Antiferromagnetic spintronics of Mn2Au: An experiment, first principle, mean field and series expansions calculations study

Antiferromagnetic spintronics of Mn 2 Au: An experiment, fi rst principle, mean fi eld and series expansions calculations study

R. Masrour

^a,b,n

, E.K. Hlil

^c

, M. Hamedoun

^d

, A. Benyoussef

^b,d,e

, A. Boutahar

^f

, H. Lassri

^f

aLaboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, 63 46000, Safi, Morocco

bLMPHE (URAC 12), Faculty of Science, Mohammed V-Agdal University, Rabat, Morocco

cInstitut Néel, CNRS et Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France

dInstitute of Nanomaterials and Nanotechnologies, MAScIR, Rabat, Morocco

eHassan II Academy of Science and Technology, Rabat, Morocco

fLPMMAT, Université Hassan II-Casablanca, Faculté des Sciences, BP 5366 Maârif, Morocco

a r t i c l e i n f o

Article history:

Received 8 April 2015 Received in revised form 24 May 2015

Accepted 28 May 2015 Available online 29 May 2015 Keywords:

Alloys

Magnetic materials Ab initiocalculations Electronic structure

a b s t r a c t

The self-consistentab initiocalculations, based on DFT (Density Functional Theory) approach and using FLAPW (Full potential Linear Augmented Plane Wave) method, are performed to investigate both electronic and magnetic properties of the Mn2Au. Polarized spin and spin–orbit coupling are included in calculations within the framework of the antiferromagnetic state between two adjacent Mn plans. Magnetic moment considered to lie along (110) axes are computed. Obtained data fromab initiocalculations are used as input for the high temperature series expansions (HTSEs) calculations to compute other magnetic parameters.

The exchange interactions between the magnetic atoms Mn–Mn in Mn2Au are given by using the ex- periment results and the meanfield theory. The High Temperature Series Expansions (HTSEs) of the magnetic susceptibility with the magnetic moments in Mn2Au (mMn) is given up to tenth order series in, 1/kBT. The Néel temperatureTNis obtained by HTSEs combined with thePadéapproximant method. The critical exponent associated with the magnetic susceptibility is deduced as well.

&2015 Elsevier B.V. All rights reserved.

1. Introduction

Since Mn carries the largest moment, among transition metals and most of the bimetallic alloys containing Mn order anti- ferromagnetically, the goals of strong magnetic anisotropy phe- nomena and of antiferromagnetic (AFM) spintronics appear to merge naturally together [1]. The effective kinetic-exchange model calculations in (Ga,Mn)As show chemical potential anisotropies consistent with experiment and ab initio calculations in transition metal systems. They suggest that this generic effect persists to high temperatures in metal ferromagnets with strong spin-orbit coupling [2]. The large magnetic anisotropy and tunneling aniso- tropic magnetoresistance in Mn/W(001) layered bimetallic na- nostructures is previously studied Ref. [3]. Mn alloys containing noble metals are often antiferromagnetic [4,5]. However, this has not been considered to be the case for Mn

2

Au. From the

fi

rst- principle Local Spin Density Approximation (LSDA) study, it was

argued that Mn

2

Au should be AFM, with a large Mn magnetic moment, approaching 4

mB

per Mn, and a Néel temperature well above room temperature [6]. Recent investigation reported in Nature Communications [7] provide an experimental con

fi

rmation of these theoretical predictions, therefore, establishing that Mn

2

Au is a particularly promising material for AFM spintronics. Recently, signi

fi

cant effort has been devoted to harnessing the potential of these materials in so-called antiferromagnetic spintronics where the ferromagnetic electrodes are replaced by antiferromagets [8].

