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