Introduction What’s interesting, and what do we want to do? Spin Dynamics Method Results

The Search for Spin-waves in Iron The Search for Spin-waves in Iron Above TAbove Tcc:: Spin Dynamics Spin Dynamics

SimulationsSimulationsX. Tao, D.P.L., T. C. Schulthess*, G. M. Stocks*X. Tao, D.P.L., T. C. Schulthess*, G. M. Stocks*

* Oak Ridge National Lab* Oak Ridge National Lab• Introduction

What’s interesting, and what do we want to do?

• Spin Dynamics Method

• ResultsStatic propertiesDynamic structure factor

• Conclusions

Croucher ASI on Frontiers in Computational Methods and Their Applications in Physical Sciences

Dec. 6 - 13, 2005 The Chinese University of Hong Kong

Iron (Fe) has had a great effect on mankind:

N S

Iron (Fe) has had a great effect on mankind:

Our current interest is in the magnetic propertiesOur current interest is in the magnetic properties

N S

The controversy about paramagnetic Fe:

Do spin waves persist aboveDo spin waves persist above TTcc??



Experimentally (triple-axis neutron spectrometer)

ORNL: Yes, spin waves persist to 1.4 Tc

BNL: No



Experimentally (triple-axis neutron spectrometer)

ORNL: Yes, spin waves persist to 1.4 Tc

BNL: No

Theoretically

What is the spin-spin correlation length for Fe above Tc?

Are there propagating magnetic excitations?

What is a spin wave?

Consider ferromagnetic spins on a 1-d latticeConsider ferromagnetic spins on a 1-d lattice

(a) The ground state (T=0 K)

(b) A spin-wave state

Spin-waves are propagating excitations with characteristic wavelength and velocity

Facts about BCC iron

• Electronic configuration Electronic configuration 3d3d664s4s22

• Tc = 1043 K (experiment, pure Fe)

• TBCC FCC = 1183 K (BCC FCC eliminated with addition of silicon)

Heisenberg Hamiltonian

N = 2 L3 spins on an L L L BCC lattice

|Sr| = 1 ,classical spins

Spin magnetic moments absorbed into J

J = Jr,r’ where is the neighbor shell

)(),(

21

rrr

r SSJ

H

Shells of neighbors

Exchange parameters J

First principles electronic structure calculations

(T. Schulthess, private communication)

Exchange parameters J (cont’d.)

T = 0.3 Tc (room temperature) BCC Fe dispersion relation

After Shirane et al, PRL (1965)

Nearest neighbors only

Least squares fit

NATURE

Theory

Experiment(Neutron scattering)

(Spin dynamics)

Simulation

Center for Stimulational PhysicsCenter for Stimulational Physics


Center for Simulated PhysicsCenter for Simulated Physics


Center for Simulated PhysicsCenter for Simulated Physics

Inelastic Neutron Scattering:Inelastic Neutron Scattering:Triple axis spectrometerTriple axis spectrometer

Elastic vs inelastic Neutron Elastic vs inelastic Neutron ScatteringScattering

Look at momentum space: the reciprocal lattice

Computer simulation methodsComputer simulation methodsHybrid Monte Carlo

1 hybrid step = 2 Metropolis + 8 over-relaxation

• Find Tc

M(T) = M0

• Generate equilibrium configurations as initial conditions for integrating equations of motion

= 1 – T/Tc 0+

M(T, L) = L -/ F ( L 1/ ) L -/ at Tc

HeffPrecess spinsmicrocanonically

Deterministic Behavior in Magnetic Deterministic Behavior in Magnetic ModelsModels

Classical spin Hamiltonians

i

zi

zj

zi

yj

yi

xj

ji

xi SDSSSSSSJ 2

),(

)()( H

exchange crystal field anisotropy anisotropy

Equations of motion

ieffii SHSS

Sdt

d

H

Integrate coupled equations numerically

(derive, e.g.: ii SSdt

d ,H , let spin value S )

Heff

Spin Dynamics Integration MethodsSpin Dynamics Integration Methods

Integrate Eqns. of Motion numerically, time step = t

Symbolically write )(tfy

Simple method: expand,

)()()()()()( 43!3

1221 tttyttyttytytty O

Improved method: Expand, - t is the expansion variable,

)()()()()()( 43!3

1221 tttyttyttytytty O

(I.)

(II.)

Subtract (II.) from (I.)

)()()(2)()( 5331 tttyttyttytty O

complicated function

Predictor-Corrector MethodPredictor-Corrector Method

Integrate

• Two step method

Predictor step (explicit Adams-Bashforth method)

)(tfy

))]3((9))2((37

))((59))((55[24

)()(

ttyfttyf

ttyftyft

tytty

Corrector step (implicit Adams-Moulton method)

)]2((

))((5))((19))((9[24

)()(

ttyf

ttyftyfttyft

tytty

local truncation error of order ( t )5

Suzuki-Trotter Decomposition Suzuki-Trotter Decomposition MethodsMethods

kk SSSdt

d }][{ Eqns. of motion

effective field

Formal solution: )()( tSettS kdt

k

rotation operator (no explicit form)

How can we solve this?How can we solve this?

Idea: Rotate spins about local field by angle || t

spin length conservation

Exploit sublattice decomposition energy conservation

ImplementationImplementation

Sublattice (non-interacting) decomposition A and B.The cross products matrices A and B where = A + B .Use alternating sublattice updating scheme.

An update of the configuration is then given by

)()( )( tyetty t BA

Operators e A

t and e B

t have simple explicit forms:

ttS

ttS

tStS

ttS

kk

kk

kk

kkkk

k

kkkk

sin

cos22

Implementation (cont’d)Implementation (cont’d)

Suzuki-Trotter Decompositions

e (A+B) t = e A

t e B t + O ( t )2 - 1st order

= e A t/2 e B

t e A t/2 + O ( t )3 - 2nd order

etc.

For iron with 4 shells of neighbors, decompose into 16 sublattices

tSttS kkkk Consequently

Energy conserved!

2/2/2/2/ 11516151 ...... tAtAtAtAtA eeeee

Types of Computer SimulationsTypes of Computer Simulations

Stochastic methods . . . (Monte Carlo)

Deterministic methods . . . (Spin dynamics)

Dynamic Structure FactorDynamic Structure Factor

dtetrrCeeqS

functionresolution

tt

t

ti

rr

rrqicutoff

cutoff

2

21

,,,

Time displaced, space-displaced correlation function

Spin Dynamics MethodSpin Dynamics Method

Monte Carlo sampling to generate initial statescheckerboard decompositionhybrid algorithm (Metropolis + Wolff +over-relaxation)

Time Integration -- tmax= 1000J-1

t = 0.01 J-1 predictor-corrector method t = 0.05 J-1 2nd order decomposition method

Speed-up: use partial spin sums “on the fly” -- restrict q=(q,0,0) where q=2n/L, n=±1, 2, …, L

00,,,,

zyzy

xx

xx rrr

rrr

rriq

rrrr

rr

rrqi StSeStSethen

Time-displacement averaging 0.1 tmax different time starting points

0 0.1 0.2 0.3 . . . 100.0 . . .t tcutoff=0.9tmax

Other averaging500 - 2000 initial spin configurationsequivalent directions in q-spaceequivalent spin components

Implementation: Developed C++ modules for the -Mag Toolset at ORNL

Static Behavior: Spontaneous Static Behavior: Spontaneous MagnetizationMagnetization

• Tc (experiment) = 1043 K

• Tc (simulation) = 949 (1) K (from finite size scaling)

Static Behavior: Correlation Static Behavior: Correlation LengthLength

Correlation function at

1.1 Tc :

( r ) ~ e -

r

/

/r 1+

2a 6Å

Dynamic Structure FactorDynamic Structure Factor

Low T sharp, (propagating) spin-wave peaks

T Tc propagating

spin-waves?

Dynamic Structure Factor Dynamic Structure Factor LineshapeLineshape

21

2

21

2

2

2

2

:

:

exp

exp

lol

ll

o

oggo

ogoo

IL

IG

IG

Lorentzian

Gaussian

• Fitting functions for S(q,)

• Magnetic excitation lifetime ~ 1 / l

• Criterion for propagating modes: 1 < o


LowLow T T = 0.3 Tc |q| = (0.5 qzb , 0, 0)


LowLow T T = 0.3 Tc |q| = (0.5 qzb , 0, 0)


AboveAbove Tc T = 1.1 Tc |q| = (q,q,0)

Q=1.06 Å-1

Q=0.67 Å-1

Dispersion curves

Compare experiment and simulation

Experimental results: Lynn, PRB (1975)

Dynamic Structure factorDynamic Structure factor

T = 1.1 Tc:

Constant E-scans

Summary and ConclusionsSummary and Conclusions

Monte Carlo and spin dynamics simulations have Monte Carlo and spin dynamics simulations have been performed for BCC iron with 4 shells of been performed for BCC iron with 4 shells of interacting neighbors. These show that:interacting neighbors. These show that:

• Tc is rather well determined

• Spin-wave excitations persist for T Tc

• Short range order is limited

• Excitations are propagating if

To learn more about To learn more about MC in Statistical MC in Statistical

Physics (and a little Physics (and a little about spin dynamics):about spin dynamics):

the 2the 2ndnd Edition is Edition is coming soon . . .coming soon . . .

now availablenow available

AppendixAppendix

Introduction What’s interesting, and what do we want to do? Spin Dynamics Method Results

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