Mean magnetic moment of polydisperse superparamagnetic nanoparticles: Correlation between grain size and magnetic moment distributions
Abstract
The field dependence of magnetization in superparamagnetic systems can be fitted using a set of Langevin contributions characterized by their particle densities and their magnetic moments. In fact, these contributions are just partial average values over the actual magnetic moment distribution of the system. The achievement of the most adequate mean magnetic moment is non-trivial and depends on the characteristics of the moment distribution and the applied magnetic field range. As an example, results have been applied to a nanocrystalline alloy.
Full text
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Copyright © 2007 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Nanoscience and Nanotechnology Vol. 7, 1043–1051, 2007 Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles: Correlation Between Grain Size and Magnetic Moment Distributions J. S. Blázquez, V. Franco, C. F. Conde, and A. Conde∗ Departamento de Física de la Materia Condensada, ICMSE-CSIC, Universidad de Sevilla, P. O. Box 1065, 41080 Sevilla, Spain The field dependence of magnetization in superparamagnetic systems can be fitted using a set of Langevin contributions characterized by their particle densities and their magnetic moments. In fact, these contributions are just partial average values over the actual magnetic moment distribution of the system. The achievement of the most adequate mean magnetic moment is non-trivial and depends on the characteristics of the moment distribution and the applied magnetic field range. As an example, results have been applied to a nanocrystalline alloy. Keywords: Nanocrystalline Alloys, Soft Magnets, Nanocrystalline Microstructure, Superparamagnetism. 1. INTRODUCTION From technological and fundamental points of view, nanostructured materials are a field of intensive research. The high percentage of atoms located at the surface of the crystallites can produce strong differences between the properties (mechanical, magnetic, etc.) of nanostructured systems and their bulk counterparts.1Particularly, Fe-based nanocrystalline alloys show outstanding properties for soft magnetic applications.2The responsible for this extreme softness is the two phase character of these systems: nano-sized ferromagnetic crystallites embedded in a residual amorphous matrix, also ferromagnetic but with a lower Curie temperature. This configuration yields an averaging out of the magnetocrystalline anisotropy due to the exchange coupling between nanocrystals at temperatures below the Curie temperature of the amorphous phase.3Above this temperature, the amorphous matrix becomes paramagnetic and, at high enough temperatures, the crystallites could behave as isolated superparamagnetic particles,45provided that the dipolar interactions between them are negligible.67 The field and thermal dependence of magnetization, M, of a set of identical superparamagnetic particles is described by: M=NLH kT (1) ∗Author to whom correspondence should be addressed. where Nis the number density of the particles, is the magnetic moment of the individual particles, Lis the Langevin function, His the applied magnetic field, kis the Boltzmann constant, and Tis the temperature. However, real samples rarely show a unique grain size but a distribution of grain sizes and, consequently, a distribution of magnetic moments. The determination of grain size distributions for this sort of samples is usually performed by transmission electron microscopy. However, the comparison with results on grain size distributions obtained by other techniques is still a subject of current interest.8–11 From the magnetic point of view, the grain size distribution is evidenced by a magnetic moment distribution. Therefore, the magnetic response of such complex superparamagnetic materials might be fitted with the help of several contributions.12–15 Although there are numerical methods to extract a quasicontinuous moment distribution from the magnetization curves,15–18 the increase of free fitting parameters can prevent a direct physical interpretation of the results. Therefore, a compromise must be reached to describe the behaviour of the real system with the less possible number of free parameters. Changing from a continuous to a discrete description implies that, for each particular physical magnitude, the continuous distribution of its values is grouped in partial average values. The simplest discrete representation of the system would be to use a mean value to describe the whole system. J. Nanosci. Nanotechnol. 2007, Vol. 7, No. 3 1533-4880/2007/7/1043/009 doi:10.1166/jnn.2007.407 1043
