- Email: [email protected]
Non-equilibrium collective dynamics of a superspin glass
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1316–1318
Non-equilibrium collective dynamics of a superspin glass S. Sahooa,*, O. Petracica, W. Kleemanna, P. Nordbladb, S. Cardosoc, P.P. Freitasc b
a Angewandte Physik, Universitat . Duisburg-Essen, Lotharstr. 1, 47048 Duisburg, Germany Department of Material Science, Uppsala University, Box 534, SE 751 21, Uppsala, Sweden c INESC, Rua Redol 9-1, 1000 Lisbon, Portugal
Abstract Magnetization relaxation measurements show that aging occurs in a discontinuous metal–insulator multilayer (DMIM) [Co80Fe20(0.9 nm)/Al2O3(3 nm)]10 below the spin glass transition temperature, Tg E44 K. Furthermore, the DMIM system memorizes the structure of a quasi-equilibrium state reached after an intermittent stop-and-wait protocol during cooling. These results unambiguously corroborate the collective nature of the low temperature superspin dynamics. r 2004 Elsevier B.V. All rights reserved. PACS: 75.10.Nr; 75.50.Lk Keywords: Magnetic nanoparticle; Superspin glass state; Aging; Memory
It is now widely accepted that three-dimensional (3D) random distribution of nearly monodisperse single domain magnetic nanoparticles (‘‘superspins’’) with appreciable interparticle interactions exhibits a magnetic phase transition from pure N!eel–Brown-type superparamagnetic to a collective superspin glass (SSG) state at low enough temperature [1–5]. The collective state has further been evidenced by non-equilibrium properties such as aging, memory, and rejuvenation similar to spin glasses [1,2,5]. In this paper, the non-equilibrium collective dynamics is further elucidated in the SSG state [3,4] of the discontinuous metal–insulator multilayer (DMIM) system [Co80Fe20(0.9 nm)/Al2O3(3 nm)]10 from zero-field-cooled (ZFC) magnetization relaxation measurements. It has already been shown by dynamic [3] as well as by static criticalities [4] that this DMIM system reveals an SSG transition below a well-defined glass temperature, Tg E44 K. We will adopt the important concept of domain growth from the droplet scaling model of spin glasses [6], which is a real space picture, in order to interpret our results. *Corresponding author. Tel.: +49-203-379-2939; fax: +49203-379-1965. E-mail address: [email protected] (S. Sahoo).
The DMIM sample was prepared by focused Xe+-ion beam sputtering on a glass substrate. Transmission electron micrographs obtained on a bilayer Co80Fe20(0.9 nm)/Al2O3(3 nm) sample show that CoFe forms well-separated and quasi-spherical particles with an average diameter d ¼ 2:8 nm within a Gaussian distribution width sv ¼ 0:95 [7]. Magnetization relaxation and magnetization (M) vs. temperature (T) measurements were performed by the use of a non-commercial low-field superconducting quantum interference device (SQUID) magnetometer. In an aging experiment, the sample is ZFC from T > Tg to a constant measurement temperature, Tm oTg ; where after a wait time tw ; a small probe field m0 H ¼ 0:04 mT was applied and the magnetization was recorded vs. time. In a memory experiment, the sample was ZFC from T > Tg to a stop temperature Ts oTg ; where the system was aged for a certain duration before further cooling down to lower temperatures and the magnetization was recorded during heating. This is referred to as a stop-and-wait protocol. Fig. 1(a) shows the relaxation of the magnetization (M) vs. log t at T ¼ 40 KoTg in an aging experiment. The employed wait times tw are indicated in the figure. The curves show a clear wait time dependence indicating that non-equilibrium phenomena play a key role for the dynamics at low temperatures. The wait time
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.091
ARTICLE IN PRESS S. Sahoo et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1316–1318
1.5 300 s 1000 10000
M [arb. units]
M [arb. units]
1.0
0.5 (a)
1.0
tw=104s
0.5
(a)
tw=10000s
0.0
1000s 300s 0.08
∆M [arb. units]
S [arb. units]
0.0 0.12
1317
(b)
0.04 -1 10
0
10
1
10
2
10
3
10
0.00
-0.04
4
10
(b)
t [s] Fig. 1. Relaxation curves M (a) and rate S vs. log t (b) at T ¼ 40 K recorded at m0 H ¼ 0:04 mT after different wait times as indicated.
