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Correlation of superparamagnetic relaxation with magnetic dipole interaction in capped ironoxide nanoparticles

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 026002 (http://iopscience.iop.org/0953-8984/27/2/026002) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 026002 (11pp)

doi:10.1088/0953-8984/27/2/026002

Correlation of superparamagnetic relaxation with magnetic dipole interaction in capped iron-oxide nanoparticles ¨ J Landers, F Stromberg, M Darbandi, C Schoppner, W Keune and H Wende Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen, Lotharstr. 1, D-47048 Duisburg, Germany E-mail: [email protected] Received 16 October 2014, revised 14 November 2014 Accepted for publication 24 November 2014 Published 15 December 2014 Abstract

Six nanometer sized iron-oxide nanoparticles capped with an organic surfactant and/or silica shell of various thicknesses have been synthesized by a microemulsion method to enable controllable contributions of interparticle magnetic dipole interaction via tunable interparticle distances. Bare particles with direct surface contact were used as a reference to distinguish between interparticle interaction and surface effects by use of M¨ossbauer spectroscopy. Superparamagnetic relaxation behaviour was analyzed by SQUID-magnetometry techniques, showing a decrease of the blocking temperature with decreasing interparticle interaction energies kB T0 obtained by AC susceptibility. A many-state relaxation model enabled us to describe experimental M¨ossbauer spectra, leading to an effective anisotropy constant Keff ≈ 45 kJm−3 in case of weakly interacting particles, consistent with results from ferromagnetic resonance. Our unique multi-technique approach, spanning a huge regime of characteristic time windows from about 10 s to 5 ns, provides a concise picture of the correlation of superparamagnetic relaxation with interparticle magnetic dipole interaction. Keywords: M¨ossbauer effect, magnetic anisotropy, superparamagnetic relaxation, nanoparticles (Some figures may appear in colour only in the online journal)

is highly dependent on interparticle interaction effects [7]. While thick protective shells directly increase the interparticle distance and may minimize or eliminate magnetic dipole interactions, thin capping layers prevent direct surface contact and exchange interactions, but keep magnetic dipole interactions. Furthermore, capping components may form a strong chemical bond to the surface of nanoparticles, modifying their surface magnetic properties. The broken symmetry at the surface gives rise to altered magnetic properties, including a surface magnetic anisotropy KS and radially or randomly distributed spin orientations. This has a great effect on nanoparticle magnetism [8–12]. For IONPs with a diameter d < 10 nm the magnetocrystalline contribution KV may be significantly exceeded by the surface anisotropy, resulting in an effective anisotropy Keff

1. Introduction

Superparamagnetic iron-oxide nanoparticles (IONPs) are subject of progressive interest in biomedical applications like hyperthermia, contrast enhancing agents for nuclear magnetic resonance imaging (MRI), cell labeling and drug delivery systems [1–4]. The usage of an iron-oxide magnetic particle core in combination with various inorganic (gold, silica) or organic (oleic acid, peptides, polymers) capping materials guarantees high saturation magnetization as well as good biocompatibility, tunable surface properties, protection against unwanted chemical reactions (first of all oxidation) and gives the possibility of further functionalization [5, 6]. For these applications it is crucial to obtain detailed insight into superparamagnetic relaxation behaviour of IONPs, which 0953-8984/15/026002+11$33.00

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given by [13]

not provide information on the precession of the particle magnetization about the easy axis of particle magnetization and on the damping dependence of the relaxation times of the nanoparticle magnetizations. In spite of these limitations, the energy barrier model is capable of reproducing experimental results at higher temperatures and over a wide temperature range [20]. The main objective of the present work is to study the superparamagnetic behaviour of interacting capped nanoparticles. Because capping of the surface of the nanoparticles is known to affect not only surface properties (KS ) but also the relaxation process itself by modifying interparticle interaction [21–23], a well-founded study of correlation between interaction forces and superparamagnetic relaxation requires the ability to distinguish both effects. Guided by this consideration, in a preceding study we investigated changes of nonaligned magnetic surface moments of magnetite nanoparticles with average diameters of d ≈ 3–9 nm capped with an organic surfactant using in-field M¨ossbauer spectroscopy [24]. Comparable results were found for particles with additional silica capping (see below), so we can assume minor influence of both capping materials on the magnetic surface properties and we conclude that changes in relaxation behaviour are caused only by direct variation of the particle distance. Effects of exchange interaction will not be discussed in detail, since in the case of spherical, ferrimagnetic particles with a small contact area they are assumed to be much smaller compared to contributions of magnetic dipole interactions [16]. Four magnetite (Fe3 O4 ) nanoparticle samples with different cappings were synthesized by a microemulsion method [24] and analyzed by M¨ossbauer spectroscopy, ferromagnetic resonance (FMR) measurements and various SQUID (Superconducting QUantum Interference Device) magnetometry protocols like field-cooled/zero-field-cooled (FC/ZFC), thermoremanent magnetisation measurements (TRM) and AC-susceptibility. The theory of Jones and Srivastava [18] was applied to temperature-dependent 57 Fe M¨ossbauer spectra in order to obtain the effective magnetic anisotropy energy EA as a function of temperature and interparticle interaction. Increased values of EA are found to be in general agreement with interaction energies kB T0 from AC-susceptibility, which could be described in terms of the Vogel–Fulcher law (equation (7)) and with the D −3 -law (equation (8)) that is valid for interacting magnetic dipoles. Our unique synergistic approach includes a huge regime of characteristic time windows τM ≈ 10 s–5 ns. Our results provide a concise picture of the correlation of superparamagnetic relaxation and interparticle magnetic dipole interaction.

6 · KS Keff ≈ KV + . (1) d This results in an increased anisotropy energy barrier between easy magnetic directions, so higher thermal energies are needed to cause fast superparamagnetic relaxation of the particles’ net magnetic moment. A well known equation (equation (2)) for the superparamagnetic relaxation time τ was established by N´eel [14], which includes the ratio β of anisotropy energy (EA ) to thermal energy (Eth ) and an attempt time τ0 of about 10−9 –10−13 s [15].  

