Dynamic hysteresis of magnetic aggregates with non-integer dimension

ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 2429–2432

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Dynamic hysteresis of magnetic aggregates with non-integer dimension M.Y. Sun a,b, X. Chen a,b, S. Dong a,b, K.F. Wang a,b, J.-M. Liu a,b,c, a b c

Nanjing National Laboratory of Microstructures, Nanjing University, Nanjing 210093, China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China School of Physics, South China Normal University, Guangzhou 510006, China

a r t i c l e in fo

abstract

Article history: Received 27 September 2008 Received in revised form 29 December 2008 Available online 4 March 2009

We investigate the dynamic hysteresis of nanoscale magnetic aggregates by employing Monte Carlo simulation, based on Ising model in non-integer dimensional space. The diffusion-limited aggregation (DLA) model with adjustable sticking probability is used to generate magnetic aggregates with different fractal dimension D. It is revealed that the exponential scaling law A(H0, o)Ha0  ob, where A is the hysteresis area, H0 and o the amplitude and frequency of external magnetic field, applies to both the low-o and high-o regimes, while exponents a and b decrease with increasing D in the low-o regime and keep invariant in the high-o regime. A mean-field approach is developed to explain the simulated results. & 2009 Elsevier B.V. All rights reserved.

PACS: 75.60.Ej 75.10.Hk 75.40.Gb Keywords: Dynamic hysteresis Magnetic DLA Scaling exponent

1. Introduction When a spin system is swept by a time-oscillating magnetic field H(t), say, H ¼ H0  sin(ot), where H0 and o are the amplitude and frequency of field H and t the time, the system cannot respond instantaneously, and the magnetization M(t) gets delayed. In particular, if the relaxation time of the system is comparable to the time period of H(t), an interesting competition takes place, leading to a hysteresis if one plots M(t) against H(t), arising from the delay in response to H(t), which is a typical nonequilibrium phenomenon observable in a broad range of condensed matters [3]. To understand this hysteretic phenomenon, it is of interest to H study how the hysteresis area A M  dH, varies with o and H0, i.e. the dynamic hysteresis A(H0,o). The hysteresis area represents the energy exchange between the system and its environment for spin domain switching driven by H(t) in one period, and thus A(H0,o) contains the dynamic details of the first-order phase transitions associated with hysteresis sequence. In the last decades, theoretical and experimental efforts have been paid along this line, resulting in the power-law scaling of the dynamic hysteresis, i.e. A(H0,o)H0a  ob over both the low-o and high-o

 Corresponding author at: Nanjing National Laboratory of Microstructures, Nanjing University, Nanjing 210093, China. Tel.: +86 25 8359 6595; fax: +86 25 8359 5535. E-mail address: [email protected] (J.-M. Liu).

0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.02.133

regimes, where a and b are the scaling exponents [6–10]. It has been established that for an N-vector model, when N-N, the system can be solved numerically, and the exponents have value a ¼ 2/3 and b ¼ 1/3 in the low-o limit in two dimensional space [11]. For other models with discrete excitations, for instance, Ising model and its extensions, numerical simulations again confirm that a power law works well, with a0.47 and b0.40 for two dimensional Ising model in the low-o limit, and a ¼ 2 and b ¼ 1 in high-o regime [12]. In this work, we intend to investigate the dynamic hysteresis for magnetic systems of non-integer dimension D. In fact, it becomes well known that the dynamics of first-order phase transitions is essentially associated with the dimensionality under consideration [1]. It is also realized that a physical quantity such as magnetization is strongly dimension dependent. One of the most attractive phenomena is that many kinds of nano-scale magnetic materials synthesized by chemical approaches or thin film deposition may show spontaneously porous or well-developed branching structures [2]. In particular, those magnetic materials may find potential applications in a number of technological opportunities. One expects that the magnetic properties of these magnetic patterns exhibit significant deviation from those in dense or bulk materials, among which, a typical and important example is the dynamic hysteresis behavior. We will study the dynamic hysteresis of Ising model with fractal dimension D, using Monte Carlo simulation and mean-field approach. As the first step, we simplify the patterns as fractal clusters mapped by diffusion-limited aggregation (DLA) model,

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Fig. 1. (Online color) Simulation generated magnetic DLA clusters with different dimension D. (a) P ¼ 1, D ¼ 1.58, (b) P ¼ 0.1, D ¼ 1.75, (c) P ¼ 0.01, D ¼ 1.90.

and then focus on the scaling property of A(H0,o). We will show that the dynamic hysteresis of these non-integer dimensional fractal aggregates obeys the power-low scaling too in both high-o and low-o regimes, and with great interests, the two exponents, a and b, change with D in the low-o regime.

