- Email: [email protected]
Computer simulation of slow magnetic relaxation
Journal of Magnetism and Magnetic Materials 140-144 (1995) 36541%
ELSEVIER
Computersimulationof slow magneticrelaxatio C. SLnchez,J.M. Gonziilez-Miranda, J. Tejada* Departament de Fkica FonamentaJ Unioersitat de Barcelona, AC. Diagonal 647, E-08028 Barceloma, Spain
Abstract We present a computer simulation study of magnetic relaxation for a system of single-domain particles. We dependence of the magnetic viscosity coefficient with the shape of the distribution function of activation ene compare our computational results with known experimental results for a ferrofluid system. ‘we
have
s&&d
the
-~‘---*:~n I IM(III”II
nf “A
*Ia S.S”
maon+atinn .I._ cT* ---.
._
in
systems of single-domain particles by means of computer simulation. Such systems ale of considerable interest because the interesting phenomenon of qUartFUm tunneling of magnetixaFion [1,2j shows up in the form of a plateau in the temperature dependence of the parameter that characterizes the slow relaxation in these systems [3,4]. Computer simulations of magnetization relaxation in sharply defined models of magnetic systems [S-7] are useful to improve the analysis and interpretation of experimants. Our main concern here is to calculate the effect of the shape of the distribution of energy barriers on the relaxation behavior of the system. We pay particular attention to the type of relaxation law, its characterization by means of an appropriate parameter, and its dependence on temperature. Our model is a system of N non-identical non-mteracting particles, each with its own magnetic moment mi than can point in either of the two opposite directions. An energy barrier characteristic of each particle, Ui, must be overcome to invert this magnetization, by thermal agitation or otherwise. Roth the magnitude of the magnetization and the energy barrier are assumed to be proportiond to the volume of the particle. The diverse energy barriers of the particles in the system, Q, are assumed to be distributed aCCQrdhIg to a log-normal law: f(U)
ies of relaxation were made when all the merits are aligned in the same direction. event is repeatedly iterated to simulate librium non-stationary process: a particle is sen and its magnetic moment is inverted wi given by its barrier by means of P(y) = exp( - Q/k,T), which has been normalized so that
acting single-domain particles ES]. The computer simulations were of N = 3.6 X lo4 particles, evolved The quantity of interest was the system, M(t)
IV c m,(r). i-
=A exp[ -a(ln(CJ/U,)}Z],
where U, is the energy value where the distribution takes its maximum; the value of this maximum depends on a, a parameter that affects the shape of the distribution in such a way that it becomes more narrow and sharply peaked as a is increased. This type if distribution function for activation energies has been reported for reai systems of singie-domain particles [S]. Our computer simulation stud’ Corresponding author. [email protected].
=
Fax:
+ 34-3-4021149;
1
We present here the results energies distributed ascording for four values of a: 1, 2,4,5. the variation of magnetization appropriate time window, can rithmic law,
for the case to a log-norm In all caw with time, when seen in be approximated by a toga-
M(t) =&[l -S(T) IWT,)], where S(T) is the magnetic viscosi teristic parameter of the retaxatiou. size of the window decreases with we present the results for magnetic
03W8853/95/$09.50 0 1995 Elsevier Science B.V. AU rights reserved sSDIO304-8853(94)01033-l
et Cpr/him&
0fiVagiwtism and Magnetic Matetiis
,--__mt
10 1
10 ’
t
10 5
etization for differem temperatures -normal distriiution of energy barriers with T, is given io units of Ue /a,.
