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Effective energy barrier distributions for random and aligned magnetic nanoparticles

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2014 J. Phys.: Condens. Matter 26 146006

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Systems of magnetic nanoparticles are studied in many different contexts. In rock magnetism the magnetization of naturally occurring magnetic minerals yields information on the environment in which they formed and their subsequent history [1]. In medicine, applications of magnetic nanoparti-cles include destroying cancer cells by radio frequency heating and use as contrast agents for magnetic resonance imaging [2]. For such nanoparticle systems, the energy barriers that oppose their magnetic reversal can be characterized through isother-mal magnetic relaxation measurements, where the magneti-zation of a sample is recorded following a change in applied field [3–6].

In this letter we report how the energy barrier distribution sampled by monodisperse cobalt-doped magnetite nanopar-ticles during zero-field magnetic relaxation depends on the alignment of their easy axes. We observe a high proportion of low-energy barriers when the nanoparticles are randomly oriented, but this low-energy tail is greatly attenuated when the particles’ easy axis is aligned with the applied field. We attribute our results to the nanoparticles’ magnetic anisotropy having both uniaxial and cubic terms. This combination can generate local as well as global energy minima as a function of the orientation of the magnetization vector [7, 8]. When the par-ticles are randomly oriented, the states corresponding to local minima are populated at the start of an isothermal magnetic

relaxation measurement, and a low-energy tail appears in the energy barrier distribution. When the easy axis is parallel to the applied field, only the states corresponding to global minima are populated, so the energy barrier distribution no longer con-tains the low-energy tail corresponding to relaxation from local minima. No low-energy tail is observed for magnetite nano-particles without cobalt doping, for which the cubic anisotropy is absent.

Even though the changes to the magnetic relaxation on alignment are clear-cut and the explanation appears simple, this phenomenon has not been observed previously. What ena-bles its observation here is not only the presence of both cubic and uniaxial terms in the nanoparticle anisotropy, but the abil-ity to fix the nanoparticle orientation when at a temperature a few times greater than the effective blocking temperature TB. In the present case, the nanoparticles are supported in a 50mM Tris–HCl buffer solution. Although the bulk of the solution freezes close to T = 273 K, the cobalt-doped magnetite nano-particles in ferritin (MGF) remain surrounded by fluid and therefore free to rotate on the timescale of seconds down to ~ 210 K. Below this temperature, the nanoparticle orientation is fixed by a glass transition [9]. For comparison, TB determined from zero-field-cooled (ZFC) magnetic susceptibility meas-urements was 62 ± 1 K and 103 ± 1 K for samples with 2.5% and 6.6% cobalt doping respectively [10].

Journal of Physics: Condensed Matter

Effective energy barrier distributions for random and aligned magnetic nanoparticles

J-C Eloi1, M Okuda1, S Correia Carreira1, 2, W Schwarzacher1, M J Correia3 and W Figueiredo3

1 H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK2 Bristol Centre for Functional Nanomaterials, Tyndall Avenue, Bristol BS8 1FD, UK3 Departamento de Física, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, SC, Brazil

E-mail: w.schwarzacher@bristol.ac.uk

Received 13 December 2013, revised 12 February 2014Accepted for publication 13 February 2014Published 20 March 2014

AbstractIsothermal magnetic relaxation measurements are widely used to probe energy barriers in systems of magnetic nanoparticles. Here we show that the result of such an experiment can differ greatly for aligned and randomly oriented nanoparticles. For randomly oriented cobalt-doped magnetite nanoparticles we observe a prominent low-energy tail in the energy barrier distribution that is greatly attenuated when the particles are magnetically aligned. Monte Carlo simulations show that this behaviour arises for nanoparticles with both cubic and uniaxial magnetic anisotropy energy terms even though for cubic or uniaxial anisotropy alone the energy barrier distribution is independent of nanoparticle orientation.

Keywords: nanoparticle, ferritin, nanomagnet, energy barrier, magnetic relaxation

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The nanoparticles used in this study were synthesized using the internal cavity (diameter ~8 nm) of the quasi-spherical cage-shaped iron storage protein ferritin as a template [11, 12]. Cobalt-doped MGF were prepared by adding cobalt sulfate (CoSO4) to the synthesis solution. The effect of cobalt addition in increasing the magnetic anisotropy energy of MGF and consequently TB has been noted previously [13] and attributed to replacement of Fe2+ ions in the magnetite Fe2+ octahedral site by Co2+ [14]. Micromagnetic calculations showed that both cubic and uniaxial anisotropy energy terms are required to fit hysteresis data for our 2.5% cobalt-doped MGF, with the magnitude of the cubic term being approximately twice that of the uniaxial, and the easy axes of the cubic and uniaxial anisotropies for a given particle being randomly oriented with respect to each other [15].

