ORGANIZACIONES DE EDUCACIÓN SUPERIOR (ASF)

User Manual.

LLG Micromagnetics SimulatorTM _{was founded in 1997 as a sole proprietorship by Michael R.}

Scheinfein, who was then a professor of physics at Arizona State University in Tempe, Arizona. In 1997, the first users of LLG were a handful of scientists working on MRAM in corporate research labs. Now, there are over 100 corporate and academic users worldwide, including Europe, Asia and North America.

CHAPTER 3

INTRODUCTION

TO USING LLG

This section provides you with an overview of LLG’s design, of how to use LLG and of LLG’s functionality. For complete details and instructions, refer to the appropriate sections found later in the Manual.

T

M

F

LLG Micromagnetics Simulator has three functional modules. These modules are specified in terms of the serial pro- cess of defining the solutions to most problems, while maintaining consistency with the Windows event-driven pro- gramming interface. These modules are listed below with their corresponding menus.

• Input phase: data specification (LLG Input Sheet)

• Simulation phase: solution of the differential equations (LLG Simulation Sheet)

• Review phase: playback of results through graphical animation (movies) (LLG Movie Viewer)

I

P

: D

S

The LLG Input Sheet is the central interface for coordinating input parameters, error checking and setting critical glo- bal parameters. In general, inputting data specifications is the most tedious aspect of numerical simulations. The pro- gram has been designed to allow you flexibility in customizing simulations; however, this makes the data specification phase time-consuming and increases the risk of input error. Although, as a counter measure, LLG performs exhaus- tive error checking to prevent floating-point exceptions, defining a structure and how well it models an actual material or device is ultimately the user’s responsibility.

Since the program solves the Landau-Lifshitz-Gilbert equations using finite differences for exchange energies and fields, as well as boundary elements for magnetostatic self-energies and fields, the structure of interest must be defined as a grid. The program uses rectangular pixels on a Cartesian grid. Although you can change the material parameters, including eliminating magnetic material altogether, this must be done on a Cartesian grid.

Once you have specified the structure and clicked the Begin Simulation button, LLG initializes all of the arrays, com- putes the demagnetization field coupling tensors and calculates the fields for any boundary conditions. Once these large arrays have been specified, the simulation phase can begin. Also, you are prompted to store the simulation parameters in several files.

S

P

: S

D

E

Once LLG has verified that you have input the data correctly, you have your first chance to set the graphical represen- tation of the data. Many features can be viewed interactively.

Chapter 3: Introduction to Using LLG

R

P

: P

R

G

A

M

Once a simulation is complete, you can review the results by replaying them through a graphically animated movie or you can view a domain or field file in the viewer control. Graphical representation of the data is essential to compre- hending the results of a simulation. The program provides complete two- and three-dimensional views in the form of bit- map images, contour maps and vector fields.

T

O

Micromagnetic structure, such as that present in surface domain walls, can be extracted with standard methods for the solution to the Landau-Lifshitz-Gilbert equation. Such methods have been given in the literature by Brown [1], LaBonte [1,2], Aharoni [3-9], Hubert [10,11], and Schabes [9,12]. The equilibrium magnetization configuration results from the minimization of the system’s free energy. The energy of a ferromagnetic system is composed of 1) the mean field

exchange energy E_ex between nearest neighbors characterized by the exchange coupling constant A (erg/cm); 2) the

magnetocrystalline anisotropy energy E_K, which describes the interaction of the magnetic moments with the crystal

field characterized by the constant K_v (erg/cm3); 3) the surface magnetocrystalline anisotropy energy E_ks,which cor-

rects for broken symmetry near surfaces in the interaction of the magnetic moments with the crystal field, and is char-

acterized by the constant K_s (erg/cm2); 4) the magnetostatic self-energy E_s, which arises from the interaction of the

magnetic moments with the magnetic fields created by discontinuous magnetization distributions both in the bulk and

at the surface; 5) the external magnetostatic field energy E_h, which arises from the interaction of the magnetic moments

with any externally applied magnetic fields; and 6) the magnetostrictive energy E_r, which arises when mechanical

stress (strains) are applied to a ferromagnetic material thereby introducing effective anisotropy into the system charac-

terized by K_m (erg/cm3).

The solution for the equilibrium magnetization distribution is a constrained boundary value problem in two or three spa-

tial dimensions with the constraint of constant magnetization M_s. The continuous magnetization distribution of a ferro-

magnet is approximated by a discrete magnetization distribution consisting of equal volume cubes (3-D) or rods (2-D). Each individual discretized magnetization cell, interior to the array, will be addressed by the (X, Y, Z) coordinates of its

centroid. There are N_x cells along X, N_y cells along Y, and Nz cells along Z interior to the structure to be modeled.

There is one plane (3-D) or column (2-D) of boundary cells bounding the discretized region. These boundary cells (con- ditions) can reflect the continuous uniform magnetization distribution present within the domains themselves on either side of the structure. If no boundary conditions are specified, the cells at the edges are free. In the absence of surface anisotropy, the normal derivative of the magnetization distribution at the surface is zero [2,13]. In the presence of sur- face anisotropy, the Rado-Weertman boundary conditions is used [13,14].

Fundamental to the solution of the micromagnetic equations is the assumption that the bulk saturation magnetization

Ms (emu/cm3) is constant microscopically throughout the ferromagnet. The parameter Ms represents saturation magne-

tization at room temperature. For most practical systems being considered (Fe, Co or Permalloy), there is little devia-

tion in M_s at room temperature from the 0 K value. The value of the magnetization vector M(r) at each point within the

ferromagnet is the saturation magnetization multiplied by the direction cosines, that is M(r) = (M_x(r), M_y(r), M_z(r)) =

M_sα(r) = M_s (α(r), β(r), γ(r)). The constraint equation implied by the constant magnetization assumption is |α(r)| = 1.

The individual contributions to the energies in this continuum model are calculated by integrating the energy expres- sions over the structure in question. The energy integrals below are integrated over the appropriate dimension, dV. The

exchange energy Eex in the continuum approximation is given by.

The exchange parameter A can be extracted from spin-wave theory [15-17], which shows that A = A'M_s2 = DS/2V,

where D is the spin-wave dispersion parameter, S is the spin per atom and V is the volume per atom. The spin-wave dispersion parameter, D, is related to the exchange constant, J, in the Heisenberg hamiltonian by D = 2JSa2, where 'a'