THERMALLY ACTIVATED REVERSAL IN MAGNETIC …THERMALLY ACTIVATED REVERSAL IN MAGNETIC NANOSTRUCTURES...

THERMALLY ACTIVATED REVERSAL IN

MAGNETIC NANOSTRUCTURES

ULRICH NOWAK

Theoretische Physik, Gerhard-Mercator-Universitat Duisburg

47048 Duisburg, Germany

Abstract

The miniaturization of magnetic structures plays an important role

for fundamental research as well as for technical applications. New

experimental techniques allow for a preparation and investigation of

magnetic systems of smaller and smaller spatial extension. This leads

to an incremental interest in the understanding of the behavior of small

magnetic particles and structures down to the nanometer scale. With

decreasing size of magnetic particles thermal activation becomes rel-

evant. The understanding of the role of a finite temperature for the

dynamical behavior and magnetic stability of ferromagnetic particles is

hence a modern subject in micromagnetism. It is interesting from a

fundamental point of view as well as for the application development of

magnetic devices.

The goal of this review is to give an overview on numerical ap-

proaches to thermal activation in magnetic systems as far as they can

be described by classical spin systems. Here, the established methods

are either a numerical solution of the Landau-Lifshitz-Gilbert equation

with Langevin dynamics or Monte Carlo simulations. Special emphasis

is put on the relation between these two methods and on the possibility

to quantify the steps of a Monte Carlo procedure in terms of realistic

time intervals.

1

2

As an application of the numerical techniques this overview includes

a description of thermally activated reversal modes in models for nano-

wires. In these systems different reversal modes can occur like coherent

rotation, nucleation, and curling, depending on the system geometry

and model parameters. Some asymptotic, analytic solutions exist for

the relevant energy barriers and for the escape times so that the models

introduced are relevant also as test tool for the numerical techniques.

CONTENTS 3

Contents

1 Introduction: Magnetism of Nanostructures 5

2 Theoretical Concepts 10

2.1 Classical spin models and the equation of motion . . . . . . . . 10

2.2 Analytical ansatz: calculation of escape rates . . . . . . . . . . . 13

3 Numerical Methods 17

3.1 Langevin dynamics simulations . . . . . . . . . . . . . . . . . . 17

3.2 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Time quantified Monte Carlo simulations . . . . . . . . . . . . . 26

3.4 Tests for the algorithms . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Calculation of the dipolar field by fast Fourier transformation . 35

4 Applications: Reversal in Extended Systems 42

4.1 Coherent rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Multidroplet nucleation . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Size dependence of the characteristic time . . . . . . . . . . . . 50

4.5 Influence of the stray field: curling . . . . . . . . . . . . . . . . 54

5 Summary and Outlook 58

LIST OF FIGURES 4

List of Figures

1 Energy of a Stoner-Wohlfarth particle . . . . . . . . . . . . . . . 14

2 Trajectories of a spin in phase space following the LLG equation

for high and low damping. . . . . . . . . . . . . . . . . . . . . . 19

3 Characteristic time vs. trial step width for a Monte Carlo sim-

ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Characteristic time vs. damping constant: comparison of Langevin

dynamics and Monte Carlo simulation. . . . . . . . . . . . . . . 31

5 Characteristic time vs. temperature: comparison of asymptotic

escape time, Langevin dynamics, and Monte Carlo simulations. . 34

6 Characteristic time vs. update interval of the dipolar fields in a

Monte Carlo simulation. . . . . . . . . . . . . . . . . . . . . . . 39

7 Efficiency of dipolar field calculation by FFT. . . . . . . . . . . 40

8 Snapshots of a spin chain during coherent rotation. . . . . . . . 43

9 Characteristic time vs. temperature for coherent rotation. . . . 44

10 Snapshots of a spin chain during soliton-antisoliton nucleation. . 46

11 Characteristic time vs. temperature during nucleation. . . . . . 47

12 Snapshots of a spin chain during multidroplet nucleation. . . . . 49

13 Diagram showing regimes of different reversal mechanisms for a

spin chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

14 Characteristic time vs. system size for a spin chain. . . . . . . . 53

15 Snapshot of an ellipsoid during switching by a curling mode. . . 57

1 INTRODUCTION: MAGNETISM OF NANOSTRUCTURES 5

1 Introduction: Magnetism of Nanostructures

Many novel physical effects occur in connection with the reduction of the

spatial extension of the systems under investigation. Magnetic materials are

now controllable down to the nanometer scale leading to a broad interest in the

understanding of the magnetism of small magnetic structures and particles [1]

due to the broad variety of industrial applications. Especially the dynamic

behavior of interacting spin systems is a topic of considerable current interest,

much of this interest being driven by the need to understand new spin electronic

devices [1, 2].

In nanostructured systems thermal activation can be relevant for the mag-

netic stability even at room temperature [3], and for theoretical investigations

numerical methods are thus desirable which treat realistic magnetic models

including the effects of thermal activation. This review focuses on the problem

of thermally activated magnetization reversal in nanoparticles, on its physical

principles as well as on appropriate numerical methods. The two most estab-

lished numerical methods in this context are Monte Carlo [4] and Langevin

dynamics [5] simulations. Special emphasis will be laid on the relation be-

tween these different methods, which directly leads to the interesting problem

of how to relate a Monte Carlo algorithm to a realistic dynamics [6]. But for a

systematic strategy let us first of all review the effects occurring in connection

with the decreasing size of ferromagnetic systems.

A macroscopic ferromagnet which might be used as permanent magnet in

general is in a multidomain state [7, 8]. Here, the magnetic behavior follows

from the atomic microstructure of the underlying material as well as from the

balance of the different occurring magnetic energy contributions. The latter are

mainly ferromagnetic exchange, crystalline anisotropies, Zeeman contributions

from an external magnetic field, and stray field contributions which depend es-

pecially on the shape of the magnetic system. The calculation of the domain

structures for a given magnetic system is a ground state optimization problem

where the biggest challenge is the inclusion of the stray field energy, or — in a

localized spin picture — the calculation of the dipole-dipole interactions. The


understanding of the arising domain structures on the basis of a continuum

theory is usually called “domain theory” [7]. The corresponding calculations

build the fundament for the understanding of magnetic material properties like

coercivity, anisotropies and hysteresis among others. Even though numerical

methods for the solution of this optimization problem exist and also commer-

cial software is available one should note that there are still many unresolved

problems. This is evidenced by the fact that different programs lead to differ-

ent results for the so-called “standard problem #2” which is a defined test tool

in this field (see e. g. [9] and references therein). For a review on domain theory

see the book of Hubert and Schafer [7] and for a review on the corresponding

numerical methods see Refs. [10] and [11].

With decreasing size of the system due to the ferromagnetic exchange a

domain structure which necessarily must contain domain walls becomes en-

ergetically unfavorable so that sufficiently small magnetic particles are in a

long-range ordered single-domain state [7]. In conventional magnetic materi-

als the single-domain limit is — depending on the specific material parame-

ters — somewhere below the micrometer scale. Single-domain particles are

expected to become very important for technical applications since they are

proposed to have good qualities for magnetic storage. Especially arrays of iso-

lated, nanometer-sized particles are thought to enhance the density of magnetic

recording [1].

On the other hand, there is an ultimate lower limit for the size of particles

that can be used for magnetic storage, and consequently there is also an upper

limit for the storage density which is called the superparamagnetic limit [3,8].

To understand the concept of superparamagnetism consider a simple Ising

ferromagnet of finite size. In such a system the two stable ordered states are

separated by a finite energy barrier, so that at non-zero temperatures a finite

probability exists to overcome this barrier by thermal fluctuations. As a result

the time averaged magnetization will be zero on time scales that are larger than

the characteristic switching time of the system. Whether the magnetization of

such a superparamagnetic particle is measured to be zero or not is a question

of the time resolution of the measurement. Therefore, superparamagnetism is


a dynamic effect and not a well defined equilibrium phase. Nevertheless, this

effect is extremely important for possible applications in every magnetic device

with nanometer-sized particles, where the magnetic state has to be stable at

finite temperatures on sufficiently long time scales.

Finally, for still smaller particles or clusters which consist only of a limited

number of atoms, quantum effects set in, leading additionally to reversal modes

which in the low-temperature regime are dominated by the tunnel effect [12,13].

However, in the following we will restrict ourselves to the understanding of the

role of thermal activation for the stability of the magnetization in nanometer-

sized structures and particles. We will call a thermally assisted magnetization

reversal a switching process, in contrast to non-thermal magnetization reversal

driven either by an external field of the order of the coercive field or by quantum

effects.

Measurements of the switching behavior of nanometer-sized particles were

often performed on powders, i. e. ensembles of particles where properties like

the size of the individual particles and the corresponding direction of the

anisotropy axis are distributed [1]. This distribution of relevant properties

complicates the interpretation of the measurements. Only recently, Werns-

dorfer et al. measured the switching time of isolated nanometer-sized parti-

cles [14, 15], and wires [16, 17]. For sufficiently small particles [14] agreement

was found with the theoretical predictions of Neel [18] and Brown [19] who

described the magnetization switching of Stoner-Wohlfarth particles by ther-

mal activation over a single energy barrier following from a coherent rotation

of the magnetization of the particle. For larger particles [15] and wires [16,17]

activation volumes were found which were much smaller than the correspond-

ing particle and wire volumes. One can conclude that here more complicated

switching mechanisms are relevant like nucleation processes with a subsequent

domain wall motion.

Most of the numerical studies of magnetization switching base on Monte

Carlo methods. Here, nucleation phenomena have been studied in Ising mod-

els [20–25] as well as thermally activated hysteresis in models for magneto-

optical materials [26–31]. Vector spin models have been used [24,25,32–34] to


investigate the magnetization reversal in systems with continuous degrees of

freedom. However, Monte Carlo methods — even though well established in

the context of equilibrium thermodynamics — do not allow for a quantitative

interpretation of the results in terms of a realistic dynamics. Only recently, a

Monte Carlo method with a quantified time step was introduced for a simple

test system [6,35] and afterwards applied also to more complex interacting spin

systems [36, 37]. The interpretation of a Monte Carlo step as a realistic time

interval was achieved by a comparison of one step of the Monte Carlo process

with a time interval of a corresponding Langevin equation, i. e., a stochastic

equation of motion which for the case of magnetic moments is based on the

Landau-Lifshitz-Gilbert equation.

Numerical methods for the direct integration of a Langevin equation are

also available [5,38–42] and in the following we will call these numerical meth-

ods Langevin dynamics simulations. Since the Langevin equation for the prob-

lem which we consider here is a stochastic differential equation with a mul-

tiplicative noise term, care has to be taken regarding the validity of different

integration schemes [39]. In general, Langevin dynamics simulations are more

computation time consuming than Monte Carlo methods but, nevertheless,

they are very important since here, naturally, a more realistic time is intro-

duced by the equation of motion.

