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Atomistic spin model simulations of magnetic nanomaterialsEVANS,
R F L, FAN, W J, CHUREEMART, P, OSTLER, Thomas , ELLIS, M O A and
CHANTRELL, R W
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EVANS, R F L, FAN, W J, CHUREEMART, P, OSTLER, Thomas, ELLIS, M
O A and CHANTRELL, R W (2014). Atomistic spin model simulations of
magnetic nanomaterials. Journal of Physics: Condensed Matter, 26
(10), p. 103202.
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Journal of Physics: Condensed Matter
J. Phys.: Condens. Matter 26 (2014) 103202 (23pp)
doi:10.1088/0953-8984/26/10/103202
Topical Review
Atomistic spin model simulations ofmagnetic nanomaterialsR F L
Evans1, W J Fan, P Chureemart, T A Ostler2, M O A Ellis andR W
Chantrell
Department of Physics, The University of York, York YO10 5DD,
UK
E-mail: [email protected]
Received 3 December 2013, revised 2 January 2014Accepted for
publication 7 January 2014Published 19 February 2014
AbstractAtomistic modelling of magnetic materials provides
unprecedented detail about theunderlying physical processes that
govern their macroscopic properties, and allows thesimulation of
complex effects such as surface anisotropy, ultrafast laser-induced
spindynamics, exchange bias, and microstructural effects. Here we
present the key methods usedin atomistic spin models which are then
applied to a range of magnetic problems. We detail
theparallelization strategies used which enable the routine
simulation of extended systems withfull atomistic resolution.
Keywords: magnetism, atomistic model, classical spin model,
Monte Carlo, spin dynamics
(Some figures may appear in colour only in the online
journal)
Contents
1. Introduction 2
2. The atomistic spin model 2
2.1. The classical spin Hamiltonian 32.2. A note on magnetic
units 4
3. System parameterization and generation 4
3.1. Atomistic parameters from ab initio calculations 43.2.
Atomistic parameters from macroscopic
properties 53.3. Atomistic system generation 6
4. Integration methods 6
4.1. Spin dynamics 64.2. Langevin dynamics 74.3. Time
Integration of the LLG equation 74.4. Monte Carlo methods 9
1 www-users.york.ac.uk/∼rfle500/.2 www.tomostler.co.uk.
5. Test simulations 10
5.1. Angular variation of the coercivity 11
5.2. Boltzmann distribution for a single spin 11
5.3. Curie temperature 12
5.4. Demagnetizing fields 12
6. Parallel implementation and scaling 15
7. Conclusions and perspectives 16
Acknowledgments 17
Appendix A. Code structure and design philosophy 17
Appendix B. Atomistic system generation in VAMPIRE 17
Appendix C. Parallelization strategies 18
C.1. Statistical parallelism 18
C.2. Geometric decomposition 18
C.3. Replicated data 20
C.4. Additional scaling tests 20
References 21
0953-8984/14/103202+23$33.00 1 c© 2014 IOP Publishing Ltd
Printed in the UK
Made open access 17 December 2014
Content from this work may be used under the terms of the
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of this work must maintain attribution to the author(s) and the
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
1. Introduction
Atomistic models of magnetic materials, where the atoms
aretreated as possessing a local magnetic moment, originatedwith
Ising in 1925 as the first model of the phase transition ina
ferromagnet [1]. The Ising model has spin-up and spin-downonly
states, and is amenable to analytical treatment, at leastin two
dimensions. Although it is still extensively used inthe study of
phase transitions, it is limited in applicability tomagnetic
materials and cannot be used for dynamic simula-tions. A natural
extension of the Ising model is to allow theatomic spin to vary
freely in 3D space [2, 3] which yieldsthe classical Heisenberg
model, where quantum mechanicaleffects on the atomic spins are
neglected [2]. Monte Carlosimulations of the classical Heisenberg
model allowed thestudy of phase transitions, surface and finite
size effects insimple magnetic systems. The study of dynamic
phenomenahowever was intrinsically limited due to the use of Monte
Carlomethods until the development of dynamic [4, 5] and
stochasticLandau–Lifshitz–Gilbert atomistic spin models [6–8].
Today atomistic simulation of magnetic materials hasbecome an
essential tool in understanding the processesgoverning the complex
behaviour of magnetic nanomaterials,including ultrafast
laser-induced magnetization dynamics[9–11], exchange bias in
core-shell nanoparticles [12–14]and multilayers [5, 15], surface
anisotropy in magneticnanoparticles [16, 17], microstructural
effects [18–20], spinvalves [21] and spin torque [22], temperature
effects andproperties [23–26] and magnetic recording media [27,
28]. Asignificant capability of the atomistic spin model is to
bridgethe gap between ab initio electronic structure calculations
andmicromagnetics by using a multiscale model [29–32]. Sucha model
is able to calculate effective parameters for largerscale
micromagnetic simulations [33], such as anisotropies,and exchange
constants [34]. The atomistic model is alsoable to interface
directly with micromagnetic simulations totackle extended systems
by calculating interface propertiesatomistically while treating the
bulk of the material with amicromagnetic discretization [35, 36].
Despite the broad ap-plicability and importance of atomistic
models, no easy-to-useand open-source software packages are
presently available toresearchers, unlike the mesoscopic
micromagnetic approacheswhere several packages are currently
available [37–39].
Today most magnetic modelling is performed using nu-merical
micromagnetics in finite difference [37] and finiteelement [38, 39]
flavours. The theoretical basis of micromag-netics is that the
atomic dipoles which make up the magneticmaterial can be
approximated as a continuous vector fieldwhere, due to the exchange
interaction, the atomic dipoles ina small finite volume are
perfectly aligned. Micromagneticshas proven to be an essential tool
in understanding a rangeof complex magnetic effects [40–42] but due
to the rapidpace of technological development in magnetic materials
thecontinuum approximation at its heart precludes its applicationto
many problems of interest at the beginning of the 21stcentury, such
as heat assisted magnetic recording [43], ultrafastlaser-induced
demagnetization [44, 45], exchange bias in spinvalves [46], surface
and interface anisotropy [47, 48] and high
anisotropy materials for ultrahigh density recording mediasuch
as FePt [49]. The common theme to these problemsis a sub-nanometre
spatial variation in the magnetizationcaused by high temperatures,
atomic level ordering (anti-and ferrimagnets), or atomic surface
and interface effects. Totackle these problems requires a formalism
to take accountof the detailed atomic structure to express its
impact on themacroscopic behaviour of a nano particle, grain or
completedevice.
Some, but not all, of these problems can adequately betackled by
next-generation micromagnetic approaches uti-lizing the
Landau–Lifshitz–Bloch equation [50–52], whichis based on a
physically robust treatment of the couplingof a macrospin to a heat
bath, allowing the study of hightemperature processes [53],
ultrafast demagnetization [54, 55]and switching [56]. However, true
atomic scale variations ofthe magnetization, as apparent in
antiferromagnets, surfacesand interfaces, still require an
atomistic approach. Additionallywith the decreasing size of
magnetic elements, finite size ef-fects begin to play in increasing
role in the physical propertiesof magnetic systems [57].
In this article we present an overview of the
commoncomputational methods utilized in atomistic spin
simulationsand details of their implementation in the open-source
VAMPIREsoftware package3. Testing of the code is an essential
partof ensuring the accuracy of the model and so we also
detailimportant tests of the various parts of the model and
comparethem to analytic results while exploring some
interestingphysics of small magnetic systems.
VAMPIRE is designed specifically with these problems inmind, and
can easily simulate nanoparticles, multilayer films,interfacial
mixing, surface anisotropy and roughness, core-shell systems,
granular media and lithographically definedpatterns, all with fully
atomistic resolution. In addition, trulyrealistic systems predicted
by Molecular Dynamics simula-tions [19, 20, 59] can also be used
giving unprecedenteddetail about the relationships between shape
and structureand the magnetic properties of nanoparticles. In
addition tothese physical features VAMPIRE also utilizes the
computingpower of multiprocessor machines through
parallelization,allowing systems of practical interest to be
simulated routinely,and large-scale problems on the 100+ nm length
scale to besimulated on computing clusters. Further details of the
VAMPIREcode and its architecture are presented in appendix A.
2. The atomistic spin model
The physical basis of the atomistic spin model is the
local-ization of unpaired electrons to atomic sites, leading to
aneffective local atomistic magnetic moment. The degree
oflocalization of electrons has historically been a
contentiousissue in 3d metals [60], due to the magnetism
originatingin the outer electrons which are notionally loosely
bound tothe atoms. Ab initio calculations of the electron density
[61]show that in reality, even in ‘itinerant’ ferromagnets, the
spinpolarization is well-localized to the atomic sites.
Essentially
3 Details available from vampire.york.ac.uk.
2
vampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.ukvampire.york.ac.uk
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
this suggests that the bonding electrons are unpolarized,
andafter taking into account the bonding charge the
remainingd-electrons form a well-defined effective localized moment
onthe atomic sites.
