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Out-of-equilibrium dynamics in superspin glass state of strongly interacting magnetic nanoparticle assemblies
Journal of Magnetism and Magnetic Materials 355 (2014) 225–229
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Out-of-equilibrium dynamics in superspin glass state of strongly interacting magnetic nanoparticle assemblies Sawako Nakamae n Service de Physique de l'Etat Condensé (CNRS URA 2464) DSM/IRAMIS, CEA Saclay, 91191 Gif sur Yvette, France
art ic l e i nf o
a b s t r a c t
Article history: Received 10 September 2013 Received in revised form 2 December 2013 Available online 17 December 2013
Interacting magnetic nanoparticles display a wide variety of magnetic behaviors ranging from modified superparamagnetism, superspin glass to possibly, superferromagnetism. The superspin glass state is described by its slow and out-of-equilibrium magnetic behaviors akin to those found in atomic spin glasses. In this article, recent experimental findings on superspin correlation length growth and the violation of the fluctuation-dissipation theorem obtained in concentrated frozen ferrofluids are presented to illustrate certain out-of-equilibrium dynamics behavior in superspin glasses. & 2013 Elsevier B.V. All rights reserved.
Keywords: Superspin glass Out-of-equilibrium dynamics Magnetic nanoparticles
1. Introduction Magnetic nanoparticles (np) are, as the name suggests, a class of nanometric particles made of magnetic elements such as iron, nickel and cobalt and their alloys. A quick search on “magnetic nanoparticles þresearch” reveals a plethora of current research efforts involving magnetic np's for their potential (and some current) use in biomedicine [1], magnetic resonance imaging [2], tomographic imaging [3], data storage [4], nanofluids [5], etc. While their unique and tunable physical properties (most generally that of superparamagnetism) make them attractive for technological applications, the np magnetism by itself has also been a focus of active fundamental research since the original works by Néel [6], Stoner and Wolfarth [7] in the 1940s. Within a single-domained nanoparticle, all atomic spins are aligned in the same direction and thus it can be considered as a small permanent magnet with a large magnetic moment, typically in the order of 103–105 μB, where μB is the Bohr magneton [8]. The energy that holds the spins aligned together is called “anisotropy energy” and the magnetization reversal of one particle requires the collective rotation of all atomic spins by overcoming the anisotropy energy barrier. If nanoparticles are widely spaced and hence non-interacting, their magnetization rotation dynamics is governed solely by thermal agitation, each behaving like a paramagnetic atom but with a much larger magnetic moment (also known as superspins) leading to the phenomenon known as superparamagnetism. In an assembly of concentrated yet isolated particles (not touching), the
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magnetic properties can be greatly influenced by the dipolar interaction energy among neighboring particles [9,10]. Dipolar interactions modify the effective energy barrier height of individual particles; i.e., the reversal of one superspin can change the energy barriers of surrounding nanoparticles. In strongly interacting nanoparticle systems, low temperature collective states are observed. These collective states, now called superspin glasses, show many magnetic features that are common among atomic spin glasses [11–25]. “Spin glass (SG)” and “superspin glass (SSG)” refer to metastable magnetic states created by quenched (frozen) and randomly interacting magnetic moments [26–28]. In reality, such systems are produced by random inclusion of magnetic impurities in a non-magnetic medium (in SG), or magnetic nanoparticles in a solid matrix such as a frozen fluid or a solid material (in SSG). The randomness induces competing interactions (frustrations) among magnetic moments such that it is difficult to find a global (super)spin configuration that minimizes the system's energy, resulting in a multitude of highly degenerate ground states. Therefore, the (super)spin dynamics of SG's and SSG's are always out-of-equilibrium; i.e., their magnetization relaxes slowly while neighboring (super)spins form ‘correlated zones’ whose size grows forever in search of a true ground state. SG experiments revealed a large collection of peculiar dynamic behaviors that cannot be explained by one universal model. Generally, SG behaviors are interpreted within the framework of two models. One is the Parisi's solution to the Sherrington– Kirkpatrick model, where the low temperature SG state consists of “infinitely many pure thermodynamic states” [29–32]. The other is the “droplet” model introduced by Fisher and Huse, based on the domain wall renormalization concepts. Here, there
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is only a single-pair of spin-flip-reversed pure states at low temperature in any finite dimension [33–37]. Despite some fundamental differences between atomic SG's and nanoparticle SSG's; e.g., short-range exchange (SG) vs. long-range dipolar (SSG) interactions, theoretical models developed for atomic SG's have so far succeeded in describing many aspects of SSG dynamics. From the experimental point-of-view, interacting magnetic np systems present certain advantages; e.g., slower superspin fluctuations and larger magnetic moments, which make them particularly advantageous for the investigation of existing theoretical models. Here, recent experimental studies on the out-of-equilibrium SSG dynamic behaviors in concentrated frozen ferrofluid samples made of maghemite nanoparticles dispersed in glycerin are presented; i.e., (1) the dynamic superspin correlation length growth and (2) the violation of the fluctuation-dissipation theorem (FDT). For the former study two types of samples were examined; one in which anisotropy-axis of np's are randomly distributed and the other where anisotropy-axis were all aligned in parallel. These samples are denoted as ‘random and ‘aligned’, respectively.
