Theory for the spin dynamics in ultrathin disordered binary magnetic alloy films: Application to cobalt-gadolinium

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Research articles

Theory for the spin dynamics in ultrathin disordered binary magnetic alloy films: Application to cobalt-gadolinium

T

D. Ghadera, , A. Khaterb,c ⁎

a

College of Engineering and Technology, American University of the Middle East, Kuwait Department of Theoretical Physics, Institute of Physics, Jan Dlugosz University in Czestochowa, Am. Armii Krajowej 13/15, Czestochowa, Poland c Department of Physics, University du Maine, 72085 Le Mans, France b

ARTICLE INFO

ABSTRACT

Keywords: Spin dynamics Disordered alloys Ultrathin films

We present the theory of a dynamic variant of the non-local coherent potential approximation (DNLCPA) to investigate the spin dynamics in ultrathin magnetic cobalt-gadolinium random alloy films [Co1 c Gdc ]n , n atomic monolayers being their thickness. This novel theoretical approach in the classical spin wave representation introduces the idea for the scattering potential of a defect as an operator built up from the phase matching of the spin dynamics on the defect site with the spin dynamics in an otherwise virtual crystal. The scattering potentials are then used to establish a dynamic formulation of the CPA employing the Dyson’s formalism. A mathematical approach is rigorously developed to solve analytically the Dyson T-matrix equation, and determine the unique physical solution for the configurationally averaged Green’s function propagator of the disordered system. The DNLCPA numerical calculations for the [Co1 c Gdc ]n systems yield the characteristic eigenmodes and energy dispersion curves of the confined spin wave excitations and their corresponding lifetimes, for arbitrary alloy concentration and diverse film thickness. Our theoretical results for the lifetimes of spin waves depend strongly on their wave vectors, and are in general agreement with recent experimental data. The developed DNLCPA theoretical method is general and can be applied to compute the spin dynamics for any magnetic random binary alloy nanostructure.

1. Introduction Spin dynamics in transition metal (TM) – rare earth (RE) ferrimagnetic alloy films constitute an active research field motivated by technological interests and the advanced experimental techniques which permit the observation of novel magnetic phenomena in these systems [1–12]. The technological interest in RE-TM magnetic systems extends to RE-TM multilayers [13–15] and nanojunctions [16–22] due to their perspective potential as spin wave (SW) filters, and elements for resonance assisted transmission in magnonic devices. Several models based on the Landau–Lifshitz–Gilbert equation (atomistic spin models) and the Landau–Lifshitz–Bloch equation (micromagnetic approaches) were developed to study the spin dynamics in these exotic systems [1–12,16–29]. In these models, the effective magnetic field acting on a particular spin is derived using the free energy approach or the mean field approximation. Notably, a great deal of effort has been made to model the disorder of spin systems due to different types of effects such as high temperatures, thermal magnetization, ultrafast laser-induced magnetic switching, and temperature-

⁎

dependent magnetization dynamics using different theoretical and numerical techniques [23–29]. The disordered nature of the spin interactions in random alloys, however, calls for an advanced and systematic disorder theory, like the Coherent Potential Approximation (CPA) [30–38]. This approach was introduced early [30–33], but despite considerable efforts the CPA continues to present a number of deficiencies. The more recent non local versions (NLCPA) of this approach are an improvement and provide systematic corrections notably for electronic systems [34–36,38]. However, the NLCPA has not been widely developed or applied particularly to the dynamics of other elementary excitations. In the present work, we develop in particular a novel dynamic variant of the non local coherent potential approximation (DNLCPA), based on the treatment of the phase of spin waves, and apply it to analyze the spin dynamics on 2D ultrathin disordered binary magnetic alloy films, using the application to CoGd ferrimagnetic random alloy films. This is motivated by the importance of the magnetic properties of ultrathin magnetic alloy films as nanojunctions and magnetic chips in

Corresponding author. E-mail address: [email protected] (D. Ghader).

https://doi.org/10.1016/j.jmmm.2019.03.006 Received 20 November 2018; Received in revised form 1 March 2019; Accepted 2 March 2019 Available online 04 March 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

magnonics [39–43], and magnon spintronics [44,45]. In particular, the question of the lifetime of magnon excitations on ultrathin films of magnetic alloys, in comparison with the ultrashort timescales that characterize dynamic effects, including those implicated in potential ultrafast data processing, as to ensure the control and utility of the underlying magnetic order, has become an important theme which is crucial for engineering future magnonic devices [6,9,15]. In parallel to this, the developed experimental technique of spin-polarized electron energy loss spectroscopy (SPEELS), [46], has been used to investigate in recent years the lifetimes of the magnetic exchange spinwaves in ultrathin magnetic films, as the two monolayers of bcc Fe on W(1 1 0), [47], and two monolayers of the Fe50Pd50 alloy on Pd(0 0 1), [48]. The results of these experiments have succeeded to establish the scale and order of magnitude for magnetic exchange spinwave lifetimes; they also show that these lifetimes decrease strongly with increasing wave vector. These results present certain generic properties which make them useful as an experimental reference. To develop the DNLCPA method for the [Co1 c Gdc ]n systems, the scattering potential of an embedded defect is built up dynamically, using the Heisenberg Hamiltonian and the classical spin representation, from the phase matching of the spin dynamics on the defect site with the spin dynamics of the virtual reference VCA crystal. Details of the phase field matching theory are given elsewhere [16–22]. The scattering potentials determined in the form of non-commuting matrices are used to establish the dynamic formulation of the DNLCPA, employing the Dyson’s formalism. A rigorous mathematical approach is then developed to solve the CPA T-matrix equation and determine the complex self-energy , and consequently the configurationally averaged Green’s function Gdis . The theory yields the exchange SW excitations in the disordered ultrathin magnetic alloy films and their corresponding lifetimes. The Heisenberg Hamiltonian is considered here with only the exchange magnetic interactions, without loss of generality as will be developed. The rest of this paper is organized as follows. In Section II, a dynamic definition is presented for the scattering potentials associated with the Co and Gd sites in the [Co1 c Gdc ]n = 1 alloy system. Note that the CoGd alloys are represented generically as [Co1 c Gdc ]n , with n atomic planes randomly occupied by the Co and Gd atoms. In Section III, we determine an analytic solution of the CPA T-matrix equation for the [Co1 c Gdc ]n = 1 system. The DNCLPA is generalized in Section IV for [Co1 c Gdc ]n 2 disordered ultrathin films. Our numerical results, which are in general agreement with experimental data for comparable ultrathin magnetic films, are presented in Section 5. The salient conclusions are given in Section 4.

