# Devouring The Centaurus A Satellites: Modeling Dwarf Galaxies with
Galacticus

Sachi Weerasooriya Department of Physics and Astronomy, Texas Christian
University, Fort Worth, TX 76109, USA  Mia Sauda Bovill Department of
Astronomy, University of Maryland, College Park, MD 20742  Department of
Physics and Astronomy, Texas Christian University, Fort Worth, TX 76109, USA
Matthew A. Taylor  University of Calgary, 2500 University Drive NW, Calgary
Alberta T2N 1N4, CANADA  Andrew J. Benson Carnegie Observatories, 813 Santa
Barbara Street, Pasadena, California, 91101 USA  Cameron Leahy  University of
Calgary, 2500 University Drive NW, Calgary Alberta T2N 1N4, CANADA

###### Abstract

For the first time, systematic studies of dwarf galaxies are being conducted
throughout the Local Volume, including Centaurus A (NGC 5128), which is the
nearest elliptical galaxy. Given Centaurus A’s mass (roughly ten times that of
the Milky Way), AGN activity, and recent major mergers, investigating these
dwarfs and their star formation physics is imperative. However, simulating the
faintest dwarfs in a massive galaxy like Centaurus A with sufficient
resolution in a hydrodynamic simulation is computationally expensive and
currently unfeasible. In this study, we seek to reproduce Centaurus A dwarfs
using the same star formation physics as the Milky Way. We employ the semi-
analytic model Galacticus to model dwarfs within a 600 kpc region. Utilizing
astrophysical prescriptions and parameters matching the Milky Way satellites,
we explore predictions for various properties and star formation histories
(SFHs) to investigate environmental effects. We also reproduce cumulative
luminosity and luminosity-metallicity relations consistent with observations
for the overall Centaurus A satellite population, while predicting half-light
radii, velocity dispersion, and SFHs for the dwarf galaxies in Centaurus A.
The agreement between our predicted SFHs for Centaurus A dwarfs and those of
the Milky Way implies the presence of universal processes governing star
formation in these galaxies. Overall, our findings shed light on the star
formation physics of dwarf galaxies in the Centaurus A system, revealing
insights into their properties and dependence on the host environment.

Dwarf galaxies (416) — Galaxy evolution (594) — Galaxy formation (595) —
Theoretical models (2107)

##  1 Introduction

Dwarf galaxies are the fundamental building blocks of larger structures. They
are the most abundant type of galaxies in the universe at all redshifts (e.g.,
Binggeli et al., 1988; Ferguson & Binggeli, 1994; Marzke & da Costa, 1997).
Sensitivity to their host environments makes them excellent objects to study
the effects of the surrounding environment and its role in dwarf galaxy
physics, as shallow gravitational potentials make them extremely sensitive to
environmental feedback (e.g., Dekel & Silk, 1986; Thoul & Weinberg, 1996;
Benson et al., 2002; Okamoto et al., 2010). The dwarf galaxies of the Local
Group are well studied both observationally (Albareti et al., 2017; Aguado et
al., 2019; Martin et al., 2016; Abbott et al., 2018; McConnachie, 2012; York
et al., 2012) and theoretically (Hopkins et al., 2018; Applebaum et al., 2021;
Pandya et al., 2020; Shipp et al., 2022; Weerasooriya et al., 2023).

While Milky Way dwarfs have been simulated hydrodynamically, simulations of
massive systems outside our Galactic neighborhood up to $z\,=\,0$ do not meet
the resolution required to resolve star formation physics of fainter dwarfs.
For example hydrodynamic simulations like TNG50 (Nelson et al., 2019) can only
resolve dwarf galaxies down to $M_{*}=10^{8}\,\mathrm{M}_{\odot}$. Studies of
dwarfs in the Local Group alone cannot give a clear picture of the role
environment plays in the physics of their evolution. With the upcoming
unprecedented amount of observations through soon to be available from
facilities like the Nancy Grace Roman telescope and Rubin Observatory, it is
essential that we explore dwarfs beyond the Local Group with theoretical
models. One method of investigation is through the luminosity- metallicity
relation, which captures the stellar feedback physics of dwarf galaxies.
Studies show that the mass–metallicity relationships of both Milky Way and M31
follow similar trends (Kirby et al., 2020). Thus, it is imperative that we
look beyond the Local Group (LG) to understand whether or not properties in
dwarf galaxies are independent of the environment.

Observational and theoretical studies have focused on dwarf galaxies that are
hosted by Milky Way and Andromeda-like environments or cluster scale
environments such as Virgo $\sim 10^{15}\,\mathrm{M}_{\odot}$ and Fornax $\sim
10^{14}\,\mathrm{M}_{\odot}$ (McConnachie, 2012; Richardson et al., 2011;
Ferrarese et al., 2012; Eigenthaler et al., 2018). Centaurus A provides the
most accessible opportunity to study dwarf galaxies within a higher mass host
and in an environment which sits between group and cluster scales. But, few
have explored dwarfs of Centaurus A (Müller et al., 2015, 2017, 2019). Thus,
the question of how dwarf galaxy physics are affected by different mass hosts
needs further exploration in these intermediate mass host environments.
Centaurus A is the closest, easily observable giant elliptical galaxy located
3.8 Mpc from the Milky Way (Harris et al., 2010) with poorly constrained mass
of $4.7\times 10^{12}\,\mathrm{M}_{\odot}$ -$1.8\times
10^{13}\,\mathrm{M}_{\odot}$ (Pearson et al., 2022; van den Bergh, 2000).

In recent years the halo of Centaurus A has been targeted by several surveys
including the Survey of Centaurus A’s Baryonic Structures (SCABS; Taylor et
al., 2016, 2017, 2018) and the Panoramic Imaging Survey of Cen & Sculptor
(PISCeS; Sand et al., 2014; Crnojević et al., 2014; Crnojevic et al., 2016).
These are the first systematic surveys of the dwarf satellites of Centaurus A.
In addition, studies by Müller et al. (2015, 2017, 2019, 2021, 2022) have also
covered both M83 and Centaurus A with DECam. As a result of these surveys the
number of known or suspected dwarf galaxies has almost doubled over the past
5-10 years.

The mass of Centaurus A is currently not well constrained. Several studies
have predicted the virial mass of Centaurus A using a variety of methods. For
example Woodley et al. (2007) has estimated the mass of Centaurus A (pressure
supported and rotational supported mass) using a globular cluster population
within $50$ kpc ($1.3\times 10^{12}\,\mathrm{M}_{\odot}$). van den Bergh
(2000) has calculated the mass of Centaurus A using the virial theorem
($1.4\times 10^{13}\,\mathrm{M}_{\odot}$) and projected mass method out to
$640$ kpc ($1.8\times 10^{13}\,\mathrm{M}_{\odot}$). Müller et al. (2022)
estimates a dynamical mass of $1.2\times 10^{13}\,\mathrm{M}_{\odot}$ within
$800$ kpc. Pearson et al. (2022) has constrained the lower limit ($4.7\times
10^{12}\,\mathrm{M}_{\odot}$) on the mass of Centaurus A using stellar stream
models. The upper range of virial masses measured for Centaurus A is $\sim
10^{13}\mathrm{M}_{\odot}$ (van den Bergh, 2000; Peng et al., 2004; Woodley et
al., 2007; Łokas, 2008; Harris et al., 2015), which falls between the masses
of Milky Way and large clusters such as Virgo and Fornax.

To date, few theoretical studies of the Centaurus A system have been carried
out. Bovill et al. (2016) used a high-resolution N-body simulation that did
not include baryons to study the Centaurus A globular clusters, and Müller et
al. (2019) compares the luminosity function within 200 kpc to Centaurus A
analogs in the TNG100 of Illustris-TNG simulations. TNG100 has a resolution of
$m_{DM}=7.5\times 10^{6}\,\mathrm{M}_{\odot}$ and
$m_{\textrm{baryon}}=1.4\times 10^{6}\,\mathrm{M}_{\odot}$ resolving galaxies
with $M_{*}>10^{8}\,\mathrm{M}_{\odot}$ (Pillepich et al., 2018). However,
this mass is only softly below an SMC-like halo mass and is inadequate to
resolve fainter dwarfs. Therefore, we need to explore more computationally
efficient techniques, such as semi-analytic models (SAMs) that provide an
efficient method to explore the star formation physics of dwarf galaxies.

In this work, we run the SAM Galacticus (Benson, 2012) on merger trees
generated with Extended Press-Schechter (EPS) and from $N$-body simulations of
Centaurus A analogs. We use the same star formation physics that reproduces
the properties of observed dwarfs of the Milky Way down to ultra-faint dwarfs
(Weerasooriya et al., 2023). The goal of this work is to test whether the same
astrophysical prescriptions and parameters of the Milky Way can reproduce the
observed cumulative luminosity function and the luminosity-metallicity
relation of the Centaurus A satellites, and if it can, make predictions for
their properties and SFHs, in an effort to investigate the effects of host
environment on these dwarfs. In Section 2, we describe the details of the
simulation and modeling. In Section 3, we describe the sample of observational
data taken from the literature. Next, in Section 4, we compare our models to
the observed properties of the known Centaurus A dwarfs before exploring their
potential star formation histories. Lastly, we present our discussions and
conclusions in Section 6.

##  2 Simulation

We use a high-resolution cosmological N-body simulation of an isolated
Centaurus A halo from Bovill et al. (2016). This simulation is run from
$z\,=\,150$ to $z\,=\,0$ with WMAP9 cosmology ($\sigma_{8}\sim
0.821,\,H_{0}\sim\,70.0\,km\,s^{-1}\,\mathrm{Mpc}^{-1},\,\Omega_{b}\sim
0.0463,\,\Omega_{\Lambda}\sim 0.721$). Initial conditions were generated with
MUSIC (Hahn & Abel, 2011) and the simulation run with Gadget 2 (Springel,
2005) and analyzed with the AMIGA (Knollmann & Knebe, 2009) and
CONSISTENT_TREES (Behroozi et al., 2013b). The Centaurus A analog is selected
to be a $\sim~{}10^{13}~{}M_{\odot}$ halo with no halos $M\geq
10^{12}\;\mathrm{M}_{\odot}$ within $3\;\mathrm{Mpc}\;h^{-1}$ at $z\,=\,0$.