EA Keff V = τ0 · exp . (2) τ = τ0 · exp Eth kB T This model is reasonable for particles of volume V with uniaxial anisotropy and an anisotropy energy given by EA = Keff V sin2 (θ ), where θ is the angle between the magnetization and the symmetry axes. The assumption of uniaxial anisotropy is generally accepted in the case of well structurally ordered nanoparticles, since even small distortions from spherical shape can lead to uniaxial anisotropy by superposition of magnetocrystalline and shape anisotropy [10, 12]. Detailed investigations of superparamagnetic effects in the presence of interparticle interaction were carried out by Dormann et al [16], leading to the interpretation of decelerated (or reduced) relaxation upon increasing interaction. These results were based on a model of a high number of metastable states of different energies caused by randomly oriented adjacent interacting particles. More recent magnetometric studies by Caruntu et al on 6.6–17.8 nm Fe3 O4 nanoparticles showed that, as the strength of the dipolar interparticle interaction decreases, the energy barrier is lowered and, consequently, the blocking temperature TB is reduced [17]. Because of the characteristic time window τM of any measurement technique, one observes discrete states of magnetization, if τM is small compared to τ and averaged states, if τM  τ . We define the blocking temperature TB by the identity of τ and τM . It is crucial to consider that the blocking temperature is dependent on the chosen measurement technique, while the relaxation time τ is a physical parameter. Using equation (2) we obtain:
 Keff V τM . (3) , where η = ln TB = kB η τ0 In M¨ossbauer spectroscopy the time constant can be approximated to τM ≈ 5 ns (η ≈ 5) using the nuclear Larmor frequency, which is small compared to the time constant of SQUID measurements of about 10–100 s (η ≈ 30). The shape of the 57 Fe M¨ossbauer spectrum depends on the superparamagnetic relaxation time τ . Therefore, it is possible to determine the temperature dependence of τ , using, e.g. the theory by Jones and Srivastava [18]. The energy-barrier model, used for the description of M¨ossbauer spectra has certain limitations. First, in the low-temperature regime, so-called collective magnetic excitations (i.e. spin-wave excitations in the nanoparticles) cannot be reproduced in that model [19]. Second, the energy-barrier model does

2. Sample synthesis and characterization 2.1. Synthesis and experimental

To synthesize magnetite (Fe3 O4 ) nanoparticles we used a water-in-oil microemulsion technique with FeCl2 and FeCl3 as iron precursors. Nanopools of water in the dispersion, 2

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stabilized by an organic surfactant (IGEPAL® CO-520) act as nanoreactors, in which the synthesis takes place. The diameters of synthesized particles can be tuned by varying the size of these nanopools through stirring or by changes in the water/surfactant ratio. A detailed description of the synthesis process of IGEPAL-capped particles was presented by Darbandi et al [24]. To produce silica (SiO2 ) capped or IGEPAL/SiO2 -capped particles TEOS-solution (tetraethylorthosilicate) is added subsequently. Four samples of IONPs were synthesized: bare particles (hereafter referred as M1), particles capped with IGEPAL (M2) and particles double-capped with IGEPAL and an additional silica shell of 3 nm (M3) or 6 nm (M4) thickness, representing functionalized IONPs as required for biomedical application. Bare particles formed a powder, whereas samples containing IGEPAL exhibited a pasty state. Transmission electron microscopy (TEM) images were analyzed to achieve information about size and shape of each set of nanoparticles, as reported in table 1. TEM characterization was carried out in a Phillips CM 12 transmission electron microscope at 120 kV. X-ray diffraction (XRD) in -2 geometry was performed with Cu-Kα radiation using a Philips PW1730 x-ray diffractometer with a graphite monochromator. Magnetometry measurements were carried out in a commercial SQUID magnetometer (Quantum Design MPMS 5S). M¨ossbauer spectra were recorded in transmission geometry at 4.2–300 K using a constant acceleration spectrometer (57 Co source, Rh matrix) and a liquid-helium bath cryostat containing superconducting coils to apply an external magnetic field up to 5 T along the γ -ray direction, if required. M¨ossbauer absorbers were nanoparticle material mixed with chemically inert boron nitride powder. The least-squares fitting procedure of the spectra1 was performed using a many state relaxation model, as described below [18]. FMR was measured at a constant frequency of 9.485 GHz using a CW Bruker E500 spectrometer with rectangular cavity in the X-band frequency regime and a helium gasflow cryostat to control the sample temperature. The signal was detected using the lock-in technique by modulating the applied magnetic field at a modulation frequency of 100 kHz. During each scan the applied field was swept from 1.6 T to 0 T.

Table 1. Mean values dTEM and dXRD of particle size distributions (magnetic cores without shell) estimated from TEM and XRD, respectively. Errors of dXRD are expected to be about 10% due to the unknown shape factor k, which is close to 1 for spherical particles.

Sample

Capping

dTEM (nm)

dXRD (nm)

M1 M2 M3 M4

— IGEPAL 3 nm SiO2 + IGEPAL 6 nm SiO2 + IGEPAL

6.3(1) 6.3(1) 6.4(1) 6.3(1)

6.2(6) 6.6(7) 6.5(7) 6.2(6)

diameter d can be described by a narrow lognormal distribution (equation (4)) with median µl and standard deviation σ . For sample M1 we obtain µl = 6.14(5) nm and σ = 0.190(8) corresponding to an average particle diameter dTEM ≈ 6.3(1) nm and a standard deviation σTEM ≈ 1.2(1) nm. 
1 (ln(d/µl ))2 p(d) = √ . (4) · exp − 2σ 2 2π σ d XRD measurements (figure 1(f )) confirm the spinel structure of the particles and provide lattice constants in the range of 0.839–0.840 nm for M2–M4 comparable to magnetite bulk samples (0.840 nm) [25] and a reduced lattice constant of about 0.837 nm for bare particles M1 being closer to that of maghemite (γ –Fe2 O3 , 0.833 nm) [26]. This can be explained by partial oxidation of sample M1. XRD patterns of M1 and M2 exhibit no other diffraction patterns but magnetite or maghemite, while silica capped particles M3 and M4 show minor additional contributions of ammonium chloride (figure 1(f ), red arrows). Based on in-field M¨ossbauer spectra (shown below), these impurities have no effect on the particles’ magnetic properties. Calculations of the particle size dXRD from XRD linewidths  were done using the Scherrer equation. Results of these calculations (table 1) are in agreement with average particle diameters obtained by TEM. The existence of Bragg reflections in figure 1(f ) proves that our IONPs are structurally well ordered. ¨ 2.3. In-field Mossbauer spectroscopy