2. Out-of-equilibrium behaviors of superspin glasses 2.1. The onset of SSG state When studying the magnetic property of a magnetic nanoparticle assembly, it is most instructive to start with the so-called “ZFC/FC measurements” as illustrated in Fig. 1 (left panel). The FC curve corresponds to the magnetization measured after the sample is cooled in the presence of a small magnetic field H, while the ZFC curve is obtained after cooling in zero-field and applying H at the lowest temperature. It should be noted that the separation between the ZFC and FC curves alone is not a signature of a low-temperature SSG state; rather, it simply indicates the existence of irreversibility. As it is well known, non-interacting superparamagnetic nanoparticles also exhibit similar splitting between the ZFC and FC curves due to the blocking of individual superspins without having any collective state. That being said, the apparent flatness, or a slight decrease in the FC curve below the freezing temperature is suggestive of a collective behavior. Thus the ZFC/FC measurements have become a staple initial test when studying could-be superspin glass samples. A commonly used “test” to distinguish between a blocked SPM state from an SSG state is the frequency dependent magnetic
susceptibility (χ) measurements. When particles are non-interacting, the ac susceptibility χ(T, ω) peak follows an Arrhenius law τ ¼ τ0exp(Ea/kBT), where τ is the inverse of the measurement frequency ω, τ0 is a microscopic limiting relaxation time, (typically 10 9 s) and Ea is the anisotropy energy of individual nanoparticles. Applying this law to concentrated nanoparticle systems, one generally obtains unphysically small τ0 values. Instead, the frequency dependent χ(ω, T) peak is best described by a critical-law of the form
τðTÞ=τo ¼ ððT–T g Þ=T g Þ zv
ð1Þ
here zv the dynamic exponent and Tg the glass transition temperature at ω ¼ 0. In the concentrated frozen ferrofluid samples used in the following studies, zv values were between 5 and 12 [38], similar to those found in atomic spin glasses [39]. 2.2. Dynamic superspin correlation length growth If an SSG sample is zero-field cooled quickly to a certain temperature To Tg and then a small magnetic field, H is applied, its magnetization will first jump to the ZFC(T) value then slowly increase toward the equilibrium value; i.e., ZFC (t, T)-FC(T) (see Fig. 1 left panel). Conversely, if the sample is cooled in H to ToTg and subsequently the field is turned off, the magnetization will decrease. The latter effect is called the “thermo-remanent” magnetization relaxation (TRM(t, T)). In spin glasses, it has been demonstrated that ZFC(T)þ TRM(t, T) ¼FC(T) and the same relation also appears to hold in superspin glasses. It is in this TRM region (shaded in blue in Fig. 1 (left) between the FC(T) and ZFC(T) curves) where slow, glassy behavior is observed in spin glasses and superspin glasses. One important feature of the out-of-equilibrium dynamics of SSG is the “ageing” effect (relaxation) that takes place at all timescales (from the microscopic to possibly geological time scales). This occurs due to the disordered and frustrated nature of interactions that prohibits superspins from quickly establishing a unique ground state. Rather, superspins form locally “correlated zones” whose size grows steadily with time. In SSG's (and SG, of course), this effect can be observed in their magnetization relaxation (ZFC(t) and TRM(t), see above). In the ZFC measurement protocol, a sample is cooled from a temperature well above Tg, to a measuring temperature, Tm o Tg in zero field and held for an experimentally chosen amount of time tw (waiting time) during which superspins form correlated zones of varying sizes. At tw, an average-sized correlated zone contains
Fig. 1. (Left): FC (filled circles) and ZFC (open circles) magnetization vs. temperature curves at 0.5 G, measured on a concentrated (35%) ferrofluid sample (γ-Fe2O3 nanoparticles suspended in water) [23]. The blue-shaded area describes the out-of-equilibrium zone. (Right): Real part of AC susceptibility as a function of temperature. The sample used: γ-Fe2O3 ferrofluid in glycerol with the magnetic volume concentration of 15%. [38]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
S. Nakamae / Journal of Magnetism and Magnetic Materials 355 (2014) 225–229
Ns(tw) correlated superspins, with a corresponding energy barrier EB(Ns(tw)); i.e., t w ¼ τno exp ðEB ðNs ðt w ÞÞ=ðkB TÞÞ
ð2Þ
where τοn is the temperature dependent microscopic flipping time of an individual superspin determined via the Arrhenius law described above. At tw, a small field, H is applied, to which the system's magnetization responds by slowly increasing, M(t', tw), toward the final value. The slow magnetization increase occurs via relaxation of individual correlated zones that require simultaneous flipping all correlated superspins within. M(t', tw) is recorded as a function of time, t' and the measurements are repeated for different values of tw and H. Lundgren et al. [40] has proposed a physical interpretation of this slow magnetization relaxation behavior demonstrating how its logarithmic derivative, S(log(t')) ¼d(M/MFC)/d log(t'), qualitatively represents the relaxation time distribution of all correlated zones, g(τ), with larger zones possessing longer relaxation times. Therefore, in the very low-field limit, the average sized zones possess τave tw (see Eq. (2)) and thus, S(log(t')) curves show maxima near t' tw as depicted in Fig. 2 (left panel). Beyond the very low-field limit, the magnetic field coupling to the magnetization of correlated zones (the Zeeman energy, Ez) reduces the barrier energy height EB [41]. As a result, the relaxation times associated with each correlated zones become faster and the peak in S(log(t')) is shifted to earlier times, teff o tw with increasing H values (see Fig. 2, right panel); i.e.,
