# First-principles Investigation of Electrides Derived from Sodalites

Byungkyun Kang  Department of Physics and Astronomy, University of Nevada, Las
Vegas, NV 89154, USA  Qiang Zhu<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Nevada, Las Vegas, NV 89154, USA

###### Abstract

Recently, the electride materials, with excess anionic electrons confined in
their empty space, have received a growing attention due to their promising
applications in catalysis, nonlinear optics and spin-electronics. However, the
utilization of electride materials is limited by their thermal instability.
Here we introduce an alternative way to achieve the localized anionic electron
states via the removal of high symmetric Wyckoff sites of anions from the
existing sodalite compounds. Using four halide sodalites as the parental
structures, our simulation reveals that the materials after the removal of
anionic halide sites exhibit typical electride behaviors that are
characterized by the existence of localized electronic states near the Fermi
level. Compared to most previously studied electrides, these materials are
expected to be more thermally stable due to the complex structural framework
and thus promising for practical applications. Among them, Na4(AlSiO4)3
manifests magnetic electronic structure. We demonstrate that this magnetism
originates from a highly localized excess electron state surrounded by
electronpositive alkaline cations. Our results suggest Na4(AlSiO4)3 could be a
promising spintronics component, thus encouraging further experimental study.

##  I INTRODUCTION

Electides represent a unique class of materials in which the excess electrons
are trapped inside crystal cavities and serve as the anions Dye (1990); Li and
Mahanti (2004); Dye (2009); Hosono and Kitano (2021); Liu  _et al._ (2020).
Localization of electrons provides the early examples of quantum confinement
Dye (2009). These trapped electrons are usually loosely bound and float around
Fermi level to form unique interstitial energy bands Inoshita  _et al._
(2021), giving rise to interesting transport and magnetic behaviors.
Therefore, electrides have attracted growing attention for various material
applications. While the first crystalline organic electride was made by Dye
and coworkers in 1983 Ellaboudy  _et al._ (1983), the use of organic
electrides for practical applications was limited by their thermal instability
Dye (2009). In 2003, Hosono and coworkers reported the synthesis of the first
thermally stable electride based on the inorganic mineral mayenite
(12CaO$\cdot$7Al2O3) Matsuishi  _et al._ (2003). The resulting electride
Ca6Al7O16 (C12A7:2e-), with the excess electrons confined in the zero-
dimension (0D) cages, exhibit excellent thermal stability and low reactivity
with air, opening a new chapter in electride synthesis and utilization. The
discovery of C12A7:2e- has stimulated many new efforts to search for other
inorganic electrides. Recently, a number of new electrides with electrons
confined in different geometries, including 2D (Ca2N Lee  _et al._ (2013), Y2C
Zhang  _et al._ (2014)), 1D (Y5Si3 Lu  _et al._ (2016), Sr5P3 Wang  _et al._
(2017), Mn5Si3 Zhang  _et al._ (2017a), La8Sr2(SiO4)6 Zhang  _et al._ (2015)),
and 0D (YH2 Mizoguchi  _et al._ (2016)), have been increasingly investigated
in experiment. In the mean time, many new materials with improved
functionality were proposed theoretically thanks to the recent advances in
computer simulation Inoshita  _et al._ (2014, 2015); Tada  _et al._ (2014);
Ming  _et al._ (2016); Zhang  _et al._ (2017b); Burton  _et al._ (2018); Wang
_et al._ (2019); Zhu  _et al._ (2019a); Qu  _et al._ (2019); Zhu  _et al._
(2019b); Li  _et al._ (2021).

Figure 1:  Two representative porous structures consisting of anions at the
center of the cages. (a) mayenite Ca${}_{1}2$Al${}_{1}4$O33 and (b) sodalite
Na4(AlSiO4)3Cl. Both structures are featured by the presence of anions
(partially) occupying the Wyckoff site around the center of cages. The left
panels highlight the overall distribution of cages, while the right panels
shows the chemical environment of each representative cage. In each cage, the
coordination environment of the center anion is also displayed by the red dash
lines. For clarity, only 6 out of 12 cages are shown for the mayenite
structure.