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles Blázquez et al. Different definitions of average magnetic moment: The transition from a continuous system to its discrete description has to be made in the most accurate way. Therefore, although there are different definitions of average magnetic moments currently used in the literature, the adequacy of each of them for accurately describing an underlying experimental system has to be tested. Although there is no strict argument to a priori select any particular definition of the average moment, a result of this study will suggest the conditions under which they are the most reliable ones. The most straightforward definition of the average magnetic moment is to base it on the number density of the particles with different magnetic moments (e.g. Refs. [19, 20]): N=Nii Ni (2) In other cases, for the calculation of the magnetic moment of the particles is weighted with their contribution to the overall magnetization (Mi=Nii) (e.g. Refs. [7, 12]): M=Mii Mi =Ni2 i Nii (3) The physical meaning of this second definition can be clarified by considering an ideal system composed of identical atoms, with magnetic moment at. The magnetic moment of a particle composed of niatoms is i=niat and, from Eq. (3), Mwould be: M=Ninii Nini (4) Therefore, while the first definition, N, takes into account the number of particles with magnetic moment i, the second one, M, considers the number of atoms in each particle with magnetic moment iand hence, the weight factor for calculating the average moment is Nini. Whereas Nhas an intensive nature, in the sense that all particles contribute equally to the average moment, M is extensive, as the contribution of each moment emerges from the number of atoms in the particles. In fact, analogous situation can be found in nonmagnetic problems when describing the properties of particulate systems in terms of mean values. For example, in TEM images, average grain size determination is usually performed by counting the particles with a specific size and making the numerical average. Therefore, all the particles, big or small, contribute equally to the average value, as for N. However, when mean grain size is estimated from the broadening of the diffraction maximum, each particle contributes to the diffracted intensity of a maximum with a width related to its size, dependent on the number of atoms which form that particle. As larger particles yield higher intensity peaks, they contribute more to the total intensity. This situation is analogous to that of M. Different criteria of selection of the most adequate average magnetic moment: In order to choose the most representative mean value describing a system it is necessary to establish some selection criteria. One criterion to determine the most adequate to be used would be to find that which better represents the magnetization curve of the polydisperse system of interest. Another criterion would be to find that moment which better makes the reduced magnetization curves collapse when represented as a function of H/kT. The aim of this work is to test the adequacy of the two proposed mean magnetic moments according to the two defined criteria. This will be done from the analysis of the magnetization curves of a superparamagnetic system. As an application, results will be correlated with those of other techniques such as differential scanning calorimetry, DSC, and transmission electron microscopy, TEM, for the same system. This paper is structured in the following sections. After this introduction, in the second section, the influence of the characteristics of the magnetic moment distribution on the adequacy of the two proposed methods of calculating (Nand M) is analyzed with the help of numerically generated magnetization curves. In the third section, an application of the previous numerical results to a real case is made. A nanocrystalline alloy (Fe685Mo5Si135B9Cu1Nb3) has been chosen because both mean magnetic moment definitions are commonly employed in the literature for this kind of system. Finally, the main conclusions achieved in this study are summarized. 