Magnetic properties of samples M1–M4 were investigated at the atomic scale using in-field M¨ossbauer spectroscopy. M¨ossbauer spectra measured at 4.2 K in the presence of an applied magnetic field of 5 T along the propagation of the γ -rays are shown in figure 2. They consist of three sextetsubspectra corresponding to Fe3+ and Fe2+ in octahedral (B-site) and Fe3+ in tetrahedral (A-site) surroundings, as expected for magnetite nanoparticles at low temperatures. The spectra were least-squares fitted with Lorentzian lineshape and resulting linewidths (FWHM) of 0.43 mm s−1 (Fe3+ on A-sites), 0.65 mm s−1 (Fe3+ on B-sites) and 0.85 mm s−1 (Fe2+ on B-sites). Resulting values (table 2) for the isomer shift δ and effective magnetic field Beff of both Fe3+ -sextets are in agreement with literature values [24, 27, 28]. M¨ossbauer parameters of the Fe2+ -sextet were obtained from measurements on larger particles with superior content of magnetite and δ and Beff for Fe2+ were fixed for the fitting of the

2.2. Structural characterization

TEM micrographs of sample M1 (figure 1(a)) show a gently agglomerated assembly of bare IONPs, while there is no indication of agglomeration effects in IGEPAL-capped particles M2 (figure 1(b)). Figure 1(c) reveals a weak contrast of the silica shell, so the shell thickness can be directly estimated from TEM micrographs. More than 60% of the particles were found to be single-core particles, so interaction forces are expected to correlate directly with the capping thickness. All samples consist of particles of spherical shape with some irregularities and nearly uniform size being independent of the capping material. The particle core 1 Calculations were performed using the ‘Pi’ program package by U von H¨orsten (www.uni-due.de/physik/wende/hoersten/home.html).

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Figure 1. TEM micrographs of bare and capped 6 nm IONPs: (a) M1, (b) M2, (c) M3 and (d) M4 (for details see table 1). Particle size distributions are described by lognormal-functions (e) for core and total particle diameter, respectively. (f ) Background-corrected XRD patterns of various IONPs with theoretical Bragg reflections (bar diagram) of bulk magnetite (gray) and bulk maghemite (black). Red arrows show additional reflections attributed to ammonium chloride as an artifact. XRD data of sample M2 are taken from [24].

spectra shown in figure 2. The nuclear quadrupole level shift 2 of the Fe2+ subspectrum was measured to be −0.59 mm s−1 in larger particles and also used as a fixed parameter, whereas the nuclear quadrupole level shift of the Fe3+ subspectra was negligible and set to zero. The relative spectral area A of Fe2+ on octahedral B-sites is small as compared to those of the other subspectra. This effect, which is more pronounced for the bare particles in M1, is attributed to beginning oxidation to maghemite of bare particles (M1), in accordance with lattice parameters determined from XRD patterns, whereas magnetite is assumed to be the predominant phase in the capped particles [29]. The fitting procedure is presented in more detail in [24]. We found δ and Beff to be comparable for M1–M4 (table 2), which proves similar structural and magnetic states. Spin canting angles θC between the orientation of the magnetic moments and the direction of the γ -ray (or the applied-field direction) were obtained from the ratio R23 of spectral areas of lines 2 and 3 using the following equation [30]:   4 − R23 θC = arccos . (5) 4 + R23

in M1–M4. Consequently, the surface anisotropy KS and, therefore, the effective magnetic anisotropies Keff are expected to be similar for all of our samples, not considering interaction effects. Furthermore, small canting angles of 15–20◦ indicate a limited number of partially disaligned surface spins, so there is no need for more complex models of the magnetic structure, e.g. spin-glass-like core-shell structures [34]. 3. Results and discussion 3.1. SQUID magnetometry (FC-ZFC, TRM, AC)

To study the particle relaxation behaviour we carried out FC-ZFC magnetization measurements in an applied magnetic field of 10 mT. Results are shown in figure 3. We observed a sharp maximum in mZFC for M3 and M4 and broadened magnetization curves with maxima shifted to higher temperatures for stronger interacting particles in samples M1 and M2. The temperature Tmax of the maximum in the ZFC-magnetization mZFC is often identified as the blocking temperature TB . This is problematic, since the distribution of particle diameters generates a shift of Tmax to higher values, so the deviation between Tmax and TB depends on the particle size distribution [15]. Tirr represents the temperature where mZFC and mFC coincide and, therefore, represents the blocking temperature of the biggest particles in the sample. At temperatures below Tmax mFC does not clearly increase for M1 and M2, which can be attributed to interaction effects,

Spin canting gives information on surface magnetic moments non-aligned towards the magnetic field [30]. Size dependent analysis of canting angles θC averaged over all subspectra verified non-aligned surface spins in our IONPs [24], contrary to volume-canting by structural disorder [17, 31–33]. The spin canting angles (table 2) have comparable values for all samples. This indicates the existence of similar states of surface disorder 4

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Table 2. M¨ossbauer parameters determined from M¨ossbauer spectra at 4.2 K in an applied magnetic field of 5 T: isomer shift δ relative to α-iron at room temperature, canting angle θC , effective magnetic field Beff and relative spectral area A. Marked values (∗ ) were fixed in the fitting procedure. Data of sample M1 and M2 are taken from [24].