Despite the growing number of reports on identifying new electride materials,
most of them are only stable at low temperatures and very sensitive to air and
moisture. To date, C12A7:2e- remains one of the only room-temperature stable
electrides from practical application perspective, with the exception of
several recent reports Liu  _et al._ (2022); Yang  _et al._ (2021); Kang  _et
al._ (2020). Therefore, it is necessary to examine the uniqueness of C12A7:2e-
from the viewpoint of structural stability. The original composition of
mayenite can be represented as Ca12Al14O33, which is an air-stable neutral
material at ambient temperature and atmosphere with a melting point of 1415 ∘C
Hosono and Kitano (2021). Its crystal structure has a cubic body centered unit
cell, consisting of four positively charged [Ca6Al7O16]+ as the main framework
with the following Wyckoff sites: Ca (24d), Al1 (16c), Al2 (12a), O1 (48e), O2
(16c) Boysen  _et al._ (2007). As shown in Fig. 1a, this framework constitutes
twelve cages with a free inner space of 4-6 Å in diameter in the unit cell. In
addition, there exist another type of oxygen atoms (O3) that are statistically
distributed inside centers of the twelve cages, which can be regarded as the
partial occupancy of the 12b Wyckoff site with a ratio of $\frac{1}{6}$.
Inside the cage, each O3 site is collinearly coordinated to two Ca atoms with
a distance of 2.77 Å, which is a bit higher than 2.40 Å in the rocksalt CaO
compound. The resulting two oxygen may be regarded as the counteranion (O2-)
to the giant framework cation. Due to the long Ca-O distance, it is expected
that O2- anions are loosely bound to the cages Matsuishi  _et al._ (2003).
When the material is thermally treated in a reductive atmosphere, part of the
loosely bonded O2- ions may escape as gaseous O2 from the lattice and leave
electrons in the cages, considered as the electron captured oxygen vacancy, or
F-center. With the proper reductive condition, all of the loosely bonded O2-
anions can be removed from the lattice. In this case, the relatively free
electrons play the role of anions and result in a high conductivity. In the
meantime, the unpaired feature of these free electrons also generate
nontrivial magnetic Matsuishi  _et al._ (2003) and superconducting properties
Miyakawa  _et al._ (2007). Thanks to its excellent stability, C12A7:2e- has
been used for ammonia synthesis Kitano  _et al._ (2012); Kuganathan  _et al._
(2014); Hayashi  _et al._ (2014) and as an electron-injection barrier material
Hosono  _et al._ (2017).

Inspired by the synthesis of C12A7:2e-, we proceed to check the feasibility of
designing new electrides from the existing minerals sharing structural
features similar to mayenite that allow the localization of anionic electrons.
Clearly, the most important characteristic of a good precursor is the presence
of anions occupying the high-symmetry Wyckoff site. In the entire materials
database, there exist many compounds meeting such a requirement. In this work,
we focus on a common type of sodalite minerals consisting of the halide anions
at the high symmetry Wyckoff sites (see Fig. 1b). Similar to mayenite, the
sodalite has a high melting point of 1079 ∘C Antao and Hassan (2002). Its
crystal structure also contains the porous framework with two cages centering
around the corner and body center. The halide anions ($X^{-}$) occupy the cage
centers at the 2a Wyckoff site, exhibiting a tetragonal coordination with the
neighboring alkaline metals $M^{+}$ (e.g., Na, K). The resulting $M$-$X$
distance is about the same bond length as that in the typical ionic compound.
For instance, the Na-Cl contact in sodalite is 2.76 Å, which is close to 2.79
Å in the rocksalt NaCl compound. Due to the large dimeter size, it is possible
to form many diffusion pathway if the $X$ anion is removed from the $M_{4}$
tetrahedron to realize the electride state. Using these materials as the
precursor, we immediately found that the so called black sodalite Na4(AlSiO4)3
has been found in our recent screening work Zhu  _et al._ (2019b). In this
work, we checked several thermal-stable sodalite compounds and derived the
likely crystalline electride structures via rigorous structural analysis.
Based on these structures, we further performed a thorough computational
investigation on their electronic and magnetic properties based on first-
principles calculations. Our results suggest that the electrides derived from
sodalite are likely to achieve similar properties as found in C12A7:2e-. We
hope this work will encourage more investigations of porous zeolites as the
precursor to synthesize more thermal stable electrides in the future.

