#  $Ab-initio$ Prediction of Mechanical, Electronic, Magnetic and Transport
Properties of Bulk and Heterostructure of a Novel Fe-Cr based Full Heusler
Chalcogenide

Joydipto Bhattacharya1,2, Rajeev Dutt1,2, Aparna Chakrabarti1,2 1 Raja Ramanna
Centre for Advanced Technology, Indore-452013, India  2Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai-400094, India

###### Abstract

Using electronic structure calculations based on density functional theory, we
predict and study the structural, mechanical, electronic, magnetic and
transport properties of a new full Heusler chalcogenide, namely, Fe2CrTe, both
in bulk and heterostructure form. The system shows a ferromagnetic and half-
metallic(HM) like behavior, with a very high (about 95%) spin polarization at
the Fermi level, in its cubic phase. Interestingly, under tetragonal
distortion, a clear minimum (with almost the same energy as the cubic phase)
has also been found, at a c/a value of $\sim$1.26, which, however, shows a
ferrimagnetic and fully metallic nature. The compound has been found to be
dynamically stable in both the phases against the lattice vibration. The
elastic properties indicate that the compound is mechanically stable in both
the phases, following the stability criteria of the cubic and tetragonal
phases. The elastic parameters unveil the mechanically anisotropic and ductile
nature of the alloy system. Due to the HM-like behavior of the cubic phase and
keeping in mind the practical aspects, we probe the effect of strain as well
as substrate on various physical properties of this alloy. Transmission
profile of the Fe2CrTe/MgO/Fe2CrTe heterojunction has been calculated to probe
it as a magnetic tunneling junction (MTJ) material in both the cubic and
tetragonal phases. Considerably large tunneling magnetoresistance ratio (TMR)
of $\approx 10^{3}$ % is observed for the tetragonal phase, which is found to
be one order of magnitude larger than that of the cubic phase.

##  I Introduction

Half-metallic (HM) ferromagnets (FM) have become a topic of active research
due to their potential for various technological applications. Theoretically,
the HMFM materials are shown to exhibit 100% spin polarization (SP) at the
Fermi level (EF), with one of the spin channels showing semi-conducting (SC)
and the other one possessing metallic nature. In 1983, half-metallicity has
been predicted in some half Heusler alloys (HHA), namely, NiMnSb and its
isoelectronic compounds, PtMnSb and PdMnSb.de Groot  _et al._ (1983) Ever
since this study has been published in the literature, the field of HM Heusler
alloys (HMHA) has attracted immense attention of the researchers, both
theoreticians and experimentalists alike. As a result, innumerable studies on
half-metallic half as well as full Heusler alloys (FHA) have come to the fore,
in order to either understand some fundamental aspects or explore their
potential for various applications.Webster (1971); Miura  _et al._ (2008a);
Hülsen  _et al._ (2009); Roy  _et al._ (2016); Enamullah and Lee (2018); Baral
and Chakrabarti (2019); Kang and Zhang (2017); Kervan and Kervan (2011); Zhu
_et al._ (2014)

It has been seen in the literature that typically many of the FHAs exhibit
metallic nature, while a large number of HHAs are SC in nature. Many works
have been carried out to show that the number of valence electrons (nv) plays
a crucial role in defining the electronic as well as magnetic properties of
materials. The Slater-Pauling rule established the relation between nv and the
magnetic moment of a transition metal.Slater (1936); Pauling (1938) Further,
for Heusler alloys, specifically for Co-based ones, Slater-Pauling behavior
has been seen, as reported in the literature.Galanakis  _et al._ (2002); Luo
_et al._ (2008) It has been observed that among the FHAs, primarily Co-based
alloys show HM behavior with a typical value of valence electrons (nv) of 26
to 28. As discussed above, searching for a new or novel HMHA, which has high
to very high (preferably 100 %) spin polarization at EF, recently become of
utmost importance, both from the points of view of fundamental understanding
and technological application.Hirohata  _et al._ (2013); Elphick  _et al._
(2021); Bainsla and Suresh (2016) To this end, we take a combination of Fe, Cr
and Te atoms, yielding a FHA, Fe2CrTe, with a nv value of 28. This alloy
contains no Co atom, on the other hand, contains a chalcogen atom,Te. In the
recent past, chalcogen atoms have been shown to be elements of interestDutt
_et al._ (2022); Pandey and Chakrabarti (2019); Pandey  _et al._ (2021); Ru
and Fisher (2006); Zhu  _et al._ (2016). From our $ab-initio$ electronic
structure calculations, it turns out that Fe2CrTe alloy is expected to show a
HM-like behavior. The HM properties have been observed to be greatly
influenced by defects, surfaces and interfaces. In this context, the study of
the effect of strain as well as substrate on various physical properties of a
HMHA can be interesting and important from practical application points of
view. Hence, we embark upon the same in the present work. We apply uniform
isotropic strain and also apply a bi-axial strain by putting the cubic alloy
on a well-known and lattice-matched SC substrate, namely, MgO.

In one of our recent works, we have probed the possibility of coexistence of
half-metallicity and tetragonal (martensite) transition in a series of Ni- and
Co-based FM FHAs, including Ni2MnGa and a few other well-known alloys.Roy  _et
al._ (2016) As a martensite phase transition (MPT) indicates occurrence of a
shape memory behavior in a magnetic alloy and a HMFM alloy has the possibility
of application in the field of spintronics, studying both these aspects is
important. Since typically it has been observed that the MPT and HM behaviors
are not seen in the same material, we have dwelved in the study of the same in
the past.Roy  _et al._ (2016) We predicted a novel Co-based FHA (Co2MoGa),
which exhibited a tendency of MPT and also a HM-like behavior. Three of the
other studied FHAs have shown a very clear local minimum in the energy versus
c/a plot, with a range of values of c/a ratio ($\sim$1.25 to 1.35). While the
clear display of a tetragonal (martensite) phase is well-known and it is well-
studied in many of the FHAsRoy  _et al._ (2016); Chakrabarti  _et al._ (2013);
Barman  _et al._ (2005); Ducher  _et al._ (2008); Siewert  _et al._ (2011);
Roy and Chakrabarti (2016), the appearance of a minimum at a c/a ratio other
than 1, in an otherwise cubic alloy, has seldom been observed.Roy  _et al._
(2016) In this work, we explore the possibility of a tetragonal distortion for
the FHA Fe2CrTe and find that along with a cubic (austenite) phase, a clear
minimum is observed for a tetragonal phase with a c/a ratio of $\sim$ 1.26,
leading to a double-minima like structure in the energy versus c/a plot.

For spintronic devices, in recent times, an extensive search for new materials
suitable for magnetic tunneling junction (MTJ) is going on.Feng  _et al._
(2022); Cui  _et al._ (2021) These heterojunctions comprise two FM electrode
materials and a non-magnetic insulator or semi-conducting spacer material in
between the two electrodes. In these systems, the tunnenling magnetoresistance
(TMR) is strongly dependent on the relative spin orientations of both the
electrodes (parallel or anti-parallel). The TMR ratio has been defined as the
difference in conductance of the MTJ in two different magnetic orientations
divided by the smaller value. This ratio can in principle (theoretically) be
infinitely large if a HMFM material is used as an electrode in the MTJ. In the
literature, many studies have been reported where Co-based HMFM alloys have
been used. Most of these studies include a thin insulating layer of MgO as the
barrier/spacer material and favorable tunneling properties have been observed
in these MTJs.Miura  _et al._ (2011); Hülsen  _et al._ (2009); Han  _et al._
(2017); Gercsi  _et al._ (2006); Roy  _et al._ (2020) As Fe2CrTe is expected
to exhibit a HM-like character, we first probe the magnetic properties of the
Fe2CrTe thin film (13 monolayers (ML)) with 5 (7) ML of MgO as a substrate
material. We find that the SP at EF is about 75% (70%), when 5 (7) ML of MgO
substrate. We further calculate the transmission properties of the
heterojunctions Fe2CrTe/MgO/Fe2CrTe to understand and explore the potential
for MTJ application.

In the next section, we discuss the method of electronic structure
calculations, which is based on density functional theory. We also briefly
discuss the calculational method related to transport properties. In the
section followed by methodology, we present our results and discuss the same.
Finally, we summarize and conclude our work in the last section.