dependence is more clearly reflected in the relaxation rate, S ¼ ð1=m0 HÞðqM=qln tÞ (see Fig. 1b), with peaks appearing at times close to the corresponding wait times similarly as observed in atomic spin glasses [8]. We have previously shown that the relaxation of thermoremanent magnetization exhibits similar aging phenomena albeit showing less clear correlations between tw and tðSmax Þ [9]. Presumably this was due to non-linearity effects owing to comparatively large field steps applied. The characteristic aging observed in the SSG phase implies that the correlation between the particle magnetic moments develops in the same way as the correlation between the spins of an atomic spin glass. In the droplet model [6], this correlation constitutes a domain whose size after an aging time ta is Rðta Þ ¼ ðT lnðta =t Þ=DðTÞÞ1=c ; where t is the relaxation time of an individual particle moment, DðTÞ sets the free energy scale, and c is a barrier exponent. When a small probe (DC) field is applied, the domain is probed via the polarization of droplets of size LðtÞ ¼ ðT lnðt=t Þ= DðTÞÞ1=c : Two limiting cases can be considered: (i) ln t5ln ta ; where LðtÞ5Rðta Þ and (ii) ln tbln ta ; where LðtÞERðta Þ: In the first case, quasi-equilibrium dynamics is probed whereas in the second case the probed length scale involves domain walls. This crossover from equilibrium to non-equilibrium response is reflected by a point of inflexion in M or, equivalently, a peak in the relaxation rate S vs. time.
-0.08 20
30
40
50
60
70
T [K]
Fig. 2. (a) Temperature dependence of the reference magnetization, Mref ðTÞ; (solid circles) and of the magnetization with a stop-and-wait protocol, MðTÞ (open circles) at a magnetic field of m0 H ¼ 0:04 mT. (b) DM ¼ MðTÞ2Mref ðTÞ vs. T:
Fig. 2 illustrates the memory and rejuvenation effects of the ZFC DC magnetization after a stop-and-wait procedure at Ts ¼ 42 K (=0.95 Tg ) for a duration of 104 s. As can be seen in Fig. 2(a), the data corresponding to the intermittent stop-and-wait protocol, MðTÞ; (open circles) lies significantly below the reference curve, Mref ðTÞ; (solid circles) at temperatures close to Ts : As seen in Fig. 2(b), the difference of the two data sets DM ¼ MðTÞ2Mref ðTÞ minimizes at Ts : This indicates that the magnetic moment configuration spontaneously rearranges toward equilibrium via growth of domains when the system is aged at Ts : These equilibrated domains become frozen-in on further cooling and are retrieved on reheating. In other words, the system shows a memory effect. The fact that Mref ðTÞ and MðTÞ curves coalesce at low temperatures and only start to deviate as Ts is approached again from below clearly indicates that rejuvenation occurs as the temperature is decreased away from Ts in the stop-and-wait protocol. Thanks are due to DFG (GK-277) for financial support.
ARTICLE IN PRESS 1318
S. Sahoo et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1316–1318
References [1] J.L. Dormann, et al., J. Magn. Magn. Mater. 187 (1998) L139. [2] H. Mamiya, et al., Phys. Rev. Lett. 82 (1999) 4332. [3] O. Petracic, et al., Phase Trans. 75 (2002) 73. [4] S. Sahoo, et al., Phys. Rev. B 65 (2002) 134406.