##  II Computational Methods

In this work, we selected four materials for detailed study, including
Na4(AlSiO4)3Cl, Na4(AlSiO4)3Br, Na4(AlGeO4)3I and Na4(GaSiO4)3I. For each of
them, we first created the new structures by removing the halide anions in the
unit cell. The structures are subsequently relaxed to obtain the optimum cell
parameters. The structural stability was then checked by phonon calculation.
Whenever the imaginary frequencies were found, we followed the largest soft
mode to obtain a new structure with a subgroup symmetryKang and Estreicher
(2014).

Figure 2:  The optimized crystal structures of four sodalite materials.

We used the projector augmented wave (PAW) method with the plane-wave code
VASP Blöchl (1994); Kresse and Furthmüller (1996); Kresse and Joubert (1999)
to carry out all calculations with the framework of density functional theory
(DFT). The Generalized Gradient Approximation (GGA) with Perdew, Burke and
Ernzerhof (PBE) functional Perdew  _et al._ (1996) was adopted. We have used a
3$\times$3$\times$3 $\Gamma$-centered k-point grid for geometric relaxation of
unit cell, which contain two formula units. The cutoff energy for all
calculations is 520 eV, and forces are converged within 0.03 eV/Å. Using the
optimized structure, we constructed a 2$\times$2$\times$2 supercell (368
atoms) for the F-center defect (with the halide anion vacancy) calculation, in
which a single $\Gamma$ point was used for sampling the Brillouin zone. Phonon
bands were calculated using the force constants, which were calculated by the
finite displacement methods using the Phonopy code Togo  _et al._ (2008).

##  III Results and Discussions

###  III.1 Pristine Sodalites

Sodalite is a common structural type for many nature-occurring minerals. The
typical crystal structure possesses a cubic symmetry in space group
$P\bar{4}3n$ (218). As shown in Fig. 2, each unit cell contain two formula
units. According to X-ray diffraction analysis Pauling (1930); McMullan  _et
al._ (1996); Hassan  _et al._ (2004), the mineral Na4(AlSiO4)3Cl is
characterized by a $\beta$-cage made of four-membered (Al, Si)O4 rings on the
(100) plane linked together in six-membered rings parallel to (100) plane.
This results in a highly symmetric framework with just five atoms in the
asymmetric unit. Other materials, including Na4(AlSiO4)3Br, Na4(AlGeO4)3I and
Na4(GaSiO4)3I, show the same packing behavior except a slight difference in
terms of lattice parameters.

Table 1: The comparison of cell parameters and band gaps for sodalite materials between simulation and experimental values. System |  Cell Parameters (Å) | Band Gap (eV)
---|---|---
DFT | Expt. |  DFT |  Expt.
Na4(AlSiO4)3Cl | 8.973 | 8.876 Nielsen  _et al._ (1991) | 4.69 | 6.1 Van Doorn  _et al._ (1972)
Na4(AlSiO4)3Br | 9.031 | 8.932 Nielsen  _et al._ (1991) | 4.61 | 5.9 Van Doorn  _et al._ (1972)
Na4(AlGeO4)3I | 9.287 | 9.174 Johnson and Weller (1997) | 3.39 |
Na4(GaSiO4)3I | 9.198 | 9.074 Johnson and Weller (1997) | 3.59 |

Table 1 summarizes the calculated lattice parameters, which agree well with
the experimental reports. Using the dispersion corrected calculations with the
DFT-D3 method Grimme  _et al._ (2010), we found that the relaxed lattice
parameter of Na4(AlSiO4)3Cl is 8.897 Å, which is approximately 0.1 Å smaller
than the value with the standard PBE functional. This indicates a minor effect
of dispersion correction in the sodalites. Our results are close to other DFT
results in which the calculated lattice parameters are 8.721Mofrad  _et al._
(2018), 8.91Thomson and Wentzcovitch (1998), and 8.78 ÅUlian and Valdrè
(2022).

