WASP 1814+48 in the Mg II $\lambda$4481 region.
Figure 3: Radial velocities of WASP 1814+48 with a fitted model. The solid
curves represent the results from a consistent light and RV curve analysis and
the dotted line denotes the system velocity of $-$25.65 km s-1. The lower
panel displays the residuals between measurements and models.  Figure 4: Five
spectral regions of WASP 1814+48 A. The open circles and the red lines
represent the reconstructed spectrum obtained with the FDBinary code and the
synthetic model spectrum with the best-fitting parameters of $T_{\rm
eff,1}=7770\pm 130$ K and $v_{1}$$\sin$$i=47\pm 6$ km s-1, respectively. The
spectra were converted from vacuum to air wavelengths using the IAU standard
conversion (Morton 1991).  Figure 5: Periodogram from the PERIOD04 program for
the entire light curve residuals. The amplitude spectra before and after
prewhitening the first two frequencies, and then 38 frequencies are shown in
the top to bottom panels. The inset box in the top panel is the amplified
spectrum of the low-frequency domain showing a series of orbital harmonics
denoted by the vertical dotted lines. The red line in the bottom panel
corresponds to four times the noise spectrum.  Figure 6: Amplitude spectra for
the light curve residuals during times of the primary (upper panel) and
secondary (lower panel) eclipses. In both panels, the five vertical lines
denote the pulsation frequencies of $f_{39}$, $f_{42}$, $f_{45}$, $f_{48}$,
and $f_{52}$ found from the entire data listed in Table 6.  Figure 7: (a) H-R
and (b) $\log T_{\rm eff}-\log g$ diagrams for WASP 1814+48 (star symbols). In
both panels, the black dotted, dashed, and solid lines are the evolutionary
tracks of low-mass He WDs with metallicities $Z$ of 0.001, 0.01, and 0.02,
respectively, for masses of 0.182 $M_{\odot}$, 0.176 $M_{\odot}$, and 0.171
$M_{\odot}$ (Istrate et al. 2016). In the upper panel (a), the green solid
line denotes the ZAMS for solar metallicity, and the colored dashed and dash-
dotted lines represent the instability strips of $\delta$ Sct and $\gamma$ Dor
stars.

Table 1: TESS Eclipse Timings of WASP 1814+48. A sample is shown here: the full version is provided as supplementary material to the online article.  BJD | Error | Epoch | $O-C$ | Min
---|---|---|---|---
2,458,683.90998 |  $\pm$0.00057 |  $-$181.5 |  $-$0.00002 | II
2,458,684.81013 |  $\pm$0.00016 |  $-$181.0 |  $+$0.00041 | I
2,458,685.70884 |  $\pm$0.00038 |  $-$180.5 |  $-$0.00060 | II
2,458,686.60921 |  $\pm$0.00025 |  $-$180.0 |  $+$0.00006 | I
2,458,687.50768 |  $\pm$0.00028 |  $-$179.5 |  $-$0.00119 | II
2,458,688.40861 |  $\pm$0.00019 |  $-$179.0 |  $+$0.00003 | I
2,458,689.30703 |  $\pm$0.00033 |  $-$178.5 |  $-$0.00127 | II