##  II Method

The FHAs are known to exist either in a conventional or an inverse Heusler
alloy structure. From the structure optimization, we find that in the lowest
energy state, Fe2CrTe alloy possesses the conventional structure with a
$L2_{1}$ phase that consists of four interpenetrating face-centered-cubic
(fcc) sub-lattices with origin at the following fractional positions: Fe atom
at (0.25, 0.25, 0.25) and (0.75, 0.75, 0.75) sites, Cr atom at (0.5,0.5,0.5)
site and Te atom at (0,0,0) site. The structure has been optimized by doing
full geometry optimization using Vienna Ab Initio Simulation Package
(VASP)Kresse and Furthmüller (1996); Kresse and Joubert (1999) with the
projector augmented wave (PAW) method.Kresse and Furthmüller (1996); Kresse
and Joubert (1999) For exchange-correlation (XC) functional, generalized
gradient approximation (GGA) over the local density approximation has been
used.Perdew  _et al._ (1996) We use an energy cutoff of 500 eV for the
planewaves. The final energies have been calculated with a $k$ mesh of
15$\times$15$\times$15 for the cubic symmetry and an equivalent number of
k-points for the tetragonal symmetry. The energy and force tolerance for our
calculations were 1 $\mu$eV and 20 meV/Å, respectively. For obtaining the
electronic properties, the Brillouin zone integration has been carried out
using the tetrahedron method with Blöchl corrections. The directional
dependencies of different mechanical properties (Young’s modulus, inverse of
bulk modulus or compressibility, shear modulus, and Poisson’s ratio) of this
alloy in both the cubic and tetragonal phases have been calculated with the
help of the ELATE software.Marmier  _et al._ (2010) For the calculation of
transport properties of the heterojunction of Fe2CrTe and MgO, we make use of
the PWCOND codeSmogunov  _et al._ (2004), which has been implemented in the
Quantum ESPRESSO (QE) packageBaroni  _et al._ (2001). The spin-dependent
tunneling conductance has been calculated using the Landauer formula:

$G^{\sigma}=\frac{e^{2}}{h}\sum_{K_{||}}T^{\sigma}(K_{||},E)$

here $\sigma(=\uparrow,\downarrow$) is the spin index and
$T^{\sigma}(K_{||},E)$ is the spin-dependent transmission coefficient at a
particular energy value $E$, with $K_{||}=(K_{x},K_{y})$, where $K$ are the
wave-vectors, x and y are directions.

In the literature, Choi and IhmJoon Choi and Ihm (1999) have given a method to
calculate $T^{\sigma}(K_{||},E)$. We perfom this calculation for the optimized
geometry using the GGA exchange functional.Perdew  _et al._ (1996) We have
taken the cut-off energy for the wave function and the charge density as 60
and 600 Ry, respectively. A mesh of 12$\times$12$\times$1 k-points have been
used for the self-consistent-field(SCF) calculation of Fe2CrTe/MgO/Fe2CrTe
heterojunction. A high tolerance (10-8 Ry) and a large k-mesh (100 points in
both $x$ and $y$ directions) have been taken, which are required to capture
the fine spikes in a transmission.Karki  _et al._ (2021) The scalar
relativistic ultrasoft pseudopotentials(USPP) with the GGA exchange-
correlation term have been used, as obtained from the PSLibrary 1.0.0. For
further details of calculation of ballistic conductance see Ref.Joon Choi and
Ihm (1999). Convergence of all the relevant parameters for VASP and QE
packages have been tested before embarking upon the calculations of the
physical properties.

##  III Results and Discussion

In this section, first we predict the energetic stability of the cubic phase
of Fe2CrTe alloy and then explore the possibility of a stable tetragonal
(martensite) phase of this alloy, where the c/a ratio (a and c being the
lattice constants along the x and z-directions) has been varied, keeping the
volume fixed. We then calculate and present the dynamical and mechanical
properties of these two phases. The mechanical properties of the well-known HA
Ni2MnGa have been discussed in some places, for the sake of comparison.
Further, we discuss the electronic and magnetic properties of the cubic and
tetragonal phases. Due to the near HM behavior of the cubic phase, keeping in
mind the practical aspects, we probe this phase further. An important aspect
in materials growth is strain. Hence, we simulate and discuss the effect of
isotropic strain on the electronic and magnetic properties of cubic Fe2CrTe
alloy. Further, to check the effect of substrate (leading to bi-axial strain),
we calculate the physical properties of the thin film (having 13 ML) of the
cubic Fe2CrTe alloy, supported by a suitably lattice-matched substrate MgO
(probed both 5 and 7 ML). Finally, the reasonably high SP at EF (75%) of the
alloy when interfaced with MgO motivated us to perform the calculation of
transmission properties of the Fe2CrTe/MgO/Fe2CrTe heterojunctions. We discuss
and analyze the results of this heterojunction to probe its MTJ properties.

Figure 1: Bulk crystal structure shown in (a) cubic and (b) tetragonal phase.
(c) shows the difference in total energy (between cubic and tetragonal phases)
with respect to c/a ratio for bulk Fe2CrTe, where the energy for the bulk
cubic phase has been taken as reference. Since we normalize the difference
with respect to the energy of the cubic phase, the cubic phase (c/a = 1)
corresponds to an energy value of 0 eV. (d) Shows the temperature dependent
Gibbs free energy, for the cubic and tetragonal phases.

###  III.1 Bulk Physical Properties

####  III.1.1 Energetic and Dynamical Stability

Energetics -  The binding energy(BE) of the cubic phase of Fe2CrTe has been
found to be -4.0252 eV per atom (for Ni2MnGa the BE is -4.3963 eV per atom)
and this indicates a stable alloy. As discussed above, the Fe2CrTe alloy
exhibits a lowest energy state in the conventional cubic Heusler alloy
structure (Figure 1(a)), like the well-known Ni2MnGa compound, and it has a
lattice constant of 5.95 Å. In order to assess if a stable tetragonal
(martensite) phase is possible in this system or not, we vary the c/a value,
by keeping the volume same as the cubic phase, since MPT is known to be a
volume conserving transition. In Figure 1(b) we show a schematic figure of the
tetragonal phase with the optimized c/a ratio of $\sim$1.26. In Figure 1(c),
we plot the energy difference between the cubic and tetragonally distorted
phase as a function of c/a. We find that the cubic (austenite) phase is very
close to the energy of the tetragonal (martensite) phase (lowest energy state)
showing a double-minima like plot and the energy difference is as small as
$\sim$1 meV per atom. In order to cross-check this interesting observation, we
carry out an all-electron calculation employing WIEN2k programme packageBlaha
_et al._ (2019) using the GGA XC term. A plot with a double-minima structure
is found in this case too. We find that the energy trend is reversed but the
energy difference between the two phases continues to be very small ($\sim$12
meV per atom). Similar observation of a reversal of energy ordering for very
small energy difference between two phases obtained from an all-electron and a
pseudopotential calculation has already been reported in the
literature.Siewert  _et al._ (2010) Since the energy difference (between the
cubic and the tetragonal phases) is very small and the difference between the
phycical properties, such as total magnetic moment and density of states (DOS)
are insignificant, when results from both the methods are compared, in this
work, we continue to consider and present the results of physical properties,
obtained from VASP.Kresse and Joubert (1999); Kresse and Furthmüller (1996)

In order to understand the energetics further and to probe the possibility of
MPT, we calculate the Gibbs free energy for both the cubic and tetragonal
states as a function of temperature (Figure 1(d)). Our results show that
interestingly, there is no crossing of the curves, which indicates that no MPT
is possible in this alloy. This would bring us to the conjecture that since
the two phases are energetically rather close and no MPT seems feasible, both
these phases will compete and will have equal possibility to form, depending
upon the growth conditions. During growth, possibility of finding a material
in two (or more) different symmetries has already been discussed by us in our
group and the experimental references therein.Baral and Chakrabarti (2019)
However, it may be noted from Figure 1(d) that, at higher temperatures, cubic
phase has a slight edge over the tetragonal phase.

Figure 2: Schematic representation of phonon dispersion relations and atom
projected vibrational density of states (VDOS) of (a) cubic and (b) tetragonal
unit cell. LA, TA1 and TA2 represent the longitudinal and transverse acoustic
branches, respectively.

