1. Dec. 6 - 13, 2005 The Chinese University of Hong Kong
Croucher ASI on Frontiers in Computational Methods and Their Applications in Physical Sciences
Dec , The Chinese University of Hong Kong
The Search for Spin-waves in Iron Above Tc: Spin Dynamics Simulations X. Tao, D.P.L., T. C. Schulthess*, G. M. Stocks* * Oak Ridge National Lab
Introduction
What’s interesting, and what do we want to do?
Spin Dynamics Method
Results
Static properties
Dynamic structure factor
Conclusions

2. Iron (Fe) has had a great effect on mankind:
N S

3. Iron (Fe) has had a great effect on mankind:
N S
Our current interest is in the magnetic properties

4. The controversy about paramagnetic Fe:
Do spin waves persist above Tc?

5. The controversy about paramagnetic Fe:
Do spin waves persist above Tc?
Experimentally (triple-axis neutron spectrometer)
ORNL: Yes, spin waves persist to 1.4 Tc
BNL: No

6. The controversy about paramagnetic Fe:
Do spin waves persist above Tc?
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?

7. What is a spin wave?  (a) The ground state (T=0 K)
Consider 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

8. Facts about BCC iron Tc = 1043 K (experiment, pure Fe)
Electronic configuration 3d64s2
Tc = 1043 K (experiment, pure Fe)
TBCC  FCC = 1183 K (BCC  FCC eliminated with addition of silicon)

9. Heisenberg Hamiltonian
Shells of neighbors
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

10. Exchange parameters J
First principles electronic structure calculations
(T. Schulthess, private communication)

11. Exchange parameters J (cont’d.)
T = 0.3 Tc (room temperature) BCC Fe dispersion relation
Nearest neighbors only
Least squares fit
After Shirane et al, PRL (1965)

12. NATURE Simulation Theory Experiment (Spin dynamics)
(Neutron scattering)

13. Center for Stimulational Physics

14. Center for Stimulational Physics Center for Simulated Physics

15. Center for Stimulational Physics Center for Simulated Physics

16. Inelastic Neutron Scattering: Triple axis spectrometer

17. Elastic vs inelastic Neutron Scattering
Look at momentum space: the reciprocal lattice

18. Computer simulation methods
Hybrid Monte Carlo
1 hybrid step = 2 Metropolis + 8 over-relaxation
Precess
spins
microcanonically
Heff
Find Tc
M(T) = M0  
 = 1 – T/Tc  0+
M(T, L) = L -/ F ( L 1/  )  L -/ at Tc
Generate equilibrium configurations as initial conditions for integrating equations of motion

19. Deterministic Behavior in Magnetic Models
Classical spin Hamiltonians
 
exchange crystal field
anisotropy anisotropy
Equations of motion
Heff
(derive, e.g.:
, let spin value S  )
Integrate coupled equations numerically

20. Spin Dynamics Integration Methods
Integrate Eqns. of Motion numerically, time step =  t
Symbolically write
Simple method: expand,
(I.)
Improved method: Expand, -  t is the expansion variable,
(II.)
Subtract (II.) from (I.)  
complicated function

21. Predictor-Corrector Method
Integrate
Two step method
Predictor step (explicit Adams-Bashforth method)
Corrector step (implicit Adams-Moulton method)
 local truncation error of order (  t )5

22. Suzuki-Trotter Decomposition Methods
Eqns. of motion 
effective field
Formal solution: 
rotation operator (no explicit form)
How can we solve this?
Idea:
Rotate spins about local field  by angle || t
 spin length conservation
Exploit sublattice decomposition  energy conservation

23. Implementation Use alternating sublattice updating scheme.
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
Operators e A  t and e B  t have simple explicit forms:

24. Implementation (cont’d)
Consequently
 Energy conserved!
Suzuki-Trotter Decompositions
e (A+B) t = e A t e B t + O ( t ) st order
= e A t/2 e B t e A t/2 + O ( t ) nd order
etc.
For iron with 4 shells of neighbors, decompose into 16 sublattices 

25. Types of Computer Simulations
Stochastic methods (Monte Carlo)
Deterministic methods (Spin dynamics)

26. Dynamic Structure Factor
Time displaced, space-displaced correlation function

27. Spin Dynamics Method Time Integration -- tmax= 1000J-1
Monte Carlo sampling to generate initial states
checkerboard decomposition
hybrid 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=2n/L, n=±1, 2, …, L

28. Time-displacement averaging
0.1 tmax different time starting points t
tcutoff=0.9tmax
Other averaging
initial spin configurations
equivalent directions in q-space
equivalent spin components
Implementation: Developed C++ modules for the -Mag Toolset at ORNL

29. Static Behavior: Spontaneous Magnetization
Tc (experiment) = 1043 K
Tc (simulation) = 949 (1) K (from finite size scaling)

30. Static Behavior: Correlation Length
Correlation function at
1.1 Tc :
 ( r ) ~ e - r /  /r 1+
   2a  6Å

31. Dynamic Structure Factor
Low T 
sharp, (propagating) spin-wave peaks
T  Tc  propagating spin-waves?

32. Dynamic Structure Factor Lineshape
Fitting functions for S(q,)
Magnetic excitation lifetime ~ 1 /  l
Criterion for propagating modes: 1 < o

33. Dynamic Structure Factor Lineshape
Low T  T = 0.3 Tc |q| = (0.5 qzb , 0, 0)

34. Dynamic Structure Factor Lineshape
Low T  T = 0.3 Tc |q| = (0.5 qzb , 0, 0)

35. Dynamic Structure Factor Lineshape
Above Tc  T = 1.1 Tc |q| = (q,q,0)
Q=1.06 Å-1
Q=0.67 Å-1

36. Dispersion curves Compare experiment and simulation
Experimental results: Lynn, PRB (1975)

37. Dynamic Structure factor
Constant E-scans
T = 1.1 Tc:

38. Summary and Conclusions
Monte Carlo and spin dynamics simulations have been performed for BCC iron with 4 shells of 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   

39. Appendix
To learn more about MC in Statistical Physics (and a little about spin dynamics): the 2nd Edition is coming soon now available