227
α
interparticle distance ξο) may be calculated from; ξ/ξo ¼Ns1/(d ), a model also developed for atomic SG's [44,45]. In these works, the exponents d¼ 3 and α ¼0.5 (1) are used for 3D Ising (Heisenberg) spin glasses. The right panel in Fig. 3 shows ξ⧸ξο determined in our ferrofluid samples along with the values found in atomic SG's (experimental) [46,47] and the numerical simulation results. The data are presented as a function of scaled time, T/Tg ln(t/τon) [46] where t corresponds the experimental waiting time. The choice of scaling parameter is based on the ξ(t) found in Ising spin glasses that follow a power law [44,45,48,49], ξðt; TÞ ¼ Aðt=τo ÞzðT=T g Þ , where, A is a constant, τo the elementary time (i.e., atomic spinflip time E 10 12 s) and z, the critical exponent. As can be seen from the graph, ξ⧸ξο data from SSG samples position themselves between the Heisenberg SG simulation curve and the experimental results from atomic Heisenberg SG's. These results strongly suggest that the ageing dynamics in SSG's (at least, in the randomly oriented case) appear to be governed by the same physical law as that of atomic Heisenberg SG's. 2.3. Violation of fluctuation-dissipation theorem The fluctuation-dissipation theorem (FDT) states that in a given system, the linear response to an external perturbation can be expressed in terms of the response fluctuation in thermal equilibrium [50]. In its simplest form, FDT is described as 2kB T
ð3Þ
Imχ^ ðωÞ ð4Þ ω where Sx(ω) is the power spectrum (noise) of the observable x as a function of frequency ω and χ^ (ω) is the imaginary component of
Thus, by repeating the measurements with various tw and H values, one can establish the relationship between tweff and Ez from Eqs. (2) and (3): ln(tweff/tw)¼ (Ez(H, M(Ns(tw)))/kBT). Knowing that Ez is simply H.M(Ns), the tw dependence of Ns(tw) can be extracted using the empirical forms of M(Ns) for both random and aligned SSG states [41–43]. In Fig. 3 (left panel), the typical number of correlated spins Ns(tw) determined at T¼ 0.7 Tg is plotted as a function of tw. As can be seen from the graph, Ns(tw) grows to about 102–103 after tw ¼105 s, much smaller than the values found in atomic spin glasses (in the order of 104–106). This reflects a slow superspin flip-time (10 6 s at 0.7 Tg, compared to 10–12 s of atomic spin-flip time). The exponent values of 0.27 (random) and 0.32 (aligned) found in the a power-law behavior of Ns(tw), on the other hand, are comparable to those found in atomic spin glasses. The physical ‘size’ of the correlated zones, often expressed in terms of the “correlation length” ξ/ξ0 (normalized to the average
the susceptibility (response to a small perturbation). The FDT has been verified in a plethora of real physical systems at thermodynamic equilibrium, such as the Brownian particle motion in a fluid. In out-of-equilibrium systems, the FD relation is expected to break down. In the “weak ergodicity breaking” model, an effective temperature Teff is introduced in the FD relation above to represent the out-of-equilibrium state [51]. The ratio Teff/T indicates the degree of “out-of-equilibrium-ness” of the system. The larger the deviation from unity of Teff/T, the further from equilibrium the system is. Experimental attempts to verify the violation of FD relation in various glassy systems, however have produced rather conflicting results [52–56]. In order to test the FDT violation in (super)spin glasses, one can measure the magnetization fluctuations at a given frequency and compare them to the corresponding imaginary part of the ac
t w ef f ¼ τno exp ðEB ðN s ðt w ÞÞ Ez ðH; MðN s ÞÞ=ðkB TÞÞ
Sx ðωÞ ¼
Fig. 2. (Left): MZFC relaxation measurements in a γ-Fe2O3 frozen ferrofluid at 0.7 Tg and an applied magnetic field of 0.5 Oe. The waiting time was varied between 3 ks and 24 ks. (Right): ZFC relaxation measurements in a γ-Fe2O3 frozen ferrofluid at 0.7 Tg. tweff shifts from about 18000 s ( ¼tw) at H¼1 Oe to just below 200 s at 8.5 Oe.
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Fig. 3. (Left): Ns as a function of tw in aligned and random ferrofluid samples. In both samples, Ns growth follows a power law behavior. (Right): Normalized correlation lengths (ξ⧸ξο) calculated from Ns in the SSG's (our experimental results) as well as in atomic SG's (experimental data found in literature [45,46]). α ¼0.5 (Ising SG and aligned SSG) and 1.0 (Heisenberg SG and random SSG) were used for the calculation. ξ⧸ξο are plotted as a function of scaled time (see text) and compared to the numerical simulations [43,44]. Symbols: Stars – Ising SG (numerical) at T ¼ Tg and 0.5Tg, solid triangles – from right to left, Heisenberg SG (numerical) at T ¼Tg, 0.875Tg, 0.75Tg, 0.5Tg and 0.25Tg, solid circles – Atomic SG's (experimental), and octagons – SSG's. (Reprinted with permission from [43]© 2012, American Institute of Physics).
Fig. 4. (Left): χ″(f, T) of a bulk sample as a function of Sf/T for frequencies, 0.08, 0.8, and 4 Hz. Each data point corresponds to χ″ and S measurements at a given bath temperature T and a frequency f. The solid straight line indicates the linear relation (FDT) in the high temperature region above Tg. (Right): The temperature dependence of Teff/T at three frequencies. The horizontal line corresponds to Teff/T ¼1.
susceptibility during the “strongly ageing” time regime; i.e., during the TRM relaxation. The FD relation for a magnetic system is written as [57]: 2kB T χ }ðωÞ o δMðωÞ2 4 ¼ ð5Þ π V μo ω o δM(ω)2 4 is the ensemble average of the power spectrum of the magnetization fluctuations (noise), χ″(ω) the imaginary component of the ac susceptibility and V the sample volume. The effective temperature Teff will replace T in Eq. (4) in the out-ofequilibrium state. Teff should be larger than T when the system is out-of-equilibrium; that is, for measurement frequencies larger than the inverse of tw, i.e., 2kB T ef f χ ″ðω; t w Þ ð6Þ 〈δMðωÞ2 〉 ¼ πV μ0 ω In an atomic spin glass, Hérisson and Ocio have observed Teff/T41 below Tg [58]. Due to experimental difficulties involved in measuring the extremely small magnetization fluctuation signals, however, their work remains the only experimental example of
FDT violation observed in spin glasses (via magnetization noise measurements) to-date. Recently, we have tested the FDT on a less than 10 pl of the same concentrated ferrofluid (as the one used in the correlation length study) via magnetic noise measurements using a two-dimension electron gas (2DEG) quantum well Hall sensor with a micrometric active surface area [59]. In this experiment, a sample was temperature quenched in zero-magnetic field from above Tg to 0.8 Tg and the magnetic noise power spectra were S(f) collected at in the frequency range between 0.08 and 8 Hz. The waiting time, tw of the system was of the order of a few 103 s for all frequencies investigated. More in-depth experimental details and data analysis methods are found in [59]. In Fig. 4 (left panel), the imaginary component of the ac magnetic susceptibility χ″(f, T) of a bulk sample is presented as a function of Sf/T at f¼0.08, 0.8 and 4 Hz. As can be seen from this figure, all noise data taken at T4Tg coincide on a single straight line; χ″ Sf/T. This frequency independent linear relationship indicates that the FDT is upheld according to Eq. (5). However, for ToTg such a relationship is clearly violated at all three measurement frequencies, as expected in out-of-equilibrium systems. The right panel in Fig. 4 shows the
S. Nakamae / Journal of Magnetism and Magnetic Materials 355 (2014) 225–229
ratio between the effective temperature and the bath temperature (Teff/T) which deviates from 1 at Tg and grows to larger as T decreases. The observed Teff/T values are comparable to those reported in an atomic SG (Teff/T¼2.8–5.3) [58] and in a Monte Carlo simulation on a Heisenberg SG (Teff/T¼2–10) [60]. It should be noted; however, that the observation of Teff 4T indicates that the system is in the ageing regime rather than in the quasi-equilibrium regime. This is rather surprising considering that the observation times tobs (the inverse of the measurement frequencies) are less than 1 s, much shorter than the waiting time tw E103 s. In another similar SSG system, Jonsson et al., did not observe FDT violation [61] where tobs/tw ¼10 5. These conflicting results indicate that, the FDT violation observation in the SSG is very much systems and experimental conditions dependent, just as in other glassy systems. 3. Concluding remarks Compared to the atomic spin glass counterpart, the experimental catalog on superspin glasses is still very small. Further experimental work taking advantage of nanoparticles' large individual magnetic moments and their associated longer flip-times can be effectively used to test some of the fundamental questions in spin glasses. For example, the correlation length growth study presented in Section 2.2 illustrates how one can effectively bridge the gap between the experiments and the numerical simulations in atomic spin glasses, by properly tailoring the experimental conditions of interacting magnetic nanoparticles. The true nature of the ground state in (super)spin glasses; i.e., “infinitely many pure thermodynamic states” vs. “disguised ferromagnets with just two spin-flip reversed states,” is also still a matter of debate. Additional investigations on the violation of FDT, i.e., the waiting time dependence of Teff, should serve to advance this particular issued in the field of glassy physics [62]. References [1] A.K. Gupta, M. Gupta, Biomaterials 26 (2005) 3995; Q.A. Pankhurst, et al., J. Phys. D: Appl. Phys. 42 (2009) 224001. [2] S. Mornet, et al., Progr. Solid State Chem. 34 (2006) 237. [3] B. Gleich, J. Weizenecker, Nature 435 (2005) 1214. [4] N.A. Frey, et al., Chem. Soc. Rev. 38 (2009) 2532. [5] J. Philip, P.D. Shima, Baldev Raj, Appl. Phys. Lett. 91 (2007) 203108. [6] L. Néel, Ann. Geophys. 5 (1949) 99. [7] E.C. Stoner, E. P Wolfarth, Philos. Trans. R. Soc. Lond. Ser. A 240 (1948) 599. [8] Here, the atomic spins are aligned ferromagnetically. It should be noted that other ordering; ferrimagnetic and antiferromagnetic, are also possible. [9] J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283; D. Fiorani, et al., J. Magn. Magn. Mater. 196–197 (1999) 143. [10] X. Battle, A. Labarta, J. Phys. D 35 (2002) R15–R42; S. Bedanta, W. Kleeman, J. Phys. D: Appl. Phys. 42 (2009) 013001. [11] P. Nordblad, J. Phys. D: Appl. Phys. 41 (2008) 134011. [12] D. Parker, et al., J. Appl. Phys. 97 (2005) 10A502. [13] W. Luo, et al., Phys. Rev. Lett. 67 (1991) 2721. [14] E. Vincent, et al., J. Magn.Magn. Mater. 161 (1996) 209. [15] H. Mamiya, I. Nakatani, T. Furubayashi, Phys. Rev. Lett. 80 (1998) 177.
