We calculate the contribution of the Néel surface anisotropy to the effective anisotropy of magnetic nanoparticles of spherical shape cut out of a simple cubic lattice. The effective anisotropy arises because deviations of atomic magnetizations from collinearity and thus the energy depends on the orientation of the global magnetization. The result is second order in the Néel surface anisotropy, scales with the particle's volume and has cubic symmetry with preferred directions [±1, ±1, ±1].PACS numbers: 61.46.+w, 75.70.Rf With the decreasing size of magnetic particles, surface effects are believed to become more and more pronounced. A simple argument based on the estimation of the fraction of surface atoms shows that for a particle of spherical shape and diameter D (in units of the lattice spacing), this fraction is an appreciable number of order 6/D. Regarding the fundamental property of magnetic particles, the magnetic anisotropy, the role of surface atoms is augmented by the fact that these atoms in many cases experience surface anisotropy (SA) that by far exceeds the bulk anisotropy. As was suggested by Néel [1] and microscopically shown in Ref.[2], the leading contribution to the anisotropy is due to pairs of atoms and can be written aswhere m i is the reduced magnetization (spin polarization) of the ith atom, e ij are unit vectors directed from the ith atom to its neighbors, and L ij is the pairanisotropy coupling that depends on the distance between atoms. Eq. (1) describes in a unique form both the bulk anisotropy including the effect of elastic strains and the effect of missing neighbors at the surface that leads to the SA. In particular, for an unstrained simple cubic (sc) lattice the bulk anisotropy in Eq. (1) The 1/D surface contribution to K V,eff is in accord with the picture of all magnetic atoms tightly bound by the exchange interaction whereas only the surface atoms feel the surface anisotropy. This is definitely true for magnetic films where a huge surface contribution to the effective anisotropy has been observed. The same is the case for cobalt nanoclusters of the form of truncated octahedrons [5] where contributions from different faces, edges, and apexes compete resulting in a nonzero, although significantly reduced, surface contribution to K V,eff . However, for symmetric particle shapes such as cubes or spheres, the symmetry leads to vanishing of this (first-order) contribution. In this case one has to take into account deviations from the collinearity of atomic spins that result from the competition of the SA and the exchange interaction J. The resulting structures (for the simplified radial SA model) can be found in Refs. [6,7,8] (see also Fig. 1 for the NSA). In the case L > ∼ J deviations from collinearity are very strong, and it is difficult if not impossible to characterize the particle by a global magnetization suitable for the definition of the effective anisotropy. On the other hand, in the typical case L ≪ J the magnetic structure is nearly collinear with small deviations tha...
We derive a generalized Ginzburg-Landau (GL) functional near the tricritical point in the (T, H)-phase diagram for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state, in 1, 2, and 3 dimensions. We find that the transition from the normal to the FFLO state is of second order in 1 and 2 dimensions, and the order parameter with one-coordinate sine modulation corresponds to the lowest energy near the transition line. We also compute the jump of the specific heat and describe in the one-dimensional case the transformation of the sine modulation into the solitonlattice state as the magnetic field decreases. In 3 dimensions however, we find that the transition into an FFLO state is of first order, and it is impossible to obtain an analytic expression for the critical temperature. In this case the generalized GL functional proposed here provides a suitable basis for a numerical study of the properties of the FFLO state, and in particular for computing the critical temperature, and for describing the transition into a uniform state.
We present a microscopic model for nanoparticles, of the maghemite (γ-Fe2O3) type, and perform classical Monte Carlo simulations of their magnetic properties. On account of Mössbauer spectroscopy and high-field magnetisation results, we consider a particle as composed of a core and a surface shell of constant thickness. The magnetic state in the particle is described by the anisotropic classical Dirac-Heisenberg model including exchange and dipolar interactions and bulk and surface anisotropy. We consider the case of ellipsoidal (or spherical) particles with free boundaries at the surface. Using a surface shell of constant thickness (∼ 0.35 nm) we vary the particle size and study the effect of surface magnetic disorder on the thermal and spatial behaviors of the net magnetisation of the particle. We study the shift in the surface "critical region" for different surface-to-core ratios of the exchange coupling constants. It is also shown that the profile of the local magnetisation exhibits strong temperature dependence, and that surface anisotropy is reponsible for the non saturation of the magnetisation at low temperatures.PACS. 75.50.Tt Fine Particle Systems -75.30.Pd Surface Magnetism -75.10.Hk Classical Spin Models
R. Evans and R. W. ChantrellDepartment of Physics, University of York, Heslington, York YO10 5DD, UKMagnetic nanoparticles with Néel surface anisotropy, different internal structures, surface arrangements and elongation are modelled as many-spin systems. The results suggest that the energy of many-spin nanoparticles cut from cubic lattices can be represented by an effective one-spin potential containing uniaxial and cubic anisotropies. It is shown that the values and signs of the corresponding constants depend strongly on the particle's surface arrangement, internal structure and elongation. Particles cut from a simple cubic lattice have the opposite sign of the effective cubic term, as compared to particles cut from the face-centered cubic lattice. Furthermore, other remarkable phenomena are observed in nanoparticles with relatively strong surface effects: (i) In elongated particles surface effects can change the sign of the uniaxial anisotropy. (ii) The competition between the core and surface anisotropies leads to a new energy that contributes to both the 2 nd − and 4 th −order effective anisotropies. We also evaluate energy barriers ∆E as functions of the strength of the surface anisotropy and the particle size. The results are analyzed with the help of the effective one-spin potential, which allows us to assess the consistency of the widely used formula ∆E/V = K∞ + 6Ks/D, where K∞ is the core anisotropy constant, Ks is a phenomenological constant related to surface anisotropy, and D is the particle's diameter. We show that the energy barriers are consistent with this formula only for elongated particles for which the surface contribution to the effective uniaxial anisotropy scales with the surface and is linear in the constant of the Néel surface anisotropy.
For nite-temperature micromagnetic simulations the knowledge of the temperature dependence of the exchange stiness plays a central role. We use two approaches for the calculation of the thermodynamic exchange parameter from spin models: (i) based on the domain wall energy, (ii) based on the spin-wave dispersion. The corresponding analytical and numerical approaches are introduced and compared. A general theory for the temperature dependence and scaling of the exchange stiness is developed using the classical spectral density method. The low-temperature exchange stiness A is found to scale with magnetization as m 1.66 for systems on a simple cubic lattice and as m 1.76 for an FePt Hamiltonian parametrized through ab initio calculations. The additional reduction of the scaling exponent, as compared to the mean-eld theory (A ∼ m 2), comes from the non-linear spin-wave eects.
We study the influence of surface anisotropy on the zero-temperature hysteretic properties of a small single-domain ferromagnetic particle, and investigate limiting cases where deviations from the Stoner-Wohlfarth model are observed due to non-uniform reversal of the particle's magnetization. We consider a spherical particle with simple cubic crystal structure, a uniaxial anisotropy for core spins and radial anisotropy on the surface. The hysteresis loop is obtained by solving the local (coupled) Landau-Lifshitz equations for classical spin vectors. We find that when the surface anisotropy constant Ks assumes large values, e.g. of the order of the exchange coupling J, large deviations are observed with respect to the Stoner-Wohlfarth model in the hysteresis loop and thereby the limitof-metastability curve, since in this case the magnetization reverses its direction in a non-uniform manner via a progressive switching of spin clusters. This characteristic value of Ks depends on the surface-to-volume ratio of exchange coupling and the angle between the applied field and core easy axis.PACS number(s): 75.50. .Hk
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