Phase diagrams of a spin-1 Ising superlattice with alternating transverse field

Journal of Magnetism and Magnetic Materials 261 (2003) 78–87

Phase diagrams of a spin-1 Ising superlattice with alternating transverse field A. Sabera,b,c,*, H. Ez-Zahraouya,d, S. Lo Russob, G. Matteib, A. Ainanea,c a

Abdus Salam International Centre For Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy b INFM, Dipartimento di Fisica, Universita" di Padova, Via Marzolo 8, 35100 Padova, Italy c D!epartement de Physique, Facult!e des Sciences, Universit!e Moulay Ismail, B.P. 4010 Mekn"es, Morocco d D!epartement de physique, Laboratoire de Magn!etisme et de Physique des Hautes Energies, Facult!e des Sciences, B.P. 1014, Rabat, Morocco Received 26 September 2002; received in revised form 25 October 2002

Abstract The effects of alternating transverse fields Oa and Ob on the critical behavior of an alternating spin-1 Ising superlattice are studied within an effective field theory with a probability distribution technique that accounts for the single-site spin correlation. Critical temperatures are calculated as a function of the thickness of the superlattice and the strength of the transverse field. Depending on the values of the transverse fields Oa and Ob ; the critical temperature can increase or decrease with increasing the thickness of the film, such result is not obtained in the uniform transverse field case (Oa = Ob ). Furthermore, for each thickness L of the film, a long range ordered phase persist at low temperature for selected values of the transverse field Oa and arbitrary values of Ob : The effects of interlayer and intralayer exchange interactions are also examined. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.70.Cn Keywords: Phase diagrams; Superlattices

1. Introduction During the last few years much effort has been directed towards the study of the critical phenomena in various magnetic layered structures and superlattices [1–7]. The basic theoretical problem is the examination of the magnetic excitation and the phase transitions in these systems. Magnetic excitations in superlattices were considered in numerous papers (see e.g. [8] for a brief review). Yet, less attention has been paid to the critical behavior and in particular to the critical temperatures in superlattices. Ma and Tasi *Corresponding author. INFM, Dipartimento di Fisica, Universit"a di Padova, Via Marzola 8, 35100 Padova, Italy. Tel.: +390498277039; fax: +390498277003. E-mail address: [email protected] (A. Saber). 0304-8853/03/$ - 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 1 4 1 7 - 8

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[9] have studied the variation with modulation wavelength of the Curie temperature for a Heisenberg magnetic superlattice. Their results agree qualitatively with experiments on Cu/Ni films [10]. Superlattice structures composed of alternating ferromagnetic and antiferromagnetic layers have been investigated by Hinckey and Mills [11,12], using a localized spin model. Magnetic properties of an alternating magnetic and nonmagnetic superlattice is studied by Benyoussef and Ez-Zahraouy [13] using finite cluster approximation. A sequence of spin transitions is found to be different for superlattices with antiferromagetic component consisting of an even or odd number of spin layers. In two earlier papers [5,6] two of the present authors have studied the effects of a uniform transverse field and surface magnetism on the critical behavior of an alternating ferromagnetic spin-12 Ising superlattice. Our aim in this paper is to study the effects of an alternating transverse field on the critical temperature of a spin-1 Ising superlattice consisting of two ferromagnetic materials with different bulk properties, with a simple cubic structure. In particular, we consider the two constituents A and B with different bulk transition temperatures, i.e. TcA aTcB : The interface is in general different in nature from both bulks, even if the bulk critical temperatures are the same. We use the effective field theory with a probability distribution technique in its simplest form [14,15]. This technique is believed to give more exact results than those of the standard mean-field approximation. In Section 2 we outline the formalism and derive the equation that determines the transition temperature. Results and discussion are given in Section 3. Section 4 is reserved for conclusion.

