Ferromagnetic Properties of Bond-Dilution and Random Positive or Negative Uniaxial Anisotropy Blume Capel Model

Transcription

1 Commun. Theor. Phys. (Beijing, China) 42 (2004) pp c International Academic Publishers Vol. 42, No. 5, November 15, 2004 Ferromagnetic Properties of Bond-Dilution and Random Positive or Negative Uniaxial Anisotropy Blume Capel Model ZHU Hai-Xia 1 and YAN Shi-Lei 1,2,3, 1 Department of Physics, Suzhou University, Suzhou , China 2 Key Laboratory of Film Material of Jiangsu Province, Suzhou University, Suzhou , China 3 CCAST (World Laboratory), P.O. Box 8730, Beijing , China (Received March 9, 2004) Abstract We study the ferromagnetic properties of spin-1 system, which is considered in the frame of the bonddilution and random positive or negative anisotropy Blume Capel model in the effective field theory and a cutting approximation. The investigation of phase diagrams displays some rich properties of the trajectory of tricritical point, reentrant phenomena at low temperatures. Under certain both bond concentrations and random negative anisotropy, there are new transition lines of double tricritical points. So special emphasis is placed on the influence of the bond dilution and random anisotropy on phase diagrams. The magnetizations of the system are also discussed. Some results have not been revealed in previous reports. PACS numbers: Dg, Cx, Ej Key words: ferromagnetic properties, Blume Capel model, bond dilution, random uniaxial anisotropy, effective field theory 1 Introduction The low temperature properties of disordered magnetic systems have been a subject of interest for some time. In a series of recent publications, the works were carried out on a variety of systems including the Ising model, XY model, and Heisenberg model, in zero and finite applied fields. [1 4] In local conditions the Ising model has been widely applied to solve many different problems mainly because it is a simple but fruitful model. Since 1966, the Blume Capel model (BCM) [5] has been an important models in the family of Ising models. The BCM is originally proposed to study spin-1 magnetic phase transition. Its phase diagram presents a line of continuous transitions and a line of first-order transitions, separated by a tricritical point (TCP). Meanwhile, the BCM is also used to study the critical behavior of 3 He- 4 He mixtures. Because the BCM can describe both ferromagnetic properties and critical behavior of 3 He- 4 He mixtures, the BCM is of a practical importance. Although introduction of the Blume Emery Griffiths model (BEGM) is of a generalization, [6] and the biquadratic exchange parameter is related to the interaction energy between 3 He- 4 He, the interaction energy is nearly independent, one can assume that the interaction energy is zero. Thus, the BEGM is reduced to the BCM. The introduction of randomness is an important development in various Ising spin systems. No matter how disorder distributions are adopted, people have learned that the disorder distributions play an important role in the order of phase transition and may lead to some new results. For example, the critical behavior of 3 He- 4 He mixtures in random media has been modeled by the BCM with a random anisotropy. [7,8] The critical behavior of the system may be changed in a drastic way. People have also noticed that disordered Ising ferromagnet describing the BCM has been studied by means of various techniques. [9 19] However, for two-dimensional systems, the results obtained by using different techniques are not in complete agreement. The TCP is eliminated suddenly when the exchange interaction is randomized according to the real-space renormalization-group method. [9] But the study of MC simulations has proved that the TCP will disappear gradually in the certain ranges of bond concentration. [10] On the other hand, the TCP should also vanish abruptly from the real-space renormalization-group method, [11] while TCP only exists in the very narrow regions and has the first-order phases inside order phases at low temperatures by means of the pair approximation due to the existence of random anisotropy. [12] But other researches have shown that a gradual elimination of the first-order phase with 2D cases studied by Kaneyoshi, [13] and Yan et al. [14,15] Although there are many possible different versions in two dimensions when randomness is introduced in the BCM, for three-dimensional case, the first-order phase transition is converted to the second-order one above a threshold of disorder no matter what different techniques are adopted. Falicov and Berker [20] found the TCP is replaced by a line segment of second-order phase transitions The project supported partly by the Natural Science Foundation of the Education Bureau of Jiangsu Province under Grant No. 03KJA and the Thin Film Materials Key Laboratory Open Foundation of Jiangsu Province under No. K slyan@suda.edu.cn