---~ 0.30
Q.OG 0.00
-
-~
0 60
--+ 0.90
T
a fuoction of temperature for four cn=t,(O) a=2,(0)a=4,aud in uoits of Cl, /it,. Inset: experld CoFe,O, hll circles). e time interval. Table I presents ime for which these magnetic in Fig. 2, the viscosity inn decreases ack
to zero. The main
peaked; (ii) the peak is and (iii) the large temper-
(t) versus
intervals, io tlmsaods of steps, inside
lr$t) was qqxoximated by a linear function. ts were about 5(10 steps for all temperatures and
a W,/k,)
Q=l
Q!=
-05 0.10 0.15 0.20 0.25 030
36.0 31.7 24.5 25.7 23.3 22.0
36.0 36.0 22.0 13.4 7.5 5.2
2
a=$
Ct=5
36.0 36.0 15.0 4.8 i.8 2.2
36.0 36.0 13.9 4.3 2.3 0.7
140-144 (1995) 365-366
ature tail decays faster. We have been unable to scale the different S(T) functions onto a single curve by multiplying S and T by convenient constants. This indicates that the behaviors are essentially different. We have not observed evidence of plateaus 6 constant while T changes> :egardless the value of (Y, for this process, which is exclusively thermaMy activated. We have compared our computational results for the magnetic viscosity with the experimental ones by Tejada et al. 181 for a ferrofluid containing an small concentration of CoFe,O, particles. These results appear in the inset of Fig. 2. One first notes that the experimental behavior is very similar to the calculated one. Although the experimenta system considered is one of randomly oriented interacting particles, this agreement should not be a surprise, because: 61 the system is very diluted, so that the interactions are expected to be weak, and (ii) the orientation and magnitudes of the magnetic moments are uncorrelated items, one can think the magnetic moments of the simulation as the projections of the magnetic moments on the direction of the initially applied magnetic field. It deserves to be noted that all the simulation results for the viscosity extrapolate to zero as the temperature goes to zero, while the experimental values converge to a small positive non-null value. As the relaxation in the simulation is exclusively thermal, this is an indication of the presence of another type of effect in the experiment, very likely quantum tunneling of the magnetization [1,8]. In conclusion, we have presented a computational model for single-domain particles that provides us with a quite realistic simulation of the reIaxation behaviors observed in experiment. This has been used to verify the reliability of the time logarithmic relaxation law, the effect of the shape of the distribution on the magnetic viscosity, and to interpret some experimental observations of quantum tunneling of the magnetization in single-domain particles. Acknowledgement: Financial support from CICYT is acknowledged.
References [l] E.M. Chudnovsky and L. Gunter, Phys. Rev. Lctt. 60 (1988) [21 rk Chudnovsky and L. Gunter, Phys. Rev. B 37 (1988) 9455. [3] J. Tejada, X.X. Zhang and EM. Chudnovsky, Phys. Rev. B47 (1993) 14977. [41 J. Tejada, Ll. Balcells and X.X. Zhang, J. Magn. Magu. Mater. 69 (1987) 106. [5] ;;7Gonz51ez-Miranda and J. Tejada, Phys. Rev. B49 (1994) [6] M.E.‘Matson, D.K. LQttis and E, Dan Dablberg. J. Appl. Phys. 75 (1994) 5475. [7j k Lyberatos. R.W. Chantrell and A. Hoare, IEEE Trans. Magn. 26 (1990) 222. 181J. Tejada, LI. Bakells, S. Linderot, R. Perzyosli. B. Rigau, B. Barbara and SC Bacri, J. Appl. Phys. 73 (1993) 6952.
191 EL Binder, Monte Carlo Methods in Statistical Physics @pringer, Berlin, 1979).
ELSEVIER
Computersimulationof slow magneticrelaxatio C. SLnchez,J.M. Gonziilez-Miranda, J. Tejada* Departament de Fkica FonamentaJ Unioersitat de Barcelona, AC. Diagonal 647, E-08028 Barceloma, Spain
Abstract We present a computer simulation study of magnetic relaxation for a system of single-domain particles. We dependence of the magnetic viscosity coefficient with the shape of the distribution function of activation ene compare our computational results with known experimental results for a ferrofluid system. ‘we
have
s&&d
the
-~‘---*:~n I IM(III”II
nf “A
*Ia S.S”
maon+atinn .I._ cT* ---.
._
in
systems of single-domain particles by means of computer simulation. Such systems ale of considerable interest because the interesting phenomenon of qUartFUm tunneling of magnetixaFion [1,2j shows up in the form of a plateau in the temperature dependence of the parameter that characterizes the slow relaxation in these systems [3,4]. Computer simulations of magnetization relaxation in sharply defined models of magnetic systems [S-7] are useful to improve the analysis and interpretation of experimants. Our main concern here is to calculate the effect of the shape of the distribution of energy barriers on the relaxation behavior of the system. We pay particular attention to the type of relaxation law, its characterization by means of an appropriate parameter, and its dependence on temperature. Our model is a system of N non-identical non-mteracting particles, each with its own magnetic moment mi than can point in either of the two opposite directions. An energy barrier characteristic of each particle, Ui, must be overcome to invert this magnetization, by thermal agitation or otherwise. Roth the magnitude of the magnetization and the energy barrier are assumed to be proportiond to the volume of the particle. The diverse energy barriers of the particles in the system, Q, are assumed to be distributed aCCQrdhIg to a log-normal law: f(U)
ies of relaxation were made when all the merits are aligned in the same direction. event is repeatedly iterated to simulate librium non-stationary process: a particle is sen and its magnetic moment is inverted wi given by its barrier by means of P(y) = exp( - Q/k,T), which has been normalized so that
acting single-domain particles ES]. The computer simulations were of N = 3.6 X lo4 particles, evolved The quantity of interest was the system, M(t)
IV c m,(r). i-
=A exp[ -a(ln(CJ/U,)}Z],
where U, is the energy value where the distribution takes its maximum; the value of this maximum depends on a, a parameter that affects the shape of the distribution in such a way that it becomes more narrow and sharply peaked as a is increased. This type if distribution function for activation energies has been reported for reai systems of singie-domain particles [S]. Our computer simulation stud’ Corresponding author. [email protected].
