|  Anionic nickel and nitrogen effects in the chiral antiferromagnetic
antiperovskite Mn3NiN
---|---
|  E. Triana-Ramíreza,b, W. Ibarra-Hernandezc and A. C. Garcia-Castroa,∗
|  Magnetic antiperovskites, holding chiral noncollinear antiferromagnetic
ordering, have shown remarkable properties that cover from negative thermal
expansion to anomalous Hall effect. Nevertheless, details on the electronic
structure related to the oxidation states and the octahedral center’s site
effect are still scarce. Here, we show a theoretical study, based on first-
principles calculations in the framework of the density-functional theory,
DFT, on the electronic details associated with the nitrogen site effect into
the structural, electronic, magnetic, and topological degrees of freedom.
Thus, we show that the nitrogen vacancy increases the values of the anomalous
Hall conductivity and retains the chiral $\Gamma_{4g}$ antiferromagnetic
ordering. Moreover, we reveal, based on the Bader charges and the electronic
structure analysis, the negative and positive oxidation states in the Ni and
Mn sites, respectively. The latter is in agreement with the expected
$A_{3}^{\alpha+}B^{\beta-}X^{\delta-}$ oxidation states to satisfy the charge
neutrality in the antiperovskites, but rare for transition metals. Finally, we
extrapolate our findings on the oxidation states to several Mn${}_{3}B$N
compounds showing that the antiperovskite structure is an ideal platform to
encounter negative oxidation states in metals sitting at the corner $B$-site.

##

††footnotetext: a School of Physics, Universidad Industrial de Santander,
Carrera 27 Calle 09, 680002, Bucaramanga, Colombia††footnotetext: b Centro de
Investigación y Estudios Avanzados del IPN - cinvestav-Querétaro, MX-76230,
Querétaro, México. ††footnotetext: c Facultad de Ingeniería, Benemérita
Universidad Autónoma de Puebla, Apartado Postal J-39, Puebla, Pue. 72570,
México.††footnotetext: *E-mail<EMAIL_ADDRESS>
##  1 Introduction:

