Surface effect on compensation and hysteretic behavior in surface/bulk

(Submitted, (2016))

Dans ce papier, nous nous sommes attachés à l’étude de l’effet de la surface sur les propriétés magnétiques et hystérétiques d’un nanocube avec la morphologie surface-volume en utilisant la simulation Monte Carlo. Nous avons rapporté les effets des champs magnétiques et cristallins ainsi que les couplages intermédiaires et le couplage de volume, la température et la taille sur le diagramme de phase, l’aimantation, la susceptibilité, les cycles d’hystérésis, la température critique et la température de compensation du modèle. Cette étude a démontré un certain nombre de comportements caractéristiques, tels que l’existence de comportement de type Q et

N dans la classification de Néel et aussi l’apparition des boucles d’hystérésis simples et triples

avec un nombre élevé de pas.

Surface effect on compensation and hysteretic behavior in surface/bulk nanocube

L. B. Drissi1,2,∗,S. Zriouel1_{, L. Bahmad}3

1- Lab-PHE, Modeling & Simulations, Faculty of Science, Mohammed V University, Rabat, Morocco 2- CPM, Centre of Physics and Mathematics,

Faculty of Science, Mohammed V University, Rabat, Morocco 3- Lab de Magn´etisme et PHE, Faculty of Science, Mohammed V

University, Rabat, Morocco and * Corresponding author: ldrissi@fsr.ac.ma

Abstract

Magnetic properties and hysteresis loops of a ferrimagnetic surface-bulk nanocube have been studied by the mean of Monte Carlo simulations. We have reported the effects of the magnetic and the crystal fields, as well as the intermediate and the bulk couplings, the temperature and the size on the phase diagram, the magnetization, the susceptibility, the hysteresis loops, the critical and the compensation temperatures of the model. The thermal dependence of the coercivity and the remanent magnetization is also discussed. This study shows a number of characteristic behaviors, such as the existence of Q- and N-types of compensation behaviors in the N´eel classification nomenclature and the occurrence of single and triple hysteresis loops with high number of step-like plateaus. The obtained results make ferrimagnetic surface/bulk nanocube useful for technological applications such as thermo-optical recording.

Keywords: Surface-bulk systems; Ferrimagnets; Monte Carlo calculations; Magnetic phases; Compensation and critical temperatures; Hysteresis cycles

I. INTRODUCTION

Recently, ferrimagnetic materials have attracted a lot of theoretical and experimental interest [1]-[5] and the refer- ences therein. The possible existence of a compensation temperature Tcomp, under certain conditions, makes the ferri- magnets promising for advanced permanent magnetic applications particularly high-density magneto-optical recording [6]. Tcompis the temperature below the critical temperature Tcrit, at which the magnetic moments located on different sublattices of the model counteract each other, giving a zero total moment [7].

A simple model exhibiting the ferrimagnetic behavior is the model composed of two sublattices with different spin directions. In these models, the magnetic reorientation phase transition (MRPT) is the result of the competing forces that favor different directions of the magnetization [8]. In [9], the study of a spin–1 two-sublattice ferrimagnetic monolayer on a simple square lattice confirms the existence of two magnetic phases in the ferrimagnets.

Surface-bulk provides another model of ferrimagnets. Several surface/bulk materials display a different magnetic behavior at the surface than in the bulk due to the different coordination number and the symmetry of the atoms at the surface [10]. Using the mean-field theory (MFT) and the Monte Carlo (MC) calculations, the occurrence of a higher critical temperature in the surface compared to the bulk is shown for the semi-infinite cubic Ising model [11]. By the mean of MC simulations, it is found that the finite size effects affect the magnetic properties of the surface and the bulk differently [13].

