The ordering temperature of coupled magnetic planes

Journal of Magnetism and Magnetic Materials 242–245 (2002) 550–552

The ordering temperature of coupled magnetic planes H. Moradi* Department of Physics, Faculty of Sciences, University of Ferdowsi of Mashhad, Mellat Park, Mashhad, Iran

Abstract The high-temperature susceptibility is evaluated within the Bethe–Peierls approximation for a two-dimensional lattice in the presence of a random magnetic field with Gaussian distribution. The susceptibility of a plane is used to obtain the transition temperature of coupled planes when the interplanar coupling is weak. It is shown that the transition temperature of the coupled planes falls below the result for uncoupled films at some values of the average spacer thickness. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Bethe–Peierls; Random magnetic field; Transition temperature; Coupled planes

The effect of a random magnetic field (RMF) on a two-dimensional Ising ferromagnet has been much studied since the proof [1] that it lies below the lower critical dimension [2]. The Bethe–Peierls approximation (BP) represents considerable improvement on the mean field theory in two dimensions [3]. This problem has been solved for an RMF given by 2-delta function [4]. The paramagnetic susceptibility, wp ðTÞ; of an Ising plane within BP in the presence of RMF with Gaussian distributions is calculated when the RMF is varying very slowly over a correlation length. The results are used in an expression that we had previously derived [5] for the ordering temperature (OT) of weakly coupled planes where the interplanar exchange coupling (IEC) has a fluctuating component. We showed that the fluctuating component gives rise to the RMF and the OT-coupled planes are given by the solution of 27J12 7wp ðTÞ ¼ 1 where J12 is the average IEC. In the BP, one retains the interaction of a single spin s0 and its surrounding q-neighbors, and takes into account the other spins only by way of an effective field H which acts upon the q-neighboring spins. Thus the Hamiltonian is

H ¼ J0

q X i¼1

q X s0 si  ðh þ h0 Þs0  ðH þ h1 Þsi ;

ð1Þ

i¼1

*Fax: +98-511-843-8032. E-mail address: [email protected] (H. Moradi).

where the first term describes the Ising interaction between s0 and its nearest neighbors, h is the external field, h0 is the RMF acting on site 0; and h1 is the RMF acting on neighbors (site 1) that is different from h0 : The field h1 is the resultant of the RMFs felt by the RMF acting directly on the neighbors. So its magnitude is different from h0 : H is the effective field which is determined by the equation /s0 S ¼ /s1 S: The hi is the thermal average of the local magnetization. The partition function is given by z¼

X s0 ;si ¼71

exp b J0

q X

s0 si þ ðh þ h0 Þs0

i¼1

! q X ðH þ hi Þsi ; þ

ð2Þ

i¼1

where b ¼ 1=kB T: For a square lattice we have: z7 ¼ 24 exp½7bðh þ h0 Þ cosh4 ½bðH7J0 þ h1 Þ ; z ¼ z þ zþ :

ð3Þ

The average of magnetization on the central site is /s0 S ¼

1 qz zþ  z ¼ bz qh zþ þ z

ð4Þ

and on site 1 is given by 1 /s1 S ¼ ðzþ tanh bðJ0 þ H þ h1 Þ z þ z tanh bðJ0 þ H þ h1 ÞÞ:

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 1 ) 0 1 1 5 0 - 7

ð5Þ

H. Moradi / Journal of Magnetism and Magnetic Materials 242–245 (2002) 550–552

The effective magnetic field, H; is equal to 3 H ¼ h þ arctanhðtanh J0 b tanh bðH þ h1 ÞÞ b þ h0  h1 ;

551

8

ð6Þ

6

ð7Þ

4

δ/J0 =0.03 δ/J0 =0.15 δ/J0 =0.3 δ/J0 =0.45

J0 /J12=5

b /sS ¼ tanh ðh  4H  4h1 þ h0 Þ: 3

We take the Gaussian distribution of RMF, we define the R þN configuration average of h1 as 0sT ¼ N dh1 pðh1 Þ/sS; where pðhÞ is the probability of RMFs. We have two Gaussian distributions, for h0 and h1 with halfwidths d and D ¼ ed; respectively. The average of /sS over the RMFs h0 and h1 is equal to Z N  Z þN dh0 pðh0 Þ dh1 pðh1 Þ/sS : ð8Þ 0sT ¼ N

