# Zero-metallicity hypernova uncovered by an ultra metal-poor star
in the Sculptor dwarf spheroidal galaxy 111Based on observations made with ESO
VLT/X-SHOOTER at the La Silla Paranal observatory under program ID 0102.B-0786

Ása Skúladóttir<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia,
Universitá degli Studi di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino,
Italy  INAF/Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125
Firenze, Italy  Stefania Salvadori  Dipartimento di Fisica e Astronomia,
Universitá degli Studi di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino,
Italy  INAF/Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125
Firenze, Italy  Anish M. Amarsi  Theoretical Astrophysics, Department of
Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden
Eline Tolstoy  Kapteyn Astronomical Institute, University of Groningen, PO Box
800, 9700AV Groningen, the Netherlands  Michael J. Irwin  Institute of
Astronomy, Madingley Road, Cambridge CB3 0HA, England  Vanessa Hill
Laboratoire Lagrange, Université de Nice Sophia Antipolis, CNRS, Observatoire
de la Côte d’Azur, CS34229, 06304 Nice Cedex 4, France  Pascale Jablonka
Laboratoire d’astrophysique, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Observatoire, CH-1290 Versoix, Switzerland  GEPI, Observatoire de Paris, CNRS,
Université de Paris Diderot, F-92195 Meudon, Cedex, France  Giuseppina
Battaglia  Instituto de Astrofísica de Canarias, Calle Via Láctea s/n, E-38206
La Laguna, Tenerife, Spain  Universidad de La Laguna. Avda. Astrofísico Fco.
Sánchez, La Laguna, Tenerife, Spain  Else Starkenburg  Kapteyn Astronomical
Institute, University of Groningen, PO Box 800, 9700AV Groningen, the
Netherlands  Davide Massari  Kapteyn Astronomical Institute, University of
Groningen, PO Box 800, 9700AV Groningen, the Netherlands  INAF - Osservatorio
di Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti 93/3, I-40129
Bologna, Italy  Amina Helmi  Kapteyn Astronomical Institute, University of
Groningen, PO Box 800, 9700AV Groningen, the Netherlands  Lorenzo Posti
Observatoire astronomique de Strasbourg, Université de Strasbourg, 11 rue de
l’Université, 67000 Strasbourg, France

###### Abstract

Although true metal-free “Population III” stars have so-far escaped discovery,
their nature, and that of their supernovae, is revealed in the chemical
products left behind in the next generations of stars. Here we report the
detection of an ultra-metal poor star in the Sculptor dwarf spheroidal galaxy,
AS0039. With [Fe/H]${}_{\rm LTE}=-4.11$, it is the most metal-poor star so far
discovered in any external galaxy. Contrary to the majority of Milky Way stars
at this metallicity, AS0039 is clearly not enhanced in carbon, with
[C/Fe]${}_{\rm LTE}=-0.75$, and A(C)=+3.60, making it the lowest detected
carbon abundance in any star to date. It furthermore lacks $\alpha$-element
uniformity, having extremely low [Mg/Ca]${}_{\rm NLTE}=-0.60$ and
[Mg/Ti]${}_{\rm NLTE}=-0.86$, in stark contrast with the near solar ratios
observed in C-normal stars within the Milky Way halo. The unique abundance
pattern indicates that AS0039 formed out of material that was predominantly
enriched by a $\sim$20$\,M_{\odot}$ progenitor star with an unusually high
explosion energy $E=10\times 10^{51}$ erg. The star AS0039 is thus one of the
first observational evidence for zero-metallicity hypernovae and provides a
unique opportunity to investigate the diverse nature of Population III stars.

Sculptor dwarf elliptical galaxy (1436) — Stellar abundances (1577) —
Population II stars (1284) — Population III stars (1285) — Hypernovae (775)

††facilities: ESO VLT/FLAMES, ESO VLT/X-SHOOTER, Gaia.

##  1 Introduction

Great advances have been made in the discovery of ancient extremely metal-poor
stars ($\rm[Fe/H]<-3$) over the last decade. Within the Milky Way, there are
now more than a dozen stars known with $\rm[Fe/H]<-4.5$ (Norris & Yong, 2019).
In dwarf spheroidal (dSph) and the ultra faint dwarf (UFD) satellite galaxies,
dozens of stars have now been discovered with $\rm[Fe/H]<-3$, with the
hitherto most metal-poor star reaching $\rm[Fe/H]=-3.92\pm 0.06$ (Tafelmeyer
et al., 2010). Of these, almost half of the most metal-poor stars
($\rm[Fe/H]\lesssim-3.4$) belong to the ancient Sculptor dSph galaxy, which is
intrinsically metal-poor, $\text{\textlangle[Fe/H]\textrangle}=-1.8$ and is
dominated by an old stellar population $>$10 Gyr old (Bettinelli et al.,
2019).

Extremely metal-poor stars in the Milky Way and its satellite dwarf galaxies
show a dichotomy of abundance patterns, namely carbon-enhanced metal-poor
(CEMP-no) stars ($\rm[C/Fe]_{\rm LTE}>+0.7$, $\rm[Ba/Fe]_{\rm LTE}\leq 0.0$),
and carbon-normal stars ($\rm[C/Fe]_{\rm LTE}\leq+0.7$). The fraction of CEMP-
no stars increases towards lower metallicity and reaches $\sim$70% at
$\rm[Fe/H]<-4$ (Lee et al., 2013; Yoon et al., 2018). The abundance patterns
of CEMP-no stars have been successfully reproduced by theoretical yields of
individual faint supernovae with low explosion energy, along with mixing and
fallback (Iwamoto et al., 2005). These faint supernovae release significant
amounts of the lighter elements, such as C, O, and Mg, but little amounts of
heavier elements such as Fe. This results in high [C/Fe] ratios in their
descendants. Often high values of other light-to-heavier element ratios such
as [Mg/Fe] are also seen in CEMP-no stars and their $\alpha$-element abundant
ratios are typically non-uniform (Norris et al., 2013).

On the other hand, the C-normal stars show a remarkable uniformity in the
abundances of the $\alpha$\- and the iron-peak elements. This also holds for
the most metal-poor C-normal stars found in the Milky Way halo, below
$\rm[Fe/H]<-4.5$ (Caffau et al., 2011; Starkenburg et al., 2018). The uniform
abundance pattern indicates that these C-normal stars formed from a well-mixed
gas polluted by many supernovae (Cayrel et al., 2004).

Here, however, we report the discovery of an ultra metal-poor star in the
Sculptor dSph galaxy, AS0039. Having both low [C/Fe] and non-uniform
$\alpha$-element abundance ratios, this star falls outside of the present
dichotomy, thereby indicating a different nucleosynthetic origin.

Figure 1: The spectrum of AS0039. Top: Comparison to other metal-poor stars in
Sculptor of similar temperature and gravity (Starkenburg et al., 2013;
Skúladóttir et al., 2015b). Bottom: Comparison of the CH G-band region of
AS0039 to synthetic spectra: Red shows the best fit, blue is a synthetic
spectrum with no CH lines, and gray represents a star born with the minimum
amount of [C/Fe]=+0.7 to qualify as a CEMP-no star (accounting for mixing;
Placco et al. 2014).

##  2 Observations and data reduction

The star AS0039 was a part of a large survey (ESO ID 0102.B-0786) of the Ca II
near-infrared triplet in Sculptor. This low-metallicty star was followed up
with the ESO VLT/X-Shooter for a more detailed abundance analysis. The radial
velocity of AS0039 from the X-Shooter spectrum was measured to be
$v_{rad}=135\pm 1$ km/s. In addition to $v_{rad}$, the star’s position, R.A.:
00:58:45.64, Dec.:$-$33:42:24.4; its location on a color magnitude diagram as,
$G=16.9$, $BP-RP=1.45$; and its proper motion, $\mu_{\alpha^{*}}=-0.01\pm
0.06$ mas/yr, $\mu_{\delta}=-0.16\pm 0.05$ mas/yr (Gaia Collaboration et al.,
2020), all confirm AS0039 to be a member of the Sculptor dwarf spheroidal
galaxy.

The X-Shooter spectra (see Fig. 1) were reduced with the most recent ESO
pipeline. The present analysis focuses on the spectra from the UVB
($\lambda=300-560$ nm, $R=5\,400$, $S/N=38$ pix-1 at 468 nm) and the VIS
($\lambda=550-1030$ nm, $R=8\,900$, $S/N=42$ pix-1 at 605 nm) arms. Details of
the observations and instrumental set-up are listed in the appendix.

##  3 Spectral Analysis

###  3.1 Atmospheric parameters and stellar models

The effective temperature, $T_{\rm eff}=4377\pm 81$ K, of the star AS0039 was
determined from Gaia eDR3 photometry (Gaia Collaboration et al., 2020;
Mucciarelli & Bellazzini, 2020). The gravity, $\log g=0.8\pm 0.1$, was
obtained through photometry, using the known distance to Sculptor (see
Skúladóttir et al. 2015b). Finally the microturbulent velocity,
$v_{mic}=2.0\pm 0.1$ km/s, was determined with an empirical calibration (Kirby
et al., 2009). The best-fitting stellar atmosphere model was taken from MARCS
(Gustafsson et al., 2008) and all abundance analyses were carried out using
TURBOSPEC (Plez, 2012), assuming local thermodynamic equilibrium (LTE), as
described in Sect. 3.2. These LTE abundances were then post-corrected, see
Sect. 3.3.

Table 1: Chemical abundances of AS0039.  Element | $\log\epsilon_{\odot}$ | $\mathrm{[X/Fe]_{LTE}}$ | $\mathrm{[X/Fe]_{NLTE}}$ |  $\sigma_{\text{NLTE}}$ [dex]
---|---|---|---|---
Fe | $7.46$ |  $-4.11$aaresults for $\mathrm{[Fe/H]}$ are shown |  $-3.95$aaresults for $\mathrm{[Fe/H]}$ are shown | $0.17$
C | $8.46$ | $-0.75$ | $-0.91$ | $0.22$
Na | $6.22$ | $+0.02$ | $-0.15$ | $0.11$
Mg | $7.55$ | $-0.09$ | $-0.16$ | $0.12$
Si | $7.51$ | $+0.06$ | $-0.15$ | $0.28$
Ca | $6.30$ | $+0.65$ | $+0.44$ | $0.11$
Sc | $3.14$ | $-0.24$ | $-0.40$ | $0.28$
Ti | $4.97$ | $+0.66$ | $+0.70$ | $0.13$
Cr | $5.62$ | $+0.04$ | $+0.58$ | $0.18$
Sr | $2.83$ | $-0.91$ | $-1.07$ | $0.22$
Li | $0.96$ |  $+3.65$bbupper limits are shown |  $+3.52$bbupper limits are shown | -
Al | $6.43$ |  $-1.02$bbupper limits are shown |  $-0.18$bbupper limits are shown | -
Mn | $5.42$ |  $-0.76$bbupper limits are shown |  $+0.08$bbupper limits are shown | -
Co | $4.94$ |  $+0.07$bbupper limits are shown |  $+0.91$bbupper limits are shown | -
Ni | $6.20$ |  $+0.21$bbupper limits are shown |  $+0.21$bbupper limits are shown | -
Ba | $2.27$ |  $-1.36$bbupper limits are shown |  $-1.52$bbupper limits are shown | -

###  3.2 LTE abundances

All measured abundances are listed in Table 1 (see also Appendix A). Not
included are the cases where only limits on the level of $\rm[X/Fe]>+1$ or
higher were obtained. The solar abundances were adopted from the recent
compilation of Asplund et al. (2021), and all literature data discussed and
shown in this paper has been put on the same scale.

