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
1 INTRODUCTION: MAGNETISM OF NANOSTRUCTURES 6
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
1 INTRODUCTION: MAGNETISM OF NANOSTRUCTURES 7
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
1 INTRODUCTION: MAGNETISM OF NANOSTRUCTURES 8
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
1 INTRODUCTION: MAGNETISM OF NANOSTRUCTURES 9
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)
2 THEORETICAL CONCEPTS 11
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
2 THEORETICAL CONCEPTS 12
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
2 THEORETICAL CONCEPTS 13
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-
2 THEORETICAL CONCEPTS 14
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
2 THEORETICAL CONCEPTS 15
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
2 THEORETICAL CONCEPTS 16
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
3 NUMERICAL METHODS 18
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
3 NUMERICAL METHODS 19
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.
3 NUMERICAL METHODS 20
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-
3 NUMERICAL METHODS 21
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
3 NUMERICAL METHODS 22
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
3 NUMERICAL METHODS 23
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.
3 NUMERICAL METHODS 24
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
3 NUMERICAL METHODS 25
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
3 NUMERICAL METHODS 26
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.
3 NUMERICAL METHODS 27
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)
3 NUMERICAL METHODS 28
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
3 NUMERICAL METHODS 29
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
3 NUMERICAL METHODS 30
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
3 NUMERICAL METHODS 31
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
3 NUMERICAL METHODS 32
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.
3 NUMERICAL METHODS 33
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
3 NUMERICAL METHODS 34
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
3 NUMERICAL METHODS 35
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-
3 NUMERICAL METHODS 36
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
3 NUMERICAL METHODS 37
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
3 NUMERICAL METHODS 38
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].
3 NUMERICAL METHODS 39
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
3 NUMERICAL METHODS 40
∼ 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.
3 NUMERICAL METHODS 41
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 43
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 44
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 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 45
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 46
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].
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 47
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 48
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 49
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].
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 50
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 51
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.
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 52
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 53
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 54
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
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 55
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.
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 56
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.
4 APPLICATIONS: REVERSAL IN EXTENDED SYSTEMS 57
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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