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Chapter 4
4.1 Introduction
Apart from the conventional Heusler alloys i.e. X2YZ (L21-type structure) and XYZ
(C1b-type structure), where X and Y are the two TMs atoms and Z as a main group
element, quaternary Heusler alloys (XX'YZ) are new structural variant of Heusler alloys
family which have come to the center attention, recently [1]. The primitive cell of L21-
type structure contains four atoms that form the base of the fcc primitive cell. The result is
a lattice with 3Fm m (space group no. 225) symmetry where the Wyckoff positions 4a
(0,0,0), 4b (½,½,½), and 8c (¼,¼,¼) are occupied by Z, Y and X, respectively, as depicted
in Fig. 4.1 (a).
Fig. 4.1 Crystal structure of (a) full-Heusler alloy or L21-type structure and (b)
quaternary Heusler alloy or Y-type structure [1]. The highlighted
spheres represent the position of atoms on principal diagonal, as
explained in the text.
Chapter 4
92
The simple-cubic sublattice is lost if one of the X atoms is replaced by a third type of
transition metal X' [1]. In this way, a new alloy can be obtained which crystallizes in Y-
type structure (Prototype LiMgPdSn) with no inversion symmetry (space group 216-
43F m ). The Wyckoff positions 4a (0,0,0), 4b (½,½,½), 4c (¼,¼,¼), and 4d (¾,¾,¾) are
occupied by Z, Y, X and X', respectively, as shown in Fig. 4.1 (b).
A large number of possible atomic combinations in this type of alloys guide a way to
design a wide range of new materials. The tuneability of electronic and magnetic properties
of these systems with different combinations of atoms can be used to engineer the novel
materials to meet the demand of spintronics. The recent studies [2-4] on quaternary
Heusler alloys demonstrated their potential in the field of spintronics.
Dai et al. [2] investigated in detail the DOSs of the three nonequivalent superstructures
for the quaternary CoFeMnSi alloy and concluded that it is favorable to generate half
metallicity when the low valence transition metal atom occupies the 4c (B) site and the
high valence transition metal atom occupies 4a (A) and 4b (C) sites. They were confirmed
this site occupation and 100% spin polarization of quaternary CoFeMnSi Heusler alloy by
XRD and magnetic measurement. Alijani et al. [1] identified the two quaternary Heusler
alloys, NiFeMnGa and NiCoMnGa, as HMF using ab-initio approach. They also
synthesized these alloys and investigated structural and magnetic properties
experimentally. Out of these two alloys, NiFeMnGa has a TC that is too low to make it
relevant for technological applications, but NiCoMnGa, which has a high spin polarization,
high magnetic moment and Curie temperature, is an interesting new material for
spintronics applications. The same group also reported Co2-xRhxMnZ (Z D Ga, Sn, Sb)
quaternary Heusler alloy by ab initio electronic structure calculations. They synthesized
the compounds by the arc melting technique and characterized by powder XRD and
SQUID. The 100% spin polarization is not realized in CoRhMnGa, CoRhMnSb, and in the
alloy Co0.5Rh1.5MnSb. The low temperature magnetic moments vary with the composition
and are in the range of 3.4–5.5 µB. The exchange of one Co in Co2MnSn by Rh results in
the stable, well-ordered quaternary Heusler alloy, CoRhMnSn. The ordered compound
CoRhMnSn shows HM ferromagnetism with a saturation magnetization of 5 µB. It,
furthermore, exhibits a high TC of 620 K allowing utilization at room temperature and
above [4]. Recently, Gao et al. [5] used the first-principles calculations to design new
FeCrMnSb Heusler alloys: Distortion effect
93
quaternary Heusler compounds CoFeCrZ (Z = Al, Si, Ga, Ge). They predicted CoFeCrAl
and CoFeCrSi as true HMF with the HM gaps of 0.16 and 0.28 eV, respectively, whilst
both CoFeCrGa and CoFeCrGe are nearly half-metals. They also showed that the half-
metallicity of CoFeCrAl and CoFeCrSi is robust against the lattice compression (up to 7%
and 4%, respectively), and argued that the Coulomb interaction is a reason behind the loss
of half-metallicity lost in CoFeCrAl and CoFeCrGa but retentive for CoFeCrSi and
CoFeCrGe. Further, their calculations also revealed that both CoFe- and CrSi-terminated
(001) surfaces with and without antisite defects lose the bulk half-metallicity in CoFeCrSi.