Lattice Dynamical Stability -  To probe this, we calculate the phonon
dispersion curve for the cubic and tetragonal phases and the results are
presented in Figure 2. For the phonon calculations, a 4$\times$4$\times$4
supercell is taken. The finite displacement method within the phonopy codeTogo
and Tanaka (2015) has been employed in order to obtain the phonon dispersion
of the material. From Figure 2 it can be clearly seen that, all the
frequencies are positive for both the phases, which is an important
prerequisite for the lattice dynamical stability of a material. If the phonon
spectra of Fe2CrTe are compared with other previously studied FHAsPaul  _et
al._ (2014); Zayak  _et al._ (2005), instability in their respective cubic
phases are observed. These instabilities are led by the anomalous behavior of
the acoustic TA2 branch along the $\Gamma$ to X direction. However, in the
present case, we do not observe clear anomalous dips in the acoustic TA2
branch in the cubic phase (Figure 2). In the cubic phase, along the $\Gamma$
to X direction, the TA1 and TA2 branches are found to be degenerate and the
degeneracy is lifted in the tetragonal phase. The atom projected vibrational
density of states (VDOS) for the cubic and tetgragonal phases (shown in Figure
2 (a) and (b) ) are consistent with our discussion on the phonon spectra. As
expected the Te atoms dominate the vibration at lower frequency range, whereas
at the higher frequency range, Cr atoms dominate the vibration for both the
phases. Hence, in Fe2CrTe, in both the phases, the sequence of the optical
vibration is regular, i.e. decreasing as the mass of the atom is increasing.
This behavior is quite different from some other well-known FHAs, which show
instability in the cubic (austenite) phase. In those systems, VDOS show
unexpected anomalous behavior in the cubic phase, where optical vibration of
the lighter atom is seen lying below the optical vibration of the heavier
atom.Paul  _et al._ (2014); Zayak  _et al._ (2005) These observations
collectively indicate that Fe2CrTe is stable in both the phases, which gets
further support from the variation of the Gibbs free energy with temperature
in these two phases, as discussed before.

####  III.1.2 Mechanical Properties

Having discussed the stability of the alloy, we now proceed to present and
discuss the mechanical properties. For the details of the mathematical
expressions for all the elastic constants and parameters, we refer to our
earlier workRoy  _et al._ (2015) and the references therein.

Table 1: Calculated values of elastic constants ($C_{ij}$ in GPa) for Fe2CrTe in cubic and tetragonal phases. Computed values of bulk modulus ($B$ in GPa), Young’s modulus ($E$ in GPa), Shear modulus ($G$ in GPa), Poisson’s Ratio ($\sigma$), Cauchy pressure ($C_{P}$ in GPa) are also tabulated. $B$ and $G$ are calculated using the formalism given by Hill.Hill (1952) Phase | $C_{11}$ | $C_{12}$ | $C_{13}$ | $C_{33}$ | $C_{44}$ | $C_{66}$ | $C^{{}^{\prime}}$ | $B$ | $E$ | $G$ | $\sigma$ | $C_{P}$ |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Cubic | 208.86 | 182.96 | - | - | 102.89 | - | 13 | 191.60 | 130.52 | 47.07 | 0.38 | 80.10 |  |
Tetragonal | 330.830 | 82.493 | 136.696 | 271.440 | 93.966 | 72.366 | 124.17 | 182.76 | 228.63 | 88.51 | 0.29 | -11.50 |  |

In order to study the mechanical stability criteria, we calculate the elastic
constants for Fe2CrTe alloy in cubic phase. It is well-known that there are
only three independent ones as $C_{11}$ = $C_{22}$ = $C_{33}$, $C_{12}$ =
$C_{13}$ = $C_{23}$ and $C_{44}$ = $C_{55}$ = $C_{66}$ in the cubic phase. We
list the three constants $C_{11}$, $C_{12}$ and $C_{44}$ in Table 1. It is
well-known that if a cubic system fulfills the following stability criteria,
then the structure is mechanically stable.Zhi-jian  _et al._ (2007)
$C_{11}$ \- $C_{12}>$ 0 ; $C_{11}+2C_{12}>$ 0 ; $C_{11}>C_{44}>$ 0

In case of Fe2CrTe, from the results presented in Table 1, we find that all
the above-mentioned criteria are satisfied, suggesting that it is a
mechanically stable alloy in the cubic phase. Further, we calculate the
tetragonal shear constant ($C^{\prime}$) and the Zener ratio (or the elastic
anisotropy parameter: $A_{e}$) which are defined as:
$C^{\prime}$ = ($C_{11}$ \- $C_{12}$)/2
$A_{e}$ = $\frac{2\times C_{44}}{C_{11}-C_{12}}$

We find that while $C^{\prime}$ has a value of $\sim$ 13, $A_{e}$ possesses a
value of $\sim$8\. It has been established in the literature that a negative
or very small positive value of $C^{\prime}$ indicates an unstable cubic
phase, as has been observed for Ni2MnGa (a value close to 5 has been obtained
both from experiments and theory)Ozdemir Kart and Cagın (2010); Roy  _et al._
(2015) which has a non-cubic ground state.Barman  _et al._ (2005) The
calculated value for Fe2CrTe alloy can be considered to be somewhat small, as
it has been seen that the typical HM systems like Co2VGaKanomata  _et al._
(2010), which exhibit cubic ground state, show much higher $C^{\prime}$
values.Roy  _et al._ (2016) Further, we find that the $A_{e}$ value turns out
to be much larger than 1. It is well-known that materials with an $A_{e}$
value much different from 1 often shows the tendency to deviate from the cubic
symmetry and may suggest instability in the cubic phase.Luan  _et al._ (2018)
Presence of a minimum in energy for a tetragonal symmetry (Figure 1(c)) seems
to be consistent with the low positive $C^{\prime}$ and large $A_{e}$ values
corresponding to the cubic bulk phase of Fe2CrTe alloy.

Now we discuss the mechanical properties, which are important and most-
discussed in the literature, namely, the bulk and shear modulii, Young’s
modulus (E) and Poisson’s ratio ($\sigma$), which are often used to describe
the ductility, mallieability and overall mechanical stability of a material.
Table 1 presents these values (all the values being rounded off up to second
decimal place). For the cubic phase, we find that the values of bulk, shear
and Young’s modulii are somewhat higher than those of Ni2MnGa.Ozdemir Kart and
Cagın (2010); Roy  _et al._ (2015) This result suggests that in case of
tensile, volumetric and shear strains, the present alloy is slightly less
compressible compared to Ni2MnGa. In other words, it is expected to provide
larger deformation resistance. However, since the $\sigma$ value is very close
to that of Ni2MnGa as well as many of the metals, Fe2CrTe is expected to
behave similar to the common metals and well-known HAs in terms of
compressibility. A similar conclusion can be drawn from the calculated Pugh’s
ratio ($B$/$G$), which has been found to be $\sim$2.77. On an empirical level,
a material with a value higher than the critical value of 1.75 can be
considered to have less inherent crystalline brittleness (ICB).Pugh (1954)
Further, according to PettiforPettifor (1992), a metal typically exhibits a
high positive value of Cauchy pressure. With a value of $\sim$80, which is
higher than that of Ni2MnGaRoy  _et al._ (2015), more metallic than
directional bonding is expected in case of Fe2CrTe alloy. The Kleinman
parameter ($\zeta$) is a dimensionless parameter which corresponds to the
relative ease of bond bending to that of bond stretching Naher and Naqib
(2021); Han  _et al._ (2018). Under given stress, bond stretching (bending)
dominates if $\zeta$ is closer to 0.0 (1.0). We find a value close to 1.
Hence, bond lengths are excepted to be largely unchanged if the system is
distorted.

Next we analyze the elastic stability and mechanical properties of the
tetragonal phase. Table 1 also lists the relevant parameters for the
tetragonal phase. The mechanical stability of tetragonal materials correspond
to the following conditions: (1) all of $C_{11}$, $C_{33}$, $C_{44}$,
$C_{66}>$ 0; (2)($C_{11}$ \- $C_{12}$) $>$ 0; (3)($C_{11}$ \+ $C_{33}$ \-
2$C_{13}$) $>$ 0; (4) (2($C_{11}$ \+ $C_{12}$) + $C_{33}$ \+ 4$C_{13}$) $>$
0.Zhi-jian  _et al._ (2007) We find all the criteria are fulfilled and hence
the tetragonal phase of the material is mechanically stable. $C^{\prime}$
value is high and positive, indicating a stable tetragonal phase is possible
from the mechanical point of view. However, larger resistance to deformation
is indicated by the increased values of $B$, $G$ and $E$, compared to the
cubic phase. Though the $B$/$G$ value exhibits similar value as the cubic
phase, the $C_{P}$ value becomes negative, indicating the presence of a less
metallic and more directional bonding in the tetragonal phase.

Table 2: Maximum and minimum values of $E$ (in GPa), $\beta$ (in TPa-1), $G$ (in GPa) and $\sigma$ for cubic and tetragonal phases of bulk Fe2CrTe alloy. Phase | $E$ | $\beta$ | $\sigma$ | $G$
---|---|---|---|---
| Max | Min | Max | Min | Max | Min | Max | Min
Cubic | 261.742 | 37.994 | 1.740 | 1.740 | 1.301 | -0.485 | 102.891 | 12.950
Tetragonal | 264.527 | 181.023 | 1.870 | 1.801 | 0.477 | 0.052 | 124.168 | 69.220

Anisotropic character of mechanical properties - It is well-known that if the
value of $A_{e}$ is 1, the Young’s modulus turns out to be isotropic in
nature. As we obtain an $A_{e}$ value much larger than 1, we calculate the
maximum and minimum values as well as study the three-dimensional characters
to probe the direction-dependent nature of various mechanical properties,
including the Young’s modulus. Table 2 gives the maximum and minimum values of
Young’s modulus ($E$), compressibility ($\beta$), shear modulus ($G$) and
Poisson’s ratio ($\sigma$). We find that except $\beta$, maximum and minimum
values for rest of the parameters differ from each other. Further, we probe
the directional mechanical properties in two (2D) and three (3D) dimensions.
The results (presented in Figures S1 and S2Bhattacharya and Chakrabarti  for
cubic and tetragonal phases, respectively) establish the anisotropic nature,
more for cubic phase.

