# Interplay between altermagnetism and nonsymmorphic symmetries generating
large anomalous Hall conductivity by semi-Dirac points induced anticrossings

Amar Fakhredine ID Institute of Physics, Polish Academy of Sciences, Aleja
Lotników 32/46, PL-02668 Warsaw, Poland  International Research Centre Magtop,
Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46,
PL-02668 Warsaw, Poland  Raghottam M. Sattigeri ID International Research
Centre Magtop, Institute of Physics, Polish Academy of Sciences, Aleja
Lotników 32/46, PL-02668 Warsaw, Poland  Giuseppe Cuono ID
<EMAIL_ADDRESS>International Research Centre Magtop, Institute of
Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw,
Poland  Carmine Autieri ID International Research Centre Magtop, Institute of
Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw,
Poland

###### Abstract

We investigate the interplay between altermagnetic spin-splitting and
nonsymmorphic symmetries using the space group no. 62 as a testbed. Studying
different magnetic orders by means of first-principles calculations, we find
that the altermagnetism (AM) is present in the C-type magnetic configuration
while it is absent for the G-type and A-type configurations due to different
magnetic space group types. The nonsymmorphic symmetries constrain the system
to a four-fold degeneracy at the border of the Brillouin zone with semi-Dirac
dispersion. In the case of large hybridization as for transition metal
pnictides, the interplay between AM and nonsymmorphic symmetries generates an
intricate network of several crossings and anticrossings that we describe in
terms of semi-Dirac points and glide symmetries. When we add the spin-orbit
coupling (SOC), we find a Néel-vector dependent spin-orbit splitting at the
time-reversal invariant momenta points since the magnetic space groups depend
on the Néel vector. The magnetic space group type-I produces antiferromagnetic
hourglass electrons that disappear in the type-III. When the Néel vector is
along x, we observe a glide-protected crossing that could generate a nodal-
line in the altermagnetic phase. The SOC splits the remaining band crossings
and band anticrossings producing a large anomalous Hall effect in all
directions excluding the Néel-vector direction.

##  I Introduction

Until a few years ago, the two collinear magnetic phases were known in
condensed matter physics as ferromagnetism and antiferromagnetism displaying
completely different properties. Very recently, a new variant of the collinear
antiferromagnetism was discovered called altermagnetism (AM) or collinear
antiferromagnets with non-interconvertible spin-structure motif pairŠmejkal
_et al._ (2022a, b); Yuan and Zunger (2023); Hayami  _et al._ (2019, 2020);
Chen  _et al._ (2023) which hosts both properties of the ferromagnets and
usual antiferromagnets. Unlike ferromagnetism, where the crystal has a net
magnetization and there is time-reversal symmetry (TRS) breaking, and unlike
antiferromagnetism, where the total magnetization is zero, altermagnetism
hosts systems in which the magnetization in the real space is zero but there
is breaking of the spin degeneracy in the reciprocal space like in
ferromagnetic compounds. The condition to observe the AM is the absence of a
translation, an inversion or a combination of both that maps the spin-up
charge to the spin-down charge. In this case, only a roto-translation or
mirror can map the spin-up charge in the spin-down charge. From the point of
view of group theory, the altermagnetic compounds must belong to type-I and
type-III magnetic space groups (MSG).Guo  _et al._ (2023) The type-I MSGs are
crystallographic space groups without any additional symmetry while the type-
III are crystallographic space groups with additional antisymmetry versions of
half of the symmetry operations.Bradley and Cracknell (2009)

The altermagnetic systems exhibit nonrelativistic spin-splitting and may
produce anomalous Hall effect (AHE)Šmejkal  _et al._ (2020); Feng  _et al._
(2022) once the relativistic effects are included. The AHE is enhanced by
avoided crossings, also called anticrossings. While further investigations
should be made in order to assure the possession of AHE , AM can produce an
AHE along the direction of the Hall vector, but a very limited number of
altermagnetic systems are metallic. Regarding technological applications,
altermagnets could be assumed as a leading role in realizing concepts in
spincaloritronicsZhou  _et al._ (2023). They can be also used in Josephson
junctionsOuassou  _et al._ (2023), room-temperature magnetoresistance in an
all-antiferromagnetic tunnel junctionQin  _et al._ (2023) and to generate
neutral currents for spintronicsShao  _et al._ (2021).

One of the space groups in which the altermagnetic phase was established is
the Pnma space groupGuo  _et al._ (2023). In a recent work, a new route to
search altermagnetic states was introduced and it was taken in consideration
the example of the Pnma perovskites Fernandes  _et al._ (2023); it was shown
that distinctive signatures on the band structure emerge from the angular
variation of magnetization components in altermagnets. These signatures
manifest as protected nodal lines along mirror planes of the crystal structure
and pinch points on the Fermi surface, which act akin to type-II Weyl nodes
Fernandes  _et al._ (2023). The Pnma presents several nonsymmorphic
symmetriesZhao and Schnyder (2016); Watanabe  _et al._ (2016); König and
Mermin (1997), which are a composition of fractional lattice translations with
point-group operations, like mirror reflection (glide plane) or rotation
(screw axis). The glide symmetry or glide reflection symmetry is a symmetry
operation that consists of a combination of a mirror reflection with respect
to a plane and then a translation parallel to that plane. The eigenvalues of
the glide symmetry are +1 and -1, and the eigenvectors are orthogonal,
therefore, the bands produce a protected band crossing. In the absence of
magnetism and SOC, the nonsymmorphic symmetries force the electronic bands to
be degenerate at the borders of the Brillouin zone with the presence of semi-
Dirac pointsWieder  _et al._ (2016); Zhao and Schnyder (2016), generating
linear magnetoresistanceCampbell  _et al._ (2021), unconventional topological
phasesWang  _et al._ (2016); Bzdušek  _et al._ (2016); Parameswaran  _et al._
(2013); Wieder and Kane (2016); Brzezicki and Cuoco (2017); Yoshida  _et al._
(2019); Daido  _et al._ (2019) with unique surface states, Fermi surfaces with
reduced dimensionalityCuono  _et al._ (2019a) and topological nonsymmorphic
crystalline superconductivity with $\mathbb{Z}_{4}$ topological
invariantYoshida  _et al._ (2019); Daido  _et al._ (2019). In the presence of
SOC and magnetism, the system presents a partial or selective removal of the
degeneracyCuono  _et al._ (2019a). The semi-Dirac points exhibit linear band
dispersion in one direction (the high-symmetry direction at least in the
space-group 62) and quadratic band dispersion in the orthogonal
directionAdroguer  _et al._ (2016). The existence of these semi-Dirac points
in the space group 62 and its relation with the nonsymmorphic symmetries has
been widely demonstrated in literature Niu  _et al._ (2017, 2019); Cuono  _et
al._ (2019a); Campbell  _et al._ (2021); Cuono  _et al._ (2019b). The
connection between semi-Dirac points and glide symmetry will be described
later in the text.

