The Size and Shape Dependence of Ferromagnetism in Nanomagnets

1 School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China 2 School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China 3 Inner Mongolia Key Laboratory for Utilization of Bayan Obo Multi-Metallic Resources, Elected State Key Laboratory, Inner Mongolia University of Science and Technology, Baotou 014010, China 4 College of Physics and Information Engineering, Henan Normal University, Xinxiang 453007, China


Introduction
The need for the increase of data-recording densities in the magnetic recording technique has driven the size of magnetic particles used down into the nanometer range [1,2].As the dimensions of magnetic materials decrease from the bulk size down to the nanometer scale, the magnetic properties will undergo a dramatic transition.The magnetism depends strongly on the shape and size of magnetic nanostructures [3][4][5][6][7][8][9][10]. It was well known that the continual decrease of the size of the magnetic nanoparticles will lead to the emergence of the superparamagnetic limit, below which the random magnetization reversal in particles will frequently occur, and consequently degrade the recorded information.On the other hand, an experimental study showed that the superparamagnetic limit of out-of-plane magnetized particles is strongly shape dependent, for example, oblong particles switch much more often than compact, almost circular particles of equal volume [9].The superparamagnetic limit is a key factor in the magnetic recording industry, and it directly determines the maximum achievable recording density.In order to extend the physical understanding of the superparamagnetic limit, it is essential to obtain a complete view of magnetic properties of nanomagnets through theoretical studies as well as experimental investigations [11][12][13].
In the present work, with the kinetic Monte Carlo method based on the transition state theory we study the dynamic behavior of magnetization reversal in the chain-shaped, oblong-shaped, and bulk-shaped nanomagnets with giant uniaxial anisotropy under the action of an external field at different temperatures.The aim of the work is first to provide a systematic understanding of the influence of shape on the magnetization reversal behavior and second to discuss the changes in coercive field with various spin array patterns of the system.Our study shows that magnetization reversal is much easier to occur in finite chain than in the oblongshaped and bulk-shaped systems.This indicates that the magnetization of the elongated particles is more sensitive to the applied field than that of the compact particles.Furthermore, it is found that the coercive field and magnetic

Model and Method
Consider the nanomagnets of size N = L x × L y × L z , where L x / = 1, L y = L z = 1 reduces to a one-dimensional magnetic chain, L x / = 1, L y / = 1, L z = 1 indicates a two-dimensional oblong-shaped system, and L x / = 1, L y / = 1, L z / = 1 is a threedimensional bulk-shaped system while the values of L x , L y , and L z are finite.In this paper the uniaxial axis of the system is labelled by the ±z direction and the single-ion anisotropy energy has a large value.An external magnetic field is applied along the easy axis.We use a Heisenberg model with an extremely large uniaxial magnetic anisotropy to describe the magnetic properties of the system.The Hamiltonian of the system can be written as where S i is the normalized spin variable at site i.In the first term, the sum runs over all the nearest-neighbor spin pairs with J being ferromagnetic exchange coupling constant.The second term denotes the uniaxial anisotropy energy with k u being the single-ion anisotropy constant.The last term is the coupling of spin magnetic moment to the applied field H, that is, Zeeman energy.The magnetization of the system is defined as m = (1/N) N i=1 S i , where N is the total number of all spin sites and • • • denotes the thermodynamical average.
Owing to the fact that each atom has a large magnetic anisotropy, two metastable states of a spin will prefer to orient along the easy axis.The spin variable S i can reduce to the s i , which takes values +1 and −1, corresponding to the spin orientation in +z and −z direction, respectively.We use an angle θ i to describe the angular deviation of the spin at site i between its current state and its initial state.A transition state begins at θ i = 0(cos θ i = 1) and ends at θ i = π(cos θ i = −1).From the Hamiltonian (1), a nonzero θ i will produce an energy increment where h i = (J j s j + μH)s i .The energy increment yields a transition-state barrier, [14][15][16].The reversal rate of the spin can be expressed as the Arrhenius law where k B is Boltzmann constant and T is temperature.For 2k u ≤ |h i |, the transition state barriers disappear for some spin reversal processes where we use the Glauber method [18] to treat the exponential factor of the rates, with the prefactor being kept [14].The expression of the rate implies that our spin processes are thermal activated.This scheme is justified for our simulation because dipolar interactions can be ignored [6,19] and quantum tunnelling comes into action at very low temperature only [20].By the kinetic Monte Carlo method, we carry out simulation on the magnetization responses to the applied field for the systems with chain, oblong, and bulk shape.The simulation uses a single-spin flip algorithm under free boundary conditions.Starting from an initial state with all spins s i = −1, we computed the local field for a randomly chosen spin and flipped the spin state according to the transition rates mentioned above.The spin flip may change the local field of the neighboring spins, thereby affecting the stability of other spins.The local fields of other spins are computed again and another spin is flipped.To reduce error each data point is averaged over at least 500 independent runs.
In our simulation, the exchange interaction between the nearest-neighbor spins is taken as J = 7 meV, and the anisotropy energy is k u = 0.3J = 2.1 meV/spin.Here, it is worth noting that the experimental studies have revealed that the magnetic anisotropy energy per Co atom can reach 2.0 meV in one-dimensional Co chains on Pt(997) surface [21], and the magnetic anisotropy energy per Fe atom is up to 1.6 meV in FePt nanoparticles embedded in Al [22].The larger the magnetic anisotropy energy, the smaller the critical particle size for stable magnetization at room temperature.So the nanomagnets with a giant magnetic anisotropy energy have been considered as the prime candidate for future data storage media application.The magnetic parameters such as J and k u remain unchanged for all systems unless specified otherwise.The magnetic field sweeping rate is taken as 132 T/s.We sweep the magnetic field, starting from a strong field H = −H 0 , at which all spin magnetic moments are aligned along the −z direction.The field strength is then gradually increased in increments of ΔH to +H 0 , followed by a decrease back to −H 0 .Thus, a magnetic field sweeping cycle is completed.

