Matteo Franchin

Joule heating in nanowires

We study the effect of Joule heating from electric currents flowing through ferromagnetic nanowires on the temperature of the nanowires and on the temperature of the substrate on which the nanowires are grown. The spatial current density distribution, the associated heat generation, and diffusion of heat is simulated within the nanowire and the substrate. We study several different nanowire and constriction geometries as well as different substrates: (thin) silicon nitride membranes, (thick) silicon wafers, and (thick) diamond wafers. The spatially resolved increase in temperature as a function of time is computed. For effectively three-dimensional substrates (where the substrate thickness greatly exceeds the nanowire length), we identify three different regimes of heat propagation through the substrate: regime (i), where the nanowire temperature increases approximately logarithmically as a function of time. In this regime, the nanowire temperature is well-described analytically by You et al. [APL89, 222513 (2006)]. We provide an analytical expression for the time tc that marks the upper applicability limit of the You model. After tc, the heat flow enters regime (ii), where the nanowire temperature stays constant while a hemispherical heat front carries the heat away from the wire and into the substrate. As the heat front reaches the boundary of the substrate, regime (iii) is entered where the nanowire and substrate temperature start to increase rapidly. For effectively two-dimensional substrates (where the nanowire length greatly exceeds the sub- strate thickness), there is only one regime in which the temperature increases logarithmically with time for large times. We provide an analytical expression, valid for all pulse durations, that allows one to accurately compute this temperature increase in the nanowire on thin substrates

Proposal for a Standard Micromagnetic Problem: Spin Wave Dispersion in a Magnonic Waveguide

In this paper, we propose a standard micromagnetic problem, of a nanostripe of permalloy. We study the magnetization dynamics and describe methods of extracting features from simulations. Spin wave dispersion curves, relating frequency and wave vector, are obtained for wave propagation in different directions relative to the axis of the waveguide and the external applied field. Simulation results using both finite element (Nmag) and finite difference (OOMMF) methods are compared against analytic results, for different ranges of the wave vector.

Proposal for a Standard Problem for Micromagnetic Simulations Including Spin-Transfer Torque

of micromagnetic simulation tools. The work is based on the micromagnetic model extended by the spin-transfer torque in continuously varying magnetizations as proposed by Zhang and Li. The standard problem geometry is a permalloy cuboid of 100 nm edge length and 10 nm thickness, which contains a Landau pattern with a vortex in the center of the structure. A spin-polarized dc current density of 10 12 A/m 2 ows laterally through the cuboid and moves the vortex core to a new steady-state position. We show that the new vortex-core position is a sensitive measure for the correctness of micromagnetic simulators that include the spin-transfer torque. The suitability of the proposed problem as a standard problem is tested by numerical results from four dierent nite-dierence and nite-element-based simulation tools.

A new approach to (quasi) periodic boundary conditions in micromagnetics: The macrogeometry

We present a new method to simulate repetitive ferromagnetic structures. This macrogeometry approach combines treatment of short-range interactions (i.e., the exchange field) as for periodic boundary conditions with a specification of the arrangement of copies of the primary simulation cell in order to correctly include effects of the demagnetizing field. This method (i) solves a consistency problem that prevents the naive application of three-dimensional periodic boundary conditions in micromagnetism and (ii) is well suited for the efficient simulation of repetitive systems of any size.

Ultrahard magnetic nanostructures

The performance of hard-magnetic nanostructures is investigated by analyzing the size and geometry dependence of thin-film hysteresis loops. Compared to bulk magnets, weight and volume are much less important, but we find that the energy product remains the main figure of merit down to very small features sizes. However, hysteresis loops are much easier to control on small length scales, as epitomized by Fe-Co-Pt thin films with magnetizations of up to 1.78 T and coercivities of up to 2.52 T. Our numerical and analytical calculations show that the feature size and geometry have a big effect on the hysteresis loop. Layered soft regions, especially if they have a free surface, are more harmful to coercivity and energy product than spherical inclusions. In hard-soft nanocomposites, an additional complication is provided by the physical properties of the hard phases. For a given soft phase, the performance of a hard-soft composite is determined by the parameter (Ms - Mh)/Kh.

