Correlated spin canting in ordered core-shell ${\mathrm{Fe}}_3{\mathrm{O}}_4/{\mathrm{Mn}}_x{\mathrm{Fe}}_{3-x}{\mathrm{O}}_4$ nanoparticle assemblies

Polarization-analyzed small-angle neutron-scattering methods are used to determine the spin arrangements and experimental length scales of magnetic correlations in ordered three-dimensional assemblies of $\sim 7.4$-nm-diam core-shell ${\mathrm{Fe}}_3{\mathrm{O}}_4/{\mathrm{Mn}}_x{\mathrm{Fe}}_{3-x}{\mathrm{O}}_4$ nanoparticles. In moderate to high magnetic fields, the assemblies display a canted magnetic structure where the canting direction is coherent from nanoparticle to nanoparticle, in contrast to the less extended, more single-particle-like behavior for similar ferrite assemblies. The observed magnetic scattering is modeled by assuming that the interparticle dipolar coupling combined with Zeeman effects in a field leads to nanoparticle domains with preferred net spin alignments relative to packing symmetry axes. Over a range of fields and temperatures, the model qualitatively explains the observed scattering anomalies in terms of clusters that vary in area and thickness, highlighting the complex structures adopted in real, dense nanoparticle systems. The clusters often have a strong two-dimensional magnetic character which is attributed to structural stacking faults and the resulting influence of interparticle dipolar interactions for these magnetically soft nanoparticles.

Figure 1

(a) Schematic of the composition of the core-shell nanoparticles, (b) organized into face-centered-cubic close-packed clusters for magnetic modeling, (c) each with different possible orientation (indicated by angles $\psi$, $\omega$, and $\mathrm{\Phi }$) relative to an applied field along $X$ (in red). Arrows within the depicted close-packed region in (c) denote expected sixfold easy symmetry axes; the thicker (yellow) pair indicates the set closest to the applied field direction in this instance. The stacking of the layers is fcc to match the observed structural ordering. The direction of magnetization, $\vec{M}$, is determined in the model by the energy minimum between alignment in the plane (black or yellow arrows) and along the field direction (red arrow). (d) Transmission electron microscopy (TEM) image of self-assembled monolayers and bilayers of the nanoparticles and the presence of dislocations and faults in the ordering.

Figure 2

Comparison of magnetization curves for the dense assemblies of nanoparticles at 10 K (dark blue), 100 K (light blue), 200 K (green), 300 K (orange), and 400 K (red). Since an absolute mass normalization is not feasible due to the unknown amount of included surfactant content, the data are normalized against the 10 K maximum value at 5 T.

Figure 3

(a) Experimental setup includes a polarizing supermirror and rf flipper to select the incident spin state and a ${}^3\mathrm{He}$ cell with in situ NMR flipper to select the scattered spin state. The angle $\theta$ shown is between the $X$ axis and the projection of $\vec{Q}$ onto the $X-Y$ plane. The scattering pattern is collected on a 2D detector with (b) showing polarization-corrected images for the ${\mathrm{Fe}}_3{\mathrm{O}}_4/{\mathrm{Mn}}_x{\mathrm{Fe}}_{3-x}{\mathrm{O}}_4$ nanoparticles at 400 K in a small remanent field. The $I^{+-}$ and $I^{-+}$ patterns have been added together.

Figure 4

Data for extracted PASANS intensity $M_X^2$ using Eq. (4) vs scattering vector $Q$ for (a) 10 K, (b) 200 K, and (c) 400 K with squares for 1.5 T, diamonds for 0.2 T, triangles for 0.1 T, and circles for remanent fields. Error bars in the plot denote standard uncertainties.

Figure 5

Data for extracted PASANS intensity $M_{X,\mathrm{net}}^2$ using Eq. (5) vs scattering vector $Q$ for (a) 10 K, (b) 200 K, and (c) 400 K with squares for 1.5 T, diamonds for 0.2 T, triangles for 0.1 T, and circles for remanent fields. Error bars in the plot denote standard uncertainties. Relative to Fig. 4, the scale is changed to show better features in the data.

Figure 6

Data for extracted PASANS intensity $M_Z^2$ (perpendicular to the field direction) using Eq. (7) vs scattering vector $Q$ for (a) 10 K, (b) 200 K, and (c) 400 K with circles for remanent fields, triangles for 0.1 T, diamonds for 0.2 T, and squares for 1.5 T. Error bars in the plot denote standard uncertainties.

Figure 7

Spin-flip horizontal-to-vertical scattering ratio (SFR) defined in Eq. (9) as a function of $Q$ for (a) 10 K, (b) 200 K, and (c) 400 K with squares for 1.5 T, diamonds for 0.2 T, triangles for 0.1 T, and circles for remanent fields. Error bars in the plot denote standard uncertainties.

Figure 8

Simulated spin-flip scattering in the (a) vertical $(\theta =\pm {90}^{\circ })$ and (b) horizontal ($\theta =0^{\circ }$ or ${180}^{\circ }$) directions, compared against PASANS data for the ${\mathrm{Fe}}_3{\mathrm{O}}_4/{\mathrm{Mn}}_x{\mathrm{Fe}}_{3-x}{\mathrm{O}}_4$ nanoparticle assemblies at 200 K in a magnetic field of 0.1 T. Error bars in the plot denote standard uncertainties. The dotted (red) lines show predicted scattering assuming no correlations and a random selection of the symmetry axis in a close-packed plate, while the solid (black) lines show the same scattering, assuming selection of the axis nearest to the applied magnetic field. All other model parameters are maintained constant, with key values listed in Table S-II of the Supplemental Material [38]. Insets indicate the sectors of data that were analyzed.

Figure 9

Simulated spin-flip scattering ratio assuming random selection of sixfold symmetry axis for no correlation (dashed red curve) vs selection of axis nearest to applied field (solid black curve) for preferred correlation, in comparison against experimental data for assemblies at 200 K in a magnetic field of 0.1 T. Error bars in the plot denote standard uncertainties.

Figure 10

(a) Averaged tilt angle of the magnetization vs magnetic field assuming model parameters listed in Table S-II of the Supplemental Material [38], with blue circles for 10 K, green squares for 200 K, and red triangles for 400 K. Zero degrees corresponds to alignment within the plane of scattering centers. Note at 400 K in remanence, the modeled “cluster” consists only of one scattering center and is thus omitted. (b) Averaged tilt angle of the magnetization vs derived magnetization tilt angle using magnetization data from Fig. 2. The relationship is nearly linear as expected, at 10 and 200 K.