###  2.1 Semi-Analytic Model (SAM)

Despite the influx of observational data, high-resolution hydrodynamic
simulations of Centaurus A analogs have not yet been run to $z\,=\,0$. They
are on the edge of current computational capabilities. Note that the studies
such as van den Bergh (2000); Peng et al. (2004); Woodley et al. (2007); Łokas
(2008); Harris et al. (2015); Pearson et al. (2022) use a variety of methods
to determine the mass of Centaurus A, while in this work we defined the virial
mass of a halo in terms of the spherical overdensity given by Bryan & Norman
(1998). Thus they are not necessarily consistent with the methods used in our
models. Due to significant uncertainties in these mass estimates of Centaurus
A, we investigate a wide range of possible halo masses within virial radius
for Centaurus A using 30 EPS trees with different merger histories.

We model the baryonic properties of the Centaurus A system using the Semi-
Analytic Model (SAM) Galacticus (Benson, 2012). We apply the same
astrophysical prescriptions and parameters that reproduced the Milky Way
satellites in Weerasooriya et al. (2023). These include a reionization
redshift of 9, filtering velocity (the scale below which halos are unable to
accrete gas efficiently from the IGM after reionization) of 25
$\mathrm{\,km\,s^{-1}}$, cooling velocity (the scale below which cooling
becomes inefficient in the circumgalactic medium) of 19
$\mathrm{\,km\,s^{-1}}$. Please refer to Weerasooriya et al. (2023) for
further details. Of these parameters, quenching from ram pressure, and tidal
stripping primarily determine how dwarf satellites are affected by a host
environment. For example, ram pressure stripping can strip gas out of galaxies
resulting in a shortage of gas supply and eventually quench star formation.
However, we would like to remind the reader that the effect of ram pressure
stripping efficiency was negligible for the Milky Way satellites (Weerasooriya
et al., 2023). Regardless, we implement a ram pressure with the same high
efficiency as in Weerasooriya et al. (2023).

We apply the astrophysical prescriptions and parameters which reproduce the
star formation physics and star formation histories of the Milky Way
satellites (Weerasooriya et al., 2023) to the N-body merger trees and Extended
Press Schechter (EPS) merger trees of the Centaurus A analog. We make this
assumption because there are very few studies exploring the star formation
histories of Centaurus A dwarfs (Côté et al., 2009; Crnojević et al., 2011).
The models run with merger trees from the N-body simulation probe only one
possible mass of Centaurus A. Therefore, we run several EPS trees with
different mass Centaurus A analogs spanning the full range of possible
Centaurus A halo masses. While the EPS trees allow us to efficiently probe a
range of $M_{vir}$ for the Centaurus A halo, they do not provide positional
information on the satellites. As such, EPS trees do not allow us to look at
the dependence of satellite populations on their distance from the host.

##  3 Observational Sample

Observations of the Centaurus A dwarfs are inhomogeneous, therefore in this
section we describe the sample of Centaurus A dwarfs taken from the literature
that we will compare to our models. We use observational data for Centaurus A
satellites from a variety of sources including Crnojević et al. (2010, 2014);
Crnojevic et al. (2016); Crnojević et al. (2019); Karachentsev et al. (2013);
Müller et al. (2015, 2017, 2019); Taylor et al. (2016, 2018). In Table 1, we
present a compilation of observations from a variety of sources used in this
study along with their references. In addition to the dwarfs in the above
sample, we also include 38 new dwarf galaxy candidates from Taylor et al.
(2023, in prep.) in our overall analysis, but do not list their properties in
Table 1. Table 1 only includes galaxies with distance or velocity measurements
that verify them as members of Centaurus A with distances $\leq
5.8\,\mathrm{Mpc}$ (Table 1).

The current sample of observations of Centaurus A is incomplete due to three
major reasons: 1.) Lack of systematic sky coverage that leads to spatial
incompleteness. For example the PISCeS survey is spatially incomplete, and
biased toward coverage of the Northeastern region of Centaurus A’s halo. These
non-uniformities in design or analysis could make completeness calculations
more complex. 2.) The detectability limits of different surveys, e.g. SCABS is
limited to $M_{V}<-7.2$ within $150$ kpc covering an area of 50 square degrees
and surface brightness limit of $27.8\,\mathrm{mag\,arcsec}^{-2}$ in the g
band (Leahy et al., private communication), while PISCeS can detect dwarfs
down to $M_{V}<-8$ within $150$ kpc and surface brightness limit of
$26.5\,mag\,arcsec^{-2}$ in the g band (Crnojević et al., 2014, 2019). 3.) The
lack of distance measurements in the outskirts also hinders the determination
of membership (Müller et al., 2017; Taylor et al., 2018). Distances are
essential to determine the membership of satellites, their shape, and their
brightness. Thus higher uncertainties in distances can cause these data to
have higher errors in their sizes and magnitude measurements. Note that while
the virial radius of Centaurus A is $\sim 409$ kpc, most of these satellites
are beyond that limit, yet within Centaurus A’s ‘splashback radius’ of
$1.1\,\mathrm{Mpc}$. The splashback radius is a physically motivated halo
boundary that eliminates spurious evolution of radius and mass caused by
standard definitions of virial radius (Diemer & Kravtsov, 2014).

Figure 1: The number of galaxies per apparent magnitude in the $g_{DES}$
filter. The sea green curve shows the luminosity function of our modeled
dwarfs within 150 kpc, the olive green curve shows the number of satellite
galaxies expected to be observed based on completeness tests run by Leahy et
al. (private communication) for SCABS, and the purple curve shows the observed
luminosity function.

The observational properties of Centaurus A satellites are still largely
unknown. Thus our knowledge of these galaxies and their properties is nowhere
near as complete as our knowledge of the Milky Way system. Quantitative
estimates of the incompleteness of Centaurus A dwarfs have not been made by
previous studies, and such exploration is beyond the scope of our current
work. A preliminary quantitative exploration of the completeness limits in
SCABS Centaurus A satellites is underway by Leahy et al. (private
communication). Their exploration of completeness using 5000 Monte Carlo dwarf
galaxy realizations reveals completeness of $96\%$ for dwarf galaxies of
$\lesssim 18\,\mathrm{mag}$ in the g-band. They report $50\%$ completeness at
g-band magnitudes of 20.01 and $50\%$ completeness at surface brightness of
$\sim 27.8\,\mathrm{mag\,arcsec}^{-2}$. In Figure 1, we compare our models
within 150 kpc to the total number of galaxies expected in the region based on
these preliminary artificial galaxy experiments. We note here that given the
unconfirmed natures of the Taylor et al. 2023 dwarf candidates, this estimate
should be considered as an upper limit with results interpreted in that
context. Based on the comparison, our galaxy models agree with the observed
luminosity function. However note that the observed luminosity function is
slightly steeper.

Figure 2: Luminosity functions for the Centaurus A satellites within $r_{vir}=600$ kpc for models and observations. Purple line shows the observational sample within 600 kpc projected radius. The predicted luminosity functions for Centaurus A satellites within 600 kpc and $M_{V}\leq-4$ in the y-z plane are shown in green (N-body). Lines shown in shades of blue represent EPS trees with different Centaurus A masses. Each blue line shows the median per $M_{V}$ bin for 30 different EPS trees, and a shaded region of minimum and maximum. Name | $RA_{h}$ | $RA_{m}$ | $RA_{s}$ | $Dec_{deg}$ | $Dec_{arcmin}$ | $Dec_{s}$ | $M_{B}$ | $M_{V}$ | Distance | References
---|---|---|---|---|---|---|---|---|---|---
| (h) | (min) | s | (deg) | (arcmin) | (arcsec) | (mag) | (mag) | (Mpc) |
NGC4945 | 13 | 05 | 26.1 | -49 | 28 | 16.0 | -20.34 | -20.6 | 3.47 | 1,7
NGC5102 | 13 | 21 | 57.8 | -36 | 37 | 47.0 | -18.24 | -20.37 | 3.66 | 1,7
E274-01 | 15 | 14 | 13.5 | -46 | 48 | 45.0 | -17.35 | -19.2 | 3.09 | 2,7
NGC5253 | 13 | 37 | 5.0 | -31 | 23 | 30.0 | -17.33 | -17.7 | 3.9 | 1,7
E383-087 | 13 | 49 | 17.5 | -36 | 03 | 48.4 | -16.83 | -17.17 | 3.19 | 1,7
NGC5206 | 13 | 30 | 41.0 | -47 | 53 | 42.0 | -16.43 | -16.97 | 3.6 | 1,7
NGC5408 | 14 | 00 | 18.0 | -41 | 08 | 11.0 | -15.91 | -17.3 | 4.81 | 2,7
E324-24 | 13 | 27 | 37.4 | -41 | 28 | 50.0 | -15.49 | -15.6 | 3.78 | 2,7
E26958 | 13 | 07 | 38.0 | -46 | 43 | 30.0 | -14.99 | -16.8 | 4.78 | 6,7
NGC5237 | 13 | 37 | 38.9 | -42 | 50 | 51.0 | -14.82 | -15.08 | 3.33 | 3,5,7
Table 1: References: (1) Lauberts & Valentijn (1989), (2) de Vaucouleurs et
al. (1991),(3) Karachentsev et al. (2003),(4) James et al. (2004), (5) Doyle
et al. (2005), (6) Sharina et al. (2008), (7) Karachentsev et al. (2013), (8)
Müller et al. (2015), (9) Müller et al. (2017), (10) Crnojević et al. (2014);
Crnojevic et al. (2016); Crnojević et al. (2019), (11) Taylor et al. (2018).
Table 1 is published in Machine-readable format. Only a portion of this table
is shown here to demonstrate its form and content.

##  4 Properties of Centaurus A dwarfs

In this section, we compare different properties of the modeled Centaurus A
dwarfs with observational data described in Section 3. We start by exploring
the properties of the Centaurus A dwarfs including luminosities, half-light
radii, and velocity dispersions.

###  4.1 Luminosity Function

In Figure 2, we plot the cumulative luminosity function of Centaurus A
satellites for observations within $r_{vir}=600$ kpc of the projected radius
and for dwarfs modeled with Galacticus out to 600 kpc in the y-z plane. For
the purpose of comparison of Centaurus A dwarfs, we analyze satellites viewed
in projection, as they would be observed. We remind the reader that this
selection can be made only when using N-body trees since positional
information is available. Since we do not know the viewing angle of our
modeled Centaurus A, we rotate the line of sight relative to the $\hat{x}$
direction of the simulation in 5-degree steps (note that the step size used
here is arbitrary). This accounts for uncertainty in the luminosity functions.
The median luminosity function over all potential viewing angles is shown in
green, while the maximum and minimum is shown by the shaded envelope.