2 790 ZHU Hai-Xia and YAN Shi-Lei Vol. 42 dominated by quenched bond randomness in the framework of renormalization-group theory. Recently, Puha and Diep [21] have shown that the TCP is suppressed gradually a correct tendency with increasing random crystal field and bond concentrations through MC simulations, respectively. However, detailed information has not been obtained concerning the phase transition properties at low temperatures according to Ref. [21]. This is because an obstacle encountered in MC simulations on disordered systems is the presence of many metastable states at low temperatures, separated by barriers that may diverge in the thermodynamic limit, making it very hard for the system to equilibrate. Hence, for disordered spin systems, the description in terms of MC simulations is generally valid and accurate only at the higher temperature status. In this paper, we will study the phase diagrams and magnetizations of three-dimensional bond dilution and random positive or negative anisotropy BCM in the same time. The global phase diagrams and magnetizations are presented on a simple cubic lattice. Our calculations show that the results obtained can overcome the difficulty of the large amount of computational time of MC simulations and gain the global phase diagrams with rich messages at low temperatures. Under certain both bond concentrations and random negative anisotropy strengths, there are new transition lines of double tricritical points. The influence of the competition between two different disorder factors and positive or negative anisotropy on physical properties of phase diagrams is also analyzed. Additionally, the occurrence of the reentrant phenomena is also closely related with the above parameters. To our knowledge, the above mentioned subjects have not been investigated yet. In the present work, we employ an effective field theory (EFT) and a cutting approximation to discuss the phase diagrams and magnetizations of the system. This approach is suitable for the disordered BCM since the global phase diagram, including first- and second-order transition lines, can be given through the numerical analysis. The outline of this paper is organized as follows. In Sec. 2, the expressions of the BCM with two random factors for evaluating the second-order transition, the TCP and magnetization are presented. In Sec. 3, the trajectory of the TCP, reentrant phenomena, complete phase diagrams, and magnetization behaviors are investigated in detail. We also include an extensive discussion of the possible physical reasons in this section. A brief conclusion is given in Sec Formulae Let us consider the nearest-neighbor interaction bond dilution and random positive or negative anisotropy spin-1 BCM, defined by the Hamiltonian H = ij J ij S z i S z j i D i (S z i ) 2, (1) where the spins S z i can be assumed to have values of 0, ±1 at each site i of a lattice. The first summation runs only over all pairs of the nearest neighbor sites. The second summation involves all sites of lattice. J ij = J is the exchange interaction between the nearest-neighbor sites and let J > 0. D i is a random anisotropy parameter of the system, assumed to be negative. We consider only the case of uniaxial anisotropy with the anisotropy (z) axis the same for all of the ions. In the most general cases, both the exchange interaction and anisotropy parameters are random functions of position. We assume that J ij and D i satisfy independent dilution and random distributions, respectively P (J ij ) = pδ(j ij J) + (1 p)δ(j ij ), (2) P (D i ) = tδ(d i D) + (1 t)δ(d i αd), (3) where p c < p 1.0, 0 t 1.0 and 1 α 1. Let p denote the concentration of bond dilution and t indicate the concentration of anisotropy, while α is a tunable parameter of positive or negative magnitude of uniaxial anisotropy. In fact, the ground state of the above BCM undergoes a phase transition at p = p c from an ordered state to a disordered state. Hence, the p c is the percolation threshold of bond dilution. As pointed out in our previous work, [15] we can perform the averaged magnetization m and quadrupolar moment q within the framework of the EFT and a cutting approximation, m = [q cosh(j ij ) r + m sinh(j ij ) r + 1 q] z F (x) x=0, (4) q =[q cosh(j ij ) r +m sinh(j ij ) r +1 q] z G(x) x=0, (5) where z is the lattice coordination