Ji , j . A random scattering potential may consequently be defined for a specific seed site i in the otherwise virtual crystal as Vi

i, j

,j

c ) SC + cSg

J (c ) = (1

c)2JCC

V' = V

(2.1a)

+ 2c (1

(2.1b)

c 2Jgg

c ) JCg +

V;

(2.2)

= C or g

The CPA T-matrix equation [31–33,34] then reads

T = (1

c )[VC

+ c [V g

][I ][I

G (VC

G (V g

)] )]

1

1

(2.3)

=0

Here denotes the complex self-energy [34–38] which constitutes an effective potential, invariant under translation from one site to another. It offers, however, a far more realistic description of the disordered system compared to the VCA approximation. G denotes the Green’s function of the virtual crystal, and the symbol I denotes the identity matrix. In our formalism, the scattering potentials VC' and V g' are derived dynamically from the phase field matching of the spin dynamics for an alloy specie seed site with the spin dynamics on the same site in the virtual crystal approximation. Three sub-systems are hence introduced for this purpose as in Fig. 1, namely the virtual crystal as a reference (Fig. 1b), and the virtual crystal seeded with an embedded Co and Gd atoms (Fig. 1c and 1d respectively). Using the Landau-Lifshitz (LL) equations [49] for spin dynamics in the classical atomic spin representation, and a proper treatment of the symmetries underlying the seeded subsystems, we derive the dynamic matrix M for the virtual crystal, and the dynamic matrices MCo, defect MC, d and Mg, defect Mg, d V in our refor the Co and Gd seeded subsystems. Since M presentation, the dynamic scattering potentials VC' and V g' are then deduced from Eq. (2.2) as

VC' = M

MC, d

(2.4a)

V g' = M

Mg, d

(2.4b)

The VCA subsystem is ferromagnetic with a 2D hexagonal symmetry. In order to match the spin dynamics in the seeded subsystems with the spin dynamics in the virtual crystal, we use an augmented basis and hence describe the spin dynamics in the virtual crystal in terms of two spin amplitudes, denoted u0 and u . In the augmented basis, the spin precession amplitude u ( r ) at a nearest neighbor position vector r in the hexagonal 2D virtual crystal is expressed as

u( r ) =

u0 u

r = 0

ei k · r

r

0

with the origin r = 0 chosen at an arbitrary site. The vector

k = (k y, k z ) denotes the wave-vector of the propagating spin wave (SW) in the 2D hexagonal lattice. The closed LL equation of motion for the spin amplitude at the origin reads

Si · Sj

,

with i , j nearest neighbor sites, and ,

S (c ) = (1

A modified random scattering potential for a specific seed on a reference site is next defined with reference to the constant VCA dynamic potential V as the difference

Cobalt and Gadolinium are known to be in a hexagonal close-packed (hcp) structure and it is adequate to assume this lattice structure for the 2D disordered magnetic [Co1 c Gdc ]1 random alloy monolayer (ML), as an adequate approximation for its known amorphous structure (Fig. 1a). The Co and Gd atoms are randomly distributed on the sites of the alloy; they carry respectively the spins SCo SC and SGd Sg , and are coupled magnetically by the set of exchange constants JCoCo JCC , JGdGd Jgg and JCoGd JCg . Neglecting here other magnetic interactions, much smaller than the exchange interaction in the present system, the Heisenberg Hamiltonian for the disordered random alloy may be written as

Ji

= C or g

In our formalism of the DNLCPA theoretical approach, we consider the magnetically ordered virtual crystal (Fig. 1b) as a first reference, characterized by the effective spin per site, S , and effective exchange interaction, J , between nearest neighbors. These are defined as average quantities using the equations

2. Scattering potentials for Co and Gd sites in the 2D ferrimagnetic [Co1 c Gdc ]n = 1 alloy monolayer

Hdis =

Vi ;

, ran

(E

{C org } . Sites in the 2D dis-

6S J ) u 0 + S J

E

ordered alloy are hence characterized by spins Si (or Sj ) randomly distributed with corresponding nearest neighbor exchange interactions

1

89

= 4cos

denotes

( ) cos ( 1 2 z

1u

3 2

the

y

(2.5a)

=0

) + 2cos(

energy

z ),

y

= k y a,

of z

the

SW,

= k z a and a denotes

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

Fig. 1. Schematic representations of the disordered [Co1 c Gdc ]n = 1 ferrimagnetic alloy (a), the virtual crystal (b), and the seeded subsystems (c and d).

the lattice constant in the 2D hexagonal crystal. Next, the equivalent LL equation for the spin amplitude u , reduced by phase matching, is

E+SJ

6+

5 6

u +

1

1 SJ 6

1 u0 = 0

|U

|U *

(2.5b)

=

u0 ; u

E

M =

6S J 1 S 6

J

1

SJ E+SJ

Jg , d (c ) = c (1

c )2JCC + c (1

c ) JCg

c ) JCg + c 2Jgg

uC

5 6 1

MC |UC = 0

3 1)

+ ei )

(2.9)

;

u*

Mg 6S *Jgd

E

S *Jgd e

6Sg Jgd ei

i

E + Sg Jgd + J *S * ( 3 + 2e i (

3 1)

+ ei )

3. The DNLPCA configurationally averaged Green’s function for the 2D [Co1 c Gdc ]1 ferrimagnetic alloy monolayer The self-energy of the disordered alloy is determined by solving the CPA T-matrix Eq. (2.3). Note that the matrices VC' , V g' and G contained in (2.3) do not commute. Eq. (2.3) can be written in terms of the T-matrices TC and Tg of the constituting species as follows

(2.7a) (2.7b)

T = (1 [V g' as

G

(3.1)

c ) TC + cTg = 0

with ][I

TC =

TC = G (V g' 1

Tg = ][I G )] and )] 1. These may be expressed after some algebra

[VC'

1 (I

+G

XC = G (VC'

(VC'

XC )

1

and Tg =

); Xg = G (V g'

1

G

1

+G

1 (I

Xg ) 1 ,

).

Consequently the T-matrix Eq. (2.3) may be expressed as

I + (1

c )(I

XC )

1

+ c (I

Xg )

1

=0

(3.2)

By defining the following matrix transformations 0

0

'

k k = + and a normalized phase factor can also be defined as = ka . Writing the LL phase matched equations in closed form for the seed site and any of its nearest neighbor sites on the 2D hexagonal structure, one can derive the dynamic matrix MC as k y2

SC JCd + J S ( 3 + 2ei (

E

At this stage, the dynamic scattering potentials VC' and V g' may be deduced from Eqs. (2.4), (2.6), (2.8) and (2.9).

r = 0

u eikr r = | r |

ug

=

The numerical values of the exchange interactions between pairs of nearest neighbor atoms are adopted from [18] as JCC = 15.6meV , Jgg = 0.48meV and JCg = 1.75meV . For the known eutectic equilibrium compositions c 0.5, for the binary Co1 c Gdc alloy, the effective exchange JC, d turns out to be positive while Jg , d is negative. In the Gd seed subsystem, the spin of the embedded Gd ion is hence aligned opposite to the spin S of the hosting ferromagnetic virtual crystal. The spin dynamics in a seed subsystem , say = C , are described in terms of two irreducible spin amplitudes, denoted uC and u . Given the circular symmetry of the seed subsystem, the spin precession amplitude u ( r ) is expressed as

u( r ) =

S JCd e

|Ug =

This formalism further permits the identification of the Green’s function propagator for the virtual crystal as G = M 1. We proceed to derive the dynamic matrices MC , d and Mg, d . The embedded Co and Gd atoms are coupled to the identifiable species of their nearest neighbors by the effective exchanges JCo, d JC, d (c ) and JGd, d Jg, d (c ) , given by