Figure 3:  The electronic band structures and atomic projected density of
states of (a) Na4(AlSiO4)3Cl, (b) Na4(AlSiO4)3Br, (c) Na4(AlGeO4)3I, and (d)
Na4(GaSiO4)3I. In each band structure, blue solid lines represent spin up
bands and red dashed lines represent spin down bands. In the plots of density
of states, two different scales of density of states are used: one below 3 eV
and one above 3 eV.

Fig. 3 shows the band structures and density of states (DOS) of all four
sodalite materials. Clearly, none of their electronic structures show spin
polarization in their band structures. This means that the electron energy
levels are not affected by the spin of the electron. In addition, all four
compounds exhibit direct band gaps at the $\Gamma$ point. The calculated band
gaps are also summarized in Table 1. For Na4(AlSiO4)3Cl, we obtained a gap of
4.69 eV, similar to other DFT studies (e.g., 4.39 eV Pan  _et al._ (2016);
Filippone  _et al._ (1995) and 5.0 eV Cano  _et al._ (2010)). All of these
values are smaller than the experimentally reported 6.1 eV Van Doorn  _et al._
(1972). This is expected since DFT systematically underestimates the energy
gap. Similar trends were also observed in other compounds.

From the DOS analysis, the computed valence bands near Fermi level in all four
sodalite structures are very flat valence bands, with the contribution of
O-$p$ orbitals. On the contrary, their conduction bands are rather dispersive,
which are very similar to that of mayenite Hosono (2013). For Na4(AlSiO4)3Cl,
its valence band maximum (VBM) consists of large O-$p$ and small Cl-$p$
orbitals, while Na4(AlSiO4)3Br has a VBM comprised of large Br and small O
orbitals. This is consistent with the fact that the Br is more electronegative
than Cl. For Na4(AlGeO4)3I and Na4(GaSiO4)3I, their highest valance bands are
mainly comprised of I orbitals and are separated from the other valence bands.
It can be explained by the lowest electronegativity of iodine among the
halides (I, Br, Cl), which leads to less hybridization with O-$p$ and a higher
energy level.

###  III.2 The Structures of Crystalline Electrides

Figure 4:  The computed phonon spectrum for three different electrides (a)
Na4(AlSiO4)3, (b) Na4(AlGeO4)3, and (c) Na4(GaSiO4)3. In (a), the left panel
shows phonon dispersion curves of relaxed Na4(AlSiO4)3, which has a soft mode
shown in the middle panel. The right panel shows phonon dispersion curves of
relaxed Na4(AlSiO4)3 after applying atomic displacements according to the
depicted soft mode.

Prior to investigation of electride, an electron trapped at the site of an
anion vacancy in ionic crystals is known as an F-center in solid state
physics. F-center is a point defect and its concentration is so small that
interaction between them is negligible. On the other hand, electrides may be
regarded as materials having a stoichiometric F-center, and the electron
concentration in the electride is high enough to allow them to interact with
each other Hosono and Kitano (2021). In the past, most works attempted to
derive the crystal structure by directly removing the anions from the parent
crystal Srdanov  _et al._ (1998); Madsen  _et al._ (2001). However, such
treatment failed to take into account the likely structural changes due to
complete removal of anions (instead of low-concentration point defects).
Therefore, it is important to examine the resulting crystalline electride in a
more rigorous manner.

Table 2: The simulated crystallographic data for three crystalline electrides.
Na4(AlSiO4)3, Space group $P$23, a = 8.98 Å
---
Atomic coordinates
Na1 |  4e |  0.72 |  0.72 |  0.72
Na2 | 4e | 0.17 | 0.17 | 0.17
Al | 6g | 0.75 | 0.00 | 0.50
Si | 6h | 0.25 | 0.50 | 0.00
O1 | 12j | 0.36 | 0.05 | 0.65
O2 | 12j | 0.86 | 0.85 | 0.45
Na4(AlGeO4)3, Space group $P$23, a = 9.28 Å
Atomic coordinates
Na1 | 4e | 0.18 | 0.18 | 0.18
Na2 | 4e | 0.70 | 0.70 | 0.70
Al | 6g | 0.25 | 0.00 | 0.50
Ge | 6h | 0.75 | 0.50 | 0.00
O1 | 12j | 0.65 | 0.93 | 0.65
O2 | 12j | 0.85 | 0.85 | 0.43
Na4(GaSiO4)3, Space group $P\bar{4}3n$, a = 9.09 Å
Atomic coordinates
Na | 8e | 0.18 | 0.18 | 0.18
Ga | 6d | 0.25 | 0.00 | 0.50
Si | 6c | 0.25 | 0.50 | 0.00
O | 24i | 0.57 | 0.87 | 0.15