Micromagnetic studies of three-dimensional pyramidal shell structures

We present a systematic numerical analysis of the magnetic properties of pyramidal-shaped core-shell structures in a size range below 400 nm. These are three-dimensional structures consisting of a ferromagnetic shell which is grown on top of a non-magnetic core. The standard micromagnetic model without the magnetocrystalline anisotropy term is used to describe the properties of the shell. We vary the thickness of the shell between the limiting cases of an ultra-thin shell and a conventional pyramid and delineate different stable magnetic configurations. We find different kinds of single-domain states, which predominantly occur at smaller system sizes. In analogy to equivalent states in thin square films we term these onion, flower, C and S states. At larger system sizes, we also observe two types of vortex states, which we refer to as symmetric and asymmetric vortex states. For a classification of the observed states, we derive a phase diagram that specifies the magnetic ground state as a function of structure size and shell thickness. The transitions between different ground states can be understood qualitatively. We address the issue of metastability by investigating the stability of all occurring configurations for different shell thicknesses. For selected geometries and directions hysteresis measurements are analysed and discussed. We observe that the magnetic behaviour changes distinctively in the limit of ultra-thin shells. The study has been motivated by the recent progress made in the growth of faceted core-shell structures.

Compression of boundary element matrix in micromagnetic simulations

A hybrid finite element method/boundary element method (FEM/BEM) is a standard approach for calculating the magnetostatic potential within micromagnetics [D. Fredkin and T. Koehler, IEEE Trans. Magn. 26, 415 (1990)]. This involves dealing with a dense N×N matrix Bij, with N being the number of mesh surface nodes. In order to apply the method to ferromagnetic structures with a large surface, one needs to apply matrix compression techniques on Bij. An efficient approach is to approximate Bij by hierarchical matrices (or H matrices). We have used HLIB [http://www.hlib.org], a library containing implementations of the hierarchical matrix methodology, together with the micromagnetic finite element solver NMAG in order to optimize the hybrid FEM/BEM. In this article we present a study of the efficiency of algorithms implemented in HLIB concerning the storage requirements and the matrix assembly time in micromagnetic simulations.

Parallel execution and scriptability in micromagnetic simulations

We demonstrate the feasibility of an “encapsulated parallelism” approach toward micromagnetic simulations that combines offering a high degree of flexibility to the user with the efficient utilization of parallel computing resources. While parallelization is obviously desirable to address the high numerical effort required for realistic micromagnetic simulations through utilizing now widely available multiprocessor systems (including desktop multicore CPUs and computing clusters), conventional approaches toward parallelization impose strong restrictions on the structure of programs: numerical operations have to be executed across all processors in a synchronized fashion. This means that from the user’s perspective, either the structure of the entire simulation is rigidly defined from the beginning and cannot be adjusted easily, or making modifications to the computation sequence requires advanced knowledge in parallel programming. We explain how this dilemma is resolved in the NMAG simulation package in s...

Analysis of Magnetoresistance in Arrays of Connected Nano-Rings

We study the anisotropic magnetoresistance (AMR) of a 2-D periodic square array of connected permalloy rings with periodicity of 1 mum combining experimental and computational techniques. The computational model consists of two parts: 1) the computation of the magnetization and 2) the computation of the current density. For 1), we use standard micromagnetic methods. For 2), we start from a potential difference applied across the sample, compute the resulting electric potential, and subsequently the corresponding current density based on a uniform conductivity. We take into account the backreaction of the magnetoresistive effects onto the current density by self-consistently computing the current density and conductivity until they converge. We compare the experimentally measured AMR curve (as a function of the applied field) with the numerically computed results and find good agreement. The numerical data provides insight into the characteristics of the AMR data. Finally, we demonstrate the importance of taking into account the spatial variation of the current density when computing the AMR

Micromagnetic simulations of magnetoelectric materials

Some magnetic materials show a magnetoelectric coupling between inhomogeneous magnetization patterns and electric polarization that is sufficiently strong to allow external control of magnetization structure by electric fields. Numerical simulations of “magnetoelectric” materials of this type require an extension of the standard micromagnetic model which conceptually parallels the introduction of spin-current interaction terms. We show how the micromagnetic simulator “Nmag” can be extended to support the inhomogeneous magnetoelectric interaction term and also give a simple self-contained example for simulating the micromagnetic dynamics of a magnetoelectric system in the presence of an external electric field.