An example of a material with the Pnma crystal phase is the CrAs compoundWu
_et al._ (2010, 2014) in the MnP-type phase, which is a rare itinerant
antiferromagnetWu  _et al._ (2010, 2014) that could exhibit AHE due to AM.
Additionally, the CrAs system is not ionic, therefore, $p$\- and $d$-bands are
strongly hybridizing with a large bandwidth of the order of 11 eVAutieri and
Noce (2017). This grants us the possibility to observe induced AM in the
$p$-bands. The magnetic ground state of the CrAs is a helimagnetic phase, with
the components of the magnetic momentWu  _et al._ (2010, 2014); Shen  _et al._
(2016); Keller  _et al._ (2015) in the $a$-$b$ plane and with a spin-orbit
coupling mostly from the $p$-states of AsAutieri and Noce (2017); Cuono  _et
al._ (2019b); Wadge  _et al._ (2022). Here we investigate the CrAs as a
testbed exclusively in hypothetical collinear magnetic orders since we are
mainly interested in the interplay between nonsymmorphic symmetries and AM.
Further investigations are necessary to extend these results to the non-
collinear phase and to understand how much AM will survive in the non-
collinear phaseZhu  _et al._ (2023). CrAs will work as a prototype for other
itinerant antiferromagnets of the same space group as MnPd2, FeP, MnP or
CuMnAs that may have a smaller AHE due to the lower spin-orbit
couplingCampbell  _et al._ (2021). Different from most studied altermagnetic
Pnma compounds until nowSattigeri  _et al._ (2023); Cuono  _et al._ (2023a);
Nguyen and Yamauchi (2023) such as LaMnO3, YVO3 and CaCrO3 that have their
magnetic atoms in the Wyckoff positions 4bAutieri  _et al._ (2023), the atoms
of CrAs are only present in the 4c Wyckoff positions, therefore, the system
belongs to a different magnetic space group that should be more symmetric and
host unexplored properties.

In this paper, we study the interplay between the altermagnetic properties of
the Pnma phase and the nonsymmorphic symmetries using first-principle
calculations. The computational details are reported in Appendix A.
Investigating different collinear magnetic configurations as the A-type,
G-type and C-type shown in Fig. 1(a-c), respectively, we obtained that the
altermagnetic spin-splitting is present only in the C-type configuration while
it is absent in the A-type and G-type ones. Even if the C-type is not the
ground stateAutieri and Noce (2017), we are interested in the interplay
between altermagnetism and nonsymmorphic symmetries that is a general aspect
that could appear in several other compounds. We find that the altermagnetic
spin-splitting can be up to 0.5 eV in the $d$-bands close to the Fermi level
while it gets reduced to 0.2 eV in the $p$-bands. Furthermore, we find that
the interplay between AM and nonsymmorphic symmetries produces an intricate
network of crossings and anticrossings in large areas of the Brillouin zone.
As a result, the presence of nonsymmorphic symmetries generates a large
anomalous Hall conductivity. The paper is organized as follows: in the next
Section, we show the main results, while the last Section is devoted to
conclusions.

Figure 1: Crystal structure and collinear magnetic orders for CrAs as (a)
A-type, (b) G-type and (c) C-type. The balls with colors blue and red
represent the Cr atoms with the opposite spin moments. Green balls represent
the As atoms. (d) Symmetries of the irreducible Brillouin zone of the
orthorhombic primitive cell for the C-type magnetic order. The C-type is the
only magnetic order that can host AM. The position of the high-symmetry
$k$-points U1, U2, R1 and R2 are highlighted in green and yellow. The dashed
magenta line indicates the high-symmetry path U1-$\Gamma$-U2 which is one of
the possible paths to show the AM in this magnetic space group.

##  II Results

This Section is divided into two subsections with the results without SOC in
the first one and the results with SOC in the second one. We report the
electronic properties focusing on the interplay between AM and nonsymmorphic
symmetries by considering different magnetic orders. The semi-Dirac points
generate a large number of crossings and anticrossings that we describe. When
we add the SOC in the second subsection, the calculation of the anomalous Hall
conductivity (AHC) confirms a relatively large value due to the several
crossings and anticrossings.

###  II.1 Electronic properties without SOC and nonsymmorphic symmetries

In the nonmagnetic phase, the bands are twofold degenerate at any $k$-vector
of the Brillouin zone due to the presence of the inversion-time reversal
symmetry. This is the Kramers degeneracy and it is protected under the action
of the spin-orbit coupling interaction. The nonsymmorphic symmetries produce
additional degeneracy at the border of the Brillouin zone. The $\Gamma$ and R
points are time-reversal invariant momenta (TRIM) points. In the Pnma space
group without magnetism, the $\Gamma$ point has only the Kramer degeneracy,
two additional nonsymmorphic symmetries along the SR line produce an eightfold
degeneracy at R and S, while X, Y, Z, U and T have degeneracy 4 due to one
nonsymmorphic symmetry. When we consider the C-type magnetism, $\Gamma$ is a
TRIM point and it still presents a twofold degeneracy. In this magnetic
configuration, the TRIM points R, S and T have fourfold degeneracy due to an
additional nonsymmorphic symmetry while all the other high-symmetry points
have degeneracy 2. The fourfold degeneracy at the TRIM point R makes this
space group one of the few where we can study the coexistence of AM and
nonsymmorphic symmetries.

Figure 2: Band structure of the C-type magnetic order along the $k$-path
R1-$\Gamma$-R2. The spin-up channel is shown in blue, while the spin-down
channel is shown in red. The grey area represents the non-relativistic spin-
splitting. The band structure is plotted between -0.5 and +0.5 eV where the
$d$-electrons dominate. The black circles represent the band crossings
protected by glide symmetry. Figure 3: Nonrelativistic spin-splitting along
the R1-$\Gamma$-R2 for the first bands above the Fermi level at the $\Gamma$
point. The spin-splitting reaches the value of 0.46 eV. For every couple of
bands producing a finite spin-splitting at the TRIM point, another couple of
bands will produce an opposite spin-splitting. Figure 4: Schematic
representation of the crossings and anticrossings network in altermagnetic
semi-Dirac fermions without SOC. The spin-up channel is shown in blue, while
the spin-down channel is shown in red. The solid line is used for the positive
eigenvalues of the glide operator, while the dashed line is used for the
negative eigenvalue of the glide operator. The black circles represent the
band crossings between band with opposite glide eigenvalues, while the green
boxes represent the band anticrossings between bands with the same glide
eigenvalues.

Figure 5: Band structure along the R1-S-R2 $k$-path for the C-type magnetic
order including SOC with the Néel vector along the (a) x-axis, (b) y-axis and
(c) z-axis. No AM is present along this $k$-path. In panel (b), we obtain the
antiferromagnetic hourglass fermions with a magnetic space group type-I. No
hourglass fermions are found in the magnetic space groups of type-III.

We find AM in the C-type magnetic order, but not in the other magnetic orders
and this is due to the following reasons. In the space group no. 62, there are
16 different magnetic space groups. 8 of these magnetic space groups can host
AM, while the other 8 groups cannot.Gallego  _et al._ (2012) In the C-type
magnetic order, the space groups that depend on the Néel vector belong to
type-I and type-III. Both of these types present AM. Changing the magnetic
configuration to G-type or A-type, the magnetic space group changes to type-II
or type-IV, therefore, different magnetic configurations can have different
altermagnetic properties. The CrAs contains 4c Wyckoff positions for both Cr
and As, therefore the altermagnetic properties of the Pnma CrAs are different
from the Pnma perovskites that have the magnetic atoms in 4b Wyckoff
positions.Nguyen and Yamauchi (2023); Cuono  _et al._ (2023a) A spin symmetry
group analysis Liu  _et al._ (2022) would provide additional understanding of
the system.