JC, d (c ) = (1

i

Mg |Ug = 0

1

6+

6SC JCd ei

6S JCd

Similarly,

(2.6)

=0

E

=

The dynamic matrix M in the augmented basis is consequently deduced via Eqs. (2.5a), and (2.5b) as

M*

uC ; MC u

|UC =

kz2 ,

= (1 =

c ) VC' + cV g' ; 0;

and

cG

''

V = VC'

=G

'

=G

V g' ; G

0

then

(2.8)

I

XC =

''

I

Xg =

''

as 90

+ (1

V+I

c) G

V+I

Substituting this result in Eq. (3.2), the equation may be expressed

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

I + (1

''

c )(

cG

1

V + I)

+ c(

''

+ (1

c) G

V + I)

1

and the matrices MCC , MCg , and Mgg for the 3 seeded subsystems. This yields 3 non-commuting 4 × 4 dynamic scattering potentials, denoted ' ' ' VCC , VCg and V gg , with

=0 (3.3)

Eq. (3.3) then yields with some algebra the quadratic matrix equation ''2

''

+

c ) ''G

+ (1

V

cG

V

''

c (1

c )(G

' VCC =M

' ' VCg = V gC =M

V )2 = 0

From this equation, it is direct to prove that commutes with the matrix G V . Collecting the terms yields consequently the following form for Eq. (3.1) ''

''2

+ (I + (1

2c ) G

''

V)

c (1

One should solve the quadratic matrix Eq. (3.4) in order to determine the matrix '' and consequently the self-energy of the disordered system under study. The study of polynomial matrix equations constitutes an active and wide research field [50–52] in view of the wide range of applications for such equations. In general, Eq. (3.4) admits many mathematical solutions, and it is necessary to determine the unique physical solution in all appropriate limits of the alloy system. The physical solution of Eq. (3.4) is derived on the basis of the basic criterion, namely the requirement that the DNLCPA gives appropriate results when the two species of atoms of the random alloy are taken to replace one by the Gd ) leading to the two pure specie limits of the alloy other (Co system. In both cases, the system under study reduces to an ordered ferromagnet, and one can formally write the following relations

MC , d

M *;

Mg, d

VC'

0;

V g'

0;

0

0;

Gdis1

T = (V '

1 G 2

1 {(1

2c) G

V+I

(G

V) 2 + 2(1

I4 + (1

G =

V + I}

1

(3.5)

(G

1 {(1

V)2

2c) G + 2(1

=

V+ I }

)] 1 .

c ) 2 (I4

XCC )

1

+ 2c (1

c )(I4

XCg )

1

+ c 2 (I4

Xgg )

1

=0

V ' (11) V ' (12) G (11) G (12) and V ' = . G (12) G (11) V ' (12) V ' (11)

This property formally extends to the matrices Z = I4 X , = (I4 X ) 1, as well as the self-energy . We consequently write (11) (12)

(12) , Z (11)

=

Z (11) Z (12) Z (12) Z (11)

and Z

1

=

Z

1

(11) Z

1

(12)

Z

1

(12) Z

1

(11)

.

Note that Z 1 (11) (Z (11)) 1. Substituting these expressions in Eq. (4.3) yields the following two equations

V+ I 2c) G

1

Z

= (1 c) MC, d + cMg, d + 2G

G (V '

)[I4

(4.3)

The configurationally averaged Green’s function is consequently derived as

Gdis1

(4.2)

c ) TCg + c 2Tgg = 0

). with X = G (V ' The alloy system under study of two monolayers should be invariant under the interchange of the monolayers, and this is reflected in the derived scattering potentials V ' (see Appendix). These matrices may hence be written in terms of 2 × 2 block sub-matrices as follows

G * 1;

2c) G

(4.1c)

Here the symbol I4 represents the 4 × 4 identity matrix. The self-energy is defined as the physical solution of the 4 × 4 matrix Eq. (4.2). Several steps are required to determine this unique physical solution. The first of these steps is to decompose Eq. (4.2) into two coupled 2 × 2 matrix equations. To start, we rewrite Eq. (4.2) in the following form

These limits consequently require that '' converges to zero. This criterion, together with the fact that '' commutes with G V , yield the unique analytic solution for the self-energy of the 2D [Co1 c Gdc ]1 ferrimagnetic spin alloy monolayer as follows 0

(4.1b)

MgC

Mgg

c ) 2TCC + 2c (1

(1

0.

=

MCg =M

The dynamic matrices are derived using the LL equations for spin dynamics and are presented in the Appendix. The T-matrix equation for the present system reads

(3.4)

V )2 = 0

c )(G

' V gg =M

(4.1a)

MCC

I2 + (1

(3.6)

c ) 2ZCC1 (12) + 2c (1

(1

for the 2D ferrimagnetic [Co1 c Gdc ]1 disordered spin alloy monolayer. Eq. (3.6) is a key result of the present paper, which also applies to 1D alloy systems and 3D bulk binary alloys, with appropriate 1D and 3D matrix relations for the constitutive terms.

c ) 2ZCC1 (11) + 2c (1

c ) ZCg1 (11) + c 2Zgg1 (11) = 0

c ) ZCg1 (12) + c 2Zgg1 (12) = 0

By adding and subtracting these equations, one gets

I2 + (1

c ) 2 [ZCC1 (11) + ZCC1 (12)] + 2c (1

c )[ZCg1 (11) + ZCg1 (12)]

+ c 2 [Zgg1 (11)+Zgg1 (12)] = 0

4. Formalism of the DNLPCA theoretical approach for the [Co1 c Gdc ]n 2 ferrimagnetic alloy ultrathin layer

I2 + (1

c ) 2 [ZCC1 (11)

(4.4a)

ZCC1 (12)] + 2c (1

c )[ZCg1 (11)

+ c 2 [Zgg1 (11) Zgg1 (12)] = 0

In this section, we develop the DNLCPA method for ultrathin alloy films composed of two or more alloy monolayers. The main ideas are conserved in this generalization: the virtual crystal is again considered as a reference system and the scattering potentials are defined dynamically as before. The underlying mathematics, however, becomes much more complicated as the number and dimensions of the scattering potentials increase. We first consider the [Co1 c Gdc ]2 ferrimagnetic spin alloy ultrathin layer of two monolayers, as presented schematically in Fig. 2a. For the present case, the scattering potentials are defined via 4 subsystems, namely the virtual crystal and three seeded subsystems presented schematically in Fig. 2b–e. The subsystems introduce four dynamic matrices: the matrix M for the virtual crystal of the two monolayers,

ZCg1 (12)] (4.4b)

Employing the formula for the inverse of a matrix in block form [53], it is possible to write the matrices Z 1 (11) and Z 1 (12) as follows

Z

1

(11) = [I2

Z

1

(12) =

Y 2 ] 1 (Z (11))

[I2

1

Y 2 ] 1 Y (Z (11))

1

with Y = (Z (11)) 1Z (12) . Then, by reducing these expressions we obtain

91

Z

1

Z

1

(11) + Z

1

(11)

1

Z

(12) = [Z (11) + Z (12)]

1

(4.5a)

(12) = [Z (11)

1

(4.5b)

Z (12)]

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

Fig. 2. Schematic representations of the disordered [Co1 c Gdc ]n = 2 ferrimagnetic random alloy (a), the virtual crystal (b), and the seeded subsystems (c, d, e). Atoms in the second atomic plane are presented in fading colors.