The abundance of C was measured from the CH molecular band at $430$ nm. The
strength of the CH molecular lines can be affected by the assumed O abundance.
Unfortunately, the O I line at $630\,\mathrm{nm}$ only gave an upper limit of
$\rm[O/Fe]<+2$. Thus a value of $\rm[O/Fe]=+0.6$ was assumed for the synthetic
spectra, since this corresponds to the average value measured at
$\rm[Fe/H]<-2$ in Sculptor (Hill et al., 2019), and is in good agreement with
the Milky Way halo (Amarsi et al., 2019). The best fitting supernova model
(Sect. 5) implies $\rm[O/Fe]\approx+0.1$; nevertheless, a change of
$\rm\Delta[O/Fe]=\pm 0.5$ dex fortunately only had a minor effect,
$\rm\Delta[C/Fe]=\mp 0.04$ dex, because the C/O ratio stays well below unity
in all cases. The N abundance could only be constrained through NH lines to
$\rm[N/Fe]<+2.20$.

The Na abundance was measured from the Na I D resonance lines at 589.0, and
589.6 nm, which gave consistent results within error bars. For the odd
elements Al and K, only upper limits could be determined, $\rm[Al/Fe]<-1$ (Al
I 396.2 nm), and $\rm[K/Fe]<+1.15$ (K I 769.9 nm). The abundance of Sc was
measured from two lines of Sc II, at 424.7, and 431.4 nm. The result between
those two lines differed by 0.4 dex, which is still consistent given the
errors.

The abundances of three $\alpha$-elements were determined: Mg, Ca, and Ti.
Four Mg I lines were measured, at 382.9, 383.8, 517.3, and 518.4 nm. The Ca
abundance was determined by two Ca I lines (422.7, 616.2 nm), and the near-
infraread Ca II triplet. In total 15 Ti II lines were used for the abundance
determination of Ti. A further three Ti I lines were measured, but were
ultimately dropped due to large and uncertain non-LTE effects.

With the exception of Cr and Fe, only upper limits for the iron-peak elements
could be determined. Four Cr I lines were used for the determination of Cr,
while 31 Fe I lines were measured. The upper limits for Mn, Co, and Ni, based
on lines of the neutral species, are listed in Table 1, but the upper limits
for other elements (V, Cu, Zn) were too high ($\rm[X/Fe]<+1.5$) to be
informative.

The only neutron-capture element that could be reliably measured was Sr, which
was based on two Sr II lines (407.8, and 421.6 nm) which agreed within error
bars. No Ba line was visible in the spectrum, but the Ba II line at 455.4 nm
gave the listed upper limit. Other upper limits of neutron-capture elements
could not be determined to better than $\rm[X/Fe]<+1$, and were in most cases
significantly higher.

The errors were determined as follows. For elements with two or more atomic or
ionic lines measured, $N_{l}\geq 2$, the error in $\log N_{X}/N_{H}$ was
determined as the standard error in the mean. For the remaining elements (C,
Si), the error was determined based on the $\chi^{2}$ of the fits of the
synthetic spectra (Skúladóttir et al., 2015a; Skúladóttir et al., 2017). The
error in the adopted solar abundances (Asplund et al., 2021) were added in
quadrature to obtain errors in [X/H]; the error in [Fe/H] was similarly folded
into the error estimate for [X/Fe]. Random errors due to the stellar
atmospheric parameters were finally folded into these estimates; these were
determined to be $\rm\Delta[Fe/H]_{stellpar}=0.16$ dex, and a typical value of
$\rm\Delta[X/Fe]_{stellpar}=0.06$ dex was obtained for abundance ratios.

###  3.3 Departure from LTE

The non-LTE corrections for Mg, Ca, and Fe were calculated specifically for
this study, using the Balder code (Amarsi et al., 2018) (our custom version of
Multi3D; Leenaarts & Carlsson 2009). The general method of calculations
follows that given in Amarsi et al. (2020); in particular, scattering by
background lines, including Rayleigh scattering in the red wing of the Lyman
series, was included as described therein. The calculations were performed on
1D spherical MARCS model atmospheres of standard chemical composition, and
used recent model atoms (Asplund et al., 2021) that adopt physically-motivated
descriptions of inelastic collisions with neutral hydrogen. The equivalent
widths were determined by direct integration, from which abundance corrections
were obtained. These were interpolated onto the stellar parameters $T_{\rm
eff}=4377\,\mathrm{K}$ and $\log g=1.0$.

The non-LTE corrections for the other measured elements were generally drawn
from the literature: for Na from Lind et al. (2011) via the INSPECT
database222http://www.inspect-stars.com, for Si from Amarsi & Asplund (2017),
and for Ti and Cr from Bergemann (2011) and Bergemann & Cescutti (2010) via
the NLTE-MPIA database333http://nlte.mpia.de (Kovalev, 2019). The corrections
for Sc and Sr were neglected for lack of a better alternative; in reality,
slightly negative corrections might be expected for these low-excitation,
ionised lines. We note that for Sc, rather minor corrections reaching down to
$-0.04$ dex have been noted in the literature (Zhang et al., 2014), while for
Sr, data in the INSPECT database based on Bergemann et al. (2012) extends down
to $\log g=2.2$, where the mean correction is just $-0.03$ dex.

For the elements with upper limits, only the non-LTE corrections for Li (Wang
et al., 2021; Barklem et al., 2021) were drawn from the literature. For Al,
Mn, and Co, non-LTE corrections in the literature are large (Nordlander &
Lind, 2017; Bergemann et al., 2019, 2010), and rather uncertain owing to
uncertainties in the atomic data and, in the case of Al, with the equivalent-
width method; consequently, a conservative estimate of $+1$ dex was adopted
here. For Ni, the mean abundance correction found for Fe was adopted ($+0.16$
dex). The correction for the ionised Ba II resonance line was assumed to be
negligible by virtue of the extremely low line strength.

Finally, we note that 3D effects are likely to bring the C abundance down
(Caffau et al., 2011; Norris & Yong, 2019), and Fe slightly up (Amarsi et al.,
2016). Consequently, the 3D non-LTE [C/Fe] ratio is expected to be even lower
than the 1D value listed in Table 1.

##  4 Results

The spectrum of the newly discovered Sculptor star AS0039 (Fig. 1) reveals it
to be an ultra metal-poor star, $\rm[Fe/H]_{\rm LTE}=-4.11$ ($\rm[Fe/H]_{\rm
NLTE}=-3.95$). The low [Fe/H] in combination with extremely low [C/H] and
[Mg/H] (Table 1) make AS0039 the most metal-poor star currently known in any
external galaxy.

While a large fraction ($\sim$70%) of ultra metal-poor stars (Lee et al.,
2013; Yoon et al., 2018) in the Milky Way are carbon-enhanced
($\rm[C/Fe]>+0.7$), this Sculptor star has extremely low $\rm[C/Fe]_{\rm
NLTE}=-0.91$ (Fig. 1). This corresponds to A(C)=+3.60, which is currently the
lowest C abundance detected in any galaxy, breaking the recent record, set by
the Milky Way star SPLUS J210428.01-004934.2, at A(C)=+4.34 (Placco et al.,
2021). The carbon abundance of AS0039 remains low even after correcting for
mixing on the upper red giant branch (RGB), which brings processed material to
the surface, lowering the C and enhancing the N abundances; following Placco
et al. (2014), the initial birth composition becomes $\rm[C/Fe]_{corr}=-0.29$.
The low C in AS0039 is in line with other stars in Sculptor, which seems to be
relatively void of C-rich stars at low metallicities, compared to the Milky
Way (Starkenburg et al., 2013; Skúladóttir et al., 2015b).

Figure 2: Comparison of the 1D LTE $\alpha$-element ratios of AS0039 to those
of the literature, [Mg/Ca] (top) and [Mg/Ti] (bottom). Classical 1D LTE
abundances for RGB stars in Sculptor (blue), other dSph galaxies (green), the
UFD galaxies (yellow), and C-normal RGB stars in the Milky Way (Suda et al.,
2008). The Milky Way star SPLUS J210428.01-004934.2 (Placco et al., 2021) is
shown with a gray open square.

One of the most notable characteristics of the AS0039 abundance pattern is the
low [Mg/Fe]${}_{\rm NLTE}=-0.16\pm 0.12$, while C-normal, extremely metal-poor
stars in the Milky Way have a tight plateau (Andrievsky et al., 2010) of
super-solar [Mg/Fe]${}_{\rm NLTE}=+0.57\pm 0.13$. In the rare cases where Mg-
poor stars have been discovered, their depletion of Mg is always followed with
low abundances of other $\alpha$-elements, such as Ca and Ti. This is not the
case for AS0039, where both the [Ca/Fe] and [Ti/Fe] are higher than what is
typical for stars at these low metallicities, both in Sculptor and in the
Milky Way (Cayrel et al., 2004; Jablonka et al., 2015).

