PHYSICAL REVIEW B VOLUME 59, NUMBER 10 1 MARCH 1999-II Magnetic irreversibility and relaxation in assembly of ferromagnetic nanoparticles R. Prozorov Loomis Laboratory of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801 Y. Yeshurun Institute of Superconductivity, Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel T. Prozorov* and A. Gedanken Department of Chemistry, Bar-Ilan University, 52900 Ramat-Gan, Israel Received 21 July 1998 Measurements of the magnetic irreversibility line and time-logarithmic decay of the magnetization are described for three Fe2O3 samples composed of regular amorphous, acicular amorphous, and crystalline nano- particles. The relaxation rate is the largest and the irreversibility temperature is the lowest for the regular amorphous nanoparticles. The crystalline material exhibits the lowest relaxation rate and the largest irrevers- ibility temperature. We develop a phenomenological model to explain the details of the experimental results. The main new aspect of the model is the dependence of the barrier for magnetic relaxation on the instantaneous magnetization and therefore on time. The time-dependent barrier yields a natural explanation for the time- logarithmic decay of the magnetization. Interactions between particles as well as shape and crystalline mag- netic anisotropies define an energy scale that controls the magnetic irreversibility. Introducing this energy scale yields a self-consistent explanation of the experimental data. S0163-1829 99 03009-X I. INTRODUCTION netic nanoparticles. This enables a study of the effect of shape and crystalline anisotropies on magnetic irreversibility A ferromagnetic particle becomes monodomain if its size and the relaxation rate. We find that, qualitatively, all three d is reduced below a critical value dcr 1 ­100 nm, deter- samples exhibit similar irreversible magnetic behavior. How- mined by the competition between dipole and exchange ever, their irreversibility lines and relaxation rates differ sig- energies.1,2 Below this critical size, the energy loss due to nificantly. creation of magnetic domain walls proportional to d2) is Irreversible magnetic behavior similar to that described larger than the gain due to the disappearance of the dipole here is also observed in other systems. A noticeable example magnetic-field energy proportional to d3). Such mon- is magnetic irreversibility in superconductors.14­16 In such odomain ferromagnetic particles can be viewed as large mag- systems the origin for irreversibility is the interplay between netic units, each having a magnetic moment of thousands of thermal energy and some energy barrier, which prevents Bohr magnetons. Usually neighboring particles are well magnetic reorganization in those materials. The microscopic separated 10­20 nm , and direct exchange between particles origin of the barrier, however, depends on the system. The may be neglected. Thus, the magnetic properties of an as- magnetic irreversibility in nanoparticles is conventionally as- sembly of nanoparticles are determined by the dipole field sociated with the energy required for a particle moment re- energy along with thermal and magnetic anisotropy energies orientation, overcoming a barrier due to magnetic shape or see, e.g., Refs. 3­6 . crystalline anisotropy. It is important to note that the barrier Experiments conducted on magnetic nanoparticles show is considered to be independent of the magnetic moment irreversible magnetic behavior below the ``irreversibility itself.8,10,17­32 In superconductors, magnetic irreversibility is line'' Tirr(H). In particular, the zero-field-cooled ZFC and due to the inevitable spatial fluctuation of the superconduct- field-cooled FC magnetization curves do not coincide, and ing order parameter caused by defects, imperfections, etc.; magnetic hysteresis appears in M vs H curves see, e.g., the barrier is the energy required to overcome the pinning Refs. 7­10 . Moreover, time-logarithmic magnetic relax- due to this disorder. ation, towards the thermodynamic equilibrium state, is ob- An important concept in the theory of irreversible mag- served below Tirr(H). Similar observations are reported here netic properties of superconductors, based on the work of for three systems of Fe2O3 nanoparticles with different shape Anderson,14­16 is that the effective barrier for magnetic re- and crystalline magnetic anisotropies. These nanoparticle laxation increases with time. This is because the supercon- samples were prepared by a sonochemical method, which ducting shielding current proportional to the magnetization produces ``regular'' amorphous nanoparticles.11­13 decays with time, causing a decrease in the Lorentz force Sonochemical irradiation carried out in the presence of a which drives the fluxons away from their positions. Lottis, magnetic field results in synthesis of acicular amorphous White, and Dahlberg33 have put forward similar arguments nanoparticles.13 Annealing of amorphous particles leads to to study slow dynamics. They noticed the close analogy be- crystallization. In this manner we have prepared regular tween ferromagnetic assemblies and superconductors. Ana- amorphous, acicular amorphous, and crystalline ferromag- lyzing the results of numerical computations they concluded 0163-1829/99/59 10 /6956 10 /$15.00 PRB 59 6956 ©1999 The American Physical Society PRB 59 MAGNETIC IRREVERSIBILITY AND RELAXATION IN . . . 