Due to large spin

–

orbit coupling on the 5d shell of its Au atoms, Mn

2

Au, a layered bimetallic material, has been proposed as an interesting candidate for these emerging antiferromagnetic spin- tronic devices [8]. In addition, Mn

2

Au, unlike other well estab- lished AFMs, has a body-centered tetragonal (bct) structure which may bene

fi

t the thin-

fi

lm growth. Mn

2

Au has been recently dis- cussed and determined, on the basis of

fi

rst-principle calculations of exchange-coupling constants, to be a robust antiferromagnet with a very large Néel temperature in excess of 1500 K [6]. How- ever, there have been no experimental studies so far that con

fi

rm the AFM nature of magnetic ordering in this material. It is inter- esting to note that, in early reports, the Mn

2

Au alloy was experi- mentally identi

fi

ed to be a non-magnetic material based on

¹⁹⁷

Au Contents lists available at ScienceDirect

journal homepage:www.elsevier.com/locate/jmmm

Journal of Magnetism and Magnetic Materials

http://dx.doi.org/10.1016/j.jmmm.2015.05.085 0304-8853/&2015 Elsevier B.V. All rights reserved.

nCorresponding author at: Laboratory of Materials, Processes, Environment and Quality, Cady Ayyed University, National School of Applied Sciences, 63 46000, Safi, Morocco.

E-mail address:rachidmasrour@hotmail.com(R. Masrour).

Journal of Magnetism and Magnetic Materials 393 (2015) 600–603

Mössbauer spectra and magnetization measurements [9]. In order to utilize Mn

2

Au as an antiferromagnet for the purposes men- tioned above and other applications, it is important to prepare Mn

2

Au thin-

fi

lm samples of high quality, to clarify its magnetic nature. An

ab initio

study of the magnetic and electronic properties of Fe, Co, and Ni nanowires on Cu(001) surface is given in Ref. [10].

The self-consistent

ab initio

calculations, based on the density functional theory (DFT) approach and using the full potential lin- ear augmented plane wave (FLAPW) method, are performed to investigate both electronic and magnetic properties of the MnAu layers [11]. In the present work, three approaches self-consistent

ab initio

calculations, mean

fi

eld and temperature series expansions (HTSEs) calculations are used to shed light on the magnetic structure. Firstly, FLAPW calculations based on DFT principle are performed on Mn

2

Au. Appropriate polarized spin and spin

–

orbit coupling as well as antiferromagnetic state are considered. Considering computed magnetic moment from FLAPW calculations as input data, we have used the mean

fi

eld theory to

fi

nd the

fi

rst, second and third exchange interactions between the magnetic atoms Mn

–

Mn in Mn

2

Au. The

fi

rst ex- change interaction between Mn

–

Mn in Mn

2

Au is obtained by using the experiment results. HTSEs of the magnetic susceptibility of Mn

2

Au combined with the

Padé

approximant [12] is studied up to tenth order series in ( β

¼

1/k

BT). Finally, the Néel temperature

and critical exponent γ associated with the magnetic susceptibility are deduced.

2. Electronic structure calculations

We used FLAPW method [13] which performs DFT calculations using the local density approximation with wave functions as a basis. The Kohn

–

Sham equation and energy functional were evaluated consistently using the Full Potential Linearized Aug- mented Plane Wave (FLAPW) method. For this method, the space was divided into the interstitial and the non overlapping muf

fi

n tin spheres centered on the atomic site. The employed basis function inside each atomic sphere was a linear expansion of the radial solution of a spherically potential multiplied by spherical harmonics. In the interstitial region, the wave function was taken as an expansion of plane waves and no shape approximation for the potential was introduced in this region which is consistent with the full potential method. The core electrons were described by atomic wave functions which were solved relativistically using the current spherical part; the valence electrons were also treated relativistically in our case. The atomic muf

fi

n-tin (MT) spheres, supposed not to overlap with each other, are taken as 2.50 and 2.0 a.u for Au and Mn, respectively. The gap energy, which de

fi

nes the separation of the valence and core state, was chosen equal to 6.0 Ry. The largest reciprocal vector G in the charge Fourier ex- pansion, G

max

, was equal to 12 and the cut-off energy corre- sponding to the product of the muf

fi

n-tin radius and the max- imum reciprocal space vector, RMT

kmax

, was set to 7. Inside the atomics spheres, the potential and charge density are expanded in crystal harmonics up to

lmax¼

6. Calculations are performed with 30 inequivalent

k-points in the irreducible Bril-

louin. Such number is suf

fi

ciently large to ensure the spin moment convergence. The convergence criterion was chosen to be the total energy and set at 10