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Out-of-equilibrium dynamics in superspin glass state of strongly interacting magnetic nanoparticle assemblies Sawako Nakamae n Service de Physique de l'Etat Condensé (CNRS URA 2464) DSM/IRAMIS, CEA Saclay, 91191 Gif sur Yvette, France
art ic l e i nf o
a b s t r a c t
Article history: Received 10 September 2013 Received in revised form 2 December 2013 Available online 17 December 2013
Interacting magnetic nanoparticles display a wide variety of magnetic behaviors ranging from modified superparamagnetism, superspin glass to possibly, superferromagnetism. The superspin glass state is described by its slow and out-of-equilibrium magnetic behaviors akin to those found in atomic spin glasses. In this article, recent experimental findings on superspin correlation length growth and the violation of the fluctuation-dissipation theorem obtained in concentrated frozen ferrofluids are presented to illustrate certain out-of-equilibrium dynamics behavior in superspin glasses. & 2013 Elsevier B.V. All rights reserved.
Keywords: Superspin glass Out-of-equilibrium dynamics Magnetic nanoparticles
1. Introduction Magnetic nanoparticles (np) are, as the name suggests, a class of nanometric particles made of magnetic elements such as iron, nickel and cobalt and their alloys. A quick search on “magnetic nanoparticles þresearch” reveals a plethora of current research efforts involving magnetic np's for their potential (and some current) use in biomedicine [1], magnetic resonance imaging [2], tomographic imaging [3], data storage [4], nanofluids [5], etc. While their unique and tunable physical properties (most generally that of superparamagnetism) make them attractive for technological applications, the np magnetism by itself has also been a focus of active fundamental research since the original works by Néel [6], Stoner and Wolfarth [7] in the 1940s. Within a single-domained nanoparticle, all atomic spins are aligned in the same direction and thus it can be considered as a small permanent magnet with a large magnetic moment, typically in the order of 103–105 μB, where μB is the Bohr magneton [8]. The energy that holds the spins aligned together is called “anisotropy energy” and the magnetization reversal of one particle requires the collective rotation of all atomic spins by overcoming the anisotropy energy barrier. If nanoparticles are widely spaced and hence non-interacting, their magnetization rotation dynamics is governed solely by thermal agitation, each behaving like a paramagnetic atom but with a much larger magnetic moment (also known as superspins) leading to the phenomenon known as superparamagnetism. In an assembly of concentrated yet isolated particles (not touching), the
n
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0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.12.018
magnetic properties can be greatly influenced by the dipolar interaction energy among neighboring particles [9,10]. Dipolar interactions modify the effective energy barrier height of individual particles; i.e., the reversal of one superspin can change the energy barriers of surrounding nanoparticles. In strongly interacting nanoparticle systems, low temperature collective states are observed. These collective states, now called superspin glasses, show many magnetic features that are common among atomic spin glasses [11–25]. “Spin glass (SG)” and “superspin glass (SSG)” refer to metastable magnetic states created by quenched (frozen) and randomly interacting magnetic moments [26–28]. In reality, such systems are produced by random inclusion of magnetic impurities in a non-magnetic medium (in SG), or magnetic nanoparticles in a solid matrix such as a frozen fluid or a solid material (in SSG). The randomness induces competing interactions (frustrations) among magnetic moments such that it is difficult to find a global (super)spin configuration that minimizes the system's energy, resulting in a multitude of highly degenerate ground states. Therefore, the (super)spin dynamics of SG's and SSG's are always out-of-equilibrium; i.e., their magnetization relaxes slowly while neighboring (super)spins form ‘correlated zones’ whose size grows forever in search of a true ground state. SG experiments revealed a large collection of peculiar dynamic behaviors that cannot be explained by one universal model. Generally, SG behaviors are interpreted within the framework of two models. One is the Parisi's solution to the Sherrington– Kirkpatrick model, where the low temperature SG state consists of “infinitely many pure thermodynamic states” [29–32]. The other is the “droplet” model introduced by Fisher and Huse, based on the domain wall renormalization concepts. Here, there
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is only a single-pair of spin-flip-reversed pure states at low temperature in any finite dimension [33–37]. Despite some fundamental differences between atomic SG's and nanoparticle SSG's; e.g., short-range exchange (SG) vs. long-range dipolar (SSG) interactions, theoretical models developed for atomic SG's have so far succeeded in describing many aspects of SSG dynamics. From the experimental point-of-view, interacting magnetic np systems present certain advantages; e.g., slower superspin fluctuations and larger magnetic moments, which make them particularly advantageous for the investigation of existing theoretical models. Here, recent experimental studies on the out-of-equilibrium SSG dynamic behaviors in concentrated frozen ferrofluid samples made of maghemite nanoparticles dispersed in glycerin are presented; i.e., (1) the dynamic superspin correlation length growth and (2) the violation of the fluctuation-dissipation theorem (FDT). For the former study two types of samples were examined; one in which anisotropy-axis of np's are randomly distributed and the other where anisotropy-axis were all aligned in parallel. These samples are denoted as ‘random and ‘aligned’, respectively.