In the modern quest for novel and exciting phenomena in new materials,
antiperovskite, or inversed-perovskite with the formulae $A_{3}BX$, has stood
out as an astonishing type of materials showing large anomalous Hall
conductivity 1, 2, 3, 4, superconductivity 5, 6, 7, 8, good performance for
new batteries 9, 10, tunable hybrid-improper ferroelectricity 11, tangible
magnetocaloric effects 12, and large spin-lattice coupling 13, 14, 15, 16, 17.
All the latter are just a few examples that can be mentioned among the vast
functionalities offered by these materials. Interestingly, in antiperovskites
18, 19 the electrostatic balance and the oxidation site occupation are
apparently reversed with respect to the known perovskites, $ABX_{3}$ 20. Most
of the mentioned phenomena in the antiperovskites owe most of their properties
to the reversed occupation of their anionic and cationic sites forming the
reversed $XA_{6}$ octahedra. This subtle change in the coordination and atomic
occupations gives rise to, for example, the triangular geometric coordination
of the magnetic sites that in turn, induces a strong magnetic frustration
converging into chiral noncollinear antiferromagnetic orderings 21. Another
example is the topologically related properties in the Sr3SnO (Sr3PbO) with
bands crossing at the Fermi level between the Sr:4$d$ and Sn:5$p$ (Pb:6$p$)
electronic states 22, 23. Here, the Sn and Pb atomic sites hold formal
negative oxidation states confirmed by Mössbauer 22 and X-ray photoemission
spectroscopy, XPS 23. Interestingly, the negative oxidation states in metals
have attracted considerable attention due to the possible new physics that may
unveil 24, 25. Despite these reports, few metallic species are known in the
literature. Some examples of negative oxidation states in metals have been
demonstrated in compounds such as CsAu 26 and Na-Au binary compounds 27, 28.
In the latter compounds, gold’s oxidation state is Au1- achieving a full
6$s^{2}$5$d^{10}$ electronic occupation in the outer shell. Other examples are
the Pt’s negative oxidation at Ba-Pt systems 29, 30 and dimethylformamide’s
surface 31, 32 as well as the negative oxidation state in Zn at octahedrally
coordinated Zn${}_{2}M_{4}$ ($M$ = Li and Na) 33. Moreover, multiple molecular
compounds have shown metals in their structures with formal negative oxidation
states 24, 25. As such, antiperovskites appear to be potential candidates to
explore possible negative oxidation states in metals, and their induced
properties, due to the expected stoichiometric relationship
$A_{3}^{\alpha+}B^{\beta-}X^{\delta-}$ in contrast to
$A^{\alpha+}B^{\beta+}X_{3}^{\delta-}$ to hold the charge neutrality. Then,
when going from the perovskite to the antiperovskite, the $B$-site switches
from $B^{\beta+}$ to $B^{\beta-}$. This is the case of the SrSnO3 and Sr3SnO
where the Sn oxidation state goes from $4+$ to $4-$ 22 to maintain the charge
neutrality in both compounds, while the Sr and O sites keep the $2+$ and $2-$
oxidation, respectively. These findings are also in agreement with other gold-
based antiperovskites oxides Cs3AuO and Rb3AuO 34. Interestingly as observed
in perovskites, the anionic vacancies, such as oxygen deficiency in perovskite
oxides, 35, 36 could alter the electronic structure by inducing an electronic
reconstruction that directly affects the presented oxidation states of the
atomic species 37. These vacancies can be present despite the advances in
growth techniques nitrogen vacancies are expected to be formed, as in the
Mn3PtNx case 38. Therefore, exploring the effect of the nitrogen deficiency in
the Mn${}_{3}B$N antiperovskites, and particularly in the Mn3NiN prototype, is
essential in order to unveil its influence on the structural and electronic
degrees of freedom that may affect the magnetic response and anomalous Hall
conductivity. Additionally, the N-site is strongly correlated with the
oxidation states at the Mn- and Ni-sites in the Mn3NiN antiperovskite. In this
paper, we study from first-principles calculations, the electronic structure
of the chiral antiferromagnetic antiperovskite Mn3NiN. Thus, we performed a
detailed study of this antiperovskite’s electronic structure and explored the
formal charges of the atomic species coupled with the understanding of the
N-site effect in the physics beneath the electronic structure of this
compound. This antiperovskite stands as a prototype in this family of
materials and we pay special attention to the influence of the nitrogen
vacancy in the topological features, such as the anomalous Hall conductivity.
In Section 2 we present all the computational details and the theoretical
approaches used for the development of this work. This section is followed by
the presentation of the obtained results and the consequent analysis, in
Section 3. Finally, we draw our conclusions, in Section 4, and highlight some
perspectives associated with our findings around the oxidation states and the
nitrogen’s effect in the Mn3NiN antiperovskite. Furthermore, we extrapolate
these analyzes and include our results in several Mn${}_{3}B$N
antiperovskites.

