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Temperature dependence of domain kinetics and magnetization processes in 3D Ising model
Journal of Magnetismand MagneticMaterials 140-144 (1995) 1515-1516
Journal of magnetic materials
,i~
ELSEVIER
Temperature dependence of domain kinetics and magnetization processes in 3D Ising model C. Mor6n *, M. Mora Dpto. S.LA., E.U. Inform{ttica (U.P.M.), 28031 Madrid, Spain Abstract Monte Carlo simulations of the Ising model in 3D systems have been carried out to study the effect of temperature on the domain kinetics and magnetization processes. The results obtained show that small closed shapes are not stable but rectangular prisms are stable indefinitely at sufficiently low temperatures.
Recently, the magnetism of structurally disordered systems has become the subject of both experimental and theoretical interest. A number of experimental and theoretical investigations leads to the result that magnetic longrange order may exist in amorphous systems. In the study of such systems, the lattice model of amorphous magnets has been used, in which the structural disorder is replaced by the random distribution of the exchange integral [1], and many interesting physical properties in it were found. In this work we extend our previous simulations [2] to study the magnetization processes and the effect of temperature on the domain kinetics. We have carried out extensive Monte Carlo simulations with different simple cubic lattice samples of size 20 × 20 X 20. The Monte Carlo technique used is the single-spin flipping procedure where in each run we discard a sufficient number of Monte Carlo steps (MCS) per spin to equilibrate the system before averaging physical quantities over a number of MCS/spin. We consider the following Hamiltonian:
H = -JEMiMj ij
+ 3'B E M i i
x.--/40 ~
C 3o1\
\
S
\
-5'
0 ....
g ....
10 . . . .
lg ....
20 . . . .
2g . . . .
30 ....
,3,5
TEMPERATURE (K) Fig. 1. Coercivefield againsttemperature.
0.6 4
(1)
....
i ....
i ....
T - 2 1 K ~ T-25K"~ . ' ~ r= 3oK "-,..,,._~,.,~
where 3/= e h / m c , with me the mass of the electron, and ij implies a summation over nearest-neighbour pairs i and j. We have used an exchange constant J / k a of magnitude 17 K, where k B is Boltzmann's constant. This gives an observed transition temperature Tc in the range 20-25 K, typical of magnetic insulators, and J is usually positive, leading to ferromagnetic alignment. In the ferromagnetic state, the hysteresis curves simulated by the model are shown that the coercive field decreases as the tempera-
J
0.2
i ....
i ....
i ....
f,
' ' 'g ....
,'5 . . . .
2's'
_
T=IO0
LLI
<>o.o ~U3-0.2
•
-0.4,
-0.6
-3g
' L~g'
' :'~.~ ' ' ' - ' i
MAGNETIC FIELD (T)
* Correspondingauthor. Fax: +34-1-336 7522.
Fig. 2. Graphs of magnetization against magnetic field for a ferromagnet with S = 1/2 at different temperatures T > Tc in the paramagnetic region.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00973-2
1516
C. Mor6n, M. Mora /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1515-1516
ture is increased, as can be seen in Fig. 1. For temperatures higher than Curie temperature, the phenomenon of hysteresis disappears, and the loops collapse to single curves, as shown in Fig. 2. If the simulation is allowed to come to equilibrium at high temperature, with randomly distributed z component of spin M i, and then thermally quenched by reducing T to well below To, domains are frequently formed (Fig. 3). These are regions in which there is a local alignment of
SWmM
(a)
zzl
•=•
z=|
•¢•
ZZ$
z=•
z=T
Z=l
zt~
zztO
Zzll
•¢12
Z=12
Z=I4
Z=I5
z=26
z--IT
x=18
z=lg
z=|O
Fig. 4. Rectangular structure at T = 0.1Tc after 100 MCS.
z~LA
z a18
z = L~
z ~ L4
Z zLU
s mMm • ~ la
• ~AT
• =lib
• 2 &gl
zzEO
z z LI
z z1.1
• •1~
••t4
zz18
(b)
opposite sign to that in neighbouring regions. In zero field, such domains may persist or may vanish with time. In general, small closed shapes are not stable, but structures involving lamellae or rectangular prisms of one sign of spins average embedded in a matrix of the opposite spin are stable indefinitely at sufficiently low temperatures (Fig. 4). It illustrates the type of visual display available from the simulation whereby any one of the 20 x - y planes can be displayed. Domain structures are found to occur spontaneously even on slow cooling in a significant number of cases, for which reason the plots of thermodynamic functions against temperature T are best obtained by starting at T = 0 with a known uniform configuration, and the increasing T. Domains can be removed by 'poling', i.e. temporarily applying a large magnetic field. The simulation provides considerable scope for further investigation of domain structures, their dependence upon T and B, and the effects of thermal quenching and annealing. References
n =16
x=LT
• =LIO
• = t9
8cmO
Fig. 3. Typical spontaneously formed domain structure for S = 1/2 and B = 0 as T ~ 0: (a) random initial configuration, (b) after 50 MCS.
[1] T. Kaneyoshi, R. Honmura, I. Tamura and E.F. Sarmento, Phys. Rev. B 27 (1984) 5121. [2] C. Mor6n and M. Mora, J. Magn. Magn. Mater. 133 (1994) 57.