Besides the two-sublattice ferrimagnetic systems and the surface/bulk materials, other morphologies such as Ising core/shell nanostructures are used to study ferrimagnets. In [14], a model explaining the formation process of a single spin core-shell nanocrystal, using a reverse micellar system is presented using the MC technique and the post-core route. The coercive force increasing as a function of the antiferromagnetic shell thickness is the result obtained for a composite spin–1/2 core-shell nanoparticles [15]. The influence of a transverse field on the thermodynamic quantities of a cylindrical core/shell spin–1 Ising nanowire shows two distinct magnetic susceptibility peaks and some magnetic reversal events [16]. Within the Green’s function technique, the size and the anisotropy effects on the static and dynamic properties of ferromagnetic spherical nanoparticles with spin S = 2 are investigated in [17].

Mixed spin ferrimagnets have also attracted great interest because they exhibit many new phenomena which cannot be observed in their single spin counterparts. The study of the nonequilibrium properties [18] and the equilibrium behavior [19] of the spin (1/2, 1) Ising model shows that the phase diagram contains three phases separated by two continuous transition lines. In [20], MC simulations are used to examine the effect of the shell and the intermediate coupling on the magnetic properties and the thermodynamic quantities of a ferrimagnetic (1/2, 1) hexagonal nanorib- bon.

More complicated ferrimagnets with higher spin, as (1, 3/2) and (2, 5/2), have also been studied employing different techniques such as the mean-field approximation (MFA) based on Bogoliubov inequality for the Gibbs free energy [21], the effective-field theory (EFT) [22], the MC calculations [23, 24] and the Glauber stochastic dynamics [25, 26]. Up to now, the tailoring of nanostructured magnetic materials has reached a high level of sophistication due to the advanced development of the experimental techniques. Special interest is devoted to magnetic nanocubes, the ideal building blocks for the formation of three-dimensional 3D nanocrystals. Superlattices of iron nanocubes are synthesized via the thermal decomposition of relevant organometallic compounds for magnetic applications because their high magnetization and their adjustable anisotropy [27]. The shape-controlled synthesis and the self-assembly of FePt nanocubes are reported as an approach for single particle recording and for maximization of energy product in an exchange-spring nanocomposite system [28]. Cobalt nanoparticles with cubic shape are prepared by the wet- chemical processing [29]. In [30], single-crystalline La1−xBaxMnO3nanocubes with an adjustable Ba doping level are fabricated using the hydrothermal synthesis for producing colossal magnetoresistivity in mixed valence manganites.

In parallel, there are extensive theoretical studies modeling nanocube structures and showing some interesting magnetic results. Using the MC simulations, the (3/2, 1) core/shell ferrimagnetic nanoparticles on a simple cubic lattice show a dynamic phase transition from paramagnetic to a dynamically ordered phase in the presence of ultrafast switching fields [31]. This core-shell system defined on a body-centered-cubic lattice is also investigated in [32, 33]. The effects of the crystal field and the exchange interactions on the thermal and the magnetic properties of the system are reported in [32]. In [33], it is found that the nearest-neighbor interactions influence clearly the compensation behavior. This compensation temperature dependence of particle size is now known as a characteristic property of ferrimagnetic systems. The investigation of the hysteresis loops of a spin (1, 3/2) core/shell cubic nanowire in terms of the transverse field, the crystal field and the exchange interactions, in the framework of EFT, leads to triple, pentamerous and heptamerous hysteresis loops [34]. The study of a mixed spin (2, 5/2) ferrimagnet for a simple cubic lattice using MFA shows the possibility of many compensation points at low temperatures depending on the values of anisotropies [35].

As reported above, the ferrimagnetic materials in their different morphologies exhibit interesting magnetic phenom- ena beside the existence of the compensation temperature which is a characteristic property of these nanostructures. Previous works on systems having surface/bulk morphology shows that the magnetic properties of the surfaces differ

drastically from those of the bulk to which they are coupled leading to novel properties. However, to date there are no reports on the magnetic behavior of ferrimagnetic surface–bulk nanoparticles with regular cubic structure even if cubic geometry is the simplest model for the study of the critical phenomena and a majority of the important engineering materials crystallize in this geometry [36].