Figure 3: Left panel: Number of luminous satellites around Centaurus A
brighter than $M_{V}\leq-6$ predicted within 150 kpc for different lines of
sight as the model is rotated in your y-z plane (shown in green). The purple
line shows the total number of satellites in the observed sample. Right panel:
luminosity functions for the Centaurus A satellites within 150 kpc in the y-z
plane are shown in green. We plot the median of the cumulative luminosity
function over all potential viewing angles in green and shade within the
minimum and maximum luminosity functions. Figure 4: We show the modeled half
light radii vs. the absolute $g_{SDSS}$ band magnitude of the Centaurus A
satellites in blue, observed half-light radii of Centaurus A dwarfs from
Taylor et al. in prep (g’ band), and Crnojevic et al. (2016); Crnojević et al.
(2019) (V band). Note that not all values here are in V band. Peak wavelengths
in $g_{SDSS}$ and V bands are $\sim 555\,\mathrm{nm}$ and $\sim
551\,\mathrm{nm}$ respectively. Given the small differences in peak
wavelengths in g’ and V bands, we plot the data available in respective bands.
The observations of McConnachie (2012) for the Milky Way (V band), and M31 (V
band) are also shown in blue stars and grey circles. Iso-surface brightness
lines are shown in grey dashed lines. These results are consistent with models
we predicted for the Milky Way satellites (Weerasooriya et al., 2023).

We also run models using merger trees generated by Extended Press Schechter
(EPS) to explain the mass range of Centaurus A’s halo. We build the EPS merger
tree using the method of Cole et al. (2000) with an accretion limit of 0.1,
merge probability of 0.1, and mass resolution $1.41\times
10^{7}\,\mathrm{M}_{\odot}$ and recalibrated merger rates from Parkinson et
al. (2008). Then we generate the EPS merger tree halo mass functions by
implementing the method by Tinker et al. (2008) available in Galacticus. These
halo mass distributions are then used to simulate galaxies. EPS models for
different mass merger trees are shown in shades of blue. These models
inherently do not have positional information. We run EPS models for
$5-9\times 10^{12}\,\mathrm{M}_{\odot}$ and $1\times
10^{13}\,\mathrm{M}_{\odot}$ halos with 30 merger trees for each halo mass.
The solid curves show the median per $M_{V}$ bin with shaded areas indicating
the minimum and maximum over 30 EPS trees. All of these models follow the
general shape of the luminosity function for observations within 600 kpc.
However, there is some variation in the shape of the luminosity functions
between N-body and EPS merger trees.

While we cannot constrain the mass of Centaurus A from the models due to
poorly understood completeness in the observations for fainter dwarfs,
comparison with models at the bright end of the luminosity function suggests
that Centaurus A is likely to have a higher mass
($10^{13}\,\mathrm{M}_{\odot}$). We find our models slightly overproduce the
number of dwarfs at all luminosities, but the observed luminosity function
within 600 kpc is still likely to be incomplete. However, our modeled dwarfs
reproduce the overall shape of the observed cumulative luminosity function for
N-body merger trees and EPS trees traced at all modeled host masses. Note that
any such inference on the mass of Centaurus A halo from this approach is
subject to the caveat that Galacticus predictions could be inaccurate.

Poorly understood completeness limits of the observations within 600 kpc of
Centaurus A make exploration of these satellites and their properties limited.
However, the observational sample is relatively complete within 200 kpc
(Müller et al., 2019) down to $M_{V}\,=\,-10$. SCABS and PISCeS surveys cover
a spatial region within a projected radius of 150 kpc down to dwarfs as faint
as $M_{V}<-7.2$ (Crnojević et al., 2014; Crnojevic et al., 2016; Taylor et
al., 2016) and we can assume completeness within 200 kpc down to
$M_{V}\sim-10.0$ (see section 3 for a detailed completeness discussion). We
plot the luminosity function within 150 kpc and match the number of luminous
satellites within 150 kpc in Figure 3 . The left panel shows the number of
luminous satellites in our model in green and observed number of satellites in
purple, and the right panel shows the luminosity function within 150 kpc for
both our $N$-body model (green) and observations (purple). Our current model
predicts a median that is lower than the observed number of dwarf galaxies
located within 150 kpc of Centaurus A, but the observed number is well within
the distribution predicted by our model (refer to the left panel in Figure 3).
The observed luminosity function demonstrates a slightly steeper slope within
the 150 kpc radius. Nevertheless, our dwarf galaxy models are consistent with
the lower boundary of the luminosity function envelope (see Figure 3). This is
consistent with other studies of the inner 200 kpc of the Centaurus A halo
(Müller et al., 2019).

###  4.2 Half Light Radii

Figure 5: Luminosity–metallicity relation for the Centaurus A satellites. The
observed values are shown by squares (Crnojević et al., 2010, 2019; Müller et
al., 2019, 2021). Metallicities of the Milky Way satellites are shown as grey
triangles (McConnachie, 2012). Predicted abundances for satellites for 30 EPS
merger trees of $10^{13}\,\mathrm{M}_{\odot}$ are shown in blue. Each
hexagonal bin may contain multiple satellites.

In Figure 4, we show the luminosity and half-light radii for the observed
Centaurus A dwarfs and dwarfs modeled with 30 EPS merger trees with $\sim
1\times 10^{13}\,\mathrm{M}_{\odot}$. We compare our results with available
Centaurus A dwarfs (Crnojevic et al., 2016; Crnojević et al., 2019), Milky Way
dwarfs, and M31 dwarfs (McConnachie, 2012). Note, the majority of observations
beyond the Local Group do not go below $0.01\;L_{\odot}\,pc^{-2}$. Galaxy
sizes are computed by finding the radius at which the galaxy is rotationally
supported against the combined gravitational potential of itself and the dark
matter halo, given the computed angular momentum content of the galaxy. The
half-light radii of the modeled Centaurus A satellite galaxies are computed in
the $g_{SDSS}$ band. Galacticus determines the half-light radii using the dark
matter profile of halos. Dark matter profile is determined from a NFW profile.
The scale radii for trees based on simulations are set from the N-body
simulations while for trees based on EPS trees, the concentrations are
calculated using the model by (Gao et al., 2008). We use SDSS filters since
SCABS data are calibrated to the SDSS photometric system (Taylor et al.,
2016).

Figure 6: Predicted $\sigma_{sat}$ of Centaurus A satellites (blue hexagons).
These $\sigma_{sat}$ are calculated at radii enclosing half of the stellar
mass of each satellite galaxy. Grey triangles show the observed velocity
dispersion of the Milky Way satellites (McConnachie, 2012). Note that the
predicted values of Centaurus A satellites are consistent with the velocity
dispersion of the observed Milky Way satellites.

At high luminosities, the modeled sizes of the dwarfs agree with observations
of the Centaurus A dwarfs. However, modeled sizes for the fainter dwarfs are
larger than the observed values. This is consistent with the systematically
larger sizes of the fainter modeled dwarfs compared to observations of the
Milky Way satellites (Weerasooriya et al., 2023). Galacticus tends to
overpredict the half-light radii with merger trees generated from EPS and
N-body simulations. EPS trees are unreliable for halos with
$~{}<~{}10^{10}M_{\odot}$ due to dynamic range limitations of EPS (Somerville
& Kolatt, 1999; Zhang et al., 2008). This may lead to inaccuracies in halo
masses and/or formation time, which may subsequently affect the sizes of the
galaxies which they contain. N-body trees over estimate half-light radii if a
halo has $N<1000$ particles (Weerasooriya et al., 2023). A study of how the
resolution affects the size of the modeled dwarfs will be a subject of future
study.

###  4.3 Metallicity

The currently available metallicities of the Centaurus A dwarfs are limited.
In Figure 5 we have shown the observations of [Fe/H] for Centaurus A
satellites from Crnojević et al. (2010, 2014, 2019) and Müller et al. (2019,
2021). Notice that Müller et al. (2019) finds a [Fe/H] - 2.25 dex metallicity
floor between $\mathrm{M}_{V}\sim$ -8 to -10. However, their measurement
errors are $\sim 0.5\,\mathrm{dex}$, consistent with our modeled values. While
spectroscopic measurements of metallicities by Müller et al. (2021) agree well
with Milky Way dwarfs, their photometric measurements are reported to have a
larger scatter. The authors state that the scatter might be due to the age-
metallicity degeneracy and incorrect assumptions of uniformly old ($\sim 10$
Gyr) stellar populations. Using the same astrophysical prescriptions and
parameters as those that reproduced the luminosity-metallicity relations of
the Milky Way dwarfs, we present the modeled metallicities of the Centaurus A
dwarfs in 30 $1\times 10^{13}\,\mathrm{M}_{\odot}$ EPS trees. The
metallicities of the modeled dwarfs in this work agree well with currently
available observations of Centaurus A satellites (Crnojević et al., 2010,
2019; Müller et al., 2019). This could potentially mean that the Centaurus A
satellites have a similar enrichment history to that of the Milky Way’s
satellites and/or that dwarf metallicities are independent of their local
environment. In Weerasooriya et al. (2023), we show that ram pressure does not
significantly affect the luminosity-metallicity relation of the Milky Way
satellites. Therefore, we do not expect a significant change in luminosity
metallicity relation as a function of halo mass.

###  4.4 Velocity Dispersion

The velocity dispersion of satellite galaxies probes the dark matter mass of
the host (Wake et al., 2012; Bogdán & Goulding, 2015; Schechter, 2015), and
consequently its gravitational potential. However, velocity dispersions of
Centaurus A satellites are unknown with the exception of KK197. As such,
additional data is required to comment on the dwarfs population as a whole. In
Figure 6, we show the velocity dispersion of KK197 (Müller et al., 2021), and
those of the Milky Way satellites (McConnachie, 2012). We calculate the
velocity dispersion of the modeled dwarfs ($\sigma_{sat}$) at stellar half-
mass radii for the modeled dwarfs using the N-body tree. While no theoretical
studies have looked at the velocity dispersions of the Centaurus A dwarfs,
most of our modeled sample falls within the observed velocity dispersions of
the Milky Way satellites (McConnachie, 2012). Given that velocity dispersion
probes dark matter halo mass and the evolution of these galaxies, this
suggests that Centaurus A dwarfs occupy similar halos, at a given stellar
mass, as their Milky Way counterparts.