2. Out-of-equilibrium behaviors of superspin glasses 2.1. The onset of SSG state When studying the magnetic property of a magnetic nanoparticle assembly, it is most instructive to start with the so-called “ZFC/FC measurements” as illustrated in Fig. 1 (left panel). The FC curve corresponds to the magnetization measured after the sample is cooled in the presence of a small magnetic field H, while the ZFC curve is obtained after cooling in zero-field and applying H at the lowest temperature. It should be noted that the separation between the ZFC and FC curves alone is not a signature of a low-temperature SSG state; rather, it simply indicates the existence of irreversibility. As it is well known, non-interacting superparamagnetic nanoparticles also exhibit similar splitting between the ZFC and FC curves due to the blocking of individual superspins without having any collective state. That being said, the apparent flatness, or a slight decrease in the FC curve below the freezing temperature is suggestive of a collective behavior. Thus the ZFC/FC measurements have become a staple initial test when studying could-be superspin glass samples. A commonly used “test” to distinguish between a blocked SPM state from an SSG state is the frequency dependent magnetic
susceptibility (χ) measurements. When particles are non-interacting, the ac susceptibility χ(T, ω) peak follows an Arrhenius law τ ¼ τ0exp(Ea/kBT), where τ is the inverse of the measurement frequency ω, τ0 is a microscopic limiting relaxation time, (typically 10 9 s) and Ea is the anisotropy energy of individual nanoparticles. Applying this law to concentrated nanoparticle systems, one generally obtains unphysically small τ0 values. Instead, the frequency dependent χ(ω, T) peak is best described by a critical-law of the form
τðTÞ=τo ¼ ððT–T g Þ=T g Þ zv
ð1Þ
here zv the dynamic exponent and Tg the glass transition temperature at ω ¼ 0. In the concentrated frozen ferrofluid samples used in the following studies, zv values were between 5 and 12 [38], similar to those found in atomic spin glasses [39]. 2.2. Dynamic superspin correlation length growth If an SSG sample is zero-field cooled quickly to a certain temperature To Tg and then a small magnetic field, H is applied, its magnetization will first jump to the ZFC(T) value then slowly increase toward the equilibrium value; i.e., ZFC (t, T)-FC(T) (see Fig. 1 left panel). Conversely, if the sample is cooled in H to ToTg and subsequently the field is turned off, the magnetization will decrease. The latter effect is called the “thermo-remanent” magnetization relaxation (TRM(t, T)). In spin glasses, it has been demonstrated that ZFC(T)þ TRM(t, T) ¼FC(T) and the same relation also appears to hold in superspin glasses. It is in this TRM region (shaded in blue in Fig. 1 (left) between the FC(T) and ZFC(T) curves) where slow, glassy behavior is observed in spin glasses and superspin glasses. One important feature of the out-of-equilibrium dynamics of SSG is the “ageing” effect (relaxation) that takes place at all timescales (from the microscopic to possibly geological time scales). This occurs due to the disordered and frustrated nature of interactions that prohibits superspins from quickly establishing a unique ground state. Rather, superspins form locally “correlated zones” whose size grows steadily with time. In SSG's (and SG, of course), this effect can be observed in their magnetization relaxation (ZFC(t) and TRM(t), see above). In the ZFC measurement protocol, a sample is cooled from a temperature well above Tg, to a measuring temperature, Tm o Tg in zero field and held for an experimentally chosen amount of time tw (waiting time) during which superspins form correlated zones of varying sizes. At tw, an average-sized correlated zone contains
Fig. 1. (Left): FC (filled circles) and ZFC (open circles) magnetization vs. temperature curves at 0.5 G, measured on a concentrated (35%) ferrofluid sample (γ-Fe2O3 nanoparticles suspended in water) [23]. The blue-shaded area describes the out-of-equilibrium zone. (Right): Real part of AC susceptibility as a function of temperature. The sample used: γ-Fe2O3 ferrofluid in glycerol with the magnetic volume concentration of 15%. [38]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Ns(tw) correlated superspins, with a corresponding energy barrier EB(Ns(tw)); i.e., t w ¼ τno exp ðEB ðNs ðt w ÞÞ=ðkB TÞÞ
ð2Þ
where τοn is the temperature dependent microscopic flipping time of an individual superspin determined via the Arrhenius law described above. At tw, a small field, H is applied, to which the system's magnetization responds by slowly increasing, M(t', tw), toward the final value. The slow magnetization increase occurs via relaxation of individual correlated zones that require simultaneous flipping all correlated superspins within. M(t', tw) is recorded as a function of time, t' and the measurements are repeated for different values of tw and H. Lundgren et al. [40] has proposed a physical interpretation of this slow magnetization relaxation behavior demonstrating how its logarithmic derivative, S(log(t')) ¼d(M/MFC)/d log(t'), qualitatively represents the relaxation time distribution of all correlated zones, g(τ), with larger zones possessing longer relaxation times. Therefore, in the very low-field limit, the average sized zones possess τave tw (see Eq. (2)) and thus, S(log(t')) curves show maxima near t' tw as depicted in Fig. 2 (left panel). Beyond the very low-field limit, the magnetic field coupling to the magnetization of correlated zones (the Zeeman energy, Ez) reduces the barrier energy height EB [41]. As a result, the relaxation times associated with each correlated zones become faster and the peak in S(log(t')) is shifted to earlier times, teff o tw with increasing H values (see Fig. 2, right panel); i.e.,