Magnetic systems are fundamentally quantum mechani-cal in nature
since the electron energy levels are quantized,the exchange
interaction is a purely quantum mechanicaleffect, and other
important effects such as magnetocrystallineanisotropy arise from
relativistic interactions of electronicorbitals with the lattice,
which are the province of ab initiomodels. In addition to these
properties at the electronic level,the properties of magnetic
materials are heavily influencedby thermal effects which are
typically difficult to incorporateinto standard density functional
theory approaches. Thereforemodels of magnetic materials should
combine the quantummechanical properties with a robust
thermodynamic formal-ism. The simplest model of magnetism using
this approach isthe Ising model [1], which allows the atomic
moments one oftwo allowed states along a fixed quantization axis.
Althoughuseful as a descriptive system, the forced quantization
isequivalent to infinite anisotropy, limiting the applicability
ofthe Ising model in relation to real materials. In the
classicaldescription the direction of the atomic moment is a
continuousvariable in 3D space allowing for finite anisotropies
anddynamic calculations. In some sense the classical spin model
isanalogous to Molecular Dynamics, where the energetics of
thesystem are determined primarily from quantum mechanics, butthe
time evolution and thermodynamic properties are
treatedclassically.
2.1. The classical spin Hamiltonian
The extended Heisenberg spin model encapsulates the essen-tial
physics of a magnetic material at the atomic level, wherethe
energetics of a system of interacting atomic moments isgiven by a
spin Hamiltonian (which neglects non-magneticeffects such the as
the Coulomb term). The spin HamiltonianH typically has the
form:
H=Hexc+Hani+Happ (1)
denoting terms for the exchange interaction, magneticanisotropy,
and externally applied magnetic fields respectively.
The dominant term in the spin Hamiltonian is the Heisen-berg
exchange energy, which arises due to the symmetry of theelectron
wavefunction and the Pauli exclusion principle [60]which governs
the orientation of electronic spins in over-lapping electron
orbitals. Due to its electrostatic origin, theassociated energies
of the exchange interaction are around1–2 eV, which is typically up
to 1000 times larger than thenext largest contribution and gives
rise to magnetic orderingtemperatures in the range 300–1300 K. The
exchange energyfor a system of interacting atomic moments is given
by theexpression
Hexc =−∑i 6= j
Ji j Si · S j (2)
where Ji j is the exchange interaction between atomic sitesi and
j , Si is a unit vector denoting the local spin momentdirection and
S j is the spin moment direction of neighbouring
atoms. The unit vectors are taken from the actual atomic
mo-mentµs and given by Si =µs/|µs|. It is important to note herethe
significance of the sign of Ji j . For ferromagnetic materialswhere
neighbouring spins align in parallel, Ji j > 0, and
forantiferromagnetic materials where the spins prefer to
alignanti-parallel Ji j < 0. Due to the strong distance
dependenceof the exchange interaction, the sum in equation (2) is
oftentruncated to include nearest neighbours only. This
significantlyreduces the computational effort while being a good
approxi-mation for many materials of interest. In reality the
exchangeinteraction can extend to several atomic spacings [29,
30],representing hundreds of pairwise interactions.
In the simplest case the exchange interaction Ji j isisotropic,
meaning that the exchange energy of two spinsdepends only on their
relative orientation, not their direction.In more complex
materials, the exchange interaction forms atensor with
components:
JMi j =
[Jxx Jxy JxzJyx Jyy JyzJzx Jzy Jzz
], (3)
which is capable of describing anisotropic exchange
interac-tions, such as two-ion anisotropy [29] and the
Dzyaloshinskii–Moriya interaction (off-diagonal components of the
exchangetensor). In the case of tensorial exchange HMexc, the
exchangeenergy is given by the product:
HMexc =−∑i 6= j
[Six S
iy S
iz] [Jxx Jxy Jxz
Jyx Jyy JyzJzx Jzy Jzz
]Sjx
S jyS jz
. (4)Obtaining the components of the exchange tensor may bedone
phenomenologically, or via ab initio methods such asthe
relativistic torque method [62–65] or the spin-clusterexpansion
technique [30, 66–68]. The above expressionsfor the exchange energy
also exclude higher-order exchangeinteractions such as three-spin
and four-spin terms. In mostmaterials the higher-order exchange
terms are significantlysmaller than the leading term and can safely
be neglected.
While the exchange energy gives rise to magnetic orderingat the
atomic level, the thermal stability of a magnetic materialis
dominated by the magnetic anisotropy, or preference for theatomic
moments to align along a preferred spatial direction.There are
several physical effects which give rise to anisotropy,but the most
important is the magnetocrystalline anisotropy(namely the
preference for spin moments to align with particu-lar
crystallographic axes) arising from the interaction of
atomicelectron orbitals with the local crystal environment [69,
70].
The simplest form of anisotropy is of the single-ionuniaxial
type, where the magnetic moments prefer to alignalong a single
axis, e, often called the easy axis and is aninteraction confined
to the local moment. Uniaxial anisotropyis most commonly found in
particles with elongated shape(shape anisotropy), or where the
crystal lattice is distortedalong a single axis as in materials
such as hexagonal Cobalt andL10 ordered FePt. The uniaxial
single-ion anisotropy energyis given by the expression:
Huniani =−ku∑
i
(Si · e)2 (5)
3
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
where ku is the anisotropy energy per atom. Materials witha
cubic crystal structure, such as iron and nickel, have adifferent
form of anisotropy known as cubic anisotropy. Cubicanisotropy is
generally much weaker than uniaxial anisotropy,and has three
principal directions which energetically areeasy, hard and very
hard magnetization directions respectively.Cubic anisotropy is
described by the expression:
Hcubani =kc2
∑i
(S4x + S
4y + S
4z
)(6)
where kc is the cubic anisotropy energy per atom, and Sx , Sy
,and Sz are the x , y, and z components of the spin moment
Srespectively.
Most magnetic problems also involve interactions be-tween the
system and external applied fields, denoted as Happ.External fields
can arise in many ways, for example a nearbymagnetic material, or
as an effective field from an electriccurrent. In all cases the
applied field energy is simply given by:
Happ =−∑
i
µsSi · Happ. (7)
2.2. A note on magnetic units
The subject of magnetic units is controversial due to
theexistence of multiple competing standards and historical
ori-gins [60]. Starting from the atomic level however, the
dimen-sionality of units is relatively transparent. Atomic
momentsare usually accounted for in multiples of the Bohr
magneton(µB), the magnetic moment of an isolated electron, with
unitsof J T−1. Given a number of atoms of moment µs in a volume,the
moment per unit volume is naturally in units of J T m−3,which is
identical to the SI unit of A m−1. However, thedimensionality
(moment per unit volume) of the unit A m−1
is not as obvious as J T−1m−3, and so the latter form is
usedherein.
Applied magnetic fields are hence defined in Tesla, whichcomes
naturally from the derivative of the spin Hamiltonianwith respect
to the local moment. The unit of Tesla for appliedfield is also
beneficial for hysteresis loops, since the areaenclosed a typical
M–H loop is then given as an energy density(J m−3). A list of key
magnetic parameters and variables andtheir units are shown in table
1.
3. System parameterization and generation
Unlike micromagnetic simulations where the magnetic systemcan be
partitioned using either a finite difference or finiteelement
discretization, atomistic simulations generally requiresome a
priori knowledge of atomic positions. Most simplemagnetic materials
such as Fe, Co or Ni form regular crystals,while more complex
systems such as oxides, antiferromagnetsand Heusler alloys possess
correspondingly complex atomicstructures. For ferromagnetic metals,
the details of atomicpositions are generally less important due to
the strong parallelorientation of moments, and so they can often
(but not always)be represented using a simple cubic discretization.
In contrast,the properties of ferrimagnetic and antiferromagnetic
materials
Table 1. Table of key variables and their units.
Variable Symbol Unit
Atomic magnetic moment µs Joules/Tesla (J T−1)Unit cell size a
Angstroms (Å)Exchange energy Ji j Joules/link (J)Anisotropy energy
ku Joules/atom (J)Applied field H Tesla (T)Temperature T Kelvin
(K)Time t Seconds (s)
Parameter Symbol Value
Bohr magneton µB 9.2740× 10−24 J T−1
Gyromagnetic ratio γ 1.76× 1011 T−1 s−1
Permeability of free space µ0 4π × 10−7 T2 J−1 m3
Boltzmann constant kB 1.3807× 10−23 J K−1
are inherently tied to the atomic positions due to
frustrationand exchange interactions, and so simulation of these
materialsmust incorporate details of the atomic structure.
In addition to the atomic structure of the material, it is
alsonecessary to parameterize the terms of the spin
Hamiltoniangiven by equation (1), principally including exchange
andanisotropy values but also with other terms. There are
generallytwo ways in which this may be done: using
experimentallydetermined properties or with a multiscale approach
usingab initio density functional theory calculations as input to
thespin model.