The $\alpha$-element ratios [Mg/Ca] and [Mg/Ti] for AS0039 are compared to the
published literature LTE values in Fig. 2. The stars of the Milky Way (Cayrel
et al., 2004) sit at around $\rm[Mg/Ca]_{LTE}=-0.03\pm 0.10$, which is in
sharp contrast with the low value of AS0039, $\rm[Mg/Ca]_{LTE}=-0.60\pm 0.15$.
Abundance measurements in other stars in dwarf galaxies, both dSph and UFD,
also have a plateau of solar, or even slightly super-solar values of
$\rm[Mg/Ca]_{LTE}$. Similarly, AS0039 is a clear outlier from the normal trend
with $\rm[Mg/Ti]_{LTE}=-0.82\pm 0.16$, while the Milky Way has an average
value (Cayrel et al., 2004) of $\rm[Mg/Ti]_{LTE}=+0.08\pm 0.12$. The ratios of
[Mg/Ti] in the Sculptor dSph differ from other galaxies, both the Milky Way
and other dwarf galaxies, with a declining trend towards lower [Fe/H].
Combined with the lack of CEMP stars in this galaxy, this is further evidence
that the earliest chemical enrichment of Sculptor was significantly different
from that of the Milky Way. The lack of known stars in other dwarf galaxies at
$\rm[Fe/H]<-3.5$ makes it currently impossible to conclude whether this trend
of [Mg/Ti] is unique for Sculptor, or a common trait of dSph, or even UFD
galaxies. To ensure a fair comparison, Fig. 2 is limited to RGB stars, but we
note that the uniformity of $\alpha$-elements in the Milky Way C-normal stars
extends down to the lowest $\rm[Fe/H]<-4.5$ (Caffau et al., 2011; Starkenburg
et al., 2018).

The extreme $\alpha$-element ratios are not the only distinctive feature in
the abundance pattern of AS0039, which also has high $\rm[Cr/Fe]_{\rm
NLTE}=+0.58\pm 0.18$ compared to the tight plateau around the solar ratio that
is observed in the metal-poor Milky Way halo (Bergemann & Cescutti, 2010),
with $\rm[Cr/Fe]_{\rm NLTE}=0.00\pm 0.06$. Quite low upper limits were
obtained for the odd elements Al, Mn and Co, comparable to or lower than the
typically measured values in Sculptor, while the odd element Na in AS0039 has
a normal abundance for extremely metal-poor stars. Finally, the extremely low
abundance of the neutron-capture elements, $\rm[Ba/H]\leq-5.5$ and
$\rm[Sr/H]=-5.02\pm 0.22$ are typical for pristine second-generation stars
(Jablonka et al., 2015).

On the whole, the abundance pattern of AS0039 is starkly different from all
other extremely metal-poor stars that have been discovered in the Milky Way or
its satellite galaxies. This suggest that AS0039 was not formed from well-
mixed material enriched by many supernovae, instead it is showing a dominant
contribution from only one single event. This is in line with theoretical
studies that have shown that a single supernovae is sufficient to reach
metallicities of $\rm[Fe/H]=-4$ in dwarf galaxies (Cooke & Madau, 2014; Rossi
et al., 2021). AS0039 therefore opens up a new window into the diverse
properties of the Population III stars.

Figure 3:  The NLTE abundance pattern of AS0039 (red pentagons) compared to
theoretical models. Blue triangles show the best fitting model of a zero-
metallicity 21 M⊙ hypernovae, of energy $10\times 10^{51}$ erg. Also depicted
are models for a PopIII faint supernova of 25 M⊙ with $E=0.7\times 10^{51}$
erg (light grey circles), and pair instability supernova of 200 M⊙ (dark grey
squares). All models are normalized to the measured [Ca/H]NLTE of AS0039. The
C abundance of AS0039 has been corrected for internal mixing (Placco et al.,
2014).

##  5 Formation scenario

To investigate the formation scenario of AS0039, its observed abundances were
compared to nucleosynthetic stellar yields of Population III supernovae over a
large range of progenitor masses ($10-100$ M⊙), explosion energies
($0.3-10\times 10^{51}$ erg), and internal mixing efficiencies (Heger &
Woosley, 2010), using the Starfit tool444http://starfit.org. The best fit
revealed that the unique abundance pattern of AS0039 shows a dominant
signature of a $M=21$ M⊙ hypernovae, with $E=10\times 10^{51}$ erg, see Fig.
3. For confirmation, we also used our own fitting routine (Salvadori et al.,
2019), which gave the best fit of a hypernovae with $M=20\pm 2$ M⊙, in good
agreement with the result obtained with Starfit.

For comparison, we also show an example of a model of a typical faint
supernova (Fig. 3), which have been very successful in explaining the origin
of CEMP-no stars (Iwamoto et al., 2005); and of a pair-instability supernovae
(PISN) of a typical mass $M=200$ M⊙. Such PISN enrich their surroundings with
large amounts of Fe, and their descendants are thus typically expected at much
higher metallicities, $\rm[Fe/H]\approx-2$ (Salvadori et al., 2019). Both
faint SNe and PISN can be excluded as having a dominant contribution to
AS0039.

The high explosion energies of hypernovae permit simultaneously low [C/Fe],
[Mg/Ca], and [Mg/Ti]. While C and Mg have hydrostatic origin, and are mainly
created in fusion throughout the lifetime of the star, both Ca and Ti are so-
called explosive $\alpha$-elements that mainly form in the supernova itself.
With the higher explosion energy of the hypernovae, relatively high amounts of
Ca and Ti are formed, resulting in a low [Mg/Ca,Ti] ratio, as is observed in
AS0039. The opposite is true in faint SN, and close to half of CEMP-no stars
have high $\rm[Mg/Ca]\gtrsim+0.5$ (Norris et al., 2013). During these faint
SN, only the outer C-rich layers are expelled, while the Fe-rich center of the
star falls into the black hole, resulting in very high [C/Fe] ratios (Iwamoto
et al., 2005). We note that none of the available models (Heger & Woosley,
2010) were able to reproduce the high [Ti/Fe] observed in AS0039. However this
is in line with other results, where theoretical models are known to
systematically under-produce Ti relative to observations at all metallicities
(Heger & Woosley, 2010; Kobayashi et al., 2020).

##  6 Discussion

With a striking signature of simultaneously low values of [C/Fe] and
[Mg/Ca,Ti], the star AS0039 is thus the first unambiguous evidence of a
hypernova imprint in any of the Milky Way satellite galaxies. In the Milky Way
halo, there are two other possible detections reported in the literature,
albeit each displaying rather different abundance signatures. We discuss these
in turn below.

Recently it was argued that the CEMP-no star HE 1327$-$2326 in the Milky Way
halo shows an imprint of a SN with rather high explosion energy, $E=5\times
10^{51}$ erg (Ezzeddine et al., 2019). However, their models depend on
asymmetrical effects, which are still poorly understood; these effects
moreover introduce three additional free parameters to be fitted. Furthermore,
more recent models struggle to produce the large [C/Fe] observed in HE
1327$-$2326 (Grimmett et al., 2021). In contrast, the abundance pattern of
AS0039 can be matched by a simple spherical hypernova, without having to
invoke any additional physics and free parameters.

Even more recently, Placco et al. (2021) presented results for SPLUS
J210428.01-004934.2. They demonstrated that its abundance pattern is best fit
with a progenitor of $30$ M⊙, that is $\sim$50% larger than that of AS0039,
but with the same explosion energy ($E=10\times 10^{51}$ erg). Interestingly,
although J210428.01-004934.2 is also somewhat low in C,
$\rm{[C/Fe]_{LTE}=-0.1}$, it displays $\rm[Mg/Ca,Ti]_{LTE}$ ratios that are
close to zero, in stark contrast to the atypically low values measured in
AS0039 (Fig. 2). Clearly the abundance signatures of PopIII hypernovae have a
strong mass dependence, and further observations are needed to fully
understand the diversity of the first generations of stars.

##  7 Conclusion

New ESO VLT/X-Shooter observations have revealed that at $\rm[Fe/H]_{\rm
LTE}=-4.11$ ($\rm[Fe/H]_{NLTE}=-3.95$), with unusually low
$\rm[Mg/Fe]_{NLTE}=-0.16$, and $\rm[C/Fe]_{corr}=-0.29$, the star AS0039 in
the Sculptor dwarf spheroidal is the most metal-poor star that has been
discovered in an external galaxy. Furthermore, it has the currently lowest
detected C abundance, $A(C)=+3.60$, in any galaxy. In addition to the overall
low metallicity, this star has a unique abundance pattern, with particularly
low ratios of hydrostatic-to-explosive $\alpha$-elements
($\rm[Mg/Ca,Ti]<-0.6$). The detailed non-LTE abundance pattern of AS0039 can
best be explained with a Population III progenitor of $M=21$ M⊙, and high
explosion energy $E=10\times 10^{51}$ erg, making AS0039 one of the first
observational evidence of a zero-metallicity hypernova.

The number of known stars in the Milky Way, at $\rm[Fe/H]<-3.5$, is almost an
order of magnitude higher than in the dwarf galaxy satellites (Suda et al.,
2008). Given the available data, it is therefore unlikely merely a coincidence
that one of two known first descendant of a zero-metallicity hypernova is
found in Sculptor. First of all, the galaxy formed as much as 80% of its stars
12-14 Gyr ago (Bettinelli et al., 2019), and thus it gives an unobscured view
to the earliest star formation. Secondly, Sculptor may lie in a sweet spot in
terms of mass. Compared to UFD galaxies it is relatively massive (Battaglia et
al., 2008), with $\rm M_{tot}\gtrsim 4\times 10^{8}$. If a rare event such as
hypernovae were to occur in a much smaller UFD galaxy, it would likely be too
small to retain the yields of such an energetic event (Cooke & Madau, 2014).
At the same time, given the only modest size and star formation rate of
Sculptor, the imprints of hypernovae will not be lost or rapidly diluted by
subsequent star formation events. The Sculptor dwarf spheroidal galaxy may
thus be the ideal system for further discoveries.

Until now, the observational evidence for Population III stars has been
completely dominated by the imprints of faint supernovae seen in CEMP-no
stars. These low-energy supernovae, however, only occupy a limited portion of
the parameter space that has been theoretically predicted for Population III
stars. With the discovery of AS0039, we are for the first time expanding this
view, opening a window into the investigation of the most energetic primordial
supernovae, i.e. Population III hypernovae. Combined with advances in
theoretical simulations of early galaxy formation and improved data from
upcoming large spectroscopic surveys (e.g., Christlieb et al. 2019), this
discovery will enable us to assemble the full picture of the elusive nature of
the first stars in the Universe.

These results are based on VLT/FLAMES and VLT/X-Shooter observations collected
at the European Organisation for Astronomical Research (ESO) in the Southern
Hemisphere under programme ESO ID 0102.B-0786. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 804240).
A.M.A. gratefully acknowledges support from the Swedish Research Council (VR
2020-03940). G.B. acknowledges financial support through the grant (AEI/FEDER,
UE) AYA2017-89076-P, as well as by the Ministerio de Ciencia, Innovación y
Universidades (MCIU), through the State Budget and by the Consejería de
Economía, Industria, Comercio y Conocimiento of the Canary Islands Autonomous
Community, through the Regional Budget. E.S. acknowledges funding through VIDI
grant “Pushing Galactic Archaeology to its limits” (with project number
VI.Vidi.193.093) which is funded by the Dutch Research Council (NWO). This
work was supported by computational resources provided by the Australian
Government through the National Computational Infrastructure (NCI) under the
National Computational Merit Allocation Scheme (NCMAS) and Australian National
University Merit Allocation Scheme (ANUMAS).