####  III.1.3 Electronic and Magnetic Properties

Figure 3: Bulk electronic structure of cubic Fe2CrTe: (a) and (c) depict
orbital projected band structure of minority and majority electrons and (b)
gives atom projected density of states. Here $\Delta_{1}(s,p_{z},d_{z^{2}})$,
$\Delta_{2}(d_{xy},d_{x^{2}-y^{2}})$ and
$\Delta_{5}(p_{x},p_{y},d_{xz},d_{yz})$ represent the orbital symmetries of
the bands. Figure 4: Bulk electronic structure of tetragonal Fe2CrTe: (a) and
(c) show orbital projected band structure of minority and majority electrons
and (b) presents atom projected density of states. Here
$\Delta_{1}(s,p_{z},d_{z^{2}})$ , $\Delta_{2}(d_{xy},d_{x^{2}-y^{2}})$ and
$\Delta_{5}(p_{x},p_{y},d_{xz},d_{yz})$ represent the orbital symmetries of
the bands.

DOS and Band Structure - In Figure 3, we plot the density of states (DOS) and
the band structures of the cubic phase. From the total density of states, we
find that there is a high SP at EF. It is found to be about 95%. The EF for
the equilibrium lattice constant is found to be located on the valence band
edge. The majority spin DOS in the occupied region near EF has dominant
contribution from the Cr and Fe d states, with a small contribution from the
Te atoms (Figure 3(b)). On the contrary, in case of the minority spin, Fe DOS
contributes much more near and at the EF than the other atoms. These Fe d
states in the minority spin channel leads to the reduction of the SP at EF
from 100 to 95%. Further, the unoccupied states in the conduction bands near
EF also has dominant Fe-d character. The orbital projected band structure of
Fe2CrTe in the cubic phase is shown in Figure 3(a), (c). The $\Delta_{1}$
symmetry is associated with the $s,p_{z},d_{z^{2}}$ orbital character. In
contrast, $p_{x},p_{y},d_{xz},d_{yz}$ orbitals specify $\Delta_{5}$ symmetry
and $d_{xy},d_{x^{2}-y^{2}}$ orbitals are assigned to the $\Delta_{2}$
symmetry. We can see the presence of highly disperssive parabolic electron
like conduction band in the both majority and minority spin channels around
the $\Gamma$ point and these bands have dominant contribution from
$\Delta_{1}$ symmetric bands, originating from Te 5s states. However, we do
not observe any bands crossing the EF along the $\Gamma$ to X ($i.e.$ along
$<001>$) direction in the majority spin channel (Figure 3(c)), which has a
great importance in the spin dependent transport properties and we discuss it
in the later part of our paper. However, in the minority spin channel, we find
two bands cross the EF, which have dominant $\Delta_{5}$ and $\Delta_{2}$
orbital symmetries. Further, in the minority spin case the top of the valence
band just touches the EF, leading to negligible DOS at EF.

In case of the tetragonal phase (Figure 4), while the majority DOS
contributions near the FF remain similar (both from Fe and Cr d states), the
minority DOS at (and also above) EF gets slightly populated by both Fe and Cr
d states, Fe states having more contribution. Due to the increased DOS at EF
for the minority channel, the SP at EF gets reduced significantly (67%). Other
than at the EF, overall DOS of the tetragonal phase for the majority and
minority spin channels also show that the peak positions are rather close to
each other unlike the cubic case. This leads to a much lower magnetic moment
for the former phase. Further, the orbital projected band structure in Figure
4(a), (c) suggests that parabolic conduction and valence bands around the
$\Gamma$ point in the minority spin channel is pushed away from the EF and
move towards the higher (lower) energy side for the conduction (valence) bands
when compared to the cubic case. From the symmetry analyses, we confirm the
presence of $\Delta_{1}$ symmetric band along the $\Gamma$ \- M direction
($i.e$ along the $<001>$ direction), in both the spin channels. We have
further found that there are spaghetti of bands around the high symmetry point
$\Gamma$ in both the spin channels, increasing the valley degeneracy at
$\Gamma$.

Figure 5: Atom and orbital projected DOS (PDOS) of Fe2CrTe: (a) cubic, (b)
tetragonal Phase, respectively. The eg states have d${}_{x^{2}-y^{2}}$ and
d${}_{z^{2}}$ orbital contributions and the t2g states have dxy, dxz and dyz
orbital contributions.

Figure 5 shows the DOS contribution of the various d states of the transition
metal elements (Fe and Cr) as these populate the DOS near and at EF. The
splitting of the eg and t2g like states in the cubic case, specifically in
case of Cr, which is in an ocatehedral symmetry, being surrounded by Fe atoms,
is clearly visible from Figure 5 (bottom panel). Although the Fe atoms are the
2nd nearest neighbor of other Fe atoms, the hybridization between them is
qualitatively more important. The t2g and eg like states for Fe atoms, which
is in a tetrahedral symmetry, due to the four transition metal atoms Cr as
neighbor can be seen from Figure 5 (middle panel). This aspect of crystal-
field splitting of the t2g and eg like states has been shown to play an
important role in yielding a HM like behavior in cubic half and full Heusler
alloys.Baral and Chakrabarti (2019); Galanakis  _et al._ (2002) The DOS of Cr
atom shows much larger energy gap ($i.e.$ larger t2g and eg splitting) around
the EF, larger than what has been observed for Fe2CrTe bulk (Figure 5 (top
panel)). However, the real gap is determined by the Fe-Fe interaction and the
t2g, eg splitting of the Fe atoms. On the contrary, for the case of tetragonal
symmetry, due to the absence of clear octahedral/tetrahedral symmetric
geometrical environment for the magnetic elements, and consequent absence of
crystal-field effect, the splitting between t2g and eg like states is not
clear (Figure 5 (b)). This might have led to the smaller SP value at EF as
compared to the cubic case.

Figure 6: Spin-polarized Fermi surface of Fe2CrTe: (a) and (b) panels
correspond to the cubic and tetragonal phases, respectively. The $\uparrow$
and $\downarrow$ represent Fermi surface due to majority and minority
electrons, respectively. We have also mentioned the band index of the bands
crossing the EF, which constitute the Fermi surfaces and those particular
bands are shown in Figures S6 and S7Bhattacharya and Chakrabarti  for the
cubic and tetragonal phase, respectively.

Effect of Hubbard U term - We have also addressed the role of onsite Coulomb
interaction (Hubbard U) on the electronic and magnetic properties of Fe2CrTe
in both the phases. The electron - electron Coulomb interaction and the self-
interaction correction are considered in the rotationally invariant way
(GGA+U) according to the Dudarev’s method Dudarev  _et al._ (1998). We have
considered the U value of Fe and Cr to be 3 eV and 2 eV, respectively. This is
in accordance with previous studies reported in the literature.Cox  _et al._
(1974); Paudel  _et al._ (2020); Chakrabarti  _et al._ (2012). The results are
found to be significantly different for both the phases. In the cubic phase
the SP at EF is found to be changed drastically with the Hubbard U parameter
(Table S1).Bhattacharya and Chakrabarti  Further analyses on atom-projected
DOS indicate that electronic and magnetic properties show significant
dependence on UFe and minimal dependence on UCr(Figure S3).Bhattacharya and
Chakrabarti  However, the SP value in the tetragonal phase shows moderate
change over the range of UFe and UCr values, considered in our calculation
(Table S1).Bhattacharya and Chakrabarti  and the electronic structure is found
to be less affected as compared to the cubic phase (Figure S3).Bhattacharya
and Chakrabarti  Usually the strength of Hubbard U for each atom in different
local environments can be easily estimated by seeking a good agreement between
the calculated and the experimental results. However, due to the predictive
nature of the present work, the present results await experimental validation.