For any numerical method, analytically solvable models are important as

test tools for the evaluation of the numerical techniques. For some simple

model systems which are analytically treatable, asymptotic formulae for the

escape rates following from corresponding Fokker-Planck equations have been

derived. These simple models include ensembles of isolated Stoner-Wohlfarth

particles [18, 19, 43–47] as well as a one-dimensional model [44, 48–50]. These

analytic solutions should be the starting point for any systematic numerical

investigation of more complex systems.

The organization of the paper is as follows: The next chapter is on the basic

theoretical concepts. The Hamiltonian of a classical spin model for magnetic

systems is introduced as well as the equation of motion for this model, the

Landau-Lifshitz-Gilbert equation with Langevin dynamics. Also, the basis for


the analytic determination of the escape rates of a switching process is dis-

cussed. In Chapter 3 the numerical methods are described, namely Langevin

dynamics simulations and Monte Carlo methods, the latter with special em-

phasis on systems with continuous degrees of freedom and on the time quantifi-

cation problem. Tests for these algorithms are discussed and the calculation of

the dipolar field by fast Fourier transformation is explained as well as the pos-

sible applicability to Monte Carlo methods. Chapter 4 discusses applications

of the different numerical methods to the problem of magnetization switch-

ing in extended systems, where different reversal mechanisms can be observed,

like coherent rotation, single and multidroplet nucleation, and curling. Finally,

chapter 5 gives concluding remarks.

2 THEORETICAL CONCEPTS 10

2 Theoretical Concepts

2.1 Classical spin models and the equation of motion

Throughout this paper we will describe a magnetic system using a model of

classical magnetic moments which are localized on a given lattice. Such a

spin model can be motivated following different lines: on the one hand it

is the classical limit of a quantum mechanical, localized spin model — the

Heisenberg model (see Refs. [8, 51, 52] for the theoretical background). On

the other hand a classical spin model can also be interpreted as the discretized

version of a micromagnetic continuum model, where the charge distribution for

a single cell of the discretized lattice is approximated by a point dipole. From

a microscopic point of view this might be plausible, but for the use of finite

element methods which is the common numerical approach to a continuum

description of the magnetostatic problem, other assumptions for the magnetic

charge distribution are often made, like the sole existence of either surface or

volume charges [7,11]. For certain magnetic systems their description in terms

of a lattice of magnetic moments is based on the mesoscopic structure of the

material, especially when a particulate medium is described [27,28,30,31] or an

ensemble of isolated particles [18, 19, 43–47]. In both cases it is assumed that

one grain or particle can be described by a single magnetic moment. Therefore,

those must be small enough so that internal degrees of freedom are not relevant

for the special problem under consideration.

In general the energy of a classical spin model describing a magnetic system

may contain contributions from exchange, crystalline anisotropies, the external

magnetic field and from dipole-dipole interaction. The latter is the microscopic

origin of the stray field energy and the shape anisotropy. For simplicity let us

assume that the spins are located on a regular lattice and let us neglect any

disorder. Then an appropriate Hamiltonian may be written in the form

H = −J∑

〈ij〉

Si · Sj − dz

∑

i

S2iz − b ·

∑

i

Si − w∑

i<j

3(Si·eij)(eij ·Sj)−Si·Sj

r3ij

, (1)


where the Si = µi/µs are three dimensional magnetic moments of unit length.

µs is the absolute value of the magnetic moment which for an atomic moment is

of the order of a Bohr magneton. The first sum which represents the exchange

of the magnetic moments is usually restricted to nearest neighbor interactions

with the exchange coupling constant J . For J > 0 this part of the Hamilto-

nian leads to ferromagnetic order. The second sum is an example for a uniaxial

anisotropy favoring the z axis as easy axis of the system for positive anisotropy

constant dz. Of course, also other anisotropy terms describing any crystalline,

stress, or surface anisotropies could be considered. The third sum is the cou-

pling of the moments to an external magnetic field with b = µsB. The last

part of the Hamiltonian above is the dipolar interaction with w = µ2sµ0/4πa3,

handled in a point dipole approximation. The rij are the distances between

moments i and j normalized to the lattice spacing a, and the eij are the unit

vectors in the direction of rij. The influence of the dipolar interaction on the

over-all behavior of the system is less obvious. The dipole-dipole interaction

of two moments depends on their distance vector which leads to an effective

anisotropy. Thus, a chain of ferromagnetically ordered spins prefers to have

the magnetization aligned with the chain. This effect is sometimes called shape

anisotropy: elongated particles have an easy axis aligned with the long axis of

the particle (as far as there are no other anisotropies like in the second sum

of the Hamiltonian above). Also, the dipolar interaction leads to the fact that

dipoles try to be aligned, without “open ends” at the surface (free surface

charges). Hence complicated domain structures may arise which minimize the

energy of the Hamiltonian.

All the parameters used in the Hamiltonian above are expressed as energies

per atom. When the Hamiltonian is interpreted as a discretization of a con-

tinuum model on a cubic lattice with lattice constant a, the transformations

to the material parameters usually used in continuum theory are J = 2aAx,

where Ax is here also called the exchange constant, µs = Msa3, where Ms

is the spontaneous magnetization, and dz = Kza3 where Kz is an anisotropy

energy density.

The equation of motion for magnetic moments coupled to a heat bath is the


Landau-Lifshitz-Gilbert (LLG) equation with Langevin dynamics [19] (see [53]

for the Landau Lifshitz equation, [54] for the Gilbert equation and [55] for the

equality of both). For electronic magnetic moments it can be written in the

form

Si = − γ

(1+α2)µs

Si ×(

H i(t) + α Si × H i(t))

, (2)

where γ = 1.76 · 1011(Tesla ∗ ses)−1 is the absolute value of the gyromagnetic

ratio and H i(t) = ζi(t) − ∂H/∂Si. The thermal noise ζ

i(t) obeys

〈ζi(t)〉 = 0 (3)

〈ζiη(t)ζjθ(t′)〉 = δi,jδη,θδ(t − t′)2αkBTµs/γ. (4)

Once again i and j denote the sites of the lattice and η and θ the Cartesian

components. The first part of Eq. 2 describes the spin precession while the

second part includes the relaxation of the moments. α is the dimensionless

damping constant describing phenomenologically the strength of the coupling

to the heat bath. As a consequence of the fluctuation dissipation theorem

it governs the relaxation aspect of the coupling to the heat bath as well as

the fluctuations via the strength of the thermal noise. Usually, values for the

damping constant α are measured to be lower than one [45, 56], but a funda-

mental theoretical derivation is still missing. Note, that the first part of Eq.

2 describing the spin precession can be derived from fundamental quantum

mechanics while the description of the coupling to the heat bath via Langevin

dynamics is phenomenologically. One can solve the LLG equation easily for

an isolated spin coupled to an external field B, neglecting the thermal fluc-

tuations. Then the first term leads to a spin precession with the precession

time τp = 2π(1 + α2)/(γB). The second part of Eq. 2 describes a relaxation

of the spin from an initial state into local equilibrium on the relaxation time

scale τr = τp/α. In other words, α sets the relation between the times scales of

precession and relaxation. Since α is a phenomenological constant the micro-

scopic evaluation of which is missing, there is a lack of knowledge concerning

the time scale of the relaxation.

In the high damping limit which in the following will be important in


connection with Monte Carlo simulations mainly the second term of the LLG

equation is relevant and the time can be rescaled by the factor (1+α2)µs/(αγ).

Hence, this factor should completely describe the α and γ dependence of any

time scale in the high damping limit.

2.2 Analytical ansatz: calculation of escape rates

The LLG equation is a stochastic equation of motion. Starting repeatedly from

identical initial conditions will lead to different trajectories in phase space due

to the influence of the noise. Hence, averages have to be taken in order to

describe the system appropriately. The basis for the statistical description of

an ensemble of systems where each one is described by a Langevin equation is

the corresponding Fokker-Planck (FP) equation. This is a differential equation

for the time evolution of the probability distribution in phase space [57]. In his

pioneering work Brown [19] developed a theoretical formalism for the descrip-

tion of thermally activated magnetization reversal based on the FP equation

which led to low temperature asymptotic formula for the escape rates in simple

magnetic systems (for an overview see [57]).

As a solvable example Brown considered an ensemble of isolated magnetic

moments with a uniaxial anisotropy in an external magnetic field. This system

is described by Eq. 1 for a single spin (thus without exchange and dipole-dipole

interaction) with dz > 0 and arbitrary B. In this model the spin is thought

to represent the magnetic moment of a whole particle being sufficiently small

so that it is always homogeneously magnetized without internal degrees of

freedom. These assumptions are similar to those made within the framework of

Stoner and Wohlfarth [58] for a basic explanation of non-thermal magnetization

reversal (hysteresis). For the simple axially symmetric case, B = −Bz z, the

energy of the particle is only a function of the angle θ between the moment

and the z axis,

E = −dz cos2 θ + bz cos θ. (5)

This function is sketched in Fig. 1. The energy barrier ∆Ecr which has to be

overcome by the magnetic moment during the reversal from an initial orienta-


θ

∆

E

Ecr

0 Π

Figure 1: Sketch of the energy of a Stoner-Wohlfarth particle in a field parallel

to the easy axis.

tion which is antiparallel to the external field to the stable state aligned with

the field is due to the anisotropy of the system. It follows from the maximum

of E with respect to θ and is

∆Ecr = dz(1 − h)2 (6)

with the reduced field h = bz/(2dz). h = 1 is the coercive field of the Stoner-

Wohlfarth model, hence for h > 1 there is no energy barrier. For h < 1

the energy barrier is finite and can be overcome by thermal activation with a

certain probability or, in other words, on a certain time scale.

This time scale is the central quantity for the understanding of thermally

activated dynamics. It can be calculated in certain limits from the so-called

escape rate, i. e. the probability of escape from the metastable state per time

unit due to thermal activation. An escape rate follows either from a calculation

of the flow of the probability current over the energy barrier — this concept

was originally introduced by Kramers for the motion of particles [59] and later

applied to magnetic moments by Brown [19] — or it can be identified with the

lowest non-vanishing eigenvalue of the Sturm-Liouville equation corresponding


to the FP equation, where the lowest eigenvalue can also be obtained numer-

ically [46, 60]. The inverse quantity is then the escape time τ , i. e. the mean

time that is needed for the magnetic moment to overcome the energy barrier.