##  2 Theoretical and computational details:

We performed first-principles calculations within the density-functional
theory (DFT) 39, 40 approach by using the vasp code (version 5.4.4) 41, 42.
The projected-augmented waves, PAW 43, 44 scheme was used to represent the
valence and core electrons. The electronic configurations considered in the
pseudo-potentials as valence electrons are Mn: (3$p^{6}$3$d^{5}$4$s^{2}$,
version 02Aug2007), Ni: (3$p^{6}$3$d^{8}$4$s^{2}$, version 06Sep2000), and N:
(2$s^{2}$2$p^{5}$, version 08Apr2002). The exchange-correlation, $E_{xc}$, was
represented within the generalized gradient approximation, GGA in the PBEsol
parametrization 45, and the $E_{xc}$ of the $d$-electrons was corrected
through the DFT$+U$ approximation within the Liechtenstein formalism 46. We
used a Coulomb on-site value of $U$ = 2.0 eV parameter. The latter optimized
to reproduce the experimentally observed lattice parameter. Also, a metaGGA
formalism 47, within the SCAN implementation 48, and the hybrid based-
functional, HSE06 49 were adopted to correlate with the Hubbard correction
within the PBEsol$+U$ calculations. The periodic solution of the crystal was
represented by using Bloch states with a Monkhorst-Pack 50 _k_ -point mesh of
13$\times$13$\times$13 and 600 eV energy cut-off to give forces convergence of
less than 0.001 eV$\cdot$Å-1 and an error in the energy less than 0.5 meV. The
spin-orbit coupling (SOC) was included to consider noncollinear magnetic
configurations 51. The phonon calculations were performed within the finite-
differences methodology 52, 53 and analyzed through the phonopy interface 54,
55. The latter calculations were performed in the 2$\times$2$\times$2
supercell to properly map the lattice dynamics at the zone-boundary. For these
calculations in the supercell, the $k$-mesh was then set to
6$\times$6$\times$6 and the noncollinear magnetic orderings were also
considered. To evaluate the anomalous Hall conductivity, and the changes in
the Berry curvature, we have used the Wannier functions methodology for which
the wannierization was performed with the Wannier90 code 56, 57 and post-
processed with the WannierBerri package 58. Here, $s$, $p$, and $d$ orbitals
were considered in the Mn and Ni cases, while $s$ and $p$ were considered at
the N site. Additionally, a 3.0 eV window was used around the Fermi level for
the wannierization. Bader charges were evaluated by the methodology developed
by G. Henkelman _et al._ 59. Finally, the atomic structure figures were
elaborated with the vesta code 60.

##  3 Results and discussion:

Fig. 1: (Color online) (a) Mn3Ni and Mn3NiN $Pm\bar{3}m$ structures where the
N-site octahedral center is shown in the latter and absent in the former. In
(b) are shown the chiral antiferromagnetic noncollinear $\Gamma_{4g}$ and
$\Gamma_{5g}$ orderings. Here, the magnetic moments per Mn atom are shown as
grey arrows by notation. In (c) are presented the full phonon-dispersion
curves obtained for the Mn3NiN, as well as the nitrogen-deficient Mn3Ni
antiperovskites. The latter were computed with $U$ = 2.0 eV. In both cases, we
consider the $\Gamma_{4g}$ chiral antiferromagnetic ordering ground state.

In what follows, we start by describing the electronic properties related to
the N-site effect in the Mn3NiN antiperovskite. In Fig. 1 are shown the Mn3Ni
and Mn3NiN cubic $Pm\bar{3}m$ (SG. 221) antiperovskites as well as the
symmetry allowed noncollinear chiral antiferromagnetic $\Gamma_{4g}$ and
$\Gamma_{5g}$ orderings. Thus, in the nitrogen deficiency antiperovskite, the
$\Gamma_{4g}$ ordering is more stable over the $\Gamma_{5g}$ explained in
terms of the MAE energy $\Delta E=E_{\Gamma_{4g}}-E_{\Gamma_{5g}}$ = $-$0.58
meV$\cdot$f.u.-1. Thus, as in the Mn3NiN case, the magnetic ground state in
the Mn3Ni is the chiral $\Gamma_{4g}$ antiferromagnetic ordering that allows
the anomalous Hall effect 61, as will be discussed further. Therefore, it is
worth recalling that all the calculations contained in this work were
performed considering the spin-orbit coupling and the noncollinear
antiferromagnetic states for Mn3NiN, as well as for Mn3Ni antiperovskites.
After fully relaxing the Mn3NiN and Mn3Ni the obtained lattice parameters are
$a$ = 3.889 Å and $a$ = 3.707 Å respectively. It can be noted that, in the
Mn3NiN case, the lattice parameter is in good agreement with the
experimentally reported value of $a$ = 3.886 Å 62. In the Mn3Ni, although
there is no experimentally reported parameter, as the exchange-correlation
correction was also considered in the Mn:3$d$ orbitals, it can be expected a
close value to the one reported here by us. When comparing the lattice
parameter, it can be observed that the inclusion of the N-site in the
octahedral center induces a tangible lattice expansion, equivalent to 0.182 Å.
However, the symmetry space groups remain the same being $Pm\bar{3}m$ (SG.
221) without considering the magnetic ground state and $R\bar{3}m^{\prime}$
(MSG. 166.101) once the chiral noncollinear antiferromagnetic ground state is
accounted into the symmetry operations. Then, only changes in the volume and
electronic structure were found. Moreover, both Mn3NiN and Mn3Ni
antiperovskite are fully dynamically stable in the cubic configuration under
the noncollinear $\Gamma_{4g}$ antiferromagnetic ordering, see Fig. 1. As it
can be observed, the high-frequency modes are absent in Mn3Ni in which case,
these modes are nitrogen driven. For instance, the antiperovskite Mn${}_{3}B$N
structure can be viewed as magnetic Mn-based kagome lattices with $B$-sites
embedded into them and separated by nitrogen layers.