Magnetic measurements were made using solutions of cobalt-doped MGF in 50mM Tris–HCl with a nanoparticle concentration of 0.9 µmol l-1. Magnetic relaxation data were obtained at different T by applying a field of 5 T and recording the sample magnetic moment M as a function of the time t for a period of ~3000 s after removing the field and quenching the superconducting magnet to remove any residual field3 . All mag-netic measurements used a commercial SQUID magnetometer (Quantum Design MPMS).

Figure 1(a) shows that when plotted as a function of τT tln( / )0 , where τ0 = 10–9 s, relaxation data taken at differ-

ent T lie on a single curve. This scaling behaviour is expected if the magnetic relaxation is thermally activated and τ1 / 0 is the attempt frequency [16]. The scaling behaviour is rela-tively insensitive to the choice of τ0. Given a single-valued energy barrier E, M would decay exponentially as = τM M e t

0- / ,

whereτ τ= eE kT0

/ and M0 is the total moment immediately after the moment of each nanoparticle has relaxed from parallel to the applied field to parallel to its easy axis [4]. By extension, for a distribution of energy barriers f(E),

∫= τ∞

−M M f E Ee ( )dt E0

0

/ ( ) . If the characteristic width of f(E)

is large compared to kBT, as is true for many experimental nan-oparticle systems, then it is a good approximation to assume that at any time t all nanoparticles with τ< =E E k T tln( / )C B 0 have relaxed but those with >E EC have not. This implies

∫=∞

M M f E E( )dE

0C

, which not only means M is a function

of EC, explaining the observed τT tln( / )0 scaling, but also that f(E) is simply found by differentiation:

τ= −f E

k

M M

T t( )

1 d( / )

d( ln( / ) ).

B

0

0

For interacting nanoparticles, the observed f(E) is the effective energy barrier distribution sampled by the particles during relaxation, which differs from f(E) at the start of the process because f(E) becomes a function of the magnetic con-figuration [17, 18].

Figure 1(b) shows the energy barrier distribution f(E) determined by differentiating the data of figure 1(a) with respect to τT tln( / )0

4. From the figure it is clear that f(E) for a ZFC sample of 2.5% cobalt-doped MGF, in which the nano-particles are randomly oriented, consists of a broad peak and a low-energy tail. Similar data were also obtained for 6.6% cobalt-doped MGF.

To understand the origin of the low-energy tail observed in figure 1(b), we compared data for randomly oriented and aligned cobalt-doped MGF. Consider the axis that gives the minimum total anisotropy energy (cubic and uniaxial) when the magnetic moment is parallel to it. We can align the cobalt-doped MGF so that this effective easy axis is parallel to the applied field by cooling the sample through the temperature at which its orientation is fixed (~210 K) in a high field (5 T) [9]. Although the alignment is not expected to be complete [19],

Figure 1. (a) Magnetic moment M normalized to the saturation value measured as a function of time t at temperatures T between 2 and 140 K for a randomly oriented (ZFC) sample of 2.5% cobalt-doped magnetite (Fe3O4) nanoparticles in ferritin (cobalt-doped MGF). The data is plotted as a function of τT tln( / )0 where τ0= 10–9 s. (b) Magnetic energy barrier distribution f(E) determined by differentiating the data of (a) with respect to τT tln( / )0 .

3t is equal to the time recorded by the magnetometer plus a constant t0 which represents the time taken to remove the field H = 5 T, quench the magnet and start recording data in zero applied field. t0 was chosen to give plots of ln M M/ 0against t that are linear, and was typically 100 – 150 s.4 When calculating the data of figures 1(b) and 3, we have approximated M0 by the remanence determined from M–H measurements at 5 K.

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it is sufficient to generate significant changes in the magnetic properties as shown in figure 2.

We emphasize that the changes shown in this figure are not due to field-induced aggregation because firstly, we only apply the field below 250 K when the solution has frozen, restricting the translational motion of the cobalt-doped MGF though not its rotation, and secondly, the changes are completely reversible on heating the sample to 250 K and cooling again in zero field. Protein aggregation is not generally reversible. The changes are also not due to the combination of field-cooling and interparticle interactions, because they are absent if the cobalt-doped MGF is cooled in zero field to 205 K and then in 5 T applied field from 205 to 5 K, but present if cooled from 250 to 205 K in 5 T and in zero field from 205 to 5 K. In other words, the changes are only generated at the temperatures at which the particles’ orientation is fixed by the glass transition in the Tris–HCl solution.

Figure 3 shows the effect on the energy barrier distribution of aligning the effective easy axis of 2.5% cobalt-doped MGF by cooling through Tg in 5 T. The low-energy tail becomes much less prominent, while the broad peak at higher ener-gies becomes more pronounced. The area under the curve remains constant, because ∫ =f E E( ) d 1. Very similar behav-

iour is seen for 6.6% cobalt-doped MGF. If the nanoparticles are warmed to a temperature at which they are free to rotate and then zero-field-cooled, the alignment is lost and the low-energy tail re-appears.

