A DFT study in bulk magnetic moment of FexCo1–x (0 £ x £ 1)

M SOLORZA-GUZMAN1, G RAMIREZ-DAMASO2,* and F L CASTILLO-ALVARADO3

1 Escuela Superior de Computo del Instituto Politecnico Nacional, C. P. 07738 Ciudad de Mexico, Mexico2 Escuela Superior de Ingenierıa y Arquitectura ‘‘Unidad Ticoman’’ del Instituto Politecnico Nacional,

C. P. 07340 Ciudad de Mexico, Mexico3 Escuela Superior de Fısica y Matematicas del Instituto Politecnico Nacional, C. P. 07738 Ciudad de Mexico, Mexico

*Author for correspondence (gramirezd@ipn.mx)

MS received 2 November 2020; accepted 28 December 2020

Abstract. In this study, we present results of the electronic density of states (DOS) and bulk magnetic moment of iron (Fe),

cobalt (Co) and their alloys (FexCo1–x; x = 1.0, 0.95, …, 0.0). Density functional theory with the generalized gradient

approximation was applied to obtain geometric and electronic properties. The methodology uses virtual crystal approxi-

mation, in conjunction with CASTEP module and the functionals PBE and PBESol of the molecular simulation program

Material Studio. We optimized the geometry of the bulk (obtaining their lattice parameters), which the structure was used to

determine the bulk magnetic moments. To determine the magnetic moment, we calculated the difference of the electronic

DOS of the electrons with spin up and spin down. The geometric optimization and magnetic moment obtained in the present

study are very similar to the experimental results, with a maximum error of 8%, which makes the present article interesting.

Keywords. Iron–cobalt alloy; bulk magnetic moment; electronic density of states.

1. Introduction

It is very well known that iron (Fe), cobalt (Co) and their

alloys have one of the major magnetic moments, with an

experimental value of 2.22 lB for Fe, 1.72 lB for Co [1] and

2.42 lB for Fe0.5Co0.5 [2]. The magnetic properties of

Fe0.5Co0.5 have been studied by different authors experi-

mentally [2–4] or theoretically [5–9]. The study of properties

of iron–cobalt equiatomic has been done in bulk [2,4,6] as a

homogenous material, or over the surface [7–9], with the

inhomogeneity or discontinuity of the surface. Bardos [2]

reported the influence of composition and the temperature

rate quenching of mean magnetic moment of iron–cobalt

alloy. Asano et al [3] reported the specific heat and the lattice

parameter of ordered and disordered FeCo alloy. Also, dif-

ferent authors have reported the value of magnetic moment on

the surface, with an increase, about bulk of the order of 30%.

This increase in the surface magnetic moment compared with

bulk magnetic moment is mainly due to the nearest number

neighbour atoms in bulk and on the surface. In previous works

[8,9], we used a phenomenological method, where we studied

magnetization as a function of temperature and aspects

related to the surface, because thin films were considered. In

this article, we present a methodology to compute their den-

sity of states and magnetic moment of FexCo1–x binary alloy

with a quantum mechanical method.

Magnetic materials represent the core of expanding

business, many industrial electronic components are based

on nanocrystalline thin-film media and magneto-optic

materials that were not available 20 years ago. At the

beginning of this century, O’Handley and Robert [10] have

described a more efficient and punctual method in the

manufacture of electronic materials and components, as

well as electrochemical and electromagnetic systems, where

FexCo1–x alloys were used. The magnetism (in bulk or on

the surface) of disordered binary alloys has been exten-

sively and experimentally studied, the different theoretical

models that have tried to explain the observed results are

based on crude approximations, such as the Bethe’s lattice.

The experimental part has shown that the magnetic prop-

erties on the surface of iron are different from those in the

bulk, while in the theory of magnetism it is not possible to

predict these variations in magnetic properties between bulk

and surface.

The methods of approximation developed by ab initiotheory can be divided in two different groups. The first one

is based on the wavefunction [11]. For example, when we

study disordered binary alloys, it is always accompanied

with other methods, concepts and assumptions, with very

large clusters of atoms as the clusters–Bethe–lattice method

(CBLM) with a strong link Hamiltonian, to calculate the

internal energy of the system, and combining it with the

cluster variation method proposed by Kikuchi [12], which

give us the configurational entropy of the alloy. However in

general, the disadvantage of this method is that their

application to non-homogeneous systems such as atoms,

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molecules, solids or surfaces has a high computational cost.

