In this paper we derive a Gilbert-Varshamov type bound for linear codes over Galois rings GR(pl; j): However, this bound does not guarantee existence of better linear codes over GR(pl; j) than the usual Gilbert-Varshamov bound for linear codes over the residue class field GR(pj): Next we derive a Gilbert-Varshamov type bound for Z4 linear codes which guarantees the existence of Z4 linear codes ...

In this paper, we discuss the accuracy and complexity of various numerical techniques to solve the stochastic Landau-Lifshitz-Gilbert-Slonczewski (s-LLGS) equation. The s-LLGS equation is widely used by researchers to study the temporal evolution of the macrospin subject to spin torque and thermal noise. The numerical simulation of the s-LLGS equation requires an appropriate choice of stochasti...

We present a first-principles theory of dynamical spin excitations in the presence of spin-orbit coupling. The broken global spin rotational invariance leads to a new sum rule. We explore the competition between the magnetic anisotropy energy and the external magnetic field, as well as the role of electron-hole excitations, through calculations for 3d-metal adatoms on the Cu(111) surface. The s...

Magnetostatic Maxwell equations and the Landau-Lifshitz-Gilbert (LLG) equation are combined to a multiscale method, which allows to extend the problem size of traditional micromagnetic simulations. By means of magnetostatic Maxwell equations macroscopic regions can be handled in an averaged and stationary sense, whereas the LLG allows to accurately describe domain formation as well as magnetiza...

We theoretically study domain wall oscillations upon the injection of a dc current through a geometrically constrained wire with perpendicular magnetic anisotropy. The frequency spectrum of the oscillation can be tuned by the injected current density and additionally by the application of an external magnetic field. Our analytical calculations are supported by micromagnetic simulations based on...

The spin-torque driven dynamics of antiferromagnets with Dzyaloshinskii-Moriya interaction (DMI) were investigated based on the Landau-Lifshitz-Gilbert-Slonczewski equation with antiferromagnetic and ferromagnetic order parameters (l and m, respectively). We demonstrate that antiferromagnets including DMI can be described by a 2-dimensional pendulum model of l. Because m is coupled with l, toge...

A new way of magnetization switching employing both the spin-transfer torque and the torque by a magnetic field is proposed. The solution of the Landau-Lifshitz-Gilbert equation shows that the dynamics of the magnetization in the initial stage of the switching is similar to that in the precessional switching, while that in the final stage is rather similar to the relaxing switching. We call the...

The magnetization dynamics of Co nanoislands on Cu(111) is studied on the atomic scale by means of the stochastic Landau-Lifshitz-Gilbert equation. The exchange and anisotropy constants of the spin Hamiltonian are computed from first principles. We focus on hysteresis loops and magnetic switching in dependence on temperature, island size, and strength of an external magnetic field. The magnetic...

A single domain ferromagnetic particle is characterized by an internal potential, having several local states of equilibrium with potential barriers between them. If the particles are small ~; 10 nm! so that the potential barriers are relatively low, the magnetization vector M may cross over the barriers due to thermal agitation. The ensuing thermal instability of the magnetization results in t...

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence...