Critical behavior in a random field classical Heisenberg model for amorphous systems

Physics Letters A 346 (2005) 128–132 www.elsevier.com/locate/pla

Critical behavior in a random field classical Heisenberg model for amorphous systems Douglas F. de Albuquerque ∗ , Sandro Roberto L. Alves, Alberto S. de Arruda Departamento de Matemática, Universidade Federal de Sergipe, 49100-000 São Cristovão, SE, Brazil Departamento de Engenharia Civil, Universidade Federal de Sergipe, 49100-000 São Cristovão, SE, Brazil Departamento de Física, Universidade Federal de Mato Grosso, 78060-900 Cuiabá, MT, Brazil Received 30 July 2003; received in revised form 7 March 2004; accepted 22 July 2005 Available online 1 August 2005 Communicated by A.R. Bishop

Abstract By using the differential operator technique and the effective field theory scheme, the critical behavior of amorphous classical Heisenberg ferromagnet of spin-1/2 in a random field is studied. The phase diagram in the T –H and T –α planes on a simple cubic lattice for a cluster with two spins is obtained. Tricritical points, reentrant phenomena and influence of the random field and amorphization on the transition temperature are discussed.  2005 Elsevier B.V. All rights reserved.

1. Introduction Various physical systems such as random field, magnetic thin films, amorphous and others have been a considerable source of researches in the last two decades. The problem of the random field Ising model (RFIM) which, was introduced by Imry and Ma [1], has become the subject of experimental and theoretical interest [2]. Many theoretical problems associated with the ferromagnetic RFIM have been studied exten* Corresponding author.

E-mail addresses: [email protected], [email protected] (D.F. de Albuquerque), [email protected] (S.R.L. Alves), [email protected] (A.S. de Arruda). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.07.060

sively by many authors [2–10], such as the phase transitions and critical behavior. Within the framework of the mean field approximation, it is well known that the bimodal random field distribution leads to a tricritical behavior, but the Gaussian distribution only exhibits a second order phase transition. On the other hand, in the mean field renormalization group (MFRG) framework the trimodal distribution (TD), introduced in the literature by Mattis [11] to simulate the Gaussian distribution, leads to a tricritical behavior as well a second order phase transition only for certain values of the critical value of the parameter that governs the random field [12–16]. In this context, various questions need to be answered (such as the exact relation between the RFIM and a dilute antiferromagnet in a uniform field).

D.F. de Albuquerque et al. / Physics Letters A 346 (2005) 128–132

Many theoretical studies in the amorphous magnetic systems have been described by the Heisenberg model with fluctuating exchange interaction and/or with random anisotropy [17]. Recently, some attention has been given to the random field Heisenberg model (RFHM) [18]. In particular, the RFHM exhibits tricritical behavior when the random field is governed by a bimodal distribution within the framework of the effective field theory. On the other hand, the RFHM can also be used to describe the essential physics of a class of experimentally accessible disordered systems, which include structural phase transitions in random alloys, binary fluid mixtures in random porous media and others. Essentially, the RFHM displays the same critical behavior exhibited by the RFIM. Nevertheless, within the context of the random media, an interesting problem is the amorphous ferromagnet behavior, due to their technological importance for magneto-optical recording devices. Theoretical and experimental investigations in structurally disordered systems have shown that may exist magnetic long range order in amorphous systems. In particular, the RFIM has been employed to describe critical behavior of amorphous magnetic systems, such as thin films and critical surface behavior of the amorphous semi-infinite systems [19,20]. Kaneyoshi and co-workers [21] studied such systems by employing the lattice model of amorphous magnets, in which the structural disorder is replaced by a random distribution of the exchange integrals. By using the framework of the effective field approximation, for a cluster with a single spin, Hai and Li [22] have obtained interesting phase diagrams in the temperature (T ) vs. field (H ) plane for an infinite amorphous ferromagnet in a bimodal random field on a simple cubic lattice. In this Letter, we study an amorphous classical Heisenberg Ferromagnet model, with a probability distribution function for the exchange couplings and a bimodal random field distribution for clusters containing two spins on a simple cubic lattice. The calculation is carried out within the effective field theory (EFT) approximation. The outline of the remainder of the Letter is as follows: in Section 2, the model is introduced, and an analytical expression for the average magnetization is found. In Section 3 an analytical expression for the second order phase transition line is

129

obtained, and the existence of tricritical points, reentrant phenomena and topology of the phase diagram are investigated.