The purpose of the present work is to study the magnetic behavior of ferrimagnetic surface–bulk nanocube made of spins S =±3/2, ±1/2 ferromagnetic square surface above the ferromagnetic tetragonal spins σ = ±1, 0 bulk with an anti-ferromagnetic interface coupling. Using the Monte Carlo simulations, we determine the role of the interaction between surface and bulk and the effect of some relevant parameters such as the bulk coupling, the crystal and external magnetic fields, the size and the temperature on the occurrence of the compensation temperature and the hysteretic properties of the model.

This paper is organized as follows: in section 2, we present the model and describe the Monte Carlo simulations. In Section 3, we report the obtained results and discussions. In the last section we give our conclusion.

II. THE MODEL AND METHOD

In this work, we study the magnetic behavior of an Ising ferrimagnetic nanocube with surface–bulk morphology. The spin S ferromagnetic surface is coupled to a spin σ ferromagnetic bulk via a ferrimagnetic surface–bulk interface coupling (see Fig.1). The sites of the surface are occupied by spins S = 3

2 taking the spin values± 3

2 and±

1 2, while the spins σ = 1 take_{±1 and 0 values at each site of the bulk.}

The classical Ising model is described by the Hamiltonian:

H =−Js ∑ ⟨i,j⟩ SiSj− Jb ∑ ⟨k,l⟩ σkσl− Jint ∑ ⟨i,k⟩ Siσk− h ( ∑ i Si+∑ k σk ) − ∆ ( ∑ i Si2+ ∑ k σ2k ) , (1)

where Jb and Js are the couplings between the first nearest neighbor magnetic atoms with spin σk and σl at the bulk and spin Si and Sj at the surface respectively. Jint is the exchange-interaction parameter between the nearest- neighbor magnetic atoms one at the surface and the other at the bulk. ∆ and h represent the crystal-field interaction terms and the external longitudinal magnetic field, respectively. _{⟨· · · ⟩ denotes the nearest neighbor interactions on} the surface/bulk nanocube.

In the Monte Carlo simulations based on Metropolis algorithm [37], we apply periodic boundary conditions in the X and Y_{−directions. Free boundary conditions are applied in the Z−direction. Configurations are generated by} selecting the sites in sequence through the lattice and making single-spin-flip attempts, which are accepted or rejected according to the Metropolis algorithm. Starting from different initial conditions to get reliable results and independent configurations, we perform the average of each parameter and estimate the Monte Carlo simulations averaging over many configurations. Each site of the system is visited each time step. Our data are generated with 5× 104 _Monte Carlo steps per spin in lattice, discarding the 5× 103 _{initial Monte Carlo simulations. Our program [38] calculates} the following parameters, namely the internal energy per site defined as,

E = 1

N ⟨H⟩ , (2)

where N = Ns+ Nb is the total number of spins in the system and Ns and Nb denote the number of spins in the surface and the bulk, respectively.

The total magnetization M and its corresponding susceptibility χ are respectively given by: M =NsMs+ NbMb

N and χ = βN

(⟨

M2⟩− ⟨M⟩2), (3)

where Ms and Mb are defined as follows

Ms= 1 Ns Ns ∑ i=1 Si, and Mb= 1 Nb Nb ∑ k=1 σk, (4)

and β = _kB1_T, T is the absolute temperature and kB is the Boltzmann’s constant, we set kB= 1.

At the compensation point, the total magnetization must disappear. Subsequently the compensation temperature Tcomp is determined by the crossing point between the absolute values of the surface and the bulk magnetizations. Notice that Tcomp< Tcrit, where Tcrit is the critical temperature i.e. N´eel temperature [39]

III. RESULTS AND DISCUSSION

Using Monte Carlo simulations within the Ising model, we study the magnetic properties of a ferrimagnetic nanocube with a surface–bulk morphology. We start by calculating the ground state diagrams required to check the reliability of the theoretical results. Then, we show the existence of the compensation temperature Tcompand we examine the effect of the crystal field ∆, the bulk coupling Jb and the intermediate coupling Jint on Tcomp and the N´eel temperature Tcrit. We study also the influence of the set of parameters, namely Jint, Jb, ∆ and T , on the magnetic hysteresis curves.