6957 TABLE I. Characteristic parameters of the samples. Sample d (nm) Surface area (m2/g) Regular amorphous 50 148 Acicular amorphous 5 50 164 Crystalline 100 88 that the decay of the demagnetizing field is the origin of the ``quasilogarithmic'' relaxation. Although the distribution of particle sizes may explain quasi-time-logarithmic relaxation in a limited time interval, it is not necessary for the explana- tion of the experimentally observed time-logarithmic relax- ation. This approach was later employed in other studies of FIG. 1. X-ray diffraction patterns for a regular amorphous, b the magnetic relaxation, for example, in thin magnetic acicular amorphous, and c crystalline samples. films.34 In this work we adopt the concept of a time varying bar- rier and derive the phenomenological model to explain mag- due, material was annealed in vacuum at 140­150 °C for 3 netic irreversibility and logarithmic magnetic relaxation in h. Heating up to this temperature was necessary to evaporate nanoparticles. The physics for the time dependence of the residua of solvents, particularly decaline which has a high barrier in nanoparticles is related to the fact that the effective boiling point (189­191 °C). The amorphous nature of the barrier for reorientation of the magnetic moment of each particles is confirmed by x-ray diffraction, differential scan- nanoparticle depends on the internal magnetic field, which ning calorimetry DSC analysis, and electron-diffraction includes the average dipole field from surrounding nanopar- patterns at selected areas as shown in Figs. 1 a , 2 a , and the ticles. This average dipole field decreases with time due to inset to 3 a , respectively. The absence of Bragg peaks in the increase of randomness in the orientation of the magnetic Fig. 1 a demonstrates the absence of the long-range order in moments of the surrounding nanoparticles. This, in turn, the atomic structure; the large endothermic peak in Fig. 2 a causes the increase of the effective barrier with time, yield- indicates an amorphous to crystalline transition at 300 °C. ing a natural explanation to the experimental observation of The electron-diffraction pattern of the inset to Fig. 3 a also time-logarithmic relaxation, and a sample-dependent irre- confirms the amorphous nature of the particles. A typical versibility line and relaxation rate. Interactions between par- particle size of 50 nm is inferred from the transmission ticles as well as shape and crystalline magnetic anisotropies electron micrography TEM picture of Fig. 3 a . define an energy scale that controls the magnetic irreversibil- Acicular amorphous particles have been prepared by per- ity. Introducing this energy scale yields a self-consistent ex- forming sonication in external magnetic field of 7 kG for 3 h. planation of the experimental data. The sonication has been carried out in the 0.25M solution of This article is organized in the following way. In Sec. II Fe(CO)5 in a flask open to air. We then repeat the wash and we describe the preparation of the three nanoparticle sys- dry procedure as described above. The amorphous nature of tems. We then describe our experimental results of irrevers- the particles was confirmed by x-ray diffraction, DSC, and ible magnetic properties at various temperatures, fields, and selected area electron-diffraction patterns as shown in Figs. times. In Sec. III we describe our phenomenological model 1 b , 2 b , and the inset to 3 b , respectively. A typical par- and derive equations for the irreversibility line and the mag- ticle length of 50 nm and diameter of 5 nm are inferred netic relaxation. In Sec. IV we compare the predictions of from the TEM picture of Fig. 3 b . our model with the experimental results. Heating of amorphous Fe2O3 up to 370­380 °C in ambi- ent atmosphere for 3­4 h resulted in crystalline -Fe2O3 II. EXPERIMENTAL A. Sample preparation and characterization Three Fe2O3 samples composed of regular amorphous, acicular amorphous, and crystalline nanoparticles were pre- pared by a sonochemical