N

The double integrals of the 0sT is difficult to evaluate in the general case with Gaussian distributions. To proceed, we first expand /sS in Eq. (7) in powers of h1  /sSh ¼ /sSh ¼0 þ bh1 qð/sSÞ=ðqbh1 Þ 1

1

J0χ

then the thermal average of magnetization is

2 0

2

3

4

5

T [J0/K] Fig. 1. The susceptibility of a plane with the RMF plotted as a function of temperature for different strengths of random fields (d=J0 =constant and e ¼ 1).

h1 ¼0

227

 ðbh1 Þ2 2 q ð/sSÞ=ðqbh1 Þ2 h1 ¼0 þL þ 2

wp ¼ q0sT=qh þ ðq0sT=qHÞ/dH=dhS;

ð9Þ

we evaluate wp with RMF when we study the problem above the Curie temperature (CT). When h and H are zero, the susceptibility is given by Fig. 1 for various values of d=J0 : We see that the RMF reduces the susceptibility as expected. It is confirmed to show a divergence at a reduced value of Tc : The OT of coupled planes depends on |J12 j; d and e: It can be higher or lower than the CT for an isolated plane. We present the theoretical results on the OT of well characterized Ni/Au/Ni multilayers which are measured by Bayreuther et al. [6]. In a multilayer, the IEC is an oscillating function of the spacer layer thickness [7]. The observed oscillation period of the IEC is Lexp ¼ ð1:1570:1Þ nm. The IEC is suppose given by J12 ¼ AegR cosðLR þ jÞ; the halfwidth of the RMF is given by    qJ12 d ¼  dR ¼ ALegR jsinðLR þ jÞdRj; qR

ð10Þ

ð11Þ

δ /J0=0.5 δ /J0=0.8

211 Tc [K]

and perform the integration over this quantity, with Gaussian distribution we have /h1 S ¼ 0 and /h21 S ¼ R þN 2 2 dh h 1 1 pðh1 Þ ¼ D : Next, we expand the resultant N expression in power of h0 before performing the second integration, then we integrate over the expanded vs h0 as h1 : These expansions are correct when bh1 and bh0 are small. We follow the way in which it was done on 0sT for /dH=dhS; that is, one integrates expression (6) over the expanding. By using the equation

δ /J0=0

Tc =184 K

195 179 162 146 0

1

2 tAu [nm]

3

4

Fig. 2. OT of Ni/Au multilayers as a function of thickness of Au for various RMF when dXJ0 =constant and e ¼ 1: The dashed–dotted line is CT of isolated plane.

where A=0.2412; R is the thickness of spacer, Au[1 1 1]; j ¼ 0:3035 rad; L ¼ 5:4636 rad/nm is the period IEC and g ¼ 0:6115/nm; the experimental IEC, Ni, is 0.5 erg/ cm2 at first antiferromagnetic coupling by the effective spin wave [6]. OT depends on the thickness of the layer, t; for a Ni layer with tNi ¼ 0:73 nm it is 183.94 K [6]. The OT of coupled planes, Tc ; is plotted as a function of thickness of spacer, tAu½1 1 1 ; in Fig. 2, for various d=J0 : When d ¼ 0; Tc oscillates with half period of J12 and its magnitude is greater or equal to the CT of uncoupled planes (tNi ¼ 0:73 nm). This result is contrary to the experimental result [6], which led us to include the effects of fluctuations in the spacer thickness.

H. Moradi / Journal of Magnetism and Magnetic Materials 242–245 (2002) 550–552

552

242

229

Exp. ε=0.5 ε =1 ε =2 T=184 K

217

2

2

2 1/2

Tc=184 K

204

191 178 166

191 178 166

153 140

Ex p.