Results and Discussions
Figure 1 shows the magnetization curves of chain-shaped and oblong-shaped systems with fixed atoms and varying arrays, that is, 40 × 1, 20 × 2, 10 × 4, 8 × 5, at 10, 16, and 25 K. Clearly, the results from the chain L x × L y = 40 × 1, that is, the mostly inner hysteresis curves, are very different from those of oblong-shaped arrays.At a fixed temperature, the magnetization of spin arrays with a larger length-to-width ratio, that is, η = L x /L y , is more sensitive to the applied field so that the value of the coercive field of elongated arrays is smaller than that of nearly square-shaped arrays.On the other hand, the difference between the magnetization curves for different spin arrays changes with temperature.
In Figure 2, we plot the variation of the coercive field with temperature for spin arrays 40 × 1, 20 × 2, 10 × 4, 8 × 5.It is clear that the crossover temperature for chain-shaped arrays is smaller than that of the oblong-shaped arrays.The relationship between the coercive field and temperature can be described by a fit function where H c and T represent the coercive field and temperature and H 0 , T 0 , and g are fit parameters whose values are shown in Table 1.In addition, we further study the magnetization reversal properties for nanomagnets of chain shape and oblong shape including 60, 80, and 120 atoms.In Table 2, we give various spin arrays of N = 40, 60, 80, and 120 atoms, their lengthto-width ratio η, their coercive fields H c (T) at T = 25 K, and standard deviations of the coercive fields.We can see that the coercive fields have a decreasing tendency with increasing η for arrays with a fixed width and varying length.This implies that the elongated particles switch more easily than the nearly square-shaped particles, which is in agreement with the results from the literature [9].Additionally, for the system with a fixed number of total spins, the coercive field of the compact-shaped array is large compared with that of the oblong-shaped array, but the coercive field exhibits a nonmonotonic increase with η decreasing, as shown in Table 2.We think that the slight decrease of the coercive field with η should be induced by standard deviations, which are given in Table 2.
In order to gain a systematical understanding of the effects of the shape (or spin array pattern) of nanomagnets on the magnetization reversal, we also study the magnetic properties of bulk-shaped systems with various spin arrays Bringing together the results from the chain-shaped, oblong-shaped, and bulkshaped systems, we found that at a fixed temperature the coercive field is closely dependent on the shape (or spin array pattern) of the system.The dependence relation can be explained in the following way.For the systems comprising of the same atoms, the transitions both from chain to oblong shape and from oblong to bulk shape lead to an increase in the number of effective nearest neighbors for each spin, which in turn raises the coercive field.Here, the number of effective nearest neighbors per spin, also called the effective coordination number Z eff , is defined as twice the ratio of  3).The corresponding fit parameters are listed in Table 1 (see text).