##  5 Star Formation Histories

Next we explore the modeled star formation histories (SFHs) of dwarfs. We
remind the reader that due to the dearth of SFHs for the known Centaurus A
satellites, we have used the star formation physics from Weerasooriya et al.
(2023). Given the lack of observations of $z=0$ properties, SFHs of the
Centaurus A dwarfs are even harder to come by. Thus, we make the assumption
that astrophysical prescriptions and parameters of the Milky Way satellites
could be applied to Centaurus A as well. Given our success reproducing the
star formation history of the Milky Way satellites in Weerasooriya et al.
(2023), overall shape of the luminosity function, and properties at $z=0$ of
Centaurus A satellites, we consider it worthwhile to examine predicted SFHs
for the Centaurus A dwarfs.

Figure 7: Cumulative star formation histories of the Centaurus A satellites
(within 600 kpc) as a function of look back time colored by their luminosity
in the DES $g$-band. The star formation histories are divided into four panels
$-8<M_{V}\leq-4$, $-10<M_{V}\leq-8$, $-14<M_{V}\leq-10$, and
$-18<M_{V}\leq-14$. Each line is colored by the galaxy’s luminosity in DES g
band magnitude at $z\,=\,0$ (faintest in blue and brightest in yellow). The
majority of the faintest dwarfs quench very early on. Most luminous galaxies
are quenched later (6 Gyrs ago or later). Note that the galaxies on the upper
left panel have not yet been observed. However, we predict their SFHs, given
observations will reach those magnitudes in the era of the Roman. Figure 8:
The distributions of $\tau_{90}$ (time taken for a particular galaxy to reach
90% of its stellar mass at $z=0$) for the panels shown in Figure 7. The
faintest satellites quenched 12 Gyrs ago, while satellites between $-14\leq
M_{V}\leq-8$ quenched after 11.2 Gyrs ago. The most luminous galaxies quench
much later $\sim 8.8$ Gyrs ago.

Unlike the Milky Way satellites, extensive observational studies on star
formation histories of Centaurus A satellites are not available, with the
exception of KK197, ESO-269066, ESO-381018 (Makarova et al., 2007) and five
dwarf irregulars KK182 (Cen6), ESO269-58, KK196 (AM1318-444), HIPASS J1348-37,
ESO384-16 (Crnojević et al., 2012). Both KK197 and ESO-269066 are dwarf
spheroidals, while ESO-269066 is dwarf irregular. Dwarf spheroidals typically
have old stellar populations whose light is dominated by their red giant
branches, while dwarf irregulars are metal-poor and have varying levels of
current star formation Crnojević et al. (2012). states KK197 and ESO-269066
have unusual RGB color scatter, which shows active star formation with high
metallicity, while ESO-381018 is a typical dwarf irregular. Two of the dwarf
irregulars (KK196 and ESO269-58) studied are within $600$ kpc (see Figure 6 of
Crnojević et al. (2012)). Positioned in the middle of Centaurus A’s southern
radio lobe, KK196 has a star formation rate of $0.0046\pm
0.0004\,\mathrm{M}_{\odot}\,yr^{-1}$ and has formed more than
$60\%_{-30\%}^{+20\%}$ of stars more than 5 Gyrs ago (Crnojević et al., 2012).
Meanwhile, ESO269-58, located $300\pm 50$ kpc from Centaurus A, has few blue
loop, red supergiants and a very broad red giant branch stars, and dense
asymptotic giant branch zone. This dwarf has a higher star formation rate
compared to KK196 with $0.07\pm 0.04\,\mathrm{M}_{\odot}\,yr^{-1}$, and has
formed $50\%_{-15\%}^{+15\%}$ of stars more than 5 Gyrs ago. While its star
formation activity has been enhanced between 3-5 Gyrs ago, Crnojević et al.
(2012) also find that Centaurus A has lowered its star formation rate in the
last 1 Gyr.

In Figure 7, we show the modeled cumulative star formation histories of the
Centaurus A satellites as a function of look-back time colored by their
absolute V band magnitude at $z\,=\,0$ for our N-body model. As expected, our
Centaurus A modeled SFHs are similar to that of the Milky Way satellites in
Weerasooriya et al. (2023). Their corresponding distributions for time taken
to gain 90% of the stellar mass observed at z=0 ($\tau_{90}$) are given in
Figure 8. In the upper left panel, we show dwarfs with $-8<M_{V}\leq-4$. These
are the faintest galaxies. Most ultra-faint dwarfs quenched 8–12 Gyrs ago as
expected since the faintest observed Milky Way satellites are the fossils of
the first galaxies (Bovill & Ricotti, 2011; Brown et al., 2012). The upper
right panel shows the satellites in the range $-10<M_{V}\leq-8$; the majority
of these satellites reached 90% of their present stellar mass 12 Gyrs ago.
Most dwarfs in the upper right ($-10<M_{V}\leq-8$) and lower left panel
($-14<M_{V}\leq-10$) reached 90% of their present-day stellar mass $\sim 11.2$
Gyrs ago. Known SFHs from observations of Centaurus A dwarfs discussed earlier
would fall in the lower left panel of Figure 7. However, our model
overpredicts the quenching times in comparison to the two known quenching
times. Most of the brightest galaxies in the lower right panel
($-18<M_{V}\leq-14$) acquired 90% of their stellar mass 8.8 Gyrs ago.

##  6 Summary & Conclusions

We have presented models of dwarf galaxies around Centaurus A analogs using
the SAM Galacticus with astrophysical prescriptions and parameters calibrated
to match the observed properties of Milky Way dwarf galaxies. In this study,
we apply the best-fit parameter space chosen to reproduce the properties of
the Milky Way satellites (Weerasooriya et al., 2023) to several N-body and EPS
realizations of Centaurus A analogs in order to study the predicted satellite
population. Our models are able to reproduce the overall properties of the
dwarf population reasonably well.

We have explored properties such as the cumulative luminosity function,
luminosity-metallicity, and half-light radii with respect to observations of
the Centaurus A system. We also predict their velocity dispersions, star
formation histories, etc. We summarize our results as follows,

  * •

The same astrophysical prescriptions and parameters that allow Galacticus to
reproduce the Milky Way are used for the global properties of the Centaurus A
satellites. While being cognizant of the lack of observations in the Centaurus
A system, we conclude that the overall properties and trends of the Centaurus
A satellite population is similar to the dwarf galaxies around the Milky Way.

  * •

Our N-body models within 600 kpc and EPS trees of Centaurus A reproduce
luminosity functions that follow the overall shape of observations.

  * •

Centaurus A satellites follow a similar metal enrichment/stellar feedback to
the Milky Way satellites. That is we obtain similar luminosity-metallicity
relations for both the Milky Way and Centaurus A as expected. This suggests
that the physics that governs the metal enrichment of dwarf galaxies is
largely independent of the environment.

  * •

Assuming identical star formation physics of the Milky Way satellites for the
Centaurus A satellites, SFHs and quenching times of the Centaurus A satellites
also follow a similar trend to the Milky Way satellites.

Software: Galacticus (Benson, 2011), ROCKSTAR (Behroozi et al., 2013a), AMIGA
(Knollmann & Knebe, 2009), CONSISTENT_TREES (Behroozi et al., 2013b), JUPYTER
(Kluyver et al., 2016), NUMPY (Harris et al., 2020), SCIPY (Virtanen et al.,
2020), and MATPLOTLIB (Hunter, 2007).

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# The 3d and 5d electronic structures and orbital hybridization in Ba- and Ca-
doped La2CoIrO6 double perovskite

J. R. L. Mardegan  Deutsches Elektronen-Synchrotron DESY, Notkestra$\beta$e
85, 22607, Hamburg, Germany  L. S. I. Veiga  Diamond Light Source, Chilton,
Didcot, Oxfordshire, OX11 0DE, United Kingdom  T. Pohlmann  Deutsches
Elektronen-Synchrotron DESY, Notkestra$\beta$e 85, 22607, Hamburg, Germany  S.
S. Dhesi  Diamond Light Source, Chilton, Didcot, Oxfordshire, OX11 0DE, United
Kingdom  S. Francoual  Deutsches Elektronen-Synchrotron DESY,
Notkestra$\beta$e 85, 22607, Hamburg, Germany  J. R. Jesus  Centro Brasileiro
de Pesquisas Físicas, 22290-180, Rio de Janeiro, RJ, Brazil  C. Macchiutti
Centro Brasileiro de Pesquisas Físicas, 22290-180, Rio de Janeiro, RJ, Brazil
E. M. Bittar  Centro Brasileiro de Pesquisas Físicas, 22290-180, Rio de
Janeiro, RJ, Brazil  L. Bufaiçal<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil

###### Abstract

Here we present a detailed investigation of the Co and Ir local electronic
structures in La${}_{1.5}A_{0.5}$CoIrO6 ($A$ = Ba, Ca) compounds in order to
unravel the orbital hybridization mechanism in these CoIr-based double
perovskites. Our results of x-ray powder diffraction, ac and dc magnetization,
Co and Ir $L_{2,3}$-edges and Co $K$-edge x-ray absorption spectroscopy and
x-ray magnetic circular dichroism suggest a competition between magnetic
interactions. A dominant antiferromagnetic coupling is found to be responsible
for the ferrimagnetic behavior observed for $A$ = Ca below $\sim$96 K, the
competing magnetic phases and the cationic disorder in this compound giving
rise to a spin-glass state at low temperatures. For the $A$ = Ba, on the other
hand, there is no evidence of long range order down to its spin-glass
transition temperature. The remarkably different magnetic properties observed
between these two compounds is discussed in terms of the structural distortion
that alters the strength of the Co – Ir couplings, with a relevant role played
by the Co 3$d$ $e_{g}$ – Ir 5$d$ $j_{eff}$ = 1/2 hybridization.

##  I Introduction

Since the discovery of ferrimagnetism (FIM) and half-metallic behavior above
room temperature ($T$) in Sr2FeMoO6 [1], double perovskites (DP) with various
compositions have been produced and characterized in view of tuning the
electronic and magnetic properties for spintronic applications. The most
interesting properties are observed when the systems contain a combination of
3$d$ and 4$d$/5$d$ transition-metal (TM) ions [2, 3]. Ir-based DPs have
recently received considerable attention due to the peculiar interplay between
the strong spin-orbit coupling (SOC), the Coulomb repulsion, and the crystal
field splitting which, in octahedral coordination, lifts the Ir $t_{2g}$
orbital degeneracy into a fourfold $j_{eff}$ = 3/2 and a $j_{eff}$ = 1/2
doublet, giving rise to unusual magnetic properties [4, 5].