δ /J0 =(0.2 +δ1 /J0 ) 2 2 2 1/2 δ /J0 =(0.8 +δ1 /J0 )

217

Tc [K]

Tc [K]

204

229

153 0

1

2

3

140

4

tAu [nm] Fig. 3. OT of Ni/Au with RMF plotted as a function of Au thickness when d1 ¼ jðqJ12 =qRÞdRj; e ¼ 1; and dR ¼ 0:5 nm. The experimental results (squares) taken from Ref. [6].

We have studied the behavior of the OT as a function of the thickness of the spacer, tAu½1 1 1 ; when d=J0 ¼ constant and e ¼ 1; for d=J0 ¼ 0; 0.5, and 0.8, the results are depicted in Fig. 2. When d=J0 is large, the value of Tc falls below that of an uncoupled plane (dashed–dotted line). However, since the RMF is assumed to arise from the fluctuations in the IEC due to variation in spacer thickness, d is related to the spacer thickness through Eq. (11) [5]. Since J12 ðRÞ is an oscillatory function of R; Eq. (10), its derivative dJ12 =dR is zero when J12 ðRÞ is a maxima or minima, and a maximum when J12 ðRÞ is zero. Hence a small variation in the spacer thickness dR will give a vanishingly small effect when jJ12 ðRÞj is large, but a large effect (RMF) when J12 ðRÞ-0: Hence, when J12 ðRÞ-0 the OT (given by the divergence of the planar susceptibility wp ) is small in the presence of an RMF determined by the spacer fluctuations. The bigger RMF the smaller the OT. The OT is shown as a function of Au thickness in Fig. 3. In these cases, d varies as 7ðqJ12 =qRÞdRj and e ¼ 0:5; 1, 2, where dR ¼ 0:5 nm. In some parts of these plots, the OT is less than the CT of uncoupled plane because of RMF, and some parts higher, because of IEC. Moreover, the depths of the valleys are different, in agreement with experiment (squares) taken from Ref. [6]. The results for the OT obtained for an RMF that varies as d ¼ ðd21 þ d22 Þ1=2 (where d1 ¼ jðqJ12 =qRÞdRj; d2 =J0 ¼ 0:2; 0.8, and dR ¼ 0:5 nm) are shown as a

0

1

2

3

4

tAu [nm] Fig. 4. OT of Ni/Au multilayers with RMF plotted as a function of thickness of spacer (Au) when d ¼ ðd21 þ d22 Þ1=2 ; that d1 ¼ jðqJ12 =qRÞdRj; d2 =J0 ¼ 0:2; 0.8, D ¼ d1 ; and dR ¼ 0:5 nm.

function of Au thickness, Fig. 4. The agreement with experimental results is better than that of Figs. 2 and 3. In this paper, it is shown that the OT varies with the half period of IEC. Without randomness, the OT of coupled planes is larger than that of uncoupled planes and that does not agree with experiment. For an RMF of constant halfwidth, d; the OT is lowered relative to that of uncoupled plane with increase in d: When d varies as jðqJ12 =qRÞdRj; the OT is less than the CT of uncoupled plane and in some part more than it. In experimental results, the depths of valleys are different, we get these. A combination of RMF with constant halfwidth and an RMF varying as jðqJ12 =qRÞdRj; gives a theoretical OT in the better agreement with experiment. The author is grateful to Prof. G. A. Gehring and Dr. Tucker for helpful discussions.

References [1] [2] [3] [4] [5] [6]

Y. Imry, S.-K. Ma, Phys. Rev. Lett. 35 (1975) 1399. T. Nattermann, J. Villain, Phase Transitions 11 (1988) 5. P.R. Weiss, Phys. Rev. 74 (10) (1948) 1493. O. Entin-Wohlman, C. Hartzstein, J. Phys. A 18 (1985) 315. H. Moradi, G.A. Gehring, in preparation. G. Bayreuther, F. Bench, V. Kotter, J. Appl. Phys. 79 (8) (1996) 4509. [7] S.S.P. Parkin, N. Moore, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304.