91 chapter 4 4.1 introduction apart from the conventional heusler

91

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


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.


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


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


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


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.


105

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91 chapter 4 4.1 introduction apart from the conventional heusler

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