Nehra et al. [6] investigated the structural, electronic and magnetic properties of quaternary
CoFeCrAl Heusler alloy both theoretically and experimentally. According to their study,
the Rietveld refinement and Mössbauer spectroscopy of samples show absence of the A2-
phase and presence of a highly ordered B2-structure. They predicted an energy gap of 0.41
eV around the EF in the MIC using full-potential calculations on the experimental lattice
constant. The partial substitution of Co by Fe in Co2CrAl showed excellent structural
ordering while retaining the high TC, spontaneous magnetization and half-metallicity.
Recently, Xu et al. [7] proposed four quaternary Heusler alloys, CoFeMnSi, CoFeCrAl,
CoMnCrSi and CoFeVSi that would be the probable spin gapless semiconductors (SGS).
In a SGS, there is a gap in MIC around the EF whereas in MAC, the EF falls within a zero-
width gap. In SGS, no threshold energy is required to excite the carriers from valence
states to conduction states owing to the zero-width gap, thus achieving considerably higher
electron mobility and more sensitive response to the external fields than the ordinary
semiconductors. Thus the quaternary Heusler alloys based SGS, which indicate some novel
transport properties, can be proved as potential candidate for applications in spintronic
devices because in these materials conducting electrons or holes are not only 100% spin
polarized but can easily be excited also [8-9].
The all discussed materials are only worth when they are incorporated in to devices.
When a device has to be made from thin films of a material, it will be strongly affected by
adjacent layer structure and chemistry. The {011} lattice spacing of MgO differs by a few
percent from the typical Heusler {001} spacing. Thus, one expects to find significant strain
in thin epitaxial films of these compounds on MgO if they are not buffered by a material
with intermediate lattice parameter (as Cr). A tetragonal distortion is the most likely to be
Chapter 4
94
occurred due to symmetry [10]. Also, the lattice mismatches with adjacent layers can lead
to strains and distortions in the crystal structure which could result in a uniform strain or
more likely, a non-uniform distortion of the Heusler lattice [11-12].
Certainly, it is the amazing predictive power of material modeling methods and
consistency which enables a material scientist to discover new novel materials and
motivates experimentalists to fabricate these for technological advancement. HM materials
are one of such novel materials. Since the discovery of ternary Heusler alloy NiMnSb [13]
as half-metal, a large amount of scientific endeavor have been devoted to this arena.
Besides theoretical prediction, the HM character in these systems has been established
experimentally at room temperature also [14].
In continuation to the wide spectrum of Heusler alloys, in this chapter, we propose a
new quaternary Heusler alloy (FeCrMnSb) which can be proved as an ideal candidate for
spintronic applications. Further, as the lattice mismatches with adjacent layers in a material
can be possible due to induced strains when it is incorporated in to a device [15], therefore,
it is necessary to analyze the effect of distortions on HM property of this alloy. Hence for
device utility of FeCrMnSb, we also studied it under the effect of uniform and tetragonal
strains.
4.2 Details of the calculations
We have employed density functional theory (DFT) [16] based full potential
linearized augmented plane wave (FPLAPW) method as implemented in WIEN2k [17] to
perform electronic structure calculations of FeCrMnSb quaternary Heusler alloy. The
exchange and correlation (XC) potentials were treated using generalized gradient
approximation (GGA) within the parameterization of Perdew–Burke–Ernzerhof [18]. In
FPLAPW calculations, the core states were treated fully relativistically, whereas for the
valence states, a scalar relativistic approximation was used. The plane wave cut-off
parameters were decided by RMTkmax = 9 (where kmax is the largest wave vector of the basis
set such that kmax controls the accuracy of the calculation) and Gmax = 12 a.u.-1 for Fourier
expansion of potential in the interstitial region. A 17×17×17 k-point mesh was used as base
for the integration in the cubic systems resulting in 405 k-points in the irreducible Brillouin
FeCrMnSb Heusler alloys: Distortion effect
95
zone (IBZ). The k-space integration was carried out using the modified tetrahedron
method. The energy and charge convergence criteria were strictly set to 10-4 Ry/cell.