2,458,690.20880 |  $\pm$0.00021 |  $-$178.0 |  $+$0.00079 | I
2,458,691.10946 |  $\pm$0.00025 |  $-$177.5 |  $+$0.00173 | II
2,458,692.00737 |  $\pm$0.00029 |  $-$177.0 |  $-$0.00008 | I
Table 2: Radial Velocities of WASP 1814+48 A.  BJD | $V_{1}$ | $\sigma_{1}$ | $O-C$
---|---|---|---
(2,450,000+) | (km s-1) | (km s-1) | (km s-1)
7,128.2085 | $-33.3$ | 0.3 | $-0.8$
8,963.1064 | $-41.4$ | 0.3 | $1.4$
8,963.1344 | $-45.9$ | 0.4 | $-2.2$
8,963.1625 | $-44.1$ | 0.4 | $0.4$
8,963.1906 | $-42.5$ | 1.0 | $2.7$
8,963.2186 | $-45.3$ | 0.3 | $0.3$
8,963.2467 | $-46.0$ | 0.3 | $-0.1$
8,963.2748 | $-47.6$ | 0.3 | $-1.7$
8,963.3029 | $-43.5$ | 1.4 | $2.3$
8,964.0968 | $-6.2$ | 0.8 | $0.3$
8,964.1249 | $-4.8$ | 1.1 | $1.3$
8,964.1876 | $-6.3$ | 0.8 | $-0.5$
8,964.2157 | $-6.8$ | 0.4 | $-0.8$
8,964.2438 | $-5.9$ | 0.7 | $0.5$
8,964.3011 | $-5.4$ | 1.2 | $2.3$
8,968.1461 | $-22.2$ | 1.2 | $-1.7$
8,968.1742 | $-22.3$ | 0.3 | $-0.1$
8,968.2023 | $-24.4$ | 0.4 | $0.0$
9,000.2267 | $-7.2$ | 0.4 | $-0.9$
9,304.1956 | $-7.2$ | 0.8 | $-0.8$
9,304.2237 | $-7.2$ | 1.1 | $-1.2$
9,304.2518 | $-5.3$ | 0.3 | $0.5$
9,304.2799 | $-7.1$ | 0.7 | $-1.3$
9,304.3081 | $-5.2$ | 0.3 | $0.8$
9,304.3362 | $-4.9$ | 0.8 | $1.5$
9,305.1875 | $-42.9$ | 0.9 | $2.9$
9,305.2156 | $-47.0$ | 1.6 | $-1.4$
9,305.2437 | $-44.4$ | 0.5 | $0.7$
9,305.2719 | $-44.1$ | 0.6 | $0.4$
9,305.3000 | $-42.5$ | 0.7 | $1.1$
9,305.3281 | $-42.2$ | 0.6 | $0.4$
Table 3: Orbital Elements of WASP 1814+48.  Parameter | Value
---|---
$T_{0}$ (HJD) | 2,459,010.5067$\pm$0.0054
$P$ (day)a | 1.79943078
$\gamma$ (km s-1) |  $-25.85\pm$0.20
$K_{1}$ (km s-1) | 20.17$\pm$0.23
$a_{1}$$\sin$$i$ ($R_{\odot}$) | 0.717$\pm$0.010
$f$(M) ($M_{\odot}$) | 0.00153$\pm$0.00010
rms (km s-1) | 1.35
a Fixed.
Table 4: Light and Velocity Parameters of WASP 1814+48.  Parameter | Primary | Secondary
---|---|---
$T_{0}$ (HJD) | 2,459,010.50670$\pm$0.00023
$P$ (day) | 1.7994280$\pm$0.0000014
$a$ (R⊙) | 7.611$\pm$0.095
$\gamma$ (km s-1) |  $-25.65\pm$0.22
$K_{1}$ (km s-1) | 20.13$\pm$0.23
$K_{2}$ (km s-1) | 193.9$\pm$2.7
$q$ | 0.1038$\pm$0.0019
$i$ (deg) | 89.27$\pm$0.13
$T$ (K) | 7770$\pm$130 | 11,964$\pm$256
$\Omega$ | 4.041$\pm$0.022 | 5.522$\pm$0.041
$\Omega_{\rm in}$a | 1.971
$f$ (%)b | 48.8 | 35.7
$X$, $Y$ | 0.664, 0.226 | 0.737, 0.075
$x$, $y$ | 0.506, 0.213 | 0.371, 0.186
$l/(l_{1}+l_{2})$ | 0.9784$\pm$0.0005 | 0.0216
$r$ (pole) | 0.2538$\pm$0.0015 | 0.0255$\pm$0.0006
$r$ (point) | 0.2570$\pm$0.0016 | 0.0255$\pm$0.0006
$r$ (side) | 0.2561$\pm$0.0016 | 0.0255$\pm$0.0006
$r$ (back) | 0.2567$\pm$0.0016 | 0.0255$\pm$0.0006
$r$ (volume)c | 0.2556$\pm$0.0016 | 0.0255$\pm$0.0006
a Potential for the inner critical Roche surface.
b Fill-out factor $f=\Omega_{\rm in}/\Omega$ $\times$ 100.
c Mean volume radius.
