Crossover From Individual to Collective Magnetism in Dense Nanoparticle Systems: Local Anisotropy Versus Dipolar Interactions

Abstract

Dense systems of magnetic nanoparticles may exhibit dipolar collective behavior. However, two fundamental questions remain unsolved: i) whether the transition temperature may be affected by the particle anisotropy or it is essentially determined by the intensity of the interparticle dipolar interactions, and ii) what is the minimum ratio of dipole–dipole interaction (Edd) to nanoparticle anisotropy (KefV, anisotropy⋅volume) energies necessary to crossover from individual to collective behavior. A series of particle assemblies with similarly intense dipolar interactions but widely varying anisotropy is studied. The Kef is tuned through different degrees of cobalt‐doping in maghemite nanoparticles, resulting in a variation of nearly an order of magnitude. All the bare particle compacts display collective behavior, except the one made with the highest anisotropy particles, which presents “marginal” features. Thus, a threshold of KefV/Edd ≈ 130 to suppress collective behavior is derived, in good agreement with Monte Carlo simulations. This translates into a crossover value of ≈1.7 for the easily accessible parameter TMAX(interacting)/TMAX(non‐interacting) (ratio of the peak temperatures of the zero‐field‐cooled magnetization curves of interacting and dilute particle systems), which is successfully tested against the literature to predict the individual‐like/collective behavior of any given interacting particle assembly comprising relatively uniform particles.

Role of Dipolar Interactions on the Determination of the Effective Magnetic Anisotropy in Iron Oxide Nanoparticles

“…[ 67 , 68 ] This gradual trend shows that the customization of SiO 2 shells makes it possible to gain control on the interaction effects; more specifically, this flattening of the FC curve can be quantified in relation to the magnetization value for T Max , i.e., for T B , through the equation FC rise = ( M plateau – M TMAX )/M TMAX enabling the estimation of the collective or individual behavior of systems based on MNPs. [ 69 ] The trends obtained, observed in Figure S10 (Supporting Information), clearly reveal a collective behavior (low FC rise values) for the MNPs that have a smaller distance between magnetic cores, i.e., smaller SiO 2 thickness, MC8‐S2, MC10‐S2, and MC15‐S5, which generally decreases as the distance increases and the interaction effects are reduced. In this sense, focusing on the general trend of each of the batches, the one with the smallest magnetic core size, MC8‐SX, presents a weaker collective behavior (higher FC Rise value) while for batches MC10‐SX and MC15‐SX with larger magnetic core sizes and higher magnetization, it appears to be stronger, indicating that larger SiO 2 thickness is needed to achieve an individual behavior of the systems.…”

The Boundary Between Volume and Surface-Driven Magnetic Properties in Spinel Iron Oxide Nanoparticles

“…Noteworthy, the comparison of interparticle interactions in the set of samples under investigation requires special care, due to the different average magnetic anisotropy in each sample. Indeed, the experimental observations are the results of the combined effect of the interparticle interaction energy and the effective single particle anisotropy energy and their specific ratio [69,70].…”

Non-Exchange Bias in Binary Nanoparticle Systems

“…Surface disorder was concluded to be absent or too thin to provide a detectable exchange bias effect in the pure -Fe2O3 particles (Supplementary Section 2) 35 . When a single type of particles is compacted into a dense assembly, an increase of TB and HC takes place due to interparticle dipolar interactions (Supplementary Section 1) 39 . However, the exchange-bias hardly changes in either of the systems (Supplementary Section 1); the small change in the Co-doped sample (from 1.20 to 1.28 kOe in the compact) may reflect small variations in the magnetic disorder at the nanoparticle/shell interface due to the growth of the SiO2 layer 36 .…”

“…(with k B the Boltzmann constant, 𝑀 𝑆 the saturation magnetization, V the particle volume and φ the particle packing fraction) for either soft-soft, hard-hard and soft-hard combination 39 affects the strong intraparticle exchange-coupling (i.e., the exchange bias) of the Co-doped particles, the latter having a much larger associated energy 26 , i.e., the ratio T ex /T dd (where the temperature T ex is proportional to the exchange-coupling energy at the interface, 𝐸 𝐸𝑋 = 𝐻 𝐵 𝑉 𝐹𝑖𝑀 𝑀 𝑆 2 ), is much larger than unity for small particles (as it is the case here). The exchange stiffness constant Aex for the corresponding ferrites (~10 -12 ) 44 is 2 orders of magnitude larger than the dipolar stiffness Adip extracted from the random anisotropy model (~10 -14 ) for the present systems 35 , confirming that the exchange energies involved are much higher 45,46 .…”

“…The exchange stiffness constant Aex for the corresponding ferrites (~10 -12 ) 44 is 2 orders of magnitude larger than the dipolar stiffness Adip extracted from the random anisotropy model (~10 -14 ) for the present systems 35 , confirming that the exchange energies involved are much higher 45,46 . Indeed, a bibliographic study presented in the SI (Supplementary Section 5) shows that T ex T dd ≫ 1 in all the nanoparticle systems considered, but those exhibiting very low exchange bias fields 26,39,47,48 . Thus, it is more appropriate to consider that the strong dipolar interactions between the two types of particles are mutually influencing the magnetic response of each population, and therefore the overall magnetization loop.…”