Calculation of anisotropy energy and temperature dependence of

anisotropyA.Meo,R.W.Chantrell,R.F.L.EvansDepartmentofPhysics,UniversityofYork,York,YO105DD,UK

1st Advanced VAMPIRE Workshop

2

ConstraintMonteCarloalgorithm

Constraint Monte Carlo (CMC) algorithm

3

1. Why Monte Carlo (MC) algorithms?• Determine macroscopic equilibrium quantities

2. Issue• Standard MC allows to determine magnetic properties at

thermal equilibrium (magnetisation)

• At thermal equilibrium magnetocrystalline energy (MAE)can’t be determined since magnetisation is parallel to easyaxis

3. Solution• Investigate the system while in quasi-equilibrium condition

CMC

4

• CMC algorithm1 allows to constrains the direction of the globalmagnetisation while it allows the single spins to vary

• This allows to calculate restoring torque acting on themagnetisation

• From the torque, MAE can be obtained

1.ConstrainedMonteCarlomethodandcalculationofthetemperaturedependenceofmagneticanisotropy,P.Asselin,R.F.L.Evans,J.Barker,R.W.Chantrell,R.Yanes,O.Chubykalo-Fesenko,D.Hinzke,andU.Nowak,Phys.Rev.B82,054415(2010)

CMC step

5

• In CMC trial move acts on 2 spins at time:1. (Assume constraint direction along +z-axis) Select 2 spins i,j2. Apply MC move to first spin 𝑆"# → 𝑆"#%

3. Move second spin 𝑆"& to compensate change ofmagnetisation in xy-components (𝑀( = 𝑀* = 0):

,𝑆"&(% = 𝑆"#( + 𝑆"&( − 𝑆"#(%

𝑆"&*% = 𝑆"#* + 𝑆"&* − 𝑆"#*%

4. Adjust z-component of new spin j

/𝑆"&0% = 𝑠𝑖𝑔𝑛 𝑆"&0 1 − 𝑆"&(6 − 𝑆"&*6

�

1 − 𝑆"&(6 − 𝑆"&*6 > 0

CMC step

6

5. Calculate new magnetisation

9𝑀0% = 𝑀0 + 𝑆"#0% + 𝑆"&0% − 𝑆"#0 − 𝑆"&0

𝑀0% > 0

6. Calculate the change in energy

9∆𝐸# = 𝐸#% − 𝐸#

∆𝐸#& = ∆𝐸# + ∆𝐸&

7. Compute the acceptance probability of the moves

𝑃 = 𝑚𝑖𝑛 1,𝑀0%

𝑀0

6 𝑆"&0𝑆"&0%

𝑒𝑥𝑝 −∆𝐸#& 𝑘C𝑇⁄

Geometricalfactor

Boltzmannprobabilitydensity

Free energy

7

• In ensemble at constant temperature and volume, energy of thesystem given by Helmoltz free energy:

𝐹 = 𝑈 − 𝑇𝑆

• How can we obtain it?Calculating the restoring torque acting on the magnetisation ofthe system

Free energy

8

• For a reversible system, the variation of free energy is:

∆𝐹 = H𝛿𝑊K

L

• In a magnetic system the workW done on a system is equivalentto the torque T that acts on the whole system:

M 𝛿𝑊 = 𝑇 𝑑𝜗𝜗 = angleformedbyspinandnetfieldactingonspins

➡ ∆𝐹 = −∫ 𝑇 𝑑𝜗KL

Torque

9

• Total torque acting on the system is given by

𝑇 =Q𝑇#

�

#

= Q𝑆"#

�

#

×𝐻# =Q𝑆"#

�

#

× −𝜕ℋ𝜕𝑆"#

• The magnitude of the average of the total internal torque T iscalculated from thermodynamic average of T:

𝑇 = 𝑇

• Once the torque is calculated, the free energy can be computed

Uniaxial anisotropy

10

• Let’s assume the material has a single axis along which themagnetic moments prefer to alignà Uniaxial anisotropy

• Internal energy U is given by:𝑈 = 𝐾W sin 𝜗6

• At zero temperature

𝑇 = −𝜕𝐹𝜕𝜗 = −

𝜕𝑈𝜕𝜗 = −𝐾W sin 2𝜗

• sin 2𝜗 holds at all temperatures, while KuàKu(T)

Uniaxial anisotropy: Torque & free energy

11

-1.0

-0.5

0.0

0.5

1.0

π / 2 3π / 4 π

Tota

l tor

que/

k u

Angle from easy axis (rad)

0K50K

100K300K

0.0

0.2

0.4

0.6

0.8

1.0

0 π / 4 π / 2 3π / 4 π

Free

ene

rgy ∆

F / k

u

Angle from easy axis (rad)

0K50K

100K300K

sin(2q) sin2(q)

Uniaxial anisotropy: Temperature scaling

12

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

K(T)

/Ku,

M(T

)/Ms

Temperature (K)

M/MsK(T)/Ku

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-1 -0.8 -0.6 -0.4 -0.2 0

ln(K

eff(T

)/ku)

ln(M(T)/Ms)

Fit, 1.0 m3.0Data

2.H.B.CallenandE.Callen,J.Phys.Chem.Solids27,1271(1966).