The use of Co as 3$d$ TM in addition to 5$d$ TM adds further complexity to the
stabilization of magnetism in these Ir-based DPs. A delicate balance between
crystal field and interatomic exchange interactions can lead to different
valences and spin states of the Co ions in the octahedral symmetry [6]. In
particular, La2CoIrO6 shows exciting properties such as magnetodieletric
effect [7] and cooperative octahedral breathing distortion [8]. There is,
however, some controversy concerning the microscopic mechanisms governing its
magnetic properties.

N. Narayanan et al., from the analysis of neutron powder diffraction (NPD)
data, ascribed the small ferromagnetic (FM)-like behavior in La2CoIrO6 to spin
canting in the two interpenetrating antiferromagnetic (AFM) Co and Ir
sublattices [9]. However, the strong neutron absorption of Ir ions prevents a
reliable determination of its magnetic structure. On the other hand, recent
studies using x-ray absorption spectroscopy (XAS), x-ray magnetic circular
dichroism (XMCD), and muon spin spectroscopy suggest an AFM coupling between
Co and Ir FM sublattices, resulting in a ferrimagnetic (FIM) behavior [10,
11]. The small magnetization observed in the system can be ascribed to the
antisite disorder (ASD) at Co/Ir sites and the competing magnetic interactions
that lead to frustration. There are also other interpretations. Min-Cheol Lee
et al., using Co and O $K$-edge XAS, claim that the spin frustration and
extended paramagnetic (PM) phase are due to the non spin-selective nature of
the Co2+ (3$d$ $e_{g}\oplus 4p$) – Ir4+ 5$d$ $e_{g}$ coupling [12]. At the
same time, recent Ir $L_{3}$-edge resonant inelastic x-ray scattering show the
importance of the 3$d$-5$d$ hybridization in the magnetism of Ir-based DPs
[13]. Additionally, x-ray photoelectron spectroscopy measurements point to a
large covalence in several DP iridates, questionning the description of these
materials by a simple ionic picture [14].

The observations reported above indicate how far we are from reaching a
consensus on the nature of the magnetic ordering in Ir-based DPs. This is also
valid when La is partially replaced by alkaline-earth in La2CoIrO6. For
La2-xSrxCoIrO6, a structural transition accompanied by changes in the Co/Ir
valences and magnetic structures is observed even for small Sr-doping [9, 15].
For La2-xCaxCoIrO6, on the other hand, there is no structural transition, and
up to 25% of Ca to La, the substitution acts mainly at changing the Co
valence, with Ir remaining nearly tetravalent [11]. The $x$ = 0.5
concentration is of particular interest due to its compensation temperatures
and spontaneous exchange bias effect [16], which seems to be directly related
to a re-entrant spin glass (RSG) phase [17, 18]. Likewise, the Sr to La
partial substitution also induces competition between magnetic phases that
leads to RSG [19], whereas the La2CoIrO6 parent compound does not show glassy
magnetic behavior.

In order to shed light on the role of the Co 3$d$ – Ir 5$d$ orbital
hybridization in stabilizing the magnetic coupling in CoIr-based DPs, we have
performed a detailed investigation of how the crystal structure affects the
electronic and magnetic structures of Co and Ir sublattices in
La1.5(Ba,Ca)0.5CoIrO6 compounds using x-ray powder diffraction (XRD), ac and
dc magnetization, XAS and XMCD. Our results show remarkably different magnetic
properties for these compounds, that are discussed in terms of the lattice
distortions which ones alter the orbital hybridization between Co and Ir.

##  II Experimental details

The polycrystalline samples here investigated were synthesized by conventional
solid-state reaction. To obtain La1.5Ba0.5CoIrO6 (hereafter called Ba0.5),
stoichiometric amounts of La2O3, BaCO3, Co3O4 and metallic Ir in powder form
were mixed and heated at $650^{\circ}$C for 24 hours in air atmosphere. Later,
the sample was re-grinded before a second step at $800^{\circ}$C for 48 hours.
Finally, the material was grinded, pressed into a pellet, and heated at
$1200^{\circ}$C for additional 48 hours. The synthesis route used to produce
La1.5Ca0.5CoIrO6 (herein called Ca0.5) is described elsewhere [11].

High-resolution XRD data were collected at room temperature using a Cu
$K_{\alpha}$ radiation operating at 40 kV and 40 mA at Centro Brasileiro de
Pesquisas Físicas (CBPF), Brazil. The XRD data was carried over the angular
range of $10^{\circ}\leq 2\theta\leq 110^{\circ}$, with a 2${\theta}$ step
size of 0.01∘. Rietveld refinement was performed using GSAS software and its
graphical interface program [20]. The ac and dc magnetic measurements were
carried out using a Quantum Design PPMS magnetometer, at CBPF.

Low-temperature XAS measurements were carried out at the Co $K$\- and Ir
$L_{2,3}$-absorption edges on beamline P09 at PETRA III at the DESY [21] and
at Co $L_{2,3}$-absorption edges on beamline I06 at Diamond Light Source
(DLS). For the hard x-ray measurements, fine powder samples were mixed and
pressed into a low-Z BN material to produce pellets for transmission
measurements. XMCD measurements were performed at 5 K by fast-switching the
beam helicity between left and right circular polarization [22]. In order to
align the magnetic domains and correct for nonmagnetic artifacts, an external
magnetic field of $\pm$ 5 T was applied parallel and antiparallel to the
incident beam wave vector $k$ using a 6T/2T/2T vector magnet. For the soft
x-ray measurements, powder samples from the same batches were spaded on
conductive carbon tapes and measured in total electron yield (TEY). The
samples were cooled at 2 K, and an external magnetic field of $\pm$ 5 T was
applied along the beam axis to reach saturation.

##  III Results and discussion

###  III.1 x-ray diffraction

Fig. 1 shows the XRD pattern of Ba0.5, which confirms the formation of a
perovskite structure with no trace of impurity phase. Previous studies on
La2CoIrO6 and LaBaCoIrO6 show that the former compound grows in the monoclinic
$P2_{1}/n$ space group whereas the latter was reported as belonging to the
triclinic $I\overline{1}$ non-standard space group [9, 11, 23]. Such non-
standard setting is commonly adopted to facilitate the comparison of this
structure with that of other DPs [3, 24]. In our case, a detailed
investigation of the XRD data of the Ba0.5 sample revealed that some of the
experimentally observed reflections are not predicted for the triclinic
symmetry. In contrast, all peaks match with that of the monoclinic structure.
Therefore, the 25% of Ba substitution at the La site is not sufficient to
induce a structural transition, with Ba0.5 belonging to the $P2_{1}/n$ space
group, the same as that of Ca0.5 [11]. The main results obtained from the
Rietveld refinement are displayed in Table 1, where the structural parameters
of Ca0.5 (some of them already reported in Ref. 11) are also shown for
comparison.

Figure 1: Rietveld refinement fitting of Ba0.5 XRD. The vertical lines
represent the Bragg reflections for the $P2_{1}/n$ space group. Inset shows
the crystal structure, in which IrO6 and CoO6 are drawn as blue and green
octahedra, respectively.

From Table 1 we observe an increase in the unit cell volume ($V$) of Ba0.5 as
compared to Ca0.5, in agreement with the Ba2+ ionic radius being larger than
Ca2+ (respectively 1.61 and 1.34 $\AA$ in XII coordination [25]).
Interestingly, although both compounds belong to the $P2_{1}/n$ monoclinic
space group, there is a tendency towards a more tetragonal symmetry for Ba0.5,
which is manifested by the closeness of the $a$ and $b$ lattice parameters and
a significant increase in the $c$ parameter. The increased crystallographic
symmetry is expected for larger rare-earth/alkaline-earth ionic radii in DPs
[2, 3], manifested in the increased Co–O–Ir bond angles and the more similar
Co–O and Ir–O bond lengths for Ba0.5. It must also be noticed in Table 1 that
the Co/Ir ASD, i.e. the amount of Co(Ir) ions lying at Ir(Co) site, is rather
similar for both compounds. Though this may look unexpected at a first glance,
since an increased crystallographic symmetry usually leads to larger ASD, we
recall that Ca0.5 was synthesized at $1000^{\circ}$C [11] while Ba0.5 was
produced at $1200^{\circ}$C. The synthesis condition was already shown to play
a role on the B/B’ cationic ordering of DPs, which commonly increases with
increasing the synthesis temperature [26].