In Fig. 2 we present the entire electronic characterization for the Mn3NiN and
Mn3Ni antiperovskites. Here, the full orbitally-projected band electronic
structure is presented, in Fig. 2(a), as well as the local density of states,
in Fig. 2(b), and the computed anomalous Hall conductivity for the
$\sigma_{xy}$, and $\sigma_{111}$ terms, in Fig. 2(c). At first glance, we can
observe from Fig. 2(a) that there is a substantial reduction of the available
states close to the Fermi energy, here located at $E_{F}$ = 0.0 eV by
notation, once the N-site is introduced in the antiperovskite. It can be
appreciated that the major contribution at and above the Fermi level is
associated with the Mn:3$d$ orbitals in both cases. As for the Ni states,
those appear to be located well below $E$ = $-$0.5 eV and are quite localized
around $-$1.5 eV as in an insulator case, see Fig. 2(b). Nevertheless, a small
contribution from the Ni states can be observed above the Fermi level. The
latter is expected because the antiperovskite structure can be understood, as
commented before, as (111) Mn-based kagome planes with Ni sites embedded and
separated by the N-sites. Importantly, as the N-site is located at the
octahedral center, the Mn3NiN and the Mn3Ni hold the same crystallographic and
magnetic symmetry. Thus, only modifications in the electronic structure are
observed, but the AHC tensor is kept fixed. In this case, the anomalous Hall
conductivity component, $\sigma_{xy}$, has been computed by the relationship:

$\sigma_{xy}=-\frac{2\pi
e^{2}}{h}\sum_{n}^{occ}\int_{BZ}\frac{d^{3}\bf{k}}{(2\pi)^{3}}f_{n}({\bf{k}})\Omega_{n,xy}(\bf{k}),$
(1)

where
$\Omega_{xy}(\bf{k})$=$\sum_{n}^{occ}f_{n}(\bf{k})$$\Omega_{n,xy}(\bf{k})$ is
the summation of all the occupied $n$-bands and $f_{n}\bf(k)$ represents the
Fermi distribution. Moreover, the symmetry-allowed AHC components within the
$\Gamma_{4g}$ ordering in the $R\bar{3}m^{\prime}$ magnetic symmetry group
are:

$\sigma_{R\bar{3}m^{\prime}}=\begin{pmatrix}0&\sigma_{xy}&-\sigma_{xy}\\\
-\sigma_{xy}&0&\sigma_{xy}\\\ \sigma_{xy}&-\sigma_{xy}&0\end{pmatrix}$ (2)

Fig. 2: (Color online) (a) Atomically projected band structure, and (b)
atomically projected density of states, DOS. Here, the Mn, Ni, and N states
are denoted in violet, blue, and green colors respectively. Additionally, in
(b) the total DOS is denoted in grey color. (c) Anomalous Hall conductivity,
$\sigma_{xy}$ and $\sigma_{111}$ components, computed at the $\Gamma_{4g}$
orderings in the Mn3NiN and Mn3Ni.

