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Colossal Anisotropy of the Dynamic Magnetic Susceptibility in Low-Dimensional Nanocube Assemblies Erik Wetterskog, Christian Jonasson, Detlef-M. Smilgies, Vincent Schaller, Christer Johansson, and Peter Svedlindh ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.7b07745 • Publication Date (Web): 12 Jan 2018 Downloaded from http://pubs.acs.org on January 15, 2018

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Colossal Anisotropy of the Dynamic Magnetic Susceptibility in Low-Dimensional Nanocube Assemblies Erik Wetterskog,∗,† Christian Jonasson,‡ Detlef-M. Smilgies,¶ Vincent Schaller,§ Christer Johansson,‡ and Peter Svedlindh† †Solid state physics, Dept. of Engineering Sciences, Uppsala University, 751 21 Sweden ‡RISE Acreo, 400 14 Gothenburg, Sweden ¶Cornell High Energy Synchrotron Source (CHESS), Cornell University, Ithaca, New York, 14853, United States. §Chalmers Industriteknik, 412 88, Gothenburg, Sweden E-mail: [email protected]

Abstract One of the ultimate goals of nanocrystal self-assembly is to transform nanoscale building blocks into a material that displays enhanced properties relative to the sum of its parts. Herein, we demonstrate that 1D needle shaped assemblies composed of Fe3−δ O4 nanocubes display a significant augmentation of the magnetic susceptibility and dissipation as compared to 0D and 2D systems. The performance of the nanocube needles is highlighted by a colossal anisotropy factor defined as the ratio of the parallel to the perpendicular magnetization components. We show that the origin of this effect cannot be ascribed to shape anisotropy in its classical sense; as such, it has no analogy in bulk magnetic materials. The temperature dependent anisotropy factors of the in-

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and out-phase components of the magnetization have an extremely strong particle size dependence and reach values of 80 and 2500, respectively, for the largest nanocubes in this study. Aided by simulations, we ascribe the anisotropy of the magnetic susceptibility, and its strong particle-size dependence to a synergistic coupling between the dipolar interaction field, and a net anisotropy field resulting from a partial texture in the 1D nanocube needles.

Keywords magnetic properties, ac-susceptibility, anisotropy, magnetic nanoparticles, arrays, supercrystals, assemblies Nanoparticle assemblies composed of e.g. magnetic, metallic, and semiconducting nanocrystals have been the subject of immense scientific activity during the last decade. 1–3 Clearly, the fabrication and design of complex nanoparticle assemblies have exploded. 1,4,5 In contrast, the exploitation of their unique collective properties has merely begun. 3 While initial bottom-up work on magnetic particle arrays targeted high density storage media, 6,7 today’s focus has shifted towards biomedical applications and functional materials. The assembly of magnetic nanoparticles into low-dimensional mesoscale arrangements is a particularly powerful strategy, where the combination of size-dependent and collective properties can yield materials with global anisotropy. 5,8,9 In this context, the term collective properties refers to ensemble properties emerging from specific spatial arrangements of an interacting nanoparticle system, clearly distinct from the properties of an isolated/non-interacting system. The classical cases of low-dimensional magnetic particle ensembles include ferrohydronamic instabilities such as rods/chains, 10–14 hexagonal arrays of pillars, 12,14,15 helices, 5,16 and labyrinthine structures. 17 Other strategies to produce low-dimensional magnetic materials include exploiting interface geometry or confinement. 9,18,19 Uses for engineered low-dimensional magnetic particle assemblies include e.g. actuators, 20 microfluidic devices, 21 and magneto-responsive materials with tunable optical bandgaps. 22,23 Arguably, the most well-known examples of low-dimensional 2