2. Model and Monte Carlo simulation We start from the DLA model [13] and magnetic DLA (MDLA) [14] model in two-dimensional (2d) square lattice because they are most similar to some actual fractal clusters [15]. Surely, an understanding of the problem may be more sufficient for a 3d simulation. Nevertheless, we pay more attention to a qualitative dependence of the dynamic hysteresis on D, and thus the space dimensionality is not a substantial issue. A DLA cluster is generated with a seed particle at the beginning, and new particles are added far from it separately through random walking sequence, and stick onto the seed/ occupied sites. The sticking probability of particle, p, is a control parameter for the value of D of generated cluster, leading to a higher D for a smaller p. For example, D1.6 if p ¼ 1.0 and 2.0 if p-0. A MDLA cluster, as its name suggests, is almost the same as a DLA one except that each particle has a value of spin. For details of the simulation, readers may refer to Refs. [11,12]. In Fig. 1 are shown three typical DLA clusters with different p. The corresponding value of D is labeled aside too. Each aggregate contains at least 6000 particles for a sufficiently reliable statistics. The color in the figure indicates the time sequence that particles join the cluster, and D is calculated using the spatial correlation function method [16]. The MDLA clusters are generated using the same algorithm as the DLA clusters and the only difference is that we assign each particle sticking to the cluster with an Ising spin in a random manner. For each MDLA cluster, our simulation on the dynamic hysteresis starts from the Ising model with the Hamiltonian X X si sj  H si , (1) E ¼ J hi;ji

i

where si ¼ 71 is the spin variable of the particle at site i, and J is the interaction between the nearest-neighboring spin-pairs. In our simulation, J is fixed to 1.0 and the time of simulation is measured in mcs (Monte Carlo step). The standard Metropolis algorithm is used to update the spin flip in our simulation, in which a random site is chosen for updating. The probability for flip of a spin is p ¼ exp(DE) in which DE is the energy change after the flipping. In the simulation process, we update the status of each spin in a random order, and change the external field over

Fig. 2. (Online color) Simulated hysteresis loops for (a) MDLA cluster of D ¼ 1.58 under various o at H0 ¼ 8.0, (b) MDLA cluster of D ¼ 1.58 under various H0 at o ¼ 0.1, (c) MDLA cluster of D ¼ 1.58 under o ¼ 5.6 and H0 ¼ 4.0, and (d) MDLA cluster of various D under o ¼ 0.1 and H0 ¼ 8.0.

time in order to obtain the magnetization data which form a hysteresis loop against field H(t). Here, we assume that the scaling law A(H0,o)H0a  ob remains valid for the objects of non-integer dimension. The simulated data presented below represent an averaging of many cycles of statistics on MDLA clusters with particle number over 6000 and generated by different seeds for random number generator. The magnetization and hysteresis are obtained using the standard procedure [17–19].

3. Results of simulation and discussion We first look at the simulated hysteresis loops at various f and H0, as shown in Fig. 2 for a typical example at D ¼ 1.58. In the low-o regime, with increasing o, the loop becomes fatter and extended due to the delayed response to H(t), as shown in Fig. 2(a). Fig. 2(b) shows the hysteresis loops obtained at different H0 for a given o. In the high-o regime, we observe loops far from saturated and one example is shown in Fig. 2(c). These results are quite similar to earlier ones on the Ising model with

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neighboring particles would dominate the dynamic hysteresis. For weaker spin interaction, the field driven spin flip becomes kinetically easier, leading to a smaller coercivity associated with the hysteresis, as demonstrated in Fig. 2(d). Thus, the dynamic hysteresis spectrum A(H0,o) favors certainly a more sensitive response to the variation of H(t). Obviously, a MDLA cluster of lower D would exhibit weaker spin interaction and thus larger exponents a and b, as revealed in our simulation (Table 1).

4. Mean-field approach

Fig. 3. (Online color) Simulated hysteresis area A(o) under various H0 for MDLA cluster of (a) D ¼ 1.58, (b) D ¼ 1.75, (c) D ¼ 1.92, and (d) D ¼ 2.0.

We further develop a mean-field approach (MFA) to explain some of the simulated results on the power-law scaling exponents a and b as a function of fractal dimension D, respectively. We will see that this MFA does not work well in the low-o regime while it works perfectly in the high-o regime. Given a kinetic Ising system whose Hamiltonian is given by Eq. (1) and a sinusoidal time-dependent external field H(t) ¼ H0  sin(2pot), one can deduce, according to the Glauber stochastic process [21], that the system evolves at a rate of 1/(2po) transitions (flips) per unit time, so the probability per unit time of flipping the ith spin at time t is [22,23] wi ¼

Table 1 Evaluated exponents a and b for MDLA of different D.

