An Introduction to Atomistic Spin Models

T. OstlerDept. of Physics, The University of York, York, United Kingdom.

December 2014

• Introduction – why atomistic modeling– Time/length scales.– What is a spin model?– When is it appropriate to use a spin model? Types of systems.

• Essentials of atomistic spin models– 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.

Overview

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

A few GB to TB’s

25TB daily log 100TB storage

2.5PB 24PB daily

330 EB demand in 2011

Estimated size of the internet 4ZB

kb

MB (10002)

GB (10003)

TB (10004)

PB (10005)

EB (10006)

ZB (10007)

Christmas Cracker Fact

• 4ZB is around 1015 IBM 350’s• The area of the IBM 350 is 1.12m2 and 1015

of them would cover 1x1015m2

• The surface area of the earth is 5.1x1014m2

Time and length scales

10-15 s (fs)

10-12 s (ps)

10-9 s (ns)

10-6 s (µs)

10-3 s (ms)

Langevin Dynamics on atomiclevel

Kinetic Monte Carlo10-0 s (s)+

10-16 s (<fs)TDFT/ab-initio spin dynamics

Tim

e

10-9 m (nm) 10-6 m (μm) 10-3 m (mm)10-10 m (Å)

Length

Micromagnetics/LLB

http://www.psi.ch/swissfel/ultrafast-manipulation-of-the-magnetization http://www.castep.org/

Superdiffusive spin transport

• As we decrease the size of devices we have to go to ever smaller and shorter timescales to describe the physics.

• It is the magnetic equivalent of molecular dynamics.

What is a spin model?

MD ASD

V(r)

r

bcc Fe (Pajda PRB 64, 174402 (2001))

Repulsive

Attractive

• 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.

FM

AFM

• 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.

What is a spin model?

Ab-initio calculations

Interaction potentials Exchange interactions

spinwavesphonons

http://www.fhi-berlin.mpg.de/~hermann/Balsac/BalsacPictures/Phono1.gif

To obtain the collective dynamics in both cases an iterative process of solving the

time-dependent equations for each

atom/spin is performed

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.

Anisotropic Exchange (Dzyaloshinskii-Moriya)Above: one monolayer of Mn on W(110)

Phys. Rev. B 79, 020403(R)

arXiv:1405.3043IrMn3

Four spin interactions in FeRh (metamagnetic phase transitions)

Interfaces/layered systems

Nat Nanotech, 8, 438-444 (2013)

• Introduction – why atomistic modeling– Time/length scales.– What is a spin model?– When is it appropriate to use a spin model? Types of systems.

• Essentials of atomistic spin models– 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.

Overview

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).

Most difficult to determine, in general long-ranged.

(can scale as N2)

Can be difficult to determine.(usually scales with N)

Easy to determine but computationally expensive

(without tricks scales with N2)

• What is the effect of these microscopic parameters on the resulting magnetic structure?

10-23 - 10-21J/at

10-25 - 10-22J/at

10-25 - 10-23J/at

Simple form Difficulty to determine/cost Energy range

Usually pretty straight forward and computationally cheap

(scales with N)10-25 - 10-22J/at

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.

B

• The minimization of the anisotropy depends on the form(s). For example first order uniaxial anisotropy.

• 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.

AFMFM

• Depends on the magnetic moments and the positions of those moments in space (the shape).

• Tends to demagnetize the system.

• 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

Equilibrium

Free energy surfaces

Constrained monte-carlo Metadynamics

Metropolis monte carlo

Time-dependent properties/dyna

mics

Time integration of the LLG equation

Time-quantified monte-carlo

ReferencesConstrained 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.

• 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 spin-

transfer torque.

IEEE Transactions on Magnetics, 6, 3443–3449 (2004)

Numerical Integration Methods

• 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.

See this link for the derivation of the correlator.[1] – PRL 102, 057203 (2009)

spin at next timestep spin at current timestep s x H … based on spin at current time s x H … for predicted spin

Heun Scheme (predictor-corrector algorithm)

Predictor step (e.g Euler scheme)

Predicted spin spin at current timestep s x H … based on spin at current timestep

Corrector step

• To simulate thermal effects we include a stochastic term. This is an additional “field” that mimics thermal fluctuations:

• 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:

How to introduce temperature effects

• In terms of implementation the noise is Gaussian distributed about zero and multiplied by:

See this link for the derivation of the correlator.[1] – PRL 102, 057203 (2009)

Typical values for different energy terms

• 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).

Table of anisotropy constants: http://www.ifmpan.poznan.pl/~urbaniak/Wyklady2012/urbifmpan2012lect5_03.pdf

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

These effective parameters appear similar but their form on the atomic level can be very different.

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.

[1] - Nature Materials, 12, 293-298 (2013). [2] – PRB 90, 094402 (2014) [3] – Nature 472, 205-208 (2011)

Kind of measurement or calculation

Possible with atomistic model (+comments)

Accessible experimentally (+comments)

Temperature dependent magnetization

✔ ✔

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 ✔ ✔ - for high resolution requires linear accelerator (~10nm) [1]

Magnetization dynamics ✔ ✔

Element resolved dynamics ✔ ✔ - High harmonic generation✔ - XMCD (synchrotron) [3]

Atomic structure of domain wall dynamics

✔ ✗

Ferromagnetic resonance ✗ - larger scale models (micromagnetics)[2]

✔

Pulsed laser excitation ✔ ✔

• Introduction – why atomistic modeling– Time/length scales.– What is a spin model?– When is it appropriate to use a spin model? Types of systems.

• Essentials of atomistic spin models– 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.

Overview

• 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.

Another Christmas Cracker Fact

Question from earlier slide: how do we determine the terms in the Hamiltonian?

B

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.

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.

http://arxiv.org/pdf/1306.3642.pdf

Semi-infinite Fe Semi-infinite Fe

FePt

• Let’s look at an Fe/FePt/Fe interface system. Potential application as exchange spring system for magnetic recording.

• How do the magnetic parameters vary across this system?

• Magnetic moment, anisotropy and exchange.

Atomic layer

Spin

mom

ent [

μ B]

Example System: Fe/FePt/Fe

http://arxiv.org/pdf/1306.3642.pdf

Semi-infinite Fe Semi-infinite Fe

FePtAnisotropy Effective Exchange (sum over all layers) acting on layer i

• So what?

K 1 (m

ev)

Atomic layer Atomic layer

J (m

ev)

So what?

http://arxiv.org/pdf/1306.3642.pdf

Semi-infinite Fe Semi-infinite Fe

FePt

• 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.

mz

Domain wall coordinate

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?

Atomic LevelSub-lattice

magnetization

Fe-Fe and Gd-Gd interactions are ferromagnetic (J>0)

Fe-Gd interactions are anti-ferromagnetic (J<0)

PRL 99, 047601 (2007)

Example GdFeCo• The samples of GdFeCo measured were amorphous so parameters very tricky to calculate ab-

initio.• 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 Model results

Nature 472, 205-208 (2011)

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:

Nature Scientific Reports, 3, 3262 (2013).

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.

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

T

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

store in memory

time loop t=0

t < num timesteps?

FFT Spins

Convolute IFFT Spins

Update spin positions

(LLG)

F

ExitIt should be noted that the spin arrays have to be zero padded (twice as long in each dimension).

[1] – More info PRB 90, 094402 (2014).

• If we consider the Zeeman energy only we can write the field:

• The LLG equation for this single spin is then:

LLG No Damping: Maple

Worksheet available at this link.

There is also a C++ version that uses the Heun scheme for numerical integration here.

Only applied field (no interactions)

Write a DE for each component

In general we solve it numerically

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.

975K

M/2

X/2

1090K FeCoGd

M/2

X/2

[1] Nature Communications, 3, 666 (2012).[2] Nature Scientific Reports, 3, 3262 (2013).

No significant change. Larger spread in across BZ around Γ-point (demagnetization)

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 togetheras one large volume of magnetization whichsubstantially reduces the number of pairwiseinteractions.

• 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)