A spin-12 model for the Curie temperature of binary superlattices containing alloy disordered interfaces

Journal of Magnetism and Magnetic Materials 247 (2002) 305–315

A spin-12 model for the Curie temperature of binary superlattices containing alloy disordered interfaces A. Khatera, M. Abou Ghantousb,*, M. Fresneaua a

Laboratoire de Physique PEC, Universit!e du Mans, F-72085 Le Mans, France b Department of Physics, American University of Beirut, Beirut, Lebanon Received 26 July 2001; received in revised form 8 March 2002

Abstract A theoretical model is presented on a simple cubic lattice to determine the Curie temperature phase diagrams for superlattices A2 ðAp1 B1p1 ÞðAp2 B1p2 ÞB2 : These are considered to consist of periodic cells containing two layers of spin-12 A atoms, two layers of spin-12 B atoms and alloy disordered ðAp1 B1p1 ÞðAp2 B1p2 Þ double layer interfaces in between, that are characterised by a random arrangement of the A and B atoms. The model is general and can be used for ferroor anti-ferromagnetic A–B exchange coupling. The A–A and B–B exchange couplings are considered ferromagnetic. An effective field theory is employed to calculate the Curie temperature TC phase diagrams as a function of the twodimensional (2D) alloy concentrations p1 and p2 : The needed superlattice exchange constants are determined by calculating the TC phase diagram for an equivalent bulk alloy and comparing with known experimental data. The theoretical phase diagram results for the considered superlattices are presented for different values of the 2D concentrations and compared with those for a spin-12 Ising multilayer superlattice A2 ðAp B1p ÞB2 with concentrations 0ppp1: The concentration dependence of the phase diagram is an important feature of this work allowing a useful experimental analysis of similar systems. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.70.Cn Keywords: Superlattices; Multilayers; Phase diagrams; Effective field theory and Ising films

1. Introduction Magnetic superlattices and layered structures are of current interest since they can be successfully prepared and may provide new and possibly technologically useful applications [1]. The preparation and study of magnetic superlattices which is composed of two different magnetic materials is, hence, motivated by the search for composite materials of controlled magnetic properties. In this respect the study of the relation between the composition and the structure of these systems on the one hand, and their magnetic properties on the other, has become an active field of research [2, see for example Ref. 3].

*Corresponding author. Fax: +961-1-351706. E-mail address: [email protected] (M.A. Ghantous). 0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 2 8 3 - 4

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There has been in recent years an effort directed towards the experimental study of critical phenomena in various binary magnetic superlattices [4,5]. Theoretical studies of some of these systems include models based on the Ising superlattice representation [6–11]. In general, the atomic layers in these superlattice systems are often considered to terminate abruptly in a clean manner. Experimentally, however, it is known that the two types of magnetic atoms in a binary superlattice may interdiffuse during the superlattice preparation process to give a thermodynamically stable alloy disordered interface [12–16]. The influence of these interfaces, whether alloy or structurally disordered, is also well know from the giant magnetoresistance effect observed in metallic superlattices (see for example Ref. [17]). The consequences of such alloy disordered interfaces to the superlattice bulk magnetic properties has been investigated in the past in an Ising representation [8,18,19], for a spin-12 system Ak ðAp B1p ÞBk’ that consists of a repeating array of k block atomic layers of A atoms, k0 block atomic layers of B atoms, and a single disordered alloy atomic layer interface (Ap B1p ) between them. It has been shown that the single atomic alloy layer interfaces may severely modify the superlattice bulk magnetic properties, especially when the numbers of block atomic layers k and k0 in a superlattice unit cell are relatively small [19]. In this paper, we investigate the influence of this alloy interface when the stoichiometric disorder is spread over two atomic layers rather than one, expressed by the superlattice formula A2 Ið1Þ Ið2Þ B2 : This is schematically presented in Fig. 1. The individual interface layers have the following compositions Ið1Þ  ðAp1 B1p1 Þ and Ið2Þ  ðAp2 B1p2 Þ; for which the alloy disorder is characterised by two concentrations p1 and p2 in the respective layers. This scheme is perhaps a more realistic configuration for the alloy disorder in the superlattice interfaces, and is a model for a wider class of material applications. It is reasonable to assume that the concentrations p1 and p2 will be different in the interface layers in general. This is discussed further in Section 4. In particular, we determine the Curie temperature phase diagrams for these superlattices. The concentration dependence of the phase diagrams is an important feature of this work allowing a useful experimental analysis of similar systems.