2. INFLUENCE OF THE MAGNETIC MOMENT DISTRIBUTION ON THE CALCULATION OF The aim of this section is to identify which of the two definitions of mean magnetic moment is the most adequate to describe ideal polydisperse systems. Using numerically generated curves, the uncertainties which could arise from the sample preparation and/or characterization techniques can be avoided. The magnetization of a quasicontinuous magnetic moment distribution could be generated as the starting point for the analysis. However, the large number of parameters involved could jeopardize the physical meaning of the results regarding . Therefore, the more simplistic approach of considering only two contributions has been used in this study: M=N11L1H kT +N22L2H kT (5) where Mis the magnetization of the system, N1and N2are the particle densities, and 1and 2are the corresponding magnetic moments. This simple magnetic moment distribution can be characterized by the ratios between its two magnetic moments and its two particle densities (2/1 and N2/N1, respectively). The Mcurves were generated in a2/1range from 0 to 1 with a step of 0.05. Regarding 1044 J. Nanosci. Nanotechnol. 7, 1043–1051,2007
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Blázquez et al. Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles the N2/N1ratio, three different values (1, 5, and 10) have been considered. This choice is justified by the typical TEM results for nanocrystalline alloys. An example is given in Section 3, showing asymmetrical grain size distributions in which the amount of small particles is enhanced. This selection of two fitting moments does not imply an assumption of a bimodal distribution of the real samples. In fact, a continuous distribution can be represented with big enough “bins” as that of two adding contributions. This low resolution in bin selection emerges from the ill-posed nature of the problem: the extraction of a moment distribution from the fitting of the global magnetic response of the sample, where the physical meaning of the fitting parameters is progressively lost as their number increases. 2.1. from the Best Fit of the Magnetization Curve to a Monodisperse System In order to identify the most adequate mean magnetic moment following the first criterion (the monodisperse system which better substitutes the polydisperse one), each of the generated magnetization curves (with different values of 2/1and N2/N1) were fitted to: M=NFITFITLFITH kT (6) where FIT is the magnetic moment and NFIT is the particle density obtained from each fitting. The values of N(numerical average value of magnetic moments of the system) and M(average value weighted with the individual magnetization of each particle type), corresponding to each generated magnetic moment distribution were also obtained. The comparison between these two mean values and FIT yields the selection of the most adequate method to calculate . As an example, Figure 1(a) shows FIT,Nand Mdependence on the 2/1ratio for N2/N1=5, the behaviour for the other particle density ratios being similar. Figure 1(b) shows the absolute values of the differences N=FIT − N and M=FIT −M as a function of 2/1 ratio for the three studied values of N2/N1(1, 5, and 10). Although the general trend of FIT is more similar to that of M, there are regions where Nis smaller than M. Therefore, according to this criterion, the identification of the most representative mean magnetic moment (in the sense that the magnetization curve of this single moment can substitute the generated one) depends on the characteristics of the moment distribution. For example, for small 2/1(i.e., broad particle size distributions) Mshould be selected. Regarding the particle densities, as the saturation magnetization of the system, MS, should be independent of the average magnetic moment selected to represent the system, the following relationship must be fulfilled: MS=Nii=N∗(7) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 (b) ∆N (N2/N1 = 1) ∆M (N2/N1 = 1) ∆N (N2/N1 = 5) ∆M (N2/N1 = 5) ∆N (N2/N1 = 10) ∆M (N2/N1 = 10) ∆N , ∆M 0.0 0.2 0.4 0.6 0.8 1.0 (a) <µ> (a.u.) N2/N1 = 5 <µN> <µM> µ FIT µ2/µ1 Fig. 1. (a) Normalized magnetic moment of the generated magnetization curve FIT and calculated average values, Mand N,asa function of 2/1for N2/N1=5. (b) Absolute values of the difference between both average values and FIT. where N∗is the effective particle