Fermi Surface -  We present the results of calculated Fermi surfaces(FS) for
the cubic and tetragonal phases of Fe2CrTe alloy, for both majority and
minority electrons in Figure 6. Further, bands with the respective band
indices are shown in Figures S6 and S7Bhattacharya and Chakrabarti , whereas
Figure.S5 shows the positions of the high symmetry k-points in the irreducible
Brillouin zone.Bhattacharya and Chakrabarti  By analyzing these figures, we
observe the following for the cubic phase. The minority spin channel has three
bands, 19, 20 and 21, which are mostly Fe derived (small contributions from
the Te atoms) and cross EF. While the former two bands give rise to only a
carrier pocket at the $\Gamma$ point, band number 21 additionally shows a
pocket at the X point. On the other hand, in the majority spin channel the
bands, 25, 26 and 27 are mostly Cr derived and have small contributions from
Fe and Te atoms, specially for band 26. The FS due to band 26 forms a open
spherical cone, indicating an electron-like behavior, whereas the FS is hole-
like for band 25. Apart from that, small electron-like pockets are also
observed at the W point, due to band 27.

In the tetragonal phase, the character of the minority FS is shared by both Fe
d and Cr d electrons (see Figure 4). The minority spins generate very small
electron-like pockets at the X point and hole-like pockets at the $\Gamma$
point. But the majority spin FS undergoes significant changes as we go from
cubic to tetragonal phase. The majority FS due to band 25, produces small
electron-like pockets at the high symmetry points X and N. Large hole-like
pockets, centered around the $\Gamma$ point, can be seen due to band 24. This
drastic change in the majority spin FS (as we go from cubic to tetragonal
phase) is also accompanied by a change in the spin magnetic moment, as seen
for the Fe atoms, which we will discuss in the next section.

Table 3: Calculated values of total and atom-projected moments (in $\mu_{B}$) for both the cubic and tetragonal phases. Phase |  |  $\mu_{T}$/f.u. | $\mu_{Fe}$ | $\mu_{Cr}$ | $\mu_{Te}$ |
---|---|---|---|---|---|---
Cubic | GGA | 3.99 | 0.89 | 2.22 | -0.003 |
Tetragonal | GGA | 1.99 | -0.23 | 2.31 | 0.004 |
Figure 7: (a) and (b) exhibit variation of total and partial magnetic moments,
respectively, plotted as a function of c/a.

Magnetic Properties -  As it is clear from Table 3, the total magnetic moment
of the system significantly reduces in the tetragonal phase. This happens due
to the change in the partial magnetic moment of the Fe atom. From an overall
ferromagnetic coupling observed in the cubic phase, due to the negative moment
of Fe, the system assumes an overall ferrimagnetic configuration in the
tetragonal phase. The partial moment of the Cr atom remains quite similar in
both the cubic and non-cubic cases. The spin-polarized DOS of the d states of
the Fe and Cr atom (Figure 5) give clear indication of these. To explore this
in more detail, in Figure 7, we plot the total and partial magnetic moments of
both the atoms, as a function of c/a values. It is observed that at about c/a
of $\sim$1.15, a transition is observed for both the total and partial atomic
moments. While the partial moment value of Fe goes from positive ($\sim$1
$\mu_{B}$) to a negative value (-0.23 $\mu_{B}$), partial moment of Cr atom
make a transition of value of about 2.1 to close to 2.4 $\mu_{B}$. Hence, it
is clear that though Cr is the main moment-carrying atom in both the phases,
it is the partial moment of Fe atom which leads to a change in magnetic
configuration of the system. Additionally, it appears that such a transition
of magnetic configuration from ferromagnetic in the cubic phase to
ferrimagnetic in the tetragonal phase is independent of the Hubbard U
parameter used in this study (Figure S4).Bhattacharya and Chakrabarti

Figure 8: Spin-polarized density of states for tetragonal distortion in
Fe2CrTe for different c/a values. (a) presents orbital projected density of
states for both the Cr and Fe atoms and (b) shows the DOS of d orbitals of the
Fe atoms corresponding to minority and majority electrons, respectively.

In order to understand the drastic change in the magnetic properties at a
particular value of c/a, in Figure 8, we plot the spin-polarized DOS for
different c/a values around the value of c/a = 1.15. We find that the Cr atom
projected DOS (PDOS) is less affected than the Fe PDOS (Figure 8 (a)). But the
overall intensity of the minority spin DOS for Fe seems to have increased.
Further, we have observed strong hybridization between the majority spin
states of Fe and Cr d electrons near EF (-0.50 to 0 eV) for the c/a values
1.12 and 1.14, which is absent for the c/a values beyond $\sim$1.15 (Figure
8(a)). This might give rise to to the difference in Fe and Cr spin moments
below and above c/a = 1.15. In Figure 8(b), we have shown the DOS of the Fe-d
states for different c/a values near the EF. The peak around $\sim$-0.10 eV in
the majority DOS is shifted to $\sim$0.10 eV as we increase the c/a ratio.
Further the peak around $\sim$ -1 eV in the majority DOS is also pushed to the
higher energy ($i.e.$ towards EF) as c/a ratio changes from 1.12, 1.14 to 1.16
and 1.18. On the contrary in the minority spin states, we see the trend is
quite opposite. The minority DOS (around $\sim$ 0.5 eV and -1 eV) are pushed
from the lower binding energy to the occupied side with increasing c/a ratio.
This is also corroborates well with decrease in exchange-splitting energy of
the Fe atoms from 0.91 eV to 0.07 eV as c/a ratio changes from 1.14 to 1.16.
On the other hand for the Cr atom the change in the exchange-splitting energy
is rather small ( 1.88 eV and 1.82 eV for c/a ratio 1.14 and 1.16
respectively). The exchange-splitting energy is obtained by calculating the
d-band centers of the atoms using VASPKIT programme Wang  _et al._ (2021).

####  III.1.4 Effect of Isotropic Strain on the Electronic, Magnetic and
Elastic Properties of Cubic Fe2CrTe

Figure 9: Effect of strain (isotropic pressure) on the (a) magnetic
properties; (b) electronic properties; (c) elastic properties of Fe2CrTe in
the cubic phase.

It is well-known that, any change in the geometrical properties of Heusler
alloy system may affect its electronic, magnetic and mechanical
propertiesZhang  _et al._ (2020); Singh  _et al._ (2013); Ray  _et al._
(2021). Hence, it is important to probe the effect of lattice constant
variation on the electronic structure and magnetic properties of the bulk
material. One of the simplest ways is to apply an external pressure or
equivalently apply a uniform strain in the system. In our case, we have
investigated the electronic, magnetic properties and mechanical stability of
the system by applying a uniform strain of $\pm 5\%$ (changing pressure from
-20 GPa to 20 GPa). In Figure 9, we have shown these effects. The total
magnetic moment of the system remains nearly integer ($\approx
3.99\leavevmode\nobreak\ \mu_{B}$), almost over the entire range of latice
constants (5.71 - 6.10 Å) and after that the total magnetic moment suddenly
increases at 6.22 Åand thereafter decreases (Figure 9(a)). Further, we do not
observe any magnetic phase transition ( ferromagnetic to anti-ferromagnetic)
over the studied range of lattice constant. Since the cubic phase of Fe2CrTe
shows high spin polarization, we have also verified its HM property under
uniform strain (Figure 9(b)) and found that it shows close to 100 $\%$ SP at
EF on applying a negative uniform strain in the system. We further study the
mechanical stability of the system, when the lattice constant changes. Under
uniform pressure (P), the elastic constants are modified according to the
following equationZhang  _et al._ (2020):

$B_{11}=C_{11}-P\leavevmode\nobreak\ ;\leavevmode\nobreak\
B_{12}=C_{11}+P\leavevmode\nobreak\ ;\leavevmode\nobreak\
B_{44}=C_{44}-P\leavevmode\nobreak\ $

and the stability criteria changes to: $B_{11}-B_{12}>0\leavevmode\nobreak\
;B_{11}+2B_{12}>0\leavevmode\nobreak\ ;B_{44}>0$

In Figure 9(c), we have shown the change of elastic constants with the lattice
constant under uniform strain. The compound is found to be mechanically stable
over almost the whole range of lattice constants studied, as it satisfies the
stability criteria as discussed above. However, when the lattice constant is
too large ($\sim$6.25 Å), the compound is no longer mechanically stable.This
is due to the fact that with increasing lattice constant, the interaction
between the atoms weakens and the stability is thus destroyed.