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##  Appendix A Literature compilation

The measured abundance ratios of [Mg/Ca] and [Mg/Ti] in AS0039 are compared to
literature values in Fig. 2. Typically only LTE abundance analysis is provided
in the literature (Suda et al., 2008). However, NLTE effects for similar stars
(in terms of $T_{\rm eff}$, $\log g$, and [Fe/H]) are expected to be in the
same direction and have comparable magnitudes. All existing stellar abundance
measurements in dwarf galaxies are for RGB stars, as main-sequence stars are
too faint for such analysis to be feasible. In Fig. 2 we therefore only
include RGB stars in the Milky Way to ensure a fair comparison. Where
provided, literature abundances based on Fe I and Ti II are chosen, to be as
consistent to the analysis of AS0039 as possible.
To show the uniqueness of [Mg/Ca] and [Mg/Ti] in AS0039, our literature
compilation was quite extensive, especially in relation to other dwarf
galaxies. Fig. 2 includes classical 1D LTE abundances for RGB stars in
Sculptor (Starkenburg et al., 2013; Jablonka et al., 2015; Tafelmeyer et al.,
2010; Frebel et al., 2010a; Simon et al., 2015), other dSph galaxies
(Tafelmeyer et al., 2010; Theler et al., 2020; Shetrone et al., 2001; Cohen &
Huang, 2009, 2010; Kirby & Cohen, 2012), the UFD galaxies (Chiti et al., 2018;
François et al., 2016; Frebel et al., 2010b, 2014, 2016; Norris et al., 2010b;
Gilmore et al., 2013; Ji et al., 2016c, b, a, 2019; Norris et al., 2010a;
Roederer & Kirby, 2014; Simon et al., 2010; Venn et al., 2017), and C-normal
RGB stars in the Milky Way (Cayrel et al., 2004; Lai et al., 2008; Yong et
al., 2013; Roederer et al., 2014). All literature data shown or discussed in
this paper have been put to the same solar abundance scale (Asplund et al.,
2021).

The general abundance pattern of AS0039, compared to other Sculptor stars, and
the Milky Way, is shown in Fig. 4.

Figure 4: Comparison of the LTE abundances of AS0039 (red pentagon) with
literature values. Abundances for other Sculptor stars are shown with blue
circles (Starkenburg et al., 2013; Jablonka et al., 2015; Tafelmeyer et al.,
2010; Simon et al., 2010; Frebel et al., 2010a), while grey squares are stars
in the Milky Way (Cayrel et al., 2004).

##  Appendix B Additional tables

Detailed information about the star AS0039 and the X-Shooter spectra used in
this analysis are listed in Table 2. List of the Fe lines used for the
abundance analysis is given in Table 3, and information for other elemental
lines are in Table 4

Table 2: The observational log for the X-Shooter spectra of AS0039. Star | AS0039
---|---
RA | 00:58:45.64
Dec. |  $-$33:42:24.4
Obs. date | 2018-12-16
Exp. time | 2 x 1800 s
Airmass (mean) | 1.1
Seeing (mean) | 0.6”
Slit width (UVB) | 1.0”
R${}_{\text{UVB}}$ | 5 400
$\lambda_{\text{UVB}}$ | 3000-5600 Å
S/N${}_{\text{UVB}}$ (4680 Å) |  38 pxl-1
Slit width (VIS) | 0.9”
R${}_{\textsl{VIS}}$ | 8 900
$\lambda_{\text{VIS}}$ | 5500-10200 Å
S/N${}_{\text{VIS}}$ (6050 Å) |  42 pxl-1
G |  16.932 $\pm$ 0.001
BP |  17.525 $\pm$ 0.010
RP |  16.073 $\pm$ 0.010
$\mu_{\alpha^{*}}$ |  $-0.01\pm 0.06$ mas/yr
$\mu_{\delta}$ |  $-0.16\pm 0.05$ mas/yr
$v_{rad}$ |  135 $\pm$ 1 km/s
$T_{\textsl{eff}}$ |  4377 $\pm$ 81 K
$\log g$ |  0.8$\pm$ 0.1
$v_{mic}$ |  2.0$\pm$ 0.1 km/s
Table 3: Measurements of individual Fe lines.  Species | $\lambda/\mathrm{nm}$ | $\chi_{\text{low}}$ | $\log{g\,f}$ | $\log\epsilon_{\text{LTE}}$ |  $\Delta_{\text{NLTE}}$ [dex]
---|---|---|---|---|---
Fe I | $381.584$ | $1.490$ | $0.237$ | $3.37$ | $0.17$
Fe I | $381.634$ | $2.200$ | $-1.196$
Fe I | $382.043$ | $0.860$ | $0.119$ | $3.21$ | $0.19$
Fe I | $382.118$ | $3.270$ | $0.198$
Fe I | $382.430$ | $3.300$ | $-0.043$ | $3.38$ | $0.18$
Fe I | $382.444$ | $0.000$ | $-1.362$
Fe I | $382.588$ | $0.920$ | $-0.037$ | $2.81$ | $0.21$
Fe I | $382.782$ | $1.560$ | $0.062$ | $2.70$ | $0.24$
Fe I | $385.082$ | $0.990$ | $-1.734$ | $3.00$ | $0.17$
Fe I | $386.552$ | $1.010$ | $-0.982$ | $3.75$ | $0.07$
Fe I | $387.250$ | $0.990$ | $-0.928$ | $3.47$ | $0.12$
Fe I | $389.789$ | $2.690$ | $-0.736$ | $3.40$ | $0.18$
Fe I | $389.801$ | $1.010$ | $-2.018$
Fe I | $390.648$ | $0.110$ | $-2.243$ | $2.79$ | $0.22$
Fe I | $392.026$ | $0.120$ | $-1.746$ | $3.16$ | $0.18$
Fe I | $392.291$ | $0.050$ | $-1.651$ | $3.91$ | $0.09$
Fe I | $400.524$ | $1.560$ | $-0.610$ | $3.80$ | $0.06$
Fe I | $404.581$ | $1.490$ | $0.280$ | $2.83$ | $0.20$
Fe I | $406.359$ | $1.560$ | $0.062$ | $3.57$ | $0.09$
Fe I | $407.174$ | $1.610$ | $-0.022$ | $3.43$ | $0.11$
Fe I | $413.206$ | $1.610$ | $-0.675$ | $3.79$ | $0.06$
Fe I | $414.342$ | $3.050$ | $-0.204$ | $3.63$ | $0.18$
Fe I | $414.387$ | $1.560$ | $-0.511$
Fe I | $419.825$ | $3.370$ | $-0.457$ | $3.64$ | $0.20$
Fe I | $419.830$ | $2.400$ | $-0.719$
Fe I | $420.203$ | $1.490$ | $-0.708$ | $3.53$ | $0.10$
Fe I | $423.594$ | $2.420$ | $-0.341$ | $3.20$ | $0.24$
Fe I | $425.012$ | $2.470$ | $-0.405$ | $3.80$ | $0.19$
Fe I | $425.079$ | $1.560$ | $-0.714$ | $3.44$ | $0.11$
Fe I | $426.047$ | $2.400$ | $0.109$ | $3.56$ | $0.20$
Fe I | $432.576$ | $1.610$ | $0.006$ | $3.32$ | $0.13$
Fe I | $438.355$ | $1.490$ | $0.200$ | $3.05$ | $0.17$
Fe I | $441.512$ | $1.610$ | $-0.615$ | $2.73$ | $0.26$
Fe I | $442.731$ | $0.050$ | $-2.924$ | $3.44$ | $0.17$
Fe I | $446.165$ | $0.090$ | $-3.210$ | $3.37$ | $0.18$
Fe I | $448.217$ | $0.110$ | $-3.501$ | $3.34$ | $0.19$
Fe I | $448.225$ | $2.220$ | $-1.482$
Fe I | $452.861$ | $2.180$ | $-0.822$ | $3.25$ | $0.23$
Table 4: Atomic data and measured abundances of individual lines. Species | $\lambda/\mathrm{nm}$ | $\chi_{\text{low}}$ | $\log{g\,f}$ | $\log\epsilon_{\text{LTE}}$ |  $\Delta_{\text{NLTE}}$ [dex]
---|---|---|---|---|---
Li I | $670.776$ | $0.000$ | $-0.002$ | $0.50$aaupper limits are shown | $0.03$
Li I | $670.792$ | $0.000$ | $-0.303$
Na I | $588.995$ | $0.000$ | $0.117$ | $2.24$ | $-0.06$
Na I | $589.592$ | $0.000$ | $-0.194$ | $2.01$ | $0.06$
Mg I | $382.936$ | $2.709$ | $-0.231$ | $3.59$ | $0.06$
Mg I | $383.829$ | $2.717$ | $-1.530$ | $3.20$ | $0.12$
Mg I | $383.829$ | $2.717$ | $0.392$
Mg I | $383.830$ | $2.717$ | $-0.355$
Mg I | $517.268$ | $2.712$ | $-0.402$ | $3.26$ | $0.11$
Mg I | $518.360$ | $2.717$ | $-0.180$ | $3.35$ | $0.07$
Al I | $396.152$ | $0.014$ | $-0.323$ |  $1.30$aaupper limits are shown | $1.00$
Si I | $390.552$ | $1.909$ | $-1.041$ | $3.46$ | $-0.05$
Ca I | $422.673$ | $0.000$ | $0.244$ | $2.59$ | $0.07$
Ca I | $616.217$ | $1.899$ | $-0.090$ | $2.82$ | $0.15$
Ca II | $849.802$ | $1.692$ | $-1.416$ | $3.01$ | $-0.28$
Ca II | $854.209$ | $1.700$ | $-0.463$ | $2.82$ | $-0.07$
Ca II | $866.214$ | $1.692$ | $-0.723$ | $2.94$ | $-0.09$
Sc II | $424.682$ | $0.315$ | $0.242$ | $-1.47$ | $0.00$
Sc II | $431.408$ | $0.618$ | $-0.096$ | $-0.95$ | $0.00$
Ti I | $441.727$ | $1.887$ | $-0.020$ |  $1.64$bbTi I excluded from the mean | $0.16$
Ti I | $447.124$ | $1.734$ | $-0.103$ |  $2.17$bbTi I excluded from the mean | $0.16$
Ti I | $501.419$ | $0.000$ | $-1.220$ | $1.43$bbTi I excluded from the mean | $0.57$
Ti I | $501.428$ | $0.813$ | $0.110$
Ti II | $390.054$ | $1.131$ | $-0.200$ | $1.51$ | $0.36$
Ti II | $391.346$ | $1.116$ | $-0.420$ | $1.43$ | $0.34$
Ti II | $428.787$ | $1.080$ | $-1.790$ | $1.74$ | $0.17$
Ti II | $430.004$ | $1.180$ | $-0.440$ | $1.38$ | $0.33$
Ti II | $430.191$ | $1.161$ | $-1.150$ | $1.62$ | $0.24$
Ti II | $431.497$ | $1.161$ | $-1.100$ | $2.04$ | $0.25$
Ti II | $433.791$ | $1.080$ | $-0.960$ | $1.44$ | $0.19$
Ti II | $439.503$ | $1.084$ | $-0.540$ | $1.58$ | $0.24$
Ti II | $439.977$ | $1.237$ | $-1.190$ | $1.73$ | $0.26$
Ti II | $444.379$ | $1.080$ | $-0.720$ | $1.09$ | $0.02$
Ti II | $445.048$ | $1.084$ | $-1.520$ | $1.83$ | $0.17$
Ti II | $446.851$ | $1.131$ | $-0.600$ | $1.37$ | $0.02$
Ti II | $450.127$ | $1.116$ | $-0.770$ | $1.26$ | $0.24$
Ti II | $457.197$ | $1.572$ | $-0.320$ | $1.07$ | $0.12$
Ti II | $518.868$ | $1.582$ | $-1.050$ | $1.68$ | $0.15$
Cr I | $428.972$ | $0.000$ | $-0.361$ | $1.35$ | $0.89$
Cr I | $520.451$ | $0.940$ | $-0.208$ | $1.90$ | $0.64$
Cr I | $520.604$ | $0.940$ | $0.019$ | $1.77$ | $0.65$
Cr I | $520.843$ | $0.940$ | $0.158$ | $1.17$ | $0.66$
Mn I | $403.075$ | $0.000$ | $-0.470$ |  $0.70$aaupper limits are shown | $1.00$
Mn I | $403.306$ | $0.000$ | $-0.618$ |  $0.40$aaupper limits are shown | $1.00$
Co I | $399.530$ | $0.923$ | $-0.220$ |  $0.90$aaupper limits are shown | $1.00$
Ni I | $547.690$ | $1.826$ | $-0.890$ |  $2.30$aaupper limits are shown | $0.16$
Sr II | $407.771$ | $0.000$ | $0.167$ | $-2.00$ | $0.00$
Sr II | $421.552$ | $0.000$ | $-0.145$ | $-2.38$ | $0.00$
Ba II | $455.403$ | $0.000$ | $0.170$ |  $-3.20$aaupper limits are shown | $0.00$