In the right-hand side of Eq. (4.5), we substitute the expressions

Z (11) = I2 + G (11)[ (11)

V ' (11)] + G (12)[ (12)

V ' (12)] + G (12)[ (11)

Z (12) = G (11)[ (12)

obtained from the definition Z 1

Z

1

Z

1

(11) + Z (11)

1

Z

(12) = (I2

X

(12) = (I2

+)

X

)

V ' (12)]

±

V

±

= G±* (V

= V (11) ± V (12);

±

I2 + (1

c ) (I2

XCC +)

1

(4.6a)

1

(11) ±

c )(I2

c )2 (I2

XCC )

1

+ 2c (1

2

+ c (I2

Xgg +)

1

+

+

) and

+0

= (1 ' +

=

VCC + =

+

+0

' (VCC +

+ 2c (1

and + 0)

'' +

' c ) VCg +

= G+

= 2c (1

+

' c ) 2 (VCC +

' V gg +)

XCg )

1

+ c 2 (I2

Xgg )

1

=0

D1 =

2I2

D2 =

I2

+ G+ VCC + + I2)

'' 2 +

+ D2

'' +

1

1

+ 2c (1

c )(

'' +

+ G+ VCg + + I2 )

1

(4.8)

=0

(4.9)

+ D3 = 0

G+ VCC + G+ VCC +

G+ VCg + G+ VCg +

D3 =

(1

c ) 2G+ VCC +

G+ Vgg + G+ Vgg +

c 2G+ Vgg +

c ) G+ VCg +

G+ VCC +G+ Vgg +

(1

c )2G+ VCC +

G+ VCg+G+ Vgg+

G+ VCC +G+ VCg + 2c (1

c ) G+ VCg +

c 2G+ Vgg +

G+ VCC +G+ VCg +G+ Vgg + [1

+

' c 2 (V gg +

(1

c )2] G+ VCg +G+ Vgg + c 2] G+

[1

[1

2c (1

c )] G+ VCC +G+ Vgg +

VCC +G+ VCg +

The physical solution for Eq. (4.9) is determined using the same criterion as for the monolayer system in section III, and is derived for + as follows

' + ' VCC +)

' V gg +)

+ G+ Vgg + + I2)

2c (1

' c 2V gg +

' c )(VCg +

'' +

c )2 ( '' +

+ D1

)

+

' c )(VCg +

with

We now proceed to solve Eqs. (4.7). Two different methodological steps are developed to solve them. □ The first step considers Eq. (4.7a). Establishing the following definitions of the matrices of interest as ' c ) 2VCC +

= (1

=0

The 4 × 4 T-matrix equation in Eq. (4.3) is now transformed into two 2× 2 matrix Eqs. (4.7) having the same structure as the original equation (4.3). The self-energy can be determined from the solutions of Eqs. (4.7a) and (4.7b), respectively, via the basic identities + and

1 (12) = ( 2

+ 0)

'' 3 + 1

(4.7b)

1 (11) = ( 2

' (V gg +

' VCg +)

Now using the explicit form of the basic 4 × 4 dynamic matrices given in the Appendix, one can prove that the matrices ' ' ' ' ' ' V gg G+ (VCC VCg V gg +) commute. This + +) , G+ (VCC + +) , and G+ (VCg + implies that the matrices G+ V + commute amongst each other, and also with +'' . Eq. (4.8) consequently yields the cubic matrix equation

(12);

XCg +)

c )(I2

Vgg + =

+ c2 (

(4.7a) I2 + (1

' ' 2 VCg +) + c (V gg +

I2 + (1

(4.6b)

+ 2c (1

' c ) 2 (VCC +

Eq. (4.7a) may be reduced to the following compact form

Finally, substituting (4.6) in (4.4) gives 2

= (1

X . This then gives

= I4

1

=

+0)

+ 2c (1

G±* = G * (11) ± G * (12);

±);

±

' (VCg +

V ' (11)]

with

X

VCg + =

+

' VCC +)

92

=

+0

+ G+

1

1 D1 3

3

2 2 (D1 + 3D2 ) M 3

1

1 M 33 2

(4.10)

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

with

M=

3

2D13

9D1 D2

+3 3

D12 .

[Co1 c Gdc ]n = 2 with that corresponding to the single monolayer system [Co1 c Gdc ]n = 1, one can deduce the contributions to the diagonal and offdiagonal entities of Gdis1;n = 2 due to the interlayer interactions in the bilayer system. These contributions may be used to build Gdis1;n 3 . Let Gn =12 denotes the 4 × 4 dynamic matrix for a system of two non-interacting Co1 c Gdc alloy planes, then one can write

27D3

D22

4D23 + 4D13. D3 + 18D1 D2 D3 + 27D32

□ The second step considers Eq. (4.7b). We start by defining the matrix ' as

I2 + G

=

Gn =12 =

1 ' ' + G (VCC + V gg ) 2

'

' ' ' ' + V gg ) V gg ) and R = 2 (VCC Putting V = 2 (VCC (4.7b) may be reduced to the following compact form '

G

+G

V)

c )2 (

I2 + (1 '

+ c2 (

' VCg , Eq.

1

1

V) 1

1

+ 2c (1

c )(

'

+ G R)

''

c )2 (

I2)

1

c )(

''

+ A)

1

+ c2 (

''

+ I2)

1

c ) 2 /det[

1

= (1

2

= 2c (1

3

= c 2/det[

''

c )/det[ ''

1

=0

R.