Table 1: Main results obtained from the Rietveld refinements of the room temperature XRD data. Some of the parameters displayed for Ca0.5 were already reported in Ref. 11. |  |  |  |  | Lattice Parameters and R factors |  |  |
---|---|---|---|---|---|---|---|---
Sample |  $a$ (Å) |  $b$ (Å) |  $c$ (Å) |  $\beta$ (∘) |  $V$ (Å3) |  $R_{p}$ (%) |  $R_{wp}$ (%)
Ca0.5 | 5.5566(1) | 5.6427(2) | 7.8808(2) | 89.98(1) | 247.09(2) | 8.5 | 11.2
Ba0.5 | 5.6326(2) | 5.6152(1) | 7.9402(3) | 90.02(1) | 251.13(1) | 4.0 | 5.6
|  |  |  |  |  | Bonds and Angles |  |  |
Sample |  Co-O1 (Å) |  Co-O2 (Å) |  Co-O3 (Å) |  $\langle$Co-O$\rangle$ (Å) |  Ir-O1 (Å) |  Ir-O2 (Å) |  Ir-O3 (Å) |  $\langle$Ir-O$\rangle$ (Å) |  Co-O1-Ir (∘) |  Co-O2-Ir (∘) |  Co-O3-Ir (∘) |  $\langle$Co-O-Ir$\rangle$ (∘)
Ca0.5 | 2.062(13) | 2.127(13) | 2.072(15) | 2.087(8) | 1.980(13) | 2.033(13) | 2.076(15) | 2.030(8) | 156.7(8) | 144.4(7) | 143.6(5) | 148.2(4)
Ba0.5 | 2.023(1) | 2.060(1) | 1.996(1) | 2.023(1) | 2.012(1) | 2.043(1) | 2.071(1) | 2.042(1) | 162.2(1) | 151.4(1) | 154.8(1) | 156.1(1)
|  |  |  | Atomic Coordinates and Thermal Displacement Parameters |  |  |  |
Sample | atom | Wyckoff position | occupancy | x | y | z |  Uiso (Å2)
| La/Ca |  4$e$ | 0.75/0.25 | 0.5103(3) | 0.5437(5) | 0.2503(3) | 0.0105(5)
| Co1/Ir1 |  2$c$ | 0.927/0.073(3) | 0 | 0.5 | 0 | 0.0121(3)
Ca0.5 | Ir2/Co2 |  2$d$ | 0.927/0.073(3) | 0.5 | 0 | 0 | 0.0121(3)
| O1 |  4$e$ | 1 | 0.2188(21) | 0.2093(23) | 0.9629(27) | 0.0045(24)
| O2 |  4$e$ | 1 | 0.3064(21) | 0.7076(24) | 0.9363(23) | 0.0045(24)
| O3 |  4$e$ | 1 | 0.3842(22) | 0.9867(18) | 0.2503(20) | 0.0045(24)
| La/Ba |  4$e$ | 0.75/0.25 | 0.5140(2) | 0.5271(1) | 0.2474(5) | 0.0104(2)
| Co1/Ir1 |  2$c$ | 0.936/0.064(2) | 0 | 0.5 | 0 | 0.0129(2)
Ba0.5 | Ir2/Co2 |  2$d$ | 0.936/0.064(2) | 0.5 | 0 | 0 | 0.0129(2)
| O1 |  4$e$ | 1 | 0.2475(26) | 0.2474(25) | 0.9608(29) | 0.0072(12)
| O2 |  4$e$ | 1 | 0.3039(30) | 0.6980(31) | 0.9645(23) | 0.0072(12)
| O3 |  4$e$ | 1 | 0.4214(28) | 0.9996(11) | 0.2548(29) | 0.0072(12)

###  III.2 ac and dc magnetization

Fig. 2(a) shows the zero-field cooled (ZFC) and field cooled (FC) dc magnetic
susceptibility ($\chi_{dc}$) data for Ba0.5, measured with a magnetic field
($H$) of 0.05 T. For comparison, Fig. 2(b) displays $\chi_{dc}$ curves for
Ca0.5 also measured with $H$ = 0.05 T, as reported in Ref. 11. For Ca0.5, the
higher-$T$ FM-like magnetic transition at $T\sim 96$ K is ascribed to
Co2+–O–Ir4+ AFM coupling that results in FIM. At the same time, the peak at
$T\sim 27$ K is related to the emergence of a spin glass (SG) phase caused by
competing magnetic interactions and ASD [11].

For Ba0.5, although there is only one clear peak at $T\sim 24$ K in the
$\chi_{dc}$ data, a bifurcation of the ZFC and FC curves at $T\sim 85$ K is
observed, at which temperature a deviation of the Curie-Weiss law is also
observed in the inverse susceptibility [inset of Fig. 2(a)]. This could
indicate that a non-percolating short-range magnetic coupling starts to
develop around this temperature, which could be in turn related to the
appearance of a spin-glass phase at much lower temperatures. The emergence of
short-range correlations and fluctuations far above the freezing temperature
is commonly observed in spin-glasses [27].

The real part of the ac magnetic susceptibility ($\chi^{\prime}_{ac}$) curves
measured for Ba0.5 in an oscillating field $H_{ac}$ = 10 Oe with six ac
frequencies ($f$) ranging from 30 to 10000 Hz is shown on Fig. 2(c). In
contrast to the ac magnetic susceptibility data for Ca0.5, where a clear cusp
is observed at $T_{C}$ [inset of Fig. 2(d)], the $\chi^{\prime}_{ac}$ curves
of Ba0.5 gives no evidence of any anomaly close to 85 K, further indicating
that the weak magnetic coupling between Co and Ir does not percolate on this
compound. At lower temperatures, it is observed a systematic decrease in the
amplitude of the $T\sim 24$ K peak and its shift to higher temperatures as $f$
increases. These are characteristic features of glassy magnetism, signaling
that this peak corresponds to the freezing temperature ($T_{f}$) of a SG-like
phase [28]. The evolution of $T_{f}$ with $f$ turns out to be well fitted by
the power-law equation of the dynamic scaling theory, commonly used to
investigate SG-like systems

$\frac{\tau}{\tau_{0}}=\left[\dfrac{(T_{f}-T_{g})}{T_{g}}\right]^{-z\nu},$ (1)

where $\tau$ is the relaxation time corresponding to the measured frequency,
$\tau_{0}$ is the characteristic relaxation time of spin flip, $T_{g}$ is the
SG transition temperature ($T_{f}$ as $f$ tends to zero), $z$ is the dynamical
critical exponent and $\nu$ is the critical exponent of the correlation length
[28, 29]. The solid line on inset of Fig. 2(c) represents the fit to Eq. 1,
yielding $T_{g}\simeq 24.4$ K, $\tau_{0}\simeq 6.3\times 10^{-12}$ s and
$z\nu\simeq 5.9$, being these values typical of conventional SG systems [16,
29, 30].

Figure 2: (a) ZFC-FC $\chi_{dc}$ curves for Ba0.5, measured with $H$ = 0.05 T.
The arrow indicates the temperature where the curves bifurcate, and the inset
shows the best fit of PM region with the CW law. (b) ZFC-FC $\chi_{dc}$ curves
measured with $H$ = 0.05 T for Ca0.5, adapted from Ref. 11. Inset shows the
$\chi^{\prime}_{ac}$ curve measured with $H_{ac}$ = 10 Oe, $f$ = 100 Hz. (c)
$\chi^{\prime}_{ac}$ as a function of $T$ for Ba0.5 at different $f$, measured
with $H_{ac}$ = 10 Oe. Inset shows $T_{f}$ as a function of $f$, where the
solid line represents the best fit to the data using Eq. 1. (d) M(H) curves
for Ba0.5 and Ca0.5 samples, measured at 5 K.

Usually, the relative shift $\delta T_{f}$ = $\Delta T_{f}/T_{f}(\Delta lnf)$
is calculated to classify the material as canonical SG, cluster SG or
superparamagnet. For Ba0.5 we find $\delta T_{f}\simeq$0.006, which is within
the range usually found for canonical SG ($\delta T_{f}$ $<$ 0.01 [30, 31,
32]). This confirms that, as for Ca0.5, the ASD and competing magnetic
interactions also lead to the emergence of a SG phase in Ba0.5.

The fit of the PM region of 1/$\chi_{dc}$ raw data with the Curie-Weiss (CW)
law, inset of Fig. 2(a), yields a CW temperature $\theta_{CW}=-58$ K for
Ba0.5. The negative sign indicates that AFM couplings dominate the magnetic
interactions in this compound, as observed for Ca0.5 [11]. The $\mu_{eff}=5.3$
$\mu_{B}$/f.u. obtained from the fit is fairly close to the value found for
Ca0.5 (5.4 $\mu_{B}$/f.u.) [11]. It is important to mention that the situation
here is quite different from that of some Ir5+-based compounds where a
$\chi_{0}$ term is usually included in the CW equation to account for a
diamagnetic contribution to the susceptibility [5, 33, 34, 35]. In our
Ir4+/Co2+,3+ system, the van-Vleck constant term and the core diamagnetic
contribution are expected to be negligible regarding the paramagnetic moments
of the TM ions.

Fig. 2(d) displays the magnetization as a function of $H$ [M(H)] curves
measured at 5 K for Ba0.5 and Ca0.5 samples. Both curves are typical of FIM
systems, with a FM-like hysteresis but a linear increase of magnetization at
high $H$, further indicating the presence of AFM interactions. It is
interesting to notice the remarkable decrease in the coercive field ($H_{C}$)
of Ba0.5 with respect to Ca0.5. This resembles other 3$d$-5$d$ DPs for which
$H_{C}$ dramatically decreases with the expansion of the unit cell [36, 37,
38, 39].

###  III.3 XAS and XMCD measurements

XAS spectra at the TM $L_{2,3}$-edges are very sensitive to the valence state,
corresponding to electronic transitions from the 2$p$ to 3$d$ (5$d$) orbitals
for ions of the first (third) row.Fig. 3(a) shows the Co $L_{2,3}$-edge
spectra for the samples of interest together with those of CoO and LaCoO3,
used here as standards for Co2+ and Co3+, respectively. The presence of Co2+
in Ba0.5 and Ca0.5 samples is confirmed by the similarity of the spectra with
that of CoO. However, for Ba0.5 it is observed the presence of pronounced
peaks at $\sim$785 and $\sim$800 eV, associated with Ba $M_{4,5}$ absorption
edges. A closer inspection of the higher energy side of the $L_{3}$-edge
($\sim$780 eV) of Ca0.5 and Ba0.5 reveals a relative increase in the intensity
of the spectra with respect to that of CoO near the energy position of Co3+
$L_{3}$-edge absorption, suggesting that some fraction of Co is in trivalent
state. This is in agreement with the effective magnetic moments obtained for
both compounds [11].

A quantitative estimate of the Co oxidation state can be made for our Ca0.5
sample by comparing the center of mass of its $L_{3}$-edge spectrum with those
of the reference samples. Assuming a linear variation in energy between the
center of mass of Co2+ and Co3+, the Co formal valence was estimated to be
$\sim$2.4 for Ca0.5, which would mean the presence of $\sim$10% of Ir5+ or
oxygen vacancy in this sample. Surely this is just a rough estimate, since
other parameters such as the crystal field and structure may play a role on
the integrated intensity of XAS curves. Nevertheless, a +2.4 formal valence is
reasonably close to the +2.5 value expected for a 25% Ca2+ to La3+
substitution, which one would act mainly at changing the Co valence, a
scenario corroborated by the magnetometry results and Ir $L_{3}$-edge XAS. For
Ba0.5, the presence of the Ba-$M_{5}$ absorption peak nearby the Co
$L_{3}$-edge curve affects the calculation of the center of mass, preventing a
reliable estimate of its valence state. Notwithstanding, the similarity of its
$L_{2,3}$-edge curves with that of Ca0.5, as well as the closeness of the
$\mu_{eff}$ observed for both samples, suggest that the Co valence for Ba0.5
is similar to that of Ca0.5.