method.11­13 For the ultrasound ir- radiation we used VC-600 Sonics and Materials sonicator with Ti horn at 20 kHz and 100 W cm 2. In Table I we summarize their features. One molar solution of Fe(CO)5 in decaline was sonochemically irradiated for 3 h in ambient pressure at 0 °C. The powder obtained was centrifuged, washed repeatedly with dry pentane 6­7 times, 8500 rpm , and dried in vacuum at room temperature for 3 h. The ma- terial obtained has been accumulated from 2­3 sonications and the total amount of Fe2O3 was mixed to ensure the reli- FIG. 2. Differential scanning colorimetry spectra for a regular ability of the results. Then, in order to remove organic resi- amorphous, b acicular amorphous, and c crystalline samples. 6958 PROZOROV, YESHURUN, PROZOROV, AND GEDANKEN PRB 59 FIG. 3. Transmission electron micrographs for a regular amorphous, b acicular amorphous, and c crystalline samples. nanoparticles. The nature and the internal structure of the zation curves we extract the irreversibility line for the three crystalline iron oxide were determined using the x-ray dif- samples. From the relaxation measurements we deduce the fraction shown in Fig. 1 c . The DSC data, Fig. 2 c , do not relaxation rate, as a function of temperature, for the three show any endothermic peak. The TEM image of Fig. 3 c samples. We then present measurements of magnetization show particles of mean size of 100 nm. loops M(T const,H) and magnetic relaxation at different The second column in Table I summarizes typical particle values of external field. The relaxation rate, as a function of size for the three samples. The third column includes the field, is then deduced for the three samples. total surface area of the particles, as measured by Brunauer- Figure 4 exhibits typical results of ZFC-FC magnetization Emmett-Teller absorption using N2 gas as absorbent. curves and magnetic relaxation at 500 G for the sample com- posed of amorphous round nanoparticles. The vertical lines B. Magnetic measurements procedure of open circles in Fig. 4 depict the relaxation measurements A Quantum Design MPMS superconducting quantum in- at different temperatures. The vertical arrow indicates the terference device magnetometer was used for all magnetic direction of the time increase. The magnetic moment relaxes measurements reported here. The irreversibility line was de- towards the equilibrium moment Mrev , determined by the termined from ZFC and FC magnetization measurements. FC curve. In the inset to Fig. 4 we zoom out at the ZFC-FC Before taking a data point, temperature was stabilized with curves and indicate by an arrow, the experimental definition 0.05 K accuracy and a 30 s pause was sustained. The tem- of Tirr . perature at which ZFC and FC merge for a constant field H is The magnetic relaxation data of Fig. 4 are replotted in defined as irreversibility temperature T Fig. 5 as a function of time. The solid lines in Fig. 5 are irr(H). We define the merging point using a criterion M linear fits for M ln(t). A qualitatively similar time- FC M ZFC 0.1 emu/g. The procedure for measurements of magnetic relaxation logarithmic decay is also observed in the other two samples. at different temperatures is as follows: The sample is cooled Quantitative differences will be discussed below. in H 2 T from room temperature larger than Tirr(2 T)] to a target temperature T, the magnetic field is then reduced to 500 G and the magnetic moment is measured for approxi- mately 2 h. The first data point is taken approximately 2 min after the field change. The field dependence of the magnetic relaxation rate is measured at T 20 K. At this temperature the field is ramped up to H 2 T and reduced back to a target field H, from where the measurements start. The same has been re- peated for negative field H 2 T with a consequent in- crease of the magnetic field to a target value. C. Results The experimental results in this section are organized as follows: we first show M(T,H const) data, and related FIG. 4. Typical ZFC-FC curves with superimposed relaxation measurements of magnetic relaxation at different tempera- for the amorphous sample measured at 500 G at different tempera- tures. From the merging point of the ZFC and FC magneti- tures. Inset: full-range ZFC-FC curve. PRB 59 MAGNETIC IRREVERSIBILITY AND RELAXATION IN . . . 