The remanence ratio in CoFe2O4 nanoparticles with approximate single-domain sizes

Approximately single-domain-sized 9-, 13-, and 16-nm CoFe2O4 nanoparticles are synthesized using the thermal decomposition of a metal-organic salt. By means of dilution and reduction, the concentration, moment, and anisotropy of nanoparticles are changed and their influence on the magnetic properties is investigated. The relation of M r/M s ∝ 1/lgH dip is observed, where M r/M s is the remanence ratio and H dip is the maximum dipolar field. Especially, such relation is more accurate for the nanoparticle systems with higher concentration and higher moment, i.e., larger H dip. The deviation from M r/M s ∝ 1/lgH dip appearing at low temperatures can be attributed to the effects of surface spins for the single-phase CoFe2O4 nanoparticles and to the pinning effect of CoFe2O4 on CoFe2 for the slightly reduced nanoparticles. Graphical Abstract Approximately single-domain-sized 9-, 13-, and 16-nm CoFe2O4 nanoparticles were synthesized and then the concentration, moment, and anisotropy of these NPs were changed. The correlation of M r/M s ∝ 1/lgH dip was observed, independent of the size, concentration, moment, and anisotropy, and especially, such correlation is more accurate for the nanoparticle systems with higher concentration or moment, i.e., stronger dipolar interaction, which has not been reported before as far as we know.


Background
Nanoscale magnetic materials often exhibit novel properties, differing from those of their bulk polycrystalline counterparts [1][2][3][4][5], as a result of several effects including the finite size effect, surface effect, and interparticle interaction [6][7][8]. These effects affect the magnetic properties and the magnetic ordering state of nanoparticles (NPs) individually, and sometimes synergetically which usually occurs in the dense magnetic NPs. One typical phenomenon of the size effect is that the coercivity (H c ) reaches the maximum as the particle size (D) decreases to a single-domain critical dimension D c , and then reduces monotonically to zero when D is further decreased to a certain size below D c [9]. Concomitantly, the NPs exhibit the superparamagnetic behavior with the theoretical remanence (M r ) to saturation (M s ) magnetization ratio (M r /M s ) being zero. Surface spin is another factor to affect the magnetic properties for the nano-sized magnetic materials. The total magnetization of a nanoparticle composes of the surface and core spins [10,11], which is known as the core-shell magnetization model. Surface spins reduce the magnetization of a magnetic nanoparticle due to the disorder of spins at the nanoparticle surface, and the disordered surface spins lower the critical magnetic ordering temperature of magnetic nanoparticles compared with that of the bulk material [12]. Below a certain temperature, the canted surface spins freeze into a spin-glass-like state and the hysteresis loops obtained after a field cooling (FC) shift, as a consequence of the unidirectional anisotropy resulting from the coupling between the disordered surface layer and core spins [6,13,14]. Furthermore, the interaction between the surface spins of different particles enhances the effective anisotropy [7,15], making the surface anisotropy constant many orders higher than that of the bulk material [16].
Apart from surface spins, the interparticle dipolar interaction (IPDI) widely exists in the magnetic NPs; it plays a complex role in magnetic properties. For example, it has been suggested that the IPDI enhances H c because of additional induced anisotropy [17], while the opposite conclusion has also been observed [18]. A strong IPDI decreases the M r /M s ratio, which has been proved both theoretically and experimentally [18][19][20][21][22][23]. Furthermore, it is well known that the strength of IPDI depends on the concentration of magnetic NPs and obviously affects the magnetic ordering states: in a heavily diluted system with low concentration of magnetic NPs, the system exhibits superparamagnetism [24], while in a dense system with high concentration, the NPs exhibit the super-spin glass (SSG) states at low temperature [25,26].
So far, the effects of IPDI on the magnetic properties have not been comprehensively investigated, instead they have been reported piecemeal by different researchers. The strength of IPDI, which can be expressed by the maximum dipolar field H dip , defined as H dip = 2 μ/d 3 , where μ is the particle moment (μ = M s × V m ; M s is saturation magnetization and V m is magnetic grain volume) and d is the distance between particles (center to center), which is inversely proportional to the NP concentration. The purpose of the present work is to systematically reveal the effects of H dip on the M r /M s ratio for the 9-, 13-, and 16-nm magnetic NPs with different concentrations, particle moments, and anisotropies.

