††thanks: Both authors contributed equally.††thanks: Both authors contributed
equally.

# Intrinsic ferromagnetism and restrictive thermodynamic stability in MA2N4
and Janus VSiGeN4 monolayers

Dibyendu Dey<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Maine, Orono, Maine 04469, USA  Avijeet Ray  Department of
Physics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667,
India  Liping Yu<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Maine, Orono, Maine 04469, USA

###### Abstract

The seminal experimental discovery of the remarkably stable MoSi2N4 monolayer
has led to a handful of predicted magnetic two-dimensional (2D) materials in
the MA2Z4 family (M = transition metals, A = Si, Ge, and Z = N, P, As). These
magnetic monolayers were predicted to be dynamically stable, but none of them
has been synthesized to date. In this Research Letter, from first-principles
thermodynamic stability analysis, we demonstrate that only the nitrides are
thermodynamically stable and this occurs under N-rich conditions. Based on
this finding, we propose two ferromagnetic, semiconducting Janus monolayers in
the family: VSiGeN4 and VSiSnN4. They are both dynamically and thermally
stable, but only the former is thermodynamically stable. Intriguingly, Janus
VSiGeN4 and VSiSnN4 monolayers show weak in-plane anisotropy compared with the
VSi2N4 monolayer. These two emerging Janus magnetic semiconductors offer
opportunities for studying 2D magnetism and spin control for spintronics
applications.

Atomically thin magnetic monolayers provide the ideal platform to study
magnetism and spintronics device concepts in the two-dimensional (2D) limit,
where magnetic properties can be effectively controlled or switched by
proximity effects and external perturbations such as the magnetic field,
electric field, defects, strain, optical doping, etc. Deng  _et al._ (2018);
Gibertini  _et al._ (2019); Huang  _et al._ (2018); Burch (2018); Chua  _et
al._ (2020); Zhao  _et al._ (2018); Jiang  _et al._ (2020); Tian  _et al._
(2019) In the last few years, a large number of 2D intrinsic magnetic
materials have been experimentally discovered and theoretically proposed. Blei
_et al._ (2021); Jiang  _et al._ (2021) The majority can be classified into
three broad groups, namely, transition metal halides, Huang  _et al._ (2017);
Zhang  _et al._ (2019a) transition metal chalcogenides Deng  _et al._ (2018);
Lee  _et al._ (2016), and MXenes and MXene analogs Anasori  _et al._ (2015);
Frey  _et al._ (2018). These materials have exhibited a wide spectrum of
magnetic and electronic properties. However, they also carry various
disadvantages for practical manufacturing applications. For example, halides
and MXenes mostly are vulnerable and reactive in the presence of ambient air
and water Shcherbakov  _et al._ (2018); Zhang  _et al._ (2017). Besides,
experimentally reported Curie temperatures $T_{C}$ are usually much lower than
the room temperature Huang  _et al._ (2017); Zhang  _et al._ (2019a); Deng
_et al._ (2018). Hence, the search for new stable magnetic 2D materials
continues to be a major research direction in this field.

An emerging group of 2D materials are septuple atomic layers of ternary
transition metal pnictides in the form of MA2Z4, where M is the transition
metal, A = Si or Ge, and Z = N, P, or As Wang  _et al._ (2021). The first
member of this group discovered is the semiconductor MoSi2N4 centimeter-scale
monolayer, which was successfully synthesized in 2020 Hong  _et al._ (2020).
This material exhibited exceptional stability to air, water, acid, and heat.
Following this discovery, first-principles calculations found tens of
dynamically stable MA2Z4 compounds Wang  _et al._ (2021); Ding and Wang (2021)
with properties of interest for various applications such as spintronics Cui
_et al._ (2021); Feng  _et al._ (2021), superconductors Yan  _et al._ (2021),
and catalysts Liu  _et al._ (2021a); Chen  _et al._ (2021). Of these, nine
were predicted to be magnetic, including five nitrides (VSi2N4, NbSi2N4,
NbGe2N4, TaGe2N4, and YSi2N4), two phosphides (VSi2P4 and VGe2P4), and two
arsenides (VSi2As4 and VGe2As4) Wang  _et al._ (2021); Ding and Wang (2021).
Remarkably, they have all been determined to be ferromagnetic and
semiconducting except YSi2N4, which is also ferromagnetic but is metallic Ding
and Wang (2021).

However, these 2D ferromagnetic MA2Z4 materials have yet to be synthesized.
They were mainly predicted based on their dynamical stability, _ab initio_
molecular dynamics, and negative heat formation energies (meaning
energetically more favorable than their constituent elemental phases). The
thermodynamic stability with respect to their stable competing phases has yet
to be determined. It is well known that those dynamically stable compounds can
often be thermodynamically unstable and be easily decomposed into their
competing phases. Such thermodynamic instability can make their experimental
synthesis very challenging especially under thermodynamic (quasi)equilibrium
conditions.

Above or near room temperature long-range magnetic ordering is necessary to
realize spintronics functionality in 2D materials. Unfortunately, none of
these proposed 2D ferromagnetic MA2Z4 compounds has been identified to possess
an intrinsic long-range magnetic order with a $T_{C}$ above the room
temperature. For example, density functional theory (DFT) and Monte Carlo
calculations suggested that monolayers VSi2N4 and VSi2P4 in the same structure
as MoSi2N4 have magnetic phase transition temperatures above room temperature,
but they are XY magnets. Feng  _et al._ (2021); Akanda and Lake (2021)
Nevertheless, the VSi2P4 monolayer in a different crystal structure is the
only reported 2D magnet with an out-of-plane easy axis, but its predicted
$T_{C}$ is merely about 90 K Wang  _et al._ (2021).

In this Research Letter, from first-principles thermodynamic stability
analysis (cf. computational methods in Supplemental Material Sec. S1) of all
the nine 2D magnetic MA2Z4 monolayers, we explain why they have not been
synthesized so far and suggest possible routes to grow them. Two-dimensional
Janus crystals and their superlattices are a new class of 2D materials that
offer extraordinary physical, chemical, and quantum properties Yagmurcukardes
_et al._ (2020); Varjovi  _et al._ (2021). Although the electronic structures
and piezoelectric properties of two non magnetic Janus MSiGeN4 (M = Mo and W)
monolayers were studied in a recent work Guo  _et al._ (2021), no magnetic
Janus monolayers within the same family have been predicted to date. Here, we
propose two ferromagnetic Janus monolayers, namely, VSiGeN4 and VSiSnN4, which
can be viewed as replacing one of the Si layers in a VSi2N4 monolayer with Ge
or Sn (Fig. 1). We demonstrate that thermodynamically stable VSiGeN4 and
dynamically stable VSiSnN4 monolayers are weak XY-type 2D ferromagnets with
transition temperatures above 300 K, and both of them are small-gap
semiconductors.