The BZ for the C-type magnetic order is reported in Fig. 1(d). With subscripts
1 and 2, we indicate the two points in the $k$-space that have opposite non-
relativistic spin-splitting. In this case, the $k$-path connecting the
$\Gamma$ point with U and R shows altermagnetic spin-splitting. In this paper,
we will take as an example for the discussion the altermagnetic properties
along the path R1-$\Gamma$-R2. Similar arguments will be valid for the
altermagnetic properties along U1-$\Gamma$-U2, however, the presence of
multiple $k$-paths where the AM is present means that there is a large area of
the k-space where the altermagnetic spin-splittings reside.

We stress the relevant role of the semi-Dirac points in the generation of the
band crossings and anticrossings. We focus on the semi-Dirac points at the R
point, but they are present at all borders of the Brillouin zone. From the
nonsymmorphic symmetries in space group no. 62, we obtain that the energy
spectrum at the point R=($\pi$,$\pi$,$\pi$) is always a semi-Dirac point with
the dispersion relations for spin-up and spin-down as :

$E_{\uparrow}(\pi-\epsilon,\pi-\epsilon,\pi-\epsilon)=\varepsilon_{0}\pm
v\epsilon$ (1)
$E_{\downarrow}(\pi-\epsilon,\pi-\epsilon,\pi-\epsilon)=\varepsilon_{0}\pm
v\epsilon$ (2)

with $\varepsilon_{0}$ being a combination of the on-site energies and $v$ a
combination of the first-neighbor hopping parametersCuono  _et al._ (2019a).
Since the Dirac velocity does not depend on the energy difference between
majority and minority electrons, it is the same for the spin-up or spin-down
channel. In the case of large hybridization between the $p$-$d$ electrons, as
happens in CrAs, the Dirac velocities are also large. The presence of multiple
orbitals and semi-Dirac bands with large Dirac velocities, which can be
positive and negative, favors the creation of several band crossings and band
anticrossings which are observable in the band structure as presented in Fig.
2. What we have shown in Fig. 2 for the Cr-$d$ bands close to the Fermi level
happens qualitatively also for the As-$p$ bands as we have described in
Appendix B. The bands with opposite Dirac velocity +$v$ and -$v$ also have
opposite glide eigenvalues and orthogonal eigenvectors. Therefore, the bands
with opposite glide eigenvalues are orthogonal and do not hybridize each
otherZhou  _et al._ (2021); Bannies  _et al._ (2021); Zhao and Schnyder
(2016), this is exactly true for the bands with linear dispersion very close
to the border of the Brillouin zone, while away from the Brillouin zone border
the bands tend to show mixing of the eigenvectors. In Fig. 2, we have reported
with black circles the band crossings protected by the glide symmetry.

Now, we move to the analysis of the nonrelativistic spin-splitting in the
presence of nonsymmorphic symmetries. If we consider the Kramers pair bands
which at the $\Gamma$ point are at 0.1 eV above the Fermi level and follow
them, we can see that the spin-up and spin-down channels have both a glide-
protected band crossing and then they reach the R points at two different
eigenvalues creating surprisingly a finite nonrelativistic spin-splitting at a
TRIM point as shown in Fig. 3. The nonzero value of the spin-splitting at the
point R apparently contradicts the concept of TRIM point, but there would be
another couple of bands that would produce the opposite nonrelativistic spin-
splitting to preserve the total spin-splitting at a given TRIM point. Due to
these exceptional conditions, the nonrelativistic spin-splitting for those
bands, reported in Fig. 3, reaches a maximum of 0.46 eV.

In Fig. 4, we plot a schematic figure with the spin-channels in blue and red
colors, with the solid (dashed) line used for the positive (negative)
eigenvalues of the glide operator. We have the expected intrachannel avoided
band crossings and the interchannel band crossings. Additionally, we have
intrachannel band crossings, namely crossings between bands with opposite
glide eigenvalues. In the next subsection, we will deliberate on how these
crossings will evolve with SOC as a function of the Néel vector orientation.

Figure 6: Band structure of the C-type magnetic order along the $k$-path
R1-$\Gamma$-R2 with Néel vector along the (a) x-axis, (b) y-axis and (c)
z-axis, respectively. The spin-up channel is shown in blue, the spin-down
channel is shown in red while the band structure with SOC is plotted in green.
The band structure is plotted between -0.3 and +0.3 eV where the $d$-electrons
dominate.

Figure 7: Magnification of the band structure in FIG. 6. Band structure of the
C-type magnetic order along the $k$-path R1-$\Gamma$ with the Néel vector
along the (a) x-axis, (b) y-axis and (c) z-axis. The band structure is plotted
between -0.11 and -0.02 eV where there are the bands linear in $k$ with
opposite glide-eigenvalues. The black circle in panel (a) describes the band
crossing between opposite glide eigenvalues not splitted by SOC when the Néel
vector is along x.

###  II.2 Electronic properties with SOC and Anomalous Hall conductivity

In the non-magnetic case, when we consider the SOC, we have shown, in a
previous work Cuono  _et al._ (2019a), a selective removal of the bands
degeneracy due to the nonsymmorphic symmetries at the TRIM points and at the
borders of the Brillouin zone. In the C-type magnetic configuration, when we
include the SOC interaction, we observe a selective removal of the spin-
degeneracy depending on the Néel vector direction and consequently on the
magnetic space group. Indeed, the variation of the Néel vector direction
changes the magnetic space group. We define the x-, y- and z-axis as parallel
to the $a$, $b$ and $c$ lattice constants, respectively, as defined in Fig. 1.
When the Néel vector is along y, the magnetic space group is 62.441, which is
a type I. When the magnetization is along x and z, the magnetic space groups
are 62.446 and 62.447, respectively, which are type III. This selective
removal of the degeneracy is valid along $k$-paths with and without AM.
Looking at Figs. 5 and 6, we can see that the spin-orbit splitting is
qualitatively and quantitatively different depending on the magnetic space
group.

It was shown that, in the non-magnetic case, the SOC acts selectively at the
TRIM points and Brillouin zone border due to the nonsymmorphic symmetriesCuono
_et al._ (2019b). We will investigate the SOC effects in the magnetic phase as
a function of the Néel vector starting from the path R1-S-R2 where no AM is
present. Without SOC, the entire RS line including the TRIM points have
degeneracy 4. The band structure with SOC along the R1-S-R2 $k$-path is
reported in Fig. 5. When the Néel vector is along x, the spin-orbit splits the
bands at R but not at S. When the Néel vector is along z, the spin-orbit
splits the bands at S but not at R. When the Néel vector is along y, the spin-
orbit splits the bands at both R and S, and therefore, in this latter case, we
find antiferromagnetic hourglass fermions. These hourglass fermions are
relativistic features already present along the RS line without magnetismWang
_et al._ (2016, 2017); Li  _et al._ (2018). Recently, the ferromagnetic and
antiferromagnetic hourglass fermions were investigated. However, we find the
antiferromagnetic hourglass in the magnetic space group 62.441 that was not
reported beforeHu  _et al._ (2022).

The spin-orbit acts selectively also along the R1-$\Gamma$-R2 path where the
nonrelativistic spin-splitting is present. The effects of the SOC for
different Néel vector orientations are reported in Fig. 6, we will highlight
in the discussion the $k$-points where the SOC is not effective. When the Néel
vector is along x, we have additional SOC splittings for all the crossing and
anticrossing points except that for the intrachannel crossing point at around
-0.1 eV, which is protected by the glide operator as shown in Fig. 6(a). A
magnification of Fig. 6(a-c) is reported in Fig. 7(a-c), respectively, to
highlight the band crossings protected by the glide against SOC when the Néel
vector is along x. Indeed, the band crossing in the black circle in Fig. 7(a)
is not splitted by SOC, while the SOC splitting is evident in 7(b,c). It was
shown that this kind of glide-protected band crossings can generate a non-
isoenergetic nodal-lineBannies  _et al._ (2021); Zhou  _et al._ (2021).
Therefore, this case would be a nodal-line in the altermagnetic phase, further
investigations in this direction using model hamiltonian calculations could be
interesting for the topological matter community. When the Néel vector is
along y, we do not have SOC splitting at the $\Gamma$ point as obtained in
Fig. 6(b). When the Néel vector is along z as shown in Fig. 6(c), we do not
have SOC splitting at the R1 and R2 points coherently with what is observed in
Fig. 5(c).