A (1)

(

A

)

I (1)

( m A(1) , m B(1) )

I (2 )

( m A( 2) , m B(2) )

B(1)

(

B

)

B(2)

(

B

)

I (2 )

( m A( 2) , m B(2) )

I (1)

( m A(1) , m B(1) )

A (1)

(

A

)

A (2)

(

A

)

I (1)

( m A(1) , m B(1) )

Fig. 1. A schematic cross-section representation of the A2 ðAp1 B1p1 ÞðAp2 B1p2 ÞB2 magnetic superlattice with partial magnetisation symbols per species and layer.

A. Khater et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 305–315

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In Section 2, we present the model formulation for the problem. The Hamiltonian for the system is given, and the analytical approach is developed to derive expressions for the transition Curie temperatures as a function of the 2D alloy disorder in the interface. The model is applied to the binary Fe2(Fep1Gd1p1) (Fep2Gd1p2)Gd2 superlattice composed of iron and of the rare earth metal gadolinium. The presented analysis employs the effective field method for Ising systems, an approach that was founded by Zernicke [20] and Callen [21], and widely developed by other workers [22–24]. Since the magnetic exchange constants JFeGd are not known in the spin-12 model, we develop in Section 3 a calculation for the phase diagrams of the Curie temperatures for the bulk alloy FecGd1c as a function of the 3D concentration c: The theoretical results are compared with available experimental data to determine the most reliable JFeGd parametric value for this exchange constant in the bulk alloy over the whole range of concentrations. This exchange constant is necessary for the numerical calculation in Section 4, in order to obtain the phase diagrams of the considered Fe2(Fep1Gd1p1)(Fep2Gd1p2)Gd2 superlattice.

2. Model formulation Consider the thermodynamically stable superlattice A2(Ap1B1p1)(Ap2B1p2)B2 system as an infinite simple cubic lattice made up of successive atomic layers. The repeating superlattice unit cell along the normal to its planes consists, as in Fig. 1, of two layers of spin-12 A atoms and two layers of spin-12 B atoms, separated by an extended interface of two alloy disordered layers. In the Ising representation the two spin species A and B are described by the generalised z component of their spin operators. These atoms are supposed to be distributed randomly on the sites of the interface layers I(1) and I(2). We consider here, with no loss of generality, that the atoms of each species couple ferromagnetically amongst themselves but that an A and a B atom couple anti-ferromagnetically. In the cubic lattice any given atom may interact with six nearest neighbours via the exchange constants JAA JA > 0; JBB JB > 0; JAB o0; for the A–A, B–B, and A–B liaisons. The Hamiltonian of the system may then be written as XX XX XX XX Z Z Z Z H ¼  JA Sm S m 0  JB SnZ SnZ0  JAB SnZ SlZ dlA  JA Sm Sm0 dlA m

 JB

m0

XX n

n

SnZ SlZ dlB

l

þ JAB ðdlA dl 0 B d þ

 JAB

n0

XX

m dlB dl 0 A Þ SlZ SlZ0 :

n

Z Z Sm Sl dlB

m

l

l

XX  ½JA dl 0 A dl 0 A þ JB dlB dl 0 B

l

l

ol 0

ð1Þ

Snz ¼ 712 is the Ising spin operator for a considered site i; and d is the Kronecker delta. The first two terms in the Hamiltonian denote the magnetic exchange interactions inside the A2 and B2 block atomic layers. The four following terms denote the exchange interactions between the atoms of one of the interface layers, I(1) and I(2), with the atoms of the adjacent layers in the A2 and B2 block layers. The last three terms denote the exchange interactions inside the interface alloy layers I(1) and I(2). The summations are carried out over all nearest neighbours and are counted only once. The thermodynamic average for a spin operator SiZ per magnetic species and layer in a superlattice unit cell, may be written as //SiZ iZSSr ¼ /Tr½SiZ expðbHÞ =Tr½expðbHÞ Sr ;

ð2Þ

where b ¼ 1=kT ; and /?Sr denotes a configuration average. Note that /dlA Sr ¼ p1 in I and /dlA Sr ¼ p2 in I(2). Similar relations may be given for the other concentrations in the interface layers. The thermodynamic and configuration average for a magnetic species per layer is also well defined. For Z example, //SnZ SSr ¼ sA1 in one of the block A2 layers, and //Sm SSr ¼ sB2 in one of the block B2 layers. (1)

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Let the partial magnetisations per species and layer in a 1D unit cell of the system be denoted by ð1Þ ð2Þ ð2Þ sA1 ; sA2 ; sB1 ; sB2 ; mð1Þ A ; mB ; mA ; mB ; as indicated in Fig. 1. By symmetry sA1 ¼ sA2 ¼ sA and sB1 ¼ sB2 ¼ sB : To simplify the theoretical expressions we use the notation cA ¼ coshðJAA rÞ; sA ¼ sinhðJAA rÞ; cB ¼ coshðJBB rÞ; sB ¼ sinhðJBB rÞ; c* ¼ sinhðJBB rÞ; s* ¼ coshðJAB rÞ;

ð3Þ

where r is the differential operator, r  q=qx; that acts on f ðxÞ ¼ tanhðbxÞ which is the characteristic generating function of the spins system for the more convenient spins operator mn ¼ 2SnZ ¼ 71 used in the analysis. The system of coupled equations for the magnetisations per species and layer is derived from Eqs. (1) and (2), using the notation of Eq. (3), and introducing the decoupling approximations Z Z Z //Sm Sn ?SkZ SlZ SSr E//Sm SSr //SnZ SSr ?//SlZ SSr

ð4Þ

for manayakal; in the multi-spin correlation functions which appear when the exact Ising spin identities are expanded. This decoupling procedure corresponds to the Zernike approximation. The system of coupled equations may then be written as sA ¼ ðcA þ sA sA Þ5 ½p1 ðcA þ sA mð1Þ c þ s* mð1Þ A Þ þ ð1  p1 Þð* A Þ f ðxÞjx¼0 ; ð1Þ ð1Þ 4 mð1Þ A ¼ ðcA þ sA sA Þ½ p1 ðcA þ sA mA Þ þ ð1  p1 Þðc* þ s* mB Þ ; ð2Þ ½ p2 ðcA þ sA mð1Þ A Þ þ ð1  p2 Þðc* þ s* mB Þ f ðxÞjx¼0 ; ð1Þ 4 mð1Þ c þ s* sA Þ½ p1 ð*c þ s* mð1Þ B ¼ ð* A Þ þ ð1  p1 ÞðcB þ sB mB Þ ; ð1Þ ½ p2 ð*c þ s* mð2Þ A Þ þ ð1  p2 ÞðcB þ sB mB Þ f ðxÞjx¼0 ; ð1Þ ð2Þ mð2Þ c þ s* mð1Þ B Þ ½ p2 ðcA þ sA mA Þ A ¼ ½p1 ðcA þ sA mA Þ þ ð1  p1 Þð* 4 þ ð1  p2 Þð*c þ s* mð2Þ c þ s* sB Þf ðxÞjx¼0 ; B Þ ð* ð1Þ 4 mð2Þ c þ s* mð2Þ c þ s* mð2Þ B ¼ ½p1 ð* A Þ þ ð1  p1 ÞðcB þ sB mB Þ ½p2 ð* A Þ

þ ð1  p2 ÞðcB þ sB mð2Þ B Þ ðcB þ sB sB Þ f ðxÞjx¼0 ; ð2Þ 5 sB ¼ ½p2 ð*c þ s* mð2Þ A Þ þ ð1  p2 ÞðcB þ sB mB Þ ðcB þ sB sB Þ f ðxÞjx¼0 :

ð5Þ

The order–disorder Curie transition temperature TC is obtained when the bulk magnetisation and the constituent partial magnetisations of the system tend simultaneously to zero. In this limit Eqs. (5) may be ð1Þ ð2Þ ð2Þ developed to first order in sA ; mð1Þ A mB mA mB sB ; to determine the following linear system of coupled equations ð1Þ sA ¼ 5½p1 k1A þ ð1  p1 Þa12 sA þ p1 k1A mð1Þ A þ ð1  p1 Þa11 mB ; ð1Þ ð1Þ ð2Þ ð2Þ mð1Þ A ¼ A1 sA þ 4A2 mA þ 4A3 mB þ A4 mA þ A5 mB ; ð1Þ ð1Þ ð2Þ ð2Þ mð1Þ B ¼ B6 sA þ 4B7 mA þ 4B8 mB þ B9 mA þ B10 mB ; ð1Þ ð1Þ ð2Þ ð2Þ mð2Þ A ¼ A10 mA þ A9 mB þ 4A8 mA þ 4A7 mB þ A6 sB ; ð1Þ ð1Þ ð1Þ ð2Þ mð2Þ B ¼ B5 mA þ B4 mB þ 4B3 mA þ 4B2 mB þ 4B1 sB ; ð2Þ sB ¼ p2 b11 mð2Þ A þ ð1  p2 Þk1B mB þ 5½p2 b11 þ ð1  p2 Þk1B sB :

ð6Þ

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309

The coefficients fAi g; fBi g; and co-factors faij g; fbij g; k1A ; and k1B ; in Eqs. (6) are given in detail in the Appendix A of the present paper. This linear system of equations may finally be expressed as ð1Þ ð2Þ ð2Þ t LV t  L½sA ; mð1Þ A ; mB ; mA ; mB ; sB ¼ 0;

ð7Þ

where 2 6 6 6 6 6 L¼6 6 6 6 4

5Pa  1

p1 k1A

ð1  p1 Þa11

0

0

0

A1 B6

4A2  1 4B7

4A3 4B8  1

A4 B9

A5 B10

0 0

0 0

A10 B5

A9 B4

4A8  1 4B3

4A7 4B2  1

A6 B1

0

0

0

p2 b11

ð1  p2 Þk1B

5Pb  1

3 7 7 7 7 7 7: 7 7 7 5

ð8Þ

Pa ¼ p1 k1A þ ð1  p1 Þa12 and Pb ¼ p2 b12 þ ð1Fp2 Þk1B : To obtain nontrivial solutions for the unknown vector V t variables the determinant of L should vanish. The condition that det L ¼ 0 yields the Curie temperature TC identified as the highest solution of the corresponding secular equation.

3. Exchange constants for the Fec Gd1c alloy In order to apply the present model to calculate the phase diagrams and Curie temperatures for a magnetic iron–gadolinium superlattice, it is necessary to know the exchange constants JFeFe JFe > 0; JGdGd JGd > 0; JFeGd o0; for these interactions. If JFeFe and JGdGd are relatively well known in the pure state, they are less so for the alloy. Furthermore, the JFeGd exchange constant is difficult to obtain, and is not known for the whole range of alloy concentrations. Previous work on this subject, which treats the FecGd1c and FecTb1c alloys, is mainly based on the mean field approximation and usually restricts the comparison between theory and experiment to limited intervals of the alloy concentration. This has given rise in the literature to significant disagreement as to the values of the exchange constants that should be attributed, even as to the exact values of JFeFe and JGdGd [25–30]. These exchange constants vary greatly in the literature as is resumed in Table 1. In a previous work [31], we have calculated for the whole range of alloy concentrations for the alloy FecGd1c a realistic JFeGd exchange constant using the effective field approach and attributing the correct spin values for iron and gadolinium, namely SFe ¼ 1 and SGd ¼ 72: In the present work, however, it is necessary to evaluate for this alloy the corresponding parametric value of JFeGd in the spin-12 representation as we apply a corresponding model for the superlattices. The magnetic properties of bulk ferromagnetic alloys have been intensively studied in the past, using different theoretical methods [32,33]. The composition of these systems corresponds to a simple stoichiometric proportion indicated by AcB1c in 3D, for which the atomic arrangement remains crystalline and orderly in general under a range of conditions. The bulk alloys such as FecTb1c and

Table 1 0.017 Ref. [28] oJTbTb =JFeFe o0:8 Ref. [27] 0.65 Refs. [25,28] oJFeTb =JFeFe o 0.26 Ref. [27] 0.03 Ref. [29] oJGdGd =JFeFe o0:15 Ref. [30] 1.0 Ref. [30] oJFeGd =JFeFe o  0:5 Ref. [29]

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310

FecGd1c are ferrimagnetic with opposite spin alignment on the iron and the rare earth atoms. In this section, we consequently study the case of a ferrimagnetic random 3D alloy. Consider an infinite simple cubic lattice on which sites A and B atoms are distributed randomly. The atoms in each species couple ferromagnetically amongst themselves, but the coupling is anti-ferromagnetic between the A and B atoms. The Hamiltonian for this magnetic alloy is given in the Ising representation by XX H¼ Jij SiZ SjZ ; ð9Þ i

ði;jÞ

where Jij is the exchange interaction coupling the nearest neighbour spins. The average magnetisation on a site is given in the effective field approach by si ¼ //SiZ SSr ¼

j¼6 Y ½coshðJij rÞ þ sinhðJij rÞ f ðxÞjx¼0 ;

ð10Þ

j¼1

where f ðxÞ ¼ tanhðbxÞ is the characteristic generating function of the spins system, employing again the spins operator mn ¼ 71: The phase diagrams of the Curie temperatures are calculated as a function of the bulk alloy concentration c; by putting the developed equations from Eq. (10) in linear form, and looking for appropriate solutions as indicated in the procedure leading to Eqs. (7) and (8). It should be emphasised that very little experimental information is available concerning the phase diagrams for the rare earth alloys FecTb1c and FecGd1c which exist in the amorphous and the crystalline phases. One of the first known papers concerning their properties is due to Heiman et al. [25], where one finds some data for the Curie temperature TC and for the compensation temperature Tcomp ; as a function of the concentration c: However, the few experimental points therein cannot be considered as leading to general trends in the whole range 0pcp1: For this reason it is preferable to work with the paper of Hansen et al. [26], which contains more available experimental data to compare with. The best agreement that we obtain for the phase diagram of the Curie temperature as a function of the concentration c of the FecGd1c alloy, between our calculations and the experimental data, is presented in Fig. 2. Even though the spin-12 is a simple model and the experimental data are limited in the whole range of 6

5

kTc JFe Fe

4

3

Theory Experiment

2

r = 0.3 s= 1.3 1

0 0.0

0.2

0.4

0.6

0.8

1.0

c

Fig. 2. The continuous curve is the calculated spin-12 model result for the phase diagram of the normalised Curie temperature kTC =JFeFe for the bulk FecGd1c alloy as a function of the concentration c; for r ¼ JGdGd =JFeFe ¼ 0:3 and s ¼ JFeGd =JFeFe ¼ 1:3: The black squares represent the available experimental data.

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311

concentrations, there is an overall qualitative agreement between the theory and the experimental data. This agreement is obtained in the present calculation by adjusting to obtain the dimensionless parameters r ¼ JGdGd =JFeFe ¼ 0:3 and s ¼ JFeGd =JFeFe ¼ 1:3: The adjustment of the end points for the limits c ¼ 0 and 1 are dictated by the values of the pure JGdGd and JFeFe exchange constants. These values for r and s here are comparable to those in Table 1. The absolute value for JFeFe in these systems, however, is not precisely defined and varies greatly in the literature. In Ref. [31] which treats both the phase diagrams for the Curie and the compensation temperatures, we obtain two slightly different values for which a mean is JFeFe E1:48 10221 J. Whilst the present spin-12 model still serves to study the limiting order–disorder transitions, note, however, that it is inadequate to account for the phase diagram of the compensation temperatures of an alloy since this depends more critically on the realistic spin values as in the case for iron and gadolinium where SFe ¼ 1 and SGd ¼ 72:

4. Numerical calculations for the Fe2 ðFep1 Gd1p1 ÞðFep2 Gd1p2 ÞGd2 superlattice Using the values r ¼ 0:3 and s ¼ 1:3 evaluated in Section 3, we can now calculate the phase diagrams for the Fe2(Fep1Gd1p1)(Fep2Gd1p2)Gd2 superlattice in the present model, where Eq. (8) yields the Curie temperature TC ; for any given p1 and p2 in the alloy disordered interface layer. The numerical results are presented as a function of these concentrations. It is reasonable to suppose in this respect that these satisfy the following conditions: 0:5op1 o1 and 0op2 o0:5 in the Ið1Þ layer;

ð11Þ

0:5op2 o1 and 0op1 o0:5 in the Ið2Þ layer:

In Fig. 3 are the phase diagrams of the normalised Curie temperatures kTC =JFeFe calculated for the Fe2(Fep1Gd1p1)(Fep2Gd1p2)Gd2 superlattice as a function of p1 : These are presented as the continuous curves for various values of s ¼ JFeGd =JFeFe running through s ¼ 0:3 to –1.3 in steps of Ds ¼ 0:2; from

5

p1 + p2 = 1 r = 0.3

kTc J FeFe

4.8

4.6

4.4

4.2 0.2

0.4

0.6

0.8

1

p, p1 Fig. 3. The continuous curves are the phase diagrams of the normalised Curie temperature for the Fe2(Fep1Gd1p1)(Fep2Gd1p2)Gd2 superlattice, calculated as a function of p1 ; for s ¼ 0:3 to –1.3 in steps of –0.2 from bottom to top. The dashed curves are the superlattice Fe2(FepGd1p)Gd2 phase diagrams calculated in comparison.

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312

bottom to top. Note that they are evaluated in the interval 0:5op1 o1 and 0:5op2 o1; in conformity with the condition of Eq. (11). The purpose of Fig. 3 is to show the variations of the superlattice phase diagram with s; if ever the 2D alloy interfaces exchange constant JFeGd is not commensurable with that calculated for the bulk 3D alloy. Fig. 3 shows that the concavity in the superlattice phase diagram becomes comparable to that for the bulk alloy only when sB  1:3: For the purpose of comparison, the dashed curves in Fig. 3, are the calculated phase diagrams for the Fe2(FepGd1p)Gd2 superlattice with only a single alloy disordered layer interface, presented in the interval 0ppp1: The concentrations p1 and p2 may vary in the interface owing to different mechanisms during growth. Nevertheless, their sum g ¼ p1 þ p2 is a characteristic superlattice parameter since it satisfies from Eq. (12) the following condition: ð12Þ

0:5ogo1:5:

It is consequently instructive to study the variation of the phase diagrams of the Curie temperatures with this parameter. The calculated results for kTC =JFeFe of the Fe2(Fep1Gd1p1)(Fep2Gd1p2)Gd2 superlattice are given in Fig. 4 as a function of p1 : The values of r and s are fixed and correspond to those calculated in Section 3. These phase diagrams are presented as the continuous curves calculated for various values of g from 0.6 to 1.4, in steps of 0.2 from bottom to top. For the purpose of comparison, the dashed curve in Fig. 4 is the calculated phase diagram for the Fe2(FepGd1p)Gd2 superlattice with a single alloy disordered layer interface, as a function of p for the same fixed r and s values. In a previous classic work, Camely and Tilley [30], investigated the magnetic-field phase transitions in magnetic superlattices which couple anti-ferromagnetically at the interfaces, using mean field theory, and applied their model to Fe/Gd superlattices. The experimental magnetic phase diagrams for some characteristic Fe/Gd superlattices [29], are given later and compared to an improved mean field model. It is, hence, interesting in principle to compare our present results with this previous work, especially since both lead to compensation effects. While ostensibly comparable and applicable to similar Fe/Gd superlattice systems, the two works differ, however, in various aspects. The Camely and Tilley paper considers idealised superlattice geometries with no disorder at the interface, but more important their theoretical model is investigated to calculate phase transitions as a function of an external field H0: Our present work considers

5

kTc J FeFe

4.8

4.6

p1 + p2 = g r = 0. 3 s = 1.3

4.4

4.2 0

0.2

0.4

0.6

0.8

1

1.2

1.4

p, p1 Fig. 4. As in Fig. 3 for the fixed values of r ¼ 0:3 and s ¼ 1:3: The continuous curves are calculated for p1 þ p2 ¼ 0:6 to 1.4 in step of 0.2 from bottom to top. The dashed curve is the superlattice Fe2(FepGd1p)Gd2 phase diagram calculated for comparison.