###  III.2 Fe2CrTe on MgO Surface

From our previous discussion, we have established that the cubic phase of
Fe2CrTe behaves like a nearly HM system. But in reality, HM properties can be
highly affected by any kind of crystal disorders; such as defects, surfaces
and interfaces. The study of a HM system on a substrate and corresponding
surface and interface-related effects on its electronic and magnetic
properties can lead to interesting results. In order to simulate an interface,
we have constructed the Fe2CrTe/MgO(001) system (Figure S8)Bhattacharya and
Chakrabarti , by placing the O atoms (a) on top of the Cr and Te atoms $i.e$
on a Cr-Te terminated interface and (b) on top of the Fe atoms of a Fe-Fe
terminated interface. The Cr-Te terminated interface is found to be
energetically more stable. This corroborates well with previous studies on
Heusler alloy and MgO based heterojunctionsMiura  _et al._ (2008b); Hülsen
_et al._ (2009); Saito  _et al._ (2010), where it has already been reported
that YZ interface of X2YZ FHA, where Y and Z atoms are situated on the top of
O atoms, is the most stable one. Hence we consider this interface for further
studies. Here we have considered 13 mono-layers (ML) of Fe2CrTe and 7 ML of
MgO with $sim$15 Å of vacuum to prevent interaction between the adjacent
surfaces in a periodic arrangement. Further, the in-plane lattice constant of
the Fe2CrTe/MgO surface, was fixed at 4.21 Å ($\frac{a}{\sqrt{2}}$, a being
the lattice constant of bulk Fe2CrTe) in the cubic phase, which has an
excellent lattice matching with bulk MgO (4.21 Å).

First we discuss the stability of this interface in the $ab-initio$ DFT
framework. We have calculated the binding energy and also the surface free
energy ($\gamma$) Zarei  _et al._ (2008), which is defined as,

$\gamma=\frac{G(T,P_{i})-\sum_{i}{n_{i}\mu_{i}}}{2A}$

where, G is the Gibbs free energy of the surface, $n_{i}$ and $\mu_{i}$ are
the number and chemical potential of the ith element and A is the surface area
of the supercell. The binding energy and surface free energy ($\gamma$) are
$\sim$ -4.3767 eV/atom and -1.9781 ev/Å2, respectively and these indicate a
stable composite system. From Table 4, we see that there is some buckling in
the interface and subsurface Cr-Te layers, where the Te atoms move towards the
substrate (MgO) side due to higher electronegativity of Te as compared to Cr.
The increase of magnetic moments of the interface atoms (Table 4) is a well-
known phenomenonBhattacharya  _et al._ (2021); Habibi  _et al._ (2013), when
the lower hybridization at the surface leads to the enhancement in exchange-
splitting of the interface atoms. This can also be confirmed from the atom
projected spin-polarized DOS (Figure 10), where we see the majority (minority)
spin states of the Cr atoms shift towards higher (lower) binding energy with
respect to EF, as compared to bulk Cr. It is also evident from Figure 10, that
the interface states of the minority spin channel are mostly localized at the
Fe atoms of the sub-surface layer. As a result, the SP of the surface is
significantly reduced as compared to bulk. However, the surface effect is able
to penetrate a few ML and rest of the layers show bulk-like properties (as is
evident from Table 4 and Figure 10).

Table 4: Calculated electronic, geometric and magnetic properties of Fe2CrTe/MgO interface. Surface buckling ($\delta$l in Å), atomic magnetic moments (in $\mu_{B}$) and the SP (in $\%$) at EF are shown in the table. Bulk values are also given for comparison. Atomic Layer |  $\delta$l | SP | Atomic Magnetic Moments |  |
---|---|---|---|---|---
|  |  | Cr | Fe | Te |
Interface (S) | 0.21 | 71 | 3.12 | – | -0.05 |
S-1 | 0.00 | 11 | – | 1.82 | – |
S-2 | 0.20 | 94 | 2.05 |  | -0.07 |
S-3 | 0.00 | 41 | – | 0.98 | – |
S-4 | 0.00 | 94 | 2.10 | – | -0.05 |
Bulk | – | 95 | 2.22 | 0.89 | -0.04 |
Figure 10: The spin-polarized, atom projected density of states are shown for
atoms at different layers for the composite system with Cr-Te/MgO interface.
Here S, S-1, S-2, and S-3 represent surface and subsurface layers as we go
away from the interface, respectively.

###  III.3 Spin-Transport Properties of Fe2CrTe/MgO/Fe2CrTe Heterojunction

Figure 11: K-resolved Transmittance in parallel magnetization case for Fe2CrTe/MgO/Fe2CrTe heterojunction with Cr-Te interface for 5 ML of with cubic phase of Fe2CrTe: Top and bottom panels are for parallel (PC) and anti-parallel spin configurations (APC), respectively. Table 5: Calculated transport properties of Fe2CrTe/MgO/Fe2CrTe junctions for the cubic and tetragonal phases. G${}^{\uparrow\uparrow}_{PC}$ (G${}^{\downarrow\downarrow}_{PC}$) are spin-up (spin-down) conductance at the Fermi energy for parallel (P) spin configuration. G${}^{\uparrow\downarrow}_{PC}$ (G${}^{\downarrow\uparrow}_{PC}$) are the spin up-to-spin down and spin down-to-spin up conductance in the anti-parallel configuration state (APC), respectively. The tunnel magnetoresistance ratio is defined as $\frac{G_{PC}-G_{APC}}{G_{APC}}$, where G$P=G^{\uparrow\uparrow}_{PC}+G^{\downarrow\downarrow}_{PC}$ and G${}_{APC}=G^{\uparrow\downarrow}_{APC}+G^{\uparrow\downarrow}_{APC}$. Phase | Layer thickness |  G${}^{\uparrow\uparrow}_{PC}$ |  G${}^{\downarrow\downarrow}_{PC}$ |  G${}^{\uparrow\downarrow}_{APC}$ |  G${}^{\downarrow\uparrow}_{APC}$ |  TMR ($\%$) |
---|---|---|---|---|---|---|---
Cubic | 5 |  3.567$\times 10^{-3}$ |  5.342$\times 10^{-6}$ |  6.678$\times 10^{-6}$ |  5.372$\times 10^{-6}$ | 300 |
Tetragonal | 5 |  7.050$\times 10^{-3}$ |  4.461$\times 10^{-5}$ |  3.048$\times 10^{-7}$ |  1.716$\times 10^{-6}$ | 3522 |
Figure 12: K-resolved Transmittance in parallel magnetization case for
Fe2CrTe/MgO/Fe2CrTe heterojunction with Cr-Te interface for 5 ML of MgO and
with tetragonal phase of Fe2CrTe: Top and bottom panels are for parallel (PC)
and anti-parallel spin configurations (APC), respectively.

From our previous discussion, we predict that the Fe2CrTe cubic alloy may be
grown on MgO(001) substrate as it forms an energetically stable surface. In
bulk Fe2CrTe the half-metallic behavior is affected due the highly disperssive
bands in the minority spin channel due to the dominant contribution from the
Fe atoms (Figure 3), which is also evident from the non-integer spin magnetic
moment ($\sim 4$ $\mu_{B}$). Thus transport in both the spin channels is
expected in case of cubic phase of Fe2CrTe. Further, the SP at EF for the
tetragonal phase has been found to be much less than that of the cubic phase.
However, we find a significantly strong SP of transport even at high
temperature, when we calculate the temperature dependence of the spin-
polarized conductivity (Figure S9)Bhattacharya and Chakrabarti , specially in
the cubic phase. This indicates even though Fe2CrTe may not completely be
half-metallic, it may be a useful for a spin-injector material even at higher
temperature.

MTJ materials with an electrode with a reasonably high SP at EF have recently
gained lots of attention, due to their high TMR ratio, where the resulting
current in the junction strongly depends on the relative magnetization of the
electrode.Hülsen  _et al._ (2009); Miura  _et al._ (2008a) Here we investigate
the Fe2CrTe/MgO/Fe2CrTe heterojunction to explore its MTJ properties, in both
the phases. It may be noted that the tetragonal phase of Fe2CrTe exhibits a
lattice mismatch of 7$\%$ with MgO (in-plane lattice constant = a/$\sqrt{2}$).
Here we have constructed our heterojunction with 13 ML of Fe2CrTe and have
taken 5 ML of MgO. In Table 5, we have shown the transmittance value and TMR
ratio of the heterojunction for cubic and tetragonal phases of Fe2CrTe, where
the TMR value is defined as $\frac{G_{PC}-G_{APC}}{G_{APC}}$, where GPC and
GAPC indicate the total conductance for the MTJs in the parallel (PC) and
anti-parallel (APC) state. Despite the fact that the cubic phase has a higher
SP value, the TMR ratio indicates a 10 fold rise in the TMR ratio value for
the tetragonal phase compared to the cubic phase (Table 5). To understand
this, we examine the majority spin band structure of both the phases along the
$<001>$ direction (Figure 3 and Figure 4). It has already been established
that the $\Delta_{1}$ band in the majority spin state is essential to obtain a
large TMR ratio for MgO-based MTJs.Miura  _et al._ (2008a); Butler (2008) We
see that the bands with $\Delta_{1}$ symmetry along the $<001>$ direction are
only present for the tetragonal phase. Here we must keep in mind the
possibility that these theoretical values can be constrained by the presence
of a variety of disorders and defects at the interface that may develop during
the sample growth. Therefore, our predicted values of the TMR ratio define the
upper limit of the same.