Figure 8: Anomalous Hall effect in altermagnetic C-type magnetic order with
Néel vector along x, along y and along z reported in panels (a), (b) and (c),
respectively. In every case, we show the AHCs $\sigma_{xy}$, $\sigma_{xz}$ and
$\sigma_{yz}$ for an Hall vector orthogonal to the Néel vector. The AHCs are
presented between -0.25 and +0.25 eV, in this energy range the maximum in
absolute value was reached by $\sigma_{xy}$ around -1000 Scm-1 with Néel
vector along y-axis. The Fermi energy is set to zero.

The Hall vector in altermagnetic systems with space group no. 62 can lie in
all possible directions in principleŠmejkal  _et al._ (2020). However, when
the structural details are added, the Hall vector has a precise orientation.
It was shown that in space group no. 62, we can expect the Hall vector in
directions orthogonal to the Néel vectorNguyen and Yamauchi (2023). Several
anticrossing points appear in the whole Brillouin zone boundary due to the
presence of the semi-Dirac points. These anticrossing points generate large
Berry curvature and consequently a large anomalous Hall effect, since, as it
is well known from the literature Nagaosa  _et al._ (2010); Xiao  _et al._
(2010); Taguchi  _et al._ (2001); Chakraborty  _et al._ (2022); Zhang  _et
al._ (2023); Jungwirth  _et al._ (2002); Yao  _et al._ (2004); Chen  _et al._
(2014); van Thiel  _et al._ (2021), the intrinsic anomalous Hall effect can be
expressed in terms of the Berry curvature. The semi-Dirac points and the
glides linked to the nonsymmorphic symmetries are key ingredients to the
generation of these several crossings and anticrossings. When we add the SOC,
avoided band crossings are obtained and a large AHC is expected. Therefore, we
can claim that the interplay between AM and nonsymmorphic symmetries generates
large anomalous Hall conductivity. We calculate the AHC between -0.25 and
+0.25 eV and we report it in Fig. 8 for different Néel vector orientations
along the principal axes. The three AHC components that we have calculated are
$\sigma_{yz}$, $\sigma_{xz}$ and $\sigma_{xy}$, the numerical details are
reported in Appendix A. We obtain large values of the AHE in the two
directions orthogonal to the Néel vector. When the Néel vector is along x, we
obtain AHCs values up to -400 S/cm for the $\sigma_{xy}$ and 300 S/cm for the
$\sigma_{xz}$ (see Fig. 8 (a)). When the Néel vector is along y, we obtain AHC
values up to -1000 for the $\sigma_{yz}$ and +600 S/cm for the $\sigma_{xy}$
(see Fig. 8 (b)), the spike that produces the large values at +0.14 eV was
verified with a denser energy grid. Finally, when the Néel vector is along z,
we obtain AHC values up to 450 for the $\sigma_{yz}$ and -200 S/cm for the
$\sigma_{xz}$ (see Fig. 8 (c)). We observe a strong change of all the AHCs
when we switch the Néel vector from the one axis to another. The compounds
with space group no. 62 and magnetic atoms in Wyckoff position 4b hosts the
AHC in one componentNguyen and Yamauchi (2023), however, compounds with the
same space group but different Wyckoff positions host an Hall vector
orthogonal to the Néel vector. These values are smaller but of the same order
of magnitude as the AHC in other altermagnetic metallic compoundsŠmejkal  _et
al._ (2020); Nguyen and Yamauchi (2023) as RuO2 and CaCrO3.

##  III Conclusions

The presence of AM varies with the magnetic configuration, since the magnetic
space group type strongly depends on the magnetic configuration. For the space
group no. 62 with magnetic atoms in 4c Wyckoff position, the non-relativistic
spin-splitting is present for the C-type magnetic order while it is absent in
G-type and A-type magnetic orders. Following a couple of bands with opposite
spin from the degenerate $\Gamma$ point through the glide-protected crossing,
we can end up with a finite spin-splitting at the TRIM points R1 and R2,
however, there would be another couple of bands that would produce the
opposite spin-splitting restoring the zero non-relativistic spin-splitting at
the TRIM points. The magnetic space group is type-I when the Néel vector is
along y and type-III when the Néel vector is along x or z. A selective removal
of the spin degeneracy acts as a function of the magnetic space group at the
TRIM points. For example, in the type-I magnetic space group, we find
antiferromagnetic hourglass electrons that are not present when the Néel
vector is along x or z. When the Néel is along x, we have a glide-protected
crossing that could generate a nodal-line in the altermagnetic phase. Due to
the semi-Dirac points and glide symmetries, the interplay between AM and
nonsymmorphic symmetries produces several band crossings and avoided band
crossings, once we apply the SOC these crossings generate a large AHC of the
same order of magnitude found in altermagnetic RuO2 and CaCrO3.

###### Acknowledgements.

We thank Tomasz Dietl, Canio Noce and Rajibul Islam for useful discussions.
The work is supported by the Foundation for Polish Science through the
International Research Agendas program co-financed by the European Union
within the Smart Growth Operational Programme (Grant No. MAB/2017/1). A.F. was
supported by the Polish National Science Centre under Project No.
2020/37/B/ST5/02299. We acknowledge the access to the computing facilities of
the Interdisciplinary Center of Modeling at the University of Warsaw, Grant
g91-1418, g91-1419 and g91-1426 for the availability of high-performance
computing resources and support. We acknowledge the CINECA award under the
ISCRA initiative IsC99 ”SILENTS”, IsC105 ”SILENTSG” and IsB26 ”SHINY” grants
for the availability of high-performance computing resources and support. We
acknowledge the access to the computing facilities of the Poznan
Supercomputing and Networking Center Grant No. 609.

Figure 9: Band structure of the C-type magnetic order along the $k$-path
R1-$\Gamma$-R2. The spin-up channel is shown in blue, while the spin-down
channel is shown in red. The band structure is plotted between +2 and +3 eV
where the As $p$-electrons dominate.

##  Appendix A Computational details

We performed density functional theory (DFT) calculations by using the VASP
packageKresse and Hafner (1993); Kresse and Furthmüller (1996, 1996). We
employed the local density approximation and the Perdew–ZungerPerdew and
Zunger (1981) parametrization of the Ceperly-Alder dataCeperley and Alder
(1980). The same parameters of previous worksAutieri and Noce (2017) as the
lattice constants and atomic positions of Ref. Shen  _et al._ , 2016 have been
used. The lattice constants are a=5.60499 Å, b=3.58827 Å and c= 6.13519 Å,
while the Wyckoff positions both in 4c are Cr: (0.0070, $1/4$, 0.2049) and As:
(0.2045, $1/4$, 0.5836). The band structure plots were obtained with 130
$k$-points for every path. Close to the Fermi level we have the Cr
$d$-orbitals sandwiched between As $p$-orbitalsAutieri and Noce (2017). We
performed the wannierizationMarzari and Vanderbilt (1997); Souza  _et al._
(2001) by using the WANNIER90 codeMostofi  _et al._ (2008) and we considered
the Cr-3$d$ and As-4$p$ orbitals as we have already done in previous
papersAutieri  _et al._ (2017, 2018); Cuono  _et al._ (2019b, a, 2018, c). For
the calculation of the Anomalous Hall conductivity we used WannierTools
code,Wu  _et al._ (2018) we have used a $k$-grid 200 $\times$ 200 $\times$
200\. The AHC calculations were verified with a $k$-grid of 300 $\times$ 300
$\times$ 300 that reproduces the same results with negligible differencesAlam
_et al._ (2023). The information about the magnetic space groups were
extracted by the Bilbao Crystallographic ServerGallego  _et al._ (2012).