As we discussed earlier, we start by removing the halide anions from the
parental crystal structures. Since Na4(AlSiO4)3Cl and Na4(AlSiO4)3Br share the
same framework, this lead to three candidate structures for further
consideration. After the structural relaxation, we found that there exist
notable imaginary frequencies in the computed phonon spectrum of Na4(AlSiO4)3
and Na4(AlGeO4)3. As shown in the left panel of Fig. 4a for Na4(AlSiO4)3, the
most negative phonon frequency is located at the $\Gamma$ point. Therefore, We
derived a new structure by displacing atoms along the eigenvector of the soft
mode (see the middle panel of Fig. 4a) with the maximum magnitude of 0.2 Å on
the Na atoms. Then, the new structure was relaxed again. We repeated this
process until the absolute minimum of the potential energy surface was reached
and no imaginary frequency phonon occurred. The similar method has been used
by the authors in a previous study Kang and Estreicher (2014). After these
treatments, we obtained new structures with the energy drops of 71 and 4 meV
per formula unit for Na4(AlSiO4)3 and Na4(AlGeO4)3, respectively. On the other
hand, Na4(GaSiO4)3 retains the same space group symmetry as found in the
original sodalite form.

Due to the soft modes, the original Na (8e) and O (24i) sites in space group
$P\bar{4}3n$ were split into the 4e and 12j sites in space group $P23$.
Therefore, the final stable structure for these two electrides has a subgroup
symmetry $P$23 (195). Table 2 summarizes the crystallographic information for
all three structures. Compared to Table 1, one can clearly found that the cell
parameters nearly remains the same regardless of either the symmetry change or
atomic removal.

###  III.3 Electronic Properties

Figure 5:  The electronic band structures and isosurfaces of partial charge
density in the range of -2 $<$ $E$ \- $E_{\mathrm{F}}$ $<$ 0 eV of (a)
ferromagnetic Na4(AlSiO4)3, (b) antiferromagnetic Na4(AlSiO4)3, (c)
Na4(AlGeO4)3, and (d) Na4(GaSiO4)3. In each band structure, blue solid lines
represent spin up bands and red dashed lines represent spin down bands.

Next, we investigated the electronic properties of the resulting three
crystalline electrides. In particular, the Cl removed electride Na4(AlSiO4)3
corresponds to the previously investigated black sodalite, in which each
periodic $\beta$-cage contains one Na43+ cluster and its color is black due to
a metallic band structure Srdanov  _et al._ (1992); Smeulders  _et al._
(1987); Blake  _et al._ (1996). As shown in Fig. 5, the three structures
manifest electride bands inside the gap of the parent sodalite. The partial
charge density obtained in the energy range $-2<E-E_{\mathrm{F}}<0$ is
centered inside the cage after the removal of halide anions, indicating that
the excess electrons are localized at the crystal cages and forming the
electride bands. There are sizable contributions of oxygen orbitals in the
charge density, giving rise to relatively dispersed electride bands in
comparison to the valence bands of the parent compounds. Therefore, the
dispersion of the electride bands can be attributed to the hybridization
between localized excess electrons in Na4 tetrahedron and oxygen orbitals.

Figure 6:  The computed wavefunctions for pristine Na4(AlSiO4)3Cl, F-center of
Cl vacancy in the supercell of Na4(AlSiO4)3Cl, and electride states in
Na4(AlSiO4)3. F and E denote the F-center and electride state, respectively.
In the left panel, the magnified views of O-$p$ and O-$s$ wavefunctions are
displayed to aid the visual analysis. Three gray lines denote states that are
triply degenerate. In the right panel, wavefunctions which consist of
electride bands are marked by black boxes.