The existence of a large number of low-energy barriers for the unaligned cobalt-doped MGF and their disappearance on aligning the sample can be explained by the presence of both cubic and uniaxial contributions to the magnetic anisotropy energy in the cobalt-doped MGF. If the anisotropy energy con-tains both cubic and uniaxial terms, in addition to the direction of the magnetization vector that minimizes the total anisot-ropy energy (the effective easy axis), there will be other direc-tions giving local minima. The energy barrier between a local

minimum and a global minimum will be less than between the two global minima (corresponding to the two possible senses of the magnetization vector parallel to the effective easy axis). Essentially, the latter barrier corresponds to the broad peak in f(E) while the former corresponds to the low-energy tail. For an unaligned sample, the magnetic moment can lie in either a local or a global minimum immediately after removing a satu-rating field, while for an aligned sample it will lie in a global minimum, because the effective easy axis is parallel to the applied field. Hence in the former case the system ‘sees’ the low-energy tail while in the latter case it does not.

Our conclusions are supported by Monte Carlo simulations performed on ensembles of non-interacting magnetic nano-particles. We consider competing cubic and uniaxial aniso-tropies for each particle [20] and for simplicity, one of the three orthogonal cubic axes is also assumed to be the axis for the uniaxial anisotropy. However, qualitatively similar results are obtained when the uniaxial anisotropy axis is randomly oriented with respect to the cubic ones.

We have employed Monte Carlo simulations along with the Metropolis algorithm [21–23]. Typically, we take =N 104

particles with a volume selected from a log-normal distri-bution, and one Monte Carlo step (MCs) means N trials to change the state of the system. To simulate a single relaxation experiment, we prepare an initial state at t = 0 with all the particle moments μi parallel to the applied field H, represent-ing saturation in this direction. Then, the field is turned off, the system relaxes, and we record M as a function of time (the number of MCs). When the easy axes of the nanoparticles are randomly oriented, there is an ultrafast relaxation from saturation to a state where μi is parallel to one of the easy axes ( =M M0) immediately after the saturating field is removed [24]. When the easy axes are aligned parallel to the applied field there is no initial ultrafast relaxation and ≈M M0 saturation. Relaxations are simulated for temperatures between T = 0.005 and T = 0.03 in steps of T = 0.001, where temperature is measured in units of K k/u B, with Ku being the mean value of the uniaxial anisotropy energy. The data from these differ-ent simulations are combined into a single plot of M/M0 as a

Figure 2. Normalized magnetic moment curves at 5 K of a 0.9µM solution in 50mM Tris–HCl of randomly oriented cobalt-doped MGF (solid circles ●) and cobalt-doped MGF with its easy axes aligned by cooling through ~ 210 K (the temperature at which the cobalt-doped MGF loses its rotational freedom) in a 5 T applied field (open circles ○). Data from [9].

Figure 3. As for figure 1(b), except that the nanoparticle easy axes were aligned by cooling through ~ 210 K in a 5 T applied field.

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function of τT tln( / )0 as in the experiments, and the energy barrier distributions f(E) obtained from this curve by numeri-cal differentiation.

Simulations confirm that for randomly oriented nano-particles with pure uniaxial or cubic anisotropy the energy barrier distribution has a single broad peak. However figure 4(a) shows that the situation changes for nanoparticles with both cubic and uniaxial magnetic anisotropy energy terms (ku = 1 and kc = 2). Note that two distinct peaks appear: one at low energies, which is high and narrow and corresponds to the low-energy tail seen in the experimental data (figure 1(a)), and the other at higher energies, which is lower and very broad. Figure 4(b) shows that in agreement with the experimental data (figure 3) the low-energy tail vanishes when the easy axes are aligned.

To summarize, we report a new observation concern-ing fine particle magnetism. The energy barrier distribu-tion sampled by randomly oriented cobalt-doped magnetite

nanoparticles on relaxation from saturation includes a low-energy tail that is heavily attenuated when the nanoparticles are aligned and absent for undoped magnetite nanoparticles for which the cubic anisotropy is absent. We argue that the low-energy barriers separate local minima resulting from a combination of uniaxial and cubic magnetic anisotropy energy terms from the global minimum, and that the system does not sample the local minima when the easy axes are aligned.

Acknowledgments

This work has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no 228673 (MAGNONICS) and was carried out with the support of the Bristol Centre for Nanoscience and Quantum Information.

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Figure 4. The solid line shows the simulated M/M0 (left-hand scale) as a function of τT tln( / )0 for the case of randomly aligned single-domain magnetic nanoparticles possessing both cubic and uniaxial anisotropies (ku = 1 and kc = 2). (a) shows randomly oriented nanoparticles and (b) shows a sample with easy axes aligned parallel to the applied field. The time t is measured in units of Monte Carlo steps (see text) and τ0 is assumed to be equal to 1. Temperature is measured in units of K k/u B, where Ku is the mean value of the uniaxial anisotropy. The dashed line represents the numerical derivative of M/M0 with respect to τT tln( / )0 (right-hand scale), which gives the magnetic energy barrier distribution.

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