##  Appendix B Induced altermagnetism in p-bands

As mentioned in the main text, the system hosts a strong $p$-$d$
hybridizationCuono  _et al._ (2019a), therefore, the AM can be induced from
the Cr $d$-bands to the As $p$-bands. The $d$-bands dominate up to 1.5 eV
above the Fermi levelCuono  _et al._ (2019b), while above 2 eV the bands are
mainly composed by As-4$p$ spectral weight. The altermagnetic spin-splitting
is of the order of 0.5 eV on the magnetic $d$-bands close to the Fermi level
and survives in the $p$-bands through the $d$-$p$ hybridization but it gets
reduced. The largest spin-splitting in the $p$-orbitals is around 0.2 eV as we
can see in Fig. 9. The Cr $d$-bands induce AM in As $p$-bands, this is
slightly different from the AM in EuIn2As2 where the magnetic $f$-bands of Eu
induce AM on the $d$-bands of the same Eu atomsCuono  _et al._ (2023b).

## References

  * Šmejkal  _et al._ (2022a) L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022a).
  * Šmejkal  _et al._ (2022b) L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 031042 (2022b).
  * Yuan and Zunger (2023) L.-D. Yuan and A. Zunger, Advanced Materials , 2211966 (2023).
  * Hayami  _et al._ (2019) S. Hayami, Y. Yanagi, and H. Kusunose, journal of the physical society of japan 88, 123702 (2019).
  * Hayami  _et al._ (2020) S. Hayami, Y. Yanagi, and H. Kusunose, Physical Review B 102, 144441 (2020).
  * Chen  _et al._ (2023) X. Chen, D. Wang, L. Li, and B. Sanyal, Applied Physics Letters 123, 022402 (2023), https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/5.0147450/18038936/022402_1_5.0147450.pdf .
  * Guo  _et al._ (2023) Y. Guo, H. Liu, O. Janson, I. C. Fulga, J. van den Brink, and J. I. Facio, Materials Today Physics 32, 100991 (2023).
  * Bradley and Cracknell (2009) C. Bradley and A. P. Cracknell,  _The mathematical theory of symmetry in solids_ , Oxford Classic Texts in the Physical Sciences (Oxford University Press, London, England, 2009).
  * Šmejkal  _et al._ (2020) L. Šmejkal, R. González-Hernández, T. Jungwirth, and J. Sinova, Science Advances 6, eaaz8809 (2020), https://www.science.org/doi/pdf/10.1126/sciadv.aaz8809 .
  * Feng  _et al._ (2022) Z. Feng, X. Zhou, L. Šmejkal, L. Wu, Z. Zhu, H. Guo, R. González-Hernández, X. Wang, H. Yan, P. Qin, X. Zhang, H. Wu, H. Chen, Z. Meng, L. Liu, Z. Xia, J. Sinova, T. Jungwirth, and Z. Liu, Nature Electronics 5, 735 (2022).
  * Zhou  _et al._ (2023) X. Zhou, W. Feng, R.-W. Zhang, L. Smejkal, J. Sinova, Y. Mokrousov, and Y. Yao, “Crystal thermal transport in altermagnetic ruo2,” (2023), arXiv:2305.01410 [cond-mat.mtrl-sci] .
  * Ouassou  _et al._ (2023) J. A. Ouassou, A. Brataas, and J. Linder, “Josephson effect in altermagnets,” (2023), arXiv:2301.03603 [cond-mat.supr-con] .
  * Qin  _et al._ (2023) P. Qin, H. Yan, X. Wang, H. Chen, Z. Meng, J. Dong, M. Zhu, J. Cai, Z. Feng, X. Zhou, L. Liu, T. Zhang, Z. Zeng, J. Zhang, C. Jiang, and Z. Liu, Nature 613, 485 (2023).
  * Shao  _et al._ (2021) D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, and E. Y. Tsymbal, Nature Communications 12 (2021), 10.1038/s41467-021-26915-3.
  * Fernandes  _et al._ (2023) R. M. Fernandes, V. S. de Carvalho, T. Birol, and R. G. Pereira, “Topological transition from nodal to nodeless zeeman splitting in altermagnets,” (2023), arXiv:2307.12380 [cond-mat.mes-hall] .
  * Zhao and Schnyder (2016) Y. X. Zhao and A. P. Schnyder, Phys. Rev. B 94, 195109 (2016).
  * Watanabe  _et al._ (2016) H. Watanabe, H. C. Po, M. P. Zaletel, and A. Vishwanath, Phys. Rev. Lett. 117, 096404 (2016).
  * König and Mermin (1997) A. König and N. D. Mermin, Phys. Rev. B 56, 13607 (1997).
  * Wieder  _et al._ (2016) B. J. Wieder, Y. Kim, A. M. Rappe, and C. L. Kane, Phys. Rev. Lett. 116, 186402 (2016).
  * Campbell  _et al._ (2021) D. J. Campbell, J. Collini, J. Sławińska, C. Autieri, L. Wang, K. Wang, B. Wilfong, Y. S. Eo, P. Neves, D. Graf, E. E. Rodriguez, N. P. Butch, M. Buongiorno Nardelli, and J. Paglione, npj Quantum Materials 6, 38 (2021).
  * Wang  _et al._ (2016) Z. Wang, R. J. Alexandradinata, A. andCava, and B. A. Bernevig, Nature 532 (2016), 10.1038/nature17410.
  * Bzdušek  _et al._ (2016) T. Bzdušek, Q. Wu, A. Rüegg, and M. Sigrist, Nature 538 (2016), 10.1038/nature19099.
  * Parameswaran  _et al._ (2013) S. A. Parameswaran, A. M. Turner, D. P. Arovas, and A. Vishwanath, Nature Physics 9 (2013), 10.1038/nphys2600.
  * Wieder and Kane (2016) B. J. Wieder and C. L. Kane, Phys. Rev. B 94, 155108 (2016).
  * Brzezicki and Cuoco (2017) W. Brzezicki and M. Cuoco, Phys. Rev. B 95, 155108 (2017).
  * Yoshida  _et al._ (2019) T. Yoshida, A. Daido, N. Kawakami, and Y. Yanase, Phys. Rev. B 99, 235105 (2019).
  * Daido  _et al._ (2019) A. Daido, T. Yoshida, and Y. Yanase, Phys. Rev. Lett. 122, 227001 (2019).
  * Cuono  _et al._ (2019a) G. Cuono, F. Forte, M. Cuoco, R. Islam, J. Luo, C. Noce, and C. Autieri, Phys. Rev. Mater. 3, 095004 (2019a).
  * Adroguer  _et al._ (2016) P. Adroguer, D. Carpentier, G. Montambaux, and E. Orignac, Phys. Rev. B 93, 125113 (2016).
  * Niu  _et al._ (2017) Q. Niu, W. C. Yu, K. Y. Yip, Z. L. Lim, H. Kotegawa, E. Matsuoka, H. Sugawara, H. Tou, Y. Yanase, and S. K. Goh, Nature Communications 8, 2041 (2017).
  * Niu  _et al._ (2019) Q. Niu, W. C. Yu, E. I. P. Aulestia, Y. J. Hu, K. T. Lai, H. Kotegawa, E. Matsuoka, H. Sugawara, H. Tou, D. Sun, F. F. Balakirev, Y. Yanase, and S. K. Goh, Phys. Rev. B 99, 125126 (2019).
  * Cuono  _et al._ (2019b) G. Cuono, C. Autieri, G. Guarnaccia, A. Avella, M. Cuoco, F. Forte, and C. Noce, The European Physical Journal Special Topics 228, 631 (2019b).
  * Wu  _et al._ (2010) W. Wu, X. Zhang, Z. Yin, P. Zheng, N. Wang, and J. Luo, Science China Physics, Mechanics and Astronomy 53 (2010), 10.1007/s11433-010-4006-1.