In the limit of low temperatures it follows a thermal activation law,

τ = τ ∗cr exp

∆Ecr

kBT, (7)

where the prefactor as well as the energy barrier depends on the reversal mech-

anism which one considers. Hence, τ ∗ is not a simple, constant attempt fre-

quency as claimed often by several authors, but a complicated function which

in general may depend on system size, field, anisotropies and even on temper-

ature. Eq. 7 yields the basis for the understanding of superparamagnetism: on

time scales much larger than τ the particles will have switched repeatedly so

that a time averaged measurement of the magnetization will effectively show a

paramagnetic behavior. However, since the dependence of τ on the energy bar-

rier is exponential with increasing particles size (and hence increasing energy

barrier) the particle will be metastable on exponentially long time scales.

After the original work of Brown, extensive calculations have been per-

formed in order to calculate the energy barrier as well as the prefactor asymp-

totically for various model systems. Improved approximations were found for

the axially symmetric case [43, 44, 47] investigating also the different regimes

imposed by the damping parameter α of the LLG equation. Coffey and co-

workers also derived formulae for the non-axially symmetric case where the field

points into an arbitrary direction [45, 46, 61]. This work represents an impor-

tant basis for the understanding of dynamic processes in single-domain parti-

cles, as new experimental techniques allowed for an investigation of nanometer-

sized, isolated, magnetic particles which confirmed this theoretical approach

to thermal activation [14]. Physically, the dynamic behavior of interacting

spin systems is even more interesting. Here, it was Braun who derived asymp-

totic formulae for an extended system, namely a spin chain where also nearest

neighbor interactions were considered [44, 48–50].

All these analytic results are important for both the physical understanding

of thermal activation in magnetic systems as well as for the test of numerical


methods. We will give and discuss the explicit form of some of these asymptotic

formulae later in connection with the comparison with numerical approaches.

In the next chapter we will first introduce the numerical methods which can

be used for the investigation of magnetization switching.

3 NUMERICAL METHODS 17

3 Numerical Methods

Realistic calculations in systems with many degrees of freedom would appear

to be impossible except by computational approaches. The two basic methods

for the simulation of spin systems with many degrees of freedom are Langevin

dynamics simulations and Monte Carlo methods. The following two sections

are devoted to these methods. The third section of this chapter discusses

the relation between these two methods. Section four is on the test of the

numerical methods and the last section is on the special problem of including

long-range interactions efficiently. Concerning numerical methods for the sole

solution of domain structure optimization problems described by continuum

theory the reader is referred to the article of Schrefl et al. [10]. However, for

an inclusion of thermal activation in a continuum theory, time and space have

to be discretized and once this has been done the basic numerical schemes are

the same as those explained in the following for a spin system.

3.1 Langevin dynamics simulations

The basic numerical approach for the description of thermal activation in spin

systems is the direct numerical integration of the Langevin equation of the

problem. Instead of solving the corresponding FP equation one calculates

trajectories in phase space following the underlying equation of motion. In

order to obtain results in the sense of a thermodynamic average one has to

calculate many of these trajectories starting with the same initial conditions,

taking an average over these trajectories for the quantities of interest. This

method is referred to as the Langevin dynamics formalism [5].

The LLG equation with Langevin dynamics (Eq. 2) is a stochastic differ-

ential equation with multiplicative noise. For this kind of differential equation

a problem arises which is called the Ito-Stratonovich dilemma [62]. As a con-

sequence different time discretization schemes may with decreasing time step

converge to different results (see [63] for a discussion of the different discretiza-

tion schemes from a physical point of view). As was pointed out in [39] the


multiplicative noise in the Langevin equation was treated in Browns original

work — and also in subsequent publications — by means of the Stratonovich

interpretation. Hence, in order to obtain numeric results which are compara-

ble to these approaches via the FP equation one has to use adequate meth-

ods. Note, that the simplest method for the integration of first order differ-

ential equations, the Euler method, converges to an Ito interpretation of the

Langevin equation. The simplest appropriate discretization scheme leading to

a Stratonovich interpretation is the Heun method [39,62,63] which is described

in the following.

For simplicity, the Heun discretization scheme is introduced here for a

one dimensional problem. We consider a first order differential equation with

multiplicative noise,

x(t) = f(

x(t), t)

+ g(

x(t), t)

ζ(t), (8)

where ζ(t) represents a noise with a distribution with moments 〈ζ(t)〉 = 0 and

〈ζ(t)ζ(t′)〉 = Dδ(t− t′). The time variable is discretized in intervals ∆t so that

tn = n∆t and xn = x(tn). Then, within the Heun method it is

xn+1 = xn +1

2

(

f(xn, tn) + f(xn+1, tn+1))

∆t (9)

+1

2

(

g(xn, tn) + g(xn+1, tn+1))

ζn.

This method is a predictor-corrector method where the predictor xn+1 is cal-

culated from an Euler integration scheme,

xn+1 = xn + f(xn, tn)∆t + g(xn, tn)ζn.

The ζn are random numbers with a distribution which is characterized by the

two first moments 〈ζn〉 = 0 and 〈ζnζm〉 = D∆tδn,m. This can be achieved by

use of random numbers with a Gaussian distribution, p(ζ) ∼ exp(−ζ2/2σ),

with width σ = D∆t. The generalization of the scheme above to Eq. 2 is

straightforward and can explicitly be found in [39].

As an example, the method above is applied in the following to a simple

test system consisting of a single spin. This spin may represent the magnetic


SxS y

0.60.40.20-0.2

0.40.20-0.2-0.4 Sx 0.60.40.20-0.2

0.40.20-0.2-0.4Figure 2: Trajectories in phase space, (Sx, Sy), for a single spin in a magnetic

field following from a Langevin dynamics simulation for low damping (α =

0.01, left) and high damping (α = 1, right).

moment of a Stoner-Wohlfarth particle, i. e. a particle which is always in a

single-domain state. The material parameters are chosen close to those of a

Co particle of size V = 8×10−24m3, with a uniaxial anisotropy energy density

Ke = 4.2×105J/m3 so that dz = V Ke = 3.36×10−18J in Eq. 1. The magnetic

moment is then µs = 1.12× 10−17J/T and the field is set to |B| = 0.2T under

an angle of 27◦ to the (negative) z axis. The temperature is kBT = 2.5×10−19J

which is 1/3 of the energy barrier so that on the short time scales presented

here the spin remains in the upper energy basin where the simulation starts.

Fig. 2 shows two examples for the time evolution of the spin in the (Sx, Sy)

plane of the phase space resulting from a numerical integration of Eq. 2 using

the Heun method. The simulation starts at Sx = Sy = 0, Sz = 1 which is close

to the local energy minimum at Sy = 0, Sx ≈ 0.22. The spin-precession time

is τp = 9 × 10−11s here and the interval of the time discretization was chosen

much shorter than this, namely ∆t = 5 × 10−13s. Fig. 2 shows simulations

for two different damping constants, both for the same time period of 5τp.


In the limit of low damping, α = 0.01, one can observe the spin precession

around the local energy minimum and the influence of thermal fluctuations.

The spin cannot reach equilibrium during the observation time since the time

for relaxation is much larger here (τr = 100τp for α = 0.01). For high damping,

α = 1, the situation is different: the spin shows no significant precession since

the precession is destroyed by the stronger thermal fluctuations leading to

a random-walk-like motion. After a time of only a few τp an ensemble of

spins would be completely uncorrelated and would reach a local equilibrium

configuration.

These observations will turn out to be important for the connection of

Langevin dynamics simulations and Monte Carlo methods since the latter do

not solve the equation of motion, but do only consider the coupling to the

heat bath, i. e. fluctuations and relaxation. As we will show later this can

be thought to correspond to the high damping limit of Langevin dynamics

simulations.

3.2 Monte Carlo methods

A fundamental physical understanding of thermal activation processes requires

dynamic studies over the whole time range. The Langevin dynamics approach,

although having a firm physical basis, is limited to rather short time scales

since the time interval needed for a single spin precession has to be discretized

sufficiently in order to obtain an acceptable numeric accuracy. Therefore, a

second numerical method turns out to be important namely the Monte Carlo

method.

Monte Carlo methods are well established in the context of equilibrium

thermodynamics [4, 64]. Here, mainly Ising systems have been investigated

due to the broad variety of applications of this class of models in statisti-

cal physics. However, for the description of ferromagnetic materials the use

of Ising models is restricted to the modeling of materials with a very large

uniaxial anisotropy [26–31], while more realistic models have to include finite

anisotropies. In the past such Heisenberg-like spin systems have been investi-


gated by means of Monte Carlo methods mainly for the simulation of critical

phenomena. Examples are the usual ferromagnetic phase transition [65], the

reorientation transition [66], spin-flop phases [67], the singular behavior of the

susceptibility in low-dimensional systems [68], or the modeling of exchange

bias [34]. Since the application of Monte Carlo methods to Ising-like systems

can be found in textbooks [4,64], in the following we will give special emphasis

on the application of Monte Carlo methods to systems with continuous degrees

of freedom and also on the interpretation of the Monte Carlo process in terms

of dynamics.

Within a Monte Carlo approach trajectories in phase space are calculated

following a master equation [69] for the time development of the probability

distribution Ps(t) in phase space,

dPs

dt=

∑

s′(Ps′ws′→s − Psws→s′). (10)

Here, s and s′ denote different states of the system and the w are the transition

rates from one state to another one which have to fulfill the condition [69]

ws→s′

ws′→s

= exp(E(S) − E(S ′)

kBT

)

. (11)

The master equation describes exclusively the coupling of the system to the

heat bath [69]. Hence, only the irreversible part of the dynamics of the system

is considered including only the relaxation and the fluctuations.

As an artefact of the Ising model, here there exists no equation of motion

and the master equation in connection with a single-spin-flip dynamics governs

the so-called Glauber dynamics [70] which is usually thought to yield a qualita-

tively realistic dynamic behavior. For a system of classical magnetic moments

the situation is different due to the existence of an equation of motion— the

Langevin equation.

However, a Monte Carlo simulation does not include the energy conserving

part of an equation of motion. Hence, no precession of magnetic moments

will be found. Having in mind Fig. 2, the spin precession scenario cannot be

simulated by means of a Monte Carlo approach but, on the other hand, the


random-walk like motion which is due to the coupling to the heat bath can

also appear. We will discuss the connection to Langevin dynamics later and

continue with a general description of Monte Carlo algorithms for vector spin

models, as far as they are different from algorithms for Ising models due to

the continuum degrees of freedom.

