Magnetization of Nano-Size Subsystem in a Two-Dimensional Ising Square Lattice ()

Liang-Ming He, Rong Wang, Cai-Liu Yin, San-Jun Peng, Dong Qian

College of Chemistry and Biological Engineering, Changsha University of Science and Technology, Changsha, China.

College of Chemistry and Chemical Engineering, Central South University, Changsha, China.

College of Science, Guangxi University for Nationalities, Nanning, China; Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning, China.

Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning, China.

**DOI: **10.4236/wjcmp.2012.24029
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College of Chemistry and Biological Engineering, Changsha University of Science and Technology, Changsha, China.

College of Chemistry and Chemical Engineering, Central South University, Changsha, China.

College of Science, Guangxi University for Nationalities, Nanning, China; Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning, China.

Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning, China.

A two-dimensional Ising square lattice is modeled as a nano-size block array to study by Monte Carlo simulation the magnetic thermal stability of nano-structure magnetic media for data storage, thereon in the blocks J_{1} > 0 is assigned for the interaction of a pair of nearest-neighbor spins, while 0 J_{0} J_{1} for that in regions between the blocks and (J_{0} + J_{1})/2 for the nearest-neighbor pairs with one in the block and the other one out of but near-most the block. We show that the magnetic thermal stability of the block accrues with the increase of J_{1} and with the decrease of J_{1} - J_{0} for a given J_{1}, but contrarily, the anchoring ability for the initial magnetic orientation in nano-size block trails off as J_{1} - J_{0} diminish. This phenomena and size dependence of such anchoring ability are discussed in detail.

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He, L. , Wang, R. , Yin, C. , Peng, S. and Qian, D. (2012) Magnetization of Nano-Size Subsystem in a Two-Dimensional Ising Square Lattice. *World Journal of Condensed Matter Physics*, **2**, 175-180. doi: 10.4236/wjcmp.2012.24029.

1. Introduction

Recently magnetic microstructure and nano-structure related to high-density magnetic recording, such as magnetic nanowires, chains, rings, planar arrays of magnetic nanoparticles and magnetic nano-structure thin film [1-3], is a topic of active research in material science. Practically, such microstructures are often imprinted on patterned substrate materials using lithography or beset in alloyed materials [4-6], which for practical applications are often imprinted using lithography on patterned substrate materials or doped with nanoparticles into alloyed ones [4-6]. That sophisticated techniques continue to be used to fabricate and characterize magnetic nano-structures assures the appearance of high density magnetic recording devices hereafter. Now, grains, as small as 7 - 8 nm, are used to store data at high signal-to-noise ratio (SNR) [7]. In the future, even smaller grains will be needed to store data at much higher areal densities. However, with the decreasing in size the particles will lose thermal stability, reaching the so called superparamagnetic limit, i.e., the magnetic orientation of particle at some size point no longer acts as magnetic bit due to thermal fluctuations [8,9]. Therefore the thermal stability in magnetic media for high density magnetic recording is still an open question.

In general, magnetic materials for data storage are composed of tiny but isolated magnetic nanocrystalline grains, but for magnetic nano-structure materials and nanoparticle materials, dipolar interactions between particles are unavoidable becoming relevant when increasing the packing density of the particles. This means that bit relevant is not a piddling issue in magnetic media for high density magnetic recording and the thermal stability of nanoparticle/unite of nano-structure materials must be considered in all of the system rather than as an isolator. Recently, a model of a two dimensional monodispersed array of single domain magnetic nanoparticles arranged in a square lattice was thus investigated to simulate thermal stability of magnetic media [10]. In that model each nanoparticle is a unite of the nano-structure material and is modeled as a giant magnetic moment, possessing saturated magnetism, that is, the magnetic moment interactions between particles and thermal fluctuation work only on the magnetic orientation, not changing the magnetic quantity of nanoparticle. Indeed, a nanoparticle is a cluster consisting of decades or hundreds of atoms/ions whose magnetic orientation ensemble is responsible for magnetization of nanoparticle. With the increase of temperature its magnetization will die away, reaching the superparamagnetic limit. Furthermore, the particles must further close up when continuing to heighten the density of magnetic recording, consequently here the interaction between particles is not taken as that of giant magnetic moments. Alike in bulk crystal, the interaction between two particles comes from the nearest-neighbor spin-spin interaction of atoms/ions respectively in their outer layers. If consider the influence of the atoms of substrate materials the correlation of two particles, based on the view of nearest-neighbor interaction, results from the interaction of their outer atoms with the atoms of substrate materials insulating them.

