Monte Carlo methods applied to magnetic nanoclusters L

Monte Carlo methods applied to magnetic nanoclusters L. Balogh, K. M. Lebecki, B. Lazarovits, L. Udvardi, L. Szunyogh, U. Nowak Uppsala, 8 February 2010 balogh@phy. bme. hu

Introduction n Deposited magnetic nanoparticles magnetic cluster, e. g. , Cr, Co, 1− 100 atoms non-magnetic host, e. g. , Cu (001), Au (111) n n Magnetic ground state? Thermal properties: magnetization, reversal? Simple description: Heisenberg-model We need the model-parameters. . . n Monte Carlo (MC) simulation based on fully relativistic Green's function method

Heisenberg-model n Classical, 3 -dimensional Heisenberg-model J < 0: ferromagnetic; n Example: L x L cubic lattice: Model: basic, well-known, fast simulation. J > 0: antiferromagnetic

+ spin-orbit coupling (S. O. C. ) n → Tensorial coupling constants Cr trimer on Au (111) isotropic symmetric antisymmetric Dzyaloshinsky−Moriya interaction: n → On-site uniaxial anisotropy Jij = 144. 9 me. V |Dij | = 1. 78 me. V di ≈ 0. 2 me. V A. Antal et. al. , Phys. Rev. B 77, 174429 (2008)

How to calculate Jij-s? atoms: potential scattering: t-operator propagation: Green's function scattering path operator (SPO) i i j k i j

Embedding Lloyd's formula coming soon. . . B. Lazarovits, Electronic and magnetic properties of nanostructures (Dissertation, 2003) L. Udvardi et. al. , Phys. Rev. B 68, 104436 (2003)

Clusters n Example: Co 16 cluster on Cu (001) surface Different coupling constants! L. Balogh et. al. , J. Phys. : Conference Series (in press)

Problem n Let us use the Heisenberg picture n Cluster-average n Simulation result:

Simple MC n n n Isotropic and uniform phase space sampling Metropolis algorithm is used "Driving force": Lloyd-energy sampling (f) starting configuration (i) SKKR ? Metropolisalgorithm

Other sampling methods n Restricted Optimization of the cone angle (not implemented yet); see: U. Nowak, Phys. Rev. Lett. 84 163 (1999) ¨ Possible use of Taylor series ¨ n n n fixed, small cone adv. : efficient at low tempetarure (ground state!) disadv. : not effective at high temperature; disadv. : unclear effect on the specific heat Multiple sampling temperature depenent simulation: does not work because of too strongly correlated states ¨ searcing for the ground state: can be efficient ¨

Summary n Instead of using an a priori model, we use the Lloydenergy of the SKKR calculation to drive a MC simulation n Temperature dependent quantities are accessible, and agree with an appropriate Heisenberg-model n Searching for the ground state can be efficient

Bonus slide n Parallelization (recent version): each temperature point on different computers adv. : easy, efficient ("poor guy's supercomputer") ¨ disadv. : vaste time on each thermalization ¨ possible solution: "Heisenberg-engine" ¨ n Future plans ¨ STM structure ground state: simulated annealing ¨ Reorganize the inversion of the τ-matrix: in-the-place inversion + changing the configuration together