density (N∗ Nor N∗ M) and is the corresponding average value of the magnetic moment (Nor M). Introducing the definitions of Nand M(Eqs. (2) and (3), respectively) inside Eq. (7), it can be shown that N∗ Nis equal to the actual particle density N∗ N=N1+N2. However, in the case of M the effective population density is: N∗ M=N11+N222 N12 1+N22 2 (8) It is worth noticing that N∗ Mis analogous to other effective densities as it occurs with the effective density involved in the Hall effect of a system with two different types of carriers. As an example, Figure 2 shows the dependence of N∗ N,N∗ M, and NFIT and the relative differences N= NFIT −N∗ N/N1+N2and M=NFIT −N∗ M/N1+N2on 2/1for N2/N1=5. Similar behaviour could be observed for the other particle density ratios. As expected, the general trend is also better represented by N∗ Mthan by N∗ N. However, for high values of 2/1, the error between the fitting and the calculated values is smaller when N∗ Nis considered. In all cases, the region where N< Mis included in the region where N< Mbut they are not exactly coincident. It must be taken into account that the N∗value was J. Nanosci. Nanotechnol. 7, 1043–1051, 2007 1045
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles Blázquez et al. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 (b) δN (N2/N1 = 1) δM (N2/N1 = 1) δN (N2/N1 = 5) δN (N2/N1 = 10) δM (N2/N1 = 10) δM (N2/N1 = 5) δN , δM 0 2 4 6 (a) N* (a.u.) N2/N1 = 5 NN NM NFIT µ2/µ1 Fig. 2. (a) Particle density of the generated magnetization curve NFIT and calculated effective values, NMand NN, as a function of 2/1. (b) Absolute values of the difference between both effective values and NFIT. just calculated imposing that the saturation magnetization of the system remains constant regardless of the selection of (Eq. (7)). 2.2. from the Collapse of Magnetization Curves After Rescaling Following the second criterion, the way to determine the most adequate mean magnetic moment is to find the which makes the reduced magnetization (M/MS) curves collapse when represented as a function of H/kT. This rescaled xaxis is related to the ratio between the average magnetic energy of a system with a distribution of magnetic moments and its thermal energy. Analogously to the previous subsection, the two previously defined mean magnetic moments Nand M can be used to rescale the generated Mcurves. Figure 3 shows the generated magnetization curves after rescaling in a range of 2/1from 0 to 0.4 for N2/N1=1. It can be observed that in this range the collapse obtained from Mis better than the one from N. Figure 4 shows the generated magnetization curves after rescaling in a range of 2/1from 0.4 to 1 for N2/N1=1. Unlike the results of Figure 3, in this case it is clear that the collapsing of the curves using Nis much better than the one using M. From this feature, it is evident that as the distribution becomes narrower (2/1tends to 1) Nshould be used. However, for small values of 2/1,Mis roughly better. Nevertheless, for different values of N2/N1, –100 –50 0 50 100 –1 0 1 –1 0 1 <µN> M/MS <µ>H/kT <µM> µ2/µ1: 0–0.4 (step 0.05) Fig. 3. Generated magnetization curves after rescaling in a range of 2/1from 0 to 0.4 for N2/N1=1, using M(above) and N(below). The inset shows an enlargement (10 times in the xaxis) of the low field region of the curves. The arrow indicates the increasing trend of 2/1. the selection of can change, although the general trend is maintained. These results are in agreement with those obtained by the first selection criterion (Section 2.1). 3. APPLICATION TO A REAL SUPERPARAMAGNETIC SYSTEM: Fe685Mo5Si135B9Cu1Nb3 NANOCRYSTALLINE ALLOY Ribbons of Fe685Mo5Si135B9Cu1Nb3alloy (∼25 m thick and ∼10 mm wide) were produced by planar flow casting. Figure 5 shows the differential scanning calorimetry (DSC) plot of the as-quenched alloy, measured in a Perkin-Elmer DSC-7. Devitrification occurs in two stages, the first one corresponds to the formation of nanocrystals ofa-Fe(Si) phase embedded in a residual amorphous matrix and the second one yields the fully crystallization of the material after the formation of boride phases.21 Transmission electron microscopy (TEM) observations of the samples were performed in a Philips CM200 electron microscope (200 kV). The field dependence of the magnetization curves, measured at different temperatures, was registered in a Lakeshore 7407 Vibrating Sample Magnetometer (VSM) equipped with a furnace, using a maximum applied field of 5 kOe. In order to develop different nanocrystalline microstructure, as-quenched samples 1046 J. Nanosci. Nanotechnol. 7, 1043–1051,2007
