Energy contributions in magnetite nanoparticles: computation of magnetic phase diagram, theory, and simulation
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(2) 7116. Magnetite, with molecular formula Fe3O4, is an interesting material belonging to the family of ferrites; compounds that exhibit complex and useful properties. Ferrites adopt spinel crystal structure with ferrimagnetic ordering (Cibert et al. 2005; Néel 1948). In this structure two ferromagnetically ordered sublattices are oriented antiparallel leading to a ferrimagnetic state below Curie temperature (about 860 K for magnetite). Owing to its complex structure and magnetic interaction scheme (Cibert et al. 2005), magnetite exhibits exotic magnetic and transport properties. These phenomena has attracted considerable attention at the field of study in electronic, magnetic, and structural characterization of bulk samples (Bimbi et al. 2008; Garcı́a and Subı́as 2004; Jeng et al. 2004; Klotz et al. 2008; Leonov et al. 2006; McQueeney et al. 2007; Nazarenko et al. 2006; Piekarz et al. 2006; Rozenberg et al. 2006; Schlappa et al. 2008; Wright et al. 2001). Furthermore, the structure and properties of fine particles attract interest of technological applications, motivated by the challenge of discovering and understanding some phenomena related to size and surface effects for applications in: spintronics (Arisi et al. 2007), new ferrofluids (Klokkenburg et al. 2006), diagnosis and cancer treatment in health sciences (Kim et al. 2006; Park et al. 2008), functionalized nanoparticles (Latham and Williams 2008) and magnetoelectronic devices (Zeng et al. 2006), among others, involving topologies where finite size effects play a crucial role. Many interesting studies focused on the electronic, structural, and magnetic properties of nanostructured magnetite, have been reported recently (Jeng and Guo 2002; Klokkenburg et al. 2007; Leonov et al. 2006; Wenzel and Steinle-Neumann 2007). The interplay between stoichiometry, particle size, and magnetic properties, has been investigated for magnetite nanoparticles by using MC simulations in the framework of a core–shell model (Mazo-Zuluaga et al. 2008). In that work, it was found that critical temperature is smaller than the one of bulk magnetite due to the presence of dangling bonds on the surface. Also, it was found that there is a diameter around 5 nm, below which offstoichiometry is more noticeable, and above which nanoparticles can be considered highly stoichiometric. Another study explored the effects of different coating environments (Mazo-Zuluaga et al. 2008a). Coatings on the nanoparticles led to different surfaceanisotropy values, consequently, to different magnetic. 123. J Nanopart Res (2011) 13:7115–7125. spin structures. The zero-field magnetic behavior of magnetite fine particles at the low-temperature regime indicates that the resulting spin structure strongly depends on the sign and magnitude of the surfaceanisotropy/bulk-anisotropy ratio. For positive values of this ratio, the magnetization decreases slightly and a ‘‘throttled’’ state, characterized by canted spins at surface, is obtained. Above certain critical surfaceanisotropy value, magnetization almost vanishes and a quasi-radial ‘‘hedgehog’’ state emerges. For negative and extremely high ratios, a two-pole (‘‘artichokelike’’) configuration is reached, in accordance with a distribution of easy planes on the surface. The effect of the surface-anisotropy on the hysteresis and exchange bias properties of different particle diameter, as well as the temperature dependences of the coercive force and magnetic susceptibility, have also been considered in a previous study (MazoZuluaga et al. 2009). Results of this study suggest that the magnitude of the surface-anisotropy constant might act as a driving force leading to strong surface spin pinning. The exchange bias behavior obtained can be attributed to the tendency of the surface spins to be more or less compensated and to the different ratio of octahedral to tetrahedral spins. Therefore, taking into account possible future technological applications; like magnetic sensors or magnetic random access memories MRAM’s, magnetization values and magnetic configurations could be controlled via synthesis routes allowing control on the size and surfaceanisotropy, depending on the coating material. In order to use the power of first-principles methods, a study of electronic structure calculations for small clusters built from magnetite structure has also been addressed (López et al. 2009). In this case, initial structures corresponding to two different space groups (cubic Fd3m and monoclinic P2/c symmetries) were considered, and the relaxation process of the atoms for these very small samples (size ranged between 33 and 113 atoms, corresponding to length scales between around 7 and 12 Å in diameter) was allowed. At these sizes, it was observed that the discrete character of the energy levels becomes dominant; and a charge transfer process between Fe and O ions, featured by an increase in the electronic charge of Fe cations on the surface, is observed. The structural analysis, based on the computation of radial and angular distribution functions, shows peaks broadening that indicates strong.