The Ising model has an enormous impact on modern physics in general and statistical physics, as well as in condensed matter physics for understanding the magnetism of materials [11,12]. In this paper, we provide a two-dimensional Ising square lattice which is modeled as a nano-size block array to study the magnetic thermal stability of nano-structure unite in magnetic media for data storage. As schematically delineated in Figure 1, in the blocks J_{1} > 0 is assigned for the interaction of a pair of nearest-neighbor spins, while for that in regions between the blocks and for the nearest-neighbor pairs with one in the block and the other one out of but near-most the block. Here J_{1} > 0 stipulates that the material in blocks/nanoparticles (acting as nanobits) is ferromagnetic, and stands for the magnetic nature of media separating the particles from each other or for the interaction of nearest-neighbor spins in grain boundaries when they being close very much. For a binary Ag-Ni alloy, commonly the pairwise potential between different species of Ag and Ni atoms is assumed to be a mathematical or geometrical average function of the monoatomic potential [13,14]. Analogously we hence take the interaction of the nearest-neighbor pairs with one in the block and the other one out of but near-most the block as.

Figure 1. Schematic description of the Ising lattice model which is partitioned into blocks with each coated by patterned substrate material.

The Hamiltonian of our model is written as

, (1)

where J_{ij} is the energy of spin-spin interaction which takes the values of J_{1}，J_{0} and, respectively corresponding to the above three types of nearestneighbor spin pairs. The spins can only be in one of two states, either spin-up, S_{i} = +1, or spin-down, S_{i} = −1. The notation restricts the sum to run over all nearestneighbor pairs. It is noted that periodic boundary condition is implemented in this model. This Hamiltonian does not contain any information about its temporal evolution. Nevertheless, one can obtain good estimates for the average value of an observable quantity such as magnetization, susceptibility, free energy and specific heat by applying Metropolis Monte Carlo (MMC) simulation.

2. Methods

MMC has been used extensively in the researches of Ising models [15], which are described by a simple recipe: 1) Choose an initial state; 2) choose a site i; 3) calculate the energy change ∆E which results if the spin at site i is overturned; 4) generate a random number r such that; 5) if, flip the spin; 6) go to the next site and go to (III). For convenience in this paper, all energy scales are in units of the Boltzmann constant, which is set to k_{B} = 1. Focusing on the magnetization of subsystem, we first investigate a Ising model which is of the size of 56 × 56 and partitioned into 4 × 4 blocks. These blocks are segregated from each other by patterned substrate material with the thickness of 4 sites and possess the size of 10 × 10. The cases of the model with other size will be discussed at the end of this paper. As an example, the block locating on the second row and second column, named as particle A in the following for convenience, is considered in the cases of J_{1} = 4.0 and. It is well-known that for ferromagnetic bulk material, i.e., J_{ij} > 0, there is a phase transition from a disordered (paramagnetic) high-temperature phase to an ordered (ferromagnetic) low-temperature phase, and a continuous but abrupt pick-up of the average magnetization per spin can be anticipated at a temperature T_{C}. In zero external field, the measured average magnetization would always be zero even for T < T_{C}, since a finite system which is necessarily performed to emulate the bulk material remains ergodic even for T < T_{C}. To surmount this puzzle, one of the simplest ways of emulating a system confined to a particular phase is to consider the average absolute magnetization per spin [11]. We found that when J_{1} ≥ 4.0 the time scaled by 10^{5} MMC steps is long enough for system relaxing towards its equilibrium from an arbitrary initial state, during which the global spin inversion of the equilibrium state has not been observed. Therefore the average magnetization is still an observable quantity measured in the above time when J_{1} ≥ 4.0. Furthermore it is a necessarily quantity to character nanoparticle in the case of nanoparticle acting as nanobit. As mentioned above, we note that here the simulation results of the average magnetization, in terms of symmetry breaking, are suitable for bulk material consisting of the interacting nanoparticles. Under the same initial condition where S_{i} takes the value randomly from +1 and −1, for the system at an arbitrary temperature T we perform MMC 10^{5} steps and sample the microscopic states after 5 × 10^{4} steps with the interval of 100 steps to obtain the average magnetization and susceptibility of particle A.