A3+

B 3+

M1 0.50(1) 22(2) 47.3(1) 60(1) M3 δ (mm s−1 ) 0.38(1) 0.51(1) θC ( ◦ ) 15(2) 18(2) 55.0(1) 47.4(1) Beff (T) A (%) 31(1) 53(1) δ (mm s−1 ) θC ( ◦ ) Beff (T) A (%)

0.37(1) 15(3) 54.9(1) 30(1)

B 2+

A3+

0.88∗ 13(13) 43.6∗ 10(1)

0.38(1) 15(3) 54.7(1) 30(1)

0.88∗ 20(6) 43.6∗ 17(1)

B 3+

M2 0.52(1) 21(2) 47.1(1) 56(1) M4 0.38(1) 0.50(1) 12(3) 19(2) 54.9(1) 47.2(1) 31(1) 55(1)

B 2+ 0.88∗ 0(16) 43.6∗ 14(1) 0.88∗ 20(7) 43.6∗ 15(1)

Figure 3. Temperature dependent FC-ZFC magnetizations mFC and mZFC of samples M1–M4 in an applied magnetic field of 10 mT, normalized to mZFC (Tmax ) (dotted lines).

Figure 2. M¨ossbauer spectra of M1–M4 measured at 4.2 K in an

applied magnetic field of 5 T parallel to the propagation of the γ -rays: experimental data (dots) and least-squares fits (red lines). The three fitted subspectra correspond to Fe3+ ions on A sites (cyan) and B sites (green) and Fe2+ ions on B sites (blue), respectively. Vertical arrows indicate the positions of the M¨ossbauer lines attributed to ( m = 0) transitions. Data of sample M1 and M2 are taken from [24].

magnetization decay, valid in the weak-interaction regime. It has been demonstrated that the distribution of anisotropy energies p(EA ) can be calculated from mTRM by a method called T ln(t/τ0 ) scaling [36], as follows: Assuming that the magnetization decay corresponds to the relaxation processes of particles with an anisotropy energy smaller than kB T ln(t/τ0 ), experimental data of mTRM measured at different times and temperatures can be normalized to their maximum value at 4.2 K and plotted versus T ln(t/τ0 ) in order to obtain continuous magnetization curves (figure 4). The inflection point of mTRM corresponds to those particular combinations of time and temperature, where the thermal energy exceeds the anisotropy energy barrier of many particles and, therefore, corresponds to the maximum of the distribution p(EA ). At this energy a high fraction of IONPs defreezes from the blocked state upon heating. We obtain p(EA ) from the derivative of this normalized magnetisation curve as described by equation (6), analogous to previous calculations for interacting and non-interacting

suppressing further magnetic orientation at low temperatures [35]. Values of Tmax and Tirr are given in table 3. For a more detailed study of anisotropy energies we performed measurements of the thermoremanent magnetization mTRM at temperatures from 5 to 35 K. Samples were field-cooled in an applied field of 10 mT to the measurement temperature in order to observe the time-dependent decrease of magnetization by relaxation when the field was switched off. Because the magnetization is monitored for t ≈ 1 h, even a slight decrease by low temperature relaxation can be detected. Therefore, TRM is a more valuable tool for demagnetization measurements at temperatures T
TB as compared to FC-ZFC magnetization. Following equation (2), τ −1 can be understood as the probability that the magnetization vector has switched within a defined time interval. Thus, τ −1 establishes a correlation between temperature, measurement time t and 5

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Table 3. Relaxation parameters obtained from different methods. Core-to-core distance D from TEM. Characteristic temperatures Tmax and Tirr were measured using FC-ZFC-magnetometry and time constants τ0 were obtained from TRM measurements. Average anisotropy energies EA were determined from TRM, FMR and M¨ossbauer (MS) results. T0 from AC-susceptibility was calculated using the Vogel–Fulcher law.

FC-ZFC

TRM

Sample

TEM D (nm)

Tmax (K)

Tirr (K)

τ0 (ps)

EA (meV)

MS EA (meV)

FMR EA (meV)

AC T0 (K)

M1 M2 M3 M4

6.3(1) 7.7(3) 13.3(1) 18.7(2)

92(2) 57(1) 40(1) 38(1)

102(2) 85(2) 70(2) 68(2)

150(120) 100(90) 110(60) 80(50)

76(2) 62(3) 57(1) 55(1)

69(10) 54(6) 43(4) 39(4)

— — — 39(2)

100(2) 43(2) 17(2) 8(2)

nanoparticles [37, 38]. d(mTRM )  .  p(EA ) = −  d kB T ln ( t+t) τ0

(6)

The correction factor t in equation (6) is attributed to the time delay of about 60 s in which the applied field is driven to zero. This correction produces no considerable change in mTRM or in the fitting parameters, since the total measurement time is much longer (about an hour). Figure 4 reveals that the experimental data can be described nicely by a lognormal function p(EA ). This is not unexpected, since we obtain a linear correlation between the particle volume V and EA , if we assume only minor variations of Keff within the sharp particle size distribution (see below). Therefore, mTRM was fitted by an integrated lognormal function (figure 4, solid lines) to calculate the corresponding function p(EA ) (dashed lines). Time constant τ0 and effective anisotropy energies EA obtained from these fits are shown in table 3. Since interaction effects are expected to influence the height of the energy barrier rather than the magnetic anisotropy of the particles themselves, it is more reasonable to discuss changes of EA rather than changes of Keff in terms of interparticle interaction [21]. Nevertheless, values of Keff ≈ 60–90 kJm−3 , calculatued from the average anisotropy energy EA , assuming a (core) particle volume of about 140 nm3 , are about one order larger than comparable magnetocrystalline anisotropies of K1 ≈ −11 kJm−3 at room temperature [39]. In figure 4 we observe broader distributions p(EA ) and increasing EA (dotted, vertical lines) with increasing interparticle interaction. The assumption above of only minor variations of Keff within our sharp particle size distribution is justified, since, for example for sample M4, the lognormal form of p(V) has an observed (TEM) width of σ = 0.46(4), while the lognormal distribution p(EA ) has an observed (TRM) width of σ = 0.43(1). Both values agree with each other within error margins. If Keff would show a remarkable distribution within our (narrow) range of particle sizes, then, as a consequence of the relation EA = Keff V , p(EA ) should be broader than p(V), which is not the case, however. Therefore, within the narrow size distribution of our nanoparticles, Keff in equation (1) can be assumed to be approximately constant. To gather information on relaxation behaviour within a wide range of blocking temperatures, the AC susceptibility was measured at temperatures of 5–200 K. The AC-field amplitude was µ0 HAC = 0.4 mT with frequencies fAC in the range of 2.3