A. Ground state diagrams

We study the ground state phase diagrams by minimizing the Hamiltonian to determine the spin states at zero temperature in the mixed S = ±3/2, ±1/2 and σ = ±1, 0 Ising model ferrimagnetic surface/bulk nanocube. In what follows, we report results for the model with fixed size N = 153_{. Later, we will examine the size-dependent} compensation behavior of the nanocube.

In Fig.2, we plot the ground state phase diagrams in three different planes, namely (a) in the( h JS,

Jint

JS

)

-plane, (b) in the plane(_J∆_S,Jint

JS

)

and (c) in the(_J∆_S,_Jh_S)-plane. In the plane (h

JS,

Jint

JS

)

, Fig.2-a shows four possible spin configurations for Jint

JS < 0. These phases are

( +3 2, +1 ) , ( −3 2,−1 ) ,(+3₂,−1)and(−3 2, +1 ) . For Jint

JS > 0 only the two configurations

(

+3₂, +1)and(−3 2,−1

)

are present. The energy equations of the four possible spin configurations and the conditions for the existence are reported in Table 1 for ∆ = 0 and Jb

JS = 0.25.

As plotted in Fig.2-b, in addition to the four phases described above in the plane(h JS,

Jint

JS

)

, four new phases arise in the plane (_J∆_S,Jint

JS

)

, namely, (+1₂, +1), (₋1₂,₋₁), (₋1₂, +1) and(+1₂,₋₁). These four new phases are present for Jint/Js> 0 as well as for Jint/Js_{≤ 0, provided that crystal field takes negative values. In Table 2, we summarize} the ground state energy equations and the conditions when the coupling in the border frustrates the bounds of the eight phases. In the plane(∆ JS, h JS ) , we find for ∆

JS > 0 only two configurations

( −3 2,−1 ) and (+3 2, +1 ) . However for ∆ JS < 0, the

two new phases(₋1 2,−1

)

and(+1 2, +1

)

are added leading to four possible spin configurations as listed in Table 3. We deduce that the ground state of surface/bulk nanocube depends on the values of the external magnetic field, the crystal field and the intermediate coupling.

B. Transition temperatures

In this section, we start by studying the temperature dependencies of the total magnetization M and its corre- sponding susceptibility χ of the ferrimagnetic surface/bulk nanocube.

Fig.3-a and 3-b represent M and χ as function of temperature T /JS for three different values of the antiferromagnetic intermediate coupling between surface and bulk spins Jint/JS with Jb/JS = 0.25 and ∆/JS = h/JS = 0. For the two specific values Jint/JS = _{−0.1 and −0.5, the magnetization curves undergoes two critical temperatures. The} first one corresponds to the compensation temperature Tcomp and the second one is the critical temperature. As depicted in Fig.3-a, the system presents a first order transition at Tcomp due to the jump discontinuity in the M curve. For T > Tcomp, the magnetization increases slowly taking negative values and vanishes at T = Tcrit. When increasing the absolute value of Jint/JS, the value of Tcomp increases while the critical temperature is not affected. For Jint/JS =_{−2.5, the total magnetization exhibits only one zero at the critical temperature of the system. The} style of total magnetization shifts from N−type behavior to Q−type behavior [7] with varying value of Jint/JS from −0.1 to −2.5.

This behavior is confirmed when examining the curves of the corresponding susceptibilities in Fig.3-b. For the two values of Jint/JS =_{−0.1 and −0.5, the curves of the total susceptibility χ exhibit two peaks. The first peak corre-} sponds to the compensation temperature Tcompand the second one is related to the critical temperature Tcrit of the system. The first peak , that is pronounced, occurs at a first-order phase transition from the ferrimagnetic phase to the antiferrimagnetic phase. When increasing the absolute value|Jint/JS| , this first peak shifts to higher values. In contrast, the second peak in χ curves is kept constant for all absolute values of Jint/JS smaller than 0.8. This second peak presents a second-order phase transition from the antiferrimagnetic phase at low temperature region to the

paramagnetic one at high temperature region. For the specific value Jint/JS =−2.5, the compensation temperature disappears and the χ curve exhibits only one peak that corresponds to the critical temperature.