  * Wu  _et al._ (2014) W. Wu, J. Cheng, K. Matsubayashi, P. Kong, F. Lin, C. Jin, N. Wang, Y. Uwatoko, and J. Luo, Nature Communications 5 (2014), 10.1038/ncomms6508.
  * Autieri and Noce (2017) C. Autieri and C. Noce, Philosophical Magazine 97, 3276 (2017), https://doi.org/10.1080/14786435.2017.1375607 .
  * Shen  _et al._ (2016) Y. Shen, Q. Wang, Y. Hao, B. Pan, Y. Feng, Q. Huang, L. W. Harriger, J. B. Leao, Y. Zhao, R. M. Chisnell, J. W. Lynn, H. Cao, J. Hu, and J. Zhao, Phys. Rev. B 93, 060503(R) (2016).
  * Keller  _et al._ (2015) L. Keller, J. S. White, M. Frontzek, P. Babkevich, M. A. Susner, Z. C. Sims, A. S. Sefat, H. M. Rønnow, and C. Rüegg, Phys. Rev. B 91, 020409(R) (2015).
  * Wadge  _et al._ (2022) A. S. Wadge, G. Grabecki, C. Autieri, B. J. Kowalski, P. Iwanowski, G. Cuono, M. F. Islam, C. M. Canali, K. Dybko, A. Hruban, A. Łusakowski, T. Wojciechowski, R. Diduszko, A. Lynnyk, N. Olszowska, M. Rosmus, J. Kołodziej, and A. Wiśniewski, Journal of Physics: Condensed Matter 34, 125601 (2022).
  * Zhu  _et al._ (2023) Y.-P. Zhu, X. Chen, X.-R. Liu, P. Liu, H. Zha, C. Hong, J. Li, Z. Jiang, X.-M. Ma, Y.-J. Hao, W. Liu, M. Zeng, J. Ding, S. Mo, Z. Liu, M. Ye, D. Shen, R.-H. He, S. Qiao, Q. Liu, and C. Liu, “Observation of nonrelativistic plaid-like spin splitting in a noncoplanar antiferromagnet,” (2023), arXiv:2303.04549 [cond-mat.mtrl-sci] .
  * Sattigeri  _et al._ (2023) R. M. Sattigeri, G. Cuono, and C. Autieri, “Altermagnetic surface states: towards the observation and utilization of altermagnetism in thin films, interfaces and topological materials,” (2023), arXiv:2307.10146 [cond-mat.mtrl-sci] .
  * Cuono  _et al._ (2023a) G. Cuono, R. M. Sattigeri, J. Skolimowski, and C. Autieri, Journal of Magnetism and Magnetic Materials 586, 171163 (2023a).
  * Nguyen and Yamauchi (2023) T. Nguyen and K. Yamauchi, Phys. Rev. B 107, 155126 (2023).
  * Autieri  _et al._ (2023) C. Autieri, M. Cuoco, G. Cuono, S. Picozzi, and C. Noce, Physica B: Condensed Matter 648, 414407 (2023).
  * Gallego  _et al._ (2012) S. V. Gallego, E. S. Tasci, G. de la Flor, J. M. Perez-Mato, and M. I. Aroyo, Journal of Applied Crystallography 45, 1236 (2012).
  * Liu  _et al._ (2022) P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, Phys. Rev. X 12, 021016 (2022).
  * Zhou  _et al._ (2021) X. Zhou, C.-H. Hsu, H. Aramberri, M. Iraola, C.-Y. Huang, J. L. Mañes, M. G. Vergniory, H. Lin, and N. Kioussis, Phys. Rev. B 104, 125135 (2021).
  * Bannies  _et al._ (2021) J. Bannies, E. Razzoli, M. Michiardi, H.-H. Kung, I. S. Elfimov, M. Yao, A. Fedorov, J. Fink, C. Jozwiak, A. Bostwick, E. Rotenberg, A. Damascelli, and C. Felser, Phys. Rev. B 103, 155144 (2021).
  * Wang  _et al._ (2017) L. Wang, S.-K. Jian, and H. Yao, Phys. Rev. B 96, 075110 (2017).
  * Li  _et al._ (2018) S. Li, Y. Liu, S.-S. Wang, Z.-M. Yu, S. Guan, X.-L. Sheng, Y. Yao, and S. A. Yang, Phys. Rev. B 97, 045131 (2018).
  * Hu  _et al._ (2022) Y. Hu, X. Wan, and F. Tang, Phys. Rev. B 106, 165128 (2022).
  * Nagaosa  _et al._ (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * Xiao  _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
  * Taguchi  _et al._ (2001) Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science 291, 2573 (2001), https://www.science.org/doi/pdf/10.1126/science.1058161 .
  * Chakraborty  _et al._ (2022) T. Chakraborty, K. Samanta, S. N. Guin, J. Noky, I. n. Robredo, S. Prasad, J. Kuebler, C. Shekhar, M. G. Vergniory, and C. Felser, Phys. Rev. B 106, 155141 (2022).
  * Zhang  _et al._ (2023) C.-P. Zhang, X.-J. Gao, Y.-M. Xie, H. C. Po, and K. T. Law, Phys. Rev. B 107, 115142 (2023).
  * Jungwirth  _et al._ (2002) T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 88, 207208 (2002).
  * Yao  _et al._ (2004) Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204 (2004).
  * Chen  _et al._ (2014) H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * van Thiel  _et al._ (2021) T. C. van Thiel, W. Brzezicki, C. Autieri, J. R. Hortensius, D. Afanasiev, N. Gauquelin, D. Jannis, N. Janssen, D. J. Groenendijk, J. Fatermans, S. Van Aert, J. Verbeeck, M. Cuoco, and A. D. Caviglia, Phys. Rev. Lett. 127, 127202 (2021).
  * Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Perdew and Zunger (1981) J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
  * Ceperley and Alder (1980) D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
  * Marzari and Vanderbilt (1997) N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
  * Souza  _et al._ (2001) I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001).
  * Mostofi  _et al._ (2008) A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Computer Physics Communications 178, 685 (2008).
  * Autieri  _et al._ (2017) C. Autieri, G. Cuono, F. Forte, and C. Noce, Journal of Physics: Condensed Matter 29, 224004 (2017).
  * Autieri  _et al._ (2018) C. Autieri, G. Cuono, F. Forte, and C. Noce, Journal of Physics: Conference Series 969, 012106 (2018).
  * Cuono  _et al._ (2018) G. Cuono, C. Autieri, F. Forte, G. Busiello, M. T. Mercaldo, A. Romano, C. Noce, and A. Avella, AIP Advances 8 (2018), 10.1063/1.5042740, 101312, https://pubs.aip.org/aip/adv/article-pdf/doi/10.1063/1.5042740/12872589/101312_1_online.pdf .
  * Cuono  _et al._ (2019c) G. Cuono, C. Autieri, F. Forte, M. T. Mercaldo, A. Romano, A. Avella, and C. Noce, New Journal of Physics 21, 063027 (2019c).
  * Wu  _et al._ (2018) Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Computer Physics Communications 224, 405 (2018).
  * Alam  _et al._ (2023) M. S. Alam, A. Fakhredine, M. Ahmad, P. K. Tanwar, H.-Y. Yang, F. Tafti, G. Cuono, R. Islam, B. Singh, A. Lynnyk, C. Autieri, and M. Matusiak, Phys. Rev. B 107, 085102 (2023).
  * Cuono  _et al._ (2023b) G. Cuono, R. M. Sattigeri, C. Autieri, and T. Dietl, Phys. Rev. B 108, 075150 (2023b).