The charge at the Ni-site is expected to be the same and only changes in the
allowed electronic states in the proximity to the Fermi level might influence
the anomalous Hall conductivity. Additionally, as the AHC is strongly
dependent on the spin-orbit coupling strength 63, the absence or presence of
the nitrogen octahedral center site has a negligible effect on $\sigma_{xy}$.
In Fig. 2(c) we show the computed anomalous Hall conductivity for the
$\sigma_{xy}$ in both compounds, as well as the $\sigma_{111}$ component in
the magnetic kagome lattice at the (111) lattice plane. The $\sigma_{111}$
component is computed as
$\sigma_{111}\equiv\frac{1}{\sqrt{3}}\left(\sigma_{xy}+\sigma_{yz}+\sigma_{zx}\right)$
and corresponds to the conductivity on the (111) kagome lattice. We then found
that in absence of the N-site $\sigma_{xy}$ = 139 S$\cdot$cm-1 ($\sigma_{111}$
= 241 S$\cdot$cm-1) whereas in the Mn3NiN is $\sigma_{xy}$ = 78 S$\cdot$cm-1
($\sigma_{111}$ = 135 S$\cdot$cm-1), both at the $E_{F}$ level,. Our findings
show a considerable increase of the $\sigma_{xy}$ in the nitrogen-deficient
antiperovskite that can be correlated to the increase of the available
electronic states close to Fermi. The latter enhancement of the $\sigma_{xy}$
component is in agreement with the electronic band structure, also shown and
discussed before. Therefore, the N-site is directly influencing the
$f_{n}(\bf{k})$ function into Eq. 1 modifying the $\sigma_{xy}$ value but
keeping the symmetry operations.

Table 1: Bader charges, in $e^{-}$ units, computed for the Mn, Ni, and N sites in the Mn3NiN and Mn3Ni considering the chiral antiferromagnetic $\Gamma_{4g}$ ordering. The latter values were extracted under several exchange-correlation representations. Additionally, we present the magnetic moment, per Mn atom, in each case. XCPBEsol | $Z_{Mn}$ | $Z_{Ni}$ | $Z_{N}$ |  $m$ ($\mu_{B}$$\cdot$Mn-1)
---|---|---|---|---
Mn3NiN |  $+$0.907 |  $-$0.723 |  $-$2.006 | 3.560
Mn3Ni |  $+$0.259 |  $-$0.704 | — | 3.583
XCSCAN | $Z_{Mn}$ | $Z_{Ni}$ | $Z_{N}$ |  $m$ ($\mu_{B}$$\cdot$Mn-1)
Mn3NiN |  $+$0.806 |  $-$0.745 |  $-$1.957 | 3.418
Mn3Ni |  $+$0.262 |  $-$0.788 | — | 3.470
XCHSE06 | $Z_{Mn}$ | $Z_{Ni}$ | $Z_{N}$ |  $m$ ($\mu_{B}$$\cdot$Mn-1)
Mn3NiN |  $+$0.973 |  $-$0.790 |  $-$2.130 | 3.854
Mn3Ni |  $+$0.324 |  $-$0.718 | — | 3.883