The FeCrMnSb quaternary Heusler alloy crystallizes in Y-type structure and the
Wyckoff positions for this structure are 4a (0,0,0), 4b (½,½,½), 4c (¼,¼,¼) and 4d
(¾,¾,¾), respectively, occupied by Sb, Mn, Fe and Cr atoms. This site occupation is
shown in Fig. 4.2.
Fig. 4.2 Unit cell of FeCrMnSb quaternary Heusler alloy (left side) and the
symmetry k-points in IBZ (right side).
4.3 Result and discussion
We have calculated some important ground state properties for the sake of
experimental studies to be performed for this alloy in future. Here we have described
important results of these properties of quaternary Heusler alloy, FeCrMnSb and discussed
the outcomes of the study.
4.3.1 Structural properties
In order to obtain the true ground state of FeCrMnSb, we performed the energy
minimization as a function of lattice constant with respect to the different possible site
occupation, namely YI, YII and YIII, as shown in Fig. 4.3. For YI-structure, the Wyckoff
Chapter 4
96
positions, 4a (0,0,0), 4b (½,½,½), 4c (¼,¼,¼ ), and 4d (¾,¾,¾) were occupied by Sb, Mn,
Fe and Cr atom, respectively. Similarly YII/YIII-structure was realized by placing Sb, Mn,
Fe and Cr at (4a, 4d, 4b and 4c)/( 4a, 4d, 4c and 4b), respectively.
Fig. 4.3 Energy (E-Emin) versus lattice parameter of three possible different
site occupations. YI: FeCrMnSb, YII: CrMnFeSb, and YIII:
FeMnCrSb.
The results of the structural optimization are summarized in Table 4.1. The
optimization of the cubic lattice parameters for all three possible configurations revealed
the lowest energy for YI structure with a ferromagnetic ground state. Hence, all the further
calculations were performed on this structure only.
FeCrMnSb Heusler alloys: Distortion effect
97
Table 4.1 Results of the structure optimization. The structure types are explained in the
text.
Site occupation E (Ry) a (Å) m (µB)
YI
-19931.914325 6.06 2.00
YII
-19931.867133 6.09 5.73
YIII
-19931.894186 6.03 6.24
After having the equilibrium lattice parameter (a = 6.067 Å) for FeCrMnSb, the
next task was to check the stability of the same. For this purpose, first of all, we have
calculated the elastic constants cij for its cubic structure. There are only three independent
components for cubic symmetry:
c11 = c22 = c33, c12 = c13 = c23, and c44 = c55 = c66
The three independent elastic constants (c11, c12, and c44), bulk modulus (B = (c11 +
2c12)/3) and elastic anisotropy (Ae = 2c44/(c11 − c12)) of this alloy were calculated by
applying isotropic strain as well as volume-conserving tetragonal and rhombohedral strains
to the optimized primitive cubic cell. The corresponding values of these properties are
listed in Table 4.2. The positive value of elastic anisotropy of FeCrMnSb, the bulk
modulus and the shear modulus (c44) is clearly a signature of structural stability of this
alloy. Further, the elastic constants (c11, c12, and c44) can also be used to investigate the
stability of the structure using cubic crystal stability conditions [19] i.e.
c11+2c12 > 0; c11-c12 > 0; c11 > 0; c44 > 0.
For FeCrMnSb, all these four conditions are satisfied which means that this alloy is
stable in YI structure. The elastic anisotropy Ae = 2c44/(c11 − c12) compares the shear
moduli and allows a decision about the structural stability. The Young’s modulus becomes
isotropic for Ae = 1. The materials with large Ae ratio show a tendency to deviate from the
cubic structure.
Chapter 4
98
Table 4.2 Three independent elastic constants (cij), elastic anisotropy (Ae), atom-resolved
and total magnetic moments (m) of FeCrMnSb.
FeCrMnSb
cij: c11 c12 c44 Ae = 2c44/(c11 − c12)
158
109
104
4.24
m (µB)
mFe
mCr
mMn
mtot
1.02
-2.07
3.13
2.00
4.3.2 Density of states (DOS)
The calculated total density of states (DOS) and band structure of FeCrMnSb are
shown in Fig. 4.3.
Fig. 4.3 Spin resolved total DOS and bandstructure of FeCrMnSb.
