An Introduction to Atomistic Spin Models
T. Ostler
Dept. of Physics, The University of York, York, United Kingdom.
December 2014
Overview
•
Introduction – why atomistic modeling
–
–
–
•
Essentials of atomistic spin models
–
–
–
–
•
Time/length scales.
What is a spin model?
When is it appropriate to use a spin model? Types of systems.
The Hamiltonian terms and typical values.
Numerical approaches and the Landau-Lifshitz-Gilbert equation.
Numerical integration methods.
Typical calculations/simulations.
Examples of where atomistic spin models are appropriate:
–
–
Fe/FePt/Fe interface.
Typical calculations by example GdFeCo.
For anyone interested the slides are available at
http://tomostler.co.uk/wp-content/uploads/2014/12/IOP2014.pptx
Why do we need atomistic models?
• Back in (computing) ancient times (1956) a hard drive was born.
• In 1953 IBM launched first commercial HHD with data transfer rate of
8,800 characters per second and drive size of around 3.75Mb.
Me
IBM 350
• Fifty 24-inch (610 mm) diameter disks with 100 recording surfaces.
We are storing more on
smaller things
kb
25TB daily log
100TB storage
MB (10002)
GB (10003)
TB (10004)
A few GB to TB’s
Christmas Cracker Fact
•
•
2.5PB
24PB daily
•
PB (10005)
4ZB is around 1015 IBM 350’s
The area of the IBM 350 is 1.12m2 and 1015
of them would cover 1x10615m2
EB (1000 )
The surface area of the earth is 5.1x1014m2
ZB (10007)
330 EB demand in 2011
Estimated size of the internet 4ZB
Time and length scales
• As we decrease the size of devices we have to go to ever smaller and
shorter timescales to describe the physics.
Length
10-10 m (Å)
Time
10-16
s (<fs)
10-9 m (nm)
Superdiffusive
spin transport
10-12 s (ps)
Langevin
Dynamics on
atomic
level
s (ns)
10-3 m (mm)
TDFT/ab-initio
spin dynamics
10-15 s (fs)
10-9
10-6 m (μm)
Micromagnetics
/LLB
10-6 s (µs)
10-3 s (ms)
10-0
s (s)+
http://www.psi.ch/swissfel/ultrafast-manipulation-of-the-magnetization
Kinetic Monte Carlo
http://www.castep.org/
What is a spin model?
•
It is the magnetic equivalent of molecular dynamics.
MD
•
•
•
ASD
•
The important variables in
MD are the positions and
velocities.
The forces arise due to the
interaction potentials.
The configuration of the
atoms is determined by
the form of the interaction
potential.
V(r)
•
•
The important variables in
ASD are the spin vectors
(atomic positions fixed).
The fields arise due to the
terms in the magnetic
Hamiltonian.
The ground state is
(mostly) determined by the
fields.
FM
Repulsive
AFM
r
Attractive
bcc Fe (Pajda PRB 64, 174402 (2001))
What is a spin model?
Ab-initio calculations
Exchange interactions
Interaction potentials
phonons
To obtain the
collective dynamics in
both cases an iterative
process of solving the
time-dependent
equations for each
atom/spin is
performed
http://www.fhi-berlin.mpg.de/~hermann/Balsac/BalsacPictures/Phono1.gif
spinwaves
Types of systems the atomistic approach is useful for
•
This type of approach is particularly important when the physics at the atomic
level is important.
Phys. Rev. B 79, 020403(R)
Interfaces/layered systems
Anisotropic Exchange (Dzyaloshinskii-Moriya)
Four spin interactions in FeRh (metamagnetic phase transitions)
Above: one monolayer
IrMn3 of Mn on W(110) Nat Nanotech, 8, 438-444 (2013)
arXiv:1405.3043
Overview
•
Introduction – why atomistic modeling
–
–
–
•
Essentials of atomistic spin models
–
–
–
–
•
Time/length scales.
What is a spin model?
When is it appropriate to use a spin model? Types of systems.
The Hamiltonian terms and typical values.
Numerical approaches and the Landau-Lifshitz-Gilbert equation.
Numerical integration methods.
Typical calculations/simulations.
Examples of where atomistic spin models are appropriate:
–
–
Fe/FePt/Fe interface.
Typical calculations by example GdFeCo.
The Hamiltonian
•
•
•
To do atomistic modelling we really need information about the Hamiltonian.
The terms have different origins and span a wide range of energies and depend on the system of interest.