^–4

eV. These FLAPW calculations were per- formed with the crystal structure parameters (a

¼

b

¼

3.329 Å and c

¼

8.537 Å) reported in Ref. [7]. Here, polarized spin, spin

–

orbit coupling as well as the antiferromagnetic state were considered for adjacent Mn plans (110) in Mn

2

Au compound. The Mn mag- netic moments were considered to lie along (110) axes as shown in Fig. 1.

3. Calculation of the exchange integrals values by experiment measurement

To determine the

fi

rst exchange interaction, an experiment result given in Fig. 1b reported in Ref. [7] has been used. The

fi

eld dependence of the magnetization can be analyzed in terms of the existing model proposed by Zhao et al. [14]. The molecular

fi

eld coef

fi

cient

nMnMn

is obtained as proposed by the model for the antiferromagnetic compound [14] from the formula:

⎜ ⎟

⎛

⎝

⎞

⎠

M H H

n 1

K MnMn M

Mn2

( ) =

+ ( )

where

H

is the applied magnetic

fi

eld in (T),

K

is the magnetic anisotropy constant in (J/m

³

) at 2 K and

MMn

is the magnetic moment of Mn.

The magnetization curve for antiferromagnetic spintronics was found to well

fi

t Eq. (1) as shown in Fig. 1b reported in Ref. [5] at

T¼

2 K, and the values of the parameters

K

and

nMnMn

as obtained from the

fi

t at 2 K are 7.2 10

⁶

J/m

³

and 1940 T/

mB

respectively.

The molecular

fi

eld coef

fi

cient introduced in Eq. (1) is related to the antiferromagnetic exchange interactions between Mn and Mn by the following relation:

J n g N

g z

2 1 2

MnMn

MnMn Mn B Mn

Mn MnMn

= ² μ ²

( − ) ( )

where

NMn

is the number of Mn atoms per unit of mass and

ZMnMn¼

8 the coordination number. We

fi

nd that the

JMnMn

value derived from Eq. (2) is 326 K.

4. Theories and models

4.1. Meanfield theory

The Hamiltonian of the system is given by:

H J m m h m

3

i j ij

i ,

Mn Mn Mn

i j i

∑ ∑

= − −

( )

< >

where,

h

is the external magnetic

fi

eld,

Jij

(J

1(Mn−Mn ,) J2(Mn−Mn and) J3(Mn−Mn)) are the fi

rst, sec- ond and third exchange interactions between the (Mn

–

Mn) atoms

Fig. 1.Magnetic structure of Mn2Au as used in calculations.

R. Masrour et al. / Journal of Magnetism and Magnetic Materials 393 (2015) 600–603 601

in Mn

2

Au (see Fig. 1). m

i

is the magnetic moment of Mn ion located on the

ith site. We have used the mean fi

eld approximation [15] which leads to a simple relations between exchange integrals

J₁(Mn−Mn ,) J₂(Mn−Mn and) J₃(Mn−Mn)), the Néel tempera-

ture T

N

, the Curie temperature

θCW

[7] and the ferromagnetic energy (E

FM¼

85460.878474 Ry obtained by

ab initio

calculation).