Within the Monte Carlo method trajectories in phase space following Eq. 10

are calculated usually using single-spin-flip dynamics. For Ising systems [71] as

well as for Heisenberg systems [72] there exist also cluster algorithms which,

depending on the details of the problem, can possibly equilibrate a system

much faster. However, for cluster algorithms the relation of the Monte Carlo

process to a realistic dynamical behavior of the system is even more unclear

so that in the following we will restrict ourselves to the simple case of single-

spin-flip dynamics.

A single-spin-flip algorithm is performed in the following way: at the be-

ginning one single spin from the lattice is chosen either randomly or in some

systematic order and a trial step of this selected spin is made (possible choices

for trial steps will be described in detail below). Then the change of the en-

ergy of the system is computed according to Eq. 1. Finally the trial step is

accepted, for instance with the heat-bath probability,

ws→s′ =w0

1 + exp(E(S′)−E(S)kBT

), (12)

which is one possible choice among others satisfying the condition Eq. 11 for

any arbitrary constant w0. Scanning the lattice and performing the procedure

explained above once per spin (on average) is called one Monte Carlo step

(MCS). It defines a quasi-time scale of the simulation. The connection to

real time — if there is one at all — is in general an open problem which is

settled up to now only in certain cases [6]. The way the trial step is chosen

is of importance for the efficiency of the algorithm as well as for the physical

interpretation of the dynamic behavior of the algorithm [73].

The important difference between the simulation of a system with contin-

uum degrees of freedom and the simulation of an Ising model is the question


of how to choose the trial step mentioned above. For an Ising system the trial

step is naturally to flip the spin. For a Heisenberg spin there are many choices.

One possible trial step which is often used in models with continuous de-

grees of freedom [4] is a small deviation from the initial state. For a spin this

could be a movement of the spin with uniform probability distribution within

a given opening angle around the initial spin direction1. Here, each spin can

only move by small steps and hence, in a model with a uniaxial anisotropy, it

has to overcome the anisotropy energy barrier for a complete reversal (remem-

ber Fig. 1). This might be a realistic choice for many model systems. But on

the other hand if one is for instance interested in the crossover from Heisen-

berg to Ising-like behavior with increasing anisotropy, one has to allow also

for large steps which are able to overcome a given anisotropy energy barrier.

Otherwise the dynamics of the system would freeze and in a system with very

large anisotropy (Ising limit) no spin flip would occur at all [73].

Another possible trial step which circumvents this problem is a step with a

uniform distribution in the entire phase space. Here, an arbitrary spin direction

is chosen at random which does not depend on the initial direction of the spin.

This step samples the whole phase space efficiently and a single spin is not

forced to overcome the anisotropy energy barrier. Instead it is allowed to

change from one direction to any other instantaneously. Both of these trail

steps are allowed choices in the sense that the corresponding algorithms lead

to correct equilibrium properties since they fulfill two necessary conditions:

they are ergodic and symmetric.

Ergodicity demands that time averages yield identical results as ensemble

averages. Thus it must be guaranteed that the whole phase space can be sam-

pled by an algorithm. An example for an non-ergodic algorithm is one which

performs only Ising-like trial steps, Sz → −Sz, in a Heisenberg model. Here,

starting from some initial condition the spin can only reach two positions out

of the whole phase space which is a unit sphere for a Heisenberg spin. Never-

1Note that there are more efficient ways to perform well-defined small trials steps in a

Heisenberg system which then have a non-uniform probability distribution. One will be

described later in connection with Fig. 3.


theless, one is allowed to perform such reflection steps as long as one uses also

other trial steps that guarantee ergodicity. These ideas lead to combinational

algorithms which — depending on the problem — can be very efficient [66,73].

The second condition which has to be fulfilled by any algorithm is a sym-

metry condition: for the probability to do a certain trial step it must be

pt(s → s′) = pt(s′ → s). Otherwise Eq. 11 is not fulfilled since the proba-

bilities to perform certain trial steps contribute to the transition rates. The

symmetry condition would for instance be violated in a Heisenberg system if

one chooses new trial spin directions by simply generating three random num-

bers as Sx, Sy, and Sz coordinates within a cube and normalizing the resultant

vector to unit length. Then before normalization the random vectors are ho-

mogeneously distributed within the cube and after the normalization they have

some non-uniform probability distribution on the unit sphere which is higher

along the diagonal directions of the cube. Hence, trial steps from any other

direction into the diagonal direction are more probable then the other way

round and the algorithm yields wrong results. A description, how to choose

random vectors on a unit sphere with constant probability distribution can be

found in [74]. To the best of our knowledge the most efficient algorithm is that

from Marsaglia [75].

However, as already mentioned above even for different algorithms which

are all correct in the sense that they lead into equilibrium the computation time

needed for a relaxation process can be very different. We illustrate this in Fig.

3 for two test systems: one is a single moment as used in the previous section

for the Langevin dynamics simulation and the other system is a chain of ten

spins interacting via a ferromagnetic nearest neighbor interaction J . In both

cases we consider a uniaxial anisotropy barrier and a field with an arbitrary

angle to the easy axis. The parameters we use are dz/J = 0.1, |b|/J = 0.095

for the chain and |b|/J = 0.022 for the single spin. The angle of the field to

the (negative) z axis is 27◦. We do not consider dipolar interaction (w = 0 in

Eq. 1).

As in the Langevin dynamics simulation of the previous section (Fig. 2) our

simulations start with the magnetic moments in z direction. The magnetic field


single spinspin chainR�2trial step width R

�[MCS]10.1

1e+071e+0610000010000100010010Figure 3: Characteristic time vs. trial step width for a Monte Carlo simulation.

The dotted lines follow a τ ∼ R−2 law.

has a negative z component so that the initial spin state is metastable and the

magnetization will reverse after some time. The time τ when the z component

of the magnetization changes its sign is the characteristic time τ which we

consider here, where averages are taken over 100 simulation runs for the chain

and 1000 for the single spin.

The trial step of our Monte Carlo algorithm used here is a random move-

ment of the magnetic moment up to a certain maximum opening angle. In

order to achieve this efficiently we first construct a random vector with con-

stant probability distribution within a sphere of radius R by use of the rejection

method [74]. This random vector is then added to the initial moment and sub-

sequently the resulting vector is again normalized. Note that the probability

distribution for these trial steps is non-uniform but isotropic, so that the sym-

metry mentioned above is guaranteed. We will call the maximum size of the

trial step, R, in the following trail step width. The trial step width of our

algorithm influences the characteristic time needed for the reversal, and we


investigate this influence by varying R and calculating τ . As usual in a Monte

Carlo procedure the time is measured in Monte Carlo steps (MCS).

As Fig. 3 demonstrates for small trial step widths2, R < 0.2, one finds a

simple power law, τ ∼ R−2. This dependence of τ on R can be understood

by considering the spins as performing a random walk during the simulation.

The width x of a random walk increases in time as x ∼ t1/2. Since R sets the

mean step width of the random walk it is also x/R ∼ t1/2. Thus the time scale

τ needed to attain a certain distance from an initial position obeys τ ∼ R−2.

For larger R the two systems in Fig. 3 behave differently. For the single spin

the switching is even faster for R > 1 since for large enough R the moment

is not forced to overcome the anisotropy energy barrier. Instead it can be

jumped over by a single step. The spin chain behaves completely different:

here with increasing trial step width τ decreases first, reaches a minimum,

and then increases. The reason for this effect is that due to the exchange

interaction large steps are usually not accepted since they are energetically

unfavorable. Therefore an algorithm which often tries to perform large steps

is unefficient here. From the viewpoint of efficiency, one should try to do Monte

Carlo simulations with a trial step width where the characteristic time is at

the minimum. But on the other hand the simple τ ∼ R−2 behavior for small

R gives the possibility to scale the step width in such a way that one MCS can

represent a certain real-time interval in the sense of Langevin dynamics. We

have to search for a relation for R, such, that one MCS corresponds to a real

time interval. We will discuss this problem in the next section.

3.3 Time quantified Monte Carlo simulations

In the following we will show that the random-walk-like high-damping scenario

shown in Fig. 2 can also be simulated by a Monte Carlo simulation since

the exact knowledge of the movement of the spins is not necessary here, in

order to describe the effects of thermal activation. We will derive a theoretical

expression for the time step ∆t which one MCS represents in terms of the trial

2Remember that the spin is normalized to |S| = 1.


step width R of the algorithm, and we will also discuss the conditions under

which this relation should be valid, namely (i) the time scales of interest are

larger than the precession time τp of the moments, (ii) the system is locally

in equilibrium on these time scales. In the high damping limit, α ≈ 1, both

conditions are fulfilled for time scales larger than the precession time since here

the energy dissipation during one cycle of the precession is considerably large

so that the system relaxes (to the local energy minimum) on the time scale of

the precession, i. e., it is τr ≈ τp. For lower values of the damping constant the

conditions above are also fulfilled as long as the relevant times scales are large

enough so that effectively also a high damping scenario is observed (see [45]

for a definition of high and low damping limits).

The main idea is to compare the fluctuations which are established in the

Monte Carlo technique within one MCS with the fluctuations within a given

time scale associated with the linearized stochastic LLG equation [76,77]. We

start with a calculation of the fluctuations following from the Langevin equa-

tion. In general, close to a local energy minimum one can expand the energy

of a system given that first order terms vanish as

E ≈ E0 +1

2

∑

i,j

AijSiSj , (13)

where the Si are now variables representing small deviations from equilibrium.

Let us go back to that system which we discussed already in section 3.1, the

single spin. We consider the simplest case here, namely that the field is either

parallel or antiparallel to the easy axis, b = ±bz z. In this system, we find

equilibrium along the z axis, leading to variables Sx and Sy describing small

deviations from the equilibrium position S = ±z. The energy increase ∆E

associated with fluctuation in Sx and Sy is then simply

∆E ≈ 1

2(AxxS

2x + AyyS

2y), (14)

with Axx = Ayy = 2dz +bz. Rewriting the LLG equation in the linearized form

without the thermal fluctuations,

Sx = LxxSx + LxySy (15)


Sy = LyxSx + LyySy,

we can identify the matrix elements

Lxx = Lyy = − αγ

(1 + α2)µs(2dz + bz)

Lxy = −Lyx =γ

(1 + α2)µs(2dz + bz).

As has been shown in [78] the correlation function for the variables describing

small deviations from equilibrium can be expressed in the form

〈Si(t)Sj(t′)〉 = µijδi,jδ(t − t′). (16)

Here, i and j denote the Cartesian components and Dirac’s δ function is an

approximation for exponentially decaying correlations on time scales t − t′

that are larger than the time scale of the exponential decay τr. The covarianz

matrix µij can be calculated from the system matrices Aij

and Lij

as [78]

µij = −kBT (LikA−1kj + LjkA

−1ki ).