3. Result and Discussion

We denote the absolute value of average magnetization per spin as and susceptibility per spin of particle A as χ_{A}. In the cases of J_{0} = 0.0, 1.5, 3.0, 4.0, the functions of the absolute value of average magnetization per spin and susceptibility per spin of particle A with respective to temperature T are plotted together in Figure 2. It is found from Figure 2 that there exists a disordered (paramagnetic) high-temperature phase and an ordered (ferromagnetic) low-temperature phase. This can be interpreted by the free energy of particle A, , where S_{A} is its entropy. At relatively high temperatures, the case of, the term TS_{A} dominates so that the free energy F_{A} is minimized by maximizing the entropy, corresponding to paramagnetic phase where the spins orientate randomly; while at relatively low temperatures, the case of, the interaction energy dominates so that the free energy is minimized by minimizing the energy, corresponding to ferromagnetic phase where the spins orienttate to the same direction; and when the thermal energy TS_{A} and the interaction energy are comparable, i.e., when T ≈ T_{C}_{,} phase transition between ferromagnetic phase and paramagnetic phase occurs. Theoretically, as T approaches the critical temperature T_{C}, the susceptibility per spin χ diverges. Therefore the temperature corresponding to the peak value of the function of χ versus T can be roughly taken as the value of T_{C}. It is well known that the susceptibility per spin χ can be obtained from the variance of the magnetization of system, formulated as follow [11],

(2)

So as shown in Figure 2(b), that T_{C} increases with the decreasing of J_{1} – J_{0} illuminates that the temperature variance range for the particle (nanobit) in ferromagnetic phase becomes larger and larger as J_{1} – J_{0} decreases,

Figure 2. The functions of the absolute value of average magnetization per spin and susceptibility per spin χ_{A} of particle A vs. temperature T, in the cases of J_{1} = 4.0 and J_{0} = 0.0, 1.5, 3.0, 4.0, respectively, where N_{A} is the number of sites in particle A, and the increment of T is 0.5.

meaning that the magnetic thermal stability of the particle accrues when J_{1} – J_{0} trails off.

What is related closely to the susceptibility is spin-spin correlation function defined as

(3)

Based on Equation (3) the susceptibility χ can be written as

which is the analogue in percolation describing how site-site correlation function is related to the average cluster size or to the average correlation length [11]. It is known that for an infinite system, in the low temperature phase the correlation length increases with increasing of the temperature and diverges as the temperature approaches the critical temperature T_{C}; while for a finite system the correlation length reaches its peak value at the critical temperature T_{C}. To explore block-block correlation, we define the correlation of total spins of particle A with those of its four nearest-neighbor particles as

where A_{j} = 1, 2, 3, 4 stand for the left, the right, the down, the up particles, respectively. Figure 3 shows that the correlation of total spins of particle A with those of its four nearest-neighbor particles is a function of J_{0}. Comparing Figure 3 with Figure 2(b), one can see that at temperature T = 9.0, g(J_{0} = 3.0) > g(J_{0} = 1.5) > g(J_{0} = 0.0) accords with χ_{A}(J_{0} = 3.0) > χ_{A}(J_{0} = 1.5) > χ_{A}(J_{0} = 0.0), as well as g(J_{0} = 4.0) versus χ_{A}(J_{0} = 4.0). It accounts for that the magnetic thermal stability of the particle A having

Figure 3. The plot for the mean value of the correlation function g in the fluctuation of total spins of particle A and those of its four nearest-neighbor particles (blocks) vs. J_{0} at temperature T = 9.0, where J_{1} = 4.0.

relationship with J_{0} indicates the relevance of the particle A with the other particles, bridged by substrate material.