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α
interparticle distance ξο) may be calculated from; ξ/ξo ¼Ns1/(d ), a model also developed for atomic SG's [44,45]. In these works, the exponents d¼ 3 and α ¼0.5 (1) are used for 3D Ising (Heisenberg) spin glasses. The right panel in Fig. 3 shows ξ⧸ξο determined in our ferrofluid samples along with the values found in atomic SG's (experimental) [46,47] and the numerical simulation results. The data are presented as a function of scaled time, T/Tg ln(t/τon) [46] where t corresponds the experimental waiting time. The choice of scaling parameter is based on the ξ(t) found in Ising spin glasses that follow a power law [44,45,48,49], ξðt; TÞ ¼ Aðt=τo ÞzðT=T g Þ , where, A is a constant, τo the elementary time (i.e., atomic spinflip time E 10 12 s) and z, the critical exponent. As can be seen from the graph, ξ⧸ξο data from SSG samples position themselves between the Heisenberg SG simulation curve and the experimental results from atomic Heisenberg SG's. These results strongly suggest that the ageing dynamics in SSG's (at least, in the randomly oriented case) appear to be governed by the same physical law as that of atomic Heisenberg SG's. 2.3. Violation of fluctuation-dissipation theorem The fluctuation-dissipation theorem (FDT) states that in a given system, the linear response to an external perturbation can be expressed in terms of the response fluctuation in thermal equilibrium [50]. In its simplest form, FDT is described as 2kB T
ð3Þ
Imχ^ ðωÞ ð4Þ ω where Sx(ω) is the power spectrum (noise) of the observable x as a function of frequency ω and χ^ (ω) is the imaginary component of
Thus, by repeating the measurements with various tw and H values, one can establish the relationship between tweff and Ez from Eqs. (2) and (3): ln(tweff/tw)¼ (Ez(H, M(Ns(tw)))/kBT). Knowing that Ez is simply H.M(Ns), the tw dependence of Ns(tw) can be extracted using the empirical forms of M(Ns) for both random and aligned SSG states [41–43]. In Fig. 3 (left panel), the typical number of correlated spins Ns(tw) determined at T¼ 0.7 Tg is plotted as a function of tw. As can be seen from the graph, Ns(tw) grows to about 102–103 after tw ¼105 s, much smaller than the values found in atomic spin glasses (in the order of 104–106). This reflects a slow superspin flip-time (10 6 s at 0.7 Tg, compared to 10–12 s of atomic spin-flip time). The exponent values of 0.27 (random) and 0.32 (aligned) found in the a power-law behavior of Ns(tw), on the other hand, are comparable to those found in atomic spin glasses. The physical ‘size’ of the correlated zones, often expressed in terms of the “correlation length” ξ/ξ0 (normalized to the average
the susceptibility (response to a small perturbation). The FDT has been verified in a plethora of real physical systems at thermodynamic equilibrium, such as the Brownian particle motion in a fluid. In out-of-equilibrium systems, the FD relation is expected to break down. In the “weak ergodicity breaking” model, an effective temperature Teff is introduced in the FD relation above to represent the out-of-equilibrium state [51]. The ratio Teff/T indicates the degree of “out-of-equilibrium-ness” of the system. The larger the deviation from unity of Teff/T, the further from equilibrium the system is. Experimental attempts to verify the violation of FD relation in various glassy systems, however have produced rather conflicting results [52–56]. In order to test the FDT violation in (super)spin glasses, one can measure the magnetization fluctuations at a given frequency and compare them to the corresponding imaginary part of the ac
t w ef f ¼ τno exp ðEB ðN s ðt w ÞÞ Ez ðH; MðN s ÞÞ=ðkB TÞÞ
Sx ðωÞ ¼
Fig. 2. (Left): MZFC relaxation measurements in a γ-Fe2O3 frozen ferrofluid at 0.7 Tg and an applied magnetic field of 0.5 Oe. The waiting time was varied between 3 ks and 24 ks. (Right): ZFC relaxation measurements in a γ-Fe2O3 frozen ferrofluid at 0.7 Tg. tweff shifts from about 18000 s ( ¼tw) at H¼1 Oe to just below 200 s at 8.5 Oe.
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Fig. 3. (Left): Ns as a function of tw in aligned and random ferrofluid samples. In both samples, Ns growth follows a power law behavior. (Right): Normalized correlation lengths (ξ⧸ξο) calculated from Ns in the SSG's (our experimental results) as well as in atomic SG's (experimental data found in literature [45,46]). α ¼0.5 (Ising SG and aligned SSG) and 1.0 (Heisenberg SG and random SSG) were used for the calculation. ξ⧸ξο are plotted as a function of scaled time (see text) and compared to the numerical simulations [43,44]. Symbols: Stars – Ising SG (numerical) at T ¼ Tg and 0.5Tg, solid triangles – from right to left, Heisenberg SG (numerical) at T ¼Tg, 0.875Tg, 0.75Tg, 0.5Tg and 0.25Tg, solid circles – Atomic SG's (experimental), and octagons – SSG's. (Reprinted with permission from [43]© 2012, American Institute of Physics).