1 ð1  tanh Þ 4po

(2)

in which e ¼ zJM(t)+H(t). Hence, the temporal evolution of magnetization M(t), i.e. the mean value of spins over the whole cluster, is

D

1.58

1.75

1.90

2

b(o-0) b(o-+N)

0.7470.05 0.9770.02

0.6570.05 0.9770.02

0.6370.03 0.9770.02

0.4570.03 0.9770.02

a(o-0) a(o-+N)

0.9070.08 1.8570.10

0.7170.06 1.9370.10

0.6870.05 2.0270.10

0.4670.03 2.1570.10

D ¼ 2 and 3 [5,7]. We calculate A(H0,o) over a broad o range for different H0 and plot them against o, as shown in Fig. 3 for four types of MDLA clusters of different D. It is shown that given a H0, all of the A(o) curves exhibit the single-peaked pattern with perfect power-low dependence over both the low-o and high-o regimes, respectively. Also, the peak frequency shows clear blueshift with increasing H0, especially for the low-H0 cases. What stimulates us most is the fact that in the low-o regime, both a and b show significant variation upon different D. By fitting the curves to the power law, we obtain the exponents as presented in Table 1, with relatively small uncertainties. In the low-o regime, b increases with decreasing D, while in the high-o regime, all curves converge to a2.00 and b1.00 with subtle deviation. To understand this effect, we consider this hysteresis behavior as a result of the competition between the native randomness of MDLA and the spin interaction among particles. While the former leads to a situation that the numbers of the particles with both spin orientations are equal, that is, M0, the latter drives the system to a state in which all spins are universally oriented. As o is so large that the system can hardly respond, M is always close to zero. In this case, the spin randomness is significantly exhibited. Since the randomness has no essential difference for MDLA with different D, they share the identical a and b in the o-N limit, to be addressed again below in the mean-field approach. On the other hand, if o is not so high that the system can be sufficiently magnetized and instantly follow the variation of field H(t), all of the spins align in parallel in most time of the period. The demagnetization process from one direction to the other driven by opposite H(t) will be completed in a short interval of time. Therefore, in the low-o regime, the spin interaction among

dMðtÞ ¼ MðtÞ þ tanh½zJMðtÞ þ HðtÞ. dt

(3)

For the high-o regime, following the standard procedure given in Ref. [4], one can linearize Eq. (3) into: dMðtÞ ¼ HðtÞ þ ðzJ  1ÞMðtÞ dt

(4)

which has a special solution M(t) ¼ u  cos(ot) + v  sin(ot). Substituting this solution into Eq. (4) one obtains parameters u and v u¼ v¼

oH0 J 2 z2  2Jz þ 1 þ o2 ðzJ  1ÞH0 J 2 z2  2Jz þ 1 þ o2

, .

(5)

The hysteresis area then can be written as A¼ /

H20

I

o

MdH ¼ puH0 ¼

1

poH20 J 2 z2

 2Jz þ 1 þ o2

if o ! 1

(6)

which confirms the simulated a2.0 and b1.0 in the high-o regime. For the low-o limit, Eq. (3) is too rough to be functional. A general solution is possible by truncating the third-order term to Eq. (3), following the procedure given in previously [19,20] and yielding: dMðtÞ Jz ¼ ðJz  1ÞMðtÞ þ HðtÞ  M 3 ðtÞ. dt 3

(7)

2/3 , This nonlinear equation gives rise to a solution of AH2/3 0 o according to Ref. [18], for J ¼ 1, indicating no dependence of exponents a and b on parameter z and thus on dimension D. However, a numerical calculation without the third-order truncation, as presented in Fig. 4, still reveals a clear dependence of b on D and then on z, which is consistent with the simulated data in Table 1, while the MFA evaluated dependence of exponent

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on fractal MDLA clusters. We have demonstrated that in noninteger dimensional space, under both high-o and low-o regimes, the exponential scaling law A(H0,o)Ha0  ob works well. In high-o regime, exponents a and b remain invariant despite the variation of the MDLA dimension, while in low-o regime, both exponents decrease with increasing dimension D. A mean-field approach is then developed in attempt to explain this behavior with the success in the high-o regime. Acknowledgements The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (10674061) and the National Key Projects for Basic Researches of China (2006CB921802).

References Fig. 4. (Online color) Numerically calculated hysteresis area A(o) by Eq. (4) at H0 ¼ 8 for various values of dimension D. It is clearly shown that o-0 produces a larger b for a smaller D.

a on z is unfortunately weaker than the simulated one. This is because of the fact that in the above MFA, as Eq. (2) shows, parameter z is considered as a coefficient instead of exponent term, which allows the MFA only for a very rough approximation to the MDLA cluster that we actually study. In fact, for a MDLA of small D, there are a large fraction of spins located on the surface sites and they have much less number of neighbors than those sites inside the clusters. In the other words, the fluctuation in term of parameter z for MDLA is much bigger than that for a 2d cluster, leading to the partial failure of the present MFA. 5. Conclusion In conclusion, we have investigated the dynamic hysteresis of non-integer dimensional magnetic aggregates using Ising model

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