# Tuning superconductivity in nanosecond laser annealed boron doped
$Si_{1-x}Ge_{x}$ epilayers

S. Nath  I. Turan  L. Desvignes  L. Largeau  O. Mauguin  Uni. Paris-Saclay,
CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France
M. Túnica  M. Amato  Uni. Paris-Saclay, CNRS, Laboratoire de Physique des
Solides, 91405 Orsay, France  C. Renard  G. Hallais  D. Débarre  Uni. Paris-
Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau,
France  F. Chiodi<EMAIL_ADDRESS>Uni. Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

###### Abstract

Superconductivity in ultra-doped $Si_{1-x}Ge_{x}:B$ epilayers is demonstrated
by nanosecond laser doping, which allows introducing substitutional B
concentrations well above the solubility limit and up to $7\,at.\%$. A Ge
fraction $x$ ranging from 0 to 0.21 is incorporated in $Si:B$ : 1) through a
precursor gas by Gas Immersion Laser Doping; 2) by ion implantation, followed
by nanosecond laser annealing; 3) by UHV-CVD growth of a thin Ge layer,
followed by nanosecond laser annealing. The 30 nm and 80 nm thick
$Si_{1-x}Ge_{x}:B$ epilayers display superconducting critical temperatures
$T_{c}$ tuned by B and Ge between 0 and 0.6 K. Within BCS weak-coupling
theory, $T_{c}$ evolves exponentially with both the density of states and the
electron-phonon potential. While B doping affects both, through the increase
of the carrier density and the tensile strain, Ge incorporation allows
addressing independently the lattice deformation influence on
superconductivity. To estimate the lattice parameter modulation with B and Ge,
Vegard’s law is validated for the ternary $SiGeB$ bulk alloy by Density
Functional Theory calculations. Its validity is furthermore confirmed
experimentally by X-Ray Diffraction. We highlight a global linear dependence
of $T_{c}$ vs. lattice parameter, common for both $Si:B$ and
$Si_{1-x}Ge_{x}:B$, with $\delta T_{c}/T_{c}\sim 50\,\%$ for $\delta a/a\sim
1\,\%$.

Nanosecond laser annealing, SiGe, Superconductivity

##  I Introduction

$SiGe$ is a key material for micro-electronics. The possibility to combine
classical SiGe technology with quantum circuits is appealing to exploit the
large-scale integration and reproducibility associated with CMOS devices [1].
Hole spin qubits have been developed in Ge/SiGe quantum dots [2] and SiGe
nanowires [3, 4], taking advantage from the control on the environment and low
nuclear spin possible in group IV materials. Furthermore, Ge and SiGe have
been incorporated in Josephson field effect transistors [5, 4], hosted
transmon qubits, and their microwave losses have been investigated [6]. The
possibility of inducing superconductivity directly in thin SiGe layers might
furthermore provide an advantage in coupling SiGe-based classical electronics
to superconducting quantum circuits.
It has been shown that Silicon displays a superconducting phase when ultra-
doped with B [7, 8, 9, 10]. An extreme boron doping concentration is required
to trigger superconductivity in $Si:B$, more than three times the solubility
limit. This concentration, impossible to reach using conventional micro-
electronic techniques, was obtained using Gas Immersion Laser Doping (GILD),
an out-of-equilibrium technique combining chemisorbtion of a precursor gas in
a Ultra-High-Vacuum environment, and nanosecond laser annealing [11, 12, 10].
In this paper, we employ this technique to ultra-dope with B thin $SiGe$
layers, demonstrating the realisation of a superconducting phase.
In addition to the intrinsic interest of $SiGe$ superconductivity, the
investigation of the evolution of the superconducting critical temperature
$T_{c}$ with both B and Ge doping allows to better understand what triggers
superconductivity in Si and SiGe. Indeed, BCS theory in the weak coupling
limit expects $T_{c}$ to exponentially increase with the electron-phonon
coupling $\lambda=N(E_{F})V_{e-ph}$, the product of the electron-phonon
potential $V_{e-ph}$ and the density of states at Fermi energy $N(E_{F})$. B
doping modifies both $N(E_{F})$ and $V_{e-ph}$: $N(E_{F})$ is related to
$n_{B}$, as evident in the frame of the free electron model where
$E_{F}\propto(3\pi^{2}n_{B})^{2/3}$. But, due to the smaller B size, an
important lattice deformation up to $\delta a/a=-3.5\,\%$ is induced at the
same time [8], affecting the phonon frequencies involved in Cooper pairing.
The incorporation of Ge makes it possible to modify, solely and independently,
the lattice deformation, allowing to elucidate the relevant parameters that
govern superconductivity.

Ge incorporation | Thickness (nm) | Laser shots Ge |  $C_{Ge}$ (at.$\%$) | Laser shots B |  $C_{B}$ (at.$\%$)
---|---|---|---|---|---
GILD - 5 | 30 | 5 | 0.27 | 30-400 | 2.3 - 10.6
GILD - 15 | 30 | 15 | 0.8 | 30-400 | 2.3 - 10.6
GILD - 200 | 30 | 200 | 10.7 | 160-475 | 8 - 11.2
GILD - Ge | 30 | 5-400 | 0.27-21.3 | 220 | 9.2
Ge CVD | 30 | - | 17.1 | 30-400 | 2.3 - 10.6
Ge implanted ($10^{15}\,cm^{-2}$) | 75 | - | 0.27 | 3 - 258 + B impl. ($5\times 10^{15}\,cm^{-2}$) | 1.5 - 10.8
Reference $SiB$ | 30 | - | - | 50-400 | 3.6 - 10.6
Table 1: Details on the $SiB$ and $SiGeB$ sample series investigated in this
work: Ge incorporation method, thickness, number of nanosecond laser annealing
repetitions, total B and Ge concentrations $C_{B}$ and $C_{Ge}$ in atomic
$\%$.

##  II Ultra doped $Si_{1-x}Ge_{x}:B$

###  II.1 Gas Immersion Laser Doping

To attain the extremely high doping levels necessary to overcome the
superconductivity threshold, we employ fast liquid phase epitaxy,
characterized by recrystallization times of a few tens of nanoseconds (see
Methods). A puff of the precursor gas ($BCl_{3}$ or $GeCl_{4}$) is injected
onto the substrate surface, saturating the chemisorption sites, so that the
supply of incorporated atoms is constant and self-limited. A pulse of excimer
XeCl laser ($\lambda$ = 308 nm, pulse duration 25 ns) melts the substrate, and
the chemisorbed atoms diffuse in the liquid. At the end of the laser pulse, an
epitaxial out-of-equilibrium recrystallization takes place from the substrate
at a speed of $\sim 4\,$m/s [13], achieving concentrations larger than the
solubility limit ($\sim 1\,at.\%$ for B in Si). The laser energy density,
tuned with an attenuator, controls the melted thickness in the 5-500 nm range.
A flat, straight, and sharp (few nm thick) interface is created between the
ultra-doped layer and the substrate. In order to control the amount of B and
Ge incorporated, the entire chemisorption-melting-crystallization process is
repeated the desired number of times (number of laser shots N). The total B
(Ge) concentration $C_{B}$ ($C_{Ge}$) is proportional to the number of GILD
process repetitions $N$ [14, 15, 16] and is calibrated by integrating the SIMS
(Secondary Ion Mass Spectrometry) concentration profiles to calculate the
average concentration in the layer.

###  II.2 Ge incorporation

We explore the low temperature electrical characteristics of thin
$Si_{1-x}Ge_{x}:B$ films, ultra-doped in boron (B) by Gas Immersion Laser
Doping. Three different methods are used to incorporate the Ge and control its
amount: 1) Gas Immersion Laser Doping with $GeCl_{4}$ as a precursor gas; 2)
Implantation of Ge and B, followed by nanosecond laser annealing; 3) Growth by
UHV-CVD of a thin Ge layer, followed by nanosecond laser annealing.
N-type Si substrates of resistivity $50\,\Omega cm$ are used for all sample
series. The substrates are introduced in the UHV chamber after an acetone and
ultrasounds cleaning to remove organic surface contamination, and 1 minute
Buffered Hydrofluoric acid etch to remove the native silicon oxide.