1

+

2

+

Gdis1;n = 3

=(

( (

(

1

+

1

+

1

+

n=

m2 m2 m2

2

+

2

+

2

+

3)

3)

3)

3 ) I2

1

=

(4.12)

A 2A

B

A 2A ]

= det[(

2

+ 2 1 ) I2

= det[(

2

+ 2 3 ) I2 + B +

= det[(

1

1

2 c2 2

3 ) I2

+(

B

2c3 n

2

2c1 m+n

2

c2 2

= det[(m + n) I2 = det[(m

= det[nI2 + (m

+ 2 G1

G2 Gdis1; n= 1 + G1

G2

1

+

A 2A ]

A 3)A

B

n) I2 + B + A 2) A

B]

A 2A ]

A 2A ]

(4.13c) 1

+

2

+

3

G2

0

0

G2

Gdis1; n= 1 + 2 G1

G2

0

0

G2

Gdis1; n= 1 + 2 G1

G2

0

0

G2

Gdis1; n= 1 + G1

The exchange interaction coefficients are considered between pairs of nearest neighbor atoms, namely: JCC = 15.6 meV , Jgg = 0.48 meV and JCg = 1.75 meV [18]. Though the exchange constants could vary with the alloy concentration, we adopt the same exchange values for the two systems [Co0.7 Gd 0.3 ]n and [Co0.6 Gd0.4 ]n , to be treated numerically here. For sufficiently low temperatures, the [Co1 c Gdc ]n ferrimagnetic alloy can be considered in its ground state, with Sg = 3.5 and SC = 1. The eigenmodes of the matrix Gdis1 , derived on the basis of the DNCLPA theory, are determined numerically and yield the energies and lifetimes of the spin wave excitations in the disordered ultrathin films. The numerical results, for the SW dispersion curves, and their lifetimes, are presented in Figs. 3–5 for the considered systems composed of one, two and three alloy monolayers, respectively. Each figure presents results for two different eutectic equilibrium concentrations, namely c = 0.3 and c = 0.4 . The SW energies , normalized with respect to JCC SC , and the lifetimes in femtosecond, are plotted as a function of the 1.3 in the lower section of the 2D BZ normalized wave vector 0 y where the classical spin representation is valid; note that z = 0 in the present numerical applications. The continuous colored curves represent the DNLCPA numerical results, whereas the dashed black curves represent the VCA results. The same color scheme is used for energy and lifetime graphics to identify each mode with its corresponding lifetimes. The numerical applications of the DNLCPA theory to the ultrathin magnetic 2D alloy films [Co1 c Gdc ]n reproduce successfully the Goldstone SW acoustic mode for the ensemble of systems under study, over c = 0.4 and 0.3, and n = 1, 2, and 3. Note that we have neglected here magnetic anisotropy and Zeeman terms in the Hamiltonian. On the basis of our numerical results, presented in Figs. 3–5, our theory predicts relatively long lifetimes for the SWs in the ultrathin 2D

(4.13b)

B]

Gdis1; n = 1 + G1

5. Numerical applications

(4.13a)

Finally, applying the change of parameters m = 1 3 , Eqs. (4.13) simplify to

m

G2

0

Gdis1; n= 4

3 2 c1

=

+ G1

G2 Gdis1;n = 1

0

In Eq. (4.12), the factors ''A and AA denote the adjugate of the matrices '' and A respectively. Adding respectively the terms ( 1 + 2 + 3 ) I2 , ( 1 + 2 + 3 ) I2 and ( 1 + 2 + 3 ) A for both sides of Eq. (4.12), and taking the determinant yields 3 scalar equations as follows 2 c3

G2

+ A]

+ I2]

''A

3)

G2 Gdis1;n = 1

Gdis1; n= 1 + G1

Using the relation between the inverse determinant and the adjugate of a 2 × 2 matrix, we derive the following equation from (4.11)

(

G2 , G2

Gdis1;n = 1 + G1

Gdis1;n = 2 =

I2] ''

G1 G2

Gn=12 =

then the 2 × 2 block matrices G1 and G2 represent the diagonal and off-diagonal contributions due to the interlayer interactions in the bilayer system. With G1 and G2 in hand, one can formally write the following identitites:

(4.11) with '' = (G V ) 1 ' , B = G V and A = V We next introduce the scalar parameters

Gdis1; n = 1

G = Gdis1;n = 2

1

=0

+ 2c (1

0

0

Next define G as

This last equation can be restated in the following form

B + (1

Gdis1; n= 1

and

(4.14a) (4.14b) (4.14c)

Eqs. (4.14) are polynomial equations inm , n , and 2 and can be solved directly using built-in functions in standard numerical software (e.g. Mathematica), with a computational cost of about 1.5 s only. Once the parameters m , n , and 2 are determined, one can determine the matrix '' from Eq. (4.12) and then deduce the physical solution for based on the same criterion as before. The mathematical development of the present theoretical approach is sufficient to determine the 2n × 2n dynamic matrix Gdis1;n for any ultrathin random spin alloy [Co1 c Gdc ]n with arbitrary nanostructure thickness of n monolayers. By comparing the configurationally averaged Green’s functions corresponding to the two monolayers system 93

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

Fig. 3. Figures (a) and (c) present respectively the numerically calculated dispersion curves for the SW propagating in [Co0.7 Gd 0.3 ]n = 1 and [Co0.6 Gd0.4 ]n= 1 disordered random spin alloys. Dashed curves correspond to the VCA results, and solid curves to the DNLCPA results. Figures (b) and (d) present the corresponding lifetimes (logarithmic plot) for [Co0.7 Gd 0.3 ]n = 1 and [Co0.6 Gd0.4 ]n= 1 alloys respectively. The SW energies, , normalized with respect to JCC SC , and the SW lifetimes, are plotted as 1.3 in the lower section of the 2D BZ where the classical spin representation is valid. a function of the normalized wave vector 0 y

Fig. 4. Same as Fig. 3 but for the [Co0.7 Gd 0.3 ]n = 2 and [Co0.6 Gd0.4 ]n= 2 disordered random spin alloys. 94

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

Fig. 5. Same as Fig. 3 but for the [Co0.7 Gd 0.3 ]n = 3 and [Co0.6 Gd0.4 ]n= 3 disordered random spin alloys.

magnetic disordered alloy film, for the lower section of the BZ. In particular, the calculated lifetimes of the SWs are observed to increase exponentially to high values when y decreases towards the Brouillon Zone (BZ) center; this BZ section corresponds to SW with wavelengths at the low nano scale (∼2 nm) and upwards. The DNLCPA theoretical results can hence validate with precision the limits where the diffusive damping of SWs can be justifiably neglected whether for theoretical or experimental needs in layered spin alloy nanostructures, with useful applications in magnonic studies of the spin alloy systems [1–12,16–29,39–45]. Further, our theoretical results demonstrate that the lifetime of a spin wave depends strongly on its wave vector, in agreement with recent experimental results [47,48]. The spin-polarized electron energy loss spectroscopy (SPEELS) technique is used in these experiments to probe the lifetimes of the magnetic exchange SWs in ultrathin magnetic films, as the two monolayers of bcc Fe on W(1 1 0) in Ref. [47], and the two monolayers of the Fe50Pd50 alloy on Pd(0 0 1) in Ref. [48]. These films, notably the latter, are comparable structurally and magnetically to the [Co1 c Gdc ]n systems in the present work. One observes remarkably a similar functional behavior for the SW lifetimes and lifetime scales for the theoretical results in Figs. 3b, d, 4b, d, 5b and d of the present work, as compared with the experimental data of Figs. 3 and 2a in respectively Refs. [47] and [48]. The DNLCPA numerical results demonstrate the strong dependence of the SWs’ energies and lifetimes on the alloy concentration of the ultrathin binary alloy, as well as on the thickness of n atomic monolayers for the film. As a result, the energies and lifetimes of the confined SWs in the 2D binary alloy film can be tuned structurally, chemically, and magnetically to meet the requirements of potential technological applications. The DNLCPA calculations show also that the energies of the SWs at low temperatures are in general higher for the ferrimagnetic [Co0.6 Gd0.4 ]n film than for the other one [Co0.7 Gd 0.3 ]n . This is due to the values of the site magnetizations which can be attributed to Gd and Co. The energies and lifetimes of the confined SWs will vary significantly