𝐾(𝑇) ∝ 𝑀(𝑇)_

InagreementwithCallen-Callentheory2

13

Handson

Main parameters for CMC simulations

14

sim:constraint-angle-theta-minimum = 0.0sim:constraint-angle-theta-maximum = 0.0sim:constraint-angle-theta-increment= 10.0

sim:constraint-angle-phi-minimum = 0.0sim:constraint-angle-phi-maximum = 90.0sim:constraint-angle-phi-increment= 15.0

sim:integrator = constrained-monte-carlosim:program = cmc-anisotropy

output:constraint-thetaoutput:constraint-phioutput:mean-magnetisation-lengthoutput:mean-total-torque

Plotting data with gnuplot

15

# In bash we can sort the data in a file based on specific keys (-k):>sort -k 1g -k 3g output

# Enter in gnuplot:> Gnuplot

#column ($) 3 is the azimuthal angle phi, column 6 is y-component of torque##In gnuplot we can have conditional statements with the following syntax:($1==0 ?$3:0/0):($6)

#Once in gnuplot, type the following commands where > plot ‘<sort -k 1g -k 3g output’ u($1==0?$3:0/0):($6) w l

FILE to plot COLUMNS to plot

PLOT with lines

Practice on CMC

16

Torque

17

-1.0

-0.5

0.0

0.5

1.0

0 π / 4 π / 2

Tota

l tor

que/

k u

Angle from easy axis (rad)

0K100K300K600K

1000K1600K

Temperature scaling

18

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

K(T)

/Ku,

M(T

)/Ms

Temperature (K)

M/MsK(T)/Ku

Scaling of anisotropy vs magnetisation

19

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

-1 -0.8 -0.6 -0.4 -0.2 0

ln(K

eff(T

)/ku)

ln(M(T)/Ms)

Fit, 1.0 m3.0Data

Practice on CMC

20

Torque

21

-1.0

-0.5

0.0

0.5

1.0

0 π / 4 π / 2

Tota

l tor

que/

k c

Angle from easy axis (rad)

θ=0 0K100K300K600K

θ=π/8 0K100K300K600K

Torque depends on both rotational (q) and azimuthal (f) angle

Temperature scaling

22

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200 1400

K(T)

/Kc,

M(T

)/Ms

Temperature (K)

M/MsK(T)/Kc

Scaling of anisotropy vs magnetisation

23

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

-1 -0.8 -0.6 -0.4 -0.2 0

ln(K

eff(T

)/kc)

ln(M(T)/Ms)

Fit, 1.0 m9.9Data

24

More complex systems

0.0

0.2

0.4

0.6

0.8

1.0

0 π / 4 π / 2 3π / 4 π

Ener

gy b

arrie

r / k

u

Angle from easy axis (rad)

0K100K200K300K

mz+1

0

-1

mz

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350 400 450

(Ene

rgy

barri

er/k

u)/V

olum

e

Temperature (K)

d=10nm15nm20nm25nm30nm35nm40nm50nm

CoFeBMgOCoFeB

-1.0

-0.5

0.0

0.5

1.0

π / 2 3π / 4 π

Tota

l tor

que/

k u

Angle from easy axis (rad)

0K50K

100K300K

More complex systems

25

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

Δf(1

0-18

J)

θ[rad]

z =1nmz =2nmz =3nmz =4nmz =5nmz =6nmz =7nmz =8nmz =9nm

z =10nmz =11nmz =12nm

ImagesanddatakindlyfromRazvan Ababei

Practice on CMC

26

Main parameters for CMC simulations

27

material[1]:constrained=true #false for other material

material[1]:constraint-angle-theta-minimum=0.0material[1]:constraint-angle-theta-maximum=0.0material[1]:constraint-angle-theta-increment=10.0

material[1]:constraint-angle-phi-minimum=0.0material[1]:constraint-angle-phi-maximum=90.0material[1]:constraint-angle-phi-increment=15.0

#------------------------------------------#------------------------------------------sim:integrator = hybrid-constrained-monte-carlosim:program = hybrid-cmc

output:material-constraint-thetaoutput:material-constraint-phi

Constraining only uniaxial material

28

-1.0

-0.5

0.0

0.5

1.0

0 π / 4 π / 2

Tota

l tor

que/

k u

Angle from easy axis (rad)

0K100K300K500K

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

K(T)

/Ku,

M(T

)/Ms

Temperature (K)