Further to understand magnetic thermal stability of the block, the fluctuation in magnetization of particle A,

with respect to temperature T are plotted on Figure 4. Here, D_{1} describes the relative deviation of magnetization M_{A} to its average value. Figure 4 shows that with increasing of temperature T the fluctuation of magnetization M_{A} becomes severe and particle A will lose its magnetization when it is in paramagnetic phase; further more, D_{1} augments in the range of 0.05 ≤ D_{1} ≤ 0.20 as reduces for a given J_{1} or increases with the increasing of J_{1}. Thus one may conclude that in fluctuation criterion of D_{1 }≤ D_{1C}, e.g., D_{1} ≤ 0.1, particles acting as nanobits can be used to storage data at higher temperature when is smaller than that when is large in the case of J_{1} being fixed.

However, in the magnetic media for data storage, what must be kept and be ensured not fluctuating intensity with the change of temperature T are particles’ initial magnetic states. After the finish of the writing information on magnetic media, i.e., with the applied field being switched off, the information bit has been recorded by the reversed spins S_{i} in particle and is represented by its total magnetization M_{A} with an orientation (say up or down). The fluctuation of magnetization D_{1} does not contain any information about the initial magnetic state of particle. Therefore instead of D_{1}, we compute the fluctuation of magnetization D_{2}, which defined as

where M_{A}(0) is the initial total magnetization of particle

Figure 4. The functions of the fluctuation in magnetization of particle A, D_{1}, with respect to temperature T in the cases of (J_{1}, J_{0}): (4.0, 0.0) , (4.0, 1.0) , (4.0, 1.5) , (4.0, 3.0), (4.0, 4.0) and (6.0, 1.0).

A. D_{2} describes the relative deviation of magnetization M_{A} to its initial total magnetization. Note that here we initialize the system with S_{i}(0)_{ }= 1 if site in particle A, otherwise S_{i}(0) = –1. D_{2} = 0 means that M_{A} = M_{A}(0), while D_{2} = 2.0 means that M_{A} = −M_{A}(0). Thus Figure 5 shows that in the cases of being large, say (J_{1}, J_{0}): (4.0, 0), (4, 1), (4, 1.5) and (6, 1), the initial magnetic state of particle A can be kept down when it is in feromagnetic phase, but the initial magnetic state will be lost with the spin in particle A S_{i} = –1 ≠ S_{i}(0) = 1 in the case of being small, e.g., (J_{1}, J_{0}): (4, 3) and (4, 4). For clarity, sixteen microstates of the system being in equilibrium states evolving from the initial state mentioned above are shown in Figure 6, corresponding respectively to different J_{1} and J_{0} at temperature T = 2, 4, 6, 12. It is consistent with Figure 5. Up to now we thus conclude that the magnetic thermal stability of the block accrues with the increase of J_{1} and with the decrease of for a given J_{1}, but contrarily, the ability for the initial magnetic orientation kept in nano-size block, named as anchoring ability of initial magnetization, trails off as diminishes.

The above discussions are based on the two dimensional Ising square lattice with size 56 × 56, in which the blocks with size 10 × 10 are segregated from each other by patterned substrate material with the thickness of 4 sites. With the decreasing of particle size the thermal fluctuations induce random flipping of the magnetic moment with time so that nanoparticles lose their stable magnetic order and become superparamagnetic. V. Skumryev and coworkers [16] had studied the systems where ferromagnetic cobalt nanoparticles of about 4 nm in diameter that are embedded in either a paramagnetic or an antiferromagnetic matrix. They found that the cobalt cores lose their magnetic moment at temperature being

Figure 5. The functions of the fluctuation in magnetization of particle A, D_{2}, with respect to temperature T in the cases of (J_{1}, J_{0}): (4.0, 0.0), (4.0, 1.0), (4.0, 1.5), (4.0, 3.0), (4.0, 4.0) and (6.0, 1.0), respectively.