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nanoparticle arrangements are found in nature. 1D structures have been found in many organisms that possess magnetoreception (e.g. salmon, termites), 24 with the most famous example being the magnetosome chains of magnetotactic bacteria. 25–27 New concepts based on low-dimensional particle assemblies are emerging also in the biomedical area. Particle chains have been exploited in a number of magnetic biosensor protocols, 28,29 while both synthetic particle low-dimensional particle arrangements 30–33 as well as magnetosome chains 26,34 have been shown to display improved magnetic hyperthermia performance relative to randomly arranged systems. Some of the key challenges in magnetic sensing and magnetoreception, revolve around the maximization of the magnetic susceptibility in the low-field range (< 1 mT). Indeed, avoiding the use of high magnetic fields simplifies device and protocol design vastly, and enables the use of magnetic ac-fields, which typically are limited in amplitude. Although the interest in low-dimensional magnetic assemblies is steadily growing, it is clear that the interplay between shape and texture at the mesoscale, and their effect on the dipolar magnetism in nanoparticle arrays is not well understood. A relatively large number of studies, have up to now attempted to harness the inherent anisotropy that these materials possess. In terms of the anisotropy of the low-field magnetization, the effects have up to now been limited to within one order of magnitude. 9,11,16,19,21,35–37

Herein, we explore the magnetic properties of a system of needle-shaped magnetic particle arrays composed of iron oxide (Fe3−δ O4 ) nanocubes of three different edge-lengths. The nanocube needles display a remarkably large anisotropic magnetic response in static and alternating (dc- and ac-) magnetic fields, in particular with respect to the dynamic magnetic susceptibility. By quantitatively comparing the magnetic properties of nanocube dispersions and arrays with different geometries, we demonstrate that the dynamic susceptibility of the 1D needles is strongly particle size dependent and orders of magnitude larger than in the other systems under study. Importantly, we show beyond doubt that the anisotropy of the dynamic magnetization cannot be ascribed to shape anisotropy in its classical sense. Using

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a combination of dc-/ac-magnetometry and magnetic Monte-Carlo simulations, we attribute the anisotropy of the needle shaped arrays to an orientation dependent dipolar interaction field, resulting from a synergistic combination of array shape and easy axis texture.

Results and Discussion

Figure 1: Characterization of the needle shaped nanocube arrays using electron microscopy, atomic force microscopy and small angle X-ray scattering. (a) TEM images of C096-C136, scale bar 10 nm. (b) Low-magnification SEM image showing the length and shape of the nanocube needles. Inset: Intermediate magnification showing the surface of two C126 nanocubes needles. (c) AFM topography image of an array of C136 nanocube needles. The height profile at the bottom corresponds to integration of the region bounded by the arrows. (d) High resolution SEM image showing the detail of the mesostructure of a C126 nanocube needle with a fast Fourier transform (FFT) inset. (e) GISAXS patterns of the C126 nanocube needles (left) and platelets (right, cf. fig. S2). (f) Sketch defining the magnetization parallel (Mlong ) and perpendicular (Mshort ) to the long needle axis. Needle shaped iron oxide nanocube arrays, from here on referred to as the nanocube needles, were produced by drop-casting monodisperse oleic acid capped Fe3−δ O4 nanocubes with edge-lengths l = 9.6, 12.6 and 13.6 nm (C096, C126, and C136; see fig. 1a and fig. S1) according to methods described in previous works. 38,39 Applying a magnetic field (B = 65 mT) in the plane of the substrate surface results in the assembly of nanocube needles 4


