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.
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) [
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 [
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
The Hamiltonian of our model is written as
where Jij is the energy of spin-spin interaction which takes the values of J1,J0 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, Si = +1, or spin-down, Si = −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.
MMC has been used extensively in the researches of Ising models [
We denote the absolute value of average magnetization per spin as and susceptibility per spin of particle A as χA. In the cases of J0 = 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
So as shown in
meaning that the magnetic thermal stability of the particle accrues when J1 – J0 trails off.
What is related closely to the susceptibility is spin-spin correlation function defined as
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 [
where Aj = 1, 2, 3, 4 stand for the left, the right, the down, the up particles, respectively.
relationship with J0 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
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 Si in particle and is represented by its total magnetization MA with an orientation (say up or down). The fluctuation of magnetization D1 does not contain any information about the initial magnetic state of particle. Therefore instead of D1, we compute the fluctuation of magnetization D2, which defined as
where MA(0) is the initial total magnetization of particle
A. D2 describes the relative deviation of magnetization MA to its initial total magnetization. Note that here we initialize the system with Si(0) = 1 if site in particle A, otherwise Si(0) = –1. D2 = 0 means that MA = MA(0), while D2 = 2.0 means that MA = −MA(0). Thus
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 [
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 T0 based on the criterion D1 ≥ 0.1, i.e., D1 ≥ 0.1 if temperature T ≥ T0. 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
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.
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 J1 while that for spins in substrate material is J0. It has been found that the magnetic thermal stability of the nano-subsystem accrues with the increase of J1 and with the decrease of for a given J1, 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 J0 to make nanoparticle meeting both thermal fluctuation criterions D1 and D2 is more available than that coating nanoparticle by nonmagnetic substrate material (i.e., the case of J0 = 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).