Figure 4. Normalized thermoremanent magnetization mTRM of samples M1–M4 (full symbols) versus relative temperature, least-squares fitted by integrated lognormal functions (according to equation (6), solid lines, left scale). Also shown are lognormal distributions of anisotropy energies p(EA ) (dashed lines, right scale) and mean values of p(EA ) (dotted vertical lines).

to 997 Hz. Figure 5 (right inset) shows the real component χ  of the AC-susceptibility at a frequency of fAC = 2.3 Hz for all samples. In accordance with our ZFC-magnetization results and theoretical calculations [40] we observe a broadening of the maximum of χ  at temperature Tmax and a shift of Tmax to higher values with increasing interparticle interaction. In order to provide additional evidence that the nanoparticles in sample M4 experience no (or weak) interparticle interactions, we prepared a reference sample with particles of M4 dispersed in isopropyl alcohol, thus increasing the core-to-core distance compared to the M4 powder sample. Comparing the AC-susceptibility χ  of the particles in solution before and after drying (figure 5, left inset) we observe only a marginal increase of Tmax for the dried solution, which directly demonstrates that the nanoparticles in M4 are sensing very weak interparticle interactions, in agreement with our assumption. The variation of Tmax for samples M1–M4 versus fAC is shown in figure 5 in an Arrhenius-like plot, i.e. ln(τAC ) −1 versus Tmax . The characteristic time window τM of the AC measurement is given [41] by τAC = (2πfAC )−1 . Using an average value of τ0 = 1.1(4)·10−10 s, as obtained from TRM measurements (table 3), least-squares fits of the AC-susceptibility data in figure 5 reveal that the experimental data follow nicely the phenomenological Vogel–Fulcher law, 6

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a consequence these techniques are unable to distinguish between different fractions of particles in a sample performing fast or slow magnetic relaxation with the same net decrease or increase of magnetization. However, 57 Fe M¨ossbauer relaxation studies provide this ability, since the spectral shape depends highly on the relaxation rate, illustrated by sextetstructures for blocked particles and broad intermediate spectra or doublet-structures for slow or fast superparamagnetic relaxation, respectively. Different models have been established to describe these structures theoretically. Under the assumption of uniaxial anisotropy it is reasonable to use as a starting point a model with only two states between which the moment flips [44] with a defined relaxation time, given, e.g. by equation (2). As a disadvantage, these models are unable to reproduce effects of collective excitation at low temperatures with small angles relative to the easy axis, which lead, however, to reduced hyperfine magnetic fields Bhf . Several such states with different occupation propabilities create a distribution of Bhf , which can only be described by a model including a larger number of possible orientations of the particles’ magnetic moment. Due to these reasons, we have reproduced temperature-dependent M¨ossbauer spectra by a many-state relaxation model developed by Jones and Srivastava [18] in combination with a lognormal distribution p(EA ) of anisotropy energies to describe effects of the particle volume distribution. The sextet linewidth and the mean value of β = Keff V /kB T were used as free parameters, while the standard deviation of p(EA ) was fixed to σβ = 0.43, as obtained from the function p(EA ) of TRM results on sample M4 (figure 4). This sample is expected (and has been shown) to exhibit negligible dipolar interaction effects and, therefore, σβ includes effects of the particle size distribution only, which is sharp enough to neglect the influence of the particle size on the effective anisotropy constant Keff (see section 3.1). Experimental results and calculated relaxation spectra are shown in figure 6. Low temperature spectra, taken between 4.2
T < 100 K without applied magnetic field, were fitted with three sextet subspectra consistent with figure 2, in order to reproduce the spectral fine structure and with linewidths as given in section 2.3. In this low temperature range, the many-state model explained above yields large values of β representing nanoparticles, which are almost magnetically blocked. The static M¨ossbauer parameters of each subspectrum were obtained from the in-field spectra (table 2). The observed asymmetric non-Lorentzian lineshapes with pronounced inner shoulders can be explained by collective excitations of the particle moments which are close to their preferred magnetic orientation at moderate thermal energies below EA [45]. Collective excitations lead to a progressive decrease of Bhf with T, depending on the particle volume. One observes reasonable agreement between experimental and calculated M¨ossbauer spectra. Magnetite is expected to indicate a structural change at the temperature TV of the Verwey transition, which is about 120 K for bulk samples. At higher temperatures one observes only two sextet subspectra corresponding to Fe3+ on tetrahedral sites and an intermediate state of Fe2.5+ on octahedral sites, which can be attributed to fast electron hopping between

−1 Figure 5. Arrhenius graph of ln(τAC /s) versus experimental Tmax

from AC susceptibility, least-squares fitted by the Vogel–Fulcher law (solid lines), using the average value of τ0 = τ0,AC = 0.11(4) ns from TRM results (table 3). The dashed line represents the calculated trend for non-interacting particles: T0 = 0 K in equation (7). Right inset: exemplary T-dependence of the real component (χ  ) of the AC-susceptibility at fAC = 2.3 Hz for samples M1–M4. The left inset shows χ  (T) of nanoparticles of sample M4 dispersed in isopropyl alcohol compared to χ  (T) of these nanoparticles in dried solution, at fAC = 11 Hz. Also shown are M¨ossbauer data points at ln(τAC /s) = −19.1 for M1–M4, as explained in the text.

describing spin glasses as well as superparamagnetic particles in the weak interaction limit [42, 43]: τ = τ0 · exp(

Keff V ). kB · (T − T0 )

(7)

T0 provides an estimate of the interparticle interaction strength, being useful for a comparison between different samples [21]. T0 values obtained from fitting the AC-susceptibility data of samples M1–M4 are given in table 3. The interparticle interaction T0 is found to increase systematically from 8(2) K for M4 (weak or negligible interaction) to 100(2) K for M1 (relatively strong interaction), as expected from the different shell thicknesses of the nanoparticles in M1–M4. In figure 5 we have also indicated data points at ln(τAC /s) = −19.1, obtained from M¨ossbauer spectroscopy (see section 3.2). These M¨ossbauer blocking temperatures were calculated using equation (7) with (i) EA = Keff V = 39(4) meV, as obtained from M¨ossbauer spectra of sample M4 (see section 3.2), representing particles without (or very weak) interparticle interaction T0 and (ii) T0 values for each sample obtained from the AC-susceptibility data (table 3). In figure 5, in view of the very different time scales τM of both methods (≈5 ns versus ≈0.1–100 ms) one observes reasonable agreement in the general trend for the relaxation-time dependence of the blocking temperature. ¨ 3.2. Zero-field Mossbauer spectroscopy