In regards to the electronic structure, formally, the oxidation states
according to the IUPAC 64, 65, 66, quantifies the oxidation degree of an atom
defined based-on the electron counting of such atomic species after the
bonding is reached. Therefore, the oxidation number can be obtained by
following a set of rules, as exposed by A. Walsh _et al._ 67, 68 that, as
mentioned before, can be ascribed to the electron counting. Then, aiming to
estimate the potential oxidation states, hold by each atomic component in the
antiperovskite, we proceded by obtaining the charges around each site. As in
the Mn3NiN and Mn3Ni antiperovskites, the electronic structure is metallic,
the Born effective charges, $Z^{*}$, are not accessible due to the ill-defined
polarization in metals***The $Z^{*}$ tensor is defined as
$Z_{\alpha\beta,\kappa}^{*}$ = $\frac{\partial
P_{\beta}}{\partial\tau_{\alpha,\kappa}}|_{\mathcal{E}=0}$ where $\alpha$ and
$\beta$ are the cartesian coordinates, $P_{\beta}$ is the polarization and
$\tau$ are the atomic displacements of the $\kappa$ atom., and therefore, the
Bader charges offer an alternative route to estimate the charges in the atomic
species. In Table 1 are condensed the results related to the Bader charges
computed in the Mn3NiN and Mn3Ni antiperovskites. These values were obtained
for the PBEsol$+U$, SCAN, and HSE06 exchange-correlation approaches. We can
observe that, independently of the exchange-correlation considerations,
following the Jacob’s ladder from the GGA$+U$ to the hybrid-functional
approach 69, the computed charges are close to $+$0.9$e^{-}$, $-$0.7$e^{-}$,
and $-$2.0$e^{-}$ for the Mn, Ni, and N sites, respectively, in the Mn3NiN
case. The previous charges are in contrast with the computed charges of
$+$0.3$e^{-}$ and $-$0.7$e^{-}$ for Mn and Ni in the Mn3Ni. These results are
suggesting a good representation of the charges with the PBEsol$+U$ approach.
Thus, the PBEsol$+U$ exchange-correlation is used for further analysis. As
expected, the Mn-sites hold, in both antiperovskite cases, a positive charge
associated with a Mnα+ oxidation state. Meanwhile, the corner Ni-site shows a
negative charge leading to a Niβ- oxidation state. In the nitrogen case, as
expected the Bader charge is negative and it is related to the Nδ-, ($\delta$
= 3) oxidation state expected in this anionic site. Interestingly, the Mn’s
Bader charge is $+$0.259$e^{-}$ in the Mn3Ni whereas is $+$0.907$e^{-}$ in the
Mn3NiN. This can be explained due to the charge localized in the nitrogen site
when incorporated in the antiperovskite and that it is transferred from the
manganese sites. Moreover, this result is in agreement with larger electronic
states, close to the Fermi level, available in the Mn3Ni in comparison to
Mn3NiN, and also explaining the AHC results. Aiming to compare with other
insulating antiperovskites, such as Ca3SnO and Ca3BiN, we have computed the
Bader charges and found that $Z_{Ca}$ = $+$1.308$e^{-}$, $Z_{Sn}$ =
$-$2.364$e^{-}$, and $Z_{O}$ = $-$1.558$e^{-}$ for the Ca3SnO oxide, and
$Z_{Ca}$ = $+$1.333$e^{-}$, $Z_{Bi}$ = $-$1.955$e^{-}$, $Z_{N}$ =
$-$2.043$e^{-}$, for the Ca3BiN nitride antiperovskite case. As for Born
effective charges, $Z^{*}$ accessible in these compounds, we observed that the
diagonal terms are $Z^{*}_{Ca}$ = $+$2.388$e^{-}$, $Z^{*}_{Sn}$ =
$-$3.023$e^{-}$, and $Z^{*}_{O}$ = $-$3.381$e^{-}$ in the Ca3SnO oxide, and
$Z^{*}_{Ca}$ = $+$2.380$e^{-}$, $Z^{*}_{Bi}$ = $-$2.899$e^{-}$, $Z^{*}_{N}$ =
$-$4.397$e^{-}$ for the Ca3BiN nitride. The deviation of the Born effective
charges, with respect to the nominal values ($Z_{Ca}$ = $+$2$e^{-}$, $Z_{Sn}$
= $-$4$e^{-}$, $Z_{Bi}$ = $-$3$e^{-}$, $Z_{O}$ = $-$2$e^{-}$ and $Z_{N}$ =
$-$3$e^{-}$,), can be explained in terms of the large polarizability of the
Sn–O and Bi–N bondings widely observed and reported in ferroelectric
perovskite oxides 70, 71. Despite the charge underestimation, shown by the
Bader analysis, and overestimation, obtained with the Born effective charges,
the latter results are in fair agreement with the expected oxidation states of
$A_{3}^{2+}B^{4-}$O2- and $A_{3}^{2+}B^{3-}$N3- compounds, respectively. As
such, these findings are consistent with the experimentally measured, by
Mössbauer spectroscopy and X-ray photoemission spectroscopy, XPS, oxidation
states of the atomic constituents in the Sr3SnO and Sr3PbO antiperovskites 22,
23. In such compounds, the oxidation state was associated with Sn4- and Pb4-
states based on the experimental results. Additionally, these results on the
oxidation states are also in agreement with the calculations in other
antiperovskite insulators such as Ba3SiO and Ba3SiO/Ba3GeO ferroelectric
superlattices in which, the $Z^{*}$ values are $+$2.396$e^{-}$,
$-$4.720$e^{-}$, $-$4.594$e^{-}$, and $-$2.801$e^{-}$ for the Ba, Si, Ge, and
O sites, respectively 11, 72.