For our problem this yields

µxx = µyy = 2kBTαγ

(1 + α2)µs(17)

µxy = µyx = 0.

Integrating the fluctuating quantities Sx(t) and Sy(t) over a finite time interval

∆t, Eqs. 16 and 17 yield

〈S2x〉 = 〈S2

y〉 = 2kBTαγ

(1 + α2)µs∆t, (18)

representing the fluctuations of Sx(t) and Sy(t) respectively, averaged over a

time interval ∆t.

For comparison, now we calculate the fluctuations 〈S2x〉 which are estab-

lished within one MCS of a Monte Carlo simulation. As algorithm we con-

sider that one used for Fig. 3, described in the previous section. For this


algorithm the probability distribution for trial steps of size r =√

S2x + S2

y is

pt = 3√

R2 − r2/(2πR3) for 0 < r < R. The acceptance probability using a

heat bath algorithm is w(r) = 1/(1 + exp(∆E(r2)/kBT )). Assuming that the

spin is close to its (local) equilibrium position, as before, ∆E(r2) for small r

can be taken from Eq. 14. In order to calculate the fluctuations within one

Monte Carlo step we have to integrate over that part of the phase space which

can be reached within one MCS,

〈S2x〉 =

∫ 2π

0dϕ

∫ R

0r dr

r2

2w(r)pt(r)

=R2

10−O

((2dz + bz)R4

kBT

)

, (19)

where the last line is an expansion for small R leading to the validity condition

R ≪ kBT/(2dz + bz). (20)

By equalizing the fluctuations within a time interval ∆t of the LLG equa-

tion and one MCS we find the relation [6]

R2 =20kBTαγ

(1 + α2)µs∆t (21)

for the trial step width R. Eq. 21 is the central result of this considerations. It

relates one MCS, performed using an algorithm as explained before, with a real

time interval of a Langevin equation. Corresponding relations for other trial

step distributions or other acceptance probabilities, as for instance following

from a Metropolis-algorithm, can be derived as well. Note, that from the

derivation above it follows that one time step ∆t must be larger than the

intrinsic time scale τr of the relaxation. This means — as already mentioned

above — that results from the Monte Carlo method can only be interpreted

on time scales that are clearly larger than the microscopic time scale of the

(local) relaxation of the spin.

In the equation above αγ(1+α2)µs

∆t is the reduced time of the LLG equation,

rescaled in the high damping limit where only the second part of Eq. 2 is

relevant. The more interesting result of Eq. 21 is the temperature dependence


since it turns out that there is no trivial assignment of one MCS to a fixed

time interval. Instead, the larger the temperature is, the larger have to be the

trial steps of the Monte Carlo algorithm in order to allow for the appropriate

fluctuations.

In principle, Eq. 21 gives the possibility to choose the trial step width for a

Monte Carlo simulation in such a way that 1 MCS corresponds to some micro-

scopic time interval, say ∆t = 10−12s. However, there are of course restrictions

for possible values of the trial step width: R must be small enough so that

the truncated expansion in Eq. 19 is a good approximation. Furthermore, R

should be in that region where the R−2 scaling of the Monte Carlo time can be

found (see Fig. 3). On the other hand R should not be too small since other-

wise the Monte Carlo algorithm needs too much computation time to sample

the phase space. Therefore, either one has to choose such a value for ∆t so

that R takes on reasonable values, or one chooses a reasonable constant value

for R and uses Eq. 21 to calculate ∆t as the real time interval associated with

1MCS. In the following we use the first method since it turns out to be very

efficient to change R with temperature. Also, in this case it is much easier

to control the fulfillment of condition 20. However, the other method yields

corresponding results as long as condition 20 is not violated.

In Fig. 4 we compare results from Monte Carlo and from Langevin dynam-

ics simulations. The simulated system is exactly the same as that used for Fig.

2 and partly for Fig. 3 — an isolated spin with a uniaxial anisotropy repre-

senting the magnetic moment of a Co nanoparticle in a field with an oblique

angle to the easy axis. All parameters are chosen as before. We calculate the

characteristic time averaged over 1000 runs for the thermally activated rever-

sal for different values of the damping constant. For the Langevin dynamics

simulation we used as before the Heun method. The Monte Carlo simulation

was performed using the trial step width R as described above following from

Eq. 21 with ∆t ≈ 6×10−12s (the inverse value of γ) so that for the parameters

we used here it is R between 0.22 and 0.06, depending on the value of α.

As Fig. 4 demonstrates the time scales calculated within the two different

numerical methods agree in the high damping limit confirming the validity of


Monte CarloLangevin

��[s]

1011e-081e-09

Figure 4: Characteristic time vs. damping constant: comparison of Langevin

dynamics and Monte Carlo simulation. The dotted line illustrates the (1 +

α2)/α dependence of the Monte Carlo data.

the time quantification approach for the Monte Carlo method. The Monte

Carlo data follow a τ ∼ (1 + α2)/α dependence since this is the general form

for the α dependence of the characteristic time in the high damping limit as al-

ready mentioned above. Note that nevertheless the complete α dependence for

the whole range of damping constants for any model can be more complicated

as is also demonstrated in Fig. 4 by the Langevin dynamics data which reveal

the correct low damping behavior. However, there exist also other examples

where the simple α dependence above seems to describe the whole range of

damping constants [37].

Even though the Monte Carlo time step quantification by Eq. 21 was de-

rived only for the simple system which we considered here [6], it turned out to

be successfully applicable also to more complicated, interacting spin systems

[36, 37] and further work following the lines above is under progress [35, 77].

However, one should note that the method rests on a comparison with Langevin

dynamics. Here, the coupling to the heat bath is added phenomenologically


to the equation of motion leading to a damping constant α, the microscopic

evaluation of which is still missing. In this sense there is still a lack of an

absolute microscopic time scale. Nevertheless, there is at least a non-trivial

connection between Monte Carlo methods and Langevin dynamics.

3.4 Tests for the algorithms

In order to verify the validity of any numerical method, analytical solutions

should be used as test tools before more complex problems are tackled which

cannot be solved analytically. Concerning magnetization switching our goal is

a comparison of characteristic times obtained numerically with those following

from analytical treatments.

During a simulation for temperatures which are low compared to the en-

ergy barrier the system is in the metastable, initial state for a very long time,

while the time needed for the magnetization reversal itself is extremely short.

In this case the characteristic time τ is comparable to the escape time fol-

lowing from a Fokker-Planck equation and also to the so-called metastable

lifetime of the classical nucleation theory3 [79]. The latter is the time required

by a system to build up a supercritical droplet which from then on will grow

systematically and reverse the system [80]. The metastable lifetime was de-

termined numerically in Ising models using similar methods as in the previous

sections [20, 23–25,80]. For low enough temperatures T all the different times

mentioned above are expected to coincide.

For the case of a system of classical magnetic moments there exist some

asymptotic solutions for escape rates for both, isolated spins and a more com-

plicated, interacting system, namely a spin chain. The first category includes

one spin with uniaxial anisotropy and a field parallel to the easy axis [19], a

field in rectangular direction [43, 47], and a field with an oblique angle to the

easy axis [45]. In all these cases approximate formulae for the energy barriers

3The characteristic time and the nucleation time are not comparable when the switching

is fast, e. g. for higher temperatures, as will be discussed in Sec. 4.3 in connection with

multidroplet nucleation.


as well as the prefactors of the thermal activation law (Eq. 7) exist. The latter

system was used as test tool for Langevin dynamics as well as for the time

quantified Monte Carlo simulations in [6]. In the following we will refer to a

slightly different model with two uniaxial anisotropies [44] which is even more

interesting due to the fact that it has been solved for a spin chain including

nearest neighbor interactions too [48]. Since the physics of the interacting sys-

tem is much richer we will discuss it in the next chapter. In the following we

will only use a simple one-spin version as test tool for the numerical methods.

We consider an isolated spin with two uniaxial anisotropy axes, an easy z

axis and a hard x axis,

E = −dzS2z + dxS

2x − µsBSz, (22)

where the field B is parallel or antiparallel to the easy axis. The corresponding

energy barrier ∆E is the same as that of a Stoner-Wohlfarth particle (Eq. 6),

since the additional hard axis does not change the energy of the optimal path in

phase space from one minimum to the other. The escape time was calculated

from the FP equation in the large damping limit [44]. It follows a thermal

activation law (Eq. 7) where the explicit form of the prefactor transformed

into the units used here is

τ ∗cr =

2π(1 + α2)

αγBc

√

d(1 + h)/(1 − h + d)

1 − h2 − d +√

(1 − h2 + d)2 + 4d(1 − h2)/α2. (23)

We introduced the coercive field Bc = 2dz/µs and the reduced quantities h =

µsB/(2dz) and d = dx/dz. Note, that the first term in Eq. 23 is the microscopic

relaxation time of one spin in the field Bc (see section 2.1), while the second

term includes corrections following from the details of the model. The equation

above should hold for low temperatures kBT ≪ ∆Ecr and for B < Bc since

otherwise the energy barrier is zero, leading to a spontaneous reversal without

thermal activation.

For the comparison of the analytic results with our numerical data in Fig.

5 we use the model parameters h = 0.75, d = 10, and α = 4, the latter

in order to be in the high damping limit where the time quantified Monte


MC, no time quantificationLangevin dynamics

Monte Carlo

asymptote

∆E/kBT

τγ/µ

s

876543210

1e+04

1000

100

10

1

Figure 5: Reduced characteristic times vs. inverse temperature. Comparison

of the asymptotic escape time with results from Langevin dynamics and Monte

Carlo simulations, the latter with and without time quantification.

Carlo method should work. Each data point of the reduced time tγ/µs is an

average over 1000 runs. The Langevin dynamics simulations were performed

as explained in Sec. 3.1 using the Heun method. The Monte Carlo simulations

were done with the algorithm using the time quantification according to Eq. 21

with ∆t ≈ 140

γ/µs. As Fig. 5 demonstrates the numerical data of the Langevin

dynamics and the time quantified Monte Carlo simulations agree in the whole

range of temperatures. Comparing with the asymptote, there is a remarkable

agreement in the low temperature limit except of a slight deviation of roughly

15% which might be due to the facts that the formula is only an asymptote,

and that the escape time is not exactly the characteristic time which was

determined numerically. For higher temperatures, kBT > ∆E, the asymptote

is no longer appropriate. The numerical data for τ tend to zero for T →∞. This is obviously correct since an infinite temperature corresponds to an

infinitely strong noise which can flip a spin instantaneously.