1) Gas Immersion Laser Doping of Ge
Three samples series are realized with varying B content and a fixed Ge
concentration, and one with a varying Ge content and fixed B concentration, by
controlling the respective B and Ge number of GILD process repetitions (see
summary table 1). The layers doped thickness is $t=30\,$nm, corresponding to
an annealing time of $30\,$ns on undoped Si by a laser energy density at the
sample level of $1000\,$mJ/cm2. B atoms have a high diffusion coefficient in
the liquid Si ($D\sim 10^{-4}$cm2/s) and a segregation coefficient near 1 at
the high crystallisation speeds attained ($k=0.95$) [17], insuring a
homogeneous B distribution within the recrystallised layer even for the longer
annealing processes ($\sim 15\,\mu$s for 500 cycles of 30 ns). Thus, a
homogeneous distribution is expected even when the B is further submitted to
the subsequent process time of the Ge incorporation. In contrast, a graded Ge
profile is expected, evolving toward the surface and depleting the bottom of
the layer, an effect of the smaller segregation coefficient ($k\sim 0.6-0.8$)
[18].
2) Nanosecond Laser Annealing of implanted Ge and B
In one sample series, a Ge dose of $10^{15}\,cm^{-2}$ is introduced before the
nanosecond laser annealing through an implantation step. The following GILD
processes serve both the purposes of B doping and Ge incorporation in
substitutional sites. This series is thicker than the rest of this work
($t=75\,$nm) to ensure that the implanted Ge atoms, including the implantation
queue, are within the melted depth. A small implanted B dose ($5\times
10^{15}\,cm^{-2}$, equivalent to 42 laser shots, i.e. to 1.25 at.$\%$) is also
present in the sample and is activated alongside the Ge. B doping is then
completed with GILD.
3) Nanosecond Laser Annealing of a thin CVD Ge layer
The epitaxial growth of Ge on Si is carried out in an UHV-CVD system with a
base pressure of $10^{-10}$ mbar. Pure $SiH_{4}$ and $GeH_{4}$ diluted at
10$\%$ in $H_{2}$ are used as gas sources. The growth time is settled at 20
min in order to achieve 6 nm of Ge [19, 20] (see Methods for further details).
During the GILD step to incorporate B, the laser energy is kept initially low
and gradually increased over the last 10 process repetitions, in order to
limit the time of Ge diffusion towards the surface and achieve a Ge profile as
homogeneous as possible.

##  III Superconductivity in $Si_{1-x}Ge_{x}:B$

###  III.1 Low temperature measurements

At the end of the B and Ge incorporation, Ti(15 nm)/Au(150 nm) metallic
contacts for 4-points measurements are deposited over the doped layers by
laser lithography and e-beam evaporation. The resistance $R$ of a region
150$\,\mu$m wide and 300$\,\mu$m long is extracted from dc V(I) measurements
of averaged positive and negative bias current ($I_{dc}$=10 to 50 nA). $R$ is
recorded as a function of temperature (300 K to 0.05 K) and perpendicular
magnetic field (0 to 55 mT) in an Adiabatic Demagnetisation Refrigerator
setup. After the demagnetisation and relative cool down, the system is left to
evolve during the slow ($\sim$1.5 hours) temperature increase, giving a
precise evaluation of the superconducting transition temperature. For the
transitions in presence of a magnetic field B, the demagnetisation is stopped
at B (instead of decreasing to zero field as in usual demagnetisation cycles),
so that the transition is recorded in presence of a constant magnetic field.
In order to access the hole carrier density, Hall bars are realized in a
separate reference $SiB$ sample series (see Methods).

Figure 1:  Typical resistance $R$ vs. temperature $T$ superconducting
transitions for a) the reference $SiB$ GILD samples series, containing no Ge
and a variable B concentration $C_{B}=9.2-10.3\,$at.$\%$; b) $SiGeB$ layers
from the GILD-Ge sample series, with constant $C_{B}=9.2\,$at.$\%$ and varying
Ge concentration $C_{Ge}=0.8-21.3\,$at.$\%$. $T_{c,h}$ and $T_{c,l}$ indicate
respectively the high and low temperature transition for $SiGeB$ at high
doping.

###  III.2 Superconductivity evolution with B in $Si:B$

Fig. 1-a shows typical $R(T)$ superconducting transitions for the reference
$SiB$ GILD samples series, containing no Ge, with varying total B
concentration $C_{B}$. The $R(T)$ curves show a single, relatively sharp
transition, of width $\Delta T\sim 0.08\,$K$\sim 16\%$. We observe that the
superconducting critical temperature $T_{c}$ and the normal state resistance
$R_{N}$ evolve with $C_{B}$, the total amount of B incorporated. Varying
$C_{B}$ results in a modification of the hole carrier density $n_{B}$. For
$C_{B}<6\,at.\%$, all B atoms are substitutional, providing a hole carrier,
and we obtain $100\%$ activation, with $n_{B}=C_{B}$ (Fig. 2-b). For
$C_{B}>6\,at.\%$, the activation progressively lowers with the gradual
increase of inactive B complexes, and $n_{B}$ increase slows down. Finally, at
$C_{B}>9.5\,$at.$\%$, $n_{B}$ saturates, as a result of the formation of B
aggregates [16, 21]. As a consequence, $R_{N}$ initially decreases with
$C_{B}$ in the full activation regime, while at higher doping it saturates and
slowly increases following the increase of disorder and formation of
aggregates [16]. In the parameter range of this work, we are close to $R_{N}$
saturation, and only little variations are observed when modifying $C_{B}$
(see Fig. 1).
The evolution of $T_{c}$ with $C_{B}$ is instead marked. $T_{c}$ initially
increases roughly linearly with $C_{B}$, to attain a maximum at
$C_{B}=8.9\,at.\%$, and then decreases, more slowly, in the saturation regime
(Fig.2-a). The similar evolution of $T_{c}(C_{B})$ and $n_{B}(C_{B})$ suggests
that $T_{c}$ is controlled by the active concentration $n_{B}$. However, we
observe that $T_{c}$ keeps increasing with $C_{B}$ even after the saturation
of the hole concentration $n_{B}$ at $C_{B}=7.8\,at.\%$.

Figure 2:  a) $SiB$ and $SiGeB$ superconducting critical temperature $T_{c}$
at mid-transition vs. total B concentration $C_{B}$ for sample series GILD-5,
GILD-15, GILD-200, Ge CVD and Reference $SiB$. All series are 30 nm thick.
$SiGeB$ samples present two transitions at the highest $C_{Ge}$. In this case,
the one leading to the zero resistance state, $T_{c,l}$, is plotted. b) Hole
carrier density $n_{B}$ (substitutional B concentration) vs. total B
concentration $C_{B}$ extracted from Hall measured in a dedicated $SiB$ sample
series with t=30 nm. $C_{B}$ is measured from SIMS (Secondary Ion Mass
Spectrometry) concentration profiles over the layer thickness $t$:
$C_{B}=\frac{\int{C_{SIMS}\,dt}}{t}$. The gray line is a guide to the eye for
$SiB$ (no Ge) evolution.

Thus, the question arises if superconductivity in silicon, besides being
controlled by the carrier density, might be tuned through the strain induced,
at the same time, by the smaller B atoms. The demonstration of
superconductivity in SiGe by ultra-doping with B, in addition to its intrinsic
interest associated to the role played by SiGe in classical (and quantum)
electronics, opens the way to an experimental answer to this question. The
independent incorporation of B and Ge allows indeed to address, independently,
the role of the carrier concentration and the strain for superconductivity.

Figure 3:  Superconducting critical temperatures of the two resistive
transitions $T_{c,l}$ (blue) and $T_{c,h}$ (red) observed in $SiGeB$ series
GILD-Ge (see Fig.1), for a fixed $C_{B}=9.2\,at.\%$ and as a function of Ge
concentration $C_{Ge}$. The dotted line corresponds to the $T_{c}$ of the
reference $SiB$ sample with $C_{B}=9.2\,at.\%$ and no Ge, realized in the same
run as the GILD-Ge series. The gray line is a guide to the eye.

###  III.3 Superconductivity evolution with Ge and B in $Si_{1-x}Ge_{x}:B$

Fig. 1-b shows the $R(T)$ superconducting transitions of typical $SiGeB$
layers from the GILD-Ge sample series, with constant $C_{B}=9.2\,$at.$\%$ and
varying Ge concentration $C_{Ge}=0.8-21.3\,$at.$\%$. While at small Ge content
the transitions are nearly on top of the reference $SiB$, single, transition,
for higher $C_{Ge}$ the curves show two transitions. The first transition,
$T_{c,h}$, at higher temperature, accounts for $23-24\,\%$ of the total
resistance. The second transition, accounting for $\sim 75\%$ of the total
resistance, is similarly sharp, with $\Delta T\sim 0.05\,$K$\sim 13\%$, and is
characterized by a lower transition temperature $T_{c,l}$. The decrease of
$T_{c,l}$ with Ge concentration observed in Fig. 1-b is a general occurrence
for all B concentrations, and not particular to the fixed $C_{B}$ of the
curves displayed. Indeed, as shown in Fig. 2, $T_{c,l}(C_{B})$ follows for
each $SiGeB$ series a dependence that mimics that of $SiB$, but shifted
towards lower $T_{c}$ values, with a shift that increases with $C_{Ge}$. A
strong disorder induced by the Ge incorporation might explain such $T_{c,l}$
reduction. However, an important disorder would be evident in the normal state
square resistance at low temperature, $R_{N,sq}\sim 3\,\Omega$, which instead
remains well below the resistance quantum (see Fig. 1). Moreover, $R_{N}$ is
only little affected by the Ge incorporation, and a slight $R_{N}$ reduction
with $C_{Ge}$ is even observed, probably related to a better carrier mobility
in SiGe despite the scattering induced by Ge random position in the lattice.
Thus, the large $T_{c}$ suppression ($\delta T_{c,l}/T_{c,l}\sim 50\,\%$)
cannot be explained by the low disorder ($\delta R_{N}/R_{N}\sim 10\,\%$).
Fig. 3 shows the evolution of $T_{c,l}$ and $T_{c,h}$ as a function of the Ge
concentration. $T_{c,h}$ is on average constant, globally independent of the
Ge content, and its value is consistent within $7\%$ with the $T_{c}$
expected, in the absence of Ge, for a $SiB$ layer of the same B doping. In
addition, the measured critical magnetic field $H_{c2,h}$, is also compatible
with the $SiB$ reference, $H_{c2}\sim 200$ to $1000\,$G in the $C_{B}$ range
examined. Fitting $H_{c2,h}(T)$ (see Methods), it is possible to extract the
superconducting coherence length $\xi_{SiGeB,h}$. For $C_{B}=9.2\,at.\%$, we
find $\xi_{SiGeB,h}=59-65\,$nm, similar to $SiB$ $\xi_{SiB}=60\,$nm but with
$\sim 10\,\%$ larger value associated to a higher diffusion coefficient (as
also observed in $R_{N}$).
In contrast, we find, for the low temperature transition $T_{c,l}$, a strong
dependence with $C_{Ge}$, a suppressed $H_{c2,l}\sim 150$ to 500 G, and a
larger $\xi=70$ to 150 nm, the result of a doubled diffusion coefficient as
compared to $\xi_{SiB}=50$ to 100 nm.
Thus, while at low Ge content we observe the behavior of a homogeneous $SiGeB$
layer, two phases are present at high Ge content: one depleted in Ge, behaving
as pure $SiB$ with only a slightly increased disorder as a result of the Ge
incorporation processes, the other deeply affected by the incorporated Ge,
with a doubled diffusion coefficient and a suppressed $T_{c}$.