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with widths of ≈ 5-10 µm, typical lengths of ≈ 1 mm (cf. fig. 1b-c), and an on average slightly flattened shape with an aspect ratio of ≈ 3 (see fig. 1c). High resolution scanning electron microscopy (HRSEM, fig. 1d) shows that the nanocube needles have high local order, as evidenced by discrete spots in the fast Fourier transform (FFT). Grazing-incidence small and wide-angle scattering (GISAXS, GIWAXS) on the self-assembled nanocube arrays were performed at the D1 station at the Cornell High Energy Synchrotron Source (CHESS). GISAXS patterns of the nanocube needles and mesocrystals are shown in fig. 1d-e. Analysis of the well-defined Debye-Scherrer rings indicate that the needles crystallize in a body centered tetragonal structure (BCT) with a random orientation of the superlattice grains. Although being globally polycrystalline, the needles exhibit a slight mesoscale texture; a slight preference towards (101)BCT is evident from several spots in the Debye-Scherrer rings (see fig. 1e and fig. S3). In contrast, assembly of the C126 nanocubes in zero field results in the formation of quasi-2D truncated triangular platelets (see fig. S2), 39 referred from hereon as the mesocrystal platelets, with a well-developed (101)BCT structure, (see fig. 1e and fig. S3). At wide angles (GIWAXS, shown in fig. S4), the scattering of the needles and platelets reveal several weak arcs, indicating that the cubes on average are oriented with the (100), and to a lesser degree the (101) and (111) planes parallel to the substrate, although with a broad orientational distribution of several degrees. Magnetization versus field measurements were performed with the field oriented both parallel, Mlong (H), and perpendicular, Mshort (H), to the long needle axes, as shown in fig. 1f. Results from low-field (equilibrium) magnetization measurements on the C126 Fe3−δ O4 nanocube needle system are shown in fig. 2a. Clearly, there is a significant anisotropy of the magnetic susceptibility, increasing towards the blocking temperature. Mlong (H)) is nonlinear and significantly larger than the Langevin magnetization: ML (H) = Ms [coth ξ − ξ −1 ] (see dashed lines in fig. 2a), where ξ = µ0 HMs V /kB T , Ms is the saturation magnetization, and V is the particle volume. Conversely, Mshort (H) is linear in the measured field range and considerably smaller than ML (H). This significant anisotropy of the magnetic suscep-

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Figure 2: Experimental and simulated M (H) curves of the needle shaped arrays. (a) Experimental M (H) curves of the C126 nanocube needles at T = 200, 250, and 300 K. The dotted lines are calculated ML -curves. (b) Comparison between simulated and experimental M (H) curves for the C126 needles with dipole-dipole interactions and a random distribution of easy axes. Error bars represent the standard deviation of 16 simulated curves. (c) Snapshot from the simulation of a nanocube needle, corresponding to a 24 × 3 (L × W ) unit cell long rod with truncated edges. The cones and their hue correspond to the direction of the magnetization vector of the nanocubes. (d) Orientational distribution functions (RaikherStepanov) of the C096 – C136 nanocubes in a field of 65 mT (left), and a comparison to the simpler Watson distributions (right). The focus of the Watson distribution function is determined by the concentration parameter κ. (e) Watson distributions projected on a unit sphere for κ = 0.5 – 4. (f) Comparison of simulated Mlong (H) curves with nanocube needles for the Watson distributions shown in (d). (g) Simulated M/Mrand versus κ for two different fields. M/Mrand corresponds to the magnetization ratio of nanocube needles with texture, and nanocube needles with a random distribution of easy axes. Note that the field dependence vanishes when dipole-dipole interactions are turned off. The lines are guides to the eye. Error bars correspond to the propagated error calculated from the ratio of the magnetization values.

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tibility only applies to the low-field range. As shown in fig. S8, the high-field properties of the nanocube needles are isotropic. In order to elucidate the origin of the anisotropy of the low-field magnetization, we simulated M (H) curves of the C126 needles using a Monte-Carlo (MC) algorithm, modeling the needles as 1D rods composed of 3 × 24 BCT unit cells (as shown fig. 2c). In the simulations, the magnetic energy of a 1D needle is calculated as the sum of the Zeeman, magnetic anisotropy, and (non-truncated) dipolar interaction energies of the individual Fe3−δ O4 nanocubes, see the Experimental section for further details. Fig. 2b shows the simulated magnetization curve for the temperature T = 300 K, using a random distribution of (uniaxial) easy axes of the nanocubes. Evidently, this simple approach already replicates the experimental data with some accuracy, with the simulations overlapping the experimental Mshort (H) data, and just slightly underestimating Mlong (H). As expected for a non-interacting system, turning off the dipolar interactions collapse the magnetization values on the ML (H) curve. In order to account for the discrepancy between the simulated and experimental Mlong (H) curves, we considered a partial alignment of the nanocrystal easy axes along the long needle axis, 36,40 as a result of the magnetic field applied over the substrate during the nanocube assembly. The GIWAXS scattering is too weak for an outright comparison of the perpendicular in-plane scattering intensities, and therefore cannot be used to assess the crystallographic orientation distribution of the nanoparticle arrays. Instead, we calculated the easy axis alignment using the statistical thermodynamic framework for magnetic (uniaxial) nanocrystal dispersions established by Raikher-Stepanov. 41 The distribution of magnetic easy axes n in a magnetic field H = Hh is given by the distribution function 40–42