For comparison we also performed a conventional Monte Carlo simulation


without time quantification. Here we used — as usual — a constant trial step

width of R = 0.1. Data from an ordinary Monte Carlo simulation do not have

any intrinsic time scale. Instead the unit of (quasi) time is the MCS. For a

qualitative comparison, we rescaled the data therefore arbitrarily so that they

fit into Fig. 5. Obviously, there is not only a lack of an absolute times scale,

even the qualitative behavior turns out to be incorrect in the high temperature

limit since τ remains finite. This behavior follows from the fact that using a

small, constant trial step width one cannot reverse the spin instantaneously.

Instead one always needs a certain minimum number of MCS for the rever-

sal. Only the time quantified algorithm with the temperature dependent trial

step can yield the correct temperature variation of τ . However, even the con-

ventional algorithm will converge to the qualitatively correct behavior in the

limit of low temperatures since the thermal activation law has an exponential

temperature dependence, while possible deviations of the time scale due to the

temperature dependence of R (see Eq. 21) are of the order of T .

Time quantified Monte Carlo methods were applied successfully to other

models as well [6, 35], including also interacting spin systems [36, 37]. We will

discuss some of these results later in connection with applications to extended

systems.

3.5 Calculation of the dipolar field by fast Fourier trans-

formation

Before we are able to apply the numerical methods which we introduced in

the previous sections to real magnetic problems, we have to reconsider the

complete Hamiltonian, Eq. 1. The most time consuming part for any numerical

method which is based on this Hamiltonian is the calculation of the dipole-

dipole interaction. An algorithm which performs a direct summation of the

dipole-dipole interaction would need of the order of N2 calculations for the

energy in an N -spin system. This system size dependence slows down the

efficiency of the algorithm dramatically.

Fortunately, much more elaborate methods based on fast Fourier trans-


formations (FFT) [81] have been developed in the last decade leading to al-

gorithms where the computational effort scales with N log N . The method

was originally worked out by Yuan and Bertram [82] and later generalized to

a scalar charge version [11] which is appropriate for micromagnetic problems

handled by continuum theory. We give here a short introduction to this method

and especially discuss in how far it can be applied to Monte Carlo methods

where certain problems arise as long as single-spin flip methods are used [37].

Optimized cluster algorithms for the Monte Carlo simulation of systems with

long-range interactions were introduced in Refs. [83, 84].

In order to keep the notation simple we consider here a one dimensional sys-

tem, i. e., a spin chain of length L. The generalization to higher dimensions is

however straightforward. First, we rewrite the dipolar part of the Hamiltonian

as

Hdip = −w

2

L∑

i=1

H i · Si. (24)

Then, the task is to calculate the dipolar field at all lattice points i,

Hi =∑

j 6=i

3(Sj · eij)eij − Sj

r3ij

.

We use now Greek indices for the Cartesian components of our vectors (α, β ∈{x, y, z}) and rewrite the dipolar field components as

Hαi =

∑

β

L∑

j=1

W αβij Sβ

j .

As an example we give here the explicit form of the interaction matrix Wij

for

two spins at sites i and j of a chain aligned along the z direction which is

Wij

=1

r3ij

−1 0 0

0 −1 0

0 0 2

with the convention Wii

= 0. Note, that Wij

is only a function of the distance

rij = |i − j|. Hence, there are only L different matrices and the dipolar field


can be calculated from a discrete convolution

Hαi =

∑

β

L∑

j=1

W αβ|i−j|S

βj

which can be performed efficiently by use of the convolution theorem. This

theorem states that the Fourier transform of H i can be expressed by the Fourier

transforms of W|i−j|

and Si as

Hαk =

∑

β

W αβk Sβ

k . (25)

An algorithm using the convolution theorem would first compute the L inter-

action matrices W|i−j|

and then calculate the Fourier transform Wk. This task

has to be performed only once before the simulation starts since Wk

depends

only on the lattice structure and remains constant during the simulation. For

each given spin configuration the dipolar field can then be calculated by first

performing the Fourier transform of the Si, second calculating the product

above following Eq. 25, and third transforming the fields Hk back into real

space, resulting in the dipolar fields H i. Using FFT techniques the algorithm

needs only of the order of L log L calculations instead of L2 [81].

However, for the use of the convolution theorem a few conditions have to

be fulfilled by the system which make the application a bit more complicated.

First, the spin system has to be periodic in space and second, the range of

interaction must be of the same size as the system, i. e. finite. In a realistic

spin system these two conditions are usually not fulfilled at the same time:

either one is interested in finite systems where the interaction has a finite

range but the system is non periodic, or one is interested in the limit of infinite

system size, where usually a finite system with periodic boundary conditions

is simulated. Here, the system is periodic but the dipole-dipole interaction is

of infinite range.

These problems can simply be overcome if one is interested in finite systems.

Then the solution of the problem is called zero padding [81]: one simulates a

system of double size (in one dimension), adding a second system with zero

spins which cuts the interaction to the appropriate range. Using zero padding


one obtains a system which is periodic on the length scale 2L but has an

interaction limited to a finite range, so that both condition above are fulfilled.

The implementation is now straight-forward following the equations above

for the zero-padded system of size 2L. A generalization of the algorithm for

systems with periodic boundary conditions is given in [85]. Note, that the

FFT method for the evaluation of the dipolar fields contains no approximation

[7, 11].

For the Langevin dynamics simulations the dipolar fields at each lattice

point have to be updated in parallel while solving the differential equation.

Hence, one needs of the order of L log L calculations for each time step of the

integration. In a Monte Carlo simulation the implementation is less straight

forward. Here, using a single-spin flip method, the spins are updated one after

the other, and the update of one spin influences the dipolar fields at every other

lattice point. An algorithm which updates the dipolar field by FFT after each

spin update would need of the order of L2 log L calculations for one whole

MCS. These are even more calculations than one needs for a direct calculation

of the changes of the dipolar field following from each (accepted) spin flip [86]

according to Eq. 24 which after the whole MCS needs only of the order of

L2 calculations4. However, in the following we will argue that under certain

conditions it can be a good approximation to recalculate the dipolar fields as

a whole after a certain number of MCS so that one can draw advantage from

the FFT method [37].

Let us consider a Monte Carlo algorithm where the trial steps which are

performed are only small random moves. Then the changes of the dipolar field

due to a spin update are also small. Also, all spins contribute to the dipolar

field and only few of them (depending on the acceptance rate) are updated

within one MCS. In this case one can assume that it is sufficient to update the

dipolar fields in parallel after a certain update interval tu (measured in number

of MCS) which will depend on the size of the trial steps. Fig. 6 investigates

the role of the update interval for the characteristic time determined by Monte

4Here, the prefactor of the proportionality depends strongly on the acceptance rate of

the Monte Carlo procedure [66].


R = 0.044R = 0.062

∆tu

τγ/µ

s

1000100101

10000

6000

2000

Figure 6: Reduced characteristic time vs. update interval of the dipolar field

in MCS for two different trial step widths [37]. The solid line represents the

numerically correct result, the vertical lines indicate the minimum number of

MCS needed for a spin reversal (see text for details).

Carlo simulation of a spin chain. The model parameters are w/J = 0.032,

b/J = −0.15z. We do not assume any crystalline anisotropy. Instead, the

anisotropy here stems exclusively from the dipolar interaction, favoring the z

axis as easy axis of the system. The system size is L = 8. As Fig. 6 shows

the results converge to the correct one already for update intervals which are

even well above 1MCS, depending of course on the trial step width R which

has to be small enough so that only small changes in the system are possible

within the update interval. For comparison one can calculate the number of

MCS needed for a spin reversal under the assumption that the spin is strongly

driven by an external field so that each update in the direction of the reversal

is accepted. Using the algorithm described in section 3.3 the mean step width

from Eq. 21 with acceptance probability w = 1 is R/√

5 so that the minimum

number of MCS for a complete spin reversal is roughly estimated to be at least√5π/R. These values are also indicated in Fig. 6 as vertical lines. As long

as the update time tu is well below this minimum reversal time the Monte


∼ L2

∼ L lnL

L

t[s]

106105104103102

10000

1000

100

10

1

0.1

0.01

Figure 7: CPU time needed for 100 MCS vs. system size [37]. The dipolar

fields are calculated by FFT methods after each MCS (lower curve) and by

direct summation (upper curve).

Carlo method with dipolar field calculation by FFT appears to be a good

approximation5 [37].

In Fig. 7 the advantage which can be drawn from the FFT method is

demonstrated. Here the need of CPU time for 100 MCS of a Monte Carlo

simulation versus system size is shown. The calculations were done on an

IBM RS6000/590 workstation. The dipolar fields are either calculated after

each MCS by the FFT method or by a direct summation of all changes of the

dipolar field following from spin updates. For large enough system sizes both

algorithms scale as expected. The advantage following from the FFT method

is roughly a factor of 5000 for the largest system simulated here (L = 218 =

262144). This is a rather impressing demonstration for the efficiency of the

FFT method which justifies also the use of Monte Carlo algorithms with small

trial steps.

5Hence, this method cannot work for Ising-like systems where the minimum time for a

spin reversal is 1MCS.


For Langevin dynamics simulations the gain of efficiency is of course the

same as demonstrated in Fig. 7. Since here the use of FFT methods is no

approximation it is even more recommendable [87].

4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 42

4 Applications: Reversal in Extended Systems

In the previous section we used very simple systems as test tool for numerical

methods — mostly isolated magnetic moments. We will now turn to physically

more interesting, interacting spin systems which can model the behavior of real

nanowires more realistic. In a nanowire different mechanisms can dominate the

switching behavior depending on the geometry, such as coherent rotation, nu-

cleation, and curling. Coherent rotation and nucleation can be modeled by

a simple spin chain — a model which is also very useful since it was treated

analytically and asymptotic results for the energy barriers as well as for the es-

cape rates are available [44,48,49]. A three dimensional model for an extended

nanowire is discussed at the end of this chapter in connection with curling.

4.1 Coherent rotation

Let us consider a chain of magnetic moments of length L with periodical bound-

ary conditions defined by the Hamiltonian

H =∑

i

[

− JSi · Si+1 − dz(Szi )

2 + dx(Sxi )2 − B · Si

]

. (26)

This is a discretized version of the one dimensional model for a magnetic

nanowire considered by Braun [48]. The material parameters are defined as

in Eq. 1. As in our test system defined by Eq. 22 the z axis is the easy axis

and the x axis the hard axis of the system with anisotropy constants dx = J

and dz = 0.1J . These anisotropy terms may contain contributions from shape

anisotropy as well as crystalline anisotropies [49]. In the interpretation as

shape anisotropy, this single-ion anisotropy is assumed to imitate the influence

of a dipolar interaction of strength w = dz/π [48]. Even though an exact

treatment of the dipolar interactions is possible numerically [37], we neglect

these here so that the results presented are comparable to the analytical work

of Braun [48, 49]. Note however that since we neglect the dipolar interaction,

there is no coupling between spin space and the real space, so that we are

free to choose the easy axis perpendicular to the chain. This simplifies the


time �!Figure 8: Snapshots of a spin chain for various times during coherent rotation.