2. Model and formulation We consider an infinite simple cubic superlattice with a unit cell consisting of arbitrary number L of magnetic layers. The transverse spin-1 Ising Hamiltonian of the system is given by XX XX H¼ Jnn0 sznr szn0 r0  On sxnr ; ð1Þ n;n0 r;r0

n

r

where sznr and sxnr denote, respectively, the z and x components of a quantum spin ~ s nr of magnitude snr ¼ 1 at site ðn; rÞ; ðn; n0 Þ are plane indices and ðr; r0 Þ are different sites of the planes, and Jnn0 is the strength of the ferromagnetic exchange interaction which is only plane dependent. We will retain only nearest-neighbor terms. For a fixed configuration of neighboring spins of the site ðn; rÞ that the longitudinal and the transverse magnetizations and quadrupolar moments of any spin at site ðn; rÞ are given by mnra ¼ /sanr S ¼ /f1a ðA; BÞS;

ð2Þ

 2 qnra ¼ / sanr S ¼ /f2a ðA; BÞS;

ð3Þ

where a ¼ z; x for the longitudinal and transverse magnetizations and quadrupolar moments, respectively, and     2 2 1=2 2sinh b A þ B A  ; f1z ðA; BÞ ¼ ð4Þ 2 ½A þ B2 1=2 1 þ 2cosh b½A2 þ B2 1=2     1=2  B2 þ 2A2 þ B2 cosh b A2 þ B2 1   f2z ðA; BÞ ¼ 2 ; ½A þ B 2  1 þ 2cosh b½A2 þ B2 1=2

ð5Þ

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and f1x ðA; BÞ ¼ f1z ðB; AÞ;

ð6Þ

f2x ðA; BÞ ¼ f2z ðB; AÞ

ð7Þ

with A¼

XX n0

Jnn0 szn0 r0 ;

ð8Þ

r0

ð9Þ

B ¼ On ;

where b ¼ 1=kB T; /yS indicates the usual canonical ensemble thermal average for a given configuration and the first and second sums run over all possible configurations of atoms environing or lying on the ðn; rÞ site, respectively. Each of these configurations can be characterized by numbers of magnetic atoms in the planes n  1; n; n þ 1: To perform thermal averaging on the right-hand side of Eqs. (2) and (3) one can now follow the general approach described in Refs. [14–16]. Thus, with the use of the integral representation method of Dirac d-distribution, Eqs. (2) and (3) can be written in the form Z Z Y  a   1 snr ¼ do f1a ðo; BÞ dt exp ðiotÞ ð10Þ exp itJn;n0 szn0 r0 ; 2p n0 r0 Z D  E Z Y   1 a 2 dtexp ðiotÞ snr ð11Þ exp itJn;n0 szn0 r0 : ¼ do f2a ðo; BÞ 2p n0 r0 In the derivation of Eqs. (10) and (11), the commonly used approximation has been made according  to whichDthe multi-spin correlation functions are decoupled into products of the spin averages. Then, as sanr E 2 and sanr are independent of r; we introduce the longitudinal magnetization and the longitudinal quadrupolar moment of the nth layer, on the basis of Eqs. (2) and (3), with the use of the probability distribution of the spin variables [13,14]:         ð12Þ P sznr ¼ 12½ðqnz  mnz Þd sznr þ 1 þ 2ð1  qnz Þd sznr þ ðqnz þ mnz Þd sznr  1 : Allowing for the site magnetizations and quadrupolar moments to take different values in each atomic layer parallel to the surfaces of the superlattice, and labeling them in accordance with the layer number in which they are situated, the application of Eqs. (2), (10) and (12) yields the following set of equations for the layer longitudinal magnetizations: N0 NX N0 NX N Nm 0 m1 X 0 m2 X XX mna ¼ 2N2N0 2mþm1 þm2 CmN CnNm CmN10 CnN1 0 m1 CmN20 CnN2 0 m2 m¼0 n¼0 m1 ¼0 n1 ¼0 m2 ¼0 n2 ¼0

 m  n ð1  2qnz Þm ðqnz  mnz Þn ðqnz þ mnz ÞNmn 1  2qn1;z 1 qn1;z  mn1;z 1  N m n  m  n qn1;z þ mn1;z 0 1 1 1  2qnþ1;z 2 qnþ1;z  mnþ1;z 2  N m n qnþ1;z þ mnþ1;z 0 2 2 f1a ðyn ; On Þ; where     yn ¼ ½Jn;n ðN  m  2nÞ þ Jn;n1 N0  m1  2n1 þ Jn;nþ1 N0  m2  2n2 :

ð13Þ ð14Þ

N and N0 are the numbers of nearest neighbors in the plane and between adjacent planes, respectively (N ¼ 4 and N0 ¼ 1 in the case of a simple cubic lattice which is considered here) and Ckl are the binomial coefficients, Ckl ¼ l!=k!ðl  kÞ!: The periodic conditions of the superlattice has to be satisfied, namely m0a ¼ mLa ; mLþ1;a ¼ m1a and q0a ¼ qLa ; and qLþ1;a ¼ q1a : The equations of the longitudinal and transverse

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quadrupolar moments are obtained by substituting the function f1a by f2a in the expressions of the layer longitudinal and transverse magnetizations, respectively. This yields qna ¼ mna ½f1a ðyn ; On Þ-f2a ðyn ; On Þ:

ð15Þ

In this work we are interested in the calculation of the ordering near the transition critical temperature. The usual argument that mnz tends to zero as the temperature approaches its critical value allows us to consider only terms linear in mnz because higher order terms tend to zero faster than mnz on approaching a critical temperature. Consequently, all terms of the order higher than linear terms in Eqs. (13) that give the expressions of mnz can be neglected. This leads to the set of simultaneous equations mnz ¼ An;n1 mn1;z þ An;n mnz þ An;nþ1 mnþ1;z

ð16Þ

~z ¼ m ~ z; Am

ð17Þ

or

~ z is a vector of components ðm1z ; m2z ; y; mnz ; y; mLz Þ and the matrix A is symmetric and where m tridiagonal with elements   Ai;j ¼ Ai;i di;j þ Ai;j di;j1 þ di;jþ1 : ð18Þ The system of Eqs. (17) is of the form ~ nz ¼ 0; Mm

ð19Þ

where     Mi;j ¼ Ai;j  1 di;j þ Ai;j di;j1 þ di;jþ1 :

ð20Þ

The only non zero elements of the matrix M are given by Mn;n1 ¼ 2N2N0

ðm1 þn1 Þ N0 NX N0 NX n1 N0X N Nm 0 m1 X 0 m2 X X XX m¼0 n¼0 m1 ¼0 n1 ¼0 m2 ¼0 n2 ¼0 i¼0



N0 ðm1 þn1 Þ CnN1 0 m1 CmN20 CnN2 0 m2 Cin1 Cj ð1

ð1Þi 2mþm1 þm2 d1;iþj CmN CnNm CmN10

j¼0 m

 rn Þ ð1  rn1 Þm1

ðN0 m Þðiþj Þ N0 m2 ð1  rnþ1 Þm2 rNm rn1 1 rnþ1 f1z ðyn ; On Þ; n Mn;n ¼ 2N2N0

ðmþnÞ N0 NX N0 NX N Nm n NX 0 m1 X 0 m2 X X XX m¼0 n¼0 m1 ¼0 n1 ¼0 m2 ¼0 n2 ¼0 i¼0

CnN1 0 m1 CmN20 CnN2 0 m2 Cin CjNðmþnÞ ð1

ð21Þ

ð1Þi 2mþm1 þm2 d1;iþj CmN CnNm CmN10

j¼0

ðiþj Þ  rn Þ ð1  rn1 Þm1 ð1  rnþ1 Þm2 rNm n m

ðN0 m Þ N0 m2 rn1 1 rnþ1 f1z ðyn ; On Þ  1;

Mn;nþ1 ¼ 2

N2N0

ð22Þ

ðm2 þn2Þ N0 NX N0 NX n2 N0X N Nm 0 m1 X 0 m2 X X XX m¼0 n¼0 m1 ¼0 n1 ¼0 m2 ¼0 n2 ¼0 i¼0 N0 ðm2 þn2Þ

CnN1 0 m1 CmN20 CnN2 0 m2 Cin1 Cj

ð1Þi 2mþm1 þm2 d1;iþj CmN CnNm CmN10

j¼0

ð1  rn Þm ð1  rn1 Þm1

ðN0 m Þ N0 m2 ðiþjÞ ð1  rnþ1 Þm2 rNm rn1 1 rnþ1 f1z ðyn ; On Þ n

ð23Þ

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where rn ¼ 2N2N0

N0 NX N0 NX N Nm 0 m1 X 0 m2 X XX

2mþm1 þm2 CmN CnNm CmN10 CnN10 m1 CmN20 CnN2 0 m2

m¼0 n¼0 m1 ¼0 n1 ¼0 m2 ¼0 n2 ¼0

ðN0 m Þ N0 m2 ð1  2rn Þm rNm ð1  rn1 Þm1 rn1 1 ð1  2rnþ1 Þm2 rnþ1 f2z ðyn ; On Þ n

ð24Þ

All the information about the critical temperature of the system is contained in Eq. (19). Up to know we did not precise the values of the exchange interactions; the terms in matrix (19) are general ones. In a general case, for arbitrary coupling constants and superlattice thickness, the evaluation of the critical temperature relies on the numerical solution of the system of linear Eq. (19). These equations are fulfilled if and only if det M ¼ 0:

ð25Þ

This condition can be satisfied for L different values of the critical temperature Tc : Here after, we will take Jaa as the unit of the energy, the length is measured in units of the lattice constant and we introduce the reduced exchange couplings R1 ¼ Jbb =Jaa and R2 ¼ Jab =Jaa : Let us begin with the evaluation of the critical temperature with an example: the critical temperature of the spin-1 Ising model for the simplest possible ‘‘bulk case’’ of a material A (i.e. N ¼ 4; N0 ¼ 1; Ji;j ¼ Jaa ; On ¼ O). Then we can reduce det M to the following form:    a b b     a b   b     b a b      y y y y y y: y y y y y y     det M ¼  b a b ; ð26Þ    y y y y y y y y y y y y     b a b       b a b    b b a  ðL;LÞ

whose value is det Mbulk ¼

L  Y k¼1

a þ 2b cos

  2pðk  1Þ ; L

where the elements in the above determinant are given by   a ¼ Mn;n Jn;n ¼ Jn;n1 ¼ Jn;nþ1 ¼ Jaa ; O ; b ¼ 14ða þ 1Þ

ð27Þ

ð28Þ ð29Þ

and L in the ‘‘bulk’’ case is an arbitrary number. Now we obtain the critical temperature from the condition given by det Mbulk ¼ 0:

ð30Þ

We apply the obtained formalism to an alternating magnetic superlattice consisting of atoms of type A and B which alternate as ...ABABAB...AB... in alternating transverse magnetic fields Oa ; Ob respectively. The

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4.0

R1 = 1

Ω /Jaa =2

3.5

Ω /Jaa =1

Tc /Jaa

Ω /Jaa =0

3.0

2.5

2.0 0.5

1.0

(a)

1.5

2.0

R2 R1 = 1

4.5 2 2

Tc / Jaa

4.0

3.5 1 1 3.0 0.5 0.5

2.5

2.0 8

(b)

12

16

20

24

28

32

36

40

L

Fig. 1. Critical temperature Tc =Jaa as a function of R2 (a) and the thickness of the superlattice (b). In (b) the number accompanying each curve denotes the value of R2 ; while the solid and dashed lines correspond to O=Jaa ¼ 0; and 1, respectively.