Recently, Stoliaroff et al. have suggested that high concentrations of
chlorine vacancies can be created by F-center point defects using DFT modeling
Stoliaroff  _et al._ (2021). It was found that ground and excited states of
the singly occupied molecular orbital and the triply degenerate lowest
unoccupied molecular orbital for the F-center Colinet  _et al._ (2020). It was
observed that the F-center is an electron trapped at an halide anion vacancy
in alkali halides Holton and Blum (1962). The trapped electron is in a set of
quantized states similar to a hydrogen atom: the ground state is the atomic
1$s$ state, and the first excited state is composed of the 2$s$ and 2$p$
states Koyama and Suemoto (2011). However, it is shown that hybridization
between metal and electride orbitals leads to large splitting between
electride associated states McRae  _et al._ (2022). Electride associated
states are split into two when hybridization between metal and electride
orbitals occurs. Otto et al. proposed that the particle-in-a-box state is
better described as a particle in a box hybridized with framework oxygen atoms
for the black sodalite Na4(AlSiO4)3 Sankey  _et al._ (1998). In this model,
the particle is confined to a box defined by the oxygen atoms, and the
electron density is spread more uniformly throughout the structure. These
results suggest that the Cl removed electride can be described by periodic
F-center and it may keep dual nature of localized and itinerant electrons.
However, all aforementioned studies on the black sodalite were based on the
dynamically unstable $P\bar{4}3n$ Na4(AlSiO4)3. Thus, we need to investigate
the co-operative localization and hybridization in the ground state $P$23
Na4(AlSiO4)3 electride.

Fig. 6 displays the computed wavefunctions for pristine $P\bar{4}3n$
Na4(AlSiO4)3Cl, F-center of Cl vacancy in the supercell structure, and
crystalline $P$23 Na4(AlSiO4)3. In the pristine structure (see the left panel
of Fig. 6), it shows that the wavefunction of VBM is comprised of O-$p$ while
the CBM is comprised of [O-$s$ \+ Cl-$s$]. The valence band right below the
VBM is comprised of [Cl-$p$ \+ O-$p$], in which the wavefunction manifests a
spherical shape inside of Na4 tetrahedron at the vicinity of Cl. This is a
typical for an ionic bonding in compound like NaCl.

As shown in the middle panel of Fig. 6, the introduction of Cl vacancy gives
rise to the formation of a defect F-center. Like the atomic orbital, the F
center exhibits a set of quantized states similar to a hydrogen atom. The
first F-center associated state appears between the VBM and CBM. In comparison
to the spherical [Cl-$p$ \+ O-$p$] state in the pristine structure, this state
is mainly comprised of localized F-$s$ state in the Na4 tetrahedron and a few
O-$p$ states centered at the O atoms. Around the CBM, we observed the
hybridization of [F-$s$ \+ O-$p$] as the first unoccupied state. At the second
lowest conduction bands, we also found triply degenerate hybridization of
F-$p$ and O-$s$, where a spherical wavefunction inside Na4 tetrahedron
contains an empty region at the center. Similar to F-$s$, the linear
combination of F-$p$ also has a spherical shape, but it has an inner hollow
due to the requirement of orthogonality. Furthermore, the higher energy
conduction bands consist of [F-$s$ \+ (O, Si, Cl)-$s$] and another triply
degenerate [F-$p$ \+ O-$p$].

Finally, the crystalline electride states (as shown in the right panel of Fig.
6) have qualitatively the same properties with F-center. Their electride
states exhibit distinct local $s$\- or $p$-like features similar to F-$s$ and
F-$p$. These E-$s$ and E-$p$ states also have notable hybridization with the
surrounding atomic orbitals. Due to the hybridization, the band width becomes
larger, suggesting a notable excess electron hopping through $\beta$ frames in
the crystalline electride Sankey  _et al._ (1998).

