- M. Morini, V.V. Slastikov, Geometrically induced phase transitions in two-dimensional dumbbell-shaped domains, submitted
Abstract
- A. Goussev, J.M. Robbins, V.V. Slastikov, Domain wall motion in ferromagnetic nanotubes: analytic results, EPL, 2014, 105, 67006
Abstract
Dynamics of magnetization domain walls (DWs) in thin ferromagnetic nanotubes subject to longitudinal external fields is addressed analytically in the regimes of strong and weak penalization.
Explicit functional forms of the DW profiles and formulas for the DW propagation velocity are derived in both regimes. In particular, the DW speed is shown to depend nonlinearly on the nanotube radius.
- A. Goussev, R.G. Lund, JM Robbins, , V.V. Slastikov, C. Sonnenberg Domain wall motion in magnetic nanowires: An asymptotic approach, accepted to Proc. Roy. Soc. London Ser. A, 2013, 469, 20130308
Abstract
We develop a systematic asymptotic description for domain wall motion in one-
dimensional magnetic nanowires under the influence of small applied magnetic fields
and currents and small material anisotropy. The magnetization dynamics, as governed
by the Landau -- Lifshitz -- Gilbert equation, is investigated via a perturbation expansion.
We compute leading-order behaviour, propagation velocities, and first-order corrections
of both travelling waves and oscillatory solutions, and find bifurcations between these
two types of solutions. This treatment provides a sound mathematical foundation for
numerous results in the literature obtained through more ad hoc arguments.
- A. Goussev, R.G. Lund, JM Robbins, , V.V. Slastikov, C. Sonnenberg , Fast domain wall propagation in uniaxial nanowires with transverse fields, Phys. Rev. B, 2013, 88, 024425
Abstract
Under a magnetic field along its axis, domain-wall motion in a uniaxial nanowire is much slower than in the fully anisotropic case, typically by several orders of magnitude (the square of the dimensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse to the wire, this behavior is dramatically reversed; up to a critical field strength, analogous to the Walker breakdown field, domain walls in a uniaxial wire propagate faster than in a fully anisotropic wire (without a transverse field). Beyond this critical field strength, precessional motion sets in, and the mean velocity decreases. Our results are based on leading-order analytic calculations of the velocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation.
- M. Morini, V.V. Slastikov, Geometrically constrained walls in two dimensions, Arch. Ration. Mech. Anal., 2012, 203, N2, 621-692
Abstract
We address the effect of extreme geometry on a non-convex variational problem, motivated by studies on magnetic domain walls trapped by thin necks. The recent analytical results of Kohn and Slastikov revealed a variety of magnetic structures in three-dimensional ferromagnets depending on the size of the constriction. The main purpose of the paper is to study geometrically constrained walls in two dimensions. The analysis turns out to be significantly more challenging and requires the use of different techniques. In particular, the purely variational point of view cannot be adopted in the present setting and is here replaced by a PDE approach.
Existence of local minimizers representing geometrically constrained walls is proven under suitable symmetry assumptions on the domains and an asymptotic characterization of the wall profile is given. The limiting behavior, which depends critically on the
scaling of length and height of the neck, turns out to be more complex than in the higher-dimensional case and a richer variety of regimes is shown to exist.
- Y. Gou, A. Goussev, J.M. Robbins, V.V. Slastikov, Stability of precessing domain walls in ferromagnetic nanowires, Phys. Rev. B, 2011, 84, 104445
Abstract
We show that recently reported precessing solution of
Landau-Lifshitz-Gilbert equations in ferromagnetic nanowires is
stable under small perturbations of initial data, applied field and
anisotropy constant. Linear stability is established analytically, while nonlinear stability is verified numerically.
- V. Slastikov, C. Sonnenberg, Reduced models for ferromagnetic nanowires, IMA J. Appl. Math, 2012, 77, N2, 220-236
Abstract
In this paper, we consider the micromagnetic variational problem for soft ferromagnetic nanowires. We show that, as the diameter of the wire is small, the magnetization inside the wire depends only on the length variable of the wire. The micromagnetic energy of the wire, in this case, is greatly simplified and in order to find the optimal magnetization distribution, one has to solve a 1D local variational problem.