number, = / x is a differential operator. The r denotes the bond dilution average for Eq. (2). Functions F (x) and G(x) are written as F (x) = P (D i )f(x, D i )dd i, (6) G(x) = P (D i )g(x, D i )dd i, (7) while f(x, D i ) and g(x, D i ) are f(x, D i ) = g(x, D i ) = 2 sinh(βx), (8) βdi 2 cosh(βx) + e 2 cosh(βx). (9) βdi 2 cosh(βx) + e From Eq. (4) we may easily separate the ferromagnetic state (m 0) from the paramagnetic state (m=0). On the critical transition line, there may exist a TCP where the transition changes from the second- to the first-order. In particular, there may exist double TCPs in a critical transition line under certain disorder conditions. In order to calculate this solution (m 0), we must combine Eqs. (4) and (5). Then the self-consistent equation with respect to averaged magnetization m can be written as m = am + bm 3 + cm 5 + (10)

3 No. 5 Ferromagnetic Properties of Bond-Dilution and Random Positive or Negative Uniaxial Anisotropy Blume Capel Model 791 When a = 1, the second-order phase transition line is found according to Landau theory. Meanwhile, we can obtain an explicit expression of the magnetization m, i.e. ( 1 a ) 1/2 m =. (11) b The coefficients a and b are given by the following forms a = z sinh(j ij ) r [q 0 cosh(j ij ) r + 1 q 0 ] z 1 F (x) x=0, (12) b = z(z 1)q 1 sinh(j ij ) r [ cosh(j ij ) r 1][q 0 cosh(j ij ) r + 1 q 0 ] z 2 F (x) x=0 z(z 1)(z 2) + sinh 3 (J ij )[q 0 cosh(j ij ) + 1 q 0 ] z 3 F (x) x=0, (13) 3! where q 0 and q 1 satisfy the solutions of q 0 = [q 0 cosh(j ij ) + 1 q 0 ] z G(x) x=0, (14) q 1 = z(z 1)/2![ sinh(j ij ) r ] 2 [q 0 cosh(j ij ) r + 1 q 0 ] z 1 G(x) x=0 {1 z[ cosh(j ij ) r 1][q 0 cosh(j ij ) r + 1 q 0 ] z 1 G(x) x=0 }. (15) If b > 0 in Eq. (13), the phase transition is of the first order. Therefore, b < 0 must be satisfied. Hence the point at which a = 1 and b = 0 determines the TCP on the transition line, distinguishing the second-order from the first-order transition. The second order transition line can be determined by a = 1 and b < 0. (16) Up to now, the system described by the BCM has been generalized to two disorder cases (bond dilution and random positive or negative uniaxial anisotropy). One does not expect to obtain results as precise as those from MC simulations. Nevertheless, the EFT procedure can compensate for the defect in the study of the low temperature properties of disordered magnetic system by means of MC simulations. Here we are interested in studying the phase diagrams and magnetizations of three-dimensional lattice because three-dimensional system is experimentally the most relevant dimension. In order to avoid too lengthy calculations, we only consider a simple cubic lattice as a three-dimensional version. In particular, there is an interesting possibility of the existence of double TCPs at which the transition changes firstly from the second- to the firstorder and then changes from the first- to the second-order again. Obviously, the important physical reasons come from the competition between disorder in both the bond and anisotropy. 3 Results and Discussions All the results that will be presented in this section are obtained from the solutions of Eqs. (11) (16) numerically. Firstly, the Curie temperatures versus uniaxial anisotropy for the present system with α = 0.5 and concentrations of bond dilution p = 1.0, 0.7, and 0.4 are depicted in Figs. 1(a) 1(c), respectively, when the values of random anisotropy concentration t are changed. For p = 1.0 and t = 1.0, it corresponds to the pure BCM and shows a TCP. From Fig. 1(a), it is clear that, with the increasing concentration of random anisotropy t, the TCP is depressed monotonously and disappears at a critical concentration t 1. The location of the TCP is the function of the concentration of random anisotropy. In other words, the second-order one above a critical concentration t 1 gradually replaces the first-order phase transition. The TCP exists only in the ranges of 1.0 t > t 1. From our calculation, t 1 = when p = 1.0. This result is in qualitative agreement with the estimates from the MC simulations at higher temperatures, like Fig. 4 in Ref. [21]. However, they do not give a real critical concentration at which the TCP just disappears. In fact, we can show a global phase diagram in a complete random concentration region. We see that the second-order phase transition temperature goes to zero at four different values of the anisotropy in the range of t 1 t t 1 = They represent four degenerate patterns of the anisotropy at ground state. In particular, we notice that the second-order phase transition lines show a rapid change at low temperatures for a very narrow region of random anisotropy concentration from t = to , while the second-order ones are almost the same at high temperatures. A tiny change of random concentration may trigger a rich variety of the phase transitions. Hence, the study is of practical signification under low temperature conditions. Additionally, the transition line between the ferromagnetic phase and the paramagnetic one extends to D for all 0.34 t < t 1 and its critical temperature is finite. This means that the system is always in ordered phase at low temperatures. This stems from the fact that the negative anisotropy acting on the given sites tends to force them to be in the states S = ±1. In this case, the infinite clusters of spin S = ±1 states will form and will be able to sustain order for 0.34 t < t 1. Certainly, there is the cluster but its critical temperature is zero at t = t 1. Therefore, t 1 is a random critical concentration at which the anisotropy

4 792 ZHU Hai-Xia and YAN Shi-Lei Vol. 42 will just go to negative infinite. Fig. 1 Uniaxial anisotropy dependencies of Curie temperature on the simple cubic lattice for α = 0.5 with different bond concentration p = 1.0 (a), 0.7 (b), and 0.4 (c). The numbers on the curves are the values of random anisotropy concentration t. However, we find clearly that the Curie temperature versus negative anisotropy turns into positive anisotropy when t < In this case, all sites of the lattice mainly occur with the positive anisotropy distribution of probabilities (1 t). In fact, the form of the phase transition lines for t < 0.34 is very similar to those discussed for t Certainly, the scope of the positive anisotropy is twice of negative anisotropy. It is reasonable since the magnitude of the positive anisotropy in all sites is only half of the negative anisotropy of probability t. Additionally, the second-order reentrant transitions exist in an appropriate range of randomness. A special type of reentrant transition line that is called zigzag one can be also observed in the phase diagram (see curves labeled with t = 0.734, 0.710, 0.290, and in Fig. 1(a)). The reentrant transitions, which may be caused by the competition between the anisotropy effect and the random overlap correlations, occur with the large values of positive or negative anisotropy. Figure 1(b) shows that the random anisotropy concentration scope that affects the TCP becomes large. The TCP exists in the ranges of 1.0 t > t 2 = at p = 0.7. This comes from the fact that the connectivity of the exchange interactions among all sites is weakened due to the introduction of bond dilution. Thus, the anisotropy correlation between sites will decline so that TCP is depressed of by a larger random anisotropy concentration. Secondly, the Curie temperature has been sharply decreased and the ordered phase can be strongly depressed. This is mainly due to the fact that the role of bond dilution will force some sites from S = +1 or S = 1 state to S = 0. If the system is always to maintain the ordered phase at low temperatures, it is necessary that there be a larger critical anisotropy concentration t 2 = to form the clusters of S = ±1 states. On the other hand, we notice that the change of transition lines is very obvious for t < 0.34 compared with t The transition lines for D + have completely vanished and their critical temperature extends to zero. This means that smaller positive anisotropy with probability (1 t) in sites is easily affected by bond dilution concentration p. In Fig. 1(c), we observe that the infinite cluster of S = ±1 disappears with further increasing bond dilution concentration whether the sites are present or absent with probability t or 1 t. The second-order transition temperature goes to zero at three values of anisotropy and one value if the random concentration t varies in the region of t 3 = t t 3 = 0.34 and 0.34 > t 0.223, respectively. That is to say that the degenerate patterns of anisotropy are decreased at ground state with further increasing bond dilution concentration. We also pay attention to the fact that the TCP can be depressed by concentration of bond dilution from comparison between Figs. 1(a) 1(c). From our calculation, the TCP can be discovered in the range of 1.0 p > p = , while the bond percolation threshold is p c = when t=1.0. The trajectory of the TCP at bond dilution status is in qualitative agreement with the result of MC simulations at higher temperatures, like Fig. 7 in Ref. [21]. So we can obtain a conclusion that the order of phase transition and other thermodynamic properties are closely related to two disorder factors.