Table 5: Absolute Parameters of WASP 1814+48.  Parameter | Primary | Secondary
---|---|---
$M$ ($M_{\odot}$) | 1.659$\pm$0.048 | 0.172$\pm$0.005
$R$ ($R_{\odot}$) | 1.945$\pm$0.027 | 0.194$\pm$0.005
$\log$ $g$ (cgs) | 4.080$\pm$0.018 | 5.098$\pm$0.026
$\rho$ ($\rho_{\odot}$) | 0.226$\pm$0.012 | 23.6$\pm$2.0
$v_{\rm sync}$ (km s-1) | 54.70$\pm$0.77 | 5.46$\pm$0.15
$v$$\sin$$i$ (km s-1) | 47$\pm$6 |
$T_{\rm eff}$ (K) | 7770$\pm$130 | 11,964$\pm$256
$L$ ($L_{\odot}$) | 12.35$\pm$0.90 | 0.69$\pm$0.07
$M_{\rm bol}$ (mag) | 2.00$\pm$0.08 | 5.13$\pm$0.11
BC (mag) | 0.03$\pm$0.01 |  $-$0.68$\pm$0.05
$M_{\rm V}$ (mag) | 1.97$\pm$0.08 | 5.81$\pm$0.12
Distance (pc) | 536$\pm$20
Table 6: Results of the multiple frequency analysis for WASP 1814+48a.  | Frequency | Amplitude | Phase | S/Nb | Remark
---|---|---|---|---|---
| (day-1) | (mmag) | (rad) |  |
$f_{1}$ | 33.70479$\pm$0.00001 | 0.618$\pm$0.018 | 3.23$\pm$0.08 | 59.44 |
$f_{2}$ | 32.80546$\pm$0.00001 | 0.504$\pm$0.018 | 4.55$\pm$0.10 | 48.02 |
$f_{3}$ | 0.55558$\pm$0.00003 | 0.221$\pm$0.026 | 2.44$\pm$0.34 | 14.62 | $1f_{\rm orb}$
$f_{4}$ | 1.66715$\pm$0.00005 | 0.136$\pm$0.024 | 5.43$\pm$0.52 | 9.61 | $3f_{\rm orb}$
$f_{5}$ | 1.10969$\pm$0.00004 | 0.156$\pm$0.025 | 4.48$\pm$0.47 | 10.58 | $2f_{\rm orb}$
$f_{6}$ | 32.59328$\pm$0.00005 | 0.104$\pm$0.018 | 4.36$\pm$0.51 | 9.92 |
$f_{7}$ | 2.77858$\pm$0.00006 | 0.104$\pm$0.022 | 0.35$\pm$0.63 | 7.95 | $5f_{\rm orb}$
$f_{8}$ | 0.45251$\pm$0.00008 | 0.090$\pm$0.026 | 4.73$\pm$0.85 | 5.93 |
$f_{9}$ | 35.86918$\pm$0.00005 | 0.096$\pm$0.018 | 1.91$\pm$0.54 | 9.21 |
$f_{10}$ | 0.49203$\pm$0.00008 | 0.088$\pm$0.026 | 0.53$\pm$0.86 | 5.82 |
$f_{11}$ | 33.91699$\pm$0.00005 | 0.091$\pm$0.018 | 3.67$\pm$0.58 | 8.59 |
$f_{12}$ | 0.39523$\pm$0.00008 | 0.087$\pm$0.026 | 5.55$\pm$0.88 | 5.71 |
$f_{13}$ | 0.50368$\pm$0.00009 | 0.078$\pm$0.026 | 1.19$\pm$0.98 | 5.11 |
$f_{14}$ | 9.44726$\pm$0.00006 | 0.079$\pm$0.019 | 2.19$\pm$0.70 | 7.15 | $17f_{\rm orb}$
$f_{15}$ | 7.22447$\pm$0.00006 | 0.074$\pm$0.018 | 6.25$\pm$0.71 | 7.03 | $13f_{\rm orb}$
$f_{16}$ | 8.33590$\pm$0.00006 | 0.074$\pm$0.018 | 5.07$\pm$0.72 | 6.98 | $15f_{\rm orb}$
$f_{17}$ | 0.47818$\pm$0.00008 | 0.083$\pm$0.026 | 4.40$\pm$0.91 | 5.49 |
$f_{18}$ | 0.93935$\pm$0.00011 | 0.061$\pm$0.025 | 3.36$\pm$1.19 | 4.21 |
$f_{19}$ | 1.11036$\pm$0.00005 | 0.129$\pm$0.025 | 4.06$\pm$0.57 | 8.72 | $2f_{\rm orb}$
$f_{20}$ | 0.23589$\pm$0.00010 | 0.069$\pm$0.026 | 3.84$\pm$1.12 | 4.48 |
$f_{21}$ | 3.89005$\pm$0.00008 | 0.063$\pm$0.020 | 4.52$\pm$0.94 | 5.32 | $7f_{\rm orb}$
$f_{22}$ | 0.98233$\pm$0.00011 | 0.064$\pm$0.026 | 5.72$\pm$1.18 | 4.26 | $2f_{10}$
$f_{23}$ | 3.33800$\pm$0.00009 | 0.062$\pm$0.021 | 1.47$\pm$1.01 | 4.96 |  $6f_{\rm orb}$(?)
