calculation of anisotropy energy and temperature
TRANSCRIPT
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
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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
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• 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
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• 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
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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
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• 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
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• 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
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• 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
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• 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
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-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
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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
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Handson
Main parameters for CMC simulations
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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
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# 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
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Torque
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-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
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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
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-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
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Torque
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-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
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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
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-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
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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
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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
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Main parameters for CMC simulations
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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
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-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
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-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
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Dipolefields
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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.
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• 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
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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)
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• 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
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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
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• 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
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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
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Handson
Main parameters
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cells:macro-cell-size=10.1 !A
dipole:solver=tensordipole:field-update-rate=100dipole:cutoff-radius=2
Practice on Dipole fields
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0
Practice on Dipole fields
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Practice on Dipole fields – final snapshot
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• 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
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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
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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
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Thankyouforyoutime!