2. Model and calculations The amorphous classical Heisenberg ferromagnet in a version of a n-vector model in a random field, whose Hamiltonian is

Kij Si · Sj + hi · Si , −βH = (1) (i,j )

i

where the summation is carried out only over pairs of nearest-neighboring sites (i, j ). The quantities Si are isotropically interacting n-dimensional classical spins √ of magnitude n localized at the sites i of a simple cubic lattice, and the Cartesian components of Si obey the normalization condition [23], n
 ν 2 Si = n.

(2)

ν

Kij (≡ Jij /kB T , kB is the Boltzmann constant and T the temperature) is the exchange interaction between the spins, which is assumed to be randomly distributed according to the probability distribution function P (Kij ) =

1 δ(Kij − K − αK) 2  + δ(Kij − K + αK) ,

(3)

and the reduced random magnetic field hi (h ≡ µB H /kB T , where µ is the Bohr magneton and H is the random magnetic field) at site i obeys the following bimodal distribution P (hi ) =

 1 δ(hi + h) + δ(hi − h) . 2

(4)

The Hamiltonian (1) reduces to the well-known S = 12 Ising, classical planar (XY), classical Heisenberg and spherical models for n = 1, 2, 3 and ∞, respectively. In this Letter, we follow the EFT procedure (see Refs. [24–27]) to study the critical properties of the amorphous ferromagnet in a random field distribution described by Hamiltonian given by Eq. (1) and employ the axial approximation [28]. The Hamiltonian for a cluster with two spins can be written as H = K12 S1 · S2 + a1 S11 + a2 S21 ,

(5)

130

D.F. de Albuquerque et al. / Physics Letters A 346 (2005) 128–132

1 where al = hl + jZ−1 =l Klj Sj , (l = 1, 2) and Z is the lattice coordination number. In order to obtain the average magnetization per spin m = 12 (S11 + S21 ) for a two-spin cluster, we employ the equation (see Refs. [7–10])

 m=

Z−1 

    Z−1  1 1 αx + Sk βx αy + Sl βy

k=1,2

l=1,2

× gn (X, Y )|X=h1 ,Y =h2 ,

(6)

αν = cosh(Kj Dν ), βν = sinh(Kj Dν ), (ν = x, y;  = ∂ 1, 2), and X(Y ) = x(y) + h1 (h2 ). Dν (≡ ∂ν ) is the differential operator [27] which satisfies the mathematical relation 2 sinh(aDx + bDy )gn (X, Y )|X=h1 ,Y =h2

dom average in Eq. (6)  1

m ¯ = dKij dhi P (Kij )P (hi ) S11 + S21 . 2

(8)

By using the properties of the differential operator and assuming translational invariance, the magnetization for the n-vector model in a random field (RFNVM), on a simple cubic lattice, is given by m ¯=

4


A2+1 (K, n, α, h)m ¯ 2+1 ,

(9)

=0

where we have used the fact that, in order to satisfy the time reversal symmetry of the Ising model, as well the properties of the differential operator technique, the coefficients A2 (K, n, α, h) (even) have been set equal to zero.

= gn (a + h1 , b + h2 ) − gn (−a + h1 , −b + h2 ), where gn (X, Y ) is given by  gn (X, Y ) = sinh(X + Y ) cosh(X + Y )

−1 + exp(−2K12 )Tn (K12 ) cosh(X − Y ) , (7)

with Tn (K12 ) = (1 − tanh(n/2−1) (nK12 )) × (1 + tanh(n/2−1) (nK12 ))−1 . Here tanh(n/2−1) (X) denotes the generalized hyperbolic tangent defined by I(n/2) (X) , tanh(n/2−1) (X) = I(n/2−1) (X) and In (X) is a modified Bessel function of the first kind. Eq. (6) is exact and will be applied here as the basis of our formalism, since it yields the cluster magnetization and the corresponding multi-spin correlation functions associated with various sites for the cluster under consideration. Here we apply the EFT approximation on both sides of Eq. (6), i.e., the thermal and random average (denoted by · · ·c ), along with the decoupling procedure which ignores all high-order spin correlations, namely Si1 Sj1 · · · Sn1 c ≈ Si1 c Sj1 c · · · Sn1 c , with i = j = · · · = p. Based on this approximation, replacing each boundary configurational spin average by the symmetry breaking meanfield parameters bi for all i, and considering that for such random system, it is necessary to perform the ran-