To contrast the obtained oxidation states in the antiperovskite Mn3NiN, we
defined a hypothetical perovskite compound as NiMnN3 by inverting the Mn and N
sites. Thus, the Mn occupies the octahedral center whereas the N sites form
the octahedra, $i.e.$ MnN6. Here, the Ni sites remain in the cell’s corner
site. In such a compound, we have fully relaxed the structural and electronic
degrees of freedom and found a metallic behavior with a tangible magnetic
response in which, the Mn holds a $m$ = 2.501 $\mu_{B}$$\cdot$Mn-1 and the Ni
is $m$ = $-$1.080 $\mu_{B}$$\cdot$Ni-1. After extracting the Bader charges we
obtained values of $Z_{Mn}$ = $+$1.981$e^{-}$, $Z_{Ni}$ = $+$0.888$e^{-}$,
$Z_{N}$ = $-$0.957$e^{-}$. As expected, the $A^{+}B^{+}X_{3}^{-}$ is kept as
Niα+Mnβ+N${}_{3}^{\delta-}$. It is worth mentioning that the Bader charges
seem to underestimate the computed charge in the atomic sites, as observed,
for example in Ca3BiN possibly due to the partitioning methodology and
exchange-correlation approach 73. Nevertheless, it can be concluded that as
perovskite and antiperovskite structures are considered, the Ni-site oxidation
state is reversed from positive, in the former, to negative in the latter.

Table 2: Computed Bader charges, in $e^{-}$ units, for the Mn, $B$-sites, and N sites in the Mn${}_{3}B$N within the chiral antiferromagnetic $\Gamma_{4g}$ ordering. Here, we also present the magnetic moment, per Mn atom in each case as well as the electronic configuration for each $B$-site with the outer valence electrons in neutral state. In the Mn and Ni cases, the outer electrons are [Ar]4s23$d^{5}$ and [He]2$s^{2}$2$p^{3}$, respectively. Mn${}_{3}B$N | $Z_{Mn}$ | $Z_{B}$ | $Z_{N}$ |  $B$-site
---|---|---|---|---
Mn3NiN |  $+$0.907 |  $-$0.723 |  $-$2.006 | Ni:[Ar]4$s^{2}$3$d^{8}$
Mn3PdN |  $+$0.884 |  $-$1.023 |  $-$1.629 | Pd:[K]5$s^{0}$4$d^{10}$
Mn3PtN |  $+$0.982 |  $-$1.351 |  $-$1.596 | Pt:[Xe]6$s^{1}$5$d^{9}$
Mn3IrN |  $+$0.946 |  $-$1.273 |  $-$1.566 | Ir:[Xe]6$s^{2}$5$d^{7}$
Mn3NiN |  $+$0.907 |  $-$0.723 |  $-$2.006 | Ni:[Ar]4$s^{2}$3$d^{8}$
Mn3CuN |  $+$0.754 |  $-$0.601 |  $-$1.661 | Cu:[Ar]4$s^{1}$3$d^{10}$
Mn3ZnN |  $+$0.682 |  $-$0.391 |  $-$1.656 | Zn:[Ar]4$s^{2}$3$d^{10}$
Mn3GaN |  $+$0.593 |  $-$0.136 |  $-$1.643 | Ga:[Ar]4$s^{2}$3$d^{10}$4$p^{1}$
Mn3SnN |  $+$0.661 |  $-$0.355 |  $-$1.630 | Sn:[Kr]5$s^{2}$4$d^{10}$5$p^{2}$