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Blázquez et al. Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles –100 –50 0 50 100 –1 0 1 –1 0 1 <µN> M/MS <µ>H/kT <µM> µ2/µ1: 0.4–1 (step 0.05) Fig. 4. Generated magnetization curves after rescaling in a range of 2/1from 0.4 to 1 for N2/N1=1, using M(above) and N (below). The inset shows an enlargement (10 times in the xaxis) of the low field region of the curves. were heated up to different temperatures, Ta(in the range 843–883 K, see Fig. 5), at 10 K/min, in a halogen-lamp furnace under vacuum, and subsequent free cooling. 3.1. Magnetization Results The temperature dependence of coercivity (Fig. 6) can be explained on the basis of the random anisotropy model 750 800 850 900 950 1000 0.1 W/g dQ/dt T (K) Fig. 5. DSC plot at 10 K/min of as-quenched sample. Lines indicate the Tarange. 300 400 500 600 700 0 2 4 6 8 HC (Oe) T (K) Ta (K) 853 863 873 883 Fig. 6. Temperature dependence of the coercivity. Data for the sample pre-heated up to 843 K are below the field sensitivity of the system. extended to two-phase systems.3For T<T am c(about 500 K) the ferromagnetic character of the matrix facilitates the exchange coupling of the nanocrystals, resulting in the averaging out of the magnetic anisotropy. As Tam c is approached, the matrix can no longer transmit the exchange. This is evidenced by the progressive increase in coercivity. The reduction in coercivity at higher temperatures is associated to the transition to a superparamagnetic regime of the -Fe(Si) nanoparticles. It has been shown that this transition is controlled by the dipolar interaction between the particles, both in Finemet6and Nanoperm7 type nanocrystalline materials. The maximum value of coercivity is related to the effective dipolar interaction field. As nanocrystallization proceeds, the increase in grain size and the reduction of the average distance between the nanocrystals enhance the dipolar coupling between the particles, as evidenced by the progressive increase in the maximum value of coercivity. For T=713 K, all the samples present zero coercivity due to the superparamagnetic behaviour of the nanocrystalline system. The MH curves registered at this temperature were fitted using two Langevin contributions as well as a lineal term which accounts for the contributions to MH from the amorphous matrix and the sample holder (Fig. 7 shows an example). Table I summarizes the fitting results as well as the corresponding calculated Nand Mvalues. Following the first criterion (Section 2.1), the experimental magnetization curves, after subtraction of the linear term, are compared to the resulting curves using N and Mcorresponding to the values obtained from the previously mentioned fittings (Fig. 8 shows an example of the typical behaviour). It is observed that the experimental curve is more similar to that obtained from N in all the studied cases. It is worth noticing that for very low magnetic fields (<100 Oe) the use of Mis better than the use of Nas it can be observed in the inset of Figure 8. This is in agreement with the fact that in the calculation of Mthe contribution of the larger particles, J. Nanosci. Nanotechnol. 7, 1043–1051, 2007 1047
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles Blázquez et al. –4000 –2000 0 2000 4000 –20 0 20 Ta = 863 K (3) (2) (1) M (emu/g) H (Oe) Fig. 7. Experimental MH curve at 713 K of nanocrystalline sample pre-heated up to 863 K (crosses) along with the resulting curve of the fit and the different contributions used: two Langevin functions, (1) and (2), and a linear term (3). which are responsible for the big slope at low fields, is enhanced. On the other hand, the slowly saturating evolution of MH at high fields is mainly due to the smaller particles, which are better taken into account by N. In fact, for all the studied samples, the values of N2/N1range from 5.5 to 8.4 and 2/1from 0.10 to 0.32, respectively, and, as these values lay on the region where N< M, the experimental magnetization curves can be better represented by N, in agreement with the results obtained in Section 2.1. Following the second criterion (Section 2.2), the xaxis is rescaled using both possible mean magnetic moments (Fig. 9). It can be observed that using Nthe collapse is better fulfilled than using M, in agreement with the results obtained above. 