. P. Jonsson, et al., Phys. Rev. B 61 (2000) 1261. D.S. Fisher, D.A. Huse, Phys. Rev. B 38 (1988) 373. S. Sahoo, et al., Appl. Phys. Lett. 82 (2003) 4116. J.A. Mydosh, Spin Glasses: an Experimental Introduction, Taylor & Francis, London, 1993. [9] S. Sahoo, et al., J. Phys.: Condens. Mater 14 (2002) 6729. [5] [6] [7] [8]
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1316–1318
Non-equilibrium collective dynamics of a superspin glass S. Sahooa,*, O. Petracica, W. Kleemanna, P. Nordbladb, S. Cardosoc, P.P. Freitasc b
a Angewandte Physik, Universitat . Duisburg-Essen, Lotharstr. 1, 47048 Duisburg, Germany Department of Material Science, Uppsala University, Box 534, SE 751 21, Uppsala, Sweden c INESC, Rua Redol 9-1, 1000 Lisbon, Portugal
Abstract Magnetization relaxation measurements show that aging occurs in a discontinuous metal–insulator multilayer (DMIM) [Co80Fe20(0.9 nm)/Al2O3(3 nm)]10 below the spin glass transition temperature, Tg E44 K. Furthermore, the DMIM system memorizes the structure of a quasi-equilibrium state reached after an intermittent stop-and-wait protocol during cooling. These results unambiguously corroborate the collective nature of the low temperature superspin dynamics. r 2004 Elsevier B.V. All rights reserved. PACS: 75.10.Nr; 75.50.Lk Keywords: Magnetic nanoparticle; Superspin glass state; Aging; Memory
It is now widely accepted that three-dimensional (3D) random distribution of nearly monodisperse single domain magnetic nanoparticles (‘‘superspins’’) with appreciable interparticle interactions exhibits a magnetic phase transition from pure N!eel–Brown-type superparamagnetic to a collective superspin glass (SSG) state at low enough temperature [1–5]. The collective state has further been evidenced by non-equilibrium properties such as aging, memory, and rejuvenation similar to spin glasses [1,2,5]. In this paper, the non-equilibrium collective dynamics is further elucidated in the SSG state [3,4] of the discontinuous metal–insulator multilayer (DMIM) system [Co80Fe20(0.9 nm)/Al2O3(3 nm)]10 from zero-field-cooled (ZFC) magnetization relaxation measurements. It has already been shown by dynamic [3] as well as by static criticalities [4] that this DMIM system reveals an SSG transition below a well-defined glass temperature, Tg E44 K. We will adopt the important concept of domain growth from the droplet scaling model of spin glasses [6], which is a real space picture, in order to interpret our results. *Corresponding author. Tel.: +49-203-379-2939; fax: +49203-379-1965. E-mail address: [email protected] (S. Sahoo).
The DMIM sample was prepared by focused Xe+-ion beam sputtering on a glass substrate. Transmission electron micrographs obtained on a bilayer Co80Fe20(0.9 nm)/Al2O3(3 nm) sample show that CoFe forms well-separated and quasi-spherical particles with an average diameter d ¼ 2:8 nm within a Gaussian distribution width sv ¼ 0:95 [7]. Magnetization relaxation and magnetization (M) vs. temperature (T) measurements were performed by the use of a non-commercial low-field superconducting quantum interference device (SQUID) magnetometer. In an aging experiment, the sample is ZFC from T > Tg to a constant measurement temperature, Tm oTg ; where after a wait time tw ; a small probe field m0 H ¼ 0:04 mT was applied and the magnetization was recorded vs. time. In a memory experiment, the sample was ZFC from T > Tg to a stop temperature Ts oTg ; where the system was aged for a certain duration before further cooling down to lower temperatures and the magnetization was recorded during heating. This is referred to as a stop-and-wait protocol. Fig. 1(a) shows the relaxation of the magnetization (M) vs. log t at T ¼ 40 KoTg in an aging experiment. The employed wait times tw are indicated in the figure. The curves show a clear wait time dependence indicating that non-equilibrium phenomena play a key role for the dynamics at low temperatures. The wait time
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.091
ARTICLE IN PRESS S. Sahoo et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1316–1318
1.5 300 s 1000 10000
M [arb. units]
M [arb. units]
1.0
0.5 (a)
1.0
tw=104s
0.5
(a)
tw=10000s
0.0
1000s 300s 0.08
∆M [arb. units]
S [arb. units]
0.0 0.12
1317
(b)
0.04 -1 10
0
10
1
10
2
10
3
10
0.00
-0.04
4
10
(b)
t [s] Fig. 1. Relaxation curves M (a) and rate S vs. log t (b) at T ¼ 40 K recorded at m0 H ¼ 0:04 mT after different wait times as indicated.
