Origin of Surface Canting within ${\mathrm{Fe}}_3{\mathrm{O}}_4$ Nanoparticles

The nature of near-surface spin canting within ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles is highly debated. Here we develop a neutron scattering asymmetry analysis which quantifies the canting angle to between 23° and 42° at 1.2 T. Simultaneously, an energy-balance model is presented which reproduces the experimentally observed evolution of shell thickness and canting angle between 10 and 300 K. The model is based on the concept of $T_d$ site reorientation and indicates that surface canting involves competition between magnetocrystalline, dipolar, exchange, and Zeeman energies.

Figure 1

(a) PASANS setup with polarizing neutron spin optics. White lines indicate regions of horizontal and vertical sector slices. At 1.2 T, 200 (b) and 300 K (c) spin-flip data are shown in black circles, while simulated data based on Table 1, entry 5 are shown with blue lines. The horizontal to vertical ratios, red triangles, are $<2$ due to the negative cross term in Eq. (1). Points around $0.075{\AA }^{-1}$ that approach zero have been removed from the ratio, and the 300 K ratio is limited to $0.11{\AA }^{-1}$ due to limited vertical angular acceptance. Simulated ratios (Table 1, entry 5) are shown with pink lines. Error bars here and in the text indicate 1 standard deviation.

Figure 2

(a) Near-edge section of a core–canted-shell model broken into cubic formula units of length 0.42 nm. Insets show constituent $T_d$ (purple) and $O_h$ (orange) Fe sites; yellow arrows indicate the net local moments moments of these constituent Fe spins. Simulated energy landscapes as a function of $t$ and $T_{d\text{tilt}}$ at 1.2 T are shown for 300 (b), 200 (c), 160 (d), and 10 K (e). Remanence at 300 K is shown in (f). Pink stars indicate global minima; white space indicates areas of high energy.

Figure 3

(a) Energy contributions at 160 K, 1.2 T as a function of shell thickness ($t$) are dominated by exchange and Zeeman energies. (b) The sum of these energies produces a minimum at $t=0.9\mathrm{nm}$ (purple circles), although without the inclusion of both dipole and anisotropy contributions the minimum would shift toward $t=0\mathrm{nm}$ (green open squares).