Computationally they vary in complexity and computational “cost” with increasing numbers of spins (N).
Simple form
Difficulty to determine/cost
Energy range
Usually pretty straight forward and
computationally cheap
(scales with N)
10-25 - 10-22J/at
Can be difficult to determine.
(usually scales with N)
•
10-25 - 10-22J/at
Easy to determine but
computationally expensive
(without tricks scales with N2)
10-25 - 10-23J/at
Most difficult to determine, in
general long-ranged.
(can scale as N2)
10-23 - 10-21J/at
What is the effect of these microscopic parameters on the resulting
magnetic structure?
Effect of the terms in the Hamiltonian
•
The magnetic moments will always try
to align with the magnetic field. The
strength depends on the moments and
the size of the field.
•
The minimization of the anisotropy
depends on the form(s). For example
first order uniaxial anisotropy.
•
Depends on the magnetic moments
and the positions of those moments in
space (the shape).
Tends to demagnetize the system.
B
•
•
In the simplest picture if J>0 the
moments align (dot product of two
spins minimizes energy).
If J<0 anti-alignment
(antiferromagnetism) is the minimum.
FM
•
•
AFM
Question for later: how are the parameters determined?
What can we do with the Hamiltonian?
•
Since we have large numbers of interacting atoms it is (in most cases) impossible
to solve the system analytically. We require a numerical approach.
What kind of
calculation
Time-dependent
properties/dynamics
Equilibrium
Free energy surfaces
Constrained montecarlo
Metropolis monte
carlo
Time integration of
the LLG equation
Time-quantified
monte-carlo
Metadynamics
References
Constrained Monte Carlo Phys. Rev. B 82, 054415 (2010)
Time-quantified Monte Carlo Phys. Rev. Lett, 84, 163 (2000)
Metadynamics Phys. Rev. E. 81, 055701(R) (2010)
The Landau-Lifshitz-Gilbert Equation
•
One of the most common methods used with this type of atomistic modeling is to integrate
the Landau-Lifshitz-Gilbert (LLG) equation.
H
SxSxH
SxH
S
•
•
•
•
The first term is the usual precession term and the second is the damping (λ).
The damping is a phenomenological parameter that ignores how the magnetization is
damped.
There are some links to simple sample programs in the slides at the end to demonstrate the
implementation.
There are a number of extensions to this equation, for example taking into account spintransfer torque.
IEEE Transactions on Magnetics, 6, 3443–3449 (2004)
Numerical Integration Methods
•
Heun Scheme (predictor-corrector algorithm)
Loop over time
• Loop over spins
• Calculate a field acting on each
spin.
• The field is the sum of the terms
that we include in our
Hamiltonian.
•
Solve using numerical integration.
Predictor step (e.g Euler scheme)
Predicted spin
spin at current timestep
s x H … based on spin at current timestep
Corrector step
spin at next timestep
spin at current timestep
See this link for the derivation of the correlator.
[1] – PRL 102, 057203 (2009)
s x H … based on spin at current time
s x H … for predicted spin
How to introduce temperature effects
•
To simulate thermal effects we
include a stochastic term. This is an
additional “field” that mimics
thermal fluctuations:
Damping
Precession
Noise
•
•
In it’s simplest form the noise is
“white”, i.e. it is uncorrelated in
both time and space (can be
coloured [1]).
The mean and variance of the
process can be shown through
fluctuation dissipation theorem to
be equal to:
See this link for the derivation of the correlator.
[1] – PRL 102, 057203 (2009)
•
In terms of implementation the
noise is Gaussian distributed about
zero and multiplied by:
Typical values for different energy terms
Fe (bcc)
Co (hcp)
Ni (fcc)
K1 [J/m3]
54,800
760,000
-126,300
[J/at]
6.43 x 10-25
8.53 x 10-24
1.38 x 10-24
•
•
•
•
•
•
•
•
L10 FePt: likely candidate for Heat
Assisted Magnetic Recording.
Zeeman in 2T field: ~5x10-23 J/at
MC Anisotropy: ~5x10-22 J/at (@ 0K).
Effective exchange: ~3x10-21 J (per nn
interaction).
Permalloy: high permeability
shields/vortex cores.
Zeeman in 2T field: ~2.5x10-23 J/at.
MC Anisotropy: ~10-25 J/at.
Effective exchange: ~4x10-21 J (per nn
interaction).
These effective
parameters appear
similar but their form on
the atomic level can be
very different.