6959 FIG. 7. Scaling of the M(T) FC-ZFC curves with irreversibility FIG. 5. Typical relaxation curves measured in the amorphous temperature. sample in 500 G at different temperatures. We define the ``normalized relaxation rate'' R we show in Fig. 9 M(H) for the amorphous nanoparticles at M/ ln(t) /Mc , i.e., the logarithmic slope of the relax- T 5 and 100 K. Magnetic hysteresis is apparent at 5 K, ation curve normalized by the magnitude of the irreversible whereas the behavior is purely reversible at 100 K. magnetization at which the relaxation starts, Mc M0 The relaxation at different values of the external magnetic Mrev . Here M0 is the initial value of the total magnetic field is shown in Fig. 10. The vertical lines represent M(t) moment and Mrev is the magnetic moment corresponding to a curves shown along with the standard magnetization loop. field cooling in 500 G. Figure 6 summarizes the values of R The field dependences of the relaxation rates for our samples as a function of temperature, for the three samples. At low enough temperatures, R is the lowest for the crystalline are shown in Fig. 11. There is an apparent change in R be- sample, intermediate for the acicular amorphous sample and tween low and high fields. At lower fields R is the largest in the largest for the regular amorphous sample. Note, that at an amorphous sample, whereas at large fields the relaxation higher temperatures it looks as if R(T) curves will cross. rate in an amorphous sample is the lowest. This is due to a large difference in the absolute values of Tirr 90, 162, and 216 K at 500 G for regular amorphous, acicular amorphous, and crystalline, respectively . As shown in Fig. III. MAGNETIC RELAXATION IN THE ASSEMBLY 7, M(T) curves scale with T OF NANOPARTICLES irr and, therefore, in the inset to Fig. 6 we plot R vs T/Tirr . In this presentation, the whole A. Time-dependent effective barrier R(T/Tirr) curve of the crystalline sample is lower than that of for magnetic reorganization the acicular amorphous sample and both are lower than the R curve of the regular amorphous sample. Magnetic relaxation is a distinct feature of systems with In Fig. 8 we compare the irreversibility lines for the three interacting particles, far from thermodynamic equilibrium. In samples. The largest irreversibility is found in a crystalline an assembly of ferromagnetic nanoparticles, the elementary sample, intermediate in the sample with acicular particles process of a change in the magnetization is the rotation of the and the lowest in the regular amorphous sample. We explain these observations below. magnetic moment of a nanoparticle or cluster of such mag- Magnetic irreversibility below T netic moments . In the following we assume that the mag- irr is also demonstrated by measuring the magnetization loops M(H). As an example, netic anisotropy of each nanoparticle is strong enough to utilize an Ising-like model, i.e., the magnetic moment of each FIG. 6. Normalized logarithmic relaxation rate R for three types of samples as a function of temperature. Inset: R as a function of a reduced temperature T/T FIG. 8. Irreversibility lines for three types of samples. irr . 6960 PROZOROV, YESHURUN, PROZOROV, AND GEDANKEN PRB 59 FIG. 9. Typical magnetization loops at T 5 K open circles and at T 100 K solid line . FIG. 11. Normalized logarithmic relaxation rate R for three samples as a function of magnetic field at T 20 K. particle is aligned only along the anisotropy axis. In Fig. 12 a we illustrate schematically the orientation of the el- the other minima (W2). The backward reorientation requires ementary magnetic moments of several of such nanopar- overcoming the energy barrier U21 . ticles. The full arrows represent the size and direction of In order to take interparticle interactions into account we each magnetic moment. The experimentally measured mag- view the field H in Eq. 1 as the internal magnetic field, netic moment is determined by the sum of the projections of which is the sum of the external field and the dipole field each individual particle's moment on the direction of the from the surrounding nanoparticles. This local magnetic field external magnetic field. Note that the directions of the easy depends on the directions of neighboring magnetic axes are randomly distributed. For such a system, the energy moments.33­36 Since the magnetic moment is a statistical av- W of each magnetic nanoparticle, neglecting for the moment erage of those moments, the local field depends, on the av- the interparticle interactions, varies with the angle as21 erage, on the total magnetic moment. This induces a feed- back mechanism: each reorientation of an individual W KV sin2 M nanoparticle decreases the total magnetic moment. This is pH cos . 