L = 20, kBT = 0.0019J . Results from Monte Carlo simulation [88].

graphical representation in the following. Nevertheless in a system with dipolar

interaction, the shape anisotropy will favor the spins to be aligned with the

chain.

In the case of small chain length the magnetic moments rotate coherently

minimizing the exchange energy while overcoming the energy barrier which is

due to the anisotropy of the system. In Fig. 8 snapshots of such a reversal

process are shown following from a Monte Carlo simulation. Due to the hard-

axis anisotropy the rotation is mainly in the yz plane. As long as all spins are

mostly parallel, they can be described as one effective magnetic moment which

behaves like the one-spin model described in Sec. 3.4. In the thermal activation

law for the escape time the energy barrier (Eq. 6) is now proportional to the

system size L,

∆Ecr = Ldz(1 − h)2, (27)

while the prefactor is the same as that of a single spin (Eq. 23) since the latter

is not volume dependent [48].

We will now compare the characteristic time τ following from a Langevin

dynamics simulation using the Heun method as well as a Monte Carlo simula-

tion with time step quantification with the asymptotic solutions for the escape

time. Fig. 9 shows the temperature dependence of τ for a given value of the


asymptoteL = 16L = 8L = 4

J/kBT

τγ/µ

s

4003002001000

1e+06

1e+05

1e+04

1000

100

Figure 9: Reduced characteristic time vs. inverse temperature in the case of

coherent rotation for different system sizes [36]. The data are from Monte

Carlo calculations (open symbols) and Langevin dynamics simulations (filled

symbols). Solid lines are the asymptotic escape times (see Eqs. 7, 27 and 23).

external magnetic field (h = 0.75) and three different system sizes. For low

temperatures the data confirm the asymptotic solutions above for smaller sys-

tem sizes. For the largest system shown here (L = 16) the numerical data are

systematically lower than the theoretical prediction. Obviously, the prefactor

τ ∗cr does depend on the system size in contradiction to Eq. 23, while the energy

barrier (Eq. 27) is still correct as follows from the slope of the data. This size

dependence of the prefactor τ ∗cr was explained in [36] with the temperature de-

pendence of the absolute value of the entire magnetic moment of the extended

system.


4.2 Nucleation

With increasing system size nucleation must become energetically favorable

since here the energy barrier is a constant, while it is proportional to the system

size in the case of coherent rotation. For the spin chain which we already

considered in the previous section switching by soliton-antisoliton nucleation

was proposed by Braun [49] for sufficiently large system size. This scenario is

shown in Fig. 10. Here, the nucleation process initiates a pair of domain walls

which splits the system into domains with opposite directions of magnetization

parallel to the easy axis. These two domain walls pass the system in the

subsequent reversal process. Due to the hard-axis anisotropy the spin rotation

is once again mainly in the yz plane. Since these two domain walls necessarily

have opposite helicities in this easy plane they are called a soliton-antisoliton

pair.

The energy barrier ∆Enu which has to be overcome during this nucleation

process is [49]

∆Enu = 4√

2Jdz(tanhR − hR), (28)

with R = arcosh(√

1/h). For vanishing magnetic field this energy barrier

has the form ∆Enu(h=0) = 4√

2Jdz which represents the well-known energy

of two domain walls [89]. The corresponding escape time follows as usual a

thermal activation law where the prefactor has also been calculated in various

limits [49]. Since time quantified Monte Carlo simulations reveal high damping

scenarios, data should be compared with the prefactor obtained in the over-

damped limit (Eq. 5.4 in [49]) which in our units is

τ ∗nu =

2π(1 + α2)

αγBc

(πkBT )1/2(2J)1/4

16L d3/4z |E0(R)| tanhR3/2 sinh R

(29)

with Bc = 2dz/µs. As in Eq. 23 the left fraction is the microscopic relax-

ation time of a spin in the coercive field Bc. The eigenvalue E0(R) has been

calculated numerically in [49]. In the limit h → 1 it is |E0(R)| ≈ 3R2. The

1/L dependence of the prefactor reflects the size dependence of the probability

of nucleation. The larger the system is the more probable is the nucleation


time �!Figure 10: Snapshots of a spin chain for various times during soliton-antisoliton

nucleation. L = 80, kBT = 0.0019. All the other parameters are the same as

in the simulation for Fig. 8. Results from Monte Carlo simulations [88].


asymptoteL = 320L = 80

single nucleus

multiple nucleation

∆Enu/kBT

τγ/µ

s

14121086420

1e+07

1e+06

1e+05

1e+04

1000

100

10

1

Figure 11: Reduced characteristic time vs. inverse temperature during nucle-

ation for two different system sizes. The data are from Monte Carlo (open

symbols) and Langevin dynamics (filled symbols) simulation [36]. Solid lines

are the asymptotic escape times for a single soliton-antisoliton nucleation pro-

cess (see Eqs. 7, 28, and 29) and for multidroplet nucleation (see Eq. 33).

process and the smaller the time scale of the relaxation. Furthermore, the

prefactor has a remarkable√

kBT dependence.

Fig. 11 shows the temperature dependence of the reduced characteristic

time for the same system parameters and field as in the previous section

but for two different, larger system sizes. The formulae above are confirmed

for sufficiently low temperatures (kBT < ∆Enu/8 for the smaller system and

kBT < ∆Enu/10 for the larger system). This is in contrast to the case of

coherent rotation where the analytic asymptotes were confirmed for all tem-

peratures kBT < ∆Enu (see Fig. 5). In the whole temperature range the

numerical data from Langevin dynamics and Monte Carlo simulations with

time step quantification agree, demonstrating once more the validity of the


time quantification approach [6]. In the range of intermediate temperatures

(∆Enu/10 < kBT < ∆Enu) the numerical data deviate from the formulae

above. Also, in this region the characteristic times do not depend on system

size in contrast to Eq. 29 where a 1/L dependence occurs due to the size de-

pendence of the nucleation probability. In the next section we will explain that

these deviations are due to a so-called multidroplet nucleation.

Concluding this section we should remark that all the results above are for

systems with periodic boundary conditions which restricts the applicability

to ”real nanowires” where nucleation processes may start at the sample ends.

Therefore, the case of open boundaries was also considered analytically [50,90]

as well as numerically [37]. Even though the prefactor of the thermal activation

law could not be obtained up to now, it was shown [50] that the energy barrier is

just halved in that case due to the fact that in systems with open boundaries

the nucleation can set in at only one end. Hence, solitons and antisolitons

do not necessarily emerge pairwise. In the case that two solitons (or two

antisolitons) nucleate at both ends these cannot annihilate easily in the later

stage of the reversal process due to their identical helicity. Instead a 360◦

domain wall remains in the system.

4.3 Multidroplet nucleation

Let us now investigate the intermediate temperature range mentioned before.

The corresponding switching behavior is depicted in Fig. 12. Due to the larger

thermal fluctuations as compared to the soliton-antisoliton nucleation pre-

sented in Fig. 10, several nuclei grow simultaneously. Obviously, depending on

temperature (and also on other quantities like system size and field) with a

certain probability many nuclei may arise during the time period of the rever-

sal process. This multiple nucleation process was investigated mainly in the

context of Ising models [80] where it is called multidroplet nucleation.

The characteristic time τmn for the multidroplet nucleation can be esti-

mated with respect to the escape time for a single nucleation process with

the aid of the classical nucleation theory [79]. Here, the following scenario is


time �!Figure 12: Snapshots of a spin chain for various times during multidroplet

nucleation. L = 120, h = 0.95, and kBT = 0.038J . The other parameters are

as before in Figs. 8 and 10. Results from Monte Carlo simulation [88].


assumed: in the first stage many nuclei of critical size arise within the same

time interval. Later these nuclei expand with a certain domain wall velocity v

and join each other. This leads to a change of magnetization

∆M(t) =∫ t

0

(2vt′)D

τnu

dt′ (30)

after a time t in D dimensions. The characteristic time when half of the system

(LD/2) is reversed is then given by [36, 80]

τmn =( L

2v

) DD+1

(

(D+1)τ ∗nu

) 1

D+1 exp∆Enu

(D+1)kBT. (31)

The domain wall velocity in a spin chain following the LLG equation can be

calculated neglecting thermal fluctuations [89]. For small fields it is

v = γB/α (32)

Hence for the one dimensional system which we consider here the characteristic

time is given by

τmn =

√

αLτ ∗nu

γBexp

∆Enu

2kBT. (33)

This means that the (effective) energy barrier for the multidroplet nucleation

is reduced by a factor 1/2 and the characteristic time does no longer depend

of the system size, since τ ∗nu for the soliton-antisoliton nucleation has a 1/L

dependence (see Eq. 29).

In Fig. 9 the asymptote above is confirmed for intermediate temperatures,

its reduced effective energy barrier as well as the value of the prefactor itself.

The latter includes also the value of the domain wall velocity taken from Eq. 32

[36]. Also, the fact that the prefactor does not depend on the system size is

directly confirmed. A similar crossover from single to multidroplet excitations

was observed in Ising models, field dependent [21–23] as well as temperature

dependent [25].

4.4 Size dependence of the characteristic time

The different reversal mechanisms mentioned in the previous sections can occur

within the same model system — the spin chain — depending on the system


size among other parameters. The crossover from coherent rotation to soliton-

antisoliton nucleation was studied in [90] for a periodic system. Here, the

value Lc of the chain length below which only uniform solutions of the Euler-

Lagrange equations of the problem exist (regarding coherent rotation) was

calculated to be

Lc = π

√

2J

dz(1 − h2). (34)

For vanishing magnetic field this crossover length scale is Lc = π√

2J/dz, a

value that is clearly related to the domain wall width δ =√

J/2d [7,89], due to

the fact that two domains walls have to fit into the system during the nucleation

process6. For a chain with open boundary conditions the crossover length scale

is halved since here only one domain wall has to fit into the system [90].

One can understand this result also from a slightly different point of view,

namely by comparing the energy barrier of soliton-antisoliton nucleation (Eq.

28) with that of coherent rotation (Eq. 27). This results in a very similar

condition for the crossover from coherent rotation to nucleation [36] which can

also be generalized to higher dimensions [24].