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the Fe/Gd superlattices in zero H ¼ 0 field. To compare, hence, the two models have to be analysed at H ¼ 0 in the aligned Fe geometry. Whereas the previous compensation effects are deducible from structural magnetic phase transitions (Fe aligned, Fe twisted, Gd paramagnetic) under the influence of the external field, our compensation effects are specifically due to the disorder at the interface and layer architecture. Further, whereas we have calculated correctly the anti-ferromagnetic JFeGd coupling as a function of the disorder at the interface, the previous model JFeGd coupling is considered fixed, and taken at strong (Fe) and weak (Gd) limits. A key step in our theory is the assumption that all sub-components of the total magnetisation vanish near the same temperature, thereby defining a Curie temperature TC : This is justifiable in view of the dominant Fe alignment effects in zero external fields. It is interesting to note that the previous work [30], considers TC to be fixed and taken as the transition temperature for bulk Fe, whereas our TC is itself variable and is calculated as a function of the interface disorder and the Fe/Gd layers architecture. Despite the basic differences between the two models, they share a common feature of a paramagnetic phase for the Gd core layers at high temperatures. It is obviously interesting in the future to reconsider the effective field model with external field effects for a comparison, looking for compensation effects that are a function of the interface disorder and the magnetic phase geometry. In this work, we have developed an effective field approach, known to be superior to the mean field theory, to determine the Curie temperature phase diagram for Ising A2(Ap1B1p1)(Ap2B1p2)B2 superlattices, based on an approximate spin-12 model. This model is applied for Fe/Gd superlattices. The needed superlattice exchange constants in this case for the 2D alloy interfaces are determined by calculating the TC phase diagram for a bulk Fe/Gd alloy and comparing with known experimental data. The range of variation of the disordered interface concentrations is taken to be limited to 0:5op1 þ p2 o1:5 from technical requirements. The primary feature of this model that one can use for the experimental analysis of these superlattices, is the Curie temperature phase diagram dependence on the interface alloy concentrations. Owing to its simplicity, this model gives useful information on the influence of the double disordered interface of these systems. The model can be extended in a direct manner for A an B species of spin greater than 12 and for different superlattices Ak(Ap1B1p1)(Ap2B1p2)Bh of arbitrary unit cells. Appendix A The coefficients A1 ; A2 ; y; A10 are given as follows: A1 ¼ p2 ½ p41 k1A þ 4p31 ð1  p1 Þa12 þ 6p21 ð1  p1 Þ2 a22 þ 4p1 ð1  p1 Þ3 a33 þ ð1  p1 Þ4 a43 þ ð1  p2 Þ½ p41 a12 þ 4p31 ð1  p1 Þa22 þ 6p21 ð1  p1 Þ2 a33 þ 4p1 ð1  p1 Þ3 a43 þ ð1  p1 Þ4 a54 ; A2 ¼ p1 p2 ½p31 k1A þ 3p21 ð1  p1 Þa12 þ 3p1 ð1  p1 Þ2 a22 þ ð1  p1 Þ3 a33 þ ð1  p2 Þ½ p41 a12 þ 3p31 ð1  p1 Þa22 þ 3p21 ð1  p1 Þ2 a33 þ p1 ð1  p1 Þ3 a43 ; A3 ¼ p1 ð1  p2 Þ½ p31 a11 þ 3p21 ð1  p1 Þa21 þ 3p11 ð1  p1 Þ2 a31 þ ð1  p1 Þ3 a41 þ ð1  p1 Þð1  p2 Þ½ p31 a21 þ 3p21 ð1  p1 Þa31 þ 3p1 ð1  p1 Þ2 a41 þ ð1  p1 Þ3 a54 ; A4 ¼ p2 ½p41 k1A þ 4p31 ð1  p1 Þa12 þ 6p21 ð1  p1 Þ2 a22 þ 4p1 ð1  p1 Þ3 a33 þ ð1  p1 Þ4 a43 A5 ¼ ð1  p2 Þ½ p41 a11 þ 4p31 ð1  p1 Þa21 þ 6p21 ð1  p1 Þ2 a31 þ 4p1 ð1  p1 Þ3 a41 þ ð1  p1 Þ4 a51 A6 ¼ p1 ½ p42 a11 þ 4p32 ð1  p2 Þa21 þ 6p22 ð1  p2 Þ2 a31 þ 4p2 ð1  p2 Þ3 a41 þ ð1  p2 Þ4 a51 þ ð1  p1 Þ½ p42 a21 þ 4p32 ð1  p2 Þa31 þ 6p22 ð1  p2 Þ2 a41 þ 4p2 ð1  p2 Þ3 a51 þ ð1  p2 Þ4 a61 ;