For a comprehensive understanding of the above spin-dependent conductance and
TMR effect, in Figure 11 we show the k-resolved transport properties at the EF
of the heterojunction with MgO (with layer thickness of 5 ML) in PC and APC
state for the cubic phase of Fe2CrTe. The majority spin transmission in PC
state is found to be negligible around the center of the BZ, though MgO has a
small decay constant ($\kappa$) around $\Gamma$ point, as can been seen from
the complex band structure of MgO (Figure S10).Bhattacharya and Chakrabarti
It is evident from the orbital projected majority spin band structure of bulk
Fe2CrTe that there are no incoming majority-spin $\Delta_{1}$ states at the EF
(Figure 3). In contrast, for the minority spin channel in the PC state, we
observe tunneling hot-spots around $\Gamma$ point, however showing a much
weaker transmission. Apart from the highly conducting channel around the
$\Gamma$ point, we observe appearance of conducting channels around the
Brillouin zone corners for the majority spin channel showing a four-fold
rotational symmetry of the Fe2CrTe electrode. These are identified as resonant
tunneling states.Miura  _et al._ (2008a); Butler (2008) In the APC state the
spin up and down channels exhibit similar transmission properties with
considerably large tunneling states around the $\Gamma$ point (Figure 11).

Now as we increase the barrier thickness the transmission around $\Gamma$
point gets highly affected for the majority spin channel, showing
astonishingly small transmission around $\Gamma$ and the resonant tunneling
states are also diminished. However, transmission for the minority states are
not largely affected with increasing barrier thickness (Figure
S10).Bhattacharya and Chakrabarti  This large decrement in the majority spin
transmission can be explained form the majority spin band structure of the
bulk cubic Fe2CrTe (Figure 3), where we do not observe any bands crossing the
EF along the $\Gamma$ \- X direction ($i.e$ along the propagation direction
(001). These bands are mainly responsible for the highly spin-polarized
transmissionButler (2008); Hülsen  _et al._ (2009); Miura  _et al._ (2008a),
provided that those bands have some preferred orbital character. Due to the
absence of such bands along the $\Gamma$-X direction in the majority spin
channel, we observe large decrement in the majority spin transmission with
increasing barrier thickness (Table 5 and Figure S10Bhattacharya and
Chakrabarti ).

To understand why larger TMR ratio is possible for the tetragonal phase as
comapared to the cubic phase, we further investigate the microscopic tunneling
process for the tetragonal phase. In Figure 12, we have plotted the spin-and
k-resolved transmission coefficients at EF, T${}^{\sigma}(E_{F},K_{||})$ for
the tetragonal Fe2CrTe phase in both PC and APC states. This provides a vivid
picture of how tunneling process can be relalized in this heterojunction. At a
first glance, all the transmission patterns (including parallel and anti-
parallel) have a four-fold rotational symmetry, which is in accordance with
the C4v symmetry of the MTJ heterostructure. We note that for the majority
spin electrons in the PC state, the very sharp transmission features around
the $\Gamma$ point domminate the transmission process. However, minority spin
electrons in the PC state show $\Gamma$-centric transmission, which are of
much weaker intensity. We observe for the tetragonal phase that there is an
overall increase in the transmission of the majority as well as of the
minority spins around the $\Gamma$ point in the parallel configuration, as
compared to the cubic phase. This is primarily due to the presence of
$\Delta_{1}$ symmetric bands in the bulk tetragonal phase, along the $<001>$
direction. In the APC state, however, we find that there is a dramatic
reduction in the transmission of both the spin channels (Figure 12), due to a
stronger suppression of the electron tunneling proceess, which leads to the
larger TMR ratio for the tetragonal phase as compared to the cubic phase.

##  IV Conclusion

In this work, we have carried out a first principles study on the structural,
mechanical, electronic and transport properties of a novel full Heusler
chalcogenide, Fe2CrTe. The compound is found to undergo a volume conserving
tetragonal distortion and a clear minimum has been observed at c/a $\sim$1.26.
The cubic phase shows a ferromagnetic behavior with a nearly half metallic
character, whereas the tetragonal phase exhibits a ferrimagnetic and fully
metallic nature. This behavior has been found to be robust against a sizable
value of Hubbard onsite electron-electron correlation term for both the
transition metal atoms. The compound is found to possess no negative phonon
frequencies in both the phases and no martensite phase transition (MPT) was
observed, as suggested from the temperature dependent free energy behavior.
Further we have also established the mechanical stability of both the phases.
We have also studied the effect of uniform strain on the electronic and
mechanical properties of the system in cubic phase. It shows 100$\%$ SP on
applying a negative uniform strain and the compound is found to be
mechanically stable over almost the whole range of lattice constant. To probe
the effect of substrate on various physical properties a thin film of 13 mono-
layers of Fe2CrTe is placed on a MgO substrate, which shows an energetically
stable composite system and the spin polarization has been found to continue
to be high for the cubic phase (above 70$\%$). We have further investigated
the transmission profile of Fe2CrTe/MgO/Fe2CrTe heterojunction in both the
cubic and tetragonal phases. Spin-transport properties for the tetragonal
phase looks promising for lower thickness of spacer layer (5 ML). Finally, in
light of all the above discussions, synthesis and characterization of the
predicted alloy seem essential to understand its structural, electronic,
magnetic and spin transport properties and our present work awaits the
experimental validation.

##  V Acknowledgements

Authors thank the director, RRCAT for facilities and encouragement. We thank
A. Banerjee, H. Ghosh and T. Ganguli for scientific discussions. The
scientific computing group, computer division of RRCAT, Indore is thanked for
the help in installing and support in smooth running of the codes. JB thanks
D. Pandey, A. Kumar for useful discussions during the work. JB and RD thank
RRCAT and HBNI for financial support.