ZFC-FC and TRM results describe relaxation behaviour with respect to changes of the total magnetization. As 7

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Figure 6. Typical spectra for a M¨ossbauer relaxation study at temperatures between 4.2–300 K. The three fitted subspectra at low temperatures (T < 100 K) correspond to the three Fe states in magnetite according to figure 2 (dots: experimental data, red curve: least-squares fit according to the many-state relaxation model). The fitting method for relaxation spectra measured at higher temperatures (T > 100 K) is explained in the text. From M4 to M1, we can observe increasing deviation between experimental and calculated spectra, which is attributed to increasing interparticle interaction.

Fe3+ and Fe2+ [46], although this interpretation is a matter of debate [47]. Since we cannot distinguish different sextet subspectra at higher temperatures (T > 100 K) due to severe line broadening by relaxation, we use a more simple model consisting of only one sextet contribution starting at 100 K with a linewidth of 0.64 mm s−1 . Spectra of samples M3 and M4 include contributions of a central superparamagnetic doublet with a quadrupole splitting of about EQ = 0.65 mm s−1 , beginning to appear at ≈50–90 K upon heating and show blocked and superparamagnetic contributions in coexistence at higher temperatures, as can be seen in the experimental data (figure 6, M3 and M4). Doublet contributions of these spectra could not be reproduced, since the applied many-state model does not include effects of quadrupole splitting. Because of this reason, components of the calculated spectrum corresponding to small particles with low anisotropy energy and low blocking temperature are represented by a central singlet, which was fitted to match the doublet contribution in the experimental spectra. Apart from this, the presented model is able to reproduce very well the lineshape of experimental spectra of M3 and M4 at all temperatures: The line asymmetry due to the collective excitations as well as the increasing spectral area of the superparamagnetic contribution are well described by the calculations. This proves the correctness of the applied lognormal standard deviation σβ = 0.43. In the calculations

the temperature dependent decrease of hyperfine magnetic fields Bhf was varied for all samples in accordance with Bhf (T ) of sample M4 (sample with negligible interparticle interaction), assuming similar intrinsic magnetic properties for all samples, as verified by in-field M¨ossbauer spectra (section 2.3). Theoretical spectra of the applied many-state model exhibit stronger deviations in the case of the samples M1 and M2, due to increased interparticle interaction in these samples, as compared to M3 and M4. Interparticle interaction is not considered explicitly in the many-state model. Upon heating samples M1 and M2, the first indications of superparamagnetic contributions, which can be attributed to the smallest particles in the samples, do not appear below 120 K (M2) and 200 K (M1). Interparticle interaction forces conserve the sextet structure, although its shape is deformed by thermal collective excitation, represented by larger values of β in the calculated spectra. Between 150–200 K the sextets start to collapse into a broad singlet-like spectrum, which is often assigned to relaxation rates of 108 –109 s−1 [48] and is similar for all samples near room temperature. Comparison of results from M¨ossbauer spectroscopy (figure 6) and TRM (figure 4), for instance for samples M1 and M2, reveals the importance of the different characteristic measurement times τM of the two methods (τM ≈ 5 ns for M¨ossbauer spectroscopy versus τM ≈ 10 s for TRM). As mentioned above, the onset of superparamagnetism was found to appear at 120 K (M2) and 200 K (M1) in 8

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Figure 8. Inverse interparticle distance D −3 (black full square) and interaction energies T0 (red open square) versus average anisotropy energy EA from M¨ossbauer spectroscopy. Solid lines represent linear regressions.

Figure 7. Average anisotropy energies EA calculated from M¨ossbauer spectra using the many-state relaxation model. Error bars are given exemplarily for few data points.

T-independent behaviour of EA was presented by van Lierop and Ryan, who also applied the model by Jones and Srivastava [18] including a particle size distribution, to study relaxation dynamics in ferrofluids [20]. Since we do not observe discontinuities at about 100 K in the data of figure 7, we can assume that fitting with only one sextet contribution at T  100 K provides reliable results in EA comparable to the fitting routine with three sextets at lower temperatures. Although results of M¨ossbauer spectroscopy and TRM both display an increase of EA with decreasing interparticle distance (table 3), calculated M¨ossbauer spectra exhibit lower anisotropy energies EA due to differences in τ0 in the applied fitting routines. In case of M¨ossbauer spectroscopy, τ0 = 1.37 ns was obtained from the relaxation fitting parameter R = 10 mm s−1 in the theory by Jones and Srivastava [18]. Comparison of interaction parameters T0 from ACsusceptibility and anisotropy energies EA from M¨ossbauer spectroscopy with the average interparticle core-to-core distance D reveals that T0 is in general proportional to D −3 (figure 8), as one would expect for the magnetic dipole interaction energy in a system of separated particles with a net magnetic moment (as given by equation (8) [15]). For estimating D, we assumed direct (shell) surface contact for M3 and M4, while we estimated D for sample M2 from the smallest core-to-core distances observed in the TEM images. Surprisingly, a similar behaviour is observed for EA , which provides evidence that T0 as a parameter of interparticle interaction energy can be determined from the change in anisotropy energy EA calculated from M¨ossbauer spectra within the weak interaction regime. Using equation (8) the particles’ net magnetic moment µ can be estimated to be about 7300 µB , corresponding to a particle volume of about 130 nm3 , in agreement with our TEM results.