with temperature since the site magnetizations do too [19]. In contrast to their energies, the SWs’ lifetimes at low temperatures are shorter for the ferrimagnetic [Co0.6 Gd0.4 ]n film than for the other one [Co0.7 Gd 0.3 ]n , and this, notably, at identical wave vectors. For further investigation, the SWs’ lifetimes are plotted in Fig. 6 as a function of the disorder strength c , for a fixed value of the normalized wavevector ( y = 0.3 ). Numerical data presented in red, blue and green respectively correspond to ultrathin films with n = 1, 2 and 3 atomic layers. In general, the disorder is found to significantly reduce the SWs’ lifetimes with a sharp drop observed between c = 0.25 and c = 0.3. Moreover, the SW’s lifetimes for the monolayer case are significantly long compared to n= 2 and n = 3 cases. To understand the spin dynamics underlying the DNLCPA eigenmodes of Figs. 4 and 5, we have calculated the corresponding eigenvectors which give valuable information about the effective spin precession amplitudes (spa) in the different atomic monolayers of the ultrathin magnetic alloy films. Schematic representations of the

Fig. 6. Spin waves lifetimes as a function of the disorder strength c in the interval 0.2 c 0.4 , with y = 0.3. Red, blue and green numerical data correspond to ultrathin films with n = 1, 2 and 3 atomic layers respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 95

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

Fig. 7. Schematic representations of the DNLCPA spin precession amplitudes corresponding to the two spin wave modes presented in Fig. 4a for the [Co0.7 Gd 0.3 ]n = 2 ultrathin alloy film. The spin precession amplitudes are calculated at y = 0 . See the text for details as regards y in general. The horizontal planes in the figures represent the atomic planes constituting the ultrathin alloy. The left and right figures correspond respectively to the red and blue dispersion curves in Fig. 4a. The figures demonstrate the phase relationship between the spin precession amplitudes in different model atomic layers of the ultrathin alloy; see text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

DNLCPA spa (at y = 0 ) are presented for the SW eigenmodes, during their lifetime, for the [Co0.7 Gd 0.3 ]n = 2 and [Co0.7 Gd 0.3 ]n = 3 systems, in respectively Figs. 7 and 8. The atomic monolayers which constitute the alloy film are represented by horizontal planes in these figures. It is worth noting that the above spa phase results are general; we have numerically verified that although the spa eigenvectors change with variable y , the phase relation between the spa in different atomic layers per each dispersion curve of eigenmodes, remain unchanged; this corresponds to constant patterns of confinement of the spinwave eigenmodes. The left and right figures of Fig. 7 correspond respectively to the red and blue dispersion curves in Fig. 4a. For both eigenmodes, the calculated spa in the two monolayers are observed to be equal and in-phase for the low energy mode (red dispersion curve), and out-of-phase for the high energy mode (blue dispersion curve). In Fig. 8 the left, middle and right figures correspond respectively to the red, blue and green dispersion curves in Fig. 5a. For the low energy eigenmode (red dispersion curve), all spa are observed to be equal and in-phase. For the middle energy eigenmode (blue dispersion curve), the spa in monolayers 1 and 3 are equal and out-of-phase, whereas the spa in monolayer 2 is zero. As for the high energy eigenmode (green dispersion curve), the spa in layers 1 and 3 are equal, in-phase, and significantly smaller than the spa in monolayer 2. In addition, always for the high energy eigenmode, one observes that the spa in layers 1 and 3 are out-of-phase relative to the spa in layer 2. These phase relationships, acting via the

binding magnetic exchange between atomic sites, characterize the energy differences between the different SW eigenmodes propagating in the ultrathin 2D magnetic alloy film. The DNLCPA method gives hence access to valuable knowledge about the spin dynamics and the localization of the SW on each layer of the ultrathin alloy, as well as the phase relationship between spin precession amplitudes in different atomic layers. 6. Conclusion and perspectives We have developed a dynamic variant of the coherent potential approximation theory (DNLCPA) to analyze the spin wave dynamics in ultrathin films of randomly disordered binary magnetic alloys. The theoretical approach is in particular applied to the system of ultrathin [Co1 c Gdc ]n spin alloy on a hexagonal lattice. The dynamic nature of the theory resides in the definition of the scattering potential of an embedded defect by the phase matching of the spin dynamics on a seed spin site, with the spin dynamics in the virtual crystal VCA considered as a first approximation of the spin alloy. The scattering potentials are then used to formulate the DNLCPA using the Dyson’s formalism. The complex self-energy and the configurationally averaged Green’s function Gdis are determined by solving the Dyson T-matrix equation. The SW excitations in the ultrathin [Co1 c Gdc ]l spin alloy are numerically determined as the eigenmodes of the matrix Gdis1 . The solutions turn out to be complex, yielding the energies and also the

Fig. 8. As in Fig. 6, but for the three spin wave modes presented in Fig. 5a for the [Co0.7 Gd 0.3 ]n = 3 ultrathin alloy film. The left, middle and right figures correspond respectively to the red, blue and green dispersion curves in Fig. 5a; see text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 96

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater

lifetimes of the spin wave excitations. This demonstrates the robustness and efficiency of the DNLCPA method as it successfully converts the alloy site disorder into the incoherent decay of these elementary excitations. The DNLCPA theory shows that this decay depends strongly on the wave vector and energy of the considered SW excitations, in agreement with available experimental results. We finally note that the developed theory is general and can be applied for nanostructures and bulk systems constituted of diverse binary magnetic alloys. Although only the exchange interaction is included in the present study, the method can readily account for other

interactions as the magnetic anisotropy and the Zeeman effect, via a proper modification of the scattering potentials. Acknowledgments The numerical calculations were performed using the Phoenix High Performance Computing facility at the American University of the Middle East (AUM), Kuwait. The author A.K. acknowledges the NPRP 4184-1-035 project grant from TAMU – QNRF.

