Dec. 6 - 13, 2005 The Chinese University of Hong Kong Croucher ASI on Frontiers in Computational Methods and Their Applications in Physical Sciences Dec. 6 - 13, 2005 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

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

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

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

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

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?

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

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)

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

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 Nearest neighbors only Least squares fit After Shirane et al, PRL (1965)

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

Center for Stimulational Physics

Center for Stimulational Physics Center for Simulated Physics

Center for Stimulational Physics Center for Simulated Physics

Inelastic Neutron Scattering: Triple axis spectrometer

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

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

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

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

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

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

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:

Implementation (cont’d) Consequently  Energy conserved! 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 

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

Dynamic Structure Factor Time displaced, space-displaced correlation function

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

Time-displacement averaging 0.1 tmax different time starting points 0 0.1 0.2 0.3 . . . 100.0 . . .t tcutoff=0.9tmax Other averaging 500 - 2000 initial spin configurations equivalent directions in q-space equivalent spin components Implementation: Developed C++ modules for the -Mag Toolset at ORNL

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

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

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

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

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

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

Dynamic Structure Factor Lineshape Above 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 factor Constant E-scans T = 1.1 Tc:

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   

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