4.2. High temperature series expansion

The statistics of our magnetic moment are studied using the HTSE whose starting point is the expansion of the correlation function

m mi j

Trm m e Tre

i j H

⟨ ⟩ = H^β

β

−

−

between spins at sites i and j , in powers of

_{k T}¹

β= B

[16]:

Trm m e

m Trm m H a

i 1

4

i j H

l m

i j m m

m

m m

0

∑ ^β ∑ ^β

) = ( − )

! =

( )

−β

∞

=

∞

with:a

m m1mTrm m Hi j m

=^{(− )}

!

which can be written on the form:

a m m m H

m

1 1

m

5

m

i j m m

ν

m

= (− )

! = ( − )

! ( )

where

νm= m m Hi j m

and

...

the average is conducted at in

fi

nite temperature

( =β 0).

Z Tre b

ii

6

H n

n n 0

∑ ^β

) = =

( )

−β

=

∞

with

b n

1

n

7

n

μ

n

= ( − )

! ( )

where

m Hm

μ = T_=∞

The correlation function is

m m a

b 8

i j T

m m m

n n n

0 0

β

= ∑ β

∑ ( )

=

∞

=

∞

The

fi

nal expression of the correlation function is

m m l

1

ij i j T

9

l l

l l 0

∑

γ = 〈 〉 = ( − ) α β

! ( )

=

∞

where

_{k T}¹

β= B

(k

B

being the Boltzmann's constant) with

C , m m H and H

l l k

l kl

k l k m i j m

T m m

0 T

α =ν − ∑⁻₌1 α μ₋ ν = ⟨ ⟩_=∞ μ = ⟨ ⟩_=∞

In our case, we have to deal with nearest-neighbor and next nearest neighbor coupling J

_ij

. The coef

fi

cient

αl

have to be ex- pressed for each topological graph as given in Refs. [17,18].

The high temperature series expansions of magnetic suscept- ibility are given by:

T m m

l 1

10

i j i j T

l l

l l

, 0

∑ ∑

χ ( ) = 〈 〉 = ( − ) α β

! ( )

=

∞

with

J J ... J

l sk

m k k m

k

m_w l

1 1

2 3

α ≈ (

2 ^ν_⊥

)[ ] α

The HTSE method is developed for the magnetic susceptibility

χ( )T

with arbitrary exchange interactions

J₁(Mn−M ,n) J₂(Mn−Mn and) J₃(Mn−Mn). The ‘‘

weight

’’[ ]α^l

of each graph is tabulated and are given in Ref. [19] and

k k1, 2, ... ,kw

represent the sites surrounding the sites i and j .

a yz x

n m

11

n

mn m n

0 6

0

∑ ∑

χ β ( ) = β ( )

( )

= =

−

with

x ^J

k T Mn Mn 1

= ⁽ B^{− )}

is the reduced temperature, and

y ^J

J Mn Mn Mn Mn 1

= 2^{( − )}

( − )

and

z _J^J1Mn_Mn ^Mn_Mn

= 3^{( − )}

( − )

. The coef

fi

cients

amn

given in magnetic suscept- ibility of Mn

2

Au are tabulated in Table 1.

The high temperature series expansions of magnetic suscept- ibility obtained in the present calculation are directly evaluated from the two rooted diagrams.

5. Results and discussion

The Density of State (DOS) of MnAu

2

deduced from band structure calculations is reported in Fig. 2. Here, the Fermi level is taken as reference. The DOS is symmetrical with respect to energy axis, pointing out to the Mn magnetic atoms are anti- ferromagnetically ordered.

On the range energy from 4 eV to

þ

4 eV, the DOS is domi- nated by the Mn atom contributions taking place in both occupied states at negative energies and unoccupied states localized at positive energies. The contributions from Au to the DOS take place

Table 1

The series coefficients for the high-temperature developed susceptibility series for Mn2Au with magnetic momentmMn¼3.93μB.

an a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

Value of (an) 1 8.3 25.81 78.26 234.29 695.52 2052.6 6030.966 17660.07 51573.25 150279.6

-8 -6 -4 -2 0 2 4

0

5 10

0 5 10

energy (eV) total

Au Mn

density of states (states/eV)

spin down

spin up

Fig. 2.Total DOS of Mn2Au from FLAPW calculations.