dependence is more clearly reflected in the relaxation rate, S ¼ ð1=m0 HÞðqM=qln tÞ (see Fig. 1b), with peaks appearing at times close to the corresponding wait times similarly as observed in atomic spin glasses [8]. We have previously shown that the relaxation of thermoremanent magnetization exhibits similar aging phenomena albeit showing less clear correlations between tw and tðSmax Þ [9]. Presumably this was due to non-linearity effects owing to comparatively large field steps applied. The characteristic aging observed in the SSG phase implies that the correlation between the particle magnetic moments develops in the same way as the correlation between the spins of an atomic spin glass. In the droplet model [6], this correlation constitutes a domain whose size after an aging time ta is Rðta Þ ¼ ðT lnðta =t Þ=DðTÞÞ1=c ; where t is the relaxation time of an individual particle moment, DðTÞ sets the free energy scale, and c is a barrier exponent. When a small probe (DC) field is applied, the domain is probed via the polarization of droplets of size LðtÞ ¼ ðT lnðt=t Þ= DðTÞÞ1=c : Two limiting cases can be considered: (i) ln t5ln ta ; where LðtÞ5Rðta Þ and (ii) ln tbln ta ; where LðtÞERðta Þ: In the first case, quasi-equilibrium dynamics is probed whereas in the second case the probed length scale involves domain walls. This crossover from equilibrium to non-equilibrium response is reflected by a point of inflexion in M or, equivalently, a peak in the relaxation rate S vs. time.
-0.08 20
30
40
50
60
70
T [K]
Fig. 2. (a) Temperature dependence of the reference magnetization, Mref ðTÞ; (solid circles) and of the magnetization with a stop-and-wait protocol, MðTÞ (open circles) at a magnetic field of m0 H ¼ 0:04 mT. (b) DM ¼ MðTÞ2Mref ðTÞ vs. T:
Fig. 2 illustrates the memory and rejuvenation effects of the ZFC DC magnetization after a stop-and-wait procedure at Ts ¼ 42 K (=0.95 Tg ) for a duration of 104 s. As can be seen in Fig. 2(a), the data corresponding to the intermittent stop-and-wait protocol, MðTÞ; (open circles) lies significantly below the reference curve, Mref ðTÞ; (solid circles) at temperatures close to Ts : As seen in Fig. 2(b), the difference of the two data sets DM ¼ MðTÞ2Mref ðTÞ minimizes at Ts : This indicates that the magnetic moment configuration spontaneously rearranges toward equilibrium via growth of domains when the system is aged at Ts : These equilibrated domains become frozen-in on further cooling and are retrieved on reheating. In other words, the system shows a memory effect. The fact that Mref ðTÞ and MðTÞ curves coalesce at low temperatures and only start to deviate as Ts is approached again from below clearly indicates that rejuvenation occurs as the temperature is decreased away from Ts in the stop-and-wait protocol. Thanks are due to DFG (GK-277) for financial support.
ARTICLE IN PRESS 1318
S. Sahoo et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1316–1318
References [1] J.L. Dormann, et al., J. Magn. Magn. Mater. 187 (1998) L139. [2] H. Mamiya, et al., Phys. Rev. Lett. 82 (1999) 4332. [3] O. Petracic, et al., Phase Trans. 75 (2002) 73. [4] S. Sahoo, et al., Phys. Rev. B 65 (2002) 134406.
. P. Jonsson, et al., Phys. Rev. B 61 (2000) 1261. D.S. Fisher, D.A. Huse, Phys. Rev. B 38 (1988) 373. S. Sahoo, et al., Appl. Phys. Lett. 82 (2003) 4116. J.A. Mydosh, Spin Glasses: an Experimental Introduction, Taylor & Francis, London, 1993. [9] S. Sahoo, et al., J. Phys.: Condens. Mater 14 (2002) 6729. [5] [6] [7] [8]