1 illustrated in Figs. 12 a and 12 b . Figure 12 a represents a Here is the angle between the easy axis K and the external snapshot of a field-cooled system of nanoparticles in which magnetic field H , and is the angle between the particle most of the individual magnetic moments are favorably ori- ented in a direction such that their projections are along the magnetic moment M p and the external field. In order to have external field. After a field decrease, as a result of thermal any magnetic irreversibility and relaxation, the KV term in fluctuations, some magnetic moments reorient so that their Eq. 1 must be larger than the MpH term and we will con- projection is antiparallel to the external field. The open ar- sider this limiting case. The reduced energy W/KV of Eq. 1 rows in Fig. 12 b represent those reoriented moments. is plotted in Fig. 13 as a function of the angle for two Since the local dipole field decreases during this process, different fields H1 2.5KV/Mp bold and H2 0.5KV/Mp the average barrier U light . Since the magnetic anisotropy has no preferable di- 12 increases. As indicated in the Intro- duction, an increase of the barrier with time is a characteris- rection, there are two minima in the angular dependence of tic of other irreversible systems, such as type-II supercon- the energy, as shown in Fig. 13. The external magnetic field ductors in the process of magnetic flux creep. fixes the direction of the lowest minima. We denote by U12 The dynamics resulting from such a scenario is sketched the barrier for reorientation from the lowest minima (W1) to FIG. 10. Magnetic relaxation at different values of magnetic FIG. 12. Schematic snapshots of magnetic moments distribution field. Vertical lines are the M(t) curves superimposed on a regular in powder sample at a beginning of the relaxation and b at later magnetization loop measured at the same temperature. time. PRB 59 MAGNETIC IRREVERSIBILITY AND RELAXATION IN . . . 6961 FIG. 13. Energy profiles after FC in high magnetic field (H1) FIG. 14. Energy profiles at the beginning and at the latter stage and after reduction of the magnetic field, when the relaxation starts of the relaxation. Dots indicate population of magnetic moments of (H2). the particular energy minima. in Fig. 14. Immediately after reducing the magnetic field, Considering the balance of forward and backward rota- individual magnetic moments are still along the direction of tions, and averaging over the volume of the sample, we show the external field, i.e., in minima W1 of Fig. 13, as depicted in the Appendix that magnetic relaxation is described by a by the dense population of the black dots. During the relax- differential equation similar to that derived for ation process magnetic moments flip to the minimum W2 in superconductors15,16 the figure. Since, as discussed above, this barrier depends on the total magnetic moment via dipole fields, it will increase M U with time as shown in the figure, with dipole fields working , 2 on the average against the external field. The total magnetic t AMc exp T moment along the magnetic field is thus decreased, as sketched in Fig. 12 b . where A is an attempt frequency and U is an effective barrier for magnetic relaxation given by B. Equations of magnetic relaxation M In a realistic sample, the directions of easy axes are ran- U U 0 1 , 3 domly distributed, the particles cannot physically rotate e.g., M0 in a dense powder of ferromagnetic nanoparticles , and di- where pole interactions are strong. We will model this situation as outlined below. Any given particle i in Fig. 12 has an anisotropy axis at a U0 2KV 4Mp H Mrev / 4 fixed angle i relative to the external magnetic field. The and M magnetic moment of this particle is then oriented at an angle 0 1/ ( KV/2M p H M rev). Here is the con- stant accounting for the strength of the dipole-dipole interac- i to the field. This angle is defined by the nonlocal energy tions, M minimization, due to dipole fields of the surrounding. It is p is the magnetic moment of an individual particle, K is the anisotropy constant and V is the particle volume. important to note that each particle interacts with a local Apparently, as 0 the energy barrier U U magnetic field H 0 , thus U0 is i which is the result of a vector sum of the the barrier in the assembly of noninteracting particles. It is external and dipole fields. At small enough external field and worth noting that in our model the barrier U depends on the large enough anisotropy i may have two values: i i or magnetic moment in the same way as that used by i i , which leads to the situation described in Fig. 13, Anderson14­16 for a description of magnetic relaxation in su- with two energy minima at Wi i 1 M pHi cos( i) and W2 perconductors. In the following we analyze magnetic relax- MpHi cos( i). Thermal fluctuations may force a particle ation described by those equations. moment in the minima Wi1 to change its direction to another If the barrier for a particle moment reorientation does not minima Wi i i 2 and vise versa. The W1 W2 rotation requires depend on the total