26 research outputs found
Fast and Robust Algorithm for the Energy Minimization of Spin Systems Applied in an Analysis of High Temperature Spin Configurations in Terms of Skyrmion Density
An algorithm for the minimization of the energy of magnetic systems is
presented and applied to the analysis of thermal configurations of a
ferromagnet to identify inherent structures, i.e. the nearest local energy
minima, as a function of temperature. Over a rather narrow temperature
interval, skyrmions appear and reach a high temperature limit for the skyrmion
density. In addition, the performance of the algorithm is further demonstrated
in a self-consistent field calculation of a skyrmion in an itinerant magnet.
The algorithm is based on a geometric approach in which the curvature of the
spherical domain is taken into account and as a result the length of the
magnetic moments is preserved in every iteration. In the limit of infinitesimal
rotations, the minimization path coincides with that obtained using damped spin
dynamics while the use of limited-memory quasi-newton minimization algorithms,
such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm,
significantly accelerates the convergence
The effect of confinement and defects on the thermal stability of skyrmions
The stability of magnetic skyrmions against thermal fluctuations and external
perturbations is investigated within the framework of harmonic transition state
theory for magnetic degrees of freedom. The influence of confined geometry and
atomic scale non-magnetic defects on the skyrmion lifetime is estimated. It is
shown that a skyrmion on a track has lower activation energy for annihilation
and higher energy for nucleation if the size of the skyrmion is comparable with
the width of the track. Two mechanisms of skyrmion annihilation are considered:
inside the track and escape through the boundary. For both mechanisms, the
dependence of activation energy on the track width is calculated. Non-magnetic
defects are found to localize skyrmions in their neighborhood and strongly
decrease the activation energy for creation and annihilation. This is in
agreement with experimental measurements that have found nucleation of
skyrmions in presence of spin-polarized current preferably occurring near
structural defects
Energy surface and lifetime of magnetic skyrmions
The stability of skyrmions in various environments is estimated by analyzing
the multidimensional surface describing the energy of the system as a function
of the directions of the magnetic moments in the system. The energy is given by
a Heisenberg-like Hamiltonian that includes Dzyaloshinskii-Moriya interaction,
anisotropy and external magnetic field. Local minima on this surface correspond
to the ferromagnetic and skyrmion states. Minimum energy paths (MEP) between
the minima are calculated using the geodesic nudged elastic band method. The
maximum energy along an MEP corresponds to a first order saddle point on the
energy surface and gives an estimate of the activation energy for the magnetic
transition, such as creation and annihilation of a skyrmion. The
pre-exponential factor in the Arrhenius law for the rate, the so-called attempt
frequency, is estimated within harmonic transition state theory where the
eigenvalues of the Hessian at the saddle point and the local minima are used to
characterize the shape of the energy surface. For some degrees of freedom,
so-called 'zero modes', the energy of the system remains invariant. They need
to be treated separately and give rise to temperature dependence of the attempt
frequency. As an example application of this general theory, the lifetime of a
skyrmion in a track of finite width for a PdFe overlayer on a Ir(111) substrate
is calculated as a function of track width and external magnetic field. Also,
the effect of non-magnetic impurities is studied. Various MEPs for annihilation
inside a track, via the boundary of a track and at an impurity are presented.
The attempt frequency as well as the activation energy has been calculated for
each mechanism to estimate the transition rate as a function of temperature
The effect of temperature and external field on transitions in elements of kagome spin ice
Transitions between magnetic states of one and two ring kagome spin ice elements consisting of 6 and 11 prolate magnetic islands are calculated and the lifetime of the ground states evaluated using harmonic transition state theory and the stationary state approximation. The calculated values are in close agreement with experimental lifetime measurements made by Farhan and co-workers (Farhan et al 2013 Nat. Phys. 9 375) when values of the parameters in the Hamiltonian are chosen to be best estimates for a single island, obtained from measurements and micromagnetic modeling. The effective pre-exponential factor in the Arrhenius rate law for the elementary steps turns out to be quite small, on the order of 109 s-1, three orders of magnitude smaller than has been assumed in previous analysis of the experimental data, while the effective activation energy is correspondingly lower than the previous estimate. The application of an external magnetic field is found to strongly affect the energy landscape of the system. Even a field of can eliminate states that correspond to ground states in the absence of a field. The theoretical approach presented here and the close agreement found with experimental data demonstrates that the properties of spin ice systems can be calculated using the tools of rate theory and a Hamiltonian parametrized only from the properties of a single island.Peer reviewe