Figure 1: (a) Top and (b) side views of the crystal structures of ternary
layered VSiXN4 (X = Ge, Sn) Janus monolayers. V atoms are depicted in pink, X
atoms are in blue, Si atoms are in green, and N atoms are in gray. For a
VSi2N4 monolayer, X is replaced by Si. N-V-N bond angles ($\sim 90^{\circ}$)
and V-N bond lengths for the top (bottom) layer are indicated as $\theta_{1}$
($\theta_{2}$) and d1 (d2), respectively. The crystal field environments: (c)
tetrahedral for Si/X and (d) trigonal prismatic for V atoms.

Thermodynamic stability is a key property that must be addressed to predict
new materials and understand their experimental synthesizability. It refers to
the relative stability with respect to all the possible competing phases in
thermodynamic equilibrium. For a material to be thermodynamically stable
during growth, the chemical potentials of its constituent elements must
satisfy a set of conditions. Taking MoSi2N4 as an example, one condition is
that

$\Delta\mu_{\textrm{Mo}}+2\Delta\mu_{\text{Si}}+4\Delta\mu_{\text{N}}=\Delta
H_{f}(\text{MoSi}_{2}\text{N}_{4}),$ (1)

where $\Delta H_{f}(\text{MoSi}_{2}\text{N}_{4})$ is the enthalpy of
formation, which can be calculated from first-principles.
$\Delta\mu_{\textrm{Mo}}$, $\Delta\mu_{\text{Si}}$, and $\Delta\mu_{\text{N}}$
are the relative chemical potentials of elemental Mo, Si, and N with respect
to their solid bulk components (extreme rich limits), respectively. These
chemical potentials depend on the experimental growth conditions and are
regarded as variable in the formalism. They are further bound by (i) the
values that will cause precipitation of solid elemental Mo, Si, and N, so that

$\displaystyle\Delta\mu_{\text{Mo}}\leq 0,\;\Delta\mu_{\text{Si}}\leq
0,\;\Delta\mu_{\text{N}}\leq 0,$ (2)

and (ii) by the values that will lead to the precipitation of competing binary
phases MoN and Si3N4, so that

$\displaystyle\Delta\mu_{\text{Mo}}+\Delta\mu_{\text{N}}<\Delta
H_{f}\text{(MoN)},$ (3) $\displaystyle
3\Delta\mu_{\text{Si}}+4\Delta\mu_{\text{N}}<\Delta
H_{f}(\text{Si}_{3}\text{N}_{4}).$ (4)

The ranges of $\Delta\mu_{\textrm{Mo}}$, $\Delta\mu_{\text{Si}}$, and
$\Delta\mu_{\text{N}}$ that satisfy Eqs.(1)$-$(4) define the thermodynamic
stability region, where possible equilibrium experimental growth conditions
are suggested. No solution to these equations means that the material
investigated is thermodynamically unstable and its equilibrium experimental
synthesis would be quite challenging or impossible as kinetic barriers between
different phases at high temperatures are often ignorable or small.

Figure 2: 2D or 3D chemical potential regions (shaded in yellow) where (a)
MoSi2N4, (b) VSi2N4, and (c) VSiGeN4 monolayers are thermodynamically stable
against the formation of constituent elements and competing binary compounds.
The shaded area inside the solid blue line (dashed blue line) in the phase
diagram of MoSi2N4 is the chemical potential region where the monolayer can be
synthesized using chemically rich (poor) N. X (marked with a red circle)
represents the vertex of the chemical potential region, up to which the
respective compounds can be synthesized using chemically poor N. (d) The
difference in chemical potential of N at X and N2 gas (hlp-N) is shown on the
right (left) axes. All the energies are in eV.

As a test, we first perform the above thermodynamic stability analysis for two
nonmagnetic monolayers MoSi2N4 and WSi2N4, which have been experimentally
synthesized Hong  _et al._ (2020). Consistent with experiment, our calculated
results shown in Fig. 2(a) and Supplemental Material Fig. S2(a) demonstrate
that these two compounds are indeed thermodynamically stable. Their stability
regions are highlighted in yellow in the plots. It is worth mentioning that
this is defined relative to the chemical potential of hexagonal layered
polymeric nitrogen (hlp-N), based on which $\Delta H_{f}$ are calculated. This
hlp-N phase represents the extreme N-rich growth condition and was
experimentally synthesized near 250 GPa Laniel  _et al._ (2019). The N2 gas
under normal conditions is about 3.05 eV lower in chemical potential (poorer)
than hlp-N; in between, there are another two metastable phases, cubic gauche
polymeric nitrogen (cg-N) Benchafia  _et al._ (2017) and a polymeric nitrogen
allotrope with black phosphorus structure (bp-N) Laniel  _et al._ (2020),
which were also experimentally synthesized. As displayed in Fig. 2(a) and
Supplemental Material Fig. S2(a), the thermodynamic stability regions of
MoSi2N4 and WSi2N4 both cover from extreme N-rich conditions to conditions
that are much poorer than the N2 gas.

We now perform the same thermodynamic stability analysis for all nine magnetic
MA2Z4 monolayers. Computational details are given in Supplemental Material
Sec. S1. Our calculated results are shown in Fig. 2(b) and Supplemental
Material Figs. S2(b)-S2(d). Although these monolayers were all predicted to be
dynamically stable and have a negative formation enthalpy, our calculations
reveal that only nitrides, except YSi2N4 are thermodynamically stable and this
occurs only under some conditions for N that are richer than N2 gas. Their
stability regions are all much narrower than those for MoSi2N4 and WSi2N4.
Furthermore, to understand the relative stability of magnetic MA2N4, the
distance between the vertex of each chemical potential region (marked as X and
shown with a red circle) and N2 gas has been calculated. $\Delta\mu_{N}$(X)$-$
$\Delta\mu_{N}$(N2) increases in the following order: VSi2N4 $\rightarrow$
TaGe2N4 $\rightarrow$ NbGe2N4 $\rightarrow$ NbSi2N4, along which the
respective regions are far from the chemical potential of N2 gas [Fig. 2(d)].
This means that VSi2N4 is the most stable among these magnetic nitrides. These
results suggest that the experimental growth of these nitrides under
thermodynamic equilibrium conditions will be difficult as extreme N-rich
conditions are necessary. VSi2N4 may be grown via the reaction

$\displaystyle 2\text{Si}_{3}\text{N}_{4}+3\text{VN}+\text{N
(solid)}\longrightarrow 3\text{VSi}_{2}\text{N}_{4},$ (5)

where the solid N reactant can be hlp-N, cg-N, or bp-N. For phosphides (VSi2P4
and VGe2P4) and arsenides (VSi2As4 and VGe2As4), we cannot find any
thermodynamically stable region. Thus, under equilibrium growth conditions,
the competing binary phases (e.g., V3P, VP, SiP2, GeP3, etc., for phosphides
and V3As, VAs, SiAs2, GeAs2, etc., for arsenides) of these materials would
form rather than the materials themselves.