-8 -6 -4 -2 0 2 4

2

0 4

0

spin down

spin up

density of states (states/eV)

energy (eV) Mn(3d)

Au(5d)

2 4

Fig. 3.Thel-decomposed DOS of of Mn(3d) and Au(5d)-like states in Mn2Au from FLAPW calculations.

R. Masrour et al. / Journal of Magnetism and Magnetic Materials 393 (2015) 600–603 602

close to 5 eV. Also, the

l-decomposed DOS ofs,p,d

and

f

like- states are calculated in order to provide more detailed picture on the electronic structure. Fig. 3 shows the dominating contributions from these like-states. They allow to conclude that both Mn con- tributions have mainly a character of 3d band while the projected DOS on Au atom is dominated by contributions from the full 5d band of Au. Magnetic moment of Mn is computed as well and found equal to 3.71

mB

. This value is slightly lower than the ex- perimental value of 4 and slightly higher than the theoretical value of 3.64

mB

reported in Nature Communications [7].

We have used the magnetic measurement of Néel temperature and Curie Weiss reported in Ref. [7] to calculate the exchange in- tegrals

J₁(Mn−Mn ,) J₂(Mn−Mn and) J₃(Mn−Mn)

by using the mean

fi

eld theory. The

fi

rst exchange interaction

J1

(K) is also ob- tained by using the experiment results reported in Ref. [7]. The obtained values are comparable with those given in Ref. [7] (see Table 2). The experiment results are comparable with those ob- tained by mean

fi

eld theory and those obtained by Ref. [7] (see Table 2). The high-temperature series expansion (HTSE) extra- polated with Padé approximants method is known to be a con- venient method to provide valid estimate of the critical tempera- tures for real systems. By applying this method to the magnetic susceptibility

χ( ), we have estimated the Néel temperatureT

T

N

for Mn

2

Au. The Padé approximant analysis of the magnetic suscept- ibility is used to estimate Néel temperature of Mn

2

Au. The ob- tained value is close to those obtained in Ref. [7]. The Néel tem- perature corresponds to the simple pole of [ ] χ . The obtained values are gathered in Table 2. In addition, the critical exponent γ asso- ciated with the magnetic susceptibility

^χ≈

(

T⁻TN

)

^−γ

is computed for different values of Padé approximant (see Table 2). The ob- tained value is given in Table 2 and comparable with those re- ported in Refs. [20, 21].

6. Conclusions

FLAPW calculations were performed to investigate both elec- tronic and magnetic structures for Mn

2

Au considering that mag- netic moments to lie along (110) axes in (ab) plans anti- ferromagnetically ordered. The projected DOS on Mn and Au point out that the total DOS is dominated by the Mn(3d) and the full Au (5d) band contributions. Magnetic moments carried by Mn atoms

were computed as well and used as input data for HTSEs calcu- lations. The

fi

rst, second and third exchange interactions are de- duced by using the mean

fi

led theory. The

fi

rst exchange interac- tion is also obtained by using the experiment results reported in Ref. [7]. The obtained values are comparable with those given by experiment results. Considering the spin antiferromagnetic on Mn

2

Au, the magnetic properties are investigated using the high- temperature series expansions of magnetic susceptibility. As re- sults, The Néel temperature

TN

(K) is estimated from the diver- gence of the magnetic susceptibility. The critical exponent asso- ciated with the magnetic susceptibility is established.

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Table 2

The exchange interactions, Néel temperature and critical exponent of Mn2Au.

Compound J1(K) J2(K) J3(K) J1(K)[7] J2(K)[7] J3(K)[7] J1(K) Experiment TN(K) HTSE method TN(K)[7] TN(K)[9] γ Mn2Au 395.77 531.38 115.10 396 532 115 326 1440 1300–1600 1610 1.24

R. Masrour et al. / Journal of Magnetism and Magnetic Materials 393 (2015) 600–603 603