Journal of magnetic materials
,i~
ELSEVIER
Temperature dependence of domain kinetics and magnetization processes in 3D Ising model C. Mor6n *, M. Mora Dpto. S.LA., E.U. Inform{ttica (U.P.M.), 28031 Madrid, Spain Abstract Monte Carlo simulations of the Ising model in 3D systems have been carried out to study the effect of temperature on the domain kinetics and magnetization processes. The results obtained show that small closed shapes are not stable but rectangular prisms are stable indefinitely at sufficiently low temperatures.
Recently, the magnetism of structurally disordered systems has become the subject of both experimental and theoretical interest. A number of experimental and theoretical investigations leads to the result that magnetic longrange order may exist in amorphous systems. In the study of such systems, the lattice model of amorphous magnets has been used, in which the structural disorder is replaced by the random distribution of the exchange integral [1], and many interesting physical properties in it were found. In this work we extend our previous simulations [2] to study the magnetization processes and the effect of temperature on the domain kinetics. We have carried out extensive Monte Carlo simulations with different simple cubic lattice samples of size 20 × 20 X 20. The Monte Carlo technique used is the single-spin flipping procedure where in each run we discard a sufficient number of Monte Carlo steps (MCS) per spin to equilibrate the system before averaging physical quantities over a number of MCS/spin. We consider the following Hamiltonian:
H = -JEMiMj ij
+ 3'B E M i i
x.--/40 ~
C 3o1\
\
S
\
-5'
0 ....
g ....
10 . . . .
lg ....
20 . . . .
2g . . . .
30 ....
,3,5
TEMPERATURE (K) Fig. 1. Coercivefield againsttemperature.
0.6 4
(1)
....
i ....
i ....
T - 2 1 K ~ T-25K"~ . ' ~ r= 3oK "-,..,,._~,.,~
where 3/= e h / m c , with me the mass of the electron, and ij implies a summation over nearest-neighbour pairs i and j. We have used an exchange constant J / k a of magnitude 17 K, where k B is Boltzmann's constant. This gives an observed transition temperature Tc in the range 20-25 K, typical of magnetic insulators, and J is usually positive, leading to ferromagnetic alignment. In the ferromagnetic state, the hysteresis curves simulated by the model are shown that the coercive field decreases as the tempera-
J
0.2
i ....
i ....
i ....
f,
' ' 'g ....
,'5 . . . .
2's'
_
T=IO0
LLI
<>o.o ~U3-0.2
•
-0.4,
-0.6
-3g
' L~g'
' :'~.~ ' ' ' - ' i
MAGNETIC FIELD (T)
* Correspondingauthor. Fax: +34-1-336 7522.
Fig. 2. Graphs of magnetization against magnetic field for a ferromagnet with S = 1/2 at different temperatures T > Tc in the paramagnetic region.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00973-2
1516
C. Mor6n, M. Mora /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1515-1516
ture is increased, as can be seen in Fig. 1. For temperatures higher than Curie temperature, the phenomenon of hysteresis disappears, and the loops collapse to single curves, as shown in Fig. 2. If the simulation is allowed to come to equilibrium at high temperature, with randomly distributed z component of spin M i, and then thermally quenched by reducing T to well below To, domains are frequently formed (Fig. 3). These are regions in which there is a local alignment of
SWmM
(a)
zzl
•=•
z=|
•¢•
ZZ$
z=•
z=T
Z=l
zt~
zztO
Zzll
•¢12
Z=12
Z=I4
Z=I5
z=26
z--IT
x=18
z=lg
z=|O
Fig. 4. Rectangular structure at T = 0.1Tc after 100 MCS.
z~LA
z a18
z = L~
z ~ L4
Z zLU
s mMm • ~ la
• ~AT
• =lib
• 2 &gl
zzEO
z z LI
z z1.1
• •1~
••t4
zz18
(b)
opposite sign to that in neighbouring regions. In zero field, such domains may persist or may vanish with time. In general, small closed shapes are not stable, but structures involving lamellae or rectangular prisms of one sign of spins average embedded in a matrix of the opposite spin are stable indefinitely at sufficiently low temperatures (Fig. 4). It illustrates the type of visual display available from the simulation whereby any one of the 20 x - y planes can be displayed. Domain structures are found to occur spontaneously even on slow cooling in a significant number of cases, for which reason the plots of thermodynamic functions against temperature T are best obtained by starting at T = 0 with a known uniform configuration, and the increasing T. Domains can be removed by 'poling', i.e. temporarily applying a large magnetic field. The simulation provides considerable scope for further investigation of domain structures, their dependence upon T and B, and the effects of thermal quenching and annealing. References
n =16
x=LT
• =LIO
• = t9
8cmO
Fig. 3. Typical spontaneously formed domain structure for S = 1/2 and B = 0 as T ~ 0: (a) random initial configuration, (b) after 50 MCS.
[1] T. Kaneyoshi, R. Honmura, I. Tamura and E.F. Sarmento, Phys. Rev. B 27 (1984) 5121. [2] C. Mor6n and M. Mora, J. Magn. Magn. Mater. 133 (1994) 57.