3. Results and discussions In this section, we are interested in the phase boundary of the model. Then we will focus our attention in the second-order transition line and tricritical point (TCP) in the RFNVM, where only the case n = 3 will be studied, corresponding to the classical Heisenberg model of spin-1/2. In the vicinity of the second-order phase transition m ¯ ∼ 0, Eq. (9) can be rewritten as m ¯2 =−

A1 (K, α, h) − 1 . A3 (K, α, h)

(10)

Since the magnetization m ¯ goes to zero continuously, a seconder-order transition line is obtained from the simultaneous solution of the equations A1 (K, α, h) = 1,

A3 (K, α, h) < 0.

(11)

On the other hand, the r.h.s of Eq. (10) must be positive. If this is not the case, the phase transition is of first-order. In this work, we have confined our calculation only to the second-order transition, including the TCP. The tricritical points are obtained by solving the equations A1 (K, α, h) = 1,

A3 (K, α, h) = 0.

(12)

Numerical solutions of Eqs. (12) provide the values of the coordinates of the tricritical points. Some selected values of the coordinates of the tricritical points

D.F. de Albuquerque et al. / Physics Letters A 346 (2005) 128–132

131

Table 1 Localization of the tricritical points TTPC , hTPC and T0 is the temperature when the magnetic random field is zero (zero point temperature) for selected values of α α

TTPC

hTPC

T0

0.00 0.10 0.30 0.50 0.60 0.65 0.70 0.7196 0.80 0.90 0.95 0.98 1.0 1.476

2.389 2.369 2.205 1.842 1.419 1.193 0.861 0.671 0.307 0.144 0.0615 ··· ··· ···

2.784 2.786 2.795 2.809 2.7855 2.7707 2.7365 2.7061 2.4872 2.2551 2.1269 ··· ··· ···

5.030 5.021 4.943 4.782 4.669 4.603 4.531 4.501 4.367 4.173 4.065 3.994 3.945 1.865

Fig. 1. Phase diagram in the T –h plane for the classical Heisenberg model. The solid curves are the second-order phase transition lines. Black diamonds represent the positions of the tricritical points. (a) α = 0.0, (b) α = 0.65, (c) α = 0.7196, (d) α = 0.8, (e) α = 0.95, (f) α = 1.0, (g) α = 1.1, (h) α = 1.3, (i) α = 1.476.

together with the temperatures of zero point (temperature at which the magnetic field is zero) are exhibited in the Table 1. By solving Eqs. (12), the numerical results are obtained and are shown in Figs. 1 and 2. When the random field and α are zero, we obtain the well-known critical temperature of classical Heisenberg model of spin-1/2, Tc = 5.030 (temperature of the zero point). In the particular case α = 0, without fluctuations in the exchange interaction, we recover all the results presented in the previous papers [18]. In Fig. 1, we sketch the curves of temperature versus

Fig. 2. The transition temperature against α for various selected values of the random field. (a) h = 0.0, (b) h = 1.0, (c) h = 1.5, (d) h = 2.0, (e) h = 2.5, (f) h = 2.8.

random field for various selected values of the parameter α. The solid curves are second-order phase transition lines. Black diamonds represent the positions of the tricritical points at which the nature of the phase transition changes from second to first order. The phase diagram shows three distinct regions. In the first region, which is located in the range 0
α
0.65, basically we have second order lines that end at the tricritical points. On the other hand, in the second region with 0.65 < α
0.95, the phase diagram shows two critical temperatures for a given value of h/J , which corresponds to the interesting reentrant phenomena. These reentrant phenomena may come from the competition between two disorder factors, the random field and structural fluctuation. However, the second order lines still finish at tricritical points. Finally, in the third region, α > 0.9, the phase diagram only presents the reentrant phenomena. The growth of the parameter α, associated with the reduction of the random magnetic field, makes the tricritical behavior disappear, once that this behavior is a consequence of the competition between the two disorder factors. Fig. 2 shows the plot of the transition temperature against α for various selected values of the random field. In the range 0
h/J < 1 there occurs the reentrant phenomenon. The transition temperature decreases smoothly to its zero-point value (see Table 1) which is zero. Here, the values of the fluctuation parameter are large, while the values of the intensity of the random magnetic field are small. In the interval 1
h/J < 1.7, the these phenomena can occur but they are soften. This is due

132

D.F. de Albuquerque et al. / Physics Letters A 346 (2005) 128–132

to the increase in the magnitude of the random magnetic field and to the decrease in the fluctuations of the exchange interaction (α). For h/J  1.7, the reentrant phenomena reappears. This fact is expected once the system is in a region where the magnetic field is large and the exchange parameter fluctuation is small, favoring the appearance of a tricritical behavior (see Table 1).