A dip in the minority DOS at EF clearly reveals the HM nature of this alloy. A high
DOS at EF in MAC advocates its strong metallic character, whereas, the absence of DOS in
FeCrMnSb Heusler alloys: Distortion effect
99
MIC make it insulator for this spin channel. The system contains an indirect minority band
gap along the Γ-Χ direction. The total DOS of this alloy is mainly governed by TM atoms.
Therefore, to have the more insight in qualitative features of DOS, we have also explored
the d-DOS for all TMs of the present alloy as shown in Fig. 4.4.
Fig. 4.4 Calculated d-DOS of all TMs and p-DOS of Sb atom of FeCrMnSb
quaternary Heusler alloy.
The bonding states in majority spin are mainly consist of Fe-d and Mn-d states,
whereas Fe-d and Cr-d electrons are main building block of minority bonding states. The
majority antibonding region is less populated and predominantly consists of Cr-d states.
The d-states of all three TM atoms contribute in the formation of minority antibonding
region. This character of TM atoms is quite obvious from their electronic distribution in
the d-subshell. In the total DOS, the Sb-s states lie deep in valence band (~ -11.0 eV), not
shown here, and remain isolated from rest of atoms of the alloy. The Sb-p states (Fig. 4.4)
Chapter 4
100
also remains low ~ -4.0 eV, nevertheless, these states are effectively hybridized with d-
states of TM atoms and determine the occupancy of p–d orbitals. This further decides the
position of EF and hence energy gap MIC. The FeCrMnSb quaternary Heusler alloy is
found to be a perfect HMF. The maximum values of minority band gap and HM gap or
spin-flip gap are to be 0.65 eV and 0.1 eV, respectively.
Fig. 4.5 Minority-spin gap and HM gap as a function of the uniform strain (blue
color) and the tetragonal strain (red color) of FeCrMnSb quaternary
Heusler alloy.
Next, we examined the sensitivity of the half-metallicity within uniform strain and
tetragonal distortion keeping the volume of unit cell constant. The variation of minority
band gap and HM gap with respect to the uniform strain are shown (Fig. 4.5) by blue color.
It is found that the HM character remains intact in this system for -6 % to 9 % uniform
strain of the lattice relative to its equilibrium volume. The minority band gap is maximum
FeCrMnSb Heusler alloys: Distortion effect
101
at the equilibrium value and decreases monotonically with both positive and negative
uniform strains. Similar trend is also observed for the effect of tetragonal distortion, as
shown Fig. 4.5 by red color. In this case, the half-metallicity sustains in the system for
relatively larger values of tetragonal strain (from -9 % to 12 %). The value of lattice
parameter, a, varies from 4.43 Å to 4.13 Å (c = 5.69 - 6.54 Å) for the tetragonal strain
which is close to Cu (a = 3.61 Å). This makes FeCrMnSb compatible with Cu for using in
multilayer spin valve applications.
4.3.3 Magnetic properties
The calculated total and atom resolved spin magnetic moments of FeCrMnSb alloy
are listed in Table 4.2. The total magnetic moment of this alloy is an integer value which is
exactly according the Slater-Pauling rule for a half-metal [20], Vm 24n= − , where m is
the total spin magnetic moment per formula unit of the system and Nv is the total number
of valence electrons accumulated in the system. It is clear from the Table I that major part
of total magnetic moment mainly localized at Mn site. It is due to the large exchange
splitting of Mn atom which results in two peaks (as shown in Fig. 4.4), one below EF in
majority spin channel which is fully occupied and other above EF in minority spin channel
remains unoccupied . The value of spin magnetic moment at Cr site is negative. This is due
to the antiparallel alignment of Cr with rest of TM atoms which arises from the strong
interatomic covalent interaction of the Cr-d, which benefits the antiferromagnetic
alignment. It may be argued that the nearest neighbors of Mn are Fe and Cr atoms. It can
therefore be supposed that the magnetic moments at the Cr atoms are induced by the
neighboring Mn spins. Alternatively, we can say that the stability of the Fe and Mn
moments, together with the Slater–Pauling rule, decides whether the moment at the Cr sites
is aligned anti-parallel or parallel to Mn.