3.2. TEM Results Figure 10 shows bright field (BF) TEM images of the studied samples, their corresponding selected area diffraction (SAD) patterns are included as insets (diameter of the selected area ∼0.5 m). For all the samples, nanocrystalline microstructure (-Fe(Si) phase embedded in residual amorphous matrix) can be observed, in agreement with SAD patterns. The amount of -Fe(Si) crystalline phase Table I. Parameters from the fit of the MH curves of nanocrystalline samples at 713 K. Average values are also shown. 12N1N2NMNNNM TaK10 −16 emu 10−16 emu 1015 g−11015 g−110−16 emu 10−16 emu 1015 g−11015 g−1 843 8.1 2.6 1.9 15.7 3.2 4.1 17.6 13.7 853 13.8 3.0 4.5 30.5 4.4 7.4 35.0 21.0 863 23.8 4.3 5.2 28.8 7.3 14.0 34.0 17.7 873 43.7 6.3 4.1 26.8 11.2 25.4 30.9 13.6 883 87.4 8.4 2.5 20.7 17.1 52.7 23.2 7.5 –100 0 100 –10 0 10 –4000 –2000 0 2000 4000 –20 –10 0 10 20 Ta = 863 K M (emu/g) H ( Oe ) exp. data from <µN> from <µM> Fig. 8. Experimental MH curve at 713 K of nanocrystalline sample pre-heated up to 863 K along with the calculated MH curves corresponding to the two different mean magnetic moments: Mand N. An enlargement of the figure in the low field region is shown as an inset. increases as Taincreases, as it could be expected from DSC results. The shape of the nanocrystals is approximately spheroidal, characteristic of FINEMET-type alloys, unlike the irregular shaped crystals observed for Cu-free NANOPERM alloys.222 Grain size distributions, nD, were obtained from BF images. Figure 11 shows the nD histograms for all the studied samples. It can be observed a continuous increase of the fraction of crystallites exceeding 10 nm, while that of the crystallites below 5 nm decreases. The average grain size Dcontinuously increases from 6 up to 10 nm. It is evident that the grain size distributions of the studied samples are not bimodal. However, the use of only two components simplifies enormously the description of the system but keeping some relevant information, as the asymmetry of the distribution. 3.3. Comparison Between DSC, TEM, and VSM Results The progress of the nanocrystallization can be followed from the results of the three characterization techniques (DSC, TEM, and VSM) used in this study. In the case of DSC results, the enthalpy change rate, dQ/dt, can 1048 J. Nanosci. Nanotechnol. 7, 1043–1051,2007
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Blázquez et al. Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles –100 –50 0 50 100 –1 0 1 –1 0 1 <µN> M/MS <µ>H/kT <µM> Fig. 9. Rescaled plot of the different experimental MH curves obtained at 713 K for the different studied samples using M(above) and N(below). The insets show an enlargement (10 times in the xaxis) of the low field region of the curves. be assumed proportional to the transformation change. This approximation is less adequate at the end of the transformation23 but in the studied case samples are heated up to ∼100 K below the end of the nanocrystallization 843 K 50 nm 853 K 863 K 873 K 883 K Fig. 10. Bright field images and selected area diffraction patterns of nanocrystalline samples pre-heated up to different Ta. 01020 0 20 40 n(D) (%) 843 K D (nm) 01020 0 20 40 853 K D (nm) n(D) (%) 01020 D (nm) 863 K 01020 0 20 40 D (nm) D (nm) 873 K n(D) (%) 01020 883 K Fig. 11. Grain size distribution of nanocrystalline samples pre-heated up to different Ta. process (see Fig. 5). Therefore, the crystallization fraction can be obtained from the integration of the DSC signal from the onset temperature up to Tanormalized to the total area of the nanocrystallization peak, Q. XDSC C=1 Q Theat Tonset 1 dQ dt dT (9) where is the heating rate. It must be taken into account that XDSC C(fraction of nanocrystallization process) is not the crystalline volume fraction, because at the end of the nanocrystallization process some amount of residual amorphous remains, which will be transformed during the second transformation stage. In the case of TEM, if a set of images obtained in the same conditions are compared, the crystalline volume (measured after the addition of the volumes of each crystal, Vi, for a statistically significant number of them) is proportional to the crystalline volume fraction of the system. XTEM C=A i Vi(10) where Ais a constant. In order to accept this assertion, it must be assumed that the explored volume in each BF TEM image is constant, which implies that the thickness of the observed regions is the same for all the studied samples. J. Nanosci. Nanotechnol. 7, 1043–1051, 2007 1049