5 No. 5 Ferromagnetic Properties of Bond-Dilution and Random Positive or Negative Uniaxial Anisotropy Blume Capel Model 793 Fig. 2 Uniaxial anisotropy dependencies of Curie temperature on the simple cubic lattice for α = +0.5 with different bond concentration p = 1.0 (a), 0.7 (b), 0.35 (c), (d). The numbers on the curves are the values of random anisotropy concentration t. Figures 2(a) 2(d) illustrate the behaviors of the phase transition as functions of anisotropy with bond dilution concentration p = 1.0, 0.7, 0.35, and for α = +0.5, respectively, when values of anisotropy concentration are changed. Figure 2(a) shows that there exists always the TCP for all ranges of random anisotropy concentration. Obviously, the difference between Figs. 1(a) and 2(a) is rather apprent. This means that although magnitudes of negative uniaxial anisotropy are random in all sites, in this case, the negative uniaxial anisotropy plays a key role in the occurrence of the TCP when α is positive. Moreover, the trajectory of the TCP displays an arc curve. Figure 2(b) also indicates the same feature except for the decrease of critical temperature and tricritical temperature. However, in Fig. 2(c), the phase diagram takes on some outstanding properties by changing random anisotropy concentration. We find that there is a line of first-order transition between two lines of continuous transition, separated by two TCPs (see curves labeled with t = 0.7, 0.6, 0.53, and 0.2). This phenomenon is not discovered in the previous works. We think that strong bond dilution makes the transition lines of single TCP crossover to those of double TCPs. Certainly, the curve labeled with t = 0.3 is of the second-order transition, while other curves have only one TCP. Hence, the transition lines of different species are affected by random anisotropy concentration t. On the other hand, we can observe the reentrant phenomena and two degenerate patterns of the anisotropy at ground state, respectively. For t = 1.0 the present system is equivalent to the bond dilution BCM, where all sites are of the value of same anisotropy. When p*=0.2954, in this case, the TCP is just depressed at all. However, the TCP appears in Fig. 2(d) again due to introduction of random anisotropy. The trajectory of the TCP dependence of the two disorder factors is of a complex relation and shows an apparent fluctuation. The double TCPs transition line can be also observed in Fig. 2(d). In order to further study the ferromagnetic properties of the disorder BCM, we plot the dependence of magnetization on the Curie temperature with a fixed negative anisotropy value D/J = 3.4, and let t = 0.6 for α = 0.5 in Fig. 3, while bond dilution concentration is changed. Here, magnetic curves increase firstly and then decrease rapidly to zero. In fact, the results shown in Fig. 3 reflect reentrant transition behavior corresponding to Fig. 1. Actually, in some other models reentrant behavior has been

6 794 ZHU Hai-Xia and YAN Shi-Lei Vol. 42 found for three-dimensional system. [22] Of course, we give a more detailed description of reentrant behavior. Finally, we need to repeat that sophisticated MC simulations leads to a correct tendency of phase transition in random BCM but does not predict many important results at low temperatures. These results display the competition between two disorder factors in bond and anisotropy. Fig. 3 Curie temperature dependencies of magnetization on the simple cubic lattice for α = 0.5 with a fixed random anisotropy concentration t=0.6 and a