$f_{24}$ | 1.23523$\pm$0.00011 | 0.063$\pm$0.026 | 4.96$\pm$1.21 | 4.13 | $2f_{5}-f_{22}$
$f_{25}$ | 0.29531$\pm$0.00011 | 0.061$\pm$0.026 | 3.10$\pm$1.26 | 3.99 |
$f_{26}$ | 0.34236$\pm$0.00010 | 0.067$\pm$0.026 | 1.22$\pm$1.14 | 4.41 |
$f_{27}$ | 2.22238$\pm$0.00011 | 0.056$\pm$0.023 | 4.33$\pm$1.23 | 4.08 | $4f_{\rm orb}$
$f_{28}$ | 10.00365$\pm$0.00010 | 0.051$\pm$0.019 | 2.32$\pm$1.12 | 4.47 | $18f_{\rm orb}$
$f_{29}$ | 8.89224$\pm$0.00010 | 0.050$\pm$0.019 | 2.22$\pm$1.09 | 4.58 | $16f_{\rm orb}$
$f_{30}$ | 10.55889$\pm$0.00010 | 0.049$\pm$0.020 | 2.51$\pm$1.17 | 4.30 | $19f_{\rm orb}$
$f_{31}$ | 11.11509$\pm$0.00011 | 0.048$\pm$0.020 | 2.89$\pm$1.20 | 4.17 | $20f_{\rm orb}$
$f_{32}$ | 43.34495$\pm$0.00009 | 0.047$\pm$0.017 | 0.48$\pm$1.04 | 4.82 |
$f_{33}$ | 28.89502$\pm$0.00010 | 0.047$\pm$0.018 | 4.15$\pm$1.12 | 4.49 | $f_{32}-26f_{\rm orb}$
$f_{34}$ | 16.11616$\pm$0.00010 | 0.046$\pm$0.017 | 5.67$\pm$1.09 | 4.60 | $29f_{\rm orb}$
$f_{35}$ | 26.67267$\pm$0.00011 | 0.045$\pm$0.018 | 4.20$\pm$1.19 | 4.20 | $f_{32}-30f_{\rm orb}$
$f_{36}$ | 4.45508$\pm$0.00011 | 0.045$\pm$0.019 | 2.07$\pm$1.27 | 3.96 |
$f_{37}$ | 18.89540$\pm$0.00010 | 0.045$\pm$0.017 | 2.30$\pm$1.10 | 4.55 | $34f_{\rm orb}$
$f_{38}$ | 21.12724$\pm$0.00010 | 0.044$\pm$0.017 | 6.11$\pm$1.10 | 4.56 |
$f_{39}$ | 203.85745$\pm$0.00010 | 0.044$\pm$0.017 | 1.85$\pm$1.13 | 4.43 |
$f_{40}$ | 15.00015$\pm$0.00011 | 0.042$\pm$0.018 | 5.37$\pm$1.23 | 4.07 |  $27f_{\rm orb}$(?)
$f_{41}$ | 23.84905$\pm$0.00011 | 0.042$\pm$0.018 | 2.34$\pm$1.23 | 4.09 |
$f_{42}$ | 128.93254$\pm$0.00011 | 0.042$\pm$0.017 | 1.40$\pm$1.19 | 4.22 |
$f_{43}$ | 31.57909$\pm$0.00011 | 0.042$\pm$0.018 | 3.19$\pm$1.25 | 4.00 |
$f_{44}$ | 22.78744$\pm$0.00011 | 0.042$\pm$0.017 | 3.01$\pm$1.21 | 4.16 |  $41f_{\rm orb}$(?)
$f_{45}$ | 135.25945$\pm$0.00011 | 0.041$\pm$0.017 | 3.53$\pm$1.22 | 4.11 |
$f_{46}$ | 36.54663$\pm$0.00011 | 0.041$\pm$0.018 | 1.60$\pm$1.25 | 4.02 |
$f_{47}$ | 23.33990$\pm$0.00011 | 0.041$\pm$0.017 | 1.90$\pm$1.23 | 4.07 | $42f_{\rm orb}$
$f_{48}$ | 287.85952$\pm$0.00011 | 0.041$\pm$0.017 | 3.45$\pm$1.18 | 4.24 |
$f_{49}$ | 17.78415$\pm$0.00011 | 0.041$\pm$0.017 | 6.20$\pm$1.22 | 4.12 | $32f_{\rm orb}$
$f_{50}$ | 42.23532$\pm$0.00011 | 0.041$\pm$0.017 | 0.07$\pm$1.22 | 4.12 | $f_{32}-2f_{\rm orb}$
$f_{51}$ | 200.42459$\pm$0.00010 | 0.040$\pm$0.016 | 2.74$\pm$1.18 | 4.26 |
$f_{52}$ | 20.00649$\pm$0.00011 | 0.040$\pm$0.016 | 3.73$\pm$1.20 | 4.18 | $36f_{\rm orb}$
a Frequencies are listed in order of detection.
b Calculated in a range of 5 day-1 around each frequency.