Table of anisotropy constants: http://www.ifmpan.poznan.pl/~urbaniak/Wyklady2012/urbifmpan2012lect5_03.pdf
Possible Calculations
•
There are a wide range of possible kinds of calculations. Here is a (not so extensive) list of
measurements/calculations and if they are accessible with the atomistic model.
Kind of measurement or
calculation
Temperature dependent
magnetization
Possible with atomistic model
(+comments)
✔
Accessible experimentally
(+comments)
✔
Free energy surfaces
✔ - metadynamics or constrained
monte carlo
✗
Spinwave dispersion
✔ - calculations time 2-4 days
✗ - low k (magnetostatic) modes
not accessible
✔ - requires a neutron source for
high k (edge of BZ)
Spin-spin correlations
✔
Magnetization dynamics
✔
Element resolved dynamics
✔
Atomic structure of domain
wall dynamics
✔
Ferromagnetic resonance
Pulsed laser excitation
[1] - Nature Materials, 12, 293-298 (2013).
✗ - larger scale models
(micromagnetics)[2]
✔
[2] – PRB 90, 094402 (2014)
✔ - for high resolution requires
linear accelerator (~10nm) [1]
✔
✔ - High harmonic generation
✔ - XMCD (synchrotron) [3]
✗
✔
✔
[3] – Nature 472, 205-208 (2011)
Overview
•
Introduction – why atomistic modeling
–
–
–
•
Essentials of atomistic spin models
–
–
–
–
•
Time/length scales.
What is a spin model?
When is it appropriate to use a spin model? Types of systems.
The Hamiltonian terms and typical values.
Numerical approaches and the Landau-Lifshitz-Gilbert equation.
Numerical integration methods.
Typical calculations/simulations.
Examples of where atomistic spin models are appropriate:
–
–
Fe/FePt/Fe interface.
Dynamics and switching in GdFeCo.
Another Christmas Cracker Fact
•
•
Assuming the write speed of 8,800
bytes/sec the IBM 350 would take
1.44x1010 years to “write” the internet.
This is 3 times longer than the age of the
earth.
Question from earlier slide: how do we determine the
terms in the Hamiltonian?
Magnetic moments and where
they are in space gives us the
Zeeman and demagnetizing terms.
The exchange to determine the
ordering (at least on a short
range).
Anisotropy for each moment.
B
Electronic structure calculations
provide direct information on the
atomic information
Can also get a lot of information
from experimental observations
Example System: Fe/FePt/Fe
•
As mentioned in the introduction one of the examples where atomistic spin
models are most powerful is at interfaces.
Semi-infinite Fe
Semi-infinite Fe
FePt
• How do the magnetic
parameters vary across this
system?
Spin moment [μB]
http://arxiv.org/pdf/1306.3642.pdf
• Let’s look at an Fe/FePt/Fe
interface system. Potential
application as exchange spring
system for magnetic recording.
• Magnetic moment, anisotropy
and exchange.
Atomic layer
Example System: Fe/FePt/Fe
Semi-infinite Fe
Semi-infinite Fe
FePt
Effective Exchange (sum over all layers)
acting on layer i
J (mev)
K1 (mev)
Anisotropy
Atomic layer
Atomic layer
•
http://arxiv.org/pdf/1306.3642.pdf
So what?
So what?
Semi-infinite Fe
Semi-infinite Fe
FePt
•
mz
•
•
Domain wall coordinate
http://arxiv.org/pdf/1306.3642.pdf
Domain wall profiles and
energies are just one
type of calculation that
is possible with the
atomistic model.
When the exchange is
calculated properly the
structure of the domain
wall is shown to have a
sharp jump due to
reduced exchange at the
interface.
This is important for the
magnetic reversal
process in exchange
spring media.
Example GdFeCo
•
Initial interest in this material came from experiments of helicity dependent, all-optical switching
(AOS).
•
•
•
Little was know from the theory point of view about the magnetic processes in AOS.
Aim: to understand more about the dynamics using a spin model.
Why do we need a spin model for this system?
Fe-Gd interactions are antiferromagnetic (J<0)
Fe-Fe and Gd-Gd
interactions are
ferromagnetic (J>0)
PRL 99, 047601 (2007)
Atomic Level
Sub-lattice
magnetization
Example GdFeCo
•
•
The samples of GdFeCo measured were amorphous so parameters very tricky to calculate abinitio.
By comparing equilibrium properties (element resolved M(T), hysteresis) could construct a model.