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles Blázquez et al. 020406080 0 10 20 30 40 0 2 4 6 8 10 MS (emu/g) Nanocrystallization fraction (%) ΣVi (a.u.) Fig. 12. Crystalline volume (from TEM) and saturation magnetization (from VSM) as a function of the completed fraction of the nanocrystallization process (from DSC), for samples pre-heated up to different Ta. Regression lines are plotted. Finally, in the case of VSM results, the saturation magnetization, MS, can be considered proportional to the transformed fraction. The ratio between each couple of crystalline fractions obtained by these three experimental methods is a constant. The good linear correlation between TEM and VSM results with DSC results observed in Figure 12 supports the approximations stated above. Whereas DSC results give information only about the total volume transformed, TEM and VSM techniques can supply average values of the volume and magnetic moment of the particles, respectively. For a single superparamagnetic particle, its magnetic moment, i, is proportional to its volume, Vi. Therefore, it is possible to check this proportionality for the average values obtained from VSM and TEM results. The two possible values of yield two mean values of V: N=Nii Ni =NiCVi Ni =CNiVi Ni =CVN(11) and M=Ni2 i Nii =NiC2V2 i NiCVi =CNiV2 i NiVi =CVM (12) where C=natat is a constant, nat is the “average atom” number density in the -Fe(Si) nanocrystals and at its magnetic moment. The first average volume is simply the 0123456 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 <VM> (103 nm3) <µM> (10–15 emu) <µN> (10–15 emu) <VN> (103 nm3) Fig. 13. Average particle volumes (VMand VN) versus the corresponding average magnetic moment (Mand N, respectively). Lines are a guide to the eye. numerical average of the particle sizes, whereas for the second average volume, the particle volume is weighted with its number of atoms (proportional to the volume). Figure 13 shows the correlation between the mean magnetic moment and the mean particle volume using both possibilities. Linearity is better fulfilled in the case of VN versus Nthan for VMversus M, in agreement with the magnetization results. Considering that the atomic density in the -Fe lattice is 2 atoms per unit cell of volume a3 (a∼0285 nm is the lattice parameter), the atomic moment obtained from the constant C(Eq. (11)) and averaged over all the studied samples, 23±07B, is of the order of the magnetic moment of Fe atoms in -Fe. The large errors of the volume measured from TEM (for an error of the grain diameter of 1 nm, V /V > 30% for D<10 nm) prevent a further discussion on this parameter. It must be taken into account that the two contributions, 1and 2(1> 2), with which it was possible to fit the experimental MH plots, are in fact average values of magnetic moments. TEM results clearly show that a continuous distribution of grain sizes exits and not only two values of Dand, therefore, 1represents an average for the bigger particles and 2represents an average 1050 J. Nanosci. Nanotechnol. 7, 1043–1051,2007
Delivered by Ingenta to: University of Arkansas IP : 130.184.152.173 Wed, 16 May 2007 20:13:33 RESEARCH ARTICLE Blázquez et al. Mean Magnetic Moment of Polydisperse Superparamagnetic Nanoparticles for the smaller ones. The evolution of both distributions (from TEM and VSM results, respectively) with Taqualitatively agrees (see Fig. 11 and Table I). The width of the distributions becomes broader as the nanocrystallization progresses and the particle density for low magnetic moment and small grain size is higher than for high magnetic moment and large grain size contributions. Finally, it would be interesting to consider some effects which would affect the relation between TEM observations and VSM results. Whereas from TEM images we can directly observe a distribution of crystal sizes, MH plots give us the global response of the system, from which the distribution information can be extracted after a fitting process. Besides, the magnetic and the TEM crystal size may differ due to some physical reasons. For example, the Fe atoms located at the surface of the nanocrystals will be affected by the high concentration of Nb atoms