4.3.4 Bandstructure
In the system, FeCrMnSb obeying the Slater-Pauling rule, Vm 24n= − , the gap in
MIC around the EF basically arises between the occupied t1u and the unoccupied eu states,
as shown in Fig. 4.6, which are exclusively localized at Fe and Cr sites [20]. The
symmetry representations of degenerated orbitals refer to the work of Galanakis et al. [20].
Chapter 4
102
The red line indicates the location of the EF, which is actually in the same position with
respect to both spin directions [7].
Fig. 4.6 The schematic illustration of possible hybridizations between the d-
orbitals of transition metal atoms at different sites in the FeCrMnSb
quaternary Heusler alloy. (a) The hybridization between Fe and Cr
atoms and afterwards the hybridization with the Mn atom (b) in
MAC and (c) in MIC.
This gap is essentially determined by the d-orbitals hybridization between the Fe
atom at the 4c site and the Cr atom at the 4c site. In the spin-up direction, the total number
of energy levels and symmetry representations are identical to those in the spin-down
direction, however, the relative position of the hybridization energy scale is moved by the
exchange splitting in atoms both inside and in between [7]. It is very interesting here to
note that in MAC, Fig. 4.6 (b) and Fig. 4.7 (majority), EF crosses the through a band
formed by Fe-t2g and Cr-t2g states but falls in a gap between occupied t1u and the
FeCrMnSb Heusler alloys: Distortion effect
103
unoccupied eu states for MIC (shown by green lines in Fig. 4.6 (c)) which are exclusively
consists of Cr-t2g and Cr-eg, repectively (Shown in Fig 4.7 (Minority)).
Fig. 4.7 Spin-resolved bandstructure of FeCrMnSb quaternary Heusler alloy.
The explanation for the minority band gap for FeCrMnSb qualitatively goes in the
same way as we have already explained for the case of NiCrZ semi Heusler alloys in
Chapter 2. Unlike in semi Heusler alloy, the minority band gap in quaternary Heusler alloy
arises from a particular band filling by 12 valence electrons [21]. This filling of the
minority bands takes place successively as shown in Fig 4.7 for minority spin channel.
The low lying a1 state is filed by one s electron, the three p electrons filled in triply
degenerated Cr-t2g states, five d electrons are filled in a combined band of doubly
degenerated Fe-eg and triply degenerated Fe-t2g states. This is followed by subsequent
Chapter 4
104
filling of remaining three d electrons in triply degenerated Cr- t2g states. It should be noted
here that the representation of partial band shown in Fig. 4.7 are only for the sake of
explanation. In reality, these bands are consists many of interatomic d states.
4.4. Conclusions
We have proposed a new quaternary Heusler alloy, FeCrMnSb as a HMF using first
principle band structure calculations. The stability of the structure has been checked for
different site occupations i.e. YI, YII and YIII. The stable structure, YI, is again confirmed
using stability conditions governed by three independent elastic constants (cij). The HM
gap appears in the minority spin channel. The alloy contains an indirect minority band gap
along the Γ-Χ direction. The bonding states in majority spin are mainly consist of Fe-d and
Mn-d states, whereas Fe-d and Cr-d electrons are main building block of minority bonding
states. The Sb-p states remains low ~ -4.0 eV, nevertheless, these states are effectively
hybridized with d-states of TM atoms and determine the occupancy of p–d orbitals which
further decides the position of EF and hence energy gap MIC. The sensitivity of the half-
metallicity of this alloy is analyzed under uniform strain and tetragonal distortion keeping
the volume of unit cell constant and it is found that the half metallicity is robust against the
uniform strain from -6% to 9% and tetragonal strain from -9% to 12%. The total spin
magnetic moment of FeCrMnSb is found to be 2.0 µB i.e. an integer value which is in
accordance with the Slater-Pauling rule for a half-metal. The major part of total magnetic
moment mainly comes from Mn site. The value of spin magnetic moment at Cr site is
negative. This is due to the antiparallel alignment of Cr with rest of TM atoms which arises
from the strong interatomic covalent interaction of the Cr-d, which benefits the
antiferromagnetic alignment. The sustainability of half-metallicity in FeCrMnSb
quaternary Heusler alloy against relatively large uniform and non-uniform strains may
stimulate the experimental research on this alloy and would make this alloy as perfect
choice for spin valves and magnetic tunnel junction applications.
FeCrMnSb Heusler alloys: Distortion effect
105
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