magnetic moment, i.e., 0 and U overcoming a barrier Ui i 12 , and a barrier U21 for backward U0 , direct integration of Eq. 2 yields rotation, see Eqs. A1 and A2 of the Appendix, respec- tively. We then assume that the field Hi can be represented M Mc exp t/ , 5 as a simple sum of the external field H and the collinear to H dipole field Hd i.e., independent of i). The amplitude of a where Mc is the initial irreversible magnetization and dipole field Hd at any given site depends upon orientations of exp(U0 /T)/A is the macroscopic characteristic relaxation the moments of the surrounding particles. If those orienta- time. This result is very similar to that derived in early works tions are totally random minima W1 and W2 are equally for classical NeŽel's superparamagnetic relaxation, see e.g., occupied the dipole field is small, whereas if all surrounding Ref. 37. This exponential decay is observed experimentally, particles are in one of the minima the resulting dipole field is for example, in the work of Wegrowe et al.19 on a single maximal. From this simple analysis, we conclude that the nanowire. magnitude of a dipole field depends upon the total magnetic If interactions are not negligible, Eq. 2 may be rewritten moment M. in dimensionless form: 6962 PROZOROV, YESHURUN, PROZOROV, AND GEDANKEN PRB 59 u Tirr(H) and R(H) in different samples provide a verification of our model on self-consistency. exp u , 6 It is interesting to note that the expression for Tirr , Eq. where u U/T and t/ t with 13 , is typical for the blocking temperature of individual noninteracting particles, which is obtained from Eq. 5 : M T 1 T T t 0 M U c U0 A 4 AM pM c A , 7 T0 0 irr , 14 where we introduced an energy scale , which, as we show ln t/t* below, determines the relaxation process and the irreversibil- where t* 1/(A ln Mc / M ) is the characteristic time and U0 ity line: is given by Eq. 4 . Energy U0 is proportional to KV for noninteracting nanoparticles, but it is reduced by a term pro- McU0 /M0 4 McMp / . 8 portional to due to interparticle interactions. This is in This energy is directly related to the strength of the interpar- agreement with previous works where ``static'' modifica- ticle interactions. tions of the barrier for relaxation were considered.17,23,38 Ir- Solving Eq. 6 we obtain reversibility temperature, Eq. 14 approaches 0 when M 0, and so does T0 . This reveals an important difference in t irr u u the physics of the irreversibility line in interacting and non- c ln 1 t , 9 0 interacting particles. In the former, there is a true irrevers- where u ibility in the limit M 0 associated with freezing of mag- c Uc /T is the reduced effective energy barrier at netic moments due to interparticle interactions. In the case of t 0, the time when the relaxation starts. The normalization noninteracting nanoparticles, the apparent irreversibility is time t0 is given by due to experimental limitations finite sensitivity, e.g., M). T It is important to stress that this is true only on a macro- t 0 t exp Uc scopic time scale t t*, such as relaxation or M(T) mea- T A exp U0 T . 10 surements. If, however, t t* is realized, for example in Now, using Eqs. 3 and 9 we get the time evolution of the Mošssbauer measurements, one may detect the irreversibility magnetic moment: temperature according to Eq. 14 .17,24,39 We also note that in any case Tirr is a dynamic crossover T t from a reversible to an irreversible state and is defined for a M t M c 1 ln 1 t . 11 particular experimental time window t. In the following 0 section we compare our experimental observations with the Normalized relaxation rate R M/ ln(t) /Mc is given model developed above. by T t IV. DISCUSSION R t Our phenomenological model provides a description of 0 t . 12 the irreversible magnetic behavior in the assembly of ferro- As we will see below, experiment shows that t0 1 s. In our magnetic nanoparticles. In particular, the model predicts the measurements typical time window t 100 s, therefore we time-logarithmic decay of the magnetization, see Eq. 11 . can assume t t0 and Eq. 12 predicts that the relaxation Also, Eqs. 12 and 13 relate both the irreversibility line rate saturates at R T/ . Thus, measurements of the nor- and the relaxation rate to a single parameter malized relaxation rate can provide a direct estimate of the 4 M energy scale governing the relaxation process. cM p / . The magnetic relaxation data of Fig. 5 reveal, indeed, time-logarithmic relaxation. Fitting these data to Eq. 11 C. Irreversibility temperature yields the parameters of Mc and . In Fig. 15 we plot the The irreversibility temperature T derived energy as a function of temperature for the three irr of the assembly of magnetic nanoparticles is defined by the condition samples and find that is the largest for a crystalline M( t,T sample, intermediate for an acicular amorphous and the low- irr) M . Here M is the smallest measured mag- netic moment and t is the time window of the experiment. est for a regular amorphous sample. The straightforward ex- Using Eq. 11 we obtain planation is that in a crystalline sample both and Mp are the largest; in an acicular amorphous sample Mp is of the 1 M/M same order as in the regular amorphous, but is much larger T c irr ln . 