Appendix We present the explicit forms of the derived 4 × 4 dynamic matrices M , MCC , MCg , and Mgg associated with the hexagonal [Co1 c Gdc ]2 ferrimagnetic spin alloy layer in Fig. 2. To simplify the presentation, we define the following expressions 1

= 2cos( z ) + 4cos

2

= 10cos2

3

=

4

= 5 + 2e i (

=

1 2

2 y

3 1)

3 1)

1 3 2

cos

z

1 2

+ 10cos

z

17 + 2e i ( 3

1 2

z

y

1 3 2

cos

+4

y

+ ei

+ ei

2 z

+

The dynamic matrices obtained from the LL equations of motion for this system have the following forms

M =

M (11) M (12) , and MCC = M (12) M (11)

MCg (11) MCg (12)

MCC (11) MCC (12) , MCg = MCC (12) MCC (11)

MCg (12) MCg (11)

, Mgg =

Mgg (11) Mgg (12) Mgg (12) Mgg (11)

with

M (11) =

E

MCC (11) =

MCC (12) =

MCg (11) =

MCg (12) =

Mgg (11) =

Mgg (12) =

9J S 1 J 6

E

S

J S E+J S

1

8JCd S JCd S e

E

1

e

i

1 J S 2 Cg C 1 J S 2 gd

e

)

1 J S 2 Cg A

i

+

1 J S 2 gd

1 J S 2 Cg g 1 J Se i 2 Cd

+

Jgd S e

Jgd S e

i

J S 1 J 18

S

e

i

3JCd SC e i

i

E

JCd SC

JCd SC ei

1

1 J 3 1 J 9

S

1

S

2

3Jgd Sg ei

7J S + Jgd Sg + 5J S ei

Jgd Sg ei

2J S

6Jgd Sg e i

8Jgd S + Jgg Sg

Jgg Sg

, M (12) =

6JCd SC ei 2JCd SC 7J S + 5J S ei

JCC SC i

8JCd S + 2 JCg Sg 1 J S 2 Cd

E

(

9+

5 6 1

2JCd SC ei 2J S

JCC SC J Se i E

1

E + 2Jgd Sg

7J S + 5J S ei

2Jgd Sg ei 2J S

These matrices serve to establish the scattering potentials.

Rev. Mod. Phys. 82 (2010) 2731. [3] T.A. Ostler, R.F.L. Evans, R.W. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, I. Radu, R. Abrudan, F. Radu, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, A. Kimel, Crystallographically amorphous ferrimagnetic alloys: comparing a localized atomistic spin model with experiments, Phys. Rev. B 84 (2011) 024407. [4] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H.A. Dürr, T.A. Ostler, J. Barker, R.F.L. Evans, R.W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th. Rasing, A.V. Kimel, Transient ferromagnetic-like state mediating ultrafast

References [1] M. Binder, A. Weber, O. Mosendz, G. Woltersdorf, M. Izquierdo, I. Neudecker, J.R. Dahn, T.D. Hatchard, J. Thiele, C. Back, M. Scheinfein, Magnetization dynamics of the ferrimagnet CoGd near the compensation of magnetization and angular momentum, Phys. Rev. B 74 (2006) 134404. [2] A. Kirilyuk, A.V. Kimel, T. Rasing, Ultrafast optical manipulation of magnetic order,

97

Journal of Magnetism and Magnetic Materials 482 (2019) 88–98

D. Ghader and A. Khater reversal of antiferromagnetically coupled spins, Nature 472 (2011) 205. [5] A. Mekonnen, M. Cormier, A.V. Kimel, A. Kirilyuk, A. Hrabec, L. Ranno, Th. Rasing, Femtosecond laser excitation of spin resonances in amorphous ferrimagnetic Gd1xCox alloys, Phys. Rev. Lett. 107 (2011) 117202. [6] T.A. Ostler, J. Barker, R.F.L. Evans, R.W. Chantrell, U. Atxitia, O. ChubykaloFesenko, S. El Moussaoui, L. Le Guyader, E. Mengotti, L.J. Heyderman, F. Nolting, A. Tsukamoto, A. Itoh, D. Afanasiev, B.A. Ivanov, A.M. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk, Th. Rasing, A.V. Kimel, Ultrafast heating as a sufficient stimulus for magnetization reversal in a ferrimagnet, Nat. Commun. 3 (2012) 666. [7] A. Kirilyuk, A.V. Kimel, T. Rasing, Laser-induced magnetization dynamics and reversal in ferrimagnetic alloys, Rep. Prog. Phys. 76 (2013) 026501. [8] A. Mekonnen, A.R. Khorsand, M. Cormier, A.V. Kimel, A. Kirilyuk, A. Hrabec, L. Ranno, A. Tsukamoto, A. Itoh, Th. Rasing, Role of the inter-sublattice exchange coupling in short-laser-pulse-induced demagnetization dynamics of GdCo and GdCoFe alloys, Phys. Rev. B 87 (2013) 180406. [9] N. Bergeard, V. López-Flores, V. Halté, M. Hehn, C. Stamm, N. Pontius, E. Beaurepaire, C. Boeglin, Ultrafast angular momentum transfer in multisublattice ferrimagnets, Nat. Commun. 5 (2014) 3466. [10] Kh. Zakeri, Elementary spin excitations in ultrathin itinerant magnets, Phys. Reports 545 (2014) 47. [11] W. He, H. Wu, J. Cai, Y. Liu, Z. Cheng, Laser-Induced magnetization dynamics of GdFeCo film probing by time resolved magneto-optic Kerr effect, SPIN 05 (2015) 1540014. [12] I. Radu, C. Stamm, A. Eschenlohr, F. Radu, R. Abrudan, K. Vahaplar, T. Kachel, N. Pontius, R. Mitzner, K. Holldack, A. Föhlisch, T.A. Ostler, J.H. Mentink, R.F.L. Evans, R.W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, A.V. Kimel, Th. Rasing, Ultrafast and distinct spin dynamics in magnetic alloys, SPIN 05 (2015) 1550004. [13] J.P. Andrés, J.L. Sacedon, J. Colinoa, J.M. Riveiro, Interdiffusion up to the eutectic composition and vitrification in Gd/Co multilayers, J. Appl. Phys. 87 (2000) 2483. [14] J.A. Gonzalez, J. Colino, J.P. Andrés, M.A. Lopez de la Torre, J.M. Riveiro, Polarized neutron study of Gd1-xCox/Co multilayers, Phys. B 345 (2004) 181. [15] J.P. Andrés, J.A. Gonzalez, T.P.A. Hase, B.K. Tanner, J.M. Riveiro, Artificial ferrimagnetic structure and thermal hysteresis in Gd0.47Co0.53/Co multilayers, Phys. Rev. B 77 (2008) 144407. [16] A. Khater, B. Bourahla, M.A. Ghantous, R. Tigrine, R. Chadli, Magnons coherent transmission and heat transport at ultrathin insulating ferromagnetic nanojunctions, Eur. Phys. J. B 82 (2011) 53. [17] D. Ghader, A. Khater, V. Ashokan, M. Abou Ghantous, Spin waves transport across a ferrimagnetically ordered nanojunction of cobalt-gadolinium alloy between cobalt leads, Eur. Phys. J. B 86 (2013) 180. [18] M. Abou Ghantous, A. Khater, V. Ashokan, D. Ghader, Sublattice magnetizations of ultrathin alloy Co1-cGdcn nanojunctions between Co leads using the combined effective field theory and mean field theory methods, J. App. Phys. 113 (2013) 094303. [19] V. Ashokan, M. Abou Ghantous, D. Ghader, A. Khater, Ballistic transport of spin waves incident from cobalt leads across cobalt-gadolinium alloy nanojunctions, J. Mag. Mag. Mater. 363 (2014) 66. [20] V. Ashokan, A. Khater, M. Abou Ghantous, D. Ghader, Pin wave ballistic transport properties of Co1-cGdclColCo1-cGdcl nanojunctions between Co leads, J. Mag. Mag. Mater. 384 (2015) 18. [21] V. Ashokan, M. Abou Ghantous, D. Ghader, A. Khater, Computation of magnons ballistic transport across an ordered magnetic iron-cobalt alloy nanojunction between iron leads, Thin Solid Films 616 (2016) 6. [22] E. Mojaes, A. Khater, M. Abou Ghantous, V. Ashokan, Magnonic ballistic transport across Fe-Ni alloy nanojunctions between Fe/Co leads using phase field matching and Ising effective field theory approaches, Acta Mater. (2018) (accepted 19 October 2018). [23] D.A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B 55 (1997) 3050. [24] S. Wienholdt, D. Hinzke, K. Carva, P.M. Oppeneer, U. Nowak, Orbital-resolved spin model for thermal magnetization switching in rare-earth-based ferrimagnets, Phys. Rev. B 88 (2013) 020406(R). [25] U. Atxitia, T. Ostler, J. Barker, R.F.L. Evans, R.W. Chantrell, O. Chubykalo-Fesenko,