Figure 6. Sixteen microstates characterizing the system in equilibrium states for temperature T = 2, 4, 6, 12 from left to right and for (J_{1}, J_{0}): (4.0, 3.0), (6.0, 1.0), (4.0, 1.0), (4.0, 0.1) from bottom to top, respectively. Sites containing upspins are white while sites containing down-spins are gray.

10 K in the first system but remain ferromagnetic up to about 290 K in the second. Similarly here for paramagnetic substrate material, we investigate theoretically the size dependence of temperature T_{0} based on the criterion D_{1} ≥ 0.1, i.e., D_{1} ≥ 0.1 if temperature T ≥ T_{0}. For this purpose, the two-dimensional Ising square lattices are set to be 200 × 200, which are partitioned by the blocks with size of 6 × 6, 16 × 16, 21 × 21, 36 × 36 and 46 × 46, respectively, segregated from each other by patterned substrate material with the thickness of 4 sites. From Figure 7 one can deduces 1) that for a given value of, T_{0} increases with increasing the particle’s size N_{A} and will reach its saturated value; 2) that for a given N_{A}, T_{0} becomes large as decreases if N_{A} is smaller and such change is not distinct anymore if N_{A} is larger. This

Figure 7. The size dependence of T_{0} larger than which when temperature T is, D_{1} ≥ 0.1 corresponding respectively to different values of J_{1} − J_{0}. The size of lattices are 200 × 200, thereon the size of blocks are N_{A} × N_{A} with N_{A} = 6, 16, 21, 36 and 46, respectively, segregated from each other by patterned substrate material with the thickness of 4 sites. Also, periodic boundary condition is implemented. The temperature increment for searching T_{0} is 0.2.

may be explained as following. Reckon the sites in patterned substrate material coating the particle as its shell, with the increasing of the size of particle, the surface sites (shell’s sites) to core sites (particle’s sites) ratio decreases and the effects induced by surface become weak, that is, the properties of particle are dominated by itself, thereby it behaves like bulk material disregard of the patterned substrate material and can be considered as a isolated bulk system.

4. Conclusions

Based on the pair interaction of nearest neighbor spins in a two-dimensional Ising square lattice, we have investigated the thermal stability of magnetization of ferromagnetic nano-size subsystems separated by paramagnetic substrate material. The interaction for spins in subsystem is denoted as J_{1} while that for spins in substrate material is J_{0}. It has been found that the magnetic thermal stability of the nano-subsystem accrues with the increase of J_{1} and with the decrease of for a given J_{1}, but contrarily, the anchoring ability for the initial magnetic orientation in nano-size subsystem trails off as diminishes. In addition, with increasing the subsystem’s size its magnetization behaves like that of bulk material, disregard of the patterned substrate material and being able to considered as a isolated bulk system. These results can be used to understand high-density magnetic nanoparticle material for data storage and superparamagnetic limit. Also, it may proposed some ways to enhance the storage density of recording magnetic media by reducing the size of particle: 1) Increasing the spin interaction in ferromagnetic nanoparticle, i.e., increasing; 2) For a fixed spin interaction in ferromagnetic nanoparticle, selecting suitable J_{0} to make nanoparticle meeting both thermal fluctuation criterions D_{1} and D_{2} is more available than that coating nanoparticle by nonmagnetic substrate material (i.e., the case of J_{0} = 0), for in the former case the nanoparticle can keep its initial magnetization state at a higher temperature.

The authors acknowledge the finance supports from the Department of Education of Guangxi (200911MS78), the Ministry of Education Science and Technology Key Project under Grant 210164 and Grants (HCIC201103) of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open fund, National Natural Science Foundation of China (Grant No. 21171174), and Hunan Provincial Science and Technology Projects (Grant No. 2011 GK3115).

Conflicts of Interest

The authors declare no conflicts of interest.

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