###  III.4 Magnetism of Na4(AlSiO4)3

Figure 7:  The calculated electronic structures of electrides. (a) The total
density of states and wavefunctions of fully occupied band at $\Gamma$ point
for Na4(AlSiO4)3 with two different lattice parameters. (b) Atomic (Ge,Si)
projected density of states of Na4(AlGeO4)3, Na4(AlSiO4)3 (a=8.98 Å), and
Na4(AlSiO4)3 (a=9.28 Å). The spin up density of states are illustrated. The
bottom panel shows isosurfaces of partial charge density in the range of -2
$<$ $E$ \- $E_{\mathrm{F}}$ $<$ 0 eV for Na4(AlSiO4)3 (a=8.98 Å) and
Na4(AlGeO4)3. (c) The electron localization functional maps at (100) plane of
Na4(AlSiO4)3 (a=8.98 Å), Na4(AlSiO4)3 (a=9.28 Å), Na4(AlGeO4)3, and
Na4(GaSiO4)3.

Of the three proposed electrides, Na4(AlSiO4)3 manifests ferromagnetic (FM)
electronic structure. Blake et al. first showed that the antiferromagnetic
(AFM) transition is associated with the incompletely doped regions of the
sample Blake  _et al._ (1996). But no such transition was observed in the
fully doped sample. However, a later experiment by Heinmaa and Lippmass
reported the AFM phase with a Neel temperature of $T_{\mathrm{N}}$ = 54 K for
heavily sodium-doped sodalite Na4(AlSiO4)3 via nuclear magnetic resonance
spectroscopy study Heinmaa  _et al._ (2000). This experiment was also
supported by following theoretical calculations. Sankey et al. reported the
magnetic ground state of Na4(AlSiO4)3 (i.e. fully doped case) by conducting
local-spin-density approximation (LSDA) simulations Sankey  _et al._ (1998).
They showed a lower energy of AFM ordering was favored over the FM ordering.
Our calculations also reveal that AFM configuration has a lower energy of 33
meV per formula unit than FM configuration. As shown in Fig. 5b, the AFM spin
ordering opens a gap, which is an indirect gap of 0.45 eV from R to X point,
which is consistent with the previous study on the $P\bar{4}3n$ (Na,
K)4(AlSiO4)3 Madsen  _et al._ (2001).

In this work, we also investigated the origin of magnetization within FM
configuration to compare existing ferromagnetic electrides Y2C Inoshita  _et
al._ (2015) and Ca5Ga2N4 Wang  _et al._ (2019). The calculated total energy of
spin-polarized Na4(AlSiO4)3 is lower by 42 meV per formula unit than the non-
spin-polarized system. The total magnetization moment is 0.97
$\mu_{\textrm{B}}$ per formula unit. These values are between $\sim$ 15 meV
and 0.38 $\mu_{\textrm{B}}$ for a weak itinerant ferromagnetic electride Y2C
Inoshita  _et al._ (2015), and 65 meV and 1.25 $\mu_{\textrm{B}}$ for Ca5Ga2N4
Wang  _et al._ (2019).

Similar to the previous study Wang  _et al._ (2019), we found that the strong
magnetization of Na4(AlSiO4)3 is due to excess electron localization inside
the Na4 tetrahedron. As shown in Fig. 7a, increasing lattice parameter (length
of Na4 tetrahedron edges) from 8.98 Å (4.51 Å) to 9.28 Å (5.71 Å) results in
less excess electron confinement in the larger Na4 tetrahedron making the
magnetic moments disappear. The electron localization can be determined by the
electron localization function (ELF), which is a measure of the likelihood of
finding an electron in the neighborhood space of a reference electron located
at a given point Becke and Edgecombe (1990). The ELF value is a dimensionless
index. ELF = 1 corresponding to perfect localization and ELF = $\frac{1}{2}$
corresponding to the electron gas. As shown in Fig. 7c, the maximum ELF value
at the cage center for Na4(AlSiO4)3 is 0.99 when a=8.98 Å, which is higher
than 0.94 when a=9.28 Å. For the reference, the corresponding maximum ELF
values are 0.85 for Na4(AlGeO4)3, and 0.96 for Na4(GaSiO4)3. The trend of ELF
values is also consistent with the results on the band width. The calculated
electride spin up band widths are 0.72 eV for Na4(AlSiO4)3 with a=8.98 Å),
1.19 eV for Na4(AlSiO4)3 with a=9.28 Å, 1.39 eV for Na4(AlGeO4)3, and 1.20 eV
for Na4(GaSiO4)3. Clearly, these localizations are strongly associate with the
electride band widths.