###  III.4 Superconductivity evolution with lattice deformation

In order to understand the role of Ge on $SiGeB$ superconductivity, and as Ge
concentration does not directly affect the carrier concentration, we focus on
the structural properties of the layer. In particular, we examine the
deformation induced by both the Ge-induced compressive strain and the opposite
B-induced tensile strain. X-Ray Diffraction maps around the (224) reflection
are realised to image both the in-plane and out-of-plane layer deformations
(see Methods). Two samples are shown in Fig. 4: a $SiB$ layer, with
$C_{B}=9.2\,at.\%$, and a $SiGeB$ layer, with the same B concentration and
$C_{Ge}=10.7\,at.\%$. The $SiB$ layer is partially relaxed, with an in-plane
lattice constant smaller than that of the Si substrate, as visible from the
larger $Q_{x}$ wavevector in $SiB$ as compared to Si. However, upon
incorporation of Ge, the layer evolves back to a nearly fully strained
configuration, with only the beginning of strain relaxation. This is the
result of Ge partially compensating the B induced strain, as
$a_{Ge}=5.6578\,\AA>a_{Si}=5.4307\,~{}\AA>a_{B}=3.74\,~{}\AA$ [22]. From the
in-plane and out-of-plane wavevectors $Q_{x}$ and $Q_{z}$, we extract the
lattice parameter of the $SiB$ ($SiGeB$) layers, reported in Fig. 4-a, with
$(Q_{Si}-Q_{SiB})/Q_{SiB}=(a_{SiB}-a_{Si})/a_{Si}$ and
$Q=\sqrt{Q_{x}^{2}+Q_{z}^{2}}$.

Figure 4:  a) XRD reciprocal maps along (224) direction for (left) a $SiB$
sample with $C_{B}=9.2\,at.\%$ and (right) a $SiGeB$ sample with
$C_{B}=9.2\,at.\%$ and $C_{Ge}=10.7\,at.\%$. Intensity is depicted in color
contrast, with cold colors for the lower signal and hot colors for higher
intensities. $Q_{x}$ and $Q_{z}$ correspond to the in-plane and out-of-plane
wavevectors. The lattice parameter extracted from the XRD measurements is
noted on each image. b) Optimized lattice constant for the ternary $SiGeB$
bulk alloy calculated within Density Functional Theory (DFT) in the
generalized gradient approximation (GGA) as a function of the Ge fraction,
$x_{Ge}$ and for different values of the B fraction, $x_{B}$. The $x_{Ge}$
value lies in the range of 0 to $25\,at.\%$ while $x_{B}$ varies between 2 and
$8\,at.\%$.

Having established the experimental lattice parameters available from the
limited number of XRD maps, we explore their dependence on B and Ge
concentration by performing Density Functional Theory (DFT) calculations in
the generalized gradient approximation (GGA) for the ternary $SiGeB$ bulk
alloy. Special Quasi Random Structure approach [23] is used to extract from
the ensemble of all the possible random configurations only those which
provide the most accurate approximation to the true random alloys. Further
details of the DFT simulations are presented in the Methods section. For the
pure SiGe alloy we find that, in the low Ge concentration regime, the behavior
of the lattice constant follows the Vegard’s law for binary semiconductors
[24, 25]. Indeed, the DFT calculated lattice parameter of the alloy linearly
increases with Ge fraction $x_{Ge}$, according to $a_{SiGe}=a_{Si}\cdot
x_{Si}+a_{Ge}\cdot(1-x_{Si})$, where aSi= 5.449 Å and aGe= 5.789 Å are the
DFT-GGA lattice parameter of pure Si and pure Ge, respectively. Once the pure
SiGe alloy case analyzed, we calculate the dependence of the lattice constant
on the B concentration for the ternary $SiGeB$ bulk alloy. We consider a B
fraction, $x_{B}$, ranging from 2$\,at.\%$ to 8$\,at.\%$ and we vary the Ge
composition from 0 to 25$\,at.\%$, matching the experimental parameter range.
As is shown in Fig. 4-b, increasing $x_{B}$ lowers the value of aSiGe while
maintaining the linear Vegard’s behavior. These results demonstrate that, in
this chemical composition regime, the use of a linear interpolation of the
three alloy components is theoretically justified and can be summarized in the
following equation:

$a_{SiGeB}=a_{B}\,\cdot x_{B}+a_{Ge}\,\cdot
x_{Ge}+a_{Si}\,\cdot(1-x_{B}-x_{Ge})$ (1)

where $a_{B}$ is determined through a constrained fit (with aSi and aGe fixed
to their GGA values) to be 3.81 Å, which is in very good agreement with the
experimental value measured in Ref. [22]. Even though, due to the well-known
GGA underbinding tendency [26], the simulated pure elements lattice parameters
are slightly overestimated if compared with the experimental values (by a few
percent difference), the theoretical results fully validate Eq.1. We thus
employ Vegard’s law (Eq. 1) as a function of $x_{B}$ and $x_{Ge}$ to predict
the lattice parameters of the samples shown in Fig. 4, by taking as input the
experimental values of $C_{B}$ and $C_{Ge}$ and the experimentally determined
$a_{Si}=5.4307$ Å, $a_{Ge}=5.6578$ Å and $a_{B}=3.74$ Å, the B and Ge
fractions $x_{j}=C_{j}/n_{Si}$ being calculated in respect to the pure Si
density $n_{Si}=5\times 10^{22}\,cm^{-3}$. For the samples analyzed in Fig. 4,
we obtain $a_{SiB}=5.2755$ Å(vs. $a_{SiB,XRD}=5.276\,\pm 0.005$ Å), and
$a_{SiGeB}=5.300$ Å(vs. $a_{SiGeB,XRD}=5.299\,\pm 0.005$ Å), an excellent
agreement, well within the error associated to ’pointing’ uncertainty on the
XRD maps.
With both experimental and numerical validation, we extend the 3-elements
Vegard’s law to calculate the lattice parameters $a$ for the samples for which
no XRD is available. To correctly estimate $a_{SiB}$ and $a_{SiGeB}$, only the
substitutional dopant concentration providing a lattice deformation is
relevant. We thus exclude from the following analysis the samples with
$C_{B}>9.2\,at.\%$, the concentration range where aggregates appear, rendering
inaccurate the estimation of the substitutional concentration [21]. The
concentration range between the fully activate regime and the saturation
regime $6\,at.\%<C_{B}\leq 9.2\,at.\%$ is however included, as the still-high
activation (ratio of active to total B concentration $>75\,\%$) is limited in
this region by substitutional inactive B complexes formed by a few atoms (B
dimers, trimers) [16]. Such complexes also induce a lattice deformation, whose
value differs however from that of isolated B atoms [27, 28]. We thus might
expect a maximum error of $\sim 20\,\%$ on the deformation estimation for
these complexes, but, as they account at most for $25\,\%$ of $C_{B}$ (and
only at the highest concentrations), the final induced error on the lattice
parameter is expected within a few $\%$. In the case of Ge, the whole $C_{Ge}$
range is considered, as for the concentrations investigated here, Ge is
expected to be fully substitutional [18].
The dependence of $T_{c}$ on the lattice parameter $a$ calculated from $C_{B}$
and $C_{Ge}$ is shown in Fig. 5. It is remarkable that all sample series
collapse in a common linear trend: $SiB$ samples see their lattice parameter
decrease with B doping and ’move’ towards higher $T_{c}$ from right to left;
$SiGeB$ samples with a fixed $C_{B}$ see the lattice parameter increase due to
the Ge incorporation, and move from left to right to lower $T_{c}$, over $SiB$
samples with smaller B concentrations. The multiple methods employed to
incorporate the Ge do not seem to affect significantly the global result, and
neither does the difference between the 30 nm and 80 nm thick samples series.
The series to series deviations from the average $T_{c}(a)$ observed in Fig. 5
are associated to uncertainties in the lattice parameter of $\delta a/a\sim
0.7\,\%$, and can be traced back to the uncertainty in the determination of
the deformation associated to the few-atoms complexes. It is noteworthy that
modifying the lattice parameter by $\delta a/a=-1\,\%$ leads to a large change
in the superconducting critical temperature of $\delta T_{c}/T_{c}=50\,\%$ . A
strong dependence of $T_{c}$ with the lattice parameter is reported for other
superconductors, such as $In_{x}Te$ [29] or covalent superconductors (like Si
and SiGe), such as superconducting B doped diamond ($\delta T_{c}/T_{c}\sim
64\,\%$ for $\delta a/a\sim 0.2\,\%$) [30] or $K_{3}C_{60}$ and $Rb_{3}C_{60}$
fullerenes ($\delta T_{c}/T_{c}\sim 83\,\%$ for $\delta a/a\sim 4\,\%$) [31].
Such strong increase of $T_{c}$ upon reduction of the lattice parameter can be
associated to the softening of phonon modes, which increases the electron-
phonon coupling $\lambda$ through the increase of the electron-phonon
potential $V_{e-ph}$ [32]. The incorporation of $Ge$ in the lattice by
nanosecond laser annealing allows thus addressing specifically the electron-
phonon potential, through a finely tuned lattice parameter.

Figure 5:  Superconducting critical temperature vs. lattice parameter
calculated with Eq.1 for $SiB$ reference series, GILD-5, GILD-15, GILD-200,
GILD-Ge and Ge implanted (see Table 1 All series are plotted, for $C_{B}\leq
9.2\,at.\%$, to avoid the region where B aggregates are present, affecting the
estimation of the lattice parameter.