# Elastic Properties and Glass Forming Ability of the
Zr50Cu40Ag10 Metallic Alloy

Ramil M. Khusnutdinoff1,2a and Anatolii V. Mokshin1,2b 1Kazan Federal
University, 420008 Kazan, Russia,
2Udmurt Federal Research Center of the Ural Branch of the Russian Academy of
Sciences, 426068 Izhevsk, Russia
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The elastic properties of the Zr50Cu40Ag10 metallic alloy, such as the bulk
modulus $B$, the shear modulus $G$, the Young’s modulus $E$ and the Poisson’s
ratio $\sigma$, are investigated by molecular dynamics simulation in the
temperature range $T=250-2000$ K and at an external pressure of $p=1.0$ bar.
It is shown that the liquid-glass transition is accompanied by a considerable
increase in the shear modulus $G$ and the Young’s modulus $E$ (by more than
$50\%$). The temperature dependence of the Poisson’s ratio exhibits a sharp
fall from typical values for metals of approximately $0.32-0.33$ to low values
(close to zero), which are characteristic for brittle bulk metallic glasses.
Non-monotonic temperature dependence of the longitudinal and transverse sound
velocity near the liquid-glass transition is also observed. The glass forming
ability of the alloy is evaluated in terms of the fragility index $m$. As
found, its value is $m\approx 64$ for the Zr50Cu40Ag10 metallic glass, that is
in a good agreement with the experimental data for the Zr-Cu-based metallic
glasses.

molecular dynamics, elastic moduli, amorphous metallic alloy, structural
transformations, glass forming ability

##  I Introduction

Interest to the amorphous metallic alloys is provoked by the obvious
advantages of their potential applications in various fields of techniques,
engineering and materials science. Amorphous alloys or metallic glasses are a
new class of materials with unique physical and mechanical properties, which
differ from the properties of their crystalline analogs tempbib1 ; tempbib2 .
So, for example, non-crystalline metallic alloys are characterized by high
strength and elasticity, as well as good ductility under strong deformation
effects. As a rule, metallic glasses are multi-component systems with high
glass forming ability, the disordered phase of which can be formed by means of
cooling the equilibrium melt with the cooling rates of $\gamma>10^{4}$ K/s
tempbib3 . Despite a significant number of experimental and numerical studies,
there is still no clear understanding of the mechanisms of formation of
structural heterogeneities and their influence on the elastic and mechanical
properties of metallic glasses.

The aim of this work was a thorough study of the elastic properties of the
Zr50Cu40Ag10-alloy in a wide temperature range, which includes “liquid-glass”
transition.

##  II Simulation Details

Molecular dynamics (MD) simulations of the Zr50Cu40Ag10 melt were carried out
with an isothermal-isobaric ($NPT-$) ensemble in the temperature range
$T=250-2000$ K at an external pressure of $p=1.0$ bar. To stabilize the
temperature and the pressure of the system, the Nosé-Hoover thermostat with a
relaxation parameter of $0.1$ ps and the barostat with damping parameter of
$1.0$ ps were applied. The system was consisted of $N=32000$ atoms enclosed in
a cubic cell with the periodic boundary conditions. The interactions between
atoms are described by means of the EAM model potential tempbib4 . Initial
configurations were equilibrated Zr50Cu40Ag10 melt at $T=2000$ K. Then, the
system was cooled down to $T=250$ K with the cooling rate $1.0$ K/ps tempbib5
. The integration of the equations of motion of atoms was carried out using
the velocity Verlet algorithm with a time step of $\tau=1.0$ fs. To bring the
system to a state of thermodynamic equilibrium, we performed $10^{6}$ time
steps and then $10^{7}$ steps to calculate the mechanical properties and
characteristics. All simulations were performed in the LAMMPS simulation
package tempbib6 .

##  III Results and Discussions

Fig. $1$ (a-c) shows the partial components of the radial distribution
function for the Zr50Cu40Ag10 melt at various temperatures. As seen from the
figure, all partial components $g_{\alpha,\beta}(r)$ (where
$\alpha,\beta\in\\{$Zr, Cu, Ag$\\}$) with decreasing temperature manifest the
structural transformations in the system due to the transition from the
equilibrium melt to the amorphous glassy state tempbib7 . The critical
temperature $T_{c}$ associated with liquid-glass transition was determined
using the translational order parameter tempbib2 . The temperature dependence
of this parameter in the semilogarithmic coordinates has two segments with an
intersection point at $T_{c}\approx 750$ K (see Fig. 1d).

Figure 1: Partial components of the radial distribution function and the
translational order parameter for Zr50Cu40Ag10 melt at different temperatures.

The elastic properties of the Zr50Cu40Ag10 metal melt were studied by means of
evaluation of the elastic (bulk and shear) moduli tempbib8 ; tempbib9 . For
the case of the $NPT$-ensemble, the isothermal bulk modulus is related to the
mean-square fluctuations $\sigma_{V}^{\;2}$ of the volume $V$ of the
simulation cell by the expression:

$B=\frac{k_{B}TV}{\sigma_{V}^{\;2}}.$

The shear modulus was calculated by the formula

$G=\frac{V}{k_{B}T}\left\langle\left|p_{xy}(0)\right|^{2}\right\rangle,$

where angle brackets mean averaging over time samples and $p_{xy}(0)$ are the
non-diagonal components of the pressure tensor.

To assess the deformation properties of materials, it is necessary to know the
equilibrium elastic modulus (the so-called Young’s modulus) and the transverse
strain coefficient (the so-called Poisson’s ratio). The Young’s modulus $E$
and the Poisson’s ratio $\sigma$ are related with the elastic moduli as
follows

$E=\frac{9BG}{3B+G},\;\;\;\;\sigma=\frac{3B-2G}{6B+2G}.$

Figure 2: Temperature dependences of the bulk modulus $B$, the shear modulus
$G$, the Poisson’s ratio $\sigma$ and the Young’s modulus $E$ for the
Zr50Cu40Ag10 metallic alloy.