3.1. Atomistic parameters from ab initio calculations
Ab initio density functional theory (DFT) calculations
utilizethe Hohenberg–Kohn–Sham theory [71, 72] which states thatthe
total energy E of a system can be written solely in terms
theelectron density, ρ. Thus, if the electron density is known
thenthe physical properties of the system can be found. In
practice,the both electron density and the spin density are used
asfundamental quantities in the total energy expression for
spin-polarized systems [73]. In many implementations
DFT-basedmethods only consider the outer electrons of a system,
sincethe inner electrons play a minimal role in the bonding and
alsopartially screen the effect of the nuclear core. These
effectsare approximated by a pseudopotential which determines
thepotential felt by the valence electrons. In all-electron
methods,however, the core electron density is also relaxed. By
energyminimization, DFT enables the calculation of a wide rangeof
properties, including lattice constants, and in the case ofmagnetic
materials localized spin moments, magnetic groundstate and the
effective magnetocrystalline anisotropy. Standardsoftware packages
such as VASP [74], CASTEP [75, 76] andSIESTA [77] make such
calculations readily accessible. Atpresent determining site
resolved properties such as anisotropyconstants and pairwise
exchange interactions is more involvedand requires ab initio
Green’s functions techniques such asthe fully relativistic
Korringa–Kohn–Rostoker method [78,79] or the LMTO method [80, 81]
in conjunction with themagnetic force theorem [62]. An alternative
approach forthe calculation of exchange parameters is the
utilization
4
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
of the generalized Bloch theorem for spin-spiral states
inmagnetic systems [82] together with a Fourier transformationof
k-dependent spin-spiral energies [83, 84].
A number of studies have determined atomic magneticproperties
from first principles calculations by direct mappingonto a spin
model, including the principal magnetic elementsCo, Ni and Fe [81],
metallic alloys including FePt [29],IrMn [31], oxides [85] and spin
glasses [86], and also bilayersystems such as Fe/FePt [87]. Such
calculations give detailedinsight into microscopic magnetic
properties, including atomicmoments, long-ranged exchange
interactions, magnetocrys-talline anisotropies (including surface
and two-ion interac-tions) and other details not readily available
from phenomeno-logical theories. Combined with atomistic models it
is possibleto determine macroscopic properties such as the Curie
tem-perature, temperature-dependent anisotropies, and
magneticground states, often in excellent agreement with
experiment.However, the computational complexity of DFT
calculationsalso means that the systems which can be simulated with
thismulti scale approach are often limited to small clusters,
perfectbulk systems and 2D periodic systems, while real materials
ofcourse often contain a plethora of defects disrupting the
longrange order. Some studies have also investigated the effectsof
disorder in magnetic systems combined with a spin modelmapping,
such as dilute magnetic semiconductors [88] andmetallic alloys
[89].
Magnetic properties calculated at the electronic level havea
synergy with atomistic spin models, since the electronicproperties
can often be mapped onto a Heisenberg spin modelwith effective
local moments. This multiscale electronic andatomistic approach
avoids the continuum approximations ofmicromagnetics and treats
magnetic materials at the naturalatomic scale.
3.2. Atomistic parameters from macroscopicproperties
The alternative approach to multiscale
atomistic/density-functional-theory simulations is to derive the
parameters fromexperimentally determined values. This has the
advantage ofspeed and lower complexity, whilst foregoing
microscopicdetails of the exchange interactions or anisotropies.
Anotherkey advantage of generic parameters is the possibility
ofparametric studies, where parameters are varied explicitly
todetermine their importance for the macroscopic propertiesof the
system, such as has been done for studies of surfaceanisotropy [17]
and exchange bias [13].
Unlike micromagnetic simulations, the fundamental ther-modynamic
approach of the atomistic model means that allparameters must be
determined for zero temperature. Thespin fluctuations then
determine the intrinsic temperature de-pendence of the effective
parameters which are usually putinto micromagnetic simulations as
parameters. Fortunatelydetermination of the atomic moments,
exchange constants andanisotropies from experimental values is
relatively straightfor-ward for most systems.
3.2.1. Atomic spin moment. The atomic spin moment µs isrelated
to the saturation magnetization simply by:
µs =Msa3
nat(8)
where Ms is the saturation magnetization at 0 K in J T−1m−3,a is
the unit cell size (m), and nat is the number of atoms per
unitcell. We also note the usual convention of expressing
atomicmoments in multiples or fractions of the Bohr magneton,µB
owing to their electronic origin. Taking BCC iron as anexample, the
zero temperature saturation magnetization is1.75 MA m−1 [90], unit
cell size of a = 2.866 Å, this gives anatomic moment of 2.22
µB/atom.
3.2.2. Exchange energy. For a generic atomistic model withz
nearest neighbour interactions, the exchange constant isgiven by
the mean-field expression:
Ji j =3kBTc�z
(9)
where kB is the Boltzmann constant and Tc is the
Curietemperature z is the number of nearest neighbours. � is
acorrection factor from the usual mean-field expression whicharises
due to spin waves in the 3D Heisenberg model [91].Because of this �
is also dependent on the crystal structure andcoordination number,
and so the calculated Tc will vary slightlyaccording to the
specifics of the system. For Cobalt with a Tcof 1388 K and assuming
a hexagonal crystal structure withz = 12, this gives a nearest
neighbour exchange interactionJi j = 6.064× 10−21 J/link.
3.2.3. Anisotropy energy. The atomistic
magnetocrystallineanisotropy ku is derived from the macroscopic
anisotropyconstant Ku by the expression:
ku =Kua3
nat(10)
where Ku in given in J m−3. In addition to the
atomisticparameters, it is also worth noting the analogous
expressionsfor the anisotropy field Ha for a single domain
particle:
Ha =2KuMs=
2kuµs
(11)
where symbols have their usual meaning. At this point it isworth
mentioning that the measured anisotropy is a free energydifference.
While the intrinsic ku remains (to a first approxima-tion)
temperature independent, at a non-zero temperature thefree energy
in the easy/hard directions is increased/decreaseddue to the
magnetization fluctuations. Thus the macroscopicanisotropy value
decreases with increasing temperature, van-ishing at Tc. The
thermodynamic basis of atomistic modelsmakes them highly suitable
for the investigation of suchphenomena, as we show later. Applying
the above, parametersfor the key ferromagnetic elements are given
in table 2.
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Table 2. Table of derived constants for the ferromagnetic
elements Fe, Co, Ni and Gd.
Fe Co Ni Gd Unit
Crystal structure bcc hcp fcc hcp —Unit cell size a 2.866 2.507
3.524 3.636 ÅInteratomic spacing ri j 2.480 2.507 2.492 3.636
ÅCoordination number z 8 12 12 12 —Curie temperature Tc 1043 1388
631 293 KSpin-wave MF correction [91, 92] � 0.766 0.790 0.790 0.790
—Atomic spin moment µs 2.22 1.72 0.606 7.63 µBExchange energy Ji j
7.050× 10−21 6.064× 10−21 2.757× 10−21 1.280× 10−21
J/linkAnisotropy energy [93] k 5.65× 10−25 6.69× 10−24 5.47× 10−26
5.93× 10−24 J/atom
3.2.4. Ferrimagnets and antiferromagnets. In the case
offerrimagnets and antiferromagnets the above methods foranisotropy
and moment determination do not work dueto the lack of macroscopic
measurements, although theestimated exchange energies apply equally
well to the Néeltemperature provided no magnetic frustration (due
to latticesymmetry) is present. In general, other theoretical
calculationsor formalisms are required to determine parameters,
suchas mean-field approaches [9] or density functional
theorycalculations [31].
3.3. Atomistic system generation
In addition to determining the parameters of the spin
Hamil-tonian, an essential part of the atomistic model is the
determi-nation of the nuclear, or atomic, positions in the system.
In themultiscale approach utilizing ab initio parameterization of
thesystem, the spin Hamiltonian is intrinsically tied to the
atomicpositions. The additional detail offered by first
principlescalculations is highly desirable even for perfect
crystals andatomically sharp interfaces, however the computational
com-plexity of the calculations limits the ability to parameterize
aspin Hamiltonian for systems of extended defects over 10 nm+length
scales, including physical effects such as vacancies,impurities,
dislocations and even amorphous materials.
For systems modelled using the nearest neighbour ap-proximation,
the atomic structures are much less restricted,allowing for
simulations of material defects such as interfaceroughness [13] and
intermixing [21], magnetic multilayers[94], disordered magnetic
alloys [9], surface [17] and finitesize effects [57]. VAMPIRE
includes extensive functionality togenerate such systems, the
details of which are included inappendix B. In addition to
crystallographic and molecular sys-tems [95, 96] it is also
possible to investigate magnetic effectsin disordered materials and
nanoparticles by incorporating theresults of Molecular Dynamics
simulations [19, 20, 97].
4. Integration methods
Although the spin Hamiltonian describes the energetics of
themagnetic system, it provides no information regarding its
timeevolution, thermal fluctuations, or the ability to determine
theground state for the system. In the following the
commonlyutilized integration methods for atomistic spin models
areintroduced.
4.1. Spin dynamics
The first understanding of spin dynamics came from
ferro-magnetic resonance experiments, where the
time-dependentbehaviour of a magnetic materials is described by the
equationderived by Landau and Lifshitz [98], and in the modern
formgiven by:
∂m∂t=−γ [m×H+αm× (m×H)] (12)
where m is a unit vector describing the direction of thesample
magnetization, H is the effective applied field actingon the
sample, γ is the gyromagnetic ratio and α is aphenomenological
damping constant which is a property ofthe material. The physical
origin of the Landau–Lifshitz(LL) equation arises due to two
distinct physical effects. Theprecession of the magnetization
(first term in equation (12))arises due to the quantum mechanical
interaction of an atomicspin with an applied field. The relaxation
of the magnetization(second term in equation (12)) is an elementary
formulation ofenergy transfer representing the coupling of the
magnetizationto a heat bath which aligns the magnetization along
the fielddirection with a characteristic coupling strength
determined byα. In the LL equation the relaxation rate of the
magnetizationto the field direction is a linear function of the
dampingparameter, which was shown by Gilbert to yield
incorrectdynamics for materials with high damping [99].