(3) J Nanopart Res (2011) 13:7115–7125. modifications of the starting crystalline structures by atomic displacements with a tendency to form onionlike structures. Despite the increasingly importance of all these features regarding surface effects on the electronic and magnetic properties of magnetite, to the best of our knowledge, no analytical expressions for energy of the different magnetic states obtained in a magnetite nanoparticle have yet been published. Also, it is important to consider systems of increasingly large sizes, but the study of big systems is limited by the computational power. An alternative approach is to develop an analytical framework that allows to expand the analysis to system sizes beyond the current computational facilities. Therefore, the main goal of this article is to present a theoretical study, and make comparisons with MC simulation results, on magnetite fine particles in order to contribute to the understanding of the magnetic transition processes taking place at nanometric scale, where effects of surface-anisotropy become dominant. The approach presented here might be useful to confirm the spin-structure proposed by MC simulations (Mazo-Zuluaga et al. 2008a, 2009). It is also possible to think about it as a guide for computing the magnetic phase diagram and studying analytically other magnetic compounds by taking into account the respective anisotropy features and spin arrangements belonging to the material of interest.. 7117. Magnetic moments are represented by classical Heisenberg spins Si. Their magnitudes have been considered by taking into account their respective oxidation states; therefore, their corresponding electronic configurations are: 2.5 for Fe3? and 2 for Fe2?. Since on B sites there are a mixture of Fe3? and Fe2? ions the observed average spin value for this sublattice is 2.25 (Degrave et al. 1993). Oxygen ions were considered as non-magnetic since they only play the role of intermediary in superexchange interactions. Spins interact via antiferromagnetic couplings when considering 3? 3? 3? the following bonds: Fe3? A - FeA , FeA - FeB , and 3? 2? FeA - FeB . The remaining couplings are considered 3? 3? 2? 2? ferromagnetic: Fe3? B - FeB , FeB - FeB , and FeB - Fe2? B . Numerical values of the integrals employed were JAA = - 0.11 meV, JBB = ? 0.63 meV, and JAB = JBA = - 2.92 meV, which were fitted by keeping constant the ratios between exchange integrals from a first principles study in the framework of the local spin density approximation (LSDA) (Uhl and Siberchicot 1995). Hence, the greater magnitude and sign of the inter-sublattice integral JAB gives rise to the anti-parallel inter-sublattice alignment. This fact, in addition to the differences in the spin values, explain the ferrimagnetic behavior observed in bulk magnetite below the Curie temperature. A classical Heisenberg Hamiltonian has been used to describe our system: X X H ¼ 2 Jij Si Sj KV ðS2x;i S2y;i þ S2y;i S2z;i hi;ji. þ S2x;i S2z;i Þ KS. X i 2 ðSk ek Þ :. ð1Þ. k. Computation details In our model, we simulate spherical magnetite samples in the core–shell scheme (Kryka et al. 2010) by implementing the inverse spinel crystalline structure of magnetite (Fe3O4) with Fd3m symmetry. This structure involves a total number of 56 ions per unit cell located as follows (Cibert et al. 2005; Cornell and Schwertmann 2003): 32 O2-ions, 8 Fe3? ions in tetrahedral sites (A-sites), and 8 Fe2? plus 8 Fe3? ions equally distributed in octahedral sites (B-sites). The iron cations belonging to B-sites are responsible for the non-resolved sextet observed by Mössbauer spectroscopy above the Verwey temperature. This sextet has been interpreted as corresponding to a Fe2.5? mixed valence state (Zhou and Ceder 2010) and is understood as a consequence of an electron hopping mechanism between Fe2? and Fe3? (Degrave et al. 1993).. The first sum involves exchange interactions of iron ions and runs over the nearest magnetic neighbors. The second term in Eq. 1 is the anisotropy energy, and KV (=0.002 meV/spin) is the first-order cubic magnetocrystalline anisotropy constant (Goya et al. 2003). The third term describes the surface-anisotropy. Easy axes at surface arise wherever the surface-anisotropy is strong enough to deform the collinear ferrimagnetic configuration. Then, the unitary vector ek is computed on every kth position Pk, by taking into account the positions Pj of the nearest magnetic neighbors (Kodama and Berkowitz 1999): P j ðPk Pj Þ : ð2Þ ek ¼ P j ðPk Pj Þ Positions over which these vectors were computed correspond exclusively to Fe-cation sites on the. 123.