f (n) = Z

−1

Z

exp[ξ(e · h) + σ(e · n)2 ]de,

(1)

where e is a unit vector describing the direction of the particle magnetic moment, R ξ 1 σu2 e du is the partition function, ξ = (µ0 Ms V H)/(kB T ) is the Langevin Z −1 = 16π 2 sinh ξ 0 parameter, and σ = KV /kB T . The above integral is not solvable by standard methods, but 7


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can be converted to a sum 41

f (n) = f (ϑ) =

∞ X 1 [1 + (4k + 1)L2k (ξ)Pe2k (σ)P2k (cos ϑ)]. 4π k=1

(2)

Here, ϑ is the angle between e and n; Lk (ξ) = Ik+ 1 (ξ)/I 1 (ξ) where Ik is the modified 2

2

Bessel function of index k; Pk is a Legendre polynomial of index k; Pe2k =

R1 0

2

P2k (u)eσu du R1 . σu2 du 0 e

Calculated orientational distributions for the C096 – C126 nanocubes are shown in fig. 2d (see details in table S1), and predict a substantial easy axis alignment in the assembly field of B = 65 mT, increasing with the nanocube moment: µ = Ms V . It should be noted that the sum appears to converge relatively quickly, the graphs shown in fig. 2d represents eq. 2 truncated at k = 8. Moreover, as eq. 2 lacks analytical solutions it cannot be directly implemented in the Monte-Carlo model. For this purpose, we sampled the easy axis 2

distribution using the similar, albeit simpler, Watson distribution: 43 h(θ, φ) = Cw eκ cos θ R1 2 with Cw = (4π 0 eκu du)−1 with θ and φ being the polar and azimuthal angles in a spherical coordinate system. For the Watson distribution, the degree of alignment is described by the focus parameter κ (larger for more focused distributions, random for κ = 0), and examples with κ = 0.5 – 2 are shown in fig. 2d and compared to distributions calculated by the RS model. For clarity, fig. 2e illustrates four discrete distributions with κ = 0.5 – 4 on the unit sphere. Simulations of 1D rods, with different degrees of orientational alignment are shown in fig. 2f. With dipole-dipole interactions turned on, we observe a large increase of the magnetic susceptibility (see fig. 2f). In fact, the experimental Mlong (H) curve for the C126 nanocube needles is well replicated assuming κ ≈ 0.5, far smaller than the degree of alignment predicted by the R-S model. As can be seen in fig. 2f, in simulations with interacting nanocubes, the magnetization increases sharply with the increase of the focus parameter (κ), relative to the randomly oriented configuration. This effect is more clearly illustrated in fig. 2g, which shows M/Mrand (H, κ), i.e. the magnetization of the textured rods divided by the magnetization of a rod with a random easy axis distribution. Conversely, turning off dipole-dipole interactions results in M (H) curves with a much weaker (and completely 8


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field-independent) linear dependence on κ. These results implicate a coupling between the dipolar interaction field and the alignment of the nanocrystal easy axes. While this effect is particularly significant for small fields (1 mT), even considering relatively weak orientational distributions (κ ≈ 1), it diminishes drastically in a field of 5 mT. Thus, our simulations show that the augmentation of the low-field magnetization of the 1D nanocube needles (in low fields) results from the synergy of two separate effects; a net magnetic anisotropy field created by a weak (partial) easy-axis alignment, and an anisotropic dipolar interaction field resulting from the 1D shape of the array.