SubsequentlyGilbert introduced critical damping, with a maximum
effectivedamping for α = 1, to arrive at the
Landau–Lifshitz–Gilbert(LLG) equation. Although initially derived
to describe thedynamics of the macroscopic magnetization of a
sample, theLLG is the standard equation of motion used in
numericalmicromagnetics, describing the dynamics of small
magneticelements.
The same equation of motion can also be applied at theatomistic
level. The precession term arises quantum mechan-ically for atomic
spins and the relaxation term now describesdirect angular momentum
transfer between the spins and theheat bath, which includes
contributions from the lattice [100]and the electrons [101]. A
distinction between the macroscopicLLG and the atomistic LLG now
appears in terms of the effectsincluded within the damping
parameter. In the macroscopicLLG, α includes all contributions,
both intrinsic (such asspin–lattice and spin–electron interactions)
and extrinsic (spin-spin interactions arising from demagnetization
fields, surface
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
defects [102], doping [103] and temperature [50]), while
theatomistic LLG only includes the local intrinsic contributions.To
distinguish the different definitions of damping we
thereforeintroduce a microscopic damping parameter λ. Although
theform of the LLG is identical for atomistic and macroscopiclength
scales, the microstructural detail in the atomistic modelallows for
calculations of the effective damping includingextrinsic effects,
such as rare-earth doping [103]. Including amicroscopic damping λ
the atomistic Landau–Lifshitz–Gilbertequation is given by
∂Si∂t=−
γ
(1+ λ2)[Si ×Hieff + λSi × (Si ×H
ieff)] (13)
where Si is a unit vector representing the direction of
themagnetic spin moment of site i , γ is the gyromagnetic ratio
andHieff is the net magnetic field on each spin. The atomistic
LLGequation describes the interaction of an atomic spin momenti
with an effective magnetic field, which is obtained from
thenegative first derivative of the complete spin Hamiltonian,
suchthat:
Hieff =−1µs
∂H∂Si
(14)
where µs is the local spin moment. The inclusion of the
spinmoment within the effective field is significant, in that
thefield is then expressed in units of Tesla, given a Hamiltonian
inJoules. Given typical energies in the Hamiltonian of 10 µeV–100
meV range. This gives fields typically in the range0.1–1000 T,
given a spin moment of the same order as theBohr magneton (µB).
4.2. Langevin dynamics
In its standard form the LLG equation is strictly only
applicableto simulations at zero temperature. Thermal effects
causethermodynamic fluctuations of the spin moments which
atsufficiently high temperatures are stronger than the
exchangeinteraction, giving rise to the
ferromagnetic-paramagnetictransition. The effects of temperature
can be taken into accountby using Langevin Dynamics, an approach
developed byBrown [104]. The basic idea behind Langevin Dynamics
isto assume that the thermal fluctuations on each atomic sitecan be
represented by a Gaussian white noise term. As thetemperature is
increased, the width of the Gaussian distributionincreases, thus
representing stronger thermal fluctuations.The established Langevin
Dynamics method is widely usedfor spin dynamics simulations and
incorporates an effectivethermal field into the LLG equation to
simulate thermaleffects [105–107]. The thermal fluctuations are
representedby a Gaussian distribution 0(t) in three dimensions with
amean of zero. At each time step the instantaneous thermalfield on
each spin i is given by:
Hith =0(t)
√2λkBTγµs1t
(15)
where kB is the Boltzmann constant, T is the system
temper-ature, λ is the Gilbert damping parameter, γ is the
absolutevalue of the gyromagnetic ratio, µs is the magnitude of
the
atomic magnetic moment, and 1t is the integration time step.The
effective field for application in the LLG equation withLangevin
Dynamics then reads:
Hieff =−1µs
∂H∂Si+Hith. (16)
Given that for each time step three Gaussian distributedrandom
numbers are required for every spin, efficient gen-eration of such
numbers is essential. VAMPIRE makes use theMersenne Twister [108]
uniform random number generatorand the Ziggurat method [109] for
generating the Gaussiandistribution.
It is useful at the this point to address the applicabilityof
the atomistic LLG equation to slow and fast problemsrespectively.
In reality the thermal and magnetic fluctuationsare correlated at
the atomic level, arising from the dynamicinteractions between the
atoms and lattice/electron system.These correlations may be
important in terms of the thermalfluctuations experienced by the
atomistic spins. In the conven-tional Langevin dynamics approach
described above, the noiseterm is completely uncorrelated in time
and space. For shorttimescales however, the thermal fluctuations
are correlatedin time, and so the noise is coloured [110]. The
effect ofthe coloured noise is to lessen the effect of sudden
temper-ature changes on the magnetization dynamics. However,
theexistence of ultrafast magnetization dynamics [11, 44], andthat
it is driven by a thermal rather than quantum mechanicalprocess
[111], requires that the effective correlation timeis short, with
an upper bound of around 1 fs. Given thatthe correlation time is
close to the integration timestep, theapplicability of the LLG to
problems with timescales ≥ 1 fsis sound. There will be a point
however where the LLG isno longer valid, where direct simulation of
the microscopicdamping mechanisms will be necessary. Progress has
beenmade in linking molecular dynamics and spin models [100,112,
113] which enables the simulation of spin–lattice inter-actions,
which is particularly relevant for slow problems, suchas
conventional magnetic recording where switching occursover
nanosecond timescales. However, it is also essentialto consider
spin–electron effects [101, 114] necessary forultrafast
demagnetization processes, although the physicalorigins are still
currently debated [115].
4.3. Time Integration of the LLG equation
In order to determine the time evolution of a system ofspins, it
is necessary to solve the stochastic LLG equation,as given by
equations (13) and (16), numerically. The choiceof solver is
limited due to the stochastic nature of the equations.Specifically,
it is necessary to ensure convergence to theStratonovich solution.
This has been considered in detail byGarcia-Palacios [105], but the
essential requirement [116]is that the solver enforces the
conservation of the magni-tude of the spin, either implicitly or by
renormalization. Themost primitive integration scheme uses Euler’s
method, whichassumes a linear change in the spin direction in a
singlediscretized time step, 1t . An improved integration
scheme,known as the Heun method [105] is commonly used, which
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
allows the use of larger time steps because of its use ofa
predictor–corrector algorithm. Other more advanced in-tegration
schemes have been suggested, such as the mid-point method [117] and
modified predictor–corrector midpointschemes [103, 118]. The
principal advantage of the midpointscheme is that the length of the
spin vector is preserved duringthe integration which allows larger
time steps to be used. How-ever for the midpoint scheme the
significant increase in com-putational complexity outweighs the
benefits of larger timesteps [118]. Modified predictor–corrector
schemes [103, 118]reduce the computational complexity of the
midpoint scheme,but with a loss of accuracy, particularly in the
time-dependentdynamics [103]. For valid integration of the
stochastic LLGequation it is also necessary for the numerical
scheme toconverge to the Stratonovich solution [105, 119].
Althoughproven for the midpoint and Heun numerical schemes,
thevalidity of the predictor–corrector midpoint schemes for
thestochastic LLG have yet to be confirmed. On balance theHeun
scheme, despite its relative simplicity, is
sufficientlycomputationally efficient that it is still the most
widely usedintegration scheme for stochastic magnetization
dynamics, andso we proceed to describe its implementation in
detail.
In the Heun method, the first (predictor) step calculatesthe new
spin direction, S′i , for a given effective field H
ieff by
performing a standard Euler integration step, given by:
S′i = Si +1S1t (17)
where
1S=−γ
(1+ λ2)[Si ×Hieff + λSi × (Si ×H
ieff)]. (18)
The Heun scheme does not preserve the spin length and so itis
essential to renormalize the spin unit vector length Si afterboth
the predictor and corrector steps to ensure numericalstability and
convergence to the Stratanovich solution. Afterthe first step the
effective field must be re-evaluated as theintermediate spin
positions have now changed. It should benoted that the random
thermal field does not change betweensteps. The second (corrector)
step then uses the predicted spinposition and revised effective
field Hi
′
eff to calculate the finalspin position, resulting in a complete
integration step givenby:
St+1ti = Si +12
[1S+1S′
]1t (19)
where
1S′ =−γ
(1+ λ2)[S′i ×H
i ′eff + λS
′
i × (S′
i ×Hi ′eff)]. (20)
The predictor step of the integration is performed on everyspin
in the system before proceeding to evaluate the correctorstep for
every spin. This is then repeated many times so that thetime
evolution of the system can be simulated. Although theHeun scheme
was derived specifically for a stochastic equationwith
multiplicative noise, in the absence of the noise term theHeun
method reduces to a standard second order Runge–Kuttamethod [120].
