1. Monte Carlo study of small deposited clusters from first principles L. Balogh, L. Udvardi, L. Szunyogh Department of Theoretical Physics, Budapest University of Technology and Economics B. Lazarovits Research Institute for Solid State Physics and Optics of the HAS

2. Outline Motivation: high density magnetic data storage Simulation possibilities  Solving a model Hamiltonian  MC simulation of a model Hamiltonian  MC simulation from first principles Investigation of ferromagnetic systems Antiferromagnetic systems Outlook

3. MC simulation Ground state Finite T properties Spin dynamics, MC simulation Fitting of the model Hamiltonian Model Hamiltonian Electronic structure calculation Simulation possibilities Energy as a function of the magnetic configuration Ground state Finite T properties

4. First principles methods to explore magnetic ground state of nanoparticles Fully unconstrained LSDA FLAPW Ph. Kurz, G. Bihlmayer, K. Hirai, and S. Blügel, Phys. Rev. Lett. 86, 1106 (2001) PAW D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B 62, 11556 (2000) H. J. Gotsis, N. Kioussis, and D. A.Papaconstantopoulos, Phys. Rev. B 73, 014436 (2006) Non-collinear real-space TB-LMTO R. Robles and L. Nordström, Phys. Rev. B 74, 094403 (2006) A. Bergman, L. Nordström, A.B. Klautau, S. Frota-Pessoa and O. Eriksson, J. Phys.: Condens. Matter 19 156226 (2007) A. Bergman, L. Nordström, A.B. Klautau, S. Frota-Pessoa and O. Eriksson, Phys. Rev. B 75, 224425 (2007) Ab initio spin dynamics with constrained LSDA B. Újfalussy, B. Lazarovits, L. Szunyogh, G. M. Stocks, and P. Weinberger, Phys. Rev. B 70, 100404(R) (2004) B. Lazarovits, B. Újfalussy, L. Szunyogh, G. M. Stocks, and P. Weinberger, J. Phys.: Condens. Matter 16, S5833 (2004) G.M. Stocks, M. Eisenbach, B. Újfalussy, B. Lazarovits, L. Szunyogh and P. Weinberger, Prog. Mat. Sci. 52, 371-387 (2007) Multiscale approaches based on a model Hamiltonian mapped from first principles: Spin-cluster expansion & LLG R. Drautz and M. Fähnle, Phys. Rev. B 69, 104404 (2004); Phys. Rev. B 72, 212405 (2005) M. Fähnle, R. Drautz, R. Singer, D. Steiauf, and D. V. Berkov, Comp. Mat. Sci. 32, 118 (2005) Torque method & MC S. Polesya, O. Sipr, S. Bornemann, J. Minár, and H. Ebert, Europhys. Lett. 74, 1074 (2006) O. Sipr, S. Bornemann, J. Minár, S. Polesya, V. Popescu, A. Simunek, and H. Ebert, J. Phys.: Condens. Matter 19, 096203 (2007) O. Sipr, S. Polesya, J. Minár, and H. Ebert, J. Phys.: Condens. Matter 19, 446205 (2007)

5. Classical Heisenberg model A. Antal et. al., PRB 77, 174429 (2008) antisymmetric (Dzyaloshinsky–Moriya) symmetric isotropic coupling J ij = 144.9 meV Q 1213 = 7.06 meV Q 1212 = -4.42 meV |D ij | = 1.78 meV K xx = -0.09 meV on-site anizotropy Cr 3 |Au(111)

6. MC simulation Fully relativistic screened KKR New approach to finite temperature simulation of magnetic structure Energy as a function of the magnetic configuration Ground state Finite T properties Lloyd formula: Derivatives: Embedded cluster technique Magnetic force theorem Frozen potential approx. 2 nd order Taylor approximation:

7. MC simulation The SKKR method provides an approx. of the free energy up to 2 nd order Restricted Metropolis algorithm:

8. MC simulation based on ab initio calculations Initial configuration etc… MC simulation controlling the temperature Ground state, finite temperature properties magnetic configuration

9. Co 9 canted states Co 36 out of plane Orientation of the magnetization depends on the size and the shape of the clusters Co 16 Ferromagnetic systems: Co n |Au(111)

10. Ferromagnetic system: Co 36 |Au(111) random configuration

11. Ferromagnetic system: Co 36 |Au(111)

12. Reorientation at about 150 K

13. Antiferromagnetic system: Cr 36 |Au(111)

14. Conclusion, outlook Ab initio cluster simulations Larger clusters  Magnetization (thermodynamic average) →  → Susceptibility (temperature dependence) →  → Critical temperature (reorientation transition temp.) Future plan:  Importance sampling → DLM method for layers

15. Thank you for your attention