The second method is based on the density functional the-

ory (DFT) [13–15], which studies the properties of each one

of the nodes with electronic density of charge qelec r~ð Þ,avoiding the explicit determination of the wavefunction

uelec; that is, uses an approximation of the Hamiltonian and

obtains an accurate density of charge, including the

dynamic correlation due to the Coulomb interactions

between electrons. This approximation presents a unique

problem: the exchange-correlation functional and potential

derivative therein, the solution requires a good approxi-

mation to the exchange-correlation functional. The most

relevant functionals are local density approximation (LDA),

the generalized gradient approximation (GGA), the

metaGGA and hybrid functionals. All these functionals are

very efficient and accurate; however, the results obtained

for metals are more accurate with the GGA [16,17].

In the present study, compositional disorder was con-

sidered as the probability of finding an atom of Fe or Co in a

site at the centre or at the corner of a bcc crystal structure.

Electronic density of states (DOS) is obtained after that bulk

FeCo was optimized, using different approximation meth-

ods developed in this program. We select the best func-

tionals of GGA as PBE and PBESol, to determine the DOS

and the magnetic moment that had better approximate to

experimental value.

2. Methodology and computational details

Virtual crystal approximation (VCA) is a technique that

considers the periodicity of the primitive cell and consider

that on each site, there is a virtual atom that interpolates

between the behaviours of both atoms. On the other hand,

the ab initio method is generally well suited for the

approximation (VCA) because it is possible to solve prob-

lems more accurately and molecules with basal non-

degenerate states. The importance in the Fe0.5Co0.5 alloy is

the difference between the kinetic energy of a node of

configurational structure with electrons of independent

valences and the first interacting neighbours due to Cou-

lomb interaction between electrons. This dynamic correla-

tion is crucial in DFT theory, which allows us an accurate

description of the interaction of many-particle systems in

terms of an effective system of non-interacting particles

with a density functional, whose formulism is not complete

with the exchange-correlation functional; requires a good

approximation in terms of this interaction correlation.

To treat this functional exchange-correlation energy

(EXC) in terms of the DOS, it is coupled with the functional

improved with GGA approximation and the computer pro-

gram developed in CASTEP of Material Studio [18–20];

this program uses the method of a pseudopotential plane

wave, allowing to develop calculations in quantum

mechanics first-principles.

2.1 Geometry optimization of bulk FexCo1–x

The first step is to construct a crystal structure of Fe (bcc) or

Co (fcc) included in the library of Material Studio program.

For FexCo1–x alloys (x = 1.00, 0.95, …, 0.30), we use a bcc

unit cell with the mentioned concentrations and for x = 0.25,

0.20, ..., 0.00, we use a fcc unit cell. Body centred cubic

Fe0.5Co0.5 crystal structures are shown in figure 1a and face

centred cubic Fe0.25Co0.75 unit cell is shown in figure 1b.

The optimized crystal structure gives us the state of

minimum energy and the electronic DOS of the alloy.

2.2 DOS and magnetic moment

The electronic DOS is a function that allows knowing the

properties of materials. For instance, knowing whether it is

a conductor, insulator or semiconductor. In this work, DOS

allows determining the magnetic moment of the FexCo1–x

alloys.

The spin up with a" of atom a (Fe, Co) and spin down b#of atom b (Fe, Co) were designated, with a cutoff energy of

300 eV of the basic set of plane waves. The density of

points in the Brillouin zone was considered with k-point

grid, corresponding to a 3 � 4 � 1 Monkhorst-Pack grid.

The convergence of the geometric optimization is carried

out with the following values: 5 9 10-6 eV per atom for

maximum energy change, 0.01 eV A-1 as the maximum

force, 0.02 GPa as the maximum stress and 5 9 10-4 A for

the maximum displacement tolerance. In this part, the

functionals PBE and PBESol were considered to obtain the

most accurate results of bulk magnetic moments.

Using the procedure outlined above, the DOS for elec-

trons with spin up qa"� �

of atoms a Fe, Coð Þ and the

electronic DOS with spin down qb#� �

of atoms b Fe, Coð Þare obtained. Then, the average number of electrons with

spin up hn"i and the number of electrons with spin down

hn#i are calculated by:

hndi ¼Z

qd eð Þde; with d ¼"; # ð1Þ

Finally, magnetic moments are calculated as:

l ¼ hn"i � hn#i ð2Þ

in units of Bohr magneton.

3. Results and discussion

As mentioned previously, the first step in this work was to

optimize the geometry of the unit cell. In this process, the

optimal parameters of the crystal lattice are obtained. In

table 1, experimental results compared with our results of

the optimized unit cell of iron, cobalt and iron–cobalt alloy

are presented.