Experimental Procedure Preparation of CoFe 2 O 4 NPs
Co(acac) 2 (97 %; acac is acetylacetonate), Fe(acac) 3 (98 %), benzyl ether (97 %), oleic acid (90 %), and oleylamine (80-90 %) were mixed in a 1000-ml three-necked round-bottom flask by magnetic stirring under a flow of nitrogen (99.999 %). The mixture was heated at 120°C for 0.5 h to remove air and moisture, at 200°C under reflux for 2 h, and then at 290°C for 1 h. After the mixture was cooled naturally to room temperature, absolute ethanol was added to produce a precipitate. The precipitate was separated via centrifugation and then washed with absolute ethanol several times to obtain CoFe 2 O 4 NPs. By varying the intermediately treating temperature, we prepared 9-, 13-, and 16-nm CoFe 2 O 4 NPs.

Dilution of CoFe 2 O 4 NPs in a SiO 2 Matrix
In order to change the interparticle distance, some of the 9-and 13-nm CoFe 2 O 4 NPs were diluted in a SiO 2 matrix with different concentrations, because the interparticle distance is inversely proportional to the concentration. CoFe 2 O 4 NPs were added to a solution of cyclohexane (400 ml), polyethylene glycol (25 ml), tetraethyl orthosilicate, and ammonia, and then stirred mechanically for 24 h. Ethanol was added to form a precipitate. The precipitate was isolated by centrifugation and then washed with ethanol and water to remove unreacted molecules. The precipitate was dried at 80°C for 6 h to obtain the diluted CoFe 2 O 4 NPs by SiO 2 . By increasing the relative content of CoFe 2 O 4 , we prepared the CoFe 2 O 4 NPs with low, moderate, and high concentrations, and the samples are referred to as L9, M9, and H9 for 9-nm NPs and as L13, M13, and H13 for 13-nm NPs.

Reduction of CoFe 2 O 4 NPs
In order to change the moment and anisotropy of NPs, the reduction reactions were performed in the H 2 /N 2 atmosphere (500 sccm, 96 % N 2 + 4 % H 2 ) to prepare a composite of CoFe 2 O 4 and CoFe 2 alloy, because the CoFe 2 alloy is a typical soft ferromagnet with high moment and small anisotropy, compared with CoFe 2 O 4 . To avoid the aggregation of NPs during reduction, the 16nm NPs were first diluted or separated by SiO 2 and then reduced at low, moderate, and high temperatures at 300, 400, and 500°C for 4 h, and the obtained samples are denoted as LT16, MT16, and HT16.

Characterization
The crystal structure of the products was determined by X-ray diffraction (XRD) using an X-ray diffractometer (DX-2000 SSC) with Cu Kα irradiation (λ = 1.5406 Å) in the scanning range 20°-80°with a step size of 0.02°. (High-resolution) transmission electron microscopy ((HR)TEM; JEOL, JEM-2100) was used to observe the morphology characteristics. Magnetic measurements were carried out using a superconducting quantum interference device PPMS system (Quantum Design, PPMS EC-II).

Results and Discussion
Crystal Structure and Morphology TEM images and size histograms in Fig. 1 show that the NPs are ca. 9, 13, and 16 nm. Most of 9-and 13-nm NPs exhibit the spherical-like morphology, while some 16-nm NPs exhibit rhombohedral and quadrate shapes. The size distribution range is ca. 2 nm for the 9-nm NPs and 4 nm for 13-and 16-nm NPs.
The XRD experiments have been performed on all samples, and herein, we show the results of several samples representatively. As shown in Fig. 2b [27]. To observe the existence of CoFe 2 in the reduced sample, the (HR)TEM was ever performed on the undiluted sample. The lattice fringes of CoFe 2 can be observed at the surface of particle, i.e., CoFe 2 exists at the outer layer while CoFe 2 O 4 exists in the inner of particle.