Figure 3: Comparison of magnetic ac-magnetization of the nanocube needles, platelets and 0 00 frozen dispersions. (a) Real (M ) and imaginary (M ) components of Mlong (T, f ) for the 00 C096 nanocube needles (f = 0.17, 1.7, 5.1, 17, 170, 510 Hz). (b) Scaling analysis of M (T, f ) corresponding to 9 different nanoparticle systems: needles, triangular platelets and frozen dispersions composed of C096, C126 and C136 nanocubes. (c) Schematic showing the geometry of the samples (not to scale) with respect to the probing ac-field. Magnetic ac-magnetization measurements were performed on a variety of C096/C126/C136

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Fe3−δ O4 nanocube systems (with one example shown in fig. 3a) in the temperature range of 4 – 300 K, using ac-field frequencies of 0.17 – 510 Hz. The various systems under study are summarized in fig. 3c and include 1D nanocube needles (long/short), 2D platelets, and 0D dispersions (zero-field frozen, and field-frozen in B = 1 T). In all cases, the measuring field was Bac = 0.4 mT except for C136 Needles/Long (Bac = 20 µT) due to a strongly non-linear response at very small fields (see Experimental section). First, we assessed the orientation-dependent spin-coupling of nanocube needles by calculating the mean interaction field (Hi = αM ) and the associated mean-field constant α. This was done by fitting the 0

high-temperature (equilibrium) part of the M (T ) curves to a modified Curie-Weiss law, 44 accounting for the temperature dependence of the Fe3−δ O4 nanocrystal single domain magnetic moments (see fig. S10). In essence, the mean-field constant provides a measure of the average sign and strength of the dipolar interaction field acting on a particle in the array: α > 0 implies an average ferromagnetic coupling and α < 0 corresponds to an antiferromagnetic coupling. For the C096 and C126 nanocube needles this reveals a ferromagnetic C096/C126

interaction in the long direction: αlong

≈ +4 × 10−2 (TB of the C136 systems is too

high to determine αC136 ). Conversely, we find an (on average) antiferromagnetic interaction C126 along the short axis, varying in magnitude with the particle size, αshort = −8.4 × 10−2 and C096 αshort = −3.1 × 10−2 , revealing the inherent anisotropy of the dipolar interaction field of the

nanocube needles. Secondly, we assessed the influence of array shape and orientation on the relaxation properties of the assemblies. Fig. 3b shows a comparison of 9 systems composed of C096, C126 and C136 Fe3−δ O4 nanocubes. Here, we define the blocking temperatures as the onset of 00

00

dissipation: (TB ) as M (TB ) = 0.5 × Mmax . Blocking temperatures vary not only with particle size, but also with the orientation and dimensionality of the nanocube assemblies. For instance, TB is ≈ 15 K (C126) and 30 K (C136) lower along the short axis compared to the long axis. In essence, the short and long axes of the nanocube needles represent the two extremes for which the blocking temperatures of all other systems fall in between: i.e. the

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zero-field, and field frozen dispersions, and the mesocrystal platelets (cf. fig. 3b). Attempting to fit the temperature dependence of the relaxation times to an Arrhenius equation: τ = τ0 eKV /kB T with τ = (2πf )−1 yields unphysical ( 10−10 s) values of the characteristic attempt times (τ0 = 10−24 − 10−51 s, see table S2), 45 as a result of dipolar interactions in the nanocube arrays and dispersions. While there are some theoretical models capable of treating the magnetization dynamics of interacting nanoparticle systems based on e.g. thermodynamical pertubation theory, 46 and Fokker-Planck formalism, 47–50 these are only applicable in the weak interaction regime, ξd =