(4) 7118. surface. Cubic magnetocrystalline anisotropy is applied to core spins and the surface-anisotropy constant is taken as variable parameter and represented by KS. The different surface effects (including for example: (i) surface relaxation due to lower iron coordination, (ii) size, shape and strain effects, (iii) coatings environments, (iv) spin canting at surface, among others) can change both the atomic positions and the magnitudes of interaction between ions at surface. Vectors ek are useful for modeling anisotropy effects due to changes in the positions of the ions, whereas variations on the KS parameter describe changes in the magnitudes of interaction. Therefore, the use of a variable KS parameter, along with the variable ek surface vectors, is useful to describe the wide phenomenology and different conditions taking place at surface. Estimations of the different energies involved, including dipole-dipole interactions over the entire volume, were initially performed on system sizes around 5 nm in diameter. These estimations led to the following orders of magnitude: *103 meV/spin for superexchange interactions, *10-2 meV/spin for surface-anisotropy, *10-4 meV/spin for cubic anisotropy, and *10-5 meV/spin for dipole-dipole interactions. Therefore, dipolar interactions were neglected in this study (Kachkachi and Dimian 2002). In order to compute equilibrium averages, we have employed a single-spin movement Metropolis-Monte Carlo dynamics, in which spin directions (h, /) are chosen randomly (unrestricted classical angular dynamics). A maximum number of 7 9 105 Monte Carlo steps per spin (MCS) were used and the first 3 9 105 steps were discarded for equilibration. The lowest temperature considered was 10 K and no special considerations were taken into account for temperatures below the Verwey temperature, TV, known to be around 120 K for bulk magnetite. This fact has been suggested by several experimental studies in which the suppression of the Verwey transition takes place for particle sizes below 20 nm (Goya et al. 2003; Lima et al. 2006; Park et al. 2008). For those results showing the dependence on the KS value at 10 K, the maximum number of MCS was 5 9 105 and the first 2 9 105 steps were discarded. Free boundary conditions were implemented in order to obtain particles of different sizes in a range 2–7 nm in diameter.. 123. J Nanopart Res (2011) 13:7115–7125. Analytical framework In order to study the KS-driven phase transition from an analytical perspective, we bring to discussion the result presented in Mazo-Zuluaga et al. (2008a), which shows that, for positive values of surface-anisotropy, there are two different kind of ground states separated by an energy barrier: (i) a ‘‘throttled’’ configuration with reduced magnetization obtained at intermediate values of KS, and (ii) a quasi-radial or ‘‘hedgehog’’ configuration with nearly null magnetization in the high surface-anisotropy regime. For negative KS, which corresponds to an easy-plane anisotropy at surface, whenever the |KS| value is increased, the collinear ferrimagnetic state becomes deformed into a two-pole configuration (also known as an ‘‘artichokelike’’ state), where the spins tend to lie tangentially on the surface (Mazo-Zuluaga et al. 2008a). Here, we focus our analysis on the positive KS regime and consider the ferrimagnetic-throttledhedgehog transition. So far, no analytical expression is known for the throttled configuration. Spin configuration for this intermediate throttled state is very difficult to express in an analytical way. Thus, as one theoretical simplification to face this problem this state is not taken into account in this study, and we consider here ferrimagnetic (F) and hedgehog (H) configurations as initial and final states. The quality of this approximation is tested by comparisons with MC results, where the ferri-throttled-hedgehog process can be followed in detail. This issue is discussed below. Then, we base our approximate analysis on equating energy for ferrimagnetic (EF) and hedgehog states (EH) at the transition point (as function of size and for positive KS values): EF ¼ E H. ð3Þ. Energy can be computed as a contribution of core anisotropy EC, surface-anisotropy ES, and exchange energy EX, which, in turn, has also core (C) and surface (S) contributions: EX = EXC ? EXS. Then, Eq. 3 reads: ECF þ ESF þ EXF ¼ ECH þ ESH þ EXH. ð4Þ. At each R value we can solve this equation for KS, obtaining thus the phase diagram for the transition. In what follows we will outline the procedure for obtaining each term in Eq. 4..