Moving forward, we applied our analysis to several members of the Mn${}_{3}B$N
family in order to extrapolate our findings. In Table 2 we present our
calculations of the Bader charges across several reported Mn-based
antiperovskites. In all the cases, the chiral antiferromagnetic $\Gamma_{4g}$
ordering was considered as the magnetic ground state in the calculations. As
expected, the N-sites remain negative with values between $-$2.0$e^{-}$ to
$-$1.6$e^{-}$ whereas the Mn-sites vary from $+$0.9$e^{-}$ to $+$0.6$e^{-}$.
Thus, we followed the trend vertically along the periodic table from the 3$d$
in Ni to 5$d$ in Pt and horizontally from Ni to Ga. We found an increase of
the charge at the $B$-site, from Ni to Pt, suggesting an increase of the
negative oxidation state, for example, $\beta$ $\sim$ 1$-$ in Ni to $\beta$
$\sim$ 2$-$ in Pt. In the Mn3IrN case, the Bader charge is close to the value
observed for the Pt site. Interestingly, the received charge could be possibly
located at the open $s$\- and $d$-orbitals. On the contrary, the charge
decreases from Ni to Ga possibly due to the tendency, in this case, to keep
the outer electronic shells closed. As for the magnetic moment value, per Mn
site, remains in values between 3.5 $\mu_{B}$$\cdot$Mn-1 to 3.8
$\mu_{B}$$\cdot$Mn-1. In contrast, we observed that the charge at the
$B$-sites decreases when going from Ni to Ga. This can be explained in terms
of the closing electronic shell limiting the space for acquired charge and
therefore, diminishing the possible negative oxidation state.

Although we are aware of the underestimated charge obtained through the Bader
charges approach, and fluctuations in the values of the charges could be
expected due to the partitioning method employed 73, 68, our findings
consistently suggest negative oxidation states in metals when located at the
antiperovskite’s corner $B$-site. Finally, it is worth noticing that as the
spin-orbit coupling, SOC, increases with the oxidation state 74, the
$B$-site’s oxidation is paramount to understanding its contribution to the
anomalous Hall conductivity. Thus, in the case of the Mn${}_{3}B$N compounds,
the SOC possibly decreases due to the negative oxidation state in the $B$-site
when compared with perovskite compounds.

##  4 Conclusions:

In this paper, we have studied the electronic structure of the Mn3NiN and
Mn3Ni magnetically chiral noncollinear antiferromagnetic antiperovskites by
utilizing first-principles calculations. We found that the N-site expands the
cell when it is located at the center of the octahedral. Nonetheless, due to
the center position, the symmetry operations and expected properties are
conserved. We observed a tangible increase of the available electronic states
close to the Fermi level that favors the conductivity, as in the case of the
anomalous Hall effect in which, $\sigma_{xy}$ = 139 S$\cdot$cm-1
($\sigma_{111}$ = 241 S$\cdot$cm-1) in absence of the N-site in contrast to
$\sigma_{xy}$ = 78 S$\cdot$cm-1 ($\sigma_{111}$ = 135 S$\cdot$cm-1) in the
Mn3NiN counterpart. Our findings suggest that the nitrogen inclusion in the
Mn3NiN system enhances a positive oxidation state, possibly $\sim$1$+$, in the
Mn whereas, and more interestingly, the Ni sites hold a negative, potentially
$\sim$1$-$, oxidation state. This behavior is observed although the overall
electronic structure remains metallic. Finally, our findings also suggest that
several transition metals may exhibit negative oxidation states when located
at the $B$-site in the Mn${}_{3}B$N antiperovskite. We thus hope that our
result will motivate further studies in antiperovskite structures that might
be ideal candidates to further investigate the negative oxidation states in
metals.

## Author contributions

All of the authors were involved in the preparation and development of the
manuscript. Moreover, all of the authors read and approved the final
manuscript.

## Conflict of interest

The authors declare no personnel or financial conflict of interests with any
person or organization.

## Acknowledgments:

The calculations presented in this paper were carried out using the Grid UIS-2
experimental testbed, being developed under the Universidad Industrial de
Santander (SC3-UIS) High Performance and Scientific Computing Centre,
development action with support from UIS Vicerrectoría de Investigación y
Extension (VIE-UIS) and several UIS research groups as well as other funding
resources. Additionally, we acknowledge the computational support extended to
us by Laboratorio de Supercomputo del Sureste de México (LNS), Benemérita
Universidad Autónoma de Puebla, BUAP, for performing heavy theoretical
calculations. A. C. Garcia-Castro acknowledge the grant No. 2677 entitled
“Quiralidad y Ordenamiento Magnético en Sistemas Cristalinos: Estudio Teórico
desde Primeros Principios” supported by the VIE – UIS.

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