For itinerant ferromagnets, magnetism can be understood by the Stoner
criterion of $D(E_{\mathrm{F}})I>1$, where $D(E_{\mathrm{F}})$ is density of
state at the Fermi level and $I$ is the exchange parameter. As shown in Fig.
7a, the sharp $D(E_{\mathrm{F}})$ of Na4(AlSiO4)3 (a=8.98 Å) suggests the
magnetism is raised from the Stoner-type instability. We showed the sharp
$D(E_{\mathrm{F}})$, which associates with the small electride band width, is
due to localization of excess electron. Then, a question is raised for the
non-magnetism of Na4(AlSiO4)3 and Na4(GaSiO4)3. We checked the possible
magnetism of Na4(AlGeO4)3 with smaller lattice parameters to try to localize
the excess electron and increase $D(E_{\mathrm{F}})$. All attempts resulted in
non-spin-polarized electronic structures. This can be attributed to the higher
electronegativity of Ge than Si. As shown in Fig. 7b, owing to the
electronegativity, the Ge’s projected DOS (PDOS) in Na4(AlGeO4)3 is greater
than the Si’s PDOS in Na4(GaSiO4)3 below the Fermi level, while there is no
notable difference in the Si’s PDOS in the two different lattices. The large
metal cation’s electronegativity leads to increasing hybridization McRae  _et
al._ (2022), which results in less localization of excess electrons and a
large band width. This hinders spin polarization of the excess electrons.
Therefore, there is no magnetism in either Na4(AlSiO4)3 or Na4(GaSiO4)3.

##  IV SUMMARY AND CONCLUSIONS

In sum, we present an approach to realize the electride material via the
crystallographic symmetry analysis. Inspired by the synthesis of C12A7:2e-, we
proposed to make potentially stable electrides from thermally stable inorganic
compounds that possess the high symmetry Wyckoff site being occupied by
anions. Using the well-known halide sodalites as the examples, we found that
the materials after the removal of anionic halide anions exhibit typical
electride behaviors that are characterized by the existence of localized
electronic states near the Fermi level. Among them, Na4(AlSiO4)3 manifests
antiferromagnetic electronic structure. We demonstrate that this magnetism
originate from a highly localized excess electron state surrounded by
electronpositive alkaline cations. While some compounds (e.g., Na4(AlSiO4)3)
have been investigated previously due to interest in F-center research Blake
_et al._ (1996); Srdanov  _et al._ (1998); Sankey  _et al._ (1998); Heinmaa
_et al._ (2000); Madsen  _et al._ (2001), our work provided more rigorous
phonon analysis to derive the truly stable ground state structures of
crystalline electrides. Compared to most previously studied electrides, these
materials are derived from the sodalite materials with high melting points.
Therefore, they are expected to be more thermally stable due to the complex
structural framework. Thanks to the improved stability, these predicted
materials are more amenable for fabrication and practical applications. In the
present work, we only investigated several halide sodalite to demonstrate the
proof of concept. The extension to other sodalite type materials with
different anions is the subject of future work. In addition, our work suggests
the connection between porous zeolite and electride since the existence of
crystalline zeolite can naturally provide the crystal cavity to accommodate
the excess electrons. While efforts on making zeolite electrides have been
attempted in the past Petkov  _et al._ (2002); Wernette  _et al._ (2003), the
extraordinary diversity in zeolite structures Baerlocher and McCusker (2017)
warrants many opportunities for future exploration.

## Acknowledgments

This research was sponsored by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, Theoretical Condensed Matter Physics
program and the DOE Established Program to Stimulate Competitive Research
under Award Number DE-SC0021970. The computing resources are provided by
ACCESS (TG-DMR180040) and National Energy Research Scientific Computing Center
(NERSC).

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