(5) J Nanopart Res (2011) 13:7115–7125. 7119. In our model, we consider spherical magnetite particles with N = NA ? NB iron atoms, where NA and NB indicate number of ions belonging to tetrahedral A and octahedral B sublattices, respectively. According to the magnetite stoichiometry, the relationship NB = 2NA holds. The total number of iron atoms as function of particle radius R is computed as N ¼ 24ð4=3ÞpR3 a3. ; where 24 is the number of Fe ions per unit cell and a ¼ 8:396 Å represents the lattice parameter for magnetite. The nominal coordination numbers under bulk conditions are: zAA = 4, zBB = zBA = 6, and zAB = 12. The expression zAB = 12 indicates that a FeA ion belonging to the core has twelve iron ions belonging to B-sites as its nearest neighbors. These numbers apply for the core, whereas the surface is defined as a shell formed by those NS = NAS ? NBS iron ions having smaller coordination numbers (see Fig. 1). As consequence of the breaking of crystal symmetry at the surface, there are some missing atoms, a reduction on the coordination number is obtained, and the surface-anisotropy appears. Thus, mean coordination numbers at surface were computed by counting the average number of neighbors for different nanoparticle sizes, yielding hzAA i ¼ 2; hzBB i ¼ hzBA i ¼ 4; and hzAB i ¼ 7: By assuming the mentioned definition for surface, number of atoms belonging to the core (C) and the surface (S) for A sublattice can be computed, respectively, as: NCA ¼ NSA. 32p ðR dAA Þ3 : 3 a3. 32p R3 ðR dAA Þ3 ¼ : 3 a3. ð5Þ. Similarly, number of atoms at C and S for B sublattice are, respectively, found as: NCB ¼. 64p ðR dAB Þ3 : 3 a3. ð7Þ. NSB ¼. 64p R3 ðR dAB Þ3 : 3 a3. ð8Þ. Here, dAA and dAB are the distances for a FeA ion with its nearest neighbors of type A and B, respectively. These parameters are used to define the surface, and take the following values: dAA ¼ 3:636 Å and dAB ¼ dBA ¼ 3:481 Å: The following paragraphs describe the procedure to evaluate all the energy terms involved in this problem (Eq. 4). In the F configuration core spins are oriented along easy axes, which correspond to diagonal directions on the cubic cell. Thus, cos2 a cos2 b þ cos2 b cos2 c þ cos2 c cos2 a ¼ 1=3; and we get: ECF ¼. ðNAC þ NBC ÞKV : 3. ð9Þ. In the surface region, easy axes are oriented along quasi-radial directions, and spins are pointing to directions of easy axes at core, then the contribution of anisotropy for spins at radius q on the surface (with surface density r = dNS/dA) should be evaluated by considering the sum: ESF ¼. NS X. KS cos2 hi. i¼1. ð6Þ. ¼. Z2p Zp. ð10Þ 2. 2. rq sin hKS cos hdhd/: 0. 0. where hi represents the angle between the easy axis at core and the magnetic moment at site i. With a uniform distribution on the surface we can write r = NS/(4pq2), and the result becomes: ESF ¼ . 1 NAS S2A þ NBS S2B KS : 3. ð11Þ. Since the F state implies collinear arrangement of magnetic moments, the exchange energy for neighboring spins at core (EXCF) can be computed as: Fig. 1 (Color on-line). Schematic view of a nanoparticle of radius R showing a generic surface shell determined by the value RC = R - dAA for A-sublattice or RC = R - dBA for B ions. 1 NAC JAA S2A zAA JAB SA SB zAB zAB 2 þ NBC JBA SA SB zBA zBA þ JBB S2B zBB ð12Þ. EXCF ¼ . 123.
