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Monte Carlo methods applied to magnetic nanoclusters

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. magnetic cluster, e.g., Cr, Co, 1 −100 atoms. Deposited magnetic nanoparticles.

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Monte Carlo methods applied to magnetic nanoclusters

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  1. 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

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

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

  4. + spin-orbit coupling (S.O.C.) • → Tensorial coupling constants Cr trimer on Au (111) antisymmetric isotropic symmetric Dzyaloshinsky−Moriya interaction: • → On-site uniaxial anisotropy A. Antal et. al., Phys. Rev. B 77, 174429 (2008)

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

  6. 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)

  7. Clusters • Example: Co16 cluster on Cu (001) surface Different coupling constants! L. Balogh et. al., J. Phys.: Conference Series (in press)

  8. Problem • Let us use the Heisenberg picture • Cluster-average • Simulation result:

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

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

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

  12. Bonus slide • 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" • Future plans • STM structure ground state: simulated annealing • Reorganize the inversion of the τ-matrix: in-the-place inversion + changing the configuration together

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