Figure 3: (a) Co $L_{2,3}$-edge XAS spectra of Ca0.5 and Ba0.5 samples,
together with those of CoO (reference for Co2+) and LaCoO3 (reference for
Co3+). The presence of Ba $M_{4,5}$-edges in Ba0.5 sample is highlighted by
green arrows. (b) and (c) show respectively the XMCD spectra of Ca0.5 and
Ba0.5 samples, taken at 2 K with $H$ = 5 T. The photon spin was aligned
parallel ($\mu^{+}$, black) and antiparallel ($\mu^{-}$, red) to $H$, and the
difference spectra are shown in blue.

Figs. 3(b) and (c) show the Co $L_{2,3}$ XMCD of Ca0.5 and Ba0.5 samples,
respectively. The same overall behavior suggests nearly the same magnetic
structure for both compounds. The black and red curves stand respectively for
$\mu^{+}$ and $\mu^{-}$, i.e., for parallel and antiparallel alignments
between the photon spin and $H$, and the difference spectra
$\Delta\mu=\mu^{+}-\mu^{-}$ corresponds to the blue line. As can be noticed,
the XMCD signal is largely negative at $L_{3}$-edge but with a very small
positive value at $L_{2}$-edge, indicating non-negligible orbital contribution
to the magnetic moment, as commonly observed for Co in octahedral coordination
[6]. For a quantitative analysis of the Co-$L_{2,3}$ XMCD spectra, we have
used the sum rules to derive the orbital and spin contributions to the
magnetization [41, 42]

$m_{l}=-\frac{4\int_{L_{3}+L_{2}}(\mu^{+}-\mu^{-})d\omega}{3\int_{L_{3}+L_{2}}(\mu^{+}+\mu^{-})d\omega}N_{h},$
(2)
$\begin{split}m_{s}&=-\frac{6\int_{L_{3}}(\mu^{+}-\mu^{-})d\omega-4\int_{L_{3}+L_{2}}(\mu^{+}-\mu^{-})d\omega}{\int_{L_{3}+L_{2}}(\mu_{+}+\mu_{-})d\omega}\times\\\
&N_{h}\left(1+\frac{7\langle T_{z}\rangle}{2\langle
S_{z}\rangle}\right)^{-1},\\\ \end{split}$ (3)

where $m_{l}$ and $m_{s}$ are the angular and spin magnetic moments in units
of $\mu_{B}$/atom, $S_{z}$ denotes the projection along $z$ of the spin
magnetic momentum, $N_{h}$ represent the number of empty 3$d$ states, $T_{z}$
denotes the magnetic dipole moment, and $L_{2}$ and $L_{3}$ represent the
integration range. Here we used $N_{h}$ = 3.5$\pm$0.5, which is an
approximated atomistic value corresponding to a mixture of Co2+ ($Nh$ = 3) and
Co3+ ($Nh$ = 4). Moreover, $T_{z}$ is estimated to be negligible against
$S_{z}$ for TM ions in octahedral symmetry [43, 44]. With these
considerations, we obtained $m_{l}$ = 0.15 $\mu_{B}$ and $m_{s}$ = 0.29
$\mu_{B}$. Considering sources of deviations such as electronic interactions,
imprecision in the integral calculations, and the $N_{h}$ value assumed, we
estimate an uncertainty of $\sim 20\%$ on these values [43, 44].

The calculated moments are similar to those reported for La2CoIrO6 ($m_{l}$ =
0.18 $\mu_{B}$, $m_{s}$ = 0.31 $\mu_{B}$) [10], which is somewhat unexpected
considering that from our XAS spectra we can expect about 40% of the Co ions
in a trivalent state, as opposed to a presumably 100% of Co2+ in La2CoIrO6.
However, in Ref. 10 a quantitative investigation of the Co formal valence was
not performed for the parent compound, whilst another study performed on a
La2CoIrO6 polycrystalline sample produced by a similar synthesis route
reported the presence of some amount of Co3+ [11]. Furthermore, the previous
XMCD study reported for La2CoIrO6 in Ref. 10 was carried out at 43 K using $H$
= 0.8 T, while element-specific hysteresis curves measured for CoIr-DPs have
already shown that Co is far from saturation at these conditions [15].

For the Ba0.5 sample, one must recall the pronounced Ba $M_{4,5}$-edge peaks
observed near the Co $L_{2,3}$ absorption edge. Although one may not expect a
XMCD signal at the Ba $M_{4,5}$-edge, the large XAS white line intensity may
influence the background of the Co $L_{2,3}$ pre- and post-edge regions, and
affect the adjustment of the XAS curve for the calculation of the
$L_{2}+L_{3}$ integral. Therefore, $m_{l}$ and $m_{s}$ were not estimated for
this compound. The $m_{l}$/$m_{s}$ ratio, however, is independent of the XAS
integrated intensity, resulting in a trustworthy value that is fairly close to
that found for Ca0.5.

Table 2: Co and Ir orbital ($m_{l}$), spin ($m_{s}$) and total ($m_{t}$) magnetic moments obtained from the sum rules calculations of Co and Ir $L_{3,2}$ XMCD spectra. Sample | Co $m_{l}$ ($\mu_{B}$) | Co $m_{s}$ ($\mu_{B}$) | Co $m_{t}$ ($\mu_{B}$) | Co $m_{l}/m_{s}$ | Ir $m_{l}$ ($\mu_{B}$) | Ir $m_{s}$ ($\mu_{B}$) | Ir $m_{t}$ ($\mu_{B}$) | Ir $m_{l}/m_{s}$
---|---|---|---|---|---|---|---|---
Ca0.5 | 0.15$\pm$0.03 | 0.29$\pm$0.06 | 0.44$\pm$0.07 | 0.52$\pm$0.11 | -0.11$\pm$0.02 | -0.11$\pm$0.02 | -0.22$\pm$0.03 | 1.00$\pm$0.21
Ba0.5 | - | - | - | 0.54$\pm$0.15 | -0.07$\pm$0.01 | -0.06$\pm$0.01 | -0.13$\pm$0.01 | 1.17$\pm$0.27
Figure 4: (a) Co $K$-edge XAS spectra of Ba0.5 and Ca0.5 samples, measured at
5 K. Inset shows a magnified view of the pre-edge region, highlighting the
$\alpha$, $\beta_{1}$ and $\beta_{2}$ peaks. (b) The XMCD spectra of these
samples.

In order to avoid the problem of Ba $M_{4,5}$-edges, we investigated the Co
$K$-edge XAS and XMCD, as shown in Fig. 4. The Co $K$-edge XAS reflects mainly
the electric dipole transitions from 1$s$ to 4$p$ level, where the edge
position in energy may probe the Co valence. At the same time, local
arrangements of neighboring ions determine the region after the absorption
threshold. The similarity between the curves displayed in Fig. 4(a) is in
accordance with the same monoclinic structure and similar Co valence states
for both compounds.

Figure 5: Ir $L_{2,3}$-edges: (a) XAS and (b) XMCD spectra for Ba0.5 and Ca0.5
measured at 5 K and under 5 T. (c) Magnified view of the $L_{3}$-edge, where
the IrO2 standard is added for comparison. (e) Element-specific XMCD
hysteresis loops of Ir $L_{3}$-edge.

The inset of Fig. 4(a) shows a magnified view of the pre-edge region. Although
the $K$-edge XAS reflects mostly the transitions to unoccupied Co 4$p$ states,
M.-C. Lee et al. showed that the pre-edge anomalies $\alpha$ and $\beta_{1}$
are related to Co 4$p$ partial electron filling via intra-site mixing with Co
3$d$ orbitals [12]. Thus, the orbital hybridization with Ir 5$d$ orbitals may
largely contribute to these pre-edge features (see Fig. 6). Interestingly, a
third anomaly, $\beta_{2}$ $\sim 7720$ eV, is observed for Ca0.5 and Ba0.5,
which may be related to Co3+ (3$d$ $e_{g}\oplus 4p$) – Ir 5$d$ $e_{g}$
hybridization via O 2$p$.

The Co $K$-edge XMCD signals measured at $T$ = 5 K and $H$ = 5 T are very
similar for both Ca0.5 and Ba0.5 [Fig. 4(b)], suggesting that although the Co
3$d$ bands present a relative extended character, the changes in the
crystallographic environment does not drastically affect the Co magnetism.

Similar scenario cannot be extrapolated to the Ir ions, since the 5$d$
orbitals are even more extended than the Co 3$d$, which makes Ir ions more
sensitive to structural and electronic changes. Fig. 5(a) shows the Ir
$L_{2,3}$-edge XAS spectra of Ba0.5 and Ca0.5 samples together with that of
IrO2, reference for Ir4+ state. The spectra of Ba0.5 and Ca0.5 are reasonably
similar to that of IrO2, indicating that the majority of Ir ions are in a
tetravalent state for both compounds. Nonetheless, the similarity between the
curves is characteristic of the $L_{2,3}$-edge spectra of 5$d$ TM ions, being
related to the more diffuse valence orbitals compared to the 3$d$ ones [15,
45]. A magnified view of the $L_{3}$-edge region depicted in Fig. 5(c)
indicates some subtle differences between the spectra. The Ca0.5 and Ba0.5
spectra are shifted of respectively $\sim$0.2 eV and $\sim$0.3 eV with respect
to the IrO2 $L_{3}$ XAS, suggesting the presence of a small amount of Ir5+.
However, these shifts lie within the instrumental resolution ($\sim$1.5 eV),
while the difference in energy between the white line positions of Ir4+ and
Ir5+ is of about 1.2 eV [34]. With these considerations, we can only conclude
that Ba0.5 and Ca0.5 presents nearly the same Ir valence state, with the
majority of ions in a tetravalent state.

Fig. 5(b) shows the Ir $L_{2,3}$-edge XMCD spectra measured at $T$ = 5 K and
$H$ = 5 T. The positive signs of the Ir XMCD signals, combined with the
negative sign for the Co signal, indicate that those two ions develop an AFM
coupling in both compounds. Table 2 shows the Ir orbital and spin moments
obtained from the sum rules, for which it was assumed $N_{h}$ = 5$\pm$0.5
(Ir4+). The $m_{l}$ and $m_{s}$ values found are similar to those previously
reported for La1.5Sr0.5CoIrO6 [15], and the large $m_{l}$/$m_{s}$ ratio
observed is typical of iridates and other 5$d$-based materials presenting
strong SOC [46, 47]. The somewhat smaller moments observed for Ba0.5 with
respect to Ca0.5 helps to explain its slightly smaller $\mu_{eff}$ obtained
from the $\chi_{dc}$ curves, also suggesting that the Ir magnetization is
closely related to its orbital hybridization with neighboring Co ions.