(6) 7120. J Nanopart Res (2011) 13:7115–7125. At the surface, due to the crystal symmetry breaking, a reduction on the coordination number is observed, therefore, an average number of nearest neighbors is considered to obtain: 1 NAS JAA S2A hzAA i JAB SA SB hzAB i 2 þ NBS JBA SA SB hzBA i þ JBB S2B hzBB i ð13Þ. EXSF ¼ . Once the surface-anisotropy is big enough, and the transition is overcome, in the hedgehog configuration, spins are oriented along quasi-radial directions. Thus, energy can be approximated by assuming that the spin at site i is given by: Si ¼ jSi j. In this H configuration, energy terms can be evaluated as indicated in the following. For surface spins, which are oriented along radial easy axes, anisotropy contribution can be expressed as: ESH ¼ NAS S2A þ NBS S2B KS : ð14Þ For the anisotropy energy at the core, we take into account the term: X ECH ¼ KV ðS2x;i S2y;i þ S2y;i S2z;i þ S2x;i S2z;i Þ i. ¼ KV. Fðhi ; /i Þ;. N N R ¼ sin hdrdhd/ 4p r 2 dr. Then, Eq. 16 reduces to: ECH ¼ . 1 ðNAC þ NBC ÞKV : 15. ð17Þ. The exchange energy contribution EXH can be evaluated by taking into account the Fig. 2, which shows two neighboring spins labeled as Si and Sj, with Si located at a distance r from the center and its nearest neighbor Sj at distance d from Si. The exchange interaction between these spins can be calculated by:. Thus, exchange energy takes the form: Z XX X Jij qSi Sj dV wmij EXH ¼ 2 fdg m¼C;S i;j¼A;B Z r2mi XX X Jij wmij ð4pÞq Si Sj r 2 dr ¼ 2 r 1mi fdg m¼C;S i;j¼A;B R r2mi 2 XX X Jij Nmi r1mi r Si Sj dr R r2mi ¼ wmij 2 2 r1mi r dr fdg m¼C;S i;j¼A;B ð18Þ. ð15Þ. i. where spin components have been expressed in spherical coordinates (r, h, /), which gives: Fðh; /Þ ¼ sin4 h sin2 / cos2 / þ sin2 h cos2 h sin2 / þ sin2 h cos2 h cos2 / :. where the first sum runs over the complete set of nearest neighbors {d} of one given atom, the second one runs over the region of the nanoparticle (core or surface) which the ion belongs to, and the third one runs over the lattice type (A or B sites). Here, the wm ij parameter represents the effective coordination of the spins: wmij ¼. Considering a continuum approximation, and taking into account the radial distribution of spins in this state, Eq. 15 can be computed as the sum of the next integrals: Z Z ECH ¼ KV qA Fðh; /ÞdV þ qB Fðh; /ÞdV Z Z 2 ¼ KV r dr ðqA þ qB ÞFðh; /Þsinhdhd/; ð16Þ where q (=dN/dV) represents the volumetric density of spins. This density is constant and can be expressed as:. 123. r2. r þ dz Si Sj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dx þ d2y þ ðd2z þ rÞ. Ri ¼ jSi jðsin h cos /; sin h sin /; cos hÞ jR i j. X. q¼R. 1. hzij i zij. if m ¼ C if m ¼ S. Fig. 2 Schematic view of a couple of ions used to compute the exchange energy at the hedgehog configuration. The ith spin is located at a distance r from the center of the particle, and its direction is taken as z axis. The jth ion, one of the nearest neighbors, is located at a distance d from the ith atom.