Journal of Nanomaterials
Table 2: The various chain-shaped and oblong-shaped spin arrays, their ratio of length to width η, the coercive field at T = 25 K, and standard deviations of the coercive field.6 in chain-shaped, oblong-shaped, and bulk-shaped systems.For bulk-shaped systems, if we treat those spins having the largest coordination number as the inner spins (or interior atoms) and other spins as outer spins (or surface atoms), Z eff is in some sense analogous but inversely proportional to the surface-to-volume ratio of nanomagnets [11].Taking N = 120 for example, in Table 3 we display the values of Z eff for various spin array patterns corresponding to chainshaped, oblong-shaped, and bulk-shaped systems.
In Figure 3 we show the coercive field as a function of Z eff for bulk-shaped arrays L x ×L y ×L z = 30×2×2, 20×3×2, 15× 4×2, 5×12×2, 6×10×2, 10×4×3, 5×8×3, 5×4×6 of N = 120 in the cases of J = 7 meV and J = 3 meV at 25 K.The anisotropy energy remains unchanged in both cases.From Figure 3 it follows that the coercive field increases with Z eff and becomes small for relatively small exchange interaction J.The Z eff dependence of H c can be described by the function expression where H c0 and Z eff0 represent the fit parameters and λ is a minus power exponent.We also find that the relation between the coercive field and Z eff is also true for those systems with larger anisotropy or smaller exchange coupling.The coercive field increases with increasing the anisotropy energy, but when the anisotropy energy increases to the extent to which the transition between two metastable states involves transition states with barriers only, the influence of Z eff on the coercive field becomes small.3. The solid line is fit to data by (4), as discussed in the text.
For the purpose of comparison, in Figure 4 we display the coercive field as a function of Z eff for chain-shaped, oblongshaped, and bulk-shaped systems containing the same atoms N = 120 with J = 7 meV.The solid line is fit to data by (4).The values of Z eff correspond to those of Table 3.It can be seen that the coercive field tends to increase with Z eff increasing.This indicates that the coercivity or the magnetic order transition temperature of nanomagnets is closely related to the effective coordination number.
For the case of bulk-shaped nanomagnets, our simulated results are qualitatively consistent with the experimental ones [10], where the order-disorder transition temperature of oblong nanoparticles (with 4 and 1.5 nm for the size and the thickness) was found to be lower than that of spherical nanoparticles (with 3 nm for the size).Besides, our results are also in qualitative agreement with the theoretical ones, where the order transition temperature decreases with the increase of surface-to-volume ratio for nanodots, rod, plate, and icosahedra [11].

Conclusions
In conclusion, we have investigated the size and shape dependence of magnetic dynamic behaviors in chain-shaped, oblong-shaped, and bulk-shaped nanomagnets.The coercive fields are found to be strongly dependent on the size and shape (or spin array patterns) of the nanomagnets.The magnetization of the elongated array is more sensitive to the applied field than the nearly square-shaped arrays and bulkshaped arrays.We propose that the coercive field is closely associated with the number of effective nearest neighbors for each spin (or effective coordination number), which has different values corresponding to different systems such as chain-shaped, oblong-shaped, and bulk-shaped systems.

Figure 4 :
Figure 4: The coercive field as a function of Z eff at 25 K for chainshaped, oblong-shaped, and bulk-shaped systems including equal atoms N = 120 with J = 7 meV.The values of Z eff are shown in Table3.The solid line is fit to data by (4), as discussed in the text.

Table 1 :
Parameters corresponding to the fit curves in Figure2.

Table 3 :
The Z eff for chain-shaped, oblong-shaped, and bulk-shaped systems.