## References

  * de Groot  _et al._ (1983) R. A. de Groot, F. M. Mueller, P. G. v. Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).
  * Webster (1971) P. Webster, J. Phys. Chem. Solids 32, 1221 (1971).
  * Miura  _et al._ (2008a) Y. Miura, H. Uchida, Y. Oba, K. Abe, and M. Shirai, Phys. Rev. B 78, 064416 (2008a).
  * Hülsen  _et al._ (2009) B. Hülsen, M. Scheffler, and P. Kratzer, Phys. Rev. Lett. 103, 046802 (2009).
  * Roy  _et al._ (2016) T. Roy, D. Pandey, and A. Chakrabarti, Phys. Rev. B 93, 184102 (2016).
  * Enamullah and Lee (2018) Enamullah and S.-C. Lee, J. Alloys Compd. 765, 1055 (2018).
  * Baral and Chakrabarti (2019) M. Baral and A. Chakrabarti, Phys. Rev. B 99, 205136 (2019).
  * Kang and Zhang (2017) X.-H. Kang and J.-M. Zhang, J. Phys. Chem. Solids 105, 9 (2017).
  * Kervan and Kervan (2011) N. Kervan and S. Kervan, J. Phys. Chem. Solids 72, 1358 (2011).
  * Zhu  _et al._ (2014) X. Zhu, Y. Wang, L. Wang, Y. Dai, and C. Luo, J. Phys. Chem. Solids 75, 391 (2014).
  * Slater (1936) J. C. Slater, Phys. Rev. 49, 931 (1936).
  * Pauling (1938) L. Pauling, Phys. Rev. 54, 899 (1938).
  * Galanakis  _et al._ (2002) I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002).
  * Luo  _et al._ (2008) H. Z. Luo, H. W. Zhang, Z. Y. Zhu, L. Ma, S. F. Xu, G. H. Wu, X. X. Zhu, C. B. Jiang, and H. B. Xu, J. Appl. Phys. 103, 083908 (2008).
  * Hirohata  _et al._ (2013) A. Hirohata, J. Sagar, L. Lari, L. R. Fleet, and V. K. Lazarov, Appl. Phys. A 111, 423 (2013).
  * Elphick  _et al._ (2021) K. Elphick, W. Frost, M. Samiepour, T. Kubota, K. Takanashi, H. Sukegawa, S. Mitani, and A. Hirohata, Sci. Technol. Adv. Mater. 22, 235 (2021).
  * Bainsla and Suresh (2016) L. Bainsla and K. G. Suresh, Appl. Phys. Rev. 3, 031101 (2016).
  * Dutt  _et al._ (2022) R. Dutt, J. Bhattacharya, and A. Chakrabarti, J. Phys. Chem. Solids 167, 110704 (2022).
  * Pandey and Chakrabarti (2019) D. Pandey and A. Chakrabarti, Physics Letters A 383, 2914 (2019).
  * Pandey  _et al._ (2021) D. Pandey, R. Gangwar, J. Bhattacharya, and A. Chakrabarti, Surface Science 714, 121910 (2021).
  * Ru and Fisher (2006) N. Ru and I. R. Fisher, Phys. Rev. B 73, 033101 (2006).
  * Zhu  _et al._ (2016) X.-G. Zhu, Y. Zhang, W. Feng, B.-K. Yuan, Q. Liu, R.-Z. Qiu, D.-H. Xie, S.-Y. Tan, Y. Duan, Y. Fang, W. Zhang, and X.-C. Lai, New Journal of Physics 18, 093015 (2016).
  * Chakrabarti  _et al._ (2013) A. Chakrabarti, M. Siewert, T. Roy, K. Mondal, A. Banerjee, M. E. Gruner, and P. Entel, Phys. Rev. B 88, 174116 (2013).
  * Barman  _et al._ (2005) S. R. Barman, S. Banik, and A. Chakrabarti, Phys. Rev. B 72, 184410 (2005).
  * Ducher  _et al._ (2008) R. Ducher, R. Kainuma, and K. Ishida, J. Alloys Compd. 463, 213 (2008).
  * Siewert  _et al._ (2011) M. Siewert, M. E. Gruner, A. Dannenberg, A. Chakrabarti, H. C. Herper, M. Wuttig, S. R. Barman, S. Singh, A. Al-Zubi, T. Hickel, J. Neugebauer, M. Gillessen, R. Dronskowski, and P. Entel, Appl. Phys. Lett. 99, 191904 (2011).
  * Roy and Chakrabarti (2016) T. Roy and A. Chakrabarti, J. Magn. Magn. Mater. 401, 929 (2016).
  * Feng  _et al._ (2022) Y. Feng, H. Ding, X. Li, B. Wu, and H. Chen, Journal of Applied Physics 131, 133901 (2022), https://doi.org/10.1063/5.0081588 .
  * Cui  _et al._ (2021) Z. Cui, H. Ding, and Y. Feng, Phys. Chem. Chem. Phys. 23, 17984 (2021).
  * Miura  _et al._ (2011) Y. Miura, K. Abe, and M. Shirai, Phys. Rev. B 83, 214411 (2011).
  * Han  _et al._ (2017) H. Han, T. Feng, L. Fan, Z. Zhao, M. Li, and K. Yao, J. Magn. Magn. Mater. 438, 95 (2017).
  * Gercsi  _et al._ (2006) Z. Gercsi, A. Rajanikanth, Y. K. Takahashi, K. Hono, M. Kikuchi, N. Tezuka, and K. Inomata, Appl. Phys. Lett. 89, 082512 (2006).
  * Roy  _et al._ (2020) T. Roy, M. Tsujikawa, T. Kanemura, and M. Shirai, J. Magn. Magn. Mater. 498, 166092 (2020).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Marmier  _et al._ (2010) A. Marmier, Z. A. Lethbridge, R. I. Walton, C. W. Smith, S. C. Parker, and K. E. Evans, Computer Physics Communications 181, 2102 (2010).
  * Smogunov  _et al._ (2004) A. Smogunov, A. Dal Corso, and E. Tosatti, Phys. Rev. B 70, 045417 (2004).
  * Baroni  _et al._ (2001) S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  * Joon Choi and Ihm (1999) H. Joon Choi and J. Ihm, Phys. Rev. B 59, 2267 (1999).
  * Karki  _et al._ (2021) S. Karki, V. Rogers, P. Jadaun, D. S. Marshall, and J. A. C. Incorvia, Adv. Theory Simul. 4, 2100309 (2021).
  * Blaha  _et al._ (2019) P. Blaha, K. Schwarz, G. K. H. Madsen, D. F. Kvasnicka, J. Luitz, R. Laskowsk, F. Tran, and L. D. Marks (2019).
  * Siewert  _et al._ (2010) M. Siewert, M. E. Gruner, A. Dannenberg, A. Hucht, S. M. Shapiro, G. Xu, D. L. Schlagel, T. A. Lograsso, and P. Entel, Phys. Rev. B 82, 064420 (2010).
  * Togo and Tanaka (2015) A. Togo and I. Tanaka, Scr. Mater. 108, 1 (2015).
  * Paul  _et al._ (2014) S. Paul, B. Sanyal, and S. Ghosh, J. Phys. Condens. Matter 27, 035401 (2014).
  * Zayak  _et al._ (2005) A. T. Zayak, P. Entel, K. M. Rabe, W. A. Adeagbo, and M. Acet, Phys. Rev. B 72, 054113 (2005).
  * Roy  _et al._ (2015) T. Roy, M. E. Gruner, P. Entel, and A. Chakrabarti, J. Alloys Compd. 632, 822 (2015).
  * Hill (1952) R. Hill, Proceedings of the Physical Society. Section A 65, 349 (1952).
  * Zhi-jian  _et al._ (2007) W. Zhi-jian, E.-j. Zhao, H.-p. Xiang, X.-f. Hao, X.-j. Liu, and J. Meng, Phys. Rev. B 76, 054115 (2007).
  * Ozdemir Kart and Cagın (2010) S. Ozdemir Kart and T. Cagın, J. Alloys Compd. 508, 177 (2010).
  * Kanomata  _et al._ (2010) T. Kanomata, Y. Chieda, K. Endo, H. Okada, M. Nagasako, K. Kobayashi, R. Kainuma, R. Y. Umetsu, H. Takahashi, Y. Furutani, H. Nishihara, K. Abe, Y. Miura, and M. Shirai, Phys. Rev. B 82, 144415 (2010).
  * Luan  _et al._ (2018) X. Luan, H. Qin, F. Liu, Z. Dai, Y. Yi, and Q. Li, Crystals 8 (2018).
  * Pugh (1954) S. Pugh, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 45, 823 (1954).
  * Pettifor (1992) D. G. Pettifor, Mater. Sci. Technol. 8, 345 (1992).
  * Naher and Naqib (2021) M. I. Naher and S. H. Naqib, Scientific Reports 11, 5592 (2021).
  * Han  _et al._ (2018) Y. Han, Y. Wu, T. Li, R. Khenata, T. Yang, and X. Wang, Materials 11 (2018), 10.3390/ma11050797.
  * (57) J. Bhattacharya and A. Chakrabarti, Supplementrary Information .
  * Dudarev  _et al._ (1998) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
  * Cox  _et al._ (1974) B. N. Cox, M. A. Coulthard, and P. Lloyd, Journal of Physics F: Metal Physics 4, 807 (1974).
  * Paudel  _et al._ (2020) R. Paudel, G. C. Kaphle, M. Batouche, and J. Zhu, International Journal of Quantum Chemistry 120, e26417 (2020), https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.26417 .
  * Chakrabarti  _et al._ (2012) A. Chakrabarti, S. D’Souza, and S. Barman, Physica B: Condensed Matter 407, 3547 (2012).
  * Wang  _et al._ (2021) V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng, Computer Physics Communications 267, 108033 (2021).
  * Zhang  _et al._ (2020) C. Zhang, H. Huang, C. Wu, Z. Zhu, Z. He, and G. Liu, Frontiers in Physics 8 (2020).
  * Singh  _et al._ (2013) M. Singh, H. S. Saini, J. Thakur, A. H. Reshak, and M. K. Kashyap, Journal of Alloys and Compounds 580, 201 (2013).
  * Ray  _et al._ (2021) R. B. Ray, G. C. Kaphle, R. K. Rai, D. K. Yadav, R. Paudel, and D. Paudyal, Journal of Alloys and Compounds 867, 158906 (2021).
  * Miura  _et al._ (2008b) Y. Miura, H. Uchida, Y. Oba, K. Abe, and M. Shirai, Phys. Rev. B 78, 064416 (2008b).
  * Saito  _et al._ (2010) T. Saito, T. Katayama, T. Ishikawa, M. Yamamoto, D. Asakura, T. Koide, Y. Miura, and M. Shirai, Phys. Rev. B 81, 144417 (2010).
  * Zarei  _et al._ (2008) S. Zarei, S. J. Hashemifar, H. Akbarzadeh, and Z. Hafari, J. Phys. Condens. Matter 21, 055002 (2008).
  * Bhattacharya  _et al._ (2021) J. Bhattacharya, D. Pandey, S. R. Barman, and A. Chakrabarti, J. Magn. Magn. Mater. 540, 168398 (2021).
  * Habibi  _et al._ (2013) P. Habibi, C. Barreteau, and A. Smogunov, J. Phys. Condens. Matter 25, 146002 (2013).
  * Butler (2008) W. H. Butler, Sci. Technol. Adv. Mater. 9, 014106 (2008).