Fig. 5(c) shows element-specific hysteresis loops obtained by monitoring the
Ir $L_{3}$-edge XMCD signal as a function of $H$. As can be noticed, the Ir
moment is nearly saturated for Ca0.5 under $H$ = 3 T, whereas for Ba0.5 the
saturation might be reach only for much higher magnetic fields. A remarkable
point in these curves is the pronounced decrease in the $H_{C}$ of Ba0.5 with
respect to Ca0.5, as also observed in the macroscopic M(H) curves measured in
PPMS [Fig. 2(d)].

###  III.4 Discussion

Figure 6: (a) Schematic diagram of the mechanisms of hybridization between
Co2+/3+ (3$d^{7}$/3$d^{6}$) and Ir4+ (5$d^{5}$), where the red arrow
represents an additional spin down electron for Co2+ $t_{2g}$, which is absent
for Co3+. The green arrows represent virtual electron hopping. The blue
double-arrow in Ir $j_{eff}$ = 1/2 is spin down for the case of AFM
interaction and spin up for the FM case. (b) (Co/Ir)O6 octahedra geometries
for Co–O–Ir bond angle $\theta$ = $180^{\circ}$ and $\theta$ $<$
$180^{\circ}$. (c) The Co $t_{2g}$ – Ir $j_{eff}=1/2$ (left) and Co (3$d$
$e_{g}\oplus 4p$) – Ir $j_{eff}=1/2$ (right) orbital hybridizations.

In general, the magnetic ordering of TM ions in DPs can be described by the
orbital hybridization mechanism, according to which the $d$ orbital states
reside close to the Fermi surface, being respectively slightly below the
chemical potential for one TM ion and slightly above for another. Then, a
hybridization between these orbitals, mediated by O 2$p$ orbitals, enables the
virtual spin-dependent electron hopping from one TM to another and can
stabilize either FM or AFM coupling [2, 3]. For HS Co2+ (Co3+), there are two
(one) completely filled and one (two) half filled $t_{2g}$ orbital(s), while
both $e_{g}$ orbitals are partially filled. The 5$d^{5}$ Ir4+ ion, on the
other hand, is at LS configuration, with the SOC lifting the $t_{2g}$
degeneracy to a fourfold $j_{eff}$ = 3/2 and a twofold $j_{eff}$ = 1/2 state.
Thus, Ir4+ has one $j_{eff}$ = 1/2 electron while its $j_{eff}$ = 3/2 orbitals
are fully occupied [4, 48].

Recent first-principles calculations for La2CoIrO6 have shown that the energy
of the Ir-$j_{eff}$ = 1/2 state resides in between those of Co $t_{2g}$ and
$e_{g}$, whereas the strong crystal field splitting on Ir puts its $e_{g}$
states $\sim$3 eV higher in energy, making these later orbitals irrelevant to
our discussion [49]. As a consequence, only the $t_{2g}$ – O 2$p$ – $t_{2g}$
$\pi$ bonding would be a relevant exchange path between Co and Ir. However,
although the $t_{2g}$ – O 2$p$ – $e_{g}$ coupling is in principle forbidden by
symmetry in a cubic perovskite (i.e. for a 180∘ B–O–B’ bond angle), it can
become possible with lattice distortion, as is the case for the samples here
investigated.

Fig. 6(a) depicts the spin-selective exchange pathways between Co2+/Co3+ and
Ir4+ in our system. There is one possible FM virtual hopping from the Co
$t_{2g}$ minority spin state to Ir $t_{2g}$ ($j_{eff}$ = 1/2), and a number of
possible AFM couplings. Here we must recall the highly covalent character of
the Ir–O bond, with an O-electron partially occupying the Ir-$t_{2g}$ orbital
[14], which in principle inhibits the virtual electron transfer from Co to Ir.
This partial population of Ir-$j_{eff}$ = 1/2 orbital with O-electrons,
together with the SOC, may explain the dramatic decrease of the Ir magnetic
moment observed here and for other Ir4+-based perovskites [9, 15, 8, 11].
Therefore, AFM charge transfer from Ir to Co seems to dominate the orbital
hybridization in these samples, which is supported by the fact that all
experimental and theoretical works reported so far for CoIr-based DPs points
towards AFM coupling between these ions. As Fig. 6(a) shows, the intermediate
charge transfer from Ir $t_{2g}$ can occur to a Co $t_{2g}$ or $e_{g}$
orbital. A transfer to Co $t_{2g}$, through the Ir $t_{2g}$ – O 2$p$ – Co
$t_{2g}$ pathway, is expected to be stronger for Ba0.5 than for Ca0.5, since
the former compound presents increased Co–O–Ir bond angles [see Figs. 6(b) and
(c)]. The fact that Ca0.5 shows a clear magnetic ordering at a relatively high
temperature, contrasting the results found for Ba0.5, suggests that the Ir
$t_{2g}$ – O 2$p$ – Co $t_{2g}$ is not the most relevant hybridization channel
in these compounds. On the other hand, the Ir $t_{2g}$ – O 2$p$ – Co $e_{g}$
channel is expected to strengthen as the Co–O–Ir angle decreases, subsequently
becoming the most relevant hybridization pathway between Co and Ir. This is
reminiscent of other 3$d$-5$d$ DPs, for which a systematic increase of the
magnetic ordering temperature is observed as the B–O–B’ angle decreases [36,
38, 39, 50]. The absence of a clear magnetic ordering for Ba0.5 is thus likely
related to the weakened Ir $t_{2g}$ – O 2$p$ – Co $e_{g}$ and strengthened Ir
$t_{2g}$ – O 2$p$ – Co $t_{2g}$ hybridization, with the competing magnetic
interactions leading to spin frustration.

Another striking difference between Ba0.5 and Ca0.5 resides in their $H_{C}$.
In other 3$d$-5$d$ DPs, as well, changes in the orbital hybridization caused
by structural distortions remarkably alter their magnetic properties. For
$AA^{\prime}$FeReO6 ($AA^{\prime}$ = Ba, Ca, Sr), the increase of $H_{C}$ and
$T_{C}$ was observed with decreasing the average $A$ radius, which was
ascribed to the increase of Re orbital moment due to lattice distortion [36].
Contrastingly, high-pressure studies in $A_{2}$FeReO6 ($A$ = Ba, Ca) showed a
remarkable increase of $H_{C}$ with increasing pressure that was not related
to the increase of Re orbital contribution. Instead, such changes in $H_{C}$
were explained in terms of reducing the strength of SOC caused by the
structural distortion [37].

For our CoIr-based samples, the dramatic increase in the $H_{C}$ of Ca0.5 with
respect to Ba0.5 is probably not related to increase of the orbital moments
since the Ir $m_{l}$ value obtained for Ca0.5 is only slightly larger than
that of Ba0.5, while the Co $m_{l}/m_{s}$ ratio is higher for the latter
compound. Here we must recall that FeRe-based DPs are somewhat different from
our CoIr-system. Re5+ has two unpaired electrons at $j_{eff}$ = 3/2 orbitals
whereas for Ir4+ these orbitals are complete and there is one unpaired spin at
$j_{eff}$ = 1/2. Moreover, a quenched orbital moment is expected for Fe3+,
whereas Co2+ is known to present a non-negligible orbital contribution, as our
XMCD results clearly indicate. Our results may find more resemblance with
those observed for NiIr-based DPs, since Ni2+ may also present orbital
contribution to the magnetic moment [51, 52]. For $A_{2}$NiIrO6 ($A$ = La-Lu)
the systematic increase of $T_{C}$ and $H_{C}$ with decreasing $A$ ionic
radius is attributed to the increased strength of the AFM Ni2+ $e_{g}$ – Ir4+
$t_{2g}$ orbital hybridization [38, 39]. The enhancement of $H_{C}$ with
bending of B–O–B’ angle in the La2-xAxCoIrO6 samples here investigated gives
further evidence that the $t_{2g}$ – O 2$p$ – $e_{g}$ hybridization plays a
major role on the CoIr-based DPs.

##  IV Summary

In summary, we thoroughly investigated the structural, electronic and magnetic
properties of La1.5Ba0.5CoIrO6 (Ba0.5) and La1.5Ca0.5CoIrO6 (Ca0.5) samples by
means of XRD, ac and dc magnetization, Co and Ir $L_{2,3}$-edges and Co
$K$-edge XAS and XMCD. Our results show that the presence of 25% of Ba2+/Ca2+
at La3+ site leads to Co2+/Co3+ mixed valence, while Ir remains mostly in a
tetravalent state. For Ca0.5 it is observed a magnetic ordering at $\sim$96 K
and a large $H_{C}$, while for Ba0.5 a magnetic ordering is not clear and the
$H_{C}$ is much smaller. Since both compounds presents nearly the same Co and
Ir valence states and effective magnetic moments, the remarkable differences
between the magnetic properties of these samples can be understood in terms of
changes in the exchange paths between Co $e_{g}$/$t_{2g}$ and Ir $t_{2g}$
($j_{eff}$ = 1/2) orbitals. For Ca0.5, the AFM Co $e_{g}$ – Ir $t_{2g}$
coupling seems to dominate due to its highly distorted structure. For Ba0.5,
the increased structural symmetry weakens such hybridization but strengthen
the Co $t_{2g}$ – Ir $t_{2g}$ pathway, with the competing interactions leading
to a magnetic frustration on this sample. Together with the ASD, the competing
magnetic couplings explain the emergence of a SG phase at low temperatures in
both compounds.

###### Acknowledgements.

This work was supported by the Brazilian funding agencies: Fundação Carlos
Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) [Nos.
E-26/202.798/2019 and E-26/211.291/2021], Fundação de Amparo à Pesquisa do
Estado de Goiás (FAPEG) and Conselho Nacional de Desenvlovimento Científico e
Tecnológico (CNPq) [No. 400633/2016-7]. We thank Diamond Light Source for time
on beamline I06 under proposal MM29620-1. We acknowledge DESY (Hamburg,
Germany), a member of the Helmholtz Association HGF, for the provision of
experimental facilities. Parts of this research were carried out at beamline
P09 under proposal I-20200348 and we would like to thank J. Bergtholdt and O.
Leupold for assistance in setting up the experiment and 6T/2T/2T vector
magnet, respectively.

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