To identify more stable intrinsic ferromagnetic 2D MA2Z4 compounds, we extend
our first-principles thermodynamic stability analysis to two Janus compounds,
VSiGeN4 and VSiSnN4. They are derived from the VSi2N4 monolayer, which has
been demonstrated to be thermodynamically stable. Janus monolayers VSiXN4 (X =
Ge, Sn) can be viewed as a VN2 layer sandwiched between Si-N and X-N layers
(Fig. 1). Due to the presence of different group-IV elements in the top and
bottom layers, such Janus monolayers lack both inversion and out-of-plane
mirror symmetries in contrast to the VSi2N4 monolayer.

Our thermodynamic stability study reveals that both VSiGeN4 and VSiSnN4
monolayers have negative formation enthalpies (using hlp-N) of $-$4.329 eV and
$-$2.132 eV, respectively. However, only monolayer VSiGeN4 is found to be
thermodynamically stable. Figure 2(c) shows our calculated stability region
for VSiGeN4 and its competing phases Si3N4, Ge3N4, and VN. As expected, this
thermodynamic stability region is smaller than that of VSi2N4 and stays 1 eV
farther from the chemical potential of N2 gas [Fig. 2(d)]. However,
advantageously, this stability region also covers the chemical potentials of
both cg-N and bp-N phases. The VSiGeN4 monolayer may thus be synthesized via

$\displaystyle\text{Si}_{3}\text{N}_{4}+\text{Ge}_{3}\text{N}_{4}+3\text{VN}+\text{cg-N}\longrightarrow
3\text{VSiGeN}_{4}.$ (6)

For the VSiSnN4 monolayer, no thermodynamic stability region is found in this
paper.

Figure 3: (a) and (b) Phonon band structures, and (c) and (d) temperatures and
total energy fluctuations after 5 ps of _ab initio_ molecular dynamics
simulations at 500 K of VSiGeN4 (top panels) and VSiSnN4 (bottom panels) Janus
monolayers.

Both Janus monolayers VSiGeN4 and VSiSnN4 are also found to be dynamically and
thermally stable. Figures 3(a) and 3(b) show our calculated phonon spectra
along the high-symmetry directions of the hexagonal Brillouin zone (BZ). No
negative frequencies are found in either case. Figures 3(c) and 3(d) show our
_ab initio_ molecular dynamics simulation results for both Janus monolayers at
500 K. No indication of bond breaking or significant structural distortions is
seen after 5 ps of simulation time (1 fs time step). Such stability features
render these Janus materials also synthesizable using non-equilibrium
techniques. Lu  _et al._ (2017); Qin  _et al._ (2022)

Figure 4: Side and top views of the effective triangular lattice formed by V
ions in VSi2N4 and VSiXN4 Janus monolayers showing (a) and (b) FM and (c) and
(d) stripe AFM states. V ions are shown as pink spheres. Up and down spins are
indicated by blue and gold arrows, respectively. (e) A schematic of the energy
levels of $d$ orbitals in a trigonal prismatic crystal field environment.
Here, only the middle layer containing V ions is displayed since the non-
magnetic top and bottom layers of these monolayers do not contribute to the
spin density distribution. The single valence electron of the V4+ ion
predominately occupies the $d_{z^{2}}$ orbital, and the corresponding spin
density distribution is shown in (f).

Now, we turn to study the magnetic properties of VSiGeN4 and VSiSnN4
monolayers. To identify the magnetic ground state, we calculate the total
energy difference between the ferromagnetic (FM) and stripe antiferromagnetic
(AFM) spin configurations as depicted in Figs. 4(a)$-$4(d) using the
generalized gradient approximation (GGA)+$U$ method Dudarev  _et al._ (1998).
To include correlation effects in V 3$d$ orbitals, we choose $U$ = 3.2 eV, at
which the GGA+$U$ calculated energy difference between the AFM and FM states
matches with the revised Heyd-Scuseria-Ernzerhof screened hybrid functional
(HSE06) Heyd  _et al._ (2003) calculated one. (cf. Supplemental Material Fig.
S3). The $U$ value determined in this way has successfully predicted correct
magnetic, structural properties in numerous systems Hong  _et al._ (2012);
Zhuang and Hennig (2016).

Our GGA+$U$ calculations find that both Janus monolayers stabilize into a FM
ground state, like the VSi2N4 monolayer. As one can see from Table 1, the FM
states in VSiGeN4 and VSiSnN4 are 65.5 and 44.8 meV lower in energy than their
AFM states, respectively. Magnetic moments $m_{V}$ obtained from our GGA+$U$
calculations agree qualitatively with the ionic description that gives a V4+:
3$d^{1}$ electronic configuration. In a trigonal prismatic crystal field
environment, V $d$ orbitals split into the low-lying $d_{z^{2}}$ orbital
followed by the ($d_{xy}$+$d_{x^{2}-y^{2}}$) and ($d_{xz}$+$d_{yz}$) orbitals
[Fig. 4(e)], respectively. Our calculated spin density distribution of the
occupied V $d$ orbital [Fig. 4(f)] shows that the top valence band in the
majority-spin channel is predominantly of $d_{z^{2}}$ orbital character. Here,
the V-N-V angle is close to 90∘, and within this orthogonal setup,
superexchange interaction between the V atoms mediated by the neighboring N
atoms is the dominating exchange mechanism for ferromagnetism. This is in
agreement with the Goodenough-Kanamori rules. Goodenough (1955); Kanamori
(1959) As a result, VSi2N4 and VSiXN4 monolayers adopt a FM ordered ground
state.

Monolayer |  $a$ |  $E_{g}$ |  $m_{V}$ |  $\Delta E_{m}$ |  MAE |  $T_{BKT}$ |
---|---|---|---|---|---|---|---
|  (Å) |  (eV) |  ($\mu_{B}$) |  (meV) |  ($\mu$eV) |  (K) |
VSi2N4 |  2.90 |  0.50 (D) |  1.10 |  88.7 |  -59 |  687 |
VSiGeN4 |  2.97 |  0.48 (D) |  1.11 |  65.5 |  -20 |  507 |
VSiSnN4 |  3.07 |  0.85 (ID) |  1.14 |  44.8 |  -10 |  347 |
Table 1: Lattice constants $a$, band gap ($E_{g}$), V magnetic moments
($m_{V}$), energy differences between the AFM and FM spin configurations
[$\Delta$Em = (EAFM $-$ EFM)/f.u.], magnetic anisotropy energy (MAE), which
includes both magnetocrystalline anisotropy (MCA) and magnetic shape
anisotropy (MSA) and transition temperatures $T_{BKT}$ of VSi2N4 and VSiXN4
monolayers. The band gap $E_{g}$ values mentioned in the third column are
obtained using HSE06 and agree qualitatively with the band gap values obtained
within the GGA+$U$ method: VSi2N4, 0.3 eV [direct semiconductor (D)], VSiGeN4,
0.15 eV (D), VSiSnN4: 0.28 eV [indirect semiconductor (ID)].