Fig. 4. (Left): χ″(f, T) of a bulk sample as a function of Sf/T for frequencies, 0.08, 0.8, and 4 Hz. Each data point corresponds to χ″ and S measurements at a given bath temperature T and a frequency f. The solid straight line indicates the linear relation (FDT) in the high temperature region above Tg. (Right): The temperature dependence of Teff/T at three frequencies. The horizontal line corresponds to Teff/T ¼1.
susceptibility during the “strongly ageing” time regime; i.e., during the TRM relaxation. The FD relation for a magnetic system is written as [57]: 2kB T χ }ðωÞ o δMðωÞ2 4 ¼ ð5Þ π V μo ω o δM(ω)2 4 is the ensemble average of the power spectrum of the magnetization fluctuations (noise), χ″(ω) the imaginary component of the ac susceptibility and V the sample volume. The effective temperature Teff will replace T in Eq. (4) in the out-ofequilibrium state. Teff should be larger than T when the system is out-of-equilibrium; that is, for measurement frequencies larger than the inverse of tw, i.e., 2kB T ef f χ ″ðω; t w Þ ð6Þ 〈δMðωÞ2 〉 ¼ πV μ0 ω In an atomic spin glass, Hérisson and Ocio have observed Teff/T41 below Tg [58]. Due to experimental difficulties involved in measuring the extremely small magnetization fluctuation signals, however, their work remains the only experimental example of
FDT violation observed in spin glasses (via magnetization noise measurements) to-date. Recently, we have tested the FDT on a less than 10 pl of the same concentrated ferrofluid (as the one used in the correlation length study) via magnetic noise measurements using a two-dimension electron gas (2DEG) quantum well Hall sensor with a micrometric active surface area [59]. In this experiment, a sample was temperature quenched in zero-magnetic field from above Tg to 0.8 Tg and the magnetic noise power spectra were S(f) collected at in the frequency range between 0.08 and 8 Hz. The waiting time, tw of the system was of the order of a few 103 s for all frequencies investigated. More in-depth experimental details and data analysis methods are found in [59]. In Fig. 4 (left panel), the imaginary component of the ac magnetic susceptibility χ″(f, T) of a bulk sample is presented as a function of Sf/T at f¼0.08, 0.8 and 4 Hz. As can be seen from this figure, all noise data taken at T4Tg coincide on a single straight line; χ″ Sf/T. This frequency independent linear relationship indicates that the FDT is upheld according to Eq. (5). However, for ToTg such a relationship is clearly violated at all three measurement frequencies, as expected in out-of-equilibrium systems. The right panel in Fig. 4 shows the
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ratio between the effective temperature and the bath temperature (Teff/T) which deviates from 1 at Tg and grows to larger as T decreases. The observed Teff/T values are comparable to those reported in an atomic SG (Teff/T¼2.8–5.3) [58] and in a Monte Carlo simulation on a Heisenberg SG (Teff/T¼2–10) [60]. It should be noted; however, that the observation of Teff 4T indicates that the system is in the ageing regime rather than in the quasi-equilibrium regime. This is rather surprising considering that the observation times tobs (the inverse of the measurement frequencies) are less than 1 s, much shorter than the waiting time tw E103 s. In another similar SSG system, Jonsson et al., did not observe FDT violation [61] where tobs/tw ¼10 5. These conflicting results indicate that, the FDT violation observation in the SSG is very much systems and experimental conditions dependent, just as in other glassy systems. 3. Concluding remarks Compared to the atomic spin glass counterpart, the experimental catalog on superspin glasses is still very small. Further experimental work taking advantage of nanoparticles' large individual magnetic moments and their associated longer flip-times can be effectively used to test some of the fundamental questions in spin glasses. For example, the correlation length growth study presented in Section 2.2 illustrates how one can effectively bridge the gap between the experiments and the numerical simulations in atomic spin glasses, by properly tailoring the experimental conditions of interacting magnetic nanoparticles. The true nature of the ground state in (super)spin glasses; i.e., “infinitely many pure thermodynamic states” vs. “disguised ferromagnets with just two spin-flip reversed states,” is also still a matter of debate. Additional investigations on the violation of FDT, i.e., the waiting time dependence of Teff, should serve to advance this particular issued in the field of glassy physics [62]. References [1] A.K. Gupta, M. Gupta, Biomaterials 26 (2005) 3995; Q.A. Pankhurst, et al., J. Phys. D: Appl. Phys. 42 (2009) 224001. [2] S. Mornet, et al., Progr. Solid State Chem. 34 (2006) 237. [3] B. Gleich, J. Weizenecker, Nature 435 (2005) 1214. [4] N.A. Frey, et al., Chem. Soc. Rev. 38 (2009) 2532. [5] J. Philip, P.D. Shima, Baldev Raj, Appl. Phys. Lett. 91 (2007) 203108. [6] L. Néel, Ann. Geophys. 5 (1949) 99. [7] E.C. Stoner, E. P Wolfarth, Philos. Trans. R. Soc. Lond. Ser. A 240 (1948) 599. [8] Here, the atomic spins are aligned ferromagnetically. It should be noted that other ordering; ferrimagnetic and antiferromagnetic, are also possible. [9] J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283; D. Fiorani, et al., J. Magn. Magn. Mater. 196–197 (1999) 143. [10] X. Battle, A. Labarta, J. Phys. D 35 (2002) R15–R42; S. Bedanta, W. Kleeman, J. Phys. D: Appl. Phys. 42 (2009) 013001. [11] P. Nordblad, J. Phys. D: Appl. Phys. 41 (2008) 134011. [12] D. Parker, et al., J. Appl. Phys. 97 (2005) 10A502. [13] W. Luo, et al., Phys. Rev. Lett. 67 (1991) 2721. [14] E. Vincent, et al., J. Magn.Magn. Mater. 161 (1996) 209. [15] H. Mamiya, I. Nakatani, T. Furubayashi, Phys. Rev. Lett. 80 (1998) 177.
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