4. Conclusion In conclusion, we have shown that the tricritical behavior appears in the classical Heisenberg model simply because the random magnetic field is governed by a bimodal distribution [18], such as in the random field Ising model. When the fluctuations in the nearest-neighbor exchange coupling (large structural fluctuations) are governed by a probability distribution function, the system exhibits tricritical points and reentrant phenomena due to the competition between these two disorder factors. The increase of the structural disorder (α) associated with the lowering in the random field (other disorder factor) makes the TCP disappear. However, by a suitable combination of the values of the parameter α (structural fluctuation) and h/J (random field) the TCP and reentrant phenomena may occur.

Acknowledgements One of the authors (S.R.L.A.) acknowledges the partial support from the program PIBIC (UFS–CNPq). The authors would like to thank professor Ernesto Raposo (LFTC–UFPE) and Wagner Figueiredo (UFSC) for the critical reading of the manuscript.

References [1] Y. Imry, S.K. Ma, Phys. Rev. Lett. 35 (1975) 1399. [2] Y. Shapir, in: D.H. Ryan (Ed.), Recent Progress in Random Magnets, World Scientific, Singapore, 1992, p. 309. [3] A.S. de Arruda, W. Figueiredo, Mod. Phys. Lett. B 11 (21–22) (1997) 973. [4] S. Galam, C.S.O. Yokoi, S.R. Salinas, Phys. Rev. B 57 (1998) 8370. [5] N. Benayad, A. Fathi, L. Khaya, Physica A 300 (2001) 225. [6] E.E. Reinehr, W. Figueiredo, Phys. Lett. A 244 (1998) 165. [7] D.F. de Albuquerque, Physica A 287 (2000) 185. [8] D.F. de Albuquerque, J. Appl. 75 (1994) 5832. [9] D.F. de Albuquerque, I.P. Fittipaldi, J.R. de Sousa, Phys. Rev. B 56 (1997) 13650. [10] D.F. de Albuquerque, J. Magn. Magn. Mater. 219 (2000) 349. [11] D.C. Mattis, Phys. Rev. Lett. 55 (1985) 3009. [12] A. Aharohy, Phys. Rev. B 18 (1978) 3318. [13] M. Kaufman, P.E. Klunzinger, A. Khurana, Phys. Rev. B 34 (1986) 4766. [14] R.M. Sebastianes, V.K. Saxena, Phys. Rev. B 35 (1987) 2058. [15] S. Galam, Phys. Rev. B 31 (1985) 7274. [16] A.S. de Arruda, W. Figueiredo, R.M. Sebastianes, V.K. Saxena, Phys. Rev. B 39 (1989) 4409. [17] R. Kishore, I.C. da Cunha Lima, M. Cristina Forti, J. Phys. Chem. Solids 43 (1982) 337. [18] D.F. de Albuquerque, A.S. de Arruda, Physica A 316 (2002) 13. [19] Y. El Amraoui, H. Arhchoui, S. Sayouri, J. Magn. Magn. Mater. 219 (2000) 89. [20] Y. El Amraoui, A. Khmou, J. Magn. Magn. Mater. 218 (2000) 182. [21] T. Kaneyoshi, R. Honmura, I. Tamura, E.F. Sarmento, Phys. Rev. B 27 (1984) 5121. [22] T. Hai, Z.Y. Li, J. Magn. Magn. Mater. 80 (1989) 173. [23] H.E. Stanley, Phys. Rev. 179 (1969) 570. [24] J.R. de Sousa, D.F. de Albuquerque, Physica A 236 (1997) 419. [25] H.B. Callen, Phys. Lett. 4 (1963) 161. [26] M. Suzuki, Phys. Lett. 19 (1965) 267. [27] R. Honmura, T. Kaneyoshi, J. Phys. C 12 (1979) 3979. [28] T. Idogaki, N. Uryu, Physica A 181 (1992) 173.