According to the Mermin-Wagner theorem Mermin and Wagner (1966), long-range
magnetic order should be absent in the 2D isotropic spin systems at any finite
temperature. However, magnetic anisotropy removes this restriction and
stabilizes magnetic order in 2D materials Huang  _et al._ (2017); Kim  _et
al._ (2019); Cai  _et al._ (2019); Liu  _et al._ (2021b). To determine the
nature of magnetic anisotropy, we first calculate magnetocrystalline
anisotropy (MCA) using the GGA+$U$ method including spin-orbit coupling (SOC).
MCA is defined as the energy difference between two FM spin configurations,
where magnetic moments are directed along the $x$ and $z$ directions,
respectively. For the VSiGeN4 monolayer, we obtain a negative MCA
($-$4$~{}\mu$eV) and a positive MCA (4$~{}\mu$eV) for the VSiSnN4 monolayer.
Since MCA is significantly small here, magnetic shape anisotropy (MSA), which
originated from dipole-dipole interactions may become relevant and contribute
to the magnetic anisotropy energy (MAE). The calculated MSA values for VSiGeN4
($-$16$~{}\mu$eV) and VSiSnN4 ($-$14$~{}\mu$eV) monolayers are found to be
greater than the magnitude of MCA values, whereas for VSi2N4, MCA
($-$42$~{}\mu$eV) remains higher than MSA ($-$17$~{}\mu$eV) (cf. Supplemental
Material Sec. S5 for details). As a result, after the inclusion of MSA, the
total MAE for Janus monolayers becomes negative (easy-plane anisotropy), and
owing to its small magnitude (Table 1), VSiXN4 monolayers can be regarded as
weak 2D-XY magnets like the CrCl3 monolayer. Kim  _et al._ (2019); Akram  _et
al._ (2021) A quasi-long-range order with a divergent correlation length could
be observed below the Berezinskii-Kosterlitz-Thouless (BKT) Kosterlitz and
Thouless (1973) transition temperature for these planar magnets. However, due
to the weak anisotropy, in-plane magnetization in both Janus monolayers could
easily be tuned to out-of-plane long-range order by perturbations such as
strains and defects.

The magnetic transition temperature $T_{BKT}=$ 1.335 $J/k_{B}$ is obtained
from the results of the Monte Carlo simulation for 2D-XY magnets on a
triangular lattice Zhang  _et al._ (2019b) (cf. Supplemental Material Sec.
S4). We find that the transition temperatures of VSiGeN4 (507 K) and VSiSnN4
(347 K) are lower than that of VSi2N4 (687 K), but they are still higher than
the room temperature.

Figure 5: (a) Comparison of magnetocrystalline anisotropy (MCA) values between
strained monolayers VSi2N4 and Janus monolayers VSiXN4 (X = Ge, Sn).
$\epsilon$ is the uniform biaxial strain with respect to the VSi2N4 optimized
lattice constant. Band structures of (b) VSi2N4, (c) VSiGeN4, and (d) VSiSnN4
monolayers within HSE06 method in the FM ground state. Red and blue lines in
the band structures represent the majority and minority spin channels,
respectively.

To understand the nature of MCA variations in Janus monolayers, we also
calculated the MCA for strained VSi2N4 monolayers. Due to the presence of
larger-size atoms, VSiGeN4 and VSiSnN4 Janus structures have lattice constants
that are 2.4 and 5.8% larger than that of monolayer VSi2N4 (Table 1),
respectively. Figure 5(a) shows a comparison of MCA values between strained
monolayer VSi2N4 and Janus monolayers. Clearly, one can observe that the MCA
values of Janus monolayers (marked as red squares) are almost identical to the
MCA values of VSi2N4 monolayers at the same lattice constants of corresponding
Janus monolayers. This means that the key factor that drives the transition
from the in-plane MCA in VSi2N4 and VSiGeN4 to the out-of-plane MCA in the
VSiSnN4 monolayer is the biaxial in-plane tensile strain. The strain effects
of MCA in VSi2N4 have already been theoretically studied in the past Cui  _et
al._ (2021). These reported results, agreeing well with ours, indicate that
switching the MCA from in-plane to out-of-plane in VSi2N4 needs more than 4%
of biaxial strains, which would be very challenging for experimental
realization and disadvantageous for device applications. Here our results
demonstrate that Janus engineering is a useful (and probably more viable)
method to tune MCA and other properties of interest.

In addition to remarkable stability and magnetic properties, these two Janus
monolayers exhibit excellent electronic properties. Figures 5(b)$-$5(d) depict
the electronic band structures of VSi2N4 and VSiXN4 monolayers as obtained
from HSE06. VSi2N4 and VSiGeN4 monolayers are direct-gap semiconductors with
band gap $E_{g}$ $\sim$ 0.5 eV. Interestingly, the nature of the band gap
changes from direct to indirect when $E_{g}$ increases to 0.85 eV for VSiSnN4.
This band gap switching and these different $E_{g}$ values make VSiXN4
monolayers suitable candidates to operate at variable voltages, frequencies,
and temperature ranges for device applications. Note that these band
structures are consistent with those determined within the GGA+$U$ method
(Supplemental Material Fig. S7).

Finally, we shed light on the valley polarization, which is a measure of spin-
valley coupling. This quantity can be estimated from the energy difference
$\Delta E_{sv}$ between valance band maxima or conduction band minima at the
$-K$ and $+K$ points. For this purpose, we calculate the band structures
within GGA+$U$+SOC (see Supplemental Material Fig. S8) with the magnetization
direction along +$x$ and +$z$ directions, respectively. A relatively large
valley polarization of 64 meV can be induced in the VSi2N4 monolayer by
changing the spin directions from +$x$ to +$z$, consistent with a previous
report Cui  _et al._ (2021). On the other hand, the valley polarization can be
induced in VSiXN4 monolayers through the valley splitting of the bottom
conduction band.