(7) J Nanopart Res (2011) 13:7115–7125. 7121. Limits of the integrals in Eq. 18 should be selected in accordance with the region which energy contributions are evaluated in. These limits are 0 and RC for core contribution, and, RC and R for surface contribution (see Fig. 1). Nm i values are giving by Eqs. 5–8. Finally, rearranging Eq. 4, and taking into account Eqs. 11 and 14, the analytical expression for the phase diagram, i.e., KS as a function of size, can be written as: KS ¼. 3 ðEXH þ ECH EXF ECF Þ : 2 NAS S2A þ NBS S2B. ð19Þ. Here, the different terms are functions of the size and are given previously in the text above. For later discussion purposes this expression can be rewritten as: KS ¼. ð3=2ÞðEXH EXF Þ ð3=2ÞðECH ECF Þ þ NAS S2A þ NBS S2B NAS S2A þ NBS S2B. ¼ DX þ DC ; ð20Þ where DX and DC represent the exchange energy difference per surface atom and the anisotropy energy difference at core per surface atom between H and F configurations, respectively.. Results and discussion Figure 3 shows the number of each type of iron atoms, according to the case (core, surface, sublattice type),. Fig. 3 (Color on-line). Number of each type of atom as a function of the particle size. Symbols stand for simulation result, whereas lines show the analytical result proposed by Eqs. 5–8. Note the good agreement obtained between analytical and simulation results. as a function of the nanoparticle radius. The symbols correspond to values coming from samples built computationally from an ideal magnetite structure, whereas lines correspond to analytical expressions proposed in this study to compute N (Eqs. 5–8). It is worthwhile highlighting the significant agreement between the datasets computed from analytical expressions and the one obtained from simulation results. This fact is very important since our analysis rests on the correct computation of the number of atoms in particles of a given size. Figure 4 shows MC results for the magnetization per spin as a function of the KS value for three different selected sizes. The sudden decrease observed in magnetization indicates the KS-driven magnetic phase transition taking place at some critical KTrans value, S which is different for different system sizes. A ferrimagnetic state is obtained at small temperature regime. As KS increases, the spins tend to point toward quasi-radial directions imposed by the surface vectors. Intermediate values of KS take the system to the throttled state. Thus, above the critical surfaceanisotropy value, the magnetic state transforms onto a hedgehog configuration, as it was previously reported (Mazo-Zuluaga et al. 2008a, 2009). As it is observed in Fig. 4, the smaller the size of the nanoparticle, the greater the KS value required to overcome the transition. At first glance, this result appears as a counter-intuitive finding. Since surface to volume ratio decreases as the system size grows, it might lead us to think that greater values of the. Fig. 4 (Color on-line). Magnetization per spin as a function of the KS/KV value for three different selected sizes. This result has been obtained from MC simulation. 123.
(8) 7122. surface-anisotropy KS would be necessary to drive the system through the transition. Figure 4 clearly indicates the opposite situation. In order to confirm this observation, we proceeded to explore, via MC simulations, the surface-anisotropy dependence of energy for the particle taken in the fixed F and H configurations, respectively. Figure 5 shows the resulting energy lines for two selected nanoparticle sizes, in which the lower energy state is the stable one. As it is observed, F states dominate at small KS values, whereas H state energy becomes smaller beyond a certain KS value, which marks the transition point (crossover of the lines). This critical KS value is greater for the smaller nanoparticle, as it was previously discussed. The simulation phase diagram can be collected by computing the crossover of these two energy contributions for a wide range of particle sizes. Figure 6 shows the analytical phase diagram (solid line) for magnetite nanoparticles computed by equating energy for the F and the H states from the analytical expression Eq. 4; which includes the terms reported in Eqs. 9, 11–14, 17, 18, or, equivalently, by resolving Eq. 19. The line separates two regions, that corresponding to hedgehog-like states above, and the one assigned to ferrimagnetic states below the curve. This figure also encloses the numerical results (open circles), obtained by taking two fixed configurations: (a) a collinear F state, and (b) a radial state, for each size, and then computing energy from Eq. 1 as. Fig. 5 (Color on-line). Energy per spin as a function of the surface-anisotropy for the both of the configurations considered here, ferrimagnetic and hedgehog states. Results for two selected nanoparticle sizes are shown. 