µ0 (Ms V )2 4πa3 kB T

1, where a is a characteristic

interparticle length scale of the system. 46,48 In our case, by setting a as the nearest neighbor center-to-center distance (aBCT , as shown in fig. 2c), one obtains ξd ≈ 0.4, 1 and 1.6 for the C096, C126 and C136 systems, respectively. 39 For strongly interacting systems, workers in the field have typically resorted to interpretations based on spin-glass laws, such as the Vogel-Fulcher law (τ = τ0 eKV /(kB (T −T0 )) ), 49,51 or critical slowing down approaches. 52 However, none of the systems in fig. 3b show any evidence for spin-glass behaviour (divergence of τ ), nor can they be soundly fitted to a Vogel-Fulcher or a power-law. This lack of spinglass characteristics of the nanocube needles becomes evident in a direct comparison with a disordered C136 paste sample, that display a divergence-like behaviour of τ (see fig. S9 and discussion in Supplementary information). Indeed, the two key ingredients in spin-glass systems are (long-range) magnetic disorder, and magnetic frustration; intuitively these elements are largely missing in the ordered nanocube mesocrystal platelets and needles. In essence, as in any other many-body problem with long-range interactions, a quantitative description of the relaxation properties of ordered and strongly interacting particle systems remains a formidable challenge 53 . Nevertheless, our results demonstrate the characteristic relaxation times in these systems vary in a highly regular fashion, and underlines the importance of considering finite-size/geometry effects in future modeling efforts. Arguably, the most exceptional property of the Fe3−δ O4 nanocube needles is found when comparing the amplitude of the dynamic (ac-) magnetization between the various

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0

Figure 4: Colossal anisotropy of the dynamic magnetic susceptibility: dependence of M 00 and M with the nanocube edge length, array shape, and orientation. (a) Temperature 0 dependence of the in-phase component (M ) of the ac- magnetization (f = 0.17 Hz) for the C126 nanocube needles, the mesocrystal platelets, a dispersion frozen in a magnetic field of B = 1 T, and a zero-field frozen dispersion. (b) Temperature dependence of the out-of-phase 00 00 component (M (T )) for the C126 systems shown in (a). (c) Variation of Mdisp (T ) with the 00 00 cube edge length, i.e. M (T ) of the frozen nanocube dispersions. (d) Variation of Mlong (T ) with the cube edge length in the nanocube needles. Note that the scale is the same as in (c), but that the probing ac-field is 20 times smaller (µ0 Hac = 400 versus 20 µT). (e) Same as in (d) but with the magnetization axis magnified ×10.

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C096/C126/C136 nanocube systems. The significant influence of the array shape and di0

00

mensionality is immediately obvious when comparing M (T ) and M (T ) of the five C126 0

nanocube systems shown in fig. 4a-b. The in-phase magnetization component M (T ) at f = 00

0.17 Hz corresponds to the ZFC branch of a typical, ZFC/FC curve, whereas M (T ) is equivalent to the magnetic dissipation and the signature of irreversible magnetic processes caused 0

00

by e.g. hysteresis loss. Evidently, both Mlong and Mlong are exceptionally large compared 0

00

0

00

to M and M of the other C126 systems, and in particular Mshort and Mshort are orders of magnitude smaller. Compared to the exceptionally strong orientation dependence of the 0

00

nanocube needles, the differences between both M (T ) and M (T ) for the zero-field, and field frozen dispersions, and the triangular mesocrystal platelets are comparably minor. This 0

00

clearly shows that the augmentation of Mlong and Mlong cannot be ascribed to texturization (cf. field frozen dispersion) or close packing alone (cf. mesocrystal platelets). Moreover, the anisotropy of the dynamic magnetization is clearly collective in origin, and thus mediated by interparticle interactions. This is evidenced by the increasing spread in magnetic properties between the the strongly interacting needles, relative to the weakly interacting dispersions, as the nanocube volume is increased. This effect is most clearly seen in fig. 4c-e, that com00

pares M (T ) of frozen dispersions and nanocube needles composed of C096, C126 and C136 nanocubes. As seen in fig. 4c, an increase of the nanocube volume only results in a small 00

00

increase in Mmax for the nanocube dispersions. In comparison, the magnitude of Mlong (T ) for the C096-C136 nanocube needles spans just short of three orders of magnitude (see fig. 4d-e). For the purpose of quantifying the anisotropy of the magnetic response we introduce 0

temperature dependent anisotropy factors A = Mlong /Mshort and Aiso = Mlong /Mdisp . A (T ) 00

and A (T ) for the C096/C126/C136 nanocube systems are shown in fig. 5a-b. Although the 0

anisotropy factor of the in-phase component of the magnetization, A (T ), is the smaller of 0

0

the two, it is strongly particle size dependent. Here, max(A )= 3.5, 16 and, 80 and max(Aiso ) = 3, 7, 46 for the C096, C126 and C136 nanocube needles, respectively. Intuitively, it is