93 Page 2 of 4 Bull. Mater. Sci. (2021) 44:93

The differences between the experimental values and the

results of the lattice of this work are less than 3%, as shown

in table 1. It means that the results of this work were almost

identical to the experimental values.

In figure 2, the results of the bulk electronic DOS with

spin up (solid) and spin down (dot) of iron (blue), cobalt

(red) and equiatomic iron–cobalt (green) are displayed. The

Fermi energy corresponding to zero value and the values of

the DOS were obtained in the interval [–5, 5] eV. The

results of the electronic DOS for the valence electrons and

the conduction electrons are presented by observing that a

high probability of finding valence electrons with spin up or

spin down is in the interval [–4, 0] eV. It is important to

underline that for the Fe0.5Co0.5 (green curve) there is a

greater variation in its DOS.

As shown in figure 2, the density of valence electrons

with spins up is greater than the density of electrons with

spin down; the difference between both areas (spin up and

spin down) is the bulk magnetic moment.

The values obtained for the magnetic moment, using

equation (2), are for iron lFe ¼ 2:16 lB, for cobalt lCo ¼1:69 lB and for iron–cobalt lFeCo ¼ 2:41 lB. Table 2

shows the magnetic moment of pure Fe (bcc), Co (fcc) and

the alloy Fe0.5Co0.5 (bcc) compared with experimental

results.

The differences between the values are for magnetic

moment of cobalt, 0.03 lB below the experimental value,

for magnetic moment of iron is 0.06 lB, below the exper-

imental value; and for Fe0.5Co0.5, 0.01 lB below the

experimental result. The average error from the calculations

in table 2 is between 0.5 and 3% when compared with the

experimental values.

Although the geometric optimizations and DOS of the

remaining alloys have been also calculated, these results are

Figure 1. Optimized crystal structure of (a) bcc Fe0.5Co0.5 and (b) fcc Fe0.25Co0.75.

Table 1. Dimensions of the unit cell (* from reference [1], ** from reference [3]).

Lattice parameters Iron (bcc) Cobalt (fcc) Iron–cobalt (bcc)

This work: a = b = c (A) 2.830 3.476 2.851

Experimental: a = b = c (A) 2.87* 3.55* 2.855**

Figure 2. Bulk density of states of Fe, Co and Fe0.5Co0.5.

Bull. Mater. Sci. (2021) 44:93 Page 3 of 4 93

not submitted. In figure 3, the results of the magnetic

moments are displayed.

In this figure, the representation of magnetic moments in

a scale from 1.5 to 2.5 on the y-axis is shown. It is observed

that results beginning with a value of 2.16 lB (pure Fe)

increase to 2.42 lB (Fe0.65Co0.35) and then decrease to 1.69

lB (pure Co). The comparison of the results with experi-

mental values [2,4] retains an error that does not exceed 8%

(the maximum error occurs for the Fe0.80Co0.20 alloy).

4. Conclusion

The results obtained of bulk lattice parameters of the unit

cell of Fe, Co and Fe0.5Co0.5 have a difference of less than

3% compared to the experimental values reported by Kittel

[1] or Asano et al [3]. The electronic DOS is an excellent

and effective tool to determine the magnetic properties of

the iron–cobalt alloy. The bulk magnetic moment of iron–

cobalt alloy reveals results with a marginal error less than

8%, compared to the experimental magnetic moments

reported by Kittel [1], Bozorth [4] or Bardos [2]. Some

results of magnetic moment obtained in this work have not

been obtained experimentally or theoretically yet, which

makes this an exceptional work.

Acknowledgements

Solorza-Guzman acknowledges the fellowship provided by

CONACyT Mexico in ESFM-IPN and the generous support

of ESCOM-IPN. Castillo-Alvarado is grateful with the

support provided by COFAA-IPN and EDD-IPN, Mexico.

Ramırez-Damaso appreciates the support provided by SIP-

Instituto Politecnico Nacional through the proyect

20190256, Mexico.

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Table 2. Experimental bulk magnetic moment compared to the results obtained in this work (* from reference [1], ** from reference

[2]).

Element Experimental magnetic moment [lB] Magnetic moment in this work [lB]

Cobalt 1.72* 1.69

Iron 2.22* 2.16

Iron–cobalt 2.42** 2.41

Figure 3. Bulk magnetic moment of FexCo1–x (x = 0.0, 0.05, …,

1.0).

93 Page 4 of 4 Bull. Mater. Sci. (2021) 44:93