Monte Carlo: a study on BCC Fe

Siebe Rossen^1,2, Phivos Mavropoulos¹, Timo Schena¹, Stefan Blügel¹ and Theo Rasing²

1 Peter Grünberg Institut and Institute for Advanced Simulation, Forschungzentrum Jülich and JARA, 52425 Jülich, Germany

2 Institute for Molecules and Materials, Radboud Universiteit Nijmegen, 6525 AJ Nijmegen, The Netherlands

DPG Regensburg 14-03-2013

Tight-binding spin dynamics and tight-binding

Monte Carlo: a study on BCC Fe

Motivation

• Study of laser-induced magnetization dynamics

• Scientific interest: excitation on the time scale of exchange

• Technological interest: magnetization reversal within 100ps

→ Accurate atomistic description outside equilibrium needed

• System studied: BCC Fe for T>0K

K. Vahaplaret al, 2009 Phys Rev Lett 103, 117201

Localized versus delocalized description

Heisenberg model

ij i j

i j i

E J m m



 



Stoner model

   

E_ k  E k 

and: ^E

   

Stoner criterium (T=0K): ID E  F ¹

Characteristics

Heisenberg model

• Reasonable T_C

• Qualitative correct above T_C

Stoner model

• Failure description for T>0K

• No Curie-Weiss law above T_C

• However Heisenberg model lacks

…

• Itinerant character of electrons around ε_F

• Large cohesive energies and specific heat

→

• Unified approach needed:

• Rigorous description of non-collinear state

• Electronic structure via tight binding Picture taken from:

NoncollinearMagnetism, David Hobbsand Jürgenhafner

ij i j

i j i

E J m m



 



Tight binding

• Free energy:

• F in terms of ψ:

     

0 , , , 0

, ,

,

ˆ ^j

k n k n i k n j i

k n i j

E f H ^

  

 

   ^{ }











0

1

2 4

LCN

i i i id i i i

i i i

F E U n n I _ m m_ B_ m_

 

 







Pinning of charge

Stoner model

Constraint field

spin independent kinetic + potential energy

    

' †

, , ,

,

i k n, F k n i k k n i

k n

f ^ S ^

  

 



i ˆi

c _{  }tr ^ _^

→

Parametrizationof Ĥ₀:M. J. Mehland D.A. Papaconstantopoulos, 1996, Phys. Rev. B, 54 andJ. C. Slaterand G. F. Koster, 1954, PhysRev94, 1498-1524

Variational Hamiltonian

• Minimalisation free energy:

• Obtained Hamiltonian:

 

0

k n i

F



 

 

 ^

   

i

 

 



  



under the constraint:



    

0

1 1 ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

4 4

j j

TB i i ij j i i i i i i

i i

H ^ H ^ U n n _ I m_{ } B S I m_{ } B

       

                

   

 

 

Description of a NC state

• Problem to be solved:

• Magnetic moment directions:

• Self-consistency for fixed moment directions!

ˆ ˆ

HTB  c ES c 

     

' † ˆ

ˆ_{i k}, _{F k k}

i i

k

f c Sc

 

  

 



 





, , ni , ,

n i

c _ n i



 



Torques

• Validity adiabatic approximation?

→ Spin wave frequencies whole BZ ~ 0.1eV/ħ – 1eV/ħ (~ k_BT_C/ħ)

→ Electron hopping frequency 3d bandwidth (W/ħ) (W_Fe~ 5eV)

→ Better look at*:

• Torque**: ^{ˆ ˆ , ˆ}

2 ⁱ 2 ^{Hub i}

d i

dt   ^H  ^



* D. M. Edwards, JMMM 45 (1984) 151-156

 

1 ˆ

2 ^{i i i}

d m B

dt       Bi

** L.M. Small and V. Heine., 1984 J. Phys. F: Met. Phys.14 3041

Spin dynamics and Monte Carlo

• Add temperature b_i and damping λ:

• Computational effort:

• Number of time steps: 10³ - 10⁶

• Per time step: ~ 10 iterations

• Per iteration: diagonalization of Ĥ ~ N_K· N_A³

• Thermodynamic properties: Monte Carlo

• Reduces number of time steps

• Metropolis criterion:

 

, 0

i

bfluc^ t 

     

,^{fluc fluc}, 2

i j ij

b _ t b _ s  D  _ t  s _with:

m T D k^B



 

     

i

i i i i i i i

de e B b e e B b

dt  ^    ^ ^     ^

      

and

 

 

exp E_old  E_new / k T_B 

Monte Carlo results

System:

• 125 atoms per unit cell

• 8 k-points

• 500MC steps per T

• 4 averages

Conclusions

• We developed a TB spin dynamics and Monte Carlo code

• Self-consistent constraining fields are the torques in SD

• Thermodynamics properties via tight – binding MC

• Outlook: use of large computer systems

Aknowledgements

Forschungzentrum Jülich:

MSc. Timo Schena dr. Phivos Mavropoulos

prof. Stefan Blügel

Radboud Universiteit Nijmegen:

dr. Johan Mentink

prof. Mikhail Katsnelson prof. Andrei Kirilyuk

dr. Alexei Kimel prof. Theo Rasing

Thank you for your attention!