13 1 t/t due to shape anisotropy. 0 ln 1 t/t0 Similarly, we derive the magnetic-field dependence of thus we can estimate the characteristic time t0 from measure- from the data of Fig. 10 and plot (H) for three samples in ments of Tirr , because the energy can be determined sepa- Fig. 16. We note that M p and should not depend on mag- rately from the measurements of the relaxation rate R netic field. It is therefore expected that the field dependence T/ . On the other hand, the irreversibility line Tirr(H) of is determined by the field dependence of Mc Ms gives the field dependence of . The latter may be obtained Mrev(H), which decreases with field, Fig. 10. Figure 16 also from R(H) measurements. Thus measurements of shows the agreement with this observation. The weak in- PRB 59 MAGNETIC IRREVERSIBILITY AND RELAXATION IN . . . 6963 FIG. 15. Temperature dependence of energy extracted from the measurements of normalized relaxation rate for the three samples. FIG. 16. Magnetic-field dependence of the energy . crease of with temperature, Fig. 15, may be related to ACKNOWLEDGMENTS some nonlinear dependence of barrier U on the magnetic moment. We thank Y. Rabin and I. Kanter for valuable discussions. Independent estimations of are derived from T This work was partially supported by The Israel Science irr of Fig. 8 using Eq. 13 . Comparing Figs. 8 and 16 we get for three Foundations and the Heinrich Hertz Minerva Center for High samples /T Temperature Superconductivity. Y.Y. acknowledges support irr 4 6. Thus, t0 0.05 0.5 s. Note that these values of t from the German Israeli Foundation G.I.F . R. P. acknowl- 0 are much larger than the ``microscopic'' values predicted by NeŽel,37 simply because they reflect col- edges support from the Clore Foundations. lective behavior of the whole assembly controlled by the effective barrier , see Eq. 10 , and not a single-particle APPENDIX EFFECTIVE BARRIER FOR MAGNETIC barrier KV. RELAXATION AND EQUATION FOR TIME EVOLUTION Let us now compare the irreversibility lines of different OF THE MAGNETIC MOMENT samples, Fig. 8. In most parts of this diagram the region of Here we consider in detail the model outlined in the text. the irreversible behavior is the largest for a crystalline We assume that magnetic moment Mp of any given particle sample. The amorphous sample containing acicular particles i can be in one of the two possible energy minima: Wi occupies the intermediate space and the amorphous sample 1 M i M embraces the smallest space in this T-H phase diagram. Such pHi cos( i) or W2 pHi cos( i). These minima are separated by the barrier of height KV M behavior is naturally explained in terms of a strength of in- pHi sin( i). In the presence of thermal fluctuations, a particle moment sitting in terparticle interactions, which are the smallest in the case of the minima Wi can spontaneously change its direction to the a regular amorphous sample, intermediate for an acicular 1 i amorphous sample due to shape anisotropy and the largest next minima W2 . The energy barrier for such reorientation is for a crystalline sample due to crystalline anisotropy. Also, Ui KV M highest irreversibility temperature of crystalline sample is 12 pHi sin i cos i . A1 understood on the basis of its largest particle size. The backward rotation is also possible and requires over- coming the barrier: i V. SUMMARY AND CONCLUSIONS U21 KV MpHi sin i cos i . A2 We presented measurements of irreversible magnetization From this point on one can conduct a self-consistent sta- as a function of temperature, time, and magnetic field in tistical average over angles i(Hi ,t, i) in order to evaluate three types of ferromagnetic nanoparticles: regular amor- the resulting magnetic moment M of the system. On the other phous, acicular amorphous, and crystalline nanoparticles. hand, we may try to simplify the problem assuming that the The results are interpreted using a developed phenomeno- internal field Hi can be represented as a simple sum of the logical approach based on the assumption that the barrier for external field H and the collinear to it dipole field Hd i.e., magnetic moment reorientation depends on the total mag- independent of i). If all easy axes