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Figure 5: Anisotropy factors corresponding to the in-phase and out-of-phase components 0 of the magnetization of the C096-C136 nanocube needles. (a) Anisotropy factor (a) A (T ) 00 and (b) A (T ) for f = 0.17 Hz on a logarithmic scale. The solid lines correspond to A = 00 Mlong /Mshort , whereas the dashed lines correspond to Aiso = Mlong /Mdisp . The Mshort signal was smoothed using a Savitzky-Golay filter in order to reduce noise. (c) Minor magnetization (Mlong and Mshort ) versus field loops near T ≈ TB for the C126 nanocube needles. tempting to ascribe the extreme difference in the low-field linear magnetization of nanocube needles to shape anisotropy (magnetostatic self-energy), but it is straightforward to show that demagnetization effects are much too small to satisfactorily explain the observed results. In contrast to bulk magnetic materials, the nanocube arrays and dispersions in this work are exchange de-coupled as a result of the oleic acid double layer separating the nanocubes (x = 3.4 – 4.1 nm). 39 First note, that the demagnetization field (N M (Hi )) is always opposite to the internal (Hi ) and applied (HA ) magnetic fields: Hi = HA − N M (Hi ). Imposing a demagnetization field on the Langevin susceptibility (χL ) yields an effective susceptibility: χA =

M HA

= χL /(1 + N χL ), always smaller than the uncorrected susceptibility. Using the

chain-of-cubes model suggested by Phatak et al., 54 and the experimentally observed face-toface distance in the bct structure, we estimate the lower limit of the demagnetization factors in the Fe3−δ O4 nanocube needles to Nk = 0.16 − 0.2, and conversely N⊥ = 0.42 − 0.4 (see section S4 and fig. S6a-b); in contrast, the demagnetization factor of an isolated cube is equal to that of an isolated sphere: N = 1/3. 54 Using χL of the C136 as a reference point, 0

we find that the maximal value of A due to shape anisotropy should barely exceed 2 for 0

the C136 needles (≈ 1.5 in the case of Aiso ), and is consequently a factor ≈ 40 smaller than 14


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the experimentally observed effect. As a final proof, we simply note that the calculated demagnetization factors of the C096/C126/C136 needles are similar in magnitude, but that these materials exhibit vastly different experimental magnetic susceptibilities. Rather, the augmented low-field magnetization is the result of a positive dipolar interaction field in the exchange decoupled nanocube needles, co-aligned with the applied magnetic field. As such, this effect is collective in nature and has no counterpart in exchange coupled bulk magnetic materials. Although not straightforward, we have estimated anisotropy factors from available literature data on similar low-dimensional nanoparticle systems. Generally, M (T ) data found in literature was either non-quantitative (or normalized to unity); 10,16 in some cases the M (H) data was acquired in relatively high magnetic fields, 10,37 obscuring the directional properties (cf. fig. S8). However, we estimate A to 2 – 8 (at T ≈ TB ) for 1D nanocluster chains formed by capillary bridge mediated assembly; 9 1.5 – 3.5 for oriented Fe microparticle/PDMS composites; 21 2 for oriented cellulose/magnetite films; 19 2.5 for a thick array of magnetic field assembled γ-Fe2 O3 nanocrystals; 36 and 2.5 for γ-Fe2 O3 needle-shaped assemblies in a polymer matrix. 11 Regarding work on more traditional ferrofluids, we found ≈ 4.5 (Aiso ) for a field-frozen (B = 3 T) ferrofluid (versus a zero-field frozen sample), 35 and ≈ 4.5 (Aiso ) based on the difference in initial susceptibility for a concentrated ferrofluid passing over its melting point. 55 In each case the particle concentration was high (15 – 32 vol%), suggesting that 1D chains formed. 14 00

The anisotropy factor of the out-of-phase component (A (T )) displays a much stronger 0

00

particle size dependence than A (T ). For the C136 Fe3−δ O4 nanocube needles both A and 00