Magnetic Properties
The dependence of the magnetization (M) of all samples on the applied magnetic field (H), i.e., M(H) loop (−70 kOe < H < 70 kOe) was measured at temperatures of 10, 50, 100, 150, 200, 250, 300, and 390 K. Representatively, the M(H) loops recorded at 10 and 390 K were shown in Fig. 3 for 9-and 13-nm NPs with different concentrations and in Fig. 4 for diluted 16-nm NPs with different reducing temperatures. From these loops, the coercivity (H c ), saturation magnetization (M s ), and remanence (M r ) to saturation magnetization ratio (M r /M s ) values at different temperatures can be obtained. In the case of the CoFe 2 O 4 NPs with moderate (Fig. 3b, e) and high (Fig. 3c, f ) concentrations, the loops recorded at 10 K show a jump around H = 0; this phenomenon is common for magnetic NPs and can be assigned to the reorientation of surface spins around NPs [28,29]. However, in the case of the CoFe 2 O 4 NPs with low concentration, as shown in Fig. 3a, d, the 10-K loops become smooth which is characteristic of a single-phase hard magnet, and the possible reason is that the surface spins are strongly pinned by the SiO 2 matrix.
For the diluted and reduced 16-nm NPs, containing CoFe 2 O 4 and CoFe 2 phases, as shown in Fig. 4, the jump around H = 0 becomes higher as the reducing temperature increases, as a result of the increase in the relative CoFe 2 content [30,31]. Herein, the jump is attributable to the different reversal fields of hard CoFe 2 O 4 and soft CoFe 2 , which is an indicative of no exchange-coupling occurring between soft and hard species, because in an exchange-coupled system, the magnetization could show an equivalent reversal The undiluted 9-and 13-nm NPs, i.e., H9 and H13, as well as the 16-nm NPs reduced at 300°C (LT16) have the H c values of 20.0, 18.8, and 20.5 kOe at 10 K, smaller than 23.8 kOe (10 K) for our previously reported 10.7-nm CoFe 2 O 4 NPs [32] which particle size is close to the critical size of a single domain [33]. These H c values are much larger than that for the bulk CoFe 2 O 4 (6.8 kOe at 10 K) [34], because on the one hand, the particles are approximately single-domain sized, and on the other hand, the interaction between surface spins induces extra anisotropy and hence enhances H c , which will be further discussed below. As the temperature increases, H c monotonically decreases. The decrease in coercivity can be attributable to thermal fluctuations of the blocked moment, across the anisotropy barrier. For an assembly of non-interacting single-domain magnetic nanoparticles with uniaxial anisotropy, the coercivity can be written in the form of simple model of thermal activation of particle moments over the anisotropy barriers (Kneller's law) as [6,35] Fig. 5. The obtained fitting parameter T B will be used to calculate the volume of magnetic grain, V m , as discussed below.
In the case of non-interacting and randomly oriented spherical particles with cubic anisotropy, H c obeys the relation of H c = 0.64 K/M s [15], where K is the anisotropy constant, so the K value can be calculated for all samples. The highest K for CoFe 2 O 4 NPs at 10 K reaches~10 7 erg/cm 3 , much larger than 1.8-3.0 × 10 6 erg/cm 3 for the bulk CoFe 2 O 4 ; this is consistent with the previous reports that in thin films and nanoparticles, surface anisotropy constant is found higher by many orders of magnitude than that of the bulk [15,16]. In narue the enhanced K value results from the interaction between the surface spins of different particles [7], and between the spins from the surface and core in a single NP [6,13]. Large K value induced by the surface spins leads to the increase of H c due to H c ∝ K. It should be mentioned that the fitting according to Kneller's law and the calculation of K from H c = 0.64 K/M s are based on an assumption that nanoparticles do not interact with each other. The real situation is that the dipolar interaction widely exists in the system of magnetic nanoparticles; however, the dipolar interaction is weak, compared with the anisotropy. Therefore, such the assumption has usually been considered to be reasonable in many previous reports [6,15,35].
Given T B and K, the magnetic grain sizes (V m ) of the nanoparticles can be estimated according to Stoner-Wohlfarth expression [36]: 25k B T B = KV m , where k B is the Boltzmann constant. Subsequently, the diameter D m of a magnetic grain for all samples can be obtained, as shown in Fig. 6. The D m values of the 9-and 13-nm NPs are smaller than D TEM below about 250 K, because of the canted surface spin layer surrounding magnetic particles [37]. Above this temperature, D m becomes larger than D TEM because the surface spins become able to thermally fluctuate, so they can be polarized by the core moments [38]. Subsequently, the collective behavior of several particle moments promoted by the IPDI leads to an increase of D m [28]. In the case of the diluted and reduced 16-nm NPs, D m of the HT16 sample is much larger than that for the LT16 and MT16 samples as a result of the increase of strong magnetic CoFe 2 .
Based on the obtained V m values, the strength of IPDI can be estimated by calculating H dip according to H dip = 2 μ/d 3 , where μ is the particle moment (μ = M s × V m ; M s is saturation magnetization and V m is magnetic grain volume). The inverse of the logarithm of H dip , i.e., 1/lgH dip values and M r /M s ratios with added error bars (5 %) are plotted against temperature (T) in Figs. 7, 8, and 9 for 9-, 13-, and 16-nm NPs, respectively.
Next, we will discuss the correlation between M r /M s and H dip . As seen in Figs. 7 and 8, the correlation between M r /M s and 1/lgH dip roughly follows M r /M s ∝ 1/ lgH dip within a reasonable error range for 9-and 13-nm NPs, even though the NPs have different size and concentration. It can be noticed from Figs. 7 and 8 that the slope of the M r /M s curve is smaller than that for the 1/lgH dip curve at low temperatures, resulting in the deviation between M r /M s and 1/lgH dip , which may result from the several competing effects, including the surface effects, finite size effects, and interparticle interactions, that are sometimes difficult to isolate in these nanoparticles, as suggested by Maaz [6]. As discussed in Fig. 6, the surface spins exist around the NP at low temperatures. Therefore, it is reasonable to suggest that the smaller slope of M r /M s may originate from the interaction between spins of surface and core and between surface spins of the neighboring particles.
In the case of the diluted and reduced 16-nm NPs which contain CoFe 2 O 4 and CoFe 2 , as seen in Fig. 9, the obvious deviation between M r /M s and 1/lgH dip for the slightly reduced LT16 and MT16, while M r /M s matches well with 1/lgH dip for the heavily reduced HT16, which can be assigned to the interaction in    [38], which is the possible reason for the deviation between M r /M s and 1/lgH dip . This suggestion may be supported by the results of the irreversible magnetization reversal field (H irr ) [27], defined as the magnetic field where the derivative (dM/dH) of the virgin curve has a peak. Figure 10 shows the field derivative dM/dH of the virgin curves for LT16 (a), MT16 (b), and HT16 (c) samples at 10  However, compared with LT16 and MT16, the heavily reduced HT16 has the higher CoFe 2 content as a result of reduction at a higher temperature 500°C. Therefore, HT16 has larger particle moment, and hence stronger interparticle dipolar interaction which overcomes other effects such as surface spins and interaction between CoFe 2 O 4 and CoFe 2 , consequently making the correlation between M r /M s and 1/lgH dip obey M r /M s ∝ 1/lgH dip .