are randomly distributed netic moment via dipole fields. This explains the time- the average barrier for flux reorientation is then given by logarithmic magnetic relaxation governed by the energy scale related to interparticle interaction. Values of 1 Nk 2 /2 found from measurements of the irreversibility line and the U k Ui k N Ui U d , relaxation rate are in perfect agreement, implying the validity ki 1 0 of our model. where k 12 or 21 denotes particle's moment flipping from We demonstrate the influence of a shape anisotropy on the minima W1 to the minima W2 , or backward, respec- magnetic properties of nanoparticles, i.e., irreversible mag- tively. Using Eqs. A1 and A2 we find netic response of acicular amorphous particles is close to that found in crystalline particles. U12 KV 4MpHi / , A3 6964 PROZOROV, YESHURUN, PROZOROV, AND GEDANKEN PRB 59 U We shall now consider direct and backward moment ro- 21 KV. A4 tation processes in a nonequilibrium state. As above, we de- Let us now consider a situation where temperature is note by N1 and N2 number of moments in energy minima 1 higher than irreversibility temperature and system is at ther- and 2, respectively. The total number of particles in the sys- mal equilibrium. The number of particles jumping per tem is N N1 N2 . The magnetic moment is proportional to unit time from one minima to another is proportional to the difference n N1 N2 . During small time t this differ- Nk exp( Uk /T). The condition for equilibrium is ence changes as N1e U12 /T N2e U21 /T. Thus U U n N 1 2 1 exp T N2 exp T t. A10 U 4M N 12 U21 pHi 2 N1 exp T N1 exp T . Using simple algebra and the above relationships between It is clear that the difference n N N1 , N2 , n, and N, we get 1 N2 determines the re- sulting magnetic moment of a system. If the total number of particles in the system is N, the difference n is n ncosh U1 U2 1 exp 4M t exp U1 U2 T T n N pHi / T 1 N2 N 1 exp N tanh 2MpHi . 4M pHi / T T A5 N sinh U1 U2 T . A11 The total reversible magnetic moment then is From this we arrive at a nonlinear differential equation gov- M erning process of magnetic relaxation not too close to equi- rev M pn M pN tanh 2MpHi T . A6 librium: This formula is similar to the expression for the Ising super- paramagnet and simply reflects the two-state nature of our M model.37,40 The difference is, however, that the physical M rev M cosh U1 U2 magnetic field is the total external dipole field H t A exp U1 U2 T T i . Dipole field Hd at any given site depends upon orienta- tions of the moments of the surrounding particles. If those M s sinh U1 U2 T , A12 orientations are totally random minima W1 and W2 are equally occupied the dipole field is small, whereas if all surrounding particles are situated in one of the minima the where A is a constant measured in s 1 and having the mean- resulting dipole field is maximal. From this simple picture, ing of attempt frequency. we conclude that the magnitude of a dipole field depends Equation A12 can be simplified considering magnetic upon the total magnetic moment of a sample M relaxation not too close to equilibrium and retaining our as- rev M , where M sumption that anisotropy contribution to the magnetic energy rev is given by Eq. A6 and M is the irreversible, time-dependent contribution to the total magnetic moment is much larger than that of magnetic field both conditions resulting from the finite relaxation time needed for a system are better satisfied at low fields . In this case, Eq. A12 may to equilibrate. Therefore, we may write H be approximated in a reduced form: i H ( M rev M). Here is the coefficient accounting for the contribu- tion of dipole fields. Now we can obtain the equation for reversible magnetization from Eq. A6 : M t AMc exp U/T , A13 tanh 2M M pH/ T rev M S 1 , A7 2M pMS / T where Mc is the total magnetic moment at the beginning of where M relaxation and U is the effective barrier: S M pN. We note that this formula is valid at small enough fields 2MpH/ T when particle moments are al- most locked along the easy axes and small enough interac- M tions i.e., H M rev). The important result is that reversible U 2KV 4M p H Mrev M / U0 1 , magnetization decreases as the interparticles interaction in- M0 A14 creases. Interestingly, Eq. A7 provides a good description of the experimental data. Thus, the barriers for moment reorientation in Eqs. A3 where U0 KV 4Mp(H Mrev)/ and and A4 can be rewritten as M0 1/ ( KV/2Mp H Mrev). We reiterate that Eq. A13 is valid only in the case when U1 KV 4Mp H Mrev M / , A8 the magnetic anisotropy is large and magnetic moment is far from equilibrium. Close to equilibrium, one ought to con- U2 KV. A9 sider Eq. A12 . 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