Effective anisotropies and energy barriers of magnetic nanoparticles with Néel surface anisotropy

Abstract

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.

Direct observation of enhanced magnetism in individual size- and shape-selected 3d transition metal nanoparticles

“…The simulations consider classical spins distributed over the lattice sites of spherical model particles with diameter D. The magnetic properties of the particles are described by an anisotropic Heisenberg Hamiltonian [69]. The effective energy landscapes of the many-spin particles are then evaluated using the Lagrangian multiplier method, as described in Refs.…”

“…Since K s is not a priori known and can significantly differ between various experimental reports, we have chosen to consider the range 0 < |K s /K 1 | < 800 in order to cover a large range of experimentally determined surface anisotropies deduced from thin film studies [74][75][76][77][78][79]. Further, K s is assumed to be constant; however, the model intrinsically takes the local environment of each atom into account and therefore yields a magnetic surface energy barrier, which depends locally on the actual coordination number [68,69]. Finally, for calculating K s /K 1 , also K s is considered on a per atom basis.…”

Core-Shell Magnetic Morphology of Structurally Uniform Magnetite Nanoparticles

“…Surface disorder can lead to a magnetically dead layer or a spin glasslike phase [6][7][8][9]. Surfaces can also introduce a cubic anisotropy term when the material otherwise has uniaxial anisotropy [10]. Surface anisotropy has led to predictions of a ''hedgehog'' configuration where the spins spike outward normal to the surface [11][12][13][14], an ''artichoke'' confirmation where they are parallel to the surface and directed from one pole to the other [13], and a ''throttled'' configuration where the spins have been tilted slightly inward at the south pole and outward at the north pole [13].…”

Magnetic anisotropy of La0.7Sr0.3MnO3 nanopowders

“…Some deviation from Eq. (6) may result from the observation in [16] that the bulk and surface anisotropies are not additive values.…”