The temperature dependences of the bulk modulus $B$, the shear modulus $G$,
the Poisson’s ratio $\sigma$ and the Young’s modulus $E$ for the Zr50Cu40Ag10
metallic alloy are plotted in Fig. 2. As seen from this figure, the
temperature dependences of the shear modulus $G$, the Poisson’s ratio $\sigma$
and the Young’s modulus $E$ are characterized by a sharp change near the
liquid-glass transition. At the same time, the bulk modulus grows
monotonically with decreasing temperature. The importance of the Poisson’s
ratio in the study of glasses was pointed out by Novikov and Sokolov tempbib10
. They found that the Poisson’s ratio is related to the fragility index $m$ of
the glass forming liquid, which can be used to account for the glass forming
ability (GFA) properties of a melt. So, for example, the Poisson’s ratio for
brittle bulk metallic glasses demonstrates a low value. It can be seen from
the Fig. 2 that glassy Zr50Cu40Ag10-alloy at the temperature $T=250$ K is
characterized by a Poisson’s ratio close to zero. The Poisson’s ratio takes
values close to zero showing very little lateral expansion when compressed.
The fragility index $m$ was calculated by the formula tempbib10

$m=17+29(B/G-1),$

where $17$ is the lowest value expected for $m$ and $B/G$ in glasses is not
expected to be lower than $1$ tempbib11 . For the Zr50Cu40Ag10-system, the
calculated value of the fragility index is $m=64\pm 4$, which is in a good
agreement with the experimental data for the Zr-Cu-based metallic glasses (see
Table 1).

Table 1: Experimental data on the fragility index for amorphous Zr-Cu-based metallic alloys tempbib12 ; tempbib13 . Alloy composition | Fragility index, $m$
---|---
Zr50Cu50 | $76\pm 10$
Zr44Cu50Ti6 | $48\pm 4$
Zr46Cu46Al8 | $63\pm 7$
Zr47Cu47Al6 | $53\pm 5$
Figure 3: Temperature dependences of the longitudinal and transverse sound
velocities for Zr50Cu40Ag10 metallic alloy.

The elastic characteristics and the velocities of the longitudinal and
transverse ultrasonic waves for an isotropic medium are related by the
expressions tempbib14 ; tempbib15 ; tempbib16

$\vartheta_{L}=\sqrt{\frac{B+4/3G}{\rho}},\;\;\;\;\vartheta_{T}=\sqrt{\frac{G}{\rho}}$

where $\rho$ is the mass density of the system. Fig. 3 depicts the obtained
temperature dependences of the speed of sound for Zr50Cu40Ag10 metallic alloy.
As seen from the figure, that the temperature dependences of the longitudinal
and transverse sound velocities for the equilibrium melt phase are monotonic.
At the same time, near the liquid-glass transition, the changes in the
temperature dependencies of both the rates $\vartheta_{L}$ and $\vartheta_{T}$
are observed.

##  IV Summary

In the present work, using molecular dynamics simulations, the elastic
properties and glass forming ability of the ternary metallic alloy are
investigated. The temperature dependences of bulk modulus $B$, the shear
modulus $G$, the Young’s modulus $E$ and the Poisson’s ratio $\sigma$, for the
Zr50Cu40Ag10 metallic alloy were obtained for the first time. It is shown that
the liquid-glass transition is accompanied by a considerable increase in the
shear modulus $G$ and the Young’s modulus $E$ (by more than $50\%$).
Temperature dependence of the Poisson’s ratio for the Zr50Cu40Ag10-system
exhibits a sharp fall from typical values for metals of approximately
$0.32-0.33$ to low values (close to zero), which are characteristic for
brittle bulk metallic glasses. It was also found that the temperature
dependence of the sound velocities undergo significant changes near the
liquid-glass transition. The calculated value of fragility index $m\approx 64$
for the Zr50Cu40Ag10-system is in a good agreement with the experimental data
for the Zr-Cu-based metallic glasses.

## Acknowledgements

This work is supported by the Russian Science Foundation (project No.
19-12-00022). The molecular dynamic simulations were performed by using the
computational cluster of Kazan Federal University and the computational
facilities of Joint Supercomputer Center of RAS.

## References

  * (1) W.H. Wang, C. Dong, C.H. Shek, Bulk metallic glasses, Mater. Sci. and Eng. 44 (2004) 45-89.
  * (2) R.M. Khusnutdinoff, A.V. Mokshin, B.A. Klumov, R.E. Ryltsev, N.M. Chtchelkatchev, Structural features and the microscopic dynamics of the three-component Zr47Cu46Al7 system: equilibrium melt, supercooled melt, and amorphous alloy, JETP. 123 (2016) 265–276.
  * (3) R.M. Khusnutdinov, A.V. Mokshin, Local structural order and single-particle dynamics in metallic glass, Bull. Russ. Acad. Sci.: Phys. 74 (2010) 640-643.
  * (4) T. Fujita, P.F. Guan, H.W. Sheng, A. Inoue, T. Sakurai, M.W. Chen, Coupling between chemical and dynamic heterogeneities in a multicomponent bulk metallic glass, Phys. Rev. B 81 (2010) 140204(1)-140204(4).
  * (5) R.M. Khusnutdinoff, A.V. Mokshin, Electrocrystallization of supercooled water confined by graphene walls, J. Cryst. Growth. 524 (2019) 125182(1)-125182(4).
  * (6) S. Plimpton, Fast Parallel Algorithms for Short-Range Molecular Dynamics, J. Comp. Phys. 117 (1995) 1-19.
  * (7) R.M. Khusnutdinoff, A.V. Mokshin, Short-range structural transformations in water at high pressures, J. Non-Cryst. Solids. 357 (2011) 1677-1684.
  * (8) A.G. Lyapin, E.L. Gromnitskaya, O.F. Yagafarov, O.V. Stal’gorova, V.V. Brazhkin, Elastic properties of crystalline and liquid gallium at high pressures, JETP. 107 (2008) 818-827.
  * (9) R.M. Khusnutdinoff, A.V. Mokshin, Electrocrystallization of supercooled water confined between graphene layers, JETP Letters. 110 (2019) 557-561.
  * (10) V.N. Novikov, A.P. Sokolov, Poisson’s ratio and the fragility of glass-forming liquids, Nature. 431 (2004) 961-963.
  * (11) A.V. Mokshin, B.N. Galimzyanov, D.T. Yarullin, Scaling relations for temperature dependences of the surface self-diffusion coefficient in crystallized molecular glasses, JETP Letters. 110 (2019) 511-516.
  * (12) S.Venkataraman, K. Biswas, B.C. Wei, D.J. Sordelet, J. Eckert, On the fragility of Cu47Ti33Zr11Ni8Si1 metallic glass, J. Phys. D: Appl. Phys. 39 (2006) 2600-2608.
  * (13) S. Pauly, J. Das, N. Mattern, D.H. Kim, J. Eckert, Phase formation and thermal stability in Cu-Zr-Ti(Al) metallic glasses, Intermetallics. 17 (2009) 453-462.
  * (14) R.M. Khusnutdinov, A.V. Mokshin, R.M. Yulmetyev, Molecular dynamics of liquid lead near its melting point, JETP. 108 (2009) 417-427.
  * (15) B.N. Galimzyanov, V.I. Ladyanov, A.V. Mokshin, Remarkable nuances of crystallization: from ordinary crystal nucleation to rival mechanisms of crystallite coalescence, J. Cryst. Growth. 526 (2019) 125214(1)-125214(4).
  * (16) A.V. Mokshin, S.O. Zabegaev, R.M. Khusnutdinoff, Dynamic Heterogeneity of a Colloidal Solution near the Sol-Gel Transition, Phys. Solid State. 53 (2011) 570-576.