the M¨ossbauer spectra. On the other hand, the TRM measurements (figure 4), which were performed in the Trange of ≈5 K to 35 K, indicate a strong reduction of mTRM by 50% (M1) and 65% (M2) upon heating from ≈5 K to 35 K, owing to superparamagnetic relaxation. The fact that no indication of superparamagnetism is observed in the M¨ossbauer spectra within that low-temperature range means that within 5 K
T
35 K the magnetizations of all particles in the size distribution p(V) show nearly static relaxation times τ for M¨ossbauer spectroscopy (τ  τM ≈ 5 ns), while a large fraction of the particle magnetizations within p(V) relax fast as compared to the characteristic TRM measurement time (τ  τM ≈ 10 s) and superparamagnetism is observed by TRM. The absence of a superparamagnetic doublet (corresponding to absence of fast relaxation processes) proves that stronger interparticle interactions in samples M1 and M2 lead to decelerated rates of magnetic relaxation, as compared to the case of samples M3 and M4 (for which the first doublets were found to appear at lower temperatures of ≈90 K (M3) and 50 K–65 K). Figure 7 shows average anisotropy energies EA versus T obtained from calculations (many-state model) of the spectra shown in figure 6. We only considered values obtained from spectra between ca. 50 K and 180 K, where considerable changes of the lineshape by superparamagnetic relaxation cannot be mistaken for the temperature-dependent decrease of Bhf described by Brillouin-like behaviour. Simulations of theoretical spectra matching the experimental results of sample M4 yield a nearly T-independent value of EA ≈ 39(4) meV (Keff ≈ 45(4) kJm−3 ) in the range of 20 K–175 K, representing an approximate value for particles with negligible interaction effects. EA shows no decrease with increasing temperature as would be expected for the magnetocrystalline anisotropy constants K1 (T) and K2 (T). This observation might be related to the dominating and peculiar effect of surface anisotropy. We observe similar T-behaviour (within the error bars) for samples M1–M3, however, with larger anisotropy energies up to 69(10) meV for sample M1. A similar approximately

T0 =

µ0 µ2 . 4π kB D 3

(8)

3.3. Ferromagnetic resonance (FMR)

Like M¨ossbauer spectroscopy, ferromagnetic resonance is a ‘fast’ technique with a characteristic time window comparable 9

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Figure 10. Resonance field µ0 Hres (circles) and calculated anisotropy field µ0 HA (squares) of sample M4. Dashed lines are a guide to the eye, solid lines designate the low temperature approximation µ0 HA ≈ 174(6) mT and the resonance field in the 0 superparamagnetic regime µ0 Hres ≈ 342(3) mT.

Figure 9. FMR spectra of sample M4 between 5.2–300 K. The shift of the resonance field µ0 HRes (red, solid line) is used to calculate the effective anisotropy constant Keff .

4. Conclusions

to τ0 . Thus, the presented M¨ossbauer results may be verified independently using FMR. Figure 9 shows a set of FMR spectra of sample M4 measured in a range of temperatures between 5.2–300 K at a microwave frequency of 9.485 GHz, where the derivative of the imaginary part of the susceptibility ∂χ  /∂H is proportional to the absorbed microwave power. We observe a pronounced shift of the resonance field, µ0 Hres , to larger values at higher temperatures. Hres was estimated from the inflection point of ∂χ  /∂H . The resonance field in 0 , the superparamagnetic regime at high temperatures, µ0 Hres is extrapolated to 342(3) mT from figure 10. Following the approach used by Antoniak et al [49], we are able to estimate the anisotropy field HA using equations (9) and (10), as shown in figure 10. 
 0.44 HA 1.25 0 , (9) Hres = Hres 1 − 0 Hres

A multi-technique approach with measurement time windows in the range of ≈5 ns–10 s was used to analyze different aspects of superparamagnetic relaxation behaviour of interacting bare and capped iron oxide nanoparticles. The interparticle interaction was tailored via the tuned shell thickness, while M¨ossbauer spectra in an applied magnetic field verified comparable properties of their magnetic cores without significant dependence on capping material or thickness. SQUID magnetization measurements (FC-ZFC, TRM) have shown that the macroscopic magnetization of a superparamagnetic system can be well described by a broadened lognormal distribution of anisotropy energies, while we concluded from zero-field T-dependent M¨ossbauer spectra and measurements of the AC-susceptibility that the Vogel–Fulcher law is a reasonable way to reproduce relaxation times. Temperatures Tmax of the AC-susceptibility plotted in an Arrhenius-graph against logarithmic τAC allowed to calculate an approximate value of the interparticle interaction parameter T0 . M¨ossbauer blocking temperatures obtained on a fast time scale of 5 ns are consistent with the AC-susceptibility data. T0 values could be described by a D −3 -law and were found to be proportional to the change of the anisotropy energies EA calculated from M¨ossbauer relaxation spectra by use of the many-state magnetic relaxation model [18]. Therefore, the interparticle interaction strength T0 may be estimated from M¨ossbauer spectra. EA values were observed to be independent of temperature. Ferromagnetic resonance data yielded an average effective anisotropy constants Keff of 44(2) kJm−3 near the blocked state, in good agreement with M¨ossbauer results of 45(4) kJm−3 for weakly-interacting particles. This observation displays a dramatic increase of the effective anisotropy Keff in the nanoparticles, as compared to magnetocrystalline anisotropy in bulk Fe3 O4 .

2Keff . (10) M One should notice that the temperature-dependent anisotropy field HA (T ) obtained from this method still includes superparamagnetic relaxation effects (unlike EA (T) in figure 7). Thus, HA (T ) exhibits a shape similar to FC- and thermoremanent magnetization (figures 3 and 4). At low temperatures, HA converges to a value of about 174(6) mT. If we assume that the fraction of magnetic moments exhibiting undisturbed relaxation behaviour is predominantly given by bulk-like core magnetic moments instead of canted surface spins, the magnetization in presence of an applied magnetic field of about 300 mT at low temperatures can be approximated by the saturation magnetization of bulk magnetite [50], MS ≈ 510 kJm−3 . Under this assumption, from equation (10) we obtain an effective anisotropy constant of Keff ≈ 44(2) kJm−3 or EA ≈ 39(2) meV for sample M4, in good agreement with the M¨ossbauer results (table 3). where