=
Fax:
+ 34-3-4021149;
1
We present here the results energies distributed ascording for four values of a: 1, 2,4,5. the variation of magnetization appropriate time window, can rithmic law,
for the case to a log-norm In all caw with time, when seen in be approximated by a toga-
M(t) =&[l -S(T) IWT,)], where S(T) is the magnetic viscosi teristic parameter of the retaxatiou. size of the window decreases with we present the results for magnetic
03W8853/95/$09.50 0 1995 Elsevier Science B.V. AU rights reserved sSDIO304-8853(94)01033-l
et Cpr/him&
0fiVagiwtism and Magnetic Matetiis
,--__mt
10 1
10 ’
t
10 5
etization for differem temperatures -normal distriiution of energy barriers with T, is given io units of Ue /a,.
---~ 0.30
Q.OG 0.00
-
-~
0 60
--+ 0.90
T
a fuoction of temperature for four cn=t,(O) a=2,(0)a=4,aud in uoits of Cl, /it,. Inset: experld CoFe,O, hll circles). e time interval. Table I presents ime for which these magnetic in Fig. 2, the viscosity inn decreases ack
to zero. The main
peaked; (ii) the peak is and (iii) the large temper-
(t) versus
intervals, io tlmsaods of steps, inside
lr$t) was qqxoximated by a linear function. ts were about 5(10 steps for all temperatures and
a W,/k,)
Q=l
Q!=
-05 0.10 0.15 0.20 0.25 030
36.0 31.7 24.5 25.7 23.3 22.0
36.0 36.0 22.0 13.4 7.5 5.2
2
a=$
Ct=5
36.0 36.0 15.0 4.8 i.8 2.2
36.0 36.0 13.9 4.3 2.3 0.7
140-144 (1995) 365-366
ature tail decays faster. We have been unable to scale the different S(T) functions onto a single curve by multiplying S and T by convenient constants. This indicates that the behaviors are essentially different. We have not observed evidence of plateaus 6 constant while T changes> :egardless the value of (Y, for this process, which is exclusively thermaMy activated. We have compared our computational results for the magnetic viscosity with the experimental ones by Tejada et al. 181 for a ferrofluid containing an small concentration of CoFe,O, particles. These results appear in the inset of Fig. 2. One first notes that the experimental behavior is very similar to the calculated one. Although the experimenta system considered is one of randomly oriented interacting particles, this agreement should not be a surprise, because: 61 the system is very diluted, so that the interactions are expected to be weak, and (ii) the orientation and magnitudes of the magnetic moments are uncorrelated items, one can think the magnetic moments of the simulation as the projections of the magnetic moments on the direction of the initially applied magnetic field. It deserves to be noted that all the simulation results for the viscosity extrapolate to zero as the temperature goes to zero, while the experimental values converge to a small positive non-null value. As the relaxation in the simulation is exclusively thermal, this is an indication of the presence of another type of effect in the experiment, very likely quantum tunneling of the magnetization [1,8]. In conclusion, we have presented a computational model for single-domain particles that provides us with a quite realistic simulation of the reIaxation behaviors observed in experiment. This has been used to verify the reliability of the time logarithmic relaxation law, the effect of the shape of the distribution on the magnetic viscosity, and to interpret some experimental observations of quantum tunneling of the magnetization in single-domain particles. Acknowledgement: Financial support from CICYT is acknowledged.
References [l] E.M. Chudnovsky and L. Gunter, Phys. Rev. Lctt. 60 (1988) [21 rk Chudnovsky and L. Gunter, Phys. Rev. B 37 (1988) 9455. [3] J. Tejada, X.X. Zhang and EM. Chudnovsky, Phys. Rev. B47 (1993) 14977. [41 J. Tejada, Ll. Balcells and X.X. Zhang, J. Magn. Magu. Mater. 69 (1987) 106. [5] ;;7Gonz51ez-Miranda and J. Tejada, Phys. Rev. B49 (1994) [6] M.E.‘Matson, D.K. LQttis and E, Dan Dablberg. J. Appl. Phys. 75 (1994) 5475. [7j k Lyberatos. R.W. Chantrell and A. Hoare, IEEE Trans. Magn. 26 (1990) 222. 181J. Tejada, LI. Bakells, S. Linderot, R. Perzyosli. B. Rigau, B. Barbara and SC Bacri, J. Appl. Phys. 73 (1993) 6952.
191 EL Binder, Monte Carlo Methods in Statistical Physics @pringer, Berlin, 1979).