123. J Nanopart Res (2011) 13:7115–7125. Fig. 6 (Color on-line). Phase diagram for magnetite nanoparticles computed from the different methodologies discussed in the text. Solid line depicts the analytical result, which is in good agreement with numerical (connected open circles) and MC simulation results (full circles). Error bars are smaller than symbol sizes. The dotted line is a guide to the eye. function of the surface-anisotropy KS. By equating energy for these two configurations at several system sizes, the numerical phase diagram can be obtained (open circles). It is evident the good agreement between these two results. The small energy differences obtained can be attributed to the approximations adopted in the analytical framework, i.e., the fact of considering the system as a continuous medium in order to compute integrals. In fact, the differences are bigger at small sizes, where granularity of the system is more evident and bigger deviations from a continuous scheme are found. In addition, Fig. 6 presents critical KS values computed from the MC simulations for some different system sizes (full circles). At this point a difference emerges: in the MC simulations the system follows a relaxation process to get an equilibrium state. It implies that, in the simulation, the transition process takes place through the sequence ferri-throttledhedgehog states. This fact is the responsible one for the obtained overestimation of the MC results as compared to the analytical and numerical ones. In this case, energy differences can be attributed to the approximations adopted in the analytical framework; i.e., the approximation involved in assuming the F configuration as the starting point for the transition instead of the actual throttled configuration. However, the behavior and main features of the result persist. In.
(9) J Nanopart Res (2011) 13:7115–7125. fact, MC results depicts transition between throttled and H states, which can not be accomplished analytically for the reason discussed in ‘‘Analytical framework’’ section above. As it is observed, behaviors obtained by the three different schemes employed (analytical, numerical, and simulation) are consistent themselves; which confirms the as-seen phenomenology previously discussed. Other researchers have reported a magnetic phase diagram for generic samples in a simple cubic lattice structure (Berger et al. 2006). Their result consists of a curve with positive concavity, with a minimum for the transition located at a radius below 5 nm. Our simulation results, for the sizes considered (\8 nm), do not show such behavior; instead the obtained behavior corresponds to a decreasing curve. One reason for this difference can be attributed to the fact that our system is a real one, and it has complex structure and magnetic behavior, as it was discussed in ‘‘Computation details’’ section. From the experimental standpoint, some studies on ferrimagnetic maghemite nanoparticles (which also crystallizes in the spinel structure) confirm the results presented here (Papefthymiou et al. 2009; Rebbouh et al. 2007). These authors reported a clear trend in the size dependence of the magnetic anisotropy, they estimated an increase in the anisotropy energy density of the nanoparticles as compared to bulk, and attributed it ‘‘to surface and strain effects that dominate the magnetic anisotropy density in small particles’’ (Papefthymiou et al. 2009). In order to explore the behavior of bigger sizes of our magnetite nanoparticles, and compare it with the reported result of the pure system, we take advantage of the analytical framework, which is free of the computational limitations of MC simulations, and then compute the phase diagram for an extended range of sizes. Figure 7 presents the extended phase diagram (solid line), which exhibits a curve with similar shape to that reported in Fig. 3 of Berger et al. 2006. In fact, the minimum in the curve KS(R) exists, and, for this case of magnetite nanoparticles, it is located at around 176 nm. To understand this behavior, we explore the different energy terms involved in the process. Figure 7 also shows: (i) the exchange energy difference per unit of surface atom between H and F configurations, DX of Eq. 20, (dotted line), and (ii) the differences between the anisotropy contributions at core of the particles, DC ; (dashed line), as functions of. 7123. Fig. 7 (Color on-line). Analytical phase diagram for magnetite nanoparticles in a wide range of KS values (solid line). DX and DC contributions (explained in the text) are also shown in discontinuous lines. the size. As it is observed, DX is a monotonously decreasing function and has predominant contribution for small particles. Therefore, it can be interpreted as a term proportional to the energy barrier to be overcome for the particle to go from F to H configuration; whereas DC is an increasing function of the size that becomes relevant for bigger systems. Then, at these sizes, the energy barrier will be proportional to DC : In short, at the small size regime, below the minimum in the phase diagram, magnetic structure of the nanoparticles is governed by DX ; for bigger particles is DC what controls the state, instead. These features are responsible for the observed behavior of the KS versus R curve. At the small size regime, since DX is the dominant one, a size increase leads to a decrease in the surface-anisotropy needed to overcome the transition. This tendency holds up to DC becomes relevant, where the minimum takes place and, beyond 176 nm approx., in what we call the big size regime, a surface-anisotropy increase is now necessary to trigger the phase transition, and drive the particle toward a hedgehog-like configuration. One important fact has to be regarded in this case, however: for building Fig. 7, we have resolved the analytical model for a real and complex system beyond its validity limits, i.e., for huge sizes. As it is known, the bigger the number of magnetic moments in the system, the less negligible the dipolar interactions. Although the result obtained for this huge range of size is in qualitative agreement with the reported behavior. 123.