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

[45] [46] [47] [48] [49] [50] [51] [52] [53]

98

Ultrafast dynamical path for the switching of a ferrimagnet after femtosecond heating, Phys. Rev. B 87 (2013) 224417. R.F.L. Evans, U. Atxitia, R.W. Chantrell, Quantitative simulation of temperaturedependent magnetization dynamics and equilibrium properties of elemental ferromagnets, Phys. Rev. B 91 (2015) 144425. U. Atxitia, P. Nieves, O. Chubykalo-Fesenko, Landau-Lifshitz-Bloch equation for ferrimagnetic materials, Phys. Rev. B 86 (2012) 104414. R.F.L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R.W. Chantrell, O. ChubykaloFesenko, Stochastic form of the Landau-Lifshitz-Bloch equation, Phys. Rev. B 85 (2012) 014433. C. Etz, L. Bergqvist, A. Bergman, A. Taroni, O. Eriksson, Atomistic spin dynamics and surface magnons, J. Phys.: Cond. Matter 27 (2015) 243202. P. Soven, Coherent-potential model of substitutional disordered alloys, Phys. Rev. 156 (1967) 809. D.W. Taylor, Vibrational properties of imperfect crystals with large defect concentrations, Phys. Rev. 156 (1967) 1017. B. Velicky, S. Kirkpatrick, H. Ehrenreich, Single-site approximations in the electronic theory of simple binary alloys, Phys. Rev. 157 (1968) 747. R.J. Elliott, J.A. Krumhansl, P.L. Leath, The theory and properties of randomly disordered crystals and related physical systems, Rev. Mod. Phys. 46 (1974) 465. M. Jarrell, H.R. Krishnamurthy, Systematic and causal corrections to the coherent potential approximation, Phys. Rev. B 63 (2001) 125102. D.A. Rowlands, Investigation of the nonlocal coherent-potential approximation, J. Phys.: Condens. Matter 18 (2006) 3179. A. Marmodoro, A. Ernst, S. Ostanin, J.B. Staunton, Generalized inclusion of shortrange ordering effects in the coherent potential approximation for complex-unit-cell materials, Phys. Rev. B 87 (2013) 125115. S.F. Edwards, A new method for the evaluation of electric conductivity in metals, Phil. Mag. 3 (1958) 1020. E.N. Economou, Green’s Functions in Quantum Physics, Springer, Berlin, 1979. A.V. Chumak, H. Schultheiss, Magnonics: spin waves connecting charges, spins and photons, J. Phys. D Appl. Phys. 50 (2017) 300201. A.V. Chumak, A.A. Serga, B. Hillebrands, Magnonic crystals for data processing, J. Phys. D Appl. Phys. 50 (2017) 244001. B. Lenk, H. Ulrichs, F. Garbs, M. Münzenberg, The building blocks of magnonics, Phys. Rep. 507 (2011) 107. V.V. Kruglyak, S.O. Demokritov, D. Grundler, Magnonics, J. Phys. D Appl. Phys. 43 (2010) 264001. S.A. Nikitov, Ph. Tailhades, C.S. Tsai, Spin waves in periodic magnetic structures magnonic crystals, J. Mag. Mag. Mater. 236 (2001) 320. Haoliang Liu, Chuang Zhang, Hans Malissa, Matthew Groesbeck, Marzieh Kavand, Ryan McLaughlin, Shirin Jamali, Jingjun Hao, Dali Sun, Royce A. Davidson, Leonard Wojcik, Joel S. Miller, Christoph Boehme, Z. Valy Vardeny, Organic-based magnon spintronics, Nat. Mater. 17 (2018) 308. A.V. Chumak, V.I. Vasyuchka, A.A. Serga, B. Hillebrands, Magnon spintronics, Nat. Phys. 11 (2015) 453. R.M. Vollmer, P.S. Etzkorn, H. Anil Kumar, J. Ibach, Kirschner Spin-polarized electron energy loss spectroscopy: a method to measure magnon energies, 272–276, J. Mag. Mag. Mater. 2126 (2004). Y. Zhang, T.-H. Chuang, Kh. Zakeri, J. Kirschner, Relaxation time of terahertz magnons excited at ferromagnetic surfaces, Phys. Rev. Lett. 109 (2012) 087203. H.J. Qin, Kh. Zakeri, A. Ernst, L.M. Sandratskii, P. Buczek, A. Marmodoro, T.H. Chuang, Y. Zhang, J. Kirschner, Long-living terahertz magnons in ultrathin metallic ferromagnets, Nat. Commun. 6 (2015) 6126. L. Landau, E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. der Sow. 8 (1935) 153. J. Dennis, J. Traub, R. Weber, The algebraic theory of matrix polynomials, SIAM J. Num. Anal. 13 (1976) 831. I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic, New York, 1982. N. Higham, H.-M. Kim, Numerical analysis of a quadratic matrix equation, IMA J. Num. Anal. 20 (2000) 499. T.-T. Lu, S.-H. Shiou, Inverses of 2 × 2 block matrices, Comput. Math. Appl. 43 (2002) 119.