In order to test the implementation of the Heunintegration scheme,
it is possible to compare the calculatedresult with the analytical
solution for the LLG. For the simple
case of a single spin initially along the x-axis in an
appliedfield along the z-axis, the time evolution [121] is given
by:
Sx (t)= sech(λγ H1+ λ2
t)
cos(γ H
1+ λ2t)
Sy(t)= sech(λγ H1+ λ2
t)
sin(γ H
1+ λ2t)
Sz(t)= tanh(λγ H1+ λ2
t).
(21)
The expected and simulated time evolution for a single spinwith
H = 10 T, 1t = 1× 10−15 s and λ= 0.1, 0.05 is plottedin figure 1.
Superficially the simulated and expected timeevolution agree very
well, with errors around 10−6. The errorgives a characteristic
trace the size and shape of which isindicative of a correct
implementation of the Heun integrationscheme.
Ideally one would like to use the largest time step possibleso
as to simulate systems for the longest time. For micromag-netic
simulations at zero temperature, the minimum time stepis a well
defined quantity since the largest field (usually theexchange term)
essentially defines the precession frequency.However, for atomistic
simulations using the stochastic LLGequation with Langevin
dynamics, the effective field becomestemperature dependent. The
consequence of this is that foratomistic models the most difficult
region to integrate is in theimmediate vicinity of the Curie point.
Errors in the integrationof the system will be apparent from a
non-converged valuefor the average magnetization. This gives a
relatively simplecase which can then be used to test the stability
of integrationschemes for the stochastic LLG model. A plot of the
meanmagnetization as a function of temperature is shown in figure
2for a representative system consisting of 22× 22× 22 unitcells
with generic material parameters of FePt with an fcccrystal
structure, nearest neighbour exchange interaction ofJi j = 3.0×
10−21 J/link and uniaxial anisotropy of 1.0×10−23 J/atom. The
system is first equilibrated for 10 ps at eachtemperature and then
the mean magnetization is calculatedover a further 10 ps.
First, comparing the effect of temperature on the
minimumallowable time step, the data show that for low
temperaturesreasonably large time steps of 1× 10−15 give the
correctsolution of the LLG equations, while near the Curie
point(690 K) the deviations from the correct equilibrium valueare
significant. Consequently for simulations studying hightemperature
reversal processes time steps of 1× 10−16 s arenecessary. It should
be noted that the time steps which can beused are
material-dependent—specifically if a material withhigher Curie
temperature is used then the usable time stepswill be
correspondingly lower due to the increased exchangefield.
From a practical perspective a significant advantage ofthe spin
dynamics method is the ability to parallelize theintegration system
by domain decomposition, details of whichare given in appendix C.
This allows the efficient simulationof relatively large systems
consisting of tens or hundreds ofgrains or nano structures with
dimensions greater than 100 nmfor nanosecond timescales, with
typical numbers of spins inthe range 106–108.
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Figure 1. Time evolution of a single isolated spin in an applied
field of 10 T and time step of 1 fs. Magnetization traces (a) and
(c) showrelaxation of the magnetization to the z-direction and
precession of the x component (the y-component is omitted for
clarity) for dampingconstants λ= 0.1 and λ= 0.05 respectively. The
points are the result of direction integration of the LLG and the
lines are the analyticalsolution plotted according to equation
(21). Panels (b) and (d) show the corresponding error traces
(difference between the expected andcalculated spin components) for
the two damping constants for (a) and (c) respectively. For λ= 0.1
the error is below 10−6, while for lowerdamping the numerical error
increases significantly due to the increased number of precessions,
highlighting the damping dependence of theintegration time
step.
Figure 2. Time step dependence of the mean magnetization
fordifferent reduced temperatures for the Heun integration
scheme.Low (T � Tc) and high (T � Tc) temperatures integrate
accuratelywith a 1fs timestep, but in the vicinity of Tc a timestep
of around10−16 is required for this system.
4.4. Monte Carlo methods
While spin dynamics are particularly useful for obtain-ing
dynamic information about the magnetic properties orreversal
processes for a system, they are often not theoptimal method for
determining the equilibrium properties, for
example the temperature-dependent magnetization. The MonteCarlo
Metropolis algorithm [122] provides a natural way tosimulate
temperature effects where dynamics are not requireddue to the rapid
convergence to equilibrium and relative easeof implementation.
The Monte Carlo metropolis algorithm for a classicalspin system
proceeds as follows. First a random spin i ispicked and its initial
spin direction Si is changed randomlyto a new trial position S′i ,
a so-called trial move. The changein energy 1E = E(S′i )− E(Si )
between the old and newpositions is then evaluated, and the trial
move is then acceptedwith probability
P = exp(−1EkBT
)(22)
by comparison with a uniform random number in the range0–1.
Probabilities greater than 1, corresponding with a reduc-tion in
energy, are accepted unconditionally. This procedure isthen
repeated until N trial moves have been attempted, whereN is the
number of spins in the complete system. Each set ofN trial moves
comprises a single Monte Carlo step.
The nature of the trial move is important due to tworequirements
of any Monte Carlo algorithm: ergodicity andreversibility.
Ergodicity expresses the requirement that allpossible states of the
system are accessible, while reversibility
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Figure 3. Schematic showing the three principal Monte
Carlomoves: (a) spin flip; (b) Gaussian; and (c) random.
requires that the transition probability between two states
isinvariant, explicitly P(Si → S′i )= P(S
′
i → Si ). From equa-tion (22) reversibility is obvious since the
probability of aspin change depends only on the initial and final
energy.Ergodicity is easy to satisfy by moving the selected spinto
a random position on the unit sphere, however this hasan
undesirable consequence at low temperatures since largedeviations
of spins from the collinear direction are highlyimprobable due to
the strength of the exchange interaction.Thus at low temperatures a
series of trial moves on the unitsphere will lead to most moves
being rejected. Ideally a moveacceptance rate of around 50% is
desired, since very high andvery low rates require significantly
more Monte Carlo steps toreach a state representative of true
thermal equilibrium.
One of the most efficient Monte Carlo algorithms for clas-sical
spin models was developed by Hinzke and Nowak [123],involving a
combinational approach using a mixture of dif-ferent trial moves.
The principal advantage of this methodis the efficient sampling of
all available phase space whilemaintaining a reasonable trial move
acceptance rate. TheHinzke–Nowak method utilizes three distinct
types of move:spin flip, Gaussian and random, as illustrated
schematically infigure 3.
The spin flip move simply reverses the direction of thespin such
that S′i =−Si to explicitly allow the nucleation of aswitching
event. The spin flip move is identical to a move inIsing spin
models. It should be noted that spin flip moves do notby themselves
satisfy ergodicity in the classical spin model,since states
perpendicular to the initial spin direction areinaccessible.
However, when used in combination with otherergodic trial moves
this is quite permissible. The Gaussian trialmove takes the initial
spin direction and moves the spin to apoint on the unit sphere in
the vicinity of the initial positionaccording to the expression
S′i =Si + σg0|Si + σg0|
(23)
where0 is a Gaussian distributed random number and σg is
thewidth of a cone around the initial spin Si . After generating
thetrial position S′i the position is normalized to yield a spin of
unitlength. The choice of a Gaussian distribution is deliberate
sinceafter normalization the trial moves have a uniform
samplingover the cone. The width of the cone is generally chosen to
betemperature dependent and of the form
σg =225
(kBTµB
)1/5. (24)
Figure 4. Visualization of Monte Carlo sampling on the unit
spherefor (a) random and (b) Gaussian sampling algorithms at T = 10
K.The dots indicate the trial moves. The random algorithm shows
auniform distribution on the unit sphere, and no preferential
biasingalong the axes. The Gaussian trial moves are clustered
around theinitial spin position, along the z-axis.
The Gaussian trial move thus favours small angular changesin the
spin direction at low temperatures, giving a goodacceptance
probability for most temperatures.
The final random trial move picks a random point on theunit
sphere according to
S′i =0
|0|(25)
which ensures ergodicity for the complete algorithm andensures
efficient sampling of the phase space at high tem-peratures. For
each trial step one of these three trial moves ispicked randomly,
which in general leads to good algorithmicproperties.
To verify that the random sampling and Gaussian trialmoves give
the expected behaviour, a plot of the calculatedtrial moves on the
unit sphere for the different algorithms isshown in figure 4. The
important points are that the randomtrial move is uniform on the
unit sphere, and that the Gaussiantrial move is close to the
initial spin direction, along the z-axisin this case.
At this point it is worthwhile considering the
relativeefficiencies of Monte Carlo and spin dynamics for
calcu-lating equilibrium properties. Figure 5 shows the
simulatedtemperature-dependent magnetization for a test system
usingboth LLG spin dynamics and Monte Carlo methods. Agree-ment
between the two methods is good, but the spin dynamicssimulation
takes around twenty times longer to compute due tothe requirements
of a low time step and slower convergence toequilibrium. However,
Monte Carlo algorithms are notoriouslydifficult to parallelize, and
so for larger systems LLG spindynamic simulations are generally
more efficient than MonteCarlo methods.