periodic conditions suggests that we only have to consider one unit cell which interacts with its nearest neighbors via the interlayer coupling. The coupling strength between nearest-neighboring spins in A and B is denoted by Jaa and Jbb ; respectively, while Jab stands for the exchange coupling between the nearestneighbor spins for all successive layers. Let us consider a simple alternating lattice of L layer (L is pair

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3.5

L = 10

3.0 Ωa /Jaa = 0

2.5

Ωa /Jaa = 2

Tc /Jaa

Ωa /Jaa = 1 Ωa /Jaa = 0.5

2.0

1.5

1.0 Ω a /J aa =Ω b /Jaa

0.5

0.0 0

1

2

3

4

5

6

7

Ω b /Jaa

Fig. 2. Dependence of the critical temperature Tc =Jaa on the transverse field Ob =Jaa for L ¼ 10:

number), n ¼ 1; 3; 5; y; L  1 consist of atoms of type A, whereas layers n ¼ 2; 4; y; L consist of atoms of type B. In this case we can represent detMab as    a1 b1 b1       b2 a2 b2     b a b 1 1 1     det Mab ¼  y y y y y y y y y y y y y  ; ð31Þ     b2 a2 b2     b1 a1 b1      b2 b2 a2 ðL;LÞ whose value is det Mab ¼ ða1 a2 ÞL

L  Y k¼1

1

   2b1 b2 2pðk  1Þ 1 þ cos ; L a1 a2

ð32Þ

where the elements in the determinant are given by   a1 ¼ Mn;n Jn;n ¼ Jaa ; Jn;n1 ¼ Jn;nþ1 ¼ Jab ; Oa     b1 ¼ Mn;n1 Jn;n ¼ Jaa ; Jn;n1 ¼ Jn;nþ1 ¼ Jab ; Oa ¼ Mn;nþ1 Jn;n ¼ Jaa ; Jn;n1 ¼ Jn;nþ1 ¼ Jab ; Oa n ¼ 1; 3; y; L  1;

ð33Þ

A. Saber et al. / Journal of Magnetism and Magnetic Materials 261 (2003) 78–87 L = 32

85

Ω a /Jaa =0.5

3.5

L = 20 3.0

L = 10

2.5

Tc /Jaa

2.0

1.5

1.0

0.5

0.0 0

1

2

4

3

5

6

7

Ω b /Jaa

(a) 4.0

Ω a /Jaa = 2

Bul k ( Ωa /Jaa = Ω b /Jaa ) 3.5

3.0

L=10

L=20

Tc /Jaa

2.5

L=32

2.0

1.5

1.0

0.5

0.0 0

1

2

(b)

3

4

5

6

7

Ω b /Jaa

Fig. 3. Dependence of the critical temperature Tc =Jaa as a function of the transverse field Ob for (a) Oa =Jaa ¼ 0:5 and (b) Oa =Jaa ¼ 2:

  a2 ¼ Mn;n Jn;n ¼ Jbb ; Jn;n1 ¼ Jn;nþ1 ¼ Jab ; Ob     b2 ¼ Mn;n1 Jn;n ¼ Jbb ; Jn;n1 ¼ Jn;nþ1 ¼ Jab ; Ob ¼ Mn;nþ1 Jn;n ¼ Jbb ; Jn;n1 ¼ Jn;nþ1 ¼ Jab ; Ob n ¼ 2; 4; y; L:

ð34Þ

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3.5 Ω b /Jaa =1 , Ω a /Jaa =2

3.0

Tc /Jaa

2.5

2.0

1.5 Ω b /Jaa= 5 , Ω a /Jaa =0.5

1.0 8

12

16

20

24

28

32

36

L Fig. 4. Dependence of the critical temperature Tc =Jaa on the thickness L of the finite superlattice.