In conclusion, through first-principles analysis, we have shown that among all
dynamically stable magnetic MA2Z4 monolayers, only four magnetic nitrides are
thermodynamically stable and this occurs under some experimentally challenging
growth conditions. In the group of 2D magnetic nitrides, VSi2N4 has been found
to be the most stable monolayer, from which two Janus ferromagnetic
monolayers, namely, VSiGeN4 and VSiSnN4, have been designed. The dynamical
stability and thermal stability of these Janus monolayers ensure their non
equilibrium growth. Alternatively, a thermodynamically stable VSiGeN4
monolayer can be synthesized in equilibrium conditions. The electronic
structures reveal that they are narrow-band-gap semiconductors. However, the
most exciting feature of VSiGeN4 and VSiSnN4 monolayers is their weak easy-
plane magnetic anisotropy, which is primarily associated with the MSA and the
contrasting semiconducting nature (direct vs indirect). The magnetic
properties of these 2D ternary layered vanadium-based Janus semiconductors
make them emerging candidates for spintronics and optoelectronics
applications. Besides, the possibility of realization of skyrmions in VSiGeN4
and VSiSnN4 monolayers can also be investigated, as it was for other Janus
monolayers as reported in the contemporary literature Liang  _et al._ (2020);
Xu  _et al._ (2020). These results will provide guidance for experimental as
well as theoretical explorations of the magnetic monolayers of the MA2Z4
family and these two Janus VSiXN4 monolayers.

This work has been supported by the U.S. DOE, Office of Science, Office of
Basic Energy Sciences, under Award No. DE-SC0021127. D.D. thanks Harrison
LaBollita for the useful discussion on MSA calculations. D.D. and L.Y.
gratefully acknowledge the assistance of the Advanced Computing Group of the
University of Maine System for providing computational resources for this
work.

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# Isospin dependence in single-nucleon removal cross sections explained
through valence-core destruction effects

M. Gómez-Ramos<EMAIL_ADDRESS>Departamento de FAMN, Universidad de Sevilla,
Apartado 1065, 41080 Sevilla, Spain.  J. Gómez-Camacho  Departamento de FAMN,
Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain.  Centro Nacional
de Aceleradores (U. Sevilla, J. Andalucía, CSIC), Tomás Alva Edison, 7, 41092
Sevilla, Spain  A.M. Moro  Departamento de FAMN, Universidad de Sevilla,
Apartado 1065, 41080 Sevilla, Spain.  Instituto Interuniversitario Carlos I de
Física Teórica y Computacional (iC1), Apdo. 1065, E-41080 Sevilla, Spain

###### Abstract

The discrepancy between experimental data and theoretical calculations in one-
nucleon removal reactions at intermediate energies (quantified by the so-
called “quenching factors”) and its dependence on the isospin asymmetry of the
nuclei has been an open problem in nuclear physics for the last fifteen years.
In this work, we propose an explanation for this long-standing problem, which
relies on the inclusion of the process of core destruction due to its
interaction with the removed nucleon. To include this effect, we extend the
commonly used eikonal formalism via an effective nucleon density, and apply it
to a series of nucleon knockout reactions. The effect of core destruction is
found to depend strongly on the binding energy of the removed nucleon, leading
to a significant reduction of the cross section for deeply bound nucleons,
which reduces the isospin dependence of the “quenching factors”, making them
more consistent with the trends found in transfer and $(p,pN)$ reactions.

Single nucleon knockout reactions with light targets (9Be, 12C) at
intermediate energies have been a key experimental tool to study the structure
of unstable nuclei Orr  _et al._ (1992); Bazin  _et al._ (1995); Simon  _et
al._ (1999); Cortina-Gil  _et al._ (2002, 2004); Stroberg  _et al._ (2015);
Gade  _et al._ (2016). These reactions can be described as
$P(C+V)+T\rightarrow C+X$, where the projectile $P$ collides with the target
$T$ so that the residual nucleus (the core) $C$ is detected, while the valence
nucleon $V$ can be detected (diffractive breakup) or is absorbed (stripping).
From the momentum distribution of the core, properties of the valence nucleon
can be extracted Hüfner and Nemes (1981); Bertulani and McVoy (1992). The
dynamics of the collision is standardly modelled within the eikonal
approximation Hansen and Tostevin (2003), which is reasonable for sufficiently
high energies ($\sim$80-90 MeV per nucleon). Other nucleon removal reactions
such as nucleon transfer Kay  _et al._ (2013) and quasifree nucleon removal
with proton targets $(p,pN)$ Jacob and Maris (1966) provide complementary
information on the properties of the removed nucleons.

A systematic study of the cross section of nucleon knockout reactions in light
and medium-mass nuclei showed an intriguing trend Gade  _et al._ (2008), where
the discrepancy between experimental cross sections and theoretical
predictions, quantified by the so-called “quenching factor”
($R_{s}=\sigma_{\mathrm{exp}}/\sigma_{\mathrm{theor}}$), shows a marked
dependence on the isospin asymmetry of the nucleus, such that for very
asymmetric nuclei, the removal of the more abundant nucleons presents a small
“quenching” ($R_{s}\sim 1$) while the removal of the less abundant ones
suffers from a large reduction ($R_{s}\sim 0.2-0.4$). This tendency has been
interpreted as the effect of short-range correlations on the less abundant and
more deeply bound nucleons. However, other systematic studies with transfer
Kay  _et al._ (2013); Flavigny  _et al._ (2013, 2018) and $(p,2p)$ reactions
Atar  _et al._ (2018); Gómez-Ramos and Moro (2018); Holl  _et al._ (2019) have
failed to find this marked dependence on isospin asymmetry, while the addition
of new data for heavy-target nucleon-knockout reactions has only reinforced it
Tostevin and Gade (2014, 2021). A recent overview on this topic can be found
in Aumann  _et al._ (2021). Whether this isospin dependence is a manifestation
of short-range correlations not included in standard, small-scale shell-model
calculations or an artifact derived from a not yet understood deficiency of
the reaction model Paschalis  _et al._ (2020) is a pressing question in
nowadays nuclear physics which calls for a careful revision of both the
structure and reaction inputs employed in these analyses.

Eikonal descriptions assume straight-line trajectories for core and valence
nucleon and ignore their mutual final-state interaction. A potentially
important effect absent from this description of knockout reactions is the
destruction of the residual core because of its interaction with the valence
nucleon after its removal from the projectile. This core destruction would
naturally lead to a reduction in the knockout cross section, an effect that
should be larger when removing more deeply bound nucleons, with stronger
interactions with the core, as illustrated in Fig. 1. In fact, intranuclear
cascade calculations using the Liege implementation (INCL) Louchart  _et al._
(2011); Sun  _et al._ (2016) point to this increased reduction for more
deeply-bound nucleons. Moreover, for exclusive breakup reactions, where both
valence nucleon and core are detected, the inclusion of this effect in the
standard Continuum-Discretized Coupled-Channel (CDCC) formalism Austern  _et
al._ (1987) has also shown a larger reduction in cross section for the removal
of the more deeply-bound species Gómez-Ramos  _et al._ (2022). A recent
publication Hebborn and Potel (2023) has presented a Green’s function
description of knockout reactions, but without numerical results.