5. Test simulations
Having outlined the important theoretical and
computationalmethods for the atomistic simulation of magnetic
materials,we now proceed to detail the tests we have refined to
ensurethe correct implementation of the main components of
themodel. Such tests are particularly helpful to those wishing
toimplement these methods. Similar tests developed for
micro-magnetic packages [124] have proven an essential benchmarkfor
the implementation of improved algorithms and codes withdifferent
capabilities but the same core functionality.
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Figure 5. Comparative simulation of
temperature-dependentmagnetization for Monte Carlo and LLG
simulations. Simulationparameters assume a nearest neighbour
exchange of6.0× 10−21 J/link with a simple cubic crystal structure,
periodicboundary conditions and 21952 atoms. The Monte
Carlosimulations use 50 000 equilibration and averaging steps,
while theLLG simulations use 5000 000 equilibration and averaging
stepswith critical damping (λ= 1) and a time step of 0.01 fs. The
valueof Tc ∼ 625 K calculated from equation (9) is shown by the
dashedvertical line. The temperature-dependent magnetization is
fitted tothe expression m(T )= (1− T/Tc)β (shown by the solid line)
whichyields a fitted Tc = 631.82 K and exponent β = 0.334 297.
5.1. Angular variation of the coercivity
Assuming a correct implementation of an integration schemeas
described in the previous section, the first test case of
interestis the correct implementation of uniaxial magnetic
anisotropy.For a single spin in an applied field and at zero
temperature,the behaviour of the magnetization is essentially that
of aStoner–Wohlfarth particle, where the angular variation of
thecoercivity, or reversing field, is well known [125]. This
testserves to verify the static solution for the LLG equation
byensuring an easy axis loop gives a coercivity of Hk = 2ku/µsas
expected analytically. For this problem the Hamiltonianreads
H=−kuS2z −µsS · Happ (26)
where ku is the on-site uniaxial anisotropy constant and Happis
the external applied field. The spin is initialized pointingalong
the applied field direction, and then the LLG equationis solved for
the system, until the net torque on the systemS×Heff ≤ |10−6| T,
essentially a condition of local minimumenergy.
The field strength is decreased from saturation in stepsof 0.01
H/Hk and solved again until the same condition isreached. A plot of
the calculated alignment of the magnetiza-tion to the applied field
(S · Happ) for different angles from theeasy axis is shown in
figure 6. The calculated hysteresis curveconforms exactly to the
Stoner–Wohlfarth solution.
5.2. Boltzmann distribution for a single spin
To quantitatively test the thermal effects in the model andthe
correct implementation of the stochastic LLG or MonteCarlo
integrators, the simplest case is that of the Boltzmann
Figure 6. Plot of alignment of magnetization with the applied
fieldfor different angles of from the easy axis. The 0◦ and 90◦
loopswere calculated for very small angles from the easy and hard
axesrespectively, since in the perfectly aligned case the net
torque is zeroand no change of the spin direction occurs.
Figure 7. Calculated angular probability distribution for a
singlespin with anisotropy for different effective temperatures
ku/kBT .The lines show the analytic solution given by equation
(27).
distribution for a single spin with anisotropy (or
appliedfield), where the probability distribution is characteristic
ofthe temperature and the anisotropy energy. The
Boltzmanndistribution is given by:
P(θ)∝ sin θ exp(−
ku sin2 θkBT
)(27)
where θ is the angle from the easy axis. The spin is
initializedalong the easy axis direction and the system is
allowedto evolve for 108 time steps after equilibration,
recordingthe angle of the spin to the easy axis at each time.
Sincethe anisotropy energy is symmetric along the easy axis,
theprobability distribution is reflected and summed about π/2,since
at low temperatures the spin is confined to the upperwell (θ <
π/2). Figure 7 shows the normalized probabilitydistribution for
three reduced temperatures.
The agreement between the calculated distributions isexcellent,
indicating a correct implementation of the stochasticLLG
equation.
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
5.3. Curie temperature
Tests such as the Stoner–Wohlfarth hysteresis or
Boltzmanndistribution are helpful in verifying the mechanical
implemen-tation of an algorithm for a single spin, but interacting
systemsof spins present a significant challenge in that no
analyticalsolutions exist. Hence it is necessary to calculate some
well-defined macroscopic property which ensures the correct
imple-mentation of interactions in a system. The Curie
temperatureTc of a nanoparticle is primarily determined by the
strength ofthe exchange interaction between spins and so makes an
idealtest of the exchange interaction. As discussed previously
thebulk Curie temperature is related to the exchange coupling bythe
mean-field expression given in equation (9). However,
fornanoparticles with a reduction in coordination number at
thesurface and a finite number of spins, the Curie temperature
andcriticality of the temperature-dependent magnetization willvary
significantly with varying size [57].
To investigate the effects of finite size and reduction
insurface coordination on the Curie temperature, the
equilibriummagnetization for different sizes of truncated
octahedronnanoparticles was calculated as a function of
temperature. TheHamiltonian for the simulated system is
H=−∑i 6= j
Ji j Si · S j (28)
where Ji j = 5.6× 10−21 J/link, and the crystal structure
isface-centred-cubic, which is believed to be representativeof
Cobalt nanoparticles. Given the relative strength of theexchange
interaction, anisotropy generally has a negligibleimpact on the
Curie temperature of a material, and so theomission of anisotropy
from the Hamiltonian is purely forsimplicity. The system is
simulated using the Monte Carlomethod with 10 000 equilibration and
20 000 averaging steps.The system is heated sequentially in 10 K
steps, with thefinal state of the previous temperature taken as the
startingpoint of the next temperature to minimize the number of
stepsrequired to reach thermal equilibrium. The mean
temperature-dependent magnetization for different particle sizes is
plottedin figure 8.
From equation (9) the expected Curie temperature is1282 K, which
is in agreement with the results for the 10 nmdiameter
nanoparticle. For smaller particle sizes the magneticbehaviour
close to the Curie temperature loses its criticality,making Tc
difficult to determine. Traditionally the Curie pointis taken as
the maximum of the gradient dm/dT [57], howeverthis significantly
underestimates the actual temperature atwhich magnetic order is
lost (which is, by definition, the Curietemperature). Other
estimates of the Curie point such as thedivergence in the
susceptibility are probably a better estimatefor finite systems,
but this is beyond the scope of the presentarticle. Another effect
visible for very small particle sizes isthe appearance of a
magnetization above the Curie point, aneffect first reported by
Binder [126]. This arises from localmoment correlations which exist
above Tc. It is an effect onlyobservable in nanoparticles where the
system size is close tothe magnetic correlation length.
Figure 8. Calculated temperature-dependent magnetization
andCurie temperature for truncated octahedron nanoparticles
withdifferent size. A visualization of a 3 nm diameter particle is
inset.
5.4. Demagnetizing fields
For systems larger than the single domain limit [33] andsystems
which have one dimension significantly differentfrom another, the
demagnetizing field can have a dominanteffect on the macroscopic
magnetic properties. In micromag-netic formalisms implemented in
software packages such asOOMMF [37], MAGPAR [38] and NMAG [39], the
calculation ofthe demagnetization fields is calculated accurately
due tothe routine simulation of large systems where such
fieldsdominate. Due to the long-ranged interaction the
calculationof the demagnetization field generally dominates the
computetime and so computational methods such as the
fast-Fourier-transform [127, 128] and multipole expansion [129]
have beendeveloped to accelerate their calculation.
In large-scale atomistic calculations, it is generally
suffi-cient to adopt a micromagnetic discretization for the
demag-netization fields, since they only have a significant effect
onnanometre length scales [7]. Additionally due to the
generallyslow variation of magnetization, the timescales
associatedwith the changes in the demagnetization field are
typicallymuch longer than the time step for atomistic spins. Here
wepresent a modified finite difference scheme for calculating
thedemagnetization fields, described as follows.
The complete system is first discretized into macrocellswith a
fixed cell size, each consisting of a number of atoms,as shown in
figure 9(a). The cell size is freely adjustablefrom atomistic
resolution to multiple unit cells depending onthe accuracy
required. The position of each macrocell pmc isdetermined from the
magnetic ‘centre of mass’ given by theexpression
pαmc =
∑ni µi p
αi∑n
i µi(29)
where n is the number of atoms in the macrocell, µi is thelocal
(site-dependent) atomic spin moment and α representsthe spatial
dimension x, y, z. For a magnetic material with thesame magnetic
moment at each site, equation (29) corrects forpartial occupation
of a macrocell by using the mean atomicposition as the origin of
the macrocell dipole, as shown infigure 9(b). For a sample
consisting of two materials withdifferent atomic moments, the
‘magnetic centre of mass’ is
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Figure 9. (a) Visualization of the macrocell approach used
tocalculate the demagnetization field, with the system discretized
intocubic macrocells. Each macrocell consists of several atoms,
shownschematically as cones. (b) Schematic of the
macrocelldiscretization at the curved surface of a material,
indicated by thedashed line. The mean position of the atoms within
the macrocelldefines the centre of mass where the effective
macrocell dipole islocated. (c) Schematic of a macrocell consisting
of two materialswith different atomic moments. Since the
magnetization isdominated by one material, the magnetic centre of
mass movescloser to the material with the higher atomic
moments.
closer to the atoms with the higher atomic moments, as shownin
figure 9(c). This modified micromagnetic scheme givesa good
approximation of the demagnetization field withouthaving to use
computationally costly atomistic resolutioncalculation of the
demagnetization field.