Aiso are on the order of ≈ 2500 around 245-250 K and remain > 1000 in a temperature window of 235-265 K. As seen in fig. 5b, these values are more than one and two orders 00

of magnitude larger than A (T ) ≈ 142 and 7.5 for the C126 and C096 nanocube needles, 00

respectively. A large A (T ) is of particular interest in the field of magnetic biosensing where 00

the spectral position and amplitude of the M peak is used for the biodetection of e.g. DNA

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00

and pathogens. 56 It is worth pointing out, that the origin of the extremely large A (T ) val00

ues of the C126 and C136 needles is in part due to a shift of the frequency dependent Mlong 00

00

curves to higher temperatures; the simple ratios max(Mlong )/max(Mshort ) are significantly smaller, ≈ 17.5 and 650 for C126 and C136, respectively. Further insights into the mech00

anisms behind the colossal anisotropy of M are found by comparing two minor hysteresis 00

00

curves of the nanocube needles near max(M (T )) (see fig. 5c). Since M is proportional to the dissipative (hysteresis) loss; this is also mirrored by the respective areas of the minor hysteresis loops at small field amplitudes. Whereas the Mlong (H) opens up near TB , Mshort (H) is reversible with no measurable hysteresis. In the framework of the Stoner-Wohlfarth (SW) model, the magnetization processes of uniaxial (and non-interacting) particles may proceed by irreversible and reversible rotations of the particle magnetic moments. For a texturized particle ensemble, applying the ac- magnetic field along the easy axis causes the hysteresis 00

curve to open (M > 0), due to irreversible rotations of the particle moments. Along the 00

hard axis, the ac response is reversible (M = 0) and corresponds to "wiggling" of the particle moment inside the minima defined by the uniaxial anisotropy. However, in contrast to the classical SW case, the nanocube needles are strongly interacting and exhibit a collective magnetic response; the needle easy axis is here jointly defined by the (field-induced) dipolar interaction field, and the magnetocrystalline anisotropy of the nanocubes, in correspondence to the synergistic combination of the array geometry and (weak) texture (cf. fig. 2g). Furthermore, we speculate that the high anisotropy factors observed in this work, compared to others, is partly the result of the narrow size- and shape distributions of the nanocubes and the high local order of the resulting mesostructures. Indeed, the broadness of the particle volume distribution, directly translates into a wide relaxation time distribution, effectively lowering the associated anisotropy factor. Furthermore, in dense and strongly interacting systems, mesoscale disorder will result in further broadening of the relaxation time distribution (as a result of a distribution of interparticle distances), as well as further broadening of the orientational distribution of easy axes. Moreover, our simulations suggest that the

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0

00

strong particle size-dependence of A and A is due to the narrowing of the orientational distribution, as predicted by the R-S model (cf. fig. 2d) and the non-linear coupling with the particle size-dependent dipolar interaction field. We conclude by noting that the relatively weak texture inferred by the MC simulations suggests that there is plenty of room for further enhancement of the magnetic properties.

Conclusions In summary, we have demonstrated that 1D needle shaped Fe3−δ O4 nanocube assemblies display an extremely anisotropic response in alternating magnetic fields. This directional dynamic response is collective in nature and not associated with the concept of shape anisotropy, in its classical sense. We quantified this effect by an anisotropy factor, defined as the ratio of the magnetization measured parallel and perpendicular to the long needle axis. We demonstrated the strong particle size dependence of this effect, resulting in anisotropy factors of 80 0

00

(M ) and 2500 (M ) for the needles composed of the largest nanocubes in this study. Aided by simulations we show that imposing a partial alignment of the magnetic easy axes in the nanocube needles, leads to a strong coupling with the dipolar interaction field. Our study is a simple demonstration of the great promise of self-assembled materials: a significant augmentation of the material properties by simply varying the size and spatial arrangement of the nanoscale building blocks.

Experimental Magnetic measurements Dispersions and nanoparticle arrays were characterized using a Quantum Design MPMS XL system. Prior to all low-field measurements the background field was cancelled using the ultra-low field option. The resulting background field was

Colossal Anisotropy of the Dynamic Magnetic ... - ACS Publications

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