It is the goal of this work to investigate the effect of these final-state-
interactions between the removed nucleon and the residual core on the survival
probabilty of the latter in knockout reactions. For that, a novel extension of
the eikonal formalism is presented that accounts for such effects and is
applied to measured removal reactions for deeply- and weakly-bound nucleons.

Theoretical framework: In the following, we will focus on the stripping
process. The development for diffractive scattering is beyond the scope of
this work. The stripping channels, although not individually resolved, can be
identified by an index $j$ which labels the complex target-nucleon state
which, along with the outgoing core, describes the final state. In particular,
$j=0$ labels the nucleon-target elastic state where the target remains in its
ground state, whereas $j>0$ correspond to states in which the nucleon excites
the target, contributing to stripping. $\vec{k}$ is the relative momentum
between the nucleon and the core. The stripping probability, for some impact
parameter ${\vec{b}}$, can be written as

$\displaystyle P_{\mathrm{str}}({\vec{b}})$ $\displaystyle=\int
d^{3}{\vec{k}}\sum_{j\neq 0}|A_{j}({\vec{b}},\vec{k})|^{2}$ $\displaystyle
A_{j}({\vec{b}},\vec{k})$ $\displaystyle=\int
d^{3}{\vec{r}}\phi_{g}({\vec{r}})^{*}S^{0}_{CT}(b_{CT})S^{j}_{VT}(b_{VT})\psi^{(-)}(\vec{k},\vec{r}),$
(1)

where $\phi_{g}$ is the bound core-valence state, $S^{0}_{CT}$ is the core-
target elastic S-matrix while $S^{j}_{VT}$ is the valence-target S-matrix for
state $j$ and $\psi^{(-)}(\vec{k},\vec{r})$ is the final unbound core-valence
state. See Fig. 2 for a representation of the impact parameters $b_{CT}$ and
$b_{VT}$. A key magnitude in this approach is the nonlocal density:

$\langle{\vec{r_{2}}}|\rho_{f}|{\vec{r_{1}}}\rangle=\int
d^{3}{\vec{k}}\;\left(\psi^{(-)}(\vec{k},{\vec{r_{2}}})\right)^{*}\psi^{(-)}(\vec{k},{\vec{r_{1}}}).$
(2)

If the core-nucleon interaction is real (i.e., if the nucleon cannot break the
core), closure can be used, and
$\langle{\vec{r_{2}}}|\rho_{f}|{\vec{r_{1}}}\rangle=\delta({\vec{r_{1}}}-{\vec{r_{2}}})$.
This allows the use of unitarity in the valence-target S-matrix,

$\sum_{j\neq 0}|S^{j}_{VT}(b_{VT})|^{2}=1-|S^{0}_{VT}(b_{VT})|^{2},$ (3)

and leads to the standard compact eikonal expression Hansen and Tostevin
(2003)

$P^{\mathrm{Ei}}_{\mathrm{str}}({\vec{b}})=\int
d^{3}\vec{r}\;|\phi_{g}({\vec{r}})|^{2}|S^{0}_{CT}(b_{CT})|^{2}\left(1-|S^{0}_{VT}(b_{VT})|^{2}\right),$
(4)

However, in a more realistic situation, where the interaction between valence
and core is taken as complex and energy dependent (for example, to describe
the excitation or break-up of the core) this is not the case. This is the key
contribution of our work, as compared to standard eikonal approximations.
Instead of assuming closure, we will use complex valence-core interactions to
get explicitly the continuum wavefunctions at all energies, and then evaluate
$\langle{\vec{r_{1}}}|\rho_{f}|{\vec{r_{2}}}\rangle$, which would be non
local.

In this work, we look for an expression as close as possible to the eikonal
derivation. The expression of the proton removal probability requires
integration over two radial variables, ${\vec{r_{1}}},{\vec{r_{2}}}$. The
integrand involves the product of two S matrices
$S^{j}_{VT}(b_{VT}({\vec{r_{1}}}))(S^{j}_{VT}(b_{VT}({\vec{r_{2}}})))^{*}$,
which are evaluated at different impact parameters, so unitarity (Eq. (3))
cannot be applied. This problem with unitarity can be avoided by approximating
the two impact parameters in the previous expressions by an average impact
parameter defined as $b_{VT}=\sqrt{(b+\alpha x)^{2}+(\alpha y)^{2}}$, where
$\alpha={A-1\over A}$, $x={x_{1}+x_{2}\over 2}$ and
$y={\sqrt{\dfrac{y_{1}^{2}+y_{2}^{2}}{2}}}$. $b_{CT}$ requires an equivalent
expression. Then we can approximate

$\sum_{j\neq
0}S^{j}_{VT}(b_{VT}({\vec{r_{1}}}))(S^{j}_{VT}(b_{VT}({\vec{r_{2}}})))^{*}\simeq
1-|S^{0}_{VT}(b_{VT}(x,y))|^{2},$ (5)

Figure 1:  Schematic illustration of one-proton and one-neutron removal
processes in the 40Si knockout reaction. Note that more channels corresponding
to an unbound core are open when a deeply bound nucleon is removed, which
leads to larger destruction of the core.

Figure 2:  Scheme of the coordinates used in this work. The beam is
perpendicular to the paper.

leading to

$\displaystyle P_{\rm{str}}({\vec{b}})$ $\displaystyle\simeq\int
dxdy\;\rho^{(2)\rm{eff}}(x,y)$
$\displaystyle\times|S^{0}_{CT}(b_{CT})|^{2}\left(1-|S^{0}_{VT}(b_{VT})|^{2}\right)$
(6) $\displaystyle\rho^{(2)\rm{eff}}(x,y)$
$\displaystyle=\int\\!d^{3}{\vec{r_{1}}}\int\\!d^{3}{\vec{r_{2}}}\;\langle{\vec{r_{2}}}|\rho_{f}|{\vec{r_{1}}}\rangle\phi_{g}^{*}({\vec{r_{2}}})\phi_{g}({\vec{r_{1}}})$
$\displaystyle\times\delta\left(x-{x_{1}+x_{2}\over
2}\right)\delta\left(y-\sqrt{\dfrac{y_{1}^{2}+y_{2}^{2}}{2}}\right),$ (7)

where $\langle{\vec{r_{2}}}|\rho_{f}|{\vec{r_{1}}}\rangle$ must be computed
without applying closure. A more detailed derivation of $\rho^{(2)}_{eff}$ can
be found in the Supplementary Material. Thus, core destruction through
interaction with the valence particle can be simply described, in standard
eikonal calculations, by using an effective two-dimensional local density
$\rho^{(2)\rm{eff}}(x,y)$, which is obtained from the nonlocal final density
$\langle{\vec{r_{2}}}|\rho_{f}|{\vec{r_{1}}}\rangle$ and the nonlocal initial
density $\phi_{g}^{*}({\vec{r_{2}}})\phi_{g}({\vec{r_{1}}})$. In the usual
eikonal approach, this two-dimensional local density is obtained by
integrating the ground state density on $z$, giving rise to:

$\rho^{(2)\rm{Ei}}(x,y)=\int dz|\phi_{g}({\vec{r}})|^{2}.$ (8)

As the impact parameters defining valence and target absorption depend on the
coordinates $(x,y)$, the densities required to do the calculations of the
stripping probabilities require the two-dimensional densities
$\rho^{(2)\rm{Ei}}(x,y)$, $\rho^{(2)\rm{eff}}(x,y)$. One may also compute one-
dimensional densities for $x$ (or $y$) as:

$\displaystyle\rho^{(1)\rm{eff}}(x)$
$\displaystyle=\int\\!dy\rho^{(2)\rm{eff}}(x,y)$ (9)
$\displaystyle\rho^{(1)\rm{Ei}}(x)$
$\displaystyle=\int\\!dy\rho^{(2)\rm{Ei}}(x,y).$ (10)

In the Supplementary Material, an expansion to optimize the calculation of
$\rho^{\rm{eff}}$ is presented. A fundamental difference between this method
and standard eikonal calculations (e.g. Sauvan  _et al._ (2004)) lies in the
consideration that valence particle and core keep interacting after the
absorption of the former by the target, while standard calculations neglect
this interaction. We believe that this interaction is still important for the
dynamics of the reaction even after the valence particle has been absorbed (as
it has not disappeared, rather it has become deeply correlated with the
internal degrees of freedom of the target), which is consistent with the
spirit and results from INCL calculations Louchart  _et al._ (2011); Sun  _et
al._ (2016).

Results: We apply this formalism to a selection of the knockout reactions
presented in Tostevin and Gade (2021), for removal of neutron and proton from
a neutron-rich nucleus (40Si), a proton-rich nucleus (24Si) and an isospin
symmetric one (12C). These nuclei were selected because both proton and
neutron removal were measured, only a few single-particle configurations of
the removed nucleon had to be considered (except for neutron removal from
40Si) and because the ingredients for the original calculations were
accessible in the literature. In order to restrict the integration in
$\vec{k}$ in the evaluation of
$\langle{\vec{r_{2}}}|\rho_{f}|{\vec{r_{1}}}\rangle$, we included a weighting
factor $e^{-k^{4}a^{4}}$ with $a=0.15$ fm-1 and expanded
$\psi^{(-)}(\vec{k},\vec{r})$ in multipoles up to $l_{\mathrm{max}}=29$ (see
Supplementary Material). For the single-particle wavefunction we used the same
geometry Gade  _et al._ (2008); Brown  _et al._ (2002); Stroberg (2016) used
in the results presented in Gade  _et al._ (2008); Tostevin and Gade (2014,
2021) for 24Si, 12C and 40Si respectively. To build the continuum
wavefunctions, an optical potential is required. For consistency and to focus
on core destruction, for the evaluation of this potential, we have considered
the imaginary part of the global energy-dependent dispersive potential by
Morillon et al. Morillon and Romain (2007) for all considered nuclei. This
potential reproduces reasonably the reaction cross sections from the ENDF
database Brown  _et al._ (2018) between $p-^{11}$B and $n-^{11}$C for energies
$>20$ MeV. As is general for optical potentials, the computed reaction cross
section (related to the imaginary part of the potential) includes the
formation of compound nucleus, which may decay into the original valence-core
channel, not resulting in the destruction of the core. Therefore, for a proper
description of the destruction of the core, the potential must be modified to
eliminate this process from the reaction cross section. In order to evaluate
the importance of this “elastic-compound-nucleus” contribution, we have
performed compound-nucleus calculations to obtain the fraction of the cross
section that results in actual destruction of the core for the different
systems in a range of relevant energies. Then, for the different energies, we
have rescaled the reaction cross sections obtained with the Morillon potential
by this factor and modified the depth of the imaginary surface term of the
potential to reproduce this core-destruction cross section (when required, the
imaginary volume term was removed) (see supplementary material). Given the
significant dispersion in compound nucleus results Blank  _et al._ (2018), we
present the results using two widely-used compound-nucleus codes: PACE Tarasov
and Bazin (2008); Gavron (1980), which will be referred to as Model I, and
GEMINI Charity (2010); Mancusi  _et al._ (2010), which will be referred to as
Model II. The effects of the neglect of elastic compound nucleus are presented
in the Supplementary Material. We note that for the deeply bound nucleons many
open channels exist even at zero relative energy (as illustrated in Fig. 1) so
the elastic channel was not significantly populated in the compound-nucleus
calculations and no potential modification was required. The same occurred for
all nuclei at valence-core energies $>30$ MeV. The computed effective density
is presented as a function of $x$ in Fig. 3, where the left panel corresponds
to the valence neutron in 40Si in the $1f_{7/2}$ orbital (bound by 4.72 MeV)
and the right panel to the valence proton in the $1d_{5/2}$ orbital (bound by
23.1 MeV). To validate the density calculation, the red line corresponds to
calculations where $\psi^{(-)}(\vec{k},\vec{r})$ were taken as plane waves,
which should coincide with the density for the eikonal calculation
corresponding to the orange line. For both cases, the plane-wave and eikonal
calculations agree very well, except for small oscillations in the interior,
which can be related to the cutoff in $k$ and $l$. When comparing the plane-
wave calculation to that with core destruction (the blue line corresponds to
model I and the green line to model II), core destruction is shown to produce
a significant reduction in the density for both cases, particularly in the
interior. This reduction is larger for the more bound case (less abundant
species), as expected due to the abundance of open channels (see Fig. 1).

Figure 3: One-dimensional effective density for the valence neutron (left) and
proton (right) in the $1f_{7/2}$ and $1d_{5/2}$ orbitals, respectively, for
40Si. The red line corresponds to the effective density without core
destruction, and the orange one to the eikonal calculation. The blue and green
curves correspond to the calculation with core destruction, for model I and
model II, respectively. The potential required no modification for the valence
proton, so both curves coincide in the right panel (see text).