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A7 ¼ p1 ð1  p2 Þ½ p32 a21 þ 3p22 ð1  p2 Þa31 þ 3p12 ð1  p2 Þ2 a41 þ ð1  p2 Þ3 a51 þ ð1  p1 Þð1  p2 Þ½ p32 a31 þ 3p22 ð1  p2 Þa41 þ 3p2 ð1  p2 Þ2 a51 þ ð1  p2 Þ3 a61 ; A8 ¼ p1 p2 ½p32 a12 þ 3p22 ð1  p2 Þa22 þ 3p2 ð1  p2 Þ2 a33 þ ð1  p2 Þ3 a43 þ p1 ð1  p2 Þ½ p32 a22 þ 3p22 ð1  p2 Þa33 þ 3p2 ð1  p2 Þ2 a43 þ ð1  p2 Þ3 a54 ; A9 ¼ ð1  p1 Þ½ p42 a21 þ 4p32 ð1  p2 Þa31 þ 6p22 ð1  p2 Þ2 a41 þ 4p2 ð1  p2 Þ3 a51 þ ð1  p2 Þ4 a61 ; A10 ¼ p1 ½ p42 a21 þ 4p32 ð1  p2 Þa22 þ 6p22 ð1  p2 Þ2 a33 þ 4p2 ð1  p2 Þ3 a43 þ ð1  p2 Þ4 a54 : The Bi coefficients are obtained directly from the Ai coefficients by replacing the k1A by k1B terms, and the aij by the bij terms, and by simultaneously permuting the interface concentrations: p1 by ð1  p2 Þ; ð1  p1 Þ by p2 ; p2 by (1  p1 ), and (1  p2 ) by p1 ; wherever they occur in the Ai coefficients. The needed co-factors are given following the notation of Eq. (3): k1A ¼ c5A sA f ðxÞjx¼0 ;

k2A ¼ c3A s3A f ðxÞjx¼0 ;

k3A ¼ cA s5A f ðxÞjx¼0

and a11 ¼ c5A s* f ðxÞjx¼0 ;

a12 ¼ c4A sA c* f ðxÞjx¼0 ;

a14 ¼ c2A s3A c* f ðxÞjx¼0 ; a21 ¼ c4A c*s* f ðxÞjx¼0 ;

a15 ¼ cA s4A s* f ðxÞjx¼0 ;

a31 ¼ c3A c*2 s* f ðxÞjx¼0 ; a37 ¼ s3A c*3 f ðxÞjx¼0 ; a42 ¼ c2A c*s*3 f ðxÞjx¼0 ;

a44 ¼ cA c*2 sA s*2 f ðxÞjx¼0 ; a47 ¼ c*s2A s*3 f ðxÞjx¼0 ;

a23 ¼ c3A sA s*2 f ðxÞjx¼0 ;

a25 ¼ cA c*2 s3A f ðxÞjx¼0 ;

a33 ¼ c2A c*3 sA f ðxÞjx¼0 ; a41 ¼ c2A c*3 s* f ðxÞjx¼0 ;

a16 ¼ s5A c* f ðxÞjx¼0 ;

a22 ¼ c3A c*2 sA f ðxÞjx¼0 ;

a24 ¼ c2A c*s2A s* f ðxÞjx¼0 ; a27 ¼ c*s4A s* f ðxÞjx¼0 ;

a13 ¼ c3A s2A s* f ðxÞjx¼0 ;

a26 ¼ cA s3A s*2 f ðxÞjx¼0 ;

a32 ¼ c3A s*3 f ðxÞjx¼0 ; a38 ¼ c*s3A s*2 f ðxÞjx¼0 ; a43 ¼ cA c*4 sA f ðxÞjx¼0 ;

a45 ¼ cA sA s*4 f ðxÞjx¼0 ;

a51 ¼ cA c*4 s* f ðxÞjx¼0 ;

a53 ¼ cA s*5 f ðxÞjx¼0 ;

a54 ¼ c*5 sA f ðxÞjx¼0 ;

a56 ¼ c*sA s*4 f ðxÞjx¼0 ;

a61 ¼ c*5 s* f ðxÞjx¼0 :

a46 ¼ c*3 s2A s* f ðxÞjx¼0 ;

a52 ¼ cA c*2 s*3 f ðxÞjx¼0 ; a55 ¼ c*3 sA s*2 f ðxÞjx¼0 ;

The kjB and bij co-factors are derived by permuting the indices in the above expressions, noting that JAB ¼ JBA : Their detailed values are derived by carrying out the indicated operations using f ðxÞ¼ tanhðbxÞ:

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