For even larger system size multiple nucleation becomes probable. Com-

paring the escape time for soliton-antisoliton nucleation with the characteristic

time for multiple nucleation, we get for the intersection of these two times the

crossover condition

Lsm =√

γBτ ∗nuLsm/α exp

∆Enu

2kBT. (35)

The corresponding time Lsm/v is the time that a domain wall needs to cross

the system. In other words, as long as the time needed for the nucleation

event itself is large compared to the time needed for the subsequent reversal

where the walls have to cross the system, one single nucleus determines the

characteristic time. In the opposite case many nuclei will appear during that

time interval, where the first soliton-antisoliton pair crosses the system result-

6Even though the domain wall profile can be derived in a one dimensional model, the

resulting domain wall width is defined either with or without a factor π in literature.


Lc

non-t

her

mal

Lsm

coherent rotation

nucleation

h

L

1.210.80.60.40.20

120

100

80

60

40

20

0

Figure 13: Diagram showing the regions where different reversal mechanisms

occur in the system size vs. field plane. The lines correspond to Eq. 34 and

35, the latter for kBT = 0.006J . The data points are results from Monte Carlo

simulations confirming Eq. 34 (see [36] for details).

ing in multidroplet nucleation. These considerations are also comparable to

calculations in Ising models [21].

A diagram showing the regions where the different reversal mechanisms oc-

cur in our model in the system size vs. field plane is presented in Fig. 13. The

crossover line Lc mentioned above separates the coherent rotation region from

that of soliton-antisoliton nucleation. For h > 1 the reversal is non-thermal.

In the nucleation region, for larger fields a temperature dependent crossover

to multidroplet nucleation sets in. The lower the temperature the more van-

ishes this region. The diagram above was confirmed in parts by Monte Carlo

simulations [36]. The corresponding data points stemming from a quantita-

tive characterization of the reversal mechanism by means of a calculation of

appropriate correlation functions are also shown.

We will now give an example for the surprising effects which can occur while


nucleusmultiple nucleation

single

coherent rotation

L

τγ/µ

s

1000100101

1e+05

1e+04

1000

100

Figure 14: Reduced characteristic time vs. system size for kBT = 0.024J

(triangles) and kBT = 0.016J (circles). h = 0.75. Solid lines are piecewise the

appropriate asymptotes and the data are from Monte Carlo simulations [36].

changing the switching mechanisms of a system by a variation of the system

size. Fig. 14 shows the system size dependence of the reduced characteristic

time when crossing the diagram of Fig. 13 for h = 0.75. Results from Monte

Carlo simulations are shown as well as the appropriate asymptotes for two

different temperatures.

For small system sizes the spins rotate coherently. Here the energy barrier

(Eq. 27) is proportional to the system size leading to an exponential increase

of τ with system size. Following Eq. 23 the prefactor of the thermal activation

law should not depend on L but as already mentioned, numerically one finds

slight deviations from the asymptotic expressions stemming probably from the

non-constant magnetization of extended systems. In the region of soliton-

antisoliton nucleation the energy barrier does not depend on the system size

but the prefactor varies as 1/L (see Eqs. 28 and 29). Interestingly, this leads

to a decrease of the characteristic time with increasing system size. Therefore,

there is a maximum characteristic time close to that system size where the


crossover from coherent rotation to nucleation occurs. This decrease ends

where multidroplet nucleation sets in, following Eq. 35. For still larger systems

the characteristic time has a constant value which is given by Eq. 33.

Note, that qualitatively the same behavior will be found in the particle size

dependence of the so-called dynamic coercivity which is the coercive field one

observes during hysteresis on a given time scale τ : solving Eq. 7 describing the

thermal activation in the three regimes explained above for h(L) at constant τ

one finds an increase of the dynamic coercivity in the coherent rotation regime,

a decrease in the nucleation regime, and at the end a constant value for multiple

nucleation. These findings are qualitatively in agreement with measurements

of the size dependence for the dynamic coercivity of barium ferrite recording

particles [91].

4.5 Influence of the stray field: curling

In the last sections we considered a model which even though it is one dimen-

sional shows properties which are far from being trivial since different switching

mechanisms can occur. Due to the fact that the system is one dimensional,

asymptotic solutions for the escape times could be derived analytically which

enhances its value [48,49]. Many of the findings obtained using this model are

relevant for real magnetic nanowires, as long as those are thin enough to be ef-

fectively one dimensional. Nevertheless, for a realistic description of magnetic

nanoparticles one needs three dimensional models, and one has to consider

the dipole-dipole interaction. In the following we will discuss in how far the

physics of the switching process changes when one considers a three dimen-

sional model including dipole-dipole interaction. Only few numerical results

exist so far, some of them we will discuss in the following.

Considering the mathematical form of the dipole-dipole interaction in Eq. 1

one notes that dipoles prefer to be aligned. Hence, spins try to build up closed

loops or vortices. On the other hand, a loop has an enhanced exchange energy.

Therefore to calculate the spin structure of an extended magnetic system is a

complicated optimization problem. Even a magnetic nanostructure which is


small enough, so that in equilibrium the system is in a single-domain state,

could reverse its magnetization by more complicated modes than coherent

rotation or nucleation. A characteristic length scale below which it cannot be

energetically favorable for the system to break the long range order and split

into domains is the so-called exchange length δx [7]. Like the domain wall

width δ =√

J/2d mentioned already before [7, 89] it is a characteristic length

scale for a given material. For a spin model it can be derived in the following

way: a twist of the direction of the spins by an angle of π over a length scale

l (number of spins) costs an exchange energy of

∆Ex = −Jl

∑

i=1

(1 − Si · Si+1) ≈ −Jl

∑

i=1

(θi − θi+1)2/2 ≈ Jπ2/2l,

assuming constant changes of the angle θ from one spin to the next one7. The

dipolar field energy of a chain of parallel oriented dipoles can be expressed via

Riemann’s Zeta function using

ζ(3) =∞∑

i=1

1

i3≈ 1.202.

Hence, the gain of dipolar energy of a chain of l spins can roughly be estimated

to be at most 3wlζ(3) (see also [66] for a similar calculation in two dimensions).

A comparison of the energies yields the exchange length

δx = π

√

J

6ζ(3)w.

Note that in a continuum theory the dipolar energy is estimated from formulae

for the magnetostatic energy of ellipsoids [7]. The results deviate slightly since

the factor 3ζ(3) is replaced by π. We prefer the expression above derived

directly for a spin model.

Let us now consider a nanowire, i. e. either a thin cylindrical system or

an extremely elongated ellipsoid. As long as the thickness of the particle is

7In the continuum limit this can also be shown to be the wall profile with the minimum

energy by a solution of the corresponding Euler-Lagrange equations.


smaller than the exchange length the magnetization will be homogenous in the

planes perpendicular to the long axis so that the system behaves effectively

one dimensional [50]. For thicknesses larger than the exchange length reversal

modes may occur where the magnetization is non-uniform in the perpendicular

planes. One of such possible reversal modes is called curling [92].

Fig. 15 depicts switching by a curling mode in an elongated rotational

ellipsoid. The surrounding cubic lattice has a size of 10 × 10 × 40. The

result is from a Monte Carlo simulation for kBT = 0.1J , with w = 0.096J

and b = 0.3J . The field was oriented along the easy axis of the particle

which follows exclusively from the dipolar interaction (shape anisotropy) since

no crystalline anisotropies were assumed. Following the equation above the

exchange length is δx ≈ 7 for the system parameters used. The thickness of

the ellipsoid is larger which explains the occurrence of a curling dominated

switching mode. In [41] results from a Langevin dynamics simulation are

presented where the reversal mode of an ellipsoid is nucleation at one sample

end. Here the thickness was below the exchange length.

The existence of the crossover from nucleation to curling was investigated

by simulations of cylindrical systems [37]. Here, for the first time FFT methods

for the calculation of the dipolar fields was combined with Monte Carlo simu-

lations with quantified time step. These methods allowed for the investigation

of particle sizes of up to 32768 spins in three dimensions. A systematic nu-

merical determination of the corresponding energy barriers and characteristic

times is nevertheless still missing.

To conclude this chapter, a nanowire appears to be a very interesting model

system, where depending on the length (compared to the domain wall thickness

δ) and width (compared to the exchange length δx ) of the system one can find

and investigate an astonishingly broad variety of reversal modes, from coherent

rotation, single- and multidroplet nucleation to more complex reversal modes

like curling.


Figure 15: Snapshot of a nanoparticle during reversal by a curling mode [93].

The reversal has started by building an outer vortex in that part where the

ellipsoid has the largest thickness. There is still an inner lengthwise axis where

all spins are pointing into the original up direction, connecting the sample ends

which are also still magnetized up.

5 SUMMARY AND OUTLOOK 58

5 Summary and Outlook

The smaller the spatial dimension of a magnetic system is, the more important

becomes thermal activation for the stability of its magnetic state. The under-

standing of the thermal as well as the dynamic behavior of a magnetic system

is thus an important topic of modern research, much of this interest being

driven by technical applications which demand a perpetual miniaturization of

magnetic particles.

In this review, the two basic numerical techniques for the investigation of

thermal activation in classical spin systems were introduced, namely Monte

Carlo methods and Langevin dynamics simulations. Considering the problem

of thermally activated magnetization reversal, we gave special emphasis on the

possibility of time step quantification of a Monte Carlo algorithm by appropri-

ate adjustment of the trial step width of the Monte Carlo procedure, and on

the fast Fourier transformation method for the calculation of the dipolar field.

The latter is the most computation time consuming part of any simulation

considering a Hamiltonian with long-range interactions.

We discussed mainly models for magnetic nanowires which are extraordi-

nary model systems where depending on the geometry a variety of switching

modes can appear, such as coherent rotation, single- and multidroplet nucle-

ation, and curling. The calculation of the characteristic time scales of the

switching process allows for a test of the validity of the numerical techniques

by comparing with analytical expressions for the escape time derived for sim-

plified model systems. Energy barriers can be obtained from the temperature

variation of the characteristic times.

The numerical treatment of the problem of thermal activation in magnetic

nanostructures where each magnetic moment of the Hamiltonian represents the

classical approximation of an atomic spin would demand to simulate systems

with sizes of the order of 106 spins. This is not far from being possible with

the techniques explained in this review and in the near future corresponding

simulations will become an important tool for the investigation of magnetic

nanostructures.

5 SUMMARY AND OUTLOOK 59

Acknowledgments

The author is grateful for discussions and collaboration with H. B. Braun, R.

W. Chantrell, D. A. Garanin, D. Hinzke, A. Hucht, E. C. Kennedy, T. Schrefl,

and K. D. Usadel. This work was supported by the Deutsche Forschungs-

gemeinschaft, and by the EU within the framework of the COST action P3

working group 4.

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