PRB 84, 024407 (2011)
Example GdFeCo
•
By varying the exchange parameters the magnetization curves the important points on the
magnetization curves can be shown to agree (Curie temperature, compensation temperature).
PRB 84, 024407 (2011)
Example GdFeCo
•
By matching the model to experiments for the static properties the time-resolved dynamics give
good agreement with experiment.
Experiment
Nature 472, 205-208 (2011)
Model results
Spinwave Dispersion
•
The spinwave dispersion is experimentally obtainable from Neutron scattering (see
Christy Kinane’s talk at 17:00 today) and can be determined by calculating the
following:
Linear spinwave
theory
•
•
The longer the runtime the better the resolution of low lying modes.
Requires large system sizes. Typical calculation time around 1 week.
Nature Scientific Reports, 3, 3262 (2013).
Summary
•
Atomistic spin models are most powerful when considering scenarios where complex exchange
interactions are required (interfaces, exotic exchange etc).
•
Can account for on-site variations in parameters at the atomic level.
•
Limitations of time and length-scale. Often for generic/static properties other models more
appropriate (micromagnetics, mean field, LLB).
•
Can include a number of effects to simulate specific systems (fluctuations in moments, spin transport
effects, laser experiments).
•
Models take into account excitations across the entire Brillouin zone which can reveal interesting
physics behind processes.
Thanks for listening
FFT Method
•
We can write the field for each spin:
•
Then the Fourier components of the components of the field can be written:
•
Where the elements of the tensor are:
•
In terms of the algorithm:
Calculate the
elements of the
tensor
FFT
time loop
t=0
store in memory
It should be noted that the spin
arrays have to be zero padded
(twice as long in each dimension).
Exit
FFT Spins
Convolute
[1] – More info PRB 90, 094402 (2014).
t < num
timesteps?
Update spin
positions
(LLG)
IFFT Spins
LLG No Damping: Maple
•
If we consider the Zeeman energy only we can write the field:
Only applied field (no interactions)
Write a DE for each component
•
The LLG equation for this single spin is then:
In general we solve it numerically
Worksheet available at this
link.
There is also a C++ version that
uses the Heun scheme for
numerical integration here.
Distribution of spinwave energies
•
The distribution of spinwave energies can be determined dynamically from:
•
•
Example: heat induced switching in ferrimagnets[1,2].
Large systems required for smooth data but only care about 10-50ps. Typical calculation
time ~1-2 days.
1090K
975K
X/2
X/2
M/2
M/2
No significant change. Larger spread in across BZ around Γpoint (demagnetization)
[1] Nature Communications, 3, 666 (2012).
[2] Nature Scientific Reports, 3, 3262 (2013).
FeCo
Gd
Excited region during switching
Basic Formalism
•
Within this approach the exchange is
written in Heisenberg form between
spins in neighbouring atoms.
•
This assumes that the magnetic
moment is localised to atomic sites.
•
OK for systems with well localised
magnetic moments, BUT what about
metallic magetic systems?
•
We have to define the atomic moment
in this case as “the integral of the spin
dependent electron density over the
atomic (Wigner-Seitz) volume”.
•
As long as we can write the terms in
the Hamiltonian we can determine the
magnetic properties.
Time-average electron spin within
the atomic volume
Time larger than electron relaxation
time (~10-15s) but less than spinwave
excitation (~10-13s).
LLG+
•
There have been some extensions to the LLG equation to incorporate different
effects:
–
spin torque (Phys. Rev. B. 68, 024404 (2003)):
–
Moments with fluctuating length (see Phys. Rev. B. 86, 054416 (2012)).
–
Models for conducting ferromagnets (see Phys. Rev. Lett. 102, 086601 (2009)).
Dipole-Dipole Interaction
•
As long as we know the magnetic moments we can calculate the dipole field.
•
Due to the double sum the calculation scales as N2. This can potentially be the
most computationally expensive part of the entire calculation.
There are a number of ways of speeding this up:
•
–
–
–
Super cell/macro cell method[2].
Discrete convolution theorem and using FFT’s[1].
Fast multipole methods[3].
•
In the super cell method spins are groups together
as one large volume of magnetization which
substantially reduces the number of pairwise
interactions.
•
Some more information on the FFT method at the end of the presentation.
[1] – JMMM, 221, 365-372 (2000)
[2] – J. Phys.: Cond. Mat 26, 103202 (2014)
[3] – JMMM, 227, 9913-9932 (2008)