Anisotropy of Land Surface Skin Temperature Derived from ...
Calculation of anisotropy energy and temperature ...
Transcript of Calculation of anisotropy energy and temperature ...
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
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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!