L is the thickness of the superlattice, it is an arbitrary number. Now we obtain the critical temperature of the system from the condition given by det Mab ¼ 0:

ð35Þ

3. Results and discussion From the numerical solution of Eq. (35), the critical temperature of the alternating superlattice is calculated as a function of the reduced exchange   interactions R1 and R2 : We assume that R1 p1 and denote by Tca =Jaa ¼ 3:5186 and by Tcb =Jaa ¼ Tca =Jaa R1 the bulk critical temperature of a uniform lattice of material A and of material B. In the case of a uniform transverse field (Oa ¼ Ob ), Fig. 1a shows the dependence of the critical temperature Tc =Jaa on the reduced interlayer exchange coupling R2 for various values of the transverse field and R1 ¼ 1: This dependence is quasi-linear, in good agreement with the results obtained in the spin-12 case [17,18]. From Fig. 1b, it is clear that the critical temperature increases with increasing the thickness of the superlattice. Besides, this, we note that the critical temperature increase with increasing R2 and/or decreasing the value of the transverse field O: Now, in order to examine the effects of the alternating transverse fields Oa and Ob ; we will neglect the effect of the competition between interlayer and intralayer exchange interactions, for this reason it is practical to choose R1 ¼ R2 ¼ 1: We note that there is a symmetry between Oa and Ob (i.e. if we change Oa by Ob and vice versa the results remain the same). The results presented in Fig. 2 show the effects of the alternating transverse field on the critical temperature Tc =Jaa of the superlattice with a fixed thickness L ðL ¼ 10Þ: However, Tc decreases with increasing Ob : Furthermore, there exists a critical transverse field Oca ; below which the system remains ordered at low temperature for any value of Ob (this is called the long range ordered phase), while for Oa sufficiently large the system exhibits an ordered–disordered phase transition at very low temperature (in particular at T ¼ 0) and the long range order disappears. Such behavior does not occur for very large

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thickness L and weak strength transverse field Oa (Fig. 3a), while for Oa sufficiently large, and for L large ðL ¼ 32Þ the long range ordered phase occurs at low temperature (Fig. 3b). These results mean that the critical transverse field Oca mentioned above depends on the thickness value of the superlattice. Depending on the values of the alternating transverse fields, the critical temperature undergoes two different behaviors as a function of the thickness of the superlattice. Indeed from Fig. 3a it is clear that Tc =Jaa decreases with increasing the thickness L of the superlattice for sufficiently weak values of Oa and large values of Ob ; while it increases with increasing the thickness L for large Oa and small values of Ob : Such results are not obtained in the uniform transverse field case (Oa ¼ Ob ). We denote that the critical temperature vanishes at the same critical transverse fields Ob for any thickness L; which is different from the bulk one (Fig. 3b). In Fig. 4 we give the behavior of Tc as a function of the thickness of the superlattice from which it is clear that Tc decreases with increasing the thickness L for weak enough Oa and large values of Ob while it increases with L in the contrary case. 4. Conclusion We have investigated the effects of alternating transverse fields on the critical behavior of a spin-1 Ising superlattice formed by two alternating ferromagnetic materials using the effective field theory. On the one hand, it is found that the ordered phase persists at low temperature for sufficiently weak values of (Oa =Jaa ), and arbitrary value of Ob =Jaa ; such behavior is not obtained in the uniform transverse field case. On the other hand, depending on the values of the transverse field Oa and Ob ; the critical temperature can increase or decrease with increasing the thickness of the superlattice. Acknowledgements The authors A.S, H. Ez and A.A would like to thank UNESCO, IAEA and the Abdus SalamInternational Centre for Theoretical Physics, Trieste, Italy, for their hospitality. The authors A.S and A.A thank the ‘‘AS-ICTP TRIL Program’’ for the financial support and the collaboration between Padova University and CNCPRSPT Morocco. The author H. Ez would like also to thank the Arab Foundation for the Scholarship Association at the AS-ICTP, Italy. This work has been partially supported by the P.F. MSTA II-CNR and MIUR (COFIN ex 40%). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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