In Fig.3-c and 3-d, we examine the effect of the bulk exchange coupling Jb/JS at Jint/JS =_{−0.5 and ∆/J}S = h/JS = 0. It is found that the magnetization curves corresponding to the values Jb/JS = 0.1 and 0.5 exhibit two vanishing points indicating the presence of a compensation and critical temperatures. The values of these two temperatures increase when increasing Jb/JS from 0.1 to 0.5. For higher exchange coupling value Jb/JS = 1 and 1.5, the magnetization curve exhibits only one vanishing point corresponding to the critical temperature of the system. The style of total magnetization shifts from N_{−type behavior to Q−type behavior with increasing value of J}b/JS from 0.1 to 1.5. The result obtained for M is manifest in the curve of the corresponding total susceptibilities in Fig.3-d. For Jb/JS = 0.1 and 0.5, the χ curve shows two peaks that are shifted to higher temperatures when increasing the exchange coupling Jb/JS, while only one peak, that increases in term of the parameter Jb/JS, is observed for the values Jb/JS = 1 and 1.5. To explain this result, we focus on how the temperature affects separately the behavior of the surface and bulk magnetizations for the bulk exchange coupling Jb/JS = 0.1, 0.5, 1 and 1.5 at Jint/JS =_−0.5. Fig.4-a and -b show that for Jb/JS= 0.1 and 0.5 where the compensation temperature exists, the bulk disorders before the surface. However, when Tcompdisappears for Jb/JS larger than 1, the surface and bulk disorder simultaneously as illustrated in Fig.4-c for Jb/JS = 1. When Jb/JS = 1.5, the surface disorders before the bulk (see Fig.4-d). We deduce that the bulk exchange interaction parameter plays a crucial role in controlling the disorder of the surface and/or bulk system. More explicitly, increasing the bulk exchange coupling Jb/JS favors the surface disorder compared to the bulk. However, the decreasing of Jb/JS leads to an inverse behavior.

In Fig.5, we study the effect of the intermediate coupling Jint/JS and the bulk exchange coupling Jb/JS on the occurrence of the compensation temperature in the absence of the crystal and the external fields: ∆/JS = h/JS = 0. At fixed value Jb/JS = 0.25, Fig.5-a shows two transition lines for |Jint/JS| < 0.8. In this region, the critical temperature of the model remains constant Tcrit= 3.04 while the compensation temperature increases linearly. The horizontal line Tcrit= 3.04 is the transition temperature of surface sublattice since the Jint/JS has no effect on Tcrit in this region and the contribution of the bulk to the transition temperature is very weak.

Up to the value_|Jint/JS_{| = 0.8, only one line transition is present corresponding to a second order transition separating} the ferrimagnetic phase, present at low temperature region from the paramagnetic phase at high temperature region. It is clear that the compensation temperature has completely disappeared in this region. Moreover, Tcrithas a linearly increasing behavior originating from the effect of the intermediate coupling that becomes dominant. This result is in good agreement with the one obtained for the semi-infinite systems [40].

In Fig.5-b, we fix Jint/JS =_{−0.5, and plot T}compand Tcritas function of the bulk exchange coupling Jb/JS. When increasing Jb/JS , the compensation temperature Tcomp increases linearly and gradually toward Tcrit. The difference between the two temperatures Tcrit and Tcomp decreases as Jb/JS increases until it vanishes at Jb/JS ≃ 0.8. This is due to the fact that it is more difficult for the spin lattice to become disordered when there is a strong interaction that tends to keep the bulk spins σ parallel. When the bulk exchange coupling Jb/JS is sufficiently large (Jb/JS > 0.8), the compensation point disappears, and the critical temperature system subsists. This result is in good agreement