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

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

4. → Tensorial coupling constants + spin-orbit coupling (S.O.C.) isotropicsymmetric antisymmetric J ij = 144.9 meV |D ij | = 1.78 meV d i ≈ 0.2 meV → On-site uniaxial anisotropy A. Antal et. al., Phys. Rev. B 77, 174429 (2008) Dzyaloshinsky−Moriya interaction: Cr trimer on Au (111)

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

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

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

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

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

10. Restricted Other sampling methods Multiple sampling  temperature depenent simulation: does not work because of too strongly correlated states  searcing for the ground state: can be efficient  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

11. 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 Summary

12. 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 Bonus slide