HA =

10

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[23] Vestal C R, Song Q and Zhang Z J 2004 J. Phys. Chem. B 108 18222 [24] Darbandi M, Stromberg F, Landers J, Reckers N, Sanyal B, Keune W and Wende H 2012 J. Phys. D: Appl. Phys. 45 195001 [25] Haavik C, Stølen S, Fjellvåg H, Hanfland M and H¨ausermann D 2000 Am. Mineral. 85 514 [26] Pecharrom´an C, Gonz´alez-Carre˜no T and Iglesias J E 1995 Phys. Chem. Minerals 22 21 ´ Sz¨ucs I, Gubicza J and [27] D´ezsi I, Fetzer Cs, Gombk¨ot¨o A, Ung´ar T 2008 J. Appl. Phys. 103 104312 [28] Roca A G, Marco J F, del Puerto Morales M and Serna C J 2007 J. Phys. Chem. C 111 18577 [29] Warland A, Antoniak C, Darbandi M, Weis C, Landers J, Keune W and Wende H 2012 Phys. Rev. B 85 235113 [30] Coey J M D 1971 Phys. Rev. Lett. 27 1140 [31] Morales M P, Serna C J, Bødker F and Mørup S 1997 J. Phys.: Condens. Matter 9 5461 [32] Lima E Jr, Brandl A L, Arelaro A D and Goya G F 2006 J. Appl. Phys. 99 083908 [33] Kubickova S, Niznansky D, Morales Herrero M P, Salas G and Veypravova J 2014 Appl. Phys. Lett. 104 223105 [34] Kodama R H, Berkowitz A E, McNiff E J and Foner S 1996 Phys. Rev. Lett. 77 394 [35] Mamiya H, Nakatani I and Furubayashi T 1998 Phys. Rev. Lett. 80 177 [36] Labarta A, Iglesias O, Balcells L and Badia F 1993 Phys. Rev. B 48 10240 [37] Iglesias O, Badia F, Labarta A and Balcells Ll 1996 Z. Phys. B 100 173 [38] Pierre T G St, Gorham N T, Allen P D, Costa-Kr¨amer J L and Rao K V 2001 Phys. Rev. B 65 024436 [39] Syono Y and Ishikawa Y 1963 J. Phys. Soc. Japan 18 1230 [40] Andersson J O, Djurberg C, Jonsson T, Svedlindh P and Nordblad P 1997 Phys. Rev. B 56 13983 [41] Gittleman J I, Abeles B and Bozowski S 1974 Phys. Rev. B 9 3891 [42] Dormann J L, Spinu L, Tronc E, Jolivet J P, Lucari F, D’Orazio F and Fiorani D 1998 J. Magn. Magn. Mater. 183 255 [43] Shtrikman S and Wohlfarth E P 1981 Phys. Lett. A 85 467 [44] Blume M and Tjon J A 1968 Phys. Rev. 165 446 [45] Mørup S 1983 J. Magn. Magn. Mater. 37 39 [46] K¨undig W and Hargrove R S 1969 Solid State Commun. 7 223 [47] Walz F 2002 J. Phys.: Condens. Matter 14 R285 [48] Wickmann H H, Klein M P and Shirley D A 1966 Phys. Rev. 152 345 [49] Antoniak C, Lindner J and Farle M 2005 Europhys. Lett. 70 250 [50] Ashcroft N W and Mermin N D 1976 Solid State Physics (Philadelphia: Saunders College)

Acknowledgments

The authors are grateful to U von H¨orsten for his expert technical assistance and Dr Carolin Schmitz-Antoniak for helpful discussions of the ferromagnetic resonance results. This work was supported by DFG (SPP 1681, WE2623/7-1 and WE2623/3-1). References [1] Dennis C L, Jackson A J, Borchers J A, Hoopes P J, Strawbridge R, Foreman A R, van Lierop J, Gr¨uttner C and Ivkov R 2009 Nanotechnology 20 395103 [2] Maier-Hauff K, Ulrich F, Nestler D, Niehoff H, Wust P, Thiesen B, Orawa H, Budach V and Jordan A 2011 J. Neurooncol. 103 317 [3] Okuhata Y 1999 Adv. Drug Deliv. Rev. 37 121 [4] Neuberger T, Sch¨opf B, Hofmann H, Hofmann M and von Rechenberg B 2005 J. Magn. Magn. Mater. 293 483 [5] Gupta A K and Gupta M 2005 Biomaterials 26 3995 [6] Fang C, Veiseh O, Kievit F, Bhattarai N, Wang F, Stephen Z, Li C, Lee D, Ellenbogen R G and Zhang M 2010 Nanomedicine 5 1357 [7] Chantrell R W, Coverdale G N, El Hilo M and O’Grady K 1996 J. Magn. Magn. Mater. 157 250 [8] Labaye Y, Crisan O, Berger L, Greneche J M and Coey J M D 2002 J. Appl. Phys. 91 8715 [9] Iglesias O and Labarta A 2001 Phys. Rev. B 63 184416 [10] P´erez N, Guardia P, Roca A G, del Puerto Morales M, Serna C J, Iglesias O, Bartolom´e F, Garc´ıa L M, Batlle X and Labarta A 2008 Nanotechnology 19 475704 [11] Kodama R H and Berkowitz A E 1999 Phys. Rev. B 59 6321 [12] Shendruck T N, Desautels R D, Southern B W and van Lierop J 2007 Nanotechnology 18 455704 [13] Bødker F, Mørup S and Linderoth S 1994 Phys. Rev. Lett. 72 282 [14] N´eel L 1949 Ann. Geophys. 5 99 [15] Mørup S, Hansen M F and Frandsen C 2011 Magnetic nanoparticles Comprehensive Nanoscience and Technology ed D Andrews, G Scholes and G Wiederrecht, vol 1 (Amsterdam: Elsevier) pp 437–91 [16] Dormann J L, Bessais L and Fiorani D 1988 J. Phys. C: Solid State Phys. 21 2015 [17] Caruntu D, Caruntu G and O’Connor C J 2007 J. Phys. D: Appl. Phys. 40 5801 [18] Jones D H and Srivastava K K P 1986 Phys. Rev. B 34 7542 [19] Mørup S and Hansen B R 2005 Phys. Rev. B 72 024418 [20] van Lierop J and Ryan D H 2001 Phys. Rev. B 63 064406 [21] Dormann J L, Fiorani D and Tronc E 1999 J. Magn. Magn. Mater. 202 251 [22] Hansen M F and Mørup S 1998 J. Magn. Magn. Mater. 184 262

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