(10) 7124. by Berger et al. 2006 in the sense the curve exhibits a certain size for which transition occurs at a certain minimum KS value, it is necessary to be aware of the validity of the model (dipolar energy negligible, see ‘‘Computation details’’ section). For size particles beyond 20 nm approx., dipolar interactions are not negligible anymore. Therefore, for the big sizes reached here, dipolar energy will affect the magnetic states; and final configurations at the transitions are not expected to be simple hedgehog-like states.. Conclusion The effect of surface-anisotropy and particle size on the magnetic properties of magnetite nanoparticles has been addressed. Analytical expressions for energy contributions at ferrimagnetic and hedgehog states have been presented. Likewise, atomistic simulations based on the Monte Carlo method, using a classical Heisenberg Hamiltonian and single-spin flip dynamics, have been carried out for comparison purposes. The phase diagram of the KS-driven magnetic phase transition is obtained, a pending issue in the study of ferrite nanoparticles. Since no analytical expression for a throttled state is available, calculation of the transition is carried out by considering the ferrimagnetic and the hedgehog-like configurations. It leads to the observed underestimation of the KS values as compared with Monte Carlo results, where the transition process is considered in all its steps as surfaceanisotropy increases: ferrimagnetic-throttled-hedgehog states. However, very good agreement is obtained for both cases. The physical mechanism for the KS-driven phase transition has been explained by considering energy barriers rising due to energy differences between hedgehog and ferrimagnetic configurations for both the exchange and core anisotropy contributions, which exhibit dominant behaviors at different range sizes. For magnetite, the energy barrier assigned to core anisotropy contributions is negligible for all those sizes where dipolar energy can be neglected. Therefore, the KS is a monotonously decreasing function of particle radius at this small size regime. This behavior agrees qualitatively with some recent experimental reports on ferrimagnetic maghemite nanoparticles with diameters less than 12.5 nm (Papefthymiou et al. 2009; Rebbouh et al. 2007). Finally, it might be worthwhile mentioning that the process presented here can be useful as a guide for studying analytically other magnetic materials in order. 123. J Nanopart Res (2011) 13:7115–7125. to avoid limitations of Monte Carlo simulations, which become computationally unmanageable for large particle sizes. To extend this theoretical scheme for computing the magnetic phase diagram to other ferrimagnetic compounds, the analytical expressions presented here can be applied by taking into account the respective set of nearest neighbors for each atom along with the surface-anisotropy and spin arrangements belonging to the material of interest. Acknowledgments This study was supported by several projects: the Chile-Colombia scientific exchange program CONICYT-COLCIENCIAS under contracts 2008-157, 2792009; the FONDECYT grant 1100365, Millenium Science Nucleus ‘‘Basic and applied magnetism’’ P10-061-F; Financiamiento basal para centros cientı́ficos y tecnológicos de excelencia FB 0807; the CODI-UdeA projects IN576CE, IN578CE, and ‘‘Sostenibilidad’’ projects of the GES and GICM Groups at the Universidad de Antioquia. We are grateful to Dr. Johans Restrepo for helpful discussions. J.M-Z. wants to thank Universidad de Antioquia for a ‘‘Dedicación Exclusiva’’ program.. 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