just outside the nanocrystals and, therefore, the magnetic moment of these Fe atoms might be smaller than the corresponding value of an Fe atom in the inner regions of the nanocrystals, yielding a fictitious decrease in the magnetic results with respect to the actual grain size. On the other hand, this effect can be partially compensated if the Fe atoms in the amorphous matrix close to the nanocrystals were polarized.24 4. CONCLUSIONS Two different methods to calculate the mean magnetic moment of a distribution of superparamagnetic particles have been employed: either considering the number density of particles (N) or considering their contribution to the magnetization of the system (M). In order to select the most adequate mean magnetic moment, two different criteria have been proposed. The first one is to find the monodisperse system which better resembles the experimental magnetization curve of the polydisperse real system. The second criterion corresponds to identify an average magnetic energy which produces the collapse of the experimental magnetization curves after rescaling the xaxis. With the help of numerical calculations, it has been shown that both criteria yield the same conclusions. The selection of the most adequate mean magnetic moment depends on both the characteristics of the magnetic moment distribution and the range of the applied magnetic field. For broad distributions or low magnetic fields (well below saturation), Mshould be selected, while for narrow distributions and almost saturating magnetic fields, Ngives a better description of the system. Experimental results of superparamagnetic magnetization curves and grain size distribution from TEM have been obtained for nanocrystalline samples of a Mocontaining Finemet-type alloy. Applying the previously described selection criteria, the most adequate mean magnetic moments have been identified. These results are in agreement with the information extracted from the grain size distributions obtained from TEM. Acknowledgments: This work was supported by the Spanish Government and EU-FEDER (Project MAT 2004- 04618) and the PAI of Junta de Andalucía. J. S. B. is grateful to Junta de Andalucía for a research contract. References and Notes 1. H. Gleiter, Adv. Mater. 4, 474 (1992). 2. M. E. McHenry, M. A. Willard, and D. E. Laughlin, Prog. Mater. Sci. 44, 291 (1999), and references therein. 3. A. Hernando, M. Vázquez, T. Kulik, and C. Prados, Phys. Rev. B 51, 3581 (1995). 4. A. Slawska-Waniewska, M. Gutowski, H. Lachowicz, T. Kulik, and H. Matyja, Phys. Rev. B 46, 14594 (1992). 5. V. Franco, C. F. Conde, A. Conde, and L. F. Kiss, J. Magn. Magn. Mater. 215, 400 (2000). 6. V. Franco, L. F. Kiss, T. Kemény, I. Vincze, C. F. Conde, and A. Conde, Phys. Rev. B 66, 224418 (2002). 7. V. Franco, C. F. Conde, A. Conde, and L. F. Kiss, Phys. Rev. B 72, 174424 (2005). 8. F. Sánchez-Bajo, A. L. Ortiz, and F. L. Cumbrera, Acta Mater. 54, 1(2006). 9. P. Scardi and M. Leoni, Acta Cryst. A58, 190 (2002). 10. S. Calvin, S. X. Luo, C. Caragianis-Broadbridge, J. K. McGuiness, E. Anderson, A. Lehman, K. H. Wee, S. A. Morrison, and L. K. Kurihara, Appl. Phys. Lett. 87, 233102 (2005). 11. R. J. Marín-Palma, L. Pascual, P. Herrero, and J. M. Martínez-Duart, Appl. Phys. Lett. 87, 211906 (2005). 12. P. Allia, M. Coisson, M. Knobel, P. Tiberto, and F. Vinai, Phys. Rev. B 60, 12207 (1999). 13. P. Allia, F. Ghigo, M. Knobel, P. Tiberto, and F. Vinai, J. Magn. Magn. Mater. 158, 319 (1996). 14. H. K. Lachowicz, K. Zaveta, and A. Slawska-Waniewska, IEEE Trans. Magn. 38, 3033 (2002). 15. A. Kákay, M. Gutowski, L. Takács, V. Franco, and L. K. Varga, J. Phys. A: Math. Gen. 37, 6027 (2004). 16. T. Weser and K. Stierstadt, Z. Phys. B 59, 253 (1985). 17. L. Takacs, J. Appl. Phys. 63, 4264 (1988). 18. M. W. Gutowski, J. Phys. A: Math. Gen. 27, 7893 (1994). 19. V. Franco and A. Conde, J. Magn. Magn. Mater. 277, 181 (2004). 20. K. Yakushiji, S. Mitani, K. Takanashi, J. G. Ha, and H. Fujimori, J. Magn. Magn. Mater. 212, 75 (2000). 21. C. F. Conde, V. Franco, and A. Conde, Phil. Mag. B 76, 489 (1997). 22. K. Hono, Prog. Mater. Sci. 47, 621 (2002). 23. J. M. Barandiarán, I. Tellería, J. S. Garitaonandía, and H. A. Davies, J. Non-Cryst. Solids 329, 57 (2003). 24. A. Hernando, I. Navarro, and P. Gorria, Phys. Rev. B 51, 3281 (1995). Received: 19 January 2006. Revised/Accepted: 28 August 2006. J. Nanosci. Nanotechnol. 7, 1043–1051, 2007 1051