- A. Goussev, J.M. Robbins, V.V. Slastikov, Domain wall motion in ferromagnetic nanowires driven by arbitrary time-dependent fields: an exact result,
Phys. Rev. Lett., 2010, 104, 147202
Abstract
We address the dynamics of magnetic domain walls in ferromagnetic
nanowires under the influence of external time-dependent magnetic
fields. We report a new exact spatiotemporal solution of the
Landau-Lifshitz-Gilbert equation for the case of soft ferromagnetic
wires and nanostructures with uniaxial anisotropy. The solution
holds for applied fields with arbitrary strength and time
dependence. We further extend this solution to applied fields slowly
varying in space and to multiple domain walls.
- V.V. Slastikov, A note on configurational anisotropy ,
Proc. Roy. Soc. London Ser. A, 2010, 466, 3167-- 3179
Abstract
We investigate an effect of configurational anisotropy in highly symmetric soft ferromagnetic nanoparticles. Using micromagnetic variational principle and methods of Gamma-convergence we show that in ferromagnetic prisms with D4 symmetry there is a finite number of preferred magnetization directions and that these directions are independent of the shape of a magnet. This result provides a rigorous justification of work by Cowburn and Welland.
- R.V. Kohn, V.V. Slastikov,
Geometrically constrained walls, Calc. Var. PDE., 2007, 28, N1, 33--57
Abstract
We address the effect of extreme geometry on a non-convex variational problem.
The analysis is motivated by recent investigations of magnetic domain walls
trapped by sharp thin necks. To capture the essential issues in the simplest
possible setting, we focus on a scalar variational problem with a symmetric
double well potential, whose spatial domain is a dumbell with a sharp neck.
Our main results are (a) the existence of local minimizers representing
geometrically constrained walls, and (b) an asymptotic characterization of
the wall profile. Our analysis uses methods similar to
Gamma-convergence; in particular, the wall profile minimizes
a certain ``reduced problem'' -- the limit of the original problem, suitably
rescaled near the neck. The structure of the wall depends critically on the
choice of scaling, specifically the ratio between length and width of the neck.
- V.V. Slastikov,
Micromagnetics of thin shells, Math. Models Methods Appl. Sci.,
2005, 15, N10, 1469-1487.
Abstract
We study a thin-shell limit of micromagnetic energy for soft small ferromagnets.
The relations between thickness of the magnet t , diameter l and magnetic exchange length w are t/l --> 0 and tl/w^2 < 1. We prove a Gamma-convergence of the original 3D problem to a nonlocal 2D problem.
- R.V. Kohn, V.V. Slastikov,
Another thin-film limit of micromagnetics,
Arch. Ration. Mech. Anal., 2005, 178, N2, 227-245.
Abstract
We consider the variational problem of micromagnetics
for soft, relatively small thin films with no applied magnetic field.
In terms of the film thickness t , the diameter l and the magnetic
exchange length w, we study the asymptotic behavior in the
small-aspect-ratio limit t/l --> 0, when
w^2/l^2 >= (t/l) | log (t/l) | .
Our analysis builds on prior work by Gioia & James and Carbou.
The limiting variational problem is much simpler than 3D
micromagnetics; in particular it is two-dimensional and local,
with no small parameters.
- R.V. Kohn, V.V. Slastikov,
Effective dynamics for ferromagnetic thin films: rigorous justification,
Proc. Roy. Soc. London Ser. A, 2005, 461, N2053, 143--154.
Abstract
In a thin-film ferromagnet, the leading-order behavior of the magnetostatic energy is
a strong shape anistropy, penalizing the out-of-plane component of the magnetization
distribution. We study the thin-film limit of Landau-Lifshitz-Gilbert dynamics, when the
magnetostatic term is replaced by this local approximation. The limiting 2D effective
equation is overdamped, i.e. it has no precession term. Moreover if the damping
coefficient of 3D micromagnetics is a then the damping coefficient of the 2D
effective equation is a + 1/a; thus reducing the damping in 3D can
actually increase the damping of the effective equation. This result was previously
shown by Garcia-Cervera and E using asymptotic analysis; our contribution is a
mathematically rigorous justification.