Introduction What’s interesting, and what do we want to do? Spin Dynamics Method Results
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Transcript of 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:
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??
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
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
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
Center for Stimulational PhysicsCenter for Stimulational Physics
Center for Simulated PhysicsCenter for Simulated Physics
Center for Stimulational PhysicsCenter for Stimulational 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
Dynamic Structure Factor Dynamic Structure Factor LineshapeLineshape
LowLow T T = 0.3 Tc |q| = (0.5 qzb , 0, 0)
Dynamic Structure Factor Dynamic Structure Factor LineshapeLineshape
LowLow T T = 0.3 Tc |q| = (0.5 qzb , 0, 0)
Dynamic Structure Factor Dynamic Structure Factor LineshapeLineshape
AboveAbove Tc T = 1.1 Tc |q| = (q,q,0)
Q=1.06 Å-1
Q=0.67 Å-1
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