M/MsK(T)/Ku

Constraining only uniaxial material

29

-1.0

-0.5

0.0

0.5

1.0

0 π / 4 π / 2

Tota

l tor

que/

k c

Angle from easy axis (rad)

0K100K300K500K

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000

K(T)

/Kc,

M(T

)/Ms

Temperature (K)

M/MsK(T)/Kc

Practice on CMC

30

31

Dipolefields

32

Dipole energy

For N interacting dipoles (macro-cells), the total dipole-dipoleenergy is given by:

𝐸`ab = −12Q𝑚c

𝜇e4𝜋 Dci𝑚i

j

cki

Where Dqp is the dipolar matrix and𝑚i,c are the moments of thedipoles p,q. In terms of magnetic induction 𝐸`ab can be writtenas:

𝐸`ab = −12Q𝑚cBc

`abj

cki

where Bqdip is the dipolar field experienced by moment q.

33

• The system is divided into cubic macro-cells

• Each cell is supposed to have uniform magnetisation

• The dipolar interaction is calculated between macro-cellsconsidering for each one a pairwise summation.

• A self-demagnetisation term is added, to include the internal fieldof the macro-cell.

Hc`ab =

𝜇e4𝜋Q

3 𝜇i r �̂�ci �̂�ci − 𝜇i𝑟ci

_

�

cki

−𝜇e3𝜇c𝑉c

Bare macro-cell approach

34

Inter-intra dipole approachWe can write the dipolar matrix as summation of the contributionfrom interaction with other cells (inter) and internal to themacrocell (intra) [G. J. Bowden et al, J.Phys.:Condens. Matter,28(2016),066001]:

B`ab = Bc`ab + Bi

`ab + Bivwxy

= Dciaz{w|𝑚i + Diiaz{|}𝑚i + ~�_ 𝑚i/𝑉i

Dciaz{w| + Diiaz{|} =QQDc&,i#az{w|��

i�

��

c�

+ Q QDi&i#az{|}��

i�

��

i�ki�

=Dci

INTER INTRA

Interaction between thespins in cell p and spins incell q (atomistic)

Interaction betweenmoments inside cell p(atomistic)

35

• Bivwxy =~�_𝑚irepresent the Maxwellian internal field of a

dipole.

• 𝑚i is the macro-moment of the cell p and Vp is the volume ofthe same cell.

• Dci#����,Dii#���L do not correspond to real dipole-dipolematrices, but their sum Dci does.

• Dci#����,Dii#���L are casted in terms of usual dipolar matrix

Inter-intra dipole approach

36

The dipolar matrix for the interaction between different macro-cells, is given by:

Dc&,i#az{w| =

1𝑟i#c&_

(3𝑥i#c&6 − 1) 3𝑥i#c&𝑦i#c& 3𝑥i#c&𝑧i#c&3𝑦i#c&𝑥i#c& (3𝑦i#c&6 − 1) 3𝑦i#c&𝑧i#c&3𝑧i#c&𝑥i#c& 3𝑧i#c&𝑦i#c& (3𝑧i#c&6 − 1)

Same in case of internal term (replace qj→pj) for Di#,i&az{|}.

Warning: Assumption is that moments in the macro-cell are allaligned along the same direction, i.e. parallel to each other.

Dipolar matrixes

37

• This approach works independently of the shape of the macro-cell.

• After ~ 2 macro-cells, contribution of intra term Daz{|}becomes negligible and a bare macro-cell method could beused.

• Since it requires parallel moments, the size of the macro-cellshould be less than the domain-wall width.

Notes on the approach

38

Test: Demagnetisation factor of ellipsoid

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8 9 10

Dem

agnetis

atio

n fact

or

Aspect ratio (c/a)

sphere

Osborn’s relationshipsmacrocell = system-size (=L)

75% L50% L25% L10% L7.5% L5.0% L

Osborn’srelationship:

Aspectratio=k0 =c/a

39

Handson

Main parameters

40

cells:macro-cell-size=10.1 !A

dipole:solver=tensordipole:field-update-rate=100dipole:cutoff-radius=2

Practice on Dipole fields

41

0

Practice on Dipole fields

42

Practice on Dipole fields – final snapshot

43

44

• Download Povray from github repository:https://github.com/POV-Ray/povray/tree/3.7-stable

• To install look at “README.md” in “unix” directory:

Povray – visualising spin cofigurations

45

Povray – visualising spin cofigurations: How to compile it

git clone https://github.com/POV-Ray/povray.git

cd unix/

./prebuild.sh

cd ../

./configure COMPILED_BY="your name <email@address>"

make

make install

46

Povray – Running povray

> /path/to/vdc/vdc

> povray spins # generate all snapshots

# or you can do

> povray +KFI0 –KFF0 spins.pov

# You can add some option as size of the image

> povray –W800 –H600 spins

# You can add some option as antialiasing

> povray +A0.3 spins

47

Thankyouforyoutime!