To evaluate the effect of this reduction on stripping cross sections, the
latter have been computed using the effective density from Eq. (Isospin
dependence in single-nucleon removal cross sections explained through valence-
core destruction effects). The values for $S^{0}_{CT}(b_{CT})$ and
$S^{0}_{VT}(b_{VT})$ have been taken from the original references. The top
panel of Fig. 4 shows ratios between the computed stripping cross sections and
those from the standard eikonal model Hansen and Tostevin (2003) as a function
of the difference between the separation energy of the removed species and its
isospin pair $\Delta S=S_{n(p)}-S_{p(n)}$ Gade  _et al._ (2008), with
$S_{n(p)}$ taken from nud . For all cases except 40Si$(-n)$, only one single-
particle configuration was dominant in the cross section. For 40Si$(-n)$, the
$1f_{7/2},2p_{3/2},1d_{5/2}$ and $2s_{1/2}$ configurations were considered and
weighted by their spectroscopic factors from the SDPF-U interaction Stroberg
(2016), which accounts for 95% of the cross section. Red squares correspond to
the effective density computed without core destruction, with a difference to
the standard calculation of at most $1.5\%$. Blue diamonds and green triangles
correspond to the calculations using models I and II, respectively. The
results show larger reduction for removal of the more deeply-bound nucleon (to
the right of the graph), $\sim 0.5$ and smaller reduction in the weakly-bound
case, $\sim 0.9$. The reduction in cross section is smaller than the one in
the norm of the density, as seen in Fig. 3, due to the peripherality of the
reaction, since in the nuclear surface the reduction due to core destruction
is smaller than in the interior.

Figure 4:  a) Ratio between computed stripping cross sections and standard
eikonal calculations Hansen and Tostevin (2003) as a function of the
difference in proton-neutron separation energy $\Delta S$. Calculations are
shown without the $N+C$ potential (red squares) and with core destruction
modelled with model I (blue diamonds) and with model II (green triangles). b)
Standard (red diamonds) and modified “quenching factors” $R_{s}$ as a function
of $\Delta S$, considering for the $N+C$ system model I (blue triangles) and
model II (green triangles). (See text).

The bottom panel of Fig. 4 shows the effect of this reduction on the
“quenching factors” $R_{s}$. Red diamonds correspond to the original values
from Tostevin and Gade (2021). Since we have only studied the effect of core
destruction in stripping, to compare to experimental data, which also include
diffractive scattering, we will assume the same reduction for diffractive
scattering. Since stripping is the main contributor to the cross section Gade
_et al._ (2008); Brown  _et al._ (2002); Stroberg (2016), we consider this
approximation to be sufficient for the purposes of this work. Therefore, the
values of $R_{s}$ with core destruction are computed through:

$R_{s}^{\mathrm{cdes}}={\sigma_{\mathrm{exp}}\over\sigma_{\mathrm{cdes}}}={\sigma_{\mathrm{exp}}\over\sigma_{\mathrm{eik}}}{\sigma_{\mathrm{eik}}\over\sigma_{\mathrm{cdes}}}\simeq{\sigma_{\mathrm{exp}}\over\sigma_{\mathrm{eik}}}{\sigma_{\mathrm{eik,str}}\over\sigma_{\mathrm{cdes,str}}}=\dfrac{R_{s}^{\mathrm{orig}}}{\dfrac{\sigma_{\mathrm{cdes,str}}}{\sigma_{\mathrm{eik,str}}}},$
(11)

(where $\sigma_{\mathrm{exp}},\sigma_{\mathrm{eik}},\sigma_{\mathrm{cdes}}$
are the experimental, standard eikonal and with-core-destruction knockout
cross sections and $\sigma_{\mathrm{eik,str}},\sigma_{\mathrm{cdes,str}}$ the
ones for stripping) and are presented in the bottom panel of Fig. 4 as blue
and green triangles, corresponding to calculations using models I and II,
respectively. These modified “quenching” factors present a significantly
smaller dependence on $\Delta S$, with a slope of $-0.004\mathrm{MeV}^{-1}$
for model I and $-0.005\mathrm{MeV}^{-1}$ for model II, which is less than
half the original value: $-0.013\mathrm{MeV}^{-1}$. Therefore, these results
indicate that a large part of the dependence of the “quenching factors” on
$\Delta S$ can be related to the destruction of the core through its
interaction with the removed particle, an effect that can be included in
standard eikonal calculations using the effective density from Eq. (Isospin
dependence in single-nucleon removal cross sections explained through valence-
core destruction effects). Including core destruction significantly reduces
the dependence on isospin asymmetry, making the trend for nucleon-knockout
reactions consistent with that in transfer and $(p,pN)$ reactions. It is
remarkable that the tendency with core destruction agrees quite well with the
reduction in spectroscopic factors found in coupled-cluster calculations for
oxygen isotopes Jensen  _et al._ (2011), which would suggest that the
remaining dependence on $\Delta S$ could be described by many-body
correlations. These results show that the low-energy interaction between
removed particle and core is fundamental to properly interpret the
measurements from nucleon-knockout experiments. Therefore, better information
on this interaction, obtained from theoretical calculations starting from
first principles or from measurement of nucleon-core reaction cross sections
(particularly for exotic species with larger $|\Delta S|$), is essential to
extract significant spectroscopic information from nucleon knockout
experiments.

Summary and outlook: In this work, we have investigated the effect of core
destruction due to its final-state interaction with the removed nucleon in
nucleon stripping reactions. The inclusion of this effect significantly
flattens the dependence of the “quenching factors” on isospin asymmetry,
making this dependence consistent with that found in transfer and $(p,pN)$
reactions. Therefore, core destruction appears as one of the key contributors
to answer the open question on this dependence. Experimental measurements that
detect the products of core destruction could be used as validation of these
results. A precedent exists for these measurements: in Sun  _et al._ (2016)
experimental results were compared to INCL calculations for nucleon removal
from 14O. As well, confirmation of these results would require more accurate
optical potentials between valence nucleon and core, which could be extracted
via ab-initio methods Idini  _et al._ (2019). Experimental measurements to
extract the core-nucleon reaction cross section would also be useful to reduce
the uncertainties in the potentials required for these calculations. Some
improvements in the formalism are also desirable, such as an extension to
diffractive scattering or a more sophisticated description of the reaction
going beyond the eikonal approximation, with proper energy and momentum
conservation, such as the Ichimura-Austern-Vincent (IAV) formalism Ichimura
_et al._ (1985); Lei and Moro (2015, 2015); Potel  _et al._ (2015); Carlson
_et al._ (2016), which could be extended to include valence-core destruction.
In addition, the inclusion of the real part of the valence-core interaction
(and its bound states), which has been neglected in this work, should be
considered. Further work on the latter points is currently underway.

###### Acknowledgements.

The authors thank A. Di Pietro for her help in the calculations with PACE4 and
GEMINI. M.G.-R. J.G.-C. and A.M.M. acknowledge financial support by
MCIN/AEI/10.13039/501100011033 under I+D+i project No. PID2020-114687GB-I00
and under grant IJC2020-043878-I (also funded by “European Union
NextGenerationEU/PRTR”), by the Consejería de Economía, Conocimiento, Empresas
y Universidad, Junta de Andalucía (Spain) and “ERDF-A Way of Making Europe”
under PAIDI 2020 project No. P20_01247, and by the European Social Fund and
Junta de Andalucía (PAIDI 2020) under grant number DOC-01006.

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