##  IV Conclusions

In conclusion, we demonstrate superconductivity in $Si_{1-x}Ge_{x}:B$
epilayers by nanosecond laser ultra-doping with B. The B concentration
reached, well above the solubility limit thanks to this out-of-equilibrium
technique, is $C_{B}=1.5$ to $11\,at.\%$, with 100% activation rate below
$C_{B}=6\,at.\%$ and over 75% up to $C_{B}=9.2\,at.\%$. The Ge fraction
explored is $x=$0 to 0.21. Ge is incorporated in $SiB$ in three different
ways: 1) through a precursor gas by Gas Immersion Laser Doping; 2) by ion
implantation, followed by nanosecond laser annealing; 3) by UHV-CVD growth of
a thin Ge layer, followed by nanosecond laser annealing. The 30 nm and 80 nm
thick $Si_{1-x}Ge_{x}:B$ epilayers display a zero resistance state, with
superconducting critical temperature $T_{c}$ varying with B and Ge
concentration from 0 to 0.6 K, a superconducting critical field $H_{c2}=150$
to 500 G, and a superconducting coherence length $\xi=70$ to 140 nm, larger
than $SiB$ layers of equal concentration due to a doubled diffusion
coefficient associated to Ge incorporation. To understand the $T_{c}$
evolution, we turn towards BCS theory, which predicts a $T_{c}$ exponential
evolution with the electron-phonon coupling constant
$\lambda=N(E_{F})V_{e-ph}$, the product of the density of state at Fermi level
and the electron-phonon interaction potential. Starting with the simplest
$SiB$ alloy, we observe an initial increase of $T_{c}$ with $C_{B}$, that can
be associated to the increase of the charge carrier density
($n_{B}=1.5-7\,at.\%$), and as a consequence of $N(E_{F})$. However, $T_{c}$
keeps increasing even when the hole concentration $n_{B}$ saturates at
$C_{B}=7.8\,at.\%$, following the formation of B aggregates. We thus explore
the role played by the structural deformations on superconductivity, by fine-
tuning the strain through the modulation of the Ge concentration at fixed
carrier density. To estimate the lattice parameter modulation with B and Ge,
we validate Vegard’s law for the ternary $SiGeB$ bulk alloy by DFT-GGA
calculations. The theory is in excellent agreement with X-Ray Diffraction
maps, allowing to measure both the in-plane and the out-of-plane lattice
deformation. By correlating the $T_{c}$ with the calculated lattice parameter,
we observe a global linear dependence, common for both $Si:B$ and
$Si_{1-x}Ge_{x}:B$ layers, and independent on the Ge incorporation method or
on the sample thickness, with $\delta T_{c}/T_{c}\sim 50\,\%$ for $\delta
a/a\sim 1\,\%$, thus highlighting the importance of structural strain, at
fixed carrier concentration.

### Acknowledgements

We are grateful for support from the French CNRS RENATECH network, the
Physical Measurements Platform of University Paris-Saclay, the French National
Research Agency (ANR) under Contract No. ANR-16-CE24-0016-01,
ANR-19-CE47-0010-03 and ANR-22-QUA2-0002-02. M.A. and M. T. acknowledge the
ANR AMPHORE project (ANR-21-CE09-0007).

### Methods

####  IV.0.1 Gas Immersion Laser Doping

GILD is performed in an ultra high vacuum (UHV) reactor of base pressure
$10^{-9}-10^{-10}$ mbar, to ensure a minimal impurity incorporation during the
melt phase. A puff of the precursor gas ($BCl_{3}$ or $GeCl_{4}$) is injected
onto the substrate surface, saturating the chemisorption sites ($p\sim
10^{-5}mbar$). A pulse of excimer XeCl laser ($\lambda$ = 308 nm, pulse
duration 25 ns, working frequency 2 Hz) melts the substrate, the light being
completely and instantly converted into thermal energy in the top 7 nm. During
the melted phase, the chemisorbed atoms diffuse in the liquid. At the end of
the laser pulse, an epitaxial out-of-equilibrium recrystallization takes place
from the substrate at a speed of $\sim 4\,$m/s [13], achieving concentrations
larger than the solubility limit. When the crystallization front reaches the
surface, the excess impurities contained in the liquid are expelled outwards,
such as Cl whose segregation coefficient is close to 0 [33]. Thanks to a
careful optical treatment of the laser beam, the energy density at the 2mmx2mm
sample level has $\sim$1.2$\%$ spatial homogeneity. This ensures the
homogeneity of the layer thickness, resulting in a flat, straight, and sharp
(a few nm thick) interface of the SiGe with the substrate. Since the laser
absorption is sensitive to the layer doping level, a fixed laser energy
results in an increasing layer depth. In order to obtain a constant doped
depth independent of the B or Ge content, we measure the time-resolved
reflectometry, and maintain a constant melt time during the doping by
decreasing progressively the laser energy.
The B doping is always performed before the Ge incorporation. Indeed, a
homogeneous distribution is expected even when the B is further submitted to
the subsequent process time of the Ge incorporation. In contrast, the Ge
profile is expected to evolve toward the surface, depleting the bottom of the
layer [18]. To keep the Ge profile the most homogeneous possible, we
incorporate Ge last, to minimise the time spent by Ge atoms in the liquid
phase.

####  IV.0.2 UHV-CVD growth of Ge/Si

The epitaxial growth of Ge on Si is carried out in an UHV-CVD system with a
base pressure of $10^{-10}$ mbar. Pure $SiH_{4}$ and $GeH_{4}$ diluted at
10$\%$ in $H_{2}$ are used as gas sources. After a modified Shiraki chemical
cleaning [34] the substrates are slowly annealed up to 700∘C, the pressure
being maintained below $7\cdot 10^{-9}$ mbar. Afterwards, the chemical surface
oxide is removed by flashing at 990∘C, maintaining the low pressure. After the
deposition of a Si buffer layer at 700∘C under a pressure of $4\cdot 10^{-4}$
mbar, the Ge heteroepitaxy at 330∘C is initiated at a total pressure of
$7\cdot 10^{-3}$ mbar. The growth time is settled at 20 min in order to
achieve 6 nm of Ge [19, 20].

####  IV.0.3 Hall measurements

The transverse voltage $V_{H}$ is measured in a magnetic field perpendicular
to the layer, at room temperature, with $V_{H}/I=\gamma\frac{B}{en_{B}t}$,
$\gamma=0.68$ the Hall mobility factor [35], $I$ the bias current (10$\mu$A),
$B$ the applied magnetic field (0 to 2 T), $e$ the electron charge and $t$ the
layer thickness.

####  IV.0.4 Measurement of the critical magnetic field $H_{c2}(T)$

$R(T)$ superconducting transitions are measured for fixed values of a
perpendicular magnetic field from 0 to 55 mT. Both the $SiB$ reference samples
series and a few selected $SiGeB$ samples are studied
[$(C_{B},C_{Ge})=(9.2,8)\,at.\%$; $(9.2,21.3)\,at.\%$; $(8,10.7)\,at.\%$;
$(10.6,10.7)\,at.\%$]. In the temperature range $T=0.2-0.5\,$K, $SiB$ follows
the expected trend for a thin superconducting film near $T_{c}$:
$\mu_{0}H_{c2}=\frac{3\Phi_{0}}{(2\pi^{2}\,\xi^{2})}(1-\frac{T}{T_{c}})$. The
extracted coherence length is $\xi=60\,$nm, in agreement with previous
measurements [15]. $SiGeB$ $T_{c,h}$, measured in the same temperature range,
follows the same law, with $\xi=59-65\,$nm. These values, in agreement with
$SiB$ results, confirm the observed absence of Ge. The slightly larger values
might be a result of a small additional disorder as the layers have been
submitted to supplementary processes to incorporate Ge. Indeed, the diffusion
coefficient $D$ affects $\xi$, as $\xi=\sqrt{(\hbar D/1.76k_{B}T_{c})}$.

####  IV.0.5 X-Ray Diffraction

The diffractograms are realized with a Rigaku Smartlab XRD system with
Cu-K$\alpha$1 radiation of wavelength 1.54056 Å, operated at 45 kV and 200 mA.
The x-ray beam is narrowed to measure only the central, homogeneous part of
the laser annealed spot. To avoid the contribution of the gold contacts in the
diffractogram, the contacts were removed by a KI Au-etch followed by 1 min dip
in a 10$\%$ HF solution to remove Ti.

####  IV.0.6 DFT simulations

DFT calculations were performed by using the SIESTA package [36] whose
numerical atomic orbitals basis set allows treating large systems with an
affordable computational cost. The exchange–correlation energy functional was
approximated using the generalized gradient approximation (GGA) as implemented
by Perdew, Burke, and Ernzerhof (PBE functional) [37]. Only valence electrons
have been taken into account with core electrons being replaced by norm-
conserving pseudopotentials of Troullier-Martins type. An optimized
double-$\zeta$ polarized basis set was used for Si while a double-zeta plus
two polarization orbitals (DZP2) basis set was chosen for both Ge and B. All
the equilibrium ground state unit cells and geometries were obtained from
conjugate-gradients structural relaxation using DFT forces through the
Hellman-Feynman theorem. The structures were relaxed until the force on each
atom was smaller than 0.01 eV/ Å. The cutoff of the grid used for the real
space integration was set to 300 Ry while the self-consistent cycle tolerance
for solving the Kohn-Sham equations was set to 10-4 eV. A uniform Monkhorst-
Pack grid with 3$\times$3$\times$3 k-points was employed to sample the
Brillouin zone. A supercell of 216 atoms was considered, which corresponds to
a 3$\times$3$\times$3 supercell of the conventional 8-atom cell of bulk Si. To
take into account the random nature of the alloy the Special Quasi Random
Structure (SQS) approach [23] was adopted. The SQS configurations were
generated using the ATAT code [38] and considering that each atom in the
supercell can be replaced with a probability depending on its concentration,
as shown in Ref. [39]. As a starting point, a pure Si supercell was considered
and the Ge fraction of atoms was varied from 0 to 0.25 for several SQS
configurations. For each of these configurations, the lattice parameter of the
alloy was averaged over the different configurations and over the three cubic
crystal axis to minimize the error due to numerical fluctuations during
optimization. Once the pure SiGe alloys case was treated, the dependence of
the lattice parameter on the B concentration was calculated. B concentrations
from 2% to 8% and Ge composition from 0 to 25% were considered. All the
studied systems are substitutional solid alloys in which Si, Ge, and B can
occupy only substitutional lattice sites (interstitial are not taken into
account).

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