The total moment in each macrocell mmc is calculatedfrom the
vector sum of the atomic moments within each cell,given by
mαmc =n∑i
µi Sαi . (30)
Depending on the particulars of the system, the macrocellmoments
can vary significantly depending on position, com-position and
temperature. At elevated temperatures close tothe Curie point, the
macrocell magnetization becomes small,and so the effects of the
demagnetizing field are much lessimportant. Similarly in
compensated ferrimagnets consistingof two competing sublattices the
overall macrocell magnetiza-tion can also be small again leading to
a reduced influence ofthe demagnetizing field.
The demagnetization field within each macrocell p isgiven by
Hmc,pdemag =µ0
4π
∑p 6=q
3(mmcq · r̂)r̂−mmcq
r3
− µ03
mmc p
V pmc(31)
where r is the separation between dipoles p and q , r̂ is a
unitvector in the direction p→ q , and V pmc is the volume of
themacrocell p. The first term in equation (31) is the usual
dipoleterm arising from all other macrocells in the system, while
thesecond term is the self-demagnetization field of the
macrocell,taken here as having a demagnetization factor 1/3.
Strictlythis is applicable only for the field at the centre of a
cube.However, the non-uniformity of the field inside a
uniformlymagnetized cube is not large and the assumption of a
uniformdemagnetization field is a reasonable approximation. The
self-demagnetization term is often neglected in the literature,
butin fact is essential when calculating the field inside a
magneticmaterial. Once the demagnetization field is calculated for
eachmacrocell, this is applied uniformly to all atoms as an
effectivefield within the macrocell. It should be noted however
thatthe macrocell size cannot be larger than the smallest
sampledimension, otherwise significant errors in the calculation
ofthe demagnetizing field will be incurred.
The volume of the macrocell Vmc is an effective volumedetermined
from the number of atoms in each cell and givenby
Vmc = namcVatom = namc
Vucnauc
(32)
where namc is the number of atoms in the macrocell, nauc is
the
number of atoms in the unit cell and Vuc is the volume of
theunit cell. The macrocell volume is necessary to determine
themagnetization (moment per volume) in the macrocell. For
unitcells much smaller than the system size, equation (32) is a
goodapproximation, however for a large unit cell with
significantfree space, for example a nanoparticle in vacuum, the
freespace contributes to the effective volume which reduces
theeffective macrocell volume.
5.4.1. Demagnetizing field of a platelet. To verify the
im-plementation of the demagnetization field calculation it
isnecessary to compare the calculated fields with some
analyticsolution. Due to the complexity of demagnetization
fieldsanalytical solutions are only available for simple
geometricshapes such cubes and cylinders [130], however for an
infiniteperpendicularly magnetized platelet the demagnetization
fieldapproaches the magnetic saturation −µ0 Ms. To test this
limitwe have calculated the demagnetizing field of a 20 nm×20 nm× 1
nm platelet as shown in figure 10. In the centreof the film
agreement with the analytic value is good, while atthe edges the
demagnetization field is reduced as expected.
5.4.2. Performance characteristics. In micromagnetic
simu-lations, calculation of the demagnetization field usually
dom-inates the runtime of the code and generally it is preferable
tohave as large a cell size as possible. For atomistic
calculationshowever, additional flexibility in the frequency of
updates of
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Figure 10. Calculated cross-section of the demagnetization
fields ina 20 nm× 20 nm× 1 nm platelet (visualization inset)
withmagnetization perpendicular to the film plane. A macrocell size
of 2unit cells is used. In the centre of the film the
calculateddemagnetization field is −2.236 T which compares well to
theanalytic solution of Hdemag =−µ0 M =−2.18 T. Note that the
2Dgrid used slightly overestimates the demagnetization field.
the demagnetization field is permitted due to the short
timesteps used and the fact that the magnetization is generally
aslowly varying property.
To investigate the effects of different macrocell sizes andthe
time taken between updates of the demagnetization fieldwe have
simulated hysteresis loops of a nanodot of diameter 40nm and height
of 1.4 nm. Figure 11(a) shows hysteresis loopscalculated for
different macrocell sizes for the nanodot. For
most cell sizes the results of the calculation agree quite
well,however, for a cell size of 4 unit cells, the calculated
coercivityis significantly larger, owing to the creation of a flat
macrocell(with dimensions 4× 4× 1 unit cells). This illustrates
that forsystems with small dimensions, extra care must be taken
whenspecifying the macro cell size in order to avoid
non-cubiccells. In general, the problem with asymmetric macrocells
isnot trivial to solve within the finite difference formalism,
sincethe problem arises due to a modification of both the
intracelland intercell contributions to the demagnetizing
field.
Figure 11(b) shows the runtime for a single update of
thedemagnetizing field on a single CPU for different macrocellsize
discretizations. Noting the logarithmic scale for the simu-lation
time, single unit cell discretizations are computationallycostly
while not giving significantly better results than largermacrocell
discretizations. Although the demagnetization fieldcalculation is
an n2mc problem, it is possible to pre-calculatethe distances
between the macrocells at the cost of increasedmemory usage. Due to
the computational cost of calculatingthe position vectors, this
method is often much faster than thebrute force calculation.
However, due to the fact that memoryusage increases proportionally
to n2mc, fine discretizations forlarge systems can require many GBs
of memory.
By collating terms in equation (31) it is possible to con-struct
the following matrix Mpq for each pairwise interaction:
Mpq =[(3rx rx − 1)/r 3pq − 1/3 3rx ry 3rx rz
3rx ry (3ryry − 1)/r 3pq − 1/3 3ryrz3rx rz 3ryrz (3rzrz − 1)/r
3pq − 1/3
](33)
Figure 11. Simulated hysteresis loops and computational
efficiency for a 40 nm× 40 nm× 1 nm nanodot for different cell
sizes (multiplesof unit cell size) ((a), (b)) and update rates
(seconds between update calculations) ((c), (d)).
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
where rx , ry , rz are the components of the unit vector in
thedirection p→ q, and rpq is the separation of macrocells.
Sincethe matrix is symmetric along the diagonal only six
numbersneed to be stored in memory. The total demagnetization
fieldfor each macrocell p is then given by:
Hmc,pdemag =µ0
4π
∑p 6=q
Mpq · mmcq− µ0
3mmc p
V pmc. (34)
The relative performance of the matrix optimization is
plottedfor comparison in figure 11(b), showing a significant
reductionin runtime. Where the computer memory is sufficiently
large,the recalculated matrix should always be employed for
optimalperformance.
In addition to variable macrocell sizes, due to the smalltime
steps employed in atomistic models and that the mag-netization is
generally a slowly varying property, it is notalways necessary to
update the demagnetization fields everysingle time step. Hysteresis
loops for different times betweenupdates of the demagnetization
field are plotted in figure 11(c).In general hysteresis
calculations are sufficiently accuratewith a picosecond update of
the demagnetizing field, whichsignificantly reduces the
computational cost.
In general good accuracy for the demagnetizing fieldcalculation
can be achieved with coarse discretization andinfrequent updates,
but fast dynamics such as those inducedby laser excitation require
much faster updates, or simulationof domain wall processes in high
anisotropy materials requiresfiner discretizations to achieve
correct results.
5.4.3. Demagnetizing field in a prolate ellipsoid. Since
themacrocell approach works well in platelets and nanodots, itis
also interesting to apply the same method to a slightlymore complex
system: a prolate ellipsoid. An ellipsoid addsan effective shape
anisotropy due to the demagnetizationfield, and so for a particle
with uniaxial magnetocrystallineanisotropy along the elongated
direction (z), the calculatedcoercivity should increase according
to the difference in thedemagnetization field along x and z, given
by:
H shapedm =+1Nµ0 Ms (35)
where 1N = Nz − Nx . The demagnetizing factors Nx , Ny ,and Nz
are known analytically for various ellipticities [131],and here we
assume a/c = b/c = 0.5, where a, b, and c arethe extent of the
ellipsoid along x , y and z respectively.
To verify the macrocell approach gives the same expectedincrease
of the coercivity we have simulated a generic ferro-magnet with
atomic moment 1.5 µB, an FCC crystal structurewith lattice spacing
3.54 Å and anisotropy field of Ha = 1 T.The particle is cut from
the lattice in the shape of an ellipsoid,of diameter 10 nm and
height of 20 nm, as shown inset infigure 12. A macrocell size of 2
unit cells is used, which isupdated every 100 time steps (0.1
ps).
As expected the coercivity increases due to the shapeanisotropy.
From [131] the expected increase in the coercivityis H shapedm =
0.37 T which compares well to the simulatedincrease of 0.33 T.
Figure 12. Simulated hysteresis loops for an ellipsoidal
nanoparticlewith an axial ratio of 2 showing the effect of the
demagnetizing fieldcalculated with the macrocell approach. A
visualization of thesimulated particle is inset.