negative anisotropy value D/J = 3.4. The numbers on the curves are the values of bond dilution concentration p. 4 Summary So far, we have studied the ferromagnetic properties of three-dimensional bond dilution and random positive or negative anisotropy BCM on simple cubic lattice and given lots of meaningful results within the framework of the EFT. The trajectory of the TCP is in qualitative agreement with the estimates from the MC simulations at higher temperatures, if the BCM only considers bond dilution or random anisotropy, respectively. [21] We have obtained critical random anisotropy concentration and bond dilution concentration, respectively, at which the TCP just disappears. When random anisotropy and bond dilution are considered in the same time, the trajectory of the TCP shows new characteristics. In particular, a line of phase transition has double TCPs, i.e. the phase transition line present two lines of continuous transitions and line of first-order transitions, separated by two TCPs. The TCP takes on a complex fluctuation phenomenon by changing random anisotropy concentration for α = +0.5 under strong bond dilution condition, as depicted in Figs. 2(c) and 2(d). At low temperatures, the second-order phase transition exhibits many rich properties, which is not performed by MC simulations, such as reentrant phenomenon and zigzag phase transition. We have found that the anisotropy has different degenerate patterns at ground state and the patterns will decrease with increasing bond dilution concentration. On the other hand, the change of random anisotropy and bond dilution concentration for α = ±0.5 may lead to a large variation of the phase diagrams at low temperatures. It is very obvious that two different disorder distributions play a key role for ferromagnetic properties of the system described the BCM. The physical reasons with respect to some results have been analyzed in Sec. 3. Finally, we need to stress that the bond dilution and random positive or negative anisotropy BCM provides many valuable messages of phase transition and magnetization, which do not discover in other research. References [1] S.L. Yan and C.Z. Yang, Sol. Stat. Commun. 100 (1996) 851. [2] G.M. Buendia and E. Machado, Phys. Rev. B61 (2001) [3] I. Chatterjee, J. Magn. Magn. Mater. 265 (2003) 363. [4] I. Avgin and D.L. Huber, Phys. Rev. B66 (2002) [5] M. Blume, Phys. Rev. 141 (1966) 517; H.W. Capel, Physica 32 (1966) 966. [6] M. Blume, V.J. Emery, and R.B. Griffiths, Phys. Rev. A4 (1971) [7] A. Maritan, M. Cieplak, M.R. Swift, F. Toigo, and J.R. Banavar, Phys. Rev. Lett. 69 (1992) 221. [8] C. Buzano, A. Maritan, and A. Pelizzola, J. Phys. Condens. Matter 6 (1994) 327. [9] K. Hui and A.N. Berker, Phys. Rev. Lett. 62(1989) [10] G.M. Zhang and C.Z. Yang, Acta Phys. Sin. 42 (1993) 128 (in Chinese). [11] N.S. Branco and B.M. Boechat, Phys. Rev. B56 (1997) [12] D.P. Lara and J.A. Plascak, Physica A260 (1998) 443. [13] T. Kaneyoshi, Phys. Stat. Sol. (b)170 (1992) 313. [14] S.L. Yan and L.L. Deng, Physica A308 (2002) 301. [15] S.L. Yan and L.L. Deng, Commun. Theor. Phys. (Beijing, China) 39 (2003) 481. [16] A. Bobak, S. Mockovcial, and M. Jurcisin, Physica A230 (1996) 703. [17] C. Ekiz, M. Keskin, and O. Yalcin, Physica A293 (2001) 215. [18] S.G. Rollan, E. Kierlik, and M.L. Rosinberg, Phys. Rev. E63 (2001) [19] J.L. Monroe, Phys. Rev. E65 (2002) [20] A. Falikov and A.N. Berker, Phys. Rev. Lett. 76 (1996) [21] I. Puha and H.T. Diep, J. Magn. Magn. Mater. 224 (2001) 85. [22] N.S. Branco, Physica A232 (1996) 477.