Monte Carlo study of small deposited clusters from first

principlesL. 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

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

MC simulation

• Ground state• Finite T properties

Spin dynamics,MC simulation

Fitting of themodel Hamiltonian

Model Hamiltonian

Electronic structurecalculation

Simulation possibilities

Energy as a function ofthe magnetic configuration

• Ground state• Finite T properties

First principles methods to explore magnetic ground state of nanoparticles

Fully unconstrained LSDAFLAPW 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)

Classical Heisenberg model

,i ij j i i i

i j i

H J K

S Aij ij ij ijJ J I J J

4

,,

ijkl i j k li jk l

H Q

Ai ij j ij i jD J

A. Antal et. al., PRB 77, 174429 (2008)

antisymmetric

(Dzyaloshinsky–Moriya)

symmetricisotropic coupling

Jij = 144.9 meV

Q1213 = 7.06 meV

Q1212 = -4.42 meV

|Dij | = 1.78 meV

Kxx = -0.09 meV

on-site anizotropy

Cr3|Au(111)

MC simulation

Fully relativisticscreened KKR

New approach to finite temperature simulation of magnetic structure

Energy as a function ofthe magnetic configuration

• Ground state• Finite T properties

1Im Tr , d

F

iF

τ

Lloyd formula:

2

1Im Tr d ,

: can be calculated similarly

F

iii

i

i j

Fm

F

τ

Derivatives:

2

0, , , ,

1

2i i ji i ji i j

F FF F

• Embedded cluster technique

• Magnetic force theorem

• Frozen potential approx.

• 2nd order Taylor approximation:

MC simulation

The SKKR method provides an approx. of the free energy up to 2nd order

1

F f F i

F i F fW i f

e F i F f

Restricted Metropolis algorithm:

MC simulation based on ab initio calculations

Initial configuration

SKKR: iτ

,i

F

2

,i j

F

etc…

MC simulation

controllingthe temperature

Ground state,finite temperature properties

:imagnetic

configuration

:i

Co9

canted states

Co36

out of plane

Orientation of the magnetization depends on the sizeand the shape of the clusters

Co16

Ferromagnetic systems: Con|Au(111)

0 50 100 150 200 250 300

-74,9

-74,8

-74,7

-74,6

-74,5

Ave

rag

e e

ne

rgy

(Ryd

)

Temperature (K)

Ferromagnetic system: Co36|Au(111)

random

configuration

Ferromagnetic system: Co36|Au(111)

0 50 100 150 200 250 300

-1

0

1

Ma

gn

etiz

atio

n (

arb

. un

it)

Temperature (K)

Mx

My

Mz

|M|

Ferromagnetic system: Co36|Au(111)

Reorientation at about 150 K

Antiferromagnetic system: Cr36|Au(111)

0 50 100 150 200 250 300-60,0

-59,9

-59,8

-59,7

-59,6

-59,5

Ave

rag

e e

ne

rgy

(Ryd

)

Temperature (K)

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

Thank you for your attention