6. Parallel implementation and scaling
Although the algorithms and methods discussed in the preced-ing
sections describe the mechanics of atomistic spin models, itis
important to note finally the importance of parallel process-ing in
simulating realistic systems which include
many-particleinteractions, or nano patterned elements with large
lateralsizes. Details of the parallelization strategies which have
beenadopted to enable the optimum performance of VAMPIRE
fordifferent problems are presented in appendix C. In generalterms
the parallelization works by subdividing the simulatedsystem into
sections, with each processor simulating part ofthe complete
system. Spin orientations at the processor bound-aries have to be
exchanged with neighbouring processors tocalculate the exchange
interactions, which for small problemsand large numbers of
processors can significantly reducethe parallel efficiency. The use
of latency hiding, where thelocal spins are calculated in parallel
with the inter-processorcommunications, is essential to ensure good
scaling for theseproblems.
To demonstrate the performance and scalability of VAM-PIRE, we
have performed tests for three different system sizes:small (10 628
spins), medium (8× 105 spins), and large (8×106 spins). We have
access to two Beowulf-class clusters; onewith 8 cores/node with an
Infiniband 10 Gbps low-latencyinterconnect, and another with 4
cores/node with a GigabitEthernet interconnect. For parallel
simulations the intercon-nect between the nodes can be a limiting
factor for increasingperformance with increasing numbers of
processors, sinceas more processors are added, each has to do less
work pertime step. Eventually network communication will
dominatethe calculation since processors with small amounts of
workrequire the data from other processors in shorter times,
leadingto a drop in performance. The scaling performance of thecode
for 100 000 time steps on both machines is presented infigure
13.
The most challenging case for parallelization is the smallsystem
size, since a significant fraction of the system mustbe
communicated to other processors during each timestep.
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
Figure 13. Runtime scaling of VAMPIRE for three different
problemsizes on the Infiniband network (a) and Ethernet network
(b),normalized to the runtime for 2 cores for each problem
size.
On the Ethernet network system for the smallest system
sizereasonable scaling is seen only for 4 CPUs due to the
highlatency of the network. However larger problems are much
lesssensitive to network latency due to latency hiding, and
showexcellent scalability up to 32 CPUs. Essentially this means
thatlarger problems scale much better than small ones, allowingmore
processors to be utilized. This is of course well known forparallel
scaling problems, but even relatively modest systemsconsisting of
105 spins show significant improvements withmore processors.
For the system with the low-latency Infiniband network,excellent
scalability is seen for all problems up to 64 CPUs.Beyond 64 CPUs
the reduced scalability for all problemsizes is likely due to a
lack of network bandwidth. Thebandwidth requirements are similar
for all problem sizes,since smaller problems complete more time
steps in a givenperiod of time and so have to send several sets of
data to otherprocessors. Nevertheless improved performance is seen
withincreasing numbers of CPUs allowing for continued reductionsin
compute time. Although not shown, initial tests on anIBM Blue Gene
class system have demonstrated excellentscaling of VAMPIRE up to 16
000 CPUs, allowing the realpossibility for atomistic simulations
with lateral dimensionsof micrometres. Additional scaling tests for
systems includingcalculation of the demagnetizing field and a
long-rangedexchange interaction are presented in appendix C.
7. Conclusions and perspectives
We have described the physical basis of the rapidly
developingfield of atomistic spin models, and given examples via
itsimplementation in the form of the VAMPIRE code. Althoughthe
basic formalism underpinning atomistic spin models iswell
established, ongoing developments in magnetic materialsand devices
means that new approaches will need to bedeveloped to simulate a
wider range of physical effects at theatomistic scale. One of the
most important phenomena is spintransport and magnetoresistance
which is behind an emergentfield of spin–electronics, or
spintronics. Simulation of spintransport and spin torque switching
is already in development,and must be included in atomistic level
models in order tosimulate a wide range of spintronic materials and
devices,including read sensors and MRAM (magnetic random
accessmemory). Other areas of interest include ferroelectrics, the
spinSeebeck effect [132], and Coloured noise [110] where
simu-lation capabilities are desirable, and incorporation of
theseeffects are planned in future. In addition to modelling
knownphysical effects, it is hoped that improved models of
dampingincorporating phononic and electronic mechanisms will
bedeveloped which enable the study of magnetic properties
ofmaterials at sub-femtosecond timescales.
The ability of atomistic models to incorporate
magneticparameters from density functional theory calculations is
apowerful combination which allows complex systems suchas alloys,
surfaces and defects to be accurately modelled. Thismultiscale
approach is essential to relate microscopic quantummechanical
effects to a macroscopic length scale accessible toexperiment. Such
a multiscale approach leads to the possibilityof simulation driven
technological development, where themagnetic properties of a
complete device can be predicted andoptimized through a detailed
understanding of the underlyingphysics. Due to the potential of
multiscale simulations, it isplanned in future to develop links to
existing DFT codes suchas CASTEP [75, 76] to allow easier
integration of DFT parametersand atomistic spin models.
The computational methods presented here provide asound basis
for atomistic simulation of magnetic materials, butfurther
improvements in both algorithmic and computationalefficiency are of
course likely. One area of potential compu-tational improvement is
GPGPU (general purpose graphicsprocessing unit) computation, which
utilizes the massivelyparallel nature of GPUs to accelerate
simulations, with speedups over a single CPU of 75 times routinely
reported. Withseveral supercomputers moving to heterogenous
computing ar-chitectures utilizing both CPUs and GPUs, supporting
GPGPUcomputation is likely to be important in future, and an
imple-mentation in our VAMPIRE code is currently planned. In
termsof algorithmic improvements it should be noted that the
Heunnumerical scheme although simple is relatively primitive
bymodern standards, and moving to a midpoint scheme mayallow for
larger time steps to be used than currently.
With the continuing improvements in computer power,atomistic
simulations have become a viable option for theroutine simulation
of magnetic materials. With the increas-ing complexity of devices
and material properties, atomistic
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J. Phys.: Condens. Matter 26 (2014) 103202 Topical Review
models are a significant and important development.
Whilemicromagnetic models remain a powerful tool for the
simula-tion and design of devices, the limitations of the
(continuum)micromagnetic formalism are increasingly exposed by its
fail-ure to deal with the complex physics of elevated
temperatures,ultrafast magnetization processes and interfaces. The
atomisticmodel also allows for the study of complex
microstructuraleffects which is pertinent to Brown’s paradox [133],
wherethere is generally a large disparity between the intrinsic
andmeasured magnetic properties arising from nucleation causedby
defects in the sample structure. While micromagnetics willremain
the computational model of choice for large-scale andcoarse-grained
applications, the ability to accurately modelthe effects of
microscopic details, temperature effects andultrafast dynamics make
atomistic models an essential tool tounderstand the physics of
magnetic materials at the forefrontof of the field.
Acknowledgments
The authors wish to thank Jerome Jackson for helpful
discus-sions particularly with regard to the parallel
implementationof the code; Thomas Schrefl for insisting on the name
ofVAMPIRE ; Matt Probert, Phil Hasnip, Joe Barker, Uli Nowak
andOndrej Hovorka for helpful discussions; and Laszlo
Szunyogh,Phivos Mavropoulos and Stefan Blügel for assistance
withthe section on the ab initio parameterization of the
atomisticmodel. We also acknowledge the financial support from the
EUSeventh Framework Programme grant agreement No. 281043FEMTOSPIN
and Seagate Technology Inc. This work made useof the facilities of
N8 HPC provided and funded by the N8consortium and EPSRC (Grant No.
EP/K000225/1) co-ordinatedby the Universities of Leeds and
Manchester and the EPSRCSmall items of research equipment at the
University of YorkENERGY (Grant No. EP/K031589/1).
Appendix A. Code structure and design philosophy
In addition to implementing the necessary computationalmethods
for magnetic atomistic calculations, it is also im-portant to
provide a framework structure for the code, wherenew additions in
the form of features or improvements canbe made with minimal
disturbance to other sections of thecode. Equally important for
intensive computational problemsis ensuring high performance of the
code so that simulationsproceed as rapidly as possible.
In VAMPIRE this is achieved through hybrid coding usinga mixture
of object-oriented and functional programmingstyles.
Object-oriented programming is widely used in modernsoftware
projects as a level of abstraction around the data, orobjects, in
the code. This abstraction makes it easy to storeinformation about
an element, for example an atom, as a singleunified data type,
known as a class. One significant caveat withobject-oriented code
is that it is generally hard to optimize formaximum performance.
High performance codes generallyutilize a different coding approach
known as functional pro-gramming, where the focus is on functions
which operate onnumerous data sets. However the organization of
data into large
blocks in functional programming generally makes it harder
toorganize the data. VAMPIRE therefore makes use of both
method-ologies to gain the benefits of object-oriented design
during theinitialization phase, while for the performance-critical
partsof the code the data is re-arranged to use a functional
stylefor maximum performance. Due to the requirements of
highperformance, object-oriented design and parallelization, theC++
programming language was chosen for all parts of thecode. The
popularity of the C++ language also allows foreasy future
integration of other libraries, such as NVIDIA’sCUDA framework for
utilizing graphics processing units. Forportability the code also
has a minimal dependence on externallibraries and also conforms to
the published standard allowingsimple compilation on the widest
possible variety of computerarchitectures.
In addition to the low-level structure described in terms
ofobject-oriented and functional programming styles, the codeis
also designed in a modular fashion so that most of themechanistic
operations (such as the para