Conclusions
Well-dispersed uniform CoFe 2 O 4 NPs with sizes of 9-, 13-, and 16-nm were synthesized. Some 9-and 13-nm NPs were diluted in a SiO 2 matrix to change their concentration that is inversely proportional to interparticle distance, and some diluted 16-nm NPs were reduced by H 2 at 300, 400, and 500°C to change the moment and anisotropy of the NPs. These samples were used as model systems to reveal the intrinsic correlation between M r /M s and IPDI, the strength of which was estimated by H dip .
For the diluted 9-and 13-nm NPs that were not reduced, the correlation between M r /M s and H dip follows M r /M s ∝ 1/lg H dip , regardless of the particle size and distance. Slight deviation from M r /M s ∝ 1/lg H dip , occurring at low temperatures, can be attributed to the effects of surface spins. In the case of the diluted and reduced 16-nm NPs, the relation between M r /M s and H dip deviates M r /M s ∝ 1/lgH dip for the slightly reduced NPs at 300 and 400°C because of the pinning effect of CoFe 2 O 4 on CoFe 2 . However, the heavily reduced NPs at 500°C follows M r /M s ∝ 1/lg H dip because the strong interparticle dipolar interaction is a dominant factor to affect M r /M s . Abbreviations (HR) TEM: (High-resolution) transmission electron microscopy; IPDI: Interparticle dipolar interaction; PDF: Standard powder diffraction file; SSG: Super-spin glass; XRD: X-ray diffraction Fig. 10 The field derivative dM/dH of the virgin curves at 10 K for LT16 (a), MT16 (b), and HT16 (c) samples. The inset in (c) shows local dM/dH data around 11.70 kOe of the HT16 sample. The pink solid line is the curve of Lorentz fit in order to find the peak position